Mechanics of Flow-Induced Vibration: Physical Modeling and Control Strategies 9811985774, 9789811985775

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Rajeev Jaiman Guojun Li Amir Chizfahm

Mechanics of Flow-Induced Vibration Physical Modeling and Control Strategies

Mechanics of Flow-Induced Vibration

Rajeev Jaiman · Guojun Li · Amir Chizfahm

Mechanics of Flow-Induced Vibration Physical Modeling and Control Strategies

Rajeev Jaiman Department of Mechanical Engineering The University of British Columbia Vancouver, BC, Canada

Guojun Li School of Mechanical Engineering Xi’an Jiaotong University Xi’an, Shaanxi, China

Amir Chizfahm Department of Mechanical Engineering The University of British Columbia Vancouver, BC, Canada

ISBN 978-981-19-8577-5 ISBN 978-981-19-8578-2 (eBook) https://doi.org/10.1007/978-981-19-8578-2 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Foreword

This book provides a comprehensive and self-contained review on the physical understanding, modeling and control of fluid–structure interaction. Using a state-of-the-art body-fitted Lagrangian–Eulerian formulation, single- and two-phase viscous incompressible flows at low and high Reynolds numbers are considered with an increasing complexity of structural representations ranging from a freely vibrating rigid body, a continuum linear and nonlinear elasticity to a fully coupled flexible multibody system. The book is organized so that the chapters are fairly independent, and each chapter can be used by researchers and graduate students for their graduate course works and projects. In Part I, the computational modeling and the physical insight are provided for various elastically mounted bodies and flexible cylindrical structures. While Part II is devoted to advanced reduced order modeling and control strategies, the flapping dynamics and self-excited synchronized vibration of thin elastic structures are presented in Part III. In this book, we combine a description of high-fidelity modeling of flow-induced vibration and synchronization response via advanced reduced-order modeling, data-driven stability analysis and control strategies. The scope of the book is restricted in that there is no attempt to review and discuss all the aspects written in recent years on flow-induced vibration and wakebody synchronization. The book is selective with a view to providing computational results of observed flow-induced phenomena and ideas for modeling them using the fully coupled differential system given by the Navier–Stokes equations and a range of representations of structural dynamics. It is our hope that this book will bridge a gap in the literature, particularly in the area of computational modeling of flow-induced vibration. The body of work will be welcomed by researchers and graduate students in engineering, physics and applied mathematics.

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Preface

Coupled multiphysics and multidomain simulations are emerging and challenging research areas, which have received significant attention during the past decade. One of the most common multiphysics and multidomain problems is fluid–structure interaction (FSI), i.e., the study of coupled physical systems involving fluid and a structure that have a mechanical influence on each other. Regardless of the application area, the investigation toward modeling of fluid–structure interaction and the underlying mechanisms in dealing with coupled fluid–structure instability with real-world applications remains a challenge to scientists and engineers. This book is designed for students and researchers who seek knowledge of computational modeling and control strategies for fluid–structure interaction. Specifically, this book is intended to provide a comprehensive review of the underlying unsteady physics and coupled mechanical aspects of the fluid–structure interaction of freely vibrating bluff bodies, the self-induced flapping of thin flexible structures and aeroelasticity of shell structures. Understanding flow-induced loads and vibrations can lead to safer and cost-effective structures, especially for light and high-aspect-ratio structures with increased flexibility and harsh environmental conditions. Using the body-fitted and moving mesh formulations, the physical insights associated with structure-to-fluid mass ratios, Reynolds number, nonlinear structural deformation, proximity interference, nearwall contacts, free-surface, and other interacting physical fields will be covered in this book. The book also discusses various passive and active techniques for controlling unsteady flow dynamics and associated coupled mechanics of fluid–structure interaction. In conjunction with the control techniques, data-driven model reduction approaches based on subspace projection and deep neural calculus will be covered for low-dimensional modeling of unsteady fluid–structure interaction. This book will present a detailed nonlinear physics of fluid–structure interaction from advanced computational physics viewpoints. Earlier books on fluid–structure interaction and flow-induced vibrations in cross-flow were primarily written under semi-empirical and experimental backgrounds to understand the coupled dynamics of a bluff body and a thin filament or a flexible panel. The present contribution is unique with regard to explaining the unsteady coupled physics through detailed

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flow kinematics and coupled dynamical characteristics related to vorticity distributions, force/amplitude relationship, energy transfer, phase relations and among others. Using the system identification and machine learning techniques, for the first time, we will present a generalized understanding of the frequency lock-in phenomenon during vortex-induced vibrations for various bluff body geometries, rounding, mass ratio and Reynolds number. Comprehensive data-driven stability analysis of frequency lock-in of transversely vibrating two-dimensional bluff bodies with smooth curves and sharp corners will be covered to demarcate the flutter- and resonance-induced regimes. New insights on the self-induced flapping dynamics of conventional and inverted flexible plates will be presented for the first time in a textbook format. To our knowledge, there exists no book which covers comprehensively the high-fidelity and reduced-order modeling techniques and control strategies for fluid–structure interactions. We have made an effort to make the chapters fairly independent of one another. As a result, we believe that readers who are interested in specific topics may often be able to settle for reading almost exclusively only the chapters that are relevant to the material they wish to pursue. While this book is essentially a compilation of various peer-reviewed papers, we have made an effort to unify the notation and to include cross-referencing among the different chapters. The objective of this work is to describe the physical aspects of fluid–structure interaction, rather than the computational details. The book is divided into three parts reflecting this distinction. The Part I (Chaps. 1–7) deals with the Flow-Induced Vibration of Bluff Body structures. We systematically review the numerical modeling and physical insight of vibrating circular and prismatic bodies in isolated and collective ways with proximity and interference effects. While Part II (Chaps. 8–10) covers the reduced-order modeling and deep learning of flow-induced vibration for the stability analysis and the development of control strategies, the Part III presents the modeling of flexible thin structures such as flag fluttering, vibrating flexible plate and membrane aeroelasticity. Vancouver, Canada Xi’an, China Vancouver, Canada

Rajeev Jaiman Guojun Li Amir Chizfahm

Acknowledgements

The authors would like to acknowledge the support and contributions by several individuals, organizations and funding agencies. The contents of book chapters are based on several Ph.D. dissertations and journal papers produced under the supervision of the first author. To begin, we would like to acknowledge the kind support from our host organizations, namely the University of British Columbia (UBC), the National University of Singapore (NUS) and the University of Illinois, Urbana-Champaign. We would like to thank many current and former students and colleagues involved in our lab research activities. Especially, we would like to express our appreciation to those individuals whose works have been used directly used in this book: Tharindu Miayanawala, Mengzhao Guan, Bin Liu, Zhong Li, Yun Zhi Law, Narendran Kumar, Vaibhav Joshi, Sandeep Reddy Bukka, Weigang Yao, Pardha S. Gurugubelli, Vaibhav Joshi and among others at the Computational Multiphysics Lab at UBC and the Fluid– Structure Interaction Group (FSIG) at NUS.

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Contents

Part I

Flow-Induced Vibration of Bluff Bodies

1

Introduction: Modeling of Flow-Induced Vibration . . . . . . . . . . . . . . . 1.1 Background and Historial Perspective . . . . . . . . . . . . . . . . . . . . . . . 1.2 Motivating Applications and Challenges . . . . . . . . . . . . . . . . . . . . . 1.3 Physical Modeling of Flow-Induced Vibration . . . . . . . . . . . . . . . . 1.4 Concept of Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Vortex-Induced Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Galloping and Flutter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Nondimensional Parameters . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Data-driven Modeling of FSI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Book Organization: Volume 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 7 11 14 16 16 17 19 21

2

VIV and Galloping of Prismatic Body . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Numerical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Free Vibrations of Single Square Prism . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Effect of Mass Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Effect of Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Effect of Damping Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Physical Investigation of Representative Cases . . . . . . . . 2.3.5 Pure Rotational Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.6 Combined Translational and Rotational Motion . . . . . . . 2.3.7 Interim Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Three-Dimensional FSI of a Square Column at High Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Problem Definition, Convergence and Validation . . . . . . 2.4.2 Energy Spectra in the Near Wake . . . . . . . . . . . . . . . . . . . 2.4.3 Response Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Vorticity Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 25 28 31 34 35 35 37 43 48 53 54 56 59 62 66

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2.4.5 Reynolds Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.6 Self-Sustained Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: Turbulent Energy Balance in the Near Wake . . . . . . . . . .

70 75 81 82

Proximity and Wake Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 FSI of Side-by-Side Square Prisms at Low Reynolds Number . . . 3.2.1 Numerical Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Problem Description and Key Parameters . . . . . . . . . . . . 3.2.3 Two-Dimensional Simulations . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Three-Dimensional Effects at Lock-in . . . . . . . . . . . . . . . 3.2.5 Interim Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 FSI of Side-by-Side Circular Cylinders . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Coupled Fluid-Structure System . . . . . . . . . . . . . . . . . . . . 3.3.2 Problem Setup and Verification . . . . . . . . . . . . . . . . . . . . . 3.4 Gap Flow Interference in Three-Dimensional Flow . . . . . . . . . . . . 3.4.1 Three-Dimensional Gap-Flow Interference . . . . . . . . . . . 3.4.2 Coupling of VIV and 3D Gap-Flow Kinematics . . . . . . . 3.4.3 Interim Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Freely Vibrating Tandem Square Prism . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Response Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Vortex Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Wake Interference of Tandem Circular Cylinder . . . . . . . . . . . . . . 3.6.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Response Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3 Decomposition of Transverse Force . . . . . . . . . . . . . . . . . 3.6.4 Pressure Distribution and Wake Contours . . . . . . . . . . . . 3.6.5 Upstream Vortex and Downstream Boundary Layer . . . . 3.6.6 Effect of Streamwise Gap . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.7 Interim Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Three-Dimensional Wake Interference of Tandem Cylinders . . . . 3.7.1 VIV Dominated Response . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.2 WIV Dominated Response . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: Side-by-Side Stationary Square Cylinders . . . . . . . . . . . . Appendix B: Tandem Cylinders Verification and Convergence Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix C: Three-dimensional Tandem Cylinders Mesh Convergence and Validation . . . . . . . . . . . . . . . . . . . . . . . .

85 85 87 90 92 94 106 109 113 116 117 120 124 128 133 134 136 138 140 146 148 151 156 161 164 165 168 171 172 180 182

Near Wall Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Dynamics of a Circular Cylinder with Wall Proximity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

197 197

185 190

199 201

Contents

4.2

Cylinder VIV in the Vicinity of a Stationary Wall . . . . . . . . . . . . . 4.2.1 Numerical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Problem Definition and Convergence Study . . . . . . . . . . . 4.2.3 Two-Dimensional Results and Discussion . . . . . . . . . . . . 4.2.4 Three-Dimensional Results and Discussion . . . . . . . . . . . 4.2.5 Interim Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effects of Wall Boundary Layer Thickness . . . . . . . . . . . . . . . . . . . 4.3.1 Problem Description and Convergence . . . . . . . . . . . . . . . 4.3.2 Two-Dimensional Results and Discussion . . . . . . . . . . . . 4.3.3 Three-Dimensional Results and Discussion . . . . . . . . . . . 4.3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

202 202 203 208 226 236 237 238 242 251 262

FIV Suppression Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 VIV Suppression by Spanwise Grooves . . . . . . . . . . . . . . . . . . . . . 5.2.1 Problem Setup and Methodology . . . . . . . . . . . . . . . . . . . . 5.2.2 Extruded Grooves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Staggered and Helical Grooves . . . . . . . . . . . . . . . . . . . . . 5.2.4 Assessment on the Performance of Staggered Groove . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.5 Interim Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Appendage Devices for VIV Wake Stabilization . . . . . . . . . . . . . . 5.3.1 Fairing, Connected-C and Splitter Plate Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Assessment at Low Reynolds Number . . . . . . . . . . . . . . . 5.3.3 Assessment at Subcritical Reynolds Number . . . . . . . . . . 5.3.4 Interim Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Near-Wake Jets for FIV Suppression . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Multi-column Offshore Platform by Near-Wake Jets . . . 5.4.2 Validation and Response Characteristics . . . . . . . . . . . . . 5.4.3 Various Configurations of Near-Wake Jets . . . . . . . . . . . . 5.4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

265 265 267 269 278 281

VIV of Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Flow-Induced Vibrations of Bluff Bodies . . . . . . . . . . . . . 6.1.2 Free Surface and Vorticity Dynamics . . . . . . . . . . . . . . . . 6.1.3 Flow-Induced Vibration of Sphere in a Close Proxmity to a Free Surface . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.4 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Numerical Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Two-Phase Flow Modeling with Moving Boundary . . . . 6.2.2 Structural Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Fluid-Structure Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Implementation Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Convergence Study and Validation . . . . . . . . . . . . . . . . . . . . . . . . . .

355 355 356 357

4.3

5

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293 299 300 303 309 320 328 329 333 334 341 353

357 359 361 361 363 364 364 365

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6.4.1

VIV of Fully Submerged Freely Vibrating Elastically Mounted Sphere . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 VIV of Submerged Elastically Mounted Sphere Close to the Free Surface . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 VIV of Elastically Mounted Sphere Piercing the Free Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Vorticity Dynamics with Free-Surface Deformation . . . . 6.5.3 Effect of Mass Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.4 Effect of Froude Number . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix: The Effect of Reynolds Number on the Mode Transition for the Freely Vibrating Elastically Mounted Sphere . . . . . . . 7

Flexible Cylinder VIV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 VIV of Flexible Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 VIV of Long Flexible Riser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Numerical Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Problem Setup and Validation . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Response Characteristics at Uniform Flow . . . . . . . . . . . . 7.2.4 Response Characteristics at Linearly Sheared Flow . . . . 7.2.5 Interim Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Flexible Cylinder with Spanwise Grooves . . . . . . . . . . . . . . . . . . . . 7.3.1 Problem Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Suppression via Spanwise Grooves . . . . . . . . . . . . . . . . . . 7.3.3 Analysis on Spanwise Correlation . . . . . . . . . . . . . . . . . . . 7.3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part II 8

367 369 370 373 375 384 385 392 394 399 399 400 401 402 402 407 410 418 419 420 423 424 425 427

Model Reduction and Control

Data-Driven Reduced Order Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Synchronization of Coupled Fluid-Structure System . . . 8.1.2 Low-Dimensional Models for Wake Features . . . . . . . . . 8.1.3 Objectives and Organization . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Numerical Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Full-Order Model for Fluid-Body Interaction . . . . . . . . . 8.2.2 Low-Order Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Problem Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Assessment of Low-Order Model for Wake Decomposition . . . . . 8.3.1 Linear POD Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Nonlinear POD-DEIM Reconstruction . . . . . . . . . . . . . . . 8.3.3 Drag and Lift Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.4 Wake Feature Interaction and Sustenance of VIV Lock-in . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

433 433 434 435 437 438 438 440 444 446 448 451 454 460

Contents

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8.3.5

8.4 9

Synchronized Wake-Body Interaction at Below Critical Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.6 Effect of Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.7 Force Decomposition Based on Modal Contribution . . . 8.3.8 Performance Comparison of POD Reconstruction Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

System Identification and Stability Analysis . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 VIV Mechanism and System Identification . . . . . . . . . . . 9.1.2 Model Order Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.4 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Numerical Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Full Order Model Formulation . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Eigensystem Realization Algorithm . . . . . . . . . . . . . . . . . 9.2.3 ERA-Based Coupled Formulation of a Cylinder VIV . . . 9.2.4 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Linear Stability Analysis for VIV of a Cylinder . . . . . . . . . . . . . . . 9.3.1 Unstable Flow Past a Stationary Cylinder . . . . . . . . . . . . 9.3.2 Assessment of ERA-Based ROM . . . . . . . . . . . . . . . . . . . 9.3.3 Effect of Mass Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.4 Effect of Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . 9.3.5 Effect of Rounding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.6 Effect of Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.7 Interim Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Stability Analysis of Tandem Cylinders . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 Assessment of ERA-Based ROM for Wake-Induced Vibrations . . . . . . . . . . . . . . . . . . . . . . . 9.4.3 WIV of Tandem Circular Cylinders . . . . . . . . . . . . . . . . . . 9.4.4 Effect of Longitudinal Spacing . . . . . . . . . . . . . . . . . . . . . 9.4.5 Effect of Sharp Corners . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.6 Interim Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Deep Learning for Predicting Frequency Lock-in . . . . . . . . . . . . . 9.5.1 Reduced-Order State-Space Model . . . . . . . . . . . . . . . . . . 9.5.2 Nonlinear DL-Based Model Reduction . . . . . . . . . . . . . . . 9.5.3 Stability Analysis via DL-Based ROM Integrated with ERA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.4 Problem Setup and Hyperparameter Analysis . . . . . . . . . 9.5.5 RNN-LSTM Training Procedure . . . . . . . . . . . . . . . . . . . . 9.5.6 Verification of DL-Based ROM Integrated with ERA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Assessment of DL-Based ROM for VIV of Sphere . . . . . . . . . . . .

464 467 470 475 476 479 479 480 482 483 484 485 485 487 488 490 491 494 495 499 499 503 508 514 516 520 521 522 526 528 531 532 536 537 541 543 546 548 555

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9.6.1 9.6.2

The Role of Structural Mode Instability . . . . . . . . . . . . . . Stability Analysis of Sphere VIV at the Onset of Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.3 Stability Analysis of Sphere VIV at Moderate Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: Derivation of Phase Angle for VIV . . . . . . . . . . . . . . . . . . Appendix B: Fluid-Structure Energy Transfer . . . . . . . . . . . . . . . . . . . . . Appendix C: Assessment with Silverbox Benchmark . . . . . . . . . . . . . . 10 Data-Driven Passive and Active Control . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 Control of Vortex-Induced Vibration . . . . . . . . . . . . . . . . . 10.1.2 Types of Reduced Order Models . . . . . . . . . . . . . . . . . . . . 10.1.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.4 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Numerical Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Full-Order Model Formulation . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Model Reduction via Eigensystem Realization Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.3 ROM for FSI Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Passive Control of VIV via Appendages . . . . . . . . . . . . . . . . . . . . . 10.3.1 Cylinder-Fairing System . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Performance of ERA-Based ROM for Passive Suppression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.3 Modal Decomposition of Wake Features . . . . . . . . . . . . . 10.3.4 Effect of Other Appendages: Splitter Plate and Connected-C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.5 Effect of Characteristic Dimensions . . . . . . . . . . . . . . . . . 10.3.6 Interim Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Active Control of FIV via Near-Wake Jet Flow . . . . . . . . . . . . . . . 10.4.1 Sphere Via Jet-Based Actuation . . . . . . . . . . . . . . . . . . . . . 10.4.2 Assessment of the ERA-Based ROM for VIV of a Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.3 Effect of Near-Wake Jet Flow . . . . . . . . . . . . . . . . . . . . . . . 10.4.4 Interim Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Feedback Control of VIV via Jet Blowing and Suction . . . . . . . . . 10.5.1 Cylinder Blowing/Suction Porous Surface Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.2 Feedback Control Via Reduced-Order Model . . . . . . . . . 10.5.3 Active Feedback Control of Cylinder Unsteady Wake Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.4 Sensitivity Study for Unsteady Wake Flow Control . . . . 10.5.5 Feedback Control of Vortex-Induced Vibration . . . . . . . . 10.5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

556 557 564 567 569 570 572 573 574 574 575 576 576 577 577 578 580 582 582 586 590 596 599 602 603 607 614 620 629 631 633 636 639 642 646 651

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Part III Flow-Induced Vibration of Thin Structures 11 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background and Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Specific Applications and Challenges . . . . . . . . . . . . . . . . . . . . . . . 11.3 Flow-Induced Vibration of Flexible Thin Structures . . . . . . . . . . . 11.3.1 Flow-Excited Instability and Synchronization . . . . . . . . . 11.3.2 Nondimensional Parameters . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Organization: Volume 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

655 656 660 662 662 665 667

12 Theoretical Background of Flexible Plate . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.1 Theoretical Studies of Flapping Foils . . . . . . . . . . . . . . . . 12.1.2 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Linear Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.2 Determination of Fluid Loading . . . . . . . . . . . . . . . . . . . . . 12.2.3 Added Mass Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.4 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

669 669 671 673 673 674 676 679 684 696

13 Isolated Conventional Flapping Foils . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.1 Experimental Studies on Flapping Foils . . . . . . . . . . . . . . 13.1.2 Numerical Simulations of Flapping Foils . . . . . . . . . . . . . 13.1.3 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Numerical Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1 Fluid-Structure Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.2 Variational Quasi-Monolithic Formulation . . . . . . . . . . . . 13.3 Two-Dimensional Flapping Dynamics . . . . . . . . . . . . . . . . . . . . . . . 13.3.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.2 Flapping Dynamics and Response Study . . . . . . . . . . . . . 13.3.3 Effect of Nodimensional Bending Rigidity . . . . . . . . . . . 13.3.4 Effect of Mass Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.5 Effect of Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . 13.3.6 Net Energy Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.7 Traveling Wave Mechanism . . . . . . . . . . . . . . . . . . . . . . . . 13.3.8 Interim Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Three-Dimensional Flapping Dynamics . . . . . . . . . . . . . . . . . . . . . 13.4.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.2 Role of Aspect Ratio on the Onset of Flapping . . . . . . . . 13.4.3 Flapping Dynamics of Foil with Spanwise Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.4 Effect of Mass Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.5 Effect of Aspect Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.6 Traveling Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

697 697 698 700 702 703 703 705 708 708 712 714 720 723 723 728 733 734 734 734 737 739 747 750

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13.4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 751 14 Proximity Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.1 Proximity Effects in Flapping Foils . . . . . . . . . . . . . . . . . . 14.1.2 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Numerical Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.1 Governing Equations for Fluid-Foil System . . . . . . . . . . . 14.2.2 Multibody Combined Field Formulation . . . . . . . . . . . . . 14.3 Side-by-Side Foil Arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.2 Effect of Gap on Coupled Dynamics . . . . . . . . . . . . . . . . . 14.3.3 Interaction Dynamics of Gap Flow with Flapping . . . . . . 14.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

753 753 753 755 756 756 758 761 761 763 779 797

15 Trailing Edge Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1.1 Effect of the Trailing Edge Shape and Flexibility . . . . . . 15.1.2 Drag-Thrust Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1.3 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Numerical Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 Flapping Foils With Varying Trailing Edge . . . . . . . . . . . . . . . . . . . 15.3.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.2 Flapping Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.3 Flow Field and Wake Structures . . . . . . . . . . . . . . . . . . . . . 15.3.4 Unsteady Momentum Transfer and Thrust Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.5 Drag-Thrust Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.6 Added Mass Effect on Thrust Generation . . . . . . . . . . . . . 15.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

799 799 800 802 803 804 805 805 807 812

16 Isolated Inverted Flapping Foils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1.1 Inverted Flapping Foils . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1.2 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Numerical Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.1 Fluid-Structure Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.2 Variational Quasi-Monolithic Formulation . . . . . . . . . . . . 16.3 Two-Dimensional Flapping Dynamics . . . . . . . . . . . . . . . . . . . . . . . 16.3.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.2 Development of Flapping Instability . . . . . . . . . . . . . . . . . 16.3.3 Effect of Bending Rigidity . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.4 Effect of Mass Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.5 Transition to Deformed Flapping Regime . . . . . . . . . . . . 16.3.6 Formation of Leading Edge Vortex . . . . . . . . . . . . . . . . . . 16.3.7 Vortex Organizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.8 Net Energy Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

833 833 833 835 836 836 838 841 841 844 849 854 855 858 859 862

818 824 826 830

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16.3.9 Interim Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4 Three-Dimensional Flapping Dynamics . . . . . . . . . . . . . . . . . . . . . 16.4.1 Problem Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4.2 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4.3 Flow Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4.4 Effect of Splitter Plate Behind Inverted Flapping . . . . . . 16.4.5 The Role of Vortex Shedding on Inverted Foil Flapping at Low Reynolds number . . . . . . . . . . . . . . . . . . 16.4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix: A Simplified Analytical Model for the Effect of Elasticity and Inertia on LAF Phenomenon . . . . . . . . . . . . . . . . . . . . . .

865 866 869 871 875 881

17 Thin Structure Aeroelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.1 Review of Studies on Morphing Membrane Wings . . . . . 17.1.2 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Partitioned Coupled Fluid-Structure Formulation . . . . . . . . . . . . . 17.3 Two-Dimensional Thin Structure Aeroelasticity . . . . . . . . . . . . . . 17.3.1 Problem Setup and Mesh Convergence Study . . . . . . . . . 17.3.2 Membrane Dynamics as a Function of Angle of Attack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.3 Effect of Flexibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.4 Effect of Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . 17.3.5 Interim Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4 Three-Dimensional Thin Structure Aeroelasticity . . . . . . . . . . . . . 17.4.1 Problem Setup and Validation . . . . . . . . . . . . . . . . . . . . . . 17.4.2 Membrane Aeroelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

899 899 899 904 904 907 907

18 Aeroelastic Mode Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1.1 Mode Decomposition and Mode Selection in Fluid-Structure Interaction . . . . . . . . . . . . . . . . . . . . . . . 18.1.2 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 Global Fourier Mode Decomposition . . . . . . . . . . . . . . . . . . . . . . . . 18.2.1 Data Collection for Aeroelastic Mode Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2.2 Global Fourier Mode Decomposition Algorithm . . . . . . . 18.3 Mode Selection of Three-Dimensional Flexible Thin Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.1 Aeroelastic Mode Decomposition . . . . . . . . . . . . . . . . . . . 18.3.2 Effect of Flexibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.3 Aeroelastic Mode Selection Strategy in Separated Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

887 890 894 895

911 914 916 918 919 919 923 926 929 930 930 933 934 934 936 938 938 943 949

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Contents

18.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954 19 Flow-Excited Instability in Thin Structure Aeroelasticity . . . . . . . . . 19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.1.1 Flow-Excited Instability in Morphing Membrane Wings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.1.2 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2 Flow-Excited Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2.1 Stability Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2.2 Effect of Mass Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2.3 Effect of Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . 19.2.4 Effect of Aeroelastic Number . . . . . . . . . . . . . . . . . . . . . . . 19.2.5 Onset of Flow-Induced Membrane Vibration . . . . . . . . . . 19.2.6 Mode Transition in Flow-Induced Vibration . . . . . . . . . . 19.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

957 957 957 962 963 963 967 971 974 978 984 987

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 989

Part I

Flow-Induced Vibration of Bluff Bodies

Chapter 1

Introduction: Modeling of Flow-Induced Vibration

Fluid-structure interactions involve the coupled behavior of fluid flow with moving or deformable structures and they are commonly observed phenomena in our daily life. The importance of fluid flow interacting with a flexible structure has been well acknowledged in science and engineering. Flow-induced vibrations are particularly one of the most ubiquitous coupled fluid-structure phenomena which are of engineering interest across a variety of mechanical fields ranging from aerospace/marine applications, such as modeling of aircraft and flying vehicles or large-scale offshore structures and long flexible risers, to biomedical applications such as modeling of blood flows in vessels. Successful prediction and control of fluid-induced loads and vibrations can lead to safer and cost-effective structures. Depending on geometric and physical conditions, there are various types of flow-induced vibrations such as vortex-induced vibration, flutter/galloping, buffeting, wake-induced vibration, etc. Understanding the mechanisms of various forms of flow-induced vibrations has tremendous importance to industrial systems. For example, the interaction of fluid flow over slender flexible structures finds special interest in offshore and wind engineering from a safety and structural reliability standpoint. Fluid-structure interactions can also pertain to flapping phenomena in fish swimming, flight of birds and insects, and fluttering of flags and leaves. The feedback process between the ambient fluid and the deformation of the flexible structure forms a coupled nonlinear dynamical system, whereby the flapping or fluttering motion is characterized by a mutual coupling of the fluid flow and the flexible body. Such flapping dynamical effects are important owing to their potential applications in designing devices for micro-energy harvesting, efficient propulsion, flow separation control, drag reduction, bio-prosthetics and sensing devices.

1.1 Background and Historial Perspective Flow-induced vibration of elastic bodies can be manifested in various forms, particularly transverse vibrations of a bluff body section in steady incident flow attracting attention in various engineering applications. While a bluff body section implies the © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Jaiman et al., Mechanics of Flow-Induced Vibration, https://doi.org/10.1007/978-981-19-8578-2_1

3

4

1 Introduction: Modeling of Flow-Induced Vibration

(b) (b)

(a) (a)

(c) Fig. 1.1 Visualization of low Re flows around a circular cylinder at: a Re = 0.16, b Re = 26, and c Re = 140 for flow from left to right. Adapted from Van Dyke (1982)

existence of flow separation accompanied by two shear layers bounding a relatively broad wake, a steady incident flow means that there exists no organized transient or oscillatory character in the flow. The mutual coupling between body motion and flow separation can occur in a particular time window determined by the vortex shedding. As the vortices shed from either side of the structure in Kármán vortex street, alternating forces at certain frequencies are induced in streamwise and transverse directions. The vortex shedding and the dynamical quantities describing the flow around a cylinder depend on the Reynolds Number, Re, which is defined as Re ≡

ρU D , μ

(1.1)

where ρ is the density of the fluid, U is the free stream velocity, D is the diameter of the cylinder, and μ is the dynamic viscosity of the fluid. As the Reynolds number is increased from zero, various flow patterns can be observed. For very low Re(Re  4), the fluid wraps around the cylinder with boundary layers fully attached, which means that the boundary layers do not separate from the cylinder at any point. One example of such flows is illustrated in Fig. 1.1a which depicts a steady flow. For 4  Re  45, the flow separation can be observed, as shown in Fig. 1.1b. Symmetric recirculation bubbles are formed immediately downstream of the cylinder. Further increase of Re

1.1 Background and Historial Perspective

5

Fig. 1.2 Vortex regimes behind circular cylinders

results in unsteady vortices and the flow bifurcates to a time-periodic state where opposite-signed vortices are periodically shed from both the top and bottom cylinder surfaces. A vortex wake, normally termed as von Kármán vortex street is observed in the wake of the cylinder, as shown in Fig. 1.1c. This periodic vortex shedding generates oscillatory forces exerting on the cylinder which can be decomposed into drag in the streamwise direction and lift in the transverse direction. Various regimes of vortex shedding from a smooth circular cylinder are given by Fig. 1.2. Further details on the mechanism of vortex shedding and various regimes can be found in [207, 401]. A pioneering study of a circular cylinder excited into vibration by a flowing fluid can be attributed to Strouhal [400] and Rayleigh [356]. According to the findings of Strouhal and Rayleigh, the frequency f vs of the aeolian tone generated by the relative motion of the cylinder (or wire) and the air varies with the diameter D and

6

1 Introduction: Modeling of Flow-Induced Vibration

with the speed U of the relative motion. The value of the dimensionless group the so called Strouhal number St = f vs D/U is dependent upon the Reynolds number, the geometry and roughness of the cylinder surface and free stream turbulence. Thus the vortex shedding frequency is determined for the flow past a bluff body given by: f vs = St

U , D

(1.2)

where f vs denotes the vortex shedding frequency, U is the free stream flow velocity approaching the cylinder, and D is the cylinder diameter. The coupling between body motion and flow separation can occur in a particular time window determined by the vortex shedding. When the cylinder is elastically mounted rather than being fixed, vibrations may occur due to the presence of the oscillatory forces imposed by the fluid flow. This vibration occurs predominantly in the transverse direction and is termed as the vortex-induced vibration since the cause of the forces is associated with the vortex shedding [113]. This interaction between the cylinder motion and the flow separation makes the coupled response very complicated. During the interaction, the cylinder displacement alters the flow field and the flow field exerts forces on the cylinder. The synchronized vibrations of a circular cylinder are well documented in the experiment of Feng [113]. The experimental set-up involved was a flexibly-mounted circular cylinder with one degree of freedom of movement in the transverse direction. Figure 1.3 shows the transverse response amplitude and the frequency of the flexibly-mounted circular cylinder subject to steady air flow. There was no vibration experienced until the velocity reaches a value of about Ur = 4, where Ur = U/D f n denotes the nondimensional velocity or reduced velocity. A characteristic feature of vortex-induced vibration is that of the lock-in phenomenon, where the vortex shedding frequency, f vs , deviates from the Strouhal’s relationship (vortex shedding frequency of a stationary cylinder) and becomes equal or close to the cylinder’s natural frequency. In the range of lock-in or synchronization, the frequency of vortex shedding, f vs , or the frequency of lift force oscillation is in sympathy with the frequency of cylinder vibration, f . They both lock into the natural frequency of the system, f n , i.e. f vs ≈ f ≈ f n . The existence of a well-defined lock-in range at which the cylinder vibrates with the highest amplitudes due to wake-body synchronization. In the figure, the lock-in can be seen in the range between 5 < Ur < 7. In this range, the vortex shedding frequency is controlled by the natural frequency, instead of the Strouhal relation. After a further increase in the velocity beyond the lock-in/synchronization range, the shedding frequency desynchronizes from the natural frequency and there appears an abrupt change to follow the Strouhal trend again (i.e., Ur ≈ 7.3). Owing to the sudden phase transition, the experiments showed hysteresis effects in the amplitude variation. The pioneering work described is typical for a flexibly-mounted cylinder exposed to airflow. When the cylinder is exposed to water flow, the lock-in/synchronization exhibits in a similar way but the coupled frequency will be different. This is illustrated in Fig. 1.4. Detailed discussions on vortex shedding modes, the frequency and phase analysis using experimental measurements can be found in the reviews of [40, 381, 455].

1.2 Motivating Applications and Challenges

7

Fy

cx

y

U

D

Fx

x

m ky

kx cy

(a)

(b) Fig. 1.3 Frequency lock-in phenomenon in a flexibly-mounted cylinder in a flow: a schematic of 2-DOF system, b experimental data for frequency and amplitude trends reported in [113]

1.2 Motivating Applications and Challenges Flow-induced vibration of elastic bodies can be manifested in various forms which can be of importance to numerous scientific and engineering systems. The applications of fluid-structure interactions are widespread ranging from flow across aerospace, biomedical and civil to marine/offshore structures such as mooring lines, risers and subsea pipelines. These applications involve highly complex multiscale and nonlinear multiphysics dynamics which are extremely challenging to predict and analyze via physical experiments and analytical techniques. Despite decades of research, many of the complex fluid-structure interaction phenomena, particularly the synchronization of the wake and the bluff body motion are still not well understood.

8

1 Introduction: Modeling of Flow-Induced Vibration

Lock-in Lock-in 2 f fn

2

f fn

1

1

Strouhal relation 2

4

6

U fnD

8

10 12

Strouhal relation 2

4

6

U fnD

8

10 12

Fig. 1.4 Schematics of lock-in behavior of a flexibly-mounted cylinder in air flow (left) and water flow (right)

The main objective of this book is to explain the mechanism of wake-bluff body synchronization in fluid-structure interaction systems and provide low-dimensional modeling with stability analysis and control strategies. This book is concerned with the numerical modeling and physical insight of coupled fluid-structure interaction problems that arise in several engineering applications. Most engineering problems for external flows can be classified as streamline and bluff body. While streamlined bodies exhibit flows following contours of the body, flow separation with vortices formed by rolling of shear layers is a common feature of bluff bodies. Flow in pipes and arteries can be characterized as internal flows. No matter how complex is the shape of a certain object, fluid forces and moments acting on the object are entirely by two factors, namely pressure distribution and the viscous shear stress over the wetted surface of the object. Pressure and shear stress act normally and tangentially on the surface of interest, respectively. Together with computational modeling, we present the underlying physics of fluid-structure interaction for a variety of canonical problems. Bluff body flows are ubiquitous in nature and engineering structures, such as wind blowing over high-rise buildings, offshore platforms and pipelines, aircraft at a high angle of attack, and suspension bridges. Coupled fluid-structure interactions can lead to a great variety of flow-induced vibrations, both in catastrophic and valuable (energy harvesting) ways in numerous engineering and scientific applications. These flow-induced vibrations of bluff bodies are particularly important for offshore structures that are increasingly deployed in the ocean environment. In this section, some of the motivating applications and challenges dealing with the FSI applications are briefly discussed. The offshore floating system is an integrated coupled dynamical system comprising an offshore floating platform, risers, and moorings interacting with ocean currents and waves. These offshore structures undergo self-excited vibrations and coupled fluid-elastic instabilities, which pose a great challenge for numerical and mathematical modeling. Figure 1.5) shows several types of offshore platforms for a

1.2 Motivating Applications and Challenges

9

Fig. 1.5 Schematics of various types of offshore oil/gas platforms (top) and offshore wind turbine configurations (bottom). The images are taken from Wikipedia

10

1 Introduction: Modeling of Flow-Induced Vibration

Fig. 1.6 Semi-submerisble operating in an ocean enviroment (left) and the physical modeling of flow-induced vibration in semi-submerisble (right). The left image is taken from Wikipedia

range of water depths for the oil and gas fields and wind turbines. One of the most popular floating platforms in marine/offshore engineering is a semi-submersible platform (Fig. 1.6) which has a specialized offshore design providing good stability and sea-keeping characteristics. A generic shape of the semi-submersible platform is composed of submerged pontoons and vertical columns. The submerged pontoons and topside deck structure are connected by vertical columns at the water-plane area. Predicting fluid-structure interaction in offshore floating structures such as semisubmersible is a challenging task due to complex wake interference, vortex-induced vibrations, galloping and other fluid-elastic instabilities. These coupled instabilities associated with rhythmic oscillations are undesirable for the riser and mooring fatigue. Furthermore, the flow interference and shielding effects of tandem and sideby-side configurations significantly alter the wake dynamics and loads on the offshore floating structure. The development of high-fidelity tools and the discovery of new physical mechanisms can naturally lead to a host of new designs and control strategies for practical use. Using full-order FSI simulations, passive and active flow control techniques can be developed for drag reduction and the mitigation of flow-induced vibration (Fig. 1.6) Transport of oil and gas from ocean floor wells to land-based production facilities requires the use of risers and subsea pipelines that can vary in length up to several kilometers. The aspect ratios (length to diameter) of these slender structures are typically in the order of thousands and are subjected to flow-induced vibrations when exposed to ocean currents. Figure 1.7 shows an integrated marine vessel and riser system together with the computational mesh and the representative flow patterns. Based on the marine/wind environment and structural properties, flow-induced vibrations can cause significant dynamic bending stresses, drag force amplification and large deflections on the pipelines and risers from a short-term perspective. From the longterm viewpoint, rapid accumulation of fatigue damage may lead to structural failure if the vibrations are left unchecked. Consequently, it is imperative for the safety

1.3 Physical Modeling of Flow-Induced Vibration

11

Fig. 1.7 Flow-induced vibration in a coupled marine vessel connnected with elastic riser pipeline: schematic of marine vessel-riser system (left), unstructured finite element mesh around ship and riser (middle-left), representative hydrodynamics solution (middle-right), vortex-induced vibration of marine riser with vortex shedding patterns (right)

of offshore operations that flow-induced vibration is adequately predicted on these long flexible structures. In addition, the physical understanding of vibrating flexible structures can help in developing devices and methods to suppress flow-induced vibrations. While experiments are expensive to perform, a high-fidelity computational framework helps to predict the coupled fluid-structure dynamics from a first principle and rigorous physics-based understanding. Fluid flowing around a solid structure can result in dynamic oscillations in the solid and/or fluid. Broadly speaking, there are two major challenges from the viewpoints of physical modeling and control: (i) understanding of the wake-body synchronization, and (ii) robust and accurate numerical tools for predicting flow-induced vibrations. In this book, we target these two challenges by providing a a comprehensive review on the physical modeling using the fully-coupled differential equations and the physical insight from the numerical results. Essentially, we aim to provide a high-fidelity computational representation of complex flow-induced vibration for canonical and large-scale practical problems.

1.3 Physical Modeling of Flow-Induced Vibration The interplay between the continuum fluid and solid fields forms a complex nonlinear dynamical system. All the motivating applications discussed above are highly nonlinear and involve strong interaction of fluid and elastic structures, and thus require fully-coupled modeling and simulation of fluid-structure interaction. Owing to the growth of computers and advanced methods, full-scale FSI modeling has begun to be realized. A coupled fluid-structure analysis represents a special class of multiphysics problems in which it is important to study the effects of fluid flow on flexible

12

1 Introduction: Modeling of Flow-Induced Vibration

Fig. 1.8 Coupled fluid-solid system: a Illustration of interface condition and domains, and b sketch of feedback mechanism between fluid and solid through exchange of dynamical and kinematical data

structures and their subsequent interactions. Throughout this book, we consider the continuum hypothesis whereas solid, liquid or gas is assumed to be continuous. We briefly discuss the underlying physical modeling of flow-induced vibration using the fully-coupled differential equations of fluid flow and structural dynamics. During the modeling of continuum fluid-structure interactions, three main computational and mathematical difficulties need special attention: (i) treatment of fluid-structure interface to satisfy the kinematic, the dynamic, and the geometric boundary conditions, (ii) conflict in the representation of the dissimilar coordinate frames for the fluid and the structural domains and their motion during the fluid-structure interactions, and (iii) numerical solution of coupled nonlinear partial differential equations for the fluid and the structural systems. Figure 1.8 provides an illustration of interface conditions and fluid-solid mechanics together with a sketch of the feedback mechanism. The coupling between the fluid and solid domains is done through exchanging traction and velocity boundary conditions along the interface. The coupled FSI equations comprise the initial-boundary value problems of the fluid and the structure complemented by the displacement (kinematic) and the traction (dynamic) boundary conditions at the fluid-structure interface. Furthermore, geometric boundary conservation also needs to be satisfied at the moving fluidstructure boundary. The first coupling condition is the kinematic condition and states that the velocity of the fluid is the same as the velocity of the structure at the fluidstructure interface. This means that the fluid will stick to the structural boundary, which is the moving interface. This model is similar to the no-slip condition in viscous fluid dynamics. The second condition is known as the dynamic condition and prescribes a balance of the forces across the fluid and the structural domain. The fluid forces on the structure include the integration of the pressure and the shear stress effects at the interface. For body-fitted discretizations of the domain, the interface and the mesh are required to move together keeping the conformity of the mesh (see Fig. 1.9). Throughout this book, we consider a body-fitted interface

1.3 Physical Modeling of Flow-Induced Vibration

13

(a)

(b) Fig. 1.9 Illustration of unstructured body-fitted meshes for Eulerian-Lagrangian FSI: a freely vibrating spherical body at free surface, and b thin elastic structures undergoing flapping motion

with exact geometry tracking using the arbitrary Lagrangian-Eulerian formulation (ALE). The methods based on ALE coordinates are Lagrangian at the interfaces and employ moving mesh strategies with Eulerian coordinates away from the interface. The challenge lies in handling the mesh motion when large deformation or motion as well as topological changes in the interfaces are involved. With regard to the solution of coupled FSI systems, one can consider monolithic or partitioned schemes. A monolithic approach assembles the fluid and structural equations into a single block and solves them simultaneously for each iteration. These schemes lack the advantage of flexibility and modularity of using existing stable fluid or structural solvers. However, they offer good numerical stability even

14

1 Introduction: Modeling of Flow-Induced Vibration

Fig. 1.10 Coupling methods for fluid-structure interaction: a monolithic versus paritioned coupling, b partitioning of fluid-solid domain

for problems involving very strong added mass effects. In contrast, a partitioned approach solves the fluid and structural equations in a sequential manner, facilitating the coupling of the existing fluid and structural program with minimal changes. This trait of the partitioned approach, therefore, makes it an attractive option from a computational point of view. An illustration of monolithic and partitioned coupling is shown in Fig. 1.10.

1.4 Concept of Synchronization Synchronization is omnipresent in nature, science and engineering, which represents an intrinsic tendency to operate in synchrony or rhythm. The process of synchronization can be observed in various oscillating structures in a flowing fluid such as

1.4 Concept of Synchronization

15

oscillating elastic bodies, flapping flags, and fluttering leaves. Owing to a mutual or feedback interaction, the adjustment of rhythms of vibrating systems (e.g., fluid flow and structure motion) is the fundamental characteristic of the synchronization process. The word synchronous implies “sharing the common time window”. Broadly speaking, in physics, such oscillatory systems are denoted by self-sustained oscillators that can transform a continuous source of energy into an oscillatory movement until the source of energy is removed. We can refer to this as an autonomous dynamical system. Once triggered, in an autonomous system, the oscillation is stable to perturbations and the form of oscillation is determined by the system parameters. During self-sustained oscillation via synchronization, fluid-structure systems can exhibit rhythms of various forms. The rhythm during synchronization is characterized by the oscillation frequency. The frequency of an isolated or uncoupled system can be referred to as the natural frequency. When fluid and solid interact weakly, a mutual coupling is established, which can lead to synchronization. That means two systems have different uncoupled oscillation frequencies, when coupled, they adjust their rhythms and trigger to oscillate with a common frequency. This phenomenon is also generally referred to as frequency lock-in or entrainment. Two systems whether they undergo synchronization depend on two factors, namely coupling strength and frequency match. While the coupling strength describes whether the interaction is weak or strong, frequency match characterizes whether one system (fluid oscillator) can tune the frequency of another system (structural oscillator). It is worth mentioning that synchronization is a self-sustained process and it is different than a well-known phenomenon in oscillating systems namely resonance. The rhythm is entirely determined by the properties of the systems. The self-sustained oscillations are maintained by an internal source of energy that overcomes the dissipative forces in the system. In other words, self-sustained oscillations are non-decaying stable oscillations in an autonomous dissipative system. In fluid-solid systems, there is a dissipation of energy due to viscosity and the continuous energy source is the uniform flow speed. While stability with respect to perturbations, and nonlinearity is essential for maintaining stable oscillations. The interplay of the viscous dissipation and supply of energy from an inflow determine the self-sustained oscillations, based on the system parameters. Analogoulsly, fluid-structure interactions can be considered as two mutually interacting coupled oscillators. Synchronization or frequency lock-in is a remarkable feature during the interaction. In general, the interaction between the fluid and solid systems is nonsymmetrical and bi-directional. Notably, the solid oscillator is more powerful than the fluid oscillator. In the FSI system, the frequency of the driven fluid system is pulled towards the frequency of the driver solid system. When the system parameters are in a certain range, frequency lock-in appears as the mutually adjusted frequency. Indeed, two systems mutually adjust their frequencies until they achieve a compromise; the resulting frequency depends on the initial detuning. In fluid-structure systems, frequency lock-in or synchronization could give rise to VIVs, flutter, or other types of instabilities. Thus, VIVs and flutter can both be viewed as manifestations of synchronization, and not two different categories.

16

1 Introduction: Modeling of Flow-Induced Vibration

1.4.1 Vortex-Induced Vibration Vortex-induced vibration is a well-known coupled fluid-structure phenomenon in which strong vortices are shed in the wake of bluff body structures. The vortexinduced transverse oscillation is a type of synchronized response to the periodic surface loading caused by the discrete wake vortex street formed from the shear layers separating from the bluff cross-section. Synchronization of the vortex frequency with the structural response frequency happens over a discrete flow speed range accompanied by relatively large amplitude vibrations and phase shift of the loading in the synchronization range. Such wake instability-induced excitation vibrations are self-limiting and the vibration amplitudes are typically around one cross-sectional unit length of the structure. For a flexibly-mounted circular cylinder, the characteristic frequency is its natural frequency and for the wake, it is termed the Strouhal frequency. During flow-induced vibration, the phenomenon of frequency lock-in or synchronization occurs for a certain range of fluid and structural parameters and the preferred frequency of wake deviates from its expected value determined by the Strouhal relation while being close to the value of the natural frequency of the structure.

1.4.2 Galloping and Flutter Flow-induced vibrations may be caused by effects other than fluid instabilities and unsteadiness such as vortex shedding. For example, galloping vibrations and aircraft flutter arise from fluid forces by structural vibration in a fluid flow. While flutter is traditionally used as a coupled torsion-plunge instability of airfoil structures, galloping is one-degree-of-freedom instability of bluff structures in winds and oceanic flows. In galloping, the body shape may be such that a small torsional motion of the body causes a flow asymmetry. This, in turn, creates a force that drives the body in the direction of its initial motion, resulting in galloping instability. The classic example of galloping is the vibration of ice-coated power lines because the ice cover normally forms an asymmetric (unfavorable) shape. Other examples include the vibrations of a group of cylinders and the vibration of a flowline attached to one leg of an offshore tower in the area of offshore engineering. We can further elaborate on some features of the galloping phenomenon from an engineering viewpoint. The motion of the structure is slow compared to the motion of the fluid hence quasi-steady assumption is reasonable for studying galloping. In other words, the flow has time to adjust to the motion of the structure and the fluid forces (lift and drag) remain nearly the same at each angle as the value measured statically. Generally, galloping occurs at higher velocities than vortex shedding frequency and the vibration amplitude increases with increasing velocity after the critical velocity. In a way, galloping is induced by the body motion rather than flow fluctuations and thus it can occur in steady and low turbulent flows if a non-circular section

1.4 Concept of Synchronization

17

is disturbed transverse the flow. Once disturbed, the transverse force increases in the direction of motion aiding movement and the body will move further; until opposing stiffness overcomes fluid loading or until the transverse force decreases with increasing movement. Similar to galloping, flutter is a self-excited oscillation that can occur in coupled fluid-structure systems. A steady-flowing fluid interacting with elastic structures can spontaneously cause periodic oscillations. Movement-induced excitation associated with asymmetric aerodynamic force oscillations usually continues to grow as reduced velocity increases. Flutter is the result of fluid forcing alone, which may have vortex shedding due to dynamic interactions with the geometry of the solid. The feedback may lie in the exchange of fluid dynamic forces exerted on a solid body, and the dynamic mechanical (elastic/inertial) forces of the solid exerted back on the fluid. Flutter is generally exhibited at a critical velocity below which the structural damping exceeds the energy received from airflow, and flutter does not occur. When damping is overcome, and oscillations increase in amplitude as a result of positive feedback between fluid and structural (inertial and elastic) forces. Flutter may not involve the existence of spatially discrete vortices that form around the structure and drive its motion. While flutter does not preclude the existence of vortex formation and shedding, the observed vibrations are not predicted or explained by the vortex dynamics. Depending on the flow speed and structural properties, VIV and galloping may occur simultaneously. Various other forms of flow-induced vibrations can occur with multiple bodies. For example, wake interference in tandem flexible bodies can lead to galloping-like phenomena. At higher reduced velocity, when a downstream tandem body lies in the upstream wake and oscillates in a large-amplitude elliptical orbit. This phenomenon can occur in an array of marine risers and electric transmission lines. Side-by-side bluff bodies and a bluff body in the close proximity can also generate some special forms of flow-induced vibration. In this book, we will systematically explore the origin of wake-induced vibration and these special forms of proximity interference using high-fidelity simulations.

1.4.3 Nondimensional Parameters Herein, we briefly highlight the key nondimensional parameters for a freely vibrating bluff body in a flowing fluid. These dimensionless parameters can be used for scaling flow-induced vibration and the coupled dynamical response. Using the governing fluid flow and structure equations, various dimensionless groups can be defined based on the relative importance of forces. Apart from the Reynolds number Re, these FIV instabilities are strongly influenced by three key nondimensional parameters, namely the mass-ratio (m ∗ ), the reduced velocity (Ur ), and the damping ratio (ζ ) defined as:

18

1 Introduction: Modeling of Flow-Induced Vibration

m∗ =

c m U , ζ = √ , , Ur = mf fN D 2 km

(1.3)

where m is the mass of the vibrating body, c and k are the damping and stiffness coefficients, respectively for an equivalent spring-mass-damper system of a vibrating structure, U and D denote the free-stream speed and the chracteristic length of the bluff body, respectively. The natural frequency of the body in a vacuum is given √ by f N = (1/2π ) k/m and the mass of displaced fluid by the structure is m f = (ρπ D 2 L c )/4 and m f = ρ D 2 L c for circular and square cross-sections respectively, where L c denotes the span of the cylinder. The dimensionless Strouhal frequency is St = f vs D/U , where f vs denotes the uncoupled vortex shedding frequency of a stationary cylinder. The mass ratio m ∗ is an indicator of the relative dominance of buoyancy and added mass effects over structural mass. Generally speaking, the strength of synchronization and flow-induced vibration increases as the ratio of solid mass to fluid mass decreases. Lightweight and buoyant structures have more propensity to undergo severe flowinduced vibrations. The reduced velocity can be linked with non-dimensional bending rigidity and aeroelastic number for elastic structures interacting with a flowing fluid. Furthermore, the Cauchy number as a dimensionless number in continuum mechanics can be defined to characterize the relative importance of fluid inertia over elasticity Ca ∼ (fluid inertia)/(solid elasticity) instead of the reduced velocity. The ability of a vibrating structure to dissipate energy plays an important role in flow-induced vibration. Aside from fluid damping, structural damping and material damping can be characterized using the structural damping factor ζ . While structural damping is generally caused by friction and contact effects, the internal energy dissipation of materials can be represented by material damping. Using mass ratio m ∗ and ζ , one can also define the combined mass-damping parameter m ∗ ζ . This is known as the Scruton number in wind engineering. The incoming turbulence and shear can also influence the bluff body flows and flow-induced vibrations. To measure turbulence relative to the free stream fluid velocity, one can define the turbulence intensity I =

u rms , U

(1.4)

and the nondimensional shear in the cross-flow direction as Sh =

D dU U dy

(1.5)

where u rms denotes the root mean square turbulence in the free stream flow and dU /dy represents the shear in the cross-flow direction. The shear can also occur in the streamwise direction. When a bluff body is exposed to an oscillatory flow, the Keulegan-Carpenter number, K C number, can be defined as

1.5 Data-driven Modeling of FSI

19

KC =

Um Tw D

(1.6)

where Um is the maximum velocity and Tw is the period of the oscillatory flow. It can be interpreted similar to the reduced velocity however the velocity Um is defined as the amplitude of velocity of an oscillatory flow with period Tw about a bluff body of diameter D. To measure the effect of compressibility, Mach √ number is defined as M = U/c and one can consider Froude number Fr = U/ (g D), here c and g denote the speed of sound and the gravitational acceleration, respectively. In this book, we consider the incompressible Newtonian fluid whereby Mach numbers are less than 0.3 at which compressibility does not play a significant role. The effects of wall proximity and geometry variations such as the aspect ratio and sharp corner can play an important role in determining the flow-induced loading and vibration. These non-dimensional parameters will be further discussed during the corresponding analysis of flow-induced vibration. The objective of the FSI analysis is to study the amplitude, frequency, and loading characteristics as a function of the governing non-dimensional parameters. Using numerical simulations, we will examine the effects of the wake interference, mass ratio, Reynolds number, reduced velocity, damping ratio and proximity effects with the upstream body and the stationary wall. Finally, we will draw some parallels between flow-induced vibrations in bluff bodies and flapping dynamics in streamline bodies as a dynamic equilibrium between the instability of the flow and the synchronized response of structures.

1.5 Data-driven Modeling of FSI Traditionally the underlying physics of fluid-structure interaction is being explored by theoretical and computational methods together with experimental measurements. For example, designing marine and offshore structures for a specified ocean environment is a complex process due to the highly nonlinear nature of fluid flow interacting with structures. Identifying the flow conditions and/or structural characteristics at which the body shows relatively significant motions and unstable loads is an essential part of this design process. The hydrodynamic behavior of such systems is governed by the Navier-Stokes equations for which the associated equations are a set of nonlinear partial differential equations that yield a wide range of dynamics, including bifurcation, limit cycle oscillation, resonance, and turbulence. Closed-form analytical solutions to the nonlinear equations for typical engineering systems with complicated geometries and complex flow regimes are nearly impossible to achieve. As a result, we tend to rely on high-performance simulations to analyze such nonlinear dynamical systems. In the past decade, the hardware and techniques in experiments and computations have progressed dramatically, which resulted in the production of huge quantities of data. This progress has promoted the development of advanced data-driven methods and machine learning to provide the fourth engineering pillar for improved understanding and controlling of fluid-structure interaction.

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1 Introduction: Modeling of Flow-Induced Vibration

Computational simulations of fluid-structure interaction are beneficial not just for investigating unusual or inaccessible situations, but also for examining a larger range of common scenarios that would be too expensive and time-consuming to investigate with traditional “build-and-test” methods using physical prototypes. While physicsbased computational tools have revolutionized the performance of engineering systems, they have significant challenges for efficient optimization and real-time control. These high-fidelity simulations provide useful information as the ground truth data. However, the massive size of modern databases creates significant analysis challenges. Furthermore, an extremely high amount of data collected from full-order physics-based tools are underutilized, e.g., the data from one simulation are not used to reduce the computational cost of subsequent simulations. These drawbacks of physics-based simulations have increased the demand for innovative data-driven techniques, which are primarily motivated by the need to develop efficient models that can characterize data in meaningful ways. Reduced-order modeling has emerged as a valuable tool in the design of largescale systems while dealing with multi-query analysis, control and optimization of fluid-structure systems. Reduced-order models attempt to replace the full-order model with a lower-dimensional representation capable of expressing the physical properties of the fluid-structure problem described by the full-order model. This is achieved by finding low-rank structures in the full-order data that describe the underlying spatial-temporal dynamics. In recent years, machine learning has witnessed a resurgence with groundbreaking successes in a wide range of technological and engineering applications [230]. Reduced-order modeling is one type of application, in which deep learning models can be used to approximate a physical system in lowdimension and make an inference [71]. For example, using deep learning algorithms, one can construct data-driven reduced-order models of fluid flow [296, 298] and nonlinear dynamical systems e.g., fluid-structure interaction [294]. By extracting fluid features such as vortices and learning the evolution of these vortices from spatialtemporal data, deep learning models can learn complex relationships and patterns for fast and efficient inference. Figure 1.11 illustrates a generic methodology for the integration of high-fidelity and data-driven modeling of fluid-structure interaction. The study begins by obtaining full-order data of the system from a state-of-the-art finite element-based high-fidelity (full-order) solver. These data are typically utilized to understand the synchronization mechanism between the bluff body and the wake. The next step is to perform a lowdimensional projection via proper orthogonal decomposition (POD). Through lowdimensional modeling, the flow fields are decomposed into large-scale features and compared to their behavior in synchronized and desynchronized wake-body systems. Deep learning-based reduced-order models are then built on the high-dimensional data to train and create offline tools that can predict the wake dynamics. Using such a hybrid methodology, one can conduct parametric studies to predict the variation of fluid loading with the change of bluff body geometry and Reynolds number.

1.6 Book Organization: Volume 1

21

Fig. 1.11 Schematic illustrating an integration of high-fidelity model with data-driven reducedorder modeling. Some images are extracted from co-authored papers: [147, 180, 276], and the internet

1.6 Book Organization: Volume 1 The book chapters in Volume 1 are divided into two parts namely Part I and Part II, which are arranged with the increasing complexity of physical modeling and underlying physics. Chapter 2 starts with the numerical modeling and the underlying physics of freely vibrating prismatic square bodies. The study begins with a numerical analysis of 2D flow past a square bluff body at low Re flows. The effects of some structural parameters on bluff body motion and forces are systematically investigated. The relationship between translational and rotational degrees of freedom is examined. We next explore a fully-coupled 3D FSI analysis of a square section in cross-flow at moderately high Re. A self-sustaining process based on quantitative analysis is introduced, together with the analysis of the flow features behind stationary and freely oscillating bluff bodies. Chapter 3 will discuss the proximity and wake interference effects on a prismatic square and circular bodies. We investigate the flow physics of the gap flow and the VIV kinematics in terms of the wake topology, the response characteristics, the force components, and the phase and frequency characteristics. To understand

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1 Introduction: Modeling of Flow-Induced Vibration

the origin of wake-induced vibration, we also consider two circular cylinders in a tandem arrangement where the upstream cylinder is fixed and the downstream cylinder is allowed to vibrate in the transverse direction. We present the characterization of the response dynamics of isolated and tandem cylinders and discuss the basic differences between the two arrangements in terms of force decompositions, phase relations, stagnation point movement, pressure coefficients and the corresponding wake contours. Detailed investigations of the upstream vortex interacting with the boundary layer of the downstream cylinder and the effects of the streamwise gap between the cylinders are provided. Chapter 4 introduces the near-wall effects for the circular cylinders. We present the characterization of the response dynamics of isolated and near-wall cylinders and discuss the basic differences between the two arrangements. The interaction mechanism is detailed from both 2D and 3D perspectives. We provide a regime map summarizing the vortex shedding modes as a function of the reduced velocity and the wall boundary layer thickness is offered in the laminar flow regime. Chapter 5 deals with various suppression devices based on surface modification, wake stabilization and synthetic jet. We start with the mechanism and the results of recently developed staggered grooves to suppress flow-induced vibration of flexible cylinders. We next move to low-drag suppression devices of various types. Detailed contours of vorticity, amplitudes, phase and frequency relations, force histories and wake characteristics provide a deeper insight into complex dynamical interactions between the wake flow and the oscillating cylinder system in the lock-in condition. Simulations of more complex jet-based control are introduced for a freely vibrating 3D semi-submersible model. Chapter 6 focuses on the FIV response of a fully and partially submerged sphere in a uniform flow. We first review the coupled two-phase fluid-structure solver based on the spatially filtered two-phase Navier-Stokes equations in the moving boundary arbitrary Lagrangian-Eulerian framework. The role of streamwise vorticity/free-surface interaction on the VIV response of a freely vibrating sphere is analyzed numerically as functions of the immersion ratio, the mass ratio and the Froude number. Coupled dynamics of unsteady wake-sphere interaction, the force and amplitude characteristics and the vorticity and pressure distributions are investigated during the oscillation. In Chap. 7, we present a systematic study on a slender flexible cylinder immersed in a turbulent flow is performed whereby the flexible cylinder is pinned at both ends. We first provide a brief overview of the governing equations and the numerical methodology for the turbulent fluid-structure interactions of flexible cylinders. Full three-dimensional simulations are carried out for the flexible cylinder exposed at moderate Reynolds number with uniform and linearly sheared profiles. Detailed validation and physical investigation are performed on the response characteristics and vorticity dynamics at various locations along the span of the flexible cylinder. We particularly investigate the standing and traveling wave responses. Finally, we investigate the groove-based suppression device to minimize vortex-induced vibrations in a flexible cylinder.

1.6 Book Organization: Volume 1

23

In Chap. 8, recent literature on the low-dimensional analysis of fluid flows is presented. We present the numerical tools utilized throughout the study, which are: the finite element-based Navier-Stokes solver, proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) techniques. The study begins with a preliminary analysis of 2D flow past a square bluff body. The full-order data of flow past the square cylinder is decomposed into large-scale flow features using POD techniques. The POD energy cascade, nonlinearity capturing and reconstruction quality of the POD methods are examined. Then, the low-dimensional projection of the data is used to explain the synchronization mechanism of the FSI system. This mechanism is further explored in the subcritical Re flows and moderately high Re turbulent flows as well. Chapter 9 presents two advanced physics-based system identification approaches via projection-based and deep-learning-based reduced-order models. The projectionbased approach includes a linear reduced-order model (ROM) for stability prediction using the eigensystem realization algorithm (ERA), which provides a low-order approximation of unsteady flow dynamics in the neighborhood of equilibrium steady state. We perform a systematic ROM-based stability analysis to understand the frequency lock-in mechanism and self-sustained FIV phenomenon by examining eigenvalue trajectories. We present a data-driven coupling for predicting unsteady forces and vortex-induced vibration (VIV) lock-in by using a long short-term memory network (LSTM) as a DL-based ROM technique. The simplicity and computational efficiency of the proposed ROMs allow investigation of the FIV mechanism for a variety of geometries and parameters, and open ways for the development of control devices and on-board and in real-time predictions. In Chap. 10, we present efficient parametric design optimization and control strategies using the model reduction techniques. We explore various suppression devices based on passive-based wake stabilization and active-based synthetic jet. A data-driven model reduction approach based on Eigensystem Realization Algorithm (ERA) is used to construct the reduced order model (ROM) in a state-space format. To establish the reliability of the ERA-based ROM, we first examine the passive-based wake stabilization techniques such as a cylinder-fairing arrangement and connectedC device. A base bleeding mechanism in the near-wake region of a sphere and its influence over the flow dynamics, the wake characteristics and the VIV response are investigated for the freely vibrating sphere system. An active feedback blowing and suction procedure via model reduction is presented for unsteady wake flow and the vortex-induced vibration of circular cylinders.

Bibliography Notes Detailed aspects of computational modeling can be found in the textbook of Jaiman and Joshi [184]. Further theoretical materials on flow-induced vibration can be found in several textbooks e.g. Blevins [56, 330] and review papers by Sarpkaya [381]. A comprehensive discussion on the concept of synchronization is provided in the textbook by Pikovsky et al. [341].

Chapter 2

VIV and Galloping of Prismatic Body

Fluid-structure interaction of prismatic bodies is omnipresent and has numerous applications in offshore, wind and aerospace engineering. Besides their practical importance, these phenomena offer a fundamental value to flow physics of vorticity dynamics and wake-body interaction. In this chapter, we present the numerical modeling and underlying physics of freely vibrating prismatic square bodies. The study begins with a preliminary analysis of 2D flow past a square bluff body at low Re flows. The effects of key fluid-structural parameters on bluff body motion and forces are systematically investigated. The relationship between translational and rotational degrees of freedom (DOF) is explored. We next consider a fully-coupled 3D FSI analysis of the flow past a square section at moderately high Re. The validation and convergence studies are presented for stationary and flow-induced vibration configurations. Several lock-in and desynchronization branches are discussed as a function of mass ratio, reduced velocity and Reynolds number. A self-sustaining process based on quantitative analysis is introduced, together with the analysis of the flow features behind stationary and freely oscillating bluff bodies.

2.1 Introduction Coupled fluid-structure interactions can lead to a great variety of flow-induced vibrations (FIV), both in catastrophic and valuable ways in numerous engineering and scientific applications. These flow-induced vibrations are particularly important for civil engineering (e.g., tall buildings, suspension bridges) and offshore structures that are increasingly deployed for hydrocarbon extraction in ultra-deepwater environments. The asymmetric vortex shedding from the bluff body induces unsteady transverse loads, which in turn may lead to vibrations when the structure is free to vibrate in the transverse direction [56, 330]. Predicting these vibrations in immersed structures such as offshore semi-submersible or suspension bridge decks is a challenging task due to vortex-induced vibrations, translational or torsional galloping and © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Jaiman et al., Mechanics of Flow-Induced Vibration, https://doi.org/10.1007/978-981-19-8578-2_2

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2 VIV and Galloping of Prismatic Body

other fluid-elastic instabilities. These coupled instabilities associated with rhythmic oscillations are undesirable and they can lead structural failure and fatigue damage. These instabilities are strongly dependent on the angle of attack, Reynolds number, mass ratio, reduced velocity, and damping ratio. Moreover, flow-induced vibrations can become more complicated if more than one body is involved. Flow interference and shielding effects of tandem side-by-side configurations can significantly alter the wake dynamics and flow-induced vibrations. There has been a growing demand to understand or optimize the fluid loads and flow-induced motions in wind and marine/offshore systems comprising such prismatic or column structures. Structures submerged in flow stream experience fluid forces and they can undergo flow-induced vibration in certain conditions [56, 330, 401]. Most of the early research on fluid-structure interactions has been carried out for smooth surface-based geometries such as airfoils and circular cylinders. In these smooth geometries, the VIV alone response can be found and they are immune to galloping when flexibly mounted in cross-flow without any interference effects. The presence of sharp corners on a square cylinder largely alters the flow characteristics as compared to the ones with circular/elliptical sections having smooth contours. Besides the angle of incidence, the sharp corners appear as a major influencing factor in the body geometry, that affects the flow separation. The location of the separation points strongly depends on the body shape which in turn governs the wake dynamics and fluid loading. The square cylinder is generally a bluffer body and has a wider wake as compared to the circular counterpart. The other key difference is that the separation points are fixed at some edges of sharp-cornered configurations, while they are free to move around for smooth bodies. The location of the separation point is a function of the angle of attack, which can affect the wake topology and force dynamics of the square cylinder. Owing to this fundamental difference concerning the sharp corners, a square-shaped bluff body immersed in a flowing stream can undergo the combination of both vortex resonance and galloping instability. In this chapter, we consider a configuration of a square-shaped bluff body immersed in a flowing stream undergoing VIV and galloping. These two FIV phenomena are fundamentally and technologically important and were observed in experiments of the freely vibrating square body in a wind tunnel conducted by Bearman et al. [41]. A summary of wind tunnel results for the flow-induced vibrations in prismatic bodies can be found in [335]. Recently, there have been some experimental investigations in [313, 488] on the FIV response regimes for the elastically mounted square cylinder in a water channel in the Reynolds number range 1400 ≤ Re ≤ 10,000. In [488], three representative values of the angle of attack (α) were considered to demarcate the synchronization regimes associated with the VIV and transverse galloping. For m ∗ = 2.64, α = 0o case, the classical response of galloping was recovered, whereby the amplitude followed approximately a linear trend with respect to the reduced velocity Ur beyond a certain critical value of the reduced velocity. A narrow VIV lock-in region was observed around reduced velocity Ur = 4.84 where the frequency of vibration and the vortex shedding is the same, i.e. 1:1 frequency synchronization was observed. The rest of the cases for α = 0o was dominated by the galloping response. The second synchronization region was

2.1 Introduction

27

identified in Ur ∈ [9, 13], with significant contributions of the frequency component at three times the vibration frequency to the total lift force and the vortex lift force. The third regime was found around Ur = 18.0, which had a single dominant frequency of vibration but the vortex lift spectra showed a significant contribution from the frequency five times the body vibration implying a 1:5 synchronization between the vortex shedding and the vibration. In contrast to the experimental investigations,there are a few numerical studies on the free vibration of a square body with sharp corners [33, 153, 194, 385]. In the recent numerical investigations [180, 181, 495] a beating-like phenomenon was observed in the time history of displacements at Ur = 5 for the laminar flow, where the maximum vibration amplitude occurs at the peak of lock-in region. In [33, 194], galloping of a single square column was numerically studied at low Reynolds numbers and found that galloping would happen for the Reynolds number larger than 140 and the amplitude decreased abruptly for decreasing values of the mass ratio m ∗ ≈ 3. There was a systematic investigation of freely vibrating sharp and rounded square cylinders for the laminar flow regime in [185]. With the considered FIV parameters, the author found the VIV response at lower Ur and galloping at the higher side of Ur for basic square configuration, whereas the coupled responses of rounded cylinders were observed to be VIV alone in the synchronization regime. In [180], the authors introduced a stable coupling technique and variational large eddy simulation (LES) formulation for fluid-structure interaction of bluff bodies at a high Reynolds number and low mass-damping parameter. The developed dynamic LES scheme was validated for both stationary and vibrating configurations of a square-section cylinder in three dimensions. The authors examined the phenomena associated with transverse vibrations of elastically-mounted the square cylinder at subcritical Reynolds number for α = 0◦ and 45◦ . The FIV characteristics were reasonably captured and the formation of hairpin-like vortical structures observed for the vibrating square cylinder at zero incidence case. The scope of this chapter is restricted to provide the numerical modeling and physical characteristics of galloping and vortex-induced vibrations in prismatic bodies. There is no attempt to review and discuss the detailed theoretical and experimental works reported for freely vibrating prismatic square bodies. A detailed review of the mathematical modeling and the phenomenon of FIV in prismatic bodies can be found in [335]. Hence the review is selective with a view to cover a numerical treatment of FIV phenomena in freely vibrating prismatic bodies. In the following section, we briefly summarize a coupled fluid-body solver based on the filtered Navier-Stokes and rigid-body equations. Further details about the numerical formulation and mathematical description can be found in [180, 185]. In this chapter, only a single square body of bluff section is considered whereby the flow separation plays an important role to sustain flow-induced vibrations.

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2 VIV and Galloping of Prismatic Body

2.2 Numerical Formulation For the sake of completeness, we briefly summarize the coupled fluid-rigid body solver based on the filtered Navier-Stokes and the rigid-body motion equations used in this numerical study. A coupled fluid-structure formulation based on PetrovGalerkin finite-element and semi-discrete time stepping is adopted in the present work to investigate the interaction of incompressible viscous flow with elastically mounted structures [180, 181]. The spatially filtered Navier-Stokes equations for the incompressible flow in the arbitrary Lagrangian-Eulerian (ALE) frame are    ∂uf  + ρ f uf − w · ∇uf = ∇ · σ f + ∇ · σ sgs + bf on Ω f (t), ρ  ∂t x

(2.1)

∇ · uf = 0 on Ω f (t),

(2.2)

f

where the spatial and temporal coordinates are denoted by x and t, respectively. Here, uf = uf (x, t) and w = w(x, t) represent the fluid and mesh velocities defined for each spatial point x ∈ Ω f (t), respectively, bf is the body force applied on the fluid, and σ f is the Cauchy stress tensor for a Newtonian fluid, written as  T   , σ f = − p I + μf ∇uf + ∇uf

(2.3)

where p denotes the filtered fluid pressure, μf is the dynamic viscosity of the fluid, and σ sgs denotes the extra stress term due to the subgrid filtering procedure for large eddy simulation. In Eq. (2.1), the partial time derivative is kept fixed with respect to the ALE referential coordinate  x . For fluid-structure interaction, the present study involves only the translational degree of freedoms of the rigid-body structures. To simulate translational motion of rigid body about its center of mass, the equation along the Cartesian axes is given by: m·

dus + c · us + k · (ηs (z0 , t) − z0 ) = Fs + bs , dt

(2.4)

where m = (m x , m y , m z ), c = (cx , c y , cz ) and k = (k x , k y , k z ) denote the mass, damping and stiffness vectors per unit length for the translational degrees of freedom. The subscripts x, y and z represent the Cartesian components of the rigid-body coefficient vectors. The rigid-body velocity us (t) at time t is given by us (z0 , t) =

∂ηs , ∂t

(2.5)

where ηs denotes the position vector mapping the initial position z0 of the rigid body to its position at time t, and Fs and bs are the fluid traction and body forces acting on the rigid body, respectively. In the present study, we set the body force bf in the fluid domain and bs on the solid body to zero. Since the solid body is

2.2 Numerical Formulation

29

constrained rotationally and thereby the conservation of angular momentum and the velocity corresponding to the rotations are ignored. The fluid and the structural equations are coupled by the continuity of velocity and traction along the fluidstructure interface. Let Γfs = ∂Ω f (0) ∩ ∂Ω s be the fluid-structure interface at t = 0 and Γfs (t) = ηs (Γfs , t) is the fluid-structure interface at time t. The coupled system requires to satisfy the continuity of velocity and traction at the fluid-body interface Γfs as follows 

uf (ηs (z0 , t), t) = us (z0 , t) ,

(2.6)

σ f (x, t) · ndΓ (x) + Fs = 0,

(2.7)

Γfs

where n is the outer normal to the fluid-body interface and dΓ denotes a differential surface area. In the above interface conditions, the fluid and the structure are coupled in such a way that the fluid velocity is exactly equal to the velocity of the body along its surface. The motion of the immersed cylinder is governed by the fluid forces, which constitute the integration of pressure and shear stress effects on the solid-body surface. The movement of the internal ALE nodes is computed by solving a continuum hyperelastic model for the fluid mesh such that the mesh quality does not deteriorate as the displacement of the body increases. The coupled fluid-structure variables are advanced explicitly and the interface force correction is applied at the end of each fluid sub-iteration [181] to handle low-mass bluff bodies subjected to strong added mass effects of incompressible fluid flow. The temporal discretization of both the fluid and the structural equations is implemented by energy conservative formulation of the generalized-α framework [94] and equal order of interpolation (collocated arrangement) is used for the primitive variables. The linear system of incremental velocity and pressure is evaluated by using the standard Generalized Minimal Residual (GMRES) solver proposed in [370]. At each time step, we perform Newton-Raphson type iterations to reduce the linearization errors between fluid and structural dynamics. We incorporate an explicit large eddy simulation filtering technique which provides a mechanism for resolving large-scale flow features while modeling subgrid-scale stresses via Smagorinsky-based functional modeling [128, 395]. In the following section, we present the convergence study and validation of our fluid-structure solver for stationary and vibrating configurations of a square-section structure immersed in a flowing flow at subcritical Re.

Dynamic Subgrid-Scale Turbulence Model Next we present the dynamic subgrid-scale (SGS) model in our variational fluidrigid body solver. The model relies on the standard explicit filtering based on a scale separation that permits to reduce the computational cost with respect to direct

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2 VIV and Galloping of Prismatic Body

numerical simulation [371]. The dynamic subgrid model applies a filter, denoted by an overline, to the incompressible Navier-Stokes equations to model the scales smaller than the spatial resolution Δ. This operation introduces an extra subgridscale stress, τi j = u if u fj − u i u fj , to the filtered Navier-Stokes equation Eq. (2.1). This stress needs to be modeled because u if u fj involves the unknown SGS quantities u if and u fj . The nonlinear SGS stress tensor can be expressed as: τi j −

δi j 2 τkk ≈ −2μt S i j − C N L 6μt 2 /τkk (S ik Ω k j + S jk Ω ki − 2S ik S k j + S nk S kn δi j ) (2.8) 3 3

f 2 where dynamic eddy  f  viscosity μt = ρ (Cs Δ) |S|, the resolved strain-rate tensor ∂u fj 1 ∂u i S i j ≡ 2 ∂ x j + ∂ xi having norm |S| ≡ (2S i j S i j )1/2 , and the filtered rate-of-rotation  f  ∂u f ∂u tensor is Ω i j ≡ 21 ∂ x ij − ∂ xij . For the Smagorinsky model [395], the theoretical constant value of Cs ≈ 0.17 can be derived by assuming a local equilibrium between the production and dissipation of energy. For C N L = 0, Eq. (2.8) recovers to the standard linear SGS stress model. The dynamic SGS model [128] then proceeds by defining two filters: the grid  The first filter with scale dimension Δ and the test filter with scale dimension Δ. filter is automatically provided by the mesh discretization and the latter may be any coarser level filter. An identity between subgrid-scale stresses generated by different filters exists:

L i j = Ti j − τi j

(2.9)

where the Leonard tensor, L i j , is the stress generated by performing the test filter on the grid filtered data: f f  f f L i j = ui u j −  ui  uj

(2.10)

Ti j =  u i u j − ui uj .

(2.11)

and the SGS stress Ti j is

By considering the Smagorinsky model for the unknown stress tensors τ and T, we arrive at the following relation for the Leonard stress:   S − Δ2 |S|S 2 |S| L i j = −2Cs2 Δ ij ij

(2.12)

2.3 Free Vibrations of Single Square Prism

31

The Lily determination of the Smagorinsky constant leads to the closure

Cds =

1 Mi j L i j

2 Mlk Mlk

(2.13)

  2 | S| S i j − Δ2 |S|S i j and indicates some type of smoothing where Mi j ≡ Δ process such as averaging, planar averaging or Lagrangian averaging and Cds now denotes the dynamic coefficient as opposed to a constant Cs in Eq. (2.8). For Cds > 0, √ we take Cs = C ds , or otherwise Cs = 0. The nodal point eddy viscosity is interpolated to the quadrature points for the evaluation of the element residual and Jacobian matrices. The Smagorinsky model S . We apply L -projection of the S and |S| requires additional test-filtered data for  ij 2 needed data from the quadrature points to the nodal points. For example, the L 2 projection for the filtered quantity φ at the nodal point a (i.e., the test-filtered data) can be given by

 a = e Ωe φ e Ωe

Na φ dΩ Na dΩ

(2.14)

The lumped mass matrix [167] approach has been employed to evaluate the above projection. To construct a smooth filtered set of quantities, we loop over the elements and quadrature points, evaluate the quantities, assemble them in a global vector, and then scale them by a lumped mass matrix. An algebraic eddy viscosity model [31] has been used to assure the correct behavior of μt at the wall. As an alternative to the dynamic Smagorinsky closure, variational multiscale (VMS) was introduced in [166, 169] as a framework to separate the resolved scales into a large- and small-scale and to rationalize stabilization techniques to circumvent numerical difficulties using the standard Galerkin method. This multiscale LES provides a decomposition of the turbulent stress tensor into large and small-scale parts. Several variants of VMS are assessed and found to be as accurate as the standard dynamic model for the turbulent channel problem [432].

2.3 Free Vibrations of Single Square Prism We first simulate a laminar flow over freely vibrating square cylinder in twodimension. The flow-induced vibration of a bluff body is primarily influenced by the four key non-dimensional parameters [381], namely mass-ratio (m ∗ ), Reynolds number (Re), reduced velocity (Ur ), and critical damping ratio (ζ ) defined as:

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2 VIV and Galloping of Prismatic Body

m∗ =

C M ρfUD U , ζ = √ , , Re = , Ur = mf μf fN D 2 KM

(2.15)

where M is the mass per unit length of the body, C and K are the damping and stiffness coefficients, respectively for an equivalent spring-mass-damper system of a vibrating structure, U and D the freestream speed and the diameter of cylinder, √ respectively. The natural frequency of the body is given by f N = (1/2π ) K /M and the mass of displaced fluid by the structure is m f = ρ f D 2 L c for square crosssection and L c denotes the span of the cylinder. In the above definitions, we make the isotropic assumption for the translational motion of the rigid body, i.e., the mass vector m = (m x , m y ) with m x = m y = M, the damping vector c = (cx , c y ) with cx = c y = C, the stiffness vector k = (k x , k y ) with k x = k y = K . In the case of VIV or lock-in range, the structural response frequency f in the immersed fluid approaches to the vortex shedding frequency f = f vs , where f vs is the dominant frequency of vortex shedding. Response in the lock-in is self-limiting and it does not increase without bound. For transverse galloping mode, the structural response frequency is much smaller than the vortex shedding frequency, i.e. f 10, the vibration amplitudes are small (ArYms ≈ 0.05D) upto the reduced velocity Ur = 50. For Re = 150, the desynchronized branch ends around Ur = 15 and the response behavior transits into the galloping mode. In the galloping mode, the vibration amplitudes keep increasing as a function of reduced velocity increases and show a monotonically increasing trend. Notably, the second synchronized kink mode does not occur for Re = 150. For Re = 200, it has shorter desynchronized branch compared with Re = 150 and the galloping mode starts to appear at Ur = 11 and the vibration amplitudes are larger than Re = 150 for similar reduced velocity. The kink mode reported by Bearman [41] appears between the reduced velocity Ur = 15 and 16, which is approximately three times of Ur where the maximum amplitude occurs in the VIV branch. The vibration frequency of transverse amplitude f AY is synchronized with the low frequency 13 f C L , i.e., the modified galloping phenomenon with the 1 : 3 synchronization.

2.3.3 Effect of Damping Ratio The damping ratio ζ is a non-dimensional parameter to describe how rapidly oscillations decay in a system. Since a damping-free system does not exist in real world, the effect of damping ratio is significant in engineering applications, especially during the study of multi-column structures. In Fig. 2.4, the black solid line represents the amplitude with ζ = 0, i.e. no damping case. From ζ = 0.01−1%, the amplitude only decreases 10%. However, the amplitude drops significantly from ζ = 1% onwards and decreases more than 60% when the damping ratio reaches 10%.

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2 VIV and Galloping of Prismatic Body

Fig. 2.3 The effect of Reynolds number on the coupled response: dependence of transverse amplitude on reduced velocity at m ∗ = 3

Fig. 2.4 Dependence of transverse amplitude on reduced velocity for a range of damping ratio at Re = 100, m ∗ = 3

2.3 Free Vibrations of Single Square Prism

37

Fig. 2.5 Representative VIV results of freely vibrating square cylinder at Ur =5 for Re = 200 and m ∗ = 3: a vibrational amplitudes, b force coefficients

Fig. 2.6 Representative galloping results of freely vibrating square cylinder at Ur = 50 for Re = 200 and m ∗ = 10: a vibrational amplitudes, b force coefficients

2.3.4 Physical Investigation of Representative Cases Figs. 2.5 shows some representative time-histories of amplitudes and forces at Ur = 5 for Re = 200 and m ∗ = 3. As expected, the amplitudes of streamwise vibration are much smaller than those in the transverse direction. The VIV response is periodic and regular in the lock-in region, with a single dominant frequency. Typical time-histories of displacements and forces for the transverse galloping are shown in Fig. 2.6 at Ur =50 for Re = 200 and m ∗ = 10. The transverse amplitude and frequency are larger than Ur = 5. Similar to the VIV response, the amplitude of in-line vibration is much smaller than those in the transverse direction. Unlike in VIV, the frequencies of vortex shedding and cylinder oscillation do not match during the galloping. The variations of lift C L and drag C D force coefficients are shown in Fig. 2.6b, which clearly show the subharmonics associated with the galloping mode. Presence of irregular oscillation and shedding frequencies in the forces will eventually lead to quasi-periodicity of galloping [185]. Figure 2.7 shows the wake vortex mode in terms of instantaneous vorticity contours at Ur = 5 and 50 during the instants when the cylinder is at the top and bottom locations. The standard 2S mode of vortex shedding exists for Ur = 5, with vor-

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2 VIV and Galloping of Prismatic Body

(a)

(b)

Fig. 2.7 Instantaneous streamwise vorticity contours for square cylinder at the bottom position (left) and at top position (right): a Ur = 5 b Ur = 50. The wake vortex mode is 2S

tices being shed from each side of the cylinder. For the galloping-dominated branch at Ur = 50, there is still the 2S vortex mode with a significant deflection of wake topology, the downward deflection when the body is at the peak top position and the upward deflection when the body is in the bottom position. At the peak displacement, the component of vortex-induced force due to the wake deflection draws the cylinder towards the mean position. We next examine three specific cases corresponding to the reduced velocity at Ur = 5 for the maximum peak in the lock-in region, Ur = 16 for the transitional kink and Ur = 40 for the galloping at Re = 200 and m ∗ = 3. In the lock-in region (Ur ∈ [3, 7]), the maximum transverse amplitude appears around Ur = 5. In Fig. 2.8, both the time history of force responses and vibration amplitudes are shown to illustrate strong beating patterns. Such beating phenomena was also recently reported in [495] for Re = 100. The beating patterns at Re = 200 are much more obvious than that of Re = 100 as shown in Fig. 2.8, where C Lr ms = 0.49, AY /D = 0.29 and St = 0.17. The vibrating frequency is locked in the range of vortex shedding frequency, therefore the lock-in phenomenon occurs when the vibration of the column resonates with the vortex shedding and results into the large transverse amplitude. The trajectory is skewed figure-8 shape where the top and bottom are stretched toward downstream. As reduced velocity increases to Ur = 16, which is about three times the reduced velocity where the peak happens in the lock-in region, the transverse response amplitude shows the second lock-in region (Ur ∈ [15, 17]) as reported experimentally in [41]. After the desynchronized branch, the characteristics of galloping mode start to dominate in the vibration of the square column. The vibration frequency begins to shift to a low value. At transitional reduced velocity Ur = 16, both VIV and galloping modes contribute to the transverse response. Figure 2.9 shows the variations of the force and vibration amplitudes. The vortex shedding frequencies have two dominating value at St1 = 0.17 and St2 = 0.055 (as shown in Fig. 2.9c), where the first one corresponds to the system natural frequency f N and the second one is one

2.3 Free Vibrations of Single Square Prism

39

Fig. 2.8 Freely vibrating single square column at Ur = 5 (VIV branch): time history plots of a lift and drag coefficients, b lift and transverse vibration response, c power spectrum analysis of lift and transverse amplitude, d X Y trajectory

third of the natural frequency. The vibration frequency is synchronized with the low frequency component corresponding to St2 = 0.055. The trajectory, as shown in Fig. 2.9d, appears to be a complex letter-V shape instead of the skewed figure-8 shape. When the reduced velocity is increased further, the vibration fully develops into galloping mode. The features of galloping mode such as large transverse vibration amplitude and low vibrating frequency can be observed. Figure 2.10 shows the key FIV characteristics at Ur = 40. The lift coefficient contains a superposition of two dominate harmonics, as shown in Fig. 2.10c. The high frequency St1 = 0.17 is the vortex shedding frequency and the low frequency St2 = 0.02 coincides with the galloping vibration frequency. The time history plot of transverse vibration amplitude is a combination of VIV and galloping where the galloping is dominating in this reduced velocity range. Figure 2.10d shows X Y -trajectories, which resemble a “butterfly wing” shape. Due to the combination of VIV and galloping modes, the trajectories have the superposition of low frequency (transverse galloping) and high frequency (vortex synchronization) motions.

40

2 VIV and Galloping of Prismatic Body

Fig. 2.9 Freely vibrating single square column at Ur = 16 (intermediate kink): time history plots of a lift and drag coefficients, b lift and transverse vibration response, c power spectrum analysis of lift and transverse amplitude, d X Y trajectory

While it is known that the branching of flow-induced vibrations has its genesis in two-dimensional low Re flow, the VIV physics of cylinder response captured via simulations at low Re, similar in nature to the upper and lower branches seen at higher Re. Therefore, the 2D laminar study is relevant in identifying the key features of flow-induced vibrations of offshore structures at higher Re.

2D FSI study of a square prism In the dynamics of coupled fluid-body interaction, the flow-induced vibration associated with frequency lock-in, galloping, flow interference may occur for a given range of fluid-structure and geometric parameters [56, 401]. The focus of this study is to present numerical results on the flow past an elastically mounted square-cross section, where the cylinder is free to oscillate in stream-wise and transverse directions and also rotate about its axis. It is known that this particular sharp- cornered geometry is susceptible to vortex-induced vibration (VIV) where the periodic vortex

2.3 Free Vibrations of Single Square Prism

41

Fig. 2.10 Freely vibrating single square column at Ur = 40 (galloping mode): temporal evolution a lift and drag force coefficients, b lift force and transverse vibration response, c power spectrum analysis of lift and transverse amplitude, d X Y trajectory

shedding and the frequency of the cylinder oscillation lock-in or synchronize. Of practical relevance to offshore floating structures, it is useful to consider the effects of yaw motion of the square cylinder as well. There are some recent studies [17, 106, 152, 153, 385, 470, 488, 495] that explore the flow-induced vibrations of a rigid square cylinder with sharp corners. The rounded cylinders undergo vortex-induced motion alone whereas the motion of the basic square is vortex-induced at low Re and galloping at high Re. Table 2.1 summarizes the results of critical analyses performed on similar problems of stationary and vibrating cylinders. During the literature survey on the multi-DOF motion of square cylinders, it has been found that the combined motion of X, Y translation and Z rotation has not been studied systematically. Hence this investigation will contribute to understand the basics of 3-DOF fluid-structure interaction. This numerical study is based on the two questions about the coupled dynamical exchanges between the translational and rotational motions of the square cylinder in a uniform flow: (i) Is there a similarity between the translational and rotational vortex induced vibration response? (ii) How will the vortex synchronization process and VIV dynamics be affected by the rotational motion?

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2 VIV and Galloping of Prismatic Body

Table 2.1 Results of previous studies on flow past circular and square sectional bluff bodies DOF Conditions Re CD C Lr ms C Lmax St Circular cylinder Stationary [59] Stationary [487] YT [59] YT [152] X T + YT [152] Stationary [495] Stationary [495] Stationary [495] Stationary [470] YT [488] YT [17] YT [152] X T + YT [152] Z R [106] Z R [152] X T + YT X T + YT + Z R

100 1.32 1000 1.170 100 2.08 100 1.8682 100 1.8409 Square cylinder 100 1.4520 22.5◦ Incidence 100 1.6600 45◦ Incidence 100 1.8770 Experimental 6300 1.86 Experimental Varies Experimental Varies 250 1.9222 90 1.8657 Aspect ratio = 4 250 Aspect ratio = 4 250 m ∗ = 20, Ur = 4 100 1.4918 m ∗ = 20, 100 1.4947 Ur = 7, Uθ = 4 3D experimental

0.32 0.335 0.2902

0.88 0.4314 0.2021

0.1908 0.5437 0.6539

0.7298

0.2016 0.2013

1.2875

0.164 0.210 0.188 0.1793 0.1647 0.1447 0.1408 0.1388 0.124 0.208 0.127 0.1610 0.1574 0.13 0.1409 0.1416 0.1514

X T , YT and Z R denote the translational DOFs in X & Y directions and rotational DOF in Z direction respectively

Most of the existing experimental observations for square bodies are at significantly higher Reynolds number. In the first part of this study, the flow considered here is a low Re laminar flow. Our results show that important aspects of the FIV dynamics observed in the experiments are also reproduced numerically at lower Re through direct numerical simulation. This finding has been confirmed in several previous numerical studies for freely vibrating cylinder, and it was corroborated that low Re simulations can capture essential elements of FIV physics. Furthermore, direct numerical simulation of high Re flows demand high computational resources and the turbulent models may introduce some assumptions and modeling uncertainties. In contrast, the low Re cases can be simulated without any assumptions and with a relatively low computational cost. Since the aim of the study is to identify the fundamental characteristics of 3DOF motion, the low Re flow will be considered to elucidate the underlying flow-induced vibration.

2.3 Free Vibrations of Single Square Prism

43

2.3.5 Pure Rotational Motion The ultimate aim of the study is to obtain the motion characteristics of a square cylinder with 3DOF (translation in X and Y directions and rotation in Z direction) when placed in a uniform flow. Before proceeding with this coupled problem, two fundamental cases (2 DOF—X & Y translation and 1 DOF—Z rotation) are considered together with the previous studies. Figure 2.1 illustrates a schematic of the two-dimensional simulation domain used for the pure translational motion. The center of the square column is located at the origin of the Cartesian coordinate system. The side length of the square column is denoted as D. The distances to upstream and downstream boundaries are 20D and 40D, respectively. The distance between side walls is 40D, which corresponds to the blockage of 2.5%. The square column is free to oscillate in in-line (stream-wise) and transverse directions. The flow velocity U∞ is set to unity at the inlet and a no-slip wall is implemented at the surface of the square column. The top and bottom boundaries are defined as slip walls. Figure 2.11b shows the schematic of the pure rotation case. Here, all the geometric parameters are kept the same as above. However, the X and Y translations are restricted and the column is allowed to rotate in Z only. Figure 2.11c shows the schematic diagram of the combined translation and rotation motion. Again, all the geometric parameters are kept the same, but, the column is allowed to move in 3DOF. Rotational motion consists of independent structural parameters which are the counterparts of the translational motion, which are inertia ratio (I ∗ ), torsional damping ratio (ζθ ), torsional natural frequency ( f θ ) & torsional reduced velocity (Uθ ) and are defined as: I Cθ 1 kθ U∞ ∗ , Uθ = , (2.18) I = f , ζθ = √ , fθ = I 2π I fθ D 2 I kθ where I and I f are the moments of inertia of the bluff body and the displaced fluid. Cθ and kθ are the torsional damping coefficient and the torsional spring constant.

2.3.5.1

Numerical Convergence and Verification

The domain is discretized using an unstructured finite-element mesh. The grid noted as M1 comprises of 173,89 elements. There is a boundary layer mesh surrounding the square column and triangular mesh outside the boundary layer region. Each side of the square column is discretized with 40 uniformly distributed nodes. The first layer of the boundary layer mesh is placed at 0.01D from the column wall. Besides, the central area surrounding the square column contains 9644 elements. Three more grids are generated where the mesh elements are successively increased by a factor of 2. They are designated as M2, M3 and M4, respectively. The discretized domain, along with a magnified view of the corners of the square column is shown in Fig. 2.12. All cases are run at Re = 100, m ∗ = 3 and Ur = 5. Results of grid convergence study

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2 VIV and Galloping of Prismatic Body

Fig. 2.11 Schematic diagrams of the different cases: a pure translation b pure rotation c translation and rotation combined

2.3 Free Vibrations of Single Square Prism

45

Fig. 2.12 Representative finite element mesh (M1) used for convergence study Table 2.2 Grid convergence study with parameters Re = 100, m ∗ = 3 and Ur = 5 Parameter M1 M2 M3 M4 No. of nodes No. of elements Time-step size Δt Average frequency f¯ ¯ Average amplitude A/L Average drag coeff. C D Lift coeff. rms C Lrms

17,622 17,389 0.01 0.1633 0.0974 1.6231 0.4845

34,302 34,027 0.01 0.1633 0.1996 (13.10%) 1.9941 (18.60%) 0.6042 (19.81%)

71,552 71,199 0.01 0.1633 0.2195 (9.07%) 2.1338 (6.54%) 0.6872 (12.08%)

145,608 145,195 0.01 0.1633 0.2211 (0.72%) 2.1377 (0.18%) 0.6893 (0.30%)

are recorded in Table 2.2 for the lock-in region. It can be seen that values recorded for mesh M3 and M4 differ by less than 1%. Therefore, mesh M3 is selected to proceed with the study.

2.3.5.2

Response Characteristics

Square columns are widely used in semi-submersibles and offshore floating structures. Pure translational analyses in literature reveal the instabilities and hydrodynamic loads of flow-induced motion. However, most of these studies neglect the effect of rotational motion which is frequently experienced by floating structures (since they are not rigidly fixed to the seabed). This section provides an analysis on flow-induced pure rotational motion around Z -axis (yaw) of a square cylinder which will be used as a reference to analyze the combined translation and rotational motion.

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2 VIV and Galloping of Prismatic Body

Fig. 2.13 a Rotational displacement variation with Uθ and b frequency of parameters for m ∗ = 20, Ur = 7.0, Re = 100 and ζθ = 0.05 of pure rotational case

In this study, the results of a pure rotation case with parameters I ∗ = 20/6, Re = 100 and ζθ = 0.05 in Uθ ∈ [3, 16] are presented. Figure 2.13a shows the variation of rotational displacement θr ms with Uθ . The motion has four reduced velocity regimes: Pre-lock-in (≤ 5.0), lock-in (6 − 8), post Lock-in (9 − 12) and galloping (≥ 12.0). The lock-in phenomena has a similar variation as of A y /D vs Ur in pure translational case. The rotational motion undergoes galloping even in this low Re = 100 case which is critical for the stability of the system. Figure 2.13b shows a comparison of the frequencies of important motion parameters together with the classical Strouhal Law. The transverse force obeys the Strouhal Law while the frequency of the Z-moment jumps the natural frequency for Uθ ≥ 13.0. The rotational motion starts to vary in the natural frequency for Uθ ≥ 10.0, which confirms the galloping phenomena. Figure 2.14 explains the variation of force and moment coefficients with Uθ . The lock-in region shows a similar variation of these coefficients as in the translational lock-in region, but in the galloping region, the coefficients escalates drastically. The frequency spectrum for the motion parameters (Fig. 2.15) gives a clear idea of the various harmonics and their effect. In the lock-in region, all the parameters are affected by a single dominant shedding frequency. In the post lock-in region, the natural frequency makes a significant contribution and in the galloping regime natural frequency dominates the behaviour. The usual practices of engineering design will almost always avoid galloping phenomena1 and due to that reason the main focus of the further analysis will be on the lock-in region of the rotational motion. The results of this and previous sections will be used as the baseline for the next study.

1

This phenomenon occasionally occurs in tension leg platforms (TLP) in high current regions.

2.3 Free Vibrations of Single Square Prism

47

Fig. 2.14 Force and moment coefficients variation with Uθ of pure rotation case for m ∗ = 20, Ur = 7.0, Re = 100 and ζ = 0 a drag and lift coefficients b moment coefficient

Fig. 2.15 Frequency spectrum for the pure rotational case a Uθ = 4 b Uθ = 7 c Uθ = 11 d Uθ = 14

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2 VIV and Galloping of Prismatic Body

Fig. 2.16 Transverse displacement amplitude variation with Uθ for m ∗ = 20, Ur = 7.0, Re = 100 and ζθ = 0.05 in combined translation and rotation

2.3.6 Combined Translational and Rotational Motion The two preceding sections separately describe the flow-induced motion of a square cylinder under pure translational motion and pure rotational motion respectively. In this section, the behaviour of flow-induced motion of a 2D square cylinder subjected to translational motion in X and Y directions and rotational (yaw) motion in Z direction are reported. The pure translational case with m ∗ = 20, Ur = 7.0, Re = 100 and ζ = 0 is used as a reference case here. The only difference between the reference case and the other cases is that they are allowed to rotate (yaw) around Z axis. Figure 2.16 shows the response amplitude A y /D vs Uθ variation for m ∗ = 20 and Re = 100. It is evident that the rotational degree of freedom has affected the translational motion. At Uθ = 8.0 it shows a variation of 153%. Figure 2.17 shows the variation of force and moment coefficients with Uθ . Again it is evident that the yaw motion has affected the C D , C L and C M values. Uθ ∈ [7, 11] region shows a secondary lock-in region for the translational motion. Figure 2.18 compares the X Y trajectories of un-yawed (reference case) and yawed motion in the secondary lock-in region. This shows a clear variation in the motion pattern of the cylinder. In the reference case (Fig. 2.18a), the bluff body follows the characteristic ‘figure 8’ shape periodically. When the cylinder is allowed to rotate, it tries to keep the ‘figure 8’ shape as in the reference case, however, throughout many shedding cycles, the fluid forces it downstream. This gives a series of offset ‘figure 8’ shapes that look like a spiral (Fig. 2.18b). These figures emphasize that, even though the translational and rotational parameters are decoupled, the rotational motion has affected the translational motion severely. The explanation for this behaviour is that the rotational motion changes the fluid-solid interaction and affects the vortex shedding and this eventually affects the translational motion.

2.3 Free Vibrations of Single Square Prism

49

a

Fig. 2.17 Force and moment coefficients variation with Uθ for m ∗ = 20, Ur = 7.0, Re = 100 and ζ = 0 for combined translation and rotation a drag coefficient b lift coefficient c moment coefficient

Fig. 2.18 X Y Trajectories for Ur = 7, Re = 100 and m ∗ = 20: a pure translation, b translation and rotation combined

50

2 VIV and Galloping of Prismatic Body

Fig. 2.19 Rotational displacement variation with Uθ for m ∗ = 20, Ur = 7.0, Re = 100 and ζθ = 0.05 for combined translation and rotation case

Figure 2.19 shows the dependence of θr ms on the reduced velocity Uθ variation. The rotational component of the combined motion shows a lock-in region at Uθ ∈ [6, 8] which is similar to the lock-in region observed for the pure rotational motion. However, in the lock-in region, the rotational displacement is much higher in the 3DOF motion. This reveals a very important fact that when the rigid body is allowed to rotate while translating, the two motions affect each other even though the structural parameters are independent. This can be explained by the difference in the vortex pattern when the cylinder is allowed to rotate. Figure 2.20 shows the vorticity plots of m ∗ = 20, Ur = 7.0, Re = 100 and ζ = 0 for the unyawed and yawed motions. It is typical for square cylinders to shed vortices from the stream-wise far edge as seen in the pure translation case (Fig. 2.20a). However, when the cylinder is allowed to rotate, this vortex expands throughout the far edge and merges with the main vortex. This is visible in the lock-in region where the angle of rotation is relatively high. Also, the strength of the vortices is significantly higher in the combined motion. Basic vortices are generated by the rotations of the fluid. Thus, the yawing of the rigid block strengthens the resulting vortices significantly and it eventually changes both the rotational and translational motion parameters. It is known from the Kutta-Zhukhovaski theorem that the increased vorticity or net circulation will promote the lift generation. The contribution of the rotational DOF can also be established by comparing the frequency spectrum of motion parameters. Figure 2.21 shows the comparison of FFTs of the reference case and the yawed case at Uθ = 15.0. The natural frequency of the rotational motion contributes remarkably to the translational parameters. The only possibility is that the fluid has interacted with the solid in the frequency of the rotational motion and that interaction created a significant contribution to the translational motion. Further, Fig. 2.22 clearly shows that in the combined translation

2.3 Free Vibrations of Single Square Prism

51

(a)

(b)

Fig. 2.20 Vorticity contours in a pure translational motion b combined translation and rotation for Re = 100, m ∗ = 20 and Uθ = 8.0 Fig. 2.21 Comparison of FFT: a pure translation b combined translation and rotation

(a)

(b)

and rotation motion, the force and moment coefficients oscillate in a combination of more than one frequencies, thus confirming the explained behaviour. Figure 2.23 shows the variations of normalized frequency of Y , C L , C M and θr ms with Uθ . There are important characteristics to be noticed in this figure. First of all, the Strouhal Number is (0.1514) different from the pure rotational problem (0.1416— Fig. 2.13b). All the parameters follow a general pattern outside the lock-in region; which can be explained as; f / f N ≈ 1 and f / f θ ≈ St. Accordingly, the vortex shedding is characterized by the relationship of relative frequency ratio as St ≈ f N / f θ .

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2 VIV and Galloping of Prismatic Body

Fig. 2.22 Variation of force and moment coefficients for Ur = 7, Re = 100 and m ∗ = 20: a pure translation, b translation and rotation combined

This implies that the shedding is affected by both the rotational and translational natural frequencies. However, in the lock-in region, there are slight deviations from this behaviour as the natural frequency of rotation tends to dominate the motion.

2.3 Free Vibrations of Single Square Prism

53

Fig. 2.23 Variation of frequency of motion parameters in the combined translation and rotation case a transverse displacement b rotational displacement c transverse force coefficient d moment coefficient

2.3.7 Interim Summary This preliminary study examined the combined translational and rotational effects on the fluid-structure interaction of a freely vibrating square column. The important results of this preliminary study are: 1. The mass ratio of a square cylinder drastically affects the motion parameters and shifts the lock-in region of the translational motion. 2. Pure rotational motion behaves similar to pure translational motion in a range of reduced velocity. It adds additional natural frequencies and instabilities to the system. Moreover, it undergoes galloping even in low Re, unlike pure translation. 3. The characteristics of combined rotational and translational motion are much different from the motions considering the DOFs individually (i.e., pure translation and pure rotation). The rotational motion adds more circulation and affects the vortex shedding process of the fluid severely. Hence the introduction of the rotational DOF changes the translational motion parameters significantly.

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2 VIV and Galloping of Prismatic Body

4. Vortex shedding patterns are different in pure translation, pure rotation, and combined translation and rotation motions. In the combined motion, the vortex shedding outside the lock-in region can be characterized by the equation St ≈ f N / f θ .

2.4 Three-Dimensional FSI of a Square Column at High Reynolds Number Next, we numerically study the flow past a 3D geometry of square body mounted elastically, whereby the flow is perpendicular to the axis of the cylinder. Most of the early research on fluid-structure interactions has been carried out for smooth surface-based geometries such as airfoils and circular cylinders, rather than a square cross-section with sharp corners. The square cylinder is a bluffer body and has a wider wake as compared to the circular counterpart. The other key difference is that the separation points are fixed at some edges of sharp-cornered configurations, while they are free to move around for smooth bodies. The location of the separation point is a function of the angle of attack, which can affect the wake topology and force dynamics of the square cylinder. Owing to this fundamental difference with regard to the sharp corners, a square-shaped bluff body immersed in a flowing stream can undergo the combination of both vortex resonance and galloping √ instability. The natural frequency of the body in vacuum is given by f N = (1/2π ) K /M and the mass of displaced fluid by the structure is m f = ρ f D 2 L c for square cross-sections, where L c denotes the span of the cylinder. Around Re ≈ 45, the flow behind a square-shaped prismatic cylinder becomes unsteady and periodic [388, 466] and vortices are shed alternately from the top and bottom surfaces. The dimensionless Strouhal frequency is St = f vs D/U∞ , where f vs denotes the uncoupled vortex shedding frequency of a stationary cylinder. Likewise the natural frequency f N and the Reynolds number Re, the mass ratio m ∗ is an important parameter for flow-induced vibration and it is defined as the ratio of vibrating structure mass M to the mass of displaced fluid m f . In the present study, we restrict to low mass ratio body interacting with a flowing water stream at a small damping value. As reported in the case of circular cylinders [138, 208, 312], the VIV alone response can be categorized into three branches, namely initial, upper and lower for low-mass ratio cylinders undergoing VIV at subcritical Reynolds numbers. While the upper branch is attributed to large irregular amplitude patterns at a frequency close to the natural frequency of the structure with two oppositely signed pairs of vortices per cycle, the initial branch has a monotonically increasing amplitude with two single counter-rotating vortices per cycle. The third branch has a lower amplitude than the upper branch but comprises two oppositely signed pairs of vortices per cycle. Here, the aero/hydroelastic behavior of the noncircular square section is considered with low mass and damping parameters whereby the body can experience combined vortex-induced and galloping oscillations.

2.4 Three-Dimensional FSI of a Square Column at High Reynolds Number

55

Similar to the experimental study of [488], we perform a systematic analysis of the turbulent wake characteristics through three-dimensional fluid-structure simulations. The experimental settings of [313, 488] were studied via numerical simulations at the identical FIV operating parameters (Re, m ∗ , ζ ). This study provides a direct follow-up to the recent experimental work and the first numerical study on the FIV of square-section cylinder in cross-flow at subcritical Reynolds number with a low mass ratio. The variation of response amplitudes, the synchronization regimes and the wake transitions are predicted via numerical simulations. We characterize the wake topology and Reynolds stress distributions behind the vibrating square cylinder through FSI simulations. Moreover, the observations from the experimental studies have inspired several fundamental questions on the FIV of a square cylinder at a low mass-damping value: What are the effects on three-dimensional vortical structures due to the free vibration of a square body? Is there a link between the 3D wake structure development with the FIV response? How are the kinetic energy and enstrophy distributed in the near wake region? Apart from the above questions, this numerical study provides a side-by-side comparison of the flow characteristics and distributions of wake quantities between the stationary and VIV lock-in configurations. By performing numerical simulations it is possible to obtain the flow velocities in all three directions, probing at extreme proximity to the moving bodies and many locations as required. This gives us the opportunity to perform the numerical analysis on the interested Ur region and obtain a longer time history to obtain detailed frequency content and dynamical properties. In this part of chapter, our objective is to establish a link between the coherent structures and the shearing process of the wake of flow past a freely vibrating square body. Through the regeneration cycle and burst, we present a self-sustaining cycle of fluid-structure interaction through a freely vibrating square body at low mass ration for the Reynolds number range 1400 ≤ Re ≤ 10,000. We employ the recently developed variational FSI solver based on the filtered Navier-Stokes and rigid body motion equations via the body-conforming treatment of the fluid-solid interface. To handle strong inertial effects in the partitioned fluid-structure framework, the non-linear iterative force correction scheme [181] is employed for a robust and stable treatment of fluid-structure interaction. Through the 3D FSI simulations, we successfully confirm the motion responses and the vortex patterns from the experiment of [488] for 1400 ≤ Re ≤ 10,000 at m ∗ = 2.64 for the zero incidence angle. We next compare the formation mechanisms of the vortex patterns and vortical structures of the flow around stationary and vibrating square cylinders. We quantitatively examine the Reynolds stress patterns of the wakes with a particular interest in their symmetrical/asymmetrical distribution over spatial and temporal domains. To further analyze the Reynolds stresses, the time histories of velocities and pressure at critical wake positions (near wake, shear layer, etc.) are recorded for each time step around the moving turbulent wake behind the vibrating cylinder. We report and examine the formation mechanism of hairpin-like structures around the freely vibrating square cylinder during the VIV lock-in process. To assess the kinetic energy and the enstrophy evolution, we introduce a 3D representative control volume in the near-wake region. Finally, we present a self-sustaining regeneration cycle of the turbulent wake embedded in a periodic motion of a freely vibrating square cylinder.

56

2 VIV and Galloping of Prismatic Body

(a)

(b)

(c)

(d)

Fig. 2.24 a Schematic of the problem setup and associated boundary condition details for flow past a square cylinder at 1400 ≤ Re ≤ 10,000 for vibrating cylinder, b full domain mesh in x-y plane, c cross-sectional mesh along z-direction, d mesh close to cylinder and near-wake region

2.4.1 Problem Definition, Convergence and Validation Before discussing the free vibration of a square body via dynamic SGS scheme, it is essential to establish the appropriate mesh resolution and time-step size for the planar wake generated behind a stationary square cylinder. The separated wake flow around a square cylinder consists of a complex flow phenomena such as highly unsteady vortical structures and wake turbulence. In particular, there is a need for special attention to capture the formation of spanwise vortices through the sufficient span of the cylinder with an appropriate spanwise resolution. In this study, we employ the guidelines and best practices developed in the previous LES studies [60, 273, 363] for a square cylinder configuration. We examine the parameters affecting the variational SGS implementation to capture the turbulent flow around a stationary square-section cylinder at Re = 22,000 and validate our results with the experimental results. The present variational LES formulation has been extensively validated for circular cylinders in the previous studies [226, 307] We consider the identical problem setup employed in [180] for this generic square cylinder configuration with sharp corners. The only difference here is the spanwise length of the cylinder, which is 10D in the present study instead of 5D.

2.4 Three-Dimensional FSI of a Square Column at High Reynolds Number

57

The problem schematic and the representative meshes are shown in Fig. 2.24. The coordinate origin is located at the geometric center of the cylinder. The streamwise, transverse and spanwise directions are denoted x, y, and z, respectively. The distances from the cylinder center to inlet and outlet boundaries are respectively set as L u = 15D and L d = 40D. The sidewalls are equidistant from the origin of the coordinate system (cylinder center) and the distance between the sidewalls is 40D, which indicates the blockage is 2.5% in the present study. No slip condition is applied on the cylinder wall, whereas slip condition is imposed on the sidewalls. Periodic boundary conditions are imposed on both ends of the cylinder. While a uniform flow stream velocity u = U∞ is specified on the inlet plane, a traction-free Neumann boundary condition is prescribed at the outlet boundary. All the flow statistics are extracted over 15 vortex shedding cycles for the stationary and vibrating cases. In the computational domain, we probe many strategically chosen points in the near cylinder region recording the time histories of the velocities and pressure variation in the shear layer and wake regions. Except stated otherwise, all positions and length scales are normalized by the cylinder diameter D, velocities with the free stream velocity U∞ , and frequencies with U∞ /D.

2.4.1.1

Temporal Convergence

In the present formulation, the implicit numerical scheme allows a larger time step for the wake flow computation of a bluff body. To resolve unsteady wake dynamics with a proper time step size, we have performed a convergence study for the flow past a stationary cylinder at Re = 22,000. The time steps are progressively divided by two until the relative error of the considered parameters becomes sufficiently small. Since a closed-form analytical solution for this physical problem is not available, we consider the values obtained at the smallest time step Δt = 0.0125 as a reference value for the error analysis. Table 2.3 summarizes the mean drag coefficient Cd , the root mean square (rms) value of lift coefficient Clr ms , and the Strouhal number St for the three-time step sizes. The percentage differences in the flow quantities for Δt = 0.05, 0.025 are listed in the parenthesis of the table. The differences in the results between Δt = 0.025 and Δt = 0.0125 are reasonably small (< 2%) for Cd , St, and −C pb , except relatively larger difference in Clr ms is observed. The predicted flow properties are also in good agreement with the experimental results of [42, 269]. From the temporal convergence study, we can conclude that the time step of ΔtU∞ /D = 0.025 is sufficient for the present study of the unsteady wake computation of vibrating square body.

2.4.1.2

Mesh Convergence

Similar to the temporal convergence, we determine the required spatial discretization for our LES implementation. The unstructured finite element mesh employed in this study is composed of six-noded wedge elements. To check the adequacy of the spatial

58

2 VIV and Galloping of Prismatic Body

Table 2.3 Temporal convergence study for square cylinder at Re = 22,000: mean drag coefficient Cd , rms value of lift coefficient Clr ms , Strouhal number St, and the base pressure −C pb ΔtU∞ /D

Cd

Clr ms

St

−C pb

Experiment 0.05 0.025 0.0125

2.1a 2.260 (2.1%) 2.213 (0.39%) 2.222

≈ 1.2b 1.444 (14.4%) 1.262 (3.89%) 1.313

0.132a 0.117 (2%) 0.120 (1.56%) 0.122

1.65b 1.360 (10.35%) 1.517 (1.88%) 1.546

a−[269] b−[42] Table 2.4 Statistics of the meshes used for convergence study M1 M2 32 × 32 32 725,505 2.35 ×106

Cylinder wall mesh Number of z-layers Total nodes Total elements

64 × 64 64 1.62×106 5.62 ×106

M3 128 × 128 128 9.41 ×106 31.25 ×106

Table 2.5 Mesh convergence study and validation for flow around a square cylinder at Re = 22,000: mean drag coefficient Cd , rms value of lift coefficient Clr ms , the Strouhal number St, and the base pressure −C pb Experiment M1 M2 M3

Cd

Clr ms

St

−C pb

2.1a

≈ 1.203 (4.7%) 1.262 (1.83%) 1.285

0.132a

1.65b 1.316 (13.27%) 1.517 (4.21%) 1.584

2.330 (5.3%) 2.213 (3.14%) 2.146

1.2b

0.117 (2.00%) 0.120 (0.83%) 0.121

a−[269] b−[42]

resolution, we have performed grid refinement tests with different resolutions while maintaining the dimensionless wall distance y + ≤ 1 in the boundary layer around the cylinder body. To determine the cylinder length and the mesh resolution along the spanwise z-direction, we adopt the guidelines suggested in [273] and refine the mesh along the cylinder wall by a factor of two until the relative error becomes reasonably small. The statistics of the different meshes used are summarized in Table 2.4. The convergence error is computed with respect to the mesh M3. The mesh convergence results are shown in Table 2.5, and the flow quantities such as Strouhal number St, the mean and fluctuating drag and lift forces, the base pressure −C pb are quantified. The absolute relative error is provided in the parenthesis. From the table, we can notice that the results obtained from the medium and fine meshes show a reasonable level of convergence and are in good agreement with the experiment results. The medium size mesh, M2 with wall mesh of 64 × 64 with 64 layers in the z-direction is selected for the present study of the freely vibrating square cylinder.

2.4 Three-Dimensional FSI of a Square Column at High Reynolds Number

2.4.1.3

59

Comparison with Experimental Data

As shown in Tables 2.3 and 2.5, the LES results of the mean drag coefficient, the rms lift coefficient, the Strouhal number and the base suction pressure are in good agreement with the experiment data of [42, 269]. Furthermore, the average vortex formation length is found to be 0.94D which is close to 1.0D reported in [41]. To further assess the implementation of the dynamic SGS, we compare the key wake characteristic data and the pressure distribution with the available experimental data for a stationary square cylinder. Good agreement of the mean streamwise velocity with the measured data of [269] can be seen in Fig. 2.25a. The overall trend of flow recovery along the wake centerline is similar for both the measurement and the present simulation. As shown in Fig. 2.25b, the cross-stream profile of the mean streamwise velocity profile at X/D = 1.0 is also well predicted by the variational dynamic model. The root mean square (rms) trends of the velocity fluctuations in the turbulent wake are shown in Fig. 2.25c, which has a good match with the measured values. The u  v  variation at the circulation length also agrees reasonably well with the experiment data. However, the circulation length of the experiment is reported to be X = 1.0D and here we observe it to be X = 0.94D. Furthermore, the pressure distribution around the cylinder compares reasonably with the experimental results of [42]. Owing to the difficulty in measurements and the lack of sufficient resolutions for the fine-scale eddies around the sharp corners, there are some discrepancies between the measured and calculated values of the pressure coefficient near the corner regions. A similar scatter in the experimental data and the discrepancy against the DNS results is recently reported by Portela et al. [343] for a square prism at Re ≈ O(103 − 104 ). We next compare the fluctuating pressure coefficient around the cylinder with experimental measurement [42] and the reference numerical results [414]. The results matched satisfactorily with both studies for the front face AB. Apart from the corners C and D, the top and bottom surfaces (BC and D A) show a good agreement with the previous studies. The slight discrepancies can be attributed to the difference in Re which is significant in the range of 10,000 ≤ Re ≤ 30,000 according to the findings discussed in [42].

2.4.2 Energy Spectra in the Near Wake Although we have reasonably validated our numerical framework in previous subsections by comparing the pressure distributions, the velocity profiles and fluctuations and the force coefficients, it is important to investigate whether the mesh used here adequately captures the turbulent wake characteristics. For that purpose, we look into the turbulence power spectra in the wake region and introduce several probes in the wake region to assess the mesh resolution for the dynamic subgrid-scale model. For locally homogeneous turbulence, Kolmogorov’s −5/3 power states that the kinetic energy and the wave number of integral scale eddies follow the relation E(κ) ∝ κ −5/3 in the inertial range. This relationship is generally applied to the spatial distribution of

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Fig. 2.25 Flow past a stationary square cylinder at Re = 22,000. Comparison of dynamic subgridscale results with measurement data: a mean streamwise velocity Uc along the wake centerline, b cross-stream profile of the mean streamwise velocity at X/D = 1.0, c rms of streamwise velocity fluctuations u r ms at X/D = 1.0, d u  v  variation along +Y direction at the recirculation length, e mean pressure distribution along the cylinder wall, f pressure fluctuation along the cylinder wall

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Fig. 2.26 Spectra of the turbulent kinetic energy at different downstream positions along the wake centerline: a stationary, b VIV cylinder. Except for the highest spectra, other spectra trends are shifted for clarity. The dashed lines show the −5/3 decay

turbulent kinetic energy (T K E ≡ u i u i /2) as a function of wavenumber spectrum. However, via the Taylor’s frozen turbulence hypothesis [413], the −5/3 decay of the kinetic energy versus frequency provides a reasonable indicator for the turbulence spectrum. It is known that the near wake region behind a body is highly nonhomogeneous, anisotropic and unstable. While the inter-scale energy budget cannot not be established by the Kolmogorov’s and Taylor’s hypotheses, Karman-HowarthMonin-Hill (KHMH) equation provides a general way to describe the energy balance in the turbulent planar wake regardless of the degrees of inhomogeneity, anisotropy and unsteadiness in the near wake region [154, 343]. The equation provides the relationship of the energy transfer associated with eddies of varying sizes and scales. With the hypotheses of small-scale isotropy and self-similarity, Kolmogorov’s −5/3 power law can be obtained from Eq. (2.27) by either dimensional analysis [215] or the skewness of the velocity differences to be constant [216]. Further detail and justification on the energy cascade derivation are provided in Appendix A. Figure 2.26 shows the FFT spectra of the turbulent kinetic energy past representative points in the wake centerline for both stationary and vibrating cylinder cases. The integral scale eddies are of the scale of D and can be expected to be in the frequency band of 0.3 ≤ f D/U∞ ≤ 3. In this range, for both configurations, the trend-line of decay is parallel to the −5/3 decay, just over a decade of frequencies. From this, we can establish that the wake mesh is adequately refined to capture the turbulence characteristics. Overall, this validation study confirms the convergence and the effectiveness of the dynamic SGS turbulence model for the fluid-structure interaction of a square-section cylinder at a subcritical Reynolds number of Re = 22,000, much higher than the considered Re range for the present FSI study. In the next section, we further establish the accuracy of our FSI framework in the range of subcritical Reynolds numbers, by validating our results with the experimental measurements of [488].

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Results and Discussion Unlike the flow past stationary cylinders, the coupled wake dynamics of freely vibrating cylinders is governed by four non-dimensional parameters; Reynolds number (Re), mass ratio (m ∗ ), reduced velocity (Ur ) and damping ratio (ζ ). As shown in Fig. 2.24, we consider the same problem setup with the transversely vibrating cylinder with mass M mounted elastically on a spring with a stiffness value of K and a linear damper with a damping value of C in the transverse direction with one-degreeof-freedom (1-DOF) motion. Hence, the coefficient vectors in Eq. (2.4) become m = (0, M, 0), c = (0, C, 0) and k = (0, K , 0). Two interacting fluid and structure systems with their respective uncoupled frequencies fvs and f N may fall into synchronization via the mode locking phenomenon and the structure can experience different oscillation and vortex shedding frequencies. Through numerical simulations, we will investigate the transverse amplitude and frequency responses, the vortex shedding patterns, the force and response histories and the synchronization regimes of a flow past a freely vibrating square cylinder. Interactions of pairs of these frequencies in the canonical cylinder problem are of deep interest to understand the coupled wake and body interaction. During the VIV lock-in process, the wake and structure modes are strongly coupled to each other and the response amplitude and the lock-in range strongly depend on bluff-body geometry and physical parameters [466]. A great variety of vortex wake modes can be formed behind the vibrating cylinder during the synchronization at a higher Reynolds number. We use the experimental data from [488] to further validate the results of our FSI solver for freely vibrating cylinders. We scale the reduced velocity U ∗ given in the experiment in water U ∗ = U∞ / f N w D, where the frequency in water f N w is determined by the potential added mass and the natural frequency f N as m A = (( f N / f N w )2 − 1)M. From this expression of the added mass, the √ ratio of reduced velocity in vacuum Ur to water U ∗ can be evaluated as: Ur /U ∗ = M/(M + m A ) = 0.807. In our discussion of vortex shedding patterns, we will adopt the classical terminology of [458] to identify the vortex shedding patterns, e.g. two vortices of opposite signs are shed per oscillation period (2S) and wakes in which two vortex pairs are shed per oscillation period (2P).

2.4.3 Response Amplitudes We begin by discussing the 1-DOF FIV response and the underlying vorticity dynamics. We will consider the range of reduced velocity Ur ∈ [3, 20] to cover VIV and galloping synchronizations for a square cylinder, in the Reynolds number range of 1400 ≤ Re ≤ 10,000. The numerical results are first compared against the experimental results of [488] for a freely vibrating square cylinder with m ∗ = 2.64. Figure 2.27 shows the amplitude and frequency responses as a function of reduced velocity Ur . The amplitude trend shows a typical galloping characteristic of

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Fig. 2.27 Transverse response of the vibrating cylinder with m ∗ = 2.64. Dependence of reduced velocity on a transverse motion response Aryms , b frequency response f ∗ . Insets represent the ∗ ) and demonstrate the three synchronization regimes frequency spectrum of the lift coefficient ( f Cl and the frequency f ∗ = f / f N

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Fig. 2.28 Time histories and frequency spectra of transverse motion (top) and force (bottom) for representative reduced velocities at each synchronization regimes for m ∗ = 2.64. Here 1:1 synchronization is at Ur = 6.0, and 1:3 and 1:5 correspond to Ur = 12.0 and Ur = 18.0 respectively. Frequency contributions are normalized by the highest spectra magnitude

a nearly linear increase of amplitude with respect to the reduced velocity. Consistent with the experimental observations [488], three synchronization regimes are found when the vortex shedding frequency is close to an odd-integer multiple of galloping vibration frequency. Experimentally observed synchronized regimes in the amplitude response are reasonably predicted by the numerical simulations. The first kink occurs in the neighborhood of Ur ∈ [5, 6], which shows the VIV lock-in response whereby the vortex shedding frequency synchronizes with the cylinder frequency. This is termed as a 1:1 synchronization, whereas the motion is dictated by the dominant vortex shedding frequency. The second synchronization ranges in a broader range of reduced velocity Ur ∈ [8, 14] and exhibits a 1:3 synchronization whereby the vortex shedding is synchronized at three times of the vibration frequency. In this regime, the vibration and the lift force acquire the contributions from both the first and third harmonics. As observed in the experiment of [488], a distinct plateau in the amplitude response can be seen in Fig. 2.27a. The third regime is found around Ur = 18, where the same, but much weaker synchronization is observed for Ur = 17 and 19. In this regime, the transverse force has marked contributions from the five times vibration frequency, implying a 1:5 synchronization between the vortex shedding and the vibration response. This can be further seen through the frequency spectra in Fig. 2.27b. All the three synchronization regimes show similar patterns with the experiments of [488]. Figure 2.28 demonstrates the contributions from the higher harmonics on the transverse force in the VIV (1:1) and galloping (1:3 and 1:5) synchronization regimes. Notably, the higher harmonics have no significant contribution to the transverse displacement. This is a clear indication that an odd number of multiples of force fluctuations occur per one motion cycle in the galloping synchronization regimes which is reflected in the vortex shedding patterns of these regimes. Each synchronization

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Fig. 2.29 Response of freely vibrating square cylinder for VIV dominated 1 : 1 synchronization at Ur = 6.0 (top) and galloping dominated 1 : 3 synchronization at Ur = 14 (bottom): (a, c) time histories of lift and transverse y-displacement, (b, d) contours of spanwise z-vorticity. While the VIV response is of 2S vortex mode, the galloping mode exhibits 3(2S) vortices

has an intrinsic preference for the vortex shedding to lock to the different vibration frequencies of the cylinder. As the harmonic of vibration frequency becomes higher, the number of vortices shed in the cycle increases. With regard to the wake mode, the 1:1 synchronization depicts a 2S mode vortex shedding which is also found in the other odd synchronization regimes of 1:3 and 1:5. However, for the synchronization regimes 1:3 and 1:5, the 2S shedding occurs three and five times during one motion cycle respectively, as shown in Fig. 2.29). This synchronization is due to the slow oscillation response of the galloping response which allows multiple shedding cycles during one motion cycle. For example, during the 1:3 synchronization, there is one and a half vortex shedding occurs during a one-half cycle of galloping oscillation and there are three vortices involved in the synchronization. For the 1:5 synchronization, there are five vortices that contribute to the transverse force during the body oscillation frequency. As discussed in [488],

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there is a need for fine-tuning to lock in for these higher synchronization modes and the stable range for synchronization for the 1:3 is larger than the 1:5 synchronization. Furthermore, the reason for only odd number synchronization is that if there is an even number synchronization (e.g., 1:2 or 1:4), it will result in a zero average lift during a one-half cycle. It will result in a de-synchronized motion and will not be able to sustain the transverse galloping with a zero mean lift.

2.4.4 Vorticity Dynamics To further explore the FIV characteristics of a freely vibrating square cylinder, we now present a detailed investigation and novel physical insights with regard to the vorticity dynamics. The concentrated vorticity in the boundary and wake interacts in a complex and nonlinear manner with the vibrating body. Free vibration of the body affects the creation, convection and diffusion of vorticity, which in turn alters the unsteady forces comprising inviscid inertia and viscous drag components. Therefore it is important to understand the vorticity field behind a vibrating body. In particular, we comparatively analyze the wake characteristics of the stationary and freely vibrating configurations. Due to the strong fluid-structure interaction, the vibration of the structure significantly affects the wake topology which in turn alters the fluid force and the energy distribution in the wake. The inviscid irrotational fluid is transported across the wake boundary behind the vibrating body, which results in the added mass effect and the modification of the structural natural frequency. For the low mass ratio m ∗ , this entrainment of fluid into the wake makes the added mass a significant fraction of the total mass of the vibrating body. Here, we investigate the wake topology and the vorticity distribution behind the freely vibrating square cylinder. Figure 2.30 shows representative vortical structures at a particular shedding cycle of the flow past the stationary and freely vibrating configurations. To analyze the three-dimensional vortical structures, we employ a vortex identification based on Qcriterion [172]. The flow structures are highly irregular and dominating streamwise vortex ribs superimposed on the spanwise vorticity structures can be seen in the plots. The large vortical structures about the order of cylinder diameter are distributed in the spanwise direction and they are reasonably resolved by the spanwise length in the present simulations. The spanwise shear layer roll-ups in the near wake region by the action of a Kelvin-Helmholtz instability mechanism can be seen for both stationary and vibrating configurations. There are complex interactions among the primary Karman vortices, the shear layer vortices and the streamwise (longitudinal) mode B type of secondary vortices, which occur along the span of the square cylinder body. In contrast to the stationary case, the so-called hairpin vortices are visible in the near cylinder region of the vibrating cylinder in both VIV lock-in and galloping cases. These structures are identified to be characteristic for flow over a flat surface as a result of a high-velocity gradient of the boundary layer. It is evident that in the lock-in phenomenon there is a sufficiently high-velocity gradient between the shear layer and the near cylinder deficit region for the creation of these structures. The mean

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(a)

(b)

(c)

(d)

Fig. 2.30 Iso-surfaces of three-dimensional instantaneous vortical structures behind a square cylinder: a stationary cylinder at peak lift, and vibrating configurations at peak motion amplitude b VIV regime at Ur = 6.0 with 1:1 synchronization, c galloping at Ur = 14.0 with 1:3 synchronization, d galloping at Ur = 18.0 with 1:5 synchronization. Top view is shown and iso-surface is defined by Q-criterion Q = 21 (Ωi j Ω ji − Si j S ji ) = 0.25 and colored by u/U∞ . Here labels A, B and C denote spanwise rollers, streamwise ribs and hairpin vortices and the flow is bottom to top

shear acts on the spanwise vortices, resulting in hairpin vortices whose legs are the streamwise vortices. In the galloping scenarios with 1:3 and 1:5 synchronizations, the spanwise rollers are distorted and have somewhat more disconnected regions. In fact, these rollers are only visible in a fraction of the span; sometimes indistinguishable in the Q criterion plots (Fig. 2.30c,d). However, the z-vorticity plots at different sections of the span revealed the presence of a few rollers in a fractional length of the span. This is an indication that in the high Ur regimes, the large vortical structures become much more disturbed and disconnected. The occurrence of the hairpin structures leads to the study of their formation mechanism. However, it is already observed that the spanwise rollers and the streamwise ribs, which are dominant features of the stationary case, appear to be of a lesser occurrence for the vibrating case with 1:1 synchronization. This can be observed via instantaneous z-vorticity and x-vorticity distributions in Figs. 2.31 and 2.32. There are several striking differences are revealed by the contour plots in the near wake of stationary and vibrating configurations. The near cylinder rollers are replaced by a set of hairpin-like vortices and the streamwise ribs are regular and more clustered for the vibrating case. In addition, in the region closer to the cylinder it is visible that the

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(a)

(b)

(c)

(d)

Fig. 2.31 Instantaneous z-vorticity distributions for a stationary, and vibrating configuration at b Ur = 6.0, 1:1 VIV lock-in, c Ur = 14.0, 1:3 synchronized galloping, d Ur = 18.0, 1:5 synchronized galloping. While solid contours denote positive vorticity ωz+ ∈ [0.5, 2.5], dashed contours represent negative vorticity ωz− ∈ [−2.5, −0.5] at 0.5 intervals. Formation of hairpin structures and inviscid regimes via coalescence of small eddies can be seen for the vibrating cases

transverse motion facilitates the coalescence of small eddies and promotes the merging of the rollers and ribs to generate hairpin-like structures. This process increases the supply of shear-layer vorticity into the near wake to counteract the vorticity diffusion. There exists a self-sustaining mechanism for the hairpin-like structures for the vibrating cylinder. A further discussion with the help of Reynolds stresses is provided in Sect. 2.4.5. While the vorticity field is well distributed in the near-wake region for the stationary cylinder (Fig. 2.31a), these large vortical regions are clustered for the vibrating cases in the streamwise direction with greater non-vortical fluid regions between them (Fig. 2.31b–d). Coalescence of small eddies closer to the body is also visible for the vibrating cases, predominantly in the VIV lock-in scenario, as shown in Fig. 2.31b. For the vibrating cases, the vortex regions smaller than D are visible at the position of hairpin vortices just downstream of the coalescing of small vortices in the roll-up regions. The streamwise ribs are further visualized by x-vorticity in Fig. 2.32. In the stationary case, the wake region is richly populated by the streamwise vorticity component. After the formation region, the streamwise ribs extend fully between the two opposite signed spanwise rollers. In the vibrating cases, there are prominent regions of irrotational flow between the streamwise ribs and the ribs are relatively shorter and does not fully extend between the adjacent rollers (Fig. 2.32b). As expected, we can also observe a transverse stretching of the streamwise ribs for the VIV lock-in case.

2.4 Three-Dimensional FSI of a Square Column at High Reynolds Number

(a)

(b)

(c)

(d)

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Fig. 2.32 Instantaneous x-vorticity distributions for a stationary, and vibrating configuration at b Ur = 6.0, 1:1 VIV lock-in, c Ur = 14.0, 1:3 synchronized galloping, d Ur = 18.0, 1:5 synchronized galloping. While solid contours denote positive vorticity ω+ x ∈ [0.5, 2.5], dashed contours represent negative vorticity ω− x ∈ [−2.5, −0.5] at 0.5 intervals

The coalescence of near-wake eddies occurs when the cylinder is in the transverse vibration. The periodic motion provides an avenue for the merging of eddies in the vicinity of the body and there are relatively more clustered rollers and ribs as compared to the stationary counterpart. The merging and coalescence mechanisms of vortices near the vibrating cylinder may also facilitate the entrainment of inviscid fluid into the wake region. This can be observed by the relatively larger non-vortical regions of inviscid irrotational fluid for the vibrating cases in Figs. 2.31b–d and 2.32b–d. This flow entrainment explains the increased added mass effect for the vibrating cases, which acts in phase with the acceleration of transverse motion. This entrainment phenomenon essentially increases the effective mass of the cylinder and to decrease the natural frequency. The surface stresses induced by the vorticity distribution affect both the inertial and the transverse force over the bluff body undergoing vortex-induced oscillations. A simple force decomposition based on Lighthill’s assumption [247] may not be able to capture the such higher harmonic effect of flow-induced vibration and complex inter-dependency between the time-dependent inertial, the viscous and vorticity generated forces over the vibrating body. The present variational subgrid-scale and the fluid-structure simulations based on the first-principle model can provide further understanding to construct a generalized force decomposition and reduced-order modeling for engineering applications. To further analyze the intermediate wake

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Fig. 2.33 Mean Reynolds stress distributions in the turbulent wake at three wake centerline positions: x = 1D, 1.25D and 1.5D; a–c stationary d–f VIV lock-in at Ur = 6.0. Time averaging is 2 performed over 15 shedding cycles and the Reynolds stress is normalized by U∞

structures, we next focus our analysis to compare the Reynolds stress distributions of the stationary and VIV lock-in cases. Here we pay particular attention to the Reynolds stress u  v  which is a measure of the flow shearing intensity.

2.4.5 Reynolds Stress As shown in Fig. 2.33a–c, the distributions of Reynolds stresses in the stationary cylinder are nearly symmetric about the wake centerline. While u  u  decreases along the streamwise direction for X/D = 1, 1.25, 1.5,  v  has a rising trend for increasing X/D. The trends of u  u  and u  v  have a typical single peak towards one side of the centerline occurring at the transverse position Y ≈ ±0.6D. The peak value for v  v  occurs at Y = 0. For the stationary cylinder configuration, a symmetric trend of the Reynolds stress is an indication that the vortices shed from the top and bottom surfaces of the cylinder have nearly equal strengths with opposite signs of rotation. In contrast, the variations of Reynolds stresses for the vibrating cylinder are asymmetric in nature (Fig. 2.33 d–f). The peak of u  u  is relatively spread out for the above side of the wake centerline (positive Y ) than the below (negative Y ) side. A similar variation is found for other X/D positions. Unlike the stationary cylinder, the fluctuations of u  u  maintain somewhat similar magnitudes for the three locations for the VIV case. The variation of shear stress u  v  for X/D = 1 shows two peaks above or below the

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wake centerline. The peak of v  v  is slightly shifted to Y ≈ 0.2D and has a small hump at Y ≈ −0.7D. This particular asymmetry of the transverse velocity can be attributed to the asymmetric sinusoidal motion of the cylinder as seen in Fig. 2.29a. Small wiggles in the negative Y -direction motion can be seen. The wake moves transversely with the cylinder, giving similar wiggles in variation for the Reynolds stress v  v  of the considered points in Y < 0 direction. We quantify the degree of spatial asymmetry of the Reynolds stresses around the wake centerline (Y = 0) for the two configurations of stationary and vibrating cylinders. For this purpose we consider a statistical variable to examine the deviation from symmetry (DS), which is given by:

 2 N 1  (u  u  |Y =yi − u  u  |Y =−yi ) , DSuu =  N i=1 (max(u  u  ))2

 2 N 1  (u  v  |Y =yi + u  v  |Y =−yi ) , DSuv =  N i=1 (max(u  v  ))2

 2 N 1  (v  v  |Y =yi − v  v  |Y =−yi ) , DSvv =  N i=1 (max(v  v  ))2

(2.19)

(2.20)

(2.21)

where N is the number of probes on one side of the wake centerline. Note that the probes are distributed symmetrically around the wake centerline along a constant X -axis. For the Reynolds normal stresses, the statistical DS variable estimates the difference between the mean stresses of equidistant probes and for the Reynolds shear stress it quantifies the difference of the values with opposite signs. All the values are non-dimensionalized by the maximum temporal average value recorded in all the probes. The variable gives the relative deviation of the spatial symmetry of the Reynolds stress distribution. Unlike the standard skewness, this statistical variable is suitable for the quantification of the degree of symmetry for both single peak and double peak variations. A larger value of DS variable suggests a higher spatial asymmetry of the Reynolds stress distribution. Figure 2.34 shows the statistical DS variable for the stationary and the vibrating configurations. The stationary configuration has the values of deviation from the symmetry (DS) parameter within 3−6% suggesting a very high degree of spatial symmetry. On the other hand, the VIV case has a very higher values of deviation from symmetry (DS) parameter (16−17% for u  u  ; 25−55% for u  v  and 5−20% for v  v  ) suggesting a higher spatial asymmetry. From this analysis, we can conclude both qualitatively and quantitatively that the spatial distribution of average Reynolds stresses is asymmetric for the VIV case. The vibration of the cylinder disturbs the spatial symmetry of the Reynolds stress components, which can be related to the vorticity distributions in the wake of vibrating cylinder. We further study these asymmetries to understand their relationship with the development of the observed vortical structures.

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Fig. 2.34 Distribution of the deviation from symmetry (DS) variable for stationary (colored symbols) and VIV (open symbols) cases for X = 0.75D, 1.0D, 1.25D and 1.5D. For stationary case the deviations from symmetry of Reynolds stresses are in the range of 3−6%. In the VIV case, the DS variable ranges from 16−17% for u  u  ; 25−55% for u  v  and 5−20% for v  v 

2.4.5.1

Quadrant Analysis of u v 

To further explain the development and sustenance of the hairpin vortex structures, we examine the symmetry/asymmetry of the Reynolds stress distribution u  v  across the wake and its evolution as a function of time. We present the quadrant analysis for u  v  which is a widely used analytical tool in high shear flow analysis such as near-wall turbulent flows. The quadrant analysis is introduced by Wallace et al. [434] where they classified u  v  into four categories called quadrants: Q1(u  > 0; v  > 0), Q2(u  < 0; v  > 0), Q3(u  < 0; v  < 0) and Q4(u  > 0; v  < 0). All these quadrants indicate different properties of the flow shearing process. Q2 and Q4 which are called ejection and sweep, quantify the mean shear growth since the fluid advects transversely from a higher streamwise momentum to a lower momentum region. Q1 and Q3 are called the out-of-wall and wallward counter-gradient motion in the wall fluid motion research. Herein, we rename them as out-of-the-wake and into-the-wake countergradient motions. However, the physical significance of these quadrants remains the same. They quantify the mean shear damping and the contribution for it from the out-of-wake and towards the wake flow motion. It is important to note that these quadrants are defined for the domain section where both the X and Y coordinates are positive. For the negative Y domain, the quadrants need to be inverted to give the same physical meaning. Then the negative Y domain quadrants become; Q1(u  > 0; v  < 0), Q2(u  < 0; v  < 0), Q3(u  < 0; v  > 0) and Q4(u  > 0; v  > 0).

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Fig. 2.35 Accumulation of u  v  quadrants along X = 0.5D, 0.75D, 1.0D, 1.25D and 1.5D lines on the center spanwise plane for a stationary, b VIV configurations over 15 shedding cycles. The values are inverted for Y < 0 range for a clear representation of the spatial symmetry. The quadrants 2 are normalized by U∞

In Fig. 2.35, we compute the accumulation of the four quadrants over 15 shedding cycles for the stationary and VIV cases. For consistency, we check the difference of the quadrant patterns for 10 and 20 cycles which show no significant variation. Here 2 instead of u  v  and we do not we non-dimensionalize the quadrant values using U∞ average them individually. In this way, we can properly calculate the contribution of each quadrant over time for the mean shear term u  v  and also the differences in their contributions for the two cases. Figure 2.35 clearly shows that the quadrants change nearly symmetrically in the stationary cylinder wake. Both Q2 and Q4 have local maxima in the region 0.3D ≤ |Y | ≤ 0.7D indicating the high shear region or the shear layer. These maximum values move towards the wake centerline downstream. The local minima of both of

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these components occur in the vicinity of the wake centerline. However, the ejection maxima occur at slightly wider wake points than the sweep. The less dominant Q1 and Q3 quadrants have the maxima near the wake centerline. The values of these maxima increase further downstream. Furthermore, the ejection (Q2) and sweep(Q4) are very dominant than the counter gradient motions (Q1 and Q3). In other words, the mean shear growth components dominate over the shear damping indicating high-intensity shearing in the wake. In summary, the flow shearing process has a high intensity and spatially symmetric distribution for the stationary cylinder. In contrast, in the VIV case, all the quadrants are rather asymmetric over the initial wake centerline. The ejection and sweep quadrants have unequal maxima at different |Y | values in the positive and negative Y sections. The Q2 is more dominant than the Q4 and their maxima occur at wider points than the stationary case indicating a wider wake as expected. The counter gradient motions (Q1 and Q3) provide a significant contribution to the u  v  term. Unlike the stationary cylinder, the Q3 has two peak behaviors throughout the near wake. Compared to the stationary cylinder, the VIV case has significantly lesser intense mean shear and the shear distribution is asymmetric around the wake centerline.

2.4.5.2

Time History of u v 

Through time histories and the frequency spectra of many probes, we further investigate the Reynolds stress u  v  distribution of the wakes of stationary and VIV configurations. The distributions of four representative points shown in Fig. 2.36; one each in the top (P1 ) and bottom (P2 ) shear layers and one each in above (P3 ) and below (P4 ) the wake centerline are presented in Fig. 2.37. It is important to note that all the probes inside the near wake region display similar variations and these particular probes are just a representation. We also performed the FFT analysis for 10 and 20 cycles which resulted in similar results as 15 cycles. The time history of u  v  component is biased to the negative region and has a wide frequency bandwidth for the stationary case, as shown in Fig. 2.37a, c. On the other hand, the VIV case has a low-frequency bias and is somewhat evenly distributed at the zero stress level (u  v  = 0), as shown in Fig. 2.37b, d. All the observations of the u  v  distributions up to this point present an interesting phenomenon: In the stationary cylinder case where a spatially symmetric, high intense and high frequency dominated shearing process occurs, we only observe spanwise rollers and streamwise ribs. In the VIV case where the shearing process is spatially asymmetric, less intense and dominated by low frequencies we observe intermediate hairpin structures developing in the wake. The motion of the cylinder disturbs the spatial symmetry of the flow shearing process; shifting the symmetry to the temporal domain. We combine these findings with the help of the self-sustaining process of coherent flow structure development in the next section.

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Fig. 2.36 Arrangement of probes and location of representative control volume (RCV) for detailed analysis in the near wake: P1 − (0.75, 0.6, 0), P2 − (0.75, −0.6, 0), P3 − (1.0, 0.4, 0), P4 − (1.0, −0.4, 0) and the control volume in Sect. 2.4.6: center at (1.0, 0.9, 0) with size 0.5D × 0.4D × 0.3D. The fixed coordinate origin is located at the initial center of the square cylinder. The wake boundary is denoted where average streamwise velocity averaged over 15 cycles equals the free stream velocity; U = U∞ . Contour lines are of Z-vorticity, ωz∗ = ωz D/U∞ ∈ [−3, −0.5] ∪ [0.5, 3] with 0.5 intervals

2.4.6 Self-Sustained Process The flow structures observed in a particular flow are directly related to the flow shearing process. This shearing is indicated by the Reynolds stress term u  v  . As explained in the previous section, the sign, magnitude and frequency of the u  v  term differ according to the motion state of the square cylinder. We establish the link between the coherent flow structures observed in Sect. 2.4.4 and flow shearing observed in Sect. 2.4.5 using the self-sustained process of wake structure development. A somewhat similar self-sustained cycle has been developed for homogeneous turbulent flows by [143, 347]. First, we look into the cyclic nature of the flow structures in relation to the kinetic energy (KE) of the incompressible turbulent fluid. The total KE of the wake flow

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Fig. 2.37 Time histories and spectra of Reynolds stress u  v  in the near wake regions; a and b shear layer: top—P1 (0.75,0.6,0) and bottom—P2 (0.75,-0.6,0), c and d near the wake centerline: top—P3 (1.0,0.4,0) and bottom—P4 (1.0,-0.4,0) for a and c Stationary, b and d VIV lock-in at Ur = 6.0

can be divided into two components, namely the narrow-band large-scale shedding energy and the broadband turbulent kinetic energy (TKE). The energy transfer associated with the broadband turbulence can be completely expressed by the KHMH equation, as indicated in Eq. (2.26). In our numerical analysis, the turbulent energy component contributes almost equally to the wake energy budget for both stationary and oscillating cylinders, which is consistent with Taylor’s frozen turbulence hypothesis. When the cylinder with the natural frequency f N is free to vibrate, a considerable portion of the large-scale energy (associated with vortex shedding frequency f vs ) is exchanged through fluid-structure interaction. While the individual contributions differ, the total KE, which is our interest herein, remains cyclic within a certain time interval. Next, we consider the locality of the energy exchanges in the phenomenology of the turbulent cascade in the near wake region. We calculate the kinetic energy and the enstrophy ξ of fluid regions where most of the coher-

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ent structures are concentrated; e.g., 0.5D ≤ X ≤ 1.5D and |Y | ≥ 0.5D region. For this purpose, we select a representative control volume (RCV) indicated in Fig. 2.36, which contains the turbulent flow field and the effects of dominating wake structures (e.g., shear layer roll-up). Slightly larger and smaller-sized RCV-based fluid objects with the same center are found to give similar trends for the total KE and ξ . Furthermore, the same-sized RCVs inside the near wake also display a similar variation of these flow parameters. Hence, this representative fluid object is considered a proper representation to investigate the wake dynamics and the structural interaction. The total kinetic energy and the enstrophy of an incompressible fluid control volume can be expressed as  1 |u f |2 d V, 2V V  1 |∇ × u f |2 d S. ξ= S KE =

(2.22) (2.23)

S

where the kinetic energy is a volumetric scalar indicative of the total (inviscid + rotational) energy components of the fluid object and the enstrophy is the quantity directly related to the dissipative effects of the kinetic energy. The 3D enstrophy is also an indicator of the coherent structure generation and the breakdown, as shown by Dascaliuc et al. [102]. While we approximate the KE integral using the standard 3 × 2 × 2 Gaussian quadrature rule, the enstrophy formula is approximated by the summation of 2 × 2 Gaussian quadrature along each surface of the control volume. These integral approximations of the kinetic energy and the enstrophy can be expressed as: 1 KE = 2V

 |u f |2 d V ≈ V

ξ=

1 S

1  Wi W j Wk |u(βi , φ j , ψk )|2 |J (βi , φ j , ψk )|, 2V i j k ⎡

 |∇ × u f |2 d S ≈ S

(2.24) ⎤

6  1 ⎣  Wi W j |∇ × u(βi , φ j )|2 |J (βi , φ j )|⎦ , S n n=1 i j n

(2.25) where Wm , m = i, j, k are the standard Gaussian weights while βi , φ j , ψk are coordinates of the standard Gauss points for standard 8-node brick and 4-node quadrilateral elements and J denotes the Jacobian of the coordinate transformation. In the enstrophy formula, n = 1, 2, ..., 6 denotes the 6 surfaces of the rectangular fluid region. Figure 2.38 shows the time histories of the kinetic energy and the enstrophy in the representative region for the stationary and VIV configurations over a few shedding cycles. Due to the three-dimensionality and the wide frequency bandwidth of the turbulent wake, the KE and the enstrophy variations do not repeat with exact

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Fig. 2.38 Time histories of the kinetic energy and the enstrophy for the representative fluid region (RCV) of size 0.5D × 0.4D × 0.3D with center (1.0, 0.9, 0) for a stationary and b VIV cases

magnitudes. However, it is clear that the variations of both quantities are cyclic for both configurations. The enstrophy, which is directly related to the dissipation effect of the fluid has a burst slightly delayed (∼ 5St) from the kinetic energy burst. This enstrophy burst is an indicator of the breakdown of coherent structures contained in the control volume [372]. By comparing the period of kinetic energy and the enstrophy variation for the stationary and VIV cases, we can determine a core reason for the lock-in phenomenon. According to Table 2.6, the average interburst period of the VIV increases to 33% as compared to its stationary counterpart. This stretched interaction period gives rise to a decrease in the frequency of the coupled wake system. This decrement in the frequency causes the wake to be synchronized to the rigid structure motion thereby

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Table 2.6 Comparison of enstrophy cycle parameters between stationary and VIV cases: t B — average interburst period, t A —average active period, duty cycle and f ens —frequency of the enstrophy cycle; averaged for 15 bursting cycles Stationary VIV t A U∞ /D t B U∞ /D Duty cycle (t A /t B ) f ens

3.09 8.35 36.99% ≈ f Cl

4.62 12.45 37.09% ≈ f Cl

Fig. 2.39 A schematic of self-sustaining process of turbulent wake structure development for stationary and VIV configurations. Here KE is the kinetic energy, u  v  < 0 denotes that shear growth dominates, u  v  ≈ 0 denotes that shear growth (u  v  < 0) and damping (u  v  > 0) coexist. The kinetic energy for the flow is continuously supplied by the steady inflow

the coupled fluid-structure system leads to the phenomenon of synchronization/lockin. During the synchronization, the cylinder reaches to the limit cycle state with a relatively large amplitude whereby nonlinear effects of the wake system tend to saturate the response. Interestingly, the duty cycle of both stationary and VIV cases are similar (∼ 37%) indicating that the flow structures break down during the same proportion of the interburst time period. Next, we provide a unified description and establish a link among the flow structure development, the shearing effect, and the kinetic energy and the enstrophy bursts through a self-sustained cycle. A general self-sustained cycle of turbulent flow structure development can be briefly explained as follows. When a highly non-linear and three-dimensional flow is disturbed by the presence of a bluff body, the coherent structures develop and

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then lead to a break-down and regeneration process. During this process, there is a continuous supply of steady kinetic energy input to sustain this cycle of flow structure development. The process of the self-sustained cycle is summarized in Fig. 2.39. Notably, this cycle has some features of the self-sustaining process of turbulent structures in wall-shear flows [372], but differs in the aspects of largescale periodic disturbances of the vortex shedding and the absence of wall effect at the wake centerline. In this self-sustaining cycle, likewise the near wall-turbulence cycle, we can start from any place but let us begin with the interaction between streamwise vortices and mean shear. When a disordered flow field is subjected to a mean shear action in the velocity gradient direction; which is indicated by a growth in the mean shear (u  v  < 0), we observe that the vortex structures with axial vorticity (streamwise ribs) are developed by the stretching of the spanwise rollers extending throughout the span. When the mean shear acts again on these spanwise rollers we find that it results into the hairpin vortices whose legs are the streamwise ribs developed earlier. Increased shearing results into the kinetic energy burst and the structures break down resulting in a disordered field. Due to the continuous kinetic energy input, this self-sustaining cycle repeats for the sheared wake flow. In relation to the stationary cylinder wake, we observe the streamwise ribs and the spanwise rollers but not the hairpin structures. In the VIV case, we observe all three flow structures. Through the Reynolds shear stress distribution, we can provide the reason for this behavior. The shearing process of the stationary cylinder wake is of higher frequency and greater intensity and leads to a sudden kinetic energy burst, which allows bypassing the development stage of the hairpin vortices. Relatively lower frequency dominance and lesser intense flow shearing during VIV gives rise to the complete cycle of the coherent structures. To summarize, the transverse vibration breaks down the spatial symmetry of the vortical wake behind the body which leads to a new type of symmetry in the time domain. This adjustment in the temporal dynamics gives rise to the frequency lock-in and the energy transfer from a surrounding flow to a vibrating structure. The dominance of low frequency prevents a sudden kinetic energy burst and results into the intermediate hairpin structures. In classical dynamics, these continuous symmetries are equivalent to Hamilton’s principle of least action and lead to the existence of conservation laws [142]. This study provides a unified understanding through the connection between the symmetry in the evolution of Reynolds stresses, the flow shearing process and the development of coherent structures during VIV lock-in. The near-wake structures responsible to supply the energy for free vibrations are regenerated autonomously by a self-sustaining process. Key dynamics of the selfsustaining process in the near-wake appear to be well described by the representative control volume. Further work is required to understand the interaction of a vibrating cylinder with a representative volume of fluid in the near-wake, which can be useful in model reduction and in developing control strategies.

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2.4.7 Summary In this chapter, we have numerically investigated the synchronization and the wake statistics of a freely vibrating square cylinder at m ∗ = 2.64 for subcritical Reynolds numbers 1400 ≤ Re ≤ 10,000. We employed the variational fluid-structure interaction formulation and the dynamic subgrid turbulence model. We first validated the results obtained by the numerical scheme with the published experimental data for the stationary configuration and established the accuracy of the numerical model for the flow-induced vibration. The results for the vibrating cylinder for zero incidence angle are then validated with the recent experimental data. Three synchronization regimes namely 1:1, 1:3 and 1:5 and the associated wake modes have been successfully predicted by our numerical simulations. The FIV response amplitudes, the regimes and frequent contents are well captured by the dynamic subgrid LES using the present Petrov-Galerkin-based fluid-structure formulation. By comparing the wake topology of stationary and the VIV lock-in, we found significant differences in the 3D vortical structures and observed hairpin-like patterns in the near-wake region for the vibrating case. The periodic vibration of the body provides an avenue for coalescing and merging of small eddies in the vicinity of the body. This eventually results in somewhat shorter streamwise ribs and smaller size of spanwise rollers in the near wake region. While the stationary cylinder exhibits a spatially symmetric shearing process, the FIV motion of the cylinder disturbs this spatial symmetry of the shearing process and shifts the wake symmetry to the time domain. As a result, there is a competition between the growth and damping components of mean shear, which makes the shearing process lesser intense and has a low-frequency shift in the spectra of Reynolds shear stress. Compared to the stationary cylinder, the average interburst period of 1:1 synchronization increases to 33%, which appears to be linked with a decrease in the wake frequency. The kinetic energy and enstrophy bursts follow the same low frequency in the case indicating that the wake structures break down and regenerate at a lower rate. This lower frequency allows the periodic vortex wake frequency to be synchronized with the structural natural frequency, thereby resulting into the phenomenon of synchronization/lock-in. This synchronization process with the matching of frequencies is associated with the high amplitude of cylinder response. Finally, we introduced the self-sustaining process of coherent flow structures and explained that the frequency difference during the shearing process gives rise to intermediate hairpin-like flow structures very close to the vibrating body. Acknowledgements Some parts of this Chapter have been extracted from the PhD thesis of Tharindu Miyanawala carried out at the National University of Singapore and supported by the Ministry of Education, Singapore.

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Appendix A: Turbulent Energy Balance in the Near Wake While the power law scaling of the energy spectrum for the near field of planar wakes of circular cylinders has been reported in previous studies [63, 321], we briefly present the various terms of the energy balance based on the Karman-HowarthMonin-Hill (KHMH) equation [343]. The energy balance via the KHMH equation incorporates the interscale and interspace transfers as well as advection, turbulence dissipation and production. Consider a six-dimensional reference frame with a position xi in physical space and the scale space ri for all separations and orientations between two points. By means of the Reynolds decomposition and using the differences of fluctuating velocity, the mean velocity and the fluctuating pressure [154], the energy balance for the incompressible turbulent flow can be expressed as [343, 426] (2.26) At = A − Π − ΠU + P + TU + T P + Dx + Dr − ε, where At = the rate of change of TKE, A = advective transport of TKE by the mean flow, Π = non-linear interscale energy transfer, ΠU = linear interscale energy transfer, P = production of TKE by mean velocity gradients, TU = transport of TKE by fluctuating velocity, T P = 21 × velocity and pressure gradient correlation, Dx = diffusion due to viscosity in physical space, Dr = diffusion in scale space due to viscosity and ε = dissipation rate. For length scales larger than the local Taylor length scale (r T ), the diffusion effects are negligible. For the length scales smaller than the local inhomogeneity length scale (r H ) the terms ΠU , P, TU and T P become insignificant [343]. Hence, for the length scale range bounded as r T < ri < r H , also termed as the inertial range, the energy balance Eq. (2.26) reduces to the Kolmogorov’s local equilibrium condition in terms of the balance between the interscale energy transfer Π and the turbulence dissipation rate per unit mass ε: Π = −ε.

(2.27)

The above relationship is valid for the well-developed homogeneous turbulent region of the wake flow, i.e. when X > 1.0D. However, in the recent DNS study [343], it has been shown that the energy cascade close to the near wake centerline region follows the −5/3 power law and the Taylor frozen turbulence hypothesis is reasonably valid. Using the orientation averaged terms of the KHMH equation, they further justified the reason of this surprising result by considering two approximate energy balances in the wake region as: Π a + εa ≈ 0 ≈ −Aa + TUa + T Pa − (ΠUa − P a ).

(2.28)

where the superscript a denotes the orientation averaging of the energy terms. Furthermore, the nearly constant variation of interscale energy transfer Π a was found in the PIV measurements of [132] for a range of separations in the very near field of inhomogeneous turbulent flow field. Thus it can be established that along the wake

2.4 Three-Dimensional FSI of a Square Column at High Reynolds Number

83

centerline, the kinetic energy spectra follow the −5/3 power scaling even in the near wake region. Consistent with the DNS study [343], our variational LES results confirm the −5/3 energy spectra of different wake centerline points over a decade range of frequencies for the inhomogeneous and anisotropic turbulence in the near wake region of a square-shaped prismatic cylinder.

Chapter 3

Proximity and Wake Interference

In this chapter, we discuss the proximity and wake interference effects on freely vibrating prismatic square and circular bodies. We investigate the flow physics of the gap flow and the VIV kinematics in terms of the wake topology, the response characteristics, the force components, and the phase and frequency characteristics. To understand the origin of wake-induced vibration, we also consider two circular cylinders in a tandem arrangement where the upstream cylinder is fixed and the downstream cylinder is allowed to vibrate in the transverse direction. We present the characterization of the response dynamics of isolated and tandem cylinders and discuss the basic differences between the two arrangements in terms of force decompositions, the phase relations, the stagnation point movement, the pressure coefficients and the corresponding wake contours. Detailed investigations of the upstream vortex interacting with the boundary layer of the downstream cylinder and the effects of the streamwise gap between the cylinders are provided.

3.1 Introduction Offshore and civil engineering structures interacting with the surrounding flow are inevitably subject to unsteady fluid forces and they may undergo flow-induced vibrations (FIV) under certain conditions. The flow past flexibly mounted bodies provides a generic FIV model setup and has been extensively studied both numerically and experimentally for understanding coupled nonlinear dynamics of flow-structure interaction. Of particular interest in this chapter is to investigate proximity and wake interference effects which are common in various marine/offshore and wind engineering systems. Such proximity and interference effects can be considered via side-by-side and tandem configurations.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Jaiman et al., Mechanics of Flow-Induced Vibration, https://doi.org/10.1007/978-981-19-8578-2_3

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Proximity interference becomes noticeable when two bodies are close to each other, e.g., 0.5D–3.5D in the transverse direction, where D is the characteristic length of a bluff body. After the boundary layer separates from the solid wall, it becomes a free shear layer in the wake or possibly sheds away as vorticity clusters. The influence of proximity interference becomes prominent when there is an interaction between the boundary layers or the separating shear layers from a solid body. Such proximity interference frequently occurs in various engineering applications. The simplest form to observe the proximity interference is the flow past a pair of side-by-side square bodies and circular cylinders, which are presented in this chapter. A detailed review of the flow dynamics of stationary side-by-side circular cylinders can be found in [402]. When an additional body is placed downstream of the first, the flow phenomena become further intricate and are now governed by the relative positioning of the two bluff bodies. Three flow interference regimes [175, 482]: proximity interference, wake interference (reattachment), and co-shedding can be identified for the tandem configuration. The range of spacing to diameter (L/D) ratio for each of these categories is problem-dependent. In the proximity interference regime for 1 ≤ L/D ≤ 1.2–1.8, negative drag is produced on the downstream cylinder and vortex shedding from the upstream cylinder is suppressed. The tandem bodies behave like a single bluff body and vortex shedding occurs behind the rear cylinder. In the wake interference or reattachment regime for 1.2–1.8 ≤ L/D ≤ 3.4–3.8, several different phenomena such as shear layer reattachment, intermittent vortex shedding, etc. can be observed as the separation distance is gradually increased. In the regime of large spacing L/D ≥ 3.8, the so-called co-shedding regime, vortex shedding occurs from both the cylinders and there is no interference effect. Further review and discussions will be provided later in chapter. The organization of the chapter is as follows. The proximity interference effects on the side-by-side prismatic columns are discussed in Sect. 3.2, along with the formulation of the problem and the key parameters. In Sect. 3.4, we present the characteristics of the two side-by-side circular cylinders for the independent vibrating configuration and compare the results with the combined vibrating counterpart. This is followed by the investigation of three-dimensional effect in Sect. 3.5 with a detailed discussion in terms of wake topology, response characteristics, force components and frequency characteristics. The wake interference effects on the tandem square prisms and circular cylinders are discussed in Sects. 3.6 and 3.7, respectively. We present the characterization of the response dynamics of isolated and tandem cases and discuss the basic differences between the two arrangements in terms of force decompositions, phase relations, stagnation point movement, pressure coefficients and the corresponding wake contours. The three-dimensional wake interference effects on the tandem circular cylinders are discussed in Sect. 3.8.

3.2 FSI of Side-by-Side Square Prisms at Low Reynolds Number

87

3.2 FSI of Side-by-Side Square Prisms at Low Reynolds Number Squared cross-section structures are commonly used as a fundamental member in a wide range of offshore, aerospace and civil engineering applications. When the structure is free to vibrate, there exists a strong fluid-structure coupling between the motion of structure and the wake dynamics [39, 138, 207, 335, 455–457]. For a squared cross-section structure immersed in the flow stream, the vibrational response usually exhibits a combination of both vortex synchronization and galloping as functions of Reynolds number, reduced velocity, mass-damping and various geometry-related parameters [41, 185, 194, 264–267, 386, 495]. The side-by-side configuration of multiple squared structures has many implications in offshore engineering applications, e.g., large-scale floating production, storage, and (FPSO) and semi-submersible platform operating in a side-by-side arrangement with wind and ocean current flow, multiple cylinders bolted together with a common pontoon base in a floating platform, tender assisted drilling along with tension leg platform. Apart from their engineering relevance, the fluid-structure interaction of multiple square-shaped cylinders offers the canonical side-by-side configuration to explore the fundamental behavior of wakes behind vibrating structures. The accurate prediction of flow-induced vibration (FIV) in a multi-cylinder configuration is a formidable task for researchers due to the complex wake interference, vortex-induced vibrations, galloping, and several other self-excited instabilities for both the side-by-side and tandem configurations of square cylinders. The large oscillations due to fluid-structure interaction can be dangerous and can result in structural failure, for example in offshore platforms at high ocean currents [431]. The coupled fluid-structure responses of multiple square cylinder systems are significantly different and are much more complex than the isolated square cylinder due to the effects of vortex-to-vortex and the interactions between the cylinder and the gap flow. The recent FIV investigations on tandem and side-by-side square cylinders by [145, 148, 180] were carried out at low Reynolds numbers Re ∈ [100, 200] for mass ratio m ∗ ∈ [2.6, 10], where ρ f is the fluid density, μf denotes the dynamic viscosity, U and D denote the free-stream speed and the diameter of cylinder, respectively. The present study considers the Reynolds number Re = 200 and the mass ratio m ∗ = 10 for the comparison of combined and independent configurations for side-by-side square cylinders. Although the configurations considered herein may seem a somewhat simplification of realistic engineering situations, they contain the important features of gap flow dynamics and flow-induced vibration. Previous experimental and numerical investigations for two side-by-side cylinder configurations have been mainly conducted in a stationary condition. In particular, several experimental works of multiple circular cylinders have been done in past decades for a side-by-side configuration [402, 481, 482]. For the two identical side-by-side square cylinders at a fixed gap ratio (g ∗ = 2), Kolar et al. [214] studied the wake flow through a two-component laser-Doppler velocimetry system at Reynolds number of Re = 23,000. Compared with the isolated square cylinder, the

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two-cylinder system was found to have a higher Strouhal number St = f vs D/U ; e.g., the single cylinder with St=0.13 and the two-cylinder with St = 0.14, where f vs denotes the vortex shedding frequency. Yen et al. [469] conducted experiments in an open-loop wind tunnel by means of a smoke-wire scheme to capture the flow profiles, and measured the surface pressure and the vortex shedding frequency through a pressure transducer and a hot-wire anemometer for the Reynolds number 2262 < Re < 28,000 and the gap ratio 0.6 ≤ g ∗ ≤ 12. The flow characteristics were classified into three regimes as a function of gap ratio, namely the single bluff-body, the gap-flow and the coupled vortex-shedding. From the classical experimental measurements of [50] for circular cylinders, the maximum value of drag coefficient was found for the single-bluff-body regime (g ∗ < 1) and the minimum at bistable/flipflop regime (g ∗ > 1) with respect to the single isolated cylinder at Re = 4.7 × 104 . The flip-flop regime is defined as the bistable regime where the gap flow between the cylinders is biased, resulting in one wide and one narrow vortex street [43]. In the event of a flip-flop, the jet-like gap flow fails to maintain its original straight path and tends to deflect intermittently with new asymmetric states. This intermittent bistable instability causes highly chaotic and irregular variations in the flow dynamic variables. The cylinder with the deflected gap flow has a narrower near-wake region than its counterpart with the wider vortex street, as discussed in [213, 254]. In the recent experiments, Refs. [5, 6] characterized the wake dynamics of two side-by-side square cylinders at Reynolds number about 47,000 and for the gap ratio 0 ≤ g ∗ ≤ 5.0. Instead of the three regimes reported by [5, 469] identified four flow regimes, where the gap flow mode has been further decomposed into two regimes. In the range of g ∗ = 0.3 − 1.2 for the gap flow, the jet develops a certain adequate strength and separates the wake into one narrow and one broad vortex street with high and low vortex shedding frequencies, respectively. The range g ∗ = 1.2 − 2.0 can be considered as the transition regime, where the three distinct vortex frequencies are observed intermittently as compared to the two-frequency mode. Most of the works of both circular and square cylinders are concentrated on the stationary condition for a broad range of Reynolds numbers. For the vibrating condition, few works can be traced in the literature and almost all of those studies are concentrated on circular cylinders. The numerical investigation of two elastically mounted coupled circular cylinders with one-degree-of-freedom (1-DOF) in the sideby-side arrangement was carried out by Cui et al. [99]. The distance between two cylinders was kept at g ∗ = 2 and the RANS equations are solved by the 2D finite element method at Re = 5000 and m ∗ = 2. From their numerical studies, the authors observed five different response regimes, namely the first-mode lock-in regime, the second-mode lock-in regime, the sum-frequency lock-in regime and two transition regimes, with symmetric and asymmetric flow profiles. Kim et al. [213] conducted the experiments to examine FIV characteristics of two identical circular cylinders in a side-by-side arrangement for gap ratio g ∗ = 0.1 − 3.2 to cover all possible flow regimes and observed four characteristic vibrational patterns as a function of the gap ratio. While Regime I (0.1≤g ∗ 30. If the mass ratio is relatively small, the kink may not occur, while if the mass ratio is relatively large, the lock-in may not appear. In the considered gap ratios g ∗ = 1.2, 1.6 and 2.0 corresponding to Regimes II and III, the flow behavior gradually changes from the gap flow regime into the coupled vortex shedding regime. Unlike the combined vibrating configuration in which no vortex synchronization occurs in both Regime II and Regime III, the vortex synchronization phenomena in the independent vibrating configuration occur in both gap ratios at relatively higher reduced velocity (Ur ≈ 7 − 9). However, the flow behaves similarly to the combined vibrating configuration in the galloping range, which shows a relatively smaller transverse vibration amplitude and tends to asymptote after Ur ≈ 25. A summary diagram on the effect of gap ratio on the VIV and the galloping amplitudes is provided at the end of this section. Understanding of these vibration regimes is useful to circumvent oscillatory behavior of offshore structures and the development of suppression devices based on the jet flow injecting into the near-wake. While the representative gap ratios are considered herein, a broad range of the reduced velocity is investigated to explore the characteristics of flow-induced vibrations. The comparison between the combined and independent configurations is mainly motivated by the practical understanding of multibody offshore systems subjected oceanic flows.

3.2.1 Numerical Methodology The governing equations for the fluid are written in an arbitrary Lagrangian-Eulerian (ALE) form while the structural equation is considered in a Lagrangian manner. The Navier-Stokes equations for an incompressible flow in the ALE frame are

3.2 FSI of Side-by-Side Square Prisms at Low Reynolds Number

91

 ∂ uf  ρ + ρ f (uf − w) · ∇uf = ∇ · σ f + bf on Ω f (t), ∂t x

(3.1)

∇ · uf = 0 on Ω f (t),

(3.2)

f

where uf = uf (x, t) and w = w(x, t) represent the fluid and mesh velocities defined for each spatial point x ∈ Ω f (t), respectively, bf is the body force applied on the as σ f = fluid and σ f is the Cauchy  stress tensor for a Newtonian fluid, written f f f T f − p I + μ (∇u + ∇u ) , where p denotes the fluid pressure, μ is the dynamic viscosity of the fluid. The spatial and temporal coordinates are denoted by x and t, respectively. In Eq. (3.1), the partial time derivative with respect to the ALE referential coordinate  x is held fixed. The present study involves streamwise and transverse translational degrees-of-freedom of the rigid-body structures concerned and the body is restricted from having a rotational degree-of-freedom. A rigid-body structure immersed in the fluid experiences unsteady fluid forces and consequently may undergo flow-induced vibrations if mounted elastically. The rigid-body motion along the Cartesian axes is governed by the following equation: m·

∂us + c · us + k · (ϕ s (z 0 , t) − z 0 ) = Fs + bs on Ω s , ∂t

(3.3)

where m, c and k denote the mass, damping and stiffness vectors per unit length for the rigid body, us (t) represents the the translational degrees-of-freedom, Ω s denotes ∂ϕ s s rigid-body velocity at time t as u (z0 , t) = ∂t , where ϕ s denotes the position vector mapping the initial position z0 of the rigid body to its position at time t, and F s and bs are the fluid traction and body forces acting on the rigid body, respectively. The solid body is rotationally constrained and thereby the conservation of angular momentum and the velocity corresponding to the rotations can be ignored. The fluid and the structural equations are coupled by the continuity of velocity and traction along the fluid-structure interface. Let Γfs = ∂Ω f (0) ∩ ∂Ω s be the fluid-structure interface at t = 0 and Γfs (t) = ϕ s (Γfs , t) is the fluid-structure interface at time t. The coupled system requires to satisfy the continuity of velocity and traction at the fluid-body interface Γfs as follows 

uf (ϕ s (z 0 , t), t) = us (z 0 , t) ,

(3.4)

σ f (x, t) · ndΓ (x) + F s = 0,

(3.5)

Γfs

where n is the outer normal to the fluid-body interface and dΓ denotes a differential surface area. Further details about the fluid-structure formulation can be found in [180, 181].

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3.2.2 Problem Description and Key Parameters The schematics of two side-by-side square cylinders are shown in Fig. 3.1 for both combined and independent configurations. The center point of the gap is located at the origin of the Cartesian coordinate system. The center points of the cylinders are collinear at the beginning of the computations. The unit side length of a square cylinder is denoted as D. The distances to upstream and downstream boundaries are 20D and 40D, respectively. The gap ratio g ∗ defines as the distance g over unit square length D as g ∗ = g/D. Similar to [144], each side of the square cylinder is discretized with 100 uniformly distributed nodes. The size of the first layer cell surrounding two square cylinders is controlled within 0.01D. The distance between the outer surface of the cylinder and the side-wall constantly remains 19D in order to keep the same side wall effect of the single square cylinder. Three different gap ratios, g ∗ = 1.2, 1.6 and 2.0 are within Regime II and Regime III, are considered to compare the differences between the two-dimensional and three-dimensional effects in the sideby-side configuration. The domain is discretized using a hybrid finite-element mesh. The grids for the gap ratio g ∗ = 1.2, 1.6 and 2.0 comprise of 161,444 elements, 169,876 and 202,256 elements, respectively. The mesh convergence study for this side-by-side configuration has been presented in our previous work [180]. The flow velocity U is set to unity at the inlet and the top and bottom boundaries are considered as slip walls. The no-slip boundary condition is imposed on the cylinder walls. The pressure outlet condition is imposed at the exit of the computational domain, as shown in Fig. 3.1. For the three-dimensional simulation, the two-dimensional domain extrudes to with 25 layers and the span-wise height is 5D (Δz = 0.2D). The convergence and validation of the adopted numerical methodology have been established in earlier studies [144, 180, 181, 254, 307]. Apart from the gap ratio g ∗ between the side-by-side cylinders, the flow-induced vibration of the coupled system is strongly influenced by four key non-dimensional parameters, namely mass-ratio (m ∗ ), Reynolds number (Re), reduced velocity (Ur ), and damping ratio (ζ ) defined as: m∗ =

m , mf

Re =

ρfUD , μf

Ur =

U , fN D

c ζ = √ , 2 km

(3.6)

where m is the mass of a single square cylinder, c and k are the damping and stiffness coefficients, respectively for an equivalent spring-mass-damper system of a vibrating structure, U and D the free-stream speed and the diameter of the cylinder, respec√ tively. The natural frequency of the body is given by f N = (1/2π ) k/m and the mass of displaced fluid by the structure is m f = ρ f D 2 L c for square cross-sections, where L c denotes the span of the cylinder. In the above definitions, we assume that the translational motion of the rigid body is isotropic, i.e., the mass vector m = (m x , m y ) with m x = m y = m , the damping vector c = (cx , c y ) with cx = c y = c, and the stiffness vector k = (k x , k y ) with k x = k y = k for the independent configuration. To maintain the same natural frequency for the combined (dumbbell-like model) vibrat-

3.2 FSI of Side-by-Side Square Prisms at Low Reynolds Number v=0,

93

σxy=0 comb

K Y = 2k comb

K x =2k

σxx=0 σyx=0

Y

u=U g v=0

X

D

Mcomb=2m

S=19D

v=0,

σxy=0

Lu=20D v=0,

(a)

σxy=0

Ld=40D

indp

K y1 =k

indp

K x1 =k

σxx=0 σyx=0

Y

u=U Mindp=m

g

v=0

X

D indp

K x2 =k indp

S=19D

K y2 =k v=0, Lu=20D

σxy=0 (b)

Ld=40D

Fig. 3.1 Schematics of freely vibrating two side-by-side square cylinders: a combined configuration with joint 2-DOF motion of both cylinders connected with a linkage, b independent configuration with each cylinder vibrating separately with 2-DOF motion. The natural frequencies of mass-spring system in vacuum are kept identical for both configurations

ing configuration, we consider the stiffness values to be k x comb = 2k x indp = 2k and k y comb = 2k y indp = 2k and the total system mass m comb = 2m indp = 2m. The damping ratio is ζ = 0 and the mass ratio is set to m ∗ = 10 for all two-dimensional and three-dimensional simulations. The fluid loading is computed by integrating the surface traction considering the first layer of elements located on the cylinder surface. The instantaneous force coefficients are defined as

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3 Proximity and Wake Interference

CL =

1 1 f 2 ρU D 2

 (σ f · n) · n y dΓ, C D = Γ

1 1 f 2 ρU D 2

 (σ f · n) · nx dΓ,

(3.7)

Γ

where ρ f , σ f denote fluid density and stress, respectively, and nx and n y are the Cartesian components of the unit normal, n. In the present study, the drag coefficient C D and lift coefficient C L are post-processed using the conservative flux extraction of nodal tractions along the fluid-body interface. The standard spectral technique, Fast Fourier Transform (FFT), will be used to analyze the steady and time-invariant signal in this section.

3.3 Results and Discussion Our investigations are conducted in both two-dimensional and three-dimensional at Re = 200, m ∗ = 10 and ζ = 0. To cover the range from the vortex synchronization to galloping, we consider the reduced velocity from Ur ∈ [1, 40] for two-dimensional simulations. The comparison of different fluid modes of flow-structure interaction is discussed. While the three representative gap ratios (g ∗ = 1.2, 1.6, 2.0) are considered for the comparison of combined and independent configurations, we examine a wide range of reduced velocities to characterize the FIV response. Essentially, the independent configuration represents the mirror symmetry with respect to the gapflow centerline (2-DOF each symmetry plane with the identical spring-mass parameters, see Fig. 3.1b). Gap flow interference and side-by-side wake interactions impose additional effects on the vibrational characteristics. Both cylinders (top and bottom) for the independent configuration are free to vibrate in two-degree-of-freedom motions in the streamwise (X ) and transverse (Y ) directions, where the mass and natural frequencies are identical in both X - and Y -directions. Each elastically-mounted cylinder in the independent configuration recovers to the isolated counterpart for a large gap ratio. As discussed in [144], the combined configuration has a rigid link between the cylinders and they interact with the gap flow interference and the sideby-side coupled wakes as a single body. After the completion of the 2D analysis, we will perform three-dimensional simulations for the representative reduced velocities whether the maximum FIV response amplitudes occur.

3.3.1 Two-Dimensional Simulations As presented in [144], the combined vibrating configuration can be classified into four vibrating regimes as a function of the gap ratio. From g ∗ < 0.4 which is in the combined vibrating Regime I, the whole system acts as a single 1D × 2D rectangular cylinder and no flow passes through the gap between the two square cylinders. The transition from the rectangular cylinder to the square cylinder occurs in the range

3.3 Results and Discussion 1.2

95

Si ng le Combined I ndependent

AY rms /D

0.9

0.6

0.3

0

0

5

10

15

20

25

30

35

40

Ur Fig. 3.2 Dependence of transverse vibration amplitude ArYms on reduced velocity Ur at (g ∗ , m ∗ , Re) = (1.2, 10, 200) for the transition regime. The response of single cylinder (g ∗ = ∞ ) is also included in the plot for comparison purpose

of g ∗ ∈ [0.4, 2.5], which correspond to Regimes II and III. The gap flow results in no-vortex synchronization for all the reduced velocities. The wake becomes regular in the vibrating Regime IV and almost recovers to single square cylinder mode.

3.3.1.1

Transition Regime

The transition between Regimes II and III for the combined configuration occurs near the gap ratio g ∗ = 1.2, which possesses the characteristics of both regimes. Compared with the combined two side-by-side cylinder system, the two independent vibrating side-by-side cylinder system exhibits the dominance of gap flow characteristics in the force and response amplitude. However, the flow profiles are similar to the vibrating coupled vortex regime whereas the biased gap flow (i.e., a tendency of the gap flow closer to one cylinder) becomes weaker. In Fig. 3.2, the root-mean-square (rms) of transverse amplitude (ArYms ) of the independent system is presented for one cylinder as the trend of the amplitudes versus reduced velocity of both cylinders are similar. Unlike the vanished desynchronization trend in the combined vibrating regime, the independent vibrating system exhibits a distinct synchronization, the transitional kink and the galloping phenomena. These phenomena have a resemblance to the singlecylinder case. However, the amplitude magnitude of each cylinder is approximately half of that of a single cylinder counterpart (g ∗ = ∞ ). The instantaneous vorticity contours for the gap ratio g ∗ = 1.2 are illustrated in Fig. 3.3 for the three cases at Ur = 7, 17 and 35 corresponding to the lock-in,

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Fig. 3.3 Instantaneous vorticity contours (left) and streamline patterns (right) of two independent vibrating side-by-side cylinders at (g ∗ , m ∗ , Re) = (1.2, 10, 200) for different reduced velocities Ur = a 7, b 17, c 35

the transitional kink and the galloping, respectively The effect of gap flow can be observed in the near-wake region, but it is not as strong as that in the small gap ratio [144]. Two narrow and broad vortex streets are nearly similar and quickly fuse into one street. Clockwise and anti-clockwise inner vortices from two cylinders interact with each other and diffuse quickly, as opposed to the co-existence of two vortices in a larger gap ratio. Therefore, only the external vortices from the extreme top and bottom surfaces survive and eventually combine into one vortex street, especially at the low reduced velocity Ur = 7. The above observations have remarkable similarities with the staggered configuration of circular cylinders [141]. Consistent with the present independent vibrating configuration, the authors found the gap flow as a dominant feature to influence the coupled dynamics of staggered circular cylinders for a cross-

3.3 Results and Discussion

97

stream offset of T /D = 1.0 at L/D = 1.4. Furthermore, the interaction dynamics and the formation of shear layers and vortices from the present study show some resemblance with the staggered vibrating arrangement of [141]. An elaborate linking and comparison should be investigated in future work. For higher reduced velocity, e.g. Ur = 35, such merging process of inner vortices does not exist due to large transverse amplitude. Figure 3.4 shows the irregular time traces of force and amplitude of both upper and lower cylinders for Ur = 7, corresponding to the peak amplitude in the vortex synchronization regime. Compared with the single cylinder behavior, a beating pattern in the lock-in does not take place in the side-by-side configuration. Since the two side-by-side cylinders are vibrating independently, the range of gap ratio is found to vary between 0.7 and 1.9. When the temporary gap ratio is less than g ∗ < 1.0, the gap flow effect is enhanced and the flip-flop phenomenon appears; and an irregular chaotic-like response becomes prominent for this gap ratio. This also leads to much interference to extract the dominant frequency of lift coefficient (C L ) and the transverse amplitude ( AY /D) and the natural frequency does not synchronize with the vortex shedding frequency. When the temporary gap ratio is greater than g ∗ > 1.2, the gap flow effect begins to weaken and the vortices from the two cylinders are coupled. However, the effect of gap flow still results into the irregular flow behavior. The trajectories of two cylinders are quite symmetric along the central-line. The external far-end trajectories are inclined to upstream, while the internal near-end ones are inclined to the downstream side. The force trends and the vibration amplitudes become more regular and periodic as the reduced velocity increases and the power spectra show only one dominating frequency in both force and vibration responses. Figure 3.5 shows the time series of force responses, the trajectories and the frequency at Ur = 17 where the kink with a large oscillation amplitude occurs. The large-amplitude kink occurs at three times of the peak of lock-in. Two dominant frequencies are found ( f 1/3 = 0.05 and f V I V = 0.16) in the lift force response, and the frequency of amplitude is synchronized with the low frequency. This finding is consistent with the experimental observations of a freely vibrating square cylinder at moderate Reynolds number [492]. At Ur = 35 in Fig. 3.6, the drag and lift force responses show a beating pattern, while the vibration amplitude response becomes more regular than those in the lower reduced velocities Ur = 7 and 15. The frequency of the vibration amplitude can be observed as one dominant low-frequency f galloping = 0.02.

3.3.1.2

Coupled-Vortex Regime

As demonstrated in [144], the two gap ratios, g ∗ = 1.6 and 2.0, are in the coupledvortex regime, whereas the vortices from the two cylinders are coupled and move in pairs behind the cylinders. The paired vortices are in the anti-phase mode and the biased gap flow no longer exists to influence this wake flow system. Each cylinder generates its own street which appears somewhat symmetric with respect to the central-line. In Fig. 3.7, the synchronization peak occurs at Ur = 7, which has a

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3 Proximity and Wake Interference

Fig. 3.4 2D results of two side-by-side system for upper cylinder (left) and lower cylinder (right) for the independent vibrating configuration of g ∗ = 1.2 at (m ∗ , Re, Ur ) = (10, 200, 7): a, b force responses, c, d transverse amplitude, e trajectory of the system, and f power spectrum of C L and AY

3.3 Results and Discussion

99

Fig. 3.5 2D results of two side-by-side system for upper cylinder (left) and lower cylinder (right) in independent vibrating configuration of g ∗ = 1.2 at (m ∗ , Re, Ur ) = (10, 200, 17): a, b force responses, c, d transverse amplitude, e trajectory of the system, and f power spectrum of C L and AY

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3 Proximity and Wake Interference

Fig. 3.6 2D results of two side-by-side system for upper cylinder (left) and lower cylinder (right) in independent vibrating configuration of g ∗ = 1.2 at (m ∗ , Re, Ur ) = (10, 200, 35): a, b force responses, c, d transverse amplitude, e trajectory of the system, and f power spectrum of C L and AY

3.3 Results and Discussion 1.2

101

Single Combined Independent

AY rms/D

0.9

0.6

0.3

0

0

5

10

15

20

25

30

35

40

Ur Fig. 3.7 Dependence of transverse vibration amplitude ArYms on reduced velocity Ur at (g ∗ , m ∗ , Re) = (2.0, 10, 200) for coupled-vortex regime

similar maximum ArYms as of the single vibrating square cylinder. In contrast to the single square cylinder, both the independent vibrating and the combined vibrating configurations do not exhibit the galloping phenomenon. In other words, the lowfrequency amplitude does not appear at higher reduced velocity, however, the amplitude increases to a large value and the synchronization exists between the vibration amplitude and the vortex shedding. Figure 3.8 illustrates the instantaneous vorticity contours for three representative reduced velocities Ur = 7, 17 and 35 at the gap ratio g ∗ = 2.0. As shown in Fig. 3.7, Ur = 7 corresponds to the maximum amplitude in the lock-in range and the transverse amplitude asymptotes nearly to a constant value for Ur > 7 due to the anti-phase response as shown by the vorticity contours. This observation has been discussed in detail in [144]. Therefore, Ur = 17 and 35 are selected according to the same value in g ∗ = 1.2 to compare with these two different ratios. The instantaneous gap ratio is between 1.2 and 2.0, the gap flow effect no longer exists when gap ratio is larger than g ∗ > 1.2. The vibrations of two cylinders show anti-phase behavior, while the two independent vortex streets are symmetric along the central-line. Although the overall flow dynamics at g ∗ = 2 has a similarity with the single square cylinder, the force coefficients are still higher than the single counterpart. In addition, as shown in Fig. 3.9, the drag trend does not exhibit a periodic pattern. The typical beating pattern is not observed (Ur = 7) at the synchronization region. There is one distinct dominant frequency in both force and vibration responses. The trajectory for each cylinder is an oblique elliptical configuration instead of the typical figure-eight shape for the single square cylinder and the two trajectories are symmetric. The relative frequency ratio of the transverse and streamwise response

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3 Proximity and Wake Interference

Fig. 3.8 Instantaneous vorticity contours of two independent vibrating side-by-side cylinders at (g ∗ , m ∗ , Re) = (2.0, 10, 200) for different reduced velocity a Ur = 7, b 17, c 35

is equal to f ∗ = f X / f Y = 1. In Figs. 3.10 and 3.11, the time traces of force and vibration amplitude of Ur = 17 and Ur = 35 are comparable with each other. There is just one dominant frequency in both force and vibration amplitudes. The trajectories show a somewhat slender airfoil-like shape instead of the oblique elliptical at the lock-in peak.

3.3 Results and Discussion

103

Fig. 3.9 2D results of two side-by-side system for upper cylinder (left) and lower cylinder (right) in independent vibrating configuration of g ∗ = 2.0 at (m ∗ , Re, Ur ) = (10, 200, 7): a, b force responses, c, d transverse amplitude, e trajectory of the system, and f power spectrum of C L and AY

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3 Proximity and Wake Interference

Fig. 3.10 2D results of two side-by-side system for upper cylinder (left) and lower cylinder (right) in independent vibrating configuration of g ∗ = 2.0 at (m ∗ , Re, Ur ) = (10, 200, 17): a, b force responses, c, d transverse amplitude, e trajectory of the system and f power spectrum of C L and AY

3.3 Results and Discussion

105

Fig. 3.11 2D results of two side-by-side system for upper cylinder (left) and lower cylinder (right) in independent vibrating configuration of g ∗ = 2.0 at (m ∗ , Re, Ur ) = (10, 200, 35): a, b force responses, c, d transverse amplitude, e trajectory of the system and f power spectrum of C L and AY

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3 Proximity and Wake Interference

3.3.2 Three-Dimensional Effects at Lock-in While the above simulations are two-dimensional at Re = 200, we next explore the possibility of the transition to three-dimensional flow at the lock-in condition. It is well known that the vortex-induced vibration can delay the three-dimensionality in a freely vibrating isolated cylinder. For the present side-by-side configurations, we wish to examine whether there exist three-dimensional patterns (e.g., streamwise ribs) and how they impact the coupled response dynamics. Three representative cases of three-dimensional simulations for each gap ratio (g ∗ = 1.2, 1.6 and 2.0) are conducted with the same physical conditions used in the two-dimensional simulations for the combined and the independent vibrating conditions. Owing to the large computational cost for 3D analysis, we have only considered the selected reduced velocities where the peak FIV amplitudes appear in the vibration response. For the reference purpose, we have also performed the simulations of stationary side-by-side configuration, as summarized in Appendix A. The three-dimensional simulation of two side-by-side cylinders in the independent vibrating condition at Ur = 7 is studied to examine the vortex-induced vibration where the frequency synchronization leads to a large vibration and a stronger threedimensional wake dynamics. This gap ratio with g ∗ = 1.2 falls under the transition region of wake flow behind the side-by-side system. Considering each independent cylinder vibration response of Δg ∗ that is found to be around +/− 0.4D about its own axis, the gap separation between the two cylinders is ranging from 0.4–2.0. Under such a range, the exhibited flow behavior shows an intermittent transition between that of gap flow regime and the coupled flow regime. Compared with the stationary counterpart in Fig. 3.12, the near-wake flow features show a stronger threedimensionality, with a significant amount of streamwise coherent vortex tubes connecting the spanwise vortices. The three-dimensional spanwise vorticity vanishes faster and the spanwise vortex rollers are somewhat twisted and irregular, while the streamwise vorticity becomes much stronger than the stationary condition. The wake region becomes longer and wider (width is about 7D instead of 5D in the stationary condition as shown in Appendix A). The three-dimensional simulations are conducted at Ur = 7 which is the peak location of the vortex synchronization region. Firstly, the beating pattern of force and amplitude responses is the most important feature of this regime. This beating pattern results in an envelope shape of the time evolution of force and vibration responses. The shape of the beating envelope is not uniform, and no clear detection of a second distinct frequency is found from this envelope. The beating response is much stronger and more obvious in the three-dimensional simulation, as shown in Fig. 3.13 than for the two-dimensional simulation in Fig. 3.4. In comparison with the two-dimensional simulation, the three-dimensional simulation shows a smoother temporal response, especially at the crest for each cycle. This is caused by the stronger three-dimensional flow when the lift force magnitude reaches the maximum value. The size of each envelope is slightly different from each other. The trajectory of the

3.3 Results and Discussion

107

Fig. 3.12 Vorticity contour for independent vibrating side-by-side configuration of g ∗ = 1.2 at (m ∗ , Re, Ur ) = (10, 200, 7): a 2D span-wise vorticity, b 3D span-wise vorticity, and c 3D streamwise vorticity

two cylinder system is within the triangular region but exhibits an irregular motion due to the beating phenomenon. The coupled vortex shedding property still remains the most important characteristic in the independent vibrating condition at g ∗ = 2.0. Similar to the stationary and the combined vibrating conditions, the vortex shedding is also synchronized to the anti-phase state. The paired anti-phase vortices from two cylinders persist for a long distance in the two-dimensional simulation in Fig. 3.14a. However, the threedimensional vorticity contours show the spanwise vortex rollers are coupled for a relatively short distance and the rollers are slightly twisted and developed to a complex three-dimensional flow features in Fig. 3.14b. The streamwise vortices from the two cylinders remain in their own street. There are small interferences between the two streets and the horizontal vortices rollers do not cross the centerline in Fig. 3.73b, which is similar to the two-dimensional simulations. According to Fig. 3.7, the VIV synchronization phenomenon occurs in the independent vibrating condition and the peak happens at Ur = 7 which is the same reduced velocity for g ∗ = 1.2, but higher than that of the single square cylinder (the peak happens at Ur = 7). It is different for the combined vibrating system whereas the VIV synchronization does not happen in the vibrating Regimes II and III [144]. The typical beating pattern phenomenon shows in both of the time history plots of force and amplitude responses. The beating envelope shape becomes smaller and smoother than the two-dimensional simulation. The size of each envelope is still not uniform, however, the frequency of the envelope is observed in both lift force and transverse amplitude. The dominating frequency of the lift coefficient is synchronized with the second frequency of the transverse amplitude in Fig. 3.15. Moreover, the dominating frequency of the transverse amplitude is synchronized with the second frequency of the lift coefficient. This is the reason why the beating envelope becomes more obvious than the envelope in g ∗ = 1.2, but it is not larger and more uniform than the single square cylinder. Also, the trajectories for both two cylinders still maintain the oblique elliptical shapes.

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Fig. 3.13 3D results of two side-by-side system for upper cylinder (left) and lower cylinder (right) in independent vibrating configuration of g ∗ = 1.2 at (m ∗ , Re, Ur ) = (10, 200, 7): a, b force responses, c, d transverse amplitude, e trajectory of the system and f power spectrum of C L and AY

3.3 Results and Discussion

109

Fig. 3.14 Vorticity contour for independent vibrating side-by-side configuration of g ∗ = 2.0 at (m ∗ , Re, Ur ) = (10, 200, 7). a 2D span-wise vorticity, b 3D span-wise vorticity, and c 3D streamwise vorticity

3.3.3 Interim Summary A summary diagram for all the three gap ratios of the two-dimensional simulations at Re = 200, m ∗ = 10 and zero damping ζ = 0 in the independent vibrating condition is shown in Fig. 3.16. The trend of the independent vibrating condition is similar to the behavior of a single square cylinder, in which the lock-in and kink happen at a specific reduced velocity accordingly. However, galloping does not occur up to Ur = 40 in this regime. Figure 3.17a,b show the mean drag coefficient and the root-mean-square lift coefficient for the single square cylinder and the upper cylinder of the two side-by-side square cylinder system for the different vibrating conditions. The behaviors of upper cylinder and lower cylinder are similar and symmetric. Therefore, the only upper cylinder is selected to compare with the single cylinder. The force responses of the single square cylinder are variant with the increasing value of reduced velocity, while the other two cases of two-cylinder system are relatively invariant as a function of the reduced velocity. In Fig. 3.17c, d, the streamwise and transverse amplitudes are compared. The vibrational amplitude of the single square cylinder in both directions is close to those for the two cylinder system in the VIV branch. However, for the higher reduced velocity, the two cylinder system for both the combined and independent vibrating conditions are more stable than the single square cylinder counterpart which remains in a low amplitude and is not affected much by the increased reduced velocity, especially for the independent vibrating condition. The vibration amplitude almost remains to a constant value of AY /D max = 0.15. The streamwise amplitude of combined vibrating condition is half that of the independent vibrating condition. In contrast, the transverse amplitude of the combined vibrating condition is doubled that of the independent vibrating condition. In Fig. 3.18, it can be noted that two-dimensional simulations over-predict the value of the force and the vibrational amplitudes. The differences in drag coefficient are within 3 and 10% in g ∗ = 1.2 and 2.0. In contrast, the differences in lift coefficient are within 8 and 4% in the transition and coupled vortex regimes, respectively. While

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3 Proximity and Wake Interference

Fig. 3.15 3D results of two side-by-side system for upper cylinder (left) and lower cylinder (right) in independent vibrating configuration of g ∗ = 2.0 at (m ∗ , Re, Ur ) = (10, 200, 7): a, b force responses, c, d transverse amplitude, e trajectory of the system and f power spectrum of C L and AY

3.3 Results and Discussion

111

Fig. 3.16 Transverse vibration amplitude ArYms as a function of reduced velocity for g ∗ = 1.2, 1.6 and 2.0 at (Re, m ∗ ) = (200, 10) for independent vibrating condition in 2D numerical simulation

the two-dimensional simulation can capture all the force vibration responses which the amplitude reduces by about 5%, the trajectories shape can still retain its shape. The relationship between the vibration amplitude versus gap ratio for the independent configuration coincides with the previously proposed parabolic equation in [144] for the combined vibrating configurations. In this present study, the lowest vibration amplitude is found between g ∗ = 1.2 and 2.0 in the couple-vortex regime. The present section examined the flow past two side-by-side square cylinders at a low Reynolds number of Re = 200 with two configurations namely combined vibrating and independent vibrating. The first part of this section compared the combined vibrating and the independent vibrating conditions via two-dimensional simulations at gap ratios g ∗ = 1.2, 1.6 and 2.0 in whereby the gap flow dominated and had strong effects on the fluid responses in these regimes. When two side-by-side cylinder vibrating systems changed from the combined vibrating condition to the independent vibrating condition, both two regimes were found to exhibit lock-in synchronization in the VIV range. While the VIV peak was located at Ur = 7 which shifted to a higher reduced velocity compared with the single square cylinder that occurred at Ur = 5 due to the interference between the two cylinders. The trajectories of two side-by-side cylinders were symmetric along the central line and the shape was an oblique ellipse in which the leading edge was upstream and the trailing edge was downstream. Regardless of the gap distance between two cylinders, the two trajectories could be viewed, upon combining them, as a typical twisted figure-eight shape. In comparison, the vibration amplitude of an individual cylinder for the independent vibrating condition was approximately half that of the combined vibrating

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3 Proximity and Wake Interference

Fig. 3.17 Comparison of a drag coefficient C D , b lift coefficient C L , c streamwise amplitude A X /D, and d transverse amplitude AY /D for single square cylinder, upper cylinder of the twocylinder system in combined vibrating condition and independent vibrating condition at synchronization peak, transition and galloping at (g ∗ , m ∗ , Re) = (1.6, 10, 200)

condition which could be considered as the reduction of the vibration amplitude of an individual cylinder in a side-by-side configuration. The investigation of the threedimensional effect of the two side-by-side cylinders for the two different regimes gap ratios in the stationary, the combined vibrating, and the independent vibrating condition was carried on at the same VIV parameters used for the two-dimensional simulations. Generally, the two-dimensional simulations over-predict the drag and lift force data, in which the overprediction of lift coefficient in Regime II was over 15%. In the independent vibrating condition, the overprediction of both force and vibration responses was reduced to less than 10%. The global time history profiles of force and vibrating responses in both two-dimensional and three-dimensional simulations follow a similar trend. However, the local characteristics at the crest and trough regions of each cycle in the two-dimensional simulation were not as smooth as observed in the three-dimensional simulation.

3.4 FSI of Side-by-Side Circular Cylinders

113

Fig. 3.18 Comparison of two-dimensional and three-dimensional simulations in the peak location of VIV branch for stationary, combined vibrating condition and independent vibrating condition a drag coefficient C D , b lift coefficient C L , c streamwise amplitude A X /D, and d transverse amplitude AY /D. While (−) denotes the equation proposed in [144], (·) represents the new proposed trend at (m ∗ , Re, Ur ) = (10, 200, 7) via 3D simulations

3.4 FSI of Side-by-Side Circular Cylinders The canonical side-by-side arrangements of circular cylinders are common and have a wide range of applications in various fields such as offshore, wind and aerospace engineering. In addition to their great practical relevance in engineering applications, a side-by-side system has a fundamental value due to the richness of nonlinear flow physics associated with the near-wake dynamics and the vortex-to-vortex interactions. There is a considerable difference between the flow dynamics of an isolated cylinder and the multiple-cylinder arrangements. Many comprehensive investigations, e.g., [251, 402–404, 482], were performed to understand and describe the mutual flow interference in the basic canonical multi-body systems, in which the importance of the wake and proximity interference was discussed as a function of the gap width between the cylinders and the Reynolds number. Among them, the flip-flopping of gap flow during the flow past two symmetrically arranged cylinders

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has received significant attention among researchers. Different from the other fundamental flow regimes in a two-dimensional laminar flow, the bi-stable character and chaos-like fluctuation of the flip-flopping pattern have intrigued the research community over the past few decades. In the event of a flip-flop, the gap flow fails to maintain its straight path and has an intrinsic tendency to oscillate intermittently between two asymmetric states. The phenomenon of flip-flopping flow was reported in several experimental works at relatively large Reynolds numbers (Re = 103 ∼ 105 ) [43, 177, 210, 451]. In particular, the flip flop was interpreted by [210] as a complex dynamical state with bi-stable stochastic characteristics and a biased deflected gap flow regime with asymmetric narrow and wide wakes with distinct predominant frequencies. The gap flow was found to switch intermittently its direction at a time scale that was a few orders of the magnitude greater than the frequency of primary vortex shedding. In addition to these experimental works, the flip flop was also observed within a narrow gap ratio range from 0.3 to 1.25 diameter in a two-dimensional laminar flow from various numerical investigations [1, 200]. In the deflected gap-flow regime, the narrow near-wake region involves an enhanced vortex-wake interaction, which results in a higher vortex shedding frequency and the mean drag force. On the other hand, a lower frequency is observed in the wide wake. The vortex shedding frequency of each cylinder dynamically changes with the gap-flow kinematics as time evolves. Although the origin of the flip-flopping was investigated by many researchers, there is no common consensus on a general understanding of the phenomenon. To begin, [8] reported that a perfect symmetric structure geometry was a critical condition which originated intermittent switching of the gap flow. However, the gap-flow flip-flop was also observed in the asymmetric VSBS arrangements from [254]. In one of the pioneering study, Ref. [177] considered the Coanda effect as the origin of the gap-flow flip-flop. Nonetheless, the flip-flop was found in the near-wake region behind a pair of side-by-side flat plates by [43, 451]. Peschard and Le Gal [340] modeled the dynamics of the deflected gap-flow regime through a system of two coupled Landau oscillators. The study illustrated that the stable deflected gap-flow regime and the flip-flop were formed by different mechanisms. Following the earlier studies, Ref. [74] reported that the flip-flop could be explained as a secondary instability through the coupling between a Hopf bifurcation (in-phase vortex synchronization) and the pitchfork bifurcation (deflected gap flow regime). This finding was subsequently supported by [254] in which the evolution of the flip-flop from the interaction of these two bifurcations was shown in a series of streamline plots as the Reynolds number increased. The flow characteristics were systematically investigated as a function of the gap distance between the two cylinders. The exact instants of the flip-flop and the instantaneous vortex shedding frequencies were visualized via the Hilbert-Huang Transform (HHT) technique of [161]. In relation to the vibrating side-by-side configuration, Ref. [254] also reported that the flip-flop was suppressed at the lock-in, in which the time-averaged streamwise velocity profile of the gap flow became asymmetric. On the other hand, the lock-in range with respect to the reduced velocity became relatively narrower, owing to the enhanced vortex-to-vortex interaction caused by the gap-flow proximity inter-

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ference. A topological description based on critical points has also shown that the in-determinant two-dimensional saddle-point regions intermittently appeared in the middle path of the gap flow. The appearance of such saddle points was found to be linked with the near-wake instability. These critical points, where the velocity is zero and the streamline slope is indeterminate, contribute to the shear stress and can provide some understanding of the three-dimensional flow structures behind bluff bodies [499]. While there exist several studies on the near-wake instability and three-dimensionality for the canonical case of a circular cylinder, such effects are not explored for stationary side-by-side cylinders. In particular, the mutual effects between the three-dimensionality and the gap-flow interference have not been examined in the past in the context of vibrating side-by-side cylinders. During the lockin/synchronization, the vibrating cylinder undergoes a complex interaction with the gap flow and the near-wake vortex system behind the two cylinders. The primary focus of the present study is to investigate the influence of 3D flow structures on the dynamics of immersed side-by-side configuration in a uniform flow. The complete three-dimensional flow field is important for interpreting the topology of flow patterns and the role of critical points in the instability process, however 3D information is generally difficult to extract from physical experiments. In a 3D flow behind an isolated circular cylinder (i.e., the large gap distance between cylinders), the formation of the ribs-like streamwise vortical structures connecting the spanwise Kármán vortices is one of the characteristic flow features of the organized motion. As reviewed in [450], there are two types of instabilities during the wake flow transition, namely mode-A and mode-B. The mode-A instability (180  Re  230) is associated with the waviness of the primary Kármán vortices induced by the elliptic instability, whereas the counter-rotating streamwise vortices are formed in the highstrain region between the main spanwise vortex rolls. The conversion of the spanwise vorticity from the Kármán vortex cores into the streamwise vortices is an outcome of the elliptic instability and forms the central element of the mode-A instability. The onset of mode-A instability manifests a hysteretic discontinuity of Strouhal number St and the Reynolds number Re relationship with a spanwise wavelength about 3 ∼ 4D. While the mode-A intrinsically triggers a vortex dislocation in the wake of a stationary isolated circular cylinder during the wake transition, the mode-B (Re  230) with a spanwise wavelength about one diameter exhibits a non-hysteretic transition. A relatively high shedding frequency occurred with more organized threedimensional state of the mode-B and vice versa. In the present study, we consider the near-wake instability and three-dimensionality for side-by-side cylinders for 100 ≤ Re ≤ 500. In spite of the above investigations, many aspects of the proximity interference and the wake interference from the gap flow remain largely unexplored. A detailed three-dimensional description of the flow dynamics and the wake-body interaction is particularly lacking in the literature. In the context of three-dimensionality associated with the elliptic instability for a single cylinder, the hyperbolic critical points were investigated by [204, 228, 286]. From the topological theory of separated flows, the two-dimensional streamline orbitals resemble hyperbolas around a hyperbolic critical point, where its central velocity magnitude is zero and all eigenvalues of

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velocity gradient have the nonzero real parts. The hyperbolic critical points in the fluid domain had been previously reported as an unstable factor by [229, 246], where the maximal perturbation growth was found precisely around these hyperbolic points near the vortex wake. Although the role of critical points has been limited to a single cylinder, they have not been examined in the context of side-by-side cylinder wakes. One of the contributions of the present study is to build some connections between the near-wake instability, the vortex wake interaction and the fluid momentum. By considering one of the cylinders as elastically-mounted in the VSBS arrangement, the coupling between the near-wake dynamics and VIV is examined in three dimensions. In the present work, we use well-resolved numerical simulations to elucidate some insights on the effects above at Reynolds numbers of 100 ≤ Re ≤ 500. Specifically, we explore the spanwise characteristics of the gap-flow and VIV kinematics at 3D flow through systematic numerical analysis. We employ the recently developed variational finite element solver for fluid-structure interaction [180, 181]. The fluid and structure equations are coupled in three dimensions by body-conforming treatment of the fluid-solid interface via arbitrary Lagrangian-Eulerian formulation. Of particular interest is to answer the following questions: How do the VIV kinematics, the gap flow instability and the hydrodynamic responses accommodate themselves in a 3D flow? How do the gap-flow kinematics influence the 3D flow features? How does the spanwise correlation respond to the cylinder’s kinematics and the gap flow instability? In most engineering applications, flexible multi-body structures subjected to proximity interference and resonant wake-body interaction are much more common. Such multi-body systems can exhibit complex spatial-temporal dynamics as functions of geometric variations and physical parameters. A fundamental understanding of such complex nonlinear coupling is essential for efficient engineering design and safer operations. The incorporation of VIV in the investigation is crucial to reflect the practical aspects of structural motion and the interference with the hydrodynamic forces.

3.4.1 Coupled Fluid-Structure System A Petrov-Galerkin finite element formulation is employed to investigate the fluidstructure interaction problem, whereby the body interface is tracked accurately by the arbitrary Lagrangian-Eulerian technique. The traction and the velocity continuity conditions are imposed on the body-conforming fluid-solid interface via the non-linear iterative force correction procedure [180, 181]. The coupling scheme relies on a dynamic interface force sequence parameter to stabilize the coupled fluidstructure dynamics with strong inertial effects of incompressible flow on immersed solid bodies. The temporal discretizations of both the fluid and structural equations are formulated in the variational generalized-α framework and the systems of linear equations are solved via the Generalized Minimal Residual (GMRES) solver. Further details can be found in [180, 185]. Detailed convergence investigations and the

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validation results of the two- and three-dimensional simulations are reported in [242, 254, 307], which support the suitability of fluid-structure solver to simulate the 3D gap flow and VIV interaction.

3.4.2 Problem Setup and Verification The basic apparatus comprises a flexibly mounted cylinder of diameter D, placed in a uniform free-stream flow stream U . For the side-by-side configuration, another cylinder with an equal diameter is placed at a gap distance g. The cylinders are placed in a three-dimensional hexahedron domain, where the flow is along the streamwise x-axis, while the axis of the cylinder is along the spanwise z-axis. The numerical setup for the 3D simulation is essentially a spanwise extension of the twodimensional setup presented in [254], where the upstream distance, the downstream distance and the overall height of the fluid domain are respectively 50D, 50D and 100D. A schematic diagram of the three-dimensional SBS arrangement is shown in Fig. 3.19a. The traction-free boundary conditions are respectively implemented along the domain boundaries Γt , Γb and Γo . The top cylinder with mass m, Cylinder1, is elastically-mounted on a linear spring (with natural frequency f n ) in the transverse direction for the VSBS arrangements. The blockage ratio is taken as 2 %. A representative (x, y)-plane sectional mesh configuration is exhibited in Fig. 3.19b. Based on the mesh convergence analysis in [254], the spatial discretization error is less than 1 % in the (x, y)-plane mesh. For the 3D flow at Re = 500, the x–y sectional mesh is further refined, particularly the mesh within the boundary layer and the

(a)

(b)

Fig. 3.19 Three-dimensional computational setup of SBS arrangement: a schematic diagram of the fluid domain and the boundary conditions; b representative unstructured mesh distribution in (x, y)-plane at g ∗ = 0.8. Here Cylinder 1 is free to vibrate in transverse direction and Cylinder 2 is stationary

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Table 3.1 Convergence of global flow quantities at different spanwise mesh resolutions for a stationary isolated circular cylinder at Re = 500 and l ∗ = 10 Spanwise resolution Cdmean Clr ms St Δz = 0.4 Δz = 0.15 Δz = 0.075

1.341 (12.2%) 1.196 (0.08%) 1.195

0.763 (118.6%) 0.357 (2.3%) 0.349

0.2197 (7.1%) 0.2051 (0.0%) 0.2051

Table 3.2 Convergence of global flow quantities at different x–y meshes for a stationary isolated circular cylinder at Re = 500, Δz = 0.15 and l ∗ = 10 Number of elements in Cdmean Clr ms St x–y plane 50 × 103 81 × 103 110 × 103

1.29 (7.9%) 1.196 (0.05%) 1.1954

0.637 (80%) 0.357 (0.8%) 0.354

0.2051 (0.0%) 0.2051 (0.0%) 0.2051

near-wake regions. Here ρ and μ denotes the fluid density and the dynamic viscosity, respectively. The dimensionless wall distance y + is kept less than one (within the viscous sublayer) for the first layer of the structural mesh around bluff bodies. The incremental ratio of element size from the boundary layer to the near-wake region and far field is less than 1.1 to reduce the effect of element skewness. Overall, there are approximately 80 × 103 elements and 120 × 103 elements on each x–y section for the isolated cylinder cases and SBS arrangement cases, respectively. The spanwise length is taken as l ∗ = 10, based on the aspect ratio analysis in the numerical simulations from [236] and the experiments from [407]. A periodic boundary condition is employed at the ends of the cylinder span to eliminate the endplate effect. The mesh convergence study along the z-axis is shown in Table 3.1. The spanwise resolution Δz = 0.15 is chosen such that the spanwise spatial discretization error is controlled within 2.5% while maintaining the computational efficiency of our parametric study. Furthermore, the x–y plane mesh convergence analysis in Table 3.2 shows that the spatial discretization error is within 1% at chosen x–y plane mesh resolution of 81 × 103 , which has 160 points along the cylinder surface. Since the gap flow instability is the key concern of the present investigation, the majority of the investigations are performed at two representative gap ratios g ∗ = 0.8 and g ∗ = 1.0. However, the investigations on the boundary circumstances, e.g., around g ∗ ≈ 0.3 and g ∗ ≈ 1.5 where the gap flow is significantly suppressed and weakened, are still incorporated to facilitate the generality of our analysis. A detailed temporal convergence study of the current numerical solver has been performed in [181] for two-dimensional cases. The L 2 norm error was reported at about 1% at a constant time step Δt = 0.05. Results for the temporal convergence of a 3D stationary isolated circular cylinder at the Reynolds number of 500 are summarized in Table 3.3. The maximum Courant number is about 3 for this stationary isolated cylinder case at Re = 500. Since the fully-implicit second-order variational formulation [181] based

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119

Table 3.3 Convergence of integrated force quantities at different time steps for a stationary isolated circular cylinder at Re = 500, Δz = 0.15 and l ∗ = 10 Time step Cdmean Clr ms Δt Δt Δt Δt

= 0.1 = 0.05 = 0.025 = 0.01

1.310 (11.6%) 1.196 (1.9%) 1.178 (0.43%) 1.173

0.634 (82.2%) 0.357 (2.6%) 0.352 (1.2%) 0.348

Table 3.4 Non-dimensionless parameters for isolated and side-by-side VIV Parameter Value Description l ∗ = L/D

5–10

g ∗ = g/D m ∗ = ρπ4m D2 L

0.3–3 10

ζ =

C 4π m f n Ur = fU nD Re = ρUμD

Dimensionless spanwise length Gap ratio Mass ratio

0.01

Damping ratio

0–10

Reduced velocity

100–500

Reynolds number

on the generalized-α time integration [187] is employed, the present fluid-structure solver is stable at relatively large Courant numbers while selecting the time-step size appropriately to resolve the spatial-temporal dynamics of the vortex shedding process. The important dimensionless simulation parameters and the post-processing quantities are listed in Tables 3.4 and 3.5, respectively. In Table 3.5, the standard quantities f vs , Aryms , φ A y , φCl , Fx , Fy and v ∗ = v/U are respectively the vortex-shedding frequency, the root-mean-squared transverse vibration amplitude, the phase angle of A y , the phase angle of Cl , the streamwise hydrodynamic force, the transverse hydrodynamic force and the dimensionless transverse velocity of the vibrating cylinder. Except stated otherwise, all positions and length scales are normalized by the cylinder diameter D, velocities with the free stream velocity U , and frequencies with U/D. To validate the numerical formulation in a three-dimensional flow, a comparison of a stationary isolated circular cylinder at Re = 300 are presented in Table 3.6. The comparison of the overall VIV response with the previous results of [54] is shown in Table 3.7. The results are in close agreement with the previous studies, and thus the computational setup is adequate for our present investigation. A total of eighty-five simulations is performed in the present investigation, comprising seven simulations for the validation of the three-dimensional FSI solvers, fifty-two cases for the investigation of the isolated, SSBS and VSBS arrangements; and twenty-six two-dimensional cases to investigate the relationship between the near-wake instability, the fluid shearing ratio and the fluid momentum. By taking into consideration of a large number of three-dimensional simulations and the involved computational resources, the selected time window is constrained at tU/D ∈ [250,

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Table 3.5 Derived dimensionless quantities for detailed analysis Parameter Description St = fvsU D √ Amax = 2 Aryms y Δφ = φ A y − φCl Cl = 1 Fx2

Strouhal number Dimensionless transverse displacement Phase angle difference Lift coefficient

Cd =

Drag coefficient

2 ρU

Ce =

DL Fy 1 ρU 2 DL 2 ∗ T Cl v dτ

Energy transfer coefficient

Table 3.6 Comparison of numerical and experimental results for a stationary isolated circular cylinder at Re = 300, where Cdmean is the mean drag coefficient, Clr ms is the root-mean-squared of lift coefficient fluctuation and St is the Strouhal number Cdmean Clr ms St Simulation

Experiment

[485] [339] Present Wieselsberger (1921) [453]

1.44 1.366 1.26 1.208

0.68 0.477 0.5 –

0.216 0.206 0.205 –

–

–

0.203

Table 3.7 Validation of transverse amplitude Amax for a freely vibrating cylinder in threey dimensional flow at m ∗ = 5.08 and ζ = 0.024 Re Ur Simulation [54] Experiment [54] Present 606.1 713.9 848.1

5.51 6.49 7.71

0.460 0.433 0.420

0.550 0.485 0.430

0.525 0.462 0.424

350], in which the fluid flow is already fully-developed for the extraction of flow statistics. In the selected time window, all fluid features such as the hydrodynamic responses and the vibration amplitude, undergo at least twenty cycles. In particular, we are interested in the behavior of the flip-flop subjected to the influence of VIV and the three-dimensionality within a short time window.

3.5 Gap Flow Interference in Three-Dimensional Flow Before proceeding to further investigation on the complex coupling between the 3D flow, the VIV and the gap-flow kinematics in the SBS arrangements, the interference of the VIV on the 3D flow dynamics is systematically examined at first. Similar

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121

to the work of [331], three-dimensional effects in the near-wake are examined via dimensionless enstrophy, as defined below: D2 ξi (t) = 2 U V

 ωi2 (x, t)dV

(3.8)

Ω

where ωi and V denote the i-component of vorticity vector and the integration volume, respectively. The enstrophy is decomposed into primary ξz and secondary components ξx y = ξx + ξ y , where ξi is the time-averaged value of ξi (t) over a time interval. The enstrophy is directly associated with the dissipative effects of the fluid kinetic energy, and the generation and breakdown of coherent flow structures. In particular, the ratio of secondary enstrophy ξx y over the total enstrophy ξt = ξx + ξ y + ξz is investigated to quantify the three-dimensionality in the near-wake region. The secondary enstrophy should become negligible for two-dimensional flow. The enstrophy values are computed within a representative control volume (V ) in the near-wake region. It incorporates the majority of the near-wake region where coherent structures are concentrated, e.g., x/D ∈ [0.6, 1.6], y/D ∈ [−4, 4] and z/D ∈ [0, 10]. The number of data-sampling probes is close to the total number of grid points in the chosen control volume. The sampling frequency is 10 times higher than the primary vortex shedding frequency. The sampling period is approximately 10 primary vortex shedding cycle starting from the instance of a peak Cl value. The enstrophy analysis for an isolated cylinder at Re = 500 is shown in Fig. 3.20a. Overall, the total enstrophy ξt values are nearly constant at the off lock-in condition. While the values of ξz slightly decrease at the post-lock-in, ξx y shows an obvious increase at the post-lock-in. Notably, a significant suppression of ξx y , ξx y /ξt ≈ 0.0, is clearly shown at the peak lock-in at Ur ∈ [4, 5]. Sudden variations of ξx y and ξz are observed at the transition from the earlier to the peak lock-in. In a nutshell, we can deduce that (i) the VIV kinematics possesses a regulation effect, whereby the forces are strongly correlated in the spanwise direction at the peak lock-in, and (ii) the streamwise vorticity is significantly suppressed at the peak lock-in. Next, we proceed with the spanwise variation of flow dynamics along the cylinder. To measure the waviness of flow properties in the spanwise z-direction, we quantify the correlation length using the spatial-temporal variations of fluid forces, which are directly dependent on the vorticity distributions in the near-wake region. A correlation length provides a statistical description to identify a representative length scale for the spanwise fluctuations associated with three-dimensional effects. To quantify the degree of spanwise correlation, we define the cross-correlation along the cylinder as follows κ(li∗ , l ∗j ) =

i  j − ¯i ¯ j σi σ j

(3.9)

where i and j indicate the spanwise locations. i , ¯i and σi refer to the scalar quantity at spanwise location li∗ , its time-averaged value and standard deviation,

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Fig. 3.20 Quantification of three dimensionality at Re = 500, m ∗ = 10, ζ = 0.01 and Ur ∈ [0, 9]: a variation of enstrophy ξ with respect to Ur value, b dependence of cross-correlation of hydrodynamic responses at Ur = 5.0

Fig. 3.21 Instantaneous vortical structures using the Q-criterion for an isolated cylinder at Re = 500, Q = 0.2, ω y = ±1 (contours) and tU/D = 300: a stationary Ur =; b 3; c 5; and d 7 at m ∗ = 10 and ζ = 0.01 for a freely transverse vibrating cylinder. Streamwise vorticity clusters vanish at the peak lock-in Ur = 5

respectively. Using Eq. (3.9), we estimate the cross-correlation κ of the spanwise Cl and Cd to extract the spanwise fluctuations. While a shorter correlation length implies dominance of spanwise fluctuations (i.e., three-dimensionality), a uniform 2D flow exhibits a longer correlation length. A higher value of κ at the peak lock-in in Fig. 3.20b illustrates the suppression of streamwise vorticity in the near-wake region, as observed in Fig. 3.20a. Such weakening effect of the spanwise force suggests that there exists a particular regulation mechanism which causes recovery of 2D hydrodynamic behavior along the cylinder. This regulation (stabilization) effect is further visualized by the iso-surfaces of the vortical structures using a vortex-identification based on Q-criterion [172] in Fig. 3.21. Consistent with these observations, the streamwise vorticity clusters at the lock-in are completely invisible in Fig. 3.21c. An additional 2D simulation at the iden-

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123

tical problem setup is also performed and the resultant hydrodynamic responses from the 2D configuration are identical with its 3D counterpart. Since there is no external energy source to perturb the flow field, we can deduce that the aforementioned recovery of two-dimensionality at the peak lock-in is an intrinsic characteristic of the fluid response, instead of an artificially restricted fluid behaviour. Huang et al. [164] recently reported that a planar shear flow could enhance the three-dimensionality in the wake behind a circular cylinder.1 In the present investigation, we further analyze the relationship between the nearwake instability and the resultant imbalanced vortex-to-vortex interaction from the planar shear flow. The fluid shearing is known to be critical to the Kelvin-Helmholtz instability. Since significant shear stresses are observed on the interface of the imbalanced counter-signed vorticity clusters, the interaction between different vorticity clusters is believed to be crucial to the near-wake instability. The critical points, e.g., streamline saddle point, can be observed in both 2D and 3D flows. To simplify our discussion, the fluid stability around a saddle point region for the 2D laminar flow is included as a supportive example. While Re  48, the symmetric counter-signed circulations are interacting and no instability is observed behind a stationary isolated circular cylinder. It is when the perturbation approaches to the brink of a critical value, the perturbation becomes non-negligible and induces the need for the extra dimension to quantify itself e.g., the introduction of a new dimension via a Hopf bifurcation and the flow transition from the laminar flow to the turbulent flow. These two factors associated with the near-wake instability facilitate the understanding of the proximity interference induced from the gap flow behaviour in a three-dimensional flow in Sect. 3.5.1. A turbulent wake flow at the off lock-in results into a smaller Cdmean value than its laminar flow counterpart in Fig. 3.22, where Cdmean is over-predicted by a relatively large two-dimensional vortex wakes at the lock-in. Overall, the response of Cl shows an increment in the transverse fluctuating lift force and an earlier onset of the VIV lock-in. As reported by [254] for the similar problem setups with two-dimensional laminar flow, the earlier onset of VIV than its corresponding case without proximity interference is attributed to the enhanced vortex interaction which leads to a higher vortex shedding frequency. At the peak lock-in, both Cd and Cl at Re = 500 show respectively about 6 and 22% amplification compared to their laminar counterparts from [254]. In contrast to the streamwise effect, these results indicate that the VIV regulation effect has a profound influence on the transverse response. A similar phenomenon was observed by [492], in which the spanwise correlations were discussed at the VIV lock-in and uniformity of Cl was observed along the span of a vibrating cylinder. Here, the primary focus is to understand the complex near-wake flow physics in the SBS arrangements.

1

Here a planar shear flow refers to an inflow with a constant velocity gradient along the y-axis.

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Fig. 3.22 Hydrodynamic forces as a function of the reduced velocity for a transversely vibrating isolated cylinder at m ∗ = 10 and ζ = 0.01: a the mean drag coefficient Cdmean with respect to the reduced velocity Ur ; b the r.m.s. lift coefficient Clr ms with respect to the reduced velocity Ur

3.5.1 Three-Dimensional Gap-Flow Interference In this section, the relationship between the gap-flow-induced proximity interference and the near-wake instability is discussed for the cylinders with SBS arrangements. An incorporation of the interference from the VIV kinematics is important to analyze the practical applications and operations in side-by-side systems. So far the three-dimensional numerical investigation of the gap flow instability in the SBS arrangements is rarely documented. Hence, the present investigation on the VSBS arrangements in a 3D flow is deemed as another step further to understand the gap flow and the VIV kinematics. From a systematic analysis viewpoint, it is desirable to first focus merely on the interaction between the gap-flow kinematics and the 3D flow by eliminating the motion of the structure. The flip-flopping pattern is frequently described as an intermittent deflection of the gap flow. As reported by [254], the switch-over of Cdmean from each cylinder in the SSBS arrangement could indicate the direction of gap-flow deflection. Since the vortex-to-vortex interaction is enhanced in the narrow near-wake region, the corresponding frequency f vs value is higher than its wide near-wake-region counterpart. To investigate the characteristics related to the gap flow features without and with the presence of 3D effects, Fig. 3.23 is plotted. In the two-dimensional laminar flow, Fig. 3.23a, c, f vs of the cylinder with the narrow near-wake region is observed possessing a larger value. However, this tendency is not confirmed in its three-dimensional counterpart, as shown in Fig. 3.23b, d. The figures show that there is no significant difference among the mean vortex-shedding frequency of two cylinders for the 3D flow, although the gap flow deflects. In addition, the flip flop

3.5 Gap Flow Interference in Three-Dimensional Flow

125

Fig. 3.23 Time traces of hydrodynamic forces for SSBS arrangement at g ∗ = 0.8: a, c Re = 100, the flip-flopping are marked at tU/D = 270, 300 and 330; b, d Re = 500

is not observed in the selected time window tU/D ∈ [250, 350] in Fig. 3.23d, since f f li p is remarkably low for the 3D flow. In Table 3.8, a comparison of secondary enstrophy is shown between a representative SSBS arrangement in the deflected gap flow regime and a stationary isolated cylinder. Overall, the mean concentration of secondary enstrophy ξx y /ξt in the nearwake region for the side-by-side arrangement is relatively smaller than the isolated counterpart. While ξx y /ξt is distinctively small in the wide near-wake region, ξx y /ξt in the narrow near-wake region is higher than the secondary enstrophy concentration for the isolated cylinder. Consequently, the three-dimensional structure is prevailing in the narrow near-wake region where the gap-flow proximity interference is significant, as visualized in Fig. 3.24.

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Table 3.8 Comparison of ξx y /ξt between isolated and side-by-side arrangement at Re = 500, g ∗ = 0.8 and tU/D ∈ [300, 350] Arrangement ξx y /ξt Mean (%) Narrow (%) Wide (%) Isolated Side-by-side

10.26 8.86

– 11.54

– 3.8

Fig. 3.24 Instantaneous vortical structures using the Q-criterion at Re = 500, tU/D = 300, Q = 0.2 and ω y = ±1 (contours): a stationary cylinder; b SSBS arrangement at g ∗ = 0.8. The streamwise vorticity concentration is higher in the narrow near wake region

A further investigation of the velocity profile in Fig. 3.25 exhibits that the threedimensional gap flow at a higher Reynolds number possesses a much larger fluid shearing than its two-dimensional counterparts. Therefore, the fluid in the narrow near-wake is more prone to be unstable, because of the high-velocity gradients associated with the gap flow. Apart from the critical factors identified in Sect. 3.5, we also notice another unstable factor in the narrow near-wake region, large adjoining interfaces of ωz clusters, as shown in Fig. 3.26. Shear stresses with a remarkable strength are present along these adjoining interfaces, which may result in significant streamwise vorticity concentration formed in the narrow near-wake region, as shown in Fig. 3.27. In the near-wake region, the locations with the intensified streamwise vorticity clusters follow closely along these interfaces, which further confirms the observation about the SSP in Sect. 3.5. The SSP lies right at the point with significant streamwise vorticity concentration along the interface of ωz clusters. Figure 3.28a shows distinctively higher and lower Cdmean values for the cylinders with the narrow and wide near-wake regions, respectively. Furthermore, the algebraic sum of Cdmean shows a base-bleeding type effect, as reported for the SSBS arrangements in [43]. Hence the overall response of Cd is diminished. However, this base-bleeding effect is weakened as the value of g ∗ increases beyond the deflected gap-flow regime. To analyze the transverse response, Clr ms is adopted to characterize the fluctuating extent of Cl as a function of the gap ratio g ∗ in Fig. 3.28b. The quantity Clr ms represents the fluctuation intensity (absolute value) of transverse force and is

3.5 Gap Flow Interference in Three-Dimensional Flow

127

Fig. 3.25 Horizontal velocity profiles of gap flow for SSBS arrangements at Re = 100 and Re = 500. Velocity profiles are extracted at the gap flow location (0.6D, 0.7D to −0.7D) where (0D, 0D) is the center between the cylinders. The time-averaging is performed from tU/D ∈ [250, 350]

measured from the Clmean value between a time interval when the gap flow stably deflects to one particular side of the SSBS arrangements. It should be noted that the in-phase and anti-phase of Cl from both cylinders have to be taken into account when computing the resultant transverse force fluctuation for SBS arrangements. Similar to Fig. 3.28a, a drastic transverse fluctuation of the lift appears along the cylinder with the narrow near-wake region. The overall transverse fluctuation of Cl is calculated as a sum of Cl from each cylinder, as shown in Fig. 3.28b. A force modulation is clearly shown at the deflected gap flow regime g ∗ ∈ [0.8, 1.5], where the overall transverse fluctuation of Cl is excited by a factor of 2.4. Since the gap flow is significantly suppressed at g ∗  0.5, the overall fluctuation of Cl is much more benign. While Clr ms along individual cylinder is drastically amplified beyond g ∗  1.5, the overall value of the entire structure system is diminished and canceled out instead, due to the dominant anti-phase vortex shedding regime at these gap ratios. The above observations show that the gap-flow instability is critical to the global stability of the SBS systems in engineering operations with a relatively small gap ratio, where stronger force modulation is observed. It is observed that the 3D flow not only modulates the hydrodynamic forces, but also the flip flop frequency f f li p . Generally, f f li p appears to be lower in 3D higher Re flow, as compared to its 2D laminar flow counterpart. Liu and Jaiman [254] visualized the flip-flopping instant as a zero phase angle difference between Cl in the SSBS arrangements. Different from a two-dimensional laminar flow, the existence of the streamwise vorticity clusters in the formation region varies f vs values along the cylinder span and results in a repetitive temporal modulation of f vs . To completely flip over the gap-flow direction, at least a few cycles of the in-phase vortex shedding are required. Due to the interference from the gap flow, the modulation of f vs on each cylinder is chaotic and intermittent and f f li p is significantly influenced by the three-dimensionality.

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Fig. 3.26 Spanwise vorticity ωz contours in SSBS arrangement at Re = 500, g ∗ = 0.8, l ∗ = 5 and ωz = ±1.0 (contours): a tU/D = 303; b tU/D = 313. Large interfaces of different vorticity concentrations are observed in the narrow near-wake regions

3.5.2 Coupling of VIV and 3D Gap-Flow Kinematics In this section, the VIV kinematics and the gap flow instability are coupled in the VSBS arrangements, where Cylinder1 is elastically-mounted in the transverse direction. To begin, the flip-flopping frequency f f li p is analyzed for the VSBS arrangements. In Fig. 3.29, f f li p is not observed at the off-lock-in conditions for the selected time window, which is similar to the SSBS arrangement. However, f f li p in the VSBS arrangements is remarkably increased during the onset and the end of the VIV lockin. In the present 3D work, f f li p is found to be VIV-dependent. Some phenomena reported by [254] for the VSBS arrangements are also observed for the 3D configurations. For instance, a quasi-stable deflected gap flow regime occurs at the peak lock-in, where the gap flow permanently deflects toward the locked-in vibrating cylinder. Liu and Jaiman [254] observed an early onset of VIV in the side-by-side arrangements for 2D laminar flow. The enhanced vortex interaction results in a higher frequency of primary vortex shedding and an early match with the natural frequency f n . In the present investigation, early onset of VIV is also observed in Fig. 3.30 for the 3D side-by-side arrangement. This observation is further supported by the analysis of the vortex-shedding frequency, the phase angle and energy transfer in Fig. 3.33, which are discussed in the subsequent paragraphs. As discussed in the previous sections, the near-wake instability is subjected to influences from the VIV kinematics and the gap-flow-induced proximity interference. When both VIV and proximity interference are present in a three-dimensional flow, the analysis of three-dimensionality becomes subtle and multifaceted. To quantify the three-dimensional effect, the concentration of ξx y is summarized in Table 3.9 for a representative VSBS arrangement at different reduced velocity values. It can be seen the recovery of two-dimensional response is still observable at the peak lock-in. A distinct difference of ξx y concentration in the narrow and wide nearwake regions are found at the stationary and peak lock-in. The concentration of

3.5 Gap Flow Interference in Three-Dimensional Flow

129

Fig. 3.27 Instantaneous contours of streamwise vorticity ωx and spanwise vorticity ωz of cylinders in SSBS arrangement at (x, y)-plane at Re = 500, g ∗ = 1.0, tU/D = 300, ωx = ±1.0 (contours), ωz = ±1.0 (solid-dash lines) in a, c and sectional streamlines in b, d: a l ∗ = 4; b l ∗ = 8

Fig. 3.28 Fluid forces as a function of gap ratio for stationary side-by-side arrangement: a mean drag coefficient, b r.m.s. lift coefficient. Here the subscripts t, n and w denote respectively the total, the narrow and the wide near-wake regions

ξx y is further visualized by the sectional contour plots in Fig. 3.31 on the (x, y)plane and Fig. 3.32 on the (y, z)-plane. The concentration of ξx y in the near-wake behind the locked-in cylinder is much higher than its isolated counterpart at the same Ur value, owing to the gap-flow proximity interference. Different from the SSBS arrangements, ξx y concentration in the narrow near-wake region behind a vibrating cylinder is relatively smaller than the wide one behind the stationary cylinder, This phenomenon is attributed to the regulation effect of VIV kinematics. The above observation implies both VIV and gap-flow proximity interference influence the nearwake instability in the VSBS arrangements. The VIV regulation effect is generally dominant at the peak lock-in.

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Fig. 3.29 Time traces of drag force in the VSBS arrangements where Cylinder1 vibrates in the cross-flow direction at Re = 500, g ∗ = 0.8, m ∗ = 10 and ζ = 0.01: Ur = a 3; b 4; c 6 and d 8

Fig. 3.30 Time-averaged maximum transverse vibration amplitude as a function of reduced velocity for the VSBS arrangements at Re = 100 and 500, m ∗ = 10 and ζ = 0.01

3.5 Gap Flow Interference in Three-Dimensional Flow

131

Table 3.9 Comparison of ξx y /ξt for VSBS arrangement at Re = 500, g ∗ = 0.8, Ur ∈ [0.0, 6.0] and tU/D ∈ [300, 350] Ur ξx y /ξt Mean (%) Narrow (%) Wide (%) 0.0 3.5 4.0 (lock-in) 5.0 6.0

8.86 19.96 5.67 18.66 23.73

11.54 19.65 0.76 16.76 21.75

3.8 20.40 12.72 21.80 25.93

To further investigate the characteristics of the VIV lock-in and the gap-flow proximity interference, the VSBS arrangements at two typical gap ratios g ∗ = 0.8 and 1.0 in the deflected-gap flow regime are considered. Consistent with Fig. 3.30, the frequency ratio plot in Fig. 3.33a confirms the early onset of the VIV lock-in. While the onsets of the VIV lock-in at various g ∗ values are different, the ends of their VIV lock-in approximately occur at an identical Ur value for the VSBS arrangements. The phase angle difference Δφ is plotted in Fig. 3.33b, whereas Δφ is computed as Δφ = φ A y − φCl using the HHT technique to study the energy transfer between the fluid flow and the vibrating cylinder. Δφ is a time-averaged value. For the VSBS configuration, the phase difference is computed within a time interval when the gap flow stably deflects toward the vibrating cylinder. For an isolated vibrating cylinder in 2D flow, Δφ keeps increasing, as the reduced velocity increases. In addition, a sharp jump is observed for the isolated vibrating cylinder at about Ur ≈ [7, 8]. A similar discontinuity of the phase angle difference was also reported by [237] at the peak lock-in, which is correlated to the VIV kinematics and the vortex wakes. When Ur exceeds 8.0, Δφ becomes completely anti-phase. A similar profile of Δφ is observed for the isolated vibrating cylinder with 3D flow, as depicted by the blue curve in Fig. 3.33b. The difference is the onset of VIV lock-in, which occurs at a relatively smaller Ur value. For the VSBS arrangements, when the gap ratio is reduced, the proximity interference becomes greater. As a consequence, the values of Δφ are stabilized at approximately 140◦ for the vibrating cylinder in the VSBS arrangement at the lock in Ur ∈ [5,8], as shown in Fig. 3.33b. In addition to the phase difference Δφ, the energy transfer coefficient Ce is also a useful quantity. It indicates the instantaneous energy transfer between the fluid flow and the vibrating structure. As defined in Table 3.5, the energy transfer coefficient Ce is expressed by the transverse velocity v for the dimensionless time scale τ = tU/D in the primary vortex shedding cycle T . While the magnitude of Ce quantifies the work done by the fluid forces, its sign indicates the direction of energy transfer. The trend of Ce computed within one cycle of the primary vortex shedding is shown in Fig. 3.33c. While the value of Ce reaches a maximum at the peak lock-in for all cases, it attains its maximum at smaller Ur for the higher Reynolds number. Specifically, the peak value of Ce becomes greater, as the gap ratio decreases in

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3 Proximity and Wake Interference

Fig. 3.31 Instantaneous contours of streamwise vorticity ωx and spanwise vorticity ωz at (x, y)plane for Ur = 4.0 (peak lock-in) at (Re, g ∗ , m ∗ , ζ ) = (500, 0.8, 10, 0.01): a l ∗ = 4; b l ∗ = 8. In a ωx = ±1.0 and ωz = ±1.0 are shown by colour contours and solid-dash, respectively

Fig. 3.32 Instantaneous contours of ωx in (y, z)-plane for the cylinders in VSBS arrangement at Re = 500, g ∗ = 0.8, m ∗ = 10.0, ζ = 0.01, Ur = 4 (the peak lock-in sage), ωx = ±1.0 (contours) and tU/D = 350: a x ∗ = 1.25; b x ∗ = 1.5

3.5 Gap Flow Interference in Three-Dimensional Flow

133

Fig. 3.33 Variations of frequency, phase relation and energy transfer as a function of reduced velocity at Re ∈ [100, 500], m ∗ = 10 and ζ = 0.01: a frequency ratio f / f n , b phase angle Δφ between A y and Cl , and c averaged energy transfer for one primary vortex shedding cycle

the VSBS arrangements, e.g., the blue curve at Ur = 4.0 in Fig. 3.33c. In contrast, the curve of Ce becomes closer to the isolated vibrating cylinder as the gap ratio increases. The amount of energy transferred over the off-lock-in Ur range is trivial compared to their lock-in counterparts. These trends of Ce imply that the amount of energy transferred is much large for the configurations of higher Reynolds number and the small gap ratio.

3.5.3 Interim Summary The dynamics of three-dimensional gap flow and VIV interaction is numerically investigated in the side-by-side circular cylinder arrangements at Reynolds numbers ranging 100 ≤ Re ≤ 500. A body-conforming Eulerian-Lagrangian technique based on the variational finite-element formulation has been applied for the fluid-structure interaction. We found that the VIV kinematics regulate the streamwise vorticity concentration in the near wake, which results into a significant recovery of twodimensional responses at the peak lock-in. The in-determinant streamline saddle point was observed to form along the interface between the imbalanced vorticity clusters in the near wake. The saddle-point regions are intrinsically associated with high local strain rates and contribute to the formation of three-dimensional vortical structures. The gap flow momentum and the interaction between the imbalanced vorticity clusters are found to be critical for the near-wake instability. In both SSBS and VSBS arrangements, the concentration of ξx y was observed to exhibit a strong dependency on the gap-flow proximity interference. We observed a distinctive concentration of ξx y in the narrow and wide near-wake regions for both SSBS and VSBS arrangements. While the narrow near-wake region has a higher ξx y concentration than the wide one for the SSBS arrangements, the narrow near-wake behind the locked-in vibrating cylinder exhibits a lower ξx y concentration for the

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3 Proximity and Wake Interference

VSBS configuration. A force modulation was observed by the deflected gap flow in the SSBS arrangements, which caused the amplification of Clr ms . While an early onset of VIV lock-in was observed for the VSBS arrangements, the flip-flopping was remarkably suppressed for both SSBS and VSBS arrangements at off lock-in. A quasi-stable deflected gap flow regime and a recovery of two-dimensionality were reported for the VSBS arrangements at the peak lock-in. By promoting the energy transfer and regulating the VIV lock-in, the gap flow was found to exert a strong proximity interference. Through the modal analysis, the third-order harmonic ωz vorticity modes were dynamically decomposed for the isolated cylinder and the SSBS arrangements. Owing to their odd-order characteristics, the third vortex modes are crucial to the stability of the dynamic fluid-structure system and represents an unstable factor. A vortex discontinuity originated from the gap-flow kinematics was observed using the DMD technique in the wide near-wake region. An additional influential ωz vorticity mode along the middle path of the gap flow in the VSBS arrangement was observed, which was related to the periodic undulation of spanwise force fluctuation along the cylinder and represented the promoted gap-flow instability. Overall, it was found that the vortex-to-vortex interaction between the imbalanced counter-signed vorticity clusters plays an important role in the near-wake stability, because of the significant fluid shearing along the vortical interfaces. In general, the intensive fluid shearings along the vortical interfaces are associated with the indeterminant streamline saddle-point regions. The saddle-point region is found in all range of Reynolds number and is interlinked with various flow dynamic events, e.g. the vortex shedding, the flip flop, and the streamwise vorticity clusters. Furthermore, the near-wake instability is found to be closely interlinked with the gap flow and the VIV kinematics. In particular, as the VIV kinematics increases and stretches the vortices, the vorticity clusters are more separated to weaken the vortex-to-vortex interaction in the near-wake region. As a result, the two-dimensional hydrodynamic responses are significantly restored along the cylinder. On the contrary, the interaction dynamics between the gap-flow proximity interference and the gap-flow instability enhances the vortex-to-vortex interaction. These observations and findings are important in multi-body systems, from both operations and design viewpoints, found in offshore and aeronautical engineering.

3.6 Freely Vibrating Tandem Square Prism Square-shaped columns are widely used in semi-submersibles and offshore floating structures. Predicting flow-induced vibrations in multi-column floaters is a challenging task due to complex wake interference and galloping, a self-excited non-linear instabilities associated with large amplitude oscillations and is undesirable to avoid structural failure. This section reports a set of numerical experiments to understand flow-induced vibrations of the square columns kept in a tandem arrangement. Results on the coupled force and response dynamics are presented for an isolated column and

3.6 Freely Vibrating Tandem Square Prism

135

Fig. 3.34 Schematic diagram of the geometric arrangement and boundary conditions for tandem square column cases

for a pair of square columns in the tandem configuration where the downstream column is elastically mounted and free to oscillate in in-line and transverse directions. We assess the combined wake-induced and sharp-corner based galloping effects on the downstream column by comparing with the isolated square column counterpart. It is known that the circular cylinders undergo vortex-induced motion alone whereas the motion of a square column is vortex-induced at low Re and galloping at high Re [28, 58]. The effects of reduced velocity on the fluid forces, amplitudes, wake contours, and phase angles will be analyzed. The two-dimensional simulation domain illustrated in Fig. 3.34 describes a setup with two square columns kept in a tandem arrangement. All the dimensions are normalized with the side length of the column, denoted as D. The columns are separated by a center-to-center distance of 4D. The downstream column has two degrees-of-freedom. The inlet boundary is placed at a distance of 10D from the upstream column and the outlet is placed at a distance of 25D from the downstream column. The top and bottom boundaries are defined as slip walls. Discretization is carried on the basis of the outcome from grid independence study for the single square cylinder. A triangular finite element mesh was generated comprising a total of 104650 elements as shown in Fig. 3.35. Reasonable resolutions of the boundary layers and the wakes of upstream and downstream columns. Region near to the structural members is discretized with approximately 20 nodes in the normal direction to the wall and the nodes are placed as close as 0.02D in the normal direction from wall. Simulations are presented for the range 2 ≤ Ur ≤ 40 with an objective to examine the response of downstream column in the initial, lock-in as well as galloping branches. The mass ratio and Reynolds number were kept constant at m ∗ = 5 and Re = 200, respectively. The damping ratio was kept zero in order to realize maximum amplitudes during oscillations.

136

3 Proximity and Wake Interference

Fig. 3.35 Unstructured finite element mesh for tandem square cylinder configuration

3.6.1 Response Characteristics From the displacement trajectories shown in Fig. 3.37, for the downstream column, we notice that the amplitudes of transverse oscillation are substantially higher than the in-line oscillations. The column traces a characteristic “8” shape in the initial branch and skewed elliptical trajectory at the higher Ur values. With the onset of galloping the trajectory develops into an open loop. This particular sequence of changes in the column trajectory is unique when compared with trajectories of an isolated square column reported by [495]. The frequency of oscillation of the downstream column is plotted against Ur in Fig. 3.36. The results demonstrate a well defined lock-in region for 6 ≤ Ur ≤ 8 where the oscillation frequency closely matches the natural frequency of the structure. However, the onset of lock-in window has shifted towards higher reduced velocities when compared to the response of isolated square column in a uniform flow [495]. Figure 3.38 shows the time histories of lift coefficient and transverse displacement of the downstream column at representative Ur values. Except in the lock-in branch, both signals exhibit a sinusoidal behavior everywhere. In the initial regime, lift force oscillations are in-phase with displacement oscillations, whereas in the galloping (quasi-periodic) regime the signals are out of phase, which is illustrated in Fig. 3.39. In the lock-in regime, the lift coefficient signal leads the displacement signal by a small angle. The force signal also possesses components at frequencies other than that of displacement signal.

3.6 Freely Vibrating Tandem Square Prism

137

10

Lift coe fficient Strouhal frequency

9 8 7

f/fN

6 5 4 3 2 1 0 0

10

20

30

40

50

Ur

Fig. 3.36 Frequency of the lift forces acting on the tandem downstream square cylinder

Figure 3.40 illustrates the variation of drag forces on the downstream column along with the corresponding in-line motions. The column experiences negative drag in the initial regime and therefore gets sucked into the wake of the upstream column. In the lock-in and the galloping regimes, the downstream column experiences a positive drag and therefore gets pushed back. Figure 3.41 shows the root mean square values of the transverse displacement experienced by the downstream column plotted against corresponding reduced velocities Ur . The results from the isolated square cylinder are also included for comparison. For the tandem square cylinder, the components of response are the initial and lower branches of VIV, desynchronization and galloping [185]. The maximum displacement amplitude is found for the tandem arrangement at Ur = 10. It is nearly 1.5 times the maximum displacement of the single square column which appears at Ur = 5. For the single column set-up, the displacement amplitude is largest in the range 4 ≤ Ur ≤ 7. For Ur ≥ 7, the transverse response y r ms suddenly drops and remains nearly constant for Ur ≥ 10. For the tandem arrangement, the displacement amplitude increases up to Ur = 10, with a small dip in the lock-in region, and decreases gradually as the Ur increases up to 15. For Ur ≥ 15, the displacement amplitude remains constant at a value nearly five times higher than that of a single-column arrangement.

138

3 Proximity and Wake Interference Ur = 2

0.004

0.2

y/D

y/D

y/D

0

0 -0.005

-0.002

-0.0035

-0.003

-0.01 -0.018

-0.0025

-0.016

-0.014

Ur = 8

y/D

y/D

y/D

-0.25

0.35

0.355

0.36

x/D

0.365

-0.5 0.54

0.37

0

0.83

-0.5 1.1

0.84

1.12

x/D Ur = 20

1.51

1.52

0 -0.25

-0.25

0.82

0.58

0.25

y/D

y/D

0

0.57

Ur = 18

0.5

0.25

-0.25

0.56

x/D

Ur = 16

0.5

0.25

0.81

0.55

x/D

Ur = 14

0.5

y/D

0

-0.5 0.04

0.07

0.25

0 -0.25

0.035

0.065

Ur = 12

0.5

0.25

0.03

0.06

x/D

Ur = 10

-0.2

0.5

-0.4 0.055

-0.01

0.5

0

-0.5 0.8

-0.012

x/D

0.2

-0.4 0.025

0 -0.2

x/D

0.4

Ur = 6

0.4

0.005

0.002

-0.004 -0.004

Ur = 4

0.01

0.5

1.14

-0.5 1.48

1.16

1.49

1.5

x/D

x/D

Ur = 25

Ur = 35 0.8

0.25

0.25

0 -0.25

-0.25 -0.5 1.87

y/D

y/D

y/D

0.7 0

1.88

1.89

1.9

-0.5 1.68

x/D

0.6 0.5

1.7

1.72

1.74

0.4 3.1

3.2

x/D

3.3

3.4

x/D

Fig. 3.37 Displacement trajectories of the tandem downstream square cylinder

3.6.2 Vortex Organization The wake of the upstream cylinder interacts with the boundary layer of the downstream cylinder, which influences the flow contours on the downstream cylinder. Due to the wake interaction, the pressure on the low suction side is reduced on the downstream cylinder, increasing the lift and in turn the response. Figures 3.42a through 3.42f illustrate the vorticity contours at selected reduced velocities. The contour plots suggest that in the initial regime the downstream column oscillates within the wake of the upstream column. As shown in Fig. 3.42a, the shear layer separated from the upstream column effectively encapsulates the downstream column and there is no individual vortex shedding from the columns. The lock-in and galloping regimes are characterized by positive drag on the downstream column resulting in an increase in

3.6 Freely Vibrating Tandem Square Prism

425

435

-0.01

-0.3

-0.02

445

405

415

tUf /D

-0.2

-0.2

-0.4

445

-0.2 -0.4 207

217

237

247

Ur = 10

Ur = 12

2

1

1

1

1

0.2

A y /D

227

tUf /D 2

2

0

0

0

0

-0.2

-0.2

-1

-1

-1

-1

-0.4

-0.4

-2

-2

-2

0

0

405

415

425

435

405

445

415

435

-2 405

445

415

425

435

445

tUf /D

Ur = 16

Ur = 14

Ur = 18

2

2

0.8

2

1

1

1

1

0.4

1

0

0

0

0

0

0

-1

-1

-1

-1

-0.4

-1

-2

-2

-2

-2

-0.8

427

437

406

447

1

1.25

0.5

0

0 -0.5 -1 415

425

tUf /D

435

445

CL

2.5

A y /D

0.5

405

426

436

CL

446

-2 408

418

tUf /D

tUf /D Ur = 20

1

416

-0.5

-2.5

-1 405

415

425

tUf /D

438

448

435

445

Ur = 35

2

2.5

1.2

1.25

0.9

1

0.6

0

-1.25

0.3

-1

-2.5

0

0

0

-1.25

428

tUf /D

Ur = 25

A y /D

417

CL

407

A y /D

2

A y /D

2

CL

A y /D

425

tUf /D

tUf /D

CL

0

2

0.4

CL

A y /D

0.2

A y /D

435

0.2

tUf /D

Ur = 8

0.4

425

0.4

0

-0.1

A y /D

415

0

-0.15

CL

405

0.2

CL

-0.006

0

0.1

Ur = 6

CL

0

-0.003

0.4

CL

0.01

0.2

CL

0.15

Ur = 4

A y /D

0

0.02

CL

Ay /D

0.003

0.3

A y /D

Ur = 2

0.006

139

-2 405

415

425

435

445

tUf /D

Fig. 3.38 Time histories of the transverse displacement (solid lines) and lift coefficients (dash lines) of the downstream column at selected values of Ur

center-to-center spacing and augmented transverse oscillations. Consequently, the extended shear layer destabilizes leading to the commencement of individual vortex shedding from the columns. The vortex shedding is occurring in a 2S mode in the lock-in regime which further transforms to a 2P-like mode in the desynchronization regime. In the galloping regime, the wake structures are somewhat wide and irregular and the phase angle is consistently close to 180◦ with the trajectories of skewed elliptical shapes. A detailed investigation of the vortex dynamics and wake topology is beyond the scope of this study.

140

3 Proximity and Wake Interference 200 180 160

Phase angle ( )

140 120 100 80 60 40 20 0 0

10

20

30

40

50

Ur

Fig. 3.39 Variation in phase angle between transverse displacement and lift coefficient with Ur for the downstream tandem square cylinder

3.7 Wake Interference of Tandem Circular Cylinder When a rigid cylinder is free to vibrate in a cross-flow direction, there is a strong nonlinear coupling between the motion and the flow dynamics. This results in a complex evolution of shedding frequency which does not follow the Strouhal law as the natural frequency of the cylinder is approached, which is referred to as lock-in or synchronization of the wake frequency to the cylinder frequency. In the lock-in range, the frequency of the perturbed wake system and the natural frequency of the cylinder are close to each other, which results in an increased amplitude of motion [330, 381]. In such self-excited cylinders in fluid flows, the amplitude of vibration grows until it becomes so large that nonlinear dynamical effects become relevant and achieve a self-limiting amplitude. The oscillatory nature of the forces arises from the vortex shedding and wake instabilities around cylinders. However, there is a considerable difference between the fluid-structure coupled response of an isolated cylinder arrangement and a tandem cylinder arrangement [22, 28, 58, 159]. The phenomena of the vortex and wake-induced vibration have applications in the offshore industry, aerospace, power transmission, energy extraction, and many more. In particular, the majority of the risers of the offshore industry and power transmission wires typically operate in bundles and the oncoming flow interacts in a complex way. Flow-induced vibrations due to vortex and wake excitations significantly affect the performance and fatigue life of the structures undergoing complex motions with nonlinear dynamics such as traveling waves [186, 277] and chaotic motions. Due to the complex nature of flow-induced vibrations, the design of riser arrays has been an

3.7 Wake Interference of Tandem Circular Cylinder

141 Ur = 8 0.3

0.07

0.675

0.035

0.25

0.03

0.2

0.525

0.025

0.15

0.45

0.02

0.6

0.065 0.06

207

217

227 tU ∞ /D

237

247

405

415

1.5125

0.36

1.425

0.35

1.3375

425 tU ∞ /D

435

445

A x /D

0.37

415

445

1.16

CD

A x /D

1.6

405

435

0.1

Ur = 16

Ur = 10 0.38

0.34

425 tU ∞ /D

1.145

1.925

1.13

1.75

1.115

1.575

1.1

1.25

2.1

406

416

426 tU ∞ /D

436

446

CD

0.055

CD

0.04

A x /D

0.75

CD

A x /D

Ur = 6 0.075

1.4

Ur = 35

Ur = 20 2.1

3.5

1.4

1.9875

3.4

1.2

3.3

1 0.8

1.881

1.875

1.873

1.7625

3.2

1.65

3.1 0.6 200 225 250 275 300 325 350 tU ∞ /D

1.865

205

215

225 235 tU ∞ /D

245

CD

A x /D

A x /D

1.889

CD

1.897

Fig. 3.40 Time histories of the in-line displacement (solid lines) and drag coefficients (dash lines) of the downstream cylinder at selected values of Ur

area of great uncertainty in recent years. The wake dynamics of the upstream riser influence the loads and the structural response of the downstream riser. Through experiments [11] and numerical simulations [397], it was shown that the case with the straked upstream riser has suppressed motion whereas the downstream riser experienced low-frequency oscillations with amplitude higher than the upstream due to continuous impingement of vorticity with the downstream structure. This suggested that when a straked riser resides in the wake of another riser, the strakes lose their ability to suppress flow-induced vibrations. To understand the wake interactions with flexible structure, the flow around two cylinders mounted elastically can be considered as an idealized model. Many researchers have attempted to characterize coupled response of stationary and vibrating cylinders in a tandem arrangement. When an additional cylinder is placed downstream of the first, the flow phenomena become intricate due to vortexbody interactions and are governed by the relative positioning of the two bluff bodies.

142

3 Proximity and Wake Interference 1 Tandem, Re = 200, m = 5 0.9 0.8

Single, Re = 100, m = 3 Single, Re = 200, m = 5

Ay rms /D

0.7

Galloping

0.6 0.5 0.4 V IV 0.3 0.2 Desynchronization 0.1 0 0

10

5

15

25

20

30

35

40

Ur Fig. 3.41 Comparison between RMS values of the transverse displacements for the downstream tandem and isolated cylinder configurations at two different mass ratio and Reynolds numbers

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 3.42 Instantaneous vorticity contours for reduced velocity Ur = a 4.5, b 7.0, c 10.0, d 16.0, e 30.0, f 40.0

3.7 Wake Interference of Tandem Circular Cylinder

143

U m cylinder

x0

k

c

y

x

Fig. 3.43 Schematic of tandem cylinder arrangement for wake-induced vibration. Here U is freestream velocity, x0 denotes streamwise distance, and m, c, k are mass, damping and stiffness parameters of the elastically mounted downstream cylinder

In the present study, this parameter has been defined in terms of the distance separating the centers of the two cylinders x0 as shown in Fig. 3.43. Three flow interference regimes [175, 478]: proximity interference, wake interference and no interference, are identified for the tandem cylinder configuration based on the ratio of spacing x0 to cylinder diameter D. In the proximity interference regime for 1 ≤ x0 /D ≤ 1.2– 1.8, negative drag is produced on the downstream cylinder and vortex shedding from the upstream cylinder is suppressed. The tandem bodies behave like a single bluff body and vortex shedding occurs behind the rear cylinder. In the wake interference or reattachment regime for 1.2–1.8 ≤ x0 /D ≤ 3.4–3.8, a number of different phenomena such as shear layer reattachment, intermittent vortex shedding, etc. are observed as the separation distance is gradually increased. In the regime of large spacing x0 /D ≥ 3.8, the so-called co-shedding regime, vortex shedding occurs from both the cylinders and there is no significant interference effect. The critical separation distance xcr for the onset of the co-shedding regime has been indicated by several researchers [306, 463, 500] both numerically and experimentally to be between 3.5D to 5D for a wide range of Reynolds numbers. The study of the stationary setup of tandem cylinders was attempted by researchers in [7, 390] to understand the forces developed on the downstream cylinder. The loads on the cylinders in the tandem arrangement were experimentally investigated in [7]. The authors reported that if the phase of flow pattern between upstream and downstream cylinders coincides, then the lift forces become maximum. The authors in [390] numerically studied the laminar flow over two tandem cylinders at Reynolds number of 100. The critical streamwise gap between two cylinders is around 4D, where there is a sudden jump in fluctuating forces and Strouhal number. They also

144

3 Proximity and Wake Interference

reported that flow contours of the downstream cylinder are characterized by only one separation point and one reattachment point for a streamwise gap greater than 4D. The dynamics of the stationary tandem cylinder arrangement near the moving wall were reported in [110] at a Reynolds number of 200. The moving wall in the vicinity of tandem cylinders reduces the critical streamwise gap. The authors in [135] numerically investigated the effect of streamwise gap on the stationary tandem cylinder arrangement at the subcritical Reynolds number. The downstream cylinder of the tandem setup experiences large displacements at reduced velocities beyond the lock-in region [22, 28, 58, 159]. The authors [58] carried out several experiments on the tandem cylinder arrangement is shown in Fig. 3.43, where the streamwise distance between two cylinders ranges from 1.099 to 3 diameter. The authors reported a large structural displacement for the downstream cylinder and the amplitudes are sensitive to the cylindrical aspect ratio. Experimental results on the similar setups at Re ≈ 3 × 104 were reported in [159]. The authors found increased transverse loads and displacement responses for the downstream cylinder compared to those of the isolated cylinder. In addition, at large reduced velocities, there exist two frequencies in the cross-flow loads of the downstream cylinder: one frequency component equal to the vortex shedding frequency of the upstream cylinder, and the other frequency component which is close to the natural frequency of the structure. In their results, the loads on the fixed upstream cylinder are not significantly affected by the movements of the downstream cylinder. More recently, the authors in [28] have stressed that wake-induced vibration is mainly due to an unsteady wake which supplies energy to the downstream cylinder resulting in the large transverse amplitude. The origin of the forces on the downstream cylinder was qualitatively linked to the wake interaction. In another recent work, [22], the concept of wake stiffness was proposed based on their experimental studies on cylinders in a tandem arrangement. The authors found that the wake stiffness dominates over the spring stiffness which characterizes the wake-induced vibration response of the downstream cylinder for the reduced velocities larger than the lock-in region. Similar experiments were done on two flexible risers for the tandem arrangement in [165]. The dynamic response of the upstream cylinder is similar to vortex-induced vibrations whereby the downstream cylinder undergoes wakeinduced vibrations for a relatively large gap x0 > 4D between the cylinders. The effect of different diameters of the tandem cylinders on the dynamic response was reported in [26]. The diameter ratio has a remarkable influence on the displacement response and however, it has no effect on the frequency response of the downstream cylinder. The same authors in [25] have reported the experimental studies of tandem cylinders in a staggered arrangement. The wake-induced vibration of two degrees of freedom of downstream cylinder is similar to that of single degree of freedom. As the lateral distance increases the wake effect on the downstream cylinder decreases with negligible effect at a lateral distance of three diameters. Direct numerical simulations of tandem cylinders at low Reynolds numbers were reported in [77, 292]. The authors in [292] studied the effect of wake-induced vibrations at low Reynolds number of 100, where both the cylinders were mounted on stiffness. The authors reported large displacement responses for the downstream

3.7 Wake Interference of Tandem Circular Cylinder

145

cylinder in tandem arrangements. Numerical experiments on isolated as well as tandem cylinder arrangements at low Reynolds numbers were performed in [77]. The authors characterized the displacement amplitudes for tandem cylinder arrangement with different gaps. They presumed that the large displacements of the downstream cylinder are due to the motion of the stagnation point which was influenced by the unsteady flow between the two cylinders. The three dimensional flow structures at Re ≤ 300 have no effect on the coupled response of the tandem arrangements. The coupled displacement response of a downstream cylinder for a streamwise gap distance of x0 /D ≤ 3 is greater compared to a case with a streamwise gap distance of x0 /D ≥ 5. The analytical approach to the wake-induced vibrations is attempted by authors in [329] with the aid of quasi-steady approximation to model the large amplitude behavior of tandem cylinder arrangements. The role of vortex formation and interaction on the coupled dynamics of flow-body interactions was investigated by [68, 379, 460]. Laminar flow dynamics of a cylinder attached with a flapping plate reported in [460]. The positive reinforcement of the secondary vortex from the flapping plate with the primary vortex results in larger drag, whereas the negative reinforcement leads to a lesser amount of drag. The authors in [68] investigated experimentally that the flexible flaplets on the airfoil near to the trailing edge can provide a regularized flow thereby delaying a non-linear transition to stall. Turbulent energy budget in the wake of a vibrating cylinder is analyzed in [379] to characterize the amplitude response. The differences between the loads acting on an isolated single cylinder and a tandem downstream cylinder have been reported in the literature. However, the mechanisms involved in the generation of the forces have not been studied in detail for flowinduced vibrations observed in an isolated and a tandem cylinder arrangement. For the tandem cylinder arrangement, the dynamical behavior of the downstream cylinder is influenced by the interaction of the wake of the upstream cylinder. To reduce flow-induced vibrations, suppression devices [477] aim to disrupt the boundary layer on the surface and disorganize the shedding of vortices due to wake synchronization. However, a similar suppression device may not be as effective for suppressing the wake-induced vibrations in tandem arrangements. An understanding of the mechanisms of load generation in tandem arrangements may pave the way to design more effective suppression devices. In this work, we consider two circular cylinders in a tandem arrangement where the upstream cylinder is fixed and the downstream cylinder is allowed to vibrate in a transverse direction as shown in Fig. 3.43. The coupling of upstream cylinder wake with a stationary and freely vibrating downstream body can lead to a wide variety of coupled dynamical phenomena. Of particular interest here is to understand the interaction of upstream cylinder wake with the boundary layer of the downstream cylinder, which results in the coupled response of the system different from the isolated single-cylinder arrangement. The interaction of the upstream vortex with the downstream cylinder can lead to large wake-induced vibrations, which can in turn lead to premature structural failure. A careful study of the generation of the fluid forces and their phase relations with cylinder velocity has been performed for the isolated and tandem configurations. The wake contours and phase angles are

146

3 Proximity and Wake Interference symmetry,

∂u/∂y = 0, v = 0

LCy

D outlet σxx = 0

inlet u=U

σyx = 0

v=0

LCy y

symmetry,

x LU x

∂u/∂y = 0, v = 0 LDx

Fig. 3.44 Schematic of computational domain and boundary conditions for isolated cylinder

discussed with respect to the transverse loads for different regimes. The constituents of the pressure and viscous loads to the total transverse load as well their effect on the vortex-body interactions are systematically studied with the aid of vorticity contours. We illustrate that the forces from continuous impinging vortices can alter the dynamics of the downstream tandem cylinder.

3.7.1 Problem Description Figure 3.44 shows the schematic of the computational domain considered in the Cartesian x–y domain for two-dimensional incompressible flow around a rigid cylinder. At the inlet of the domain the freestream velocity is fixed at a value u = U and at the outlet boundary the traction vector is set to zero. The side boundaries are considered symmetric. The length of the domain L x is set to 40D and width L y is set to 20D. The center of the cylinder is stationed from the inlet boundary L U x = 10D and from the outlet boundary L Dx = 30D. Illustration of the tandem cylinder arrangement with an elastically mounted downstream cylinder is shown earlier in Fig. 3.43. The downstream cylinder is displaced by x0 in streamwise direction from the upstream cylinder and with no shift in the transverse direction. For simplicity, the downstream cylinder is only allowed to move in the transverse direction. Flow-induced vibration of a cylinder is influenced by three key non-dimensional parameters, namely, mass-ratio, m ∗ ; Reynolds number, Re and reduced velocity, U ∗ defined as

3.7 Wake Interference of Tandem Circular Cylinder

m∗ =

4m , ρ f π D2

Re =

ρf U D U , , Ur = μf fn D

147

(3.10)

√ where f n = (1/2π ) k/m is the dimensional natural frequency, m is the cylinder mass, k is the spring constant. In the above definitions, we make the isotropic assumption for the translational motion of the rigid body, i.e., the mass vector m = (m x , m y ) with m x = m y = m, the stiffness vector k = (k x , k y ) with k x = k y = k. In the absence of structural damping, the dynamical behavior of isolated single cylinder and tandem cylinder setup is studied. The derived quantities that we will extensively study include lift and drag coefficients. The force coefficients are computed by carrying out an integration, that involves elementwise contributions of the pressure and viscous stresses, for elements located on the cylinder surface. The lift coefficient C L and the drag coefficient C D are defined as  1 CD = 1 f 2 (σ f · n) · nx dΓ, (3.11) ρ U D 2 Γ

1 CL = 1 f 2 ρU D 2

 (σ f · n) · ny dΓ.

(3.12)

Γ

Here nx and ny are the Cartesian components of the unit normal vector n. In the present study, the force coefficients C D and C L are calculated as the derived quantities using the direct evaluation of the Cauchy stress on the boundary. The transverse lift force is a crucial parameter in flow-induced vibrations and there exists a complex nonlinear relationship between the amplitude vibration and the transverse force. We can further decompose the total cross-flow force coefficient into its pressure C L p and viscous C Lμ constituents which are defined as below CLp

1 = 1 f 2 ρU D 2

C Lμ =

1 1 f 2 ρ U D 2

 (σ p · n) · ny dΓ,

(3.13)

(σ μ · n) · ny dΓ.

(3.14)

Γ

 Γ

Notably, the periodic transverse force on the cylinder can also be decomposed into the added-mass force induced by relative fluid acceleration and the wake force due to the effect of vorticity in the boundary layer and wake of the cylinder. Relationship between the amplitude vibration and the transverse force components can be constructed for a known frequency [392, 455]. For the sake of completeness, the detailed verification of the numerical method and convergence studies are summarized in Appendix B. We next consider the tandem cylinder problem for investigating the wake-body dynamics and the origin of wake-

148

3 Proximity and Wake Interference

induced vibration in a systematic manner and the results are compared with the isolated cylinder. It is known that the complexity of the coupled physical phenomena involved in a vibrating isolated cylinder is enhanced when two cylinders are placed in a tandem arrangement. For a representative systematic study of wake-induced vibration, we consider the streamwise distance x0 between the two cylinders to be 5D, which is beyond the critical distance of 4D [390]. The unsteady wake of the upstream cylinder interacting with the downstream cylinder influences the coupled response of the tandem arrangement different from the isolated cylinder arrangement. For this large streamwise separation case, the greater amplitude vibrations observe to be sustained for higher reduced velocity beyond the lock-in range. To analyze the origin of wake-induced vibrations, we assess dynamic response characteristics along with the movement of stagnation point for the isolated and tandem cylinder arrangements. The pressure and viscous contributions to the transverse loads are thoroughly investigated to understand the development of forces on the cylinder. We also study the wake contours to analyze the importance of the upstream wake-interaction with the downstream cylinder and associate the interaction with the total transverse force acting on it. The numerical experiments for the isolated and tandem cylinder arrangements are conducted at Reynolds number of Re = 100 and the mass ratio of m ∗ = 2.6 with no structural damping.

3.7.2 Response Characteristics We first present the dynamic characteristics of the coupled response of isolated as well as tandem cylinder configurations. Figure 3.45 shows the variation of the transverse amplitude and force with respect √ to the reduced velocity. The transverse displacement amplitude, A ymax,r ms = 2 A y,r ms , is observed to be larger for the downstream cylinder in tandem arrangement than the isolated cylinder as shown in Fig. 3.45a. Here, A y,r ms is the root mean square (rms) of the transverse displacement of the cylinder. The maximum displacement amplitude for the tandem arrangement at Ur = 6 is nearly two times the value of the isolated cylinder, which is consistent with the results of Carmo et al. [77]. For the isolated configuration, the displacement amplitude is largest in the range 4 ≤ Ur ≤ 7 which is referred to as the lock-in region. For Ur > 7, the displacement amplitude suddenly drops and remains nearly constant for Ur ≥ 10. For the tandem arrangement, the displacement amplitude increases, as Ur is increased to 6, and decreases gradually as the reduced velocity increases up to 15. For Ur ≥ 15, the displacement amplitude remains constant at a value larger than that of the isolated cylinder arrangement. At higher Reynolds numbers, the transverse displacement amplitude of the downstream cylinder is observed to be larger compared to the laminar flow at reduced velocities greater than the lock-in region in tandem arrangement [22, 28, 58, 159]. Figure 3.45b shows the variation of C L ,r ms with respect to Ur for the isolated and tandem cylinder setups. C L ,r ms value remains almost constant for the upstream cylin-

3.7 Wake Interference of Tandem Circular Cylinder

149 1.5

1 isolated cylinder tandem cylinder - downstream

0.9

isolated cylinder tandem cylinder - upstream tandem cylinder - downstream

0.8 1

0.6

CL,rms

Aymax, rms

0.7

0.5 0.4

0.5

0.3 0.2 0.1

0

0 0

5

10

15

20

reduced velocity, Ur

(a)

25

30

0

5

10

15

20

25

30

reduced velocity, Ur

(b)

Fig. 3.45 Effects of reduced velocity Ur on a transverse amplitude A ymax,r ms and b rms lift coefficient C L ,r ms

der which is stationary. C L ,r ms of the downstream cylinder is larger than the upstream cylinder, as well as the isolated cylinder. The pressure and viscous contributions to transverse loads and wake interactions is investigated in the forthcoming sections to study the difference in the loads of two setups. Figure 3.46 shows typical force and displacement responses for the isolated and downstream cylinders along with their frequency spectra. There exist multiple frequencies in the temporal response of C L whereas only one frequency is dominant in the temporal response of transverse displacement y(t). The results are shown for both the isolated and tandem configurations in Fig. 3.46a, c, respectively. The frequency spectra of C L are shown in Fig. 3.46b, d. Two distinct peaks in the frequency spectra of C L can be observed. The major contribution comes from the lower harmonic f L , which is equal to the shedding frequency of the vibrating cylinder. The higher harmonic f H is about three times that of the lower harmonic f L , i.e. f H ≈ 3 f L . The frequency of the isolated cylinder is known to lock in with the structural frequency in the synchronization range 4 < Ur < 8, deviating from the Strouhal frequency as shown in Fig. 3.47a. The non-dimensionalized frequency f / f n of the upstream stationary cylinder is mostly unaffected by the movements of the vibrating downstream cylinder, thus the frequency ratio varies linearly with the reduced velocity. The unsteady wake flow coming from the upstream cylinder interacts with the downstream cylinder enforcing the shedding frequency in the response of the vibrating downstream cylinder. Thus, the dominant frequency ratio of the downstream cylinder is similar to that of the upstream cylinder, varying linearly with respect to the reduced velocity Ur . There is no low-frequency component compared to the Strouhal frequency of the upstream cylinder in the transverse load, contrary to the experimental observations at higher Reynolds numbers [22, 58, 159].

150

3 Proximity and Wake Interference

0.8

0.1 CL

0.6

displacement response y(t)

0.08 0.4

|CL(f)|

0.2 0

0.06

0.04

-0.2 0.02 -0.4 -0.6 60

80

0

100

0.5

1

1.5

2

f/fn

(a)

(b)

1.4 1.2

0

time, tU/D

2.5

3

3.5

4

2.5

3

3.5

4

0.4

CL displacement response y(t)

1

0.35

0.8

0.3

0.6 0.25

|CL(f)|

0.4 0.2 0 -0.2

0.2 0.15

-0.4 0.1

-0.6 -0.8

0.05

-1 -1.2 100

0 120

140

0

0.5

1

1.5

2

time, tU/D

f/fn

(c)

(d)

Fig. 3.46 Temporal variations of lift coefficient (left) and frequency spectrum (right) at Ur = 6: a, b isolated cylinder, c, d downstream cylinder of tandem configuration

Similar to C L ,r ms , the average drag coefficient C D,avg remains constant for the upstream cylinder as shown in Fig. 3.47b. The average drag coefficient for the downstream tandem cylinder and the isolated cylinder is higher at large displacements in the lock-in region. The lower wake velocity magnitude aft the upstream cylinder interacts with the downstream cylinder, which results in a relatively lower drag coefficient than the isolated cylinder counterpart.

3.7 Wake Interference of Tandem Circular Cylinder 4.5

2.4 isolated cylinder upstream Cylinder downstream Cylinder

4

isolated cylinder tandem cylinder - upstream tandem cylinder - downstream

2.2

3.5

2

3

1.8

C d,avg

frequency ratio, f/f n

151

2.5 2

1.6 1.4 1.2

1.5 1 1

0.8

0.5

0.6 5

10

15

20

25

30

0

reduced velocity, Ur

(a)

5

10

15

20

25

30

reduced velocity, Ur

(b)

Fig. 3.47 Variation of a primary frequency of C L and b Cd,avg with respect to Ur

3.7.3 Decomposition of Transverse Force To understand the coupled response of both configurations, we analyze the pressure and viscous contributions of the transverse force. Figure 3.48 shows the variation of C L due to the pressure and viscous contributions at various reduced velocities for the isolated and the downstream tandem cylinders. In both the configurations, it is observed that C L p,r ms for Ur ≥ 5 is lower compared to Ur < 5. For the downstream tandem cylinder, a larger value of C L p is observed than the isolated cylinder counterpart. A similar qualitative trend as that of C L p,r ms is found for the variation of C Lμ,r ms with respect to the reduced velocity for the isolated cylinder which is illustrated in Fig. 3.48b. At reduced velocity Ur = 6, where excitation frequency ratio is near to one, C L p,r ms is found to be less than that of C Lμ,r ms for both the configurations. We next analyze the ratio C Lμ,r ms /C L p,r ms to assess the pressure and viscous forces for the range of reduced velocities as shown in Fig. 3.48c. The maximum of this ratio exists around Ur = 6, where the frequency ratio is nearly one, and C L p is nearly in-phase with the velocity as shown in Fig. 3.49a. Thus, the ratio C Lμ,r ms /C L p,r ms indicates the well-known phenomenon of so-called self-limiting of cross-flow cylinder vibrations where the pressure force supplies energy to the system and the viscous force dissipates the energy. The amplitude of stagnation point movement along the circumference of the cylinder with respect to the mean θ = π is plotted in Fig. 3.48 (d). Qualitatively, the rms of viscous and pressure forces follow the amplitude of stagnation point movement. It implies that the stagnation point location on the circumference of the cylinder plays a major role in the load generation mechanism. To understand the energy transfer in the coupled system, we further analyze the phase differences of C L p and C Lμ with respect to the velocity of the vibrating cylinder. Figure 3.49 shows the phase difference of C L p and C Lμ with respect to the cylinder velocity, at various reduced velocities for both configurations. It can be seen in

152

3 Proximity and Wake Interference 1.5

0.4 isolated cylinder tandem cylinder - downstream

isolated cylinder tandem cylinder - downstream

0.35 0.3

1

CLμ, rms

CLp, rms

0.25 0.2 0.15

0.5

0.1 0.05 0

0 0

5

10

15

20

25

0

30

5

10

(a)

20

25

30

(b) 0.4π

stagnation point - amplitude

1.5

CLμ, rms/CLp, rms

15

reduced velocity, Ur

reduced velocity, Ur

single cylinder tandem cylinder - downstream

1

0.5

0.35π

isolated cylinder tandem cylinder - downstream

0.3π 0.25π 0.2π 0.15π 0.1π 0.05π

0

0 0

5

10

15

20

25

30

0

5

reduced velocity, Ur

10

15

20

25

30

reduced velocity, Ur

(c)

(d)

Fig. 3.48 Dependence of coupled response a C L p,r ms , b C Lμ,r ms , c ratio C Lμ,r ms /C L p,r ms , and d amplitude of stagnation point movement on Ur 0.6π

2π 1.8π isolated cylinder tandem cylinder - downstream

1.6π

phase difference φ

phase difference φ

0.4π 0.2π 0 -0.2π

1.4π 1.2π 1π 0.8π 0.6π isolated cylinder tandem cylinder - downstream

0.4π

-0.4π

0.2π -0.6π 0

5

10

15

20

reduced velocity, Ur

(a)

25

30

0

0

5

10

15

20

25

reduced velocity, Ur

(b)

Fig. 3.49 Phase difference relationships between a C L p and velocity, b C Lμ and velocity

30

3.7 Wake Interference of Tandem Circular Cylinder

0.15π

0.1

0.05π

0

0

-0.05π

-0.1

-0.1π

-0.2

-0.15π

temporal variation

0.1π

CLp ~ CLp1 + CLp2 CLp1 CLp2 Velocity/2

0.2

stagnation point

0.2

CL , Velocity

0.3

0.2π

stagnation point location CL Velocity/2

0.3

153

CLp, max

0.1 0 -0.1 -0.2

-0.3 1

2

3

4

5

6

-0.2π

-0.3 0

1

2

3

4

time, tU/D

time, tU/D

(a)

(b)

5

6

Fig. 3.50 Isolated cylinder arrangement at Ur = 6: temporal variation of a C Lμ , velocity and stagnation point, b pressure force coefficient C L p , the two leading components C L p1 , C L p2 and cylinder velocity

Fig. 3.49b that the phase difference between C Lμ and velocity lies between π/2 and 3π/2. Clearly, the viscous forces dissipate the energy of the coupled system, thereby limiting the coupled vibrational response. At the reduced velocity Ur = 6, for the isolated cylinder arrangement where the frequency ratio f / f n is near to one, the viscous force is nearly out-of-phase with the cylinder velocity. A similar trend is observed for the downstream tandem cylinder. As the viscous forces dissipate the energy, the supply has to come from the pressure component. At 1 ≤ Ur ≤ 5 for the isolated cylinder arrangement, the frequency ratio is less than one, the velocity of the cylinder lags C L p by approximately 0.4π as shown in Fig. 3.49a. For Ur = 6, the velocity is near in-phase with C L p , where the maximum energy is added into the coupled system, and we see a large displacement at Ur = 6. For Ur greater than 6, the inertia force dominates over the stiffness force, thus the velocity lags C L p by nearly π/2. For both configurations, we further study the temporal variation of transverse loads due to the pressure and viscous components and the net power transfer at Ur = 6. Figure 3.50 shows the temporal variation of C L due to the pressure and viscous forces as well as the temporal variation of velocity at Ur = 6 for the isolated cylinder arrangement. The stagnation point movement along the circumference of the cylinder about the mean position θ = π is plotted in Fig. 3.50a. The upward movement of the stagnation point with respect to θ = π is considered positive and vice-versa. The stagnation point movement on the circumference of the cylinder is nearly in-phase with the velocity. Temporal variations of C Lμ vary with a phase lag of 0.764π with the velocity. To analyze the effect of the harmonics of lift forces, the pressure lift coefficient C L p can be represented as Fourier series

154

3 Proximity and Wake Interference

CLp =



C L0 pi cos(2π f i t + φi ),

(3.15)

i

where C L0 pi is the force amplitude of the component with frequency f i , and φi is the corresponding phase of the signal. The pressure lift coefficient C L p can be further decomposed into two dominating parts, namely, C L p1 and C L p2 as C L p ≈ C L p1 + C L p2 ,

(3.16)

where C L p1 = C L0 p1 cos(2π f 1 t + φ1 )

and

C L p2 = C L0 p2 cos(2π f 2 t + φ2 ). (3.17)

Here f 1 denotes the frequency for the low-harmonic with the phase φ1 and f 2 is a frequency for the high-harmonic with the phase φ2 . A representative quantitative values at Ur = 6 are given as: f 1 = 0.16, f 2 = 3 f 1 , φ1 = 1.27, φ2 = −0.04, C L0 p1 = 0.115 and C L0 p2 = 0.021. These two dominant components in the frequency spectrum of C L p are shown in Fig. 3.46b. It is observed that the velocity is in-phase with the low frequency component, C L p1 , as the low-harmonic frequency nearly equals the natural frequency of the structure. We next look at the flow contours to understand the origin of harmonics in the transverse lift force. The vorticity contours at six equal intervals over a half-time period are plotted in Fig. 3.51 for the isolated cylinder at Ur = 6. It is worth noting that the reverse flow vorticity regions present in Fig. 3.51c, d, e on the aft part −π/2 < θ < π/2 of the cylinder but are absent in Fig. 3.51a, b, f. The absence of vorticity regions due to the reverse flow in the aft part of the cylinder signifies the presence of only one separation point and one reattachment point, which is similar to the observations made on the stationary downstream cylinder of tandem arrangement in [390]. The high-frequency component, C L p2 , may be attributed to the intermittent presence of vorticity regions in the aft part of the cylinder as shown in Fig. 3.51. Figure 3.52 shows the temporal variation of C L p , C Lμ , the stagnation point location and the velocity of the downstream tandem cylinder at Ur = 6. The temporal variation of the viscous component of the force in the transverse direction is shown in Fig. 3.52a for the downstream tandem cylinder, which is somewhat similar to the isolated cylinder arrangement. Figure 3.52b shows the temporal variation of pressure lift coefficient C L p and velocity for the downstream cylinder at Ur = 6. The lowfrequency component C L p1 , which is approximately in-phase with the velocity and supplies the energy to the cylinder for a self-sustaining vibration. To characterize the energy transfer, we evaluate the instantaneous work done by the product of force components with the velocity. If the relative phase between the force and the velocity is φ ≈ 0, the self-excited oscillator keeps drawing energy from the surrounding fluid flow. At Ur = 6, Fig. 3.53 shows the temporal variation of coefficient of energy rate C Et due to the viscous and pressure forces for the isolated and tandem cylinder arrangements. The coefficient of energy rate due to the pressure

3.7 Wake Interference of Tandem Circular Cylinder

(a)

(b)

(c)

(d)

(e)

(f)

155

0.2π

stagnation point location CL Velocity/2

0.1π 0.05π 0π -0.05π -0.1π -0.15π 1

2

3

4

5

6

-0.2π

CLp ~ CLp1 + CLp2 CLp1 CLp2 velocity/2

0.6

0.15π

temporal variation of forces

0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6

stagnation point

CL , Velocity

Fig. 3.51 Wake contours of isolated cylinder at Ur = 6 during half period: a 0.0833T , b 0.1666T , c 0.2499T , d 0.322T , e 0.4165T , f 0.5T

0.4

CLp, max

0.2 0 -0.2 -0.4 -0.6

2

4

time, tU/D

time, tU/D

(a)

(b)

6

8

Fig. 3.52 Response of downstream tandem cylinder at Ur = 6: temporal variation of a C Lμ , velocity and stagnation point and b pressure force coefficient C L p , the two leading components C L p1 , C L p2 and cylinder velocity

156

3 Proximity and Wake Interference

0.1

CEt CEtp1 CEtp2

0.08

CEt CEtp1 CEtp2

0.2

0.06 0.04 0.02

0 0 -0.02 -0.04

-0.2 -0.06 -0.08 0

1

2

3

4

5

6

0

1

2

3

time, tU/D

time, tU/D

(a)

(b)

4

5

6

Fig. 3.53 Temporal variations of power coefficients due to pressure and viscous forces over the cylinders: a isolated, b downstream

force C Et p and the viscous force C Etμ are defined as follows: ¯ C Et p (t) = C L p (t)v(t),

(3.18)

¯ C Etμ (t) = C Lμ (t)v(t),

(3.19)

where v¯ = v/U is the non-dimensional transverse velocity of the cylinder. Here, C Et p1 and C Et p2 are coefficients of energy rate due to the pressure lift coefficients C L p1 and C L p2 , respectively. In both the cylinder arrangements, the viscous forces dissipate the energy, while the pressure forces feed energy into the vibrations, which can be observed in Fig. 3.53a, b for the isolated and downstream cylinders, respectively. The high-frequency pressure force component, on average, over a time period, results in a very low energy transfer in both the arrangements as compared to the low-frequency components of pressure and viscous effects. We next consider the effects of vortex-body interactions on the pressure distribution around the cylinder and analyze the resultant wake contours for a range of reduced velocities.

3.7.4 Pressure Distribution and Wake Contours To understand the nature of the fluid forces around the cylinders, we now look into the pressure coefficient, C p = 2 p/ρU 2 , over the circumference of the cylinder for different reduced velocities. Figure 3.54 shows C p distributions for the isolated cylinder and the downstream tandem cylinder at the instance when the maximum lift due to pressure C L p,max is observed. At C L p,max , the suction pressure for the top

3.7 Wake Interference of Tandem Circular Cylinder 2

2 stationary Ur = 2 Ur = 4 Ur = 5 Ur = 6 Ur = 30

1.5 1

θ

stationary Ur = 2 Ur = 4 Ur = 5 Ur = 6 Ur = 30

1.5 1

0.5

0.5

0

0

Cp

Cp

157

-0.5

-0.5

-1

-1

-1.5

-1.5

-2

-2

-2.5

-2.5

0

0.5π

1π

1.5π

along the circumference of circle

(a)

2π

0

0.5π

1π

1.5π

2π

along the circumference of circle

(b)

Fig. 3.54 Distribution of C p with angular position θ over the circumference of cylinder at maximum lift due to pressure: a isolated cylinder, b downstream tandem cylinder

region of cylinder 0 ≤ θ ≤ π is larger than that of the suction pressure in bottom region of cylinder π < θ < 2π . Here, the top region is denoted as the high suction region and the bottom region as the low suction region and vice-versa at C L p,min . For all the cases of isolated cylinder arrangement, the stagnation point lies nearly at θ = π except for the reduced velocity Ur = 5, 6 as shown in Fig. 3.54a. At Ur = 5, 6, the stagnation point is moved to the high suction pressure region 0 < θ ≤ π . The pressure force at the stagnation point in the transverse direction acts opposite to the pressure force due to the suction. Thus, there is a sudden drop of C L p,r ms at Ur = 5, 6. The suction pressure at Ur = 30 in the high suction region is smaller compared to that found in a stationary case, thus limiting the displacement of the cylinder, in turn, limiting the amplitude of stagnation point movement. For the tandem cylinder arrangement, as shown in Fig. 3.54b, at maximum C L p , the stagnation point has been shifted to the low suction region π < θ < 3π/2 for all the cases except Ur = 6. At Ur = 6, the stagnation point lies in the high suction region similar to the isolated cylinder case. It suggests that the flow contours over the downstream tandem cylinder, are influenced by the wake of the upstream cylinder which is reflected in the shifting of the stagnation point for all the cases, except for the reduced velocity Ur = 6. The suction pressure in the low suction region π < θ < 2π of the downstream cylinder is small compared to the isolated cylinder. The suction pressure of the downstream stationary cylinder is greater than that of the isolated stationary cylinder in the high suction region. Larger values of C L p,max are caused by the shift in the stagnation point position, away from θ = π to the low suction region, an increase in the suction pressure in the region 0 ≤ θ ≤ π/2 and decrease in the suction pressure in the region π ≤ θ ≤ 2π . The flow contours can further assist to understand this point more clearly. We plot the vorticity contours of both the setups when the value of C L p is approximately minimum, zero and maximum as shown in Figs. 3.55 and 3.56. For the vibrating

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3 Proximity and Wake Interference

Fig. 3.55 Wake contours of isolated cylinder setup, stationary case (a–c), Ur = 4 (d–f), Ur = 6 (g–i), Ur = 30 (j–l): first column—minimum C L p , second column—C L p ≈ 0, third column— maximum C L p

3.7 Wake Interference of Tandem Circular Cylinder

159

cases, a mean horizontal dashed line is drawn which passes through the center of the stationary upstream cylinder. Based on the phase difference between C L p and the cylinder velocity, as shown in Fig. 3.49a, the reduced velocities Ur =4, 6, 30 are representatively selected, whose phase difference is approximately π/2, 0, −π/2 respectively. The vorticity contours at the different reduced velocities are compared with the stationary cylinder counterpart. For all the cases of the isolated cylinder shown in Fig. 3.55, two single vortices (2S) are shed per one time period. The vortex is stretched in the streamwise direction at Ur = 4 and in the transverse direction for the remaining cases. The in-line distance between the two consecutive shed vortices is the smallest for the reduced velocity Ur = 4 and the clock-wise vortices are distinctly separated from the anti-clockwise vortices by the horizontal mean line. The maximum and minimum values of C L p are observed at the mean line at Ur = 6 as shown in Fig. 3.55a–c, whereas for Ur = 4 they are observed at the maximum and minimum values of displacement of cylinder as shown in Fig. 3.55d–f. The inertia forces dominate over the stiffness forces for Ur = 30 and therefore the minimum C L p occurs at the maximum displacement of cylinder and vice-versa as seen through Fig. 3.55j–l. The vorticity regions due to reverse flow on the aft part of the cylinder −π/2 ≤ θ ≤ π/2 are present throughout the time period for the stationary cylinder and Ur = 30. There is no reverse flow vorticity region observed for Ur = 4, where only one separation point and one reattachment point exist similar to the observations made for the stationary downstream tandem cylinder in [390]. As discussed in the previous subsection, there is an intermittent presence of the reverse flow vorticity regions for Ur = 6. It can be observed through the contour plots that the stagnation point movement along the circumference of the cylinder with θ = π as the mean position is nearly in-phase with the velocity. As the stagnation point movement, as well as the transverse load, are in-phase with the velocity at the reduced velocity of Ur = 6, the stagnation point moves to the high suction region at C L p,max and C L p,min . The shed vortex from the upstream cylinder interacts with the boundary layer of the downstream cylinder, affecting the flow features along the circumference of the cylinder as shown in Fig. 3.56. The upstream vortex, after interaction with the boundary layer of the downstream cylinder coalesces with the shed vortex to form a larger vortex. Thus, after the downstream cylinder, there exists only two single vortices which are being shed per one time period. The vortex after the downstream cylinder is stretched in the streamwise direction at Ur = 30 and in the transverse direction for the remaining cases. At the maximum lift due to pressure, a clockwise vortex from the upstream cylinder interacts with the boundary layer in the high suction region 0 ≤ θ ≤ π of the downstream cylinder, except for Ur = 6. There exists only one separation point and one reattachment point on the downstream cylinder of the stationary case and Ur = 4 similar to the vibrating isolated cylinder at Ur = 4. When C L p is zero for Ur = 4, the downstream cylinder is approximately at the mean position and the upstream vortex has reached the downstream cylinder. Further in time, the downstream cylinder moves towards the convecting upstream vortex at Ur = 4, increasing the interaction with the convecting upstream vortex. We define this case as a converging interaction, where the velocity leads C L p by approximately

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3 Proximity and Wake Interference

Fig. 3.56 Wake contours of tandem cylinder setup, stationary case (a–c), Ur = 4 (d–f), Ur = 6 (g–i), Ur = 30 (j–l): first column—minimum C L p , second column—C L p ≈ 0, third column— maximum C L p

3.7 Wake Interference of Tandem Circular Cylinder

161

Fig. 3.57 Close-up view of the instantaneous flow for the tandem downstream cylinder at Ur = 30: a streamlines and b vorticity contours

π/2. For the converging interaction case, there is an increase in suction pressure in the high suction region compared to the stationary isolated cylinder. The stagnation point is moved to low suction region for the downstream stationary cylinder as well as for Ur = 4 at the maximum and minimum lift due to pressure. At Ur = 6 where the velocity is near in-phase with C L p , the downstream cylinder is found to be at minimum displacement by the time the upstream clockwise vortex reaches the downstream cylinder center. The interaction of the downstream cylinder with the upstream vortex is limited as shown in Fig. 3.56a–c. Subsequently, the location of the stagnation point at the maximum C L p of the downstream cylinder is similar to that of the isolated cylinder at Ur = 6. For the case Ur = 30, the downstream cylinder is nearly at the mean position when C L p is zero and the convecting upstream vortex has reached near the downstream cylinder as shown in Fig. 3.56. Further in time, the downstream cylinder moves away from the convecting upstream vortex, which we define as a diverging interaction. The diverging interaction is characterized by a decrease in suction pressure in the high suction region compared to a stationary isolated cylinder case. The stagnation point is moved to the low suction region and the amplitude of the stagnation point is lesser than that of Ur = 4. The vorticity regions due to the reverse flow are present on the aft part of the downstream tandem cylinder throughout at Ur = 30 as shown in Fig. 3.56j–l. A close view of the flow over the downstream cylinder at Ur = 30 is shown in Fig. 3.57, where the corresponding vorticity contours is shown in Fig. 3.56. The recirculation blob due to reverse flow on the aft part of the cylinder can be seen in Fig. 3.56a.

3.7.5 Upstream Vortex and Downstream Boundary Layer We next investigate the interaction of vortices from the upstream cylinder colliding with the transversely vibrating downstream cylinder. Continuous vorticity shed from the upstream cylinder impinges on the downstream cylinder can influence boundary layer development and consequently unsteady force dynamics of the downstream cylinder. This interference can in turn alter the pressure force induced through the

162 Fig. 3.58 Upstream vortex and the movement of stagnation point: a diagram showing the interaction of vortex with the stagnation point, b instantaneous streamline and vorticity contours

3 Proximity and Wake Interference

upstream vortex

downstream cylinder HS S1 S2 LS

complex mechanism between the cylinder motion and approaching vortices. This mechanism can be further illustrated through the movement of the stagnation point. The approaching upstream vortex as shown in Fig. 3.58 displaces the stagnation point from the initial point S1 to the new point S2 which is nearer to the low suction pressure L S. Since the fluid particle has to travel a lesser distance for the case S2 − L S, the flow has to accelerate as a contrast to S1 − L S which results in a higher pressure (i.e., lower suction force) at L S with respect to the isolated cylinder counterpart. This process is illustrated through the diagram in Fig. 3.54. We now consider the interaction of the approaching vortex with the downstream cylinder motion. An illustration of converging and diverging interactions between approaching vortex and cylinder motion is shown in Fig. 3.59. The velocity u(y) at streamwise distance x = x0 in the wake can be represented approximately as u(y) = U∞ + u (y),

(3.20)

where u vanishes at infinity [225]. The boundary layer on the top of downstream cylinder B due to U∞ is U (y) as shown in Fig. 3.59. In the case of converging interaction, the cylinder below the mean line travels towards the clockwise vortex as time progresses. The point B of cylinder is above A A as shown in Fig. 3.59a by the time center of the clockwise vortex reaches point B. The resulting boundary layer at point B on the cylinder is approximately the summation of the U¯ (y) and the boundary layer due to u (y). The point B is above A A for the converging interaction Ur = 4 as well as for the stationary case. At Ur = 30 as the cylinder moves away from the upstream vortex, the point B of the downstream cylinder lies below A A by the time the center of the upstream vortex has reached point B as shown in Fig. 3.59b. The flow velocity u (y) which is negative below A A interacts with the boundary layer U (y) at point B on the cylinder results in a lesser favorable pressure gradient for Ur = 30 as shown in Fig. 3.59c compared to Ur = 4 and stationary case. This lesser favorable pressure gradient leads to a lower suction pressure at the high suction side for Ur = 30 as compared to the stationary and vibrating cylinder at Ur = 4, which is illustrated in Fig. 3.54b.

3.7 Wake Interference of Tandem Circular Cylinder

(a)

163

(b) 

U (y) B

A

A

A

U  (y) A B

u (y)

u (y)

2

downstream, stationary downstream, Ur = 4 downstream, Ur = 30

normal distance

1.5

1

0.5

0

0

0.5

1

1.5

tangential velocity

(c) Fig. 3.59 Upstream vortex and the boundary layer of vibrating downstream cylinder: a converging interaction, b diverging interaction, c boundary layer profiles for three different reduced velocities

To summarize, the upstream vortex displaces the boundary layer of the downstream cylinder, thus shifting the stagnation point towards the low suction region. Consequently, the shift of stagnation point causes the increment of viscous contribution to C L as the viscous component acts in the direction of C L at θ = π . The maximum pressure at the stagnation point in the low suction region contributes to the lift force. The shifting of the stagnation point to the low suction region causes a further reduction in suction pressure at the low suction region. The suction pressure at the high suction region for the diverging interaction is lower compared to the converging interaction and the stationary case of tandem arrangement. As shown earlier, the phase difference between the pressure force and the velocity indicates the underlying feedback process of the upstream wake interaction with the downstream cylinder and also the forces acting on the cylinder. Such continuous interactions of the upstream vortex with the downstream cylinder lead to larger transverse forces and larger periodic vibrations in the post-lock-in region than the isolated cylinder

164

3 Proximity and Wake Interference

counterpart. It will be worth investigating the development of a boundary layer on the circumference of the cylinder with respect to the phase relation between the forces and velocity as well as the interaction with the upstream wake. In the next subsection, we extend our parametric study to investigate the effects of streamwise gap x0 on the transverse force and response amplitudes of the downstream cylinder.

3.7.6 Effect of Streamwise Gap Due to viscous effects, the strength of the vortex weakens as it convects further downstream. Of interest in this subsection is to report the effect of streamwise gap x0 on the dynamical response of the downstream cylinder at Re = 100. The transverse displacement amplitude and C L ,r ms variation with respect to the reduced velocity for five streamwise gaps x0 = 4D, 5D, 6D, 8D, 10D are shown in Fig. 3.60. The transverse displacement amplitude and C L ,r ms are plotted against the streamwise gap x0 for reduced velocities 4, 6, 30 in Fig. 3.61. For the streamwise gap x0 ≤ 4D, the displacement amplitude is negligible in the pre-lock-in region Ur ≤ 5 which is reported in [77]. In the lock-in and post-lock-in regions Ur > 5 the amplitudes at x0 ≤ 4D are considerable. As the streamwise gap increases beyond the critical distance 4D, the displacement amplitude tends to recover the isolated cylinder counterpart. A similar response is observed for the transverse lift force C L ,r ms . In the pre-lock-in region for the streamwise gap less than critical distance x0 < xcr ≈ 4D, the upstream vortex attaches to the downstream cylinder suppressing the periodic vortex shedding in both the cylinders as shown in Fig. 3.62a. For the coshedding regime x0 > 4D, we observe a sudden jump in the amplitude response for Ur in the pre-lock-in region as shown in Fig. 3.61. There is no upstream vortex attachment with the downstream cylinder in the lock-in and post-lock-in regions at x0 = 4D as shown in Fig. 3.62b, c. In the lock-in region, a small amplitude of force results in large displacements as the shed frequency locks with the natural frequency of the cylinder, which leads to the upstream and downstream vortex shedding as shown in Fig. 3.62b. In the post-lock-in region the velocity of the cylinder, as well as the stagnation point displacement, lags the transverse force by π/2. We suspect the phase difference between the transverse load and velocity plays a major role in the vortex motion and the vorticity shedding process for the streamwise gap x0 ≤ 4D at Ur = 30. In the post-lock-in region, as the streamwise gap increases the displacement amplitude decreases linearly from x0 > 5D onwards. Almost a similar trend is found for the rms transverse lift C L ,r ms . There is no particular trend observed for the effect of the streamwise gap on the dynamic response in the lock-in region as shown in Fig. 3.61.

3.7 Wake Interference of Tandem Circular Cylinder 1

2 single cylinder downstream cylinder - 4D downstream cylinder - 5D downstream cylinder - 6D downstream cylinder - 8D downstream cylinder - 10D

0.9 0.8 0.7 0.6 0.5

single cylinder downstream cylinder - 4D downstream cylinder - 5D downstream cylinder - 6D downstream cylinder - 8D downstream cylinder - 10D

1.5

C L, rms

A ymax, rms

165

1

0.4 0.3 0.5 0.2 0.1 0

0 0

5

10

15

20

25

30

0

5

10

reduced velocity, U r

15

20

25

30

reduced velocity, U r

(a)

(b)

Fig. 3.60 Effect of streamwise gap x0 on the dynamic response of downstream cylinder a transverse displacement amplitude, b transverse force C L ,r ms 1.5

1 0.9 0.8

0.5

Ur = 4 Ur = 6 Ur = 30

1

Ur = 4 Ur = 6 Ur = 30

0.6

CL, rms

A ymax, rms

0.7

0.4 0.5

0.3 0.2 0.1

0

0 2

4

6

streamwise gap x 0

(a)

8

10

2

4

6

8

10

streamwise gap x 0

(b)

Fig. 3.61 Effect of streamwise gap x0 on the dynamic response of downstream cylinder a transverse displacment amplitude, b transverse force C L ,r ms

3.7.7 Interim Summary This work is concerned with the modeling of the interaction of fluid flow with isolated and tandem circular cylinders for a range of reduced velocity Ur . An implicit Petrov-Galerkin technique has been employed for fluid-rigid body interaction problems. Due to the wake interaction from the upstream cylinder, elastically-mounted downstream cylinder stationed at center-to-center distance x0 = 5D experiences a larger transverse force C L . It is known that the frequency f / f n of C L deviates from the Strouhal frequency in the lock-in region for a vibrating isolated cylinder arrange-

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3 Proximity and Wake Interference

Fig. 3.62 Instantaneous wake contours of tandem cylinder setup at x0 = 4D for a Ur = 4, b Ur = 6, c Ur = 30

ment. The upstream cylinder is stationary and the shedding frequency is constant which is not affected by the oscillatory motion of downstream cylinder. The oscillatory upstream wake interacting with the downstream cylinder enforces the shedding frequency on the downstream cylinder as the primary frequency content in C L . The major component of the frequency f / f n in the transverse load of downstream cylinder varies linearly with the reduced velocity. There is no low frequency component present in C L of the downstream tandem cylinder, contrary to the observations made in experimental studies at Reynolds number greater than 5 × 103 . The appearance and disappearance of vorticity regions due to reverse flow are present in the isolated

3.7 Wake Interference of Tandem Circular Cylinder

167

as well as the tandem downstream case where the large displacements are observed, characterized by the high frequency component in C L which is three times that of the shedding frequency. The phase difference between the force components and the structural response plays a major role in the coupled behavior of both cylinder arrangements. The phase difference between transverse pressure load and the velocity lies between −π/2 and π/2, thus feeds energy into the vibration. Whereas, the phase difference between the transverse viscous load and the velocity lies between π/2 and 3π/2, which dissipates the energy. The viscous forces are inherently large for an vibrating cylinder near the excitation ratio of unity compared to the pressure forces, thus proving a self-limiting response. The ratio C Lμ,r ms /C L p,r ms is maximum when the frequency ratio is unity. The stagnation point movement with respect to θ = π is nearly in-phase with velocity for the isolated cylinder, which has been confirmed through the vorticity contours. If the velocity is in-phase with C L p , the stagnation point is on the side of high suction region opposing the transverse force due to suction for the isolated cylinder setup. Thus low C L p is observed for the excitation frequency ratio of unity, increasing the ratio C Lμ,r ms /C L p,r ms . If the velocity either leads or lags C L p by π/2, the stagnation pressure does not contribute to the transverse force at maximum lift, as the stagnation point is found to be at θ = π . The wake of the upstream cylinder interacts with the boundary layer of the downstream cylinder, which influences the flow patterns on the downstream cylinder. The interaction of the convecting upstream vortex with the downstream cylinder differs based on the phase difference between the transverse load and the cylinder motion. If the velocity leads C L p by approximately π/2 or −π/2, the downstream cylinder is nearly at the mean position by the time upstream vortex reaches the downstream cylinder. Further in time the downstream cylinder moves towards the upstream vortex for the velocity leading C L p by π/2, which we define as the converging interaction. Diverging interaction is the case where velocity lags C L p by π/2 in which the cylinder moves away from the approaching upstream vortex. At frequency ratio of one, where the velocity is nearly in-phase with C L p , the downstream cylinder is at the maximum or minimum position and on the other side of mean line by the time upstream vortex reaches the downstream cylinder. Thus, the interaction of the upstream vortex is minimal in this case. The upstream wake interaction with the downstream cylinder leads to: (i) the shifting of the stagnation point to the low suction region; (ii) decrease in the suction pressure in the low suction region of the cylinder compared to isolated stationary cylinder; and (iii) high (low) suction pressure in the high suction region of the cylinder based on the converging (diverging) interaction. Such phenomena associated with vortex-body interactions lead to larger transverse force and transverse vibrations of the downstream tandem cylinder in the post-lock-in region. Further investigations are underway in three-dimensions for studying the origin of wake-induced vibrations and force decompositions at sub-critical Re turbulent flows.

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3 Proximity and Wake Interference

3.8 Three-Dimensional Wake Interference of Tandem Cylinders Riser pipelines, heat exchangers, and transmission lines generally operate in groups, which can result in nonlinear dynamical motions due to unsteady flow interaction with the structures. These complex motions significantly affect the system performance and fatigue life of the constituent structure. In particular when riser pipelines are exposed to high-speed ocean current, they interact in a very complex manner due to vortex-induced vibration (VIV), wake-induced vibration (WIV), jet switching, turbulent buffeting and fluid-elastic instability. Apart from the significance in various engineering applications, such flow interference phenomena and associated flowinduced vibrations are of fundamental importance in fluid mechanics. Due to complex wake-body interactions, there is a considerable difference between the fluid-structure interaction of an isolated cylinder arrangement and multiple cylinder arrangements. Although in the last few decades, a vast body of works on flow-induced vibrations is reviewed by many authors [40, 328, 381, 446, 455], it is still challenging for engineers to determine a range of optimal parameters and conditions to avoid flowinduced vibrations and collisions, owing to the complexities pertaining to fluidstructure interaction and operating flow environment. To understand the flow-induced vibration of an array of flexible bluff bodies, the flow around multiple elastically mounted cylinders can be considered as an idealized model, which serves as a generic problem to analyze the interactions of bluff bodies. More specifically, we consider a pair of two cylinders in a tandem arrangement to understand the upstream vortical wake excitation mechanism from a fundamental viewpoint [28]. It is well known that the dynamic response of an isolated cylinder due to vortex-induced excitation is characterized by a vortex shedding frequency via a resonant phenomenon [455], whereas the downstream tandem cylinder exhibits a low frequency and high amplitude vibration during the interaction with vortices coming from the upstream cylinder [22, 28, 58, 159]. This subharmonic low-frequency dominant motion is an intrinsic action of turbulent von Kármán vortex street for the Reynolds number approximately greater than 2000 [22, 28, 58, 307]. At higher reduced velocity, the such subharmonic frequency appears due to the self-excited nonlinear behavior of elastically mounted downstream cylinder. The physical complexity of WIV excitation lies in the nonlinear interaction dynamics of the convecting upstream von Kármán vortex with the downstream cylinder surface undergoing complex motion. Asymmetric oncoming wake vortices generate unbalanced loads, which can trigger the self-induced instability of an elastically mounted downstream cylinder in the post-lock-in region. In the past literature, there have been considerable experimental works reported about the vortical wake interfering motion of downstream cylinder for a prototypical tandem arrangement of two identical circular cylinders. In the majority of studies, the upstream cylinder is kept stationary and the downstream cylinder is mounted on a linear spring and is allowed to move in a transverse direction only, as shown in Fig. 3.63. One of the pioneering experimental studies by [58] reported the response

3.8 Three-Dimensional Wake Interference of Tandem Cylinders

U

169

D

upstream

downstream

Lx

θ

m y

k

c

x Fig. 3.63 Schematic of tandem cylinder arrangement in steady incident flow. Here the downstream cylinder with mass m is mounted on linear spring-damper with constant stiffness k and damping c. Center of upstream cylinder is the origin of Cartesian coordinate system. The angle θ along the downstream cylinder surface is measured from the back of the downstream cylinder and the forward stagnation point is located at θstg = π for the stationary configuration

of the downstream cylinder for a streamwise gap of L x ≤ 3D. The downstream cylinder was found to experience a relatively large dynamic response in the postlock-in region (i.e., high reduced velocities), as compared to the isolated cylinder arrangement. Due to similarity with the galloping response of a sharp cornered square cylinder, this peculiar behavior of downstream tandem cylinder was termed as wakeinduced galloping by [58]. The critical streamwise gap L xcr , where the onset of the co-shedding regime for stationary tandem cylinders, occurs somewhere between 3.5D to 5D for a wide range of Reynolds numbers as reported both numerically and experimentally [135, 306, 463, 500]. For a streamwise gap of L x = 4.7D in the co-shedding regime, the vibrational characteristics of downstream cylinder response were investigated experimentally by [159]. At higher reduced velocities in the postlock-in region, the response of the downstream cylinder was characterized by the combination of vortex shedding frequency and a lower frequency, whereby the lower frequency became closer to the natural frequency of the downstream cylinder. Through a series of physical experiments for a pair of cylinders in a tandem arrangement, Assi et al. [28] provided a detailed mechanism of vortical wake-induced excitation of the downstream the cylinder at subcritical Reynolds number flow. The authors suggested that a build of the amplitude of downstream cylinder for high reduced velocities is due to the unsteady vortex-structure interaction between the body and the vortical wake. In the absence of unsteady vortices, the authors demonstrated that the steady shear flow has a similar effect to that of uniform flow-inducing vortex-induced vibration for an isolated cylinder case. Specifically, the upstream von Kármán vortex, which has rotational velocity and induces the high-speed velocity in the gap region is the key to the sustained wake-induced vibration of the downstream cylinder. Recently, Assi et al. [22] proposed a concept of wake stiffness based on the experimental results of tandem cylinder arrangement for L x ≥ 4D and for Re  2000. The characteristic wake stiffness concept has been deduced based on the response of the downstream cylinder with and without elastically mounted spring, where the restoration force was extracted from the stiffness-like behavior of wake region. In another recent study of [25], the wake effect on the downstream cylin-

170

3 Proximity and Wake Interference

der was found to decrease with an increase in the lateral distance between a pair of two cylinders. Similar numerical and experimental studies were conducted on two flexible riser sections in a tandem arrangement [165, 397]. While the importance of stagnation point on the coupled dynamics of tandem cylinders has been suggested by [77], the origin of WIV response of the downstream cylinder for various streamwise gaps in a laminar flow has been systematically investigated by [307]. The numerical study of [307] attempted to provide a detailed local behavior of vortical wake-induced vibration and the interaction dynamics of upstream von Kármán vortex with the downstream cylinder for low Reynolds number laminar flow. Of particular interest was the stagnation point movement, which played a major role in the transverse load and vibrational characteristics of the downstream cylinder. The forward stagnation point on the cylinder surface is where the local velocity of the fluid approaches zero (i.e., maximum pressure) and the oncoming flow divides into upward and downward boundary layers along the cylinder. Due to the disturbance of the forward stagnation point by the oncoming vortex interaction, the boundary layer on the downstream cylinder will lead to different amounts of vorticity into the associated shear layers and the vortex rolls. It has been found that the upstream wake upon interaction with the downstream cylinder displaces the boundary shear layer on the downstream cylinder and significantly alters the generation and the composition of transverse force in a laminar WIV configuration [307]. The coupled WIV response at moderate Reynolds number with the turbulent wake [22, 58, 159] is different from the laminar flow [307], where the low-frequency component in the displacement response of downstream tandem cylinder is absent in the laminar flow. In the present section, the upstream von Kármán vortex interaction with the downstream cylinder is analyzed numerically at subcritical Reynolds numbers 5000 ≤ Re ≤ 10,000. The physical setup for the elastically mounted downstream cylinder is identical to the experimental configuration of [28]. The effective mass of the oscillating downstream cylinder gives a mass ratio (the oscillating mass of the cylinder divided by the mass of displaced water) m ∗ = 2.6 and the structural damping of the system is set to ζ = 0.7 %. We employ the recently developed partitioned iterative procedure to couple the filtered Navier-Stokes solver with elastically mounted structure [181]. The coupled fields are advanced explicitly using the bodyconforming treatment of fluid-solid interface and the interface force correction is constructed at the end of each fluid subiteration [181] and the coupled scheme can handle strong added-mass effects associated with the low mass-damping parameter. We consider an explicit large eddy simulation (LES) filtering technique, which provides a mechanism of resolving large-scale flow features while modeling subgridscale (SGS) stresses using the Smagorinsky-based functional modeling [180]. The filtering technique aims at modeling only the impact of the subgrid scales on the evolution of the resolve scales within the filtered Navier-Stokes equations [371]. For representative Reynolds numbers and the reduced velocities in the post-lock-in region, the developed dynamic LES scheme is validated against the measured WIV response of [28]. This numerical study aims to provide a detailed investigation of the unsteady flow fields and vibrational characteristics of the downstream cylinder subjected to

3.9 Results and Discussion

171

the upstream wake interference. Similar to the recent experimental study of [28], we consider a representative longitudinal separation L x /D = 4.0 in the co-shedding regime to investigate the WIV excitation mechanism via numerical computations. This separation distance allows the full development of vortices between the cylinders and provides the largest amplitudes of vibration that increase as a function of reduced velocity. Coupled dynamics of unsteady vortex-structure interaction, the motion of the downstream cylinder, the instantaneous energy transfer and the movement of the boundary layer along the downstream cylinder is investigated during the oscillation. Through the enforced periodic the motion of the cylinder in the transverse direction, we study the role of stagnation point movement over an isolated cylinder and contrast the response with the downstream cylinder in the tandem arrangement. We deduce that the WIV excitation of the downstream cylinder is directly related to the local interaction between the upstream von Kármán vortex and the stagnation point over the front surface of the body. From the time evolution of instantaneous energy transfer and the stagnation point movement, we show that the local unsteady movement of the stagnation point opens up the door to the energy supply from the surrounding fluid flow to the downstream body. A fundamental understanding of interaction dynamics between the upstream vortex and the downstream body may guide to develop effective suppression devices for preventing the development of self-induced vibrations due to the wake excitation.

3.9 Results and Discussion To analyze the local characteristics of unsteady vortex-structure interaction, we assess the dynamic response and the movement of forward stagnation point at the longitudinal distance L x = 4D corresponding to the co-shedding regime. Consistent with the experimental setup of [28], we consider the Reynolds number Re = 10,000 and the mass damping parameter m ∗ ζ = 0.018 for the fundamental analysis of upstream Kármán vortex interaction with the vibrating downstream cylinder. As discussed earlier, two regimes can be observed in Fig. 3.79, indicated by the difference in slopes of the displacement curve. One of them is the VIV resonance regime (4.0 ≤ Ur < 7.0), where the VIV phenomenon is dominant; the other regime is the combined VIV and WIV regime (7.0 ≤ Ur ≤ 14.0), where the WIV with the lower branch and postlock-in phenomenon is dominant. While we choose Ur = 5.0 as the representative case for the VIV regime, Ur = 14.0 is considered as the representative case for the combined WIV regime.

3.9.1 VIV Dominated Response In this regime, the vibration of the downstream cylinder is dominated by the VIV resonance/lock-in behavior of elastically mounted cylinder. Lock-in is characterized

3 Proximity and Wake Interference CL y/D

Lift coe/ cient, CL and displacement, y/D

1.5 1 0.5 0 -0.5 -1 -1.5 210

220

230

240

250

Power spectra of lift force |CL (t)|

172 0.6 0.5 0.4 0.3 0.2 0.1 0

2

4

6

Time, tU/D

Frequency ratio f /fN

(a)

(b)

8

10

Fig. 3.64 WIV response of downstream cylinder in the VIV upper branch at Re = 3, 700, Ur = 5, m ∗ ζ = 0.018: a temporal variation of transverse force C L and amplitude y/D, b frequency spectra in the transverse force, demonstrating the presence of low harmonic frequency, flow = f 1 / f N = 0.8057

by the matching of the vortex shedding frequency f vs with the oscillation frequency of the downstream cylinder. This lock-in phenomenon leads to a large amplitude of oscillations of the elastically mounted cylinder. In the VIV-dominated regime, the underlying phenomenon and the response curve of the vibrating downstream cylinder are similar to those of a freely vibrating isolated cylinder in the transverse direction. We choose Ur = 5.0 as a representative case to investigate the response in this VIVdominated regime. Figure 3.64 shows the temporal variation of transverse force and amplitude, as well as the frequency of the transverse force. It can be observed in Fig. 3.64a that the transverse force and amplitude are in-phase. This observation is consistent with the finding of experiment [28]. As shown in Fig. 3.64b, there is only one frequency observed in the transverse force. This observed frequency is induced by the vortex shedding from the downstream cylinder itself and it synchronizes with the oscillation frequency. The oscillation frequency is quite close to the natural frequency of a cylinder in a vacuum. Therefore, this regime of vibrating downstream cylinder is dominated by the lock-in process associated with the matching of vortex shedding frequency and cylinder oscillation frequency. We next turn our attention to the combined VIV and WIV regime, which is the central focus of the present study.

3.9.2 WIV Dominated Response To investigate the underlying dynamics of WIV, we attempt to establish a relationship between the movement of forward stagnation point with the force dynamics and the energy transfer during the interaction of upstream Kármán vortex with the vibrating downstream cylinder. While we consider a representative Ur = 14.0, the presented

3.9 Results and Discussion

173

findings on the stagnation point movement and low-frequency characteristics are applicable to other cases in the WIV-dominated regime. To start with, we first explore the frequency relations via temporal variations of the transverse displacement and the lift force of the downstream cylinder.

3.9.2.1

Frequency Characteristics of Transverse Displacement and Lift Force

It is known that the dynamical complexity of the coupled physical phenomena involved in a vibrating isolated cylinder increases dramatically due to flow interference when two cylinders are placed in a tandem arrangement. As shown in experiments [28], a typical wake excitation is characterized by an increasingly build up of amplitude for high reduced velocities, whereby vortex-induced excitation occurs in the lock-in range and has a self-limited behavior. Coherent strong vortices generated from the upstream cylinder interfere with the downstream cylinder, which gives rise to the complex interaction of the oncoming vortex with the boundary and the merging of vortices. These events cause considerable changes in the magnitude and spectral content of the transverse force. To quantify the lift and the amplitude response from our numerical analysis, Fig. 3.65 shows the time traces of transverse force and amplitude together with the frequency content of the transverse force. The displacement response is primarily dominated by the single frequency while there exists two dominant frequencies in the transverse load, as shown in Fig. 3.65a. The low-frequency component flow of the transverse load is dominant in the displacement response, as it is near to the natural frequency f N of the cylinder. The high-frequency component is associated with the vortex shedding frequencies of both cylinders. The subharmonic low frequency component flow is nearly 1/2 of the vortex shedding frequency f vs of the upstream cylinder. The oscillation frequency in the wake sustained regime is greater than the natural frequency and lower than the shedding frequency for Ur > 7. This signifies the response of downstream tandem cylinder to be inertia dominant for the higher reduced velocity. As suggested by [28], we confirm in our numerical study that the WIV response is not vortex synchronization; instead it is a synthesis of the interaction of upstream vortices with the self-excited system of an elastically mounted downstream cylinder. Furthermore, continuous impingement of upstream vortices on the wall boundary layer of the downstream cylinder induces additional vorticity into the shear layers, which contributes to the subharmonic low-frequency component in the transverse load. We next show the process of upstream Kármán vortex interaction with the downstream cylinder in relation to the energy transfer and the sustained low-frequency component.

3 Proximity and Wake Interference

1.5

0.5 CL y/D

Power spectra of lift force |CL(t)|

Lift coefficient, CL and displacement, y/D

174

1 0.5 0 -0.5 -1 -1.5 200

0.4

0.3

0.2

0.1

0 210

220

230

Time, tU/D

(a)

240

250

0

1

2

3

4

5

6

7

8

9

10

Frequency ratio, f/fN

(b)

Fig. 3.65 Response of downstream cylinder in the WIV dominated regime at Re = 10,000, Ur = 14, m ∗ ζ = 0.018: a temporal variation of transverse force C L and amplitude y/D, b frequency spectra in the transverse force, demonstrating the presence of low harmonic frequency, flow = f 1 / f N = 1.23, f vs = f 2 / f N = 2.425

3.9.2.2

Interaction of Upstream Vortex with Vibrating Downstream Cylinder

As expected, the load exerted on the downstream cylinder is considerably affected by the interaction of the upstream wake with von Kármán vortices [7, 307, 390]. From the potential theory [288], the global dynamical effects of surrounding fluid flow over a rigid body can be carried out using the complex integral theorem of Blasius. From a local dynamics viewpoint, the interaction of upstream von Kármán vortex with the downstream cylinder causes the boundary layer to shift to a new position based on the induced velocity vector of the oncoming vortex. The clockwise von Kármán vortex displaces the forward stagnation point to the downward direction along the downstream cylinder and vice versa. Such vortex-induced disturbance in the stagnation region can significantly affect the boundary layer development and the vorticity generation process. The displacement of the forward stagnation point, in turn boundary layer development, affects the transverse load and will be investigated in detail with respect to the large motion of downstream cylinder. To deduce the role of stagnation point movement, we briefly summarize the flow past an isolated cylinder with respect to the angle of attack (α) in freestream flow. When an isolated cylinder is located in a stream of fluid, the flow has to divide near the front stagnation point and pass along the upper and the lower surfaces of the cylinder. The position of the stagnation point moves with the angle of attack and the streamline at a stagnation point is perpendicular to the surface of the cylinder. For an isolated cylinder, which is axisymmetric, the mean lift is zero [151] thereby the resultant mean force acts in the direction of freestream velocity. The angle of attack α for a

3.9 Results and Discussion

veloctiy of cylinder stagnation point location

0.5

stg in

radians

1

Velocity, u/U and

Fig. 3.66 Representative prescribed periodic motion results of an isolated cylinder in freestream flow at Re = 10,000: time traces of the cylinder velocity and the stagnation point location with the prescribed forcing frequency f p D/U = 0.09 and the amplitude of cylinder velocity u max /U = 1

175

0

-0.5

-1 100

102

104

106

108

110

Time, tU/D

transversely oscillating cylinder has a negative relation through the inverse tangent of the ratio of the time-dependent cylinder velocity u c (t) to the freestream velocity, i.e. α = tan−1 (−u c /U ). Thus the variation of α with respect to time for an oscillating cylinder is reflected during the motion of the boundary layer around the cylinder. The resultant movement of the boundary layer around the top and bottom surfaces of the cylinder can be equivalently represented by the motion of the forward stagnation point along the front surface of the cylinder. Figure 3.66 shows the movement of the forward stagnation point on the cylinder at Re = 10,000 together with the velocity of cylinder u c . Here, the cylinder is forced to move transversely with a prescribed periodic motion as y p (t) = A sin(2π f p t), where f p is the prescribed forcing frequency which is set to f p = 0.09 and A denotes the maximum displacement amplitude with a value equal to U/(2π f p ). As expected, the stagnation point moves in-phase with the velocity of cylinder [307]. The pressure force close to the stagnation point on the isolated cylinder always acts opposite to the velocity, thus opposing the motion of the cylinder. For the tandem cylinder setup, due to the interaction of the upstream von Kármán vortex with the downstream cylinder, the stagnation point movement is not in-phase with the cylinder velocity, as shown in Fig. 3.67. There is a remarkable change in the stagnation point movement, which clearly manifests the interaction of upstream von Kármán vortex with the downstream cylinder. Due to this movement of the singular point over the front surface of the downstream cylinder, there is a positive transfer of energy to the vibrating cylinder in the stagnation region (i.e., maximum pressure region), as shown in Fig. 3.68. In other words, this energy transfer via the interaction upstream von Kármán vortex with the front surface of downstream cylinder sustains the oscillation. The unsteady movement of the stagnation point provides an avenue for the energy transfer from the surrounding fluid flow. The region where the stagnation point suddenly changes, the

176 4.5

velocity stagnation point location maximum pressure

stg

in radians, and pressure

4

Velocity,

Fig. 3.67 WIV response of downstream cylinder at Re = 10,000 and m ∗ ζ = 0.018: temporal variation of cylinder velocity u¯ c , stagnation point movement θstag and maximum pressure C p,max . The stagnation point location and the maximum pressure are evaluated via spanwise averaging along the downstream cylinder

3 Proximity and Wake Interference

3.5 3 2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5 200

210

220

230

240

250

Time, tU/D 1

instantaneous energy stagnation point location on cylinder

0

stg

in radians

0.5

Energy and

Fig. 3.68 Temporal variation of instantaneous energy transfer C E and the spanwise average stagnation point location θstg on the downstream cylinder at Re = 10,000, Ur = 14 and m ∗ ζ = 0.018. The stagnation point movement θstg and the maximum pressure C p,max represent a spanwise averaged value

-0.5

-1

-1.5 200

210

220

230

240

250

Time, tU/D

stagnation pressure is found to be low. This can be attributed to the decrease in the streamline velocity due to the continuous impingement of von Kármán vortices. In one of the physical experiments of [28], if the von Kármán vortices are removed from the steady wake flow of the upstream body then there is no WIV excitation. Thus the upstream von Kármán vortex is central to sustain the low-frequency oscillation of the downstream cylinder and the movement of the stagnation point plays a role as an indicator for the vortex interaction with the downstream cylinder and the associated energy transfer process. We next provide a link to the stagnation point movement to the pressure and the spanwise vorticity distributions.

3.9 Results and Discussion

3.9.2.3

177

Evolution of Spanwise Vorticity and Pressure Distribution

The local analysis of the pressure distribution on the cylinder along with the vorticity contours will help in understanding the nature of forces acting on the cylinder, which sustains the low frequency and the large displacement response. To understand the upstream von Kármán vortex wake interaction with respect to the energy transfer, the evolution of vorticity contours and the pressure distribution on the cylinder at equal intervals in the non-dimensional time range tU/D = 222–233.25 are plotted in Figs. 3.69 and 3.70. The time range considered approximately equals to one time period T of oscillation. In Fig. 3.69, the downstream cylinder moves from the maximum position to the minimum position. In the process, two upstream shed vortices interact with the downstream cylinder, firstly the clockwise von Kármán vortex and later the anti-clockwise vortex. Corresponding to the vorticity contours illustrated in Fig. 3.69b–f, the spanwise averaged pressure distributions acting on the downstream cylinder are shown in Fig. 3.69a. Here, the stagnation point is the location of the cylinder where maximum pressure C p,max occurs. In contrast to the isolated single cylinder where the stagnation point lies at θ = π at the maximum displacement as shown in Fig. 3.66, the stagnation point is located at 0 < θ < π (upper part of downstream cylinder), as shown in Fig. 3.69a, b. As it moves downwards, the interaction of upstream clockwise vortex shifts the stagnation point to the region π < θ < 2π (lower part of downstream cylinder), as shown in Fig. 3.69c. In the process of von Kármán vortex interaction, the stagnation pressure C p,max on the downstream cylinder is lessened. As the clockwise vortex convects away, the upstream counter-clockwise vortex reaches the downstream cylinder, as illustrated in Fig. 3.69d. The counter-clockwise vortex on interaction with the downstream cylinder shifts the stagnation point towards θ = π opposite to the direction of velocity, reducing its contribution to the transverse load C L which is anti-phase with the velocity of the cylinder. A further reduction in the stagnation point pressure can be seen Fig. 3.69e at 4T /10. The suction pressure in the region π < θ < 2π is larger compared to the region 0 < θ < π , causing the lift force to act in the direction of velocity. As shown in Fig. 3.67, when the stagnation point suddenly changes in the opposite direction to the velocity due to the interaction of the upstream vortex, there is positive energy supplied to the vibrating cylinder. Due to three-dimensional turbulent wake effects, there exists a clockwise vortex in the core of the counter-clockwise vortex as shown in sub-plot of Fig. 3.69f, which is absent in the laminar WIV response of tandem cylinder arrangement [307]. This reverse direction flow in the core interacts with the downstream cylinder at the minimum position, which makes the stagnation point to remain in the region π < θ < 2π . Since the WIV response is not perfectly symmetric with respect to the centerline mean position, the vortical flow structures and the pressure distributions are not identical during the upward and downward cycles. The next half cycle of the downstream cylinder is shown in Fig. 3.70, where the cylinder moves from the minimum position to the maximum position. Similar to the previous cycle, the spanwise pressure contours on the cylinder is plotted in Fig. 3.70a, which correspond to the vorticity snapshots shown in Fig. 3.70b–f. By the time the

178

3 Proximity and Wake Interference

pressure coefficient Cp

4

1/5 (T/2) 2/5 (T/2) 3/5 (T/2) 4/5 (T/2) T/2

3 2

θ

1 0 -1 -2 -3 -4

0

0.5π

1π

1.5π

θ along the circumference of the cylinder

2π

(b)

(a)

(c)

(d)

(e)

(f)

Fig. 3.69 Evolution of flow field for WIV response of downstream tandem cylinder at Re = 10,000, Ur = 14 and m ∗ ζ = 0.018 during the first half cycle in downward direction: a pressure coefficient C p distribution on the downstream cylinder from the maximum to minimum position; instantaneous spanwise vorticity snapshots along mid-plane cross-section at b t = 222 ≈ T /10, c t = 223.25 ≈ 2T /10, d t = 224.5 ≈ 3T /10, e t = 225.75 ≈ 4T /10, f t = 227 ≈ T /2. Angle θ = 0 in a represents the base location behind the downstream cylinder and the flow is from left to right in b–f

cylinder reverses its direction and starts to move in the downward direction, the clockwise von Kármán vortex whose center lies above the mean line approach the downstream cylinder. The downstream cylinder moves up without the effect of the convecting clockwise vortex, which can be seen through Fig. 3.70b–d. The suction pressure in the region 0 < θ < π is larger compared to the suction pressure in the region π < θ < 2π at time instants 7T /10 and 8T /10, as illustrated in Fig. 3.70a. This may be attributed to the favorable pressure gradient developed onto the upper

3.9 Results and Discussion

179

pressure coefficient Cp

4 6/10 T 7/10 T 8/10 T 9/10 T T

2 0 -2 -4 -6

0

0.5π

1π

1.5π

θ along the circumference of the cylinder

2π

(b)

(a)

(c)

(d)

(e)

(f)

Fig. 3.70 Evolution of flow field for WIV response of downstream tandem cylinder at Re = 10,000, Ur = 14 and m ∗ ζ = 0.018 during the second half cycle in upward direction: a C p distribution on the downstream cylinder from the minimum position to maximum position; instantaneous spanwise vorticity snapshots at b t = 228.25 ≈ 6T /10, c t = 229.5 ≈ 7T /10, d t = 230.75 ≈ 8T /10, e t = 232 ≈ 9T /10, f t = 233.25 ≈ T

side of the cylinder due to the larger velocity of the moving cylinder. The suction pressure contribution to the lift force dominates over the stagnation pressure contribution which acts in the direction of the velocity, thus supplying the energy for a major part of the second half cycle, which is reflected in Fig. 3.68. With further increase in time during the oscillation cycle, the stagnation pressure tends to dominate over the suction pressure contribution at 9T /10 and T , as shown in Fig. 3.70a, which results in a negative supply of energy to the vibrating system. This can be further confirmed in Fig. 3.68. The continuous supply of energy in both half cycles allows sustaining the

180

3 Proximity and Wake Interference

low-frequency component during the coupled response and the load generation process. A similar qualitative observation to the other cycles is obtained, as the upstream von Kármán vortex interaction with the downstream cylinder is not exactly same for the turbulent vortical wake flow. Furthermore, the aforementioned analysis is found to be valid for other reduced velocities in the WIV-dominated region.

3.9.3 Summary For the WIV response of downstream cylinder, the movement of wall boundary layer region due to the upstream von Kármán vortex interaction provides a channel to transfer the energy to the body from surrounding fluid flow. This local vortexbody interaction at the front surface, where the boundary layer starts to develop, has a global effect on the vorticity dynamics of downstream cylinder. For the forced isolated case, the fluid force near the stagnation point flow acts opposite to the velocity of the cylinder, suppressing the low-frequency component if it is excited. The movement of stagnation point can be considered as the varying angle of attack for the forced isolated cylinder. In the case of tandem cylinder arrangement, however, the upstream von Kármán vortex interacts with the downstream cylinder boundary layer and alters the in-phase relation of the stagnation point with the velocity of the cylinder. As a result, this interaction dynamics produces the larger amplitude of downstream cylinder in the post-lock-in region associated with the high reduced velocities. In the limit of a small curvature and without the upstream wake vortices, we can consider the local region of stagnation point flow of the downstream cylinder as the unsteady viscous flow impinging a plate situated in the transverse plane. According to the boundary layer theory for a periodically oscillating plate [365], we can deduce that the boundary layer flow is independent of the superimposed oscillating (periodic) flow component and the boundary layer recovers to well-known Hiemenz solution in the absence of periodic motion [365]. Since the oscillating downstream cylinder is immersed in the upstream wake, the presence of the upstream wake vortices will induce the coupling between the periodic cylinder motion and the unsteady boundary layer. Analogous to the linear forced oscillator model [284], we can consider the vibrating downstream cylinder as a simple linear mass-spring model [336] with a periodic force excitation from its own vortices and the unbalanced rotating excitation arising from the impingement von Kármán vortex from the upstream body. Stagnation point flow in the neighborhood of the front surface of an oscillating cylinder can be considered as unsteady rigid rotary motion [95] via boundary layer theory, as studied for a plate oscillating transversely in unsteady viscous flow [365]. Unbalanced rotational effects allow to trigger the self-induced vibration of elastically mounted downstream the cylinder at a higher reduced velocity in the post-lock-in region. From this analogy with the rotor excited model, the steady state solution of the spring-mass system will not decay to zero for high frequency ratio f / f N 1 due to the extra rotating effects

3.9 Results and Discussion

181

attached to the body. The response amplitude of an isolated cylinder system decays in the post-lock-in region when the frequency ratio exceeds sufficiently f / f N > 1. However, due to the continuous material rotation (i.e., von Kármán vortex) impinging on the downstream cylinder provides a large transverse force and a sustained vibration in the post-lock-in region with high reduced velocities when f / f N > 1. Due to the fact that vortices (angular momentum) are being steadily added to the downstream cylinder body and the vortex displaces the stagnation point, the resulting local and global vortex-structure interaction lead to a sustained vibration of the downstream body. As shown experimentally [28], the strength of vortex rotation and vortex-structure interaction increases as the Reynolds number of oncoming flow increases and the vibration amplitudes become larger for higher reduced velocities in the WIV-dominated region. To summarize this section, we numerically examined the interaction dynamics of the unsteady vortex generated from the upstream cylinder with the elasticallymounted downstream cylinder at m ∗ ζ = 0.018 for subcritical Reynolds number flow 5000 ≤ Re ≤ 10,000. To demonstrate the capability of the developed dynamic SGS and the coupled fluid-structure solver, we have validated the key characteristics of wake-induced interaction with the experimental data. Two response regimes namely the VIV-resonance dominated (4.0 ≤ Ur ≤ 7.0) and the combined VIV and WIV (7.0 < Ur ≤ 14.0) have been successfully predicted by numerical simulations. The response amplitudes and the frequencies in both regimes are generally captured by the variational subgrid LES using the Petrov-Galerkin based fluid-structure formulation. We confirmed the recent experimental findings that the wake-excitation mechanism is indeed sustained by the interaction of upstream vortical wake with the freely downstream cylinder. To further investigate the underlying dynamics of WIV, we examined the temporal relationship of stagnation point movement with the force dynamics and the energy transfer during the interaction of upstream Kármán vortex with the vibrating downstream cylinder at representative reduced velocity of Ur = 14. Our flow visualizations allow us to pinpoint the essential features of the upstream vortex interacting with the vibrating downstream cylinder. We have found that the large displacement of the downstream cylinder is due to the appearance of lowfrequency component in the transverse load, which is closer to the natural frequency of the downstream cylinder. It was shown that the boundary layer movement on the downstream cylinder plays a major role in sustaining the low-frequency component in the transverse load. Notably, the low frequency in the stagnation point movement was found to be explicitly related to the origin of low frequency in the transverse load. Due to a continuous upstream vortex interaction, the low-frequency component in the boundary layer of the downstream cylinder is sustained for higher reduced velocities in the post-lock-in region. The stagnation point movement is proportional to the prescribed velocity magnitude of the cylinder for the isolated cylinder in freestream flow. The low-frequency component is not sustained for the isolated cylinder as the force developed by the movement of the viscous boundary layer acts against the cylinder velocity. In the case of tandem arrangement at a sufficiently large Reynolds number, the upstream vortex breaks the proportionality relation between the bound-

182

3 Proximity and Wake Interference

ary layer movement and the cylinder velocity, thus causing the fluid force to act in phase with the cylinder velocity. This event plays an essential role in the energy input from the fluid flow to the structure and the sustaining process of the low-frequency component for a freely vibrating downstream cylinder. These dynamical characteristics of the stagnation point for the downstream tandem cylinder have been identified as a generic feature of the WIV response. Here, our results could serve as an effective guide to the develop a simplified WIV model and help to build a theoretical understanding of the vortex-body interaction. Furthermore, effective manipulation of the interaction between the upstream vortex and the elastically mounted cylinder may allow controlling the vibrational characteristics. Further research is required for a large parametric investigation for a range of distance and reduced velocities. The extension to multiple flexible bodies interacting in proximity will be of interest from a practical viewpoint. Finally, a rigorous link between the local stagnation point movement with the vortical wake dynamics should be explored theoretically in the future studies.

3.10 Appendix A: Side-by-Side Stationary Square Cylinders Here, we present some background results for the side-by-side stationary square cylinders for the two representative gap ratios of g ∗ = 1.2 and 2.0. The stationary configurations serve as the reference results for the non-lock-in scenario (pre- and post-lock-in conditions). The gap ratio g ∗ = 1.2 is in the transition region between Regime II and Regime III, which exhibits both the flip-flop effect and the coupled vortex shedding mode. The gap between the cylinders is large enough such that the deflection of the gap flow becomes weak and the two streets begin to couple with each other. The intermittent switches between two cylinders turn into a low frequency, and the flip-flop instability almost vanishes. The intermittent switching is evident in the time histories of force coefficient responses in Fig. 3.72 for the 2D and 3D cases. For the 2D simulations, the time histories exhibit the sudden change from the low values to the high values for both the cylinders at the same instants (Fig. 3.72a, b). On the other hand, for the 3D cases it is quite opposite trend. The intermittent switching for the 3D cases is not similar to the 2D cases due to the three-dimensional effects, as shown in Fig. 3.72d, e. The difference in the widths of narrow and broad streets becomes quite similar. As shown in Fig. 3.71a, two vortex streets remain coupled, while the spanwise vortex rollers are found to advect further downstream. The nearwake flow shows a clear three-dimensional character in Fig. 3.71b, c. The difference in the values of both drag and lift coefficients between the narrow streets (C¯D 1 and C Lr ms 1 ) and the broad streets (C¯D 2 and C Lr ms 2 ) are approximately 6 and 9%, respectively. The drag coefficients of the two-dimensional and the threedimensional simulations are very close and the profiles of time histories are similar with each other (Fig. 3.72). The discrepancy of lift coefficients is about 16%. This

3.10 Appendix A: Side-by-Side Stationary Square Cylinders

183

1 −1 410

420

430

440

450

−3 400

410

420

430

440

450

U pper Lower

0 −2 −4 400

410

420

430

tU/D

tU/D

(a)

(b)

(c)

CD CL

3 1 −1 410

1 −1

4 2

tU/D

5

−3 400

CD CL

3

420

430

440

450

5

CD CL

3 1 −1 −3 400

410

tU/D

420 430 tU/D

(d)

(e)

440

450

CL of Both Cylinders

−3 400

5

CL of Both Cylinders

CD CL

3

CD , CL of Lower Cylinder

5

CD , CL of Lower Cylinder

CD , CL of U pper Cylinder

CD , CL of U pper Cylinder

Fig. 3.71 Instantaneous vorticity contours for two stationary side-by-side configuration of g ∗ = 1.2 at Re = 200: a 2D span-wise vorticity, b 3D span-wise vorticity and c 3D stream-wise vorticity

4

440

450

U pper Lower

2 0 −2 −4 400

410

420 430 tU/D

440

450

(f)

Fig. 3.72 Force responses for two stationary side-by-side configuration of g ∗ = 1.2 at Re = 200. 2D (top) and 3D (bottom) simulations: a, d upper cylinder, b, e lower cylinder and c, f lift coefficients of both cylinders

difference can be attributed to the three-dimensional flow dynamics in the near wake region, which contributes to the transverse oscillating forces on the cylinders. In addition, the two-frequency mode is observed in both two-dimensional and threedimensional simulations. The low- and high-frequency effects are associated with the narrow and broad streets, respectively. As shown in Table 3.10, the two-dimensional simulation is generally found to over-predict the drag and lift coefficients and the two frequencies are underestimated by around 3% by the two-dimensional simulation. For Regime III, referred to herein as the coupled vortex shedding regime, the vortex shedding is found to be highly synchronized regardless of the in-phase or anti-phase shedding modes. The biased gap flow is completely dissolved and the two streets become nearly equivalent to each other instead of one narrow and one wide.

184

3 Proximity and Wake Interference

Table 3.10 Comparison between 2D and 3D simulations at g ∗ = 1.2 Parameter 2D 3D ¯ Avg. frequency f 1 0.145 0.149 Avg. frequency f¯2 0.154 0.157 Avg. drag coeff. C¯D 1 1.97 1.95 Avg. drag coeff. C¯D 2 1.86 1.83 Avg. drag coeff. Diff RMS. lift coeff. C Lr ms 1 RMS. lift coeff. C Lr ms 2 RMS. lift coeff. Diff

5.6% 0.84 0.76 9.5%

6.6% 0.72 0.65 9.7%

Diff (%) 2.6 1.9 −1.0 −1.6 – −16.7 −16.9 –

Fig. 3.73 Vorticity contour for two stationary side-by-side configuration of g ∗ = 2.0 at Re = 200. a 2D span-wise vorticity, b 3D span-wise vorticity and c 3D stream-wise vorticity

The vortices from the two cylinders are coupled and they appear as the in-phase or anti-phase modes in the near wake. While for the in-phase mode, the vortices emanating from the top and bottom cylinders follow the same pattern, the vortices are shed in the opposite manner for the anti-phase mode. In Fig. 3.73, the upper and lower cylinders are in the anti-phase condition, where the green vortex from the light grey cylinder and the blue vortex from the dark grey cylinder in the inner sides are paired together. However, the paired vortices from the inner sides are misaligned and interacted at the far wake region. When the gap ratio reaches to g ∗ = 2.0, the threedimensional flow structure virtually disappears in the spanwise vorticity and shows the two-dimensional behavior. Although the flow structure shows two-dimensional behavior, the streamwise vortices still exist and can be observed with a small value (ωz = ±0.0004). As the gap ratio increases further, the force response differences between the two cylinders approach close to a negligible level gradually (less than 4%). Meanwhile, the discrepancy of the drag and lift coefficients between two-dimensional and threedimensional simulations is also reduced by 2 and 4%, respectively in Table 3.11. The three-dimensional effect remains relatively stronger in the transverse direction than the streamwise direction. The time traces of lift coefficient for both cylinders in

3.10 Appendix A: Side-by-Side Stationary Square Cylinders

185

Table 3.11 Comparison between 2D and 3D simulations at g ∗ = 2.0 Parameter 2D 3D ¯ Avg. frequency f 0.16 0.16 Avg. drag coeff. C¯D 1 1.79 1.73 Avg. drag coeff. C¯D 2 1.75 1.71

CD CL

3 1 −1 −3 150

160

170

180

190

200

5

CD CL

3 1 −1 −3 150

160

tU/D

CD CL

3 1 −1 −3 150

160

170 180 tU/D

(d)

190

200

CD , CL of Lower Cylinder

CD , CL of U pper Cylinder

(a) 5

170

180

190

200

U pper Lower

2 0 −2 −4 150

160

170

180

tU/D

(b)

(c) CD CL

3 1 −1 160

4

tU/D

5

−3 150

CL of Both Cylinders

5

– −3.4 −2.3 – −3.6 −3.7 –

1.3% 0.83 0.80 3.6%

170 180 tU/D

(e)

190

200

CL of Both Cylinders

2.2% 0.86 0.83 3.4%

CD , CL of Lower Cylinder

CD , CL of U pper Cylinder

Avg. drag coeff. Diff RMS. lift coeff. C Lr ms 1 RMS. lift coeff. C Lr ms 2 RMS. lift coeff. Diff

Diff (%)

4

190

200

U pper Lower

2 0 −2 −4 150

160

170

180

190

200

tU/D

(f)

Fig. 3.74 Force responses for two stationary side-by-side configuration of g ∗ = 2.0 at Re = 200. 2D (top) and 3D (bottom) simulations: a, d upper cylinder, b, e lower cylinder and c, f lift coefficients of both cylinders

Fig. 3.74c, f show a symmetric behavior along Y = 0 between the upper and lower cylinders.

186

3 Proximity and Wake Interference Y

Y Z

X

Z

X

Fig. 3.75 Unstructured finite element mesh for tandem cylinder arrangement. A zoomed-view of the mesh is also shown

3.11 Appendix B: Tandem Cylinders Verification and Convergence Study 3.11.1 Verification To establish validity of the numerical method and accuracy of VIV computations, we compare our results with [238] for the isolated cylinder arrangement. Figure 3.75 shows a typical unstructured grid used in the numerical experiments. The grid is finer in the wake of the cylinder and around the cylinder as shown in a close-up view of Fig. 3.75. The cylinder is assumed to have a single degree of freedom to move only in the cross-flow y-direction and its movement is constrained with a spring with the stiffness of k. The Reynolds number Re = 200, mass ratio m ∗ = 10, and mass-damping parameter m ∗ ξ = 0.1 are chosen to compare the results of [238] for the isolated cylinder setup. Here ξ is the damping coefficient. The spring stiffness is adjusted to vary the reduced velocity ranging from 3.8 to 7.0. Characterizations of maximum transverse displacement A y,max , maximum lift coefficient C L ,max , frequency ratio f / f n and the phase difference φ between C L and displacement with respect to reduced velocity Ur are shown in Fig. 3.76. Results from [238] are included for a comparison purpose. There is a well defined lock-in range, where the vortex shedding frequency locks on to a value near to the natural frequency of the structure. Branching behavior consisting of an initial branch 3 ≤ Ur ≤ 4.5 and a lower branch 4.6 ≤ Ur ≤ 6 of the lock-in region can be observed in the amplitude response in 3.76a. After crossing the lower branch for Ur > 6, the amplitude of vibration is very low and the phase angle approaches 180◦ . Furthermore, we can observe that the maximum lift force experienced in the range 4.1 ≤ Ur < 4.6 and there is

3.11 Appendix B: Tandem Cylinders Verification and Convergence Study 0.6

187

3

0.55

Present simulation Leontini et. al

0.5

2.5

0.45 2

C L, max

A y, max

0.4 0.35 0.3 0.25

1.5 1

0.2 0.15

0.5

0.1 0.05

0 4

4.5

5

5.5

6

6.5

7

4

4.5

5

5.5

Ur

Ur

(a)

(b)

1.5

6

6.5

7

6

6.5

7

180

1.4 150

1.3 1.2

120

1

φ

f/f n

1.1 90

0.9 60 0.8 0.7

30

0.6 0

0.5 4

4.5

5

5.5

6

6.5

7

4

4.5

5

5.5

Ur

Ur

(c)

(d)

Fig. 3.76 Verification of vortex-induced vibration results for isolated cylinder arrangement: a transverse displacement amplitude A y,max ; b maximum coefficient of lift C L ,max ; c frequency ratio f / f n ; d average phase difference φ between C L and the transverse displacement

a discontinuous variation of peak lift force C L ,max in the upper limit of this range. At Ur = 4.1 and Ur = 4.6, as reported in [238], there is no significant variation in the maximum cylinder amplitude but there is a sudden change in the peak transverse force and the frequency ratio. This sudden behavior characterizes the upper branch observed in high Reynolds number flows [206]. Despite different underlying formulations and discretization methods, our numerical results agree with the reference results within 5% of difference for all the characteristic response values. Furthermore, these results for the isolated cylinder will serve as a baseline for the comparisons with the tandem cylinder arrangement cases. To have a representative comparison with experimental VIV response of [57] at Re = 204, the 2D grid employed for the study at Re = 200 is extended to a 3D grid with spanwise length of 4D. Figure 3.12

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Table 3.12 Comparison of maximum amplitude with experiment [57] at Re = 204, m ∗ = 6.4, ξ = 0.006 A y,max Present 3D simulation Experiment [57]

0.52 0.48

Fig. 3.77 Comparison of downstream cylinder amplitude plotted against reduced velocity at Re = 150, m ∗ = 2.0, ξ = 0.0007 for a streamwise gap x0 = 5D

1

Carmo et. al. 2011 Present Simulation

0.9 0.8 0.7

A y, max

0.6 0.5 0.4 0.3 0.2 0.1 0

5

10

15

20

25

30

Ur

shows the maximum displacement A y,max at Re = 204 with a percentage difference of 7.6%. We next consider the tandem cylinder problem of wake-induced vibration where the downstream cylinder is free to move in transverse direction with a continuous impingement of upstream vortices. The amplitude response with respect to reduced velocity are compared against the results of [77] in Fig. 3.77 at Re = 150, m ∗ = 2.0 and ξ = 0.0007 for the streamwise gap x0 = 5D. Detailed discussions of larger amplitude vibrations of downstream cylinder is reported in Fig. 3.7.1.

3.11.2 Convergence Studies for the Validation Cases The convergence studies for the validations cases are performed for both isolated cylinder and tandem cylinder arrangements. The mesh size MVI1 used for isolated cylinder at Re = 200 is 18.6 × 104 , whose results are compared against the results in [238]. The convergence studies with respect to grid sizes MVI1 and MVI2 = 35 × 104 are shown in Table 3.13 and the percentage error is found to be well within the acceptable range.

3.11 Appendix B: Tandem Cylinders Verification and Convergence Study

189

Table 3.13 Grid convergence study for isolated cylinder at Re = 200, m ∗ = 10 Mesh C L ,max C D,avg A y,max MVT 1 MVT 2

0.149 0.152

1.4500 1.4645

0.354 0.360

Table 3.14 Grid convergence study for downstream cylinder at Re = 150, m ∗ = 2 Mesh C L ,r ms C D,avg A y,max MVT 1 MVT 2

0.3991 0.4068

1.8351 1.8334

Table 3.15 Grid convergence study for isolated cylinder arrangement Mesh C L ,r ms C D,avg MI1 MI2

0.0732 0.0749

1.8443 1.8403

0.872 0.8725

A y,max 0.4885 0.4882

We next address the the adequacy of resolution for the tandem arrangement. For the downstream cylinder of the tandem arrangement, the influence of the grid resolutions at Re = 150 are shown in Table 3.14. The results of grid size MVT 1 = 32 × 104 , which is used to generate data in Fig. 3.77, are compared with the grid size MVT 2 = 68 × 104 . The error percentage in the dynamic response of the both meshes is found to be less than 1.9%.

3.11.3 Convergence Studies at Re = 100 The parametric values for the detailed analysis of the isolated and tandem cylinder arrangements which is discussed in Fig. 3.7.1 are Re = 100 and μ = 2.6 with no damping. Mesh convergence and domain independence studies are performed to determine a suitable number of finite elements and computational boundaries to be used in the analysis of the problem. Table 3.15 shows the mesh convergence of isolated cylinder arrangement at Ur = 6, m ∗ = 2.6 and Re = 100. The mesh MI1 has 16 × 104 elements where the mesh MI2 has 27 × 104 elements. The differences in the displacement amplitude A y,max and C D,avg are less than 1%, whereas the convergence error is 2.2% for C L ,r ms . The mesh MI1 is employed for further studies of the isolated cylinder configuration. Results of the convergence study for the tandem cylinder setup at Ur = 6, M ∗ = 2.6 and Re = 100 are shown in Table 3.16. Two mesh grids MT 1 with 23 × 104 elements and MT 2 with 40 × 104 elements are considered for the convergence test. The differences between the two cases for average drag C D,avg and

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Table 3.16 Grid convergence study for tandem cylinder arrangement Mesh C L ,r ms C D,avg MT 1 MT 2

0.2535 0.2448

1.8563 1.8575

A y,max 0.9260 0.9266

Table 3.17 Domain sensitivity study for 5% blockage for tandem cylinder arrangement with streamwise gap x0 = 10D Mesh domain size C L ,r ms C D,avg A y,max MDT 1 MDT 2

40D × 20D 80D × 20D

0.3472 0.3511

1.5722 1.5719

0.7412 0.7412

maximum displacement A y,max are found to be less than 1%. For C L ,r ms the difference between the two cases MT 1 and MT 2 is about 3.4%. The grid MT 1 is considered to be adequate for the detailed investigations of tandem cylinder arrangement. The sensitivity study of domain size effects on the results are presented in the Table 3.17. The tandem cylinders with maximum streamwise gap of 10D is chosen for the study as the wake effects along the length of the domain is largest for this case. The domain size L x = 40D is compared with double its size L x = 80D with the identical blockage of 5%. The grid structure similar to MI1 and MT 1 is employed for constructing the grid distribution for the case M DT 1 with L x = 40D and the case M DT 2 whose L x is 80D. The percentage differences between the two cases for the fluid forces and transverse amplitude results are less than 1.1%, which confirm the adequacy of the computational domain.

3.12 Appendix C: Three-dimensional Tandem Cylinders Mesh Convergence and Validation Since the turbulent wake dynamics between the two cylinders is quite complex, we determine the required spatial discretization for our LES implementation. The unstructured finite element mesh employed in this study is composed of six-noded wedge elements. The mesh used in the simulation for Re = 10,000 contains 1.6 million nodes and 4.8 million wedge elements (Fig. 3.78). The element length in the spanwise direction is uniform with Δz = 0.125D. To evaluate the adequacy of the spatial resolution, we have carried out extensive grid refinement tests with different resolutions while keeping the dimensionless wall distance y + < 1. As depicted in Fig. 3.19b, both cylinders are surrounded by a relatively finer grid to capture vortex shedding and the overall mesh quality of deformed mesh around the vibrating downstream cylinder is preserved. To check the adequacy of the mesh resolution in the wake and around cylinders, we have performed mesh convergence study for three

3.12 Appendix C: Three-dimensional Tandem Cylinders Mesh Convergence and Validation 191 Table 3.18 Comparison of WIV response of the downstream cylinder m ∗ ζ = 0.018, Ur = 14.0 for three levels of mesh resolutions. Percentage value in the bracket is computed with respect to the mesh M3 Grid Total Nodes (×106 ) A ymax,r ms f/ fN M1 M2 M3

0.61 1.58 5.43

0.8079 (13.5%) 0.9409 (0.8%) 0.9339

1.26 1.26 1.26

Y

Y

Z

X

Z

X

Fig. 3.78 Representative mesh for wake-induced vibration of tandem cylinders: X Y -sectional view of deformed unstructured mesh around two circular cylinders in tandem arrangement. A close-up view of the boundary layer is shown in the right corner inset and the right side shows Y Z -sectional view in the spanwise direction

different meshes, while maintaining the dimensionless wall distance y + ≤ 1 in the boundary layer around the cylinder body. The characteristic WIV results of amplitude and frequency are shown in Table 3.18 for the three representative meshes. We consider the mass damping parameter m ∗ ζ = 0.018 and the reduced velocity Ur = 14 corresponding to √ Re = 10,000. The values of the root mean square (rms) amplitude A ymax,r ms = 2 Ar ms and the non-dimensional frequency f / f N are tabulated, where Ar ms is the root mean square of harmonic amplitude arising from the WIV excitation. The convergence error in the transverse amplitude A ymax,r ms is computed with respect to the finest mesh M3. The mesh M2 is found to be adequate to obtain reasonable results and therefore it is employed for our simulations in the present study. To further validate the implementation of the dynamic SGS, we compare the WIV characteristic response with the recent experimental data of [22] for of the

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Table 3.19 Details of meshes used in convergence study M1 M2 Spanwise layers Total nodes Total elements

32 0.61 × 106 0.59 × 106

32 1.58 × 106 3.05 × 106

M3 32 5.43 × 106 10.51 × 106

downstream cylinder located at L x /D = 4.0 behind the stationary upstream cylinder. As shown in Fig. 3.79, the values of the rms amplitude A ymax,r ms and the nondimensional frequency f / f N agree reasonably with the experimental measurements [22] for the mass ratio m ∗ = 2.6 and ζ = 0.7%. Similar to the experimental measurements, two branches corresponding to a VIV resonance (Ur < 7.0) and a combined VIV (lower branch) and WIV (7.0 ≤ Ur ≤ 14.0) responses can be observed in Fig. 3.79a. Owing to higher computational costs, we do not simulate the third WIV branch as reported experimentally by [22] thus we restrict our WIV investigation for Ur ≤ 14.0 or Re < 10,000. In the VIV resonance regime, there is good agreement of the rms amplitudes for Ur = 4 and 6, but there is some under-prediction of the predicated rms amplitude around Ur ≈ 5.0. A similar under-prediction of response amplitude in the upper branch for an isolated cylinder, VIV was observed in a previous LES study [226]. In the WIV regime, for higher reduced velocities (Ur ≥ 7.0), the predicted response amplitude increases monotonically, which is consistent with the experimental measurements. There is a good quantitative match between the predicted and measured values at Ur = 9.0, however some under-predictions at Ur = 12 and 14 can be seen which may be perhaps due to the numerical modeling or resolution errors. To investigate these discrepancies in the response amplitudes, there is a need for systematic direct a numerical simulation study, which may increase the cost of computations by two to three orders of magnitudes O(102−3 ). In addition, further investigations on the sensitivity of experiment measurements and numerical simulations on the separation point location and the shear layer transition should be explored. Interestingly, we can observe an excellent prediction of the dominant frequency of the downstream cylinder for both VIV resonance and the combined VIV and WIV regimes, as shown in Fig. 3.79b. Similar to the experimental trend, the frequency of vibration in the VIV regime regime varies nearly with St ≈ 0.2 as a function of reduced velocity and after Ur ≈ 6.0 it deviates to follow the lockin/synchronization behavior similar to an isolated cylinder VIV. Owing to the complexity of turbulent wake and vortex-structure interaction, we have a reasonable prediction of wake-induced excitation to investigate (Table 3.19). Figure 3.80 shows the vorticity distribution of the middle Z -plane, which indicates the wake topology generated by the cylinder. The dynamic model has reasonably resolved the opposite sign vortex islands within the traditional von Kármán vortices. In Fig. 3.80, instantaneous vorticity contours at maximum lift is compared qualitatively with the corresponding flow fields using particle-image velocimetry (PIV) [28]. As shown in Fig. 3.80a, the maximum lift occurs due to high-velocity flow by

1.6 1.4 1.2

Experiment Present

WIV Dominated Regime

VIV Dominated Regime

1 0.8 0.6 0.4 0.2 0

4

6

8

10

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Reduced velocity, Ur

(a)

14

2

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Amplitude, Aymax,rms

3.12 Appendix C: Three-dimensional Tandem Cylinders Mesh Convergence and Validation 193 Experiment Present

1.5 1 0.5 0

4

6

8

10

12

14

Reduced velocity, Ur

(b)

Fig. 3.79 Comparison of WIV response of the downstream cylinder as a function of reduced velocity Ur at m ∗ ζ = 0.018 with experimental results of [28]: a transverse displacement amplitude A ymax,r ms , b dominant oscillation frequency ratio f / f N . For 4.0 < Ur < 7.0 the response is VIV dominated, while Ur ≥ 7.0 represents WIV dominated response of the downstream cylinder. The errorbar is set to 5% to account for the uncertainty in collecting the statistics of amplitude data. It is worth pointing that the reduced velocity Ur is defined using the natural frequency in vacuum

vortex A2 on the inner side of the cylinder and the vortex, A1 induces vortex B3 to shed closer to the downstream cylinder surface. The instantaneous vorticity contours at the maximum lift using the present simulation is plotted in Fig. 3.80b, which is qualitatively similar to that of Fig. 3.80a. Owing to the dynamical complexity of vortices in the turbulent wake, a precise comparison of wake topologies between the numerical and experimental data is beyond the scope of the present work. The flow contours and the pressure coefficient on the downstream cylinder will be discussed in detail to determine the underlying unsteady vortex-structure interaction behind the wake excitation. Figure 3.81a–c show the capability of the SGS model to capture the fine turbulence details of the fluid flow. When the velocity gradients and inertia effects are sufficiently large, wake turbulence leads to the generation of a broad range of spatial and time scales with vigorous vortex stretching associated with the rotational flow in the wake. The vortical structures generated using a vortex identification based on Qcriterion [172] are shown in Fig. 3.81a–c at three instances corresponding to the top, mean and bottom positions of the downstream cylinder. The isosurfaces of quantifying Q are shown at a constant positive value and the contour surfaces are colored by the normalized streamwise velocity. The flow structures are complex due to three-dimensional wake and turbulence effects and dominating streamwise vortex ribs superimposed on the spanwise vorticity structures can be seen in the plots. The large vortical structures about the order of cylinder diameter are distributed in the spanwise direction and they are reasonably resolved by the spanwise length. In Fig. 3.81c, the stagnation region over the vibrating downstream cylinder can be seen. As it is known that turbulence contains a wide range of spatial and time scales and the fluctuation of the energy is generated at the level of large eddies associated with low wave numbers. The vortex stretching mechanisms transform the large eddies into smaller and smaller eddies and the energy continuously cascades down to higher wave

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A3 A2

A1 B4 B3

B1 B2

(a)

(b)

Fig. 3.80 Instantaneous spanwise vorticity contours at maximum lift around a pair of cylinders at Reynolds number Re = 10,000, m ∗ = 2.6 and ζ = 0.7%: a schematic of particle-image velocimetry (PIV) data from Fig. 25a in [28], b contours of spanwise vorticity from the present numerical study. Detail description of vortical regions A1–A3 and B1–B4 extracted from PIV can be found in [28]

numbers in the inertial spectrum. In this range, the slope of the energy spectrum as a function of the wave number remains constant. As discussed earlier, the wake of a bluff body contains turbulent fluctuations, which are nonlinearly superimposed on the periodic vortex shedding motion and the turbulent kinetic energy is distributed among the eddies of different sizes behind the vibrating cylinder. From the instantaneous velocity uf = (u,  2  v, w) data, we compute the instantaneous kinetic energy as E = 1 u + v 2 + w 2 ≡ 21 u if u if . To assess this energy cascade process, Fig. 3.81d shows 2 the single-sided spectra of kinetic energy at a centerline point x = 3D, y = 0 along the wake of the downstream cylinder. The inertial frequency region shows a good agreement with the Kolmogorov’s −5/3 rule. When the integral scale eddies follow the relation E(κ) ∝ κ −5/3 , it gives an indication of a properly resolved LES grid, where κ denotes the wave number. While this relationship is typically considered for the spatial distribution of turbulent kinetic energy, the −5/3 decay of the kinetic energy versus frequency can also provide a reasonable indicator for the turbulence spectrum via the frozen-in turbulence hypothesis [413].

3.12 Appendix C: Three-dimensional Tandem Cylinders Mesh Convergence and Validation 195

(a)

(b)

(c)

spectra of kinetic energy |E(k)|

10 0

10 -1

slope ~ -5/3

10 -2

10 -3

10 -4

10-2

10-1

100

frequency

(d) Fig. 3.81 Representative results of vibrating tandem cylinder at Re = 10,000, Ur = 14, m ∗ ζ = f ∂u f ∂u

0.018: vortical wake structures based on the instantaneous isosurfaces of Q(= − 21 ∂ x ij ∂ xij ) value for three sample configurations: a peak negative displacement, b mean position, c peak positive displacement, and d frequency spectra of instantaneous kinetic energy E(= 21 u if u if ) at x = 3D, y = 0, z = 0, where the origin is the center of the upstream cylinder. Isosurfaces of non-dimensional Q + ≡ Q(D/U )2 = 0.25 are colored by the normalized streamwise velocity u/U

Acknowledgements Some parts of this Chapter have been adapted from the PhD dissertations of Mengzhao Guan and Liu Bin carried out at the National University of Singapore and supported by the Ministry of Education, Singapore.

Chapter 4

Near Wall Effects

In this chapter, we introduce the near-wall effects on the mechanics of freely vibrating circular cylinders. The complexity of the coupled physical phenomena involved in a vibrating cylinder close to a plane stationary wall is enhanced by wake/boundary layer and cylinder/wall interactions. The unsteady wake of the cylinder interacting with the boundary layer makes the coupled response of the near-wall configuration different from the isolated-cylinder arrangement. We present the characterization of the response dynamics of isolated and near-wall vibrating cylinders and discuss the basic differences between the two arrangements. In terms of the wall distance and the upstream boundary layer, we systematically analyze various aspects of coupled fluid-structure dynamics such as the wake topology, the response characteristics, the force components, the phase relations and frequency characteristics. The interaction mechanism is covered for both 2D and 3D configurations. We provide a regime map summarizing the vortex-shedding modes as a function of the reduced velocity and the boundary layer thickness is offered in the laminar flow regime.

4.1 Introduction Marine pipelines and subsea cables are two representative motivating applications of the near-wall effects on flow-induced vibrations. The uneven nature of seafloor or possible seabed scouring may cause free spanning along the pipeline. The span length can be easily 100 times that of the pipeline diameter, with a gap from the seabed which can range from essentially zero to more than 2–3 times the pipeline diameter [401]. When exposed to flow action, such free-span pipelines may undergo flow-induced vibrations. There have been several serious incidents in the past when pipelines floated to the surface due to lost protective concrete coatings as a result of flow-induced vibrations. Damage associated with the fatigue of a pipeline undergoing vibrations is known to be proportional to the product of A4 f where A is the vibration © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Jaiman et al., Mechanics of Flow-Induced Vibration, https://doi.org/10.1007/978-981-19-8578-2_4

197

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amplitude and f is the vibration frequency, as indicated in [423]. In the short term, small amplitude vibrations with high frequencies may not be detrimental. In the long run, however, they can naturally result in serious consequences due to fatigue. Therefore, the understanding of the VIV of a free-span pipeline is of significant importance for the offshore industry. The problem of a free-span along the pipeline can be modeled by the configuration in which the flow past on an elastically mounted circular cylinder with two-degree-of-freedom (2-DoF) in proximity to a stationary plane wall. The numerical studies of VIV of a circular cylinder near a fixed plane wall serve as a foundation to improve pipeline design and installation guidelines. Flow past an isolated cylinder in a free-stream beyond a critical Reynolds number manifests itself as alternate vortex shedding in the cylinder’s wake. This leads to the cylinder being subject to fluctuating lift and drag components. When such a cylinder is placed in proximity to a plane wall, the symmetry of the flow domain is broken as the wall interferes with the symmetric vortex shedding. An isolated pipeline in the free oceanic stream, for example, will experience vortex-induced vibrations if such alternate vortex shedding exists. However, if the pipeline is installed on the seabed itself, the interaction with the oceanic currents will be very much different. In particular, the wall-induced lift force is due to two competing mechanisms. First, the presence of a nearby wall breaks the symmetry of the wake vorticity distribution which results in an effective lift force that tends to move the cylinder away from the wall. Second, from inviscid theory one can argue that the flow relative to the cylinder will accelerate faster in the gap between the cylinder and the wall. The resulting low pressure in the gap will induce a lift force directed toward the wall. Most of the previous studies on VIV were generally focused on an isolated circular cylinder placed in a uniform cross-flow without wall-proximity effects. There are a number of comprehensive review works on this topic in recent years, such as [40, 381, 449, 455]. In [455], a map of vortex shedding modes (2S, 2P and P+S) was presented. Sarpkaya [381] summarized the intrinsic nature of VIV of circular cylindrical structures subjected to steady uniform flow. In [449], the authors summarized some fundamental results on VIV with low mass and damping with new numerical and experimental techniques. The effects of Reynolds number, Re, on VIV responses of both isolated and tandem cylinders were reviewed in [40], in which the Reynolds number based on cylinder diameter D is defined as Re = U∞ D/ν, where U∞ denotes the freestream velocity and ν is the kinematic viscosity of the fluid. Investigations at low Reynolds numbers include the experimental study conducted by Anagnostopoulos and Bearman [18] in laminar flow at Re from 90 to 150. Direct numerical simulations carried out in low Re regime include [150, 238, 345, 392]. In the work by Sherdan et al. [392], the dimensionless peak transverse oscillation amplitude is 0.59 for a massless cylinder with 1-DoF motion, achieving a good agreement with experimental results [18]. The vortex sheddings of a forced streamwise oscillating cylinder in water at rest with Re = 100 and K C = U∞ T /D = 5 (where T denotes the period of oscillation) and a forced transversely oscillating cylinder in a uniform flow at Re = 185 were numerically investigated in [150]. In [238], the authors performed 2D simulations at Re = 200 and found that the genesis of the higher-Re flow behavior is also present in low-Re 2D flow in terms of regimes of

4.1 Introduction

199

cylinder response, frequency and phase response of the cylinder. The authors [345] numerically studied the VIV of two circular cylinders in both tandem and staggered arrangements using a stabilized finite element method in 2D at Re = 100. The authors [206] found that the cylinder response can be characterized by two types of VIV behavior. With low mass damping, there are three distinct branches in the response curve with the variation of reduced velocity. The three branches are termed as the initial, upper and lower branches. With high mass damping, the upper branch does not exist. The distinct branches in the response curve are associated with different vortex shedding modes at the wake region of the cylinder. Gopalan and Jaiman [136] found that for 1-DoF vibrating cylinder with high mass damping, two branches of the response are found, namely the initial and lower branches. The vortex wake on the initial branch comprises a 2S mode while a 2P mode is shown on the lower branch. Jensen et al. [191] found that in 2-DoF VIV the streamwise displacement inhibits the formation of the 2P vortex shedding mode.

4.1.1 Dynamics of a Circular Cylinder with Wall Proximity The proximity of a wall introduces complex interactions between the wall boundarylayer and the shear layer over the circular cylinder. The proximity effects of a cylinder to a plane wall at a given Reynolds number can be characterized first by defining the gap ratio e/D as the ratio of the spacing between the cylinder and the moving plane wall and the diameter of the cylinder. One of the earliest experiments studying ground effect on a circular cylinder was reported by Taneda’s experiment [410], where the flow behind a circular cylinder towed through stagnant water close to a fixed ground was visualised at Re = 170. The water and ground moved together relative to the cylinder and thus there was essentially no boundary layer formed on the ground. Regular alternate vortex shedding occurred at a gap ratio, e/D where e denotes the gap distance, of 0.6, while only a weak single row of vortices were shed at e/D = 0.1. When the wall in proximity is stationary, a boundary layer forms along the wall. The development of three shear layers are involved, namely the two separated from the upper and lower sides of the cylinder, as well as the wall boundary layer. Bearman et al. [37] showed that the vortex shedding is suppressed if e/D is small enough. Studies by Zdravkovich [480] and Lin et al. [249], carried out at Re = 3550 and 780, respectively, showed the cessation of regular vortex shedding for a stationary cylinder near a fixed plane wall. Other studies of stationary cylinder near a fixed wall by Lei et al. [234] and Wang and Tan [439] showed that Re, e/D and boundary layer thickness, δ/D, are parameters affecting the flow for a cylinder near a fixed wall. Ong et al. [322] applied the standard high Reynolds number k − ω model at Re = 1.0 × 104 − 4.8 × 104 with δ/D = 0.14 − 2, finding that under-predictions of the essential hydrodynamic quantities were observed in the subcritical flow regime due to the limited capacity of the k −  model in capturing the vortex shedding correctly. Ong et al. [323] carried out numerical studies on stationary near-wall

200

4 Near Wall Effects

cylinder in the turbulent regime. For the stationary near-wall cylinder, it was found that the C D increases as e/D increases for small e/D, reaching a maximum value before decreasing to a constant. When the wall in proximity is moving at the freestream velocity, no wall boundary layer is formed. Huang and Hyung [163] used immersed boundary method (IBM) to study the critical gap ratio for Re = 300, 400, 500 and 600, observing an inverse relation between the critical e/D and Reynolds number. They further concluded that the critical gap ratio corresponds to a local minimum of the streamwise maximum mean velocity in the gap. In [355], the researchers conducted numerical investigations on the dynamics and stability of the flow past two tandem cylinders sliding along a wall for 20 ≤ Re ≤ 200. Li et al. [241] numerically investigated shear-free wall proximity effects at low Reynolds numbers for flow past a circular cylinder placed in the vicinity of a plane moving wall. The dynamics of the tandem cylinder arrangement near to a moving wall was reported at low Re in [110] . Due to the effect of wall proximity, the dynamics of VIV is much more complex than freely vibrating cylinder in a freestream flow. For vibrating near-wall cylinder, the vortex shedding may occur even at very small gap ratio as compared to the stationary counterparts, according to Fredsøe et al. [120] and Raghavan et al. [349]. Investigations for vibrating cylinder near a plane wall were mainly conducted in the moderate to high Re regime, including those by Kozakiewicz et al. [220] and Tsahalis and Jones [424], which showed that the X Y -trajectory of a near-wall cylinder is an oval-shape, instead of the common figure-of-eight shape observed in vibrating isolated cylinder. Fredsøe et al. [120] found that the transverse vibration frequency is close the vortex shedding frequency of a stationary cylinder when reduced velocity, Ur = U∞ / f n D where f n is the natural frequency, Ur < 3 and e/D > 0.3. When 3 < Ur < 8, the transverse vibrating frequency deviates considerably from the vortex shedding frequency of a stationary cylinder. Taniguchi and Miyakoshi [412] reported the fluctuations of lift and drag on the cylinder under the wall proximity effects by looking into e/D and δ/D at Re = 9.4 × 104 . It is assumed that Kármán vortex streets are formed by concentrations of vorticity due to the rolling-up of separated shear layers which issued from both sides of the cylinder. They pointed out that the concentration of vorticity was reduced and formation of the Kármán vortex streets was interrupted by the gap flow at small gap ratios based upon their flow visualizations. Brørs [67] and Zhao et al. [493] utilized k −  model at Re = 1.5 × 104 and a k − ω model at Re = 2.0 × 104 , respectively. Their results agreed well with the experimental data available. Zhao and Cheng [491] studied numerically 2-DoF VIV of near-wall cylinder in the turbulent regime. Their study investigates low gap ratios of e/D = 0.002 and 0.3, where effects of bounce-back from the plane boundary on the VIV are studied. Wang and Tan [440] conducted an experimental study for a 1-DoF vibrating cylinder near a plane wall at 3000 ≤ Re ≤ 13,000 and a low mass ratio m ∗ = (4m)/(πρ D 2 ), where m denotes the mass of cylinder and ρ denotes the density of fluid, of 1.0 for 1.53 ≤ Ur ≤ 6.62. It was demonstrated that the nearby wall not only affects the amplitude and frequency of vibration, but also leads to non-linearities in the cylinder response as evidenced by the presence of super-harmonics in the drag

4.1 Introduction

201

force spectrum. The vortices shed that would otherwise be in a double-sided vortex street pattern are arranged into a single-sided pattern, as a result of the wall. Tham et al. [416] presented a numerical study on VIV of a freely vibrating 2-DoF circular cylinder in close proximity to a stationary plane wall at Re = 100. They reported that the effect of wall proximity tends to disappear for e/D ≥ 5 and proposed new correlations for characterizing peak amplitudes and forces as a function of the gap ratio. The literature review above indicates that previous research on near-wall stationary and vibrating cylinder focused primarily on moderate to high Re regime. However, few works on numerical studies of 2-DoF VIV of an elastically supported circular cylinder in proximity to a wall in the low Re regime can be found in the literature. As it is known that the branching of cylinder response for VIV has its genesis in twodimensional low Re flow and the essential aspects of VIV dynamics can be captured numerically at lower Re, as reported in [238] as well as [307]. The primary aim of this study is to investigate the effects of the wall proximity on VIV of an elastically mounted circular cylinder with 2-DoF by characterizing the hydrodynamic forces, the vibration responses, the phase differences between the hydrodynamic forces and displacements as well as the vortex shedding patterns for both the near-wall and isolated cases in both 2D and 3D. In the neighbourhood of a stationary wall, the vortex dynamics and response characteristics are quite different from that of the isolated cylinder vibrating in a freestream flow. In particular, we investigate the origin of enhanced streamwise vibrations of near-wall cylinder as compared to the freely vibrating isolated cylinder configuration.

4.1.2 Organization In this chapter, the flow around a single vibrating circular cylinder placed in proximity to a horizontal plane boundary is investigated. Firstly, when the cylinder is elastically mounted, vortex-induced vibrations will probably occur and play a role in fluid-structure interactions. The flow past a near-wall circular cylinder with two degrees of freedom (2-DoF) is studied in Sect. 4.2 to reveal how wall proximity affects the cylinder response characteristics and vortex shedding patterns in both 2D and 3D. The governing equations and the numerical method employed in this work are presented in Sect. 4.2.1. The problem definition and the convergence study are described in Sect. 4.2.2. This is followed by the results for 2D simulations in Sect. 4.2.3. In this section, we present the characterization of the response dynamics of isolated and near-wall cylinders and discuss the basic differences between the two arrangements in terms of wake topology, response dynamics, force components, phase and frequency characteristics. The results for 3D near-wall VIV simulations are presented in Sect. 4.2.4 for representative reduced velocities and contrasted against the isolated counterpart. Lastly, the major conclusions of this study are provided in Sect. 4.2.5.

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4 Near Wall Effects

Secondly, the flow past a fixed cylinder placed in the vicinity of a moving plane wall is investigated in Sect. 4.3 to study the shear-free wall proximity effects on the hydrodynamics involved. The problem definition and the convergence study are described in Sect. 4.3.1. This is followed by the results and discussion for both 2D and 3D simulations in Sect. 4.3.2. Lastly, the major concluding remarks are provided in Sect. 4.3.4.

4.2 Cylinder VIV in the Vicinity of a Stationary Wall 4.2.1 Numerical Formulation The Navier-Stokes equations governing an incompressible Newtonian flow in an arbitrary Lagrangian-Eulerian reference frame are ρf

  ∂uf + ρ f uf − w · ∇uf = ∇ · σ f + bf on Ω f (t), ∂t ∇ · uf = 0 on Ω f (t),

(4.1) (4.2)

where uf and w represent the fluid and mesh velocities, respectively, bf is the body force applied on the fluid and σ f is the Cauchy stress tensor for a Newtonian fluid written as   T  . (4.3) σ f = − p I + μf ∇uf + ∇uf A cylinder body submerged in the flow stream experiences transient vortex-induced forces and consequently may undergo rigid body motion if mounted elastically. The rigid-body motion of the cylinder in the two directions along the Cartesian axes, is governed by the following equation: m·

∂us + c · us + k · (ϕ s (z0 , t) − z0 ) = Fs + bs on Ω s , ∂t

(4.4)

where m, c and k denote the mass, damping and stiffness vectors per unit length for the translational degrees of freedom, Ω s denotes the rigid body, us (t) represents the rigid-body velocity at time t, Fs and bs are the fluid traction and body forces acting on the rigid body, respectively. Here ϕ s denotes the position vector mapping the initial position z0 of the rigid body to its position at time t. The spatial and temporal coordinates are denoted by x and t, respectively. The coupled system requires to satisfy the no-slip and traction continuity conditions at the fluid-body interface Γ as follows

4.2 Cylinder VIV in the Vicinity of a Stationary Wall

203

uf (ϕ s (z0 , t), t) = us (z0 , t) ,  σ f (x, t) · ndΓ + Fs dΓ = 0 ∀γ ∈ Γ,

 ϕ(γ ,t)

(4.5) (4.6)

γ

where n is the outer normal to the fluid-body interface, γ is any part of the fluid-body interface Γfs in the reference configuration, dΓ denotes a differential surface area and ϕ s (γ , t) is the corresponding fluid part at time t. In Eq. (4.6), the first term represents the force exerted by the fluid on ϕ s (γ , t), while the second term represents the net force by the rigid body. A solver using Petrov-Galerkin finite-element and semi-discrete time stepping has been employed to investigate the interaction of incompressible viscous flow with rigid-body dynamics of freely vibrating cylinder. To account for fluid-rigid body interaction, a partitioned iterative scheme based on nonlinear interface force correction [181] has been employed for stable and accurate coupling with strong added mass effects. The temporal discretization of both the fluid and the structural equations is embedded in the generalized-α framework by making use of classical Newmark approximations in time [94]. Throughout this study, the incremental velocity and pressure are computed via the matrix-free implementation of the restarted Generalized Minimal Residual (GMRES) solver proposed in [370]. The GMRES uses a diagonal preconditioner and a Krylov space of 30 orthonormal vectors. In the current formulation, we perform Newton-Raphson type iterations to minimize the linearization errors per time step. The fluid loading is computed by integrating the surface traction considering the first layer of elements located on the cylinder surface. The instantaneous force coefficients are defined as   1 1 (σ f .n).n y dΓ, CD = 1 f 2 (σ f .n).nx dΓ, (4.7) CL = 1 f 2 ρ U D ρ U D 2 2 Γ

Γ

where ρ f , σ f denote fluid density and stress, respectively, and nx and n y are the Cartesian components of the unit normal, n. In the present study, C D and lift coefficient C L are post-processed using the conservative flux extraction of nodal tractions along the fluid-body interface.

4.2.2 Problem Definition and Convergence Study 4.2.2.1

Model Description

The 2D computational domain is depicted in Fig. 4.1 to illustrate the domain and boundary conditions details. A circular cylinder is placed in the vicinity of a plane wall and the center of the cylinder is at a distance of 8D to the inflow boundary and

204

4 Near Wall Effects

Fig. 4.1 Schematic of 2D computational domain with details of boundary conditions

of 26D to the outflow boundary, allowing sufficient distances in capturing the vortex dynamics in the downstream wake. The adequacy of similar boundary locations was demonstrated in earlier works in [180, 312, 344]. The width of the computational domain is 20D, thereby the resulting blockage based on the diameter of the cylinder and the lateral dimension of the domain is 5%. The same blockage was also used in [312]. For the isolated configuration, the lateral boundaries are each 10D away from the cylinder center, with slip boundary conditions applied on both top and bottom walls, i.e. ∂u x /∂ y = 0. For the near-wall configuration, in which the bottom boundary serves as a stationary rigid wall, the distance between the lower surface of the cylinder and the bottom wall is governed by the gap ratio, denoted by e/D, with e being the initial gap distance. The center of the near-wall cylinder is then located at (0.5D + e) to the bottom boundary, as shown in Fig. 4.1. The boundary conditions are described as follows for the nearwall configuration: At the inlet, a Dirichlet velocity for the steady incoming flow is given with u x = 1 and u y = 0 to represent the free stream; at the outlet, a traction free condition is imposed as ∂u x /∂ x = 0 and ∂u y /∂ y = 0 for velocities and p = 0 for the pressure equation; at the top a free-slip boundary condition is applied by ∂u x /∂ y = 0 and u y = 0; for the cylinder surface and the bottom wall, a no-slip boundary condition is imposed by setting u x = 0 and u y = 0. The mesh movement method involved has been designed in such a way that the mesh surrounding the cylinder moves along with it like a rigid body while there is an outer boundary remaining stationary, shown in Fig. 4.2. A few simulations are performed using three-dimensional stabilized finite element formulation to investigate the 3D effects on freely vibrating cylinders. The 3D domain, as shown in Fig. 4.3, is simply an extrusion of the 2D plane along the cylinder axis (z-axis). In addition to the boundary conditions applied at the 2D domain, for the 3D domain, a periodic boundary condition is applied at the two ends of the cylinder (front and back sides of the 3D domain) to model a cylinder of infinite length. The 3D calculations are conducted for a cylinder with a spanwise length (in z direction) of 5D, which is larger than the spanwise length of 4D used by Navrose and Mittal [312] for the same Reynolds number.

4.2 Cylinder VIV in the Vicinity of a Stationary Wall

205

Fig. 4.2 Two-dimensional mesh used in simulations for the near-wall configuration with the zoomin view on the cylinder

Fig. 4.3 Schematic of the 3D computational domain with details of boundary conditions

4.2.2.2

Cylinder VIV Modelling

A rigid circular cylinder is mounted elastically by springs and dampers and allowed to vibrate freely in both transverse (y-axis) and streamwise (x-axis) directions. The structural property of the cylinder is depicted in the dashed circles in both Figs. 4.1 and 4.3. Both the transverse and streamwise springs are assumed to be linear and homogeneous in the stiffness, yielding identical natural frequencies in both directions, so the ratio of f nx to f ny is unity. Computations are carried out for various reduced velocities for a fixed mass ratio of the cylinder, m ∗ = (4m)/(πρ D 2 ) = 10 where m is the cylinder mass and ρ is the fluid density, at given Re. The reduced

206

4 Near Wall Effects

velocity is defined as Ur =

U∞ fn D

=

1 2π

U∞ √ . Therefore, Ur is adjusted by varying k m

D

the spring stiffness, k, and thus varying f n of the cylinder. One characteristic feature of VIV is frequency lock-in, and the key parameter for the frequency lock-in phenomenon in VIV is the natural frequency of the system f n . In a recent work [486], the authors pointed out that the mechanism of frequency lock-in at low Re can be divided into two modes: flutter-induced (coupled wake mode and structure mode instability) and resonance-induced (only wake mode instability) using numerical simulations. The definitions of the VIV quantities used in the present analysis are described as follows. The coefficients of mean lift C L and mean drag C D are evaluated as CL =

i=n 1 C L ,i , n i=1

CD =

i=n 1 C D,i n i=1

(4.8)

whereby their root-mean-square (rms) counterparts are

(C L )r ms

 i=n 1  =

(C L ,i − C L )2 , n i=1

(C D )r ms

 i=n 1  =

(C D,i − C D )2 n i=1

(4.9)

and the maximum and rms amplitudes are defined as 1 (A y )max /D = [|(A y )max − (A y )min |/D], 2

  1 i=n  (A x )r ms /D =

(A x,i − A x )2 /D n i=1

(4.10) where n denotes the number of samples taken.

4.2.2.3

Grid Convergence Study

A mesh independent study is carried out in this section. The domain is discretized using an unstructured finite-element mesh, which provides a flexibility to design finer grid resolutions in the high gradient regions and a coarse grid in the far-field regions. All the meshes evaluated are given in Tables 4.1 and 4.2. The mesh convergence study involves two steps: firstly, a 2D mesh is chosen by assessing different 2D meshes; secondly, the 3D mesh is generated by layering the 2D mesh along the spanwise direction (z-axis). The effects of the spanwise resolution, Δz /D, are then taken into account to determine a suitable mesh to be used for 3D simulations. The first step evaluates three 2D meshes of different resolutions. In order to quantify the dependency of the numerical results on the mesh density, numerical simulations are conducted at three meshes with different densities as shown in Table 4.1. The calculations are performed for an isolated circular cylinder of m ∗ = 10 with

4.2 Cylinder VIV in the Vicinity of a Stationary Wall

207

Table 4.1 Grid convergence study of 2D simulations for an isolated cylinder at Re = 200 and Ur = 5 Mesh Ncyl Ne2D CD (C L )r ms (A x )r ms /D (A y )max /D A

128

21,612

B

168

25,988

C

200

32,477

2.1311 (3.70%) 2.0645 (0.46%) 2.0551

0.0924 (3.47%) 0.0901 (0.89%) 0.0893

0.0092 (5.75%) 0.0088 (1.15%) 0.0087

0.5693 (3.55%) 0.5548 (0.91%) 0.5498

∗

Ncyl : number of points around the cylinder circumference; Ne2D : total number of elements in 2D domain

2-DoF at Re = 200 and Ur = 5. At this particular reduced velocity, high vibration amplitudes are expected along with the sensitive nature of amplitude response on numerical discretization errors. For different meshes, we have calculated the representative hydrodynamic forces and cylinder responses undergoing free vibrations subjected to the incoming flow. Results from different meshes are summarized in Table 4.1. With Mesh C being the reference, the differences between results from Mesh A and Mesh B and those from Mesh C are thus calculated and noted in the corresponding brackets. It is shown in Table 4.1 that differences between the results are approximately within 1% for the two finer meshes, Mesh B and Mesh C. Considering the computational efficiency, Mesh B is then selected as the mesh to perform our 2D simulations. The accuracy of our implementation using Mesh B is assessed by comparing the present results with [238] in Sect. 4.2.3.2. By a similar process, the second step for the grid convergence study is conducted for a 3D modelling of VIV of an isolated cylinder at Re = 1000 and Ur = 5. Using the 2D mesh selected, different spanwise resolutions are assessed, namely Δz /D = 0.5, 0.2 and 0.1. The results obtained for different Δz /D are given in Table 4.2. Comparing the results in Table 4.2, it is seen that the spanwise resolution has an obvious influence on the results. Also, the guidelines of spanwise resolution used in 3D studies can be found in relevant references. Navrose and Mittal [312] used a spanwise resolution of 0.08 for the same Re and m ∗ . In [492], a spanwise resolution of 0.1 is utilized for VIV of a vertical circular cylinder with a mass ratio of m ∗ = 2 at Re = 1000. Thus, following the guidelines, for the 3D calculations in the present study, Mesh B3 with the finest spanwise resolution of Δz /D = 0.1 is then considered for this study. It is also noted that the time step size used in both 2D and 3D simulations is Δt = 0.05. A systematic temporal convergence study has been performed for VIV in [181], which utilizes the same numerical scheme.

208

4 Near Wall Effects

Table 4.2 Grid convergence study of 3D simulations for an isolated cylinder at Re = 1000 and Ur = 5 Mesh Δz /D Ne3D CD (C L )r ms (A x )r ms /D (A y )max /D B1

0.5

313,168

B2

0.2

709,276

B3

0.1

∗

1,366,576

1.6671 (5.70%) 1.6105 (2.11%) 1.5772

0.1066 (6.28%) 0.1027 (2.39%) 0.1003

0.0089 (8.54%) 0.0084 (2.44%) 0.0082

0.6050 (5.54%) 0.5849 (1.32%) 0.5773

Ne3D : total number of elements in 3D domain

Table 4.3 Key VIV parameters for 2D simulations Parameter Symbols Definitions Mass ratio Reduced velocity Damping ratio Reynolds number Initial gap ratio

m∗ Ur ζ Re e/D

4m/(πρ D 2 ) U∞ /( f n D) √ c/2 km U∞ D/ν e/D

Value 10 3–9 0, 0.01 200 0.9, ∞

4.2.3 Two-Dimensional Results and Discussion Two-dimensional simulations of VIV of an elastically mounted circular cylinder of m ∗ = 10 with 2-DoF have first been performed at Re = 200 for Ur ranging from 3 to 9 with an interval of 0.5. Definitions of the governing parameters and corresponding values for the 2D studies are listed in Table 4.3. For the near-wall cylinder, the initial gap ratio is e/D = 0.90, which is large enough to prevent the cylinder from contacting the bottom wall during VIV. For the purpose of studying the wall proximity effects, VIV of an isolated cylinder, i.e. e/D = ∞, is compared with the results of the near-wall configuration. All 2D cases have been simulated until dimensionless time tU/D = 500. The results presented in the following are extracted from the dimensionless time range tU/D = 250−500.

4.2.3.1

Fluid Forces

For a self-excited cylinder undergoing free vibrations, the vortex shedding due to complex interaction between the shear layers and the near wake leads to oscillating hydrodynamic forces which serve as the exciting forces. It is important to understand the unsteady fluid loads over vibrating bodies. The mean and root-mean-squared lift coefficient, C L and (C L )r ms , as well as the mean and root-mean-squared drag coefficient, C D and (C D )r ms , for both the near-wall cylinder and the reference isolated cylinder with the variation of Ur are presented in Fig. 4.4a, b respectively. It is apparent that the wall proximity greatly increases the C L for all Ur values considered

4.2 Cylinder VIV in the Vicinity of a Stationary Wall

209

Fig. 4.4 Force coefficients as a function of reduced velocity Ur for both isolated and near-wall cylinders at Re = 200: a mean lift coefficient C L , root-mean-squared lift coefficient (C L )r ms , b mean drag coefficient C D , root-mean-squared drag coefficient (C D )r ms

herein. This is because the increase in the transverse force coefficient is associated with the breaking of the symmetric flow fields by the wall presence which yields a non-zero C L on the cylinder. This phenomenon has already been reported in [416] for the low Re regime and in [323] for the subcritical regime. As seen in Fig. 4.4a, the largest increase in C L due to wall proximity takes place over the lock-in region where the cylinder undergoes vibrations of large amplitude. The increase of C L in the lock-in is also a result from the increased transverse vibration in the lock-in region, seen in Fig. 4.5a where (A y )max /D peaks at Ur = 4.5, and so does C L in Fig. 4.4a. As the transverse vibration amplitude peaks, the cylinder vibrates closest to the wall, leading to the most unsymmetrical flow field at the smallest instantaneous gap thereby C L becomes the largest. It can also be observed that the wall proximity does not affect (C L )r ms much as the trend of the near-wall cylinder follows closely with that of the isolated configuration. It is also noted that (C L )r ms has a small jump from the lock-in regime to the post-lock-in. This jump in (C L )r ms is also reported in [416] for the isolated cylinder. In the lock-in range, it is found that wall proximity causes a substantial increase in C L while a slight decrease in both C D and (C D )r ms in general. For both configurations, the drag forces are relatively much larger in the lock-in as compared to the lock-in free regime. As reported earlier in [190, 479], in general, a neighbouring wall has a decreasing effect on the mean drag, which is consistent with the present study. Shown in Fig. 4.4b, the decrease in C D is most obvious in the lock-in range. To be specific, C D for the near-wall case is approximately 90% that of the isolated cylinder. However, with the presence of the plane wall, although C L has a large difference, the change in transverse displacement is small, shown in Fig. 4.5a. On the contrary, although C D and (C D )r ms have little difference, the streamwise response is greatly enlarged by the wall proximity, as shown in Fig. 4.5b. This suggests that stark differences in the streamwise oscillations between near-wall and isolated cases

210

4 Near Wall Effects

Fig. 4.5 Vibration amplitudes as a function of reduced velocity Ur for both isolated and near-wall cylinders at Re = 200: a normalized maximum transverse displacement (A y )max /D, b normalized root-mean-squared streamwise displacement (A x )r ms /D

are not due to the difference in the streamwise forces. Reasons for the enhanced streamwise oscillations in Fig. 4.5b are discussed in Sects. 4.2.3.3 and 4.2.3.4.

4.2.3.2

Cylinder Response Amplitudes

Figure 4.5 summarizes the normalized maximum transverse displacements, (A y )max /D, and the normalized root-mean-squared streamwise displacements, (A x )r ms /D, for both the isolated and near-wall cylinders as a function of Ur . In Fig. 4.5a, to verify the accuracy of our implementation, (A y )max /D for the VIV of an isolated cylinder with the mass damping of m ∗ ζ = 0.1 (i.e. m ∗ = 10 and ζ = 0.01) is first compared with the results from [238], and our (A y )max /D agrees well with the reference data. For the case of m ∗ ζ = 0, the results of (A y )max /D for both isolated and near-wall configurations are also shown in Fig. 4.5a. To further assess the accuracy, the peak normalized maximum transverse displacement over the whole range of Ur , [(A y )max /D]∗ , in the present 2D simulations is also benchmarked with some other studies. The comparison of [(A y )max /D]∗ is tabulated in Table 4.4. It is shown in Fig. 4.5a that the lock-in region approximately ranges from Ur = 4 to 6.5, where the cylinder undergoes large-amplitude vibrations. The lock-in state of the freely oscillating rigid cylinder is first described by a two-branch response as documented in the experiments of [64] as well as [205]: an upper branch that corresponds to large amplitude and low values of reduced velocity, and a lower branch that corresponds to low amplitudes and large values of reduced velocity. Prasanth and Mittal [344] found that, unlike high Re VIV, low Re VIV only exhibits two branches, namely initial and lower. In Fig. 4.5a, in the lock-in region, the cylinder response is clearly characterized by two branches: initial and lower. The range of Ur ≤ 3.5 represents the pre-lock-in regime, where the vibration amplitudes are negligibly small. As Ur increases, the vortex shedding frequency synchronizes with the cylinder

4.2 Cylinder VIV in the Vicinity of a Stationary Wall

211

Table 4.4 Comparison of [(A y )max /D]∗ with reference studies at low Re Category

Study

Re

m∗ζ

[(A y )max /D]∗

Numerical

Blackburn and Karniadakis [52] Newman and Karniadakis [314] Fujarra et al. [121] Leontini et al. [238] Guilmineau and Queutey [149] Shiels et al. [392] Tham et al. [416] Anagnostopoulos and Bearman [18] Present Present

200

0.012

0.64

200

0

0.65

200

0.015

0.61

200

0.1

0.51

100

0.0179

0.54

100 100 90–150

0 0 0.179

0.59 0.57 0.64

200 200

0 0.1

0.57 0.49

Experimental

vibration frequency, and the lock-in range is then entered. In the initial branch of the lock-in, the vibration amplitude increases dramatically as Ur increases before reaching a peak value. The maximum amplitude (A y )max /D then decreases gently as Ur increases prior to decreasing sharply in the lower branch of the lock-in range. As Ur continues increasing when Ur ≥ 7, the post-lock-in region is then entered. It is also observed in Fig. 4.5a that the (A y )max /D of the near-wall cylinder behaves similarly with its isolated counterpart, thus the wall proximity does not significantly alter the behaviour of the transverse response. On the contrary, in Fig. 4.5b, (A x )r ms /D of the near-wall cylinder is by far larger than that of the isolated cylinder, particularly in the lock-in region. The maximum amplitude (A x )r ms /D of the near-wall cylinder is as large as approximately 17 times that of the isolated cylinder. It can be easily deduced here that the wall proximity has a much larger effect in the streamwise direction than the transverse direction, and this wall proximity effect is most pronounced in the lock-in regime. Further, the influence of the wall proximity on the cylinder response shows an inverse trend with that of the hydrodynamic forces. In the earlier discussion, the wall proximity largely enhances the hydrodynamic force on the transverse direction, i.e. C L , but has little effect in the streamwise direction, i.e. C D , as shown in Fig. 4.4. In summary, the major influence of wall proximity is a large enhancement in the streamwise response with little effect on the transverse response; as well as a large increase in the transverse force with little effect on the streamwise force. This shows that the large enhancement in the streamwise response by the wall proximity is not because of the increase in the hydrodynamic force in the streamwise direction.

212

4 Near Wall Effects

Therefore, in the following section, we will investigate why the streamwise response of the near-wall cylinder is largely enhanced compared to its isolated counterpart.

4.2.3.3

Phase Relations Between Forces and Displacements

To offer an explanation of why wall proximity enhances the streamwise vibration amplitude but has little impact on the transverse vibration, the phase angle φC L −Y between the drag force and the streamwise displacement and the phase angle φC D −X between the lift force and the transverse displacement have been obtained by Hilbert transform, given in Fig. 4.6a, b, respectively. As shown in Fig. 4.6a, φC L −Y remains mostly unchanged with the presence of a plane wall in proximity. For both the isolated and the near-wall cases, φC L −Y is close to 0◦ (in-phase) in the pre-lock-in and the initial branch of the lock-in regions, and dramatically increases to 180◦ (out-of-phase) on the lower branch of lock-in with a sudden jump occurring at the demarkation between initial and lower branches. Blackburn and Henderson [51] and Carberry and Sheridan [73] pointed out that the phase shift between the lift force and the transverse displacement of the cylinder is associated with a change in the direction of the energy transfer. Based upon this, in Fig. 4.6a there exists net energy transfer from the fluid to the cylinder in the pre-lockin and initial branch of the lock-in regions, and vice versa in the lower branch of the lock-in and post-lock-in regions. Here, we can define the energy-in phase as the prelock-in and initial branch of the lock-in regions; and the energy-out phase as the lower branch of the lock-in and post-lock-in regions. The positive energy transfer partly leads to the large increase in peak transverse amplitude in corresponding regimes, shown in Fig. 4.5a. The sudden jump in phase difference to 180◦ afterwards, meaning there is no net energy transfer from the fluid to structure, leads to the sudden decrease in peak transverse vibration amplitude, shown in Fig. 4.5a. It is worth pointing out that at Ur = 6 while φC L −Y = 180◦ , meaning the cylinder is damping energy out to the fluid, (A y )max /D is still relatively large, as shown in Fig. 4.5a. At Ur = 6, it is still inside the lock-in range in which both the vortex shedding frequency and the cylinder vibration frequency lock into the natural frequency of the system. This gives rise to the resonance which leads to large vibration amplitude although work done on the cylinder is negative at this moment. Zhang et al. [486] pointed out that as Ur increases within the lock-in range, flutter-induced lock-in turns into resonance-induced lock-in across a critical value of Ur . Thus, the root cause of the large vibration amplitude at Ur = 6 with φC L −Y = 180◦ is the resonance-induced frequency lock-in by VIV. That is to say, the large vibration amplitude in the lock-in range is caused by two superimposed factors: frequency lock-in and energy transfer. As shown in Fig. 4.6b, φC D −X has a remarkable difference between the isolated and the near-wall configurations. The difference is noted in the pre-lock-in and the initial branch of the lock-in regions. For the isolated cylinder, φC D −X is 180◦ throughout all values of Ur considered. For the near-wall cylinder, the trend of variation of φC D −X with Ur is very similar to φC L −Y . Small phase angle (in-phase) is observed in the pre-lock-in and initial branch of lock-in regions, as Ur increases to the lower branch,

4.2 Cylinder VIV in the Vicinity of a Stationary Wall

213

Fig. 4.6 Phase relations as a function of reduced velocity Ur for isolated and near-wall cylinders at Re = 200: a φC L −Y between lift and transverse displacement, b φC D −X between drag and streamwise displacement

φC D −X suddenly jumps to 180◦ . This shows that there exists net energy transfer from the fluid to the structure in the pre-lock-in and initial branch of the lock-in regimes, which largely enhances the streamwise oscillations, as shown in Fig. 4.6b. In contrast to the near-wall cylinder, φC D −X for the isolated cylinder remains 180◦ (out-of-phase), meaning the cylinder is damping energy to the fluid. This leads to the fact that the vibration amplitude is almost negligible for the isolated cylinder in the streamwise direction. From the energy transfer viewpoint, this explains why the wall proximity can largely enhance the streamwise oscillation for an elastically mounted circular cylinder with 2-DoF undergoing VIV. A similar trend of the φC D −X is also reported in [416]. Similarly with the transverse direction, it is also worth pointing out that though φC D −X turns into 180◦ at 5 ≤ Ur ≤ 6 for the near-wall cylinder, (A x )r ms /D is still relatively large, as shown in Fig. 4.6b. The same argument that the frequency lock-in is the root cause of large amplitude vibration and a more predominant factor than energy transfer can also be applied here. The streamwise frequency lock-in observed in the near-wall case is due to the frequency reduction in the streamwise direction, which is initially caused by the suppression of bottom shear layer roll-up. The details of the streamwise frequency lock-in and cylinder-bottom vortex shedding suppression for the near-wall configuration are discussed in Sects. 4.2.3.4 and 4.2.3.6, respectively.

4.2.3.4

Response Frequencies

In the range of lock-in, the frequency of vortex shedding, f vs , or the frequency of lift force oscillation, f C L , is in sympathy with the frequency of cylinder vibration, f , according to Williamson and Roshko [457]. They both lock into the natural frequency of the system, f n , i.e. f vs ≈ f ≈ f n . The dominant frequencies normalized by the

214

4 Near Wall Effects

natural frequency f n as a function of Ur are shown for both the transverse and streamwise directions in Fig. 4.7a, b, respectively. A characteristic feature of VIV is that of the lock-in phenomenon, where the vortex shedding frequency diverges from the Strouhal’s relationship (vortex shedding frequency of a stationary cylinder) and becomes equal or close to the cylinder’s natural frequency. The vibration frequency generally increases with reduced velocity Ur . In the transverse direction, shown in Fig. 4.7a, the response frequencies for an isolated cylinder with mass damping of m ∗ ζ = 0.1 are first compared with the results from Leontini and Stewart [238] to assess the accuracy. For both the isolated and near-wall configurations, the response frequencies behave similarly, following the Strouhal’s relationship in the pre-lock-in and post-lock-in regions but departing from the Strouhal’s relationship and vibrating with the natural frequency in the lock-in region. In the streamwise direction, shown in Fig. 4.7b, it is evident that the response frequencies of the near-wall cylinder behave very differently from the isolated case. As for the isolated configuration, it is intuitive that the vibration frequency in the streamwise direction is twice of the transverse direction owing to the two alternate vortices shed from the cylinder each cycle. Nonetheless, for the near-wall configuration, the vibration frequency in the streamwise direction does not really differ from that of the transverse direction. This is caused by the fact that the counter-clockwise vortices shed from the bottom surface of the cylinder are suppressed by the wall proximity, as discussed in Sect. 4.2.3.6. To better identify the difference between the transverse and streamwise directions, the ratios of the response frequencies in the streamwise direction to the transverse direction are plotted for both configurations with the variation of Ur in Fig. 4.7c. The ratio of the response frequencies in the streamwise direction to the transverse direction for the isolated cylinder is approximately 2, which is consistent with the physics of alternate vortex shedding. However, the ratio becomes unity for the nearwall cylinder. It is observed that in the streamwise direction the vibration frequency of the near-wall cylinder locks into the natural frequency in the lock-in range, whereas the isolated cylinder vibrates with twice the natural frequency. This means that the frequency lock-in is attained in the streamwise direction for the near-wall cylinder, but not for the isolated cylinder. The streamwise frequency lock-in of the near-wall cylinder is the major reason for enlarged streamwise oscillation as compared to the isolated configuration. On top of this, positive energy transfer in the energy-in phase also plays an important role towards enhanced streamwise oscillations. Therefore, at this point, we may conclude that the wall proximity effects not only strongly enhance the streamwise oscillation but also reduce the streamwise vibration frequency by half. Besides the net energy transfer, the streamwise frequency lock-in is the root cause of higher vibration amplitudes for the vibrating near-wall cylinder than that of the wall-free cylinder. We further investigate what underlying physical mechanism induces the streamwise frequency lock-in.

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215

Fig. 4.7 Response frequencies as a function of reduced velocity Ur for both isolated and near-wall cylinders at Re = 200: a transverse, b streamwise and c ratio of streamwise direction to transverse direction

4.2.3.5

Motion Trajectories

The X Y -trajectory for the isolated cylinder takes the form of a classical figure of eight, as reviewed in [381]. The trajectories of both isolated and near-wall cylinders are taken from the cylinder displacements during the dimensionless time range tU/D = 250−500. As shown in Figs. 4.8 and 4.9, with the presence of a plane wall, the trajectory of the freely vibrating cylinder is changed from the typical figure-eight shape into an oblique elliptical configuration. The shape of an oblique ellipse possessed by the near-wall cylinder is similarly reported by Tsahalis and Jones [424]. This indicates a change in the relative frequency ratio of the transverse and streamwise response, which is consistent with our observation in Fig. 4.7. The figure-eight shape for the isolated cylinder case is associated with a frequency ratio of f y / f x ≈ 2, while the oblique elliptic trajectory for the near-wall case corresponds to the frequency ratio of f y / f x ≈ 1.

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4 Near Wall Effects

Fig. 4.8 2-DoF motion trajectories of isolated cylinder for different reduced velocities Ur at Re = 200

One interesting point to note here is that the trajectories at Ur = 4 are not as repeatable as what is shown for other reduced velocities. This is because at this particular reduced velocity, VIV undergoes a transition from the pre-lock-in to the lock-in regime, thus the cylinder responses become increasingly sensitive to the change in reduced velocity. As Zhang et al. [486] pointed out, the cylinder vibration mode transitions from the wake mode (weak interaction) to a combined wake and structure mode (strong interaction) across the critical value of Ur between the prelock-in and lock-in. This mode switching gives rise to unstable and non-repetitive trajectories. Details and explanations of this phenomenon are given in Sect. 4.2.3.11.

4.2.3.6

Flow Fields

In this section, we investigate the flow field and vortices associated with the VIV of the cylinder for both isolated and near-wall configurations. We select three representative reduced velocities, Ur = 3, 5 and 8, in the pre-lock-in, lock-in and post-lock-in regions, respectively. To investigate how the wall proximity affects the vortex dynamics, we look into the time histories of the force coefficients and cylinder displacements at some specific time steps and connect them with the corresponding vorticity contours. The detailed mechanism for vortex shedding suppression of bottom shear layer roll-up is also described in the following subsections.

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217

Fig. 4.9 2-DoF motion trajectories of near-wall cylinder for different reduced velocities Ur at Re = 200

4.2.3.7

Pre-lock-in Region at Reduced Velocity Ur = 3

As shown in Fig. 4.10, for the isolated cylinder, the transverse force and vibration are in-phase (Fig. 4.10a) but the force and amplitude are out-of-phase in the streamwise direction (Fig. 4.10b). For the near-wall cylinder, the hydrodynamic force and the vibration amplitude are in-phase for both transverse and streamwise directions. This is consistent with what we have observed in Fig. 4.6. The phase difference between the lift force and the transverse displacement is approximately 0◦ for both isolated and near-wall cylinders and the phase differences between drag and streamwise displacement are 180◦ and 0◦ for the isolated and near-wall cylinders, respectively. Corresponding vorticity contours are presented in Figs. 4.11 and 4.12. The black dot marked in the plot represents the initial location of the cylinder as a reference. At Ur = 3, the displacements in both transverse and streamwise directions are negligible since it is still in the pre-lock-in region. In Fig. 4.11, regular alternate vortices are shed from the cylinder undergoing VIV, clockwise (negative) vortices from the upper surface and counter-clockwise (positive) from the lower surface of the cylinder. In Fig. 4.12, the clockwise vortices shed from the upper surface of the cylinder coalesce with the clockwise wall boundary layer vortices. This coalescing action strengthens the negative vortices, suppressing the positive ones shed from the lower surface of the cylinder. To better illustrate this mechanism of the bottom shear layer roll-up suppression in the near-wall configuration, a schematic diagram is presented in Fig. 4.13. The wall boundary layer is forced to separate by the counter-clockwise Vortex B shed from the bottom surface of the cylinder. The counter-clockwise Vortex B induces an upward velocity onto the separated wall boundary layer, producing a secondary

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4 Near Wall Effects

Fig. 4.10 Selected time histories of a, c lift coefficient and transverse displacement b, d drag coefficient and streamwise displacement for isolated cylinder (upper) and near-wall cylinder (lower) in pre-lock-in region at Ur = 3, Re = 200

clockwise Vortex C from the wall vorticity layer which eventually merges with the clockwise Vortex A shed from the upper surface of the cylinder. The coalescence of Vortex A and Vortex C in turn suppresses counter-clockwise Vortex B. A similar mechanism for the interaction of wake vortices with ground effects was summarized by Puel and Victor [346]. The suppression of the counter-clockwise vortices shed from the bottom of the cylinder is the reason why the ratio of streamwise vibration frequency to transverse vibration frequency becomes unity rather than 2, which leads to streamwise frequency lock-in and eventually enhanced streamwise oscillations.

4.2.3.8

Lock-in Region at Reduced Velocity Ur = 5

At Ur = 5, the cylinder undergoes VIV in the synchronization regime whereby large vibration amplitudes are expected. The time traces of the force coefficients and displacements for both isolated and near-wall cylinders at Ur = 5 at some specific time steps are illustrated in Fig. 4.14. A relatively small lift force and large transverse displacement can be observed for both configurations. However, large streamwise displacement can be found for the near-wall cylinder in comparison to the isolated counterpart whereas the drag force of the two configurations is quite similar. Corresponding vorticity contours are presented in Figs. 4.15 and 4.16. To analyze the relationship between the vortices shed from the cylinder and the force acting on the

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219

Fig. 4.11 Vorticity contours for isolated cylinder at Ur = 3, Re = 200 for discrete time instants from tU/D = 481 to 486 for flow coming from left to right. An unsteady 2S wake mode is evident

cylinder, as shown in Fig. 4.14c, the movement of the near-wall cylinder is considered from its most negative to the most positive displacement from tU/D = 485 to 487. At tU/D = 485, a strong clockwise vortex starts to form on the upper surface of the cylinder, and this vortex is rolling clockwise from tU/D = 485 to 487 and exerting a large upward force vertically onto the cylinder. The clockwise vortices shed from the upper surface of the cylinder coalesce with the clockwise vorticity layer over the wall. This coalescing phenomenon strongly strengthens the negative vortices, suppressing the positive ones shed from the lower surface of the cylinder. This mechanism of cylinder bottom vortex suppression is much more pronounced in the lock-in region at Ur = 5 as compared to the pre-lock-in region at Ur = 3. Also shown in Fig. 4.15, the vortices shed from the cylinder are much wider in the transverse direction than those for Ur = 3 (see Fig. 4.11) owing to the larger vibration amplitude.

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4 Near Wall Effects

Fig. 4.12 Vorticity contours for near-wall cylinder at Ur = 3, Re = 200 for discrete time instants from tU/D = 481 to 486 for flow coming from left to right. Coalescing of clockwise vortex with wall boundary layer can be seen

4.2.3.9

Post-lock-in Region at Reduced Velocity Ur = 8

At Ur = 8, the cylinder undergoes VIV in the post-lock-in range, where the hydrodynamic force and the displacement are out-of-phase in both transverse and streamwise directions for both isolated and near-wall cylinders, observed in Fig. 4.17. The time traces of the lift coefficient and displacement exhibit harmonic behavior for both configurations. This is also consistent with the observation from Fig. 4.6. The flow field in the post-lock-in region, shown in Figs. 4.18 and 4.19, looks similar with that of at Ur = 3 in the pre-lock-in region.

4.2.3.10

Vortex Shedding Modes

The instantaneous vorticity fields for the fully developed flow for both isolated and near-wall cylinders undergoing VIV at four selected reduced velocities of Ur = 3, 4, 5 and 8 are illustrated in Fig. 4.20. The 2S and C(2S) vortex shedding modes

4.2 Cylinder VIV in the Vicinity of a Stationary Wall

221

Fig. 4.13 An illustration for interaction of vortices shed from the near-wall cylinder with the wall boundary layer

are described in [457] and similarly reported in the literature of laminar VIV of isolated cylinder such as [393, 411, 497]. At low Reynolds number, in the laminar regime, Singh and Mittal [393] found that the mode of vortex shedding is primarily 2S, but C(2S) is observed when the cylinder undergoes high-amplitude oscillations. They also reported the existence of a P+S mode of vortex shedding at Re ≥ 300 Govardhan and Williamson [136] and Khalak and Williamson [206] reported the 2S mode of shedding for the initial branch and 2P mode of shedding in the upper and lower branches. For the isolated cylinder, the vortex shedding possesses a 2S mode for all reduced velocities considered, shown in the left column of Fig. 4.20. To be more specific, as Ur increases from 3 to 5, the separation distance between the positive and negative vortex rows becomes increasingly larger as VIV transforms from the pre-lock-in to the lock-in state. For the values of Ur where the cylinder executes high-amplitude transverse vibration, e.g. Ur = 5, the vortex shedding develops into two parallel rows. The distance between the parallel rows is larger than that of Ur = 4 owing to the larger displacement in the lock-in region. When Ur = 8 in the post-lock-in region, the 2S mode recovers to what is shown for the pre-lock-in region at Ur = 3.

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4 Near Wall Effects

Fig. 4.14 Selected time histories of a, c lift coefficient and transverse displacement b, d drag coefficient and streamwise displacement for isolated cylinder (upper) and near-wall cylinder (lower) in lock-in region at Ur = 5, Re = 200

When the cylinder is placed in proximity to a plane wall, some drastic changes occur in the flow around the cylinder, the break-down of symmetry in the flow and the suppression of bottom shear layer roll-up in general, shown in the right column of Fig. 4.20. The suppression of bottom vortex shedding is closely connected with the asymmetry in the development of the vortices on both sides of the cylinder. The shear layer on the wall-side of the cylinder will not develop as strongly as the shear layer on the freestream side. The combined VIV and wall-induced oscillations due to the asymmetric wake dynamics have also been reported in [401]. Vortex shedding mode 1S is observed at Ur = 3 and 4, in which the positive vortices are strongly suppressed due to the presence of the plane wall since almost no positive vortices can be seen downstream. As Ur increases to 5, weak 2S mode can be seen, where weakly developed positive vortices can be observed downstream but become weaker as it goes downstream due to the wall proximity effects which suppress the development of positive vortices on the lower surface of the circular cylinder. When Ur = 8, the 1S mode recovers to what is shown for Ur = 3 as it transfers to the post-lock-in region. The suppression of counter-clockwise vortices or the shedding of mere clockwise vorticity from the upper surface of the cylinder leads to the earlier observation that the ratio of streamwise vibration frequency to transverse vibration frequency becomes unity when the cylinder is placed in the vicinity of a plane wall.

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223

Fig. 4.15 Vorticity contours for isolated cylinder at Ur = 5, Re = 200 for discrete time instants from tU/D = 482 to 487 for flow coming from left to right. Stretching of vortices can be seen due to large amplitude motion

4.2.3.11

Beating Phenomenon

The beating oscillations are described in this section for two selected reduced velocities of Ur = 4 and 6.5, which correspond to the critical values for VIV enters and leaves the lock-in range, respectively. In previous studies, Leontini and Stewart [238] reported the intermittent responses of the cylinder at Ur = 4.6, whereas Prasanth and Mittal [344] described the intermittent regime and the mode switching during the transition from the initial to the lower branch. The intermittent regime of the upperlower branch transition for 4.7 < Ur < 4.8 is reported in [312].

4.2.3.12

Time Histories of Ur = 4 and 6.5

The traces of lift force and transverse displacement as well as the drag force and streamwise displacement with respect to Ur = 4, the critical Ur at which VIV enters

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4 Near Wall Effects

Fig. 4.16 Vorticity contours for near-wall cylinder at Ur = 5, Re = 200 for discrete time instants from tU/D = 484 to 489 for flow coming from left to right. Stronger coalescing phenomenon strengthens the negative vortices while suppressing the positive ones shed from the lower surface of the cylinder

the lock-in range, for both isolated and near-wall cylinders are displayed in Fig. 4.21. It is evident that both the hydrodynamic force and the displacement on both transverse and streamwise directions clearly exhibit beating oscillations. This kind of beating phenomenon has been reported in [238]. As for the isolated cylinder, shown in Fig. 4.21a, b, long-period oscillations are evident in both forces and cylinder amplitudes, giving rise to the classification of this case as weakly chaotic, as found in [51] for forced cylinder motion in transverse direction. Notably, we observe that when the cylinder is brought close to the plane wall, the absence of periodical repetition, which is clearly observed for the isolated case, reveals the characteristic of random oscillation, as shown in Fig. 4.21c, d. Particularly in the streamwise direction, the drag force and cylinder displacement exhibit nonperiodic and chaotic beating patterns with variable amplitudes and frequencies. This brings the need to utilize Hilbert-Huang transform to conduct the frequency analysis

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225

Fig. 4.17 Selected time histories of a, c lift coefficient and transverse displacement b, d drag coefficient and streamwise displacement for isolated cylinder (upper) and near-wall cylinder (lower) in post-lock-in region at Ur = 8, Re = 200

in the following subsection. It can be inferred here that the wall proximity breaks up the periodicity of the time histories of force and displacement presented by the isolated cylinder. Further, comparing Fig. 4.21c, d, the beating oscillation in force is much more sensitive in the streamwise direction than the transverse direction. The traces of lift force and transverse displacement as well as the drag force and streamwise displacement with respect to Ur = 6.5, the critical Ur at which VIV leaves the lock-in range, for both isolated and near-wall cylinders are displayed in Fig. 4.22. Beating oscillation can still be observed for lift force acting on the isolated cylinder, shown in Fig. 4.22a. When the cylinder is put close to the wall, illustrated by Fig. 4.22c, d, the non-periodicity of force and displacement does not clearly occur due to the wall proximity, as compared to Ur = 4 discussed above. This shows that the hydrodynamic forces and the cylinder responses are more sensitive to the wall proximity when entering the lock-in range than leaving the lock-in range. This can be further deduced to the conclusion that the cylinder at the energy-in phase (at Ur = 4) is more sensitive to the wall proximity than the cylinder at the energy-out phase (at Ur = 6.5).

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Fig. 4.18 Vorticity contours for isolated cylinder at Ur = 8, Re = 200 for discrete time instants from tU/D = 481 to 486 for flow coming from left to right. A standard 2S wake mode can be seen

4.2.4 Three-Dimensional Results and Discussion Our discussion of the VIV dynamics of a circular cylinder with 2-DoF has thus far been based upon a key assumption that for Re = 200 the underlying flow phenomena can be adequately described by 2D simulations. The flow around an isolated cylinder, however, has been shown to transit to 3D at Re = 188, according to Karniadakis and Triantafyllou [201]. Nonetheless, Re = 200 is widely considered in the literature to be the upper threshold for which the wake flow remains 2D, and such examples are [75, 76, 331, 416]. Thus, such 2D simulations for Re = 200 presented in the previous section should be appropriate. To fully examine the wall proximity effects on VIV of an elastically mounted circular cylinder with 2-DoF, we perform 3D simulations by considering circular cylinders of finite span in both isolated and nearwall configurations. VIV of an elastically mounted circular cylinder with 2-DoF at Re = 1000, the beginning of the subcritical regime, in 3D is simulated for Ur ranging from 3 to 9 with an interval of 1. In order to capture long and short timescale variations in

4.2 Cylinder VIV in the Vicinity of a Stationary Wall

227

Fig. 4.19 Vorticity contours for near-wall cylinder at Ur = 8, Re = 200 for discrete time instants from tU/D = 484 to 489 for flow coming from left to right. Merging of wall vorticity layer with clockwise vortices from top cylinder surface can be seen

the fluid flow, at least 50 periodic variations of the cylinder vibration are simulated till the dimensionless time tU/D = 250 with a time step size of Δt = 0.05 in the present 3D studies. For the purpose of direct comparison, 2D VIV simulations are also conducted at Re = 1000 for both configurations, and the 2D results are presented together with the 3D results in the following cylinder responses and the hydrodynamic force coefficients plots. We show the invalidity of 2D numerical simulations for VIV in the subcritical regime in terms of the computations for the cylinder responses and hydrodynamic force coefficients.

4.2.4.1

Amplitudes and Forces

For 3D simulations, although it is not possible to cover as many cases as what we have done in 2D due to high computational cost, it can be seen that the 3D results follow

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4 Near Wall Effects

Fig. 4.20 Vortex shedding modes at a, b Ur = 3 c, d Ur = 4 e, f Ur = 5 g, h Ur = 8 at Re = 200 for isolated (left) and near-wall (right) cylinders

similar trends presented by the 2D data. In Fig. 4.41a, from the 3D simulations, the normalized maximum transverse vibration amplitude, (A y )max /D, of the near-wall cylinder is a bit smaller compared to that of the isolated cylinder. Our (A y )max /D of the isolated cylinder agrees well with [312]. For 2D simulations for the isolated cylinder at Re = 1000, (A y )max /D are quite comparable with 3D results in the prelock-in and post-lock-in regimes where the vibration amplitude is relatively small. However, results for (A y )max /D in 2D are far away from the 3D results in the lockin region in which relatively large vibration amplitudes are observed. Similar large discrepancies can be observed for the 2D results of the near-wall cylinder in the lockin range, but fairly close to 3D results in the pre-lock-in and post-lock-in regimes.

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Fig. 4.21 Time traces of a, c lift coefficient and transverse displacement b, d drag coefficient and streamwise displacement for isolated (upper) and near-wall (lower) cylinders from tU/D = 250 to 500 at Ur = 4, Re = 200

With regard to 3D results for the streamwise direction in Fig. 4.41b, the maximum normalized root-mean-squared streamwise displacement at Ur = 6, (A x )r ms /D, of the near-wall cylinder is approximately 6 times that of the isolated cylinder, compared to 17 times in the 2D studies at Re = 200. This means that the wall proximity still largely enhances the streamwise oscillations in 3D, but to a lesser extent compared to 2D. (A x )r ms /D for the isolated cylinder at Re = 1000 by 2D simulations is overpredicted to a very large extent compared to the 3D results. In terms of the hydrodynamic forces calculated by 3D simulations, the mean lift force coefficient, C L , is enlarged by the wall proximity, shown in Fig. 4.42a. In Fig. 4.42b, we find that C D for the near-wall cylinder is larger than that of the isolated cylinder at every Ur considered. The mean drag C D is enhanced by a factor of 2 at Ur = 4 and by approximately 10% at other Ur due to the wall proximity. However, this is not the case for 2D results for Re = 200, as shown in Fig. 4.4b, where C D are very similar for both configurations. Also, it can be observed in Fig. 4.24 that 2D simulations for Re = 1000, at the beginning of the subcritical regime, are not adequate in capturing the hydrodynamic quantities accurately. Force coefficients are in general over-predicted by 2D simulations, and the most extreme case is (C L )r ms . In [323], they also showed that the mean drag coefficient and root-mean-squared lift coefficient are not correctly calculated by 2D numerical simulations in the subcritical regime (Re = 1.31 × 104 in [323]). That is to say, 2D numerical simulations for 2-

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Fig. 4.22 Time traces of a, c lift coefficient and transverse displacement b, d drag coefficient and streamwise displacement for isolated (upper) and near-wall (lower) cylinders from tU/D = 250 to 500 at Ur = 6.5, Re = 200

Fig. 4.23 Vibration amplitudes as a function of reduced velocity Ur for both isolated and near-wall cylinders at Re = 1000 in both 2D and 3D: a normalized maximum transverse vibration amplitude (A y )max /D, b normalized root-mean-squared streamwise vibration amplitude (A x )r ms /D

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231

Table 4.5 Comparison of present 3D simulations at Re = 1000 and Ur = 6 with Navrose and Mittal [312] at Re = 1000 and Ur = 6.2 m∗ m∗ζ (A y )max /D (A x )r ms /D (C L )r ms CD Reference Present

10 10

0 0

0.62212 0.6147

0.00650 0.0134

0.09944 0.09

1.58270 1.52

Table 4.6 Comparison of [(A y )max /D]∗ by present 3D simulations with experimental studies at similar Re Re m∗ζ [(A y )max /D]∗ Fujarra et al. [122] Angrilli et al. [20] Present

1000–2500 2500–7000 1000

0.023 0.049 0

0.78 0.54 0.6147

DoF VIV at Re = 1000 in the subcritical regime are not valid since the flow is strongly three-dimensional at this particular Reynolds number. Comparisons of the cylinder responses and hydrodynamic forces from our present 3D computations with the results from the numerical study by Navrose and Mittal [312] for an isolated cylinder undergoing VIV at Re = 1000 are tabulated in Table 4.5. As can be seen in the table, (A y )max /D, (C L )r ms and C D in our present simulations at Ur = 6 agree well with those in [312] at Ur = 6.2. It is also worth mentioning that (A x )r ms /D at Ur = 6 in our present study is approximately as large as twice the value at Ur = 6.2 from [312]. One possible reason for the discrepancy at Ur = 6.2 is that (A x )r ms /D reaches the maximum and becomes sensitive to parameter settings at this particular Ur . At other reduced velocities considered, (A x )r ms /D in our study agrees reasonably well with the results in [312]. For instance, at Ur = 5, (A x )r ms /D is approximately 0.008 in [312], compared to (A x )r ms /D = 0.0082 in the present study. For the sake of completeness, the peak normalized maximum transverse vibration amplitude across all reduced velocities considered, [(A y )max /D]∗ , is also compared with two experimental studies conducted in similar Re regimes. The experiments are conducted using an elastically-mounted rigid cylinder and a pivoted cylinder in [20] and [122], respectively. The comparison is shown in Table 4.6.

4.2.4.2

Flow Fields

To gain further insight into the flow structures, the three-dimensionality of the wake is further visualized in the following figures in terms of the isosurfaces of spanwise and streamwise vorticities in Figs. 4.26, 4.27 and 4.28. For the wake structures, similarities and differences can be clearly observed for the isolated and near-wall cylinders. It is worth noting that there is turbulent activity, particularly in the isosurfaces of spanwise vorticity, ωz , downstream of the cylinder, observed in Fig. 4.25a, b.

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Fig. 4.24 Force coefficients as a function of reduced velocity Ur for both isolated and near-wall cylinders at Re = 1000 in both 2D and 3D: a mean lift coefficient C L , b mean drag coefficient C D , c root-mean-squared lift coefficient (C L )r ms , and d root-mean-squared drag coefficient (C D )r ms

However, in the near-wake, the isosurfaces of ωz maintain laminar, yielding an almost two-dimensional wake. This is consistent with the fact that at Re = 1000, in the subcritical regime, the boundary layer remains laminar and the wake becomes turbulent. For the spanwise vorticity, ωz , the counter-clockwise vortices are largely suppressed for the near-wall cylinder, compared to the isolated configuration, owing to the wall proximity effects. Three-dimensional alternate vortex shedding can be observed for the isolated cylinder in Fig. 4.25a, whereas in the near-wall configuration, the counter-clockwise vortices are suppressed and stretched by the wall in Fig. 4.25b. Similarly, with the mechanism of suppression of bottom shear layer rollup in 2D, the clockwise vortices shed from the upper surface of the cylinder coalesce with the clockwise wall boundary layer vortices. This coalescing action strengthens the clockwise negative vortices, suppressing the positive ones shed from the lower surface of the cylinder. We also present the isosurfaces of spanwise vorticity, ωz , from the top view with two specific values of the contour in Fig. 4.26, with symmetric values of ωz = ±0.5

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Fig. 4.25 Isometric view of spanwise vorticity ωz at Ur = 5, Re = 1000 for a isolated and b nearwall cylinders at dimensionless time tU/D = 250. Merging of wall vorticity layer with clockwise vortices from top cylinder surface can be seen in the near-wall configuration

plotted in Fig. 4.26b and asymmetric values of ωz = −3 and 0.5 plotted in Fig. 4.26c. As can be seen from the top view in Fig.4.26a, regular alternate vortex shedding rollers are observed for the isolated case. However, for the near-wall cylinder, the positive spanwise vortices are largely covering the weak positive ones due to the vortex suppression by the plane wall, shown in Fig. 4.26b. If the lower limit of the streamwise vorticity is taken at ωz = −3 to reduce the intensity of negative vorticity, positive vorticity rollers can then be observed under the suppression of negative ones when the asymmetric values of ωz are taken, as illustrated in Fig. 4.26c. The isosurfaces of streamwise vorticity ωx for both isolated and near-wall cylinders in the isometric view are plotted in Fig. 4.27. The 3D effects in the wake also manifest themselves in the form of streamwise vorticity blobs. Observed from the top view of ωx in Fig. 4.28, where two specific values of ωx = ±0.5 are presented, there exists approximately 4.5 wavelengths of the streamwise vorticity blob across the span of the isolated cylinder, i.e. L/λiso ≈ 4.5 where L denotes the span length and λiso denotes the wavelength of streamwise vorticity blob for the isolated case. As for the near-wall cylinder, approximately 1.5 wavelengths of the streamwise vorticity blob across the span can be seen, i.e. L/λnw ≈ 1.5 where λnw denotes the wavelength of streamwise vorticity blob for the near-wall case. It can be concluded here that the wall proximity largely increases the streamwise vorticity blob by a factor of 3, i.e. λnw /λiso = 3. This is because the neighboring plane wall hinders the communication between the streamwise rib vortices, leading to a much larger wavelength of the blob in the near-wall configuration. A summary of the wall proximity effects on VIV of an elastically mounted cylinder with 2-DoF in 3D at Re = 1000 can be given in the following. The wall proximity increases the mean lift force to a lesser extent compared to 2D results at Re = 200, while also enhances the mean drag unlike in 2D at Re = 200. The wall proximity also enhances the streamwise oscillation, similar to 2D. In terms of the flow field,

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Fig. 4.26 Top view of spanwise vorticity ωz at Ur = 5, Re = 1000 for a isolated and b, c nearwall cylinders at dimensionless time tU/D = 250. Alternate vortex rollers are shed in the isolated configuration. The negative vortices are suppressing positive vortices in the near-wall configuration

Fig. 4.27 Isometric view of streamwise vorticity ωx at Ur = 5, Re = 1000 for a isolated and b near-wall cylinders at dimensionless time tU/D = 250

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235

Fig. 4.28 Top view of streamwise vorticity ωx at Ur = 5, Re = 1000 for a isolated and b near-wall cylinders at dimensionless time tU/D = 250. The wavelength of the streamwise vorticity blob is enlarged in the near-wall configuration

the wall proximity increases the wavelength of streamwise vorticity blob as the wall hinders the communication between the streamwise ribs. Similarly with the mechanism of bottom vortex suppression in 2D, wall boundary layer vortices strengthen the clockwise vortices shed from upper surface of cylinder, stretching and suppressing the counter-clockwise vortices shed from the bottom surface of cylinder.

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4 Near Wall Effects

4.2.5 Interim Summary In the present study, 2D and 3D numerical simulations of flow past an elastically mounted circular cylinder with 2-DoF undergoing VIV have been carried out at laminar Re = 200 and subcritical wake flow Re = 1000. In our 2D simulations, comparisons have been established between isolated and near-wall cylinders with a gap ratio of e/D = 0.9 at Re = 200 in terms of the hydrodynamic forces, the cylinder responses, the phase relations between forces and displacements, the response frequencies, the cylinder motion trajectories as well as the vorticity fields. We have found that (1) the wall proximity enlarges the mean lift force (transverse force) but has little effect on mean drag force (streamwise force); (2) the wall proximity promotes the streamwise oscillation by approximately 17 times in comparison to the isolated configuration due to the streamwise frequency lock-in and net energy transfer from the fluid to the cylinder in the pre-lock-in and initial branch of lock-in; (3) the wall proximity reduces the streamwise vibration frequency by half due to the suppression of bottom shear layer roll-up; (4) the mechanism of vortex shedding suppression is a cyclic process where the counter-clockwise vortex shed from the bottom surface of the cylinder forces the wall boundary layer to separate and induces a secondary clockwise vortex from the wall boundary layer vorticity which eventually merges with the clockwise vortex shed from the upper surface of the cylinder, strongly suppressing the counter-clockwise vortex roll as a result; (5) in the energy-in phase (at Ur = 4) the wall proximity breaks up the periodicity of traces of forces and amplitude responses as found in the isolated configuration, and the resulting beating oscillation is more sensitive in the streamwise direction than the transverse direction, confirmed by detailed HHT analysis; and (4) in the energy-out phase (at Ur = 6.5) the wall proximity does not lead to an apparent non-periodicity of traces of forces and responses, showing that VIV is more sensitive to the interference of the neighbouring wall during energy-in phase than energy-out phase. With regard to 3D simulations, similar comparisons have been established on nearwall VIV. We have found that (1) the wall proximity increases the mean lift force to a lesser extent compared to 2D, and also slightly increases the mean drag force which is different from the 2D scenario; (2) the wall proximity promotes the streamwise oscillation by approximately 6 times, a much lesser extent compared to 2D; (3) the wall proximity increases the wavelength of streamwise vorticity blob by a factor of 3 as the neighboring wall hinders the communication between the streamwise rib vortices; and (4) similar suppression mechanism of bottom shear layer roll-up is also observed in 3D. Further studies will be considered to evaluate the effects of oblique flows and associated hysteresis effects due to the wall proximity on the phase transitions, namely, the energy-in (initial branch) and the energy-out (lower branch). While the current study has focused on the large gap ratio e/D ≥ 0.9, there is a need for further study in the small gap ratio range where a rebound of a vibrating cylinder with the plane wall occurs.

4.3 Effects of Wall Boundary Layer Thickness

4.3

237

Effects of Wall Boundary Layer Thickness

Due to the effects of wall proximity, the dynamics of VIV is much more complex than an isolated cylinder in a uniform flow. For a vibrating near-wall cylinder, the vortex shedding may occur even at a very small gap ratio as compared to the stationary counterparts, as experimentally observed by Fredsøe et al. [120]. Investigations for vibrating cylinders near a plane wall were mainly conducted in the moderate to high Re regime. Fredsøe et al. [120] found that the transverse vibration frequency is close to the vortex shedding frequency of a stationary cylinder when the reduced velocity, Ur < 3 and e/D > 0.3. When 3 < Ur < 8, the transverse vibrating frequency deviates considerably from the vortex shedding frequency of a stationary cylinder. Wang and Tan [440] conducted experimental study for a 1-DoF vibrating cylinder near a plane wall at 3000 ≤ Re ≤ 13,000 with a low cylinder mass ratio (m ∗ ≡ 4m/πρ f D 2 L where m is the cylinder mass, ρ f is the density of the fluid, and L is the cylinder spanwise length). It was demonstrated that the neighboring wall not only affects the amplitude and frequency of vibration but also leads to non-linearities in the cylinder response as evidenced by the presence of super-harmonics in the drag force spectrum. The vortices shed that would otherwise be in a double-sided vortex street pattern are arranged into a single-sided pattern, as a result of the stationary wall. Tham et al. [416] presented a comprehensive numerical study on the VIV of a freely vibrating 2-DoF circular cylinder in close proximity to a stationary plane wall at Re = 100. They reported that the effect of wall proximity tends to disappear for e/D ≥ 5 and proposed new correlations for characterizing peak amplitudes and forces as a function of the gap ratio. In this section, a regime map summarizing the vortex shedding modes as a function of the reduced velocity and the wall boundary layer thickness is offered in the laminar flow regime. We also take a closer look at the cylinder bottom shear layer roll-up suppression mechanism, which was reported in Li et al. [242]. In Li et al. [242], the authors proposed a vortex shedding suppression mechanism which is a cyclic process. The counter-clockwise vortices shed from the bottom surface of the cylinder force the wall boundary layer to separate and induce secondary clockwise vortices which merge with clockwise vortices shed from the upper surface of the cylinder. The amalgamating process strengthens the clockwise vortices, suppressing the development of the counter-clockwise vortices from the cylinder bottom. As such, this cyclic process entails the shedding of the cylinder’s bottom shear layer which hinders its self-development. This vortex shedding suppression mechanism reduces the cylinder vibration frequency in the streamwise direction by half and attains the frequency lock-in in the streamwise direction, which in turn largely enhances the streamwise oscillations as compared to the isolated configuration. In this section, we aim to move forward to assess in detail the coupled dynamics between the cylinder wake shear layers and the bottom wall boundary layer by employing the POD analysis. We further investigate this cycling mechanism from the perspective of 3D vortical structures. Lastly, the relations between the cylinder stagnation point movement and the force distributions on the bottom wall are also examined with the aid of the flow field visualizations and force calculations.

238

4 Near Wall Effects

4.3.1

Problem Description and Convergence

4.3.1.1

Computational Setup

The problem of a free-span subsea pipeline is modeled by the configuration in which the flow passes an elastically mounted circular cylinder with two-degree-of-freedom (2-DoF) placed in the vicinity of a plane boundary. The 2D computational domain is depicted in Fig. 4.29 with boundary conditions details. A frame of reference is defined with its origin placed at the center of the circular cylinder and its axes are referred to as (x, y). The circular cylinder mounted in a spring-damper system is placed in proximity of a plane wall. The cylinder center is placed e + 0.5D above the bottom wall. The width of the computational domain is fixed at 20D, or a 5% of blockage. This is large enough to avoid the effects from the top boundary, with the same blockage ratio used in Navrose and Mittal [312], Tham et al. [416] and Li et al. [242]. In this study, e/D = 0.9 is chosen as the representative gap ratio in both 2D and 3D simulations. The reason for this is that the wall proximity effects are pronounced as Tham et al. [416] showed that the root-mean-squared (RMS) maximum transverse displacement peaks at e/D = 0.9. Also, e/D = 0.9 is a sufficiently large gap ratio to avoid the collision between the cylinder and the bottom wall. In the streamwise direction, the cylinder center is placed at an upstream distance, L U , from the inlet and at a downstream distance, L D , from the outlet. In 2D studies, L U is varied to change the boundary layer thickness which is calculated based on classical Blasius’ solutions for flow past a flat plate

without the presence of the cylinder, as in δ =

U) U ∞D = 5 ( URe )( UL∞U ) = 5 DL . The corresponding L U , L D and δ/D used in 5 ν(L U∞ Re the 2D simulations at Re = 100 are listed in the Table 4.7. The boundary conditions are summarized as follows. At the inlet, a Dirichlet velocity for the steady incoming flow is given by u x = U∞ and u y = 0 to represent the free stream. At the outlet, a traction-free condition is imposed. While the slip boundary conditions are applied at the top, no-slip boundary conditions are imposed on the cylinder surface and the bottom wall. For simplicity, the system damping coefficient, ζ , is considered to be zero for both 2D and 3D simulations. In the 2D simulations for Re = 100, at each δ/D, 11 representative Ur are selected with Ur ∈ [2, 20] at two mass ratios of m ∗ = 10 and 2. In this study, Ur is defined as Ur = Ufn∞D , where f n is the natural frequency of

1 k . In the 3D simulations for Re = 200, the the system and defined as f n = 2π m upstream distance is fixed as L U = 10D with δ/D = 1.581, and 7 representative Ur is used for calculations in the range of Ur ∈ [3, 9], with the cylinder mass ratio of m ∗ = 10 and the spanwise length of L = 4D.

4.3 Effects of Wall Boundary Layer Thickness

239

Fig. 4.29 Schematic diagram of 2D computational domain with details of boundary conditions Table 4.7 Length of computational domain and corresponding wall boundary layer thickness ratio for 2D simulations at Re = 100 LU LD δ/D 5D 10D 20D 30D 40D

4.3.1.2

20D 20D 20D 30D 30D

1.118 1.581 2.236 2.739 3.162

Convergence Study

A representative finite element mesh for the case of L U = 10D is shown in Fig. 4.30. As can be seen, finer resolutions are considered close to the cylinder surface and the bottom wall to capture the boundary layer and the wake dynamics accurately. The grid convergence and time-step convergence studies are then carried out to determine the mesh density and the size of time step used in the following study. The hydrodynamic quantities involved are first defined here. The coefficients of mean lift C L and mean drag C D are evaluated as i=n 1 CL = C L ,i , n i=1

i=n 1 CD = C D,i , n i=1

(4.11)

240

4 Near Wall Effects

Fig. 4.30 Representative 2D mesh used in simulations for the near-wall VIV configuration. A close-up view of the mesh around the cylinder in shown in the inset

whereby their root-mean-squared (RMS) counterparts are  i=n 1  =

(C L ,i − C L )2 , n i=1

 i=n 1  (C D )r ms =

(C D,i − C D )2 . (C L )r ms n i=1 (4.12) The maximum transverse displacement, (A y )max , and the RMS streamwise displacement, (A x )r ms , are evaluated as   i=n 1   √ √ 1 i=n

2 (A y,i − A y ) , (A x )r ms =

(A x,i − A x )2 . (A y )max = 2(A y )r ms = 2 n n i=1

i=1

(4.13) where n denotes the number of samples taken. Note that the mean value is subtracted in computing the RMS quantities so that the RMS quantities are direct indicators to measure the fluctuating amplitudes, without being affected by their non-zero mean values. The grid convergence study is first conducted for the VIV of a near-wall 2-DoF circular cylinder of m ∗ = 10 placed at e/D = 0.90 and L U = 10D subjected to the flow at Re = 100 and Ur = 5. Here, the dimensionless time step is taken as ΔtU∞ /D = 0.05. Representative hydrodynamic quantities are considered to assess the mesh independence. Table 4.8 shows that the solutions converge with the increase of mesh density. The results by relatively coarser meshes M1 and M2 are compared with those of the finest mesh M3, and their differences are marked in the respective brackets. Therefore, the mesh M3 is selected to perform the 2D simulations. The time-step convergence study is then conducted using the mesh M3 and four dimensionless time steps of ΔtU∞ /D = 0.100, 0.075, 0.050 and 0.025 for the same

4.3 Effects of Wall Boundary Layer Thickness

241

Table 4.8 Grid convergence study: VIV of a cylinder of m ∗ = 10 placed at e/D = 0.90 and L U = 10D subjected to the flow at Re = 100 and Ur = 5 Mesh Nodes (K) Elements CD (C L )r ms (A y )max /D (A x )r ms /D (K) M1

32.8

32.5

M2

50.5

50.5

M3

63.1

62.5

2.1745 (2.19%) 2.1390 (0.53%) 2.1278

0.4982 (6.16%) 0.4771 (1.66%) 0.4693

0.4905 (4.41%) 0.4788 (1.92%) 0.4698

0.0710 (3.80%) 0.0693 (1.32%) 0.0684

Table 4.9 Time-step convergence study: VIV of a cylinder of m ∗ = 10 placed at e/D = 0.90 and L U = 10D subjected to the flow at Re = 100 and Ur = 5 ΔtU∞ /D CD (C L )r ms (A y )max /D (A x )r ms /D 0.100 0.075 0.050 0.025

2.1047 (1.44%) 2.1148 (0.97%) 2.1278 (0.36%) 2.1355

0.4503 (4.98%) 0.4579 (3.38%) 0.4693 (0.97%) 0.4739

0.4575 (3.52%) 0.4644 (2.07%) 0.4698 (0.93%) 0.4742

0.0652 (5.51%) 0.0666 (3.48%) 0.0684 (0.87%) 0.0690

numerical test as discussed above. The results are summarized in Table 4.9. The results by ΔtU∞ /D = 0.100, 0.075, 0.050 are compared with those of ΔtU∞ /D = 0.025, and their differences are marked in the respective brackets. The time histories of the transverse displacement are also presented in Fig. 4.31. It is evident that the solutions converge as the non-dimensional time-step size ΔtU∞ /D is reduced. Since the differences between ΔtU∞ /D = 0.050 and 0.025 for all hydrodynamic quantities are within 1%, the dimensionless time step ΔtU∞ /D = 0.050 is chosen for the present study to save computational cost without compromising accuracy to a large extent. To verify the accuracy of the numerical technique, the peak normalized transverse displacement [(A y )max /D] peak is benchmarked with other studies in the low Re flow regime and shown in Table 4.10. The present result agrees reasonably well with the reference data. As for the 3D studies, the 3D mesh is simply generated by layering the 2D mesh along the spanwise direction (z-axis). Following Li et al. [242] using the same numerical technique, 40 layers of mesh are used to discretize along the spanwise length of 4D with a spanwise resolution of Δz /D = 0.1. The same numerical methodology has also been previously validated by comparing the numerical results with published experimental and numerical data in Mysa and Jaiman [307], Li et al. [242] and Liu and Jaiman [254].

242

4 Near Wall Effects

Fig. 4.31 Time-step convergence study: time traces of the transverse displacements of a 2-DoF circular cylinder of m ∗ = 10 placed at e/D = 0.90 and L U = 10D subjected to the flow at Re = 100 and Ur = 5 with four dimensionless time steps Table 4.10 Comparison of [(A y )max /D] peak with reference studies at low Re e/D

m∗ζ

Shiels et al. [392] 100

∞

0

0.59

Leontini et al. [238]

∞

0.1

0.51

Category

Study

Numerical

Experimental

∗ e/D

Re 200

[(A y )max /D] peak

Tham et al. [416] 100

0.90

0

0.48

Li et al. [242]

0.90

0

0.49

Anagnostopoulos 90–150 and Bearman [18]

∞

0.179

0.64

Present

0.90

0

0.47

200

100

= ∞ refers to the isolated configuration

4.3.2

Two-Dimensional Results and Discussion

4.3.2.1

Cylinder Response Characteristics

A characteristic feature of VIV is that of the lock-in phenomenon, where the vortex shedding frequency diverges from Strouhal’s relationship (vortex shedding frequency of a stationary cylinder) and becomes close to the natural frequency of the cylinder. With regards to the overall vibrational characteristics, Fig. 4.32 summarizes the normalized maximum transverse displacement, (A y )max /D, and the normalized RMS streamwise displacement, (A x )r ms /D, as a function of Ur with five δ/D for the cylinder placed at e/D = 0.90 with m ∗ = 10 at Re = 100. The general feature observed

4.3 Effects of Wall Boundary Layer Thickness

243

Fig. 4.32 Vibrational characteristics of a freely vibrating circular cylinder as a function of reduced velocity Ur for δ/D ∈ [1.118, 3.162] at m ∗ = 10 and Re = 100: a maximum transverse displacement, b RMS streamwise displacement

in Fig. 4.32a, b, is that the lock-in range shifts towards the direction of larger Ur with the increase of δ/D. In Fig 4.32a, for example, the lock-in regime roughly ranges from Ur = 3 to 8 for δ/D = 1.118 and from Ur = 5 to 10 for δ/D = 3.162. The reduced velocity Ur at which the peak normalized maximum transverse displacement, [(A y )max /D] peak , occurs increases with the increase of δ/D. The [(A y )max /D] peak is recorded at approximately 0.5 for δ/D = 1.118, 1.581, 2.236 and 2.739, slightly larger than that of δ/D = 3.162. In Fig. 4.32b, a similar trend that Ur at which the peak normalized RMS streamwise displacement, [(A x )r ms /D] peak , occurs increases with the increase of δ/D is noticed. [(A x )r ms /D] peak is the largest at δ/D = 2.739 amongst all the cases of δ/D, reaching slightly more than 0.15. In all cases, for Ur ≥ 10, both (A y )max /D and (A x )r ms /D almost remain constant in the post-lock-in region. Li et al. [242] conducted a systematic comparison between the near-wall (e/D = 0.90) and isolated configurations and reported that the wall proximity effect largely enhances the streamwise oscillations as compared to the isolated counterpart and that [(A x )r ms ] peak reaches approximately 0.12 for the near-wall cylinder at Re = 200. In the present work, [(A x )r ms ] peak ranges from approximately 0.07 to 0.16 for different δ/D. For a cylinder undergoing flow-induced vibrations near a stationary plane boundary, the complex interaction of the shear layers in the cylinder near wake with the bottom wall boundary layer strongly alters the periodic vortex shedding, thus affecting the hydrodynamic forces during VIV. Fig. 4.33 summarizes the hydrodynamic force coefficients as a function of Ur with five δ/D. The mean lift and drag coefficients, C L and C D , are depicted in Fig. 4.33a, b, respectively. The RMS lift and drag coefficients, (C L )r ms and (C D )r ms , are depicted in Fig. 4.33c, d, respectively. As observed in Fig. 4.33, the most noticeable trend is, similar to what is shown in Fig. 4.32, that the lock-in range shifts towards the right-hand side with the increase of the wall boundary layer thickness. It is also worth noting that, in Fig. 4.33b, C D

244

4 Near Wall Effects

Fig. 4.33 Force variations as a function of reduced velocity for δ/D ∈ [1.118, 3.162] at m ∗ = 10 at Re = 100: a mean lift coefficient, b mean drag coefficient, c RMS lift coefficient, and d RMS drag coefficient

decreases with the increase of δ/D. The major reason for this is that the effective velocity of the boundary layer flow in which the cylinder is submerged decreases with the increase of the boundary layer thickness, leading to a smaller drag. Lei et al. [235] and Ong et al. [324] both reported the neighboring wall has a decreasing effect on the mean drag. In other words, it is found that C D decreases with the decrease in the gap ratio. In fact, a decrease in the gap ratio at which the cylinder is placed with a given boundary layer thickness is equivalent to an increase in the boundary layer thickness with a given gap ratio. Therefore, the findings on the variation of C D are consistent with previous experimental and numerical studies. A similar trend is also observed for (C D )r ms as shown in Fig. 4.33d. Fig. 4.34 illustrates the cylinder vibration frequencies normalized by the system natural frequency as a function of Ur with five δ/D. Note that the cases with (A y )max < 2 × 10−4 are considered negligible vibration, and thus they are not marked in the plots. In Fig. 4.34a, the normalized transverse vibration frequency, f y / f n , as a function of Ur follows the Strouhal’s relationship in the pre-lock-in

4.3 Effects of Wall Boundary Layer Thickness

245

Fig. 4.34 Dependence of response frequency on reduced velocity for δ/D ∈ [1.118, 3.162] at m ∗ = 10 and Re = 100: a transverse component, b streamwise direction. Cases with (A y )max < 2 × 10−4 are not marked in the plots

and post-lock-in regimes and approximately equals to unity in the lock-in regime. Fig. 4.34b depicts the normalized streamwise vibration frequency, f x / f n , as a function of Ur , which resembles what is shown for f y / f n . This means that the ratio of the cylinder vibration frequency in the streamwise direction to the transverse direction, f y / f x = 1. From numerous previous works and the basic understanding of von Kármán vortex street in the wake of an elastically mounted isolated cylinder, alternating vortices shed from both upper and lower surfaces of the cylinder are shed in each cycle, leading to f y / f x = 2. However, in the near-wall configuration, the ratio becomes unity and the frequency lock-in is attained in the streamwise direction, i.e. f x / f n = 1, in the lock-in range. This has been reported in several previous works on near-wall VIV, such as Tham et al. [416] and Li et al. [242]. The present results are consistent with the earlier findings. In Fig. 4.34, it is also worth mentioning that almost no noticeable vibrations can be observed in the pre-lock-in and post-lock-in regimes at both δ/D = 2.739 and 3.162. The mass ratio of the cylinder is then reduced to m ∗ = 2 to investigate the effects of a smaller mass ratio on the relevant hydrodynamic quantities. Fig. 4.35 summarizes (A y )max /D and (A x )r ms /D as a function of Ur with five δ/D for the cylinder placed at e/D = 0.90 with m ∗ = 2 at Re = 100. To begin, it is worth highlighting the widening of the lock-in range as compared to m ∗ = 10. For instance, with the largest boundary layer thickness of δ/D = 3.162, the lock-in range covers from Ur = 5 to 10 for m ∗ = 10 while the lock-in ranges from Ur = 4 to 15 for m ∗ = 2. The widening of the synchronization regime for decreasing mass is an effect that was first shown by Griffin and Ramberg [140]. Govardhan and Williamson [136] also reported the effect of a mass reduction (from m ∗ = 8.63 to 1.19) can dramatically increase the width of the synchronization regime. In Fig. 4.35a, [(A y )max ] peak is approximately 0.55 at δ/D = 2.236, which is 10% higher than that of m ∗ = 10 at δ/D = 2.236. Similar to m ∗ = 10, as δ/D increases the lock-in range shifts towards the right hand

246

4 Near Wall Effects

Fig. 4.35 Vibrational characteristics of a freely vibrating circular cylinder as a function of reduced velocity Ur for δ/D ∈ [1.118, 3.162] at m ∗ = 2 and Re = 100: a maximum transverse displacement, b RMS streamwise displacement

side for both (A y )max /D and (A x )r ms /D. Fig. 4.35b shows that [(A x )r ms ] peak is the largest at δ/D = 2.739 amongst all δ/D considered, which is 20% larger than that of m ∗ = 10 at δ/D = 2.739. Figure 4.36 summarizes the hydrodynamic force coefficients as a function of Ur with five δ/D. C L and C D are depicted in Fig. 4.36a, b, respectively. (C L )r ms and (C D )r ms are depicted in Fig. 4.36c, d, respectively. As observed in Fig. 4.36, the general trend is that, similar to what is shown in Fig. 4.35, the lock-in range shifts towards the right with the increase of δ/D. Similar to m ∗ = 10 shown in Fig. 4.33a, Fig. 4.36a shows that C L is negative in both the pre-lock-in and post-lock-in regimes at δ/D ≥ 2.236, implying that the cylinder is pressed towards the plane wall in terms of the transverse hydrodynamic force. Also, similar to the larger m ∗ case, the increasing δ/D has a decreasing effect on C D , as shown in Fig. 4.36b. This means that, at a given Ur , C D decreases with the increase of δ/D. It can also be observed in Fig. 4.36c that the maximum (C L )r ms generally decreases with the increase of δ/D. Fig. 4.37 illustrates the cylinder vibration frequencies normalized by the system natural frequency as functions of Ur with five δ/D. Note that the cases with (A y )max < 2 × 10−4 are considered negligible vibration, and thus they are not marked in the plots. In Fig. 4.37a, the normalized transverse vibration frequency, f y / f n , as a function of Ur follows the Strouhal’s relationship in the pre-lock-in and post-lock-in regimes and approximately equals to unity in the lock-in regime. Fig. 4.37b depicts the normalized streamwise vibration frequency, f x / f n , as a function of Ur , which resembles what is shown for f y / f n . As found, the frequency lock-in in the streamwise direction is also attained for m ∗ = 2, like for the case of m ∗ = 10 as discussed earlier. The critical reduced velocity for the onset of VIV, (Ur )crit , is another important parameter in the VIV analysis, which is also highly relevant in practical offshore applications. In the present study, according to DNV RP-F105 [430], Zang and Gao [475] and Zang and Zhou [476], (Ur )crit is defined as Ur at which (A y )max /D = 0.15

4.3 Effects of Wall Boundary Layer Thickness

247

Fig. 4.36 Force variations as a function of reduced velocity for δ/D ∈ [1.118, 3.162] at m ∗ = 2 at Re = 100: a mean lift coefficient, b mean drag coefficient, c RMS lift coefficient, and d RMS drag coefficient

Fig. 4.37 Dependence of response frequency on reduced velocity for δ/D ∈ [1.118, 3.162] at m ∗ = 2 and Re = 100: a transverse component, b streamwise direction. Cases with (A y )max < 2 × 10−4 are not marked in the plots

248

4 Near Wall Effects

Fig. 4.38 Dependence of critical reduced velocity for the onset of VIV for m ∗ = 10 and 2 at Re = 100

in the initial branch. The variation of (Ur )crit as a function of δ/D for both m ∗ = 10 and 2 is displayed in Fig. 4.38. The general trend is that (Ur )crit increases with the increase of δ/D at a given mass ratio, which shows that the increasing δ/D delays the onset of VIV. It also shows that the onset of VIV is delayed with a larger mass ratio at a given δ/D. To be specific, (Ur )crit increases by approximately 30% with an increase in m ∗ from 2 to 10. It is also interesting to recognize the approximate linear variation of (Ur )crit with δ/D for both mass ratios at Re = 100 and e/D = 0.9. It is found that (Ur )crit normalized by (m ∗ ) p can be expressed by a simple equation as a function of δ/D as follows: (Ur )crit = a(δ/D) + b, (m ∗ ) p

(4.14)

with the constants p = 1/6, a = 0.7, and b = 1.63. These constants may vary for a different gap ratio. Two fitted functions are also plotted with the results of (Ur )crit for both mass ratios with the respective value of R 2 which indicates the accuracy of the fitted function. Therefore, Eq. 4.14 can be used to evaluate the (Ur )crit for the onset of VIV in the laminar flow regime of a circular cylinder placed in proximity to a plane wall at the gap ratio of e/D = 0.9. This analysis on (Ur )crit for the onset of VIV further reaffirms the earlier observation that the lock-in regime shifts towards to the right with the increase of δ/D and that the lock-in regime widens at a smaller cylinder mass ratio.

4.3 Effects of Wall Boundary Layer Thickness

4.3.2.2

249

Vortex Shedding Modes

In this section, we aim to classify the vortex shedding patterns in different scenarios and offer regime maps to demonstrate different wake modes. The vortex shedding modes are well documented for the free vibration of an isolated cylinder. Williamson and Roshko (1988) provided the map of regimes for vortex wake modes showing primarily the 2S, 2P and P+S mode regimes. Khalak and Williamson [206] reported the 2S mode for the initial branch and 2P mode in the upper and lower branches. Govardhan and Williamson [136] then confirmed that the initial and lower branches correspond to the 2S and 2P modes by the vorticity measurements for free vibrations. However, to the best of our knowledge, a systematic classification of wake modes for the near-wall VIV in the laminar flow regime is not yet available in the literature. In this work, we first define four different patterns of vortex shedding mode, and then offer two regime maps for both m ∗ = 10 and 2.

4.3.2.3

Terminology for Vortex Shedding Patterns

For the purpose of illustration, four different shedding wake modes, namely W2S(A), W2S(B), 1S and NS, are shown in Fig. 4.39 which depicts the vorticity fields for the case of L U = 20D with δ/D = 2.236 and are defined as follows. • Weak 2S (Mode A)—W2S(A) As shown in Fig. 4.39a, the W2S(A) mode is characterized by the weakly shed counter-clockwise vortices which are stretched and elongated by the clockwise vortices shed from the top surface of the cylinder and the bottom wall boundary layer vortices. The counter-clockwise vortex cut off is so weak that it is almost damped out as it is convected to approximately x/D = 5. • Weak 2S (Mode B)—W2S(B) Shown in Fig. 4.39b, the W2S(B) mode is characterized by the weakly shed counter-clockwise vortices and the clockwise clockwise vortices which are located on both sides of the cylinder downstream. Another obvious difference from W2S(A) mode is that the W2S(B) mode possesses a shorter wavelength of ωz . Further, the cylinder with vortex shedding patterns in the W2S(B) mode generally has a larger vibration amplitude as compared to the W2S(A) mode. • 1S Mode—1S Shown in Fig. 4.39c, the 1S mode is featured with the counter-clockwise vortices which are not cut off by the clockwise vortices, forming a shear layer between the cylinder and the bottom wall. The 1S Mode is formed in the downstream of cylinder for x/D > 5. • No Shedding Mode—NS Shown in Fig. 4.39d, the NS mode is featured with no vortex shedding, which resembles a steady state. The shear layer from the top surface of the cylinder is almost parallel to the bottom wall, and there is almost no interaction between

250

4 Near Wall Effects

Fig. 4.39 Classification of vortex shedding patterns: a Weak 2S (Mode A), b Weak 2S (Mode B), c 1S Mode, and d NS Mode

the upper and the lower shear layers from the cylinder. Further, the vibration amplitudes of the cylinder are almost negligible with this mode.

4.3.2.4

Vortex Shedding Mode Regimes

Figure 4.40a, b present the vortex shedding mode regime maps for near-wall VIV of a cylinder with a mass ratio of m ∗ = 10 and 2, respectively. The most striking feature, as seen in Fig. 4.40, is that the W2S(A) mode dominates the relatively small boundary layer thickness cases of δ/D = 1.118 and 1.581. This shows that when the elastically mounted cylinder is subjected to relatively large effective flow velocity the wake pattern is classified under W2S(A) mode in which the weakly shed counterclockwise vortices from the lower surface of the cylinder are stretched by the merging of the clockwise vortices from the upper surface and the wall boundary layer vortices. Secondly, the NS mode dominates the top left and right corners of the regime maps for both mass ratios. This implies that when the effective flow velocity is relatively small there is no vortex shedding in the pre-lock-in and post-lock-in regimes. In such a case, there barely exists any vibrations for the near-wall cylinder, resembling a scenario where the flow past a near-wall stationary cylinder with the vortex shedding suppressed. For δ/D = 2.236, 2.739 and 3.162, the wake mode typically transits from the NS mode in the pre-lock-in back to the NS mode in the post-lock-in. In Fig. 4.40a for m ∗ = 10, at these three particular δ/D, W2S(B) is the dominant mode in the initial branch of the lock-in regime up to the Ur at which the peak amplitude occurs. The mode then transits to W2S(A) in the lower branch. The mode transition from W2S(B) to W2S(A) is generally associated with the branching of (A y )max /D shown in Fig. 4.40a. In other words, W2S(B) is associated with large amplitude vibrations at a relatively large boundary layer thickness, while W2S(A) is seen in cases where smaller amplitude vibrations are observed. In Fig. 4.40b for m ∗ = 2, this mode

4.3 Effects of Wall Boundary Layer Thickness

251

Fig. 4.40 Regime map of the vortex shedding modes as a function of reduced velocity for five representative wall boundary layer thicknesses of δ/D ∈ [1.118, 3.162] at Re = 100 for the cylinder mass ratio of a m ∗ = 10, b m ∗ = 2

transition from W2S(B) to W2S(A) becomes more apparent due to the widening of the lock-in range. It is also interesting to note that the 1S mode is an intermediate mode state between the W2S(A) and NS in the high Ur range. Prasanth and Mittal [344] also described the intermittent regime and the mode switching during the transition from the initial to the lower branch. Therefore, in the context of near-wall laminar VIV, a mode switching between two sub-categories of W2S modes has been proven to exist from the initial to the lower branch within the lock-in regime, which is consistent with the previous findings with regard to the mode transition.

4.3.3

Three-Dimensional Results and Discussion

After discussing the 2D results, we aim to present the results of 3D simulations for VIV of an elastically mounted 2-DoF circular cylinder with the spanwise length of L = 4D and the mass ratio of m ∗ = 10 placed at the upstream distance of L U = 10D and the gap ratio of e/D = 0.90 for the representative reduced velocities Ur ∈ [3, 9] in this section. By means of the hydrodynamic quantities, the flow fields and the force distributions on the bottom wall, the coupled dynamics involved in the cylinder bottom shear layer roll-up suppression mechanism is examined from a 3D perspective. Further, special attention is paid to how the cylinder response is associated with the force distributions on the bottom wall, which has important practical implications.

4.3.3.1

3D Response Characteristics

The cylinder amplitudes from the 3D simulations are first compared with previously published 2D results found Li et al. [242]. As can be seen in Fig. 4.41, both

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4 Near Wall Effects

Fig. 4.41 Vibrational characteristics of a freely vibrating circular cylinder of m ∗ = 10 as a function of reduced velocity Ur placed at L U = 10D at Re = 200 by 3D simulations: a maximum transverse displacement, b RMS streamwise displacement

(A y )max /D and (A x )r ms /D generally agree well with the reference data, following the same trend of the 2D results. This verifies the accuracy of the present 3D simulations. Nonetheless, differences can be observed at critical Ur entering the lock-in region. Approximately 25% and 18% of differences in (A y )max /D between the 2D and 3D are recorded for Ur = 4 and 4.5, respectively. This is mainly because streamwise vorticity in 3D contributes to the highly sensitive VIV responses at critical Ur entering the lock-in range. This is also attributed to the fact that the near-wall cylinder responses exhibit weakly chaotic beating oscillations at Ur = 4, according to Li et al. [242], where accurate cylinder responses are hard to capture. In the following, the case at Ur = 5 is taken as an example to investigate the flow field, the vortical structures and the force distributions around the cylinder and the bottom wall.

4.3.3.2

3D Coupled Response of Cylinder VIV and Stationary Wall

Further insight into the 3D results can be gained by studying the flow field and the forces acting on both the cylinder and the bottom wall by considering Ur = 5 as a representative. The time histories of the force coefficients, transverse displacement and streamwise displacement are displayed in Fig. 4.42. Figure 4.43 shows the zoomin time traces of the hydrodynamic force coefficients and the cylinder displacement for two cycles, and nine specific time instants in the time range of t ∈ [0T, 1T ] are marked in one cycle. Particular attentions will be paid to these time instants for a detailed investigation of the coupled interaction between the vibrating near-wall cylinder with the stationary wall.

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Fig. 4.42 3D results of a freely vibrating circular cylinder for L U = 10D at Ur = 5,m ∗ = 10, and Re = 200: Time traces of a force coefficients, b cylinder displacements

Fig. 4.43 3D results of a freely vibrating circular cylinder for L U = 10D at Ur = 5,m ∗ = 10, and Re = 200: Zoom-in plot of two cycles of a hydrodynamic force coefficients, b cylinder responses

4.3.3.3

Interaction Dynamics Between Cylinder Wake and Plane Wall

We first aim to establish a link between the wake flow field and the force acting on the bottom wall to gain further insight into the mechanism of the vortex shedding suppression in 3D. At the commencement of the cycle 0T , the cylinder is located at its positive maximum transverse displacement, as shown in Fig. 4.43a. The spanwise vorticity field, denoted by ωz = ±1, illustrated in Fig. 4.45a with the trajectory of the cylinder center and its corresponding location depicted on the top right corner, shows the flow is actually two-dimensional since Re = 200 is not a sufficiently large Re to yield the spanwise variation. To prove that, the instantaneous streamwise vorticity field, denoted by ωx , is not apparent until ωx is set to ωx = ±0.0001 in Fig. 4.44, showing that the streamwise vorticity is so weak that it can be neglected in the following analysis. On the other hand, Re = 200 is widely considered in the

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Fig. 4.44 Isosurfaces of instantaneous streamwise vorticity of ωx = ±0.0001 (yellow contour: ωz = 0.0001; blue contour: ωz = −0.0001) of a 3D freely vibrating cylinder of m ∗ = 10 placed at L U = 10D at Re = 200 and Ur = 5. The flow is from left to right

literature to be the upper threshold for which the wake flow remains 2D, and such examples are found in Carmo and Meneghini [75] and Papaioannou et al. [331]. The trajectory shown here is in a slated oval shape, which is in contrast with the classical figure-of-eight trajectory of the isolated cylinder. This is consistent with Kozakiewicz et al. [220], Tsahalis and Jones [424] and Zhao and Cheng [491] which reported that the X Y -trajectory of a near-wall cylinder is an oval. The 3D flow field at 0T is depicted in Fig. 4.45. In Fig. 4.45a, at the instant of 0T , the counter-clockwise vortex (positive) has just been cut off and shed in the near wake. Adjacent to this positive vortex, there is a clockwise (negative) vorticity roller coupled with the weak positive vortex. The intensity of this negative vorticity roller is much stronger than the positive vorticity roller, which is evidenced by the fact that further downstream the weak positive vortex shed in the earlier cycle is already damped out while the negative vorticity roller is still in existence. More details of the flow field involved within one vibration cycle will be discussed later. In this study, the pressure coefficient, C p , is defined as follows Cp =

p − p∞ , 1 f 2 ρ U∞ 2

(4.15)

where p∞ = 0 is the reference pressure taken at the outlet. It is also interesting to calculate the force acting on the bottom wall, as shown in Fig. 4.45b, c for the pressure distribution and the shear stress distribution, respectively. In this case, the shear stress, τ , is evaluated for its magnitude as follows  τ = μf

∂u ∂y

2

 +

∂w ∂y

2 ,

(4.16)

where u and w are the flow velocity components in the streamwise and spanwise directions, respectively. Therefore, C p and τ distributions on the wall represent the normal force and the tangential force acting on the bottom wall. These two force components are critical in prediction of the scour profile of the deformable seabed beneath the offshore pipelines, see Brørs [67], Liang et al. [243, 244], Lu et al. [262], Zhao and Cheng [490] and Ong et al. [323]. For instance, the calculation of τ

4.3 Effects of Wall Boundary Layer Thickness

255

is important when evaluating the Shields parameter θ ∗ , which is a physical property of the sandy seabed, using θ∗ =

gρ f (s

|τ | , − 1)d50

(4.17)

where g is the gravitational acceleration, s is the relative density of the sediment, and d50 is the mean grain diameter. Fig. 4.45b, c show that the cylinder does not disturb the force distribution on the bottom wall in the downstream of the cylinder to a very large extent. In Fig. 4.45b, it shows that C p is minimum right underneath the cylinder on the bottom wall. Slightly downstream of the trough, C p reaches a local maximum between 0 < x/D < 5. This means that there exists an adverse pressure gradient along the wall leading to the separation of the wall boundary layer. The induced secondary clockwise (negative) vortex separated from the wall boundary layer then merges with the clockwise vortex shed from the upper surface of the cylinder. This amalgamation is pertaining to the vortex shedding suppression mechanism which will be further discussed later. In Fig. 4.45c, it shows that the shear stress also reaches the maximum underneath the cylinder on the bottom wall. The fact that the cylinder center moves in a counter-clockwise direction at Ur = 5 is evident in the X Y -trajectory shown in Fig. 4.47, which is consistent with Zhao and Cheng [491]. As the cylinder moves downwards at 18 T as displayed in Fig. 4.46a, the cylinder arrives at approximately half of the positive maximum transverse displacement. The weakly shed counter-clockwise vortex starts dying out as it is convected downstream. A more important feature at this instant is that the merging starts between the separated wall boundary shear layer and the shear layer shed from the upper surface of the cylinder. It is shown in Fig. 4.48b that the suction pressure (C p < 0) below the cylinder increases as C p becomes more negative. As the gap becomes narrower, the suction pressure increases, and the shear stress right below the cylinder also increases as shown in Fig. 4.48c. As the cylinder moves further downwards at 28 T , as displayed in Fig. 4.46b, the cylinder arrives at approximately its transverse equilibrium position. The weak counter-clockwise vortex continues dwindling as it is convected further downstream. It is evident that |C p | and τ keep increasing as the gap narrows, as shown in Fig. 4.48b, c. As the cylinder continues moving downwards in this half vibration cycle, the cylinder arrives at approximately half negative maximum and negative maximum transverse displacement at 38 T and 48 T , respectively, shown in Fig. 4.23c, d. At 48 T , it is obvious that the coalescence of the induced secondary clockwise vortex and the clockwise vortex shed from the upper surface of the cylinder strongly suppresses the shedding of the proceeding clockwise vortex which has not left the cylinder yet. This strong suppression weakens the positive vorticity. At this instant, it is also conspicuous that the suction pressure and the shear stress right below the cylinder both reach the maximum, as shown in Fig. 4.48b, c, since the gap is the minimum at this moment. The fact that the maximum suction pressure and maximum shear stress

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Fig. 4.45 Flow field characteristics of a 3D freely vibrating cylinder of m ∗ = 10 placed at L U = 10D at Re = 200 and Ur = 5 at the time instant 0T : a isosurfaces of spanwise vorticity of ωz = ±1 (yellow contour: ωz = 1; blue contour: ωz = −1), b pressure coefficient distribution, and c shear stress distribution around the bottom plane wall in the domain of x ∈ [−5, 20] and z ∈ [0, 4]. The flow is from left to right

occur right below the cylinder leads to the equilibrium morphology of the scour hole formed which has important implications for the offshore industry. The maximum scour depth of the equilibrium morphology frequently occurs approximately below the cylinder, see Brørs [67]. As the cylinder starts bouncing back and moving upwards, the cylinder arrives at approximately half negative maximum at 58 T , shown in Fig. 4.23e. At 58 T , the most striking feature of the flow field is that the weak counter-clockwise vortex is almost damped out. At this instant, the proceeding clockwise vortex has just been shed and its intensity has been strengthened by the separated wall boundary shear layer due to the merging process, as discussed earlier. It is also interesting to note that the counter-clockwise vortex in the near wake is largely stretched and suppressed. Fig. 4.48b, c show that the suction pressure and the shear stress start decreasing as the gap between the cylinder and the bottom wall widens at 58 T .

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257

Fig. 4.46 Isosurfaces of spanwise vorticity of ωz = ±1 (yellow contour: ωz = 1; blue contour: ωz = −1) of a 3D freely vibrating cylinder of m ∗ = 10 placed at L U = 10D at Re = 200 and Ur = 5 with its corresponding location in the trajectory at the time instant: (a) 18 T , (b) 28 T , (c) 38 T , (d) 48 T

As the cylinder continues moving upwards in this half vibration cycle, the cylinder arrives at a transverse equilibrium position and approximately positive maximum transverse displacement at 86 T and 78 T , respectively, shown in Fig. 4.23f, g. In Fig. 4.23f, the previously shed weak counter-clockwise vortex completely damped out. While its predecessor already dies out, the current visible counter-clockwise vortex continues being stretched and weakened. It is understandable that both the suction pressure and the shear stress continue decreasing as the cylinder is moving away from the wall, as shown in Fig. 4.48b, c. Figure 4.23h illustrates that the cylinder, at this instant, returns to its start position, the positive maximum transverse displacement. A new weakly shed counterclockwise vortex has been cut off and starts being convected downstream. Also, the C p and τ distributions around the bottom wall at 1T , as displayed in Fig. 4.48b,

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Fig. 4.47 Isosurfaces of spanwise vorticity of ωz = ±1 (yellow contour: ωz = 1; blue contour: ωz = −1) of a 3D freely vibrating cylinder of m ∗ = 10 placed at L U = 10D at Re = 200 and 5 6 7 Ur = 5 with its corresponding location in the trajectory at the time instant: a 8T , b 8T , c 8T and d 1T

c, respectively, resemble the situation at 0T . The cycle restarts from this moment onwards.

4.3.3.4

Relationship Between Cylinder Pressure Distribution and Plane Wall

In Fig. 4.48a, C p distributions around the cylinder mid-plane are depicted at these particular time instants. Their corresponding 2D ωz contour plots with streamlines are presented in Fig. 4.49. At 18 T , when the cylinder is approximately located at half of the positive maximum transverse displacement, the stagnation point is located at around − 56 π . The stagnation point then gradually moves to the lower half of the frontal surface of the cylinder as the cylinder moves downwards. At 48 T when the

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259

cylinder is at its negative maximum transverse displacement, the stagnation point is approximately at π , the centerline of the frontal surface. From 18 T to 48 T , the vortex dynamics are described by ωz contour for −1 ≤ ωz ≤ 1 displayed in Fig. 4.49a– d. The stagnation point, manifested by the location where the flow bifurcates on the cylinder frontal surface, slightly moves from the lower half to the centerline. Another interesting feature is that there exists a local flow reversal, manifested by a small recirculation zone on the bottom wall, approximately located at x/D = 2, due to the separation of the wall boundary layer which is initiated by the shedding of the counter-clockwise vortex from the cylinder bottom. This small recirculation zone underneath the separated wall boundary layer becomes increasingly diminished as the cylinder moves downwards. At 48 T , the recirculation is pushed slightly downstream to x/D = 2.5. Further, the recirculation contributes to the suction pressure on the bottom wall as shown in Fig. 4.49b. The location where the minimum of C p occurs, or the suction pressure, is gradually forced downstream from 18 T to 38 T at approximately x/D = 2, and relocates to nearly x/D = 0 right below the cylinder at 48 T as the cylinder reaches its negative maximum transverse location. In Fig. 4.49c, this small recirculation owning to the separated wall boundary layer is also manifested by a secondary peak of τ distribution on the bottom wall at about x/D = 2. As the cylinder enters the second half of the vibration cycle traveling upwards from 48 T to 1T , the stagnation point moves onto the upper half of the cylinder’s frontal surface. This is evident by Fig. 4.49a, e–h. As the cylinder moves from the lowest to the highest position, the strength of the recirculation underneath the separated wall boundary layer increases and becomes more obvious at x/D = 2. As shown in Fig. 4.49b, c, during this process, both the suction pressure and the shear stress on the bottom wall right below the cylinder decrease as the effects of the cylinder become less pronounced. Therefore, the relation between the stagnation point location and the forces acting on the bottom wall within one vibration cycle can be seen with the aid of the vorticity contour plots. In the first half of the vibration cycle when the cylinder moves from positive to negative maximum transverse displacement, the stagnation point gradually relocates to the lower half of the frontal surface of the cylinder and returns to the centerline at 48 T . During this half cycle, both the suction pressure and the shear stress on the bottom wall right underneath the cylinder increase to the maximum. In the second half of the vibration cycle when the cylinder moves from negative to positive maximum transverse displacement, the stagnation point gradually relocates to the upper half of the frontal surface of the cylinder and returns to the centerline at 1T . During this half cycle, both the suction pressure and the shear stress on the bottom wall right underneath the cylinder decrease to the minimum.

4.3.3.5

3D Vortical Structures

To summarize the vortical structure relevant to the cylinder bottom shear layer rollup suppression, Fig. 4.50 demonstrates three critical instantaneous isosurfaces of spanwise vorticity of ωz = ±1. During every single vibration cycle, a bundle of

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4 Near Wall Effects

Fig. 4.48 Normal and tangential force distributions on the plane wall at Ur = 5: a pressure coefficient distributions around the cylinder mid-plane (θ is the stagnation point angle measured from the centerline of the cylinder rear surface), b pressure coefficient distributions on the bottom wall mid-plane, and c shear stress distributions on the bottom wall mid-plane within one cycle

vortices is convected downstream. This bundle consists of three vortices: (1) Vortex A: clockwise (negative) vortex shed from the upper surface of the cylinder; (2) Vortex B: counter-clockwise (positive) vortex shed from the lower surface of the cylinder; and (3) Vortex C: induced clockwise (negative) vortex due to the upwash of Vortex B. As can be seen in the left image, Vortex B from Bundle 1 (Vortex B1) has already been damped out and disappeared because of vorticity diffusion due to the fluid viscosity while Vortex A1 and Vortex C1 still exist. At this moment, a new Vortex B2 has just been shed from the upper surface of the cylinder but it is strongly stretched and suppressed by the coalescence of Vortex A2 and Vortex C2. The clockwise vorticity is intensified due to the supply of Vortex C2, and the merging of Vortex A2 and Vortex C2 weakens Vortex B2 leading to the shorter life span of Vortex B2. The shorter life span of Vortex B is evident by the fact that it dies out first as the bundle of vortices is convected downstream, as shown by the middle and right images. It is

4.3 Effects of Wall Boundary Layer Thickness

261

Fig. 4.49 Two-dimensional spanwise vorticity ωz contours with streamlines of a 3D freely vibrating cylinder of m ∗ = 10 placed at L U = 10D at Re = 200 and Ur = 5 at the time instant: a 18 T , b 2 3 4 5 6 7 8 8T , c 8 T , (d) 8 T , e 8 T , f 8 T , g 8 T , h 1T . The flow is from left to right. The stagnation point moves upwards with respect to the centerline of the frontal surface and back to the centerline at 1T

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4 Near Wall Effects

Fig. 4.50 Three-dimensional instantaneous vortical structures of a 3D freely vibrating cylinder of m ∗ = 10 placed at L U = 10D at Re = 200 and Ur = 5. The vorticity packets A(−), B(+) and C(−) are coupled and convected downstream. B will die out first due to the cylinder bottom shear-layer roll-up suppression

also worth noting in the right image that a new bundle of vortices starts being shed. The coalescence of Vortex A3 and Vortex C3 hinders the development of Vortex B3. This gives rise to the result that Vortex B3 does not contribute to the streamwise vibration of the cylinder. It further leads to the reduction of the streamwise vibration frequency by half. Ultimately, this streamwise vibration frequency reduction is the reason for the frequency lock-in in the streamwise direction which is the root cause of the enhanced streamwise oscillation for the VIV of a near-wall cylinder. Hence, the cylinder bottom shear layer roll-up suppression, described as a cyclic process in Li et al. [242], is now visualized and further investigated in 3D in this work. This vortical structure is in stark contrast with Nishino et al. [315] where the flow around a circular cylinder in the vicinity of a moving plane wall is studied. In Nishino et al. [315], without the supply from the wall boundary layer, the 3D wake structures are almost symmetric when the gap ratio is relatively large; and the lower shear layer is almost parallel to the moving wall when the gap ratio is relatively small.

4.3.4

Summary

In this study, the VIV of a 2-DoF elastically mounted circular cylinder placed at a gap ratio of e/D = 0.9 subjected to the boundary layer flow is numerically studied in both 2D and 3D. In our 2D investigations, the effects of the wall boundary layer thickness, δ/D, and the cylinder mass ratio, m ∗ , are taken into consideration for the study of the wall proximity effects on the hydrodynamic quantities involved in VIV. It is firstly found that the lock-in regime shifts to the right with the increase of δ/D and that the increasing δ/D delays the onset of VIV. With the reduction of

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263

m ∗ from 10 to 2, the lock-in regime widens as compared to the case of high mass ratio. Secondly, two vortex shedding mode regime maps are then provided for both m ∗ = 10 and 2 as a function of Ur and δ/D, in which four vortex shedding modes are classified, namely W2S(A), W2S(B), 1S and NS. It is shown that the W2S(A) mode dominates the small δ/D cases, i.e. δ/D = 1.118 and 1.581. As δ/D increases, more different shedding modes appear in the regime map: (1) NS mode dominates both the pre-lock-in and post-lock-in regimes. (2) There exists a mode transition from W2S(B) to W2S(A), and this transition is associated with branching. (3) 1S is an intermediate mode between W2S and NS modes. Thirdly, the POD analysis is employed to assess the cylinder bottom shear layer roll-up suppression mechanism and shows that the relative dominance between the shear layer from the cylinder top and the induced shear layer from the wall boundary layer is associated with the boundary layer thickness. In our 3D investigations, the cylinder responses using 3D simulations are first compared with 2D results to ensure the accuracy of the numerical methodology implemented in 3D. By investigating the flow field, the stagnation location on the cylinder frontal surface and the force distributions on the bottom wall within one vibration cycle, the relations between the stagnation point location, the pressure distributions around the cylinder and the forces acting on the bottom wall within one vibration cycle are revealed. The cylinder bottom shear layer roll-up suppression mechanism is also investigated in the perspective of the 3D vortical structures. It is found that a combination of vortices is convected downstream in every single cylinder vibration cycle. This bundle consists of three vortices: the amalgamation of the clockwise vortex shed from the cylinder top and the induced clockwise vortex due to the disturbance from the counter-clockwise vortex shed from the cylinder bottom shortens the lifespan of the vortex from the cylinder bottom. Acknowledgements Some parts of this Chapter have been taken from the PhD thesis of Zhong Li carried out at the National University of Singapore and supported by the Ministry of Education, Singapore.

Chapter 5

FIV Suppression Devices

In this chapter, we introduce various suppression devices we introduce various suppression devices via surface modification, wake stabilization and synthetic jet for flow-induce vibrations. We start with the mechanism and the results of recently developed staggered grooves to suppress the flow-induced vibration of flexible cylinders. We next move to low-drag suppression devices of various types. Detailed contours of vorticity, amplitudes, phase and frequency relations, force histories and wake characteristics provide a deeper insight into complex dynamical interactions between the wake flow and the oscillating cylinder system in the lock-in condition. Simulations of more complex jet-based control are introduced for a freely vibrating 3D semi-submersible model.

5.1 Introduction The role of the vortex-induced vibration (VIV) phenomenon is well-recognized during structural designs in offshore, aeronautics and civil engineering. In particular, effective suppression of VIV and the reduction of fluid loading can lead to safer, sustainable and cost-effective structural design for a broad range of operational conditions. Broadly speaking, the vibrations can be suppressed by controlling the reduced velocity, the vortex shedding and by adjusting the mass and damping of the system. In general, it is desirable to control the vortex shedding and unsteady flow loading so that the excitation forces are eliminated or weakened. In the past several decades, numerous passive control techniques [30, 226, 327, 473, 477] have been investigated via surface modification and by adding auxiliary surfaces to modify the vortex-wake flow dynamics hence reducing the vibration amplitude.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Jaiman et al., Mechanics of Flow-Induced Vibration, https://doi.org/10.1007/978-981-19-8578-2_5

265

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5 FIV Suppression Devices

These passive devices can be classified into two categories based on their mechanisms, namely near-wake stabilizer and surface-geometry modifier [401]. The former device relies on the alterations of the near-wake and the shear layers around the vibrating structure, which results in the wake stabilization and the suppression of VIV. Examples of such wake stabilizing devices are the fairing [12], the splitter plate [24], the guided foil [127], and the recently developed connected-C device [226] and near-wake jet-based actuators. While these devices are effective in suppressing the VIV and reducing the drag force significantly, they need to be aligned with the direction of the oncoming flow. This requires the device to rotate around the structure smoothly as the flow direction changes, which can be somewhat challenging from a mechanical design standpoint, especially for a long deepwater riser and a tall chimney, where the current or wind flow changes its direction irregularly. Owing to the mechanical design considerations, these fairing-type devices may lead to a relatively higher installation and maintenance cost for deepwater risers operating in a harsh ocean environment. These devices are also known to undergo low-frequency galloping instability at higher reduced velocities. Furthermore, the fairing-type devices are difficult to implement over square-shaped multicolumn offshore platforms [79] and subsea pipelines undergoing VIV in the proximity with seabed floor [401]. The second category of the passive control device, namely the surface-geometry modifier, suppresses the VIV phenomenon by manipulating the boundary-layer vorticity distribution and the separation points over the vibrating structure via surface variations, e.g., protrusions and grooves. The well-known examples of the such type of control devices are helical strakes [13, 384, 498], dimples [36], and bumps [35, 327]. Among these devices, helical strakes are the most common and widely used in offshore and wind engineering applications. With the helical strakes on the cylinders, flow separation positions are typically fixed at the sharp edges of the strakes, which may prevent the correlation of vortex shedding along the span. Owing to the helical profile, the performance is independent of the oncoming flow direction, thus it can be mounted on the structure without much mechanical design considerations. The simplicity of mechanical design makes the helical strakes quite economical and robust option for deepwater risers, subsea pipelines, tall chimneys and various circular cross-section towers. However, helical strakes are known to increase the mean drag loading along the span of structure, thereby increases the bending moments on the structure. Moreover, the higher drag loading can dramatically increase the mean deflection and the loads at the top and bottom ends, which pose serious concerns for the design and operation of ultra-deepwater risers. The higher drag of the helical strakes inspires the necessity of alternative design solution, which aims to reduce both the amplitude and drag force experienced by the vibrating cylinder. Apart from the high-drag concern, the difficulty of installing helical strakes on buoyancy modules provide another reason to develop an alternative design from a practical aspect, as discussed in [162, 278]. The modules are made of composite syntactic foam, and are generally used to provide buoyancy and thereby reducing the riser tension as well as creating insulation for the riser pipe. Instead of having protrusions like helical strakes over the buoyancy modules, the surface molded grooves can offer a better choice from a practical viewpoint.

5.2 VIV Suppression by Spanwise Grooves

267

This chapter is structured as follows. Firstly, the VIV responses of the groovedcylinders will be presented at the lock-in condition to compare the performance of VIV suppression against the plain cylinder. The wake structures and the spanwise correlations of sectional hydrodynamic forces will be analyzed to provide some physical insights on the suppression mechanism of the spanwise grooves over the vibrating cylinder. The physical understanding of spanwise grooves has a direct impact to design low-drag and improved VIV suppression solutions for deepwater risers, tall chimneys and towers. Secondly, the two- and three-dimensional VIV simulations of the elastically mounted cylinder-device system are presented in Sect. 5.3. For both 2D low Re and 3D subcritical Re cases, the effects of reduced velocity on the force variation, the vibration amplitudes and the vorticity dynamics are investigated to understand the underlying VIV suppression physics. Detailed contours of vorticity, amplitudes, phase and frequency relations, force histories and wake characteristics provide a deeper insight of complex dynamical interactions between the wake flow and the oscillating cylinder system in the lock-in condition. Finally, comprehensive investigations for various near-wake jet configurations are performed in Sect. 5.4. Test cases are conducted for various reduced velocities, such as pre-lock-in, lock-in and galloping regions. The response characteristics and the flow dynamics of the semi-submersible with proposed near-wake jets are discussed.

5.2 VIV Suppression by Spanwise Grooves The motivation of this study is to develop a surface-modification based device to stabilize the wake flow thereby reducing the VIV amplitude and the mean drag of a freely vibrating cylinder. We consider the spanwise grooves as the surface-geometry modifier, which has been shown to be effective in reducing the drag force for stationary circular cylinders. One of the groove’s geometries investigated is circumferential grooves, where the grooves are aligned perpendicularly to the axis of the cylinder with two different shapes namely U-shaped [248, 348] and V-shaped [233, 239]. Both of them have a small groove depth to the cylinder diameter ratio, which is less than 3%. In these studies, it has been found that U-shaped groove has a higher drag reduction compared to V-shaped grooved. On the other hand, longitudinal (streamwise) groove configuration, where the grooves are aligned to the axis of the cylinder, has been studied [257, 496]. A significant reduction in the drag force via the longitudinal groove was reported in these studies. Longitudinal striations or groove surfaces the so-called riblets are widely studied for skin friction reduction in aeronautical applications [111, 435] and the reduction up to 9.9% in the turbulent skin friction drag is experimentally reported [44]. Of particular interest in this work is to understand the role of these longitudinal grooves over vibrating circular cylinders. In the recent experimental study by [162], the author has considered a triplestarting helical groove for the VIV suppression and the drag reduction of an elastically mounted two-degree-of-freedom circular cylinder. The experiments were conducted at subcritical Reynolds numbers, and the groove tested had the depth of 0.16D, the

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width of 0.2D, and the pitch of 6D, where D is the diameter of the main cylinder. It was found that the peak vibration amplitude and the drag force were reduced up to 64% and 25%, respectively, in comparison to the freely vibrating plain cylinder counterpart. As a follow up to the investigation of helical grooves, a practical design termed as Longitudinal Groove Suppression (LGS), which was inspired by the surface of Saguaro Cacti, was recently investigated and demonstrated for subcritical and post-critical Reynolds number flows [278]. The LGS design consists of ridges and troughs along the circumference of the cylinder, without any mechanical complexity of protrusions or auxiliary moving parts. In addition, the positions are staggered in the spanwise direction of the cylinder. A variety of the catci-like cross sections of LGS designs have been studied experimentally, and it was found that the LGS models can reduce the drag up to 50% when compared to the vibrating plain cylinder. The authors conjectured that the ridges modify the flow in the boundary layer, thus affecting the vorticity dynamics and reducing hydrodynamic forces on the cylinder. The optimized LGS design with low drag performance was found to be quite promising for drilling riser buoyancy modules from manufacturing, deployability and reliability standpoints. In the present work, we focus on the VIV suppression mechanism of grooves with different spanwise alignments. A new suppression device, namely a staggered groove is proposed for the VIV suppression and drag reduction. The device is designed in such a way that it has a regular jump pattern in the cross-section geometry along the spanwise direction, as shown in Fig. 7.18. The staggered pitch ps and the staggered angle θ are the key design parameters, which control the continuous geometric variation. We also study the helical spanwise grooves as a counterpart of the staggered spanwise groove configuration. Unlike the staggered groove, the crosssection geometry of a helical groove transited smoothly along the spanwise direction with the given helical pitch ph , as shown in Fig. 5.1b. We hypothesize that regular and properly designed geometric jumps in the staggered groove may decorrelate the vortex shedding process along the spanwise direction, which can increase the three-dimensionality of the separated flows. We investigate the underlying physical mechanisms by conducting a series of numerical experiments for both the staggered and the helical grooves at sub-critical Reynolds number with low mass and damping values. The key aspect of these simulations is to investigate the percentage changes in the peak VIV amplitude and the mean drag for the grooved cylinders. We employ a partitioned iterative variational framework [180] for the modeling of fluid-structure interaction (FSI) of low-mass bluff bodies. The turbulent bluff-body wake flow is modeled by an explicit large eddy simulation (LES) filtering technique to resolve the large-scale flow features while modeling subgrid-scale stresses [180]. The dynamic LES-based coupled FSI solver has been extensively validated in our previous studies [180, 226, 295]. Here, we further validate the numerical methodology with the available experimental measurements for a plain cylinder with two-degreeof-freedom motions in the streamwise (X ) and transverse (Y ) directions, where the mass and natural frequencies are identical in both X - and Y -directions. To begin with, the characteristic responses of extruded grooves are examined to serve as a reference for our study. Various depths and widths of grooves are simulated to inves-

5.2 VIV Suppression by Spanwise Grooves

269

Fig. 5.1 Variation of cross-section of staggered and helical groove along spanwise direction: a Regular jump in pattern for staggered groove, b smooth transition for helical groove. Here θ is the staggered angle, while ps and ph are the pitch for staggered and helical grooves, respectively

tigate their influence on the control effectiveness of the VIV amplitude and the drag reduction. Among the extruded grooves simulated, the optimal dimensions of depth and width are selected to construct the helical grooved-cylinder and the staggered grooved cylinder. All the investigations are carried out at subcritical Reynolds number where the grooved cylinders are elastically mounted and allowed to vibrate in both streamwise and transverse directions.

5.2.1 Problem Setup and Methodology A schematic diagram is shown in Fig. 5.2 to simulate the flow over groove surfaces. An elastically mounted cylinder with spanwise grooves is immersed in a flowing incompressible fluid stream. The fluid enters the domain from the inlet boundary at horizontal velocity u = U , and exits the domain from the outflow boundary with the traction free boundary condition, σx x = σ yx = 0. The cylinder-groove system is placed 20D away from the inlet boundary and 40D away from the outflow boundary. Its distance to either side is 25D, which results in a blockage ratio of 2%. Both sides are implemented with the symmetry boundary condition, where v = 0 and ∂u/∂ y = 0. The cylinder is elastically mounted on springs with a stiffness value of k and linear dampers with a damping value of c in both streamwise (in-line) and transverse (cross-flow) directions. The resulting frequency ratio between the streamwise and the transverse directions is thus equal to one. It is extruded in the spanwise direction for L = 5D, where its ends are attached to a pair of periodic wall boundaries. The no-slip condition is implemented on the surface of the cylindergroove system using the body-fitted Eulerian-Lagrangian formulation. The fluid and the vibrating cylinder-groove system are coupled such that the fluid velocity is exactly equal to the velocity of the cylinder-groove system. This is particularly important for the chaotic nature of VIV whereby infinitesimally small perturbations to the motion can result in large changes in the transverse force.

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Fig. 5.2 Schematic diagram of computational setup and boundary conditions

Fig. 5.3 Schematic diagram of a cylinder with three uniformly distributed extruded grooves: a cross-section, and b isometric view. Representative width and depth of the groove are w = 0.2D and d = 0.16D, respectively and the grooves are extruded in the spanwise direction

There are several parameters that can characterize the geometry of grooves. Here we mainly focus on the width w and the depth d of the sharp-cornered grooves at the cross-sectional plane. A schematic diagram of the descriptions of these parameters is shown in Fig. 5.3. The reference geometry is selected based on the cross-section of the grooves considered in the experiment study [162], where w = 0.2D, d = 0.16D. We aim to investigate the effect of groove dimensions on the flow dynamics and the VIV characteristics. As discussed for the feedback control via blowing/suction strategy in [465], the actuator placements near the shoulder of the cylinder (before the separation) have a larger impact on reducing mean drag and suppressing fluctuating forces.

5.2 VIV Suppression by Spanwise Grooves

271

Fig. 5.4 Representative cross-sectional geometries of grooves. Unless specified, the width and the height of the grooves are w = 0.2D and d = 0.16D respectively and the grooves are extruded for 5D in the spanwise direction

We consider similar configurations for the placement of passive groove geometries. The moderate adjustments of near-wall vorticity distributions via passive grooves can have a great impact on the boundary layer, the separation point and the wake characteristics. The spanwise grooves with various sizes of depth and width are extruded in the spanwise direction, and their cross-section geometries are listed in Fig. 5.4. The spanwise length of these grooves is set to 5D, which is similar to our previous work [226]. This spanwise length is sufficient to capture the wake dynamic when the cross-sections of the grooves do not vary in the spanwise direction. The optimal width and depth of grooves that give the highest VIV suppression and drag reduction are chosen to construct a helical groove and a staggered groove. Their schematic diagrams are shown in Fig. 5.5. There are three grooves along the circumference for both geometries. Their spanwise lengths are set to 18D in order to capture the three-dimensionality effect induced by the varying geometries along the spanwise direction. The pitch of the helical grooves is set to 6D, which is equal to the one tested in the experiment [162]. The pitch ps of the staggered grooves is set to ps = 2D, which is approximately one-third of the pitch ph of the helical groove configuration. This makes both geometries comparable, as their cross-sections repeat themselves at every 2D in the spanwise direction. Apart from the groove geometry, there are four key dimensionless parameters used to characterize the fluid-structure interaction, namely Reynolds number (Re),

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Fig. 5.5 Two configurations of the spanwise grooves fitted on the vibrating circular cylinder: staggered arrangement (left) and helical profile (right). Three grooves along the circumference span over the cylinder length L = 18D. The pitch of helical groove is 6D, while the pitch of staggered groove is 2D

reduced velocity (Ur ), mass ratio (m ∗ ), and damping ratio (ζ ), which are defined as follows Re =

c U 4m ρfUD , m∗ = , ζ = √ , Ur = f f 2 μ fn D πρ D L 2 mk

 1 k where ρ f is the density of fluid, μ f is the dynamic viscosity of fluid, f n = 2π m is the natural frequency of the spring-mass system in vacuum, and m is the mass of the cylinder. For a given Reynolds number, we can also define dimensional shedding frequency, the so-called Strouhal number St = f vs D/U , where f vs denotes the vortex shedding frequency. In our numerical analysis based on the coupling of incompressible Navier-Stokes and rigid body equations, we use the natural frequency in a vacuum for the purpose of non-dimensionalization. During this fluid-structure

5.2 VIV Suppression by Spanwise Grooves

273

coupling cycle, the added mass effect is implicitly accounted in the coupled formulation and response results are appropriately adjusted to match the experimental conditions [226, 295]. In this study, a spatially-filtered incompressible Navier-Stokes equations with an arbitrary Lagrangian-Eulerian (ALE) description is employed:    ∂uf  + ρ f uf − w · ∇uf = ∇ · σ f + ∇ · τ + bf on Ω f (t), ρ  ∂t x

(5.1)

∇ · uf = 0 on Ω f (t),

(5.2)

f

where uf = uf (x, t) and w = w(x, t) are the fluid and mesh velocities defined for each spatial point x ∈ Ω f (t), respectively, bf is the body force applied on the fluid and σ f is the Cauchy stress tensor for a Newtonian fluid, and τ represents the extra stress term due to the subgrid filtering procedure for large eddy simulation. The spatial and temporal coordinates are denoted by x and t, respectively. In Eq. (5.1), the ALE referential coordinate  x is kept fixed. The structural equation of the translational motion of the elastically mounted cylinder is given by m·

dus + c · us + k · (ϕ s (t) − ϕ s (0)) = F, dt

(5.3)

where us and ϕ s (t) are the velocity vector and the position vector of the center of the cylinder; m, c, and k are the mass, damping and stiffness coefficient matrices; F is the integrated hydrodynamic forces along the surface of the cylinder wall. The fluid and the structural equations are coupled by the continuity of velocity and traction along the fluid-structure interface. To account for the fluid-body interaction, the recently proposed partitioned iterative scheme based on the nonlinear interface force correction is employed [180, 181]. The adopted fluid-structure interaction solver has been extensively validated for a wide range of fluid-structure interaction problems at subcritical Reynolds number [180, 226, 295].

Convergence Study and Validation An unstructured finite element mesh is used to discretize the computational domain for all configurations. A representative mesh for the cylinder with four grooves is shown in Fig. 5.29. A finer mesh is created at the region around the cylinder and the grooves to capture the dynamics of the boundary layer and the vortical wake flow. To resolve the boundary layer dynamics accurately, the dimensionless firstgrid point from the wall surface y + value is controlled such that it is smaller than 1. The temporal and spatial discretizations are selected based on our previous studies [226, 295]. The grid convergence is conducted by simulating a representative case using meshes with different number of nodes and elements. A representative case is chosen to be a cylinder subjected to a flow at Re = 4800, Ur = 5.6. A summary of

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Fig. 5.6 Representative computational mesh: a mesh of the full domain, and b mesh around the cylinder with four grooves Table 5.1 Characteristic responses for different mesh sizes at Re = 4800, Ur = 5.6 Mesh Nodes (×106 ) A∗x,max A∗y,max Cl,r ms Cd,mean M1 M2 M3

0.405 0.848 1.683

0.341 (13.4%) 1.145 (2.2%) 0.308 (2.3%) 1.117 (0.3%) 0.301 1.120

1.655 (6.7%) 1.770 (0.2%) 1.773

2.615 (4.9%) 2.779 (1.0%) 2.750

The percentage in the bracket indicates the deviation from the corresponding value of M3 mesh

the meshes and their corresponding response results is listed in Table 5.1. By taking the mesh M3 as a reference, the discrepancy between amplitude and force results for M1 and M2 are quantified inside the brackets in the table. As apparent, the response results of the mesh M2 are very close to that of M3 thus it is adequate to consider the mesh M2 from a computational efficiency viewpoint. To validate the accuracy of our numerical solver, the simulation results are compared against the measurement data of [189] for a freely vibrating circular cylinder at low mass and damping. The cylinder is allowed to vibrate freely in two-degreesof-freedom (2-DOF) motion, which includes both the in-line (X ) and the cross-flow (Y ) directions. The mass and natural frequencies of the spring-mass system are identical in both directions. Consistent with the experimental set-up, a low mass ratio of m ∗ = 2.6 and a damping ratio of ξ = 0.0036 are used to match the experimental setting and the 2-DOF VIV simulations are performed for the Reynolds number Re ranging from 2000 to 10,000. In particular, the aim of this validation study is to establish the predictive capability of our numerical solver in the various regimes of VIV at low mass and damping. The maximum amplitudes of both in-line and cross-

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275

Fig. 5.7 Comparison of in-line A∗y and cross-flow A∗x amplitudes with experimental data from Jauvtis and Williamson [189] at m ∗ = 2.6, ξ = 0.0036, and Re ∈ [2000, 10,000]. The error bars are set to 5%, which is an upper bound for the coefficient of variation computed from the amplitude collected

flow directions are extracted from 10 oscillation cycles when the system reaches a steady state. Figure 5.7 shows the comparison of the in-line A∗y and the cross-flow A∗x amplitudes against the experimental data. Furthermore, the normalized crossflow frequency, f y∗ = f y / f N is also compared in Fig. 5.8. Overall, both the response amplitudes and the frequency trend agree reasonably well with the experimental data. Based on the vortex wake modes and the response amplitudes, there are five branches identified in the measurement of [189] as a function of normalized reduced velocity, namely the symmetry (SS), the antisymmetric (AS), the initial, the super-upper, and the lower branch. Our numerical results reasonably predict the five-branch response pattern for the 2-DOF VIV at low mass and damping parameters. Notably, our solver predicts the super-upper branch (A∗y ∼ 1.5) with a high transverse amplitude, which is a distinctive phenomenon of a freely vibration of the cylinder at the low mass ratio with 2-DOF motion. With regard to the phase angle and the vortex formation, this supper-upper branch is different from the upper branch of the transverse 1-DOF motion [189, 226]. We next briefly investigate the vortex dynamics that play an important role to sustain the vibration amplitudes through the work done by the vortex-induced force. From each of the five branches, a representative reduced velocity is selected to investigate the vortex wake patterns behind the freely vibrating cylinder. The instantaneous z-vorticity contours at the maximum cross-flow displacement are shown in Fig. 5.9. The observed wake patterns remarkably agree with the observation in the experiment for the AS branch and the initial branch with the 2S wake mode and the lower branch with the 2P mode. While the 2S wake mode exhibits two single

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Fig. 5.8 Comparison of the normalized cross-flow frequency, f y∗ = f y / f N with the experimental data for 2-DOF plain cylinder. The Strouhal number (St) is taken to be 0.2 in the plot

counter-rotating vortices per cycle, the 2P mode consists of two pairs of counterrotating vortices per cycle. In the case of the SS branch, a symmetric 2S mode is not distinctly observed, which may be due to its sensitivity whereby any deviation to the initial condition may break up the symmetry of vortices shed. For the super-upper branch, the 2T mode can be somewhat seen very close to the upper part of the cylinder in Fig. 5.9d. The 2T mode comprises a triplet of vortices forming in each half-cycle, which is different than the 2P mode (observed in the upper branch). The 2T mode has additional opposite-signed vorticity together with the vortex pair P. In Fig. 5.9d, the 2T vortex wake modes become highly irregular away from the vibrating cylinder. It is known that the upper (super-upper) regime of VIV circular cylinders can exhibit low-dimensional chaos [303, 489], whereby the interaction between the body and vortex wake system can be extremely sensitive. Owing to the rapid divergence of trajectories in the phase space for low dimensional chaos, a small perturbation to the movement of the cylinder may result in a large variation in the vorticity distribution, which in turn impacts the force due to the vortex shedding. A detailed investigation of vortex wake modes is beyond the focus of the present study. The present validation study deems sufficient to serve as the reference to examine the VIV characteristics of the cylinder-groove system. In the following section, we present the coupled dynamical response of the grooved cylinders immersed in a uniform flow.

Results and Discussion To assess the vibrational characteristics and the vorticity dynamics of the staggered and helical groove configurations, the numerical simulations are performed at identical physical conditions corresponding to Re = 4800, Ur = 5.6, m ∗ = 2.6, and ξ = 0.0036. These conditions are selected based on the validation conducted in the previous section and the reduced velocity corresponds to the lock-in condition at the super-upper branch. The characteristic responses are collected for each simulation, which includes the maximum normalized cross-flow amplitude ( A∗y,max ), the

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277

Fig. 5.9 Vortex wake modes at different reduced velocities for 2-DOF motion of elastically mounted circular cylinder at m ∗ = 2.6, ξ = 0.0036 . The snapshots are taken when the cross-flow displacement reaches maximum. The indices A1, A2, and A3 specify the vortex shedded in a half cross-flow oscillation cycle. The cross indicates the initial position of the center of the cylinder

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maximum normalized in-line amplitude (A∗x,max ), the root-mean-square of lift coefficient (Cl,r ms ), and the mean drag coefficient (Cd,mean ). We first present the results of the extruded grooves (shown in Fig. 7.19b) with various depths and widths, which will be utilized to determine the effective dimensions of spanwise groove configurations. Next, the dynamic response characteristics of both the helical and the staggered grooves are presented. We quantify the spanwise correlation to understand the role of cross-sectional variation along the cylinder-groove system. In particular, we consider the cross-correlations of the sectional lift and drag forces to estimate the spanwise correlation. For consistent comparisons, we keep the size of the computational domain, the mesh distribution and the time step nearly similar for all configurations. We run all simulations until the statistically stationary states are achieved for the post-processing.

5.2.2 Extruded Grooves Before we proceed to the study of staggered groove configuration, a brief parametric investigation of the depth and the width of the grooves is performed to understand the sensitivity of groove dimensions with regard to their effectiveness in suppressing the VIV and reducing the drag force. These grooves are extruded in the spanwise direction, hence they do not have a variation in the cross-section geometry along the spanwise direction. Three different sets of depth and width are considered for the sensitivity study. To examine the performance of staggered and helical configurations, the dimensions which provide the suppression of more than 10% in the cross-flow amplitude and drag forces compared to the plain cylinder will be considered in the following section. This particular selection of the dimension is to ensure that it has a relatively significant effect on the characteristic responses of the grooved-cylinder system.

5.2.2.1

Depth of Grooves

The characteristic responses of grooves with different depths are investigated and compared against the plain cylinder counterpart. Three depths are investigated, which include d = 0.08D, 0.12D, and 0.16D. Their results are summarized in Table 5.2. The table shows that all these grooves reduce the VIV cross-flow amplitude and the drag coefficient, while increasing the in-line amplitude and the lift coefficient. However, both the reduction and increment of characteristic responses are not larger than 10% (except for d = 0.16D), which suggests that they have similar near-wake dynamics. Among the grooves simulated, it is found that the cross-flow amplitude and the drag coefficient reduce as the depth of the groove increases. Among the depth simulated, d = 0.16D has the best performance, in which the cross-flow amplitude and the mean drag coefficient are decreased by 15% and 10%, respectively. Figure 5.10 shows the time traces of the normalized cross-flow displacement (y ∗ = y/D) and the lift coefficient for the plain cylinder and the cylinder-groove configurations. It

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279

Table 5.2 Comparison of the characteristic responses among the plain cylinder and the grooves with different depths Cylinder d = 0.08D d = 0.12D d = 0.16D A∗y,max A∗x,max Cl,r ms Cd,mean

1.12 0.31 1.77 2.78

1.03 (−8.04%) 0.33 (6.45%) 1.90 (7.34%) 2.63 (−5.40%)

1.03 (−8.04%) 0.32 (3.23%) 1.94 (9.60%) 2.61 (−6.12%)

0.95 (−15.18%) 0.32 (3.23%) 1.90 (7.34%) 2.50 (−10.07%)

The value in bracket indicates its percentage difference to the corresponding value of plain cylinder’s response. Negative value indicates a reduction, while positive value indicates an increment

Fig. 5.10 Time histories of cross-flow displacement and lift coefficient cylinder and grooves with different depth at Re = 4800, Ur = 5.6

can be seen that the time histories of these grooves are similar to one of the plain cylinders. The cross-flow displacement is always in-phase with the lift coefficient in all cases simulated. We shall now consider the qualitative relationship between the vorticity dynamics and the groove geometry. The spanwise z-vorticity contours of these grooves when their displacement reaches maximum are shown in Fig. 5.11. It is observed

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Fig. 5.11 Instantaneous z-vorticity contour around cylinder and grooves with different depths at Re = 4800, Ur = 5.6. The snapshots are taken when the crossflow displacement of the cylinder reaches maximum. The cross indicates the initial position of the center of the cylinder

that the presence of grooves promotes the flow to separate at the edges of grooves. This induces a perturbation in the boundary layer, which results in a modified local vorticity distribution in the wake of the grooved-cylinder system. The vortex force on the body is dependent on the change of impulse of the vorticity distribution in the wake. A small difference in the vorticity distribution will influence the work done on the cylinder body due to the vortex force. At a larger distance, the vortex shed by grooves has a similar wake mode as the plain cylinder.

5.2.2.2

Width of Grooves

The characteristic responses of the grooves with different widths are summarized, together with the comparison with the plain cylinder counterpart. Three values of

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281

Table 5.3 Comparison of characteristic responses among the plain cylinder and the grooves with different widths Plain cylinder w = 0.10D w = 0.15D w = 0.20D A∗y,max A∗x,max Cl,r ms Cd,mean

1.12 0.31 1.77 2.78

1.11 (−0.89%) 0.30 (−3.23%) 1.89 (6.78%) 2.69 (−3.24%)

1.08 (−3.57%) 0.34 (9.68%) 1.94 (9.60%) 2.64 (−5.04%)

0.95 (−15.18%) 0.32 (3.23%) 1.90 (7.34%) 2.50 (−10.07%)

The value in the bracket indicates its percentage difference to the corresponding value of the plain cylinder’s response. A negative value indicates a reduction, while a positive value indicates an increment

widths are considered namely w = 0.10D, 0.15D, 0.20D, and their results are summarized in Table 5.3. It can be seen that, as the width of the grooves become smaller, it resembles the responses of the plain cylinder. Moreover, as the width gets larger, it shows a relatively higher reduction in the cross-flow amplitude and the mean drag coefficient, as well as a relatively lower increment in the rms of lift coefficient. Overall, among the width examined, it can be deduced that grooves with w = 0.20D have the best performance in terms of VIV suppression and drag reduction. The time histories of the normalized cross-flow displacement and the lift coefficient are shown in Fig. 5.12. Similar observations are found, where the displacement of the grooves is close to the one of the plain cylinder. The differences in the widths of the grooves do not significantly change the phase relation between the cross-flow amplitude and the lift coefficient. As discussed earlier, the complete description of the vorticity field provides the vortex-induced lift force through the vorticity impulse relation. Figure 5.13 shows the spanwise vorticity contours of grooves with different widths. Similar to the observations in Fig. 5.11, the presence of grooves introduces a perturbation in the vortex shedding by the grooved-cylinder configuration. Due to the narrow opening for the groove with w = 0.10D, the fluid in the grooves has a relatively lesser interaction with the flow around the cylinder. Therefore, the grooves with smaller widths resemble the profile of the plain cylinder, which agrees with the observed responses in Table 5.3. Next we investigate the staggered and helical grooves at (Re, Ur , m ∗ , ξ ) = (4800, 5.6, 2.6, 0.0036).

5.2.3 Staggered and Helical Grooves Among the dimensions investigated in the parametric investigation conducted in the previous section, it is found that grooves with w = 0.20D and d = 0.16D have a reduction of more than 10% in the cross-flow amplitude and the drag reduction compared to the plain cylinder. These dimensions are used to construct the helical and staggered groove configurations. The spanwise length L is set to 18D to capture the three-dimensional effect in the spanwise direction. The plain cylinder counterpart

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Fig. 5.12 Time traces of cross-flow displacement and lift coefficient of cylinder and grooves with different widths at Re = 4800, Ur = 5.6 Table 5.4 Comparison of characteristic responses among the plain cylinder, the helical and the staggered groove configurations Plain cylinder Staggered groove Helical groove A∗y,max A∗x,max Cl,r ms Cd,mean Cd,r ms

1.10 0.28 1.75 2.86 1.20

0.69 (−37.27%) 0.07 (−75.00%) 0.12 (−93.14%) 2.13 (−25.52%) 0.29 (−75.83%)

1.05 (−4.55%) 0.15 (−46.43%) 1.49 (−14.86%) 2.31 (−19.23%) 0.72 (−40.00%)

The value in the bracket indicates its percentage difference from the corresponding value of the plain cylinder. A negative value indicates a reduction, while a positive value indicates an increment

is also simulated to serve as a reference case. Their characteristic VIV responses are summarized in Table 5.4, while their time histories are shown in Fig. 5.14. It is found that the staggered groove is quite effective in suppressing the VIV, whereby the cross-flow and the in-line amplitudes are reduced by 37% and 75% respectively compared to the plain cylinder counterpart. Moreover, the lift and drag

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283

Fig. 5.13 Instantaneous z-vorticity contour around cylinder and grooves with different depth at Re = 4800, Ur = 5.6. The snapshots are taken when the displacement of the cylinder reaches maximum. The cross indicates the initial position of the center of the cylinder

coefficients are also reduced by 93% and 25%. On the other hand, while the helical groove does not have a significant reduction in the cross-flow amplitude, it reduces the in-line amplitude up to 46%. Now there is a need for a deeper understanding of the VIV phenomenon for a freely vibrating cylinder-groove system. To investigate the underlying physical mechanisms, let us consider a series of questions, namely: What is the cause of the VIV amplitude reduction for the staggered groove? What are the differences in the energy transfer between the plain cylinder and the grooved cylinder? What is the relationship between the spanwise correlation and the energy transfer during the fluid-structure coupling? What is the role of the broadening of frequency spectra on VIV suppression? Do the grooves promote three-dimensionality in the near-wake region behind the vibrating cylinder-groove system? How do the

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Fig. 5.14 Time traces of response amplitudes and force coefficients for the plain and groove cylinders at the identical VIV parameters

distributions of the vorticity appear visually for the groove configurations in contrast to the plain cylinder counterpart? We attempt to address these questions in the below subsections.

5.2.3.1

Energy Transfer and Response Amplitudes

To begin, we first explore the energy transfer between the fluid flow and the vibrating cylinder-groove system. Figure 5.15 shows a time history of the instantaneous power transfer from the fluid flow to the elastically-mounted cylinder. The energy transfer from the fluid dynamics to the body motion is instantaneous and represents the work done by the fluid force through a cycle. Qualitatively, when the body moves in the upward direction, there is positive work done due to the vorticity dynamics and vice versa. As discussed in [189], a peak energy transfer into the vertical motion will occur when the dominance of clockwise vorticity happens as the body moves downwards. From Fig. 5.15, it is observed that the staggered groove has the lowest power transferred, followed by the helical groove and the plain cylinder. This result is consistent with the amplitude results summarized in Table 5.4. The staggered groove has the lowest vibration amplitude, hence it has a relatively lesser energy transfer from the fluid flow to the vibrating cylinder-groove system. There exists a complex interaction between the cylinder-groove system and the wakes dynamics. It has been known that there are sectional forces that are correlated

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285

Fig. 5.15 Time history of instantaneous power transferred to the plain cylinder, helical groove, and staggered groove. The power is directly computed as an inner product of instantaneous hydrodynamic force and instantaneous velocity

with the cylinder motion, and the displacement of the cylinder can increase the spanwise correlation of the fluid forces over the cylinder, as reported experimentally in [53, 55]. Due to the changes in the cross-section of the grooved cylinder in the spanwise direction, the hydrodynamic force experienced in each section may have different frequencies and phase angles. We extend our investigation by relating the power transferred to the spanwise correlation of hydrodynamic forces. A detailed derivation of the relationship between the sectional forces and the energy transfer is presented in the next section. We next investigate the relationship between the sectional forces and the cylinder amplitudes. In particular, we estimate the spanwise correlations for the staggered and the helical groove configurations, which measure the waviness of flow properties in the spanwise z-direction. The correlation length can be quantified using the spatial-temporal variations of fluid forces, which are directly dependent on the vorticity distributions in the near-wake region. A correlation length provides a statistical description to characterize a representative length scale for the spanwise fluctuations associated with three-dimensional effects.

5.2.3.2

Sectional Forces and Energy Transfer

Here, we present some analytical relations for the sectional forces, the energy transfer and the spanwise correlation for a vibrating cylinder. The extraction method of the sectional forces is first presented, followed by establishing its analytical relation to the power transferred [303]. The computation of the spanwise correlation is briefly presented, and its connection with energy transfer is discussed. The fluid and the vibrating cylinder are coupled through the velocity continuity condition and the cylinder motion is driven by the hydrodynamic force integrating the pressure

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Fig. 5.16 Schematic diagram to illustrate the extraction of sectional force. Grooved-cylinder are divided into M segments along the spanwise direction, whereby each segment has a length of z. The position and the fluctuating component of force collected on each segment are labeled as z m and Fm , respectively. Sectional per unit fluctuating force s(z m ) are obtained by dividing Fm with z

and shear stresses on the cylinder surface. The hydrodynamic force F, as given in ¯ and Eq. (5.3), can be decomposed into two components: the mean component, F the periodic component, F  . While the mean component is related to the average pressure difference between the stagnation point and the base point, the periodic component is induced by the vortex shedding process. Let d = ϕ s (t) − ϕ s (0) be the displacement of the center of the cylinder, its solution to Eq. (5.3) can thus be written in the following form: d = d¯ + d 

(5.4)

¯ where d¯ = F/k is the mean displacement caused by the mean force, and d  is the periodic displacement induced by the periodic force. We next measure the forces along the grooved-cylinder by dividing it into M segments, as depicted in Fig. 5.16. To obtain the sectional forces, forces collected at each segments are divided by the length of the segment, whereby F  can be written as a sum of sectional forces over each segment as follows F =

M  m=1

 Fm =

M  m=1

s(z m )z

(5.5)

5.2 VIV Suppression by Spanwise Grooves

287

where s(z m ) is the sectional (per unit length) fluctuating force extracted from m-th segment at position z m . Here z is the length of each segments, which is set to 0.2D for all segments. Assuming that the sectional force is sinusoidal, it can be further written as: M 

Fi =

si0 (z m ) sin(2π f i (z m )t + i (z m ))z

(5.6)

m=1

where i = 1, 2 indicates the degree-of-freedoms, si0 (z m ) is the maximum sectional force at position z m , f i (z m ) and i (z m ) are the corresponding frequency and phase angle. With this, it can be shown that the steady-state solution of d  is: di =

M 

di0 (z m ) sin(2π f i (z m )t + i (z m ) − ϕi (z m ))

(5.7)

m=1

where di0 (z m ) is the amplitude induced by the sectional fluctuating force σ i (z m ), ϕi (z m ) is the phase lag between the cylinder displacement and the fluid force. Both di0 (z m ) and ϕi (z m ) are related to the sectional force through the following expressions: di0 (z m ) =

si0 (z m )z



2π 4π 2 M 2 ( f i (z m )2 − f N2 )2 + c2 f i (z m )2  c f i (z m ) ϕi (z m ) = tan−1 2π M( f i (z m )2 − f N2 )2

(5.8)

(5.9)

 1 k where f N = 2π is the natural frequency of the system. The mean power transfer M from the fluid to the system due to the periodic motion is then given by: 1 P = T

t+T

F ·

d ˙ ( d )dt dt

(5.10)

t

where T is the common period of all sectional forces. It can be taken as a large value, such that the period of each sectional force is small compared to it. After some algebraic manipulations, we arrive at the following relationship: P = π

M  i=1,2 m=1

si0 (z m )di0 (z m ) f i (z m ) sin ϕi (z m )z

(5.11)

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5 FIV Suppression Devices

where sin ϕi (z m ) = sin ϕi ( f i (z m )) =

(5.12) c f i (z m )

4π 2 M 2 ( f i (z m )2 − f N2 )2 + c2 f i (z m )2

(5.13)

Equation (5.11) gives the analytical relation between the power transferred and the sectional hydrodynamic forces. It can be seen that the power contributed by the sectional force at frequency f i (z m ) is scaled down by a factor of sin ϕi ( f i (z m )). This factor is the largest when f i (z m ) = f N , and it gets smaller as f i (z m ) is further away from f N . Henceforth, the reduction of the mean power transferred can be achieved by either deviating f i (z m ) from f N or broadening it along the spanwise direction. The frequency broadening effect can be quantified by discerning the spanwise correlation of the sectional force coefficients [53, 55].

5.2.3.3

Sectional Forces and Spanwise Correlation

We shall estimate the force cross-correlations to understand the three-dimensional flow characteristics of near-wake flow structures. To estimate the degree of spanwise correlation of fluid forces, it is convenient to define the correlation of lift and drag force coefficients extracted at position z ∗ and position z ∗ + z ∗ along the groovedcylinder as follows [53] ρcl (z ∗ ) =

cl (z ∗ , t)cl (z ∗ + z ∗ , t) σcl (z ∗ ) σcl (z ∗ +z ∗ )

(5.14)

ρcd (z ∗ ) =

cd (z ∗ , t)cd (z ∗ + z ∗ , t) σcd (z ∗ ) σcd (z ∗ +z ∗ )

(5.15)

where cl (z ∗ , t) and cd (z ∗ , t) denote the sectional fluctuating lift and drag coefficients at position z ∗ at time t with their corresponding standard deviations σcl (z ∗ ) and σcd (z ∗ ) ) respectively. Here, z ∗ = z/D is the dimensionless parameter that relates to the position in the spanwise direction, which provides a reference for the computation of the cross-correlation. In this analysis, we take z ∗ = 6 and consider the maximum of z ∗ as 6 for the movable location in the spanwise direction. This covers a midportion of the vibrating cylinder away from the end effects. To ignore the influence of the phase difference across the spanwise direction, which does not contribute in Eq. (5.11), the correlation is taken as the maximum over a time lag parameter, τ for the natural period of the system. If the displacement of the cylinder-groove system is well correlated along the cylinder span at each instant, then the correlation factor becomes near unity. With the definition of spanwise correlations in Eq. (5.14) and (5.15), a smaller value effectively also indicates that a broader range of frequency is presented in the sectional hydrodynamic forces along the spanwise direction. As

5.2 VIV Suppression by Spanwise Grooves

289

Fig. 5.17 Spanwise correlation of lift and drag coefficients measured at interval of z ∗ = 0.2. The reference position for correlation computation is located at z ∗ = 6

discussed in the Sect. 5.2.3.2, this scales down the factor sin ϕi (z m ) on the average, thus leading to a lower power transferred, whereby ϕi (z m ) is the phase lag between the displacement and the fluid force at m-th segment at position z m . In a nutshell, a lower correlation will lead to a lower power transfer, thereby suppressing VIV. The computed spanwise correlations are shown in Fig. 5.17 for the plain and the grooved cylinders.

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5 FIV Suppression Devices

It is observed that the staggered groove significantly reduces the spanwise correlation of the sectional force coefficients. This indicates a greater variation of frequency or phase of the forces along the spanwise direction. A relatively smaller correlation also implies that the frequency of the sectional forces span a broader range, which reduces the power transferred to the vibrating system. On the other hand, the helical groove has a similar correlation profile to the plain cylinder, where the correlation is close to one for the lift coefficient, and around 0.9 for the drag coefficient. This shows that the flow around them is relatively two-dimensional and correlated, where the vortices are coherent in the spanwise direction. We further confirm our statement by computing the frequency of the force coefficients along the spanwise direction using the discrete fast Fourier transform. Figure 5.18 shows the frequency spectra of the sectional force coefficients for the plain cylinder, the staggered groove, and the helical groove. It is observed that both helical grooves and plain cylinders have similar frequency spectra along the spanwise direction. From this, we can deduce that the coherent vortex patterns are induced in the wake region. On the other hand, the frequency spectra of staggered grooves have a larger variation along the spanwise direction. This is consistent with the lower spanwise correlation obtained in Fig. 5.17, which contributes to a lower power transfer. This results in a lower amplitude as well as the hydrodynamic forces on the vibrating system. Since the fluctuations of fluid forces are induced by the vortex shedding process, we next examine the vorticity dynamics of the plain and the grooved cylinders.

5.2.3.4

Vorticity Dynamics

Figure 5.19 shows the spanwise vorticity distribution of the plain cylinder, the staggered groove, and the helical groove. The snapshots are taken when the clockwise (blue) spanwise vorticity detaches from the cylinder. It is found that the plain cylinder has vortices that span in the spanwise direction for the entire length of the cylinder. Similar vortices with relatively shorter lengths are found in the near-wake region of the helical grooves. However, this is not the case for the staggered groove, whereby the vortices are segmented along the spanwise direction. The segmentation of vortices is primarily due to the presence of the grooves at both sides of the grooved-cylinder system. This mechanism can be further evident in Fig. 5.20, where the isosurfaces of spanwise vorticity are shown. While the spanwise vortices remain connected after they detach from the plain cylinder, the spanwise vortices are generally segmented from the surface of grooved cylinders. In particular, they are segmented according to the position of grooves on the cylinder. This suggests that the presence of grooves on the cylinder surface causes the segmentation of vortices. The spanwise correlations have provided some quantifications of the three-dimensional flow characteristics of the vortical structures in the spanwise direction. Here, we also look into the streamwise vorticity distribution of the plain cylinder and the groove-cylinder configurations to observe the three-dimensionality in the separated flows. Figure 5.21 shows the instantaneous isosurfaces in the separated flow, while Fig. 5.22 shows the

5.2 VIV Suppression by Spanwise Grooves

291

Fig. 5.18 Frequency spectra of sectional lift (left) and drag (right) coefficient measured at interval of z ∗ = 0.2 along spanwise direction. The reference position for correlation computation is located at z ∗ = 6

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5 FIV Suppression Devices

Fig. 5.19 Instantaneous isosurfaces of λ2 = 1 around cylinder and grooves at Re = 4800, Ur = 5.6 with spanwise length L = 18D. The isosurfaces are colored by z-vorticity

instantaneous contours of streamwise vorticity in the wake. It is observed that the streamwise vorticity alternates its sign along the spanwise direction more often in the staggered grooves and the helical grooves compared to the plain cylinder. This further suggests that the vorticity distributions are more disorganized in the spanwise direction for the staggered and helical grooves, which implies that a relatively greater three-dimensionality is present. It is known that the increased level of threedimensionality promotes the increment of base pressure, hence lesser drag on the cylinder [450]. Next, we relate the vorticity distributions to the force induced by vortices from the vibrating cylinder configurations. It has been well established in [189, 247] that the force induced by vortices in a volume V acting on an immersed cylinder, FV can be expressed as the vorticity impulse: 1 d FV = ρ f 2 dt

(ω A × x)d V

(5.16)

where ω A is the additional vorticity at the position x. The additional vorticity is defined as the vorticity field minus a distribution of vorticity attached to the boundary in the form of a vortex sheet allowing exactly the tangential velocity associated with the potential flow [247], which is essentially the vorticity shed by the immersed cylinder. A direct implication from Eq. (5.16) is that, the vortex force will be higher for an aligned vorticity field in comparison to a disorganized vorticity field, as the latter may lead to some cancellation of forces in different directions. For the contiguous spanwise roll observed in the plain cylinder’s wake where the vorticity is aligned along the spanwise direction, larger lift and drag forces are induced (via the BiotSavart law) on the cylinder by the vortex filaments. On the other hand, the segmented spanwise rolls observed in the wakes of staggered and helical groove configurations have relatively disordered vorticity, which leads to forces acting in different directions on the body and can have some cancellation effect.

5.2 VIV Suppression by Spanwise Grooves

293

Fig. 5.20 Instantaneous isosurfaces of spanwise vorticity ωz = ±1 around cylinder and grooves at Re = 4800, Ur = 5.6

5.2.4 Assessment on the Performance of Staggered Groove As shown the effectiveness of staggered groove for the VIV suppression, we next explore whether a similar performance will be observed under different conditions. Specifically, we explore the effectiveness of staggered grooves as functions of reduced velocities and staggered pitches.

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5 FIV Suppression Devices

Fig. 5.21 Instantaneous isosurfaces of streamwise vorticity ωx = ±1 around cylinder and grooves at Re = 4800, Ur = 5.6 with spanwise length L = 18D. They are taken when the cylinder reaches at the lowest cross-flow displacement

5.2.4.1

Effect of Reduced Velocity

We assess the performance of the staggered groove for a range of reduced velocity Ur ∈ [3.5, 11]. The characteristic response amplitudes are shown in Fig. 5.23 for the staggered groove and the plain cylinder configurations. Compared to the plain cylinder, it is observed that the maximum vibration amplitude across the range of reduced velocity investigated is reduced remarkably in both in-line and cross-flow directions. This is because the presence of staggered grooves prevents the occurrence of the super-upper branch. It is also notable that the in-line amplitude is suppressed to less than 0.05D for all values of reduced velocity examined.

5.2 VIV Suppression by Spanwise Grooves

295

Fig. 5.22 Instantaneous contour of streamwise vorticity ωx = ±1 on Y = 0 plane at Re = 4800, Ur = 5.6 with spanwise length L = 18D. They are taken when the cylinders reach y ∗ = 0 and move toward −Y direction (out of paper)

5.2.4.2

Staggered Pitch

It is known that the staggered groove resembles the geometry of an extruded groove as the staggered pitch becomes more than two times its spanwise length. On the other hand, it is expected that it recovers the performance of the plain cylinder as its staggered pitch approaches zero. This is because the grooves become so small that it does not significantly affect the flow at the limiting case. Using the pitch of the staggered groove shown in Fig. 5.5 as the reference configuration, we consider

296

5 FIV Suppression Devices

Fig. 5.23 Comparison of response amplitude of staggered groove with plain cylinder at m ∗ = 2.6, ζ = 0.0036 for the reduced velocity range Ur ∈ [3.5, 11]: a in-line amplitude, and b cross-flow amplitude

two additional staggered grooves with two different pitches of 1D and 4D. All other parameters are kept identical for the comparative assessment. Geometries of staggered grooves with three representative pitches of ps = 1D, 2D and 4D are shown in Fig. 5.24. Similar to the previous analysis, we compare the spanwise correlation of these staggered grooves in Fig. 5.25. Their characteristic responses are compared against the plain cylinder and the extruded cylinder counterpart and results are summarized in Table 5.5.

5.2 VIV Suppression by Spanwise Grooves

297

Fig. 5.24 Representative geometries of staggered grooves with different staggered pitches: a ps = 1D, b ps = 2D, and c ps = 4D

It is observed that the responses of staggered grooves with different pitches are reduced at similar percentages. This indicates that the performance of the staggered groove is robust at different pitches and exhibits reasonable effectiveness towards the suppression of VIV. For instance, the difference of A y,max , A x.max and Cl,r ms between the pitch configurations investigated are marginal. Notably, a decreasing trend is observed in the mean drag Cd,mean as the staggered pitch increases. It is observed that the spanwise correlation of the drag coefficient for the staggered groove with the pitch 4D is significantly lower when compared to the other two pitches. As discussed earlier, the lower correlation indicates a higher threedimensionality in the flow around the grooved-cylinder configuration, thereby reducing the drag force experienced by the coupled fluid-structure system. This finding conforms with the results presented in Table 5.5. On the other hand, the influence on the lift force spanwise correlation is less obvious due to the pitch variation. Overall, the proposed staggered groove design has a promising influence on the VIV suppression and the mean drag reduction for practical engineering applications. Unlike helical grooves or strakes, the arrangement of grooves such as pitch and angle can be easily adjusted for desired performance towards the VIV response and the drag reduction. Furthermore, the adjustment of grooves can be incorporated with some feedback mechanical control as well as can be combined with active blowing/suction

298

5 FIV Suppression Devices

Fig. 5.25 Spanwise correlations of lift and drag coefficient for staggered grooves with different pitches. The correlations are measured at interval of z ∗ = 0.2, and the reference position for correlation computation is located at z ∗ = 6

mechanisms, as recently demonstrated for low Reynolds number in [465]. The hybrid passive groove with the active blowing/suction control will eliminate VIV while achieving the desired level of mean drag loading associated with the VIV-induced drag amplification and due to the intrinsic bluff-body wake flow.

5.2 VIV Suppression by Spanwise Grooves

299

Table 5.5 Characteristic responses of staggered grooves with different pitches Pitch 1D 2D 4D Extruded A∗y,max A∗x,max Cl,r ms Cd,mean

0.65 (−40.91%) 0.07 (−75.00%) 0.24 (−86.29%) 2.29 (−19.93%)

0.69 (−37.27%) 0.07 (−75.00%) 0.12 (−93.14%) 2.13 (−25.52%)

0.56 (−49.09%) 0.06 (−78.57%) 0.25 (−85.71%) 1.64 (−42.66%)

1.06 (−3.64%) 0.34 (21.43%) 1.89 (8.00%) 2.58 (−9.79%)

The case where pitch equals to 2D is the reference case simulated in the previous section. Value in the bracket indicate its percentage different to the plain cylinder counterpart

5.2.5 Interim Summary In this section, the effect of spanwise alignment of grooves on controlling the vortexinduced vibrations has been numerically investigated. Two configurations of grooves, namely staggered and helical groove, have been examined, which differ by their transition of cross-section geometry along the spanwise direction, whereby the former has a jump in pattern and the latter has a smooth transition. The depth and the width of the grooves were set to a local optimum by conducting a systematic parametric investigation on the extruded grooves. The characteristic responses, the spanwise correlation, and the vortex-wake dynamics of both the staggered and the helical grooves were analyzed. It was found that the staggered groove has the best performance among the designs tested in suppressing VIV, where the cross-flow amplitude and the drag force are suppressed up to 37% and 25%, respectively. As we have hypothesized, the staggered groove is observed to have a lower spanwise correlation in comparison to the helical groove, owing to its regular jump pattern the cross-section geometry along the spanwise direction. This appears to detune or de-correlate the spanwise correlation, thereby broadens the frequency range of the hydrodynamic force generated and reduces the energy transferred to the system. We extended our investigation on the staggered grooves by examining the influence of staggered pitch to the VIV suppression. We found that the VIV responses of the staggered groove are not monotonic with respect to the staggered pitch. Among the pitches that we have tested, we found that their performances of VIV suppression are similar. In summary, the proposed staggered groove configuration offers a promising design for the suppression of VIV while reducing the drag force. Moreover, it has a relative simplicity with regard to mechanical design, the fabrication and the installation on deepwater drilling riser compared to the state-of-the-art helical strakes. Further research should be done to examine its robustness at high Reynolds number, as well as to optimize the design of the staggered groove for a broad range of physical and geometric conditions in conjunction with experimental measurements.

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5 FIV Suppression Devices

5.3 Appendage Devices for VIV Wake Stabilization A deepwater floating system is a coupled nonlinear dynamical system consisting of a floater, risers and moorings that interact with the harsh ocean environment in a complex way. Due to the existence of vortices with alternating signs, elastically mounted rigid cylinders or elastic structures can develop self-excited vibrations in the transverse direction when they are immersed in an ocean stream. There is a net work done by the fluid flow to the body when the frequency of vortex shedding approaches the natural frequency of the vibrating body. This phenomenon is referred to as lock-in or vortex synchronization [56, 330]. The existence of a well-defined synchronization range at which the cylinder vibrates with the highest amplitude indicates that vortex-induced vibration (VIV) is a type of nonlinear resonant response. The lock-in causes a large amplitude oscillatory motion of the bluff body, which is of practical importance in offshore and marine structures [40, 381]. In addition, the drag force amplification experienced by the riser undergoing VIV is another issue that affects the structural performance of risers and the operation of a floating offshore system. These practical issues associated with VIV have motivated the development of various methods and devices [222, 477], such as fairings [29, 473], helical strakes [57], splitter plate [24, 34, 364], control cylinders [502]. For a successful offshore operation, an effective VIV suppression method should meet the following requirements: (1) robust in eliminating vibrations for a wide range of current speeds; (2) low drag force on the bluff-body structure; (3) ease of handling and provision for a pre-installation; (4) prevention from global coupled mode instabilities such as galloping and flutter; and (5) free from marine growth and inexpensive to produce and maintain. Among these suppression devices, neutrally buoyant fairing devices are generally found to be effective to minimize VIV and to reduce drag force by preventing the interaction of shear layers with the near wake and extending the shear layers further downstream to avoid alternate oppositely-signed shed vortices in the afterbody region. An experimental evaluation has been conducted between the fairing and helical strakes in [12]. The experimental study showed that while the helical strakes suffered from a higher drag force, the fairing reduced the drag force significantly. Several numerical [342, 462, 473] and experimental [29] investigations have been followed to investigate the performance of fairings for a greater range of geometric parameters and physical conditions. In a recent numerical work [462], free-torotate U-shaped fairings were studied numerically to investigate their performance, as shown in Fig. 5.26a. The fairings with various rotational friction coefficients were simulated at Reynolds number up to Re = 1000. The numerical results found that there exists a critical small value of friction coefficient at low Reynolds number, at which the fairings can undergo large amplitude oscillation. For the higher friction coefficient, the fairings were able to suppress both VIV and drag force effectively. In a recent experimental study [29], various designs of fairings have been tested at subcritical Reynolds numbers (Re ∈ [7 × 103 , 105 × 103 ]). In the experiment setup, only transverse direction movement of cylinder-fairing system was allowed and the

5.3 Appendage Devices for VIV Wake Stabilization

301

fairing was free to rotate. Among the fairing devices tested, the short-crab-claw (SCC) configuration was found to be effective and robust in both VIV suppression and the reduction of drag force. Due to improper design, however, the short fairings may rotate in an undesirable manner and can lead to dynamical instability associated with asymmetrical (non-circular) cross-section. Such dynamic galloping-type instability can cause a greater transverse lift force and slowly varying large motion, which may not be self-limiting [232]. Owing to the intrinsic unstable nature of the shear layers and the feedback from the wake to shear layers and vice versa, the vortex motion induces a mean streamwise drag force and the transverse lift force over a bluff body. The fluctuating transverse force is associated with the pressure gradient in the downstream part of the body, i.e. downstream with respect to the separation points. Therefore, the shape of the downstream afterbody and the pressure gradient is explicitly interlinked. By applying external passive (with no energy input) or active (with an external source of energy) strategies, one can offset the rolling up of shear layers further downstream to reduce unsteady periodic force and suppress vortex-induced vibration of a bluff body. This modification based on the delay of vortex formation stabilizes the near wake region, which leads to the reduction of transverse fluctuating pressure gradient along the afterbody surface. This wake stabilization process also helps to reduce the drag force over the body. A similar mechanism is recently observed in the experimental study of [24]. In the experiment, several devices were tested namely a fixed splitter plate, a free-to-rotate splitter plate, and a pair of parallel plates. The authors found that in a steady state, the free-to-rotate splitter plate aligns itself such that its tip intercepts a line that is parallel to the flow direction. At such alignment, the shear layer leaving the shoulder of the cylinder reattaches to the tips of the splitter plate. The VIV is suppressed and the drag force is reduced significantly at such alignment. Recently, the reattachment of the shear layer was observed in the work of [307]. When the shear layer of the leading cylinder reattaches to the downstream cylinder, there is a lower drag force experienced by the tandem cylinder arrangement. Given that the VIV suppression effectiveness relies on both extension and reattachment of shear layers, we would like to numerically assess the difference between these two mechanisms by modifying the near-wake region behind a vibrating body. Based on these mechanisms, we propose two new configurations of suppression devices in this section. Representative sketches of four suppression devices are shown in Fig. 5.26. The conventional fairing, as shown in Fig. 5.26a, serves as a baseline device which relies on the extension of shear layer and offsetting of the formation of vortical patterns. The proposed device, connected-C is illustrated in Fig. 5.26b. This device is designed to allow the shear layer to reattach to the tips of the C-shaped device after it detaches from the cylinder surface. The C-shaped geometry is attached with the cylinder by a connector plate. To investigate the significance of each part of the connected-C, two additional configurations are considered in our numerical study. The first device is disconnected-C, which is illustrated in Fig. 5.26c. It is essentially a connected-C device without a plate between the main cylinder and the C-shaped geometry. By the placement of C-shaped geometry behind the cylinder, this configuration allows to investigate the role of the connector plate in the wake

302

5 FIV Suppression Devices

Fig. 5.26 Sketches of low-drag VIV suppression devices: a Conventional U-shaped fairing around a pipe [342], b novel connected-C device, c disconnected-C without plate, and d splitter plate

stabilization of the vibrating cylinder. The second device is the classical splitter plate, which is shown in Fig. 5.26d. This configuration is considered to illustrate the role of the connector plate in the proposed connected-C device. In the present work, in order to focus on the dynamics of shear layers and vortical patterns, we allow the cylinder and the device to move in the transverse direction only with one degree-of-freedom (1-DOF) motion. We do not consider the relative motion (in particular rotational motion) between the attached device and the cylinder. By setting zero relative rotational effect, we can solely focus on the wake stabilization and the suppression mechanism of VIV. Therefore, the present numerical study is an idealization to understand the role of reattachment and the extension of shear

5.3 Appendage Devices for VIV Wake Stabilization

303

Fig. 5.27 Schematic diagram of numerical setup and boundary conditions

layers on the flexibly mounted cylinder-device system. Since there are no effects of rotational stiffness and friction, the observed VIV suppression characteristics may not translate directly to the practical configurations of devices when free rotation and in-line motion are allowed. We divide our analysis into two parts: low Reynolds number (Re = 100) and subcritical Reynolds number (Re = 6150−7400). For the low Re laminar flow, two dimensional (2D) simulation is conducted for all devices. The cylinder-device geometry is simplified by using their mid-plane cross sections. Detailed low Re simulations are considered an important first step towards understanding the important aspects of VIV suppression. In the second part, we carry out three-dimensional (3D) VIV simulations via a dynamic large eddy simulation (LES) model at subcritical Reynolds numbers. For demonstration purposes, we consider the fairing and connected-C configurations for the three-dimensional analysis. This numerical study focuses on the wake flow fields and vibration characteristics of the combined cylinder-device system. Of particular interest here is to investigate the free vibration response of the connected-C device and to provide physical insights with regard to VIV response dynamics.

5.3.1 Fairing, Connected-C and Splitter Plate Configurations Figure 5.27 shows a schematic diagram of the setup used in our simulation study for an elastically mounted cylinder with various suppression devices in a flowing water stream. A stream of incompressible fluid enters the domain from an inlet boundary at horizontal velocity u = U . A cylinder with diameter D is mounted on a spring with a stiffness value of k and a linear damper with a damping value of c in the transverse direction. The cylinder body is attached to different devices, which will be introduced in the later part of this section. The computational domain and the

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5 FIV Suppression Devices

boundary conditions are illustrated in Fig. 5.27. The coordinate origin is located at the geometric center of the circular cylinder. The streamwise and transverse directions are denoted x and y, respectively. Each device is attached to the cylinder, thus there is no relative motion between the device and the cylinder. In all simulations, the control device is symmetrically arranged with respect to the wake centerline and the cylinder-device the system is free to vibrate in the transverse direction. The cylinder is placed 20D away from the inlet boundary and 40D away from the outflow boundary. The distance from the cylinder to either side is 50D, which results in a blockage ratio of 1%. No-slip wall condition is implemented on the surfaces of the cylinder and the suppression devices, while the slip wall condition is implemented on the top and bottom boundaries. For 3D configurations, periodic boundary conditions are imposed on both ends of cylinder-device systems. Except stated otherwise, all positions and length scales are normalized by the cylinder diameter D, velocities with the free stream velocity U , and frequencies with U/D. For the low Reynolds number analysis, the mid-plane cross-sections of the devices are simulated in two dimensions and their geometries are shown in Fig. 5.28. The characteristic longitudinal length of all devices is measured from the front of the cylinder (stagnation point) to the end of the device. The first geometry is fairing, which is shown in Fig. 5.28a. The fairing used in our simulation is adopted based on the SCC fairing configuration tested in the experiment of Baarholm et al. [29], where the fins are slightly curved towards each other. The opening of the fairing, a is kept constant and set to 0.95 throughout our simulation study. The second device is connected-C, as shown in Fig. 5.28b. It consists of two geometric components, namely a splitter plate-like connector and a C-shaped foil. For consistency with the fairing, the opening of the C-shaped device is kept equal to the opening of the fairing. The third device is the mid-plane cross-section of disconnected-C. Its geometry is shown in Fig. 5.28c. It has the same opening as the one of connected-C to ensure that the geometry scales are comparable. The fourth device is a standard splitter plate, as shown in Fig. 5.28d. For all the devices, the characteristic longitudinal length is set to L c = 2.0 for consistency in the streamwise geometric scale. For the subcritical Reynolds number analysis, the geometries of fairing and connected-C used in the simulation are just the extrusion of their two-dimensional geometries, with the spanwise length to the diameter ratio, L/D = 5.0. Apart from the longitudinal length scale L c , the other key dimensionless VIV parameters for the cylinder-device system are the mass ratio (m ∗ ), the reduced velocity (Ur ), Reynolds number (Re) and the damping ratio (ζ ), which are defined as follow: m∗ =

ρfUD 4m U c , U , Re = = , ζ = √ r f 2 f πρ D L fn D μ 2 mk

where f n is the natural frequency measured in vacuum, c is the damping coefficient. The value of U is fixed at 1.0 throughout the numerical study. By varying the value of dynamic viscosity μ f and the natural frequency f n , the Reynolds number Re and the reduced velocity Ur can be adjusted accordingly.

5.3 Appendage Devices for VIV Wake Stabilization

305

Fig. 5.28 Mid-plane cross section of cylinder with various devices attached (refer to Fig. 5.26). The characteristic length L c is set to be 2.0 for all devices for consistent streamwise geometric scale. The opening of the fairing a is set to 0.95D and it is identical to the opening of connected-C and disconnected-C devices

Convergence Study and Validation The computational domain is discretized using unstructured finite element mesh. Different meshes are used for two-dimensional and three-dimensional simulations. A typical mesh of the U-shaped fairing with L c /D = 2.0 that is used in the twodimensional simulation is shown in Fig. 5.6. A finer mesh is created at the region close to the cylinder and the fairing, while the boundary layer mesh is considered on the surface of the cylinder and fairing fins. The size of the meshes surrounding the cylinder and devices are controlled such that y + is not larger than 1, where y + is the distance to the solid surface in wall units based on instantaneous friction velocity. To study the grid convergence in our simulation study, three grids with different number of nodes per layer are created, which are denoted as M1, M2, M3. The flow at Re = 100 and Ur = 4 with fairings of L c /D = 2.0 is simulated using these grids, and their results are summarized in Table 5.6. The convergence study shows that the maximum difference in the values obtained using M1 and M2 is about 1.7%, while M2 and M3 are about 0.23%. Hence, our simulations are conducted with M2 mesh, which is adequate to provide the assessment of VIV suppression devices. A detailed validation and convergence of the numerical solver is reported in [242, 254, 307] for freely vibrating isolated, side-by-side and tandem arrangements.

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Fig. 5.29 Representative meshes at cross-sectional plane: a close-up around cylinder and fairing, b zoom view of cylinder and connected-C device for L c /D = 2.0 Table 5.6 Grid convergence at Re = 100, Ur = 4.0 Mesh Nodes Elements A∗y,max M1 M2 M3

1.4 × 104 2.7 × 104 5.3 × 104

2.8 × 104 5.5 × 104 10.5 × 104

0.0895 0.0884 0.0882

Cl,r ms

Cd,mean

0.3740 0.3678 0.3679

1.3349 1.3331 1.3311

Table 5.7 Summary of meshes used in three-dimensional simulation Cylinder Cylinder Fairing (Ur < 6.0) (Ur ≥ 6.0) Nodes per layer Elements per layer Layer thickness Number of layers Total nodes

Connected-C

9.7 × 103 1.9 × 104

16.6 × 103 3.3 × 104

22.0 × 103 4.35 × 104

22.5 × 103 4.45 × 104

0.1D 50 4.9 × 105

0.1D 50 8.4 × 105

0.15D 33 7.5 × 105

0.15D 33 7.6 × 105

The span of each device is equal to 5D, where D is the diameter of the cylinder

For the 3D simulations of turbulent wake flow, due to a high computational cost, an adequate mesh is considered for the dynamic subgrid LES computations based on our previous work in [180]. The 3D domain is simply an extrusion of the 2D cross-sectional plane in the cylinder axis direction. The span of the cylinder is fixed to 5D for all three-dimensional simulations. The details of the 3D mesh used in the simulations are summarized in Table 5.7. Periodic boundary conditions are applied at the two ends of the cylinder (front and back sides of the 3D domain). A time step of t = 0.1 is employed for the LES computations. To establish the accuracy of our numerical framework for subcritical Reynolds numbers, we validate our results with the experimental measurements of [23] for VIV of a plain circular cylinder. The plain cylinder is mounted elastically and is only free to vibrate in a cross-flow direction. The simulation is conducted in three dimensions using the mesh specified in Sect. 5.3.1, while the dimensionless parameters are chosen to be m ∗ = 1.92, ζ = 0.007, Re = 6150−7400 to match the experimental setting.

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Fig. 5.30 Validation of the response amplitude for transversely vibrating circular cylinder with free vibration experimental data from Assi et al. [23] at m ∗ = 1.92, Re = 4000−14,000. The amplitude A∗y is computed using Hilbert transform, as described in [206]

The VIV amplitudes are obtained by computing the instantaneous amplitude using Hilbert transform, as explained in [206]. The comparison of the amplitude response against the free vibration experiment of [23] is shown in Fig. 5.30. We observe that the VIV amplitudes from our numerical simulations agree well with the experimental data. The comparison shows that the overall agreement is remarkable in terms of the three-branch amplitude response pattern, the amplitude peak value, and the lock-in region. The upper branch associated with the low mass ratio is reasonably captured in our simulations. The wake mode and the phase relation between the normalized displacement (y ∗ = y/D) and the lift coefficient (Cl ) of a representative case for each branch are shown in Fig. 5.31. At the initial branch with reduced velocity Ur = 3.25, the displacement is in-phase with the lift coefficient, and the 2S wake mode with two single counter-rotating vortices per cycle is observed, as shown in Fig. 5.31a. When the reduced velocity is increased to Ur = 4.88 in the upper branch, the phase relation between displacement and lift coefficient remains the same, but the wake mode is changed to 2P vortex pattern consisting of two pairs of counter-rotating vortices per cycle (as shown in Fig. 5.31b). Such wake mode transition is observed in the previous measurements [206, 208] and they influence the amplitude and frequency content of the fluid forces on the body. As Ur continues to increase, the cylinder VIV mode transits into the lower branch, where the phase angle between the transverse displacement and the lift coefficient changes to π [208]. The wake mode remains to be 2P. These VIV characteristics are captured in our 3D simulations, as shown in Fig. 5.31c for a representative case of lower branch at Ur = 7.32. This validation study will serve a reference to analyze the dynamic response of an elastically mounted connected-C device immersed in a freestream flow.

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Fig. 5.31 Time histories of displacement y ∗ and lift coefficient Cl (left); and instantaneous spanwise z-vorticity contours at different phase in a cycle (right) for three VIV branches: a initial branch with in-phase y ∗ versus Cl and 2S vortex wake, b upper branch with in-phase y ∗ versus Cl in-phase and 2P vortex wake, c lower branch with anti-phase y ∗ and Cl and 2P vortex wake mode. These responses in three branches agree with measurements of [208]

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Results and Discussion To assess the effectiveness of the devices on vortex-induced vibration, we now present our results in the following two subsections. In the first section, the characteristic VIV response as well as the vorticity dynamics of various suppression devices are investigated at low Reynolds number Re = 100. In the second section, we perform the same analysis at the subcritical Reynolds number in a three-dimensional setting. In both 2D and 3D simulations, we consider only transverse one-degree-of-freedom (1-DOF) motion to investigate the effect of attached devices on VIV dynamics. For all configurations, the size of the computational domain, the density and distribution of the mesh and the time step size are kept identical for a consistent comparison. All simulations are run until the statistically stationary states are achieved.

5.3.2 Assessment at Low Reynolds Number It is well known that the dynamics and response characteristics of VIV have their root in two-dimensional low Re flow, which provides a good avenue to identify the key VIV characteristics and the suppression mechanisms at low Reynolds number [238, 307, 462]. Apart from the smaller computational cost for parametric investigations, the low Re computation can eliminate many uncertainties related to turbulence modeling and mesh resolution. The four devices specified in Sect. 5.3.1 are simulated at Re = 100, m ∗ = 2.6, ζ = 0.001 in two dimensional space. To ensure that the results are comparable, the characteristic longitudinal length of these devices is set to be L c = 2.0D.

5.3.2.1

Amplitude Response

We start our analysis with the VIV amplitude response curves of the suppression devices, which are shown in Fig. 5.32. For reference purposes, the response curve of the plain cylinder is also included in the figure. We consider the range of reduced velocity Ur ∈ [2, 15] to cover the pre-lock-in, the lock-in and the post-lock-in regions of vortex synchronization. As shown in Fig. 5.32, the attachments of the fairing, the connected-C and the disconnected-C devices suppress the VIV effectively for the whole range of reduced velocity, while the splitter plate is effective until Ur < 6.0. The splitter plate experiences large amplitude at Ur ≥ 6.0. The amplitude is reduced up to 83.8% for fairing, 89.6% for connected-C, 90.6% for disconnected-C, and 89.8% for splitter plate (at Ur < 6.0). The large amplitude response of the splitter plate at Ur ≥ 6.0 is attributed to the galloping instability of the splitter plate, which is well known and reported in previous experimental studies [24, 364]. The amplitude response shows that, both the shear layer extension and the reattachment are effective in modifying the near wake

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Fig. 5.32 Amplitude response curves of various suppression devices at Re = 100, m ∗ = 2.6, ζ = 0.007. Suppression of VIV from all four devices can be seen, except for reduced velocity Ur > 6, splitter plate undergoes higher amplitude of motion

and thereby reducing the VIV amplitude. Moreover, the presence of the C-shaped device is essential in the connected-C device, as it may undergo high amplitude galloping without the stabilization effect of the C-shaped profile. In addition, the absence of the connector in the disconnected-C does not have a significant effect on the amplitude response, as both connected-C and disconnected-C devices are found to have a similar VIV response curve. From Fig. 5.32, another notable observation is the sharp increment of amplitude for the plain cylinder at Ur = 4.0 at a lock-in, which is not observed for all the devices. This implies that both shear layer extension and reattachment help to prevent the lock-in phenomenon.

5.3.2.2

Response Frequencies and Phase Relations

To further explore the lock-in phenomenon observed in Fig. 5.32, we look into the frequency ratio f / f n,w and the phase difference between the amplitude and lift coefficient φ A∗y ,Cl , which are shown in Fig. 5.33. The frequency ratio is defined as the ratio between oscillation frequency f and the natural frequency of cylinder in still water f n,w while the phase difference between the amplitude A∗y and the lift coefficient Cl is computed by taking the average time interval between two neighboring peaks of A∗y and Cl divided by their periods. There are several important features of the VIV suppression mechanism are revealed by diagrams in Fig. 5.33. The frequency ratio f / f n increases as Ur for the plain cylinder, the fairing and the C-shaped devices. For the splitter case, the frequency increases until Ur ≈ 5, then it decreases to f / f n ≈ 0.5. One of the notable

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Fig. 5.33 Dependence of dominant response frequency f and phase difference on reduced velocity for all devices and plain cylinder at Re = 100, m ∗ = 2.6, ζ = 0.007: a frequency ratio f / f n,w , b phase difference φ A∗y ,Cl between response amplitude A∗y and lift Cl . Here f n,w denotes the natural frequency of cylinder in still water

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observations from the figure is that the frequency ratio of plain cylinder is not lockin at f / f n = 1. In other words, the oscillation frequency of the system does not exactly match the natural frequency and instead the frequency ratio value stays around f / f n ≈ 1.15 for the reduced velocity Ur ∈ [4, 6]. This non-classical lock-in phenomenon can be attributed to the low mass-damping parameter of the cylinder body. This is in agreement with the previous experimental observations for low mass-damping parameter [206]. Such lock-in behavior is not observed for the fairing, connected-C, and disconnected-C devices. This indicates that both shear layer extension and reattachment can prevent lock-in. The prevention of lock-in helps to reduce the VIV amplitude. On the other hand, the occurrence of galloping is signified by the frequency ratio of the splitter plate, where it stays low at around 0.5. The galloping of the splitter plate is also shown in Fig. 5.33b, where its average phase angle between amplitude and lift coefficient remains zero for all reduced velocities. For all other devices, the average phase is close to zero for Ur ≤ 6.0, and close to π for Ur > 6.0. Such a jump in the phase angle indicates the transition to the lower branch, as pointed out in an experimental study [206] at the subcritical Reynolds number. Except for the splitter plate, the phase angle for all the devices jumps sharply at a similar reduced velocity as the plain cylinder. For the plain cylinder, this sharp transition in the phase angle is related to the change in the timing of vortex shedding. Apparently, the same mechanism plays a role in the cylinder-device system as well. To further elucidate the dynamic response of the system, we explore the forces experienced by each device-cylinder configuration.

5.3.2.3

Lift and Drag Coefficients

The lift and drag forces are computed on the entire geometry of the device-cylinder system. The variations in the rms lift Cl,r ms and mean drag Cd,mean coefficients as a function of the reduced velocity are shown in Fig. 5.34. Compared to the plain cylinder, the lift coefficient is reduced in all devices, especially during the lock-in region of cylinder. Consider Ur = 4.0 as an example, it is reduced by 67.7% in fairing, 80.3% in connected-C, and 86.3% disconnected-C, 79.6% in splitter plate. However, in the case of the splitter plate, the lift coefficient increases sharply at Ur = 6.0. This agrees with the galloping observed in the amplitude response. On the other hand, as pointed out in the experimental study [206], the sharp peak in the lift coefficient is an evidence of the transition from the initial branch to the upper branch completely. It occurs at Ur = 4.0 in all cases except disconnected-C, where it is shifted to Ur = 4.5. This shows that, both shear layer extension and reattachment do not eliminate the occurrence of upper branch. The mean drag force is also reduced in all devices. At Ur = 4.0 as an example, it is reduced by 27.9% in the fairing, 32.4% in the connected-C, and 36.3% the disconnected-C, 36.1% in the splitter plate. Moreover it is worth mentioning that the drag coefficient is positively related to the amplitude for plain cylinder, which can be observed by comparing the response profile of plain cylinder in Figs. 5.32 and 5.34b. This is because, the higher amplitude effectively

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Fig. 5.34 Fluctuating lift Cl,r ms and mean drag Cd,mean forces as a function of reduced velocity Ur for plain cylinder and various devices at Re = 100, m ∗ = 2.6, ζ = 0.007: a r.m.s of lift, b mean drag coefficient

increases the characteristic length in the transverse direction, which in turn increases the drag force experienced by the vibrating body. However, this explanation does not apply on the splitter plate, as observed in Fig. 5.34b. Despite the large amplitude response of the splitter plate at Ur > 6.0, the drag coefficient remains lower than the cylinder. This is because the splitter plate stabilizes the wake region by preventing the interaction of shear layers in the near wake region [34, 364].

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Fig. 5.35 Mean pressure coefficient C¯ p along the cylinder for each device at Re = 100, m ∗ = 2.6, ζ = 0.001. Angle is measured from the stagnation point of cylinder to back of cylinder. For fairing, the angle between 90◦ and 100◦ is covered by fins while for connected-C and splitter plate, the angle above 174◦ is covered by connector and splitter plate

5.3.2.4

Pressure Distributions

The pressure distribution along the cylinder has a direct relation with the VIV dynamics, as it affects the lift and drag forces acting on the cylinder. We choose Ur = 5.0 as the representative case to investigate the pressure distribution for each device-cylinder system. The mean pressure coefficient, C¯ p along the cylinder for each device is shown in Fig. 5.35. The mean quantity is obtained by averaging the pressure over five cycles. As shown in Fig. 5.35, the mean pressure distribution along the cylinder is modified significantly by attaching the devices. In addition these devices share similar C¯ p profiles until θ ≤ 90◦ . A notable difference is a discontinuous jump of C¯ p for the fairing at the attachment point on the cylinder surface. In the rear surface of the cylinder, the pressure distribution remains fairly flat for the fairing configuration. Some similarity in the pressure profiles indicates that the flow dynamics in the near wake region are quite similar at Ur = 5.0 for these devices. The presence of these devices prevents the shear layer to roll from one side to the other immediately after the cylinder. This stabilizes the shear layer around the shoulders of the cylinder, which leads to higher pressure on both sides of the cylinder. Therefore, the lift force experienced by the cylinder is lower with the devices attached, as observed in Fig. 5.34. By attaching the devices with the flexibly mounted cylinder, the mean pressure distribution tends to recover the stationary plain cylinder counterpart. From the above 2D low Reynolds number analysis, it is evident that both fairing and the C-profile-based devices have high effectiveness in the suppression of VIV while reducing the drag force.

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5.3.2.5

315

Wake Characteristics

To analyze the intrinsic feature of the wake behind the vibrating bluff body system, we quantify the wake width to illustrate the topology of wake structures for all configurations. The modification of wake topology induces a significant influence on the variation of forces, the base pressure and the vortex shedding frequency. The measurement of wake width, w and the vortex formation length, L f follows the definition used in the study of Park et al. [333]. A diagram to define the vortex formation length and wake width behind a cylinder is shown in Fig. 5.36. To quantify streamwise length scales in the wake region, the parameter δ(x) is defined as the distance between local maxima of streamwise velocity fluctuation, u r ms in transverse direction at different position x, where x is the streamwise distance measured from the center of the cylinder. The wake width, w is defined as the minimum value of δ(x), while the vortex formation length, L f is defined as the streamwise distance x which gives the minimum value of δ(x). Using such definitions, we show the profiles of δ(x) for each device at Ur = 5.0 in Fig. 5.37. The result of a static plain cylinder is also included in the graph for a comparison purpose. It is observed that, while the moving cylinder has a larger wake width due to its high amplitude oscillation, the wake width of the fairing, the connected-C, the disconnected-C, and the splitter plate are close to the diameter of the cylinder. This implies that these devices are able to reduce the wake region remarkably, thus increases the base pressure and decreases the mean drag over the cylinder-device system. On the other hand, compared to the static cylinder, the vortex formation length is increased for all cylinder-device configurations. As discussed in the study of Williamson [450], the vortex formation length is roughly inversely proportional to the shedding frequency. This indicates that these devices induce a lower vortex shedding frequency as compared to the plain cylinder counterpart in both stationary and vibrating scenarios. This can be observed in Fig. 5.33a. Among these devices, the vortex formation length is nearly similar for the fairing, the connected-C and the disconnected-C, while it is somewhat longer for the splitter plate. Owing to the similarity in the wake width and vortex formation length among the fairing, connected-C, and disconnected-C devices, it can be concluded that the wake topology for the shear layer extension and the reattachment have a similar effects in the near wake region. Due to the control devices, the amount of fluctuating kinetic energy in the near wake region is much smaller to sustain the transverse free vibration of the combined cylinder-device system. It is evident that the connector in the connected-C device does not have a significant effect on the wake topology.

5.3.2.6

Vortex Patterns

The flow fields in terms of instantaneous vorticity, streamlines and pressure fields are presented to elucidate the wake stabilization mechanisms. In our discussion, we will adopt the classical terminology of [458] to identify the vortex shedding patterns (e.g., 2S, 2P). The flow structures of these devices at Ur = 5.0 are presented. Figure 5.38

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Fig. 5.36 Illustration for the estimation of wake width w and vortex formation length L f [333]. Solid curve lines on both sides of the wake are the line connecting the local maxima of streamwise velocity fluctuation u r ms along transverse y-direction at different streamwise position x. The distance function δ(x) is defined as the transverse distance between the two mentioned lines at position x, where x is the streamwise distance measured from the center of cylinder. The width w is defined as the minimum value of d(x) over x, while L f is the value of x which gives the minimum value of δ(x)

Fig. 5.37 Normalized transverse distance between local maxima of u r ms , δ(x)/D at different streamwise distance x/D. Values are extracted for each device at Ur = 5.0. To avoid the discontinuity due to mesh discretization, δ(x) is extracted every 0.5D from the center of the cylinder in streamwise direction. The markers are the value obtained from the extraction

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shows the snapshot of vorticity for each device when it reaches maximum transverse displacement. A regular 2S mode with two single Karman vortices is observed for all the devices. The distance between the vortices formed is nearly equal. It is known that the shed vortex with accumulated vorticity is fed by the separated shear layer until the vortices are not sufficiently strong to enter into the wake flow. All the controlled devices extend the shear layer until the characteristic longitudinal distance L c /D = 2.0. Therefore the extension of the shear layer and the interaction between the separated shear layers and the vorticity strength generated on the device play a dominant role in the VIV dynamics. The attachment of the devices provides a desired redistribution of the vorticity, which has a positive influence on the dynamic feature of the near wake. We next examine the modification of near wake flow via instantaneous flow visualizations of streamlines and vorticity contours. At a representative oscillation cycle, the vorticity contours, the streamline, and the pressure contours around the devices are plotted in Figs. 5.39, 5.40, 5.41, 5.42. Instantaneous vorticity contours with superimposed streamlines (left panel) and the pressure distribution in the wake region (right panel) are shown at the interval of t = T /4, where T denotes the oscillation period. The unsteady and asymmetric wake is evident through vorticity contours and streamlines. In the case of the cylinderfairing system, as discussed in the previous section, the underlying wake suppression mechanism relies on the extension of shear layer from the cylinder body. As shown in Fig. 5.39, the shear layer separates from the tips of fairing fins instead of the cylinder surface. These shifts in the shear layer roll-up delay the formation of vortex shedding. Moreover, the vortices formed do not penetrate into the region surrounded by the fins, in turn this helps to stabilize the near wake region behind the cylinder. For the connected-C device, two closed circulations with opposite directions are formed in the region between the cylinder and the C-shaped device, as shown in Fig. 5.40. The presence of a C-shaped device prevents the circulation from moving into the wake region. Due to the cancellation effect of oppositely signed vortices, the center of the region behind the cylinder tends to stabilize the near wake region. Other than reducing base suction pressure, the recirculating flow in the gap also serves as a barrier to prevent the rolling of the shear layer. Hence, the shear layer is observed to be stretched from the cylinder surface to the C-shaped device. Such extension of the shear layer, as discussed in the case of U-shaped fairing, induces suppression of VIV and reduces the drag force on the body. The wake structures of disconnected-C and connected-C are quite similar. Even with the absence of a connector, closed circulations are formed on both sides of the connector, as shown in Fig. 5.41. These circulations serve as a barrier to prevent the shear layer from rolling from one side to the other. This helps in the reattachment of the shear layer. As shown in Fig. 5.42, the closed circulation is also observed at both sides of the plate in the case of the splitter plate. However, the circulation moves into the wake region and forms a vortex eventually, which is different from the trapped circulation observed in the connected-C and the disconnected-C devices. Before the circulation moves into the wake, it is opposed by the opposite vorticity built at the surface of the plate and the C-foil.

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Fig. 5.38 Comparison of spanwise z-vorticity of all devices at maximum transverse displacement for representative reduced velocity Ur = 5.0 at Re = 100, m ∗ = 2.6, ζ = 0.001

Fig. 5.39 Results of flow pattern in terms of instantaneous spanwise z-vorticity contour, streamline trace and pressure distribution for U-shaped fairing in single oscillation cycle at Re = 100, m ∗ = 2.6, ζ = 0.001. Here T is the period of the oscillation of cylinder-fairing system

This weakening of the vorticity before the vortex enters into the wake region contributes to the suppression of VIV. Moreover, as the region behind the cylinder is stabilized, the drag force is also reduced. It is worth pointing that as the circulation on both sides moves into the wake region periodically, it creates a pressure difference in the transverse direction periodically. Such periodic oscillation in the pressure causes the splitter plate to vibrate. At higher reduced velocity (Ur ≥ 6), the pressure difference becomes larger, as shown in the increase of Cl,r ms in Fig. 5.34, which gives rise to the galloping of splitter plate.

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Fig. 5.40 Evolution of spanwise z-vorticity, streamline trace and pressure distribution of connectedC in a cycle at Re = 100, m ∗ = 2.6, ζ = 0.007. Here T is the period of the oscillation and the flow is from left to right

Fig. 5.41 Evolution of spanwise z-vorticity, streamline trace and pressure distribution of disconnected-C in a cycle at Re = 100, m ∗ = 2.6, ζ = 0.007. Here T is the period of the oscillation and the flow is from left to right

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Fig. 5.42 Evolution of spanwise z-vorticity, streamline trace and pressure distribution of splitter plate in a cycle. Here T is the period of the oscillation and the flow is from left to right

To summarize, there are two factors that affect the effectiveness of VIV suppression and drag force reduction for the aforementioned devices. The first one is the extension of the shear layer from the cylinder body to farther into the wake. It is well known that a shear layer is much weaker and has relatively lower fluctuation compared to Karman vortices. Therefore by extending the shear layer and delaying the formation of vortices, the pressure fluctuation in the transverse direction can be reduced. This process can be observed from the flow fields of U-shaped fairing. The second factor is the formation of closed circulations on both sides of the device. Such symmetric circulation prevents the formation of alternate Karman vortices from one side to the other in the immediate near wake region. In turn, it stabilizes the near wake region and delays the formation of Karman vortices. This type of mechanism is observed in the wake structures of the disconnected-C, the connected-C, and the splitter plate. In the case of the splitter plate, as the circulation moves into the wake region periodically, the galloping instability is triggered at a higher reduced velocity (Ur > 6).

5.3.3 Assessment at Subcritical Reynolds Number In the previous section, two-dimensional simulations at a low Reynolds number are conducted to investigate the interaction of shear layers with the near wake and its implication to suppress VIV and thereby reducing the drag force. Here, we extend our assessment to three dimensions at the subcritical Reynolds number. Since the 3D simulations are computationally intensive, we simulate only fairing and

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Fig. 5.43 Dependence of transverse amplitude on reduced velocity for various suppression devices at Re = 3000−13000, m ∗ = 1.92, ζ = 0.007. The proposed connected-C and the fairing exhibit comparable performance to suppress transverse VIV amplitude

connected-C. The dimensionless VIV parameters are similar to those in Assi et al. [23], where the mass ratio m ∗ = 1.92, the damping ζ = 0.007, and Reynolds number Re ∈ [4000, 14,000]. We focus our discussion on the lock-in region. The representative reduced velocities are chosen to be Ur = 4.07, 4.47 and 4.88. To resolve the wake turbulence in the variational coupled solver, we employ a large-eddy simulation filtering based on dynamic subgrid procedure [180]. We consider the finite span cylinders attached with the devices, mounted elastically between the two vertical periodic walls located 5D distance apart in the spanwise direction z-axis. The results for each configuration are presented and analyzed in the following sections. 5.3.3.1

Amplitude Response

Figure 5.43 shows the transverse amplitude A∗y and the lift force Cl of both devices against the plain cylinder. Both fairing and connected-C devices provide effective suppression of VIV for the three reduced velocities corresponding to subcritical Reynolds numbers. Compared to the plain cylinder, the maximum amplitude is reduced by 93.3% in fairing and 93.9% for the connected-C device. The profiles of both fairing and connected-C devices are also similar in the range of reduced velocities simulated, whereby the high amplitude associated with the upper branch is not observed in both fairing and connected-C devices. This suggests that both shear layer extension and reattachment are able to prevent the transition of cylinder VIV to the upper branch. Further insight into the three-dimensional results can be gained by studying the time histories of forces and the pressure distribution along the cylinder body.

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Fig. 5.44 Time history plots of plain cylinder (black solid line), U-shaped fairing (blue dashed line) and connected-C (red dash-dotted line) at Re = 6150−7400 m ∗ = 1.92, ζ = 0.007

5.3.3.2

Force Dynamics

Figure 5.44 shows the lift and drag forces experienced by fairing and connected-C in the upper branch. The expected results from the vibrating plain cylinder in the transverse-only direction can observed from the time traces of amplitude and forces. In the lock-in region, the lift force has a dominant fluctuating component with the vortex shedding, while the streamwise drag force contains a steady component and a fluctuating component with half the vortex shedding period. From the figure, it is evident that both fairing and connected-C are effective in the reduction of lift and drag force for all three reduced velocities as compared to the plain cylinder. We further elaborate this in Fig. 5.45, which shows the time histories as well as frequencies of the transverse amplitude and the lift coefficients for the plain cylinder, the fairing, and the connected-C at Ur = 4.07 in the upper branch. It is observed that the lift coefficient of the cylinder is in-phase with its amplitude, an expected phase relation in the upper branch. On the other hand, the lift coefficient and transverse amplitude of connected-C are out-of-phase. This indicates that the connected-C is at its lower

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Fig. 5.45 Time histories (left) and frequency spectra (right) of transverse displacement (solid line) and lift coefficient (dashed line) for plain cylinder, fairing and connected-C device at Ur = 4.07, Re = 6150

branch, and the reattachment of the shear layer helps to prevent the cylinder from transiting into the upper branch. In the case of the fairing, it does not show any obvious phase difference between the amplitude and the lift coefficient. There appears to be irregular aperiodic evolution of lift force due to complex interactions of shear layer vortices and the gap flow between the two parallel fairings. From the frequency spectra of y ∗ and Cl in Fig. 5.45, the standard lock-in of amplitude and lift force can be observed for all three configurations at the dominant synchronization frequency. The lift force fluctuates with similar frequency components as the body motion for the plain cylinder and connected-C device. In the case of the fairing, it is not obvious, as the fairing has a broadband response due to complex interactions of tip vortices of the fins with the shed Karman vortices. This suggests that the shear layer reattachment provides suppression of highly irregular and secondary vortices as compared to the shear layer extension.

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Fig. 5.46 Mean pressure coefficient C¯ p along the cylinder at Ur = 4.07. The angle is measured from the front most point of the cylinder to the back of the cylinder. For fairing, the angle between 90◦ and 115◦ is covered by fins while for connected-C, the angle above 174◦ is covered by connector Table 5.8 Base pressure coefficient of different devices at Re = 6150, Ur = 4.07, m ∗ = 1.92, ζ = 0.007 Device Mean, C¯ p Standard deviation, σCb Plain cylinder Fairing Connected-C

−2.468 −0.500 −0.948

0.7757 0.08673 0.02441

To further elucidate the reduction of fluid forces, we plot the pressure distribution along the cylinder for each device at Ur = 4.07 in Fig. 5.46. The pressure distribution is obtained by taking the average of the pressure coefficient when the cylinder is at its maximum displacement. Similar to the low-Reynolds number study, both fairing and connected-C have similar profiles of the pressure distributions. The pressure gradient along the cylinder has been reduced via a redistribution of the vorticity, thereby decreasing the lift force experienced by the system. As discussed earlier for the low Reynolds number analysis, such reduction is due to the shear layer stabilization through extension and reattachment. We further look into the mean base pressure coefficient, C¯ p experienced at Ur = 5.0, which is summarized in Table 5.8. In the case of cylinder and fairing, C¯ p is measured at the back of the cylinder, while in the case of connected-C, it is obtained by averaging the C¯ p at both connection points between the connector plate and the cylinder body.

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5.3.3.3

325

Flow Visualization

Comparisons of the instantaneous spanwise vorticity contours for the U-shaped fairing and the connected-C device are shown in Fig. 5.47, while their base pressure coefficients are summarized in Table 5.8. For the purpose of reference, the numerical results of vibrating plain cylinder with the same configuration is also included for Ur = 4.07 in the upper branch. In the case of plain cylinder, as shown in Fig. 5.47a, the shear layer roll ups in the near wake region by the action of a Kelvin-Helmholtz instability mechanism. As the near wake region is continuously disturbed by the vortex rolls, the standard deviation of the base pressure is found to be large. In addition, the transverse pressure gradient across the plain cylinder is also relatively high i.e., a greater lift force is experienced by the cylinder body. As observed in the experiments, two pairs of counter-clockwise vortex rolls (2P mode) can be seen for the vibrating plain cylinder at lock-in with Ur = 4.07. Differing from what was observed in two-dimensional simulation, the shear layer detaches at the shoulder of the cylinder and reattaches to the fins before the vortex sheds at the tips. Upstream boundary layers continuously supply vorticity which feeds into the shear layer vortices. Oppositely signed wall vorticity layer can be observed over the fins, which joins with the cylinder shear vorticity layer and forms the alternating vortex rolls at the tip of the fin. The flow between both fins is irregularly disturbed by the vortex shedding in the afterbody region. Despite the vortex-induced disturbances, the pressure distribution between the fins is rather stabilized. This is reflected in the mean and standard deviation of the base pressure of each device, as shown in Table 5.8. The fairing has the highest mean base pressure among all devices. This is due to the wake stabilization between the fins, which increases the base pressure in the wake region. In the case of connected-C, the wake region is partly shielded by the C-shaped foil. As shown in Fig. 5.47c, the pressure at such region is affected by the vortex shedding. This leads to a relatively higher magnitude of base pressure. From the spanwise vorticity contours, two symmetric closed recirculations are observed in the trapped regions between the cylinder and the C-shaped foil. The pressure distributions in both recirculation regions are qualitatively similar to each other, as shown in Fig. 5.47c. Thus, the pressure gradient in the transverse direction across the connected-C is smaller. This causes the smaller lift force experienced by the connected-C device. On the other hand, the pressure behind the C-shaped foil is also stabilized. However, as the region is partly disturbed by the vortex roll-ups, the standard deviation of base pressure is somewhat higher compared to the fairing. The drag force experienced by the C-hinged device is remarkably smaller compared to the plain cylinder counterpart. However, we observe a bit larger streamwise drag force for the proposed device as compared to the fairing device. Through some optimization and by employing perforations in the C-shaped foil may relieve the drag force, which can be considered for a future study. To further analyze the three-dimensional vortical structures, we employ a vortex identification based on Q-criterion [172]. Figure 5.48 shows the Q-based vortical structures for the plain cylinder, the fairing and connected-C devices. For the plain

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5 FIV Suppression Devices

Fig. 5.47 Flow field results for subcritical Reynolds number at center slice (z = L/2) of spanwise vorticity contours (left) and pressure coefficient (right) around cylinder and suppression devices at Re = 6150, Ur = 5.0, m ∗ = 1.92, ζ = 0.007. The snapshots are taken at the maximum transverse displacement and + denotes the mean cylinder position

5.3 Appendage Devices for VIV Wake Stabilization

327

∂u

Fig. 5.48 Instantaneous isosurfaces of Q (− 21 ∂∂ux ij ∂ xij ) at Q = 0.5, around plain cylinder and suppression devices at Re = 6150, Ur = 5.0, m ∗ = 1.92, ζ = 0.007. It is colored by Z-vorticity and both devices have L c /D = 2.0

cylinder, the flow structures are highly irregular due to large amplitude oscillation. Dominating streamwise vortex ribs superimposed on the spanwise vorticity structures can be seen in the plots. The large vortical structures about the order of cylinder diameter are distributed in the spanwise direction and they are reasonably resolved by the spanwise length. The attachment of fairing and C-foil has a strong influence on the wake topology and provides a regularity into the vortical structures. The width of the wake is reduced for the fairing and the connected-C as compared to the plain cylinder for identical conditions. According to the freestreamline model from a simple momentum consideration [364], the reduction in the width of wake has a direct influence on the drag force. While the extended shear layers with the offset of vortex rolls can be seen for the fairing through the spanwise vorticity contours, symmetric

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5 FIV Suppression Devices

recirculations are present between the cylinder and C-foil. The streamwise vorticity blobs are well organized and somewhat stretched along the fairing. In comparison with the plain cylinder, there are finer flow structures over the cylinder body for both devices, as illustrated by the instantaneous streamwise vorticity patterns. In general, the wavelengths of streamwise vorticity patterns for the fairing and the connected-C devices are much shorter than the plain cylinder counterpart. A detailed investigation of the wake topology will be an object of future investigation.

5.3.4 Interim Summary In this section, we have proposed a novel connected-C device for VIV suppression and studied the wake stabilization mechanism. Through numerical simulations, we assessed the performance of the proposed connected-C device against the conventional U-shaped fairing and other variants namely the splitter plate and disconnectedC. The numerical evaluation was conducted for two regimes of Reynolds numbers: laminar flow at Re = 100 in two dimensions and the turbulent flow for subcritical Reynolds number ranging Re = 4000−14,000 in three dimensions. In both low and subcritical Re flow regimes, the cylinder and the attached suppression device were allowed to move in a transverse direction only. The wake stabilization and VIV suppression mechanisms were investigated in two aspects: extension and reattachment of shear layers in the near wake region. The conventional U-shaped fairing, the connected-C, the disconnected-C, and the splitter plate were simulated for low mass-damping parameter and their performance was assessed for a range of reduced velocities. It was found that, while the splitter plate experienced galloping at higher reduced velocity, the other three devices were robust at higher reduced velocity for two-dimensional laminar flow. By combining a splitter plate with the C-shaped profile, the galloping instability disappeared in the postlock-in region associated with the higher reduced velocity. For the connected-C and the disconnected-C devices, the reattachment of the shear layer was observed. Two closed circulations were formed when the shear layer was attached to the C-shaped foil, regardless of the presence of the connector plate. These counter-rotating circulations provided a force cancellation effect by symmetrizing the pressure gradient in the transverse direction, which favored the near wake stabilization and thereby reduced the amplitude of vibration. Due to the wake stabilization and the symmetric flow field around the connector plate, there was no galloping-like instability observed at a higher reduced velocity in the post-lock-in regime. Similar performance was observed for the U-shaped fairing and the connected-C at subcritical Reynolds number, where the VIV amplitude and the drag force were reduced as compared to the plain cylinder counterpart. Compared to the transversely vibrating plain cylinder, the maximum VIV amplitude is reduced by 93.3% in fairing and 93.9% for the connected-C device. The mechanism of shear layer reattachment was also found to be effective in reducing fluctuating lift force and the mean drag on the vibrating system. The mechanism based on the free shear layer

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329

attachment is successfully employed for wake stabilization and VIV suppression. We believe that the systematic analysis presented can help to improve the understanding of wake stabilization and VIV suppression. Further research is required towards a parametric investigation of the connected-C device to fine-tune its geometric variables and arrangement for a wide range of flow conditions. In particular, the rotation effects of the disconnected-C device should be accounted for from a practical viewpoint for different current incidences with a larger range of Reynolds numbers to fully understand the dynamically coupled instabilities of the proposed design. To avoid any galloping-like instabilities of the connected-C device, a practical range of mass-damping parameters of the device should be explored with additional degrees of freedom of rotation and streamwise motion. Lastly, the proposed connected-C device should be tested through a physical experiment.

5.4 Near-Wake Jets for FIV Suppression Floating offshore structures such as semi-submersibles are generally used in deepwater and ultra-deepwater regions for oil/gas production and exploration, floating wind turbines and ocean space utilization. The semi-submersible platform is a selfstabilized structure with improved motion characteristics and good resistance to the wind, wave and current in any directions [389] and comprises of a submerged pontoon and four vertical columns, with square geometry cross-section. The vertical columns connect the submerged pontoon and the deck structure. The semi-submersible platforms have relatively lesser sensitivity to the environmental loads due to winds, waves and currents when compared to ship-shaped FPSO’s (Floating Production, Storage, and Offloading). This stable characteristic of semi-submersible is beneficial for deep-water operations and hence it is widely deployed. Due to the existence of strong ocean currents, the structure undergoes flow-induced motions causing large transverse oscillations termed vortex-induced motion (VIM). The phenomenon of VIM is similar to vortex-induced vibration (VIV) with regard to the underlying principle, which arises when the vortex shedding frequency matches the natural frequency of the structure. The VIM occurs for large floating bodies with large natural periods. Hence, the vortex-induced oscillation of floating structures is commonly referred to as VIM. The semi-submersibles deployed in harsh sea conditions experience large unsteady loads and six degree-of-freedoms (dofs) motions due to complex nonlinear interaction with ocean currents and waves. In recent years, investigations of VIM for semi-submersibles have gained importance due to the presence of strong currents in ultra-deepwater environments. In the VIM measurements of semi-submersible model by Waals et al. [433], the maximum transverse amplitude was measured to be 0.48D and 0.41D for 0◦ and 45◦ angle of incidence at Ur = 11 and 7, respectively at Re = 6 × 103 to 7 × 104 . Here D denotes the characteristic column diameter, Re = U D/ν is based on this diameter, U is the oncoming velocity and ν denotes the

330

5 FIV Suppression Devices

kinematic viscosity. The existence of lock-in and post-lock-in responses was exhibited from the field measurements of prototype [360]. The lock-in and post-lock-in responses are defined as VIM and galloping regimes. Galloping is a self-excited instability that exhibits high-amplitude and low-frequency oscillation that typically occurs in elastic structures with non-circular cross-sections [56]. Comprehensive studies of [173, 464] have led to the development of a new generation of semi-submersibles, followed by the measurements of Goncalves et al. [133], which provided insights into the flow dynamics of semi-submersibles. The flow profile on the wake side of the semi-submersible is more complex and wider when compared to the circular cylinder [133]. This complexity of the flow profile is associated with the consequences of wake interaction of vortices shed from multiple columns. Owing to the intrinsic presence of vertical columns for semi-submersibles in the fluid flow, complex wake flow patterns are generated. The flow interference and shielding effects of tandem and side-by-side square cylinder alter the wake patterns and hydrodynamic loads. Different types of regimes exist for tandem square cylinders depending upon the spacing (L/D) between them. The wake interference comes into the existence for square cylinders at L/D = 3–4 and becomes irrelevant at L/D = 27 [374]. As most of the offshore structures are designed for L/D ≈ 2 to 3, a clear understanding of the wake interference regime is necessary. The geometry of the semi-submersibles is generally square-shaped and therefore it is susceptible to VIM and galloping motions [181] and is essential to control VIM of floating platforms. It is well known that the frequency of vortex shedding is associated with switching of the confluence point (saddle point) from one side to another side, with respect to the wake centerline of the square cylinder and only the analogy is represented for the floating platform. The saddle point is defined as the point where the two entrainment layers coming from opposite sides of the square cylinder meet and indicate the end of the vortex formation region [269]. Zdravkovich [477] stated that, by disturbing the shear layers or preventing the switching of confluence points, the VIM could be completely attenuated. Many flow control methods have been proposed in the last few decades to suppress the VIM of bluff bodies. The flow control techniques are broadly classified into active and passive techniques. Each of these techniques is further classified and its details are presented in [125]. The key difference between active and passive techniques is that the former requires auxiliary power to operate and the latter does not consume power. The manifestation of passive flow control techniques are appendages and auxiliary geometries over the bluff body. The prominent passive flow control techniques which are used in present scenarios are fairings, helical strakes, splitter plate, control rods, slits parallel to flow and surface protrusions [477]. Implementation of strakes to mitigate VIM for floating structures such as truss spars were already reported by researchers such as [176, 362, 441, 442]. The effectiveness of the splitter plate and its role in reducing vortex-induced forces were investigated by [19, 21, 34] for bluff bodies. Implementation of surface protrusions, small control rods, parallel slits over the cylinder surface and its influence over VIM and drag reduction were examined by [30, 483, 501]. Although the passive flow control techniques are most sought after due to their zero power input, the active flow control techniques are sometimes

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331

necessary for suppressing the VIM and the vortex-induced forces. The active flow control techniques such as blowing or suction from the wall surface, injection of micro-bubbles or particles, acoustic excitation, periodic rotation, and wall motion are some examples. Investigations related to the active flow control techniques, which include rotating cylinders at the wake side [293] or at the shear layer side of the bluff body [291] were all carried out using the finite element method with the Petrov-Galerkin technique [168, 170, 415]. A number of studies, related to the injection of steady fluid for attenuating vortex-induced loads and VIM of circular cylinders have been reported earlier. Only a handful of investigators have focused on the blowing technique for square-shaped columns. Measurements conducted by [119], for the porous cylinder at Re = 8 × 103 and 8 × 104 , observed that suction and blowing of fluid at the base side of the body influence the separation point. Through numerical computations, Dong et al. [109] demonstrated the elimination of the vortex streets of circular cylinders at Re = 100 and 500 via a combined suction and blowing strategy. Intermittent breakage of vortices along the span-wise length of the cylinder was noticed in their studies. Suppression of VIM by acoustic means, conducted by [123], observed 30% reduction of drag forces at Re = 9000. Significant changes in the wake velocity profile were visualized with the increase in the acoustic strength. Numerical investigations conducted by [256], for a circular cylinder with synthetic jets at the mean separation point, observed a remarkable attenuation of the transverse lift force. The injection provides additional momentum to the fluid surrounding the body, causing a net pressure increment on the top and bottom surfaces of the body and lowering pressure on the front and rear sides of the body [98]. Furthermore, from the numerical studies of [447], it was examined that the blowing jet increases the base pressure influencing the vortex shedding process and the drag force. The prominent effects of a splitter plate are to reduce the strength of the vortex formation region and delay the interaction of free shear layers until the end of the plate [10]. As an analogy to a splitter plate, we will consider near-wake streamwise jets at the base side of the columns of semisubmersible since both the splitter plate and the streamwise jet tend to stabilize the near-wake region by delaying the interaction of shear layers [92]. The experimental investigations conducted by [98] for a square cylinder is the prominent work for Re in the order of 104 . From their investigations, they observed that the pressure and drag forces around the square cylinder change significantly as the positions of the jet flow change. The pressure deficit at the base side of the cylinder tends to get filled up with the injection of fluid [98]. The near-wake jet at the center position of the square cylinder prevents the shear layer interaction. Interaction of separated layers is also delayed due to the presence of a near-wake jet at the center position of the square cylinder. Saha et al. [373] have made these observations in their numerical investigations for a square cylinder with the near-wake jet at the center position of the body at Re =100. They also concluded that the near-wake jet mimics like a splitter plate, the higher the jet flow velocity lesser the undulations of shear layers. The visualization flow experiments and pressure measurements of [3] have advocated the fact that injection of fluid in the near-wake region results in pressure recovery at the base side of the body. Therefore, based on earlier investigations

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5 FIV Suppression Devices

Fig. 5.49 A conceptual sketch of a square cylinder subjected to uniform flow with near-wake jet flow at the base side of the body. Sketches of vortices, streamlines and presence of saddle point marked by X. Adapted from [269]

available in the literature it is evident that active flow control methods are efficient and applicable for cases where passive flow control methods are not feasible. Recent works on the implementation of active damping systems in semisubmersible have achieved 75% of VIM reduction with 25% of critical damping coefficient [398]. Among various flow control techniques, the most prominent active flow control techniques are positioning of actuators in the sensitive regions, where the shear layers are generated, mean separation points, injection of jet flow into the near-wake region, and suction of fluid over the body walls. Several investigators have advocated the benefits of injecting jet flow in the near-wake region of the body. Generally, the near-wake jets at the center portion of the bluff body mimic like a virtual splitter plate preventing the interaction of alternating vortices, influencing the saddle points and pushing the vortex formation region. Therefore, near-wake jets at the center portion of the column will act as near-wake stabilizers, such as splitter plates, guiding plates, base-bleed, and slits cut along the column, influencing the saddle point [477]. The objective of this study is to suppress VIM and vortex-induced loads acting on the semi-submersible, by injecting prescribed steady jet flow on the wake side of the semi-submersible. A conceptual sketch of a square cylinder with the steady jet flow downstream is elaborated in Fig. 5.49. The near-wake jets induce high momentum fluid in the wake region, which prevents the shear layers to interact, as illustrated in Fig. 5.49. The velocity of the fluid jet is denoted as V jet injected in the near-wake region as depicted in the figure. The jet flow prevents the interaction of alternating vortices A and B, thereby preventing the shedding of vortices in the near-wake region close to the body and the vortex formation region is shifted farther downstream. The near-wake jet performs a similar function as that of the base bleed or guiding plates at the base side of the body, by affecting the saddle points [477].

5.4 Near-Wake Jets for FIV Suppression

333

5.4.1 Multi-column Offshore Platform by Near-Wake Jets The 3D CAD model of deep-draft semi-submersible is shown in Fig. 5.50a. The model is scaled at 1:70 and the dimensions are similar to the case considered by [433]. The diameter of the column is considered as D, the height of the column and length of the pontoon are 2.6D and 4.4D, respectively. The computational domain employed in our simulation is presented in Fig. 5.50b. The streamwise and transverse directions are denoted as x and y, respectively. The position of the model is at the origin of the domain, upstream and downstream length of the domain is 85D and 175D. The width and depth of the computational domain considered is 175D and 40D, respectively. No-slip conditions are implemented on the walls of the semi-submersible, slip wall conditions are implemented on the top, bottom and side walls of the computational domain as elucidated in Fig. 5.50b. Uniform flow velocity (U ) is specified at the inflow plane and pressure outlet is imposed at outflow plane. The model is free to vibrate in the transverse direction only, other degree of freedoms are restricted in the present study. The width of the computational domain is sufficiently large, corresponding to the blockage ratio of 0.06%. Three-dimensional unstructured mesh is generated using open source software GMSH [129] and its details are presented in Fig. 5.51. The mesh consists of 1.7 million grid points and the time step (t) considered for the computation is t = 0.05. Figure 5.51a presents the top and close-up view of the discretized model and the three-dimensional isometric view of the unstructured finite element mesh is presented in Fig. 5.51b. To capture the turbulent wake dynamics 10 million linear P1 tetrahedral elements are employed. The distance (yw ) between the model wall and first grid line is maintained to be 0.03D, and 30 mesh layers are generated along the length of the pontoon and column length. A reasonable fine grid has been designed for the dynamic subgrid LES model. The VIM of the floating platform is influenced by key parameters, such as mass ratio (m ∗ ), Reynolds number (Re), reduced velocity (Ur ) and damping ratio (ζ ), which are expressed as m∗ =

M , mf

C , ζ = √ 2 MK

Re =

UD , ν

Ur =

U , fN D

(5.17)

and the corresponding dimensional parameters in Eq. (5.17) are summarized in Table 5.9. Based on the column diameter D, the Reynolds number (Re) considered for our study is similar to the model tests conducted by Waals et al. [433]. Typically for floating platforms, the mass ratio (m ∗ ) is less than unity and the damping ratio (ζ ) is about 1%, which is comparable to the model test. Furthermore, it is found that the damping ratio up to 10% does not significantly affect the VIM response [359]. Similar observations with regard to the damping effect are reported in [133, 272, 360]. Consistent with the model test, we set the values of m ∗ and ζ to 0.83 and 0.01, respectively. For the dynamical analysis, the lift and drag force coefficients are defined as

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Table 5.9 Definition of dimensional parameters Parameters Definition M mf ρ C ν K fN = U

1 2π



K M

Units

Mass per unit length of the body Mass of fluid displaced by the body Density of fluid Damping constant Kinematic viscosity of fluid Spring stiffness

kg/m3 Ns/m m2 /s N/m

Natural frequency of the body

Hz

Fluid flow velocity

m/s

Cl =

2Fl , ρU 2 L H

Cd =

kg kg

2Fd ρU 2 L H

(5.18)

where Fl and Fd are the lift and drag forces. The fluid loading is evaluated by integrating the surface traction considering the first layer of elements located on the surface of the semi-submersible. Here the L and H are the length and height of the semi-submersible, which is 4.4D and 2.6D as shown in Fig. 5.50a, respectively.

5.4.2 Validation and Response Characteristics To establish the accuracy of our numerical model, we compare our predictions with the experimental measurements of [433] for the semi-submersible model at 0◦ angle of incidence. The spatial and temporal convergence studies are presented in our earlier works [180, 297]. A reasonable fine grid for 3D scaled model with the complete geometry has been designed in the wake of the dynamic subgrid LES model, as detailed in [146]. The comparison of the computed values of transverse amplitude with the measurement data is presented in Fig. 5.52a, whereby the amplitude is evaluated from the time-series of displacement data: Amax y D

=

max(A y ) − min(A y ) D

(5.19)

where max(A y ) and min(A y ) are maximum and minimum amplitude values in the displacement evolution. Figure 5.52a also shows the transverse amplitude values obtained from two-dimensional (2D) and three-dimensional (3D) simulations. The 2D simulation comprises a simple four-column configuration without a pontoon and the 3D simulation corresponds to the complete offshore platform with multi-columns and pontoon. Our numerical model is capable of capturing all the three regions, such

5.4 Near-Wake Jets for FIV Suppression

335

(a)

(b)

Fig. 5.50 3D model of deep-draft semi-submersible: a CAD model of deep-draft semi-submersible floater in the scale of 1:70, b computational domain with boundary conditions

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Fig. 5.51 3D unstructured mesh of the computational domain: a top view of the discritized computational domain and close-up view of tetrahedral elements over the semi-submersible wall, b isometric view of the computational domain

as pre-lock-in, lock-in and galloping regions. The computed values are observed to have a close match with experiments as shown in Fig. 5.52a. The measurements performed by Waals et al. [433] is for the deep-draft semi-submersible model in the scale ratio of 1:70. Due to the 3D effect, as expected, the maximum response amplitudes (Amax y /D) of the 3D scaled model are much closer to measurements in comparison to the 2D simulations. The hydrodynamic forces (lift and drag) acting on the 3D scaled model are presented in Fig. 5.52b, as a function of the reduced velocities. The values in Fig. 5.52b represent the root mean square of lift coefficient and the mean drag force coefficient, denoted as Clrms and Cd , respectively. The Clrms

5.4 Near-Wake Jets for FIV Suppression

337

0.6

0.5

4

Waals et al., 2007 Present 2D - four-column configuration Present 3D - scaled model

Cd Cl

0.5

0.4 3

0.3

2 |

Cd

0.3 Clrms

max

Ay /D

0.4

0.2 0.2 1 0.1

0.1 0

0

5

10

15

20 Ur

25

30

35

0

40

0

5

10

15

(a)

20 Ur

25

30

35

0 40

(b)

Fig. 5.52 Response characteristics: a maximum transverse amplitudes for 2D four-column configuration, 3D scaled model and measurements [433], b lift and drag force coefficients of semisubmersible with respect to Ur

0

1.6 0

-0.35

Power spectrum

Ay/D

0.35

-0.7 100

101

3.2 Ay/D Cl

Cl

0.7

-1.6

200

tU/D

300

-3.2 400

10

fAy fCl

0

10

-1

10

-2

10

-3

10-4 10

-5

0

0.1

0.2

0.3

0.4

0.5

0.3

0.4

0.5

0.3

0.4

0.5

fD/U

(a) 1.6

0

0

-0.35 -0.7 100

Power spectrum

0.35

101

Cl

3.2

Ay/D

0.7

-1.6

200

tU/D

300

-3.2 400

10 10

0

-1

10-2 10

-3

10

-4

10

-5

0

0.1

0.2

fD/U

(b) 1.6

0

0

-0.35 -0.7 100

Power spectrum

0.35

101

Cl

3.2

Ay/D

0.7

-1.6

200

tU/D

300

-3.2 400

10

0

10

-1

10

-2

10-3 10

-4

10

-5

10

-6

0

0.1

0.2

fD/U

(c) Fig. 5.53 Time histories (left) and frequency spectra (right) of transverse displacements (A y /D) and lift coefficients (Cl ) for semi-submersible at: a Ur = 6, b Ur = 10 and c Ur = 30

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5 FIV Suppression Devices

f/fN

135

φ

Present simulations Strouhal frequency

3

180

90

Cy-Ay/D

2

f = fN

1

45 0

0 0

5

10

15

20

25

30

35

40

0

5

10

15

20

Ur

Ur

(a)

(b)

25

30

35

40

Fig. 5.54 Response results of semi-submersible with respect to Ur : a phase angle between lift force coefficient and transverse amplitude, b frequency ratio

2

1.5 1 100

Cd

3 2.5

Cd

3 2.5

Cd

3 2.5

2

1.5

200

300

400

500

1 100

2

1.5

200

300

400

500

1 100

200

300

tU/D

tU/D

tU/D

(a)

(b)

(c)

400

500

Fig. 5.55 Time histories of drag force coefficients for semi-submersible at Ur : a 6, b 10 and c 30

is observed to be maximum of 0.4 at Ur = 6 and reaches to a minimal value of about 0.1 at Ur = 10, and eventually levels out for higher Ur . Similarly, the Cd achieves a maximum value of 0.26 at Ur = 10 and decreases gradually for higher Ur cases. The time traces and frequency spectra of the amplitude ratio and the lift force coefficient are presented in Fig. 5.53a–c for Ur = 6, 10 and 30. From the time histories, it is observed that for Ur = 6 the lift and the displacement responses are in phase with each other and the amplitude increases until the maximum at Ur = 10. For Ur = 10 and 30 there is a phase lag of about 45◦ and 182◦ , which implies that the lift force leads the amplitude. The amplitudes and lift force responses are regular for Ur = 6 and 10, unlike the cases of Ur = 30, where the responses are irregular due to the presence of a galloping phenomenon. To further elucidate the dynamical responses of the semi-submersible for various Ur cases, the frequency spectra of the amplitudes and the lift force are presented in Fig. 5.53, adjacent to the time histories of the respective reduced velocities. The frequency spectra for Ur = 6 exhibit a single dominant frequency of 0.09 for both the amplitude and the lift force. Similarly, the dominant frequencies for Ur = 10 and 30 are estimated as 0.086 and 0.040, respectively. The frequency of the lift force is broadband and for the amplitude, it is single dominant at Ur = 30. This phenomenon is attributed to the presence of multiple columns in the geometry. Compared to the single cylinder/column, the wake generated from each column and its periodic vortex shedding leads to the combined wake effects for the semi-submersible. Through the phasing and the frequency analysis, we next explore how the combined wake behind the multi-column transfers the energy to sustain the platform motion.

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339

Typically, the phase angle between the transverse oscillation and the lift force coefficient of a structure and the frequency response can give some insight into the energy transfer from the fluid to the structure. Therefore, the phase angle ( C y −A y /D ) between the transverse amplitude and the lift force coefficient is presented in Fig. 5.54a and the frequency response is shown in Fig. 5.54b. The phase angle is calculated based on the averaged time lag between the local maxima points of lift force and the amplitude. We have considered 20 oscillation cycles for time averaging. The averaged time lag is then multiplied by the frequency of the response to estimate the phase angle. From the figure, it is observed that the C y −A y /D increases as a function of reduced velocity. The C y −A y /D begins to increase gradually starting from Ur = 5 and reaches 180◦ for Ur ≥15, indicating a jump in the amplitude response. A similar observation was made by [41] in their measurements. It is to be noted that the observations in aspects of phase angles for semi-submersibles are not reported elsewhere. The frequency ratio ( f / f N ) is defined as the ratio of oscillating frequency ( f ) to the natural frequency of the structure ( f N ). The positive energy transfer signifies that the energy is transferred from the fluid flow to the cylinder when the lift force leads the position in the range of 0◦ to 180◦ . For an asymmetric sharp-cornered shaped body such as a square cylinder, the separation points are fixed and can undergo VIV and galloping. From the studies of [313], the energy transfer could be associated with the frequency response. The frequency ratio of the floating platform in the present study reaches the value of one and the corresponding transfer of energy from fluid to structure is marked, as stated by [313] and hence this region is termed as lock-in [41]. At the lock-in region i.e., at Ur = 10 and 11, the frequency ratio is approximately equal to 1, indicating that the oscillating frequency and the natural frequency of the structure are equal. For Ur = 5, 6 and 7 the values are less than 1, attributing to the fact that the oscillating frequency is less than the structural natural frequency and indicating that it is in the de-synchronized region. For Ur ≥15, the oscillating frequency is no more equal to the natural frequency since the structure lies in the galloping region, where the structural oscillation is not self-limiting as stated by [433]. The time histories of drag force coefficients for Ur = 6, 10 and 30 are shown in Fig. 5.55a–c. The trends of the drag force coefficient show an irregular trend for all the Ur cases, which is associated with the periodic vortex shedding of the combined wake effects due to the multiple columns.

Results and Discussion The large oscillation of a bluff body with a complex geometry undergoing VIM in a uniform flow could be attenuated by modifying the near-wake flow dynamics. In the present study, this is achieved by injection of near-wake jets on the wake side of the bluff body. The steady jet blowing method at the base side of the sharp corner geometry like a square cylinder is related to shifting of vortex formation region farther downstream and prevention of shear layer interaction. Since the bluff body considered in the present study is a multi-column system, complex wake profiles are

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generated, resulting in large amplitude oscillations. Therefore, this section presents the influence of near-wake jets in different configurations. Systematic investigations are performed to identify the efficacy of each near-wake jet configuration. The positions of streamwise jet locations are determined based on the 2D simulations of four-square cylinder configuration and 3D simulations of single square column [147]. Simulations are carried out for different blowing angles ranging from 0◦ to 90◦ which correspond to the normal streamwise jet and parallel to the column surface, respectively. From the previous studies [147], it was observed that hydrodynamic forces and kinematic responses were lower for jet flows normal to the surface compared to other angles. In addition, the simulations were extended to a 3D square column to understand the influence of jet locations. Square column models with the jets located at the center and corner (staggered) positions of the column were analyzed. The studies demonstrated a reduction of forces and VIM for both cases but the reduction percentage for the jets located at the center position is relatively higher than the staggered positions. This could be explained by the influence of jet flow on the free shear layers. For the staggered jet locations, we observed that the jet flow mixes with the free shear layers and makes the vortex roll-up relatively stronger. The streamwise jets at the center position act as a splitter plate thereby inhibiting the shear layer interaction and also repositioning the vortex formation region. In order to understand the effectiveness of near-wake jets, we present the response characteristics and the flow dynamics of semi-submersible with near-wake jets. In the first section, we discuss the unsteady response characteristics and the flow profiles for various types of near-wake jets, proposed in this study. The second section reveals the influence of the mass flow rate (Cμ ) as a function of reduced velocity (Ur ).

Response Characteristics Substantial alterations of kinematic and dynamic responses are observed in the aspects of the number of near-wake jets and its geometrical shape. From our comprehensive analysis, we present the time histories of amplitude ratio, the lift and drag force coefficients at various reduced flow velocities and for various near-wake jet configurations. The nondimensional parameter of the jet flow velocity (Vjet ) is represented as mass flow rate coefficient (Cμ ) defined as Cμ =

A j V j2 0.5U 2 Aref

(5.20)

where Ajet , is the area of the jet flow, Aref is the reference area which typically represents the projected area of the semi-submersible.

5.4 Near-Wake Jets for FIV Suppression

341

Fig. 5.56 Near-wake jets of different configurations at the wake side of the floater: a configuration 1: near-wake jet at column, b configuration 2: near-wake jet at column and pontoon, and c configuration 3: elongated near-wake jet at column

5.4.3 Various Configurations of Near-Wake Jets Figure 5.56 shows various locations and the number of jets on the wake side of the columns. Configuration 1, depicted in Fig. 5.56a consists of four jets at each column downstream and is positioned at its center. The length and the width of the jet are 0.2D and 0.1D as illustrated in the zoom-in view of Fig. 5.56a. The fluid, coming out from the jet area prevents the interaction of shear layers. Similarly, Fig. 5.56b shows configuration 2, typically representing 14 jets, positioned along the columns and pontoon downstream of the semi-submersible. The concept of positioning the jets along the columns and pontoon is to prevent the shear layers separated from both column and pontoon. Configuration 3 as shown in Fig. 5.56c, consists of a jet in an elongated configuration positioned at the center of columns. The length and width of the jet are 2.2D and 0.1D, respectively. The semi-submersible with various nearwake jet configurations are simulated at Ur = 10 (lock-in), and the corresponding amplitude time history is shown in Fig. 5.57. It is worth noting that the velocity ratio (Vjet /U ) for each near-wake jet is 10. Semi-submersible with the maximum response is observed for the model with no jet case, and it decreases with the injection of near-wake jets with various configurations. The reduction of transverse amplitude is associated with the decrease in lift forces exerted over the model, and the values are summarized in Table 5.10. The root mean square (rms) of transverse amplitude, the lift force coefficient (Cl ) and the mean drag force coefficient Cd are summarized in Table 5.10.

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5 FIV Suppression Devices 0.4

No jet Configuration 1 Configuration 2 Configuration 3

Ay/D

0.2

0

-0.2

-0.4

350

375

400 tU/D

425

450

Fig. 5.57 Time history of transverse motion of semi-submerisble with different near-wake (N-W) jet configurations at Ur = 10. The Cμ for the cases considered are as follows: No jet—Cμ = 0; configuration 1—Cμ = 4.30 (N-W jet at the column); configuration 2—Cμ = 7.30 (N-W jet at the column and pontoon); configuration 3—Cμ = 11.60 (elongated N-W jet at the column) Table 5.10 Statistical values of semi-submersible model with various near-wake jet configurations at Ur = 10 and Vjet /U = 10 Arms y /D Clrms Cd

No jet

Configuration 1

Configuration 2

Configuration 3

0.15 0.05 0.89

0.11 0.04 1.18

0.12 0.04 1.58

0.09 0.03 1.35

From the studies, it is examined that there is a reduction of Arms y /D for semisubmersible with the near-wake jet, of configurations 1, 2 and 3, by 26, 20 and 40%, in comparison to the no jet case (Cμ = 0). The transverse amplitude of elongated jet (configuration 3) is lesser by 18% and 25%, as compared to configurations 1 and 2. Similarly, the Clrms of configuration 1, 2 and 3 is 20%, 20% and 40% lower than no jet cases. With respect to the configurations 1 and 2, the Clrms for configuration 3 is 25% lesser. However, the mean drag force coefficient (Cd ) increases, and is associated with the increase in the mass flow rate coefficient (Cμ ). The flow visualizations, as shown in Figs. 5.58 and 5.59, exhibit the instantaneous near-wake patterns for various jet configurations. The contours as shown in Fig. 5.58, illustrate the pressure distributions for the zero mass flow rate and the different mass flow rates. The decrease in the drag force coefficient for the body with near-wake jets compared to the no-jet case could be related to the pressure contours presented in Fig. 5.58a–d. The pressure contours for the no-jet case show that the pressure values at the downstream columns are comparatively higher than the near-wake jets. This implies that the shear layers are shed immediately at the base side of the downstream columns. For the cases with the near-wake jets, the shear layers separating from the downstream columns are pushed away by the jet flows, thereby preventing them from shedding immediately

5.4 Near-Wake Jets for FIV Suppression

343

Fig. 5.58 Pressure contours at tU/D = 510 for various near-wake jet configurations at Ur = 10: a no jet, b configuration 1, c configuration 2 and d configuration 3

at the near-wake region. Therefore, the pressure values are minimal compared to the no-jet cases as observed in Fig. 5.58b–d. Since the difference between upstream and downstream pressure results in the estimation of the total drag force of the system. Hence, the minimal pressure value for the body with a near-wake jet attributes to an increase in drag forces. These observations comply with the observations made by [114], where the synthetic jets at the base side of a circular cylinder pushes the vortex formations region away from the near-wake region. The vorticity contours for no mass flow rate and the different mass flow rates are presented in Fig. 5.59. Figure 5.59a exhibits the shedding of vortices immediately at the base side of the downstream columns, generating complex flow profiles. Unlike the cases of Fig. 5.59b–d a clear absence of elongated vortices are seen for semisubmersible with zero mass flow rate at the wake side of the columns. For the models with mass flow rate, Fig. 5.59b–d the alternating vortices which are shed at the nearwake side of the columns are pushed away by the injecting fluids. In comparison to the zero mass-flow rates, the vortices which are shed at each of the columns in the wake side, are observed to interact with one another causing large VIM and vortex-induced loads. To understand the vortical structures on the wake side of semi-submersible, we employ a vortex identification method based on the Q-criterion [172]. Figure 5.60 presents the Q-criterion based vortical structures for the semi-submersible without and with the near-wake jets. The isosurfaces are shown at a constant positive value and the contour surfaces are colored by the stream-wise fluid flow. The semi-submersible

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5 FIV Suppression Devices

Fig. 5.59 Spanwise z-voricity contours at tU/D = 510 for various near-wake jet configurations at Ur = 10: a no jet, b configuration 1, c configuration 2 and d configuration 3

configurations with the mass flow rates Cμ = 0, 4.30, 7.30 and 11.60 are shown in Fig. 5.60a–d, respectively. Figure 5.60a–d, illustrate the generation of shear layers from the sides of the upstream columns of semi-submersible. The shear layers rolling from upstream columns re-attach with the downstream columns as shown in Fig. 5.60a. The shear layer reattachment occurs when the gap between the inner surfaces of the square columns in the tandem arrangement is 2.4D. According to the investigations of [374], who had performed measurements for various gap ratios, observed that for gap ratio less than 3D, the drag force coefficient and the wake frequency are in decreasing trend. Further, they confirmed the existence of shear layer reattachment for a gap ratio less than 3D. Therefore similar observations are also made in our study. The shear layers then roll into vortices and shed immediately on the wake side of the column, generating a complex flow profile, which is shown in Fig. 5.60a. These vortical structures, which are shed at the downstream columns, interact with one another causing huge vortex-induced loads and VIM. Figure 5.60b presents the semi-submersible with near-wake jets of configuration 1, where the jet flows are induced only at the columns, as shown. Similarly, near-wake jet flows are observed for semi-submersibles with configurations 2 and 3 of respective mass flow rates 7.30 and 11.60 in Fig. 5.60c, d. The vortices which are shed immediately on the wake side of the columns are pushed away by the near-wake jets. The stronger the near-wake jet flow farther the vortices are pushed. Figure 5.60d exhibits the presence of stronger near-wake jet flow for semi-submersible with a mass flow rate of 11.60. Therefore, the near-wake jet causes a reduction in the lift forces and the corresponding transverse amplitudes of the semi-submersible.

5.4 Near-Wake Jets for FIV Suppression

345

Fig. 5.60 Top view representation of Q-criterion of semi-submersible for different near-wake jet configurations at Ur = 10, with Q value = 1 at the time instant tU/D = 510: a no jet, b configuration 1, c configuration 2 and d configuration 3; A-reattachment of shear layers., B-shedding of vortices at the downstream columns., C-near-wake jet flow., D-stronger near-wake jet flow

Dominating streamwise vortex ribs, superimposed over spanwise vorticity structures is shown in the isometric views of the semi-submersibles without the jet in Fig. 5.61a. An immediate shedding of the vortices at the wake side of the downstream columns accompanied by the complex wake flow profile is also shown in Fig. 5.61a. Similar figures of semi-submersibles with near-wake jet configurations are shown in Fig. 5.61b–d, respectively. The alternating shear layers generated at the upstream columns reattach themselves with the downward columns, generating another set of alternating shear layers with less intensity. These shear layers usually roll up and are shed as vortices in the near wake region. This process leads to huge VIM for floating platforms like semi-submersibles. The near-wake jets prevent the shear layers to shed immediately at the near-wake region and pushes the vortex formation region away from the floating body. This process reduces lift forces and transverse amplitudes to a large extent. Figure 5.61b represents the near-wake jets at columns, which comprises four jets at each column and the corresponding mass flow rate is 4.30. The vortices engulfing the near-wake jets are evident in Fig. 5.61b, which is associated with lesser drag force but results in increased lift force and VIM when compared to other near-wake jet configurations. Figure 5.61c represents the model

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Fig. 5.61 Isometric view representation of Q-criterion of semi-submersible for different nearwake jet configurations at Ur = 10, with Q value = 1, at the time instant tU/D = 510: a no jet, b configuration 1, c configuration 2 and d configuration 3; A-complex flow profile in near-wake region; B-vortices dissipating the momentum of the near-wake jet flow; C-near-wake jet flow along the entire column

with jets positioned at column and pontoon, which in total consists of 14 numbers of near-wake jets and its corresponding mass flow rate, is 7.30. The Q-criterion for vortical structures of the semi-submersible with elongated near-wake jets at each column is presented in Fig. 5.61d, where the fluid is blown along the entire length of the downstream column. In our study, the elongated near-wake jets (configuration 3) attenuates the formation of vortices along the entire column length effectively, when compared to configuration 1 and 2, respectively. Moreover, the observations from Fig. 5.61b, shed further insight about vortices dissipating the momentum of near-wake jets at the downstream columns, unlike Fig. 5.61d. Therefore, it is essential to blow high-velocity fluids to push the vortices away from the wall. Hence, configurations 1 and 2, requires a higher amount of energy to blow high-velocity fluids to effectively prevent the formation of vortices and suppress VIM. Hence, considering this fact, configuration 3 is effective in mitigating the VIM and the vortex-induced loads. Therefore, it is necessary to extend the study of configuration 3 for various reduced velocities.

5.4 Near-Wake Jets for FIV Suppression

347

Fig. 5.62 Spanwise wake structures at column and pontoon level of semi-submersible with and without elongated near-wake jets at Ur = 10 at the time instant tU/D = 510: a no jet, b configuration 1, c configuration 2 and d configuration 3; A - jet flow at column and pontoon levels

Similar to the Q-criterion as illustrated in Figs. 5.60 and 5.61, the evolution of spanwise vortices at column and pontoon levels for the model without and with nearwake jets is presented in Fig. 5.62. Figure. 5.62 is a manifestation of the injection of jet flows into the near-wake region with an illustration of instantaneous jet flow velocity in the z-vorticity field. Figure 5.62a is the representation of semi-submersible without near-wake jets, while (b), (c) and (d) represents the model with various nearwake jet configurations. Figure 5.62b–d exhibit the jet flows at the column, column and pontoon, and at the column in an elongated form. The near-wake jet flow at the column as shown in Fig. 5.62b is observed to be of lesser efficient and does not act as a barrier to the shear layer interaction. Instead, the alternating vortices, dissipate the momentum of the jet flows, thereby resulting in the suppression of VIM in an inefficient way. The application of near-wake jets at the column and pontoon as shown in Fig. 5.62c, is comparatively more efficient than the former configuration. However, the vortex formation length near the column levels is not adequately pushed farther away from the body. Figure 5.62d, is the typical instantaneous jet flow profile for an elongated configuration. The near-wake jets act as a shear layer interaction barrier along the entire spanwise configuration, i.e., at the column and pontoon planes of the model. Moreover, the jet flows at the entire column length acts like a splitter plate fitted at the base side of the downstream column, suppressing VIM effectively when compared to other configurations.

348

5 FIV Suppression Devices 0.8 Measurements (Waals et al., 2007) Cμ= 0 Cμ = 0.73 Cμ = 3.00 Cμ = 11.7

0.6

max

Ay /D

Fig. 5.63 Maximum amplitude response of semi-submersible with various mass flow rate coefficients for elongated near-wake jets (Configuration 3) and comparison with no jet flow case

0.4

0.2

0

0

5

10

15

Ur

20

25

30

35

From earlier investigations, it is evident that the impact of elongated near-wake jets at the wake side of the column is high when compared to other near-wake jet configurations. Therefore, further numerical investigations are carried out for various Ur and different mass flow rates (Cμ ). The values of mass flow rates (Cμ ) are calculated based on the ratio of jet velocity (Vjet ) to the free stream velocity (U ). The velocity ratios considered are 2.5, 5 and 10, and the corresponding Cμ is 0.73, 3.00 and 11.7, respectively.

5.4.3.1

Response Characteristics and Flow Field for Elongated Near-Wake Jets

The maximum transverse amplitude with respect to Ur is presented in Fig. 5.63 for various mass flow rates (Cμ ). The simulations are conducted for Ur = 5, 10, 15, 20 and 30, representing initial branch, lock-in region (Ur = 10 and 15), and galloping region, respectively. From our studies, 66, 70 and 77% reduction in the maximum amplitude response is noticed for semi-submersible with Cμ = 0.73, 3.00 and 11.70, respectively at Ur = 10. The reduction in amplitude is associated with the prevention of shear layers interaction due to the injection of fluid in the near-wake region, which attenuates VIM. A similar observation was reported earlier by [109], at Re = 500. The root mean square (rms) of lift force and mean drag force coefficients for the semi-submersible model with elongated near-wake jets is presented in Fig. 5.64a, b and Fig. 5.65. For the purpose of comparison, Clrms and mean Cd of semi-submersible without jets are also illustrated in the figure. From the figure it is observed that there is considerable decrease in Clrms at Ur = 10 as shown in Fig. 5.64b. The mean drag force coefficients of the semi-submersible with and without near-wake jets are presented in Fig. 5.65. The model with a higher value of near-wake jet flow will exhibit higher

5.4 Near-Wake Jets for FIV Suppression

349

0.4

0.1 Cμ = Cμ = Cμ = Cμ =

rms

0.2

Cl

Cl

rms

0.3

0 0.73 3.00 11.7

0.1

0

0

5

10

15

20

25

30

0

35

10

15

20 Ur

Ur

(a)

25

30

(b)

Fig. 5.64 Variation of lift force coefficient (Clrms ) with respect to Ur for various mass flow rate coefficient (Cμ ): a an overall view of Clrms , b zoom in view of Clrms in-between 9.5 ≤ Ur ≤ 12 Fig. 5.65 Variation of mean drag force coefficient (Cd ) with respect to Ur for various mass flow rate coefficient (Cμ )

3 Cμ = Cμ = Cμ = Cμ =

0 0.73 3.00 11.7

|

Cd

2

1

0

0

5

10

15

Ur

20

25

30

35

drag force when compared to the lower values of Cμ . The drag force increase for the body with near-wake jets is due to the minimum pressure values as elaborated in Sect. 5.4.3. The spanwise vortices at column and pontoon planes for near-wake jets of different mass flow rates are shown in Fig. 5.66. From the figures, it is observed that the presence of near-wake jets prevents the shear layers interaction generated from the alternating sides of the column and pontoon planes. The semi-submersible model without a near-wake jet is shown in Fig. 5.66a. Figure 5.66b–d represent the model with near-wake jets for mass flow rates 0.73, 3 and 11.7, respectively. From the figures, it is evident that the near-wake jets with higher Cμ prevent the shear layer interaction effectively, but in the aspects of VIM responses, the values

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5 FIV Suppression Devices

Fig. 5.66 Spanwise wake structures at column and pontoon level of semi-submersible with and without elongated near-wake jets at Ur = 10: Cμ = a 0, b 0.73, c 3.0 and d 11.70; A-near-wake jets preventing the vortices interaction; B-stronger prevention of vortices interaction

are close to one another, exhibiting a 12% difference at Ur = 10. Therefore, from our investigations, the semi-submersible model with near-wake jets of Cμ = 0.73 benefits in suppressing VIM of the floating structure and also exhibits lower Cd and Cl when compared to other mass flow rates. The corresponding velocity ratio (Vjet /U ) is 2.5. Moreover, the strength of near-wake jets is exhibited in Fig. 5.66b– d. Strong prevention of vortices interaction is observed as the fluid blown from the near-wake jets (Cμ ) increases.

5.4.3.2

Flow Field for VIM and Galloping Regimes for Cμ = 0.73

From the earlier sections, in our study, it is evident that the value of mass flow rate, appropriate for suppressing VIM, is 0.73. The increase in mass flow rate does not benefit in reduction of amplitudes largely compared to lower values of Cμ , but results in higher drag forces. Hence, it is significant to observe the flow profiles for the model with near-wake jets with a mass flow rate 0.73 at Ur = 10 and 30. The pressure contours with velocity magnitude lines projected over it for column and pontoon planes are shown in Fig. 5.67a, b at resonance and galloping regimes, i.e., Ur = 10 and 30. The three-dimensional aspect of the contour plots is presented in the figure, where the pressure difference between the upstream and downstream sides of the model depicts a large difference. It is to be noted that the flow from the near-wake jets benefits in suppressing VIM irrespective of the huge pressure difference at resonance and galloping regime. The near-wake jets represented in Fig. 5.67a depict the strength of the jet flow at resonance and galloping regimes. In the galloping regime, the strength of the near-wake jets is greater than in the

5.4 Near-Wake Jets for FIV Suppression

351

Fig. 5.67 Pressure contours at column and pontoon levels for Cμ = 0.73 at VIM and galloping regimes: Ur = a 10 and b 30

resonance regime. As mentioned in the earlier discussions, the velocity ratio between flow velocity and jet flow velocity is 2.5 for both resonance and galloping regimes and the corresponding Cμ is 0.73. Similar observations are exhibited in the contour plots of velocity magnitude which are shown in Fig. 5.68a, b for Ur = 10 and 30, respectively. A clear difference in the flow velocity and jet flow velocity is illustrated in the figures for both cases. Hence, injection of near-wake jets in the near-wake region benefits in suppressing VIM for wide operating flow conditions effectively. A complete summary of VIM control for floating platforms using near-wake jets are presented in this section, for Re = 20,000, m ∗ = 0.83 and ζ = 0.01. In the present study, the galloping (higher reduced velocity) and VIM regimes correspond to higher and lower fluid flow velocities, respectively. Figure 5.69 illustrates the variation of

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5 FIV Suppression Devices

Fig. 5.68 Contours of velocity magnitude at column and pontoon levels for Cμ = 0.73 at VIM and galloping regimes: Ur = a 10 and b 30

maximum amplitude response (Amax y ) with respect to mass flow rate (C μ ) for various reduced velocities (Ur ). From the figure, the lowest amplitude values are observed at Cμ =3.00, when compared to other mass flow rates for all the cases of reduced velocities. The mass flow rate 3.00 corresponds to the jet velocity ratio (V jet /U )=5. At VIM and galloping regimes, the responses at Cμ = 0.73 is similar to the case of mass flow rate 3.00. Since, the jet velocity is 2.5 times the flow velocity for Cμ = 0.73, the energy spent in generating jet flow is comparatively less than the energy required for Cμ = 3.00. Hence, for an optimum mass flow rate maximum amplitude reduction is observed for Cμ = 0.73. From the contour plots, it is evident

5.4 Near-Wake Jets for FIV Suppression

0.3 Ur=5 Ur=10 Ur=15 Ur=20 Ur=30

* X

Amax y

0.2

Vjet/U=10

Vjet/U=5

Vjet/U=2.5

Fig. 5.69 The influence of elongated near-wake jet flow for pre lock-in, lock-in and galloping regimes

353

*

* *X

X

0.1

X

0

0

2

4

6

Cμ

8

10

12

14

that at Cμ = 0.73 the, near-wake jets act as a shear layer interaction barrier, similar to splitter plates for bluff bodies. Since the VIM is largely reduced for Cμ = 0.73 when compared to no jet case, it is therefore concluded that the bluff body with a mass flow rate of 0.73 is more appropriate to suppress VIM efficiently.

5.4.4 Summary In the present section, a numerical study was performed to suppress vortex-induced motion of semi-submersible platform by means of a continuous blowing-based flow control method. Streamwise near-wake jets at the base side of the floating platform are considered to demonstrate the effectiveness of the proposed flow control method for a semi-submersible model mounted as a spring-mass-damper system. The floating platform considered in our study is a semi-submersible in the scale ratio of 1:70 and the numerical investigations are conducted at Re = 20,000 for the mass ratio m ∗ = 0.83 and the damping ratio ζ = 0.01. The coupled responses of the semisubmersible model with and without the control method are analyzed in terms of vibration characteristics, the force coefficients, the frequency and phase relationships. Overall, the blowing-based flow control provides a good control performance to suppress the VIM of the platform model by decreasing the transverse amplitude. The jet flow of blowing-based control acts as shear layers interaction barrier, pushing the vortex formation region farther downstream, alike splitter plates. The interaction between the shear layers generated at the alternating sides of the semisubmersible columns is prevented by inducing the streamwise jet in the near-wake region. We investigated various near-wake jet configurations for assessing the effec-

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5 FIV Suppression Devices

tiveness of VIM reduction. From our systematic analysis, it is observed that elongated near-wake jets at the center portion of the column benefits in suppressing VIM for semi-submersible, when compared to other near-wake jet configurations. The reduction of transverse amplitude is estimated to be 18% to 25%. For the cases of configuration 1, the alternating shear layers attenuate the momentum of the jet flows from the individual near-wake jets thereby, reducing its effect. We further investigate the effect of VIM with respect to mass flow rate coefficient (Cμ ) from the elongated near-wake jet. The values of Cμ considered in our study is 0.73, 3 and 11.7. Considering the power requirement for the injected jet flows, the optimum mass flow rates were estimated to be 0.73 and 3. The transverse amplitude and the lift coefficient of the semi-submersible with the elongated near-wake jet were observed to decrease and the drag force increases with increasing mass flow rate. From our analysis, the difference of transverse amplitude for various mass flow rates was examined to be just 12%. Furthermore, the hydrodynamic coefficients Cd and Cl of the semi-submersible with higher near-wake jets are much higher when compared to lower Cμ . It is observed from our studies that the elongated near-wake jets at the columns with Cμ = 0.73 is more effective in suppressing VIM. In the near future, investigations of multi-column platform with a combined suction and blowing would be of an interest to improve the performance of VIV control and for various angle of incidences.

Chapter 6

VIV of Sphere

In this chapter, we focus on the FIV response of a fully and partially submerged sphere in a uniform flow. We first review the coupled two-phase fluid-structure solver based on the spatially filtered two-phase Navier-Stokes equations in the moving boundary arbitrary Lagrangian-Eulerian framework. The role of streamwise vorticity/freesurface interaction on the VIV response of a freely vibrating sphere is analyzed numerically as functions of immersion ratio, the mass ratio and Froude number. Coupled dynamics of unsteady wake-sphere interaction, the force and amplitude characteristics and the vorticity and pressure distributions are investigated during the oscillation.

6.1 Introduction Fluid-structure interaction (FSI) involves the coupled behavior of fluid flow with deformable/moving solid structures. Fluid mechanics and its interactions with the structure play a pivotal role in various engineering applications ranging from aerospace, biomedical, power transmission to marine and offshore systems. Oil and gas platforms, risers, and vessels, for example, are designed to operate in harsh fluid environments. These offshore systems should perform safely and stably in the ocean environment. Fluid-structure interaction poses tremendous challenges to the safety and reliability of such offshore systems. Especially, flow-induced vibration (FIV) can cause catastrophic failure and fatigue damage to marine/offshore and civil engineering systems. Successful prediction and control of fluid-induced loads and vibrations can lead to safer and cost-effective structures. Engineering structures in a freestream can form a great variety of vortex wake modes that can have a profound role in the performance of structural dynamics. An unsteady fluid flow exerts an unstable force on a structure, which can cause severe © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Jaiman et al., Mechanics of Flow-Induced Vibration, https://doi.org/10.1007/978-981-19-8578-2_6

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6 VIV of Sphere

flow-induced vibrations. Flow-induced vibration of a structure can potentially create a positive feedback loop, amplifying and even driving the oscillatory nature of the flow. Depending on geometric and physical conditions, there are various types of flow-induced vibrations such as vortex-induced vibration, flutter/galloping, buffeting, wake-induced vibration, etc. Of particular interest is vortex-induced vibration (VIV), which is related to lock-in/synchronization of the vortex shedding frequency with the structural natural frequency. The lock-in phenomenon leads to large oscillations of the structure. Therefore, a fundamental understanding of VIV for bluff-body flows is required to identify the lock-in regimes and stability properties.

6.1.1 Flow-Induced Vibrations of Bluff Bodies Fluid-structure interaction (FSI) of spherical bodies is omnipresent and has numerous applications in marine and offshore engineering. For example, flow-induced vibrations (FIV) of an elastically mounted or tethered spherical configuration can be useful for power generation and wave energy harvesting while such vibrations are undesirable on spherical marine/offshore structures such as low-aspect-ratio escort tugboats connected with ships [399]. Physical understanding of these problems poses serious challenges due to the richness and complexity of nonlinear coupled FIV phenomenon together with vorticity/free-surface interactions. A prototypical geometry of the sphere (i.e., axisymmetric bluff body) close to the free surface or piercing the free surface can be considered as an idealized model, which serves as a generic problem to examine the coupled fluid-structure and free-surface interactions. A freely vibrating sphere in the vicinity of the free surface can exhibit complex spatial-temporal dynamics and synchronization as functions of physical and geometric parameters. Synchronization or lock-in is a general nonlinear physical phenomenon in fluidstructure systems whereby the coupled system has an intrinsic ability to lock at a preferred frequency and amplitude. The phenomenon of lock-in and vortex-induced vibrations are extensively reviewed for cylindrical structures in [40, 381, 455]. When the natural frequency of an elastically-mounted configuration of a spherical body approaches to the frequency of the unsteady vortex shedding, the sphere can undergo vortex-induced vibration similar to two-dimensional bluff bodies [139, 454, 455]. In contrast to two-dimensional circular cylinder wakes, the vortex topology and shedding process are significantly different for a three-dimensional configuration of an elastically mounted sphere. Furthermore, due to the vorticity/free-surface interactions, the vortex-induced vibration of the sphere can be significantly altered as reported recently in [377]. Through high-fidelity numerical simulations, the central intent of this chapter is to explore the effect of free surface on the vortex-induced vibration of an elastically mounted sphere at subcritical Reynolds numbers (based on the freestream velocity and the sphere diameter).

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357

6.1.2 Free Surface and Vorticity Dynamics Many have studied the vortex dynamics of circular cylinders in close proximity to a free surface. The free surface deforms to satisfy the fluid-fluid (i.e., air-water) interface conditions such that the tangential stress maintains zero stress condition and the normal stress remains constant. The vorticity generation process at a free surface is different in contrast to a no-slip solid surface, whereby the flow velocity at the surface can slip freely to satisfy the zero stress condition and the surface can deform or distort. Owing to complex dynamical interactions between the deformed free surface and the flow field, free-surface boundaries can act as sources or sinks for vorticity. To characterize the free-surface dynamics, the ratio of the inertial force to the gravitational force and surface deformation is defined by Froude number √ Fr = U/ g D, where U is the freestream velocity, D is the characteristic length and g is the gravitational acceleration. The experimental work on flow past a cylinder close to the free surface in [391], investigated the effect of the Froude number at a high range of Fr ∈ [0.47, 0.72]. They reported that for high Froude numbers, the wake of a cylinder close to the free surface is fundamentally different than the wake of a cylinder far beneath the free surface. The generation of a vorticity layer from the free surface was observed in their experiments. The authors found that for low Froude number (Fr ≤ 0.3) with small free-surface deformation, the problem is analogous to the flow past a cylinder close to the no-slip wall. The free surface was found to act like a rigid free-slip boundary at a low Froude number. However, for higher Froude numbers, the surface can deform significantly, giving rise to larger surface vorticity that can defuse or convect into the main flow and modify the wake dynamics. The numerical study on the two-dimensional flow past a cylinder close to the free surface at Re = 180 was performed in [357]. It was found that the free-surface curvature can lead to a relatively larger diffusion of vorticity. A series of numerical studies considered the flow past a cylinder piercing the free surface [203, 472]. The study in [472] on the piercing cylinder, considered the effect of the free surface on the vortex pattern in the near wake for the Froude number up to Fr = 3. They found that the free surface prevents vortex generation in the near wake, and therefore reduces the vorticity and vortex shedding. At Fr = 0.8, 2-D vortex structures were spotted in the deep wake while in the proximity of the free surface, the vortex structures showed strong 3-D features. At higher Froude number Fr = 2, the effect of the free surface propagated throughout the wake and prevented the regular vortex shedding and vortices with less intensity dominated the region below the free surface.

6.1.3 Flow-Induced Vibration of Sphere in a Close Proxmity to a Free Surface While there are a plethora of publications on the FIV of a cylindrical body, a few studies focusing on a freely vibrating sphere are available in the literature. The first

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study on the FIV of a sphere was performed in [137, 454]. The authors conducted an experimental study on a fully submerged tethered sphere in a steady fluid flow. Their experimental observation on the tethered sphere, uncovered the vigorous vibration of the sphere similar to the cylinders when exposed to the uniform flow stream. The authors experimentally observed that a tethered sphere will undergo a large peakto-peak amplitude of about two diameters of the sphere over a wide range of flow velocities. The motion trajectory of the sphere was found to form a “figure-eight” trajectory and by the increase in the mass ratio (m ∗ = m/m d , where m is the mass of the structural body and m d is the mass of the displaced fluid), the motion trajectory transformed to the “crescent shape”. Further investigations on the effect of the mass ratio on the amplitude response for the tethered and elastically mounted sphere configurations were studied in [139]. They showed the maximum peak amplitude as a function of the mass-damping parameter (m ∗ ζ ) on the Griffin plot. The plot exhibited a good collapse of data where a saturation maximum amplitude of around (0.9D) was recorded for all cases. The existence of multiple modes of vibration was reported for the tethered sphere configuration in the experimental studies performed in [137, 139, 188, 454]. These modes were identified based on the amplitude response curve ( A∗ -U ∗ ), where A∗ is the non-dimensional amplitude defined as A∗ = A/D, and U ∗ is the reduced velocity defined as U ∗ = U/ f n D ( f n is the natural frequency of the system in a vacuum). For the range of reduced velocities U ∗ ∈ [5 ∼ 10], the authors identified mode I and mode II of vibrations that correspond to the lock-in region where the sphere vibration synchronized with the shedding frequency and the natural frequency of the system [137, 454]. Mode III of vibration was found to exist at a higher reduced velocity range U ∗ ∈ [20 ∼ 40], where the shedding frequency is three to eight times higher than the sphere vibration frequency [139]. The existence of mode IV (intermittent mode) at much higher reduced velocities (U ∗ ≥ 100) for a tethered sphere was reported in [188], where non-periodic large amplitude response was observed. The experimental and numerical studies in [231, 354] further explained the existence of non-stationary chaotic dynamic response similar to the mode IV in [188], at a lower reduced velocity range for low inertia systems. Unlike a vast amount of literature available on the two-dimensional geometry of elastically-mounted circular cylinders, there are a handful of numerical studies on a three-dimensional geometry of a sphere undergoing flow-induced vibration. A numerical study on the VIV of a freely oscillating sphere in all three spatial directions has been performed in [45] at Re = 300 and the reduced velocity range of U ∗ ∈ [4, 9]. The authors observed two distinct VIV response and wake modes, termed as the hairpin mode and the spiral mode, at the same reduced velocity. It was found that the motion trajectory corresponding to the hairpin mode followed a linear path, while for the spiral mode, the sphere shifted to a circular orbit [45, 46]. In another recent numerical study of [353] on 3-DOF elastically mounted sphere, the authors reported the instability of the hairpin mode for the range of Re ∈ [300, 2000]. They observed that the hairpin mode was always followed by the spiral mode during the transformation from a transient state to a stationary state for the Reynolds number range studied. The VIV response of elastically mounted spheres restricted to move

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359

in the transverse direction was carried out experimentally and numerically in [352]. The authors provided additional insight into the experimental study of the sphere transverse motion [376], and explored some distinctions from the numerical and experimental studies with specific constraints such as the tethered configuration [454] or 3-DOF elastically mounted [45]. All the aforementioned studies on the FIV of the sphere were performed for the flow past a fully submerged structural body with no free-surface effect. However, the VIV response of the elastically mounted sphere close to the free surface or piercing it could be very different due to the complications of vorticity/free-surface interactions. The experimental study in [289], was conducted for a tethered sphere in shallow water. The authors reported the reduction of the amplitude response due to the presence of the free surface when the sphere is fully submerged. In the recent experimental investigation of [377], the authors systematically studied the effect of the free surface on the vortex-induced vibration of fully and semi-submerged elastically mounted sphere for the range of Reynolds number Re ∈ [5000, 30,000], the reduced velocity U ∗ ∈ [3, 20], and the immersion depth ratio h ∗ = h/D ∈ [−0.75, 2.5], where h denotes the distance from the top of the sphere to the free surface. When the sphere came closer to the free surface for a range of immersion ratio (0.185 ≤ h ∗ ≤ 1), the authors observed the reduction in the peak amplitude response and the transverse fluctuating force acting on the sphere. However, by further decreasing the immersion ratio in the range (0 ≤ h ∗ ≤ 0.185) for the submerged sphere and in the range (−0.375 ≤ h ∗ ≤ 0) for the piercing sphere, the peak amplitude response of the oscillations was found to increase significantly. Several modes of vibrations were categorized in their study for the VIV response of fully and partially submerged cases based on force measurement, the total phase (phase difference between the fluid force and the body displacement) and the vortex phase (phase difference between the vortex force and the body displacement). Due to the limitation on the PIV imaging set-up in their experiments, capturing the vorticity formation close to the sphere was found to be challenging. The reasons for the large-amplitude response for the piercing sphere cases could not be fully explained. The effect of the free surface on the wake dynamics, the vortex forcing and the unsymmetrical geometry of the semi-submerged immersed body once part of it lies above the waterline added to the complexity of the problem.

6.1.4 Organization In the present study, the role of streamwise vorticity/free-surface interaction on the VIV response of a freely vibrating sphere is analyzed numerically as functions of immersion ratio, the mass ratio and Froude number. The physical setup for the elastically-mounted sphere along the free surface is similar to the experimental configuration of [377]. Figure 6.1 shows a schematic of the problem setup for a piercing sphere case together with the geometric and flow parameters. Building upon the experimental investigations, we attempt to understand the origin of the large ampli-

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6 VIV of Sphere Free-surface level

ky

Deformed Free-surface h

Z

U D

cy Y

U X

D

X

ky Side view

cy

Top view

Fig. 6.1 Schematic illustrating a side view of sphere piercing a viscous free surface (left) and a top view of 1-DOF elastically mounted sphere in steady incident flow (right). Here D denotes the sphere diameter, U is the freestream velocity, k is the spring stiffness and c is the structural damping. h measures the distance from top of the sphere to the free-surface level

tude vibration when the sphere pierces the free surface. We consider a representative h ∗ = −0.25 to investigate the large-amplitude excitation mechanism via numerical computations. We employ a first-principle based fully-coupled continuum mechanics formulation for solving a multiphase fluid-structure interaction at sub-critical Reynolds number [198]. The free-surface effects are taken into account by modeling the air-water interface with the aid of the phase-field Allen-Cahn equation and the turbulence is modeled via dynamic large eddy simulation (LES) [180]. Successful validation of the 1-DOF vibrating sphere by considering the effect of the free surface is established through detailed quantitative and qualitative comparisons with the experiments. We systematically examine the effect of Reynolds number on the mode transitions at VIV regime for a freely vibrating sphere, and its substantial effect on the coupled dynamical behavior and the motion trajectories. The central intent of this work is to perform a numerical investigation of the unsteady flow fields and the vibrational characteristics of the elastically mounted sphere subjected to the vorticity/free-surface interactions. The insight gained is used for identifying the wake modes and the coupled dynamical interactions that lead to vortex-induced vibration with a large amplitude response for the sphere configuration piercing the free surface. Coupled dynamics of unsteady wake-sphere interaction, the force and amplitude characteristics and the vorticity and pressure distributions are investigated during the oscillation. We deduce that the extra vorticity generation at the free surface for the piercing sphere has a significant impact on the synchronization of the vortex shedding and the vibration frequency, and could be the major cause of the large-amplitude VIV response. Such physical insight on the VIV and free-surface interactions may guide the development of effective active or passive suppression devices. In Sect. 6.2, we summarize the two-phase FSI framework which is followed by brief implementation details in Sect. 6.3. While Sect. 6.4 presents the validation of the FSI solver, Sect. 6.5 focuses on the FIV response of a fully and partially submerged sphere with a particular emphasis on the origin of the large-amplitude response of the piercing sphere. The major conclusions of this work are reported in Sect. 6.6.

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361

6.2 Numerical Methodology A brief description of the coupled two-phase fluid-structure solver based on the spatially filtered two-phase Navier-Stokes equations in the moving boundary arbitrary Lagrangian-Eulerian (ALE) framework and six-degrees of freedom structural equation is presented in this section along with its variational form.

6.2.1 Two-Phase Flow Modeling with Moving Boundary The spatially filtered Navier-Stokes equations in an ALE framework for an incompressible flow are given as  ∂ u¯ f  ρ + ρ f (u¯ f − um ) · ∇ u¯ f = ∇ · σ¯ f + ∇ · σ sgs + bf on f (t), ∂t xˆ f f

∇ · u¯ f = 0 on f (t),

(6.1) (6.2)

where u¯ f = u¯ f (x f , t) and um = um (x f , t) represent the fluid and mesh velocities defined for each spatial point x f ∈ f (t) respectively. The fluid density is denoted by ρ f and bf represents the body force acting on the fluid and σ¯ f is the Cauchy stress tensor for a Newtonian fluid which is given as σ¯ f = − p¯ I + μf (∇ u¯ f + (∇ u¯ f )T ),

(6.3)

where p¯ is the filtered fluid pressure, I denotes the second-order identity tensor, μf represents the dynamic viscosity of the fluid, and σ sgs is the extra turbulent stress term based on subgrid filtering procedure for large eddy simulation. Details about the dynamic subgrid model utilized in the present formulation can be found in [180]. The partial derivative with respect to the ALE referential coordinate xˆ f is kept fixed in Eq. (6.1). The density ρ f and viscosity μf for two-phase flows in Eq. (6.1) depend on the phase-indicator φ as 1+φ f 1−φ f ρ1 + ρ2 , 2 2 1+φ f 1−φ f μf (φ) = μ1 + μ2 , 2 2 ρ f (φ) =

(6.4) (6.5)

where ρ1f , μf1 and ρ2f , μf2 represent the densities and viscosities of the two phases respectively. In contrast to the traditional approaches to evolve the fluid-fluid interface of the two-phase flow like volume-of-fluid (VOF) and level-set which require some kind of geometric manipulation which can be computationally expensive in three-dimensions [195], we utilize the diffuse interface description which originate from thermody-

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6 VIV of Sphere

namically consistent theories of phase transitions and avoid any kind of geometric manipulations. The diffused interface has a finite thickness (O(ε)) and is evolved by the minimization of the Ginzburg-Landau energy functional,   E(φ) = f (t)

 ε2 2 |∇φ| + F(φ) d, 2

(6.6)

where ε is the representative length scale of the finite thickness of the fluid-fluid interface. It represents a balance between the interfacial energy and the bulk energy of the two-phase system. The convective form of the conventional Allen-Cahn equation [15] is considered in the present formulation with a Lagrange multiplier for mass conservation. The order parameter φ distinguishes between the two phases, being 1 in water and −1 in air. The order parameter φ changes continuously but steeply across the phase interface from one phase to the other in the interface thickness ε. The conservative Allen-Cahn equation is written on f (t) as:     ∂φ  + (u¯ f − um ) · ∇φ − γ ε2 ∇ 2 φ − F (φ) + β(t) F(φ) = 0,  ∂t xˆ f

(6.7)

where u¯ f is the convection velocity which is coupled with the Navier-Stokes equations and γ is a relaxation factor with units of [T −1 ] which is selected as 1 for the present study. The term F (φ) denotes the derivative of F(φ) with respect to φ, with F(φ) = (1/4)(φ 2 − 1)2 being a double well potential function which has minima at φ = ±1 indicating the two stable phases. The parameter β(t) is the time dependent part of the Lagrange multiplier given as β(t) =

F (φ)d . √ F(φ)d f (t) f (t)

(6.8)

The convection-diffusion-reaction form of the convective Allen-Cahn equation (Eq. 6.7) can be written as, ˆ + sˆ φ − fˆ = 0, ∂t φ + uˆ · ∇φ − ∇ · (k∇φ)

(6.9)

ˆ sˆ and fˆ are the modified convection velocity, diffusion where uˆ = (u¯ f − um ), k, coefficient, reaction coefficient and source terms respectively, the expressions of which are given in detail in [196, 197].

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363

6.2.2 Structural Modeling The structure is modeled as a rigid body having six degrees of freedom of motion. Consider a mapping function ϕ s (x s , t) which denotes the position vector and maps the reference configuration of the rigid body x s at t = 0 to its position at the deformed state s (t). If ηs (x s , t) is the displacement of the rigid body, the position vector is given by ϕ(x s , t) = ηs (x s , t) + x s . Let the center of mass of the body in the reference configuration x s and the current configuration ϕ s be x s0 and ϕ s0 respectively and ηs0 denote the displacement of the center of mass due to translation of the body. Therefore, the rigid body kinematics is given by ϕ s = Q(x s − x s0 ) + ϕ s0 = Q(x s − x s0 ) + x s0 + ηs0 ,

(6.10)

where Q is a rotation matrix. The displacement can be expressed as, ηs = ( Q − I)(x s − x s0 ) + ηs0 , ∂Q s ∂ηs ∂ηs = (x − x s0 ) + 0 , ∂t ∂t ∂t

(6.11) (6.12)

where I is the identity matrix and Eq. (6.12) is obtained by differentiating Eq. (6.11) with respect to time. Suppose the rotational degrees of freedom for the body are given by θ s . Equation (6.12) can be restructured in terms of the angular velocity of the body denoted by ωs = ∂θ s /dt as ∂ηs ∂ηs = ωs × (ϕ s − ϕ s0 ) + 0 ∂t ∂t

(6.13)

Therefore, the six degrees-of-freedom rigid body motion is governed by the matrix form, ∂ 2 ηs0 ∂ηs + C η 0 + K η ηs0 = f s , on s , 2 ∂t ∂t s 2 s ∂ θ ∂θ + K θ θ s = τ s , on s , Is 2 + Cθ ∂t ∂t

Ms

(6.14) (6.15)

where M s , C η and K η denote the mass, damping and stiffness matrices for the translational degrees of freedom respectively, I s , C θ and K θ represent the moment of inertia, damping and stiffness matrices for the rotational degrees of freedom respectively, and f s and τ s denote the forces and the moments applied on the body respectively.

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6.2.3 Fluid-Structure Interface It is imperative for a fluid-structure interaction problem that the kinematic and dynamic equilibrium are satisfied at the fluid-structure interface fs . Mathematically, these relations can be written as 

u¯ f (ϕ s (x s , t), t) = us (x s , t), σ¯ f (x f , t) · nd + f s = 0

(6.16) (6.17)

fs

where ϕ s is the position vector mapping the initial position x s of the rigid body to its position at time t, f s is the fluid force acting on the body and us is the structural velocity at time t given by us = ∂ϕ s /∂t. Here, n is the outer normal to the fluid-body interface fs in the reference configuration. Owing to the body-fitted ALE formulation, the fluid velocity is exactly equal to the velocity of the body along the interface. The motion of the immersed body is governed by the fluid forces which include the integration of pressure and shear stress effects on the body surface. The coupling algorithm between the two-phase fluid and the rigid-body structural equations is based on the nonlinear iterative force correction (NIFC) scheme presented in [198].

6.3 Implementation Details The continuum equations with their variational form presented in the previous section are coupled in a nonlinear partitioned iterative manner. The movement of the internal ALE nodes is evaluated by considering a continuum hyperelastic model for the fluid mesh such that the mesh quality does not deteriorate as the displacement of the body increases during wake-induced vibration. For fluid-structure interaction with strong added mass effects (m ∗ ≈ O(1)), a partitioned iterative scheme based on nonlinear iterative force correction has been employed [181]. The temporal discretization of both the fluid and the structural equations is embedded by energy conservative implementation of the generalized-α framework [94]. The equations are linearized via the Newton-Raphson technique and are then solved in a predictor-corrector format. Further details about the coupling procedure for the two-phase fluid-structure interaction problems can be found in [198]. While the displacement after solving the structure equations forms a predictor step, the transfer of corrected forces via the NIFC algorithm is a corrector step in a particular nonlinear iteration. The increments of the velocity and pressure fields in the linearized system of the equations are then evaluated by the Generalized Minimal Residual (GMRES) algorithm [370]. The left-hand side matrix is not constructed explicitly for this procedure, but we perform matrix-vector products of each block matrix for the GMRES algorithm. On the other hand, the mesh equation is solved by the conjugate

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365

gradient method owing to the symmetric property of the left-hand side matrix. The NIFC scheme [180, 181] increases the stability of the fluid-structure coupling for low structure-to-fluid mass ratio regimes. All the variables are interpolated using first-order Lagrange polynomials. For parallel computing, the solver relies on a standard master-slave strategy for distributed memory clusters by message passing interface (MPI) which depends on the domain decomposition strategy of the computational domain. The partition of the mesh is generated by the master process into different subgrids with the help of an automatic graph partitioner [202]. The master process performs the operations at the root subgrid and all other subgrids behave as the slave processes. The matrices at the element level are computed by the slave processes and then the system of equations is then solved across different compute nodes. The adopted fluid-structure interaction solver has been extensively validated for a wide range of fluid-structure interaction problems at subcritical Reynolds number without free-surface effects [180, 227, 242, 295, 308]. Systematic validation of the free-surface interaction with floating objects is provided in [198].

6.4 Convergence Study and Validation Before we proceed to a detailed analysis of sphere VIV with free-surface effects, we first perform a convergence study and validate the solver by comparing with the experimental and available numerical data. The definitions of some relevant important non-dimensional parameters are summarized in Table 6.1. The non-dimensional is defined as A∗ = A/D and f ∗ denotes the normalized amplitude response A∗

frequency and f n =

1 2π

k m

is the natural frequency of the spring-mass system in

Table 6.1 Definition of the non-dimensional parameters and post-processing quantities Parameter Definition Reynolds number Reduced velocity Mass ratio Damping ratio Froude Number Non-dimensional amplitude Immersion ratio Normalized horizontal force Normalized transverse force Normalized vertical force Normalized frequency

Re = ρ f U D/μf U ∗ = U/ f n D m ∗ = m/m d √ ζ = c/2 mk √ Fr = U/ g D √ A∗rms = 2 Arms /D h ∗ = h/D C x = f xs /( 21 ρU 2 S) C y = f ys /( 21 ρU 2 S) C z = f zs /( 21 ρU 2 S) f ∗ = f / fn

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vacuum, where m is the mass of the sphere and k is the spring stiffness. The mass ratio is given by m ∗ = m/m d , where m is the mass of sphere and m d is the mass of displaced fluid. For the cases of partially submerged bodies in the flow field, the mass ratio would be increased as the mass of the displaced fluid is reduced. In our numerical analysis based on the coupling of incompressible Navier-Stokes and rigid body equations, we use the natural frequency f n in a vacuum for the purpose of non-dimensionalization. During this fluid-structure coupling cycle, the added mass effect is implicitly accounted for in the coupled formulation and the response results are appropriately adjusted to match the experimental conditions [377]. The normalized horizontal and transverse forces are evaluated from the fluid traction, acting on the structural body, where C x is the normalized horizontal force, C y and C z are the normalized transverse forces in y and z directions, respectively. The normalized force coefficients are evaluated as follows  1 (σ¯ f · n) · n x d (6.18) Cx = 1 2S ρU 2  1 Cy = 1 (σ¯ f · n) · n y d (6.19) 2S ρU 2  1 Cz = 1 (σ¯ f · n) · n z d (6.20) 2S ρU 2

where n x , n y and n z are the Cartesian components of the unit normal n to the sphere surface, and S is the relevant surface area which is defined as S = π D 2 /4. The transfer of energy between the flow and oscillating sphere can be characterized by means of the normalized transverse force in phase with the sphere velocity. Thus the non-dimensional time-averaged quantity of fluid-structure energy transfer E over a period T of motion can be expressed as t+T C y (t)

E=

u y

U

dt

(6.21)

t

where u y is the transverse component of the sphere velocity. To identify the frequencies and the direction of the energy transfer, we make use of the time-dependent energy coefficient (C E ), which can be given as C E (t) = C y (t)u¯ y ,

u¯ y =

u y

U

(6.22)

The sign of C E demonstrates the relative direction between the transverse force and velocity, hence the direction of energy transfer between the rigid body and the fluid flow. While the positive value (C E > 0) represents the supply of energy from the flow to the structure, the flow damps the body oscillations for the negative value (C E < 0).

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367

6.4.1 VIV of Fully Submerged Freely Vibrating Elastically Mounted Sphere At very low Reynolds numbers (Re ≤ 200), the flow past a stationary sphere is steady and axisymmetric but it loses the axisymmetry first and then the steadiness with increasing of Reynolds number [193, 290]. While a pair of streamwise vortices are formed behind the sphere without shedding (210 ≤ Re ≤ 270), hairpinshaped vortices are periodically shed with the same strength in a fixed orientation for the unsteady planar-symmetric flow (280 ≤ Re < 375). As the Reynolds number is increased further, the strength and shedding orientation of the hairpin vortices vary in time and thus the flow becomes asymmetric (375 ≤ Re < 800) [421]. In the case of the subcritical flow over a stationary sphere (non-lock-in condition), there exists the high-frequency mode associated with the small-scale instability of the separating shear layer and the low-frequency mode related to the large-scale instability of the wake due to the vortex shedding [409]. For the present VIV study, a representative case of a fully submerged sphere that is free to translate in all spatial directions is considered for the grid convergence study. Figure 6.2 depicts a three-dimensional computational domain of the size (50 × 20 × 20)D set up with a sphere of diameter D placed at an offset of 10D from the inflow surface. The origin of the coordinate system is fixed at the center of the sphere. We consider the x-axis as streamwise flow direction, the y-axis in horizontal and perpendicular to the flow direction, and the z-axis is the vertical direction. While the streamwise motion corresponds to the freestream (x-direction), the transverse motion is parallel to the y-direction. A uniform freestream flow with velocity U is along the x-axis. At the inlet boundary, a stream of water enters into the domain at a horizontal velocity (u, v, w) = (U, 0, 0) where u, v and w denote the streamwise, transverse and vertical velocities in x, y and z directions, respectively. The sphere is elastically mounted on springs with a stiffness value of k and linear dampers with a damping value of c in all three spatial directions. We have considered the slip-wall boundary condition along the top, bottom and side surfaces, in addition to the Dirichlet and traction free Neumann boundary conditions along the inflow and outflow boundaries, respectively. Figure 6.3 shows a representative computational mesh for the simulations which contains structured prismatic six-node wedge elements at the boundary layer region and unstructured four-node tetrahedral elements elsewhere. For the spatial convergence, to maintain the accuracy of the boundary layer dynamics, the refinement is kept such that the non-dimensional wall unit y + remains less than 1. In this study, we have performed a grid convergence study for a freely vibrating sphere in all spatial directions. The maximum amplitude response (A∗ ) of the 3-DOF fully submerged vibrating sphere at the VIV regime is observed at U ∗ ≈ 9 for m ∗ = 3.82 [353]. The grid convergence study is carried out at U ∗ = 9 and Re = 2000 for four different grid size domains. The spatial convergence with the mesh details is given in Table 6.2. By considering M4 as the reference, the differences between the amplitude and force resulting from M1, M2, and M3 and those from M4 are thus calculated and noted in

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Fig. 6.2 Schematic and associated boundary conditions of the fluid flow past a fully submerged elastically mounted sphere with 3-DOF

Fig. 6.3 A representative computational mesh employed for an elastically mounted sphere in a uniform steady flow: a two-dimensional slice of the mesh for the entire domain along the X –Y plane, and b zoomed view of mesh in the vicinity of the sphere

the corresponding brackets. The differences between the results are approximately within 1% for the two finer meshes, M3 and M4. To consider the blockage effect from the domain boundaries, an enlarged domain size,  ∈ (60 × 40 × 40)D, is established with the upstream and downstream boundaries corresponding to 20D and 40D, respectively. The distance between side walls is 40D, which corresponds to the blockage of 2.5%. It can be seen that values recorded for both size fields differ by less than 1%. Considering computational efficiency, M3 mesh is selected for validation with the experimental results and all the simulations in this study. For the sake of completeness, the VIV of the 3-DOF fully submerged sphere (without free-surface effects) for a higher range of Reynolds number up to 30,000 is presented in the appendix. In particular, we briefly assess the numerical results and examine the effect of the Reynolds number on the mode transition.

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369

Table 6.2 Mesh statistics and convergence study of the flow parameters for VIV of a 3-DOF elastically mounted sphere with the mass ratio m ∗ = 3.82 at Re = 2000 and U ∗ = 9 Mesh Nodes (×106 ) A∗y,rms C y,rms Cx M1 M2 M3 M4 Domain size ∈ (50 × 20 × 20)D ∈ (60 × 40 × 40)D

0.637 1.330 2.963 5.947 Nodes (×106 ) 2.963

0.758 (13.6%) 0.681 (2.1%) 0.671 (0.7%) 0.667 A∗y,rms 0.671 (0.44%)

0.029 (25.6%) 0.0383 (1.7%) 0.0392 (0.6%) 0.0390 C y,rms 0.0392 (0.25%)

0.947 (9.4%) 0.885 (2.3%) 0.876 (1.2%) 0.865 Cx 0.876 (1.15%)

4.054

0.674

0.0393

0.866

The error deviation is evaluated based on the corresponding value of M4 mesh for the grid convergence study, and the enlarged domain size  ∈ (60 × 40 × 40)D for the spacial convergence study. The r.m.s. value of the sphere dimensionless amplitude response in y-direction, A∗y,rms , the r.m.s. value of the normalized transverse force in y direction, C y,rms and normalized mean horizontal force, C x are also recorded

6.4.2 VIV of Submerged Elastically Mounted Sphere Close to the Free Surface To validate the accuracy of our two-phase FSI solver, we next consider a fully submerged elastically mounted sphere restricted to move in the transverse y-direction. Our simulation results are compared against the measurement data of [377] at the immersion ratio of h ∗ = 1. Consistent with the experimental set-up, a low mass ratio of m ∗ = 7.8 and a damping ratio of ζ = 0.002 are considered. The 3D VIV simulations are performed for the Reynolds number Re in the range 5000 ≤ Re ≤ 30,000, which corresponds to the reduced velocity U ∗ range of 3 ≤ U ∗ ≤ 20. The goal of this validation study is to establish the predictive capability of our solver in the two regimes (mode I and mode II) of sphere VIV. The maximum amplitude is extracted from 10 oscillation cycles when the system reaches a steady state. Figure 6.4 shows the time histories of the amplitude response for several selected U ∗ values at mode I (U ∗ = 6), transition mode (∼ U ∗ = 8.7) and mode II (U ∗ = 13, U ∗ = 20) response at h ∗ = 1. A comparison of the r.m.s. values of the amplitude response, the normalized transverse (y-direction) force and the total phase difference (the phase difference between the sphere vibration frequency and the transverse force frequency) with that of the experiment data in [377] and the numerical data in [352] is shown in Fig. 6.5. Our results show a good agreement with the experimental study in [377] and follow a similar trend. Through our simulation, it can be inferred that the noticeable difference between the numerical simulations in [352] at fixed Reynolds number (Re = 300 and Re = 800) with that of the experiment is due to the significant effect of Reynolds number on the VIV response.

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(a) U ∗ = 6

(b) U ∗ = 8.7

(c) U ∗ = 13

(d) U ∗ = 20

Fig. 6.4 Time histories of the amplitude response (A∗ ) and the normalized transverse force (C y ) with non-dimensional time for 1-DOF sphere at h ∗ = 1 at different reduced velocities

We next briefly study the streamwise vortex dynamics which plays a crucial role to sustain the vibration amplitudes through the work done by the transverse force. A sketch to illustrate the formation of the streamwise vortex pairs and the visualization planes is shown in Fig. 6.6. Figure 6.7 shows the streamwise x-vorticity in a plane normal to the flow at 1.5D downstream of the sphere center, which enables us to measure the dominant counter-rotating streamwise pairs for both modes I and mode II. The distinct differences in the timing of vortex pair formation for modes I and II in Fig. 6.7, are consistent with the differences in the total phase φtotal between the modes, which is quantified in Fig. 6.5c. Our numerical results are qualitatively comparable with the experimental observation in [139] for the transversely oscillating sphere at lower Reynolds number Re = 3000. Detailed investigation of vortex wake modes and VIV characteristics is beyond the focus of the present study. The present validation study deems sufficient to serve as the reference to examine the VIV characteristics with the free-surface effect.

6.5 Results and Discussion The complexity of the coupled physical phenomena involved in a freely vibrating sphere close to the free surface is enhanced by the wake dynamics and sphere/freesurface interactions. The unsteady wake of the sphere interacting with the free-surface

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371

(a)

(b)

(c)

Fig. 6.5 Variation of VIV response parameters as a function of the reduced velocity U ∗ at h ∗ = 1, m ∗ = 7.8 and ζ = 0.002: a r.m.s. amplitude response (A∗rms ), b r.m.s. normalized transverse fluid force (C y,rms ) and, c total phase difference (φtotal ). The results are compared with the experimental data of [377] Fig. 6.6 Sketch illustrating a sphere in steady flow with streamwise vortex pairs and the transverse force on the sphere. Two representative Y –Z planes in the sphere wake at 0.5D and 1.5D from the centre are shown for the plotting of streamwise x-vorticity contours. Dashed line shows the formation of hairpin vortex loop

D/2

Z

U D

Transverse force

Y X

D

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Fig. 6.7 Streamwise x-vorticity (ωx D/U ∈ [−3.3, 3.3]) contours showing the dominant counterrotating vortex pair for mode I and mode II at two representative time locations: a mode I at U ∗ = 6, Re = 9400, and b mode II at U ∗ = 12, Re = 18,800. Blue and red contours show clockwise and anti-clockwise vorticity, respectively. x-vorticity contours are plotted on Y –Z plane at 1.5D downstream from the center of the sphere

makes the coupled response of the piercing case configuration fundamentally different from the fully submerged sphere counterpart. To understand the coupled dynamics of free-surface VIV, we investigate the effects of immersion ratio h ∗ = h/D for the piercing case at h ∗ = −0.25 and contrast the VIV behavior with the submergedsphere counterpart at h ∗ = 0 and h ∗ = 1. We explore the vibration response and the wake dynamics through the range of immersion ratios. We then proceed to study the sensitivity of large-amplitude oscillation as functions of the mass ratio m ∗ and Froude number Fr at the lock-in range.

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Fig. 6.8 Problem setup of a piercing sphere at a free surface: a sketch showing a cross-section view for the 1-DOF sphere, restricted to move in the transverse direction (Y ), while piercing the free surface at h ∗ = −0.25, b unstructured finite element mesh in the cross-sectional plane with a close-up view of the boundary layer. The grid size is refined along the air-water interface to capture the free-surface deformation. The flow is in the normal direction (X )

6.5.1 VIV of Elastically Mounted Sphere Piercing the Free Surface A schematic of the setup is provided in Fig. 6.8. To be consistent with the literature, identical parameters are used for the current simulations with the experimental work carried out in [377] at subcritical Reynolds numbers. A sphere of diameter D = 0.080 m is placed initially at an offset of (−0.25D) from the free surface in a computational domain  ∈ [0, 50D] × [0, 20D] × [0, 20D]. The physical properties of the two phases are ρ1f = 1000, ρ2f = 1.225, μf1 = 1 × 10−3 and μf2 = 1.983 × 10−5 and the mass ratio considering the submerged volume of the sphere is m ∗ = 9.2 at h ∗ = −0.25. Figure 6.9 shows the mesh motion and the free-surface deformation for the case of the piercing sphere at h ∗ = −0.25. Noticeable standing wave structures are formed from the sides of the sphere as it pierces the free surface, similar to the observation in [377]. Figure 6.10 shows the variation of maximum peak amplitude, A∗max , with immersion ratio h ∗ , for the experiments performed in [377]. The maximum VIV response in this study for h ∗ = 0 and h ∗ = −0.25, is reported for the reduced velocity range U ∗ ∈ [7, 15]. Different regimes were identified, where specific features were dominant [377]. Table 6.3 compares the amplitude response, the normalized transverse force in y-direction and the total phase of the present simulations with experiments in [377] at the lock-in regime (U ∗ = 10) for the fully and partially submerged cases. It is quantified that the amplitude response for the piercing case is greater than all submerged cases in both experiments and our numerical results at the lock-in state. Through our results, similar to the observation in [377], we find that the peak amplitude response of the submerged sphere at h ∗ = 0 is decreased by almost 30% compared to the case at h ∗ = 1. It is found that as the sphere pierces the free surface at h ∗ = −0.25, the amplitude response increases substantially with the maximum

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Fig. 6.9 Illustration of mesh motion at the free surface for the hybrid ALE/phase-field formulation: a free-surface deformation of 1-DOF elastically mounted sphere piercing the free surface at h ∗ = −0.25 at the lock-in state, and b zoomed view from a side of the sphere Fig. 6.10 The variation of the maximum amplitude response (A∗max ) as a function of the immersion ratio (h ∗ )

peak-to-peak amplitude ∼ 2D. Based on the total phase difference in Table 6.3, the amplitude response for the submerged cases at U ∗ = 10 corresponds to mode II. For the piercing case, the maximum amplitude response at U ∗ = 10 corresponds to mode I of vibration. It can be deduced that the lock-in region is shifted toward higher reduced velocities for the piercing sphere case. Figure 6.11a shows the time histories of the amplitude response and the normalized transverse force for the sphere at h ∗ = −0.25, 0, 1 at U ∗ = 10. For the fully submerged cases, the amplitude response decreases by changing the immersion ratio from h ∗ = 1 to h ∗ = 0 at the lock-in state. The amplitude response is increased significantly as the sphere pierces the free surface at h ∗ = −0.25. The maximum peak-to-peak amplitude response of ∼ 2D is observed at the stationary state which is larger than all the submerged cases studied (Fig. 6.5). Further analysis of the frequency spectrum is shown in Fig. 6.11. The only dominant vortex shedding frequency for the fully submerged cases (h ∗ = 1 and h ∗ = 0) is at f ∗ = 1 corresponding to the VIV response. However, two dominant frequencies of the vortex shedding at f 1∗ = 0.99 and f 2∗ = 2.98 are found for the piercing sphere case. The existence of a third-harmonic behavior for the piercing sphere case is related to the vorticity/free-surface interaction. Figure 6.12 shows the time traces

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375

Table 6.3 Comparison of the r.m.s. amplitude response, A∗rms , the r.m.s. of the normalized transverse force, C y,rms , and the total phase difference, φtotal for the piercing sphere case with m ∗ = 9.2, and fully submerged cases with m ∗ = 7.8 at U ∗ = 10, m ∗ ζ = 0.017 and Fr = 0.22 with the experimental data in [377] Parameters Case Sareen et al. [377] Present study ∈ ∈ (50 × 20 × 20)D (60 × 40 × 40)D

A∗rms C y,rms φtotal

h∗ h∗ h∗ h∗ h∗ h∗ h∗ h∗ h∗

=1 ∼0 = −0.25 =1 ∼0 = −0.25 =1 ∼0 = −0.25

0.81 0.65 0.88 0.08 – – ∼ 149 ∼ 175 ∼4

0.87 0.68 1.016 (1.9%) 0.12 0.14 0.104 (1.8%) ∼ 179 ∼ 178 ∼ 1 (0.0%)

– – 0.997 – – 0.106 – – ∼1

The error deviation is evaluated based on the corresponding value of the enlarged domain size  ∈ (60 × 40 × 40)D for the spatial convergence study

and the corresponding frequency spectrum of the amplitude response and the normalized transverse force at h ∗ = −0.25 for two different reduced velocities at mode I (U ∗ = 10) and mode II (U ∗ = 12.7). The third harmonic behavior is observed for both modes of vibration at the VIV regime. As a baseline for our study, where the large-amplitude oscillation is found around h ∗ ∼ −0.25, we aim to focus on the VIV response and the wake dynamics for the piercing sphere case and compare with the submerged cases in the next subsection.

6.5.2 Vorticity Dynamics with Free-Surface Deformation In this section, the effects of the free surface on the vibration response and the wake dynamics of a fully and partially submerged sphere are investigated. On a free surface, the primary driving mechanism of vorticity creation is the balance between the shear stress (measured by tangent vorticity) and the tangent components of the surface-deformation stress. For an incompressible viscous Newtonian fluid, an analytical relationship between the tangential stress and the surface vorticity at a free surface was derived in [461]. To analyze the parallel surface vorticity on a curved free interface S, a vector and its derivatives, including gradient operator ∇ can be , decomposed into components tangent π and normal n to S, e.g. ∇(·) = ∇π (·) + n ∂(·) ∂n As derived in [461], the tangent vorticity ωπ right on the free surface is solely balanced by the tangent components of the surface stress as follows: where K ≡ −∇π n is the surface curvature tensor. This analytical expression clearly underlines the relationship between the surface parallel vorticity and the curvature of a deformed free surface.

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Fig. 6.11 Time histories of the amplitude response (A∗ ) and the normalized transverse force (C y ) with non-dimensional time and their corresponding frequency spectrum with normalized frequency ( f ∗ ), at U ∗ = 10 for 1-DOF sphere at a h ∗ = 1, b h ∗ = 0 and c h ∗ = −0.25. The mass ratio for the fully submerged cases (h ∗ = 1 and h ∗ = 0) is m ∗ = 7.8 and for partially submerged case (h ∗ = −0.25) is m ∗ = 9.2

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Fig. 6.12 Time histories of the amplitude response (A∗ ) and the normalized transverse force (C y ) and their corresponding frequency spectra for 1-DOF sphere piercing the free surface at h ∗ = −0.25 at two reduced velocities: a U ∗ = 10 and, b U ∗ = 12.7

The vorticity of different signs is created in the flow field whenever there is a curvature in the free surface. The free surface deforms as it interacts with the vorticity field and vice versa. The interaction of the initial vorticity field along the free surface may lead to the generation of additional vorticity by the deformation of the free surface. In the present study, the free surface deforms as it interacts with the spherical body and there is a complex nonlinear interaction between the sphere wake and the vorticity flux at the free surface. We attempt to explore the complex vorticity interactions with the free surface in the context of piercing sphere VIV response. Through some quantitative and qualitative comparison with the experiment in [377], we study the VIV wake dynamics of the sphere at h ∗ = 1 (fully submerged case), h ∗ = 0 (when the top of the sphere touches the free surface) and h ∗ = −0.25 (when the sphere is piercing the free surface). The temporal evolution of streamwise vorticity in a plane normal to the flow can provide important insight into wake

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Fig. 6.13 Streamwise x-vorticity (ωx D/U ∈ [−3.3, 3.3]) contour on Y –Z plane at 1.5D downstream from the centre of the sphere for: a the fully submerged sphere at h ∗ = 1, b h ∗ = 0 and c the piercing sphere case at h ∗ = −0.25 at mean amplitude position

dynamics for the sphere as the streamwise vortex loops pass through the cross-plane. Hence, in the current study, we have measured the streamwise vorticity in a crossplane and will compare with the experiments in the literature [139, 377]. Figure 6.13 shows the x-vorticity contour plots for the submerged sphere at h ∗ = 1, h ∗ = 0 and the piercing sphere at h ∗ = −0.25 at 1.5D downstream, while the sphere is at the end of its stroke. The plots correspond to the reduced velocity of U ∗ = 10, where the peak amplitude is obtained at the VIV regime. The top boundary in the plots represents the free-surface boundary. The streamwise vorticity for the submerged case at h ∗ = 1, Fig. 6.13a, consists of two dominant opposite sign vortex pair that is symmetric across the horizontal plane. These vortex loops formation is consistent with the observation in [139, 377]. Figure 6.13b, c shows the change in the formation of the vortex pairs when the sphere moves closer to the free surface at h ∗ = 0 and when it pierces the free surface at h ∗ = −0.25. As it is evident from the plots, the vortex structures change significantly due to the effect of the free surface and the horizontal plane through the sphere center can also no longer act as a plane of symmetry. Stretched vorticity formation (compared to Fig. 6.13a) is observed due to the effect of the free surface. Figure 6.14 shows the streamwise x-vorticity contour plots for one complete oscillation cycle for the piercing case at h ∗ = −0.25 at U ∗ = 10. The stretched vortex patterns consist of both clockwise and anti-clockwise vorticity loops. When the sphere moves from one side to the other side, the vorticity changes sign accordingly (Fig. 6.14d, h show a clear contrast). In Fig. 6.14d, where the sphere is at its mean position during half-stroke, the blue vortex is trapped with the red vortex loop. On the other hand, when the sphere moves across the opposite side at its mean position, Fig. 6.14h, the vorticity changes sign with the red vortex now trapped with the blue vortex loop. This confirms the existence of the hairpin loops that form from the opposite sides of the sphere and are shed into the downstream wake. The stretched vortex formation structures for the piercing sphere case were also observed in the experiments in [377]. However, during each stroke, only one sign vortex loop was

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Fig. 6.14 Streamwise x-vorticity contour (ωx D/U ∈ [−3.3, 3.3]) for the piercing sphere at h ∗ = −0.25 and U ∗ = 10 for one complete oscillation period. Vorticity plot is taken at 1.5D downstream on the Y –Z plane

captured by the PIV in the cross-plane and therefore, no specific wake mode was identified for the piercing sphere case. To further analyze the three-dimensional vortical structures, we employ a vortex-identification based on the Q-criterion. Figure 6.15 shows the Q-criterion based vortical structures for the sphere at h ∗ = 1, h ∗ = 0 and 2 h ∗ = −0.25. The iso-surfaces of quantity ( Q¯ = Q UD2 ) are shown at a constant positive value and the contour surfaces are colored by the non-dimensional streamwise velocity. The existence of hairpin wake loops for all cases, even when the sphere is piercing the free surface, is observed in our numerical analysis. We find that the vortex loop patterns that were observed for the submerged case, Fig. 6.15a, are slightly stretched to elliptical loops for the cases at h ∗ = 0, Fig. 6.15b, and the piercing case at h ∗ = −0.25, Fig. 6.15c. Figure 6.16 compares the free-surface deformation for the submerged case at h ∗ = 0 and the piercing case at h ∗ = −0.25 at the mean position. It can be seen that the surface deformation for the piercing case is significantly larger compared to the submerged case, where a part of the sphere lies above the water line and causes a noticeable deformation. To understand the vortex dynamics and the force exerted on the sphere, we plot the vorticity formation and the pressure distribution

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6 VIV of Sphere

Fig. 6.15 Underwater view of the iso-surface wake structures formed behind the 1-DOF sphere at U ∗ = 10 and Re = 15,700 at stationary state: a fully submerged with the immersion ratio of h ∗ = 1, b the top of the sphere touches the free surface at h ∗ = 0 and c piercing the free surface with the immersion ratio of h ∗ = −0.25. Iso-surfaces are plotted by the Q-criterion ( Q¯ = 0.33)

on the cross-flow plane located at 0.5D downstream from the center of the sphere for the submerged and the piercing cases. The plots provide an overview of different types of wake behaviors due to the existence of the free surface. Figure 6.17 shows the change in the vorticity field and the pressure distribution at different immersion ratios for the sphere at the mean position. As can be seen in Fig. 6.17a1, b1, the vorticity for the piercing sphere at h ∗ = −0.25, and the vorticity for the submerged sphere when the top of the sphere touches the free surface at h ∗ = 0, is significantly different to that of flow past sphere with no free-surface effect at h ∗ = 1, Fig. 6.17c1. At h ∗ = 1, the x-vorticity pattern has feasible symmetry about the horizontal centerline when there is no effect of the free surface. When the sphere comes closer to the free surface at h ∗ = 0, the free-surface affects the vortex dynamics substantially. The induced surface distortion due to the sphere motion in the proximity of the freesurface boundary, causes large opposite sign stretched vortex loops at the top region, shown in Fig. 6.17b1. The vorticity generated due to the free surface makes the wake asymmetric about the horizontal center-line. In Fig. 6.17a1, the vorticity pattern for the piercing case at h ∗ = −0.25 is remarkably different from the former case at h ∗ = 0, where the surface distortion is considerably larger as 25% of the sphere lies above the waterline. In Fig. 6.17a2, the corresponding pressure distribution for the piercing case shows the high-pressure region on the left side of the sphere due to the induced surface curvature. To further analyze this behavior, we plot the evolution of the vorticity and the pressure distribution for one complete oscillation period for the sphere close to the free surface at h ∗ = 0 and the piercing case at h ∗ = −0.25 in Figs. 6.18 and 6.19, respectively. In Fig. 6.18, for the evolution of the vorticity at h ∗ = 0, throughout the whole oscillation cycle, a flux of vorticity due to the induced surface distortion appears on the top region. This secondary vorticity flux causes diffusion of the vorticity that is induced due to VIV. The free-surface diffusive vorticity flux acts as a sink of energy and leads to the reduction in the corresponding transverse force acting on the spherical body and hence lower amplitude response. In Fig. 6.19, for the evolution of the vorticity at h ∗ = −0.25, the surface distortion is considerably larger and consequently strong flux of vorticity is supplied to the wake due to the induced surface distortion. This strong vorticity at the top, induces opposite sign vortex loops immediately below and cross-annihilated small-scale vortex structures. This phenomenon

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Fig. 6.16 The contour plot of the phase-field order parameter φ to quantify the free-surface deformation at 41 D downstream on the Y –Z plane: a submerged sphere at h ∗ = 0, and b piercing sphere case at h ∗ = −0.25

Fig. 6.17 Streamwise x-vorticity (ωx D/U ∈ [−3.3, 3.3]) and pressure distribution contours ( p/ρU 2 ∈ [−0.78, 0.25]) plotted at 0.5D downstream on the Y –Z plane at U ∗ = 10, for the sphere at a h ∗ = −0.25, b h ∗ = 0, and c h ∗ = 1

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Fig. 6.18 Evolution of streamwise x-vorticity and pressure distribution plotted at 0.5D downstream at U ∗ = 10 for the submerged sphere at h ∗ = 0. Top of the sphere touches the free surface and one complete oscillation period is considered

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Fig. 6.19 Evolution of streamwise x-vorticity and pressure distribution plotted at 0.5D downstream at U ∗ = 10 for the piercing sphere at h ∗ = −0.25 for one complete oscillation period

alters the vorticity pattern and the synchronization of the vortex shedding. This extra vorticity generation acts as a source of energy supplied to the wake of the piercing sphere and makes the associated vorticity stronger. This leads to a larger transverse force acting on the body and therefore large amplitude oscillations. Figure 6.20 compares the variation of the instantaneous energy transfer C E for the piercing case at h ∗ = −0.25 and the submerged cases at h ∗ = 0 and h ∗ = 1. It is quantified that the non-dimensional time-averaged quantity of the energy transfer (E) over each period of motion T for the piercing sphere case at h ∗ = −0.25 is significantly more than the

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Fig. 6.20 Temporal variation of instantaneous energy transfer C E for the piercing sphere case at h ∗ = −0.25 and the submerged sphere cases at h ∗ = 0 and h ∗ = 1 at lockin state with U ∗ = 10, Re = 15,700, and m ∗ ζ = 0.017

submerged cases at h ∗ = 0 and h ∗ = 1. This energy transfer sustains the large amplitude oscillations for the piercing sphere case more than all the submerged cases. It can be deduced that the extra vorticity generation at the free surface for the piercing sphere has a significant impact on the synchronization of the vortex shedding and the energy transfer.

6.5.3 Effect of Mass Ratio It is known that the non-dimensional parameter mass ratio m ∗ has a strong influence on the flow-induced vibration. When the sphere pierces the free surface, this ratio varies significantly due to the rapid decrease in the mass of the displaced fluid (m d ). However, identifying the effect of the mass ratio on the FIV response for the piercing cases while the immersion ratio is changing cannot be clearly explained because the geometry of the submerged portion of the sphere changes. Here in this subsection, we aim to understand the effect of mass ratio on the FIV characteristics of the sphere piercing the free surface at the fixed immersion ratio h ∗ = −0.25 and zero damping ratio. Figure 6.21a, b shows the variation of the amplitude response A∗ and the normalized transverse force C y , with a range of mass ratio m ∗ ∈ [1, 20] at identical Reynolds number Re = 15,700 and the reduced velocity U ∗ = 10. The results show that a small variation of the mass ratio does not have a significant effect on the amplitude response. The variation of the amplitude response with the mass ratio is found to be less than 3%, consistent with previous experimental and numerical investigations for VIV of low-mass-damped fully submerged spheres [139, 352]. The experimental study in [377], considered two different mass ratios at h ∗ = −0.25 where the considerable effect of the mass ratio was reported. In their work, the r.m.s. amplitude response has a noticeable reduction by increasing the mass ratio. This difference may be due to different parameter setup for the mass-damping parameter in the experiments and zero-damping in our numerical simulations in this subsection. Periodic

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385

(a)

(b)

(c)

(d)

Fig. 6.21 Variation of VIV response parameters as a function of the mass ratio for the sphere piercing the free surface at h ∗ = −0.25, U ∗ = 10 and Re = 15,700: a r.m.s. amplitude response, b r.m.s. normalized transverse force, c total phase difference and, (d) normalized cross-flow frequency. The normalized cross-flow frequency is defined as f y∗ = f y / f n , where f y is the frequency of the oscillations and f n is the natural frequency of the system

and large amplitude vibrations are observed over a range of mass ratio m ∗ ∈ [1, 20] in our numerical results. In Fig. 6.21c, it is quantified that the vibrations for all the mass ratio ranges correspond to mode I. The oscillation frequency of the piercing sphere for the entire range of the mass ratio at U ∗ = 10 is close to the natural frequency of the system, consistent with the observation for the submerged sphere in [139]. Figure 6.22 shows the time traces of the amplitude response and the normalized transverse force with their corresponding power spectrum for the piercing case at h ∗ = −0.25 with m ∗ = 1 and m ∗ = 20. The existence of the third-harmonic behavior of the transverse force is observed for all the mass ratio cases. It can be deduced that the FIV response at h ∗ = −0.25 is relatively insensitive to the mass ratio in the range m ∗ ∈ [1, 20], where the free-surface effect sustains large amplitude oscillations.

6.5.4 Effect of Froude Number To investigate the effect of the free-surface deformation on the transverse VIV response, we next explore the influence of the Froude number Fr . All the previous simulations, for validation purposes (Fig. 6.5), are carried out with the range

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(a1)

(a2)

(b1)

(b2)

Fig. 6.22 Time histories of the amplitude response (A∗ ) and the normalized transverse force (C y ) and their corresponding frequency spectra for the piercing sphere at h ∗ = −0.25 at two representative mass ratios: a m ∗ = 1 and b m ∗ = 20

of Froude numbers Fr ∈ [0.05, 0.45] with the experiments performed in [377]. In the experimental study in [377], a relatively insensitive effect of the Froude number on the VIV response of the sphere is reported in the range of Fr ∈ [0.05, 0.45] and U ∗ ∈ [3, 20]. We further investigate the effect of the Froude number on the FIV response at the lock-in state for the piercing sphere case at h ∗ = −0.25, U ∗ = 10 and Re = 15,700, with the mass ratio m ∗ = 9.2. The Froude number is investigated over a range of Fr ∈ [0.22, 4] by changing the acceleration due to gravity. Figure 6.23 shows the variation of the amplitude response A∗ , the r.m.s. transverse force C y , the r.m.s. and mean streamwise force as a function of Froude number. The results indicate a significant effect of the Froude number on the VIV response. To further analyze the effect of the Froude number, Fig. 6.24 shows the time histories of the amplitude response for the piercing sphere case at Fr = 0.22 and Fr = 0.44, where no significant surface deformation is expected [377]. The r.m.s. amplitude response decreases

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Fig. 6.23 Variations of r.m.s. amplitude response, r.m.s. normalized force in y- and x-directions and normalized mean streamwise force as a function of Froude number for the piercing the sphere at h ∗ = −0.25, U ∗ = 10 and Re = 15,700

by about ∼ 30% by doubling the Froude number from Fr = 0.22 to Fr = 0.44 at identical reduced velocity U ∗ = 10 and Reynolds number Re = 15,700. Figure 6.24b2 shows the frequency spectrum at Fr = 0.44 compared to Fig. 6.24a2 at Fr = 0.22. The third harmonic frequency at Fr = 0.44 is found to be the only dominant force frequency on the sphere. This higher harmonic behavior is expected due to the free-surface effects. Despite the case at Fr = 0.22, the first harmonic frequency at Fr = 0.44 has almost disappeared. To have a better understanding, Fig. 6.25 shows the surface deformation, the normalized vorticity and the pressure distribution plots at Fr = 0.22 and Fr = 0.44. From Fig. 6.25a1, b1, it can be seen that the surface deformation at higher Froude number Fr = 0.44 is considerably larger than the case at Fr = 0.22. Through the vorticity plots in Fig. 6.25, it is found that the vorticity supplied to the wake by the free surface at Fr = 0.22 is much stronger than Fr = 0.44. In Fig. 6.25a2 for the case at Fr = 0.22, the strong negative sign vorticity (blue vortex loop) at the top left corner, generated due to the free-surface distortion, has completely disappeared for the case at Fr = 0.44 as shown in Fig. 6.25b2. Since the free-surface boundary is allowed to deform due to the stress-free condition, the vorticity at the top region for the higher Froude number case causes a large surface deformation and dissipates the energy. It can be deduced that at a higher Froude number Fr = 0.44, the strength of supplied vorticity due to the free surface is reduced significantly. For Fr = 0.22, Fig. 6.25a3 shows the existence of the high-pressure region on the top left corner due to the induced surface curvature and the extra supplied vorticity. In Fig. 6.25b3 at Fr = 0.44, it is found that the pressure level on top left region is decreased substantially compared to the case at Fr = 0.22.

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Fig. 6.24 Time histories of the normalized amplitude response ( A∗ ) and transverse forces (C y ) and their corresponding frequency spectra for the piercing sphere (h ∗ = −0.25) at two representatives Froude numbers: a Fr = 0.22, and b Fr = 0.44. Streamwise forces (C x ) are also included in a1 and b1

Figure 6.26 shows the three-dimensional wake structures along with the surface deformation at four different Froude numbers. By comparing the cases at Fr = 0.22 and Fr = 0.44, where the surface deformation is not substantial, it can be seen that at lower Froude number Fr = 0.22, the hairpin type structures at the near wake region are generated, although the sphere pierces the free surface. The upper vortex loops at downstream flow are detached from the lower loops and lose their strength through diffusion into the free surface. At higher Froude number Fr = 0.44, the upper vortex loops of the hairpin vortex structures get diffused into the free surface immediately in the near wake region and cause larger surface deformation. The upper vortex loops completely disappear downstream, which results in the reduction of the circulation and corresponding transverse force on the sphere.

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389

Fig. 6.25 Flow visualizations for the piercing sphere case at h ∗ = −0.25 and U ∗ = 10 for two representative Froude numbers, a Fr = 0.22 and, b Fr = 0.44: Free-surface deformation quantified with the order parameter (φ) at 14 D downstream (top), normalized streamwise x-vorticity (middle) and pressure distribution (bottom) contours plotted at 0.5D downstream

At higher Froude number cases Fr ∈ [0.88, 4], the surface deformation becomes substantial. As can be seen in Fig. 6.26, for the case at Fr = 0.88, the large surface deformation covers the front side of the sphere and the back side of the sphere is exposed to air. Therefore, the extreme surface distortion breaks the synchronization of vortex shedding and prevents the formation of the hairpin vortex loops completely. This causes a large reduction in the hydrodynamic transverse force on the sphere and a significant decrease in amplitude response, as shown in Fig. 6.27a. By further increasing the Froude number to Fr ≥ 1.76, the free surface covers the entire part of the sphere that is above the undisturbed free-surface level. The hairpin vortex structures shed behind the sphere while the centreline of the wake is directed downwards in the vertical z-direction. The transverse hydrodynamic force on the sphere recovers the strength and the amplitude response is increased. The FIV response for Fr ≥ 1.76 is comparable with the fully submerged cases where the only dominant shedding frequency matches with the oscillation frequency of the sphere, as can be seen in Fig. 6.27b. The VIV response for Fr > 2.4 is found to reach a saturated state with similar behaviour as plotted in Fig. 6.23. The energy transfer between the flow and the oscillating sphere for different Froude numbers is characterized by the time-dependent energy coefficient (C E ). Figure 6.28a compares the variation of the instantaneous energy transfer C E for the

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6 VIV of Sphere

(a) Fr = 0.22

(b) Fr = 0.44

(c) Fr = 0.88

(d) Fr = 1.76

Fig. 6.26 Iso-surface of wake structures formed behind the sphere and the free-surface deformation for the piercing sphere at h ∗ = −0.25 for different Froude numbers. Iso-surfaces of the 3D vortical structures are plotted by the Q-criterion ( Q¯ = 0.33), and the iso-surfaces of the free-surface deformation are plotted by the order parameter (φ = 0)

6.5 Results and Discussion

391

Fig. 6.27 Time histories of the normalized amplitude response ( A∗ ) and transverse forces (C y ) and their corresponding frequency spectra for the piercing sphere (h ∗ = −0.25) at two representatives Froude numbers: a Fr = 0.88, and b Fr = 1.76. Streamwise forces (C x ) are also included in a1 and b1

piercing case. The non-dimensional time-averaged quantity of energy transfer (E) is extracted for 10 oscillation cycles when the system reaches a stationary state. E denotes the mean quantity of the non-dimensional energy transfer over the 10 cycles, which is shown in Fig. 6.28b. It is found that by increasing the Froude number in the range Fr ∈ [0.22, 0.88], the amount of net energy transfer per oscillation cycle decreases. The reduction in the net energy transfer leads to a significant reduction of the amplitude response. By a further increase in the Froude number for Fr ≥ 1.4, the amount of net energy transfer per cycle is increased as the free-surface deformation becomes substantial, covering the entire sphere surface at h ∗ = −0.25. In summary, we find that the large-amplitude VIV response is strongly sensitive to the Froude number at the range of Fr ∈ [0.22, 4]. For the piercing sphere case, by increasing the Froude number in the lower range of Fr ∈ [0.22, 0.44], wherein the surface deformation is not substantial, the near-surface vorticity diffuses into the deformable

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Fig. 6.28 Dependence of energy transfer on Froude number for the piercing sphere at h ∗ = −0.25, U ∗ = 10, and Re = 15,700: a temporal variation of instantaneous energy transfer C E , and b mean energy transfer over 10 oscillation periods

(a)

(b)

free surface which in turn leads to significant energy dissipation and reduction in the amplitude response. At a higher Froude number range Fr ∈ [0.8, 4], the surface deformation becomes substantial and the free surface covers the entire part of the sphere surface above the undisturbed free-surface level for Fr ≥ 1.4, altering the wake dynamics and increasing the VIV amplitude. For Fr > 2.4, the amplitude response is observed to reach a saturated state.

6.6 Summary A numerical study has been performed to investigate the effect of the free surface on the FIV response of a transversely vibrating sphere in proximity to a free surface. We employed the recently developed three-dimensional fluid-structure-free-surface interaction solver to explore the FIV response of fully and partially submerged sphere configurations. To begin, we first examined the VIV phenomenon and the wake modes of a fully submerged freely vibrating sphere in a wide range of Reynolds number Re ∈ [300, 30,000] at the lock-in state. We found that the sphere begins to move along a linear trajectory with hairpin vortex-shedding mode, eventually transforming into a circular trajectory with the spiral mode in its stationary state for Re ∈ [2000, 6000].

6.6 Summary

393

By examining the mode transitions and the motion trajectories we found that the mode transition is strongly sensitive to the Reynolds number. We observed that the motion trajectories at the higher Reynolds number range Re ∈ [12,000, 30,000] show a chaotic response with a combination of linear motions and circular-type motions at the periodic state, where the vortex structure modes transform frequently from the hairpin mode to the spiral mode and vice versa. The FIV response of a sphere in the proximity of the free surface was investigated at three stages of immersion ratios at h ∗ = 1, h ∗ = 0 and h ∗ = −0.25 at the lockin regime. Successful validation of the sphere by considering the effect of the free surface has been established through quantitative and qualitative comparisons with the experiments. The important findings of the investigation can be summarized as follows: • For the fully submerged cases, the amplitude response of the sphere vibration when it touches the free surface at h ∗ = 0 is decreased by ∼ 20% compared to the case at h ∗ = 1. The vorticity plot in the cross-plane 0.5D downstream at h ∗ = 0 revealed a diffusion of the vorticity flux due to induced free-surface distortion on the top of the sphere. The free surface changes the vorticity structure significantly and causes the vorticity pattern to become asymmetric along the horizontal plane. The free surface acts as a sink of energy which leads to the reduction in the transverse force and the amplitude response of the elastically mounted sphere. • The amplitude response for the piercing case at h ∗ = −0.25 is increased dramatically with the maximum peak-to-peak amplitude of ∼ 2D, larger than all the submerged cases studied. We observed that the free-surface distortion for the piercing case is considerably larger due to the complex interaction of the free surface with the piercing sphere geometry and the sphere wake. It was found that a strong flux of vorticity is supplied due to the piercing sphere/free-surface interaction. The existence of third-harmonic in the transverse force is related to the extra free-surface vorticity flux. Increased streamwise vorticity gives rise to a relatively larger transverse force to the piercing sphere at h ∗ = −0.25, resulting in a relatively greater positive energy transfer per cycle to sustain the large-amplitude vibrations. • The effect of the mass ratio on the amplitude response for the piercing sphere case at h ∗ = −0.25 was studied over a range of m ∗ ∈ [1, 20] at the lock-in state. The FIV response was found to be relatively insensitive to the mass ratio m ∗ in the range studied, although increasing the mass ratio led to a slight reduction in the peak amplitude. The existence of the third-harmonic behavior of the transverse force was observed for all the mass ratios. • Lastly, it is found that the FIV response is strongly sensitive to the Froude number for the piercing sphere case at h ∗ = −0.25 over a range of Fr ∈ [0.22, 4] at the lock-in state. For the Froude number range Fr ∈ [0.22, 0.44], where surface deformation was not substantial, we observed that the amplitude response is decreased by ∼ 30% at Fr = 0.44 compared to the case at Fr = 0.22. It was found that at Fr = 0.44, the strength of vorticity flux due to the free-surface distortion is considerably lower than the case at Fr = 0.22. At a higher Froude number range

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Fr ∈ [0.8, 4], the surface deformation becomes substantial. At Fr = 0.88, the large surface deformation covers the front side of the sphere and the backside of the sphere is exposed to air. The extreme surface distortion breaks the synchronization of vortex shedding and prevents the formation of the hairpin vortex loops. For Fr ≥ 1.4, the free surface covers the entire part of the sphere surface above the undisturbed free-surface level, altering the wake dynamics and the FIV response. Hairpin vortex structures form behind the sphere with the centreline of the wake slightly shifted downwards in the vertical z-direction. This results in an increase in the transverse hydrodynamic force and the amplitude response of the sphere. Further research is required to systematically decompose the vorticity contributions of the sphere/free-surface and the wake/free-surface interactions for a broader range of physical parameters.

6.7 Appendix 6.8 The Effect of Reynolds Number on the Mode Transition for the Freely Vibrating Elastically Mounted Sphere Here we present the VIV response for the fully submerged sphere that is allowed to move in all three spatial directions and examine the effect of Reynolds number in the range of Re ∈ [300, 30,000] on the mode transition. Figure 6.29 compares the results obtained for the sphere r.m.s. amplitude response as a function of Reynolds number in the transverse direction (A∗y ) and the vertical direction ( A∗z ) with the numerical results of [46] and [353] for the Reynolds number range of Re ∈ [300, 2000]. Our results show a similar trend with the results of [353], and the response amplitudes increase as a function of Reynolds number. As elucidated by [353], the simulations of [46] did not reach the asymptotic state and the response amplitudes are not comparable with our numerical prediction. Figure 6.30 shows time histories of the amplitude response in the transverse (A∗y ) and vertical ( A∗z ) directions at Re = 6000, 12,000

Fig. 6.29 Dependence of transverse (A∗y ) and lateral (A∗z ) amplitudes on Reynolds number for elastically mounted 3-DOF sphere at reduced velocity U ∗ = 9 and mass ratio m ∗ = 3.82. The response amplitudes are contrasted with the numerical results of [46] and [353]

6.8 The Effect of Reynolds Number on the Mode Transition ...

395

and Re = 15,000 at the periodic state. A noticeable difference in the amplitude response is observed for the case at Re = 6000 compared to the cases at higher Reynolds number at Re = 12,000 and 15,000. At Re = 6000 the variation of the peak amplitude oscillations is small (Fig. 6.30a1) and the frequency of the transverse oscillation ( f A y ) and vertical oscillation ( f Az ) is matched (Fig. 6.30a2), therefore, the phase difference between (A∗y ) and (A∗z ) does not change with time (φ ≈ π/2). This represents the circular type motion in the transverse plane similar to the cases in the range of Reynolds number Re ∈ [300, 2000]. While at higher Reynolds number (Re = 12,000 and Re = 15,000), the amplitude response in both transverse and vertical directions (A∗y and A∗z ) are found to be harmonic with multiple frequencies, similar to beating-type behaviour (Fig. 6.30b1, c1). The phase difference between the transverse and vertical motion is found to change with time due to the difference in the oscillation frequencies of f A y and f Az , Fig. 6.30b2, c2. Figure 6.31 shows the trajectory response of the sphere motion exposed to the unsteady flow field at the lock-in regime (U ∗ = 9) for eight cases in the Reynolds number range of Re ∈ [300, 30,000] at their periodic states. As it can be seen, the motion trajectories for the Reynolds number range of Re ∈ [2000, 6000] show circular-type motion. However, at higher Reynolds numbers Re ∈ [12,000, 30,000] the behavior of the motion trajectories consists of a combination of linear and circular-type motions. To further look into the vortex formation and the wake structure, we employ the Q-criterion [172] which is given as Q=

 1  2 − S 2 2

(6.23)

In Fig. 6.32, we observe that at each Reynolds number, hairpin vortex loops from the opposite sides of the sphere forms in the wake at the initial state. The sphere initially begins to vibrate in a linear path as shown in Fig. 6.32a. For the range of Reynolds number Re ∈ [2000, 6000], the hairpin mode is found to be unstable and the wake mode transforms to spiraling vortical structure behind the sphere at the final state. The sphere motion merges to circular trajectory orthogonal to the flow as shown in Fig. 6.32b. The wake mode transition for the higher Reynolds number range Re ∈ [12,000, 30,000] is found to be quite different, where both the hairpin mode and the spiral mode were identified as unstable states. For this higher range of Reynolds number, it is found that the vortical structures transform frequently from the hairpin mode to the spiral mode and vice versa.

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6 VIV of Sphere

Fig. 6.30 Time histories of the stationary-state amplitude response of the sphere versus nondimensional time at U ∗ = 9, and the corresponding frequency spectrum of the oscillations in transverse direction ( f A y ) and vertical direction ( f Az ) at Re = 6000, 12,000 and 15,000

6.8 The Effect of Reynolds Number on the Mode Transition ...

397

X -0.7791 Y 0.0341

Fig. 6.31 The sphere trajectories in the Y –Z plane at U ∗ = 9 and m ∗ = 3.82 for a range of Re ∈ [500, 30,000]

(a) Initial state

(b) Final state

Fig. 6.32 Iso-surface of three-dimensional wake structures formed behind the 3-DOF sphere at U ∗ = 9 and Re = 2000: a initial state with linear path (tU/D = 60), and b final stationary state with circular motion (tU/D = 240). Iso-surfaces are plotted by the non-dimensional Q-criterion 2 ( Q¯ = Q UD2 = 0.001), colored with the normalized streamwise velocity

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6 VIV of Sphere

Acknowledgements Some parts of this Chapter have been taken care from the MASc thesis of Amir Chizfahm carried out at the University of British Columbia and supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).

Chapter 7

Flexible Cylinder VIV

In this chapter, we present a systematic study on a slender flexible cylinder immersed in a turbulent flow is performed whereby the flexible cylinder is pinned at both ends. We first provide a brief overview of the governing equations and the numerical methodology for the turbulent fluid-structure interactions of flexible cylinders. Full three-dimensional simulations are performed on the flexible cylinder exposed at moderate Reynolds number with uniform and linearly sheared profiles. Detailed validation and physical investigation are performed on the response characteristics and vorticity dynamics at various locations along the span of the flexible cylinder. We particularly investigate the standing and traveling wave responses. Finally, we investigate the groove-based suppression device to minimize vortex-induced vibrations.

7.1 Introduction Fluid-structure interactions of flexible cylindrical structures are prevalent in nature and occur in many engineering systems. The understanding and control of flowinduced vibrations are important in aerospace engineering, civil engineering, ocean and offshore industry, wind engineering, power generation and transmission lines. This chapter is particularly motivated by the application marine risers and pipelines undergoing flow-induced vibrations. In recent years, oil explorations and drilling operations are moving towards greater depths with harsh environmental conditions. The increased drilling depth imposes big challenges and requirements to the equipment due to the extreme hydrostatic pressures and distance between the drilling unit and the seabed. Greater depths means larger and heavier drilling riser pipelines, more top tension and large quantities of drilling mud. Safety becomes more important due to the difficulties in solving the problem if something goes wrong in deep waters. These risers are subjected to vortex-induced vibrations (VIV) when exposed to ocean currents, especially in the deep/ultra-deep waters. The vibrations are a result of oscillating fluid forces that arise from flow separation and vortex shedding. For a wide © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Jaiman et al., Mechanics of Flow-Induced Vibration, https://doi.org/10.1007/978-981-19-8578-2_7

399

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7 Flexible Cylinder VIV

range of reduced velocity, the vortex-shedding is locked to or synchronized with the flexible cylinder vibration. The existence of a well-defined synchronization range at which the cylinder vibrates with the highest amplitude indicating VIV. Subject to flow characteristics and structural properties, VIV may inflict significant dynamic stresses on structures, resulting in drag force amplification and rapid accumulation of fatigue damage. If these nonlinear dynamical effects due to VIV are left unchecked, they may lead to structural failure. Consequently, it is imperative for the safety of offshore operations that VIV response is adequately predicted.

7.1.1 VIV of Flexible Cylinder Vortex-induced vibration is observed in many fields of engineering and is summarized in the comprehensive reviews of Sarpkaya [381], Williamson and Govardhan [455], and Gabbai and Benaroya [124]. However, most of these works are focused on the VIV of short and rigid cylinders with an aspect ratio (length to diameter) of less than 100. The VIV of large aspect ratio flexible cylinders is quite complex due to the nonlinear dynamics of wake interactions with numerous flexible modes. As compared to small aspect ratio flexible cylinders, the large aspect ratio structures respond to higher harmonics and a wider range of wake excitations. In addition, the flexible cylinder placed in a uniform current generally exhibits a narrow-band standing wave response, whereas the non-uniform sheared current can cause wide-band excitation and multi-frequency response with a complex phase-locked traveling wave pattern along the span of the cylinder. In standing wave response, the alternating regions of energy-gain and energy-loss are formed with well-behaved motion trajectories and amplitudes. On the other hand, the phase-locked traveling wave has highly distributed regions of energy transfer and damping associated with high amplitudes and motion trajectories. The energy transfer from fluid flow to the structure occurs at a natural structural frequency close to the local Strouhal frequency along a portion of the flexible cylinder and then the energy is moved to the dissipative region where the local Strouhal frequency is different than the structural frequency. With regard to predicting VIV response, available methods include fully-coupled computational fluid dynamics, wake-oscillator models [320] and semi-empirical models [418, 419]. Owing to the ease of use and computational efficiency, the most widely used riser VIV prediction tools in the industry are semi-empirical models, which combine theoretical equations with experimental data to simplify the VIV problem. The weakness of these methods is that their formulation restricts prediction to stationary flows; they are unable to account for non-linear structural behavior, interaction between different response frequencies, unsteady flows such as waves, and the relationship between cross-flow and in-line frequencies and motion. While there are several simplified semi-empirical solvers available in the community, there is a need for improved time-domain VIV solver which can provide interaction of inline and cross-flow responses via fully-coupled fluid-structure interaction. Nonlinear effects such as vortex-shedding, turbulent wake dynamics and flow interference, large

7.1 Introduction

401

deformation due to fluid-structure coupling are discarded in these semi-empirical models. For a given ocean current environment, a long flexible riser can respond to higher-mode frequencies as compared to low aspect ratio riser. It is well known that there exists a strong coupling between the in-line (IL) and cross-flow (CF) response and force dynamics and there are complex vibrations patterns along long riser due to wake-body resonance. Due to VIV, traveling or standing-like wave patterns are observed along the riser in sheared/uniform flow, as discussed by [429]. Both standing wave and traveling wave patterns of vibration responses have been realizable in field and laboratory experiments. Vandiver [428] and Alexander [9] observed traveling wave responses for cables. Various experiments have been performed by the Norwegian Deepwater Programme [422] and British Petroleum [420] for understanding the behavior of risers with uniform and shear currents. The focus of the experiments was to obtain data from an instrumented riser model to provide an understanding of traveling wave and higher mode VIV excitations to improve semiempirical codes for long flexible cylinders. However, these model tests and field experiments could not establish physical relations among the phase-locked traveling wave along the span, the figure-8 trajectory motion and vortex shedding dynamics behind the vibrating flexible cylinder. Due to the complexity of hydro-elasticity associated with cylinder-wake interaction [478], theoretical and semi-empirical models remain incomplete for vibrating flexible cylinders in the fluid flow. In particular, the complex interactions among various response modes of vibrations are not accounted properly in the semi-empirical models, thus necessitating the use of fully-coupled Navier-Stokes-based VIV analysis with reasonable turbulence modeling.

7.1.2 Organization This chapter focuses on the modeling of flexible cylindrical structures subjected to nonlinear wake dynamics and interaction of the wake with the deforming elastic structures and the development of effective suppression devices. We simulate a full three-dimensional flexible riser exposed to uniform and linearly sheared inflow current using the fluid-structure solver [180]. Of particular interest is understanding the mechanism of standing and traveling wave patterns. The structure is modeled with the help of linear modal analysis based on the Euler-Bernoulli beam theory. A positivity preserving variational method is used for the DDES formulation to better capture the turbulence effects while maintaining positivity in the solution. The response amplitudes show standing wave response for the uniform current case. Spectral analysis suggests the dominant frequency in the in-line direction to be twice that in the crossflow direction. Detailed analysis of the orbital trajectories along the riser span is carried out to further assess the fluid-structure interactions. For the linearly sheared flow case, a traveling wave response is visible. A complex multi-modal response is also observed in this case. The content of this paper is organized as follows. We first provide a brief overview of the governing equations for turbulent fluid-structure interactions. We then present

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the variational turbulent fluid-structure formulation based on the positivity preserving scheme. We describe the problem setup and computational domain considered for the numerical analysis of the flexible riser. Then, we present the detailed numerical analysis of the coupled kinematics and dynamics of a flexible riser pinned at both ends for two different inflow conditions. Finally, we present a VIV suppression mechanism of the staggered groove on a flexible cylinder.

7.2 VIV of Long Flexible Riser 7.2.1 Numerical Framework In this section, a brief overview of the proposed coupled fluid-elastic solver based on the Navier-Stokes and linear structural equations for the flexible body in its semidiscrete Petrov-Galerkin variational form. The turbulent stress term in the unsteady Navier-Stokes equations in the proposed variational form is modeled using the S-A based DDES with a positivity preserving variational method [196].

7.2.1.1

Governing Equations

The unsteady Reynolds averaged Navier-Stokes equations in an Arbitrary Lagrangian-Eulerian (ALE) framework for an incompressible flow are given as  ∂ u¯ f  ρ + ρ f (u¯ f − um ) · ∇ u¯ f = ∇ · σ¯ f + ∇ · σ des + bf on f (t), ∂t xˆ f f

∇ · u¯ f = 0 on f (t),

(7.1) (7.2)

where u¯ f = u¯ f (x f , t) and um = um (x f , t) represent the fluid and mesh velocities defined for each spatial point x f ∈ f (t) respectively. ρ f is the fluid density, bf denotes the body force acting on the fluid and σ¯ f is the Cauchy stress tensor for a Newtonian fluid which is given as σ¯ f = − p¯ I + μf (∇ u¯ f + (∇ u¯ f )T ),

(7.3)

where p¯ is the averaged fluid pressure, I denotes the second-order identity tensor, μf represents the dynamic viscosity of the fluid, and σ des is the extra turbulent stress term. The partial derivative with respect to the ALE referential coordinate xˆ f is kept fixed in Eq. (7.1). Due to unsteady fluid forces, an immersed flexible body may undergo deformation and flow-induced vibration. The long flexible riser can be considered as a slender pipe with a constant tension and relatively small lateral motion. The Euler-Bernoulli theory is applicable for

7.2 VIV of Long Flexible Riser

403

describing the dynamics of such long flexible riser. The tensioned Euler-Bernoulli beam equation that is excited by the distributed unsteady fluid force F s comprising of external force F sext (z, t) and internal force F sint (z, t) is given as ρs A

∂ 2 ws (z, t) ∂ 4 ws (z, t) ∂ 2 ws (z, t) + E I − P = F sext (z, t) + F sint (z, t). (7.4) ∂t 2 ∂z 4 ∂z 2

Here, w s (z, t) denotes the lateral displacement vector of the flexible riser at the spanwise coordinate z, P is axial tension, ρ s is the structural density, and A is the cross-sectional area of the riser. E and I are the elastic modulus and second moment of area respectively. In addition to the initial condition w(z, 0), following boundary conditions at top and bottom ends are considered w s (z, t)|z=0 = 0,  ∂ 2 w s (z, t)  = 0,  ∂z 2 z=0

w s (z, t)|z=L = 0,  ∂ 2 w s (z, t)  = 0.  ∂z 2

(7.5) (7.6)

z=L

The Eq. (7.4) is solved by considering the mode superposition procedure for dynamic response of the riser structure [56], where the natural frequency of the nth-mode for a pinned-pinned beam of length L under axial tension P is given as     π4  4 E I n4 + 1 L fn = 2π m

 n2 P L 2 π2

(7.7)

where m is the mass per unit length of the beam. The eigenmodes are assumed to be sinusoidal so that the eigenvectors S can be written as: S n (z) = sin

 nπ z L

(7.8)

where S n denotes the eigenvector associated with the nth mode. The tension is assumed constant along the riser span and there is no internal flow effect inside the riser tube. The coupling between the fluid and structural equations is achieved by the kinematic and traction continuity along the fluid-structure interface. Mathematically, these relations can be written as u¯ f (ws (x s0 , t), t) = us (x s0 , t),



f f f σ¯ (x , t) · ndΓ (x ) + F s dΓ = 0 w s (γ ,t)

(7.9) (7.10)

γ

where ws is the position vector mapping the initial position x s0 of the flexible body to its position at time t, F s is the fluid traction acting on the body and us is the

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structural velocity at time t given by us = ∂ws /∂t. Here, n is the outer normal to the fluid-body interface, γ is any part of the interface Γfs in the reference configuration. The coupling algorithm between the fluid and structural equations is based on the non-linear iterative force correction (NIFC) scheme presented in [180, 181]. At each time step, the fluid surface tractions on the riser surface are projected onto the eigenvectors to find the values of generalized modal forces. The resulting modal forces are then used to determine the modal amplitudes and the displacements for the next time step. The mesh motion required to accommodate the changes in riser geometry is accomplished by explicitly controlling the motion of each node in the mesh while satisfying the kinematic consistency of the discretized interface. The movement of the internal finite element nodes should be chosen such that the mesh quality does not deteriorate as the displacements of the solid structure become large. For this purpose, the fluid mesh is simply assumed to represent a hyperelastic solid model. A standard Lagrangian finite element technique can be used to adapt the mesh to the new geometry of the domain. The coupling of the fluid and structural response can be achieved numerically in different ways, but the interface conditions of displacement continuity (motion transfer) and traction equilibrium (momentum transfer) along the fluid-structure surfaces must be satisfied across non-matching discrete fluid-solid interface [182, 183]. The turbulent stress term in Eq. (7.1) is modeled using the Boussinesq approximation given by σ des = μT (∇ u¯ f + (∇ u¯ f )T )

(7.11)

where μT is the turbulent dynamic viscosity. The turbulent viscosity, νT = μT /ρ f is modeled using the SA one equation turbulence model. νT and related parameters are given as, νT = ν˜ f v1 ,

f v1 =

χ˜ 3 , 3 χ˜ 3 + cv1

where ν is the molecular viscosity defined as ν = equation:

μf . ρf

χ˜ =

ν˜ ν

(7.12)

ν˜ is solved using the transport

 ∂ ν˜ 1 + (u¯ f − um ) · ∇ ν˜ = cb1 S˜ ν˜ + ∇ · (ν + ν˜ )∇ ν˜ ∂t σ

2 cb2 ν˜ + (∇ ν˜ ) · (∇ ν˜ ) − cw1 f w σ d˜

(7.13)

where S˜ = S + (˜ν /(κ 2 d 2 )) f v2 , S being the magnitude of vorticity, cb1 , cb2 , σ , κ, cw1 and cv1 are the constants defined for the S-A model in [396], d˜ = d − f d max(0, d − CDES Δ), d is the distance to the closest wall, CDES is the DES coefficient and Δ is based on the largest dimension of the grid element. The quantity f d is given by f d = 1 − tanh([Ard ]p ), which is designed to be 1 in the LES region and the constants

7.2 VIV of Long Flexible Riser

405

A = 8 and p = 3. The quantity rd is defined as: rd = √u i, j uν˜i, j κ 2 d 2 , where u i, j is the velocity gradient and κ is the Karman constant. Setting f d = 0 yields RANS setting (d˜ = d) while when f d = 1, d˜ = min(d, CDES Δ) which corresponds to the DES model. The empirical constant CDES was calibrated to 0.65. The value of S˜ is limited using the method given in [16] to avoid numerical problems.

7.2.1.2

Variational Formulation

We next present the stabilized variational forms for the governing equations discussed in the previous sub-section with equal order interpolations for velocity, pressure and turbulent viscosity variable. We have considered a stabilized Petrov-Galerkin finite element framework for the spatial discretization of the flow and turbulence equations. The governing equations are integrated in time from t ∈ [t n , t n + 1] using the Generalized-α method [94] which is unconditionally stable and second-order accurate for linear problems. Let S h be the space of trial solution, the values of which equal the given boundary condition on the Dirichlet boundary and V h be the space of test functions which vanish in the Dirichlet boundary, the variational form of the flow equation will be: f ¯ hf,n + 1] ∈ S h such that ∀[φ fh , qh ] ∈ V h : find [u¯ f,n h + αm , p

m,n f f f f ¯ f,n ρ f (∂t u¯ f,n + α f ) · ∇ u¯ f,n h + αm + ( u h + α − uh h + α ) · φ h d



f,n f f + σ¯ f,n + α : ∇φ d + σ des h + α f : ∇φ fh d h h e e 

 f,n f − ∇qh · u¯ h + α d e

e

+ + − −

nel

 e=1 nel

 e=1

e

e=1

e

e=1

e

nel

 nel





=

e

e

m,n f e τm (ρ f (u¯ f,n + α f ) · ∇φ fh + ∇qh ) · R m (u¯ f , p)d ¯ h + α − uh

f e ∇ · φ fh τc ∇ · u¯ f,n h + α d

f e τm φ fh · (R m (u¯ f , p) ¯ · ∇ u¯ f,n h + α )d

e ∇φ fh : (τm R m (u¯ f , p) ¯ ⊗ τm R m (u¯ f , p))d ¯

bf (t n + α f ) · φ fh d +

Γh

hf · φ fh dΓ

(7.14)

where the terms in the first line correspond to the transient and convective terms of the momentum equation, second line consists of the viscous and turbulent stress

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terms, third line is the variational form of the continuity equation, fourth and fifth lines represent the Galerkin/least-squares stabilization terms for the momentum and continuity equations respectively. With the use of linear finite element spaces, the higher derivatives of the weighting function related to the viscous stress tensor will be very small and hence, they have been neglected. The two residual terms in the sixth and seventh lines are introduced as the approximation of the fine scale velocity on the element interiors and its corresponding convective stabilization based on the multiscale argument [4, 160, 171]. hf is the corresponding Neumann boundary condition for the Navier-Stokes equation. The element-wise residual of the momentum equation is given by m,n f f f f ¯ f,n ¯ = ρ f ∂t u¯ f,n + α f ) · ∇ u¯ f,n R m (u¯ f , p) h + αm + ρ ( u h + α − uh h +α f n f f f ¯ des −∇ · σ¯ f,n h , n + α − b (t + α ) h +α −∇ ·σ

(7.15)

The stabilization parameters τm and τc are the least-squares metrics added to the element level integrals in the stabilized formulation [66, 118, 387, 415]. The S-A based DDES equation can be simplified as convection-diffusion-reaction equation. Let S h to be the space of trial solution, the values of which equal the given boundary condition on the Dirichlet boundary and V h be the space of test functions which vanish in the Dirichlet boundary, the variational form of the turbulence equaf ∈ S h such that ∀ψh ∈ V h : tion will be: find ν˜ hf,n + αm

f,n+α f

f

f

m (∂t ν˜ h + u · ∇ ν˜ hf,n+α + s ν˜ hf,n+α )ψhf d e 

f + k∇ ν˜ hf,n+α · ∇ψhf d

e

+

nel

 e=1

+ +

nel



e e=1 

n el 

e=1



=

e

Γh

e

(u · ∇ψhf + |s|ψhf )τt Rt (˜ν )de χ χ

|Rt (˜ν )| f |∇ ν˜ hf,n+α |

|Rt (˜ν )| f

|∇ ν˜ hf,n+α |

ψhf gdΓ

 ksadd ∇ψhf · kcadd ∇ψhf

u⊗u |u|2



f

· ∇ ν˜ hf,n+α de

  f u⊗u · ∇ ν˜ hf,n+α de · I− |u|2 (7.16)

where the terms in the first and second line correspond to the Galerkin terms, third line is the linear stabilization term which converts the method to Galerkin/leastsquares (GLS) when s > 0 and to subgrid scale (SGS) when s < 0, thus maintaining the phase of the exact solution. Terms in the fourth and fifth lines are the nonlinear stabilization terms in the streamline and crosswind directions [196] while the term in the right-hand side of the equation is the Neumann boundary condition. The residual of the transport equation is given by

7.2 VIV of Long Flexible Riser f f,n+αm

Rt (˜ν ) = ∂t ν˜ h

407 f

f

f

+ u · ∇ ν˜ hf,n+α − ∇ · (k∇ ν˜ hf,n+α ) + s ν˜ hf,n+α .

(7.17)

More details regarding the formulation and implementation can be found in [196].

7.2.2 Problem Setup and Validation Understanding the coupled dynamic response of a long flexible offshore riser under tension is a complex problem since vortex shedding can excite multiple modes. The dynamics of such complex dynamical systems depends on four nondimensional parameters KB =

EI ρ f U02 D 4

P∗ =

P ρ f U02 D 2

Re =

ρ f U0 D μf

m∗ =

ms π 2 D Lρ f 4

(7.18)

where the first and second nondimensional terms represent the nondimensional bending rigidity and nondimensional tension respectively. Re and m ∗ denote the Reynolds number and mass ratio. U0 represents the free-stream velocity, E is the Young’s modulus, I is the second moment of cross-sectional area and m s is the mass of the flexible riser. In this study, we have made an attempt to understand the response dynamics of the flexible riser under tension for two different inflow current configurations. In the first configuration, we consider a uniform inflow, while in the second configuration we assume a shear flow velocity profile. Figure 7.1a shows the typical schematic for the flexible riser of length L and diameter D under constant axial tension P. The riser is pinned at both ends and its axis is perpendicular to the incoming flow. U (z) in this figure represents the incoming velocity profile. Figure 7.1b presents the schematic of the computational domain considered for this numerical analysis. The riser is located at 10D and 25D from the inflow and outflow boundaries respectively. The distance between the riser and its sides is 10D. We have considered the slip-wall boundary condition along the top and bottom surfaces. The slip boundary condition is also implemented along the sides in addition to the Dirichlet and traction-free Neumann boundary conditions along the inflow and outflow boundaries respectively. The incoming flow is devoid of any turbulence, i.e., ν˜ = 0 at the inflow boundary. Table 7.1 presents the summary of nondimensional and computational domain parameters for both configurations. Figure 7.2 shows the two-dimensional (2D) mesh in the X-Y plane employed for the computational study. The three-dimensional (3D) computational mesh for the first and second configurations is made up of 800 and 2350 2D layers in the spanwise direction. The number of layers in the spanwise direction is selected based on a detailed mesh convergence study.

408 Fig. 7.1 A long flexible riser model in current flow along the Z -axis: a Pinned-pinned tensioned riser with current flow, b Sketch to illustrate the computational setup and boundary conditions for the PPV-based DDES on flow past a flexible riser. Here, u¯ f = (u, v, w) denotes the fluid velocity components

7 Flexible Cylinder VIV

7.2 VIV of Long Flexible Riser

409

8

4

Y/D

0

-4

-8 -10

-5

0

5

10

15

20

25

X/D

Fig. 7.2 Two-dimensional mesh in the X-Y plane employed in the computation of flexible riser VIV. This mesh is extruded in the third dimension to get the three-dimensional mesh Table 7.1 Summary of the nondimensional and computational domain parameters for both the configurations Configuration 1 Configuration 2 ExxonMobil NDP Data set (MIT VIV repository) KB P∗ Re m∗ L/D U (z) No. fluid elements No. nodes nodes

(Case no. 1103) 2.1158 × 107 5.1062 × 104 4000 2.23 481.5 U0 12.80 M 12.95 M

(Case no. 2350) 2.2995 × 106 1.1197 × 104 18900 1.6 1407.4 U0 z/L 38.11 M 38.48 M

Before we present our numerical analysis, we first perform a validation study of the coupled fluid-structure formulation with that of the experiment performed by ExxonMobil. For this validation study, we have considered U (z) = 0.2 which corresponds to case no. 1103 in ExxonMobil. Figure 7.3 presents the comparison between the current simulations and experiments of ExxonMobil for the root mean square amplitude profile along the span of the riser for both cross-flow and in-line directions. One can clearly observe that our results provide a qualitative agreement in the oscillation modes. While the simulated cross-flow amplitude is in good agreement with the experimental measurements, there is some over-prediction of the in-line response A x,rms /D. The difference in the peak in-line displacement between numerical and

410

7 Flexible Cylinder VIV Present Experiment

0.8

0.8

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0

0

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0.4

0.5

Present Experiment

0.9

z/L

z/L

0.9

0

0

0.1

0.2

0.3

0.4

0.5

Ax,rms /D

Fig. 7.3 Uniform flow across a flexible riser at Re = 4000 (Configuration 1): variation of the root mean square amplitude along the span of the riser in: a cross-flow, b in-line directions

experimental measurement is < 15 % with respect to the total peak rms amplitude  A/D, where A = (A2x,rms + A2y,rms ). Numerical prediction and measurement of the in-line response are very sensitive to the precise lock-in range and the boundary layer characteristics around the flexible riser. From a practical point of view, the in-line response is several factors smaller than the cross-flow amplitude, therefore a good estimate of the cross-flow response is generally sufficient for the riser design study. A comparison of the cross-flow vibrational amplitude between the simulation and the experiment in Fig. 7.4 shows a good agreement. Nonetheless, there is a need for further study to understand the role of nonlinearity in structural dynamics, the effects of variable tension along the riser, and the roughness on the laminar flow separation over the riser surface.

7.2.3 Response Characteristics at Uniform Flow Figure 7.5 presents the riser’s displacement envelops in both cross-flow and in-line directions for tU/D ∈ [400, 470]. From this figure, it can be inferred that the riser vibrates with a dominated second mode in the in-line (IL) and the first mode in the cross-flow (CF) directions. Since the dominant frequency of the in-line vibration is twice that of cross-flow vibration for the flow across the cylinder. Hence, we can say that the vibration amplitudes of the flexible riser are coupled to the flow physics and the hydrodynamic forces. This is further reflected in the dominant response modes of the riser observed in Fig. 7.5. We plot the space-time contours for the cross-flow and in-line displacements in Fig. 7.6. One can observe that the in-line vibration frequency

7.2 VIV of Long Flexible Riser

411

1 Present

Experiment

(Ay − Ay )/D

0.5

0

-0.5

-1 400

420

440

460

480

500

tU/D

1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

z/L

z/L

Fig. 7.4 Uniform flow across a flexible riser at Re = 4000 (Configuration 1): comparison between the time history of the cross-flow amplitude at z/L = 0.55. The amplitude agrees quite well with that of the experiment

0.5

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0

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(Ay − Ay )/D

0.4

0.6

0.8

0 -0.3

-0.2

-0.1

0

0.1

0.2

0.3

(Ax − Ax )/D

Fig. 7.5 Uniform flow across a flexible riser at Re = 4000 (Configuration 1): riser response envelope with spanwise resolution of 800 layers for cross-flow (top) and in-line (bottom) directions

is twice that of the cross-flow response. We also notice the standing wave pattern from the space-time contours which agrees with the observations made in [96]. We observe the figure-8 configuration for most of the locations along the riser. Figs. 7.7 and 7.8 depict the orbital trajectories for two locations along the riser (z/L = 0.33 and 0.66) for the time window [400, 470] and [600, 670] respectively.

412

7 Flexible Cylinder VIV

Fig. 7.6 Uniform flow across a flexible riser at Re = 4000 (Configuration 1): standing wave response of flexible riser VIV with spanwise resolution of 800 layers for cross-flow (top) and in-line (bottom) directions

For the first time window (Fig. 7.7), the trajectory is observed to be clockwise on the upper half of the riser at locations z/L ∈ [0.55, 0.88] while it is counter-clockwise for z/L ∈ [0.11, 0.44]. The clockwise and counter-clockwise configurations are defined on the basis of the direction of trajectory on the upper circle of figure-8. Similarly, the trajectories get reversed with time which is evident from the plot for the second time window (Fig. 7.8). Such alternating clockwise and anticlockwise trajectories were also reported by ExxonMobil. It was discussed in [429] that for cross-flow and in-line standing wave response, one may observe the alternating clockwise and anticlockwise trajectories which are unfavorable and favorable for VIV respectively. Due to the observation of alternating trajectories along the riser and also in time, we will focus on two-time windows for subsequent analyses: tU/D ∈ [400, 470] where the lower part of the riser (0 < z/L < 0.5) is undergoing favorable VIV trajectory and tU/D ∈ [600, 670] where the upper part (0.5 < z/L < 1) has a counter-clockwise trajectory to the incoming flow. A comparison of the orbital trajectories with that of the experiment ExxonMobil has been carried out in Figs. 7.9 and 7.10 for the time windows tU/D ∈ [440, 460] and tU/D ∈ [600, 620] respectively. A good agreement is observed between the trajectories across the simulation and the experiment. Furthermore, the change in the direction of the trajectory (clockwise to counter-clockwise and vice-versa) along the span of the riser and with time is also observed in the experiment. Spectral analysis of the vibration amplitude provides insight into the dominant frequencies in the vibration. Figures 7.11 and 7.12 present the frequencies power spectral plots for the cross-flow and in-line oscillations at z/L = 0.66 and 0.33 pertaining to the two time windows tU/D ∈ [400, 470] and [600, 670] respectively. From these figures, we make the following observations. First, for every location along the riser, we observe a dominant non-dimensional frequency ( f D/U ) of 0.166

7.2 VIV of Long Flexible Riser z/L = 0.66

1 0.8

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1

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413

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0.6

-0.4

-0.2

0

0.2

0.4

0.6

(Ax − Ax )/D

(Ax − Ax )/D

Fig. 7.7 Uniform flow across a flexible riser at Re = 4000 (Configuration 1): orbital trajectories for z/L = 0.66 (left) and 0.33 (right) for the time window tU/D ∈ [440, 460] z/L = 0.66

0.8

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0

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1

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1

0.2

(Ax − Ax )/D

0.4

0.6

-1 -0.6

-0.4

-0.2

0

0.2

0.4

0.6

(Ax − Ax )/D

Fig. 7.8 Uniform flow across a flexible riser at Re = 4000 (Configuration 1): orbital trajectories for z/L = 0.66 (left) and 0.33 (right) for the time window tU/D ∈ [600, 620]

for the cross-flow vibration and 0.342 for the in-line vibration. Hence, the in-line vibrational frequency is twice that of the cross-flow frequency. Moreover, the values of the frequencies are much closer to the experimental value than that in [437]. Second, for the cross-flow vibration, in almost all the cases, we observe the second and third modes with the frequency of 0.342 and 0.508 respectively. Third, analyzing the in-line vibration spectra, similar observations are made for most of the locations having a frequency twice that of the dominant frequency. Furthermore, a frequency of 1.5 times the dominant frequency is also observed.

414

7 Flexible Cylinder VIV z/L = 0.77

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1

-0.2

0.4

0.6

-1 -0.6

-0.4

-0.2

0

0.2

(Ax − Ax )/D

Fig. 7.9 Orbital trajectory at different locations along the span of the riser (Configuration 1): comparison of experimental trajectory (left column) with the simulation in the time window tU/D ∈ [440, 460] (right column)

7.2 VIV of Long Flexible Riser z/L = 0.77

1

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415

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0

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(Ax − Ax )/D

Fig. 7.10 Orbital trajectory at different locations along the span of the riser (Configuration 1): comparison of experimental trajectory (left column) with the simulation in the time window tU/D ∈ [600, 620] (right column)

416

7 Flexible Cylinder VIV z/L = 0.66

0.6

z/L = 0.33

0.6 In-line Cross-flow

In-line Cross-flow

0.5 Power spectrum of (A − A)/D

Power spectrum of (A − A)/D

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0

0

1

2

0.4

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0

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f D/U

2

3

f D/U

Fig. 7.11 Uniform flow across a flexible riser at Re = 4000 (Configuration 1): oscillation frequencies at z/L = 0.66 (left) and 0.33 (right) in the time window tU/D ∈ [400, 470] z/L = 0.66

0.6

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0.6 In-line Cross-flow

In-line Cross-flow

0.5 Power spectrum of (A − A)/D

Power spectrum of (A − A)/D

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0

1

2 f D/U

3

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0

0

1

2

3

f D/U

Fig. 7.12 Uniform flow across a flexible riser at Re = 4000 (Configuration 1): oscillation frequencies at z/L = 0.66 (left) and 0.33 (right) in the time window tU/D ∈ [600, 670]

Figure 7.13 summarizes the 3D-contours of the fluid-structure characteristics along the riser and the mid-plane for tU/D = 460. Figure 7.13a shows the riser surface flooded with the fluid pressure acting on it and the cross-sectional view of the velocity vector magnitude at the mid-plane. In Fig. 7.13b, we flood the riser surface with the magnitude of vibration amplitude. Additionally, Fig. 7.13b also shows the z-vorticity patterns at various locations along the riser (z/L ∈ [0.11, 0.88]). From Fig. 7.13b, one can observe that the magnitude of vibration amplitude is slightly biased and the maximum displacement amplitude occurs in the lower half region of the riser which is undergoing the counter-clockwise trajectory for

7.2 VIV of Long Flexible Riser Pressure along 2 the riser (p/ U ):

-0.5 0.5

1.5

2.5

3.5

417 Vibration amplitude along the riser (|A|/D): 0.05 0.15 0.25 0.35 0.45 0.55 0.65

4.5

Y

Y X Z

X Z

P+S 2P

Wide 2S

Velocity magnitude (|U|/U):

0.5

1

1.5

Fig. 7.13 Flow visualization along the riser under uniform flow at Re = 4000 (configuration 1) for tU/D = 460 : pressure distribution along the riser with velocity magnitude contour on the mid-section (top), vibration amplitude along the riser with Z-vorticity contours in various sectionsred and blue colour indicate the positive and negative vorticity respectively (bottom)

tU/D ∈ [400, 470]. This type of counter-clockwise trajectory is opposite to the incoming flow direction, which in turn produces more shear on the riser surface thereby the fluid does net work on the structure [100]. On the other hand for the second time window tU/D ∈ [600, 670], the counter-clockwise trajectories are observed in the upper half of the riser. The 3D vortex patterns resulting from the cross-flow and in-line oscillations are complex and exhibit multiple vortex modes at different spanwise locations. We observe the 2S mode of vortex shedding for most of the spanwise locations and a wider 2S with two rows configuration near the locations where large vibration amplitudes are found. At some of the places, a 2P vortex mode is also observed. Further analysis is required to better assess the vortex dynamics in detail. The isosurfaces of the Q-criterion are presented in Fig. 7.14. It can be inferred that the vortical structures are more intense in the lower half of the riser for tU/D = 460. These high-intensity and multi-scale vortices are produced by the counter-clockwise motion in the figure-8 trajectory due to the large shear produced by the incoming current. This in turn leads to a high amplitude of vibration as observed in the lower half of the riser in Fig. 7.13b. Similar observations can be made for the response of the riser at tU/D = 650 where the high intensity of vortices is seen at the upper half of the riser.

418

7 Flexible Cylinder VIV

Velocity magnitude (|U|/U):

0.5

0.8

1.1

1.4

Fig. 7.14 Uniform flow across a flexible riser at Re = 4000 (Configuration 1): vortical wake structures of iso-surfaces of Q + = 0.25 coloured by velocity magnitude at tU/D = 460 along the mid-span of the riser

7.2.4 Response Characteristics at Linearly Sheared Flow In this subsection, we analyze the VIV of a pinned-pinned flexible riser in a linearly varying shear flow. Figure 7.15 shows the flexible riser’s displacement envelope in both cross-flow and in-line directions as a function of time. Similar to the uniform flow, even here we observe that the in-line oscillation mode is twice that of the cross-flow vibration mode. Unlike the uniform flow, where the in-line and crossflow vibrations have exhibited second and first modes respectively, here, we observe higher oscillation modes. However, we would not directly attribute this effect to only shear flow because of the differences in the other nondimensional parameters. In [437], the authors have shown that increase in Re would result in higher response modes.

7.2 VIV of Long Flexible Riser

419

Fig. 7.15 Shear flow across a flexible riser at Re = 18,900 (Configuration 2): riser response envelope with spanwise resolution of 2350 layers for cross-flow (top) and in-line (bottom) vibrations

In Fig. 7.16, we plot the space time contours for the cross-flow and in-line oscillations. Interestingly, unlike the uniform flow where we have observed the standing wave, here, we observe the traveling wave like phenomenon in both cross-flow and in-line directions. From Fig. 7.16 (top), one can clearly see that the cross-flow disturbance travels from the top to bottom. A similar, traveling wave like phenomenon can be found even for the in-line oscillations traveling from the top point to the bottom point. To better understand the traveling wave mode and frequency involved a more detailed space time spectral analysis [61] is required to decompose the oscillation modes involved. Figure 7.17 shows the response trajectories along the span. The transition of trajectories from clockwise to counter-clockwise and vice-versa along the riser span becomes more complex. For this case we observe multiple transition along the span. However, interestingly we did not observe any transition in time like we observed for the uniform flow configuration. In the upper half of the riser, i.e. for z/L ≥ 0.6, the trajectories are predominantly counter-clockwise. In the lower half for z/L < 0.6, we observe that the trajectory changes its direction intermittently.

7.2.5 Interim Summary We have presented a variational partitioned turbulent fluid-structure formulation based on the positivity preserving scheme. The turbulent viscous shear stress term in the Navier-Stokes equation is modeled using the SA-based delayed detached eddy simulation. The partitioned iterative scheme relies on the nonlinear iterative force correction scheme. Using the proposed formulation, we have studied the standing

420

7 Flexible Cylinder VIV

Fig. 7.16 Shear flow across a flexible riser at Re = 18,900 (Configuration 2):traveling wave response of flexible riser VIV with spanwise resolution of 2350 layers for cross-flow (top) and in-line (bottom) vibrations

and traveling wave-like phenomenon in the cross-flow and in-line vortex-induced vibrations of a long flexible riser pinned at both ends for two different inflow configurations namely, uniform and linearly sheared. For uniform inflow conditions, we showed that the flexible riser exhibits a standing wave in both cross-flow and in-line directions. Multi-directional figure-8 configurations are observed along the riser span which changes their direction with time. On the other hand, for the linearly sheared configuration, we observed a traveling wave-like phenomenon in the cross-flow and in-line responses which travel from the top point to the bottom point. Additionally, we have also observed that the riser trajectories are predominantly counter-clockwise for z/L ≥ 0.6 and for z/L < 0.6, we have noticed an intermittent change in the direction of trajectory rotation. In addition to inflow characteristics, the dynamics of standing and traveling waves in long flexible risers can also be affected by nondimensional parameters Re, m ∗ , nondimensional bending rigidity, nondimensional tension and aspect ratio L/D of the riser. It would be important to investigate the effect of these parameters on the coupled kinematics and dynamics of flexible riser VIV in the future. For further progress, it would also be interesting to look into the underlying physical mechanism behind the formation of traveling waves.

7.3 Flexible Cylinder with Spanwise Grooves An offshore structure forms a complex coupled nonlinear dynamical system due to its interaction with the harsh ocean environment. In particular, deepwater risers and mooring system are subjected to the vortex-induced vibration (VIV) due to the

7.3 Flexible Cylinder with Spanwise Grooves z/L = 0.77

1 0.8

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421

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Fig. 7.17 Shear flow across a flexible riser at Re = 18,900 (Configuration 2): orbital trajectories for various locations along the flexible riser span for the time window tU/D ∈ [540, 560]

shedding of alternating signed vortices. The amplitude of the vibration becomes significant when its frequency is close to the natural frequency of the structure. This is referred to as lock-in or vortex synchronization [56, 330], where the cylinder vibrates with high amplitudes in a well-defined synchronization range. On top of that, the hydrodynamic forces, in particular, the drag force that acts on the structure can cause a large deformation. Both the VIV and drag force have a very significant impact on the lifespan and integrity of the offshore structure, hence suppression devices are installed to minimize the fatigue damage and to improve the structural reliability [40, 381]. Many suppression devices have been studied and investigated [381]. They can be broadly classified into two main categories based on their suppressing mechanism, namely near-wake stabilizer and surface-geometry modifier [401]. The former sup-

422 Fig. 7.18 Schematic diagram of proposed staggered groove. The alignment of grooves is alternated along the spanwise direction to create a jump pattern in cross-section. The stagger angle, θ and staggered pitch, ps are two geometric parameters

7 Flexible Cylinder VIV

ps

θ

presses the VIV and the drag force by streamlining the flow and by stabilizing the near-wake region. These devices include splitter plate [24], guided foil [127], fairing [12], and connected-C device [226]. While they perform well in reducing VIV and drag force, they usually come at a cost of high installation and maintenance due to their weathervane feature. Hence, such devices may not be attractive for a deepwater structure subjected to the harsh ocean environment. On the other hand, the surface-geometry modifier manipulates the distribution of vorticity in the boundary layer and the separation point of the vibrating structure, thereby desynchronizing the vortices shed and reducing vibration [227]. These includes helical strakes [13, 384, 498], dimples [36], and bumps [35, 327]. Among these devices, helical strakes are the most widely used device owing to their high performance for VIV suppression. Nevertheless, the helical strakes have the disadvantage of increasing the drag force, which can impact the lifespan of the structure. This drawback of the helical strakes motivates us to further research on a suppression device that is capable of reducing both the VIV amplitude and the drag force. In the recent experimental study by [162], a triple-starting helical groove has been considered for the VIV suppression and the drag reduction of an elastically mounted two-degree-of-freedom circular cylinder. It was found that the peak vibration amplitude and the drag force were reduced up to 64% and 25%, respectively at the subcritical Reynolds number. Another study on the groove was conducted by [278], where a design termed as Longitudinal Groove Suppression (LGS) was recently investigated and demonstrated for subcritical and post-critical Reynolds number flows. The LGS design was inspired by the surface of Saguaro Cacti, which consists of ridges and troughs along the circumference of the cylinder. Moreover, the positions of the grooves are staggered in the spanwise direction of the cylinder. A variety of LGS designs have been studied experimentally, and it was observed that the LGS models could reduce the drag up to 50% when compared to the vibrating plain cylinder. The authors conjectured that the flow in the boundary layer is modified by the ridges, thereby affecting the vorticity distribution and reducing hydrodynamic forces on the cylinder. It was found to be quite promising for drilling riser buoyancy modules due to its deployability and reliability. In the present work, we consider a new suppression device, termed as staggered groove [227], which will be shown to be effective in VIV suppression and drag

7.3 Flexible Cylinder with Spanwise Grooves

423

d

w

D

(a)

(b)

Fig. 7.19 Schematic diagram of a cylinder with three uniformly distributed extruded grooves: a cross-section, and b isometric view. Representative width and depth of the groove are w = 0.2D and d = 0.16D, respectively and the grooves are extruded in the spanwise direction

reduction. The device consists of an alternating alignment of grooves along the spanwise direction to create a jump pattern in cross-section. Its schematic diagram is shown in Fig. 7.18, where the staggered angle, θ and the staggered pitch, ps are two parameters that define the geometry of the staggered groove. We hypothesize that the jump pattern in the cross-section may decorrelate the vortex shedding along the spanwise direction, thereby reducing the resultant forces acting on the structure and suppressing VIV. To investigate the suppression mechanism, a numerical study has been conducted to assess the characteristic response of the staggered groove on an elastically mounted cylinder and compare it with a plain cylinder counterpart. A demonstration of its performance on a flexible cylinder is also conducted to ensure the robustness of the device when it is deployed on flexible offshore structures, such as mooring and riser.

7.3.1 Problem Setup There are several parameters that can characterize the geometry of grooves. In this work, we mainly focus on the sharp-cornered grooves at the cross-sectional plane. The width w and the depth d of the groove are set to w = 0.2D, d = 0.16D, according to the cross-section of the grooves studied in the experiment [162] and the analysis performed in [227]. A schematic diagram of the geometry of the grooves is shown in Fig. 7.19. We aim to investigate the effect of the spanwise variation of grooves on the flow dynamics and the VIV characteristics. Thus, the grooves are configured such that it is staggered along the spanwise direction, as shown in the schematic diagram in Fig. 7.20. Its spanwise length is set to 18D to capture the three-dimensionality effect induced by the varying geometries along the spanwise direction. Three different pitches ps of the staggered grooves are investigated, where ps = 1D, 2D, 4D.

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7 Flexible Cylinder VIV

Fig. 7.20 Staggered arrangement of the spanwise grooves fitted on the vibrating circular cylinder. Three grooves along the circumference span over the cylinder length L = 18D. The pitch of staggered groove ps = 2D

L = 18D

Staggered Pitch ps

Apart from the groove geometry, there are four key dimensionless parameters used to characterize the fluid-structure interaction, namely Reynolds number (Re), reduced velocity (Ur ), mass ratio (m ∗ ), and damping ratio (ζ ), which are defined as follows Re =

c U 4m ρfUD , m∗ = , ζ = √ , Ur = f f 2 μ fn D πρ D L 2 mk

where ρ f is the density of the fluid, μ f is the dynamic viscosity of the fluid, m is the mass of the cylinder, and f n is the natural frequency of the spring-mass system in a vacuum. In our numerical analysis that is based on the coupling of incompressible Navier-Stokes and rigid body equations, we use the natural frequency in a vacuum for the purpose of non-dimensionalization. During this fluid-structure coupling cycle, the added mass effect is implicitly accounted in the coupled formulation and the response results are appropriately adjusted to match the experimental conditions [226, 295].

7.3.2 Suppression via Spanwise Grooves Most of the offshore structures that suffer from VIV, such as moorings and risers, are flexible. Therefore, it is necessary to assess the effectiveness of a staggered groove in suppressing the VIV of a flexible cylinder. The cylinder is modeled as an Euler-Bernoulli beam, as the structure is assumed to have relatively small lateral movements. The flexible cylinder is described by the following equation of motion:

7.3 Flexible Cylinder with Spanwise Grooves

425

Table 7.2 Characteristic responses of bare cylinder and staggered grooves with different pitches on a flexible cylinder at Re = 4800. The responses are collected by selecting the maximum value along the spanwise direction. Value in the bracket indicate its percentage different to the bare cylinder counterpart Bare ps = 1D ps = 2D ps = 4D A∗y,max A∗x,max Cl,r ms Cd,mean

0.908 0.198 1.312 2.290

m

0.557 (−39%) 0.046 (−77%) 0.251 (−81%) 2.001 (−13%)

0.567 (−38%) 0.042 (−79%) 0.264 (−80%) 1.9767 (−14%)

0.521 (−43%) 0.040 (−80%) 0.266 (−80%) 1.784 (−22%)

∂ 2 ws (z, t) ∂ 4 ws (z, t) ∂ 2 ws (z, t) + E I − P = F s (z, t) ∂t 2 ∂z 4 ∂z 2

(7.19)

where m is the mass of the beam per unit length, P is the applied axial tension on the beam, E is the elastic modulus, I represents the second moment of area of the beam and ws is the structure’s displacement. The structure’s coordinates, F s is the fluid load. Note that, P is set to 0 in this study. Both ends of the cylinder are pinned, and the spanwise length of the cylinder is identical to the cases presented in the previous section. Due to the high computational cost, the simulation is only conducted at the following dimensionless parameters: Re = 4800 , m ∗ = 2.6 , K B = 2.76 × 103 ,

(7.20)

whereby four cases have been simulated in total, which includes a bare cylinder and staggered grooved cylinder with staggered pitches ps = 1D, 2D, 4D. Their characteristic responses are summarized in Tab. 7.2. The numerical results show that the staggered grooves remain the effectiveness in suppressing VIV and reducing drag force for a flexible cylinder. Moreover, among the pitches investigated, it is found that ps = 4D has the best performance, where the cross-flow amplitude is reduced by 43%, the in-line amplitude is reduced by 80%, and the drag coefficient is reduced by 22%. Notably, the reduction in drag coefficient is significantly larger in ps = 4D compared to others. In the next section, we analyze the spanwise correlation of hydrodynamic forces to provide insight into the suppression mechanism of the staggered groove.

7.3.3 Analysis on Spanwise Correlation Given the promising performance of staggered grooves in suppressing VIV and reducing drag force, we would like to further our analysis of its suppressing mechanism. We hypothesize that the suppression of VIV is due to the jump pattern of the cross-section geometry when the alignment of grooves is alternated along the

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7 Flexible Cylinder VIV

spanwise direction. This will affect the spanwise correlation of the hydrodynamic forces, thereby reducing the energy transferred to the grooved cylinder. We define the spanwise correlation of hydrodynamic forces as: ρcl (Δz ∗ ) =

cl (z ∗ , t)cl (z ∗ + Δz ∗ , t) σcl (z ∗ ) σcl (z ∗ +Δz ∗ )

(7.21)

ρcd (Δz ∗ ) =

cd (z ∗ , t)cd (z ∗ + Δz ∗ , t) σcd (z ∗ ) σcd (z ∗ +Δz ∗ )

(7.22)

where cl (z ∗ , t) and cd (z ∗ , t) denote the sectional fluctuating lift and drag coefficients at position z ∗ at time t with their corresponding standard deviations σcl (z ∗ ) and σcd (z ∗ ) ) respectively. Here, z ∗ = z/D is the dimensionless parameter that relates to the position in the spanwise direction, which provides a reference for the computation of the cross-correlation. Recall that the spanwise length of the cylinder is 18D, we take z ∗ = 6 and consider the maximum of Δz ∗ as 6 for the movable location in the spanwise direction. This covers a mid-portion of the vibrating cylinder away from the end effects. A correlation of 1 implies that the forces are perfectly in-phase, which will exert a large resultant force on the cylinder and induce a larger vibration. To examine our hypothesis, we study the spanwise correlation of the cases we presented in the previous section. Without loss of generality, the flexible grooved cylinders with ps = 2 at Re = 4800 are chosen. The spanwise correlation between lift and drag forces is shown in Fig. 7.21. It is observed that the spanwise correlation of the staggered grooved flexible cylinder is significantly lower compared to the flexible plain cylinder counterpart. This suggests that the correlation of the hydrodynamic forces is reduced by the staggered alignment of the grooves. To further illustrate the lower correlation observed in the staggered groove, the normalized frequency spectrum of the sectional hydrodynamic forces is plotted in Fig. 7.22, and the vortex distribution is shown in Fig. 7.23. It is shown in Fig. 7.22 that the staggered grooved cylinder has a broader spectrum in the frequency of the forces, which is consistent with the result obtained for the rigid cylinder in our previous study [227]. The broader spectrum implies that the spanwise correlations will be lower, which is also suggested by the segmented vortex observed in the wake of the staggered groove in Fig. 7.23. This agrees with our hypothesis: Due to the jump pattern introduced in the staggered alignment of grooves, the vortex shedding is de-synchronized in the spanwise direction. This causes the broadening of the frequency spectrum of the sectional hydrodynamics forces, thereby reducing the spanwise correlation and resultant force acting on the cylinder, which in turn suppresses the vortex-induced vibration and the drag force.

7.3 Flexible Cylinder with Spanwise Grooves

427

(a) Lift coefficient

(b) Drag coefficient Fig. 7.21 Spanwise correlation of lift and drag coefficients for flexible cylinder measured at interval of Δz ∗ = 0.2. The reference position for correlation computation is located at z ∗ = 6

7.3.4 Summary In this work, the effectiveness of our proposed novel VIV suppression device, termed as staggered grooved, has been numerically investigated for flexible long cylinders at moderate Reynolds number. Stabilized finite element based methods is employed to discretize the coupled partial differential equations that arise from the mathematical modelling of nonlinear fluid-structure interaction. The proposed groove-based device successfully reduces the cross-flow VIV by de-correlating the spanwise hydrodynamic force and broadening the frequency spectrum [227]. Due to its simplicity and the ease of installation, the staggered groove is well suited to offshore structures, especially deep-water riser with and without buoyancy modules. The characteristic responses of the staggered grooved cylinder has been investigated at Reynolds number ranges Re ∈ [3000−9500], which showed reductions up to 37% in the cross-flow

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7 Flexible Cylinder VIV

(a) Flexible plain cylinder

(b) Flexible staggered groove

Fig. 7.22 Frequency spectra of sectional lift (left) and drag (right) coefficient measured at interval of Δz ∗ = 0.2 along spanwise direction. The spectra are normalized at each Δz ∗ . The reference position for correlation computation is located at z ∗ = 6

amplitude and 25% in the drag force. The effect of the staggered pitch ps was also investigated by simulating staggered grooved cylinder for ps = 1D, 2D and 4D, wherein ps = 4D was found to have an improved performance in the VIV suppression compared to others. An application of staggered groove on a flexible cylinder was also performed at moderate Reynolds number, which demonstrated the VIV reduction of 43% in the cross-flow amplitude and 22% reduction in the drag force. Based on these findings, the proposed staggered groove concept offers a practical alternative for the suppression of VIV while reducing the drag force. With regard to the application of a full-scale offshore riser in the ocean environment, further numerical or experimental tests can be explored to assess the performance of staggered groove at high Reynolds numbers (e.g., Re > 100,000). In addition, a detailed parametric optimization of the staggered groove will be useful for the improved effectiveness of the VIV suppression and the drag reduction.

7.3 Flexible Cylinder with Spanwise Grooves

Contiguous spanwise roll

(a) Flexible plain cylinder

429

Segmented spanwise roll

(b) Flexible staggered groove

Fig. 7.23 Instantaneous isosurfaces of Q-criterion at Q = 1 around flexible cylinder and grooves at Re = 4800 with spanwise length L = 18D. The isosurfaces are colored by z-vorticity

Acknowledgements Some parts of this Chapter have been taken from the Ph.D. thesis of Vaibhav Joshi carried out at the National University of Singapore and supported by the Ministry of Education, Singapore.

Part II

Model Reduction and Control

Chapter 8

Data-Driven Reduced Order Models

In this chapter, we present the low-dimensional and data-driven analysis of unsteady fluid flow and fluid-structure interaction. Using the high-fidelity data from the fullorder FSI model, we analyze the underlying dynamics using proper orthogonal decomposition (POD) techniques. The full-order data of flow past the square cylinder is decomposed into large-scale flow features using POD techniques. The energy cascade, nonlinearity capturing and reconstruction quality of the POD methods are examined. The low-dimensional projection of the data is used to explain the synchronization mechanism of the FSI system. This mechanism is further explored in the subcritical Re flows and moderately high Re turbulent flows as well.

8.1 Introduction Unsteady flows involving fluid-structure interactions are widespread in numerous engineering applications and their fundamental understanding poses serious challenges due to the richness and complexity of nonlinear coupled physics. Even a simple configuration of a coupled fluid-structure system can exhibit complex spatialtemporal dynamics and synchronization as functions of physical parameters and geometric variations. Synchronization is a general nonlinear physical phenomenon in fluid-structure systems whereby the coupled system has an intrinsic ability to lock to a preferred frequency and amplitude. For example, the phenomenon of frequency lock-in is a major concern in offshore, marine and aeronautical engineering, whereby structures are designed to avoid the large-amplitude vibrations by selecting optimal system parameters and/or installing active and passive devices to control the intensity of fluid-structure interaction. Apart from the synchronization, the turbulent motion of fluid flow poses a serious challenge owing to its strongly non-linear and multiscale character with random spatial-temporal fluctuations of broad range scales. New tools and techniques for active surface morphing in highly unsteady turbulent flows © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Jaiman et al., Mechanics of Flow-Induced Vibration, https://doi.org/10.1007/978-981-19-8578-2_8

433

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8 Data-Driven Reduced Order Models

are useful for future designs of micro-air- and unmanned aerial vehicles, efficient locomotive systems, large-scale wind turbines and various flow control devices. As discussed earlier, when a bluff body immersed in a cross-flow is flexible or mounted elastically, there exists a strong coupling between the bluff body and the vortices forming in its wake. In particular, as the natural frequency ( f n ) of the bluff body approaches the frequency of the wake system, typically the frequency of vortex shedding ( f vs ), the wake-body frequency lock-in behavior is observed which plays a crucial role in establishing the synchronization. During this frequency lockin, the bluff body experiences a large self-limiting vibration [206] and a dynamical equilibrium between the energy transfer and dissipation exists. This wake-body synchronization has been a major topic of research to understand the mechanism of this energy transfer and the sustenance of self-excited vibrations. In the present study, we consider a prismatic square geometry to understand the wake-body synchronization and to perform the decomposition of wake dynamics during the fluid-structure interaction.

8.1.1 Synchronization of Coupled Fluid-Structure System The phenomenon of frequency lock-in is a major concern in offshore, marine and aeronautical engineering, whereby structures are designed to avoid the largeamplitude vibrations by selecting optimal system parameters (e.g., geometric dimensions, stiffness, damping) and/or installing active and passive devices to control the intensity of fluid-structure interaction. In particular, several studies have been conducted to control the wake-body interaction via passive and active devices [147, 226, 309] with the physical insight based on the reliance of frequency lock-in on the largescale features of the wake. These studies were found to be remarkably successful in suppressing large-amplitude motion of the body by avoiding the interaction between the major organized features of the wake. However, the mechanism of the interactions among the wake features and their impact on the free movement of the bluff body is not properly explained. Moreover, the available experimental and numerical data can be used to provide a deeper understanding and a new insight into the kinematics and dynamics of synchronized wake-body interaction. This chapter aims at explaining how different organized flow features (i.e., near-wake structures) amplify the bluff body motion and sustain the energy transfer from the fluid flow to the vibrating body. Specifically, we examine the formation of the dominant coherent structures and their nonlinear interactions during the wake-body synchronization. The vortex shedding pattern is undoubtedly the most prominent wake feature behind a bluff body. It is omnipresent in almost all of the separated wake flows and has been studied extensively in the literature. This primary wake feature begins at a much lower Re: for example in a circular cylinder wake, at Re ≈ 49, exhibits a classical Kármán vortex street and develops the three-dimensional vorticity patterns when Re  190. In addition to the vortex street, a free shear layer (not attached to a solid surface) is an important dynamical feature that represents a separating

8.1 Introduction

435

high-gradient layer behind a bluff body and it arises between the higher free stream velocity and the smaller velocity occurring in the wake region. The shear layer behaves like a perturbed vortex sheet and is highly sensitive and unstable to small disturbances, giving rise to alternating thickening and thinning of the vortex sheet. The characteristic vortex structures develop when the thickening of the shear layer occurs. For the unsteady 2D regime (49 ≤ Re ≤ 190 for a circular cylinder) the roll-up of the shear layers with the formation of the vortex street can be observed [450]. These shear layers are predominantly elongated in the streamwise direction and have a high gradient in the cross-flow direction. Behind a moving or stationary bluff body, the region of a recirculating region with the rotational flow is present due to the fluid viscosity. Owing to nonlinear flow separation and turbulence, complex interactions occur in the mean recirculation region, which is also referred to as the near-wake bubble. Several previous studies have explored the dynamical features inside the wake region (with the vortex shedding and the shear layer) using experimental [72], numerical [62] and both [38, 108] techniques. In our present analysis, we consider the near-wake bubble as a distinct feature from the vortex street and the shear layer. The near-wake region accounts for the complex interactions of the mean circulation region, which can be considered a general feature and can be identified separately from the other two features. Hence, we divide the wake into three dominant organized coherent structures: the vortex street, the shear layer, and the near-wake bubble. These organized features have intrinsic dynamics of their own and influence each other in a nonlinear manner over a wide range of space and time scales. A primary goal of this chapter is to employ low-dimensional models to extract the organized wake features and to examine their roles during wake-body synchronization.

8.1.2 Low-Dimensional Models for Wake Features To extract the large-scale organized/coherent wake features, it is required to decompose the dynamic flow fields by scales into different constituent kinematical regions. The concept of decomposition by scale has been prevalent in many fluid dynamics research ranging from a low-dimensional projection of flow field to turbulence modeling by ensemble averaging, temporal or spatial averaging. A general decomposition technique can be considered to separate the space-time data for representing different characteristics of the field. For example, the proper orthogonal decomposition (POD) extracts the most energetic modes in an optimal way and provides structural information from the wake data. The POD is a popular method for constructing low-order modeling from the data [158], and it is often referred to as the Karhunen-LoJeve expansion or the principal component analysis. The key idea behind the Karhunen-Loeve ` expansion is to determine a low-dimensional affine subspace from the high-dimensional data while retaining the important dynamics of the fullorder model. After the determination of the best approximating low-dimensional subspace, a Galerkin projection is employed to project the dynamics onto it. In

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8 Data-Driven Reduced Order Models

this work, we will employ this low-dimensional subspace projection procedure for extracting the large-scale wake features from the high-dimensional flow dynamics data. In the context of the present study, the POD-Galerkin projection method is quite attractive to capture the synchronized dynamics such as the vortex shedding and the near-wake interactions [316, 358]. In addition, it has been the prominent empirical model reduction technique incorporated for the standard flow around a stationary circular cylinder for the past few decades. For example, in one pioneering study, [104] reproduced the flow dynamics of the laminar wake by employing merely an eight-dimensional model, which was further generalized to generate reduced spaces for the 3D velocity field by [270] using direct numerical simulation (DNS) data. In general, the empirical POD-Galerkin models are capable of reconstructing the reference dynamics with higher accuracy than the standalone mathematical or physical Galerkin methods, while capturing the physically most significant modes [316]. With regard to the applications of the POD-Galerkin to bluff body wake flows, these modes correspond to the organized wake features such as the vortex street, the shear layer and the near-wake bubble. Although there exists a significant body of work on the wake modes for a stationary cylinder, they have not been examined in the context of wake-body interaction and the lock-in process. One of the contributions of the present study is to build some connections between the wake features and the lock-in process. During the lock-in/synchronization, the vibrating body undergoes a highly nonlinear-wake interaction with self-sustained oscillations. In the early studies of POD application to fluid flows [263, 394], the dynamic flow field is reconstructed by a linear combination of the most significant modes. Hence, it has a considerable local error in the highly nonlinear regions of the organized wake motions and the evaluation of the projected nonlinear term has a direct dependence on the large dimension of the original system. This problem is mitigated to a certain extent by increasing the sampling frequency and/or refining the spatial discretization of the reference data. However, these temporal and spatial refinements increase the cost of model reduction without directly addressing the nonlinear nature of the flow. To introduce the nonlinearity, Petrov-Galerkin projections to the Navier-Stokes formulation or Koopman operators are incorporated in some studies [367]. Instead of such explicit models, we employ the recently developed discrete empirical interpolation method (DEIM) [81] for dynamical systems, which reconstructs the fields as a nonlinear combination of the POD modes. Apart from the POD basis subspace, the method relies on the additional POD basis to enrich the low-rank approximation of the nonlinear terms. In the POD-DEIM, a set of best points are selected using a greedy selection and the reconstruction is based on the time history of the field data of those points. This reduces the computational cost of the technique and further allows to capture of nonlinearities during the reconstruction of highly nonlinear dynamic wake fields [367].

8.1 Introduction

437

8.1.3 Objectives and Organization For the past few decades, the studies on the low-dimensional decomposition of wake features have been primarily focused on flow past stationary bodies, particularly on a circular cylinder [104, 316, 367, 408]. This may be due to the fact that the flow exhibits a diverse set of complex phenomena despite its simple geometry. However, very few studies [245, 466] are found on unsteady fluid-structure interaction systems. Here, we provide a modal reduction study on the flow past a freely vibrating sharpcornered square cylinder with two-degree-of-freedom motions. We consider a configuration of a square cylinder for our numerical study of wakebody synchronization because: (i) this configuration has fixed and perfectly symmetric separation points at the leading sharp corners, (ii) entirely resonance-induced lock-in exists [466]. The physical investigation is general for any fluid-structure system involving the interaction dynamics of flexible structures with an unsteady wake-vortex system. We hypothesize that the solution space of wake-body interaction attracts a low-dimensional manifold, which allows for building a set of basis vectors for a low-dimensional representation of the high-dimensional space. The low-dimensional subspace is constructed by means of the samples collected from the high-dimensional solutions via projection-based model order reduction. We utilize linear and nonlinear POD-based reduced-order reconstructions to understand the most significant features in the wake flow. To extract the modes for the dynamics of wake-body synchronization, the POD in conjunction with the nonlinear POD-DEIM is applied to a set of samples collected from the full-order simulations. We exploit the obtained POD modes to answer the following intriguing questions that are prevalent in the field of fluid mechanics: (i) How do the large-scale features contribute to the unsteady forces acting on the bluff body? (ii) How do the wake features interact when the structural frequency and vortex shedding frequency are locked-in, such that the vortices remain very energetic even though the fluid has transferred energy to the structure? (iii) Will the wake and bluff body undergo synchronized motion below the critical Re due to the structural flexibility? (iv) What role does the wake turbulence play when we attempt to decompose the wake into its large-scale features? In relation to the first question, we quantify the force contribution from each wake feature mode to the streamwise (drag) and transverse (lift) forces and explain the observed variation. We further investigate the modal contribution of different wake features in the pre-lock-in, lock-in and post-lock-in regimes and propose a cycle explaining the sustenance of lock-in phenomena of the wake-body synchronization. We then explore the below critical Re flows to examine whether the bluff body and the wake can undergo synchronization via flexibility-induced unsteadiness. Finally, we apply POD decomposition to the three-dimensional flow at moderate Re = 22,000, whereas the wake is fully turbulent after flow separation. A well-established dynamic large-eddy simulation (LES) is employed for generating full-order data for the turbulent wake. At this sub-critical Reynolds number, we explore the role of turbulence during the reconstruction of flow-field data and extend the wake-body synchronization cycle to the turbulent flow.

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8 Data-Driven Reduced Order Models

The chapter is structured as follows. In Sect. 8.2, we briefly review the fullorder model (FOM) for the coupled fluid-structure system, which follows by the formulation of modal reduction via linear POD and nonlinear POD-DEIM. Section 8.2.3 discusses the problem setup and the mesh convergence study performed for the full-order analysis. In Sect. 8.3, the reduced order reconstruction of fluid fields using the linear and nonlinear POD methods is presented together with the analysis of the role of wake features in generating the forces. The section follows by the mode energy contributions from different flow features under lock-in conditions are investigated and a self-sustaining cycle is proposed to explain the wake interaction with the bluff body. We further investigate the wake-body synchronization phenomenon at below critical Re and the application of modal decomposition for moderate-Re flows.

8.2 Numerical Methodology We first briefly summarize our high-dimensional full-order model to simulate the coupled fluid-body interaction using the incompressible Navier-Stokes equations and the rigid body dynamics.

8.2.1 Full-Order Model for Fluid-Body Interaction We employ a variational formulation based on the arbitrary Lagrangian-Eulerian (ALE) to solve the following coupled fluid-body system ρf

  ∂uf + ρ f uf − w · ∇uf = ∇ · σ f + bf on Ω f (t) ∂t ∇ · uf = 0 on Ω f (t) ∂us + Cus + K (ϕ s (z0 , t) − z0 ) = Fs on Ω s M ∂t

(8.1) (8.2) (8.3)

where superscripts f and s denotes the fluid and structural domains, and Ω f (t) and Ω s represent the fluid and solid domains, respectively. Here ρ f is the fluid density, uf and w are the fluid and mesh velocities at a spatial point x ∈ Ω f (t), and bf denotes the body force in the fluid domain. For the structural system, M, C and K are the mass, damping and stiffness matrices of the bluff body and Fs is the external force acting on the body. The function ϕ s (z0 , t) maps the initial position vector of the center of mass (z0 ) to its position at time t, and σ f is the Cauchy stress tensor for a Newtonian fluid given by:  T   , σ f = − pI + μf ∇uf + ∇uf

(8.4)

8.2 Numerical Methodology

439

where p is the fluid pressure. In addition to the initial conditions and the standard Neumann/Dirichlet conditions, the coupled system incorporates the velocity and traction continuity conditions at the fluid-body interface Γ as follows:  Γ (t)

uf (t) = us (t),

(8.5)

σ f (x, t) · n dΓ + Fs = 0,

(8.6)

where n is the outer normal to the fluid-body interface. The above fluid-body interface conditions are satisfied by the body-conforming Eulerian-Lagrangian treatment, which provides accurate modeling of the boundary layer and the vorticity generation over a moving body. While Eqs. (8.1–8.3) of the coupled fluid-body system are directly solved for low-Re flows, we consider the well-established dynamic subgrid-scale model for high-Re turbulent flow. The spatially-filtered Navier-Stokes and continuity equations are solved in the variational form. Details of the dynamic subgrid-scale model are provided in [180]. The weak variational form of Eq. (8.1) is discretized in space using equal-order iso-parametric finite elements for the fluid velocity and pressure. In the present study, we utilize the nonlinear partitioned staggered procedure for the full-order simulations of fluid-structure interaction [181]. The motion of structure is driven by the traction forces exerted by the fluid flow, whereby the structural motion predicts the new interface position and the geometry changes for the moving fluid domain at each time step. The movements of the internal ALE fluid nodes are updated such that the mesh quality does not deteriorate as the motion of solid structure becomes large. To extract the transient flow characteristics, we solve the Navier-Stokes equations at discrete timesteps which lead to a sequence of linear systems of equations via NewtonRaphson type iterations. We employ the Conjugate Gradient (CG), with a diagonal preconditioner for the symmetric matrix arising from the pressure projection and the standard Generalized Minimal Residual (GMRES) solver based on the modified Gram-Schmidt orthogonalization for the non-symmetric velocity-pressure matrix. The above coupled variational formulation completes the presentation of the fullorder model for the fluid-structure interaction. From a model reduction viewpoint, the coupled system of the nonlinear differential equations for the fluid-body interaction can be written in the following form: dy = F(y), dt

(8.7)

where y is the column state vector describing the unknown degrees of freedom and F is a vector-valued function describing the spatially discretized governing equations. In the present fluid-body system, the state vector comprises of the fluid velocity and the pressure as y = {uf , p} and the structural velocity involves the three translational degrees-of-freedom. For a discretized domain of m elements and n timesteps, the fullorder simulation outputs a high-fidelity data set y ∈ Rm×n×q , where q is the number

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8 Data-Driven Reduced Order Models

of variables in y. This dataset is extremely valuable to determine the instantaneous physics of the fluid-body system and to construct a low-order representation that preserves the behavior of the original system.

8.2.2 Low-Order Models We now turn to the data-driven model reduction technique whose goal is to decompose the aforementioned high-dimensional data set into a set of low-dimensional modes. For that purpose, we can consider the decomposition of the nonlinear mapping F of Eq. (8.7) as (8.8) F(y) = f + Ay + F (y), where f denotes a constant column vector with m rows, A and F are the linear and nonlinear terms. For the ease of explanation, consider the solution vector y(x, t) ∈ Rm×1 comprising a single quantity of interest which have been determined at discretized locations x of the spatial domain and for a particular time t and the matrix operator A is an m × m matrix which captures the linear dynamics while F (y) is a nonlinear function of y. Using the projection-based model reduction, we can represent the state vector y by an element in a low-rank vector subspace spanned by the column vectors of an m × k matrix V = [v1 v2 ... vk ], where k  m. The state vector y can be approximated by V yˆ , where yˆ is a reduced column vector with k entries. Since the columns of V are orthonormal (i.e., V T V = I), via the Galerkin projection onto the basis V , we get the following reduced dynamics: d yˆ = V T f + V T AV yˆ + V T F (V yˆ ). dt

(8.9)

Next, we have to choose a suitable subspace for the mode decomposition. Using the reduced singular value decomposition (SVD), the above state vector y can be expressed as k  Y = V ΣW T = σ j v j w Tj , (8.10) j=1

where the vectors v j are the POD modes of the matrix Y with rank k, W is an orthonormal matrix with n × k , and Σ is a k × k diagonal matrix with diagonal entries σ1 ≥ σ2 ≥ ... ≥ σk ≥ 0. For any r ≤ k, the subspace spanned by {v1 , ..., vr } provides an optimal representation of y in the subspace of dimension r using the SVD process. The total energy contained in each POD mode v j can be computed by the singular value σ j2 . Note that V and W are the orthonormal eigenvectors of YYT and YT Y, respectively.

8.2 Numerical Methodology

8.2.2.1

441

Proper Orthogonal Decomposition

The POD method provides an algorithm to decompose a set of data into a minimal number of modes. We give a brief outline of this projection-based model reduction for the dynamical analysis of wake-body interaction. The general POD algorithm can be expressed as follows. Here, the eigenvectors of YYT are determined instead of performing the SVD. The algorithm adopted from [408] is summarized in Algorithm 1. The standard linear POD is almost in the same order expensive as the full-order ˜Y ˜ T matrix which has the size of m × m. In a typical analysis since it is using the Y time-dependent flow analysis, it is unnecessary to generate m POD modes for comparison as the POD mode energy decays exponentially. Hence an alternative method the so called snapshot POD [394] is applied to extract the most significant modes. ˜ ∈ Rk×k ˜ TY In the snapshot method, the eigenvalue decomposition is performed on Y T ˜ ˜ which is significantly smaller than YY as k  m. Let the eigenvalues and eigen˜ be given by, ˜ TY vectors of Y Algorithm 1: Snapshot POD Input: Snapshots of spatial field expressed as Y(x, t) where Y ∈ Rm×k (m number of spatial points, k - number of snapshots.) Output: Significant r POD modes V = [v1 , v2 , ..., vr ] ˜ 1. Develop the fluctuation matrix by subtracting the mean: Y(t) = Y(x, t) − Y(x) ˜Y ˜ T ∈ Rm×m 2. Construct the covariance matrix R = Y 3. Find the eigenvalues and eigenvectors of R by RV = ΛV   4. Determine the number of required POD modes (r ) using rj=1 λ j / mj=1 λ j ≈ 1.0, where λ j are the eigenvalues given by Λ. ˜ T YW ˜ Y = ΛW ,

(8.11)

˜ a maximum ˜ T Y, ˜Y ˜ T and Y then using the relationship between the eigenvectors of Y of k significant POD modes can be extracted by ˜ Λ−1/2 . V = YW

(8.12)

Throughout the study, every POD decomposition will be performed via the snapshot POD method due to its low computational cost and memory usage. After extracting the significant POD modes, the constant and linear components of the instantaneous state vector can be recovered as a linear combination of the identified significant modes r  yˆ j (t)v j , (8.13) Y(x, t) ≈ Y(x) + j=1

442

8 Data-Driven Reduced Order Models

where r is the number of significant POD modes. The temporal coefficients of the linear combination are determined by the L 2 inner product  . , . between the fluctuation matrix and the modes as follows ˆ Y(t) = Y − Y, V .

(8.14)

This summarizes the process of POD by performing the SVD on the snapshots of the sampled solutions at certain timesteps. While the above POD-Galerkin process can reconstruct the linear term to the expected error threshold, the nonlinear term will not be reconstructed properly in the context of nonlinear incompressible flow which involves quadratic nonlinearity. The linear POD reconstruction requires a higher number of modes and/or a smaller sampling interval for snapshots to obtain the required local domain accuracy. In other words, the spatial and/or temporal discretizations of the POD method have to be so small that the nonlinearities behave almost linearly. Subsequently, the POD reconstruction may result in a similar order of computational expense as full-order simulation. This issue can be handled by employing the discrete empirical interpolation method (DEIM). The DEIM introduces nonlinearity by supplementing an additional basis for a low-order representation of nonlinear terms. This gives rise to the reduction in the requirement of POD modes, hence decreasing the computational cost while capturing the nonlinear regions properly.

8.2.2.2

Discrete Empirical Interpolation Method

To overcome the difficulty in the linear POD, [81] proposed the discrete empirical interpolation method to reconstruct the full-order variable as a nonlinear combination of the POD modes. The aim of DEIM is to design a low-order representation for the nonlinear terms by introducing an additional basis. Consider U as a basis generated from the leading l modes of the POD, which is attracted to a low-dimensional subspace. We can approximate the nonlinear term in Eq. (8.8) by the sequence of nonlinear snapshots as F (V yˆ (t)) ≈ U cˆ . The coefficients cˆ can be selected based on Algorithm 2, which relies on a greedy approximation of nonlinear function. In Algorithm 2, ρˆ and ℘1 denote the assigned value and the assigned index of max{|v1 |}, and e℘i = [0, ..., 0, 1, 0, ...0]T ∈ Rm is the ℘i th column of the identity matrix of size m × m. The accuracy of DEIM approximation depends on the error induced by the POD projection and the estimation of (P T U )−1 . Further details of the DEIM process can be found in [81].

8.2 Numerical Methodology

443

Algorithm 2: POD-DEIM Output: Indices of l best points ℘ = [℘1 , ℘2 , ...., ℘l ]T Input: Most significant l POD modes [v1 , v2 , ..., vl ] 1. [ρˆ ℘1 ] = max{|v1 |} 2. U = [v1 ], P = e℘1 , ℘ = [℘1 ] 3. for i=2 to l do a. b. c. d.

Solve (P T U )ˆc = P T vi for cˆ Compute residual rˆ = vi − U cˆ Assign [ρˆ ℘i ] = max{|ˆr|} Augment U ← [U vi ], P ← [P e℘i ], ℘ ← [℘ ℘i ]T

4. end for Here, a set of entries ℘ ⊂ {1, 2, ..., l} often called optimal (best) points are selected to determine cˆ by the following relation cˆ = (P T U )−1 P T F (V yˆ (t)).

(8.15)

Assuming that F is a component-wise function P T F (V yˆ (t)) = F (P T V yˆ (t)), we can rewrite Eq. (8.7) as d yˆ (t) = (V T AV )ˆy(t) + V T U (P T U )−1 F (P T V yˆ (t)). dt

(8.16)

In the present work, we perform the nonlinear POD on the same fluctuation matrix y˜ m×k without separating the linear and nonlinear components. Consider the approximation of y˜ as a nonlinear combination of the POD modes: y˜ (t) ≈ V θ (t).

(8.17)

The coefficients θ (t) are calculated by the conditions imposed by the POD-DEIM. While the POD modes are linearly independent, we can obtain a unique number of DEIM points if P T U matrix is invertible. By using just the ℘ rows of V and U , we can establish the following relationship:

which further gives

V ℘ θ(t) = y˜ ℘ (t)

(8.18)

˜℘. y˜ (t) ≈ V V −1 ℘ y

(8.19)

If the number of points used is higher than the number of significant modes, i.e. l > r , which is often the case, V ℘ becomes a rectangular matrix. This makes the coefficients θ (t) given by the gappy POD reconstruction .

(8.20)

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8 Data-Driven Reduced Order Models

The solution to the least square problem (Eq. 8.20) gives the result ˜℘, y˜ (t) ≈ V V + ℘y

(8.21)

where V + ℘ is the Moore-Penrose pseudoinverse of V ℘ . The POD-DEIM provides a way to introduce nonlinearity to the POD reconstructions, however, due to this nonlinear behavior, it is not guaranteed to converge to the full-order results. In other words, the use of more POD modes or DEIM points does not assure an improvement in the result. Therefore, determining the optimal sizing of the low-dimensional representation is critical when using POD-DEIM for reconstruction. In the next section, we present the full-order model for generating high-dimensional data.

8.2.3 Problem Setup In this section, we give an overview of full-order simulations for a freely vibrating structure immersed in a viscous incompressible fluid flow. Specifically, the focus of this section is to present numerical results on the flow past an elastically mounted square cylinder, whereby the cylinder is free to oscillate in the streamwise (X ) and the transverse (Y ) directions. The mass and natural frequencies are identical in both X and Y -directions. The translational flow-induced vibration of a cylinder is strongly influenced by the four key non-dimensional parameters, namely mass-ratio (m ∗ ), Reynolds number (Re), reduced velocity (Ur ), and critical damping ratio (ζ ) defined as m∗ =

M , mf

Re =

ρ f U∞ D , μf

Ur =

U∞ , fn D

C ζ = √ , 2 KM

(8.22)

where M is the mass of the body , C and K are the damping and stiffness coefficients, respectively for an equivalent spring-mass-damper system of a vibrating structure, U∞ and D denote the free-stream speed and the diameter of √ cylinder, respectively. The natural frequency of the body is given by f n = (1/2π ) K /M and the mass of displaced fluid by the structure is m f = ρ f D 2 L c for a square cross-section, and L c denotes the span of the cylinder. In the above definitions, we make the isotropic assumption for the translational motion of the rigid body, i.e., the mass vector M = (m x , m y ) with m x = m y = M, the damping vector C = (cx , c y ) with cx = c y = C, the stiffness vector K = (k x , k y ) with k x = k y = K . The fluid loading is computed by integrating the surface traction considering the first layer of elements located on the cylinder surface. The instantaneous lift and drag force coefficients are evaluated as

8.2 Numerical Methodology

445

(b)

(a)

Fig. 8.1 Full-order problem setup for fluid-structure interaction: a schematic diagram of the computational domain and boundary conditions, and b representative Z -plane slice of the unstructured mesh. The top right inset displays the near cylinder mesh

 1 CL = 1 f 2 (σ f .n).n y dΓ, ρ U∞ DL c Γ 2  1 CD = 1 f 2 (σ f .n).nx dΓ. ρ U DL Γ c ∞ 2

(8.23) (8.24)

Here nx and n y are the Cartesian components of the unit outward normal n. In this study, we focus on the lift and drag forces due to the pressure field. Hence, we evaluate the pressure-induced drag (C Dp ) and lift (C L p ) forces given by: C Dp =

1 1 f 2 ρ U∞ D 2

 Γ

(σ p · n) · nx dΓ, C L p =

1 1 f 2 ρ U∞ D 2

 Γ

(σ p · n) · n y dΓ, (8.25)

Figure 8.1a illustrates a schematic of the two-dimensional simulation domain used for the fluid-body interaction problem. The center of the square column is located at the origin of the Cartesian coordinate system. The side length of the square column is denoted as D. The distances to the upstream and the downstream boundaries are 20D and 40D, respectively. The distance between the side walls is 40D, which corresponds to a blockage of 2.5%. The flow velocity U∞ is set to unity at the inlet and a no-slip wall is implemented at the surface of the square column. While the top and bottom boundaries are defined as slip walls, the computational domain is assumed to be periodic in the spanwise direction for the 3D simulations.

8.2.3.1

Mesh Convergence Study

For the high-dimensional approximation of the full-order model, the computational domain is discretized using an unstructured finite-element mesh, wherein a boundary layer mesh surrounding the body and three-node triangle (2D) and six-node wedge

446

8 Data-Driven Reduced Order Models

Table 8.1 Grid convergence study at Re = 100, m ∗ = 3 and Ur = 5.0 M1 M2 M3 Number of nodes Number of elements Time-step size Δt Shedding frequency f / f n rms amplitude Aryms /D Mean drag C D rms lift C Lrms

17,622 17,389 0.025 0.9798 0.097 (56.0%) 1.623 (24.1%) 0.485 (29.7%)

34,302 34,027 0.025 0.9798 0.200 (9.7%) 1.994 (6.7%) 0.604 (12.4%)

87,120 86,631 0.025 0.9798 0.220 (0.72%) 2.134 (0.18%) 0.687 (0.30%)

M4 145,608 145,195 0.025 0.9798 0.2211 2.1377 0.6893

(3D) elements outside the boundary layer region. Three more grids are generated where the mesh elements are successively increased by approximately a factor of 2, designated as M2, M3 and M4. The discretized domain, along with a close-up view of the corners of the square column is illustrated in Fig. 8.1b. Results of the grid convergence study are recorded in Table 8.1 for the lock-in region. All cases for the mesh convergence are simulated at Re = 100, m ∗ = 3 and Ur = 5.0. The mesh convergence error is computed by considering the finest mesh M4 as the reference case. The force coefficients, the shedding frequency and the root mean square (rms) of the transverse amplitude are analyzed. It can be seen that values recorded for mesh M3 and M4 differ by less than 1%. Therefore, the mesh M3 is adequate for the present study. Furthermore, the adopted full-order solver and the numerical discretizations have been extensively validated in several earlier studies for both low-Re [180, 297] and moderate-Re [180, 295] flows. In the next section, the modal decomposition of the pressure field is presented for a representative reduced velocity of Ur = 6.0 in the lock-in region at (Re, m ∗ , ζ ) = (100, 3.0, 0). The snapshots of the FOM performed for the flow past a vibrating square cylinder are utilized to recover the POD modes and the DEIM points. The accuracy of the linear POD and POD-DEIM is systematically assessed with regard to their effectiveness to extract the flow features.

8.3 Assessment of Low-Order Model for Wake Decomposition As described earlier, we incorporate the snapshot POD method described to obtain the low-dimensional decomposition of the wake dynamics. As found in [295], the laminar bluff body flow involves simply a few significant features. It will be ineffective to generate the entire set of POD modes, e.g., the order of the mesh points of 87,120 for this particular problem. Hence, we use the snapshot POD technique and obtain just the most significant POD modes, which are a few orders of magnitude smaller. We reconstruct the pressure field using linear and nonlinear techniques and compare their effectiveness to capture the organized wake features. In the present analysis,

8.3 Assessment of Low-Order Model for Wake Decomposition

(a)

447

(b)

Fig. 8.2 Distribution of modal energy for a laminar flow past a freely vibrating square cylinder: a energy decay of POD modes, and b cumulative energy of POD modes

the unsteady pressure field values for all the mesh points, are collected to a m × k matrix P where m (mesh count) = 87, 120 and k (number of snapshots) = 320. The snapshots are sampled at every 50 time steps, i.e at 1.25D/U∞ intervals (sampling frequency = 0.8U∞ /D). Further details about the determination of the sampling frequency and the adequate number of samples are presented in appendix A. The fluctuation matrix y˜ m×k is then generated by subtracting the mean value (P) of each ˜ = P − P. point over the snapshots Y The POD modes are extracted using the eigenvalues Λk×k = diag[λ1 , λ2 , ..., λk ] ˜ ∈ Rk×k given by ˜ TY and eigenvectors W = [w1 w2 ... wk ] of the covariance matrix Y T ˜ ˜ Y YW = ΛW . As presented earlier, the POD modes V = [v1 v2 ... vk ] are related ˜ Λ−1/2 . Each eigenvalue represents the energy/strength of to Λ and W by V = YW the POD mode. Since the mean pressure distribution is initially removed from the pressure field, the relative strength of the mode directly expresses the contribution from each mode to the pressure fluctuations. Figure 8.2a displays the energy of these modes normalized by the total energy of the 320 modes obtained. It is clear that this energy decays exponentially and the most energetic mode has 56% of the total energy. In fact, the first 9 most significant modes contain 99% of the total energy of the modes, as shown in Fig. 8.2b. Initially, these 9 significant modes are used to recover the pressure field in the linear POD reconstruction. We refer to these modes as mode 1, mode 2, etc. and they are in the descending order of mode energy (λi ). We first incorporate the linear reconstruction method wherein we assume the final flow field is a linear combination of the flow features captured by the POD modes.

448

8 Data-Driven Reduced Order Models

Fig. 8.3 The mean field and the first 9 significant POD modes. The energy fraction of the POD 2 . The flow is from left to mode is mentioned in brackets. The values are normalized by 1/2ρ f U∞ right

8.3.1 Linear POD Reconstruction In the linear POD reconstruction method, the instantaneous pressure field is recovered by the mean and a linear combination of the identified significant modes. In this analysis, r is set to 9, which represents the most energetic modes containing ∼ 99% of the total contribution to the pressure fluctuations. The temporal coefficients yˆ j are determined by the L 2 inner product between the fluctuation matrix and the modes as expressed in Eq. (8.14). The mean pressure distribution and the first 9 POD modes

8.3 Assessment of Low-Order Model for Wake Decomposition

449

Fig. 8.4 Absolute value of the time invariant contribution from each mode to the in-line(|Fxi |) and cross-flow(|Fyi |) forces. Modes 1, 3 and 9 capture the vortex shedding, modes 2 and 8 correspond to the effect of the shear layer and modes 4, 5, 6 and 7 capture the near-wake bubble effects

are displayed in Fig. 8.3. Note that, throughout this chapter, the time-independent fields such as the mean pressure field and POD modes correspond to the initial flow field with zero bluff body motion. The mean field is symmetric around the X -axis along the wake centerline. This is expected as the time-averaged distribution of the flow past a symmetrical bluff body should be symmetrical. Furthermore, modes 2, 4, 5, 6, 7 and 8 are symmetric around the wake centerline while modes 1, 3 and 9 are anti-symmetric with equal values and opposite signs about the wake centerline. As shown in Fig. 8.5, the POD time coefficients of these modes have the same frequency as the lift coefficient. It is evident that the first, third and ninth modes correspond predominantly to the Karman vortex street with alternating positive and negative pressure regions about the X -axis and the pressure contours resulting from a staggered vortex street. By examining Fig. 8.5, the time coefficients of the symmetric modes have the same frequency as the dragforce. Of these modes, the modes 2 and 8 have a high transverse gradient ∂∂y  ∂∂x behavior in the near-wake (0.5D − 5D) region almost parallel to the top and bottom edges of the square cylinder suggesting that this mode represents the influence of the shear layer. The modes 4, 5, 6 and 7 originate from the near-wake region and diffuse symmetrically towards the far   ∂ ∂ wake. These modes have a dominant streamwise gradient ∂ x  ∂ y compared to the transverse gradient. We can attribute these contributions to the near-wake bubble and its local dynamical property. For ease of explanation, we refer to these modes as vortex shedding, the shear layer and the near-wake. Figure 8.4 quantifies the time-invariant contributions (F ji ) from each mode to the drag and lift forces. For the definition of F ji , j = (x, y) is the direction of the force and i is the mode number. These values are calculated based on the fluid-solid boundary values of the mode fields displayed in Fig. 8.3. It is clear that the vortex shedding modes (modes 1, 3 and 9) contribute entirely to the lift force, while the shear

450

8 Data-Driven Reduced Order Models

(a)

(b) Fig. 8.5 Time history and FFT comparisons: a temporal coefficients of first 5 POD modes, b force coefficients. Note that modes 2, 4 and 5 have the same frequency (2 f n ) as the drag and modes 1 and 3 has the same frequency ( f n ) as the lift

layer and the near-wake modes contribute entirely to the drag force. Further details on the force decomposition procedure using the modal contributions are presented in Appendix B. Due to this directionally independent contribution of the bluff body features for the forces, the time coefficients ( yˆ j (t)) of these modes should display the same frequencies of the lift and drag forces. The time histories and the FFT spectra of the first 5 POD modes are shown in Fig. 8.5a. The first and third mode coefficients have a low-frequency sinusoidal variation with the natural frequency ( f n ). The second, fourth and fifth mode coefficients have a non-zero mean with a frequency ≈ 2 f n . Interestingly, as presented in Fig. 8.5b,

8.3 Assessment of Low-Order Model for Wake Decomposition

451

f n and 2 f n coincide with the frequencies of lift and drag, respectively. Hence, we can further confirm that modes 1 and 3 make their sole contribution to the fluctuating lift while modes 2, 4 and 5 contribute to the drag force. From these observations, we can further confirm that the vortex shedding process contributes exclusively to the lift force and the near-wake and the shear layer phenomena influence the drag force. Using the POD procedure, we successfully decompose the flow field into physically significant features. We reconstruct the same field by combining these modes in the linear POD technique: such that, utilizing Eq. (8.13) for the pressure field,  gives P(t) ≈ P + rj=1 yˆ j (t)v j . Figure 8.6 illustrates the pressure distribution at tU∞ /D = 100 using the linear POD reconstruction. The recovered POD mode is compared with the result obtained from the full-order model. A good match with a maximum relative local error < 2% can be seen in Fig. 8.6. To quantify the accuracy of the entire flow field recovery, the normalized root mean square (rms) error of the entire distribution is considered. The rms error εr ms is given by:  ε

r ms

=

(PF O M − PP O D )2 /n c |P100 |

× 100,

(8.26)

where PF O M and PP O D are the pressure values of the mesh nodes extracted from the full-order model and the POD reconstruction, respectively, n c is the node count of the mesh and |P100 | is the mean pressure of the field. When 9 modes are used, this error is εr ms = 3.78%. In this linear reconstruction, the highest error is observed at the regions known to exhibit a nonlinear variation, such as the near-wake region, the shear layer and the vortex cores. Next, we analyze the POD-DEIM technique to improve the accuracy in these nonlinear flow features using the snapshot sequence and their respective DEIM points.

8.3.2 Nonlinear POD-DEIM Reconstruction The linear POD reconstruction has the highest error in the nonlinear regions. To reduce this error, more POD modes should be added to the reconstruction which makes the POD-based reconstruction very expensive. Instead, when the DEIM technique is used, it reduces the calculation load while properly capturing the nonlinearity of the field variable. The DEIM utilizes two POD bases using the snapshot method, namely a first POD basis V from the snapshot sequence, and a second basis U from the nonlinear snapshots via the DEIM points. However, unlike the linear POD reconstruction, the accuracy does not necessarily improve with the number of DEIM points and the number of POD modes employed. Using many DEIM points results in adding contributions from some non-significant indices. In Fig. 8.7a, it is clearly seen that for 100 DEIM points there are few mesh points which lie away from the significant nonlinear region taken into the calculation. Further in Table 8.2, we quantify the number of points in the nonlinear-wake region as we increase the number of DEIM points. It is clear that the percentage of points in the

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8 Data-Driven Reduced Order Models

Fig. 8.6 Comparison between POD reconstruction and FOM result: pressure distribution obtained by a full-order model, b linear POD reconstruction, and c the relative error (%). Maximum error percentage is less than 2%. The highly nonlinear near-wake region and the vortex cores have the 2 . The flow is from left to right highest error. The pressure values are normalized by 21 ρ f U∞

8.3 Assessment of Low-Order Model for Wake Decomposition

453

2

100 Points 1

70 Points 0

50 Points -1

30 Points -2 0

2

4

6

8

10

12

Y/D

5

0

-5 -5

0

5

10

15

20

25

30

35

40

X/D (a)

(b)

Fig. 8.7 a DEIM best points, the top right inset illustrates the near-wake DEIM points. b Performance of POD-DEIM compared with linear POD, solid lines denote the linear POD error. The least error is observed when 70 DEIM points are used with 7 POD modes

critical region decreases as we include more points for the DEIM calculation. Owing to the nonlinear combination of the POD modes, including additional insignificant modes can increase the total error. As shown in Fig. 8.7b, the lowest εr ms = 4.05% can be obtained when 70 DEIM points are used with 7 POD modes. It further establishes that the linear POD reconstruction is generally accurate in a global sense, i.e. the entire flow field reconstruction, in contrast to the nonlinear POD-DEIM. However, Fig. 8.8 demonstrates the reconstructed pressure distribution at tU/D = 100

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8 Data-Driven Reduced Order Models

Table 8.2 Distribution of DEIM points in the near cylinder wake Number of DEIM points Number of points in near % cylinder wake 30 50 70 100

21 31 41 54

70.0 62.0 58.6 54.0

using the optimum number of points and POD modes. There is a significant reduction in the local error as it allows to capture the nonlinear regions more accurately. Apart from the global and local accuracy, we further assess the computational time consumed by the linear POD and nonlinear POD-DEIM reconstructions. The detailed analysis is presented in Appendix C. Theoretically, the DEIM reconstruction process should be ≈ 64 times faster than linear POD reconstruction and the total DEIM process should be ≈ 3.14 times faster. In the actual computations, when just the reconstructions are considered, the DEIM is 9.28 times faster than the linear POD. When the total processes are compared, DEIM has a speedup of 3.98. In terms of accuracy, DEIM is more accurate in a local sense since it captures the nonlinearities better than the linear POD reconstruction. However, when the entire fluid domain is considered, the linear POD method is more accurate than the DEIM. It is likely that the DEIM introduces unnecessary nonlinearities to the potential regions slightly changing the reconstructed field values. When decomposing and reconstructing the laminar flow fields, both linear and nonlinear methods perform to a satisfactory level. Both methods are capable of reaching the required threshold in a reasonable computational time while accurately capturing the flow features of the wake. Here onwards, we employ the POD-DEIM since it has an improved accuracy when capturing the nonlinearities in the flow field at a lower computational cost.

8.3.3 Drag and Lift Modes In this section, we analyze the behavior of different modes in the near cylinder region and explain the exclusive nature of their contributions to the pressure-induced drag and lift forces exerting on the oscillating cylinder in a uniform flow. Herein, the vortex shedding modes are referred to as the lift modes, while the shear layer and the near wake represent the drag modes. Figure 8.9 displays the combined variation of the lift modes during a single cycle of lift. Note that the motion of the cylinder is not shown for this reconstruction as the POD modes are time-invariant. The lift modes vary in an alternating manner in four quadrants. The variation is anti-symmetric about the streamwise centerline. In the maximum lift case (Point B in Fig. 8.9a), the positive pressure force difference (i.e. +Y direction) in the downstream quadrants dominates the small negative difference in the upstream quadrants and vice versa for the minimum lift (Point D). In the zero lift cases (Point A and C), the upstream and downstream pressure force differences tend to become equally strong and they cancel each other. The lift modes vary in such a way that the force on the top 2

8.3 Assessment of Low-Order Model for Wake Decomposition

455

Fig. 8.8 Comparison of POD-DEIM and FOM results: pressure distribution obtained by a full-order model, b POD-DEIM reconstruction; and c the relative error of the reconstruction (%). Maximum error percentage is less than 1.5%. The error in the highly nonlinear regions has reduced compared 2 . The flow is to the linear-POD reconstruction. The pressure values are normalized by 1/2ρ f U∞ from left to right

quadrants is equal in magnitude and opposite in direction to the force on the bottom 2 quadrants. Due to this force cancellation, the vortex shedding (lift) modes have no contribution to the drag force. Hence, the FFT of the drag force does not contain the corresponding harmonic of the natural frequency ( f n ).

456

8 Data-Driven Reduced Order Models

(a)

(b) Point A: zero lift (increasing)

(c) Point B: maximum lift

(d) Point C: zero lift (decreasing)

(e) Point D: minimum lift

Fig. 8.9 Reconstruction of the lift modes in the near cylinder region for Ur = 6.0: a lift variation, b zero lift (increasing) at tU∞ /D = 203.75, c maximum lift at tU∞ /D = 205.25, d zero lift (decreasing) at tU∞ /D = 206.85, and e minimum lift at tU∞ /D = 208.50. The pressure values 2 . The contours levels are from −0.4 to 0.4 in increments of 0.1. The are normalized by 1/2ρ f U∞ flow is from left to right

8.3 Assessment of Low-Order Model for Wake Decomposition

457

(a)

(b) Point A: minimum drag fluctuation

(c) Point B: zero drag fluctuation (increasing)

(d) Point C: maximum drag fluctuation

(e) Point D: zero drag fluctuation (decreasing)

Fig. 8.10 Reconstruction of the drag modes in the near cylinder region for Ur = 6.0: a fluctuation of drag, b minimum drag fluctuation at tU∞ /D = 205.05, c zero drag fluctuation (increasing) at tU∞ /D = 205.85, d maximum drag fluctuation at tU∞ /D = 206.70, e zero drag fluctuation 2 . The contours (decreasing) at tU∞ /D = 207.40. The pressure values are normalized by 1/2ρ f U∞ levels are from −0.34 to 0.16 in increments of 0.025. The flow is from left to right

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8 Data-Driven Reduced Order Models

Figure 8.10 describes the variation of drag modes with the fluctuation of the  = C Dp (t) − pressure-induced drag force. The drag fluctuation is defined as C Dp C Dp . The drag modes vary symmetrically around the wake centerline and hence offer no contribution to the lift. Similar to the lift modes, this explains the absence of a 2 f n harmonic in the lift. The maximum drag fluctuation (Point C) is higher than the minimum drag fluctuation (Point A). Further, at the zero drag fluctuation points (B and D), the magnitude of the drag fluctuation remains a positive value. This further confirms that the drag modes exert a non-zero mean drag on the bluff body apart from the drag force due to the base flow. With the aforementioned observations, the decomposition of force due to pressure field on a moving bluff body based on the contributions from different POD modes can be expressed as nr j  0 bij (t)F ji , (8.27) F j (t) = F j + i=1

where F j is the force in a particular direction ( j = x for in-line and j = y for transverse). F j0 is the time-independent contribution from the mean-field and F ji is the time-independent pressure fluctuation contribution calculated for the ith mode. While bij (t) is the time-dependent coefficient of the i th mode for the force in direction j, n r j is the number of POD modes with a significant contribution to the particular force. Using the snapshot data, we can determine bij (t) for the streamwise and transverse forces as F j (t) − F j0 bij (t) = . (8.28) F ji n r j Table 8.3 summarizes the qualitative analysis of the contributions from the mean field and the modes to the pressure drag and lift forces. The mean-field has a symmetric pressure distribution about the wake centerline, hence contributes solely to the time-independent component of the drag force. The vortex shedding modes have an anti-symmetric pressure distribution throughout the time history, hence they have no drag force contribution. We observe that these lift force contributions have a near zero mean (similar to the lift variation) as well. The shear layer and near-wake modes have the same qualitative properties of the mean field, however their contributions to the drag force are time-dependent. The POD modes provide a deep insight into important flow features and their contribution to the wake dynamics. It is important to investigate their variation with different flow conditions, i.e. the parameters mentioned in Eq. (8.22). The variation of POD modes and their contribution to wake dynamics with the reduced velocity is examined in the next section, with the goal to explain the role of the wake features in sustaining the synchronized wake-body motion.

8.3 Assessment of Low-Order Model for Wake Decomposition

459

0.2

rms

CCDrms L CLrms

0.15

CDrms

Ayrms /D

0.1

0.1

0.3

CLrms

0.4

0.15

0.2

0.05

0.05

0.1 0

0

0.5

4

6

8 Ur

10

12

0 4

8 Ur

6

10

12

(b)

(a)

(c)

(d)

(e)

(f)

Fig. 8.11 Response characteristics and decomposition of wake dynamics for a freely vibrating square cylinder at m ∗ = 3.0, Re = 100 and ζ = 0: a transverse displacement, b drag and lift force variations, mode energy contributions from different flow features c vortex shedding, d shear layer, e near-wake, and f total mode energy contributions. E vs , E sl , E nw denote the relative mode energy of the wake features as a percentage of the total mode energy. The first 9 modes which contain 99% of the total mode energy are considered. For all Ur values, mode 1 and mode 3 correspond to the vortex shedding, while mode 2 corresponds to the shear layer. The flow features of modes 4–9 depend on the Ur value (e.g., mode 4 is a shear layer mode for Ur = 4, 8, 10 and 12 and a near wake mode for Ur = 5, 6 and 7)

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8 Data-Driven Reduced Order Models

Table 8.3 Force contributions from the mean field and POD modes Near cylinder field

Schematic pressure force distribution

F ji , bij

Mean field

Fx0 = 0, Fy0 ≈ 0

Vortex shedding modes

Fxi ≈ 0, Fyi = 0,

Shear layer and near-wake modes

biy ≈ 0

Fxi = 0, bix = 0, Fyi ≈ 0

bij denotes the time averaged force coefficient

8.3.4 Wake Feature Interaction and Sustenance of VIV Lock-in In this section, we investigate the relative contributions from different features to the pressure fluctuations and eventually the forces due to the pressure field on the freely oscillating bluff body. When a bluff body is free to oscillate in a current flow it undergoes the lock-in phenomenon: the oscillation amplitude significantly increases when the natural frequency of the bluff body approaches the vortex shedding frequency. In [297], the lock-in phenomenon for a square cylinder immersed in a laminar flow at Re = 100 is systematically studied. Figure 8.11a and b summarize the bluff body dynamics of a freely vibrating square cylinder. The cylinder undergoes wake-body synchronized lock-in in the range Ur ∈ [4.5, 7] and the peak oscillation occurs at Ur = 5.0. Figure 8.11c–f elucidate the variation of relative contributions from different modes as a function of the reduced velocity (Ur ). It is interesting to note that the three most energetic modes correspond to the same flow features throughout the Ur range namely the first and third modes (vortex shedding) and the second mode (shear layer). However, the modes 4–10 vary in this regard, where most of these modes correspond to the near-wake phenomena.

8.3 Assessment of Low-Order Model for Wake Decomposition

461

We quantify the relative energy contribution from each wake feature by summing the mode energy of the corresponding modes, i.e. n j E j = i=1 k i=1

λi λi

,

(8.29)

where E j is the relative energy contribution from the wake feature ( j = vs for the vortex shedding, j = sl for the shear layer and j = nw for the near-wake), n j is the number of significant modes corresponding to a particular flow feature and k is the total number of modes. As shown in Fig. 8.11c, the total contribution from the vortex shedding increases in the lock-in region. However, the first mode becomes more energetic while the third mode is relatively less energetic in this region. Figure 8.11d exhibits that the contribution from the shear layer modes reduces significantly in the lock-in region. All the individual shear layer modes also follow a similar trend. The near-wake modes depicted in Fig. 8.11e become more energetic in the lock-in region relative to the pre- and post-lock-in regions. Unlike the vortex shedding and shear layer, the primary and secondary near-wake modes have remarkably similar contributions. A summary of the contributions from the 10 most energetic POD modes corresponding to different physical phenomena is illustrated in Fig. 8.11f. Note that these 10 modes capture ≈ 99% of the total mode energy. It is clear that the shear layer contributions decrease while the vortex shedding and the near-wake contributions increase in the lock-in region. In the post-lock-in region, the relative contributions from the flow features remain almost constant. These observations pose an important question: why does the vortex shedding and near wake bubble are energized and the shear layer is weakened during the lock-in? In that relation, we propose a cycle to explain this counter-intuitive behavior of the decomposed wake features. Figure 8.12a elaborates the interaction between the wake features and the bluff body motion. When the vortex shedding synchronizes with the bluff body motion, it causes the bluff body to undergo a relatively high-amplitude motion. This widens the wake and eventually the shear layer, decreasing the velocity gradients. This causes the shear layer to give away vorticity flux to the vortex shedding process, intensifying the vortices and the near-wake bubble. The strengthening of vortices increases the in-phase forces with the motion, i.e., the surrounding fluid flow tends to supply higher energy to the structure. As illustrated in [189], the force Fv and the energy transfer rate e˙v due to the principal vortices can be analyzed using the following simple analytical relations: Fv = ρ f Γ Uv , e˙v = ρ f Γ Uv y˙ ,

(8.30)

where Γ is the vortex strength, Uv is the streamwise velocity of the predominant vortex relative to the bluff body and y˙ is the transverse velocity of the bluff body. It is clear that the increase in the vortex strength will increase the forces and energy

462

8 Data-Driven Reduced Order Models

Fig. 8.12 a Schematic of the interaction between bluff body motion, vortex shedding and shear layer instability in the lock-in region. b–e Primary shear layer mode variation for different Ur regimes. The flow is from left to right. In the lock-in regime, the high gradient region shrinks in the in-line (X ) and expands in the transverse (Y ) direction. f Cylinder vibration frequency variation with Ur . g Phase difference (φ) between the fluid force and bluff-body motion. The onset of phase jump from 0◦ to 180◦ coincide with the sign change of the primary shear layer mode (c–d)

8.3 Assessment of Low-Order Model for Wake Decomposition

463

transfer to the bluff body. The widening of the high gradient shear layer region in the lock-in regime can be seen in Fig. 8.12b–e, which demonstrate the primary shear layer mode for the different Ur cases. In the pre-lock-in regime, the near-wake region is positive compared to the shear layer region. When Ur = 5.0, the maximum amplitude case, it is clear that the high gradient region has shrunk in the streamwise direction and expanded in the transverse direction. Consequently, the near-wake region and the shear layer region interchanges the distribution when Ur = 6.0, i.e. the near-wake region is negative compared to the shear region. This sign change continues to the post-lock-in regime, where the high gradient region extends to the streamwise direction and becomes narrower in the transverse direction. We further generalize this variation of the wake feature contribution for Re > Recr (= 44.7 [466]). Figure 8.13 demonstrates the bluff body motion response and the modal energy contribution from the large-scale features of the wake. The cylinder motion follows a similar trend for Re = 100, 125 and 150 where the lock-in region is detected as Ur ∈ [4.5, 7]. This region is slightly shifted to Ur ∈ [5.5, 8] for Re = 70. In all cases, we observe a maximum of Aryms ≈ 0.2D. Regardless of Re, the wake features exhibit a similar trend in terms of modal energy. As displayed in Fig. 8.13b and d, the vortex shedding and the near-wake modes become more energetic during the lock-in and the shear layer modes become less energetic. This further confirms the proposed interaction cycle for the coupling of the wake features and the bluff body motion. Using the modal decomposition, we have quantitatively explained the interaction dynamics of the flow features which have been conjectured by many previous studies. For example, many successful VIV suppression techniques are proposed by passive [226] and active [147, 309] methods with the experience based presumption that preventing the interaction between the shear layer, the vortex street and the near-wake will suppress the synchronized wake-body lock-in phenomena. The cycle proposed above provides a proper physical mechanism for the success of those methods: they prevent the vorticity transfer between the shear layer and the vortex shedding and/or near-wake bubble, which breaks the self-sustenance of the wake interaction cycle. This understanding of the wake features and their interactions will be vital for the development of effective suppression methods and devices for flow-induced vibrations. In this analysis, we observe that the synchronization of the wake and bluff body weakens the shear layer and intensifies the vortices and the near-wake bubble. In this Re regime, there exists a periodic vortex shedding for the stationary and pre/post-lock-in cases. Hence, it is difficult to state whether the flexibility of the bluff body or the unsteadiness of the fluid flow led to the wake-body synchronization. In what follows, we investigate whether the perturbation of the near-wake bubble via flexibility can sustain the synchronized wake-body interaction at below critical Re flow wherein the well-defined periodic vortex shedding does not exist for the stationary counterpart.

464

8 Data-Driven Reduced Order Models

Arms y /D

0.2

Re = 150 Re = 125 Re = 100 Re = 70

0.15 0.1 0.05 0

3

4

5

6

7

8 Ur

9 10 11 12

(b)

(a)

(c)

(d)

Fig. 8.13 a Transverse amplitude Aryms of square cylinder as a function of Ur for Re ∈ [70, 150] at m ∗ = 3 and ζ = 0.0 and the mode energy contributions from wake features: b vortex shedding, c shear layer and d near-wake bubble at different Reynolds numbers. The vortex shedding and near-wake is energized at the lock-in region while the shear layer mode energy is reduced

8.3.5 Synchronized Wake-Body Interaction at Below Critical Reynolds Number At very low Reynolds number, the flow past a bluff body is two-dimensional, steady and symmetric with respect to the wake centerline. The near-wake bubble attached to its surface is the essential feature below the critical Reynolds number Recr , which is formed by the steady separation from the sharp corners of a square cylinder. Two symmetric and counter-rotating recirculation zones are present in the wake bubble. As Re increases above the critical value, a Hopf bifurcation sets in and the flow becomes periodic via vortex shedding process. For circular and square cylinders [332] demonstrated these values to be Recr = 46.8 and 44.7 respectively, which were further confirmed by [466]. Interestingly, when the bluff body is free to vibrate, [283] predicted for a circular cylinder that this unstable boundary will hold when m ∗ > 1000 and the Hopf bifurcation will occur at much lower Re for low m ∗ values. The authors further conjectured that for a circular cylinder there will be a limiting Re ∼ 32, below which the wake flow will be 2D and steady regardless of the mass ratio m ∗ . Herein, we observe that for some Re < Recr the spring-mounted square cylinder undergoes significant synchronized wake-body motion for a specific range of Ur . Figure 8.14a illustrates a variation of high amplitude motion for Re = 30, 35 and 40. When Re becomes closer to Recr , the synchronization regime widens and the

8.3 Assessment of Low-Order Model for Wake Decomposition Re = 40 Re = 35 Re = 30

Arms y /D

0.1

0.05 0

0.2 0.15

0.9

Ur = 7 Ur = 8

0.85 f / fn

0.2 0.15

465

0.1 0.05

4

5

6

7

8 Ur

(a)

9

10 11 12

0 20

0.8

Ur = 7 Ur = 8

0.75 25

30 Re

35

40

25

30

35

40

Re

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Fig. 8.14 Response characteristics of wake-body interaction for Re < Recr : dependence of transverse amplitude on Ur and Re a reduced velocity range Ur ∈ [4, 12] for Re ∈ [30, 40], b Re ∈ [20, 40] for Ur = 7, 8 c vibration frequency of the bluff body, d the spanwise Z -vorticity for a non-synchronized case (Ur = 4.0, Re = 40), e unsteady wake in a synchronized case (Ur = 7.0, Re = 40) at tU∞ /D = 120. f–i POD modes containing ∼ 99% of the mode energy of the synchronized wake-body case: Ur = 7.0, Re = 40. The flow is from left to right

466 50

Unsteady wake 44.7

Flexibility-induced unsteadiness

40

Re

Fig. 8.15 Demarcation of wake unsteadiness for a freely vibrating square cylinder at m ∗ = 3. For the stationary square cylinder, the wake is steady for Re < 26 and for Re > 44.7 the wake becomes unsteady, as shown by dashed lines

8 Data-Driven Reduced Order Models

35 30 26

Steady wake 20

4

5

6

7

8

9

10

11

12

13

14

Ur

highest amplitude Ur shifts from Ur = 8 to Ur = 7. In contrast to Re > Recr cases, we observe no motion of the cylinder in pre- and post-synchronization regimes. We further examine the conjecture of [283] and demonstrate that for a square cylinder this synchronized motion is present when Re ≥ 26 (Fig. 8.14b). Additional analysis on these synchronized motion cases revealed that the wake-body system synchronizes to a frequency slightly less than the natural frequency f n of the bluff body similar to the Re > Recr lock-in regime (Fig. 8.14c). We observe that, for all synchronized motion cases, the wake is unsteady with some vortex shedding patterns. For example, we can compare the representative Z -vorticity contours for the zero motion and the synchronized motion cases displayed in Fig. 8.14d and e, respectively. The zero motion case is almost identical to the stationary cylinder counterpart, while the synchronized motion case is similar to the lock-in scenario for Re > Recr . Notwithstanding, the vortex formation length is considerably high for this below Recr configuration. We further decompose the unsteady wake of the synchronized motion case and examine similar features as Re > Recr cases, i.e. the vortex shedding (Fig. 8.14f and h), the shear layer (Fig. 8.14g) and the near-wake bubble (Figs. 8.14i and 8.15). These observations constitute the basic requirement for the wake-bluff body synchronized motion: the bluff body should have an optimal amount of flexibility (i.e., not too rigid nor too flexible) and the flow needs to have sufficiently large inertia (i.e., higher Re) to trigger the unsteadiness in the near-wake bubble. This particular Re is lower than the Recr for a fixed bluff body. This means that the flexibility of the solid body provides an avenue for the wake and the spring-mounted body to synchronize eventually causing the wake to be unsteady. From this numerical experiment, we can deduce that the flexibility of the bluff body is the primary factor driving the synchronized wake-body motion, neither the vortex shedding nor the shear layer roll-up. Hence the most critical wake feature for the onset of wake-body synchronization is the near-wake bubble. When the Re is very low (< 26) this bubble remains steady and the counter-rotating recirculation zones behind the bluff body are stable. For 26 ≤ Re ≤ 44.7, it remains same if the bluff body is either too rigid or

8.3 Assessment of Low-Order Model for Wake Decomposition

467

flexible. However, in this Re regime, when the bluff body is appropriately flexible, slight perturbations cause distortions in the steady wake. These distortions become periodic and begin to synchronize with the bluff body. This synchronization leads to a relatively higher amplitude oscillations at a frequency slightly less than the natural frequency of the solid body in a vacuum. At the same time, due to the vorticity generation by the unsteadiness developed in the wake, the vortices are shed from the downstream end of the wake bubble. With the aforementioned findings, we deduce that the root cause of wake-body synchronization is the frequency lock-in between the natural frequency of the bluff body and the near-wake bubble. This further demonstrates that the unsteadiness of the wake can be induced by the flexibility of the bluff body. Moreover, the unsteady wake alone cannot induce high amplitude bluff body oscillations (e.g., pre- and post-lockin in Re > Recr ). Hence, we can further infer that the wake-body synchronization is induced by the synchronization of the bluff body motion with the near-wake bubble, not with the vortex street. In the next section, we generalize our findings to threedimensional turbulent flows at moderately high Reynolds number.

8.3.6 Effect of Turbulence In this section, we investigate the dynamic decomposition of the wake behind a three-dimensional oscillating square cylinder at Re = 22,000, wherein the wake is fully turbulent. Our aim is to understand the role of turbulence when we extend the wake feature interaction cycle to turbulent flow. To retrieve the high-fidelity data at this Re, we employ a well-established dynamic subgrid-scale turbulence model in our finite-element formulation. The filtered Navier-Stokes formulation and the determination of the subgrid stress term via the dynamic subgrid-scale model is provided in [180]. We incorporate this full-order model to generate 3D snapshots of the flow fields and the POD-Galerkin projection is applied on this high-fidelity data set. At high-Re turbulent wake flow, the aforementioned large-scale organized flow features are fragmented into smaller scales until the scales are fine enough to dissipate by the fluid viscosity. Therefore, small-scale modes can have a significant impact on the overall dynamics for the high-Re turbulent condition, in contrast to the lowRe study. Figure 8.16 demonstrates the mode energy distribution for Re = 22,000. Compared to the low-Re cases, the modal energy is much more distributed among the modes. For instance, the most energetic mode of the Re = 100 case contains 56% of the total energy while it is 32.88% for the high-Re case at Re = 22,000. Due to this broadening of the mode energy, the energy decay is less steep. For the low-Re cases, the first 5 modes contain 95% of the total mode energy and the first 9 modes contain 99%. On the other hand, for the high-Re case, a total of 123 modes is required to capture 95% of the energy and 211 modes are required for 99%. We further investigate this distribution of the mode energy with the presumption that the presence of broadband turbulence is the key factor.

468

8 Data-Driven Reduced Order Models

(a)

(b)

Fig. 8.16 The energy distribution of the POD modes for Re = 22,000, m ∗ = 3.0, Ur = 6.0: a energy decay of POD modes, and b cumulative energy of POD modes. The dashed line represents 95% of the total mode energy

Figure 8.17 displays the mean pressure field and 10 most energetic POD modes obtained using the same snapshot technique for the moderate-Re case. All the modes exhibit distorted and scattered patterns compared to the low-Re cases. However, the POD modes further illustrate general large-scale features. For example, modes 1, 3, 4, 5, 6, 7 and 9 exhibit the flow patterns related to vortex shedding while mode 2 is related to the shear layer and modes 8 and 10 are of the near-wake phenomena. The distortions occur in each mode throughout the spatial domain due to the broadband and multiscale effects of turbulence, which are not decomposed by the singular value decomposition. Turbulence distributes the modal energy across the modes which makes it requires significantly more modes to reconstruct the flow field and the underlying wake dynamics. Hence, the POD based reconstruction becomes computationally more expensive in turbulent flows due to the broadband and multiscale character. Similarly, Fig. 8.18 demonstrates the broadband nature of turbulence in the temporal domain. Even with the multiscale spatial distortions, the first mode of the moderate-Re has a similar temporal contribution as the first mode of the low-Re case with a single dominant frequency close to the natural frequency of the system. However, the temporal coefficients of the other modes have multiple harmonics. Some of the modes exhibit predominant frequencies among the broadband FFTs. For example, the mode 2 has a dominant 2 f n frequency behavior, the mode 3 has f n , 2 f n and 3 f n harmonics and mode 4 has f n and 2 f n harmonics. These multiple harmonics occur due to the bombardment of turbulence on the corresponding flow features of the POD modes. Figure 8.19 displays the pressure field reconstruction using the POD-DEIM technique. The actual instantaneous field contains some distortions and fine near-wake variations, which are not completely captured by the reconstruction process. How-

8.3 Assessment of Low-Order Model for Wake Decomposition

469

(a) Mean Pressure

(b) Mode 1 (32.88%)

(c) Mode 2 (10.58%)

(d) Mode 3 (6.94%)

(e) Mode 4 (5.37%)

(f) Mode 5 (2.73%)

(g) Mode 6 (2.46%)

(h) Mode 7 (2.16%)

(i) Mode 8 (1.83%)

(j) Mode 9 (1.65%)

(k) Mode 10 (1.54%)

Fig. 8.17 The mean field and the first 10 significant POD modes for a oscillating square cylinder at Re = 22,000. The energy fraction of the POD mode is mentioned in brackets. The plots are of the mid Z -plane

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8 Data-Driven Reduced Order Models

Fig. 8.18 Time dependent contributions of the 10 most energetic modes and their normalized FFT spectra for Re = 22,000 case

ever, the general large-scale variations are properly reconstructed with a maximum local error of ≈ 2%. The broadband energy distributing nature of turbulence has reduced the contribution of significant POD modes to the wake dynamics. Due to this behavior, many flow features are not captured when few of the most energetic modes are considered for the reconstruction. Hence, the inclusion of many POD modes is required for an accurate reconstruction which makes the POD reconstruction computationally expensive and time-consuming. However, this can be mitigated by selective reconstruction of a few required timesteps instead of the entire time history. Using this to our advantage, we investigate the validity of the wake-body interaction cycle at this moderate-Re. In summary, the presence of turbulence distorts the spatially symmetric/antisymmetric nature of POD modes and distributes the mode energy throughout many POD modes. The reconstruction requires many modes and the classification of features is more complex at moderately high-Re. Despite this complexity, the fundamental wake-body interaction process proposed using low-Re analysis is observed to be valid for three-dimensional turbulent flows. Hence, we can conclude that the proposed wake-body interaction cycle in this study is a general cycle for coupled fluid-structure systems.

8.3.7 Force Decomposition Based on Modal Contribution The aim of this appendix is to present a general decomposition of the force due to pressure field exerted on a moving body in an incompressible viscous flow. To begin, we provide some background on existing force decomposition techniques that

8.3 Assessment of Low-Order Model for Wake Decomposition

471

Fig. 8.19 FOM and POD-DEIM reconstructed pressure field comparison for Re = 22,000 case at tU∞ /D = 100: pressure distribution obtained by a full-order model and b 123 POD modes and 200 DEIM points and c the relative error (%). The plots are of the mid Z -plane. The pressure values 2 . The flow is from left to right are normalized by 1/2ρ f U∞

472

8 Data-Driven Reduced Order Models

60

0.6

% Mode Energy

0.5

Arms y

0.4 0.3 0.2

Evs

Esl

Enw

20

0.1 0

40

3

8

5 6

10

3

5

8

6

Ur

Ur

(a)

(b)

10

Fig. 8.20 Response and energy distribution for a freely vibrating square cylinder at Re = 22,000: a transverse amplitude as a function of reduced velocity, b POD mode energy contribution of wake features. Note that Ur = 3.0 represents the pre-synchronization, Ur = 5.0 and 6.0 denote the wakebody synchronization regime and Ur = 8.0 and 10.0 cases are in the galloping regime. For the mode energy analysis, the most energetic modes containing 95% of the total mode energy are considered

characterize the fluid inertial and the viscous forces on a moving body. In one of the pioneering works, [302] proposed a force decomposition for the in-line force acting on a cylindrical object which is widely used in many engineering applications. This semi-empirical decomposition can be written as a linear sum of a velocity squareddependent drag force and an acceleration-dependent inertial force: F(t) =

π D 2 dU 1 Cd D|U |U + ρCm , 2 4 dt

(8.31)

where Cd and Cm represent the averaged drag and inertia coefficients, which can be determined by experiments or numerical computations. Owing to the nonlinear dependency of these coefficients on the evolution of vorticity field, [380] argued that “It does not perform uniformly well in all ranges of K , β and k/D", where K denotes the Keulegan Carpenter number, β = Re/K and k/D is the relative roughness. In [247], a different approach is taken by the assertion that the viscous drag and the inviscid inertia force operate independently, by re-writing Eq. (8.31): F(t) =

dU 1 Cd ρ A p U 2 + Cm∗ ρ Vb 2 dt

(8.32)

and for a flow defined by U (t) = −Um cos ωt it reduces to C F = −Cd | cos ωt| cos ωt + Cm∗

π2 sin ωt, K

(8.33)

8.3 Assessment of Low-Order Model for Wake Decomposition

473

where A p and Vb denote the projected area and the volume of the body, respectively and Cm∗ is the ideal value of the inertia coefficient. Notwithstanding, many studies demonstrate that it is difficult to represent the actual force with this relation as long as a constant value Cm∗ is considered. In particular, [380] clearly demonstrated that the viscous drag force and the inviscid inertia force are not completely independent and it is impossible to decompose the unsteady drag force to an inviscid and a vorticity-drag component. The decomposition of the total force into inviscid and viscous components by Lighthill’s relation (Eq. 8.32) can be considered as an effort to lump the effects of the complex generation and evolution of vorticity field into mutually independent forces related to the inviscid inertia and the viscous effects. In such force decomposition techniques, the characteristics vorticity patterns and their dynamics generated during the motion of a body are not included. In what follows, we propose an alternative force decomposition for the in-line (drag) and transverse (lift) pressure forces applied on a bluff body which extends the above decompositions to incorporate significant features of unsteady separated flow. In particular, the unsteady force is decomposed to include the nonlinear generation and evolution of vorticity field around a moving body in a fluid flow (Fig. 8.20). This decomposition is based on the contributions from different POD modes to the forces and can be written as nr j  0 bij (t)F ji , (8.34) F j (t) = F j + i=1

where F j is the force on a particular direction ( j = x for the in-line and j = y for transverse). While F j0 is the time-independent contribution from the mean field, F ji is the unsteady pressure fluctuation contribution associated with i th mode. Here, bij (t) is the time-dependent coefficient of the i th mode for the force in direction j and n r j is the number of POD modes with a significant contribution for the particular force. Using the snapshot data, we can exactly determine bij (t) for the in-line and transverse forces by the following relation bij (t) =

F j (t) − F j0 F ji n r j

.

(8.35)

The magnitude of the modes, the time coefficients and the force contributions introduced in this decomposition slightly fluctuate when flow parameters and the bluff body geometry are changed. Similar to the above methods, we can create databases of F ji for different bluff bodies. These databases can then be used to determine the total forces as well as the contribution from each flow feature to the bluff body dynamics. To further generalize the force decomposition, deep learning techniques [298] for parametric predictions can be employed.

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8 Data-Driven Reduced Order Models

(a)

(b) Fig. 8.21 Time dependent coefficients of pressure force contributions from a drag, and b lift modes

Figure 8.21a and b present the reconstructed values of the time dependent coefficients for drag and lift modes. Note that the relevant six out of the first 9 modes are considered for the drag and the rest for the lift (i.e. n r x = 6 and n r y = 3). In a nutshell, the force component represented by the modal decomposition implicitly characterizes the three constituent components involving an inviscid inertial force, the dynamics of the vorticity field, and a skin friction force.

8.3 Assessment of Low-Order Model for Wake Decomposition

475

Table 8.4 Number of floating points operations (FLOPs) required for linear and nonlinear POD reconstruction Operation FLOPs Order Modes generation YT Y YT YW = ΛW V = YW Λ−1/2 Total Linear POD ˆ Y(t) = P − P, V ˆ VY r j=1 yˆ j (t)v j Total DEIM (P T U )ˆc = P T vi rˆ = vi − U cˆ

V+ ℘

Y−Y=VV+ ℘ Y℘ Total

Matrix multiplication Eigenvalue solution Matrix multiplication

2k 2 m − k 2 k3 2k 2 m

2k 2 m k3 2k 2 m 4k 2 m

Inner products Matrix multiplication Summation

r k(2m − 1) 2r km − mk r k2m

2r km 2r km r k2m r k2m

Matrix solution

(n p − 1)n p (n p + 6m − 1)/6 (n p − 1)n p m 4n 2d n p − n 2d − n d n p k(2n d n p + 2mn d − n d − m)

n 2p m

Matrix subtraction Matrix multiplication Matrix multiplication

n 2p m 4n 2d n p 2mkn d 2n 2p m + 2mkn d

Mesh count (m) = 87,120, number of snapshots (k) = 320, number of significant modes for linear POD (r ) = 9, number of significant modes for DEIM (n d ) = 7 and number of DEIM points (n p ) = 70

8.3.8 Performance Comparison of POD Reconstruction Methods Herein, we briefly compare the number of floating point operations (FLOPs) required for the linear and the nonlinear DEIM based POD reconstructions in Table 8.4. The generation of the POD modes which is essential for both reconstructions requires ∼ O(4k 2 m) FLOPs. In this study, we estimate this value to be 3.571 × 1010 . The linear POD reconstruction needs ∼ O(r k 2 m) computational steps where the equal contributions are from the multiplication between the time coefficient and the POD modes, and the summation of the multiplied POD contributions. With the use of 9 POD modes, the estimated FLOP count is 8.029 × 1010 . The DEIM technique consumes fewer computational steps than POD as it requires ∼ O(2n 2p m + 2mkn d ) FLOPs. Performing the matrix solution in the DEIM first step is the most expensive operation as it needs more FLOPs for the last DEIM points. With the use of 7 POD modes and 70 DEIM points, the POD-DEIM reconstruction needs 1.244 × 109 FLOPs. According to this estimation, the linear POD reconstruction demands ≈ 64.54 times more FLOPs than the DEIM reconstruction. However, the total linear POD process needs ≈ 3.14 times more FLOPs than the total DEIM process.

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8 Data-Driven Reduced Order Models

Table 8.5 Performance comparison between linear POD and POD-DEIM for the most accurate reconstruction Linear POD (10 modes) DEIM (7 modes + 70 points) POD mode generation Reconstruction Total time elapsed Speedup (reconstruction) Speedup (total) Maximum local error Cumulative domain error

35.1s 113.2s 148.3s – – 1.43% 3.85%

35.1s 12.2s 37.3s 9.28 3.98 1.24% 4.05%

The actual performance of the linear POD and the POD-DEIM reconstruction techniques is compared in Table 8.5. All the calculations are performed using the same quad-core Intel Xeon 3.50 GHz × 1 CPU with 16 GB memory. When just the reconstructions are considered, the nonlinear DEIM is 9.28 times faster than the linear POD. When the total processes are compared, the DEIM has a speed gain of 3.98 and the DEIM is more accurate in the nonlinear flow regions. However, the linear POD method is more accurate when the cumulative error of the fluid domain is considered.

8.4 Summary Despite the prevalence of SVD-based modal reduction techniques, there are very few studies on their application to fluid-structure interaction systems. In this chapter, we considered the 2-DOF free vibration of a square cylinder under laminar and turbulent flows. We explored the capability of POD decomposition to interpret the most significant wake features and their contributions to the forces on the vibrating body interacting with fluid flow. When the linear and nonlinear POD-DEIM reconstructions are contrasted, we found that the DEIM method is faster and has a higher local accuracy since it captures the nonlinearity of the principle vortices and the near-wake region. For the low-Re cases, every POD mode represents one of the large-scale flow features: vortex shedding, shear layer or near-wake bubble. In these cases, we further observed that the nine most energetic modes contain ≈ 99% of the energy. Further, we identified that the vortex shedding modes solely contribute to the transverse (lift) force while the shear layer and the near-wake modes solely contribute to the drag force. Based on these observations, we proposed a novel force decomposition for the drag and lift forces which is different from the conventional force decomposition based on the added mass and the viscous force contributions. We examined the POD decomposition for a range of Ur values and we proposed the mechanism of the sustenance of synchronized wake-body lock-in. This further

8.4 Summary

477

provided the explanation to the observation: in the lock-in region, even though the kinetic energy is transferred from the fluid to the bluff body, the principle vortices are much more energetic than the pre- and post-lock-in regimes. It is seen that the bluff body motion widens the wake and it causes a vorticity transfer from the shear layer to the near-wake and vortices. We proposed the wake feature interaction cycle based on these observations. We further confirmed that this mechanism is valid for the laminar Re > Recr range. For below critical Re flows, we observed that the bluff body and the wake still synchronize and undergo large amplitude motion at some Ur values when Re ≥ 26. Decomposition of these wakes further exhibited similar behavior as the synchronized large amplitude motion cases at Re > Recr . This revealed that the flexibility of the bluff body induced the unsteadiness in the near-wake bubble causing it to break and generate the vortices. With this observation, we can conclude that the fundamental requirements for the wake-body synchronized motion are, large enough flow inertia and appropriate flexibility of the structure. When the moderate-Re turbulent bluff body flow is decomposed, we observe that all the dominant wake modes are bombarded with different scales of turbulence. The broadband nature of turbulence resulted in a wide mode energy distribution, which required up to 123 modes to reach the 95% mode energy threshold. Further analysis of the time coefficients of the modes confirmed that the large-scale features are battered by the multiple frequency turbulence. However, they generally correspond to a large-scale wake feature similar to the laminar cases. The wake decomposition of turbulent flows for Ur ∈ [3, 10] confirmed that the wake interaction cycle proposed for laminar cases is valid for turbulent flows as well. Acknowledgements Some parts of this Chapter have been taken care from the PhD thesis of Tharindu Pradeeptha Miyanawala carried out at the National University of Singapore and supported by the Ministry of Education, Singapore.

Chapter 9

System Identification and Stability Analysis

In this chapter we present two advanced physics-based system identification approaches via projection-based and deep-learning-based reduced-order models. The projection-based approach includes a linear reduced-order model (ROM) for stability prediction using the eigensystem realization algorithm (ERA), which provides a low-order approximation of unsteady flow dynamics in the neighbourhood of equilibrium steady state. We perform a systematic ROM-based stability analysis to understand the frequency lock-in mechanism and self-sustained FIV phenomenon by examining eigenvalue trajectories. At high Reynolds number flows and near realtime feedback control, this goal can only be achieved through the recent advances in nonlinear model reduction and deep learning (DL) algorithms. To demonstrate this idea, we have developed a data-driven coupling for predicting unsteady forces and vortex-induced vibration (VIV) lock-in by using a long short-term memory network (LSTM) as a DL-based ROM technique. The structure of the LSTM has the format of a nonlinear state-space model (NLSS) and provides a nonlinear mapping of input-output dynamics that can potentially predict the dynamics for a longer horizon utilized for the stability predictions. The simplicity and computational efficiency of the proposed ROMs allow investigation of the FIV mechanism for a variety of geometries and parameters, and open ways for the development of control devices and on-board and in real-time predictions.

9.1 Introduction Vortex shedding from a bluff body and the vortex-induced vibrations (VIV) are ubiquitous and have a broad range of applications in numerous fields such as offshore, wind and aerospace engineering. Apart from their great practical importance, these phenomena in fluid mechanics have a fundamental value due to the vast richness of their vorticity dynamics and coupled nonlinear physics. Asymmetric vortex shedding shed from a bluff body causes a large unsteady transverse load, which in turn may © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Jaiman et al., Mechanics of Flow-Induced Vibration, https://doi.org/10.1007/978-981-19-8578-2_9

479

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9 System Identification and Stability Analysis

lead to structural vibrations when the structure is free to vibrate in the transverse direction [40, 381, 455]. These large vortex-induced vibrations can lead to damage and potential risk to the structures, in particular for ocean structures such as marine risers, subsea pipelines and cables. When the natural frequency of the structure is close to the vortex shedding frequency, the phenomenon of VIV results in a complex evolution of the shedding frequency, which deviates from the Strouhal relation of stationary counterpart. In this frequency lock-in regime, the vortex formation locks on to the natural frequency of the body within a range of the Strouhal frequency and there exists a strong coupling between the fluid and the structure [381]. Such frequency lock-in phenomenon leads to high amplitude and self-sustained vibrations, thus there is a need to understand the origin and different regimes during the lock-in process. The lock-in process is self-excited and is characterized by the matching of the frequency of periodic vortex shedding and the oscillation frequency of the body [206]. The flow over a single elastically mounted two-dimensional bluff body has served as a generic VIV model for both numerical and experimental investigations. In this canonical configuration, it is often convenient to consider the elastically mounted cylinder as two coupled oscillators whereby one system is the oscillating body and the other one is the wake. Numerous studies have been conducted to understand the frequency lock-in phenomenon for this simplified fluid-structure system. This VIV model problem manifests a complex dynamical behavior, which is still the subject of active research over the past decade [40, 455]. Apart from the fundamental physics of a single cylinder VIV [51, 238, 392, 393], the topics for numerical investigations range from the development of coupling procedures for the Navier-Stokes and the structural equations [153, 180, 181], to the modeling of near wall proximity effects [416], multiple-cylinder arrangements [254, 307], and suppression devices [226, 473]. High-fidelity computational fluid dynamics (CFD) can reveal a vast amount of physical insight in terms of vorticity distribution, the force dynamics, the frequency characteristics and phase relations, and the shape of the VIV trajectory. Despite improved algorithms and powerful supercomputers, the state-of-art CFD-based VIV simulation is less attractive with regard to parametric optimization and the development of control strategies. The primary motivation behind the present work is: (i) to develop an efficient low-order model for the VIV lock-in of a circular-shaped bluff body, and (ii) to generalize the eigenvalue analysis of VIV lock-in mechanism for other two-dimensional bluff bodies.

9.1.1 VIV Mechanism and System Identification A simple interpretation of frequency lock-in during VIV is attributed to the classical resonance or synchronization with a well-defined frequency. Structural response amplitude gradually should grow as the structure natural frequency f N approaches the alternate vortex shedding frequency f vs , and should attain its maximum value when f N / f vs ≈ 1. However, VIV simulations [393, 416] at Re = 100 reveal that the circular cylinder acquires the maximum amplitude at f N / f vs ≈ 1.3 or in the vicinity

9.1 Introduction

481

of VIV lock-in onset, which is not consistent with the simple resonance interpretation. Therefore, the classical resonance is not adequate to interpret the underlying VIV lock-in mechanism and the large amplitude during the lock-in process. Through a linear global stability analysis of the flow past an elastically-mounted cylinder, [97] identified two modes in the fluid-structure system, namely nearly structural mode and the von Karman wake mode. [103] investigated the mechanism of lock-in to a coupled mode flutter by using a simple linear wake-oscillator model for a transversely vibrating circular cylinder. The VIV analysis in [103] was performed by considering an empirical wake oscillator model while neglecting nonlinear and viscous terms. Analogous to plunging and pitching instability of airfoil in the classical aeroelasticity, [103] attributed the root cause of VIV lock-in to the mode coupling between the transverse periodic motion and the continuous rotation of the separation point along the smooth contour of a circular cylinder. Using a standard asymptotic analysis, [283] confirmed the existence of the two modes identified by Cossu et al. [97] and termed them as the wake mode (WM) and structure mode (SM). For weak fluid-structure interaction in the limit of large solid-to-fluid mass ratio, the eigenvalue of wake mode was found to be similar to the leading eigenvalue computed for the flow past a fixed cylinder whereas the eigenvalue of the SM approached the natural eigenvalue of the cylinder-only system. Inspired by the semi-analytical finding of [103, 486] recently employed a linear ROM-based CFD method to study the frequency lock-in phenomenon of a circular cylinder at Re = 60, and two regimes have been termed in the VIV response, namely resonanceinduced lock-in and flutter-induced lock-in. The resonance regime is related to the vorticity dynamics of wake flow, whereas the flutter regime may be interpreted as an inertial coupling between the structure and global wake flow. In another recent work of [311], the lock-in phenomenon has been investigated via linear stability and direct time integration and two leading eigenmodes referred to as the fluid mode and the elastic mode were classified for a transversely vibrating circular cylinder. These two leading modes were found to have a strong coupling for low mass ratios and a clear demarcation of the fluid (wake) mode or elastic (structure) mode was found to be non-trivial. As opposed to the decoupled modes (WM and SM) for high mass ratios, these modes were termed as coupled modes for low mass ratios [311]. Owing to the complexity of VIV with regard to fluid-structure interaction, a unified description of the frequency lock-in still remains unclear for arbitrarily shaped bluff bodies and general physical conditions. Of particular interest to this study is to understand some elementary aspects of the self-sustained VIV oscillations by considering a linear aspect of the lock-in process. The linear instability plays a key role in the origin of self-sustain VIV oscillation arising from the coupled fluid-structure system. Once the fluid-structure system rises to a high-amplitude VIV response, the nonlinearity begins to dominate and the system transforms into a fully developed (self-limiting) limit-cycle state. Some key questions with regard to the generality of VIV lock-in process have remained unexplained, such as: How does the geometry of the bluff body influence the frequency lock-in in VIV? Why the VIV behavior of a square cylinder is different from its circular counterpart? Do the resonance and flutter regimes exist always or the regimes are actually influenced by the Reynolds

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9 System Identification and Stability Analysis

number and the geometry of the bluff body? In this section, we attempt to answer these questions and understand more general aspects of the linear VIV mechanism via our proposed ERA-based ROM procedure. An understanding of the VIV mechanism can help in developing flow control techniques based on both passive and active control schemes, whereby the passive schemes require no energy input and the active schemes rely on continuous energy input. Owing to the complexity of VIV, the control schemes are generally ad-hoc and a good understanding of the dynamical behavior with respect to the flow and structure parameters is required. Although a high-fidelity CFD model is able to resolve physical feature of interest, a linear model based on the model reduction provides a way to perform stability analysis for the flow past a bluff body and to design active control strategies [116, 279, 285, 417]. Two ways exist to derive a linear model of original nonlinear system. While the first one is to derive a linear governing equation and then discretize the system of equations, the second approach is to discretize the nonlinear model first and then to obtain the linear model from it. The latter method is widely used in the aeroelastic research community to construct the linear model by automatic differencing method. However, both types of methods are expensive and are not attractive for parametric study and the development of VIV control strategies. A low-order model based on minimal state-space dimension has a potential to become a practical alternative to understand the VIV mechanism and to design a proper control procedure. A model-based control design can help to regulate and stabilize alternate vortex formation and the near-wake dynamics. Such model relies on the smallest state-space dimension of realized systems that have the similar input-output relations within a specified degree of precision. As shown in [155] that the minimum problem represents the problem of identifying the sequence of real matrices, also known as the Markov parameters, based on the impulse response of a dynamic system.

9.1.2 Model Order Reduction The model order reduction (MOR) technique is to approximate the original full order (high dimensional) system with a low order model, which retains the significant dynamics of the original system and provides an order of magnitude efficiency improvement to construct the essential dynamics of the system. As discussed in [116], we can categorize the previous studies on the linear model order reduction into two main approaches. The first ROM construction approach is based on Galerkin projection of full order system onto a small subspace spanned by mode vectors. The mode vector can be obtained by proper orthogonal decomposition (POD), balanced truncation [301], or dynamic mode decomposition (DMD) [368, 382]. One of the drawbacks of conventional POD/Galerkin models is that while they capture the most energetic modes based on a user-defined energy norm, low-energy features may be crucial to the dynamics of an underlying problem. As compared to the POD method, which only extracts modes from the snapshots of the primary system, the balance

9.1 Introduction

483

truncation method derives the modes by collecting the snapshots of both primary and adjoint systems. This feature of the balance truncation method allows for identifying the modes which are dynamically important. Based on the work of [301, 448] and [366] further extended the balance truncation concept to a large system by approximating the system observable and controllable Gramians via two sets of snapshots from a linearized forward simulation and a companion adjoint simulation. This algorithm is usually referred to as the balanced proper orthogonal decomposition (BPOD) and provides two sets of modes, namely primal and adjoint modes. The second approach is based on the system identification method, which only requires input and output information and considers the original system as a black box via input-output dynamical relationship. From a time-domain formulation and the realization of a state-space model, a ROM of a dynamic system can be constructed on the basis of input-output data. One of the widely used system identification methods is the eigensystem realization algorithm (ERA) introduced by [199] for the model reduction using a Hankel matrix based decomposition. ERA essentially extends the well-known algorithm of [155] in control theory and creates a minimal realization that follows the evolution of system output when it is subjected to an impulse input. In a recent theoretical study, [271] proves that the ERA constructs a ROM which is mathematically equivalent to the BPOD method. With regard to recent fluid dynamics applications, the ERA has been considered for unsteady problems by [116, 468]. The aforementioned methods are originally developed for stable linear systems. Extensions have been made to circumvent this restriction of the model reduction for unstable systems by either partitioning the unstable and stable subspace or inverting the large linear system [2, 32, 105]. In one of the recent works by Flinois et al. [117], a theoretical analysis was presented to show that the unmodified balanced truncation (designed for stable systems) method can be applied to an unstable system. Following this analysis and the work of [271], the ERA is recently employed for the active control of unstable wake behind a bluff body [116]. Compared to the ROM method used in [486], which lacks a mathematical rigor and is highly sensitive to training trajectory, the ERA has a theoretical foundation for unstable linear systems generated by the unsteady wake dynamics and vortex-induced vibrations. Therefore, following [117] and the finding of [271], the ERA is adopted in this chapter to construct the low-order fluid model.

9.1.3 Objectives In this work, we present a physical insight and the underlying mechanism of vortexinduced vibration by exploiting a unified description of frequency lock-in during elastically mounted cylinders. We introduce the ERA-based ROM to capture just enough physics to extract the stability properties of the fluid-structure system of twodimensional bluff bodies consisting of sharp corners and smooth curves. Of particular interest is to provide a generalized description of these frequency lock-in regimes at low Reynolds numbers via the model reduction technique. Unlike the wake-oscillator

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9 System Identification and Stability Analysis

model, the present technique does not rely on any empirical formulation and captures naturally the physical effects related to the added mass and damping forces through the solution of the Navier-Stokes equations. We first employ the ERA-based ROM for unstable wake flow over a stationary circular cylinder and predict the critical Reynolds number Recr of vortex shedding. We then perform the stability analysis of fluid-structure system via the ERA-based ROM to analyze the effects of Reynolds number Re, the mass ratio m ∗ and the rounding of a square cylinder. To examine the accuracy and reliability of the low-order model, we assess the ROM results against the full-order simulations performed by the variationally coupled Navier-Stokes and rigid body equations. We will show in this section that the two frequency lock-in regimes associated with resonance and flutter characteristics only exist when certain conditions are satisfied. These regimes have a strong dependence on the shape of the bluff body, the Reynolds number and the mass ratio. The presence of sharp corners on a square cylinder largely alters the VIV lock-in characteristics as compared to the circular counterpart with smooth curves. We report that the frequency lock-in of the square cylinder is found to be dominated by the resonance regime without any coupled mode flutter at a low Reynolds number (Re ≤ 80). This indicates that the previous theoretical finding by de Langre et al. [103] on the root cause of frequency lock-in due to the coupled flutter does not hold for a transversely vibrating sharp-cornered square cylinder. Apart from the frequency lock-in regimes, we qualitatively visualize the spatio-temporal evolution of vortex shedding and leading eigenmodes to link the lock-in process with the intrinsic wake dynamics. To understand the influence of geometry on the frequency lock-in regimes, we present a stability phase diagram for five two-dimensional bluff bodies namely, circle, square, ellipse, forward triangle and diamond. Compared to the circular cylinder, we show that the flutter mode is pronounced in the elliptical cylinder while the lock-in/synchronization is gallopingdominated for the forward triangle configuration. The proposed ERA-based ROM is general and efficient for fluid-structure systems without the need for a linearized flow or an adjoint solver, which allows the method to be even applicable for physical experiments.

9.1.4 Organization The chapter is structured as follows: Sect. 9.2 introduces the full order model, the state-space formulation for the model reduction and the eigensystem realization algorithm for the wake flow and vortex-induced vibration. Section 9.3 describes the VIV problem setup for VIV of a single cylinder in the flow field, along with the systematic analysis of the frequency lock-in mechanism as a function of Reynolds number and the effects of rounding and geometry. Next, Sect. 9.4 describes the problem set-up of the ERA-based ROM for the VIV of tandem cylinder arrangement, along with the WIV results with the effects of longitudinal spacing and the role of sharp corners.

9.2 Numerical Methodology

485

Finally, Sect. 9.5 provides a nonlinear system identification technique based on deep learning, the state-space formulation for the model reduction, and the idea of stability predictions for the wake flow and VIV of a 3D sphere. We further describe the VIV problem setup and present the numerical verification of the DL-based ROM model. A systematic analysis of the frequency lock-in mechanism and the effects of mass ratio (m ∗ ) and Reynolds number (Re) are provided in Sect. 9.6.

9.2 Numerical Methodology For the sake of completeness, we first summarize the formulation for highdimensional full order model (FOM) and describe the implementation of the numerical schemes used for the coupled variational fluid-structure solver. Later we present the ERA for the construction of reduced-order model (ROM) using the Navier-Stokes and rigid-body equations.

9.2.1 Full Order Model Formulation To study the interaction of elastically mounted cylinder with the fluid, we consider a variational fluid formulation based on the arbitrary Lagrangian-Eulerian (ALE) description and the semi-discrete time stepping [181, 255]. Consider the fluid domain f (t) with the spatial and temporal coordinates denoted by x and t, respectively. The Navier-Stokes (NS) equations governing an incompressible flow in the ALE reference frame are     f  ∂uf  f f + u − w · ∇u = ∇ · σ f + bf on f (t), (9.1) ρ ∂t χ ∇ · uf = 0 on f (t),

(9.2)

where ρ f , uf , w, σ f , and bf are the fluid density, the fluid velocity, the ALE mesh velocity, the Cauchy stress tensor and the body force per unit mass, respectively. For the partial time derivative in Eq. (9.1), the ALE referential coordinate χ is held fixed and for a Newtonian fluid σ f is defined as  T   , (9.3) σ f = − pI + μf ∇uf + ∇uf where p, μf and I are the pressure, the dynamic viscosity of the fluid and an identity tensor, respectively. A rigid-body structure submerged in the fluid experiences unsteady fluid forces and consequently may undergo flow-induced vibrations if the body is mounted elastically. To simulate translational motion of a two-dimensional rigid body about its center of mass, the Lagrangian motion along the Cartesian axes

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9 System Identification and Stability Analysis

is given by: m·

dus + c · us + k · (ϕ s (t) − ϕ s (0)) = Fs , dt

us (t) =

∂ϕ s , ∂t

(9.4)

(9.5)

where us (t) represents the velocity of immersed rigid body in the fluid domain, ϕ s (t) denotes the position of the center of the rigid body at time t, and m, c, k are mass, damping and stiffness coefficient matrices for the translational motions. Here, Fs is fluid force on the rigid body. Let (t) is the fluid-structure interface at time t, the coupled system requires to satisfy the continuity of velocity and the force equilibrium at the fluid-body interface  as follows uf (t) = us (t) ,

(9.6)

σ f (x, t) · nd + Fs = 0,

(9.7)

(t)

where n is the outer normal to the fluid-body interface. In Eq. (9.7), the first term represents the force exerted by the fluid and the second term is the solid load vector applied in Eq. (9.4). The ALE mesh nodes on the fluid domain  f (x, t) can be updated by solving a linear steady pseudo-elastic material model ∇ · σ m = 0,

σ m = (1 + km )

 T      + ∇ · ηf I , ∇ηf + ∇ηf

(9.8)

where σ m is the stress experienced by the ALE mesh due to the strain induced by the rigid-body movement, ηf represents the ALE mesh node displacement and km is a mesh stiffness variable chosen as a function of the element area to limit the distortion of small elements located in the immediate vicinity of the fluid-body interface. The weak variational form of Eq. (9.1) is discretized in space using Pn /Pn−1 isoparametric finite elements for the fluid velocity and pressure, where Pn denotes the standard n th order Lagrange finite element space on the discretized fluid domain. To satisfy the inf-sup condition, P2 /P1 finite element mesh is adopted and the secondorder backward scheme is used for the time discretization of the Navier-Stokes system [255]. In the present study, a partitioned staggered scheme is considered for the fullorder simulations of fluid-structure interaction [178]. The motion of the structure is driven by the traction forces exerted by the flowing fluid at the fluid-structure interface , whereby the structural motion predicts the new interface position and the geometry changes for the ALE fluid domain at each time step. The movement of the internal ALE fluid nodes is accommodated such that the mesh quality does not deteriorate as the motion of the solid structure becomes large. The above coupled

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487

variational formulation completes the presentation of the full order model for the direct numerical simulation of fluid-structure interaction. We next present a statespace formulation of the model reduction using a system identification technique based on the input-output dynamics.

9.2.2 Eigensystem Realization Algorithm The linear time-invariant (LTI) and multiple-input multiple-output (MIMO) model represented in a state-space form at discrete times t = kt, k = 0, 1, 2, ..., with a constant sampling time t reads xr (k + 1) = Ar xr (k) + Br u(k)



yr (k) = Cr xr (k) + Dr u(k)

,

(9.9)

where xr is an n r -dimensional state vector, u denotes a q-dimensional input vector and yr is a p-dimensional output vector. The integer k is a sample index for the time stepping. The system matrices are (Ar , Br , Cr , Dr ), whereby the transition matrix Ar characterizes the dynamics of the system. Here, Br , Cr and Dr denote the input, output and feed-through matrices, respectively. For the given output vector yr , the statement of system realization is to construct the system matrices (Ar , Br , Cr , Dr ), such that the vector yr is reproduced by the state-space model. In a discrete-time setting, the state-space realization matrices (Ar , Br , Cr , Dr ) of the dynamical system are constructed by the ERA, in which only the impulse response function (IRF) of the original full order system is required for the system realization. The impulse 

response of the full-order linear system is first defined as y = y1 , y2 , y3 , ..., yni , where n i represents the length of the impulse response and yi denotes IRF with the dimension p × q . Based on the impulse response, the generalized block Hankel matrices r × s can be constructed as ⎡ ⎤ yk+1 yk+2 ... yk+s ⎢ yk+2 yk+3 ... yk+s+1 ⎥ ⎢ ⎥ (9.10) H(k − 1) = ⎢ . ⎥. .. .. .. ⎣ .. ⎦ . . . yk+r yk+r +1 ... yk+(s+r −1) From the partitioned SVD of the Hankel matrix H(0), we can have ∗

H(0) = U σ V = [U1



σ1 0 U2 ] 0 σ2



 V1∗ , V2∗

(9.11)

where the diagonal matrix  are the Hankel singular values (HSVs) σi , which represents the dynamical significance through sorting such that σ1 ≥ . . . ≥ σn ≥ 0. The block matrix 2 contains the zeros or negligible elements. By truncating the dynami-

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9 System Identification and Stability Analysis

cally less significant states, we estimate H(0) ≈ U1 σ 1 V1∗ . The reduced system matrices (Ar , Br , Cr , Dr ) are then defined as Ar Br Cr Dr

−1/2

−1/2 ⎫

= 1 U1∗ H(1)V1 1 1/2 = 1 V1∗ Em 1/2 = Et ∗ U1 1 = y1

⎪ ⎪ ⎬ ⎪ ⎪ ⎭

.

(9.12)



 Here, Em ∗ = Iq 0 q×N , Et ∗ = I p 0 p×M , where N = s × q, M = r × p, I p and Iq are the identity matrices. We next present the ERA to construct the fluid ROM, which relies on the incompressible NS equations to represent the dynamics of a small amplitude perturbation around the equilibrium base flow.

9.2.3 ERA-Based Coupled Formulation of a Cylinder VIV In the present work, we only consider the transverse motion of a cylinder in a flowing stream for the sake of simplicity. However, the formulation of ERA-based ROM is general for any fluid-structure system. The cylinder is mounted on a spring system in the cross-flow direction, which allows the cylinder to vibrate through an unsteady lift comprising of the pressure and shear stresses of the fluid. Owing to the direct solution of the Navier-Stokes equations, the effects of added-mass and added damping forces are implicitly captured in the present model. To perform linear stability analysis, the fluid ROM constructed by ERA is coupled with the linear structural model. The nondimensional structural equation for a transversely vibrating cylinder with onedegree-of-freedom can be written as as Y¨ + 4ζ π Fs Y˙ + (2π Fs )2 Y = ∗ Cl , m

(9.13)

where Y is the transverse displacement; Cl is the lift coefficient, m ∗ and ζ are the ratio of the mass of the vibrating structure to the mass of the displaced fluid and the damping coefficient, respectively; Fs is the reduced natural frequency of the structure defined as Fs = f N D/U = 1/Ur , where Ur is the reduced velocity which is an alternative parameter to describe the frequency lock-in phenomenon. Mass ratio m ∗ is a key parameter for VIV lock-in and it is defined as the ratio of vibrating structure to the mass of displaced fluid. The characteristics length scale factor as is related to the geometry of the body. For example, the values are as = π2 for a circular cylinder, and as = 0.5 for a square cylinder. There exists a complex dynamical relation between the transverse amplitude Y and the lift force Cl . One of the main objectives of this work is to construct a state-space relationship between the transverse force and the amplitude directly from the NS equations subject to an impulse. We next proceed to the model reduction of fluid-structure system.

9.2 Numerical Methodology

489

The non-dimensional structural Eq. (9.13) can be cast into a state-space formulation as (9.14) x˙s = As xs + Bs Cl , where the state matrices and vectors are       Y 0 0 1 , B , x As = = = . s s a 2 s −(2π Fs ) −4ζ π Fs Y˙ m∗ It is straightforward to solve Eq. (9.14) using a standard time-integrator [467]. In the present work, an exact state-space discretization of Eq. (9.14) is considered as follows: xs (k + 1) = Asd xs (k) + Bsd Cl (k), , (9.15) Y (k) = Csd xs (k) where the state matrices are Asd = eAs t , Bsd = As −1 (eAs t − I)Bs at discrete times t = kt, k = 0, 1, 2, ..., with a constant sampling time t, I is an identity matrix with the same size of As , and Csd = [1 0]. Through the input-output dynamics, the fluid ROM is derived by the ERA method as described in Eq. (9.9). The input for the ROM is the transverse displacement Y , and the output is the lift coefficient Cl . The ERA-based ROM with the single-input and single-output (SISO) can reformulated as: xr (k + 1) = Ar xr (k) + Br Y (k), (9.16) Cl (k) = Cr xr (k) + Dr Y (k). Substituting Eq. (9.16) to Eq. (9.15), the resultant ROM can be expressed as  xfs (k + 1) =

 Asd + Bsd Dr Csd Bsd Cr xfs (k) = Afs xfs (k), Br Csd Ar

(9.17)

where (Ar , Br , Cr , Dr ) are the ROM matrices defined by the ERA method as given in Eq. (9.9) and Afs denotes the coupled fluid-structure matrix in the discrete state-space form, and xfs = [xs xr ]T . The present ERA-based ROM reproduces the the input-output dynamics of the full order system. The linear stability analysis of the VIV system can be expressed into an eigenvalue problem of Eq. (9.17). The eigenvalue distribution of coupled x) correfluid-structure matrix Afs characterizes the stability of VIV system. (λ, spond to continuous-time eigenvalue/eigenvectors of Afs , whereby the spatial structure is characterized by the complex vector field  x and their temporal behavior by the complex scalar λ. The stability analysis can be easily accomplished by tracing the trajectory of complex eigenvalue λ in the complex plane, whereby x provides the spatial global modes of the ROM. Based on the leading global mode or least damped eigenvalue of ERA-based ROM, we define growth (amplification) rate σ = Re(λ) and frequency f = Im(λ/2π ). The construction of the above ERA-based ROM model is

490

9 System Identification and Stability Analysis

computationally efficient as it only relies on the impulse response of the FOM. While the aforementioned formulation is presented for the transverse-only vibration of the structure, it is general for any coupled fluid-structure system. After reviewing the mathematical formulation and ERA-based ROM technique, we next present the numerical set-up and verification of our solver.

9.2.4 Problem Definition Fig. 9.27 shows a schematic diagram of the setup used in our simulation study for an elastically mounted bluff body with various cross-sections in a flowing stream. The coordinate origin is located at the geometric center of the bluff body. The streamwise and transverse directions are denoted x and y, respectively. A stream of incompressible fluid enters the domain from an inlet boundary in at a horizontal velocity (u, v) = (U, 0), where u and v denote the streamwise and transverse velocities, respectively. The bluff body with mass m and characteristic diameter D is mounted on a linear spring in the transverse direction. The damping coefficient ζ is set to zero in the present work. The computational domain and the boundary conditions are also illustrated in Fig. 9.27. No-slip wall condition is implemented on the surfaces of the bluff body, and a traction-free boundary condition is implemented along the outlet out while the slip wall condition is implemented on the top top and bottom bottom boundaries. The numerical domain extends from −10D at the inlet to 30D at the outlet, and from −15D to 15D in the transverse direction. Except stated otherwise, all positions and length scales are normalized by the characteristic dimension D, velocities with the free stream velocity U , and frequencies with U/D. The Reynolds number Re of flow is based on the characteristic dimension D, the kinematic viscosity of the fluid and free-stream speed U (Fig. 9.1).

Fig. 9.1 Schematic diagram of a representative bluff body of elastically mounted cylinder in uniform horizontal flow. Computational domain and boundary conditions are shown

9.3 Linear Stability Analysis for VIV of a Cylinder

491

9.3 Linear Stability Analysis for VIV of a Cylinder We verify the validity of our ERA-based ROM by investigating the stability of laminar wake behind a two-dimensional circular cylinder. To study the mesh convergence for our simulation study, we consider three representative discretizations M1, M2 and M3 consisting of 9517, 22,004, and 41,132 P2 /P1 iso-parametric elements. A representative M2 mesh is shown in Fig. 9.28 and the corresponding central square represents the fine mesh region around the cylinder body. The mesh in the cylinder wake is appropriately refined to resolve the alternate vortex shedding. The quality of mesh convergence is determined by the prediction of growth rate σ and the frequency f of fluid ROM for the flow past a circular cylinder at Re = 60. The nondimensional time step size is t = 0.05. Based on the procedure described in the previous section, we next briefly describe the process of extracting ROMs from the incompressible NS equations. The ERA-based ROM is constructed in the neighborhood of the base flow, which is computed via fixed-point iteration without the time-dependent term in Eq. (9.1). At Re = 60 with M2 mesh, Fig. 9.3 shows the streamwise velocity contours of the base flow with a symmetric recirculation bubble. The bubble length (measured from the center of the cylinder) is L w = 4.1, which agrees reasonably well with the literature (L w = 4.2, [131]; L w = 4.1, [274]). The Hankel matrix shown in Eq. (9.10) is obtained from the output lift signal (Cl ) subject to the impulse input of transverse displacement Y . A sufficiently small amplitude (δ = 10−4 ) is considered such that the flow evolves linearly for a relatively large time. To ensure that the unstable modes start to dominate the essential dynamics of the input-output relationship, an adequate number of cycles is required to capture the linear dynamics of the system. However, the excessive simulation time should be avoided before the nonlinearity appears via exponential growth of the unstable modes.

15 2 10 1

y

y

5 0 -5

-1

-10 -15 -10

0

-2 -5

0

5

10

15

20

25

30

-2

0

x

x

(a)

(b)

2

Fig. 9.2 A representative finite element mesh with P2 /P1 discretization: a full domain discretization and b close-up view of finite element mesh in the vicinity of the cylinder. All other meshes for different bluff body geometries are similarly created

492

9 System Identification and Stability Analysis

y

2

0

-2 0

2

x

4

6

8

Fig. 9.3 Base flow of a stationary circular cylinder at Re = 60; streamwise velocity contours are shown. Contour levels are from −0.1 to 1.2 in the increment of 0.1 and the flow is from left to right

The linearity of the unstable system is confirmed by comparing the response subject to the two impulse inputs with δ = 10−4 and δ = 10−3 . A set of 1000 impulse response outputs (Cl ) is then stacked at each time step t = 0.05, resulting the final simulation time tU/D = 50. The adequate simulation cycles are then determined by examining the convergence of the fluid unstable eigenvalues computed from the Hankel matrices with the dimensions of 500 × 20, 500 × 50, 500 × 150, and 500 × 200. It was found that the Hankel matrix 500 × 150 is sufficient, which corresponds to 32.5 convective time units. The order of ERA-based ROM is then determined by examining the singular values (HSVs) of the Hankel 500 × 150 matrix. As shown in Fig. 9.30, the output lift signal Cl gradually diverges as a function of convective time tU/D at RRe = 60. This asymptotic divergence behavior indicates that the wake flow begins to lose its stability via a Hopf bifurcation, which breaks the symmetry of the flow field and gives rise to the periodic vortex street. The first 30 singular values are shown in Fig. 9.4b. The fast decaying singular values indicate that a low order ROM can be constructed. To further confirm the accuracy, the ERA-based ROM with the number of modes n r = 25 is compared with the FOM result in Fig. 9.30. A good match between the impulse response of the ROM and FOM can be seen in the figure. It is worth noting that the Hankel matrix is not necessarily to be a square matrix for the suitability of ERA-based ROM [199]. As pointed by [199], further consideration is required to determine optimal r and s in Eq. (9.10). Therefore, the Hankel matrix can be tall (r > s), wide (r < s) or square (r = s). In the present study, the dimension r is fixed while s is tuned to obtain a reasonably converged unstable eigenvalue for a good match between the impulse response of FOM and ROM. Table 9.1 summarizes the comparison of the growth rate and frequency, which shows that the difference between M2 and M3 is less than 1%. Therefore, the mesh M2 is adequate for the stability analysis of VIV. This study also confirms the convergence property of our ERA-based ROM procedure for unstable wake flow. For further verification, we next show that the ERA-based ROM is able to accurately

9.3 Linear Stability Analysis for VIV of a Cylinder

493 104

1.5E-05

100 HSV

Cl

FOM ROM (nr=25) 0

10-4 10-8

-1.5E-05 0

10

20

30

40

50

0

10

20

tU/D

HSV index

(a)

(b)

30

Fig. 9.4 ERA-based ROM for the unstable wake behind a stationary circular cylinder at Re = 60: a lift history of full order model and the ROM based on linearized dynamics subject to the impulse response, and b Hankel singular values (HSVs) distribution corresponding to 500 × 150 Hankel matrix Table 9.1 Mesh convergence study: comparison of growth rate and frequency for meshes M1, M2 and M3 for the flow past a stationary circular cylinder at Re = 60 Mesh Nodes Elements σ f M1 M2 M3

4834 11,119 20,731

9514 22,004 41,132

0.0479 0.0484 0.0483

0.1207 0.1207 0.1207

predict the onset of the unstable wake state due to a Hopf bifurcation. The onset of the unstable wake is commonly determined by the linearized NS equations in the literature [131, 279, 285]. The growth rate σ and the frequency f are plotted as a function of Reynolds number in Fig. 9.5. The instability of the base flow occurs when the growth rate crosses σ = 0 line at the critical Reynolds number Recr ≈ 46.8, which is in a good agreement with the numerical prediction of [131, 279] and the experimental measurement of [450]. The frequency predicted by the ERA-based ROM at this critical Reynolds number is f = 0.119, which matches quite well with the results of [131, 279]. However, it is worth noting that the frequency given by the linear model is only valid in the vicinity of critical Reynolds number and fails to capture the frequency in the region far away from the critical point, where nonlinear effects start to dominate the wake dynamics. To extract the most energetic structures via the POD method, the snapshots of the flow solutions are stored during the process of the ROM construction i.e., the flow solution is recorded at each physical time step subjected to the impulse response. For the unstable wake case at Recr ≈ 46.8, the first POD mode corresponding to the streamwise and cross-flow velocity is shown in Fig. 9.6. The spatial structures are antisymmetric and they travel downstream with the formation of Kelvin-Helmholtz instabilities.

494

9 System Identification and Stability Analysis

0.1 0.12

0

f

σ

0.05

-0.05

0.115

-0.1

50

75

Re

50

(a)

75

Re (b)

5

5

0

0

-5

y

y

Fig. 9.5 Prediction of critical Reynolds number via ERA-based ROM for the flow past a stationary circular cylinder: a growth rate σ and b frequency f . The cylinder wake becomes unstable when the growth rate crosses σ = 0 line at the critical Recr ≈ 46.8 and the vortex shedding emanates

0

10

x (a)

20

-5

0

10

x (b)

20

Fig. 9.6 First POD mode at Recr ≈ 46.8: a streamwise velocity and b cross-stream velocity. The flow is from left to right

9.3.1 Unstable Flow Past a Stationary Cylinder As discussed in Sect. 9.3, the wake flow becomes unstable through a Hopf bifurcation when Re > Recr and the vortex shedding appears behind a stationary cylinder at the frequency f vs . The unstable flow finally reaches a fully nonlinear state with the alternate time-periodic vortex shedding. The flow field in the whole domain behaves like a global oscillator, which causes unsteady lift and drag forces on the immersed body. To further establish the validity of the numerical method and the desired accuracy for the VIV simulation, the dimensionless vortex shedding frequency St = f vs D/U and the root mean square (rms) value of lift coefficient Cl are compared with the works of [452, 484] for the Reynolds number Re < 180. The results are summarized in Table 9.2, which demonstrates a good agreement with the literature. This further confirms that the numerical methodology and the mesh discretization are adequate to capture the vortex dynamics and the stability characteristics of the VIV response.

9.3 Linear Stability Analysis for VIV of a Cylinder

495

Table 9.2 Comparison of the rms value of lift coefficient (Cl ) and Strouhal number (St) obtained with previous studies for a range of Reynolds numbers Reynolds number (Re) 60 80 100 120 Lift coefficient rms (Cl ) Strouhal number (St)

0.1 0.1 0.137 0.136 0.142

0.17 0.16 0.156 0.152 0.159

0.24 0.25 0.166 0.164 0.172

0.29 0.31 0.176 0.172 0.182

Present [484] Present [452] [484]

A constant time step size t = 0.05 is employed for the present computations

9.3.2 Assessment of ERA-Based ROM In this section, the constructed ERA-based ROM is first applied to analyze the stability properties of the transversely vibrating circular cylinder at (Re, m ∗ ) = (60, 10). Consistent with the previous literature of [283, 486], we use the terminology of structure mode (SM) and the wake mode (WM) to classify the distinct eigenvalue trajectories of the linear fluid-structure system governed by Eq. (9.17). When the eigenvalue of the fluid-structure system approaches that of the stationary cylinder, the resulting mode is defined as WM. The fluid-structure mode is considered as SM if the eigenvalue comes close to the natural frequency of the cylinder-alone system. As discussed in [486], VIV lock-in may result either from the instability of WM alone or via simultaneous existence of unstable SM and WM. In the event of the first scenario, the lock-in occurs due to the closeness of the frequencies of WM and SM. This type of VIV branch is termed as the resonance-induced lock-in. For the second scenario, the instability to sustain the VIV lock-in occurs via combined mode instability of SM and WM, which is referred to as the flutter-induced lock-in or the coupled-mode flutter [103]. In the present study, the wake mode is also considered as the leading mode of the unstable fluid system. We consider the continuous-time eigenvalues in the context of linear stability analysis via the ERA-based ROM. Using the fluid-structure matrix Eq. (9.17), the eigenvalue is obtained by λ =log(eig(Afs ))/t, where t = 0.05 is the time step. To graphically depict the dynamical behavior, we study the eigenvalue distribution of the system in the complex plane via root locus. The positions of the eigenvalues provide information about the stability of the fluid-structure system. For example, roots in the right half plane depict the unstable modes of the system, whereas the roots on the real axis characterize the asymptotic (non-oscillatory) behavior. Roots that are closest to the right half plane are lightly-damped oscillatory modes. Figure 9.7a shows the eigenvalue trajectory of the fluid-structure system as a function of the reduced natural frequency Fs with 0.05 ≤ Fs ≤ 0.25 and the increment is Fs = 0.025. In the figure, the WM branch originates in the vicinity of the eigenvalue of stationary cylinder (uncoupled WM) and loops back as the reduced natural frequency Fs increases. It is expected that the WM finally recovers to the eigenvalue of the stationary cylinder as Fs approaches infinity or the cylinder becomes

496

9 System Identification and Stability Analysis 0.1

1.2

1

Resonance

0

Fs=0.179

Im(λ)

Flutter

Re(λ)

SM WM stationary

0.1

0.15

0.2

0.15

0.2

0.8

Im(λ)

1

Fs=0.147

f=2πFs

0.6 0.5

-0.05

0

0.05

0.1

0.1

Re(λ)

Fs

(a)

(b)

Fig. 9.7 Eigenspectrum of the ERA-based ROM for a circular cylinder at (Re, m ∗ ) = (60, 10): a root loci as a function of the reduced natural frequency Fs , where the unstable right-half (Re(λ) > 0) plane is shaded in grey color, and the hollow arrow indicates increasing Fs , b real and imaginary parts of the root loci, whereby the lock-in region is shaded in grey color. Two branches of lock-in namely resonance and flutter can be seen in (b)

stationary (i.e., without elastically mounted). It is also evident that WM remains unstable (σ > 0) throughout the sweeping as Re = 60 > Recr . On the other hand, the SM branch arises from the bottom of the complex plane (low-frequency regime) to the upper complex plane (high-frequency regime). As elucidated in Fig. 9.7b, the SM becomes unstable only when 0.147 < Fs ≤ 0.179, which is determined by the real part of eigenvalue. As mentioned earlier, the unstable SM phenomenon can be considered as the coupled-mode flutter or the combined mode instability. As shown in Fig. 9.7b, the imaginary part of eigenvalue as a function of Fs reveals that the two distinct frequencies (WM and SM frequencies) co-exist in the combined mode instability. In addition, Fig. 9.7b also indicates the frequencies of WM and SM come closer when 0.11 ≤ Fs ≤ 0.147, which is recognized as the resonance mode. Note that the lower left boundary of the resonance mode cannot be pinpointed precisely from the ROM due to the overlapping of SM and WM trajectories. Thus we determine the frequency at the lower left boundary from the FOM simulation, which is found to be Fs = 0.11. To further verify the stability results predicted by the ERA-based ROM, the VIV response is computed by direct numerical simulation using FOM. As shown in Fig. 9.8a, the vortex shedding frequency begins to synchronize with the structure natural frequency at Fs = 0.11 and recovers to the vortex shedding frequency at Fs = 0.179. As the nonlinearity starts to dominate the VIV response, the two distinct frequencies of WM and SM corresponding to the flutter mode are eventually synchronized with the structure natural frequency Fs . Figure 9.8b suggests that the

9.3 Linear Stability Analysis for VIV of a Cylinder Resonance Flutter

0.6

f=Fs

Ymax 0.5 Cl

Ymax

f

0.4 0.15

0.3

Cl

0.2

497

0.2 0.1 0.1

0.1

0.15

0.2

0.25

0

0.1

0.15

0.2

Fs

Fs

(a)

(b)

0.25

Fig. 9.8 VIV results as a function of reduced natural frequency Fs using FOM at (Re, m ∗ ) = (60, 10): variation of a normalized vortex shedding frequency f and b rms value of lift coefficient (Cl ) and normalized maximum amplitude (Ymax ). The lock-in is shaded in grey color

cylinder rises to the peak VIV amplitude at Fs = 0.179 (lock-in onset Ur ≈ 5.59), which compares accurately with the upper boundary of flutter mode predicted by the present ERA-based ROM. It is worth mentioning that the cylinder acquires the maximum amplitude at Fs = 0.179 or Fs /St = 1.31, which is not at Fs /St ≈ 1, as expected from the classical resonance interpretation of VIV lock-in. This phenomenon suggests that the onset of VIV lock-in results from the amplification of energy input as a consequence of unstable SM, in which the structure is able to optimally absorb energy from the surrounding fluid system. It is analogous to the pitch and plunge flutter observed in the aeroelastic airfoil configuration. The flutter mode of VIV lock-in results from the coupling of periodic vortex shedding and the structural transverse displacement. In the flutter regime (1.07 < Fs /St ≤ 1.31), the unstable SM and WM jointly sustain VIV lock-in, whereas the wake mode dominates the resonance regime (0.8 ≤ Fs /St ≤ 1.07) until the VIV goes into the lock-out region. More quantitative insights into the VIV lock-in mechanism can be obtained by Fig. 9.9a, which shows the phase angle φ estimated by the ERA-based ROM (see Appendix A). The phase angle φ of the ROM is function of (Fs , λ) and its sign is determined by the real part of eigenvalues. The instantaneous phase angle of FOM is obtained by the Hilbert transformation of lift and displacement, as described in [416]. In Fig. 9.9a, we present the phase angles of the FOM and ROM as functions of Fs at (Re, m ∗ ) = (60, 10). As compared to the FOM result, the WM trajectory yields a continuous transition from 0◦ to 180◦ as Fs decreases in the lock-in region. In contrast, the phase angle of SM remains positive only within the flutter mode (0.147 < Fs ≤ 0.179). It is also interesting to note in Fig. 9.8b that the lift coefficient is significantly amplified in the vicinity of VIV lock-in onset reduced velocity (Ur ≈ 5.59). A gradual decrease and eventual recovery to the value of stationary cylinder counterpart as Ur increases (Fs decreases) can be seen in the figure. To further assess the behavior of lift coefficient in the flutter and resonance regimes, the lift histories for two representative reduced frequencies Fs = 0.177 and 0.140 are compared in Fig. 9.9b. The minimum rms value of lift coefficient Cl occurs at Fs ≈ 0.13 or Fs /St ≈ 0.95,

498

9 System Identification and Stability Analysis

180

FOM SM WM

60

Cl

φ

120

1.0

Fs=0.177 Fs=0.140

0.0

0 -60

-1.0 0.1

0.15

0.2

0.25

150

300

Fs

tU/D

(a)

(b)

450

Fig. 9.9 VIV results as a function of reduced natural frequency Fs at (Re, m ∗ ) = (60, 10): a comparison of phase angle difference φ between the ERA-based ROM and FOM, where the lock-in is shaded in grey color and b lift Cl history at two representative frequencies Fs = (0.14, 0.177). In (a), ( ) represents Fs = 0.13

which coincides with the phase angle jump, as shown in Fig. 9.9a. Therefore, we can infer that the reduction in the rms lift coefficient has a direct relation with the phase angle. In Sect. 9.3.4, we further confirm that the lift reduction is not associated with the resonance mode. Geometric and physical parameters such as cross-sectional shape and mass ratio play an important role with regard to the coupling strength of fluid-structure interaction. A classification of the fluid-structure eigenmodes as WM and SM is suitable for weak fluid-structure interaction (e.g., very large mass ratio), which is elucidated in Fig. 9.7 by two distinct eigenvalue branches of WM and SM. Owing to weaker fluidstructure coupling at large m ∗ (i.e., in the limit of m ∗ → ∞), the eigenfrequency of WM approaches to that of the stationary cylinder wake for all values of Fs and the frequency of SM comes close to the natural frequency of the cylinder-only system. However, the root loci of WM and SM can coalesce and form coupled modes at certain conditions, such as in the limit of low mass ratio [283, 311] and for different geometrical shapes, as demonstrated in Sect. 9.3.6. These coupled modes do not offer distinct characteristics of the WM and SM, since both branches exchange their roles when the coalescence of eigenvalue occurs. Similar to [311], we term these mixed or coupled modes as the wake-structure mode I (WSMI) and the wake-structure mode II (WSMII). For the higher value of Fs , WSMI behaves as WM and WSMII recovers to SM. On the other hand, for the smaller Fs range, WSMI and WSMII represent the SM and WM, respectively. A characteristic anticrossing with a frequency splitting can be also observed at low mass ratio, which is one of the traits of strongly coupled harmonic oscillators [319]. Next, we demonstrate the effect of the mass ratio, with further details of the coupled modes WSMI and WSMII.

9.3 Linear Stability Analysis for VIV of a Cylinder

499

9.3.3 Effect of Mass Ratio For the illustration of ERA-based ROM, the effect of mass ratio is shown in Fig. 9.10 for m ∗ = (5, 7.6, 20) at Re = 60. Figure 9.10b shows the real and imaginary parts of the root loci as a function of reduced frequency Fs . It indicates the lock-in onset starts to move to the lower reduced velocity (Ur = 1/Fs ) regime as the mass ratio m ∗ decreases. As expected, owing to a weaker fluid-structure coupling for larger mass ratio m ∗ = 20, the eigenfrequency of WM recovers to the frequency of stationary cylinder and the frequency of SM approaches to the natural frequency of the cylinderonly system as Fs increases. As the mass ratio decreases further, Fig. 9.10a shows that the root loci of SM and WM gradually coalesce and form a coupled mode due to the increased strength of fluid-structure coupling. The approximate threshold mass ratio is m ∗ ≈ 7.6 for the coupled mode, which is very close to the predicted m ∗ = 7.3 in [486]. The phenomenon is termed as mixed WM/SM [283] or coupled fluid-elastic mode [311]. As illustrated in Fig. 9.10b, for the mass ratio m ∗ = 5, the coupled wake-structure modes WSMI and WSMII resemble the SM and WM, respectively for Fs ≤ 0.175, whereas WSMI and WSMII resemble the standard WM and SM, respectively for Fs > 0.175. This finding suggests that the stability roles of WM and SM switch at a specific value of Fs , where the two growth rate Re(λ) curves of WSMI and WSMII intersect. The flutter regime for m ∗ = 5 is then defined by 0.165 < Fs ≤ 0.197 (5.08 ≤ Ur < 6.07), as shown in Fig. 9.10b, which matches well with the results of [311] (5.0 ≤ Ur < 6.0) at identical conditions. In Fig. 9.10b (bottom), the two curves of Im(λ) for m ∗ = (5, 7.6) no longer cross and there is a characteristic anticrossing with a frequency splitting ( f ) between the WSMI and WSMII. This phenomenon of anticrossing is an intrinsic property of strong coupling at the low mass ratio, as reported for generic coupled mechanical oscillators in [319]. In this section, the ERA-based ROM is successfully employed to perform the linear stability analysis of the vortex-induced vibration of transversely vibrating circular cylinder. Consistent with the previous study of [486] on the mechanism of VIV, we clearly observe two distinct lock-in patterns of the flutter and the resonance from our eigenmode analysis. However, the regime classification in [486] was only based on the VIV linear analysis at Re = 60. In the next section, we extend the existence and dependence of the two distinct lock-in modes for a larger parameter space of Reynolds number in the laminar flow regime.

9.3.4 Effect of Reynolds Number As shown in Fig. 9.5a, the growth rate amplifies as Re increases, which indicates that the coupling between the WM and SM tends to become stronger for higher Re. To further investigate the effect of the Reynolds number, the VIV ROMs for Re = (70, 100) and m ∗ = 10 are constructed and the stability analysis similar to Re = 60 is carried out. The root loci as a function of natural frequency Fs are shown

500

9 System Identification and Stability Analysis

1.25

*

WSMII (m =5) * WSMII (m =7.6) * WM (m =20)

Re(λ)

0.1

0 0.1

Im(λ)

1

Im(λ)

1.5

0.75

0.5 -0.05

0

0.05

Re(λ)

(a)

0.1

0.15

0.15

0.2

0.8

2πΔf

1

0.5

0.1

0.15

f=2πFs

0.1

0.15

0.2

Fs (b)

Fig. 9.10 Effect of mass ratio on the eigenspectrum of ERA-based ROM for a circular cylinder at m ∗ = (5, 7.6, 20) and Re = 60: a root loci as a function of the reduced natural frequency Fs , where the unstable right-half (Re(λ) > 0) plane is shaded in grey color and the hollow arrow indicates increasing Fs , b real and imaginary parts of root loci. In (Reproductive), growth rate Re(λ) curves of WSMI and WSMII intersect at Fs = 0.175 () and the flutter regime is defined between Fs = 0.197 () and Fs = 0.165 () for m ∗ = 5. The frequency anticrossing is shown in the inset of Im(λ) plot. The WSMI and SM branches are denoted by the filled symbol with the same shape as WSMII and WM in (a, b)

in Fig. 9.2b. As compared to the root loci at Re = 60, Fig. 9.2b shows that the range of unstable SM or flutter mode slightly increases to 0.106 ≤ Fs ≤ 0.187 or 0.71 ≤ Fs /St ≤ 1.25 and covers the entire lock-in region. It is also evident by the FOM results, as shown in Fig. 9.12, where the lock-in region is 0.11 ≤ Fs ≤ 0.192 or 0.73 ≤ Fs /St ≤ 0.128. This new finding from the present work suggests that the extents of flutter and resonance modes are highly sensitive to Reynolds number. This can be further elucidated by looking into the stability region where the magnitude of velocity leading eigenmode is large [131]. The complex modal velocity components, the streamwise velocity  u and the transverse velocity  v, are computed from the linearized NS equations around the base flow. To visualize the magnitude of the leading modal velocity, we first compute the amplitude of the complex modal velocity components u | and | v|) and then evaluate the pointwise modal velocity magnitude  (| | = | v|2 . As shown in Fig. 9.13, the stability region shifts gradually to u |2 + | as |U the bluff body, which indicates the coupling effect between the unstable wake and the bluff body is enhanced as Re increases. Figure 9.11 shows that the unstable SM branch is gradually pronounced and covers the lock-in region as the Reynolds number increases, whereas the size of the WM

9.3 Linear Stability Analysis for VIV of a Cylinder

1.2

WM (Re=60) WM (Re=70) WM (Re=100)

Re(λ)

0.1

1

Im(λ)

501

0 0.1

0.15

0.2

0.15

0.2

0.8 1

Im(λ)

f=2πFs

0.6 -0.05

0

0.05

0.1

0.15

0.5

0.1

Re(λ)

Fs

(a)

(b)

Fig. 9.11 Effect of Reynolds number on the eigenspectrum of ERA-based ROM at Re = (60, 70, 100) and m ∗ = 10: a root loci as a function of the reduced natural frequency Fs , where the unstable right-half (Re(λ) > 0) plane is shaded in grey color, and hollow arrow indicates increasing Fs , b real and imaginary parts of the root loci. SM data points are denoted by the filled symbol with the same shape as WM in (a, b). The boundary of Re(λ) > 0 for SM at Re = 70 is defined at Fs = 0.106 ( ), and Fs = 0.187 ( ), in the real parts of root loci in (b), respectively f=Fs

0.6

0.7 0.5

Cl

0.15

Ymax Cl

0.4

Ymax

f

0.2

0.3

0.2

0.1 0.1

0.1

0.15

0.2

0.25

0

0.3

0.1

0.15

0.2

Fs

Fs

(a)

(b)

0.25

0.3

200 150 Fs=0.135

φ

100 50 0

0.1

0.15

0.2

0.25

0.3

Fs

(c) Fig. 9.12 VIV results as a function of reduced natural frequency Fs by FOM at (Re, m ∗ ) = (70, 10): variation of a normalized vortex shedding frequency f , b rms value of lift coefficient (Cl ) and maximum amplitude (Ymax ), and c phase angle (φ). The lock-in is shaded in grey color

9 System Identification and Stability Analysis 5

5

0

0

-5

y

y

502

0

10

x (a)

20

-5

0

10

x (b)

20

Fig. 9.13 Influence of Reynolds number on the stability regions defined by the pointwise modal | of leading eigenmodes for Re =: a 60, b 70. The flow is from left to right velocity magnitude |U Table 9.3 VIV lock-in regimes of circular cylinder for Reynolds number in the range 30 ≤ Re ≤ 100 and mass ratio m ∗ = 10. For Re > Recr , flutter regime comprises both unstable eigenvalues (Re(λ) > 0) of WM and SM, whereas resonance regime has only unstable WM VIV regimes Flutter Resonance Re ≤ Recr Recr < Re < 70 Re ≥ 70

  



loop becomes smaller. The threshold Reynolds number is approximately Re = 70 when the resonance-mode ceases to exist for m ∗ = 10. This result suggests that the frequency lock-in VIV is pure flutter mode instability for Re ≥ 70, which is consistent with the theoretical analysis of [103] using the wake oscillator model. However, due to the simplification in the wake oscillator model and without nonlinear and dissipative terms, a general statement on VIV lock-in as a coupled flutter mode may not be valid for all flow conditions. On the other hand, the numerical study of [486] is only valid for Recr < Re < 70. Table 9.3 summarizes the existence of flutter and resonance modes at different Reynolds numbers for a vibrating circular cylinder. It is also interesting to note in Fig. 9.12b that the reduction in the rms value of lift coefficient also appears, although the resonance regime does not exist at Re = 70. This observation suggests that the reduction of rms lift force does not interlink with the flutter or resonance regime, which is different from the conclusion by Zhang et al. [486] that the rms lift coefficient is attenuated in the resonance regime but amplified in the flutter region. As shown in Fig. 9.12b and c, the minimum lift coefficient appears at Fs ≈ 0.135 or Fs /St ≈ 0.92, where the phase angle changes from 0◦ to 180◦ . It further confirms that the lift reduction phenomenon is explicitly linked with the phase angle. Furthermore, Fig. 9.14 shows the instantaneous patterns of vortex shedding are investigated for a broad range of reduced natural frequencies. We adopt the classical terminology of [458] to identify the vortex shedding patterns. In the 2S mode, a single vortex is alternately shed from each side of the cylinder per shedding cycle, whereas the vortices start to coalesce in the far wake in the C(2S) mode. As reported by Zhang et al. [486], the 2S mode is observed in the resonance regime, whereas C(2S) mode appears in the flutter region. However, we argue that the vorticity topology changes gradually from the C(2S) to the 2S mode as Fs decreases from 0.19 to 0.13, which indicates the topology variation associated with the vibration amplitude. The C(2S)

0

0

y

5

-5

y

503

5

0

10

x (a)

-5

20

5

5

0

0

-5

y

y

9.3 Linear Stability Analysis for VIV of a Cylinder

0

10

x (c)

20

-5

0

10

x (b)

20

0

10

x (d)

20

Fig. 9.14 Full order results for the circular cylinder at (Re, m ∗ ) = (70, 10): instantaneous vorticity contours at Fs = a 0.13, b 0.15, c 0.17, d 0.19. Contour levels are from −0.5 to 0.5 in increment of 0.077 and the flow is from left to right

mode starts to appear at VIV lock-in onset Ur ≈ 5.21 (Fs = 0.192) and gradually transits to the 2S mode as the amplitude decreases. To further generalize our ERAbased ROM for the VIV lock-in regime, we next investigate the influence of rounding in the VIV lock-in mechanism of a square cylinder.

9.3.5 Effect of Rounding In the previous section, the effects of Reynolds number and mass ratio have been considered for the transverse VIV of circular cylinder. It is interesting to explore whether the aforementioned VIV lock-in regimes of circular cylinder is still applied to an elastically mounted square cylinder. As it is known that the presence of sharp corners on a square cylinder vastly alters the flow dynamics as compared to the ones with circular/elliptical section having smooth curves. Besides the angle of incidence, the sharp corners are important contributing factor in the geometry of bluff body, as they affect the flow separation points which in turn dictates the wake dynamics. By gradual removal of sharp corners of square cylinder, a circular cross-section can be recovered. As reported in [181], the VIV response of square cylinder is dramatically different in comparison to its circular cylinder counterpart. For example, the lock-in region of square cylinder is narrower and the amplitude is smaller for the similar VIV operational parameters (Re, m ∗ , ζ ), as extensively discussed in [180, 181]. Recently, a rounded square is also numerically studied in terms of primary and secondary instabilities [332], which shows that the sharp corner alters the flow topology significantly, subsequently changing the stability properties of wake dynamics. It is therefore important to consider the effect of rounding the corners of a square cylinder for the VIV mechanism. The VIV stability properties of five different cross-sections including circle and square are explored in the context of eigenvalue distributions.

504

9 System Identification and Stability Analysis

Fig. 9.15 Square-section bluff body with projected width D and rounding radius rs in uniform flow. Rounding is introduced by inscribing a quarter circle with rs at each edge of the square geometry. The square and circular cylinders correspond to the rounding radius of rs = 0 and rs = 0.5D, respectively

WM (rs=0) WM (rs=0.1D) WM (rs=0.2D) WM (rs=0.4D) WM (rs=0.5D)

Im(λ)

1

Re(λ)

0.1

0 0.1

0.15

0.2

0.15

0.2

0.8 f=2πFs

Im(λ)

0.9

0.6 -0.05

0

0.05

0.1

0.15

0.6

0.1

Re(λ)

Fs

(a)

(b)

Fig. 9.16 Effect of rounding rs on eigenspectrum of ERA-based ROM at (Re, m ∗ ) = (60, 10): a root loci as a function of the reduced natural frequency Fs , whereby the unstable right-half (Re(λ) > 0) plane is shaded in grey and the hollow arrow indicates increasing Fs , b real and imaginary parts of root loci. In (a), the unstable eigenvalues of stationary square cylinder (uncoupled WM) with different rounding values are connected by black curve with solid arrow indicating increasing rs . SM is denoted by the filled symbol with the same shape as WM in (a, b)

Figure 9.15 schematically depicts the square cylinder with a rounding radius rs . The edge length of the square with a sharp corner is denoted by D. Rounding is introduced by inscribing a quarter circle with rs at each edge of the square geometry, whereby rs = 0.5D corresponds to a circular shape and rs = 0 recovers to the basic square shape. The characteristic dimension D of all the cross-sections is identical, where D is the maximum dimension of the cylinder normal to the flow. The origin of the Cartesian coordinate system coincides with the center of the square. To begin with, the VIV linear analysis is conducted for a square cylinder with sharp corners at (Re, m ∗ ) = (60, 10). It is evident from Fig. 9.16a that the SM is stable, which suggests the lock-in is entirely dominated by the resonance mode.

9.3 Linear Stability Analysis for VIV of a Cylinder 0.3

f=Fs

f

Ymax

0.14 0.12 0.1

Ymax 0.3 Cl

0.2

Cl

0.16

505

0.1

0.1

0.15

0.2

0.1

0 0.05

0.25

0.1

0.15

Fs

0.2

0.25

Fs

(a)

(b) 200 150 Fs=0.125

φ

100 50 0

0.1

0.15

0.2

0.25

Fs

(c) Fig. 9.17 VIV results of square cylinder with sharp corners using FOM at (Re, m ∗ ) = (60, 10): variation of a normalized vortex shedding frequency f and b rms value of lift coefficient (Cl ) and maximum amplitude (Ymax ), and c phase angle (φ). The lock-in is shaded in grey color

Due to the absence of lock-in via flutter mode, the onset reduced velocity (Ur ) for a square cylinder may not be clearly recognized as compared to the circular cylinder counterpart. Furthermore, the region of the WM loop for the square cylinder is smaller than the circular cylinder counterpart, implying the coupling between the fluid and structure is reduced due to the sharp corners. In Fig. 9.16, the root loci for the fluidstructure system of the square cylinder provide an explanation that the lock-in only consists of resonance mode and the fluttering state disappears due to the presence of sharp corners for (Re, m ∗ ) = (60, 10). As expected, the sharp corners suppress the continuous movement of separation points, whereby the smooth circular cylinder promotes the movement of separation points. Figure 9.17 shows the frequency, the VIV response and the lift coefficient from the FOM simulation for the vibrating square cylinder. The extent of lock-in region can be observed as 0.11 ≤ Fs ≤ 0.154 or 0.87 ≤ Fs /St ≤ 1.21. The results illustrate that the maximum amplitude also acquires at the lock-in onset (Ur ≈ 6.49) approximately even when no flutter regime exists. It is notable that the maximum amplitude is smaller and the lock-in region is narrower than its circular cylinder counterpart, which is observed earlier for the square cylinder [181, 495]. Similar to its circular counterpart, the lift coefficient for a square cylinder also experiences an amplification in the vicinity of lock-in onset Ur ≈ 6.49 , and gradually recovers to the stationary counterpart as Fs decreases (Ur increases). The maximum value of rms lift coefficient is Cl = 0.314, as shown in Fig. 9.17b, which is approximately 3.1 times larger than the stationary square cylinder value (Cl = 0.1). For the VIV of a circular cylinder, on the other hand, the amplification of the rms lift

9 System Identification and Stability Analysis 5

0

0

y

-5

y

5

0

10

x (a)

-5

20

5

5

0

0

-5

y

y

506

0

10

x (c)

20

-5

0

10

x (b)

20

0

10

x (d)

20

Fig. 9.18 Full order results for the square cylinder at (Re, m ∗ ) = (60, 10): instantaneous vorticity contours at Fs = a 0.12, b 0.13, c 0.14, d 0.15. Contour levels are from −0.5 to 0.5 in increment of 0.077 and the flow is from left to right

coefficient is approximately 5.9 times than the stationary circular cylinder, as shown in Fig. 9.8b. To understand the direction of energy transfer between the fluid and the square cylinder, the phase angle is shown in Fig. 9.17c. Similar to the circular cylinder, there is a sudden jump from 0◦ to 180◦ during the lock-in region around Fs ≈ 0.125 or Fs /St ≈ 0.98, where the rms lift coefficient acquires the minimum value. For the range of Fs considered, two counter-rotating vortices (2S wake mode) are released every oscillation cycle from the rear of each cylinder, as shown in Fig. 9.18. From the vorticity fields, separations along the front corner of lateral edges for the square with sharp corners can be observed for the considered cases. While the C(2S) mode is observed in the vicinity of lock-in onset for the circular cylinder, the classic 2S wake mode is dominated for the square cylinder. This is consistent with our previous hypothesis that the wake pattern is closely related to the vibration amplitude. We next elucidate the influence of rounding on the distribution of eigenspectrum and the unstable global modes. As shown in Fig. 9.16a, the SM trajectory moves gradually toward the positive real axis (Re(λ) > 0) as the rounding radius rs increases. Meanwhile, the flutter region starts to appear and achieves the maximum extent for the rounding radius rs = 0.5D. As expected, the rounding of the corners delays the onset of separation hence, the rounding aids in reducing the bluffness of the square cylinder. It is also worth noting that unstable SM and WM loop are both more pronounced indicating that the coupling effect between the fluid and structure becomes stronger. However, the growth rate of uncoupled WM does not increase monotonically from the square (rs = 0) to the circular (rs = 0.5D) configuration, as shown by a black curve with solid arrow in Fig. 9.16a. The growth rate first decreases, then increases and eventually recovers to the growth rate of the circular cylinder.

0

0

y

5

-5

y

507

5

0

10

x (a)

-5

20

5

5

0

0

-5

y

y

9.3 Linear Stability Analysis for VIV of a Cylinder

0

10

x (c)

20

-5

0

10

x (b)

20

0

10

x (d)

20

Fig. 9.19 Stability regions shown by the spatial distribution of pointwise modal velocity amplitude | at Re = 60 for different rounding parameters rs = a 0.0, b 0.1D, c 0.2D, and d 0.4D. The flow |U is from left to right

By examining the real part of root loci in Fig. 9.16b, the onset of lock-in starts to move towards the low reduced velocity (high Fs ) as the rounding radius rs increases. |, In Fig. 9.19, from the comparison of stability regions through the contours of |U we observe that the rounding has significant effects on the wake topology, subsequently alerting the stability properties. The separation point can move widely as the rounding radius increases, indicating the flutter mode is more pronounced. It also shows that the stability region gradually moves close to the rear of the bluff body as rs increases, which is similar to the effect of Re, as discussed in Sect. 9.3.4. Consequently, the level of fluid and structure interaction is enhanced. This trend is monotonic as compared to the growth rate, suggesting that the flutter mode is pronounced gradually. It explains why the SM trajectory moves towards the right half plane monotonically. The WM branch, on the other hand, moves toward the imaginary axis positive or higher frequency direction, monotonically. For the similar VIV operating parameters, the circular cylinder is much easier to perturb as compared to its square counterpart at the same Reynolds number. While the rounding generally stabilizes the wake flow for a stationary square cylinder, it promotes the movement of separation points along the smooth rounded surface of the vibrating cylinder. Analogous to the aeroelastic flutter with plungetorsion mode coupling for an airfoil configuration, as pointed by [103], the transverse periodic displacement and the movement of separation points can form a similar dynamics for a transversely vibrating circular cylinder. The square cylinder with sharp corners restricts the free motion of separation points and is relatively stable in the sense that the lock-in onset reduced velocity Ur is greater than its circular counterpart. Moreover, as compared to the circular cylinder at (Re, m ∗ ) = (60, 10), the lock-in range of the square cylinder is narrower and only the resonance regime exists. To further generalize our findings, we next examine the lock-in regimes from the eigenvalue distributions for additional bluff bodies of a smooth curve geometry of elliptic cylinder and two sharp corner shapes of forward triangle and diamond cylinders.

508

9 System Identification and Stability Analysis

Fig. 9.20 Schematics of bluff-body geometries with relevant dimensions. Representative elliptical cylinder (left) has aspect ratio A R = 0.5, forward triangle (middle) is equilateral with angle 60◦ , and diamond (right) represents a square cylinder with sharp corners at 45◦ flow incidence

9.3.6 Effect of Geometry In this section, a set of three representative two-dimensional geometries is assessed to elucidate the frequency lock-in regimes and to demonstrate the ability of the developed ERA-based ROM. The three additional geometries namely ellipse as a smooth curve and forward triangle and diamond with sharp corners are shown in Fig. 9.20. The major axis with length D of the elliptic cylinder is placed normal to the flow direction and the aspect ratio A R = 0.5 is defined by the ratio of minor D/2 to major axis length. The forward triangle is equilateral and has the edge with length D normal to the flow and the peak corner is on the leeward side. Similar to [495], the diamond geometry is considered as a square cylinder with 45◦ flow incidence and the Reynolds number is defined by the edge length D as the characteristic length scale. Similar to the circular and square cylinders, the new geometries of the ellipse, the forward triangle and the diamond cylinders undergo the unsteady wake transition via a Hopf bifurcation at a critical Reynolds number Recr , which is responsible for the onset of the time-periodic vortex-shedding phenomenon. Table 9.4 shows the critical Reynolds number Recr of different geometries computed by our ERA-based ROM, which matches reasonably well with previous studies. It is worth noting that the forward triangle has the lowest critical Reynolds number for the initial wake transition from steady to unsteady flow. The unsteady transition of the ellipse with the aspect ratio A R = 0.5 and the diamond with sharp corners occurs lower than their circular and square counterparts. Due to unsteady lift and drag forces, the three geometries can undergo flow-induced vibration if mounted on the elastic supports. To further elucidate the lock-in mechanism, we plot the root loci and the real and imaginary parts of eigenvalues for the additional three bluff bodies in Fig. 9.21. The figure clearly shows that the geometry of the bluff body has a significant impact on the eigenvalue trajectory. The elliptical configuration has the lowest lock-in onset Ur followed by the diamond and the forward triangle configurations as shown in Table 9.4. Compared to the circular cylinder at the identical condition of (Re, m ∗ ) = (60, 10), the root loci of SM and WM coalesce for the diamond, ellipse and forward triangle configurations. Similar to the low mass ratio effect during the VIV of a

9.3 Linear Stability Analysis for VIV of a Cylinder

509

Table 9.4 Comparison of critical Reynolds numbers Recr between the available literature and the predicted values by ERA-based ROM for different topologies of bluff bodies Ellipse Circular Square Forward Diamond triangle Present Recr [417] [332] Lock-in onset Ur

38.0 38.8 – 4.18

46.8 47.2 46.7 5.59

44.7 – 44.7 6.49

35.5 – – 5.43

38.9 – – 4.63

The onset reduced velocity Ur of VIV lock-in from the present study is also outlined in the last row

circular cylinder, both the branches exchange their roles and no distinction can be made between the WM and SM for the three geometries. Therefore, we consider the coupled modes WSMI and WSMII to classify the stability characteristics of these geometries. When the intersection of growth rate curves corresponding the WSMI and WSMII occurs in Fig. 9.21b (top), the stability roles of WSMI and WSMII switch at a specific value of Fs for the three geometries. As shown in Fig. 9.21b (bottom), the two curves of Im(λ) for the three geometries no longer cross, in comparison to the circular cylinder counterpart at identical conditions. Owing to stronger coupling, there is a characteristic anticrossing with a frequency splitting between the WSMI and WSMII for the three geometries. In addition, the forward triangle branch departs further away from the line f = 2π Fs as compared to the diamond and elliptical cylinders. In contrast to the square cylinder (0◦ degree flow incidence) at (Re, m ∗ ) = (60, 10), the diamond configuration has a flutter-dominated VIV lock-in. This difference in the lock-in behavior can be attributed to the boundary layer movement over the front edges of the diamond cylinder, whereby the square cylinder has flow separations at the upstream corners and inhibits the co-existence of flutter and resonance regimes. For the forward triangle configuration, it is notable that the WSMI and WSMII remain unstable for Fs ≤ 0.184 or Ur ≥ 5.43, as predicted by the ERA-based ROM, which indicates that flutter-dominated-VIV persists. Therefore, the forward triangle is of particular interest for the present study. These linear stability results have been confirmed by the FOM simulations, as shown in Fig. 9.22. The amplitude grows continually for Fs ≤ 0.184 or Ur ≥ 5.43, and the lift coefficient reaches to the maximum value at the lock-in onset Ur , which is similar to the circular and square cylinders. The vortex shedding frequency starts to synchronize with the structure’s natural frequency and there exists 1:1 frequency synchronization whereby the body is synchronized with the vortex shedding frequency. The transverse amplitude grows continually as Fs decreases (Ur increases), which is referred to as galloping-dominated flowinduced vibration. This galloping regime is characterized by a low frequency and a high amplitude vibration, whereas a circular cross-section is not susceptible to galloping. For Fs < 0.085 (Ur > 11.76), the amplitude experiences a small but distinct increase in the amplitude, and the 1:3 synchronization gradually appears as shown

510

9 System Identification and Stability Analysis

1.6

0.15

Re(λ)

WSMII (diamond) WSMII (ellipse) WSMII (triangle)

1.2

Im(λ)

0 0.05

0.1

0.15

0.2

0.25

0.15

0.2

0.25

0.8 f=2πFs

Im(λ)

1

0.4

0.5 0

0.05

0.1

0.05

0.15

0.1

Re(λ)

Fs

(a)

(b)

Fig. 9.21 Effect of geometry on the eigenspectrum of ERA-based ROM at (Re, m ∗ ) = (60, 10): a root loci as a function of the reduced natural frequency Fs , where the unstable right-half (Re(λ) > 0) plane is shaded in grey color and the hollow arrow indicates increasing Fs , b real and imaginary parts of root loci. WSMI data are denoted by the filled symbol with the same shape as WSMII in (a, b). The onset Ur is computed on Fs = 0.184 (forward triangle ), Fs = 0.216 (diamond ), and Fs = 0.239 (ellipse ), in the real parts of the root loci in b, respectively Ymax Cl

1.6

f

Ymax

2.1

0.05

1.1 1.1

0.1

0.1

0.15

0.2

0.25

0.1 0.05

Cl

f=Fs

0.2

0.6 0.1

0.15

Fs

Fs

(a)

(b)

0.2

0.1 0.25

Fig. 9.22 VIV results of the forward triangle configuration using FOM at (Re, m ∗ ) = (60, 10): variation of a normalized vortex shedding frequency f and b rms value of lift coefficient (Cl ) and maximum amplitude (Ymax ). The lock-in is shaded in grey color

in Fig. 9.23. In this regime, there is a net energy transfer from the base flow with the frequency component at three times the body oscillation frequency. During this energy transfer, the fluid force performs work on the body, which stores the energy in the form of kinetic energy as well as the potential energy in the spring. To further elucidate the high harmonic response for the forward triangle, Fig. 9.23 depicts motion and lift force traces with their corresponding spectra. In the figure, there is a clear third-harmonic frequency in the lift force whereby the body oscillates with a dominant frequency. Figure 9.24 shows the instantaneous vorticity contours at different values of reduced natural frequency Fs . The wake mode is 2S for Fs = 0.17, which

9.3 Linear Stability Analysis for VIV of a Cylinder

511

4.0 1

P

Y

2.0 0.0

0.5 -2.0 -4.0 250

0 300

350

400

1

2

3

4

5

3

4

5

*

f (b)

tU/D

(a) 2.0

P

Cl

1

0.0

0.5

-2.0 250

0 300

350

tU/D

(c)

400

1

2

*

f (d)

Fig. 9.23 Full order VIV results for forward triangle with 1:3 synchronization: temporal variation of a transverse amplitude, and c lift coefficient; normalized power spectrum P versus f ∗ of: b transverse amplitude, d lift coefficient at Fs = 0.05, where f ∗ = f /Fs is the frequency of lift and transverse displacement normalized by reduced natural frequency Fs . A third-harmonic frequency is evident in Cl

remains 2S for Fs = 0.15 with somewhat increased spacing between the vortices shed alternately from each side of the cylinder. By further decreasing Fs to 0.1, the strong SM and WM interactions result in larger vibration amplitude and the flow diverges to a wide vortex street. The geometry of the bluff body alters the flow structures significantly in the vicinity of base flow, resulting in different root loci and subsequently changing the stability properties. For example, as shown in Fig. 9.16, the sharp corner of the square stabilizes the flow and the resonance mode dominates the entire lock-in for (Re, m ∗ ) = (60, 10). It can be further elucidated by looking into the magnitude of | for the different geometrical configurations. As illustrated leading modal velocity |U in Fig. 9.25, the stability regions of the ellipse, diamond and triangle geometries shift upstream in comparison to circular geometry, indicating the coupling effect is enhanced. The stationary configurations of ellipse, diamond and forward triangle have quite similar stability regions. The forward triangle is easier to perturb from the flow unsteadiness, or by a Hopf bifurcation for alternate vortex shedding. However, the sharp corners generally stabilize the fluid-structure system, as the separation point

9 System Identification and Stability Analysis 5

0

0

y

-5

y

5

0

10

x (a)

-5

20

5

5

0

0

-5

y

y

512

0

10

x (c)

-5

20

0

10

x (b)

20

0

10

x (d)

20

5

0

0

y

-5

y

5

0

10

x (a)

-5

20

5

5

0

0

-5

y

y

Fig. 9.24 Full order results for the forward triangle cylinder at (Re, m ∗ ) = (60, 10): instantaneous vorticity contours at Fs = a 0.05, b 0.1, c 0.15, d 0.17. Contour levels are from −0.5 to 0.5 in increment of 0.077 and the flow is from left to right

0

10

x (c)

20

-5

0

10

x (b)

20

0

10

x (d)

20

Fig. 9.25 Stability regions shown by the spatial distribution of pointwise modal velocity amplitude | at Re = 60: contours of modal velocity for a circle, b ellipse, c diamond, and d forward triangle |U configuration. The flow is from left to right

is constrained and there is lesser freedom for the interaction of flow and structural modes. Due to this attribute in the forward triangle configuration, there is a delay in lock-in onset Ur with the identical operating parameters (Re, m ∗ ) as compared to the elliptical cylinder. Figure 9.26 summarizes the stability regimes of transversely vibrating bluff bod) ies for Reynolds number range 30 ≤ Re ≤ 100. In this figure, the solid curve ( depicts the trend for the critical Reynolds number. The following observations can be made from the stability phase diagram. The geometries of circle, ellipse and diamond exhibit the flutter and mixed resonance-flutter modes. In contrast to the square counterpart, the diamond geometry has a movement of asymmetric boundary layers on the front lateral edges which allows the co-existence of flutter and reso-

9.3 Linear Stability Analysis for VIV of a Cylinder

513

100 90 80

Re

70 60 50 40 30

Fig. 9.26 Stability phase diagram of VIV lock-in for transversely vibrating two-dimensional bluff bodies with smooth curves and sharp corners for 30 ≤ Re ≤ 100, m ∗ = 10 and 0.05 ≤ Fs ≤ 0.25. Here the solid curve ( ) represents the critical Reynolds number (Recr ) of the fixed bluff body and flutter- and resonance-induced regimes are demarcated, where  represents the co-existence of flutter and resonance regimes; denotes resonance regime;  represents flutter regime. For Re > Recr , flutter regime comprises both unstable eigenvalues (Re(λ) > 0) of WM and SM, whereas resonance regime has only unstable WM

nance modes. For the elliptic cylinder, the mixed flutter-resonance regime occurs at a lower Reynolds number, in comparison to the circular cylinder. Notably, the forward triangle configuration only shows the flutter-induced lock-in regime for this range of Reynolds number and the edges are in the leeward side with the separated flow. Finally, the square-section body shows a predominant resonance regime for 30 ≤ Re ≤ 80, approximately, which turns into the flutter state for Re > 80. The present ERA-based ROM study has been concerned with two-dimensional bluff body configurations for which only two directions in space are resolved. All the notions of ROM, such as base flows and eigenvalue realization can be easily extended to full three-dimensional settings. Thus the present method does not pose any theoretical limitation except there may be a numerical one with respect to memory requirements and CPU time to solve the generalized eigenvalue problem.

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9 System Identification and Stability Analysis

9.3.7 Interim Summary In this study, we presented the ERA-based model reduction for the coupled fluidstructure analysis to investigate the stability characteristics of the vortex-induced vibrations of bluff bodies. The ERA-based ROM relies on the singular value decomposition of a Hankel matrix constructed from the impulse response of the NavierStokes equations. The present study has remarkably demonstrated the effectiveness of ERA-based ROM for predicting the unstable wake flow behind a stationary circular cylinder. The critical Reynolds number and the flow dynamics were well predicated and an excellent agreement was found with the full order model and the available literature. We next employed the ROM for a unified description of the lock-in phenomenon as a function of Reynolds number Re, the mass ratio m ∗ , and for the investigation of the effect of rounding and various geometrical shapes. To investigate the VIV mechanism, the ERA-based ROM has been extended to construct the fluid ROM and coupled with a linear structure to form a reduced fluid-structure system in the state space format. Two distinct lock-in patterns of flutter- and resonance-induced regimes were investigated by the ERA-based ROM for a transversely vibrating circular cylinder at the baseline parameters of (Re, m ∗ ) = (60, 10). While the resonance state has the unstable WM together with the stable SM in the range 0.11 ≤ Fs ≤ 0.147, the flutter regime has the co-existence of the unstable SM and WM in the range 0.147 < Fs ≤ 0.179. In comparison to the linear ROM used in [486], which is sensitive to the training trajectory, the proposed ERA-based ROM is sufficiently accurate and only requires the impulse response of an unstable fluid system. To generalize the proposed ERA-based ROM for the VIV linear stability analysis, the effects of the Reynolds number, the rounding of a square cylinder and the geometry has been systematically examined and compared against the full-order simulations. Based on the systematic parametric study, the following conclusions can be drawn: 1. The study on the effect of the Reynolds number demonstrates that the flutter and resonance regimes do not always exist during the lock-in phenomenon. For m ∗ = 10, it was found that the flutter and resonance regimes co-exist for Recr < Re < 70. The flutter-induced regime gradually dominates the entire lock-in region when Re ≥ 70. The finding is consistent with the theoretical analysis of [103] for a high Reynolds number. The stability region provides an explanation that it shifts upstream as Re increases, indicating the coupling between unstable wake and bluff body is enhanced. Another observation is that the reduction of rms value of lift force is not associated with either resonance or flutter state, but is closely related to the phase angle jump from 0◦ to 180◦ , which is also demonstrated during the lock-in of a square cylinder. 2. The analysis of the rounding effect of the square cylinder study at (Re, m ∗ ) = (60, 10) shows that rounding has a remarkable impact on the flutter and resonance regimes. The flutter regime can be promoted by gradually removing the sharp corners. The sharp corners suppress the continuous movement of separation points and the WM loop of the square cylinder is smaller than the circular cylinder counterpart. As the rounding radius rs increases, the SM trajectory moves

9.3 Linear Stability Analysis for VIV of a Cylinder

515

gradually towards the positive real axis and the flutter region starts to appear. From the comparison of leading modes, the stability region shifts downstream as the rounding radius rs decreases. There is a reduction in the coupling strength between the fluid and structure due to the presence of sharp corners, which inhibit the movement of separation points and the susceptibility of inertial coupling. In comparison to the VIV of the circular cylinder, this study also explains why the lock-in onset Ur is larger and the lock-in region is narrower for a square cylinder. 3. The geometry study reveals that the cross-sectional shape significantly alters the VIV and galloping instability. The ERA-based ROM can effectively capture the stability properties and the lock-in regimes for the elliptical, forward triangle and diamond-shaped configurations. It is found that root loci of WM and SM coalesce and form coupled modes for these geometries. We have provided further insights into the phenomena of flow-induced vibration by the ERA-based ROM for these bluff-body geometries. The elliptical cylinder was found to have the lowest Ur for the lock-in onset followed by the diamond and the forward triangle configurations. Of particular note for the forward triangle VIV, the ROM predicts the flutter-dominated VIV persists for Fs ≤ 0.184 or Ur ≥ 5.43. A low-frequency galloping instability and a kink in the amplitude response associated with 1:3 synchronization were observed in the forward triangle configuration. We presented a summary phase diagram to characterize the effects of geometry on the VIV stability regimes based on the eigenspectrum distribution. Such a phase diagram based on the linear dynamics of the lock-in process provides insights to develop a unified description of flow-induced vibration. The phase diagram shows that the resonance mode only exists for a certain range of Reynolds numbers. The VIV lock-in mechanism is eventually dominated by the flutter mode as the Reynolds number increases. The proposed ERA-based ROM is demonstrated to be accurate and efficient for the VIV linear stability analysis of bluff bodies, which has relevance in the development of flow control strategies. By shifting the unstable eigenvalues of WM and SM to the stable left half complex plane, suppression of vortex street and VIV can be achieved by the model. The simplicity of the model permits the investigation of a range of geometries and parameters on VIV mechanism and paves the way for a bottom-up approach to develop control devices. We want to emphasize that we have considered only linear dynamical systems in this study, possible future direction should include an extension of the system identification process to nonlinear systems. In the future, there is also a need for further investigation of higher Reynolds numbers to expand the proposed model reduction approach for a generalized lock-in description with a wider parameter space of mass-damping parameter.

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9 System Identification and Stability Analysis

9.4 Stability Analysis of Tandem Cylinders A fundamental understanding of flow-induced vibration in an array of flexible bodies is important in various engineering applications ranging from offshore riser pipelines, heat exchangers, transmission lines to civil engineering structural designs, and the aeroelasticity of aircraft wing-tail systems at a high angle of attack. The flow around multiple elastically mounted cylinders can be considered as an idealized model to understand the fundamental physics and nonlinear dynamics of flow-induced vibration. Owing to complex wake-flexible body interactions, there is a significant dissimilarity between the underlying fluid-structure interaction of an isolated cylinder and multiple cylinder arrangements [27, 28, 58, 307, 308]. While there have been numerous experimental and numerical investigations, there is still not a clear understanding to determine an effective suppression technique for multibody arrangements and to suggest a range of optimal design parameters for avoiding severe flow-induced vibrations and collisions. The primary challenges are associated with the complexity of wake and proximity interferences on the coupled nonlinear fluid-elastic response for a wide range of physical and geometric parameters. For the isolated cylinder, vortex-induced vibration (VIV) represents a nonlinear synchronized/lock-in between a structure and the flow dynamics, which often results in large transverse motion and reduces the structural fatigue life. This nonlinear dynamical phenomenon has an impact in the broad fields of offshore, wind, aerospace and energy harvesting engineering due to its richness in fluid physics [101, 282, 405]. In the VIV lock-in regime, the vortex shedding frequency deviates from the Strouhal law and locks on to the structural natural frequency. This frequency lock-in phenomenon is typically characterized by high structural amplitude and therefore has been an active topic over the past decades [40, 381, 455]. A conventional interpretation of VIV is attributed to the classical resonance, which depends on the closeness of the vortex shedding frequency ( f vs ) with the structural natural frequency ( f N ). However, several studies have shown that the peak amplitude arises in the vicinity of lock-in onset rather than at f vs / f N ≈ 1. Reduced-order model (ROM) constantly remains a popular topic for turbulence modeling [443, 444], active flow control [337], flow stability analysis [304] and unsteady vortex shedding problem [317]. Recently, [318] proposed a recursive dynamic mode decomposition (RDMD) methodology for Galerkin projection-based models, which attempt to preserve the orthonormality and low truncation error of the proper orthogonal decomposition (POD) and the frequency-distilling features of dynamic mode decomposition (DMD) [382]. To develop physical insight of the VIV lock-in mechanism, linear ROM has also been adopted for the stability analysis of a transversely vibrating cylinder in crossflow. For example, the authors in [486] reported a linear ROM for the VIV lock-in mechanism, and the lowest Reynolds number of the instability boundary is determined in [219]. Recently, [466] developed an efficient VIV ROM using the eigensystem realization algorithm (ERA) and successfully performed stability analysis of generalized lock-in process for a wide range of physical and geometric parameters at low Reynolds numbers. The authors

9.4 Stability Analysis of Tandem Cylinders

517

have shown that the lock-in onset of bluff bodies is triggered by the unstable structure mode (SM), and the low frequency galloping-like response is characterized by a persistently unstable eigenvalue branch. Furthermore, the understanding of the lock-in mechanism by the ERA-based low-dimensional model paves the way to develop a feedback control strategy for the VIV suppression [465]. The work extends our previous stability analysis of the lock-in mechanism for the VIV to the wake-induced vibration at low Reynolds numbers. From a fundamental viewpoint, the nonlinear dynamics of WIV phenomenon can be modelled by considering a pair of two cylinders in a tandem arrangement to understand the upstream vortical wake excitation [27, 28, 58, 307, 308]. The physical complexity of WIV excitation lies in the nonlinear interaction dynamics of the upstream vortex with the elastically mounted downstream cylinder and the associated self-induced instability in the post-lock-in region at high reduced velocity. Based on the longitudinal spacing to the cylinder diameter ratio (L/D), the tandem cylinder can be characterized by three interference regimes [174, 307, 482] for the stationary configuration: proximity interference (1 ≤ L/D ≤ 1.2 to 1.8), wake interference (1.8 ≤ L/D ≤ 3.4 to 3.8) and no interference (L/D ≥ 3.8). In the proximity regime, the vortex shedding from the upstream cylinder is suppressed and the tandem bodies behave like a single bluff body, whereas the flow becomes intricate in the wake interference and the shear-layer reattachment, the intermittent vortex shedding gradually appears as L/D increases. The no-interference regime is dominated by co-shedding, where vortex shedding occurs separately from both cylinders. Several researchers [306, 500] have studied the critical separation distance for the onset of the co-shedding regime. In the pioneering experimental study by Bokaian et al. [58], the downstream cylinder was observed to exhibit a large response at higher reduced velocities corresponding to the post-lock-in region. Owing to some similarity with the galloping response of a sharp-cornered square cylinder, this particular behavior of the downstream circular cylinder was referred to as wake-induced galloping in [58]. In one of the recent works, the authors [28] dissociate the WIV from the conventional galloping concept. Here, galloping implies a self-excited non-linear instability associated with large amplitude and low-frequency oscillations. The authors argued that the WIV is sustained by the unsteady vortex shedding that provides energy to the WIV system, and has an intrinsically different mechanism than the VIV with the resonance phenomenon. In another recent work [27], the authors provided a physical explanation for the WIV by the wake-stiffness concept. The authors [27, 28] insist that the WIV is a non-resonant flow-induced vibration (FIV) with increasing amplitudes beyond frequency lock-in regime, and is mainly due to unsteady vortical wake. In the context of energy harvesting applications, the authors [49, 156] experimentally explored the tandem cylinders and the WIV phenomenon to understand the degree of energy transfer from the vortical wake flow as functions of the longitudinal spacing and the fluid-structure parameters. The present study builds upon the numerical studies of [307, 308], where the shifting of stagnation point on the downstream cylinder and the external unbalance rotating effects induced by the interaction between the upstream vortex shedding

518

9 System Identification and Stability Analysis

and the downstream cylinder are explored. [307] investigated the origin of WIV at low Reynolds number (Re, m ∗ , L) = (100, 2.6, 5D). Here m ∗ and Re denote the mass ratio and the Reynolds number based on the cylinder diameter D. The authors attributed the large transverse vibration in the post-lock-in region to the continuous vorticity shed from the upstream cylinder impinging on the downstream. The interaction can in turn shift the stagnation point of the downstream cylinder closer to a low suction region and decrease the suction pressure compared to the isolated cylinder counterpart, which leads to a relatively larger transverse force and amplitude for the higher reduced velocity in the post-lock-in region. An analogous to the linear force oscillator, the same authors [308] idealize the WIV system as a linear mass-spring model with a periodic force and a rotation effect attached to the mass-spring system. The steady solution of the spring-mass with the rotation excitation does not decay for high frequency ratio ( f / f N >> 1) or in the post-lock-in region. The aforementioned physical insights were deduced at Re > Recr , i.e. when the unsteady vortex shedding is fully sustained. As shown in Sect. 9.4.3, the large transverse vibration (galloping-type response) also appears at Re < Recr , i.e. when the flow is stable for the uncoupled stationary condition. still worthwhile to explore the origin of the WIV via eigenvalue analysis. Recently, the authors [240] performed a linear stability analysis for tandem circular cylinders WIV, and provided some insight on WIV. Systematic stability analysis of the WIV response of tandem circular and square cylinders has not been reported in the literature. It is known that sharp corners on a square cylinder affect the wake dynamics and fluid loading significantly in contrast to a circular cylinder with a smooth contour. It is interesting to examine the role of sharp corners on the stability characteristics of WIV. To achieve highfidelity physical data around the sharp-cornered body, the full-order model relies on a variational finite-element formulation with semi-discrete time stepping and the body-conforming Lagrangian-Eulerian treatment for the fluid-body interface. Following the stability analysis of [466] for the lock-in mechanism of a single isolated cylinder, we reveal a novel explanation of the WIV mechanism via eigenvalue distribution and the coupled wake and structural modes. We assess the reliability of ERA-based ROM results against the full-order fluid-structure model for the tandem circular cylinder configuration. In the previous section for the VIV stability analysis, we demonstrated that the forward triangle cylinder galloping type response is sustained by the sustained unstable eigenvalue branch. In the present work, we extend this physical insight to the WIV of tandem both circular and square cylinders by employing the linear stability analysis based on the ERA-based ROM. We will show that the WIV response shares a similar linear stability mechanism as the lock-in/synchronization found in the transversely vibrating forward-triangle cylinder configuration. Compared with a circular cylinder counterpart, we will illustrate that the sharp corner of square cylinders also stabilizes the WIV system or fluid-structure coupling in the sense that the onset reduced velocity for the WIV delays and the transverse amplitude decreases significantly in contrast to the circular cylinder counterpart. The present stability analysis and the physical insights have a profound impact on the development of WIV control strategies. The primary goal of the present work is to investigate the origin of the nonlinear WIV phenomenon via

9.4 Stability Analysis of Tandem Cylinders

519

Fig. 9.27 Schematic diagram of a representative bluff body of elastically mounted cylinder in the wake of stationary cylinder. Computational domain and boundary conditions are shown 2

15 10

Y

Y

5 0

0

-5 -10 -15

-10

0

10

X (a)

20

30

-2

-2

0

2

X (b)

Fig. 9.28 Representative finite element mesh with P2 /P1 discretization for a pair of tandem circular cylinders: a full domain discretization, a close-up view of the mesh in the vicinity of the tandem cylinders

the ERA-based ROM, instead of developing a new model reduction procedure. We next describe the problem set-up and the numerical assessment of the ERA-based ROM for the stationary tandem cylinder arrangement. The WIV results are discussed then for the circular cylinder arrangement together with the effects of longitudinal spacing and the role of sharp corners. Before we proceed to a detailed physical mechanism of WIV, we present the problem setup and the assessment of our ERA-based ROM predictions for the downstream tandem cylinder interacting with the laminar wake flow.

520

9 System Identification and Stability Analysis

Y

2

0

-2 -2

0

2

4

6

X Fig. 9.29 Base flow of tandem circular cylinder at (Re, L) = (60, 2D); the streamwise velocity contours are shown. The contour levels are from −0.1 to 1.2 in increments of 0.1

9.4.1 Problem Definition Similar to [466], Fig. 9.27 depicts a schematic diagram of the problem setup used for the present WIV linear stability analysis. While the origin of the Cartesian coordinate is located at the center of the two bluff bodies, the streamwise and transverse directions are denoted x and y, respectively. At the inlet boundary in , a horizontal uniform flow stream (u, v) = (U, 0) is specified, where u and v denote the streamwise and transverse velocities, respectively. To model the WIV phenomenon, we consider a downstream bluff body with a characteristic diameter D and mass m, which is elastically mounted on a linear spring and is free to vibrate only in the transverse direction. For simplicity, the damping coefficient ζ is set to zero in the present study. The domain size is consistent with the problem set-up employed in [466]. While slip wall conditions are specified on the top top and bottom bottom boundaries, a no-slip wall condition is implemented on the surfaces of the bluff body. A traction-free boundary condition is specified along the outlet out . Throughout our analysis, we normalize all length scales by the characteristic dimension D, velocities with the free stream velocity U , and frequencies with U/D. Herein, the Reynolds number Re of flow is defined using the free-stream speed U , the kinematic viscosity of fluid ν, and the characteristic dimension D. Similar to the mesh convergence study in [466], we consider the mesh consisting of 20,912 P2 /P1 iso-parametric elements are sufficient for our current WIV study. A representative mesh is shown in Fig. 9.28 and the corresponding central rectangle represents the fine mesh region around the cylinder bodies. The mesh in the cylinder wake is appropriately refined to resolve the alternate vortex shedding and the near-wake region. Further details on the mesh convergence and the verification of our full-order results can be found in [466]. Next we assess our ERA-based ROM for the wake dynamics of stationary tandem circular cylinders in a laminar flow.

9.4 Stability Analysis of Tandem Cylinders

521

−5

−4

x 10

1

FOM ROM Cl

Cl

1 0

−1 0

20

40 60 tU/D

(a)

80

100

x 10

0

−1 0

20

40 60 tU/D

80

100

(b)

Fig. 9.30 Time traces of lift coefficient for the downstream tandem cylinder: Comparison of 25th ROM subject to the impulse response with the FOM response. a Re = 60. b Re = 120

9.4.2 Assessment of ERA-Based ROM for Wake-Induced Vibrations In this section, we provide a brief assessment of ERA-based model reduction for predicting the wake-body dynamics. We consider a representative configuration with Reynolds number and longitudinal spacing of (Re, L) = (60, 2D) for the tandem configuration of circular cylinders. Next, we use the ERA-based ROM to perform the stability analysis. Using a fixed point iteration for the NS equation without the time-dependent term, we compute the base flow, as shown in the Fig. 9.29. Similar to the ERA-based ROM construction procedure in [466], 1000 impulse outputs (Cl ) are stacked by imposing δ(t) = 10−4 on the transverse displacement Y of downstream cylinder using a non-dimensional time step size t = 0.05. Further details about the application of the ERA-based ROM for an unsteady vortex shedding problem are summarized in [466]. Through the analysis, 25th ROM is determined by examining the decaying singular value of the Hankel matrix for Re = 60 and 120. A good match between the FOM solution and the 25th ROM prediction is found in Fig. 9.30 for predicting the lift force over the downstream tandem cylinder. The impulse response gradually diverges at Re = 120 and indicates that the real part of the least damped eigenvalue is positive and the uncoupled fluid system is unstable. On the other hand, the decaying impulse response is observed for Re = 60 and indicates a stable fluid system. The least damped eigenvalue predicted by the ERA-based ROM is contrasted against the FOM result in Fig. 9.31. It is evident that the flow becomes unstable when Recr ≈ 90, i.e. the real part of the least damped eigenvalue approaches to zero. This analysis corroborates the adequacy of the numerical methodology and the ERA-based ROM for predicting the dynamics of the downstream tandem cylinder. Next, we turn our attention to the WIV response of the downstream tandem cylinders.

522

9 System Identification and Stability Analysis 0.8

0.04

FOM ROM

0.02 0.75 0

δ

ω

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0.7

−0.04 0.65 −0.06 −0.08

60

80

100

120

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Re

60

80

100

120

Re

Fig. 9.31 The growth rate and frequency of the least damped eigenvalue for the flow past a tandem circular cylinder with L = 2D: a growth rate σ , and b frequency ω. The downstream cylinder wake becomes unstable when the growth rate crosses σ = 0 line at the critical Recr ≈ 90 and the vortex shedding emanates

9.4.3 WIV of Tandem Circular Cylinders To examine the WIV mechanism via the ERA-based model reduction, we present the results corresponding to the circular tandem cylinder arrangement, the effect of longitudinal spacing and the role of sharp corners. In particular, we assess the stability characteristics of tandem circular and square-shaped cylinder undergoing WIV. We consider the continuous-time eigenvalues via the ERA-based ROM. From the coupled matrix system in Eq. (9.17), the eigenvalue is extracted by λ =log(eig(Afs ))/t, where t = 0.05 is selected as an appropriate time step to resolve the vortex shedding process and eigenfrequency the dynamical system. We examine the eigenvalue distribution of the fluid-structure system to understand the dynamical behavior. Specifically, we plot the eigenspectrum in the complex plane via root locus, whereby the eigenvalue locations characterize the stability of the dynamical system. The roots in the right half-plane represents the unstable modes of the system. While the roots on the real axis illustrate the non-oscillatory asymptotic behavior, the roots represent lightly-damped oscillatory modes that are closest to the right half plane. Figure 9.32 shows the root loci of the WIV ERA-based ROM as a function of reduced velocity Ur with 2 ≤ Ur ≤ 40. Following the earlier work of [466], the terminologies of WMI and WMII are adopted to classify the eigenvalue branches. As shown in [466], a classification of the fluid-structure eigenmodes as WM and SM are suitable for a weak fluid-structure interaction (e.g., very large mass ratio). Owing to a weaker fluid-structure coupling at large m ∗ (i.e., in the limit of m ∗ → ∞), the eigenfrequency of WM approaches to that of the stationary cylinder wake for all

9.4 Stability Analysis of Tandem Cylinders 1

523 0.1 Re(λ)

WMII m = 2 WMII m = 20 uncoupled mode

0.9

0.8

0 −0.1

Im(λ)

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5

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20

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1 0.6

0.8 0.6

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0

Re(λ)

(a)

0.1

0.2

0.4

Ur

(b)

Fig. 9.32 Eigenspectrum of the ERA-based ROM at (Re, L) = (60, 2D): a root loci as a function of the reduced velocity Ur , where the unstable right-half (Re(λ) > 0) plane is shaded in grey color. The WMI is denoted by filled symbols with the same shape as those for the WMII. The uncoupled wake mode λ = −0.042 + 0.69i. b Real and imaginary parts of the root loci at (Re, L) = (60, 2D). The WIV region at m ∗ = 20 is shaded in grey color, which is defined by 7.44 ≤ Ur ≤ 11.0. and represent WIV onset Ur ≈ 6.1 and Ur ≈ 7.44 for m ∗ = (2, 20), respectively

values of Fs and the frequency of SM comes closer to the natural frequency of the cylinder-only system. Notwithstanding, the root loci of WM and SM can coalesce and form coupled modes at certain conditions, such as in the limit of low mass ratio [283] and for different geometrical shapes [466]. In the present context of WIV, we term these mixed or coupled modes as the wake-structure mode I (WSMI) and the wake-structure mode II (WSMII). For the higher value of reduced frequency Fs , the mode WSMI behaves as the WM while the mode WSMII recovers to the SM. On the other hand, for the smaller Fs range, the modes WSMI and WSMII represent SM and WM, respectively. As can be seen, the WMII branch arises from the high-frequency region (the top of the complex plane), while the WMI branch originates from the uncoupled mode and travels down to the low frequency (bottom of the complex plane). As elucidated in Fig. 9.32, the real part of the WMII branch remains unstable when Ur ≥ 6.1 and the WMI branch is stable, which indicates the WIV might persist at m ∗ = 2. The finding is further confirmed by the FOM result in Fig. 9.33, where the WIV onset Ur = 5.9 and the WIV remains for Ur ≥ 5.9. Figure 9.32 shows that the WMI branch is only unstable when 7.44 ≤ Ur ≤ 11.0 at m ∗ = 20, which is evident from Fig. 9.33. Furthermore, in Fig. 9.32b (bottom), the trend of Im(λ) shows the characteristic anti-crossing between the WMI and WMII for the low mass ratio m ∗ = 2. This intrinsic property of a strongly-coupled system is also found for a circular single-cylinder VIV system at a low mass ratio [466].

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9 System Identification and Stability Analysis

0.5

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0.4 0.2 0

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Fig. 9.33 Normalized maximum amplitude Ymax and rms value of lift coefficient Cl as a function of reduced velocity Ur at (L , Re) = (2D, 60). The WIV region 7.25 ≤ Ur ≤ 10.87 at m ∗ = 20 is shaded in grey color. and represent WIV onset Ur = 5.9 and Ur = 7.25, respectively. is the uncoupled mode frequency predicted by ROM

-2

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(b)

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Fig. 9.34 Instantaneous vorticity contours of WIV onset at (Re, L) = (60, 2D): a m ∗ = 2, Ur = 5.9; b m ∗ = 20, Ur = 7.25

We next provide some qualitative analysis of vortex patterns. The two counterrotating vortices are released at onset reduced velocity as shown in Fig. 9.34. We classify the vortex shedding patterns by the classical terminology of [458]. While a single vortex is alternately shed from each side of the cylinder per shedding cycle for the 2S mode, the vortices begin to coalesce in the far wake during the C(2S) mode. As elucidated in [466], the vortex shedding patterns are directly related to the vibration

9.4 Stability Analysis of Tandem Cylinders

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(b) Fig. 9.35 FOM results of circular WIV at (Re, L) = (60, 2D). Temporal variation of Y and Cl : a m ∗ = 2, and b m ∗ = 20 ; Normalized power spectrum P versus f ∗ of Y and Cl : where f ∗ = f /Fs is the frequency of lift and transverse displacement normalized by reduced natural frequency Fs . A third-harmonic frequency is evident in Cl

1 0.5 0 -0.5

5

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20

Fig. 9.36 Dependence of energy transfer coefficient on the reduced velocity for two representative mass ratios. The WMI is denoted by filled symbols with the same shape as those for the WMII. Here and represent WIV onset Ur ≈ 6.1 and Ur ≈ 7.44 for m ∗ = (2, 20), respectively

amplitude and the phase relations. The amplitudes acquired at the WIV onset are similar at m ∗ = (2, 20), thus the 2S mode is observed for both configurations. Apart from the unstable eigenvalue branches, it is also interesting to examine the WIV from the energy transfer viewpoint. The energy transfer over one period T for the WIV system is derived as Eq. (9.43) in Appendix B. The coefficient E c is defined by excluding the exponential growth/decay rate. Figure 9.36 shows a variation of energy

526

9 System Identification and Stability Analysis 1.2

0.1

1.1

Re(λ)

WMII Re = 60 WMII Re = 100 uncoupled mode

Im(λ)

1

0 −0.1 −0.2

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Im(λ)

0.7 0.6 0.5 −0.1

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Re(λ)

(a)

0.4

Ur

(b)

Fig. 9.37 Eigenspectrum of the ERA-based ROM at (m ∗ , L) = (2, 2D): a root loci as a function of the reduced velocity Ur , where the unstable right-half (Re(λ) > 0) plane is shaded in grey color. The WMI is denoted by filled symbols with the same shape as those for the WMII. The uncoupled wake mode λ = 0.01 + 0.759i. b Real and imaginary parts of the root loci at (Re, L) = (100, 2D). The WIV region at (Re, m ∗ ) = (100, 2) is shaded in grey color, which is defined by Ur ≥ 5.4

transfer coefficient E c as a function of the reduced velocity Ur for two representative mass ratios m ∗ = (2, 20). The figure suggests that the energy source to sustain the WIV is essentially unstable WMII at (Re, L) = (60, 2D). Figure 9.35 depicts the transverse displacement and the lift traces with the corresponding spectrum, which confirms the high harmonic response of WIV. The results also signify the nonlinearity of WIV as the transfer of energy occurs to higher harmonic components. As shown in Fig. 9.37, the uncoupled mode moves to the right half-complex-plane at Re = 100. Compared to the root loci at Re = 60, Fig. 9.37 shows that the eigenvalue branches moves to high frequency regime or upper part of complex plane. The onset Ur of WIV predicted by the ERA-based ROM is at Ur ≈ 5.4.

9.4.4 Effect of Longitudinal Spacing The longitudinal spacing (L/D) is a key parameter to influence the wake-induced vibration of tandem cylinders. Of interest in this subsection is to briefly investigate the effect of longitudinal spacing on the stability characteristics of WIV. We consider two representative spacing L/D = 2 and 4 for the analysis. Figure 9.38 shows that the uncoupled fluid mode moves toward the right half complex plane as L/D increases from L/D = 2 to 4. It implies that the critical Reynolds number Recr reduces and the coupling is enhanced in the sense that the WIV onset reduced velocity starts to move towards the low reduced velocity (high Fs ) as the spacing L/D increases. The

9.4 Stability Analysis of Tandem Cylinders 1

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WMII L/D = 2 WMII L/D = 4 uncoupled mode

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Fig. 9.38 Eigenspectrum of the ERA-based ROM at (Re, m ∗ ) = (60, 2): a root loci as a function of the reduced velocity Ur , where the unstable right-half (Re(λ) > 0) plane is shaded in grey color. The WMI is denoted by filled symbols with the same shape as those for the WMII. The uncoupled wake mode λ L/D=2 = −0.042 + 0.69i and λ L/D=4 = −0.015 + 0.64i b Real and imaginary parts of the root loci at (Re, L) = (60, 2D). and represent WIV onset Ur ≈ 6.1 and Ur ≈ 5.45 for L/D = (2, 4), respectively

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Fig. 9.39 Instantaneous vorticity contours of WIV at (Re, L) = (60, 4D) for tandem circular arrangement: a Ur = 5.55 WIV onset; b Ur = 6.25

instantaneous vortex shedding pattern computed by the FOM is shown in Fig. 9.39, where the WIV onset reduced velocity is Ur = 5.55 and a good match is found with the ERA-based ROM analysis (WIV onset Ur = 5.45). The longitudinal spacing effects can be further elucidated by looking into the stability region where the magnitude of velocity leading eigenmode is large [131]. The complex modal velocity components, the streamwise velocity  u and the transverse velocity  v, are evaluated from the linearized NS equations around the base flow. To qualitatively visualize the magnitude of the leading modal velocity components, we first extract the amplitude of the complex modal velocity components(| u | and | v|) | = | u |2 + | v|2 . and then compute the pointwise modal velocity magnitude as |U Figure 9.40 shows the stability regions of single and tandem L/D = (2, 4, 6) cylinder

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Fig. 9.40 Stability regions of the circular cylinder configurations at Re = 60: a single cylinder (L/D → ∞), b L/D = 2, c L/D = 4, d L/D = 6. The contour levels are from 0.005 to 0.03 in increments of 0.0036 and the flow is from left to right

configurations. The stability region tends to move closer to the rear of the downstream cylinder as L/D increases, suggesting the fluid and structure coupling becomes stronger, which also explains that the onset reduced velocity becomes lower as L/D increases. As the spacing increases beyond the critical spacing L/D > 4 in the coshedding regime, the WIV response recovers to the isolated cylinder counterpart.

9.4.5 Effect of Sharp Corners In this section, we further explore the WIV of a tandem square cylinder to understand the role of sharp corners with respect to smooth contours of circular geometry. The sharp corners greatly influence the wake dynamics and fluid loading on a bluff body via separation points. Here, we extend the stability analysis to understand flowinduced vibrations of the square cylinders kept in a tandem arrangement, where the downstream cylinder is elastically mounted and free to oscillate in the transverse direction. Similar to [181], we examine the combined wake-induced and sharp corner based galloping effects on the downstream column and compare the response against the tandem circular arrangement. To begin, the sharp corner effects are elucidated in Fig. 9.42 by plotting the stability regions for the single square cylinder and the three representative tandem configurations L/D = (2, 4, 6). Similar to its circular cylinder counterpart shown in Fig. 9.40, the stability region of the tandem square cylinder shifts closer to the downstream cylinder as L/D increases. However, the sharp corner restricts the stability region shifting. For example, the stability region of the circular cylinder is much closer to the downstream cylinder than its square counterpart at L/D = 6.

9.4 Stability Analysis of Tandem Cylinders WMII m = 2 WMII m = 20 uncoupled mode

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Fig. 9.41 Eigenspectrum of the ERA-based ROM at (Re, L) = (60, 2D) for a tandem square cylinder arrangement: a root loci as a function of the reduced velocity Ur , where the unstable righthalf (Re(λ) > 0) plane is shaded in grey color. The WMI is denoted by filled symbols with the same shape as those for the WMII. The uncoupled wake mode λ = −0.027 + 0.643i. b Real and imaginary parts of the root loci at (Re, L) = (60, 2D). The WIV region at m ∗ = 20 is shaded in grey colour, which is defined by 8.4 ≤ Ur ≤ 11.7. and represent WIV onset Ur = 6.75 and Ur = 8.4 for m ∗ = (2, 20), respectively

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Fig. 9.42 Stability regions of the square cylinder configurations at Re = 60: a single cylinder (L/D → ∞), b L/D = 2, c L/D = 4, d L/D = 6. The contour levels are from 0.005 to 0.03 in increments of 0.0036 and the flow is from left to right

9 System Identification and Stability Analysis 4

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Fig. 9.43 Instantaneous vorticity contours of square WIV at (Re, L) = (60, 2D): a WIV onset Ur = 6.76; b Ur = 8.0 Fig. 9.44 Normalized maximum amplitude Ymax and rms value of lift coefficient Cl as a function of reduced velocity Ur at (L , Re, m ∗ ) = (2D, 60, 2). The WIV region is shaded in grey colour. and represent circle and square cylinder WIV onset Ur = 5.9, and Ur = 6.76 . is the square cylinder uncoupled WM frequency predicted by ROM

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In our previous work [466], we demonstrated that the sharp corner has stabilizing effects for fluid-structure coupling. Sharp corners tend to suppress the continuous movement of separation points, whereby a smooth circular cylinder promotes the movement of separation points. Therefore, it is interesting to examine the WIV mechanism of the square cylinder by means of our proposed ERA-based ROM. To understand the role of separation points in a qualitative sense, Fig. 9.43 shows the instantaneous vorticity contours at different Ur . The wake mode is 2S with somewhat increased spacing between vortices compared with its circular tandem counterpart, which further confirms that the sharp corners restrict the movement of separation points.

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As elucidated in Fig. 9.41, the real part of the WMII branch remains unstable when Ur ≥ 6.75 and WMI branch is stable for m ∗ = 2, which indicates that the WIV might persist. This finding is confirmed by the FOM results shown in Fig. 9.44. The WIV onset reduced velocity predicted by ROM is Ur = 6.75, which matches well with the FOM results Ur = 6.76. Whereas, the unstable WMII is bounded as 8.4 ≤ Ur ≤ 11.7 for m ∗ = 20. It is also evident by Fig. 9.44 that the sharp corner has the stabilization effect, where the WIV onset Ur , the peak amplitude Ymax and the rms value of Cl are reduced as compared to the tandem circular cylinder. This is a remarkable and counterintuitive result of the comparative WIV response of a square and a circular cylinder in the tandem arrangement. For the first time, we show that sharp corners stabilize the WIV response of tandem square cylinder, owing to the constraint of the separation point in contrast to the circular cylinder counterpart. From intuitive thinking, the self-excited galloping instability of a square cylinder should have enhanced the WIV response due to the upcoming vortex excitation. However, the sharp corners provide a substantial stabilization to the WIV excitation as compared to the circular cylinder counterpart. The present stability analysis clearly demonstrates the role of sharp corners on a bluff body undergoing WIV excitation. While the present ERA-based ROM is employed for the tandem cylinders at low Re flow, the effects of high Reynolds number and three-dimensionality should be a subject of future study for the present WIV study and to extend the proposed model reduction approach.

9.4.6 Interim Summary The wake-induced vibration is a highly nonlinear response of a tandem downstream cylinder characterized by a relatively higher amplitude motion in the post-lock-in region. In this study, we presented the ERA-based model reduction to investigate the origin of WIV phenomena in the elastically-mounted tandem circular and square cylinders. The ERA-based ROM only relies on the impulse response of the NS equations and is efficient for the stability analysis of the nonlinear fluid and structure coupling system. Based on the systematic parametric study, the following conclusions can be drawn: 1. The persistent unstable eigenvalue branch sustains the WIV in the post-lock-in region. The results also show that the WIV can occur even at Re < Recr , whereas the unsteady vortex shed from the upstream cylinder impinging on the downstream cylinder is considered the origin of WIV in [307, 308]. 2. The investigation of the spacing (L/D) effect in the tandem cylinder study at (Re, m ∗ ) = (60, 2) shows that the least damping eigenvalue moves to the righthalf complex plane. Therefore, the critical Reynolds number Recr reduces as the longitudinal spacing L/D increases. The stability region shifts to the downstream cylinder as the longitudinal spacing (L/D) increases. Therefore, the coupling

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strength between the fluid and structure is enhanced, and the WIV onset reduced velocity (Ur ) moves to a lower value. 3. The analysis of the square cylinder shows that the sharp corner has a stabilizing effect on the WIV in the sense that the onset reduced velocity is postponed and the vibration amplitude is reduced significantly. The sharp corner restricts the stability region from shifting toward the downstream cylinder, hence reducing the interaction level between the fluid flow and the structure. This is a novel finding on the stabilization role of sharp corners to the wake-induced excitation of an elastically-mounted downstream cylinder. While the WIV is a nonlinear response, the present work suggests that it has a linear origin and the ERA-based ROM can successfully predict the onset of WIV. Hence, the present stability analysis lays a foundation to develop a feedback control strategy to stabilize the WIV system by shifting the unstable eigenvalues to the left-half complex plane.

9.5 Deep Learning for Predicting Frequency Lock-in Predictions and control of the spatial-temporal dynamics of fluid-structure systems are crucial in various engineering disciplines ranging from marine/offshore, aerospace to biomedical and energy harvesting. The two-way coupling between fluid and structure exhibits rich unsteady flow dynamics and flow-induced vibration such as vortex-induced vibrations and flutter/galloping [184, 381, 455]. While the coupled physics models based on coupled nonlinear partial differential equations are readily available, the analytical solutions of these differential equations are intractable. Numerical simulations are central for modeling such complex fluid-structure interactions. Using powerful numerical algorithms and large-scale computing resources, high-fidelity simulations can provide accurate predictions and a vast amount of physical insight [181, 254, 299]. However, such simulations are very expensive for extensive parametric analysis and stability predictions for emerging technologies such as digital twins [425]. This work is motivated by the need of making coupled physics simulations efficient for the digital twin technology whereby multi-query analysis, design optimization and control can be achieved through the recent advances in nonlinear model reduction and deep learning algorithms [86]. We consider a prototypical fluid-structure interaction problem of an elastically mounted three-dimensional bluff body undergoing vortex-induced vibration and frequency lock-in phenomenon [89, 352, 376]. During the frequency lock-in/synchronization for a certain range of physical parameters, the vibrating body undergoes a nonlinear coupled flow-structural instability with self-sustained oscillations and the vortices are forced to shed at a frequency equal to the oscillation frequency rather than the Strouhal frequency of the stationary body. Effective real-time control strategies are required to handle these oscillations and the undesired effects such as noise and structural failure. In this work, we are interested in the development of a data-driven reduced-order model that can learn the dynamical system well enough to efficiently predict the fluid-structure stability using full-order or measurement data. During the lock-in, the

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vibration response is limited by nonlinearity either from fluid or structure hence a nonlinear reduced-order model is desired. The goal of reduced-order modeling is to provide a simplified mathematical representation of the fluid-structure model that captures the dominant dynamics while significantly reducing the number of degrees of freedom. A wide majority of reduced-order models (ROMs) are projection-based as they provide low-dimensional representations of an underlying high-dimensional system [47, 224, 367]. The basic assumption of these models is that a lower-order representation of a higher-order model may exist and can be identified efficiently within reasonable accuracy. Using Galerkin-type projections of the full-order system onto a small subspace spanned by mode vectors, one can construct the mode vectors or the optimal subspace via proper orthogonal decomposition [48], balanced truncation [301] or dynamic mode decomposition (DMD) [382]. Conventional POD/Galerkin models capture the most energetic modes based on a user-defined energy norm, whereby low-energy features may be crucial to the dynamics of the underlying problem. While the POD extracts modes from snapshots of the primary system, the balanced truncation method derives the modes by collecting snapshots of both the primary and the adjoint systems. This feature of the balanced truncation method allows identification of the modes that are dynamically important. This algorithm is usually referred to as the balanced proper orthogonal decomposition (BPOD) and provides two sets of modes, namely primal and adjoint modes [367]. While these linear projection-based ROMs enjoy numerous attractive properties to construct an input-output model for control and parametric analysis, these methods are difficult to generalize for high-dimensional nonlinear systems. Petrov-Galerkin projections or Koopman operators can be incorporated to introduce the nonlinearity [367]. Through a nonlinear combination of the POD modes, one can also use the discrete empirical interpolation method (DEIM; [81]). The DEIM method relies on the additional POD basis to enrich the low-rank approximation of the nonlinear terms [299]. An alternative approach to projection-based ROM is based on the system identification via input-output dynamical relationship [474]. Using input and output data, system identification methods attempt to build mathematical models of dynamical systems and consider the original system as a black box [259]. In a system identification process, one needs to collect the data, identify any model structures, estimate the parameters of the model structure, and then validate the model. While the nonlinear auto-regressive with exogenous input (NARX) models rely on the static inference function (i.e., the regression vector) between input and output data, the nonlinear state-space models provide a general nonlinear dynamical system form whereby the information in the state can sustain longer in the horizon by the feedback process. One of the popular system identification methods is the eigensystem realization algorithm (ERA) introduced by [199]. The ERA is a non-intrusive linear state-space model and generates a minimal realization that follows the evolution of the system output when it is subjected to an impulse input. In a theoretical study by Ma et al. [271], the authors demonstrated the equivalence of the ERA-based model reduction with the BPOD method for which the observability and controllability gramians are the same. The ERA-based methodology is primarily data-driven and is used to analyze the stability of dynamical systems.

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In the context of flow-induced vibration and control problems, [465, 466], [70, 89] explored the ERA-based model reduction. A unified description of frequency lock-in for 2D elastically mounted cylinders has been provided by Yao et al. [466] along with the generalized stability properties of the fluid-structure system as functions of Reynolds number, mass ratio and the geometry of the bluff body. While [70] studied the passive suppression mechanism, [465] proposed a feedback control via ERA-based ROM. [89] applied the ERA-based to three-dimensional geometry of the sphere and provided physical insight on the VIV stability properties. They showed that although impulse input considers equal excitation across a broad range of frequencies, it can only identify resonance frequencies at the onset of instability at low Reynolds numbers. In addition, the significance of the impulse input is not sufficient enough to realize the system for turbulent flows which motivates the development of nonlinear system identification techniques. A more general idea of ROM in nonlinear data-driven dynamics can be constructed using ERA, DMD and their variants. In that regard, Koopman theory has got a lot of attention for which it is mainly used for linear control [217, 338] and modal decomposition [258, 287]. An interesting connection between DMD and Koopman theory is presented by Korda et al. [218]. Recently, [252] suggested a data-driven model reduction within the framework of KoopmanMori-Zwanzig formalism and its relationship with Nonlinear Auto-Regressive Moving Average with eXogenous input (NARMAX) for randomly forced dynamical systems. A harmonic balance (HB) technique for the reduced-order computation of vortex-induced vibration is presented by Yao et al. [467]. Although the HB model is not robust and general in predicting all the complex nonlinear dynamics of VIV, it appears to be quite effective to extract the basic features of wake dynamics and response characteristics in the lock-in range. Another challenge for the HB-based ROM procedure is to handle the inherently chaotic behavior of vortex-induced vibration at a high Reynolds number. In recent years, deep learning techniques have gained significant attention in the fluid mechanics community [298, 300, 438, 445]. Deep learning is a sub-field of machine learning that refers to the use of highly multilayered neural networks to analyze a complicated dataset in order to predict certain characteristics in the dataset [134, 230]. Deep neural nets provide parametric nonlinear function approximations that can fit datasets to learn functions from input vectors to output vectors. This process generates a low-dimensional subspace to represent the underlying behavior of the system. Recently, convolutional neural networks (CNNs) were utilized to develop a nonlinear modal decomposition method, which performed superior to the traditional POD Miyanawala et al. [299], Murata et al. [305]. A recent study performed by Bukka et al. [71] presents a review of deep learning-based reducedorder models for the prediction of unsteady fluid flow where these hybrid models rely on recurrent neural networks (RNNs) to evolve low-dimensional states of unsteady fluid flow. ML has been used to learn the hidden governing equations of fluid flow directly from field data in various cases in the literature e.g., [69, 369], [80, 350], [93, 261]. These deep neural network types have been recently adopted in fluid mechanics by Bukka et al. [71, 280]. Several other works that employ deep learningbased models for nonlinear dynamical systems by building on the mathematical

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framework of Koopman theory, are investigated by Yeo et al. [471], Otta et al. [325] and Lusch et al. [268]. Predicting the evolution of the parameter of interest can be categorized under sequence modeling problems in machine learning. While neural networks need persistence and retention of information dealing with sequence prediction problems, they vary from other canonical learning problems in machine learning [253]. Traditional neural networks lack a mechanism for information persistence and retention [91, 406]. Recurrent neural networks (RNNs) alleviate information retention during training and inference. RNNs contain recursive hidden states and learn functions from an input sequence to an output sequence. The internal state of the model is preserved and propagated by adding a new dimension to recurrent neural networks. Despite the success of RNN, several stability issues have been observed. The most prevalent is the vanishing gradient as they are unable to learn long-term dependencies in the data. In the present study, long short-term memory networks are employed to address the issue of long-term dependence in the unsteady dynamical data [157]. In contrast to feedforward nets such as the NARX model, the LSTM networks retain the information for a relatively long horizon. During prediction, the LSTM net utilizes the information based on its relevance and context. For example, they can be presented one observation at a time from a sequence and can learn relevant features using previous observations. Notably, both deep learning and system identification techniques attempt to address the same fundamental problem i.e., the construction of inference models from observable data to emulate the underlying system dynamics. There are many connections between the two techniques as illuminated by Ljung et al. [260]. In the present work, we will employ the RNN-LSTM methodology for the DL-based model reduction of the nonlinear fluid-structure system. In this work, we present for the first time a complete data-driven stability analysis via a deep learning model for the flow past a freely vibrating sphere at moderate Reynolds numbers. We study the underlying mechanism of transverse flow-induced vibration by exploiting a unified description of frequency lock-in for an elastically mounted sphere. We propose a methodology that connects the DL-based model reduction with the ERA to capture just enough dynamics to extract the stability properties of the fluid-structure systems. The resulting reduced-order model is nonlinear and makes use of data from full-order numerical simulations [89, 352] to identify the dynamics relevant to the input-output map of the dynamical system. It is of particular interest to provide a generalized description of these frequency lock-in regimes at both low and high Reynolds numbers where the wake is laminar (Re = 300) and turbulent (Re = 2000) respectively via our proposed model reduction technique and the eigenvalue selection process. The results from the ROM are compared with the FOM simulations based on the incompressible Navier-Stokes equations. Using nonlinear force and motion data, we employ the DL-based ROM integrated with ERA to predict stability through the eigenvalue distribution in the complex plane. The present study is based on the following questions pertaining to nonlinear stability predictions of the coupled fluid-structure system. (i) Can we characterize the frequency lock-in regimes of a transversely vibrating sphere by utilizing the proposed nonlinear DL-based ROM methodology? (ii) How does the coupled structural

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mode predicted by the stability analysis behave in conjunction with the interaction of complex wake features? (iii) How can we train the network in order to enhance its performance based on the general underlying physics of the problem? (iv) Once the network is trained, can we employ the constructed DL-based ROM to perform parameter space exploration for a range of mass ratios (m ∗ ) and reduced velocities (U ∗ )? (v) Does nonlinear DL-based ROM integrated with ERA allow stability prediction at a high Reynolds number (Re) where the wake is turbulent? In this section, we attempt to answer these questions via our DL-based model reduction procedure which can be useful for the development of emerging digital twin technologies requiring real-time control and structural health monitoring. In this section, we first provide an overview of forward and inverse representations of the nonlinear dynamical system. For the sake of completeness, we briefly summarize our high-dimensional FOM to simulate the fluid-structure interaction using the incompressible Navier-Stokes equations and the rigid body dynamics. We next introduce our DL-based model reduction for the nonlinear fluid-structure system of an elastically-mounted bluff body coupled with the incompressible Navier-Stokes equations. We finally provide our methodology to integrate ERA with the constructed DL-based ROM for the stability prediction.

9.5.1 Reduced-Order State-Space Model From a state space dynamical system perspective, the forward problem for a general nonlinear dynamics of a system can be written in a discrete form as x(t + 1) = F(x(t), u(t)), y(t) = H(x(t)),

(9.18)

where x ∈ R M is the state vector for a coupled FSI domain with a total of M variables in the system. For our fluid-body system in the current study, the state vector involves the fluid velocity and the pressure as x = {uf , p f } and the structural velocity includes the three translational degrees-of-freedom. Note that the pressure p f can be written as some function of density ρ f via the state law of a fluid flow. The right-hand side term F represents a dynamic model and can be associated with a vector-valued differential operator describing Eq. (9.1). The resultant spatial-temporal dynamics of the fluidstructure interaction are driven by the inputs u such as the modeling parameters and boundary conditions. y represents the observable quantities of interest, and H is a nonlinear function that maps the state and input of the system to the quantities of interest. The goal of the forward problem is to compute the functions F and H via principles of conservation and numerical discretizations. Next, we briefly review data-driven reduced-order modeling based on traditional projection and deep learning-based recurrent neural nets.

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9.5.2 Nonlinear DL-Based Model Reduction From the perspective of a data-driven approach, the idea is to build the functions F and H using projection-based model reduction, system identification or machine learning method as the inverse problem. The inverse problems begin with the available data and aim at estimating the parameters in the model. As a general way, a model (M (θ )) can be considered as a predictor of the next output based on prior input-output data. The output data may be written as yˆ (t|θ ), where θ is a parameter that spans a parameter set according to the model M (θ ). The inference in this inverse problem is purely based on the dynamical data. A general nonlinear state-space (NLSS) model can be written as x(t + 1) = F (x(t), y(t), u(t), θ ),

(9.19a)

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where F and H can be parametrized by θ in many different ways according to the model structure. The inverse problem aims to construct transfer functions F and H as close approximations to the functions F and H. The estimation of the parameters is essentially done by minimizing the fit between observed outputs y(t) and predicted model outputs yˆ (t|θ ), which can be expressed as min θ

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where N is the number of data points in the data set. In practice, the ground truth data are obtained via forward numerical simulations, experiments or field measurements. The ground truth data are obtained from numerical simulations in the current work. The functions F and H represent the approximate (surrogate) reduced models that generate the predicted data. We investigate the application of the inverse problem to the canonical problem of the flow past a sphere. The next section presents the mathematical formulation of nonlinear DL-based ROM.

9.5.2.1

Nonlinear System Identification via LSTM

Here we provide a brief description of the RNN-LSTM which is a well-established architecture in deep learning. In particular, we intend to illustrate the connection between DL-based ROM using RNN-LSTM with nonlinear system identification. The RNN-LSTM based ROM model can be considered an extension of the ERAbased system identification technique for predicting the unsteady forces and the VIV lock-in, as presented earlier by Yao et al. [466], Bukka [70] and Chizfahm et al. [89]. The goal of system identification is to construct mathematical models

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of dynamical systems based on pure input and output signals (Eq. 9.19). Our DLbased model reduction approach aims to combine the features of the RNN-LSTM and eigenvalue realization algorithm, thereby developing a new method of model reduction for nonlinear systems. To address the vanishing gradient problem, LSTM employs gating mechanisms during the dataflow. A single LSTM layer consists of four gates and two states to boost the recurrent computation. In particular, the lth layer has input gate it(l) , forget (l) (l) gate ft(l) , cell gate c˜ t(l) , output gate o(l) t , cell state ct , and hidden state ht . The mapping (l) (l) (l) from the input u˜ t to the output y˜ t = ht of the lth layer is as follows: (l) , u˜ t(l) ] + b(l) it(l) = σ (Wi(l) · [ht−1 i ), (l) , u˜ t(l) ] + b(l) ft(l) = σ (Wf(l) · [ht−1 f ), (l) (l) ˜ t(l) ] + b(l) c˜ (l) t = tanh(Wc · [ht−1 , u c ), (l) (l) ˜ t(l) ] + b(l) o(l) t = σ (Wo · [ht−1 , u o ),

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where σ (.) is a sigmoid function operating element-wise and (∗) is the element-wise product. The weight matrices W and the bias vectors b include the parameters of the LSTM layers. The Number of Units in an LSTM layer is the dimension of the hidden and the cell states and defines the dimensions of the matrices W and b. The gates layout and the data control are the main reasons for the robustness of LSTMs with regard to the vanishing gradient problem that plagues traditional RNNs. As a result, LSTMs can serve as a powerful tool for time-series predictions of fluid-structure interaction. The LSTM layers are connected by taking the output of a layer as the input to the next layer: u˜ (l) = h(l−1) for l = 2, . . . , L. The output of the last LSTM layer is (l) (L) ht + bfc . It is important to the input to a fully connected linear layer: yt = Wfc mention that an LSTM network is a NLSS model [Eq. (9.19)], since a state vector (l) T T T x(t) that collects the cell and hidden states in all layers xt(l) = [(c(l) t ) (ht ) ] and (1) T (L) T T x(t) = [(xt ) . . . (xt ) ] , input u(t) =˜u(l) , output yˆ (t|θ ) = yt , and a parameter vector θ that collects all the elements in the weight matrices W and the bias vectors b. More explicitly, the output yt in Eq. (9.19b) is a function of htL , which is part of the state vector x(t). Moreover, for the state equation, Eq. (9.19a), we can observe from (l) depends on the states in the same layer one time step earlier xt(l) Eq. (9.21) that xt+1 (l) (l) and on the input u˜ t+1 to the layer. For l = L, L − 1, . . . 2, this input u˜ t+1 = h (l−1) t+1 is (l−1) part of the state of the previous layer xt+1 , which in turn depends on the states of that (l−1) (l) . Ultimately, xt+1 depends on layer one time step earlier xt(l−1) , and on the input u˜ t+1 (l−1) (1) . . . xt , which is part the states of all the previous layers one time step earlier xt (1) of x(t), and on the input u˜ t+1 to the first layer, which is u(t). A schematic of a closedloop recurrent neural network is depicted in Fig. 9.45. The decoder is infused by the

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final summary vector hT(l) generated by the encoder. The decoder then proceeds in an autoregressive fashion for the prediction. At each step of Rdec , the input is generated by the predicted output from the previous step. The so-called summary vector or the bottleneck vector provides an emphasis to store all relevant information in the input sequence. The summary vector hT(l) is illustrated with the transparent lines in Fig. 9.45. Therefore, the LSTM is used through recursive feedback as a type of RNN with a gating mechanism through time. Further details about the dataflow of the LSTM can be found in [65]. The work procedure becomes quite similar when the deep learning model structures (O y(t|θ )) are integrated with the model objects of system identification. Once the model is estimated, it can be validated and used for simulation and prediction. A benchmark example of a nonlinear dynamical system is provided in Appendix C to evaluate the performance of the LSTM network toolbox for system identification. In the next section, we present the basic state-space formulation and DL-based model reduction for the coupled fluid-structure system. We then provide our methodology for the training process of the RNN-LSTM as a nonlinear system identification method for fluid-structure interaction problems in Sect. 9.5.4.

9.5.2.2

Coupled Nonlinear Fluid-Structure Formulation

Of particular interest in this work is the rigid-body equation for the freely vibrating sphere in a flowing stream. The sphere is mounted on a spring system in a cross-flow direction, which allows the sphere to vibrate through unsteady lift comprising of the pressure and shear stresses. We adopt a partitioned coupled formulation for the reduced fluid-structure problem. The LSTM-based ROM for the fluid subsystem is integrated with the linear structural model in a partitioned manner. The non-dimensional structural equation for a transversely vibrating sphere with 1-DOF can be expressed as 1 A¨ y + 4ζ π Fs A˙ y + (2π Fs )2 A y = ∗ C y m

(9.22)

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9 System Identification and Stability Analysis

where A y is the transverse displacement and C y is the normalized transverse force acting on the structural body due to fluid traction, defined as C y = f ys /( 21 ρU 2 S). m ∗ and ζ are the mass ratio (i.e. ratio of the mass of the structural body to the mass of the displaced fluid) and the damping coefficient respectively. Fs is the reduced natural frequency of the structure, defined as Fs = f n D/U = 1/U ∗ , where U ∗ is the reduced velocity which is an alternative parameter to characterize the frequency lockin phenomenon. The definitions of the key parameters can be found in Table 9.5. The non-dimensional structural equation can be presented into a state-space formulation as x˙s = As xs + Bs C y where the state matrices and vectors are       A 0 1 0 Bs = xs = ˙ y As = −(2π Fs )2 −4ζ π Fs as /m ∗ Ay

(9.23)

(9.24)

The characteristic length scale factor as is related to the geometry of the body which is as = 3/8 for a spherical body. The above formulation can be transformed into discrete state-space form as follows, xs (t + 1) = Asd xs (t) + Bsd C y (t) A y (t) = Csd xs (t)

(9.25)

where the state matrices are Asd = eAs t , Bsd = As −1 (eAs t − I )Bs at discrete times t = kt; k = 0, 1, 2 . . . with a constant sampling time t. I is the identity matrix and Csd = [1 0]. The fluid ROM is derived by the RNN-LSTM method as described in Eq. (9.19) and Eq. (9.21) through the input-output dynamics. The input for the ROM is the transverse displacement A y , and the output is the normalized transverse force C y . The DL-based ROM with the single input and single output (SISO) can be reformulated as Cˆ y (t|θ ) = H (C y (t), A y (t), θ ).

(9.26)

In this work, we aim to characterize a complex dynamical relation between the transverse amplitude A y and the transverse force C y . A state-space relationship between the transverse force and the amplitude is constructed directly from the NS equations subject to random prescribed motion. Therefore, now we proceed to the formulation of the coupled fluid-structure system. By substituting Eq. (9.26) into Eq. (9.23), the resultant ROM can be expressed as  xfs (t + 1) =

As Bs Hs H f

 xfs (t) = A fs xfs (t)

(9.27)

9.5 Deep Learning for Predicting Frequency Lock-in

Cy (t + 3) Cy (t + n)

Coupled FSI

Ay (t + 3)

ERA-based ROM

Decoder

Ay (t + 2)

Cy (t + 2)

RNN LSTM

(Output)

(Input)

Encoder

Ay (t + 2)

Ay (t + 1)

Cy (t + 1)

Ay (t + 1)

yr

ŷ

DL-based ROM Ay (1) = δ

541

Ay (t + n)

Fig. 9.46 Schematic of proposed hybrid RNN-LSTM framework with ERA for stability prediction of coupled fluid-structure system

where Hs and H f are nonlinear functions that can be parametrized according to the black-box DL-based model structure, A fs denotes the coupled nonlinear fluidstructure matrix in the discrete state-space form, and xfs = [xs C y ]T . The input-output dynamics of the full-order system is emulated through the DL-based ROM.

9.5.3 Stability Analysis via DL-Based ROM Integrated with ERA Here, we present the methodology to integrate DL-based ROM with ERA system identification for stability analysis. Figure 9.46 shows the schematic of the overall process, where the predictive RNN-LSTM as a nonlinear DL-based ROM is integrated with the ERA-based ROM to provide a linear approximation of the nonlinear model for the stability prediction. Later in this section, we present the eigenvalue selection process as an auxiliary mechanism to interpret the results for the stability prediction.

9.5.3.1

Linear Approximation via ERA

We assume that the dynamic system of interest can be modeled as xfs (t + 1) = A fs xfs (t) + ηt , t = 0, 1, 2, . . . .

(9.28)

where xfs (t) presents the state of the system at time t. The function A fs : Rn → Rn maps the state at time t to the state at time t + 1, and ηt is a small perturbation. The perturbation ηt might include modeling errors, such as slowly changing operating conditions (unmodeled dynamics) or discretization errors as investigated by Erichson et al. [112]. If ηt is small, the dynamics simply specify that the state xfs (t + 1) depends only on the value of the previous state xfs (t). Hence, through the transformation A fs ,

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9 System Identification and Stability Analysis

the state xfs (t) provides all information needed for predicting the future state at xfs (t + 1). To perform the stability analysis, we use a linear time-invariant approximation of the nonlinear system as of the following form xfs (t + 1) = Afs xfs (t) + ηt , t = 0, 1, 2, . . . .

(9.29)

where Afs : Rn → Rn presents a linear map. Linear models provide a reasonably good approximation for the underlying mechanism in many applications. Using the linear dynamics, the stability of the origin can be checked by utilizing an eigenvalue analysis. For this purpose, we use the ERA method as a system identification technique that makes a reduced model through a linear projection of the original system on the most observable and controllable subspaces. The sample data from the DL-based ROM is passed through ERA to generate a low dimensional linear space. The generated latent space from ERA is designed to have a lower dimension than the original input and output space, thus achieving compression of the data as an autoencoderdecoder-like framework. The ERA-based ROM is then used for the prediction of the stability properties of the coupled FSI system. However, despite the simplicity of this procedure, it often turns out to be a challenge to find a reliable estimate for the coupled system Afs . Therefore, we aim to formulate the basic state-space formulation to estimate the desired output of the coupled nonlinear system using the eigensystem realization algorithm. The input for the ERA-based ROM is a tiny perturbation of amplitude and the output is the temporal evolution of the transverse displacement A y . The continuoustime eigenvalues and eigenvectors of Ar , denoted by (λ, xˆ ), characterize the temporal behaviour by the complex scalar (λ), and the spatial structure by the complex vector field (ˆx). The stability analysis can be established by tracing the trajectory of the complex eigenvalue in the complex plane, whereby xˆ provides the spatial global modes of the ERA-based ROM. Through the eigenvalues of the linear ERA-based approximation of the nonlinear model, the growth rate and the frequency of the corresponding global modes are identified by Re(λ) and Im(λ/2π ), respectively.

9.5.3.2

Eigenvalue Selection Process

In this section, we present the eigenvalue selection process for stability prediction. We aim to identify the unstable region corresponding to VIV lock-in for the coupled FSI system. In Sect. 9.5.3.1, the ERA methodology is applied to construct a linear approximation of the nonlinear DL-based ROM. Consistent with the previous literature by [466], we utilize the methodology of variation of unstable modes to classify the distinct eigenvalue trajectories of the fluid-structure system. In their work, the fluid-structure mode is considered as a structural mode (SM) if the eigenvalue of the coupled linearized system comes close to the natural frequency of the structural body in a vacuum. The ERA method is applied to the desired output from the cou-

9.5 Deep Learning for Predicting Frequency Lock-in

543

pled nonlinear DL-based ROM, which is the transverse displacement ( A y ). We can plot the eigenvalue distribution corresponding to the transverse displacement ( A y ), which is equivalent to structural mode, by selecting the most unstable mode. The stability prediction can be performed by estimating the growth rate and the frequency of the most unstable mode. Based on the methodology described in the previous sections, the process of constructing the ROM for the coupled FSI system contains the following vital components: • The first step involves the extraction of the amplitude response (A y ) from the nonlinear DL-based ROM. By imposing a tiny amplitude perturbation (δ = A y /D = 10−3 ), the equilibrium position is perturbed and we can assess the growth of the disturbance and the stability of the system. The sensitivity of the unstable system to the initial condition is confirmed by comparing the response subject to two small perturbations with δ = 10−4 and δ = 10−3 . A small amplitude perturbation is applied such that the transverse force (C y ) evolves for a relatively long time. An adequate number of cycles is required to capture the dynamics of the system to ensure that the unstable modes start to dominate the essential dynamics of the input-output relationship. However, an excessively long simulation time should be avoided that leads to an increase in the error norm associated with the RNN-LSTM. • The second step is to construct the linear approximation through the ERA methodology. This procedure consists of the formation of the Hankel matrix by stacking the transverse displacement ( A y ) at each timestamp, applying the SVD and evaluating the reduced system state matrices. The dimensions of the Hankel matrix can be determined by examining the convergence of the unstable eigenvalues. The linear ROM is calculated through Eq. (9.9), where Ar characterizes the dynamics of the system. • In the final step, the eigenvalues of the matrix Ar are calculated and the most unstable mode is selected. In this study the continuous eigenvalues, λ = log (eig(Ar ))/t are considered for the stability analysis. The eigenspectrum can be constructed by plotting the root loci of selected modes as a function of the reduced natural frequency (Fs ). The position of the eigenvalues on this spectrum provides information about the stability of the coupled system. The above process with the three main components forms a general procedure to be followed while constructing the nonlinear DL-based ROM integrated with ERA for the stability prediction of the VIV lock-in. We aim to identify the lock-in regions and to predict the unseen dynamics by utilizing the eigenvalue selection process on an amplitude response of the sphere as the desired output.

9.5.4 Problem Setup and Hyperparameter Analysis This section first deals with the problem setup of the FOM for flow past a sphere at a uniform flow field. The numerical verification of the DL-based ROM integrated with

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9 System Identification and Stability Analysis

ERA is presented. We assess the generality of the DL-based ROM by evaluating the error associated with the trained model and by testing the model on different test datasets. Later, we perform hyper-parameter sensitivity analysis for the training process to assess the robustness of the model. Finally, we check the reliability of the ERA for the stability prediction.

9.5.4.1

Full-Order Problem Setup

While the NLSS (Eq. 9.19) formulation is general for a fluid-structure system, as a prototypical problem, we consider the transverse motion of a fully submerged threedimensional canonical geometry of a sphere exposed to a uniform flow field. The sphere is mounted on a spring-damper system in the cross-flow direction and the unsteady fluid force makes the sphere vibrate in the transverse direction. Figure 9.47 shows a schematic of the problem setup used in our simulation study for a sphere, both stationary and elastically mounted cases. A three-dimensional computational domain of the size (50 × 20 × 20)D with a sphere of diameter D placed at an offset of 10D from the inflow surface is considered, which is sufficient enough to reduce the effects of artificial boundary conditions around the fluid domain. The origin of the coordinate system is fixed at the center of the sphere. We consider the x-axis as the streamwise flow direction, the y-axis in the transverse direction, and the z-axis represents the vertical direction. While the streamwise motion corresponds to the freestream (x-direction), the transverse motion is parallel to the y-direction. A uniform freestream flow with velocity U is along the x-axis. At the inlet boundary, a stream of water enters into the domain with velocity (u, v, w) = (U, 0, 0) where u, v and w denote the streamwise, transverse and vertical velocities in x, y and z directions, respectively. The sphere is elastically mounted on springs with a stiffness value of k and linear dampers with a damping value of c in the transverse direction. The damping coefficient ζ is set to zero in the present work. We have considered the slip-wall boundary condition along the top, bottom and side surfaces, in addition to the Dirichlet and traction-free Neumann boundary conditions along the inflow and outflow boundaries, respectively. The definitions of some relevant important non-dimensional parameters are summarized in Table 9.5. The non-dimensional amplitude response  A∗ is defined as

1 k A∗ = A/D and f ∗ denotes the normalized frequency and f n = 2π is the natural m frequency of the spring-mass system in vacuum, where m is the mass of the sphere and k is the spring stiffness. The mass ratio is given by m ∗ = m/m d , where m is the mass of the sphere and m d is the mass of displaced fluid. The value of U is considered to be 1 throughout the numerical study. The Reynolds number Re and the reduced velocity U ∗ are varied by changing the values of dynamic viscosity μf and the natural frequency f n . The normalized forces are evaluated from the fluid traction, acting on the structural body, where C x is the normalized drag force, C y and C z are the normalized transverse and vertical forces in y and z directions, respectively.

9.5 Deep Learning for Predicting Frequency Lock-in

545

Fig. 9.47 Schematic and associated boundary conditions of the fluid flow past a 1-DOF fully submerged elastically mounted sphere Table 9.5 Definition of the non-dimensional parameters and post-processing quantities Parameter Definition Re = ρ f U D/μf U ∗ = U/ f n D m ∗ = m/m d √ ζ = c/2 mk √ A∗rms = 2 Arms /D C x = f xs /( 21 ρU 2 S) C y = f ys /( 21 ρU 2 S) C z = f zs /( 21 ρU 2 S) f ∗ = f / fn

Reynolds number Reduced velocity Mass ratio Damping ratio Non-dimensional amplitude Normalized horizontal force Normalized transverse force Normalized vertical force Normalized frequency

1 Cx = 1 ρU 2 S 2 1 Cy = 1 2S ρU 2 1 Cz = 1 ρU 2 S 2

(σ¯ f · n) · ex d

(9.30)

(σ¯ f · n) · e y d

(9.31)

(σ¯ f · n) · ez d

(9.32)







where ex , e y and ez are the Cartesian components of the unit normal e, and S is the relevant surface area  which is defined as S = π D 2 /4. The normalized lift coefficient

is obtained as C L = C y 2 + C z 2 . A comprehensive mesh convergence study along with the validation of the solver by comparing with the experimental and available

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9 System Identification and Stability Analysis

numerical data are provided in [87, 89]. In the current study we use the same mesh configuration used in our aforementioned studies.

9.5.5 RNN-LSTM Training Procedure The formulation of the LSTM network as a well-established standard architecture along with its general nonlinear state-space (NLSS) form (Eq. 9.19) are provided in Sect. 9.5.2.1. While the NLSS formulation applies to any fluid-structure system, we choose the transverse motion of a fully submerged three-dimensional canonical geometry of a sphere exposed to a uniform flow field as a prototype problem (see Fig. 9.47). The main objective is to identify the VIV lock-in states for this coupled FSI problem for a range of parameter space (i.e., Reynolds number, mass ratio, reduced velocity and damping ratio). To this aim, we need to construct a ROM model which is based on the input-output data to make a relationship between system observables (i.e., the displacement of the bluff body and the corresponding hydrodynamic force applied on the bluff body). We train a network using the RNN-LSTM framework as a system identification toolbox to extract temporal feature relationships from highfidelity numerical solutions. For that purpose, we need to generate a large training dataset with useful information across a wide range of frequencies. Since the VIV has an intrinsic lock-in character, we can probe the lock-in at preferred frequencies where a self-sustained vibration can be observed. This can provide us with an efficient lowdimensional representation of the lock-in process and the self-sustained vibration. To acquire an efficient training dataset from FOM and to identify the resonant frequencies, we next provide our methodology to impose excitation through a userdefined function (UDF). There can be many different ways to construct the training dataset for the lock-in mechanism. For example, we consider a sequence of prescribed input displacement input (A y ) to the bluff body that contains a range of frequencies and amplitudes. The function can be given as follows = αe−βt sin(ω(t)t + φ) Ainput y output

(9.33)

The normalized transverse force (C y ) is recorded for every time step (t = 0.15) as the output response from the FOM. For a given VIV configuration at a specific Reynolds number and geometry, one can use the injection of a range of frequencies to the bluff body to extract the frequency response function (FRF). For example, the UDF input function (Eq. 9.33) that specify the range of input training dataset ) is general in terms of the parameter selection (α, β, ω, φ). To frequencies ( f Ainput y construct a training dataset, the parameters can be adjusted based on the underlying system dynamics and geometry. It is worth mentioning that a training procedure can be domain specific. In our study, the training procedure is proposed in a way to obtain the most useful information based on the underlying physics of the problem and the nature of the VIV

9.5 Deep Learning for Predicting Frequency Lock-in

547

phenomenon, therefore the parameters of the input function should be defined properly. For instance, if the parameter related to the amplitude of the input function (α) is extremely small, it may have a minimal effect on the hydrodynamic output response and the input-output relationship cannot be realized properly. On the other hand, extremely large (α) may lead to divergence of the output response. The amplitude decay/growth rate parameter (β) is another important parameter to consider the effect of change in the amplitude in the process of learning and to avoid large impulsive displacements at the higher frequency domain. The rate of change of frequency in the input function is defined by ω(t). For our FSI problem setup, we have adjusted ω(t) to have a minimum of 40 time steps (t) per fluctuation cycle at the highest frequency range. It is important to specify the range of input frequencies sufficiently until the dominant output response ( f C youtput ) is captured. We consider a range of non-dimensional frequencies f ∗ ∈ [0.02 − 0.2] where we expect VIV lock-in to happen. Based on the insight gained from the underlying physics of the problem at Re = 300, we know that at higher frequency ratios (i.e., f ∗ > 0.2) we do not expect high amplitude vibrations corresponding to VIV and the fluctuations are most probably negligible. To obtain a useful training dataset, we aim to examine different scenarios for the input UDF input which contain forced displacement (A y ) with both low amplitudes and ∗ high amplitudes at f ∈ [0.02 − 0.2]. In Fig. 9.48a, the input UDF is associated with low amplitudes at f ∗ ∈ [0.02 − 0.2]. By inspecting the frequency spectrum shown output ) has relatively low in Fig. 9.48b, we can observe that the power spectrum of (C y energy and does not properly capture the dominant output frequencies for f ∗ < 0.13. Hence the strength of the perturbations (i.e., the UDF amplitude at low frequencies) is not strong enough to realize the dynamics of the system. On the other hand, Fig. 9.49a shows a UDF input with high amplitudes at f ∗ ∈ [0.02 − 0.2]. Through the frequency spectrum in Fig. 9.49b, we observe a divergence of the output response output ) at high frequencies. This observation is not consistent with the self-limiting (C y character of the VIV phenomenon due to the nonlinearity and hydrodynamic damping effects. As a result, we propose an input UDF that takes into account both significant perturbations at lower frequencies and hydrodynamic damping effects at higher frequencies as a form of damped transient excitation, as shown in Fig. 9.50. The input to the system passes through the resonant frequencies and the output exhibits frequency preferences, where the resonant peaks appear. Resonant sphere-wake modes produce relatively larger responses when driven by inputs near their resonant frequency and smaller responses at other frequencies. The input-output dataset from FOM is shown in Fig. 9.50a, where a set of 5000 responses are stacked for a total simulation time of tU/D = 750. The RNN-LSTM model is only dependent on the Reynolds number and the geometry of the bluff body. The system dynamic associated with the RNN-LSTM model is decoupled and there is a partitioned coupling between the fluid and structural solvers. Using this decoupled FSI idea for the relationship between the displacement and the hydrodynamic force, we next verify the methodology for the coupled FSI cycle.

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9 System Identification and Stability Analysis

(a)

(b) Fig. 9.48 Input-output dataset for a sphere exposed to uniform flow at Re = 300: a time histories of input output the forced displacement (A y ) via UDF and the normalized transverse force (C y ) from FOM as the output response, b corresponding frequency spectra of the displacement and the transverse force. The displacement is computed by Eq. (9.33) with α = 0.01, β = 0, ω(t) = (1.42 × 10−2 )t 0.6 , φ = 0, and tU/D ∈ [0 − 450]

9.5.6 Verification of DL-Based ROM Integrated with ERA Based on the methodology described in Sec 9.5.2.1, the LSTM network is applied to construct the nonlinear ROM based on a pure input-output dataset for a transversely vibrating sphere at Re = 300. The FOM data are used to fit a model. Figure 9.51a compares the output normalized force signal calculated from the FOM (C y ) and the ROM (Cˆ y ), where we can see good performance (Fit = 97) for the training dataset. The model is simulated using the validation data input and the error is determined by

9.5 Deep Learning for Predicting Frequency Lock-in

549

(a)

(b) Fig. 9.49 Input-output dataset for a sphere exposed to uniform flow at Re = 300: a time histories of input output the forced displacement (A y ) via UDF and the normalized transverse force (C y ) from FOM as the output response, b corresponding frequency spectra of the displacement and the transverse force. The displacement is computed by Eq. (9.33) with α = 0.3, β = 0, ω(t) = (1.42 × 10−2 )t 0.6 , φ = 0, and tU/D ∈ [0 − 700]

the discrepancy between the model output and the measured validation data output as a system identification task. Figures 9.51 (b, c) show the predicted output (Cˆ y ) and the error for two different test datasets. It is worth noting that the simulation is done without any access to the measured validation output. As shown in the figures, the character of the validation dataset is different from the training datasets. The input for test dataset-1 (Fig. 9.51b) has a growing amplitude trend with a frequency range from higher to lower values, whereas the input for test dataset-2 (Fig. 9.51c) has a decaying amplitude trend with a frequency range from lower to higher values. Here for the acceptable hyper-parameter set, the main benchmark considered is the root mean square error (RMSE) and the

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9 System Identification and Stability Analysis

(a)

(b) Fig. 9.50 Input-output dataset for a sphere exposed to uniform flow at Re = 300: a time histories of input output the forced displacement (A y ) via UDF and the normalized transverse force (C y ) from FOM as the output response, a corresponding frequency spectra of the displacement and the transverse force. The displacement is computed by Eq. (9.33) with α = 0.3, β = 3.5 × 10−3 , ω(t) = (1.42 × 10−2 )t 0.6 , φ = 0, and tU/D ∈ [0 − 750]

fit for the identified model output and measured output corresponding to the nondimensional transverse force given by    C (t) − Cˆ (t|θ )  y   y RMSE =     N

(9.34)

9.5 Deep Learning for Predicting Frequency Lock-in

551

(a)

(b)

(c) output

Fig. 9.51 Time traces of the normalized transverse force (C y ) for an impulsively disinput placed sphere at Re = 300: a Train dataset: (A y = 0.3 × e−0.0035t sin(0.0142t 0.6 )t, tU/D ∈ input [0 − 750]), b Test-1 dataset: flipped A y , (t − 540)U/D ∈ [0 − 300], and c Test-2 dataset: shifted input A y , (t − 750)U/D ∈ [0 − 150]

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9 System Identification and Stability Analysis

Table 9.6 The root mean square error (RMSE) and the fit corresponding to train and test datasets at Re = 300 Dataset RMSE Fit Number of dataset (N) Non-dimensional time domain Train Test-1 Test-2

0.19% 1.13% 0.47%

97.1 82.9 87.6

5000 2000 1000

tU/D ∈ [0 − 750] tU/D ∈ [0 − 300] tU/D ∈ [0 − 150]

 ⎞   C y (t) − Cˆ y (t|θ ) ⎠. ⎝   Fit = 100 × 1 − C y (t) − C¯ y (t) ⎛

(9.35)

We calculate the RMSE and the fit for the training set and the two aforementioned test datasets, which are shown in Table 9.6. The results show good performance with the root mean squared error less than 1% for all datasets and hence provide an acceptable fit. While the model constructed using LSTM-RNN is validated by two aforementioned test datasets, the model cannot be tested with a good performance for any arbitrary input. By increasing the training dataset, one can create a relatively accurate model. Our methodology aims to construct an efficient training dataset for the VIV lock-in process.

9.5.6.1

Hyper-Parameter Sensitivity Analysis

Extreme refining of the learning rate and overuse of the hidden cells may cause the RNN-LSTM to overfit the training dataset and make it incapable of predicting the forces for perturbed geometries of the training dataset. The underutilization of hyper-parameters, on the other hand, will increase the prediction error. Here, we provide an empirical sensitivity analysis to determine the hyperparameter values for the optimum RNN-based learning performance. In particular, we study the sensitivity of the number of epochs, the number of hidden cells, and the learning rate. We begin by assuming that the RNN-LSTM method with the small number of hidden cells (# of Hidden units = 8) is the most appropriate for making predictions. Figure 9.52a shows the root mean square error (RMSE) for different learning rates (α) while the number of hidden cells is fixed (# of Hidden units = 8) during the evolution of the neural network as it trains after each epoch through recursive feedback. The minimum error is found when (α=0.025) after 20,000 iterations. Now, by fixing the learning rate to (α=0.025), the variation of RMSE of the training network with a lower and higher number of hidden units is depicted in Fig. 9.52b. The best convergence is observed for (# of Hidden units = 8). Figure 9.52c and d, show the %Error with the variation of learning rate and several hidden units respectively. Figure 9.52c shows that if the learning rate is set extremely small, the loss function might lead to a local minima issue, and if the learning rate is set too large, the loss function may exhibit

9.5 Deep Learning for Predicting Frequency Lock-in

553

(a)

(b)

(c)

(d)

Fig. 9.52 Hyper-parameter analysis: a, b Root mean square error (RMSE) variation with the number of epochs for different values of learning rate (α) and hidden units. c, d error variation of train and test datasets with different values of α and hidden units Table 9.7 Network specifications RNN-LSTM hyper-parameters Number of layers Number of hidden unit Optimizer Learning rate

Specifications 1 fully connected layer 8 Adam 0.050 @Iter ∈ [0 − 3000] 0.025 @Iter ∈ [3001 − 6000] 0.010 @Iter ∈ [6001 − 9000] 0.005 @Iter ∈ [9001 − 12000]

undesired divergent behavior. To overcome this issue our approach is to decrease the learning rate manually after specific iterations to obtain the global minima and fast convergence. As shown in Fig. 9.52a by the black line, we can get the fastest convergence with the smallest error while decreasing the learning rate after 3000 iterations. Table 9.7 shows the specifications of the selected network for our FSI problem. Next, we conduct the error analysis with a different number of hidden cells which is shown in Fig. 9.52d. We find that the RNN shows a good performance with (# of Hidden units = 8) while increasing the number of hidden units would lead to over-fitting issues and more computational cost.

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9 System Identification and Stability Analysis

(a)

(b)

Fig. 9.53 The DL-based ROM for the VIV of a sphere at Re = 300: a time history of the transverse amplitude (A y ) due to small perturbation calculated from the DL-based ROM and the linear approximation via ERA and b HSV distribution corresponding to 500 × 250 Hankel matrix

9.6 Assessment of DL-Based ROM for VIV of Sphere

9.5.6.2

555

Reliability of ERA

Here we aim to assess the reliability of the linearized approximation through the ERA. The process with the three main components forms a general procedure to be followed while constructing the ERA-based ROM as presented in Sects. 9.5.3.1 and 9.5.3.2. It should be noted that ERA is only applied to the displacement vector A y as a desired output from the DL-based ROM for the stability predictions. Based on the tiny amplitude perturbation (δ = 10−3 ) given to the ERA-based ROM as an input, the transverse displacement A y is recorded for every time step t = 0.15. The linearity of the unstable system is confirmed by comparing the response subject to two impulse inputs with δ = 10−3 and δ = 10−4 . A set of 750 responses are stacked resulting in a total simulation time of tU/D = 112.5. Figure 9.53 shows the output amplitude signal A y calculated from the DL-based ROM and the linearized ERA-based ROM at Re = 300. The Hankel matrix may not be necessarily square but can be tall, wide or square based on the problem setup. The Hankel matrix with dimension 500 × 250 is found to be appropriate by examining the convergence of unstable eigenvalues computed from Hankel matrices with dimensions of 500 × 125, 500 × 250, and 500 × 500. The order of the ERA-based ROM is selected by examining the HSV distribution. The fast-decaying singular values as depicted in Fig. 9.53b suggest that the ERA-based ROM with order n r = 12 is sufficient. This is further confirmed by the accurate reconstruction of the impulse response by the ERA-based ROM displayed in Fig. 9.53a. It is worth mentioning that the computational effort associated with the development of the DL-based ROM integrated with ERA is extremely efficient in comparison to FOM simulations. To demarcate the lock-in range from the FOM, the long-term unsteady simulation is required to construct the relationship between the reduced natural frequency (Fs ) and the normalized transverse amplitude (A∗y ) at the stationary state. The DL-based ROM integrated with ERA completely avoids such expensive simulations. Moreover, construction of the eigenspectrum is trivial and fast since it relies on the SVD procedure. In many situations, linear models give a reasonable approximation of the underlying process. Here we utilize the ERA methodology for the linear approximation of the coupled nonlinear model. By examining the HSV distribution, we select enough modes to obtain the error corresponding to most of the cases by less than 1%. However, there exist some cases for which the linear approximation cannot be made even by considering the full-rank Hankel matrix.

9.6 Assessment of DL-Based ROM for VIV of Sphere We next analyze the stability properties of a transversely vibrating sphere at Re = 300 and Re = 2000 through the DL-based ROM integrated with ERA. The eigenvalue analysis for the sphere-wake synchronization is systematically performed for the first

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9 System Identification and Stability Analysis

time. Specifically, we provide physical insight about the role of coupled structural mode instability using the proposed DL-based ROM and our full-order simulation results.

9.6.1 The Role of Structural Mode Instability The vortex-induced vibration is a coupled fluid-structure instability and the frequency lock-in plays an important role to sustain the vibration. The frequency lock-in arises through a complex interplay between the wake features and the bluff-body motion. A self-sustained cyclic mechanism has been discussed in [299]. Through a linear global stability analysis of the flow past an elastically mounted cylinder, [97] identified two modes in the fluid-structure system, namely a nearly structural mode and the von Kármán wake mode. [283] employed a standard asymptotic analysis to confirm the existence of the two modes identified by [97] and termed them the wake mode (WM) and the structure mode (SM). They found that for weak fluid-structure interaction at a relatively large solid-to-fluid mass ratio, the eigenvalue of the WM is equivalent to the leading eigenvalue computed for the flow past a fixed cylinder (i.e., shedding frequency of a stationary cylinder), whereas the eigenvalue of the SM approaches the characteristic eigenvalue of the cylinder-only system in the absence of fluid. Using the ROM-based stability analysis, [466, 486] examined the interplay of the wake mode with the structural mode for varying physical and geometry conditions and confirmed the existence of two regimes namely resonance-induced lock-in and flutter-induced lock-in. In the recent study of [70], the authors particularly considered the stability analysis of passive suppression for VIV of the cylinder and found that wake stabilization does not appear to be necessary for the suppression of the structural vibration during fluid-structure interaction. They suggested that the SM is an appropriate indicator for the stability analysis of suppression devices while WMs may remain unstable. In another follow-up study of [87] on the passive suppression of 3D sphere VIV using the ERA-based ROM, the authors examined the stabilization of all complex wake modes using a jet-based actuation at a moderate Reynolds number. While the synchronization of the shedding process is inhibited, the instability of the SM is found to be present in the stabilized wake modes due to the asymmetry of the 3D wake structures. In this work, we further explore the role of structural mode as a potential indicator for the stability analysis of fluid-structure interaction problems using the DLbased ROM. Consistent with the previous literature by Yao et al. [466], we utilize the methodology of variation of unstable modes to classify the distinct eigenvalue trajectories of the fluid-structure system governed by Eq. (9.9). As discussed in Sects. 9.5.3.1 and 9.5.3.2, the ERA-based ROM can be applied to the desired output of the coupled FSI system. Therefore, we can extract the structural modes by considering the transverse displacement (A y ) as the desired output of the coupled DL-based ROM system. During the coupled FSI cycle, when the shedding process (unstable WMs) is dominated by the structural mode, a strong coupling between the

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557

Fig. 9.54 Eigenspectrum of the DL-based ROM (black dots) at (Re, m ∗ ) = (300, 2.865). The predicted lock-in branches, shaded with a grey color are corresponding to resonance-induced regions (branch A and B)

fluid and the structure is established close to the natural frequency of the body. In this work, the VIV branch is termed as resonance dominated if the imaginary part of the most unstable eigenvalue corresponding to the transverse displacement (A y ) gets close to the natural frequency of the structure in a vacuum. The stability of this region is determined by the value of the real part of the eigenvalues corresponding to A y . Based on the methodology presented in Sect. 9.5.3.2, we aim to identify the lock-in regions and predict the unseen dynamics by utilizing the eigenvalue selection process on the sphere transverse vibration amplitude ( A y ) through the structural mode decomposition.

9.6.2 Stability Analysis of Sphere VIV at the Onset of Instability In this section, we perform a data-driven eigenvalue analysis to investigate the synchronization of a transversely vibrating sphere at Re = 300. We aim to explore the role of structural mode instability via our proposed DL-based ROM and full-order simulations. Figure 9.54 shows the extracted eigenvalue trajectory corresponding to A y (SM) as a function of the reduced natural frequency (Fs ) with 0.03 < Fs < 0.25 and the increment is Fs = 0.002. As elucidated in Fig. 9.54 (top), through the real

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part of the eigenvalues, we evaluate the growth rate of the SM. By changing from higher Fs to lower Fs , we observe that the SM becomes unstable at Fs = 0.17, which is determined by the real part of the eigenvalues (Re(λ) > 0). Figure 9.54 (bottom) shows the imaginary part of the eigenvalue trajectory (Im(λ)) as a function of Fs that involves information with regards to the frequency of the SM. By varying from the higher Fs to the lower Fs , we can observe a change in the behaviour of the SM frequency as it intersects the WM = 2π St (i.e., the vortex shedding frequency behind a stationary sphere). In addition, we observe that the frequency of the SM remains close to the natural frequency of the sphere in a vacuum (uncoupled structural mode) for 0.092 < Fs < 0.17 while Re(λ) > 0. Through these observations, we find that the structural mode dominates the shedding process (unstable WMs), and a strong connection between the fluid and the structure is developed close to the natural frequency of the body. This is termed as branch (A) in Fig. 9.54. Another interesting finding is that the lower-left boundary of the resonance mode can be pinpointed from the extracted eigenvalue trajectory at Fs ≈ 0.092. It is worth mentioning that our previous study using purely linear ERA-based approach for the stability predictions [87] were unable to predict the lower-left boundary from the ROM. By further moving to lower Fs range, we can observe that the SM frequencies are detached from the natural frequency at Fs = 0.09 and bounce back once more to the natural frequency at Fs ≈ 0.085 close to the intersection with the half of the shedding frequency (St/2). We find that the SM frequency remains close to the natural frequency of the body for 0.07 < Fs < 0.085 while Re(λ) > 0. Therefore we can identify another resonance-induced branch at lower frequency range (branch B) for which the structural mode dominates the low-frequency wake mode in the neighbourhood of (St/2). To verify the dominance of the structural mode in the sphere-wake synchronization, the VIV amplitude is computed by direct numerical simulation using the FOM. Figures 9.55 and 9.56 show time histories of the sphere VIV oscillation (A∗y ) and the normalized transverse hydrodynamic force (C y ) along with their frequency spectrum for different values of U ∗ or Fs . As can be seen in Fig. 9.55a, for the sphere pre-lock-in response at (U ∗ = 5, Fs = 0.2), we observe that the fluid force (C y ) on the sphere body cause negligible oscillations of the structure. Through the frequency spectrum plot, we find that the fluid force frequency at ( f C y = 0.136), as a sign of the most unstable wake mode owing to the shedding process (St = 0.136), cannot dominate the structural oscillations. This finding is in line with the presence of stable SM (Re(λ) < 0) retrieved from DL-based ROM at Fs = 0.2, as shown in Fig. 9.54 (top). On the other hand, in Fig. 9.55b, for the sphere VIV response at (U ∗ = 6, Fs = 0.166), we find that the intrinsic tendency of the sphere motion drives the wake features. Hence, we observe a resonance-induced lock-in due to the structural mode instability at f ∗ = 0.149. This sphere-wake synchronization is driven by the intrinsic motion-induced instability of a sphere until the lower-left boundary of the lock-in regime. This lower-left boundary is predicted at (Fs = 0.092, U ∗ ≈ 10.8) through the DL-based ROM eigenspectrum plot in Fig. 9.54 (top) corresponding to branch (A). It is further shown in Fig. 9.55c where the structural mode instability persists for the sphere VIV in the lock-in regime at (U ∗ = 9, Fs = 0.111). The large sphere

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559

amplitude response in the order of (≈ 0.3D) is observed at the lock-in frequency at f ∗ = 0.107. As shown in Fig. 9.55d, the synchronization is terminated at (U ∗ = 11, Fs = 0.90) where we find that the structural mode is not strong enough to drive the oscillatory unstable sphere-wake modes and therefore the sphere oscillations remain small in the order of (≈ 0.02D) in the stationary state. These small amplitude oscillations at (U ∗ = 11, Fs = 0.90) confirm the existence of unstable SM (Re(λ) > 0) extracted from our DL-based ROM outside the lock-in region [(Fig. 9.54 (top)] with a frequency different from the natural frequency of the body [Fig. 9.54 (bottom)]. Figure 9.56 shows the coupled VIV response along with the associated frequency spectrum corresponding to several cases at lower non-dimensional natural frequency range Fs < 0.07. As can be seen through the frequency spectrum plots for all cases in the range Fs < 0.07, the structural mode is not strong enough to dominate the sphere-wake synchronization, and through the oscillation response we observe minimal sphere fluctuations in the stationary state. These observations verify the predicted eigenspectrum extracted from the DL-based ROM in Fig. 9.54, for which there exist unstable structural modes with (Re(λ) > 0) for Fs < 0.07 with a frequency distinct from the natural frequency of the body. To further explore the sphere-wake synchronization, we plot the iso-surface of non-dimensional Q-criterion and vorticity contours from the FOM calculations for several reduced velocities corresponding to the pre-lock-in (U ∗ = 5, Fs = 0.2), the lock-in (U ∗ = 6, Fs = 0.166) and (U ∗ = 9, Fs = 0.111), and the post-lock-in (U ∗ = 11, Fs = 0.90) regimes at (Re, m ∗ ) = (300, 2.865), as shown in Fig. 9.57. Through the figure, we can observe the oscillatory wake and vortex shedding for all pre-lockin, lock-in, and post-lock-in regions, while it has been shown that the large-amplitude oscillations of the sphere only occur at the lock-in region. From Fig. 9.57b and c, we find that in the lock-in regime the intrinsic tendency of spring-mounted sphere motion drives the sphere-wake synchronization, where the oscillatory wake forms the fully detached hairpin-type vortices that periodically shed with a fixed plane of symmetry along the transverse motion direction (y-axis). However, in the prelock-in (U ∗ = 5, Fs = 0.2) and the post-lock-in (U ∗ = 11, Fs = 0.90) regions, it is found that the structural mode is not dominant to drive the oscillatory wakes, and therefore the sphere-wake is naturally shedding with a desynchronized rhythm either in a tilted plane of symmetry along the transverse motion direction (y-axis), as shown in Fig. 9.57a corresponding to the pre-lock-in state, or it becomes asymmetry with multiple shedding modes, as shown in Fig. 9.57d corresponding to the post-lock-in state. In Fig. 9.58, we extract the sphere oscillation frequencies from the FOM data of [352] and compare them with our eigenvalue analysis based on the DL-based ROM integrated with ERA. While there exist some discrepancies between the DL-ROM and the FOM predictions, the overall trends of eigenvalue distributions and bounds are captured reasonably. We next explore the effectiveness of the DL-based ROM to identify the aforementioned VIV lock-in regimes for the elastically mounted sphere at different mass ratios through eigenvalue analysis. Figure 9.59 shows the structural mode extracted from DL-based ROM for the coupled system at (m ∗ = 2.865, 5, 10)

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9 System Identification and Stability Analysis

(a)

(b)

(c)

(d) Fig. 9.55 Time history of sphere VIV oscillations (A y ) and the normalized transverse force (C y ) from FOM, and the corresponding frequency spectrum at (Re, m ∗ ) = (300, 2.865) and U ∗ = a 5, b 6, c 9, and d 11

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561

(a)

(b)

(c)

(d) Fig. 9.56 Time history of sphere VIV oscillations (A y ) and the normalized transverse force (C y ) from FOM, and the corresponding frequency spectrum at (Re, m ∗ ) = (300, 2.865) and U ∗ = a 14, b 16, c 20, and d 30

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9 System Identification and Stability Analysis

Fig. 9.57 Iso-surface of three-dimensional wake structures formed behind an elastically mounted sphere, and z-vorticity contours from FOM at stationary state at (Re, m ∗ ) = (300, 2.865) and 2 U ∗ = a 5, b 6, c 9, and d 11. Iso-surfaces are plotted by the Q-criterion ( Q¯ = Q UD2 = 0.001)

9.6 Assessment of DL-Based ROM for VIV of Sphere

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Fig. 9.58 Eigenspectrum of the DL-based ROM (black dots) at (Re, m ∗ ) = (300, 2.865). The predicted lock-in branches, shaded with a grey color, are compared with the FOM results of [352], shaded with a blue color, corresponding to resonance-induced regions (branch A and B)

and Re = 300, where the real and imaginary parts of the root loci are plotted as a function of the reduced frequency Fs . Through the real part of the eigenvalues, we find that as the mass ratio m ∗ increases, the unstable structural mode corresponding to the lock-in onset, which is the right-higher VIV boundary corresponding to branch (A), shifts to the lower reduced natural frequencies (Fs ). In addition, by increasing the mass ratio m ∗ , the structural mode ability to dominate the wake synchronization at the lower-left lock-in boundary of branch (A) shifts to higher reduced natural frequencies (Fs ). We observe that the lock-in boundaries corresponding to branch (B) become narrower as the mass ratio increases and shift to the lower reduced frequency, and we identify another unstable resonance-induced lock-in region (branch C) for a higher mass ratio (m ∗ = 10) at lower reduced frequency Fs . To assess the predictive ability of our DL-based ROM, the high-fidelity VIV response at two different mass ratios (m ∗ = 2.865 and m ∗ = 10) from FOM is shown in Fig. 9.60. In Fig. 9.60a, the sphere oscillation frequency FOM data is extracted and transformed for the eigenvalue analysis. We find that the intrinsic tendency of the structural motion to dominate the sphere-wake synchronization is reduced as the mass ratio increases due to a relatively weaker fluid-structure coupling. Figure 9.60b shows the amplitude response of the sphere VIV based on direct numerical simulations, where we observe that the lock-in boundaries corresponding to branch (A) get narrower. We confirm that the bounds of lock-in from FOM response reasonably match with the DL-based ROM predictions. Furthermore, the amplitude response for the case at m ∗ = 2.865 in Fig. 9.60b shows that the lock-in onset (higher-right boundary) with high amplitude response corresponding to branch A starts at (Fs ≈ 0.18, U ∗ = 5.5) and terminates at lower-left boundary at (Fs ≈ 0.095, U ∗ = 10.5). From Fig. 9.60a, we can observe that the sphere oscillation frequency for branch A is close to the system natural frequency ( f n ) which compares well with the resonance-induced lock-in predicted by the present DL-based ROM in Fig. 9.54-bottom. In addition, the FOM simulations at m ∗ = 2.865 in Fig. 9.60b show noticeable oscillations at branch (B). From Fig. 9.60a it can be seen that the lock-in frequency corresponding to branch B remains close to the natural frequency of the body ( f n ) that is consistent with the identified region predicted by the DL-based ROM (Fig. 9.54-bottom).

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9 System Identification and Stability Analysis

Fig. 9.59 Eigenspectrum of the DL-based ROM at Re = 300 and m ∗ = 2.865, 5, 10. The resonance-induced lock-in branches are shaded with grey, blue, and red colors corresponding to m ∗ = 2.865, 5, and 10 respectively

9.6.3 Stability Analysis of Sphere VIV at Moderate Reynolds Number In this section, we assess the performance of the DL-based ROM for stability predictions of a transversely vibrating sphere at Re = 2000. We investigate the sphere-wake synchronization through the coupled structural mode instability. The previous studies on the stability analysis using pure linear-type projections [70, 87, 466] can only be performed close to the critical Reynolds number (Recr ) which provides a low-order representation of the unsteady flow dynamics in the neighborhood of the equilibrium steady state. However, in higher Reynolds number cases where the wake is turbulent, more complexity is added to the problem setup and the linear ERA-based ROM cannot be reliable (i.e., eigenvalues do not decay exponentially). In addition, for 3D turbulent flows at a high Reynolds number, a brute-force time integration may not be easy to converge towards the stable flow state (base flow) that is required for the stability analysis via the ERA approach. Therefore, other nonlinear-type system identification techniques are required to overcome these complexities.

9.6 Assessment of DL-Based ROM for VIV of Sphere

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(a)

(b) Fig. 9.60 The FOM VIV results as a function of the reduced natural frequency Fs at (m ∗ = 2.865, 10) and Re = 300, a variation of the oscillation frequency f , and b the r.m.s. value of the normalized amplitude (A∗rms )

For this purpose, we utilize the proposed methodology of the DL-based ROM integrated with ERA (Fig. 9.46) for a canonical sphere at Re = 2000. To generate the training datasets, we impose the excitation through a damped transient function input as a sequence of forced transverse displacement input A y to the sphere in a flow field at Re = 2000. The training dataset involves a broad range of non-dimensional frequencies f ∗ ∈ [0.02 − 0.2] to identify the VIV resonant frequencies. The output output is recorded for every time step t = response as a normalized transverse force C y 0.15. As shown in Fig. 9.61, we collect a set of 4000 responses resulting in a total simulation time of tU/D = 600. Figure 9.61a compares the input-output training and test datasets computed by the direct numerical simulations from FOM and RNNLSTM reduced model at Re = 2000. The FOM data are used to fit a model, and the validity of the model is checked by how well it reproduces the validation data as shown in Fig. 9.61a. The simulation shows a good performance with the (Train Fit = 73, Test Fit = 62). The stability prediction and eigenvalue analysis are assessed

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9 System Identification and Stability Analysis

(a)

(b) Fig. 9.61 The unstable wake behind a forced displaced sphere at Re = 2000: a time history of the normalized transverse force (C y ) from the FOM and the temporal prediction from the DL-based ROM LSTM network, b the corresponding frequency spectrum of the forced displacement of the sphere and the corresponding normalized transverse force (C y ) from FOM

via partitioned coupling through DL-based ROM integrated with ERA. The process of constructing the linearized approximation through ERA is presented in detail in Sects. 9.5.3.1 and 9.5.3.2. Figure 9.62 shows the structural mode eigenvalue trajectory extracted from the coupled DL-based ROM as a function of the reduced natural frequency (0.03 < Fs < 0.3; Fs = 0.002) for three different mass ratios m ∗ = (3, 5, 10). By considering the imaginary part of the eigenvalues (Im(λ)) in Fig. 9.62b, we can find the strong connections between the fluid and the structure to identify the lock-in regions (branches A, B, and C). We observe that the structural mode leads the sphere wake-synchronization with a frequency close to the natural frequency of the structure during the lock-in branches. The structural instability of these branches is determined

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567

by the real part of the eigenvalues (Re(λ) > 0), as shown in Fig. 9.62a. Using our full-order simulations and the proposed DL-ROM analysis, we observe the lower-left boundary of the resonance region branch A shifts to higher reduced natural frequencies (Fs ) by increasing the mass ratio m ∗ at Re = 2000. In addition, other unstable resonance-induced regions (branches B and C) are identified for higher mass ratios (m ∗ = 5 and 10) at lower reduced natural frequencies. In Fig. 9.62b, to further cross-validate the DL-based eigenvalue analysis, we extract the sphere oscillation frequency FOM data from [351] for a sphere VIV response at (Re, m ∗ ) = (2000, 3). We depict both the DL-base ROM and the FOM data in the eigenspectrum frequency diagram. In addition, Fig. 9.62c shows the FOM amplitude response for an elastically mounted sphere at two configurations (Re = 300, m ∗ = 2.865) and (Re = 2000, m ∗ = 3 with h ∗ = h/D = 1, where h is the submergence depth from a free surface). The FOM simulations for the case at Re = 2000 show a large amplitude oscillation region in the range of 0.07 < Fs < 0.24, and as shown in Fig. 9.62b, the vibration frequency corresponding to these large amplitude oscillations is very close to the natural frequency of the system. This lock-in region is predicted almost accurately by the DL-based ROM where the SM is found to be the dominant synchronization mode in the range of 0.078 < Fs < 0.25 at (Re, m ∗ ) = (2000, 3). We can observe that the bounds of lockin are reasonably matched with the full-order simulations. In addition, through the DL-based ROM predictions we find that the lock-in region corresponding to branch A at Re = 2000 (Fig. 9.62) is larger than the lock-in range at Re = 300 (Fig. 9.59) for the studied mass ratios. The present DL-based ROM integrated with ERA methodology has been concerned with fully submerged three-dimensional bluff-body configurations for which all three directions in space are resolved [90]. All of the notions of the ROM can be easily extended to more complicated settings such as realistic geometries, multiphase flows, and free surface effects. Thus, the present method does not pose any theoretical limitation except that there may be numerical ones with respect to memory requirements and CPU time to obtain the FOM-trained dataset and to train the generalized neural network.

9.6.4 Summary To summarize, we introduced a new deep learning-based reduced-order model for nonlinear system identification and stability predictions of 3D fluid-structure systems. The proposed DL-ROM relies on the LSTM recurrent neural network and the eigenvalue realization algorithm. We presented an effective training strategy to construct an input-output relationship for a reduced-order approximation of a fluid-structure system using full-order simulations as a temporal series of force and displacement measurements. The input function provided a range of frequencies and amplitudes based on prior knowledge of the VIV lock-in process, allowing for efficient DL-ROM without the requirement for a large training dataset for low-

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9 System Identification and Stability Analysis

Fig. 9.62 Eigenspectrum of the DL-based ROM at Re = 2000 and m ∗ = (3, 5, 10), and comparison of the results with the FOM response amplitude as a function of the reduced natural frequency Fs

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569

dimensional modeling. We developed a methodology to integrate ERA as a linear encoder-decoder-like framework with the DL-based ROM to extract the coupled fluid-structure dynamics modes. We provided a novel eigenvalue selection process to investigate the underlying mechanism and stability characteristics of VIV. Systematic stability analysis of sphere VIV has been numerically presented at Re = 300 and Re = 2000. We performed the data-driven eigenvalue analysis to investigate the sphere-wake synchronization. The role of structural mode during the coupled FSI cycle is explored and we find that the intrinsic motion-induced instability of a sphere dominates the wake dynamics and drives the sphere-wake synchronization. We demonstrate the change in the formation of the vortical structures in the lock-in regions due to the structural mode instability. The effectiveness of the DLbased ROM has been remarkably demonstrated for predicting the resonance-induced lock-in and self-sustained VIV. For Re = 300, multiple unstable regions were identified that showed strong coupling of the structural mode with shedding frequency and other low-frequency wake modes resulting in different resonance-induced regimes. These coupled modes were categorized into three distinct lock-in branches, namely A, B, and C, and cross-validated successfully with the FOM and the available literature. We next employed the DL-based ROM to investigate the effect of mass ratio (m ∗ ). By increasing the mass ratio, the higher frequency domain (branch A) becomes smaller and the lower-left boundary of the resonance-induced branch (A) shifts to higher reduced natural frequencies. Owing to the weaker coupling of the fluid and structure, the intrinsic tendency of the structural motion to dominate the sphere-wake synchronization is reduced as the mass ratio increases. For Re = 2000 corresponding to turbulent flow, we were able to identify unstable regions due to the strong coupling of the structural mode with complex dominant wake modes, and the results are cross-validated successfully with the full-order simulations. The simplicity and computational efficiency of the DL-based ROM allow investigation of the VIV mechanism for high Reynolds numbers and more complex problem setups and a variety of geometries and parameters and pave the way for the development of control devices. For complex turbulent flows and fluid-structure interactions, it is worth exploring nonlinear autoencoders instead of ERA in the proposed DL-based ROM architecture. Acknowledgements Some parts of this Chapter have been taken care from the PhD thesis of Weigang Yao and Zhong Li carried out at the National University of Singapore and supported by the Ministry of Education, Singapore, and the MASc thesis of Amir Chizfahm at the University of British Columbia supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).

Appendix A: Derivation of Phase Angle for VIV By considering the cylinder motion and fluid forcing as sinusoidal functions, the displacement and lift coefficient can be obtained for the VIV linear system as

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9 System Identification and Stability Analysis

Y = Yˆ eλr t cos(λi t)

⎫ ⎬

Cl = Cˆl eλr t cos(λi t + φ) ⎭

,

(9.36)

where λ = λr + iλi is eigenvalue with real λr and imaginary λi components, Yˆ and Cˆl denote the magnitudes of eigenmodes. The phase angle φ can be derived by plugging Eq. (9.41) into the structural Eq. (9.13) as ˆ

λr t

[Yˆ eλr t (λr2 − λi2 + 4π ζ Fs λr + (2π Fs )2 ) − as Cl em ∗ cos φ ] cos λi t ˆ λr t +[Yˆ eλr t (−2λr λi − 4π ζ Fs λi ) + as Cl em ∗ sin φ ] sin λi t = 0.

(9.37)

From equating the coefficients of cos(λi t) and sin(λi t) to zero, we obtain the following relations: ˆ λr t Yˆ eλr t (λr2 − λi2 + 4π ζ Fs λr + (2π Fs )2 ) − as Cl em ∗ cos φ = 0, ˆ λr t Yˆ eλr t (−2λr λi − 4π ζ Fs λi ) + as Cl em ∗ sin φ = 0.

(9.38)

By solving Eq. (9.38) and setting ζ = 0, sin φ and cos φ can be obtained as 2Yˆ λi λr m ∗ , as Cˆl Yˆ m ∗ (λr2 −λi2 +(2π Fs )2 ) . as Cˆl

sin φ = cos φ =

(9.39)

Through the trigonometric identity cos2 φ + sin2 φ = 1, the term sin φ can be further simplified in terms of (λ, Fs ) as follows: 2λr λi . sin φ =  (λr2 + (2π Fs )2 + λi2 )2 − (4π λi Fs )2

(9.40)

Appendix B: Fluid-Structure Energy Transfer Following our previous work on VIV [466], the displacement and lift coefficient can be obtained for the linear FSI system as: Y = Yˆ eλr t cos(λi t)

⎫ ⎬

Cl = Cˆl eλr t cos(λi t + φ) ⎭

,

(9.41)

where λ = λr + iλi is eigenvalue with real λr and imaginary λi components, Yˆ and Cˆl denote the magnitudes of eigenmodes. The phase angle difference is derived by plugging Eq. (9.41) into Eq. (9.13):

9.6 Assessment of DL-Based ROM for VIV of Sphere

571

Fig. 9.63 Comparison of the experimental data and the LSTM model training and testing outputs. The training fit is 95.2% and the testing fit is 91.6%

2λr λi sin φ =  . (λr2 + (2π Fs )2 + λi2 )2 − (4π λi Fs )2

(9.42)

Refer to [466] for further details. Next, the energy transfer per cycle is evaluated as: t+ 2π λ



E(t) =

i

Y˙ Cˆl dt

t

(9.43)

t+ 2π λ

=

1 (λi sin(φ) + λr cos(φ) + λr cos(2λi t + φ)) 2



i

e2λr t dt t

Using Eq. (9.43), the energy transfer coefficient E c can be defined by excluding the exponential growth/decay rate λr E c = π Yˆ Cˆl sin(φ)

(9.44)

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9 System Identification and Stability Analysis

Appendix C: Assessment with Silverbox Benchmark To demonstrate the performance and the accuracy of the LSTM network as a tool for system identification, we consider a data set of Silverbox benchmark representing a nonlinear dynamical system. The dynamical system represents a forced Duffing oscillator and includes a mechanism with a cubic hardening spring [383]. Two sets of data with different responses are collected for the study. The training data set is used to construct a model, and the validity of the model is determined by how well the validation data are reproduced as a standard system identification task. Figure 9.63 depicts the performance of the LSTM model while the training and testing data sets have different characteristics consistent with the simulation performed by [260]. The results show a good performance with the train fit = 95.2% and the test fit = 91.6%, which corroborates the reliability of the LSTM network utilized in this study for the stability prediction of the coupled fluid-structure system.

Chapter 10

Data-Driven Passive and Active Control

In this chapter, we present efficient parametric design optimization and control strategies using data-driven model reduction techniques. In particular, we explore various suppression devices based on passive-based wake stabilization and active-based synthetic jet. A data-driven model reduction approach based on Eigensystem Realization Algorithm (ERA) is used to construct the reduced order model (ROM) in a statespace format. The ERA-based ROM provides a low-dimensional representation of the unsteady flow dynamics in the neighborhood of the equilibrium steady state. The stability analysis based on the ERA-based ROM provides the eigenvalue distribution and the estimation of a possible lock-in region of the coupled FSI system. To establish the reliability of the ERA-based ROM, we first examine the passivebased wake stabilization techniques such as a cylinder-fairing arrangement and connected-C device. Then, a base bleeding mechanism in the near-wake region of a sphere and its influence over the flow dynamics, the wake characteristics and the VIV response are investigated for the freely vibrating sphere system. Finally, active feedback blowing and suction procedures via model reduction are presented for unsteady wake flow and the vortex-induced vibration (VIV) of circular cylinders. The actuation is considered via vertical suction and blowing jet at the porous surface of a circular cylinder with the body-mounted force sensor. While the optimal gain is obtained using the linear quadratic regulator (LQR), Kalman filtering is employed to estimate the approximate state vector. The feedback control system shifts the unstable eigenvalues of the wake flow and the VIV system to the left half complex plane and subsequently results in the suppression of the vortex street and the VIV in elastically mounted structures. The resulting controller designed by linear low-order approximation is able to suppress the nonlinear saturated state of wake vortex shedding from the circular cylinder.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Jaiman et al., Mechanics of Flow-Induced Vibration, https://doi.org/10.1007/978-981-19-8578-2_10

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10.1 Introduction The phenomenon of vortex-induced vibration (VIV) is ubiquitous and is relevant to various applications in offshore, underwater, automobile, aerospace, and wind engineering. A relatively large motion and the mean drag force due to the VIV phenomenon can be hazardous and cause severe damage to a submerged structure [40]. During the VIV phenomenon, the frequency of vortex shedding locks onto the oscillation frequency of the body, which is termed as lock-in. The lock-in is a selfexcited process, which is identified with the equivalence of the frequency of vortex shedding and the oscillation frequency of the body [206, 466].

10.1.1 Control of Vortex-Induced Vibration Control of VIV is of paramount importance to many engineering applications and poses a great deal of physical complexity due to the richness of vorticity dynamics and the coupled fluid-structure interaction. The control can be achieved by using active feedback control techniques such as blowing and suction or passive techniques. As described in [56], passive techniques can be generally categorized into four groups which are, avoiding resonance, streamlining the cross-section, increasing damping and adding a vortex suppression device. A systematic understanding of VIV suppression can facilitate a more effective design of control devices. Until now, experimental and numerical methods have been the most popular and rigorous approaches to understand the frequency lock-in and VIV phenomenon. These studies have primarily explored the effects of mass ratio, Reynolds number and structural damping on the lock-in regime. For this purpose, the flow past a single elastically mounted cylinder has served as a generic VIV model for both numerical and experimental investigations. As a next step to the fundamental analysis of VIV physics of a single cylinder, the investigations based on the full-order model for the multiple cylinders [254, 307] and the suppression devices [226] are recently reported in the literature. Appendages such as fairings, helical strakes, splitter plate, and control cylinders are generally considered as vortex suppression devices. Neutrally buoyant fairing devices are found to be effective in minimizing VIV by preventing the interaction of shear layers with near wake and extending the shear layers further downstream to avoid the alternate periodic vortex shedding. Recently, a novel device termed as connected-c has been designed based on the principle of shear layer reattachment by [226], which was demonstrated to be a similar performance as the conventional fairing. The primary challenges concerning the adaptation of suppression devices are the optimal placement and geometric variations. Several experimental and numerical techniques have been used for optimal design and placement of suppression devices. [29] carried out experimental investigations on various designs of fairings and found short-crab-claw (SSC) configuration to be effective and robust in both VIV suppression and reduction of drag force. The exper-

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imental study conducted by [14] showed that while the helical strakes suffered from higher drag force, the fairing reduced the drag force significantly. [209, 293] performed direct numerical simulations to find a suitable optimal position for a control cylinder to be placed in the wake of the main cylinder which resulted in the suppression of vortex shedding. A systematic theoretical analysis has been developed for the same problem by [279]. Recently, shape optimization of fairings has been undertaken by integrating CFD with genetic algorithm (GA) to obtain a high-performance riser fairing and by parametrizing the riser profile via Beizer curves [82]. The control points were chosen as design variables and the lift coefficient was selected as the objective function to obtain optimum fairing profile. High-fidelity full-order simulations provide detailed physical insight and quantitative information such as force trends, vorticity distribution, flow patterns, frequency characteristics and phase relations. Despite these benefits of the full-order analysis, when it comes to the development of control strategy and the parametric optimization, the cost associated with the full-order modeling is exceptionally high even with the current state-of-art supercomputers. Due to the complexity of FSI problems, these direct experimental and numerical investigations also pose difficulty in addressing the underlying physical mechanisms of the frequency lock-in during VIV.

10.1.2 Types of Reduced Order Models In the context of these limitations of the detailed full-order analysis, model order reduction (dimensionality reduction) serves as an attractive alternative to address the aforementioned shortcomings. Specifically, to address the problem of developing effective control strategies, a stability analysis is required via an efficient model reduction process. The important first step in this approach is to obtain a ROM of the original nonlinear system. There are generally two main approaches to obtain a ROM, the first one uses the Galerkin projection of the full-order system onto a small subspace spanned by mode vectors. Proper orthogonal decomposition (POD), balanced truncation and dynamic mode decomposition (DMD) are some of the methods by which the mode vector can be obtained. A comprehensive review of various model reduction techniques is presented in [367, 408]. The second approach is a purely data-driven model, which is widely used in the system identification community. This data-driven approach deals only with inputoutput data while considering the original system to be a black box model for generating data sets. A dynamical relationship between input and output data is realized in a state-space formulation to construct a ROM. Eigensystem realization algorithm (ERA) is a system identification approach [199], which can be employed to construct a minimal realization of the system that tracks the evolution of system output when subjected to an impulsive input. Mathematically equivalent to the balanced POD [271], the ROM resulting from the ERA is a linear projection of the original system on the set of modes possessing the most observable and controllable flow structures. Recently, the ERA-based ROM has been used for the stability analysis

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and flow control problems by fluid mechanics community in various instances [116, 117, 271, 465]. The work presented by [466] attempts to construct a unified description of frequency lock-in for elastically mounted cylinders. The ERA-based ROM technique was used to extract generalized stability properties of the fluid-structure system of two-dimensional bluff bodies. The effects of Reynolds number, mass ratio and also the geometry of the bluff body were analyzed by performing the stability analysis of a coupled system. A unified phase diagram to characterize the effects of geometry on the VIV stability regimes was discovered.

10.1.3 Objectives The objectives of this chapter lie in the ability to understand the mechanism behind the passive suppression of VIV using a data-driven stability analysis via ERA-based ROM. To the best of the authors’ knowledge, the stability analysis of passive suppression devices has not been explored in the literature. The modal analysis of the cylinder-appendage system via POD is another novel aspect of the present work. In particular, the contributions of the present work are driven by the following research questions: (i) What are the salient differences in the stability characteristics of a freely vibrating plain cylinder when compared to the cylinder-appendage system? (ii) What is the essential ingredient which characterizes the suppression of VIV via passive means? (iii) How do the wake features interact in the presence of suppression devices? (iv) How do the WM and SM predicted by the stability analysis behave in conjunction with the interaction of wake features? In addition to addressing the above fundamental research questions, a novel stability function is introduced which is of practical importance. To answer the aforementioned research questions, three representative appendage configurations namely fairing, connected-c and splitter plate are considered in the present study. The stability analysis via ERA-based ROM will be carried out on the above cylinder-appendage configurations and a detailed comparison with a plain cylinder will be presented. Full-order model results will also be reported to corroborate the findings from the stability analysis via ERA-based ROM. Modal decomposition of the wake features through the POD technique is carried out in addition to the stability analysis to gain a detailed physical insight into the fundamental suppression mechanism.

10.1.4 Organization The chapter is organized as follows. Section 10.2 deals with the numerical methodology detailing the full order model formulation and model reduction technique for the coupled FSI system. Section 10.3 presents the problem setup and the numerical verification of the ERA-based ROM on the laminar wake behind cylinder-fairing and connected-C systems. We further discuss the results of stability analysis, the modal

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decomposition and the effect of other appendages and characteristic dimensions. In Sect. 10.4 we present the problem setup on the laminar wake behind a sphere system along with the results of stability analysis, the effect of near-wake jet actuation and parameter space exploration through ERA-based ROM. Finally, in Sect. 10.5 we introduce the active feedback control based on the vertical suction and blowing for the unstable wake flow of a stationary cylinder and vortex-induced vibration. A sensitivity study on the number of suction/blowing actuators, the angular arrangement of actuators and the combined versus independent control architecture are also investigated.

10.2 Numerical Methodology This section deals with the numerical formulations used for the high-dimensional full-order model followed by the implementation of ERA to develop the ROM model for the coupled FSI problem. The numerical methodology presented in this section follows closely the one outlined in [466].

10.2.1 Full-Order Model Formulation In the present work, a numerical scheme employing the Petrov-Galerkin finite element and the semi-discrete time stepping is adopted [180, 181]. The incompressible Navier-Stokes equations are used in the Arbitrary Lagrangian-Eulerian (ALE) reference frame and formulated in the following form:  ρf

  ∂uf |x + uf − w · ∇uf ∂t

 = ∇ · σ f + f on f (t),

∇ · uf = 0 on f (t),

(10.1) (10.2)

where uf = uf (x, t) and w = w(x, t) are the fluid and mesh velocities, respectively. In Eq. (10.1) the partial time derivative with respect to the ALE referential coordinate  x is constant. Here f represents the body force per unit mass and σ f is the Cauchy stress tensor for a Newtonian fluid which is defined as   σ f = −pI + μf ∇uf + (∇uf )T ,

(10.3)

where p, μf and I are the hydrodynamic pressure, the dynamic viscosity of the fluid, and the identity tensor, respectively. A rigid-body structure submerged in the fluid experiences unsteady fluid forces and consequently may undergo flow-induced vibrations if the body is mounted elastically. To simulate the translational motion of a rigid body about its center of mass, the equation along the Cartesian axes is given by:

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m·

  ∂us + c · us + k · φ s (z0 , t) − z0 = Fs + bs on s , ∂t

(10.4)

where m, c, k, Fs and bs are the mass, damping coefficient, and stiffness coefficient vectors for the translational motions, fluid traction, and body forces on the rigid body, respectively. Here s represents the domain occupied by the rigid body and us (t) represents the velocity of the immersed rigid body. The fluid and structural equations are coupled by the continuity of velocity and traction along the fluidstructure interface. The new position of the rigid body is updated through a position vector φ s , which maps the initial position z0 of the rigid body to its new position at time t. Let γ be the Lagrangian point on  and its corresponding mapping position vector to the new position after the motion of the rigid body is φ(γ , t) at time t. Since the position and the flow field around the moving rigid body is updated continuously, the no-slip and traction continuity conditions should be satisfied on the fluid-body interface   φ(γ ,t)

uf (φ s (z0 , t), t) = us (z0 , t),  σ f (x, t) · nd + Fs d = 0 ∀ γ ∈ ,

(10.5) (10.6)

γ

where n is the outer normal to the fluid-body interface. The characterization of the moving fluid-body interface is constructed by means of the ALE technique [107]. The movement of the internal ALE nodes is constructed by solving a continuum hyperelastic model for the fluid mesh such that the mesh quality does not deteriorate as the displacement of the body increases. For the spatial and material mapping problem, we use classical Neo-Hookean material properties for the ALE variational formulation [221] which does not entail any additional user-defined re-meshing parameter. The weak variational form of Eq. (10.1) is discretized in space using P2 /P1 isoparametric finite elements for the fluid velocity and pressure. The second-order backward-differencing scheme is used for the time discretization of the Navier-Stokes system [255]. A partitioned staggered scheme is considered for the full-order simulations of fluid-structure interaction [179]. The above coupled variational formulation completes the presentation of a full-order model for high-fidelity simulation. The employed in-house FSI solver has been extensively validated in [254, 307].

10.2.2 Model Reduction via Eigensystem Realization Algorithm The basic premise of the ROM is to represent the high-dimensional non-linear dynamics governed by the Navier-Stokes equation via a low-dimensional model. We con-

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sider this model to be invariant in time and with multiple inputs and multiple outputs. It can be represented in the state-space form at a discrete time as  X(k + 1) = AX(k) + BU(k) , Y(k) = CX(k) + DU(k)

(10.7)

where X is a n-dimensional state vector, U denotes a q-dimensional input vector and Y is a p-dimensional output vector. The integer k is the sample indicator in a discrete time setting with a specified sampling period. The matrix A characterizes the dynamics of the system. The problem of identifying the system matrices (A, B, C, D) is part of the eigensystem realization algorithm (ERA). For any given impulse response Y, the ERA realizes the system by constructing the system matrices such that the response Y is reproduced by the state-variable equations. The algorithm consists of the following main steps: • Record the impulse response of the system using numerical solvers or experiments for a finite time horizon. The length of this impulse response should be sufficient to capture the linear dynamics of the system. Upon appropriate scaling to unit impulse, the response will be in the following form   CB, CAB, CA2 B, · · · , CA L i −1 B ,

(10.8)

where L i is the length of the impulse response and the terms CAk B are generally referred to as Markov parameters of the system. • Construct a generalized Hankel matrix H of size (r + 1, s + 1) ⎡

⎤ · · · CAs−1 B · · · CAs B ⎥ ⎥ ⎥, .. .. ⎦ . . r −1 r r +s CA B CA B . . . CA B

CB CAB ⎢ CAB CA2 B ⎢ H=⎢ .. ⎣ .

(10.9)

• Compute the singular value decomposition (SVD) of H ∗

H = UV = [U1

   ∗ 1 0 V1 ≈ U1 σ 1 V∗1 , U2 ] 0  2 V2∗

(10.10)

where the diagonal matrix  is invertible and contains the Hankel singular values (HSVs). The block matrix  2 contains elements with negligible values and therefore it can be truncated to estimate H ≈ U1  1 V∗1 .

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• The reduced system matrices (Ar , Br , Cr , Dr ) are calculated as −1/2

Ar =  1 Br =



−1/2 ⎫

U∗1 H V1  1

1/2  1 V∗1 Em 1/2 E∗t U1  1

⎪ ⎪ ⎪ ⎪ ⎭

Cr = Dr = Y(1) where

⎡

CAB CA1 B 2 3 ⎢  ⎢CA B CA B H =⎢ .. ⎣ . CAr B CAr +1 B

⎪ ⎪ ⎪ ⎪ ⎬

,

⎤ . . . CAs B . . . CAs+1 B ⎥ ⎥ ⎥, .. .. ⎦ . . r +s+1 . . . CA B

(10.11)

(10.12)

E convenience and are given by E∗m =  for mathematical  m , Et are defined ∗ Iq 0 q×(s+1)q , Et = I p 0 p×(r +1) p . I p and Iq are identity matrices. In the current work, we focus on the construction of ROM for a single input and single output (SISO) model. However, the above procedure is quite general and can be applied to MIMO models. Using the above algorithm, a linear reducedorder model corresponding to the fluid part of the FSI problem can be developed. The Hankel matrix is obtained by recording lift force coefficient (Cl ) subjected to impulse input of transverse displacement Y. Following the above procedure, the fluid ROM constructed will be of the form  X f (k + 1) = A f X f (k) + B f Y(k) , (10.13) Cl (k) = C f X f (k) + D f Y(k) where the subscript f is used to indicate the fluid part of the FSI system.

10.2.3 ROM for FSI Problem The non-dimensional structural equation for a transversely vibrating cylinder with one-degree-of-freedom can be written as ˙ + (2π f s )2 Y = ¨ + 4ζ π f s Y Y

2 Cl , π m∗

(10.14)

where Y is the transverse displacement, Cl is the lift coefficient, m ∗ and ζ are mass ratio and the damping coefficient respectively, f s is the reduced natural frequency which is defined as f s = f N D/U = 1/Ur , where Ur is the reduced velocity. The above equation can be written in the state-space form as below

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X˙s = As Xs + Bs Cl ,

(10.15)

where Xs is a state vector and As , Bs are the state matrices and are given by      Y 0 0 1 , Bs = 2 , Xs = ˙ , As = −(2π f s )2 −4ζ π f s Y ∗ πm 

(10.16)

The notation s is used to indicate the structure part of the FSI system. The continuous state-space equation can be converted to the discrete form and is given by  Xs (k + 1) = Asd Xs (k) + Bsd Cl (k) , Y (k) = Csd Xs (k)

(10.17)

As t − I )Bs , t is constant sampling time, I is the where Asd = eAs t , Bsd = A−1 s (e identity matrix and Csd = 1 0 . The discrete state-space form of non-dimensional structural Eq. (10.17) is coupled with the fluid ROM Eq. (10.13) to give rise to a linear ROM for the coupled FSI problem

X f s (k + 1) = A f s X f s (k), A f s

  Asd + Bsd D f Csd Bsd C f , = B f Csd Af

(10.18)

T  where X f s = Xs X f and A f s denotes the coupled fluid-structure matrix in discrete state-space form. The eigenvalues of the coupled fluid-structure matrix A f s contain information about the stability of the system. The growth rate is defined as the real part of the most unstable eigenvalue σ = Re(λ) and the corresponding frequency is defined by f = I m(λ/2π ). With the help of the growth rate, one can assess the stability characteristics of the system for different structural parameters such as m ∗ , ζ , f s and geometric variations. The numerical formulation in the above two sections describes the highdimensional (full-order) and low-dimensional (reduced-order) models used in the current work. While the high-dimensional model will provide us with detailed insights into the vorticity dynamics and VIV, the goal of the ERA-based reducedorder model is to capture essential dynamical stability properties of the coupled linearized system dynamics. The ROM resulting from the ERA provides a projection of the original system on the set of modes through a set of impulse responses from the input-output dynamics of VIV. As mentioned earlier, we have chosen the problem of passive suppression of VIV in the current work. The mechanism behind the passive suppression of VIV is examined via ERA-based stability analysis and a detailed analysis of the coupled dynamical behavior is presented in Sect. 4. Together with the POD analysis, the trends of eigenvalues are considered to gain insights into the suppression mechanism via a physically realizable framework. The ERA-based ROM employed herein has been extensively verified for cylinder vortex dynamics and VIV in [465, 466]. In the next section, we present some additional verification of the ERA-based ROM with more appropriate cases pertaining to the passive suppression of VIV.

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10.3 Passive Control of VIV via Appendages This section deals with the problem setup and the verification of the ERA-based ROM for a 2D laminar flow at low Reynolds number.

10.3.1 Cylinder-Fairing System 10.3.1.1

Problem Definition

The schematic diagram of the setup used in the current study for an elastically mounted cylinder with various suppression devices is displayed in Fig. 10.1. The coordinate origin is located at the geometric center of the circular part of the bluff body, and x and y denote the streamwise and transverse directions, respectively. A stream of incompressible fluid enters into the domain from an inlet boundary at a horizontal velocity (u, v) = (U, 0), where u and v denote the streamwise and transverse velocities respectively. A cylinder with mass m and diameter D is mounted on a linear spring with stiffness k and damping coefficient c in the transverse direction. The cylinder body is attached with different passive suppression devices, which are introduced further in the later sections. The characteristic longitudinal length L of all the devices are measured from the front of the cylinder to the end of the device. In the current section, the fairing is used as the suppression device, and its characteristic longitudinal length is set to L = 2.0D. The fins in the fairing are slightly curved towards each other. The opening of fairing a is kept constant and set to 0.95D

Fig. 10.1 Schematic diagram of a representative bluff body of elastically mounted cylinder with fairings subjected to a uniform horizontal flow. Computational and boundary conditions are shown

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throughout the study. The details about the other passive suppression devices are discussed in later sections. The total mass m of the body includes the cylinder with appendage mass. All the suppression devices are attached to the cylinder, and there is no relative motion between the appendage and cylinder. The cylinder-appendage system is free to vibrate in the transverse direction. The inlet boundary is located 20D away from the center of the cylinder and the outlet boundary at 40D. The blockage ratio is maintained to be 1% with sides located at 50D from the center of the cylinder. No-slip wall condition is implemented on the surfaces of the cylinder and the suppression devices. The top and bottom boundaries are considered with slip wall condition. All the positions and length scales are normalized by the characteristic cylinder diameter D, velocities with free stream velocity U and frequencies with U/D. The key dimensionless VIV parameters of the cylinder-appendage system considered in the current study are mass ratio (m ∗ ), the reduced velocity (Ur ), Reynolds number (Re) and they are defined as follows: m∗ =

ρfUD 4m U , U , Re = = r πρ f D 2 L fn D μf

(10.19)

where f n is the natural frequency measured in vacuum. The value of U is considered to be 1 throughout the numerical study. The Reynolds number Re and the reduced velocity Ur are varied by changing the values of dynamic viscosity μ f and the natural frequency f n . The damping is set to be zero in all the numerical simulations and therefore damping ratio is not considered among the key dimensionless parameters.

10.3.1.2

Stability Analysis

The reliability of the ERA-based ROM is verified by assessing the stability of the laminar wake behind a two-dimensional circular cylinder-appendage system. The appendage is considered to be fairing for the analysis presented in this section. Four mesh configurations are considered for the mesh convergence study. The quality of the mesh convergence is determined by the prediction of the growth rate σ and the frequency f of the fluid ROM for the flow past a cylinder with fairing system at Re = 100. The results of this study are tabulated in Table 10.1. The difference between M3 and M4 is found to be less than 1%. Therefore, Mesh M3 with 27,724 Table 10.1 Mesh convergence study: comparison of growth rate and frequency for different mesh configurations for the flow past a stationary circular cylinder at Re = 100 Mesh Nodes Elements σ f M1 M2 M3 M4

4860 10,161 27,724 39,884

9720 20,322 55,448 79,768

0.0547 0.0552 0.0550 0.0551

0.1216 0.1216 0.1216 0.1216

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Fig. 10.2 Finite element mesh with P2 /P1 discretization: a full domain discretization and b closeup view in the vicinity of the cylinder-fairing system

Fig. 10.3 Base flow of a stationary cylinder-fairing system at Re = 90; the streamwise velocity contours are shown

nodes and 55,448 elements is considered to be sufficiently refined for the current study and is depicted in Fig. 10.2. Based on the methodology described in the previous section, the process of constructing the ROM for the coupled FSI system contains the following vital components: • Baseflow: The baseflow is computed via fixed point iteration by dropping the timedependent term in Eq. (10.1). This baseflow can be considered as an equilibrium position of the FSI system. We essentially disturb this equilibrium position and study the growth of the disturbance and then assess whether the system is stable or unstable. • Impulse response: The disturbance to the base flow is provided via an impulse input of the transverse displacement Y. The response is recorded for a sufficient horizon. A small amplitude in the order of 10−3 or 10−4 is applied such that the flow evolves linearly for a relatively long time. The linearity of the impulse is verified by comparing the response with different inputs. To ensure that the unstable

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modes start to dominate the essential dynamics of the input-output relationship, an adequate number of cycles is required to capture the linear dynamics of the system. However, an excessively long simulation time should be avoided before the nonlinearity appears via exponential growth of the unstable modes. • ERA: The ERA algorithm outlined in the previous section is followed to construct the fluid ROM. This procedure involves the calculation of the Hankel matrix, by performing the SVD and calculating the reduced system state matrices. The dimensions of the Hankel matrix can be determined by examining the convergence of the fluid unstable eigenvalues. • Eigenspectrum: The linear ROM for the coupled system is calculated by Eq. (10.17). As discussed earlier, the matrix A f s characterizes the linear dynamics of the coupled system. The continuous eigenvalues, λ = log(eig(A f s ))/ t are considered for the stability analysis. The eigenvalues of the matrix A f s are calculated and the leading unstable modes are identified with a growth rate and corresponding frequency. The eigenspectrum can be constructed by plotting the root loci of leading unstable modes as a function of the reduced natural frequency f s . The position of the eigenvalues on this spectrum provides information about the stability of the coupled system. The above process with the four main components forms a general procedure to be followed while constructing the ERA-based ROM. It should be noted that ERA is only applied to the fluid system and the resulting matrices are used to compute the reduced order model of the coupled system. The specific details of each component corresponding to the current study are discussed in the upcoming paragraphs. Figure 10.3 shows the streamwise velocity contours of the base flow at Re = 90. A disturbance of δ = 10−4 is given as an impulse input to the transverse displacement Y. The lift coefficient Cl is recorded for every time step t = 0.05. A set of 1000 responses are stacked resulting in a total simulation time of tU/D = 50s. Figure. 10.4a shows the output lift signal Cl calculated from the FOM and the ERA-based ROM. The signal gradually diverges as a function of convective time tU/D at Re = 90. This asymptotic divergence behavior of the lift coefficient indicates that the wake flow begins to lose its stability via a Hopf bifurcation. Hankel matrix with dimension 500 × 250 is found to be appropriate by examining the convergence of unstable fluid eigenvalues computed from Hankel matrices with dimensions of 500 × 25, 500 × 50, 500 × 150, 500 × 250 and 500 × 300. The order of ERA-based ROM is selected by examining the Hankel singular value distribution. The fast decaying singular values as depicted in Fig. 10.4 suggests that ERA-based ROM with order nr = 30 is sufficient. This is further confirmed by the accurate reconstruction of the impulse response by ERAbased ROM displayed in Fig. 10.4a. According to [199], the Hankel matrix may not be necessarily square but can be tall, wide or square depending on the problem under consideration. The ERA-based ROM is further used to identify the onset of Hopf bifurcation for the cylinder-fairing system. The growth rates σ and f of the fluid ROM are calculated by considering the unstable eigenvalues of eig(A f ). The ERA-based ROM is constructed by following the aforementioned procedure at different Reynolds number Re ≤ 100. The growth

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Fig. 10.4 ERA-based ROM for the unstable wake behind a stationary cylinder at Re = 90: a lift history due to impulse response calculated from FOM and ERA-based ROM and b Hankel singular distribution corresponding to 500 × 250 Hankel matrix

rate σ and frequency f are plotted against Re and are depicted in Fig. 10.5. The critical Reynolds number where σ = 0 is found to be Re ≈ 68.5. This result of the cylinderfairing system has been identified for the first time. It is further confirmed by our FOM calculations. The vorticity contours from the FOM calculations are displayed in Fig. 10.6 for Re = 68 and 69. Since Re = 68 < Recr , the vortex shedding is not observed and the flow is steady. In the case of Re = 69 > Recr , there is a clear presence of the vortex shedding resulting from the wake instability. The growth rate σ appears to be growing steadily with respect to Reynolds number Re suggesting an increase in the vorticity flux and thereby the intensity of vortex shedding as Re increases. In this section, the validity of the ERA-based ROM is presented via a systematic study on the stability analysis of laminar wake behind a cylinder-fairing system. A generic process to construct ERA-based ROM for any fluid system has been discussed. The specific conditions to be followed in the process of building ERAbased ROM are presented. The critical Reynolds number Recr is identified for the cylinder-fairing system and is further validated by the full-order model computations. The following sections will explore the stability characteristics of the coupled FSI systems.

10.3.2 Performance of ERA-Based ROM for Passive Suppression In this section, the performance of ERA-based ROM for identifying the effect of passive suppression devices is presented. For this purpose, the eigenspectrum plot of plain cylinder and cylinder with fairings is systematically examined. From here

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Fig. 10.5 Growth rate and frequency of most unstable eigenvalue of fluid ROM: a growth rate σ and b frequency f . The growth rate σ ≥ 0 indicates the instability in system which is manifested as vortex shedding. c Root loci as a function of Reynolds number, where the unstable region Re(λ) > 0 is shaded in grey color. The arrow depicts the increasing order of Reynolds number

Fig. 10.6 Vorticity contours from FOM: a Re = 68 and b Re = 69

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on, the cylinder with fairings is simply referred to as fairing. Consistent with the previous literature of [283, 466, 486], we utilize the terminology of SM and WM to classify the distinct eigenvalue trajectories of the fluid-structure system governed by Eq. (10.18). These modes are identified based on the magnitudes of the real part of eigenvalues (Re(λ)). The two most unstable modes are recognized and are classified as SM and WM depending on their frequencies. When the frequency of the eigenvalue i.e., (Im(λ/2π )) approaches that of the vortex shedding of a stationary cylinder, the corresponding mode is termed as WM. The fluid-structure mode is identified as SM if the frequency of the eigenvalue comes close to the natural frequency of the cylinder-alone system. The instability of the WM alone or simultaneous existence of unstable WM and SM can induce the VIV lock-in [466, 486]. In the case of the first scenario, the closeness of the frequencies of the WM and SM is the reason for the lock-in, which is termed as the resonance-induced lock-in. In the event of the second scenario, the combined mode instability of SM and WM jointly sustains the VIV lock-in and is referred to as the flutter-induced lock-in or the coupled mode flutter. Continuous-time eigenvalues are considered for depicting the eigenspectrum in the context of the stability analysis. The eigenvalues are calculated from the coupled fluid-structure matrix (λ = log(eig(A f s ))/ t). The eigenvalue distribution of the coupled fluid-structure system is graphically depicted in a complex plane via root locus. The positions of the eigenvalues on this complex plane indicate the stability characteristics of the fluid-structure system. The roots in the right half-plane depict the unstable modes of the system, whereas the roots on the real axis characterize the asymptotic (non-oscillatory) behavior. Roots that are closest to the right half-plane are tightly damped oscillatory modes. Figure 10.7a shows the eigenvalue trajectory of the fluid-structure system as a function of the reduced natural frequency fs with 0.05 ≤ f s ≤ 0.25 and the increment is f s = 0.0025. The trajectories of both the plain cylinder and fairing are depicted in the same figure for a comparison purpose. Mass ratio and Reynolds number for both the plain cylinder and the fairing are considered to be Re = 100, m ∗ = 10.0. While the SM is represented by open circles, the WM is depicted by filled circles in the Fig. 10.7. In the case of the plain cylinder, one can observe the combined mode instability on both SM and WM which is considered as the flutter-induced lock-in and the same result is also reported in a phase diagram published in [466]. The presence of fairing has created a significant shift in both SM and WM when compared to the plain cylinder and resulted in the SM being completely on the left half of the complex plane (i.e., stable region). The WM is still on the right-hand side of the complex plane and is a clear indication that the vortex shedding is still prevalent in the case of fairing resulting in the wake instability. A similar comparison between the plain cylinder and the fairing is presented for m ∗ = 2.6 in Fig. 10.7b. It is evident from Fig. 10.7b that SM and WM coalesce and form a coupled mode for the case of a plain cylinder. This is due to the increase in the strength of fluid-structure coupling which is a consequence of low mass ratio condition. Further details on the effects of low mass ratio on eigenspectrum are discussed in [466]. It is also evident from Fig. 10.7b that both SM and WM do not

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Fig. 10.7 Root loci as a function of f s , where the unstable plain Re(λ) > 0 is shaded in grey color. Blue triangle represent the WM of the cylinder alone system. i.e., it is the most unstable eigenvalue of the matrix A f . a Effect of fairings on eigen spectrum at Re = 100, m ∗ = 10, b Re = 100, m ∗ = 2.6, c Normalized maximum transverse amplitude for plain cylinder and fairing at m ∗ = 2.6, Re = 100

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Fig. 10.8 Comparison of transverse vibration A y at lock-in for plain cylinder (Ur = 5.0) and fairing (Ur = 4.0). Re = 100, m ∗ = 2.6

coalesce for the case of fairing indicating a weak fluid-structure coupling at low mass ratio condition. The SM is found to be completely on the left half of the complex plane indicating the suppression of transverse vibration due to the presence of fairing. The location of WM implies that it is unstable and also the presence of vortex shedding at low mass ratio condition. The transverse vibration A y calculated from the full order model at lock-in for both plain cylinder and fairing is presented in Fig. 10.8. There is a sharp decrease in the amplitude of A y in the case of fairing when compared to the plain cylinder. This result from the full order model provides additional support to the conclusions obtained from the stability analysis of ERA-based ROM. To summarize in this section, the ERA-based ROM is successfully employed to perform the stability analysis of the vortex-induced vibration of a transversely vibrating plain circular cylinder and a cylinder with the fairing. The presence of fairing created a suppression of transverse vibration at both high and low mass ratio conditions and is indicated by the stability of SM depicted in Fig. 10.7. Instability of the wake did not hinder the suppression achieved by the fairing. The wake stabilization does not appear to be necessary for the suppression of the structural vibration during fluid-structure interaction. The SM is an ideal indicator for the assessment of suppression devices and its behavior will be further discussed for other appendage configurations in the following section. It should be noted here the wake patterns observed in the visualizations depicted in Fig. 10.11 for the cylinder-fairing system also reinforce the above point that the wake stabilization does not appear to be a necessary condition for the suppression of the structural vibration.

10.3.3 Modal Decomposition of Wake Features The stability analysis on ERA-based ROM has shown a clear distinction between the plain cylinder and the fairing through the root loci of WM and SM. The purpose of this section is to provide detailed physical insights into the distinction observed by the ERA-based ROM. The VIV lock-in is the result of a complex interplay between the wake features and the bluff body motion. A self-sustainable cyclical mechanism has

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Table 10.2 Reduced velocity Ur considered in FOM for POD analysis Plain cylinder Fairing Pre lock-in Lock-in Post lock-in

2 5 16

2 4 16

been proposed in [299] to explain the wake interaction with the bluff body motion. In the case of the plain cylinder, when the vortex shedding synchronizes with the bluff body motion, it causes the bluff body to undergo high amplitude motion. This widens the wake and eventually the shear layer, decreasing the velocity gradients. This causes the shear layer to give away vorticity flux to the vortex shedding process, intensifying the vortex shedding process. Intensified vortices increase the in-phase forces with the motion, i.e., the fluid supplies higher energy to the structure. Due to the presence of fairing, the shear layer is extended from the cylinder, thus preventing the interaction of vortices formed by the roll-up of the shear layer in the near wake. This disconnect between the near wake and the vortex shedding creates a stable near wake environment with fewer fluctuations in the kinetic energy and stops the energy transfer from the fluid flow to the structure, thus resulting in the suppression of transverse vibration. The self-sustenance of the wake interaction cycle is broken down in the case of the fairing by preventing the interaction of vortex shedding with the near wake. The effect of the fairing, when compared to the plain cylinder, has also been reported in [226] in terms of vortex formation length and the wake width. The dominant wake features are identified by a modal decomposition method. Snapshot POD developed by [394] is used to extract the most significant modes. The details of the POD algorithm used for the current analysis can be found in [299]. The modal decomposition is carried out for both plain cylinder and fairing. The snapshots were collected from the full order model calculations at m ∗ = 2.6 and Re = 100 for both plain cylinder and faring. The analysis is carried out at different Ur as indicated in Table 10.2 to cover pre-lock-in, lock-in and post-lock-in regions. The pressure field has been used to calculate the significant POD modes. The unsteady pressure field values for all the mesh points are collected as m × k matrix P where m = mesh count and k = number of snapshots = 350. The fluctuation matrix y˜ m×k is ¯ of each point over the snapshots: then generated by subtracting the mean value (P) ¯ y˜ = P − P.

(10.20)

The POD modes are extracted using the eigenvalues k×k = diag[λ1 , λ2 , . . . , λk ] and eigenvectors W = [w1 , w2 , . . . , wk ] of the covariance matrix y˜ T y˜ ∈ Rk×k given by, (10.21) y˜ T y˜ w j = λ j w j .

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Fig. 10.9 Energy distribution of POD modes for plain cylinder and cylinder with fairings: a eigenvalues of POD modes, b energy decay of POD modes, and c cumulative energy of POD modes at Re = 100, m r = 2.6, Ur = 5.0, 4.0 (plain cylinder, fairing)

Here, the POD modes V = [v1 , v2 , . . . , vk ] are related to  and W by V = y˜ W −1/2

(10.22)

The eigenvalues capture the contribution of each mode in the sense of the least square L 2 norm of the dataset and this contribution is referred to as a modal strength in the current section. Since the mean pressure distribution is initially removed from the pressure field, the relative strength of the mode directly expresses the contribution from each mode to the pressure fluctuations. Figure 10.9b displays the strength of these modes normalized by the total strength of the 350 modes. It is evident that this strength decays

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Fig. 10.10 Logarithmic contribution from each mode for plain cylinder and cylinder with fairings: a absolute inline force |Fx |, and b cross flow force |Fy | at Re = 100, m r = 2.6, Ur = 5.0, 4.0 (plain cylinder, fairing)

exponentially in both plain cylinder and fairing. The cumulative strength plot given by Fig. 10.9c indicates that around 70% of strength is concentrated in the first mode in the case of the plain cylinder and around 45% of strength in the first mode for fairing. In fact, over 95% of strength is concentrated in the first 3 and 4 modes for the plain cylinder and fairing, respectively. Figure 10.9a outlines the absolute eigenvalue without any normalization for a direct comparison between the plain cylinder and the fairing. There is a clear reduction in the magnitude of eigenvalues obtained for fairing when compared to the plain cylinder and this indicates a presence of lower strength levels in the fluid-structure system. The lower modal strength for the case of fairing configuration can be viewed as a natural consequence of the suppression mechanism, whereby the presence of a suppression device such as fairing in this case prevents the interaction between the wake features (e.g., vortex shedding and near wake) and breaks the self-sustenance behavior in the wake interaction cycle. Figure 10.10 quantifies the logarithmic absolute value of the time-independent pressure fluctuation contribution from each mode to drag (|Fx |) and lift (|Fy |). The contribution of modal forces is lower in the case of fairing when compared to the plain cylinder at all the significant modes. Considering the first four significant modes, the contribution to the lift is from modes 1,3 and modes 2,4 contribute towards the drag force. This result is another demonstration of the quantification of the suppression mechanism created by the presence of fairing in terms of the reduction in the modal force contributions. The mean pressure distribution and the first four significant POD modes are shown in Fig. 10.11 for both plain cylinder and fairing at lock-in conditions. The mean field is symmetric around the X -axis or the wake centerline. This is expected as the time-averaged distribution of a flow past a stationary symmetrical bluff body should be symmetrical. Modes 2 and 4 are also nearly symmetric around the wake centerline while modes 1 and 3 are nearly anti-symmetric with equal values and opposite signs about the wake centerline. There is reasonable evidence that the first and third modes correspond to the von Karman vortex street with the alternating positive and negative pressure regions about the X -axis and the pressure contours

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Fig. 10.11 Comparison of mean field and the first four significant modes of plain cylinder and cylinder with fairings at Re = 100, m r = 2.6, Ur = 5.0, 4.0 (plain cylinder (left), fairing (right)) (lockin). The strength contribution of each mode in both the cases is mentioned in the brackets. The white dashed line in the mode 2 of cylinder-fairing case indicates the region of shear layer

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resulting from the staggered vortex street. The high transverse gradient behavior observed in mode 2 in the near wake region suggests that it represents the contribution from the shear layer. The fourth mode originates from the near-wake region and diffuses symmetrically towards the far wake and can be attributed to the near-wake bubble and its interaction effects. The characterization of modes 1,2,3 and 4 into the vortex shedding, the shear layer and the near wake is applicable in both configurations of the plain cylinder and the fairing. The normalized strength contribution of each mode is presented in the Fig. 10.11 for both plain cylinder and fairing, respectively. In both cases, the dominant mode is due to the vortex shedding. However, its contribution to the fairing is reduced when compared to the plain cylinder. Figure 10.7c displays the normalized maximum transverse amplitude for the plain cylinder and fairing. There is an apparent reduction in the amplitude for the case of fairing but a peak at Ur = 4, suggests a lock-in condition i.e., the vortex shedding frequency matches with the oscillating frequency of the bluff body. This can be the reason why vortex shedding is the dominant mode in the case of the fairing. Although the vortex shedding is dominant, the lower strength of the mode can be attributed to a disconnect between the vortex shedding and the near wake due to the fairing. This is also manifested in the reduction of amplitude at the lock-in. The shear layer extension in the case of the fairing as mentioned earlier in the self-sustenance wake interaction cycle can be observed in the second mode. The dominant mode for a plain cylinder in the pre-lock-in is coming from the vortex shedding, which is depicted in Fig. 10.12a. In the case of the fairing, it shifts to the shear layer and the vortex shedding is the second most dominant. Due to the lower vibration of the bluff body in the pre-lock-in for the case of the fairing,

Fig. 10.12 Comparison of first two significant modes for a plain cylinder and b cylinder with fairings at Re = 100, m r = 2.6, Ur = 2.0 (Pre-lock-in).The strength contribution of each mode in both the cases is mentioned in the brackets

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Fig. 10.13 Comparison of first two significant modes for a plain cylinder and b cylinder with fairings at Re = 100, m r = 2.6, Ur = 16.0 (Post-lock-in). The strength contribution of each mode in both the cases is mentioned in the brackets

the energy transfer from the shear layer to the vortex shedding is reduced, thereby resulting in the shear layer being dominant. The post-lock-in results are similar to the case of the plain cylinder with the vortex shedding being superior. In the case of the fairing, both the shear layer and the vortex shedding are almost equal in terms of the strength contribution, while the vortex shedding is slightly higher than the shear layer (Fig. 10.13). This result indicates that at much higher Ur the vortex shedding will recover the dominant mode for the fairing. In this section, the modal decomposition of wake features is presented for both plain cylinder and fairing configurations. The distinction observed in the ERA-based ROM between the plain cylinder and the fairing is supported by detailed physical insights into the suppression mechanism. The effect of the fairing on the selfsustenance of the wake interaction cycle is quantified through the strength and force contributions from the POD modes. The results are presented across pre-lock-in, lock-in and post-lock-in to demonstrate the variation in the interaction between the wake features. A disconnect between the vortex shedding and the near wake is observed due to the presence of fairing.

10.3.4 Effect of Other Appendages: Splitter Plate and Connected-C The stability analysis via ERA-based ROM is further applied to other cylinderappendage systems and presented in this section. The configurations of the appendages considered for the purpose of the current study include the splitter plate and the

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L

L

r

D

D

(a)

(b)

Fig. 10.14 Schmematic of other passive suppression devices: a splitter plate and b connected-c

connected-c. The schematics of these appendages are presented in Fig. 10.14. While Fig. 10.14a displays a standard splitter plate, the connected-c portrayed in Fig. 10.14b consists of two geometric components, namely a splitter plate-like connector and a c-shaped foil with a radius r . The characteristic longitudinal length of both the appendages is denoted by L and the diameter of the cylinder is represented by D. The mesh configuration similar to M3 is used for these cases with slight modifications. The same process is followed to construct the ERA-based ROM for both the cylinder-appendage systems. Figure 10.15 displays the eigenvalue trajectories of both splitter and connected-c in comparison with the plain cylinder. All the results are plotted at Re = 100 and m ∗ = 2.6. Similar to the notation used in the previous section, the filled symbols denote WM, and hollow symbols represent SM. The behavior of connected-c is similar to the fairing. The SM is completely shifted into the stable region due to the presence of connected-c, and the trajectory of SM in the case of connected-c is identical to the one with the fairing. The WM is present in the unstable region and is a clear indication of the presence of vortex shedding due to wake instability. In the case of splitter plate, as depicted in Fig. 10.15a, the SM is not completely stable and goes into unstable region at f s = 0.22, Ur = 4.54 and remains in that state, indicating a galloping instability. The FOM results displayed in Fig. 10.15c also corroborate the above observation. It is worth noting that the ERAbased ROM is able to accurately capture the onset of galloping instability in the case of the splitter plate. As expected, the WM is located completely in the unstable region for the case of splitter indicating the presence of wake instability. In Fig. 10.16, the time histories of normalized transverse vibration Y/D calculated from the FOM for both splitter and connected-c are compared with the plain cylinder at the lock-in and galloping conditions. The suppression of transverse vibration as pointed out by the ERA-based ROM through the stability of SM is clearly reflected in the FOM calculations. The effect of galloping by the splitter has resulted in a sharp increase in the amplitude compared to the plain cylinder, which is evident in Fig. 10.16. The eigenvalues calculated from the POD analysis quantify the quadratic contribution of the field variable considered in the analysis. To get an estimate of the kinetic

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Fig. 10.15 The effect of appendages on the eigenspectrum of cylinder-appendage system at Re = 100, m ∗ = 2.6. Root loci as a function of f s , where the unstable plain Re(λ) > 0 is shaded in grey color. Blue triangle represent the WM of the cylinder alone system. i.e., it is the most unstable eigenvalue of the matrix A f . a Splitter plate, b connected-c. SM is denoted by open circles and WM is represented by filled circles in both the cases. c Normalized maximum transverse amplitude for plain cylinder, splitter and connected-c for a range of Ur at m ∗ = 2.6, Re = 100

energy from the different POD modes in the flow field, the streamwise velocity u is considered as the field variable for the POD analysis. Since the mean velocity is initially removed from the velocity field, the eigenvalue of each mode directly expresses the contribution from each mode to the kinetic energy fluctuations of streamwise velocity. The modal contributions to the kinetic energy fluctuations of streamwise velocity are plotted for the plain cylinder and the cylinder-appendage systems and are displayed in Fig. 10.17. There is a clear reduction in the kinetic energy fluctuations

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Fig. 10.16 Comparison of normalized transverse amplitude calculated from FOM for cylinderappendage system with plain cylinder Fig. 10.17 Eigenvalues of stream-wise velocity (u) POD modes for plain cylinder (Ur = 5.0) and cylinder-appendage system: connected-c (Ur = 4.0), splitter plate (Ur = 4.0), splitter plate (Ur = 16.0) at Re = 100, m r = 2.6

u 2 observed for the case of connected-c and the splitter as depicted in Fig. 10.17. The galloping instability for the case of the splitter at Ur = 16.0 is manifested in terms of the increase in the kinetic energy fluctuations of streamwise velocity.

10.3.5 Effect of Characteristic Dimensions In this section, the effect of the characteristic dimensions of the cylinder-appendage systems is studied via the stability analysis of ERA-based ROM. All the appendages have a characteristic longitudinal length denoted by L which is measured from the front of the cylinder to the end of the device. Apart from the longitudinal length, the other characteristic dimension of interest to the current study is the radius of the cshaped foil r in the case of connected-c. A limited parametric study is carried out by varying the values of these characteristic dimensions to demonstrate the robustness of the stability analysis via ERA-based ROM.

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Fig. 10.18 The effect of characteristic longitudinal length L on the eigenspectrum of cylinderappendage system at Re = 100, m ∗ = 2.6. a fairing b splitter plate. SM is denoted by open circles and WM is represented by filled circles in both the cases. Blue triangle represent the WM of the cylinder alone system. i.e., it is the most unstable eigenvalue of the matrix A f

Fig. 10.19 The effect of characteristic dimensions L , r on the eigenspectrum of cylinderconnected-c system at Re = 100, m ∗ = 2.6. a r = 0.95, L is varied b L = 2.0, r is varied. SM is denoted by open circles and WM is represented by filled circles in both the cases. Blue triangle represent the WM of the cylinder alone system. i.e., it is the most unstable eigenvalue of the matrix A f

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Three values for characteristic longitudinal length L = {1.0, 1.5, 2.0} are considered for the fairing configuration and corresponding eigenvalue trajectories are calculated and displayed in Fig. 10.18a. While the WM is completely located in the unstable region for all the cases as expected, the SM is fully stable for L = 2.0, 1.5 and enters into the unstable region for a range of f s in the case of L = 1.0. The SM is gradually shifted towards the unstable region as the characteristic longitudinal length is decreased. There will be a critical length in between L = 1.5 − 1.0, where the instability begins to appear for a small range of f s . Four configurations of the characteristic longitudinal length are considered L = {1.25, 1.5, 2.0, 2.5} for the case of splitter. Figure 10.18b portrays the eigenvalue trajectories for all the configurations. The galloping behavior is observed in the case of splitter when L = 2.0, 2.5. In the case of L = 1.5, the SM and WM coalesce and form a coupled mode and the instability in the system is jointly sustained by this coupled mode for a range of f s . The stability roles of SM and WM switch at a particular f s . This is an indication of strong fluid-structure coupling for the splitter at this characteristic longitudinal length. As the length L is further decreased to 1.25, the cylinder-splitter system behaves as an isolated plain cylinder with weak fluidstructure interaction which is prevalent from the distinct characterization of SM and WM. It is worth noting that the splitter plate exhibits a diverse behavior for different values of L. The connected-c device is described with two characteristic dimensions longitudinal length L and a radius of c-shaped foil r . The effect of longitudinal length L = 1.625, 1.75, 2.0 is reported in Fig. 10.19a. The radius of the c-shaped foil is fixed at r = 0.95D while varying the L. The connected-c behaves similarly to fairing and the eigenvalue trajectories look identical. The SM is completely stable for all the values of L considered and WM is completely stable. Figure. 10.19b depicts the effect of r = 0.95, 0.65, 0.35 with longitudinal length fixed at L = 2.0. The connected-c embodies the characteristics of the splitter at r = 0.65, 0.35 by displaying a galloping behavior. As r is increased gradually to 0.95, it achieves superior suppression by completely shifting the SM to the left half of the complex plane. It is intuitive to note that as the radius of c-shaped foil in the connected-c tends to zero, the geometry of connected-c naturally resembles the cylinder-splitter arrangement and therefore the galloping behavior of connected-c at low r is recovered. A stability function  is defined by considering the real part of the eigenvalue with a maximum magnitude as  = sgn{max[Re[λ S M ( f s )]]},

∀ fs

(10.23)

where λ S M ( f s ) corresponds to the eigenvalue of structure mode (SM) at a given f s . The output of the stability function is either + or − indicating instability/stability of the structural vibration. This function  serves as an indicator and also as a tool to assess the stability of the system. The values of this function for the fairing and the splitter are tabulated in Table 10.3 for various characteristic length L. The effect of L and R on the stability function for connected-c is outlined separately in Table 10.4. By defining such a general stability function based on structure mode (SM), it is

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Table 10.3 Stability function  for fairing and splitter at various characteristic length L L Fairing Splitter ⊕ ⊕   

1.0 1.25 1.5 2 2.5

⊕ ⊕ ⊕ ⊕ ⊕

⊕ indicates the presence of instability in the structural vibration and  represents complete suppression of vibration Table 10.4 Stability function  for connected-c at various L and R L 2 1.75 R

0.95 0.65 0.35

 ⊕ ⊕

 ⊕ ⊕

1.625  ⊕ ⊕

possible to construct an assessment of a given parameter space. The effect of other structural parameters such as the mass ratio m ∗ and the damping ratio ζ can also be assessed with minor computational efforts.

10.3.6 Interim Summary The stability analysis of passive suppression of VIV has been presented in this work. A data-driven approach based on ERA was used to construct the reduced order model (ROM). The ERA-based ROM was successfully applied to study the laminar wake behind the stationary cylinder-fairing arrangement. The critical Reynolds number for this case calculated from ERA-based ROM was found to be Recr = 68.5, which agreed well with the FOM results. The performance of the ERA-based ROM in identifying the suppression due to fairing was investigated by comparing the eigenspectrum of the plain cylinder. The presence of the fairing shifted the SM completely to the left half of the complex plane indicating a clear suppression of transverse vibration at both high and low mass ratio conditions. The FOM results also support the ERA-based ROM predictions by reporting a clear reduction in the normalized transverse amplitude. The instability of WM observed in the eigenspectrum which characterizes the periodic vortex shedding in the system did not hinder the suppression achieved by the fairing. Therefore, complete wake stabilization is not necessary for the suppression of the structural vibration in fluid-structure systems. The problem of suppression can be characterized by shifting the SM from an unstable to a stable region. The occurrence of suppression albeit the presence of wake instability was explained via the effect of the fairing on self-sustenance of the wake interaction

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cycle. The presence of fairing creates a disconnect between the vortex shedding and the near wake, thereby breaking the self-sustenance of the wake-body interaction cycle. This results in the stable near-wake environment manifesting in the suppression of transverse vibration. The effect of the fairing on the complex interaction between the wake features is quantified through the energy and the force contribution from the POD modes. The results from the POD analysis further provide physical insights into the distinction observed in the ERA-based ROM between the plain cylinder and the cylinder-fairing system. The results from the POD analysis are presented across pre-lock-in, lock-in and post-lock-in to highlight the differences of interaction between the wake features. The effect of other appendage configurations such as the splitter and the connectedc was discussed. The behavior of connected-c is found to be similar to the fairing. Galloping behavior is exhibited by the splitter at higher Ur . The geometry of connected-c naturally resembles that of the splitter at lower r . It is intuitive to expect a similar galloping behavior at lower r for the connected-c and the ERA-based ROM is able to identify such a phenomenon. The diversified behavior of splitter at lower L (1.5, 1.25) and higher L (2.0, 2.5) was also reported from stability analysis. A general stability function  was introduced to assess the stability of the fluid-structure system. Overall, the stability analysis via ERA-based ROM serves as an attractive alternative in identifying the lock-in regions for different suppression devices. The approach is data-driven and would be realizable in physical experiments. The modes SM and WM provided by the stability analysis encompass the effects of a complex interaction between wake features and behave distinctly for different geometries. The methodology provided in the current work is not limited to a single body and can be easily extended to multiple body arrangements. However, preliminary investigations in the case of side-by-side cylinder arrangement with a gap flow revealed the limitations of the current approach in terms of constructing a suitable ROM. Addressing these limitations is one of the main driving factors for future work. Currently, the suitability of the present approach for high Reynolds number cases has not been explored and it will be the other direction to continue this work.

10.4 Active Control of FIV via Near-Wake Jet Flow The vortex-induced vibration (VIV) phenomenon is well-recognized in a diverse range of offshore and marine engineering applications. In particular, effective suppression of VIV and the reduction of hydrodynamic forces on the structural body can lead to safer, sustainable and cost-effective structural design for a broad range of operational conditions. This practical concern, along with an interest in fundamental fluid mechanics, has resulted in a broad literature considering the physical mechanisms behind VIV. When the natural frequency of the structure is close to the vortex shedding frequency, a complex evolution of the shedding frequency occurs due to the VIV phenomenon. This causes the deviation of the shedding frequency from the Strouhal relation corresponding to a stationary bluff body. During the lock-in regime,

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the shedding process is dominated by the natural frequency of the body and a strong coupling between the fluid and the structure is established. The lock-in phenomenon induces high-amplitude and self-sustained vibrations hence understanding the origin of these large oscillations and identifying different regimes during the lock-in process is important. Extensive studies are carried out on the phenomenon of lock-in and vortex-induced vibrations for cylindrical structures by several researchers [40, 381, 455]. Despite the vast practical importance of the three-dimensional canonical geometry of a sphere, limited studies consider VIV mechanisms of the sphere bluff body compared to its two-dimensional cylindrical counterpart. Flow-induced vibrations (FIV) of an elastically mounted or tethered spherical configuration are beneficial for power generation in the ocean environment, while such vibrations are undesirable on spherical marine/offshore structures such as assistive low-aspect-ratio escort tugboats connected with ships presented in the study by Chizfahm et al. [88]. The experimental works by Williamson and Govardhan [139, 454, 455] show that when the natural frequency of an elastically-mounted sphere body approaches the frequency of the unsteady vortex shedding, vortex-induced vibration happens similar to twodimensional bluff bodies. In contrast to two-dimensional circular cylinder wakes, the vortex topology and shedding process of a three-dimensional sphere are significantly different. Multiple modes of vibration for the tethered sphere configuration are reported in the experimental studies performed by Williamson and Govardhan [454], Govardhan and Williamson [137, 139], Jauvtis et al. [188]. These modes were categorized based on the amplitude response curve ( A∗ - U ∗ ), where A∗ is the nondimensional amplitude defined as A∗ = A/D, and U ∗ is the reduced velocity defined as U ∗ = U/ f n D ( f n is the natural frequency of the system in a vacuum). Mode I and mode II were defined as the regimes where the equivalent fixed-body vortex shedding frequency remains close to the natural frequency of the oscillating system and resonance was observed. Mode III and mode IV regimes occurred over higher reduced velocity ranges of U ∗ ∈ [20 − 40] and U ∗ > 100, respectively. The VIV response of elastically mounted sphere restricted to move in the transverse direction is carried out numerically by Rajamuni et al. [352] through fixing the Reynolds number at Re = 300 over the reduced velocity ranges U ∗ ∈ [3.5 − 100]. Distinct VIV branches were identified corresponding to the oscillation behavior. The sphere showed a highly periodic large-amplitude response in the order of ∼ 0.4D over a lower reduced velocity range of U ∗ ∈ [5.5 − 10]. The sphere oscillation frequency was found to lock with the vortex shedding frequency (Strouhal frequency) which is designated as branch A. Other non-negligible small oscillations were found to occur in the order of ∼ 0.05D within the reduced velocity range of U ∗ ∈ [13 − 16] and U ∗ ∈ [26 − 100], named by branch C and branch E respectively. It was found that the sphere oscillation frequency locked with half of the Strouhal frequency (∼ St2 ) for these branches. A recent experimental study by Sareen et al. [377] and a numerical study by Chizfahm et al. [89] were carried out on the flow-induced vibration (FIV) of fully submerged and semi-submerged sphere configurations close to and piercing the free-surface. The wake modes and the coupled dynamical interactions of multiphysics-multiphase setup that lead to flow-induced vibrations with large ampli-

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tude response were systematically investigated in a wide range of Reynolds number Re ∈ [300 − 30 000] and reduced velocities U ∗ ∈ [3 − 20]. Large amplitude oscillations in the order of ∼ 1D were observed for the sphere piercing the free-surface level which was greater than all the submerged cases. For the piercing sphere configurations in the study by Chizfahm et al. [89], an additional third-harmonic frequency of the transverse force due to the free-surface effect found to be a major cause of the large-amplitude vibrations in the lock-in VIV regime. Such physical insight into the VIV response may guide the development of effective active or passive suppression devices. Given the implications of structural failure, or conversely the potential to harvest energy from VIV, there is a clear motivation to manipulate the vibration response of an object to our advantage. In the last few decades, several flow control devices to suppress the vortex-induced forces and vortex shedding process of bluff bodies have been explored. The review study by Zdravkovich [477] has classified various vortex suppression devices. The mechanism to suppress the shedding process can be categorized into three crucial aspects which are exploiting the shear layers by implementing surface protrusion, influencing the entrainment layers using shrouds, and disturbing the confluence point through near-wake stabilizers. These techniques could be applied through passive and active control strategies. Some comprehensive discussion on flow control strategies can be found in the study of Gad et al. [126] and the review article of Choi et al. [92]. Passive control techniques are mainly achieved through the modification of geometry. Passive suppression devices have been investigated in many studies to suppress vibration. Some examples that have been employed for the circular cylinders are helical strakes by Zhou et al. [498], control rods by Wu et al. [459] and spanwise grooves by Law and Jaiman [227]. For the three-dimensional spherical body, Chae et al. [78] employed a passive control device using a moving ring to adjust the separation point for drag reduction on a stationary sphere. On the other hand, active flow control strategies are attractive and advantageous over passive type techniques as they could be adaptively adjusted depending upon the flow environment and indeed they require additional energy input to the system to exploit the shear layers or the confluence points. The first attempt to control the VIV of a sphere has been performed experimentally by van Hout et al. [427]. They exploit acoustic excitation at frequencies much larger than the shear flow instabilities by using speakers mounted to wind-tunnel walls. They were successful to suppress the vibrations at dominant VIV modes (Mode I and II), however, vibrations corresponding to higher reduced velocities at Mode III were amplified. The recent experimental study by Sareen et al. [378], has implemented imposed rotary oscillation on an elastically mounted sphere to control the FIV response. By applying angular forcing frequency close to the natural frequency, they found a ’rotary lock-on’ regime where the forcing frequency dominates the vibration frequency. They were able to attenuate the vibrations close to but not at the natural frequency of the system in the mode I regime. More recently, McQueen et al. [281] have extended the previous experimental work by implementing a proportional feedback controller. In particular, the sphere was rotated in proportion to its transverse displacement. They found it possible to suppress vibrations of modes

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I, II and III by adjusting the gains corresponding to each specific scenario. However, there was no de-synchronization between vortex shedding and the sphere motion. One of the well-known types of active flow control techniques is the near-wake jet at the base side of the bluff body or base bleed flow. This technique imitates connective vortex suppression devices at the base side of the bluff body such as the splitter plate for two-dimensional cylinders by Kwon et al. [223] and can effectively de-synchronize the vortex shedding mechanism. The experimental study performed by Lin et al. [250] clearly showed the inhibition of vortex formation structures along the span of a cylinder by implementing jet flows in a helix pattern on the cylinder surface. A recent numerical study by Narendran [310] on a freely vibrating square cylinder with steady bleeding at its base side showed the reduction of peak transverse VIV amplitude by 90% through the shifting vortex formation region further downstream for a laminar flow condition (Re = 100, m ∗ = 1). Several studies have been conducted on an active control using blowing and suction actuators on a stationary sphere surface for drag reduction purposes. Jeon et al. [192] have implemented a local time-periodic blowing and suction from a slit on a sphere surface and obtained a significant reduction of the drag coefficient at forcing frequencies much higher than the vortex-shedding frequency at subcritical Reynolds number (Re = 105 ). Another experimental study by Findanis and Ahmed [115], used localized synthetic jets on the sphere surface and showed the effect of delaying the separation point and extending it further downstream that resulting in a significant reduction of drag force at (Re = 5 × 104 ). In this section, our focus is to examine the base bleed concept for a transversely vibrating sphere to stabilize the shedding mechanism and attenuate the VIV response. High-fidelity full-order simulations are useful for generating ground-truth data and physical insight. Despite these benefits of the full-order analysis, the FOM simulations can be assessed with extremely high computational efforts even with the current state-of-the-art supercomputers. Therefore these simulations may not be practical enough for parameter space exploration and control strategies. To overcome these shortcomings, model order reduction techniques are practical alternatives to perform stability analysis and develop effective control strategies. Reduced-order model (ROM) can be obtained through two broad approaches. The first one is to project the full-order system onto low-rank subspaces such as Galerkin projection, proper orthogonal decomposition, etc. Rowley and Dawson [367] have performed a comprehensive review of various model reduction techniques. The second approach is based on system identification methods by collecting input-output data and identifying a low-rank model by using data-driven techniques. Eigensystem realization algorithm (ERA) is a system identification technique developed by Juang and Pappa [199] that construct a minimal realization of the system based on the impulse response, and make a ROM through a linear projection of the original system on the most observable and controllable subspaces. The ERA-based ROM has been used for stability analysis and flow control problems in various recent studies (Ma et al. [271]; Flinois and Morgans [116]; Yao and Jaiman [466]; Bukka et al. [70]). Yao and Jaiman [465] provide a unified description of frequency lock-in for elastically mounted two-dimensional cylinders using the ERA-based ROM technique. By performing the stability analysis

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607

of a coupled system, they investigated the effects of Reynolds number, mass ratio and the change in the geometry of the bluff body on the VIV response. The recent work by Bukka et al. [70] presents a stability analysis of passive suppression devices (fairings and connected-C devices) for the VIV of two-dimensional cylinders in the laminar flow condition. They systematically identified two dominant VIV modes and assessed the stability characteristics of the passive suppression devices. The objectives in this section lies in a systematic understanding of the sphere VIV mechanism through a data-driven stability analysis via the ERA-based ROM. The modal analysis of a sphere with the near-wake jet active suppression device is another novel aspect of the present work. Unlike many papers on the two-dimensional cylinder cases, there exists no study on a three-dimensional sphere VIV in the aspects of elimination of vortex shedding and stabilization of flow profiles in the near-wake region by the base bleed concept. Herein, we explore the base bleed concept for reducing VIV of a freely vibrating sphere. Our present study derives motivation from this, to investigate the flow features, corresponding kinematic and hydrodynamic parameters of elastically mounted sphere subjected to a uniform flow with near-wake jet (i.e., base bleeding). In this paper, we have consistently employed the term near-wake jet to describe the base bleed flow. Our study focuses on the following questions: Can we characterize the frequency lock-in regimes associated with resonance and galloping-type characteristics of a freely vibrating sphere using the ERA-based ROM? What are the salient differences in the stability characteristics of a freely vibrating sphere when compared to the sphere with a near-wake jet actuation system? How do the wake modes (WMs) and the structure mode (SM) predicted by the stability analysis behave in conjunction with the interaction of wake features? How the injected jet interacts with the laminar-wake region? How much reduction in the VIV amplitude and fluid forces can be achieved via the near-wake jet control?

10.4.1 Sphere Via Jet-Based Actuation This section deals with the numerical formulations used for the high-dimensional FOM followed by the implementation of ERA to develop the ROM model for the coupled FSI problem. The numerical methodology presented in this section follows closely that outlined in Yao and Jaiman [466].

10.4.1.1

Full-Order Problem Set-Up

Figure 10.20a shows a schematic of the problem setup used in our simulation study for a sphere, both stationary and elastically mounted cases. A three-dimensional computational domain of the size (50 × 20 × 20)D with a sphere of diameter D placed at an offset of 10D from the inflow surface is considered, which is sufficient enough to reduce the effects of artificial boundary conditions around the fluid domain. The origin of the coordinate system is fixed at the center of the sphere. We consider

608

10 Data-Driven Passive and Active Control Y Z

v = 0, σxy = 0, σzy = 0

X

10D

Inflow u=U v=0 w=0

ky

m

ky

cy

Outflow σxx = 0 σyx = 0 σzx = 0

D

10D

cy 10D

v = 0, σxy = 0, σzy = 0 10D

10D

40D

(a)

U

s

Uj

S

(b) Fig. 10.20 a Schematic and associated boundary conditions of the fluid flow past a 1-DOF fully submerged elastically mounted sphere. b Schematic of the sphere with the base bleed jet

the x-axis as the streamwise flow direction, the y-axis in the transverse direction, and the z-axis represents the vertical direction. While the streamwise motion corresponds to the freestream (x-direction), the transverse motion is parallel to the y-direction. A uniform freestream flow with velocity U is along the x-axis. At the inlet boundary, a stream of water enters into the domain with velocity (u, v, w) = (U, 0, 0) where u, v and w denote the streamwise, transverse and vertical velocities in x, y and z directions, respectively. The sphere is elastically mounted on springs with a stiffness value of k and linear dampers with a damping value of c in the transverse direction. The damping coefficient ζ is set to zero in the present work. We have considered the slip-wall boundary condition along the top, bottom and side surfaces, in addition to the Dirichlet and traction-free Neumann boundary conditions along the inflow and outflow boundaries, respectively. The definitions of some relevant important non-dimensional parameters are summarized in Table 10.5. The non-dimensional amplitude response A∗ is defined as A∗ = A/D and f ∗ denotes the normalized frequency and f n =

1 2π

k m

is the natural

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609

Table 10.5 Definition of the non-dimensional parameters and post-processing quantities Parameter Definition Re = ρ f UD/μf U ∗ = U/ f n D m ∗ = m/m d √ ζ = c/2 mk √ A∗rms = 2 Arms /D h ∗ = h/D C x = f xs /( 21 ρU 2 S) C y = f ys /( 21 ρU 2 S) C z = f zs /( 21 ρU 2 S) Cq = Uj s/U S f ∗ = f / fn

Reynolds number Reduced velocity Mass ratio Damping ratio Non-dimensional amplitude Immersion ratio Normalized drag force Normalized transverse force Normalized vertical force Bleed coefficient Normalized frequency

frequency of the spring-mass system in vacuum, where m is the mass of the sphere and k is the spring stiffness. The mass ratio is given by m ∗ = m/m d , where m is the mass of the sphere and m d is the mass of displaced fluid. The value of U is considered to be 1 throughout the numerical study. The Reynolds number Re and the reduced velocity U ∗ are varied by changing the values of dynamic viscosity μf and the natural frequency f n . The normalized forces are evaluated from the fluid traction, acting on the structural body, where C x is the normalized drag force, C y and C z are the normalized transverse and vertical forces in y and z directions, respectively. Cx =

1 1 ρU 2 S 2

1 Cy = 1 2S ρU 2 1 Cz = 1 ρU 2 S 2

 (σ¯ f · n) · n x d

(10.24)

(σ¯ f · n) · n y d

(10.25)

(σ¯ f · n) · n z d

(10.26)



 

 

where n x , n y and n z are the Cartesian components of the unit normal n to the sphere 2 surface, and S is the relevant surface area which  is defined as S = π D /4. The

normalized lift coefficient is obtained as C L = C y 2 + C z 2 . The schematic of the near-wake jet actuator as a suppression device positioned at the center part of the base side of the sphere is shown in Fig. 10.20b. The jet is imparted by altering the sphere geometry via a small patch at the rear side of the sphere. A bleed coefficient parameter (Cq = Uj s/U S) is defined as the ratio of near-wake jet outflow rate to the freestream inflow rate, where Uj is the jet flow velocity and s is the area of an injecting

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Table 10.6 Mesh statistics and convergence study of the flow parameters for stationary sphere at Re = 300 Mesh Nodes (×106 ) C x CL St M1 M2 M3 GCI12 GCI23 ARC

1.730 2.810 5.156 – – –

0.665 (0.9%) 0.662 (0.4%) 0.659 2.26% 1.81% 0.9964

0.076 (10.1%) 0.070 (1.4%) 0.069 2.14% 0.36% 0.9857

0.1367 0.1367 0.1367 – –

The error deviation is evaluated based on the corresponding value of M3 mesh. The normalized mean drag force (C x ), the normalized mean lift force (C L ) and the Strouhal number are also recorded

patch on the base side of the sphere surface.The grid convergence indices (GC I ) for the normalized mean drag force (C x) and the normalized mean lift force (C L) are provided in (Table 10.6). Convergence indices (GC I ) for the normalized mean drag force (C x ) and the normalized mean lift force (C L ). The differences between the sensitivity analyses are within 5% with the asymptotic range of convergence reasonably close to (≈ 1) and the selected grid resolution is therefore considered suitable for the physical significance of the study presented in the manuscript hereafter. The mesh M2 is adopted in the present study, as it confirms the adequacy of the numerical results. Based on our earlier studies, the time step size with t = 0.05 is considered in the present study1 . In order to assess the adopted numerical scheme, the results are compared against the available data in the literature. To begin with, flow past a stationary sphere is conducted for Re = 300. The values of the normalized mean drag force coefficient (C x ), the normalized mean lift force (C L ) and the Strouhal frequency (St), which is the dominant frequency of the lift force have a satisfactory match with the available literature considered in our study as shown in Table 10.7. The accuracy of our FSI solver is further validated for a fully submerged elastically mounted sphere restricted to move in the transverse y-direction in the study performed by Chizfahm et al. [89], where the simulation results are compared against the measurement data of Sareen et al. [377] for the Reynolds number Re in the range 5 000 ≤ Re ≤ 30 000 corresponds to the reduced velocity U ∗ range of 3 ≤ U ∗ ≤ 20.

10.4.1.2

ERA-Based ROM Problem Set-Up

In this section, we aim to verify the ERA-based ROM for a three-dimensional flow at Re = 300. The reliability of the ERA-based ROM is verified by assessing the stability of the wake behind a three-dimensional sphere system. Afterwards, the jet injection as a base bleed flow is considered for the near-wake jet control stability 12 Asymptotic range of convergence that is defined as r pGCI ×GCI23 ; where r is the grid refinement ratio and p is the order of convergence presented by Roache [361].

1

10.4 Active Control of FIV via Near-Wake Jet Flow

611

Table 10.7 Validation of flow past stationary sphere at Re = 300 Study Cx CL St Present study Giacobello et al. [130] Kim et al. [212] Johnson and Patel [193]

0.662 0.658 0.657 0.656

0.070 0.067 0.067 0.069

0.137 0.134 0.137 0.137

Table 10.8 Mesh convergence study: comparison of growth rate and frequency for different mesh configurations for the flow past a stationary sphere at Re = 300 Mesh Nodes (×106 ) Elements σ f (×106 ) M1 M2 M3

1.730 2.810 5.156

7.969 16.223 30.026

0.0016 0.0025 0.0027

0.1369 0.1358 0.1355

analysis. Three mesh configurations are considered for the mesh convergence study. The quality of the mesh convergence is determined by the prediction of the growth rate σ and the frequency f of the fluid ROM for the flow past a stationary sphere system at Re = 300. The results of this study are tabulated in Table 10.8. The difference between M2 and M3 is found to be less than 1%. Therefore, mesh M2 is considered to be sufficiently refined for the current study and is depicted in Fig. 10.21. Here we describe the process of constructing the ROM for the coupled FSI system in four steps based on the methodology explained in the previous section. In the first step, the base flow is computed via fixed-point iteration without considering the time-dependent term in Eq. (10.2) as an equilibrium position of the FSI system at low Reynolds number. Using our fully implicit solver presented by Jaiman et al. [181], the converged base flow solution is achieved by marching to the steady state solution. The second step involves the extraction of the impulse response of the system. By imposing a small input impulse of the transverse displacement Y , the equilibrium position is perturbed and we can assess the growth of the disturbance and the stability of the system. Small amplitude of the order of 10−3 or 10−4 is applied such that the flow evolves linearly for a relatively long time, and the linearity is verified by comparing the response for different inputs. An adequate number of cycles is required to capture the linear dynamics of the system to ensure that the unstable modes start to dominate the essential dynamics of the input-output relationship. The third step is to construct the fluid ROM through the ERA methodology. This procedure consists of the formation of the Hankel matrix, applying the singular value decomposition (SVD) and evaluating the reduced system state matrices. The dimensions of the Hankel matrix can be determined by examining the convergence of the fluid unstable eigenvalues. In the final step, the linear ROM for the coupled system is calculated by Eq. 10.18, where Afs characterizes the coupled dynamics. In

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10 Data-Driven Passive and Active Control

20

10

0 0

10

20 (a)

30

40 (b)

Fig. 10.21 A representative computational mesh employed for an elastically mounted sphere in a uniform steady flow: a two-dimensional slice of the mesh for the entire domain along the X –Y plane, b close-up view of the constructed mesh around the sphere

Fig. 10.22 Base flow of a stationary sphere system at Re = 300 along the X -Y plane for the elastically mounted sphere system; the streamwise velocity contours are shown

this study the continuous eigenvalues, λ = log(eig(Afs ))/ t are considered for the stability analysis. The eigenvalues of the matrix Afs are calculated and the leading unstable modes are identified with a growth rate and corresponding frequency. The eigenspectrum can be constructed by plotting the root loci of leading unstable modes as a function of the reduced natural frequency of Fs . The position of the eigenvalues on this spectrum provides information about the stability of the coupled system. The above process with the four main components forms a general procedure to be followed while constructing the ERA-based ROM. It should be noted that ERA is only applied to the fluid system and the resulting matrices are used to compute the ROM of the coupled system. The specific details of each component corresponding to the current study are discussed in the upcoming paragraphs. Figure 10.22 shows the streamwise velocity contours of the base flow at Re = 300 for the sphere system. A disturbance of δ = 10−3 is given as an impulse input to the transverse displacement Y . The normalized transverse force C y is recorded for every time step t = 0.05.

10.4 Active Control of FIV via Near-Wake Jet Flow

(a)

613

(b)

Fig. 10.23 The ERA-based ROM for the unstable wake behind a stationary sphere at Re = 300: a Time history of the normalized transverse force (C y ) due to impulse response calculated from the FOM and the ERA-based ROM and b HSV distribution corresponding to 1500 × 500 Hankel matrix

A set of 2 000 responses are stacked resulting in a total simulation time of tU/D = 100. Figure 10.23 shows the output normalized force signal C y calculated from the FOM and the ERA-based ROM at Re = 300. The Hankel matrix with dimension 1500 × 500 is found to be appropriate by examining the convergence of unstable fluid eigenvalues computed from Hankel matrices with dimensions of 1500 × 150, 1500 × 300, 1500 × 500 and 1500 × 750. The order of the ERA-based ROM is selected by examining the HSV distribution. The fast-decaying singular values as depicted in Fig. 10.23 suggest that the ERA-based ROM with order nr = 100 is sufficient. This is further confirmed by the accurate reconstruction of the impulse response by the ERA-based ROM displayed in Fig. 10.23a. The Hankel matrix may not be necessarily square but can be tall, wide or square depending on the problem under consideration. It should be highlighted that the development of the ERA-based ROM is computationally efficient since it depends entirely on the impulse response of the FOM, which is also for a short period (i.e., tU/D = 100 in our case). Moreover, constructing the eigenspectrum is trivial since it relies on the SVD procedure for the coupled matrix (Afs ). To demarcate the lock-in range from the FOM, the long-term unsteady simulation is required to construct the relationship between the reduced natural frequency (Fs ) and the normalized transverse amplitude ( A∗y ) at the stationary state. Through the ERA-based ROM, the expensive unsteady simulations are completely avoided. The ERA-based ROM is further used to identify the onset of Hopf bifurcation for the sphere system. The growth rates Re(λ) and the frequency Im(λ) of the fluid ROM are calculated by considering the unstable eigenvalues of eig(Ar ). The ERA-based ROM is constructed by following the aforementioned procedure at different Reynolds numbers. The growth rate Re(λ) and frequency Im(λ) are plotted against Re and are depicted in Fig. 10.24. The onset of unsteadiness of the flow that leads to transverse

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10 Data-Driven Passive and Active Control

lift force on the sphere ( f ys ) is found to be in the range of Reynolds number between Re ∈ [270 − 280] where Re(λ) changes the sign from a negative value to a positive value. The studies by Johnson and Patel [193], and Mittal and Rajat [290] showed that the flow past a stationary sphere at very low Reynolds numbers (Re < 200) is steady and axisymmetric but with increasing the Reynolds number it loses the axisymmetry first and then the steadiness. They observe that for (210 < Re < 270) a pair of streamwise vortices are formed behind the sphere without shedding, and for (280 < Re < 375) hairpin-shaped vortices are periodically shed with the same strength with a fixed planar-symmetric orientation. Similar observations are reported in the study by Sansica et al. [375] in the compressible framework for a nearly incompressible flow at Mach number M = 0.1, where breaking of the axisymmetry (Re ∈ [210 − 212]) and the onset of unsteadiness (Re ≈ 277) are explored as the flow exhibit the regular and Hopf bifurcations respectively. As the Reynolds number is increased further (375 < Re < 800) the strength and shedding orientation of the hairpin vortices vary over time and thus the flow becomes asymmetric as illustrated by Tomboulides et al. [421] These observations are further confirmed by our FOM calculations. The isosurface of non-dimensional Q-criterion and vorticity contours from the FOM calculations are displayed in Fig. 10.25 for Re = 270, 280, 300 and 400. We observe that at Re = 270, the flow becomes unsteady but with a fixed plane of symmetry with respect to (x-z) plane. By further increasing the Reynolds number to Re = 280, the hairpin-type vortices are periodically shed while the plane of symmetry slightly rotates along the x axis. The rotation of the plane of symmetry amplifies the transverse lift force accordingly, where the change in the sign of Re(λ) from a negative value to a positive value occurs. In the case of Re = 300, there is a clear presence of vortex shedding resulting from the wake instability with a fixed planar-symmetric orientation. The frequency of the vortex shedding appears to be increasing as the Reynolds number increases, which verifies the ROM frequency response (Im(λ)) as a function of the Reynolds number (Re) in Fig. 10.24. By further increasing the Reynolds number to Re = 400, we can clearly see that the shedding orientation of the hairpin-type structures loses symmetricity and the strength of vortices varies with time. In this section, the specific conditions in the process of building the ERA-based ROM and the validity of the reduced-order model via a systematic study of the stability analysis of the laminar wake behind a sphere system at Re = 300 are presented. The following sections will explore the stability characteristics of the coupled FSI systems of the sphere VIV and active near-wake jet control.

10.4.2 Assessment of the ERA-Based ROM for VIV of a Sphere In this section, the constructed ERA-based ROM is first applied to analyze the stability properties of a transversely vibrating sphere at Re = 300. For this purpose, the

10.4 Active Control of FIV via Near-Wake Jet Flow

615

Fig. 10.24 Growth rate and frequency of most unstable eigenvalue of fluid ROM: (Top figure) Root loci as a function of Reynolds number, where the unstable region Re(λ > 0) is shaded in grey. The arrow depicts the increasing order of Reynolds number. (Bottom figures) growth rate Re(λ) and frequency Im(λ) as a function of Reynolds number. The growth rate Re(λ > 0) indicates the instability in the system, which is manifested as vortex shedding

Fig. 10.25 a Iso-surface of three-dimensional wake structures formed behind stationary sphere at ¯ = Q D22 = 0.001), stationary state τ = tU /D = 100. Iso-surfaces are plotted by the Q-criterion (Q U b z-Vorticity contours from the FOM, (Re = 270, 280, 300, 400)

eigenspectrum plot for a canonical sphere is systematically examined. Consistent with the previous literature by Yao and Jaiman [466], we utilize the terminology of variation of unstable modes to classify the distinct eigenvalue trajectories of the fluid-structure system governed by Eq. 10.18. These modes are identified based on the magnitudes of the real part of eigenvalues (Re(λ)). The most unstable modes are recognized and are classified based on their frequencies (Im(λ/2π )). We use the terminology of SM and WMs to classify the distinct eigenvalue trajectories of the linear fluid-structure system governed by Eq. 10.18. The fluid-structure mode is considered as an SM if the eigenvalue comes close to the natural frequency of the sphere-alone system in vacuum. When the eigenvalue of the fluid-structure system approaches that of the stationary sphere shedding frequency (St), the resulting modes are defined as WMI, and if the eigenvalue gets close to half of the stationary shedding frequency, ( St2 ), the modes are described as WMII. Another unstable low

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10 Data-Driven Passive and Active Control

frequency wake mode may be identified for low mass ratio configurations which is close to ( St2 ) and is defined as WMIII. VIV lock-in may result either from the instability of the WMs alone or via the simultaneous existence of unstable SM and WMs. The lock-in occurs due to the closeness of the frequencies of the WMs and SM. This type of VIV branch is termed as resonance-induced lock-in. The instability to sustain the VIV lock-in also occurs via combined mode instability of the SM and WMs, indicating a galloping-type instability. Continuous-time eigenvalues are considered for depicting the eigenspectrum in the context of the stability analysis. The eigenvalues are calculated from the coupled fluid-structure matrix (λ = log(Afs / t)). The eigenvalue distribution of the coupled fluid-structure system is graphically depicted in a complex plane via root locus. The positions of the eigenvalues on this complex plane indicate the stability characteristics of the fluid-structure system. The roots in the right half-plane depict the unstable modes of the system, whereas the roots on the real axis characterize the asymptotic (non-oscillatory) behavior. Roots that are closest to the right half-plane are tightly damped oscillatory modes. Figure 10.26 shows the eigenvalue trajectory of the fluid-structure system as a function of the reduced natural frequency (Fs ) with 0.03 < Fs < 0.25 and the increment is Fs = 0.005. In the figure, the WMI branch originates in the vicinity of the eigenvalue of the stationary sphere (uncoupled WM) and loops back as the reduced natural frequency Fs increases. The WMII branch shows a noticeable variation although in the stable left-half plane which indicates a new coupled FSI behavior. On the other hand, the SM branch rises from the bottom of the complex plane (low-frequency regime) to the upper complex plane (high-frequency regime). As elucidated in Fig. 10.26b, the SM becomes unstable for when 0.04 < Fs < 0.14, which is determined by the real part of the eigenvalue. In order to identify the resonancebased regimes, we need to consider the closeness of the SM and WMs frequencies. As shown in Fig. 10.26b, the imaginary part of the eigenvalue as a function of Fs reveals the closeness and the intersection of two sets of distinct frequencies (WMI and SM frequencies, WMII and SM frequencies). Figure 10.26b indicates that the frequencies of the WMI and SM come closer when 0.125 < Fs < 0.14, which is recognized as the resonance mode (branch A). It should be noted that the lower-left boundary of the resonance mode cannot be pinpointed precisely from the ROM. Thus, we determine the frequency at the lower-left boundary from the FOM simulation. On the other hand, the frequencies of the WMII and SM get closer at 0.075 < Fs < 0.09, however since WMII is stable for the entire range, branch B cannot be defined as an unstable region through ERA-based ROM terminology. To further verify the stability results predicted by the ERA-based ROM, the VIV response is computed by direct numerical simulation using the FOM. Figure 10.27 shows the eigenvalue trajectory for the sphere coupled dynamics at a lower mass ratio m ∗ = 2.875. We found that the root loci of the WMI and SM can coalesce and form coupled modes. This behavior is observed in certain conditions, such as in the limit of low mass ratio (Yao and Jaiman [466]) and for different geometrical shapes for two-dimensional cylindrical cases (Bukka et al. [70]). These coupled modes do not offer distinct characteristics of the WMI and SM, since the

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617

(a)

(b) Fig. 10.26 Eigenspectrum of the ERA-based ROM for a sphere at (Re, m ∗ ) = (300, 10), a root loci as a function of the reduced natural frequency Fs , where the unstable right half-plane Re(λ > 0) is shaded in grey colour and the arrows indicate increasing Fs ; b real and imaginary parts of the root loci, where the lock-in regions are shaded in grey colour

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10 Data-Driven Passive and Active Control

(a)

(b) Fig. 10.27 Eigenspectrum of the ERA-based ROM for a sphere at (Re, m ∗ ) = (300, 2.865), a root loci as a function of the reduced natural frequency Fs , where the unstable right half-plane Re(λ > 0) is shaded in grey colour and the arrows indicate increasing Fs ; b real and imaginary parts of the root loci, where the lock-in regions are shaded in grey colour

10.4 Active Control of FIV via Near-Wake Jet Flow

619

two branches exchange their roles when coalescence of eigenvalue occurs. Similarly to Yao and Jaiman [466], we term these mixed or coupled modes as wake-structure mode I (WSMI) and wake-structure mode II (WSMII). For higher values of Fs , WSMI behaves as the WMI and WSMII recover to the SM. On the other hand, for a smaller Fs range, WSMI and WSMII represent the SM and WMI respectively. Through Fig. 10.27b (top), it is found that the growth rate curve corresponding to WSMI become unstable, Re(WSMI) > 0 at Fs ≈ 0.175 which is identified as the higher-right boundary of the resonance mode (branch A) for the sphere with m ∗ = 2.865, and right immediately after the growth rate curve corresponding to WSMII become unstable, Re(WSMII) > 0, the stability roles of WSMI and WSMII switch at Fs ≈ 0.165. Besides, through the imaginary part of eigenvalues in Fig. 10.27b, the resonance mode (branch A) can be verified through the closeness of the WSMI and WSMII frequencies. We observe that the WMII branch again shows noticeable variation as the corresponding growth rate amplifies when WMII and WSMI frequencies get closer at 0.065 < Fs < 0.85. However since WMII is stable for the entire range, branch B cannot be recognized as an unstable region. Another unstable wake mode is identified at a low frequency close to ( St2 ) which is defined as WMIII. A new unstable branch via combined mode instability is identified through the intersection of the corresponding frequencies of WSMI and WMIII. Notably, WSMI and WMIII remain unstable for Fs < 0.05 or Ur > 20, as predicted by the ERA-based ROM, which indicates that galloping-type instability persists. These linear stability results have been confirmed by the FOM simulations. As shown in Fig. 10.28, the vortex shedding frequency begins to synchronize with the natural frequency of the structure at Fs ≈ 0.175 and recovers to the vortex shedding frequency at Fs ≈ 0.1. Figure 10.28b suggests that the VIV region with the high amplitude response starts at Fs ≈ 0.175 (lock-in onset U ∗ ≈ 5.7), which compares almost accurate with the right-higher boundary of the resonance-induced lock-in predicted by the present ERA-based ROM. In Fig. 10.28b through our FOM simulations, the amplitude response at branch B is close to zero at stationary state, and the dominant oscillation frequency is close to stationary sphere ( f = St) at Fs = 0.07. This observation verifies the stability of branch B which is identified and characterized as a stable region by the ERA-based ROM in Fig. 10.27b. However, we observe frequency lock-in at f = St2 at branch B with noticeable amplitude response in the transient state as can be seen in Fig. 10.29. The numerical study of Rajamuni et al. [352] also reported these non-negligible oscillations in branch B in the order of (0.05D) with the frequency lock-in at St2 which is consistent with the response in our simulations in the transient state. The proficiency of the ERA-based ROM to explore the mass ratio effects on the VIV response is shown in Fig. 10.30 for (m ∗ = 2.865, 5, 10) at Re = 300. Figure 10.30b shows the real and imaginary parts of the root loci as a function of the reduced frequency Fs . It indicates that the lock-in onset which is the right-higher VIV boundary corresponding to branch A, shifts to the lower reduced natural frequencies (Fs ) as the mass ratio m ∗ increases. As expected, due to weaker fluid-structure coupling for larger mass ratio m ∗ = 10, the eigenfrequency of the WMI recovers to the frequency of the stationary sphere and the frequency of the SM approaches the

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10 Data-Driven Passive and Active Control

Fig. 10.28 The FOM VIV results as a function of the reduced natural frequency Fs , at Re = 300; variation of the oscillation frequency f , and the r.m.s. value of the normalized amplitude ( A∗rms )

natural frequency of the sphere-only system as Fs increases. By reducing the mass ratio as shown in Fig. 10.30a, the root loci of the SM and WMI gradually coalesce and form a coupled mode due to the increased strength of the fluid-structure coupling. It is further found that increasing the mass ratio tends to shift unstable SM modes towards a stable left-half plane.

10.4.3 Effect of Near-Wake Jet Flow In this section, the performance of the ERA-based ROM for identifying the effect of the near-wake jet actuator as a suppression device is presented. For this purpose, it is essential to understand the near-wake jet interaction and its influence on hydrodynamic force and vibrations of the sphere system. Hence, we define a bleed coefficient parameter (Cq ) which is the ratio of the near-wake jet outflow rate to the freestream inflow rate. Figure 10.31 presents streamwise velocity contours of the base flow of a sphere with the near-wake jet. To assess the effect of near-wake jet actuation on the VIV response, a parametric analysis of the bleed coefficient (Cq ) is performed on a representative case in branch

10.4 Active Control of FIV via Near-Wake Jet Flow

621

(a-1)

(a-2)

(b-1)

(b-2)

Fig. 10.29 Time history of sphere displacement, A∗y , versus non-dimensional time, tU/D and the corresponding frequency spectrum at a U ∗ = 8 (branch A), and b U ∗ = 14 (branch B)

A (VIV region). Time histories of the non-dimensional amplitude response (A∗y ), normalized transverse force (C y ), and drag force (C x ) for the sphere with control (near-wake jet) at (U ∗ = 8, Re = 300, m ∗ = 2.865) for different bleed coefficients are shown in Fig. 10.32. By comparing the response for the sphere without a jet and with a jet at Cq = 0.9%, it is found that the amplitude of the oscillations reduces by more than 90%, and the oscillations are completely suppressed for Cq = 1%. In addition, it can be seen that the oscillations of the normalized hydrodynamic transverse and drag forces applied to the sphere are completely damped for Cq = 1%, indicating the inhibition of the shedding process. To further analyzing the results, in Fig. 10.33, the root-mean-square (rms) of non-dimensional amplitude response (A∗rms ), the rms of normalized transverse lift force (C y, rms ), the mean value of normalized drag force (C x, mean ), and the mean value of transverse lift force (C y, mean ) are presented for different bleed coefficients. In the top plot, it can be seen that rms values of both amplitude response and normalized transverse force decrease significantly with increasing Cq in a tight threshold Cq ∈ [0.75% − 1%]. This suppression persists for higher bleed coefficient values (Cq > 1%). We observe that VIV causes an increase in the mean drag force by comparing (C x ) corresponding to the stationary sphere (Table 10.7) and a representative case for the sphere without jet at branch A (Fig. 10.32). In Fig. 10.33 (bottom plot), it is

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(a)

(b) Fig. 10.30 The effect of the mass ratio on the eigenspectrum of the ERA-based ROM for a sphere at (m ∗ = 2.865, 5, 10) and Re = 300: a the root loci as a function of the reduced natural frequency Fs , where the unstable right half-plane Re > 0 is shaded in grey colour and the arrows indicate increasing Fs ; b the real and imaginary parts of the root loci. In (b), VIV branches are separated for different mass ratios, from the darkest gray to lightest gray correspond to m ∗ = 10, m ∗ = 5 and m ∗ = 2.865 respectively

10.4 Active Control of FIV via Near-Wake Jet Flow

623

Fig. 10.31 Base flow of a stationary sphere system at Re = 300 along the X -Y plane for the sphere system with base jet flow (Cq = 1%) ; the streamwise velocity contours are shown

found that through base bleed jet actuation with (Cq = 1%), the mean drag force at the VIV region decreases by more than 14% and recovers back to the mean drag force corresponding to the stationary sphere. Further increase of the bleed coefficient (Cq > 1%) increases the mean drag force. On the other hand, we find that the mean normalized transverse force (C y, mean ) increases slightly at Cq = 1% and then becomes negligible by further increasing the bleed coefficient (Cq > 1%). To further analyze the effect of the near-wake jet on the hydrodynamic forces, pressure and z-vorticity contours are plotted in Fig. 10.34 for different bleed coefficients. Through the pressure plots, we observe that the decrease in the normalized transverse lift force is attributed to the interaction of near-wake jet flow into the near-wake region and shear layers. This interaction reduces the lift force component acting on the sphere. The near-wake jet exploits the oncoming shear layers, thereby reducing the lift forces. Through the vorticity plots in Fig. 10.34c, it is found that the increase of (C y, mean ) at Cq = 1% is due to the asymmetry of the steady wake formed behind the sphere. Further increasing the bleed coefficient drives the near-wake to be symmetric which gives rise to C y, mean ≈ 0. However, increasing the bleed coefficient more than Cq = 1%, not only requires more actuation effort but also leads to a low-pressure region at the base side of the sphere (i.e., higher drag force). This can be qualitatively observed through the pressure contour plots in Fig. 10.34(d-1, e-1). The iso-surface of the Q-criterion and the full domain z-vorticity contours for the sphere system with and without the near-wake jet actuation are plotted in Fig. 10.35. As can be seen, the shedding mechanism is completely suppressed for the jet-based setup with Cq = 1%. This indicates that the jet inflow in the near-wake with Cq = 1% interferes with the synchronization of the three-dimensional vortical structures and stabilizes the wake modes. Now, we aim to assess the performance of the ERA-based ROM for identifying the effect of the near-wake jet suppression technique. The same process presented earlier is adopted to construct the ERA-based ROM for the sphere VIV with the near-wake jet actuation. Figure 10.36 compares the eigenvalue trajectories of the near-wake jet actuated sphere with different bleeding coefficients with the sphere alone system.

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Fig. 10.32 Time histories of the transverse amplitude response (A∗y ), normalized transverse force (C y ), and normalized drag force (C x ) with different Cq for the sphere at (Re, m ∗ ) = (300, 2.865) at U ∗ = 8 corresponds to branch A at the lock-in region

The stability data are plotted at Re = 300 and m ∗ = 2.865. We observe that for the sphere with the near-wake jet flow, the SM and WM do not coalesce anymore and the strength of fluid-structure coupling is reduced. The WMs are completely shifted into the stable region due to the presence of the near-wake jet and the vortex shedding is stabilized. The instability of the SM is found to be due to the asymmetry of the non-oscillatory steady wake, and not due to the VIV and the vortex shedding. Since all WMs are shifted to the stable left-half plane.

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625

Fig. 10.33 Influence of base bleed actuation on the VIV response dynamics of a sphere at (Re, m ∗ ) = (300, 2.865) at U ∗ = 8 corresponds to branch A at the lock-in region

To validate the ROM predictions, the FOM simulations of the sphere with and without near-wake jet is presented for representative cases in Fig. 10.37a. As can be seen, the jet inflow with Cq = 1% is effective to suppress the vibrations for all branches A, B and C. The time-histories of the amplitude response for the jetbased sphere system are shown in Fig. 10.37b. The amplitude response shows small oscillations at the transient state, while the oscillations are damped out entirely at the stationary state. These small oscillations at the onset of vibration are found to be due to the asymmetry of the wake as shown in Fig. 10.34c. This asymmetry of the wake leads to structural instability while the wake is stabilized and the shedding process is suppressed. This instability can be verified through the ERA-based ROM as the SM eigenvalues remain on the unstable plain and all WMs are shifted to the stable left-half plain. In addition, the mean position of the sphere slightly moves away from its initial position because of the asymmetry of the wake in the direction of the movement. In Fig. 10.38, the effect of the mass ratio on the sphere system with near-wake jet actuation is investigated via ERA-based ROM. Three representative values of the mass ratios of m ∗ = (2.865, 5, 10) are considered for the sphere jet-based configuration with Cq = 1%, and the corresponding eigenvalue trajectories are calculated and displayed in Fig. 10.38a. While the SM is located in the unstable region for all cases, the WMs are stabilized completely. The WMI and WMII are shifted to the stable

626 Fig. 10.34 Pressure contours and z-vorticity contours at the wake side for the sphere VIV at the stationary state at (Re, m ∗ ) = (300, 2.865) and U ∗ = 8 corresponds to branch A at the lock-in region: Cq = a 0, b 0.9%, c 1%, d 2%, and e 5%

10 Data-Driven Passive and Active Control

(a-1)

(a-2)

(b-1)

(b-2)

(c-1)

(c-2)

(d-1)

(d-2)

(e-1)

(e-2)

10.4 Active Control of FIV via Near-Wake Jet Flow

627

¯ = 0.001), and b z-Vorticity conFig. 10.35 a Iso-surface of three-dimensional wake structures (Q tours for the sphere VIV at the stationary state at (Re, m ∗ ) = (300, 2.865) and U ∗ = 8 corresponds to branch A at the lock-in region with Cq = 0%, 0.9%, and 1% respectively

left-half plane, and the WMIII has completely disappeared. The SM is gradually shifted towards the stable region as the mass ratio is increased. Using the ERA-based ROM, the effect of other parameters such as the damping ratio and the geometry variation can be carried out with small computational efforts.

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(a)

(b) Fig. 10.36 The effect of near-wake jet actuator on the eigenspectrum of the sphere VIV system at (Re, m ∗ ) = (300, 2.865): a the root loci as a function of the reduced natural frequency Fs , where the unstable right half-plane Re > 0 is shaded in grey colour; b the real and imaginary parts of the root loci. In (b), shaded regions show VIV branches for the sphere without jet actuation

10.4 Active Control of FIV via Near-Wake Jet Flow

(a)

629

(b)

Fig. 10.37 a The r.m.s. value of the normalized amplitude ( A∗rms ) for the sphere with and without near-wake jet actuation at (Re, m ∗ ) = (300, 2.865). b Time histories of the transverse amplitude response for the jet-based sphere system with Cq = 1%

10.4.4 Interim Summary To summarize, a systematic stability analysis and the near-wake jet (base-bleed) control of sphere VIV have been numerically presented in the laminar flow condition. A data-driven approach based on the eigensystem realization algorithm (ERA) was employed to construct the reduced-order model (ROM). The effectiveness of the ERA-based ROM has been remarkably demonstrated for predicting the unstable wake flow behind the stationary sphere system. The onset of unsteadiness of the flow as a function of Reynolds number was well predicted and showed a great agreement with the FOM and the available literature. Multiple unstable wake modes were identified that showed strong coupling with the structural mode resulting in different resonance-induced regimes and combined mode instabilities. These coupled modes were categorized into three lock-in branches, namely A, B and C, and cross-validated successfully with the FOM and the available literature. The effects of the mass ratio (m ∗ ) have been examined through ERA-based ROM for the VIV lock-in analysis. A base-bleed control technique for suppressing VIV and reducing the drag force has been explored on a freely vibrating sphere for the first time. Systematic data-driven stability analysis and high-fidelity numerical simulations were performed for a range of bleed coefficients Cq ∈ [0.5% − 5%]. Based on our high-fidelity FOM simulations, we found the minimum hydrodynamic drag coefficient at Cq = 1%, where the drag force reduced by more than 14% and recovered to its stationary counterpart. The trend of (C x, mean ) was found to increase linearly for Cq ≥ 1%, due to the low pressure at the near-wake region attributing to the high jet flow velocity at the base side of the sphere. At Cq = 1%, we observed a pressure recovery at the sphere surface and stabilization of the flow profile at the wake region of the sphere. We found that by implementing a small jet actuation with Cq = 1%, the transverse sphere oscillations are completely suppressed for all VIV branches and de-synchronization of the shedding process is achieved. The effectiveness of the near-wake jet has been shown for all branches A, B and C. The VIV oscillations were completely suppressed in all regimes. The performance of the ERA-based ROM in identifying the suppression

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(a)

(b) Fig. 10.38 The effect of mass ratio (m ∗ ) on the eigenspectrum of the sphere VIV system with near-wake jet actuation at Re = 300

10.5 Feedback Control of VIV via Jet Blowing and Suction

631

due to the base bleed was investigated by comparing the eigenspectrum of the sphere alone system. The presence of the near-wake jet shifted all unstable wake mode (WMs) into the left half of the complex plane, indicating a clear suppression of transverse vibration corresponding to shedding. The full-order model (FOM) results have confirmed the ERA-based ROM predictions by reporting a consistent reduction in the normalized transverse amplitude. The simplicity and computational efficiency of the ERA-based ROM allow investigation of the VIV mechanism for a variety of geometries and parameters and open ways for the development of control devices. For high Reynolds number flows with realistic geometries, nonlinear system identification with advanced physics-based machine learning will be of interest for future study.

10.5 Feedback Control of VIV via Jet Blowing and Suction Successful control of vortex-induced vibration (VIV) can lead to safer and costeffective structures in offshore, aeronautical and civil engineering. In the past several decades, various passive control techniques [30, 92, 226, 326, 473] have been explored via geometry modification and by adding auxiliary surfaces to alter the flow dynamics without any energy input. While the passive VIV control methods offer some simplicity, they do not have the ability to work on-demand and may not be effective from the perspective of wake stabilization, drag reduction and VIV suppression in a wide range of operational conditions. More importantly, the passive devices e.g. strakes, splitter plates or fairings are not easy to implement for certain situations such as square-shaped multicolumn offshore platforms [79] and subsea pipelines undergoing VIV in the proximity with seabed floor [401]. The motivation of this study is to develop a feedback active control algorithm based on a reduced-order model and to demonstrate the algorithm to stabilize the wake flow and the VIV for a canonical two-dimensional circular cylinder problem. To investigate the proposed control scheme, we consider the flow past a stationary circular cylinder at low Reynolds number, Re = U D/ν and the vibrating cylinder as a function of reduced velocity, Ur = U/( f N D) and mass ratio, m ∗ = 4m/(ρπ D 2 ) with zero mass-damping parameter, m ∗ ζ = 0.0. Here U , ρ, m, D, f N , ζ and ν are the freestream velocity, the density of the fluid, the mass per unit length of the cylinder, the diameter of the cylinder, the structural natural frequency, the damping ratio and the kinematic viscosity, respectively. Through external input of small tunable energy into the surrounding flow, active VIV control techniques offer a better alternative due to their adaptive and efficient performance. [211] utilized blowing/suction slots placed on the top and bottom of the circular cylinder to stabilize the unstable wake. The applied forcing in the slots was sinusoidal along the spanwise direction but kept steady in time. In another interesting study of [109] the combined steady windward suction and leeward blowing (WSLB) was found as effective strategy to eliminate the vortex street and to suppress vortexinduced vibration in cross-flow direction. This WSLB method in [109] is essentially

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an active flow control (i.e. external energy is required to maintain steady suction and blowing), but can be customized as a passive technique by deploying porous surfaces to form connecting channels between the windward and leeward stagnation points of a circular cylinder. The aforementioned blowing/suction control schemes provide only open-loop alteration of unsteady flow whereby the control input is prescribed and is independent of the flow states. In another recent study, [436] utilized a windwardsuction and leeward-blowing feedback control to suppress VIV in both inline and cross-flow directions at a specific condition (m ∗ , Ur ) = (2, 5). However, it is not certain whether the system is free of VIV for a range of reduced velocity Ur and mass ratio m ∗ . Moreover the simplified closed-loop control strategy based on the proportional (P), integral (I) and proportional-integral (PI) schemes were applied to manipulate the WSLB velocities through the standard deviation of surrounding flow velocity. The PI control was found to outperform the P and I control schemes with respect to the effectiveness of VIV suppression. To the best of our knowledge, adaptive feedback control of VIV via vertical suction and blowing has not been studied earlier. Furthermore, adaptive feedback control of VIV based on the model reduction is not explored in earlier studies, which can be important for both numerical and experimental settings. Various types of active flow control strategies have been explored for the flow past a circular cylinder, such as full-state feedback control, neural networks, and proportional closed-loop feedback control. In particular, active feedback or closedloop control of unsteady flow past over a bluff body has been recently investigated via numerical simulations [2, 116, 334]. To implement the optimal linear control in an efficient manner, it is imperative to develop a low-order linear model by retaining the significant dynamics of the original system ([2, 116]). Linear ROM provides a way to trace the eigenspectrum of dynamical systems, while maintaining about an order of magnitude efficiency improvement to construct the essential dynamics of the system. Instead of deriving the system model directly from the linearized governing equations of the fluid flow, the system identification method is particularly desirable, because it is non-intrusive and can be used directly into an existing Navier-Stokes (NS) solver to generate approximate system matrices using only the input-output data sequences. Eigensystem realization algorithm (ERA) [199] is a well-established system identification method to construct linear ROM from stable system linear impulse response. [271] proved that the ROM constructed by ERA is mathematically equivalent to balanced truncation. Recently, [117] provided a mathematical rigor that unmodified balanced truncation (designed for the stable system) is applicable for the unstable system. Based on the work of [117, 271], the ERA is used for the stabilization of unstable wake flow [116]. Although the suction/blowing control strategy has been extensively studied for unstable flow and VIV past a circular cylinder both numerically [109, 211, 275, 436] and experimentally [84, 85, 119], the earlier research relies on an open-loop control strategy, except the recent study of [436] via simplified non-adaptive control procedure. [2, 116] designed a feedback control law to stabilize unstable wake flow over a plate and bluff body using model reduction method, however, the actuation is modeled as a body force and may not be implemented as a practical actuator. The

10.5 Feedback Control of VIV via Jet Blowing and Suction

633

recognition of linear mechanism during the self-sustaining behavior of VIV in our recent study [466] inspires us to develop a feedback suppression strategy based on a linear control theory. Motivated by the insight into the frequency lock-in process during VIV, it is possible to design a linear controller to minimize the unsteady vortex-shedding and the VIV effects. For that purpose, the control input is based on vertical blowing and suction at the surface can be utilized [211]. The key objectives in this section are to develop an active feedback blowing and suction (AFBS) procedure based on ERA-based ROM to control the wake instability and the vortex-induced vibration. Using the ERA method, a low-order fluid model is constructed and coupled with the structure via the LQR optimal control scheme. The proposed AFBS procedure ensures a VIV-free system within a large parameter space of reduced velocity (Ur ) and mass ratio (m ∗ ) and it can handle both one degree-of-freedom (1-DOF) and two-degree-of-freedom (2-DOF) VIV. The results from the ROM solver are compared with those of full-order simulations based on the incompressible Navier-Stokes equations. We employ the ROM model to predict the performance of the AFBS procedure through the eigenvalue distribution in the complex plane. The present study is based on two questions on the VIV physics and control: (i) can we suppress self-sustaining VIV by assuming a linear lock-in mechanism? (ii) how much additional energy is required for the suppression of VIV compared to the stationary nonlinear vortex shedding state?

10.5.1 Cylinder Blowing/Suction Porous Surface Configuration Figure 10.39a shows a schematic of the problem setup used in our simulation study for a flexibly mounted circular cylinder in a flowing stream. At the inlet boundary in , a stream of incompressible fluid enters into the domain at a horizontal velocity (u, v) = (U, 0), where u and v denote the streamwise and transverse velocities in x and y directions, respectively. For the VIV configuration, the circular cylinder with mass m is elastically mounted on a linear spring and is allowed to vibrate only in the transverse direction. No-slip wall condition is implemented on the surfaces of the bluff body, and a traction-free boundary condition is implemented along the outlet out , while the slip wall condition is implemented on the top top and bottom bottom boundaries. Except stated otherwise, all length scales are normalized by the cylinder diameter D and velocities with the free stream velocity U . The numerical domain extends from −10D at the inlet to 30D at the outlet, and from −15D to 15D in the transverse direction. For the flow control, we consider blowing and suction on the porous cylinder surface through fluidic actuators. It is known that the moderate levels of suction/blowing into the surrounding flow can have a great impact on the boundary layer, the separation point and wake characteristics [84, 85, 119]. Through the active feedback control, the suction mechanism can delay the separation of the boundary layer (i.e., narrower wake width and reduced drag),

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Fig. 10.39 Problem setup for feedback control of unsteady wake and vortex-induced vibrations: a computational domain and boundary conditions for the flow past a freely vibrating cylinder in uniform flow, b proposed new actuator configuration B S0 through blowing/suction over the porous surface of circular cylinder. The positive control input is defined as suction from bottom and blowing at the top of cylinder

whereas blowing can have the opposite effect. On the other hand, continuous blowing generally tends to decrease the Strouhal number for the flow around a porous cylinder, continuous suction has the opposite influence on the vortex shedding frequency [119]. In the present study, we propose a feedback control based on a configuration with three pairs of suction/blowing actuators, as depicted in Fig. 10.39b. In this proposed configuration, termed as B S0, there are a total of six suction and blowing slots distributed over the cylinder surface, with a pair of slots at the windward θ = (135◦ , 225◦ ), at the midward θ = (90◦ , 270◦ ) and at the leeward θ = (45◦ , 315◦ ) sides. Here θ is the deviation angle between the centerline of each suction/blowing actuation with respect to the base suction point. As shown in Fig. 10.39b, the positive control input is defined as suction from the bottom of the cylinder and blowing at the top surface. Similar to [337], we consider the actuation slot width to be σc = π D/72, whereby the energy supply from the actuation is characterized by the momentum coefficient as Cμ = 2NρVc2 σc /(ρU 2 D), where N is the number of slots, Vc denotes the time-dependent suction and blowing velocity. Owing to the body-conforming Lagrangian-Eulerian coupling for fluid-structure interaction, the actuation conditions for the blowing and suction are accurately enforced by the Dirichlet boundary condition. The present full-order fluid-structure model relies on a variational finiteelement formulation and a semi-discrete time stepping. While the Navier-Stokes equations are discretized in space using Pn /Pn−1 iso-parametric finite elements for the fluid velocity and pressure, the second-order backward scheme is used for time discretization, where Pn denotes the standard n th order Lagrange finite element space on the discretized fluid domain. Details of the numerical techniques, the verification and the mesh convergence study based on P2 /P1 iso-parametric elements are documented in [466].

10.5 Feedback Control of VIV via Jet Blowing and Suction

635

For the sake of completeness, we first present the full-order model (FOM) based on the NS equations for the moving incompressible viscous fluid domain f (t) as  ρ

 ∂u  + (u − w) · ∇u = ∇ · σ on f (t), ∂t x

(10.27)

∇ · u = 0 on f (t),

(10.28)

where the time derivative is taken with the referential coordinate  x held fixed and the Cauchy stress tensor for a Newtonian fluid is σ = − pI + μ ∇u + (∇u)T . Here p, u, w, μ and I denote the fluid pressure, the fluid velocity, the mesh velocity, the dynamic viscosity and the identity tensor, respectively. The mesh nodes on the fluid domain  f (t) are updated by solving a linear steady pseudo-elastic material model ∇ · σ m = 0,

(10.29)

where σ m is the stress experienced by the ALE mesh due to the strain induced by the rigid-body movement, which is defined as, σ m = (1 + km )



T      + ∇ · ηf I , ∇ηf + ∇ηf

(10.30)

where ηf represent the ALE mesh node displacement and km is a mesh stiffness variable chosen as a function of the element area to limit the distortion of small elements located in the immediate vicinity of the fluid-body interface [255]. Given a base flow u0 , the corresponding linearized NS equations can be written in a semi-discrete form as dQ = FQ + GVm , E (10.31) dt where the matrices and vectors in Eq. (10.31) are  F=  G=

   − () · ∇u0 − u0 · ∇ () + μ ∇ () + ∇ T () −∇ () , ∇ · () 0

       w I0 u () · ∇u0 0 ,E = . ,Q = , Vm = 0 0 0 00 p

(10.32a)

(10.32b)

The linearized NS equation for the stationary cylinder is obtained by setting mesh velocity w = 0. After the discretization, the generalized eigenvalue problem of the linearized NS equation can be written as (Af + σ Bf )p = 0, where the non-symmetric matrices Af and Bf results from the spatial and temporal discretizations, σ denotes the eigenvalue of the discretized system, p is the right eigenvectors (forward modes). The corresponding discrete adjoint problem can be obtained as q(Af + σ Bf ) = 0, where q is the left eigenvector of the discrete system and represents the approximation of

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10 Data-Driven Passive and Active Control

the adjoint modes [131]. The linearized NS equations are solved by the semi-discrete variational procedure, which is employed for the nonlinear fluid-structure equations in [181, 255]. The fluid-structure coupling is achieved through a partitioned staggered procedure [178]. We next present the ERA-based ROM via input-output dynamics of FOM.

10.5.2 Feedback Control Via Reduced-Order Model The ERA-based ROM is constructed by linear input/output dynamics of the NS equations given in Eq. (9.1) and Eq. (10.28). The input vectors u = [Y, Vc ]T for the fluid system are transverse displacement Y and the suction and blowing vertical velocity Vc , while the output is the total lift coefficient Cl over the structural body. The fluid ERA-based ROM formulated in the state-space form at discrete times t = k t, k = 0, 1, 2, ..., with a constant sampling time t reads xf (k + 1) = Axf (k) + Bu(k) Cl (k) = Cxf (k) + Du(k)

 (10.33)

where xf is an n r -dimensional state vector, and the integer k is a sample index for the time stepping. The system matrices are (A, B, C, D), which are obtained by ERA method. To construct the ERA-based ROM, the impulse response of NS equation is first defined as y, based on which the generalized block Hankel matrix r × s can be constructed as ⎡ ⎤ yk+1 yk+2 ... yk+s ⎢ yk+2 yk+3 ... yk+s+1 ⎥ ⎢ ⎥ (10.34) H(k − 1) = ⎢ . ⎥, .. .. .. ⎣ .. ⎦ . . . yk+r yk+r +1 ... yk+(s+r −1)

and by applying singular value decomposition (SVD) of Hankel matrix H(0) as ∗

H(0) = Uσ V = [U1



σ1 0 U2 ] 0 σ2



V1∗ V2∗

 (10.35)

where the diagonal matrix  are the Hankel singular values (HSVs). The block matrix 2 contains the zeros or negligible elements. By truncating the dynamically less significant states, we estimate H(0) ≈ U1 σ 1 V1∗ . The state space matrices (A, B, C, D) are obtained by −1/2 −1/2 ⎫ A = 1 U1∗ H(1)V1 1 ⎪ ⎪ ⎬ 1/2 B = 1 V1∗ Em (10.36) 1/2 ⎪ C = Et ∗ U1 1 ⎪ ⎭ D = y1

10.5 Feedback Control of VIV via Jet Blowing and Suction

637

    Here, Em ∗ = Iq 0 q×N , Et ∗ = I p 0 p×M , where N = s × q, M = r × p, and I p,q are the identity matrices. The matrices B and D can be rewritten as B = [BY , BVc ] and D = [DY , DVc ], where the subscripts denote the input components defined in the vector u. The ERA-based ROM is constructed in the vicinity of a given base flow at t = 0 (k = 0), and the impulse signal starts from t = t (k = 1). The VIV system is simplified to a transversely vibration circular cylinder with one degree-of-freedom (1-DOF), and the nondimensional structural equation in the state-space form is written as [467] x˙s = As xs + Bs Cl .

(10.37)

The state matrices and vectors are       Y 0 0 1 , B , x As = = = , s s 2 −(2π Fs )2 −4ζ π Fs Y˙ ∗ πm where Y is the transverse displacement, Cl is the lift coefficient, Fs is a nondimensional reduced structural frequency defined as Fs = f N D/U = 1/Ur . Based on the ERA-based ROM, the resulting closed-loop system for VIV can be formulated by coupling of the structure Eq. (10.37) and the fluid system Eq. (10.33) as xsf (k + 1) = Asf xsf (k) + Bsf uc (k) ysf (k + 1) = Csf xsf (k) + Dsf uc (k)

 (10.38)

where 

 Asd + Bsd DY Csd Bsd C = , Asf A BY Csd (nr +2)×(nr +2)  Csf

3×(nr +2)

=



0 Bsd DVc Bsf = 0 BVc (nr +2)×2



   I 0 0 0 . , Dsf = DY Csd C 0 DVc 3×2

The structural time discrete matrices are defined as Asd = eAs t , Bsd = As −1 (eAs t − I)Bs and Csd = [1, 0], where the state vector xsf = [xs , xf ]T is a (n r + 2)dimensional vector, and the output vector is defined as ysf = [xs , Cl ]T . The present ERA-based ROM allows to provide the access to most controllable and observable modes of the coupled fluid and structure system. With regard to control design, the ROM is valid in the neighbourhood of the unstable steady state. As shown in Fig. 10.40, the optimal feedback control based on the linear quadratic regulator (LQR) is utilized to calculate the optimal gain matrix K such that the state-feedback law uc (k) = −Kxsf (k) minimizes the quadratic cost function for the discrete system as ∞   ∗ J= xsf Qxsf + uc∗ Ruc (10.39) k=1

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Fig. 10.40 Feedback control of VIV using reduced-order model: schematic of closed-loop control G C L with ERA-based ROM, where δ and K represent impulse input and gain matrix, respectively, Kalman denotes the filter defined in Eq. (10.40)

where Q and R set the relative weights of state deviation and input usage, respectively, and the asterisk ∗ denotes the transpose of the matrix. The Q is chosen as the identity matrix I for simplicity, whereas R = cI > 0 provides input to the cost function J . The coefficient c > 0 gives a relative weighing of output and input norms and can be tuned for an optimal tradeoff between the efficiency of VIV regulation and the energy input effort. The control input is defined as the blowing and suction through vertical velocity magnitude uc = (0, Vc ) over the surface of a two-dimensional cylinder. For the LQR algorithm, the full state vector xsf (k) should be observable to calculate the control input uc , while it is not always possible in practice. By assuming the measurements of force, velocity or acceleration on the body-mounted sensors, we can estimate the full state vector via the Kalman filter as xˆ sf (k + 1) = Asf xˆ s f (k) + Bsf uc (k) + L[ysf (k) − Csf xˆ sf (k) − Dsf uc (k)] (10.40) where xˆ sf is the minimum mean-square estimator of the state vector and L is the filter gain matrix. After that the control input vector uc can be determined as uc (k) = −Kxˆ sf (k), resulting a closed-loop VIV system as given in Eq. (10.38). Although the formulation presented here is general for any vibrating structure, results are presented for a canonical bluff body of circular cylinder. We next demonstrate the designed controller for the unstable flow past a stationary circular cylinder and the feedback control of a freely vibrating cylinder.

10.5 Feedback Control of VIV via Jet Blowing and Suction

639

y

4 0 2

-4

0

5

10

15

20

x

y

(a)

0

y

2 0 -2 -5

-2 0

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5

3

6

x (c)

Fig. 10.41 Spatial distribution of flow field: a forward, b adjoint velocity amplitudes, and c wavemker region for circular cylinder at Re = 60. In (c), actuation slots as triangles and bodymounted force sensor as red line over the cylinder are shown for B S0 configuration. In (a) and (b), Contour levels are from 0.002 to 0.018 in increment of 0.002. In (c), contour levels are from 0.02 to 0.2 in increment of 0.01

10.5.3 Active Feedback Control of Cylinder Unsteady Wake Flow We first demonstrate the feedback control scheme for the flow past a stationary circular cylinder at Re = 60 with B S0. The system input is the blowing and suction vertical velocity Vc , and the output is the fluctuating lift coefficient Cl . The corresponding fluid ROM can be obtained by setting BY = 0 and DY = 0 in Eq. (10.33). To start with the ERA-based ROM construction, the unstable steady state (i.e. the base flow) is first computed by a fixed point iteration without the time dependent term in Eq. (9.1). For that purpose, 800 impulse response outputs (Cl ) are stacked at each time step t = 0.05 by imposing an impulse of δ(t) = 10−4 to the blowing and suction vertical velocity Vc . Subsequently, the ERA-based ROM is obtained by performing a singular value decomposition of a 500 × 200 Hankel matrix. The order of ERAbased ROM is determined by examining the singular values of the Hankel matrix. The linearity of the impulse response outputs (Cl ) is confirmed by comparing the impulse response of two different values δ(t) = 10−4 and 10−3 . To visualize the most excited flow structures, the leading POD mode shown in Fig. 10.42, is extracted via the proper orthogonal decomposition (POD) method. For this purpose, the snapshots of the flow field are stacked at each time step during the impulse response modeling. The aforementioned ERA-based ROM construction procedure can be considered as an open-loop identification process. Based on the wavemaker region [131], we determine the locations with high sensitivity and strong response, which are computed by taking the pointwise product of the forward and adjoint global modes as shown in Fig. 10.41a, b. The modes

10 Data-Driven Passive and Active Control 4

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8

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-4

18

-2

3

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x

x

(a)

(b)

18

Fig. 10.42 Leading POD mode at Re = 60: a streamwise velocity, and b the cross-stream velocity. Contour levels are from −0.01 to 0.01 in increments of 0.0025

Cl

0.005 0

-0.005 0

50

100

150

200

tU/D (a)

FOM ROM (nr=33)

Vc

0.02 0 -0.02 0

50

100

150

200

tU/D (b)

Fig. 10.43 Impulse response of stationary cylinder with B S0 configuration: temporal variation of a lift coefficient Cl , b control input Vc predicted by ROM and compared with the FOM at Re = 60 and c = 102 . The controller is switched on after tU/D = 50 convective time units

are obtained directly by solving a generalized eigenvalue problem of the linearized NS Eq. (10.31) in the neighborhood of the base flow. As shown in Fig. 10.41c, the body-mounted force sensor and the suction and blowing actuators are within the wavemaker region, which allows the feedback control to produce the largest drift of unstable observable and controllable modes. Once the ERA-based ROM is constructed, the LQR method is employed to design the optimal feedback gain K for the different values of weighting parameters c = 102 , 103 , 104 . As shown in Fig. 10.40, the closed-loop model can be directly constructed through the ERA-based ROM. The effectiveness of closed-loop ROM with c = 102 is examined by comparing temporal variations of Cl and Vc against the FOM counterpart, as illustrated in Fig. 10.43. An excellent match is found between the ROM with the number of modes n r = 33 and the FOM, thereby confirming the accuracy and convergence of the present ERA-based ROM. The impulse response of the open-loop system is rapidly attenuated once the controller is switched on at t  50.

10.5 Feedback Control of VIV via Jet Blowing and Suction

641

Cl

0.2

OL 2 c=10 c=103 c=104

4

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Im( )

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c=10 3 c=10 4 c=10

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-2 (c) 1

Vc

-4 -2

-1.5

-1

-0.5

Re( ) (a)

0

0 -1 0

100

200

tU/D

(d)

Fig. 10.44 Feedback control of stationary cylinder at Re = 60 with B S0: a eigenspectrum for open and closed-loop with different values of c, time variation of b Cl , c Cd (base flow drag subtraction), and (d) control input Vc corresponding to closed-loop response of full nonlinear system. While the system has an impulse at t = 0, the controller is switched on after tU/D = 175.5

The eigenvalues are calculated by λ =log(eig(Asf − Bsf K))/ t, where t = 0.05 is used for all the computations. Figure 10.44a shows the comparison of eigenvalue distribution between the open-loop (OL) and closed-loop (CL) systems. As expected, a pair of the eigenvalue of an open-loop system is at the unstable right half-plane as Re = 60 > Recr (where the critical Reynolds number is Recr ≈ 46.8). On the other hand, the eigenvalues of the closed-loop system are all in the stable left half-plane, which further confirms that the closed-loop system is stable via the process of placing the poles in the stably-damped locations in the complex plane. As c decreases, Fig. 10.44a also indicates that the eigenvalue moves further leftward and the closed-loop system becomes more stable. However, a smaller value of c introduces a larger control input Vc thus there is a tradeoff between the aggressive control input and the rapid suppression of wake instability. To demonstrate the AFBS controller to stabilize the saturated vortex street, Fig. 10.44b, c show the effective attenuation of the fluctuating lift Cl and the drag Cd with the control turned on at tU/D = 175.5. It is expected that c = 104 requires the smallest control input Vc and the longest time scale to reach the target state, followed by c = 103 and c = 102 . The corresponding values of Cμ are 0.086, 0.22 and 0.52 for c = 104 , 103 , 102 , respectively. The flow field evolution during the suppression process is illustrated in Fig. 10.45.

10 Data-Driven Passive and Active Control 2

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Fig. 10.45 Stabilization effect on the vortex shedding of stationary circular cylinder at Re = 60 with B S0 and c = 102 . Snapshots of spanwise vorticity contours at: tU/D = a 175, b 225, c 250, d 325. Contour levels are from −1 to 1 in increments of 0.1

10.5.4 Sensitivity Study for Unsteady Wake Flow Control Before proceeding to the application of the proposed ERA-based active jet control for the VIV system, we present a parametric study of a set of representative actuator configurations based on suction/blowing pairs, the effect of the angular arrangement of actuators, and the combined versus independent control system architectures.

10.5.4.1

Effect of Actuator Configurations

To compare the effectiveness of the reference case B S0, we consider three additional configurations as depicted in Fig. 10.46 to analyze the active feedback control based on ERA-based ROM. The configuration B S1 has only midward actuation slots, whereby the configuration B S2 has the slots on the leeward side. By removing windward slots from B S0 at θ = (135◦ , 225◦ ), the new configuration of B S3 is recovered. Figure 10.47a shows the comparison of the eigenvalues of closed-loop between B S0 and other three actuation configurations with c = 102 . While B S0, B S1 and B S3 provide similar damped eigenvalues, the configuration B S2 is the least effective actuator, which is further confirmed by the closed-loop response of the full nonlinear system in Fig. 10.47b, c. The figures also show that the fastest suppression of vortex street is achieved by B S0, while the B S2 actuator takes the longest time to eliminate vortex shedding. On the other hand, the actuation configurations of B S1 and B S3 behave similarly with respect to the suppression of the vortex street. The baseline B S0 with θ = 45◦ is considered to be the most effective actuator configuration. By comparing B S1 and B S2 configurations, the midward B S1 actuation slots have an improved control performance than the leeward B S2 configuration. This implies that

10.5 Feedback Control of VIV via Jet Blowing and Suction

643

Fig. 10.46 Additional three (B S1, B S2, B S3) configurations of actuators as blowing and suction slots over the surface of circular cylinder. The positive control input is defined as suction from bottom and blowing at the top of the cylinder. While B S1 configuration forms a symmetric configuration of suction-blowing pairs, B S2 and B S3 are asymmetric with respect to the quadrants of cylinder

Cl

0.2

BS0 BS1 BS2 BS3

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Im( )

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0.1 0 0

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(c)

-4 Vc

1

-1

-0.5

Re( ) (a)

0

0 -1 0

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tU/D (d)

Fig. 10.47 Effect of the placement of fluidic actuators based suction-blowing pairs on the cylinder surface: a distribution of eigenspectrum of closed-loop system for different actuation slots placements (B S0 − B S3), temporal variations of fluctuating b Cl , c Cd , and d control input Vc for the closed-loop response of full nonlinear system at (Re, c) = (60, 102 )

the active suction/blowing control in the boundary layer (before the separation) is relatively efficient. Based on the above study, the configuration B S0 is found to be more effective in reducing mean drag and suppressing fluctuating forces. Next, we investigate the effect of suction/blowing angle θ in the baseline B S0 configuration.

644

10.5.4.2

10 Data-Driven Passive and Active Control

Effect of Angle for Suction-Blowing Pairs

In this section, the sensitivity of the actuation angle for B S0 is investigated by varying it from the baseline θ = 45◦ to θ = (30◦ , 60◦ ). We keep the midward pair of suction/blowing actuation at θ = (90◦ , 270◦ ), but we only change other two diametrically pairs of suction and blowing actuations. The distribution of eigenspectrum, as shown in Fig. 10.48a, suggests that B S0 configuration with θ = (30◦ , 45◦ , 60◦ ) provides similar effectiveness with regard to the least damped eigenvalues. Next, the configuration B S0 with a different angle θ is applied to stabilize the vortex shedding at the nonlinear saturated state. The results in Fig. 10.48b–d show similar suppression trends for the lift Cl and the drag Cd signals and the control input Vc was obtained for the three angles. Figures 10.48b and 10.48c also show that B S0 with θ = 60◦ has a slightly larger overshot when the controller is switched on followed by θ = 30◦ and θ = 45◦ . In all our previous studies, the combined controller is designed as the macro-manipulator for controlling the global dynamics. In other words, the actuators are not allowed to vary their control input speed Vc independently, therefore the same control input Vc in all suction-blowing pairs is generated. It is interesting to study the effect of control system architecture, whereby there is no coupling between the suction/blowing subsystems and the actuators can generate independent control input.

10.5.4.3

Combined Versus Independent Controller

To understand the effect of control architecture, we decouple the pairs of suction/blowing actuators and compare the performance against the combined controller counterpart. The decoupled controller configuration with different blowing and suction velocities, termed as B S0D, is shown in Fig. 10.49. As compared with the combined controller configuration B S0 with single DOF control input velocity Vc , the controller of B S0D has three independent control inputs Vc1 , Vc2 and Vc3 . As demonstrated in Fig. 10.50a, the combined configuration B S0 is more effective to damp the unstable eigenvalues as compared to the decoupled B S0D counterpart. Moreover, the configuration B S0D is found to be lesser sensitive for the least damped eigenvalues for the identical range of c values. To further assess the performance of B S0 and B S0D controllers, Fig. 10.50b, c illustrate the nonlinear saturated state suppression through the force time histories. After removing the same matching control input constraint in the LQR algorithm for B S0D controller, the force trends suggest that the controller becomes less effective when using different blowing and suction velocities at the same value of c. It is further confirmed in Fig. 10.50d, which shows that the control velocity inputs (Vc1 , Vc2 , Vc3 ) become smaller at the same c = 102 , resulting in a larger time for the suppression of vortex street. As shown in the previous section, the baseline B S0 configuration with θ = 45◦ is the most effective suction/blowing controller configuration. Therefore, we will employ this configuration for the active feedback control of VIV. As reported in [326], passive control techniques that work well for suppressing loads for fixed cylin-

10.5 Feedback Control of VIV via Jet Blowing and Suction

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(d)

Fig. 10.48 Effect of angle of fluidic actuators on the cylinder surface: a distribution of eigenspectrum of closed-loop system for different actuation slots placements, time variations of fluctuating b Cl , c Cd , and d control input Vc for the closed-loop response of full nonlinear system at (Re, c) = (60, 102 )

Fig. 10.49 Schematics of combined and independent (decoupled) suction/blowing control system architectures. In contrast to the combined controller B S0 (left), B S0D (right) partitions the controller pairs into different pieces with different blowing and suction velocity denoted by Vc1 , Vc2 , and Vc3 , respectively

646

10 Data-Driven Passive and Active Control BS0 BS0D

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(c)

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(d)

Fig. 10.50 Comparison between combined versus decoupled control system architectures: a distribution of eigenspectrum. c = 102 for B S0, and c = (10, 102 , 103 ) for B S0D, time variations of fluctuating b Cl , c Cd , and d control input Vc for the closed-loop response of full nonlinear system at c = 102 and Re = 60. In (c), the vector (Vc1 , Vc2 , Vc3 ) is the control input of the B S0D configuration

ders may not be effective for elastically mounted configurations. We next demonstrate our proposed active feedback control scheme for elastically mounted circular cylinders, which are free to vibrate in transverse only (1-DOF) and coupled streamwise/transverse directions (2-DOF).

10.5.5 Feedback Control of Vortex-Induced Vibration In [466], we discussed that the onset of VIV lock-in is related to the instability exchange between the structural mode (SM) and the fluid mode (WM), where the SM and WM represent two distinct eigenvalue branches of the ERA-based VIV ROM system. The critical reduced natural frequency Fs or onset reduced velocity Ur = 1/Fs can be pinpointed by the SM instability branch. The objective of the present active feedback control is to drive all unstable eigenvalues of the VIV system to the stable left half complex plane. While the open loop VIV system (without feedback control) can be obtained by simply setting uc = 0 in Eq. (10.38), the optimal gain is computed for the VIV system at (Re, m ∗ ) = (60, 10) using the similar procedure as discussed for the stationary case.

10.5 Feedback Control of VIV via Jet Blowing and Suction

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Cl

0.01 0 -0.01

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0.005

FOM ROM (nr=32)

0

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tU/D (c) Fig. 10.51 Impulse response of closed-loop VIV system based on AFBS control: temporal evolution of a lift coefficient Cl , b transverse displacement Y , and c control input Vc predicted by ROM and compared with the FOM for (Re, m ∗ ) = (60, 10), Fs = 0.176, and c = 102 with controller switched on at tU/D = 0

As shown in Fig. 10.51, the closed-loop VIV system response subject to an impulse is completely stabilized and the ROM results nearly overlap with the FOM counterpart. The linear stability is then determined by the eigenvalue analysis for both open- and closed-loop systems, as illustrated in Fig. 10.52a. Both the WM and a part of SM (0.147 < Fs ≤ 0.179) are unstable (Re(λ) > 0) for the open-loop system, while the eigenvalue of the closed-loop system are all in the stable left half-plane for 0.005 ≤ Fs ≤ 0.5 (2 ≤ Ur ≤ 200) with c = 102 , 103 . The results show that the unstable eigenvalues are driven to the lower stable left half-plane as the coefficient c decreases, as indicated by the dash-dot arrow in Fig. 10.52a. It is important to note that the closed-loop VIV system with c = 104 only delays the VIV onset and remains unstable for Fs ≤ 0.117 (Ur ≥ 8.55), which indicates that the amplitude may grow continually as Fs decreases. In Fig. 10.52b, the root loci of different mass ratios are plotted indicating the unstable eigenvalues of lower mass ratio damped more effectively with c = 103 . The controller with c = 103 is also able to suppress the saturated VIV response as shown in Fig. 10.53. The lift coefficient Cl and the transverse displacement Y are

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10 Data-Driven Passive and Active Control

SM Fs=0.179 WM Im( )

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Im( )

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*

m*=5 m*=10 m =15 m*=20 m*=100

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Fig. 10.52 Root loci of open and closed-loop VIV systems (0.005 ≤ Fs ≤ 0.5): a sensitivity of c parameter on eigenspectrum at (Re, m ∗ ) = (60, 10), b effect of mass ratio m ∗ on the control performance at (Re, c) = (60, 103 ). The unstable right half-plane is shaded in gray. The parameter c ∈ [102 , 103 ] ensures the VIV free system for the whole range of reduced frequency

-2

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Fig. 10.53 Suppression of VIV response of circular cylinder using AFBS procedure: time variation of a Cl for saturated (thin-dash) and suppressed (thick-solid) states, b transverse displacement Y and actuator input Vc with controller switched on at tU/D = 150, c = 103 and Fs = 0.176, snapshots of spanwise vorticity contours at: tU/D = c 155, d 400, (e) 600, (f) 800. Contour levels are from −1 to 1 in increments of 0.1

10.5 Feedback Control of VIV via Jet Blowing and Suction

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tU/D (d) Fig. 10.54 Open loop saturated state of 2-DOF VIV system at (Re, m ∗ ) = (60, 10): a figure-8 trajectory of X and Y displacements, b normalized power spectrum P versus frequency f ∗ of lift coefficient Cl and transverse displacement Y , where f ∗ = f /Fs . Suppression of 2-DOF VIV response: c transverse Y and inline X response trends, d lift coefficient Cl and actuator input Vc . The controller is switched on after tU/D = 150 with c = 103 and Fs = 0.176

10 Data-Driven Passive and Active Control 2

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Fig. 10.55 Adaptive feedback control of 2-DOF VIV of circular cylinder at (Re, m ∗ ) = (60, 10). Snapshots of spanwise vorticity contours at: tU/D = a 175, b 200, c 400, d 500. Contour levels are from −1 to 1 in increments of 0.1 Table 10.9 Comparison of control inputs among stationary cylinder, 1-DOF VIV system and 2DOF VIV with B S0 at c = 103 Control input Stationary 1-DOF 2-DOF (Vc )max (Cμ )max (Vc )r ms

0.65 0.22 0.10

2.91 4.40 0.77

3.00 4.70 0.79

The root mean square (r ms) and maximum (max) are computed for t ∈ [t0 , t0 + 200], and t0 is when the controller is switched on after t0 = 175.5 and 150 for stationary and VIV systems, respectively

suppressed once the controller is switched on after tU/D = 150, when the VIV response reaches the saturated state. As compared to the saturated vortex shedding state of stationary cylinder, which requires the maximum Vc ≈ 0.65, the VIV case requires approximately 4.5 times larger maximum control input Vc ≈ 2.91 with Cμ ≈ 4.40. Figures 10.53c and 10.53d show snapshots of the spanwise vorticity contours to illustrate the attenuation of vortex shedding when the controller is switched on t  150. We can infer from the success of the proposed linear AFBS for VIV that there exists a key linear mechanism during the self-sustained VIV oscillation that can be controlled effectively at low Reynolds numbers. Nonlinear effects of the fluid flow, which attempt to saturate vortex shedding and form a limit cycle in VIV, become dominant in the later stage. It is worth pointing out that identifying and controlling the linear mechanism in nonlinear VIV is not equivalent to predicting nonlinear VIV with a linear model. To demonstrate whether the designed controller for the 1-DOF VIV system can be utilized for 2-DOF VIV system feedback control, we introduce free vibration in the streamwise (X ) direction along with the transverse (Y ) direction. A typical figure8 VIV saturated trajectory is shown in Fig. 10.54a at Fs = 0.176. The frequency

10.5 Feedback Control of VIV via Jet Blowing and Suction

651

plot shown in Fig. 10.54b confirms the frequency lock-in occurrence. As shown in Fig. 10.54c, d, similar suppression is achieved for the 2-DOF VIV system with the same controller designed for 1-DOF VIV system (Fig. 10.53). As compared with 1DOF, which requires a maximum Vc ≈ 2.91, the same controller requires a maximum Vc ≈ 3.0 for 2-DOF VIV saturated response suppression or approximately 3% larger than its 1-DOF VIV counterpart. Figure 10.55 further illustrates the similar flow field evolution during the suppression process for the 2-DOF system. Overall, the results suggest the proposed feedback control scheme based on vertical blowing/suction is able to suppress the 2-DOF VIV system effectively with approximately the same amount of control energy as the 1-DOF system. The control input is summarized in Table 10.9 for a stationary cylinder and VIV configurations with both 1-DOF and 2-DOF response at c = 103 . As shown in the table, the 2-DOF VIV system only requires a slightly larger control energy to suppress the saturated VIV state than its 1-DOF counterpart. The results suggest that the transverse response is the most dynamically significant for the isolated circular cylinder VIV system. Therefore, the proposed controller designed for the 1-DOF VIV system can be applied to reduce the 2-DOF VIV response.

10.5.6 Summary Through the ERA-based dimensionality reduction, an active feedback blowing and suction concept is proposed to suppress the vortex street and VIV for flexibly mounted structures. A variational finite element formulation has been considered for the full order fluid-structure model and the generalized eigenvalue problem of linearized Navier-Stokes system. The control scheme relies on the optimal feedback gain by the LQR synthesis and the state estimation by the Kalman filter. Based on the combined vertical blowing and suction, we employed the feedback control to obtain suitable gains that stabilize the wake instability and the vortex-induced vibration. We found that the essential elements in the self-sustaining VIV process are linear, and are subject to active feedback control. From the ERA-based computations and feedback control of unstable eigenmodes, it can be deduced that the increase in the VIV amplitude of the cylinder occurs primarily through the linear instability process. By applying the AFBS procedure to the vortex shedding of the stationary cylinder, we observed a remarkable reduction in the time-dependent fluctuating components of both lift and drag forces. By means of eigenspectrum distributions via ERA-based simulations, we have explored several configurations to confirm the generality and sensitivity of the AFBS procedure with respect to a range of parameters. While the configuration BS0 with six actuation slots performed efficiently, followed by B S1, B S3, the configuration B S2 exhibited a poor performance as having the actuators on the leeward side of the cylinder. In general, the performance of B S0 configurations is not much sensitive to the jet angle θ . When applying for the nonlinear saturated state, a similar suppression is achieved with θ = (30◦ , 45◦ , 60◦ ), while a slightly larger overshot is found when the controller is switched on for the angle θ = 60◦ .

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10 Data-Driven Passive and Active Control

The performance of B S0 becomes less effective when removing the same control input velocity constraint, thereby suggesting an improved performance of combined control architecture than the decoupled independently designed control system. With respect to the VIV suppression, the designed controller has performed well for a range of mass ratio m ∗ ∈ [5100] and the reduced natural structural frequency Fs ∈ [0.005, 0.5] at Re = 60. In contrast to the stationary vortex shedding, the suppression of VIV requires about four times large control input for the same Reynolds number Re = 60 to suppress both the fluctuating lift and the VIV amplitude. The controller designed for transversely VIV system can be also adapted to 2-DOF VIV suppression with only approximately 3% larger control input than the transverse 1-DOF VIV system. Since the present reduced-order model does not require any adjoint solver, the proposed ERA-based feedback control can be directly used for actual physical problem and experimental setting. The ERA-based ROM approach is relatively straightforward and computationally efficient. Furthermore, the model offers a reasonable accuracy for the development of stabilized feedback controller design for unstable flows and VIV. Acknowledgements Some parts of this Chapter have been taken from the PhD thesis of Sandeep Reddy Bukka carried out at the National University of Singapore and supported by the Ministry of Education, Singapore, and the MASc thesis of Amir Chizfahm at the University of British Columbia supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).

Part III

Flow-Induced Vibration of Thin Structures

Chapter 11

Introduction

Interactions between unsteady fluid flows and flexible thin structures are widely observed in our daily life and have numerous applications in engineering. Some common examples of fluid-body interactions include fish swimming, bird and insect flight, fluttering of flags and leaves in the wind flow. When a cantileveed flexible sheet or panel is aligned with a fluid flow, it can undergo self-induced flapping oscillations or fluttering. The intrinsic feedback process between ambient fluid and the deformation of a streamline flexible body forms a coupled nonlinear dynamical system. The complex interactions between the flexible thin structures and their surrounding fluid flow are important not only because they represent a rich coupled dynamical phenomena but also due to their wide range of practical applications. For example, such coupled dynamical effects have applications in the areas of energy harvesting devices, efficient propulsive devices, flow separation control, drag reduction and bio-prosthetic devices such as heart valves. Differing from the flow-induced vibration phenomena covered for bluff-body structures in Volume 1, flexible thin structures can deform passively along their chordwise and spanwise directions and exhibit multiple vibration modes. Through the fluid-structure feedback process, the surrounding fluid flows can be adapted by the continuous deformations of the flexible thin structures. Natural flyers and swimmers are known to exploit the flowing fluid for their efficient locomotion by morphing or undulating their flexible bodies. It is important to understand the physical mechanisms that govern the fluid-structure interactions in fliers and swimmers. The benefits of fluid-structure coupling (e.g., efficiency and maneuverability) can be engineered in bio-inspired soft robotics via passive and active control. Moreover, the flow-excited instability of the flexible thin structures can cause flow-induced vibrations. While one can harness the flow-induced flapping vibrations for electric power via piezoelectric materials, such vibrations should be suppressed to avoid structural failure and fatigue in aerospace, civil and marine engineering systems. This volume aims to provide a survey of mathematical modeling and numerical simulations of flowinduced vibration in flexible thin structures. The physics of conventional flapping © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Jaiman et al., Mechanics of Flow-Induced Vibration, https://doi.org/10.1007/978-981-19-8578-2_11

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11 Introduction

foils, inverted flapping foils, proximity and trailing edge effects on foil flapping, and flexible thin-structure aeroelasticity are systematically reviewed. We employ a robust and accurate body-fitted variational framework for analyzing the fluid-structure interactions. We provide physical insights into the mechanisms of flow-induced vibration and synchronization of flexible structures.

11.1 Background and Literature Review The interaction of a flexible thin structure with the surrounding fluid represents a strong fluid-structure interaction problem. The flexible structure undergoes a continuous deformation due to the fluid dynamic forces, which in turn alters the flow field around it and thereby affects the forces acting on the structure. In nature and engineering systems, one can find different kinds of flexible thin structures with various boundary conditions, geometries, spatial arrangements, and material properties. These parameters govern the coupled fluid-structure dynamics and manipulate the performance of flexible structures. Flying animals and swimming fishes can actively change the geometry shapes, flexibility and flapping kinematics of their bodies and adapt to complex environments to achieve optimal performance. The leaves of different trees have evolved in varying spatial arrangements and geometrical shapes, which are connected to living environments and local wind conditions. The underlying physical questions behind these phenomena can be associated with the selfexcited instability of the flexible structures. The understanding of the flow-induced vibration of flexible structures can facilitate the development of optimized design and intelligent control strategies to use the power of flexibility in engineering systems. For example, the interaction between a flexible thin structure and the surrounding flow field can be used to harvest electric energy from the fluid kinetic energy [602, 738]. The interaction between the wind and leaves influences the orientation of the leaves on branches [730] which in turn affects the photosynthesis rate and water inception in the leaves [717]. Even large civil and offshore structures undergo flexible deformations due to interactions between wind/current and the structures [56, 666]. Birds and insects generate a propulsive force and transverse lift by flapping their flexible wings in the air stream as shown in Fig. 11.1a. These unsteady flapping motions lead to a continuous transfer of vortices into the wake of the body. Similarly, fishes utilize their flexible bodies to perform an undulatory motion to swim in the water and they can synchronize their motions with the oncoming vortices. These animals not only use active flapping motion but also adjust their flexible body passively to generate an improved performance of thrust and lift [526, 573, 771]. While flapping flags and fluttering leaves (Fig. 11.1b) involve only passive and intrinsic interactions [665] without any external energy input, birds spend their energy flapping or rotating their flexible wings during the active flapping motion. The coupled dynamics and the unsteady flow features of bluff bodies and streamlined flexible thin structures share some commonalities, while some significant differences exist in these coupled fluid-structure systems. Figure 11.2 presents illustra-

11.1 Background and Literature Review

657

(a)

(b)

(c) Fig. 11.1 Three common examples of interaction between fluid flow and flexible structure that can be observed in nature: a birds perform a combination of active and passive flapping of their flexible wings to fly, b fluttering of leaves due to the passive interaction in a wind and c swimming fishes. The images are taken from Wikipedia

tions of the vortex patterns behind a vibrating circular cylinder, swimming fish and a flapping flag. As discussed in Volume 1, the elastically supported bluff body can exhibit flow-induced vibration via a feedback coupling process with the vortex shedding. With regard to the vortex-body interaction, one can relate this problem with a fluttering of a thin structure or flag in a flowing fluid. There exist a synchronized motion of the structure with the shed vortices in the wake. When swimming fish and flapping flag experience large-amplitude oscillation, vortex patterns similar to those of the bluff body are produced behind a flapping structure. The onset of the flowinduced vibration and the flow features of flexible structures show some similarities to the vibrating bluff body. However, the coupled dynamics of the flexible structures exhibit fundamental differences from the bluff body due to their geometrical shapes, the wake dynamics and the continuously deforming surface. The flexible body can possess multiple structural modes and exhibit deformed shapes in the unsteady fluid flows through the coupling effect. These features allow the flexible thin structures to manipulate the unsteady fluid flows by morphing their shapes. Owing to a feedback coupling between a flowing fluid and a flexible structure, one can observe self-excited vibration or flutter by simply blowing air over a thin piece of paper while holding it in a hand [605, 767, 768]. The self-excited flapping motion of the flexible foils immersed in flowing fluid occurs at sufficiently high fluid

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Fig. 11.2 Illustration of vortex patterns behind a spring-mounted cylinder, swimming fish and flapping flag

Fig. 11.3 Biomedical applications of passive fluid-body interactions: a snoring due to flapping of soft palate, b blood flow from the heart is regulated by flexible tissues such as atrioventricular valve. The images are taken from Wikipedia

velocities due to a complex nonlinear interaction between the two fields. Particularly, a frequency synchronization of the vortex shedding frequency ( f vs ) and the frequency of the structure vibrations ( f s ) may be established in the coupled fluid-structure systems under specific conditions, resulting in the well-known frequency lock-in phenomenon [689, 750]. During the interaction, while the fluid-dynamic pressure and the structural inertia have destabilizing effects, the bending rigidity and viscous drag-induced tension provide stabilizing effects to the flapping motion. By proper harnessing of such flow-excited instability, one can regulate the structure kinematics and the vortex shedding features through the fluid-structure coupling effect [542, 594, 721]. Figure 11.3a presents a study on the flapping instability in a flexible plate clamped at its leading edge to investigate the snoring problem due to the flapping of a soft palate in the open-airways of the human respiratory system [605]. The study explored the effects due to traditional surgical techniques like trimming the length of the soft palate and proposed an alternative treatment technique of stiffening the soft palate to mitigate the problem of flapping. Other biological applications include the development of customized bio-prosthetic heart valves for the replacement of diseased ones (Fig. 11.3b). It is observed that the bio-prosthetic heart valve leaflets

11.1 Background and Literature Review

(a)

659

(b)

Fig. 11.4 Engineering applications of passive interaction of a flexible foil with surrounding fluid flow: a piezoelectric energy harvesting devices [513], b paper flutter in the paper printing industry [768]

exhibit flapping motion due to interactions between the blood jet and the valve leaflets while they are opening [535]. The interactions between the central jet flow and the valve leaflets play a significant role in defining flow structures which in turn affect the wall shear stresses in the aorta and the forces experienced by the blood cells. Figure 11.4a shows an investigation of the interaction between flexible piezoelectric membrane and the wake behind a bluff body wake and shows a periodic flapping motion of the piezoelectric membrane due to a vortex-induced instability. Such flapping motions can be used to harness fluid kinetic energy to obtain electric power. Figure 11.4b presents the flapping response, which limits the paper printing speed by tearing the paper [767, 768]. The flapping phenomenon over a wide range of paper sizes along with the quality is analyzed to improve the speed of paper printing without tearing or causing wrinkles. The coupled fluid-elastic interactions of a flexible thin structure in a micro-channel for its possible application to enhance the mixing were investigated [632]. More recently, the passive flapping dynamics have attracted a lot of attention for their application in the development of next-generation flexible propulsive devices. The ability of the combined active heaving with a passive flapping motion of a flexible foil to generate the thrust was widely explored [509, 604, 674, 685, 686, 734]. The combined motion of the foil is consistent with the anguilliform motion observed in eels. The efficiency of the heaving flexible foils decreases with an increase in flow speed and optimum propulsive efficiencies can be achieved for lower flow speeds with greater flexibility. The seminal works of [641, 642, 774] have shown that fishes exhibit traveling wave-like adulatory fin motion to generate the required thrust. [732] have shown that the phase velocity of such traveling wave-like motion should be greater than the freestream velocity to generate some thrust. More recently, [522] have considered similar traveling wave-like motion for a drag reduction in the underwater propulsive device.

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11.2 Specific Applications and Challenges Flying animals including insects, birds, and bats have evolved over millions of years to possess particular morphing and flapping wing systems to keep themselves aloft, generating thrust and maneuvering in complex environments [512, 539, 599, 652, 775]. Compared to a conventional fixed-wing flight vehicle, the flapping flight of flying creatures (e.g., bats and owls) involves active morphing and adaptive flexible as well as serrated wing configuration, which may offer some unique benefits in flight efficiency improvement, noise suppression, structural weight reduction and high maneuverability. In the past decades, there has been a large amount of research works towards the development of low-speed and small-sized vehicles, i.e., micro air vehicles. These small-sized vehicles represent the opposite end of the flight spectrum compared to most of the military and civilian aircraft that are currently in use. While a relatively small size of the MAV offers potential benefits in maneuverability, surveillance and operational robustness, the small size with low inertia make these systems difficult to operate concerning mid-air control and stability. The MAVs that possess the flight agility of natural flyers remain a challenge, especially in complex terrain with very confined spaces. Engineers and designers seek new sources of inspiration from bat flight to design novel air vehicles with light and controllable structures. A natural extension in the design of MAVs and UAVs is to develop a hybrid aerial-aquatic vehicle that can traverse in complex multiphase conditions and perform search-and-rescue operations, surveillance and environment exploration. Such a hybrid unmanned vehicle needs to handle locomotion in air and water while overcoming the challenges associated with the transition across the air-to-water interface (see Fig. 11.5). Proper analysis and design of such locomotion (flying and swimming) require flexible multibody and multiphase FSI capabilities. Flapping-wing vehicles and wing kinematics should be adapted to both air and water environments. Aerial vehicle demands a large wingspan to stay aloft, while aquatic vehicle needs to minimize the surface area to reduce the hydrodynamic drag. For example, to address such conflict, bio-inspirations from the flying of a flying-fox bat and the swimming of a bat ray can be considered to realize a hybrid aerial-aquatic locomotion. The evolution of the aerospace industry has motivated a large number of researchers to study coupled fluid-elastic instabilities. However, the focus of these studies [562, 563, 629, 646, 681, 682] was mostly limited to the self-excited motion of elastic panels clamped at both the leading and trailing edges, as shown in Fig. 11.6a. In these studies, the fluid flow is assumed to be compressible and interacts only on one side of the panel. Both analytical and experimental have revealed that the elastic panel undergoes either static deformation or limit-cycle oscillations depending on whether the flow is subsonic or supersonic respectively [563]. However, this panel motion for both the edges fixed is distinctly different from that observed on a flexible cantilevered plate. A cantilevered elastic panel that is clamped to the leading edge and with the free trailing edge presented in Fig. 11.6b exhibits flapping motion for subsonic flows and static divergence for supersonic flows [630]. A cantilevered flexible thin structure that is clamped to the trailing edge and with the free lead-

11.2 Specific Applications and Challenges

661

(a)

(b)

Fig. 11.5 Fluid-structure interactions of inverted elastic flags for energy harvesting: Large amplitude flapping of an inverted elastic flag with piezoelectric patches (left) and multiphase fluidstructure interaction of a conceptual hybrid avian-aquatic vehicle (right)

(a)

(b)

(c) Fig. 11.6 Schematic of a a flexible thin structure clamped at both ends, b a flexible thin structure clamped at the leading edge with its trailing edge free to vibrate and c a flexible thin structure clamped at the trailing edge with its leading edge free to vibrate

ing edge presented in Fig. 11.6c experiences static deformation, periodical flapping motion and chaotic flapping motion under different circumstances. The advancements made in the field of aeroelastic panel flutter and the impetus to understand the underlying dynamics of the complex interactions in the flexible deformable flaglike cantilevered structures with their surrounding incompressible fluid flow have motivated a considerable number of researchers in the last few decades. As mentioned earlier, fluid-structure interactions can exhibit complex coupled dynamics both in useful and destructive manners. In the context of sustainable clean energy, harvesting power from flapping and vibrating structures are gaining a resurgence of interest in recent years for various engineering applications. Coupled with piezoelectric materials, the flapping and vibrating structures are capable to transform wind and ocean kinetic energy into usable energy at small-to-medium scale systems. The efficiency and performance of these devices are strongly dependent on the precise

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11 Introduction

design and adaptive tuning of multiphysical properties (e.g., structures and piezoelectric materials) during nonlinear dynamic excitations. In this context, fully-coupled fluid-structure-piezoelectric modeling becomes crucial for the efficient design and control of such energy harvesting systems. An elastic foil interacting with a uniform flow with a clamped trailing edge (the inverted foil) is considered as a prototypical problem, which has a profound impact on the development of energy harvesting devices via self-sustained large-amplitude flapping (Fig. 11.5). By coupling with piezoelectric materials, the structural strain energy can be transformed into electric energy, which can be useful for low-powered onboard systems and sensors. To capture the nonlinear interaction of a flexible thin structure with a flowing fluid, there is a need for fully-coupled fluid-structure simulations. These coupled simulations based on the first-principle model system can provide profound insights into a broad array of fluid-elastic instabilities and flapping dynamics. Apart from the importance from a fundamental perspective, these simulations are relevant to applications such as energy extraction by the flow-induced flapping of flexible structures and flexibility-driven propulsive locomotions of organisms or mechanical devices. These nonlinear simulations are also critical from a structural integrity viewpoint for many engineering systems, such as the flutter of a thin elastic structure and the prediction of airplane wing stability, etc. These nonlinear fluid-structure simulations pose serious challenges in three dimensions over a wide range of parameter space.

11.3 Flow-Induced Vibration of Flexible Thin Structures Flexible thin structures lose their stability to statically deform or dynamically vibrate through the fluid-structure coupling effect. The coupled fluid-structure system exhibits various flapping dynamics under different boundary conditions, structural material properties and incoming flow conditions. The fundamental physical mechanics and the underlying mechanism of these various flow-induced vibration phenomena share some commonalities. Flow-excited instability and flexibility effects play a key role in the flow-induced vibration of thin flexible structures. The coupled dynamics of flexible thin structures are governed by several nondimensional parameters. In the following sections, we will introduce the flow-excited instability and the nondimensional parameters of flow-induced vibration of flexible thin structures.

11.3.1 Flow-Excited Instability and Synchronization When the unsteady fluid flows past the flexible thin structures, fluid loading is applied on the structure surface to force the structure to deform through the coupling effect. In turn, the surrounding flow features and the pressure distributions on the surface are altered due to the change in the structure profiles. Herein, we consider a flexible thin structure clamped at the leading edge with the free trailing edge as an example

11.3 Flow-Induced Vibration of Flexible Thin Structures

663

and present a schematic of the different flapping regimes as shown in Fig. 11.7 for demonstration. A flow-excited instability can arise and manifests itself as a selfsustained flapping motion of the structure when a flowing stream passes over the body surfaces, leaves the trailing edge and goes into the wake. This phenomenon includes complex dynamical effects such as relative fluid-structural inertial effects, vorticity generation along the foil surface, vortex shedding emanating at the trailing edge, restoring effects due to the bending rigidity and variable flow-induced tension along the foil. As we can observe from daily life, the leaves or flags will statically deform and then exhibit limit-cycle flapping dynamics when the wind speed becomes higher. Finally, the oscillation of the leaves or flags transitions to non-periodic flapping motion in the high-speed turbulent wind as shown in Fig. 11.7. The coupled dynamics of the flexible thin structure in fluid flow with different velocities can be governed by a key nondimensional parameter, namely the Reynolds number, which is defined as Re =

ρ f U∞ L μf

(11.1)

where ρ f is the fluid density and U∞ denotes the oncoming flow velocity. L represents the characteristic length of the flexible thin structure and μ f is the dynamic viscosity of the fluid. The Reynolds number is the ratio of inertial forces to viscous forces. Similar to the flow features and wake pattern behind a bluff body at different Reynolds numbers introduced in Volume 1, the fluid flows behind the flexible thin structure gradually become unstable and generate a shedding vortex. The flexible thin structures lose their static stability and produce flapping motion when coupled with unstable fluid flows, resulting in the flow-induced vibration phenomenon. Particularly, the synchronization is established between the vortex shedding frequency and the structure vibration frequency, resulting in the well-known frequency lock-in phenomenon. The flow-induced vibration of the flexible thin structures shares some common mechanisms with the bluff body. When the flexible thin structures are placed in the unsteady fluid at high angles of attack or oscillate in large amplitude, the flow features and the wake patterns behind the structures exhibit similar characteristics as those of the bluff body. It is worth noting that some inherent differences can be distinguished between the coupled systems of the flexible thin structures and the bluff body. The flexible thin structures possess infinite numbers of natural modes and frequencies. The flexible thin structures can vibrate in a single mode or multiple modes through flow-excited instability. The flow features and the wake patterns become more complex when multiple vibration modes are excited. These phenomena mentioned above greatly increase the difficulty of physical analysis of the flow-excited instability for different kinds of flexible thin structures. In flow-induced vibration of the flexible thin structures, as the physical parameters vary, the coupled system can transition between a fixed point state and a flapping state. Figure 11.8 presents a stability phase diagram of a flexible structure with clamped leading edge immersed in unsteady flow flows. During fluid-structure interaction, the competition and balance between the aerodynamic force, the elasticity force and the

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Fig. 11.7 Schematic of the different flapping regimes with qualitative vorticity contours

Fig. 11.8 Stability phase diagram of a flexible thin structure clamped at the leading edge

inertia force can finally stabilize or destabilize the coupled system. The flow features associated with the structural modes are altered during the transition process, thereby influencing the aerodynamic performance and the stability of the coupled system. As presented in Fig. 11.7, the wake behind the fixed-point stable structure exhibits a steady velocity deficit. In the limit-cycle flapping regime, the structure vibrates periodically in a traveling wave-like manner. The wake exhibits a typical vortex street associated with drag production and the power spectrum shows a distinct peak for the frequency of oscillation, that is frequency synchronization. Eventually, when the flow velocity exceeds a certain threshold, these oscillations become chaotic. The resulting power spectrum exhibits multiple frequencies and the wake pattern is irregular. The transition from the fixed point state to limit-cycle oscillation to non-periodic flapping motion is closely related to the flow-excited instability and the synchronization. The

11.3 Flow-Induced Vibration of Flexible Thin Structures

665

understanding of the flow-excited instability and synchronization in coupled fluidflexible thin structure systems can provide deeper insight into the mechanism of the flow-induced vibration of flexible thin structures. Passive flexibility is one of the practical manners to achieve the morphing function [637]. Flexible thin structures possess various degrees of flexibility along the chordwise and spanwise directions [566, 711]. For example, a flexible membrane can deform up and change its camber and twist under aerodynamic loads due to the chordwise and spanwise flexibility. These features enable the morphing structures to adapt to different operating conditions while maintaining better performances just with one type of wing structure, compared to the traditional rigid structures designed for a specific operational mission. During the flapping flight, the flexibility effect can amplify the structural motion and achieve a larger twist, which helps orienting more generated force components along the forward direction to produce a larger thrust. Generally speaking, the flexibility behaves like a coordinator to correlate the unsteady fluid flows and the morphing structures to induce various phenomena during the fluid-structure interaction. Different from the fluid-structure interaction of bluff bodies, the interaction between the unsteady fluid flows and the flexible thin structures is achieved through two factors related to flexibility effect, namely (i) static deformation and (ii) flapping motion. As shown in Fig. 11.7, the flexible thin structure maintains a streamlined shape under the action of fluid loading in the fixedpoint stable regime. Once the coupled system exceeds its critical state, self-excited vibration can be triggered through the flow-excited instability and the structure starts to flap.

11.3.2 Nondimensional Parameters Herein, we briefly highlight the key nondimensional parameters for a flapping flexible thin structure in a flowing fluid. Flexible structures can involve different boundary conditions, which can lead to various forms of coupled response during fluid-structure interaction. As shown in Fig. 11.6, the coupled system can be regarded as a membrane immersed in unsteady fluid flows when both the leading and trailing edge are clamped. A conventional foil is formed when the leading edge is clamped and an inverted foil is produced by clamping the trailing edge. In addition to the boundary conditions, the material properties (e.g., thickness, structural density) and the oncoming flow conditions, and the spatial arrangements can result in different coupled response dynamics. The dimensionless parameters can be used for scaling the coupled dynamical response of fluid-structure systems. As introduced in Sect. 11.3.1, a variation of the Reynolds number Re leads to different flow features and coupled dynamics of the flexible thin structures. Apart from the Reynolds number Re, the structure-to-fluid mass ratio m ∗ is another important parameter that governs the coupled dynamics of the flexible structure. Similar to the flow-induced vibration of the bluff body introduced in Volume 1, the mass ratio for flexible structures can be defined as

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m∗ =

ρs h ρfL

(11.2)

where ρ s is the structural density and h denotes the thickness of the flexible thin structure. The mass ratio gauges the relative influence of each medium, solid or fluid, in the inertial balance of the system. As shown in Fig. 11.8, a heavy conventional foil is prone to flap in unsteady fluid flows, compared with a lightweight structure. The inertial force becomes a dominating factor in the coupled system and the flow-induced vibration is triggered by flow-excited instability. The FIV instabilities of flexible thin structures are also strongly influenced by three interrelated key nondimensional parameters linked to the structural rigidity, namely the reduced velocity (Ur ), the bending stiffness (K B ) and the aeroelastic number (Ae) defined as: Ur =

U∞ , f nL

KB = s 3

B ρ

f U2 ∞

L3

,

Ae =

Esh 1 f 2 ρ U∞ 2

L

(11.3)

E h where flexural rigidity B = 12(1−(ν s )2 ) characterizes the flexibility of the thin strucs s n ture. E , ν and f represent Young’s modulus, the Poisson’s ratio and the first mode natural frequency in a vacuum of the thin structure, respectively. The reduced velocity is generally employed in the studies of the flexible tubes and pipelines which are cross-flow VIV dominated. While the bending stiffness is usually considered for a flapping foil or panels, the aeroelastic number is often applied in the coupled fluid-membrane system. The FIV instabilities occur when the reduced velocity, the bending stiffness or the aeroelastic number exceeds the critical value. In other words, decreasing the structural rigidity and thus lower natural frequencies will destabilize the coupled system. The dimensionless Strouhal frequency is St = f vs L/U∞ , where f vs is the vortex shedding frequency of a flexible thin structure. In this volume, we consider the incompressible Newtonian fluid with the Mach number less than 0.3 and the compressibility can be neglected. The effects of boundary conditions, spatial arrangements, coupled fluid-structure parameters and geometry variations such as the trailing edge shape and the aspect ratio can play an important role in governing the flow-excited instability and flow-induced vibration. The effects of the nondimensional parameters and the related conditions on the coupled dynamical responses (e.g., amplitude, frequency and fluid loading) of the flexible thin structures will be investigated and discussed in the following chapters. The objective of this volume is to examine the coupled dynamics and the flow features as a function of the governing nondimensional parameters for conventional foils, inverted foils and flexible membranes through a numerical simulation approach. The onset of flow-induced vibration related to flow-excited instability and synchronization, the energy transfer during fluid-structure interaction and the proximity and trailing edge effects are further studied. Finally, the similarities and differences of flow-induced vibrations between bluff bodies and flexible thin structures are discussed and analyzed to understand the flow-excited instability and synchronization in flow-induced vibrations of different structures.

11.4 Organization: Volume 2

667

11.4 Organization: Volume 2 Volume 2 covers mathematical modeling and numerical simulations for flow-induced vibration of different flexible thin structures. The present chapter starts with an introduction and application of flow-induced vibration of the flexible thin structures. Chapter 12 introduces the theoretical background of flapping flags and plates before proceeding to the detailed numerical studies of different thin structures. The purpose is to build a simplified understanding of flow-induced vibration of thin structures with linear theoretical analysis. Chapter 13 will present the investigations of conventional flag flapping physics through nonlinear numerical simulations. The quasi-monolithic coupled fluidstructure formulations are introduced for numerical simulation. The flapping dynamics of two-dimensional and three-dimensional flexible flags and the net energy transfer during flow-induced vibration are examined in detail. In Chap. 14, we explore the proximity effect in flag flapping. A side-by-side foil arrangement is considered in this chapter to investigate the proximity effect. We present the effects of the gap between the foils on coupled flapping and vortex dynamics. The phase diagram comparing the distribution of coupled modes as a function of the gap from the numerical simulations with the stability analysis is then investigated. We further present parametric analysis to study the effects of m ∗ and K B on the stability and coupled flapping modes of two side-by-side elastic foils and the transition mechanism behind the evolution of the initial out-of-phase coupling between the foils into stable in-phase coupled flapping. In Chap. 15, the trailing edge effect on flapping dynamics is explored. The problem set-up for the pitching plate is described. Then, we study the effect of trailing edge shape and flexibility on the propulsive performance and the drag-thrust transition mechanism is explored with the aid of the dynamic decomposition method, the momentum-based equation and the analytical added mass model. In Chap. 16, the flapping dynamics of isolated inverted flap flapping are examined. We start from the studies on a two-dimensional inverted flapping flag in Sect. 16.3. We first present the problem statement, computational domain and mesh convergence study. After that, the detailed results and analysis of inverted foil, namely, the onset of flapping instability, the effects of bending rigidity and mass-ratio, the transition mechanism to the unsteady deformed-flapping mode, the regimes of flapping dynamics, the wake topology and net energy transfer are systematically investigated. In Sect. 16.4, we extend the studies to a three-dimensional inverted flapping flag. The three-dimensional effect on vortex shedding and energy transfer are explored. In Chap. 17, we present the partitioned coupled fluid-structure formulations employed in the numerical simulation of membrane aeroelasticity. In Sect. 17.3, we start from the studies on two-dimensional membrane aeroelasticity from low to high angles of attack. We then extend the studies to three-dimensional membrane aeroelasticity in Sect. 17.4. In Chap. 18, we present an aeroelastic mode decomposition framework. This aeroelastic mode decomposition framework is applied to extract the frequency-

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ranking aeroelastic modes from the coupled fluid-structure system. The flexibility effect is examined by comparing the flow features between the flexible thin structure and their rigid counterpart. The aeroelastic mode section strategy is discussed to explore the mechanism of fluid-induced vibration in separated flows. In Chap. 19, we further explore the flow-excited instability of a three-dimensional flexible membrane. The mechanism of onset of flow-induced membrane vibration and mode transition is investigated over a wide range of mass ratios, Reynolds numbers and aeroelastic numbers. Bibliography notes Detailed aspects of computational modeling can be found in the textbook of Jaiman and Joshi [184]. Further theoretical materials on flow-induced vibration can be found in several text books e.g. Blevins [56, 330] and review papers by Sarpkaya [697]. Further details can be found in the dissertations of Pardha Gurugubelli and Guojun Li carried out at the National University of Singapore.

Chapter 12

Theoretical Background of Flexible Plate

In this chapter, we present a review and theoretical study of added-mass and aeroelastic instability exhibited by a linear elastic plate immersed in a mean flow. We first present a combined added-mass result for the model problem with a mean incompressible and compressible flow interacting with an elastic plate. Using the Euler-Bernoulli model for the plate and a 2D viscous potential flow model, a generalized closed-form expression of added-mass force has been derived for a flexible plate oscillating in a fluid. A compressibility correction factor is introduced in the incompressible added-mass force to account for the compressibility effects. We next present a formulation for predicting critical velocity for the onset of flapping instability. The formulation considers tension effects explicitly due to viscous shear stress along the fluid-structure interface. In general, the tension effects are stabilizing in nature and become critical in problems involving low mass ratios. We further study the effects of mass ratio and channel height on the aeroelastic instability using the linear stability analysis. It is observed that the proximity of the wall parallel to the plate affects the growth rate of the instability, but these effects are less significant in comparison to the mass ratio or the tension effects in defining the instability. Finally, we conclude this chapter with the validation of the theoretical results with experimental data presented in the literature.

12.1 Introduction The determination of added-mass effects has a wide application in the transient analysis of an elastic plate subjected to fluid flow, e.g., flutter analysis [556, 630], vortex-induced vibration [56, 673]. The added mass is an inherent characteristic of fluid loading and the understanding of the added mass will provide the fluid loading to be obtained in the most suitable form for wide-ranging applications. Added-mass effects become especially relevant for hydroelastic problems and light-weight structures as the mass of the entrained fluid by the dynamical structure is a significant part of the total mass. The added-mass effect also has an implication in the numerical © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Jaiman et al., Mechanics of Flow-Induced Vibration, https://doi.org/10.1007/978-981-19-8578-2_12

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modeling of fluid-structure interaction and is an important design parameter for the stability and convergence of underlying coupling scheme, see [186, 540, 544, 580, 613, 754]. The flapping of the elastic plate problem is of interest not only for its prevalence but also for the simplicity of the problem statement and the richness of the coupled fluid-structure behavior. Consequently, there have been numerous investigations of this problem [564, 593, 605, 617, 630, 646, 708, 777]. The aeroelastic instability with such a configuration can be easily seen by blowing air over a thin piece of paper and by waving of the flag like structures. In this chapter, we are particularly interested in the very initial stage of instability, where the linearized flow-structure theory is sufficient. The effects of added mass and channel confinement are investigated in the context of coupled fluid-elastic instability. The existing knowledge regarding the evaluation of added mass is limited to the elastic plate in incompressible flows. Typically the importance of added-mass force is proportional to the fluid to structure density ratio and is justified to be ignored in compressible flows due to a low density of a gas. However, for lightweight structures, it has significance due to strong unsteady inertial effects. The fluid loading on a flexible plate in an axial flow may be considered as the sum of a noncirculatory part, and a circulatory part [630]. The noncirculatory part of the loading involves the added mass, the fluid dynamic damping, the added stiffening of the plate. The earliest treatment of the flow compressibility effect on an infinite elastic plate motion in [660] was aimed at the the stability analysis of thin panels. Recently [540] derived an approximate expression for the added mass of a compressible inviscid flow on an infinite elastic plate for short times. The focus of work in [540] was to establish the difference in the compressible and incompressible added-mass force during early times of motion by considering an independent set of equations for compressible and incompressible flow. For the model elastic plate, the added mass of a compressible flow system is proportional to the length of time interval, whereas the added mass of an incompressible system approaches a constant asymptotically. This finding has an implication for the design of fluid-structure coupling algorithms and the stability and convergence properties of the sub-iterations [186, 580, 613]. The added-mass force has been extensively studied in multiphase flows. In an incompressible flow, added-mass force is proportional to the mass of the fluid displaced by the particle times the relative particle acceleration. An important finding is that added-mass force is independent of Reynolds number Re, see [545, 653]. In the compressible flow the dependence of added-mass force on the instantaneous relative acceleration is no more valid ([659, 678, 680]). Due to the finite speed of propagation of sound, the added-mass force depends on the history of particle relative acceleration and is expressed in an integral form. The integral is expressed as a convolution between acceleration history and a decaying particle response kernel. The extension of added-mass force in the compressible non-linear regime has been carried out in [679].

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12.1.1 Theoretical Studies of Flapping Foils In parallel to the foregoing experimental studies, there have been many theoretical studies of fluid-foil interactions over the past decades. Most of these theoretical studies aimed to determine the onset of foil instability and flapping in two dimensions. To begin, [630] theoretically investigated the coupled fluid-elastic instability of the conventional foil configuration for subsonic flows. In this study, the elastic foil is modeled as an Euler-Bernoulli beam and the aerodynamic fluid load acting on the elastic structure is determined by assuming the flow as incompressible, inviscid and irrotational. The unsteady aerodynamic fluid load acting on the cantilevered foil is defined as a combination of circulatory and non-circulatory components. The circulatory component is introduced to satisfy the Kutta condition at the trailing edge. The general solution of the flexible foil under the fluid loading is considered as a combination of the first two orthogonal beam modes. The validation of the analytical model with the experiments performed on a cantilevered foil with a splitter plate at the trailing edge has shown a big difference in Ucr value. The large difference in Ucr is attributed to the inaccuracies in the natural frequencies of the vacuo-modes considered. Interestingly, the experimental Ucr and flapping frequency are found to be in agreement with values calculated based on the non-circulatory fluid loading. One possible reason for this observation could be the presence of a splitter plate at the trailing edge. In this study, the authors also emphasized the role of two independent nondimensional parameters namely (Ur )2 =

ρ f U02 L 3 B

and

μ=

1 ρf L , = m∗ ρsh

(12.1)

where Ur is the reduced velocity and μ characterizes the fluidto solid inertial effects.  B is the flexure rigidity of the foil defined as B = Eh 3 /12 1 − (ν s )2 where ν s is the Poisson’s ratio. Using an analytical model similar to the one presented by [605, 630] investigated the effects of foil length and stiffness on the onset of flapping instability. The author analyzed the energy transfer from the fluid to the flexible foil in terms of the fluid loading components i.e. non-circulatory inertial effect and circulatory vorticity field. The author has reported that the energy transfer is negative for the non-circulatory and positive for the circulatory fluid loading. He has also concluded based on this observation that works done by the non-circulatory part are always negative and it plays the role of damping the instability. Similarly, the author concluded that it is the circulatory part that does positive work and is the key to sustain the flapping instability. These observations from the author do not clearly generalize to infinitely long foils [551, 614, 707]. To analyze the influence of leading and trailing edge boundary conditions on the stability of the coupled fluid-elastic system, [593] performed linear stability analysis by considering a 2D potential flow in Fourier space and neglecting the circulatory effects. In this analysis, the authors considered different combinations of

672

12 Theoretical Background of Flexible Plate

the boundary conditions such as clamped-clamped, pinned-pinned, clamped-pined, free-free, clamped-free, pinned-free, pinned-clamped, free-clamped and free-pinned. The free-free configuration presents an interesting case if the rigid translation of the foil is neglected, the foil exhibits a coupled-mode flapping. The clamped-free, which is the conventional foil configuration, does not exhibit the first mode of static divergence. Instead, it experiences a single-mode flapping in its second mode. The authors also studied the energy transfer mechanism from the fluid to the solid and vice-versa to provide insight into the coupled fluid-elastic instability. The authors in [519, 767] further extended the analytical model presented by Kornecki etal. [630] to include the effects of nonlinear drag induced tension. With the aid of the analytical models, the authors showed that m ∗  1 loses its stability to perform flapping in higher modes. The authors also found that with the increase in m ∗ the flexible foil begins to exhibit even the lower flapping modes. Watanabe et al. [767] have reported that the flapping frequency vs. m ∗ presents a step-like pattern with sudden jumps in the flapping frequency due to changes in the flapping modes. [519] reported that their linear model cannot capture bistable phenomenon observed experimentally by Shelley et al. [707], Watanabe et al. [768] , Zhang and Hisada [707] also made an attempt to generalize the Theodorsen unsteady aerodynamic theory to investigate the 3D effects by introducing a scaling function as a function of the aspect ratio of the finite width foil. Connell and Yue [551], Shelley et al. [707] considered a simplified analytical model to demonstrate the effects of strong added-mass effect on the onset of flapping instability. Both of these works emphasized the stabilizing effect of tension-induced by viscous boundary layer on the onset of flapping instability and considered three independent nondimensional parameters namely m ∗ , K B and Re that can influence flapping instability m∗ =

ρsh , ρf L

KB =

B , ρ f U02 L 3

Re =

ρ f U0 L , μf

(12.2)

where K B is the nondimensional bending rigidity. By employing a 3D potential flow in Fourier space, Eloy et al. [569] studied the effects of the aspect ratio on the stability and dynamics of a flexible foil. The analytical model has shown a good agreement with the slender body theory and twodimensional approximation for foils with small and large aspect ratios respectively. In this study, the authors reported that the flexible foils with small aspect ratios tend to be more stable compared to the foils with larger aspect ratios. The authors observed that the aspect ratios significantly affect the onset of flapping for m ∗  1. However, for m ∗ < 1 the effect of aspect-ratio is less significant and have a critical reduced velocity (Ur )cr ≈ 10. This phenomenon possibly explains why [768] did not observe any difference in the Ucr between the long foil with a large aspect ratio and the flag type foil with a small aspect ratio. [567] further analyzed the effects of aspect ratio on the bistable hysteresis observed both experimentally [707, 768, 784] and numerically [508, 551]. The authors reported that for small aspect ratios

12.2 Linear Stability Analysis

673

the onset of instability is super-critical and it does not exhibit any bistable hysteresis. However, for large aspect ratios, the onset is subcritical bifurcation with bistable hysteresis. The authors have noticed that the width of the bistable hysteresis region is significantly smaller than the width observed from the experiments [707, 784]. The authors attributed the large difference in the hysteresis region width to the 2D approximation made. More recently, [560, 655] performed both local and global stability analysis on the coupled fluid-elastic-electrical system to realize the impact of piezoelectric coupling on coupled fluid-elastic instability and the efficiency of energy transfer into electric energy. From the structure point of view, the dissipative piezoelectric circuit plays the role of external damping. It is known from the literature on elastic panels [553] that structural damping destabilizes the instabilities characterized by negative energy wave [543]. The authors reported that a piezoelectric circuit destabilizes the fluidelastic-electrical system compared to a fluid-elastic system. The authors also reported that energy transfer is more efficient for lower m ∗ values. More recently, the authors [776] have shown that the coupled fluid-elastic-electrical system can experience resonance if the flapping frequency lock-in with the natural frequency of the electrical circuit. The authors reported that due to a coupled resonance, the system can undergo flapping even for fluid velocities below critical velocity and it can also improve the efficiency of energy harvesting by increasing the flapping amplitude.

12.1.2 Organization The present work builds upon earlier studies of added-mass force in the context of particle interaction with the mean flow [677, 678, 680]. In the present work, we consider a 2D flexible plate in a rectangular fluid domain. We first perform theoretical analysis to estimate a closed-form for added-mass force which includes the effects of fluid compressibility, viscosity and also finite height of the channel. The new generalized formulation reduces to the correct forms in the limiting cases discussed in the literature. A phenomenon of acoustic stiffening is discussed which occurs when the spatial wave number of the plate deflection approaches acoustic wave number in the still fluid. A linear stability analysis is carried out and effects of tension, mass ratio and the height of the channel are discussed.

12.2 Linear Stability Analysis In this section, we present a theoretical study of added mass and aeroelastic instability exhibited by a linear elastic plate immersed in a mean flow. This study aims to introduce some prior knowledge to understand the aeroelastic instability caused by flow-induced vibration for thin structures from fundamental aspects.

674

12 Theoretical Background of Flexible Plate

12.2.1 Problem Statement We consider an elastic plate of length L, width W and thickness h embedded in a parallel flow with a freestream velocity U0 as shown in Fig. 12.1. Let t be the time, x and y be the coordinates in the horizontal and vertical directions and L and h be the size of the plate in the two directions as shown in the figure. The solid is modeled as a plate of negligible thickness under the Euler-Bernoulli approximation, i.e., h/L  1. Without loss of generality, let the plate be located at y = 0 plane. The governing equation per unit length for the vertical displacement of the plate α(x, t) is as follows Eh 3 αx x x x − T αx x + ρs h αtt = −Δp , 12(1 − σ 2 )

(12.3)

where E denotes Young’s modulus, ρs is the density of plate, σ is the Poisson ratio of the material, T is the tension acting along axial flow direction and Δp is the net pressure acting on the plate. We take plate deflection α(x, t) to be of the harmonic form α(x, t) = αe ˆ ikx x−iωt ,

(12.4)

√ where i = −1, ω and k x are complex quantities representing the frequency and the wave number in the x-direction, respectively. The above relation is a good approximation away from plate ends. The fluid flow is confined in a distance H above and

Fig. 12.1 Depiction of the problem of two-dimensional elastic plate interacting with mean flow

12.2 Linear Stability Analysis

675

below the plate as shown in Fig. 12.1. We want to find the fluid loading for a small ˆ  1. deflection, i.e., |Hα| In the absence of any plate deflection, the fluid can be described by a uniform parallel flow with density, pressure and velocity in the x-direction as ρf , p0 , and U0 , respectively. The assumption of a small plate deflection gives rise to a disturbance flow described by the linearized compressible Navier-Stokes equations as follows ∂ρf ∂ρf + U0 + ρf ∇ · u = 0 , ∂t  ∂ x    ∂u ∂u + U0 + ∇ p − μu − μb + 13 μ ∇∇ · u = 0 , ρf ∂t ∂x

(12.5) (12.6)

where ρf , u= (u, v) and p are perturbations in fluid density, velocity and pressure, respectively, with u and v as the velocity components in x and y directions, μ is the dynamic viscosity and μb is the bulk viscosity of the fluid. In this work, we restrict the size of the disturbance such that the disturbance of flow can be linearized. In particular, we consider the case of negligibly small perturbations in the temperature field. The undisturbed speed of sound c0 is used as a closure, where c02 = ρp . The boundary conditions at the surface of the elastic plate and the horizontal wall at y = ±H are essentially the no-penetration condition. For small deflections of the elastic plate considered herein, the boundary condition at the top and bottom surface of the elastic plate can be written as v(x, 0, t) =

Dα(x, t) = Dt



∂ ∂ + U0 ∂t ∂x

 α(x, t) ,

(12.7)

and at the upper/lower wall we have v(x, ±H, t) = 0 .

(12.8)

Because of the inclusion of the viscosity in the governing equations, another set of boundary conditions is needed. Here we will make use of slip boundary condition on wall surfaces. This is consistent with the viscous potential theory used by [540]. Viscous stress parallel to the plate (y = 0) and the wall surfaces (y = ±H ) is taken to be zero, i.e.,   ∂v ∂u + = T = 0. μ ∂y ∂x Using Eqs. (12.7) and (12.8) in above results, we get ∂u (x, ±H, t) = 0 , ∂y   ∂u ∂ ∂ ∂ α(x, t) . (x, 0, t) = − + U0 ∂y ∂ x ∂t ∂x

(12.9) (12.10)

676

12 Theoretical Background of Flexible Plate

As described above, viscous shear stress in a potential flow along the flat plate is zero. From the classical Blasius boundary layer theory, it is known that flat plate experiences a drag force equal to 1.328ρf U02 Re1/2 [523]. This drag force introduces a tension effect along the plate and the magnitude of this tension along the plate varies as [662]: T (x) =

1 C T ρf U02 Re− 2



  x 1− L

(12.11)

where C T = 1.328 and x is the distance from the leading edge. The above equation for the tension force clearly shows that the trailing edge experiences a zero tension, which explains the small amplitude oscillations for the velocity less than the critical velocity [605]. Full body oscillation, however, is observed for the velocity greater than the critical velocity. The proposed model accounts for the viscous drag effects explicitly on the structural side by assuming that the plate is pre-tensed and the tension is assumed as the average drag force experienced along the plate. Therefore, the average tension can be expressed as: T = C T ρf U02 Re− 2

1

where C T = C3T denotes the average tension coefficient and Re = Reynolds number.

(12.12) U0 L ν

is the

12.2.2 Determination of Fluid Loading This section concerns with getting a closed form solution for the fluid loading. We can define a scalar potential φ and a vector potential Ψ as follows u = ∇ φ + ∇ × Ψ such that ∇ · Ψ = 0 .

(12.13)

For the two-dimensional flow considered here we can define Ψ using a scalar potential ψ as follows Ψ = ψ kˆ , (12.14) where kˆ is the unit vector in z-direction. The velocity components can be expressed as u=

∂ψ ∂φ ∂ψ ∂φ − and v = + ∂x ∂y ∂y ∂x

(12.15)

12.2 Linear Stability Analysis

677

The perturbation density and pressure can be obtained as follows p=ρ

c02

    4 ∂ ∂ ν− + U0 φ, = ρf λ + 3 ∂t ∂x

(12.16)

where λ = μb /μ is the ratio of bulk to shear viscosity and ν is the kinematic viscosity. It is a simple matter to show that φ(x, y, t) and ψ(x, y, t) satisfy following governing equations        1 ∂ ∂ ∂ 2 4 iν ∂ + U0 φ = 2 + U0 1+ λ+ φ, 3 c02 ∂t ∂x ∂x c0 ∂t   1 ∂ ∂ ψ = + U0 ψ. ν ∂t ∂x

(12.17) (12.18)

A general solution for the scalar potentials can be written as   φ(x, y, t) = eikx x−iωt β1 e|kφ |y + β2 e−|kφ |y ,   ψ(x, y, t) = eikx x−iωt γ1 e|kψ |y + γ2 e−|kψ |y .

(12.19) (12.20)

where kφ and kψ are related to ω and k x as follows (ω − U0 k x )2 ,    4 iν c02 1 + λ + k − U (ω ) 0 x 3 c02 i kψ2 = k x2 + (ω − U0 k x ) . ν kφ2 = k x2 −

(12.21)

(12.22)

|kφ | and |kψ | denotes those values of kφ and kψ such that their real parts are nonnegative. For complex kφ and kψ the solution given by Eqs. (12.19) and (12.20) represents a traveling wave solution consisting of incoming and outgoing waves. For an unconfined domain i.e., H → ∞, the solution consists of only the outgoing waves. A specific solution can be obtained using the two boundary conditions on the surface of the elastic plate Eqs. (12.7) and (12.10)). For the case of a confined channel one can use the four boundary conditions on the surface of the elastic plate and upper wall of the channel, i.e., Eqs. (12.7–12.10), to obtain the remaining four unknowns β1 , β2 , γ1 and γ2 . Thus, we get the following expression for φ and ψ for the solution in the upper half domain. φ(x, y, t) = −i αˆ

2 (ω − U0 k x ) kψ + k x2 cosh(|kφ |(y − H )) ikx x−iωt e , |kφ | sinh(|kφ |H ) kψ2 − k x2

ψ(x, y, t) = 2 αˆ (ω − U0 k x )

kψ2

sinh(|kψ |(y − H )) ikx x−iωt kx e . 2 sinh(|kψ |H ) − kx

(12.23) (12.24)

678

12 Theoretical Background of Flexible Plate

The solution for the lower domain can be similarly obtained. In general kφ and kψ are complex numbers. Taking the limit of an incompressible flow (c0 → ∞) and semi-infinite domain (H → ∞), above solution Eqs. (12.23) and (12.24) reduces to that represented by Eq. (12.32) of [540]. The expression for the pressure perturbation on the surface of the elastic plate taking into account the effects of compressibility, finite height and viscosity can be obtained as follows ρf αˆ coth(|kφ |H )   p(x, 0, t) = − 4 iν 1+ λ+ (ω − U0 k x ) 3 c2



(ω − U0 k x ) (ω − U0 k x )2 + i 2 ν k x2 |kφ | |kφ |



0

eik x x−iωt .

(12.25)

In the limit of incompressible flow, Eq. (12.21) reduces to kφ2 = k x2 and we get  p(x, 0, t) = −ρf αˆ coth(|k x |H )

 (ω − U0 k x )2 (ω − U0 k x ) ikx x−iωt + i 2 ν k x2 . e |k x | |k x | (12.26)

Further taking the limit of H → ∞, we get  p(x, 0, t) = −ρf αˆ

 (ω − U0 k x ) ikx x−iωt (ω − U0 k x )2 e , + i 2 ν k x2 |k x | |k x |

(12.27)

which is the same as Eq. 12.33 in [540]. The net pressure Δp acting on the flexible plate is  Δp = 2ρf αˆ

 (ω − U0 k x ) ikx x−iωt (ω − U0 k x )2 + i 2 ν k x2 e |k x | |k x |

(12.28)

It is clear that the convective and the viscous part in Eq. (12.27) are proportional to ω, whereas the added-mass part is proportional to ω2 . This implies that the addedmass effect can dominate the effects of convection and viscosity in the limit of ω → ∞. Furthermore, we observe that the added mass effect in the compressible flow has a localized effect, i.e., the effect of displacement perturbation on the compressible fluid is confined to a region within distance c0 t of the interface. On the other hand, the effect is global and throughout the entire domain for an incompressible flow. This result is consistent with the observation reported in [540]. In the next section, we present the results of combined added mass for an oscillating flexible plate.

12.2 Linear Stability Analysis

679

12.2.3 Added Mass Force An accelerating body in an inviscid fluid drags a certain amount of fluid due to no-penetration condition on its surface resulting in the added-mass force. The total force on an oscillating elastic plate under a uniform flow has been derived in the previous section. The added-mass force can be extracted from the total force as the component proportional to acceleration αtt = −ω2 αˆ eikx x−iωt . For an oscillating plate under quiescent compressible inviscid fluid, the added-mass force per unit length can be written as Fam (t) = −2

ρf W coth(|kφ |H ) αtt , |kφ |

(12.29)

where kφ2 = k x2 − ka2 and ka = ω/c0 is the acoustic wave number. Using the definition for the added-mass from [660]

m am

 ⎧ ρf W coth( k x2 − ka2 H ) ⎪ ⎪ ⎪  , if k x > ka , ⎨ −2 |k | 2 2 x 1 − ka /k x = ⎪ ρf W cot( ka2 − k x2 H ) ⎪ ⎪  , if k x < ka . ⎩ −2 |k x | ka2 /k x2 − 1

(12.30)

In the limit of a flat plate in the infinite domain, i.e., H → ∞, and an incompress2ρf W , ible flow, i.e., ka → 0, the added-mass tends to the well-known result m am = |k x | see [777]. The finite height effect appears as a multiplicative factor of coth(k x H ) 2ρf W coth(k x H ) . In Eq. (12.30), the effect of compressibility resulting in m am = |k x | appears as a correction factor as  coth( k x2 − ka2 H )  ξ= . 1 − ka2 /k x2

(12.31)

For very slow oscillations, i.e., ω → 0, the incompressible result is again recovered. In an incompressible flow, the added-mass force is proportional to the plate acceleration αtt irrespective of frequency. In a compressible flow, however, the added-mass force and hence the added-mass coefficient is frequency dependent. For ka < k x or ω < k x c0 , compressibility increases the added-mass effect. The compressibility correction factor is shown in Fig. 12.2. As can be seen, there is a stiffening effect for ω → k x c0 . For large frequencies, i.e., ω > k x c0 , acoustic wavenumber dominates plate deflection wavenumber. The effect of plate deflection is overpowered by acoustic propagation and in the limit of ka /k x → ∞, locally the problem asymptotes to the behavior of a flat plate oscillating vertically with k x = 0 in a confined channel. Let us consider the work done by flat plate surface to increase the kinetic energy of the fluid

680

12 Theoretical Background of Flexible Plate 2

10

k H=1 x

1

10 ξ

kx H = 2

0

10

−1

10

−1

0

10

10

ω/(k c ) x 0

ξ

5

0

−5

1

2

3

4

5

6

7

8

9

10

ω/(k c ) x 0

Fig. 12.2 Compressible correction factor to added-mass as given in Eq. (12.31) for k x H = {1, 2}

dE = dt



 Fam αt d S = m am

αtt αt d S .

(12.32)

where E is the kinetic energy, S is the surface bounding the fluid region, αt is the velocity of the plate in the normal direction. The added mass m am represents a mass per unit length of the oscillating plate for the rate of change of kinetic energy in time. When m am is real, the work done is proportional to αt αtt . If the fluid force is given as a complex quantity, its real part is associated with the added mass and the imaginary part is the damping coefficient. For an unsteady problem, the Fourier (Laplace) transform of the unsteady fluid force will have the so-called memory effect, which is related to simply not only the added mass but also to all frequency-dependent forces i.e., both the added mass and damping effects. For the case of a flat plate oscillating in a confined domain, we find that m am is always real, i.e., there is no dissipation of energy. In the start-up process, there is a net work done by the plate on the fluid to set the fluid in the oscillatory motion which

12.2 Linear Stability Analysis

681

appears as kinetic energy of the fluid. After both the plate and fluid system is set in oscillatory motion there is no net work needed to keep the system in motion. A plate in an infinite domain would, however, lose energy due to acoustic radiation. Here we draw an analogy with added mass force on an oscillating particle in compressible inviscid fluid. An oscillating particle in a compressible fluid loses energy to the far-field due to the radiation (see discussion in [680]). To get an expression for non-oscillatory motion it is useful to cast the above equation in the Laplace space. Using −iω = s, where s is the Laplace variable, Eq. (12.29) can be rewritten in the Laplace space as 2ρf W c0 L [αtt ] . L [Fam ] = −  k x2 c02 + s 2

(12.33)

Defining ⎡

⎤

2ρf W c0 ⎦ 2ρf W K (t) = L −1 ⎣  J0 (k x c0 t) , = kx k x2 c02 + s 2

(12.34)

where Jn is the cylindrical Bessel function of the first kind and K (t) is the history kernel relating acceleration history αtt to the added-mass force is as follows t Fam (t) =

2ρf W K (t − ξ )αtt d ξ = kx

0

t J0 (k x c0 (t − ξ )) αξ ξ k x c0 d ξ . (12.35) 0

Due to the finite speed of the propagation of sound, the direct proportionality of added mass force to acceleration is not valid. The force now depends on acceleration history convoluted with the response kernel K (t). See [680] for a similar interpretation for unsteadily moving particle and [754] in the context of iterative coupling schemes for fluid-structure interaction. The response kernel for a spherical particle is given by exp(−tc0 /a) cos(tc0 /a) where a is the particle radius. The kernel for a spherical particle decays exponentially while the kernel for an oscillating flat plate decays over a longer time due to reflections from the walls of a confined channel. For constant acceleration, the force can be shown to be equal to the integration of the kernel. The integration of the Kernel K (t) is shown in Fig. 12.3. Using J0 (0) = 1, early time response k x c0 t → 0 can be obtained as t lim Fam (t) = 2ρf W c0

αtt d ξ ,

k x c0 t1

0

(12.36)

682

12 Theoretical Background of Flexible Plate 0.5

0.4

∫ K(t)

0.3

0.2

0.1

0

0

0.1

0.2

0.3

0.4

0.5

t 1.5

∫ K(t)

1

0.5

0 −2 10

−1

0

10

10

1

10

2

10

3

10

4

10

t

Fig. 12.3 Integration of Kernel K (t) given by Eq. (12.34), where early time response (top) and the long time behavior (bottom) are shown

which for constant acceleration (αtt ) is lim Fam (t) = 2ρf W c0 αtt t .

k x c0 t1

(12.37)

For early time the added-mass force in a compressible flow increases linearly with time as can be seen in Fig. 12.3 (top). This behavior has also been observed in [540, 754].

12.2 Linear Stability Analysis

683

Similarly for a constant acceleration one can obtain a very long time behavior of added-mass as follows ⎞ ⎛∞  lim Fam (t) = 2ρf W c0 ⎝ K (0, k x c0 (ξ )) d ξ ⎠ αtt , (12.38) k x c0 t1

0

k x c0 t  1 can be satisfied if either of k x or c0 or t is large, i.e., lim Fam (t) =

k x c0 t1

2ρf W αtt , kx

(12.39)

Thus above result is also valid in the incompressible limit, i.e., c0 → ∞, for all times. As shown in Fig. 12.3 (bottom), the long time response of a compressible system approaches incompressible response. Similar behavior was noted in [679, 680] in the context of added mass force on spherical and cylindrical particles in compressible flow. In the context of linearized governing equations, at large times the acoustic propagation saturates and the flow field asymptotes to an incompressible one. Thus, the incompressible added mass force is recovered. One can define added-mass coefficient Cam for the incompressible case and long time behavior of compressible case as m am , ρf L

(12.40)

λA R 2W = , kx L nπ

(12.41)

Cam = which can be further expressed as Cam =

where n is the mode number, λ = 2π/k x denotes wavelength and A R = W/L is the aspect ratio. The added mass is thus a function of the non-dimensional wave number and dependent on the flapping mode, which can be interpreted as a layer of fluid with thickness λ/π around the 2D thin plate case. This is an important difference from the long cylindrical beam problem, in which the added mass is estimated only from the cylinder cross-section. The dependence of the added mass, Cam , diminishes as the flapping mode number n becomes very large. The added mass coefficient increases monotonically as the aspect ratio A R = W/L increases, especially for plates vibrating in the fundamental mode. The results obtained can be also compared with those of thin-wing theory [617]. The dependence of added mass on various modes and aspect ratios can be further studied in a similar manner as [777]. Next, we turn our attention to the linear stability analysis of fluid-elastic flapping modes. Our primary interest is in the onset of instability where the linear theory is sufficient. The questions we want to ask are: (a) what are the factors that lead to the onset of the instability? (b) what is the role of mass ratio and channel height? and (c) at what flow speed does the plate loses its stability?

684

12 Theoretical Background of Flexible Plate

12.2.4 Stability Analysis The stability properties of traveling waves propagating in an infinite long medium in a mean axial flow are now studied. The objective is to predict the existence of instability before doing any detailed analysis, and to gain insight into the instability mechanism. In its non-dimensional form, the dispersion relation depends on the mass ratio, the non-dimensional mean flow velocity, and the non-dimensional channel height. One may classify two cases of wave instability, depending on the long time impulse response of the plate. If the exponentially growing wave packet is advected by the flow, the instability is said to be convective. Conversely, if it grows in place, so that the entire space is ultimately dominated by the instability, the latter is referred to absolute. The concepts of convective and absolute instabilities, and the underlying theory, have been initially introduced in the domain of plasma physics and successfully applied in the particular domain of fluid-structure interaction [553]. Introducing the Fourier transform in space and time   α (k, ω) =

1 2π

2 ∞ ∞

α(x, t)eiωt e−ikx x dtd x

(12.42)

−∞ −∞

a dispersion relation for the full system can be derived from the governing equation eq. (12.3), deflection relation Eq. (12.4) and solution for fluid loading Eq. (12.25) as follows Eh 3 1 k x4 − ρs h ω2 + C T ρf U02 Re− 2 k x2 2 12(1 − σ )   ρf coth(|kφ |H ) (ω − U0 k x )2 2 (ω − U0 k x )   +i 2 ν k x . −2 4 iν |kφ | |kφ | 1+ λ+ k − U (ω ) 0 x 3 c02 (12.43)

D(k x , ω) =

The deflection relation amplifies in time for all positive imaginary parts and decays for all negative imaginary parts. The value of the imaginary part of the deflection wave/angular frequency describes the the disturbance growth rate in time. To determine the temporal instability of the coupled system, we solve the dispersion relation Eq. (12.43) for the variable ω for all possible real value of k x and find if the relation admits any imaginary values of ω. Therefore, the criterion for temporal instability is that it admits at least one complex ω for any real value k x . In order to understand the nature of disturbance in space, we follow the method presented in [538]. In space, a disturbance can grow in the place where it is introduced or grow by advecting downstream. The former is described as absolute instability and the latter as convective instability. Starting with Im(ω) large and positive, the motion of the spatial roots of Eq. (12.43) is followed as Im(ω) becomes zero. Absolute instability occurs if two roots originate from different halves of the k-plane coalesce for Im(ω).

12.2 Linear Stability Analysis

685

If Im(ω) can be reduced to zero without coalescing of the roots, the system is at worst convectively unstable. Before moving further, let us consider the non-dimensional form of the dispersion relation Eq. (12.43). By defining length and time scales  L ref = L , Tref =

ρf L 5 W , EI

(12.44)

one can define non-dimensional quantities as follows μ=

ρs h ρf L

k x = k x L ref ,

ω = ωTref , U0 = U0 c0 = c0

Tref , L ref

Tref Tref , ν = ν 2 , L ref L ref

(12.45)

where (·) notation is used for non-dimensional quantities. The non-dimensionalized dispersion relation is obtained as follows coth(|kφ |H )   4 iν 1+ λ+ (ω − U0 k x ) 3 c02   (ω − U0 k x ) (ω − U0 k x )2 + i 2 ν k x2 . (12.46) × |kφ | |kφ | 3

1

D(k x , ω) = k x4 − μω2 + C T U02 ν 2 k x2 − 2

In the above equation and the remaining equations in this section, (·) notation for non-dimensional quantities is dropped for brevity. For an incompressible flow Eq. (12.46) simplifies to

3

1 D(k x , ω) = k x4 − μω2 + C T U02 ν 2 k x2 − 2 coth(|k x |H )

(ω − U0 k x )2 (ω − U0 k x ) + i 2 ν k x2 |k x | |k x |

.

(12.47) and for an unconfined flow the above dispersion relation further simplifies to D(k x , ω) =

k x4

3 2

− μω + C T U0 ν 2

 (ω − U0 k x )2 2 (ω − U0 k x ) + i 2 ν kx −2 |k x | |k x | (12.48) 

1 2

k x2

which is equivalent to the dispersion relation presented in [681] when the tension effects are neglected. By setting the dispersion relation for waves on the complete system D(k x , ω) = 0, a second order polynomial in ω will be given by

686

12 Theoretical Background of Flexible Plate 30 Tension considered Tension neglected Huang (1995)

25

15

cr

U ,m/s

20

10

5

0 0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12

Length, m

Fig. 12.4 Comparison between the experimental values observed and the values predicted using Eq. (12.51). Dots represent the Huang’s experimental values and the solid and dashed lines coincide in the figure

 μ+

 2 U0 k x coth (|k x | H ) ω2 − 4 coth (|k x | H )ω + 2U02 |k x | coth (|k x | H ) |k x | |k x | 3

1

− C T U02 ν 2 k x2 − k x4 = 0

(12.49) The stability approach consists of solving the dispersion relation for the frequency variable ω associated with real values of k. The condition for instability, for at least one real value of k, the the imaginary part of the one root ω is positive, the amplitude of the associated wave exp[i (kx − ωt)] grows exponentially with time and the system is considered locally unstable. For a given wave number k, the critical frequency satisfies the quadratic eigenvalue Eq. (12.49)) where the discriminant is as follows: 3  1  dk = μk x4 + C T U02 ν 2 μk x2 + 2k x coth (k x H ) + coth (k x H )(2k x3 − 2μk x U02 ) (12.50)

Based on the dispersion relation, the instability exists if dk < 0 and the system is neutrally stable if dk ≥ 0. A critical dimensionless velocity is defined by solving Eq. (12.50)   3 3 1 1 2μ coth (k x H )Ucr2 = μk x3 + μk x C T Ucr2 ν 2 + 2 coth (k x H ) C T Ucr2 ν 2 + k x2 (12.51) This shows that it is possible to predict the instabilities of a system of this sort simply by looking at the dispersion properties of waves on the interface. For an

12.2 Linear Stability Analysis

687

250 Tension considered Tension neglected Zhang (2000)

Frequency, Hz

200

150

100

50

0 0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Length, m

Fig. 12.5 Comparison between the experimental values observed and the values predicted using Eq. (12.47)

incoming flow velocity U < Ucr the system is expected to be stable and for U > Ucr the disturbance grows in time resulting in an unstable system. In the limit of incompressible flow, unbounded domain and neglecting the tension effects, the dispersion relation reduces to the classical problem analyzed extensively in the literature (see [525, 553, 633, 681]).

12.2.4.1

Effect of Tension

Fig. 12.4 compares the critical velocity predicted using Eq. (12.51) for a plate of length L with the experiment conducted by [605] in a wind tunnel. Herein, we consider k x = π for the first fundamental mode. A unique critical flow velocity exists for each specific length of the flexible plate. We observe that velocity predicted using Eq. (12.51) is very close to the experimentally observed values. We also observe that the effects of tension due to the viscous shear stress is negligible for air, the critical velocity predicted by considering the tension effects and neglecting tension is nearly identical. In [605], a simple ad hoc procedure based on the growth exponent was used to account for the differences between the experimental results and the formulation given by the potential flow theory. Figure 12.5 shows the calculated and the experimental flapping frequency observed by Zhang et al. [782] for a soap film experiment. Earlier theoretical data without tension effects from [707] are also included in the figure. The experiment uses cylindrical filaments as flowing soap film in a laminar two-dimensional flow tunnel. The fluid

688

12 Theoretical Background of Flexible Plate

wets the filament, with surface tension forces constraining the filament to lie always in the plane of the film. The length of the filament can be changed incrementally through the holder at the upstream end, which abuts the film perpendicularly. At a fixed flow rate, there exists a critical filament length, below which the filament is stretched straight and aligned with the flow. If the length is sufficiently large, the stable stretched-straight state disappears, and only the flapping state remains. The critical frequency predicted by Eq. (12.48) closely matches with the critical frequency observed in the experiment [782]. We also observe that there is a substantial difference in the predicted values when the tension effects are incorporated into the dispersion relation. The stabilizing effects of tension forces can be also explained as a part of the reason why there was no observable flapping with the water tunnel experiment in [707]. One can also observe that the frequency of the system decreases as the length of the plate increases, which eventually asymptotes to the flapping frequency of about 50 H z. This value of the flapping frequency is comparable with the experiment in [782] for identical physical parameters. In [782], a thin vortex street was observed that resulted from a Kelvin-Helmholtz instability. However, the resulting vortical field does not develop eddies of a size comparable to the flapping. Instead due to the traveling wave and the in-extensibility of the plate, the free end executes a figure-of-eight trajectory.

12.2.4.2

Effect of Mass Ratio

As we know that the mass ratio determines which physics dominates the coupled dynamics of fluid-structure system. In particular, for μ  1 the coupled fluidstructure dynamics is dominated by the fluid inertia and vice versa, for μ  1 i.e., structural dynamics dominates the coupled dynamics. Figure 12.6 shows the effect of mass ratio on the temporal growth rate of the instability at various mass density ratios μ. Because our aim is to understand the effects of mass ratio on stability, the dispersion relation is simplified by neglecting the tension effects and the flow domain is assumed to be unbounded. We observe that as the mass ratio increases the disturbance grows at a faster rate. A similar transient response of flexible baffle was observed in the fully-coupled fluid-structure simulations for varying mass ratio [186]. As the mass ratio increases, the growth rate of tip displacement increases in a given time interval for the identical conditions. Figure 12.7 describes the spatial instability of the dispersion relation for different values of mass density ratio. This plot is created by assuming real values of angular frequency ω and calculating the complex wave numbers k x for each of these wave ∂ω describe the numbers by solving the dispersion relation. The slope of the contours ∂k x group velocity of the disturbance (i.e.,the advection behaviour of the disturbance). The group velocity is the velocity of energy propagation of fluid-elastic waves and it loses its meaning for unstable waves. We observe from Fig. 12.7 that the group velocity of the disturbance is approximately similar for all the mass ratios. Increasing the mass ratio, the frequency range increases over which the plate is convectively unstable.

12.2 Linear Stability Analysis

689

0.004

μ=0.1 μ=0.2 μ=0.4 μ=0.8 μ=1.6

0.003

ω

0.002

0.001 Unstable

Stable

kc

0 0

0.05

0.1

0.15

0.2

kx Fig. 12.6 Effects of mass ratio μ on the instability growth rate where H = ∞ and U0 = 0.05. For μ = 1.6, the stability boundaries are shown

12.2.4.3

Effect of Channel Height

This section investigates the effect of the channel height on the aeroelastic instability. To understand the effects of channel height on the stability, the stabilizing effects of tension are neglected. Figure 12.8 shows the Ucr versus μ for different values of nondimensional channel height, H . We observe that with increase in the finite height H , there is an increase in the value of critical velocity Ucr at a given mass ratio μ. This behaviour can be explained with the aid of added mass effects. As the height or channel confinement increases, the added mass force acting on the plate reduces as a multiplier function of coth (k x H ) (see Eq. (12.30)). In other words, the more the plate is confined, the larger is the added mass. Hence the smaller is the critical value of the velocity for a given mass ratio μ. However, for a very small confinement one may also wonder whether the viscosity has a role on the critical instability limits. Due to the viscous drag of both parallel walls, it may be necessary to consider a boundary layer of a Poiseuille-type flow. Traditionally, the growth of flapping disturbances are associated with vorticity or curvature effects on velocity due to the instability of an interface between two distinct parallel streams (i.e., Kelvin-Helmholtz instability). However, the instability phenomenon reported here are not caused by the vorticity effects. They seem to be associated with the dynamic interaction of the plate with the mean flow, with the major role of plate tension, elasticity and the mass ratio. Near the onset point to flapping,

690

12 Theoretical Background of Flexible Plate

(a) 0.004

μ=0.1 μ=0.2 μ=0.4 μ=0.8 μ=1.6

ω

0.003

0.002

0.001

0.05

0.15

Re(kx)

(b)

0.004

0.1

Stable

ω

0.003

0.002 Convective Instability

0.001

0

0

0.05

0.1

0.15

Re(kx) Fig. 12.7 a Effects of mass ratio μ on the convective growth rate of the instability where H = ∞ and U0 = 0.05. Plotted are the solutions k x (ω) of D (k x , ω) = 0 for real positive ω. b The spatial dispersion diagram for μ = 0.8 and U0 = 0.05. The dashed line denotes the complex modes and the solid line denotes the neutral modes

12.2 Linear Stability Analysis 10

691

3

Critical Velocity, Ucr

H=0.0001 H=0.001 H=0.01 H=0.1 H=1 10

2

Unstable

101

100

Stable

2

4

6

8

10

μ Fig. 12.8 The stability limit for flow velocity, critical velocity Ucr Vs μ for various heights of the channel

the mass of the plate can balance with the surrounding mass of the interacting fluid, while the elastic energy of the plate balances with the kinetic energy of the mean flow.

12.2.4.4

Effect of Flexibility

To realize the nondimensional parameters that can influence the onset of flapping instability, we consider a 2D flexible foil of length L and thickness h with density ρ s placed in a uniform axial flow with velocity U0 in the x-direction and density ρ f as shown in Fig. 12.1. Here we consider h  L, E and ν s denote Young’s modulus and Poisson’s ratio, respectively. The fluid flow is considered to be bounded on both sides and the height of the fluid column on both sides of the foil is H . A small perturbation α(x, t) = αe ˆ ikx x−iωt + c.c. ,

(12.52)

is provided to the flexible foil in the y-direction where αˆ is the amplitude of a harmonic disturbance with complex ω and k x as wave frequency and wave number. c.c. represents the complex conjugate of the preceding expression. This perturbation develops the velocity potentials as

692

12 Theoretical Background of Flexible Plate

φ(x, y, t) =

 iα (ω − U0 k x ) ik x x−iωt  k x (y−H ) e + e−k x (y−H ) + c.c. for y ≥ h/2 e k x sinh(k x H )

(12.53)

 iα (ω − U0 k x ) ik x x−iωt  k x (y+H ) e φ(x, y, t) = − + e−k x (y+H ) + c.c. . for y ≤ −h/2 e k x sinh(k x H )

(12.54) in the upper and lower fluid flow regions, respectively. Assuming that the motion of the elastic foil due to the small perturbation can be modeled by 2 ∂ 2α ∂ 4α Eh 3 s ∂ α  − T + ρ h = Δp , 2 4 2 ∂x ∂t 2 12 1 − (ν s ) ∂ x



(12.55)

where Δp denotes the net fluid loading on the foil due to the disturbance and T is the tension experienced by the foil due to viscous wall shear stress. We can represent T as C T ρ f U02 L Re−0.5 where C T is the tension coefficient and Re is Reynolds number defined as ρ f U0 L/μf . For simplicity, in this study, we assume that C T is independent of x. The linearized dispersion relation describing the perturbed system is given by D(k x , ω) =

Eh 3   k x4 + C T ρ f U02 L Re−0.5 k x2 − ρ s hω2 − 2ρ f coth(k x H ) 12 1 − (ν s )2 (12.56)  2 (ω − U0 k x ) . (12.57) kx

We can non-dimensionalize the above dispersion relation by introducing the length, mass and time scales as L ref = L ,

Mref = ρ f L 3 and tref = L/U0

(12.58)

respectively to get 1

3 ! " 2  ! "  ! " 3 Eh ρsh 4 −0.5 2  (ω )2 D(k x , ω ) =  (k ) + C (k ) − Re T x x ρf L 12 1 − (ν s )2 ρ f U02 L 3  2

ω − k x − 2 coth(k x ! H . (12.59) " ) k x



4

where k x = k x L ,

ω =

ωL , U0

t =

tU0 L

and

H =

H . L

(12.60)

12.2 Linear Stability Analysis

693

In addition to the two wave properties k x and ω , one can observe that the nondimensional dispersion relation in Eq. (12.59) is a function of four nondimensional parameters ρ f LU0 ρsh H ∗ and H = . , m = f f μ ρ L L (12.61) The first nondimensional parameter is the nondimensional bending or flexural rigidity which will be denoted from here on as K B . The second one is the Reynolds number which is the ratio of inertial to viscous effects. The third one represents the ratio of the mass of the foil per unit length to that of the unit fluid column with which it is interacting which will be from here on denoted by m ∗ . In simple terms, m ∗ is referred to as the mass ratio. The mass ratio is an important parameter for any fluidbody interaction problem. The inverse of m ∗ which represents the nondimensional virtual added-mass that a structure experiences due to the surrounding fluid flow can also be used as an independent nondimensional parameter [568, 569, 614, 736]. The last nondimensional parameter is the gap ratio between the foil and the channel wall. We will be having only the first three non-dimensional parameters for a foil in an unbounded open flow. In addition to the four independent nondimensional parameters one can also find√a dependent nondimensional parameter known as reduced velocity (Ur ) defined as m ∗ /K B [736]. The dispersion relation in Eq. (12.59) presents the relationship between k x and ω as a function of the four nondimensional parameters. The linear stability analysis consists of solving the dispersion relation for ω associated with real values of k and the four nondimensional parameters in Eq. (12.61). The condition for the instability is that there exists at least one root ω whose imaginary component is positive so that the amplitude associated with the wave exp[i (kx − ωt)] grows exponentially in time and the system is considered locally unstable. For a given wavenumber k and the four nondimensional parameters, the quadratic dispersion relation in ω will have an imaginary component if the discriminant, dk , of the quadratic equation is less than zero i.e. dk < 0. Therefore, the condition for the system to be unstable is KB =

Eh 3   , 12 1 − (ν s )2 ρ f U02 L 3

Re =

    m ∗ C T Re−1/2 k x − 2 coth (k x H ) + K B k x3 + 2 C T Re−1/2 + K B k x2 coth (k x H ) < 0. (12.62) In the above equation we have dropped (·) from H and k x for simplicity. From the above equation, we can come up with the expressions for critical (m ∗ )cr and (K B )cr for which the flexible foil experiences flapping instability. The expressions for (m ∗ )cr and (K B )cr are as follows (m ∗ )cr >

(C T Re−1/2 + K B k x2 ) coth (k x H ) coth (k x H ) − C T Re−1/2 k x /2 − K B k x3 /2

(12.63)

694

12 Theoretical Background of Flexible Plate 0.35 0.3 0.25

-4

KB =1.0

10

KB =2.5

10 -4

KB =5.0

10 -4

K =7.5

10

B

-4

*

0.2 Unstable

0.15 0.1 0.05 Stable

0

0

1000

2000

3000

4000

Fig. 12.9 Effect of nondimensional parameters m ∗ , K B and Re on the onset of flapping instability for H/L = ∞ and k x = 2π

and (K B )cr
7.0 × 10−4 ; (ii) periodic limit-cycle for 7.0 × 10−4 ≥ K B ≥ 1 × 10−4 ; and (iii) non-periodic limitcycle for K B < 1 × 10−4 . In the fixed-point stable regime, the flexible foil aligns with the surrounding flow and behaves like a rigid flat plate. However, on decreasing K B , the flexible foil loses its stability to perform regular flapping motion with a sinusoidal trailing edge tip-displacement response characterized by a unique flapping frequency. This regime is referred to as the limit-cycle oscillation regime. In the non-periodic flapping regime, the flexible foil experiences non-periodic flapping motion and the trailing edge tip-displacement response is characterized by varying amplitudes with multiple frequencies. Fig. 13.8 shows the trailing edge cross-stream (—) tip-displacements for six different non-dimensional bending rigidity values in a decreasing order. The trailing edge tip-displacements for the fixed-point regime are shown in Fig. 13.8a for K B = 10−3 . By reducing the K B values below 1 × 10−3 , the flexible foil loses its

13.3 Two-Dimensional Flapping Dynamics

717

0.1

0.1

0.05

0.05

0.05

0

-0.05

0.15

δy /L

δy /L

0.15

0.1

δy /L

0.15

0

-0.1 -0.15 0

5

10

15

20

0

-0.05

-0.05 -0.1

-0.1

-0.15

-0.15

25

0

tU0 /L

5

10

(a)

tU0 /L

15

20

0

25

0.1

0.05

0.05

0.05

0

-0.05

0

-0.1

-0.1

-0.1

-0.15

-0.15

15

20

25

20

25

0

-0.15

tU0 /L

15

-0.05

-0.05

10

tU0 /L

0.15

δy /L

δy /L

0.1

δy /L

0.15

0.1

5

10

(c)

0.15

0

5

(b)

0

5

10

tU0 /L

15

20

0

25

5

10

15

(e)

(d)

20

25

tU0 /L

(f)

Fig. 13.8 Time history of cross-stream (—) tip-displacement at Re = 1000, m ∗ = 0.1 for conventional foil for K B = a 1 × 10−3 , b 3.75 × 10−4 , c 3.27 × 10−4 , d 1.8 × 10−4 , e 1.32 × 10−4 and f 5 × 10−5 i

i

ii

0.05

0.05

Y /L

0.1

Y /L

0.1

0

0

-0.05

-0.05

-0.1

-0.1

-0.15

0

0.2

0.4

X/L

(a)

0.6

0.8

1

iii

ii

-0.15

0

0.2

0.4

X/L

0.6

0.8

1

(b)

Fig. 13.9 The full body profile of conventional flexible foil performing flapping motion at Re = 1000, m ∗ = 0.1 over one full cycle for K B = a 3.27 × 10−4 , b 1.8 × 10−4

stability for a critical non-dimensional bending rigidity (K B )cr = 7 × 10−4 . For the values below (K B )cr , the flexible foil exhibits a regular sinusoidal cross-stream and stream-wise tip-displacements i.e., the limit-cycle oscillations regime. Typical limitcycle oscillations corresponding to K B = 3.75 × 10−4 , 3.27 × 10−4 , 1.8 × 10−4 and 1.32 × 10−4 are shown in Fig. 13.8b–e. For K B < 6.25 × 10−5 the regular sinusoidal tip-displacement response develops into non-periodic oscillations as shown in Fig. 13.8f for K B = 5 × 10−5 . To demonstrate the effect of flapping modes on the necking phenomenon, we plot the full body profile for K B = 3.27 × 10−4 and 1.8 × 10−4 in Fig. 13.9a and b, respectively. From these figures, one can observe that the flapping flexible foil

718

13 Isolated Conventional Flapping Foils

0.12

0.12

0.12

( 0.7812 , 0.09775 ) ( 0.7812 , 0.08234 ) 0.08

δy /L

δy /L

δy /L

0.08

0.08

0

0.04

0.04

0.04

0

2

4

6

8

0

10

0

4

f L/U0

(a)

(b)

0.12

0.12

6

8

0.12

2

4

f L/U0

6

8

10

6

8

10

( 0.8789 , 0.1073 )

( 0.9766 , 0.1062 )

0.08

δy /L

δy /L

0

(c)

0.08

( 0.9766 , 0.0682 )

0

10

δy /L

0.08

2

f L/U0

0.04

0.04

0.04

0

0

0

( 1.465 , 0.01944 )

0

2

4

f L/U0 6

(d)

8

10

0

2

4

f L/U0

(e)

6

8

10

0

2

4

f L/U0

(f)

Fig. 13.10 Amplitude-frequency spectrum of the cross-stream tip-displacement at K B = a 10−3 , b 3.75 × 10−4 , c 3.27 × 10−4 , d 1.8 × 10−4 , e 1.32 × 10−4 and f 5 × 10−5

exhibits two distinct flapping modes. The full body profile for K B = 3.27 × 10−4 shown in Fig. 13.9a exhibits the necking phenomenon at two different locations, whereas the full body profile for K B = 1.8 × 10−4 in Fig. 13.9b shows the necking phenomenon at three different locations. To further examine the effect of flapping modes on the flapping frequency, we show the amplitude-frequency spectrum in Fig. 13.10 for the cross-stream tipdisplacements discussed above. Figure 13.10a shows that the amplitude-frequency spectrum for K B = 10−3 for the fixed-point regime, which does not display a distinct frequency peak. The limit-cycle regime for K B ∈ [5.6 × 10−4 , 3.27 × 10−4 ] displays a single distinct non-dimensional flapping frequency f L/U0 = 0.7812 as shown in Fig. 13.10b, c for K B = 3.75 × 10−4 and 3.27 × 10−4 , respectively. We observe a sharp rise in the limit-cycle flapping frequency which stabilizes to f L/U0  0.9766 for K B < 3.27 × 10−4 , as shown in Figs. 13.10d, e for K B = 1.8 × 10−4 and 1.32 × 10−4 , respectively. An identical non-dimensional frequency f L/U0 = 0.9766 was also reported by [645] for m ∗ = 0.1, Re = 1000 and K B = 10−4 . The non-periodic flapping response for K B = 5 × 10−5 exhibits multiple frequencies, which can be seen in Fig. 13.10f. Fig. 13.11 shows the phase relation for K B = 3.27 × 10−4 and 1.8 × 10−4 at Re = 1000 and m ∗ = 0.1. The phase difference between the streamwise and cross-stream displacement responses remains unaffected due to the change in the flapping modes. For both these cases, we observe figure eight which indicates that the phase difference between streamwise and cross-stream displacement response is π/4 with respect to the streamwise response. Figure 13.12a, b illustrate the instantaneous vorticity contours at tU0 /L = 25 for K B = 10−3 and 3.27 × 10−4 respectively. Figure 13.12a shows the typical steady symmetric wake exhibited by the fixed-point stable regime.

13.3 Two-Dimensional Flapping Dynamics

719 0.15

0.05

0.05

δy /L

0.1

δy /L

0.1

0

0

-0.05

-0.05

-0.1

-0.1

-0.15

-0.15 -0.2

-0.1

0

0.1

-0.2

-0.1

0

0.1

δx /L

δx /L

(a)

(b)

1

1

0.5

0.5

Y/L

Y/L

Fig. 13.11 Phase relation between the cross-stream and stream-wise tip-displacements for conventional foil at K B = a 3.27 × 10−4 and b 1.8 × 10−4

0 -0.5 -1

0 -0.5

0

2

4

X/L

(a)

-1

0

2

4

X/L

(b)

Fig. 13.12 Instantaneous vorticity contours at tU0 /L = 25 for a fixed point stable mode at K B = 10−3 , b limit-cycle flapping mode at K B = 3.27 × 10−4 . Other non-dimensional parameters are Re = 1000 and m ∗ = 0.1. Dashed lines denote the negative vorticity

The regular 2S von Kármán vortex street in Fig. 13.12b corresponds to the limit-cycle flapping regime. Fig. 13.13 summarizes the rms cross-stream tip-displacement for a range of K B ∈ [10−5 , 10−3 ] or Ur ∈ [10, 45] for m ∗ = 0.1 and Re = 1000. For Ur ≤ 12.5 the flexible foil remains in the steady state condition and thus corresponds to the fixed-point stable regime. For Ur > 12.5, the flexible foil loses its stability to exhibit self-sustained flapping instability with a unique flapping frequency. The reduced velocity value of Ur = 12.5 represents the critical reduced velocity, (Ur )cr , above which the conventional flexible foil experiences the flapping motion. The amplitude of the flapping response exhibits a sharp rise as Ur is increased from 12.5 to 21.1. Interestingly, on further increasing the Ur we observe a sudden drop in the flapping amplitudes and it is attributed to the transition from lower flapping mode to higher flapping mode. For Ur ∈ [12.5, 39], the trailing edge tip-displacement response is characterized by a regular sinusoidal response with a single distinct flapping frequency. Hence, Ur ∈ [12.5, 39] corresponds to the limit-cycle oscillation regime. For Ur ≥ 40 or K B < 7.5 × 10−5 , the flexible foil flapping is characterized by variable amplitudes and multiple frequencies representing the non-periodic flapping regimes.

720

13 Isolated Conventional Flapping Foils 0.1

III

I

II

0.1

0.06

0.06

0.04

0.04

0.02

0.02

0 0

III

II

0.08

(δ/L)rms

(δ/L)rms

0.08

I

0.2

0.4

KB

0.6

0.8

(a)

1 −3

x 10

0

10

15

20

25

Ur

30

35

40

45

(b)

Fig. 13.13 Root-mean-squared trailing edge displacement (δ/L)rms for conventional foil as a function of: a bending rigidity K B , b reduced velocity Ur . Flapping response regimes are demarcated at Re = 1000 and m ∗ = 0.1 as: (I) fixed-point stable, (II) limit-cycle oscillations and (III) non-periodic flapping

13.3.4 Effect of Mass Ratio We next present the effects of mass ratio on the flapping dynamics of a flexible foil for Re = 1000. For this study, we consider the following nondimensional parameters: K B = 0.0001, h = 0.01L and m ∗ ranging between 0.01 and 0.2. We initialize the simulations by introducing a small transverse perturbation of magnitude 0.1L at the trailing edge of the foil. Similar to K B , m ∗ also exhibits three dynamic flapping regimes. Figure 13.14 shows the time histories of nondimensional crossstream displacement of the trailing edge for different m ∗ values at Re = 1000 and K B = 0.0001. From the cross-stream tip displacement curves shown in Fig. 13.14, it is clear that there exist three distinct regimes of response. For m ∗ ≤ 0.025, the perturbed flexible plate stabilizes and realigns itself with the flow field. This regime is known as the fixed-point stable. However for m ∗ ≥ 0.05, the perturbed flexible foil no longer stabilizes to the stable-straight configuration, instead it undergoes a regular flapping motion. The time history of the cross-stream trailing edge displacement presents a regular sinusoidal response with a unique flapping frequency and amplitude. The mass ratio at which the coupled system losses its stability to perform flapping motion for a given K B and Re is described as critical mass ratio (m ∗ )cr . By further increasing the m ∗ , for the mass ratios m ∗ ≥ 0.125, the regular flapping motion starts to become irregular and non-periodic. The trailing edge response of such flapping motion exhibits multiple frequencies and amplitudes. The range of m ∗ values for which the flapping phenomenon is regular with a single unique amplitude and frequency is referred to as the limit-cycle oscillations (LCO) regime. On the other hand, the range of m ∗ values for which the foil exhibits non-periodic flapping is called as a nonperiodic flapping regime. To further characterize the trailing edge tip displacement response, we perform FFT and plot the amplitude-frequency spectrum in Fig. 13.15 for a range of m ∗ between 0.05 and 0.125. A single unique frequency f 1 = 0.97656

13.3 Two-Dimensional Flapping Dynamics

721

0.1

0.1

δy /L

0.2

δy /L

0.2

0

0

-0.1

-0.1

-0.2

-0.2 0

5

10

tU0 /L

15

20

0

25

tU0 /L

15

20

25

0.2

0.2

0.1

δy /L

0.1

δy /L

10

(b)

(a)

0

0

-0.1

-0.1

-0.2

-0.2 15

5

20

15

25

20

25

tU0 /L

tU0 /L

(c)

(d)

0.2 0.2

0.1

δy /L

δy /L

0.1

0

-0.1

-0.1

-0.2

-0.2 15

0

20

tU0 /L

(e)

25

15

20

tU0 /L

25

(f)

Fig. 13.14 Nondimensional cross-stream displacement histories of the trailing edge in axial flow at Re = 1000 for K B = 0.0001 and m ∗ = a 0.01, b 0.025, c 0.05, d 0.075, e 0.10 and f 0.125

characterizes the limit cycle flapping regime (0.05 ≤ m ∗ ≤ 0.1), whereas the nonperiodic flapping regime i.e. for m ∗ ≥ 0.125 is associated with multiple frequencies. Fig. 13.16 plots the instantaneous vorticity contours in each of the three stability regimes. For m ∗ corresponding to the fixed-stable flapping regime, a narrow and steady wake is observed marked by no oscillation in the wake vortices. A periodic 2S von Kármán vortex pattern is observed for m ∗ = 0.075 and m ∗ = 0.1 for the limit-cycle flapping regime. An irregular non-periodic vortex street characterizes the non-periodic flapping regime for m ∗ ≥ 0.125.

13 Isolated Conventional Flapping Foils

0.12

0.12

0.1

0.1

0.08

0.08

δy /L

δy /L

722

0.06

( 0.9766 , 0.09667 )

0.06

( 0.9766 , 0.04315 ) 0.04

0.04

0.02

0.02

0

0 0

2

4

f L/U0

6

8

10

0

2

4

f L/U0

6

8

10

6

8

10

(b)

(a) 0.12

0.12

( 0.9766 , 0.1048 ) 0.1

0.1 ( 0.8789 , 0.08441 ) 0.08

δy /L

δy /L

0.08 0.06

0.06

0.04

0.04

0.02

0.02

0

0

2

4

f L/U0

6

8

0

10

( 1.367 , 0.04907 )

0

2

4

f L/U0

(d)

(c)

Fig. 13.15 Amplitude-frequency spectrum of the cross-stream displacement at Re = 1000 for K B = 0.0001 and m ∗ = a 0.05, b 0.075, c 0.10 and d 0.125 0.5

Y/L

Y/L

0.5

0

-0.5

-0.5 0

1

2

3

0

1

2

X/L

X/L

(a)

(b)

0.5

3

0.5

Y/L

Y/L

0

0

-0.5

0

-0.5 0

1

2

3

0

1

2

X/L

X/L

(c)

(d)

3

Fig. 13.16 Instantaneous vorticity contours at tU0 /L = 25 in the wake for different mass-ratios at Re = 1000 for K B = 0.0001. Dashed lines denote the negative vorticity

13.3 Two-Dimensional Flapping Dynamics

723

We next present the fluid-dynamic force coefficients for the three dynamic flapping regimes. Figure 13.17 shows the time series of drag coefficient Cd and lift coefficient Cl for m ∗ = 0.025, 0.1 and 0.125 for Re = 1000, K B = 0.0001 and t ∈ [15, 20]. For m ∗ = 0.025 (fixed-point stability), as shown in Fig. 13.17a, it can be seen that the drag coefficient Cd converges to 0.084, which is equal to the value of Cd for a flat plate at zero degree incidence obtained from the classical boundary layer theory. It is apparent from Fig. 13.17b, Cd and Cl are non-periodic for m ∗ = 0.1. We note snapping-like phenomena for the non-periodic flapping regime at m ∗ = 0.125 in Fig. 13.17c. A similar snapping phenomenon was also observed in [551].

13.3.5 Effect of Reynolds Number Fig. 13.18 presents the time evolution of the cross-stream tip displacement for Re = 500, 750 and 1000 at K B = 0.0001 and m ∗ = 0.1. With increasing Re, the system leads to either a limit-cycle or a non-periodic flapping mode. The maximum amplitude of cross-stream tip displacement increases with increasing Re. The stability of the system at lower Re can be explained by the stabilizing effect of the tension induced due to the viscous shear stresses [614]. The nonlinear stability results as a function of Re and m ∗ are summarized in Fig. 13.19. The diagram shows the three dynamical flapping regimes observed numerically by changing the mass density ratio and Re. The diagram shows the three distinct dynamic flapping regimes: (i) fixed-point stable, (ii) periodic limit-cycle, and (iii) non-periodic limit-cycle. The curves (—) and (· · · ) represent the transition from stable to limit-cycle and the transition from limit-cycle to non-periodic flapping regimes respectively. We find a reasonable fit of the numerical predictions of critical mass ratio with the theoretical transition curves given by Eq. (12.62) . We construct an empirical relationship for m ∗non−periodic based on the numerical simulation results as m ∗non−periodic = a m ∗cr

(13.18)

where the a = 2.5 is a multiplier. For m ∗ values above m ∗non−periodic the foil exhibits non-periodic flapping regime.

13.3.6 Net Energy Transfer To further explore the flapping characteristics of conventional foil, we now present a quantification of energy transfer between the fluid flow and flexible body. Due to the strong fluid-flexible body interaction, the flapping of structure significantly affects the wake topology which in turn alters the fluid force. There is a certain amount of flow, energy is transferred to the flapping foil which represents the work done by the

724

13 Isolated Conventional Flapping Foils

0.25

0.3

Current computation Theoretical value

0.2

0.2

0.1

Cl

Cd

0.15 0

0.1 −0.1 0.05

−0.2

0 0

20

15

10

5

20

21

22

tU0 /L

tU0 /L

(a)

(b)

0.4

23

24

25

23

24

25

23

24

25

0.3 0.2

0.3

Cl

Cd

0.1 0

0.2 −0.1 −0.2

0.1 20

21

22

23

24

20

25

21

22

tU0 /L

tU0 /L

(c)

(d) 0.3

0.4

0.2 0.3

Cl

Cd

0.1 0

0.2 −0.1 −0.2

0.1 20

21

22

23

24

25

20

21

22

tU0 /L

tU0 /L

(e)

(f)

Fig. 13.17 Time histories of the drag (left) and lift coefficients (right) at Re = 1000 for K B = 0.0001 and m ∗ = a 0.01, b 0.1, and c 0.125

fluid force to sustain the flapping during fluid-structure interaction. We estimate the bending strain energy E s and work done Ws on a flapping foil as follows: 1 2

Es =

l

t

E I κ 2 dl

0

ρ f U02

L2

and

Ws =

0

   f  f σ · n · u d dt 

ρ f U02 L 2

,

(13.19)

13.3 Two-Dimensional Flapping Dynamics

725

0.2

Re=500 Re=750 Re=1000

0.15 0.1

δy /L

0.05 0

-0.05 -0.1 -0.15 -0.2 15

20

25

tU0 /L Fig. 13.18 Evolution of cross-stream displacement at Re = 500, 750 and 1000 with m ∗ = 0.10, K B = 0.0001 0.4 Fixed point stable Limit cycle oscillation Non-periodic

0.35

mass ratio, m∗

0.3 0.25 0.2 0.15 0.1

Non-periodic Flapping

0.05

Limit Cycle Oscillation Fixed point Stable

0 0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Reynolds number, Re

Fig. 13.19 Stability phase diagram for Reynolds Number Re versus m ∗ . The plot is divided into three parts by m ∗cr and m ∗non−periodic . m ∗cr curve denotes the Eq. (12.62) and is represented by (—). m ∗non−periodic is defined by the Eq. (13.18) and is indicated by (· · · )

726

where

13 Isolated Conventional Flapping Foils

  2 ∂ f (x)/∂ x 2 

κ= 3/2 , 1 + (∂ f (x)/∂ x)2

(13.20)

is the curvature of the deformed flexible foil and f (x) is a piecewise polynomial function of 6th order that has been constructed to define the deformed foil profile at each time instant. Figure 13.20 shows the evolution of the non-dimensional bending strain energy (E s ) and the net work-done (Ws ) on a conventional foil for K B = 3.27 × 10−4 , Re = 1000 and m ∗ = 0.1. The strain energy E s of the flexible foil oscillates about a nonzero mean with a maximum value of O(10−4 ). The frequency of E s oscillations are twice the flapping frequency of the foil. The full body flapping profiles of the foil shown in Fig. 13.20c–f correspond to the points (i)-(iv) indicated in Fig. 13.20a. The strain energy E s of the flexible foil attains a local minimum for the maximum deformation of the trailing edge. In contrast, the flexible foil reaches the maximum E s as the trailing edge is about to cross the centerline. E s increases from local minimum to local maximum due to the work done by the fluid force on the flexible foil. The flexible foil eventually loses this attained strain energy E s by performing work on the surrounding fluid. Interestingly for the conventional foil flapping, we observe that the work done by the fluid in developing the structural strain energy is greater than the work done by the structure on the fluid. Therefore, the work done by the fluid per oscillation is greater than zero i.e. Ws > 0 and the net work done by the fluid continuously increases with time at a constant rate as shown Fig. 13.20b. This observation reveals that a conventional foil configuration behaves like a non-conservative system. The reason for Ws > 0 per oscillation cycle can be attributed to the non-conservative (dissipative) nature of fluid force and the path followed by the conventional flexible foil. From Figs. 13.20c–f, it can be realized that the conventional flexible foil does not follow an identical path during its upstroke and downstroke. Since the work done by a nonconservative force is path dependent, the work done by the fluid is different from the work performed by the structure. Figure 13.21 summarizes the maximum and rms non-dimensional strain energies as a function of K B ∈ [10−5 , 10−3 ] for Re = 1000 and m ∗ = 0.1. For sufficiently large K B corresponding to the steady stable mode, the maximum and rms bending strain energies are close to zero. However, for K B corresponding to the unsteady flapping regime i.e. K B < 7.0 × 10−4 , the maximum and rms strain energy values are of O(10−4 ). The sudden drop in the strain energy for K B < 3.27 × 10−4 can be attributed to the transition from the fundamental mode k x = 2π to the higher 3π flapping mode. By comparing local maximum strain energies for K B ∈ [7.5 × 10−5 , 3 × 10−4 ] and [3 × 10−4 , 7 × 10−4 ], we can conclude that lower flapping modes can extract greater bending strain energy as compared to the higher bending mode.

13.3 Two-Dimensional Flapping Dynamics

727

−4

x 10

0.18 iV

ii

0.16 4

ES

Ws

0.14 0.12

3

0.1 2

0.08 iii

i

1 20

21

22

23

24

0.06

25

tU0 /L

30

40

50

60

(b) 0.15 0.1

0.05

0.05

0.05

0.05

0

0.6

0.4

0.8

1

0

0.2

0.6

0.4

0.8

1

0

0

−0.05 −0.1

−0.1

−0.1 0.2

0

−0.05

−0.05

−0.1 0

0

Y /L

0.1

Y /L

0.15

0.1

Y /L

0.15

0.1

Y /L

0.15

−0.05

70

tU0 /L

(a)

0.8

0.6

0.4

0.2

1

0

0.2

0.6

0.4

X/L

X/L

X/L

X/L

(c)

(d)

(e)

(f)

0.8

1

Fig. 13.20 Comparison between the time evolution of a bending strain energy and b work done by the fluid on the flexible foil for K B = 3.27 × 10−4 , Re = 1000 and m ∗ = 0.1. c–f denote the full body profiles of the conventional foil at i − iv, respectively −4

6

x 10

5 4

Es

Fig. 13.21 Dependence of non-dimensional elastic strain energy on bending rigidity K B in a conventional flexible. Here (—) and (- - -) denote the maximum and rms energy at Re = 1000 and m ∗ = 0.1

3

2 1 0 0

1

2

3

KB

4

5

6 −4 x 10

(a)

In order to determine the effectiveness of the flapping phenomenon to extract energy, we define the ratio of the maximum strain energy to the total available fluid kinetic energy as ( E s )max L 2 , (13.21) R = 1 max δ T 2 y

728

13 Isolated Conventional Flapping Foils 1

0.8

0.8

0.6

0.6

s/L

s/L

1

0.4

0.4

0.2

0.2

0 20

21

22

23

24

25

0 20

21

22

23

tU0 /L

tU0 /L

(a)

(b)

24

25

Fig. 13.22 Space-time contours of transverse displacement (left) and pressure (right) along the foil over five flapping cycles for m ∗ = 0.1, K B = 0.0001 and Re = 1000

where ( E s )max is the maximum change in the strain energy per oscillation, δ max y represents the maximum tip displacement and T is the time taken to complete one half cycle. As a typical case, we consider K B = 3.27 × 10−4 , which exhibits maximum non-dimensional bending strain energy for examining the effectiveness of con= 0.12, E smax = 2.3 × 10−4 , ventional flexible foils at low m ∗ . Substituting δ max y f ρ = 1000 and T = 0.4 observed from the DNS results for K B = 3.27 × 10−4 , we get R = O(10−7 ), which is negligibly small. Therefore, we show that even though conventional foil can be used to harvest the energy from the surrounding fluid flows, the effectiveness of the conventional flexible foil for low m ∗ < 1 to extract energy from the surrounding fluid flow is very low. A similar observation has been made by [655] for m ∗ < 1.

13.3.7 Traveling Wave Mechanism A traveling wave is defined as a disturbance in the physical quantities such as displacement, velocity or pressure that either propagates upstream or downstream from its point of origin. In previous sections, we have shown that when a conventional foil is subjected to an infinitesimally small disturbance depending on the nondimensional parameters it can lose its stability to undergo a flapping motion which is nothing but a disturbance wave in the foil displacement. Earlier studies based on an inviscid reduced order model [654] and nonlinear viscous simulations [61] have shown that the flapping motion of a flexible foil exhibits kinematic waves traveling downstream from the leading edge and the reverse dynamic pressure waves traveling back upstream from the trailing edge. In this section, we want to characterize such a wave phenomenon in the flexible foil displacements and the fluid pressure for three distinct dynamic stability regimes: (a) limit-cycle oscillations, (b) fixed-point stable and (c) bistable. With the aid of the traveling wave phenomenon and the energy

13.3 Two-Dimensional Flapping Dynamics

729

2.2

2.2 0.025 0.015

0.01

1.8

2

ω/π

ω/π

2

0.02 0.015

1.8 0.01

0.005

1.6 -10

-5

0

5

p-

1.6 -10

p+

-5

0

k/π

k/π

(a)

(b)

0.005

5

Fig. 13.23 Space-time spectral contours of transverse displacement (left) and pressure (right) along the foil over ten flapping cycles for m ∗ = 0.1, K B = 0.0001 and Re = 1000

transfer from the fluid to the foil, we present the underlying physical mechanism for the bistability of a conventional flexible foil. Figure 13.22 presents the transverse displacement and pressure space-time contours along the foil over five flapping cycles for m ∗ = 0.1, K B = 0.0001 and Re = 1000 corresponding to the limit-cycle oscillation regime. The space-time contours for transverse displacement clearly show that the disturbance wave propagates downstream from the leading edge to the trailing edge. This observation is in agreement with the observations reported by [654]. On the other hand, the space-time contours for pressure in Fig. 13.22b show an interesting phenomenon. Unlike the transverse displacement, two distinct traveling pressure waves can be seen in the figure. The first wave travels downstream towards the trailing edge starting from s/L ≈ 0.7 and the second one can be seen traveling upstream for s/L  0.7, where s is the curvilinear coordinate along the foil with its origin at the leading edge. To further verify the observation found from the space-time contours, we plot the space-time spectral contours for the transverse displacement and pressure along the foil in Fig. 13.23. In Fig. 13.23b, p + and p − represent the pressure waves traveling downstream and upstream respectively. The space-time spectral contours confirm the observations reported above. The dominant wavenumber and frequency corresponding to the transverse displacement from Fig. 13.23a is 2.15π and 1.904π respectively. Therefore, the phase velocity of the traveling wave in transverse displacement is approximately 0.88. The pressure waves traveling downstream and upstream have identical frequency as the transverse displacement. However, the wavenumbers for the forward and reverse dynamic pressure waves are 1.55π and 6.19π respectively. Therefore, the phase velocities will be 1.2284 and 0.3076 for the forward and reverse pressure waves respectively. To understand the role of the pressure waves p + and p − on the flapping instability, we plot the time evolution of the net work done by the fluid on the complete foil, for the part of the foil where the pressure wave travels upstream towards the leading edge i.e. s/L ≤ 0.7 and for the rest of the foil i.e. s/L > 0.7 where the pressure wave travels towards the trailing edge in Fig. 13.24. The figure presents some interesting results which show that for a major part of the foil where the pressure wave travels upstream, i.e. p − pressure wave, it is the fluid that performs work on the foil. On the contrary, for the part of the foil with p + pressure wave, it is the foil that performs the

730

0.05 p-

0.04 0.03

Ws

Fig. 13.24 Comparison between the work done by the fluid on the complete foil (—), on a part of the foil for s/L ≤ 0.7 (− · −) and the rest of the foil for s > 0.7 (- -). The nondimensional parameters used are m ∗ = 0.1, K B = 0.0001 and Re = 1000

13 Isolated Conventional Flapping Foils

p + +p

-

0.02 0.01 0 -0.01 p+

-0.02 -0.03 15

20

25

tU0 /L

net work on the fluid. It should be noted that even though a part of the foil performs work on the fluid, the net work done by the fluid on the complete foil still remains positive. Therefore, we can say that the pressure wave p − absorbs energy from the surrounding fluid flow, whereas the pressure wave p + transfers energy to the fluid. We next investigate the traveling wave behavior for a damped flapping case corresponding to the fixed-point stable regime, to characterize the traveling wave phenomenon on the flapping instability. Figure 13.25a presents the trailing edge transverse displacement of a conventional foil for m ∗ = 0.1, K B = 0.00075 and Re = 1000 with an initial transverse disturbance of 0.2L. The initial disturbance at the trailing edge slowly damps in time. In Fig. 13.26a and b, we plot the spacetime spectral contours for the transverse displacement and pressure respectively for tU0 /L ∈ [10, 30]. Figure 13.26a clearly show that the initial disturbance at the trailing edge develops a transverse displacement which mainly propagates along the foil from the leading edge to the trailing edge. In addition to the forward traveling wave in transverse displacement, we also have a weak reverse wave due to the reflection at the trailing edge. Interestingly, even though we observe a traveling wave phenomenon for the transverse displacement, the space-time spectral contours for pressure in Fig. 13.26b does not show the existence any traveling wave. The dominant wavenumber and frequency from the space-time spectral contour is zero. Therefore, the foil loses all the potential energy attained due to the initial disturbance to attain the fixed-point stable state, which can be seen in Fig. 13.27. We finally analyze the traveling wave phenomenon in a conventional flexible foil that is bistable i.e. the stability of the foil depends on the magnitude of the initial disturbance. For this analysis, we have simulated m ∗ = 0.15, K B = 0.002 and Re = 1000 for two different initial conditions. In the first case, no initial disturbance is given, whereas in the second case we provide an initial transverse displacement with a magnitude of 0.2L. Figure 13.28 presents the trailing edge transverse tip displacements for both cases. One can clearly observe from this figure that the foil remains steady when no disturbance is provided. On the contrary, the foil performs

13.3 Two-Dimensional Flapping Dynamics

731

0.2

δy /L

0.1

0

-0.1 0

10

20

30

40

50

60

tU0 /L (a)

Fig. 13.25 Time histories of the trailing edge transverse displacements for m ∗ = 0.1, K B = 0.00075 and Re = 1000 with an initial transverse displacement of 0.2L 0.035

0.6

0.03

15

1

0.4

0.02 0.015

0.2

ω/π

ω/π

0.025

10

0.8

0.01 0.005

0 -6

×10 -3

1.2

-4

-2

0

k/π

(a)

2

4

6

5

0.6 -10

-5

0

5

10

k/π

(b)

Fig. 13.26 Space-time spectral contours for the damped flapping dynamics of a conventional foil with an initial transverse disturbance of 0.2L a transverse displacement and b pressure. The nondimensional parameters considered are m ∗ = 0.1, K B = 0.75 and Re = 1000

regular periodic flapping when a large initial disturbance is applied by an external source. In Fig. 13.29, we now plot the space-time spectral contours for the case with the initial external disturbance. The foil exhibits the traveling wave phenomenon in the transverse displacement that is propagating from the leading edge to the trailing edge. In this case, we do not have two traveling pressure waves. Instead, we have only one wave p − which is traveling upstream from the trailing edge to the leading edge. For the bistable case, since no p + pressure wave is present all the energy required for having a self-sustained flapping has to be provided by the surrounding fluid flow or by an external disturbance. Figure 13.30 shows the time evolution of the work done by the fluid on the foil for the case with the initial disturbance. This figure proves our conjecture, one can clearly see that the foil loses some of the potential energy

732

13 Isolated Conventional Flapping Foils 0

×10 -3

-1

Ws

-2

-3

-4

-5

10

20

30

40

50

60

tU0 /L Fig. 13.27 Work done by the fluid on the complete foil for m ∗ = 0.1, K B = 0.0001 and Re = 1000 corresponding to the fixed-point stable regime

0.2

0.2

0.15

0.15

δy /L

0.25

δy /L

0.25

0.1

0.1

0.05

0.05

0

0

-0.05

0

5

10

15

20

25

30

-0.05

0

5

10

15

tU0 /L

tU0 /L

(a)

(b)

20

25

30

Fig. 13.28 Time histories of the trailing edge transverse displacements for m = 0.15, K B = 0.002 and Re = 1000 without any initial disturbance (left) and with an initial transverse disturbance of magnitude 0.2L (right) 1.8

1.8

0.02 0.015

1.4

ω/π

0.025

1.6

ω/π

0.02

0.03

p-

1.6

0.015

1.4

0.01

0.01

1.2 -10

0.005

-5

0

k/π

(a)

5

10

0.005

1.2 -10

-5

0

5

10

k/π

(b)

Fig. 13.29 Space-time spectral contours of transverse displacement (left) and pressure (right) calculated over twenty flapping cycles for m ∗ = 0.15, K B = 0.002 and Re = 1000

13.3 Two-Dimensional Flapping Dynamics Fig. 13.30 Work done by the fluid on the complete foil for m ∗ = 0.15, K B = 0.002 and Re = 1000 with an transverse disturbance of amplitude 0.2L

0

733

×10 -3

-1 -2

Ws

-3 -4 -5 p-

-6 -7

5

10

15

20

25

30

tU0 /L

due to the initial disturbance to trigger the flapping instability. Unlike the limit-cycle oscillations, the net work done by the fluid per flapping cycle is zero and thereby it does not increase in time.

13.3.8 Interim Summary In this chapter, we have investigated the flapping instability and nonlinear post-critical flapping dynamics of a thin flexible foil interacting with viscous fluid flow, with its leading clamped and trailing edge left free to vibrate. We first performed a simplified linear stability analysis to identify the three nondimensional parameters that can affect the flapping of a thin flexible foil namely: (i) mass-ratio, m ∗ ; (ii) Reynolds number, Re; and (iii) nondimensional bending rigidity,  (K B ). We then performed a  series of numerical simulations for K B ∈ 10−5 ; 10−3 ; Re ∈ [500; 2000] and m ∗ ∈ [0.01; 0.125] to understand the effects of the nondimensional parameters on the postcritical flapping dynamics. The foil exhibited three distinct dynamical regimes as a function of nondimensional parameters m ∗ , Re and K1B : (i) fixed-point stable, (ii) periodic limit-cycle and (iii) non-periodic limit cycle flapping. In the fixed-point stable region, the foil remained steady and aligned in the direction of flow. For nondimensional parameters greater than the critical nondimensional parameters, the flexible foil loses its stability from the fixed point stable state to perform periodic limit cycle flapping with a distinct flapping frequency. Additionally, we observed that the periodic limit-cycle flapping is characterized by the regular von Karman vortex street. On further increasing the nondimensional parameters, we observed the flexible foil no longer exhibits a single unique flapping frequency. This flapping regime is called a non-periodic flapping regime. We then

734

13 Isolated Conventional Flapping Foils

examined the energy transfer mechanism to understand the physical mechanism behind the self-sustained flapping of flexible foils. We found out that the flexible foil continuously exchanges energy with the surrounding fluid. We showed that only a part of the energy that is transferred to the foil from the fluid is transferred back and the rest of the energy is dissipated. Our analysis also shows that the transverse disturbance propagates upstream from the leading edge to the trailing edge, thus confirming the existence of a traveling wave-like phenomenon. We also found out that the self-sustained flapping foil exhibits a coalescence of two pressure waves p + and p − traveling downstream and upstream respectively. The p + wave performs work on the fluid, whereas the p − wave requires work to be done by the fluid. For a fixed-point stable regime with damped oscillation, we still observed the transverse disturbance propagating upstream. However interestingly, in this case, no pressure waves were found.

13.4 Three-Dimensional Flapping Dynamics 13.4.1 Problem Statement To study the 3D effects on the onset of flapping instability and post critical flapping dynamics of a conventional foil with low m ∗ ∈ O(10−2 − 10−1 ), we consider a 3D computational domain of size [22L × 10L × (2L + H )] with f and s as shown in Fig. 13.31. The aspect ratio of the flexible foil s is H/L and a gap of L separates f . In addition to the the flexible foil’s sides from the computational domain sides sides boundary conditions considered for 2D simulations, a slip-wall boundary condition f . Figure 13.32 shows the isometric view of the is implemented along the sides sides finite-element fluid mesh consisting of approx. 4.3 million six-node wedge elements. Of the 49 layers 21 layers are distributed over the foil and the remaining 28 layers are distributed equally on either side of the flexible foil.

13.4.2 Role of Aspect Ratio on the Onset of Flapping The flapping dynamics of conventional foil is strongly affected by the ratio of width H to length L ratio. While a very large aspect ratio H/L >> 1 leads to a twodimensional elastic sheet, the small aspect ratio H/L 0.025 at Re = 5000, K B = 0.0001 and H/L = 0.5 to perform flapping motion. For m ∗ = 0.05, the streamwise and transverse displacements for the corner points (L , 0, 0) and (L , 0, 0.5L) overlap while the spanwise displacements are in outof-phase with each other. The midpoint on the trailing edge remains relatively steady in the spanwise direction. The spanwise displacements shown in Fig. 13.38b (right) are slightly different from the one presented by [606]. In [606], the foil is considered to be inextensible, whereas in our numerical study the foil has a Poisson’s ratio of 0.3 which gives a slight extension of the O(10−3 L) in both streamwise and spanwise directions. The Fourier transform of the trailing edge responses in Fig. 13.39a shows that there exists a single unique frequency for the flapping in streamwise, transverse and spanwise directions. The fundamental frequency of the flapping in streamwise and spanwise directions is found to be identical and twice the transverse flapping frequency. The flapping response in Fig. 13.38c for m ∗ = 0.075, exhibits similar flapping characteristics to that of m ∗ = 0.05 with a slight increase in the flapping amplitudes in the streamwise, transverse and spanwise directions. The flapping frequencies in both the streamwise and transverse are identical to m ∗ = 0.05, which can be observed from Fig. 13.39b. However, the spanwise flapping motion for m ∗ = 0.075 exhibits higher 3/2rd and 2nd harmonics in addition to the fundamental frequencies.

740

13 Isolated Conventional Flapping Foils ×10

0.2

0.04 z/L=0 z/L=0.25 z/L=0.5

z/L=0 z/L=0.25 z/L=0.5

0.1

-0.04

z/L=0 z/L=0.25 z/L=0.5

0.5

δz /L

δy /L

δx /L

0

-3

1

0

0

-0.5

-0.1 -0.08

-1 0

2

4

tU0 /L

6

8

10

-0.2

0

2

4

tU0 /L

6

8

5

10

6

7

(b)

(a)

8

9

10

(c) 2

z/L=0 z/L=0.25# z/L=0.5

0.02

tU0 /L

z/L=0 z/L=0.25 z/L=0.5

0.2

×10

-3

z/L=0 z/L=0.25 z/L=0.5

1

δy /L

δx /L

δz /L

0.1

-0.02

0

0

-0.1

-0.06

-1 -0.2 -0.1

5

6

7

tU0 /L 8

9

10

-0.3

5

6

7

tU0 /L 8

9

10

-2

5

6

7

(e)

(d)

9

10

(f)

0.02

2 z/L=0 z/L=0.25 z/L=0.5

tU0 /L 8

×10 -3

z/L=0 z/L=0.25 z/L=0.5

0.2

z/L=0 z/L=0.25 z/L=0.5

1 -0.02

-0.06

δz /L

δy /L

δx /L

0.1 0

0

-0.1 -1 -0.2

-0.1

5

6

7

tU0 /L 8

9

10

-0.3

5

6

7

(g)

tU0 /L 8

9

10

-2

5

0

z/L=0 z/L=0.25 z/L=0.5

0.2

tU0 /L 8

9

10

×10

-3

z/L=0 z/L=0.25 z/L=0.5

2

-0.04

δz /L

0.1

-0.02

δy /L

δx /L

7

(i) 4

z/L=0 z/L=0.25 z/L=0.5

0.02

0

0

-0.1

-0.06

-2 -0.2

-0.08 -0.1

6

(h)

7

8

9

10

11

12

-0.3

7

8

9

10

11

12

-4

7

8

9

10

tU0 /L

tU0 /L

tU0 /L

(j)

(k)

(l)

11

12

Fig. 13.38 Trailing edge tip-displacement histories in streamwise (left), transverse (center) and spanwise (right) directions for three points with coordinates (x, y, z) = (L , 0, 0), (L , 0, 0.25L) and (L , 0, 0.5L) on the trailing edge. The other nondimensional parameters corresponding to the tip-displacements are Re = 5000, K B = 0.0001 and H/L = 0.5

From Fig. 13.38b and c (left) and (center), one can observe that the flapping amplitudes of the two corner points in the streamwise and transverse directions are slightly greater than the midpoint (L , 0, 0.25L). This observation is distinctly different from the one reported by [606] where the authors have observed that the trailing edge midpoint exhibits higher flapping amplitude compared to the trailing edge corners. The reason behind this observation can be attributed to the 3D bending of the foil. To understand this physical phenomenon, we first plot the full body profile and its closeup view at the trailing edge in Fig. 13.40. Figure 13.40b clearly

13.4 Three-Dimensional Flapping Dynamics 0.1

8

×10

8

0.06

0.15

6

0

0.04

0

1

2

3

4

Ay /L

2

5

Az /L

( 1.953, 40.004542 )

Ax /L

0.2

-3

-4

×10

6

0.08

0.1

0.05

0

2

4

f L/U0

6

8

( 1.953, 0.0003666 )

4

( 0.9766, 0.06681 )

0.02 0

741

2

0

10

0

2

4

(a)

f L/U0

6

8

0

10

0

2

4 f L/U06

(b) 8

0.2

0.1 0.08

8

10

8

10

(c)

0.15

-4

×10

6

Az /L

Ay /L

Ax /L

( 0.9766, 0.12 )

0.06

0.1

0.04

0

( 1.831, 0.00036 ) ( 3.784, 0.0003065 )

0.05 0.02

4

2

( 2.808, 0.0002157 )

( 1.953, 0.01289 )

0 0

2

4

f L/U06

8

10

0

2

4

6

8

0

10

f L/U0

(d)

0

2

4

f L/U0 6

(f)

(e)

0.08

0.2

0.06

0.15

8

×10 -4 ( 1.831, 0.0006921 )

6

( 0.8545, 0.1308 )

Az /L

Ay /L

Ax /L

( 2.563, 0.0004846 )

0.04

0.1

( 3.54, 0.000369 )

4

( 0.9766, 0.0003017 )

0.02

0.05

( 0.7324, 0.01599 )

2 ( 4.395, 0.0001722 )

( 1.831, 0.01567 )

0

0

2

4

f L/U0

6

8

10

0

0

2

4

(g)

f L/U0

6

8

0

10

0

2

(h)

4

f L/U0 6

8

10

(i)

Fig. 13.39 Frequency-amplitude spectrum for the trailing edge response in streamwise (left), cross-stream (center) and spanwise (right) directions for m ∗ = a 0.1, b 0.075 and c 0.05 at K B = 0.0001, Re = 5000 and H/L = 0.5

0.07

z

Y /L

0.065

Y

0.06

0.055

x

0.5

Z/L

0.05 0.97

(a)

0.98

X/L

0.99

1

0

(b)

Fig. 13.40 Full body profile and the closeup view of the foil at the trailing edge when the trailing edge is at its maximum deformation for tU0 /l = 7.8. The corresponding nondimensional parameters are m ∗ = 0.05, Re = 5000 and K B = 0.0001

742 ×10 -3

0.045

0.05

0.065

Y /L

Y /L

0.04

0

-5

0

0.2

0.4

0.035

0.6

0.04

Y /L

0.045

5

Y /L

10

13 Isolated Conventional Flapping Foils

0.06

0.035

0.055 0

0.2

0.4

0

0.6

0.4

0.03

0.6

0.2

0.4

0.6

0.4

0.6

Z /L

(d)

-0.066

-0.015

0

(c)

(b)

(a)

0.2

Z /L

Z /L

Z /L

0.035 -0.042

-0.067

-0.025

0.03

-0.069

Y /L

-0.046

-0.068

Y /L

Y /L

Y /L

-0.02

-0.044

-0.048 -0.05

0.025

-0.052

-0.07

-0.054 -0.03

0

0.2

Z /L

(e)

0.4

0.6

-0.071

0

0.2

Z /L

(f)

0.4

0.6

0

0.2

Z /L

(g)

0.4

0.6

0.02

0

0.2

Z /L

(h)

Fig. 13.41 Equally spaced instantaneous trailing edge profiles projected onto the y−plane over a flapping cycle for m ∗ = 0.05, Re = 5000, K B = 0.0001 and H/L = 0.5

shows the foil bends upwards in the spanwise direction when it is at the maximum deformation. Therefore, the two corner points have greater transverse displacements. At the same time, as a result of this spanwise bending, the foil sides experience enhanced longitudinal bending which results in increased streamwise displacements at the foil sides compared to the midpoint. To further demonstrate the above physical phenomenon, we plot the projection of the trailing edge profiles onto the y−axis for m ∗ = 0.05 in Fig. 13.41 over a flapping cycle. It should be noted that the cross-sectional projections presented here are not uniformly scaled to clearly show the spanwise bending phenomenon along the foil. Figure 13.41a shows the trailing edge profile for m ∗ = 0.05 and corresponding to the time instant tU0 /L = 7.5. At this time instant, the trailing edge of foil is just about to cross the centerline on its upstroke as shown in Fig. 13.38a. One can observe at this time instant that the trailing edge of the foil is bent downwards. Figure 13.38b depicts the trailing edge profile of the foil after it has crossed the mean position during its upstroke. Therefore, the foil flips its spanwise bending direction once the foil crosses its mean position. Figure 13.38c and d show that the spanwise bending remains upwards until the foil crosses the mean position during its downstroke for tU0 /L ≈ 8. Figure 13.38 clearly displays that the spanwise bending direction at the trailing edge changes twice per flapping cycle. To explain why the foil bends upwards, we plot the spanwise velocity and pressure contours at the cross-sectional plane X = 0.98L in Fig. 13.42. When the foil is at its maximum deformation, we have a high and low pressure regions above and below the foil respectively. As a results of this pressure difference one can clearly observe a spanwise cross flow from the high pressure region above the foil to the low pressure region below the foil. Due to this cross flow, the pressure above the foil near the sides is relatively lower than at the spanwise centerline. Hence, the pressure difference which is acting against the foil inertia is less at the side than at the spanwise center.

13.4 Three-Dimensional Flapping Dynamics

743

Fig. 13.42 Cut-sectional view of the spanwise velocity (left) and pressure (right) distribution at X/L = 0.98 for tU0 /L = 7.7 when the foil is on its way to the maximum transverse displacement. The corresponding nondimensional parameters are m ∗ = 0.05, Re = 5000, K B = 0.0001 and H/L = 0.5

Therefore, the foil sides experience greater transverse and streamwise deformation compared to the foil centerline. Fig. 13.38d presents the trailing edge displacements for m ∗ = 0.1. The spanwise oscillations exhibit an interesting biased flapping. Even though Fig. 13.38d (right) shows that it is biased more towards the negative z−region, the biased flapping can also be towards the positive z−region which we will demonstrate later. The Fourier transform of the complex spanwise response in Fig. 13.39c shows that it is made up of multiple frequencies corresponding to 1/2-subharmonic, 3/2, 2 and 5/2 harmonics. We attribute this loss of symmetry in the spanwise flapping to a very small asymmetry in the spanwise velocity field. Therefore, there exists a critical m ∗ above which the spanwise flapping loses its symmetry. A similar biased flapping response in the spanwise direction has been demonstrated by [606] with increasing Re. Unlike the flapping response for m ∗ = 0.05 and 0.075 where the flapping motion of the two corner points in the streamwise and transverse direction was the same, for m ∗ = 0.1, the streamwise and transverse displacements of the corner point (L , 0, 0) exhibits slightly greater flapping amplitudes than the other corner point (L , 0, H/L). The difference in the streamwise and transverse flapping amplitudes of the corner points can also be attributed to this biased flapping in the spanwise direction. Additionally, the Fourier transform of the streamwise and transverse flapping responses in Fig. 13.39c shows that there is a small drop in the flapping frequencies compared to m ∗ = 0.05 and 0.075. Such a drop in the flapping frequencies typically signifies the change in the flapping mode. However, in this case, we do not observe any change in the flapping mode, which can be seen in Fig. 13.43. Instead, the only difference we observe is the biased flapping for m ∗ = 0.1. Fig. 13.44 shows the instantaneous isometric and bottom view of the 3D vortical structures shed from the flexible foil as a function of m ∗ ∈ [0.05, 0.1] when the foil is at its maximum transverse displacement. The Q-criterion described in [611] has been used to identify the 3D structures. In this figure, we show the iso-surfaces

744

13 Isolated Conventional Flapping Foils

0.1

0.1

0.05

0.05

Y /L

0.15

Y /L

0.15

0

0

-0.05

-0.05

-0.1

-0.1

-0.15

0

0.2

0.4

0.6

X/L (a)

0.8

1

-0.15

0

0.2

0.4

0.6

0.8

1

X/L (b)

Fig. 13.43 Full body profile of the foil spanwise centerline over a flapping cycle for m ∗ = 0.075 (left) and 0.1 (right) at Re = 5000, K B = 0.0001 and H/L = 0.5

corresponding to Q = 1 and flooded with the streamwise velocity. Figure 13.44a shows the instantaneous vortical structures for m ∗ = 0.05. As shown earlier in Fig. 13.42a, the flapping of finite aspect-ratio foils exhibits a cross-flow due to the pressure difference between the flow field above and below the foil. As a result of this cross flow, we can observe two counter-rotating streamwise vortices as shown in Fig. 13.45 (left) are formed on the lower surface of the foil when it is at the maximum transverse displacement. Since the 3D vortical structures are the combination of the streamwise vortex shedding from the foil side edges and the spanwise vortex shedding from the trailing edge, the difference in the strength of streamwise and spanwise vortices has a significant effect on the 3D structures. The spanwise vorticity is relatively very weak compared to the streamwise vorticity, which can be seen in Fig. 13.45 (right). As a result of which, we do not observe distinct 3D vortical structures for m ∗ = 0.05. By increasing the mass ratio from m ∗ = 0.05 to 0.075, due to the increase in the flapping amplitudes, we can observe from Fig. 13.46 that both the streamwise and spanwise vortices are strengthened, thereby resulting in the formation of the so-called hairpin-like vortical structures in Fig. 13.42b. However, the spanwise vorticity is still relatively weak compared to the streamwise component. Hence, the spanwise vortex dissipates as they travel downstream and we only observe the legs of the hairpin vortex. Two hairpin vortices are shed per flapping cycle. For m ∗ = 0.1, the spanwise vorticity further strengthens thereby resulting in the stable hairpin-like structures that can seen in Fig. 13.42c. The spanwise vorticity contours in Fig. 13.47 (right) show that the spanwise vortices elongate and breaks down into two vortices with an identical sign of rotation but unequal vorticity. Before we move to the next subsection on the effect of aspect ratio on the flapping dynamics, we summarize the findings of this subsection: • For m ∗  1, the two corner points on the trailing edge exhibits greater flapping amplitudes than the mid point on the trailing edge.

13.4 Three-Dimensional Flapping Dynamics

745

Fig. 13.44 Isometric (left) and bottom (right) view of the instantaneous 3D vortical structures with Q = 1 for the time instant when the foil is at its maximum transverse displacement for various m ∗ at K B = 0.0001, Re = 5000 and H/L = 0.5. The vortical structures are colored by the streamwise velocity

746

13 Isolated Conventional Flapping Foils

Fig. 13.45 Streamwise vorticity (left) along the streamwise cross-sectional slice at X/L = 0.98 and spanwise vorticity (right) along the spanwise center slice at Z /L = 0.25 for tU0 /L = 7.7 which corresponds to the time instance when the foil is at its maximum transverse displacement and nondimensional parameters m ∗ = 0.05, K B = 0.0001, Re = 5000 and H/L = 0.5

Fig. 13.46 Streamwise vorticity (left) along the streamwise cross-sectional slice at X/L = 0.96 and spanwise vorticity (right) along the spanwise center slice at Z /L = 0.25 for tU0 /L = 9.25 which corresponds to the time instance when the foil is at its maximum transverse displacement and nondimensional parameters m ∗ = 0.075, K B = 0.0001, Re = 5000 and H/L = 0.5

Fig. 13.47 Streamwise vorticity (left) along the streamwise cross-sectional slice at X/L = 0.94 and spanwise vorticity (right) along the spanwise center slice at Z /L = 0.25 for tU0 /L = 10 which corresponds to the time instance when the foil is at its maximum transverse displacement and nondimensional parameters m ∗ = 0.1, K B = 0.0001, Re = 5000 and H/L = 0.5

13.4 Three-Dimensional Flapping Dynamics

747

• A flexible foil with finite aspect ratio losses its stability to perform symmetric flapping in the spanwise direction. • There exists a critical m ∗ above which the foil starts exhibiting biased flapping in the spanwise direction which affects the flapping dynamics in both streamwise and transverse directions. • The foils bends upwards in the spanwise directions when it crosses the mean position during the upstroke. It reverses it bending direction again when the foil crosses the mean position during the downstroke. • The flapping motion generates two hairpin like 3D vortical structures per flapping cycle. The 3D hairpin like vortical structure is made up of the two counter rotating streamwise vortices at the foil sides and the spanwise wise vortex at the trailing edge.

13.4.5 Effect of Aspect Ratio Inertial swimmers or passive flapping motions have a strong dependence on the geometric aspect ratio H/L. Varying the aspect ratio results in a significant added mass effect and an impact on the dissipation of energy. A precise understanding of these local reactive or inertial components arising during the flapping plays an important role to understand the behavior of locomotion in nature but also for future conceptions of inertial artificial swimmers. For instance, slender body swimmers use a combination of lateral drag and inertial forces to generate thrust, whereas high aspect ratio swimmers achieve propulsion using mainly the added mass effects. Here, we explore the effect of aspect-ratio on the flapping dynamics for H/L = [0.5, 0.25, 0.125] at m ∗ = 0.1, Re = 5000 and K B = 0.0001. For the physical range of aspect ratios considered in the present study, the value of the added mass coefficient is linearly proportional to the width H of rectangular foil. This range is typically used in the elongated body theory, whereby the added mass effect has a dominant effect. The increment of inertial added mass reduces the natural flapping frequency of a cantilevered foil. In Fig. 13.48, we plot the streamwise, transverse and spanwise trailing edge displacements for H/L = 0.25 and 0.125 at three points (L , 0, 0), (L , 0, 0.5H ) and (L , 0, H ) along the trailing edge. From this figure and Fig. 13.38d, it can be observed that the flapping amplitude decreases as a function of aspect ratio. As the aspect ratio decreases, the cross-flow in the spanwise direction and the two counterrotating streamwise vortices come closer to the foil spanwise centerline as shown in Fig. 13.49, thereby reducing the maximum pressure difference across the foil at the foil spanwise cross-section. The reduction of pressure at the centerline reduces the lift acting on the foil which is quantified in Fig. 13.50 and therefore, flapping amplitudes decrease with aspect ratio. As the streamwise vortices come closure to the spanwise centerline, the pressure difference between the foil spanwise centerline and its sides also decreases. Hence, the difference between the two corner nodes and mid-point for the streamwise and transverse flapping amplitudes decreases. This can

748

13 Isolated Conventional Flapping Foils 0.4 z/L=0 z/L=0.125 z/L=0.25

0.02

2

z/L=0 z/L=0.125 z/L=0.25

0.2

×10

-3

z/L=0 z/L=0.125 z/L=0.25

1

δz /L

δy /L

δy /L

-0.02 0

0

-0.06 -0.2

-0.1

5

6

7

tU0 /L 8

9

10

-0.4

-1

5

6

7

tU0 /L

8

9

10

5

6

7

(b)

(a)

tU0 /L 8

9

10

(c)

0.4 z/L=0 z/L=0.0612 z/L=0.125

0.02

-2

2

z/L=0 z/L=0.0612 z/L=0.125

0.2

×10 -3 z/L=0 z/L=0.0612 z/L=0.125

1

δz /L

δy /L

δy /L

-0.02 0

0

-0.06 -0.2

-0.1

5

6

7

tU0 /L 8

(d)

9

10

-0.4

-1

5

6

7

tU0 /L

(e)

8

9

10

-2

5

6

7

tU0 /L 8

9

10

(f)

Fig. 13.48 Trailing edge tip-displacement histories for H/L = a 0.25 and b 0.125 in streamwise (left), transverse (center) and spanwise (right) directions for three points with coordinates (x, y, z) = (L , 0, 0), (L , 0, H/2L) and (L , 0, H/L) on the trailing edge. The other nondimensional parameters corresponding to the tip-displacements are m ∗ = 0.1, Re = 5000 and K B = 0.0001

Fig. 13.49 Streamwise vorticity for aspect ratios H/L = 0.25 (left) and 0.125 (right) along the streamwise cross-sectional slice at X/L = 0.96 for the time instant when the foil is at its maximum transverse displacement and nondimensional parameters m ∗ = 0.1, K B = 0.0001 and Re = 5000

be clearly observed from Figs. 13.48a and b, for both H/L = 0.25 and 0.125, all the three points have overlapping streamwise and transverse displacements. Figure 13.48a (right) clearly shows that the spanwise oscillation for H/L = 0.25 are biased. However, unlike the spanwise oscillation for H/L = 0.5 where the foil was biased downwards, for H/L = 0.25, it is biased upwards. Even though from Fig. 13.48b (right) it is not clear because the flapping amplitudes are of O(10−5 ), the spanwise oscillations for H/L = 0.125 are biased upwards. The Fourier transform of the flapping responses in both streamwise and transverse directions (Fig. 13.51) shows that the aspect ratio has little or no effect on the fundamental flapping frequencies. However, the secondary flapping frequencies are found to be attenuated

749

0.2

0.2

0.2

0.1

0.1

0.1

Cl

0

-0.1

Cl

Cl

13.4 Three-Dimensional Flapping Dynamics

0

-0.1

-0.2

-0.2

7

8

9

10

11

12

0

-0.1

-0.2

5

6

7

8

9

10

5

6

7

8

9

10

tU0 /L

tU0 /L

tU0 /L

(b)

(a)

(c)

Fig. 13.50 Time history of drag (left) and lift (right) coefficients for a range of aspect-ratios at m ∗ = 0.1, Re = 5000 and K B = 0.0001 0.2

8

0.15

6

0.1

×10 -4 ×10 -5

4

0.08

3

( 1.953, 2.249e-05 ) 2

Az /L

Ay /L

Ax /L

( 0.8545, 0.1236 )

0.06

0.1

0.04

0.05 0.02

0

0

1

2

3

4

5

2

( 0.7324, 0.009888 ) ( 1.831, 0.01144 )

0

1

4

0

2

4

0 6

f L/U0

8

10

0

2

4

6

8

0

10

f L/U0

0

2

4

f L/U0

0.2

0.1

8

8

10

×10 -4 4

0.08

6

(c)

(b)

(a)

×10 -5

3

0.15

6

0.04

0.1

Az /L

Ay /L

Ax /L

2

0.06

( 0.8545, 0.08905 )

0.05

0.02

( 1.587, 1.119e-05 ) 1

4

0

0

1

2

3

4

5

2

( 1.587, 0.009249 )

0

0

2

4

f L/U0

(d)

6

8

10

0

0

0

5 f L/U 0

(e)

10

0

2

4

f L/U0 6

8

10

(f)

Fig. 13.51 Frequency-amplitude spectrum for the trailing edge response in streamwise (left), crossstream (center) and spanwise (right) directions for H/L = a 0.075 and b 0.05 at m ∗ = 0.1 K B = 0.0001 and Re = 5000

with the aspect ratio. Additionally, no real trend could be observed for spanwise flapping frequencies mainly due to very small flapping amplitudes of O(10−5 ) for H/L = 0.125. Fig. 13.52 shows the instantaneous 3D vortical structures based on the Q-criterion for Q = 1 and flooded with the streamwise velocity for H/L = 0.25 and 0.125. The 3D vortical structures for H/L = 0.25 in Fig. 13.52a reiterate the hairpin structures observed for H/L = 0.5. However, one can observe from the bottom view that the hairpin vortical structures reduce to two cylindrical lobes representing the two counter-rotating streamwise vortices at the foil sides. This further indicates that the spanwise vorticity from the trailing edge weakens with aspect ratio and it becomes distinct from the 3D structures for H/L = 0.125 where we observe just two streamwise lobes.

750

13 Isolated Conventional Flapping Foils

Fig. 13.52 Isometric (left) and bottom (right) view of the instantaneous 3D vortical structures with Q = 1 for the time instant when the foil is at its maximum transverse displacement for various H/L at m ∗ = 0.1, K B = 0.0001 and Re = 5000. The vortical structures are colored by the streamwise velocity

13.4.6 Traveling Wave In Sect. 13.3 of this chapter, we have shown that the transverse displacement exhibits a traveling wave-like phenomenon that propagates from the leading edge to the trailing edge. In this subsection, we show that a flexible foil with a finite aspect ratio exhibits a traveling wave-like phenomenon for both transverse and spanwise displacements which propagates from the leading edge to the trailing edge at the sides. Figure 13.53 presents the space-time contours for the transverse and spanwise displacements along the foil length at the foil sides over five flapping cycles, where s is the curvilinear coordinate with s = 0 and s = L corresponding to the leading and trailing edges respectively. Figure 13.53a clearly shows that the transverse displacement propagates from the leading edge to the trailing edge. Even though the space-time contours in Fig. 13.53b for the spanwise displacement is distinct for s/L < 0.5, for s/L > 0.5 the figure clearly shows that the displacement propagating downstream. To further confirm the existence of the traveling we plot the space-time spectral contours for the spanwise displacements at the sides in Fig. 13.54.

13.4 Three-Dimensional Flapping Dynamics

751 1

0.8

0.8

0.6

0.6

s/L

s/L

1

0.4

0.4

0.2

0.2

0

5

6

7

8

9

10

0

5

6

7

8

tU0 /L

tU0 /L

(a)

(b)

9

10

Fig. 13.53 Space-time contours for the transverse (left) and spanwise (right) displacements at the foil sides over five flapping cycles corresponding to tU0 /L ∈ [5, 10] for m ∗ = 0.05, K B = 0.0001, Re = 1000 and H/L = 0.5 ×10 -3

16

2.4

14

ω/π

2.2

12 10

2

8 6

1.8

4 2

1.6 -10

-5

0

5

k/π Fig. 13.54 Space-time spectral contours for the spanwise displacements at the foil sides over five flapping cycles corresponding to tU0 /L ∈ [5, 10] for m ∗ = 0.05, K B = 0.0001, Re = 1000 and H/L = 0.5

13.4.7 Summary In this chapter, we presented the 3D flapping dynamics of both conventional and inverted foil configurations. We first began with the numerical analysis of the conventional foil for low m ∗ = 0.1 and relatively large Re = 5000. We first examined the flapping dynamics using a computational setup with spanwise periodicity to realize the 3D effects due to the foil three-dimensionality and large amplitude flapping motion. However surprisingly, the flapping remains purely twodimensional. We then studied the effects of finite foil width on the onset of flapping to find out that the foils with finite width were observed to be stable compared to their 2D counterpart. We attributed this physical phenomenon to the 3D periodic pressure distribution in both the streamwise and spanwise directions.

752

13 Isolated Conventional Flapping Foils

Following this, we investigated the effect of finite width on the flapping dynamics of a conventional foil. We observed that the foil curves outward in the spanwise direction due to the formation of streamwise vortices at the edges. As a result of this outward bending of the foil in the spanwise direction, the corner nodes on the trailing edge exhibit limit cycle oscillations in the spanwise direction and the flapping amplitudes of the corner nodes is slightly greater than the flapping amplitudes of the trailing edge midpoint for both the transverse and inline direction. It is found that as the width of the foil is reduced the flapping amplitude decreases due to a reduction in the pressure difference across the foil attributed to the movement of streamwise vortices towards the foil spanwise centerline. We then showed that the 3D flapping dynamics of a conventional foil are characterized by a pair of hairpin-like vortical structures per flapping cycle which are made up of two counterrotating streamwise vortices and a spanwise vortex. As the width of the foil is reduced, due to the movement of the streamwise vortices toward the spanwise centerline and decrease in the flapping amplitudes, the spanwise vortex gets suppressed thereby generating only two pairs of counter-rotating vortices per flapping cycle. Acknowledgements Some parts of this Chapter have been taken from the PhD thesis of Pardha Gurugubelli carried out at the National University of Singapore and supported by the Ministry of Education, Singapore.

Chapter 14

Proximity Effect

In this chapter, the proximity effects of flapping foils are investigated to understand how the flapping dynamics of flexible thin foils behave with a side-by-side arrangement. For the proximity effect, we examine a specific case when two flexible foils clamped at their leading edges are performing flapping adjacent to each other. We first study the effect of the gap between the foils on the coupled flapping modes, amplitudes, forces, frequency and vortex dynamics. We demonstrate the effects of proximity on the onset of flapping instability and then examine the role of the gap flow between the foils on the coupled flapping modes. We also visualize the back effects of the coupled flapping motion on the gap-flow rate and the velocity profiles inside the gap. Lastly, we report the effect of coupled flapping modes on energy harvesting efficiencies.

14.1 Introduction 14.1.1 Proximity Effects in Flapping Foils Apart from the conventional foil, there have been investigations on the variants of flexible foil flapping to understand the fluid-body interactions. In particular, foils do not operate in isolation and they undergo interactions with neighboring foils. The interactions of multiple flexible foils in various configurations pose a new set of challenges with regard to the coupled dynamics of fluid-body interactions. In the scope of this chapter, we consider the interactions of two parallel foils placed in a side-by-side arrangement. When two or more flexible foils are flapping in proximity to each other, the flapping phenomenon can experience a complex coupling through fluid-structure and structure-structure interactions. The soap film experiment of [784] is one of the first to visualize the coupling between two flexible filaments as a function of the gap between the foils. The flexible foils perform in-phase oscillations when the © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Jaiman et al., Mechanics of Flow-Induced Vibration, https://doi.org/10.1007/978-981-19-8578-2_14

753

754

14 Proximity Effect

U0

U0

(a)

(b)

Fig. 14.1 A representative vortex organizations for two flexible parallel plates aligned parallel to each other in open axial flow: a S + S (in-phase oscillation), b 2S + 2S (out-of-phase oscillation). The plates are pinned at the leading edge and free at the trailing edge

gap between the foils is small; otherwise, they exhibit out-of-phase oscillations. Figure 14.1 shows the typical in-phase and out-of-phase coupled flapping modes with S + S and 2S + 2S vortex modes respectively in the wake. They have also reported that the out-of-phase flapping exhibits approximately 35% higher flapping frequency when compared with its in-phase counterpart. This work [784] motivated several numerical studies [570, 787] to verify the experimental observations made. Zhang and Hisada [784] experimental work is followed by soap film and wind tunnel experiments of [616, 700], respectively, to understand the effects of the gap and foil length on the coupled flapping modes. The authors in [700] have also performed experiments on three and four parallel side-by-side foils and reported three distinct coupling modes as a function of the gap between them. In the first mode, all the foils exhibit in-phase coupled flapping. In the second mode, the two outer foils exhibit out-of-phase coupling while the coupling of the intermediate foils will depend on if we have three or four foils. In the case of three foils, the foil in the middle remains stable. On the other for the case of four foils, the two intermediate foils exhibit out-of-phase coupling with each other and in-phase coupling with the adjacent outer foil. In the third mode, every foil is in out-of-phase with its adjacent foils. More recently, [762] investigated the transition from the in-phase to out-of-phase coupling in a wind tunnel. The authors reported the existence of a transition mixed mode between in-phase and out-of-phase coupled modes. In this mode, the top and bottom do not have the same flapping frequency. Jia et al. [616] and Schouveiler and Eloy [700] extended the simplified analytical model presented by Shelley et al. [707] for a single flexible foil to two or more foils. Both studies have shown the existence of both in-phase and out-of-phase coupled modes and gave a good level of accuracy in predicting the onset of flapping instability. However, these analytical models could not predict the transition from the in-phase to the out-of-phase accurately. Similar to conventional flexible foil, coupled flapping dynamics of two sideby-side conventional foils have also been studied using both inviscid and NavierStokes solvers. Tang and Paidoussis [737] presented the coupled dynamics of the two cantilevered elastic foils in an axial open flow using the so-called lumped vortex model presented in [735]. In addition to the basic lumped vortex model, the authors have also presented a second model where two virtual springs connect both the foils to capture the in-phase coupled flapping. With the aid of both these inviscid

14.1 Introduction

755

models, the authors have shown that in-phase coupled modes are relatively more stable than their counterparts. The authors reported that both in-phase and out-ofphase coupled modes are not sensitive to the initial disturbance. Michelin [654] presented the double-wake approach based on [569, 593] to perform the coupled stability analysis on N parallel plate system. The authors found that the presence of the second foil destabilizes the parallel plate system and these destabilizing effects become significant with heavier foils. They also reported that for heavy foils i.e. for large m ∗ out-of-phase is the most dominant one. [763] also followed a similar approach [654] to study the coupled flapping modes in two side-by-side conventional foils. In [507], the author investigated the effects of wake synchronization on the flapping dynamics of two side-by-side elastic foils through numerical simulations based on a vortex-sheet model [506] in the limit of infinite Reynolds number. The author observed an erratic flapping behavior, where the flexible plates exhibit alternative in-phase and out-of-phase coupled modes for different time intervals. The author attributed it to a near collision behavior between the trailing edges of the foil. Recently [663] analyzed the coupled flapping modes in two parallel slender foils by generalizing Lighthill’s elongated body theory. The authors concluded that nonlinear interactions during the evolution of coupled modes play a significant role in the selection of the coupled flapping modes. In [570, 787], the authors performed 2D numerical simulations by employing the immersed boundary method and a weakly coupled fluid-structure solver respectively to demonstrate the coupled flapping modes observed experimentally by Zhang and Hisada [784]. Farnell et al. [570] treated the flexible foil as a combination of N rigid-elements. Both numerical studies have shown the existence of in-phase and out-of-phase coupled flapping as a function of the gap between them. Along with the in-phase and out-of-phase coupled modes, the numerical simulations of [570] found a quasi-periodic out-of-phase flapping mode for intermediate gaps. In [744], the author simulated a complex problem of three flexible foils arranged parallel to each other using the immersed boundary technique. In addition to the coupled flapping modes observed experimentally [700], the authors reported an erratic flapping regime, where the frequencies and amplitudes of all the three foils vary irregularly with time. Favier et al. [572] presented a similar numerical analysis on the various coupled flapping modes observed for two and three parallel foils using the lattice Boltzmann method. More recently, [743] performed 2D simulations on two parallel foils with zero mass to observe that unlike the single conventional foil with zero mass which remained steady [786] the parallel foil case undergoes a periodic flapping motion.

14.1.2 Organization In Sect. 14.3 of this chapter, we present a detailed two-dimensional numerical analysis of the coupled flapping dynamics of two parallel cantilevered flexible foils interacting

756

14 Proximity Effect

with a uniform axial flow. The two foils are separated by a gap and are clamped at their leading edges. To analyze the non-linear flapping dynamics and stability of the coupling modes, a high-order multi-body coupled fluid-structure solver has been employed based on the variational combined field formulation. The flapping of two side-by-side elastic foils presents a unique problem where the gap flow between the foils influences and gets influenced by the flapping motion of the flexible foils. Of particular interest is to understand the role of viscous boundary layer gap flow on the flapping response and dynamical interactions between the two foils. The flapping dynamics and unsteady gap flow are solved directly using the NavierStokes and the nonlinear elasticity equations. Unlike the earlier numerical studies [570, 787] with the bounded computational domain, we consider an unbounded open domain (low blockage effect) in this study. Moreover, our quasi-monolithic coupled model relies on a body-conforming treatment of the fluid-body interface to satisfy the kinematic constraints. This type of interface tracking allows accurate handling of viscous boundary layer effects and gap flow physics between the foils. In order to answer several outstanding questions related to the flapping of side-by-by foils, the objectives of this study are to: • Explore the coupled flapping modes as a function of the gap between the foils for a fixed m ∗ , K B and Re; • Analyze the dynamical effects and bistability of the coupled flapping modes; • Visualize the vortex modes generated due to the coupled flapping modes and understand the evolution of the coupled flapping modes; • Examine the role of the gap flow between the foils on the flapping instability and the coupled flapping modes; • Assess the effect of coupled flapping modes on the traveling wave phase velocity and energy harvesting ability.

14.2 Numerical Methodology 14.2.1 Governing Equations for Fluid-Foil System Let us consider two parallel elastic foils Ω top and Ω bot with their leading edges fixed and trailing edges free to perform flapping motion in a uniform open axial flow field Ω f (t). Figure 14.2 shows a typical schematic of two parallel flexible foils of length L and thickness h separated by a gap of d p interacting with an incompressible fluid flowing at U0 . The problem definition describes a flexible multibody flapping dynamics problem in an open axial flow field.

14.2 Numerical Methodology

757

y Ωtop Ωf U0 , ρf

dp

Ωbot

x

h L Fig. 14.2 Schematic of two parallel flexible foils of length L and thickness h separated by a spacing of d p in an axial flow with its leading edge fixed and trailing edge free

The Navier-Stokes equations governing an incompressible flow in an arbitrary Lagrangian-Eulerian (ALE) reference frame are  ρ

f

  ∂ uf  f f + u − w · ∇u = ∇ · σ f + f f on Ω f(t) , ∂t ∇ · u = 0 on Ω f

f(t)

(14.1)

,

  where uf = uf (x, t) = u f (x, t) , v f (x, t) and w = w(x, t) represent the fluid and mesh velocities defined for each spatial point x ∈ Ω f(t) respectively. u f (x, t) and v f (x, t) denote the component of fluid velocity along and perpendicular to the foil length. f f is the body force applied on the fluid and σ f is the Cauchy stress tensor for a Newtonian fluid, written as σ f = − pI + T , 1  f  f T  T = 2μf  f (uf ) and  f (uf ) = ∇u + ∇u . 2

(14.2)

where p is fluid pressure, I denotes second order identity tensor and T represents the fluid viscous stress tensor. Deformation of a flexible foil defined over Lagrangian reference frame is given by ρis

∂ usi = ∇ · σ si + f si in Ωis , ∂t

(14.3)

where subscript i represent the properties of the ith flexible foil, where us = us (z, t) is the structural velocity defined for each material point z ∈ Ωis , f s represents the external forces applied on the solid and σ s denotes the first Piola-Kirchhoff stress tensor. In this study, the structural stress tensor is modeled using the Saint VenantKirchhoff model [517].

758

14 Proximity Effect

The fluid and structural solutions are coupled through their interface boundary conditions given by 

 σ f (ηs (z, t), t) · nf dΓ +

σ s (z, t) · ns dΓ = 0 ∀ z, γi ⊂ Γi ,

(14.4)

γi

ηs (γi ,t)

uf (ηs (z, t), t) = us (z, t) ∀ z ∈ Γi ,

(14.5)

Here, Eqs. (14.4) and (14.5) represent the traction continuity and velocity continuity along the fluid-structure interface. nf and ns are, respectively, the outward normals to the deformed fluid and the undeformed solid interface boundaries, Γi represents the interface between the fluid and the ith flexible foil at t = 0, γi is an edge on Γi and ϕ s is the function that maps each Lagrangian point z ∈ Ωis to its deformed position at time t. To examine, the fully nonlinear flapping dynamics of flexible parallel plates, the following section describes the coupled fluid-structure scheme based on the Navier-Stokes equations and the geometrically nonlinear structural equations.

14.2.2 Multibody Combined Field Formulation In this section, the combined field with explicit interface (CFEI) formulation proposed for a single flexible body in [645] has been extended to solve the flexible multibody flapping dynamics. Liu et al. [645] proved the numerical stability of the CFEI formulation for very low structure-to-fluid mass ratio through the energy estimates and demonstrated the ability of the formulation to simulate the flapping dynamics accurately. More recently, Bourlet et al. [61] employed this CFEI formulation to analyze the laminar boundary layer on a flexible foil performing a coupled flapping motion. The weak form of the Navier-Stokes equations from Eq. (14.1) can be written as 

    ρ f ∂t uf + uf − w · ∇uf · φ f dΩ +

Ω f(t)



 f f · φ f dΩ +

Ω f(t)

Γnf (t)

 T f · φ f dΓ + Γ (t)

 σ f : ∇φ f dΩ =

Ω f(t)

 f  σ (x, t) · nf · φ f (x)d,

(14.6)

 q∇ · uf dΩ = 0.

(14.7)

Ω f (t)

Here ∂t denotes partial time derivative operator ∂ (·)/∂t, φ f and q are weighting functions for fluid velocity and pressure, respectively. Γ (t) = i=1 Γi (t) is the interface between the fluid and all the flexible foils. Γnf (t) represents the fluid Neumann

14.2 Numerical Methodology

759

boundary along which σ f (x, t) · nf = T f . The weak form of the structural dynamics equation in Eq. (14.3) can be written as 

 ρis ∂t usi · φ s dΩ + Ωis

 σ si : ∇φ s dΩ =

Ωis



f si · φ s dΩ+ Ωis



 s  σ i (z, t)·nsi · φ s (z)d,

T si · φ f dΓ +

(14.8)

Γi

s Γi,n

where φ s denotes the structural velocity weighting function. Similar to the fluid s represents the solid Neumann boundary and σ si (z, t) · nsi = T si . Further, domain, Γi,n the weak form of the traction continuity along ith fluid-structure interface in Eq. (14.4) can be represented as 

 f  σ (x, t)·nf · φ f (x)d +



 s  σ i (z, t)·nsi · φ s (z)d = 0.

(14.9)

Γi

Γi (t)

In the above equation, one may observe that uf and usi are defined on different domains Ω f(t) and Ωis , respectively, and the condition φ f (ηs (z, t)) = φ s (z)

∀z ∈ Γi

(14.10)

can be realized by considering a conforming mesh along the interface Γi with ηs (z, t) being the position vector of the deformed solid [183]. Therefore, the weak form of the combined field formulation obtained by combining Eqs. (14.6)–(14.9) will be 



    ρ f ∂t uf (x, t) + uf − w · ∇uf · φ f (x)dΩ +

Ω f(t)

σ f : ∇φ f dΩ

Ω f(t)



− + f · φ dΩ + f

Ω f(t)

f

Γnf (t)

ρis ∂t usi · φ s dΩ +

i=1 Ω s





 i

T ·φ d+ f

f

 i=1 Ω s



∇ · uf qdΩ

Ω f (t)

σ si : ∇φ s dΩ =

i=1 Ω s

 i

f si

· φ dΩ + s

T si · φ s dΓ . (14.11)

i=1 Γ s n

In this quasi-monolithic formulation, the solid positions and arbitrary LagrangianEulerian mesh velocities are decoupled from the remaining variables (i.e. fluid velocity, pressure and structural velocity) and are treated explicitly at the start of each time step. Moreover, the continuity of velocities across the fluid-structure interface is enforced in the function spaces and the continuity of traction across the interface

760

14 Proximity Effect

is enforced in the weak formulation. Consequently, there is a significant decrease in the size of the linear system required to be solved for each time step when compared to other monolithic schemes [186]. The solid position, ηs,n i , for the nth time step are determined using the second order Adam-Bashforth method ϕi = ϕi +

3Δt s,n−1 Δt s,n−2 u u − , 2 i 2 i

(14.12)

The fluid mesh nodes nodes represented by anj (j = 1, ..., G) on the fluid mesh τ f (t n ) with G nodes can be updated for the deformed interface locations using pseudoelastic material model [725] given by, ∇ · σm = 0

(14.13)

where σ m is the stress experienced by the fluid mesh due to strain induced by the interface deformation. Assume the fluid mesh as a linearly elastic material, we can represent the stress experienced as σ m = (1 + τm )

 T     + ∇ · ηf I , ∇ηf + ∇ηf

(14.14)

where ηf denotes the ALE mesh nodal displacement satisfying the boundary conditions ηf = ϕ(z, t) − z on Γi , ηf = 0 on Γ f (t)\Γ (t),

(14.15)

Γ f (t) is the fluid domain boundary and Γ f (t)\Γ (t) denotes the non-interface fluid boundary. τm is a mesh stiffness variable chosen as a function of the element size to limit the distortion of the small elements located in the immediate vicinity of the fluid|−mini |Ti | , structure interface. The mesh stiffness variable τm is defined as τm = maxi |Ti|T j| f n where Tj represents jth element on the mesh τ (t ). From the anj (j = 1, ..., G) data, the mesh velocities are determined using the second order approximation given by

wnj

 1 1  1  n n−1 n−1 n−2 n−2 n−3 aj − aj a a + − . = − aj − aj Δt 2 j 2 j

(14.16)

The weak variational form in Eq. (14.11) can be discretized in space using Pn /Pn−1 /Pn iso-parametric finite elements for the fluid velocity, pressure and solid velocity, respectively. In this paper, we employ the second order accurate backward difference technique to discretize the time domain and the stable P2 /P1 /P2 iso-parametric finite element meshes which satisfy the inf-sup condition for wellposedness.

14.3 Side-by-Side Foil Arrangements

761

14.3 Side-by-Side Foil Arrangements 14.3.1 Problem Statement To accomplish the above objectives, we consider two parallel flexible foils Ω top and Ω bot of length L and thickness h = 0.01L separated by a gap d p placed in a uniform axial flow Ω f with freestream velocity U0 as shown in Fig. 14.2. The flexible foils are fixed at their leading edges and the trailing edges are left free to perform the flapping motion. We perform a series of parametric DNS with identical physical parameters for both the foils are performed as a function of gap between the foils d p ∈ [0.02L , 2L], K B ∈ [1 × 10−4 , 3 × 10−3 ] and m ∗ ∈ [0.05, 0.2] for a fixed Re = 1000. Figure 14.3a shows the two-dimensional computational domain with two flexible cantilevered foils of length L and thickness h = 0.01L placed parallel to the flow direction. The leading edges of the top and bottom foils are clamped along x = 0 with their centers located at (0, d p /2) and (0, −d p /2) respectively. The size of the computational domains considered for this study is [−2L , 20L] × [−5L , 5L]. A uniform flow velocity U0 has been considered along the inlet boundary, Γinf . The traction-free f f f . Γtop and Γbottom represent boundary condition is imposed at the exit boundary, Γout

Γftop, uf = (U0, 0) Γfin uf = (U0 , 0)

h

Ωstop Ωsbottom

dp

Ωf (t)

Γfout σ f · nf = 0

L Γfbottom, uf = (U0 , 0)

(a) 5 4 0.4

3 2

0.2

1 0

0 -1

-0.2

-2 -3

-0.4

-4 -5 -2

0

2

4

6

8

10

12

14

16

18

20

-0.6 -0.5

0

0.5

1

1.5

(b) Fig. 14.3 Two parallel foils flapping: a schematic of parallel foil arrangement, b typical representation of high-order iso-parametric finite element mesh M3 for the complete fluid domain (left) and close-up view of boundary layer mesh at foil interface for two parallel foils system (right)

762

14 Proximity Effect

Table 14.1 Top and bottom foil mesh convergence study over the meshes M1, M2 and M3 for d p = 0.6L , m ∗ = 0.1, Re = 1000 and K B = 0.0001 Mesh

M1

M2

M3

Nodes Elements Top-foil: |δ/L|max

25297 12486

51809 25710

103085 51284

δ rms /L Clrms Bottom-foil: |δ/L|max δ rms /L Clrms

1.391 × 10−1 (1.40%) 1.424 × 10−1 (0.92%) 1.411 × 10−1 8.518 × 10−2 (0.61%) 8.581 × 10−2 (0.13%) 8.570 × 10−2 9.650 × 10−2 (0.10%) 9.544 × 10−2 (1.20%) 9.660 × 10−2 1.410 × 10−1 (2.08%) 1.452 × 10−1 (0.83%) 1.440 × 10−1 8.510 × 10−2 (1.13%) 8.555 × 10−2 (0.60%) 8.607 × 10−2 9.554 × 10−2 (0.56%) 9.565 × 10−2 (0.45%) 9.608 × 10−2

the lateral sides of the computational domain and the free-stream condition have been implemented. We nondimensional quantities of interest by choosing L , L/U0 and ρ f L 3 as the reference scales for length, time and mass. To perform a detailed mesh convergence study, we construct three high-order finite element meshes M1, M2 and M3 consisting of 12486, 25710 and 51284 P2 /P1 triangular elements respectively. Figure 14.3b shows the typical high-order finite element fluid mesh M3 along with a close-up image of the region around the foils. The wake mesh is sufficiently resolved to capture the wake and vortexinduced effects on the flapping. A boundary layer mesh has been provided at the fluid-structure interfaces to capture the effects of the laminar boundary layer. A typical two-foil system described above with d p = 0.6L has been considered for this study. Other non-dimensional parameters that have been considered for this study are m ∗ = 0.1, Re = 1000 and K B = 0.0001. Table 14.1 summarizes the results for the mesh independence test of the top and bottom foils. The table reports the maximum tip displacement, rms tip-displacement and rms lift coefficient for the meshes M1, M2 and M3. The values within the bracket denote the percentage difference in numerical solutions with respect to the mesh M3. of the cross-stream

The rms value  2 1 rms tip-displacement is calculated using δ /L = n t=15→25 δ ∗ − δ¯∗ , where δ ∗ is the non-dimensional tip-displacement defined as δ/L and δ¯∗ denotes the nondimensional mean tip-displacement. Similar definition has been used to calculate Clrms . All the numerical simulations have been carried out using the M2 mesh resolution.

14.3 Side-by-Side Foil Arrangements

763 0.6

top plate bottom plate

0.2 0.1

0.2

Y /L

Y /L

top plate bottom plate

0.4

0

0 −0.2

−0.1

−0.4 −0.2 0

0.2

0.4

0.6

0.8

1

−0.6

0

0.2

0.4

0.6

X/L

X/L

(a)

(b)

0.8

1

Fig. 14.4 Full body profiles of two parallel cantilevered flexible foils performing: a in-phase (d p = 0.1L); and b out-of-phase (d p = 0.6L) flapping

14.3.2 Effect of Gap on Coupled Dynamics In this section, we present the effects of the gap between the two parallel flexible foils on the coupled modes and flapping dynamics through a systematic numerical study as a function d p ∈ [0.1L , 2.0L]. The non-dimensional parameters considered for this study are m ∗ = 0.1, Re = 1000 and K B = 0.0001. Unlike the earlier DNS studies [570, 787], where the computational domain considered is a closed channel flow, here we consider a two-dimensional open axial flow with coordinates x in the direction of flow and y normal to the flow direction. The simulations are initialized with both the flexible foil straight. As the simulations proceed, the flexible foils experience the flapping motion due to the coupled fluid-structure instability. Interestingly, depending on the gap between the top and bottom foils the flapping motion exhibits a phase lock-in phenomenon. Four distinct phase lock-in regimes are identified as a function of an increasing gap: (i) in-phase for small gaps of 0.1L ≤ d p ≤ 0.25L; (ii) mixed-mode transition for 0.3L ≤ d p ≤ 0.4L; (iii) out-of-phase for 0.5L ≤ d p ≤ 0.8L; and (iv) independent for relatively large gaps of d p ≥ 0.9L. For the in-phase flapping regime, the the lock-in phase difference between crossstream displacements of the top and bottom is zero. Figure 14.4a demonstrates the full-body profiles of the top and bottom foils in-phase coupled flapping mode for d p = 0.1. On the other hand for the out-of-phase flapping regime, the cross-stream displacements of the top and bottom foils are in anti-phase i.e. the phase difference between the top and bottom foil responses is π . Figure 14.4b presents the full-body profiles for d p = 0.6L where the phase difference between top and bottom foils is π . The in-phase and out-of-phase coupled modes are in agreement with experimental studies [616, 700, 784] for varying gaps between the foils. For the case of a mixed-phase flapping regime, the transverse displacement of the top and bottom foils does not exhibit any phase lock-in i.e. no fixed phase difference between the top and bottom foils is observed. Foils can be in-phase or out-of-phase

764

14 Proximity Effect

0.1

0.05

0.05

δy /L

0.15

0.1

δy /L

0.15

0

-0.05

0

-0.05

-0.1

-0.1

-0.15

-0.15

15

20

25

25

tU0 /L

30

(a)

(b)

0.15

0.15

0.05

δy /L

0.1

0.05

δy /L

0.1

0

0

-0.05

-0.05

-0.1

-0.1

-0.15

-0.15

20

25

30

35

40

45

50

40

tU0 /L

(c)

50

tU0 /L

55

60

0.15

0.1

0.1

0.05

0.05

δ/L

δy /L

45

(d)

0.15

0

-0.05

0

−0.05

-0.1

−0.1

-0.15

−0.15

15

20

15

25

(e)

25

(f)

0.15

0.15 0.1

0.05

0.05

δy /L

0.1

0

-0.05

0

-0.05

-0.1

-0.1

-0.15

-0.15

15

20

tU0 /L

tU0 /L

δy /L

35

tU0 /L

20

tU0 /L

(g)

25

15

20

tU0 /L

25

(h)

Fig. 14.5 Cross stream tip-displacement evolutions of top foil (—) and bottom foil (- - -) for spacing between the foils a d p = 0.1L, b d p = 0.2L, c d p = 0.325L, d d p = 0.4L, e d p = 0.6L, f d p = 0.8L, g d p = 1.0L, h d p = 1.5L with m ∗ = 0.1, Re = 1000 and K B = 0.0001

14.3 Side-by-Side Foil Arrangements

765 top plate bottom plate

0.1

i

ii

iii

iv

δyrms /L

0.09

0.08

0.1 0 -0.1

0.07

15

20

15

20

25

0.1 0 -0.1

0.06 0

0.5

25

1

1.5

2

dp /L Fig. 14.6 Root-mean-squared value of the trailing edge tip displacement, δ rms y /L, plotted for a range of d p ∈ [0.1L , 2.0L] and m ∗ = 0.1 and Re = 1000. Four distinct flapping regimes are: (i) in-phase; (ii) mixed; (iii) out-of-phase and (iv) independent

depending on the time instant when we are measuring the phase difference. The independent flapping regime is observed for sufficiently large spacings between the foils. In this regime, the transverse displacement of the top and bottom foils exhibit a constant phase difference between them. However, unlike the in-phase and outof-phase we do not observe any unique phase difference over a range of spacing between the foils. The phase difference changes with the gap between the foils. We plot the time evolution of the top and bottom foil trailing edge tip-displacements for a range of nondimensional gaps between the foils in Fig. 14.5. Figure 14.5a, b show the typical trailing edge tip-displacement responses observed for the in-phase flapping with d p = 0.1L and 0.2L respectively. Figure 14.5e, f display the out-ofphase coupled mode responses for d p = 0.6L and d p = 0.8L respectively. In both the in-phase and the out-of-phase coupled modes, both the top and bottom foils flap with identical flapping frequencies. Additionally, the flapping response of the top (bottom) foil is biased towards the upward (downward) direction. This biased behavior of the tip-displacement responses becomes more lucid for the in-phase coupled modes. We attribute this outward biasing of the foils to high pressure at the gap inlet and inside. Figure 14.5c, d present the time history of the trailing edge cross-stream displacements for the mixed-mode flapping regime. Unlike the in-phase and out-of-phase regimes, the mixed-phase flapping regime exhibits different flapping frequency and amplitude for the top and bottom foils. The flapping response of both the foils exhibits a beats-like phenomenon due to the interference of two distinct frequencies. From Fig. 14.5c, d one can also observe that maximum and minimum response amplitudes occur when the trailing edge tip-displacement responses are in out-of-phase and in-

766 1.2

top-plate bottom-plate

1.1 i

ii

iii

iv

1

f L/U0

Fig. 14.7 Dependence of normalized frequency, f L/U0 , on spacing d p /L. The four flapping response regimes: (i) in-phase, (ii) mixed, (iii) out-of-phase and (iv) independent

14 Proximity Effect

0.9 0.8 0.7 0.6

0

0.5

1

1.5

2

dp /L

phase respectively. Similarly, a mixed mode response has been earlier observed by Tian et al. [744] for three parallel flexible foils. Figure 14.5g, h show the trailing edge tip-displacement time history for the independent flapping regime for d p /L = 1.0 and 1.5. From both these figures, one can clearly see that the phase difference between the top and bottom foil responses is different in both cases. Unlike the mixed mode where we do not have a constant phase difference between the top and bottom foils, in the independent foil regime, we have a constant phase difference but this phase difference is not unique and it keeps changing with the gap. It is expected that as the spacing between the foils is increased, both the foils should recover the tip-displacement response of the single foil. However, even for a large spacing of 2.0L the flapping responses of the foils are still influenced by the second foil. Figure 14.6 summarizes the rms value of cross-stream tip-displacement, δ rms y /L, as function d p ∈ [0.1L , 2.0L]. In this figure, we also demarcated the flapping response regimes as a function of d p /L to understand the effects of coupled flapping modes on the flapping amplitudes. For very large gaps, δ rms y /L values of both the foils is 0.084 which is approximately equal to their single foil counterpart. As the gap decreases, the foils start exhibiting the out-of-phase coupled flapping regime for d p /L < 0.9. In this regime, the rms tip-displacement values increases with decrease in gap and reaches a maximum for d p = 0.45L. For d p < 0.45L, a sudden drop in δ rms /L can be observed indicating the transition from out-of-phase coupled flapping. δ rms /L attains a minimum at d p = 0.2L, which corresponds to the in-phase coupled mode. The δ rms y /L values for the out-ofphase coupled modes are about 27% greater than that of the in-phase flapping. This observation of lower flapping amplitudes for open axial flows is distinctly different from the earlier numerical simulation studies [570, 787] in a closed channel flow with blockage effects. Due to the blockage, the flow outside the gap experiences a larger acceleration and resulting in a greater pressure difference between the gap and the outside. The blockage effects are less significant in an open axial flow domain.

14.3 Side-by-Side Foil Arrangements

767

The maximum blockage observed in this study is approximately 2.5%. A possible reason for the lower amplitude of oscillations in the case of in-phase can be due to the suppression of vortex shedding between the foils and a more detailed analysis of the vortex wakes. The flapping amplitudes of the top foil in the mixed-mode flapping regime are observed to be greater than the bottom foil. Even though our simulations have consistently shown that the top foil has a greater amplitude than the bottom foil, we believe that there is an equal possibility for the bottom foil to exhibit greater flapping amplitude. This motivates us to perform the effect of the initial condition on the mixed-mode flapping response later. Figure 14.7 illustrates the effects of the gap on the normalized flapping frequency which is defined as f L/U0 , where f represents the trailing edge response frequency. In Fig. 14.7, one can clearly observe that the normalized flapping frequency remains almost constant for d p ≥ 0.5L. This constant response frequency for d p ≥ 0.5L is approximately equal to its single foil counterpart. As the spacing drops below 0.5L, the frequency of flapping starts to decrease through a complex mode switching process from the out-of-phase flapping regime. For the mixed-mode flapping, we have only considered the primary flapping frequency. The in-phase flapping response for d p = 0.1L is 37.5% less than the out-of-phase flapping regime response. Similar observations have been reported by Zhang and Hisada [784] for two flexible filaments in the soap film experiment. Similar to the amplitude response, the top foil in mixed mode is observed to experience a greater frequency than the bottom foil. This phenomenon can be explained by the biased flow in the gap, which leads to a delay in the vortex roll-up of the outer shear layer from the bottom foil.

14.3.2.1

Sensitivity of Mixed-Mode and Independent-Mode

Earlier [737] have reported that both the in-phase and the out-of-phase coupled modes are not affected by the initial conditions i.e. the solution is distinctly unique irrespective of the initial conditions. In this subsection, we investigate the effect of initial conditions on the mixed-mode flapping regime. A typical mixed-mode flapping corresponding to d p = 0.35L at m ∗ = 0.1, Re = 1000 and K B = 0.0001 is considered for this investigation. As a part of this investigation, we simulate an identical problem with two different initial conditions. In the first case, we initialize the simulations with both the foils having an out-of-phase trailing edge displacement amplitude of 0.05L. While in the second case, we initialize the simulation with inphase trailing edge displacements. Figure 14.8 shows the time history of the trailing edge tip displacements. In both these conditions, the trailing edge tip displacements tend to follow the initial mode. This proves that the mixed-mode coupling is sensitive to the initial conditions and can exhibit either in-phase or out-of-phase depending on the initial condition. Similar to the sensitivity-analysis performed for the mixed-mode, we now examine the effect of initial conditions on the independent mode. For this examination we have considered d p = 1.5L , m ∗ = 0.1, Re = 1000 and K B = 0.0001 with two different

768

14 Proximity Effect 0.2

0.2

0.15

0.15

δy /L

0.1 0.05

δy /L

0.1 0.05 0

0

-0.05

-0.05 -0.1

-0.1

-0.15

-0.15

-0.2

0

5

10

15

20

25

-0.2 0

5

10

15

tU0 /L

tU0 /L

(a)

(b)

20

25

30

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05

δ/L

δ/L

Fig. 14.8 Time history of trailing edge tip-displacements of the top (—) and bottom (- - -) foils corresponding to the mixed-mode coupling with two different initial conditions: a out-of-phase and b in-phase. The nondimensional simulation parameters are d p /L = 0.35, m ∗ = 0.1, Re = 1000 and K B = 0.0001

0

−0.05

0

−0.05

−0.1

−0.1

−0.15

−0.15

−0.2

−0.2 0

5

10

15

20

25

0

5

10

15

tU0 /L

tU0 /L

(a)

(b)

20

25

Fig. 14.9 Time history of trailing edge tip-displacements of the top (—) and bottom (- - -) foils for the independent-mode coupled flapping with two different initial conditions: a out-of-phase and b in-phase. The simulation parameters are d p /L = 1.5, m ∗ = 0.1, Re = 1000 and K B = 0.0001

initial conditions. Figure 14.9 shows the trailing edge tip-displacement responses for both the out-of-phase and in-phase initial conditions. From this figure, it can be clearly seen that the independent-mode is also sensitive to the initial conditions.

14.3.2.2

Vortex Organization

In Chap. 13, we have shown that a single foil performing flapping motion develops a von Kármán vortex street with a pair of alternating counter-rotating vortices per flapping cycle. However, in parallel side-by-side flexible foils, the existence of a second vortex wake in the near proximity of the first one may strongly influence the shedding patterns which in turn can affect the force dynamics and the response characteristics. As a function of the gap, four distinct vortex mode regimes are observed: (i) S+S

14.3 Side-by-Side Foil Arrangements

769

1

Y/L

Y/L

1

0

A

0

B -1

-1 0

1

2

X/L

3

4

0

5

B

A 1

2

3

4

5

3

4

5

(b)

(a) 1

Y/L

1

Y/L

X/L

0

0

O

O -1

-1 0

1

2

X/L

(c)

3

4

5

0

1

2

X/L

(d)

Fig. 14.10 Time history of S+S vortex shedding modes over one flapping cycle for d p = 0.2L, Re = 1000, K B = 0.0001 and m ∗ = 0.1

vortex mode for d p ∈ [0.1L , 0.25L], (ii) mixed vortex mode for d p ∈ [0.3L , 0.4L], (iii) synchronized anti-phased 2S+2S vortex mode for d p ∈ [0.45L , 0.8L], and (iv) two independent wakes consisting of 2S vortex mode for d p > 0.8L. In Fig. 14.10, we plot the temporal evolution of the S+S vortex mode corresponding to d p = 0.2L. Figure 14.10a–d show four equally spaced instantaneous vorticity snapshots over a flapping cycle. Figure 14.10a shows the two counter-rotating positive circulation vortices (A) and (B) are shed from the trailing edges of the top and bottom foils respectively. Due to the partial suppression of the shear layer between the gap the size of the vortex (A) shed from the top foil is significantly lower than the vortex (B) shed from an unsuppressed shear layer. These two vortices as they convect downstream coalesce with each other to form a much bigger vortex (O), which can be seen in Fig. 14.10b, c. Due to the partial suppression of the shear layers in the gap, the up-stroke of the top foil and the down-stroke of the bottom foil are affected, which can be the reason for the low amplitude oscillations observed for the in-phase flapping. Moreover, due to the suppression of the shear layers, the vortex-induced lift forces will no longer remain symmetric and this also explains the biased oscillation observed earlier. Figure 14.11 shows the S+S mode time history observed for d p = 0.1L. The points marked in Fig. 14.11a denote the time instances for which vortex patterns are shown in Fig. 14.11b–d. Unlike the S+S mode observed for d p = 0.2L, in this case we do not observe any vortex shedding from the gap. The wake for S+S mode

770

14 Proximity Effect 0.2 tU0 /L: 51.4 δ/L: 0.1636

tU0 /L: 49.9 δ/L: 0.1631

1

Y/L

δ/L

0.1

0

0

tU0 /L: 50.7 δ/L: -0.07333

−0.1

−0.2

-1

49.5

50.5

50

51

51.5

0

52

1

2

X/L

3

4

5

3

4

5

tU0 /L

(a)

(b) 1

Y/L

Y/L

1

0

-1 0

0

-1 1

2

3

4

5

0

1

2

X/L

X/L

(c)

(d)

Fig. 14.11 Trailing edge displacement of the top (—) and bottom (- - -) foils a and the time history of S+S vortex mode b, c and d over one flapping cycle for d p = 0.1L , m ∗ = 0.1, Re = 1000 and K B = 0.0001

at d p = 0.1L resembles the von Kármán wake consisting of two counter-rotating vortices shed alternatively. Additionally, it can also be observed that the S+S vortex patterns are elongated when compared to the other vortex patterns. Figure 14.12 shows the evolution of the synchronized anti-phase 2S+2S vortex mode for d p = 0.6L over a flapping cycle. From this figure, one can observe the von Kármán vortex streets from the top foil and the bottom foil arranges themselves in an anti-phase i.e. two oppositely signed vortices are shed simultaneously from the top and bottom foils. 2S+2S anti-phase vortex mode is observed for an intermediate d p ∈ [0.45L , 0.8L]. Figure 14.13 presents the mixed vortex mode observed for the mixed mode flapping at d p = 0.325L. As the name suggests the wake pattern exhibits synchronized in-phase and out-of-phase vortex modes periodically. Figure 14.13a shows the tipdisplacement history for tU0 /L ∈ [24, 45] and it can be observed that the top and the bottom foil responses are in-phase at tU0 /L = 25 and 39, and out-of-phase at tU0 /L = 32. Figure 14.13b–d show the vortex patterns at tU0 /L = 25, 32 and 39, respectively. A synchronized in-phase vortex shedding can be observed when both the foils are in-phase. This vortex mode is distinctly different from the S+S mode observed for 0.1L ≤ d p ≤ 0.25L, where the shedding between the foils is sup-

14.3 Side-by-Side Foil Arrangements

771

1

Y/L

Y/L

1

0

-1

0

-1 0

1

2

X/L

3

4

5

0

1

2

(a)

3

4

5

3

4

5

(b)

1

Y/L

1

Y/L

X/L

0

0

-1

-1 0

1

2

X/L

3

4

5

0

1

(c)

2

X/L

(d)

Fig. 14.12 Time history of 2S+2S vortex shedding modes over one flapping cycle for d p = 0.6L, Re = 1000, K B = 0.0001 and m ∗ = 0.1

pressed. Figure 14.13c shows the synchronized anti-phase vortex mode when the foils are in out-of-phase. Although Fig. 14.13b, d do not provide a clear idea about the biased flow in the gap in the downstream wake, Fig. 14.13c clearly shows that the bottom foil has a wider wake than the top foil. This observation indicates that the flow is more biased towards the bottom foil. This could possibly be the reason for the low amplitude and frequency oscillations of the bottom foil corresponding to the mixed-mode flapping regime. For a sufficiently large gap between the foil, we do not observe any overlapping of wakes. Hence, we have two independent von Kármán wakes consisting of 2S vortex modes. Both these wakes closely resemble the single flexible foil. By comparing the vortex mode regimes with the flapping response regimes defined above, one can understand that the in-phase flapping regime exhibits S+S vortex mode, whereas the synchronized 2S+2S anti-phase vortex mode can characterize the out-of-phase flapping response. The independent flapping response develops two von Kármán wake patterns identical to the single flexible foil flapping.

14.3.2.3

Effects of Spacing on Lift and Drag Forces

In this subsection, we present the effects of the gap on the force dynamics. In Fig. 14.14a, we compare the lift coefficients corresponding to the in-phase and out-of-

772

14 Proximity Effect 0.15

1

0.1

0.75 0.5 0.25

Y/L

δ/L

0.05 0

0

-0.25

−0.05

-0.5

−0.1

-0.75 −0.15

-1 25

30

35

40

0

1

2

(b)

(a) 1

1

0.75

0.75

0.25

Y/L

0.5

0.25

Y/L

0.5

0

-0.25

0

-0.25

-0.5

-0.5

-0.75

-0.75

-1

3

X/L

tU0 /L

0

1

X/L

2

-1

3

0

1

X/L

(c)

2

3

(d)

Fig. 14.13 Trailing edge tip displacement of top (—) and bottom (- - -) foils a and vortex patterns for mixed-mode flapping (b, c and d) for d p = 0.325L at m ∗ = 0.1, Re = 1000 and K B = 0.0001 0.25 0.2 0.2

Cd

Cl

0.1 0

0.15 −0.1 0.1

−0.2 20

21

22

23

24

25

20

21

22

23

tU0 /L

tU0 /L

(a)

(b)

24

Fig. 14.14 Comparison of: a lift and b drag for in-phase (- - -) and out-of-phase (—) flapping

25

14.3 Side-by-Side Foil Arrangements

773 top plate Crms

0.14

l

bottom plate Crms l

0.12

top plate mean Cl bottom plate mean C

0.1

l

Cl

0.08 0.06 0.04 0.02 0 −0.02

0

0.5

1

1.5

2

dp /L Fig. 14.15 Dependence of rms and mean lift coefficients on the foil spacing ratio d p /L

phase flapping for d p = 0.1L and 0.6L respectively with m ∗ = 0.1, K B = 0.0001 and Re = 1000. From this figure, it can be observed that the out-of-phase flapping experiences greater peak-to-peak lift when compared to the in-phase flapping. Similarly, Fig. 14.14b presents a comparison between the drag experienced by the in-phase flapping and out-of-phase flapping at d p = 0.1L and 0.6L respectively for m ∗ = 0.1, K B = 0.0001 and Re = 1000. The figure distinctly shows that the inphase flapping motion experiences lesser drag. Figure 14.15 shows the dependence of rms lift (Clrms ) and mean lift (C¯l ) on the gap, where Clrms and C¯l account for the vortex and proximity-induced effects respectively. For very large gaps, Clrms values for the two foil systems are identical to the single foil. As the gap between the foils is reduced, the two foil system tends to experience greater lift for d p ∈ [0.45L , 0.9L]. The increase in the lift is mainly due to proximity-induced effects. The two foil system experiences maximum lift for gaps d p ∈ [0.45L , 0.5L]. For gaps d p < 0.45L, the Clrms experiences a sudden drop signifying the transition from the out-of-phase to the in-phase. One of the reasons for the reduction of Cl during the transition phenomenon can be attributed to the suppression of the vortex shedding from the gap. On the other hand, for the gaps d p < 0.45 the |C¯l | continues to increase and becomes significantly large for for d p = 0.1L.

14.3.2.4

Side-by-Side Versus Single Foil

In Fig. 14.16, we present a comparative investigation of the flapping dynamics for the two side-by-side foils with their single foil counterpart. Figure 14.16a shows the trailing edge tip-displacements for the in-phase (—) at d p = 0.1L and the single foil (- - -) for m ∗ = 0.1, Re = 1000 and K B = 0.0001. The in-phase flapping motion of

774

14 Proximity Effect 0.15

0.1

0.1

0.05

0.05

δ/L

δ/L

0.15

0

−0.05

0

−0.05

−0.1

−0.1

−0.15

−0.15

15

20

25

15

20

tU0 /L

tU0 /L

(a)

(b)

25

Fig. 14.16 Comparisons of in-phase (left) and out-of-phase (right) tip-displacement responses for the top foil (—) with single foil (- - -) response

the top foil exhibits upward biasing and lower flapping frequency compared to its single foil counterpart. On the other hand, Fig. 14.16b presents the trailing edge tipdisplacement responses for the out-of-phase coupled mode at d p = 0.6L and single foil for m ∗ = 0.1, Re = 1000 and K B = 0.0001. Both out-of-phase and single foil flapping exhibit similar flapping frequency and slightly greater flapping amplitudes when compared to single foil. Figure 14.17a compares the time history of the lift and drag coefficients for the in-phase flapping at d p = 0.1L with its single foil counterpart for m ∗ = 0.1, K B = 0.0001 and Re = 1000. The in-phase flapping experiences lesser lift in comparison to the single foil due to the suppression of the shear layer and vortex shedding from the gap. Due to smaller flapping amplitudes, the in-phase flapping experiences lower drag in comparison to the single foil. On the other hand, Fig. 14.17b shows the trailing edge tip-displacement histories for the out-of-phase flapping at d p = 0.6 and its single foil counterpart for m ∗ = 0.1, K B = 0.0001 and Re = 1000. This figure shows that the out-of-phase flapping experience greater lift and drag when compared to single foil.

14.3.2.5

Phase Diagram

We summarize the coupled flapping modes obtained from the parametric simulation study performed as a function of gap for m ∗ = 0.1, Re = 1000 and K B = 0.0001 in Fig. 14.18. The figure also shows the stability curves determined from the linear stability analysis. The dispersion relation for two parallel foils in a uniform axial flow subjected to small perturbation ηtop = η¯ top ei(kx x−ωt) , ηbot = η¯ bot ei(kx x−ωt) ,

(14.17)

14.3 Side-by-Side Foil Arrangements

775 0.25

0.2 0.2

Cd

Cl

0.1 0

0.15 −0.1 0.1

−0.2 20

23

22

21

24

25

20

21

23

22

tU0 /L

24

25

24

25

tU0 /L

(b)

(a) 0.25 0.2

0.2

Cd

Cl

0.1 0

0.15

−0.1 −0.2 20

21

22

23

24

0.1 20

25

21

22

23

tU0 /L

tU0 /L

(c)

(d)

Fig. 14.17 Comparisons of lift (left) and drag (right) coefficients for the case of: a in-phase (—) and b out-of-phase (—) flapping with single foil (- - -) flapping

where k x and ω are the complex wave number and frequency is given by Din = a1 + a2

and

Dout = a1 − a2 .

(14.18)

where ∗

C T Re−1/2 k x2

+ a1 = −m ω +   csch k x d p a2 = (k x − ω)2 . kx 2

K B k x4

  1 + coth k x d p − (k x − ω)2 , kx

(14.19) (14.20)

A detailed derivation of the above dispersion relation is provided in Appendix 14.17. Equation (14.18) (left) represents the dispersion relation for the in-phase coupled mode and the dispersion relation in Eq. (14.18) (right) corresponds to the out-ofphase coupled mode. The main challenge here is to identify which of the two coupled modes is the dominant one. To identify the most dominant coupled flapping mode, we have considered the least dispersive wave theory presented by Tam et al. [731] to predict the in-phase and out-of-phase tones in jet flows for a given Mach number.

776

14 Proximity Effect 2.2 2 1.8

Independent mode

1.6 1.4

dp

1.2 1

fixed-point stable

0.8 out-of-phase mode

0.6 0.4 0.2

in-phase mode 0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

m∗

Fig. 14.18 Stability phase diagram showing the (i) fixed-point stable, (ii) in-phase, (iii) out-ofphase and (iv) independent flapping modes observed from DNS and stability analysis for m ∗ = 0.1, K B = 0.0001 and Re = 1000. in-phase, () out-of-phase, () mixed and (∗) independent modes represent the DNS results

The least dispersive wave theory is based on the idea that dispersive waves while propagating upstream travel with different velocities for different k x and hence they disperse in space in course of time. For a positive wave number k x , the wave is dispersive when ∂ 2ω = ω

(k x ) ≡ 0 ∂k x2

(14.21)

and it is referred to as non-dispersive if ω

(k x ) ≡ 0 [527]. In order to maintain a stable feedback loop with a unique coupled mode, the fluid-elastic disturbance should be least dispersive as it propagates in space and time. This implies that the instabilities with frequencies that are close to those of the minimally dispersive wave will be most amplified. Therefore, to determine the most dominant coupled mode, we consider the unstable frequency mode which exhibits the least dispersion  2 out   2 one    2 in  ∂ ω ∂ ω ∂ ω , Re , Re ∀ k ≥ 2π | Im (ω) > 0, (14.22) ωdom = min Re ∂k 2 ∂k 2 ∂k 2

where ωone , ωin and ωout represent the eigenfrequency corresponding to the single foil (Eq. 12.58), in-phase (Eq. 14.18 left) and out-of-phase (Eq. 14.18 right) coupled modes. In Eq. (14.22), we also include the single flexible foil to account for the independent-mode which is observed experimentally [787].

14.3 Side-by-Side Foil Arrangements

777

0.19 top bottom

0.185

Ws

0.18

0.175

0.17

0.165

15

20

25

tU0 /L Fig. 14.19 Time evolution of work done by the fluid on the top (—) and bottom (- - -) foils in out-ofphase coupled flapping for d p /L = 0.6 and nondimensional parameters m ∗ = 0.1, K B = 0.0001 and Re = 1000

These curves demarcate the four coupled modes: (i) fixed-point stable, (ii) inphase, (iii) out-of-phase and (iv) independent. One can observe from Fig. 14.18 that our DNS results provide a very good match with the analytical linear stability analysis. This phase diagram provides insight into the passive flapping modes in a pair of flexible foils as a function of the gap ratio. We next explore the role of gap ratio on the energy transfer and work done, which has a profound impact on energy harvesting and propulsive effectiveness.

14.3.2.6

Effect of Gap on Net Energy Transfer

Here, we investigate the effects of the gap between the foils on the energy transfer and the strain energy developed due to the flapping motion for m ∗ = 0.1, K B = 0.0001 and Re = 1000. We first plot the net work done by the fluid on the top and bottom foils in Fig. 14.19 for the out-of-phase coupled flapping at d p /L = 0.6. The figure shows that the rate at which the net work done by the fluid increases at a much slower rate when compared to its single foil counterpart in Fig. 13.24. Additionally, one can also observe from this figure that the net work done by the fluid on the foils for tU0 /L = 15 is much greater than its counterpart at an identical time. This phenomenon can be attributed to the high static pressure between the foils which performs a large initial work on the foils so that the foils undergo a static outward deformation. We next plot the time evolution of net work done by the fluid on the top and bottom foils for the in-phase coupled flapping corresponding to the gap d p /L = 0.1 in Fig. 14.20. The time evolution of the net work done by the fluid on the top and bottom foil exhibits a phase difference of 2π . The phase difference can be attributed to the suppression of the shear layer which results in the biased flapping. As a result of which, for an in-phase flapping, the work done by the fluid on the top

778 0.135

top bottom

0.13

0.125

Ws

Fig. 14.20 Time evolution of work done by the fluid on the top (—) and bottom (- -) foils in in-phase coupled flapping for d p /L = 0.1 and nondimensional parameters m ∗ = 0.1, K B = 0.0001 and Re = 1000

14 Proximity Effect

0.12

0.115

0.11 20

22

24

26

28

30

tU0 /L 2.2

top-plate bottom-plate

2

(E s )rms × 104

Fig. 14.21 Root-meansquared value of strain energy developed (E s )rms as a function of gap d p ∈ [0.1L , 2.0L]for m ∗ = 0.1, K B = 0.0001 and Re = 1000

1.8 1.6 1.4 1.2 1

0

0.5

1

1.5

2

dp

foil during the upstroke is similar to the work done on the bottom foil during the down stroke, thereby resulting in a phase difference of 2π in the work done. Figure 14.21 summarizes the effect of d p /L on the rms of E s . The rms value of E s for d p /L > 0.5 is approximately equal to its single foil counterpart. There is a sharp rise in (E s )rms for d p /L ∈ [0.4, 0.5] can be attributed to the increase in flapping amplitudes due to proximity-induced effects. An approximately 7.6% increase in (E s )rms has been observed due to the proximity of a second flapping body. For d p /L < 0.4, the (E s )rms values exhibit a sudden drop indicating the transition from the out-phase to in-phase flapping where the flapping amplitudes are approximately 27% lesser than its out-of-phase counterparts. The (E s )rms for the in-phase flapping at d p /L = 0.1 is 47% lower than its single foil counterpart. Therefore, we can conclude that we can enhance the energy harvesting ability of a flexible foil by placing the foils at an optimal gap so that they can develop maximum periodic strain energy which can be converted into electric energy.

14.3 Side-by-Side Foil Arrangements 3

2.5

×10

779

-3

Stable

KB

2

1.5

1

0.5

0

unstable flapping 0.05

0.075

0.1

0.125

m∗

0.15

0.175

0.2

0.225

Fig. 14.22 Summary of coupled mode distribution showing (∗) fixed point stable, () out-of-phase, () mixed and ( ) in-phase for d p /L = 0.2 and Re = 1000. The curve (—) represents the stability boundary for two parallel foil system given by Eq. 14.23 and (- - -) is the stability boundary for a single foil system given by Eq. 14.24 [596]

14.3.3 Interaction Dynamics of Gap Flow with Flapping We perform a series of direct numerical simulations for a fixed gap d p /L = 0.2 and Re = 1000 over a range of K B ∈ [10−4 , 3 × 10−3 ] and m ∗ ∈ [0.05, 0.2] to analyze the effects of gap flow between two side-by-side elastic foils on the stability of the coupled fluid-foil system, the coupled flapping modes and the return effects on the gap flow due to the coupled flapping modes. Our numerical simulations show that the two foils exhibit in-phase, out-of-phase and mixed-mode coupling as a function of K B and m ∗ . These coupled modes further corroborate the experimental observations reported by Jia et al. [616], Schouveiler and Eloy [700] for a constant gap between the clamped leading edges. Figure 14.22 summarizes the coupled modes observed as a function of K B and m ∗ . Figure 14.23 shows a typical tip-displacement response for the mixed-mode. There was a conjecture that the random in-phase and out-ofphase coupling between the top and bottom elastic foils may be due to collision or near the contact between the foil [507]. However, our simulations show that the two parallel foils can exhibit a stable mixed-mode even when there is no collision or near-collision between the foils. The minimum instantaneous gap observed between the trailing edges of the top and bottom foils for the tip displacement reported in Fig. 14.23 is 0.054L. In our analysis two different types of mixed modes are observed, the first type is observed during the transition from the out-of-phase to in-phase coupling for

780

14 Proximity Effect 0.2 0.15 0.1

δ/L

0.05 0 -0.05 -0.1 -0.15 25

30

35

40

45

50

tU0 /L Fig. 14.23 Time history of trailing edge tip-displacements for the top (- - -) and bottom (—) foils for K B = 5 × 10−4 , m ∗ = 0.15 and Re = 1000

m ∗ = 0.2. This type of mixed mode conforms with the experimental observations of [616]. The second type of mixed-mode is observed during the transition between a lower in-phase flapping mode and a higher in-phase flapping mode. To demonstrate the second type of mixed-mode observed for K B = 5 × 10−4 at m ∗ = 0.15 we plot the trailing edge tip-displacement and the full body profile of the elastic foil over a flapping cycles in Fig. 14.24 for K B = 7.5 × 10−4 and 3 × 10−4 at m ∗ = 0.15. The arrows in Fig. 14.24 indicate the location of the necking observed. Figure 14.24a, b clearly show that flapping at K B = 7.5 × 10−4 and 3 × 10−4 exhibit necking at two and three locations respectively, which indicates a change in flapping mode [605]. Additionally, we observe that this second type of mixed mode is bistable in nature and is strongly influenced by the initial condition. The curve (—) in Fig. 14.22 depicts the stability boundary between the fixedpoint stable and coupled flapping modes for the two side-side elastic foil system. The empirical relation for this stability boundary is parallel = a(m ∗ )2 + bm ∗ + c, (K B )cr

(14.23)

which is constructed based on the DNS results for d p /L = 0.2 and Re = 1000, where a = −0.005, b = 0.01735 and c = −0.0007625. The dashed-line (- - -) in Fig. 14.22 is the neutral curve, which represents the stability boundary, for a single elastic foil clamped at its leading edge is given by Gurugubelli and Jaiman [596] = (K B )single cr

2 + m ∗kx 2 m∗ − 1.328 ∗ 3 Re−1/2 , 2 + 2k x ) (m k x + 2k x2 )

(m ∗ k x3

(14.24)

14.3 Side-by-Side Foil Arrangements

781 0.3

top foil bottom foil

0.2

0.2 0.1

Y/L

δy /L

0.1

0

0

-0.1 -0.1

-0.2 80

-0.2

82

84

86

88

-0.3

90

tU 0 /L

0

0.2

0.4

0.6

0.8

1

0.6

0.8

1

X/L (a) 0.3 top foil bottom foil

0.2

0.2 0.1

Y/L

δy /L

0.1

0

0

-0.1

-0.1

-0.2 40

-0.2 -0.3 42

44

46

tU /L 0

48

50

(b)

0

0.2

0.4

X/L

Fig. 14.24 Trailing edge tip-displacement history (left) and the full body flapping profile over a flapping cycle (right) for the in-phase flapping mode corresponding to: a K B = 7.5 × 10−4 and b K B = 3 × 10−4 . ↑ indicates the necking location along the flapping foil

where k x = 2π . From Fig. 14.22 one can clearly observe that the stability boundary for the two parallel foil system is always above the single foil stability boundary. Hence, we can conclude that two side-by-side foils placed close to each other exhibit flapping instability more readily then its single foil counterpart. A similar finding has been made earlier using a linear stability analysis by Michelin and Smith [657]. Four important observations from the coupled mode distribution in Fig. 14.22 are: 1. Two side-by-side configuration is more prone to flapping instability compared to its single foil counterpart. single 2. While we only observe out-of-phase coupling for K B > (K B )cr , in-phase coupled mode is relatively more stable than the out-of-phase and is observed single only for K B < (K B )cr . 3. For low mass-ratio, m ∗ = 0.1, the out-of-phase coupling is more stable than the in-phase coupling. However for higher mass-ratios, m ∗ > 0.1, the parallel foil single system exhibits predominantly in-phase coupling for K B < (K B )cr .

782

14 Proximity Effect 0.15

0.4 single two top rigid two bottom rigid two top elastic two bottom elastic

d

0.1

0.2

C

Cd

0.3

single upper single lower two top elastic inside two top elastic outside two bottom elastic inside two bottom elastic outside

0.05 0.1

0

0

2

4

6

0

0

2

4

tU 0 /L

tU /L

(a)

(b)

6

0

Fig. 14.25 Time history of the drag experienced by single and two side-by-side elastic foil for nondimensional parameters K B = 2.75 × 10−3 , Re = 1000 and m ∗ = 0.2: a total drag and b drag contributed by the upper and lower surfaces to the total drag experienced

(a)

(b)

Fig. 14.26 Instantaneous fluid pressure contour around the flexible foil for a single and b two parallel foil system corresponding to K B = 2.75 × 10−3 , Re = 1000 and m ∗ = 0.2. The solid (—) and dashed (- - -) lines represent positive and negative pressure contours, respectively

4. The mixed mode is observed during the transition between the out-of-phase and in-phase coupled modes. We also observe the mixed mode when there is a transition from the lower flapping mode to the higher flapping mode.

14.3.3.1

Role of Gap Flow on Flapping Stability and Coupled Modes

We next make an attempt to explain why two side-by-side foils with a gap flow are more prone to flapping instability compared to their single foil counterpart. In order to understand the underlying physical mechanism, we plot the horizontal drag force experienced by the two side-by-side elastic foils and its single foil counterpart in Fig. 14.25a for K B = 2.75 × 10−3 and m ∗ = 0.2 corresponding to the fixed-point stable regime. Interestingly, one can observe that the wall shear stress force expe-

14.3 Side-by-Side Foil Arrangements

783

rienced by the elastic foils in the side-by-side configuration is lesser than the shear stress on an isolated elastic foil and two side-by-side rigid foils. This observation again shows that the dynamics of flexible bodies is distinctly different from their rigid body counterpart. Previously, [687] have shown that two elastic foils performing flapping in tandem exhibit anomalous drafting when compared to their rigid counterpart. However, in our case it is the static elastic foils which experience the abnormal drag behavior. From the experimental [707], numerical [551] and analytical [551, 614] studies, it is well known that the viscous drag induced tension plays a significant role in stabilizing the flapping instability. Therefore, due to the reduction in the viscous drag induced stabilization effects the two side-by-side elastic foils are more prone to flapping instability compared to the single foil counterpart. To explain the reason behind the abnormal drag on the elastic foils in the sideby-side configuration, we decompose the net drag on an elastic foil into the drag experienced by the surfaces inside and outside the gap. Figure 14.25b shows the decomposed drag components on the upper and lower surfaces in the side-by-side and single elastic foil configurations. This figure provides interesting details about the effects of gap flow on the drag. Firstly, the drag experienced by the surfaces inside the gap is greater than the drag on the single elastic foil surfaces. This effect causes the fluid entering the gap to recover the pressure to balance for the high pressure losses inside the gap by losing its velocity in order to satisfy the Kutta condition. Consequently, a higher pressure is developed at the gap inlet which results in an asymmetrical pressure distribution across each elastic foil. To demonstrate this phenomenon and to provide a comparison against the single foil counterpart we plot the pressure contours for K B = 2.75 × 10−3 and m ∗ = 0.2 in Fig. 14.26. Secondly, the drag experienced by the external surfaces in the side-by-side configuration is significantly lower than that of the single elastic foil counterpart. This lesser drag on the external surfaces can be attributed to the static outward deformation of the side-by-side foils due to the asymmetric pressure distribution across the foil which can be seen in Fig. 14.26b. As a result of this static outward deformation not all the wall shear stress acting on the elastic foil contributes to the drag and the pressure gradient force which acts normal to the elastic foil now has a component opposite to the drag. Lastly, the increase in the drag on the gap flow surfaces is less than the reduction in the drag on the surfaces outside the gap thereby resulting in a lower net drag on the elastic foils in a side-by-side configuration as compared to the single foil counterpart. The high pressure inside the gap between two side-by-side elastic foils results in an equal and opposite lift on both the foils. Due to this biased lift, the two side-by-side elastic foils with gap flow always lose their stability through out-of-phase coupling irrespective of whether the fully developed coupled mode is out-of-phase or in-phase. In Fig. 14.27, we plot the evolution of the trailing edge tip-displacements for the top and bottom foils corresponding to the in-phase flapping mode with nondimensional parameters K B = 1 × 10−4 , Re = 1000 and m ∗ = 0.1. It can be clearly observed from Fig. 14.27b that the elastic foils initially exhibit out-of-phase coupled flapping. The out-of-phase coupling for tU0 /L < 10 slowly transforms into the stable in-phase coupling through a complex transition phenomenon for 10 < tU0 /L < 20. For the

784

14 Proximity Effect top bottom

0.2

0.1

δy /L

y

δ /L

0.1

0

0

-0.1

-0.1

-0.2

top bottom

0.2

0

10

20

tU /L 0

(a)

30

-0.2

0

2

4

6

8

10

12

tU 0 /L

(b)

Fig. 14.27 Time history of the trailing edge tip-displacements of the top(—) and bottom (- - -) foils for K B = 1 × 10−4 , m ∗ = 0.1 and Re = 1000 over tU0 /L ∈ [0, 35] (left) and tU0 /L ∈ [5, 14]

initial out-of-phase coupling to sustain, the flow field should be symmetric about the gap centerline [737] so that the elastic foils experience equal and opposite lift as shown in Fig. 14.28a for K B = 5 × 10−4 , m ∗ = 0.1 and Re = 1000. The pressure contours in Fig. 14.28b qualitatively show that the fluid pressure field is indeed symmetric about the gap centerline. Herein, it should be noted that the symmetric flow field is the reason for the sustenance of the out-of-phase coupled mode created by the high-pressure gap flow and it is not just a characteristic of the out-of-phase coupled mode. Figure 14.28b shows that the fluid flow continues to exhibit higher pressure inside the gap even when the foils are performing out-of-phase flapping. However, unlike the gap flow pressure for the fixed-point stable regime where the pressure gradually decreases from the inlet to the exit (Fig. 14.26b), the gap flow pressure between two foils performing out-ofphase flapping exhibit alternating positive and negative pressure zones. In order to understand the effect of out-of-phase flapping motion on this alternating positive and negative pressure zones, we plot the space-time contours for the gap centerline in Fig. 14.29 for K B = 5 × 10−4 , m ∗ = 0.1 and Re = 1000. Unlike the pressure field on an elastic foil which comprises both forward and backward traveling waves [61], the gap flow pressure field exhibits only a forward traveling wave with a frequency twice that of the flapping frequency. In the above paragraph, we have conjectured that the initial out-of-phase coupling is related to the higher pressure created by the gap flow. In order to further corroborate our conjecture, we demonstrate a special case whereby the gap flow is absent for a very small gap of d p = 0.02L. Since the gap flow typically represents a pressure driven flow, for very small gaps the pressure at the gap inlet will not be sufficient to develop any flow through the gap. Figure 14.30a summarizes the coupled flapping modes observed for two side-by-side elastic foils with and without gap flow. In this figure, it can be clearly observed that there is no out-of-phase coupled mode for d p /L = 0.02. Even the in-phase coupled mode observed here (Fig. 14.30b) does

14.3 Side-by-Side Foil Arrangements

785

0.15 top bottom P:

0.1

-80

26.5303

200

400

600

0.4

0.05

Y/L

Cl

0.2

0

-0.05

-0.2

-0.1 40

0

-0.4

41

42

43

44

45

tU 0 /L

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

X/L

(b)

(a)

Fig. 14.28 Out-of-phase coupled flapping at K B = 5 × 10−4 , m ∗ = 0.1 and Re = 1000: a lift acting on the top (—) and bottom (- - -) foils and b instantaneous fluid pressure contours inside and outside the gap at tU0 /L = 40, where The solid (—) and dashed (- - -) lines represent positive and negative pressure contours, respectively

0.9 0.8 0.7 0.6

X/L

Fig. 14.29 Space-time pressure contours along the gap centerline for the out-of-phase flapping corresponding to K B = 5 × 10−4 , m ∗ = 0.1 and Re = 1000

0.5 0.4 0.3 0.2 0.1 41

42

43

44

45

tU /L 0

not exhibit the initial out-of-phase coupling as observed previously for the in-phase coupling with a gap flow in Fig. 14.27.

14.3.3.2

Feedback Effect of Flapping on the Gap Flow

Having discussed the effects of gap flow on the coupled flapping modes, we now present the feedback effects of the coupled flapping modes on the gap flow. The gap flow between two flapping elastic foils typically represents a pressure-driven flow, where the upstream flow field entering the gap can adjust its pressure and velocity depending on the flapping motion of the elastic foils and viscous drag inside the gap. We plot the time history of volumetric flow rate through the gap for the out-

786

14 Proximity Effect ×10

2.5

-3

0.3 top bottom

0.2

2

0.1

δy /L

KB

1.5 1

0

-0.1

0.5 0 0.01

-0.2 0.02

0.1

d /L

0.2

0

5

10

15

20

tU 0 /L

p

(a)

(b)

Fig. 14.30 Effect of gap flow on the coupled flapping modes. a Summarizes the (∗) fixed point stable, () out-of-phase, () mixed and ( ) in-phase modes for two side-by-side elastic foils with gap flow at d p/L = 0.2 and without gap flow at d p /L = 0.02 for m ∗ = 0.1 and Re = 1000. b Typical trailing edge tip-displacement for d p /L = 0.02, K B = 6 × 10−4 , m ∗ = 0.1 and Re = 1000 0.15

0.9

top bottom

0.1 0.88

y

Q

δ /L

0.05 0.86

0 0.84

-0.05 -0.1 40

41

42

43

44

45

0.82 40

41

42

tU 0 /L

43

44

45

tU 0 /L (b)

(a)

Fig. 14.31 Out-of-phase flapping for m ∗ = 0.1, K B = 5 × 10−4 and Re = 1000. a Trailing edge tip-displacement of top (—) and bottom (- - -) foils over tU0 /L ∈ [40, 45]. b Time history of the volumetric flow rate entering the gap between the two elastic foils

of-phase coupled mode at K B = 5 × 10−4 , m ∗ = 0.1 and Re = 1000 in Fig. 14.31. For an incompressible flow, the volumetric flow rate through the gap is equal to the volumetric flow rate entering the gap. The non-dimensional flow rate entering the gap is given by ⎡ ⎤ dp

⎣ Q(t) =

2

uf (x, y, t)dy ⎦

d − 2p

x=0

d p U0

.

(14.25)

14.3 Side-by-Side Foil Arrangements

787

0.09

0.16 top gap bottom gap

0.08

0.14

Cd

Cd

0.07 0.12

0.06 0.1 0.05 0.04 42

43

44

45

tU 0 /L

(a)

46

47

0.08 42

43

44

45

46

47

tU 0 /L

(b)

Fig. 14.32 Drag experienced by the individual gap flow surfaces (left) and the net drag on both the gap flow surfaces (right) at m ∗ = 0.1 and Re = 1000 for the out-of-phase coupled mode at K B = 5 × 10−4

For the out-of-phase coupled flapping, the gap between the foils varies along the foil i.e. different locations along the foil encounter different gaps at the same instant. However, from Fig. 14.31a, b it can be observed that the flow rate through the gap depends strongly on the gap between the trailing edges of the foils. The gap flow rate increases with the trailing edge gap and decreases as the gap decreases. For the intermediate locations between the leading and trailing edges, the local gap between the foils will only affect the local flow acceleration and deceleration depending on the phase difference between the local gap and the gap at the trailing edge. Figure 14.32 shows the individual and the net drag on the surfaces inside the gap between two side-by-side elastic foils performing out-of-phase coupled flapping. Surprisingly, from Fig. 14.32 one can observe that the maximum flow rate through the gap also corresponds to the time instance when the net drag is maximum. At this point, we believe that the effect of drag on the volumetric flow rate is considerably smaller than the effect of the trailing edge gap so the volumetric flow rate depends mainly on the trailing edge gap. We will show in the next paragraph that this conjecture holds. Figure 14.33 shows the time history of the trailing edge tip-displacements and volumetric flow rate for the in-phase coupled mode with K B = 1 × 10−4 , m ∗ = 0.1 and Re = 1000. As the maximum variation in the gap between the foils is less than 5% for tU0 /L ∈ [30, 35], we do not observe any large fluctuations in the flow rate. However, interestingly, we observe two small peaks in the flow rate per flapping cycle. The time instances of both these peaks correspond to the time instance when the net drag on the surfaces inside the gap is minimum, which can be seen in Fig. 14.34 (right). In Fig. 14.34, we plot the individual drag and the net drag acting on the gap flow surfaces for the in-phase coupled mode. The gap flow between two elastic foils performing in-phase flapping provides us with a special case where the effect of the gap at the trailing edges on the volumetric flow rate is negligible. Unlike the out-ofphase coupled mode where the variation in the gap flow rate was 7%, the in-phase flapping experiences only a small variation of about 0.5% in the gap flow rate which

788

14 Proximity Effect

0.2

0.9

top bottom

0.15

0.88

0.1

y

δ /L

Q

0.05 0.86

0

-0.05

0.84

-0.1 -0.15 30

31

32

33

34

0.82 30

35

31

32

33

tU /L

tU /L

(a)

(b)

0

34

35

0

Fig. 14.33 In-phase flapping for m ∗ = 0.1, K B = 1 × 10−4 and Re = 1000. a Trailing edge tipdisplacement of top (—) and bottom (- - -) foils over tU0 /L ∈ [30, 35]. b Time history of the volumetric flow rate entering the gap between the two elastic foils 0.16

top gap bottom gap

0.1

0.14

C

d

Cd

0.08 0.12

0.06 0.1

0.04 30

31

32

33

34

35

0.08 30

31

32

33

tU /L

tU /L

0

0

(a)

(b)

34

35

Fig. 14.34 Drag experienced by the individual gap flow surfaces (left) and the net drag on both the gap flow surfaces (right) at m ∗ = 0.1 and Re = 1000 for the in-phase coupled mode at K B = 1 × 10−4

is mainly caused by the drag acting on the surfaces inside the gap. Hence, we can conclude that the flow rate through the gap predominantly depends on the gap at the trailing edge when compared to the drag on the internal surfaces. Further from Figs. 14.32 and 14.34, one can observe that the phase difference between the drag acting on the individual surfaces inside the gap for the out-ofphase and in-phase coupled modes is 0 and π , respectively. As a result of this, the out-of-phase coupled mode experiences about 10% greater drag than the in-phase coupled mode.

14.3 Side-by-Side Foil Arrangements

s1 = 0

789

s1

a

s1 = L

U0 s2 = 0

s2

b

s2 = L

Fig. 14.35 Schematic of the two side-by-side elastic foils with their respective curvilinear coordinate systems

14.3.3.3

Gap Flow Velocity Profiles

In [61], we have shown that the boundary layer on an elastic foil performing flapping exhibits three distinct regimes over a flapping cycle: (i) uniformly decelerating, (ii) uniformly accelerating and (iii) mixed accelerating and decelerating. The gap flow velocity profiles between two elastic foils performing flapping will be distinctly different from the boundary layer profiles outside the gap and will significantly influence the drag on the surfaces inside the gap. Therefore, we analyze the gap flow profiles for the coupled modes reported in Fig. 14.22. As the location of the foils change continuously in time due to the flapping motion, we introduce two curvilinear abscissas s1 and s2 along the top and bottom elastic foils as shown in Fig. 14.35, where s1 = s2 = 0 represent the leading edges and s1 = s2 = L are the corresponding trailing edges. To demonstrate the velocity profiles inside the gap, we consider the cross section joining two points a and b along the top and bottom elastic foils respectively and corresponding to s1 = s2 = s. Figure 14.36 presents the evolution of the gap flow velocity profiles for the outof-phase coupled flapping along the cross-section a − b for s = 0.75L. The nondimensional simulation parameters corresponding to the out-of-phase coupled mode are K B = 5 × 10−4 , m ∗ = 0.1 and Re = 1000. Figure 14.36a shows the displacement time history for the points a and b for s = 0.75. The annotations () and () denote the top and bottom foil locations corresponding to the time instances for which velocity profiles are plotted in Fig. 14.36b–f. The η1 and η2 in Fig. 14.36 represent the nondimensional transverse distance y/δ from the points a and b respectively, 1 where δ = (νs/U0 ) 2 . From Fig. 14.36, one can observe that the velocity profiles plotted from point a and from point b for the cross-section a − b perfectly overlap with each other thereby indicating that the local gap flow velocity profile is symmetric about the gap centerline. Intuitively, one would have expected the maximum local flow velocity for the minimum local gap between the foils. Instead, from Figs. 14.36b–f it can be observed that the local gap flow is maximum for an intermediate local gap at tU0 /L = 44.09 that corresponds to the time instance when the gap between the top and bottom foils at the trailing edge is maximum (Fig. 14.31a). The reason behind

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this observation can be attributed to the difference between the rate at which the gap at the trailing edges (tan (θT )) and the rate at which the local gap (tan (θ L )) at s1 = s2 = 0.75L changes. Figure 14.37 demonstrates the evolution of the local gap (—) at s1 = s2 = 0.75 and the gap at the trailing edges (- - -) over a flapping cycle. One can clearly observe from this figure that (tan (θT )) > (tan (θ L )). As we have shown in the previous subsection the flow rate through the gap increases with the gap and reaches a maximum for tU0 /L = 44.09. For tU0 > 44.09, the gap at the trailing edge decreases at a much faster rate than the local gap at s1 = s2 = 0.75L. Hence, the fluid flow entering the gap that is proportional to the trailing edge gap is also decreasing at a much faster rate than the rate at which the cross-section gap at s1 = s2 = 0.75L is decreasing. Therefore, local flow velocity will have maximum at tU0 /L = 44.09 and decreases for tU0 /L > 44.09 until it reaches the minimum for tU0 /L = 44.6. The gap flow between the elastic foils performing out-of-phase flapping exhibits four regimes as a function of θT , θ L and the phase difference between them. Figure 14.37 summarizes the four regimes observed in the out-of-phase coupled flapping mode. In regime I, both the local gap and the gap at the trailing edges increase. Since the trailing edge gap increases at a greater rate than the local gap, the local flow accelerates. Following this, in regime II the trailing edge gap continues to increase while the local gap starts to decrease resulting in a much stronger local flow acceleration compared to the regime I. In the subsequent regime (III), the trailing edge gap decreases at a much faster rate than the local gap and therefore results in

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the local gap flow deceleration. Finally, in the last regime (IV) the local gap begins to increase while the trailing edge gap is still decreasing, thereby resulting in a greater local flow deceleration than the flow deceleration observed in regime III. Figure 14.38 presents the gap flow velocity profiles along the cross section a − b for s = 0.75L for the in-phase coupled motion over a flapping cycle tU0 /L ∈ [32.05, 33.21]. Figure 14.38a depicts the local transverse displacement history corresponding to the points a and b for s = 0.75L. The simulation parameters corresponding to the in-phase flapping presented in Fig. 14.38 are K B = 1 × 10−4 , m ∗ = 0.1 and Re = 1000. From Fig. 14.38, one can observe that the gap flow velocity profiles plotted from point a do not overlap with the gap flow velocity profiles plotted from point b. This observation indicates that the gap flow field is not symmetrical about the local gap centerline. Interestingly, the asymmetric gap flow field is alternatingly biased towards the top and bottom gap flow surfaces. At the beginning of the downstroke, the fluid flow is biased towards the bottom foil. As the points a and b cross their mean positions, the biased gap flow switches its direction from the bottom surface to the top surface. The gap flow switches back to the bottom foil surface again when the points a and b cross the mean position during the upstroke. At the onset of the downstroke, the point a lies on the top surface which has deformed locally to attain a concave shape and the bottom surface containing the point b has deformed locally to a convex shape. This tendency of the gap flow to remain biased towards the convex surface locally is observed throughout the gap i.e. starting from the leading edge to the trailing edge. Hence, we may attribute the asymmetric biased gap flow to the Coanda effect as the

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gap flow appears to be more attached to convex-shaped surfaces. It is worth noting that we attribute the Coanda effect for the biased flow inside the gap, but do not credit it for the transition from the out-of-phase to in-phase coupled mode. From Fig. 14.38b–f, we can demarcate two distinct gap flow regimes: in the first regime, the upper half of the gap flow experiences a uniformly accelerating flow and the lower half of the gap flow exhibits a deceleration. Conversely in the second regime, it is the lower half that experiences a uniform acceleration and the upper half undergoes a deceleration. The maximum velocity inside the gap at a cross-section is almost constant and the maximum variation is less than 5%. As a result of this, unlike the out-of-phase coupled flapping where we have shown that gap flow exhibits a forward traveling pressure wave the in-phase gap flow exhibits a standing wave along its centerline as shown in Fig. 14.39. Figure 14.40 demonstrates the gap flow velocity profiles for the mixed-mode flapping along the cross-section a − b at s = 0.75L for K B = 5 × 10−4 , m ∗ = 0.15, Re = 1000 and tU0 /L ∈ [37, 38.21]. From Fig. 14.40b–d it can be clearly observed that it shares the traits of both in-phase and out-of-phase coupled modes. The gap flow velocity profiles exhibit both asymmetric gap flow and variable gap. From Fig. 14.40d, it can be observed that even though the foils are instantaneously out-of-phase the velocity profiles do not overlap. Figure 14.40b depicts the velocity profile inside the gap when the top foil is at it maximum deformation and the bottom foil is in its downstroke. This figure depicts the velocity profile inside the gap for

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a special case when the gap flow is asymmetric and we have two concave surfaces. Interestingly, the flow is slightly biased towards the bottom surface which is having lesser curvature. Similarly, Fig. 14.40c depicts the case with two convex surfaces and in this case, the gap flow is biased towards the top foil having greater curvature. To summarize, we have observed three distinct gap flow boundary layer profiles: (i) unsteady symmetrical gap flow with a variable gap for out-of-phase, (ii) unsteady alternating biased asymmetrical gap flow with a steady gap for in-phase and (iii) unsteady alternating biased asymmetrical gap flow with a variable gap for mixedmode.

14.3.3.4

Evolution of In-Phase

In the earlier sections, we have shown that two side-by-side elastic foils with a gap flow always lose their stability through out-of-phase coupling for both the in-phase and out-of-phase coupled flapping modes. The flow field inside the gap is perfectly symmetrical about the gap centerline for the out-of-phase coupled mode. Whereas, for the in-phase coupled mode the flow field inside the gap is asymmetrical and biased towards the convex surfaces. We have also shown that for higher m ∗ , the coupled system exhibit predominantly in-phase flapping. In this subsection, we will try to answer some of the questions like what is the transition mechanism behind the initial out-of-phase coupling to the final in-phase coupled modes and how this transition mechanism differs with m ∗ . To begin with, we plot the instantaneous velocity streamlines of the jet flow from the gap in Fig. 14.41 for m ∗ = 0.1, K B = 0.0001, Re = 1000 and d p /L = 0.2 which exhibits out-of-phase coupling for tU0 /L < 10, in-phase coupling for

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tU0 /L > 20 and transition mode for 10 ≤ tU0 /L ≤ 20. The velocity streamlines in Fig. 14.41a for tU0 /L = 8.0 corresponding to the out-of-phase coupling are symmetric about the gap centerline. However, for tU0 /L = 10, when the transition process begins, Fig. 14.41b shows that the streamline at the gap centerline starts to deflect upwards in the wake. At this point in time, even though the gap flow and foil shapes are perfectly symmetric the wake-induced forces on both the foils are no longer symmetric. The effect of this asymmetric force distribution on the foils due to the deflection of the jet flow from the gap between the foils can be distinctly seen from the streamlines plotted in Fig. 14.41d, e for tU0 /L = 12 and 14 respectively. Figure 14.41c–f clearly show that the jet flow from the gap meanders and affects the wake-induced loading on the foils which in turn breaks down the flow symmetry inside the gap. Additionally, the jet meandering also disturbs the stable out-of-phase 2S + 2S vortex mode in the wake. To understand the effects of this jet meandering on wake, we plot the vorticity contour for m ∗ = 0.1, K B = 0.0001, Re = 1000 and d p /L = 0.2 in Fig. 14.42. One can clearly observe the out-of-phase 2S + 2S vortex mode which characterizes the out-of-phase coupling for tU0 /L = 8. Even though the jet from the gap has deflected for tU0 /L ≥ 10, we did not observe any big differences in the wake structures until the flow field inside the gap loses its symmetry for tU0 /L  14. Once the gap flow field has lost symmetry, the wake structures and foils realign themselves to form the stable S + S vortex mode when the top and bottom foils are performing in-phase flapping. The fundamental mechanism behind the transition from the out-of-phase to inphase coupling is similar for both small and large m ∗ values. In both cases, the jet flow from the gap between the foils loses its stability to undergo oscillations. However, one main difference in the transition mechanism for m ∗ ≥ 0.15 at Re = 1000 and d p /L = 0.2 is that the transition phenomenon begins even before vortices

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are shed from the foil trailing edge. Figure 14.43a, b show the trailing edge transverse tip-displacement response over tU0 /L ∈ [0.75] and [0, 15] respectively for m ∗ = 0.15, K B = 0.001, Re = 1000 and d p /L = 0.2. While the trailing edge response exhibits out-of-phase coupling for tU0 /L < 10, the in-phase coupling occurs for tU0 /L > 15 and the transition state for 10 ≤ tU0 /L ≤ 15. Figure 14.43 presents the instantaneous vorticity contours at tU0 /L = 7.67 before the beginning of transition phenomenon. Hence, we hypothesize based on the observations that for relatively large m ∗ the inertial effects of the flexible foil plays a significant role in destabilizing jet flow from the gap between the foils. On the other hand, for low m ∗ values since the destabilizing inertial effects will be small, the jet flow does not lose its stability until the flapping amplitudes are large enough to destabilize the jet. This hypothesis tends to explain the predominant in-phase coupled mode observed for higher m ∗ . To prove this hypothesis, we need to perform a detailed stability analysis on the jet flow from the gap between two foils performing flapping motion which is beyond the scope of the present work.

14.4 Summary

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14.4 Summary In this chapter, we investigated the 2D coupled flapping dynamics of two side-by-side flexible foils clamped at their leading edges and interacting with an incompressible viscous flow. The flapping dynamics of two side-by-side foil is studied with the aid of the high-order quasi-monolithic formulation for multiple flexible structures presented in Sect.3.4. The numerical analysis showed that as the gap between the foils is reduced, the flapping motion of both the foils exhibited a phase lock-in phenomenon due to the proximity of the second foil. As a function of the gap between the foils, four distinct phase lock-in response regimes are identified for m ∗ = 0.1; Re = 1000 and K B = 0.0001: (i) in-phase for small gaps of 0.1L ≤ d p ≤ 0 : 25L; (ii) mixedmode transition for 0.3L ≤ dp ≤ 0.4L; (iii) out-of-phase for 0.5L ≤ dp ≤ 0.8L; and (iv) independent for relatively large gaps of d p ≥ 0.9L. Unlike the in-phase and out-of-phase coupled modes, we found that the mixed and independent modes are bistable, and the coupled flapping modes strongly depend on the initial condition. We also showed that the out-of-phase flapping exhibits approximately 35% and 27% greater frequency and transverse RMS response, respectively, compared to their in-phase counterparts. The out-of-phase coupled flapping is characterized by two synchronized von Kármán vortex streets (2S + 2S) with a phase difference of π . On the other hand, the in-phase coupled mode is characterized by two large counterrotating vortices (S + S) due to the suppression of the vortex shedding process from the gap. We then investigated the effect of proximity on the onset of flapping instability by performing a series of numerical simulations as a function of KB for a constant m ∗ = 0.1; Re = 1000 and d p = L = 0.2. These simulations revealed the second type of mixed-mode flapping that can occur even during the transition from the lower to higher in-phase flapping mode. We also showed that side-by-side elastic foils exhibit predominantly out-of-phase coupled flapping mode for m ∗ ≤ 0.1 and in-phase coupled flapping mode for m ∗ ≥ 0.1. The analysis showed that two sideby-side elastic foils with a gap are more prone to flapping instability when compared to their single foil counterpart due to the lower drag-induced stabilization effects. The two side-by-side elastic foils can experience significantly lower drag compared to their single foil and two side-by-side rigid foils by undergoing static outward deformation due to strong static pressure inside the gap. Due to this strong static pressure inside the gap, we showed that side-by-side elastic We then presented the feedback effects of the coupled flapping modes on the gap flow rate. We showed that the gap flow rate through the gap exhibits a strong and weak dependence on the gap at the trailing edge and drag experienced by the gap flow respectively. As a result of this, the out-of-phase coupled flapping mode exhibits large fluctuations in the gap flow rate due to large variations in the trailing edge gap. On the contrary, the in-phase coupled flapping does not exhibit any large fluctuations in the gap flow rate because the gap at the trailing edges almost remains constant. The small fluctuations in the gap flow rate for in-phase are attributed to the drag experienced by the gap flow.

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We then find that the gap flow between the elastic foils exhibited three distinct gap flow velocity profiles: (i) unsteady symmetrical gap flow with a variable gap for the out-of-phase, (ii) unsteady alternating biased asymmetrical gap flow with a uniform gap for the in-phase and (iii) unsteady alternating biased asymmetrical gap flow with a variable gap for the mixed mode. The gap flow was found to be biased towards the convex-shaped surface in the case of alternating biased asymmetrical gap flow with a uniform gap, thereby conforming to the existence of Coanda effect. In the case of mixed mode with biased asymmetrical gap flow with a variable gap, the flow can experience special cases like both the gap flow surfaces are concave or convex locally. For such special cases, the flow is biased towards the concave surface with the least curvature or the convex surface with greater curvature. For the outof-phase flapping mode, due to the symmetrical gap flow, the drag experienced the surfaces inside the gap was found to be in-phase and identical. On the other hand, the asymmetrical gap flow for the in-phase flapping results in a phase difference between the drag forces acting on the gap flow surfaces. As a result of this phase difference between the drag-acting gap flow surfaces, the gap flow experiences lesser drag for the in-phase flapping than the out-of-phase. Finally, we presented the evolution of the in-phase flapping mode from the initial out-of-phase flapping with the aid of velocity streamlines and vorticity plots. It was found that during the transition phenomenon, the jet from the gap between the foils losses its stability and begins to meander. As a result of this, the fluid forces acting on the foils became asymmetric thereby leading to the loss of gap flow symmetry until the foils rearrange themselves in phase. The stable 2S + 2S corresponds to the out-of-phase realign to form the S + S vortex mode corresponding to the in-phase flapping. The loss of the gap flow jet stability was found irrespectively of m ∗ . Acknowledgements Some parts of this Chapter have been taken care from the PhD thesis of Pardha Gurugubelli carried out at the National University of Singapore and supported by the Ministry of Education, Singapore.

Chapter 15

Trailing Edge Effect

In this chapter, we investigate the flapping dynamics of flexible foils or panels in a steady-flowing fluid. Specifically, we examine the effects of trailing edge shape on the flapping dynamics. We employ our recently developed body-conforming, fluidstructure interaction solver, for this high-fidelity numerical study. To eliminate the effect of other geometric parameters, only the trailing edge angle is varied from 45◦ (concave plate), 90◦ (rectangular plate) to 135◦ (convex plate) while maintaining the constant area of the flexible plate. For a wide range of flexibility, three distinctive flapping motion regimes are classified based on the variation of the flapping dynamics: (1) low bending stiffness K Blow , (2) moderate bending stiffness K Bmoderate , high and (3) high bending stiffness K B . We examine the impact of the frequency ratio ∗ f defined as the ratio of the natural frequency of the flexible plate to the actuated pitching frequency. Finally, the effect of trailing edge shape and flexibility on the propulsive performance and the drag-thrust transition mechanism is explored with the aid of the dynamic decomposition method, the momentum-based equation, and the analytical added mass model.

15.1 Introduction Biological species in nature have evolved over millions of years to possess superior propulsive performance and high maneuverability for locomotion. These traits are achieved by various flapping-wing-like surfaces with a wide range of shapes and flexibility in different fliers and swimmers [622, 634]. These natural fliers and swimmers can inspire the design of highly-efficient self-propelled propulsors and human-made vehicles by searching for the optimal combination of wing geometry and fluid-structure parameters. Towards this goal, a vast body of work has been carried out during the past decades [548, 574, 650, 664, 764]. Particularly, flexibility and trailing edge (TE) shape were found to play important roles in improving propulsive performance by affecting the surrounding flow features [598, 706, 709, © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Jaiman et al., Mechanics of Flow-Induced Vibration, https://doi.org/10.1007/978-981-19-8578-2_15

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711, 714]. However, the large physical parameter space poses a serious challenge to characterize the impact of each parameter on the propulsive performance. The thrust-generating mechanism and the efficiency gain of flapping wings with varying trailing edge shapes and flexibility by correlating with the flow features are not fully understood, which motivates the present computational study. During the past decades, a plethora of early research on the flapping rigid wings has been performed to characterize the effects of various geometric and physical parameters on the thrust generation and the propulsive efficiency [515, 561, 752, 781, 785]. To simplify the flapping dynamics, rigid wing models were used to understand the thrust-generating mechanism of actual biological wings [515, 650, 752]. In reality, biological wings have a variety of flexibility and wing shapes that can meet the desired performance requirements. Inspired by intelligent and efficient biological flight, a series of studies considering the shape and flexibility of the wings were carried out to optimize the performance of the rigid counterparts with simplified shapes [546, 650, 686]. However, when examining the impact of wing shape or flexibility, in most cases one of the parameters was fixed. This limitation results in a poor understanding of the optimal combination of the wing shape and flexibility that can maximize the propulsive efficiency. In flapping flight, the vortical structure properties are closely connected with the propulsive performance and thrust generation [555, 648, 783]. The flapping wing accelerates the unsteady flow to form vortices containing high velocities, thereby generating thrust by transferring momentum to the fluid [592, 628, 676]. The creation and the transport process of vortices are strongly influenced by the wing shape and flexibility. While most studies focused on the variation of the vortical structures induced by the flapping wing, very few of them have explored the thrust-generating mechanism by directly correlating the temporal and spatial evolution of the vortical structures and the time-dependent thrust forces, and even fewer examined the effects of wing shape and flexibility from this perspective.

15.1.1 Effect of the Trailing Edge Shape and Flexibility To gain further insight into the role of trailing edge shape played in the thrust generation and the optimal propulsive efficiency achievement, a number of studies have been carried out for flapping rigid plates with non-flat trailing edge. One of the pioneers in this research area focused on the lunate tails with varying trailing edge shapes to determine the optimal combination with maximum efficiency [549, 550]. Liu et al. [643] investigated the hydrodynamic performance and wake patterns for various caudal fin shapes over a wide range of Strouhal numbers. Krishnadas et al. [631] numerically studied the propulsive efficiency of biomimetic trapezoidal wings with different trailing edge angles undergoing pitching and heaving motion. The effect of the trailing edge shape on the wake behaviors for a trapezoidal pitching panel was explored through a series of experiments [625, 626]. Although the studies mentioned above aimed to investigate the impact of trailing edge shape on the propulsive performance, other geometric parameters (e.g., wing length, span and leading edge shape)

15.1 Introduction

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were modified simultaneously. Such simultaneous modifications hinder to isolate the only effect of the trailing edge shape on the propulsive performance and the wake topology of flapping plates. Recently, Van Buren et al. [755] reported their experimental work on the effect of trailing edge shape on the wake evolution and the propulsive performance of pitching rigid plates. To eliminate the impact of other geometric parameters on propulsive performance, a constant area, S = 0.1 m2 , and the same mean aspect ratio, A R = 1 were maintained, but only the trailing edge angle Φ was modified from 45◦ to 135◦ in intervals of 15◦ for different plates. A schematic of three representative plate geometries with varying chevron-shaped trailing edges is shown in Fig. 15.1. It can be concluded from this experiment that pitching plates with convex shape (Φ > 90◦ ) exhibited larger thrust production and superior propulsive efficiency than the plates with rectangular (Φ = 90◦ ) and concave (Φ < 90◦ ) trailing edge angles. However, the wake topology was only measured at a Reynolds number of Re = 6000 and the propulsive performance was available at Re = 10,000 in the experiment. Hence, it is hard to establish a connection between the wake structures and the propulsive performance. Further mechanism study on the thrust transition was restricted to the existing experimental results. To avoid this limitation, some recent numerical studies [600, 601] were performed to relate the thrust generation, the wake topology and the instantaneous flow features around a moving rigid plate. With respect to the self-propelled plates with different trailing edge shapes, the experimental work done by Van Buren et al. [755] was restricted to a rigid plate. However, the caudal fin of aquatic animals and the flapping wing of flying species exhibit various flexible properties [574]. Through numerous studies on the role of flexibility, it was found that passive flexibility can help in redistributing the pressure gradient on the plate surface and regulating the vortical structures [603, 651, 675, 714]. As a result, the thrust coefficient and the propulsive efficiency of a flexible plate were enhanced to different degrees compared to its rigid counterpart [576, 578, 656]. The resonance between the actuated frequency and the natural frequency of the coupled flapping system governed by flexibility was found to be beneficial to the thrust generation [591, 675, 686]. However, the optimal propulsive efficiency did not strongly depend on the occurrence of resonance [558, 575, 577, 661]. Considering the benefits of flexibility and trailing edge shape for enhanced propulsion performance, the participation between these two factors should be considered when designing an effective propulsion system. Only a handful of publications on the joint effects of trailing edge shape and flexibility on propulsive performance can be found in the literature. Zhang et al. [780] numerically investigated the self-propulsive performance for flexible plates undergoing heaving motion as a function of trailing edge angle Φ. As the flexibility increased, the optimal performance was achieved for the concave, convex and square panels, respectively. In the present study, we investigate the propulsive performance for pitching flexible plates with varying trailing edge shapes over a wide range of flexibility based on the experimental plate models [755]. Three representative plates with a concave shape (Φ = 45◦ ), a square shape (Φ = 90◦ ) and a convex shape (Φ = 135◦ ) are considered for simplicity in the current paper.

802

15 Trailing Edge Effect

15.1.2 Drag-Thrust Transition The transition between drag and thrust is widely observed in the self-propelled plates with different propulsive parameters. Biological species and bio-inspired vehicles may choose proper parameter combinations to achieve the goal of moving forward, backward and rapid turn (i.e., maneuverability). Godoy-Diana et al. [586] experimentally investigated the vortex streets formed behind a flapping rigid foil. The results revealed that the drag-thrust transition mechanism was governed by the transition from a Bénard-von Kármán (BvK) to a reverse BvK. Andersen et al. [514] reported a new drag-thrust transition mechanism caused by the formation of two vortex pairs per oscillation period for rigid foils undergoing heaving motion with high amplitude and low frequency. Most of the studies on the drag-thrust transition were limited to rigid foils. Recently, Marais et al. [651] discovered that flexibility suppressed the symmetry-breaking process of the reverse BvK to improve thrust production. Tzezana et al. [753] investigated the drag-thrust transition phenomenon for flapping compliant membrane models with different values of wing compliance, flapping kinematics and inertia. Flapping wings with larger flexibility were prone to trigger the thrust-drag transition earlier at a higher flapping frequency, compared to the wings with smaller wing compliance. A handful of studies can be found to investigate the effect of the trailing edge shape on the drag-thrust transition. The wake structures behind flexible wings with varying wing shapes may become complex and cannot be simply regarded as the typical von Kármán wakes or other common regular wake structures. The discovered drag-thrust transition mechanism based on the identification of the typical wake patterns is not suitable for the coupled fluid-flexible plate system with complex wake structures [579]. Some advanced approaches are desirable to establish a direct correlation between the flexible plate deformation, the temporal and spatial evolution of the vortical structures and the time-dependent fluid loads to reveal the thrustgenerating mechanism. The drag-thrust transition process is usually accompanied by complex changes in vortical structures, which are potentially related to the transition mechanism. The velocity/pressure method [670] and the vorticity/added-mass approach [772] as well as their variants [671] derived from the momentum balance equation offer a practical way to reveal the thrust-generating mechanism by connecting with the instantaneous flow features. These non-intrusive methods were successfully adopted to evaluate the time-dependent forces exerted on a body by integrating the fluid variables within the control volume obtained from experimental measurements [624, 757] or numerical simulations [514, 773]. Herein, we apply the velocity/pressure method to the fluid-pitching flexible plate coupled system with varying flexibility and trailing edge shapes. The main purpose is to explore the relationship between the instantaneous produced thrust and the induced vortical structures. Through the decomposition of the resulting total thrust into four terms with obvious physical significance, the role of flexibility and trailing edge shape in the thrust generation is quantitatively examined. From the perspective

15.1 Introduction

803

of the unsteady force dynamics, two types of the thrust-generating mechanisms have been summarized based on a vast body of works [536, 565, 597, 720, 772]: (1) the lift-based mechanism and (2) the added mass mechanism. The thrust generated by the lift-based mechanism primarily relies on the circulatory forces due to vorticity generation. However, for a pitching flexible plate immersed in the unsteady flow, the reactive force associated with the acceleration due to the pitching motion and the passive deformation due to the added mass effect can play an important role in the thrust generation. An approximate analytical formulation of the added mass coefficient evaluation [614, 777] is employed to investigate the added mass effect on the thrust generation for pitching flexible plates with varying flexibility and the physical conditions.

15.1.3 Organization In this study, we numerically investigate the propulsive performance of flexible plates undergoing prescribed pitching motion with different trailing edge shapes and flexibilities. A recently developed three-dimensional partitioned aeroelastic framework is adopted to simulate the pitching flexible plates [639]. With the aid of the combined mode decomposition technique, the momentum-based thrust evaluation approach and the analytical added mass model, the following key questions concerning the thrust generation, the propulsive efficiency and the drag-thrust transition are addressed: (1) How do flexibility and trailing edge shape affect the propulsive performance and the wake topology of pitching flexible plates? (2) What is the optimal combination of the trailing edge shape and flexibility to maximize the efficiency of flexible propulsors? (3) What is the drag-thrust transition mechanism for pitching flexible plates with varying flexibility and trailing edge shapes? To address (1), we perform a series of numerical simulations for the pitching flexible plates with three representative trailing edge shapes shown in Fig. 15.1 and varying bending stiffnesses at a moderate Reynolds number of Re = 1000 and a fixed Strouhal number of St = 0.3 that St approaches the optimal efficiency range. The comparisons of the propulsive performance, the flapping dynamics and the flow features are performed to understand the role of trailing edge shape and flexibility. The optimal combination of these two parameters for the pitching plates with maximum thrust and efficiency is determined from the propulsive performance map. The natural frequency of the flexible plate immersed in the unsteady flow is evaluated to explore the effect of flexibility on the propulsive performance. The relationship between the unsteady momentum transfer and the thrust generation is established via the momentum-based thrust evaluation approach. The added mass force contributions are quantified to assess the effects of flexibility and the trailing edge shape on the generated thrust force. To understand the feedback connection between the pitching motion and the flow features, we finally examine the mechanism of the drag-thrust transition for various pitching plates by the decomposition of the time-averaged thrust terms.

804

15 Trailing Edge Effect

c Φ b

aΦ

LΦ

(a) Φ = 45◦

(b) Φ = 90◦

(c) Φ = 135◦

Fig. 15.1 Schematic of the wing geometry with the varying trailing edge angles of Φ = a 45◦ , b 90◦ and c 135◦

15.2 Numerical Methodology The governing equations for the incompressible unsteady viscous flow with an arbitrary Lagrangian-Eulerian (ALE) reference frame are discretized using the stabilized Petrov-Galerkin variational formulation. The generalized-α method is utilized to integrate the ALE flow solution in the time domain and it can ensure unconditionally stable and second-order accuracy for linear dynamics problems. The motion equations for a flexible structure are discretized using the standard Galerkin finite element method. The kinematic joints are considered as constraints on the displacement field. A hybrid RANS/LES model based on the delayed detached eddy simulation treatment is employed to simulate the separated turbulent flow [618]. A partitioned iterative coupling algorithm is adopted to integrate the fluid equations and the multibody structural equations. A typical predictor-corrector scheme is used to solve the coupled fluid-structure governing equations. The velocity and traction continuity along the coupling interface with non-matching meshes is satisfied in the fluid-structure coupling procedure. An efficient interpolation scheme via the compactly-supported radial-basis function (RBF) with a third-order of convergence is implemented to exchange coupling data through the fluid-structure interface and to update the mesh in the fluid domain [619]. To avoid the numerical instability caused by significant added mass effect, a recently developed nonlinear interface force correction scheme [181] is utilized to correct and stabilize the fluid forces at each iterative step. The fluid-multibody structure interaction equations and the variational partitioned formulations for the fluid-structure interaction framework have been described in [639] in detail. This fluid-structure interaction solver has been validated for an anisotropic flexible wing with supporting battens and covered membrane components [639].

15.3 Flapping Foils With Varying Trailing Edge

805

15.3 Flapping Foils With Varying Trailing Edge 15.3.1 Problem Statement In the current study, we consider a series of flexible wings with varying trailing edge shapes and flexibility to investigate their effects on propulsive performance. The identical wing sizes are adopted as those in the experiment done by Van Buren [755] as shown in Fig. 15.1. The mean chord of the wing is c = 0.1 m and the width is set to b = 0.1 m, resulting in a wing area of S = bc = 0.01 m2 . The thickness of this thin wing is h = 2.54 × 10−3 m. The thin wing is placed in the unsteady fluid medium with a uniform oncoming flow. As illustrated in Fig. 15.2a, the wing is clamped at the leading edge to restrict the displacement of the leading edge at any direction, but allow the relative rotation around the X -axis with a prescribed pitching angle of θ p (t), which is defined as follows θ p (t) = Aθ p sin(2π f p t)

(15.1)

where Aθ p represents the amplitude of the pitching angle and f p is the pitching frequency. Due to the flexibility effect, the flexible wing deforms during the prescribed pitching motion, which leads to a deflected bending angle β(t) with respect to its rigid counterpart (15.2) β(t) = Aβ sin(2π f p t − γ ) where Aβ is the amplitude of the deflected bending angle and γ represents the phase lag between the pitching angle and the deflected bending angle. Thus, the effective pitching angle is defined as βe f f = θ p + β, which is measured as the angle between the initial reference wing and the trailing edge of the flexible wing shown in Fig. 15.2a. The complex dynamics of the pitching wing are mainly governed by four key non-dimensional fluid-structure interaction parameters, namely Reynolds number Re, mass ratio m ∗ , bending stiffness K B and Strouhal number St [551], which are defined as Re =

ρs h ρ f U∞ c , m∗ = f , f μ ρ c

KB =

ρ

Bk , f U 2 c3 ∞

St =

2 f p c sin(Aθ p ) . U∞

(15.3)

where ρ f represents the fluid density and U∞ denotes the oncoming flow velocity. μ f is the dynamic viscosity of the fluid and ρ s is the wing density. The flexural Eh 3 s rigidity Bk = 12(1−(ν s )2 ) characterizes the flexibility of the wing, where E and ν represent Young’s modulus and Poisson’s ratio, respectively. To quantitatively evaluate the propulsive performance and the dynamics of the pitching wing, we calculate the thrust coefficient C T , the input power coefficient C pow and the propulsive efficiency η from the numerical simulations, which are given as

806

15 Trailing Edge Effect Γslip uf · nf = 0, σ f · nf = 0

Flexible plate

β(t) δz,tip Rigid plate θp (t)

Initial reference plate θp (t) = Aθp sin(2πfp t)

Γno−slip Γin |u | = U∞

H Γout σ f · nf = 0 ∇˜ ν · nf = 0

b

f

h

θp (t)

Z

X

Y

Γslip uf · nf = 0, σ f · nf = 0

B

L

(a)

(b)

Fig. 15.2 Problem set-up for a pitching wing: a illustrations of prescribed pitching motion at the leading edge θ p (t), the deflected bending angle β(t) and the wing tip transverse displacement δz,ti p at the middle point of the trailing edge and b three-dimensional computational setup and boundary condition for the uniform flow past a pitching wing. The wings in (a) are colored by the transverse displacement δz

 Pinp T 1 (σ¯ f · n) · n y d, C pow = 1 f 3 , = − 1 f 2 1 f 2 ρ U∞ S ρ U∞ S ρ U∞ S 2 2 2 Γ  CT 1 . (15.4) CL = 1 f 2 (σ¯ f · n) · nz d, η = ρ U∞ S C pow 2

CT =

Γ

where T is the thrust force of the pitching wing. n y and nz represent the Cartesian  components of the unit outward normal n to the wing surface Γ . Pinp = Γ f s F f · s ∂d dΓ represents the instantaneous input power of the pitching wing. F f is the ∂t driving force acting on the surrounding fluid by the pitching wing along the interface Γ f s. A schematic of the 3D computational domain constructed for the flexible pitching wing is shown in Fig. 15.2b. The midpoint of the leading edge of the wing is located at the origin of the computational domain. The length L, the width B and the height H of the computational domain are all set to 40c. The unsteady fluid flows into the computational domain through the inlet boundary Γin with a uniform velocity |u f | = U∞ . A traction-free boundary condition is considered at the outlet boundary Γout . We apply the slip-wall boundary conditions on the four sides of the computational domain and set the no-slip boundary condition for all wing surfaces. The wing filled with gray color and surrounded by the black color edge in Fig. 15.2b represents the neutral position of the pitching wing. The rigid wing indicated by the red dash line denotes the instantaneous position with a pitching angle of θ p (t). To study the impact of trailing edge shape and flexibility, we select 13 groups of bending stiffness over a wide range of parameter space of K B ∈ [9.86, 1.53 × 105 ] for a concave wing with Φ = 45◦ , a rectangular wing with Φ = 90◦ and a convex wing with Φ = 135◦ . To isolate the effect of other physical parameters, we examine the propulsive performance for a moderate Reynolds number of Re = 1000 at a fixed Strouhal number of St = 0.3 for simplicity, which are close to some smallscale biological and human-made flapping systems.

15.3 Flapping Foils With Varying Trailing Edge

807

Fig. 15.3 a Mean net thrust coefficient C T and b propulsive efficiency η as a function of bending stiffness K B for pitching wings with varying trailing edge angles Φ = 45◦ , 90◦ and 135◦ at Re = 1000 and St = 0.3

15.3.2 Flapping Dynamics The mean net thrust coefficient C T and the propulsive efficiency η produced by the pitching flexible wings with three representative trailing edge angles Φ as a function of bending stiffness K B are shown in Fig. 15.3. Three classified flapping motion regimes, namely (1) low bending stiffness K Blow , (2) moderate bending stiffness high K Bmoderate near frequency synchronization and (3) high bending stiffness K B , are added in Fig. 15.3 to study the characteristics of the produced thrust and propulsive efficiency. It can be seen from Fig. 15.3a that the mean net thrust coefficient grows up rapidly to achieve its peak value at moderate K B and then decreases to a plain when the wings become stiffer. In Fig. 15.3b, the propulsive efficiency changes from negative values to the optimal values and then reduces gradually to almost constant values as K B increases. The rectangular wing can produce the largest thrust within the low bending stiffness regime. The thrust generated by the convex wing is the largest among the three types of wings at moderate and high K B values. Regarding propulsive efficiency, the convex wing is the most efficient propulsion system within the low and high bending stiffness regimes. The rectangular wing achieves the optimal efficiency gain at moderate K B values. It is worth noting that the concave wing has the poorest ability in thrust generation and propulsive efficiency gain within the studied K B range. The bending stiffness value corresponding to the overall largest thrust is smaller than that related to the optimal efficiency. The transition between thrust and drag is observed in Fig. 15.3a when the flexible wing with a moderate K B value becomes more flexible or more rigid. The convex wing shows the largest transition region with positive thrust within the studied K B range. The mechanism of the thrust generation and the drag-thrust transition will be discussed in the next sections. The generated thrust and the propulsive efficiency of the flexible wings with varying trailing edge angles are closely associated with flexibility. In this section, three distinctive flapping motion regimes are firstly classified based on the flapping dynam-

808

15 Trailing Edge Effect

ics. We further discuss how flexibility affects propulsive performance by examining the interplay between the natural frequency of the flexible wing and the pitching frequency.

15.3.2.1

Classification of Flapping Motion Regime

The pitching flexible wings with varying trailing edge shapes exhibit different types of flapping motions as a function of flexibility. Based on the variation of the propulsive performance and the related dynamic responses of the flexible wings, three distinctive regimes are classified from the coupled system: (1) low bending stiffness K Blow , (2) moderate bending stiffness K Bmoderate near frequency synchronization and (3) high high bending stiffness K B . From the analysis of propulsive performance as a function of flexibility, it can be seen that propulsive performance is strongly affected by flexibility. The generated thrust and the propulsive efficiency can be enhanced at moderate bending stiffness values. Regardless of the trailing edge shape, the flexible wing exhibits similar flapping motion modes within the same regime. Here, we plot the instantaneous deformations of the pitching wing with a trailing edge angle of Φ = 45◦ in Fig. 15.4 briefly for the regime classification purpose. It can be seen from Fig. 15.4a that the flexible wing exhibits a chordwise second flexural mode during pitching motion for low K B values. The deformation of the trailing edge shows an opposite direction to the pitching motion at the leading edge. As a result, the whole motion of the pitching wing can be divided into two parts by an almost fixed passive rotation axis (PRA) along the spanwise direction. The front portion follows the pitching motion applied along the leading edge, and the rear portion deforms passively under the action of inertial, elastic and aerodynamic forces. As K B further increases to moderate values, the dominant structural mode changes from the chordwise second flexural mode to the chordwise first flexural mode as shown in Fig. 15.4b. Moreover, the deformation amplitude increases dramatically. In Fig. 15.4c, the whole wing follows the prescribed pitching motion within the high bending stiffness regime. The passive deformation is significantly suppressed by the elastic forces at higher K B values.

15.3.2.2

Synchronization and Role of Flexibility

Based on the classification of the flapping motion regime, the passive deformation of the pitching wing is dramatically enhanced within a certain range of bending stiffness values. In this range, the natural frequency of the flexible wing immersed in the unsteady fluid approaches the fixed pitching frequency. A natural question to ask is whether the ratio between these two frequencies plays an important role in the dynamic characteristics. Here, we first calculate the frequency ratio, and then examine the connection between the flapping dynamics and the frequency ratio for flexible wings with different bending stiffnesses. As suggested in Van Eysden et al.

15.3 Flapping Foils With Varying Trailing Edge

809

Fig. 15.4 Instantaneous deformation of pitching plate with trailing edge angle of Φ = 45◦ at K B = a 9.86, b 51.8 and c 73,998

[756], the relationship between the natural frequencies of the chordwise first flexural mode in the fluid f 1s and that in vacuum f 1vac can be considered  −0.5 πρ f b Ω(κ) f 1s = f 1vac 1 + , 4ρ s h

(15.5)

The approximate hydrodynamic function Ω(κ) as a function of the coefficient κ = 1.8751 bc is given as Ω(κ) =

1 + 0.74273κ + 0.14862κ 2 , 1 + 0.74273κ + 0.35004κ 2 + 0.058364κ 4

(15.6)

In Eq. (15.5), the natural frequency of the wing in vacuum f 1vac is directly calculated from the structural motion equations via the modal analysis. The frequency ratio f ∗ between the first natural frequency of the wing in the fluid and the fixed pitching frequency is defined as  −0.5 πρ f b f 1s f 1vac 1+ Ω(κ) = , f = fp fp 4ρ s h ∗

(15.7)

As a function of the nondimensional frequency ratio f ∗ , the mean net thrust coefficient, the root-mean-squared value of the lift coefficient fluctuation, the mean input power coefficient and the propulsive efficiency are summarized in Fig. 15.5 for the concave, rectangular and convex wings. The vertical black dash line is located at f ∗ = 1, which indicates the frequency synchronization between the first natural frequency of the wing and the actuated frequency. It can be seen from Fig. 15.5a that all three types of flexible wings with different Φ exhibit the global maximum mean net thrust forces when the frequency ratio gets close to f ∗ = 1. The amplitude of the lift force is greatly enhanced within the near synchronization regime.

810

15 Trailing Edge Effect f∗ = 1

20 Φ = 45◦ Φ = 90◦ Φ = 135◦

0.4

0

CL rms

CT

0.8

f∗ = 1 Φ = 45◦ Φ = 90◦ Φ = 135◦

15 10 5

-0.4

100

f∗

0

101

0

10

(a)

(b) 0

30

C pow

1

10

f∗ = 1

f∗ = 1

Φ = 45◦ Φ = 90◦ Φ = 135◦

20 10

η

40

f∗

Φ = 45◦ Φ = 90◦ Φ = 135◦

-0.5

0 100

f∗ (c)

101

-1

100

f∗

101

(d)

Fig. 15.5 a Mean thrust coefficient C T , b r.m.s of the lift coefficient fluctuation C L r ms , c mean input power coefficient C pow and d propulsive efficiency η as a function of the frequency ratio f ∗ for pitching wing with varying trailing edge angles Φ = 45◦ , 90◦ and 135◦ at Re = 1000 and St = 0.3

As observed in Fig. 15.4, a large amplitude of passive deformation relative to the active pitching motion is excited by the frequency synchronization. According to the formulation of the input power, the flexible wing requires more input power to maintain the prescribed pitching motion due to the increased fluid loads acting on the wing surface and the improved velocity of the wing motion under the frequency synchronization condition. Although the flexible wing can produce the largest thrust forces at K B = 51.8 (close to f ∗ = 1), the optimal propulsive efficiency is achieved at a higher K B of 98.66. This is mainly because the pitching flexible wing with K B = 51.8 can only improve the thrust forces at most four times, but it needs at least six times the input power than the wing with K B = 98.66. As a result, the propulsive efficiency for the flexible wing with the largest vibration amplitude under the frequency synchronization condition is reduced and its optimal value is achieved for the flexible wing with moderate passive deformations. We further examine the role of flexibility in the dynamic responses of pitching flexible wings with varying trailing edge angles. In Fig. 15.6a, the amplitude of the effective pitching angle grows up rapidly and reaches its peak when the frequency ratio f ∗ approaches 1. As f ∗ further increases, the amplitude decreases sharply and finally maintains a value close to the pitching angle amplitude of 12◦ applied at the

15.3 Flapping Foils With Varying Trailing Edge f∗ = 1

(b) 1.2 Φ = 45◦ Φ = 90◦ Φ = 135◦

60 40 20

f∗ = 1 Φ = 45◦ Φ = 90◦ Φ = 135◦

0.8

γ/π

Aβef f (◦)

(a) 80

811

0.4 0

0 10

0

f

∗ 10

Aδz,tip/c

(c) 1.2

-0.4

1

100

f∗

101

f∗ = 1 Φ = 45◦ Φ = 90◦ Φ = 135◦

0.8

0.4

0

100

f∗

101

Fig. 15.6 a Amplitude of effective pitching angle Aβe f f , b phase lag γ and c amplitude of nondimensional transverse displacement Aδz,ti p /c as a function of the frequency ratio f ∗ for pitching wing with varying trailing edge angles Φ = 45◦ , 90◦ and 135◦ at the middle point of the TE at Re = 1000 and St = 0.3

leading edge for high K B cases. The variation of the phase lag γ between the deflected bending angle and the prescribed pitching angle as a function of the frequency ratio is presented in Fig. 15.6b. The phase lag is related to the relative direction of the motion at the trailing edge and the required input power. The flexible wing shows a phase lag close to π within the low bending stiffness regime. This is caused by the excited chordwise second flexural mode at lower K B values. When the synchronization between the first natural frequency of the flexible wing and the pitching frequency is exited at moderate K B values, the phase lag reduces rapidly. Synchronization between the pitching angles at the leading and trailing edges is maintained by reducing the phase lag to zero at larger K B values. The amplitude of the transverse displacement at the trailing edge exhibits a similar trend to the effective pitching angle as a function of f ∗ . A large displacement amplitude is excited near f ∗ = 1 by the frequency synchronization. It can be concluded that the enhanced thrust forces and the increased input power are strongly associated with the passive deformation of the flexible wing and its relative phase lag to the prescribed pitching motion at the leading edge. Since the passive deformation can mutually affect the flow field, the generated fluid forces are directly related to the flow features and wake structures around the wing. In the next section, we further analyze the relationship between the flow characteristics and the propulsive performance of the flexible wings with varying bending stiffness and trailing edge shapes.

812

15 Trailing Edge Effect

15.3.3 Flow Field and Wake Structures In this section, the time-averaged and the instantaneous flow features are analyzed to understand the thrust generation mechanism.

15.3.3.1

Time-Averaged Flow Features

To explore the effect of trailing edge shape and flexibility on the unsteady momentum imparted by the wing into the wake, the iso-surfaces of the time-averaged streamwise velocity v/U ¯ ∞ are plotted for three types of flexible wings with four representative bending stiffness values in Fig. 15.7. For the extraction of time-averaged streamwise velocity, the numerical results on the body-fitted moving mesh are projected to a reference stationary mesh via the RBF method, and then averaged over five pitching cycles. In Fig. 15.7, the iso-surfaces in the gray color with a threshold of v/U ¯ ∞ = 0.95 indicate the deceleration flow region. The iso-surfaces in the blue color thresholded at 1.15 represent the high-velocity jet produced by the pitching wing. The wing in the black color is plotted at the neutral position. The iso-surface is cut within y/c ∈ [1, 5] to concentrate on the velocity distribution in the near wake behind the wing. In Fig. 15.7a, e, i, the flexible wing produces small regions of high-velocity jet flows in the wake within the low bending stiffness regime. It can be inferred from the dynamic response shown in Fig. 15.4a that the very flexible wing with the chordwise second mode is unable to produce large thrust. As K B increases to 51.8 to meet the frequency synchronization condition, the near-wake flow which contains high streamwise velocities accelerated by the pitching flexible wing expands wider in the transverse direction and farther in the streamwise direction. Consequently, the generated thrust is significantly enhanced when more energies are transferred to the fluid. The inclination and the length of the jet behind the flexible wing become smaller gradually when the coupled system enters the desynchronization states. Meanwhile, the passive deflection of the wing reduces significantly, compared to the case under the frequency synchronization condition. As a result, the net thrust decreases sharply. It can be observed from Fig. 15.7d, h, l that the high-velocity region behind the wing is suppressed dramatically as K B further increases to 73,998. The passive deformation of the wing is almost negligible and less momentum is transferred to the fluid, resulting in lower thrust generation. The trailing edge shape also affects the topology of the high-velocity region to further govern the thrust generation. The deceleration flow changes from a quadfurcated shape to a compressed shape when the trailing edge angle increases from 45◦ to 135◦ at K B = 19.73, 197.3 and 73,998, respectively. Meanwhile, the shape of the high-velocity jet shows an opposite change. The inclination of the low-velocity and high-velocity flow becomes larger when the wing changes from a concave shape to a convex shape. When frequency synchronization is established, the high-velocity jet presents a compressed shape without obvious bifurcation for all three types of wings

15.3 Flapping Foils With Varying Trailing Edge

813

Fig. 15.7 Isosurfaces of time-averaged streamwise velocity v/U ¯ ∞ thresholded at 0.95 (gray) and 1.15 (blue) of a pitching wing at K B = a, e, i 19.73, b, f, j 51.8, c, g, k 197.3 and d, h, l 73,998 with trailing edge angle of Φ = a–d 45◦ , e–h 90◦ and i–l 135◦

but shows a larger inclination for the convex wing. As a result, the convex wing can produce the overall largest thrust within the studied K B range and the concave wing is the least efficient propulsion system. A schematic of a monitoring surface in the gray color for plotting the energetic jet flows is presented in Fig. 15.8. This monitoring surface normal to the streamwise direction is located at y/c = 2.5 behind the wing. The uniform oncoming flow U∞ is accelerated or decelerated by the pitching flexible wing to modulate the flow with a redistributed velocity profile u f (x, y, z, t) in the wake, which is directly related to the thrust generation. We extract the time-averaged streamwise velocity distribution on the monitoring plane to quantitatively compare the thrust-generating momentum. In Fig. 15.9, the high-velocity region (v/U∞ >1) expands both in the transverse and spanwise directions and the magnitude increases significantly when the frequency synchronization condition is achieved at K B = 51.8. With the further increase of K B , the jet region reduces and the magnitude decreases continuously. According to Newton’s third law, the maximum thrust is produced under the frequency synchronization condition and the propulsive performance deteriorates when the wing becomes more rigid. Except for the propulsive system near frequency synchronization, the jet changes from a compressed shape with one peak in the center to a

814

15 Trailing Edge Effect

bifurcated shape with four peaks as the trailing edge shape alters from concave to convex. The bifurcated jet contains more imparted momentum, which can improve the thrust generation for a convex wing.

15.3.3.2

Instantaneous Flow Features Associated with Fluid Loads

The time-averaged flow features reflect the connection with the time-averaged net thrust, but it lacks information about the instantaneous generated fluid loads. To fully understand the effect of trailing edge shape and flexibility on propulsive performance, we further examine the instantaneous flow features associated with the instantaneous resulting fluid loads. It can be seen from Fig. 15.5 that the force statistics show similar trends as a function of K B for flexible wings with different trailing edge shapes. Thus, the convex wings with four representative flexibility values are selected for comparison purposes to examine the role of flexibility in the temporal and spatial evolution of the flow features. Figure 15.10 presents the comparison of the instantaneous lift and thrust coefficients within one completed pitching cycle. The lift coefficient of the too-flexible wing at K B = 19.73 shows an opposite phase to the wing with higher K B values, which is affected by the chordwise second mode. As K B increases to 51.8, the amplitudes of the lift and thrust coefficients are significantly increased due to the excited large passive deformation under the frequency synchronization condition. This type of burst aerodynamic force is helpful for bio-inspired locomotion with high maneuverability. The amplitude of the fluid loads keeps decreasing when the wing becomes more rigid, resulting in lower propulsive performance. To gain further insight into the fluid load generation, the instantaneous pressure coefficient distribution around the wing and on the wing surfaces at the moment with the largest thrust, the lowest thrust and the largest lift is presented in Fig. 15.11. It can be seen from Fig. 15.11a–d that the too-flexible wing shows opposite pressure distributions on the upper and lower surfaces compared to the wings with high K B values. This is mainly caused by the inversed phase difference between the applied

Fig. 15.8 Schematic of monitoring surface for plotting time-averaged streamwise velocity and control volume enclosed the flexible wing. The control surface in gray color is used to extract the streamwise velocity profiles

15.3 Flapping Foils With Varying Trailing Edge

815

Fig. 15.9 Time-averaged streamwise velocity v/U ¯ ∞ on a plane at y/c = 2.5 normal to streamwise of a pitching wing at K B = a, e, i 19.73, b, f, j 51.8, c, g, k 197.3 and d, h, l 73,998 with trailing edge angle of Φ = a–d 45◦ , e–h 90◦ and i–l 135◦

pitching motion at the leading edge and the deflected motion at the trailing edge when the chordwise second mode is excited. When the wing flaps under the frequency synchronization condition at K B = 51.8, the wing produces much larger positive and negative pressures on the surfaces due to the higher acceleration of the fluid by the wing. The large pressure difference between the upper and lower surfaces leads to the maximum thrust value near frequency synchronization. Furthermore, the large passive deformation of the flexible wing helps in orienting the decomposed component of the pressure gradient in the chordwise direction, which further enhances the generated thrust. The pressure difference becomes weaker and the pressure gradient component that contributes to the thrust generation decreases when the passive deformation of the wing is suppressed at higher K B values. As a result, the largest thrust value reduces for a stiffer wing. Similar conclusions can be drawn for the variation of the lowest thrust and the largest lift as a function of K B . The instantaneous vortical structures indicated by the iso-surfaces of the Q criterion behind convex wings at different K B values at the moment with the largest thrust are plotted in Fig. 15.12. The iso-surfaces are colored by the normalized streamwise velocity. The wake behind the convex pitching wing shows horseshoe-like structures, but the size and direction of the vortical structures are varied with flexibility. The horseshoe-like structures are elongated in the transverse and streamwise directions when K B increases from 19.73 to 51.8, which is caused by the strong acceleration near frequency synchronization. The inclination between the formed vortical struc-

816

15 Trailing Edge Effect 40

KB KB KB KB

= 19.73 = 51.8 = 197.3 = 73998

KB KB KB KB

10

CT

CL

20 0

= 19.73 = 51.8 = 197.3 = 73998

0

-20 -10 -40

0

0.2

0.4

0.6

t/T

(a)

0.8

1

0

0.2

0.4

0.6

0.8

1

t/T

(b)

Fig. 15.10 Comparison of a instantaneous lift coefficient and b instantaneous thrust coefficient of pitching wing with trailing edge angle of Φ = 135◦ within one completed pitching cycle. t/T = 0 corresponds to the pitching upward from the neutral position

tures and the centerline becomes smaller for stiffer wings due to the suppressed passive deformation. It is worth noting that the largest thrust is generated when the trailing edge vortex detaches from the trailing edge and convects downstream by transferring momentum to the wake. To understand the effect of the trailing edge shape on the propulsive performance, the comparison of the instantaneous lift and thrust coefficients within one completed pitching cycle for a concave wing, a rectangular wing and a convex wing is shown in Fig. 15.13. It can be seen from Fig. 15.3 that the convex wing produces the overall largest thrust and the concave wing is the least efficient in thrust generation at most K B values. Thus, we choose a representative bending stiffness of K B = 98.66 which is a slightly far away from the frequency synchronization condition for simplicity. In Fig. 15.13a, the convex wing and the concave wing produce similar amplitudes of the lift coefficient. The rectangular wing is less efficient in generating lift. The convex wing produces the largest instantaneous positive thrust at t/T = 0.36. The largest instantaneous positive thrust values of the rectangular wing and the concave wing are similar. On the contrary, the rectangular wing has the smallest instantaneous drag and the other two wings show more drag penalties. As a result, the convex wing generates slightly larger net thrust, compared to the rectangular wing. The net thrust produced by the concave wing is the smallest one among the three wings. The comparison of the instantaneous pressure coefficient distributions around the pitching wing with different trailing edge shapes is shown in Fig. 15.14. It can be seen from Fig. 15.14a–c that the pressure difference between the upper and lower surfaces of a convex wing is the largest. Because the convex wing and the concave wing have the longest local chord at the mid-span location and the sides, the flows are strongly accelerated at these locations to induce the largest pressure gradient. Compared to the pressure distribution of the concave wing, the flow with bifurcated four jets on both sides of the convex wing can produce the highest pressure difference. Consequently, the convex wing can generate the largest instantaneous thrust. It can be inferred from Fig. 15.14d–f that the rectangular wing produces the smallest instantaneous drag due to the less pressure difference. In Fig. 15.14g–i, the rectangular wing shows smaller pressure gradients in the transverse direction. Since the effective projection area of

15.3 Flapping Foils With Varying Trailing Edge

817

Fig. 15.11 Instantaneous pressure coefficient contour on the spanwise symmetry plane and pressure coefficient distribution on the upper and lower surfaces of pitching wing with trailing edge angle of Φ = 135◦ at K B = a, e, i 19.73, b, f, j 51.8, c, g, j 197.3 and d, h, l 73,998 at the moment with a–d the largest thrust, e–h the lowest thrust and i–l the largest lift

 f ∂u Fig. 15.12 Instantaneous vortical structures based on the iso-surfaces of Q = − 21 ∂ xij

f

∂u j ∂ xi

 value

of a pitching wing with trailing edge angle of Φ = 135◦ at K B = a 19.73, b 51.8, c 197.3 and d 73,998 at the moment with the largest thrust. Iso-surfaces of non-dimensional Q + = Q(c/U∞ )2 = 0.25 are colored by the normalized streamwise velocity v/U∞

15 Trailing Edge Effect 10

2

5

1

CT

CL

818

0 Φ = 45◦ Φ = 90

-5

0 Φ = 45◦

◦

-1

◦

Φ = 90◦

Φ = 135

-10

0

0.2

0.4

Φ = 135◦

0.6

0.8

-2

1

t/T

0

0.2

0.4

0.6

0.8

1

t/T

(a)

(b)

Fig. 15.13 Comparison of a instantaneous lift coefficient and b instantaneous thrust coefficient of pitching wing with K B = 98.66 within one complete pitching cycle. t/T = 0 corresponds to the pitching upward from the neutral position

these three wings is similar, the amplitude of the lift coefficient of the rectangular wing is smaller than that of the concave and convex wings. The wake structures of these three wings are compared in Fig. 15.15. At the moment with the largest thrust, the convection of the detached vortices from the trailing edge is observed for all wings. The wake structures behind the concave wing behave as a reversed horseshoe-like structure. Conversely, the horseshoe-like structure is observed behind the rectangular wing and the convex wing. The vortical structures behind the convex wing are stretched to form the widest vortex ring in the transverse direction, which is caused by the largest acceleration at the longest local chord location. Furthermore, the inclination between the vortical structure and the centerline is largest for the convex wing, which helps in orienting the pressure gradient to generate more thrust.

15.3.4 Unsteady Momentum Transfer and Thrust Generation 15.3.4.1

Decomposition of Unsteady Momentum

To establish a quantitative connection between the flow dynamic behavior and the time-dependent force mechanisms, Noca et al. [670, 671] derived a velocity/pressure equation to evaluate the time-dependent forces acting on a moving body immersed in an incompressible and viscous flow. The instantaneous force F(t) exerted by the unsteady fluid is calculated by integrating the momentum change within an arbitrary time-dependent control volume V f (t) and on the control surface S c (t) as well as the body surface S s (t) F(t) = − ∂t∂ −

 S c (t)



ρ f u f dV −

V f (t)

pndS +



S c (t)

τ ndS −

 S c (t)



S s (t)

n · (u f − uis )ρ f u f dS

n · (u f − uis )ρ f u f dS.

(15.8)

15.3 Flapping Foils With Varying Trailing Edge

819

Fig. 15.14 Instantaneous pressure coefficient contour on the spanwise symmetry plane and pressure coefficient distribution on the upper and lower surfaces of pitching wing with K B = 98.66 for Φ= a, d, g 45◦ , b, e, h 90◦ and c, f, i 135◦ at the moment with a–c the largest thrust, d–f the lowest thrust and g–i the largest lift

 f ∂u Fig. 15.15 Instantaneous vortical structures based on the iso-surfaces of Q = − 21 ∂ xij = a 45◦ , b 90◦

and c 135◦

f

∂u j ∂ xi

 value

at the moment with the largest of a pitching wing with K B = 98.66 for Φ thrust. Iso-surfaces of non-dimensional Q + = Q(c/U∞ )2 = 0.25 are colored by the normalized streamwise velocity v/U∞

820

15 Trailing Edge Effect P1

15

P2 P3

P4

P5

CT n CT u

CT

10

CT c

5

CT p

0

CT s

-5

CT w

-10 -15

0

0.2

0.4

t/T

0.6

0.8

1

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 15.16 a Thrust decomposition based on the momentum equation for flexible wing with trailing edge angle of Φ = 135◦ at K B = 51.8 as a nominal case. The flow features are plotted at nondimensional time instants of b P1, c P2, d P3, e P4 and f P5. The instantaneous X -vorticity ωx on the mid-span plane is presented in the first column. The instantaneous rate of change of momentum ∂(ρ f v) on the mid-span plane is presented in the second column. The instantaneous non-dimensional ∂t streamwise velocity v/U∞ on the control surface at y/c = 2.5 is presented in the third column. The instantaneous pressure coefficient C p on the control surface at y/c = 2.5 is presented in the fourth column. t/T = 0 corresponds to the pitching upward from the neutral position

15.3 Flapping Foils With Varying Trailing Edge

821

where n represents the unit vector normal to the integral surface. uis denotes the velocity of the integral surface. p and I are the pressure and unit tensor, respectively. τ = μ f (∇u f + (∇u f )T ) is the viscous stress tensor and μ f is the dynamic viscosity coefficient. In Eq. (15.8), the first term on the right-hand side represents the rate of change of momentum within the control volume V f (t). The second term, the third term and the fourth term are the net momentum flux across the control surface, the instantaneous pressure force and the instantaneous viscous stresses acting on the control surface S c (t), respectively. The fifth term denotes the momentum flux across the body surface and it equals zero due to the no-through boundary condition of the body surface. In this study, a 3D control volume with a size of 10c × 10c × 7.5c enclosed the pitching wing is constructed in a fixed frame to evaluate the time-dependent fluid loads. Thus, the velocity uis of the control surface S c (t) is zero for a stationary control volume. The instantaneous thrust coefficient C T w evaluated from the wake information in the control volume can be decomposed into four distinct terms written as CT w = − 1

Fy f U2 S ∞

= CT u + CT c + CT p + CT s , ρ  1 ∂ (ρ f u f ) · n y dV, CT u = 1 f 2 ∂t ρ U S ∞ 2 V f (t) 1 CT c = 1 f 2 [ρ f u f (n · u)] · n y dS, ρ U S ∞ 2 S c (t) 1 CT p = 1 f 2 ( pn) · n y dS, ρ U∞ S 2 c S (t) 1 CT s = − 1 f 2 (τ n) · n y dS. ρ U∞ S 2 2

(15.9)

S c (t)

where C T u , C T c , C T p and C T s are the unsteady term, the convective term, the pressure force term and the shear stress term, respectively. The flow variables obtained from the numerical simulation on each point of the body-fitted moving mesh are mapped to each point of the fixed reference mesh of the rectangular control volume shown in Fig. 15.8 to calculate each term in Eq. (15.9). The flexible convex wing immersed in the unsteady flow with the global maximum thrust under the frequency synchronization condition is considered as the nominal case to demonstrate and validate the momentum-based thrust evaluation formulation. Fig. 15.16a presents the decomposition of the four distinct thrust coefficient terms and the comparison of the total evaluated thrust coefficient C T w and the total thrust coefficient C T n calculated from Eq. (15.4) in the numerical simulation within one completed cycle. It can be seen that the evaluated thrust coefficient C T w from the integration of the momentum equation within the control volume is consistent with the thrust coefficient C T n . In other words, the momentum-based control volume

822

15 Trailing Edge Effect

approach can accurately calculate the instantaneous thrust force based on the flow variables around and inside a control volume. Each decomposed thrust coefficient term has physical significance to correlate the instantaneous vortical structures and the resulting fluid loads. The unsteady term C T u behaves as an almost symmetrical feature with respect to the zero-thrust condition due to the symmetrical pitching motion. The variation of the unsteady term makes significant contributions to the instantaneous total thrust but leads to negligible contributions to the time-averaged net thrust. The convective term C T c shows positive momentum flux across the control surface, which is related to the vortical structures induced by the pitching wing convecting downstream with high velocities. The pressure term C T p presents comparable variations to the unsteady term and it plays a negative role in the mean thrust generation. The variation of the shear stress term C T s is negligible compared to other terms due to the large control volume far from the moving wing. The unsteady term and the pressure term show a phase synchronized with the total thrust, while the convection term shows an opposite phase. To shed light on the relationship between the instantaneous force and the evolution of the wake structures, the flow features within the control volume and on the control surface related to the decomposed thrust terms are plotted at five selected time instants (P1–P5) in Fig. 15.16b–f. It can be observed from the figures in the first column that the maximum total thrust is achieved at the time instant of P3 when the trailing edge vortex detaches from the TE of the wing during downward movement. Meanwhile, the detachment and convection of the trailing edge vortex induce the large time derivative of the momentum around the wing surface within the control volume shown in the second column, resulting in the optimal unsteady term. The low-velocity region presented in the third column expands on the control surface located at y/c = 2.5. Thus, the convective term reduces to a lower value. In the fourth column, the greatest suction pressure difference is formed between the control surface at y/c = −5 and 2.5 to lead to the largest pressure term. As the wing continues to move downward from P3 to P5, the detached trailing edge vortex convects further downstream and the vortex is gradually formed at the LE. Thus, the region with negative rates of change of momentum expands on the wing surface during the movement of the vortices from the leading edge to the trailing edge, leading to the continuous reduction of the unsteady term. Conversely, the convective term grows up as the vortex rings containing high velocities reach the control surface at y/c = 2.5. The suction effect between the front and back control surfaces is reversed when the low-pressure region on the back control surface becomes larger. As a result, the pressure term is reduced and makes negative contributions to the instantaneous thrust generation.

15.3.4.2

Effect of Flexibility on Unsteady Momentum

It can be concluded from the control volume analysis for the nominal case that the unsteady term, the convective term and the pressure term are strongly related to the instantaneous thrust generation. The contribution of the shear stress term can be

15.3 Flapping Foils With Varying Trailing Edge KB KB KB KB

6

= 19.73 = 51.8 = 197.3 = 73998

KB KB KB KB

4

CT c

CT u

4 0 -4

10

= 19.73 = 51.8 = 197.3 = 73998

2

0.2

0.4

t/T

(a)

0.6

0.8

1

0

-10

-2 0

= 19.73 = 51.8 = 197.3 = 73998

-5

0

-8

KB KB KB KB

5

CT p

8

823

0

0.2

0.4

t/T

(b)

0.6

0.8

1

0

0.2

0.4

t/T

0.6

0.8

1

(c)

Fig. 15.17 Comparison of a unsteady term, b convective term and c pressure term of pitching wing with trailing edge angle of Φ = 135◦ within one completed pitching cycle. t/T = 0 corresponds to the pitching upward from the neutral position

neglected. To further explore the effect of flexibility on the thrust generation, the instantaneous decomposed unsteady, convective and pressure terms evaluated by the momentum-based approach are compared for convex wings with four representative bending stiffness values in Fig. 15.17. The convex wing with K B = 51.8 shows the largest amplitude of the unsteady term in Fig. 15.17a . The large passive deformation excited under the frequency synchronization condition leads to the drastic change of the fluid acceleration (unsteady term) near the wing. Once the deformation amplitude reduces for flexible wings with low bending stiffness or high bending stiffness values, the variation of the unsteady term becomes weak. With respect to the convective term, the horseshoe-like vortical structures flowing across the control surface contain much higher velocities at K B = 51.8 observed in Figs. 15.7 and 15.9, resulting in the largest momentum flux under the frequency synchronization condition. Compared to the highly rigid wing with K B = 73,998, the flexible wing with moderate passive deformation is able to improve the momentum flux to enhance the thrust generation. However, the highly flexible case lacks the ability to impart large momentum to the wake. The pressure term is significantly enhanced under the frequency synchronization condition and shows negative contributions to the mean thrust. The difference between the pressure terms of the flexible wings is small under the desynchronization condition.

15.3.4.3

Effect of Trailing Edge Shape on Unsteady Momentum

The comparison of the instantaneous decomposed three thrust terms for the flexible wing at a moderate bending stiffness of K B = 98.66 with different trailing edge shapes is shown in Fig. 15.18. Compared to the concave wing, the absolute values of the minimum and maximum unsteady term of the rectangular wing become smaller, due to the lower acceleration caused by the shorter local chord. The stronger compression effect of the wake behind the convex wing leads to larger local maximum and minimum values of the unsteady thrust term than those of the concave wing. The concave wing produces the overall smallest convective term while the convex wing is beneficial to enhancing the momentum transfer. The surrounding flow can

824

15 Trailing Edge Effect

0

0.6

0

0.4

-0.5 -1

0.5

Φ = 45◦ Φ = 90◦ Φ = 135◦

CT c

CT u

0.8

Φ = 45◦ Φ = 90◦ Φ = 135◦

CT p

1 0.5

0

0.2

0.4

t/T

0.6

0.8

1

0.2

-0.5 Φ = 45◦ Φ = 90◦ Φ = 135◦

-1

0

0.2

(a)

0.4

t/T

(b)

0.6

0.8

1

-1.5

0

0.2

0.4

t/T

0.6

0.8

1

(c)

Fig. 15.18 Comparison of a unsteady term, b convective term and c pressure term of pitching wing with K B = 98.66 within one completed pitching cycle. t/T = 0 corresponds to the pitching upward from the neutral position

be strongly accelerated through the coupling effect near the mid-span location of the convex pitching wing, compared to the concave and rectangular wings. The pressure term of the rectangular wing has the smallest amplitude but the convex wing shows the largest amplitude. However, the pressure term of the rectangular wing has less contribution to the mean drag. The concave wing experiences the maximum mean drag penalty related to the pressure term.

15.3.5 Drag-Thrust Transition Considering the complex vortical structures caused by the pitching wing, it is quite difficult to explore the mechanism of the drag-thrust transition by directly investigating the evolution of the wake structures. In addition to identifying the transition boundary, the variation of the mean thrust data as a function of trailing edge shape and flexibility is limited to provide further insight into the physical mechanisms related to the vortical structures. The momentum-based thrust evaluation approach breaks the barrier of the connection between the time-dependent or mean resulting forces and the vortical structures. With the aid of this effective method, the mechanism of the drag-thrust transition is quantitatively examined based on the variation of the mean decomposed thrust terms correlated to the time-averaged flow features. The variation of the mean unsteady term, the mean convective term and the mean pressure term as a function of trailing edge shape and flexibility is summarized in Fig. 15.19. The classification of the three distinctive regimes is added to the figures to help characterize the role of flexibility. It can be seen from Fig. 15.19a that the contribution of the mean unsteady term to the total mean thrust is quite small, compared to the convective term and the pressure term. The time-averaged unsteady term is small. It is worth noting that the unsteady term reaches peak values under the frequency synchronization condition due to the symmetry breaking of the flow field. In Fig. 15.19b, the mean convective term of the flexible wing shows a peak near frequency synchronization, and then it reduces sharply and maintains an almost constant value finally when the wing becomes more rigid. By linking to the time-

15.3 Flapping Foils With Varying Trailing Edge i

ii

0.02

i

ii

iii

0

i

1

-0.02 1

10

2

10

3

10

KB

(a)

4

10

10

5

iii

-1

Φ = 45◦ Φ = 90◦ Φ = 135◦

-2

0 -0.04

ii

0

Φ = 45◦ Φ = 90◦ Φ = 135◦

2

CT c

CT u

3

iii Φ = 45◦ Φ = 90◦ Φ = 135◦

0.04

CT p

0.06

825

1

10

2

10

3

10

KB

(b)

4

10

10

5

101

102

103

KB

104

105

(c)

Fig. 15.19 a Mean unsteady thrust coefficient C T u , b mean convective thrust coefficient C T c and c mean pressure thrust coefficient C T p as a function of bending stiffness K B for pitching wings with varying trailing edge angles Φ = 45◦ , 90◦ and 135◦ at Re = 1000 and St = 0.3

averaged velocity field in the wake shown in Figs. 15.7 and 15.9, the surrounding flows are strongly accelerated by the large passive deformation under the frequency synchronization condition to induce the enhancement of the mean convective term. Since the passive deformation is suppressed for stiffer wings, the momentum flux convected downstream is reduced. With respect to the effect of trailing edge shape, it is observed that the rectangular wing can produce the largest mean convective thrust within the low bending stiffness regime. The concave and convex wings have similar contributions within this regime. As shown in Fig. 15.4, the excited chordwise second mode reduces the effective areas related to the momentum transfer, which restricts the mean convective thrust generation. The convex wing has the largest contribution to the mean convective thrust at higher K B values. The concave wing is the most inefficient shape to produce mean momentum flux at moderate K B values. As the trailing edge angle increases, the improvement of the mean convective thrust is caused by the enhancement of the bifurcated jets containing high velocities observed in Fig. 15.7. However, the effect of the trailing edge shape on the mean convective thrust is small when the wing becomes stiff enough. Similar trends of the pressure term as a function of flexibility are observed in Fig. 15.19c for flexible wings with different trailing edge shapes. The contribution of the mean pressure term to the drag force becomes larger near frequency synchronization and then reduces to almost constant values for stiffer wings. The rectangular wing, the convex wing and the concave wing show the largest contributions to the drag penalty within the low bending stiffness regime, the moderate bending stiffness regime and the high bending stiffness regime, respectively. Among the three decomposed terms, the convective term makes positive contributions to the mean thrust generation and the pressure term leads to drag. The mean unsteady term is slightly larger than zero for most of the cases. The shear stress term is generally negligible. Consequently, the drag-thrust transition is mainly governed by the relative values of the convective term and the pressure term on the control surfaces. Increasing flexibility close to the frequency synchronization condition is able to largely improve the convective term. Within the optimal thrust region, the trailing edge shape has an opposite effect on the convective term and the pressure

826

15 Trailing Edge Effect

term. From Fig. 15.3, two drag-thrust transition boundaries are observed as a function of flexibility. When varying from the low bending stiffness regime to the moderate bending stiffness regime near frequency synchronization, the transition from drag to thrust is mainly caused by a significant increase in the momentum flux. The variation of the dominant structural mode from the chordwise second mode to the first mode improves the effective areas and acceleration. As a result, this produces vortical structures containing higher velocities. When the flexible wings approach their rigid counterparts, the transition is triggered by reduced passive deformation. With regard to the effect of trailing edge shape, the convex wing broadens the drag-thrust transition region as a function of flexibility by enhancing the momentum transfer to the wake. The concave wing has the narrowest transition region due to the negative unsteady thrust term within the moderate bending stiffness regime. The thrust-generating mechanism and the drag-thrust transition mechanism can also be explained from the view of the added mass effect.

15.3.6 Added Mass Effect on Thrust Generation The added mass force plays an important role in thrust or lift generation for aquatic swimming and bird/insect flying [536]. To gain further insight into the effect of flexibility and trailing edge shape in the current study, the generated thrust force due to the added mass effect is estimated via an analytical formulation of a pitching flexible wing [777]. Due to the added mass effect, the adjacent fluid can be accelerated by the flapping wing to generate a reaction force F a . The added mass force has a component in the thrust direction, thereby contributing to the thrust generation during flapping motion [516, 597]. Fig. 15.20a illustrates a schematic of the thrust generation contributed by the added mass force F a at a selected time instant tn for the flexible wing with an instantaneous pitching angle of θ p (tn ). The added mass force F a can be decomposed into the normal component Fan and the tangential component Faτ . In the current study, the tangential force Faτ can be neglected for a relatively thin flexible wing. The normal force Fan of the added mass force can be evaluated by [597, 777] Fan = − m a (as · nc )dS, (15.10) S s (t) f

where m a = 2ρm sl(x) is the added mass coefficient of a surface wing per unit area, π which is obtained from the analytical added mass tensor [777]. Here m s denotes the chordwise mode number and l(x) is the local chord length at different spanwise locations. as is the acceleration of the wing at each Lagrangian point and nc represents the unit vector normal to the chord of the moving wing surface S s (t). In the current formulation, the variation of the chordwise mode is observed and the influence of the

15.3 Flapping Foils With Varying Trailing Edge

827

S s (t)

Flexible plate

Fan Faz

−Fay

Initial reference plate θp (ti ) = Aθp sin(2πfp ti )

(a)

(b) Fig. 15.20 Schematic of a force decomposition of the added mass force at time instant ti and b integral of the local added mass force m a (as · nc ) at each Lagrangian point over the whole wing surface S s to calculate the normal component of the added mass thrust. T = −Fay indicates the thrust force and Faz is the lateral force. θ p (ti ) represents the instantaneous pitching angle at the leading edge at time instant ti . Δx denotes the local span length in the integral and l(x) is the local chord length. L Φ and aΦ represent the chord length at the side and at the center, respectively

mode shape on the added mass effect is considered. As illustrated in Fig. 15.20a, the thrust force due to the added mass effect T = −Fay is calculated by projecting the normal component of the added mass force Fan on the inversed freestream direction. Thus, the thrust coefficient due to the added mass effect C T a can be written as Fay Fan (nc · n y ) =− 1 f 2 , f 2 ρ U∞ S ρ U∞ S 2 2

CT a = − 1

(15.11)

The thrust coefficient due to the added mass effect C T a reflects a connection with the acceleration and geometry of the flexible wing. The residual thrust coefficient C T r is calculated by subtracting C T a from the total thrust coefficient C T , which is defined as C T r = C T − C T a . The residual thrust coefficient C T r does not depend on the acceleration but depends on the variation of the spatial vorticity and the fluid velocity in the fluid domain [597].

828

15 Trailing Edge Effect

With the aid of the analytical formulation of the added mass force, we evaluate the mean thrust coefficients contributed by the added mass effect and the residual term for pitching wings with varying flexibility and trailing edge angles, as shown in Fig. 15.21. It can be seen that the mean thrust coefficient due to the added mass effect shows overall positive values in the studied parameter space. As illustrated in Fig. 15.20b, the geometry and acceleration distributions of the upper half part of the concave wing in gray color are the same as those of the lower half part of the convex wing in cyan color. Although the concave and convex wings in the current study show different trailing edge shapes, the integrated added mass thrust forces of the concave and convex wings using Eqs. (15.10) and (15.11) are similar. The acceleration near the trailing edge of the rectangular wing is smaller than the nonflat wing due to the shorter local chord length, resulting in the overall lower thrust due to the added mass for the rectangular wing. Regardless of the trailing edge shape, the generated thrust due to the added mass force achieves peak values near the frequency synchronization and decreases when the wing becomes more rigid. We observe from Fig. 15.6 that the wing exhibits significantly increased amplitude and acceleration under the frequency synchronization condition. Thus, the added mass effect becomes stronger when frequency synchronization is established. It is worth noting that the variation of the trailing edge shape is limited to the three selected cases in the current study. The connection between the trailing edge shape and the added mass effect should be explored in detail over a wider range of trailing edge shapes in future studies. The mean residual thrust coefficient related to the fluid velocity and the spatial vorticity distribution as a function of trailing edge shape and flexibility is shown in Fig. 15.21b. The non-added mass effect has negative contributions to the mean residual thrust in the studied parameter space. We observe that the mean residual thrust coefficient of the concave wing is smaller than that of the convex wing within the studied bending stiffness range. As shown in Fig. 15.7, the distributions of the fluid velocity and the spatial vorticity are strongly affected by the trailing edge shape. Thus, the mean residual thrust significantly depends on the trailing edge shape. Although the mean added mass thrust between these two wings is similar, the difference in the mean residual thrust leads to the overall large mean total thrust of the convex wing than the concave wing. The mean residual thrust coefficient also shows peak values under the frequency synchronization condition. The reason can be attributed to the strong variation of the flow features near frequency synchronization. To characterize the added mass effect, Fig. 15.22 presents the comparison of the time-varying added mass thrust coefficient and the residual term as a function of bending stiffness for the convex wing. It can be seen from Fig. 15.22a that the frequency synchronization can significantly amplify the added mass thrust due to the induced large acceleration. The contribution of the added mass to the thrust generation is suppressed when the wing becomes more rigid or softer under the desynchronization condition. The distributions of the fluid velocity and the spatial vorticity are significantly affected by interacting with the large-amplitude flapping wing near frequency synchronization. Thus, the residual thrust coefficient shows drastic changes under the frequency synchronization condition.

15.3 Flapping Foils With Varying Trailing Edge 6

i

ii

2

iii

i

ii

iii

0

CT r

Φ = 45◦ Φ = 90◦ Φ = 135◦

4

CT a

829

2

-2

Φ = 45◦ Φ = 90◦ Φ = 135◦

-4 0 1

2

10

10

10

3

10

KB

4

10

-6

5

1

10

10

(a)

2

10

3

10

KB

4

10

5

(b)

Fig. 15.21 a Mean thrust coefficient due to added mass C T a and b mean residual thrust coefficient C T r as a function of bending stiffness K B for pitching wings with varying trailing edge angles Φ = 45◦ , 90◦ and 135◦ at Re = 1000 and St = 0.3 10

KB KB KB KB

= 19.73 = 51.8 = 197.3 = 73998

CT r

5

CT a

KB KB KB KB

10

0

= 19.73 = 51.8 = 197.3 = 73998

0

-10 0

0.2

0.4

t/T

0.6

0.8

1

0

0.2

(a)

0.4

t/T

0.6

0.8

1

(b)

Fig. 15.22 Comparison of a thrust coefficient due to added mass and b residual thrust coefficient of pitching wing with trailing edge angle of Φ = 135◦ within one completed pitching cycle. t/T = 0 corresponds to the pitching upward from the neutral position 2.5

1

Φ = 45◦ Φ = 90◦ Φ = 135◦

2

0

CT r

CT a

1.5 1

-1

0.5

Φ = 45◦ Φ = 90◦ Φ = 135◦

-2 0 -0.5

0

0.2

0.4

(a)

t/T

0.6

0.8

1

-3

0

0.2

0.4

t/T

0.6

0.8

1

(b)

Fig. 15.23 Comparison of a thrust coefficient due to added mass and b residual thrust coefficient of pitching wing with K B = 98.66 within one completed pitching cycle. t/T = 0 corresponds to the pitching upward from the neutral position

830

15 Trailing Edge Effect

Figure 15.23 presents the comparison of the time-varying thrust coefficient due to the added mass and residual terms as a function of trailing edge shape for the flexible wing with K B = 98.66. The added mass thrust coefficient of the convex wing has a slightly larger amplitude than that of the concave wing. It can be seen from Fig. 15.6 that the convex wing exhibits a larger pitching amplitude at the trailing edge. Consequently, the difference in the added mass thrust between the convex and concave wings is caused by the larger acceleration of the convex wing. The added mass thrust of the rectangular wing is the smallest due to the lower acceleration and the short local chord lengths. The time-varying residual thrust coefficient of the rectangular wing presents the largest values among the three wings near t/T = 0.15, while the concave wing shows the smallest values among all wings at t/T = 0.4.

15.4 Summary In this chapter, we systematically explored the effect of flexibility and trailing edge shape on the propulsive performance of pitching flexible plates with a fixed actuated frequency. Based on the variation of the structural motions and the propulsive performance, three distinctive flapping motion regimes were classified as a function of flexibility, namely (1) low bending stiffness K Blow , (2) moderate bending stiffness high K Bmoderate , and (3) high bending stiffness K B . To examine the role of flexibility, ∗ the frequency ratio f between the natural frequency of the flexible plate immersed in the unsteady flow and the pitching frequency was calculated. By examining the flapping dynamics and the flow features, we found that the maximum mean thrust was achieved about f ∗ ≈ 1 corresponding to frequency synchronization. The large passive deformation governed by the flexibility effect can redistribute the pressure gradient to enhance the thrust generation. The optimal propulsive efficiency was observed for flexible plates with moderate passive deformation around f ∗ =1.54. Since maintaining the large passive deformation under the frequency synchronization condition required more input power, the efficiency was reduced compared to the system at a higher K B value. Based on the configurations undertaken in this work, the high convex shape can help improve the mean thrust within K Bmoderate and K B regimes, and achieve the optimal efficiency at low and high K B values. The rectangular shape has shown the largest mean thrust at low K B values, and was the most efficient propulsive system near frequency synchronization. The concave shape presented the poorest propulsive performance within the studied K B range. We employed a momentum-based thrust evaluation method to quantitatively examine the contributions of the evolution of the vortical structures to the produced thrust forces. Typically, the instantaneous maximum thrust was achieved when the vortex detached from the trailing edge. From the perspective of the unsteady momentum transfer, the reason was attributed to the large rate of change of the fluid momentum by the accelerating plate and the momentum convection process. The moderate flexibility near the frequency synchronization can greatly accelerate the surrounding

15.4 Summary

831

fluid, thereby imparting more momentum to the fluid to enhance the thrust generation. The convex shape can generate vortical structures with higher velocities due to the longer local chord at the midspan location. Thus, the wake containing more momentum was convected downstream to improve the produced thrust. To shed light on the drag-thrust transition mechanism, we examined the variation of the decomposed thrust terms as functions of flexibility and trailing edge shape. By adjusting the flexibility value to make f ∗ close to 1, the mean momentum convection was enhanced to promote the transition from drag to thrust. The increase of the trailing edge angle can further help the momentum convection process to broaden the range of flexibility that can generate positive mean thrust. Trough the investigation of the generated thrust force due to the added mass effect, we found that the moderate flexibility corresponding to frequency synchronization can enhance the added mass thrust. The non-flat trailing edge shape has more contributions to the added mass thrust, compared to the rectangular shape with the same area. By examining the evolution of the temporal and spatial vortical structures and the time-dependent thrust, we found that the variation of the propulsive performance cannot be simply determined from the wake topologies when the flow features became complex and cannot be regarded as the typical reverse von Kármán vortex street. Some often cited thrust-generating explanations based on the vortex spacing and the typical wake patterns may not be suitable for the propulsion system with complex wake structures and varying Reynolds numbers. In this study, we employed a non-intrusive velocity/pressure momentum-based thrust evaluation approach and an analytical added mass model to reveal the thrust-generating mechanism for the coupled flapping system with complex wake structures. The unsteady momentumbased approach allows establishing a direct correlation between the flexible plate deformation, the temporal and spatial evolution of the vortical structures and the time-dependent fluid loads, while the added mass model estimates the reactive force due to the acceleration of the plate. The proposed nonlinear FSI study on the unsteady momentum transfer and the added mass effect provides a comprehensive understanding of the thrust-generating mechanism for efficient bio-inspired propulsion systems. This fundamental mechanism offered an effective way to design an optimal propulsion system with flexible wings. Some passive or active control methods can be used to transfer more momentum to the fluid and adjust the accelerations to enhance the thrust generation and the propulsive performance. Acknowledgements Some parts of this Chapter have been taken from the Ph.D. thesis of Guojun Li carried out at the National University of Singapore and supported by the Ministry of Education, Singapore.

Chapter 16

Isolated Inverted Flapping Foils

In this chapter, we present a 2D numerical investigation of the flapping phenomenon observed in a flexible cantilevered foil clamped at the trailing edge. Unlike the conventional flexible foil flapping where its leading edge is clamped, the inverted elastic foil is fixed at the trailing edge and the leading edge is allowed to oscillate freely. We first examine the fundamental mechanism behind the flapping instability of a flexible foil clamped at the trailing edge and summarize the stability regimes observed. We perform a series of parametric simulations to identify the flapping regimes as a function of nondimensional bending rigidity, mass ratio and Reynolds number. We then present the wake topology and net energy transfer as a function of nondimensional bending rigidity. In the second part, we present some 3D simulation results for an inverted flexible foil and ratify the physical mechanism behind the onset of instability demonstrated through 2D simulations. We then show the time evolution of the 3D leading edge vortex and 3D vortex modes for the large amplitude flapping.

16.1 Introduction 16.1.1 Inverted Flapping Foils An inverted foil is a flexible foil clamped at the trailing edge and the leading edge free to oscillate [596, 623]. This configuration has been of interest for its ability to harness fluid kinetic energy more efficiently than its conventional foil counterpart [623, 655]. Guo et al. [593] employed a potential flow in Fourier space to report that the inverted foil configuration is unstable for all nonzero flow velocities. However, this finding clearly contradicts the real-life observations where an inverted flexible foil can remain stable even for nonzero fluid velocity. Buchak et al. [541] performed © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Jaiman et al., Mechanics of Flow-Induced Vibration, https://doi.org/10.1007/978-981-19-8578-2_16

833

834

16 Isolated Inverted Flapping Foils

an interesting experimental study by placing a book in a wind-tunnel and analyzed the clapping interactions among the stack of papers. This study closely represents the dynamics of an inverted foil flapping. A small difference between this experiment and inverted foil is that there is no flow field below the paper before they lift up from the stack. The authors reported that there exists a critical velocity above a page that lifts up from the stack and remain held in the deformed state. More recently, [623] investigated the dynamics of an inverted foil in both wind and water tunnels. The authors reported the existence of three dynamic regimes consisting of two quasi-steady and a limit-cycle flapping mode. The authors observed that an inverted foil is more prone to flapping instability compared to a conventional foil, i.e. the critical velocity for an inverted foil is lower. Very recently, [694] investigated the effect of aspect ratio on the flapping modes of inverted foil in a wind tunnel. The authors reported that with the decrease in the foil aspect ratio, the limit-cycle flapping disappears completely and the critical flow above which the foil loses its stability. The authors also reported that the Strouhal number (St) is independent of Re as no change in St is observed even when Re increases six folds. The experiment of [623] has motivated [596, 692, 733] investigate the flapping dynamics of an inverted foil numerically. All the numerical simulations have confirmed the three coupled modes observed by Kim et al. [623] experimentally. In addition to the three flapping modes, the authors have also reported a steady deformed state [596], biased flapping [596, 692] and flipped flapping [596, 733]. [710] performed fluid-structure-electric coupled numerical simulations to assess the suitability of the inverted foil flapping for the purpose of energy harvesting. The authors reported that the inverted foil configuration is more suitable for the development of energy harvesters because it is insensitive to the initial condition and is relatively more efficient than its conventional foil counterpart. Section 16.3 of this chapter presents a two-dimensional analysis of the selfexcited, self-sustained nonlinear flapping dynamics of an inverted foil in a uniform axial flow. Unlike the conventional flexible foil presented in Chap. 13 where the leading edge of the foil is clamped and the trailing edge is free to oscillate, in an inverted foil, it is the trailed edge that is clamped, and the leading edge is allowed to oscillate freely. We gain our motivation for this study from the ability of the selfsustained flapping to extract energy from the surrounding fluid flow and generate electric power [505, 738] from the vast and untapped source of energy available in the form of ocean currents and tidal flows. A large number of energy harvesting models [505, 513, 655, 738] have been proposed based on the idea of converting the fluid kinetic energy into strain energy through the flapping and bending process of a conventional foil. This strain energy in turn, can be converted to electric potential using capacitive, conductive or piezoelectric methods. In Chap. 13, we have shown that the conventional flexible foil configuration exhibits very low energy harvesting efficiency for m ∗ ∈ O(10−1 ). This leads to the requirement of developing new energy harvesting devices with an alternative configuration to extract maximum possible kinetic energy from the fluid flow even for m ∗  1, which typically represents the case of a flexible foil flapping in flowing water. The objective of this chapter is to investigate the underlying physical mech-

16.1 Introduction

835

anism of flapping instability in an inverted flexible foil. The inverted foil exhibits remarkably different flapping dynamics with respect to conventional flapping. Guo et al. [593] employed a linear stability theory neglecting the viscous damping effects and reported that an inverted foil is unstable for flow velocities. This observation clearly contradicts the physical observations where an inverted flexible foil can exhibit stability for nonzero fluid velocity. [541] performed an interesting experimental study by placing a book with its spine at downstream in a wind tunnel and analyzed the clapping interaction among a stack of papers. The experimental setup typically represents a stack of inverted foils with the fluid flow on one side. More recently, [623] performed wind and water tunnel experiments to show that an inverted foil is more prone to flapping instability compared to a conventional foil, i.e., the critical velocity for an inverted foil was lower than that of a conventional foil. The authors reported the existence of three flapping regimes: Two quasi-steady modes and a limit-cycle flapping mode. However, the physical phenomenon underlying this observation requires some attention through nonlinear viscous simulations of fluid-structure interactions.

16.1.2 Organization In Sect. 16.3, to realize the evolution flapping instability and nonlinear post-critical flapping dynamics of an inverted foil, we perform a series of coupled numerical simulations as a function of K B and m ∗ for a fixed Re = 1000. We then present the stability phase diagrams of K B versus Re and m ∗ through DNS results and simplified analytical solutions. With the aid of a simplified analytical model, we show that the foil loses its stability through static divergence instability. We demonstrate that an inverted foil is more vulnerable to static divergence than a conventional foil. As a function of decreasing K B , we observe that the post-critical flapping of an inverted foil exhibits three distinct flapping regimes: inverted limit-cycle, deformed-flapping, and flipped-flapping. We characterize the transition to the deformed-flapping regime through a quasistatic equilibrium analysis between the structural restoring and the fluid forces. We present the effects of m ∗ on the post-critical flapping dynamics for the fixed Re = 1000. We visualize the evolution of vortex patterns as a function of K B for the flapping regimes. We finally analyze the ability of inverted foil to convert the available fluid kinetic energy to the structural strain energy during the flapping and bending process. In Sect. 16.4, we consider the inverted foil by just reversing the orientation of the conventional foil whereby oncoming flow impinges on its free edge instead of the clamped end in the conventional foil. We carry out 3D simulations as a function of K B for the inverted foil. The flapping parameters such as Re and m ∗ correspond to the experimental study discussed in Sect. 16.3. With the aid of 3D simulations, we identify the flapping regimes for K B ≥ 0.2, and ratify the evolution of flapping instability demonstrated using 2D simulations in Sect. 16.3. We then examine the evolution of the 3D leading edge vortex (LEV) and the 3D vortex modes for large amplitude inverted LCO regimes. We study the coupled interaction of flapping motion

836

16 Isolated Inverted Flapping Foils

and the separated wake flow with alternate vortex shedding. Analogous to the vortexinduced vibration of a bluff body, we observe the periodic vortex shedding when the inverted foil goes into the deformed equilibrium flapping mode. Due to the highly deformed shape, the periodically flapping foil sees the oncoming flow like a smooth circular cylinder, whereby there is a continuous shedding of vortices from the edges of the foil. These vortices behind the flapping foil will periodically exert forces on the flexible foil.

16.2 Numerical Methodology To simulate the nonlinear fluid-structure interactions of thin, flexible flag-like structures at high Reynolds number with large deformation, the stability of the coupled formulation for low mass ratio and its ability to capture the boundary layer effect are two crucial factors. In the present coupled formulation, we adopt a quasi-monolithic scheme for the stable fluid-structure coupling whereas the fluid system is solved on a deforming mesh that adapts to the Lagrangian flexible body in a body-fitted manner via arbitrary Lagrangian-Eulerian (ALE) description. The body-conforming treatment of fluid-structure interface provides an accurate modeling of the boundary layer and the vorticity generation over deformable surfaces. Before proceeding to the quasi-monolithic variational formulation, we first present the governing fluidstructure equations employed for the numerical methodology.

16.2.1 Fluid-Structure Equations Let Ω f (t) ⊂ Rd be a fluid domain at time t, where d is the space dimension. The incompressible flow is governed by the Navier-Stokes equations and the equations in an arbitrary Lagrangian-Eulerian (ALE) reference frame are expressed as ρf

  ∂ uf + ρ f uf − w · ∇uf = ∇ · σ f + f f on Ω f (t), ∂t ∇ · uf = 0 on Ω f (t),

(16.1)

(16.2)

where uf = uf (x, t) ≡ (u, v, w) and w = w(x, t) represent the fluid and mesh velocities respectively at a spatial point x ∈ Ω f (t), f f denotes the body force applied on the fluid and σ f is the Cauchy stress tensor for the Newtonian fluid, written as σ f = − pI + T , T = 2μf  f (uf ),  f (uf ) =

1  f  f T  ∇u + ∇u , 2

(16.3)

16.2 Numerical Methodology

837

where p is fluid pressure, I denotes second-order identity tensor and T represents the fluid viscous stress tensor. Let Ω s ⊂ Rd be the reference domain for a flexible elastic structure. The dynamics of the inverted foil is then governed by the structural momentum equation ∂ us in Ω s , (16.4) = ∇ · σs + f s ρs ∂t us = us (z, t) is the structural velocity at a material point z ∈ Ω s deforms to position ϕ s (z, t) at time t, f s denotes the external forces acting on the solid and σ s is the first Piola-Kirchhoff stress tensor. While the structural displacement vector is given by ηs (z, t) = ϕ s (z, t) − z, the structural velocity us is related to the deformation ϕ s (z, t) as: us (z, t) = ∂t ϕ s (z, t), where ∂t denotes the partial temporal derivative. In this study, the structural stresses are modeled using the St. Venant-Kirchhoff model and the constitutive equation is given by σ s (ϕ s ) = 2μs F E + λs [tr (E)] F,

(16.5)

where μs and λs are the Lamé coefficients, tr(·) denotes the tensor trace operator and F is the deformation gradient tensor which is given by F = ∇ηs = (I + ∇ηs ) ,

(16.6)

and E represents the Green-Lagrangian strain tensor defined as E=

 1 T F F−I , 2

(16.7)

where the term F T F denotes the right Cauchy-Green deformation tensor [740]. Eventually, the fluid and structural solutions are coupled through the velocity and traction continuity along the fluid-solid interface, given by 

 σ f (x, t) · nf da(x) + ϕ(γ ,t)

σ s (z, t) · ns da(z) = 0 ∀γ ⊂ Γ,

(16.8)

γ

uf (ηs (z, t), t) = us (z, t) ∀z ∈ Γ,

(16.9)

where nf and ns are, respectively, the outward normals to the deformed fluid and the undeformed solid interface boundaries, Γ represents the interface between the fluid and the inverted foil at t = 0, γ is an edge on Γ , da denotes differential surface area and ϕ s is the function that maps each Lagrangian point z ∈ Ω s to its deformed position at time t.

838

16 Isolated Inverted Flapping Foils

16.2.2 Variational Quasi-Monolithic Formulation We next briefly present the variational fluid-structure formulation based on the Navier-Stokes and the nonlinear elasticity equations. We extend the 2D variational body-conforming quasi-monolithic technique presented by Liu et al. [645] for flow-structure interactions to large-scale 3D turbulent flow-structure problems. The employed quasi-monolithic coupling technique is numerically stable even for very low structure-to-fluid mass ratio and has been used for simulating the flapping dynamics of thin flexible foil in [61, 596]. An energy-based mathematical proof for the numerical stability of the coupling formation for any mass ratio is presented in [645]. One of the main features of this quasi-monolithic formulation is that the kinematic and dynamic continuity conditions which are to be satisfied along the common interface are absorbed into the formulation and are satisfied implicitly. Another attractive feature of the quasi-monolithic formulation is that the mesh motion is decoupled from the rest of the FSI solver variables by updating the structural positions at the start of each iteration. Decoupling of the mesh motion decreases the size of the linear system and enables us to linearize the nonlinear Navier-Stokes, i.e., the coupled system of equations is now solved only once per time step without losing stability. This makes the quasi-monolithic formulation computationally efficient in contrast to traditional monolithic formulations. To derive the weak (variational) form of the Navier-Stokes Eqs. (16.1) and (16.2), we consider a trial function space S f that satisfies the Dirichlet conditions and a test function space V f that is null along the Dirichlet boundaries. The variational form of the Navier-Stokes Eqs. (16.1) and (16.2) in the ALE reference frame can be stated as Find {uf , p} ∈ S f such that ∀{φ f , φ p } ∈ V f : 

    ρ f ∂t uf + uf − w · ∇uf · φ f dΩ +

Ω f (t)



 f f · φ f dΩ +

Ω f (t)

f Γh(t)

 σ fb · φ f dΓ + Γ (t)

 σ f : ∇φ f dΩ =

Ω f (t)

 f  σ (x, t) · nf · φ f (x)d,

(16.10)

 ∇ · uf φ p dΩ = 0,

(16.11)

Ω f (t)

where ∂t denotes the partial time derivative operator ∂(·)/∂t, Γhf denotes the Neumann boundary along which σ f · nf = σ fb . Similar to the flow equations, to construct the weak variational form of the structural dynamics Eq. (16.4) we consider the trial and test functional spaces S s that satisfy the Dirichlet conditions and V s that is null along the Dirichlet boundaries respectively. The weak form for the nonlinear elasticity of deformable foil is defined as follows. Find us ∈ S s such that ∀φ s ∈ V s :

16.2 Numerical Methodology

839



 ρ s ∂t us · φ s dΩ + Ωs



 f s · φ s dΩ +

Ωs

σ s : ∇φ s dΩ = Ωs



σ sb · φ s dΓ +

(σ s (z, t) · ns ) · φ s (z)d,

(16.12)

Γ

Γhs

where Γhs denotes the Neumann boundary along which σ s · ns = σ sb . One of the key features of the quasi-monolithic formulation is that the kinematic and dynamic interface continuity conditions in Eqs. (16.9) and (16.8) are absorbed at the variational level by enforcing the condition φ f = φ s along the interface Γ and are satisfied implicitly. Such a condition can be realized by considering a conforming interface mesh. The weak form of the traction continuity condition along the fluid-structure interface Eq. (16.8) is given as 

 f  σ (x, t) · nf · φ f (x)d +

 (σ s (z, t) · ns ) · φ s (z)d = 0.

(16.13)

Γ

Γ (t)

Let the fluid domain Ω f be discretized into nelf number of three-dimensional el el Ω e and ∅ = ∩ne=1 Ω e . A variational Lagrange finite elements such that Ω f = ∪ne=1 multi-scale (VMS) based turbulence modeling for LES has been considered in this work, wherein the trial function space S f is decomposed into coarse-scale ¯ ∈ S¯f and fine-scale space {(u )f , p } ∈ (S )f [524, 609]. Likewise, space {u¯ f , p} we can also decompose the test function space V f into the coarse and fine scale f p spaces {φ¯ , φ¯ } ∈ V¯ f and {(φ )f , (φ )p } ∈ (V )f respectively. Using the weak forms of Eqs. (16.11) and (16.10) and applying the VMS decomposition, then combining them with Eqs. (16.12)–(16.13), we construct the weak form of coupled fluidstructure system as follows: 

  f   ρ f ∂t uf + uf − w · ∇uf · φ dΩ +

Ω f (t)



f

σ f : ∇φ dΩ −

Ω f (t)

+

n el  e=1

+



n el   e=1

    f τm ρ f uf − w · ∇φ + ∇q · R m (uf , p)dΩ ¯

Ωef (t) f

∇ · φ τc ∇ · uf dΩ

Ωef (t)

 + Ωs

⎫ ⎪ ⎬ ⎪ ⎭

C

 ρ s ∂t us · φ s dΩ +

σ s : ∇φ s dΩ Ωs

⎫ ⎬ ⎭



D

⎫ ⎪ ⎬ ⎪ ⎭

Ω f (t)

B

∇ · uf qdΩ

⎫ ⎪ ⎬ ⎪ ⎭

A

840

16 Isolated Inverted Flapping Foils

+

nel  

f

τm uf · ∇φ R m (uf , p)dΩ ¯

e=1 f Ωe (t)

−

nel 



⎫ ⎪ ⎬ ⎪ ⎭

E

f

∇φ : (τm R m (uf , p) ¯ ⊗ τm R m (uf , p))dΩ ¯ =

e=1 f Ωe (t)



f



f f · φ dΩ + Ω f (t)

f



σhf · φ dΩ + Γhf

⎫ ⎪ ⎬ ⎪ ⎭

F

 f s · φ s dΩ +

Ωs

Γhs

⎫ ⎪ ⎬

σhf · φ s dΩ, G ⎪ ⎭

(16.14)

In the above equation, the terms A and G represent the Galerkin weak-form of the coarse scale component for the Navier-Stokes Eqs. (16.11) and (16.10). The term B represents the element level Galerkin least-squared (GLS) stabilization consists of both the convection and pressure stabilizations terms to damp the spurious oscillations and to circumvent the inf-sup condition, respectively. The term C is the least square stabilization on the incompressibility constraint, which provides additional stability for large Re problems. The term D represents the Galerkin weak-from of the nonlinear structural dynamics. The terms E and F correspond to the cross-stress and small-scale Reynolds stress terms, respectively. The term F accounts for the turbulent kinetic energy dissipation through the formation of small scales [610]. Unlike the traditional GLS stabilization technique, which accounts for only one cross-stress term ¯ (term B), the VMS formulation accounts for both the cross-stress terms. R m (u¯ f , p) is the residual of the momentum equation at the element level, and the stabilizing parameters τm and τc in the terms (B − E) are the least squares metrics [581, 608]. As mentioned earlier, the quasi-monolithic formulation decouples the fluid mesh motion and explicitly determines the mesh velocity w, which is carried out by determining the interface between the fluid and the structure explicitly for nth time step using the second order Adam-Bashforth method ϕ s,n = ϕ s,n−1 +

3Δt s,n−1 Δt s,n−2 u u − , 2 2

(16.15)

where ϕ s,n is the interface position between the fluid and the flexible body for any time t n . The fluid mesh nodes on the domain Ω f (t) can be updated for the interface locations using a pseudo-elastic material model given by ∇ · σ m = 0,

(16.16)

where σ m is the stress experienced by the fluid mesh due to the strain induced by the interface deformation. Assuming that the fluid mesh behaves as a linearly elastic material, its experienced stress can be written as

16.3 Two-Dimensional Flapping Dynamics

841

α

Y LE

Flow (U0, ρ , μ ) f

f

TE

h

X

Elastic foil (E, ρs, ν)

L

Fig. 16.1 Schematic of an inverted flexible foil of length L and thickness h in uniform axial flow U0 with leading edge (L E) free to oscillate and trailing edge (T E) clamped

σ m = (1 + τm )

  T    + ∇ · ηf I , ∇ηf + ∇ηf

(16.17)

where τm is a mesh stiffness variable chosen as a function of the element size to limit the distortion of the small elements located in the immediate vicinity of the fluid-structure interface. The mesh stiffness variable τm has been defined as τm = maxi |Ti |−mini |Ti | , where Tj represents jth element on the mesh T . |Tj |

16.3 Two-Dimensional Flapping Dynamics 16.3.1 Problem Statement We consider a two-dimensional thin flexible foil Ω s of length L and thickness h as shown in Fig. 16.1 interacting with an incompressible uniform axial flow Ω f (t). The trailing edge (T E) of the flexible foil in the direction of the flow is clamped and the leading edge (L E) is free to perform flapping motion. U0 is the magnitude of the uniform axial flow exciting the flapping instability. While ρ f and ρ s are the densities of the fluid and structure, μf denotes the fluid dynamic viscosity, α represents the leading edge angle of attack, E is Young’s modulus and ν is Poisson’s ratio of the foil. We consider two different computational domains to investigate the onset of flapping instability and the nonlinear post-critical flapping dynamics. The first computational domain and setup are similar to the one shown in Fig. 12.1 used for studying the flapping dynamics of the conventional flexible foil in Chap. 3, except that the trailing edge of the foil is clamped and the leading edge is left free to oscillate. This computational domain will be used for examining the initial onset and evolution of the flapping instability.

842

16 Isolated Inverted Flapping Foils

Table 16.1 Inverted foil mesh convergence study: comparison of leading edge tip-displacement and lift statics over tU0 /L ∈ [20, 30] for the meshes M1, M2 and M3 at m ∗ = 0.1, Re = 1000 and K B = 0.01 Mesh M1 M2 M3 Nodes Elements δyrms δ¯y /L δxrms δ¯x /L Clrms C¯l

23,505 11,643 2.086 × 10−1 (7.09%) 2.771 × 10−1 (8.58%) 6.173 × 10−1 (0.98%) 11.93 × 10−1 (0.27%) 2.472 × 10−1 (1.19%) −3.23 × 10−1 (1.40%)

46,709 23,197 1.950 × 10−1 (0.11%) 2.563 × 10−1 (0.43%) 6.118 × 10−1 (0.08%) 11.98 × 10−1 (0.19%) 2.469 × 10−1 (1.04%) −3.32 × 10−1 (1.39%)

92,646 46,104 1.948 × 10−1 2.552 × 10−1 6.113 × 10−1 11.96 × 10−1 2.660 × 10−1 −3.28 × 10−1

Figure 16.2a shows a typical schematic of the second computational domain, which will be used to simulate the post-critical flapping dynamics of the inverted foil in Sects. 16.3.3 and 16.3.4. In this figure, Hu is the distance between the foil centerline and the computational domain’s top side. Similarly, Hl is the distance from the foil centerline to the bottom side of the computational domain. Unlike the first computational domain, which is symmetric about the foil centerline, this domain is asymmetrical about the foil centerline i.e. Hu = Hl . The size of the computational domain is [24L × 10L] with Hu = 5.25L and Hl = 4.75L. Since an inverted flexible foil is neutrally stable in its undeformed state, it can bend towards either side of the centerline. In order to have a numerically repeatable solution, we have considered a non-symmetrical computational domain. To study the influence of domain size for K B = 0.2, Re = 1000 and m ∗ = 0.1, two different domain widths 10L and 40L has been selected for assessing the flapping response dynamics. The difference in the maximum tip displacement δmax has been found to be less than 1%. The maximum blockage ratio, which is defined as δmax /(Hu + Hl ), has been observed to be approximately 8.5% and 2.125% for two domains, respectively. We next perform a detailed mesh convergence study for the asymmetrical computational domain over the finite-element meshes M1, M2 and M3 consisting of 11,643, 23,197 and 46,104 P2 /P1 triangular elements. Typical undeformed highorder finite element fluid mesh M3 and the corresponding central block consisting of boundary layer mesh can be seen in Fig. 16.2b. Figure 16.2c shows a close-up view of the deformed mesh around the flexible foil for three-time instants. For the purpose of mesh convergence study, we consider a flexible foil with m ∗ = 0.1 and K B = 10−4 interacting with the fluid flow at U0 = 1 corresponding to Re = 1000. A constant time step size Δt = 0.001 has been employed. Table 16.1 summarizes the results of mesh independence test for the inverted foil. The values within the brackets indicate the percentage difference in the numerical solutions with respect to the mesh M3. Therms value of the cross-stream tip-displacement is calcu 2  lated using δ rms /L = 1 t=15→25 δ ∗ − δ¯∗ , where δ ∗ is the non-dimensional n

16.3 Two-Dimensional Flapping Dynamics

843

Γftop, uf = (U0, 0)

Γftop uf = (U0 , 0)

Hu Ωsbottom

Γfout σf n = 0

Ωf (t)

L Hl

h = 0.01L Γfbottom, uf = (U0 , 0) Lu = 4L

Ld = 20L

(a) 5

0.3

4 3

0.2

2

0.1

1 0

0 −1

−0.1

−2

−0.2

−3

−0.3

−4 20

15

10

5

0

−0.2

0.2

0

0.8

0.6

0.4

1

1.2

1.6

1.8

(b) 0.8

0.8

0.8

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0

0

0

−0.2 0.6

−0.2

−0.2 0.8

1

1.2

1.4

1.6

1.8

2

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.6

0.8

1

1.2

1.4

2

(c) Fig. 16.2 Inverted foil system: a representative schematic of inverted foil computational setup cantilevered at trailing edge in a uniform axial flow, b typical representation of high-order isoparametric finite element mesh M3 for the complete fluid domain (left) and close-up view of the central block and boundary layer mesh at interface (right), c close-up view of the deformed meshes around the foil for tU0 /L = 7.3, 12 and 17

tip-displacement defined as δ/L and δ¯∗ denotes the non-dimensional mean tip displacement. A similar definition has been used to calculate Clrms . Therefore, the mesh M2 will be employed to simulate the post-critical flapping dynamics of the inverted flexible foil.

844

16 Isolated Inverted Flapping Foils 0.1 K =0.5600 B

K =0.5500 B

0.08

KB=0.5475 K =0.5450 B

KB=0.5425

δy /L

0.06

0.04

0.02

0

−0.02 0

50

100

150

200

250

300

tU0 /L Fig. 16.3 Time history of the leading edge cross-stream tip-displacements for small changes in the non-dimensional bending rigidity K B at fixed m ∗ = 0.1 and Re = 1000

16.3.2 Development of Flapping Instability To present the underlying physical mechanism of the flapping instability for an inverted flexible foil, we consider the symmetrical computational domain with Hu = Hl = 5L. We first examine the evolution of the flapping instability as a function of K B for a fixed m ∗ = 0.1 and Re = 1000 by gradually decreasing K B from 0.8 to 0.2. Figure 16.3 shows the time history of the cross-stream leading edge displacements for K B = 0.56, 0.55, 0.5475, 0.545 and 0.5425. From this figure, three distinct stability regimes can be identified: (i) fixed-point stable, (ii) deformed steady state and (iii) unsteady flapping regime. In the first regime, for K B ≥ 0.56 the flexible foil remains steady in its initial configuration with negligibly small leading edge cross-stream displacements. In the second regime, for 0.56 < K B < 0.5425, the flexible foil deforms slowly to achieve a steady state. This steady state deformation increases with a decrease in K B . In the unsteady flapping regime, for K B ≤ 0.5425, the flexible foil loses its stability to perform regular sinusoidal oscillations with a constant frequency and amplitude. The transition from the fixed-point stable to the deformed steady-state regime can possibly be attributed to the static divergence instability [528, 582]. The deformed steady-state regime can be modeled as the static deformation of an inverted foil due to a small normal impulsive force P at its leading edge. Let us consider that the flexible foil undergoes a small leading edge cross-stream displacement of δ y with a leading edge angle of attack α as shown in Fig. 16.1 due to the static deformation. An inverted foil will experience static divergence instability when the aerodynamic

16.3 Two-Dimensional Flapping Dynamics

845 3.1

0.4

3

∂Cl /∂α

0.5

0.3

Cl

m = 2.87 0.2

2.9

2.8

2.7

0.1

2.6

0

0

0.05

0.1

0.15

0.2

200

400

600

α

Re

(a)

(b)

800

1000

Fig. 16.4 a Variation of Cl as function of leading edge angle of attack α at a fixed Re = 400 and m ∗ = 0.1 corresponding to the deformed steady state regime and b effect of Re on ∂Cl /∂α, where curve (—) represents the empirical relation aRe2 + bRe + c

moment acting on it exceeds the elastic restoration moment [582]. The aerodynamic moment can be evaluated by the lift force acting at the aerodynamic center of the inverted foil. The elastic restoration moment of the inverted foil can be idealized as a spring connected at the leading edge. Therefore, the condition for the static divergence will be 1 (16.18) kδ y L < ρ f U02 Cl (α, Re)L xac , 2 where k is the spring constant of an elastically mounted foil, Cl is the lift coefficient expressed as function of Re and α, and xac denotes the distance between the aerodynamic center and the trailing edge. From the DNS results corresponding to the deformed steady-state regime, we construct an empirical formulation for Cl as a function of Re and α. Figure 16.4a shows the relationship between Cl and α obtained from the DNS simulations for a typical Re = 400. From this figure, it can be noticed that Cl is directly proportional to α and can be represented as Cl (Re, α) =

∂Cl (Re)α, ∂α

(16.19)

l where ∂C (Re) denotes the slope of the curve in Fig. 16.4a. Figure 16.4b shows the ∂α l variation of ∂C for a range of Re and the curve (—) represents the best-fit curve ∂α given by the equation ∂Cl (Re) = aRe2 + bRe + c, (16.20) ∂α

where a = −5.524 × 10−7 , b = 1.139 × 10−3 and c = 2.492. Therefore, Eq. (16.18) will be

846

16 Isolated Inverted Flapping Foils

0.9 0.58 0.56

0.7

0.54

0.6

0.52

KB

KB

0.8

0.5

0.5 0.48

0.4

0.46 0.3

0.44

0.2 0.1

0.42 0.4 0

100 200 300 400 500 600 700 800 900 1000

0

100 200 300 400 500 600 700 800 900 1000

Re

Re

(a)

(b)

Fig. 16.5 Stability phase diagram: non-dimensional bending rigidity K B versus Reynolds number Re for the inverted flexible foil at m ∗ = 0.1 (left) and a closeup view of the phase diagram for K B ∈ [0.4, 0.6]. The curves (—) and (- - -) represent the critical non-dimensional bending rigidity for static divergence and flapping instability given by equations Eqs. (16.23) and (16.24), respectively. (O), (∗) and (+) denote the simulation results corresponding to the unstable flapping, steady deformed and fixed-point stable regimes

kδ y L
K B ≥ 0.2; (ii) deformed-flapping for 0.075 < K B ≤ 0.1; and (iii) flipped-flapping for K B ≤ 0.075. The above flapping response regimes have been classified based on the tip-

850

16 Isolated Inverted Flapping Foils

displacement responses and the full body profiles. Similar response regimes have been observed experimentally by [623] for mass-ratios of O(1) and O(10−3 ). In Fig. 16.9, we present the full-body profile results for K B ∈ [10−3 , 0.56]. Figure 16.9a shows the fixed-point stable foil profile in its undeformed state for K B = 0.56. For K B ∈ [0.5425, 0.2], the flexible foil experiences the onset of flapping instability that develops into large-amplitude periodic oscillations and this regime is defined as the inverted limit-cycle oscillations (LCO). Figure 16.9b–d show typical full-body profiles for K B = 0.5425, 0.4 and 0.2 over one oscillation cycle. During the initial stages of the inverted-LCO regime for K B ∈ [0.50, 5425], the flexible foil does not perform flapping about the mean position as shown in Fig. 16.9b. However, for K B ∈ [0.5, 0.2], the flexible foil performs flapping about its mean position. Moreover, from these figures, it can be observed that the inverted-LCO mode is predominantly the first mode and does not exhibit the typical necking phenomenon, which is observed in a conventional foil [605, 645]. By decreasing the K B value, the flexible foil completely deforms to one direction and performs relatively small peak-to-peak amplitude oscillations as shown in Fig. 16.9e. Due to the small peakto-peak amplitudes, this regime has been described as the deformed mode by Kim et al. [623]. Since the peak-to-peak amplitude for this regime is of the same order as the amplitudes of a conventional foil flapping, we describe this regime as the deformed-flapping regime. By further decreasing K B value, due to the low flexibility, some part of the foil curves around the fixed edge to form a semi-circular arc and the remaining part of the foil becomes parallel to the free-stream velocity as shown in Fig. 16.9i. Figure (16.9f, i show the full-body profiles for the inverted flexible foil for a range of K B < 0.1. The full body profile is shown in Fig. 16.9i closely resembles the full body profile of the conventional foil shown in Fig. 16.10a for K B = 3.27 × 10−4 , Re = 1000 and m ∗ = 0.1. Additionally, the necking phenomenon, which is generally observed in the case of conventional foil, flapping has been observed in this regime. The maximum Reynolds numbers with respect to the semi-circular arc diameter Reδ is around 380 for K B = 0.05, which signifies that our current two-dimensional simulations might not be able to capture the three-dimensional flow structures. To further compare the flapping phenomenon in the flipped-flapping regime to the conventional foil, we plot the phase relation between the stream-wise and the crossstream tip-displacements as shown in Fig. 16.11. The phase plot for the flippedflapping regime at K B = 10−3 forms an eight-shaped Fig. 16.11a, which is similar to the phase relationship shown in Fig. 16.10b for the conventional foil with K B = 3.27 × 10−4 , Re = 1000 and m ∗ = 0.1. The phase plot for the inverted-LCO presents a unique parabolic profile, i.e. the leading edge follows the same path during the upward and downward strokes, which can be seen in Fig. 16.11c for K B = 0.2. Moreover, no distinct phase plot has been observed for K B corresponding to the flipped-flapping regime as shown in Fig. 16.11a, b for K B = 0.01 and 0.001, respectively. Figure 16.12 shows the time history of stream-wise (- - -) and cross-stream (— ) displacements along with the cross-stream amplitude-frequency spectrum. In this figure, () represents the time instance for which vorticity contours will be presented

16.3 Two-Dimensional Flapping Dynamics

851

0.15

0.15 i

iii

ii

0.05

0.05

δY /L

0.1

Y /L

0.1

0

0

−0.05

−0.05

−0.1

−0.1

0

0.8

0.6

0.4

0.2

−0.2

1

−0.1

0

X/L

δX /L

(a)

(b)

0.1

Fig. 16.10 a Full body profiles over one full cycle and b phase relation between the stream-wise and cross-stream trailing edge tip displacements for a conventional foil with K B = 3.27 × 10−4 , Re = 1000 and m ∗ = 0.1 1

0.4 0.3

0.5

0.2

0.4

0.1

δY /L

δY /L

δY /L

0.5

0.3

0

0.2

−0.1

0.1

0

−0.5

−0.2

1.4

1.6

1.8

0 1.2

1.4

1.6

1.8

−1 −0.5

0

0.5

δX /L

δX /L

δX /L

(a)

(b)

(c)

1

1.5

Fig. 16.11 Phase plot of leading edge cross-stream and stream-wise displacements for K B = a 0.001, b 0.01 and c 0.2 at Re = 1000 and m ∗ = 0.1

in Sect. 5.6. The leading edge tip-displacements for the inverted-LCO regime exhibit a sudden rise when K B is decreased from 0.545 to 0.4 as shown in Fig. 16.12a, b. Figure 16.12c shows the leading edge tip-displacement history for the deformedflapping regime at K B = 0.1. In this regime, the leading edge remains steady for a significant time period, which is followed by non-periodic oscillations with relatively small peak-to-peak amplitudes ranging between 0.2 and 0.4. Figure 16.12d–g show the tip-displacement response history of the flippedflapping regime. The flapping amplitudes for this regime are significantly smaller than those observed in the inverted-LCO regime and are approximately equal to those observed in the conventional foil. However, the flapping frequencies are smaller than that observed in conventional foil flapping. The transition from the deformedflapping to the flipped-flapping regime is predominantly characterized by two frequency oscillations as shown in Fig. 16.12d. The first low-harmonic corresponds to the flapping frequency and the second frequency may be attributed to an additional higher-harmonic frequency of shear layer roll-up over the semicircular configura-

16 Isolated Inverted Flapping Foils 1

0.02

0.5

0.015

A/L

δ/L

852

0

−0.5

−1

f L/U0 : 0.03906 A/L : 0.008113

0.01

0.005

0

50

100

150

200

250

0

300

0

0.5

1

1.5

tU0 /L

f L/U0

(a)

(b)

2

2.5

2

2.5

2

2.5

2

2.5

1

0.8 0.5

A/L

δ/L

0.6 0

−0.5

−1

f L/U0 : 0.1465 A/L : 0.4025

0.4

0.2

0

10

20

30

40

50

0

60

1.5

1

0.5

0

tU0 /L

f L/U0

(c)

(d)

1.5

0.8

0.6

A/L

δ/L

1

0.5

0

−0.5

0.4 f L/U0 : 0.2441 A/L : 0.1673

0.2

0

10

20

30

40

50

0

60

0

0.5

1

1.5

f L/U0

tU0 /L

(e)

(f)

1.5

0.8 1

A/L

δ/L

0.6 0.5

f L/U0 : 0.4395 0.2 A/L : 0.1313

0

−0.5

0.4

0

5

10

15

20

25

30

0

0

f L/U0 : 0.6836 A/L : 0.09044

0.5

1

1.5

tU0 /L

f L/U0

(g)

(h)

Fig. 16.12 Time history of the cross-stream (—) and stream-wise (- - -) leading edge tipdisplacement (left) and amplitude-frequency spectrum of the cross-stream leading edge tipdisplacement (right) for K B = a 0.5425, b 0.4, c 0.1, d 0.05, e 0.01, f 0.005 and g 0.001 at Re = 1000 and m ∗ = 0.1. () represents the time instance for which vorticity contours will be presented in Sect. 5.6

16.3 Two-Dimensional Flapping Dynamics

853

2

0.8 1.5

A/L

δ/L

0.6 1 0.5

f L/U0 : 0.5371 A/L : 0.1834 f L/U0 : 1.025 A/L : 0.0604

0.2

0 −0.5

0.4

0

5

10

15

20

25

0

30

0

0.5

1

1.5

2

2.5

2

2.5

2

2.5

f L/U0

tU0 /L

(i)

(j)

2

0.8 1.5

A/L

δ/L

0.6 1 0.5

f L/U0 : 0.5859 A/L : 0.1792

0.2

0 −0.5

0.4

0

5

10

15

20

25

0

30

0

0.5

1.5

1

tU0 /L

f L/U0

(l)

(k) 2

0.8 1.5

A/L

δ/L

0.6 1 0.5

f L/U0 : 0.6348 A/L : 0.2167

0.2

0 −0.5

0.4

0

5

10

15

20

25

30

0

0

0.5

1.5

1

tU0 /L

f L/U0

(m)

(n)

Fig. 16.12 (continued)

0 tion. According to Strouhal’s law, it may be estimated as f H = St DUfoil , where Dfoil is an effective diameter of the semicircular body and St is the Strouhal number. The higher harmonics 0.6838 and 1.025 appeared in Fig. 16.12d, e approximately coincide with the Strouhal frequencies corresponding to the vortex shedding of the semicircular body. In addition, we also observe complex vortex-foil interactions and a reversed flow inside the deformed cavity-like region. A detailed investigation of these phenomena is beyond the scope of this study. The wake topology will be further illustrated in Sect. 16.3.7 with the aid of vorticity evolution in a flapping period. The high-harmonic frequency gets further damped as K B is lowered from 0.01 to 0.001 and the flapping response for K B = 0.001 exhibits a single distinct frequency

854

16 Isolated Inverted Flapping Foils 1

δymax/L

0.8

0.6

0.4

0.2

0 0

0.2

0.6

0.4

0.8

1

KB Fig. 16.13 Maximum cross-stream displacement of the inverted foil’s leading edge center over tU0 /L ∈ [20, 30] for m ∗ = 0.1, Re = 1000 and K b ∈ [0.001, 1]

Fig. 16.12h. Figure 16.13 summarizes the effect of K B on the maximum leading edge cross-stream tip-displacement. For large K B values, the flexible foil remains stable. A sharp rise in the maximum tip-displacement can be observed for K B ∈ [0.3, 0.5]. The maximum flapping amplitudes stabilize to an approximate value of 0.85, which is similar to the experimental observation made by [623].

16.3.4 Effect of Mass Ratio To analyze the effects of mass-ratio on the inverted flexible foil, we perform a series of numerical simulations for m ∗ ∈ [0.1, 10]. We first examine the effects of m ∗ on the inverted-LCO regime for a fixed K B = 0.4 and Re = 1000. Figure 16.14a shows the leading edge cross-stream tip-displacement history for m ∗ = 0.5 and 8. From this figure, it can be seen that the flapping frequency is strongly influenced by the increase in m ∗ , while the cross-stream flapping amplitudes are weakly affected. It may be noted that with an increase in m ∗ , the unsteady flapping instability takes a greater time to develop due to the inertial effects. Figure 16.14b, c summarize the effect of m ∗ on the flapping amplitudes and frequencies, respectively. To understand the effects of m ∗ on the flipped-flapping regime, we consider K B = 0.01 and Re = 1000 for m ∗ ∈ [0.1, 2]. Figure 16.15 shows the leading edge crossstream displacement histories for m ∗ = 0.1, 0.5, 1 and 2. From Fig. 16.15a and b, we observe regular periodic oscillations for m ∗ = 0.1 and 0.5, which transform into the non-periodic oscillations by increasing m ∗ to 1. These non-periodic oscillations are characterized by variable amplitudes and frequencies. Figure 16.15c, d show the

16.3 Two-Dimensional Flapping Dynamics

855

1 0.8 0.6 0.4

δy /L

0.2 0

−0.2 −0.4 −0.6 −0.8 −1

0

10

20

30

40

50

60

tU0 /L

0.86

0.26

0.84

0.22

f L/U0

δymax/L

(a)

0.82

0.18

0.8

0.14

0.78

0.1 0

2

4

6

8

10

0

2

6

4

m

m

(b)

(c)

8

10

Fig. 16.14 Effect of mass-ratio m ∗ on: a cross-stream leading edge tip-displacements for m ∗ = 0.5 (—) and 8 (- - -), b maximum cross-stream leading edge tip-displacements and c frequency, at a fixed K B = 0.4 and Re = 10,000

non-periodic oscillations for m ∗ = 1 and 2. A similar observation of the transition from regular periodic to non-periodic oscillations with an increase in m ∗ was made by [551] for conventional foils. Mass-ratio m ∗ can also affect the transition from the inverted-LCO to the deformed-flapping regime. Figure 16.16 shows the leading edge cross-stream displacements for m ∗ = 0.1 and 0.5 for a fixed K B = 0.1. With an increase in m ∗ from 0.1 to 0.5, a delay in the transition from the inverted-LCO to the deformed-flapping, can be seen in Fig. 16.16.

16.3.5 Transition to Deformed Flapping Regime In the deformed-flapping regime, an inverted flexible foil deforms completely to one side and performs a flapping motion about the deformed position. To understand

16 Isolated Inverted Flapping Foils 1

1

0.8

0.8

0.6

0.6

δy /L

δy /L

856

0.4

0.4

0.2

0.2

0

0

0

10

20

30

40

0

10

20

tU0 /L

(a)

40

50

30

40

50

(b)

1

1

0.8

0.8 0.6

0.6

0.4

δy /L

δy /L

30

tU0 /L

0.4

0.2

0.2

0

0

−0.2

0

10

20

30

40

−0.4 0

50

20

10

tU0 /L

tU0 /L

(c)

(d)

Fig. 16.15 Time history of the cross-stream displacement leading edge displacements for m ∗ = a 0.1, b 0.5, c 1 and d 2, at a fixed K B = 0.01 and Re = 1000 1.2

1

1 0.5

0.6

δy /L

δy /L

0.8

0.4 0.2

0

−0.5

0 −0.2 0

5

10

15

20

25

30

35

−1 0

5

10

15

20

tU0 /L

tU0 /L

(a)

(b)

25

30

35

Fig. 16.16 Time history of the leading edge cross-stream displacements for m ∗ = a 0.1 and b 0.5 at a fixed K B = 0.1 and Re = 1000

16.3 Two-Dimensional Flapping Dynamics

857

0.9

s=L

0.8 0.7 0.6

t Y /L

U0

θ n

0.5 0.4 0.3 0.2

Y

0.1

s=0

0.6

X

0.8

1

1.2

1.4

1.6

X/L

(b)

(a)

Fig. 16.17 a A typical schematic of the inverted flexible foil in curvilinear coordinate system with s = 0 and L at the trailing and leading edge, b comparison between the simulated (- - -) foil profiles over one complete flapping cycle and the quasistatic equilibrium foil profile predicted (—) foil profile for K B = 0.1, Re = 1000 and m ∗ = 0.1

the transition from the inverted-LCO regime to the deformed-flapping regime, we perform a simple quasistatic equilibrium analysis. Figure 16.17a shows a typical schematic of an inverted flexible foil interacting with a uniform axial flow U0 . We consider the curvilinear coordinates (s, θ ) with s = 0 and L at the trailing and the leading edge, respectively, and θ represents the local angle of incidence between the fluid flow and unit tangent vector t. A flexible foil in quasistatic equilibrium should satisfy the moment and force balance, which can be written in the dimensionless form as [541, 701] M dθ = , ds KB

Q

=

dM , ds

dQ = F + N, ds

(16.25)

where the reference scales for length, time and mass are chosen as L , L/U0 and ρ f L 3 . M, Q, F, and N denote the internal moment, shear, inviscid and viscous fluid forces, respectively. The non-dimensional fluid loading [701] can be written as 1 (Cd sin2 θ + 4Re−1/2 sin3/2 θ ), 2 dθ F = −1.189 cos2 θ, ds

N=

(16.26)

where Cd denotes the drag coefficient and 1.189ρ f L represents the quasistatic fluid mass coefficient for a two-dimensional plate per unit width [641, 668]. The quasistatic equilibrium foil profile can be predicted by solving Eq. (16.25) along with the bound-

858

16 Isolated Inverted Flapping Foils

KB=0.1 K =0.0883

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ary conditions θ = 0 at s = 0, M = 0 and Q = 0 at s = L. Figure 16.17b shows the predicted foil profile along with the DNS profiles for K B = 0.1, Re = 1000 and m ∗ = 0.1. Cd = 2.12. From this figure, we can see that the flexible foil performs flapping about the quasistatic equilibrium state. Figure 16.18 presents superimposed quasistatic equilibrium foil profiles as a function of K B for a fixed Re = 1000 and Cd = 2.12. This demonstrates the existence of a critical K B value of 0.125 above which the flexible foil can no longer be supported by the fluid force. The quasistatic equilibrium between the restoring structural and fluid forces plays a role in the transition from the inverted-LCO to the deformed-flapping regime.

16.3.6 Formation of Leading Edge Vortex The static divergence of an inverted flexible foil explains why is the inverted foil more prone to flapping instability? However, this does not fully explain the cause for the large amplitude oscillations observed in the case of inverted limit-cycle flapping regime presented in Sect. 16.3.3. The peak-to-peak amplitudes observed for this regime as function of K B , for a fixed Re = 1000 and m ∗ = 0.1 are approximately six times greater than the peak-to-peak amplitudes observed for the conventional foil counterpart.

16.3 Two-Dimensional Flapping Dynamics

859

1

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Fig. 16.19 Evolution of leading edge vortex (LEV) for the inverted-LCO regime at K B = 0.4, Re = 1000 and m ∗ = 0.1

Since vortex structures play a key role in determining the periodic loading on a flexible body, we analyze the wake topology for K B = 0.4 as shown in Fig. 16.19. A leading-edge vortex (LEV) slowly develops behind the flexible foil during the upstroke. This leading edge vortex remains attached throughout the stroke and grows continuously till the leading edge reaches the maximum displacement prior to the stroke reversal. At the stroke reversal, the leading edge vortex pairs up with a counterrotating vortex from the lower surface of the foil at the trailing edge and is shed into the wake. Meanwhile, a rotational starting vortex (RSV) is being formed at the leading edge, as shown in Fig. 16.19c, d. The fact that the leading-edge vortex produces a large low-pressure region behind the flexible foil and resulting in large amplitude oscillations. Similar leading edge vortices have been observed in the cases of insect flight [765] and delta wing [672]. These leading-edge vortices are generally attributed to the effective aerodynamic behavior observed in the case of insect flight.

16.3.7 Vortex Organizations Vortex organization and formation are key to the understanding of many phenomena in fluid-structure interactions. Earlier studies have shown that a conventional flexible foil generates a regular von Kármán vortex street with two counter-rotating vortices.

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16 Isolated Inverted Flapping Foils

Fig. 16.20 Time history of 2P + S vortex wake over a half cycle for K B = 0.4, Re = 1000 and m ∗ = 0.1. The non-dimensional time period T0 = 10 for full oscillation

However, the flapping inverted foil exhibits a wide range of vortex patterns depending on the non-dimensional bending rigidity K B . For the stable fixed-point regime i.e. for K B ≥ 0.545 the flow field typically represents the boundary layer flow over a flat plate. As the flow travels, a symmetric boundary layer flow can be observed on either side of the inverted foil, of the fixed trailing edge, to form a narrow and steady wake. This wake is very similar to the one observed for the case of the fixed-point regime of conventional flexible foil. By decreasing K B < 0.545, the inverted flexible foil experiences large amplitude oscillations with very low frequency. Figure 16.20 shows a sequence of plots showing the vorticity contours in the fluid for the case of K B = 0.4 over one down-stroke cycle corresponding to time tU0 /L ∈ [24.7, 29.7]. In this figure, the solid and dashed lines represent the positively and negatively signed vortices. Figure 16.20a displays the instantaneous vorticity contour at tU0 /L = 24.7 as the foil is about to cross the line Y/L = 0. In this figure, the leading edge vortex (B), which is shed during the previous upstroke, can be seen forming a vortex pair with the oppositely signed vortex (C) shed from the lower surface at the trailing edge. The rotation starting vortex (D) formed during the upstroke reversal is still attached to the foil as it crosses the centerline. In Fig. 16.20b,

16.3 Two-Dimensional Flapping Dynamics

861

vortex pair (B)+(C) is shed into the wake, meanwhile the rotation starting vortex (D) travels downstream along the flexible foil. The vortex pairs (B)+(C) rotate about each other and convect further downstream in the wake. As the flexible foil bends downwards, the leading edge vortex (G) can be seen developing in Fig. 16.20c. In the meantime, the rotation starting vortex (D) is shed into the wake from the trailing edge and draws an oppositely signed vortex (E) from the lower surface of the flexible foil to form a vortex pair (D)+(E). In Fig. 16.20d, the flexible foil is at maximum displacement and the leading edge vortex (G) continues to grow behind the flexible foil on the lower surface and is just about to be shed into the wake. The boundary layer on the upper surface of the flexible foil consists of a significant amount of vorticity and this vorticity is shed into the wake as the elongated vortex (F) as shown in Fig. 16.20d. As the flexible foil rebounds from the maximum displacement the leading edge vortex (G) is shed into the wake as shown in Fig. 16.20e. The approach of vortex (G) cuts the supply of vorticity to the vortex (F) from the boundary layer. Meanwhile, a new rotation starting vortex (H) can be seen developing behind the leading edge. The leading edge vortex (F) draws a negatively signed vortex (I) from the upper surface of the flexible foil at the trailing edge to form a vortex pair, similar to the vortex pair between the leading edge vortex (B) and (C), in Fig. 16.20f. Figure 16.20f at tU0 /L = 29.7 corresponds to the flexible foil crossing the line Y/L = 0 in the opposite direction which completes the half-flapping cycle. In addition, we can also observe the position of vortices (F), (I) and (H) around the flexible foil in Fig. 16.20f are similar to position of the vortices (B), (C) and (D) in Fig. 16.20a except that all the vortices are reversed. In summary, we therefore see two vortex pairs (B)+(C), (D)+(E) and a single vortex (F) are shed into the wake i.e., a total of five vortices are shed in a half cycle. As explained earlier, in the case of a flipped-flapping regime, the leading edge of the flexible foil curves around the trailing edge and performs a flapping motion behind the fixed trailing edge. In order to avoid any confusion, we refer to the trailing and leading edges as the fixed and flipped edge, respectively. With regard to the wake patterns, the flipped-flapping regime is very much similar to the limit-cycle regime of conventional flexible foil. Figure 16.21 shows the time history of the steady von Kármán wake of the inverted flexible foil for the flipped-flapping regime with a single distinct frequency at K B = 10−3 . This figure shows the instantaneous vorticity at six different points over one periodic oscillation. Figure 16.21a represents the instantaneous vorticity at tU0 /L = 19.72 corresponding to the condition when the leading edge is at its highest position, three vortices (A), (B) and (C) close to the flipped leading edge and in the immediate downstream are shown. In Fig. 16.21b, the positively signed vortex (A) and negatively signed vortex (B) convect further downstream. Meanwhile, the counter-rotating vortex (C) shed from the lower rounded surface gets reattached to the bottom surface of the flipped flexible foil. Two counterrotating vortices are totally shed over one cycle. Figure 16.22 illustrates the time development of wake topology consisting of a 2S+2S vortex pattern for the case of the flipped-flapping regime with the two distinct frequencies at K B = 0.05. In contrast to the flipped-flapping regime with a single frequency, the wake topology does not represent the von Kármán vortex street. However, a periodic vortex shedding repeats after two complete cycle. The vortex wake in

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16 Isolated Inverted Flapping Foils

Fig. 16.21 Time history of regular von Kármán vortex street for K B = 0.001, Re = 1000 and m ∗ = 0.1 over one oscillation of the flexible foil. The time period of the oscillation T0 is 1.6

Fig. 16.22f corresponding to tU0 /L = 23.38 is very much similar to that of the wake at tU0 /L = 18.98 and 4 vortices (B), (C), (D) and (E) are shed over two cycles. In the flipped-flapping regime, we observe complicated flow features namely rolled-up shear layer over the semicircular configuration, vortex-foil interaction and a reversed flow inside the deformed cavity-like region, which are not shown here in detail.

16.3.8 Net Energy Transfer In order to analyze the ability of an inverted flexible foil to extract energy from the surrounding fluid flow, we provide a comparison between the energy harvesting estimates of the conventional and the inverted flexible foil configurations for low m ∗ = 0.1. Figure 16.23a, b show the evolution of the non-dimensional bending strain energy (E s ) and work done by fluid force (Ws ) on the inverted flexible foil for tU0 /L ∈ [0, 40] for K B = 0.4, Re = 1000 and m ∗ = 0.1 and 8, respectively. Interestingly, we observe that Ws closely follow E s in both cases. However, the contribution of the structural kinetic energy to the bending strain energy increases with m ∗ . The

16.3 Two-Dimensional Flapping Dynamics

863

Fig. 16.22 Time history of periodic wake containing 4 vortices over two complete oscillations for K B = 0.05, Re = 1000 and m ∗ = 0.1. The non-dimensional time period for two complete oscillations T0 is 3.4

magnitude of strain energy developed in the inverted foil is typical O(103 ) times the values observed for the conventional foil. In contrast to E s of the conventional foil, E s of the inverted foil attains the local maximum for the maximum deformation of the leading edge and the local minimum while the leading edge crosses the centerline. Since the inverted-LCO regime of the inverted foil flapping follows the identical path for both the upstroke and the downstroke Fig. 16.11c, the work performed by the fluid will be the same as the work done by the structure. Therefore, we can observe from Fig. 16.23a, b that ΔWs ≈ 0 per cycle for tU0 /L ≥ 30 when m ∗ = 0.1 and for tU0 /L ≥ 48.6 when m ∗ = 8. The inverted-LCO flapping regime represents a conservative system i.e. ΔWs ≈ 0 per cycle and is more effective in extracting the energy from the surrounding fluid flow in comparison to the conventional foil. Figure 16.24 summarizes the maximum strain energy developed in a flexible foil as a function of K B ∈ [10−3 , 1] for Re = 1000 and m ∗ = 0.1. The experimental data [623] corresponding to a higher mass-ratio of O(1) and Reynolds numbers of O(104 ) have also been shown for a qualitative comparison. It can be seen that both the experimental data and the current numerical simulation show constant maximum strain energy for the inverted-LCO regime. However, the qualitative comparison shows

16 Isolated Inverted Flapping Foils

0.35

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Fig. 16.23 Comparison between (—) bending strain energy (E S ) and (- - -) work-done by the fluid (Ws ) over tU0 /L ∈ [0, 40] at K B = 0.4 and Re = 1000 for m ∗ = a 0.1 and b 8 0.9 0.8 0.7 0.6

Esmax

Fig. 16.24 Maximum strain energy for the inverted flexible foil as a function of the non-dimensional bending rigidity for tU0 /L ∈ [0, 30]. () denotes the experimental data reported in [623]

0.5 0.4 0.3 0.2 0.1 0 −3 10

−2

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that the current two-dimensional numerical simulations overestimate the maximum strain energy values. We attribute this difference to three-dimensional effects, which have been neglected in the current numerical study. The aspect ratio of the flexible foil used in [623] was in the range of 1-1.3. In addition, we attribute the discrepancy in the maximum strain energy to the large difference in the mass ratio and the Reynolds numbers of the experimental study and the current numerical investigation. Another observation from Fig. 16.24 is that the maximum strain energy developed in the flipped-flapping regimes are relatively smaller when compared to the invertedLCO regime. However, the maximum strain energy developed in the flipped-flapping regime is still approximately O(103 ) times the maximum value developed in the conventional foil. Figure 16.25 shows the ratio R of the maximum bending strain energy to the total kinetic energy of the incoming fluid flow as given in Eq. (13.21). It should be noted that Eq. (13.21) considers the maximum change in the E s per oscillation i.e. (ΔE s )max instead of total stored E s . Usage of total E s can include a large nonoscillatory part, however, that would not be useful for energy recovery. The effects of this consideration can be explained for the case of K B = 0.1, corresponding to

16.3 Two-Dimensional Flapping Dynamics 0.7 0.6 0.5 0.4

R

Fig. 16.25 Ratio of maximum change in strain energy to the total available fluid energy R for an inverted flexible foil as a function of non-dimensional bending rigidity K B at Re = 1000 and m ∗ = 0.1. () denotes the experimental data reported in [623] for m ∗ = O(1) and Re = O(104 )

865

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the deformed-flapping regime, where it has large stored strain energy, yet the total energy available for the energy harvesting is much less than that of K B = 0.2 and 0.3. A comparison with [623] experimental data has also been presented in the figure. The energy efficiency ratio R shows a good agreement with the experimental data for K B ∈ [0.1, 0.2]. We also observe a significant difference in the values for K B ∈ [0.2, 0.3]. This difference in the values of R is mainly due to the difference in the maximum bending strain energy. Based on E smax and R values, we confirm that the inverted-LCO regime is the most relevant for the energy harvesting applications among the three flapping regimes of the inverted foil.

16.3.9 Interim Summary In this section, we investigated the 2D flapping dynamics and the fundamental physical mechanism behind the onset of flapping instability for an inverted foil that is clamped at its trailing edge and the leading edge free to vibrate in an unbounded axial flow. We first studied the evolution of the flapping instability as a function of the nondimensional parameters K B , Re and m ∗ , and identified three distinct stability regimes namely: (i) fixed-point stable, (ii) deformed steady and (iii) unsteady flapping state. An inverted foil placed in a uniform viscous flow loses its stability for a critical nondimensional parameter to attain the deformed steady state due to the static divergence phenomenon. The viscous flow interacting with the deformed foil can separate at the leading edge to develop the unsteady flapping when the static deformation amplitude reaches a critical value. The effects of Re and m ∗ on the stability regimes are analyzed to construct two semi-empirical relations for predicting the onset of the deformed steady state and the unsteady flapping regime. We showed that inverted foil is more prone to coupled fluid-elastic instability compared to its conventional foil counterpart due to the static divergence phenomenon.

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16 Isolated Inverted Flapping Foils

In contrast to the conventional foil, the inverted flexible foil loses its stability through static divergence for K B ≤0.55 for Re = 1000 and m ∗ = 0.1. We observe that the static deformation gradually increase with decreasing K B = 0.55 to K B = 0.545. We then performed a series of numerical simulations to find three distinct unsteady flapping response regimes as a function of decreasing KB namely: (i) inverted limit-cycle oscillation (LCO), (ii) deformed-flapping, and (iii) flippedflapping regimes. The inverted-LCO typically represents large-amplitude flapping regime of an inverted foil in its first mode. With the help of a theoretical model, we then showed that as the KB value is reduced the transition from the inverted-LCO to the deformed-flapping regime is marked by the existence of a quasistatic equilibrium state between the fluid force and the structural restoring force. On further reducing the KB, the foil flips around the trailing edge and becomes parallel to the inflow. In the flipped flapping regime, the flapping dynamics of the inverted closely resembles the conventional foil. We then analyzed the effect of m ∗ on the flapping dynamics and onset of flapping instability. Interestingly, we find that m ∗ has little effect on the onset of flapping instability. However, m ∗ strongly affects the flapping frequency and the transition from the deformed to the flipped flapping regime. The invertedLCO regime generates 4P + 2S (10 vortices) wakes, whereas The flipped flapping response regime is characterized by either a regular von Kármán vortex street or a periodic wake consisting of 4 vortices. We finally showed that the large amplitude flapping of inverted flexible foil corresponding to the inverted-LCO regime could generate O(103 ) times more strain energy in comparison to the conventional flexible foil flapping for identical m ∗ and Re.

16.4 Three-Dimensional Flapping Dynamics Over the past few years, the self-induced flapping of an inverted foil immersed in an external flow field with its trailing edge (TE) clamped and leading edge (LE) free to vibrate has been a subject of active research experimentally [623], analytically [694, 695] and numerically [585, 590, 596, 692, 710]. This particular configuration of flexible foil has superior abilities to harvest electrical current by converting the fluid kinetic energy into structural strain energy efficiently in comparison to its traditional counterpart with LE clamped [655, 738]. Apart from a practical relevance, the flapping dynamics of inverted foil have a fundamental value due to the richness of coupled fluid-structure physics associated with the complex interaction of wake dynamics with the flexible structure undergoing large deformation. Furthermore, the formation of a leading-edge vortex (LEV) and the passive self-sustained large-amplitude motion can be important to understand the flight dynamics and locomotion of small birds, bats and insects [526, 696, 771]. The formation of LEV is a very common unsteady aerodynamic feature found in biological locomotion. As the wing moves with a certain angle of incidence, the airflow rolls up to produce a stable vortex at LE and the streamlines tend to curve around the wing body due to the presence of LEV dynamics. During the flight, these living organisms stabilize the

16.4 Three-Dimensional Flapping Dynamics

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LEV over their wings to recapture the energy from vortex-flexible foil interaction to enhance the lift generation. While a pioneering theoretical analysis using inviscid potential flow was carried out by Guo et al. [593] to predict the onset of linear flapping stability, experimental [623, 694] and viscous numerical simulation studies [585, 596, 692, 710, 733] are performed to understand the self-induced flapping dynamics of an inverted foil. The self-induced flapping dynamics of an inverted foil in a steady uniform flow can be characterized by four nondimensional parameters, namely Reynolds number Re = ρ f U0 L/μf , mass-ratio m ∗ = ρ s h/ρ f L, bending rigidity K B = B/ρ f U02 L 3 , and aspect-ratio A R. Here U0 , ρ f , μf , L, h, ρ s , E, and ν, denote the freestream velocity, the density of the fluid, the dynamic viscosity of the fluid, the foil length, and the foil thickness, the solid density, Young’s modulus, Poisson’s ratio, respectively and B represents the flexural rigidity defined as B = Eh 3 /12(1 − ν 2 ). The experimental and numerical studies have shown that for a given Re there exists a critical nondimensional bending rigidity (K B )cr below which an inverted foil placed in an external flow loses its stability to undergo static deformation. Kim et al. [596, 623, 694] have attributed this phenomenon to the divergence instability. Sader et al. [694] presented an experimental and theoretical treatment to investigate the effects of foil aspect ratio and have shown that the foils with lower aspect ratios are more stable towards the divergence instability compared to the foils with higher aspect ratios. Large-eddy simulations (LES) of [585] with the aid of 3D curvilinear immersed boundary technique have shown the existence of twisted flapping modes about the longitudinal axis. An inverted foil placed in a uniform flow exhibits three distinct dynamic flapping regimes as a function of reducing K B namely large amplitude flapping (LAF), deformed flapping and flipped flapping. The transition from the static deformed state to the dynamic flapping modes is attributed to the flow separation at LE and the evolution of a large leading edge vortex (LEV) [596, 623]. The experiments of [623] first showed the existence of LAF and deformed flapping mode for aspect-ratios in the range [1,1.3] at Re = 30,000 for two representatives m ∗ corresponding to water and air. This experimental work has been followed up by a number of numerical studies [585, 596, 692, 710, 733] which have confirmed the dynamic flapping modes reported by [623]. In addition to the dynamic flapping modes observed by Kim et al. [623], the two-dimensional (2D) [596, 710] and three-dimensional (3D) [733] numerical studies have demonstrated a new flipped mode for very low K B . Large variation in m ∗ can influence the flapping frequency, vortex shedding modes and transition from LAF to the deformed flapping [596, 710]. On the other hand, m ∗ has a relatively small influence on the transverse flapping amplitudes. The experiments of [694] have shown that the transition from the LAF to the deformed flapping state is characterized by a chaotic flapping regime. The 2D simulations of [692, 710] for m ∗ = O(1) have further shown that the unsteady flapping motion ceases for Re < 50 and the foil undergoes a static deformation instead. Recently, [590] observed that an inverted foil can exhibit LAF even at Re < 50 for large m ∗ ≥ 5, wherein the flapping frequency no longer synchronizes with the vortex shedding frequency.

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16 Isolated Inverted Flapping Foils

The LAF regime is of significant interest for its energy harvesting abilities compared to its conventional foil counterpart. The 2D numerical simulations of [596, 692, 710] have shown that LAF can generate complex vortex structures. Interestingly, all vortex structures consist of counter-rotating vortex pairs generated from the LE and TE. [694] studied the physical mechanism behind the LAF motion and has drawn an analogy to the vortex-induced vibration (VIV) in an elastically-mounted circular cylinder. The authors have attributed the LAF phenomenon to the synchronized vortex shedding from LE and TE. Sader et al. [694] constructed a hypothesis based on a number of similarities such as LAF requires vortex shedding, Strouhal number range (St ≈ 0.1 − 0.2) and the synchronization of the flapping motion with the periodic vortex shedding from the leading and trailing edges. Flow over elastically mounted cylinders can be characterized by counter-rotating vortices shed from its top and bottom surface for Re > 47 [40, 455, 697]. These counter-rotating vortices generate periodic vortex-induced forces on the bluff body. In VIV, a bluff body undergoes large amplitude oscillation when the periodic vortex-induced forces acting on it synchronizes with its motion. Modifications of the flow patterns can significantly impact the VIV amplitudes. One of the common ways to suppress VIV is to inhibit the interaction between the counter-rotating vortices by introducing a splitter plate in the downstream wake region [19, 520]. In another related study on flapping dynamics, [541] studied the clapping behavior exhibited by a stack of papers placed on the floor of a wind tunnel with their TE clamped. The experimental analysis showed that the papers lose their stability to exhibit only the deformed flapping regime and the stack of papers remains in the deformed state until the stack size is large enough for the wind to support. Interestingly, in this experimental study, there is no trailing edge vortex (TEV) behind the stack of papers. This work raises questions about the role of TEV on the large inverted foil deformation. Counter-rotating LE and TE vortex pair is one of the fundamental building blocks of the vortex patterns observed in LAF. The number of counter-rotating vortex pairs per flapping cycle can vary from two to four depending on the nondimensional parameters [596, 710]. Despite significant progress recently, the role of complex LE and TE vortex shedding patterns on the vortex-induced forces and their impact on the synchronization with flapping motion is not well established. The objective of this work is to understand the role of counter-rotating vortex pair on the LAF response dynamics with the aid of 3D numerical simulations on an inverted foil assuming spanwise periodicity for nondimensional parameters K B = 0.2, Re = 30,000 and m ∗ = 1 that closely represent the conditions used in the experiments of [623]. It should be noted that this work is not about parametric analysis or generalization of various vortex patterns generated by LAF as a function of K B , Re and m ∗ . To simulate the complex fluid-structure interaction of LAF, we employ the recently developed variational body-conforming fluid-structure formulation based on the 3D Navier-Stokes and the nonlinear elasticity for large deformation. For a robust and stable treatment of strong inertial effects on a very light and thin flag-like structure, the interactions between fluid and flexible foil are coupled using the quasi-monolithic approach described in [645]. To resolve the separated turbu-

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lent wake regime behind the inverted foil, the numerical formulation considers the variational multiscale (VMS) based large-eddy simulation (LES) model. To understand the impact of the counter-rotating vortices on LAF, we introduce a configuration with a fixed splitter plate at TE. The splitter plate inhibits the formation of the TE vortex and breaks the counter-rotating vortex pair. We then characterize the coupled flapping dynamics by the wake and streamline topologies, the force and response amplitudes, and the frequency characteristics of the LAF for the inverted foil with and without the splitter. To understand the impact of the vortex shedding phenomenon on the LAF, we investigate the inverted foil for the laminar low-Re regime. In particular, we determine how an inverted foil continues to undergo staticdivergence with a relatively large static deformation even for low-Re flow. Finally, we present a list of similarities and differences between the LAF phenomenon of inverted foil and the VIV of an elastically-mounted circular cylinder and provide an improved understanding of the mechanism of LAF. A simplified linkage between the LAF dynamics of a flexible inverted foil with the flow-induced oscillation of a flat plate mounted elastically on a torsional spring at the trailing edge and immersed in a flowing stream is presented to demonstrate the significance of structural elasticity and inertia. The structure of the section is as follows: the problem set-up is described firstly. Then, we present the validation of the flow past an inverted foil at Re = 30,000. We next present the flow fields to elucidate the generation of LEV and the interaction of LEV and TEV for both inverted foil configurations with and without splitter plate. We examine the vorticity dynamics, the 3D flow structures, the frequency and response characteristics. We also expand our results by considering low Re vortex shedding simulations and the comparison of LAF with the well-known VIV phenomenon of circular cylinders.

16.4.1 Problem Set-Up We study the flapping dynamics of an infinitely wide inverted foil with 3D flow field assuming spanwise periodicity for two different configurations. Figure 16.26 presents representative schematics of the two configurations. The first configuration is a simple inverted foil interacting with the 3D incompressible viscous flow Fig. 16.26a. In the second configuration, we introduce a long fixed splitter plate at the trailing edge (TE) as shown in Fig. 16.26b to inhibit the vortex-vortex interaction between the two counter-rotating vortices generated from TE and LE. In this study, we have considered spanwise periodicity to ignore the foil edge effects for simplicity and also from the viewpoint of computational efficiency. Figures 16.27a and 16.27b present the 3D spanwise periodic computational setup of an inverted foil with length L and thickness H = 0.01L clamped to a fixed splitter plate of length L s in the Cartesian coordinate system. The size of the computational domain is [22L × 10L × W ], where W is the width of the computational domain. The foil is placed along the x-plane with its LE along the y-plane. At the inlet boundary

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16 Isolated Inverted Flapping Foils

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Fig. 16.26 A typical representative schematics of an inverted foil interacting with a uniform axial flow: a without splitter plate and b with a splitter plate. Dash-line (− − −) in a represents a typical LAF motion of inverted foil Γftop

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(c) Fig. 16.27 3D computational domain along with the boundary conditions: a front view, b side view, and c a close-up view of the body-conforming boundary layer mesh around the inverted foil without splitter

16.4 Three-Dimensional Flapping Dynamics

871

Γinf , a stream of incompressible fluid enters into the domain at a uniform velocity U0 . The slip-wall boundary condition (v = 0, ∂u/∂ y, ∂w/∂ y) is implemented along f f and Γbottom , respectively. The tractionthe top and bottom boundary surfaces Γtop f free condition is specified at the outflow plane Γout and the computational domain is assumed to be periodic in the spanwise direction. The spanwise periodicity of the computational domain is considered for both fluid and solid domains. The body conforming quasi-monolithic formulation with exact interface tracking enables us to enforce the no-slip Dirichlet boundary condition along the deformable foil interface, which allows accurate modeling of the boundary layer on the foil.

16.4.2 Validation Before discussing the physical insight of LAF of an inverted foil, it is essential to establish the appropriate mesh resolution and validate the coupled fluidstructure solver. We validate the flapping dynamics of an inverted foil clamped at the TE without the splitter plate, i.e. L s = 0, for the nondimensional parameters m ∗ = 1, K B ∈ [0.15, 0.5] and Re = 30,000, which are consistent with the experimental conditions of [623]. The 3D simulations are performed on the finite element mesh with 1.9 million six-node wedge elements consisting of 21 layers of 2D finite element meshes with 46924 nodes each. The flow field is assumed to be at rest for tU0 /L ≤ 0. Fig. 16.27c presents a close-up view of the fluid boundary layer mesh around the foil at its maximum transverse displacement for K B = 0.2, Re = 30,000 and m ∗ = 1. To check the adequacy of the spatial resolution, we have performed extensive grid refinement tests with different resolutions. The width of the computational domain W = 0.5L has been selected after a detailed analysis. Figures 16.28a and 16.28b compare the time histories of the transverse and streamwise displacements of inverted foil with W/L = {0.25, 0.5, 1.0, 2.0}. The figures show that the effect of W/L ∈ [0.25, 2] on the transverse and streamwise flapping amplitudes of LAF is marginal. We also compare the time histories of the lift and the drag forces acting on the flexible foil in Fig. 16.28b. Even the forces acting on the foil as a function of W/L show similar time histories barring small variations, which can be attributed to the complex 3D turbulent flow structures since the forces acting on the foil are more sensitive toward the vortex structures than their bulk kinematic response. We next look at 3D flow structures to see if the width of the domain is large enough not to influence the flow structures. To investigate the effect of W on the large scale 3D flow structures generated by the LAF, we plot the λ2 [615] iso-surfaces in Fig. 16.29 at W/L = 0.25 and 2.0 for identical nondimensional parameters K B = 0.2, Re = 30,000 and m ∗ = 1. We can observe similar flow structures for both W values. The figures show the existence of the streamline ribs SLR1 in the wake behind the deformed foil and SLR2 along the foil surface. The spanwise roll SWR1 represents the LEV vortex formed behind the foil. SWR2 and SWR3 denote the TEV and LEV vortices, respectively, formed behind the foil during the previous downstroke.

872

16 Isolated Inverted Flapping Foils 1

0.5

0 W/L=0.25 W/L=0.5 W/L=1.0 W/L=2.0

-0.5

-1 10

12

14

16

18

20

22

(a)

(b) Fig. 16.28 Effect of computational domain width on the inverted foil at Re = 30,000, K B = 0.2, m ∗ = 1 on: a LE transverse (left) and streamwise (right) displacements, and b transverse lift (left) and streamwise drag (right) forces acting on the foil

SLR3 is the streamlined ribs formed behind the deformed foil during the previous downstroke. The streamline ribs SLR3 encloses the LEV+TEV pair of SWR3 and SWR2. It is worth mentioning again that we have ignored the spanwise end effect by considering an infinitely wide inverted foil. Such effects can become dominant for finite foils. There exists a certain low aspect ratio of foil whereby the influence of the vortices from sides can become significant for a combination of K B , m ∗ and Re [694]. Figure 16.30 (top) and (bottom) show the variation of the maximum LE transverse max flapping amplitude (Amax y /L) and the strain energy (E s ) with the bending rigidity K B . The numerical results are also compared against the experimental data [623]. The strain energy E s is evaluated as a function of curvature κ using 1 Es = 2

l

   3/2 E I κ 2 dl/ρ f U02 L 2 and κ = ∂ 2 f (x)/∂ x 2 / 1 + (∂ f (x)/∂ x)2 ,

0

(16.27) where f (x) is a a piecewise polynomial function of 6th order that has been constructed to define the deformed foil profile at each time instant. Figure 16.30 clearly

16.4 Three-Dimensional Flapping Dynamics

(a) W /L = 0.25

873

(b) W /L = 2

Fig. 16.29 Iso-surfaces of three-dimensional instantaneous vortical structures at K B = 0.2, Re = 30,000 and m ∗ = 1 for two representative domain widths: W/L = a =0.25 and b 2.0. Here SLR and SWR denote streamline ribs and spanwise rolls and the iso-surface λ2 = 30 colored by the fluid velocity magnitude is plotted

shows that our 3D simulations correctly predict the onset of flapping instability and max exhibit similar trends to that of the the post-critical variations of Amax y /L and E s max experiment. Some differences in A y /L and E smax can be attributed to the spanwise end effects of the inverted foil in the experiment. Figure 16.31 summarizes the coupled flapping modes exhibited by an inverted foil as a function of K B . The foil loses its stability for K B ≈ 0.3 to perform the flapping motion that is primarily biased towards one side. This observation is consistent with the previous 2D analysis presented in [596]. The onset of flapping instability can be attributed to the combination of divergence and flow separation at LE. By further decreasing K B , the foil begins to exhibit the LAF motion Fig. 16.31c and the foil profiles for K B = 0.2 in Fig. 16.31c show that the foil does not recoil at its maximum transverse displacement. Instead, the foil deforms further downstream before it recoils and goes through two humps-like patterns in the vicinity of maximum transverse displacement, which can be seen from the LE transverse displacement history for K B = 0.2, Re = 30,000 and m ∗ = 1.0 in Fig. 16.32. Although we have validated our numerical framework by comparing the time history of LE displacement with the experimental data, it is also essential to verify if the mesh used here adequately captures the turbulent wake characteristics. For that purpose, we look into the turbulence power spectra in the wake region. Figure 16.33 shows the spectral distribution of the turbulent kinetic energy (TKE) at eight points (P1 − P8 ) in the wake behind the inverted foil for tU0 /L ∈ [0, 8] for which the deformation of the relatively inverted plate is tiny. Also, the deformation of the mesh where the probes

874

16 Isolated Inverted Flapping Foils 2 experiment present

1.5

1

0.5

0

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.8 experiment present

Large amplitude flapping 0.6

Esmax

Steady-state 0.4

0.2

0

0.15

0.2

0.25

0.3

0.35

0.4

0.45

KB Fig. 16.30 Comparison of peak-to-peak LE transverse amplitude (top) and maximum strain energy (bottom) as a function of K B ∈ [0.15, 0.35] against experimental measurement [623] at Re = 30,000 and m ∗ = 1

are placed is negligible. The figure shows that the kinetic energy decays with a slope of −5/3. For locally homogeneous turbulence, Kolmogorov’s −5/3 power states that the kinetic energy and the wave number, k, of integral scale eddies, follow the relation T K E(k) ∝ k −5/3 in the inertial range. This relationship is generally applied to the spatial distribution of turbulent kinetic energy (T K E ≡ u i u i /2) as a function of wavenumber spectrum. The −5/3 decay of the kinetic energy can be extended to the frequency domain for the turbulence spectrum via Taylor’s frozen turbulence hypothesis [739]. Therefore, Kolmogorov’s −5/3 spectral decay of TKE versus the frequency in Fig. 16.33 confirms the adequacy of the fluid mesh for the VMS-based turbulence model.

16.4 Three-Dimensional Flapping Dynamics

875

max. δy

(a) KB = 0.35

(b) KB = 0.3

(c) KB = 0.2

Fig. 16.31 3D inverted flexible foil profiles over a flapping cycle at m ∗ = 1 and Re = 30,000 for K B = a 0.35, b 0.3, and c 0.2 1.5

1

0.5

0

-0.5

-1 0

0.5T

1T

1.5T

2T

2.5T

3T

3.5T

Fig. 16.32 Time history of LE transverse displacement for an inverted foil at K B = 0.2, Re = 30,000 and m ∗ = 1. Here T represents the flapping time period

16.4.3 Flow Field The LCO flapping of inverted foil forms a large periodic obstacle to the impinging flow, which alters the fluid motion and generates a separated wake flow. Similar to a bluff body structure, the separated wake flow consists of vortices shed periodically from the deformed foil. The interaction between the vortices from the LE and TE, and the flexible structure are strongly coupled to each other. The periodic vortex shedding in the separated wake flow can synchronize with the flapping motion. To realize the mechanism of LAF, we will investigate the onset of LAF, the response histories, the

16 Isolated Inverted Flapping Foils 10 0 P 1

10 0

10 -2 P 2

10 -2 P 6

10 -4 P 3

10 -4

P5

TKE

TKE

876

P7

10 -6 P 4

10 -6 P 8

10 -8

10 -8

10 -10 10 0

f L/U0

10 1

10 -10 10 0

f L/U0

10 1

Fig. 16.33 Spectral distribution of turbulent kinetic energy (TKE) at different positions in the wake for K B = 0.2, m ∗ = 1 and Re = 30,000: P1 = (2L , 0.75L), P2 = (2L , 0.5L) P3 = (2.5L , 0.75L), P4 = (2.5L , 0.5L), P5 = (2L , −0.75L), P6 = (2L , −0.5L), P7 = (2.5L , −0.75L) and P8 = (2.5L , −0.5L). Except for the highest spectra, other spectral trends are shifted down for clarity. The dashed lines indicate the Kolmogorov’s −5/3 power law

frequency characteristics, the force dynamics, the vortex shedding patterns and the synchronization between the vortices shed during the flapping phenomenon. Since the separated vortex structures play a key role in determining the periodic loading on a flexible body, we analyze the wake topology, more specifically, we focus on the interaction of the LEV and the TEV developed behind the deformable flexible foil. The 2D simulations by Gurugubelli et al. [596] have shown that the onset of LAF is characterized by the LEV due to the flow separation at LE for relatively large deformation of the inverted foil. We first performed a series of 3D simulations at Re = 1000 for K B = [0.4, 0.25], m ∗ = 0.1 and W/L = 0.5 to find out the impact of flow 3D effects on the LAF response. Unlike the 2D simulations of [596] for the identical nondimensional parameters, the 3D simulations at K B = 0.4 do not show any LAF phenomenon. The foil loses its stability only for K B ≤ 0.3, confirming that the foil 3D effects play a significant role in the onset of LAF by stabilizing the onset of divergence instability and then flow separation at the leading edge. To visualize the role of 3D flow structures during the evolution of LAF, in Fig. 16.34, we plot the evolution of the streamwise and spanwise vorticity of LEV at K B = 0.2, Re = 30,000, m ∗ = 1 and W/L = 0.5. As the foil deflects from its initial state due to divergence instability, at a certain angle of deformation, a small 2D LEV develops behind the foil, which can be seen in Fig. 16.34a for tU0 /L = 7. This 2D LEV results in a low-pressure region behind the foil, which enhances the downward forces acting on the foil that, in turn, increases the foil’s downward deformation. The 2D LEV grows in size with the foil deformation and the first signs of 3D vortex structures are observed in Fig. 16.34b for tU0 /L = 7.5. Figures 16.34b (left) and 16.34c (left) also show the occurrence of similar counter-rotating streamwise vortex pairs. To further confirm this behavior, we plot the streamwise x-vorticity along the sectional planes at x = 0.2L and 0.45L for tU0 /L = 8 in Fig. 16.35. The figure shows the formation of 3D streamwise counter-rotating vortices along the foil span. The formation of vertical streamwise vortices at the sharp leading of a

16.4 Three-Dimensional Flapping Dynamics

877

(a) tU0 /L = 7

(b) tU0 /L = 7.5

(c) tU0 /L = 8 Fig. 16.34 Temporal evolution of 3D-LEV over a deformable inverted foil in uniform flow: Isosurfaces of streamwise ωx (left) and spanwise ωz (right) vorticity at K B = 0.2, m ∗ = 1 and Re = 30,000. Here blue and red colors denote positive and negative vorticity respectively

foil requires a further numerical investigation with a much higher mesh resolution, which is beyond the scope of the present work. We next present the 3D vortex organization for the LAF regime corresponding to (K B, Re, m ∗ ) = (0.2, 30,000, 1.0) over a half-cycle. Figure 16.36 presents the vortex mode exhibited by the LAF over a one-half cycle of the periodic oscillation for the downstroke. Figure 16.36a–c show the LEV ‘A’ and an oppositely signed TEV ‘B’, which are formed due to the roll of a shear layer via Kelvin-Helmholtz instability, attract each other to form a pair of counter-rotating vortices. The counter-rotating vortex ‘B’ grows in size and cuts off the vortex ‘A’ from the LE. Meanwhile, as the LEV ‘A’ is shed, a new LEV ‘C’ can be seen developing at the LE in Fig. 16.36c and this phenomenon is marked by a rise in the lift acting on the foil. However, this time foil inertia overcomes the lift acting on the foil and as the foil crosses the mean position, the LEV ‘C’ convects along the foil surface and sheds into the wake.

878

16 Isolated Inverted Flapping Foils

Fig. 16.35 Instantaneous view of streamwise ωx along the sectional planes at x = 0.2L and 0.45L for K B = 0.2, Re = 30,000 and m ∗ = 1

A new LEV ‘D’ can be seen developing on the opposite surface of the foil once the foil crosses certain deformation, which in turn pulls an oppositely signed TEV ‘E’. A somewhat similar to the downstroke, even the upstroke generates a pair of counterrotating vortices. However, during the upstroke, we do not observe a secondary LEV. The LAF mode for K B = 0.2 exhibits a combination of P + S and P vortex modes per flapping cycle due to the transition from the pure 2P mode for K B = 0.25 to the pure 2P + 2S mode for K B = 0.15. Similar 2P and 2P + 2S vortex modes per flapping cycle have been observed by Ryu et al. [692], Gurugubelli et al. [596], Mittal et al. [710] through 2D simulations. The vortex mode notation used here is consistent with the notation introduced by Williamson and Roshko et al. [457] for an oscillating circular cylinder. For K B = 0.25 exhibiting 2P vortex pattern per flapping cycle, we would observe the flow structures similar to the ones shown in Fig. 16.36g–i for the upstroke and mirror image of the flow structures presented in Fig. 16.36g–i with respect to the x-axis during the downstroke. On the contrary, the 2P + 2S flow structures for K B = 0.15 appear similar to Fig. 16.36a–f and mirror image of Fig. 16.36a-f with respect to x-axis for the down and upstroke respectively. To investigate the influence of the vortex modes on the flapping motion and forces acting on the inverted foil, we plot the time traces of the LE displacements in both transverse and streamwise directions and contrast them against the lift and drag coefficients in Fig. 16.37. The figure shows that the lift acting on the foil reaches the maximum at point ‘a’ even before it reaches the maximum transverse displacement. Between points ‘a’ and ‘b’, while the lift force acting on the foil experiences a sudden drop, the drag force, streamwise displacement and transverse displacements continue to increase. We can attribute this observation to the large foil deformation like in

16.4 Three-Dimensional Flapping Dynamics

879

(a) tU0 /L = 14

(b) 15

(c) 15.75

(d) 16.25

(e) tU0 /L = 17

(f) tU0 /L = 17.5

(g) 18.55

(h) tU0 /L = 19.25

(i) tU0 /L = 20

Fig. 16.36 Generation and interaction of LEV and TEV behind an inverted foil over a half flapping oscillation: Time evolution of nondimensional spanwise ωz distribution (blue: ωz = 15 and red: ωz = −15) at K B = 0.2, m ∗ = 1 and Re = 30,000

Fig. 16.36a, where the majority of the fluid force acting on the foil contributes to the horizontal drag rather than to the vertical lift force. In other words, the deformed foil develops a maximum projected area to the oncoming flow stream. Above a critical foil deformation, the flapping dynamics is dominated by the drag acting on the foil. Point ‘b’ in the figure represents the time instance at which the inverted foil recoils and the LEV ’A’ separates from the LE. The shedding of LEV ’A’ is characterized by a sudden drop in the drag acting on the foil. This observation can be attributed to the increase in the pressure field behind the foil near the LE after the LEV ’A’ is

880

16 Isolated Inverted Flapping Foils 1

δx /L

δy /L

1 0

0.5 0

-1 4

Cd

Cl

e

0 b

15

tU0 /L

b

6

2

-2 10

8

a'

c

a

4

a'

2

f

0

20

-2 10

d

e

c a

d

15

f

20

tU0 /L

Fig. 16.37 Time histories of response amplitudes (top) and force coefficients (bottom) for the inverted foil at nondimensional parameters K B = 0.2, Re = 30,000, m ∗ = 1 and W/L = 0.5: In this plot points (a,b,c,d,e,f,a’) correspond to the time instances tU0 /L=(12.6, 13.5, 14.65, 15.65, 17.5, 18.6, 19.8)

separated from the foil as shown in Fig. 16.38a–c. Figure 16.38 presents the pressure difference across inverted for the time instances a-d shown in Fig. 16.37. During the initial stages of the downstroke, i.e., between points ‘b’ and ‘c’, the lift acting on the foil begins to increase because at the point of recoil, the foil is curved inwards Fig. 16.38b and the relative increase in pressure behind the foil near the LE would result in an increase in the lift near the LE until the foil reaches point ’c’ where the foil shape close to LE is almost vertical Fig. 16.38c. Point ‘c’ corresponds to the time instance after the LE of inverted foil crosses the maximum transverse displacement. Once the LE crosses the maximum transverse displacement, the lift acting on the foils will also experience a sudden drop. The drag acting on the foil reaches the minimum as it crosses the mean position. On the other hand, around the same time, the lift acting on the foil starts to increase slightly due to the formation of the secondary LEV ‘C’ at ‘d’. While no sharp changes in the lift even when the LEV ‘D’ is formed, one can observe oscillations in the lift plot between the points ‘d’ and ‘e’. This is because the secondary LEV ‘C’ in front of the foil minimizes the effect of the LEV ‘D’ behind the foil and thus, we do not observe any sharp changes in the lift plot. Thereby resulting in a lower streamwise flapping amplitude and drag for the downstroke compared to the upstroke. The inverted foil again recoils back at point ‘e’. Similar to the recoil phenomenon during the downstroke, the lift force acting on the foil is recovered before the LEV ‘D’ separates from the LE. The separation is again characterized by a sharp change in the lift curve at point ‘f’. Since no secondary LEV is observed during the upstroke, the lift acting on the foil increases until the point ‘a ’. This phenomenon continues and repeats itself over each flapping cycle.

16.4 Three-Dimensional Flapping Dynamics

881

Fig. 16.38 Pressure distribution around the inverted foil at K B = 0.2, Re = 30,000 and m ∗ = 1 for the time instances tU0 /L =: a 12.6, b 13.5, c 14.65, and d 15.65

The formation of the counter-rotating vortex pair of ‘A+B’ in Fig. 16.36a represents a close resemblance to the flow over a circular cylinder wherein the interaction between the counter-rotating asymmetric vortices plays a significant role in the vortex-induced vibration when the frequency of vortex formation is relatively close to the natural frequency of the structure. The VIV lock-in of an elastically mounted vibrating circular is characterized by the matching of the periodic vortex shedding frequency and the oscillation frequency of the body [697]. The VIV response characteristics of transversely vibrating circular bodies become significantly influenced if we disturb the synchronization between the shedding phenomenon or suppress either of the two alternate vortices. The prevention of vortex-vortex interaction and the frequency lock-in can reduce the large transverse amplitudes. However, the role of the vortex organization and the interaction between two counter-rotating vortices on the LAF has not been systematically explored. To understand the impact of the vortices on the LAF, we introduce a splitter plate at the trailing edge to suppress the TEV.

16.4.4 Effect of Splitter Plate Behind Inverted Flapping We replicate the 3D simulation at Re = 30,000, K B = 0.2, m ∗ = 1 and W/L = 0.5 by considering a typical splitter plate of length L s = 4L at TE. Similar to the circular cylinder, the splitter plate will inhibit the vortex-vortex interaction between the counter-rotating vortices from the LE and the TE. Figure 16.39 presents the isometric vortex mode observed for an inverted foil with a splitter plate at the trailing edge. The figure shows that unlike the inverted foil without a splitter plate which exhibited 2P + S vortex mode per flapping cycle, in this case, only two counterrotating vortices ‘A’ and ‘B’ are shed over the flapping cycle i.e. 2S vortex mode. Additionally, by comparing Figs. 16.36 and 16.39, it can be observed that the time taken over a one-half cycle of the inverted foil with a splitter is 20% more than that of a simple inverted foil without a splitter and this increase in the time period is mainly

882

16 Isolated Inverted Flapping Foils

(a) tU0 /L = 15

(b) tU0 /L = 16

(c) tU0 /L = 17

(d) tU0 /L = 18

(e) tU0 /L = 18.8

(f) tU0 /L = 19.8

Fig. 16.39 Generation and interaction of LEV and TEV behind an inverted foil with splitter plate over a half flapping oscillation: Time evolution of ωz at nondimensional parameters K B = 0.2, m ∗ = 1 and Re = 30,000. Here blue and red denote positive and negative vorticity respectively and the flow is from left to right

due to the increase in time taken by the foil to reach the mean position. However, for both the inverted foil configurations, the time taken by the foil to deform from its mean position to the point of recoil remains nearly identical. Lower time periods during the first half of the downstroke in the case of inverted foil without a splitter plate can be attributed to the greater induced velocity due to the interaction between the counter-rotating vortices ‘A+B’. As a result of this, the drag force, which is the main source of the inverted foil bending, reduces more rapidly than over the foil with the splitter plate. Detailed analysis of the impact of the vortex-vortex interaction on the flapping amplitudes and the forces is presented in the following paragraphs. To begin, Fig. 16.40a, b present the frequency characteristics of the LE displacements for the inverted foil configurations with and without the splitter plate. The figures confirm the reduction in the flapping frequency due to the inhibition of the vortex-vortex interaction by the splitter plate and the flapping frequency of the foil without splitter is 14.26% greater than its counterpart with the splitter. Figure 16.40c, d compare the LE transverse and streamwise displacements of the inverted foil with and without the splitter plate, respectively. It is evident from Fig. 16.40c that the transverse flapping amplitudes remain similar for both the inverted foils with and without the splitter plate.

f1 = 0.117 1

883

f1 = 0.156

Normalized PSD

Normalized PSD

16.4 Three-Dimensional Flapping Dynamics

0.8 0.6 0.4 0.2 0

1 0.8 0.6 0.4

0.2

0.4

0.6

0.8

f L/U0

(a) transverse frequency

(c) transverse displacement

1

f = 0.273

0.2

0.4

1

0.2 0

0

f1 = 0.234

0

0.6

0.8

1

f L/U0

(b) streamwise frequency

(d) streamwise displacement

Fig. 16.40 Frequency spectra and the time histories of LE response amplitudes for inverted foil: a transverse frequency, b streamwise frequency, c transverse displacement response and d streamwise displacement response for the inverted foil with (- - -) and without (—) the fixed splitter plate for identical physical conditions Re = 30,000, m ∗ = 1 and K B = 0.2

From Fig. 16.40d, the inhibition of the interaction between the vortices reduces the maximum streamwise flapping amplitudes and makes them more regular. To explain the reduction in the streamwise flapping amplitudes, we examine the nondimensional pressure distribution along the foil ( p(s) ˆ = Δp(s)/0.5ρ f U02 ) for both the inverted foils presented in Fig. 16.41a, b, where s is the nondimensional curvilinear abscissa along the foil length with s = 0 and 1 at the leading and trailing edges, respectively, and Δp(s) denotes the net pressure acting on any point s ∈ [0, 1] on the foil surface. Due to the formation of TEV, pˆ at the TE is significantly lower for the inverted foil without a splitter plate. As a result of this, the foil without a splitter experiences 11.5% lower and 2.4% greater mean drag compared to the foil with a splitter for the regions 0.5 ≤ s ≤ 1 and 0 ≤ s < 0.5, respectively. Due to the greater drag close to the LE, the inverted foil without a splitter experiences greater bending moments compared to its counterpart. The reduction in the streamwise flapping amplitudes due to the inhibition of the vortex-vortex interaction by the splitter affects the curvature of the foil. Figure 16.41c presents the variation of curvature along the foils for both configurations. The figure clearly shows the area under the curvature curve for the inverted foil without a splitter plate is greater than the area under the foil with a splitter. Since the strain energy is proportional to the square of the curvature (Eq. 16.27), we can deduce that the energy harvesting ability of an inverted foil can be manipulated by controlling the vortex-vortex interaction without directly altering the foil properties.

884

16 Isolated Inverted Flapping Foils

0.8

y/L

0.6

6

5

5

4

4

3

3

0.4

2

0.2 1

TE

0 0.8

(a)

1

κ

pˆ LE

without splitter with splitter

2 1 0 -1 0

0.2

0.4

0.6

x/L

s

(b)

(c)

0.8

1

Fig. 16.41 Contours of nondimensional pressure distribution along two inverted foil configurations: a with, b without splitter plate at their maximum streamwise deformation, and c comparison of the variation of curvature (κ) along the foils at maximum deformation. s represents the curvilinear coordinate with 0 at L E and 1 at T E

Fig. 16.42 Time histories of force coefficients and frequency spectra for inverted foil with and without splitter plate for the identical physical conditions of Re = 30,000, m ∗ = 1 and K B = 0.2: a lift, b drag, frequency response of lift c and d drag. Here inverted foil with (- - -) and without (—) a splitter plate are shown

16.4 Three-Dimensional Flapping Dynamics

885

1

δx /L

δy /L

1 0

0.5 0 8

-1 5 a'

c

0 b d

-5 10

6

e

Cd

Cl

a

15

e

c

a

d

f

2 0

f

20

tU0 /L

4

b

25

-2 10

a'

15

20

25

tU0 /L

Fig. 16.43 Comparison of the evolution of LE transverse and streamwise displacements against the lift (Cl ) and drag (Cd ) coefficients for the inverted foil with splitter at nondimensional parameters K B = 0.2, Re = 30,000, m ∗ = 1 and W/L = 0.5. In this plot, points (a,b,c,d,e,f,a’) correspond to the time instances tU0 /L = (13.95, 15.0, 16.4, 18.5, 19.8, 20.8, 23)

Inhibition of the vortex-vortex interaction has a profound impact on the drag and lift forces Fig. 16.42. Due to the formation of the counter-rotating vortex at TE of the inverted foil without a splitter, both the drag and lift forces acting on the foil drop more rapidly compared to its counterpart with the splitter plate. As a consequence of this, the foil without the splitter travels more quickly from its maximum deformation to the mean position, thereby resulting in a lower flapping period and higher flapping frequency compared to the foil with the splitter plate. The lift forces exhibit an additional secondary frequency f 2 , which has 3rd harmonic component in the flapping frequency in addition to the fundamental frequency f1 corresponding to the transverse response. Unsurprisingly, Fig. 16.42c shows f 2 as the dominant frequency for the inverted foil without splitter because three vortices (‘A+B’, ‘C’ in Fig. 16.36) are shed over the one-half cycle. However, there is no synchronization between the periodic vortex-induced dominant f 2 and transverse flapping. On the other hand, the dominant vortex-induced frequency f 1 synchronizes with the transverse flapping frequency for the inverted foil with the splitter plate because there is only one vortex (‘A’) shed over a one-half cycle of flapping oscillation and no secondary LEV are observed. For complex vortex modes like 2P + S observed in our study for inverted foil without the splitter and 2P + 2S or 4P modes observed in [596, 692], the dominant vortex-induced forces may not synchronize with the transverse flapping frequency. To further understand the impact of the suppression of TEV on the force dynamics, we plot the time histories of LE transverse and streamwise displacement responses and contrast them against the lift and drag acting on the foil over a flapping cycle in Fig. 16.43. Qualitatively if we compare Figs. 16.37 and 16.43, both the figures share similar characteristics. The primary difference between the two figures is the point

886

16 Isolated Inverted Flapping Foils

Fig. 16.44 Streamline topology for the inverted foil without (left) and with (right) splitter plate attached at TE for the identical nondimensional parameters K B = 0.2, Re = 30,000 and m ∗ = 1

‘d’. In Fig. 16.37, the point ‘d’ represents the time instant at which the lift acting on the foil increases due to the formation of the secondary LEV ‘C’. Since there is no secondary LEV for the inverted foil with the splitter, the lift acting on the foil drops until the foil attains a large enough deformation so that drag can sustain the deformed foil. The point ‘d’ in Fig. 16.43, represents the point of transition from the lift dominated to the drag-dominant foil deformation. Therefore, the mechanism of the foil motion from the maximum transverse deformation to the mean position depends on the foil inertia due to the elastic restoring forces and shedding of the LEV ‘A’. The TEV only enhances this motion of the foil and hence the foil travels faster when the TEV is present as compared to the foil where it is suppressed. Figure 16.44 shows the streamlined topology for the inverted foil with and without the splitter, where LEV1 (TEV1) and LEV2 (TEV2) represent the LEV (TEV) shed during current and previous half-cycles, respectively. The figure conforms with the vortex modes presented in Figs. 16.36 and 16.39. In addition to the suppression of TEV, the figure also presents one more distinct feature. In the case of the inverted foil without a splitter, (LEV+TEV) vortex pair travel away from the foil at an angle inclined to the freestream velocity. Similar observations can be seen in 2D vortex modes presented in [596, 692]. On the other hand, the LEV in the inverted foil with the splitter travels along the splitter plate and grows in size. The wall surfaces of the splitter plate act as the source of vorticity, which feeds into the LEV. To further demonstrate the role of vortex shedding on the LAF, we next perform numerical simulations for low-Re flows at identical K B and m ∗ .

887

1

2

0.5

1.5

δx /L

δy /L

16.4 Three-Dimensional Flapping Dynamics

0 Re=0.1 Re=1 Re=10 Re=20 Re=50

-0.5 -1 0

5

1

0.5

10

15

20

25

0 0

5

10

15

tU0 /L

tU0 /L

(a)

(b)

y/L

0.8 Re=20

20

25

Re=10 Re=1

0.6 0.4

Re=0.1

0.2

0.5

1

x/L

1.5

(c) Fig. 16.45 Time traces of the LE response amplitudes of inverted foil at K B = 0.2 and m ∗ = 1 for low-Re laminar flow: a transverse, b streamwise, and c steady-steady foil profiles

16.4.5 The Role of Vortex Shedding on Inverted Foil Flapping at Low Reynolds number We perform a series of numerical experiments on the simple inverted foil without the splitter for low-Re ∈ [0.1, 50] regime at a relatively low m ∗ = 1.0 to realize the effects of vortex shedding phenomenon on the LAF and the onset of flapping instability. As a function of Re based on the initial foil length, Fig. 16.45 summarizes the time traces of LE displacements and the steady-state deformed foil profiles as a function of Re. From the LE displacements in Fig. 16.45a and b, we can see that there exists a critical Re for m ∗ = 1.0 below which the LAF motion ceases to exist. However, the inverted foil continues to exhibit the static-divergence phenomenon even for very small Re and undergoes a large static deformation. Similar large static deformations for Re ∈ [20, 50] have been reported by Zhang and Hisada [692, 710] for m∗ = O(1). This phenomenon is distinctly different from the flexible foil with its LE clamped where the foil remains stable for very low-Re [708]. For Re < 10, the inverted foil flips about the clamped TE and exhibits a flipped state as shown in Fig. 16.45c. For the flipped state, the LE of the foil aligns itself with the flow direction. To examine how an inverted foil continue to undergo static-divergence with large static deformation even for low Re, we plot the time evolution of lift and drag during the onset of instability in Fig. 16.46a, b respectively for Re = 30,000 and 10 at

888

16 Isolated Inverted Flapping Foils 6 Re=10 Re=30000

3

4

|Cl |

Cd

2 2

1 0 0 -2 0

5

10

15

20

0

5

10

tU0 /L

tU0 /L

(a)

(b)

15

20

Fig. 16.46 Comparison of the time histories of lift (left) and drag (right) during the onset of instability at K B = 0.2 and m ∗ = 1 for Re = 30,000 and 10

K B = 0.2 and m ∗ = 1. Even though the maximum lift acting on the foil for Re = 10 is significantly lower than the lift for Re = 30,000 Fig. 16.46a, the maximum transverse amplitude for Re = 10 and 30,000 is approximately same Figs. 16.40c and 16.46a. For low Re, the large static deformation can be attributed to the relatively large static drag (C¯ d = 2.78) acting on the foil which is greater than the mean (C¯ d = 2.19) and the root-mean square (Cdrms = 2.68) drag at Re = 30,000. We have earlier shown in Sect. 16.4.3 that the large deformation of the foil depends on both the lift and drag forces. For low Re, the lift acting on the inverted foil is just sufficient to break its symmetry and the drag bends it through large deformations. To understand the role of viscous stress and LEV on the large drag at low Re, we plot the viscous and pressure drag components on top and bottom foil surfaces in Fig. 16.47a. The viscous and pressure drag components acting on the foil, respectively, are computed using  Cdμ =

 (T f · nf ) · nx dΓ

Γ (t)

and

Cdp =

(− pI · nf ) · nx dΓ , Γ (t)

where T is the viscous fluid stress and nx represents the unit vector in the streamwise direction. Both top and bottom surfaces experience a large viscous drag initial until the foil achieves sufficiently large transverse amplitude and the flow separates at LE to form the LEV. After the formation of LEV, the viscous drag acting on the foil drops and the pressure-induced drag due to LEV increases. Therefore, both the viscous and pressure drag components act jointly to produce the large foil deformation at low Re. Figure 16.47b summarizes the viscous and pressure drag components as a function of Re. As Re decreases both viscous and pressure drag components acting on the foil increase. Three key points to highlight from this low Re study are: (i) vortex shedding at LE of an inverted foil is necessary for sustaining unsteady periodic flapping for a relatively low m ∗ ; (ii) unlike the flexible foil with LE clamped which remains stable for the low-Re regime, the inverted foil loses its stability to undergo large static

16.4 Three-Dimensional Flapping Dynamics C

2.5

C

top

Cdμ bottom

2

Cdp bottom

Cd

40

1

20

0.5 0 -0.5

dμ

Cdp

60

Cdp top

1.5

Cd

dμ

889

0 0

5

10

15

20

25

tU0 /L

10-1

100

101

Re

(a)

(b)

Fig. 16.47 Decomposition of drag force of the inverted foil for low Re laminar flow at K B = 0.2 and m ∗ = 1: a time evolution of drag decomposed into viscous and pressure components acting on the top and bottom inverted foil surfaces for Re = 10, b summary of viscous and pressure components of drag as a function of Re Z Vorticity

1

1.0 0.5

Y/L

Y/L

Y/L

0.5 0.5

0

0

0.0

0.6 0.8

1

1.2 1.4

-0.5

0.8

1

1.2 1.4 1.6

-0.5

1

1.5

X/L

X/L

X/L

(a)

(b)

(c)

9 8 7 6 5 4 3 2 1 0 -1 -2

Fig. 16.48 Streamline topology and spanwise vorticity distributions at K B = 0.2 and m ∗ = 1 for: Re = a 10, b 1 and c 0.1

deformation for K B < (K B )cr ; and (iii) as flow speed decreases further (Re < 1) the foil undergoes 180◦ static deformation. Figure 16.48 shows the topology of the streamlines and spanwise vorticity distribution behind the deformed foil. Similar to the vortex shedding phenomenon in bluff body flows, we no longer observe any LEV shedding for the inverted foil at a relatively low m ∗ for Re < 50 and there exists two counter-rotating steady vortices for 10 ≤ Re < 50, as shown in Fig. 16.48a. In contrast to the flow around a circular cylinder, the mirror-symmetry concerning the incoming flow is not present in the formation of vortex pair behind the inverted foil. At the flipped state, the steady vortex at the LE no longer exists, as illustrated in Fig. 16.48b, c.

890

16 Isolated Inverted Flapping Foils

16.4.6 Discussion The LAF response of an inverted foil involves large structural deformations with complex interactions between 3D vortical structures shed from the LE and TE and the foil properties such as inertia and elasticity. The underlying physical mechanism behind the LAF phenomenon can be summarized as follows: • The onset of LAF is characterized by the breakdown of symmetry through the divergence instability [596, 623]. Once the foil deformation becomes large enough, the flow separates at the LE to form an LEV behind the foil. • LEV and the foil deformation depend on each other, i.e. the initial deformation of the foil creates an LEV behind the foil and the formation of LEV will result in a low-pressure region behind the foil, which will, in turn, increase the lift, drag and the foil deformation [623]. For high-Re turbulent wake flow, the initial deformation of the foil from the mean position is dominated by the lift and inertial forces until the drag force on the foil becomes large enough to sustain the deformation. On the other hand, for the low-Re laminar flow, the large foil deformation is predominately due to the drag acting on the foil. • Shedding of LEV is necessary but not enough to sustain the LAF response of inverted foil in a uniform flow. To demonstrate this phenomena let us first consider a case where there is no shedding for Re ≤ 20, K B = 0.2 and m ∗ = 1.0. The inverted foil no longer exhibits LAF; instead, an equilibrium between the fluid viscous forces and the structural restoring forces is reached Fig. 16.45a. The second possibility is where we have unsteady vortex shedding, but the combined foil elastic restoring and the inertial effects are not sufficient to overcome the fluid forces acting on it. To demonstrate this case, we present the LE transverse response history at K B = 0.075, Re = 30,000, m ∗ = 1.0 and W/L = 0.5 in Fig. 16.49. The figure shows that the inverted foil no longer exhibits LAF; instead, it performs the flapping motion in a deformed state because the combined elastic and inertial effects due to the foil recoil cannot overcome the fluid forces acting on the foil. A similar deformed flapping mode was observed both experimentally [623] and numerically [596, 692, 710]. • Mass ratio has a weak influence on the transition of LAF to a deformed flapping regime. The 2D simulations of [596] have shown that as the m ∗ increases, the transition from LAF to the deformed flapping can be delayed, i.e., the foil with greater inertia can overcome the flow-induced forces acting against the foil with more ease in comparison to the foil with lower inertia. • Unlike the results presented in Sect. 5.3 wherein the vortex-shedding is suppressed for m ∗ = 1.0 and Re < 50, at a relatively high m ∗ ≥ 5, [590] observed that vortex shedding is not suppressed even for Re = 20 and the inverted foil still exhibits the flapping phenomenon. Interestingly, the vortex-shedding frequency in this regime does not synchronize with the flapping frequency. To further demonstrate the significance of the foil’s elastic and inertial effects on the physical mechanism behind the LAF, we present a simplified analytical model

16.4 Three-Dimensional Flapping Dynamics

891

0

δy /L

-0.2 -0.4 -0.6 -0.8 -1 0

10

20

30

40

tU0 /L

Fig. 16.49 Transverse LE response history (left) and foil profiles over a flapping cycle (right) for the inverted foil without splitter at K B = 0.075, Re = 30,000, m ∗ = 1.0 and W/L = 0.5. Owing to a relatively small elastic recoil at K B = 0.075, the inverted foil does not exhibit LAF albeit unsteady vortex shedding

based on the elastically mounted rigid plate, which is free to undergo single-degreeof-freedom rotation about the TE in a uniform stream in the appendix. In this simplified model, we estimate 2D steady force and moment on the plate at a certain rotation angle by means of a simple potential theory and the mean drag force is acting on the projected length. We assume that the formation and shedding of vortices do not result in a time-dependent moment on the flat plate at a given angle of incidence. The simplified quasi-steady model reasonably predicts the LAF response amplitude and elucidates a direct connection between the large-amplitude oscillation and the LAF response amplitude. It should be noted that the simplified model given in the appendix is not an exact representation of the LAF phenomenon, instead helps us in understanding the impact of structural effects (e.g., elasticity and inertia) on the LAF phenomenon. We next present the linkage between the dynamics of the LAF with the vortex-induced vibration of a circular cylinder. With regard to the underlying fluid-structure interaction and the large amplitude response, the LAF holds both similarities and differences with the VIV of an elastically mounted circular cylinder. Although both configurations are geometrically dissimilar, they exhibit large periodic amplitudes perpendicular to the oncoming flow stream. When an inverted foil deforms, the trajectory of LE forms a circular-arc-like shape around the TE in a given flapping cycle. For the oncoming flow stream, this time-varying geometrical shape can resemble an effective semi-circular body traversing between the two peak transverse locations and emanating two counter-rotating vortices from the leading edge. Figure 16.50 shows the typical schematics of the flow structures exhibited by an elastically mounted circular cylinder, an inverted foil and an inverted foil with a splitter plate. The UVK and LVK represent the upper von Kármán and lower von Kármán vortices formed due to the roll-up of the separated shear layers from the upper and lower cylinder surfaces respectively. The LEV1 and LEV2 in Fig. 16.50b, c denote the LEV shed during the current and previous half-

892

16 Isolated Inverted Flapping Foils Elastically mounted cylinder UVK

LE Elastic foil

LEV1

TEV1

U0 TE TEV

LVK

TEV2

LEV2

(a) (b) Elastic foil

LE LEV1 Fixed splitter plate TE

Small eddies (No TEV)

Thin wake

LEV2

(c) Fig. 16.50 Illustration of typical dominant flow structures during the uniform flow past over: a elastically mounted circular cylinder undergoing VIV with upper von Kármán (UVK) and lower von Kármán (LVK) vortices, b elastic inverted foil performing LAF with LEV and TEV, and c elastic inverted foil clamped at TE to a fixed splitter plate performing LAF with LEV

cycles, respectively. Similarly, the TEV1 and TEV2 in Fig. 16.50b denote the TEV shed during the current and previous half-cycles. Based on the results and observations presented in the above subsections, we can deduce the following differences and similarities: • The fundamental difference between the LAF in an inverted foil without a splitter and the transverse VIV of an elastically mounted cylinder is that the LAF frequency may not necessarily be the same as that of the vortex-induced forces. The VIV of an elastically mounted cylinder synchronizes with the vortex-induced force frequencies [56, 666]. On the contrary, for the LAF at K B = 0.2, Re = 30,000 and m ∗ = 1 the dominant Cl frequency does not synchronize with the flapping frequency in the transverse direction (see Fig. 16.40). • The strong coupling between the UVK and LVK vortices is critical for the selfsustaining trait of the VIV phenomenon [520, 635]. On the other hand, the synchro-

16.4 Three-Dimensional Flapping Dynamics

•

•

•

•

893

nization between the LEV and TEV vortices has an insignificant influence on the LAF phenomenon. Even the complete suppression of the TEV vortex Fig. 16.50c has little influence on the flapping amplitude, which is typically not the case for the VIV of circular cylinders. The LAF dynamics of an inverted foil is a complex interplay of force dynamics associated with the unsteady LE vortex shedding, the foil inertia and the elastic recoil of the flexible structure. While both UVK and LVK vortices of similar strengths play an equal role into the VIV of an elastically mounted cylinder, the LAF of inverted foil is predominantly influenced by the LEV1 and LEV2, as illustrated in Fig. 16.50b, c. The TEV1 and TEV2 have little effect on the flapping amplitude of inverted foil. However, TEV1 and TEV2 can modulate the flapping frequency. The suppression of the vortices from the TE reduces the flapping frequency. The wake behind an elastically mounted cylinder vibrating in the transverse direction Fig. 16.50a and the wake behind a deformable inverted foil with a splitter Fig. 16.50c will look similar if the splitter is ignored. Similar to the VIV of an elastically mounted cylinder, which depends on the synchronization between the upper von Karman (UVK) and the lower von Karman (LVK) vortices, the flapping dynamics of an inverted foil depend on the synchronization between the two leading-edge vortices. The loss due to the absence of hydrodynamic interaction between the two leading-edge vortices is counter-balanced by the foil elastic and inertial effects. Likewise VIV, LAF ceases if we can suppress by shedding of LEV. Furthermore, the shedding of LEV is not sufficient to have a large amplitude flapping if the elastic recoil-induced inertia is not sufficient to counteract the fluid dynamic forces and the inverted foil will no longer exhibit the large amplitude flapping Fig. 16.49. As far as the interaction between LEV1 and TEV1 is concerned, the large amplitude flapping mechanism does not depend on this interaction. However, if present it will enhance the foil recoil motion from the maximum transverse displacement to the mean position. Unlike the VIV of an elastically mounted cylinder, the LAF response in inverted foil has its origin in the coupled dynamics of the flexible body and the LEV formed by the roll-up of the separated shear layer (Kelvin-Helmholtz instability) at the LE of the inverted foil. In comparison to the strongly coupled asymmetric vortices formed via Bénard-von Kármán instability behind circular cylinders, the interaction between LEV and TEV in the inverted foil is relatively weak and the placement of a splitter plate has a marginal effect on the overall flapping response amplitudes. This implies that the LAF of the inverted foil arises from the intrinsic fluid-body interaction between the elastic foil and the vortex shedding process behind the deformed foil, wherein the vortex forces acting on the foil perform a net work, thereby resulting in the energy transfer from the fluid flow in the form of kinetic and strain energies of the foil. One of the similarities between LAF and VIV is that both the physical phenomena require the existence of vortex shedding. Suppression of the shedding ceases the flapping, and the formation of LEV is necessary for the large amplitude deformation.

894

16 Isolated Inverted Flapping Foils

• In both the LAF of an inverted foil with a splitter and the VIV of an elastically mounted circular cylinder, the vortex-induced forces synchronize with the flapping response and exhibit‘2S’ vortex mode. However, the LAF with splitter does not involve any interaction between the counter-rotating vortices LEV1 and LEV2. On the other hand, the interaction between UVK and LVK is necessary for the self-sustained large-amplitude oscillations during VIV. • Finally, for the VIV of an elastically mounted cylinder, the response amplitude and the synchronization range strongly depend on the mass ratio of immersed body in a flowing stream. In the case of LAF, the inverted foil’s mass marginally influences the response regime range [596, 623], flapping frequencies [710] and vortex modes.

16.4.7 Summary The large amplitude limit-cycle flapping of an inverted foil is numerically investigated to elucidate the role of the unsteady counter-rotating vortices shed from the leading and trailing edges of a flexible inverted foil in a uniform flow stream. The coupled 3D fluid-structure interaction formulation relies on the quasi-monolithic formulation with the body-conforming interface and the variational multiscale turbulence model is employed for the separated wake flow at a high Reynolds number. We first explored the detailed flow field around the inverted foil undergoing the LAF and examined the influence of the vortices shed on the response dynamics. From the evolution of 3D separated flow structures, we find that the vortical structures are characterized by a large spanwise vortex and multiple pairs of small counter-rotating streamline vortices behind the foil. Owing to three-dimensional effects, we also observed streamlined ribs along the front side of the deformed foil surface. We have analyzed the relationship between the inverted foil response and the force dynamics. There exists a critical deformation above which the foil deformation is dominated by the drag until the foil recoils due to the elastic restoration forces of the inverted foil. The elastic recoil is followed by the LEV shedding phenomenon. We introduced an inverted foil configuration with a fixed splitter plate at TE to suppress the vortex shedding from the TE and thereby realize the impact of the TEV and the interaction between the TEV-LEV vortex pair. We have identified that unlike the self-excited VIV in circular cylinders, where synchronized periodic vortex shedding from the top and bottom surfaces is essential for the self-sustained response, the large amplitude periodic flapping of inverted foil does not depend on the synchronized periodic vortex shedding from the leading and trailing edges. Notably, the elimination of the TEV via a splitter plate reduces the flapping frequency and modulates the streamwise flapping amplitudes. Instead, the foil recoil motion from the maximum transverse displacement depends on the foil inertia attained due to the periodic vortex shedding from LE and the elastic restoring forces. We also investigated the interaction between an inverted flexible foil without a splitter and low-Re flow to generalize the impact of the LEV shedding on the large-amplitude periodic flapping.

16.4 Three-Dimensional Flapping Dynamics

895

Similar to the flow over the circular cylinder, the vortex shedding phenomenon ceases for Re < 50 and the foil no longer exhibits unsteady flapping motion. However, the foil exhibits a large static deformation that increases with a decrease in Re. The foil eventually flips about the leading edge to align its leading edge along flow for Re ≤ 1. For low-Re, the large foil deformation is primarily sustained by the large static drag acting on the foil. We have shown that the shedding of LEV is necessary but not just enough to sustain the LAF response of inverted foil in a uniform flow. Instead, the LAF response is the outcome of strong fluid-structure interaction associated with the combined effects of unsteady shedding of LEV and the flexible foil’s structural elasticity and inertia effects. Based on the aforementioned investigations, we also presented a list of similarities and differences between the LAF phenomenon of inverted foil and the VIV of a circular cylinder. With the aid of a simplified analytical model, we demonstrated the importance of foil’s elasticity and inertial effects by considering a rigid plate mounted on a torsion spring and a quasi-steady force estimation via potential flow theory and an empirical drag force model. The connection between the foil’s elastic and inertial effects with the LAF response can help to develop an improved understanding of underlying fluid-structure interaction, with significance on energy harvesting and propulsive systems. Acknowledgements Some parts of this Chapter have been taken from the PhD thesis of Pardha Gurugubelli carried out at the National University of Singapore and supported by the Ministry of Education, Singapore.

Appendix: A Simplified Analytical Model for the Effect of Elasticity and Inertia on LAF Phenomenon Here, we attempt to establish the importance of foil’s elasticity and inertial effects on LAF with the aid of a simplified spring-mass mode and an empirical force model. As shown in Fig. 16.51a, the LAF involves complex interactions between the fluiddynamic forces induced due to the periodic shedding of LEV and TEV vortices from the LE and TE of the foil, the nonlinear structural elastic restoring forces and the foil inertia. Figure 16.51b illustrates the schematic of the simplified pendulum-like model undergoing single-degree-of-freedom pitching motion or rotation in a uniform flow U0 , where K θ is the torsional spring constant. In this model, we have simplified the complex nonlinear structural deformations of inverted foil by assuming the flexible foil as a rigid plate of identical mass and the elastic nature of the foil is modeled through an equivalent linear torsional spring. The fluid forces acting on the plate are represented by potential flow lift and the empirical drag based on the projected plate height. The equation of motion for the nonlinear pendulum (i.e., elastically mounted rotating plate) for the rotation angle (θ s ) can be written as

896

16 Isolated Inverted Flapping Foils FN

LEV

LE O U0

θs Kθ

TE

(a)

(b)

Fig. 16.51 a Spanwise vorticity contours during LAF motion from 3D FSI simulation of an inverted foil without splitter, b sketch of simplified nonlinear pendulum model consisting of a rigid plate mounted on a torsional spring at TE and immersed in a uniform flow stream at freestream velocity U0 . K θ is the torsion spring constant and θ s is twist angle from the initial equilibrium state. In (a), LEV and TEV vortical structures during the LAF motion can be clearly seen

(I + Ia )θtts + Cθ θts + K θ θ s = Mzf ,

(16.28)

where θtts and θts represent ∂ 2 θ s /∂t 2 and ∂θ s /∂t respectively, Cθ is the torsional damping constant, I = m L 2 /3 and Ia = 9πρ f L 4 /128 [533, 698] are the moment of inertial and added moment of inertia, respectively for the rigid plate rotating about LE, and m is the mass per unit length. Mzf denotes the steady moment acting on the plate due to the fluid loading FN acting normal to the plate at the aerodynamic center O, where the pitching moment does not vary as a function of twist angle. By considering a quasi-static loading on the rotating plate, the normal force via the potential flow model and the force decomposition assumption can be given as FN =

  1 f 2 ρ U0 L(Cd sin θ s sin θ s  + Cl cos θ s ), 2

(16.29)

where Cl = 2π sin (θ s ) is the potential flow lift for an inclined flat plate in a uniform flow at the angle of incidence θ s and Cd = 1.28 represents the drag coefficient of a flat plate in a cross flow. In Eq. (16.29), the terms cos θ s and sin θ s appear due the geometric projection of the lift and drag force normal to the plate. The term |sin θ s | projects the inclined plate length in the direction normal to the flow and the absolute operator accounts for the sign of projected length. In the above analytical form, the potential flow is based on the assumption that the flow remains attached even for a large rotation angle. Using the Lighthill’s force decomposition, we also assume that drag force and the inviscid inertia force can be treated independently. For small deformations, the inverted foil behaves as a linear torsional spring and exhibits a constant K θ in that range. The equivalent linear torsional constant (K θ )

16.4 Three-Dimensional Flapping Dynamics

897 2

ΔMzf

(ΔMzf )

Δα

0

0.02

0.04

Δα

0.06

/Δα

50

1.5

ΔMzf

α

100

ΔMzf

× 1000

150

1

≈ 1.2

0.5 0.08

0.1

0.05

0.1

0.15

0.2

α

KB

(a)

(b)

0.25

0.3

0.35

 ∗ Fig. 16.52 a Variation of nondimensional moment Mzf = Mzf /(ρ f U02 L 2 W ) acting on inverted ∗ foil with respect to the foil rotation angle α for tU0 /L ∗ ∈ [0, 7] at K B = 0.2, Re = 3000, m = 1 and W/L = 0.5, and b relationship between Δ Mzf /Δα and K B for Re = 30,000, m ∗ = 1 and W/L = 0.5. The dashed line represents the best fit line given by Eq. (16.4.7)

corresponding to the inverted foil with K B = 0.2 is computed  ∗ from the FSI simulations by plotting the nondimensional torsional moment Mzf = Mzf /(ρ f U02 L 2 W ) acting on the foil as a function of the foil LE angle (α) over tU0 /L ∈  of rotation  [0, 7] in Fig. 16.52a. Here, α is defined as tan−1 δ y /δx and we consider repreRe = 30,000, m ∗ = 1.0 and W/L = 0.5. The slope sentative   f ∗values:  K Bf =2 0.2, 2 Δ Mz /Δα × ρ U0 L W in Fig. 16.52a defines the torsional spring constant K θ .  ∗ Figure 16.52b summarizes the Δ Mzf /Δα values for different K B and the dashed  ∗ line represents the linear correlation between K B and Δ Mzf /Δα, which is given by  ∗ Δ Mzf = a K B + b, Δα where a = 4 and b = 0.4. In Fig. 16.53a, we compare the time traces of the LE transverse displacement by solving the Eq. 16.29 for K θ = 1200Nm2 /rad and I = 333.33 kg m2 against the equivalent full-scale 3D simulations at K B = 0.2 and m ∗ = 1.0. We have considered ρ f = 1000 kg/m3 , L = 1m and U0 = 1m/s for the analytical model. For comparison purposes, we have also included with-splitter case for the identical nondimensional parameters. The elastically mounted plate rotates to a large pitch angle and there is a continuous interplay between the fluid force, the inertia and the restoring torsional spring force. A small damping coefficient of 0.001 has been considered for the analytical model of the elastically mounted plate. The figure shows that the response dynamics simplified analytical model qualitatively represents the complex LAF kinematics and the flapping frequency of the simple analogous model lies in between the flapping frequency of the inverted foil with and without a splitter. In Fig. 16.53b, we compare the maximum transverse amplitude of the LE as a function of K B obtained from the analytical model against the 3D FSI simulations at m ∗ = 1.0 and Re = 30,000. We have also included 2D simulations at m ∗ = 0.1 and Re = 1000 presented in [596]. The simplified model exhibits greater

898

16 Isolated Inverted Flapping Foils 2 analytical model without splitter with splitter

1.5

current 3D simulations 2D simulations analytical model

2.5 2

δy /L

1 1.5 0.5 1

0

0.5

-0.5

0

-1 0

0.5T

1T

(a)

1.5T

2T

0.1

0.2

0.3

0.4

0.5

0.6

(b)

Fig. 16.53 a Evolution of LE transverse displacement for nonlinear analytical pendulum model mounted on a torsional spring at K θ = 1200 Nm2 /rad and Iθ = 333.33 kg m2 and 3D FSI simulations of inverted foil with and without splitter at nondimensional parameters K B = 0.2, Re = 30,000 and m ∗ = 1.0 b comparison the maximum transverse amplitude as a function of K B for the simplified nonlinear pendulum model against 3D FSI simulations at m ∗ = 1.0, Re = 30,000 and W/L = 0.5, and 2D simulations from [596] at m ∗ = 0.1 and Re = 1000

maximum transverse amplitudes because a flexible inverted foil needs to bend about the leading edge, as shown in Fig. 16.50a, whereas the rigid plate in the simplified model just rotates about the TE. Overall, the semi-analytical model with just the torsional elasticity and inertial exhibits the characteristics of the LAF phenomenon observed in an inverted foil. The above analysis confirms the importance of foil’s elastic and inertial effects on the LAF response of a flexible inverted foil in a uniform flow stream.

Chapter 17

Thin Structure Aeroelasticity

This chapter examines the coupled dynamics of two-dimensional flexible thin structures with membrane components from low to very high angles of attack. Of particular interest is to explore the role of flexibility in the variation of aerodynamic performance and structural dynamics. We further extend the studies to 3D wings to examine the 3D effect on coupled dynamics.

17.1 Introduction In nature, flying animals consisting of various wing structures usually exhibit superb flying skills with high maneuverability and efficient performance. Based on the wing structures and their characteristics in biological flight, flying animals can be divided into three specific groups, namely insects, birds, and bat-like flying mammals. These flying animals can actively alter the wing shapes and structures through muscles and bones or adapt to aerodynamic loads to deform passively, the so-called morphing wings. The biological morphing wings possess varying degrees of flexibility, which play an essential role in biological flight. In this chapter, our study on morphing wings is mainly motivated by bat wings with membrane components. Morphing wing structures are the keys to the efficient flight of flying animals. Recent works carried out for morphing wings with membrane components are reviewed in the following subsections.

17.1.1 Review of Studies on Morphing Membrane Wings The term “morph” in a biological morphing wing is defined according to its ability to change wing geometry and material properties actively or passively. Various types of morphing wings can be observed in flying animals. The key feature of the morphing © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Jaiman et al., Mechanics of Flow-Induced Vibration, https://doi.org/10.1007/978-981-19-8578-2_17

899

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17 Thin Structure Aeroelasticity

wing is that the surrounding flow characteristics can be changed to meet different aerodynamic performance requirements by altering the wing shape. For example, the flying animals can experience a broad range of Reynolds numbers ranging from 0 up to 105 to perform take-off, gliding, hovering, diving and hunting as well as landing. By contrast, specific traditional fixed wings in human-made air vehicles exhibit poorer adaption to a wide range of flight envelopes. Although the human-made fixed wing can also operate within the Reynolds number range of Re ∈ [0, 106 ], the optimal wing shape with the best aerodynamic performance can be just achieved at a small flight range, generally in the cruise phase. To take care of the aerodynamic performance at low flight speed and high angles of attack during take-off, landing and stall maneuver, some auxiliary morphing mechanisms (e.g., slat, flap and variable sweep mechanism) are designed to improve the adaption of the fixed wing to various flight environments. For example, the famous Grumman F-14 Tomcat can change the wing sweep and improve the corresponding aerodynamic performance according to its combat mission. However, these humanmade morphing mechanisms are too complex and occupy additional space and weight of the air vehicles. This problem is more prominent in small-scale air vehicles and MAVs. Due to the strong viscous effect at the low Reynolds number range, these air vehicles usually need special wing structures and moving kinematics to provide enough aerodynamic forces to overcome drag and gravity. Meanwhile, the wing structures should be light to improve flight energy usage rate. The morphing wings of bats provide a good example for engineers and designers to learn from. Bats are the only mammals that can fly stably and possess a special flight mode with efficient aerodynamic performance. Compared with insects and birds, bats have a pair of high degrees-of-freedom membrane-like wings consisting of elastic fibers, muscles, blood vessels and connective tissues. The highly-controllable skeletons with joints can support and stretch the membrane during flight to adapt to the environment. The combination of skeletons and membranes could greatly reduce the wing’s weight while providing sufficient flight power. This type of membrane wing can achieve good aerodynamic performance both for non-flapping and flapping motions. These advantages offer a potential way to explore new design methods for MAVs with lightweight and high aerodynamic efficiency. Recently, the investigations of bat flight [607, 761] and fixed-wing MAVs with membrane components [510, 612, 640, 688, 766] have been carried out. The results demonstrated that the membranelike wing deformed compliantly under aerodynamic loads to generate a higher liftto-drag ratio and suppressed separated flows to some extent by adjusting the wing camber. Without loss of generality, different types of membrane structures can be simplified into simple membrane models with supporting frames, as shown in Fig. 17.1. Three key non-dimensional physical parameters, namely aeroelastic number Ae, structure-to-fluid mass ratio m ∗ and Reynolds number Re, govern the dynamic responses of the flexible membrane, which are defined as

17.1 Introduction

901 U∞ Z O

Frame

TV

X

LEV

Shear layer Travelling wave

U∞

φωx

α

Membrane

Boundary reflection

TEV

(a)

(b)

Fig. 17.1 Flow past a flexible membrane wing: a schematic of fluid-membrane interaction, b illustration of three-dimensional membrane wing with free-stream velocity U∞ and a given angle of attack α. In (a), while the red dashed line (- - -) indicates the time-averaged membrane shape, the black solid line (—) represents the instantaneous membrane shape. In (b), φωx denotes the angle between the membrane and the core of the tip vortex. The red and blue contours are the positive and negative membrane displacements, respectively

Ae =

Esh ρs h ∗ , = , m 1 f 2 ρfc ρ U∞ c 2

Re =

ρ f U∞ c . μf

(17.1)

where E s is Young’s modulus and h is the membrane thickness. c denotes the character length of the membrane. ρ f and ρ s represent the density of the fluid and the membrane, respectively. U∞ is the free-stream velocity and μ f denotes the fluid’s dynamic viscosity. The aeroelastic number represents the balance between the aerodynamic loads and the membrane tension [760]. Structure-to-fluid mass ratio defines the inertial ratio between structure and fluid. Reynolds number is the ratio between inertial forces and viscous forces of moving fluid. When the unsteady fluid flows past through the morphing membrane, the membrane can deform passively under pressure difference or vibrate through flow-excited instability. As shown in Fig. 17.1 b, when the 3D effect is taken into account, the leading edge vortex (LEV), the trailing edge vortex (TEV) and the tip vortex (TV) can induce complex and rich vibrational modes along both chordwise and spanwise directions. The flow-induced deformation and the flow-induced vibration can in turn, change the flow features and pressure distributions around the membrane, thereby affecting the aerodynamic performance. These simplified membrane models can help us gain further insight into the fluid-membrane coupling mechanism, which can be extended to the physical understanding of the real morphing membrane wings. During the past decades, a series of wind tunnel experimental studies were performed for 2D membrane wings [689, 690] and 3D membrane wings [688, 691, 748–751] in Professor Gusul’s research group in the University of Bath. The purpose is to further tap the potential of the morphing membrane structures in improving aerodynamic performance. In these experimental studies, the 2D membrane model was fixed at the leading edge (LE) and the trailing edge (TE). The 3D membrane

902

17 Thin Structure Aeroelasticity

model made of rubber latex was glued to a rigid frame to support the flexible structures. The coupled dynamics were measured over a wide range of AOAs up to stall conditions. It was found that the membrane deformed and started to vibrate when the AOA exceeded a specific critical value. Once the membrane vibration was excited, as the AOA increased, the dominant structural mode and the corresponding vibration frequency transitioned from one type to another type. They observed that a chordwise second mode was always excited at high AOAs with significant vortex shedding phenomenon. The reason was attributed to the tight coupling with the von Karman vortex shedding phenomenon in the wake. By examining the location of the shear layer fluctuation as a function of AOA, the distance between the high fluctuation region became far away from the membrane surface at higher AOAs, leading to weaker membrane vibrations. The flexible membrane can deform and reduce the size of the separation and recirculation region compared to its rigid counterpart. These conclusions were consistent for the 2D and 3D membrane wings. When the 3D effect is taken into account, the interaction between the generated tip vortices and the 3D membrane wing can result in more complicated membrane vibrations compared to the 2D membrane wing. The aerodynamic characteristics and the membrane vibrations of rectangular and triangle membranes exhibited significant differences. Once the flexible membrane was actuated with a prescribed motion in a gusty condition, the flexible membrane experienced the mode transition phenomenon during the pitching motion. The frequency of the self-excited roll vibrations can be synchronized with the actuated pitching frequency at some specific conditions through fluid-membrane interaction, resulting in a frequency lock-in phenomenon. Almost during the same period, some experimental [583, 721] and theoretical [721, 753, 760] studies as well as active control works [534, 554] were carried out in the Brown University. These works were mainly inspired by long-term research on bat flight problems. Both 2D and 3D membrane models were considered in these research works. Song et al. [721] observed the mode transition phenomenon as the AOA and Reynolds numbers changed. The explanation of the mode transition phenomenon still remains a puzzle. They thought that the flexible membrane could react to the pressure pulsations caused by the vortex shedding process to vibrate as a resonator. Meanwhile, they stated that the membrane vibration was strongly related to the eigenmodes and the natural frequency of the flexible membrane stretched by the aerodynamic loads. However, the natural frequency estimated in [721] was limited to a simple linear model, which neglected the geometric nonlinearity at large deformation and the dynamic stress due to membrane vibration. Besides, they found that the aerodynamic performance could be improved when certain pretension was applied to the flexible membrane. In the following theoretical work [760], a self-consistent theoretical framework with a nonlinear membrane model was developed to predict the membrane deformation. Tzezana et al. [753] investigated the thrust generation and the drag-thrust transition of flapping membrane structures based on a theoretical framework. The resonance between the flapping frequency and the natural frequency of the flexible membrane was found to govern the transition between thrust and drag. A series of wind tunnel experiments have been performed at the University of Southampton to investigate the ground effect [530, 532] and the aspect ratio effect

17.1 Introduction

903

[529] on the aerodynamic performance of morphing membrane wings. Bleischwitz et al. [529] reported that the lift can be improved by 57% at moderate AOAs due to the camber effect and the passive shape adaption in contrast to the rigid wings. A larger aspect ratio of the flexible membrane can further improve the lift. A freeto-rotate boundary condition at the LE and TE can improve the lift by 30–35%, compared to a clamped boundary condition. Through the studies on the ground effect, the aerodynamic performance can be improved for both rigid and flexible wings at low to moderate AOAs. The ground effect can be equivalent to a modification of freestream angle, which triggers earlier leading-edge flow separation to improve lift. Besides, they suggested that the fluid-membrane interaction in separated flows can be controlled by tightening or relaxing the morphing membrane structure to change its natural frequency. In parallel to experimental research, some numerical studies have been carried out during the past decades. An earlier physical investigation on morphing membrane structures originates from the study on the aerodynamic performance and luffing phenomenon of boat sails, which can be dated back to 1950–1960s [669, 741, 759]. These studies considered an inviscid steady flow model coupled with a linear membrane model with slight camber and incidence. To better predict the coupled membrane dynamics, Le Maitre et al. [636] developed an inviscid unsteady flow model coupled with a linear membrane model to examine the effect of harmonic perturbations of the trailing edge on the sail dynamics. Initial efforts to couple the incompressible Navier-Stokes equations with the flexible membrane dynamics were reported in [715, 718]. This series of research works inspired the following works on the application of fixed membrane wings for various MAVs, which were reviewed in [640, 724]. Recently, a series of studies o 2D and 3D membrane dynamics based on numerical simulations were carried out. Several aeroelastic numerical solvers were developed to capture the flow features and the membrane responses by coupling the NavierStokes equations with the membrane models [588, 703, 728]. As reported in [587, 588], the effects of AOA, membrane flexibility, pretension and Reynolds number on the flow features and the membrane responses were investigated. Sun et al. [726– 729] performed a series of numerical simulations to investigate the evolution of the nonlinear dynamics and bifurcations of the flow-induced vibration of a 2D membrane. The onset of the flow-induced vibration, the mode transition phenomenon and the evolution to chaos were observed. In [704], the effect of the leading and trailing edge shapes on the coupled dynamics of a 2D membrane was examined. It was found that an aerodynamic shape of the mounts can improve the lift of the membrane. The mechanism of aerodynamic efficiency reduction of flexible membrane wings at a low AOA of α = 8◦ was explored in [705]. It was found that the introduced upstreampropagating pressure waves in the coupled fluid-membrane system cause decreased aerodynamic efficiency.

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17 Thin Structure Aeroelasticity

17.1.2 Organization To begin, we construct simplified 2D membrane wing models to understand the role of flexibility in the coupled dynamics in Sect. 17.3. Although the dynamic behaviors of a simplified model and a real biological wing have some differences, the fundamental physics and the underlying mechanism are universal. We consider a 2D flexible membrane model with fixed leading and trailing edges immersed in unsteady flows. The coupled fluid-membrane system is simulated over a wide range of AOAs of [4◦ , 90◦ ]. The purpose is to understand how the flexible membrane responds to the unsteady surrounding flows via the coupling effect at various AOAs and Reynolds numbers. The flexibility effect is examined by comparing the aerodynamic characteristics to a rigid flat wing and a rigid cambered wing. This 2D coupled system can be a good entry point for us to grasp the key features of fluid-membrane interaction problems. Furthermore, the studies on the 2D membrane can be the cornerstone of further research on more complex 3D morphing and flapping wing systems. To generalize our findings to a broader range, we further a rectangular membrane immersed in uniform fluid flows at moderate and high AOAs with separated flows at moderate Reynolds numbers in Sect. 17.4. To reduce the computational cost while achieving a reasonable accuracy for a separated flow, the hybrid RANS/LES turbulence model based on the delayed detached eddy simulation treatment is employed to capture the coupled dynamics in the separated flows.

17.2 Partitioned Coupled Fluid-Structure Formulation The governing equations for an incompressible viscous fluid are unsteady Reynolds averaged Navier-Stokes equations with the DDES model given by  ∂ u f  ρ + ρ f (u f − um )] · ∇u f = ∇ · σ f + ∇ · σ des + b f on Ω f (t) ∂t xˆ f (17.2) f

∇ · u f = 0 on Ω f (t)

(17.3)

where ρ f is the density of the fluid and b f is the body force applied on the fluid. u f = u f (x f , t) and um = um (x f , t) represent the fluid and mesh velocities defined for each spatial fluid point x f ∈ Ω f (t) respectively. x f and t is the spatial and temporal coordinates. σ f is the Cauchy stress tensor for a Newtonian fluid, which can be written as σ f = − p f I + μ f (∇u f + (∇u f )T )

(17.4)

17.2 Partitioned Coupled Fluid-Structure Formulation

905

where p f is the time averaged fluid pressure and μ f represents the dynamic viscosity of the fluid. The first term in Eq.(17.2) is the partial derivative of u f with respect to time and the ALE referential coordinates xˆ f is always fixed. The turbulent stress term σ des in Eq. (1) is modeled using the Boussinesq approximation given by f

σ des = μT (∇u f + (∇u f )T ) f

(17.5) f

where μT is the turbulent dynamic viscosity defined as μT = νT ρ f . The turbulent viscosity νT solved from the Spalart-Allmaras one-equation turbulent model is defined as νT = ν˜ f v1 ,

f v1 =

χ˜ 3 ν˜ , χ˜ = 3 3 ν χ˜ + cv1

(17.6)

where ν = μρ f represents the molecular viscosity and ν˜ can be determined by the transport equation f

 2 ν˜ ∂ ν˜ f m 2 ˜2 + (u − u ) · ∇ ν˜ = cb1 (Sv + (˜ν /(κ d )) f v2 )˜ν − cw1 f w ∂t d˜ cb2 1 (∇ ν˜ )2 + ∇ · [(ν + ν˜ )∇ ν˜ ] + σ σ

(17.7)

where cb1 , cb2 , σ , cw1 , cv1 and κ are the constants defined for the Spalart-Allmaras model in [722]. Sv represents the magnitude of vorticity. d˜ is defined as the distance to the closest wall, which can be used to adjust RANS mode in the attached boundary layer region and LES mode in the separated flow region [723]. The NavierStokes equations are discretized by the Petrov-Galerkin finite element method and the generalized-α scheme is used to update these flow variables in time. The detailed descriptions of the variational formulation for the fluid equations can be found in [618]. The governing equation for a flexible multibody Ωis is given as ρs

∂2 ds s (x , t) + ∇ · σ s (E(d s )) = bs ∀x s ∈ Ωis ∂t 2

(17.8)

where x s is the Lagrangian material points in curvilinear coordinate system and i is the i th structural component in the flexible multibody system. Here ρ s represents the structural density and d s is the structural displacement. σ s is defined as the first Piola-Kirchhoff stress tensor and bs denotes the body force on the multibody Ωis . The Cauchy-Green Lagrangian strain tensor E(d s ) can be written as E(d s ) =

1 [(I + ∇d s )T (I + ∇d s ) − I] 2

(17.9)

906

17 Thin Structure Aeroelasticity

The constraint equation of the kinematic joints used to connect different parts in a flexible multibody system can be written as c(d s ) = 0

(17.10)

The governing equation is solved by a time discontinuous Galerkin method and an energy decaying scheme is employed to achieve numerical stability. Further details can be found in [595]. A brief description of coupling interface conditions for a 3D aeroelastic problem with a multibody system is presented. These conditions should be satisfied to ensure the fluid velocity is exactly equal to the velocity of the flexible multibody along the interface and the motion of the flexible multibody can be determined by the fluid forces, which contains shear stress effects on the structure surfaces and the fs integration of pressure. Γi = Ω f (0) ∩ Ωis is defined as the fluid-flexible interface fs fs for the i th part at t = 0 and Γi (t) = ϕ s (Γi , t) is defined as the interface at time fs t. The interface boundary conditions on each interface Γi can be written as fs

 ϕ s (γ ,t)

u f (ϕ s (x s , t), t) = us (x s , t) ∀x s ∈ Γi  f f σ (x , t) · nd + t s d = 0 ∀γ ∈ ifs

(17.11) (17.12)

γ

where ϕ s is the position vector which maps the initial position x s of the flexible multibody to its position at time t and ϕ s = x s + d s (x s , t). t s is the fluid traction on the body and us = ∂ϕ s /∂t is defined as the structural velocity at time t. n is the outer normal to the fluid flexible multibody interface, γ represents any part of the fs interface Γi and ϕ s (γ , t) denotes the associated fluid domain at time t. f n u(x , t ), p(x f , t n ) and ν˜ ( x f , t n ) are the fluid variables at time t n , while s d (x s , t n ) denotes the structural displacement caused by aerodynamic load at time t n . The NIFC-based multibody aeroelastic coupling procedure can be considered as a typical predictor-corrector process by coupling the predicted structural displacements and the corrected aerodynamic forces in an iterative manner. In the first step, the flexible multibody structural equations are solved under the action of aerodynamic forces to obtain the structural displacement for each node. In the second step, the solved structural displacements are interpolated to fluid surface nodes via the RBF method while satisfying the interface boundary conditions. In order to satisfy the displacement consistency between the two domains, the structural and fluid mesh configurations must have no gap or overlap. The displacement equation is shown as d m,n+1 = d s on Γ

fs

(17.13)

where d m,n+1 is the aeroelastic interface mesh displacement at time t n+1 . Furthermore, the continuity condition of velocity along aeroelastic interface should be satisfied by equating velocity on fluid surface node with interface mesh velocity at f temporal location t n+α :

17.3 Two-Dimensional Thin Structure Aeroelasticity f

u f,n+α = um,n+α

907 f

(17.14)

whereby the mesh velocity can be solved as f

um,n+α =

d m,n+1 − d m,n on Γ Δt

fs

.

(17.15)

In the third step, the aerodynamic forces are solved from the Navier-Stokes equations in the ALE form and the closure DDES turbulent model. In the last step, the NIFC scheme is used to correct the solved aerodynamic forces via a successive approximation approach to achieve numerical stability and the corrected forces are interpolated through the interface to the multibody structural solver based on the RBF method at each sub-iteration step k. These four steps are the main procedures for one aeroelastic sub-iteration. In order to achieve iteration convergence, such a sub-iteration procedure will continue to be performed until the criterion has been met. Once the sub-iteration steps are completed, the aeroelastic coupling simulation will update to the next time step t n+1 to repeat the same procedures again until the final aeroelastic simulation convergence criterion has been achieved. In this numerical simulation framework, the Generalized Minimal RESidual (GMRES) algorithm proposed by Saad et al. [693] is used to calculate the incremental velocity, pressure, and eddy viscosity in the finite element discretized equations for the fluid system. The Krylov subspace iteration and the modified Gram-Schmidt orthogonalization approaches are implemented in this algorithm and a Krylov space with 30 orthonormal vectors is adopted to solve the fully coupled matrix form comprising the ALE flow and the turbulent model. The Newton-Raphson iteration scheme is employed at each time step in the framework to minimize the linearization error and the generalized-α approach is used to update the variables in the flow field. The multibody structural equations discretized by the time discontinuous Galerkin method are solved by a classical skyline solver based on the factorization of the system matrix. Considering the high computational costs for numerical simulation with a large mesh size, parallel computing based on the message passing interface (MPI) technique is employed to speed up the numerical simulation.

17.3 Two-Dimensional Thin Structure Aeroelasticity 17.3.1 Problem Setup and Mesh Convergence Study In this section, the description of the 2D coupled fluid-membrane system is introduced. A mesh convergence study is performed to select proper fluid and structural meshes with sufficient mesh resolution for this 2D coupled system. A flexible membrane wing geometry model employed in the experimental study [689] is considered as the numerical simulation model to investigate the coupled fluid-

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17 Thin Structure Aeroelasticity

Rigid mount

Rigid mount

Flexible membrane

c Deformed

T

U∞

L

2r

δnmax Undeformed

h

α

Z O

X

T

d

Fig. 17.2 Schematic of the 2D flexible membrane wing geometry and the details of leading and trailing edge mounts

membrane dynamics. In the experiment, this flexible membrane wing made of a black latex rubber sheet with a thickness of h = 0.2 mm is attached to the rigid airfoilshaped mount at the leading and trailing edges. The schematic of this membrane wing geometry and the fixed mounts is presented in Fig. 17.2. The chord length of the membrane wing is c = 136.5 mm with a span length of b = 450 mm. The rigid mount consists of a triangle part with a length of d = 5 mm and circle support with a diameter of 2r = 1.5 mm, which results in a total length of L = 150 mm for the wing. In the current numerical study, the freestream velocity is set to U∞ = 0.2886 m/s and the air density is chosen as ρ f = 1.1767 kg/m3 . The flexible membrane has Young’s modulus of E s = 3346 Pa and the structure density of ρ s = 473 kg/m3 . The structure-to-fluid mass ratio is m ∗ = ρ s h/ρ f c = 0.589 and the aeroelastic number is 2 c) = 100.04. Ae = E s h/( 21 ρ f U∞ The schematic of the 2D computational fluid domain is displayed in Fig. 17.3. The flexible membrane wing is clamped at the leading and trailing edges to form an angle of attack of α between the uniform freestream and the initial chord directions. The distance between the inlet (Γin ) and outlet (Γout ) boundaries is 20L and the same size is set for the top (Γtop ) and bottom (Γbottom ) sides of the computational domain. A uniform freestream velocity U∞ is applied along the inlet boundary and the tractionfree boundary condition is employed at the outlet boundary. The slip-wall boundary condition is implemented at both sides of the computational domain and the no-slip wall condition is exposed at the surface of the membrane wing. The LES model is considered here to capture the unsteady flow features around the flexible membrane at low Reynolds numbers. The mesh convergence study is performed to choose an appropriate mesh resolution for the numerical simulation of the flexible membrane wing. Three different meshes M1, M2 and M3 are well designed to discrete the computational fluid domain by 50 616, 87 278 and 135 624 triangular elements, respectively. The boundary layer

17.3 Two-Dimensional Thin Structure Aeroelasticity

909

Fig. 17.3 Schematic of the computational domain for a uniform flow past a 2D flexible membrane wing at low Reynolds number with α = 8◦

mesh is built to maintain y + < 1 with a stretching ratio of 1.15 for these three meshes. The structural model of the flexible membrane is discretized by 30, 50 and 70 structured four-node rectangular finite elements along the chord direction and one element in the span direction. This multibody structural model consists of the rigid mounts at the leading as well as the trailing edges and the flexible membrane component, which are simulated by the rigid body elements and the geometrically exact co-rotational shell elements, respectively. A clamp condition is applied to the rigid mounts and the flexible membrane is attached to the mounts. There is no pretension applied to this 2D flexible membrane. The non-dimensional time step size, ΔtU∞ /c is set as 0.00423 in the simulation. The mean membrane deformation, the forces acting on the membrane wing and the pressure distribution on the wing surface are evaluated from the simulation data. The air load is calculated by integrating the surface traction taking into account the first layer of elements on the membrane surface. The instantaneous lift, drag and normal force coefficients are defined as  1 f (σˆ · n) · nz d (17.16) CL = 1 f 2 ρ U S Γ 2 CD =

1 1 f 2 ρ U S 2

 f

Γ

(σˆ · n) · nx d

(17.17)

910

17 Thin Structure Aeroelasticity

Table 17.1 Mesh convergence of a 2D flexible membrane wing at Re = 2500 and α = 8◦ Mesh CL CD C L /C D Present (M1) Present (M2) Present (M3)

0.9793 (−1.54%) 0.9948 (0.02%) 0.9946

0.1114 (1.09%) 0.1097 (0.45%) 0.1102

8.7895 (−2.62%) 9.0684 (0.47%) 9.0257

The percentage differences are calculated by using M3 results as the reference -2

0.1

M1 M2

0.08

M3 Sun et al.

M1

0.04

0

M2 M3

0.02 0

Cp

δ¯n /c

-1 0.06

Sun et al.

0

0.2

0.4

0.6

0.8

1

1

0

0.2

0.4

ξ/c

ξ/c

(a)

(b)

0.6

0.8

1

Fig. 17.4 Comparison of membrane dynamics between different meshes at Re = 2500 and α = 8◦ for: a time-averaged normalized membrane surface displacement normal to the chord, b timeaveraged pressure coefficient distribution on the membrane surface

Cn =

1 1 f 2 ρ U S 2

 f

Γ

(σˆ · n) · nc d

(17.18)

where nx and nz are the Cartesian components of the unit normal n to the membrane f surface and nc is the unit normal to the chord line. σˆ is the fluid stress tensor with Γ being the surface boundary of the membrane. The pressure coefficient is defined as p − p∞ (17.19) Cp = 1 f 2 ρ U∞ 2 where p and p∞ are the pressure at the concerned point and the pressure at the far-field, respectively. A comparison of the mean lift coefficient, the mean drag coefficient, and the mean lift-to-drag ratio for three different meshes are summarized in Table 17.1. The percentage differences for M1 and M2 are calculated with respect to M3. It can be observed that the absolute differences in the time-averaged drag coefficient and the lift-to-drag ratio between M1 and M3 are greater than 2%, and the differences for all values between M2 and M3 are less than 1%. For further comparison, the timeaveraged normalized membrane displacement normal to the chord and the mean pressure coefficient distribution on the surface is evaluated, which are shown in Fig. 17.4. The mean membrane displacement and the mean pressure coefficient are converged when the mesh is refined to M2. Hence, the mesh M2 is adequate and

17.3 Two-Dimensional Thin Structure Aeroelasticity

911

will be considered for further numerical study. The membrane dynamics are further compared with the numerical results obtained from [726] for verification purposes and a good agreement is observed between them.

17.3.2 Membrane Dynamics as a Function of Angle of Attack In this section, the time-averaged membrane dynamics and the unsteady membrane dynamics from low to very large AOAs of α ∈ [4◦ , 90◦ ] are investigated. To examine how the flexible membrane responds as the AOA changes, we also simulate a rigid flat wing and a rigid cambered wing for comparison purposes. The rigid flat wing has the identical geometry to the undeformed membrane wing. The rigid cambered wing model is constructed based on the mean shape of the flexible membrane immersed in unsteady flows. We analyze the flexible membrane dynamics as a function of the AOA in this section. In Sect. 17.3.3, the role of flexibility will be discussed in detail by comparing these three wings.

17.3.2.1

Time-Averaged Membrane Dynamics

Fig. 17.5 presents the comparison of the time-averaged aerodynamic forces of a rigid flat wing, a rigid cambered wing and a flexible membrane at different AOAs. In this section, we focus on the discussion of the coupled dynamics of the flexible membrane as a function of AOA. It can be observed that the time-averaged lift coefficient grows up rapidly when the AOA increases to 12◦ . Then, the mean lift coefficient increases slowly to the maximum value at α = 45◦ and decreases continuously to nearly zero at α = 90◦ . The mean drag coefficient keeps growing as the AOA increases. The mean normal force coefficient increases continuously and changes slightly when the AOA is larger than 45◦ . The flexible membrane achieves the best mean lift-to-drag ratio at α = 8◦ and then decreases to nearly zero at α = 90◦ . We further examine the time-averaged membrane displacement normal to the chord and the time-averaged pressure coefficient difference between the upper and low surfaces along the membrane chord. Figure 17.6 plots the contour of the timeaveraged coupled dynamics at different AOAs. It can be seen that the maximum membrane displacement increases quickly when the AOA changes from 4◦ to 45◦ . As the AOA further increases, the variation of the mean membrane displacement becomes somewhat saturated. The pressure coefficient difference is directly related to the normal force. The negative pressure coefficient difference means a suction force. We notice that the large suction area expands from the LE to the TE significantly as the AOA increases from 4◦ to 45◦ . Then, the pressure coefficient difference distribution exhibits a slight change as the AOA further increases to the vertical state.

912

17 Thin Structure Aeroelasticity 3

4 3

CD

CL

2 1 Rigid camber

0

Flexible

20

40

Rigid flat

1

Rigid flat

0

2 Rigid camber Flexible

0 60

α (◦ )

80

100

0

20

40

10

3

8

2 Rigid flat

1

Rigid camber

0

20

40

◦60

α()

100

80

Rigid flat Rigid camber

6

Flexible

4 2

Flexible

0

80

(b)

4

CL /CD

CN

(a)

60

α (◦ )

100

(c)

0 0

20

40

60

α (◦ )

80

100

(d)

Fig. 17.5 Comparison among a rigid flat wing, a rigid cambered wing and a flexible membrane at various AOAs for a time-averaged lift coefficient, b time-averaged drag coefficient, c time-averaged normal force coefficient and d time-averaged lift-to-drag ratio

(a)

(b)

Fig. 17.6 Contour of a time-averaged membrane displacement normal to the chord and b timeaveraged pressure coefficient difference between the upper and lower surfaces along the membrane chord at different AOAs

17.3.2.2

Unsteady Membrane Dynamics

Figure 17.7 presents the contour of the time history of the membrane displacement fluctuations along the membrane chord at six selected AOAs. The first case is selected at α = 12◦ when the flexible membrane exhibits a large lift slope. We consider a typical case at α = 20◦ to represent the light stall condition. Two cases at α = 35◦ and 45◦ correspond to the further improvement of the lift conditions. The case at

17.3 Two-Dimensional Thin Structure Aeroelasticity

913

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 17.7 Contour of time history of membrane displacement fluctuations along the membrane chord at α = a 12◦ , b 20◦ , c 35◦ , d 45◦ , e 60◦ and f 90◦

α = 60◦ represents the deep stall condition and α = 90◦ is the vertical state. It can be seen from Fig. 17.7 that the intensity of the membrane fluctuations increases continuously as the AOA reaches the deep stall condition at 60◦ . As the AOA further grows to the vertical state, the fluctuation intensity decreases slightly. It is worth noting that the flexible membrane vibration maintains a chordwise second mode at the deep stall condition. It is related to the alternative shedding vortices behind the bluff-body-like membrane wing at very large AOAs. In Fig. 17.8, the instantaneous streamlines colored by the pressure coefficient at different AOAs are plotted. A chain of small-scale vortices can be observed flowing across the entire membrane surface before the light stall (smaller than 20◦ ), thereby inducing vibrations. As the AOA further increases, the vortex size becomes larger and the negative pressure region following the movement of the vortices gradually expands to the whole surface. As a result, the aerodynamic force normal to the membrane chord increases. However, the projected surface area along the Z -direction becomes smaller once the AOA exceeds 45◦ . Thus, the force component along the Z -direction, i.e., lift force, reduces at the deep stall conditions. The flow features at the deep stall conditions exhibit alternative shedding vortices from the LE and TE, which shares similarities as the vortical structures behind a cylinder.

914

17 Thin Structure Aeroelasticity

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 17.8 Instantaneous streamlines colored by pressure coefficient of flexible membrane at α = a 12◦ , b 20◦ , c 35◦ , d 45◦ , e 60◦ and f 90◦

17.3.3 Effect of Flexibility Based on the discussion of the membrane dynamics at different AOAs, we can notice that the flexible membrane deforms up under the aerodynamic loads to form the wing camber. In addition, the flow-induced membrane vibration is excited when the AOA exceeds the critical value. The vibrating membrane couples with the passing through vortical structures to form a tightly coupled system. Thus, the aerodynamic performance is affected by both the camber effect and the flow-induced vibration, which are closely related to the flexibility effect. To gain deeper insight into the role of flexibility, we simulate a rigid flat wing and a rigid cambered wing and compare them with the flexible membrane. By comparing the aerodynamic characteristics between the rigid flat wing and the rigid cambered wing, the camber effect can be

17.3 Two-Dimensional Thin Structure Aeroelasticity

915

examined. The effect of the flow-induced vibration can be investigated by comparing the flow features between the rigid cambered wing and the flexible membrane. The comparison of the aerodynamic forces among these three wings is presented in Fig. 17.5. It can be seen from Fig. 17.5 a that both the rigid cambered wing and the flexible membrane produce larger mean lift forces than the rigid flat wing at all examined AOAs. The lift reduction near the light stall angle α = 20◦ is alleviated due to the flexibility effect. The deep stall angle is delayed from 40◦ to 45◦ for the flexible membrane. The flexible membrane produces a similar mean lift force as the rigid cambered wing except for the range of α ∈ [20◦ , 45◦ ]. When the flow-induced vibration is taken into account, the flexible membrane has a smaller mean lift than the rigid cambered wing in this range. As presented in Fig. 17.5b, both the rigid cambered wings and the flexible membrane generate smaller drag than the rigid flat wing at small AOAs. The mean drag produced by the rigid cambered wing becomes larger than that of the rigid flat wing when the AOA is larger than 16◦ . The mean drag of the flexible membrane exceeds that of the rigid flat wing when the flexible membrane enters into the deep stall conditions at α = 45◦ . The rigid cambered wing produces more mean drag than the flexible membrane within the transition range from the light stall to the deep stall conditions. From Fig. 17.5c, we notice that the rigid flat wing has the smallest mean normal force. The mean normal forces of the rigid cambered wing and the flexible membrane are similar before the stall occurs. During the transition from the light stall to the deep stall conditions, the rigid cambered wing exhibits a larger mean normal force. The mean normal force of these two wings approaches each other after the deep stall. Finally, the flexible membrane produces more mean normal force when the wing tends to be vertical. It can be observed from Fig. 17.5d that the rigid flat wing shows the poorest mean lift-to-drag ratio. Both the rigid cambered wing and the flexible membrane achieve the best mean lift-to-drag ratio at α = 8◦ . Before the deep stall condition, the flexible membrane showed better performance than the cambered wing. The mean lift-todrag ratio of the flexible membrane becomes slightly smaller than that of the rigid cambered wing when the AOA is larger than 45◦ . Based on the above observations, the variation of the aerodynamic characteristics can be categorized into three ranges: (i) before stall (4◦ ≤ α < 20◦ ), (ii) transition from light to deep stall (20◦ ≤ α < 45◦ ) and (iii) deep stall (45◦ ≤ α < 90◦ ). To further understand the role of flexibility in coupled fluid-membrane dynamics, we select three typical cases corresponding to these three ranges to compare their aerodynamic characteristics. Here, we consider α = 12◦ , 35◦ and 60◦ for a comparison purpose. Figure 17.9 presents the comparison of the instantaneous streamlines of the rigid flat wing and the rigid cambered wing at the three selected AOAs. It can be seen that the rigid cambered wing and the flexible membrane can suppress the vortex size to some extent at α = 12◦ , compared to the rigid flat wing. The generated vortices near the LE can cause a large suction force, which contributes to lift improvement. As the AOA further increases to 35◦ , the vortex size becomes larger for both three wings. Due to the camber effect, the rigid camber wing and the flexible membrane have

916

17 Thin Structure Aeroelasticity

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 17.9 Instantaneous streamlines colored by pressure coefficient of (a, c, e) rigid flat wing and (b, d, f) rigid cambered wing at α = a, b 12◦ , c, d 35◦ and e, f 60◦

smaller local AOAs, resulting in longer attached LEVs compared to the rigid flat counterpart. The flow-induced vibration can regulate the vortical structure behind the flexible membrane via the coupling manner. When the wing enters the deep stall condition at α = 60◦ , the difference of the vortical structures among these three wings becomes smaller.

17.3.4 Effect of Reynolds Number To investigate the coupled membrane dynamics at different AOAs over a broad range, we further perform a series of numerical simulations in the parameter space of α-Re. Six groups of Reynolds numbers are selected and the AOA range is selected from

17.3 Two-Dimensional Thin Structure Aeroelasticity 4

3 Re=100 Re=300 Re=600 Re=1000 Re=2500 Re=3500

3

CD

CL

2 1

CN

5 4

40

60

α (◦ ) (a)

80

2

20

100

Re=100 Re=300 Re=600 Re=1000 Re=2500 Re=3500

40

60

α (◦ ) (b)

3

80

100

Re=100 Re=300 Re=600 Re=1000 Re=2500 Re=3500

2

CL /CD

6

Re=100 Re=300 Re=600 Re=1000 Re=2500 Re=3500

1

0 20

917

1

2 0

1 20

40

60

α (◦ ) (c)

80

100

20

40

60

α (◦ ) (d)

80

100

Fig. 17.10 Comparison of time-averaged aerodynamic forces of flexible membrane at various AOAs as a function of Reynolds number for a time-averaged lift coefficient, b time-averaged drag coefficient, c time-averaged normal force coefficient and d time-averaged lift-to-drag ratio

moderate to very high AOAs. Figure 17.10 summarizes the time-averaged aerodynamic forces in the studies parameter space. In Fig. 17.10a, it is noticed that the variation of the mean lift coefficient at different Reynolds numbers exhibits a similar trend as a function of AOA. The deep stall AOA corresponding to the maximum mean lift coefficient is observed at lower values when Re increases. The mean lift coefficient at Re = 100 shows the smallest values under all AOAs. As Re further grows up, the mean lift coefficient presents an upward trend when the AOA is smaller than 55◦ . The mean lift coefficient has small variations as a function of Re after entering the deep stall conditions. It can be seen from Fig. 17.10b and c that both the mean drag and the mean normal force exhibit an overall growing trend as a function of AOA. The flexible membrane produces the smallest mean drag and the mean normal force at Re = 100. When the flexible membrane reaches the deep stall conditions, the increase of the mean drag and the mean normal force becomes slower. The aerodynamic characteristics at the vertical state (90◦ ) are not significantly affected by Re when the value exceeds 300. In Fig. 17.10d, the mean lift-to-drag ratio decreases continuously as AOA increases regardless of the variation of Re. The increase of Re has a positive influence on the mean lift-to-drag ratio at a fixed AOA.

918

17 Thin Structure Aeroelasticity

(a)

(b)

(c)

Fig. 17.11 Instantaneous streamlines colored by pressure coefficient of flexible membrane at α = 45◦ for Re = a 100, b 600 and c 2500

To examine the effect of the Reynolds number on the coupled dynamics, we compare the instantaneous streamlines for three selected Reynolds numbers at a typical AOA close to the deep stall condition in Fig. 17.11. It can be observed that the flexible membrane presents larger wing camber and vibrates severely as Re increases. In addition, multiple small-scale vortices are noticed at higher Re. The suction force on the upper surface of the membrane is enhanced as Re grows up, thereby improving the lift and drag forces.

17.3.5 Interim Summary In this section, we investigated the variation of the aerodynamic characteristics and the coupled dynamics from 4◦ to 90◦ . The main purpose was to study how the flexible membrane behaved in unsteady flows as a function of AOA. Based on the numerical simulation results, we found that the flexible membrane experienced a light stall near α = 20◦ and entered the deep stall condition at α = 45◦ with the maximum mean lift coefficient. After the deep stall occurred, the mean lift coefficient reduced sharply and the mean normal force showed a slowly increasing trend. The maximum mean lift-to-drag ratio was achieved at α = 8◦ and it then decreased continuously to nearly zero at the vertical state. The flexible membrane exhibited a chordwise second mode after a deeper stall. To explore the flexibility effect, a rigid flat wing and a rigid cambered wing were further simulated for comparison purposes. We found that the camber effect can improve the mean lift and the mean lift-to-drag ratio over the studied AOA range but enhance the mean drag after a stall occurs. The flow-induced vibration can weaken

17.4 Three-Dimensional Thin Structure Aeroelasticity

919

b

d

c

h Frame

2r Membrane

Frame

Membrane

Fig. 17.12 Membrane wing geometry and section of supporting frame

the produced mean lift and drag forces within the transition range from the light stall to the deep stall condition. The effect of the flow-induced vibration became weak at large AOAs after the deep stall happened. Overall, the flexibility was beneficial to the improvement of the aerodynamic performance, especially for the low to moderate AOAs. The increase of Re had a positive influence on the aerodynamic characteristics of the flexible membrane. The effect of Re on the membrane dynamics became very weak for the vertical flexible membrane at α = 90◦ . The studies in this chapter were focused on a 2D flexible membrane, which opened the door to the investigation of the fluid-membrane interaction problems. A 3D rectangular membrane will be further examined to seek the underlying mechanisms associated with flow-excited instability and the role of flexibility in the coupled dynamics in the following sections.

17.4 Three-Dimensional Thin Structure Aeroelasticity 17.4.1 Problem Setup and Validation The 3D rectangular membrane wing with a rigid supporting frame was conducted in the wind tunnel experiments by Rojratsirikul et al. [688]. The geometry information and the section of the supporting frame is presented in Fig. 17.12. The membrane has a chord length of c = 68.75 mm and an aspect ratio of A R = 2. The thickness of the membrane is h = 0.2 mm. The flexible membrane is made of latex rubber with the material density of ρ s = 1000 kg/m3 and Young’s modulus of E s = 2.2 MPa. The aerofoil-like section of the supporting frame has a length of d = 5 mm and the diameter of the rod is 2r = 2 mm. In the current study, the membrane wing is simulated at the same Reynolds number of Re = 24300 as that in the experiment. Before we proceed to investigate the coupling mechanisms of the flexible membrane, the membrane aeroelasticity at several AOAs are compared against the results obtained from the experiments for validation purpose.

920

17 Thin Structure Aeroelasticity Γslip uf · nf = 0, σ f · nf = 0

Γno−slip

Γin |uf | = U∞

H Γout σ f · nf = 0 ∇˜ ν · nf = 0

b α Z Y X

Γslip uf · nf = 0, σ f · nf = 0

B

L

(a)

(b)

Fig. 17.13 3D computational set-up for fluid-membrane interaction: a schematic diagram of the computational domain and the boundary conditions and b representative mesh distribution in (Y, Z)-plane in the fluid domain

Figure 17.13 a depicts the computational domain and boundary conditions for a 3D flexible membrane immersed in an unsteady flow with a fixed AOA. The length, width and height of the computational domain are all set to 50c. A stream of oncoming flow with uniform velocity of u f = (U∞ , 0, 0) enters the domain through the inlet boundary Γin . The slip-wall boundary condition is applied on four side boundaries (Γslip ). The boundary condition on the membrane surface (Γno−slip ) is set to the no-slip boundary condition. The outlet boundary Γout has a traction-free boundary condition. In the numerical simulation, all the degrees of freedom of the rigid frame are fixed, and the passive deformation of the flexible membrane is allowed under aerodynamic loads. To evaluate the aerodynamic characteristics of the membrane wing, we integrate the surface traction for the first layer of elements on the membrane surface to obtain the instantaneous lift, drag and normal force coefficient, which are defined below  1 (σ¯ f · n) · nz d, 1 f 2 ρ U S Γ ∞ 2  1 CD = 1 f 2 (σ¯ f · n) · nx d, ρ U S Γ ∞ 2  1 CN = 1 f 2 (σ¯ f · n) · nc d ρ U S Γ ∞ 2 CL =

(17.20) (17.21) (17.22)

where U∞ is the freestream velocity and ρ f represents the air density. The area of the membrane surface is denoted as S = bc. nx and nz are the projection of the unit normal n to the membrane surface on the X -axis and Z -axis, respectively. The unit normal vector nc is perpendicular to the membrane chord. σ¯ f denotes the fluid stress tensor. The deformation of the flexible membrane is mainly driven by the pressure acting on its surface, and the pressure coefficient is given as

17.4 Three-Dimensional Thin Structure Aeroelasticity

921

Table 17.2 Mesh convergence of a 3D rectangular flexible membrane wing at Re = 24,300 and α = 15◦ with non-dimensional time step size ΔtU∞ /c = 0.0364 Mesh M1 M2 M3 Fluid elements Structural elements Mean lift C L Mean drag C D r.m.s. lift fluctuation C L r ms r ms r.m.s. drag fluctuation C D Dominant shedding frequency f vs c/U∞ Maximum mean deflection max δ n /c Maximum r.m.s. deflection fluctuation δn r ms Dominant vibration frequency f s c/U∞

341,821 160 0.9022 (0.49%) 0.2289 (−2.97%) 0.0788 (−16.17%) 0.0289 (−10.25%) 0.9668 (−2.70%)

823,864 228 0.8902 (−0.85%) 0.2346 (−0.55%) 0.0928 (−1.28%) 0.0313 (−2.80%) 0.9937 (0%)

1,304,282 352 0.8978 0.2359 0.0940 0.0322 0.9937

0.03391 (−1.14%)

0.03415 (−0.44%)

0.03430

0.000979 (−2.88%) 0.000994 (−1.39%) 0.001008 0.9668 (−2.70%)

0.9937 (0%)

0.9937

The percentage differences are calculated by using M3 results as the reference

Cp =

p − p∞ 1 f 2 ρ U∞ 2

(17.23)

where p and p∞ represent the local pressure and the far-field pressure, respectively. To choose a proper mesh with sufficient resolution for the following numerical simulations, we conduct a mesh convergence study for the 3D flexible membrane by designing three sets of meshes namely M1, M2 and M3. The unstructured finite element is adopted to discretize the 3D fluid domain and the structure domain is discrete by the structured finite element. These three sets of meshes consist of 341821, 823864 and 1304282 eight-node brick elements in the fluid domain and the corresponding element numbers in the structure domain are 160, 228 and 352, respectively. A stretching ratio of Δy j+1 /Δy j = 1.25 is set within the boundary layer mesh to maintain y + < 1.0. The representative mesh distribution in the (Y,Z)-plane at the mid-chord position is presented in Fig. 17.13b. The non-dimensional time step size is set to ΔtU∞ /c = 0.0364 and the Reynolds number is given as Re = 24,300 with a corresponding freestream velocity of U∞ = 5 m/s in the numerical simulation. The flexible membrane at an AOA of α = 15◦ with obvious vortex shedding phenomenon is considered for the mesh convergence study. Table 17.2 summarizes the aerodynamic forces, the vortex shedding frequency statistics and the structural dynamics for the three sets of meshes. We calculate the percentage differences for M1 and M2 with respect to the finest mesh M3 to evaluate the mesh convergence discrepancy. It can be observed that the discrepancy of the mean lift and the mean drag is less than 1% and the maximum difference of the force fluctuation is 2.8%. The discrepancy of the mean membrane deflection and its root-mean-squared value of the membrane deflection fluctuation for M2 relative to M3 is smaller than 2%. The

922

17 Thin Structure Aeroelasticity

Table 17.3 Convergence of the aerodynamic forces and the dynamic responses at different time step sizes for a 3D rectangular flexible membrane wing at Re =24,300 and α = 15◦ based on mesh M2 Non-dimensional time step size 0.0728 0.0364 0.0182 ΔtU∞ /c Mean lift C L Mean drag C D r.m.s. lift fluctuation C L r ms r ms r.m.s. drag fluctuation C D Dominant shedding frequency f vs c/U∞ Maximum mean deflection max δ n /c Maximum r.m.s. deflection fluctuation δn r ms Dominant vibration frequency f s c/U∞

0.8336 (−7.8386%) 0.2223 (−4.6332%) 0.0736 (−22.20%) 0.0252 (−20.75%) 0.9399 (−5.41%)

0.8902 (−1.58%) 0.2346 (0.6435%) 0.0928 (−1.90%) 0.0313 (−1.57%) 0.9937 (0%)

0.9045 0.2331 0.0946 0.0318 0.9937

0.0351 (1.45%)

0.03415 (−1.30%)

0.0346

0.000830 (−18.79%)

0.000994 (−2.74%)

0.001022

0.9399 (−5.41%)

0.9937 (0%)

0.9937

The percentage differences are calculated by using ΔtU∞ /c = 0.0182 results as the reference

dominant frequencies of the vortex shedding process and the membrane vibration for M2 are the same as those for M3. Thus, we choose mesh M2 as the reference mesh for further validation study due to its adequate resolution. A time step size convergence study is further conducted to select a proper time step size for the numerical simulations. Three non-dimensional time step sizes of ΔtU∞ /c = 0.0728, 0.0364 and 0.0182 are chosen for the time step size convergence study. The statistical data related to the aerodynamic forces and the structural vibrations for different time step sizes are summarized in Table 17.3. The percentage differences are calculated by using the results of ΔtU∞ /c = 0.0182 as the reference. It can be seen that the differences are less than 3% for the non-dimensional time step size of ΔtU∞ /c = 0.0364. Thus, a non-dimensional time step size of 0.0364 is selected to validate the coupled framework and investigate the aeroelasticity of the flexible membranes at different AOAs. The 3D membrane immersed in the unsteady flow at different AOAs is simulated to compare against the experimental data [688, 691] for the validation purpose as shown in Fig. 17.14. The maximum magnitude of the normalized time-averaged membrane deformation is presented in Fig. 17.14 a. A difference of the time-averaged normal force coefficient between the flexible membrane wing and a rigid wing counterpart is considered for the plotting purpose in Fig. 17.14 b. It can be observed that the overall trends of the membrane deformation and the aerodynamic forces are well predicted. The frequency spectrum of the membrane vibration at the point with the maximum standard deviation in our numerical simulation is analyzed via the fast Fourier transform technique. The frequency values at the frequency peaks are plotted in Fig. 17.14 c together with the frequency spectra obtained from experiments for comparison purposes. The

17.4 Three-Dimensional Thin Structure Aeroelasticity

923

color of the frequency spectra changing from white to black corresponds to the increasing intensity of the frequency. The red circle in Fig. 17.14 c represents the dominant vibration frequency computed from our numerical simulations. The dominant frequency contents of the membrane vibration at various AOAs show similar distributions to the experimental results. We further compare the circulation of tip vortices on a cross-flow plane near the trailing edge in Fig. 17.14d. Consistent with the experimental measurements [691], the circulation of the tip vortices is calculated m by integrating the vorticity distributions over the measuring surface Ω S covering the tip vortices on the cross-flow plane. In our numerical simulation, the integration of the vorticity is evaluated using the Gaussian quadrature  Γ =

Ω Sm

∇ × u f · dSm =

np n el  

ωx (η p , ξ p )det Je (η p , ξ p )W p ,

(17.24)

e=1 p=1

where n el and n p are the number of the finite elements of the measuring surface and the number of Gauss points, respectively. η p and ξ p denote the coordinates of the Gauss nodes. ωx is the vorticity in the freestream direction. Je is the jacobian of the e-th element and W p represents the Gauss weight. It can be seen from Fig. 17.14 d that the normalized circulation obtained from our numerical simulation shows good agreement with the experimental results.

17.4.2 Membrane Aeroelasticity An overview of the membrane aeroelastic responses is displayed to provide a brief impression of the fluid-membrane interaction problems. In Figs. 17.15a, c, e, we summarize the time histories of the lift coefficient and the normalized membrane displacement normal to the chord at the membrane center over a wide time range for the flexible membrane at three different AOAs. The gray shaded region in the plots represents the time range selected to collect the snapshots for the mode decomposition. It can be seen from Fig. 17.15 that the aeroelastic responses of the flexible membrane at α = 15◦ show almost periodic dynamics. The dominant frequencies of the lift coefficient and the membrane displacement at the center are synchronized. As shown in the phase portrait in Fig. 17.16 a, the flexible membrane displacement responses at the center exhibit a period two state. From the time history responses in Fig. 17.15c, e and the phase portrait in Fig. 17.16b, c, the aeroelastic responses tend to be non-periodic at higher AOAs. To gain further insight into membrane aeroelasticity, we compare the structural vibration characteristics and the flow features of the flexible membrane at different AOAs. Figure 17.17 shows the standard deviation analysis of the normalized membrane displacement δnsd /c over several cycles to reflect the dominant structural vibration modal shapes. A typical chordwise second mode is observed for the elastic membrane at α = 15◦ . However, the dominant structural mode of the 3D mem-

924

17 Thin Structure Aeroelasticity 0.06

0.3 Present Experiment Rig

0.04

Fle

CN − CN

¯ max /c d

Present Experiment

0.02

0

0

5

10

15

α (◦ )

20

25

0.2

0.1

0

30

0

10

(a)

α (◦ )

20

30

(b)

Γ/U∞ c

0.6

0.4

0.2

0

(c)

Present Experiment

0

5

10

15

α (◦ )

20

25

30

(d)

Fig. 17.14 Comparison between the present simulations and the data obtained from experiments [688, 691] for: a the magnitude of time-averaged normalized maximum membrane deformation ¯ max /c), b the time-averaged normal force coefficient difference (C Fle − C Rig ) between the ( d

N N flexible membrane wing and rigid wing, c the membrane vibration frequency spectra at the location with maximum standard deviation of the membrane deflection, and d normalized circulation (Γ /U∞ c) of the vortices at the wingtip on a plane normal to the freestream direction

brane wing cannot be distinctly identified from the standard deviation analysis at α = 20◦ and 25◦ . Some high-order modes can be observed near the leading edge and the wingtip. It is hard to identify and isolate these several influential modes from the time-averaged standard deviation analysis of the multi-modal mixed responses. Similarly, Tregidgo et al. [751] found that a disturbing membrane in the gusty flow exhibited a chordwise and spanwise first mode based on the standard deviation analysis. However, a chordwise first and spanwise second mode and a chordwise first and spanwise third mode were observed from the instantaneous membrane vibration responses. The standard deviation analysis is not reliable in reflecting the structural modes due to the mode overlap in a time-averaged sense. The time-varying instantaneous pressure coefficient difference between the upper and lower surfaces C dp and the fluctuation contours of the membrane displacement δn −δ n c

can provide an intuitive understanding of the evolution of the membrane aeroelasticity. For that purpose, we select four equispaced time instants to plot the instantaneous membrane aeroelastic dynamics. These selected time instants are indicated by the black dash-dot lines in the time history plots as shown in Fig. 17.15b, d, f.

17.4 Three-Dimensional Thin Structure Aeroelasticity

0.04

1

1

0.035 0.8

0.5 70

80

tU ∞ / c

90

0.03 92

93

94

(a) 2

i ii iii iv

1.2

0.06

CL

0.05

0.07

δn/c

1.5 1

0.05 1

0.5 40

50

60

70

tU ∞ / c

80

90

100

0

0.8

0.04 0.03 80

82

(c)

tU ∞ / c

84

86

(d) 0.1

1.5 0.05

0.08 0.07

1.2

0.06

CL

Snapshots for mode decomposition

i ii iii iv

1.4

δn/c

2

0.05 1

1 0.5

0 30

40

50

tU ∞ / c

(e)

0.025

0.08

0.1

Snapshots for mode decomposition

CL

96

(b) 1.4

CL

95

tU ∞ / c

δn/c

60

0.6

60

70

0.8

δn/c

50

0

δn/c

0.05

0.045

CL

1.5

i ii iii iv

1.2

0.1

Snapshots for mode decomposition

δn/c

CL

2

925

0.04 0.03 62

64

tU ∞ / c

66

68

(f)

Fig. 17.15 Time histories of the lift coefficient C L and the normalized membrane displacement δn /c at the membrane center for U∞ = 5 m/s at α = a, b 15◦ , c, d 20◦ and e, f 25◦

The instantaneous membrane aeroelastic dynamics for flexible membranes at three AOAs are summarized in Fig. 17.18. All the pressure difference distributions on the membrane surface show complex evolutions over time and overlapping modal shapes in space. The instantaneous membrane displacement fluctuations at α = 15◦ exhibit an obvious chordwise second mode and varied spanwise modes. We can observe the chordwise second, third and high-order modal shapes that appear occasionally for α = 20◦ and 25◦ as shown in Fig. 17.18d and f. Due to fluctuating displacement and pressure pulsations, the dominant structural motion will be covered up in the standard deviation contours due to the time-averaged sense of the second and third modes. Thus, the traditional standard deviation analysis is not a suitable indicator to reflect the correct dominant modes of the whole membrane vibrations with temporal-spatial overlapping modal shapes. Figures 17.19 and 17.20 present the time-averaged velocity magnitude and the turbulent intensity on five equispaced slices along the spanwise direction at three AOAs, respectively. It can be observed from Fig. 17.19 that the low-velocity region is larger on the slice of the mid-span plane than those on the slices near the wingtip. Similarly, the unsteady flow near the mid-span location shows higher turbulent intensity. From Figs. 17.17 and 17.18, we see that the region close to the mid-span location of the membrane has the largest vibration amplitude. Due to the displacement con-

926

17 Thin Structure Aeroelasticity 4

-2

d(δz /c) dt

0

d(δz /c) dt

d(δz /c) dt

10

10

2

0

-10

0.03

0.035

0.04

-20 0.02

0 -5

-4 -6

5

0.04

0.06

-10 0.03

0.04

0.05

δz /c

δz /c

δz /c

(a)

(b)

(c)

0.06

0.07

Fig. 17.16 Phase portrait of the membrane center for the 3D flexible membrane at α = a 15◦ , b 20◦ and c 25◦ Y

Z

δnsd /c

X

(a)

0

0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

(b)

(c)

Fig. 17.17 Standard deviation analysis of normalized membrane displacement normal to the chord δnsd /c for U∞ = 5 m/s at α = a 15◦ , b 20◦ and c 25◦

straints of the membrane at the wingtip, the vibration amplitude near the wingtip becomes smaller. Thus, the flow fluctuations contributed by the membrane vibration are weaker at the wingtip than those near the mid-span location. As the AOA increases, both the low-velocity region and the high turbulent intensity region expand further.

17.4.3 Summary In this section, we examined flow-induced vibrations of three-dimensional flexible membrane wings at different angles of attack. The flexible membrane exhibits periodic oscillations at 15◦ . The membrane vibration responses tended to be non-periodic

17.4 Three-Dimensional Thin Structure Aeroelasticity

927

Fig. 17.18 Flow past a 3D rectangular membrane wing: (a, c, e) pressure coefficient difference between the upper and lower surfaces and (b, d, f) fluctuation of membrane displacement at four selected instantaneous time instants plotted in Fig. 17.15 at α = a, b 15◦ , c, d 20◦ and e, f 25◦

(a)

(b)

(c)

Fig. 17.19 Flow past a 3D rectangular membrane wing: time-averaged velocity magnitude on five slices along the spanwise direction at α = a 15◦ , b 20◦ and c 25◦

928

17 Thin Structure Aeroelasticity

(a)

(b)

(c)

Fig. 17.20 Flow past a 3D rectangular membrane wing: turbulent intensity on five slices along the spanwise direction at α = a 15◦ , b 20◦ and c 25◦

when the AOA is larger than 20◦ . The fluctuations of the structural displacements and surface pressure distributions become larger as the AOA increases. The oscillations of the flow in the space present larger amplitudes and regions behind the membrane. These studies provide a basis to further explore the relationship between flow fluctuations and membrane vibrations, which will be discussed in detail in the next chapter. Acknowledgements Some parts of this Chapter have been taken from the PhD thesis of Guojun Li carried out at the National University of Singapore and supported by the Ministry of Education, Singapore.

Chapter 18

Aeroelastic Mode Decomposition

In this chapter, we present a global Fourier mode decomposition framework for unsteady fluid-structure interaction. We apply the framework to isolate and extract the aeroelastic modes arising from a coupled three-dimensional fluid-membrane system. We investigate the frequency synchronization between the vortex shedding and the structural vibration via mode decomposition analysis. We explore the role of flexibility in the aeroelastic mode selection and perform a systematic comparison of flow features among a rigid flat wing, a rigid cambered wing and a flexible membrane. The camber effect can enlarge the pressure suction area on the membrane surface and suppress the turbulent intensity compared to the rigid flat wing counterpart. With the aid of our mode decomposition technique, we find that the dominant structural mode exhibits a chordwise second and spanwise first mode at different angles of attack. The structural natural frequency corresponding to this mode is estimated using an approximate analytical formula. By examining the dominant frequency of the coupled system, we show that the dominant membrane vibrational mode is selected via the frequency lock-in between the dominant vortex shedding frequency and the structural natural frequency. From the fluid modes and the mode energy spectra at α = 20◦ and 25◦ , the aeroelastic modes corresponding to the noninteger frequency components lower than the dominant frequency are observed, which are associated with the bluff body vortex shedding instability. The non-periodic aeroelastic behaviors observed at higher angles of attack are related to the interaction between aeroelastic modes caused by the frequency lock-in and the bluff-body-like vortex shedding. Using the mode decomposition analysis, we suggest a feedback cycle for flexible membrane wings undergoing synchronized self-sustained vibration. This feedback cycle reveals that the dominant aeroelastic modes are selected through the mode and frequency synchronization during fluid-membrane interaction to exhibit similar modal shapes in the membrane vibration and the pressure pulsation.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Jaiman et al., Mechanics of Flow-Induced Vibration, https://doi.org/10.1007/978-981-19-8578-2_18

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18.1 Introduction 18.1.1 Mode Decomposition and Mode Selection in Fluid-Structure Interaction During the past decades, morphing wings with flexible membrane components have received substantial attention from the aerospace engineering community in the context of bio-inspired flying vehicles [511, 640, 684, 713]. A flexible membrane can passively deform and vibrate by highly interacting with an unsteady flow, thereby forming a coupled fluid-membrane system. The coupled system exhibits a variety of correlated vibrational and fluid modes with a wide range of spatial and temporal scales. These correlated aeroelastic modes and their scales are closely related to the aerodynamic performance of the membrane wings and play an important role in efficient flight and control strategies. Hence, identifying and isolating the most influential aeroelastic modes from the coupled system is essential to further understand the aeroelastic mode selection mechanism and to promote the design of active or passive control strategies. Numerous experimental and computational studies on fluid-membrane interaction have been carried out during the past years. Song et al. [721] examined the aeromechanics of membrane wings as a function of aspect ratio, flexibility and prestrain value. It can be observed from the phase map of membrane mode that the dominant mode switched from the first mode to higher modes as the angle of attack and Reynolds number increased. Rojratsirikul et al. [688–690] performed a series of experiments to study the dynamic behaviors of flexible membrane wings at moderate Reynolds numbers. Different types of dominant vibrational modes and vortical structures have been observed in the coupled fluid-membrane system. Bleischwitz et al. [529] investigated the effect of aspect ratio on membrane dynamics in wind tunnel experiments. The frequencies corresponding to the dominant structural modes were found to be correlated with the frequencies of the force fluctuations. In view of the limitations of collecting physical data of interest in wind tunnel experiments, numerical simulation methods have become effective tools to gain further insight into the coupled mechanism of the flexible membrane. With the aid of numerical simulations, Sun et al. [726–729] systematically studied the nonlinear dynamic behaviors of flexible membrane wings. The vibration of the flexible membrane excited by the unsteady flow gradually transitioned from the periodic state to the non-periodic state as the relevant aeroelastic parameters varied. From the frequency spectra analysis of the membrane responses in the aforementioned literature, it was found that the membrane vibration usually exhibited multiple frequency peaks, which were closely correlated with the vortical structures with a variety of temporal and spatial scales. These multi-modal mixed responses of the coupled fluid-membrane system pose a challenge in identifying and isolating the correlated aeroelastic modes of interest from the system. The understanding of the underlying mechanism of how a specific aeroelastic mode is selected in the coupled system is limited.

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931

With respect to the dominant mode identification in a coupled fluid-membrane system, some ingenious methods have been adopted to distinguish the most influential modes. Standard deviation analysis has been widely applied to the fluid-membrane interaction problem to determine the dominant structural modes [688, 689, 748]. As Bleischwitz et al. [529] pointed out, the excited structural modal shapes overlapped together, which increased the difficulty of isolating the structural modal shapes of interest from the coupled system. The standard deviation analysis of the membrane deflection could reflect the dominant structural modal shape to some extent. However, occasionally appearing modes or overlapping modes with small energies may be covered by the dominant modes, which makes them hard to be identified from the overall membrane vibration responses. Additionally, the frequency spectra and the vibration state analysis at a single point are not reliable indicators to reflect the dynamic characteristics of the whole membrane structure [728]. Therefore, global mode identification methods are naturally desirable to capture the dynamic behaviors of the entire physical field of interest. The relevant dynamic information corresponding to each mode, such as mode energy and mode frequency, could help us gain further insight into the whole dynamic characteristics when performing the global mode identification. Data-based mode decomposition techniques have been widely used in the analysis of flow features and structural vibrations to identify coherent structures. These mode decomposition techniques separate the temporal-spatial data into energy-ranked or frequency-ranked modes with physical meaning to represent different characteristics of the field. The space-only proper orthogonal decomposition (POD) method [48, 394, 647] can decompose the collected physical fields of interest into a set of energy-ranked POD modes by diagonalizing the spatial correlation matrix. These decomposed modes are orthogonal in space. Multiple frequency components can be observed for each space-only POD mode when the physical fields collected for mode decomposition are complex. The spectral mode decomposition methods such as the Fourier mode decomposition (FMD) [649] and the dynamic mode decomposition (DMD) [382] project the spatial-temporal physical data into the spatial-frequency space to obtain the decomposed modes in frequency ranking. The FMD method is based on the discrete Fourier transform, which is a superposition of harmonic modes. The mode energy and the phase information are included in the transformed Fourier coefficients. The DMD method and its variants (e.g., Optimized DMD, Sparsity Promoting DMD) can extract the dynamic modes with growth rates and oscillation frequencies from nonlinear systems by approximating the modes of the Koopman operator. Chen et al. [547] mathematically demonstrated that the DMD will be reduced to the FMD when the fluctuations of the physical variables are performed in the mode decomposition process. A variant of POD, the so-called spectral proper orthogonal decomposition method, is recently studied to obtain decomposed modes evolving coherently both in space and time. These POD variants attempt to bridge the gap between energetically optimal space-only POD (with spatial orthogonalization) and the spectral mode decomposition technique (with temporal orthogonalization). Sieber et al. [716] developed a spectral proper orthogonal decomposition method to build a connection between the

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energetically optimal POD and the spectrally pure FMD. To achieve this, a low-pass filter was applied along the diagonals of the correlation matrix to enforce the diagonal similarity. The filtered correlation matrix was employed to perform the eigendecomposition as the space-only POD to obtain mode energy and temporal coefficients. The proposed spectral POD method converges to the space-only POD when the filter length is set to zero, and it changes to the FMD as the filter length becomes the maximum value. To identify spatial-temporal coherent structures, Towne et al. [747] presented another spectral POD method derived from a space-time formulation of POD. In the spectral POD algorithm, the discrete Fourier transform combined with Welch’s method is first employed to obtain the temporally orthogonal discrete Fourier modes and estimate the cross-spectral density tensor. Through the eigendecomposition, the spectral POD modes are energetically ranked and spatially orthogonalized within each frequency [747]. Similar to the spectral POD, the FMD approach proposed in the current study combines the discrete Fourier transform and Welch’s method to obtain the discrete Fourier modes and then evaluates the global FMD modes for the coupled fluid-structure system. By comparing these two mode decomposition techniques, the spectral POD mode is orthogonal both in time and space, while the FMD mode is orthogonal in time. Aforementioned data-based mode decomposition techniques have been extensively employed to identify the physically meaningful modes from the physical fields, like flow past a cylinder [382, 547, 649], wall-bounded flows [48] and cavity flows [702]. Regarding the application of the mode decomposition methods for the fluidstructure interaction problems, the mode decomposition methods were employed to extract modes from the fluid or structure fields and usually treated these modes separately in the data analysis [254, 299, 537, 654]. Specific to the fluid-membrane interaction problems, the POD method [531], the DMD method [382] and the FMD method [705] were employed to identify the dominant modes of interest in the fluid domain. The dominant structural modal shapes were successfully identified from the whole structural responses with multimodal mixed responses by the POD method [530] and the DMD method [746], which avoided the drawbacks of the standard deviation analysis. While these applications of mode decomposition approaches to the fluid-membrane interaction problems have attempted to address some relevant questions, a fully-coupled relationship between the fluid modes and the structure modes can be lost due to individual treatments. Only a handful of literature can be found to build a bridge between the decomposed modes in both fluid and structure fields to explore the coupled mechanism. Recently, Goza et al. [589] developed a combined framework based on the POD and DMD methods for the mode decomposition of flapping flags immersed in an unsteady flow. In this combined formulation, the collected data in both fluid and structure fields were decomposed in a unified matrix, which naturally ensured the inherent correlation between the dynamic modes in both fields. To gain further insight into the coupled mechanism during fluid-membrane interaction, a global mode decomposition framework for extracting the aeroelastic modes in a unified manner is highly desirable in the mode analysis of the coupled systems.

18.1 Introduction

933

In this chapter, we present an effective framework based on the radial basis function (RBF) interpolation method and the FMD method for the aeroelastic mode decomposition of the coupled fluid-membrane system. Of particular interest is to present physical insight into the underlying mechanism of how the unsteady turbulent flow interacts with the extensible 3D membrane to excite particular wake patterns and select specific vibrational modes in a frequency-synchronized way. The main purpose is to present the utility of an RBF-FMD-based mode decomposition framework to extract the frequency-ranked aeroelastic modes, rather than compare the advantages against other specific mode decomposition techniques. The proposed procedure can be used to interpret the frequency lock-in phenomenon and the associated mechanisms during fluid-membrane interaction. Using the Fourier-decomposed modes of the coupled fluid-structure system, we attempt to answer the specific questions that are relevant to membrane aeroelasticity: (i) Which types of membrane vibrations and wake patterns are dominant during fluid-membrane interaction and how do we identify these dominant aeroelastic modes in the coupled system? (ii) How does membrane flexibility affect the aeroelastic membrane characteristics? (iii) What is the aeroelastic mode selection mechanism during fluid-membrane interaction? To address these questions, we extend the original FMD method for fluid-only analysis to the coupled fluid-membrane system. To facilitate the mode decomposition, we develop an efficient data projection to a stationary reference grid via RBF, which allows the handling of physical data at time-varying grids in the fluid domain. The physical variables of interest in both the fluid and structure fields are then stored into a total vector and form a time sequence to compute the global Fourier modes via FMD. The contribution of the dominant aeroelastic mode to the overall membrane dynamics is calculated quantitatively. For reliable dominant mode frequencies, Welch’s method [770] combined with a proper window function is applied to suppress the aliasing effect and to reduce noise in the spectrum analysis. We then present the correlated Fourier modes in the fluid and structure fields. A comparison of the Fourier modes between a rigid flat wing, a rigid cambered wing and a flexible membrane wing is conducted to investigate the role of flexibility in membrane aeroelasticity. We explore the connection between the flowexcited vibration and the natural frequency of the flexible membrane immersed in an unsteady flow. For that purpose, an approximate analytical formula of the nonlinear membrane natural frequency is derived to estimate the natural frequency corresponding to a specific structural mode of interest. Based on the mode decomposition analysis for three types of wings, the relationship between the aeroelastic response and the bluff-body-like vortex shedding is investigated to interpret the non-periodic responses observed at higher angles of attack.

18.1.2 Organization In Sect. 18.2, the FMD algorithm are introduced. We then perform FMD analysis for the coupled membrane dynamics shown in Chap. 17 to extract the dominant Fourier

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18 Aeroelastic Mode Decomposition

modes and frequency spectrum. The effect of flexibility is investigated by comparing the flow features of the flexible membrane wing with their rigid counterpart. The aeroelastic mode selection strategy is revealed with the aid of the FMD analysis.

18.2 Global Fourier Mode Decomposition In this section, the algorithm of Fourier mode decomposition (FMD) for fluid-flexible structure interaction problems is presented. To identify and extract the global spatial modes at specific frequencies of interest, the FMD method projects the physical data of interest in a selected spatial domain from the temporal space to the frequency space based on the discrete Fourier transform.

18.2.1 Data Collection for Aeroelastic Mode Decomposition Before we perform the FMD analysis for the fluid-structure interaction problem, the physical data of interest in the fluid and structural domains is firstly collected from experiments or numerical simulations in a time-discrete way. To maintain the temporal consistency, the snapshot-based data is equispaced in time with the same sampling frequency f sam . Considering the deformation of the flexible structures under aerodynamic loads, the body-fitted fluid grid is updated at each iterative step by following the motion of the morphing or flapping structures. To avoid the difference of the time-varying fluid grid locations at each snapshot, the physical variables in the fluid domain Ω f are decomposed via the FMD method at a stationary grid in the f Euler coordinate as a reference for simplicity. Hence, we project the physical data ys f f f at the spatial points x s in the moving grid as sources onto M p projected points x p f in the stationary grid to collect the projected data y p for the mode decomposition. An efficient point-to-point projection method based on the RBF approach is employed to perform the projection between these two sets of nonmatching grids. Figure 18.1 illustrates this projection process based on the RBF method at a specific n for a 2D slice of the n-th snapshot for demonstration. sampling time instant t n = fsam The dimension of the extracted domain is chosen according to the specific problems. In this work, we are interested in the flow features in the near field and the wake of the coupled fluid-structure system. Hence, only the physical data within a box in red dashed line with a size of L × H as shown in Fig. 18.1 is projected onto the stationary grid with the same size. The physical data ys on the moving structure surface Γ s is collected at M s discrete points x s in a Lagrangian coordinate for the mode decomposition. In the current study, we collect the projected spanwise vorticity ω y n = ω y n f f f (x p , t n ) ∈ R M p ×1 and the projected spatial pressure coefficient C p n = C p n (x p , t n ) ∈ f f f R M p ×1 at the M p discrete points x p of a stationary reference grid and for the sam-

18.2 Global Fourier Mode Decomposition

935

Fig. 18.1 Illustration of projection process from moving grid to stationary reference grid based on radial basis function method for physical variable collection of the mode decomposition procedure for fluid-structure interaction

pling time instant t n . These two physical variables are related to the chordwise vortex structures and the spatial flow perturbations in the fluid domain. The structure s displacement normal to the chord δnn = δnn (x s , t n ) ∈ R M ×1 and the pressure coefs ficient difference between the upper and lower surfaces C dp n = C dp n (x s , t n ) ∈ R M ×1 at the M s discrete points x s and the same sampling time instant t n are stored in vector forms. These two variables can reflect the structural modal shapes and the flow perturbations applied on the structure surface. For simplicity, we store the collected physical data in the fluid and structural domains at t n into a total state vector written as ⎤ ⎡ ⎤ ⎡ f ω yn yn1 (x p , t n ) ⎢ y (x f , t n )⎥ ⎢ C p ⎥ M×1 p ⎥ ⎢ n⎥ n2 , (18.1) yn = yn (X, t n ) = ⎢ ⎣ y (x s , t n ) ⎦ = ⎣ δnn ⎦ ∈ R n3 d C pn yn4 (x s , t n ) f

where M = 2M p + 2M s denotes the total number of the discrete points. The collected snapshot physical data at N time instants is stored in a matrix form Y = [ y0 y1 . . . yn . . . y N −1 ] ∈ R M×N ,

(18.2)

In this matrix, the organized data in each column represents the physical state at a specific sampling time instant and the data in each row provides the time history for a selected physical variable at one spatial point.

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18 Aeroelastic Mode Decomposition

18.2.2 Global Fourier Mode Decomposition Algorithm The fluctuation component yn can reflect the perturbation based on the mean physical values in the time-varying global physical field. In the current study, we transfer the fluctuations of the collected physical data to the spatial-frequency domain due to their connection with the aeroelastic modes. Following the idea of the Fourier series, the fluctuation function yn can be written as a discrete Fourier series in an exponential form N −1 1  2πk yn = ck ei N n , (18.3) N k=0 where ck denotes the Fourier complex coefficient vector at M discrete spatial points X for a specific discrete frequency f k . The Fourier complex coefficient vector consists of four components corresponding to the four selected physical variables, which can be expressed as ⎤ ⎤ ⎡ f ω yk ck1 (x p , f k ) ⎢ c (x f , f )⎥ ⎢ C p ⎥ M×1 p k2 k ⎥ ⎢ k⎥ , ck = ck (X, f k ) = ⎢ ⎣ ck3 (x s , f k ) ⎦ = ⎣ δnk ⎦ ∈ C d C pk ck4 (x s , f k ) ⎡

(18.4)

By employing the discrete Fourier transform, the Fourier complex coefficient vector ck is expressed as N −1  2πk yn e−i N n , (18.5) ck = F ( yn ) = n=0

In the global FMD analysis, we perform the discrete Fourier transform to transfer the fluctuation function from the time domain to the frequency space f = [ f 0 f 1 . . . f k . . . f N −1 ] ∈ R1×N to obtain the global spatial mode matrix C = [c0 c1 . . . ck . . . c N −1 ] ∈ C M×N . The fast Fourier transform algorithm is used to speed up the Fourier transform process. The global spatial mode matrix C consists of the decomposed global spatial mode at different frequencies f k . The real and imaginary parts of the spatial mode show an explicit physical significance for the spatial disturbance structures (modal shapes) and their intensity as well as initial phase difference. We define the global amplitude spectrum Ak and the global phase spectrum θ k at M discrete points for a specific frequency f k as ⎡

⎤ ⎡ ⎤ f f Ak1 (x p , f k ) |ck1 (x p , f k )| ⎢ A (x f , f )⎥ ⎢|c (x f , f )|⎥ M×1 p k2 k ⎥ ⎢ k2 p k ⎥ Ak = Ak (X, f k ) = ⎢ , ⎣ Ak3 (x s , f k ) ⎦ = ⎣ |ck3 (x s , f k )| ⎦ ∈ R s s Ak4 (x , f k ) |ck4 (x , f k )|

θ k = θk i j = arg(ck i j ) ∈ R M×1 ,

(18.6)

(18.7)

18.2 Global Fourier Mode Decomposition

937

where | · | represents the modulus of the Fourier complex coefficient. The argument of the Fourier complex coefficient represents the phase angle. We define the Frobenius norm · of the global amplitude spectrum matrix corresponding to each physical variable in the whole spatial domain. The purpose is to calculate the global power spectrum and identify the dominant modes from the entire decomposed mode matrix. The global power spectrum matrix sk corresponding to each physical variable field at a discrete frequency f k is expressed as ⎤ ⎡ ⎤ f  Ak1 (x p , f k ) sk1 ( f k ) ⎢sk2 ( f k )⎥ ⎢ A (x f , f )⎥ 4×1 ⎥ ⎢ k2 p k ⎥ sk = sk ( f k ) = ⎢ ⎣sk3 ( f k )⎦ = ⎣  Ak3 (x s , f k ) ⎦ ∈ R , sk4 ( f k )  Ak4 (x s , f k ) ⎡

(18.8)

where sk1 ( f k ), sk2 ( f k ), sk3 ( f k ) and sk4 ( f k ) denote the global power spectrum for the spatial modes ω y k , C p k , δnk and C dp k , respectively. The contribution of individual spatial mode to the overall dynamic responses at a discrete frequency f k is defined as the global mode energy ek written as ⎡

⎤

ek1 ( f k ) ⎢ek2 ( f k )⎥ ⎥= ⎢ ek = ek ( f k ) = ⎣ ek3 ( f k )⎦ ek4 ( f k )

⎡

⎤

sk1 ( f k )  N −1 sk1 ( f k ) ⎥ ⎢ k=0 k2 ( f k ) ⎥ ⎢  Ns−1 ⎢ k=0 sk2 ( fk ) ⎥ ⎢ sk3 ( fk ) ⎥ ⎥ ⎢  N −1 ⎣ k=0 sk3 ( fk ) ⎦ sk4 ( f k )  N −1 k=0 sk4 ( f k )

∈ R4×1 ,

(18.9)

After that, the global power spectrum and the global mode energy spectrum in the transformed frequency space f = [ f 0 f 1 . . . f k . . . f N −1 ] ∈ R1×N can be stored in the matrix form as follows: S = [s0 s1 . . . sk . . . s N −1 ] ∈ R4×N ,

(18.10)

E = [e0 e1 . . . ek . . . e N −1 ] ∈ R4×N ,

(18.11)

By employing the global FMD analysis for the fluid-structure interaction problems, the aeroelastic modes are calculated by projecting the simultaneously collected aeroelastic responses from the spatial-temporal space to the spatial-frequency space (Fig. 18.2). The obtained aeroelastic modes in the fluid and structural domains and their global mode energy spectrum are stored in frequency ranking. The dominant modes can be determined as the modes with the most mode energies in the mode energy spectrum. Subsequently, these dominant modes are extracted from the frequency-ranked global mode sequence at the selected frequencies with relatively large mode energies. In the current study, we are interested in four physical variables (ω y , C p , δn and C dp ) in the coupled fluid-structure system due to their close connections with the aeroelastic modes. Hence, the aforementioned mode decomposition procedure is demonstrated for these four selected physical variables. The proposed global FMD approach can be naturally extended to the mode decomposition for

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18 Aeroelastic Mode Decomposition

Fig. 18.2 Illustration of transform of the physical data in temporal space and frequency space via Fourier transform and inverse Fourier transform

other physical variables of interest (e.g., flow velocity and structural velocity) in the fluid and structural domains. The collected physical variables can be stored together into the total state vector form shown in Eq. (18.1) for further mode decomposition process. The global FMD method avoids the limitation of the traditional Fourier transform analysis at a single point. It provides a global view to reflect the dynamics of the entire physical field by containing the information of the decomposed modes at each spatial point. The global FMD analysis can establish direct correspondences of the decomposed fluid and structural Fourier modes by choosing the modes in both domains at the same selected frequency. Hence, it is helpful to build an intrinsic relationship between the flow-induced vibrations and the coherent flow structures to reveal the physical mechanism of fluid-structure interaction because of the explicit physical interpretation of the FMD results. In the aeroelastic mode decomposition via FMD, Welch’s method is also considered to improve the detection performance to help identify the most influential aeroelastic modes.

18.3 Mode Selection of Three-Dimensional Flexible Thin Structure 18.3.1 Aeroelastic Mode Decomposition In this section, we apply the proposed global FMD method to decompose the coupled system into frequency-ranked aeroelastic modes. The influential modes are identified by detecting the frequency peaks in the mode energy spectrum. For simplicity, we first demonstrate the decomposition process of the aeroelastic mode decomposition framework for the flexible membrane at the AOA of α = 15◦ . Detailed explanations of the decomposed aeroelastic modes are then provided. Subsequently, we summarize the influential modes for the flexible membrane at two higher AOAs.

18.3 Mode Selection of Three-Dimensional Flexible Thin Structure

18.3.1.1

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Mode Decomposition at α=15◦

To perform the mode decomposition, we collect 1024 equispaced time-varying samples with a sampling frequency of f sam = 2000 Hz within the region indicated by the gray color as shown in Fig. 17.15. The pressure coefficient and the vorticity along the spanwise direction at the body-fitted grids in the fluid domain are projected onto a stationary reference mesh via the RBF method. The membrane displacement and the pressure difference on the membrane surface are collected in the Lagrangian coordinate. All the physical variables are collected simultaneously to ensure the correlation between the modes corresponding to each physical variable at a specific frequency. The global mode energy spectra calculated from Eq. (18.9) are presented in Fig. 18.3 a. Two obvious frequency peaks at f c/U∞ = 0.99 and 1.96 are observed in the computed mode energy spectra. It is noticed that the energetic frequencies are consistent for the decomposed structural and fluid Fourier modes. This indicates that the membrane vibrations and the flow fluctuations are excited in a frequency-synchronized way, resulting in the well-known frequency lock-in phenomenon. The decomposed aeroelastic modes colored by the real part of the Fourier transform coefficients based on the displacements and the pressure difference distributions of the membrane surface at the selected frequency of f c/U∞ = 0.99 and 1.96 are plotted in Fig. 18.3b–e, respectively. We notice that a typical chordwise second mode is excited at f c/U∞ = 0.99 and a high-order mode both in the chordwise and spanwise directions is observed at a higher frequency of f c/U∞ = 1.96. Except for the decomposed surface pressure modal shapes near the leading edge, the overall modes present similar modal shapes as the decomposed surface displacement modes at both energetic frequencies. We extract the dynamic Fourier modes in the spatial pressure and Y -vorticity fields on the mid-span plane. These Fourier modes are used to study the spatial flow structures correlated with the membrane vibration. In the plots of the decomposed fluid Fourier modes, the structural modal shape corresponding to the same frequency is added to help understand the correlation between the Fourier modes in the fluid and structural domains. Due to the small values of the structural Fourier modes, these structural modal shapes indicated by the black line are constructed by amplifying the corresponding structural Fourier modes based on the time-averaged membrane shape for visualization purposes. The real part Re(F (C p − C¯ p )) and the amplitude F (C p − C¯ p ) of the decomposed pressure fluctuation fields at the non-dimensional frequency of f c/U∞ = 0.99 corresponding to the chordwise second mode are shown in Fig. 18.4a, b, respectively. The real part of the transformed coefficient reflects the spatial structure of the mode. The amplitude represents the intensity distributions of the decomposed physical variables. Two small-scale pressure fluctuation regions are observed near the leading edge on the upper membrane surface. These pressure fluctuations are mainly caused by the rolled-up vortices at the leading edge in Fig. 18.5a. Two larger pressure fluctuation regions on the upper surface are generated during the periodic leading edge vortex shedding process. From the amplitude contour of the decomposed pressure field in Fig. 18.4b, the large-scale pressure pulsations with high values are noticed on the upper surface. The severe vorticity fluctuations are

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18 Aeroelastic Mode Decomposition

(%) Mode energy

30

f c/U∞ = 0.99

 δn Cp ωy

20

10

f c/U∞ = 1.96

0.5

1

1.5

2

2.5

f c/U∞

(a)

(b)

(c)

(d)

(e)

Fig. 18.3 Aeroelastic mode decomposition of 3D flexible membrane at α=15◦ : a mode energy spectra of the surface displacement fluctuations, the pressure coefficient fluctuations and the Y vorticity fluctuations based on the FMD analysis; the decomposed membrane displacement modes at f c/U∞ = b 0.99 and d 1.96 and the surface pressure difference modes at f c/U∞ = c 0.99 and e 1.96

mainly formed at the periodic vortex shedding regions near the leading and trailing edges in Fig. 18.5b. As the increase of the non-dimensional frequency to f c/U∞ = 1.96 with the high-order mode, the pressure wavelength and the flow scales become smaller. The high-intensity pressure pulsations still keep close to the membrane surface as shown in Fig. 18.4d. However, the amplitude values in this region are far less than those at f c/U∞ = 0.99 due to the weaker mode energy of the high-order mode. Meanwhile, the small-scale vortices originating from the leading edge move backwards to merge

18.3 Mode Selection of Three-Dimensional Flexible Thin Structure

Re(F (Cp − C¯p ))

: -0.05 -0.04 -0.03 -0.02 -0.01

0

0.01

0.02

0.03

0.04

0.05

 



 F (Cp − C¯p )

:

0

0.02

0.18

0.2

0.003 0.006 0.009 0.012 0.015 0.018 0.021 0.024 0.027

0.03

(a)

Re(F (Cp − C¯p ))

: -0.01 -0.008 -0.006 -0.004 -0.002

941

0.04

0.06

0.08

0.1

0.12

0.14

0.16

(b)

0

0.002 0.004 0.006 0.008 0.01

   ¯p ) : F (Cp − C

0

(c)

(d)

Fig. 18.4 Aeroelastic mode decomposition of 3D flexible membrane at α=15◦ : contours of the real part (a, c) and the amplitude (b, d) of the Fourier transform coefficients of the pressure coefficient fluctuation field corresponding to the non-dimensional frequency of f c/U∞ = (a, b) 0.99 and (c, d) 1.96

Re(F (ωy − ω ¯ y )): -30 -27 -24 -21 -18 -15 -12 -9 -6 -3 0

3

6

9 12 15 18 21 24 27 30

|F (ωy − ω ¯ y )|

:

0

12 24 36 48 60 72 84 96 108 120 132 144 156 168 180 192

(b)

(a)

Re(F (ωy − ω ¯ y )): -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

(c)

1

2

3

4

5

6

7

8

9 10

|F (ωy − ω ¯ y )|:

0

2

4

6

8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

(d)

Fig. 18.5 Aeroelastic mode decomposition of 3D flexible membrane at α=15◦ : contours of the real part (a, c) and the amplitude (b, d) of the Fourier transform coefficients of the Y −vorticity fluctuation field corresponding to the non-dimensional frequency of f c/U∞ = (a, b) 0.99 and (c, d) 1.96

942

18 Aeroelastic Mode Decomposition

with the trailing edge vortices behind the membrane in Fig. 18.5c. It can be observed from Fig. 18.5d that the high-intensity vorticity fluctuation region shrinks and the amplitude value in this region is reduced, compared to the decomposed vorticity field at the dominant frequency of f c/U∞ = 0.99.

18.3.1.2

Mode Decomposition at α=20◦ and 25◦

The aeroelastic responses at α = 20◦ and 25◦ are also decomposed via the proposed FMD technique. It can be seen from Figs. 17.15 and 17.16 that the aeroelastic responses are non-periodic and contain some noise components at α = 20◦ and 25◦ . To improve the detection performance, Welch’s method is employed in the mode energy spectrum estimation for these two cases. As shown in Fig. 17.15c and e, we extract 1024 equispaced snapshots within the selected time range indicated by the gray color from the coupled fluid-membrane system. The membrane displacements, the surface pressure coefficient, the spatial pressure coefficient and the spanwise Y vorticity are collected in the same sampling frequency of f sam =2000 Hz. The data sequence is split into two overlapping data segments and each segment is windowed with a Hamming window. Figure 18.6 presents the aeroelastic mode spectra and the energetic structural modes at four selected frequencies. It can be seen from Fig. 18.6a that the flexible membrane is highly coupled with the unsteady flow and responds in a frequencysynchronized manner, resulting a frequency-lock phenomenon. Different from the mode energy spectra at α = 15◦ , some low frequency components within the range of f c/U∞ ∈ [0, 0.6] are observed from the mode energy spectra at α = 20◦ . The aeroelastic modes at f c/U∞ = 0.122 are chosen to examine the aeroelastic characteristics caused by the low frequency components. The aeroelastic modes at f c/U∞ = 0.727 are the dominant modes in the coupled system, which exhibit the largest mode energies. A second harmonic frequency of the dominant frequency is observed at f c/U∞ =1.45. We also investigate the aeroelastic modes at a higher frequency of 2.82. As shown in Fig. 18.6d, the dominant structural mode is a chordwise second and spanwise first mode. In Fig. 18.6f and h, higher order modes both in the chordwise and spanwise directions are noticed. We observe some occasionally occurring high-order modes from the instantaneous structural displacement fluctuations shown in Fig. 17.18d. However, these high-order modes are covered by the dominant modes. With the aid of the mode decomposition techniques, these overlapping modes with lower mode energies can be separated from the coupled system. The correlated modal shapes based on the surface pressure difference are also extracted together with the structural modes at the selected frequencies. The surface pressure difference presents overall similar modal shapes as the structural displacement fluctuation except for some differences near the leading edge. The fluid modes based on the real parts of the Fourier transform coefficients of the pressure coefficient fluctuation Re(F (C p − C¯ p )) and the Y -vorticity fluctuation Re(F (ω y − ω¯ y )) at the four selected frequencies are presented in Fig. 18.7. Large size vortices are noticed on the whole membrane surface at f c/U∞ = 0.122 in

18.3 Mode Selection of Three-Dimensional Flexible Thin Structure

943

Fig. 18.7b. Periodic vortex shedding can be observed near the leading and trailing edges at the dominant frequencies of f c/U∞ = 0.727 from Fig. 18.7 d. The vortices at the second harmonics are shed in a similar way to those at the dominant frequency but with smaller sizes. In the decomposed vorticity modes shown in Fig. 18.7h, some complex vortex structures are noticed, which are different from the vortex structures at the dominant frequency and its second harmonics. When the vortices flow past through the flexible membrane, pressure pulsations are induced. The spatial pressure modes frequency-synchronized with the vortex shedding modes are plotted in Fig. 18.7a, c, e, g. These pressure pulsations are coupled with the membrane vibrations with similar modal shapes. The mode energy spectra and the decomposed modes in the structural domain for the flexible membrane at α=25◦ are summarized in Fig. 18.8. The coupled system responds over a broadband frequency range. Similar to the coupled system at α=20◦ , some non-harmonics of the dominant structural frequency are also observed in the mode energy spectra as shown in Fig. 18.8 a. To understand the non-periodic aeroelastic responses, we select two energetic modes at f c/U∞ = 0.0403 and 0.2. The aeroelastic modes at the dominant frequency of 0.764 and a higher frequency of 2.866 are also extracted from the coupled system. The modal shapes of the structural displacement fluctuations and the surface pressure difference fluctuations are presented in Fig. 18.8b–i, respectively. The dominant structural mode exhibits a chordwise second and spanwise first modal shape. The structural vibration modes and the correlated surface pressure difference modes at the same frequency show overall similar modal shapes. The Fourier modes of the spatial pressure and the Y -vorticity in the fluid domain at four selected frequencies are presented in Fig. 18.9. The vorticity modes at f c/U∞ = 0.0403 and 0.2 consist of large size vortices shed from the whole surface. These vortex shedding patterns are coupled with the chordwise first structural mode. The vortices are shed from the leading edge at the dominant frequency of 0.764 and a chordwise second structural mode is observed. The vortex structures become complex and the vortex sizes are reduced significantly at f c/U∞ = 2.866, which are associated with a chordwise third structural mode.

18.3.2 Effect of Flexibility From the observation of the aeroelastic responses and the spatial flow structure of the flexible membrane, several intertwined modes are excited through fluid-membrane interaction at higher AOAs. The vortex shedding frequency is synchronized with the membrane vibration frequency, leading to the frequency lock-in phenomenon. As the AOA increases, the aeroelastic response tends to be non-periodic. The results indicate that the membrane flexibility plays an important role in selecting particular aeroelastic modes via an underlying mechanism. To explore the role of flexibility in the aeroelastic mode selection process, we further simulate rigid flat wings and rigid cambered wings at three AOAs.

944

18 Aeroelastic Mode Decomposition

(%) Mode energy

30  δn Cp ωy

20

10

0

0

0.5

1

1.5

2

2.5

3

3.5

4

f c/U∞

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Fig. 18.6 Aeroelastic mode decomposition of 3D flexible membrane at α=20◦ : a mode energy spectra of the surface displacement fluctuations, the pressure coefficient fluctuations and the Y vorticity fluctuations based on the FMD analysis; b, d, f, h the decomposed membrane displacement modes and c, e, g, i the surface pressure difference modes at f c/U∞ = b, c 0.122, d, e 0.727, f, h 1.45 and h, i 2.82

18.3 Mode Selection of Three-Dimensional Flexible Thin Structure

Re(F (Cp − C¯p )): -0.02

-0.012

-0.004

0.004

0.012

0.02

Re(F (ωy − ω ¯ y )) : -30

-24

945

-18

(a)

Re(F (Cp − C¯p )): -0.04

-0.024

-0.008

0.008

0.024

0.04

Re(F (ωy − ω ¯ y )) : -30

-24

-18

(c)

Re(F (Cp − C¯p )): -0.02

-0.012

-0.004

-12

-0.012

-0.004

(g)

0

6

12

18

24

30

-12

-6

0

6

12

18

24

30

-2

0

2

4

6

8

10

1.2

1.8

2.4

(d)

0.004

0.012

0.02

Re(F (ωy − ω ¯ y )) : -10

-8

-6

-4

(e)

Re(F (Cp − C¯p )): -0.02

-6

(b)

(f)

0.004

0.012

0.02

Re(F (ωy − ω ¯ y )) :

-3

-2.4

-1.8

-1.2

-0.6

0

0.6

3

(h)

Fig. 18.7 Aeroelastic mode decomposition of 3D flexible membrane at α=20◦ : contours of the real part of the Fourier transform coefficients of a, c, e, g the pressure coefficient fluctuation field and b, d, f, h the Y −vorticity fluctuation field corresponding to the non-dimensional frequency of f c/U∞ = a, b 0.122, c, d 0.727, e, f 1.45 and g, h 2.82

946

18 Aeroelastic Mode Decomposition

(%) Mode energy

20  δn Cp ωy

15 10 5 0

0

0.5

1

1.5

2

2.5

3

3.5

4

f c/U∞

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Fig. 18.8 Aeroelastic mode decomposition of 3D flexible membrane at α=25◦ : a mode energy spectra of the surface displacement fluctuations, the pressure coefficient fluctuations and the Y vorticity fluctuations based on the FMD analysis; b, d, f, h the decomposed membrane displacement modes and c, e, g, i the surface pressure difference modes at f c/U∞ = b, c 0.0403, d, e 0.2, f, h 0.764 and h, i 2.866

18.3 Mode Selection of Three-Dimensional Flexible Thin Structure

Re(F (Cp − C¯p )): -0.03 -0.024 -0.018 -0.012 -0.006

0

0.006 0.012 0.018 0.024

0.03

947

Re(F (ωy − ω ¯ y )): -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0

(a)

Re(F (Cp − C¯p )): -0.05 -0.04 -0.03 -0.02 -0.01

0

0.01

0.02

0.03

0.04

0.05

Re(F (ωy − ω ¯ y )): -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0

(c)

Re(F (Cp − C¯p )): -0.02 -0.016 -0.012 -0.008 -0.004

0

(g)

4

6

8 10 12 14 16 18 20

2

4

6

8 10 12 14 16 18 20

2

4

6

8 10 12 14 16 18 20

(d)

0

0.004 0.008 0.012 0.016

0.02

Re(F (ωy − ω ¯ y )): -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0

(e)

Re(F (Cp − C¯p )): -0.02 -0.015 -0.01 -0.005

2

(b)

(f)

0.005

0.01

0.015

0.02

0.025

0.03

Re(F (ωy − ω ¯ y ))

:

-2

-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

2

(h)

Fig. 18.9 Aeroelastic mode decomposition of 3D flexible membrane at α=25◦ : contours of the real part of the Fourier transform coefficients of a, c, e, g the pressure coefficient fluctuation field and b, d, f, h the Y −vorticity fluctuation field corresponding to the non-dimensional frequency of f c/U∞ = a, b 0.0403, c,d 0.2, e, f 0.764 and g, h 2.866

948

18 Aeroelastic Mode Decomposition

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Fig. 18.10 Comparison of flow features between rigid flat wings, rigid cambered wings and flexible membrane wings at α = a, d, g 15◦ , b, e, h 20◦ and c, f, i 25◦ : a, b, c membrane profiles along the chord at the mid-span location, d, e, f time-averaged pressure coefficient difference between the upper surface and the lower surface and g, h, i turbulent intensity on the mid-span plane

Figure 18.10a–c present the comparison of the membrane profiles along the chord at the mid-span location between these three types of wings at different AOAs, respectively. The rigid flat wing has the same wing geometry as the undeformed geometry of the flexible membrane. The rigid cambered wing shares the same wing shape as the mean wing shape of the flexible membrane under aerodynamic loads. Similar to the study of the 2D membrane in Sect. 17.3, the flexibility affects the membrane dynamics from two aspects, namely (i) camber effect and (ii) flow-induced vibration. This investigation provides a perspective to explore how the flexible structure is coupled with the unsteady flow to alter the flow features and excite particular structural modes. To further investigate the effect of flexibility on the flow features, we compare the time-averaged pressure coefficient difference on the membrane surface and the turbulent intensity on the mid-span plane for the three types of wings at different AOAs. It can be observed from Fig. 18.10d–f that the wing camber can enlarge the suction area. When the membrane vibration is introduced, the suction area is further extended to the trailing edge. As shown in Fig. 18.10g–i, compared to the rigid flat

18.3 Mode Selection of Three-Dimensional Flexible Thin Structure

949

wing, the wing camber can suppress the turbulent intensity at α = 15◦ , but exhibits smaller influences on the turbulent intensity at higher AOAs. When the membrane vibration is coupled with the unsteady flow, the high turbulent intensity region gets closer to the membrane surface.

18.3.3 Aeroelastic Mode Selection Strategy in Separated Flow The comparison of the flow features among the rigid flat wing, the rigid cambered wing and the flexible membrane wing offers an opportunity to explore the aeroelastic mode selection strategy when the coupling effect is considered. With the aid of the mode decomposition technique, we further examine the relationship between the membrane aeroelasticity and the bluff body vortex shedding instability. We analyze the Fourier modes and the corresponding mode energy spectra in the fluid domain for the rigid flat wing and the rigid cambered wing. The spanwise Y -vorticity is selected to perform the mode decomposition, which reflects the vortex shedding along the chord direction. As Rojratsirikul et al. [688] reported in their study, the vortex shedding frequency of various finite wings at different AOAs was observed vs ∈ [0.15, 0.2]. The modified within a modified Strouhal number range of f cUsin(α) ∞ Strouhal number is scaled by the AOA, which reflects the standard bluff body vortex vs shedding frequency. In Fig. 18.11, the summarized Strouhal number range of Uf ∞c ∈ 0.15 0.2 [ sin(α) , sin(α) ] indicated by a gray region is added to the plots to explore the connection between the bluff body vortex shedding instability and the membrane aeroelasticity. It can be seen from Fig. 18.11 that the dominant vortex shedding frequency of a vsrigid flat wing at different AOAs is close to or falls into the frequency range of f c 0.15 0.2 ∈ [ sin(α) , sin(α) ] related to the bluff body vortex shedding instability. Except for U∞ the dominant frequency detected in the mode energy spectra, some smaller frequency components with lower mode energies are also observed. As the AOA increases, the turbulent vortex structures behind the rigid flat wing become more complex. When the camber effect is taken into account for the rigid cambered wing, we observe that the dominant frequency is reduced slightly than that of the rigid flat wing, but it is still close to the bluff body vortex shedding frequency range. In Fig. 18.12, we compare the Fourier modes in the Y -vorticity field associated with the bluff body vortex shedding phenomenon between the rigid flat wing and the rigid cambered wing at three AOAs. These Fourier modes are selected at the specific frequencies near or within the standard bluff body vortex shedding frequency range. The flow features of these two wings exhibit some similarities at α = 15◦ . The wing camber changes the vortical structures at higher AOAs. Compared to the vortical structures at α = 15◦ , the vortical structures become more complex as the AOA increases regardless of the camber effect. As shown in Fig. 18.11, the mode energy spectra of the flexible membrane wings based on the spanwise Y -vorticity and the membrane displacement are also added to the plots. The purpose is to investigate the role of membrane flexibility in membrane

950

18 Aeroelastic Mode Decomposition 0.15 sin(20◦ )

0.2 sin(15◦ ) n f21

30

Rigid flat (ωy ) Rigid cambered (ωy )

20

Flexible Flexible

(ωy ) (δn )

10 0

0

0.5

1

0.2 sin(20◦ )

30

1.5

2

(%) Mode energy

(%) Mode energy

0.15 sin(15◦ )

Rigid flat (ωy ) Rigid cambered (ωy ) Flexible (ωy ) Flexible (δn )

10 0

2.5

n f21

20

0

0.5

1

1.5

2

2.5

3

3.5

4

f c/U∞

f c/U∞

(b)

(a) 0.15 0.2 sin(25◦ ) sin(25◦ )

(%) Mode energy

20 n f21

15

Rigid flat (ωy ) Rigid cambered (ωy ) Flexible (ωy )

10

Flexible (δn )

5 0

0

0.5

1

1.5

2

2.5

3

3.5

4

f c/U∞

(c)

Fig. 18.11 Comparison of global mode energy spectra of the spatial Y -vorticity fluctuation (ωy ) between a rigid flat wing, a rigid cambered wing and a flexible membrane at α = a 15◦ , b 20◦ , and c 25◦

aeroelasticity. Based on the analysis in previous section, we observe that the flexible membrane vibration locks into a chordwise second and spanwise first mode at the three AOAs. A natural question to ask is whether the dominant frequency of the coupled system is dependent on the natural frequency of a chordwise second and spanwise first mode. We employ the derived approximate analytical formula of the nonlinear natural frequency to estimate the natural frequency of the corresponding structural mode for a rectangular membrane immersed in an unsteady flow. The approximate analytical formula of the nonlinear natural frequency f inj corresponding to a chordwise i-th and spanwise j-the mode is given as f inj =

1 2π

 3βκωi j0 2 ωi j0 + (g0 + h 20 ) , 8

(18.12)

where ωi j0 is the circular frequency corresponding to the initial tensions applied to the membrane. β denotes the perturbation parameter in the Poincar´e-Lindstedt perturbation method and κ represents the coefficient of the vibration equation. g0 and h 0 are the initial conditions of the membrane displacement and velocity of the vibrating membrane. In Eq. (18.12), the second term is caused by the geometric nonlinearity and is related to the membrane vibration amplitude. The dynamic stress caused by the membrane transverse displacement can increase the natural frequency. The added mass m am makes the natural frequency smaller than that in a vacuum. Based on our high-fidelity numerical simulation results, we can determine the n correlevant parameters to calculate the nonlinear structural natural frequency f 21

18.3 Mode Selection of Three-Dimensional Flexible Thin Structure

Re(F (ωy − ω ¯ y )): -30 -27 -24 -21 -18 -15 -12 -9 -6 -3 0

3

6

9 12 15 18 21 24 27 30

Re(F (ωy − ω ¯ y )):

951

-20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0

Re(F (ωy − ω ¯ y ))

: -10

-8

-6

-4

-2

0

2

4

6

8

10

Re(F (ωy − ω ¯ y ))

: -10

-8

-6

(c)

Re(F (ωy − ω ¯ y ))

: -10

-8

-6

-4

(e)

2

4

6

8 10 12 14 16 18 20

(b)

(a)

-4

-2

0

2

4

6

8

10

-2

0

2

4

6

8

10

(d)

-2

0

2

4

6

8

10

Re(F (ωy − ω ¯ y ))

: -10

-8

-6

-4

(f)

Fig. 18.12 Comparison of the Fourier modes in the Y −vorticity fluctuation field associated with the bluff body vortex shedding phenomenon between a, c, e rigid flat wings and b, d, f rigid cambered wings at α = a, b 15◦ , c, d 20◦ and e, f 25◦ . These Fourier modes are selected at the non-dimensional frequency of f c/U∞ = a 0.5239, b 0.4297, c 0.5103, d 0.4431, e 0.4286 and f 0.3633

responding to the chordwise second and spanwise first mode of the coupled system. n for different AOAs is indicated by The estimated structural natural frequency f 21 a purple long dash line in Fig. 18.11. By comparing with the estimated nonlinear n , it can be seen from Fig. 18.11 that both the vortex shedding natural frequency f 21 frequency and the membrane vibration frequency are close to the estimated natural n . It can be inferred that the vortex shedding frequency locks into the frequency f 21 n to sustain the flow-excited vibration in a chordwise structural natural frequency f 21 second and spanwise first mode. The frequency corresponding to the bluff-body-like vortex shedding process is not observed in the coupled system at α = 15◦ . The vortex structures are regulated to get closer to the membrane surface and shed into the wake in a dominant frequency of f c/U∞ = 0.99 via the frequency lock-in. As the AOA increases, the flexible

952

18 Aeroelastic Mode Decomposition

Re(F (ωy − ω ¯ y ))

: -10

-8

-6

-4

-2

0

2

4

6

8

Re(F (ωy − ω ¯ y ))

10

: -10

-8

-6

-4

(a)

-2

0

2

4

6

8

10

(b)

Re(F (ωy − ω ¯ y )) : -10

-8

-6

-4

-2

0

2

4

6

8

10

(c) Fig. 18.13 Comparison of the Fourier modes in the Y −vorticity fluctuation field associated with the bluff body vortex shedding phenomenon between a rigid flat wings, b rigid cambered wings and c flexible membrane wing at α = 20◦ . These Fourier modes are selected at the non-dimensional frequency of f c/U∞ = a 0.122, b 0.135 and c 1.22

Re(F (ωy − ω ¯ y ))

: -10

-8

-6

-4

-2

0

2

4

6

8

Re(F (ωy − ω ¯ y ))

10

: -20

-16

-12

(a)

-8

-4

0

4

8

12

16

20

(b)

Re(F (ωy − ω ¯ y )): -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0

2

4

6

8 10 12 14 16 18 20

(c) Fig. 18.14 Comparison of the Fourier modes in the Y −vorticity fluctuation field associated with the bluff body vortex shedding phenomenon between a rigid flat wings, b rigid cambered wings and c flexible membrane wing at α = 25◦ . These Fourier modes are selected at the non-dimensional frequency of f c/U∞ = a, b, c 0.2

18.3 Mode Selection of Three-Dimensional Flexible Thin Structure

953

Fig. 18.15 Illustration of a feedback cycle of fluid-membrane coupled mechanism

membrane also responds at several low frequency components to vibrate. These low-frequency components are also observed in the mode energy spectra of the rigid flat wing and the rigid cambered wing, which are associated with the bluff body vortex shedding instability. Except for the dominant vibrational modes caused by the frequency lock-in, we are also interested in the aeroelastic modes at these low frequencies. The Fourier modes in the Y -vorticity field corresponding to similar frequencies in the low frequency range at α = 20◦ and 25◦ are compared for the rigid flat wing, the rigid cambered wing and the flexible membrane shown in Figs. 18.13 and 18.14. By investigating the fluid Fourier modes at these low frequencies for the three types of wings, we find that the camber effect and the flow-excited vibration can change the flow features to some extent. The Fourier modes of the Y -vorticity field of the rigid cambered wing and the flexible membrane exhibit some similarities in the modal shapes. It can be inferred that the aeroelastic modes corresponding to the low frequency components in the coupled fluid-membrane system are associated with the bluff body vortex shedding instability behind a cambered up wing. The aeroelastic modes caused by the bluff body vortex shedding instability are intertwined with the aeroelastic mode that depends on the frequency lock-in, leading to the non-periodic aeroelastic responses at α = 20◦ and 25◦ . The investigation of the decomposed aeroelastic modes suggests a feedback cycle between the vibration mode and the unsteady aerodynamics, as shown in Fig. 18.15. The cycle reveals the mode selection mechanism for fluid-membrane interaction problems with obvious vortex shedding phenomenon. In this coupled system, the vibrational modes excite the separated shear layer to roll up earlier and then form large-scale vortices. As these vortices detach from the membrane surface and are convected downstream, relatively stronger surface pressure fluctuations are induced due to the passing-by vortical structures. Subsequently, the flexible membrane is synchronously driven by the pressure pulsations to excite particular vibrational modes with similar modal shapes. The modal shapes can be observed from the dominant

954

18 Aeroelastic Mode Decomposition

decomposed surface pressure and vibrational modes at the same frequency. Eventually, the unsteady flow and the membrane vibration enter a strongly coupled state and the frequency synchronization to select the particular dominant aeroelastic modes.

18.4 Summary We presented an aeroelastic mode decomposition framework based on the radial basis function interpolation method and the global Fourier mode decomposition technique. We extracted and identified the Fourier modes of interest both in the fluid and structure fields in a unified manner. The three-dimensional membrane aeroelasticity was simulated by a high-fidelity fluid-structure interaction solver at three angles of attack. The flexible membrane exhibited a periodic aeroelastic response at α = 15◦ . The aeroelastic response became non-periodic at higher angles of attack. By comparing the dominant modal shapes observed from the standard deviation analysis and the instantaneous displacement, it was found that the standard deviation was not a reliable indicator to reflect the dominant modes from the coupled system with overlapping modes due to the time-averaged sense. With the aid of the aeroelastic mode decomposition framework, the correlated dominant fluid and structure modes were successfully extracted from the coupled system by detecting the frequency peaks in the mode energy spectra. Based on the mode decomposition analysis, we observed a frequency synchronization between the vortex shedding process and the membrane vibration. The flexible membrane exhibited a similar modal shape with a chordwise second and spanwise first mode at different angles of attack. To explore the role of flexibility in membrane aeroelasticity, we assessed the flow features and the dynamic modes of a rigid flat wing, a rigid cambered wing and their flexible counterpart at three angles of attack. An approximated analytical formula of the nonlinear natural frequency was employed to estimate the natural frequency n corresponding to the chordwise second and spanwise first mode. By comparing f 21 with the mode energy spectra of the coupled system, it was observed that the vorn . Through the tex shedding frequency locked into the structural natural frequency f 21 comparison of fluid modes corresponding to non-integer low frequency components, the fluid modes of the flexible membrane showed some similarities to those of the rigid cambered wing at α = 20◦ and 25◦ . The aeroelastic modes corresponding to the low-frequency components can be attributed to the bluff body vortex shedding instability. The non-periodic aeroelastic responses were caused by the interaction between the aeroelastic modes associated with the frequency lock-in and the bluff body vortex shedding instability. Based on the modal analysis, we suggested a feedback cycle between the membrane vibration mode, the vortex shedding process and the surface pressure fluctuations. This feedback cycle revealed that the membrane flexibility acted as a coordinator between the flexible membrane and the unsteady flow to form a frequency lock-in phenomenon to select the dominant mode and sustain the membrane vibration. This mode decomposition method has the potential to be extended to the data analysis of other fluid-structure interaction problems. A

18.4 Summary

955

combined application with other mode decomposition techniques could offer better physical insight and causal inference for fluid-structure interaction problems. Acknowledgements Some parts of this Chapter have been taken care from the Ph.D. thesis of Guojun Li carried out at the National University of Singapore and supported by the Ministry of Education, Singapore.

Chapter 19

Flow-Excited Instability in Thin Structure Aeroelasticity

This chapter examines the coupled dynamics of three-dimensional flexible thin structures with membrane-like (of small bending rigidity) components at a moderate angle of attack. The coupled dynamics involve fully-coupled interactions of unsteady separated flow with the deformable flexible structure. The incompressible Navier-Stokes and the structural equations are integrated via a partitioned iterative procedure. The present study investigates the synchronized coupling between the vortex shedding and the flexible membrane at a moderate Reynolds number. In particular, we explore the self-induced vibration and the lock-in phenomena during nonlinear unsteady fluid-membrane interaction for varying parameters such as Reynolds number, mass ratio and aeroelastic number. We analyze the coupled dynamics in separated flows associated with flow-excited instability over the parameter space. The instability boundary is demarcated based on the proposed stability phase diagram. The onset of flow-induced vibration and the mode transition phenomenon is examined by monitoring the variation of the structural natural frequency relative to the vortex shedding frequency.

19.1 Introduction 19.1.1 Flow-Excited Instability in Morphing Membrane Wings A fluid-membrane interaction between a flexible membrane structure (with negligible bending rigidity) and a surrounding unsteady flow is ubiquitous in nature and engineering systems. In the past decades, morphing fins/wings with flexible membrane components have received substantial attention from the aerospace engineering community in the context of bio-inspired swimming/flying vehicles at moderate Reynolds numbers [521, 712, 769]. During the biological movement, with the aid of the morphing structures, a flexible membrane can deform up passively and display complex © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Jaiman et al., Mechanics of Flow-Induced Vibration, https://doi.org/10.1007/978-981-19-8578-2_19

957

958

19 Flow-Excited Instability in Thin Structure Aeroelasticity

spatial-temporal vibrational dynamics through the flow-excited instability as functions of wing shapes, physical parameters and the boundary conditions [688, 726]. Flow-excited instability of a flexible membrane can lead to self-sustained vibrations, the so-called flow-induced vibration when the fluctuating flows are strongly coupled with the elastic thin membrane structures. Notably, a frequency synchronization of the vortex shedding frequency ( f vs ) and the frequency of the membrane vibrations ( f s ) may be established in the coupled fluid-membrane systems under specific conditions, resulting in the well-known frequency lock-in phenomenon [689, 750]. By properly harnessing such flow-excited instability, one can regulate the membrane kinematics and the vortex shedding features through the fluid-membrane coupling effect [542, 594, 721]. Understanding the coupled fluid-membrane dynamics can facilitate the development of effective control strategies to explore the potential for improved aerodynamic performance and smooth adaptability in a gusty flow. During fluid-membrane interaction, the self-sustained vibration associated with the flow-excited instability is found to appear under certain conditions [531, 688, 721]. As illustrated in Fig. 17.1a, the perturbation caused by the pressure fluctuations in the shear layer propagates backward and interacts with the flexible membrane to vibrate. The induced traveling waves and their boundary reflections are interacted to form intertwined vibrational modes. These multiple vibrational modes further increase the complexity of the flow-induced vibrations. Figure 17.1b shows a representative schematic of the unsteady separated flow past a 3D rectangular flexible membrane wing vibrating in a chord-wise second and spanwise first mode. Different from the two-dimensional (2D) membrane, the vibrations and the unsteady flows along the spanwise direction of a 3D membrane become important in the coupled system. These 3D effects further make the coupled membrane dynamics more complex and rich. From the perspective of the unsteady separated flow, the leading edge vortex (LEV), the trailing edge vortex (TEV) and the tip vortex (TV) are highly coupled with the membrane vibrations. The vibrational energy and the flow kinetic energy are redistributed through the fluid-membrane coupling effect. In the observed frequency lock-in phenomenon, these redistributed energies were mainly concentrated in several specific frequencies of the coupled fluid-membrane system, compared to its rigid counterpart [705]. As the governing fluid-structure parameters (e.g., Reynolds number, mass ratio and aeroelastic number) vary, a transition of the dominant aeroelastic modes associated with the vibration and the vortex shedding process may be observed [529, 531, 688, 726, 728]. Through numerous investigations, the natural frequency of the flexible membrane was found to play an important role in the flow-induced vibration [530, 588]. However, the processes of flow-induced vibrations of the membrane and the aeroelastic mode transition are not fully understood. The highly nonlinear dependence of the coupled dynamics on the fluid-structure parameters restricts us to gain a deeper insight into these phenomena. This paper aims to explain how the flow-excited instability governs the 3D fluid-membrane interaction characteristics and can be linked with the mode transition. Specifically, we examine the coupled fluid-membrane dynamics and the dominant dynamic modes during fluid-membrane interaction. The variation of the natural frequency of the flexible membrane relative

19.1 Introduction

959

to the vortex shedding frequency is monitored by properly changing the physical parameters. Recent studies based on experiments and numerical simulations suggested that the characteristics of the flow-induced vibration and the underlying dynamic modes were strongly related to the membrane performance. Some of these investigations primarily focused on the mean dynamic responses of the flexible membrane exposed to an unsteady flow. However, the unsteady dynamic behaviour induced by the flowexcited instability depicted in Fig. 17.1a plays an important role in the membrane performance [689, 705]. [721] examined the effects of aspect ratio, compliance and pre-strain values on the unsteady membrane dynamics for a low aspect ratio wing in wind tunnel experiments. The results indicated that the compliant wing can produce larger lift forces and delay stall, compared to its rigid counterpart. Secondary vortices induced by local separation at higher vibrational modes weakened the aerodynamic performance. Moreover, the flow-induced vibrations in the coupled system were found to exhibit different characteristics for membranes with different geometries of the leading and trailing edges [518, 704], different excess lengths and pre-strain values [690] and at various flow conditions (e.g., angle of attack and Reynolds number) [689]. A series of wind tunnel experiments were conducted to study the aeromechanics and near-wake characteristics of flexible membrane wings in and out of ground effect [530–532]. In the experiments, the flexible membrane wings were found to produce a variety of flow features related to different types of flow-induced vibrations. In nature, for example, bats are capable of dynamically changing the material properties via miniature muscles and the high degree-of-freedom skeleton system. Therefore, the material properties also play an important role in the flow-induced vibration and the membrane performance. However, only a few investigations on the effects of membrane material properties have been performed to understand the flow-induced vibration in the coupled system. Concerning the flow-induced vibration phenomenon, various vibrational modes and wake patterns are excited in the coupled system through the flow-excited instability. Mode transition between different aeroelastic modes was observed as the angle of attack (AOA) changes [529, 721] or under forced pitching motion [751]. The reason for the mode transition phenomenon during fluid-membrane interaction remains unclear and has not been systematically studied. Once the mode transition is triggered, the altered dominant modes might be fixed at a certain mode within a range of physical parameter space. For example, a typical chord-wise second mode with varied vibrational frequencies was always observed for a 2D membrane wing with the fixed leading and trailing edges when the AOA exceeded the stall angle at different free-stream velocities [689]. Similar conclusions have been drawn in wind tunnel experiments for 3D membrane wings at high incidences [691, 748]. Moreover, the occurrence of the chord-wise second mode was independent of the geometries at the leading and trailing edges [518]. [748] found that the membrane vibrating in the chord-wise second mode was related to the frequency lock-in behaviour of a rigid airfoil. The possible reason was attributed to the strong coupling of the membrane oscillations and the particular vortex shedding process in the wake.

960

19 Flow-Excited Instability in Thin Structure Aeroelasticity

In parallel to experimental research, some numerical studies have been carried out to gain further insight into the fluid-membrane interaction phenomenon. Some of the earliest relevant works focused on the 2D linear elastic membrane model with the small deformation assumption in a potential flow [667] or in a laminar flow [719]. The recently rapid development in computational fluid dynamics, computational structural dynamics and fluid-structure coupling algorithms provided a practical and reliable way to investigate various nonlinear membrane dynamics for a wide range of physical parameters. [588] developed a computational simulation solver by coupling a high-order Cartesian Navier-Stokes solver and a nonlinear membrane solver to study the 2D fluid-membrane interaction at low Reynolds numbers. The effects of AOA, the membrane rigidity, the membrane pretension and Reynolds number on the membrane dynamic responses have been investigated. The results confirmed the tight coupling between the membrane oscillations and the vortical structures in the wake, which was similar to the phenomena observed in the experiments [689, 721]. The evolutions of the nonlinear dynamic behaviour of a 2D membrane under periodic load [728] and the membrane with the perimeter reinforcement [726, 729] have been studied systematically. The results indicated that the membrane dynamics were closely dependent on the initial conditions applied to the membrane wing and the material properties. Besides, the membrane inertia enlarged the AOA range of a 2D flexible membrane in a stable state in a laminar flow [746]. However, the study on the effect of membrane inertia focused on the fluid-membrane interaction at small AOAs without obvious vortex structures. Only a handful of publications on fully coupled 3D fluid-membrane interaction analysis of flexible membrane dynamics can be found in the literature. Recently, [778] developed a fluid-membrane interaction solver based on the coupling between a fluid solver and a finite element structural solver and conducted a systematic validation for 3D membrane dynamics. The numerical results achieved a good agreement with the experimental data [689]. In previous studies on the membrane dynamics, the relationship between the natural frequency of the tensioned membrane and the vortex shedding frequency has been found to play an important role in the coupled system [529, 689, 721, 760]. Different from the fluid-structure interaction problems for freely vibrating rigid bodies with the fixed natural frequency, the natural frequency of the flexible membrane can be changed when the aerodynamic loads are applied to a stretchable membrane surface. A simple linear natural frequency model of a 3D rectangular flexible membrane with all fixed edges can be given as [627]:  1 f inj = 2

Ns ρs h

   2  21 i 2 j + , c b

(19.1)

where f inj is the linear natural frequency of the membrane with the chord-wise i and span-wise j mode. The membrane chord length and the span length are c and b, respectively. N s = E s hεs denotes the membrane tension per unit length. E s , h and εs represent Young’s modulus, the membrane thickness and the strain of the stretched

19.1 Introduction

961

membrane. ρ s is the membrane density. For a membrane with fixed geometries, f inj is only dependent on the structural properties (ρ s and E s ) and the strain εs caused by the membrane deformation under aerodynamic loads. The natural frequency variation estimated from the linear natural frequency equation was examined for flexible membranes with different membrane rigidities, AOAs, Reynolds numbers [588] and undergoing pitching motion [751]. The dominant structural mode of the vibrating membrane and the vortex shedding pattern were also changed, a phenomenon known as mode transition. [721] reported a series of wind tunnel experiments on membrane wings with different aspect ratios, compliance and prestrain values. The mode transition from the first mode to the higher-order mode was observed. They argued that this mode transition phenomenon was possibly caused by the nonlinear resonance mechanism between the constant vortex shedding frequency and the reduced natural frequency of the membrane or influenced by the increasing forcing frequency due to the vortex shedding process as the Reynolds number increased. However, the employed linear natural frequency model in Eq. (19.1) neglects the dynamic stress caused by the nonlinear vibration and the added mass of the vibrating membrane immersed in uniform flow. Thus, this simplified model is not suitable for the analysis of the nonlinear dynamics and the mode transition in a coupled fluid-membrane system with moderate and large amplitude vibrations. A nonlinear natural frequency model that considers the added mass effect is required for further analysis. While some studies have been done to examine the physics of the flow-induced vibration and the mode transition, the underlying relationship between these phenomena and the role of the natural frequency is not yet fully understood in these coupled systems. The complexity and nonlinearity of the 3D coupled fluid-structure dynamics pose a severe challenge in the analysis. In addition, a large physical parameter space that governs the flow-induced vibration increases the difficulty of exploring the coupled dynamics in a comprehensive manner. This chapter presents physical insights into the coupled fluid-membrane dynamics of an extensible 3D membrane immersed in unsteady separated flows at moderate Reynolds numbers. Of particular interest is to examine the role of the flow-excited instability and the natural frequency of the 3D flexible membrane in the coupled fluidmembrane characteristics and the mode transition phenomenon. To shed light on the variation of the flow-induced vibration, we establish the stability phase diagrams in the m ∗ -Re space and the m ∗ -Ae space through a series of coupled numerical simulations. Our numerical simulations determined new empirical equations of the flow-excited instability boundary demarcated between the DSS and DBS regimes. The natural frequency of the tensioned membrane and the vortex structures are essential to understanding the onset of the flow-induced vibration and the mode transition. We employ the global Fourier mode decomposition (FMD) technique [638] to identify the dominant aeroelastic modes and classify the vibrational membrane states from the vibrating membrane responses. To isolate the impact of the relevant physical parameters on the flow-induced vibration and the membrane dynamics, we investigate the coupled fluid-membrane dynamics as a function of mass ratio m ∗ , Reynolds number Re and aeroelastic number Ae. We further compare the mode fre-

962

19 Flow-Excited Instability in Thin Structure Aeroelasticity

quency spectra and the energy distributions in the fluid and structural domains based on the FMD analysis. An approximate analytical formula of the nonlinear natural frequency of a simply supported 3D rectangular membrane in uniform flow considering the added mass is derived using the nonlinear structural dynamic equation based on large deflection theory. The vortex shedding frequencies are measured at the monitoring line in the wake via the Fourier-based signal analysis. The natural frequency of the membrane and the vortex shedding frequency are compared as a function of m ∗ , Re and Ae. We systematically examine the onset of the flow-induced vibration and the mode transition by monitoring the natural frequency variation. The findings and conclusions can be used to promote the development of active/passive control strategies for producing superior aerodynamic performance.

19.1.2 Organization To explore the variation of the membrane aeroelasticity in a broad range, we perform a series of numerical simulations for 3D membrane wings immersed in the unsteady flows at a representative AOA of α = 15◦ with apparently separated flows in this chapter. The coupled dynamics are simulated within a moderate Reynolds number range, which approaches to the operating Reynolds number range of biological flight and human-made air vehicle flight. There are several goals to be achieved in this chapter • Systematically explore the flow-excited instability of a 3D membrane in separated turbulent flows; • Classify distinct stability regimes and various vibrational states from the proposed stability phase diagrams; • Determine empirical relationships for the flow-excited instability boundary; • Estimate the membrane stretching and explore its relationship with the structural natural frequency through the areal strain; • Examine the onset of the flow-induced vibration and the dependence of the mode transition on the frequency lock-in. To achieve these goals, we first design the parameter space based on three key nondimensional parameters, namely mass ratio m ∗ , Reynolds number Re and aeroelastic number Ae, to establish the stability phase diagram. The design criterion is to sufficiently characterize the variation of the flow-excited instability, including the onset of membrane vibration and mode transition phenomena. New empirical equations of the flow-excited instability boundary demarcated between the stationary and vibration regimes are determined based on our numerical simulations. Similar to the studies in Chap. 18, we employ the global Fourier mode decomposition technique to identify the dominant aeroelastic modes and classify the membrane vibrational modes from the vibrating membrane responses. As discussed in Chap. 18, the structural natural frequency acts as a bridge to link the fluid dynamics with the

19.2 Flow-Excited Instability

963

flexible membrane dynamics to govern the flow-excited instability. The structural natural frequency of a vibrating membrane immersed in unsteady flows is highly dependent on the material properties (membrane density and Young’s modulus) and the fluid loads acting on the membrane. The membrane tension force is affected by the applied fluid loads to influence the structural natural frequency. The coupled dynamics with varying parameters offer a good entry point to examine the variation of the structural natural frequency relative to the vortex shedding frequency. The key idea here is to change the relevant physical parameters (m ∗ , Re and Ae) and make the structural natural frequency close to the vortex shedding frequency. The variation of the flow-excited instability and the onset of the membrane vibration are further investigated based on the systematic numerical simulation results. Furthermore, we explore the mode transition by monitoring the variation of the natural frequency.

19.2 Flow-Excited Instability 19.2.1 Stability Phase Diagram 19.2.1.1

Classification of Membrane States

Before we proceed to explore the flow-induced membrane vibration and the mode transition, we simulate the coupled 3D rectangular membrane dynamics as functions of mass ratio and Reynolds number. Six groups of mass ratios (M1→6) within [0.36, 9.6] and nine sets of Reynolds numbers (R1→9) ranging from 2430 to 97200 are chosen to form the parameter space to ensure that the flow-excited instability can be fully characterized. Figure 19.1 shows the stability phase diagram and the classification of the membrane states over the full parameter space. Two distinct stability regimes are observed from the stability phase diagram termed a deformedsteady state (DSS) and a dynamic balance state (DBS). These two stability states are termed from the perspective of the tension force involved in the nonlinear natural frequency model presented in Eq. (18.12). The membrane within the DSS regime refers to the membrane that is statically deformed under the aerodynamic loads and finally reaches a steady state. Meanwhile, the inherent tension force N s = E s hεs caused by membrane deformations is statically balanced with the aerodynamic loads Δp. The dynamic balance state is achieved when the dynamically changing tension force is balanced with the unsteady aerodynamic forces and the inertial forces. The membrane vibrates with a limited amplitude within the DBS regime. It can be seen from Fig. 19.1 that the flexible membrane maintains a static equilibrium state at low Re for light membrane structures. For brevity, the Young-Laplace equation can be employed here to describe the deformed-steady state of the membrane [721, 760], which can be written as Δp (19.2) E s h + s s = 0, εκ

964

19 Flow-Excited Instability in Thin Structure Aeroelasticity

Re

M1

10

5

10

4

M2

M3

M4

M5

M6

Static C1S2 C2S1 C2S5 C2S9

R9

Filled - Attached

DBS

Non-filled - Shedding 80000

10

R1

3

Re

60000

DSS

40000

Boundary

20000

2

m∗

4

6 8 1012

0 0

2

4

6

m∗

8

10

Fig. 19.1 Stability phase diagram: non-dimensional Reynolds number Re (R1→9) versus mass ratio m ∗ (M1→6) for the 3D flexible membrane at Ae = 423.14 for α = 15◦ . Here, the dashed line (− − −) is plotted to distinguish the flow-excited instability boundary.  denotes the simulation results corresponding to the deformed-steady state. , , ♦ and ◦ represent the chordwise first and spanwise second mode (C1S2), the chordwise second and spanwise first mode (C2S1), the chordwise second and spanwise fifth mode (C2S5) and the chordwise second and spanwise ninth mode (C2S9) in the dynamic balance state. The label with or without filled color is the flexible membrane with attached vortices or vortex shedding, respectively

where κ s is the curvature of the deformed membrane. In the current simulation, the first term in Eq. (19.2) remains a constant value over the full m ∗ -Re parameter space since the Young’s modulus E s and the thickness h are fixed. The values of the aerodynamic loads Δp, the membrane strain εs and the curvature κ s of the deformed membrane are dependent on Reynolds number Re and mass ratio m ∗ . Thus, the ratio R in the second term can be written as a function of Re and m ∗ , ∗ expressed as R = εΔp s κ s = f (Re, m ). In other words, the ratio R remains constant as ∗ Re and m vary. Through our high-fidelity numerical simulations, the flow-excited instability boundary between the DSS regime and the DBS regime can be observed in Fig. 19.1, which is plotted as a dashed curve line (− − −). The flow-excited instability boundary described by the function of Re and m ∗ should satisfy the relationship shown above. Based on the simulation results, a new empirical relation that governs the flow-excited instability boundary curve can be expressed as Recr = c0 + c1 (m ∗ )n ,

(19.3)

The above relationship is constructed using the numerical simulation results for a fixed α = 15◦ , where c0 = 0, c1 = 27,550 and n = −0.9495. The onset of the membrane vibration related to the flow-excited instability will be discussed further in Sect. 19.2.5. As Reynolds number and mass ratio exceed the boundary, it can be inferred from Fig. 19.1 that the flexible membrane has a greater tendency for flow-induced vibration and a more substantial inertial effect on the fluid-structure coupling. Consequently,

19.2 Flow-Excited Instability

965

the inertia effect caused by the membrane vibration should be considered. The flexible membrane exhibits complex vibrational behavior with overlapping structural modes within the DBS regime. With the aid of the proposed FMD method, the most influential aeroelastic mode is identified from the completed membrane responses. To represent the structural mode shapes conveniently, the chordwise i and spanwise j mode of the 3D membrane is termed as CiS j in the following discussions. Four typical dominant membrane modes are further classified within the DBS regime. We observe almost attached vortices on the membrane surface within the parameter space (M1→3,R6→9), which is indicated by the filled labels in Fig. 19.1. The obvious vortex shedding process is noticed at the rest parameter space for labels without filled color. The dominant mode gradually transitions from the chordwise second mode to the chordwise first mode as the membrane inertia becomes significant and the Reynolds number increases.

19.2.1.2

Coupled Fluid-Membrane Dynamics

The coupled fluid-membrane dynamic characteristics of the flexible membrane over the given parameter space are summarized in Fig. 19.2. The contours are colored by the mean lift coefficient, the mean lift-to-drag ratio, the mean normalized maximum deflection, the maximum root-mean-squared value (r.m.s.) of the deflection fluctuation, the non-dimensional dominant frequencies of the fluid domain and the structural domain, respectively. Both X and Y axes are displayed on a logarithmic scale. The dashed line is given by Eq. (19.3) to distinguish the instability boundary between the DSS and DBS regimes. In the DSS regime, the mean lift coefficient maintains similar values from Fig. 19.2a. We notice from Fig. 19.2b and c that the mean lift-to-drag ratio and the mean membrane deflection is almost independent of m ∗ at a fixed Re. As Re increases, the aerodynamic efficiency improves substantially. The reason is attributed to the drag reduction caused by the cambered-up membrane shape. A boundary is notice in Fig. 19.2d to separate the DSS and DBS regimes. Since the deformed-steady membrane shape almost keeps constant as m ∗ changes, the dominant frequency of the unsteady flow presented in Fig. 19.2e does not dependent on m ∗ . The dominant frequency of the structural vibration plotted in Fig. 19.2f is set to blank within the DSS regime due to the steady state. Unlike the deformed-steady membrane, the aerodynamic characteristics of the oscillating membrane are significantly affected by the interaction of the unsteady separated flows and the membrane vibration. The mean lift coefficient is further improved when the flexible membrane maintains the dynamic balance state. The optimal aerodynamic performance is observed in the parameter space of (M1→3, R6→9). Meanwhile, we notice the largest membrane deflections and mild membrane vibrations in this region. Through the comparison of the flow features of the flexible membrane for two selected cases (M3, R6) and (M5, R4), the attached vortices at the leading and trailing edges in Fig. 19.3a enlarge the suction area on the membrane surface shown in Fig. 19.3c. When the light membrane deforms up to some extent

966

19 Flow-Excited Instability in Thin Structure Aeroelasticity

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 19.2 Coupled fluid-membrane dynamic characteristics of the 3D flexible membrane at Ae = 423.14 for α = 15◦ over the full m ∗ -Re parameter space: a time-averaged lift coefficient, b timeaveraged lift-to-drag ratio, c time-averaged non-dimensional maximum membrane deflection, d maximum r.m.s. membrane deflection fluctuation, e non-dimensional dominant frequency in fluid domain via FMD approach and f non-dimensional dominant frequency in structural domain via FMD approach. The dashed line (− − −) is the flow-excited instability boundary given by the empirical solution Eq. (19.3).  represents the membrane with a deformed-steady state. , , ♦ and ◦ denote the dominant modes with the C1S2, C2S1, C2S5 and C2S9 mode shapes, respectively. The blank in f represents the deformed-steady membrane without vibration

at a relatively high Re without high-intensity oscillation, the large separated flow region is suppressed, and the attached vortices are formed on the membrane surface. It can be seen from Fig. 19.3 b that the vortices cannot keep attached to the membrane surface. These vortices are coupled with the vibrating membrane and shed into the wake alternatively. As a result, the excited separated flow and the vortex shedding process degrade the aerodynamic performance as compared to the coupled system with attached vortices. A similar distribution of the dominant frequency of the unsteady flow and the membrane vibration is observed in Fig. 19.2e and f, resulting in the frequency lock-in phenomenon.

19.2 Flow-Excited Instability

967

(a)

(b)

(c)

(d)

Fig. 19.3 Comparison of flow features of the flexible membrane with a, c attached vortices (M3, R6) and b, d vortex shedding (M5, R4) for a, b instantaneous streamlines coloured by pressure d coefficient and c, d time-averaged pressure coefficient difference C p between upper and lower surfaces

19.2.2 Effect of Mass Ratio We observe from Fig. 19.1 that the membrane exhibits various dynamic behaviors over the studied parameter space. It is found that the mass ratio plays an important role in flow-induced vibration. We present and compare the evolution of the membrane dynamics as a function of mass ratio at a fixed Reynolds number of 24,300 (R5). The purpose is to investigate the effect of mass ratio on the coupled fluid-membrane characteristics and to isolate the impact of the Reynolds number. This representative Reynolds number selected for this study covers all the typical dynamic states. To fully understand the flow-induced vibration and the mode transition phenomena, extra cases were run for the membrane dynamics near the flow-excited instability boundary and the mode transition region in the range of m ∗ ∈ [0.36, 24]. Figure 19.4 summarizes the evolution of the force characteristics, the membrane displacements and the membrane vibration intensities as a function of m ∗ . It can be seen from Fig. 19.4 that the aerodynamic force characteristics and the membrane deflections almost keep constant within the range of m ∗ ∈ [0.36, 1.2], leading to no fluctuation in the membrane responses. Similar conclusions can be drawn for the membrane dynamics of the coupled system within the DSS regime at different Reynolds numbers in Fig. 19.2. We can infer that the inertia effect is neglected during the fluid-membrane interaction when m ∗ is less than the critical value. Consequently, a static equilibrium state of the cambered-up membrane is achieved when the tension force is balanced with the aerodynamic loads acting on the membrane surface.

968

19 Flow-Excited Instability in Thin Structure Aeroelasticity DSS

DBS

i ii

DBS

0.06

0.04 0.02

0.748

0

0.747

-0.02

0.746

0

5

0

0.5

10

1

0.203

0

0.202

0.2

25

0

5

0

10

(a)

DSS ii

DSS 0.05 i ii 0.04

iii

/c max

3.676 3.672

3.6

0

0

5

10

m∗

(c)

20

m

DBS iii

25

0.006

0.5

1

0.037

0.03

15

0.001 0.0005

0.0365

0.02

0

20

25

0 0

0.002

-0.0005

1.5

0.036

0.01 3.4

0

-0.02 1.5

1

∗15

0.004

3.68

3.8

0.5

(b)

DBS

δn

CL /CD

i

0.02

0 -0.01

20

m

0.02 0.01

0.22

0.1

-0.04 1.5

∗15

0.04 0.204

δn rms

0.75

0.75 0.749

0.24

 rms CD

0.2

0.8

4

0.08

iii

0.3

0.85

CL rms

CL

DSS i ii

0.9

0.7

0.26

0.4

iii

CD

0.95

0

0.5

1

-0.001 1.5

0 5

10

∗15

m

20

25

(d)

Fig. 19.4 Evolution of 3D membrane dynamics as a function of m ∗ : a mean lift coefficient and r.m.s. lift coefficient fluctuation, b mean drag coefficient and r.m.s. drag coefficient fluctuation, c mean lift-to-drag ratio and d maximum time-averaged normalized membrane deflection and r.m.s. membrane deflection fluctuation

When the membrane vibration occurs, the flexible membrane enters a dynamic balance state. The inertia effect participates in the dynamics of the coupled system in this regime. With the aid of the global FMD method, three distinct vibrational modes are further identified from the membrane dynamic response, namely (i) chordwise first mode, (ii) transitional mode, and (iii) chordwise second mode. The mean lift coefficient and the mean lift-to-drag ratio increase dramatically from the constant values at the deformed-steady state to the overall peak values when the membrane vibrates in the chordwise second mode. As the dominant structural mode gradually transitions from the chordwise second mode to the chordwise first mode, the aerodynamic performance begins to degrade. A kink is noticed at the boundary between two types of vibrational modes. The mean lift coefficient and the mean lift-to-drag ratio keep decreasing until approximately constant values are reached at m ∗ = 9.6 in the dominant first mode regime. The drag coefficient jumps to a plateau when m ∗ approaches the transitional region, and more drag penalties are achieved as m ∗ further increases. Finally, no obvious increment of the drag coefficient is noticed for a relatively heavy membrane. The mean-membrane camber decreases continuously within the range of m ∗ ∈ (1.2, 24]. The root-mean-squared values of the aerodynamic forces and the membrane displacement fluctuations show an overall growing trend in the simulated parameter space, which indicates higher oscillation intensities for a heavier membrane.

19.2 Flow-Excited Instability 2

0.2

m∗ =1.2 m∗ =2.4 m∗ =3.6 m∗ =14.4

m∗ =1.2 m∗ =2.4 m∗ =3.6 m∗ =14.4

0.1

δz /c

1.6

CL

969

1.2

0 0.8 0.4 40

42

44

46

tU∞ /c (a)

48

50

-0.1 40

42

44

46

tU∞ /c

48

50

(b)

Fig. 19.5 Time histories of instantaneous: a lift coefficients and b normalized displacements at the membrane center along the Z -direction for four selected cases with m ∗ = 1.2, 2.4, 3.6 and 14.4 at a fixed Re = 24,300

To further investigate the effect of mass ratio, we compare the time histories of the instantaneous lift coefficients and the instantaneous displacement at the membrane center along the Z direction in Fig. 19.5. We choose one case at m ∗ = 1.2 corresponding to the DSS state and three cases at m ∗ = 2.4, 3.6 and 14.4 for the second mode, the transitional mode and the first mode in the DBS regime, respectively. It can be seen that the time-varying lift coefficient and the time-varying membrane displacement show steady responses at m ∗ = 1.2. As m ∗ further increases, both the time-varying lift coefficient and displacement exhibit growing oscillation amplitude and decreasing oscillation frequency. Meanwhile, the dynamic responses gradually change from periodic oscillation to quasi-periodic oscillation. We further compare the membrane dynamics and the flow features in Fig. 19.6. Figure 19.6a–d show the full-body profile responses at the mid-span location for these four representative cases. The red dashed line represents the mean membrane shape. We observe that the membrane deforms up and remains a static, steady wing shape at m ∗ = 1.2. The flexible membrane exhibits a dominant chordwise second mode at m ∗ = 2.4. As m ∗ further increases to 3.6, both the alternatively occurring chordwise first and second modes are observed in the superimposed membrane responses. Meanwhile, the mean membrane camber becomes smaller compared to that at a lower m ∗ . The chordwise first mode dominates the membrane vibration at m ∗ = 14.4. Different from the membrane vibration responses in the second mode and transitional mode regimes, the flexible membrane vibrates on both sides of its rigid flat counterpart in the first mode regime. Related to the membrane dynamic state, the flow features of the coupled system are also strongly affected by m ∗ . As the flexible membrane leaves its static equilibrium position to vibrate at m ∗ = 2.4, the suction force in the proximity of leading edge in the pressure coefficient difference contour becomes larger as shown in Fig. 19.6b, compared to the suction area plotted in Fig. 19.6a at m ∗ = 1.2. As the dominant structural mode transitions to the chordwise first mode, the region with the low negative pressure near the trailing edge expands to the leading edge, reducing the

970

19 Flow-Excited Instability in Thin Structure Aeroelasticity

(a)

(b)

(e)

(c)

(f)

(g)

(d)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

(p)

Fig. 19.6 Flow past a 3D rectangular membrane wing: a–d full-body profile responses of 3D flexible membrane wing at mid-span location, e–h time-averaged pressure coefficient difference between upper and lower surfaces, i–l turbulent intensity and m–p time-averaged normalized velocity magnitude on five slices along the spanwise direction at (Re, m ∗ ) = a, e, i, m (24,300, 1.2), b, f, j, n (24,300, 2.4), c, g, k, o (24,300, 3.6), d, h, l, p (24,300, 14.4). (− − −) denotes the time-averaged membrane shape in a, b, c, d

lift coefficient at a higher m ∗ . It can be observed from Fig. 19.6a and b that the flow fluctuations in the shear layer become significant and the high-intensity regions get closer to the membrane surface when the flexible membrane loses its deformed-steady state. The turbulent kinetic energy (TKE) keeps growing as m ∗ further increases. This increasing flow instability is highly coupled with the membrane vibrations with larger amplitudes. Furthermore, the low-velocity area within the separation flow regions decreases for heavier membranes in Fig. 19.6m–p. The unsteady separated flows with various spatial-temporal scales and the intertwined structural modes motivate us to build a connection between the dynamic characteristics of the fluid and structural domains. The proposed global FMD technique offers an effective approach to mapping the membrane vibrations and the unsteady flow together from the spatial-temporal space to the spatial-frequency space. The aeroelastic mode energy is quantitatively calculated to identify the dominant mode. Using the FMD method, we analyze the dynamic responses of the 3D coupled system in the studied parameter space. The mode energy spectrum for both full 3D fluid and structural domains is presented in Fig. 19.7. The contour is colored by the mode energy of the decomposed Fourier modes of the Y -vorticity field in the fluid domain at various mass ratios.

19.2 Flow-Excited Instability

971

(%) Mode energy

40

ii

iii

30 20 Chord-wise first mode Chord-wise second mode

10 0

(a)

i

0

5

10

m∗

15

20

25

(b)

Fig. 19.7 Mode frequency and energy map of coupled fluid-membrane system based on FMD analysis as a function of m ∗ : a comparison of aeroelastic mode frequencies between decomposed Fourier modes of the Y -vorticity field in the fluid domain and the structural modes with obvious mode energy, and b mode energies of the structural first and second modes

Only the chordwise first, second and some high-order structural modes with relatively large mode energies are shown in the mode energy map. The other modes are neglected due to their weak contributions to the membrane vibrations. Because the membrane achieves a deformed-steady state in the range of m ∗ ∈ [0.36, 1.2], no vibrational mode is excited in this regime. We find that the flexible membrane maintains a similar cambered-up wing shape in the deformed-steady state, which leads to the same vortex shedding frequency contents. As m ∗ grows up higher than 1.2, some structural modes are excited in the coupled system. Meanwhile, the non-dimensional dominant vortex shedding frequency jumps to f c/U∞ = 1.02 near the flow-excited instability boundary. In the DBS regime, the dominant vortex shedding frequency ( f vs ) is found to lock into the membrane vibration frequency ( f s ). The frequency contents of the coupled system show an overall downward trend as m ∗ increases. The dominant mode frequency in the fluid domain switches from the higher branch to the lower branch in the transitional mode region when the mode energy of the lower branch exceeds that of the higher branch. From the structural mode energy map shown in Fig. 19.7b, the chordwise first mode becomes the dominant mode in the membrane vibration responses as its corresponding mode energy is larger than the mode energy of the second mode. Thus, the mode transition phenomenon occurs. Further investigation of the mode transition phenomenon will be discussed in Sect. 19.2.6.

19.2.3 Effect of Reynolds Number In this section, we further investigate the effect of Re on the flow-induced vibration and characterize the evolution of the dominant mode and the corresponding mode energy. We choose a series of numerical studies in the parameter space of (M5,R1→9) in Fig. 19.1, which consists of the concerned flow-induced vibration and the mode

972

19 Flow-Excited Instability in Thin Structure Aeroelasticity 2

1.2 0.8 0.4 40

Re=2430 Re=12150 Re=24300 Re=72900

0.15

δz /c

1.6

CL

0.2

Re=2430 Re=12150 Re=24300 Re=72900

0.1 0.05 0

42

44

46

tU∞ /c

(a)

48

50

-0.05 40

42

44

46

tU∞ /c

48

50

(b)

Fig. 19.8 Time histories of instantaneous: a lift coefficients and b normalized displacements at the membrane center along the Z -direction for four selected cases with Re = 2430, 12,150, 24,300 and 72,900 at a fixed m ∗ = 4.2

transition phenomena. It can be seen from Fig. 19.2 that the aerodynamic performance improves and the mean membrane camber as well as the vibration intensity show an overall upward trend as Re increases. To gain further insight into the effect of Re, we compare the instantaneous aerodynamic forces, the unsteady flow features and the membrane vibrations at different Reynolds numbers. In Fig. 19.8, we present the comparison of the time histories of the instantaneous lift coefficients and the instantaneous vertical displacement at the membrane center at four representative Reynolds numbers (R1 = 2430, R4 = 12,150, R5 = 24,300, R8 = 72,900). The membrane displacement remains steady at Re = 2430. The oscillation amplitudes of the lift coefficient and the membrane displacement increase at higher Re values within the DBS regime. Meanwhile, the dominant frequency of the system gradually transitions to the lower frequency branch. The coupled dynamics tend to be non-periodic oscillations. We further compare the membrane profiles, the mean pressure coefficient difference, the turbulent intensity and the mean velocity magnitude for the selected four cases in Fig. 19.9. The flexible membrane achieves a deformed-steady state at Re = 2430. The membrane exhibits a chordwise second mode, and then transitions to the chordwise first mode as Re further increases. Compared to the pressure difference distribution of the membrane in the DSS regime in Fig. 19.9a, the suction force area in the proximity of the leading edge expands when the membrane vibration is excited. It can be attributed to the longer attached vortices interacting with the membrane vibration. It is observed from Fig. 19.9d that the membrane with large camber at Re = 72,900 enhances the suction area and further improves the aerodynamic performance. The flow fluctuations in the shear layer become stronger and get closer to the membrane surface when the membrane vibration occurs at a higher Re. The flow fluctuations become weaker at Re = 72,900 as shown in Fig. 19.9d due to the significantly suppressed separated flow for the largely cambered membrane. The low-velocity region becomes smaller at a higher Re.

19.2 Flow-Excited Instability

973

(e)

(d)

(c)

(b)

(a)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

(p)

Fig. 19.9 Flow past a 3D rectangular membrane wing: a–d full-body responses of 3D flexible membrane wing at mid-span location, e–h time-averaged pressure coefficient difference between upper and lower surfaces, i-l turbulent intensity and m-p time-averaged normalized velocity magnitude on five slices along the span-wide direction at (Re, m ∗ ) = a, e, i, m (2430, 4.2), b, f, j, n (12, 150, 4.2), c, g, k, o (24, 300, 4.2), d, h, l, p (72, 900, 4.2). (− − −) denotes the time-averaged membrane shape in (a, b, c, d)

(%) Mode energy

30 Chord-wise first mode

10

0

(a)

Chord-wise second mode

20

0

20000

40000

60000

Re

80000 100000

(b)

Fig. 19.10 Mode frequency and energy map of fluid-membrane coupled system based on FMD analysis as a function of Re: a comparison of aeroelastic mode frequencies between decomposed Fourier modes of the Y -vorticity field in the fluid domain and the structural modes with obvious mode energy, b mode energies of the structural first and second modes

974

19 Flow-Excited Instability in Thin Structure Aeroelasticity

We observe the frequency lock-in phenomenon at various mass ratios in the DBS regime in Fig. 19.7. A natural question is to ask whether the frequency lock-in phenomenon also occurs at various Reynolds numbers. To examine the frequency lock-in, we extract the mode energy spectra from the coupled system in the fluid and structural domains based on the FMD method. Similar to Fig. 19.7, we only plot the most energetic structural modes for simplicity. The comparison of the mode energy spectra of the fluid and structural Fourier modes is presented in Fig. 19.10a. The variation of the first and second mode energies as a function of Re is plotted in Fig. 19.10b. Similar to the studies in Sect. 18.3, we add the summarized standard bluff-body vortex shedding frequency range of ( f c sin α)/U∞ ∈ [0.15, 0.2] to Fig. 19.10a. The two red dashed lines shown in Fig. 19.10 a represent the upper and lower limits of the non-dimensional frequency. We observe the non-dimensional vortex shedding frequency shows few changes within Re ∈ [2430, 12, 150]. The dominant nondimensional vortex shedding frequency in this parameter space for a deformedsteady membrane falls into the summarized non-dimensional frequency range. This observation is consistent with the conclusion of a flat plate with an almost constant vortex shedding frequency in the range of Re ∈ [103 , 105 ] [621]. When the membrane vibration occurs, the dominant non-dimensional vortex shedding frequency jumps from 0.6 to 1.02. The frequency lock-in is noticed in Fig. 19.10a in the range of Re ∈ [12, 150, 97, 200]. In Fig. 19.10b, the mode energy of the chordwise first mode exceeds the mode energy of the chordwise second mode at Re = 24,300. The mode transition from the second mode to the first mode is triggered. Thus, the first mode dominates the membrane vibration at a higher Re. Through the studies of the effect of m ∗ and Re on the coupled fluid-membrane dynamics, we find similar phenomena and common characteristics for the onset of the membrane vibration and the mode transition at various mass ratios and Reynolds numbers. The results show that the role of the flow-excited instability in the coupled dynamics at various physical parameters shares some commonalities. In addition to Re and m ∗ , the aeroelastic number Ae related to the membrane flexibility is another critical parameter to govern the coupled dynamics. To make the findings more general, we also examine the evolution of the coupled fluid-membrane dynamics as a function of Ae. The purpose is to explore whether the membrane flexibility similarly influences the flow-induced vibration to mass ratio and Reynolds number.

19.2.4 Effect of Aeroelastic Number We perform a series of numerical simulations to investigate the role of flexibility in flow-excited instability. Four sets of aeroelastic numbers (A1→4) are selected in the simulations and the values are 84.6, 423.14, 1269.42 and 2115.7, respectively. To make the conclusion more comprehensive, we choose three mass ratios (M3, M4 and M6) to expand the parameter space, representing a light wing, a medium weight wing and a heavy wing. The Reynolds number is fixed at Re = 24,300 and the AOA is set to α = 15◦ causing the membrane interaction with unsteady separated flows.

19.2 Flow-Excited Instability

975

M3

M4

M6

2500 A4

2000

Ae

DSS

DBS

Static C1S2 C2S1 C2S2

1500 1000 500 0

A1

2

4

m∗

6

8 10 12

Fig. 19.11 Stability phase diagram: non-dimensional aeroelastic number Ae versus mass ratio m ∗ for the 3D flexible membrane at Re = 24,300 for α = 15◦ . Here, the dashed line (− − −) is plotted to distinguish the flow-excited instability boundary.  denotes the simulation results corresponding to the deformed-steady state. , ♦ and  represent the C1S2 mode, the C2S1 mode and the C2S2 mode in the dynamic balance state, respectively

Figure 19.11 presents the stability phase diagram for the flexible membrane in the parameter space of m ∗ -Ae. The dashed line plotted in the figure denotes the flowexcited instability boundary. We can suggest an empirical solution of the boundary for the critical aeroelastic number (Aecr ) as Aecr = c0 + c1 (m ∗ )n ,

(19.4)

where c0 , c1 and n are the coefficients determined by numerical simulations or experiments. Two stability regimes, namely DSS and DBS, are also identified from the membrane responses at different parameter combinations, which is the same as the stability phase diagram presented in Fig. 19.1. The membrane has a tendency to maintain a deformed-steady state when the wing is lighter and more rigid. The flexible membrane leaves its static equilibrium position to vibrate as the effect of the membrane inertia and flexibility becomes significant in the coupled system. The mode transition between different modes is also observed in the DBS regime in the studied parameter space of m ∗ -Ae. Figure 19.12 presents the aerodynamic force characteristics and the membrane deflections as well as their fluctuation intensities and the non-dimensional dominant frequencies of the fluid and structural domains via the FMD method in the studied parameter space. We observe from Fig. 19.12a and b that the vibrating membrane exhibits overall superior aerodynamic performance than the deformed-steady membrane. The optimal aerodynamic performance and the maximum membrane deflection are noticed for the lightest and most flexible membrane (M3, A1). The dominant frequency distributions summarized in Fig. 19.12e and f demonstrate that the frequency of the unsteady flow is synchronized with the frequency of the membrane vibration within the DBS regime. The above observations show that the cou-

976

19 Flow-Excited Instability in Thin Structure Aeroelasticity

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 19.12 Coupled fluid-membrane dynamic characteristics of the 3D flexible membrane at Re = 24,300 for α = 15◦ over the m ∗ -Ae full parameter space: a time-averaged lift coefficient, b timeaveraged lift-to-drag ratio, c time-averaged non-dimensional maximum membrane deflection, d maximum r.m.s. membrane deflection fluctuation, e non-dimensional dominant frequency in fluid domain via FMD approach and f non-dimensional dominant frequency in structural domain via FMD approach. The blank in f represents the deformed-steady membrane without vibration

pled membrane responses exhibit rich characteristics as functions of m ∗ and Ae. The effect of mass ratio on the coupled fluid-membrane dynamics has been discussed in Sect. 19.2.2. In this section, we focus on studying the role of membrane flexibility in the coupled fluid-membrane dynamics and the flow-excited instability for the membranes with three groups of representative mass ratios, respectively. For the light membrane with m ∗ = 1.2 (M3), no mode transition is observed in the studied aeroelastic number range. As the aeroelastic number decreases (A4→1), the mean lift coefficient, the mean lift-to-drag ratio and the mean membrane displacement grow up continuously. The membrane oscillation intensity increases sharply to 0.017 at A1 when the membrane is coupled with the unsteady flow to vibrate. For the medium-weight membrane with m ∗ = 2.4 (M4), the membrane loses its aeroelastic static stability at higher Ae, compared to the light membrane. It can be inferred that a more substantial inertia effect can reduce the static stability range. The membrane remains a dominant chordwise second mode in the DBS regime as Ae decreases.

19.2 Flow-Excited Instability 2

0.2

Ae=84.63 Ae=423.14 Ae=1269.42 Ae=2115.7

Ae=84.63 Ae=423.14 Ae=1269.42 Ae=2115.7

0.1

δz /c

1.6

CL

977

1.2

0 0.8 0.4 38

40

42

44

46

48

-0.1 38

40

42

44

46

48

tU∞ /c

tU∞ /c

(b)

(a)

Fig. 19.13 Time histories of instantaneous: a lift coefficients and b normalized displacements at the membrane center along the Z -direction for four selected cases with Ae = 84.63, 423.14, 1269.42 and 2115.7 at a fixed m ∗ = 9.6

(b)

(a)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

Fig. 19.14 Flow past a 3D rectangular membrane wing: a, b, c, d full-body responses of 3D flexible membrane wing at mid-span location, e, f, g, h time-averaged pressure coefficient difference between the upper and lower surfaces, i, j, k, l time-averaged streamlines and normalized velocity magnitude on the mid-span plane at (Ae, m ∗ ) = a, e, i (84.63, 9.6), b, f, j (423.14, 9.6), c, g, k (1269.42, 9.6), d, h, l (2115.7, 9.6). (− − −) denotes the time-averaged membrane shape in (a, b, c, d)

When the membrane becomes more flexible, the mean lift coefficient, the mean liftto-drag ratio and the mean membrane displacement are improved. Meanwhile, the membrane vibrates more violently. Different from the dynamical response of the light membrane and the medium weight membrane, the dominant mode transitions between different structural modes in the studied Ae space for the heavy membrane with m ∗ = 9.6 (M6). It is observed from Fig. 19.12 that the aerodynamic forces and the membrane deflection fluctuate with the change of Ae. The overall most significant mean lift coefficient and mean lift-to-drag ratio are achieved for the most flexible membrane. Figure 19.13 plots

978

19 Flow-Excited Instability in Thin Structure Aeroelasticity

the time histories of the instantaneous lift coefficient and the instantaneous vertical displacement at the membrane center at different Ae values. The time-varying lift coefficient and the time-varying membrane displacement exhibit variations of the amplitude and the dominant frequency as a function of Ae. Figure 19.14 summarizes the membrane vibration responses, the pressure difference distributions and the flow features of the heavy membrane at different aeroelastic numbers. Both instantaneous chordwise first and second modes are observed from the full-body profile responses of the most flexible membrane (A1) in Fig. 19.14a. The dominant mode transitions to the chordwise first mode in Fig. 19.14 b when the aeroelastic number changes to A2. The mean wing camber reduces and the vibration amplitude grows up after the mode transition occurs. In Fig. 19.14c, the membrane vibrates in the chordwise second mode with reduced vibration intensity at Ae = 1269.42 (A3). For the most rigid membrane (A4), the chordwise first mode becomes the dominant mode in the coupled system. The flow features are also significantly affected by the membrane vibrations at different dominant modes. A suction area is observed at the leading edge in Fig. 19.14a, which is related to the leading edge vortex noticed in Fig. 19.14a. The high suction area in blue color at the leading edge reduces slightly and the low suction region in red color near the trailing edge expands to the leading edge, which is caused by the decreased membrane camber and the wake pattern. We notice that the high suction area in the proximity of the leading edge reduces dramatically in Fig. 19.14c. Meanwhile, a large vortex is formed on the whole membrane surface in Fig. 19.14c. Similar pressure distributions and flow features are observed between the membranes vibrating in the chordwise first mode at Ae = 423.14 and 2115.7, resulting in similar mean lift forces. By exploring the coupled fluid-membrane dynamics as functions of mass ratio, Reynolds number and aeroelastic number, we can see that the flexible membrane gets coupled with the unsteady separated flow and vibrates when the relevant physical parameters exceed the flow-excited instability boundary. Besides, the transition of the dominant mode from a specific mode shape to another mode shape depends on the variation in the corresponding modal frequency. An underlying process should dictate the onset of the membrane vibration and the mode transition through specific combinations of the physical parameters. In the following section, the underlying process is explored based using the bluff-body vortex shedding and the nonlinear natural frequency of the tensioned membrane in the studied parameter space.

19.2.5 Onset of Flow-Induced Membrane Vibration From the aeroelastic mode analysis at different physical parameters, one can see that the frequency of the fluid Fourier mode jumps to the membrane vibration frequency once the self-sustained vibration is established. The fluid-structural parameters determine the onset of the membrane vibration (e.g., m ∗ , Re, Ae, and AOA) and the underlying coupled dynamics phenomena such as the vortex shedding, the shear

19.2 Flow-Excited Instability

979

layer fluctuation, the flow-induced tension. It is shown earlier that the membrane can maintain a constant deformed shape in the DSS regime regardless of m ∗ when other physical parameters remain fixed. The tension force and the flow features of the stretched steady membrane are not affected by the mass ratio. Among the physical parameters related to the fluid-membrane interaction, only the natural frequency of the membrane is varied as a function of m ∗ when the membrane remains a constant cambered shape. Once the mass ratio exceeds the critical value, the membrane starts to vibrate through the frequency lock-in with the vortex shedding process. The natural question is to ask whether the natural frequency of the tensioned membrane plays an important role in the onset of the membrane vibration and how it affects the flow-excited instability. The variation of m ∗ in the coupled fluid-membrane system provides a good starting point to explore the relationship between the natural frequency and the flow-induced vibration. To generalize our understanding, the variation of the natural frequency and the flow-induced vibration at different Reynolds numbers and aeroelastic numbers is further studied. We further shed light on the onset of membrane vibration. The local vortex shedding frequency along a monitoring line near the trailing edge in the wake is calculated to compare with the natural frequency of the membrane in this section. Figure 19.15 presents a schematic of the positions of the monitoring lines and points. The monitoring line for measuring the local frequency of the leading edge vortex is placed 0.2c behind the leading edge. The monitoring line corresponding to the trailing edge vortex frequency measurement is located at the half-chord behind the trailing edge. These two vertical lines are chosen at the location, which can significantly characterize the local flow dynamics. Three monitoring points are placed uniformly on the membrane surface along the membrane chord to collect the local membrane dynamic responses. We next explore the mode transition phenomenon. The cross-correlation analysis between the flow fluctuations along the monitoring lines and the membrane vibrations at three points is conducted in the next section. Similar to [620], the areal strain is employed to quantize the whole surface distortions under aerodynamic loads. The areal strain is defined as the ratio of the surface area change caused by the flow-induced deformation to the initial surface area, which is given as Sde f or med − S0 , (19.5) εas = S0 where Sde f or med denotes the surface area of the deformed membrane and S0 is the initial surface area of the membrane without deformation. A schematic of a rectangular membrane as well as its deformed shape and the Gaussian quadrature is plotted in Fig. 19.16. The surface area is calculated by integrating the finite element area using the Gaussian quadrature over the whole surface, which is given as S=

np n el   e=1 p=1

det Je (η p , ξ p )W p ,

(19.6)

980

19 Flow-Excited Instability in Thin Structure Aeroelasticity Line1

Z 0.145c

O

Point1

U∞

X

Point2

0.2c 0.25Lc

Point3

Line2

0.5Lc 0.75Lc

0.43c

Lc =cos(α)c

0.5c

(a)

Fig. 19.15 Schematic of the positions of the monitoring lines and points. The red dashed line (- - -) indicates the time-averaged membrane shape. The black solid line (—) represents the instantaneous membrane shape

where n el and n p represent the number of the membrane elements and the Gauss integration points. η p and ξ p are the local coordinates of the p-th Gauss node. Je denotes the jacobian of the e-th membrane element and W p is the Gauss weight. The dynamic membrane responses can be divided into two components: (a) mean deformed shape under aerodynamic loads and (b) vibration around its mean position. The areal strain of the mean shape (εas ) reflects the stretch ratio of the mean cambered membrane to its initial shape, which can be equivalent to the initial stretching strain applied to the membrane edges. The root-mean-squared value of the areal strain fluctuation (εas r ms ) is regarded as the vibration intensity of the oscillating membrane. To gain further insight into the onset of the membrane vibration, it is important to calculate the natural frequency of the membrane with large amplitude vibration immersed in an unsteady flow. The natural frequency predicted based on the linear model expressed in Eq. (19.1) neglects the dynamic stress caused by the geometric nonlinearity of the vibrating membrane. Furthermore, the added mass due to the oscillating surrounding air is missed in the equation. Here, we employ the derived nonlinear natural frequency formula shown in Eq. (18.12) to estimate the structural natural frequency of the vibrating membrane. Jaiman et al. [614] presented a theoretical study of the flow-excited instability of an elastic plate oscillating in the fluid. The critical Re showed a downward trend with an increasing mass ratio, which is consistent with our conclusion for the 3D membrane. Similar to the theoretical study in [614], the above analytical formulation can be extended to construct the stability boundaries as a function of tension force and mass ratio, which can be a scope for future study. The local non-dimensional frequencies of the dominant vortex and the secondary vortices are calculated based on the Fourier analysis of the collected time-varying vorticity signals along the monitoring Line2. These frequencies reflect the frequency

19.2 Flow-Excited Instability

981

Z

Membrane element

η Ny0

U∞

ξ

X δn

Nx0

b

Element node Integration point Initial membrane

Y Areal strain: εsa =

c Sdef ormed −S0 S0

Deformed membrane

Area: S =

np n el   e=1 p=1

detJe (ηp , ξp )Wp

Fig. 19.16 Schematic of a rectangular membrane with all fixed edges undergoing large amplitude vibration in a uniform flow. A deformed shape and the Gaussian quadrature procedure are illustrated

of the aerodynamic load perturbations applied to the flexible membrane. The natural frequency of the tensioned membrane is predicted by Eq. (18.12). The initial tens s and N y0 along two directions can be equivalent to the tensions of the mean sions N x0 deformed membrane shape. The physical parameters required for the prediction of the added mass can be determined from the numerical simulations. The vibration frequency, the vibration amplitude and the related parameters in the second nonlinear term in Eq. (18.12) are obtained from the numerical simulations. In the current study, the natural frequency of the first and second modes along the chordwise and spanwise directions (C1S1, C1S2, C2S1 and C2S2) are calculated because the excited dominant structural modes are close to these four modes in the examined cases. Figure 19.17a presents the calculated areal strain of the membrane as a function of m ∗ . The comparison of the vortex shedding frequencies and the nonlinear natural frequencies of the tensioned membrane as a function of m ∗ is plotted in Fig. 19.17a. It can be observed that ε as remains constant and εas r ms keeps zero in the DSS regime. The second term in Eq. (18.12) related to the membrane vibration and the dynamic stress vanishes. Moreover, there is no added mass for the static steady membrane. Thus, the natural frequency model of the statically deformed membrane can be reduced to Eq. (19.1). Since the tension force keeps constant in this regime, the non-dimensional natural frequency of each mode is only dependent on m ∗ , which gives a relationship as below

982

19 Flow-Excited Instability in Thin Structure Aeroelasticity DSS

0.006

DBS i ii

f c/U∞

0.004

0.002

εasrms

εsa

0.006

DBS i

ii

iii Dominant vortex Secondary vortex C1S1 C1S2 C2S1 C2S2

2

0.008 0.004

DSS

2.5

0.01

iii

1.5

0.002

1

0.5

0 0

0

5

10

15

m∗

20

0

25

0

5

10

15

m∗

(a) 0.025

DSS

20

25

(b) DSS

DBS

0.012

2

0.008

1.5

εsa

0.015 0.01

0.004

f c/U∞

εsarms

0.02

0.005

DBS Dominant vortex Secondary vortex C1S1 C1S2 C2S1 C2S2

1 0.5

0

0 0

40000

Re

0

80000

0

50000

(c) DBS

100000

Re

(d)

DBS

DSS

f c/U∞

εsarms

0.002

0.005

Dominant vortex Secondary vortex C1S1 C1S2 C2S1 C2S2

3

0.004

εsa

0.01

DSS

2 1

0 0

0

1000

Ae

0

2000

0

1000

Ae

2000

(f)

(e)

Fig. 19.17 Areal strain of mean membrane shape and r.m.s. of areal strain a, c, e and comparison of natural frequencies of tensioned membrane and vortex shedding frequencies measured along Line2 b, d, f as a function of a, b m ∗ at fixed (Re, Ae) = (24,300, 423.14), c, d Re at fixed (m ∗ , Ae) =(4.2, 423.14) and e, f Ae at fixed (m ∗ , Re) = (1.2, 24,300). The areal strain is calculated by integrating the finite element area from the structural responses. The nonlinear natural frequency is predicted by Eq. (18.12) with the aid of numerical simulations

f inj c U∞

 ∼

1 . m∗

(19.7)

In the DSS regime, the membrane shape maintains a fixed static equilibrium position. Thus, the flow features around the membrane keep similar and exhibit constant vortex shedding frequency regardless of m ∗ . The natural frequency of the membrane decreases when the membrane becomes heavier. It is observed from Fig. 19.17a that the structural natural frequency decreases and gets close to the harmonics of the dominant vortex shedding frequency near the flow-excited instability boundary. The flexible membrane is coupled with the vortex shedding process to vibrate. Meanwhile,

19.2 Flow-Excited Instability

983

the dominant vortex shedding frequency jumps from a lower frequency component associated with the bluff-body vortex shedding frequency to a frequency component locked into the membrane vibration. When the membrane is coupled with the separated flow to vibrate in the DBS regime, εas exhibits an overall downward trend and εas r ms shows an opposite trend as the inertia effect of the thin structure becomes more significant. It can be observed from Fig. 19.17 a that the dominant vortex shedding frequency locks into the membrane vibration with the chordwise second mode in the range of m ∗ ∈ [1.2, 3.6]. The dominant aeroelastic mode transitions to the chordwise first mode when the natural frequency of the C1S2 mode becomes close to the dominant frequency of the bluff-body vortex shedding frequency in the transitional mode regime. These approaching natural frequencies in the fluid and structural domains form a new frequency synchronization state through the coupling effect. The common mode transition phenomenon at various physical parameters will be uniformly discussed in Sect. 19.2.6. The variations of the areal strain and the nonlinear natural frequency of the coupled system as a function of Re are presented in Fig. 19.17b and b, respectively. Similar to [614], we observe from the numerical simulations that the deformation-induced tension is a function of Re, given as N s ∼ Ren , (0 < n < 2). The predicted nondimensional natural frequency decreases with increasing Re and fixed m ∗ and Ae within the DSS regime f inj c U∞

∼

1 √ s N ∼ Re−q , where q = 1 − n/2 (0 < q < 1). Re

(19.8)

The relationship between the non-dimensional natural frequency and Re given in Eq. (19.8) shows a similar conclusion reported in [721]. The areal strain of the membrane exhibits an upward trend in the DSS regime when Re increases. Although the membrane slightly deforms up in this regime, the non-dimensional dominant vortex shedding frequency at various Re falls into the range of the modified bluff-body vs ∈ [0.15, 0.2] summarized by [688]. According vortex shedding frequency f cUsin(α) ∞ to Eq. (19.8), the non-dimensional natural frequency of the membrane in the DSS regime becomes smaller at a higher Re. Once the natural frequency gets close to the harmonics of the vortex shedding frequency, the membrane starts to vibrate. As Re further increases, the areal strain of the mean shape grows up slightly, and then rapidly increases at Re = 48,600 within the DBS regime. Meanwhile, εas r ms increases continuously. Under the influence of the increasing Re, εas and εas r ms , the non-dimensional natural frequency of the vibrating membrane keeps decreasing finally. The dominant vortex shedding frequency transitions to the lower component and locks into the reduced structural natural frequency. The variations of the areal strain and the nonlinear natural frequency of the coupled system as a function of Ae are presented in Fig. 19.17c and d, respectively. Based on the simulation results within the DSS regime, the relationship between the areal strain and the aeroelastic number can be determined as εas ∼ Ae−n , (0 < n < 1). As m ∗ and Re keep fixed, the non-dimensional natural frequency can be expressed as

984

19 Flow-Excited Instability in Thin Structure Aeroelasticity

f inj c U∞

∼



Ae · εas ∼ Aeq , where q = (1 − n)/2 (0 < q < 0.5).

(19.9)

In Fig. 19.17c, εas increases as Ae reduces within the DSS regime. According to Eq. (19.9), the non-dimensional natural frequency of the membrane within this regime exhibits a downward trend at a lower Ae. The membrane vibration is excited when the structural natural frequency approaches the harmonics of the vortex shedding frequency and the frequency lock-in is formed. Meanwhile, the areal strain of the mean membrane and its oscillating intensity increase dramatically when the membrane vibration occurs at the lowest Ae. From the aforementioned analysis, we can infer that the frequency lock-in between the structural natural frequency and the vortex shedding frequency governs the selfsustained membrane vibration. Although the vortex originating from the leading edge convects downwards and sheds into the wake, the membrane still maintains a deformed-steady state at (Re, m ∗ ) = (2430, 4.2). In this scenario, the structural natural frequency is far away from the vortex shedding frequency and its harmonics. Meanwhile, the flow fluctuations in the shear layer are far away from the membrane surface. Through a proper combination of physical parameters, the natural frequency of the membrane can get closer to the dominant vortex shedding frequency or its harmonics. The membrane is coupled with the unsteady separated flow and builds up the self-sustained vibration via frequency lock-in. It can be easily inferred from Fig. 19.17 that the tension force due to the flow-induced deformation affects the structural natural frequency. Specifically, increasing tension force can result in a higher structural natural frequency and vice versa. Therefore, the tension force is key to connecting the coupled dynamic characteristics of the fluid flow and the flexible membrane. For an active control of the frequency lock-in, stretching or relaxing the membrane can suppress or excite the membrane vibration by tuning the natural frequency of the membrane.

19.2.6 Mode Transition in Flow-Induced Vibration Through the investigation of the flow-induced vibration characteristics, the mode transition from one specific aeroelastic mode to another mode is observed in Figs. 19.1 and 19.11 as the physical parameters change. The mode transition can impact the aerodynamic performance, the flow features around the membrane and the membrane vibration response. It is vital to understand the mode transition phenomenon in the coupled system. The results shown in Sect. 19.2.5 demonstrate that the onset of the membrane vibration is dependent on the frequency lock-in phenomenon. In this section, we explore the mode transition from the perspective of the frequency lock-in phenomenon. Specifically, the variation of the structural natural frequency relative to the vortex shedding frequency is investigated and the correlation with the dominant aeroelastic modes is established.

19.2 Flow-Excited Instability

985

In Fig. 19.17a, the natural frequency of the C1S2 mode becomes closer to the dominant frequency of the bluff-body vortex shedding process at m ∗ = 2.99. As m ∗ further increases, the dominant mode gradually transitions from the C2S1 mode to the C1S2 mode. As shown in Fig. 19.7, the mode energy corresponding to the C1S2 mode exceeds that of the C2S1 mode. It can be observed that the size of the dominant vortex enlarges as m ∗ reaches the transitional region, accompanied by the reduced mean camber and the growing vibration amplitude. The variation of these membrane dynamics enhances the bluff-body vortex shedding process. The flow fluctuations related to the bluff-body vortex shedding instability become more energetic in the coupled system, which is coupled with the structural C1S2 mode to dominate the system. In Fig. 19.17b, the non-dimensional natural frequency of the C1S2 mode is reduced and approaches the dominant bluff-body vortex shedding frequency within vs ∈ [0.15, 0.2] as Re increases. The frequency lock-in between the range of f cUsin(α) ∞ the frequency of the C1S2 mode and the bluff-body vortex shedding frequency is established. The dominant structural mode transitions to the C1S2 mode via the coupling effect. In Fig. 19.17c, the reduction of the aeroelastic number governs the non-dimensional natural frequency to approach the harmonic of the bluff-body vortex shedding frequency. It can be observed from Figs. 19.11 and 19.14 that the dominant mode transitions between different aeroelastic modes as Ae changes. It can be concluded that the mode transition process is dependent on the variation of the natural frequency relative to the vortex shedding frequency and its harmonics in the coupled system. The original mode synchronization process is interrupted, and a new mode synchronization state is then formed at the dominant coupled frequency, resulting in the mode transition phenomenon. To gain deeper insight into the mode transition phenomenon, the cross-correlation analysis is performed based on high-fidelity numerical data. The time-varying velocity fluctuation signals with 1024 samples are collected at the point in the middle of the monitoring Line1 and Line2, respectively. The time-varying membrane deflection fluctuation signals with 1024 samples are obtained at three equispaced points along the membrane chord shown in Fig. 19.15. The cross-correlation analysis is conducted between the membrane deflection and the local velocity information near the leading edge and the trailing edge, respectively. The purpose is to examine how the correlation between the unsteady flow and the membrane vibration changes in the mode transition phenomenon. Figure 19.18 presents the cross-correlation coefficients as functions of m ∗ , Re and Ae. The cross-correlation between the membrane deflection and the velocity is zero in the DSS regime because of the deformed-steady membrane shape. When the membrane vibration occurs, it is found that the flow fluctuations in the wake are strongly correlated with the membrane vibration responses. In Fig. 19.18a and b, the cross-correlation coefficients at monitoring Point1 and Point3 are close to each other and show much higher values than the cross-correlation coefficients at monitoring Point2 in the range of m ∗ ∈ [1.2, 3.6]. The results indicate that the unsteady separated flow has a high correlation with the chordwise second mode. As m ∗ further increases to 4.8, the cross-correlation coefficients at the monitoring Point2 exceed those values

986

19 Flow-Excited Instability in Thin Structure Aeroelasticity DSS

0.01

DBS i ii

DSS

0.008

iii

0.008

Rδn u

Rδn u

0.004

0.004 δn Point1 − uMid-Line1 δn Point2 − uMid-Line1  δn Point3 − uMid-Line1

0.002 0 0

5

10

15

m∗

20

25

0.002

 δn Point1 − uMid-Line2  δn Point2 − uMid-Line2 δn Point3 − uMid-Line2

0 -0.002

0

(a) DSS

0.03

 δn Point1 − uMid-Line1  δn Point2 − uMid-Line1  δn Point3 − uMid-Line1

Rδn u

0.03

5

10

15

m∗

20

25

(b)

DBS

DSS

0.02

0.02

DBS δn Point1 − uMid-Line2 δn Point2 − uMid-Line2  δn Point3 − uMid-Line2

Rδn u

0.04

iii

0.006

0.006

-0.002

DBS i ii

0.01

0.01 0 0

50000

Re

0 0

100000

0.004

DBS

0.006

δn Point1 − uMid-Line1 δn Point2 − uMid-Line1  δn Point3 − uMid-Line1

0.004

Rδn u

Rδn u

0.006

Re

100000

(d)

(c)

0.008

50000

DBS  δn Point1 − uMid-Line2  δn Point2 − uMid-Line2  δn Point3 − uMid-Line2

0.002

0.002

0

0 -0.002 0

1000

Ae

(e)

2000

-0.002 0

1000

Ae

2000

(f)

Fig. 19.18 Cross-correlation between the membrane deflection fluctuation δn collected at three equispaced points along the membrane surface and the local flow velocity fluctuation u measured at the point in the middle of: a, c, e monitoring Line1 near the leading edge and b, d, f monitoring Line2 in the proximity of the trailing edge as a function of a, b m ∗ at fixed (Re, Ae) = (24,300, 423.14), c, d Re at fixed (m ∗ , Ae) = (4.2, 423.14) and e, f Ae at fixed (m ∗ , Re) = (9.6, 24,300)

at Point1 and Point3. A stronger correlation between the vortex shedding process and the chordwise first mode can be observed after the mode transition. It can be seen from Fig. 19.18c and d that the cross-correlation coefficients at monitoring Point2 are smaller than those at monitoring Point1 and Point3 at Re = 12,150. The cross-correlation coefficients at the monitoring Point2 become the largest among the three points as Re further increases. The chordwise first mode dominates the vibration in the range of Re ∈ [24,300, 97,200]. In Fig. 19.18e and f, the cross-

19.3 Summary

987

correlation coefficients at monitoring Point2 are the largest among the three points at Ae = 423.14 and 2115.7 when the chordwise first mode dominates the membrane vibrations. It can be concluded that the cross-correlation between the flow fluctuations and the membrane vibration is dependent on the dominant aeroelastic mode shape. Once the mode transition occurs, we observe the changes in the cross-correlation coefficients at different positions. Combined with the analysis of the flow features and the membrane vibration responses, this strong correlation in the coupled system can help in designing effective active/passive control strategies to switch the aeroelastic modes by governing the mode transitions.

19.3 Summary In this chapter, we systematically investigated the flow-excited instability of a supported 3D rectangular membrane in separated turbulent flows at moderate Reynolds numbers. Stability phase diagrams were proposed to characterize the variation of the flow-excited instability. Two distinct stability regimes, namely DSS and DBS, were classified from the membrane dynamic responses over a wide range of the selected parameter space. In the parameter space of m ∗ -Re and m ∗ -Ae, new empirical relationships for the flow-excited instability boundary are suggested from the proposed stability phase diagrams via high-fidelity numerical simulations. The frequency lock-in phenomenon at various physical parameters in the DBS regime was observed based on the Fourier mode energy spectra between the fluid and structural domains. Mode transition phenomenon between different aeroelastic modes was observed in the stability phase diagrams, which were related to the variation of the aerodynamic performance and the membrane vibrations. In the DSS regime, the aerodynamic forces and the membrane deformation were almost independent of m ∗ . The increase of Re and membrane flexibility enhances the aerodynamic performance due to the growing membrane camber. Compared to the deformed-steady membrane, the vibration caused strong flow fluctuations close to the membrane surface. It forced the vortices to attach longer near the leading edge, which enlarged the suction area and improved the aerodynamic performance. The onset of the flow-induced membrane vibration and the mode transition phenomenon was further examined in the selected parameter space. The variation of the natural frequency of the tensioned membrane relative to the vortex shedding frequency was monitored as a function of m ∗ , Re and Ae. Under the aerodynamic loading, the areal strain was calculated to estimate the membrane stretching and its relationship with the natural frequency of the membrane. By varying fluid-structure parameters, the comparison between the predicted nonlinear natural frequency of the tensioned membrane and the measured vortex shedding frequency and its harmonics was carried out. The increase in the membrane inertia, the fluid inertia, and the membrane flexibility reduced the non-dimensional structural natural frequency, making it closer to the non-dimensional vortex shedding frequency or its harmonics. Consequently, the vortex shedding frequency locked into the natural frequency of the

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19 Flow-Excited Instability in Thin Structure Aeroelasticity

tensioned membrane, which provided the self-sustained vibration of a flexible membrane. The newly formed coupling between the varied structural natural frequency and the vortex shedding frequency interrupted the original mode synchronization process. It then generated a new mode synchronization state at the dominant coupled frequency, resulting in the mode transition phenomenon. Acknowledgements Some parts of this Chapter have been taken from the Ph.D. thesis of Guojun Li carried out at the National University of Singapore and supported by the Ministry of Education, Singapore.

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