Global Stability Analysis of Shear Flows 9811995737, 9789811995736

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Springer Tracts in Mechanical Engineering

Rameshkumar Bhoraniya Gayathri Swaminathan Vinod Narayanan

Global Stability Analysis of Shear Flows

Springer Tracts in Mechanical Engineering Series Editors Seung-Bok Choi, College of Engineering, Inha University, Incheon, Korea (Republic of) Haibin Duan, Beijing University of Aeronautics and Astronautics, Beijing, China Yili Fu, Harbin Institute of Technology, Harbin, China Carlos Guardiola, CMT-Motores Termicos, Polytechnic University of Valencia, Valencia, Spain Jian-Qiao Sun, University of California, Merced, CA, USA Young W. Kwon, Naval Postgraduate School, Monterey, CA, USA Francisco Cavas-Martínez , Departamento de Estructuras, Universidad Politécnica de Cartagena, Cartagena, Murcia, Spain Fakher Chaari, National School of Engineers of Sfax, Sfax, Tunisia Francesca di Mare, Institute of Energy Technology, Ruhr-Universität Bochum, Bochum, Nordrhein-Westfalen, Germany Hamid Reza Karimi, Department of Mechanical Engineering, Politecnico di Milano, Milan, Italy

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Rameshkumar Bhoraniya · Gayathri Swaminathan · Vinod Narayanan

Global Stability Analysis of Shear Flows

Rameshkumar Bhoraniya Department of Mechanical Engineering Marwadi University Rajkot, India

Gayathri Swaminathan Airbus-India Operations Pvt. Ltd. Bengaluru, India

Vinod Narayanan Department of Mechanical Engineering Indian Institute of Technology Gandhinagar Palaj, India

ISSN 2195-9862 ISSN 2195-9870 (electronic) Springer Tracts in Mechanical Engineering ISBN 978-981-19-9573-6 ISBN 978-981-19-9574-3 (eBook) https://doi.org/10.1007/978-981-19-9574-3 © 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

Dedicated to almighty God

Preface

The shear flow instability and transition are a wide field of research due to their significant impact on fundamental physics and applications. The viscous drag offered by such fluid flows highly depends on the regime of the flow (boundary layer). It is now well known that the laminar flow (boundary layer) offers less viscous drag than turbulent flow (boundary layer). The boundary layer transition from laminar to turbulent state is a highly complex phenomenon yet not understood well. The global stability analysis is a useful tool to predict the state of the flow and whether the small perturbation will amplify. Alternatively, decay in the flow field (boundary layer) includes a variation of the base flow in all directions. The flow (boundary layer) control aims to maintain it in the laminar state to reduce viscous drag and to improve the energy efficiency of the vehicles like submarines, torpedoes, etc. The flow control (boundary layer) is done by modifying the base flow velocity profile, and knowledge of stability/transition point is essential information to implement it. The primary motivation is to explain the stability characteristics of the spatially developing flows. The instability of the base state triggers the transition, and understanding the stability characteristics has a significant implication for flow control. This book will provide the latest scientific information in the shear flow (boundary layer) stability field, which will be useful to the global scientific community—researchers, scientists, academicians and professional engineers working in the field of fluid dynamics. This book explains the state of the art of the topic, the methodology and the investigation results of a global stability analysis for the different configurations of the internal and external shear flows. The topics covered are diverging channels, converging-diverging channels, axisymmetric boundary layer developed on a circular cylinder, cone and inclined flat-plate boundary layer and wall jets. The effect of non-parallel base flow, divergence, convergence, convergence-divergence, transverse curvature and favourable and adverse pressure gradients is explained in the global stability of the different configurations of shear flows. The book will be a valuable reference for beginners, researchers and professionals working in the fields of aerodynamics and marine hydrodynamics. The book content includes schematic diagrams, differential equations, boundary conditions, contour plots, graphs and tables to present the computational results of the stability analysis. vii

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Most of the earlier work on flow stability used the local approach, wherein the stability characteristics are determined based on the local velocity profile. Many of these studies aim at finding the critical Reynolds number (Recrit ) above which the flow will become unstable. This approach is valid when the base flow profile does not vary much between locations or does not vary at all. Typical examples are flowing through a straight channel and flat-plate boundary layer at high Re, which are a parallel flow and a weakly non-parallel flow, respectively. However, in the most reallife flow situations, the non-parallelism is not weak, i.e. the profiles and flow vary drastically in the streamwise direction. While the former type of flow can be studied using a local approach, the latter needs to be studied using a global approach, as is increasingly being done today. This book considers different internal and external flow configurations for the modelling and simulation of global stability analysis. An infinitely diverging channel, namely the Jeffery–Hamel flow, considered the simplest non-parallel flow. A channel with a series of divergent and convergent sections with large wall-waviness amplitudes and average width constant is considered to achieve low Reynolds number instabilities and achieve good mixing. An axisymmetric boundary layer developed on a circular cylinder is considered to study the effect of transverse curvature. A boundary layer on a circular cone is considered to study the effect of transverse curvature and pressure gradient. The boundary layer on an inclined flat plate is considered to study favourable and adverse pressure gradients only. A wall jet flow is considered to study the combined effect of the boundary layer and shear layer. The boundary layer becomes unstable due to the viscous effect, while the shear flow is due to the inflectional point in the velocity profile. The book is expected to be useful to undergraduate and postgraduate students and research scholars working in the field of fluid dynamics as it provides a detailed understanding of the mathematical modelling and numerical solution aspects of internal and external shear flows. This book is expected to become a valuable reference for those wishing to research the stability analysis of real-life shear flow problems using a global approach. We would like to thank our publisher Springer for the continuous support and interest in our book project. In particular, we would like to thank Editor Priya Vyas and Radhakrishanan Madhavamani for their commitment and engagement in this book project. We gratefully acknowledge the support of our past and present M.Tech. and Ph.D. students and postdoctoral fellows. We are also thankful to various researchers and International Journal publishers for permitting us to reuse certain parts of their research work. Finally, our special thanks go to our family members for supporting us during the preparation of the entire book. Rajkot, India Bengaluru, India Gandhinagar, India

Rameshkumar Bhoraniya Gayathri Swaminathan Vinod Narayanan

Contents

1 Introduction to Flow Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Physics of Flow Stability and Transition . . . . . . . . . . . . . . . . . . . . . . 1.3 Flow Transition Stages in Boundary Layer . . . . . . . . . . . . . . . . . . . . 1.4 Methods of Instability Investigation . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Overview of the Present Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 3 7 8 9

2 Global Stability Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Linear Stability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Stability of Parallel Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Stability of Weakly Non-parallel Flows . . . . . . . . . . . . . . . . . . . . . . . 2.5 Global Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 1D Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 2D Discretization-Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 2D Discretization-X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.4 Coordinate Transformation and Grid Stretching . . . . . . . . 2.6.5 Domain Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Baseflow Calculation and Boundary Conditions . . . . . . . . . . . . . . . . 2.8 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Issues with Global Stability Computations . . . . . . . . . . . . . . . . . . . . 2.11 Grid Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.12 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 12 14 16 17 21 22 23 25 26 27 29 31 32 33 34 35 37

3 Diverging Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Diverging Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Base Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 41 42 45 ix

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3.3.1 JH Base Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 SDS Base Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Comparison of Base Flow—JH and SDS . . . . . . . . . . . . . . . . . . . . . . 3.5 Sensitivity of the Critical Reynolds Number to Divergence . . . . . . 3.6 Global Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3 Sensitivity Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.4 Comparison of JH and SDS Spectra . . . . . . . . . . . . . . . . . . 3.6.5 JH Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.6 SDS Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Upstream Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 46 48 49 52 52 53 55 56 60 67 70 72 72

4 Converging-Diverging Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Base Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Numerical Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Global Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 The Standard Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 The Instability Ratchet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Spatio-Temporal Interplay . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4 Numerical Floquet Study and Its Limitations . . . . . . . . . . . 4.4.5 Reverse Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.6 Symmetric Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.7 Comparison of Forward, Reverse and Symmetric Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.8 Effect of Amplitude of Wall Waviness . . . . . . . . . . . . . . . . 4.4.9 Eigenvalue Sensitivity and Pockets of Transient Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.10 Other Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75 75 81 81 82 84 85 85 89 92 93 95 96

101 104 105 106

5 Axisymmetric Boundary Layer on a Cylinder . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Mathematical Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Solution of General Eigenvalues Problem . . . . . . . . . . . . . 5.3 Base Flow Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Validation of Global Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Axisymmetric Mode (N = 0) . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Helical Mode, N = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

109 109 110 111 114 115 118 118 121

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5.5 5.6

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Comparison with the Local Stability Analysis . . . . . . . . . . . . . . . . . Global Stability Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Grid Convergence Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Effect of Wall-Normal Domain Size . . . . . . . . . . . . . . . . . . 5.6.3 Effect of Streamwise Domain Size . . . . . . . . . . . . . . . . . . . 5.6.4 Sponging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.5 Effect of Outflow Boundary Conditions . . . . . . . . . . . . . . . 5.6.6 Axisymmetric Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.7 Effect of Transverse Curvature . . . . . . . . . . . . . . . . . . . . . . . 5.6.8 Helical Mode N = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.9 Helical Mode N = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.10 Helical Mode N = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Temporal Growth Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Spatial Amplification Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

121 123 124 124 125 126 130 131 135 138 138 140 146 147 148 149

6 Axisymmetric Boundary Layer on a Circular Cone . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Solution of General Eigenvalues Problem . . . . . . . . . . . . . 6.3 Base Flow Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Code Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Boundary Conditions Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Influence of the Domain Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Global Stability Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.1 Semi-cone Angle α = 2◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.2 Semi-cone Angle α = 4◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.3 Semi-cone Angle α = 6◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Temporal Growth Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9 Spatial Amplification Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

151 151 153 155 157 157 159 162 164 165 167 168 170 171 174 176 176

7 Boundary Layer on an Inclined Flat Plate . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Base Flow Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Code Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Grid Convergence Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Effect of Streamwise Domain Length . . . . . . . . . . . . . . . . . 7.4 Global Stability Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Favourable Pressure Gradient . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Adverse Pressure Gradient . . . . . . . . . . . . . . . . . . . . . . . . . .

179 179 183 184 187 189 191 191 193 193 197

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7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 8 Wall Jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Base Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Local Stability Analysis of Wall Jets . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Global Stability Analysis of Wall Jets . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.3 Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.4 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

205 205 206 207 209 210 211 213 214 221

About the Authors

Dr. Rameshkumar Bhoraniya is currently working as Associate Professor in Mechanical Engineering at Marwadi University, Rajkot, India. He obtained his B.E. and M.E. in Mechanical Engineering from Gujarat University, India, and Ph.D. in Mechanical Engineering from the Indian Institute of Technology (IIT) Gandhinagar, India. He has a teaching experience in Mechanical Engineering over a decade. His research interests include flow instability and transition of fluid flows. He has published more than 18 research articles and chapters in reputed journals and books. He received a research grant of about INR 22 lakhs from the DST-SERB, Government of India, in 2020 for three years to investigate the effect of wall suction and injection of the global stability of boundary layer. Dr. Gayathri Swaminathan is currently working at Airbus India as Head of Aircraft and Flight Analytics Group and has been associated with the group since 2010. She is Aeronautical Engineer by background and has completed her Ph.D. from Fluid Dynamics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research (JNCASR), Bengaluru, in 2010. Her major research area has been global stability analysis of non-parallel flows. She has six conference/journal publications and patents to her credit. Dr. Vinod Narayanan is currently working as Associate Professor in Mechanical Engineering Department at the Indian Institute of Technology (IIT) Gandhinagar, India. He obtained his B.Tech. (Mechanical Engineering) from MG University, India; M.Tech. (Thermal Science) from Kerala University; and Ph.D. from Fluid Dynamics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research (JNCASR), Bengaluru, in 2005. He has a total work experience of over 15 years. His major research interests include flow instability and transition, aerodynamics and compressible flows. He has published over 50 papers in reputed international journals and conference proceedings. He has completed two research projects funded by the Aeronautical and Research Board, Government of India. At present, he is working on

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About the Authors

a research project funded by the DST-SERB, Government of India. He has successfully guided two Ph.D. students, and at present, three students are pursuing their Ph.D. under his supervision.

Chapter 1

Introduction to Flow Instabilities

1.1 Introduction Most shear flows are spatially developing, i.e. their velocity profile evolves as the flow proceeds downstream. Typically, as the Reynolds number increases, the laminar shear flow undergoes a linear instability, followed by an often complicated and not completely understood route to turbulence. Often, the flow might not undergo the different stages of instability but directly become turbulent through a process known as bypass transition. The route by which a flow becomes turbulent depends upon many parameters like geometry, free-stream disturbances, etc. As with any work on flow stability and transition, this thesis also starts by referring to the work of Osborne Reynolds in 1883, which was the first systematic study on flow stability through pipes. The review paper by [10] discusses the two essential papers by Reynolds and his interactions with the referees that greatly influenced Engineering Fluid Mechanics’ development over the past century. To perform a flow stability analysis, the laminar state is perturbed, and the evolution of the disturbances is monitored to see if the disturbances grow or decay. A growing disturbance indicates an unstable system, while the disturbances decay in a stable system. It is essential to understand the stability characteristics of a flow, as it plays a significant role in flow control. When we require the flow to be laminar, the disturbance energy growth has to be curbed. In contrast, in situations where we prefer a turbulent flow, the disturbance energy growth is enhanced. This suppression or enhancement of the disturbance energy could be performed in many ways, like blowing and suction, heating and cooling, vortex generators, etc. The techniques mentioned above fall under active flow control methods. The disturbance energy can also be tuned by passive options where, for example, the geometry is optimally designed. To perform flow control, we need information about what to control, where and how to control it. This information is obtained from the disturbance, and it is obtained a-priori by doing a stability analysis. Thus, stability analysis is the first step in flow control. In addition, stability analysis gives us information about the underlying physics of the transition process in the less-understood (in many cases). The main aim of the present study is to © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Bhoraniya et al., Global Stability Analysis of Shear Flows, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-19-9574-3_1

1

2

1 Introduction to Flow Instabilities

try to understand flow characteristics through non-parallel but not very complicated geometries. Laminar flow is one in which fluid particles flow in an ordered manner, whereas random and chaotic motion characterizes turbulent flows. In the laminar flow, fluid particles only move in a well-predictable direction. The presence of temporal and spatial scales, large diffusivity and vorticity fluctuations are some of the silent features of turbulence. A more general description of the laminar flow is that it is with strong momentum diffusion and weak momentum convection while turbulent flow is with high dissipation and randomness The flow’s regime, whether laminar or turbulent, significantly affects the quantities, such as skin friction, formal drag and heat transfer rate. One of the most essential characteristics of aerodynamics and underwater vehicles is its drag force. The reduction in the drag forces can save a considerable quantity of fuel consumption. On the other hand, turbulence increases the mixing properties of the flow. Hence, it is desired in some applications, e.g. mixing fluids in a combustion chamber, industrial flows for better heat exchange, etc. Sound knowledge of the onset of transition is essential in designing and operating efficient vehicles. Thus, the fundamental study of flow instability is motivated by the need to delay a transition to turbulence to increase the performance of the vehicles like aeroplanes, rockets, missiles, cars, torpedoes, trains and boats that are propagating through a viscous fluid. When a vehicle re-enters from space to earth, it encounters a very high velocity in the atmosphere. The high velocity increases the drag force along with heat load also. Therefore, a better understanding of the transition onset is essential in computing additional heat load on the vehicles.

1.2 Physics of Flow Stability and Transition A sequence of events characterizes the physical mechanism of laminar to turbulent transition. The first step is the amplification of a small perturbation in the base flow. The amplification of the small disturbances is well studied using linear stability analysis. The physical mechanisms that lead to the transition of a laminar flow to a turbulent one are usually associated with more than one hydrodynamic instability process. An unstable laminar flow may break down and be replaced by a stable laminar flow or undergo a transition to a turbulent flow [6]. Once the amplitudes of the perturbations reach a certain threshold, the nonlinear interaction may trigger Furthermore, eventually, it breaks down into rough spots. In general, instability is triggered by some disturbance that upsets the flow system’s momentum balance. Viscous forces generally dissipate the energy of disturbances, thus acting in a stabilizing fashion. However, friction diffuses momentum, which may destabilize a flow. Laminar and turbulent flows are distinguished based on their structure. In laminar flow, fluid particles flow orderly, whereas, in turbulent flow, fluid particles are chaotic, random and irregular. A more general description of the laminar flow is that it has strong momentum diffusion and weak convection, while turbulent flow has high dissipation and randomness.

1.3 Flow Transition Stages in Boundary Layer

3

The knowledge of flow instability and transition mechanism is essential from an industrial and environmental point of view. A boundary layer is formed near the body surface of such vehicles where the fluid velocity is zero at the surface and fluid accelerates in an outward direction. The Reynolds number increases with the increase in velocity or boundary layer thickness, which turns the boundary layer from laminar to turbulent. Research in the area of flow instability plays a central role in the long search to identify the deterministic routes that lead to a laminar flow through the transition into turbulence. Understanding the physics of laminar–turbulent flow transition has been initially motivated by aerodynamic applications. Flow instability analysis starts with the decay/amplification of small amplitude perturbation waves. A typical flow instability analysis can be done by solving linearized Navier–Stokes equations with proper boundary conditions, which leads to an eigenvalue problem. Flow stability only involves a small part of all mechanisms that lead to transition. Several other approaches have emerged to investigate flow transition. Computational methods like direct numerical simulations (DNS) are expensive. large eddy simulation (LES) and Reynolds averaged Navier–Stokes (RANS) are expensive. The transition is typically initiated by the onset of instability of the laminar flow. There are many difficulties in the proper numerical analysis of the flow transition. There is no straightforward way to tackle the problem. Thus, the stability analysis becomes interesting as it is linear, and the approach can easily be adjusted to a given problem. Linear stability theory describes the eigenmode growth mechanism. Although this yields a restriction because additional mechanism plays a role too, the eigenmode growth phase establishes a significant base in practical applications. However, linearization provides a considerable step in simplifying the analysis, while the stability theory can be adapted according to the structure of the given base flow.

1.3 Flow Transition Stages in Boundary Layer The destabilization of the laminar boundary layer is the first step in the process of transition to turbulence [6]. As suggested by Kachanov [11], the laminar to turbulent transition process can be split roughly into the following stages. Receptivity of the disturbances from the free-stream is the first stage of the flow transition. Various noises like wall roughness, wall vibration and free-stream turbulence generate small disturbances in the base flow. The small perturbations penetrate the boundary layer as a steady or unsteady to the base flow. This process is known as receptivity. It forms an initial condition of disturbance amplitudes, frequency and phase for the breakdown of laminar flow. In the most steady state, the free-stream consists of stochastically occurring perturbation, typically of small amplitudes. The transition can vary dramatically with the external noise and its receptivity. The freestream supports both discrete and continuous disturbance eigenmodes. Such studies proved that the amplitudes of continuous modes decay slowly inside the boundary layer, whereas discrete modes decay slowly. Continuous modes of free-stream tur-

4

1 Introduction to Flow Instabilities

bulence are unable to penetrate the boundary layer, termed ‘shear sheltering.’ The various aspects of the receptivity of the boundary layers have been extensively studied by Morkovin [18], Goldstein and Hultgren [8], Kerschen [14], Reshotko [20], Choudhari and Streett [3], Crouch [5], Saric et al. [22] and Wlezien [25]. The different aspects of three-dimensional wave motion and nonlinearities are studied by Breuer et al. [2], Kobayashi et al. [16], Kato et al. [13], Kachanov [11], Saric et al. [23], Schrader et al. [24]. The next stage is the amplification of the small disturbances. The amplitude of these disturbances grows exponentially in the narrow frequency range depending on the local Reynolds number. The linear hydrodynamic theory describes this stage of transition. The two-dimensional flows have been studied extensively by this theory. The Reynolds number, in this case, would be Re = U∞ν L , where U∞ is the free-stream velocity, ν is the kinematic viscosity, and L is the length scale [it could be boundary layer thickness (δ) or the displacement thickness (δ ∗ )]. Once the primary disturbances have grown to a certain amplitude, they can destabilize the three-dimensional secondary modes, giving rise to  vortices. As a thumb rule, the nonlinearity becomes detectable when the amplitude of the primary disturbances become an order of 1 % of the base flow. The next stage in the flow transition process is the nonlinear growth of the disturbances. In this stage, the amplitudes are significant enough, more than one percent of the free-stream velocity. The different modes interact with each other and generate oblique modes through nonlinear interactions. The modes saturate at high amplitudes up to 25% of free-stream velocity, making the flow three-dimensional. This consists of a succession of events that cause the flow to break down into occasional isolated patches of turbulence, known as turbulent spots. The streamwise location where turbulent spots first originate can be defined as the onset of transition. The spots grow both in the streamwise and spanwise directions as they convect downstream with the flow; in this process, they often merge with neighbouring spots until the flow asymptotically becomes fully turbulent. In the tertiary instability region, these high-shear layers associated with instantaneous inflectional velocity profiles generate high-frequency fluctuations, forming turbulent spots. The generated turbulent spots accumulate and form a fully turbulent region. It is not necessary to be involved above discussed all the stages in the flow transition process. Thus, the flow transition is not a unique process. The process depends specifically on the given flow situation; some steps can be more significant than others, and some may be bypassed. Even in a two-dimensional boundary layer, it cannot be expected to happen all these stages in the same sequence where the dominant instability mechanism is not the exponential growth of the twodimensional Tollmien–Schlichting waves. A typical example is a transition induced by the high level of free-stream turbulence level. At higher free-stream turbulence levels (typically above 0.5–1 % of the free-stream velocity), the transition process may not follow this sequence, and one or more steps may be bypassed. Such a mechanism is known as bypass transition. Monokrousos et al. [17] studied the feedback control of the bypass transition experimentally and compared it with the simulation results. Finally, in his study of the secondary instability mechanism for the incompressible

1.3 Flow Transition Stages in Boundary Layer

5

boundary layer, Herbert [9] reviewed it as K-type, C-type, and N/H-type. Floquet analysis was used in his investigation for secondary instability, and he validated his results with the experiments. When all the waves of the same phase speed propagate together, it enables energy transfer from the primary to the secondary waves. This energy transfer results in the secondary waves’ rapid growth and finally in turbulence breakdown. Thus, secondary instability occurs in the flow. Secondary instability occurs when all waves involved propagate at the same phase speed, thus enabling energy transfer from the primary to the secondary waves which results in the rapid growth of the secondary waves and eventually in breakdown to turbulence. K-Type Klebanoff et al. [15] found this type of instability in their experiments on the flat-plate boundary layer. They found high-frequency pulsations using oscilloscope traces of Tollmien–Schlichting (T-S) waves. These were due to the interaction of the finite amplitude two-dimensional Tollmien–Schlichting (T-S) wave with a small amplitude of three-dimensional waves at the same frequency. The results obtained are different from linear stability analysis because of the three-dimensional nature of the disturbance wave dynamics. They pointed out that a shear layer in which growing two-dimensional disturbances exist has a stronger ability to amplify a small threedimensional disturbance. Thus, the mechanism proposed by Klebanoff et al. is known as K-type transition. In this situation, patches of  shaped vortices, which are aligned one behind the other in the streamwise direction, form and convect downstream. The periodic array of  vortices has a streamwise wavelength equal to the initial twodimensional T-S wavelength. It is characterized by the appearance of peaks and valleys in the spanwise velocity distribution and is commonly referred to as ‘peakvalley splitting’. Klebanoff et al. [15] discovered this instability in their flat-plate experiments [15]. A finite amplitude two-dimensional Tollmien–Schlichting (T-S) wave interacts with a small amplitude (linear) three-dimensional wave at the same frequency as the primary wave(fundamental breakdown). The finite amplitude of the primary wave is responsible for a phase locking with the secondary (generated) waves, enabling this resonance. This breakdown results in aligned streamwise vortices producing a peak-valley formation in a spanwise direction with a similar wavelength to the two-dimensional fundamental wave λz = λx. C-Type Craik [4] found a resonant triad in which fundamental two-dimensional and pair of oblique secondary waves are traveling at the same phase speed. The frequency of the primary two-dimensional wave is double that of the secondary oblique waves. He showed that a quadratic interaction between the oblique waves could produce a resonance with the primary wave. In this C-type breakdown, there is again a staggered arrangement with the streamwise wavelength of  vortices, the same as that of N-type. However, the spanwise wavelength is almost double that of the N-type. This resonance is not as high as the N/H-type breakdown. Craik [4] investigated a resonant triad where all three waves involved, i.e. a fundamental two-dimensional

6

1 Introduction to Flow Instabilities

wave and a pair of oblique secondary waves, travel at the same phase speed [4]. The fundamental two-dimensional wave has twice the frequency of the two secondary oblique (subharmonic) waves. To achieve equality of the phase speed, the oblique waves travel at a particular wave angle λ z, subh =2 λ x, f, and because of the equality of the speeds, it is not necessary for the fundamental two-dimensional waves to reach finite amplitudes in order to transfer energy to the oblique waves, thus causing early rapid amplification. Because there exists only one specific wave triad for each frequency of the primary waves, this resonance is not as robust as the N/H-type breakdown. N-Type or H-Type In the experiments carried out in Novosibirsk by Kachanov, Kozlov and Levchenko [12], where the  vortices were suggested. Later, Herbert [9] gave a theoretical explanation for this staggered pattern of  vortices in the term of subharmonic instabilities. This type of breakdown is known as N-type /H-types (for Novosibirsk/Herbert). In this staggered arrangement, the streamwise wavelength of  vortices equals twice the initial two-dimensional T-S wavelength. In this case, the angle of the oblique wave can be different than in a C-type breakdown. Thus, it makes C-type breakdown a particular type of N-type/H-type secondary instability. In the case of a flat-plate boundary layer, for low-level disturbances, sub-harmonic and high-level disturbances, the fundamental breakdown is the more feasible mechanism. The review of the physical mechanism involved can be found in Kachanov [11]. In a subharmonic breakdown, a finite amplitude two-dimensional wave leads to resonance with a small amplitude (linear) three-dimensional wave with half the frequency of the two-dimensional (primary) waves. These secondary oblique waves can be at different wave angles than in a C-type breakdown making the C-type a the particular case of the N-/H-type secondary instability. The flow field shows staggered streamwise  vortices with a spanwise wavelength of roughly half of the stream-wise wavelength of the fundamental wave (2 λ z = λ x). Experiments by Kachanov and Levchenko [12] and theoretical work by Herbert [9] revealed the associated spanwise variations of the subharmonic disturbance waves [9, 12]. For an incompressible flatplate boundary layer, the subharmonic breakdown is the more viable mechanism than a fundamental (K-type) breakdown when the disturbance level is very low. In contrast, the fundamental breakdown (K-type) is more likely for high disturbance levels. In his numerical investigations of supersonic transition, Fasel and Thumm [7] discovered this mechanism which is not secondary instability in the usual sense. A pair of oblique waves traveling with the same wave angle in opposite directions relative to the flow (ψ) nonlinearly generates a pair of steady streamwise structures with twice the spanwise wavenumber which grow rapidly in the streamwise direction. Linear stability theory is concerned with the transition of fluid motion from one state to another, particularly from laminar to turbulent flow. The celebrities like Lord Kelvin, Lord Rayleigh and Osborn Reynolds developed the fundamental concept of the theory in the 20th century. It is concerned with the initial stage of turbulence, its generation from a steady flow, but not with turbulent motion.

1.4 Methods of Instability Investigation

7

Under the effect of free-stream turbulence, the flow in the boundary layer develops streamwise elongated regions of low and high streamwise velocity. Brandt et al. [1] performed numerical experiments by varying the energy spectrum of the incoming perturbations. They found that the transition point moves to a lower Reynolds number with increasing the length scale of the free-stream disturbances. If low-frequency disturbances diffuse in the boundary layer, the streaks are induced by streamwise vortices through the lift-up mechanism. If the free-stream perturbations are above the boundary layer nonlinear mechanism is required to produce streamwise vortices in the shear layer.

1.4 Methods of Instability Investigation Mack, in 1984, split the total flow into a steady base flow and unsteady disturbance flow for his extensive numerical investigation. He then linearized the equations and applied the normal-mode approach for disturbance waves which assumes that the amplitudes vary in the wall-normal direction only. i(αx+βz−ωt) ˆ φ(y, t) = φ(y)e

(1.1)

In the above equation, α and ω are generally complex. In the temporal instability analysis, α is real while ω is complex and in spatial instability, ω is real, and α is complex. The real parts of αr and ωr are the streamwise wavenumber and frequency, respectively. The imaginary parts of αi and ωi describe the spatial and temporal ˆ amplification of the disturbance waves. The disturbance amplitude φ(y) is the function of wall-normal coordinate only, and it is not varying in a streamwise direction, so it is called local stability analysis. Mathematically it forms an eigenvalue problem (EVP) expressed by Orr–Sommerfeld or Squire equation [6]. However, the local stability analysis with the parallel base flow assumption is useful for the base flows with a slow streamwise variation like a boundary layer. The base flow velocity varies in the streamwise direction in the spatially developing boundary layers with the considerable wall-normal velocity component. The non-parallel effects are very strong at low and moderate Reynolds numbers and cannot be neglected. Thus, the parallel flow assumption is not valid for such flows where substantial non-parallel effects are present. Therefore, the local stability concept only has limited applications for parallel and weakly non-parallel (WNP) flows. In the last decade, the full non-parallel flow stability analysis has received much attention via global stability analysis. It concerns the instability of the flows developing in two inhomogeneous and one homogeneous spatial direction or three inhomogeneous spatial directions. The scope of applications of the linear stability theory is dramatically broadened compared to older instability analysis methodologies. It is a natural extension of the Orr–Sommerfeld/Squire equations; it is the solution of the partial differential equations based on two-dimensional or three-dimensional EVP describing linear growth/decay of the small amplitude disturbances. This defines the

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1 Introduction to Flow Instabilities

term global linear stability analysis. Results obtained using global linear instability analysis are slowly emerging in all areas of fluid mechanics, following the pace of hardware and algorithmic developments. In the case when a basic state is weakly dependent on the streamwise direction, a WKBJ approach is applied. The global stability analysis can analyse weakly non-parallel and fully non-parallel base flows. The concept of global stability analysis based on the solution of multidimensional EVP is broader than that based on the assumption of a weakly varying basic state. Biglobal and triglobal terms describe the instability analysis of two-dimensional and threedimensional basic states, respectively. The disturbance amplitudes are dependent on either two or three coordinates accordingly.

1.5 Overview of the Present Work This book is organized as follows: The overview and general introduction to flow instabilities are discussed in this chapter. Chapter 2 explains about global stability analysis, along with an introduction to non-parallel flows. It will be seen that any type of assumption does not restrict this approach, and it can be used to study the stability characteristics of flow through complicated geometries too. In this work, the global stability approach has been used to study both bounded and semi-bounded flows, namely channel flows and wall jets. This approach can also easily extend to unbounded flows like wakes and mixing layers without violating assumptions. The reader will find certain sections of this book containing much detail, for example, the section on numerics. The objective is that a new student may use this book as an aid. The first change in geometry considered is wall divergence, which is discussed in Chap. 3. Linear stability theory predicts that the flow through a straight channel becomes unstable for Reynolds numbers greater than 5772. However, in real life, this flow becomes unstable at a much lower Reynolds number, and this phenomenon is called subcritical transition. Nishioka et al. [19] showed that by reducing the freestream disturbances and making the walls of the channel extremely smooth, this flow could be maintained in a laminar state well above the critical Reynolds number. This hints at the effect of surface roughness and the change in geometry on flow stability. It is well known that a channel’s divergence and convergence dramatically affect flow stability. It was shown by [21] that a pipe with any nonzero wall divergence has a finite critical Reynolds number. To our knowledge, such flows have not been studied using a global approach. This motivates us to study the effect of change in geometry in channel flows. Chapter 4 covers the global stability characteristic of the converging-diverging channels. Divergence and convergence have opposite stability characteristics, and it will be interesting to study the flow through a channel with a series of alternating convergence and divergence. Such geometries have smaller critical Reynolds numbers compared to a straight channel and are widely used in heat exchangers. Chapter 5 presents a linear biglobal stability analysis of the axisymmetric boundary layer formed on a circular cylinder. The linearized Navier–Stokes (LNS) equations are derived in the polar, cylindrical coordinates. The two-dimensional

References

9

general eigenvalue problem is formulated from the discretized LNS. The results of the global stability approach are validated against the local stability approach. The temporal and spatial properties of different azimuthal waves and Reynolds numbers are reported. The axisymmetric and flat-plate boundary layer stability results are compared to show the effect of transverse curvature for the axisymmetric boundary layer. It shows that transverse curvature has stabilizing effect. The spatial growth rate (A x ) of the disturbance amplitudes is computed to show the overall effect of all the disturbances in the streamwise direction for different azimuthal wave numbers and Reynolds numbers. Chapter 6 presents a linear biglobal stability analysis of the axisymmetric boundary the layer formed over a circular cone. The governing stability equations (LNS) are derived in the spherical coordinates. The formulated two-dimensional eigenvalue problem is solved using an iterative algorithm. To validate the global stability approach, the blunt cone with a very small semi-cone angle α is considered in spherical coordinates. The global modes are computed for different semi-cone angles, azimuthal wave number (N) and Reynolds number of studying the effect of cone angle on temporal and spatial stability properties. Chapter 7 presents a linear biglobal stability analysis of the flat-plate boundary layer. The plate makes an incident angle (β) with the incoming flow. The stability equations are written in the body-fitted coordinates x and y, where x is the streamwise and y is the wall-normal direction. The global modes are computed with favourable and adverse pressure gradients to study the effect of pressure gradient on the temporal and spatial stability properties. Wall jets have many industrial applications, especially in heat and mass transfer. The applications depend on the parameters under which it will remain laminar or turbulent, and hence it is important to understand its stability characteristics. This flow, whose critical Reynolds number is very small, has been studied using local stability approaches. We study this using a global stability approach and discuss it in Chap. 8.

References 1. Brandt, L., Schlatter, P., Henningson, D.S.: Transition in boundary layer subject to free stream turbulence. J. Fluid Mech. 517, 167–198 (2004) 2. Breuer, K.S., Dzenitis, E., Gunnarsson, J., Ullmar, M.: Linear and nonlinear evolution of boundary layer instabilities generated by acoustic receptivity mechanism. Phys. Fluids 8, 1415 (1996) 3. Choudhari, M., Streett, C.: Theoretical prediction of boundary layer receptivity. In: AIAA Fluid Dynamics Conference, Colorado, CO, USA (1994) 4. Craik, A.A.D.: Nonlinear resonant instability in boundary layers. J. Fluid Mech. 50, 393–413 (1971) 5. Crouch, J.D.: Localized receptivity of boundary layer. Phys. Fluids 4, 1408 (1992) 6. Drazin, P.G., Reid, W.H.: Hydrodynamic Stability. Cambridge University Press, Cambridge (2004) 7. Fasel, H., Thumm, A.: Direct numerical simulation of three-dimensional breakdown in supersonic boundary layer transition. Bull. Am. Phys. Soc 36, 2701 (1991) 8. Goldstein, M.E., Hultgren, L.S.: Boundary layer receptivity to long-wave free-stream disturbances. Ann. Rev. Fluid Mech. 21, 137–166 (1989)

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9. Herbert, T.: Secondary instability of boundary layers. Ann. Rev. Fluid Mech. 20, 487–526 (1988) 10. Jackson, D., Launder, B.: Osborned Reynolds and the publication of his papers on turbulent flow. Ann. Rev. Fluid Mech. 39, 19–35 (2007) 11. Kachanov, Y.S.: Three-dimensional receptivity of boundary layers. Eur. J. Mech. B/Fluids 19, 723–744 (2000) 12. Kachanov, Y.S., Kozlov, V.V., Levchenko, V.Y.: Nonlinear development of a wave in a boundary layer. Fluid Dyn. 12, 383–390 (1977) 13. Kato, T., Fukunishi, Y., Kobayashi, R.: Artificial control of the three-dimensionalization process of t-s waves in boundary layer transition. JSME Intl. J. Ser. 40, 536–541 (1997) 14. Kerschen, E.J.: Boundary layer receptivity theory. Appl. Mech. Rev. 43, 152–157 (1990) 15. Klebanoff, P.S., Tidstrom, K.D., Sargent, L.M.: The three-dimensional nature of boundary layer instability. J. Fluid Mech. 12, 1–34 (1962) 16. Kobayashi, R., Fukunishi, Y., Nishikawa, T., Kato, T.: The Receptivity of Flat-Plate Boundary Layers with Two-Dimensional Roughness Elements to Free-Stream Sounds and Its Control, pp. 507–514. IUTAM Symposia, Springer, Berlin, Heidelberg (1995) 17. Monokrousos, A., Lundell, F., Brandt, L.: Feedback control of boundary layer bypass transition: comparison of simulations and experiments. AIAA J. 48, 1848–1851 (2010) 18. Morkovin, M.V.: On the many faces of the transition, in viscous drag reduction. In: Wells, C. (ed.) Viscous Drag Reduction, pp. 1–31. Plenum, 1967 (1969) 19. Nishioka, M., Iida, S., Ichikawa, Y.: An experimental investigation of the stability of plane Poiseuille flow. J. Fluid Mech. 72, 731–751 (1975) 20. Reshotko, E.: Boundary layer instability, transition and control, vol. 94. In: 32nd AIAA Aerospace Science Meeting and Exhibit, Reno, NV, USA (1994) 21. Sahu, K.C., Govindarajan, R.: Stability of flow through a slowly diverging pipe. J. Fluid Mech 531, 325–334 (2005) 22. Saric, W.S., Reed, H.L., Kerschen, E.J.: Leading edge receptivity to sound: experiments, DNS and theory. In: AIAA Fluid Dynamic Conference, Colorado, US (1994) 23. Saric, W.S., White, E.B., Reed, H.L.: Stability and transition of three dimensional boundary layers. Ann. Rev. Fluid Mech. 35, 413–440 (2003) 24. Schrader, L., Brandt, L., Henningson, D.S.: Receptivity mechanism in three dimensional boundary layer flows. J. Fluid Mech. 618, 209–241 (2009) 25. Wlezien, R.W.: Measurement of acoustic receptivity. In: AIAA Fluid Dynamics Conference, Colorado, USA (1994)

Chapter 2

Global Stability Approach

2.1 Introduction A flow can be defined entirely by the functional dependence of the flow quantities on the three coordinate directions, x, y, z and time t. Quantities such as the velocity field, pressure, temperature, density, viscosity, etc., are required for the complete flow description. For our purposes, flows can be classified in two ways, based on their dependence on time and space. First, they can be steady or unsteady, based on their time dependence. An unsteady flow varies with time, requires the flow variables expressed at every instant for a complete description and is beyond the scope of this thesis. A steady flow is independent of time. The second type of classification is based on the spatial dependence of the flow quantities. Under this classification, flows can be parallel or non-parallel. A parallel flow is one whose flow quantities depend only on one direction, usually a cross-flow direction. Typical examples are fully developed pipe flow and channel flow. Here, the flow velocity is described as a function of the wall-normal coordinate alone. Most other shear flows, however, are spatially developing, where the flow quantities change from one streamwise location to the next. Examples of non-parallel flows are wakes, jets, boundary layers, free shear layers, the flow behind a backwards-facing step, separated flows, and flow through complex geometries. So strictly parallel flows in real-life situations are very few. Even in the case of a straight channel or pipe, the flow is often not fully developed. High Reynolds number flows have a very long entry region where the flow quantities vary significantly with distance, and it can be classified as a non-parallel flow. The entire length of the pipe or channel may be part of this entry region. A non-parallel flow depends on more than one spatial coordinate, which can be two-dimensional (2D) or three-dimensional (3D). Under special circumstances, the variation of the flow quantity will be mild from station to station, and these flows will be classified as weakly non-parallel flows as the non-parallel effects are weak. For example, a flatplate boundary layer at high Reynolds numbers and flow through diverging channels at small divergence angles are common examples of weakly non-parallel flows. Also, many flows whose laminar profiles obey self-similarities, such as wakes and jets, are © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Bhoraniya et al., Global Stability Analysis of Shear Flows, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-19-9574-3_2

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2 Global Stability Approach

assumed to exhibit only weak non-parallelism at very high Reynolds numbers. The classifications discussed above, based on time and space, are independent of each other. For example, a steady flow can be parallel or non-parallel, whereas a nonparallel flow can be steady or unsteady.

2.2 Linear Stability Theory An incompressible flow, steady or unsteady, parallel or non-parallel, can be defined by the Navier–Stokes equations and continuity equation, given below. ∂u ∂u ∂u −1 ∂ p 1 ∂u +u +v +w = + ∂t ∂x ∂y ∂z ρ ∂x Re ∂v ∂v ∂v ∂v −1 ∂ p 1 +u +v +w = + ∂t ∂x ∂y ∂z ρ ∂y Re ∂w ∂w ∂w ∂w −1 ∂ p 1 +u +v +w = + ∂t ∂x ∂y ∂z ρ ∂z Re

 

∂ 2u ∂ 2u ∂ 2u + + ∂x2 ∂ y2 ∂z 2 ∂ 2v ∂ 2v ∂ 2v + + ∂x2 ∂ y2 ∂z 2



 .

(2.1)

.

(2.2)



 ∂ 2w ∂ 2w ∂ 2w . + + ∂x2 ∂ y2 ∂z 2

∂u ∂v ∂w + + = 0. ∂x ∂y ∂z

(2.3)

(2.4)

Here x, y, z are the streamwise, cross-stream and spanwise directions of the flow, respectively, u, v, w are the respective velocity components in those directions, p is the pressure, ρ is the density, t is the time, and Re is the Reynolds number. This set of equations coupled with suitable boundary conditions must be solved to define the flow field for a given situation. Dropping the time derivatives, we may obtain steady solutions satisfying the above equation and certain classes of boundary conditions. These solutions are termed base flow. We distinguish the term ‘base flow’ from ‘mean flow’, although the two are often used interchangeably in this context. A base flow satisfies the steady Navier–Stokes equations, but a mean flow may not since it is merely the average of many realizations of the unsteady equations. The stability characteristics of the base flow are determined by adding time and space perturbations and tracking the perturbations in time and space to check if they grow or decay. The perturbation’s growth and decay define the base state’s instability and stability, respectively. Flows studied in this work are two-dimensional (2D) in nature and are defined by W = 0 and ∂(U, V )/∂z. The concepts of stability are best explained using 2D base flows. Their extension to 3D base flows straightforward. Also, we confine ourselves to the addition of 2D perturbations. 3D perturbations can cause qualitatively different instabilities and are much richer. A flow quantity can be expressed, for example, as

2.2 Linear Stability Theory

u = U + u, ˆ

13

v = V + v, ˆ

p = P + p, ˆ

(2.5)

where the uppercase letters stand for the base flow and the quantities with a hat (ˆ) indicate the perturbations. Substituting Eq. 2.5 in Eqs. 2.1, 2.2 and 2.4, neglecting nonlinear terms in the perturbation and subtracting out the base flow equation, we get the evolution equations for the perturbations as, ∂ uˆ ∂U ∂ uˆ ∂U −1 ∂ pˆ 1 ∂ uˆ +U + uˆ +V + vˆ = + ∂t ∂x ∂x ∂y ∂y ρ ∂x Re ∂ vˆ ∂ vˆ ∂V ∂ vˆ ∂V −1 ∂ pˆ 1 +U + uˆ +V + vˆ = + ∂t ∂x ∂x ∂y ∂y ρ ∂y Re ∂ uˆ ∂ vˆ + = 0. ∂x ∂y

 

∂ 2 uˆ ∂ 2 uˆ + ∂x2 ∂ y2 ∂ 2 vˆ ∂ 2 vˆ + ∂x2 ∂ y2

 .

(2.6)

.

(2.7)



(2.8)

This widely used approach is referred to as linear stability theory (LST). As the transition to turbulence is characterized by instability followed by rapid nonlinearization in many shear flows, researchers criticize the use of LST, see for example [70], as it would at best be able to give information only about the initial stages of transition. However, an LST study is crucial in many flow situations. It gives many essential pointers about the initial stages of transition, signatures of which are often clearly visible even in fully developed turbulence. Besides this, linearly stable flow can undergo large transient growth and reach a completely different state. Alternatively, nonlinear instability could be the first to occur, creating a ‘bypass’ route to transition to turbulence. The remarkable success of LST is the accurate prediction of the critical Rayleigh number of 1708 for a Rayleigh–Benard flow. Its failure to match with experimental data for a plane Poiseuille flow has been explained using non-modal stability theory, see for example [56]. We are specifically interested in understanding the initial stages of transition in flows where LST is among the most significant phenomena taking the flow towards turbulence. A major portion of this book is devoted to channelling flow, where a change in geometry changes the dominant transition mechanism from one of transient growth to one of exponential and a non-exponential, but still modal, spatial growth. Our question is, what triggers instability and what are the parameters involved? Henningson [28] has shown that the nonlinear terms of the disturbance equation are conservative, and if the energy of the disturbance has to grow, it has to be through a linear process only, thus supporting LST. Note that this observation is strictly true for a parallel flow only since an assumption is made about the spatial periodicity of disturbance. It is nevertheless instructive about the importance of linear terms. To quote him, the disturbance energy produced by linear mechanisms is the only energy available. This implies that the disturbance energy produced by transient mechanisms in sub-critical transition causes the total E (energy) to increase during the transition process. This can be understood by considering the evolution equation for the kinetic disturbance energy; see for example [57].

14

2 Global Stability Approach

ui

∂u i ∂Ui 1 ∂u i = −u i u j − ∂t ∂x j Re ∂ x j  1 ∂ − ui ui U j − + ∂x j 2

∂u i ∂x j  1 1 ∂u i . u i u i u j − u i pδi j + ui 2 Re ∂ x j

(2.9)

This equation is obtained by substituting Eq. 2.5 in Eqs. 2.1–2.3 and expressing them in the Einstein notation, multiplying them with u i . Now, the left-hand side is the rate of change of the kinetic energy of the disturbance, E. Remember that the nonlinear terms are retained. We now assume that the disturbances are localized or spatially periodic. This assumption may not hold true for a non-parallel flow. Integrating the equation over a control volume V and using Gauss’s theorem, the derivative terms drop out, giving, dE V =− dt



∂Ui 1 ui u j dV − ∂x j Re



∂u i ∂u i dV. ∂x j ∂x j

(2.10)

This equation is called the Reynolds–Orr equation. We can see here that the terms remaining on the right-hand side are obtained from the linear terms of the original equation, and the terms obtained from the nonlinear terms of the original equation have dropped out. This equation says that the kinetic energy of the disturbance grows purely by linear terms, and nonlinear terms do not contribute to the energy growth.

2.3 Stability of Parallel Flows Considering 2D perturbations allows us to express the disturbance in terms of the streamfunction φ which automatically satisfies the continuity equation. Eliminating pressure between the Eqs. 2.6 and 2.7, we get the equation which determines the evolution of perturbations as,       2  2 ∂ ∂ ∂ 2U ∂ ∂ U ∂2 V ∂ 1 4 ∂ V 2 U +V ∇ + − + − − ∇ φ ∂x ∂y ∂ x∂ y ∂ y2 ∂ x ∂ x∂ y ∂x2 ∂y Re ∂ = ∇ 2 φ. ∂t (2.11) Here, uˆ and vˆ are replaced with ∂φ/∂ y and −∂φ/∂ x respectively, ∇ 2 = (∂ 2 /∂ x 2 + ∂ 2 /∂ y 2 ). This perturbation equation is a partial differential equation. Solving this equation numerically is difficult and time-consuming. A few further simplifications are possible based on the type of flow considered, and this leads to a classification based on the different stability approaches, as discussed below. As its name suggests, this method is applicable for strictly parallel flows. Over the decades, non-parallel flows have, however, often been studied using this approach. When the streamwise variation is slow, as in many high Reynolds number shear flows, the idea is that the

2.3 Stability of Parallel Flows

15

flow may be treated as being locally parallel. As the steady base flow depends only on the wall-normal direction (y) and is independent of the streamwise direction (x) and time t, the disturbance can be Fourier transformed in the independent coordinates as follows, (2.12) φ(x, y, t) = φ(y)eι(αx−ωt) . Here, α is the wavenumber in x, and ω is the frequency. We perform spatial or temporal stability analysis, depending on whether we choose a complex α or complex ω. The imaginary part of α and ω will give information about the disturbance’s spatial and temporal growth/decay rate, respectively. Assuming a complex ω and real α in Eq. 2.12, and substituting it in Eq. 2.11, we get an ordinary differential eigenvalue problem for ω as, [(U − c)(D 2 − α 2 ) − U  ]φ =

1 [D 2 − α 2 ]2 φ. iαRe

(2.13)

Here, the eigenvalue c = ω/α is the phase velocity of the disturbance, D = ∂/∂ y and the prime denotes differentiation with respect to y. In a temporal framework, the equation is solved for the eigenvalue c by supplying a real α and Re. In general, the eigenvalue obtained is complex, and a plot of the frequency ωr versus the growth rate ωi is called the frequency spectrum. A sample spectrum obtained for a plane Poiseuille flow at a Reynolds number of 5770 and α = 1.02 is shown in Fig. 2.1. As we can see from the figure, this flow is stable for this α as all the eigenvalues have a negative decay rate ωi , meaning the disturbances decay exponentially in time. Also, note that this flow is near-neutral at this Reynolds number. A small increase in Reynolds number to 5772.3 will push this mode towards the unstable half plane to exhibit exponential instability with a growth rate of ωi . The equation discussed above is the famous Orr–Sommerfeld equation, derived independently at the beginning of the 20th century by the Irish mathematician William McFadden Orr and the German theoretical physicist Arnold Johannes Wilhelm Sommerfeld. Soon after, many people developed methods for approximate solutions to this equation using expansion methods, a discussion of which is given in [17]. Tollmien [68] first solved the Orr–Sommerfeld equation for the Blasius boundary layer and obtained a neutral curve. Most of the earlier literature on hydrodynamic stability considers very simple flow configurations like plane Poiseuille flow and plane Couette flow. This is because stability studies need information on the base flow state, and it is possible to get the base state information analytically for these flows. Even though this method is restricted to parallel flows strictly, many researchers have applied this approach to study flows which evolve slowly downstream and were often able to get a reasonable agreement with experiments. However, it must be noted that this agreement is case-dependent and does not hold good for various flow geometries. Hence, a different approach was evolved to study the stability of flows which vary downstream, as discussed below.

16

2 Global Stability Approach

Fig. 2.1 Spectrum obtained by solving Orr–Sommerfeld equation at Re = 5770 with α = 1.02 for a plane Poiseuille flow. As can be seen, this flow is near-neutral in that one eigenvalue, shown in the box, is on the verge of moving into the unstable half-plane (ωi > 0)

0 ωi -0.5

-1

0

0.5

ωr

1

2.4 Stability of Weakly Non-parallel Flows As mentioned before, the Orr–Sommerfeld equation neglects the streamwise variation of the base flow and the eigenfunction and solves the flow locally. This initially was thought to be the reason for the mismatch between theory and experiments. Hence, many researchers were involved in developing theories which would incorporate non-parallel effects. They attacked it as a perturbation problem of the Orr– Sommerfeld equation or the resulting solutions, for example, see [11] in which the authors studied the stability of plane Couette flow with small variations in the base flow and found that this flow is destabilized even though the unperturbed flow is linearly stable for all Reynolds numbers. Gaster [23] studied the effect of non-parallel terms on the stability of boundary layers and did not find drastic differences from the parallel flow results. The main difference between parallel and weakly non-parallel (WNP) theory is that parallel theory neglects all variations in x. In contrast, WNP retains terms up to order Re−1 in both mean and perturbations and neglects higher orders. The disturbance thus takes the form,

φ(x, y, t) = φ(x, y)eι(

α(x)dx−ωt)

.

(2.14)

Here, the wavenumber α accounts for the fast variations in x, and φ(x) is assumed to vary slowly with x, i.e. ∂/∂ x ∼ 1/Re and ∂ 2 φ/∂ x 2 ∼ 0. The parabolized stability equations developed by [8, 30] and the minimal composite theory developed by [26, 27] are examples of the WNP approach. This approach has been widely used in many flows like boundary layers—[7, 8, 21], mixing layers—[10, 43] and diverging pipes—[55]. Since both parallel and weakly non-parallel approaches use the local velocity profiles and other local quantities to determine the stability characteristics at a given streamwise location, they are called local approaches.

2.5 Global Stability Analysis

17

2.5 Global Stability Analysis The applications of the local approaches are limited to parallel and weakly nonparallel flows. A global approach is necessary when the streamwise change of the flow is not negligible. More significantly, as will be shown later in the thesis, even in apparently weakly non-parallel flows which obey self-similarity, a global approach reveals results inaccessible to the local approaches. Even a flow between two parallel plates might exhibit non-parallelism in the region very close to the inlet where the flow is not fully developed. In such flows, the disturbance is not Fourier transformable in x and is left as an arbitrary function of x and y as, φ(x, y, t) = φ(x, y)e−ιωt .

(2.15)

Substitution of 2.15 in Eq. 2.11 results in a partial differential eigenvalue problem in ω as,       2  2 ∂ ∂ ∂ 2U ∂ ∂ U ∂2V ∂ 1 4 ∂ V U φ +V ∇2 + − + − − ∇ ∂x ∂y ∂ x∂ y ∂ y2 ∂ x ∂ x∂ y ∂x2 ∂y Re = iω∇ 2 φ. (2.16) The only difference between Eqs. 2.11 and 2.16 is in the time derivative. This approach gives a global picture of the disturbance in terms of the eigenfunction as against a local approach. This approach can be termed as biglobal and triglobal based on whether the base flow considered is 2D or 3D, following [63]. The present work, as mentioned before, is restricted to 2D flows. Extension to 3D base flows is straightforward but computationally very costly, as we will see later. The only difference between Eqs. 2.11 and 2.16 is in the time derivative. This approach gives a global picture of the disturbance in terms of the eigenfunction against a local approach. This approach can be called biglobal and triglobal based on whether the base flow is 2D or 3D, following [63]. The present work, as mentioned before, is restricted to 2D flows. Extension to 3D base flows straightforward but computationally very costly, as we will see later. The first global study in the present context dates back to work by [51], where he studied two-dimensional disturbances in inviscid vortices. First calculations on viscous flows were reported by [33], who studied the flow past variously shaped bodies, and [71], who studied flow past a circular cylinder. This approach slowly became popular with researchers in the study of free surface flows— [14], rectangular ducts—[34, 61] and boundary layers—[35, 36]. However, applying this method to large domains and at high Reynolds numbers was hindered by the large computational costs involved in solving the global stability equation. This is because the matrices emerging from the discretization process are large, dense and non-symmetric. All the researchers, as mentioned above (except [14]), have used the traditional QR algorithm to solve the resulting matrix. This algorithm’s shortcoming, and sometimes the strength, is that it must compute all the eigenvalues. Since very few ‘dangerous’ modes trigger the instability in many shear flows, it

18

2 Global Stability Approach

would be computationally far more economical to calculate only those dangerous eigenvalues. Hence, many researchers were involved in developing efficient methods to attack the global stability problem by solving only those physically relevant eigenvalues. Many algorithms like the minimal residual algorithm and the conjugate gradient method were tried. Since these methods were restricted to symmetric, sparse matrices, researchers were involved in extending these methods to handle the nonsymmetric generalized matrix systems arising out of the global stability approach, for example, see [14, 53]. Iterative techniques like the Arnoldi iteration together with ‘shift-invert strategy’ and Lanczos iteration are explained in [45, 52]. Other examples are the low-dimensional Galerkin methods (see in [46, 47]) and Simultaneous (Subspace) Iteration technique used in [16]. The techniques developed were used for a variety of fluid dynamical problems, like Rayleigh–Benard flow by [16], channel flow over riblets by [18], attachment-line boundary layer by [62] and the flow around a circular cylinder by [44], where Krylov subspace methods were used to compute only a part of the physically relevant eigenvalues. Since the beginning of the 21st century, this global stability approach has been used to study a wider variety of complex flow geometries. Theofilis et al. [66] studied 2D steady laminar separation bubble using WNP and DNS and got an excellent matching of the stability characteristics for both 2D and 3D. However, by performing a global stability study, they show the existence of new instability modes inaccessible to either approach. They propose that flow control studies, which generally consider the T-S waves and DNS frequencies, should consider the global mode frequencies for better control. Barkley [5] performed a biglobal stability study on a backwards-facing step and found that the critical eigenmode is localized in the recirculation regions behind the step. Schmid and Henningson [58] performed a global stability analysis on a falling liquid curtain. They show that while a single global mode cannot match the experimental results, an optimal superposition of many global modes was able to agree very well with the experimentally observed frequencies. Theofilis et al. [65] studied a swept attachment-line boundary layer flow using both DNS and global stability study and showed that the temporal and spatial solution of this problem could be obtained by a three-dimensional extension of the Gortler–Hammerlin model at a lower computational cost. A detailed review of global stability analysis is given in [63]. In [64], the authors studied the stability characteristics of four flow types using a global stability study. The global eigenvalue spectrum of a rectangular duct has been obtained and compared with that of a plane Poiseuille flow. It was shown that the flow through a rectangular duct stabilizes in the limit of the geometry going towards a square duct. They also calculated the global spectrum of a bounded Couette flow and found only stable modes. A lid-driven cavity was also studied, and critical Reynolds numbers were obtained. This study was able to achieve excellent agreement with experimental results. Furthermore, it explained the error in the stability prediction of wall-bounded Couette flows based on the in-plane velocity. In [67], the authors studied the global stability of separated profiles in three different flow configurations. In these flows, they show that the amplitude of the global modes is less in the separated region than in the wake region or shear layer region, depending on the problem under consideration. They also hint at the necessity to consider information from the

2.5 Global Stability Analysis

19

global modes for flow control studies. Mittal and Kumar [40] studied the flow past a rotating cylinder and gave critical values for the occurrence of instability. Mittal and Singh [42] used a finite element method to study the vortex shedding behind cylinders at sub-critical Reynolds numbers. They found that vortex shedding is possible at very low Reynolds numbers and obtained a good match between global stability results and numerical simulations. Ehrenstein and Gallaire [19] studied the global stability characteristics of a flat-plate boundary layer, which had hitherto been studied locally. By an optimal superposition of the global temporal modes, the authors could simulate the convective nature of the instability of the boundary layer. Chedevergne et al. [13] studied solid rocket motors with fluid injection and found that the global eigenspectrum is discrete for this case. The obtained global mode frequencies compare very well with experimental results, and the authors propose that these global modes give insight into the thrust oscillations in solid rocket motors. Gonzalez et al. [24], for the first time, developed a finite element method with unstructured meshes for biglobal stability applications. Mittal and Kumar [41] have developed a new approach for global stability in which the equations are written in a moving frame of reference, which travels with the disturbance. This approach thus determines the global convective instability of the modes at a particular instant, as against the large volume of temporal global instability studies. This method was used to study flow past a circular cylinder, and excellent agreement was obtained with the direct numerical simulations. Akervik et al. [3] performed a biglobal stability analysis on a flat-plate boundary layer and obtained a perfect match with the global stability results and the weakly non-parallel results. These authors could also show the convective nature of instability of the boundary layer using these global modes. They also study the transient amplifying behaviour of the global modes and discuss them in detail. Akervik et al. [1] perform a global stability study on the flat-plate boundary layer and calculate the maximum energy growth possible with these global modes using a reduced order model. They show that even with global modes, the optimal energy growth is not obtained with just a few least stable modes. This explains the need to consider a few stable global modes to capture the disturbance dynamics, which is crucial in flow control, as we will see below. In the later part of the last decade, global modes have been predominantly used in flow control. To achieve better control, the disturbance modes should have good observability and controllability. However, the best observable modes need not always be the best controllable modes. Therefore, a balance should be achieved in observability and controllability while selecting the modes for flow control. In active flow control, the disturbance energy is sensed at one streamwise location, and an appropriate response is actuated at another streamwise location, generally an upstream location. In local stability analysis, the modes at the sensor and actuator locations are not connected to each other. Whereas, while using global modes, which extend throughout the domain, including the sensor and actuator locations, the information is contained well within the global mode. Hence, these modes work much better in designing the desired control compared to local modes. The observability and controllability of global modes are determined by the maximum amplitude of the direct and adjoint global modes, respectively. A global adjoint mode is a global mode obtained by solving the adjoint global stability equation.

20

2 Global Stability Approach

A direct global mode is obtained by solving directly the global stability equation, which has been just referred to as a ‘global mode’ in the previous discussions. While using global modes for control, the sensor is placed at the location where the direct global modes have their maximum amplitude, and the actuators are placed where the adjoint global modes have their maximum amplitude. See [2] for a controlled study of a separated boundary layer in a cavity using global modes. In [29], the authors have developed reduced order models from the global modes to perform flow control on three flow configurations, namely, a falling liquid sheet, a Blasius boundary layer and a boundary layer flow along a shallow cavity. A reduced-order model does not consider all the global modes, but extracts the essential information from the global modes, converts them into a reduced form and uses it to perform flow control. Proper orthogonal decomposition (POD) is an example of a reduced order model; see [4] for example. This paper discusses in detail using global modes, POD modes and balanced POD modes to control a separated boundary layer. POD modes are best controllable, whereas balanced POD modes are both observable and controllable. Marquet et al. [38] studied a smoothed backwards-facing step using global modes and identified the non-normality associated with the governing equations in two forms—a lift-up non-normality resulting from the transport of the base flow by the perturbation and a convective non-normality resulting from the transport of the perturbations by the base flow. By computing the adjoint global modes, the authors were able to identify the optimal location of the sensors and actuators, in terms of both controllability and observability, for this flow. In the previous sections, the term global stability analysis has been used to refer to the partial differential eigenvalue problem where more than one direction is taken as the eigendirection. In the weakly non-parallel framework, this term has often been used in a slightly different context, like in [39, 43]. Primarily to clarify the terminology, we briefly discuss the concepts of convective and absolute instability to understand the global stability definition used here. These concepts were first developed in the early 1950s in the field of plasma physics; see [6, 60] for detailed reviews. In contrast to the spatial or temporal analysis, where we consider a complex α or ω, respectively, a convective-absolute stability analysis considers both α and ω to be complex. This approach is a spatio-temporal analysis, giving information about the disturbance evolution in both space and time; hence, this approach was widely called the global stability analysis. Like the WNP, this approach uses the (Wentzel–Kramers–Brillouin–Jeffreys) WKBJ approximation, details of which, along with a detailed review of the absolute/convective and local/global instabilities, are available in [32]. In a convectively unstable but absolutely stable flow, a disturbance introduced at a localized region grows in time. However, it gets convected downstream, thus leaving the base flow at that location free of a disturbance at a later time. Plane Poiseuille flow—[15], circular jets and flat-plate boundary layers—[20, 22], and mixing layers (for coflow or small counter flow)—[31] are examples of convectively unstable flows. Convectively unstable flows are called globally stable flows as the flow becomes free of disturbance as it gets convected away, and the flow eventually becomes stable. In an absolutely unstable flow, a disturbance introduced at a given streamwise station grows at that station, propagating both upstream and downstream, sometimes contaminating the entire flow field. Examples are bluff body

2.6 Discretization

21

wakes—[9, 50], mixing layers (for large counter flow)—[31], boundary layer on a rotating disc—[37, 49]. It is important to remember that the presence of absolute instability does not necessarily imply global instability. It has been shown in [43] that a region of absolute instability is a necessary but not sufficient condition for global instability. This is because if a flow has a pocket or finite region of absolute instability, like a bluff body wake, then the flow can sustain temporally growing modes only inside that region. Elsewhere, the flow may only be convectively unstable. The review paper, [32] contains an appendix which lists the various convective and absolute stability studies done in wall-bounded shear flows, jets, wakes and mixing layers. Even though global stability studies undertaken in the present work have existed since the 1980s, the term global stability was used to represent this spatio-temporal absolute/convective approach until the early 21st century. However, the current definition of global stability considering the streamwise direction also as an eigendirection (biglobal or triglobal) has gradually gained currency in the past decade. The first symposium exclusively on the global stability approach was conducted in 2001, with an increase in the number of researchers using this method. In fact, when I was a week-old student in JNC in 2005, my advisor attended one of these exclusive symposia on global stability in Crete, Greece and that laid the foundation of this research work.

2.6 Discretization Discretization is the art of expressing a continuous quantity defining a flow field at discrete points in the domain. This is necessary because the flow equation cannot be solved numerically at every point in the domain but can be solved at many points. Different types of discretization methods exist, e.g. finite difference (FD), finite element method (FEM), finite volume method (FVM) and spectral method (SP). One commonly used spectral method is the Fourier method, which has trigonometric functions as the basis functions and an exponential convergence rate. However, the drawback is that it can be used only for periodic functions. On the other hand, the Chebyshev method uses Chebyshev polynomials as the basis functions. Legendre and other polynomials may also be used in spectral methods, but they are not discussed here. The discretization scheme used in this work is Chebyshev-spectral collocation. Srinivasan et al. [59] has a very easy-to-understand introduction to Chebyshev spectral method and how to use it to solve fluid dynamical problems. The authors learnt to use spectral methods from this paper and would recommend this reference to anyone new to SP discretization. When a differential equation is discretized, we get a discretized equation in matrix form. The discretization is explained first for an ODE that governs 1D flow, then for a PDE that governs a 2D flow.

22

2 Global Stability Approach

2.6.1 1D Discretization Let u be the flow parameter of a 1D flow, governed by the equation u  − αu = 0. Here a prime denotes differentiation with respect to y. Let us discretize this equation with m grid points, with the value of u at each point expressed as u 1 , u 2 , ..., u m−1 , u m . Now the equation has to be solved at each grid point and thus for m grid points, we will have m equations as, d2 u i = αu i , dy 2

i = 1, 2, ...m.

If we define a differentiation matrix D to represent the discretized form of d/dy and D 2 for d/dy 2 , the above set of equations can be written in matrix form as, ⎛ ⎜ ⎜ ⎜ D2 ⎜ ⎝

⎞⎛

⎞ ⎛ u1 α ⎟ ⎜ u2 ⎟ ⎜ 0 ⎟⎜ ⎟ ⎜ ⎟⎜ . ⎟ = ⎜0 ⎟⎜ ⎟ ⎜ ⎠⎝ . ⎠ ⎝0 0 um

0 α 0 0 0

0 0 α 0 0

0 0 0 α 0

⎞⎛ ⎞ u1 0 ⎜ u2 ⎟ 0⎟ ⎟⎜ ⎟ ⎜ ⎟ 0⎟ ⎟ ⎜ . ⎟. 0⎠⎝ . ⎠ α um

The coefficients of the differentiation matrix D depend on the type of discretization used. FD methods are derived from Taylor’s expansion and its accuracy depends upon the number of terms retained in the Taylor’s expansion. For example, by considering up to the third term of the Taylor’s expansion, we may derive a second order accurate FD formula as, f (x + h) − f (x − h) f  (x) = . (2.17) 2h To improve the accuracy, one would retain more and more terms in Taylor’s expansion and get information from more grid points. Given the m number of points, the best one could do is to consider all the m points, and that is what the SP method does. SP methods are known to exhibit higher levels of accuracy compared to FD methods. This is shown schematically in Fig. 2.2. However, the price to pay for higher accuracy

Fig. 2.2 Schematic representation showing the dependence of the derivative at a point on the other grid points. As can be seen, SP methods have information from all the grid points, which gives high accuracy

SP

u1 u2 FD

’

ui ui-1 u u i+1 i ’

ui ui-1 ui ui+1

un-1 un

2.6 Discretization

23

with SP methods is dense in the resulting matrix. A sample D matrix from FD and SP discretizations will look like this, ⎛ ⎛ ⎞ ⎞ d11 d12 d1. d1. d1m d1 d1+1 0 0 0 ⎜ d21 d22 d2. d2. d2m ⎟ ⎜ di−1 di di+1 0 0 ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ d.1 d.2 d.. d.. d.m ⎟. ⎜ 0 di−1 di di+1 0 ⎟ ⎜ ⎜ ⎟ ⎟ ⎝ d.1 d.2 d.. d.. d.m ⎠ ⎝ 0 0 di−1 di di+1 ⎠ 0 0 0 dm−1 dm dm1 dm2 dm. dm. dmm We can note that for a discretization with ‘m’ grid points, the D matrix is of size m × m. In FD method, the D matrix is sparse (l-diagonal for a scheme which is usually accurate up to the order (l − 1)/2). Occasionally higher-order accuracy can be obtained by clever algebraic manipulations for a given number of nonzero elements. For a given m, this requires less memory for matrix storage and less computational time to operate on. On the contrary, the SP method gives a matrix where all the elements are nonzero, requiring large memory and computational time. But accurate results are obtained with less number of grid points in SP than would be required to get the same level of accuracy using FD. In Chebyshev spectral discretization, the grid points are not equally spaced but are defined by a cosine function given as, y j = cos

jπ , m

j = 1, 2, ..., m − 1, m.

This defines the value of y between +1 and −1. A transformation can be enforced to make y vary between physically relevant values based on the problem. This will be explained in Sect. 2.6.4. For 1 > y > −1, the D matrix is defined as follows: ck (−1)k+ j c j (yk − y j ) yk D(k, k) = − 2(1 − yk 2 )

D(k, j) =

1 ≤ k, j ≤ m, k = j

(2.18)

2≤k ≤m−1

(2.19)

2m 2 + 1 6

(2.20)

D(0, 0) = −D(m, m) =

with c1 = cm = 2, c j = 1 for 2 ≤ j ≤ m − 1. Higher derivatives are calculated by operating the D matrix onto itself, like D 2 = D ∗ D.

2.6.2 2D Discretization-Y Now, for simplicity, let us consider the flow variable u in a 2D domain and choose a rectangular geometry in the x − y plane. Let us discretize this domain with n points in x and m points in y, as shown in Fig. 2.3. The value of u at each grid station is

24

2 Global Stability Approach

Fig. 2.3 Nomenclature used for the grids locations in a 2D domain

y=m xu

xu

xunm

u y=2 x 12

u x 22

xun2

u u y=1 x 11 x 21 x=1 x=2

xun1 x=n

1m

2m

indicated as u i j , where i corresponds to the x grid number and j corresponds to the y grid number, as shown in Fig. 2.3. Here, the total number of grid points will be n × m = nm. For demonstration, let us discretize the following equation, ∂u ∂u + = 0. ∂x ∂y

(2.21)

As before, the value of u at all the nm grid locations can be written as a column vector with the first m values corresponding to the first x location, the following m values for the following x location and so on, like, ⎛

⎞ u 11 ⎜ u 12 ⎟ ⎜ ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ u 1m ⎟ ⎜ ⎟ ⎜ u 21 ⎟ ⎜ ⎟ ⎜ u 22 ⎟ ⎜ ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ u 2m ⎟ . ⎜ ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ u n1 ⎟ ⎜ ⎟ ⎜ u n2 ⎟ ⎜ ⎟ ⎜ . ⎟ ⎜ ⎟ ⎝ . ⎠ u nm Here, each x location is indicated with a different colour for ease of viewing. The discretization matrix in y (dy) can be defined using the formula 2.20 for each

2.6 Discretization

25

x location of size m × m and hence the D y matrix for the 2D domain will take the form, ⎛

dy11 ⎜ dy21 ⎜ ⎜ dy.1 ⎜ ⎜ dy.1 ⎜ ⎜ dym1 ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

dy12 dy22 dy.2 dy.2 dym2

dy1. dy2. dy.. dy.. dym.

⎞⎛

dy1m dy2m dy.m dy.m dymm dy11 dy21 dy.1 dy.1 dym1

dy12 dy22 dy.2 dy.2 dym2

dy1. dy2. dy.. dy.. dym.

dy1m dy2m dy.m dy.m dymm

dy11 dy21 dy.1 dy.1 dym1

dy12 dy22 dy.2 dy.2 dym2

dy1. dy2. dy.. dy.. dym.

⎞ u 11 ⎟ ⎜ u 12 ⎟ ⎟⎜ ⎟ ⎟⎜ . ⎟ ⎟⎜ ⎟ ⎟⎜ . ⎟ ⎟⎜ ⎟ ⎟ ⎜ u 1m ⎟ ⎟⎜ ⎟ ⎟ ⎜ u 21 ⎟ ⎟⎜ ⎟ ⎟ ⎜ u 22 ⎟ ⎟⎜ ⎟ ⎟⎜ . ⎟ ⎟⎜ ⎟ ⎟⎜ . ⎟ ⎟⎜ ⎟ ⎟ ⎜ u 2m ⎟ . ⎟⎜ ⎟ . ⎟⎜ ⎟ ⎟⎜ . ⎟ ⎟⎜ ⎟ ⎟⎜ . ⎟ ⎟⎜ ⎟ ⎟⎜ . ⎟ ⎟⎜ ⎟ ⎜ ⎟ dy1m ⎟ ⎟ ⎜ u n1 ⎟ ⎜u ⎟ dy2m ⎟ ⎟ ⎜ n2 ⎟ ⎜ ⎟ dy.m ⎟ ⎟⎜ . ⎟ ⎝ ⎠ dy.m . ⎠ dymm u nm

2.6.3 2D Discretization-X The differentiation matrix in x (dx) can be defined using Eq. 2.20 for x defined between 1 and −1. Since the column vector is arranged in a particular manner, the differentiation matrix Dx of the 2D domain will take the form: ⎛

dx11 0 0 0 0 dx12 0 ⎜ 0 dx11 0 0 0 0 dx12 ⎜ ⎜ 0 0 . 0 0 0 0 ⎜ ⎜ 0 0 0 . 0 0 0 ⎜ ⎜ 0 0 0 0 0 dx11 0 ⎜ ⎜ dx21 0 0 0 0 dx22 0 ⎜ ⎜ 0 dx21 0 0 0 0 dx22 ⎜ ⎜ 0 0 . 0 0 0 0 ⎜ 0 0 . 0 0 0 ⎜ 0 ⎜ 0 0 0 0 dx21 0 ⎜ 0 ⎜ . 00 0 0 . ⎜ 0 ⎜ 0 0 . 0 0 0 0 ⎜ ⎜ 0 0 0 . 0 0 0 ⎜ ⎜ 0 0 00 . 0 0 ⎜ ⎜ dx ⎜ n1 0 0 0 0 dxn2 0 ⎜ 0 dx 0 0 0 0 dxn2 n1 ⎜ ⎜ 0 0 . 0 0 0 0 ⎜ ⎝ 0 0 0 . 0 0 0 0 0 0 0 0 dxn1 0

0 0 . 0 0 0 0 . 0 0 0 . 0 0 0 0 . 0 0

0 0 dx1m 0 0 0 0 dx1m 0 0 0 0 . 0 0 0 0 dx12 0 0 0 0 dx2m 0 0 0 0 dx2m 0 0 0 0 . 0 0 0 0 dx22 0 0 0 0 0 . 0 0 0 0 . 0 0 0 0 . 0 0 0 0 dxnm 0 0 0 0 dxnm 0 0 0 0 . 0 0 0 0 dxn2 0 0

0 0 . 0 0 0 0 . 0 0 0 . 0 0 0 0 . 0 0

⎞⎛ ⎞ u 11 0 0 ⎜ ⎟ 0 0 ⎟ ⎟ ⎜ u 12 ⎟ ⎜ . ⎟ 0 0 ⎟ ⎟⎜ ⎟ ⎜ ⎟ . 0 ⎟ ⎟⎜ . ⎟ ⎜ u 1m ⎟ 0 dx1m ⎟ ⎟⎜ ⎟ 0 0 ⎟ ⎜ u 21 ⎟ ⎟⎜ ⎟ 0 0 ⎟ ⎜ u 22 ⎟ ⎟⎜ ⎟ 0 0 ⎟⎜ . ⎟ ⎟⎜ ⎟ . 0 ⎟⎜ . ⎟ ⎟⎜ ⎟ 0 dx2m ⎟ ⎜ u 2m ⎟ . ⎟⎜ ⎟ 0 0 ⎟⎜ . ⎟ ⎜ . ⎟ 0 0 ⎟ ⎟⎜ ⎟ ⎜ ⎟ . 0 ⎟ ⎟⎜ . ⎟ ⎜ ⎟ 0 . ⎟⎜ . ⎟ ⎟ ⎜ ⎟ 0 0 ⎟ ⎟ ⎜ u n1 ⎟ ⎜ ⎟ 0 0 ⎟ ⎜ u n2 ⎟ ⎟ ⎜ ⎟ 0 0 ⎟ ⎟⎜ . ⎟ . 0 ⎠⎝ . ⎠ 0 dxnm u nm

26

2 Global Stability Approach

As discussed, the calculation of higher-order derivatives is straightforward once we have the basic Dx and D y matrices. As mentioned before, extending to a 3D basic flow is straightforward but will become too large a matrix. In addition to the above-performed discretization, if we have p points in the third direction, then the leading dimension of the differentiation matrices in all the three directions, Dx , D y , Dz will be n × m × p.

2.6.4 Coordinate Transformation and Grid Stretching To be remembered is the fact that these matrices are derived for 1 > x > −1 and 1 > y > −1 (Note that x(1) = 1 and x(n) = −1; y(1) = 1 and y(m) = − 1). To apply this method to any real world application, say x going from 0 to some length L, we do the following transformation: Let x be the set of spectral collocation points extending between 1 and −1 and xreal be the real x coordinate in the physical system. Then, xreal = (x ∗ (−0.5) + 0.5) ∗ L .

(2.22)

Then, the derivative matrix in the physical space can be obtained by the transform, Dxreal = Dx/(−0.5)/L .

(2.23)

1 is called the Jacobian of transformation. Similar transformation −0.5 ∗ L can be done in the y coordinate also from the spectral space to the real space, using the appropriate Jacobian of transformation. It is worth mentioning here yet another Jacobian which could arise in the derivative matrices, namely the Jacobian of stretching. The collocation points obtained from the Chebyshev method are not uniformly spaced but are distributed according to the cosine function. This clusters the grid points close to the start and end of the coordinate direction. In bounded flows like a channel flow or pipe flow, this type of grid clustering close to the wall in the y-direction is very advantageous, as the gradients of the flow are very close to the wall, and a clustered grid in this region is necessary to capture these steep gradients. Whereas in unbounded or semi-bounded flows, like in jets, wakes and boundary layers, clustering the grid points in this fashion is not desirable. In boundary layers, it is desirable to have the clustering close to the wall region, whereas for wakes and jets, we need more clustering around the centre region. Similarly, in the x-direction, Chebyshev discretization clusters the grid close to the inlet and exit of the domain, which is undesirable. We could use suitable stretching to cluster the grids at a given region depending on the flow configuration. We use the following stretching function (see in [25]), a [sinh((xc − x0 )b) + sinh(bx0 )], (2.24) xj = sinh(bx0 ) Here,

2.6 Discretization

27

  (1 + (eb − 1)a) 0.5 log , x0 = b (1 + (e−b − 1)a)

(2.25)

Here, xc is the collocation point, a is the x location around which clustering, relative to the collocation points, is required, and b is the degree of clustering. The values of a and b can be fixed depending upon the flow configuration and the amount of clustering required. This can be used in both directions to achieve the desired grid clustering. On account of this stretching function, there will be a Jacobian matrix multiplying the Dx and Dy matrices to account for the stretching. The values of the stretching coefficients, a and b, used for the different flow configurations are given in the appropriate sections.

2.6.5 Domain Transformation Many researchers follow the idea of domain transformation in which a complicated geometry is mapped onto a rectangular geometry for ease of computation and the easy application of boundary conditions, [12]. An example of one of the domains studied in the present work is shown in Fig. 2.4, where the physical and computational domain grids are shown. This transformation from the physical x − y plane to the computational ζ − η plane is achieved using the following formulae, dζ = dx,

(2.26)

y , h(x)

(2.27)

η=

where, h(x) is the local semi-height of the physical domain. For this transformation, we get the differentiation matrices in the physical x − y domain as, ∂ ∂ = + ∂x ∂ζ



−ηh  h

∂ 1 ∂ = . ∂y h ∂η



∂ ∂η

(2.28)

(2.29)

It can be noted that even though Dη and Dζ matrices thus constructed are sparse, higher derivatives of these matrices are dense. A schematic representation of the density of these matrices is shown in Fig. 2.5, where a blue dot represents a null element, and a red star represents a non-zero element. The density of the colour red in the figure is directly proportional to the density of the matrix. As can be seen, the Dx , D y and D yy matrices are sparse, whereas the Dx x and Dx y matrices are dense. Thus, using spectral discretization in both directions with the coordinate transformation of the above kind will make the final A and B matrices dense. This density of the matrices will restrict the solution method type, as seen in Sect. 2.9.

28 Fig. 2.4 Grids on the physical domain and the computational domain

2 Global Stability Approach 5

0

−5

0

5

10

15

20

0

5

10

15

20

1

0

−1

Fig. 2.5 Schematic representation showing the density of the matrices (a) Dx (b) D y (c) Dx x (d) D yy (e) Dx y . The blue dots represent zero entry and red stars represent non-zero elements

120 100 80 60 40 20

20

40

60

80

100

120

20

40

60

80

100

120

20

40

60

80

100

120

120 100 80 60 40 20

120 100 80 60 40 20

2.7 Baseflow Calculation and Boundary Conditions Fig. 2.5 (continued)

29

120 100 80 60 40 20

20

40

60

80

100

120

120 100 80 60 40 20

20

40

60

80

100

120

2.7 Baseflow Calculation and Boundary Conditions Having discussed the various derivative terms in the global stability Eq. 2.16 and the ways to calculate them, we are left with the base flow terms in the equation. One important reason why global stability analysis did not exist much earlier than it actually did is the need to know the base flow accurately. It is well known that small errors in the base flow can produce considerable errors in the growth or decay rates. We will see in later chapters how sensitive the stability is too small changes in the base flow. The base flow, especially in complex geometry, is often not straightforward, and we need substantial computational resources to calculate it. Even in a weakly nonparallel flow, and even with a very efficient numerical technique, it is remarkable that obtaining the base flow can be the slowest step in the proceedings. This is one of the shortcomings of the global stability analysis, where studying complex flow geometries becomes problematic. In the present work, the base flow profiles are obtained from a self-similar solution, if it exists, or by solving the Navier–Stokes equations directly, using a code developed by [54]. I thank him for sharing this code with us, which has been used here for a channel with a finite diverging region, discussed in Chap. 3, and a converging-diverging channel, discussed in Chap. 4. Most of the base flow calculations for the diverging channel flow for the required parameters are made by him. In addition, he makes a few modifications to the converging-diverging channel flow code to suit the different parametric requirements. A comparison of the two types of geometries this code handles is given in Fig. 2.6. The details of

30

2 Global Stability Approach

Fig. 2.6 Typical geometries handled by the base flow code in the present work. Top: A finite diverging channel. Bottom: A converging-diverging channel. The streamwise boundary conditions differ in the two cases

H(x) y x H(x)

the numerical method are given below. The steady two-dimensional Navier–Stokes equations are solved in the stream function-vorticity formulation. This can become very time-consuming, so a full multigrid technique is used to accelerate the convergence. A fast parallel solver is incorporated, details of which are available in [69]. The governing dimensionless equations are ∂ 1 2 + (U.∇) = ∇ ∂t Re

= −∇ 2 ψ,

where U is the velocity vector, t is time, and and ψ are the mean vorticity and stream function, respectively. The solution is facilitated by a transformation of coordinate, defined by dζ = dx/H (x), and η = y/H (x), where H is the local half-width (for both diverging channel and converging-diverging channel, as shown in Fig. 2.6). Since the geometries considered are top-down symmetric, the equations are solved only for half the domain. Symmetry boundary conditions ψ = = V = ∂U/∂ y = 0 are used at the centreline; no-slip and impermeability boundary conditions, U = V = 0, are imposed at the wall. There is also a set of boundaries studied that are not top-down symmetric, and the flow here is solved over the entire domain. The boundary conditions for that case will be no slip and penetration at both walls. The streamwise boundary conditions for the channel with a finite diverging region is Neumann at the inlet and exit. A parabolic flow is prescribed at the inlet. For the converging-diverging channel with periodic units in series, the equations are solved for just one periodic unit, with periodic boundary conditions at the inlet and the exit. Except for the boundary conditions and the domain boundary, the solution procedure is the same for both cases. We begin with a guess solution: usually a parabolic velocity profile at every streamwise location, and march in pseudo-time until a steady-state solution is obtained. The vorticity distribution at each new time step is calculated adopting first-order accurate forward differencing in time and second-order accurate central differencing in space. This vorticity distribution is used in a Jacobi iterative scheme to solve the Poisson equation for the stream function. A six-level full-multigrid technique achieves numerical acceleration. The procedure is repeated until the cumulative change in vorticity reduces to below 10−8 . The grid sizes required for each case depend on the parameters under consideration, which are given in the relevant chapters. The base flow is obtained

2.8 Boundary Conditions

31

on equally spaced grid points while the stability analysis is performed on a spectralspectral grid. Hence, the base flow is interpolated onto the spectral-spectral grid using a cubic spline interpolation. This code, in addition to fitting a cubic spline between the given points, has a unique feature of minimizing the length of the spline being fit, thus avoiding spurious oscillations of the curve. Furthermore, while cubic spline interpolation is performed at the interior grid points, linear interpolation is used at the domain’s boundaries. The base flow thus obtained after interpolation is checked for spurious values and compared with the original (equidistant) data before proceeding further.

2.8 Boundary Conditions The global stability Eq. 2.16 is a generalized eigenvalue problem of the form Ax = λBx. So far, we have seen the techniques involved in calculating the A and B matrices. Before solving this eigenvalue problem, we need to enforce the boundary conditions in these matrices, as discussed below. In addition to the necessity mentioned above to know the base flow accurately, there is yet another hurdle for the global stability approach: not knowing the streamwise boundary conditions for the disturbance. There are a few guidelines for implementing these boundary conditions based on the flow configuration. In the wall-normal direction, no-slip and no-penetration boundary conditions are used in solid walls (like in channel flow); decaying boundary conditions are given in far-field boundaries (like in wakes and boundary layers). In the streamwise direction, periodic boundary conditions are most commonly used. The justification for their use is often not clear. It has been assumed that they are valid, particularly in periodic geometries. We shall see, however, that periodicity in geometry is not the reason for expecting the same periodicity in the perturbations. In localized flows such as separation bubbles and cavity flows, homogeneous Dirichlet boundary conditions are used because the disturbance is expected to be localized within the flow. Therefore, assuming the incoming and outgoing flow is free of disturbance is valid. Inhomogeneous Neumann and Dirichlet boundary conditions are applied when some property of the disturbance is known beforehand, like the frequency of the mode. A special type of Neumann boundary condition is the Robin boundary condition, in which the derivative of the disturbance is specified based on the wavenumber of the disturbance. This is expressed as ∂u = iαu, ∂x

(2.30)

where α is the wavenumber. This boundary condition has its roots in the parallel stability approach; see Eq. 2.12. Equation 2.30 is just a restatement of Eq. 2.12. In fact, due to its nature, this boundary condition has played a major role in the validation of the global stability code developed during my PhD, details of which are given in Sect. 2.12. We shall find the limitations as well of these boundary conditions. When the streamwise dependence of the disturbance is not known a priori, the use of any

32

2 Global Stability Approach

of the above-mentioned boundary conditions is not valid. Homogeneous Neumann boundary conditions are used in such situations, which do not allow for any change in the streamwise direction of the disturbance at the inlet and exit. Few researchers also propose to use the homogeneous second derivative conditions at the boundary, which would mean that the slope of the disturbance does not vary in x. However, we consider this also as a restrictive boundary condition and use ‘Extrapolated Boundary Condition’ (EBC), see [63], as it is the least intrusive boundary condition. According to this, all disturbance quantities at the boundary are prescribed linear extrapolations of their values in the domain’s interior. For example, the flow quantity u at the inlet, say (1), is written as a linear function of the values at the next two grid locations (2, 3) as, (2.31) u 1 [x3 − x2 ] − u 2 [x3 − x1 ] + u 3 [x2 − x1 ] = 0. Similarly, the value at the exit can be written as a linearly extrapolated function of the values prior to the exit. For most of the results presented in this thesis, EBC is used. For the sake of comparison, periodic and Robin boundary conditions are also used, and this is mentioned then and there. Once the boundary conditions are decided, they can be implemented by replacing a few rows of the operator matrices (A and B matrices). This is very clearly explained in [59]. If the boundary conditions are homogeneous Dirichlet, then common practice is to ignore (remove) the rows corresponding to these boundary conditions and solve for the remaining matrix. This will reduce the matrix’s size and the computational cost to a certain extent.

2.9 Numerical Method The numerical discretization using the spectral method in both x and y along with the coordinate transformation results in a generalized eigenvalue problem with dense non-symmetric matrices. However, the final form of the matrices obtained depends on the type of discretization and transformation used, if any. For example, FVM or FD in x and spectral in y will give a sparse matrix. In such cases, iterative solvers to solve for only part of the eigenspectrum is very successful. One commonly used iterative solver is ARPACK (since 1996), which uses the iterative Arnoldi algorithm. Other eigenvalue solvers include LAPACK, ScaLAPACK, EISPACK, LINPACK, UMFPACK and BiCGStab. The choice of the solver depends on the type of matrix we get, which again depends on the discretization scheme we use. In the present work, we started working with the package LAPACK, a direct solver solving all the eigenvalues using the QZ algorithm. Since this was a very time-consuming process, we resorted to the iterative solver ARPACK, which solves only for a few eigenvalues. In this, we used the ‘shift-invert strategy’, where the eigenvalues are solved in the vicinity of the ‘shift’ supplied. However, for large matrices of the order used in this work, ARPACK took more time than LAPACK, against our expectations. The iterative process is carried out through a series of matrix-vector multiplications. For a sparse matrix, this operation is fast, giving faster convergence. However, this matrixvector multiplication is not very fast for a dense matrix, and ARPACK takes much

2.10 Issues with Global Stability Computations Fig. 2.7 Comparison of the spectra obtained for a typical non-symmetric generalized dense matrix system, using LAPACK (entire spectrum) and ARPACK (for 20 eigenvalues in the vicinity of the shift provided). The values of the shift are given in the inset

33 0

ωi

-0.1

LAPACK ARPACK; shift=(0.2,0) ARPACK; shift=(0.3,0)

-0.2

-0.3

0

0.2

ωr

0.4

0.6

longer than LAPACK to calculate just a few eigenvalues. Thus, the use of ARPACK is not helpful for our work. A comparison of the spectra obtained using LAPACK and ARPACK for a few eigenvalues with a shift is shown in Fig. 2.7. Also, as the first global stability study on the problems considered, the choice of shift and the number of eigenvalues to ask for is not straightforward. LAPACK, a direct solver to solve all the eigenvalues, could be speeded up using the parallelized version of LAPACK, called ScaLAPACK. However, this software does not have an inbuilt subroutine for non-symmetric matrices of the kind we get. Hence, we resort to the long, timeconsuming, computationally costlier way of solving the equation, using LAPACK. Another point to note about the global stability equation (2.16) is that it employs complex variables. A complex system will have twice the memory requirement and a corresponding increase in computational cost compared to a real system. We convert this complex system of equations into a real system by the transformation mentioned in [63]. This is done by defining λ = iω, thus making A and B matrices real and solving for complex λ. This real system is solved using the inbuilt subroutine of LAPACK called dggev (formerly dgegv).

2.10 Issues with Global Stability Computations With great increases in processor speed, RAM and memory, global stability analysis is widely adopted to solve complex fluid dynamical problems. Even though this analysis is excellent in solving complex problems, it has a few issues regarding implementation. They are listed as follows: • A global stability study can be conducted on any complex 2D or 3D field provided the base flow is known precisely. Unfortunately, obtaining the base flow for many complex flows is often not straightforward, and even when it is, it requires much computational effort. • As mentioned before, the x boundary conditions depend on the flow’s physics. For localized flows, the x boundary conditions are fairly straightforward. In contrast, for many convectively unstable flows like the flat-plate boundary layer and

34

2 Global Stability Approach

other flow configurations, the boundary conditions are still not very clear. Many researchers agree that if a sufficiently long enough domain is considered in the streamwise direction, then the effect of the boundary conditions will not be ‘felt’ in the interior of the domain. A homogeneous Dirichlet or Neumann boundary condition would be considered safe in such a case. However, to consider a very long domain, we need to consider many grid points, which will increase the computational cost. Thus, the issue of x boundary conditions in a ‘finite’ domain is still very unclear for most flows. • Even though many iterative techniques which solve for very few physically relevant eigenvalues exist for global stability studies, usage of such techniques is restricted to a class of flows which do not require a coordinate transformation. In addition, if we use the FD method in one direction to achieve a sparse matrix, then more points must be considered to achieve reasonable accuracy. A dense matrix is not a good candidate for iterative algorithms. Thus global stability study, in general, requires high computational resources. • Howard’s semi-circle theorem limits the modes obtained from the parallel analysis, which states that the disturbance cannot move faster or slower than the base flow velocity. This theorem has been used to eliminate physically irrelevant modes in a system. However, this theorem is not necessarily valid for non-parallel flows. Hence, it is difficult to pin down the physically relevant disturbance modes and eliminate the spurious modes. Furthermore, even if we assume that the disturbances cannot propagate faster/slower than the maximum/minimum base flow speed, there is no ‘direct’ method to estimate the disturbance wave speed. Time evolution of the mode (in a movie form) can give us quantitative information, but that is not easy to implement on all the modes calculated numerically.

2.11 Grid Sensitivity As mentioned before, the size of the matrices obtained is big, so the computational cost is huge. This limits the maximum grid size studied and hence might affect accuracy. A sample grid size used by many researchers studying the global stability of different flows is given below. The grid sizes give a representative number and must be fixed for each problem depending on the domain and flow characteristics. Author Ehrenstein et al. Ehrenstein et al. Casalis et al. Theofilis et al. Theofilis et al.

Flow Separated boundary layer Boundary layer Solid rocket motor Duct, Couette flow Square lid driven cavity

Method Chebyshev × Chebyshev Chebyshev × Chebyshev Chebyshev × Chebyshev Chebyshev × Chebyshev Chebyshev × Chebyshev

Size 350 × 65 180 × 45 120 × 120 72 × 40 48 × 48

In the present work, care has been taken to ensure grid insensitivity in each of the results presented. The grid size required and the grid sensitivity results are presented in each chapter, in the relevant subsections.

2.12 Validation

35

2.12 Validation This section took the most extended duration of the entire code development phase. The code is written in Fortran language with a feature called dynamic memory allocation, which helps to handle large memory requirements encountered with large-size matrices. The idea is to allocate the required amount of memory for a variable until the values dependent on it are calculated. After the dependent values are calculated, the memory is freed to be used for another variable. With this, an extensive memory requirement program can run efficiently with less computer memory. For validation, we consider one of the benchmark problems in hydrodynamic stability, the stability of a plane Poiseuille flow. The fully developed parabolic flow through a straight channel becomes linearly unstable at a Reynolds number (based on the channel half-width and centreline velocity) of 5772 for a disturbance of wavenumber α = 1.02, see [48]. Since the present global stability formulation does not have a wavenumber, we need to force a wavelike nature of the disturbance to compare with the parallel approach Orr–Sommerfeld results. It is worth noting that replacing φ(x) ∼ eiαx in the global stability equation and forcing V = 0 and dU /dx = 0 (corresponding to a parallel flow), we get the Orr–Sommerfeld equation. For validation, we consider a 2D rectangular domain corresponding to a finite domain of the plane Poiseuille flow. The base flow is the fully developed parabolic profile of the plane Poiseuille flow (U = 1 − y 2 , V = 0). No-slip and no-penetration boundary conditions are forced at the top and bottom walls of the domain. The boundary conditions in the streamwise direction are chosen as follows: to force a wavelike nature of the disturbance, we fix the non-dimensional length of the domain equal to the wavelength of the wave under consideration (2π /α) and force periodic boundary conditions at inlet and exit. However, the results obtained were not as expected. There was something more we needed to do. One of the underlying assumptions of the parallel approach is the wavelike nature of the disturbance with a wavenumber, say α. Even though a wave is periodic over one period of its wavelength, the boundary condition we had given will allow for any shape of the eigenfunction which is periodic over the prescribed length. We will thus not be restricted to Orr–Sommerfeld-like modes. The only way to fix the disturbance with a single wavenumber is to encourage this behaviour at the inlet and exit by applying Robin boundary conditions, stated as dφ/dx = iαφ. Thus, this boundary condition has its roots in the parallel approach (see Sect. 2.8). As mentioned in the later part of Sect. 2.9, the global stability equation is made ‘real’ by solving for iω. However, enforcing Robin boundary conditions will make the entire system of equations complex again. To avoid this, we supply a modified form of d4 φ d2 φ = −α 2 φ and = α 4 φ. Since the global the Robin boundary condition as, 2 dx dx 4 stability equation is fourth order in x, we need to give four boundary conditions in x. The four boundary conditions used in this validation are, (i) φ is periodic, i.e. φ1 = φn d2 φ1 (ii) = −α 2 φ1 dx 2

36 Fig. 2.8 Comparison of spectra obtained using the Orr–Sommerfeld equation and the global stability equation for a plane Poiseuille flow at a Reynolds number of 5772, α = 1.02. The x boundary conditions in global approach are Robin + periodic

2 Global Stability Approach Global Orr-Sommerfeld

0

ωi -0.3

-0.6

0

0.5

1

ωr

d2 φn = −α 2 φn dx 2 d4 φn (iv) = α 4 φn . dx 4 We can reproduce the parallel results of [48] on a plane channel flow. A sample spectrum obtained at a Reynolds number of 5772 with α = 1.02 is shown in Fig. 2.8. Here also, we do not get an exact match with the entire spectrum of the Orr–Sommerfeld results because the Robin boundary conditions will allow for higher harmonics of α over the same wavelength. In contrast, the Orr–Sommerfeld results hold good for a single α. This is the reason we see some additional eigenvalues in the global approach. Nevertheless, we can get a good match for the least stable eigenmode. The structure of the eigenfunction is also matched with very good accuracy, as shown in Fig. 2.9. We also get a value of critical Reynolds number (Recrit ) of 5772 for an α of 1.02, as shown in Fig. 2.10. The growth rate (ωi ) of the least stable mode is displayed here versus the Reynolds number. In addition to the above checks, the following checks were also done. Since each chapter deals with a different geometry, stretching function and coordinate transformation, the final differentiation matrices obtained are checked as follows: A known function of x or y is defined, and the derivatives of the function are checked with the analytical values. For example, the fourth derivative of a function defined as y 4 is checked to be 24. Similar checks were done for the x derivative too. In the cases where a similarity solution does not exist, the derivatives of the base flow obtained numerically, and that obtained by the operation of the derivative matrices on the base flow are cross-checked. This validates the code and the approach to the extent possible. In addition, we have also made sure that the results obtained are insensitive to the compiler used (f77, f95, gfortran, ifort) and the processor configuration (like precision, accuracy, processor speed, RAM, etc.).

(iii)

References Parallel approach Global approach

1

Eigenfunction

Fig. 2.9 Comparison of the least stable eigenfunction shown in Fig. 2.8. Plotted here is the eigenfunction versus the wall-normal coordinate. The global equation is solved for full channel and parallel equation is solved for half channel. Each curve represents different x locations spanning over a wavelength of the wave

37

0.5

0

-0.5

-1 -1

0

-0.5

1

0.5

y Global approach Parallel approach

0 5

ωi=0

ωi x 10

Fig. 2.10 Graph of Reynolds number versus ωi for plane channel flow obtained from the global stability code with Robin boundary conditions. α = 1.02. Only the least stable mode is shown, which is seen to cross the imaginary axis at a Reynolds number of 5772. The results obtained from the Orr–Sommerfeld equation are also shown for comparison

Re=5772

-2

-4

5750

5760 5770 Reynolds Number

5780

References 1. Akervik, E., Ehrenstein, U., Gallaire, F., Henningson, D.S.: Global two-dimensional stability measures of the flat plate boundary-layer flow. Euro. J. Mech. B/Fluids 27, 501–513 (2008) 2. Akervik, E., Hoepffner, J., Ehrenstein, U., Henningson, D.S.: Optimal growth, model reduction and control in a separated boundary-layer flow using global eigenmodes. J. Fluid Mech. 579, 305–314 (2007) 3. Alizard, F., Robinet, J.C.: Spatially convective global modes in a boundary layer. Phys. Fluids 19, 114105 (2007) 4. Barbagallo, A., Sipp, D., Schmid, P.J.: Closed-loop control of an open cavity flow using reducedorder models. J. Fluid Mech. (2009) 5. Barkley, D., Gomes, M.G.M., Henderson, R.D.: Three-dimensional instability in flow over a backward-facing step. J. Fluid Mech. 473, 167–190 (2002) 6. Bers, A.: Space-time evolution of plasma instabilities—absolute and convective, vol. 1. In: Rosenbluth, M.N., Sagdeev, R.Z. (eds.) Handbook of Plasma Physics, pp. 451–517 (1983) 7. Bertolotti, F.P., Herbert, T.: Analysis of the linear stability of compressible boundary layer using the PSE. Theor. Comp. Fluid Dyn. 3, 117–24 (1991)

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8. Bertolotti, F.P., Herbert, T., Spalart, P.R.: Linear and nonlinear stability of Blasius boundary layer. J. Fluid Mech. 242, 441–474 (1992) 9. Betchov, R., Criminale, W.O.: Spatial instability of the inviscid jet and wake. Phys. Fluids 9(2), 359–362 (1966) 10. Bhattacharya, P., Manoharan, M., Govindarajan, R., Narasimha, R.: The critical Reynolds number of a laminar incompressible mixing layer from minimal composite theory. J. Fluid Mech. 565, 105–114 (2006) 11. Bottaro, A., Corbett, P., Luchini, P.: The effect of base flow variation on flow stability. J. Fluid Mech 476, 293–302 (2003) 12. Cebeci, T., Shao, J.P., Kafyeke, F.: Computational Fluid Dynamics for Engineers: From Panel to Navier-Stokes Methods with Computer Programs. Springer (2005) 13. Chedevergne, F., Casalis, G., Feraille, T.: Biglobal linear stability analysis of the flow induced by wall injection. Phys. Fluids 18, 014103 (2006) 14. Christodolou, K.N., Scriven, L.E.: Finding leading modes of a viscous free surface flow: an asymmetric generalized eigenproblem. J. Sci. Comput. 3(4), 355–406 (1988) 15. Deissler, R.J.: The convective nature of instability in plane Poiseuille flow. Phys. Fluids 30(8), 2303–2305 (1987) 16. Dijkstra, H.A., Molemaker, M.J., Ploeg, A.V.D., Botta, E.F.F.: A efficient code to compute non-parallel steady flows and their linear stability. Comput. Fluids 24(4), 415–434 (1995) 17. Drazin, P.G., Reid, W.H.: Hydrodynamic stability. Cambridge University Press (2004) 18. Ehrenstein, U.: On the linear stability of channel flow over riblets. Phys. Fluids 8(11), 3194– 3196 (1996) 19. Ehrenstein, U., Gallaire, F.: On two-dimensional temporal modes in spatially evolving open flows: the flat-plate boundary layer. J. Fluid Mech. 536, 209–218 (2005) 20. Gaster, M.: Growth of disturbances in both space and time. Phys. Fluids 11(4), 723–727 (1968) 21. Gaster, M.: On the effect of boundary layer growth on flow stability. J. Fluid Mech. 66, 465–480 (1974) 22. Gaster, M.: A theoretical model of a wavepacket in the boundary layer on a flat plate. Proc. R. Soc. London A 347, 271–289 (1975) 23. Gaster, M.: On the growth of waves in boundary layers. J. Fluid Mech. 424, 367–377 (2000) 24. Gonzalez, L.M., Theofilis, V., Gomez-Blanco, R.: Finite element methods for viscous incompressible biglobal instability analysis on unstructured meshes. AIAA J. 45(4), 840 (2007) 25. Govindarajan, R.: Effect of miscibility on linear instability of two fluid channel flow. Int. J. Multiphase flow 30, 1177–1192 (2004) 26. Govindarajan, R., Narasimha, R.: Stability of spatially developing boundary layers in pressure gradients. J. Fluid Mech. 300, 117–147 (1995) 27. Govindarajan, R., Narasimha, R.: A low order theory for stability of non-parallel boundary layer flows. Proc. Math. Phys. Eng. Sci. 453(1967), 2537–2549 (1997) 28. Henningson, D.S.: Comment on “transition in shear flows: non-linear normality versus nonnormal linearity” [phys. fluids 7,3060 (1995)]. Phys. Fluids 8, 2257–2258 (1996) 29. Henningson, D.S., Akervik, E.: The use of global modes to understand transition and perform flow control. Phys. Fluids 20, 031302 (2008) 30. Herbert, T.: Parabolized stability equations. Annu. Rev. Fluid. Mech. 29, 245–283 (1997) 31. Huerre, P., Monkewitz, P.A.: Absolute and convective instabilities in free shear layers. J. Fluid Mech. 159, 151–168 (1985) 32. Huerre, P., Monkewitz, P.A.: Local and global instabilities in spatially developing flows. Ann. Rev. Fluid Mech. 22, 473–537 (1990) 33. Jackson, C.P.: A finite-element study of the onset of vortex shedding in flow past variously shaped bodies. J. Fluid Mech. 182, 23–45 (1987) 34. Lee, N.Y., Schultz, W.W., Boyd, J.P.: Stability of fluid in a rectangular enclosure by spectral method. J. Heat Mass Transf. 32(3), 513–520 (1989) 35. Lin, R.S., Malik, M.R.: On the stability of attachment-line boundary layers. part 2. The incompressible swept Hiemenz flow. J. Fluid Mech. 311, 239–255 (1996)

References

39

36. Lin, R.S., Malik, M.R.: On the stability of attachment-line boundary layers. part 2. The effecting of leading-edge curvature. J. Fluid Mech. 333, 125–137 (1997) 37. Lingwood, R.J.: Absolute instability of the boundary layer on a rotating disk. J. Fluid Mech. 299, 17–33 (1995) 38. Marquet, O., Lombardi, M., Chomaz, J.M., Sipp, D., Jacquin, L.: Direct and adjoint global modes of a recirculation bubble: lift-up and convective non-normalities. J. Fluid Mech. 622, 1–21 (2009) 39. Martin, D., Carriere, P., Monkewitz, P.A.: Three dimensional global instability modes associated with a localized hot spot in Rayleigh-Benard-Poiseuille convection. J. Fluid Mech. 551, 275–301 (2006) 40. Mittal, S., Kumar, B.: Flow past a rotating cylinder. J. Fluid Mech. 476, 303–334 (2003) 41. Mittal, S., Kumar, B.: A stabilized finite element method for global analysis of convective instabilities in nonparallel flows. Phys. Fluids 19, 008105 (2007) 42. Mittal, S., Singh, S.: Vortex induced vibrations at subcritical re. J. Fluid Mech. 534, 185–194 (2005) 43. Monkewitz, Huerre: Chomaz: global linear stability analysis of weakly non-parallel shear flows. J. Fluid Mech 251, 1–20 (1993) 44. Morzynski, M., Afanasiev, K., Thiele, F.: Solution of the eigenvalue problems resulting from global non-parallel flow stability analysis. Comput. Methods Appl. Mech. Eng. 169, 161–176 (1999) 45. Nayar, N., Ortega, J.M.: Computation of selected eigenvalues of generalized eigenvalue problems. J. Comput. Phys. 108, 8–14 (1993) 46. Noack, B.R., Eckelmann, H.: A global stability analysis of the steady and periodic cylinder wake. J. Fluid Mech. 270, 297–330 (1994) 47. Noack, B.R., Konig, M., Eckelmann, H.: Three-dimensional stability analysis of the periodic flow around a circular cylinder. Phys. Fluids A 5(6), 1279–1281 (1993) 48. Orszag, S.A.: Accurate solution of the Orr-Sommerfeld stability equation. J. Fluid Mech. 50, part 4, 689–703 (1971) 49. Pier, B.: Primary crossflow vortices, secondary absolute instabilities and their control in the rotating-disk boundary layer. J. Eng. Math. 57, 237–251 (2007) 50. Pier, B.: Local and global instabilities in the wake of a sphere. J. Fluid Mech. 603, 39–61 (2008) 51. Pierrehumbert, R.T.: Universal short-wave instability of two-dimensional eddies in an inviscid fluid. Phys. Rev. Letters 57(17), 2157–60 (1986) 52. Saad, Y.: Variations of Arnoldi’s method for computing eigenelements of large unsymmetric matrices. Linear Algebra Appl. 34, 269–295 (1980) 53. Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7(3), 856–869 (1986) 54. Sahu, K.C.: Numerical computation of spatially developing flows by full-multigrid technique. Master’s thesis, Jawaharlal Nehru Centre for Advanced Scientific Research (2003) 55. Sahu, K.C., Govindarajan, R.: Stability of flow through a slowly diverging pipe. J. Fluid Mech 531, 325–334 (2005) 56. Schmid, P.J.: Nonmodal stability theory. Ann. Rev. Fluid Mech. 39, 129–162 (2007) 57. Schmid, P.J., Henningson, D.S.: Stability and Transition in Shear Flows. Springer (2001) 58. Schmid, P.J., Henningson, D.S.: On the stability of a falling liquid curtain. J. Fluid Mech. 463, 163–171 (2002) 59. Srinivasan, S., Klika, M., Ludwig, M.H., Ram, V.V.: A Beginner’s Guide to the Use of the Spectral Collocation Method for Solving Some Eigenvalue Problems in Fluid Mechanics. Tech. Rep. National Aerospace Laboratories, Bangalore, India (1994) 60. Sturrock, P.A.: Kinematics of growing waves. Phys. Rev. 112, 1488–1503 (1958) 61. Tatsumi, T., Yoshimura, T.: Stability of the laminar flow in a rectangular duct. J. Fluid Mech. 212, 437–449 (1990) 62. Theofilis, V.: On the verification and extension of the Gortler-Hammerlin assumption in threedimensional incompressible swept attachment-line Boundary layer flow. Tech. report. IB 22397, A 44, DLR (1997)

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63. Theofilis, V.: Advances in global linear instability analysis of non-parallel and threedimensional flows. Progr. Aerospace Sci. 39, 249–315 (2003) 64. Theofilis, V., Duck, P.W., Owen, J.: Viscous linear stability analysis of rectangular duct and cavity flows. J. Fluid Mech. 505, 249–286 (2004) 65. Theofilis, V., Federov, A., Obrist, D., Dallmann, U.C.: The extended Gortler-Hammerlin model for linear instability of three-dimensional incompressible swept attachment-line boundary layer flow. J. Fluid Mech. 487, 271–313 (2003) 66. Theofilis, V., Hein, S., Dallmann, U.: On the origins of unsteadiness and three-dimensionality in a laminar separation bubble. Phi. Tran. R. Soc. Lond. A 358, 3229–3246 (2000) 67. Theofilis, V., Sherwin, S.J., Abdessemed, N.: On global instabilities of separated bubble flows and their control in external and internal aerodynamic applications. NATO RTO-AVT-111, 21–1 to 21–9 (2005) 68. Tollmien, W.: Über die entstehung der turbulenz. 1. mitteilung, nachr. ges. wiss. göttingen. Math. Phys. Klasse, 21–44 (1929) 69. Venkatesh, T.N., Sarasamma, V.R., Rajalakshmy, S., Sahu, K.C., Govindarajan, R.: Super-linear speedup of a parallel multigrid Navier-stokes solver on Flosolver. Curr. Sci. 88(4), 589–593 (2006) 70. Waleffe, F.: Transition in shear flows: non-linear normality versus non-normal linearity. Phys. Fluids 7(12), 3060 (1995) 71. Zebib, A.: Stability of viscous flow past a circular cylinder. J. Eng. Math. 21, 155–165 (1987)

Chapter 3

Diverging Channel

3.1 Introduction In the previous chapter, we saw that a plane Poiseuille flow is linearly stable up to a Reynolds number of 5772. However, in a real-life situation, this flow becomes transitional at a Reynolds much lower than the critical Reynolds number. This is called sub-critical transition. Many shears flow exhibit sub-critical instability, classic examples being the Poiseuille flow through a circular pipe and plane Couette flow. These flows are linearly stable for all Reynolds numbers. The discrepancy between linear theory and experiments was attributed to the failure of LST, and several attempts to explain this discrepancy using nonlinearity were made; see [40] for a review on nonlinear stability theory. Experiments in channels and pipes were conducted in cleaner (reduced disturbance) environments to reduce nonlinear effects. It was then possible to maintain the flow in a laminar state for very high Reynolds numbers of 104 in channels, [28] and 105 in pipes; see [21]. This ability was enhanced further by making the walls of the channel/pipe smoother. This shows the sensitivity of these flows to free-stream disturbances and the surface roughness at the wall. Hof et al. [22] have shown experimentally that the amplitude of disturbance required to trigger a transition in a pipe flow scales inversely with the Reynolds number. There is also a huge volume of work dedicated to studying surface roughness’s effect in pipes and channels; see, for example, [20, 24]. Sub-critical transition in channels was also explained based on the non-normality of the operator; see [3]. There has been a lot of debate about the reason for sub-critical transition in many shear flows, as to non-normality or nonlinearity; see for example [19, 30–32, 45]. The balance is now tipped in favour of non-normality occurring first and enabling linear disturbances to become large enough to go nonlinear. Moreover, [19] showed that since the nonlinear terms are energy conserving, the energy growth of the disturbances has to be enhanced by a linear mechanism and thus explains the sub-critical transition of channels using non-normality. Note, however, that this argument for energy growth requires spatially localized perturbation or periodic, which is not necessarily true for non-parallel flows. Much work has been done to study the effect of many other © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Bhoraniya et al., Global Stability Analysis of Shear Flows, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-19-9574-3_3

41

42

3 Diverging Channel

parameters, like wall porosity, wall heating, viscosity variations, wall flexibility, etc., on the stability of channel and pipe flows. Below is a very limited, representative survey related to channels (except the first reference, which relates to a tube). Shankar and Kumuran [39] in a series of studies have shown that flow through a flexible tube is destabilized at higher Reynolds numbers as against its rigid wall counterpart. References [14, 15] have shown that a channel flow can be stabilized by having small viscosity variations near the critical layer. Tilton and Cortelezzi [41, 42] studied flow through channels with one or two porous walls and found that even very small amounts of wall permeability decrease the stability of the flow. Govindarajan [11] studied the effects of miscibility of two fluids in a channel flow. The effect of wall heating in a channel flow is studied by [46] and others. Sameen and Govindarajan [36] discuss the effects of viscosity, buoyancy and heat diffusivity. Sahu et al. [34] show that wall slip, which hugely stabilizes flow through a straight channel, actually has a minor destabilizing effect in a divergent channel. However, the transient growth obtained by a parallel stability study is unaffected by either slip or divergence. Sahu et al. [35] studied the stability characteristics of a two-layer fluid in a channel, with one layer as a non-Newtonian fluid. Apart from all the parameters mentioned above affecting the stability of flow through a channel, wall divergence has a large effect. This alone is studied here and is discussed in detail below.

3.2 Diverging Channel

The steady laminar two-dimensional flow of incompressible fluid within an infinite wedge driven by a line source/sink situated at the intersection of the rigid planes that form the wedge (Fig. 3.1) was first described by Jeffery [23] and Hamel [17] (see, e.g., [37]). Such a flow is called Jeffery–Hamel (JH hereafter) flow. The stability of JH flows was first studied by [6], who showed that divergence has a destabilizing effect. He calculated the critical Reynolds number, Recrit , as a function of the divergence angle using the Orr–Sommerfeld equation and showed that Recrit falls rapidly with wall divergence. Many other researchers also studied the stability of JH flows. Nakaya and Hasegawa [27] studied the flow through diverging plates by analytic expansions and found critical values for the onset of instability. The base flow considered here is not the JH solution but is derived analytically as a function of the diverging angle. Eagles and Weissman [8] studied the flow through straight walled diverging channels using the WKB method. The base flow considered here is the JH profile. Eagles and Smith [7] studied the flow through a channel with a finite diverg-

3.2 Diverging Channel

43

ing curved wall region confined between straight regions using the WKB method. The base flow considered here is solved numerically. Allmen and Eagles [1] numerically solved the two problems mentioned above and obtained a very good match with the earlier predictions. Georgiou and Eagles [10] studied the stability of flow through curved walled channels using the WKB method. The base flow is calculated as a perturbation for the JH flows. This study discusses and compares the parameters governing the stability characteristics of channels with straight walls, curved walls and walls changing the divergence angle. Banks and Drazin [2] studied linear and weakly nonlinear perturbations to the JH solutions and obtained critical angles necessary for disturbance growth. Hamadiche and Scott [16] showed that the critical Reynolds numbers for JH flows based on both the volume flux and the axial velocity decrease rapidly with diverging angle (θ ). A quantity defined as a product of these Reynolds numbers with (θ ) stays constant or scales linearly with (θ ). Uribe and Bravo [44] studied the stability of JH flows using the finite element method and obtained critical parameters for the stability of unidirectional and bi-directional flows. Dennis et al. [4] studied the flow numerically in a diverging channel enclosed between two arcs and found that the inlet and outlet conditions strongly influence the nonlinear development of the flow. Drazin [5] has a brief review on flow instability through diverging channels. To mention a related work, [33] studied the stability of flow through a slowly diverging pipe and showed the destabilizing effect of divergence. This result is special because, according to linear stability theory, flow through a pipe is linearly stable for all Reynolds numbers. Even a small amount of divergence in the pipe makes the critical Reynolds number finite very interesting. References [2, 26] showed that while divergence drastically destabilizes the flow, convergence causes a huge stabilization. [29] studied JH flows experimentally and demonstrated that a symmetric unidirectional outflow JH solution if it exists, is always stable. This seems to contradict the results of [44]. Kerswell et al. [25] studied, using a theoretical model, the nonlinear evolution of two-dimensional spatial waves in JH flows. In all of the numerical work mentioned above, the researchers have assumed locally parallel flow, and the dramatic dependence of Recrit on wall divergence angle has been captured to a great extent. The papers are all, however, silent on why the dependence should be so dramatic. We provide a mathematical explanation in Sect. 3.5. In the theoretical JH flow, the velocity profile is self-similar, and the velocity scales inversely with distance x, while the channel width scales linearly with x as shown in Fig. 3.1; see [37]. Hence, the Reynolds number, defined based on the centreline velocity and half channel width, does not vary downstream. One would therefore expect a disturbance of constant dimensionless wavelength, combined with a selfsimilar amplitude function φ(y), to satisfy the stability equations. Given a large amount of work on the stability of JH flows using local stability approaches, why do we need to study this geometry as a global stability problem? We study it here to show how a global analysis can reveal fundamental instability characteristics that are not accessible to the parallel or WNP approaches. None of the global instability modes resembles parallel or WNP modes. The point is that if the geometry, the pressure or other relevant parameters were varying in a complicated fashion with x, one would not be surprised that global stability studies give results very differ-

44

3 Diverging Channel

L

y

θ

H(x)

U(y)

x

x/H(x)=1/tan(θ)

Fig. 3.1 Schematic of Jeffery–Hamel flow originating from a point source at the origin. θ is the semi-divergence angle in degree and x is the streamwise coordinate. Note that the axes are nonorthogonal. For global stability computations, the inlet of the two-dimensional domain is fixed at a distance of 1/ tan(θ), as the length scale of the problem is the inlet half-width. The two-dimensional domain over which the global stability computations are made is shown by the rectangular box

ent from the parallel. The fact that global modes can look qualitatively different in this, perhaps the simplest non-parallel flows one could construct, is more interesting. This finding in JH may be contrasted with recent studies on boundary layers, e.g. [9]. There it is seen that global stability results differ from WNP quantitatively by a small percentage. However, a given mode is still described by a basic WKB structure of a wave with a slowly changing wavelength streamwise. We highlight here only a few important aspects of the effect of two-dimensionality on the disturbance eigenfunction. In addition to the JH, we consider a more realistic geometry, as shown in Fig. 3.2, where the mean flow is obtained numerically. This flow is referred to as SDS hereafter for a channel with Straight-Diverging-Straight geometry. Similar geometries were studied before by [7], Nakayama (see in [5]), [43]. The reasons for choosing such geometry are twofold: (1) It is easily realizable in real life and hence experimental validation is possible; (2) streamwise boundary conditions will become straightforward; Neumann boundary conditions are applicable at the inlet and exit straight regions. Many shear flows are spatially developing, where, as we proceed with the flow in x, there are changes in the local Reynolds number and the local flow profile. JH flows are unique as they evolve spatially, but (i) the local Reynolds number does not change with x, and (ii) the mean flow is self-similar. This feature of JH flows makes it a good candidate for a global stability study as it would give information about the effect of spatial development alone and not of streamwise changes in Reynolds number. One reason for the global approach displaying a much richer variety of modes than the parallel is that the disturbance eigenfunction obtained from solving the Orr–Sommerfeld equation can be multiplied by an arbitrary function of x and still satisfy the equation. In the case of global stability, the prefactor function of x is well defined and expressed in the eigenfunction solution.

3.3 Base Flow Fig. 3.2 Schematic of the channel with the finite divergent section, referred to here as SDS for straight-divergent-straight. The exit straight region is kept sufficiently long to achieve parabolic velocity profile at the exit

45

y L=200 H=1

θ=5

x=9.37

x

x=91

3.3 Base Flow As mentioned before, we consider both the infinitely diverging JH flow and the finite diverging SDS flow. The base flow considered for both these cases are explained separately below. The JH base flow is self-similar while the base flow for SDS is obtained numerically.

3.3.1 JH Base Flow The steady laminar two-dimensional flow of incompressible fluid within an infinite wedge driven by a line source/sink situated at the intersection of the rigid planes that form the wedge (Fig. 3.1) was first described by Jeffery (1915) and Hamel (1916) (see e.g. [37]), by the similarity equation U  + 2 SUU  + 4θ 2 U  = 0,

(3.1)

U (+1) = U (−1) = 0; U (0) = 1. Here U is the mean velocity in the similarity coordinate η = y/H (x), θ is the semi-divergence angle as shown in Fig. 3.1, H is the channel half-width, the primes stand for differentiation with respect to η, S ≡ θ Re, x is the streamwise coordinate and y is the wall-normal coordinate. As mentioned before, the Reynolds number (defined as Re ≡ Uc (x)H (x)/ν, where the subscript ‘c’ stands for the channel centreline and ν for kinematic viscosity) is constant downstream, in contrast to most other developing shear flows. The flow displays a separated region for S greater than 10.3. A plot of U obtained for different values of S is shown in Fig. 3.3. The domain over which global stability computations are performed is shown in Fig. 3.1. As the half-channel width at the inlet Hi is the length scale, the starting point in the x-direction of this domain obeys the relation xstart /H1 = 1/ tan(θ ). The domain extends up to xend = xstart + L, where L is kept sufficiently long. The velocity profile at the inlet is obtained by solving Eq. 3.1 using a fourth-order Runge– Kutta method with 10,000 grid points. Even though this equation is solved in η, this corresponds to solving it in y as the velocity and length scales at the inlet coincide.

46

3 Diverging Channel

Fig. 3.3 Self-similar JH Eq. 3.1, plotted for different values of S ≡ θRe. As can be seen clearly from the inset, this flow becomes separated for values of S greater than 10.3

1 1

0.5 0

0.5

η

0.1

4 8.72 10.3 13

0

-0.5 -1

0

0.2

0.4

0.6

0.8

U

Velocity profiles at other x locations are obtained using the similarity scaling relation U ∼ x −1 , which is given explicitly as, U (x(i), y) =

U (η) x(1) x(i)

(3.2)

U  (x(i), y) =

U  (η) x(1) x(i)

(3.3)

U  (x(i), y) =

U  (η) x(1) x(i)

(3.4)

η    η H (i)x(1) 0 U (η)dη x(1) V (x(i), y) = H (i) ηU (η) − U (η)dη + (3.5) x(i) x(i)2 0 

where the wall-normal velocity is obtained from continuity. Here x(1) is the first x location and x(i) corresponds to any x location with 1 ≤ i ≤ n, where n is the number of grids in x. As per the scaling relation given in Eq. 3.2, U decreases with distance. This is schematically shown in Fig. 3.4. Typical plots of U , V and the stream function ψ of JH flow at a particular streamwise station are shown in Fig. 3.5 for an unseparated case (left) and a separated case (right). Contours of U and V for a Reynolds number of 100 and semi-divergence angle θ = 5 are shown in Fig. 3.6.

3.3.2 SDS Base Flow The mean velocity profile for the channel with the finite divergent section (SDS, Fig. 3.2) is obtained by a numerical solution of the stream function-vorticity formulation of the two-dimensional Navier–Stokes equation on a 512 × 32 grid. The

3.3 Base Flow

47

2 1 0 −1 −2 14

12

18

16

24

22

20

Fig. 3.4 Local streamwise velocity profiles of JH flow at Re = 100, θ = 5. The velocity variation is given in Eq. 3.2 1

0.5

η

1

ψ U V

ψ U V

0.5

η

0

0

-0.5

-0.5

-1

-1 -1

0

-0.5

0.5

-1

1

0

1

Fig. 3.5 (Left) Plots of streamfunction, streamwise velocity and wall-normal velocity of JH flow for Re = 100, θ = 5. (Right) Same as the left figure, but for Re = 150. The flow at this Reynolds number is separated. Note that the values are scaled with their maximum value for ease of viewing

10

0.8 0.6 0.4 0.2

0 −10 20

40

60

80

100

120

140

160 0.02

10 0

0

−10 20

40

60

80

100

120

140

160

−0.02

Fig. 3.6 Contours of streamwise (U) and wall-normal (V ) velocity of JH flow at Re = 100, θ = 5. This flow has S = 8.72 and is not separated

48

3 Diverging Channel

numerical method is explained in Sect. 2.7. At each Re and θ , it is ensured that the final straight section is long enough for the flow to attain a parabolic profile well before the exit. This requirement must be met to apply Neumann boundary conditions at the exit. The length of this exit straight section increases approximately linearly with increasing Reynolds number. In the present computations, the divergence starts at x = 9.37 and ends at x = 91, as shown in Fig. 3.2. This flow is solved at the same parameters as the JH flow. A detailed comparison is given in the next section.

3.4 Comparison of Base Flow—JH and SDS The computations are conducted at many Reynolds numbers and different angles of divergence. To give a representative comparison of the base flow of JH and SDS flows, we consider the flow for a half-angle of divergence of 5◦ and a Reynolds number of 100. The base flow profiles obtained for the JH flow from the similarity Eq. (3.1) and the SDS channel numerically are compared in Fig. 3.7. Here is the streamwise velocity U versus the non-dimensional coordinate η. The lines shown in the figure are obtained for SDS at three different x locations, and the symbols are the solutions of Eq. (3.1) for S = Reθ = 8.72. We note that the JH profile is not separated at this S value, whereas the SDS profile is separated downstream due to centreline acceleration caused by the divergent section being finite. A domain length of L = 200 is sufficient to get a parabolic flow at the exit. The local velocity profiles obtained for the SDS channel in the exit straight region are shown in Fig. 3.8. It can be seen that the profile matches well with the parabolic profile at the exit of the domain. Contours of the mean streamwise velocity distribution are shown in Fig. 3.9. A region of weak separated flow extending over most of the divergent portion can be discerned in the SDS. 1

x=14 x=21 x=46 x=79 similarity

U 0.5

0

0

0.5 η

1

Fig. 3.7 Comparison of the mean velocity profiles of JH flow (shown in Fig. 3.1) with those of SDS flow (Fig. 3.2) at a few streamwise locations. The symbols are for a similarity solution with S = 8.72 (Re = 100, θ = 5◦ ). The lines are for the SDS channel at different x locations. It can be seen that the JH profile matches with the numerical profile for the SDS channel at x = 21

3.5 Sensitivity of the Critical Reynolds Number to Divergence x=100 x=130 x=150 x=200

1 0.5 η

49

0

-0.5 parabolic

-1

0

1

0.5 U

Fig. 3.8 Mean velocity profiles of SDS flow (Fig. 3.2) at a few streamwise locations in the exit straight region of the domain. As can be seen, the velocity profile at the exit of the domain is almost parabolic, corresponding to the fully developed straight channel profile Fig. 3.9 Comparison of mean streamwise velocity contours of JH (top) and SDS (bottom) flows, for the case shown in Fig. 3.7, with Re = 100, θ = 5◦ . Note the long region of weak separation in the SDS case

10 0 −10 20

40

60

80

100

120

140

160

5 0 −5 0

50

100

150

200

3.5 Sensitivity of the Critical Reynolds Number to Divergence While it is well known that the critical Reynolds number of this flow is dramatically sensitive to wall divergence, the cause of this sensitivity is not explained in the literature. We begin by asking why this happens and propose a scaling argument, which gives a good approximation for variation in the Recr with wall divergence. To explain this, we expand the mean flow at small divergence as a perturbation of the Poiseuille solution. At S 0) modes, respectively. The stationary mode has a complex frequency ω = 0 − 0.01818i. This global mode is temporally stable because ωi < 0. Moreover, disturbances decay over time. Figure 5.24 shows the two-dimensional spatial structure of the streamwise (u) and wall-normal (v) disturbance amplitudes for stationery mode ω = 0 − 0.01818i. The magnitudes of the velocity disturbances are zero at the inlet, as it is the inlet boundary condition. As the fluid particles move towards the downstream, the disturbances evolve monotonically in time within the domain and progress towards the downstream. The magnitude of u disturbance amplitudes is one order higher than v disturbance amplitudes. The amplitude structure has an opposite sign for u and v. The spatial structure of the amplitudes grows in size and magnitude towards the downstream. Figure 5.25 shows the variation of u and v disturbance amplitudes in streamwise direction at different radial locations. The magnitude of the disturbance amplitudes is very small near

132

5 Axisymmetric Boundary Layer … 0 -0.02

ωi

-0.04 -0.06 -0.08 -0.1 -0.12 0

0.02

0.04

0.08 0.06 ωr

0.1

0.12

Fig. 5.23 Eigenspectrum for azimuthal wavenumber N = 0 (axisymmetric mode) and Re = 383 20 -0.01

r

15

-0.02 10

(a) 200

-0.03 250

300

350 x

400

450

500

20 0.0004

r

15 10

0.0002

(b) 200

250

300

350 x

400

450

500

0.0000

Fig. 5.24 Contour plot of stationary mode a streamwise velocity and b normal velocity for the eigenvalue ω = 0.0 − 0.01818i, as marked by circle in Fig. 5.23

the cylinder surface due to the viscous effect, increases in the radial direction and vanishes at the far-field. The most unstable oscillatory mode has an eigenvalue ω = 0.003689 − 0.01874i. The flow is stable for this global mode because of ωi < 0. The spatial structure of the above global eigenmode for u and v disturbance amplitudes is not monotonic. The distribution in spatial directions is shown in Fig. 5.26. The amplitudes of the velocity disturbances are zero at the inlet as it is the the imposed boundary condition at the inlet. However, in the streamwise direction, the disturbances grow in size and magnitude. Moreover, it contaminates the flow field downstream. The magnitudes of the disturbance amplitudes grow slowly as they move further downstream. Moreover, flow is spatially unstable. However, its variation in the normal direction is different from that of a stationary mode, as seen in the respective figures. The magnitude of the normal disturbance amplitudes is one order less than that of the streamwise component.

5.6 Global Stability Results 0 -0.005 -0.01

u

Fig. 5.25 Variation of disturbance amplitudes in streamwise direction for stationary mode a streamwise velocity (u) and b normal velocity (v) for the eigenvalue ω = 0.0 − 0.01818i, as marked by circle in Fig. 5.23

133

-0.015

r=5.23 r=5.31 r=5.58 r=6.23 r=9.63 r=17.3

-0.02 -0.025 -0.03

200

250

(a) 300

350 x

400

450

500

-4

x 10 15

r=5.23 r=5.31 r=5.58 r=6.23 r=9.63 r=17.3

v

10

5

0

(b) 200

250

300

350 x

400

450

500

Figure 5.27 presents variation of disturbance amplitudes in the streamwise direction at various radial locations. It shows that most of the disturbances grow in magnitude as they move towards downstream. The magnitude of the disturbance amplitudes is very small near the wall due to the viscous effect, gradually increases in the radial direction and finally vanishes in the far-field. It shows that disturbances evolve within the flow field in time and grow in magnitude and size while moving downstream. Figure 5.28 shows the contour plot of the real part of streamwise disturbance velocity of two different eigenmodes with frequency ωr = 0.003689 and ωr = 0.1234 for N = 0 and Re = 383. The streamwise domain length is 345. The disturbances evolve in the vicinity of the wall and increase the amplitudes when moving downstream. The typical length scale of the wavelet structure decreases with the increases in frequency (ωr ).

134

5 Axisymmetric Boundary Layer …

20 0.0020

r

15 10

0.0000

(a)

-0.0020

200

250

300

350 x

400

450

500

20

-0.0006

r

15 10

-0.0008

(b) 200

250

300

350 x

400

450

500

-0.0010

Fig. 5.26 Contour plot of oscillatory mode a streamwise velocity and b normal velocity for the eigenvalue ω = 0.003689 − 0.01874i, as marked by square in Fig. 5.23 -3

12

x 10

r=5.23 r=5.31 r=5.57 r=6.23 r=9.64 r=17.33

10 8

u

6 4 2 0 -2 -4

(a) 200

250

300

350 x

400

450

300

350 x

400

450

500

-4

2

x 10

0 -2

v

-4 r=5.23 r=5.31 r=5.57 r=6.23 r=9.64 r=17.33

-6 -8 -10 -12

200

250

(b) 500

Fig. 5.27 Variation of disturbance amplitudes in streamwise direction for oscillatory mode a streamwise velocity (u) and b normal velocity (v) for the eigenvalue ω = 0.003689 − 0.01874i, as marked by square in Fig. 5.23

5.6 Global Stability Results

135

20 10

r

15 10

5

(a)

0 -3 x 10

200

250

300

350 x

400

450

500 0.02

20 15

0

r

10

(b) 200

250

300

350 x

400

450

500

−0.02

Fig. 5.28 Contour plot of the real part of u for the two different eigenmodes associated with the frequencies a ωr = 0.003689, b ωr = 0.1234. The corresponding Reynolds number and azimuthal wavenumber are Re = 383 and N = 0, respectively

5.6.7 Effect of Transverse Curvature The body radius of the cylinder is another important length scale in the case of the axisymmetric boundary layer, in addition to the boundary layer thickness (δ ∗ ). The inverse of the body radius is called the transverse surface curvature (S). The flatplate boundary layer is a case of a zero transverse curvature. It is normalized with the displacement thickness (δ ∗ ) at the inflow boundary. In the case of an axisymmetric boundary layer, it has a significant effect on the characteristics of base flow as well as disturbance flow quantities. The boundary layer along the flat plate is self-similar in the absence of a streamwise pressure gradient, while that along the cylinder is non-similar. Self-similarity is possible only for the geometry in which there is no inherent length scale. The cylinder has a characteristic length radius of r. This is because the boundary layer thickness δ grows in the streamwise direction while the cylinder radius remains constant. The transverse curvature (S) affects the stability of the boundary layer direct through the curvature term which appears in the linearized Navier-Stokes equations in the cylindrical coordinates and indirectly through the streamwise evolution of the non-similar base flow [14]. Thus, the transverse curvature increases in the streamwise direction, which in turn destroys self-similarity. In order to understand the transverse curvature effect on the stability characteristic of the axisymmetric boundary layer, the stability characteristics are compared with the flat-plate boundary layer (zero transverse curvature) at same Reynolds number and domain size. The Reynolds number is computed based on the displacement thickness and free-stream velocity at the inlet. The eigenspectrum, spatial amplification rate, effect of the domain length on the distribution of the spectrum and eigenfunctions are compared at the same Reynolds numbers. Figure 5.29 shows the comparison of the eigenspectrum for three different Reynolds numbers 383, 557 and 909 (based on

136

5 Axisymmetric Boundary Layer … Flat plate axisymmetric

-0.01

-0.01

-0.02

-0.02

-0.03

-0.03 -0.04

-0.04

-0.05

-0.05 -0.06

Flat plate axisymmetric

0

ωi

ωi

0

(a) 0

0.02

0.04

ωr

0.06

-0.06

0.08

(b) 0

0.01

0.02

0.03

0.04 ωr

0.05

0.06

0.07

0 -0.01

ωi

-0.02 -0.03 -0.04 -0.05 -0.06 0

Flat plate axisymmetric

0.01

(c) 0.02

ωr

0.03

0.04

0.05

Fig. 5.29 Comparison of eigenspectrum for axisymmetric (N = 0) and 2D flat-plate (β = 0) boundary layer for L x = 345 for three different Reynolds numbers a Re = 383, b Re = 557 and c Re = 909

the body radius of the cylinder these are 2000, 4000 and 10,000) with L x = 345 for axisymmetric and flat-plate boundary layer. The comparison is limited to the discrete part of the spectrum only. It has been observed that at a small Reynolds number, the damping rates of eigenmodes are higher (ωi is smaller) for axisymmetric boundary layer than that of a flat-plate boundary layer. The difference in the damping rate of eigenmodes for axisymmetric and flat-plate boundary layers reduces with the increase in Reynolds number. The eigenmodes of the axisymmetric boundary layer approach to eigenmodes of the flat-plate boundary layer at a higher Reynolds number. This is primarily due to the effect of transverse curvature only. The effect of transverse curvature is significant at a low Reynolds number. The effect of transverse curvature reduces with the increase in Reynolds number. Thus, it proves that the global modes are more stable at a low Reynolds number for the axisymmetric boundary layer. Figure 5.30 shows the comparison of spatial amplification rates ( A x ) at three different Reynolds numbers for both boundary layers. The least stable mode is selected to compute the spatial amplification rate. The comparison shows that at a low Reynolds number, A x is higher for the flat-plate boundary layer. The A x for the axisymmetric boundary layer also increases with the increase in Reynolds number and approaches to flat-plate boundary layer at a higher Reynolds number. Three different families of eigenspectrums are shown in Fig. 5.31 for the axisymmetric and flat-plate boundary layer. The discrete part of the spectrum is only shown

5.6 Global Stability Results 0.4 flat plate axisymmetric

0.35 0.3

Ax

0.25 0.2 0.15 0.1 0.05 0

(a) 200

250

300

350 x

400

450

500

0.4 flat plate axisymmetric

0.35 0.3

Ax

0.25 0.2 0.15 0.1 0.05

(b)

0

500

400

300

600

x 0.4 flat plate axisymmetric

0.35 0.3 0.25

Ax

Fig. 5.30 Comparison of spatial amplification rate of axisymmetric (N = 0) and 2D flat-plate (β = 0) boundary layer for three different Reynolds number a Re = 383, b Re = 557 and c Re = 909

137

0.2 0.15 0.1 0.05 0

(c) 400

500

600

700 x

800

900

1000

138

5 Axisymmetric Boundary Layer …

in the figures for comparison. The difference between the spectrum depends on the domain length. The distance between two consecutive frequencies reduces with the increase in domain length. To quantify this discretization of the eigenmodes for Re = 323, different domain lengths were considered with L x1 = 345, L x2 = 475 and L x3 = 605. Figure 5.32 shows the comparison of the modulus of the streamwise and wall-normal disturbance velocity u and v, respectively, for Re = 383 at x = 445. The magnitude of the modulus of disturbances is higher for the flat-plate boundary layer.

5.6.8 Helical Mode N = 1 The eigenspectrum for azimuthal wavenumber N = 1 and Re = 383 is shown in Fig. 5.33. Stationary eigenmode is also found corresponding to ωr = 0 in this non-axisymmetric case with N=1. The corresponding growth rate ωi is −0.02661. Figure 5.34 shows the spatial structure of disturbance amplitudes for stationary mode. The oscillatory mode with the largest imaginary part is ω = 0.03654 − 0.01763i. This global mode is also temporally stable because of ωi < 0. On the other hand, the eigenmode corresponding to N = 1 is non-axisymmetric; therefore, the disturbance velocity also has an azimuthal component (w). The spatial structure for streamwise (u), radial (v) and azimuthal (w) disturbance velocity amplitudes is shown in Fig. 5.36. The disturbance amplitudes grow while moving in the streamwise direction towards the downstream. The magnitude and size of the disturbance amplitudes also increase in the streamwise direction. Wave-like nature of the disturbance amplitudes is also found for the least stable mode for helical mode, N = 1 (Figs. 5.35 and 5.37).

5.6.9 Helical Mode N = 2 The eigenspectrum is shown in Fig. 5.38 for the helical mode N = 2 and Reynolds number Re = 383. The stationary eigenmode is also found for N = 2 with ωr = 0. The temporal growth rate ωi corresponding to the stationary wave is −0.02704. The oscillatory mode with the least stable eigenmode is ω = 0.0703 − 0.0209i. The imaginary part ωi for all the global modes is negative. Hence, the computed all global modes are temporally stable. The spatial structure for the least stable eigenmode is shown in Fig. 5.39. The 2D spatial structure of the disturbance amplitudes is qualitatively similar to that of axisymmetric and helical mode N = 1. The wavelet structure of the disturbance amplitude is found which grows in size and magnitude while moving downstream. Figure 5.40 shows variation of disturbance amplitudes in the streamwise direction at different radial locations. Figures 5.39 and 5.40 clearly indicate that disturbances grow in the streamwise direction towards the exit boundary.

5.6 Global Stability Results Lx=345 :Axisymmetric Lx=345: Flat plate −0.01

ωi

−0.015 −0.02 −0.025 −0.03 0

(a) 0.02

0.04

0.06 ωr

0.08

0.1

0.12

Lx=475:Axisymmetric Lx=475: Flat plate −0.01 −0.015

ωi

Fig. 5.31 Comparison of eigenspectrum for different streamwise domain lengths a L x = 345, b L x = 475 and c L x = 605 for Re = 383 for axisymmetric (N = 0) and 2D flat-plate boundary layer (β = 0)

139

−0.02 −0.025 −0.03 0

(b) 0.04 ωr

0.02

0.08

0.06

Lx=605: Axisymmetric

Lx=605 : Flat plate -0.01

-0.015

-0.02

-0.025

-0.03 0

(c) 0.01

0.02

0.03

0.04

0.05

0.06

0.07

140

5 Axisymmetric Boundary Layer …

Fig. 5.32 Comparison of modulus of eigenfunction a u and b v at streamwise location x = 445 for Re = 383 for axisymmetric (N = 0) and 2D flat-plate boundary layer (β = 0)

20 Flat plate Axisymmetric

y

15

10

5

(a) 0 0

1

2

3

4

5 −3

|u|

x 10

20 Flat−plate Axisymmetric

y

15

10

5

(b) 0 0

1

0.5

|v|

1.5 −3

x 10

5.6.10 Helical Mode N = 3 The eigenspectrum for the helical mode N = 3 for Re = 383 is shown in Fig. 5.41. The imaginary parts of all the global modes are negative; hence, the flow is temporally stable. The least stable eigenmode has a complex frequency ω = 0.0673 − 0.0246. It has been observed that for the same Reynolds number (Reδ∗ = 383), the temporal damping rate increases with the higher azimuthal wave number from N = 1 to N = 3. Thus global eigenmodes are more stable with higher azimuthal wavenumber N.

5.6 Global Stability Results

141

0 −0.01 −0.02

ωi

−0.03 −0.04 −0.05 −0.06 −0.07 −0.08 0

0.02

0.04 ωr

0.06

0.08

Fig. 5.33 Eigenspectrum for azimuthal wavenumber N = 1 and Re = 383 20 −0.0300

r

15 10

−0.0400

(a) 200

−0.0500 250

300

350 x

400

450

500

20 0.0010 r

15 10

0.0005

(b) 200

250

300

350 x

400

450

500

20

0.0010 0.0000 −0.0010 −0.0020 −0.0030

r

15 10

(c) 200

0.0000

250

300

350 x

400

450

500

Fig. 5.34 Contour plot of a streamwise (u), b radial (v) and c azimuthal (w), for disturbance velocity for oscillatory mode N = 1 and Re = 383. The associated eigenvalues are ω = 0.0 − 0.0266i, as marked by circle in Fig. 5.33

142

5 Axisymmetric Boundary Layer … 0.02

(a)

5

0.01

r=2.96 r=3.04 r=3.31 r=3.96 r=7.37 r=15.03

3

−0.01

2

v

u

−3

4

0

r=2.96 r=3.04 r=3.31 r=3.96 r=7.37 r=15.03

−0.02 −0.03 −0.04

x 10

200

250

1 0

300

350 x

400 x 10

450

500

−1

(b) 200

250

300

400

450

500

350 x

400

450

500

−3

r=2.96 r=3.04 r=3.31 r=3.96 r=7.37 r=15.03

4

w

2

0

−2

(c) −4

200

250

300

350 x

Fig. 5.35 Variation of disturbance amplitudes in streamwise direction for a streamwise (u), b radial (v) and c azimuthal (w) disturbance amplitudes at different radial location ω = 0.0 − 0.0266i, as marked by square in Fig. 5.33 20

0.03 0.02 0.01 0 -0.01

r

15 10

(a) 200

250

300

350 x

400

450

500

20

0.0005 0.0000 -0.0005 -0.0010 -0.0015

r

15 10

(b) 200

250

300

350 x

400

450

500

20 0.0000

r

15 10

-0.0020

(c) 200

-0.0040 250

300

350 x

400

450

500

Fig. 5.36 Contour plot of a streamwise (u), b radial (v) and c azimuthal (w), for disturbance velocity for oscillatory mode N = 1 and Re = 383. The associated eigenvalues are ω = 0.03654 − 0.01763i, as marked by square in Fig. 5.33

5.6 Global Stability Results

143 -3

1

r=5.23 r=5.31 r=5.57 r=6.23 r=9.64 r=17.33

0.025 0.02 0.015

(b)

0.5 0

v

u

0.01

x 10

0.005 0

-0.5

r=5.23 r=5.31 r=5.57 r=6.23 r=9.64 r=17.33

-1

-0.005 -0.01 -0.015

-1.5

(a) 200

250

300

350 x

400

450

-2

500

200

300

250

350 x

400

450

500

-3

8

x 10

r=5.23 r=5.31 r=5.57 r=6.23 r=9.64 r=17.33

6 4

w

2 0 -2 -4 -6

(c) 200

250

300

350 x

400

450

500

Fig. 5.37 Variation of disturbance amplitudes in streamwise direction for a streamwise (u), b radial (v) and c azimuthal (w) disturbance amplitudes at different radial location ω = 0.03654 − 0.01763i, as marked by square in Fig. 5.33 Fig. 5.38 Eigenspectrum for azimuthal wavenumber N = 2 and Re = 383

0

ωi

−0.02 −0.04 −0.06 −0.08 0

0.02

0.04

ωr

0.06

0.08

0.1

The contour plots and variation of disturbance amplitudes in streamwise direction are shown in Figs. 5.42 and 5.43 and indicate that size and magnitude of the disturbances increase in the streamwise direction.

144

5 Axisymmetric Boundary Layer … 20 0.02

r

15 10

0

(a) 200

250

300

350 x

400

450

−0.02

500

20

−0.0005

15

−0.0010

r

10

−0.0015

(b) 200

250

300

350 x

400

450

−0.0020

500

0.0000

15

−0.0020

r

20

10

−0.0040

(c)

−0.0060

200

250

300

350 x

400

450

500

Fig. 5.39 Contour plot of a streamwise (u), b radial (v) and c azimuthal (w), for disturbance velocity components of oscillatory mode for N = 2 and Re = 383. The associated eigenvalue is ω = 0.0703 − 0.0209i, as marked by square in Fig. 5.38 -3

1.5

r=5.23 r=5.31 r=5.57 r=6.23 r=9.64 r=17.33

0.02 0.015 0.01

1

0

0

-0.5

-0.005

-1

-0.01

-1.5

-0.015 -0.02

r=5.23 r=5.31 r=5.57 r=6.23 r=9.64 r=17.33

-2

(a) 200

(b)

0.5

v

u

0.005

x 10

250

300

350 x

400

450

-2.5

500

200

250

300

350 x

400

450

500

-3

10

x 10

r=5.23 r=5.31 r=5.57 r=6.23 r=9.64 r=17.33

w

5

0

-5

(c) 200

250

300

350 x

400

450

500

Fig. 5.40 Variation of velocity disturbance amplitudes in streamwise direction for a streamwise (u), b radial (v) and c azimuthal (w) disturbance amplitudes at different radial location ω = 0.0703 − 0.0209i, as marked by square in Fig. 5.38

5.6 Global Stability Results

145

0

ωi

−0.02 −0.04 −0.06 −0.08 −0.1 0

0.02

0.04

0.06 ωr

0.08

0.1

Fig. 5.41 Eigenspectrum for azimuthal wavenumber N = 3 and Re = 383 20 0.02

r

15 10

0

(a) 200

−0.02 250

300

350 x

400

450

500 −0.0005

15

−0.0010

r

20

10

−0.0015

(b) 200

250

300

350 x

400

450

500

20

0.0060 0.0040 0.0020 0.0000 −0.0020 −0.0040

r

15 10

(c) 200

250

300

350 x

400

450

500

Fig. 5.42 Contour plot of a streamwise (u), b radial (v) and c azimuthal (w), for disturbance velocity components of oscillatory mode for N = 3 and Re = 383. The associated eigenvalue is ω = 0.0673 − 0.0246i, as marked by square in Fig. 5.41

146

5 Axisymmetric Boundary Layer … 0.025 r=5.23 r=5.31 r=5.57 r=6.23 r=9.64 r=17.33

0.02 0.015 0.01

1.5

x 10

−3

(b)

1 0.5

0.005

r=5.23 r=5.31 r=5.57 r=6.23 r=9.64 r=17.33

v

u

0

0

−0.5

−0.005

−1 −0.01 −0.015 −0.02

−1.5

(a) 200

−2 250

300

350 x

400

6

x 10

500

200

250

300

350 x

400

450

500

−3

r=5.23 r=5.31 r=5.57 r=6.23 r=9.64 r=17.33

4 2

w

450

0 −2 −4 −6

(c) 200

300

250

350 x

400

450

500

Fig. 5.43 Variation of velocity disturbance amplitudes in streamwise direction for a streamwise (u), b radial (v) and c azimuthal (w) disturbance amplitudes at different radial location ω = 0.0673 − 0.0246i, as marked by square in Fig. 5.41 Fig. 5.44 Variation in temporal growth rate (ωi ) with Reynolds number for different azimuthal wavenumbers (N)

-0.01 -0.015

ωi

-0.02 -0.025 N=0 N=1 N=2 N=3 N=4 N=5

-0.03 -0.035 -0.04 -0.045 -0.05

300

350

400

450 500 Re

550

600

650

5.7 Temporal Growth Rate Figure 5.44 shows the temporal growth rate of the eigenmodes for different Reynolds numbers. The growth rate increases with the increase in Reynolds number for all the azimuthal wavenumbers. As the largest imaginary part is negative for different Reynolds number, the flow is temporally stable. The transverse curvature varies inversely with the Reynolds number. At a low Reynolds number, the temporal growth rate is small, and with the increase in Reynolds number, the temporal growth rate

5.8 Spatial Amplification Rate

147

increases, proving that the transverse curvature has a significant damping effect on the global temporal mode. Global modes with higher wavenumbers, N = 3, 4 and 5 are more stable than that of axisymmetric and helical mode N = 1 and 2. At a low Reynolds number, the axisymmetric mode has higher growth rate than that of N = 1 and 2. As Reynolds number increases, the growth rate of the helical mode N = 1 and N = 2 increases at a higher rate than that of axisymmetric mode.

5.8 Spatial Amplification Rate The global temporal modes exhibit spatial growth/decay streamwise direction when moving downstream. The disturbances in the streamwise direction at a particular radial location may decay or amplify. However, the overall effect of all disturbances together at each streamwise location can be quantified by computing the spatial amplitude growth, A(x) [4]. 

rmax

A(x) =

(u ∗ (x, r )u(x, r ) + v ∗ (x, r )v(x, r ) + w ∗ (x, r )w(x, r ))dr , (5.38)

1

where ∗ denotes the complex conjugate. Figure 5.45 shows the spatial amplitude growth of the disturbance waves for different azimuthal wavenumbers at various

0.5

Ax

0.3

N=0 N=1 N=2 N=3 N=4 N=5

N=0 N=1 N=2 N=3 N=4 N=5

0.5 0.4

Ax

0.4

0.3

0.2 0.2 0.1

0.1

(a) 140

160

200

180

220

240

0

(b) 150

200

250

x 0.5 0.4

Ax

300

350

400

x N=0 N=1 N=2 N=3 N=4 N=5

0.3 0.2 0.1 0 150

(c) 200

250

300

350 x

400

450

500

Fig. 5.45 Spatial amplification rate A x in streamwise direction x for a Re = 261, b Re = 477 and c Re = 628 for different azimuthal wavenumbers. The most unstable temporal mode is considered to compute the spatial amplitude growth A x

148

5 Axisymmetric Boundary Layer …

Reynolds numbers. Most unstable oscillatory (ωr = 0) temporal modes are considered to compute the spatial growth of the disturbance amplitudes. The spatial growth increases in the streamwise direction towards the exit boundary. At a low Reynolds number, the growth of disturbances starts at an early stage.

5.9 Summary The stability of the incompressible axisymmetric boundary layer formed on a circular cylinder has been studied using a global stability approach. For the first time, the two-dimensional global modes are computed for the axisymmetric boundary layer. The Reynolds numbers from 261 to 693 based on the displacement thickness and azimuthal wavenumbers from 0 to 5 are considered. The linearized Navier-Stokes equations are derived in cylindrical coordinates, discretized using Chebyshev spectral collocation method, and two-dimensional eigenvalue problem is solved using ARPACK, which employs Arnoldi’s iterative algorithm. The Robin and periodic boundary conditions are considered in the streamwise direction for code validation. However, all stability results presented here use Dirichlet boundary conditions at the inlet and extrapolation type conditions at the outlet. The base velocity profile is obtained by solving the steady Navier-Stokes equations using the finite volume code Ansys Fluent software. The transverse curvature ( aδ ) increases in the streamwise direction for the given Reynolds number and reduces with the increase in Reynolds number at a given streamwise location. The global stability results of axisymmetric mode (Re = 12,439) and helical mode N = 1 (Re = 1060) obtained with the Robin and periodic boundary conditions are compared with the local stability results of Tutty et al. [21]. The domain length equal to one wavelength is considered. The eigenspectrum and eigenfunctions are in good agreement. Thus, the method of global mode computations is validated. The streamwise wavenumber (αr ) and spatial amplification rate (αi ) are computed for Re = 15,000 from the global mode computed with the Dirichlet and extrapolationtype boundary conditions at the inlet and outlet, respectively. The (αr ) and (αi ) are also computed for Re = 15,000 and same frequency (ωr = 0.075) using local spatial stability analysis. Both the results are in good agreement. The eigenspectrum is computed for the different Reynolds numbers from 261 to 693 and azimuthal wavenumbers from 0 to 5. The largest imaginary part of the eigenvalues (ωi ) is negative in all these cases. Thus, the global modes are temporally stable. The twodimensional spatial structure of the eigenmodes shows the wave-like nature of the disturbance amplitudes. The disturbance amplitudes grow in size and magnitude in the streamwise direction while moving towards downstream. The spatial growth rate (A x ) also increases in the streamwise direction towards the downstream. This indicates that the flow is spatially convectively unstable. The typical length scale of the wavelet structure of the disturbance amplitudes reduces with the increase in frequency (ωr ). The monotonic variation of the disturbance amplitudes is found in the streamwise direction for the stationary eigenmode (ωr = 0) of the axisymmet-

References

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ric case, while for helical mode (N = 1), it is not monotonic. The magnitude of u disturbance amplitudes is found almost one order higher than that of v disturbances. The two-dimensional eigenvalue problem is solved numerically for axisymmetric and flat-plate boundary layer for three different Reynolds numbers 383, 557 and 909. The discrete part of the eigenspectrum (T-S waves) is compared for both the boundary layer and found that the temporal growth rate (ωi ) is higher for the planner boundary layer. Thus, the axisymmetric boundary layer is more stable than that of the planner boundary layer, and it is due to the effect of transverse curvature. The spatial growth rate (A x ) in the streamwise direction is also compared for axisymmetric and planner boundary layers and found that (A x ) is higher for the planner boundary layer. The axisymmetric boundary layer’s temporal and spatial growth rate approaches the planner boundary layer at a higher Reynold number because transverse curvature reduces at a higher Reynolds number. Thus, the transverse curvature has a significant damping effect on the disturbances. The temporal growth rate is computed for Reynolds numbers from 261 to 693 and azimuthal wavenumbers from 0 to 5. At low Reynolds number (Re < 577), the axisymmetric mode (N = 0) is least stable one, while at higher Reynolds number (Re > 577), the helical mode (N = 2) is least stable one. The helical modes N = 3, 4 and 5 are more stable. The spatial growth rate increases in the streamwise direction, showing that the flow is spatially convectively unstable.

References 1. Akervik, E., Ehrenstein, U., Gallaire, F., Henningson, D.: Global two-dimensional stability measure of the flat plate boundary layer flow. Eur. J. Mech. B/Fluids 27, 501–513 (2008) 2. Alizard, F., Robinet, J.C.: Spatially convective global modes in a boundary layer. Phys. Fluids 19, 114105 (2007) 3. Costa, B., Don, W., Simas, A.: Spatial resolution properties of mapped spectral Chebyshev methods. In: Proc. SCPDE: Recent Progress in Scientific Computing, pp. 179–188 (2007) 4. Ehrenstein, U., Gallaire, F.: On two-dimensional temporal modes in spatially evolving open flow?:the flat-plate boundary layer. J. Fluid Mech. 536, 209–218 (2005) 5. Ehrenstein, U., Gallaire, F.: Two dimensional global low-frequency oscillations in a separating boundary layer flow. J. Fluid Mech. 614, 315–327 (2008) 6. Fasel, H., Rist, U., Konzelmann, U.: Numerical investigation of the three-dimensional development in boundary layer transition. AIAA J. 28, 29–37 (1990) 7. Garnaud, X., Lesshafft, L., Schmid, P.J., Huerre, P.: Modal and transient dynamics of jet flows. Phys. Fluids 25, 044103 (2013) 8. Glauert, M.B., Lighthill, M.J.: The axisymmetric boundary layer on a long thin cylinder. Proc. Roy. Soc. London Ser. A 230, 1181 (1955) 9. Kurz, H.B., Kloker, M.J.: Mechanism of flow tripping by discrete roughness elements in a swept-wing boundary layer. J. Fluid Mech. 796, 158–194 (2016) 10. Loiseau, J.C., Robinet, J.C., Cherubini, S., Leriche, E.: Investigation of the roughness -induced transition: Global stability analyses and direct numerical simulation. J. Fluid Mech. 760, 175– 211 (2014) 11. Malik, M.R.: Numerical methods for hypersonic boundary layer stability. J. Comput. Phys. 86(2), 376–412 (1990)

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12. Marquet, O., Sipp, D., Jacquin, L.: Sensitivity analysis and passive control of cylinder flow. J. Fluid Mech. 615, 221–252 (2008) 13. Nichols, J.W., Lele, S.K.: Global modes and transient response of a cold supersonic jet. J. Fluid Mech. 669, 225–241 (2011) 14. Pruell, C.D., Streett, C.L.: A spectral collocation method for compressible, non-similar boundary layer. Int. J. Num. Mech. Fluids. 13, 713–737 (1991) 15. Rao, G.N.V.: Mechanics of transition in an axisymmetric laminar boundary layer on a circular cylinder. J. Appl. Math. Phys. 25, 63–75 (1974) 16. Roache, P.J.: A method for uniform reporting of grid refinement studies. J. Fluid. Eng. 116(3), 405–413 (1994) 17. Sipp, D., Lebedev, D.: Global stability of base and mean flows: a general approach to its applications to cylinder and open cavity flow. J. Fluid Mech. 593, 333–358 (2007) 18. Swaminathan, G., Shahu, K., Sameen, A., Govindarajan, R.: Global instabilities in diverging channel flows. Theor. Comput. Fluid Dyn. 25, 25–64 (2011) 19. Theofilis, V.: Advances in global linear instability analysis of nonparallel and three dimensional flows. Prog. Aerosp. Sci. 39, 249–315 (2003) 20. Theofilis, V.: Global linear instability. Ann. Rev. Fluid Mech. 43, 319–352 (2011) 21. Tutty, O.R., Price, W.G.: Boundary layer flow on a long thin cylinder. Phys. Fluids 14, 628–637 (2002) 22. Vinod, N.: Stability and transition in boundary layers: Effect of Transverse Curvature and Pressure Gradient. Ph.D. Thesis, Jawaharlal Nehru Center for Advanced Scientific Research (2005)

Chapter 6

Axisymmetric Boundary Layer on a Circular Cone

6.1 Introduction In this chapter, the effect of transverse curvature and favourable pressure gradient on the stability characteristics of the spatially developing incompressible boundary layer formed on a circular cone. An incompressible flow along the circular cone is considered with inflow parallel to the axis of a cone. The angle of attack is zero, and hence, the base flow is axisymmetric. The favourable pressure gradient develops in the streamwise direction due to the geometry of a cone. A semi-cone angle of 2◦ , 4◦ and 6◦ are considered for the instability analysis. The two-dimensional eigenvalue problem is formulated in the spherical coordinates and solved numerically using Arnoldi’s iterative algorithm to compute leading eigenvalues and eigenfunctions. The boundary layer grows continuously in the spatial directions, and due to the transverse curvature of the body, the base flow is fully non-parallel and non-similar. As discussed in the previous chapter, the transverse curvature (δ/a) increases in the streamwise direction, and it has an overall stabilizing effect on the global modes. For the range of semi-cone angles considered here (2◦ , 4◦ and 6◦ ), the body radius of the cone increases at a higher rate than that of a boundary layer thickness (δ), and hence, the transverse curvature (δ/a) reduces in the streamwise direction as shown in Fig. 6.4. This effect (reduction in transverse curvature) is larger for higher semi-cone angle α as shown in Fig. 6.4. As the semi-cone angle reduces, the ratio δ/a reduces at any streamwise location and very small angle α, and the boundary layer thickness may be larger than the cone radius. In such a case, the transverse curvature may increase in the streamwise direction. The transverse curvature is computed for small values of semi-cone angles, α = 0.75◦ , 0.5◦ and 0.25◦ with U∞ = 0.02 m/s, and it is found that the transverse curvature yet reduces. However, the δ/a is very large for small angle α as shown in Fig. 6.5. A favourable pressure gradient develops in the streamwise direction due to the geometry of the cone. Thus, it becomes interesting to study the combined effect of transverse curvature and pressure gradient on the stability characteristics.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Bhoraniya et al., Global Stability Analysis of Shear Flows, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-19-9574-3_6

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The laminar–turbulent transition in boundary layers has been a subject of interest to many researchers in the past few decades. It is essential to understand the onset of transition in boundary layers as the flow pattern, and its effects are very different in laminar and turbulent flows. For example, the drag force in a turbulent flow is much higher than that of a laminar flow. The boundary layers transition through the classical T-S wave mechanism at low free-stream levels. The amplification of the disturbance waves is the primary step in the transition process, which is studied in linear stability analysis. The results from stability analysis and the prediction of transition onset are very useful in hydrodynamics and aerodynamics applications like submarines, torpedoes, rockets and missiles. The linear stability analysis of shear flows with parallel flow assumption is well understood by the solution of the Orr–Sommerfeld equation [4]. It is known that the stability characteristics in a boundary layer are strongly influenced by factors such as pressure gradient and surface curvature free-stream turbulence level. The boundary layer formed over a right circular cone is axisymmetric and qualitatively different from a flat-plate boundary layer. The boundary layer on a circular cone develops continuously in spatial directions. Hence, the parallel flow assumption is not valid. Furthermore, due to the wall-normal velocity component, the boundary layer is non-parallel. The instability analysis of such two-dimensional base flow is called global stability analysis [19]. The literature on the instability analysis of the axisymmetric boundary layer is very sparse. The available literature on the axisymmetric boundary layer is limited to local stability analysis. Rao [14] first studied the stability of the axisymmetric boundary layer. He found that non-axisymmetric disturbances are less stable than that two-dimensional disturbances. Tutty [20] investigated that for non-axisymmetric modes, critical Reynolds number increases with the azimuthal wavenumber. The critical Reynolds number was 1060 for N = 1 mode and 12,439 for N = 0 mode. The axisymmetric mode is found to be the least stable fourth mode. Vinod [13] investigated that higher non-axisymmetric mode N = 2 is linearly stable for only a small range of curvature. The helical mode N = 1 is unstable over a significant length of the cylinder, but never unstable for curvature above 1. Transverse curvature has an overall stabilizing effect over mean flow and perturbations. Malik [11] studied the effect of transverse curvature on the stability of the incompressible boundary layer. They investigated that the body and streamline curvature have significant damping effects on disturbances. The secondary instability of an incompressible axisymmetric boundary layer is also studied by Vinod [21]. They found that laminar flow is always stable at high transverse curvature to secondary disturbances. The global stability analysis for hypersonic and supersonic flow over a circular the cone is reported in the literature. In supersonic and hypersonic boundary layers, higher acoustic instability modes with higher frequencies exist in addition to the first instability mode [9]. The experiments have confirmed that the two-dimensional Mack mode dominates in the hypersonic boundary layer [12, 16, 17]. Experimental results show that leading bluntness affects the transition location significantly in circular cones [8]. In the case of the axisymmetric boundary layer on a circular cylinder, transverse curvature effect increases in the streamwise direction, which helps in stabilizing the flow. In the case of an axisymmetric boundary layer on a circular cone, the body radius of the cone increases at a higher rate than a boundary

6.2 Problem Formulation

153

layer thickness (δ), and hence, the transverse curvature effect (δ/a) reduces in the streamwise direction as shown in Fig. 6.4. However, the favourable pressure gradient develops in the streamwise direction due to the cone angle (α). Thus, studying the combined effect of transverse curvature and pressure gradient on the stability characteristics becomes interesting. The two-dimensional global modes are also computed for the flat-plate boundary layer by some investigators [1, 5]. However, this is the first attempt to carry out Global stability analysis of incompressible boundary layer over a circular cone. The main aim of this paper is to study the Global stability characteristics of the axisymmetric boundary layer on a circular cone and the effect of the transverse curvature and pressure gradient on the stability characteristics. Ehrenstein and Gallaire [5], Alizard and Robinet [2] and Akervik et al. [1] have solved the two-dimensional eigenvalue problem for the flat-plate boundary layer in which transverse curvature and streamwise pressure gradient are absent. Even in the previous chapter, we studied the instabilities of the axisymmetric boundary layer developed on a cylinder where only transverse curvature was present. However, this is the first attempt to carry out the global stability analysis of the incompressible boundary layer over a circular cone.

6.2

Problem Formulation

The standard procedure is followed for the derivation of the Linearized NavierStokes equations (LNS) for the disturbance flow quantities. The Navier-Stokes (NS) equations for the base flow and instantaneous flow are written in the spherical coordinate system (r, θ , φ). The equations are non-dimensionalized using free-stream velocity U∞ and body radius of the cone (a) at the inlet. The LNS for disturbance flow quantities are obtained by subtraction and subsequent linearization. The base flow is two-dimensional, and disturbances are three-dimensional in nature. This will determine whether the small amplitudes of the disturbances grow or decay for a given steady laminar flow. The Reynolds number is defined as, Re =

U∞ a ν

(6.1)

where a is the surface radius of cone at inlet and ν is the kinematic viscosity. The flow quantities are presented as sum of the base flow and the perturb quantities as, Ur = Ur + u r ,

Uθ = Uθ + u θ ,

Uφ = 0 + u φ ,

P=P+p

(6.2)

The disturbances are considered in normal mode form and varying in radial (r) and polar (θ ) directions. Thus, the disturbances having periodic nature in the azimuthal (φ) direction. (6.3) q(r, θ, t) = q(r, ˆ θ )e[i(N φ−ωt)]

154

6 Axisymmetric Boundary Layer …

Fig. 6.1 Schematic diagram of axisymmetric boundary layer on a circular cone

U

r

X

where, q = [u r , u θ , u φ , p], Q = [Ur , Uθ , P], Q = [U r , U θ , P], r is radial coordinate, θ is polar coordinate, φ is azimuthal coordinate, ω is circular frequency and, N is azimuthal wavenumber (Fig. 6.1). ∂u r Uθ ∂u r u θ ∂Ur ∂p ∂u r ∂Ur 2Uθ + Ur + ur + + − uθ + ∂t ∂r ∂r r ∂θ r ∂θ r ∂r  1 2 cot θ ∂u ∂u 2u 2 2 r θ φ − 2 u r − 2 − 2 − =0 uθ − 2 Re r r ∂θ r2 r sinθ ∂φ ∂u θ Uθ ∂u θ u θ ∂Uθ ∂u θ ∂Uθ Ur Uθ + Ur + ur + + + uθ + ur ∂t ∂r  ∂r r ∂θ r ∂θ r r 1 ∂p 2 cot θ 1 u ∂u ∂u 2 r θ φ + − 2 u θ + 2 − 2 2 − 2 =0 r ∂θ Re r ∂θ r sinθ ∂φ r sin θ ∂u φ Uθ ∂u φ 1 ∂p ∂u φ Ur Uθ cot θ + Ur + + uφ + uφ + ∂t ∂r r ∂θ r r r sinθ ∂φ   1 ∂u u 2 cot θ ∂u 2 φ φ φ 2 −  uφ + 2 − 2 + 2 =0 Re r sinθ ∂φ r sinθ r sinθ ∂φ ∂u r 2u r 1 ∂u θ u θ cot θ 1 ∂u φ + + + + =0 ∂r r r ∂θ r r sinθ ∂φ

(6.4)

(6.5)

(6.6)

(6.7)

where, 2 =

1 ∂ 1 ∂ ∂ 2 ∂ cot θ ∂ + 2 2+ 2 + 2 2 + 2 ∂r r ∂r r ∂θ r ∂θ r sin θ ∂φ 2

(6.8)

6.2 Problem Formulation

155

6.2.1 Boundary Conditions No-slip and no-penetration boundary conditions are considered on the surface of the cone. The magnitude of all the disturbance velocity components are zero at the solid surface of the cone due to viscous effect. u r (r, θmin ) = 0,

u θ (r, θmin ) = 0,

u φ (r, θmin ) = 0

(6.9)

It is expected to vanish the amplitudes of disturbance flow quantities at free-stream far away from the solid surface of the cone. The Homogeneous Dirichlet conditions are applied to all the velocity and pressure disturbances at free-stream. u r (r, θmax ) = 0,

u θ (r, θmax ) = 0,

u φ (r, θmax ) = 0,

p(r, θmax ) = 0 (6.10) The boundary conditions are not straightforward in the streamwise direction for global stability analysis. As suggested by Theofilis [19], Homogeneous Dirichlet boundary conditions are considered for the velocity disturbances at the inflow boundary. Here, we are interested in the disturbances evolved within the basic flow field only. The boundary conditions at the outflow boundary can be applied based on the incoming and outgoing wave information [6]. Such conditions impose wave-like nature on the disturbances, and so it is more restrictive in nature which is not appropriate from the physical point of view. Even streamwise wavenumber α is not known initially in the case of the global stability analysis. Alternatively, one can impose such numerical boundary conditions which extrapolate the information from the interior of the computational domain. Linear extrapolated conditions are applied by several investigators in such cases. Moreover, the literature review on global stability analysis suggests that the linear extrapolated boundary conditions are the least restrictive [18, 19]. Thus, we considered linear extrapolated conditions at the outlet for the numerical solution of the general eigenvalues problem. u r (rin , θ ) = 0, u θ (rin , θ ) = 0, u φ (rin , θ ) = 0

(6.11)

u rn−2 [rn − rn−1 ] − u rn−1 [rn − rn−2 ] + u rn [rn−1 − rn−2 ] = 0

(6.12)

Similarly, one can write extrapolated boundary conditions for polar and azimuthal disturbance components u θ and u φ , respectively. The boundary conditions for pressure do not exist physically at the wall. However, compatibility conditions derived from the LNS equations are collocated at the solid wall of the cone [19].   1 Uθ ∂u r ∂u r 2 ∂u θ ∂p = 2 u r − 2 − Ur − ∂r Re r ∂θ ∂r r ∂θ

(6.13)

  1 ∂p 1 Uθ ∂u θ ∂u θ 2 ∂u r = 2 u θ + 2 − Ur − r ∂θ Re r ∂θ ∂r r ∂θ

(6.14)

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6 Axisymmetric Boundary Layer …

As discussed in the previous chapter, the base and disturbance velocity components are zero on the wall. The streamwise (radial) and azimuthal derivatives are also zero on the cone surface. Thus, the Eqs. 6.13 and 6.14 reduces as,   1 1 ∂u r 2 ∂u θ cotθ ∂u r ∂p = − 2 + 2 ∂r Re r 2 ∂θ 2 r ∂θ r ∂θ

(6.15)

  1 ∂p 1 1 ∂u θ 2 ∂u r cotθ ∂u θ = − 2 + 2 r ∂θ Re r 2 ∂θ 2 r ∂θ r ∂θ

(6.16)

The LNS equations are derived in the spherical coordinates. Chebyshev discretization is used in radial (r) and polar (θ ) direction in similar manner to previous chapter.  rcheb = cos 

θcheb

πi n

πj = cos m

 , where i = 0, 1, 2, 3, . . . , n

(6.17)

where j = 0, 1, 2, 3, . . . , m

(6.18)



In the region near the solid wall, lot of activities of the boundary layer is taking place. The stretching function defined by Eq. 6.19 is used in the polar (θ ) direction, to cluster the more number of collocation points near the wall [10]. θreal =

θi × L θ × (1 − θcheb ) + θmin L θ + θcheb × (L θ − 2θi )

(6.19)

Equation 6.19 transforms the spectral scale θcheb = [1 −1] to real physical scale θreal = [θmin θmax ], where L θ is the computational domain in the polar direction in radians (L θ = θmax − θmin ). The equal number of collocation points is distributed in the region from θmin to θi and θi to θmax . The non-uniform nature of the collocation points in the radial direction makes poor spatial resolution at the centre of the domain. The small distance between collocation points gives rise to Gibbs oscillations. The grid mapping is applied in the radial direction using the Eq. 6.20 [3]. rmap =

sin−1 (αm rcheb ) sin−1 (αm )

rreal = (1 − rmap ) ∗ L r /2 + rmin

(6.20) (6.21)

The value of αm was carefully selected to improve the central region’s grid resolution. The coordinate transformations, along with the effect of grid stretching and mapping, were incorporated into the Chebyshev differential matrix by multiplying a Jacobian matrix. Matrix A and B generated by the LNS operator are very large, real, square and non-symmetric.

6.3 Base Flow Solution

⎡

A11 ⎢ A21 ⎢ ⎣ A31 A41

A12 A22 A32 A42

A13 A23 A33 A43

157

⎤⎡ ⎤ ⎡ A14 ur B11 ⎢ uθ ⎥ ⎢ B21 A24 ⎥ ⎥ ⎢ ⎥ = iω ⎢ ⎣ B31 A34 ⎦ ⎣ u φ ⎦ A44 p B41

B12 B22 B32 B42

B13 B23 B33 B43

⎤⎡ ⎤ B14 ur ⎢ uθ ⎥ B24 ⎥ ⎥⎢ ⎥ B34 ⎦ ⎣ u φ ⎦ B44 p

[A][φ] = iω[B][φ]

(6.22)

(6.23)

where A and B are square matrix of size 4 nm × 4 nm, iω is an eigenvalues and φ is a vector of unknown disturbance amplitude functions u r , u θ , u φ and p. The abovementioned boundary conditions are properly incorporated in the matrix A and B.

6.2.2

Solution of General Eigenvalues Problem

In the modal analysis of the shear flow, the least stable eigenvalue makes the flow unstable. Thus, the leading eigenmodes and associated eigenfunctions near the least stable eigenmode are only important. The computations of full spectrum for Eq. 6.23 is not desirable. Arnoldi’s iterative algorithm, which uses the Krylov subspace method, is employed to compute a few selected eigenvalues only, as discussed in the previous chapter.

6.3 Base Flow Solution The base flow velocity profile is obtained by the numerical solution of the steady incompressible axisymmetric Navier-Stokes equations using finite volume code Ansys Fluent. The incoming flow is parallel to the cone’s axis at the inlet. Therefore, the axisymmetric domain is modelled with 1 m and 0.5 m in the axial U

∂U ∂P 1 ∂U +V =− + ∂x ∂r ∂x Re

∂V ∂V ∂P 1 U +V =− + ∂x ∂r ∂r Re

 

∂ 2U ∂ 2U 1 ∂U + + 2 ∂x r ∂r ∂r 2

∂2 V ∂2 V 1 ∂V + + ∂x2 r ∂r ∂r 2

∂U ∂V V + + =0 ∂x ∂r r

 (6.24)  (6.25)

(6.26)

Appropriate boundary conditions are applied to close the formulation of the above problem. The uniform inlet velocity U∞ is imposed at the inflow boundary. No-slip and no-penetration conditions are applied on the surface of a cone. A slip boundary condition is applied far away from the solid surface. The outflow boundary conditions = 0, ∂∂Vx = 0 and P = 0. The steady are applied at the outlet, which consists ∂U ∂x Navier-Stokes equations are solved using a SIMPLE algorithm with under-relaxation to get the stable solution. The spatial discretization of the N-S equations is done using

158

6 Axisymmetric Boundary Layer …

a QUICK scheme, a weighted average of the second-order upwind and second-order central scheme. Thus, obtained base velocity profile is interpolated to spectral grids using cubic spline interpolation. The simulation was started initially with the coarse grid size of 250 and 125 grid points in axial and radial directions, respectively. The grid was refined with a factor of 1.4142 in each direction. The discretization error was calculated through Grid Convergence Index (GCI) study on four consecutive refined grids [15]. The monotonic convergence is obtained for all these grids as expected. The error and GCI are computed for two different field values U (x = 0.5, r = 0.038965) and V (x = 0.5, r = 0.038965) near the solid boundary where the gradient is higher. The GCI and error were computed for four different grids, as shown in Table 6.1. The error between Mesh 1 and Mesh 2 is too small. Moreover, the GCI has also reduced with the refinement of the grids. Thus, the solution has converged monotonically towards the grid-independent one. Further refinement in the grids will hardly improve the accuracy of the solution while increasing the time for the computations. The distribution of the grid is geometric in both directions. The computed convergence order for U and V is 1.98 and 2.01 in good agreement with the second-order discretization scheme used in the finite volume code Ansys Fluent. Sufficient large domain height is selected in the radial direction, i.e. 0.5 m. Thus, Mesh 1 is used in all the results presented to calculate the velocity profile for a basic state. Figures 6.2 and 6.3 show the contour plot and velocity profile on the transformed coordinates ξ and η. The base velocity profile computed is qualitatively similar to that of Garratt [7]. The governing equations for instability analysis are already derived in spherical coordinates. To perform the global instability computations, the base velocity the field is transformed from cylindrical (U, V ) to spherical coordinates (Ur , Uθ ). U ∂ x + V ∂∂rR (6.27) Ur = ∂ R

∂r 2 ∂x 2 + ∂R ∂R ∂r U ∂ x + V ∂θ Uθ = ∂θ

∂r 2 ∂x 2 + ∂θ ∂θ

(6.28)

Table 6.1 The grid convergence study for the base flow is obtained, using U (x = 0.5, r = 0.038965) and V (x = 0.5, r = 0.038965) for U∞ = 0.1 m/s Mesh U (%) GC I (%) V (%) GC I (%) #1 #2 #3 #4

0.087225 0.087213 0.087188 0.087143

0.008 0.0287 0.0510 –

0.038 0.080 0.144 –

0.005893 0.005892 0.005890 0.005886

0.017 0.034 0.068 –

0.051 0.1028 0.2037 –

The grid refinement ratio (α) in each direction is 1.4142. The relative error ( ) and Grid Convergence Index (GCI) are calculated using two consecutive grid size. The  j and j +1 represents course f −f f −f and fine grids, respectively. = j f j j+1 × 100. GC I (%) = 3 f j+1j (α nj+1 −1) × 100, where n = f −f

j J +1 log[ f j+1 − f j+2 ]/log(α) Where, #1 = 707 × 355,

#2 = 500 × 250,

#3 = 355 × 177,

#4 = 250 × 125

6.4 Code Validation

159

0.05

(a) 0.8

0.03

0.6

0.02

0.4

0.01

0.2

η

0.04

0 0.2

0.4

ξ

0.6

0.8

1

0.05 0

(b)

0.04

-0.01 -0.02

0.03

η

-0.03

0.02

-0.04 -0.05

0.01

-0.06

0

0

0.2

0.4

ξ

0.6

0.8

1

Fig. 6.2 Base flow contour plot for a streamwise Uξ and b wall-normal Uη velocity components for semi-cone angle α = 4◦ and Re = 698. The Reynolds number is calculated based on the body radius of the cone (a) at inlet of the domain. The velocity profile is interpolated to spectral grids for stability analysis. The actual domain height is taken 0.5 m in wall-normal direction for base flow computations. Here, ξ and η are the coordinates in the streamwise and wall-normal directions, respectively

6.4

Code Validation

To validate the global stability results for the incompressible flow over a circular cone, a blunt cone with a very small semi-cone angle, α = 10−12 (in degree) is considered. Thus, the geometry of the circular cylinder and a blunt is nearly similar due to very small semi-cone angle α as shown in Fig. 6.6. A rectangle epqr shows computational domain for the cylindrical coordinates and efgh for spherical coordinates. L x and L r are the streamwise domain length for the cylindrical and spherical coordinates, respectively. The stability equations for the axisymmetric flow over a circular cylinder are derived in polar cylindrical coordinates while that of a circular cone in spherical coordinates. The Reynolds number is computed based on the body radius (a) at the inflow and free-stream velocity for both cases. The domain size in streamwise and wall-normal directions are also the same. Therefore, Dirichlet boundary conditions are applied on disturbance amplitudes in normal direction at wall surface and free-stream. The global stability results for the axisymmetric boundary layer on a circular cylinder are already validated against the local stability results for axisymmetric (N = 0) as well as for helical mode (N = 1) with the Robin and periodic

160

6 Axisymmetric Boundary Layer … 0.02

ξ=0.2 ξ=0.4 ξ=0.6

0.015

η

ξ=0.8

0.01

0.005

(a) 0 0

0.2

0.6

0.4

0.8

1

Uξ 0.02

η

0.015

0.01 ξ=0.2 ξ=0.4

0.005

ξ=0.6

(b)

0 -0.04

ξ=0.8

-0.03

-0.02 Uη

-0.01

0

Fig. 6.3 Base flow velocity profile at different streamwise locations for a streamwise velocity Uξ and b wall-normal velocity Uη for same as shown in Fig. 6.2

boundary conditions. So here, first, we compared global stability results of cone and cylinder with the Robin and periodic boundary conditions. We also compared the global stability results of circular cone and cylinder with Dirichlet and extrapolation conditions at the inlet and outlet. Robin and periodic boundary conditions are considered for the blunt cone with very small semi-cone angle α with the same Reynolds number, streamwise wavenumber and streamwise location of a circular cylinder. Figure 6.7 shows the comparison of eigenspectrum, and Fig. 6.8 shows the comparison of the real parts of eigen functions for the least stable eigenmode. The resulting eigenspectrum and eigenfunctions are in agreement for the axisymmetric boundary layer formed on a circular cylinder and cone. Figure 6.9 shows the comparison of the eigenspectrum for the global stability analysis of the axisymmetric boundary layer over a circular cone and cylinder for N = 0 and Re = 1000 with the extrapolation conditions at outflow and Dirichlet conditions at the inflow. The discrete and continuous part of the spectrum is in excellent agreement with each other. Figure 6.10 shows the comparison of the real part of the eigenfunctions for the least stable eigenmode at the same streamwise location. The eigenfunctions are also in good agreement with each other.

6.4 Code Validation

161

3 α=2o α=4o

2.5

α=6o α=8o

S

2 1.5 1 0.5 0

(a) 0.2

0.6

0.4

1

0.8

x 3 α=2o α=4o

2.5

α=6o α=8o

S

2 1.5 1 0.5 0

(b) 0.2

0.4

0.6

0.8

1

x

Fig. 6.4 Variation of the transverse curvature (S = δ/a) in the streamwise direction for different semi-cone angles (α) for a Re = 174 and b Re = 872 40 α=0.25o α=0.50o

δ/a

30

α=0.75o

20

10

0

0.2

0.6

0.4

0.8

1

x

Fig. 6.5 Variation of the transverse curvature (S = δ/a) in the streamwise direction for a small semi-cone angles (α) for U∞ = 0.02 m/s

162

6 Axisymmetric Boundary Layer … g

f

q

p

h r

e a

X

O

Lx Lr

Fig. 6.6 Computational domain for global stability analysis of axisymmetric boundary layer over a circular cylinder and cone with very small semi-cone angle α. The domain is modelled in cylindrical coordinates (dashed-line) for flow over cylinder and spherical coordinates (solid line) for flow over a cone 0

ωr

−0.2 −0.4 −0.6 −0.8 −1 0.2

Cylincer Cone 0.4

0.6 ωi

0.8

1

Fig. 6.7 Comparison of the eigenspectrum for the global stability analysis of axisymmetric boundary layer over a circular cone and cylinder for axisymmetric mode(N = 0) with Robin and periodic boundary conditions at inlet and outlet. The Reynolds number based on the body radius at inflow is Re = 12,439

6.5 Boundary Conditions Evaluation Global stability analysis is performed for an axisymmetric mode (N = 0) at Re = 349 with the same inflow boundary conditions (Dirichlet type) and two different outlet conditions, Linear extrapolation type and Neumann boundary conditions. The streamwise domain length is taken as L r = 214.9. Figure 6.11 compares the spectrum’s discrete part for two different outlet boundary conditions. The spectra with both boundary conditions are not overlapping with each other. Thus as discussed in the previous chapter, obtained eigenvalues are dependent on the type of exit boundary conditions. It shows that the temporal growth rate ωi is the same for the least stable

6.5 Boundary Conditions Evaluation

163

5

5 Cylinder Cone

4

3

r, arc

r, arc

4

3

2

2

1

1

0 0

(a) 1

0.5 u , ur

Cylinder Cone

0 0

(b) 0.05

0.1 v, uθ

0.15

Fig. 6.8 Comparison of the real parts of the eigenfunctions a streamwise u and u r and b wallnormal v and u θ for axisymmetric boundary layer on circular cone and cylinder cone cylinder

0 -0.1 -0.2

ωi

Fig. 6.9 Eigenspectrum comparison of the global stability analysis for the incompressible boundary layer flow over the circular cylinder and cone. The Reynolds number is Re = 1000 based on the body radius

-0.3 -0.4 -0.5 -0.6 0.1

0.2

0.3 ωr

0.4

0.5

eigenmode and the variation in ωi value increases with the increase in frequency ωr . The least stable eigenmodes near the frequency ωr = 0.005 as marked by the square in Fig. 6.11 is selected, and spatial eigenmodes u r and u θ are plotted at the same streamwise location at r = 90 as shown in Fig. 6.12. It shows that the qualitative features are the same with both the exit boundary conditions and we have opted for Linear extrapolated type condition because it is less restrictive, as Theofilis [19] suggested.

164

6 Axisymmetric Boundary Layer …

5

5 cone cylinder

3

3

2

2

1

1

0

(a) 0

cone cylinder

4

r,arc

r,arc

4

(b)

0.01 0.02 u,u r

0.03

0 −3

−2

−1 v,uθ

0

1 −3

x 10

Fig. 6.10 Eigenfunction comparison of the global stability analysis for the incompressible boundary layer flow over the circular cone and cylinder with extrapolation boundary condition at outflow for Re = 1000 at streamwise distance x = r = 90 a streamwise disturbances, u and u r and b wallnormal disturbances, v and u θ . Here u, v and u r , u θ are in cylindrical and spherical coordinates, respectively −0.03 Neumann bc Extrapolated bc

−0.04

ωi

−0.05 −0.06 −0.07 −0.08 0

0.05

ωr

0.1

0.15

Fig. 6.11 Comparison of the eigenspectrum resulting from the global stability analysis of the boundary layer over a circular cone for semi-cone angle α = 2◦ , N = 0 and Re = 349 for different outlet boundary conditions

6.6 Influence of the Domain Length Figure 6.13 represents the discrete spectrum (TS modes) resulting from the global stability analysis for axisymmetric mode (N = 0), semi-cone angle α = 2◦ and Re = 349. The difference in the spectrum depends on the streamwise domain size [2]. The discretization of the frequency comes from the truncation of the domain. Figure 6.13 shows that the space between the frequency decreases with the increase in domain length and approaches zero ideally with infinity domain length. In order to check

6.7 Global Stability Results

165

15

15 Neumann bc Extrapolated bc

Neumann bc Extrapolated bc

10

arc

arc

10

5

5

0

(a) 0

0.5 ur

1

0

(b) −0.1

−0.05 uθ

0

Fig. 6.12 Eigenfunctions comparison for the global stability analysis of the incompressible boundary layer over a circular cone for α = 2◦ , N = 0 and Re = 349 at streamwise location r = 92 for a u r , b u θ as marked by square in Fig. 6.11

Lr=214.9 Lr=243.6

−0.03 −0.04

ωi

Fig. 6.13 Comparison of the eigenspectrum resulting from the global stability analysis of the boundary layer over a circular cone for semi-cone angle α = 2◦ , N = 0 and Re = 349 for different domain length

−0.05 −0.06 −0.07 0

0.05

0.1 ωr

0.15

the effect of domain length on the global stability results, two different streamwise domains L r = 214.9 and L r = 243.6, are selected. Figure 6.13 shows that there is a variation in both spectra. Thus, the obtained eigenvalues are dependent on the streamwise domain size (Fig. 6.14).

6.7 Global Stability Results In the present analysis, the Reynolds number varies from 174 to 1047 and azimuthal wavenumber from 0 to 5 for different semi-cone angles α. The Reynolds number is computed based on a cone radius(a) at the inlet and free-stream velocity (U∞ ). The streamwise length is normalized by cone radius (a). The domain size in polar(wall-

166

6 Axisymmetric Boundary Layer …

Lr=214.9 Lr=243.6

20

Lr=214.9 Lr=243.6

20 15

arc

arc

15

10

10

5

5

(a) 0 0

0.5 |ur|

(b)

1

0 0

0.05 0.1 |uθ|

Fig. 6.14 Eigenfunction comparison for the global stability analysis of the incompressible boundary layer flow over a circular cone for α = 2◦ , N = 0 and Re = 349 at streamwise location r = 141 for a u r , b u θ as marked by square in Fig. 6.13 Table 6.2 The Grid Convergence study for two leading eigenvalues ω1 and ω2 for Re = 349, azimuthal wave number N = 1 and semi-cone angle α = 2◦ for different grid size Mesh n×m ω1 ω2 Error (%) #1

121× 121

#2

107× 107

#3

93× 93

0.01528– 0.03609i 0.01528– 0.03610i 0.01530– 0.03616i

0.02869– 0.04375i 0.02874– 0.04374i 0.02881– 0.04379i

0.174 0.243 –

The grid refinement ratio in each direction is 1.14. The maximum relative error is shown here The dimensions of computational domain in spherical coordinates are L r = 214.9 and L θ = 12◦ for global stability computations

normal) direction is taken as L θ = 12◦ . The streamwise (radial) length is taken as 214.9. The number of collocation points taken in radial and polar directions is n = 121 and m = 121, respectively. The general eigenvalues problem is solved using Arnoldi’s iterative algorithm. The computed eigenvalues are accurate up to three decimal points. The additional lower resolution cases were also run to verify the monotonic convergence of the solution. Arnoldi’s iterative algorithm is used to solve the general eigenvalues problem. Heavy sponging is applied at the outflow boundary to prevent spurious reflection. The selected global eigenmodes are also checked for the spurious mode. A grid convergence study was performed to check the accuracy level of the solution and the appropriate grid size. Table 6.2 shows the values of two leading eigenvalues computed for Re = 349, N = 1 and semi-cone angle α = 2◦ using three different grid

6.7 Global Stability Results

167

sizes. The grid resolution was successively improved by a factor of 1.14 in radial and polar directions. The real and imaginary parts of the eigenvalues show monotonic convergence of the solution with the increased resolution. Table 6.2 n and m indicates the number of collocation points in the radial and polar directions. The relative error is computed between two consecutive grid sizes for real and imaginary parts. The largest associated error among both the eigenmodes is considered. The Mesh #1 is used for all the computations for stability analysis results reported here.

6.7.1

Semi-cone Angle α = 2◦

Figure 6.15 shows the eigenspectrum of axisymmetric mode (N = 0) for Re = 349 and semi-cone angle α = 2◦ . The eigenmodes marked by squares and circles are called stationary and oscillatory mode. The stationary mode has a complex frequency ω = 0 − 0.02667i. The global mode is temporally stable because of ωi < 0 Moreover, hence the amplitudes of the disturbances decay over time. Figure 6.16a, b presents the two-dimensional spatial structure of the eigenmodes for radial (u r ) and polar (u θ ) velocity disturbances. At the inflow, the magnitude of the disturbance amplitudes is zero. The disturbance amplitudes evolve with the time in the flow domain and move in the streamwise direction towards the downstream. The size and magnitude of the disturbances grow as they move towards downstream. The magnitudes of the u r disturbances are one order higher than that of u θ . The polar disturbances u θ contaminates the flow field up to a large extent than that of u r disturbances. However, its magnitude is very small in comparison with u θ . Figure 6.17 shows the spatial structure of the oscillatory eigenmode. The associated complex frequency for this mode is ω = 0.008154 − 0.03937i. This oscillatory global mode is also temporally stable, because ωi < 0. The variation of the amplitude is not monotonic for this eigenmode. The wave-like nature of the disturbances is found in this mode. The oscillatory modes evolved in the flow field grow in size and magnitude while moving downstream (Fig. 6.18). Figure 6.19 shows the two-dimensional mode structure of the u r and u θ for the ωr = 0.1775. The streamwise domain length is 214.9. The

Fig. 6.15 Eigenspectrum for axisymmetric mode (N = 0) and Re = 349 for semi-cone angle α = 2◦

0

ωi

-0.05

-0.1

-0.15

0

0.05

0.1 ωr

0.15

0.2

168

6 Axisymmetric Boundary Layer … (a)

0.0000

y

50

-0.0100

40

-0.0200

30

-0.0300

20

-0.0400

10

-0.0500 -0.0600 50

100

150

200

x

(b)

0.0020

50

y

0.0010 40

0.0000

30

-0.0010 -0.0020

20

-0.0030 10

-0.0040 50

100

150

200

x

Fig. 6.16 Contour plot of the real parts of a streamwise u r and b wall-normal u θ velocity disturbances for stationary eigenmode, ω = 0 − 0.02667i for semi-cone angle α = 2◦ marked by square in Fig. 6.15

disturbances are observed to evolve near the wall surface and decay exponentially at free-stream. The typical length scale of the wavelet structure decreases with the increase in the frequency ωr . It also demonstrates that the region of contamination reduces with the increase in frequency. The wavelet structure’s length scale is small for the increased frequency, as shown in Fig. 6.19.

6.7.2

Semi-cone Angle α = 4◦

Figure 6.20 shows the spectrum for helical mode N = 1, Re = 523 and semicone angle α = 4◦ . The most unstable oscillatory mode has an eigenvalue ω = 0.01942 − 0.05512i. The global mode is temporally stable because of the largest ωi < 0. Figure 6.21 shows the two-dimensional mode structure for the N = 1, Re = 523 and semi-cone angle α = 4◦ . It has been observed that the disturbances evolved in the flow field at an earlier stage than that of a cone with a semi-cone angle α = 2◦ . The magnitudes of the disturbance velocity components and the region of contamination in the polar direction are higher than that of α = 2◦ . The largest ωi has reduced with the increased semi-cone angle α for a given Reynolds number makes the global modes more stable (Fig. 6.22).

6.7 Global Stability Results

169

(a)

0.0080

50

0.0060 0.0040

40

y

0.0020 30

0.0000 -0.0020

20

-0.0040 10

-0.0060 50

100

150

200

x

(b)

-0.0008

50

y

40

-0.0010

30 -0.0012 20 -0.0014

10 50

100

150

200

x

Fig. 6.17 Contour plot of the real parts of a streamwise u r and b wall-normal u θ velocity disturbances for oscillatory eigenmode, ω = 0.008154 − 0.03937i for semi-cone angle α = 2◦ marked by ellipse in Fig. 6.15 −3

x 10

(a)

(b) 5

5

0

−10

θ=2.004 θ=2.06 θ=2.18 θ=2.68 θ=4.78 θ=11.11 160

uθ

ur

0

−5

−4

x 10

−5

−10 180

200 r

220

240

θ=2.004 θ=2.06 θ=2.18 θ=2.68 θ=4.78 θ=11.11 160

180

200 r

220

240

Fig. 6.18 Variation of disturbance amplitudes in streamwise direction for a streamwise u r and b wall-normal u θ for oscillatory eigenmode, ω = 0.008154 − 0.03937i and Re = 349 for semi-cone angle α = 2◦ marked by circle in Fig. 6.15

170

6 Axisymmetric Boundary Layer … -0.0300

(b) 50

40

40

-0.0400

30

0.0150 0.0100 0.0050 0.0000

y

y

(a) 50

30 -0.0050

20

20

-0.0100

10

10

-0.0150

-0.0500 50

100

150

-0.0200 50

200

100

150

200

x

x

Fig. 6.19 Contour plot of the real parts of a streamwise u r and b wall-normal u θ velocity disturbances for oscillatory eigenmode, ω = 0.1775 − 0.07512i for semi-cone angle α = 2◦ Fig. 6.20 Eigenspectrum for helical mode (N = 1) and Re = 523 for semi-cone angle α = 4◦

0 -0.05 -0.1

ωi

-0.15 -0.2 -0.25 -0.3 -0.35 0.05

6.7.3

0.1

0.15

0.2 ωr

0.25

0.3

0.35

Semi-cone Angle α = 6◦

Figure 6.23 shows the spectrum for helical mode N = 1, Re = 698 and semicone angle α = 6◦ . The most unstable oscillatory mode has an eigenvalue ω = 0.02539 − 0.07707i. The global mode is temporally stable because of the largest ωi < 0. Figure 6.24 shows the two-dimensional mode structure for the N = 1, Re = 698 and semi-cone angle α = 6◦ . The structure of the discrete part of the spectrum disturbs with the increase in semi-cone angle α. The magnitudes of the u r disturbance amplitudes are one order higher than that of u θ and u . With the increase in semi-cone angle α, the disturbances start to grow at an early stage. It has been observed that the disturbances evolved in the flow field at an earlier stage than that of a cone with a semi-cone angle α = 2◦ . The spatial structure of the disturbances is of similar nature. However, the damping rate of the global mode increases with the increase in the semi-cone angle (Fig. 6.25).

6.8 Temporal Growth Rate 30

171

(a)

-0.0060

y

25 -0.0080

20 15

-0.0100

10 5

-0.0120

20 30

40

60

x

80

100

120

(b) -0.0004

y

25 20

-0.0006

15 -0.0008

10 5

-0.0010

20

30

40

60

x

80

100

120

(c)

0.0040 0.0030

25

y

0.0020

20

0.0010

15

0.0000

10

-0.0010

5

-0.0020

20

40

60

x

80

100

120

Fig. 6.21 Contour plot of the real parts of a streamwise u r and b wall-normal u θ velocity disturbances for stationary eigenmode, ω = 0.01942 − 0.05512i for semi-cone angle α = 4◦ marked by square in Fig. 6.20

6.8

Temporal Growth Rate

Figure 6.26 shows the temporal growth rate of the least stable eigenmodes for different Reynolds number and semi-cone angles α. The growth rate of eigenmodes increases with the increase in Reynolds number for all azimuthal wavenumbers (N) and semi-cone angles (α). The least stable eigenmodes are negative for the Range of Reynolds, number and semi-cone angles. Hence, the global modes are stable. The global modes with helical mode, N = 1 are the least stable for α = 2◦ , α = 4◦ and α = 6◦ . The damping rate of global modes with N = 2 is higher than N = 0 for α = 2◦ . At smaller Re, N = 0 is more stable than N = 2; however, at higher Re, N = 2 is more stable then N = 0 for semi-cone angle α = 4◦ . The helical modes

172

6 Axisymmetric Boundary Layer … x 10

6 4

ur

2

−3

−3

1

θ=4.005 θ=4.039 θ=4.114 θ=4.440 θ=5.208 θ=10.70

0.5 0

0

θ=4.005 θ=4.039 θ=4.114 θ=4.440 θ=5.208 θ=10.70

−0.5

−2 −4

−1

−6 −8 60

x 10

uθ

8

(a) 70

80

4

x 10

3 2

−1.5 60

(b) 120

100

80 r

−3

θ=4.005 θ=4.039 θ=4.114 θ=4.440 θ=5.208 θ=10.70

uφ

1

120

110

100

90 r

0 −1 −2 −3 60

(c) 70

80

90 r

100

110

120

Fig. 6.22 Variation of disturbance amplitudes in streamwise direction for a streamwise u r , b wall-normal u θ and c azimuthal u φ for oscillatory eigenmode, ω = 0.01942 − 0.05512i and Re = 523 for semi-cone angle α = 4◦ marked by circle in Fig. 6.20 Fig. 6.23 Eigenspectrum for helical mode (N = 1) and Re = 698 for semi-cone angle α = 6◦

0.1 0 -0.1

ωi

-0.2 -0.3 -0.4 -0.5 0

0.1

0.2

0.3 ωr

0.4

0.5

0.6

6.8 Temporal Growth Rate 25

173

(a)

0.0500

y

20 0.0400

15

0.0300

10

0.0200

5

0.0100 0.0000

20 25 20

40

x

60

80

(b)

-0.0005

y

-0.0010

15

-0.0015

10

-0.0020

5

-0.0025

20 25

40

x

60

80

0.0000

(c)

-0.0005

20

y

-0.0010 -0.0015

15

-0.0020

10

-0.0025 -0.0030

5

-0.0035 -0.0040

20

40

x

60

80

Fig. 6.24 Contour plot of the real parts of a streamwise u r and b wall-normal u θ velocity disturbances for stationary eigenmode, ω = 0.02539 − 0.07707i for semi-cone angle α = 6◦ marked by square in Fig. 6.23

N = 3, 4 and 5 have larger damping rates than that of N = 0, 1 and 2 for all Re and α. The global eigenmodes are more stable at a higher semi-cone angle α. The transverse curvature reduces in the streamwise direction with the increase in Reynolds number for the same α. The favourable pressure gradient is higher at a higher semi-cone angle α. The global modes are more stable at a higher semi-cone angle, proving that a favourable pressure gradient dampens the global modes.

174

6 Axisymmetric Boundary Layer … −3

0.04

θ=6.004 θ=6.039 θ=6.114 θ=6.44 θ=7.20 θ=12.70

0.035 0.03 0.025 0.02

3

x 10

(b)

2 1

uθ

ur

0

0.015

−1

0.01

−2 0.005 0

−3

(a)

−0.005 40

50

60 r

70

0.5

x 10

80

θ=6.004 θ=6.039 θ=6.114 θ=6.44 θ=7.20 θ=12.70

−4 40

50

60 r

70

80

−3

(c)

0 −0.5

uφ

−1 −1.5 −2 −2.5 −3 −3.5 40

θ=6.004 θ=6.039 θ=6.114 θ=6.44 θ=7.20 θ=12.70 50

60 r

70

80

Fig. 6.25 Variation of disturbance amplitudes in streamwise direction for a streamwise u r , b wall-normal u θ and c azimuthal u φ for oscillatory eigenmode, ω = 0.02539 − 0.07707i and Re = 698 for semi-cone angle α = 6◦ marked by circle in Fig. 6.23

6.9

Spatial Amplification Rate

The global temporal modes exhibit growth/decay in the streamwise direction while moving towards downstream. This growth/decay at a different streamwise station can be quantified by spatial amplification rate (A x ). The spatial amplification rate (A x ) shows the growth of all the disturbances together in the streamwise direction. Figure 6.27 shows the A x for different azimuthal wavenumbers (N) and semi-cone angles (α) for Re = 349. The spatial growth rate (A x ) increases with the increase in semi-cone angle (α) for all the azimuthal wavenumbers for given Reynolds number. We computed it for Re = 174 to Re = 1047; however, the result is presented for Re = 349 only. As the semi-cone angle (α) increases, the growth rate of the disturbances increases in the streamwise direction. Hence, the flow becomes spatially more unstable at a higher semi-cone angle. For N = 0, 1 and 2, the disturbances exhibit a higher spatial growth rate at a small Reynolds number; however, for N = 3, 4 and 5 the spatial growth rate is high for higher Reynolds numbers.

6.9 Spatial Amplification Rate

175

-0.02

−0.08 −0.1

-0.04 -0.06

ωi

ωi

−0.12

-0.1 -0.12

−0.14

N=0 N=1 N=2 N=3 N=4 N=5

-0.08

(a) 200

400

600 Re

800

N=0 N=1 N=2 N=3 N=4 N=5

−0.16 −0.18 −0.2 −0.22

1000

(b) 200

400

600 Re

800

1000

-0.08 -0.1

ωi

-0.12 -0.14 -0.16

N=0 N=1

-0.18

N=2

-0.2 -0.22 -0.24

N=3 N=4

(c)

N=5

1000

800

600 Re

400

200

Fig. 6.26 Variation in temporal growth rate ωi with the Reynolds number for different semi-cone angles a α = 2◦ , b α = 4◦ , c α = 6◦ -3

2.5

0.014

x 10

α=2o

α=2o α=4o

0.012

o

α=4

2

α=6o

0.01

α=6o

A(x)

A(x)

1.5 1

0.008 0.006 0.004

0.5

0.002

(a)

0 0.3

0.4

0.5 r

0.6

(b)

0 0.3

0.7

0.4

0.5 r

0.6

0.7

0.6

0.7

-3

0.05 4

α=2o

x 10

α=2o

α=4o

0.04

α=4o

o

α=6

3

α=6o

A(x)

A(x)

0.03 0.02

2

1

0.01

(d)

(c)

0 0.3

0.4

0.5 r

0.6

0.7

0 0.3

0.4

0.5 r

Fig. 6.27 Variation of spatial growth rate A x in the streamwise direction for different semi-cone angles α for Re = 349. a N = 0, b N = 1, c N = 2 and d N = 3

176

6.10

6 Axisymmetric Boundary Layer …

Summary

The linear global stability analysis of the boundary layer formed on a circular cone is performed. The combined effect of transverse curvature and pressure gradient has been studied. The global temporal modes are computed for the axisymmetric boundary layer on a circular cone for the range of Reynolds the number from 174 to 1046 with azimuthal wavenumbers, N from 0 to 5 and semi-cone angles α = 2◦ , 4◦ and 6◦ . The largest imaginary part (ωi ) of the computed global modes is negative for all the Reynolds numbers (Re), azimuthal wavenumbers (N) and semi-cone angles (α) considered here. Hence, the global modes are temporally stable. The wave-like behaviour of the eigenmodes is found in the streamwise direction. The wavelength of the wavelet structure reduces with the increase in the amplitudes frequency (ωr ). The 2D spatial structure of the global modes shows that the disturbance amplitudes size and magnitude increase while moving downstream. The damping rate of the disturbances increases with the increase in the semi-cone angle (α) from 2◦ to 6◦ . Thus, the global modes are more stable at the higher semi-cone angles (α). At the same time, with the increase in α from 2◦ to 6◦ , the spatial growth rate ( A x ) also increases at a given Reynolds number; thus, flow becomes spatially more unstable at higher semi-cone angles. The azimuthal wavenumbers N = 3, 4 and 5 have less temporal growth (ωi ) and spatial growth (A x ) compare to N = 0, 1 and 2. The azimuthal wavenumber N = 1 is the least stable for all the Re and α. The increase in the semicone angle (α) develops a favourable pressure gradient and reduces the transverse curvature effect. Thus, the favourable pressure gradient stabilizes the global modes, and the effect of transverse curvature reduces. Thus, the role of a favourable pressure gradient is more effective than that of transverse the curvature on flow stability.

References 1. Akervik, E., Ehrenstein, U., Gallaire, F., Henningson, D.: Global two-dimensional stability measure of the flat plate boundary layer flow. Eur. J. Mech. B/Fluids 27, 501–513 (2008) 2. Alizard, F., Robinet, J.C.: Spatially convective global modes in a boundary layer. Phys. Fluids 19, 114105 (2007) 3. Costa, B., Don, W., Simas, A.: Spatial resolution properties of mapped spectral Chebyshev methods. In: Proc. SCPDE: Recent Progress in Scientific Computing, pp. 179–188 (2007) 4. Drazin, P.G., Reid, W.H.: Hydrodynamic Stability. Cambridge University Press, Cambridge (2004) 5. Ehrenstein, U., Gallaire, F.: On two-dimensional temporal modes in spatially evolving open flow: The flat-plate boundary layer. J. Fluid Mech. 536, 209–218 (2005) 6. Fasel, H., Rist, U., Konzelmann, U.: Numerical investigation of the three-dimensional development in boundary layer transition. AIAA J. 28, 29–37 (1990) 7. Garrett, S.J., Peake, N.: The absolute instability of the boundary layer on a rotating cone. Eur. J. Mech. B/Fluids 26, 344–353 (2007) 8. Horvath, T.J., Berry, S.A., Hollis, B.R., Chang, B.A.: Boundary layer transition on slender cones in conventional and low disturbance mach 6 wind tunnels. AIAA J. 2743, 0153 (2002) 9. Mack, L.M.: Boundary layer linear stability theory. AGARD Report 709 (1984)

References

177

10. Malik, M.R.: Numerical methods for hypersonic boundary layer stability. J. Comput. Phys. 86(2), 376–412 (1990) 11. Malik, M.R., Poll, D.I.A.: Effect of curvature on three dimensional boundary layer stability. AIAA J. 23, 1362–1369 (1985) 12. Maslov, A.A., Mironov, S., Shiplyuk, A.A.: Hypersonic flow stability experiments. AIAA J., p. 0153 (2002) 13. Narayanan, V.: Stability and Transition in Boundary Layers: Effect of Transverse Curvature and Pressure Gradient. Ph.D. thesis, Jawaharlal Nehru Center for Advanced Scientific Research (2006) 14. Rao, G.N.V.: Mechanics of transition in an axisymmetric laminar boundary layer on a circular cylinder. J. Appl. Math. Phys. 25, 63–75 (1974) 15. Roache, P.J.: A method for uniform reporting of grid refinement studies. J. Fluid. Eng. 116(3), 405–413 (1994) 16. Stetson, K.F., Kimmel, R.: On the breakdown of a hypersonic laminar boundary layer. AIAA J. 93, 0896 (1993) 17. Stetson, K.F., Kimmel, R.L.: On hypersonic boundary layer stability. AIAA J. 92, 0737 (1993) 18. Swaminathan, G., Shahu, K., Sameen, A., Govindarajan, R.: Global instabilities in diverging channel flows. Theor. Comput. Fluid Dyn. 25, 25–64 (2011) 19. Theofilis, V.: Advances in global linear instability analysis of nonparallel and three dimensional flows. Prog. Aerosp. Sci. 39, 249–315 (2003) 20. Tutty, O.R., Price, W.G.: Boundary layer flow on a long thin cylinder. Phys. Fluids 14, 628–637 (2002) 21. Vinod, N., Govindarajan, R.: Secondary instabilities in incompressible axisymmetric boundary layers: Effect of transverse curvature. J. Fluid Eng. 134 (2012)

Chapter 7

Boundary Layer on an Inclined Flat Plate

7.1 Introduction In this chapter, global stability analysis of the flat-plate boundary layer with a favourable and adverse pressure gradient is studied. The plate makes an angle (β) with the incoming flow at the inflow boundary. The streamwise pressure gradient (d p/dx) is zero for β = 0, favourable for β > 0 and adverse for β < 0. The schematic diagram of the boundary layer with a favourable and adverse pressure gradient is shown in Figs. 7.1 and 7.2, respectively. The effect of pressure gradient on laminar flow instability is particularly interesting because it is the passive device for controlling the boundary layer. The important engineering applications where streamwise pressure gradient exists are turbomachinery, compressors, aeroplane wings, etc. The flow transition on compressor blades and aeroplane wings occurs in a higher adverse pressure gradient region. The amplification rate of disturbance waves is higher in a boundary layer with an adverse pressure gradient. However, in the case of a low-pressure turbine and sometimes on compressor blades, where a high level of free-stream turbulence is produced upstream, a transition occurs in the favourable pressure gradient. The boundary layers developing on the solid surfaces are generally laminar near the leading edge. The Reynolds number also increases as the distance increases from the leading edge. The Reynolds number at which the transition to turbulence occurs is the critical Reynolds number. The region over which the flow transition occurs is known as the transition zone or transition length. Tollmien-Schlichting (T-S) wave mechanics characterizes the transition at a very low free-stream turbulence level. The process of transition initiates with the random amplification of the small disturbances. The standard procedure to predict the transition onset is to compute the growth of the small disturbances within the laminar region. The instability/growth of the small disturbances is the very first step of the transition process. Many factors affect the transition process, e.g. free-stream turbulence, streamwise pressure gradient (PG), streamwise curvatures, surface roughness, etc. Knowledge of the transition process is beneficial where the turbulence is necessary to be avoided, but also where the turbu© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Bhoraniya et al., Global Stability Analysis of Shear Flows, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-19-9574-3_7

179

180

7 Boundary Layer on an Inclined Flat Plate

Fig. 7.1 Schematic diagram of boundary layer on a inclined flat plate. The incident angle β develops favourable pressure gradient in the streamwise direction

Fig. 7.2 Schematic diagram of boundary layer on a inclined flat plate. The incident angle β develops adverse pressure gradient in the streamwise direction

lence might be desirable to promote, for instance, better fluid mixing to manage turbulence efficiently. There is a strong relationship between transition and flow separation, a non-desirable phenomenon in most engineering applications. Therefore, understanding the transition may lead to better flow separation control. The study of the PG effect on flow instability is of specific interest because it is one of the passive flow control devices. The previous study on the effect of PG has shown that FPG increases the critical Reynolds number of incompressible boundary layer while APG have the opposite effect. Flow through turbomachinery and aeroplane wings are the engineering applications where streamwise PG exists. The necessary engineering applications wherever streamwise PG exists are turbomachinery, aeroplane wings, etc. The linear stability analysis of parallel flow shows that the disturbances amplify faster in the case of boundary layers with adverse pressure gradients (APG). Sometimes, the transition can jointly happen within a favourable pressure gradient (FPG) zone, i.e. low-pressure turbines and compressors blades with high free-stream turbulence made upstream. Obremski et al. have shown that an FPG stabilizes an incompressible boundary layer while APG produces an opposite effect [32]. Flight test data reported by Driest [7] indicates that an increase in FPG increases the transition Reynolds number. The flow with very strong APG is known to have an inflexion point and is inviscid unstable. In inviscid flows, the velocity profile with inflexion point can always be unstable by the Rayleigh criterion. The streamwise PG profoundly affects the growth of small disturbances. An FPG leads to a fuller velocity profile with a relatively lower shape factor and thus has a stabilizing impact. In contrast, an APG results in a more

7.1 Introduction

181

significant amplification rate of disturbance waves. Saxena and bose [33] investigated that the FPG stabilizes the flow and APG destabilizes the flow. A strong APG promotes boundary layer separation and speeds up the process of transition to turbulence. Such boundary layers are much more unstable than flatplate boundary layers. Corke and Gruber’s [10] have experimentally studied the resonant growth of a triad of instability waves consisting of a plane T-S modes and a pair of oblique modes with equal and opposite wave angles are undergoing a sub-harmonic transition in Falkner-Skan boundary layers with APG parameter in the range 0 >= β H >= −0.09. Although large instabilities did not occur in their experiments, the transition process differed considerably from the Blasius case in many respects. For example, the streamwise extent of the amplitude saturation was extremely short compared to the Blasius layer, and the maximum amplitudes reached by the sub-harmonic mode in their cases were twice as large as those in a Blasius layer with comparable initial conditions. The simulations of Liu and Maslowe [23] reveal sub-harmonic three-dimensional waves to be the most dangerous in the APG. Here also, amplification rates are much higher than in the Blasius flow. Consequently, transition sets in at a far lower Reynolds number in the decelerating boundary layers. The transition zone is much shorter, and the transition onset was upstream in APG boundary layers. Abu and Gostelow et al. [1, 17] have found experimentally higher spot inception and spreading rates in the decelerated flows. Vinod and Govindarajan [29, 40] showed that there exists a direct connection between the pattern of a breakdown of turbulent spots and laminar instability characteristics under the strong APG boundary layers. Narasimha and Seifert and Hodson [30, 35] experimentally found that the growth of the turbulent spots in the downstream direction is self-similar irrespective of the type of PG. It maintains an arrowhead shape in the top view and follows self-similar growth of the spot size. Maslowe and Spiteri [28] reported the behaviour of the Eigen solutions of a continuous spectrum for a boundary layer subjected to a PG in the flow direction. They found larger disturbance amplitudes compare to Blasius flow in an APG boundary layer. The possibilities of the secondary instabilities are higher close to the edge of the boundary layer because of the massive shear rate. The spatial amplification rate (αi ) is smaller than that of Blasius flow. The magnitude of eigenfunctions could be larger for Blasius flow even once the PG is favourable. However, the transition Reynolds number is higher; thus, αi is also significant. Zurigat et al. [43], in their investigation of compressible boundary layers, found that FPG has stabilization and APG has a destabilization effect on the 2D second waves. The effectiveness of the FPG reduces at hypersonic Mach number. It is clear that FPG pressure gradients are stabilizing, whereas APG destabilizes the boundary layers. Thus, boundary layers subjected to FPG and APG are stable and unstable, respectively. The band of frequencies widens to amplify the disturbances with APG for incompressible boundary layers. Franko and Lele [14] found completely different transition eventualities with APG for high-speed boundary layers. They applied APG through the free-stream condition and found that APG failed to modify the transition method. However, it quickens the transition method and ends up in the upper rate for first and second mode instability. The transition was analogous to ZPG for the soft APG boundary layers. Zhang et al. [42] performed global

182

7 Boundary Layer on an Inclined Flat Plate

stability computations for 2D flow past an inclined triangular cylinder. They found that the spatial structure of the disturbances is nearly similar for α 0. Figures 7.11 and 7.12 show the two-dimensional spatial structure of the eigenmodes for β = 0 (zero

194

7 Boundary Layer on an Inclined Flat Plate −0.02 −0.022

ωi

−0.024 −0.026 −0.028 −0.03 −0.032 0

β=0 β=0.0222 β=0.0444 β=0.0667 0.05

ωr

0.1

0.15

Fig. 7.10 Eigenspectrum resulting from the global stability analysis of the 2D flat-plate boundary layer for Re = 340 with favourable pressure gradients. β = 0 is for zero pressure gradient and β = 0.0222, β = 0.0444 and β = 0.0667 is for favourable pressure gradient

r

20 −0.0300 −0.0400 −0.0500

10 0

150 200 250 300 350 400 450 500 x 0.0150 0.0100 0.0050 0.0000 −0.0050

r

20 10 0

150 200 250 300 350 400 450 500 x

Fig. 7.11 Contour plots of the real part of u (upper) and v (lower) disturbance velocity components for Re = 340 and β = 0. The associated frequency ωr = 0.0708

pressure gradient) and β = 0.0667 (positive pressure gradient). The Re and ωr are 340 and 0.0708, respectively, the same for both the eigenmode. The two-dimensional spatial structure of the disturbance amplitudes is similar. The magnitudes of the wavelet structure of disturbance amplitudes are larger for zero pressure gradient (β = 0) than for favourable pressure gradient (β = 0.0667). The wavelet structure of the disturbance amplitudes has been found for u and v disturbance velocity components. The magnitude of the u disturbance component is almost one order higher than that of the v disturbances. The disturbances are observed to evolve near the wall surface and decay exponentially in the wall-normal direction towards the free-stream. The amplitudes of u and v disturbance velocity components evolve in the flow field and grow in size and magnitude while moving towards the downstream with the base flow. It suggests that the flow is convectively unstable. The disturbance amplitudes’ spatial structure is similar to zero and positive pressure gradients.

r

7.4 Global Stability Results

195

20

−0.0100

10

−0.0150

0

−0.0200

10

−0.0040 −0.0060 −0.0080 −0.0100

r

150 200 250 300 350 400 450 500 550 x 20

0

150 200 250 300 350 400 450 500 550 x

Fig. 7.12 Contour plots of the real part of u (upper) and v (lower) disturbance velocity components for Re = 340 and β = 0.0667. The associated frequency ωr = 0.0708 4

x 10

5

β=0 β=0.0222 β=0.0444 β=0.0667

2

0

v

u

−3

x 10

−5

β=0 β=0.0222 β=0.0444 β=0.0667

0

−2

(a)

−4 400

(b)

450

500 x

−5 400

450

500 x

Fig. 7.13 Variation of the disturbance amplitudes in the streamwise direction for various plate angles β at y = 0.114 for Re = 340. Here, β=0 is for zero pressure gradient and β = 0.0222, 0.0444 and 0.0667 for favourable pressure gradient. a u disturbance, b v disturbance

Figure 7.13 shows the variation of disturbance amplitudes in the streamwise direction at y = 0.114 for Re = 340 and ωr = 0.0707 with different plate angle β. The magnitudes of the disturbance amplitudes increase in the streamwise direction. The wave-like nature of the disturbances is found in the streamwise direction. The comparison for different angle β shows that the magnitude of the disturbance amplitudes decreases as plate angle β increases from 0◦ to 6◦ , which increases favourable pressure gradient. Figure 7.14 shows the variation of the amplitudes of disturbance velocity in y-direction at streamwise location x = 461. The nature of variation for the zero and positive pressure gradient is almost similar. At wall, the magnitudes of the disturbances amplitudes are zero due to viscous effect, then gradually increase in the y-direction and finally vanish at the far-field. It has been observed from Fig. 7.14 that for u and v disturbances the magnitudes of the disturbances in the streamwise

196

7 Boundary Layer on an Inclined Flat Plate

25

20

25

(a)

β=0 β=0.0222 β=0.0444 β=0.0667

(b)

β=0 β=0.0222 β=0.0444 β=0.0667

20

15

y

y

15

10

10

5

5

0 −2

−1

0 u

1 −3

x 10

0

−5

10

5

0 v

−4

x 10

Fig. 7.14 Variation of the disturbance amplitudes in the y-direction at streamwise location x = 461 for various plate angles β for Re = 340. Here, β=0 is for zero pressure gradient and β=0.0222, 0.0444 and 0.0667 for favourable pressure gradient. a u disturbance, b v disturbance

direction reduce with the increase in favourable pressure gradient. Thus, the development of the favourable pressure gradient in the streamwise direction reduces the spatial growth rate in the streamwise direction. The global eigenmodes show spatial growth/decay in the streamwise direction while moving towards the downstream. The overall effect of all the disturbances together in the streamwise direction can be quantified by computing the spatial amplification rate ( Ax) [3].  A(x) =

ymax

(u ∗ (x, y)u(x, y) + v ∗ (x, y)v(x, y))dy

(7.29)

0

where ∗ denotes the complex conjugate. Figure 7.15 shows the variation of spatial amplification rate (Ax) for velocity disturbances at different plate angles β and different Reynolds numbers. As shown in Fig. 7.15, the spatial amplification rate A x increases in the streamwise direction towards the downstream. It is observed from 7.15 that as plate angle β increases from 0◦ (zero pressure gradient) to 6◦ (favourable pressure gradient), the A x reduces in the streamwise direction. It proves that the development of the favourable pressure gradient in the streamwise direction reduces the spatial growth of the disturbances, and hence, it has an overall stabilizing effect. Figure 7.10 shows the comparison of eigenspectrum for Re = 340 and angle β = 0◦ , 2◦ , 4◦ and 6◦ . The favourable pressure gradient increases with the angle β. The least stable eigenmodes marked by a large rectangle for each case have the same frequency, ωr = 0.0708. It proves that the temporal growth ωi reduces with

7.4 Global Stability Results

197 0.2

0.5 β=0 β=0.0222 β=0.0444 β=0.0667

0.4

0.15

Ax

Ax

0.3

β=0 β=0.0222 β=0.0444 β=0.0667

0.1

0.2 0.05

0.1

(a) 0 400

(b)

420

440

460 x

480

0 400

520

420

440

460 x

480

500

520

β=0 β=0.0222 β=0.0444 β=0.0667

0.08 0.07 0.06

Ax

500

0.05 0.04 0.03 0.02 0.01

(c)

0 400

420

440

460 x

480

500

520

Fig. 7.15 Variation in spatial amplification rate(A x ) for the favourable pressure gradient in the streamwise direction x for different plate angle(β) for a Re = 340, b Re = 416 and c Re = 480. The least stable global mode is considered to compute the spatial amplification rate (A x )

the increased favourable pressure gradient. Figures 7.13, 7.14 and 7.15 show the comparison of spatial growth of disturbance for least stable eigenmode as marked by large square in Fig. 7.10. It indicates that the spatial growth of the disturbances decays in the streamwise direction with the increased favourable pressure gradient. Thus, a favourable pressure gradient has an overall stabilizing effect on the flow.

7.4.2 Adverse Pressure Gradient Figure 7.16 shows the eigenspectrum resulting from the global stability analysis of the 2D flat-plate boundary layer with adverse pressure gradients for the Reynolds number Re = 340. The discrete part of the eigenspectrum corresponding to T-S waves is only shown here for different plate angles β. The square marks the least stable eigenmodes for different angles β. The largest imaginary part ωi , which is the least stable one, is negative for the least stable eigenmodes for various plate angles β; hence, the global modes are temporally stable. Figure 7.16 shows the comparison of the discrete

198

7 Boundary Layer on an Inclined Flat Plate

ωi

−0.02

−0.025

−0.03 0

β=0 β=−0.0222 β=−0.0444 β=−0.0667 0.05

ωr

0.1

0.15

Fig. 7.16 Eigenspectrum resulting from the global stability analysis of the 2D flat-plate boundary layer for Re = 340 with adverse pressure gradients. β = 0 is for zero pressure gradient and β = −0.022, β = −0.044 and β = −0.0667 is for adverse pressure gradient

r

20 10 0

150 200 250 300 350 400 450 500 x

r

20

0.0050 0.0000 −0.0050 −0.0100 −0.0150

10 0

0.0100 0.0000 −0.0100 −0.0200

150 200 250 300 350 400 450 500 x

Fig. 7.17 Contour plots of the real part of u (upper) and v (lower) disturbance velocity components for Re = 340 and β = 0.0667. The associated frequency is 0.0000

part of the eigenspectrum, which corresponds to Tollmien-Schlichting (T-S) waves for adverse pressure gradient (β > 0). It is observed that increase in the adverse pressure gradient (angle β) reduces the damping rate/increases the temporal growth rate (ωi ) of the least stable eigenmode, which makes the global mode temporally less stable. However, the distribution of frequency for the T-S waves is not affected by the effect of the streamwise pressure gradient. It suggests that the adverse pressure gradient in the streamwise direction has an overall destabilizing effect. The least stable eigenmodes are selected to study the spatial evolution of the two-dimensional disturbance amplitudes. Figures 7.17 and 7.18 show the contour plots of the real part of the streamwise (u) and wall-normal (v) disturbance velocity component with adverse pressure gradient for Re = 340. These eigenmodes are oscillatory in nature because ωr > 0.

7.4 Global Stability Results

r

20

−0.0200 −0.0300 −0.0400 −0.0500

10 0

100 150 200 250 300 350 400 450 500 x

r

20

0.0000 −0.0020 −0.0040 −0.0060

10 0

199

100 150 200 250 300 350 400 450 500 x

Fig. 7.18 Contour plots of the real part of u (upper) and v (lower) disturbance velocity components for Re = 340 and β = 0.0667. The associated frequency is 0.0000

The associated frequency in four eigenmodes is near zero. The adverse pressure gradient increases with an increase in β. The two-dimensional spatial structures of the eigenmodes shown in Figs. 7.17 and 7.18 are similar; however, the magnitudes of the disturbance amplitudes are higher for β = 0.0667. The two-dimensional spatial structure of eigenmodes shows wavelet structure for u and v disturbance amplitudes. The magnitude of u disturbance is almost one order higher than that of the v disturbances. The disturbances are observed to evolve near the wall surface and decay exponentially towards the free-stream. The amplitudes of u and v disturbance velocity components evolve in the flow field and grow in size and magnitude while moving towards the downstream with the base flow. Thus, the flow is convectively unstable. The two-dimensional spatial structure of the disturbance amplitudes is similar to that of the boundary layer with zero and favourable pressure gradients. Figure 7.19 shows the variation of disturbance amplitudes in the streamwise direction at y = 0.114 and Re = 340 with different plate angle β. It shows that the magnitudes of the disturbances increase as they move downstream in the streamwise direction. The wave-like nature of the disturbances is found in the streamwise direction. The magnitude of the disturbance amplitudes increases as plate angle β increases from 0◦ to 6◦ . It proves that the development in the adverse pressure gradient increases the spatial growth rate of the disturbance amplitudes. Thus, the development of the adverse pressure gradient in the streamwise direction makes the flow convectively unstable. Figure 7.20 shows the variation of the velocity disturbance amplitudes in the ydirection at the streamwise location x = 461. The nature of variation for the zero and adverse pressure gradient is almost similar. At the wall, the magnitudes of the disturbances amplitudes are zero due to the viscous effect, then gradually increase in the y-direction and finally vanish at the far-field. It has been observed from Fig. 7.20 that magnitude of u and v disturbance amplitude increases with the development of an adverse pressure gradient. Figure 7.21 shows the variation of spatial amplification

200

7 Boundary Layer on an Inclined Flat Plate −3

6

β=0 β=−0.0222 β=−0.0444 β=−0.0667

4 2

β=0 β=−0.0222 β=−0.0444 β=−0.0667

6 4 2

0

v

u

−5

x 10

x 10

0 −2 −2 −4 −4 −6 400

420

440

460 x

480

400

500

420

440

460 x

480

500

Fig. 7.19 Variation of the disturbance amplitudes in the streamwise direction for various plate angles β at y = 0.114 for Re = 340. Here, β = 0 is for zero pressure gradient and β = −2◦ , −4◦ and −6◦ for adverse pressure gradient. a u and b v disturbance amplitudes 25

(a)

25

β=0 β=−0.0222 β=−0.0444 β=−0.0.667

20

(b)

β=0 β=−0.022 β=−0.044 β=−0.0667

20

15

y

y

15

10

10

5

5

0 −2

0

2

4 u

0

6 x 10

−3

−2

0

2 v

4 −3

x 10

Fig. 7.20 Variation of the disturbance amplitudes in the y-direction at streamwise location x = 461 for various plate angles β and Re = 340. Here, β = 0 is for zero pressure gradient and β=0.022, −0.044 and −0.0667 for adverse pressure gradient. a u and b v disturbance amplitudes

rate (A x ) for the amplitudes of the disturbance velocity for different plate angles β for different Reynolds numbers. As shown in Fig. 7.21, the spatial amplification rate A x increases in the streamwise direction towards the downstream. For a given Reynolds number, the increase in adverse pressure gradient increases the spatial growth rate (A x ). Thus, the adverse pressure gradient in the streamwise direction makes the flow convectively more unstable.

7.5 Summary

201 0.5

0.25

β=0 β=0.0222 β=0.0444 β=0.0667

0.2

0.3

Ax

Ax

0.15

β=0 β=0.0222 β=0.0444 β=0.0667

0.4

0.2

0.1

0.1

0.05

(a)

0 300

(b)

350

400 x

0 300

450

350

400

450

500

x

0.5

β=0 β=−0.0222 β=−0.0444 β=−0.667

0.4

Ax

0.3 0.2 0.1

(c)

0 300

350

450

400

500

550

x

Fig. 7.21 Variation in spatial amplification rate (A x )for adverse pressure gradient in the streamwise direction x for different plate angle (β) for a Re = 340, b Re = 416 and c Re = 480. The least stable global mode is considered to compute the spatial amplification rate(A x )

7.5 Summary Global stability analysis is performed for an incompressible boundary layer on an inclined plate in the presence of a streamwise pressure gradient. The various incident angle β considered are 0.0222, 0.0444 and 0.0667. The streamwise pressure gradient is not zero in this case. The global temporal modes are computed using Arnoldi’s iterative algorithm to solve the two-dimensional eigenvalues problem. The largest imaginary part ωi of the computed global modes is negative; hence, the global modes are temporally stable. The size and magnitudes of the disturbance amplitudes increase as they move in the streamwise direction towards the downstream. As the positive angle, β increases from 0 to 0.0667, the frequency distribution ωr remains almost the same, but the temporal growth rate ωi reduces, and global modes become more stable. The spatial growth rate ( A x ) also reduces in the streamwise direction. Thus, the development of the favourable pressure gradient makes global modes stable and convectively less unstable. As the negative value of β increases, from 0 to −0.0667 the frequency distribution ωr remains almost the same, but the temporal growth rate ωi increases and global modes become less stable. Overall, a favourable pressure gradient has damping, and an adverse pressure gradient amplifies the disturbances. Thus, the disturbances amplify subjected to APG and decay subjected to FPG. Furthermore, the spatial

202

7 Boundary Layer on an Inclined Flat Plate

growth rate (A x ) also increases in the streamwise direction. Thus, the adverse pressure gradient development makes global modes less stable and more unstable. The effect of domain length is also studied in the analysis, and it found that global stability heavily depends on the extent of streamwise domain size.

References 1. Abu-Ghannam, B.J., Shaw, R.: Natural transition of boundary layers—the effects of turbulence, pressure gradient, and flow history. J. Mech. Eng. Sci. 22, 213–228 (1980) 2. Akervik, E., Ehrenstein, U., Gallaire, F., Henningson, D.: Global two-dimensional stability measure of the flat plate boundary layer flow. Eur. J. Mech. B/Fluids 27, 501–513 (2008) 3. Alizard, F., Robinet, J.C.: Spatially convective global modes in a boundary layer. Phys. Fluids 19, 114105 (2007) 4. Bhoraniya, R., Vinod, N.: Global stability analysis of axisymmetric boundary layer over a circular cone. J. Phys. Conf. Ser. 822, 012018 (2017) 5. Bhoraniya, R., Vinod, N.: Global stability analysis of axisymmetric boundary layer over a circular cone. Phys. Rev. Fluids 02, 063901 (2017) 6. Bhoraniya, R., Vinod, N.: Global stability analysis of axisymmetric boundary layer over a circular cylinder. Theor. Comput. Fluid Dyn. 32, 425–449 (2018) 7. Blumer, C.B., Van Driest, E.R.: Boundary layer transition: free-stream turbulence and pressure gradient effect. AIAA J. 1, 1303–1306 (1963) 8. Chonghui, L.: A numerical investigation of instability and transition in adverse pressure gradient boundary layers. Ph.D Thesis, McGill University, Montreal (1997) 9. Corbett, P., Bottaro, A.: Optimal perturbations for boundary layers subject to streamwise pressure gradient. Phys. Fluids 12, 120–131 (2000) 10. Corke, T.C., Gruber, S.: Resonant growth of three-dimensional modes in Falkner-Skan boundary layers with adverse pressure gradient. J. Fluid Mech. 320, 211–233 (1996) 11. Costa, B., Don, W., Simas, A.: Spatial resolution properties of mapped spectral Chebyshev methods. In: Proceedings of SCPDE: Recent Progress in Scientific Computing, pp. 179–188 (2007) 12. Ehrenstein, U., Gallaire, F.: On two-dimensional temporal modes in spatially evolving open flow: the flat-plate boundary layer. J. Fluid Mech. 536, 209–218 (2005) 13. Fasel, H., Rist, U., Konzelmann, U.: Numerical investigation of the three-dimensional development in boundary layer transition. AIAA J. 28, 29–37 (1990) 14. Franko, K.J., Lele, S.: Effect of adverse pressure gradient on high speed boundary layer transition. Phys. Fluids 26, 24106 (2014) 15. Garnaud, X., Schimd, P.J., Huerre, P.: Modal and transient dynamics of jet flows. Phys. Fluids 25, 044103 (2013) 16. Gostelow, J.P., Blunden, A.R.: Investigation of boundary layer transition in an adverse pressure gradient. ASME J. Turbomach. 111, 366–374 (1989) 17. Gostelow, J.P., Blunden, A.R., Walker, G.J.: Effect of free-stream turbulence and adverse pressure gradients on boundary layer transition. ASME J. Turbomach. 116, 392–404 (1994) 18. Govindarajan, R., Narasimha, R.: Stability of spatially developing boundary layers in pressure gradients. J. Fluid Mech. 300, 117–147 (1995) 19. Igarashi, S., Sasaki, H., Honda, M.: Influence of pressure gradient upon boundary layer stability and transition. Acta Mechanica 73, 187–198 (1988) 20. Itoh, N.: Effect of pressure gradients on the stability of three-dimensional boundary layers. Fluid Dyn. Res. 7, 37–50 (1991) 21. Johnson, M.W., Pinarbasi, A.: The effect of pressure gradients on boundary layer receptivity. Flow Turbulence Combust. 93, 1–24 (2014)

References

203

22. Kimmel, R.L.: The effect of pressure gradients on transition zone length in hypersonic boundary layer. Flight Dynamics Directorate (1993) 23. Liu, C., Maslowe, S.A.: A numerical investigation of resonant interactions in adverse pressure gradient boundary layers. J. Fluid Mech. 378, 269–289 (1999) 24. Mack, L.M.: A numerical study of temporal eigenvalue spectrum of the Blasius boundary layer. J. Fluid Mech. 73, 497–520 (1976) 25. Malik, M.R.: Numerical methods for hypersonic boundary layer stability. J. Comput. Phys. 86(2), 376–412 (1990) 26. Marquet, O., Sipp, D., Jacquin, L.: Sensitivity analysis and passive control of cylinder flow. J. Fluid Mech. 615, 221–252 (2008) 27. Masad, J.A., Zurigat, Y.H.: The effect of pressure gradients on first mode of instability in compressible boundary layer. Phys. Fluids 6 (1994) 28. Maslowe, S.A., Spiteri, R.J.: The continuous spectrum for a boundary layer in a streamwise pressure gradient. Phys. Fluids 13, 1294 (2001) 29. N, V., Govindarajan, R.: The signature of laminar instabilities in the zone of transition to turbulence. J. Turbulence 8, 1–17 (2007) 30. Narasimha, R.: The laminar-turbulent transition zone in the boundary layer. Prog. Aero. Sci. 22, 29–80 (1985) 31. Nichols, J.W., Lele, S.K.: Global modes and transient response of a cold supersonic jet. J. Fluid Mech. 669, 225–241 (2011) 32. Obremski, H.J., Morkovin, M.V., Landahl, M.: A portfolio of stability characteristics of incompressible boundary layer. AGARDograph 134 (1969) 33. Saxena, S.K., Bose, T.K.: Numerical study of effect of pressure gradient on stability of an incompressible boundary layer. Phys. Fluids 17, 1910–1912 (1974) 34. Schlichting, H.: Concerning the origin of turbulence in a rotating cylinder. Math. Phys. Klasse. 2, 160–198 (1932) 35. Seifert, A., Hodson, H.P.: Periodic turbulent strips and calmed regions in a transitional boundary layer. AIAA J. 37, 1127–1129 (1999) 36. Sipp, D., Lebedev, D.: Global stability of base and mean flows: a general approach to its applications to cylinder and open cavity flow. J. Fluid Mech. 593, 333–358 (2007) 37. Swaminathan, G., Shahu, K., Sameen, A., Govindarajan, R.: Global instabilities in diverging channel flows. Theor. Comput. Fluid Dyn. 25, 25–64 (2011) 38. Theofilis, V.: Advances in global linear instability analysis of nonparallel and three dimensional flows. Prog. Aerosp. Sci. 39, 249–315 (2003) 39. Tumin, A., Ashpis, D.E.: Optimal disturbances in boundary layers subject to streamwise pressure gradient. In: 33rd AIAA Fluid Dynamic Conference (2003) 40. Vinod, N., Govindarajan, R.: Pattern of breakdown of laminar flow into turbulent spots. Phys. Rev. Lett. 93, 114501 (2004) 41. Walker, G., Gostelow, J.P.: Effect of adverse pressure gradients on the nature and length of boundary layer transition. In: Gas Turbines and Aeroengine Congress and Exposition (1989) 42. Zhang, W., Yang, H., Hua-Shu, D., Zuchao, Z.: Flow unsteadiness and stability characteristics of low-re flow past an inclined triangular cylinder. J. Fluids Eng. 139, 121203 (2017) 43. Zurigat, Y.H., Nayfeh, A.H., Masad, J.A.: Effect of pressure gradient on the stability of compressible boundary layers. AIAA J. 30, 2204–2211 (1992)

Chapter 8

Wall Jet

8.1 Introduction A wall jet is produced when fluid is blown tangentially across a flat surface. A radial wall jet occurs when a fluid jet hits a wall perpendicularly and spreads radially outwards. Wall jets were first studied by [6], who derived a similarity solution valid far downstream of the jet impingement, both for the planar and radial cases. The jet of fluid can be the same as or different from the surrounding fluid, as mentioned in [6]. If the fluid in the jet is different from the surrounding fluid, then the equations of motion will have to consider the interface between the two fluids, and such a flow can form a hydraulic jump. Thus, we can say that hydraulic jumps occur in a special sub-class of wall jets with an interface. Wall jets are used for efficient heat transfer like turbine blades and heat exchangers. More than its applications, a special property of a wall jet has motivated many physicists to study this interesting flow, as discussed below. Two common hydrodynamic flows are boundary layers and shear layers. Wall jets are unique as they can be seen (roughly) as a combination of these two distinct types of flows, a boundary layer very close to the wall and a free shear layer far away from the wall. The stability characteristics of a boundary layer and a shear layer are independently studied extensively. Important to note are the differences in their stability characteristics. For example, a boundary layer profile on a flat plate is inviscid stable because there is no inflexion point, and the viscous modes cause instability. On the other hand, a shear flow is inviscid unstable as it is inflectional. A wall jet will have dual stability behaviour as its velocity profile is inflectional and has strong viscous effects close to the wall. Many researchers have recognized this twin characteristic of wall jets. The two distinct effects have been possible to glean even by making a parallel flow approximation. It will be interesting to study these characteristics using a global stability approach. We believe (and have shown in the chapters on non-parallel channels, Chaps. 3 and 4) that a global stability study can reveal many important physical characteristics which are not accessible to parallel approaches. Hence, we undertake this global study on wall jets. The first stability calculations on the wall jet were done by [3], who studied the fully © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 R. Bhoraniya et al., Global Stability Analysis of Shear Flows, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-19-9574-3_8

205

206

8 Wall Jet

developed similarity profile far downstream under the parallel flow approximation using the Orr–Sommerfeld equation. Their main results are (i) the presence of an additional instability mode together with the least stable mode, (ii) the presence of two critical layers and (iii) the dominance of the outer region in instability as disturbance production is maximum there. Experimental investigations by [1, 2] confirmed the presence of the similarity solution predicted by [6] and proved the dominance of the outer region in both natural and forced transitions. Finally, it was [9] who confirmed the presence of two distinct types of instability modes, an inflectional mode peaking in the outer region and a viscous mode that peaks near the wall. In contrast, the inflectional mode dominates in the transition process. The co-existence of the two modes was experimentally confirmed by [4]. They also show that the relative roles of the inflectional (outer) and viscous (inner) modes can be controlled by blowing or suction. Scibilia [11] studied the effect of heating, roughness experimentally and forcing on the transition characteristics of wall jets. Seidel and Fasel [12] showed that heat transfer due to a wall jet is increased by forcing. Close to its origin, a wall jet may often be approximated by combining an inviscid constant velocity layer and a Blasius boundary layer. Levin et al. [8] studied the stability of this developing wall jet using the Parabolized Stability Equations (PSE), in which the non-parallel effects are considered up to O(1/Re). The present work aims at studying the fully developed wall jets using a global stability study. The ultimate aim is to study the stability characteristics of developing wall jets close to the origin, where self-similarity is not yet achieved. In future, it will also be interesting to study the heat transfer characteristics of wall jets using a global study.

8.2 Base Flow As mentioned before, the similarity equation governing a wall jet flow was first given by [6]. The equation is given as, f  + f f  + 2 f 2 = 0, 



f (0) = f (0) = 0, f (∞) = 0.

(8.1) (8.2)

Here, f is the stream function. The primes are with respect to η, where the wallnormal dimensionless coordinate is defined as η = y/δ(x). The length scale in the problem is the wall jet thickness δ(x), defined as the farthest distance from the wall at which the velocity is half the maximum velocity. A wall jet profile is inflectional away from the wall and has two locations at which the velocity is half the maximum velocity. The velocity scale is Umax , the maximum streamwise velocity. Thus, the Reynolds number will be defined as, Re = Umax δ/ν.

(8.3)

8.3 Local Stability Analysis of Wall Jets

207

The following relations hold for Umax and δ with the streamwise coordinate x, see [10] or [7], Umax ∼ x −1/2 , and δ ∼ x 3/4 .

(8.4)

From the above, we derive a relation between the dimensionless streamwise distance x/δ and Re as 1 F 2 U =a , xν 1  3 3 x ν 4 , δ=b F 1  Re Fx 4 = , ab ν3 Re x = 2, δ ab 

(8.5)

(8.6)

(8.7) (8.8)

where a = 0.498 and b = 3.23, and F is the wall jet constant introduced by Glauert. See [10] for the definition of F and Eqs. 8.5 and 8.6. Equation 8.8 gives the relation between the non-dimensional x and Re, which fixes the x coordinate at which a particular Re is obtained. The mean flow is calculated by discretizing a domain of length of y/δ = 8 into 2000 equidistant points and solved using the fourth-order RungeKutta method. Since one of the boundary conditions is required to be satisfied at ∞, a sufficiently large domain height has to be considered. A sensitivity study was conducted to find the minimum height at which the far-stream boundary condition can be imposed without losing accuracy. It was concluded that a height of 8δ and above gives results independent of the domain size, Fig. 8.1. For all the computations presented, a height of 8δ is considered tall enough to apply the free-stream boundary conditions. This work was done in collaboration with Dr. A Sameen, IIT Madras, India, and Dr. Tamer Zaki, Imperial College, London. I first thank Dr. Sameen for proposing such an interesting problem. I also thank Dr. Zaki for pointing out the mistake in the base flow calculation. The base flow calculated from the similarity equation is validated with the direct numerical simulations of Dr. Zaki.

8.3 Local Stability Analysis of Wall Jets Researchers [3, 9] have earlier used the parallel flow assumption on Glauert’s profile far downstream, where the non-parallel effects are negligible. The neutral curve reported by [3] using the Orr–Sommerfeld equation is shown in Fig. 8.2. As can be seen, this flow becomes unstable at a Reynolds number of 56.7 for an α of 1.16.

208

8 Wall Jet 7 5 10 20 25 30 40

6 5

y

4 3 2 1 0

0

0.2

0.4

0.6

0.8

1

U Fig. 8.1 Sensitivity of the base flow to the height at which the outer boundary is placed. Glauert’s profile calculated for different domain heights, y. Plotted here are the streamwise velocity U versus y. It can be noted that for domains taller than y = 20, the profiles coincide and the wall jet thickness δ lies at y = 2.5. Thus, domains at least as high as 8δ need to be used

Note that the Recrit predicted is relatively small, i.e. instability occurs closer to the origin than on a boundary layer. At this low x, it is unlikely that a similarity profile is attained as yet, which questions the validity of using the Glauert profile in the stability computations at this x. Also, according to Fig. 8.2, there is a small stable region starting at Re = 378 inside the unstable region of the neutral curve (shown in red) and this extends till Re = 785. This stable bubble is confined within wavenumbers 1 and 1.3. This sudden stabilization of the flow for a particular set of wavenumbers is not easy to explain on physical grounds. Many researchers considered a second instability mode to be the reason for this. The sudden appearance of the second mode in the neutral curve was explained by [13] using Spatio-temporal analysis. They mention that behind the stable bubble, two unstable modes co-exist, and there are two neutral boundaries in that region. This is represented by two lines extending from the stable region. We show below that this tail region of the stable bubble is unrealistic. We recomputed the neutral curve using the Orr–Sommerfeld equation, shown in Fig. 8.3. The critical Reynolds number, critical α and the location of the stable bubble match precisely with the results of [3]. We note that the tail region of the stable bubble is not seen here. In addition, a new unstable bubble is predicted for large Reynolds numbers. For the region behind the stable bubble, we have used the following approach to obtain the neutral curve. The decay/growth rate of the mode (ωi ) is plotted against the wavenumber (α) for a given Reynolds number, and it is repeated for a range of Reynolds numbers in the region behind the stable bubble. The results are shown in Fig. 8.4. It can be noted that there is no neutral region behind the stable bubble; hence, the tail region of the bubble is unrealistic. Computations were repeated for Reynolds numbers up to 17,000, and the results are shown in Fig. 8.5. From these figures, it can be deduced that there is an unstable bubble for large wavenumbers outside the neutral curve. The blue curve represents this in Fig. 8.3.

8.4 Global Stability Analysis of Wall Jets

209

3

2

α 1.5

785

378

2.5

1 0.5 0 0

200

400

600

800

1000

1200

1400

Reynolds Number

3

2.5

2.5

2

2

α 1.5

α 1.5

1

1

0.5

0.5

0

0

4000

8000

Reynolds Number

0

785

3

378

Fig. 8.2 Critical Reynolds number obtained by [3]. Note a ‘tail’ region of the stable bubble extending unrealistically in the unstable region

0 12000 500

1000

3000

6000

9000

12000

Reynolds Number

Fig. 8.3 (Left) The neutral curve obtained using the Orr–Sommerfeld equation. It can be noted that a small stable bubble exists (shown in red) within the unstable region and an unstable region exists (shown in blue) in the stable region of the neutral curve. (Right) Same as the left figure, zoomed to show the stable and unstable bubbles. Note that the x axis of the two plots are of different scales

8.4 Global Stability Analysis of Wall Jets In real life, since the critical Reynolds number is small, the flow might become unstable before reaching a similarity profile. This is the motivation for a future study on the stability characteristics of the developing wall jet flow well before it reaches the similarity profile. If the flow is still in the developing region, the non-parallel effects are not negligible, and they cannot be studied using parallel theory. This developing flow was approximated by a combination of the Blasius boundary layer and a free shear layer away and was studied using PSE by [8] as stated before. We study here the fully developed self-similar wall jet profile. Similar to a global stability study

210 Fig. 8.4 Plot of temporal growth rate (ωi ) against the wavenumber α, for a few Reynolds numbers just behind the stable bubble shown in Fig. 8.3. It can be seen that the wavenumber 1.2 does not cross the ωi = 0 line and does not have the neutral tail found by [3] for Reynolds numbers greater than 785

8 Wall Jet 0.15

Re 810 Re 900 Re 1500

0.1

ωi

0.05

0

-0.05 0

2

1.5

α

12000 10000 9000 8000 7000 6000 5000 4500

0.12

0.08

ωi 0.04

3

2.5

Increasing Re

Fig. 8.5 Same as Fig. 8.4, for large Reynolds numbers. The unstable bubble shown in Fig. 8.3 has been obtained by considering the zero-crossings shown at the bottom-right region of this figure

1

0.5

0 0

0.5

1

1.5

α

2

2.5

3

on self-similar JH flows (Sect. 3), the ‘finite domain size’ effects are unavoidable in these flows too. Hence, a detailed sensitivity study has to be undertaken to arrive at reliable results.

8.4.1 Numerical Method Chebyshev spectral discretization is used in x and y, similar to the previous chapters. Since free-stream boundary conditions are used in the top boundary of the domain, a sufficiently long domain is considered in y. Since much activity of the wall jet is close to the wall, the stretching function defined by Eq. 2.24 is used in y also to cluster the grid points close to the wall, for which the constant stretching values are fixed to be a = 0.1, b = 6.0. The constant stretching values in x remain unchanged

8.4 Global Stability Analysis of Wall Jets

211

30 25

y

20 15 10 5 0

10

15

20

x

25

30

35

Fig. 8.6 A typical distribution of grid points. The red stars indicate the location of the local wall jet thickness, δ. The grids are clustered close to the wall in y and more or less uniformly distributed in x. The grid size shown here is 201 × 71

from the previous studies (Refer to Sect. 2.6.4 for the stretching function used and a discussion on grid stretching). A typical grid is shown in Fig. 8.6, where the red stars indicate the location of δ. The inflection point lies close to y = δ.

8.4.2 Validation To compare with the results of [3], we supply the same self-similar Glauert profile at every x location, along with V = 0 in the global stability equation. Note that this is an artificial flow where the wall jet thickness and, therefore, the Reynolds number are not allowed to change downstream. Using Robin boundary conditions, the wavenumber at the inlet and the domain exit are enforced, as explained in Sect. 2.12. The spectra obtained using parallel (Orr–Sommerfeld equation) and global stability (with Robin boundary conditions), for a Reynolds number of 80 and α of 2, are compared in Fig. 8.7. A grid sensitivity study is also conducted. A sample is shown in Fig. 8.8. The Reynolds number indicated as 57 is the Reynolds number at the domain inlet. Since the length of the domain is restricted in x by the wavelength of the wave under consideration, a relatively small number of grids are needed in x. It is seen that 41 points in y are sufficient to capture the most unstable mode. Even with this small number of grid points, the critical Reynolds number calculated matches with [3]s results, as shown in Fig. 8.9. Next, to consider a realistic case, we include the wall-normal velocity and allow for Reynolds number and δ variations across the domain. The wall-normal velocity is given as,

212

8 Wall Jet

Fig. 8.7 Sample spectra obtained from the global approach, imposing Robin boundary conditions (in black circles), and the parallel approach (in red squares). The Reynolds number is 80, and the α is 2. It can be noted that the least stable mode is well captured by the global analysis

0

global parallel

ωi -0.5

-1

Fig. 8.8 Grid sensitivity study. It can be noted that the least stable eigenvalue is quite insensitive to the grid in the range considered

0

1

0.5

ωr

α = 1.16

Re=57 0

41x51 31x41 21x41

ωi -0.1

-0.2

V (x) =

0

1

ωr

  δ1 U1 − f + 3η f η . 0.75 0.25 4x1 x

2

(8.9)

Here, the subscript 1 corresponds to the value at the inlet x location (x1 ). This appears because the length scale and velocity scale are chosen as δ and Umax at the inlet. f is the streamfunction, and η is the non-dimensional y coordinate. It can be seen from Eqs. 8.3 and 8.4 that the Reynolds number increases with x as, Re ∼ x 1/4 .

(8.10)

Consider for example, a case where the inlet Reynolds number is 50, the domain length is 10. Then, from Eq. 8.8, the domain starts at x = 9.623 and ends at x = 19.623. The exit Reynolds number for this case will be, according to Eq. 8.10,

8.4 Global Stability Analysis of Wall Jets Fig. 8.9 Critical Reynolds number calculated for validation

213

Re=56.7 Re=58

α = 1.16 0.01

0

ωi -0.05

ωi

0

-0.01

-0.1

0

1

ωr

2

-0.15

-0.2

0

0.5

 1/4 Re2 x2 = , Re1 x1 i.e. Re2 = 50 ∗ 1.1949 = 59.79.

1

ωr

1.5

2

(8.11) (8.12)

A critical Reynolds number is thus not a well-defined quantity in this global study. This is because the Reynolds number varies significantly across the domain. First, a domain starting from a sub-critical Reynolds number and ending at a supercritical Reynolds number might not be able to capture the least stable eigenvalue corresponding to the inlet Reynolds number. Secondly, the least stable wavenumber at the inlet would differ from the one at the exit. Since the change in Reynolds number and the least stable wavenumber in the domain is too large, with Robin boundary conditions, we could not converge to a single neutrally stable eigenvalue for the entire domain.

8.4.3 Sensitivity We next perform a global stability analysis with extrapolated boundary conditions (EBC), as used in the previous chapters. It is, however, expected that if a considerably long domain is considered, the type of boundary conditions at the inlet/exit will not affect the solutions. The results will change quantitatively depending upon the domain’s length since the Reynolds number range increases with length L, but their qualitative picture remains the same. This is because the domain length will restrict the length of the disturbance captured by this analysis. We perform a series of tests to study the sensitivity of the spectra to the number of grids and length of the domain considered. A few comparisons are given in Figs. 8.10, 8.11 and 8.12.

214

8 Wall Jet

L 30 L 40 L 50 L 25

0

ωi -0.5

-1 -2

-1

0

ωr

1

2

Fig. 8.10 Spectra obtained at Re = 100 for different lengths of the domain. It can be seen that the qualitative picture of the spectra remains unchanged with change in domain length. In the results shown, it is ensured that the number of grid points considered over a given length is kept the same. Some quantitative effect is of course expected, since the range of Reynolds numbers over which a common global mode is being sought increases with L

81x61 101x61 121x61 141x61

0

ωi -0.5

-1 -4

-2

0

ωr

2

4

Fig. 8.11 Sensitivity of the spectra to the number of grid points in the x-direction, for L = 30 and Re = 100. It can be noted that the spectra contain distinct branches and the second limb of the branches gradually disappear with increase in the number of grid points in x. The sensitivity of the second limb of the branch is further decreased by increasing accuracy in y, as explained in the next figure

8.4.4 Main Results As mentioned before, one of the primary purposes of the present study is to understand how different the global modes are from the parallel modes and how valid a parallel assumption is for wall jets. For this, we first compare the global modes with parallel

8.4 Global Stability Analysis of Wall Jets

215

161x71 181x71 201x71

0

ωi -0.5

-1 -4

-2

0

ωr

4

2

Fig. 8.12 In continuation to the previous figure, it can be noted that with 71 points in y the second limb of the branch disappears and 161 points in x are sufficient for a length of 30, at a Reynolds number of 100. It can also be noted that the two very unstable modes which were present with other grid resolutions (in Figs. 8.10 and 8.11 and enclosed with a rectangle in Fig. 8.11) have disappeared with increase in m Fig. 8.13 Spectra obtained for different starting Reynolds numbers for a domain length of 30. We can see that the upper branch becomes more destabilized with increase in Reynolds number

R 50 R 80 R100 R200

0

ωi

-0.5 -1

0

ωr

1

modes. The global stability computations are done for different inlet Reynolds numbers of 30, 40, 50, 80, 100 and 200 for different domain lengths. A comparison of the spectra obtained is given in Fig. 8.13. As expected, the modes are destabilized with an increase in Reynolds number. A portion of this figure near the imaginary axis is shown in Fig. 8.14. It can be seen from this figure that the flow becomes globally unstable for a Reynolds number between 50 and 80. It is worth remembering that the Recrit obtained from the parallel approach for this flow is 57. Now, let us try to understand the characteristics of the global modes. The spectra obtained for a Reynolds number of 100 with domain length 30 and a grid size 201 × 71 is given in Fig. 8.15. Plots of the streamwise velocity of typical modes from each branch are shown in Figs. 8.16 and 8.17. We can see that branch 1 has a set of modes completely different

216 Fig. 8.14 Same as Fig. 8.13, zoomed close to the axis. It can be seen that this flow is globally unstable at a Reynolds number of 80

8 Wall Jet

ωi

0

-0.1 -0.5

Fig. 8.15 Spectrum obtained for a Reynolds number of 100 with domain length 30 with a grid size 201 × 71. The spectrum has distinct branches, named as branches 1–4 as shown

R 50 R 80 R100 R200

0.1

0

ωr

0.5

0

B1

ωi

B2

-0.35

B3

B4 -0.7

1

ωr

2

from the modes of the other branches. On the other hand, branches 2–4 have similarlooking modes. Also, the modes of branches 2–4 have extended regions of positive and negative velocities, thus giving them the appearance of arrowheads. These modes shall be contrasted with the modes obtained from local stability analysis, shown in Fig. 8.18. The richer variety of disturbances obtained using a global approach is immediately apparent. In addition, a comparison between a local mode and a global mode with ‘similar’ α and ω at the same Reynolds number is shown in Fig. 8.19. The global restructuring of global modes is apparent in this figure. Next, we compare the modes within one branch, say branch 1. The following trend is observed: modes with large frequency have different structures compared to the low-frequency modes, a comparison shown in Fig. 8.20. We can note here that the peak of the low-frequency mode is very close to the wall, whereas the high-frequency mode peaks at a slightly lifted location. Therefore, these two modes from the same branch may be described as wall mode (or inner mode) and outer mode, respectively. The second mode does not exactly peak at the inflexion point, so that we may describe it as a mixed mode.

8.4 Global Stability Analysis of Wall Jets

217 Branch 2

Branch 1

10

10

5

5

0

10

15

20

25

30

35

0

10

15

20

25

30

35

Fig. 8.16 Typical set of modes from branch 1 (left column) and branch 2 (right column). Plotted here are the contours of streamwise velocity. We can see that the branches exhibit qualitatively different modes. Especially, the modes in branch 1 show different wavelengths close to and away from wall

These two modes have been reported earlier in literature; see [9] for example. A similar trend is seen in other branches. Also, a sample is given in Fig. 8.22, which shows modes from branch 2. Another interesting behaviour to be noted about the low-frequency branch one mode is a new manifestation of non-wave-like behaviour. See top of Fig. 8.20. Towards the beginning of the domain, this global mode displays three blobs at each streamwise station. As we proceed downstream, we notice that the top two blobs merge so that we see just two blobs at the end of the domain. This is another non-wave-like feature revealed by a global study inaccessible to a local study. This particular feature is seen in wall jets; others were seen in previous chapters. In addition, the variation of the maximum amplitude of this mode is different at different wall-normal locations. This is shown in Fig. 8.21, where the maximum amplitude variation of one of the global modes shown in Fig. 8.20 is shown. The corresponding two wall-normal locations are also shown on the right side of the figure. We can see that within a single global mode, the amplitude of the modes can vary differently at different y, similar to what was reported in the previous chapter. To add to this, we perform a wavelet transform of a typical global mode, shown in Fig. 8.23. The variation of the dominant length scale is almost nil for a large region in the middle at both the wall-normal locations. But towards the tail end of the figure, we see that the dominant length scale varies more close to the wall rather than away from it. For the mode just discussed, the wavenumber variation is not very significant.

218

8 Wall Jet Branch 3

Branch 4

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Fig. 8.17 Typical set of modes from branch 3 (left column) and branch 4 (right column). Plotted here are the contours of streamwise velocity. We can see that these two branches have modes looking similar to those in branch 2. These modes exhibit extended regions of positive and negative velocities, thus giving them the shape of a ‘arrow head’

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Fig. 8.18 Typical modes obtained from the local stability analysis. The local profile obtained is extended in x with the given wavenumber, indicated as α in the figure. The inflectional modes (the two plots to the left) have their maximum amplitude away from the wall, and the wall modes (the two plots to the right) have their maximum amplitude close to the wall

8.4 Global Stability Analysis of Wall Jets

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Fig. 8.19 A specific comparison of the modes obtained from local analysis (top) and global analysis (bottom) for a Reynolds number of 80, L = 30. Both the modes have α = 0.84 and are near-neutral with a frequency of 0.3 5

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Fig. 8.20 Typical low-frequency global mode (top) and high-frequency global mode (bottom) from branch 1 of Fig. 8.15. It can be seen that the low-frequency mode peaks closer to the wall, whereas the high-frequency mode peaks slightly further from the wall. The black lines indicate the location of the local δ, which is close to the inflection point

An interesting feature of these global modes is their growth in space. Except for the low-frequency modes of branch 1 (see Figs. 8.16 and 8.17), we see that the other modes exhibit a spatial growth, in which the amplitude of the mode increases with x. These modes are temporally more stable. This behaviour is similar to the situation discussed in the many periodic units of the channel. However, the spatial growth we discuss here is local spatial growth in the amplitude of the global modes. Previous chapter is on converging-diverging channels. The difference is that the spatial growth

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Fig. 8.22 Typical low-frequency mode (top) and high-frequency mode (bottom) from branch 2 of Fig. 8.15. It can be seen that the low-frequency mode’s peak is close to the wall whereas the high-frequency mode’s peak is close to the inflection point. The black lines indicate the location of the local δ, which is close to the inflection point

discussed in chapter 4 is the growth over. In the global study of [5] on a boundary layer, it was found that the global study produced only a quantitative rather than a qualitative change. In the wall jet, the differences between parallel and global results are smaller than those we witnessed for the channel flows. In particular, the global instability Reynolds number is not too far from the critical Reynolds number given by parallel studies. However, some qualitative features are the arrowhead shape of the modes and some new non-wave-like behaviour. These studies indicate that stability behaviour is modified on a global basis for enclosed flows through channels. At the same time, the structure modification is local for wall jets and negligible for boundary layers.

References

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Grid number Fig. 8.23 (Top) Contours of streamwise velocity of a typical global mode from branch 1 of Fig. 8.15. (Bottom and centre) Wavelet transforms of the global mode at the two wall-normal locations indicated by the black dashed lines. For this mode, the wavenumber variation is not significant

References 1. Bajura, R.A., Catalano, M.R.: Transition in a two-dimensional plane wall jet. J. Fluid Mech. 70, 773–799 (1975) 2. Bajura, R.A., Szewczyk, A.A.: Experimental investigation of a laminar two-dimensional plane wall jet. Phys. Fluids 13, 1653–1664 (1970) 3. Chun, D.H., Schwarz, W.H.: Stability of the plane incompressible viscous wall-jet subjected to small disturbances. Phys. Fluids A 10, 911–915 (1967) 4. Cohen, J., Amitay, M.: Laminar-turbulent transition of wall-jet flows subjected to blowing and suction. Phys. Fluids (A) 4(2), 283–289 (1992) 5. Ehrenstein, U., Gallaire, F.: On two-dimensional temporal modes in spatially evolving open flows: the flat-plate boundary layer. J. Fluid Mech. 536, 209–218 (2005) 6. Glauert, M.B.: The wall jet. J. Fluid Mech. 1, 625–643 (1956) 7. Kundu, P.K., Cohen, I.M.: Fluid Mechanics, 8 edn. Elsevier (2004) 8. Levin, O., Chernoray, V.G., Lofdahl, L., Henningson, D.: A study of the Blasius wall jet. J. Fluid Mech. 539, 313–347 (2005) 9. Mele, P., Morganti, M., Scibilla, M.F., Lasek, A.: Behaviour of wall jet in laminar-to-turbulent transition. AIAA J. 24, 938–939 (1986) 10. Schlichting, H.: Boundary Layer Theory, 8 edn. Springer (2000) 11. Scibilia, M.F.: Variation of transition characteristics in a wall jet. J. Thermal Sci. 12(2) (2003) 12. Seidel, J., Fasel, H.: Numerical investigation of heat transfer mechanisms in the forced laminar wall jet. J. Fluid Mech. 442, 191–215 (2001) 13. Tumin, A., Aizatulin, L.: Instability and receptivity of laminar wall jets. Theoret. Comput. Fluid Dynamics 9, 33–45 (1997)