Transition to Turbulence: A Dynamical System Approach to Receptivity 1108490417, 9781108490412

Citation preview

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Transition to Turbulence Present understanding of the subject of transition to turbulence is based on studies performed over the last one hundred and fifty years. The path the studies have taken posed it as a modal eigenvalue problem. Some researchers have suggested alternative models without being specific. First-principle based approach of receptivity is the route to build bridges among ideas for solving the Navier–Stokes equation for specific canonical problems. This book highlights the mathematical physics, scientific computing, and new ideas and theories for nonlinear analyses of fluid flows, for which vorticity dynamics remain central. This book is a blend of classic with distinctly new ideas, which establish different dynamics of flows, from genesis to evolution of disturbance fields with rigorously developed methods to tracing coherent structures amidst the seemingly random and chaotic fluid dynamics of transitional and turbulent flows. Tapan K. Sengupta is Visiting Professor of Mechanical Engineering at IIT Dhanbad. Prior to this he was Professor of Aerospace Engineering at IIT Kanpur. He has also been Senior Visiting Fellow at NUS, Singapore, and Senior Associate at ICTP, Trieste. He is the author of nine books on Aerodynamics and Acoustics, Scientific Computing and Transition and Turbulence. He has co-edited two CISM monographs and an IUTAM symposium proceedings. His research areas include transition/turbulence, aerodynamics, fluid mechanics at all speed regimes, flow control, and high performance scientific computing including computational fluid dynamics.

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Transition to Turbulence A Dynamical System Approach to Receptivity

Tapan K. Sengupta

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University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314 to 321, 3rd Floor, Plot No.3, Splendor Forum, Jasola District Centre, New Delhi 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781108490412 ©

Tapan K. Sengupta 2021

This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2021 Printed in India A catalogue record for this publication is available from the British Library Library of Congress Cataloging-in-Publication Data Names: Sengupta, Tapan Kumar, 1955- author. Title: Transition to turbulence : a dynamical system approach to receptivity / Tapan Sengupta. Description: Cambridge, UK ; New York, NY : Cambridge University Press, 2021. | Includes bibliographical references. Identifiers: LCCN 2020040215 (print) | LCCN 2020040216 (ebook) | ISBN 9781108490412 (hardback) | ISBN 9781108780889 (ebook) Subjects: LCSH: Turbulence–Mathematical models. | Transition flow. Classification: LCC TA357.5.T87 S464 2021 (print) | LCC TA357.5.T87 (ebook) | DDC 620.1/064–dc23 LC record available at https://lccn.loc.gov/2020040215 LC ebook record available at https://lccn.loc.gov/2020040216 ISBN 978-1-108-49041-2 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

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To my loving wife, Soma, without whose lifelong sacrifices this work would never have been possible

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Contents

Preface

xvii

1 Receptivity, Instability, and Transition: A Perspective 1.1 Historical Introduction 1.1.1 1.1.2 1.1.3 1.1.4 1.1.5 1.1.6

Introduction to flow instability Inviscid instability theory Role of dissipation on instability Viscous instability: Linear equations Temporal, spatial, and spatio-temporal instability studies Similarity profile for equilibrium flow: Tollmien–Schlichting waves from the Orr–Sommerfeld equation 1.1.7 Instability theory, experiments, and some unanswered questions

1 1 2 2 3 3 4 5 6

1.2 Introduction to Receptivity Analysis

7

1.3 Simple Concepts in Instability Studies

8

1.3.1 Kelvin–Helmholtz instability 1.3.2 Taylor–Green vortex instability 1.3.3 Equilibrium solution of the two-dimensional Taylor–Green vortex problem 1.3.4 Results of the Taylor–Green vortex problem simulation 1.4 Closing Remarks 2 Dynamical System Theory and Role of Equilibrium Flows 2.1 Conservation Equations 2.1.1 2.1.2 2.1.3 2.1.4 2.1.5

Conservation of mass or continuity Conservation of translational momentum Navier–Stokes equations in derived variables ~ ω ~ )-formulation Governing equations for rotational form of the (V, ~ ω ~ )-formulation Evolution equation for solenoidality error in (V,

2.2 Boundary Layer Theory for Equilibrium Flow

8 14 17 19 23 30 31 33 34 38 40 41 43

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2.2.1 2.2.2 2.2.3 2.2.4

Thin shear layer equation for two-dimensional steady flow Boundary layers integral properties Displacement thickness Momentum thickness

46 48 50 50

2.3 Limitations of Boundary Layer Equation and Steady Separation

51

2.4 Solving Boundary Layer Equation and Similarity Transformation

54

2.4.1 Similarity transform and analysis 2.4.2 Zero pressure gradient boundary layer 2.4.3 Stagnation point or the Hiemenz flow 2.5 Closing Remarks 3 Fundamentals of Scientific Computing

54 58 59 61 62

3.1 Computing Space–Time Dependent Flows

62

3.2 The Bi-Directional Wave Equation

64

3.2.1 Three-dimensional plane waves 3.2.2 Requirements for spatial discretization 3.3 Upwind Schemes for Higher Reynolds Number Flows 3.3.1 General compact schemes 3.3.2 First derivatives obtained by compact scheme 3.3.3 Selection of compact schemes 3.4 Global Spectral Analysis: Resolution 3.4.1 Spectral accuracy of some compact schemes

68 69 70 72 73 75 76 77

3.5 Time Integration Schemes

81

3.6 Analysis of Convection–Diffusion Equation

84

3.6.1 Spectral analysis of numerical schemes 3.6.2 RK4 -OUCS3-CD2 Scheme 3.6.3 RK4 -NCCD Scheme 3.7 Role of Diffusion and Relation to Rotationality 3.7.1 Enstrophy transport equation 3.8 Spatial and Temporal Scales in Turbulent Flows

89 91 93 94 98 101

3.8.1 Spatial scales in turbulent flows 3.8.2 Temporal scales in turbulent flows

101 106

3.9 Two- and Three-Dimensional Turbulent Flows

107

3.10 Time-Averaged and Unsteady Flows

109

3.11 Closing Remarks

113

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4 Instability and Transition

114

4.1 Introduction

114

4.2 Inviscid Instability of Parallel Flows

115

4.2.1 Inviscid instability mechanism 4.2.2 Is there spatial inviscid instability? 4.2.3 Role of viscous terms: Early developments 4.3 Linear Viscous Stability Theory

117 119 120 122

4.4 Properties of the Orr–Sommerfeld Equation: Developing Solution Method 128 4.4.1 Compound matrix method (CMM) 4.5 Instability Analysis with the Orr–Sommerfeld Equation 4.5.1 Grid search method: Eigen-Spectrum 4.6 Other Linear Instability Theories 4.6.1 4.6.2 4.6.3 4.6.4

Role of Fourier–Laplace transform: Abel and Tauber theorems Temporal, spatial, and spatio-temporal growth routes Signal problem assumption: Progress or impediment? Temporal instability theory: A case-study with CMM

4.7 Instability Properties Using the Orr–Sommerfeld Equation 4.7.1 Effects of pressure gradient on instability of boundary layers 4.8 Closing Remarks 5 Receptivity Analysis: Relation with Instability Experiments

130 136 136 141 142 144 144 147 148 150 153 155

5.1 Introduction

155

5.2 Linear Receptivity of Boundary Layer: Bromwich Contour Integral Method

157

5.2.1 Receptivity to wall excitation: Frequency response for signal problem 5.2.2 Near-field response of localized excitation 5.2.3 Receptivity to free stream excitation

160 164 171

5.3 Receptivity to Free Stream Excitation: Upstream Propagating Modes 5.3.1 Low frequency free stream excitation: Klebanoff mode 5.3.2 Coupling between wall and free stream modes 5.3.3 Exciting TS waves by pulsating free stream vortex: Linear receptivity 5.3.4 Instability control by wave cancellation with wall and free stream excitation

172 177 180 182 185

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5.4 Linear Receptivity Analysis: Spatio-Temporal Wave Front

187

5.5 Closing Remarks

192

6 Dynamical System Theory of Linear Receptivity

194

6.1 Introduction

194

6.2 Case Studies of Spatio-Temporal Growth

197

6.3 Formulation of Impulse and Frequency Responses

199

6.4 Impulse, Frequency, and other Responses

201

6.4.1 Time scales and meaning of transient growth 6.4.2 Transient growth for smooth start-up 6.5 Non-Oscillatory Start-Up 6.5.1 Non-impulsive wall excitation: Ramp and error function start-up 6.5.2 Mechanism for the formation of STWF 6.6 Closing Remarks 7 Nonlinear, Nonparallel Effects on Receptivity, Instability, and Transition

206 207 207 208 208 210 212

7.1 Introduction

212

7.2 Nonparallel and Nonlinear Effects on Receptivity and Instability

214

7.3 STWF Created by Impulsive Start

223

7.3.1 Effects of excitation amplitude of wall-normal velocity 7.4 STWF Created via Non-Impulsive Start 7.4.1 Receptivity to high amplitude wall excitation

229 233 238

7.5 Bypass Transition by Free Stream Excitation: By Pulsating Stationary Vortex

242

7.6 Closing Remarks

247

8 Three-Dimensional Routes of Transition to Turbulence

248

8.1 Introduction

248

8.2 Different Routes of Three-Dimensional Transition

251

8.3 Three-Dimensional Receptivity of Boundary Layer to Wall Excitation

253

8.3.1 Governing equation 8.3.2 Boundary conditions 8.3.3 Initial condition for disturbance field

253 255 256

8.4 Grid System

257

8.5 Numerical Methods

258

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8.5.1 Three-dimensional route of transition by harmonic wall excitation 264 8.5.2 Spectrum of inhomogeneous turbulent flow over flat plate 271 8.6 Effects of Frequency and Wavenumber on Three-Dimensional Transition 273 8.6.1 8.6.2 8.6.3 8.6.4

Computational domain Grid parameters Boundary conditions Spanwise modulated wall excitation

274 275 275 275

8.7 Effects of Frequency and Wavenumbers on Three-Dimensional Transition 277 8.7.1 8.7.2 8.7.3 8.7.4

Different frequency and wavenumber cases of evolving STWF Evolution of disturbances from laminar to turbulent flow Energy propagation speed and group velocity Maximum disturbance amplitude growth for high and low frequencies 8.7.5 Near-field solution and receptivity routes

8.8 Closing Remarks 9 Receptivity to Free Stream Excitation: Theory, Computations, and Experiments

277 278 281 281 284 286

288

9.1 Introduction to Free Stream Excitation

288

9.2 Experiments on Transition without TS Waves

293

9.3 Direct Numerical Simulation of Three-Dimensional Vortex-Induced Instability

298

9.3.1 Problem set-up and numerical methods 9.3.2 Three-dimensional equilibrium flow: Initial condition 9.3.3 Imposed free stream excitation

299 300 300

9.4 Validation of Direct Numerical Simulation for Vortex-Induced Instability 303 9.4.1 Strong vortex-induced instability caused by counter-clockwise free stream vortex 303 9.4.2 Weak vortex-induced instability for counter-clockwise free stream vortex 305 9.4.3 Weak induced perturbation by clockwise free stream vortex 306 9.5 Vorticity Dynamics during Vortex-Induced Instability 9.5.1 Evolution of three-dimensional response during vortex-induced instability 9.5.2 Evolution of three-dimensionality during vortex-induced instability

307 307 316

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9.6 Nonlinear Instability Theory Based on Disturbance Energy and Enstrophy 9.6.1 Nonlinear instability study by disturbance mechanical energy

320 320

9.7 Disturbance Enstrophy Transport Equation: Three-Dimensional Vortex-Induced Instability

328

9.8 Dependence of Instability on Speed and Strength of Free Stream Vortex

336

9.9 Closing Remarks

342

10 Nonlinear Receptivity Theories: Hopf Bifurcations and Proper Orthogonal Decomposition for Instability Studies 343 10.1 Introduction

343

10.2 Nonlinear Effects and Stuart–Landau Equation

346

10.2.1 Nonlinear stability: Stuart–Landau equation

349

10.3 Receptivity of Vortex-Dominated Flows

351

10.4 Multimodal Dynamics of Vortex-Dominated Flows

352

10.5 Proper Orthogonal Decomposition and Nonlinear Instability

352

10.5.1 Instability modes and multimodal Stuart–Landau equation 10.5.2 Formulation and modeling of a multimodal Stuart–Landau equation 10.5.3 Reconstruction of instability modes using multimodal Stuart–Landau equations 10.5.4 Modeling the anomalous mode of the first kind (T 1 -mode)

361 367 369 373

10.6 Reduced Order Model Using Multimodal Reconstruction by POD Analysis

374

10.7 Universality of POD Modes

376

10.7.1 Role of disturbances in triggering instabilities 10.8 Forced LDC Flow and Sub-Critical Instability 10.8.1 Threshold amplitude for primary instability 10.8.2 Frequency spectrum of sub-critical excitation

379 382 382 384

10.9 Multiple Hopf Bifurcations for Different Grids

388

10.10 Towards Grid-Independence and Universality

390

10.11 Closing Remarks

397

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11 Mixed Convection Flow

399

11.1 Instability of Mixed Convection Flows

399

11.2 Mixed Convection Flow: Governing Equations

401

11.3 Mixed Convection Boundary Layer Flows

403

11.3.1 Similarity transform of boundary layer flow

407

11.4 Equilibrium Solution for Isothermal Wedge Flows

421

11.4.1 Boundary layer equation for flow over isothermal wall 422 11.4.2 Governing equations and boundary conditions for DNS of mixed convection flows 423 11.4.3 Boundary and initial conditions 424 11.4.4 Linear modal theory for viscous instability: Spatial and temporal routes 434 11.4.5 Linear spatial and temporal theories for mixed convection boundary layer 438 11.4.6 Temporal viscous growth rates for mixed convection boundary layer flows 441 11.5 Strict Spatial and Temporal Linear Instability Theories: Re-evaluating Mixed Convection Flows 11.5.1 DNS of mixed convection flow instability: New theorems of instability 11.6 Closing Remarks 12 Baroclinic Instability: Rayleigh–Taylor Instability

442 443 457 459

12.1 Baroclinic Instability

459

12.2 Numerical Simulation of Rayleigh–Taylor Instability

464

12.2.1 Physical domain and auxiliary conditions

468

12.3 Numerical Methods and Validation 12.3.1 12.3.2 12.3.3 12.3.4 12.3.5

Validation of numerical results for Rayleigh–Taylor instability How sensitive is baroclinic instability? Numerical amplification rate and error dynamics Creation of pressure and density fronts during primary instability Baroclinic vorticity generation during primary instability

12.4 Entropy and Vorticity Creation during Rayleigh–Taylor Instability 12.4.1 Relationship between entropy and vorticity 12.5 Role of Bulk Viscosity on Rayleigh–Taylor Instability 12.5.1 Ash et al.’s model for bulk viscosity

469 469 470 473 478 478 481 482 485 486

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12.5.2 Entropy of the system: Role of bulk viscosity 12.6 Pressure Waves: Outcome of Non-Periodic Boundary Conditions and Compressible Formulation 12.6.1 Compression and rarefaction fronts during Rayleigh–Taylor instability 12.7 Closing Remarks 13 Coherent Structure Tracking in Transitional and Turbulent Flows

487 488 492 496 497

13.1 Introduction

497

13.2 Definition of a Vortex and Coherent Structures

502

13.2.1 Circular streamlines and path-lines 13.2.2 Vorticity magnitude 13.2.3 Pressure minimum

503 504 505

13.3 Definitions of Coherent Structures Based on Velocity Gradient and Its Invariants 13.3.1 13.3.2 13.3.3 13.3.4

∆-Criterion Q-Criterion λ2 -Criterion Relation between Q- and λ2 -criteria

505 506 506 508 510

13.4 Alternative Methods to Detect Coherent Structures in Transitional and Turbulent Flows

512

13.5 Wall and Free Stream Excitation for DNS

514

13.5.1 Wall excitation case 13.5.2 Free stream convected vortex case

516 516

13.6 Vorticity Dynamics of Flows Caused by Wall and Free Stream Excitations 518 13.7 Disturbance Tracking Using DME

521

13.8 Disturbance Tracking by DETE

522

13.9 Comparing Coherent Structure Detection Methods

525

13.9.1 13.9.2 13.9.3 13.9.4

Correlation for wall excitation case Correlation for free stream excitation case Comparison of DETE method and λ2 -criterion: Wall excitation case Free stream excitation case: Comparing DETE method and λ2 -criterion 13.9.5 Comparing DME method and Q-criterion: Wall and free stream excitation 13.10 Closing Remarks

528 530 530 535 535 539

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14 The Route of Transition to Turbulence: Solution of Global Nonlinear Navier–Stokes Equation

541

14.1 Introduction

541

14.2 Three-Dimensional Impulse Response of the Boundary Layer over Semi-Infinite Flat Plate by Different Wall Excitations

544

14.3 Governing Equations, Boundary Conditions, and Numerical Methods for Subduction Motion

546

14.4 Direct Numerical Simulation of Dip-Slip and Strike-Slip Events

548

14.5 Nonmodal Disturbance Growth and Squire’s Theorem

556

14.6 Linear Stability Analysis Caused by Different Spanwise Excitation

565

14.6.1 Results and discussion for linear stability analysis

566

14.7 Nonmodal, Nonlinear Global Route for Free Stream Excitation

569

14.8 Comparing Linear and Nonlinear Global Instability Problem

573

14.9 Closing Remarks

582

References

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Preface

The subject of this book has kept fluid dynamicists occupied for nearly two centuries, ever since the governing equation for fluid motion was developed by C. L. Navier and G. G. Stokes. While mathematicians are concerned with exploring the existence and uniqueness of a solution to the Navier–Stokes equation, physicists are fascinated by the fact that some “exact” laminar solutions are not observable. Early pioneers attributed this facet of the solution to the lack of stability of such exact or equilibrium solutions. This is how instability of fluid flows became a major subject of investigation, with near-unanimity among scientists that base flow is capable of feeding omnipresent background disturbances leading to transition. As there are many equilibrium flows, the routes by which disturbances grow can also be vastly different. For example, the classical pipe flow experiment of Osborne Reynolds demonstrated that the transition of laminar flow to seemingly chaotic turbulent flow depends on the flow velocity and size of the pipe. Thus, the experiment highlighted the relationship of the instability with physical parameters. Now we know that the transition to turbulence in pipe flow depends upon the Reynolds numbers and background disturbances. The fact that quantitative description of transition to turbulent pipe flow still eludes us will encourage potential readers to embark upon research in this challenging field. One of the early forays in flow instability studies has been the development of the eigenvalue analysis. It was adopted by Kelvin and Helmholtz to qualitatively explain interfacial instabilities such as those arising during the creation of surface gravity waves in lakes and oceans. The eigenvalue analysis remains the pedagogical tool to explain the phenomenon of instability and introduce the dispersion relation between spatial and temporal scales. In a similar vein of studying disturbance growth as an inviscid phenomenon, with equal ingenuity, Rayleigh developed the governing stability equation and a theorem to explain instabilities for jets afflicted by temporally growing disturbances. The failure of this inviscid theory in explaining flow over a flat plate prompted the development of viscous linear instability theory, known as the Orr–Sommerfeld equation. From the wave-like solutions obtained from this equation grow, in space, the well-known Tollmien–Schlichting (TS) waves. Experimental demonstration of the TS wave was heralded as a landmark event for external flow transition. This remained the main focus of researchers for decades following the Second World War. The eigenvalue or normal mode approach was extensively used in many empirical studies.

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Unfortunately, this approach did not solve most problems of transition to turbulence, including the flow over a flat plate. Pipe and Couette flows were predicted to be unconditionally stable by the linear theory. Also, the binary approaches of either temporal or spatial instability theories, overlooked the possibility of exploring spatio-temporal growth of disturbances, until this was attempted by researchers at IIT Kanpur in early 1990s. This was the cornerstone for performing receptivity studies using the well-established Bromwich contour integral method of solving the Orr– Sommerfeld equation. The approach led to the discovery of the spatio-temporal wave front (STWF) as the basic unit for spatio-temporal growth of disturbances. Later on, the existence of this STWF would also be verified by solving the Navier–Stokes equation. While the STWF was being established by the author’s group, there were numerous contemporary efforts to explain the inadequacy of the Orr–Sommerfeld equation in explaining instability and transition. Apart from pointing out the inadequacy of normal mode analysis, presence of multiple time scales in the transitional flow prompted researchers to look for transient growth of energy or algebraic initial growth of disturbance field. The basic idea behind such approaches rested on nonnormal interactions among multiple present modes. The only shortcoming of these approaches was the absence of any evidence for such extraordinary claims. The lack of progress by these approaches is due to two misconceptions, apart from the fact that these approaches tried to synthesize transient growth at onset, followed by spatial growth. There are no mathematical foundations, as Schmid, Henningson, Brandt, et al., erroneously state that disturbance quantities cannot be expressed by Fourier–Laplace transform at early times, “as it only describes the asymptotic fate of the perturbation and fails to capture short term characteristics.” Such damning indictment of a universal tool is unfortunate, as the existence of the well-founded Abel’s and Tauber’s theorems in Operational Calculus, establishes that Fourier– Laplace transforms are equally suitable to provide disturbance field universally for all times. Additionally, all such nonmodal approaches use matrix eigenvalue analyses, which are plagued with spurious modes. It is known that for flow over a flat plate at moderate Reynolds number, only a few eigenmodes exist (only two/three eigenvalues are obtained from solution of the Orr–Sommerfeld equation with the compound matrix method). If the same problem is solved with matrix eigenvalue analysis using N points, it returns N eigenvalues. There are no well-established methods known to remove these numerous spurious eigenvalues. Furthermore, interactions of nonorthogonal modes proposed in these approaches can only demonstrate “transient growth” for a three-dimensional disturbance field. There has been recent progress in developing nonlinear, nonmodal approaches based on dynamical system theory which show potential for studying transition to turbulence. These seek out finite amplitude solutions disconnected from the equilibrium solution in the phase space, populated by solutions obeying the governing Navier–Stokes equation. These nonlinear instability studies, under the action of large

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disturbances, take the flow beyond the “basin of attraction” of the equilibrium state. In contrast, this book advocates dynamical system approaches following receptivity route developed in HPCL, IIT Kanpur relating cause (input) and effect (output). The members of the author’s research team have contributed greatly in producing much of the contents in the book. The treatment used in the book is kept at an easy pace accessible to senior undergraduate students. The book contains fourteen chapters, specifically designed to explain the fundamentals of receptivity and its connection to classical instability theories. This is done by considering the flow over a flat plate in the first nine chapters. The materials would show how the classical instability theories advocating solely spatial or temporal growths must be replaced by the generic spatio-temporal theory in the receptivity framework. Adopting the Bromwich contour integral, which is solved by the compound matrix method, is shown to be essential to liberate readers from the spurious modes of other methods. Understanding the contents requires some familiarity with operational calculus, often used in electrical science. Its use in fluid mechanics is testimony to the author’s willingness to take the readers out of their comfort zone of following traditional routes. A firm grip on the concept of Fourier–Laplace transform is necessary, and so is focus on details such as start-up process of fluid dynamical systems. This liberates one from the legacy of the spatial theory of instability that fixes the time scale to excitation frequency. This assumption is shown to be surprisingly inadequate. To study instability, one must retain all possible complex frequencies for possible spatiotemporal growth of response field. By questioning such well-entrenched dogma of eigenvalue analysis, the author shows the existence of the STWF, a major contribution in receptivity theory. This is one of the main results discussed in Chapters 1 and 6. Receptivity in the framework of dynamical system theory is introduced in Chapter 2, and its use is expanded in Chapters 5 to 9 and 14. The book also explores facets of classical results, which have previously been unexplored, to gain newer insights. For example, results attributed to G. I. Taylor on the role of free stream turbulence have been used in Chapters 9 and 14, to explain receptivity of the equilibrium flow by highlighting its inhomogeneous structure. The classical eigenvalue analysis is incapable of incorporating input disturbances and distinguishing between wall and free stream excitations. This is a central issue in mathematical physics: the way eigenvalue problems involve only solving homogeneous differential equations with homogeneous boundary conditions. In the receptivity framework, excitation applied at the wall or in the far field produces qualitatively different responses. Chapter 10 deals with the topic of Hopf bifurcation, Landau equation and its extension. Efforts are made to provide a new outlook by not only deriving the Stuart–Landau equation, but also by explaining the special features of the results reported by F. Homann (1936). This evidence is convincing in grasping multiple Hopf bifurcations and performing multimodal analysis by extending the eigenfunction expansion approach of Ekhaus (1965). Here it is called the Stuart–Landau–Ekhaus equation, something the reader may not note elsewhere. Similarly, the explanation

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provided for the Klebanoff mode in Chapter 5 is a unique presentation for the onset of some bypass transition problems. Chapter 11 is related to instability of mixed convection flows and provides a fresh perspective on the topic. The author explores stability of this flow, which does not possess any canonical equilibrium flow with the exception of two solutions used in the chapter. The isothermal case considered is for a wedge flow, while the adiabatic flat plate with a singular heat transfer is pedagogic. These two cases solved by direct numerical simulation (DNS) show that either intense heating or cooling can destabilize the flow, while the linear viscous theory indicates cooling to stabilize the flow. In discussing these perplexing results, the author’s group found out a new inviscid mechanism of instability, explained by two theorems (called simply as Theorems I and II). These theorems are more generic than what is advocated by Rayleigh’s and Fjfrtoft’s theorems in hydrodynamics. Recent trends in the study of transition and turbulence emphasize on direct simulation of flows. This is also evident in this book, with the fundamentals highlighted early in Chapter 3. The aim is to tie up the principal concepts of transition and scientific computing together in the form of the dispersion relation. The readers would appreciate the centrality of dispersion relation in ensuring accuracy of computing, as noted in Chapters 7 to 12. A specific type of interfacial instability, namely the Rayleigh–Taylor instability, is explored in great detail in Chapter 12. The results act as evidence for the requirement of high accuracy computing in tracing disturbance structures of spikes, bubbles and pressure fronts from the onset. Results are also presented, which avoid using the well-known Stokes’ hypothesis, by incorporating nontrivial bulk viscosity for the governing equations of motion for compressible flows. The associated effects on entropy and vorticity creation during the baroclinic instability are explained. Recently developed theoretical methods, based on the concept of disturbance mechanical energy and disturbance enstrophy transport, are introduced in this book. These serve the dual purpose of not only diagnosing receptivity in Chapter 9 for incompressible Navier–Stokes equation without any approximation, but also for tracing coherent structures in Chapter 13. These exact methods can be used to study unsteady equilibrium flows, for linear and nonlinear instabilities. The method of proper orthogonal decomposition has also been used to study nonlinear instabilities in Chapter 10 for vortex-dominated flows. The emphasis of the book is to explain transition of wall-bounded flow and few recent results are elaborated in Chapter 14. These reveal the state-of-art for DNS of transitional flows. One such case shows through the solution of the Navier–Stokes equation that a nonmodal, nonlinear approach is the only correct approach to explain free stream excitation problem. The writing of such a book with very recent results should make it abundantly clear that this is a fruit of labor of all the, past and present, research colleagues of the author at HPCL, IIT Kanpur. The author gratefully acknowledges help by all the contributors. Many figures and tables have been provided by the recent students and associates. I

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would fail in my duty if I do not single out help from Prasannabalaji, S., Suman, V. K., Soumyo Sengupta, Aditi Sengupta, Pushpender Sharma, and M. B. Raj, who have put up with my requests for material and participated in many useful discussions. Among the other colleagues and students, I must mention Sandip Nijhawan, Manish Ballav, A. P. Sinha, U. S. Chauhan, K. Venkatasubbaiah, A. Kameshawar Rao, Swagata Bhaumik, Yogesh Bhumkar, Manoj T. Nair, Vivek Rana, Manojit Chattopadhyay, T. T. Lim, K. S. Yeo, Y. T. Chew, Anurag Dipankar, V. Mudkavi, and A. K. Ray. On various occasions, I have benefitted from discussions and advice from colleagues: Wilhelm Schneider, Michel Deville, Herbert Steinrueck, Pravir Dutt, Pierre Sagaut, J. M. Kendall, O. N. Ramesh, J. S. Mathur, V. Ramesh, Thierry Poinsot, S. K. Lele, K. R. Sreenivasan, Sakir Amiroudine, P. Spalart, J. Jimenez, P. A. Davidson, T. W. H. Sheu, Francois Gallaire, Mejdi Azaiez, Eric Laurendeau, Joshua Brinkerhoff, H. D. Vo, J. Soria, R. Friedrich, D. V. Gaitonde, A. Ooi, Kalyanmoy Deb, and S. Girimaji. The contents of this book have been carefully collated by the author and students, spread over more than three decades, in HPCL. The character and vibrancy of HPCL have been enriched from contributions by Gaurav Ganeriwal, Prasoon Suchandra, Manish Kumar, D. Patidar, Amit Kasliwal, Jyothi K Puttam, Ankit Bhadouria, Himanshu Singh, Lucas Lestandi, S. B. Krishnan, R. Roy Chowdhury, Pramod Bagade, Manoj Rajpoot, Kartikeya Asthana, Unnikrishnan S., Sreejith, N. A., Vijay Vedula, Sahil Bhola, Nidhi Sharma, Jyothi Sangwan, G. Pallavi, S. I. Haider, K. Saurabh, M. K. Parvathi, Akhil Mulloth, Ashvin V. M., Sriramakrishnan M., Z. A. Shabab, A. Mittal, Atchyut S., Rikhi Bose, V. K. Sathayanarayana, Vishal Garud, Neelu Singh, Ashish Bhole, Shameem Usman, K. Sai Theja, Nishat Hussain, Shakti Saurabh, Roshan J. Samuel and last but not the least by Shrikanth Babu Talla. During this long sojourn my wife, Soma, son, Soumyo, and daughter, Aditi, have provided me strong support and they are my source of joy. They have witnessed my various moods of satisfaction, elation and frustration. We have gone the distance together and I admire their patience with my preoccupation with research and writing books. Without the strong support of the family and my students this journey would not have been possible, specifically in a strange place like IIT Kanpur. The book has been patiently and carefully typed by Mrs. Baby Gaur, who also helped in drawing many figures for the book, apart from doing other work related to it. It is also my pleasant duty to acknowledge the help and co-operation provided by the Cambridge University Press editorial team, copy-editing team and for the overall supervision by Ms. Taranpreet Kaur, Ms. Qudsiya Ahmad and other colleagues. Despite all the careful scrutiny and efforts, if there are any errors, I own up the responsibility for the mistakes and would gratefully acknowledge readers in advance for bringing them to our attention.

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Receptivity, Instability, and Transition: A Perspective

1.1 Historical Introduction In many natural or engineering fluid flows, turbulence is the natural state. However, in the eighteenth century, when fluid dynamicists were divided into two distinct and separate schools of thought belonging to hydrodynamics and hydraulics, they did not have the benefit of the Navier–Stokes equation (NSE, not developed at that time) that governs all incompressible fluid flow behavior. The hydrodynamics practitioners did not envisage the importance of viscous flow, as it was thought to be confined to a very narrow region near the body placed in a flowing fluid, while most of the flow region was considered to be inviscid. This was the justification for the use of Euler’s equation. Hydraulics practitioners approached their problems with charts and tables obtained empirically from actual observations. This segregation of thought continued for another century until the advent of the boundary layer theory, proposed by Prandtl [338], which we will visit later in the book. After the derivation of the viscous flow equations [311, 501] by introducing the constitutive relation between the stress and rate of strain to obtain the Navier–Stokes equation [20, 412], Stokes tested the equations using pipe flow experiments. There was absolutely no match between the “exact” solution of the Navier–Stokes equation and the experimental observations. There could have been various reasons for this: for example, the analytical solution of the Navier–Stokes equation is obtained after many simplifying assumptions; moreover, the correctness of the constitutive relation and no-slip condition has not been rigorously established even today, and so what is an “exact” solution? We will revisit the constitutive relation between the stress and rate of strain while discussing the Rayleigh–Taylor instability problem.

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It is pertinent to note that, in the absence of rigorous proof, even today, the no-slip condition is considered as a modeling approximation. Although Batchelor [20] noted that for Newtonian fluid flow, the absence of slip at a rigid wall is now amply confirmed by direct observation and by the correctness of its many consequences under normal conditions. In microfluidics, continuum equations are often solved with slip boundary conditions, while solving the Navier–Stokes equation [217].

1.1.1 Introduction to flow instability In retrospective, one can observe now that the analytical solution obtained by Stokes for pipe flow was acutally for a laminar flow, while the real flow was turbulent for the operating conditions. This prompted researchers such as Reynolds, Kelvin, Helmholtz, Rayleigh, and others to conclude that the analytical solution (the equilibrium flow) is not stable with respect to omnipresent ambient disturbances, however small those might be. Mathematically, this required any solution to possess a new condition, apart from existence and uniqueness. Landau and Lifshitz [255] summed up the situation by stating that the equilibrium solution is not observable due to instabilities caused by the action of ambient disturbances. In other branches of continuum mechanics, such instability problems have been studied earlier, for example, the problem of Euler column buckling as static instability. This contributed to the development of the science of hydrodynamic instability. According to Betchov and Criminale [28], the first major contribution to the study of hydrodynamic stability can be found in the theoretical papers of Helmholtz [177], which apparently reports the first attempt to quantitatively describe instability of flows. Helmholtz [177] and Kelvin [522] were the pioneers to characterize, what is now known as the Kelvin–Helmholtz instability, by considering two dissimilar fluids streaming over each other, with the interface experiencing unstable breakdown, something akin to the creation of surface gravity waves in oceans and lakes. However, such a pedagogic analysis produces results with wave parameters which are distinctly different from physical surface gravity waves [119, 412].

1.1.2 Inviscid instability theory A major difficulty faced at the beginning of instability studies was the identification of the equilibrium state, even for the canonical zero pressure gradient flow over a semi-infinite flat plate. As the instability of flow, by definition, is due to the growth of imperceptible disturbances, this prompted researchers to develop a small perturbation theory for the disturbance flow field. The most prominent development in this regard is due to the study of flow instability using a linear theory resulting in Rayleigh’s stability equation and a theorem was proposed for the problem, noted in multiple papers by Rayleigh [351, 353, 356], with a focus on inviscid temporal instability. Although Rayleigh’s equation is a non-stiff ordinary differential equation, it can only be solved for velocity profiles provided at any streamwise station. However, this

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was beyond the scientific expertise of those days, and instead Rayleigh developed his theorem to provide a necessary condition for temporal instability, that is, when disturbances grow in time. It has been noted, however, that for external wall bounded flows, disturbances do not merely grow in time. Therefore, the theorem was not a successful attempt; it only showed good qualitative agreement for jet flows [350]. In Rayleigh’s stability equation, the equilibrium flow and its second wall normal derivatives are needed, implying again the importance of obtaining the viscous equilibrium flow. Rayleigh’s instability criterion states a necessary condition for instability: the velocity profile should have an inflection point inside the range of the flow profile. However, going by this criterion, Couette flow and flow in a pipe, will be stable. There are other notable failures in the applicability of inviscid instability theory. Interestingly, a notable success of the application of inviscid instability theory is in the area of mixed convection flows [423]; this is explained in great detail in Chapter 11.

1.1.3 Role of dissipation on instability The inviscid theory started with the incorrect assumption that viscous action in fluid flow is dissipative, and can be neglected to obtain a more critical instability limit. There are many examples of mechanical non-dissipative systems that exhibit steady equilibrium state which become unstable upon addition of a small dissipation. This has been shown by Thomson and Tait [523]’s study, wherein a conservative statically unstable system was stabilized by gyroscopic forces which again was destabilized by addition of a small amount of dissipation. In fluid dynamical systems, dissipation is one component of viscous diffusion, even though there is a common misconception that they can be treated as identical (See Doering and Gibbon [114] and Sengupta et al. [468]). Thus, Thomson (Lord Kelvin) was aware of the role of viscous dissipation for instability, even though he used an inviscid model to explain Kelvin–Helmholtz instability. A good account of the paradoxical role of dissipationinduced destabilization is given by Kirillov and Verhulst [234] with examples drawn from purely mechanical systems relating to what is also known as the Ziegler paradox [575] and the Whitney umbrella singularity [554]. All of these led to applications in the field of flutter analysis related to aeroelasticity [47, 182, 576].

1.1.4 Viscous instability: Linear equations After the initial setback of Rayleigh’s equation for inviscid analysis of hydrodynamics, scientists came out with a viscous governing equation for disturbance quantities, known as the Orr–Sommerfeld equation [321, 495]. In Rayleigh’s theorem, the governing equation is not solved and instability is related to the presence of an inflection point in the streamwise component of the equilibrium flow for temporal growth. There is no such analysis and theorem for the study of the Orr–Sommerfeld equation. It is now well known that the solution of the Orr–Sommerfeld equation is far more difficult than solving Rayleigh’s equation because the former is a stiff, variablecoefficient ordinary differential equation. It is even more difficult than solving the

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governing Navier–Stokes equation [119, 412], which impeded immediate successful solution of the Orr–Sommerfeld equation. It is sobering to realize that while the Orr–Sommerfeld equation was being derived there were reports of the Navier–Stokes equation being solved by a desk calculator [369]; this is despite the fact that the former is an outcome of linear analysis of the latter, which is a nonlinear equation.

1.1.5 Temporal, spatial, and spatio-temporal instability studies The readers should note that Rayleigh’s equation is for temporal growth; as shown by Betchov and Criminale [28], this equation does not support disturbances which grow in space. It is this failure which prompted researchers to look for spatial growth of the disturbance field with the Orr–Sommerfeld equation for different equilibrium flows, while ignoring a more generic approach for spatio-temporal growth. This situation prevailed until the publishing of the results in [418, 451, 452], where the authors reported for the first time on spatio-temporal growth of disturbances. It is perhaps relevant to mention that Brillouin [67] has investigated spatio-temporally growing disturbances in electromagnetic wave propagation and Bers [27] has attempted the same for plasma physics. This aspect of disturbance in wave packet propagation is the central theme of this book, and will be described in subsequent chapters. The initial absence of tangible progress regarding linear instability theory, based on growth of small perturbations [396], led to the pronouncement that the small perturbation theory had failed to provide any useful results concerning the origin of turbulence [317]. This observation came as a direct consequence of the inability to solve the Orr–Sommerfeld equation. The supposed failure of small perturbation theory was reported after the demonstration [511] that viscous action causes atmospheric flows as a consequence of instability. This work did not attract sufficient attention as the problem solved was related to the dynamic stability of atmosphere, with the equilibrium flow being given by the diffusion equation for temperature and moisture content (a problem not directly related to the origin of turbulence). The phenomenological “supporting experimental evidences” were drawn from the measurement of temperature by a kite off the coast of Labrador in the North Atlantic Ocean; or to measure wind velocity in Salisbury Plain! We draw the readers’ attention to a short, coherent self-consistent description based on thermodynamic considerations for the dynamic instability of the atmosphere [412, 521]. The heuristics employed in [511] settled the issue of the no-slip condition inherent in viscous flows and its absence in inviscid flows; the study also reconciled works on flow instability by Rayleigh and Reynolds. It is interesting to note that the author [511] addressed the following case as perhaps the most important result of Lord Rayleigh’s investigation. A particular case of laminar motion in which d2 U/dy2 has the same sign throughout the fluid is that of an inviscid fluid flowing with the same velocity, or a viscous liquid moving under pressure between two parallel planes. In this case, unstable motion should be impossible by the inflection point requirement. Osborne Reynolds [365], however, working from an entirely different angle, had come

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to the conclusion that a viscous fluid moving in a pipe was unstable if the coefficient of viscosity was less than a certain value, which depended on the diameter of the pipe and on the velocity of the fluid. Reynolds’ result is in accordance with our experimental knowledge of the behavior of actual fluids. It is evident that there is a fundamental disagreement between the two results, as according to Reynolds, the more nearly inviscid the fluid, the more unstable it is likely to be; while according to Rayleigh, instability is impossible when the fluid is inviscid. We note that the misunderstanding is due to the interpretation of the role of Reynolds number in arriving at the correct disturbance flow, which requires a no-slip condition for viscous flows to fix a relation between time and spatial scales of the disturbance field; this will be described later as the dispersion relation! In conclusion though, Taylor noted that the complete absence of slip assumed in Reynolds’ work enables the necessary amount of momentum to escape; hence, a type of disturbance may be produced that is dynamically impossible under the condition of perfect slip at the boundaries, that is, implying the centrality of viscous action.

1.1.6 Similarity profile for equilibrium flow: Tollmien–Schlichting waves from the Orr–Sommerfeld equation With the advent of the boundary layer theory, Blasius solved the Navier–Stokes equation with the help of similarity solution, giving rise to the Blasius profile used as the equilibrium flow for innumerable instability studies, apart from being used in many engineering applications for drag estimation [337]. Yet the Orr–Sommerfeld equation remained unsolved for more than two decades, until Tollmien reported the first neutral eigenvalues and critical Reynolds number [527]. This work was facilitated by the earlier appearances of a solution of the Orr–Sommerfeld equation by Heisenberg [176] and Tietjens [524]. While the work of Heisenberg was abstract, the work of Tietjens was severely restricted, as he tried to solve the Orr–Sommerfeld equation using a linear piecewise collection of velocity profiles. Tollmien solved the Orr–Sommerfeld equation in terms of Airy functions [25], an approach that can handle the stiffness of the governing equation [119]. This development accelerated the progress of instability research, and was reported by Tollmien and Schlichting [392, 393, 394, 395, 529]. The approach followed is for the spatial growth of disturbances, which requires fixing a real temporal frequency for the excitation and calculating a complex spatial eigenvalue from the Orr–Sommerfeld equation, posed as an eigenvalue problem. A complex wavenumber as an eigenvalue implies wavy solution as the eigenfunctions; these spatial disturbances are known as Tollmien–Schlichting (TS) waves. The imaginary part of the complex wavenumber would determine amplification or attenuation of the progressive TS wave it would be in the form of a wave packet [412]. This theoretical result was not embraced immediately due to the absence of experimental verification, which appeared after almost two decades; the credit for the experiment goes to the NBS Washington team, and was reported by Schubauer and Skramstad [405].

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All these results have been obtained for the canonical zero pressure gradient boundary layer, given by the similarity solution due to Blasius. The inadequacy of the linear theory, as in any instability study, is based on the theory not requiring information about the disturbance environment. Mack [285] has emphatically noted that such a theory tells nothing about turbulence, or about the details of its initial appearance, but it does explain why the original laminar flow can no longer exist. For these reasons, and mainly because of the failure to verify TS waves experimentally, there was poor acceptance of the linear instability theory.

1.1.7 Instability theory, experiments, and some unanswered questions One of the continuing legacy of the simplification used in linear theory was the assumption that the equilibrium flow is parallel; this assumption continued till the solution of the Navier–Stokes equation appeared. In the context of TS waves obtained by the spatial theory, the appearance of the wave is associated with an imposed time scale. The imposed time scale is equivalent to performing a frequency response study by exciting the equilibrium boundary layer in a moderate frequency range. Unfortunately, Taylor [514] vibrated a diaphragm at a very low frequency and failed to detect the TS wave. This was enough for him to consign the linear theory to the background [514]! The logic was simply that if the onset of instability itself is not predicted by linear theory, then it is not possible to arrive at the nonlinear stage of turbulence via disturbance growth. However, around the same time, a team led by Dryden [121] performed the classical vibrating ribbon experiment, reported later by Schubauer and Skramstad [405], to show the existence of TS waves experimentally; they used oscillograph traces at fixed locations. Having scored this symbolic success in detecting TS waves, the research community concluded that the TS wave is responsible for the eventual transition of smooth flows to turbulence. However, in this euphoria, lots of pertinent observations and questions have been left unanswered: (i) Why have no definitive experiments been performed and reported that relate the TS waves with actual transition? (ii) Why is it necessary to use a vibrating ribbon inside the boundary layer for the experiment? Can there be other means of exciting TS waves? (Note that Schubauer and Skramstad [405] failed to excite TS waves by acoustic excitation from the free stream.) (iii) If the boundary layer is excited at a given frequency, why is the response also at the same frequency (this is the signal problem assumption and has never been demonstrated experimentally)? (iv) In a growing boundary layer, how do the signal properties continuously keep changing as the TS wave moves downstream? (v) Why should the boundary layer not be investigated for supporting spatio-temporal growth of disturbances? (vi) What happens to the other routes of transition which do not create TS waves? These are known as bypass transition events [298, 412]. In this book, we will note that such classification of transition via natural and bypass routes has outlived its utility and new classification systems will be proposed based on new perspectives [509].

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One of the major difficulties in establishing the instability theory for flows has been in experimentally verifying it. The definition of instability itself is ambiguous; this states that the system will exhibit finite response for an arbitrarily small input, which need not even be measurable! Interestingly, both the earliest experiments by Taylor [514] and Dryden’s group [121, 405] followed the theoretical eigenvalue formulation for generating TS wave using a fixed frequency vortical excitation inside the Blasius boundary layer. While Taylor did not observe the excited TS waves due to wrong choice of frequency of excitation [412, 448], Schubauer and Skramstad succeeded by performing their experiments in an ultra quiet wind tunnel. The principal idea was not to reproduce instability by unquantifiable background disturbances, but to observe TS waves caused by a deterministic and definitive input excitation. This approach of experimental demonstration of TS waves also requires a specific theoretical approach, which has been developed systematically in [32, 34, 412, 419, 449], while many other experiments have been performed, all of which can be termed as receptivity studies. In this approach, the fluid dynamical system is viewed in the framework of the dynamical system theory, as introduced in the next section. One of the major unanswered questions remaining in the field of instability has been the choice of modal and nonmodal approaches which have been used by researchers. Modal analysis poses the problem as one of seeking eigenvalues. However, it has been noted that eigenvalues and eigenfunctions do not provide a complete set of basis function; so that any arbitrary function can be represented by the set. This has prompted researchers to look for alternatives in the form of continuous spectrum. Moreover, some researchers have used the observation that the eigenvalues are not independent of each other or these fail to form orthogonal set to define transients that arise when such non-orthogonal modes interact with each other. A complete synthesis between the modal and nonmodal point of view can be presented using a more generic description of the disturbance field by the Fourier– Laplace transform along Bromwich contours. It is early in the stage of our discussion; however, an introduction to these ideas are presented in Appendix A of this chapter. The associated ideas are further developed in Chapters 3–6 of this book.

1.2 Introduction to Receptivity Analysis Most of the questions in the previous subsection can be answered if the transition experiments are simulated as receptivity problems. This is necessary in view of the first successful experimental demonstration of TS waves [405]. As mentioned earlier, the authors reported success with a vibrating ribbon excited inside the boundary layer, while acoustic excitation from outside the shear layer failed to trigger TS waves. This is one of weakest aspects of the eigenvalue approach used in linear stability theory: The application of a homogeneous boundary condition for the homogeneous differential equation to trace the eigenvalues. In the receptivity approach, the fluid dynamic system is excited by appropriate boundary and /or initial conditions; in this approach,

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it is possible to show that even the simplest possible zero pressure gradient boundary layer is selectively responsive to an imposed disturbance field. In the same context, one can also distinguish between modal and nonmodal growth alluded to in the literature [398, 400]. The disturbance field associated with eigenmodes obtained from linear spatial instability analysis has been termed as modal components of the disturbance field. Any other growth in terms of early transients are associated with instability without eigenvalues [532] or global instability with nonnormality [93]. One of the main aims of this book is to show spatio-temporal growth of the disturbance field which is governed by the Orr–Sommerfeld equation in the linear, parallel flow framework and by the Navier–Stokes equation in the nonlinear, nonparallel flow framework.

1.3 Simple Concepts in Instability Studies We begin this section by looking at highly idealized flow cases. In the first case, a model problem of Kelvin–Helmholtz instability is described, for which we dispense with the need to specify the equilibrium solution. This is followed by a study for the case of the Taylor–Green vortex (TGV), for which one can obtain an exact solution of the governing Navier–Stokes equation. This exact solution is similar to the Poiseuille flow in a channel, for which also one gets an exact solution of the simplified Navier–Stokes equation; the time independent solution has been the basis of many instability studies, without any study being able to find a final answer! However, the TGV solution is time dependent; only very recently has the instabilities of two-dimensional and three-dimensional TGVs explained by a new theory based on disturbance enstrophy [465, 475, 480].

1.3.1 Kelvin–Helmholtz instability In this model problem, the interface between two fluids in relative motion is perturbed initially and its subsequent evolution traced. Essentially, the problem is supposed to mimic the creation of surface gravity waves in oceans or lakes due to movement of air over water. This problem arises when two fluids (of dissimilar species or density) are in relative motion, as is depicted in Figure 1.1. This is therefore a problem of interfacial instability initiated by an imposed perturbation. Helmholtz [177] noted that the interface as a surface of separation is not stable; it is liable to tear apart. Kelvin [522] formulated and solved the problem as an instability problem, as described next by considering a simplified equilibrium flow, which is treated as inviscid and incompressible with constant but different values of properties on either side of the interface. Thus, the two parallel streams on either side have different density and velocity values, as indicated in Figure 1.1.

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Receptivity, Instability, and Transition

z

U2

ρ2, U2, p2

x

Initial deformation ^

η(x, y, t = 0)

U1 y

ρ1, U1, p1

Interface

Figure 1.1 Kelvin–Helmholtz instability at the interface of two flowing fluids. In the absence of any perturbation applied at t = 0, the interface is essentially aligned with the (x, y)-plane, that is, at z = 0. The subsequent displaced interface is expressed by the parametric equation as follows z s = η(x, ˆ y, t) = η(x, y, t)

(1.1)

where  is a small parameter used in the linearized perturbation analysis. One can also view the interface as an infinitesimally thin shear layer of vanishing thickness. As the flow is considered to be inviscid and irrotational, we define velocity potentials in the two fluids on either side of the interface as follows (l = 1 and 2) φ˜ l (x, y, z, t) = Ul x + φl (x, y, z, t) for l = 1; 2

(1.2)

For the inviscid, irrotational flows on either side of the interface, the governing equations are given by the Laplace’s equation, ∇2 φ˜ l = 0

(1.3)

the solutions of which must satisfy the following boundary conditions at the far stream: φ0l s are bounded as z → ±∞

(1.4)

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The other set of boundary conditions is to be applied at z = 0, which is nothing but the no fluid through the interface condition, i.e., ∂ˆη ∂φ˜ l ∂ˆη ∂φ˜ l ∂ˆη ∂φ˜ l − =− − ∂t ∂z ∂x ∂x ∂y ∂y

(1.5)

If we ignore waves of very small wavelengths, then additionally the surface tension effects can be ignored such that the pressure is continuous across the interface. Linearizing this interface boundary condition of Eq. (1.5), one obtains the simplified condition as ∂η ∂η ∂φl + Ul − =0 ∂t ∂x ∂z

for l = 1, 2

(1.6)

where φ˜ l and φl are as related in Eq. (1.2). Defining the pressure on either side by applying unsteady Bernoulli’s equation, one can write pl = Cl − ρl

 ∂φ˜ l

 1 ˜ 2 + (∇φl ) + gηˆ ∂t 2

(1.7)

Linearizing and retaining up to 0() terms, one gets the following 1 1 0(1) condition : C1 − ρ1 U12 = C2 − ρ2 U22 2 2

0() condition : ρ1

 ∂φ

1

∂t

+ U1

  ∂φ  ∂φ1 ∂φ2 2 + gη = ρ2 + U2 + gη ∂x ∂t ∂x

(1.8)

(1.9)

We consider a general interface perturbation at t = 0 described subsequently in the spectral plane by a bilateral Laplace transform as η(x, y, t) =

Z Z

F(α, β, t) ei(αx+βy) dα dβ

(1.10)

with α and β as the wavenumbers in the x- and y-directions, and F as the spectral amplitude. Correspondingly, the disturbance velocity potential is given by φl (x, y, z, t) =

Z Z

Zl (α, β, z, t) ei(αx+βy) dα dβ

(1.11)

Using k2 = α2 + β2 and Eq. (1.11) in Eq. (1.3), one gets the solution satisfying the far-stream boundary conditions of Eq. (1.4) as Zl = fl (α, β, t) e±kz for l = 1 and 2

(1.12)

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Using Eq. (1.10) in the interface boundary condition, Eq. (1.6), one gets F˙ + iαU1 F − k f1 = F˙ + iαU2 F + k f2 = 0

(1.13)

with dots denoting differentiation with time. Moreover, denoting a density ratio ρ = ρ2 /ρ1 , one can linearize the pressure continuity condition of Eq. (1.9) to get ∂φ1 ∂φ2 ∂φ1 ∂φ2 −ρ + U1 − ρU2 + (1 − ρ)gη = 0 ∂t ∂t ∂x ∂x

(1.14)

Using expressions for η and φl from Eqs. (1.10) and (1.11) in Eq. (1.14) to eliminate φl s, one gets f˙1 − ρ f˙2 + iαU1 f1 − iαρU2 f2 + (1 − ρ)gF = 0

(1.15)

Next, one can eliminate f1 and f2 from Eq. (1.15) using Eq. (1.13) to finally get (1 + ρ)F¨ + 2iα(U1 + ρU2 )F˙ − {α2 (U12 + ρU22 ) − (1 − ρ)gk}F = 0

(1.16)

This ordinary differential equation for interface displacement F with respect to time is interpreted better using Fourier transform defined by F(., t) =

Z

¯ ˆ ω) F(., ¯ eiωt dω ¯

(1.17)

Using this ansatz, one can relate the circular frequency with wavenumber by what is known as the dispersion relation, which is obtained by substituting Eq. (1.17) in Eq. (1.16) as −ω ¯ 2 (1 + ρ) − 2αω(U ¯ 1 + ρU2 ) + (1 − ρ)gk − α2 (U12 + ρU22 ) = 0

(1.18)

This being a quadratic equation for ω, ¯ one obtains the roots as ω ¯ 1,2

α(U1 + ρU2 ) =− ∓ (1 + ρ)

p gk(1 − ρ2 ) − α2 ρ(U1 − U2 )2 (1 + ρ)

(1.19)

The importance of the dispersion relation is that it relates the space–time dependence of the dependent variables by their functional form in the spectral plane. Note that in this model, the governing equation is not explicitly dependent on time (as governed by the Laplace equation in Eq. (1.3)) as no time derivative appears. Yet the spatial and temporal scales are related due to the time-dependent boundary conditions. This is the physical significance of the dispersion relation, which integrates the space and time dependence of the problem by considering the governing equation along with boundary and initial conditions.

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From the aforementioned expression, one notices that if the second quantity remains real and is smaller than the first quantity, the ensuing perturbation will always be damped. Conversely, if the second quantity becomes imaginary, then there is a possibility that the interface disturbance will grow, as suggested by Helmholtz. All these possibilities are described next based on the dispersion relation for the model Kelvin–Helmholtz problem. Following sub-cases can be observed: CASE A: If the interface is disturbed in the spanwise direction only, i.e., α = 0, then s ω ¯ 1,2 = ∓

gβ

(1 − ρ) (1 + ρ)

(1.20)

Curiously, for this case the unsteadiness does not depend upon the streaming velocities U1 and U2 . If ρ > 1, i.e., the heavier liquid is flowing over a lighter liquid, then the buoyancy force due to interface disturbances creates temporal instability (for real spanwise wavenumber β). This is a toy model for the Rayleigh–Taylor instability, which will be treated later in Chapter 12 with the solution of the compressible Navier– Stokes equation specifically with different constitutive relations between stress and rate of strain. CASE B: Instead of CASE A type, if we apply any arbitrary interface perturbation, and if the quantity under the radical sign in Eq. (1.19), i.e., gk(1 − ρ2 ) − α2 ρ(U1 − U2 )2 < 0, then the subsequent interface displacement will grow with time, which follows from Eq. (1.17). This instability condition  can be rearranged in terms of shear between the two mediums as (U1 − U2 )2 >

gk 1−ρ2 ρ α2

.

Therefore, for the given shear at the interface (U1 − U2 ), and for an oblique propagation direction of perturbation at the interface given by the wavenumber k, instability will occur for those wavenumbers kcr , which satisfy the inequality given by kcr >

 k 2 cr

α

ρ g ρ2  1 − (U1 − U2 )2 ρ2 ρ1

The wavenumber vector makes an angle γ1 with the x-axis, given by cos γ1 = the aforementioned condition can be rewritten as ρ g ρ2  1 kcr > − (U1 − U2 )2 cos2 γ1 ρ2 ρ1

α kcr ,

and

(1.21)

The lowest value of wavenumber (kcr = kmin ) would occur for two-dimensional disturbances, i.e. when cos γ1 = 1 and this is given by ∗ kmin =

ρ g ρ2  1 − (U1 − U2 )2 ρ2 ρ1

(1.22)

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For wind blowing over still water, U1 = 0, consider a wind speed of U2 = 6 m/s, then with ρ1 = 103 kg/m3 and ρ2 = 1.178 kg/m3 , one gets from the aforementioned, a value ∗ of kmin = 231.5 m−1 , which in turn creates excited wavelengths equal to and below the value of λmax = 2.71 cm! A truly, unphysical value, showing the limitation of this model. CASE C: Finally, one can consider the effects of shear if both the domains contains the same fluid, i.e., for the case of ρ = 1. The characteristic exponents given by Eq. (1.19) then simplify to ω ¯ 1,2 = −α

U1 + U2 iα ∓ (U1 − U2 ) 2 2

(1.23)

In this case, the presence of an imaginary part with a negative sign in Eq. (1.23) implies temporal instability for all wavelengths due to the expression of interface given in Eq. (1.17). Moreover, one can differentiate the circular frequency in Eq. (1.23) with respect to β to obtain the group velocity in the y-direction to note that this is identically ω ¯ zero. The same is true for the phase speed given by cy1,2 = β1,2 . Therefore, the Kelvin– Helmholtz instability for pure shear will always create two-dimensional instability. Moreover, note that the temporal growth rate is given by Imag[ω ¯ 1,2 /k] = −

α (U1 − U2 ) 2k

As k > α for three-dimensional disturbances, the largest growth rate will be found when k = α for the two-dimensional case and the maximum growth rate is −(U1 − U2 )/2. If one considers the case of a recirculating bubble, then U1 and U2 are of opposite signs; that would lead to maximum growth for two-dimensional bubbles. CASE D: (Effects of surface tension): For two dissimilar fluids at the interface, an additional force tangential to the deformed interface will arise due to surface tension. Thus, on a differential element, ds, there will be additional normal stresses acting on both sides of the interface to maintain equilibrium of the element. The normal stress due to surface tension (Tˆ per unit length) will be determined by the two principal radii of curvature. Denoting these as R1 and R2 , the normal stress discontinuity at the interface is given by −Tˆ

1 1 + R1 R2

These radii of curvatures can be written in terms of interfacial displacement ηˆ as −Tˆ

 ∂2 ηˆ ∂x2

+

∂2 ηˆ  ∂y2

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Using this in Eq. (1.9), one gets ρ1

 ∂φ

  ∂φ   ∂2 ηˆ ∂2 ηˆ  ∂φ1 ∂φ2 2 ˆ + U1 + gη − ρ2 + U2 + gη = −T + ∂t ∂x ∂t ∂x ∂x2 ∂y2 1

(1.24)

One can substitute the various Fourier–Laplace transforms in Eq. (1.24) to rewrite Eq. (1.15) as ρ1 f˙1 − ρ2 f˙2 + iαρ1 U1 f1 − iαρ2 U2 f2 + (ρ1 − ρ2 )gF = Tˆ k2 F

(1.25)

We leave it as an exercise for the readers to show the altered dispersion relation given by the following (as compared to Eq. (1.19)), ω ¯ 1,2

α(U1 + ρU2 ) ∓ =− (1 + ρ)

p gk(1 − ρ2 ) − α2 ρ(U1 − U2 )2 + k3 Tˆ (1 + ρ)/ρ1 (1 + ρ)

(1.26)

It is to be noted that the action of surface tension is to stabilize the interface instability. This is treated in detail in [84].

1.3.2 Taylor–Green vortex instability Evolution of the Taylor–Green vortices is an interesting problem of fluid dynamics as it brings forward multiple issues needing resolution and interpretation: (a) The governing Navier–Stokes equation admits a periodic solution in both the directions for two-dimensional problems. (b) Despite the problem being periodic in two-dimension, how such a flow becomes unstable was not very clear till recent times. For such two-dimensional periodic problems, it has been shown in [114] that the integrated enstrophy (vorticity squared) over the full domain is a monotonic decaying function of time, implying that the viscous diffusion terms do not contribute to instability. However, all numerical solutions show the disintegration of the initial coherent Taylor–Green vortex into small scale structures. (c) The Taylor–Green problem was originally solved analytically by Taylor and Green [515], who furthermore showed, by a perturbation series expansion in time, that such solutions encounter a finite time singularity. This was interpreted as a reason for small scale structure creation, leading to transition to turbulence for this problem. (d) We also note that for twodimensional problems, the absence of a vortex stretching mechanism does not help in creation of small scale structures by cascade mechanism in the energy spectrum [335]. (e) The perturbation analysis used by Taylor and Green was extended by Goldstein [159], with the perturbation series formed as a function of the Reynolds number (Re). For this perturbation expansion analysis with the help of Re, Goldstein reported similar singularity appearing as Re was increased. These singularities in linear perturbation analysis with respect to time (temporal instability) and Re

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(spatio-temporal instability?) do not allow us to treat such instabilities as synonymous with transition to turbulence. Equilibrium flow, which allows a spatially periodic analytical solution, is also a decaying function of time. Thus, there is a unique time dependent equilibirum state, which is not amenable to classical linear stability analysis. This situation changed with the appearance of a pair of receptivity theories [431, 475] providing both a linear and nonlinear approach to the instability of the Taylor–Green vortex in [465, 480]. A study on modal and nonmodal analysis for the two-dimensional Taylor–Green vortex problem was attempted by Gau and Hattori [150]. However, this is severely limited, as the authors made the gross assumption of treating equilibrium flow to be time independent, with results provided only up to tmax = 15. In this chapter, we relate the breakdown of this spatially periodic solution via a receptivity to various components of a numerical error. This is achieved using high accuracy computing methods, with results produced using different grids to quantify the effects of truncation errors in [465]. As will be described in subsequent chapters, the compact schemes developed for high accuracy computing methods have nearspectral accuracy in the wavenumber plane [413], and thus allows one to control truncation errors to a very low value; the round-off error has to be high to trigger breakdown of the symmetries of the initial equilibrium flow. There is one aspect used in [465] which is worth pointing out: it shows that the Taylor–Green vortex problem has a common relation with vortex shedding behind bluff bodies. In both these cases, there is a full saddle point, whose criticality has not been analyzed systematically. This prompted the authors in [465] to study the instability of periodic Taylor–Green vortex by considering periodic boundary conditions across a minimal unit of four cells instead of a single vortex with periodic conditions enforced strictly at the boundaries. This allow one to study the natural dynamics of the system by including the saddle point (hyperbolic stagnation point [150]) in the interior of the domain, with its dynamics governed by the NSE. An analytic initial solution in a three-dimensional periodic space has been used by Orszag [322] to show early time breakdown of the symmetries prevalent in the initial solution. This was made possible by discretizing the problem in Fourier spectral space, while the initial solution is advanced in time by the Adams–Bashforth scheme. Although it has been termed as a direct numerical simulation (DNS), it will be noted that such pseudo-spectral methods suffer from spurious numerical modes [413, 464], which leads to catastrophic focusing phenomenon noted in long-term computations [453]. Focusing implies a sudden breakdown of a numerical solution when a nominally working code is executed for very long times. It was not understood clearly for a long time; however, very recently, some light have been shed on this phenomenon [39, 453, 472]. Focusing has been shown to be due to various forms of numerical anti-diffusion accentuated by spurious modes, poor boundary closure schemes, and instigation of numerical viscous instabilities. The main issue is that the focusing phenomenon is due to a linear mechanism of numerical instability. For the

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Taylor–Green vortex problem, this may not be catastrophic as the equilibrium flow monotonically decreases with time. Results presented by Orszag [322] show the instability numerically; one is not certain if the instability follows the physical route or if it is strongly affected by the numerical mode of the three-time level integration method. However, the computed results show a typical energy spectrum and dissipation of the three-dimensional turbulent flows at early stages. After some time, the created turbulence decays with time. Such flow behavior has subsequently also been shown by Brachet [55] and Brachet et al. [56, 58], by considering inviscid and viscous dynamics of the Taylor– Green vortex. The breakdown of the analytical initial solution for the Taylor–Green vortex is essentially due to instability of the time-dependent equilibrium solution due to omnipresent disturbances in the numerical framework. The two-dimensional study by Brachet et al. [57] attempted to explain the dynamics of decaying turbulence associated with the Taylor–Green vortex flow. However, the two-dimensional turbulence was caused by using Gaussian-random data and not the analytical solution of Taylor and Green [515]. The authors could show the inertial range energy-spectrum exponent changing from –4 to –3 with increasing time. The exponent value of –4 was ascribed to isolated vorticity-gradient sheets (as explained by Saffman [381]); the exponent value of –3 is as proposed for two-dimensional turbulences by Kraichnan [248] and Batchelor [21] using an enstrophy cascade. The enstrophy cascade is described by Doering and Gibbon [114]. The numerical study of the two-dimensional Taylor–Green vortex problem allows one to trace evolution of the flow as a departure of the computed solutions from the initial condition. The two-dimensional study is preferred physically and numerically due to the ease of: (i) taking fine grid points compared to three-dimensional problems; in consequence, truncation and aliasing errors are reduced, and (ii) using a timedependent equilibrium flow for the Taylor–Green vortex problem without any of the assumptions used in the Navier–Stokes equation. This approach also enables one to judge the results of the numerical methods early, by comparing analytical with the numerical solution before the onset of physical instability. Novel high accuracy compact schemes have been developed in [457, 481] so that one can use non-uniform grids in the physical plane with higher resolution as compared to the uniform grids used earlier by the Fourier spectral methods [55, 56, 322]. For high accuracy computing in DNS and large eddy simulation (LES), one must characterize the methods for error propagation. Errors are always present in any computation; one needs to minimize such errors to maintain high accuracy. This epistemological shortcoming of error dynamics can be addressed using global spectral analysis (GSA) developed for the model convection equation [413], the convection– diffusion equation [506], and the convection–diffusion reaction equation [455]. We choose appropriate model equations based on physical processes relevant to the actual governing Navier–Stokes equation. The results of the Taylor–Green vortex problem with time-dependent equilibrium flow obtained analytically will help understand the

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error dynamics of the Navier–Stokes equation. More importantly, one understands the onset and growth of the physical instability. This enables one to study error dynamics as the dynamics of epistemic error, without resorting to uncertainty quantification. The aleatoric uncertainty caused by stochastic variation due to intrinsic dynamics originating with round-off error can also be responsible for the physical instability. This implies that the physical instability is related to error dynamics affected by both epistemic and aleatoric components of error. Despite the authors in [114] showing that for two-dimensional periodic flows, the viscous diffusion is strictly dissipative (for the full domain), in [468], the authors extended this principle by viewing diffusion locally at any instant to establish clearly that diffusion is not strictly dissipative for inhomogeneous flow. They first derived the enstrophy transport equation from the vorticity transport equation for general inhomogeneous flows to explain the role of diffusion in creating rotationality. The concept has been extended to explain the growth and decay of the disturbance enstrophy, for which detailed discussion and derivation are presented in [474, 475].

1.3.3 Equilibrium solution of the two-dimensional Taylor–Green vortex problem The two-dimensional incompressible flow governed by the Navier–Stokes equation can be solved using the stream function (ψ) and vorticity (ω) formulation due to its high accuracy, and the fact that it provides vorticity (as one of dependent variables) in a solution. In the pressure–velocity formulation, one obtains the velocity and taking the numerical curl of this field, one obtains the vorticity. With the numerical curl operation, the computed vorticity field is often severely filtered. Thus, the governing equations are given by the stream function equation (SFE) and the vorticity transport equation (VTE) as follows: ∂2 ψ ∂2 ψ + 2 = −ω ∂x2 ∂y ∂ω ∂ω ∂ω 1 ∂2 ω ∂2 ω +u +v = + 2 ∂t ∂x ∂y Re ∂x2 ∂y

(1.27)

! (1.28)

In the same way, the stream function and velocity are related by v = ∇ × ψ, with ψ = [0 0 ψ]T ; the vorticity (ω) is notationally expressed in terms of the velocity by ω = ∇× v. The components of the velocity vector are given by the following u=

∂ψ , ∂y

v=−

∂ψ ∂x

(1.29)

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The two-dimensional Taylor–Green vortex problem is solved in a periodic domain 0 ≤ (x, y) ≤ 2π with the following initial conditions, ψ(x, y, 0) = sin x sin y,

ω(x, y, 0) = 2 sin x sin y

(1.30)

In the periodic domain shown in Figure 1.3, there are two rows and two columns of vortical cells. The derivations of the time-dependent analytical results that satisfy Eqs. (1.27) and (1.28) are reported in [465]. The analytical solution in terms of ψ and ω are given in terms of a time-dependent amplitude function, G(t) as follows: ψ(x, y, t) = sin x sin y G(t) ω(x, y, t) = 2 sin x sin y G(t)

(1.31)

The amplitude G(t) is determined by using Eq. (1.31) in Eq. (1.28), which upon simplifying yields 2t

G(t) = e− Re

(1.32)

It is now well known that this equilibrium solution is not stable with respect to omnipresent numerical disturbances if the Navier–Stokes equation is numerically simulated [322, 465]. Now, the numerical solution’s behavior, as compared to the analytical solution, is explained and the deviations between the two are explained. Consider a square domain of length 2π; one notes the basic unit of four TGV cells, with a full-saddle point [525] at the center of the domain, having twofold, top–bottom and left–right symmetries. This symmetric arrangement experiences a breakdown due to growth of background disturbances. The combined actions of vorticity gradient terms and growth of omnipresent disturbances cause the disturbance vorticity to grow with time, while the base flow vorticity decays with time, as seen from Eq. (1.31). The grid is defined with a tangent-hyperbolic double-sided distribution [413] in the x- and the y-directions. Given an uniform distribution of points between 0 and l = π/2 by ηi in the transformed plane, with i = 1 to N/4 and βh as the stretching parameter, the nodes xi in the physical plane are given by " # l tanhβh (1 − 2ηi ) xi = 1− 2 tanhβh

(1.33)

This same grid spacing is used as an image in the adjacent segment, that is, from i = N/4 + 1 to N/2, with this sequence repeated up to i = N, so that the grid spacing is fine enough to resolve high solution gradients for the solution of Eqs. (1.27) and (1.28). Thus, one requires fine grid near the boundary of the vortex cells, at the core of the vortices, and at the saddle point. An identical grid distribution is used in the y-direction. A typical topology of the grid with (201×201) points and βh = 1.5 is shown in Figure 1.2. A small time-step (∆t) of 10−4 is used to ensure high accuracy.

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Grid = 201 × 201 βh = 1.5 6 5

y

4 3 2 1 0

0

1

2

3 x

4

5

6

Figure 1.2 Grid used in the simulations, shown here with 201 × 201 points for ease of viewing. All computations are done using 401 × 401 grid points in a tangent-hyperbolic stretched manner with the stretching parameter βh = 1.5. [Reproduced from “Non-linear instability analysis of the two-dimensional Navier–Stokes equation: The Taylor–Green vortex problem”, T. K. Sengupta, N. Sharma and A. Sengupta, Phys. Fluids, vol. 30, pp 054105 (2018), with the permission of AIP Publishing.]

1.3.4 Results of the Taylor–Green vortex problem simulation In Figure 1.3, the numerical solution (represented by solid lines) is compared with the analytical solution (indicated by dashed lines) for non-dimensional times t = 50 and 150, for Re = 2000, showing near-perfect match at these times. The root mean square error for ψ and ω are also shown in the figure to quantify error. Expressions for the root mean square quantities are presented for stream function, with ψA and ψNum representing analytical and numerical values, respectively, in the following s ErrRMS =

P j=N Pi=N j=1

i=1

(ψNum (i, j) − ψA (i, j))2 N×N

(1.34)

This error is of the order 10−5 for ψ and 10−6 for ω, as noted in Figure 1.3, with the solution experiencing loss of symmetry above t ≈ 160.

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(a) Grid = 401 401 h = 1.5

Dashed = Analytical Solid = Numerical ErrRMS = 2.9065e-05

t = 50

6

(b) Grid = 401 401 h

4

3

3

Y

4

Y

5

2

2

1

1 0

1

2

(c) Grid = 401 401 h = 1.5

4

3X

t = 150

5

Dashed = Analytical Solid = Numerical ErrRMS = 2.7870e-05 - contours Re = 2000

h

3

Y

3

Y

4

2

2

1

1 X

4

6

= 1.5

X

4

t = 150

6 Dashed = Analytical Solid = Numerical ErrRMS = 9.0968e-06

6

4

2

2

(d) Grid = 401 401

5

0

- contours Re = 2000

0

5

0

Dashed = Analytical Solid = Numerical ErrRMS = 4.0146e-06

0

6

6

t = 50

6

- contours Re = 2000

5

0

= 1.5

- contours Re = 2000

0 0

2

X

4

6

Figure 1.3 Numerical results (shown by solid lines) compared with the analytical solution (shown by dashed lines) for Re =2000 at : (a)–(b) t = 50 and at : (c)–(d) t = 150. Contours of ψ are shown in frames (a), (c), and ω-contours are in frames (b), (d). [Reproduced from “Non-linear instability analysis of the two-dimensional Navier– Stokes equation: The Taylor–Green vortex problem”, T. K. Sengupta, N. Sharma and A. Sengupta, Phys. Fluids, vol. 30, pp 054105 (2018), with the permission of AIP Publishing.] By the time t = 220, the four vortices start moving and at the same time losing coherence, as can be seen in Figure 1.4. These vortices eventually break down into small vortical structures. Similarly, the magnitude of the maximum error for ω, denoted by (ωd )max = maximum (|ωA − ωN |), is shown in log-scale plotted as a function of time for Re = 500, 2000, and 10000 in Figure 1.5. The maximum error remains almost constant up to a time t = 160 for high Re cases, while for Re = 500, it remains constant up to t = 310. Thereafter, these cases show that error increases with Re. For low Re cases, the diffusion dominates over convection, specifically at early times, and thereby, the convection is suppressed by relatively stronger diffusion. This dominance of diffusion over convection keeps the flow stable for a significantly long time for low Re cases. The deviation of the numerical solution from the analytical one during initial stages is

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Figure 1.4 Vorticity (ω) contours shown at the indicated times for the case Re = 2000. [Reproduced from “Non-linear instability analysis of the two-dimensional Navier– Stokes equation: The Taylor–Green vortex problem”, T. K. Sengupta, N. Sharma and A. Sengupta, Phys. Fluids, vol. 30, pp 054105 (2018), with the permission of AIP Publishing.] higher for Re = 500 because of numerical reasons, which will be understood in terms of the numerical Peclet number, (Pe = ∆t/(Re h2 ) range (with h denoting smallest spacing), which is the highest for the lowest Re. From Figure 1.5, one notes that for all the cases, the numerical solution exhibits stability up to t = 160, after which the solution is physically susceptible to the noise present in computing and becomes unstable. This is evident from the loss of symmetry of the original vortical cells. The phase space trajectory of dω dt versus ω for the full saddle point is shown in Figure 1.6, with marked points indicating the various stages of instability shown by the vorticity. The break down of symmetry of the Taylor–Green vortices and nonlinear saturation of vorticity at the saddle point are shown in Figure 1.5. Point A indicates the onset of −5 instability with dω dt of the order of 10 , after which this magnitude rapidly increases. Point C is an intermediate point where one notices a small dip in the temporal growth; the corresponding point can also be seen in Figure 1.5 at around t = 252.60. After this time, the magnitude again increases, with fluctuations at points D, E, F, G, and H.

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Transition to Turbulence Grid = 401× 401 βh = 1.5

|ωd| Re = 500 |ωd| Re = 2000 |ωd| Re = 10000

100 C

10−1

GHI F E D

B

Max error

−2

10

10−3

−4

10

A

−5

10

200

100

300

400

500

Time

Figure 1.5 Amplitude of (ωd )max plotted in semi log-scale as a function of time for Re = 500, 2000, and 10000. [Reproduced from “Non-linear instability analysis of the two-dimensional Navier–Stokes equation: The Taylor–Green vortex problem”, T. K. Sengupta, N. Sharma and A. Sengupta, Phys. Fluids, vol. 30, pp 054105 (2018), with the permission of AIP Publishing.] Phase space plot at saddle point A : t =150.01 B : t = 237.29 C : t = 252.60 D : t = 259.19 E : t = 262.84 F : t = 268.55 G : t = 278.99 H : t = 299.70 I : t = 310.35

t = 150.01–234.99 t = 235.00–250.00 t = 250.00–274.989 t = 274.99–330.99

0.2 0.15 0.1 dω dt

E

0.05

H I

0

C

G F A D

–0.05

B

–0.1 –0.15 –0.2 –0.25 –0.5

0

ω

0.5

Figure 1.6 Phase space trajectory of dω dt vs ω at the saddle point for Re =2000. [Reproduced from “Non-linear instability analysis of the two-dimensional Navier– Stokes equation: The Taylor–Green vortex problem”, T. K. Sengupta, N. Sharma and A. Sengupta, Phys. Fluids, vol. 30, pp 054105 (2018), with the permission of AIP Publishing.]

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At point I, the nonlinear saturation of growth is noted such that the magnitude of dω dt remains constant. It is to be emphasized that although the instabilities are physical in origin, the onset is triggered by extant numerical errors.

1.4 Closing Remarks In investigating problems of physical events, we make use of theoretical, experimental, and numerical investigations. While the event is replicated by adopting simplified models, theoretical analysis explains the physical mechanisms involved. In that sense, a theory provides both qualitative and quantitative description of the event. This must be equally true of experimental and numerical investigations. However, in the theoretical investigation of instability, dependence on eigenvalue analysis makes the approach less comprehensive as it does not require any knowledge of the background disturbances as input and hence, cannot quantify the response of the dynamical system. In this approach, one is often made to interpret the response as the natural vibration of the space–time dependent dynamics. To circumvent this inadequacy, demonstration of instability by experiment is considered mandatory. This also makes the experiment ambiguous, if one were to depend on background disturbances only. Such experimental demonstration will make the outcome vary from one realization to another! At the same time, in the spatial theory of instability, one assumes that the time dependence is fixed via the choice of monochromatic excitation. This fact was realized in the earliest successful experimental demonstration. In providing a framework for repeatable experiments, the experimental device was made as noiseless as possible to remove background disturbances. Thereafter, a deterministic monochromatic excitation is imposed to create instability. However, then, we have migrated from natural excitation to forced excitation of the dynamical system! Implicit in following these theoretical and experimental approaches lie the assumption of the signal problem, which ruled the theoretical development of the subject erroneously for a long time till it was rectified by the author’s group [509]. It can now be definitively shown that the signal problem assumption is not needed, and can even be hazardous for any dynamical system that displays instability. This has been made possible by renewed theoretical approach of receptivity and high accuracy numerical investigations. In the latter part of the chapter, we use two examples of Kelvin–Helholtz and Taylor–Green vortex instabilities to bring out the simple concepts used in theory. Instability is always present in an equilibrium flow, and it is necessary to identify this before embarking on such studies. In the first example, the Klevin–Helmholtz instability is studied by considering two uniform flows conjoined at a flat interface, which is perturbed for the ensuing instability. Within the linear context, this instability is shown using inviscid disturbance flow. The shear between the two fluids is a strong mechanism to destabilize the inviscid flow. Even when shear is absent, if the top fluid is heavier than the bottom fluid, the buoyancy force destabilizes the interface and is known as the Rayleigh–Taylor instability. This is a baroclinic instability. Finally,

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for the Taylor–Green vortex instability, one can obtain a time-dependent equilibrium flow. Even having an equilibrium flow does not allow one to solve the disturbance field from the Navier–Stokes equation, and one has to resort to numerical simulation, as in [322, 465, 480]. While the initial coherent Taylor–Green vortices are seen to evolve according to the analytical solution, after some time, these vortices lose their symmetries due to instabilities. This discrepancy was not explained for a long time till a set of new theories based on growth of disturbance mechanical energy and disturbance enstrophy were proposed and used for this problem in [465, 480].

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Appendix A Group Velocity and Energy Flux A.1 General disturbance field In the course of studying transition from laminar to turbulent state due to growth of omnipresent background disturbances, one can represent the disturbance mathematically in terms of different length and time scales. To understand this better, denote an arbitrary disturbance as a function of single spatial variable using the Fourier–Laplace transform [412, 537] f (x, t) =

Z

Z Brα

¯ F(α, ω)e ¯ i(αx−ωt) dαdω ¯ Brω¯

This is a representation that allows the space–time dependency to be as general as possible. If the function is constrained to be periodic in space (time) due to the problem description, the corresponding integral will be replaced by the summation performed over that spectral variable. For example, if the unknown is constrained to be periodic in space with the basic wavelength defined as λ0 = 2π/α0 , then the aforementioned unknown will be represented by the following f (x, t) =

∞ Z X n=1

¯ F(nα0 , ω)e ¯ i(nα0 x−ωt) dω ¯ Brω¯

One can also consider a function that is periodic in time. Such periodic description of an unknown is termed as modal representation, as the various harmonics of the fundamental modes (nα0 ) for different values of n are orthogonal to each other. In transition research, representation of disturbance quantities by orthogonal modes have been thought to be implicitly adopted, and the analysis method has been termed as normal mode analysis. It should be understood that the eigenmodes one comes across in instability studies are not necessarily super- or sub-harmonic of the fundamental mode. Thus, it is not incumbent upon the practitioner to think of using normal mode analysis only. For example, the spatial stability analysis performed on a Blasius boundary layer would show the modes to be nonorthogonal to each other. Moreover, one should keep in mind the fact that the number of eigenmodes of such viscous instability studies are finitely only a few, and should never be considered as a complete set to represent any arbitrary disturbance in terms of such incomplete sets. By contrast, if the constituents of the generic description of the non-periodic disturbance field given above are not orthogonal for any one choice of α with another, then such a representation will be termed as the nonmodal representation, and one talks of nonmodal analysis. Irrespective of whether one is adopting modal or

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nonmodal representation of the unknown, it is at the disposal of the analyst to consider linearized or fully nonlinear analysis. If not stated otherwise, one is free to adopt the most generic description of the unknown and adopt general analysis. Moreover, irrespective of the fact whether one is performing modal or nonmodal analysis, one can always associate any such unit of (α, ω) ¯ with an unit wave whose phase speed will be given by c = ω/α, ¯ where the relationship between the wavenumber (α) and circular frequency (ω) ¯ is the dispersion relation, and forms the main theme of instability studies. It is also worth remembering that despite the fact that the phase speed has the dimension of speed, it may not be a physical entity or speed with which energy or mass transports itself through a medium. Of specific interest is the speed with which the disturbance energy will be propagated in the medium; this is described next. Complete description of wave parameters and their definitions and inter-relationships are provided in Chapter 3.

A.2 Energy propagation and group velocity In the previous discussion, it is noted that any dynamical system can be described in the most general framework using the Fourier–Laplace transform to express the unknowns. In such a case, we can define the dynamics of the system in terms of all modal and nonmodal components of the response field. Consider the disturbance field given by the disturbance stream function, experienced by the boundary layer forming over a flat plate that is excited by a time-harmonic excitation source, given as follows ψd (x, y, t) =

Z

Z Brα

φ(y, α, ω0 )ei(αx−ω0 t) dαdω0 Brω0

Assume that a parallel two-dimensional equilibrium flow described by the streamwise velocity component U(y) over a flat plate supports a two-dimensional disturbance field that follows the linearized Navier–Stokes equation and is given by the Orr– Sommerfeld equation written as follows 00

00

00

φiv − 2α2 φ + α4 φ = iRe[(αU − ω0 )[φ − α2 φ] − αU φ]

(1.35)

where primes indicate differentiation with respect to wall-normal coordinates, and ω0 is the complex circular frequency. However, in the celebrated experiment of the vibrating ribbon, the exciter is placed inside the boundary layer of the flat plate, where the Reynolds number is defined in terms of the displacement thickness of the boundary layer at the exciter location [405]. Such an experimental arrangement supports a disturbance field whose dynamics is given by the Orr–Sommerfeld equation. This benchmark problem is described in detail in various chapters; only a brief description is given here to explain certain basic concepts. The spectral plane representation is shown for the solution of the Orr–Sommerfeld equation, following two different paths as Bromwich contours in the complex α-plane

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Receptivity, Instability, and Transition 15

Re = 1000, ω0 = 0.14, t0 = 900

αi Br = 0.008 αi Br = 0.001

10

|φ(α, t0)| 5

αr = 0.2915 0

0.1

0.2

0.3 αr

αr = 0.3743 0.4

0.5

0.6

Figure A.1 Solution of the Orr–Sommerfeld equation shown along two Bromwich contours in the complex α-plane. Although there is a vibration source inside the boundary layer oscillated with frequency, ω ¯ 0 = 0.1, the response will display all possible circular frequencies due to the impulsive start of the exciter. in Figure A.1. For the illustrated case, the Reynolds number is given by Re = 1000 and exciter frequency is given by, ω ¯ 0 = 0.14. The concept and choice of Bromwich contours for the Orr–Sommerfeld equation is given in [412]; the same has been described as an operational calculus tool in [324, 537]. A solution of Eq. (1.35) is obtained along two Bromwich contours given by: (i) Brα along a straight line parallel to the αr -axis, and below αiBr = −0.008, and (ii) Brα along the line parallel to the αr -axis, located below αiBr = −0.001. The details of the solution methods, etc. is provided in Chapters 4 to 6. The reader’s attention is drawn to the features of the solution, φ(α, t0 ) plotted as a function of αr , along the Bromwich contours. It is important to realize that the formulation of the Orr–Sommerfeld equation and its solution at t0 = 900 displayed in Figure A.1, contain both the modal and nonmodal components as two maxima. It is interesting to note that the modal component (given by the Tollmien–Schlichting (TS) wave, with peak at αr = 0.3743) is higher than the peak of the nonmodal component (with peak at αr = 0.2915); yet, the nonmodal solution will be shown as dominant in the long run, as compared to the modal TS wave, which is a spatially decaying mode. This aspect can be understood if one observes the mutual interference of the solution spectrum obtained as the continuum shown in Figure A.1, and explained by looking at the constructive interference of two neighboring components of the continuous spectrum noted in the figure. If one notices the energy spectrum (E(k)) of diverse dynamical systems, then it is apparent that a continuum of wavenumbers is present, as noted in Figure A.1. For such a space–time dependent system, one can observe the transport of energy. This can be adequately explained by observing at how neighboring elements in the spectrum interact. Let the spectrum and the dispersion relation ω ¯ = ω(k) ¯ be continuous functions of their arguments. Tracking two close neighboring wavenumbers k1 and k2 (= k1 + dk), one therefore reasons that the corresponding circular frequencies are equally close with values given by ω ¯ 1 and ω ¯ 2 (= ω ¯ 1 + dω). ¯ Furthermore, consider that the amplitudes of these

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k1 = 0.1, c = 1.0 dk = 0.01

150 100 50 y 0

–50 –100 –150

–1000

–500

0 x

500

1000

Figure A.2 Phenomenon of modulation in a group of waves. two components are also the same with a value a. The superposition of these two components are given as the waveform y = a cos(k1 x − ω ¯ 1 t) + a cos(k2 x − ω ¯ 2 t)   dk  dω ¯   dk  dω ¯ = 2a cos x− t cos k1 + x− ω ¯1 + t 2 2 2 2

(1.36)

The superposed solution is noted to have an amplitude component and a phase component, both of which are space–time dependent. The phase part given by the second factor resembles the phase of either of the interacting components, with the same wavenumber and circular frequency. The first factor, representing a slowly varying amplitude, is of prime importance, representing an amplitude varying slowly in space (with wavelength 4π/dk) and time (with time period 4π/dω). ¯

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This slow modulation of the resultant amplitude occurs via its phase variation and the corresponding x/t = constant line moves with a speed, defined as the group velocity, for obvious reasons, and is given by the following Vg =

dω ¯ dk

(1.37)

This is the slope of the dispersion relation. A typical sketch for the variation of this group composed of k1 and k2 is shown in Figure A.2. Note that there is a constructive interference of these two neighboring components of the continuum, such that the maximum amplitude is twice the individual components. It will be further emphasized in Chapter 6 that the most effective constructive interference would take place when the constituents have near-identical wave properties, and the slope of the dispersion relation will determine the group velocity. A necessary condition for the generation of such nonmodal components is the presence of a maximum in the spectrum.

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In Chapter 1, we have stated that in this book, the study of flow instability will be performed using dynamical system theory. For flow instability, we will follow the schematic shown in Figure 2.1 for flows undergoing transition to turbulence from a laminar state. The idea behind this path dates back to the famous pipe flow experiment of Osborne Reynolds, who understood that the phenomenon of transition depends upon the prevalent background disturbances. For this reason, Reynolds designed the experimental setup with utmost care to minimize sources of disturbances. The time of performing experiments were also so chosen that the disturbances were further minimized. Thus, the transition phenomenon significantly depends on the input to the system, referred to as the receptivity of the system. The concept of receptivity is reflected in Figure 2.1, where the dynamical system is identified by the box with thick borders and input to this system is marked on the top. There are alternative processes which are marked inside the box indicating various mechanisms responsible for transition. The output of the system is the turbulent flow, shown at the bottom of the schematic. It has been noted in [405] that for experimentally generating Tollmien–Schlichting waves, a vibrating ribbon excited time-harmonically at a single frequency was successful, while acoustic excitation of a free stream was not effective. Readers should note that the eigenvalue analysis (as we will describe in Chapter 4) is not only incapable of distinguishing between wall and free stream excitation, but also incapable of distinguishing between vortical and acoustic excitations. This prompted researchers to initiate studies about the propensity of equilibrium flows to be more receptive to one type of input excitation over the other. This is the essence of ‘receptivity’, a term coined by Morkovin [298], whose study not only discusses amplitude of input excitation, but also quality, that is, different types of physical input excitations.

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Input (Background / Impressed Disturbances)

Receptivity

(B) Linear Convecting Instability (A) High Amplitude Bypass Route

(i)

Spatio-Temporal Wave Front

(C) Non-Convecting and Other Routes of Instability

(ii) Viscous Tuned Wave-Packet (iii) Local Solution D

D

Secondary and Non-Linear Instability

Late Stages of Transition

Output/ Turbulence

Dynamical System

Figure 2.1 Schematic of flow transition to turbulence. Paths followed in experiments and computations. The initial state of the dynamical system is represented by an equilibrium flow. In this chapter, we will look at a few representative equilibrium flows with the help of which certain transition mechanisms will be explained in the book. The equilibrium flows are obtained from different levels of hierarchy of conservation equations for fluid flows. Readers are encouraged to peruse the books [412] and [551] for a range of equilibrium states for internal and external flows.

2.1 Conservation Equations The physical laws of fluid flow are based on conservation principles for mass, translational momentum and energy; these principles lead to quasi-linear partial differential equations. The following steps are taken to obtain any governing equation: • The physical principles necessary are identified depending on operational conditions. For example, if one is interested in studying fluid flow at low speed, then one can adopt the incompressible flow model with or without heat transfer. For flows without heat transfer at low speeds, one can use mass and momentum conservation, without the need to consider energy conservation. However, if one is interested in studying acoustic fields in a very low speed fluid flow, then ideally one should consider compressible fluid flow models.

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• Based on the identified physical principles, one selects an appropriate model for the flow. For example, when considering the Kelvin–Helholtz instability, we assumed a uniform flow in each phase, whereas, the disturbance field was based on incompressible, irrotational flows. As we consider inflow over a flat plate experiencing no pressure gradient in the following chapter, the viscous action has to be considered to understand flow instability, irrespective of whether we consider the disturbance field as inviscid or viscous. • Combining the aforementioned steps, one obtains the mathematical equations which will be computed if analytical solutions are not available. In the process, we may simplify the equations, as one does for obtaining velocity profiles for unseparated flows, using boundary layer assumption. This innovation in fluid mechanics by Prandtl [336] revolutionized not only understanding in fluid mechanics, but also spawned many areas of studies. For separated flows, the situation is drastically different and to solve transitional and/ or turbulent flows computationally, one must use high accuracy computing methods that are distinctly different from the engineering CFD (computational fluid dynamics) methodologies. Choice of a suitable model is decided by incorporating as many conservation principles as possible. However, irrespective of the problem and model chosen, mass conservation can never be compromised. For transitional flows, one obtains equilibrium flow by considering the Navier–Stokes equation or some simplified form of it. While these approaches provide insights, recent research efforts have used the Navier–Stokes equation for equilibrium flows in the context of nonlinear receptivity analysis of basic time-dependent equilibrium flows. Once the equilibrium flow is obtained, its stability is investigated with respect to applied disturbances in receptivity studies. In the context of infinitesimally small perturbations, the governing Navier– Stokes equation is often linearized. This leads to typical eigenvalue problems which has been the focus of most of the earlier studies and is also called the modal approach of analysis, as one studies each eigenmode in isolation. One of the drawbacks of the modal approach is that one does not have to characterize the disturbance environment. However, if one is interested in studying the response of the flow to a specific disturbance, as happens to be the case of a physical experiment, then one requires to adopt receptivity analysis, which specifically needs identification of ambient disturbances. Receptivity analysis has been performed in a linearized framework in [418, 451, 452] by the Bromwich contour integral method (BCIM), which is described in Chapters 5 and 6 for studying the canonical zero pressure gradient boundary layer, using parallel flow assumption. The implication of linearization is that the amplitude of input to the dynamical system has to negligibly small, and to circumvent this restriction, one can also use the full Navier–Stokes equation for receptivity studies. Moreover, use of the Navier–Stokes equation allows one to remove the parallel flow assumption along with the implicit requirements in interpreting the results of linearized analysis. Receptivity analysis is undertaken in Chapters 5–10.

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Thus, it is imperative that readers appreciate the nuances of solution techniques of various space–time dependent equations. Fluid dynamics problems require large numbers of degrees of freedom for characterization due to the governing Navier–Stokes equation having a very small coefficient of diffusion. As a consequence, the flow experiences a very low physical dissipation in the domain overall. However, completely ignoring it for the full flow domain, as in Euler’s equation, ignores the existence of a very thin shear layer. Within this thin shear layer, very high flow gradient is noted, which explains the observation of maximum dissipation in the spectral plane occurring at higher wavenumbers. Thus, for simulating such flows, resolving very high wavenumbers is mandatory. This important aspect is often overlooked by practitioners performing large eddy simulations (LES) of flows who only focus on the energy spectrum. In direct numerical simulation (DNS), no models are used and all scales are resolved, therefore no such conceptual difficulties arise in this approach. As mentioned earlier, in studying instability of a flow, we must first obtain the equilibrium solution by stating the conservation principles. Such solutions must be physically stable to be perceived. However, we note that the equilibrium solution can be unsteady; finding it is itself a challenge. For example, undisturbed flow past a circular cylinder above Re = 65 is computationally impossible, even with the most accurate method. On the contrary, some highly diffusive methods report steady flow which actually is noted to be unsteady by non-diffusive methods due to the extremely high receptivity of such flows to the ever-present machine round-off error. This aspect will be highlighted in Chapter 10.

2.1.1 Conservation of mass or continuity The physical principle of mass conservation states that the net mass entering and leaving an identified macroscopic control volume is zero. Balancing these two mass fluxes through the control surfaces with the mass “created or destroyed” inside the control volume provides the desired mass conservation for unsteady compressible flows Dρ ~ = ∂ρ + ∇ · (ρ V) ~ =0 +ρ∇·V Dt ∂t

(2.1)

~ denotes the velocity vector at the center of the where ρ is the density and V infinitesimally small control volume. Equation (2.1) is obtained in the limit of mathematically vanishing size of the control volume, which is distinctly different from the physically vanishing size of the control volume. Even if the sides of this elementary unit is in microns for a cubical shape, the number of molecules inside will be in millions for the continuum model to hold at sea level. In the present book, we will exclude consideration of rarefied gas dynamics.

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Furthermore, if we consider the flow density to remain invariant in space and time, as in incompressible flow models, that is ρ = constant, then Eq. (2.1) simplifies to ~ =0 ∇ · (V)

or

~ =0 Div(V)

(2.2)

A vector field satisfying the condition given in Eq. (2.2) is called the divergence-free or solenoidal field. A direct consequence of mass conservation for incompressible flows is the definition of streamlines and stream functions. For example, the solenoidality of a ~ such that velocity field can be immediately satisfied if we define a vector potential (Ψ) ~ ~ =∇×Ψ V

(2.3)

This also helps in defining streamlines as those whose tangent everywhere is parallel ~ along the streamline, then ~ at all times. If we define an infinitesimal segment ds to V ~ ~ this definition implies that V × ds = 0. In a Cartesian frame, this is equivalent to defining the streamline from this relation as follows dx dy dz = = u v w as

dx dt

= u;

dy dt

= v and

(2.4) dz dt

=w

In fluid flows, a material element experiences a force even if the flow field is steady ~ is inhomogeneous. In general, V ~ is a function of space and time, so that its total but V rate of change with time is given by ~ ~ ∂V ~ dx ∂V ~ dy ∂V ~ dz dV ∂V = + + + dt ∂t ∂x dt ∂y dt ∂z dt

(2.5)

~ ~ ∂V ~ ~ ~ dV ∂V ∂V ∂V = + u+ v+ w dt ∂t ∂x ∂y ∂z

(2.6)

The left-hand side of these equations is the substantive or total derivative of the velocity field, which is equal to the sum of local acceleration (given by the first term on the right-hand side) and the convective acceleration term (given by the rest of the product terms on the right-hand side). As we will see later, the convective acceleration terms play a dominant role in instability and receptivity mechanisms (however, not by itself!).

2.1.2 Conservation of translational momentum Conservation of translational momentum is nothing but Newton’s second law of motion applied to a fluid control volume. If one follows this control volume in motion,

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Dynamical System Theory and Role of Equilibrium Flows

then it is equivalent to tracing a control mass system (for which the system mass must be invariant with time). For this control mass system, the rate of change of translational momentum is equal to mass times the acceleration experienced by the fluid in the control volume. For example, if only the x-component of this is considered, then it can be written as F x = ma x

(2.7)

with F x as the x component of force and a x as the corresponding acceleration given by Eq. (2.6). One notes that the force experienced is a sum of body force on the element and surface forces acting on the control surfaces. The surface forces are divided into normal and shear forces associated with the generalized stress system in a flow. For a control volume (dx dy dz) with local density as ρ, the body force in the x direction is given by Fbx = ρ f x (dx dy dz)

(2.8)

with f x as the associated acceleration contributing to body force. A fluid at rest develops uniform hydrostatic pressure, which can also be termed as the thermodynamic pressure. In comparison, a fluid in motion experiences a nonuniform pressure and one defines deviatoric normal stresses given by σ0i = σi + p, along with the symmetric shear stress components. These deviatoric stresses depend on velocity gradients. To maintain isotropy, the relationship between stress and velocity gradients is not only linear; the principal axes of stress and strains are also aligned. This is the definition of a Newtonian fluid. It can be shown that the linear relationship between stress and rates of strain would involve two factors of proportionality, which are properties of the fluid medium. Written for any arbitrary coordinate system, the   ∂u ∂u ∂u j ~ + 2µ i and τi j = µ i + , with no implied stress system is written as σ0 = λ Div V i

∂xi

∂x j

∂xi

summation for repeated indices. Here, τi j is a shear stress tensor. In general, the stress system can be written as τi j = −pδi j + λ∂l ul δi j + 2µ∂i u j , where µ and λ are known as the coefficients of dynamic and bulk viscosity, and δi j is the Kronecker delta with summation implied. It is apparent that µ is associated with shear action in the fluid, while λ accounts for the bulk motion in the fluid. The surface forces are written with τi j , the stress tensor indicating force in the j direction, acting on a plane with normal in the i direction. Accounting for all contributory stresses to surface force in the x direction, one obtains  ∂p ∂τ ∂τyx ∂τzx  xx − + + + dx dy dz ∂x ∂x ∂y ∂z

(2.9)

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Thus, the force terms are obtained by summing the body and surface forces. As the mass of the element is (ρ dx dy dz) and the acceleration of the moving element is given by the substantive derivative, one can rewrite Eq. (2.7) as ρ

Du  ∂p ∂τ xx ∂τyx ∂τzx  + ρ fx = − + + + Dt ∂x ∂x ∂y ∂z

(2.10)

Obtaining the equations for all components, one also assumes the stress tensor to be symmetric. Such an assumption requires neglecting unsteadiness of the vorticity field. Using the same approach, the y and z components of the momentum equation are obtained as ρ

Dv  ∂p ∂τ xy ∂τyy ∂τzy  + ρ fy = − + + + Dt ∂y ∂x ∂y ∂z

(2.11)

ρ

Dw  ∂p ∂τ xz ∂τyz ∂τzz  = − + + + + ρ fz Dt ∂z ∂x ∂y ∂z

(2.12)

Equations (2.10), (2.11) and (2.12) are the Cauchy equations, written in tensor notation as ρ

Dui ∂p ∂τi j + + ρ fi =− Dt ∂xi ∂x j

(2.13)

We express the x component of the Cauchy equations in conservative form by using the continuity equation   ∂τ ∂(ρu) ~ = − ∂p + ∂τ xx + yx + ∂τzx + ρ f x + ∇ · (ρ uV) ∂t ∂x ∂x ∂y ∂z

(2.14)

Similarly, the other components of the linear momentum equation can be written in conservative form. However, these cannot be used in the absence of information on the general stress system in the flow. In fluid flows, this is solved by stating that the strain rates are linearly related to velocity gradient; such fluids are said to be Newtonian. In this book, non-Newtonian fluids are not considered. An elaborate discussion on the relation between stress and strain rates are to be found for Newtonian flows in [396] and [414]. For Newtonian fluids, Stokes [501] assumed no losses for quasi-static dilatation and compression of a fluid element and thus one relates stress and strain by the simplified form, τi j = λδi j

 ∂v ∂v j  ∂vk i − pδi j +µ + ∂xk ∂x j ∂xi

(2.15)

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If pm is the mechanical pressure, then by definition pm = τ3ii . Stokes hypothesized equality between thermodynamic and mechanical pressures to relate µ and λ as 3λ + 2µ = 0

(2.16)

The ratio of µ with ρ defines the kinematic viscosity (ν). Using the constitutive relation in Eq. (2.15), one obtains the Navier–Stokes equation for compressible flow as ∂(ρu) ∂(ρu2 ) ∂(ρuv) ∂(ρuw) ∂p + + + =− + ∂t ∂x ∂y ∂z ∂x        ∂ ~ + 2µ ∂u + ∂ µ ∂v + ∂u + ∂ µ ∂u + ∂w + ρ f x λ∇ · V ∂x ∂x ∂y ∂x ∂y ∂z ∂z ∂x

(2.17)

∂(ρv) ∂(ρuv) ∂(ρv2 ) ∂(ρvw) ∂p + + + =− + ∂t ∂x ∂y ∂z ∂y        ∂ ~ + 2µ ∂v + ∂ µ ∂v + ∂u + ∂ µ ∂v + ∂w + ρ fy λ∇ · V ∂y ∂y ∂x ∂x ∂y ∂z ∂z ∂y

(2.18)

∂(ρw) ∂(ρuw) ∂(ρwv) ∂(ρw2 ) ∂p + + + =− + ∂t ∂x ∂y ∂z ∂z        ∂ ~ + 2µ ∂w + ∂ µ ∂w + ∂u + ∂ µ ∂v + ∂w + ρ fz λ∇ · V ∂z ∂z ∂x ∂x ∂z ∂y ∂z ∂y

(2.19)

These equations are further simplified for incompressible flows. In the aforementioned equations, the terms multiplied by the bulk viscosity coefficient drop out due to the continuity equation for incompressible flows. If heat interaction is absent, then µ can be treated as a constant, and one finally obtains the vector form of the Navier–Stokes equation for incompressible flow as ~ ∂V ~ · ∇)V ~ = − ∇p + ν ∇2 V ~ + F~ + (V ∂t ρ

(2.20)

These are also called the equations in primitive variables. It is not necessary to use Stokes’ hypothesis, which actually removes the bulk viscosity term. If instead, one models λ with the help of experimental data, as in acoustic attenuation, then it is possible to capture the unsteady pressure field caused by the bulk action. This has been done successfully for the Rayleigh–Taylor instability problem, as will be discussed in Chapter 12.

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2.1.3 Navier–Stokes equations in derived variables Direct numerical simulation (DNS) and large eddy simulation (LES) of incompressible flows were initially formulated using primitive variables. Apart from the Marker and Cell or MAC-method [175], fractional step method has been set up with three-time level Adams–Bashforth scheme for convection terms in [232] to solve the Navier– Stokes equation. It has, however, been shown in [413, 416, 433] that the use of three-time level methods, including the Adams–Bashforth scheme is responsible for a spurious numerical mode, which removes high frequency components of the initial solution. Thus, fractional step methods are not suitable for DNS, as has been claimed at the time of its introduction in [232, 233]. Moreover, fractional step method requires specifying wall boundary conditions at intermediate stages. It has been noted in [33] that this is usually achieved via extrapolation from values obtained in previous time steps. Perot [330] has shown that while this fix may provide numerical stability, it also gives rise to a large time splitting error. To minimize this error, one is again forced to take small time steps. An approach that avoids this problem by establishing appropriate conditions is given in [238], but it is very memory intensive [292]. In [212], fourth-order accurate explicit ~ and compact finite difference methods for incompressible Navier–Stokes equation in (p, V)formulation is presented where four-stage, fourth-order Runge–Kutta (RK4 ) time integration scheme is used. In this formulation, the solenoidality constraint on velocity is enforced for each Runge–Kutta stage by solving respective Poisson equations for pressure correction; this is also computationally very intensive. To avoid pressure–velocity coupling problems, one can use formulations that eliminate pressure from the governing equation altogether. This is achieved elegantly in two-dimensional flows by stream function–vorticity (ψ, ω)-formulation, which allows the use of less number of unknowns as compared to three in the primitive variable formulation. It also exactly satisfies the mass conservation in the flow field, and one obtains vorticity as a solution of the governing equation, not by numerical differentiation of velocity. The extension of this solution to three-dimensional flows using vector potential–vorticity formulation is difficult due to the complexity of coupled boundary conditions. Although in recent times this formulation has been used to study transition and turbulence for three-dimensional periodic array of Taylor–Green vortices in [480], it is not readily useful for non-periodic problems. ~ ω ~ )-formulation can be considered as an alternative derived variable Thus, the (V, ~ = Dv = 0, formulation. One can also strictly enforce the numerical requirement of ∇ · V ~ ~ ~ )-formulation. This will be a significant improvement compared to (p, V)in the (V, ω formulation. ~ ω ~ )-formulation has been used for external [291] and The velocity–vorticity or (V, internal flow problems [563]. Mainly, two types of formulations have been used: (i) The non-conservative or convective form [274, 349] and (i) the Laplacian form [308, 563]. These two approaches differ in the way the convective acceleration and vortex stretching terms (shown in Eq. (2.21)) are handled in the vorticity transport equations (VTEs). In the non-conservative or convective formulation, the transport

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equations for vorticity are obtained by taking curl of the Navier–Stokes equation given in the primitive variables of Eq. (2.20). After the application of the curl operator and readjustment, one obtains the transport equation as [413] ∂~ ω ~ · ∇)~ ~ + 1 ∇2 ω ~ + (V ω = (~ ω · ∇)V ∂t Re

(2.21)

The vortex stretching term in Eq. (2.21) is the first term on the right-hand side. ~ and the nabla operator are Such a term is absent for two-dimensional flows, as ω perpendicular to each other. By contrast, one can rewrite the VTE in the Laplacian form by first casting the Navier–Stokes equation in primitive variables in the rotational form [412, 413], and then taking a curl of it to obtain the following ∂~ ω ~ = 1 ∇2 ω ~ ω × V) + ∇ × (~ ∂t Re

(2.22)

~ ω ~ )-formulation (presented in [163]) is often preferred, as The Laplacian form of (V, ~ = 0), this form also satisfies the solenoidality condition for the vorticity (Dω = ∇ · ω as has been shown using its evolution equation in [413]. In [163], it has been noted that this form implicitly satisfies the solenoidality condition for velocity, i.e., Dv = 0, by the way the governing equations are written. The Laplacian form has also been used in [105, 308]. In [105], the divergence free condition for the vorticity vector (Dω = 0) is satisfied by integrating the wall-normal component of vorticity (ωy ) from free stream to the wall, where the authors have been solving the stability problem for the flow over a rotating circular disk. This procedure works for external flows when the ~ ω ~ )vorticity components are zero at the free stream. Non-conservative forms of (V, formulation have been used in [273, 308]. The evolution equation for Dω is given in [33, 413, 563], identifying certain constructed schemes in a specific way to satisfy ~ divergence-free by adding the gradient of a scalar Dω = 0. The authors in [273] kept ω ˜ function φ to vorticity to form an auxiliary variable ~ tr = ω ~ + ∇φ˜ ω

(2.23)

~ tr is the unknown divergence-free vorticity vector. Taking divergence of Eq. where ω (2.23), one obtains the governing equation for the scalar φ˜ as ~ ∇2 φ˜ = −∇ · ω

(2.24)

When this approach is used, as in [268], one ends up solving four Poisson equations at each time step, a very computationally expensive and time-consuming method. Additionally, such procedures force one to encounter the problem of specifying ˜ as given in [563]. boundary conditions for φ,

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~ ω ~ )-formulation in conservative form has been proposed in [33] A new rotational (V, for DNS and LES of transitional and turbulent flows. To show the effectiveness of the proposed method, the authors derived evolution equations for Dω for the rotational, Laplacian and non-conservative forms of velocity-vorticity formulation. It has been established that enforcing Dω = 0 is easiest while using the rotational form. The proposed method has been developed for its use with an optimized compact scheme for spatial discretization [344] along with an optimized Runge–Kutta scheme [450] for time integration. This generic method uses the concept of treating spatial and temporal discretizations together to preserve the physical dispersion relation, when the numerical dispersion relation is considered. This is a general method, allowing possible applications for both external and internal flows. ~ ω ~ )-formulation 2.1.4 Governing equations for rotational form of the (V, For two-dimensional flows, it is easier to decide which derived formulation is ~ ω ~ )-formulation from the perspective of preferable. One can use either (ψ, ω)- or (V, solenoidality of vorticity. As the only non-zero component of vorticity lies in a plane perpendicular to the two-dimensional flow, it is ensured that the vorticity must satisfy the solenoidality condition, (Dω = 0), as the divergence operator is in the plane of the flow, while the vorticity vector is perpendicular to it automatically. In contrast, for three-dimensional flows, Dω , 0. More importantly, one must investigate to ensure the solenoidality of velocity, i.e., Dv  0, otherwise one will face similar problems ~ faced using primitive variable formulations. In (p, V)-formulation, the effects of nonzero Dv is studied in methods like MAC [175] and projection methods [232, 292, 330]. ~ ~ does not appear explicitly in (p, V)-formulations, However, as ω no reports on the effects of Dω are available. Evaluating the vortex stretching term in VTE given by Eq. (2.21) is important, as it defines the energy cascade by eddies for turbulent flows. Use of the vector identity ~ × B) ~ = A(∇ ~ · B) ~ − B(∇ ~ · A) ~ + (B ~ · ∇)A ~ − (A ~ · ∇) B ~ and solenoidality conditions, Dv = 0 ∇ × (A and Dω = 0, the Laplacian form of VTE given in Eq. (2.22) is obtained. One can further modify the viscous term of the Laplacian form by noting that ~ = ∇Dω − ∇ × (∇ × ω ~) ∇2 ω ~ = −∇ × (∇ × ω ~ ), if and only if ∇ · ω ~ = 0. Thus, ∇2 ω Modifying the right-hand side of Eq. (2.22), one gets the rotational form of VTE as ∂~ ω ~ + 1 ∇ × (∇ × ω ~) = 0 + ∇ × (~ ω × V) ∂t Re

(2.25)

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~+ which can be written in a concise form by denoting F~ω = (~ ω×V

1 Re ∇

~ ) in [33] as ×ω

∂~ ω + ∇ × F~ω = 0 ∂t

(2.26)

Having obtained the transport equations for the vorticity components, one is left with the task of obtaining relations between vorticity and velocity in terms of the Poisson equation for the velocity vector. This is obtained by taking the curl of the vorticity vector as ~ = ∇(∇ · V) ~ − ∇2 V ~ ~ = ∇ × (∇ × V) ∇×ω

(2.27)

If the velocity field is divergence-free, then the following vector Poisson equation for velocity components is obtained ~ = −∇ × ω ~ ∇2 V

(2.28)

Note that this equation explicitly uses the solenoidality condition for the velocity in its derivation. ~ ω ~ )-formulation 2.1.5 Evolution equation for solenoidality error in (V, To ensure the high accuracy needed for DNS and LES, it is essential to study evolution of Dv and Dω for any formulation. It is important that the solenoidality condition is not satisfied to minimize errors that may be caused by (Dv , Dω ) , 0. Taking divergence of Eq. (2.28), one obtains an equation for the divergence of velocity as ∇2 Dv = 0

(2.29)

~ = 0 is used, which is analytically In deriving this equation, the vector identity ∇·(∇× G) ~ However, there is a fallacy, as using the velocity true for any arbitrary vector G. ~ and then using the same vector identity would lead to Dω = 0. This is vector as G one aspect that the potential reader must be aware of. Whatever is given analytically, is not necessarily satisfied numerically. In the next chapter, it will be shown that it is not even easy to solve one-dimensional convection equation for which exact solutions exist! Serious practitioners of high accuracy computing methods are aware of this pitfall [413]. ~ ω ~ )-formulation, Dv achieves its Thus, if Eq. (2.29) is assumed to hold in a (V, maximum/ minimum value on the boundary, as given by the min–max theorem of the Laplace equation. This theorem also implies that Dv will be ensured to be zero everywhere, if one can ensure that the boundary condition for it is forced to be zero. Alternatively, implication of solenoidal vorticity does not ensure accuracy, as keeping the divergence of velocity at the boundary is a non-trivial exercise. One also notes that the time dependence of Dv is through the kinematic boundary conditions. Thus,

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to ensure Dv = 0, one needs to ensure zero divergence of velocity for the boundary at ~ ω ~ )-formulation. Authors in [546] have shown all times. This must be valid for any (V, ~ that for (p, V)-formulations nonzero Dv leads to spurious generation of turbulence. The evolution equation for Dω is obtained for the rotational form of VTE by taking divergence of Eq. (2.25) and using the same vector identity used in the derivation of Eq. (2.29). This is easily done as the convective and diffusive terms are written as a curl of a vector. Hence, one gets the following equation as ∂Dω =0 ∂t

(2.30)

With this evolution equation assumed to hold, one can ensure that Dω = 0 in rotational form by satisfying the condition that at t = 0 : Dω (0) = 0 in the computational domain. The evolution equation for Dω in Laplacian form is obtained analytically from Eq. (2.22) as 1 2 ∂Dω = ∇ Dω ∂t Re It has been reasoned [33] that to ensure Dω = 0 in the Laplacian form, one must satisfy the following conditions: (i) Dω = 0 at t = 0 in the full domain and (ii) Dω = 0 on the boundary of the domain subsequently for t ≥ 0. The situation is even worse for the non-conservative form of VTE. In this form, the evolution equation of Dω is derived from Eq. (2.21) to obtain the condition ∂Dω ~ 1 2 ~ · (∇Dv ) + + V · (∇Dω ) = ω ∇ Dω ∂t Re

(2.31)

This equation for Dω is the convection–diffusion–reaction equation with the source ~ · (∇Dv ). To have a vanishing source term (S ω = 0), Dv must be term given by S ω = ω zero at all points in the computational domain. Therefore, the necessary conditions to have Dω = 0 in the domain for all t are: (i) Dω = 0 at t = 0 everywhere in the domain, (ii) Dω = 0 on the boundary of the domain for all t > 0 and (iii) Dv = 0 for all t > 0 in the domain. Maintaining Dω = 0 computationally while using non-conservative VTE ~ ω ~ )-formulation described here. For is the most difficult among the three variants of (V, the non-conservative form, Dω grows due to the source term S ω which in turn often leads to numerical instability, as is the case with primitive variable formulation. Based on observations in [33], satisfying Dω = 0 is relatively easy for the rotational ~ ω ~ )-formulation. The authors furthermore suggested [33] that for Laplacian form of (V, formulation, one can replace one of the VTE by Dω = 0 to derive that component of vorticity. However, there may be practical difficulties to do this operation for both internal and external flows [33]. This suggests that the rotational form is best for high accuracy computations of physically unstable and transitional flows.

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In the reported computations for receptivity of external flows in [33], Dv = 0 is explicitly satisfied numerically by computing one of the velocity components directly using the continuity equation. This not only saves computational time, but also satisfies solenoidality of the velocity. The procedure works for external flows [29].

2.2 Boundary Layer Theory for Equilibrium Flow In [412, 552], various cases have been discussed for which simplifications can be made to the governing equations due to symmetries and/ or similarity considerations of the flow. This enables one to obtain a closed form solution that can be used for flow parameter predictions as well as obtaining equilibrium solutions to study instability. Next, we explore another very powerful concept which shows the possibility of simplifying the governing equation into different classes of partial differential equations for which easy solutions have been obtained. These can also be used as equilibrium solutions, whose receptivity and instability can be studied. In general, the governing fluid dynamical equation as developed by Navier and Stokes is a statement of dynamical equilibrium of various constituent terms. For example, at high Reynolds numbers, it can be reasoned that the convective and local acceleration terms can predominate over viscous diffusion terms. In the absence of a body force term, Eq. (2.20) simplifies to the well-known Euler equation given by ~ ∂V ~ • ∇V ~ = −∇p/ρ +V ∂t

(2.32)

This is the equation of inviscid motion used in early hydrodynamic studies. The rationale behind this simplification being that the viscous diffusion is effective only in a very narrow part of the flow domain, outside which the aforementioned equation will reproduce the actual flow behavior for high Reynolds numbers (Re → ∞). However, there are two major conceptual problems with this overtly simple modification. First, it does not reduce the number of independent variables, apart from converting the mixed partial differential equation, elliptic in space and parabolic in time, to an incomplete parabolic differential equation system, which is still nonlinear, and does not provide a closed-form analytic solution. Second, removal of the viscous terms causes the highest order derivative terms to disappear from the momentum conservation equation. This is a major hurdle, as reduction of the order of the partial differential equation will allow satisfaction of fewer boundary conditions compared to that required for the Navier–Stokes equation. Disregarding viscous diffusion at high Reynolds numbers may suggest that one can remove the noslip boundary condition and retain the impermeability condition to be satisfied. The satisfaction of impermeability is imperative for the sake of mass conservation. While the Navier–Stokes equation depends continuously upon the Reynolds number, the no-slip boundary conditions is independent of the Reynolds number. Thus, we must

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retain the viscous diffusion terms to satisfy the boundary conditions. It is true that the region where viscous diffusion is effective will change with the Reynolds number. One will never encounter a situation when the viscous region disappears entirely. For flows over a solid boundary, a very small region would exist close to the boundary within which viscous effects will be significant and dominate over inertial acceleration terms. Next, we probe the existence of such region(s) and examine the flow properties inside the region. As in many flows, this region exists as a thin layer next to the boundary, and is called the boundary layer. Outside this layer, the governing equation simplifies to that given in Eq. (2.32). This apparently simple observation has changed the understanding of fluid mechanics and spawned many fields of study. The observation was made by Prandtl [336] and is one of the major cornerstones of fluid mechanics. Boundary layer theory propounded the correct way of taking the high Reynolds number limit of the Navier– Stokes equation by simplifying the governing equation. For the Navier–Stokes equation at high Reynolds numbers, the viscous terms are multiplied by (1/Re) in the non-dimensional form, resulting in the formation of a thin shear layer (TSL) inside of which viscous terms remain relevant. In boundary layer theory terminology, this is the inner layer, where the Navier–Stokes equation can be simplified, whereas the Euler equation will drive the flow inside the inner or boundary layer given by the inner solution. The existence of a boundary layer, or more generally a TSL, depends on the smallness of the parameter (1/Re). With Re, we introduce a length scale without adequate justifications. Most often, it is the integral dimension of the fluid dynamical system, indicating an input energy that causes the fluid flow. However, in high Reynolds number flows, additional multiple length scales exist, including the thickness of the boundary layer. The existence of a shear layer is related to the structure of the solution, and this is modeled by the singular perturbation technique. When a TSL exists, it introduces a new length scale to the problem. This length scale is much smaller than the integral dimension of the flow field. One estimates it from the Navier–Stokes equation by an order of magnitude analysis. The integral length scale L defines the geometry that is maintained in an external flow to create the flow field. Thus, L is the diameter of a cylinder or the chord of an airfoil, the flow around which is to be investigated. This integral length scale defines the inertial terms in the momentum equation as ~ • ∇V ~ ∼ U 2 /L V

(2.33)

where U is similarly the velocity scale associated with the outer flow. The reason for creating an inner layer within which the viscous forces are non-negligible for high Reynolds numbers, is satisfaction of wall boundary conditions for the physical body. The developed inner length scale is much smaller than L, which we indicate as the viscous length scale, δ. The viscous forces have the following order of magnitude based on this new length scale

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~ ∼ νU/δ2 ν∇2 V

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(2.34)

Inside the TSL, the inertial and viscous forces are of the same order of magnitude as obtained in Eqs. (2.33) and (2.34), i.e., U 2 /L ∼ νU/δ2 which provides a relative comparison of these two length scales as  UL −1/2 δ ∼ = Re−1/2 L ν

(2.35)

The expression shows that the TSL progressively becomes thinner as the Reynolds number increases. This does not immediately tell us anything about the structure of the TSL, which has to come from experimental/ numerical observations. For the case of an infinitesimally thin flat plate placed in an uniform steady flow (U∞ ) in the xdirection, Euler’s equation, [Eq. (2.32)], shows that the velocity everywhere above the plate remains unaltered, as the flow is caused by inertial action in the absence of pressure gradient. In an actual flow, this will be observed, except for the fact that very near the plate, one has predominant shear stress (µ ∂u ∂y ) for the flow field to satisfy the no-slip boundary condition on the surface (y = 0). On either side of the plate, formation of TSL is noted. As one approaches the edge of the TSL, the shear stress 2 progressively become negligible, signifying that inside the TSL, ∂∂yu2 is very high. The streamwise flow asymptotically merges to the free stream value as the edge of the TSL is approached. A mathematical definition of the TSL thickness is the value of y where the streamwise velocity reaches 99% of its free stream value. For a semi-infinite flat plate, there is no natural choice of the integral length scale other than taking it as the distance of any point from the leading edge of the plate. Thus, according to Eq. (2.35), one has  U x −1/2 δ ∞ ∼ x ν

(2.36)

Equation (2.35) or (2.36) shows the existence of a TSL at high Reynolds numbers. Another way of interpreting the same is noting that outside the shear layer, ∂u ∂y is of ∂u the same order as ∂x , but within the TSL, ∂u ∂u >> ∂y ∂x

(2.37)

This is an alternative TSL assumption and is useful, when we consider flows without the presence of a solid body yet there is a narrow region inside the flow where the inequality in Eq. (2.37) holds, such as in mixing layers or jets. There is another TSL dδ approximation that is given as dx 0, then the inertial acceleration becomes weaker as the wall is approached, while the effects of pressure gradient remains same at all heights. This counteraction of pressure gradient on inertial acceleration can negate the condition of information propagating downstream as one approaches the wall. There would be circumstances when the fluid retarded by pressure gradient can cause the flow near the wall to point upstream over a small height. As a consequence of mass balance, the TSL thickness will increase abruptly, with the appearance of the shear layer being pushed into the main stream. These string of events are named as the separation of the flow from the wall. Retarded fluid cannot proceed in the local streamwise direction in the presence of sufficiently adverse pressure gradient. The fluid packet will be retarded by ddxp > 0, which will be aided by the drag. Thereafter, the shear layer next to the wall is deflected sideways in the wall-normal direction and can be graphically demonstrated by the streamline. We say that the flow has separated [414]. The point of separation is defined as the limit between forward and reverse flow inside the shear layer at the wall when ∂u ∂y = 0, i.e. the wall shear vanishes. w The close-up view of streamlines near the point of separation is shown in Figure 2.3. Mathematically, the wall streamline bifurcates at the point of separation, where the

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Sτ

τw > 0

w

τw < 0

0

Point of separation

Figure 2.3 Boundary layer evolution in the presence of adverse pressure gradient. The point of separation is denoted by vanishing wall shear stress. flow field is singular and is known as the half-saddle point. In fact, instantaneous wall streamline contours are better indicators of the phenomenon of separation. Such separation of streamlines will also cause rapid increase of the wall-normal component of velocity, implying shear layer thickening. These events invalidate the two alternate ∂ ∂ definitions of TSL, namely v 0, the wall value shows (uyy )w > 0 from Eq. (2.53). y

y

y

PI

u

uy

uyy

Figure 2.5 Adverse pressure gradient boundary layer flow shown by the velocity profile and its higher derivatives.

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In Figure 2.5, the velocity profile and its first two derivatives are shown under the influence of adverse pressure gradient. Here, uyy for y → ∞ is again negative. Thus, uyy changes from a positive value at the wall to a negative value in the free stream; it can only occur if uyy is equal to zero at some intermediate height. Hence, uy must be an increasing function of y in the vicinity of the wall following the Taylor series expansion  ∂u   ∂2 u  ∂u (y) ∼ +y 2 ∂y ∂y wall ∂y wall

(2.54)

However, uy will have to decay to zero as y → ∞, implying that the shear will have a maximum in the interim. The location where τ = uy is maximum must have uyy = 0. Such a point is known as the point of inflection, marked by PI in Figure 2.5, for the velocity profile and its two derivatives for the retarded flow. For the retarded flow, τ increases from its wall value to a maximum, and thereafter decreases to its negligible free stream value, as shown in the figure. We will show that the presence of a point of inflection at an intermediate height is a cause for probable temporal instability according to Rayleigh and Fjφrtoft’s theorem by an inviscid mechanism in Chapter 4. For zero pressure gradient boundary layer, the point of inflection is at the wall itself following Eq. (2.53) and is not a case for temporal instability by inviscid mechanism, which prompted the study of spatial instability and search for a viscous mechanism using the Orr–Sommerfeld equation.

2.4 Solving Boundary Layer Equation and Similarity Transformation Although in the present day context it is easy to solve the boundary layer equation by solving a set of nonlinear partial differential equations, it is to be noted that for certain cases, we can also transform the governing equations to ordinary differential equations, which are easily solved numerically. One makes use of similarity of velocity profiles by a similarity transformation. Such solutions are useful, nontrivial examples of the underlying physical processes. The similarity transformation also helps in numerically solving non-similar flows.

2.4.1 Similarity transform and analysis Consider a two-dimensional laminar flow without body force, governed by Eqs. (2.42) to (2.44). For external flows, using Eq. (2.32), one replaces ρ1 ddxp by the edge velocity term −Ue

dUe dx .

Thus, in general, one can write the solution of these equations as

u = u1 (x, y) Ue

(2.55)

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subject to the boundary conditions at y = 0; u = v = 0 and at y = δ; u = Ue where δ indicates the edge of the TSL. There are cases when Eq. (2.55) can be rewritten as u = u1 (η), Ue

(2.56)

where η is the similarity variable, a special combination of x and y. The ‘similar’ solutions are defined such that u has two velocity profiles u1 (x, y) located at different streamwise coordinates, x, which are similar and related to each others by a similarity transformation. Such velocity profiles at all x can be made congruent when these are plotted in terms of η. The existence of η as a single independent variable helps to reduce the dimensionality from two independent variables x and y. The existence of such a transformation converts the governing partial differential equations to an ordinary differential equation [82]. A brief description of the similarity transformation is given here via an example, by replacing Eqs. (2.42) to (2.44) in terms of stream function ψ. Existence of ψ automatically satisfies Eq. (2.42); Eq. (2.43) can be rewritten as ∂ψ ∂2 ψ ∂ψ ∂2 ψ dUe ∂3 ψ − = U + ν e ∂y ∂x ∂y ∂x ∂y2 dx ∂y3

(2.57)

We consider special flows past a wedge, for which the TSL edge velocity is presented as Ue = c˜ xm .

(2.58)

Such wedge flow results have been used for engineering calculations of flow past airfoils and other curvilinear bodies by obtaining the edge velocity by panel methods [414] and finding an equivalent local representation, with m as a function of x, given as, x dUe Ue dx = m. For the wedge flow, one can define the similarity coordinate by choosing c˜ = ν such that η=

y x(1−m)/2

=

y x1/2

xm/2 =

 U 1/2 e

νx

y

(2.59)

We also introduce a non-dimensional function f (η) by defining ψ = (Ue νx)1/2 f (η)

(2.60)

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This is the Falkner–Skan transformation to be used in the boundary layer equation, resulting in an ordinary differential equation for similar flows. For non-similar flows, the dependence of solution on x remains, and this is reflected in the choice of ψ as follows ψ = (Ue νx)1/2 f (x, η)

(2.61)

We carry out the transformations by moving from the (x, y)- to (x, η)-plane. The derivatives in the two planes are related by  ∂   ∂  ∂η ∂ = + ∂x y ∂x η ∂η x ∂x

and

(2.62)

 ∂  ∂η ∂ = ∂y x ∂η x ∂y

(2.63)

 d ∂f  ∂η So that − v = f (Ue νx)1/2 + (Ue νx)1/2 + (Ue νx)1/2 f 0 dx ∂x ∂x

and

As

Thus,

v

u = (Ue νx)1/2 f 0

∂η η =− , ∂x 2x

u

 U 1/2 e

νx

= Ue f 0

∂ η  Ue ν 1/2 0 1/2 v = − [(Ue νx) f ] + f ∂x 2 x

∂u ∂f0 dUe η Ue2 0 00 = Ue2 f 0 + Ue f 02 − f f ∂x ∂x dx 2x

Ue dUe ∂u Ue2 η 0 00 ∂ f Ue = f f − Ue2 f 00 + Ue f f 00 + f f 00 ∂y 2x ∂x 2x 2 dx

and finally ν

∂2 u Ue2 000 = f x ∂y2

So for Eq. (2.43), the left-hand side simplifies to LHS = Ue2 f 0

dUe 02 ∂ f Ue2 Ue dUe ∂f0 + Ue f − Ue2 f 00 + f f 00 + f f 00 ∂x dx ∂x 2x 2 dx

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xm

Ue x

Wedge

U

π 2

β

Figure 2.6 Flow over a wedge, with semi-wedge angle as defined in the figure. Similarly, the right-hand side of the equation is given by RHS = Ue

dUe Ue2 000 + f dx x

Thus, the boundary layer equation simplifies to Ue2 f 0

∂f0 dUe 02 dUe Ue2 000 ∂ f  Ue2 Ue dUe  00 + Ue f − Ue2 f 00 + + f f = Ue + f ∂x dx ∂x 2x 2 dx dx x

Multiplying both sides by x f0

x , Ue2

(2.64)

one gets

∂f0 x dUe 02 ∂ f 1 x dUe  00 x dUe + f − x f 00 + + ff = + f 000 ∂x Ue dx ∂x 2 2Ue dx Ue dx

(2.65)

This is simplified as  ∂f0 m+1 ∂f  f f 00 = x f 0 − f 00 2 ∂x ∂x

f 000 + m(1 − f 02 ) +

(2.66)

This non-similar boundary layer equation can be shown to be parabolic [413], and can be marched in the x direction. The aforementioned equation is solved subject to the wall boundary condition (y = 0), u=0 v = vw (x)

)

As y = 0 implies η = 0, the boundary condition given here is same as f 0 = 0 and ∂ψ ∂x = −vw (x). The latter is written in integral form as fw = −

1 (Ue νx)1/2

x

Z

vw dx, 0

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The other boundary condition at the edge of the boundary layer (y = Ye ) is given for y → Ye : as u → Ue , or for

η = ηe :

f0 = 1

where a finite but large value of ηe corresponding to Ye is taken as the far-field. It is easy to simplify the aforementioned equation for similar flows, for which f is not a function of x. Therefore, the governing boundary layer equation simplifies to f 000 +

m+1 f f 00 + m[1 − f 02 ] = 0 2

(2.67)

For similar flows, the boundary conditions cannot depend on x, and m is a constant. Similarly, fw must also be a constant. The equation given by Eq. (2.67) has been originally called the Falkner–Skan equation for flow over a wedge, with the flow 2m deflected by an angle of πβ/2, where β = m+1 . A sketch of the flow past a wedge is shown in Figure 2.6. For similarity, m is constant, so it is easy to see that the sign of m determines whether the flow is accelerating or decelerating. For decelerating flows, m takes a negative value, and it can take values up to m = −0.0904. Below this value of the Falkner–Skan parameter, flow separates, which is indicated by fw00 = 0 from the solution of Eq. (2.67). The wall skin friction coefficient can be obtained by using the transformed coordinate as 2 f 00 Cf = √ w Rx

(2.68)

whereas the displacement and momentum thicknesses are given as, Z

θ 1 = √ x Rx

Z

ηe

δ∗ 1 = √ x Rx

0

0

(1 − f 0 ) dη =

δ∗ ηe − fe + fw = √1 √ Rx Rx

θ1 f 0 (1 − f 0 ) dη = √ Rx

(2.69)

(2.70)

where the subscripts ‘w’and ‘e’ refer to wall and shear layer edge values, respectively. The velocity profiles for wedge flows depend on β, called the Hartree parameter.

2.4.2 Zero pressure gradient boundary layer The zero pressure gradient boundary layer develops over a flat plate at zero incidence, with Ue independent of x, such that m = 0, and Ue = U∞ . The governing boundary layer equation is given by

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f 000 + f f 00 = 0

(2.71)

This can be numerically solved by using appropriate boundary conditions. q νx Subsequently, one obtains the boundary layer thickness as δ = 5 U∞ , where the local velocity is 0.99155 of U∞ . In the top frame of Figure 2.7, the velocity profile ( f 0 ) is shown as function of η; shear ( f 00 ) is shown in the middle and u00 ( f 000 ) is shown in the bottom frame. The displacement and momentum thicknesses of the Blasius boundary layer are obtained as r δ∗ = 1.7208

r θ = 0.664

νx U∞

(2.72)

νx U∞

(2.73)

The shape factor for the Blasius boundary layer is H = δθ = 2.5911. We note that u00 and H are two differential and integral quantities that determine the instability of flow field. To be more precise, if u00 is the cause, then the final effect is noted in H! There is one very strong reason for which Blasius profile is used as the canonical flow in the studies of receptivity, instability and transition. It is easy to set up in the laboratories and use as the equilibrium flow for theoretical/ numerical investigations. However in recent times, researchers have reported that instead of taking the similarity solution for instability/ receptivity studies numerically, one must also include the leading edge of the plate in obtaining the equilibrium flow. Thus, the Navier–Stokes equation is also solved for equilibrium flow to include the non-similar region near the leading edge of the plate. Computing such equilibrium flows from the solution of the Navier–Stokes equation is also a delicate activity; this aspect is addressed later. ∗

2.4.3 Stagnation point or the Hiemenz flow This is another example of the shear layer equation developed for flow past a wedge, which is valid in the neighborhood of the stagnation point. Such a flow turns by an angle of π/2 upon impinging on the stagnation point and from Figure 2.6, one must 2m have β = 1. As β = m+1 , m = 1 for the stagnation point flow. The governing equation is obtained from Eq. (2.67) by substituting m = 1, to obtain f 000 + f f 00 − f 02 + 1 = 0

(2.74)

One notices that the stagnation point flow corresponds to a highly favorable pressure gradient. Such a pressure gradient is known to stabilize equilibrium flows without heat transfer [412]. This case was originally studied by Hiemenz [396, 412, 552] and

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(a) 1 0.8

f′

0.6 0.4 0.2 0

0

2

4

6

8

2

4 η

6

8

2

4 η

6

8

η

(b)

f ′′

0.3 0.2 0.1 0

0 (c)

0

f ′′′

-0.05

-0.1 0

Figure 2.7 Blasius profile and its derivatives with respect to similarity coordinates are shown for a zero pressure gradient boundary layer. was developed from the Navier–Stokes equation directly [188]. This flow has been shown to be highly stable with respect to linear theory [412], with the equilibrium flow having a shape factor of H = 2.22 lower than that for the Blasius profile. Similarly, the heuristic reason for the effects of blowing and suction on flow instability can be provided by noting that wall-normal blowing introduces a new mass of fluid with zero streamwise momentum into the boundary layer. This is akin to the flow due to adverse pressure gradient, resulting in reduced streamwise momentum of flow and at separation, this value of streamwise momentum is zero.

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2.5 Closing Remarks In this chapter, we have introduced the readers to a few equilibrium flows; there are other very good definitive texts providing more details about viscous fluid flows [552]. From the perspective of instability and receptivity studies, the focus is kept on two-dimensional equilibrium flows, which will be used in greater detail. Other equilibrium flows of interest for receptivity and instability studies are provided in [412]. Attention is drawn to the fact that there is a misconception among practitioners that by definition, equilibrium flows are necessarily steady, which then become unsteady due to instability. This is not necessarily so, as we have described in Chapter 1, the Taylor–Green vortex flow, admitted an unsteady two-dimensional equilibrium flow which becomes unstable in accurate numerical computations. In a later chapter, more details about the instability of the Taylor–Green vortex flow will be described. In this chapter, we have also familiarized ourselves with the process of obtaining simplified equilibrium flows via similarity transformations. In this context, it should be noted that almost all studies of external flows begin with the instability of zero pressure gradient Blasius boundary layer. It has also been noted that such flows are good approximations for flows over a flat plate at zero incidence, in regions away from the leading edge. While many studies abound for two-dimensional and threedimensional routes of transition for this velocity profile, there are severe limitations to these studies which exclude the leading edge of the plate. In recent times, various two-dimensional and three-dimensional transition routes have been studied [32, 419] where the leading edge is included inside the investigated zone. This is done by computing the Navier–Stokes equation for the equilibrium flow to exclude the limitations of traditional linear instability theories. It should also be pointed out that sometimes the equilibrium flows obtained by quasi-similarity transformations cannot be obtained from the solution of the Navier–Stokes equation, as in cases of mixed convection flows [423]. Thus, obtaining an equilibrium flow is of vital importance to study receptivity and instability.

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3.1 Computing Space–Time Dependent Flows It has been highlighted that the equilibrium flow described in the previous chapter requires high accuracy computations so that the subsequent investigation of receptivity and instability is not affected by numerical artifacts. It is imperative that all spatial and temporal scales are resolved accurately. For example, for the same equilibrium flow, if the imposed excitation level is increased, one may observe different types of transition with different wavenumbers and frequency spectra in the disturbance field. Thus, the behaviors perceived for different perturbed flows are not due to difference in the governing principles – rather these are due to altered boundary and initial conditions. These auxiliary conditions, in general, are given as either Dirichlet or Neumann boundary conditions. In convective heat transfer, one may have to deal with Robin or mixed boundary conditions. There are multiple aspects in computing governing equations for transitional and turbulent flows accurately. For example, one has to resolve all the space–time scales. Specifically, in numerically treating the space–time dependence of the problem, the discretization or integration process must be handled simultaneously. This last aspect is often overlooked, and spatial and temporal discretizations are treated separately. In Chapter 1, we have noted that in studies of instability and receptivity, the dispersion relation plays a central role. A poorly constructed numerical method will not follow this relation; in certain cases, the choice of numerical parameters are such that one incurs large dispersion errors. This aspect has to be studied carefully before one initiates numerical activity in studying instability and transition, which has been highlighted also in [412, 413].

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Speaking about spectra and resolution of space and time variation of variables, one would require a good understanding of waves. Although waves were introduced in the context of hyperbolic partial differential equations [540], dispersive waves are present for any flows governed by other types of partial differential equations too [412, 553]. We have noted in Chapter 1, that waves are created during the Kelvin– Helmholtz instability, for which the equilibrium flow is given by a uniform velocity profile and the disturbance field is governed by the Laplace equation (elliptic partial differential equation). In developing high accuracy computing methods in [413], it has been noted that numerical treatment of parabolic and elliptic partial differential equations requires that the discrete equations have the same formalism used for hyperbolic partial differential equations. Thus, in general, we represent any arbitrary variable in terms of its bilateral Fourier–Laplace transform [412, 537] as u(x, t) =

Z Z

U(k, ω0 ) ei(kx−ω0 ) dk dω0

where k is the wavenumber, ω0 is the circular frequency, and the integrals are performed along carefully chosen Bromwich contours [412] in the spectral planes. This chapter exposes the readers to wave attributes of many problems, which support the idea that waves are the building blocks of the perturbation field. Waves are characterized by a signal or information which is located at one point. Subsequently, another coherent and closely related signal or information is observed in its vicinity. The waves therefore carry information in space and time that may not involve any movement of the intervening medium. The perception of motion associated with a wave is related to the motion of phase and energy. These are represented by the phase speed and group velocity, both of which will be introduced shortly. As noted already, one excludes convection or advection of the medium through which waves propagate. In the presence of convection of the medium, one must include the Doppler effect in calculating signal propagation. Waves always arise due to a competition between a restoring force and inertia. A perfect balance between the two results in self-sustained oscillations noted in idealized physical systems. As noted by Whitham [553], there are no precise definitions of a wave, but intuitively we can define a wave as a perceived signal that propagates from one part to another part of a medium with definitive velocity of propagation, even in the absence of physical convection in the medium. One example of a wave is when the restoration is due to compressibility or elasticity of the medium, causing the particles to oscillate in the propagation direction of the wave. These are compression, elastic or pressure waves. For small amplitudes of pressure, one obtains sound waves. Another example is that of a surface gravity wave on a large expanse of water, with gravity trying to restore the motion. Gravity waves can also be seen in the interior of a fluid; they are known as internal gravity waves. For gravity waves, individual particles in an inviscid analysis show an orbital motion about a mean location. These orbits can take the form of a circle or an ellipse, for small perturbations of the interface.

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We define mathematically two main classes of waves, with the first obtained from a hyperbolic partial differential equation. The second class is due to the space–time dependence represented by the relation between wavenumber and circular frequency. This is the dispersion relation, with its real part admitting wave solutions, called dispersive waves. The governing equation can be time independent, but the time dependent boundary conditions gives rise to the dispersion relation, as was noted for the Kelvin-Helmholtz instability.

3.2 The Bi-Directional Wave Equation This section is based on the theory reported in [413]. The bi-directional wave equation is an example of hyperbolic waves given by utt = c2 ∇2 u

(3.1)

If, u represents fluctuating pressure, then Eq. (3.1) represents an acoustic wave with c as the speed of sound. For p elastic wave propagation in a bar, u is the longitudinal ρ/E, with E as Young’s modulus and ρ as the density. displacement and c = Moreover, one can replace the Laplacian by a partial second derivative with respect to the direction of propagation. For electromagnetic wave propagation, u is the electric √ or magnetic field and c = 1/ µ1 1 , with µ1 as the relative permeability and 1 the permittivity of the medium. One of the distinctive features of Eq. (3.1) is the presence of the second derivative of the dependent variable with respect to time. Due to this, the numerical procedures used for the Navier–Stokes equation are not readily usable for wave propagation problems. This has been highlighted in [304]. Such a second derivative with respect to time also appears in the solution of the linearized rotating shallow water equation [8, 344]. To understand the properties of the bi-directional wave equation [304], we consider the disturbance to propagate in the x direction; Eq. (3.1) simplifies to utt = c2 u xx

(3.2)

Consider the solution to be obtained subject to the following initial conditions u(x, 0) = f (x) and ut (x, 0) = gg(x) for

−∞< x |c|, i.e., the components are larger than the resultant and vector rules of addition and subtraction do not apply to phase speed.

Figure 3.2 Schematic of an oblique wave shown in the Cartesian frame.

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3.2.2 Requirements for spatial discretization While solving the Navier–Stokes equation, one must resolve all the present spatial and temporal scales. As the Navier–Stokes equation contains first and second order spatial derivatives, one must ensure adequate resolution of the discretization scheme to estimate these derivatives. First, we present various spatial discretization schemes and their resolution. Before we embark on this discussion, it is imperative to identify the derivatives with the associated physical processes. For example, the first derivative appears in the convective acceleration term of the inertial process, while the second derivatives are needed for the viscous diffusion process. Spatial discretization can be performed in many ways, but we will only use finite difference methods, which has been successful so far in reporting DNS of flows from receptivity to the fullydeveloped turbulent flow stage in [32, 415, 508, 559]. We begin by considering numerical discretization of the first derivative by explicit and implicit methods, shown for a uniform grid with spacing h. Let u0j = ( ∂u ∂x ) j represent the first derivative of the variable u = u(x) at the jth node. The second-order explicit central difference (CD2 ) scheme to evaluate u0j is given as u j+1 − u j−1 (3.23) 2h The order of accuracy can be ascertained by the Taylor series expansion of the righthand side about the jth node. The explicit fourth- and sixth-order accurate (CD4 and CD6 )-schemes for discretization of the first derivative are similarly obtained as −u j+2 + 8u j+1 − 8u j−1 + u j−2 u0j = (3.24) 12h u0j =

u j+3 − 9u j+2 + 45u j+1 − 45u j−1 + 9u j−2 − u j−3 (3.25) 60h There are certain aspects that are apparent for these explicit schemes; for example, the expressions on the right hand side of Eqs. (3.24) and (3.25) are the stencils which represent central schemes, which do not have terms with u j and the neighboring symmetric points have coefficients which are the same pairwise, but with opposite signs. Moreover we note that as the order of accuracy increases, the number of points required in the stencil keeps increasing. Apart from using the Taylor series expansion to characterize the discretization, there is another way of looking at discretization, which is particularly suited for the point of view we have adopted to study receptivity, instability and transition to turbulence. To evaluate the resolution of a particular scheme for discretization, we can express the function in terms of its bi-lateral Laplace transform given by Z uj = U(k)eikx j dk (3.26) u0j =

From Eq. (3.26), the exact first derivative is given by Z (u0j )exact = ikU(k)eikx j dk

(3.27)

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For CD2 -scheme, u0j can be written in spectral form as Z Z sin(kh) (u0j )CD2 = i U(k)eikx j dk = ikeq U(k)eikx j dk h

(3.28)

where the quantity (ikeq U(k)) is used instead of (ikU(k)) to evaluate the first derivative numerically. Thus, keq is called the equivalent wavenumber and the effectiveness of the derivative evaluation is represented by keq /k, which defines the resolution of any spatial discretization scheme plotted as a function of wavenumber. For CD2 and CD4 schemes, this ratio is obtained as k  sin(kh) eq = (3.29) k CD2 kh k   (4 − cos(kh)  sin(kh) eq = (3.30) k CD4 3 kh Note that for both Eqs. (3.29) and (3.30), (keq /k) −→ 1 as kh −→ 0, and this consistency condition must be satisfied by all discretization schemes. For kh −→ 0, one approaches the continuum from the discrete limit, and naturally, one must expect that (keq /k) −→ 1. Similarly, one can obtain keq /k for other schemes, like CD6 scheme etc. Note that for CD2 - and CD4 -schemes, (keq /k) is a real quantity, which is true for all central schemes using uniform grids. In the past, while simulating flows at low Reynolds number, one could solve the Navier–Stokes equation using the central scheme. However, as the Reynolds number increased, it was noted that the computed solutions exhibited oscillations even for flows that are known to be steady. The problem accentuated with further increase in Reynolds number and researchers concluded that the numerical methods based on central schemes are unstable. A potential solution to this problem was sought by using upwinded schemes. The essential concept behind these schemes is explored next; these schemes were used in an attempt to suppress numerical oscillations [411].

3.3 Upwind Schemes for Higher Reynolds Number Flows It is noted that oscillations arising out of numerical issues can be suppressed by using numerical diffusion in the same way the diffusion term added to the wave equation attenuates the wave amplitude with time. For example, instead of writing the second order central scheme, if one writes a one-sided scheme for the first derivative as follows u j+1 − u j u0j = (3.31) h the Taylor series expansion shows this representation to have first order accuracy. As the stencil is biased on the upstream side, it is referred to as the first order upwind scheme (UD1 ). Spectral analysis of the stencil provides (keq /k) for the UD1 -scheme given as

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k  eq

k

UD1

=

k  sin(kh) 2 sin2 (kh/2)  keq  eq +i −i = kh kh k real k imag

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(3.32)

The real and imaginary parts of (keq /k) are indicated by (keq /k)real and (keq /k)imag . The Taylor series expansion of Eq. (3.31) shows that upwinding is equivalent to adding a diffusion term proportional to h. Similarly, one can introduce a second order upwind (UD2 ) scheme [420] as u0j =

3u j − 4u j−1 + u j−2 2h

The resolution of this upwind scheme is obtained as k  sin(kh)(2 − cos(kh)) 4 sin4 (kh/2) eq = −i k UD2 kh kh

(3.33)

(3.34)

From Eqs. (3.32) and (3.34), one cannot easily discern any pattern by which upwind schemes can be constructed. However, in [262, 411] a generic scheme has been proposed by which the (2n − 1)th order upwind scheme can be obtained as a combination of the (2n)th order central scheme with the (2n)th order derivative of the same function. To demonstrate the construction of a third order upwind scheme, we first obtain the central scheme for the fourth derivative given by  d4 f  dx4

CD

= a( f j+2 + f j−2 ) + b ( f j+1 + f j−1 ) + c f j

(3.35)

The stencil comprises a symmetric pairing of terms to retain only even derivatives on the right-hand side. Collating the coefficients of even derivatives of orders up to four, we obtain 2a + 2b + c = 0

(3.36a)

22 a + b = 0

(3.36b)

24 a + b = 12/h4

(3.36c)

Solving for a, b and c, we get the fourth derivative as  d4 f  dx4

CD

=

f j+2 − 4 f j+1 + 6 f j − 4 f j−1 − 6 f j−2 h4

(3.37)

It has been reasoned [413] that accuracy demands higher resolution uncontaminated by numerical diffusion – a rationale for developing higher order central schemes, as for convection terms, these make the numerical scheme neutrally stable under ideal

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conditions. However, for actual computations at high Reynolds number, sources of error from nonlinearity, such as aliasing, will make the method numerically unstable. Such errors can be suppressed by adding dissipative terms. In the Navier–Stokes equation, the physical diffusion has a second derivative term, and thus, one would like to add numerical diffusion of a higher order. It was with this idea, that third order upwind schemes were introduced [223, 262, 458]. Other examples of using fifth order upwind schemes are noted in the literature [307, 342]. A third order upwind scheme for the first derivative is obtained from d f  dx

UD3

=

d f  dx

CD4

+θ

h3  d4 f  4! dx4 CD

where θ is a parameter to be controlled by the user, depending on various parameters in the problem. Although the blending of central schemes with even derivatives is shown with ordinary derivatives, in fluid dynamical applications, these will be partial derivatives. The value and sign of θ is dependent on the convection direction and can be decided locally, so that effectively diffusion is added locally. Up till now, we have discussed some higher order explicit central and upwind methods for spatial discretization. Next, we will discuss compact schemes, which are essentially implicit methods that have tremendous potential for solving the Navier– Stokes equation. The derivatives at all nodes are expressed as auxiliary implicit equations involving the unknown, and the methods have features of very high spectral accuracy, while using very compact stencils. Additionally, these schemes provide near-spectral accuracy and robustness, with less computational effort. Originally, these schemes were developed as Pad´e schemes for ordinary differential equations [243]. Application to partial differential equations was introduced in [1]. It has been made popular by the resolution analysis of the compact schemes in [261]. However, a major impetus for the use of compact schemes came with proper analysis and boundary closure schemes, starting with the appearance of works in [411, 438, 541]. These methods are truly an alternative to the spectral method and can solve non-periodic problems, using non-uniform mesh spacing while working in the transformed plane. In recent times, compact schemes have been developed using nonuniform grids [457, 481], which have potential applications in immersed boundary methods, to harness the power of compact schemes for flow past complex deforming bodies. As the representations are compact, resulting matrices are band-limited and the linear algebraic system is easily solved.

3.3.1 General compact schemes Consider again the function u whose derivatives are obtained using compact schemes in a uniform grid spacing of size h. To evaluate the general nth derivative denoted by u(n) j at the jth node, the general stencil is given by

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N2 X

ak u(n) j+k =

k=−N1

i

M2 1 X bk u j+k hn k=−M

(3.38)

1

This scheme is usable for uniform grids and is generally applied to solving governing equations in the transformed plane. In this equation, the derivative bandwidth to the left and right of the node is N1 and N2 , respectively. The derivatives are related to a function involving M1 points to the left of the jth node and M2 points to the right. If additional restrictions on ak s and bk s are made such that the left-hand side is symmetric about a0 and b0 , then the compact scheme will be central only if N1 = N2 and M1 = M2 . The coefficients ak and bk are determined by fixing the order of the scheme in the Taylor series expansion.

3.3.2 First derivatives obtained by compact scheme In this sub-section, the use of Eq. (3.38) is demonstrated for evaluation of the first derivative. Consider a central compact scheme obeying the following conditions [261] in Eq. (3.38): n = 1; N1 = N2 = 2; and M1 = M2 = 3

Also

a−2 = a2 = β; a−1 = a1 = α; a0 = 1

b3 = −b−3 =

b a c ; b2 = −b−2 = ; b1 = −b−1 = ; b0 = 0 6 4 2

Thus, the stencil is now given by 0

0

0

0

0

βu j−2 + αu j−1 + u j + αu j+1 + βu j+2

=c

u j+3 − u j−3 u j+2 − u j−2 u j+1 − u j−1 +b +a 6h 4h 2h

(3.39)

The terms are clubbed in such a way that only odd derivative terms are present. As there are five unknown coefficients, it is possible to equate the coefficients of the first to ninth derivatives, enabling one to construct a compact scheme up to the tenth order. As there are five points involved on the left-hand side for the derivative evaluation, one is required to solve a pentadiagonal linear algebraic set of equations. The aforementioned stencil works from j = 4 to (N − 3), when there is a total of N points. If one is solving a periodic problem, then the same stencil can be used for all points. However, solving a periodic pentadiagonal matrix equation is time consuming. For non-periodic problems, apart from the above aforementioned stencil,

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one would require what is called the boundary closure schemes (this will be described later). Unknown coefficients of the compact scheme are obtained by the Taylor series expansion of terms on right- and left-hand sides of Eq. (3.39) about the jth node. Matching coefficients of various orders, one generates the following relations. The first unmatched coefficient of derivative determines the formal order of accuracy by the Taylor series representation. To generate compact schemes, the following relations signify the order where one stops. For example, equate the coefficients by the following relations up to the indicated order, as denoted in [413]: 2nd Order :

a + b + c = 1 + 2α + 2β

(3.40)

4th Order :

a + 22 b + 32 c = 6(α + 22 β)

(3.41)

6th Order :

a + 24 b + 34 c = 10(α + 24 β)

(3.42)

8th Order :

a + 26 b + 36 c = 14(α + 26 β)

(3.43)

10th Order :

a + 28 b + 38 c = 18(α + 28 β)

(3.44)

The aforementioned equations can be interpreted in the following way: To use a sixth order scheme, one satisfies Eqs. (3.40), (3.41) and (3.42). Thus, one can choose any two unknown coefficients; the solution of these three equations fixes the other three unknowns in the stencil. Assembling all the equations (including the boundary closure schemes) for different nodes, one obtains a linear algebraic equation given by 0

[A]{u j } = [B]{u j }

(3.45)

In this equation, [A] is a pentadiagonal matrix. Despite the complex appearance of the right-hand side, it is essentially a vector, as {u j } are known and [B] is a constant coefficient matrix. For periodic problems, [A] matrix is a periodic pentadiagonal matrix and no boundary closures are needed. In Eq. (3.39), if we fix β = 0, then the [A] matrix in Eq. (3.45) will be a tridiagonal / periodic tridiagonal matrix depending on whether we are solving a non-periodic or periodic problem.

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Exercise: A Sixth Order Compact Scheme Here, a sixth order scheme due to Lele [261] is considered, for which one takes β = c = 0 in Eq. (3.39) to get the stencil as 0

0

0

αu j−1 + u j + αu j+1 =

a b (u j+1 − u j−1 ) + (u j+2 − u j−2 ) 2h 4h

(3.46)

Once again the coefficients are fixed by equating the Taylor series of the right- and left-hand sides of the aforementioned equation, about the jth node. Expanding and 0 equating the coefficients of u j gives 1 + 2α = a + b

(3.47)

000

For the coefficient of u j :

For the coefficient of u(v) j :

a + 4b 6

(3.48)

α a + 16b = 12 120

(3.49)

α=

With the three unknown coefficients, one can at the most obtain a sixth order or lower order scheme by equating successively the coefficient of the Taylor series expansion for Eq. (3.46). In such schemes, one must satisfy Eq. (3.47); this is the consistency condition for the scheme. Solving Eqs. (3.47) to (3.49) provides α = 1/3, a = 14/9 and b = 1/9 This is the sixth order scheme of Lele [261]; the leading truncation error term is 7!4 h6 u(7) . A fourth order accurate scheme can be obtained using Eqs. (3.47) and (3.48) with a one-parameter family of schemes shown with α as a = 23 (α + 2) and b = 13 (4α − 1).

3.3.3 Selection of compact schemes There are diverse compact schemes from which we can choose according to our requirements. For example, one may like to enquire about the significance of order for explicit and implicit schemes, or for the same order, we may want to determine which scheme is better. Additionally, one can also ask the following questions: (i)

What is the physical basis for choosing one compact scheme over another?

(ii)

Are the coefficients of the stencil discontinuous functions of the order of scheme?

Thus, instead of using the fixed values of α, a and b in Eq. (3.46), if one were to satisfy only the consistency condition (Eq. (3.47)), while changing the coefficients, one would want to determine if the representation would be poor in terms of accuracy.

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One would also like to determine if a small departure from the discrete representation would cause a small change for the numerical derivative. These do not have unique answers as the coefficients are not unique. Each term in the discretization must be consistent for h → 0. With model equations, one can show the connection between the discretization of individual derivatives and the discrete equation for consistency, along with other properties. Determining whether changing the coefficients would affect accuracy is equally important in terms of leading order truncation error. Haras and Ta’asan [174] obtained optimal values of the coefficients in Eq. (3.46), minimizing the error of the discrete approximation of the compact scheme from the Fourier spectral approximation. This error is measured over the Nyquist range of kh, i.e., (kh)max = π. The authors in [174] focused on a second order scheme. These observations indicate that the accuracy of the truncation error cannot be the sole criterion for solution error. A better depiction would be via resolution in the spectral plane, given by keq /k. In Eq. (3.45), discrete methods have different [A] and [B] matrices. For explicit schemes, one needs [A] = [I], the identify matrix. Apart from the interior points, there are other sources of error originating from boundary closure schemes, necessary for compact schemes used in non-periodic problems. Special stencils are used for the boundary and near-boundary points. For the first derivative, one requires closure at boundary nodes, j = 1 and N, and also for points at j = 2 and j = N − 1, for the compact schemes given by Eq. (3.46). The main stencil is used for interior nodes from j = 3 to (N − 2). Adams [1] suggested using the following boundary closure schemes along with the sixth order interior stencil given by 0

−5u1 + 4u2 + u3 h

0

j = 1 : 2u1 + 4u2 =

0

0

0

j = 2 : u1 + 4u2 + u3 =

0

3 (u3 − u1 ) h

0

0

3 ≤ j ≤ N − 2 : u j−1 + 3u j + u j+1 =

(3.50)

(3.51)

−u j−2 − 28u j−1 + 28u j+1 + u j+2 12h

(3.52)

The stencils for j = N − 1 and N are similar to those given by Eqs. (3.50) and (3.51). The stencil for j = 2 is of fourth order accuracy, while the interior points have sixth order accuracy.

3.4 Global Spectral Analysis: Resolution The Taylor series expansion provides the order and indicates the leading truncation error of the mathematical approximation. Resolution, in contrast, shows our ability to resolve length scales present in a physical problem. The resolution is measured

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in terms of the largest wavenumber (kmax ), represented without any implicit filtering, which is distinctly different from the maximum wavenumber resolved according to the Nyquist criterion: km = π/h, for the uniform grid. For a numerical scheme, one can perform Fourier analysis by representing the unknown u(x j ) with its transform U(k) by the following relation u j = u(x j ) =

Z

km

U(k) eikx j dk

(3.53)

−km

For the general stencil given in Eq. (3.45), one can rewrite the general scheme as 0

{u } = [C]{u}

(3.54)

with [C] = [A]−1 [B]. It is easy to determine the difference between explicit and compact schemes. In explicit schemes, [A] is the identity matrix and the size of the stencil is fixed by the [B] matrix. The stencil for explicit central scheme is one point wider than the order; hence, stable boundary and near-boundary one-sided stencils are also needed. For this reason, rarely more than five-point stencils are used for explicit schemes. For compact schemes, [A] and [B] matrices are band-limited, but [C] will be a wideband matrix in Eq. (3.54). Despite the compact size, compact schemes have very large equivalent explicit stencils, which guarantees tremendously high resolution.

3.4.1 Spectral accuracy of some compact schemes One of the most common sixth order compact schemes is due to Lele [261]; it is given by Eq. (3.46), with α = 1/3, a = 14/9 and b = 1/9 for the desired sixth order of the scheme. Another useful compact scheme (OUCS3) is obtained via optimization, given by the following interior stencil pi−1 v0i−1 + v0i + pi+1 v0i+1 =

 1 q−2 vi−2 + q−1 vi−1 + q0 vi + q1 vi+1 + q2 vi+2 h

(3.55)

η η η Here, the coefficients in Eq. (3.55) are given by pi±1 = D± 60 ; q±2 = ± F4 + 300 ; q±1 = ± E2 + 30 11η and q0 = − 150 with D = 0.3793894912; E = 1.57557379; F = 0.183205192. For the upwind version of this scheme, one uses η , 0. Formally, this is only second order accurate; yet in the spectral plane, this scheme has a resolution which is superior to many higher order compact schemes. It has been shown [420] that this second order scheme is significantly more accurate than the sixth order scheme of [261]. When second derivatives are required to be evaluated numerically, there are specific schemes, such as the one given in [261]; or one can use the first derivative given in Eqs. (3.46), (3.55) twice to evaluate the derivative.

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There are also combined compact difference (CCD) schemes, one of which is given in [95, 443, 476], to evaluate first (variables indicated with a prime) and second derivatives (indicated with double prime) simultaneously. The interior stencils of this scheme are given by h 15 7 0 (vi+1 + v0i−1 ) + v0i − (v00i+1 − v00i−1 ) = (vi+1 − vi−1 ) 16 16 16h

(3.56)

1 3 9 0 (v − v0i−1 ) − (v00i+1 + v00i−1 ) + v00i = 2 (vi+1 − 2vi + vi−1 ) 8h i+1 8 h

(3.57)

The global spectral analysis (GSA) for the derivatives is performed for combined compact difference (CCD) schemes to solve non-periodic problems in [413]. One can rewrite the aforementioned for global analysis as [A1 ]{v0 } + [B1 ]{v00 } = [R1 ]{v} [A2 ]{v0 } + [B2 ]{v00 } = [R2 ]{v} These equations can be simplified to equivalent explicit expressions for the first and second derivatives as {v0 } =

1 [C]{v} h

{v00 } =

1 [C2 ]{v} h2

where [C] = ([A1 ] − [B1 ][B2 ]−1 [A2 ])−1 ([R1 ] − [B1 ][B2 ]−1 [R2 ]) h

(3.58)

and [C2 ] = ([B2 ] − [A2 ][A1 ]−1 [B1 ])−1 ([R2 ] − [A2 ][A1 ]−1 [R1 ]) h2

(3.59)

All the three compact schemes shown here are central in nature. Thus, for each compact scheme, there is an equivalent explicit scheme that can be constructed. If one uses Eq. (3.26), then the exact derivative of the function is given by Eq. (3.27). The derivative at the point x j is evaluated by the phase at x j on the right-hand side, as can be noted from Eq. (3.53). In Eq. (3.54), on the right-hand side, the function values at all nodes are involved. However, one can relate the function at the jth and lth node by

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Z

U(k) eikxl dk =

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Z U(k) eik(xl −x j ) eikx j dk

(3.60)

We introduce a projection operator given by Pl j = ei(l− j)kh = Rl j + iIl j

(3.61)

which is split into real and imaginary parts. One represents a derivative at the jth node by the variables projected at the same node from the following 0

uj =

Z X N

C jl Pl j U(k) eikx j dk

(3.62)

l=1

with the entries of the [C] matrix represented by C jl . Thus, one rewrites the right-hand side of Eq. (3.62) as 0

uj =

Z

km

ikeq U(k) eikx j dk

(3.63)

−km

where the discrete form of the derivative at the jth node is related by ikeq (x j ) =

N X

C jl Pl j

(3.64)

l=1

For any scheme, one needs to obtain its [C] matrix; then using Eq. (3.64), one obtains keq globally in the domain. This global analysis was introduced in [438]. real ) showing the resolution of the scheme and keq is complex, with the real part (keq imag the imaginary part (keq ) showing the introduced numerical diffusion/ anti-diffusion [413]. In representing various schemes, (keq /k) shows efficiency of the scheme with respect to the Fourier spectral method known to produce the best resolution, plotted against nondimensional wavenumber, kh, ranging between zero and π. For symmetric interior stencils, the schemes are non-diffusive. However, one is forced to use asymmetric stencils for non-periodic problems, for the boundary and near-boundary points. As compact schemes are global, such asymmetric boundary and nearboundary stencils will have an effect on many interior nodes. One can characterize the spatial discretization schemes by viewing (keq /k) plotted as a function of kh. First, in Figure 3.3, the real and imaginary parts of (keq /k) are plotted for explicit spatial discretization schemes. Here, the real part (keq /k)real indicates the resolution of the chosen discretization scheme, while the imaginary part (keq /k)imag signifies the added numerical diffusion of the scheme.

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(a)

(b) 1.2

1.5

(k eq /k) imag

1

(k eq /k) real

CD2 CD4 CD6 UD1 UD2

1

0.8 0.6

CD2 CD4 CD6 UD1 UD2

0.4 0.2 0

0.5

1

0.5 0 0.5 1

1.5 kh

2

2.5

1.5

3

0.5

1

1.5 kh

2

2.5

3

Figure 3.3 a) (keq /k)real and b) (keq /k)imag plotted for CD2 , CD4 , CD6 , UD1 , and UD2 spatial discretization schemes as a function of kh. In Figure 3.4, for few representative compact schemes, the real and imaginary parts of (keq /k) are plotted as a function of kh. From the real part, it is clearly evident that the compact schemes are far superior in providing resolution for the first derivative. Moreover, one can see that Lele’s sixth order scheme is more accurate than (b)

(a)

NCCD 1

0

0.8

–0.2

0.6

–0.4

Lele 6th order

(keq/k)real

(keq/k)real

OUCS3(η = –2)

0.4 OUCS3 NCCD Lele 6th order

0.2

0

–0.6

–0.8 –1

0

0.5

1

1.5 kh

2

2.5

3

0

0.5

1

1.5 kh

2

2.5

3

Figure 3.4 a) (keq /k)real and b) (keq /k)imag plotted for the compact spatial discretization schemes indicated in the frames as a function of kh. the CD6 explicit scheme (not shown), even though both the schemes are of the same formal order. It is equally relevant to note that the optimized formally second order accurate OUCS3 scheme is more accurate than Lele’s sixth order accurate scheme. The

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New-CCD (NCCD) scheme is marginally less accurate as compared to the OUCS3 scheme, but it also evaluates the second derivative, which is necessary for solving the Navier–Stokes equation. However, one of the disadvantages of this method is that the NCCD method can only be used in the Cartesian frame. In the transformed plane, the diffusion term cannot be evaluated by the NCCD scheme. The value of (keq /k)imag plotted on the right frame shows that since Lele’s and NCCD schemes are central, these do not add numerical diffusion, while the OUCS3 scheme has a builtin mechanism for upwinding, which in this case has η = −2. As this parameter is controlled by the user, it can be used judiciously; the reduced value will not attenuate the larger scales in the flow, while controlling the smaller scales, where physical dissipation should be dominant anyway. The added numerical diffusion or anti-diffusion given by (keq /k)imag will effectively decide upon the discretization of the space–time dependent problem. However, it can also be qualitatively understood by considering semi-discrete analysis (where time-integration is considered as exact and spatial discretization is as given for the chosen numerical schemes) of the model 1D convection equation. The model convection–diffusion equation is considered next to analyze a few spatial and temporal discretization schemes. Before embarking upon this exercise, some typical time discretization schemes are described next.

3.5 Time Integration Schemes Before we proceed further, it is necessary that we discuss time integration, as this is important to preserve the physical dispersion relation property. A more detailed and nuanced discussion about spatial and temporal discretizations is given in [413, 420] from the perspective of the dispersion relation preservation (DRP) property. For a partial differential equation, one can perform spatial discretization to render the original partial differential equation to an ordinary differential equation at each and every node. To understand the various time integration approaches, one must consider the model semi-discretized partial differential equation as du = f (t, u) dt With an appropriate initial condition, this equation will be time advanced. In the following, two time levels, and multistage higher order time discretization methods are described, which are known to have the desired levels of accuracy, stability and DRP property. Here, we refer to only explicit Runge–Kutta methods, which are designed to work with weighted average of slopes in the time interval [tn , tn+1 ], by finding m increments for an mth order method, written in the following manner

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K1 = h f (tn , un )

(3.65)

K2 = h f (tn + c2 h, un + a21 K1 )

(3.66)

K3 = h f (tn + c3 h, un + a31 K1 + a32 K2 )

(3.67)

.. .

(3.68)

Km = h f (tn + cm h, un + am1 K1 + am2 K2 · · · + am,m−1 Km−1 )

(3.69)

Each of the K j s are evaluated from its predecessors in the explicit method. The Runge– Kutta method is written as a weighted sum of the aforementioned increments given by un+1 = un + W1 K1 + W2 K2 + · · · + Wm Km

(3.70)

These parameters of the method given by c j , a ji and W j , with i ≤ ( j − 1) are obtained by expanding the Taylor series on both sides and matching the terms in Eq. (3.70). We truncate the terms up to the order of the method. Some of these methods are described next.

(a) Explicit two-stage, second order Runge–Kutta (RK2) method This method is given by K1 = h f (tn , un ), K2 = h f (tn + c2 h, un + a21 K1 ) The solution at the advanced time is given by un+1 = un + W1 K1 + W2 K2

(3.71)

A commonly used method has the following parameter values a21 = 2/3, W1 = 1/4 and W2 = 3/4. It has been shown [416] that this time integration method displays unconditional instability, even when used with the Fourier spectral spatial discretization method for solving the one-dimensional convection equation (see Figure 3 of the reference). In the following, for the three-stage and four-stage Runge–Kutta methods, we will consider a single first order autonomous ordinary differential equation.

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By autonomous, we imply that the right-hand side of the following equation will not be a function of time explicitly du = f (u) dt

(3.72)

which requires an initial data noted as u(to ) = uo

(3.73)

(b) Three-stage, third order Runge–Kutta (RK3) scheme For the autonomous ordinary differential equation in Eq. (3.72) with a time step h from tn to tn+1 , following the general Runge–Kutta methods, we evaluate the three slopes to time-advance the solution by the following weighted sum of slopes for the third order Runge–Kutta method un+1 = un + W1 K1 + W2 K2 + W3 K3

(3.74)

with the slopes written as K1 = h f (un ),

K2 = h f (un + a21 K1 ),

K3 = h f (un + a31 K1 + a32 K2 )

One can look up any standard text to note the following parameters for the third order RK3 method for a non-autonomous ordinary differential equation c2 = 1/2, c3 = 1, a21 = 1/2, a31 = −1, a32 = 2, W1 = 1/6, W2 = 4/6 and W3 = 1/6 In [413], another set of values are also given for the autonomous ordinary differential equation as a21 = 1/3, a31 = −5/12, a32 = 5/4, W1 = 1/10, W2 = 1/2 and W3 = 2/5 We note that this RK3 method does not suffer from the instability problem, as does the RK2 method used with the Fourier spectral method for solving the one-dimensional convection equation shown in Figure 3 of [416].

(c) Fourth order Runge–Kutta (RK4) method for autonomous systems Consider again the autonomous system given in Eq. (3.72) subject to the initial conditions given by Eq. (3.73). To develop an RK4 explicit method to integrate this equation from tn to tn+1 , we require four increment functions:

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K1 = h f (un ), K2 = h f (un + a21 K1 ), K3 = h f (un + a31 K1 + a32 K2 ), K4 = h f (un + a41 K1 + a42 K2 + a43 K3 ) such that the time advancement yields un+1 = un + W1 K1 + W2 K2 + W3 K3 + W4 K4

(3.75)

The standard RK4 method uses the following parameters a21 = 1/2, a31 = 0, a32 = 1/2, a41 = 0, a42 = 0, a43 = 1 and W1 = 1/6, W2 = 1/3, W3 = 1/3, W4 = 1/6 It is to be noted that three or higher multistep time integration schemes are never to be used for DNS and LES to compute flow receptivity and transition. It has been shown in researched articles [413, 433, 453, 464] that such methods invoke spurious modes, and the results can be unreliable. Even though the demonstrations are made with model one-dimensional convection equations, some additional results are also shown for the Navier–Stokes equation to emphasize this aspect of spurious modes and their effects.

3.6 Analysis of Convection–Diffusion Equation It is to be noted that the Navier–Stokes equation represents a dynamical description of any fluid system showing the competing physical processes of convection and diffusion, apart from local acceleration and actions of body forces. In general, for a stable flow, one can consider the system to be in dynamical equilibrium; only if this equilibrium state is not stable with respect to omnipresent infinitesimal, background disturbances, will the flow instability can be triggered through a sequence of events starting with receptivity, followed by propagation of disturbances till the flow reaches another equilibrium state. For example, for flow past a cylinder, primary instability can take the steady state to a state where one can observe periodic vortex shedding. Similar situations have been noted for internal flow inside a cavity, where the background disturbances take the system to a periodic limit cycle from the original stationary flow. In both these examples, if the flow is computed, then the background round-off error is strong enough to initiate the process, even if the discretization error is negligibly small. In fact, for less accurate methods, transition from one equilibrium state to another and then to another, will be quicker. Conversely, for more accurate methods, the onset of unsteadiness will be delayed in both real time and in parameter space. For these examples, there are experimental analogues. Thus, while computing such a flow field which may be experiencing physical instabilities, it is necessary that the numerical method should mimic the physical scenario. However, as the quantification of background disturbances may not be

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possible, it is difficult to assess the numerical method. It is in this context, that a computational practitioner should look for model equations that represent the competing physical processes of convection, diffusion and unsteadiness, which at the same time admit exact solutions. Pure convection equation (in any dimension) provides the ideal benchmark for calibrating any space–time discretization process, as the exact solution is nondissipative and non-dispersive, i.e., the initial condition simply moves with the phase speed. As the system is non-dispersive, the energy propagation speed also will be equal to this phase speed. Although it may appear as one of the simplest equations, computing the flow field has been proven to test the very best of methods. There are many references in the literature, but we will cite only a few references that are often consulted, such as [502, 531] for numerical instability studies using von Neumann analysis or [78, 165] for time-stability analysis. It has been noted in [506] that these approaches have limitations with the most notable one being their inability to perform a full domain analysis in the spectral space using the actual time discretization method. A full domain spectral analysis with appropriate error metrics [413] reveals considerable information about stability/instability, dispersion and dissipation errors for all length and time scales. This type of analysis is absolutely critical to design DRP schemes for DNS and LES of transitional and turbulent flows. Furthermore, this spectral analysis demonstrates the non-intuitive property that the convection equation shows signal and error to have different dynamics [436] for any discretization scheme. The cited reference shows that for this linear system, the error and signal are governed by different equations. Similarly in [425], error dynamics of the linear diffusion equation is studied to quantify the error of discrete schemes. It is difficult to accept that the numerical phase speed in the convection equation and the coefficient of diffusion in the heat equation are not equal to the constant value in the numerical computations. To comprehend the numerical issues and error in computing, we reproduce here some of the salient points of the analysis results reported in [506] with the help of the linear convection–diffusion equation. One can obtain the convection or diffusion equation from this equation as special cases; one can also use this equation as a linearized model of the Navier–Stokes equation. Such a spectral analysis is reported for the first time in [506], identifying the appropriate metrics. Different spatial discretization schemes in conjunction with the fourth order Runge–Kutta (RK4) scheme for time integration is used for the analysis. There are specific reasons to consider the finite difference method (FDM) over the finite volume method (FVM) and the finite element method (FEM). Earlier efforts in solving one-dimensional convection and Euler equations with the same high accuracy compact scheme for FVM and FDM in [441] have established that for the Riemann problem, FDM is superior. Spectral analysis in [427] have also analyzed 1D convection equation using FDM, FVM and FEM, where the authors have shown that FEM prevents formation of upstream propagating waves or q-waves, but at the cost of using excessive diffusion preventing the recommendation of such FEMs for accurately solving instability problems.

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Consider the one-dimensional linear convection–diffusion equation given by ∂u ∂u ∂2 u +c =α 2 ∂t ∂x ∂x

(3.76)

where c and α are the constant phase speed and constant coefficient of diffusion, respectively. First, to perform global spectral analysis (GSA), one represents u(x, t), in a hybrid form [413, 438] as u(x, t) =

Z ˆ t)eikx dk U(k,

(3.77)

where Uˆ is the bilateral Laplace amplitude. Using this in Eq. (3.76), one obtains the spectral plane representation dUˆ + ickUˆ = −αk2 Uˆ dt This equation can be solved for any initial condition u(x, 0) = f (x) = obtain the exact solution ˆ t) = F(k) ˆ e−αk2 t e−ikct U(k,

(3.78) R

ikx ˆ F(k)e dk to

(3.79)

To obtain the physical dispersion relation, one represents the unknown now using ω ˆ ω)ei(kx−ωt) dk dω, yielding the the bilateral Fourier–Laplace transform as u(x, t) = U(k, dispersion relation given by ω = ck − iαk2

(3.80)

The phase speed and group velocity can be directly obtained from the dispersion relation. For solving convection–diffusion dominated problems, any combination of space and time discretization scheme must satisfy the physical dispersion relation numerically to remove or reduce phase and dispersion error [413, 436]. A precise explanation of the numerical dispersion relation for multiple time level schemes for the convection equation is given in [464]. The physical phase speed is obtained from the physical dispersion relation as cphys =

ω = c − iαk k

(3.81)

Moreover, the physical group velocity is defined as Vg,phys =

∂ω = c − 2iαk ∂k

(3.82)

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Rearranging this equation, one obtains the coefficient of diffusion as α = which when split into real and complex parts yields α=

i (Vg,phys )imag i h (Vg,phys )real − c) − 2k 2k

i 2k (Vg,phys

− c),

(3.83)

As α is real, (Vg,phys )real = c is the condition for a physically diffusive system with the aforementioned expression on the right-hand side strictly real. As the right-hand side (V )imag of Eq. (3.76) is strictly diffusive (for all k), one must have g,phys < 0 to ensure that 2k there is no anti-diffusion. The physical amplification factor Gphys is defined using the exact solution in Eq. (3.79) as Gphys =

ˆ t + ∆t) U(k, 2 2 = e−αk ∆t e−ikc∆t = e−iω∆t = e−Pe (kh) e−iNc (kh) ˆ t) U(k,

(3.84)

α∆t In this expression, Nc (= c∆t h ), is the CFL number and Pe(= h2 ), is the Peclet number – the relevant non-dimensional parameters of this equation. The absolute value of G is known as the amplification factor. Physical amplification factor for two values of Peclet numbers, Pe = 0.01 and 0.5 are shown in Figure 3.5. With increase in the Peclet number, the rate of diffusion increases according to Eq. (3.84). From the real and imaginary parts of Gphys , one can calculate the phase shift per time step, and hence the phase speed of the signal. For the physical problem, this will be found to be equal to c for all k, the constant value prescribed in the differential equation. Similar to the physical amplification factor, one can obtain the corresponding amplification factor Gnum for any combinations of discretization schemes for this convection–diffusion equation. We will shortly note that from this Gnum , we can define the numerical dispersion relation for the numerical method used to solve the convection–diffusion equation. Thus, for high accuracy computing, one must ensure that to accurately reproduce the physical attributes of the governing equations, Gnum must be very close to Gphys . For the 1D linear convection–diffusion equation, the numerical dispersion relation for convection and diffusion can be written directly by analogy from [436] and [425]

ωnum = kcnum − iαnum k2

(3.85)

We emphasize that cnum and αnum are not constants; instead, these are functions of k for any numerical method as demonstrated here. Using the numerical dispersion relation, it is easy to write down the numerical amplification factor as Gnum = e−iωnum ∆t = e−αnum k

2

∆t −ikcnum ∆t

e

(3.86)

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3

0.925

2.5

Pe = 0.01 0.95

2

kh

0.97

1.5 1 0.5

0.99

0.999 0.9999

0

1

2

3

4

5

Nc |G phys| 0.01

3 2.5

0.05

Pe = 0.5

2

kh

0.2

1.5

0.4 0.6

1 0.8

0.5

0.9 0.99

0 1

2

Nc

3

4

5

Figure 3.5 Physical amplification factor |Gphys | contours for the 1D convection–diffusion equation. The top and bottom panels represent contours for the Peclet numbers 0.01 and 0.50, respectively. As noted earlier, numerical phase shift per time step ∆t is obtained from the real and imaginary parts of Gnum as (Gnum )Imag tan(β) = − (Gnum )Real

! =⇒ β = kcnum ∆t

(3.87)

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which provides the numerical phase speed as (Gnum )Imag cnum β 1 = =− tan−1 c kc∆t (kh)Nc (Gnum )Real

! (3.88)

From the ωnum expression as a function of k, one can derive the numerical group ∂ velocity as, Vg,num = ∂k (ωnum ), which is simplified further to yield the following Vg,num 1 dβ = Vg,phys Nc d(kh)

(3.89)

where Vg,phys is the physical group velocity, which is already noted to be equal to c for the 1D convection–diffusion equation. The numerical diffusion coefficient (αnum ) has been obtained from Eq. (3.86) by 2 noting that |Gnum | = e−αnum k ∆t . As a consequence, the non-dimensional ratio of the diffusion coefficient is expressed as ln |Gnum | αnum =− α Pe (kh)2

! (3.90)

The aforementioned quantity indicates the diffusion in terms of the physical diffusion. However, as the physical diffusion is always positive, a negative value of αnum would indicate anti-diffusion and lead to catastrophic breakdown of the numerical solution. This is an important consideration and has been shown in the context of compact scheme analysis and role of boundary closure in [438]. It is emphasized here that to obtain meaningful solutions for the convection– diffusion equation, the most important properties are related to numerical diffusion, numerical phase speed and numerical group velocity. Investigations for any numerical Vg,num cnum method examine the ratios, αnum α , c , and Vg,phys ; for high accuracy, these should be as close to unity as possible. In the following, estimates of Gnum will be provided for a cnum couple of high accuracy combinations of discretization schemes to present αnum α , c , Vg,num . and Vg,phys

3.6.1 Spectral analysis of numerical schemes The first step in GSA is to transform the discrete governing equation from the physical to the spectral space. One achieves this by using the representation in Eq. (3.77) for the dependent variable at any node for the discrete equation. Thereafter, one obtains the numerical amplification factor [413, 438] (G j for the jth node), shown for two numerical schemes. Both the displayed analysis use the

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|Gnum | cnum g,num Figure 3.6 Contours of (a) |G , (b) αnum α , (c) c and (d) Vg,phys for RK4 -OUCS3-CD2 scheme phys | with Pe = 0.01. [Reproduced from “Spectral analysis of finite difference schemes for convection diffusion equation”, V. K. Suman, Tapan K. Sengupta, C. Jyothi Durga Prasad, K. Surya Mohan and Deepanshu Sanwalia, Comput. Fluids, vol. 150, pp 95-114 (2017), with the permission of Elsevier.] V

four stage, fourth order Runge–Kutta (RK4 ) method for time integration. The G j for RK4 time integration for an equation of the form ∂u ∂t = L(u) is given in [413] by

Gj = 1 − Aj +

A j2 A j3 A j4 − + 2 6 24 (3.91)

ˆ

where A j = − L(UˆU) and Uˆ is as defined by Eq. (3.77).

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3.6.2 RK4 -OUCS3-CD2 Scheme Next a case is investigated, for which the compact scheme is used for discretizing the convection term and the RK4 scheme for time integration. Compact schemes provide higher resolution and are known for high accuracy in many DNS and LES [17, 32, 222, 250, 333, 383, 422, 426, 536]. Here, the OUCS3 scheme is used to discretize the convection term, as reported for convection dominated problems [413]. The internal stencil is given in Eq. (3.55). The upwinding coefficient used is η = −2. For the boundary closure, the following explicit schemes are used: 1 [−3u1 + 4u2 − u3 ] 2h " #  8β 1   8β 1  2β 1  2β 1  u1 − u2 + (4β + 1)u3 − u4 + u5 − + + u´2 = h 3 3 3 2 3 6 3 " #  8β 1   8β 1  2β 1 2β 1  − uN − + uN−1 + (4β + 1) uN−2 − + uN−3 + uN−4 uN−1 ´ =− h 3 3 3 2 3 6 3 u´1 =

u´N =

1 [3uN − 4uN−1 + uN−2 ] 2h

β = −0.025 ( j = 2), β = 0.09 ( j = N − 1)

Following Eqs. (3.45) and (3.54), as given in [413], the compact scheme can be represented in an equivalent explicit form using the matrix notation with [A], [B] and [C] as constant matrices. For the present case, Al is given by using Eq. (3.54) as Al = Nc Cl j eik(x j −xl ) + 2 Pe [1 − cos(kh)]

(3.92)

which can be used to evaluate Gnum from Eq. (3.91). Following which, the ratios αnum α , Vg,num cnum c , and Vg,phys are calculated for this specific numerical method. The use of GSA allows one to obtain the numerical properties of any node, however, for ease of understanding, representative performance of the method is given for an interior node in Figs. 3.6 and 3.7 in which the contours of the indicated properties are shown for the Pe values 0.01 and 0.5, respectively. Accuracy of the scheme can be ascertained from the figures by the choice of appropriate numerical parameters. For a pair of Pe = 0.01 and very small values of Nc , the numerical and physical diffusions are almost the same for a large range of wavenumbers, as can be observed in Figure 3.6. When the Peclet number is increased to Pe = 0.5, one notes that for lower wavenumbers, there is higher diffusion, whereas higher wavenumbers have

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max:3.47e+03 min:8.01e-02

3 0

35

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max:1.63e+02 min:-8.95e+01

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|Gnum | g,num Figure 3.7 Contours of |G (top left), αnum (top right), cnum (bottom left) and Vg,phys α c phys | (bottom right) for RK4 -OUCS3-CD2 scheme for Pe = 0.5. [Reproduced from “Spectral analysis of finite difference schemes for convection diffusion equation”, V. K. Suman, Tapan K. Sengupta, C. Jyothi Durga Prasad, K. Surya Mohan and Deepanshu Sanwalia, Comput. Fluids, vol. 150, pp 95-114 (2017), with the permission of Elsevier.] V

lower diffusion. Such scale dependent diffusion will also effectively show splitting of the coherent part of the signal by this apparent dispersion. This numerical method combination shows better properties for numerical phase speed and numerical group velocity as compared to low order explicit spatial schemes for even very low Pe values.

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3.6.3 RK4 -NCCD Scheme Having noted the anomalous diffusion depending on length scales for the RK4 OUCS3-CD2 scheme at even small values of Nc , one presumes that the properties can be only improved if the diffusion term is discretized by a more accurate method than the CD2 method. If one decides to work on a Cartesian grid, then there is the possibility of using the CCD scheme, which has been already introduced. The CCD scheme is preferred as both the first and second order derivatives are evaluated simultaneously [95, 574]. Here, an improved CCD scheme (NCCD) proposed by Sengupta et al. [443, 476] is used to avoid problems with the boundary closure suggested in [95, 574]. For the sake of clarity of the present analysis, the stencil is given here along with the boundary closure u01 =

1 [−3u1 + 4u2 − u3 ] 2h

1 [u1 − 2u2 + u3 ] h2 " # 1 2β 1 8β 1 8β 1 2β 0 ( − )u1 − ( + )u2 + (4β + 1)u3 − ( + )u4 + u5 u2 = h 3 3 3 2 3 6 3

u001 =

u002 =

1 [u1 − 2u2 + u3 ] h2

7 0 h 15 (u + u0j−1 ) + u0j − (u00j+1 − u00j−1 ) = (u j+1 − u j−1 ), 16 j+1 16 16h

j = 3, · · · , N − 2

3 1 9 0 (u j+1 − u0j−1 ) − (u00j+1 + u00j−1 ) + u00j = 2 (u j+1 − 2u j + u j−1 ), j = 3, · · · , N − 2 8h 8 h " #  8β 1   8β 1  1  2β 1  2β u0N−1 = − − uN − + uN−1 + (4β + 1)uN−2 − + uN−3 + uN−4 h 3 3 3 2 3 6 3 u00N−1 =

1 [uN − 2uN−1 + uN−2 ] h2

u0N =

1 [3uN − 4uN−1 + uN−2 ] 2h

u00N =

1 [uN − 2uN−1 + uN−2 ] h2

β = −0.025 ( j = 2), β = 0.09 ( j = N − 1)

(3.93)

Primes indicate derivatives with respect to argument, as defined before. Following [476], one writes Eq. (3.93) in linear algebraic notation as [A]{du} = {b}

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where the matrix [A] and the vectors {du}, {b} are as shown in the reference. The solution of these simultaneous equations can be expressed as 1 [D1 ]{u} h 1 {u00 } = 2 [D2 ]{u} h {u0 } =

(3.94)

where the matrices [D1 ] and [D2 ] are presented in [443]. Block tridiagonal matrix algorithm is employed to calculate the inverses needed in [D1 ] and [D2 ] matrices. After this, we follow the procedure described for the RK4 -OUCS3-CD2 scheme. The numerical amplification factor at a node l is evaluated with Al given by Al = Nc

N N X X [D1 ]l j eik(x j −xl ) − Pe [D2 ]l j eik(x j −xl ) j=1

(3.95)

j=1

An interior node is considered again for investigating the accuracy of the NCCD Vg,num |Gnum | αnum cnum , α , c and Vg,phys for Pe = 0.01 scheme. Figures 3.8 and 3.9 show the contours of |G phys | and 0.25, respectively. Excellent error behavior is noted for this scheme for the two Pes, with the lower Pe giving better results.

3.7 Role of Diffusion and Relation to Rotationality In a coherent description, the authors in [468] introduced the role of diffusion: Investigation of the true role of diffusion has remained as a problem starting from its stated role in stabilizing fluid flow by damping disturbances, attributed to Kelvin, Helmholtz and Rayleigh [412]. This confusion of equating diffusion with dissipation was the reason that all early instability studies ignored diffusion, as discussed in [119, 412]. When flow instability over flat plate was successfully investigated by solving Orr–Sommerfield equation (OSE) by the authors in [176, 396, 528], it was thought that retaining the diffusion is equivalent to producing an appropriate phase shift for a positive feedback, which leads to flow instability. The concept of rotationality comes from the vorticity content of flow, as governed by the vorticity transport equation given by Eq. (2.21) for three-dimensional flows. This vector equation can also be cast in rotational form as given in Eq. (2.22). A measure of vorticity field (~ ω) is provided by enstrophy, which will be defined by the dot product of the vorticity with itself as the local variable, ~ ·ω ~ Ω1 = ω A global description of enstrophy is defined by integrating the local form of enstrophy over a complete flow domain as ||ω||22 =

Z

..

Z

Ω1 dd~r

(3.96)

D

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|Gnum | cnum g,num Figure 3.8 Contours of |G (top left), αnum α (top right), c (bottom left), and Vg,phys (bottom phys | right) for RK4 -NCCD scheme for Pe = 0.01. [Reproduced from “Spectral analysis of finite difference schemes for convection diffusion equation”, V. K. Suman, Tapan K. Sengupta, C. Jyothi Durga Prasad, K. Surya Mohan and Deepanshu Sanwalia, Comput. Fluids, vol. 150, pp 95-114 (2017), with the permission of Elsevier.] V

where d in the superscript indicates dimensionality of the problem (2 for twodimensional and 3 for three-dimensional problems) and D defines the complete domain. Although we will mostly be concerned with the point property of enstrophy for inhomogeneous flows, we begin by stating some well-known results for a special case of homogeneous periodic flows. Considering two-dimensional periodic flows, Doering and Gibbon [114] have shown the enstrophy transport for the evolution of global enstrophy to be given by  d 1 ||ω||22 = −ν||∇ω||22 dt 2

(3.97)

where ν is the kinematic viscosity. Thus, the effect of diffusion is strictly dissipative

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for periodic homogeneous flow globally. Generalizing this point of equating diffusion with dissipation has been used in DNS of homogeneous turbulent flows in [567]. However, in terms of local enstrophy, the effect of diffusion is not strictly dissipative. This misinterpretation about diffusion being equal to dissipation may also have been due to the fact that for the global kinetic energy of turbulent flows, the effect of diffusion is again dissipative. Interested readers can consult Eq. (4.34) of Mathieu and Scott [287], which shows that the time-averaged diffusion term has two contributions: (i) a strictly dissipative part and (ii) another viscous transfer term. The authors note that the viscous transfer term integrates to zero over the whole flow by the divergence theorem; the term is sometimes also referred to as diffusive, because it is zero for homogeneous turbulence..... the viscous transfer term is negligible at high Reynolds numbers, except within

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the thin viscous layers very near any solid surfaces ; whereas, the dissipative term is of crucial importance to turbulence energetics everywhere. However, if one does not consider only the kinetic energy, and instead consider total mechanical energy, then the action of diffusion is again not strictly dissipative. One can construct an equation for the total mechanical energy [252] to clarify this situation about the role of diffusion. We write the Navier–Stokes equation in rotational form as p V ~ ·V ~ ~ ∂V ~ ×ω ~ ~ = −∇ + + ν∇ × ω −V ∂t ρ 2

(3.98)

The diffusion term is written as the curl of the vorticity vector multiplied by ν. Defining the total mechanical energy (E) as E=

~ ·V ~ p V + ρ 2

and taking divergence of Eq. (3.98) yields the governing equation for E as ~ ×ω ~) ∇2 E = ∇ · (V

(3.99)

The diffusion term gets canceled due to a vector identity, with terms on the right-hand side arising from the rotational part of the nonlinear convection term. The right-hand side is further simplified using the vector identity ~ ×ω ~ · (∇ × ω) ~) = ω ~ ·ω ~ −V ∇ · (V ~ ·ω ~ , the total mechanical energy equation is As the enstrophy at a point is Ω1 = ω written as ~ · (∇ × ω ~) ∇2 E = Ω1 − V

(3.100)

Equation (3.100) displays the importance of Ω1 and diffusion as central in distributing E, due to the form of Eq. (3.98). The authors in [567] use an equation for the static pressure (see Eq. (1.2) of the reference) which appears similar. Using present notations of the variables, this is written as ! 2 p ∇ = (Ω1 − /ν)/2 (3.101) ρ where  = 2ν si j si j , where the symmetric part of the strain tensor is indicated by si j . This equation led the authors to relate enstrophy with dissipation. Equation (3.101) is generally valid for all flows, not only for homogeneous isotropic turbulence [567].

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The DNS in [567], used the Fourier series representation in space, along with the two stage, second order Runge-Kutta time integration scheme. This space–time dependent discretization scheme is not recommended due to its numerical instability, and high dispersion error, shown by GSA in [416]. Comparing the Poisson equations, Eqs. (3.100) and (3.101), one can infer that the right-hand side terms are due to convective acceleration. As our intention is to highlight the importance of diffusion and enstrophy for inhomogeneous flows, a transport equation for enstrophy is described next.

3.7.1 Enstrophy transport equation There is a correspondence of the role of enstrophy in fluid flow for rotationality with kinetic energy describing translational motion. Enstrophy describes the specific energy spent by the system to create and sustain rotationality. Flows experiencing physical instabilities take an equilibrium state to another by redistributing the energy of the system among rotational and translation degrees of freedom. Thus, enstrophy is a natural dependent variable for the study of transitional and turbulent flows. One can explain instabilities and pattern formations with the help of the enstrophy transport equation (ETE) derived from the non-dimensional vorticity transport equation, which for 3D flows is given in tensor notation by ∂ui ∂ωi 1 ∂2 ωi ∂ωi = ωj + + uj ∂t ∂x j ∂x j Re ∂x j ∂x j

(3.102)

where subscripts, i, j = 1, 2 and 3, represent Cartesian axes and the repeated index implies summation. Taking a dot product of Eq. (3.102) with ωi and using Ω1 = ωi ωi as the local enstrophy, one obtains its transport equation, the ETE as ∂ui ∂Ω1 ∂Ω1 1 ∂2 Ω1 2 ∂ωi − 2ωi ω j = − + uj ∂t ∂x j ∂x j Re ∂x j ∂x j Re ∂x j

!

∂ωi ∂x j

! (3.103)

The implications of this transport equation are as follows. The last term on the lefthand side (LHS) is contributed by vortex stretching (corresponding to the first term on the right-hand side (RHS) of Eq. (3.102)), present only for three-dimensional flows. Vorticity diffusion contributes to the RHS terms in Eq. (3.103), where the first term of RHS is for the diffusion of Ω1 and the last term on the RHS is essentially due to the loss or dissipation for the transport of Ω1 . It has been noted in [468] that a global view of any two-dimensional homogeneous and periodic flow problem simplifies this transport equation. For inhomogeneous and nonperiodic flows, one can analyze such general problems by adopting the local approach as noted earlier.

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Looking at two-dimensional flows with non-zero component of vorticity (ω) in the plane perpendicular to the flow, the ETE is written in vector calculus notation as " # DΩ1 2 1 2 2 = ∇ Ω1 − (∇ω) Dt Re 2

(3.104)

One notes that the first term on RHS of Eq. (3.104) is missing in Eq. (3.96) for periodic, homogeneous flows. As viewed globally in Eq. (3.97), the RHS is strictly negative, implying the dissipative nature of diffusion on enstrophy for periodic and homogeneous flows. For the inhomogeneous flow viewed locally in Eq. (3.104), the first term on the RHS is indeterminate, as it can be either positive or negative. Diffusion in the enstrophy transport equation can either create or destroy enstrophy, and hence, rotationality, determined from the sign of the RHS in Eq. (3.104). Both the terms taken together indicate net generation or dissipation of Ω1 . Despite this, diffusion and dissipation have often been used interchangeably [567] (due to the behavior of homogeneous flows), as viewed globally. In the ETE, Ω1 is by definition always positive, and would indicate convective growth when the RHS is positive. The RHS would act as a sink of Ω1 , when it is negative. This indicates a rotationality creation mechanism for different length scales by diffusion. The creation mechanism is completely different from the creation mechanism of smaller scales by vortex stretching, popularly known for generating smaller eddies. This role of diffusion in creating new length scales is valid for two- and three-dimensional flows. Effects of diffusion in Eq. (3.104) at multiple scales have been investigated in [468] by deriving transport equations for higher powers of Ω1 . Multiplying Eq. (3.104) with n−1 Ω1 and defining Ωn = Ω21 , one obtains for Ω2 , the following transport equation " # DΩ2 2 2 −1 1 2 = 2Re ∇ Ω2 − (∇Ω1 ) − 2Ω1 (∇ω) Dt 2

(3.105)

DΩ1 2 Noting further that DΩ Dt = 2Ω1 Dt , one can write the aforementioned equation as an evolution equation for Ω1 . Multiplying this equation with Ω2 , and simplifying, one can obtain a transport equation for Ω3 , which can be used to rewrite the ETE involving Ω1 , Ω2 and Ω3 . This process can be generalized to obtain the transport equation for Ωn as

" # DΩn 2 −1 1 2 = 2Re ∇ Ωn − (∇Ωn−1 ) − C Dt 2

(3.106)

where C=

n−2 X k=0

2

n−k−1

 n−1   Y   Ω j  (∇Ωk )2 

(3.107)

j=k+1

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P Q and Ω0 = ω with indicating summation over all ks and indicating the product of all the jth elements. The substantive derivative of Ωn can be simplified as DΩn DΩn−1 = 2Ωn−1 Dt Dt DΩn−2 = 2Ωn−1 × 2Ωn−2 Dt  n−1  Y  DΩ1 Ω j  = 2n−1  Dt j=1 Thus, DΩn DΩ1 = 2n−1 Ω−β Dt Dt

(3.108)

where β = 1 − 2n−1 . Thus, with the help of these relations, one rewrites Eq. (3.106) as the enstrophy transport equation given by # β " DΩ1 Re−1 Ω1 1 2 2 (∇Ω ) − C = ∇ Ω − n n−1 Dt 2n−2 2

(3.109)

In writing the transport equation for Ω1 , one notes the presence of three terms. The leading diffusion term proportional to Ωn is indeterminate in its sign; the following term is proportional to Ωn−1 which is positive and due to the negative sign will constitute strict dissipation action. The last term is C, which consists of products of terms given by Ωn−2 downwards to Ω0 . As all the constituents are positive definite and C has negative sign, this contribution of C is also negative. From Eqs. (3.106) and (3.107), one can draw similar conclusions for all n, i.e., all order of moments of enstrophy shares the same property for general inhomogeneous flows, making it impossible to state that the diffusive contribution of the Navier– Stokes equation is strictly dissipative, as many researchers have shown for periodic, homogeneous flows, including homogeneous turbulent flow [567]. As shown in Eq. (3.109), the enstrophy is noted to involve higher order squared terms of enstrophy, as if the process can cascade indefinitely. The natural query that can arise is: up to which order, are the terms relevant in determining enstrophy transport? One aspect of this is that up to any order, the corresponding transport equation does not show strict dissipation due to the presence of leading diffusion terms, the rest being strictly dissipative. It is obvious that higher order terms of enstrophy will progressively contribute more for higher wavenumbers, which indicates the order of terms retained is restricted by the amount of energy supplied to the flow at the largest scale. For three-dimensional flows as well, the RHS of Eq. (3.109) will determine growth or decay by these forcing terms. However, additionally, the vortex stretching term will

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also participate in determining the transport of enstrophy. The enstrophy transport equation for a three-dimensional flow is same as given by Eq. (3.103). Similarly, deriving the transport equation for Ωn for three-dimensional flows will be given by " # ! n−1 Y DΩn ∂ui 2 −1 1 2 n = 2Re ∇ Ωn − (∇Ωn−1 ) − C + 2 Ωk ωi ω j Dt 2 ∂x j k=1

(3.110)

with C obtained from the same expression given in Eq. (3.107). Thus, with Eq. (3.108), one can rewrite the enstrophy transport equation for three-dimensional flows as # β " DΩ1 Re−1 Ω1 ∇2 Ωn ∂ui 2 = − (∇Ωn−1 ) − C + 2ωi ω j n−2 Dt 2 ∂x j 2

(3.111)

Hence, one concludes that diffusion gives rise to an enstrophy cascade for both twoand three-dimensional flows. The contribution at higher wavenumbers depends on n, which in turn is fixed upon the energy supplied to the flow. In three-dimensional flows, vortex stretching is an additional mechanism of the energy redistribution process. Thus, in three-dimensional flows, vorticity at different scales is caused by enstrophy cascade by diffusion and vortex stretching terms. The presence of the vortex stretching term and energy cascade is therefore implicit to the convection process. For two-dimensional flows, the diffusion term alone gives rise to enstrophy (and hence vorticity) at different scales.

3.8 Spatial and Temporal Scales in Turbulent Flows Practical flows at high Reynolds number are either transitional or turbulent. Unlike laminar flows, which are classified as steady or unsteady, classification of transitional or turbulent flows is not straightforward as these exhibit space–time scales with wideband spectra. These are sources of problems in computing these flows, and we must be able to resolve these scales. We note that these scales are related via the dispersion relation. First, we discuss the scales separately.

3.8.1 Spatial scales in turbulent flows Following the accepted definition of turbulence as an aggregate of eddies of different sizes, the scales in turbulent flows are related to eddy sizes. The largest length scale cannot exceed the size of the body that creates the flow and this is associated with the dimension of the fluid dynamical system, denoted by l0 . We define turbulent flows with the help of the kinetic energy density i.e., kinetic energy per unit volume (E(k, )) in the wavenumber magnitude (k) plane. Using Parseval’s theorem, it has been shown [266] that the total energy remains the same viewed either in the physical plane or in the wavenumber plane.

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In computing high Reynolds number flows, one can first postulate an integral scale, l0 , which is of the order of the largest dimension of the flow. The corresponding velocity scale is denoted as u0 , and can be approximately related to turbulent kinetic √ energy (kT ) by 2kT /3. Conversely, for extreme length scales, one can invoke Kolmogorov’s hypothesis of local isotropy. If the energy dissipation rate is given by ε (which has SI unit of m2 s−3 ), then the integral length scale is shown by Kolmogorov’s 3/2 theory as l0 ∝ kTε . So that one can define a turbulent Reynolds number by ReL = kT1/2 l0 ν

k2

= νεT . Kolmogorov’s theory is given in detail in [139, 335]; a brief summary is provided here. As described earlier, homogeneous turbulence has the turbulent kinetic energy kT = 12 (u¯0 2 + v¯0 2 + w¯0 2 ), which is same everywhere. Additionally, if the turbulent flow is isotropic, then the eddies behave the same in every direction, i.e., u¯0 2 = v¯0 2 = w¯0 2 . In his theory, Kolmogorov argued that the directionality of the turbulent flow at the largest scales will be lost at the smallest scale, as energy is transferred from larger to smaller eddies by various mechanisms of vortex stretching and others defined before in terms of the enstrophy transport equation. Thus, Kolmogorov’s hypothesis of local isotropy for flows at sufficiently high Reynolds numbers suggests that the small-scale (l) turbulence is dictated by (l lEI ) and the small-scale isotropic eddies for which (l < lEI ). For some high Reynolds number flows, the intermediate length scale can be estimated as lEI ≈ l0 /6 [335]. At this stage, following Kolmogorov’s first similarity hypothesis, not only is directional information lost during energy cascading to a smaller scale, but information about the geometry of the flow is also lost. This leads to the proposition that the statistics of the small-scale motions are universal: these are the same for all high Reynolds number turbulent flows, irrespective of the mean flow and associated boundary conditions. The limit on the size of small-scale eddies is decided by the rate at which the energy is received from larger scales (which is approximately equal to the dissipation rate ε) and the viscous dissipation, represented by the kinematic viscosity ν. Thus, the first similarity hypothesis can also be interpreted by the statistics of the small-scale motions to have universality uniquely determined by ε and ν. A schematic of various length scales for high Reynolds number turbulent flows are shown in Figure 3.10. This directly leads us to the Kolmogorov scales from a dimensional argument to define the following ε unique scales: Length scale : η = (ν3 /ε)1/4 Velocity scale : uη = (νε)1/4 Time scale : τη = (ν/ε)1/2

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Thus, one can define a Reynolds number based on this scale as, Reη = uη η/ν = 1, i.e., at this scale, the viscous terms are of the same order as of the inertial terms. These are the smallest scales present in the flow at which the energy is dissipated and converted to heat. The scales that are smaller than lEI are said to be in universal equilibrium state due to the associated eddies’ ability to reach dynamic equilibrium quickly. These scales are statistically isotropic if the flow quantities are scaled by Kolmogorov’s scale. The ratios between large and small scale eddies’ length, velocity and time scales are given by η/l0 ∼ Re−3/4 L uη /u0 ∼ Re−1/4 L τη /τ0 ∼ Re−1/2 L where we have used the relation given before as l0 ∝ kT3/2 / and the turbulent Reynolds number as ReL = kT2 /νε. At high Reynolds numbers, the velocity and timescales of the smallest eddies are small compared to those of the largest eddies. We also note that η/l0 decreases with ReL , and for high Reynolds number, one can think of a range of intermediate scales which is smaller compared to l0 , although being larger compared to η. Thereafter, Kolmogorov stated his second similarity hypothesis by which another intermediate scale is predicted, which lies in the range, l0 >> l >> η, where an universal form is dependent on the supplied energy in the bulk (given by ε), while being unaffected by viscous actions, i.e., ν. At this stage, one can formalize this intermediate scale as lDI , with a numerical value given by, lDI = 60η. This length scale sub-divides the universal equilibrium range further into an inertial sub-range (so that this scale is spanned between lEI (above) and lDI (below)), where inertial effects dominate while viscous effects are sub-dominant; and another dissipation sub-range which is in between lDI and η. Thus, for high Reynolds number flows, most of the energy resides in the range: lEI (l0 /6) < l < 6l0 . Next, we define the Taylor microscale, indicated by λ, which in essence defines the inertial sub-range in terms of two-point auto-correlation function. This was essentially introduced by Taylor [512] to explain homogeneous isotropic turbulence. Due to homogeneity, such a correlation function Ri j (r, t) ≡ hui (x+r, t)u j (x, t)i is independent of x, and therefore, at the origin, one obtains 0

Ri j (0, t) = hui u j i = u 2 δi j

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Due to isotropy, Ri j can be defined in terms of longitudinal ( f (r, t)) and transverse auto-correlation (g(r, t)) functions as (see [335] for details)  ri r j  0 Ri j (r, t) = u 2 g(r, t)δi j + ( f (r, t) − g(r, t) 2 r such that for a longitudinal spacing, r = e1 r, one obtains these as 0

R11 /u 2 = f (r, t) = hu1 (x + e1 r, t)u1 (x, t)i/hu21 i 0

and R22 /u 2 = g(r, t) = hu2 (x + e2 r, t)u2 (x, t)i/hu22 i For isotropic turbulence, this two-point correlation is completely defined by f (r, t). Longitudinal Taylor microscale (λ f ) is obtained from f (r, t), which being an even function, must have the first derivative f 0 (0, t) as zero and the second derivative as negative. These properties are used as dimensional argument, λ f (t) = [− f 00 (0, t)/2]  This can be related to the velocity derivative as h

∂u1 ∂x1

2

i =

0

2u 2 . λ2f

Since the dissipation  2 1 function is also related to the velocity derivative by ε = 15νh ∂u ∂x1 i, one obtains by  1/2 T equating these two relations, λ f = 10νk . ε The Taylor-scale Reynolds number, Rλ is based on λ f and the corresponding velocity scale, u0 , as Rλ = u0 λ f /ν. According to Pope [335], the Taylor-scale has no definitive physical meaning, despite having a specific value. R∞ By definition, the turbulent kinetic energy is given by kT = ∞ E(k)dk. One can develop an expression for E(k) in the inertial sub-range by dimensional analysis. Noting that the dimension of kT , ε and k are given by the units, m2 s−2 , m2 s−3 and m−1 , one immediately notes that E(k) should be represented by, m3 s−2 . In terms of ε and k then, E(k) ∝ ε2/3 k−5/3 Or E(k) = Ck ε2/3 k−5/3

(3.112)

This is the well-known three-dimensional spectrum according to Kolmogorov’s theory. The constant is experimentally obtained as Ck = 1.5. In Figs. 4.14 and 4.15 in [413], the energy spectra have been shown for experimental pipe flow data given in [256] and for turbulent boundary layer [380]. The longitudinal

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Figure 3.10 Schematic of length scales noted for high Reynolds number turbulent flows. spectrum for the pipe flow (E1 (k1 )) is obtained at different radial locations for the pipe flow experiment conducted at Re = 500, 000 based on diameter and centerline velocity. The energy spectrum shows the maximum energy to be at low axial wavenumber (k1 ) corresponding to the diameter of the pipe, whereas the dissipation of energy () peak is located at k1 which is three orders of magnitude higher. It is conjectured that DNS of turbulent flow should resolve at least up to the Kolmogorov scale. One notes that all practical flows are inhomogeneous, as described in the previous section for the enstrophy transport equation. Still, as a rule of thumb, this estimate for resolving flows computationally is the holy grail of DNS! Resolution lower than this, when computed with any model will be described as a LES. Dissipation peak is defined from the energy budget of fluctuation which provides the dissipation as ν||∇u||22 . In general, the energy spectrum depends on k, dissipation (ε) and molecular viscosity (ν). For turbulent boundary layers at high Reynolds numbers, there is a region very close to the wall where the velocity varies linearly with the distance from the wall. This region is called the viscous sublayer and its thickness ξ, is related to l0 by l0 = Re ξ

(3.113)

If we represent the cut-off wavenumber by kc (related to η as, kc = 2π/η), then the relation given between integral length scale and Kolmogorov length scale can also be written as 3

kc l ≈ Re 4

(3.114)

Equation (3.114) is used for setting a limit for grid size in performing DNS. Thus, for 3D DNS, the resolution requirement should scale as (Re3/4 )3 or roughly about Re2 . In traditional LES, the flow is computed by resolving all the way up to the inertial sub-range; anything smaller than this is modeled via sub-grid scale (SGS) stress models. With modern day high accuracy compact schemes and use of numerical

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filters, under-resolved simulation of DNS, without using any explicit model, is termed as implicit LES or ILES. There are many such reported results in recent times.

3.8.2 Temporal scales in turbulent flows Temporal scales are determined based on spatial scales by the dispersion relation. Despite such dependence, one can view temporal scales as an independent variable. It is then determined by the frequency spectrum of the dynamical system, with effects induced by external agencies or those triggered as a consequence of instability. This latter aspect is also known as self-induced unsteadiness. Even for pure hydrodynamic applications at high Reynolds numbers, these two sources of unsteadiness may be coupled. In aeroelastic application or more generally for flow-induced structural vibration events, there is a coupling between structural and fluid dynamical modes. In general, unsteadiness can be stochastic, highly organized or of mixed interacting nature, as in turbulence. From the point of view of numerical computing, one is most interested in the highest frequency, as resolving that temporal scale will be the critical time step. It is not merely the Nyquist criteria; as we have already noted, there are other sources of numerical errors, which will cause even stricter limits on the time step. For unsteady flows characterized by a dominant frequency, given by f, we define a non-dimensional parameter called the Strouhal number as, S t = ful00 , where the flow speed has the velocity scale, u0 . For unsteady flows, such as in rotor wing aerodynamics or turbomachines, there is an imposed time scale corresponding to the rotation rate. As the dynamics are governed by nonlinear equations, the response is multiperiodic with imposed frequency (ω0 ) and its higher harmonics. Furthermore, the blades/aerofoils in their rotary motion pass through trailing vortices (wakes) of the preceding blades/ aerofoils. Such flows are characterized by another non-dimensional parameter defined as k f = ω2π0 c , where c is the chord of the airfoil section. Some typical S t for unsteady flows are listed here: (i)

For a body oscillating in its own plane, a shear layer forms whose thickness is of p the order of O( ν/ f ), where ν is the kinematic viscosity.

(ii)

For flow past a stationary circular cylinder in uniform flow with Reynolds number in the range, 103 < Re < 5 × 105 , the unsteadiness is approximately given by S t ≈ 0.2 and in the range 5 × 105 < Re < 107 , the unsteadiness is defined in the range S t ≈ 0.2 – 0.4. The scales for lower frequency events are related to the Ben`ard–K`arm`an vortex shedding. For higher Reynolds numbers, even when vortices are not easily discernible, one observes lower frequencies in the time series for measured quantities in the near wake.

(iii) For any impulsive changes in a boundary layer of thickness, δ, a Stokes layer is formed at the wall, which remain buried inside the boundary layer. Its time variation is governed by the viscous time scale O(δ2 /ν) for the diffusion of momentum and vorticity from the body surface where it is created.

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(iv) When a physical body suddenly √ starts moving in a fluid, the Stokes layer is formed whose thickness is O( νt), valid at very early times.

3.9 Two- and Three-Dimensional Turbulent Flows Due to stretching, the cross sectional area of a vortex element decreases for 3D flows. To conserve angular momentum, the strength of vorticity increases in the direction of stretching. This also helps transfer energy in the two orthogonal directions, producing a cascade of energy to smaller scales, until viscous dissipation intervenes and converts the mechanical energy to heat. Thus, the enstrophy of the flow increases (See [114], pages 49–53) due to R vortex stretching, if one considers inviscid flow. R The enstrophy 2 is proportional to k E(k) dk, where the kinetic energy density ( E(k) dk) remains conserved for an inviscid flow. In this limit, the requirements are met together if E(k) transfers energy to higher values of k, a process known as the forward scatter of energy due to vortex-stretching in three-dimensional flows, while the actual energy spectrum is as given in Eq. (3.112). A schematic sketch of the energy spectrum is shown in the Figure 3.11(a), where the universal nature of the inertial range is highlighted, while for other ranges of k, the spectrum depends on the problem considered. Given that the dissipation function is given by D(k) = ν||∇V||2 , one can use Eq. (3.112) to show that the dissipation is proportional to k1/3 in the inertial subrange. This is noted in Figure 3.11(b). It has been shown in [114, 412] that there are important two-dimensional flows that display turbulent structure which are totally different from three-dimensional turbulent flows. For two-dimensional inviscid flows, the forward scatter due to vortex stretching will be absent. Such flows will show smaller scale eddies produced due to physical instabilities, as has been shown for the periodic two-dimensional Taylor– Green vortex problem. For this problem, the presence of full saddle points in the flow makes the flow unstable at high Reynolds number. In the absence of such instabilities, both the R energy andR the enstrophy remains conserved. In terms of energy spectrum, if both E(k) dk and k2 E(k) dk are constants of motion, then any forward scatter must be compensated by an inverse scatter or backscatter. Thus, the existence of backscatter for two-dimensional high Reynolds number flows suggests that backscatter must also be present for three-dimensional flows. Logically, if two eddies merge together to form a larger eddy, then the corresponding process will show migration of energy from small to large scale. In general, for wall bounded flow, enstrophy increases due to vorticity generation at the wall to ensure no-slip. This increase is not compensated instantaneously by dissipation leading to an unsteady growth of enstrophy. Thus, enstrophy can be created or destroyed when large eddies break by Kelvin–Helmholtz instabilities and/or interactions among vortices and vorticity field or vortex merger. It has been reasoned in [247] that enstrophy in two-dimensional flows takes up the same role as energy does for three-dimensional flows. Doering and Gibbon [114] have reasoned that analogous to three-dimensional flows, one can formulate a high

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(a)

In E ( k )

LES Limit

C as

~ k 5/3 cad e

“Super-Grid Area” SGS Sink

(b)

In( k ) Dissipation Band

Energy Inflow Dissipation Spectrum

In k 2 E (k )

~ k1/3

In( k ) In (2π /L)

In (2k/λk)

Figure 3.11 (a) Schematic of energy spectrum in Kolmogorov’s theory; (b) Dissipation spectrum as in Kolmogorov’s theory.

wavenumber dissipation scaling for two-dimensional turbulent flows, which show the high wavenumber energy spectrum to be given by E(k, χ) = χ2/3 k−3

(3.115)

where χ is the enstrophy dissipation rate given by ||∇ω||22 in [114]. A proposed energy spectrum for two-dimensional turbulent flow is sketched in [114], and is reproduced in Figure 3.12, in which k−5/3 spectrum is followed by k−3 dependence at higher wavenumbers. This does not seem probable, as can be seen from the atmospheric data collected in [309]. This is reproduced in Figure 3.13 which shows the k−3 spectral dependence at smaller wavenumbers, while the k−5/3 spectrum follows at higher wavenumber, i.e. the two-dimensional nature is at lower wavenumbers, followed by three-dimensional spectrum. Moreover, as Figure 3.13 is shown in logarithmic scale, the energy contained by the two-dimensional part of the spectrum accounts for more than 98% of the turbulent kinetic energy; whereas the three-dimensional counterpart accounts for less than 2%; this has been explicitly discussed in [422]. It is well-known that the atmospheric flow

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Energy cascade 5/3

log (E(k)

~k

1/3

~k

Enstrophy cascade

log(k)

Figure 3.12 Sketch of two-dimensional turbulent flow energy spectrum. is predominantly two-dimensional, which is according to the geostrophic assumption and the Taylor–Proudman theorem [533]. This theorem requires that the Coriolis term in the governing equation written in [413] as Eq. (2.48) for the vorticity transport equation expressed in the moving frame of reference must be dominant over inertial and viscous terms. Stated in terms of non-dimensional parameters, this requires that the equivalent Rossby and Ekman numbers [508, 533] of the flow must be negligibly small.

3.10 Time-Averaged and Unsteady Flows Consider for the moment incompressible flow at very high Reynolds number, for which the flow is fully turbulent. The governing equations are given by ~ =0 ∇.V

(3.116)

~ ∂V ~ ~ · ∇)V ~ = −∇p/ρ + ν∇2 V + (V ∂t

(3.117)

These are written in the absence of body force for the sake of simplicity. If flow variables show unsteadiness at significantly higher frequencies, then the flow field is double-decomposed, with the unsteady velocity vector given by ~ X, ~ t) = U(X) ~ + v(X, ~ t) V(

(3.118)

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Figure 3.13 Power spectra of zonal and meridional wind and potential temperature in atmosphere measured at tropopause by aircraft. [Reproduced from “High Accuracy Computing Methods Fluid Flows and Wave Phenomena”, T. K. Sengupta, (2013), with the permission of Cambridge University Press.] On the right-hand side of Eq. (3.118), the upper case variable is time-independent or can be interpreted as the time-average of the unsteady velocity field given on the lefthand side. The lower case velocity components on the right-hand side are true random quantities, i.e., these have zero time-averages. This type of splitting for turbulent flows was first introduced in [365] and the process is called Reynolds’ time averaging. Here, time averaging is defined as 1 T →∞ T

~ t)i = U(X) ~ = lim hV(X,

T

Z

~ X, ~ t) dt V(

(3.119)

0

The angular brackets used on the left-hand side denote time averaging operation. If we split the velocity field given by Eq. (3.118), then the mean field U satisfies the ~ X, ~ t) = V ~ b (X), ~ and v(X, ~ t), time-independent boundary conditions for the velocity: V( satisfies the homogeneous boundary conditions, irrespective of whether the boundary ~ is prescribed as periodic in some conditions are of Dirichlet or of Neumann type. If V direction, then ~v is also periodic in that direction.

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If time averages of time derivatives vanish and time averaging commutes with spatial derivative, then one averages Eqs. (3.116) and (3.117) to obtain ∇·U =0

(3.120)

U · ∇U + h~v · ∇~vi = −∇p/ρ + ν∇2 U

(3.121)

In Eq. (3.121), on the left-hand side, there is additional momentum flux terms h~v · ∇~vi. This is a consequence of the presence of the fluctuation terms; even if the fluctuating velocity components are considered purely random, any product term as it is here, will have a non-zero time averaged contribution. Such terms in the time averaged Navier–Stokes equation (which is also called the Reynolds averaged Navier–Stokes (RANS) equation) are also called the Reynolds stresses. These stresses constitute a symmetric stress tensor, which is easily seen by inspection. We note that for a general stress system for fluids in motion, the symmetry of stress tensors is a consequence of an assumption that there are no distributed couples in the flow domain. A major distinction of Reynolds stresses from viscous stresses is that the former do not depend on fluid property. Instead these stresses are properties of the flow and hence, they are not isotropic. This feature will be useful to users in developing closure models for Reynolds’ stresses, which will be necessary to solve the RANS equation. Such closure models form the basis of turbulence models, a subject requiring a great deal of experience if useful information from the RANS solution is to be obtained. The time averaged Navier–Stokes equation, i.e., the RANS equation forms the basis of many present day CFD procedures used for applications in industry. There are a few disadvantages with this approach. First, we do not know how we can model the Reynolds stresses in a general way. There are no additional physical principles that exist in general for these flow-dependent stresses. In a quest for simplification, these are usually modeled or parametrized by the important area of development known as turbulence modeling. The second problem relates to defining the time averages of turbulent signals in the RANS solution. In using the double decomposition given in Eq. (3.118), an implicit assumption is made that the time dependence of the flow is solely due to the purely random fluctuating velocity ~ t), which are at very high frequencies, so that the procedure adopted components ~v(X, in Eq. (3.119) is a valid one. This approach works for flows having time-independent mean, as may be seen for flows without separation. Most often the problem arises for unsteady flows, where turbulent unsteadiness is not different from the imposed unsteadiness due to self-imposed instabilities. In other words, RANS procedure will produce poor results if the averaging time interval (T ) is less than the period of mean unsteady motion. Therefore, when turbulent structures are unaffected by externally induced unsteadiness, there is a critical frequency below which turbulence models can be used for predicting unsteady (in the mean) turbulent flows. For such flows, the unsteadiness present at low frequencies will be far removed from turbulent

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fluctuations, allowing such time scale separation that defines a time-average, as in Eq. (3.119). This is the area of unsteady RANS or URANS. It is equivalent to filtering out the effects of high frequency fluctuations by the eddy diffusivity of the turbulence model. Thus, the same effect can be achieved by performing conceptually an ensemble averaging. Of course, adopting RANS and URANS is for the sole purpose of economy of computations, whereas ensemble averaging DNS solution is beyond the present day computing power for practical geometries. For about last six decades, researchers have reported coherent structures in various flows [69, 231, 240, 543]. It has been noted that such organized structures in turbulent flows can carry about 20% of the total turbulent kinetic energy and hence, their role in determining turbulent flow dynamics should be studied with care. Coherent structures are noted as peaks and valleys in the near-wall region. In terms p of wall units based on a velocity scale (uτ ) derived from the wall shear (τw ) as (uτ = τw /ρ), these have lengths between 100 to 2000 units in the streamwise direction and have a spacing of about 50 units in the spanwise direction. Although one is talking of linear dimensions, these are essentially Reynolds numbers defined by taking the dimension as the length scale; with velocity scale already defined here, one can construct the Reynolds number. These high energetic events occur at a height of about 20 to 50 units, as statistical estimates. Additionally, these near-wall coherent structure formation events are interspersed by bursting events. After a burst, new intermediate coherent structure formation starts, which are interpreted as streamwise Λ and/ or hair-pin vortices. The unsteadiness of turbulent boundary layer is characterized by this bursting frequency for the shear layer. For zero pressure gradient boundary layer, this critical frequency is roughly between 20 to 100% of the turbulence burst frequency (Fb ), where Fb = U5δ∞ , with δ as the shear layer thickness. For adverse pressure gradient flows, this critical frequency is between 6 to 28% of Fb . A very recent review on coherent structures has been provided by Jimenez [206], where the author has noted that coherent structures larger than the Corrsin scale are a natural consequence of the shear. In wall-bounded turbulence, they can be classified into coherent dispersive waves and transient bursts. The former are found in the viscous layer near the wall, and as very large structures spanning the entire boundary layer. Although they are shear-driven, these waves have enough internal structure to maintain a uniform advection velocity. Conversely, bursts exist at all scales, are characteristic of the logarithmic layer, and interact almost linearly with the shear. While the waves require a wall to determine their length scale, the bursts are essentially independent from it. Therefore it is not surprising that for flows past bluff bodies dominated by adverse pressure gradient on the lee-side, the results produced by RANS with the turbulence model is not so effective. Moreover, in view of the existence of coherent structures in shear layer, double decomposition used in Reynolds averaging may not be the correct procedure. In the presence of a single dominant wavenumber/ frequency, triple or multiple decomposition as given in [366, 517] would be more appropriate. It is recommended that one performs LES for high Reynolds number flows using the high accuracy methods highlighted in this chapter. LES uses spatial average over a

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cell, with sub-grid scale effects parametrized in the traditional approach. However, using high accuracy methods, along with appropriate filters can be viewed as implicit LES or ILES and is a viable alternative, as detailed in [413, 534].

3.11 Closing Remarks In providing an introduction to scientific computing, emphasis must be given to the central theme of instability and receptivity analysis via dispersion relation. We note that the dispersion relation essentially maintains the space–time dependence of the problem given by the governing equation and the auxiliary conditions, and the problem can be solved either in the spectral plane (albeit with many restrictions) or in the physical plane, with or without limiting assumptions. Concomitant with the desired accuracy of the dispersion relation preservation, one must also pay similar attention to spatial and temporal discretizations for solving the problem in the physical plane. Each and every process making their appearances in the problem, require retaining the accuracy of the processes in the mathematical physics sense. This is noted in this chapter for convection, diffusion and their roles in explaining transition to turbulence and accordingly emphasized. The chapter in essence captures the spectra in the wavenumber and circular frequency planes for the modal and nonmodal components of the response field. Resolution of spatial and temporal scales during transition, and thereafter in the fully developed turbulent flows is noted with care. It is also noted that such scales and their separation for transitional and turbulent flows allow one to solve the problems as an averaged field or as a completely unsteady flow field, with respect to time variation. Similarly in space, one can introduce the concept of spatial averaging and associated subgrid scale modeling in the rich field of large eddy simulation.

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Instability and Transition

4.1 Introduction In Chapter 1, we have discussed the historical development of the field of instability and receptivity. Helmholtz [177] first provided some theoretical ideas regarding hydrodynamic instability. About a decade later, the works of Reynolds [365], Rayleigh [350, 351] and Kelvin [224] produced experimental and theoretical results that laid the foundation of stability theory. According to Betchov and Criminale [28], stability is defined as the property of the flow describing its resistance to grow due to small imposed disturbances. We note that the background disturbances do not have to be small (as noted experimentally by Reynolds [365]); we will also see in this chapter that the growth noted experimentally in [405] for the zero pressure gradient boundary layer occurs over a short streamwise distance. The original question of transition to turbulence was not addressed directly in theoretical studies, as most of these were related to finding conditions for growth of background disturbances by developing the linear stability theory. This theory investigated the ability of an equilibrium state to retain its undisturbed laminar state for stability. Instability studies began by a linear theory resulting in Rayleigh’s stability equation and a corresponding theorem, [351, 353, 356], with focus on inviscid temporal instability. This theorem was based on an incorrect assumption that viscous action in fluid flow is dissipative and can be neglected to obtain a more critical instability limit. It was strange for fluid dynamicists to accept this, as researchers in other disciplines of mechanics and electrical sciences, geophysics and engineering were aware of the role of resistive instability, which can arise in fluid flow only by viscous action. Viscous action can give rise to phase shift or time delay. A basic oscillator is governed by an ¨ + θ(t − τ) = 0. This is equivalent to providing equation with time delay (τ) as θ(t) ¨ − τθ˙ + θ(t) = 0, with the second term destabilizing the anti-diffusion as noted in θ(t)

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oscillator via the time delay, τ. Despite this rudimentary observation, only when scientists failed to explain disturbance growth for zero pressure gradient boundary layer, were alternatives sought [321, 495] via the Orr–Sommerfeld equation, which has viscous diffusion included for disturbance equations. Although we understand the importance of viscous diffusion, we begin by describing inviscid instability, as it demonstrates the logic behind Rayleigh’s early works and his theorem to explain the concept of flow instability. He did not solve the governing equation; instead instability was shown to relate to the presence of an inflection point for the streamwise velocity component of the equilibrium flow. There are no such results for the Orr–Sommerfeld equation. It is now well known that the solution of the Orr–Sommerfeld equation is far more difficult to obtain than solving Rayleigh’s equation. This is because the Orr–Sommerfeld equation is a stiff ordinary differential equation and is more difficult to solve than the governing Navier–Stokes equation [119, 412]. It is sobering to realize that while the Orr–Sommerfeld equation was being written down for the first time, there were reports of the Navier–Stokes equation being solved by a desk calculator [369].

4.2 Inviscid Instability of Parallel Flows Instability studies involve obtaining an equilibrium flow, following which we find condition(s) for infinitesimally small perturbations to grow in space and time. This is the rationale for performing linear instability studies, i.e., we calculate only the linearized terms of the perturbation equations derived for the disturbance field. The reader should note that while the perturbation equations are linearized, the equilibrium flow can be obtained by solving nonlinear equations, for example, for boundary layers, the governing parabolic partial differential equations are nonlinear. It is also noted that although the disturbance field is considered inviscid, the equilibrium flow can be a shear flow. We have noted in Chapter 1 for the Klevin– Helmholtz instability that the equilibrium flow considered was given by constant values across the interface. We want to consider more general class of flows to understand instability. For example, linear instability studies consider only those flows for which the shear layer grows very slowly; so that the streamlines within the shear layer can be approximated to be parallel to each other by the parallel flow approximation. This is required as one solves such problems with Fourier–Laplace transforms in a linear framework, ensuring that the disturbance can be considered homogeneous in planes parallel to bodies, for example, for flow over flat surfaces. The streamwise and spanwise components of velocity may be considered to be solely functions of wall-normal coordinates. Such assumptions for equilibrium flows hold true for Couette and plane Poiseuille flows, both of which are assumed to be fully developed. In recent times, there have been some criticisms against the use of the Fourier–Laplace technique, terming as an exponential growth model. Some

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researchers have instead advocated the so-called algebraic growth instead. However, we will establish the generality of the Fourier–Laplace transform methods with the help of two fundamental theorems due to Abel and Tauber [537]. Before that, a formal introduction of the instability theory will facilitate discussion on the mathematical framework required to avoid ad-hocism of various semi-empirical approaches requiring doubtful assumptions. We consider a boundary layer developing under mild or zero pressure gradient without separation, which can be approximated as a quasi-parallel flow, to study instability, even when this equilibrium flow can be obtained without such restrictions. Most linear and weakly nonlinear instability studies are based on this quasi-parallel flow assumption [177, 351, 522]. Consider the inviscid instability of a two-dimensional parallel flow with an associated two-dimensional disturbance field, such that the instantaneous flow is given by u( x˜, y˜ , t˜) = U(˜y) +  u˜ ( x˜, y˜ , t˜)

(4.1)

v( x˜, y˜ , t˜)

(4.2)

=

 v˜ ( x˜, y˜ , t˜)

p( x˜, y˜ , t˜) = P( x˜, y˜ ) +  p( ˜ x˜, y˜ , t˜)

(4.3)

The upper case quantities denote the equilibrium flow, while the perturbation field is indicated by a tilde. The independent variables with tilde may be different from the scales used for obtaining the mean flow or the convection scales from direct numerical simulations, used for global linear or nonlinear analysis. This will be specified later. For constant density flows, linearized governing equations for the disturbance field are obtained by collating terms of the order of  as: ∂˜u ∂˜v + =0 ∂ x˜ ∂˜y

(4.4)

∂˜u ∂˜u dU 1  ∂ p˜  +U + v˜ =− ∂t˜ ∂ x˜ dy˜ ρ ∂ x˜

(4.5)

∂˜v ∂˜v +U ∂t˜ ∂ x˜

(4.6)

=

−

1  ∂ p˜  ρ ∂˜y

It is apparent by inspecting the aforementioned equations why making parallel flow approximation helps one to use the Laplace–Fourier transform for the perturbation quantities given as

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" u˜ ( x˜, y˜ , t˜) =

u(˜y; α, ω0 ) ei(α x˜−ω0 t˜) dα dω0

(4.7)

v(˜y; α, ω0 ) ei(α x˜−ω0 t˜) dα dω0

(4.8)

Br

" v˜ ( x˜, y˜ , t˜) = Br

p( ˜ x˜, y˜ , t˜) = ρ

" p(˜y; α, ω0 ) ei(α x˜−ω0 t˜) dα dω0

(4.9)

Br

where Br indicates the Bromwich contours one needs to take to perform inverse transforms in these equations, in complex wavenumber α, and complex circular frequency ω0 planes. These representations (in Eqs. (4.7)–(4.9)) for the transforms can be used in the perturbation Eqs. (4.4)–(4.6). This can be algebraically manipulated to obtain a single equation for v as  U−

 d2 U ω0  d2 v 2 − α v − 2v=0 α dy˜ 2 dy˜

(4.10)

This is the well-known Rayleigh’s equation. If this homogeneous ordinary differential equation is integrated across the whole range of y˜ , then one of the possibilities will be the trivial solution. The other possibility is the classical eigenvalue problem studied for the inviscid instability of the disturbance field. In general, a flow should admit spatio-temporal growth of infinitesimal perturbations, with α and ω0 both being complex. However, Eq. (4.10) was studied by Rayleigh only for temporal growth. There is a good reason for its use in jets and mixing layers, even though the heat transfer effects are be pronounced (this will be discussed in Chapter 11). We first explore temporal instability by considering α as real and ω0 as complex for the present study. As c = ω0 /α, the complex phase speed (c = cr + ici ) will indicate the temporal instability from the eigenvalue of the equation given by  d2 v  d2 U (U − c) 2 − α2 v − 2 v = 0 dy˜ dy˜

(4.11)

4.2.1 Inviscid instability mechanism The temporal growth mechanism was described by Rayleigh by using v∗ , the complex conjugate of v, such that v v∗ = |v|2 . Multiplying Eq. (4.11) by v∗ and integrating over any limit (say, −∞ to +∞), we get the following +∞

Z

−∞

v∗

 d2 v dy˜ 2

− α2 v −

U 00  v dy˜ = 0 U −c

(4.12)

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Here U 00 is the second derivative with respect to normal to stream direction, and is written simply for convenience. For ci , 0, this equation cannot be singular, as noted from the denominator of the third term. Using integration by parts, the first term simplifies and Eq. (4.12) yields the following Z Z  2  U 00 |v|2 dv + α2 |v|2 dy˜ + (U − c)∗ dy˜ = 0 dy˜ |U − c|2

(4.13)

Out of the two set of terms in Eq. (4.13), the first set is real and positive; the imaginary part arises from the second set of terms for the temporal instability as Z ci

U 00 |v|2 dy˜ = 0 |U − c|2

(4.14)

As ci , 0, this will be true, if and only if the integral of Eq. (4.14) vanishes. This can happen, if and only if the integrand changes sign in the interval of integration. This is a possibility only when U 00 changes sign, requiring the existence of a location in the interior of the range, where U 00 = 0 for some y = y s within the limits of integration. This location is the inflection point which led Rayleigh to enunciate the following: Rayleigh’s Inflection Point Theorem: A necessary condition for instability is that the basic velocity profile should have an inflection point. The inflection point is within the shear layer (say at y = y s , where the local velocity is U s ) and the second derivative of U(˜y) vanishes. By incorporating the real part of Eq. (4.13), a stronger version of the inflection point theorem was stated in [136] as, Fj∅rtoft’s Theorem: A necessary condition for instability is that U 00 (U − U s ) is less than zero somewhere in the flow field. For a monotonically growing velocity profile with a single inflection point, this last necessary condition for instability is written as U 00 (U − U s ) ≤ 0 for the entire range of integration, with equality only at y = y s , where U(y s ) = U s . Both Rayleigh’s and Fj∅rtoft’s theorems are necessary conditions, but they do not provide a sufficient condition for instability. For flows with strong adverse pressure gradients or in free shear layers, these conditions are always satisfied, and these flows are prone to strong temporal growth. Such local temporal growth of the disturbance field can also be termed as absolute instability. A fundamental difference exists between flows having an inflection point (free shear layer, jets and wakes; cross flow component of three-dimensional boundary layers) and flows without inflection points (in Couette and Poisseuille flows or in boundary layers), with the former strongly receptive to temporal instabilities at extremely low Reynolds numbers. Detailed accounts of classical inviscid temporal instability theories can be found in [28, 119]. For mixed convection flows, augmented

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studies for linear inviscid instabilities will be provided for both high heating and cooling from the wall in Chapter 11.

4.2.2 Is there spatial inviscid instability? In Eq. (4.13), we can investigate the case of spatial instability by considering α = αr + iαi , such that c=

ω0 αr ω0 αi ω0 (αr − iαi ) ω0 = 2 −i 2 = 2 2 2 αr + iαi αr + αi αr + αi αr + α2i

Hence, U −c=

U − ω α  ω0 αi 0 r +i 2 2 2 αr + αi αr + α2i α2 = (α2r − α2i ) + 2iαr αi

and

Therefore, Eq. (4.13) is given by Z Z  2  d¯v + (α2 − α2 )|¯v|2 dy˜ + 2iα α |¯v|2 dy˜ r i r i dy˜

+

Z

ω0 αi  U”|¯v|2  U − ω0 αr − i dy˜ = 0 |U − c|2 α2r + α2i α2r + α2i

Looking at the imaginary part of this equation, we must have, 2αr αi

Z

ω0 αi |¯v| dy˜ − 2 αr + α2i 2

Z

U”|¯v|2 dy˜ ≡ 0 |U − c|2

This equation can hold for one of the possibility when αi ≡ 0, which implies that for the Blasius boundary layer, linear inviscid instability theory does not allow spatial growth due to this requirement. For this reason, linear inviscid instability allows only temporal growth for which, the velocity profile must have an inflection point in the interior of the domain as the necessary condition for instability, according to Rayleigh’s and Fj∅rtoft’s theorem. In [113], the author has tried to provide an alternative interpretation by rewriting the aforementioned equation as Z 2αr

ω0 |¯v| dy˜ = 2 αr + α2i 2

Z

U 00 |¯v|2 dy˜ |U − c|2

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and stating an alternate necessary condition that the sign of curvature of the velocity profile (U 00 ) must be the same as that for the phase speed (ω0 /αr ). This is an incomplete observation, because the phase speed is outside the integral on the right-hand side, while U 00 appears inside the integral. This is not pursued further as the central theme of the current research practice is not restricted to either temporal or spatial instability theory.

4.2.3 Role of viscous terms: Early developments Despite the demonstration by Taylor [511] that viscous action is responsible for flow in the atmosphere due to a dynamic instability, it did not immediately encourage the inclusion of viscous terms for the instability theory. Taylor’s demonstration did not receive sufficient attention as the problem solved was related to instability of the atmosphere, with the equilibrium flow given by the diffusion equation for the temperature and moisture content—not directly related to basic hydrodynamics. This was not related to the origin of turbulence and “supporting experimental evidences” drawn from measurements of temperature by a kite off the coast of Labrador in the north Atlantic and wind velocity in the Salisbury Plains in UK did not evoke interest. The issue related to no-slip condition inherent in viscous flows discussed in [511] was to reconcile works on flow instability by Rayleigh and Reynolds. Taylor [511] addressed the important result of Rayleigh’s investigation, which states that if d2 U/dy˜ 2 does not change sign in the space between two bounding planes, unstable motion is impossible. A particular case of laminar motion in which d2 U/dy˜ 2 has the same sign throughout the fluid is for an inviscid fluid flowing with same velocity or a viscous flow under favorable pressure gradient between two parallel planes. Such cases would not be unstable according to Rayleigh’s theorem. Reynolds, however, looked at this instability in a different way experimentally, to conclude that viscous fluid moving in a pipe was unstable if the coefficient of viscosity was less than a certain value, and that the instability depended on the diameter of the pipe and the velocity of the fluid. Reynolds’ result is in accordance with our present knowledge about the behavior of actual fluids. It was evident that there was an apparent mismatch between these two points of view. According to Reynolds, the more nearly inviscid the fluid is, the more unstable it will be; whereas according to Rayleigh, instability is impossible if the fluid is inviscid over a flat plate. Present day readers will be quick to note that the fallacy is due to misinterpretation of the role of the Reynolds number in arriving at the correct disturbance flow, which requires a no-slip condition for viscous flows. In conclusion though, Taylor noted that “the complete absence of slipping assumed in Reynolds’ work enables the necessary amount of momentum to escape, and so a type of disturbance may be produced that is dynamically impossible under the condition of perfect slipping at the boundaries,” implying the centrality of viscous action. It is pertinent to note that even today, in the absence of rigorous proof, the no-slip condition is a modeling approximation. It has been noted in [20] that for Newtonian fluid flows “the absence of slip at a rigid wall is now amply confirmed

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by direct observation and by the correctness of its many consequences under normal condition.” The effects of viscosity on flow instability have been highlighted by Prandtl [337] in anticipation of the solution of the Orr–Sommerfeld equation. Tollmien and Schlichting [392, 393, 394, 395, 527, 529] solved the Orr–Sommerfeld equation numerically in terms of the Airy functions. They predicted a wavy solution for the eigenfunctions, which is known as the Tollmien–Schlichting (TS) wave. Moreover, Heisenberg produced results for his doctoral dissertation based on solution of the Orr–Sommerfeld equation [176]. All these results have been obtained for the canonical zero pressure gradient boundary layer, given by the similarity solution due to Blasius [396]. As temporal instability is ruled out by Rayleigh’s theorem for this equilibrium flow, subsequent instability analysis by the German schools assumed the disturbance flow to be spatially developing. It must be emphasized that the classification of disturbance growth to either temporal or spatial route is itself arbitrary. There are other reasons that highlight the inadequacy of the linear instability theory, the primary one being based on not requiring information about the disturbance environment for subsequent disturbance growth. This was the reason why it was very difficult to experimentally verify the instability theory. There was no quantifier as to how imperceptible background disturbances can be related to deterministic TS waves. The failure to verify TS waves experimentally led to poor acceptance of the linear instability theory in the initial stages. After the initial failure by Taylor [513], a team led by Dryden [121] performed the classical vibrating ribbon experiment, reported by Schubauer and Skramstad [405] to show the existence of the TS wave with oscillograph traces at fixed locations. For the TS wave obtained by the spatial theory, an imposed time scale is required continually at a fixed location inside the boundary layer. Another perpetual legacy of the simplification used in linear theory is the parallel flow assumption for the equilibrium flow, i.e., the information gleaned is for a local instability, while the disturbance field grows in space and time globally. The absence of tangible progress by the prevalent linear viscous instability theory based on the growth of small perturbation concept led to the pronouncement [317] that the small perturbation theory has failed to provide any useful results concerning the origin of turbulence. It is pertinent to note that the situation did not change very much as Mack [285] noted that over and above, such a linear instability theory “tells nothing about turbulence, or about the details of its initial appearance, but it does explain why the original laminar flow can no longer exist.” The imposed time scale in spatial theory is equivalent to performing a frequency response study by exciting the equilibrium boundary layer at a moderate frequency. Unfortunately, Taylor [514] vibrated the diaphragm at a very low frequency (2 Hz) and failed to detect the TS wave. Interestingly, even when Gaster et al. [145] tried to reproduce the failed experiment of Taylor [513], the authors noted that “a proper mathematical account of these disturbances has not yet appeared.” This was written in 1994, and speaks of the uneven developments in the subject. Even the symbolic success in detecting TS waves by Schubauer and Skramstad [405] was greeted with overenthusiastic acclaim by the

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research community, who concluded that the TS wave is responsible for eventual transition to turbulence! In this euphoria, lots of pertinent questions were left unanswered: (i) No definitive reported experiments have been performed that relate the TS waves with actual transition as noted in [285]; (ii) Why is it necessary to use a vibrating ribbon inside the boundary layer for the experiment? Can there be other means of exciting TS waves? (iii) It has not been demonstrated experimentally that if the boundary layer is excited at a given frequency, the response is also going to be at the same frequency (the socalled signal problem assumption); (iv) In a growing boundary layer, how do the signal properties continuously keep changing as the disturbance moves downstream? (v) Why should not the boundary layer be investigated for supporting spatio-temporal growth of disturbances? (vi) What about other routes of transition, which do not create TS waves? These are known as bypass transition events [298, 412]. Most of these questions were conceived when theories for linear and nonlinear receptivity made their appearance after the 1990s. The need for developing such theories was very eloquently stated in [405], where it was noted: “In the search for schemes to excite oscillations in the boundary layer, a number of devices were tried before completely satisfactory results were obtained. Methods using sound, both pure notes and random noise, were none too satisfactory because of resonance effects and the complexity of the wave pattern in the tunnel.” Despite the growing importance of the subject of receptivity, it is necessary to provide the details of linear viscous stability theory, as both of these have the common element of the Orr–Sommerfeld equation. It is pertinent to note some of the misconceptions that still bedevil the understanding of the difference between stability and receptivity. One notes in [399] the authors, in talking about instability and receptivity of general fluid dynamical system, wrongly characterize these two aspects for small perturbation excitation as “eigenvalue analysis for stability problems, and the resonance concept for receptivity problem”; as if the vibrating ribbon experiment [405] created a resonant excitation of the boundary layer! This is completely misleading and in this book, the correct relationship between instability and receptivity is explained.

4.3 Linear Viscous Stability Theory In classical linear analysis, one looks for the growth of infinitesimally small background disturbances in terms of amplitudes of individual modes. Such modal analysis also requires making restrictive assumptions of parallel flow and solving the problem in spectral plane. These issues can be circumvented by solving the Navier– Stokes equation in linearized form, without making the parallel flow assumption. However, solving a set of three-dimensional linearized partial differential equation is as difficult as solving the nonlinear Navier–Stokes equation. For this reason, one can adopt the parallel flow assumption to solve the disturbance equation in the spectral plane. Parallel flow assumption allows one to define an equilibrium flow

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as a function of the wall-normal coordinate (y), so that the Fourier–Laplace transform can be introduced for the perturbation quantities for instability study of viscous flows with the governing equation given by the Orr–Sommerfeld equation. If U0 , V0 and W0 are the streamwise (x), wall-normal (y) and spanwise (z) equilibrium flow velocity components, then with parallel flow approximation, the simplified form of the equilibrium solution can be represented by U0 = U0 (y), V0 = 0, W0 = W0 (y). This viscous equilibrium flow is driven by the local pressure gradient which can be determined by the edge velocity (Ue ) for the boundary layer. The displacement thickness δ∗ and the momentum thickness θ of the boundary layer at a particular location are obtained from Z ∞ U0 (y)  dy 1− δ∗ = Ue 0 Z ∞ U0  U0 (y)  θ= 1− dy (4.15) Ue 0 Ue For the zero pressure gradient Blasius boundary layer, it has been shown that [82] 0.664x 1.72x and θ ' √ δ∗ ' √ Re x Re x

(4.16)

where, Re x = U∞ x/ν is the Reynolds number based on current length. It can be shown for the Blasius profile that the boundary layer thickness δ¯ ' 3δ∗ , where δ¯ is defined ¯ ' 0.99U∞ . Although one uses a zero pressure gradient boundary layer such that Ue (x, δ) profile to study linear and nonlinear receptivities, it is restrictive and instead, one can obtain the mean flow from the solution of the Navier–Stokes equation. The governing equations for the evolution of small perturbations in the shear layer are obtained again from the linearized Navier–Stokes equation [412] with parallel flow approximation along with the equation of continuity as ∂˜u ∂˜u ∂˜u dU0 ∂ p˜ + U0 + W0 + v˜ =− + ν∇2 u˜ ∂t ∂x ∂z dy ∂x

(4.17)

∂˜v ∂˜v ∂˜v ∂ p˜ + U0 + W0 =− + ν∇2 v˜ ∂t ∂x ∂z ∂y

(4.18)

∂w˜ ∂w˜ dW0 ∂ p˜ ∂w˜ + U0 + W0 + v˜ =− + ν∇2 w˜ ∂t ∂x ∂z dy ∂z

(4.19)

∂˜u ∂˜v ∂w˜ + + =0 ∂x ∂y ∂z

(4.20)

˜ and p˜ are the perturbation components of velocity and pressure, where (˜u, v˜ , w) respectively. These linearized governing equations for perturbation quantities can be

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further simplified by using formal regular perturbation theory. In this approach, any total quantity (q) is split into time independent mean flow and a space–time dependent perturbation quantity given as, q(x, y, z, t) = Q0 (x, y, z)+ q(x, ˜ y, z, t), where  is the formal small parameter representative of the amplitude of input disturbances. It is noted that using such a splitting, in Eqs. (4.17)–(4.20) for the linearized equations, the factor  is present linearly for all the terms and can be factored out. Apart from the need that  be very small for the linearization to hold, there is no other role of this parameter in linear stability analysis. Conversely, receptivity analysis relates the response of the dynamical system with , as the input excitation parameter. Noting that U0 and W0 are functions of y only, one can obtain the nondimensional perturbation equations by taking Ue and δ∗ as the velocity and length scales and retaining the nondimensional variables with tilde. One can introduce the local analysis by expressing the disturbances in the spectral plane using the Fourier–Laplace transform, as x˜ and z˜ are the homogeneous directions. One therefore writes a single mode in terms of the streamwise wavenumber, α, the spanwise wavenumber, β, and circular frequency, ω0 as [˜u, v˜ , w, ˜ p] ˜ T = [¯u(˜y), φ(˜y), w(˜ ¯ y), p(˜ ¯ y)]T ei(α x˜+β˜z−ω0 t˜) where, x˜ = x/δ∗ , y˜ = y/δ∗ , z˜ = z/δ∗ and t˜ = tUe /δ∗ . This analysis for one mode at a time is called the normal mode analysis due to the implicit assumption that such modes are orthogonal and act independently. Substituting these in Eqs. (4.17)–(4.20) and further simplifying, one obtains the Orr–Sommerfeld equation given as [285, 412]  ˜ (αU0 (˜y) + βW0 (˜y) − ω0 )(D2 − γ2 )φ (D2 − γ2 )2 φ = iRe  −αD2 U0 (˜y)φ − βD2 W0 (˜y)φ (4.21) ˜ = Ue δ∗ /ν, γ2 = α2 + β2 and wall-normal derivatives are also where, D = ddy˜ , Re indicated by primes. In the strongest criticisms of normal mode analysis, researchers [398, 400, 532] have instead looked for nonmodal stability analysis. It is indeed true that the eigenmodes to be obtained by solving Eq. (4.21) need not be orthogonal to each other. Thus, the disturbances need not be given only in terms of the least stable mode; other present modes with multimodal interactions, as given in Eqs. (4.7) to (4.9) will automatically take in both modal and nonmodal components of the disturbance field. Hence, it is premature to abandon the Orr–Sommerfeld equations. However, linear stability analysis also do not use the information contents for different modes from the applied input to the equilibrium flow due to homogeneous boundary conditions used at the wall and free stream. In linear stability analysis at the wall, one uses no-slip boundary conditions and perturbation quantities are expected to decay in the free-stream (i.e., as y˜ → ∞). Thus, there is no way of partitioning the input energy into different instability modes for the boundary value problem. Implicit in this description is that some perturbations are applied at the wall, which decay as

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one moves away from the wall. This is the equivalent wall perturbation problem and eigenvalue analysis only addresses instability excited from inside the shear layer. A boundary layer destabilized by free stream perturbation is explained in Chapter 9 from the receptivity perspective. Back to the linear stability analysis, the homogeneous boundary conditions used for equivalent wall excitation are as follows u˜ (0) = φ(0) = w(0) ˜ =0 u˜ (˜y), φ(˜y), w(˜ ˜ y) → 0

(4.22) (4.23)

as y˜ → ∞

y

Edge of shear layer

U∞

U(y) O

Exciter x

Figure 4.1 Harmonic excitation of a parallel boundary layer at the location of the exciter. The classical eigenvalue approach involves obtaining solutions of Eq. (4.21) for the homogeneous boundary conditions given by Eqs. (4.22) and (4.23), which exist only ˜ This is symbolically defined by the for particular combinations of α, β, ω0 , and Re. dispersion relation from the eigenvalue problem to be satisfied at the wall as ˜ =0 ˆ β, ω0 , Re) D(α,

(4.24)

It is noted that the governing equation is an expression of how the dependent variables depend on space–time variables; the dispersion relation is a relation between the same in the spectral plane. For some physical problems, the governing equation may not depend upon space and time simultaneously, yet the solution is space–time dependent due to time-dependent boundary conditions. This was noted for Kelvin–Helmholtz and Rayleigh–Taylor instabilities in Chapter 1. Thus, the dispersion relation originates either from the governing equation or from satisfying boundary conditions and is expressed in the wavenumber–circular frequency plane. The dispersion relation given by Eq. (4.24) corresponds to the homogeneous wall boundary condition: u¯ = φ = w¯ = 0, and different methods of solving the Orr–Sommerfeld equation will implement it in

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a different way. We will use the compound matrix method (CMM), which is a special convenient way of solving stiff differential equations [312, 410, 412]. As noted here, the governing equation for the disturbance field in classical instability theory is derived from the Navier–Stokes equation with the small perturbation assumption. The equilibrium boundary layer can be obtained by making simplifications; it can be steady, even though the equation is nonlinear. The existence of a dispersion relation for the instability problem implies that the perturbation field will always be unsteady for instability to occur. Very rarely, one would see simultaneous existence of stationary disturbance streaks during some threedimensional receptivity problems. In the dispersion relation given by Eq. (4.24), α, β, and ω0 can all be complex quantities. For mathematical convenience, in classical approaches, the following have often been adopted: (i) Temporal amplification theory, which considers complex ω0 = ω0r + iω0i with real α and β so that the perturbation quantities are written as q( ˜ x˜, y˜ , z˜, t˜) = q(˜ ¯ y)eω0i t˜ei(α x˜+β˜z−ω0r t˜) so that the growth in time is exponential, if ω0i is positive. (ii) Spatial amplification theory considers α (= αr + iαi ) and β (= βr + iβi ) to be complex while keeping ω0 as real; the perturbation quantities are written as q( ˜ x˜, y˜ , z˜, t˜) = q(˜ ¯ y)e−αi x˜−βi z˜ ei(αr x˜+βr z˜−ω0 t˜) so that if we construct a modal amplitude as A( x˜, y˜ , z˜) = q(˜ ¯ y)e−αi x˜−βi z˜ , then the modal 1 dA growth in streamwise and spanwise directions are given as −αi = A1 dA d x˜ and −βi = A d˜z , respectively. However, such modal growths are not necessarily what one would observe in actual flows, as the actual growth of perturbation field depends upon not only the growth of the amplitude of the most unstable mode, but also on how the different modes interact with each other, depending upon the phase of each mode with respect to others. This will be explained further when we adopt Fourier–Laplace transforms to represent any perturbation quantity. Such growths can also be due to nonmodal action. Essentially, it is the combined action of multiple modes viewed together; various nonnormal theories have been proposed for these growths, which will be described in Chapters 6 and 14. As shown in appendix of Chapter 1, the Orr– Sommerfeld equation captures both the modal and nonmodal components during receptivity and transition. To further develop the foundation of the linear receptivity theory, the mean flow will be represented as U = U(y) for the two-dimensional mean field experiencing

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two-dimensional disturbance growth (to understand the wall and free stream modes of the Orr–Sommerfeld equation) given by Eq. (4.21) that reduces to   ˜ (αU(˜y) − ω0 )(D2 − α2 )φ − αD2 Uφ (D2 − α2 )2 φ = iRe (4.25) The associated dispersion relation for the 2D disturbance field is ˜ =0 ˆ ω0 , Re) D(α,

(4.26)

As explained earlier for the three-dimensional perturbation field, here the dispersion relation is equivalent to enforcing the wall boundary conditions, u¯ = φ = 0. For the fourth order variable coefficient ordinary differential equation given by Eq. (4.25), in general, the perturbation field can be represented in terms of the four fundamental solution modes (φ1 , φ2 , φ3 , φ4 ) as φ = a1 φ1 + a2 φ2 + a3 φ3 + a4 φ4

(4.27)

Out of these four modes, two will grow with wall-normal height (say, φ2 and φ4 ), and the other two decay with height (φ1 and φ3 ). If one implicitly considers wall excitation, then the disturbance field will decay in the free stream (˜y → ∞). At the free stream, U = 1 and all other mean flow derivatives with respect to y˜ are zero, simplifying Eq. (4.25) to ˜ (D2 − α2 )2 φ = iRe(α − ω0 )(D2 − α2 )φ

(4.28)

The solution of Eq. (4.28) for y˜ → ∞, for the wall excitation is then φ∞ = a1 e−α˜y + a3 e−Q˜y

(4.29)

p ˜ where Q = α2 + iRe(α − ω0 ), considering the cases where real parts of α and Q are positive. It is to be emphasized that α (inviscid mode) and Q (viscous mode that ˜ can have both positive and negative values for the real part. Classical depends on Re) theories consider the real part to be negative for the case of implicit wall excitation because of the wall modes. The positive real parts of α and Q correspond to modes which grow with height, and are referred to as the free stream mode [412, 429]. It is apparent that if phase of the wall modes represent downstream propagation, then the phase of the free stream mode indicates upstream propagation. The definitive direction of wave propagation is determined by the sign of the group velocity, with a positive sign indicating downstream propagation of disturbance, whereas a negative value indicates upstream propagation. Hence, the general solution of the Orr– Sommerfeld equation is given by Eq. (4.27), and for wall excitation written in terms of φ1 and φ3 as φ = a1 φ1 + a3 φ3

(4.30)

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From Eq. (4.28), the retained fundamental modes obey φ1∞ = e−α˜y and φ3∞ = e−Q˜y , which decay exponentially at two widely different rates in the free stream. This is symptomatic of Eq. (4.25) being a stiff ordinary differential equation, which can be overcome by using special methods like CMM [3, 312, 412] for external problems.

4.4 Properties of the Orr–Sommerfeld Equation: Developing Solution Method In stability analysis of a boundary layer, with the equilibrium solution given by a parallel or quasi-parallel flow, one solves the Orr–Sommerfeld equation for the mean flow for two- or three-dimensional disturbance fields. For a two-dimensional problem, wall (˜y = 0) boundary conditions are given as u¯ , φ = 0

(4.31)

At the far field (˜y → ∞), u¯ , φ → 0

(4.32)

To solve Eq. (4.28), the wall boundary conditions are written in terms of φ alone, with primes indicating wall-normal derivatives. For a two-dimensional disturbance field, the first derivative is obtain from the continuity equation in two dimensions as φ0 = −iα¯u

(4.33)

Thus, one needs to satisfy the homogeneous boundary conditions for φ and φ0 at the wall for the boundary layer. The consequence of far stream boundary conditions, given by Eq. (4.32), is easy to understand using mean flow information at y˜ → ∞ : U(˜y) = 1 and U 00 (˜y) ≡ 0 in Eq. (4.28). This equation then reduces to a constant coefficient ordinary differential equation at y˜ → ∞ as ˜ [(α − ω0 )(φ00 − α2 φ)] φiv − 2α2 φ00 + α4 φ = i Re

(4.34)

The solution of this equation can be obtained by substituting φ ∼ eλ˜y , so that the ˜ − characteristic roots are obtained as λ1,2 = ∓α and λ3,4 = ∓Q, where Q = [α2 + iα Re(1 c)]1/2 . It is apparent that α = 0 is one of the branch points, while Q = 0 will provide the other set of branch points. These points have to be avoided for any contour integral operation performed while evaluating the Fourier–Laplace transform, and will be discussed later. For boundary layer instability problems, the Reynolds number is very large, and then |Q| >> |α|. This is the source of stiffness, which makes obtaining a numerical solution of Eq. (4.25) a difficult job. Thus, the fundamental solutions of the Orr–Sommerfeld equation are seen to vary by completely different orders of

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magnitude. This makes it a stiff differential equation that suffers from the growth of parasitic errors during numerical solution. For the Orr–Sommerfeld equation, a fourth order ordinary differential equation, one has four fundamental solutions, whose far field variation for y˜ → ∞ is given by the characteristic exponents of Eq. (4.34), i.e., φ = a1 φ1 + a2 φ2 + a3 φ3 + a4 φ4

(4.27)

at the far field (˜y → ∞) : φ1∞ ∼ e−α˜y ; φ2∞ ∼ eα˜y ; φ3∞ ∼ e−Q˜y and φ4∞ ∼ eQ˜y . To satisfy the far field condition given in Eq. (4.32), we must set, a2 = a4 = 0 for real (α, Q) > 0. Then, the general solution can be simplified as φ = a1 φ1 + a3 φ3

(4.30)

This is an admissible solution of Eq. (4.25), which automatically satisfies the far field boundary conditions. Two remaining constants in Eq. (4.30) are fixed by satisfying the wall boundary conditions given in Eq. (4.31) as ˜ + a3 φ30 (˜y = 0, α ; ω0 , Re) ˜ =0 a1 φ10 (˜y = 0, α ; ω0 , Re)

(4.35)

˜ + a3 φ030 (˜y = 0, α ; ω0 , Re) ˜ =0 a1 φ010 (˜y = 0, α ; ω0 , Re)

(4.36)

Here, the additional subscript 0 indicates the fundamental solutions to be evaluated at the wall. Non-trivial solutions for a1 and a3 are obtained, if and only if the characteristic determinant of the associated linear algebraic system is zero, i.e., (φ10 φ030 − φ010 φ30 ) = 0

(4.37)

This is the dispersion relation of the eigenvalue problem posed by the Orr– ˜ so Sommerfeld equation. Thus, one needs to obtain a set of α and ω0 for a given Re, that the solution of the Orr–Sommerfeld satisfies Eq. (4.37). The stiffness of the Orr– Sommerfeld equation causes the numerical solution to lose the linear independence of different components following the different fundamental solutions. This is the source of the aforementioned parasitic error growth while solving a stiff differential equation. The parasitic error can be elegantly removed by an analytic formulation used in the CMM. Various methods of solving stability problems have been compared by Bridges and Morris [66]. There are three primary classes of methods to solve linear stability problems: (a) matrix methods based on finite difference or spectral discretizations; (b) shooting techniques along with orthogonalization of the fundamental solutions and (c) shooting techniques with CMM. All these methods have their advantages and disadvantages. For example, it has been noted [3] that while obtaining eigenvalues in an infinite domain (as may be the case for boundary layers), matrix methods

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based on finite difference and spectral collocations produce spurious eigenvalues, with the total number of eigenvalues determined by the number of points taken to discretize the domain. This leads to not only obtaining many spurious values, but also does not satisfy boundary conditions correctly, which can cause fracturing of continuous spectra. Shooting methods based on orthogonalization are difficult to program, requiring comparatively large computer memory while producing nonanalytic solutions as the computed fundamental solutions have to be orthogonalized or orthonormalized at each step of integration. Considering these drawbacks, CMM appears to be the best method for stability problems. In this method, the stiffness problem is removed at the formulation stage itself, by introducing a new set of variables, called the second compounds. We will explain these variables in terms of the boundary layer stability problem. Once the stiffness problem is removed at the formulation stage, any standard method can be used to solve for the eigenvalues. The CMM has been reformulated in [4] using exterior algebra to define the method in a coordinate-free context. The advantages of CMM have been variously described [119, 312, 313, 410, 470]. There are other methods based on an ordinary differential equation solver in the physical plane [407] or spectral collocation methods that involve Chebyshev discretization of the Orr–Sommerfeld equation, as in [400].

4.4.1 Compound matrix method (CMM) The CMM has been in use for stability calculations for some time now, since the original method appeared in [104, 312, 313]. This method goes to the heart of the problem of solving the Orr–Sommerfeld equation, which has a longer history of attempts at solving before success was achieved, as compared to even solving the Navier–Stokes equation. We have already noted the various available methods of solving the Orr–Sommerfeld equation. It is apparent that while other methods solve the stiff differential equation and remove the effects of parasitic error after every step of integration, CMM works at one of the problems of stiffness in the governing Orr–Sommerfeld equation, by reformulating it and remove the stiffness in the new formulation. Thus, the main strength of CMM lies in its simplicity of application and in its interpretation of the results. CMM yields satisfactory results for the evaluation of eigenvalues and eigenfunctions of the Orr–Sommerfeld equation by incorporating the analytic structure of the fundamental solutions. Although, at present, it is restricted to the Orr–Sommerfeld equation and its few variations, it is highly recommended that one is familiar with CMM. Due to its analytic structure, it is used to solve both stability and receptivity problems without substantial coding effort. Some essential modifications needed for appropriate choice of equation in CMM for eigenfunctions are reported in [410], wherein its versatility in solving Orr–Sommerfeld equation is also explained. Furthermore, we will also note how CMM has helped to solve certain unanswerable questions in instability studies, related to the breathing mode [236], upstream propagating modes [446], etc.

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In this method, instead of solving for φ, we introduce a new set of variables based on combinations of the fundamental solutions, say φ1 and φ3 for wall excitation problems. These introduced new variables vary with y˜ at an identical rate which is the key to the removal of the stiffness problem. For the Orr–Sommerfeld equation for wall excitation, these new variables ({z j }, for j = 1 to 6) are obtained from all the (2 × 2) minors of the fundamental solutions of the admissible modes [119, 410, 412],    φ1 φ3   φ0 φ0   1 3   φ001 φ003    000 φ1 φ000 3

(4.38)

Thus in algebraical form, the new variables or the second compounds are given by (first variable using rows one and two; second variable using rows one and three, and so on), z1 = φ1 φ03 − φ01 φ3 z2 = φ1 φ003 − φ001 φ3 000 z3 = φ1 φ000 3 − φ1 φ3

z4 = φ01 φ003 − φ001 φ03 000 0 z5 = φ01 φ000 3 − φ1 φ3 000 00 z6 = φ001 φ000 3 − φ1 φ3

(4.39)

One can easily verify with the free stream asymptotic solution for Eq. (4.34) that z1 to z6 have identical growth rates with y˜ . From the expressions given here in Eq. (4.39), one can construct differential equations for the second compounds as the following: z01 = φ01 φ03 + φ1 φ003 − φ01 φ03 − φ001 φ3 = z2

(4.40)

000 0 00 00 0 z02 = (φ1 φ000 3 − φ1 φ3 ) + (φ1 φ3 − φ1 φ3 ) = z3 + z4

(4.41)

0 000 000 0 iv z03 = φ1 φiv 3 + (φ1 φ3 − φ1 φ3 ) − φ3 φ1

From the Orr–Sommerfeld equation which is equally valid for all the fundamental modes 2 00 4 3 00 φiv 1 = {2α + iα Re (U − c)} φ1 − {α + iα Re(U − c) + iα Re U }φ1

or in short notation, using d1 = 2α2 +2iα Re(U −c) and d2 = α4 +iα3 Re(U −c)+iα Re U 00 , 00 one can write, φiv 1 = d1 φ1 − d2 φ1

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iv A similar relation for φiv 3 is obtained and after substitution, one obtains, φ1 φ3 − = d1 z2 . Thus, for the third variable of the second compound, the governing equation is

φiv 1 φ3

z03 = d1 z2 + z5

(4.42)

00 00 00 00 000 0 z04 = φ01 φ000 3 + φ1 φ3 − φ1 φ3 − φ1 φ3 = z5

(4.43)

00 000 iv 0 000 00 z05 = φ01 φiv 3 + φ1 φ3 − φ1 φ3 − φ1 φ3

Again using the Orr–Sommerfeld equation, one can simplify the aforementioned equation to get 000 00 0 00 0 00 z05 = [φ001 φ000 3 − φ1 φ3 ] + φ1 (d1 φ3 − d2 φ3 ) − φ3 (d1 φ1 − d2 φ1 )

z05 = z6 + d1 z4 + d2 z1

(4.44)

000 000 000 000 iv 00 z06 = φ001 φiv 3 + φ1 φ3 − φ1 φ3 − φ1 φ3 = d2 z2

(4.45)

Equations (4.40)–(4.45) are the necessary first order ordinary differential equations for the six unknowns z1 to z6 . However, CMM has a certain disadvantage; the order of the system increases from four for the Orr–Sommerfeld equation to six in CMM. One can easily see that this number arises from using all possible fundamental solutions from the admissible fundamental modes. Thus, in Eq. (4.38), from the two columns of the admissible solution vectors, we can construct 4C2 numbers of (2 × 2)-matrices. In [3], the stability of the Ekman boundary layer interacting with a compliant surface has been studied using CMM, which involves solving a sixth order system. Similarly, the spatial stability for mixed convection boundary layer has been studied in [470] using CMM, which also involves solving a sixth order augmented Orr–Sommerfeld equation. Considering that there are only three admissible solutions for the equivalent wall excitation problem, then for both these examples, one is required to form 6C3 or 20 second compounds. Interested readers can refer to [470] for details of the CMM for a sixth order system, with detailed equations for second compounds provided in the appendix of [470] for the spatial stability of mixed convection problems. It is also necessary to highlight the fact that the advantages of CMM are lost if all the fundamental solutions are admissible, then the number of second compounds reduces to nCn or one! Thus, CMM will not result in any savings for such problems. This is noted in solving the free stream excitation problem for boundary layers.

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One of the advantages of CMM is the use of the analytical structure of the problem, where the solution decays with height for all wall excitation problems. This information is used in identifying the fundamental solutions that are admissible, which in this case means retaining φ1 and φ3 . This is equivalent to transforming the original boundary value problem given by the Orr–Sommerfeld equation to an initial value problem by the CMM. To solve CMM with these equations, we need to create initial conditions for the second compounds. From the knowledge of the property of the fundamental solutions for y˜ → ∞, we can construct the initial conditions for z1 to z6 . For y˜ → ∞ : φ1 ∼ e−α˜y and φ3 ∼ e−Q˜y ; and hence, we can obtain the order of the second compounds at the free stream as z1 ∼ (−Q + α) e−(α+Q)˜y

(4.46)

z2 ∼ (Q2 − α2 ) e−(α+Q)˜y

(4.47)

z3 ∼ (−Q3 + α3 ) e−(α+Q)˜y

(4.48)

z4 ∼ (−αQ2 + α2 Q) e−(α+Q)˜y

(4.49)

z5 ∼ (αQ3 − α3 Q) e−(α+Q)˜y

(4.50)

z6 ∼ (−α2 Q3 + α3 Q2 ) e−(α+Q)˜y

(4.51)

In Eqs. (4.46)–(4.51), all the second compounds have the same exponential rate of growth or attenuation, which is the specific feature of CMM, indicating that the issue of stiffness is removed via the introduction of the second compounds. Thus, to integrate Eqs. (4.40)–(4.45), one will start with the initial conditions given by Eqs. (4.46)–(4.51), using normalized values with respect to one of the second compounds. If we normalize with respect to z1 , then the initial conditions for solving Eqs. (4.40)– (4.45) are as follows: z1 = 1.0

(4.52)

z2 = −(α + Q)

(4.53)

z3 = α2 + αQ + Q2

(4.54)

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z4 = αQ

(4.55)

z5 = −αQ(α + Q)

(4.56)

z6 = α2 Q 2

(4.57)

Operationally, we solve Eqs. (4.40)–(4.45), from the free stream to the wall, with the initial conditions given in Eqs. (4.52)– (4.57). Arriving at the wall, satisfaction of the dispersion relation in Eq. (4.37) is the same as finding the combinations of (α, ω0 ) ˜ This is exactly equivalent to enforcing the homogeneous boundary for a given Re. conditions at the wall; it is apparent that the dispersion relation is equivalent to satisfying z1 = 0

at y˜ = 0

(4.58)

The quickest way of finding the eigenvalues is by reformulating the eigenvalue problem using CMM. The CMM uses the grid search method in [418] to find the set of eigenvalues. Note also that for any type of stability analysis, be it the spatial, temporal or spatio-temporal approach, the dispersion relation given in Eq. (4.58) must be satisfied. In [418], the spatial eigen spectrum is reported for a specific case, whereas we will show it in the context of the upstream propagating method in next chapter. Having devised a quick method for finding the eigen spectrum of a boundary layer, one would be interested in finding the eigen vector or eigen functions. This also can be found using CMM, as we describe in the following. We note that the eigenvector φ is a linear combination of φ1 and φ3 for the present problem such that φ = b1 φ1 + b3 φ3

(4.59)

φ0 = b1 φ01 + b3 φ03

(4.60)

φ00 = b1 φ001 + b3 φ003

(4.61)

000 φ000 = b1 φ000 1 + b3 φ3

(4.62)

It is easy to eliminate b1 and b3 from Eqs. (4.59)–(4.62) using the second compounds in Eq. (4.39). This process of elimination provides the following differential equations for φ z1 φ00 − z2 φ0 + z4 φ = 0

(4.63)

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Instability and Transition

z1 φ000 − z3 φ0 + z5 φ = 0

(4.64)

z2 φ000 − z3 φ00 + z6 φ = 0

(4.65)

z4 φ000 − z5 φ00 + z6 φ0 = 0

(4.66)

We note that in [410], it has been shown that out of these four alternatives, only the first and the last governing equations for φ are the correct ones, whereas the second and third equations will lead to spurious growing solutions and hence are not admissible. As the admissible fundamental solutions satisfy far stream boundary conditions, in principle, any one of the other two equations can be solved from the wall to the free stream. The governing equations are homogeneous, and if in addition, the necessary boundary conditions are also homogeneous, as in case of Eq. (4.63) for the eigenvalue problem, then it cannot be used as well. This leaves us with only the fourth equation as the admissible option for obtaining the eigen function. The situation for the receptivity problem is different, as in such cases, the boundary conditions are inhomogeneous and either the first or the fourth equation can be used. The overall advantage of CMM lies in replacing a stiff ordinary differential equation as a boundary value problem by two relatively easier non-stiff initial value problems. All the four alternatives, Eqs. (4.63)–(4.66), reduce to constant coefficient equations in the free stream (˜y → ∞). Then using the normalized conditions in Eqs. (4.52)–(4.57), one can find the characteristic roots for Eqs. (4.63)–(4.66) as: • For Eq. (4.63): The characteristic roots are –α and –Q, as required. • For Eq. (4.64): The characteristic roots are –α, –Q and (α + Q). • For Eq. (4.65): The characteristic roots are –α, –Q and

αQ α+Q .

• For Eq. (4.66): The characteristic roots are 0, –α and –Q. ˜ |Q| >> |α| and the third characteristic root of Eq. (4.65) simplifies to For high Re, ' α. This is a mode similar to φ2 , which grows with y˜ and is not admissible. The aforementioned result has been overlooked in [104, 143], where results have been presented using Eq. (4.65). It has been suggested in [312] that the problem of Eq. (4.63) at the wall can be avoided by integrating the Orr–Sommerfeld equation for few steps before switching over to Eq. (4.63). For eigenvalue problems, at y˜ → 0, it is seen that φ and φ0 are zero, while φ00 is indeterminate. In [470], φ00 is arbitrarily set to any value and then either Eq. (4.63) or Eq. (4.66) is used, avoiding the suggestion in [312]. ˜ due Another problem is noted with CMM in obtaining eigenfunctions for high Re, to large dynamic ranges of the second compounds inside the boundary layer. In Eq. (4.63), one requires an accurate ratio of zz21 and zz41 , which exhibits very large excursions near the wall for which z1 is close to zero. This results in an oscillatory eigensolution. αQ α+Q

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In contrast, for Eq. (4.66), one notices that zz54 and zz64 are regular and of order one. Thus, the solution obtained from Eq. (4.66) is noted to be regular, as reported in [418] for a receptivity problem.

4.5 Instability Analysis with the Orr–Sommerfeld Equation The Orr–Sommerfeld equation has been solved either as a temporal or as a spatial instability problem; it was not firmly established for a long time which perspective is more physical. In resolving this issue, an alternative point of view has been proposed by some researchers, who posed the flow instability problem as one growing in space and time. This debate has been furthermore complicated by other competing ideas originating from other models. We have noted the inability of the inviscid mechanism to cause temporal growth for an attached boundary layer following Rayleigh’s and FjØrtoft’s theorems. However, when linear viscous theory is studied, it is possible to also pose the problem for temporal growth. For the experimental arrangement of [405], where the disturbance was created by a localized excitation inside the boundary layer, the disturbance appeared to convect downstream. Hence, physically, a temporal growth of the disturbance field does not appear realistic. This is observed experimentally for unseparated flows; instability arises, which further convects. The phenomena prompted the development of the Orr–Sommerfeld equation as a tool for spatial instability. Interestingly, temporal theory is easy to use; in the early results reported by Tollmien and Schlichting, they performed temporal analysis of the Orr– Sommerfeld equation, and converted the disturbance growth rates to spatial rates by invoking (without explicitly mentioning) the concept of group velocity. This has been ‘explained’ in [141] by considering the eigenvalues as analytic functions of wavenumber and circular frequency, which allows use of the Cauchy–Riemann relation to relate spatial and temporal growth. However, as the eigenvalues are singular points, these cannot be treated as analytic functions (despite the existence of a dispersion relation). At the same time, this discussion about using either spatial or temporal theory can be bypassed, if we consider the most generic approach of spatiotemporal growth of disturbances. This spatio-temporal approach used for the Orr– Sommerfeld equation has seen immense success in recent times, and will be a central theme of this discourse.

4.5.1 Grid search method: Eigen-Spectrum Let us first discuss the grid search method used for finding the eigenvalue spectrum for the spatial instability problem for the most generic problem of zero pressure gradient boundary layer. For the purpose of local analysis, we fix the station by prescribing the ˜ = Ue δ∗ , with ν Reynolds number based on displacement thickness (δ∗ ) defined by Re ν as the kinematic viscosity and Ue as the boundary layer edge velocity (for the Blasius

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boundary layer, this is the free stream speed). For the two-dimensional disturbance field and equilibrium flow, the wall-normal disturbance velocity is governed by the two-dimensional Orr–Sommerfeld equation, given by Eq. (4.25). For the spatial instability, the boundary layer is excited by a fixed frequency excitation source, ω ¯ 0, and one looks for complex wavenumber, α. In the grid search method used here, one identifies a user specified range of real and imaginary parts of α. In this rectangular zone, one sets up a grid of points, where each point corresponds to a complex value of α. For each of these grid points, one solves Eqs. (4.40)–(4.45) starting from the free stream with the initial conditions given by Eqs. (4.52)–(4.57) by four stage, fourth order ˜ and ω Runge–Kutta method up to the wall (˜y = 0). For fixed Re ¯ 0 , if the chosen α in the grid corresponds to an eigenvalue, then the dispersion relation will be automatically satisfied, i.e., Eq. (4.58), which is given by, z1 = 0 at y˜ = 0. In general, this will not be the case, and one would store the real and imaginary parts of z1 as z1 (˜y = 0) = z1r + iz1i . Going through this exercise for all the points in the complex α-plane, one obtains the two-dimensional data sets for z1r and z1i as functions of real and imaginary parts of α, which we denote as αr and αi . Finally, we plot the contour-lines for which, z1r = 0 and z1i = 0. In Figure 4.2, the spatial eigenvalue spectrum for the Blasius boundary layer is ˜ = 1200 and ω shown for Re ¯ 0 = 0.1. This graphical construction should be followed by fine-tuning the eigenvalues up to desired degree of precision by using the Newton– Raphson search. One of stronger aspect of CMM over matrix methods is the near absence of spurious modes. In matrix methods, if one uses N number of points in the wall-normal direction, then the method returns N eigenvalues. One needs to assess the accuracy of the estimated eigenvalues. Comparatively, in CMM, fewer, most probable eigenvalues are obtained, and in Figure 4.2, there are only four eigenvalues marked as P1, P2, P3 and P4. CMM produces spurious modes along the line c = 1 (λ1,2 = λ3,4 ), which are easy to detect and eliminate from the definition of phase speed explained in Figure 4.2. One of the easiest way of testing is to take the polished eigenvalues by the Newton–Raphson search, and use those values to move from free stream to the wall by solving Eqs. (4.40)–(4.45). If the eigenvalue is genuine, then one would see that z1 = 0 at y˜ = 0. In Figure 4.3, the spatial eigenvalue spectrum for the Blasius boundary layer is shown for the same frequency of excitation, but at an increased value of ˜ = 1500 and ω ˜ cases Reynolds number given by Re ¯ 0 = 0.1. For both the Re shown in Figures 4.2 and 4.3, the first mode, P1 has negative imaginary part of α, which indicates the primary mode obtained to be spatially unstable. In the literature, this is identified as the TS mode. Having identified a spatial mode (α∗ ) for a frequency of excitation, ω ¯ 0 , one can identify its direction of propagation by numerically estimating its group velocity. This velocity can be found by altering the frequency of excitation by an infinitesimally small amount to ω ¯ 0 + ∆ω0 and repeating the process as noted here to identify the eigenvalue spectrum once again. If an altered spatial mode becomes equal to α∗ + ∆α∗ ,

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~ = 1200 Re ω 0 = 0.1

Z1r = 0 Z1i = 0

0.4

0.2

P2

αi

P3 0

P1

P4

–0.2

–0.4

–0.4

–0.2

0 αr

0.2

0.4

˜ = 1200 and ω Figure 4.2 Spatial eigenvalues of Blasius boundary layer for Re ¯ 0 = 0.1 by grid search method. One notes the intersection of the zero contour lines for z1r and z1i marked as P1, P2, P3, and P4. then the group velocity can be obtained from the real part of the right-hand side of the following expression Vg = Real

 ∆ω  0

∆α∗

The phase speed is simply obtained as c ph = ω ¯ 0 /α∗ . For the time being, we will exclude P4 from the discussion, apart from noting that this mode has negative phase speed, a hint that this is the upstream propagating mode. This can be verified by calculating the group velocity of P4, which turns out to be negative, implying that this is indeed an upstream propagating mode. The justification for using the same CMM formulation of wall mode for the upstream propagating mode will be explained in the next chapter, when we discuss about linear receptivity. In Figure 4.4, the wave properties, αr , αi , c ph , and Vg are shown for the case of ˜ = 1200 for a range of frequencies of excitation. As ω Re ¯ 0 is decreased, one notices

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~

Re = 1500 ω 0 = 0.1

Z 1r = 0 Z 1i = 0

0.4

P2

0.2

αi

P3

0

P4

P1

–0.2

–0.4

–0.4

–0.2

0 αr

0.2

0.4

˜ = 1500 and ω Figure 4.3 Spatial eigenvalues of Blasius boundary layer for Re ¯ 0 = 0.1 by grid search method. One notes the intersection of the zero contour lines for z1r and z1i marked as P1, P2, P3, and P4. that the downstream propagating modes (shown by solid lines) disappear one after the other, and at the time of disappearance, the phase speed of that mode becomes equal to one. These curves are drawn by using the Newton–Raphson procedure in locating the eigenvalues, as ω ¯ 0 is progressively reduced. Values of ω ¯ 0 at which these three downstream propagating modes disappear are listed in Table 4.1. Table 4.1 Values of ω ¯ 0 for which 2D modes of the Blasius boundary layer disappear for indicated Re. Mode Number

˜ = 1200 Re

˜ = 1500 Re

1 2 3

0.0026365 0.034114 0.024197

0.001855 0.027111 0.0192988

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~

Re = 1200 0.3

0.25

αr

0.1

0.1 1

2

0 0.2

αi

0.2

3

0.05

0.15

a_r

–0.2

3

0.02

0.1

0

4

2

0.04

ω0

0.06

4

0.08

0.1

0.02

0.04 0.06 ω0

0.08

0.1

1.5

3

0.8 1

1

0.05

0.6

0.5

2

3

1

0

0

Vg

Cph

2 1

–0.1

0.4

ωcr = 0.04775

0.15

0.2

–0.5

0

–1

-0.05

–0.2

0.02

–0.4 0.02

0.04

ω0

0.06

4 0.04 0.08

0.1

4

–1.5

0.06

w0 –2

0.08 0.02

0.04

0.06 ω0

0.1 0.08

0.1

Figure 4.4 Variation of wave properties of the spatial eigenvalues of Blasius boundary layer ˜ = 1200 with frequency of excitation ω0 for all the modes obtained by grid for Re search method and polished by Newton-Raphson search. The properties of the downstream propagating modes (P1, P2, P3) are shown by solid lines and the upstream propagating mode, P4 is shown by the dashed line. In Figure 4.5, the wave properties, αr , αi , c ph , and Vg , are shown for the case of ˜ Re = 1500 for the same range of frequency of excitation used in Figure 4.4. As ω ¯0 is decreased, one again notices that the downstream propagating modes (shown by solid lines) disappear abruptly, one after the other, and at the value of disappearance, the phase speed of that mode becomes equal to one. Values of ω ¯ 0 at which these three downstream propagating modes disappear are listed in Table 4.1 in the second column. If the Blasius boundary layer is excited below the lowest of the three values of ω ¯0 in Table 4.1, then CMM detects an eigenvalue, whose wave properties are shown in Figures 4.4 and 4.5 by dotted line, which is located in the left half of the α-plane, as noted in Figures 4.2 and 4.3 by P4. It is emphasized that the eigenvalue in the left half of the α-plane is located by CMM due to its analytical feature and was first reported in ˜ = 1200, this threshold value is given by ω [448]. From Table 4.1 for Re ¯ 0 = 0.00263 and ˜ = 1500, this is lower at ω for Re ¯ 0 = 0.00185; any excitation below this critical ω ¯ 0 shows

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~ = 1500 Re 0.3 0.25 0.2 0.15 0.1 0.05 0 –0.05 –0.1 –0.15 –0.2

ωcr = 0.04040 0.15

1

0.1 αi

2

αr

3

0

4 0.02 0.04

ω0

0.06

4 0.08

0.02

0.1

0.04

ω0

0.06

0.08

0.1

1.5

2

1

1

0.6

0.5

0.4

0

2

3

1

Vg

Cph

2

1

3 0.8

3

0.05

0.2

–0.5

0

–1

–0.2

–1.5 4

–0.4 0.02

0.04

0.06

ω0

0.08

0.1

–2

4

0.02

0.04

ω0

0.06

0.08

0.1

Figure 4.5 Variation of wave properties of the spatial eigenvalues of Blasius boundary layer ˜ = 1500 with frequency of excitation ω for Re ¯ 0 for all the modes obtained by grid search method and polished by Newton-Raphson search. The properties of the downstream propagating modes (P1, P2, P3) are shown by solid lines and the upstream propagating mode, P4 is shown by the dashed line. the presence of only a single upstream propagating mode noted by the dotted line in Figures 4.4 and 4.5. The fact that this is indeed the upstream propagating mode is readily apparent by noting the value of Vg being negative in the bottom right frames of Figures 4.4 and 4.5. For TS waves, αr is always positive, i.e., these are in the right half of the α-plane and have positive group velocity and phase speed. As a consequence, P1 is a growing mode, while P2 and P3 are damped.

4.6 Other Linear Instability Theories Although, it has been noted in the previous section that semi-infinite flat plate should support convective instability, there have been occasional efforts to use temporal and other variants of instability theories to explain some of the features of flow instability,

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which remains poorly understood. For example, Trefethen et al. [532] have mentioned in their work in explaining hydrodynamic instability without eigenvalues using other variants of linear stability theories by noting that “for other flows, notably those driven by shear forces, the predictions of eigenvalue analysis fail to match most experiments. We consider the two most studied examples of this kind: (plane) Couette flow .... and (plane) Poiseuille flow. Other examples for which eigenvalue analysis fails include pipe Poiseuille flow (in a cylindrical pipe) and, to a lesser degree, Blasius boundary layer flow (near a flat wall).” According to these authors, eigenvalue analysis for a semi-infinite flat plate is also inadequate. This is not disputed, as predicting instabilities by discrete eigenvalues is not the same as showing the eventual transition to turbulence. The authors in [532] relied on perturbation theory for linear operators and their application in partial differential equations [220, 325], where the operator is related to ‘pseudospectra’ obtained using matrix methods associated with concepts in linear algebra. According to this point of view, even when all eigenvalues of the linear temporal theory are noted to be stable, small perturbations can amplify by factors on the order of 105 by a linear mechanism [532] for three-dimensional disturbance field. As eigenvalues are not involved in these approaches, it has also been reported as nonmodal instability theory. The authors also note that “an essential feature of this nonmodal amplification is that it applies to three-dimensional (3D) perturbations of the laminar flow field” suggesting that the emphasis on twodimensional perturbations is misplaced. However, using the same method for twodimensional perturbations, only some amplifications are noted, which is far too weak to cause eventual transition.

4.6.1 Role of Fourier–Laplace transform: Abel and Tauber theorems There is also a confusion in instability theory about the roles of discrete and continuous modes, which define the perturbation field. If we represent the perturbation field by the Fourier–Laplace transform, then we can write it as q( ˜ x˜, y˜ , z˜, t˜) =

Z Z Z

q(˜ ¯ y)ei(α x˜+β˜z−ω0 t˜) dα dβ dω0

(4.67)

This representation nowhere requires that the amplitudes given by q¯ are orthogonal to each other. In fact, this also does not require that there should be only distinct modes. Thus, in the most general case, for the instability modes, (whenever present as distinct eigenvalues), the corresponding eigenfunctions need not be orthogonal. Apart from the distinct eigenvalues, there will also be continuous modes, whose evolution has been designated in the literature as nonmodal growth. Thus, the overall perturbation field is given by mutual interactions of modal and nonmodal components of the aforementioned representation. Therefore, we can understand when Trefethen et al. [532] noted: “It is a fact of linear algebra that even if all of the eigenvalues of a linear

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system are distinct and lies well inside the lower half plane, inputs to that system may be amplified by arbitrary large factors if the eigenfunctions are not orthogonal to one another.” The use of Eq. (4.67) in describing the perturbation field has been wrongly questioned by some researchers. See for e.g. [398], where the author has noted that “for most wall-bounded shear flows the spectrum is a poor proxy for the disturbance behavior as it only describes the asymptotic (t → ∞) fate of the perturbation and fails to capture short-term characteristics [400]. The many decades-long concentration on eigenvalues in hydrodynamic stability theory has accordingly resulted in a disregard for short-time perturbation dynamics and its consequences on scale selection and transition scenarios.” The author furthermore asserted that for accurate description of disturbance field for all times, one needs to introduce ‘finite time horizon’ tools to explain the observed instability!. The correct assessment of the Fourier–Laplace transform in Eq. (4.67) is provided by two theorems in operational calculus [537], and have been used for flow instability and transition very effectively in [409, 412, 418]. These theorems provide the basis for near-field (short-term) and far-field (asymptotic in time) solution using the Fourier– Laplace transforms. These are due to Tauber and Abel, whose utility are highlighted in [537] for exploring the relationship of the original solution in physical space with bilateral and unilateral transforms in spectral space. Although these are stated in the context of the receptivity problem, as shown in Figure 4.1, these can be understood in general for instability as well. They are interpreted as [412]: The solution in the physical space in the neighborhood of the exciter (near-field/ short-term solution) are decided by the behavior of the transform in the spectral space (α → ∞ for near-field and ω0 → ∞ for short-term behavior), as per the Tauber theorem. In the same way, if we are interested in the far-field/ timeasymptotic solution in the physical space, then we should look in the spectral plane, i.e., α → 0, far-field; and ω0 → 0 for time-asymptotic behavior, according to Abel’s theorem. Most of the time, the far-field/ time-asymptotic solution is determined by the eigenvalue in a stability theory or poles in the receptivity theory. The near-field or short-term solution is determined by the essential singularity of the transform given by the Orr–Sommerfeld equation, e.g., by Eq. (4.25) for two-dimensional perturbation problem [9]. It is worthwhile to note that Trefethen et al. [532] have noted that for “instabilities driven by thermal or centrifugal forces, the predictions of eigenvalue analysis match laboratory experiments. Examples are the Rayleigh– B´enard convection (a stationary fluid heated from below) and the Taylor–Couette flow (between a stationary outer and a rotating inner cylinder).” This actually vindicates the use of the Fourier–Laplace transform to describe the disturbance field, where the spatial periodicity changes the Fourier–Laplace transform to Fourier–Laplace series, thereby converting the present nonmodal spectrum to discrete eigenvalues. The Orr–Sommerfeld equation captures these discrete modes effectively as eigenvalues, showing excellent match for these spatially periodic problems. Noting that the

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Fourier–Laplace series is a special case of the Fourier–Laplace transform, the use of the latter for studying disturbance field is the most generic approach, which is also highlighted in this book. Apart from the Orr–Sommerfeld equation, we do not require any additional modeling, such as transient growth, or finite time energy growth, etc.

4.6.2 Temporal, spatial, and spatio-temporal growth routes Thus, the problem of instability theories is not in the use of the Fourier–Laplace transform, but rather in the way this is used while implementing various versions of instability theories. First and foremost is the question of which version of theory should we adopt: temporal or spatial theory. Just because temporal theory failed to show growth for zero pressure gradient flows, the question remains whether one should adopt spatial theory with all its artifacts. The major drawback in spatial theory is the forcible imposed time scale and then looking for distinct modes that can grow in space. Although Tollmien and Schlichting [119, 412] found the wave solution for ˜ and ω a specific combination of Re ¯ 0 , it was also detected in the same framework by a vibrating ribbon inside the boundary layer by Schubauer and Skramstad [405]. In interpreting these experimental results, a serious mistake has been committed by most of the early researchers, as evidenced from the statement in [399]: “Due to linearity of the governing equations, the output g responds with the same frequency and can also be represented as g = gˆ exp(iω ¯ 0 t)” where the external forcing frequency is also given by ω ¯ 0 . This is called the signal problem assumption, and is the source of all problems, including the failure by researchers who were forced to look for an additional model to account for transient growth. Physically, this assumption is wrong for the fluid dynamical system, which can be unstable for other discrete frequencies or can have nonmodal growth, which can make the response at the forcing frequency sub-dominant. In the next chapter, we will provide the details of correct formulation and results. Here, it is just enough to understand that any finite time start is capable of exciting other time scales, apart from the forcing frequency. The main point that has been overlooked in these efforts is the genuine effort to look for at most a generic scenario, where growth in space and time will be automatic. This was enforced in Sengupta et al. [418] and expanded in many possible variations to show the uniqueness of this concept of spatio-temporal approach in the study of receptivity and instability in [34, 36, 419, 451, 452, 509], where unequivocally it has been shown that the role of TS wave is marginal as the precursor for transition to turbulence.

4.6.3 Signal problem assumption: Progress or impediment? The first important limitation of using discrete eigenvalues to explain flow instability was noted in the original formulation laid out by Orr [321]. Later, Case [80] pointed out the role of the initial condition in triggering destabilization, once again using the Fourier–Laplace transform. The first attempt to explain the experiment in [405] was in [141] using the Orr–Sommerfeld equation and by introducing the role of discrete

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eigenvalues, whereas less detailed emphasis was placed on the roles of branch points, continuous spectrum and other probable singularities like Stokes’ phenomenon [25]. The branch points have been identified following Eq. (4.34), related to the analytical structure of the Orr–Sommerfeld equation in the far-field. These are given by the conditions, α ≡ 0 and Q ≡ 0. The first branch point is the origin of the complex α-plane. The other points are shown for the signal problem (with ω ¯ 0 as real) from Q2 = reiθ = p¯ 1 + iq¯ = (α2r − α2i − Reαi ) + i(2αr αi + Re(αr − ω ¯ 0 )) = 0 √ √ Therefore, Q = reiθ/2 and the real part of Q is given by r cos θ/2, which becomes zero at the branch point. As tan θ = q/ ¯ p¯ 1 , cos θ = p¯ 1 /r, and therefore, cos θ/2 = √12 (1 + q 1/2 √ cos θ)1/2 = √12 (1 + p¯ 1 /r)1/2 . Hence, the real part of Q is ( p¯ 1 +r) , p¯ 21 + q¯ 2 )1/2 = 0, i.e., ( p ¯ + 1 2 which is equivalent to q¯ = 0 and p¯ 1 = 0. There are certain features of the branch points that help in the choice of Bromwich contours for performing the inverse transform. Thus, the branch points are found by solving the two equations (to locate these at A and B in Figure 4.6), α2r − α2i − Reαi = 0 and 2αr αi + Re(αr − ω ¯ 0) = 0 Upon solving these two equations, we get the following locations of the other two branch points (at A and B) as  Re  q 1/2 Re2 + 16ω20 − Re 8

αr |A,B = ±

and αi |A,B = −

Re  ω ¯0  1− 2 αr |A,B

In Figure 4.6, the contours of p¯1 = 0 and q¯ = 0 are drawn in the complex α-plane, whose intersections locate the branch points at A and B. For the ease of viewing, the figure is ˜ = 10 and ω drawn using Re ¯ 0 = 0.5. Note that the point O’ is located on the imaginary ˜ All these coordinates help one understand the placement of the axis, at αi = −Re. Bromwich contour in the α-plane. Having already discussed the limitations of the signal problem, these branch points are of marginal interest. In claiming improvements over the signal problem formulation given in [141], the authors in [14] provided these branch points. Interestingly, they also suggested the negative q¯ = 0 axis as the possible branch cut. Similarly for the branch point at the origin, the αi -axis has been suggested as the other branch cut. These are not necessary as will be explained here.

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12

p-1 = 0

8 q- > 0

-

p1 < 0 4 -20

-15

-10

0 o

-5

A

5

10

20

15

αr

q- = 0

-4 q- = 0 -8 B

α i = -Re/2

6E-06 α i

o’

4E-06

-12

2E-06 o

q- > 0

-16

-

-0.01 -0.005

0

αr 0.005

0.01

p1 < 0

-20

-24 p-1 = 0

Figure 4.6 Locating branch points for the signal problem, with these located in the complex α-plane at the origin and at the points A and B. The pair of points at A and B are located at the intersection of the curves given by the zero contours plotted for the real and imaginary parts of Q2 . There seems to be some confusion about these enforced mandatory branch cuts with the Stokes phenomenon, see for e.g., pp 116 of [25]. These phenomena arises in connection with the subdominant and dominant nature of different modes of an asymptotic representation. This is very relevant to the present problem of solving the Orr–Sommerfeld equation. Here, the αi -axis is the line across which the fundamental solutions exchange identities, and is called the Stokes line. With respect to the branch point α = 0, as the argument (α) approaches the branch point along the imaginary axis, the asymptotic values are nearly equal and oscillatory. The leading behavior of the fundamental modes are most unequal, on the contrary, when the argument is purely real. As defined in [25], the line along which the leading behavior is most unequal is called the anti-Stokes line. Thus, the αr -axis is the anti-Stokes line. Furthermore, it is

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noted [25] that this phenomenon is a property of the exponential function. The Stokes and anti-Stokes line are a local property of the asymptote and are meaningful in the immediate vicinity of α = 0. This is one of the main reasons why one should not choose the Bromwich contour going through the branch points. There is no reason to not cut across the imaginary axis, which can be the Stokes line. Moreover, in using Fourier series and Fourier transforms (as well as Laplace transforms), one can analytically handle piecewise jump discontinuity, if there are any, by taking an average of the lefthand and right-hand limits of the transform obtained from the solution of the Orr– Sommerfeld equation. We will see from the solution of the Orr–Sommerfeld equation that such discontinuities are not present across the Stokes line (αi ) axis. In [14], it has been conjectured that the discrete spectrum and other singularities are responsible for the far-field solution of the signal problem, while the near-field contributions originate from the aforementioned enforced “branch cuts.” However, there are no analytic or computational results to back up such claims. On the contrary, Bromwich contour integral method has been used for the signal problem in [409] to show the presence of near- and far-field of the disturbance field caused by a localized delta function excitation at the wall. An analytical explanation for the near-field of this problem is also derived in [412]. These will be briefly described as a case-study in the next section.

4.6.4 Temporal instability theory: A case-study with CMM At the outset, we note that for boundary layers, the disturbance convects downstream with the flow. Hence, the meaning of strict temporal growth for unseparated flow is ambiguous. For separated flows, it is possible to experience additional temporal growth, apart from spatial growth and convection of disturbance field. Hence, the presented results in Figure 4.7 for the Blasius boundary layer, showing the temporal spectrum obtained by the grid search technique using CMM is for the purpose of demonstration of the utility of CMM, as compared to some results presented in [400] using the Chebyshev spectral collocation method. Once again, we solve the CMM equations given by Eqs. (4.40)–(4.45) from the free stream using the initial conditions given by Eqs. (4.52)–(4.57). We note that now the circular frequency is complex, while the wavenumber, α is real and takes the value of 1. The dispersion relation in Eq. (4.58) is satisfied where the real and imaginary parts of z1 at the wall (˜y = 0) are simultaneously equal to zero. These zero contour lines are plotted in the figure, and one can trace distinctly nine eigenvalues, which matches with the same given for the two-dimensional Orr–Sommerfeld equation in Table A.4 of Schmid and Henningson [400] obtained using spectral collocation method. One also notices very few spurious intersections by CMM, with only two of these located very close to ω0r = α line. It is very easy to identify and remove these modes, and they are not shown in Figure 4.7 in the plotted range. One notices that all the nine temporal modes are stable.

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Re = 800, α = 1 0.2 0 –0.2 ω0 i

–0.4 –0.6 –0.8 –1

0

0.2

0.4

0.6

0.8

ω0 i

Figure 4.7 Temporal spectrum showing discrete eigenvalues for Blasius boundary layer, for ˜ = 800 and α = 1.0, obtained by grid search technique using compound matrix Re method.

4.7 Instability Properties Using the Orr–Sommerfeld Equation We investigate the spatial stability of the semi-infinite flat plate given by the similarity solution of the Blasius equation using CMM. In performing CMM for the Blasius boundary layer equation, the CMM equations given by Eqs. (4.40)–(4.45) are integrated from the free stream using the initial conditions given by Eqs. (4.52)–(4.57). The eigenvalues are obtained by satisfying the dispersion relation given by Eq. (4.58), by finding the location where real and imaginary parts of z1 at the wall (˜y = 0) are simultaneously equal to zero. Among all the eigenvalues found, the least stable one ˜ ω is termed as the TS mode and its wave properties are shown in the (Re, ¯ 0 )-plane in terms of real and imaginary parts of α. Results obtained are plotted as contours of ˜ ω constant amplification rates (negative αi only) in the (Re, ¯ 0 )-plane in Figure 4.8, for the Blasius boundary layer. The outermost contour is for αi = 0, implying no growth in space, following the ˜ and ω spatial theory. Thus, for all combinations of Re ¯ 0 along this contour, there is neither any growth or decay in space, nor with time. This neutral curve is essential for the most general case of the spatio-temporal theory as well. It is customary to call the lower part of the neutral curve as branch I, while the upper part of the neutral curve is called branch II. The critical value of the Reynolds number for this zero pressure gradient flow is indicated as Recr = 520, below which one will not see any spatiotemporal growth. The parameter values corresponding to points inside the neutral

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~

(a) 0.14

Critical Re =520 0

Constant f 2 - lines 0

0.12

ω0

-0.00 5 0

0.1

-0 .

00

0.08

5

-0 . 0

0 -0 .

12

0

B

1

-0 .0 0

. -0

0.06

Constant frequency (f1 )- lines

D

01

. -0

0

5

A 0.02

0

0

2000

4000

0

5

-0 .0 -0 .0 1 1 2

25 -0.01 2 5 -0 .0 12 -0 .0 1 -0 .0 05

00

C

0.04

-0 .0 1

6000

~

8000

0

10000

Re (b)

Constant α r - lines 0.38

0.14

0.36 0.34

0.12

0.32

0.1

0.3

ω0

0.28

0.08

0.2 6

0.24

0.06

0.22

0.2 0.18

0.04

0.16 0.14 0.12

0.02

0.1 0.08

0. 06

0

2000

4000

~

6000

8000

10000

Re

Figure 4.8 Contour plots of (a) asymptotic growth rate and (b) the real part of wavenumber ˜ ω ˜ = Recr for TS waves are shown in the (Re, ¯ 0 )-plane. Note the critical value of Re equal to 520, below which there will be no spatio-temporal growth in the parallel flow framework for the zero pressure gradient boundary layer.

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curve indicate spatial growth. Thus, the highest unstable ω ¯ 0 for this zero pressure gradient boundary layer is close to a value of 0.14. In frame (b) of Figure 4.8, the ˜ ω corresponding αr contours are drawn in the (Re, ¯ 0 )-plane. For the plotted range of ω ¯ 0 shown in this frame, the maximum value of the wavenumber is roughly about αr,max ≈ 0.40, which corresponds to wavelength of the order of 5πδ∗ . As noted before, the information contents of Figure 4.8 relate to a discrete eigenvalue which is the least stable, but the neutral curve is special and indicates the ˜ below which one would not streamwise location, in terms of the critical value of Re, see the boundary layer supporting the spatio-temporal wave front, about which we will learn more in the next chapter.

4.7.1 Effects of pressure gradient on instability of boundary layers The stability properties of discrete eigenvalues may not be useful for transition, as there is no evidence that the most unstable discrete mode would also grow spatially to cause transition. Even then, the information contained in the neutral curve is very vital in terms of nonmodal growth. Thus, it is important to study the trend of TS wave properties with respect to other background information which affect the stability of the equilibrium flow. One of the most important passive parameters in controlling flow instability is the pressure gradient. In Figure 4.9, we show the stabilizing effects of a favorable pressure gradient, for which the Falkner–Skan parameter is given by m = 0.02564. Frame (a) of Figure 4.9 once again shows the unstable region of the TS waves, with the outer contour showing the neutral curve. As compared to the zero pressure gradient case, here the critical Reynolds number increases to the value of 860, indicating the efficacy of favorable pressure gradient to significantly delay both the modal and nonmodal growth of disturbances. We also note that the maximum growth contours of the discrete modes are significantly lower, even as compared to the zero pressure gradient flow. The highest value of unstable circular frequency significantly reduces from about ω0,max = 0.14 to 0.108 and the corresponding wavelength increases marginally to αr,max ≈ 0.34. Having seen the stabilizing effects of favorable pressure gradient on the stability of the Falkner–Skan boundary layer, it is apparent that adverse pressure gradient would have the opposite effects. In Figure 4.10, we show the αi contours for the Falkner– Skan similarity profiles for progressively worsening adverse pressure gradients in the ˜ ω four frames. In frame (a), one notes the unstable value of αi in the (Re, ¯ 0 )-plane for m = −0.02539, with the following important observations: (i) The value of Recr reduces to 316; (ii) the maximum critical circular frequency increases to about 0.186 and (iii) the maximum growth rate becomes almost double the value for m = +0.02564. Moreover, one notices the important property of the shape of the neutral curves with adverse pressure gradient, as compared to the cases of favorable and zero pressure gradient cases. For the latter, the neutral curve width in the ω ¯ 0 direction keeps decreasing ˜ which is known to shrink to a point at very high value of Re. ˜ with increasing Re,

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(a)

~

Critical Re =860 0.12 Constant α i - lines 0.1

0.08 -0 .0 0 2

ω0

00 0 2

-0.005

0

-0 .

-0 .0 0 -0 02 -0 .0 .0 0 5 07

0.06 . -0 00 7

0

-0 . 0

0.04

-0 . 0

-0 .0 0

05

-0 .0 0 9

9

-0 .0 09 5

02

-0 .0 0 7

-0 .0 05 -0.00 2

0

0.02 2000

-0 .0 -0 .0 07 0

-0 .00 95

4000

6000

~

8000

10000

8000

10000

Re (b)

Constant α r - lines

0.12 0.34

0 .3 2

0.1

0.3 0.28

0.08

ω0

0.26 0.24

0.06

0.22

0.2 0.18

0.04

0.1 6

0.1 4

0.02

0.12 0.1

0.08

0.06

2000

4000

~

6000

Re

Figure 4.9 Contour plots of (a) asymptotic growth rate and (b) the real part of wavenumber ˜ ω for TS waves are shown in the (Re, ¯ 0 )-plane for Falkner-Skan velocity profile experiencing favorable pressure gradient with m = 0.02564. Note the critical value ˜ = Recr equal to 860, below which there will be no spatio-temporal growth of Re in the parallel flow framework for this flow.

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(a)

~ Critical Re = 316

0

5 .0 -0

03

-0 .0 3

0

-0.005 -0.01

5000 ~ Re

(d)

-0.0 18

0.4

-0 .1

5000 ~ Re

10000

-0.02

-0.06

-0.1 -0.12

4

-0.15 -0.1 58

-0.1 55 -0.15 -0.14

-0 . 0

-0.1

8

2

0

0.1

-0.015

-0.04

-0 .0

-0 .0 1 -0.00 5

~

0

-0 .0 8

0.2

10000

Critical Re = 67

0.5 0.3

-0.018

-0.03

-0.02 -0.01 0

-0.1-0 .1 2

25 02 -0 .0 -0 . -0 .0 15

-0.015 -0 .0 2

-0.03

-0.03 8 -0.035

0

0.7

-0.038

-0.039

9

-0.0 2

-0.018 -0.01

-0 .

-0.035 -0.039

0.6

0.05 0

0

10000

~ Critical Re = 199

(b)

38

0.05

-0.01 5

5000 ~ Re

0

ω0

0.1 -0 .0 15

-0.005 -0.0 1 0

-0.005

0.1

ω0

-0 .0 1

0

0.15

0.15

0

5

ω0

ω0

-0 .0 0

-0.02

-0.03

3 .0 4 -0 .0-042 -0.041 -0.035 -0 .0 2-0 .001

5

-0 . 0 -0 . 1 5 018

0

0.2

-0.01 -0.02

0.2 -0 .0

1

01 -05. -0 .0 0

01

0

-0.0 3

0

0

-0 .

0.25

~ Critical Re = 139

0 -0.01

00

0.25

0.1

0.05

(c)

0.3

-0 .

0.15

0.35

0 -0.04 -0.06

0.2

0

0

-0.06 -0.04

5000 ~ Re

-0.12

-0. -0.02

10000

Figure 4.10 Contour plots of asymptotic growth rate (αi ) for the Falkner–Skan boundary layer for adverse pressure gradient cases shown for (a) m = −0.02539, (b) m = −0.04762, (c) m = −0.06542 and (d) m = −0.0904. Note that the last case is for the Falkner–Skan boundary layer which is about to separate following the Prandtl’s empirical criterion of incipient steady separation. The critical value of Recr has been noted in each frame for the pressure gradient parameters. Thus, viscous linear instability for the Falkner–Skan boundary layer for zero and favorable pressure gradient cases are noted for intermediate Reynolds number. The situation changes qualitatively for adverse pressure gradient cases. In these cases, ˜ actually widens, implying the neutral curve, instead of shrinking in width with Re, that the discrete modes, as well as the spatio-temporal nonmodal growth continue to thrive as the disturbance packets propagate downstream. In frame (b), the value of Recr decreases to 199 for m = −0.04762, with spatial maximum growth rate increasing further in excess of exponent value of 0.025. The critical frequency increases to about 0.25. When the pressure gradient is more adverse, while the flow is still attached, the unstable region is shown in frame (c) for m = −0.06542. The value of Recr further reduces to 139, critical frequency increases to above 0.31 and almost the full range ˜ of frequency remains unstable with increase in Re. Finally, the adverse pressure gradient case for the Falkner–Skan profile is considered with m = −0.0904 for which

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the equilibrium flow is about to separate, as the wall shear vanishes for this profile. For this case, not only does the asymptotic growth rate quadruple as compared to the previous case shown in frame (c), but also the full frequency band starting from 0 to a value of about 0.68 displays sustained instability. Thus, one can conclude that while nonmodal growth is important for favorable and zero pressure gradient boundary layers, for adverse pressure gradient flows, the discrete modes will have equal roles, if not more, in promoting transition.

4.8 Closing Remarks In this chapter, we start from the inviscid instability analysis of parallel flow, enunciating Rayleigh’s and FØrtoft’s theorem for the disturbance field which can only grow in time. We note the reason why such disturbance fields cannot support growth in space for zero pressure gradient boundary layer. Moreover, the roles of viscous action is discussed based on early works of Taylor, where the creation of dynamic instability of atmospheric motion is discussed. In the process, role of no-slip condition in wall bounded flow is described as a model that can reconcile the failure of Rayleigh’s equation to explain linear flow instability theoretically, while the experimental observation of Reynolds on pipe flow establishes the role of Reynolds number on the viscous action in defining the disturbance field. Having laid the framework for investigating viscous disturbance flow field for parallel flows with the Orr–Sommerfeld equation, the distinction and similarities of instability and receptivity of laminar flow are elaborated. It is possible to use the Orr–Sommerfeld equation in describing both linear temporal and spatial instabilities. It is noted historically that the latter is favored, and as a consequence exciting the flow with a fixed frequency exciter becomes necessary. With this legacy of spatial instability model, the signal problem assumption dominated the discourse, and in the process spawned many unnecessary consequences (even when the normal mode analysis is set aside by using Bromwich contour integral method). In describing flow instabilities, it is noted that artificially compartmentalizing the subject into either temporal or spatial instability problems is not necessary, and instead one should include spatiotemporal instabilities. In developing the spatio-temporal analysis, the role of the Bromwich contour integral method cannot be overemphasized. The critique of normal mode analysis, and instead using nonmodal analysis, with and without transient, algebraic growth can be solved by the Bromwich contour integral method, without any other additional modeling effort. It has been shown using Abel’s and Tauber’s theorem that the Fourier–Laplace transform based methods are all inclusive. This approach is solved with the compound matrix method, which helps both eigenvalue and receptivity analysis. The most important issue of connection between instability and receptivity analysis is demonstrated. It is shown that the eigenvalues of instability analysis are nothing but the poles for the transfer function of receptivity analysis for the wall excitation problem. Moreover, the compound matrix method allows

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one to track eigen spectrum in the full complex wavenumber plane, simultaneously showing the existence of downstream and upstream propagating modes. It is also shown that the upstream propagating modes support free stream excitation. The developed concept of uniform analysis tool is finally used to explain the effects of pressure gradient on instability and receptivity studies.

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Receptivity Analysis: Relation with Instability Experiments

5.1 Introduction In Chapter 1, we have discussed chronologically the theory of instability, starting with the works of Helmholtz [177], Reynolds [365], Rayleigh [351], and Kelvin [224], purely as a discourse about the different facets of the phenomena that we identify as the transition to turbulence. Although laminar flow can be stationary, background disturbances grow in space and time to create turbulent flow. Thus, the main issue in transition research is in identifying how disturbances display spatio-temporal growth. The research built upon the idea of imperceptible disturbances feeding upon the equilibrium flow and the resultant growth being so overwhelming that it takes the initial equilibrium state to another state that will be space–time dependent. Although it is imperative to explain how turbulence comes into being, there remain a few unexplored steps following which the laminar flow becomes transitional and turbulent. In the previous chapter, we emphasized the role of instability studies in this search, but one of the central issues of establishing any instability theory posed as an eigenvalue problem, lies in the difficulty of physically verifying such a theory. There was a major roadblock in experimentally verifying instability theories that continued till very recently. The subject started with the erroneous concept that viscous action is dissipative, and hence, an inviscid theory was considered appropriate and the theorems due to Rayleigh and FjØrtoft came into existence to explain temporal growth of inviscid disturbances. Prior to this investigation, Rayleigh [350] was successful in explaining the motion of jets by this temporal theory. However, this could not explain the instability of flow over a flat plate, and two new concepts came to the forefront. First, the significance and importance of viscous actions was seized upon,

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as diffusive actions were known to create instabilities for various mechanical and electro-mechanical systems. Second, as a reaction to the failure of temporal growth of disturbances for the zero pressure gradient boundary layer, researchers started on the path of spatial instability theory. In this theory, one has to fix a time scale and look for complex wavenumbers which satisfies the basic requirements of eigenvalue analysis. We have noted the discovery of Tollmien–Schlichting (TS) waves from this theory, proved to be difficult to verify experimentally. Once this was shown experimentally in [405], in the ensuing success, the question of whether such TS waves indeed cause transition was forgotten. Some researchers realized the importance of determining why and how the experimental verification was possible. The distinction between natural and forced excitations was not emphasized. In instability theory, one presumes that background disturbances grow owing to natural excitation of the dynamical system. This is not satisfactory from an experimental perspective. The reproduction of instability is not in the control of the experimentalist, even if the spatial theory requires a fixed frequency excitation. Thereafter, to demonstrate the TS waves, one needs to provide deterministic excitation, and as a consequence, the experiment becomes an exercise of forced excitation. Unfortunately in [405], sufficient quantitative details are not provided about the experiment, but an important observation has been noted: vortical disturbance from inside the boundary layer triggers the creation of TS waves, whereas acoustic excitation from the free stream did not produce the same effects. In trying to test the existence of TS waves, Taylor [514] vibrated a diaphragm on a flat plate at a low frequency, but could not detect these waves. These type of issues, where any vanishingly small perturbation is supposed to cause flow instability, are considered as topics for instability theories. Apart from flat plate boundary layer transition, there are many flows that suffer transition to turbulence via different mechanisms. This aspect of flow transition was noted collectively by Morkovin [298], who coined the term “receptivity” to emphasize the role of that property of any fluid dynamical system, which makes the equilibrium flow sensitive to different types of background disturbances. This is the main issue that is discussed in this chapter; it is shown that a proper framework of receptivity analysis is generic enough to explain instability of fluid flow also. Stability is defined as that property of the flow which explains its resistance to small imposed disturbances to grow [28]. However, the background disturbances need not be small, as originally noted in reporting the pipe flow experiment by Reynolds [365]. The original question of transition to turbulence has not been addressed directly in instability studies by the linear theory, nor has the ability of an equilibrium state to retain its undisturbed laminar state to different ambient disturbance sources for stable flows. In the previous chapters, the following questions have been posed: (i) Are there definitive experiments that relate the TS waves with actual transition? (ii) Apart from using a vibrating ribbon inside the boundary layer, can there be other means of exciting TS waves? (iii) Is there any proof, experimentally or otherwise, that if a boundary layer is excited at a given frequency, the response is going to be at that same

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frequency? (iv) In a growing boundary layer, how do signal properties continually change as the TS wave moves downstream? Boundary layer supports a progressive TS wave packet and not TS waves, and the important query; (v) Why the boundary layer has not been investigated for spatio-temporal growth of disturbances, before it was done in [418]? (vi) What about other flows for which quasi-parallel linear theory do not work? These questions are addressed in this chapter.

5.2 Linear Receptivity of Boundary Layer: Bromwich Contour Integral Method As we noted earlier, although the experiment by Schubauer and Skramstad [405] demonstrated the existence of the TS wave packet and neutral curve by a forced excitation problem, it is necessary to show that these two approaches of instability analysis and receptivity experiments are similar. In the process of doing so, a simplification is adopted: instead of vibrating a ribbon near the wall, it is placed exactly at the wall. Such a formulation in the linear context has been suggested as the signal problem in [14, 141]. The presented methodology here is not restricted to the signal problem, instead the complete description of the spatio-temporal approach is used. The governing Orr–Sommerfeld equation is the requisite formulation for both the eigenvalue and receptivity problem. The only difference between the signal problem and the complete spatio-temporal approach lies in the applied initial and boundary conditions. An explanation of receptivity of the parallel shear layer is presented here with essential details of the method based on materials reported in the literature [418, 451, 452]. To explain the linear receptivity theory, let us consider only two-dimensional mean flow experiencing two-dimensional disturbance growth (to make it easy to understand the roles of wall and free stream modes of the Orr–Sommerfeld equation) given in the previous chapter which is rewritten as   ˜ (αU(˜y) − ω0 )(D2 − α2 )φ − α(D2 U)φ (D2 − α2 )2 φ = iRe (5.1) This disturbance equation is written in nondimensional form, taking δ∗ as the length scale; U∞ as the velocity scale, and the time scale derived from these two scales. The Reynolds number appearing in the Orr–Sommerfeld equation is, thus, defined ˜ = U∞ δ∗ /ν. In the experimental in terms of the displacement thickness given as Re arrangements of [405], the flat plate boundary layer mimicking a zero pressure gradient was excited by a vibrating ribbon near the wall. This also presents another problem of eigenvalue analysis that originates from a boundary value approach, i.e. the exciter will be replaced by a localized harmonic exciter at the boundary and not in the interior. In Chapter 7, it will be demonstrated that for a semi-infinte flat plate, the pressure gradient near the leading edge will be rapidly changing, both with respect to streamwise and wall-normal distance from the leading edge. However in this chapter, it will be assumed that there is zero pressure gradient boundary layer that is excited at

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the wall. The wall-excitation, with streamwise and wall-normal velocity perturbation components, is given by u0 ( x˜, 0, t˜) = 0 ¯ x˜) eiω¯ 0 t˜ H1 (t˜) v0 ( x˜, 0, t˜) = δ(

(5.2)

¯ x˜) is the Dirac delta function in space representing the localized excitation where δ( at x˜ = 0, and H1 (t˜) is the Heaviside function, indicating finite impulsive start-up of the harmonic excitation with the circular frequency ω ¯ 0 at t˜ = 0. The excitation is a real quantity. In Figure 4.1, the equivalent parallel boundary layer at the location of the harmonic exciter is shown by the dotted line. Two possible approaches are adopted in linear receptivity theory: With the signal problem assumption, one enforces that the response of the system is exactly at the frequency of excitation (ω ¯ 0 ), which implies that the excitation is present forever and the Heaviside function is not used. This assumption has been followed and solved for the first time in [409], by using the Bromwich contour integral method (BCIM) in the α-plane. In the spatio-temporal receptivity approach, one avoids making the signal problem assumption, instead the response of the fluid dynamical system is also evaluated following the Bromwich contour in the complex circular frequency plane (ω0 ) following the BCIM introduced in [410] for wall excitation of Blasius boundary layer. The second approach is more general for fluid dynamical systems, which also holds a clue not only for study of instabilities [412], but also for solving the turbulence problem. In both these approaches, contour integrals are performed by solving the Orr–Sommerfeld equation for various combinations of complex wavenumbers and circular frequencies. The generic process is also called BCIM and is developed by the author and his team. The boundary conditions given in Eq. (5.2) have to be expressed in the spectral plane for BCIM. Although one can consult the texts [324, 339] for Fourier–Laplace transforms, here we report the same in brief for flow instability problems, with a short tutorial provided in [412]. The Heaviside function is equal to zero for its negative argument, whereas it is equal to one for its positive argument. Thus, a convenient way of expressing the Heaviside function is H1 (t˜) =

1 1 + sgn(t˜) 2 2

where the Signum function (sgn(t˜)) is equal to +1 when t˜ > 0, and equal to –1 for negative values of the argument. In the aforementioned expression for H1 (t˜), the first constant part can be transformed to the Dirac delta function in circular frequency plane, i.e., 1 ⇔ π δ(ω0 ) 2

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Similarly, the Signum function has the following Fourier transformation given by sgn (t˜) ⇔

2 iω0

Thus, the Heaviside transform is related to its Fourier transform as H1 (t˜) ⇔ π δ(ω0 ) +

1 iω0

The Fourier transform also obeys the frequency shift theorem [324] for any real frequency, ω ¯ 0 for the function ( f (t˜)) and its transform (F(ω0 )) as ¯ 0) eiω¯ 0 t˜ f (t˜) ⇔ F(ω0 − ω Thus, in Eq. (5.2), one can use the following transform pair ¯ 0) + eiω¯ 0 t˜H1 (t˜) ⇔ π δ(ω0 − ω

1 i(ω0 − ω ¯ 0)

The boundary conditions at the wall for the spatio-temporal approach of BCIM given by Eq. (5.2) translate into the wall-boundary conditions in the spectral plane as 1 + πδ(ω0 − ω ¯ 0) i(ω0 − ω ¯ 0) φ0 (0) = 0 φ(0) =

(5.3)

The general solution of the Orr–Sommerfeld equation, subject to homogeneous boundary conditions at the free stream is written in terms of φ1 and φ3 for the wall excitation problem as φ = a1 φ1 + a3 φ3

(5.4)

These retained fundamental solutions at the far stream obey the following analytical structure: φ1∞ ∼ e−α˜y and φ3∞ ∼ e−Q˜y , which decay exponentially at two widely different rates in the free stream owing to the fact that |α| 0), whereas modes below the contour are for upstream propagating modes (with Vg < 0). These criteria have been followed in [412, 418, 451, 452] to fix the Bromwich contours in the α- and ω0 -planes for receptivity analysis of a parallel boundary layer excited by wall excitation in linearized framework. In general, a Bromwich contour in α-plane is usually taken as parallel and below the αr -axis. This provides the advantage of using discrete fast Fourier transform (DFFT) along this contour for the numerical evaluation of the integral given in Eqs. (5.7) and (5.8). Some guidelines for the choice of Bromwich contour with respect to branch points, Stokes’ and anti-Stokes lines are already given with respect to Figure 4.6 of the previous chapter.

5.2.1 Receptivity to wall excitation: Frequency response for signal problem The response of the parallel mean flow, defined by the Blasius profile, to time harmonic localized disturbance source at frequency, ω ¯ 0 is explained next when one makes the signal problem assumption [146, 409]. This assumption is re-emphasized as conceptually is not correct if the wall exciter starts at a finite time requiring one to use the boundary

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αi

(a) −plane

Pα Pα Pα

Pα αr

αBr

Pα Pα

(b) ω −plane

ω 0i

ω 0, Br

ω 0r

Pω

Pω

Figure 5.1 Bromwich contour along with the sketch of the strip of convergence shown in (a) α- and (b) ω0 -plane. conditions given by Eqs. (5.7) and (5.8), instead of the boundary conditions stated in Eq. (5.10).

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In the simplest representation of the response, the perturbation stream function is defined for the signal problem as ψ( x˜, y˜ , t˜) =

1 2π

Z

¯ 0 ) ei(α x˜−ω¯ 0 t˜) dα φ(˜y, α; ω

(5.9)

Br

Two-dimensional disturbance in a two-dimensional parallel mean flow with the velocity profile U(˜y) is considered for easy explanation. The input disturbance is located at the origin of a Cartesian system for this problem. The disturbance field in general can move in both downstream and upstream directions. Hence, for the bilateral Laplace transform, the Bromwich contour has to be located in the strip of convergence of φ. Substituting Eq. (5.9) in the linearized Navier–Stokes equation, with parallel flow assumption leads to the Orr–Sommerfeld equation. For the receptivity problem, the boundary conditions are inhomogeneous. For the case of receptivity to localized wall excitation, a delta function is used at x = 0; which is also referred to as the impulse response (not a strictly correct nomenclature, as we will note in Chapter 6) of the excitation on the boundary layer [409]. Boundary conditions applied at the wall are given as y˜ = 0:

u = 0 and ψ( x˜, 0, t˜) = δ( x˜) e−iω¯ 0 t˜

(5.10)

Far from the wall (˜y → ∞): u, v → 0. The decaying free stream boundary condition excludes two fundamental modes of the Orr–Sommerfeld equation, and with the other retained modes, one defines φ as φ = a1 φ1 + a3 φ3

(5.11)

Constants a1 and a3 are fixed by the wall boundary conditions as follows: a1 φ010 + a3 φ030 = 0

(5.12)

a1 φ10 + a3 φ30 = 2π

(5.13)

The subscript 0 has been used to indicate the quantities to be evaluated at the wall. Equations (5.12) and (5.13) are solved for a1 and a3 to obtain ψ( x˜, y˜ , t˜) =

Z Br

φ1 (˜y, α, ω ¯ 0 )φ030 − φ010 φ3 (˜y, α, ω ¯ 0 ) i(α x˜−ω¯ 0 t˜) e dα 0 0 φ10 φ30 − φ10 φ30

(5.14)

The denominator in this expression evaluated at the wall, once again, is the characteristic determinant of the instability problem, with eigenvalues as the zeros of the characteristic determinant in the denominator of Eq. (5.14). These are the poles for the receptivity problem. Some typical results for the impulse response of a semi-infinite flat plate to timeharmonic excitation (with ω ¯ 0 = 0.09) at the wall are shown in Figure 5.2 for the

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˜ of the parallel mean flow defined with boundary layer displacement indicated Res thickness. One notes that following the schematic shown in Figure 4.1, the location ˜ prescribed, i.e., for Re ˜ = 400, the exciter will of the exciter is indicated in terms of Re be closest to the leading edge of the flat plate, which will be followed by the case of ˜ = 1000, and finally, the case of Re ˜ = 4200. However, the solutions are superposed Re to exhibit the similarity of the near-field and bring out the very interesting feature of receptivity problem, which is not addressed in eigenvalue analysis. 0.15

~

~ y = 1.2 _ ω0 = 0.09

0.1

Re = 400 ~ = 1200 Re ~ = 4200 Re

0.05 ψd 0 –0.05 –0.1 0

40

80

~x

120

160

200

Figure 5.2 Disturbance stream function plotted versus streamwise disturbance shown for the ˜ s at y˜ = 1.2 for indicated Re ˜ and ω indicated Re ¯ 0 = 0.09. The response field created by the time-harmonic wall exciter in the experiment can be measured where the signal strength is maximum. However, the plotted results are instead shown at the outer maximum (for which the signal strength is lower than that at the inner maximum) owing to the fact that the signal in the neighborhood of this maximum remains flat over a perceptible wall-normal distance, and thus, is easy to measure experimentally. Here the Bromwich contour is located below and parallel to the real wavenumber axis, along αi = −0.009. The solutions presented in this figure demonstrate far-field correspondence to TS modes obtained by spatial linear stability ˜ = 1000 and ω analysis. For Re ¯ 0 = 0.09, the calculated response displays a TS wave with αr = 0.255919 and αi = −0.00737. Results are shown at a height of y˜ = 1.205, which is in the range of location for the outer maximum of the eigenfunction of the TS mode. Considering stability properties of the Blasius profile, one expects stable responses for ˜ = 400 and 4200, with the latter case showing higher damping than the former, as Re seen in Figure 5.2. Details and other issues of experimental measurements are given in [409, 412], and can be consulted. The Reynolds numbers in this figure are widely separated and the asymptotic part of the solution corresponds to that obtained from linear spatial theory. In that respect, the asymptotic part of the receptivity solution has one-toone correspondence with the linear spatial theory [409, 418]. However, the most ˜ remarkable feature of Figure 5.2 is the near-field solution for these three Re’s, which appears to be identical to each other. This identical nature is discussed in the following section.

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The streamwise variation of the disturbance stream function at the inner maximum shown in Figure 5.2, displays a local component of the solution that decays rapidly, in either directions from the exciter. The local solution and details of selecting the Bromwich contours are described in [412, 451], where the near-field response by wall excitation is shown to be due to the essential singularity of the bilateral Laplace transform of the disturbance stream function. Although the experiments in [405] verified spatial instability theory, all the aspects of the experiment are not directly explained. Linear receptivity analysis is important as it explains the nearfield [412] and the early growth of the STWF, which eventually leads to turbulence [419, 425], when the signal problem assumption is given up in favor of the full spacetime dependent version of the BCIM.

5.2.2 Near-field response of localized excitation Receptivity analysis can explain why acoustic excitations could not create TS waves in the vibrating ribbon experiment [405]. There have been efforts to formulate receptivity of the boundary layer from the initial boundary value point of view [14, 80, 141]. The first receptivity calculations appeared subsequently [146, 409]. It is noted that the discrete spectrum and branch points can contribute to the asymptotic solution. Some authors [14] conjectured that the near-field is due to branch cuts from the fixed branch points of the Orr–Sommerfeld equation. However, this has never been proven from first principles. Conversely, the formal integral of φ in the α-plane by the Fourier– Laplace integral along chosen Bromwich contour provides the near- and far-field solutions, as shown in Figure 5.2. Here the near-field solution is explained with the help of fundamental theorems due to Abel and Tauber, whose utility has been highlighted for operational calculus based on the bilateral Laplace transform [537]. The original solution in the physical plane and the image or transform in the spectral plane are related by Tauber’s theorem for ψ in the neighborhood of the exciter (near x˜ = 0), which depends on φ for α → ∞, the essential singular point [9]. The solution far away from the exciter ( x˜ → ∞), is determined by the transform in the neighborhood of α = 0. This is Abel’s theorem, which explains the role of eigenvalues/poles located near the origin of the complex αplane, whose effects are noted far away from the exciter. Instability theory uses Abel’s theorem implicitly. but by considering only the right half of the α-plane. Thus, for the near-field, one is interested in the behavior of ψ in the neighborhood of the exciter ( x˜ → 0), as determined by the image φ(˜y, α) for α → ∞, according to Tauber’s theorem explained in the previous chapter. The Orr–Sommerfeld equation is, in general, not solvable analytically for any arbitrary α. However, it is possible to determine the contribution of α → ∞ for the original-image pair of Eq. (5.9). It is easy to visualize that this essential singular point α → ∞ represents the complete circular arc of infinite radius. It is relevant to discuss the role of Jordan’s lemma [9], which states that the contour integral along this semicircular arc approaches zero for many integrands, implying zero contribution from

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essential singularity for the integrals in the complex plane for the evaluation of the inverse transform, given as Z 1 0 u ( x˜, y˜ , t˜) = −iαφ0 (˜y)ei(α x˜−ω¯ 0 t˜) dα (5.15) 2π αBr Z 1 (5.16) v0 ( x˜, y˜ , t˜) = φ(˜y)ei(α x˜−ω¯ 0 t˜) dα 2π αBr It is important to emphasize that the vanishing contribution from essential singularity can occur only under special conditions for the integrand. Recall that the integral along C shown in Figure 5.3 is given by IC ( x˜, y˜ ) =

Z

φ(α, y˜ ) e

iα x˜

dα =

Z

N1 (˜y, α) iα x˜ e dα D1 (˜y, α)

(5.17)

αi C2 P2 P1

P3

C1

C3

ε

ε

θ αr

Figure 5.3 Bromwich contour used to evaluate the integral in Eq. (5.27) for the signal problem. This integrand would vanish iff the degree of the denominator (D1 ) of Eq. (5.17) is at least two orders higher than the degree of the numerator (N1 ), i.e., |φ(˜y, α)|
1 , one gets φi (Y) = 0, which is a trivial solution, failing to satisfy the wall boundary condition.

(iii) The distinguished limit δ2 = 1 simplifies Eq. (5.22) to 2 00 4 φiv i − 2β φi + β φi = 0

The solution of which is given by φi (Y) = A eβY + BY eβY + C e−βY + DY e−βY As β is a complex constant, for βr > 0, the inner solution satisfying Eq. (5.20) is given by φi (Y) = (1 + βY) e−βY

(5.23)

Similarly, the inner solution for βr < 0 is φi (Y) = (1 − βY) eβY

(5.24)

The other two distinguished limits, δ22 = 13 and δ22 = 14 produce only the trivial solution. Therefore, the only possible distinguished limit is δ2 = 1 , which implies 1 . In terms that the inner layer of the Orr–Sommerfeld equation is of thickness δ2 = |α| of physical variables, the asymptotic value of φ is then given by For αr > 0: φ = (1 + α˜y) e−α˜y

(5.25)

and for αr < 0: φ = (1 − α˜y) eα˜y

(5.26)

This is the analytical solution of the Orr–Sommerfeld equation, for the essential singularity, α → ∞. It is not to be confused with the inviscid mode, as the inner layer is next to the wall, evaluated for the essential singularity, and originate from the diffusion operator of the Navier–Stokes equation. The contribution of the essential singularity to ψ coming from the semicircular contour, is obtained by considering three segments of the contour shown in Figure 5.3, with φ being discontinuous across the αi -axis. The imaginary axis demarcates the spectral plane in terms of the required

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subdominance of the fundamental solutions, the consequence of which is observed in Eqs. (5.25) and (5.26). To evaluate the contour integral, the Bromwich contour is chosen along the αr -axis, with φ being analytic, and circumventing the branch point at the origin. Thus, 1 ψ( x˜, y˜ , t˜) = 2π

Z

where along

C1 :

along

φ(˜y, α) = (1 − α˜y) eα˜y

C2 :

and at P2 :

φ(˜y, α) ei(α x˜−ω¯ 0 t˜) dα

φ(˜y, α) = (1 + α˜y) e−α˜y

φ(˜y, iρ1 ) =

1 iρ1 y˜ (e + e−iρ1 y˜ ) 2

(5.27)

(5.28)

(5.29)

(5.30)

The last relation is due to discontinuity of φ across P2 . For the contributions from the neighborhood of point P2 , we consider a sector of the contour around P2 terminating at P1 and P3 , as shown in Figure 5.3, defined by the small angle . One fixes the value of β at P1 as π β1 = ei( 2 −) = i + , for small values of . The value of β corresponding to P3 is π β2 = ei( 2 −) = i − , for small values of . The contribution from C1 for the essential singularity is obtained as Z 1 −iω¯ 0 t˜ β1 (1 + ρ1 β˜y) eiβρ1 z ρ1 dβ I1 = e 2π 0 i˜y (1 + ρ1 y˜ + i˜zy )  e−iω¯ 0 t˜  iρ1 β1 z (1 + ρ1 β1 z + z ) iρ1 z = e +e 2π iz z

The contribution from contour C2 , in the limit of  → 0, is I2 =

e−ρ1 x˜−iω¯ 0 t˜ cos ρ1 y˜ 2π

Finally, the contribution from contour C3 is I3 =

e−iω¯ 0 t˜  −iρ1 z¯ (1 + ρ1 y˜ − i y˜ /¯z) (1 + ρ1 β2 y˜ − i y˜ /¯z)  e − eiρ1 β2 z¯ 2π i¯z i¯z

where z = x˜ + i˜y and z¯ = x˜ − i˜y is its complex conjugate.

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Collecting various contributions, one obtains the perturbation stream function from the semi-circular contour of radius ρ1 as  ieiρ1 z  i˜y  ie−iρ1 z¯  i˜y  ψ( x˜, y˜ , ρ1 , t˜) = e−ρ1 x˜ cos ρ1 y˜ + 1 + ρ1 y˜ + − 1 + ρ1 y˜ − z z z¯ z¯   −ρ1 z¯−i¯z  ρ1 z+iz  −iω ¯ 0 t˜ i˜y i˜y ie ie e 1 + + iρ1 y˜ + y˜ + 1 − − iρ1 y˜ + y˜ − z z z¯ z¯ 2π

(5.31)

The correctness of this is checked from the solution at y˜ = 0, where the delta function wall excitation is applied. Here ψ( x˜, 0, ρ1 , t˜) simplifies to ψ( x˜, 0, ρ1 , t˜) =

 sin x˜  e−iω¯ 0 t˜  −ρ1 x˜ 2 sin ρ1 x˜ e − + 2e−ρ1 x˜ 2π x˜ x˜

(5.32)

In the limit, ρ1 → ∞, the first and third terms do not contribute, while the second term is the Dirichlet function, which is an approximation of the delta function, δ( x˜), as given in [537], as one of the various approximate representations of the Dirac delta function. Thus, one clearly recovers the applied boundary condition at y˜ = 0, namely the delta function; this is a remarkable result which shows that the delta function is totally supported by the essential singularity of the kernel of the contour integral. There are many ramifications of this analysis for near-field solution: A) It has been shown that the Blasius boundary layer supports only very few eigenmodes [409, 412, 418]. For example, for ω ¯ 0 = 0.1 with the exciter located at ˜ = 1000, there are only three eigenvalues obtained by compound matrix method Re (CMM). These few eigenvalues preclude the possibility of expanding any arbitrary disturbances, using the eigenfunctions as the basis functions. Moreover, it is apparent that these eigenfunctions are not necessarily orthogonal for all equilibrium flows. For example, the Taylor–Couette flow and the Rayleigh–Benard convection problem produces orthogonal eigenfunctions, while the Blasius boundary layer does not. However, the present analysis shows that the essential singularity itself is adequate to support a delta function. As any arbitrary function can be shown as a convolution of shifted delta functions, the question of completeness of basis functions in terms of eigenvectors is no longer relevant. It has been shown for internal flows, as in plane channel flow, which has denumerable eigenmodes [284], so that any arbitrary applied disturbance can be expressed in terms of this basis set. There is the added confusion, with the use of the spectral collocation method to evaluate eigenvalues that produces as many eigenmodes as the number of points taken to solve the Orr– Sommerfeld equation [397]. In the limit of continuum, the number of evaluated eigenfunctions become unbounded and one is tempted to believe that one can use these eigenfunctions to represent any arbitrary response field. The fact is that there are many spurious modes obtained by the spectral collocation method and weeding those out is non-trivial. The present analysis indicates that any arbitrary disturbances

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can be expressed in terms of a few discrete eigenvalues and the essential singularity. In any flow, in addition to these singularities there are contributions from continuous spectra apart from the nonmodal part of the response field and branch points. Readers can relate this discussion with the material given in Appendix A, at the end of Chapter 1. B) Despite the differences in the values of αr and αi for the cases in Figure 5.2, one notices the remarkable similarity of the near-field solution, specially in the upstream part of the solution. Minor differences on the downstream side of the near-field are due to the effects of the asymptotic part of the solution, namely the TS wave having specific αr and αi . This is consistent with the observed properties of φ, for the limit of α → ∞. Specifically, the near-field solutions obtained in Eqs. (5.31) and (5.32) are ˜ as shown in Figure 5.2. The local solution originates in the inner independent of Re, layer whose governing differential equation is given by φiv − 2β2 φ00 + β4 φ = 0

(5.33)

The general properties of the near-field solution, in conjunction with the kinematic equation ∇2 ψ = −ω, when substituted in the vorticity transport equation yield 1 4 D 2 ∇ψ ∇ ψ= ˜ Dt Re

(5.34)

It is readily apparent that Eq. (5.33) is the right-hand side of Eq. (5.34), showing that ˜ → 0) the near-field solution is the corresponding Stokes problem (in the limit of Re given as ∇4 ψ = 0

(5.35)

The near-field of the exciter represents a highly viscous flow and for this reason, the near-field solution does not penetrate very far upstream and downstream of the exciter. C) Presented results bring forth an important aspect of real fluid flows. Despite the mathematical requirement of Jordan’s lemma (that we must have |φ(˜y, α)| → 0 for α → ∞), in real flows there always exists a cut-off wavenumber, dictated by the amount of energy supplied to the fluid dynamic system, as small waves have large strain-rates and require larger energy to support such length scales. Thus, an infinite range of α is always precluded for an excitation problem driven by a finite source of energy. At the cut-off wavenumber, the kinetic energy of the wave would be converted to heat—a concept that has been traditionally employed to fix the cut-off wavenumber in Kolmogorov’s theory of energy cascade for homogeneous isotropic turbulence. The present analysis shows that the motion in the very small scale of any flow is governed by Stokes equation which is highly dissipative.

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5.2.3 Receptivity to free stream excitation One of the most restricting aspects of the instability theory is the way it is posed with homogeneous boundary conditions. However, for most of the eigenvalue problems solved using the Orr–Sommerfeld equation, one implicitly looks for the receptivity to wall excitation, as the disturbance field is stated to decay in the far field. This rules out the study of instability due to free stream excitation by the usual linear stability analysis. The main aspect of instability by free stream disturbances has not been investigated as successfully as has been the case of wall excitation, supported by early experiments [405]. The authors did not detect TS waves when they excited the flow with acoustic waves from the free stream, despite transition being observed. Absence of TS waves is due to the following reasons: (i) an acoustic wave is three-dimensional, while the eigenvalue analysis was for two-dimensional fields at that time; (ii) The excitation is unlike that formulated in spatial classical theory with monochromatic waves. Convected vortices do not excite a fixed time scale, instead the excitation is by many time scales. Later on, in experiments with a vibrating ribbon in the free stream, Dietz [109] reported measuring eigenfunction that is usually associated with TS waves; (iii) Receptivity to weak convected free stream disturbance has been shown to have vanishingly small coupling in an early analysis [374]; (iv) Strong coupling of convecting free stream excitation on the Blasius boundary layer has been experimentally investigated [225, 226] by jet-induced free stream turbulence and a train of convecting periodic pulses which provided indirect evidence of wave-fronts; (v) Morkovin [297] proposed that the response to free stream excitation occurs in two stages. In the first stage, the external perturbations are internalized as unsteady fluctuations giving rise to a seed of TS wave packet. In the second stage, these internalized excitations grow and cause transition. However, the experimental results [260] had clearly demonstrated earlier that this coupling is of order one. Another experiment [2] demonstrated that an introduction of a surface roughness element increases the receptivity linearly with forcing amplitude and roughness height. Most of the theoretical developments were based on the triple deck theory [157, 378] for free stream excitation. The receptivity to free stream excitation [158], theoretically talks about scale adjustment mechanism to (a) rapid streamwise variations in the mean boundary-layer flow and (b) sudden changes in surface boundary conditions. As a result, the region near the leading edge of the plate needs correct asymptotic approximation to the Navier–Stokes equation to be unsteady boundary-region equations, which are the Navier–Stokes equation with streamwise derivatives neglected for the viscous and pressure gradient terms. In modern day research terminology, such modifications of the equilibrium flows are noted as mean flow distortion effects on nonmodal growth via eigenvalue analysis as given in [42, 43]. However, neglecting streamwise pressure gradient near the leading edge of such flows, is a serious compromise in specifying the equilibrium flow and this is highlighted in Chapter 7.

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Kerschen [228] used the asymptotic method to calculate vortical receptivity, and showed it to vary with the convection speed of vortices, which has been explained later in [429] from normal mode analysis, and also from solutions of the Navier– Stokes equation. Kendall [225] performed experiments in which a circular cylinder was rotated in a circular trajectory above a flat plate shear layer, to create a convecting train of periodic disturbance pulses. The speed of convection of these vortices was controlled in the experiments, and it was demonstrated that the underlying shear layer was strongly receptive to imposed disturbances in a narrow range of convecting speeds around c = 0.3U∞ . When Lim et al. [267] performed experiments, with a single captive vortex made to convect with a fixed speed at a constant height over the zero pressure gradient boundary layer, a similar preferential range of convection speeds was reported for the receptivity to aperiodic vortex in the free stream. This is discussed in Chapter 9 to explain free stream excitation in greater detail. Thus, receptivity to free stream excitation demonstrates simultaneous action of modal and nonmodal growth of disturbance field, and this is highlighted. Liu and Rodi [270] repeated Kendall’s experiment, but with the periodic disturbance directed towards the boundary layer with a large wall-normal velocity component. This experiment was designed to mimic the physical events in the flow inside turbomachinery. Experiments such as these [225, 270], exhibited very strong receptivity, whereas contemporary linear modal theories based on different models displayed lower receptivity. Some of these discrepancies have been explained [267] as due to constraints of the theoretical models on the speed of free stream disturbances. It is emphasized that in turbomachinery, in flows over helicopter rotor blades, or that is due to free stream turbulence, the vortices or small scale structures travel with local speed, dictated by various interactions among vorticity fields and vortices. Thus, such structures travel at speeds, which can be much lower than the free stream speed. For example, experimental data and their correlation [396] reveals that the vortices in the far wake of a single bluff body convect at 14% of the free stream speed. Sengupta et al. [225] showed this receptivity link by producing results simultaneously by solving the Orr–Sommerfeld equation and two-dimensional DNS of the Navier–Stokes equation. In recent times, this aspect of transition is studied under the subject finding optimal perturbation [229].

5.3 Receptivity to Free Stream Excitation: Upstream Propagating Modes Here a unit process of free stream turbulence is investigated by considering a convecting vortex, i.e., the receptivity to free stream excitation, with governing parameters being the vortex strength, height over the underlying boundary layer, convection speed of the vortex, among many others. To understand the ramification of linear theories governed by the Orr–Sommerfeld equation, one can consider the case of simultaneous wall and free stream excitation, as shown in Figure 5.4. It has

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already been highlighted that instability theories posed as eigenvalue problems do not distinguish between wall and free stream excitations. However, in the choice of retained fundamental solutions in classical modal analysis that decay with height, this implicitly uses information only for the wall modes (that propagate downstream as one focuses on the right half of α-plane). t

=0

Γ c

U∞

~

y

Y

Boundary Layer Edge δ (x)

x0

x

Figure 5.4 Flat plate boundary layer excited simultaneously at the wall and the free stream. The flow field over the flat plate is excited simultaneously at the wall (˜y = 0) and ˜ as shown in Figure 5.4, with Y˜ significantly larger than δ∗ . at the free stream (˜y = Y), At the wall, a time-harmonic blowing-suction device is placed at x˜ = x0 , defined in a coordinate system fixed at the leading edge of the plate. The circular frequency of excitation of the wall device is ω ¯ 0 such that the wall excitation is set up as u = 0,

v = vw δ( x˜ − x0 ) e−iω¯ 0 t˜

(5.36)

The line vortex (of strength, Γ) convects in the free stream with speed c, at a height Y˜ over the boundary layer. Let the instantaneous location of this irrotational vortex be given by x¯ from the origin of the boundary layer. Imposed inviscid stream function at any field point ( x˜, y˜ ) created by this free stream vortex is ψ∞ =

˜ 2 Γ ( x˜ − x¯)2 + (˜y + Y) ln ˜ 2 4π ( x˜ − x¯)2 + (˜y − Y)

(5.37)

where the vortex is located at x¯ = xv0 − ct˜, with xv0 as its initial location. The term in the denominator of Eq. (5.37) is due to the image system created by the free stream vortex. In response to this input, one can define the time-dependent perturbation stream function by ψ( x˜, y˜ , t˜) =

1 (2π)2

Z Z

φ(˜y, α, ω0 ) ei(α x˜−ω0 t˜) dα dω0

(5.38)

Br

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The Laplace–Fourier transform φ, in terms of all the four fundamental solutions is φ(α, y˜ , ω0 ) = a1 φ1 + a2 φ2 + a3 φ3 + a4 φ4

(5.39)

Here all the four fundamental solutions (modes) have to be retained for this simultaneous excitation case. To satisfy the no-slip condition, one must have the following a1 φ010 + a2 φ020 + a3 φ030 + a4 φ040 = 0

(5.40)

with the prime indicating derivative with respect to y˜ and the subscript 0 indicating the condition at the wall. For the wall-normal velocity boundary condition, the ‘timeshift’ theorem of the Laplace–Fourier transform [412] is used as vw δ( x˜ − x0 ) e−iω¯ 0 t =

1 (2π)2

Z Z

¯ 0 ) dα dω0 vw ei[α( x˜−x0 )−ω0 t˜] δ(ω0 − ω

(5.41)

Br

Therefore, the aforementioned condition can be stated in the spectral plane as a1 φ10 + a2 φ20 + a3 φ30 + a4 φ40 = +

vw −iαx0 δ(ω0 − ω ¯ 0) e iα

(5.42)

Similarly, the free stream condition of Eq. (5.37) can be written notationally as a1 φ01∞ + a2 φ02∞ + a3 φ03∞ + a4 φ04∞ = U F (α, ω0 )

(5.43)

a1 φ1∞ + a2 φ2∞ + a3 φ3∞ + a4 φ4∞ = VF (α, ω0 )

(5.44)

The free stream excitation fixes the appropriate U F (α, ω0 ) and VF (α, ω0 ) for the imposed streamwise and wall-normal velocity components, with the additional subscript (∞) indicating conditions at the free stream. The constants a1 to a4 are solved from Eqs. (5.39) and (5.42) to (5.44). All these boundary conditions can be collated as the linear algebraic equation [Φ]{ai } = {bci }  where, {bci } = 0

vw 2πiα

(5.45) e−iαx0 δ(ω0 − ω ¯ 0 ) U F (α, ω0 ) VF (α, ω0 )

T represents the forcing by

the boundary conditions. Thus, one can obtain the constants ai from {ai } = [Φ]−1 {bci }

(5.46)

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where   φ010   φ10 [Φ] =   −αe−αY˜  ˜ e−αY

φ020 φ20 ˜ αeαY αY˜ e

φ030 φ30 ˜ −Qe−QY −QY˜ e

φ040 φ40 ˜ QeQY QY˜ e

     

It is easy to directly obtain the eigenvalues from the characteristic determinant of the corresponding stability problem given by Det[Φ] = 0, which can locate the poles of the ˜ two sets of transfer function for the receptivity problem. One notes that for large Y, terms in the third and fourth rows are sub-dominant in the matrix, [Φ]. It is also noteworthy that the aforementioned linear receptivity analysis helps identify the upstream propagating modes, as was reported for the first time in [446] for the Blasius boundary layer. This was found by locating eigenvalues in both the halves of the complex α-plane using CMM. Researchers were not aware of eigenvalues in the left-half plane of α [14, 400], with the authors sketching Bromwich contours for signal problem, and marking unknown eigenvalues in the left-half of α-plane with question marks. It is easy to note that any mode present in the left-half plane would represent negative phase speed for the signal problem. To locate these upstream propagating modes, consider the case for which real (α) < 0 and real (Q) > 0. Now as Y˜ → ∞, φ2 (∼ eα˜y ) and φ3 (∼ e−Q˜y ) are the modes that decay with height in the free stream, while free stream boundary conditions are supported by φ1 and φ4 . Therefore, Det [Φ] = z1r + iz1i = 0 implies that the following expression provides the dispersion relation, which upon expansion of the determinant gives ˜

−[α + Q] e(Q−α)Y [φ020 φ30 − φ20 φ030 ] = 0

(5.47)

Thus, the eigenmodes in the left-half of the α-plane are once again obtained by the wall modes, φ2 and φ3 in Eq. (5.47). For general excitation including those from the free stream, one requires retention of all the four fundamental modes of the Orr–Sommerfeld equation. The decaying or the wall modes determine the same dispersion relation for both the wall and free stream excitation cases. In Figure 4.2, the spatial eigen-spectrum for the Blasius boundary layer is shown for the parameter ˜ = 1200 and ω combinations: Re ¯ 0 = 0.1. The figure is constructed by plotting the zero contour-lines of the real (dashed lines) and imaginary parts (solid lines) of the dispersion relation (z1r and z1i , respectively) following the grid search method described in Chapter 4, using CMM. Results of the grid search method in Figure 4.2 show only few eigenvalues, indicated as P1, P2, P3 and P4. Out of the four modes shown in Figure 4.2, P4 is the mode whose phase shows upstream propagation. These ˜ = 1000 and 1196 results can be compared to that given by Sengupta [412] for Re (corresponding to an experiment [145]). In Figure 4.3, the grid search method results ˜ = 1500. For both the Re ˜ cases show discrete modes for the same ω ¯ 0 , for higher Re

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shown in Figures 4.2 and 4.3, the mode P1 is the spatially growing mode, and is the TS mode. The group velocity determines whether any mode travels upstream or not, as was investigated in [446], while also detecting upstream propagating modes for the Blasius profile successfully. The upstream propagating modes show the property that the amplitude increases with y˜ , i.e., this mode will be able to support any excitation at the free stream. Henceforth, we will distinguish between these two classes of disturbances as the wall and the free stream modes. The free stream mode can support disturbances due to free stream turbulence or convected vortices in the free stream. Detailed wave properties of the detected four modes in Figures 4.2 and 4.3, are ˜ = 1200 and 1500 cases. plotted as a function of ω ¯ 0 in Figures 4.4 and 4.5 for these two Re These upstream propagating waves are strongly decaying (as α for P4 is in the third quadrant with αi having a large magnitude) and the group velocity clearly indicates that P4 propagates upstream (implying x˜ is negative for this mode). These two facts together imply strongly stable upstream propagating modes as obtained by the linear spatial instability here and shown first time in [446]. However, for mean flows with adverse pressure gradient, upstream propagating modes can become unstable. ˜ = 1200 and 1500 cases, one notes some In Figures 4.4 and 4.5 for these two Re features those require highlighting. As ω ¯ 0 is decreased, the three downstream propagating modes shown by solid lines disappear one after the other, when the phase speed of that mode becomes equal to one. Values of ω ¯ 0 at which these three downstream propagating modes disappear are listed in Table 4.1. If the Blasius boundary layer is excited below the lowest of the three values of ω ¯ 0 , then CMM detects an eigenvalue whose properties are shown in Figures 4.4 and ˜ = 1200, this 4.5 by dotted lines, and this is the mode P4. From Table 4.1 for Re ˜ = 1500, this is at ω threshold value is given by ω ¯ 0 = 0.00263 and for Re ¯ 0 = 0.00185; any excitation below this critical ω ¯ 0 shows the presence of only the single upstream propagating mode, whose wave properties are noted by the dotted line in Figures 4.4 and 4.5. The fact that this is indeed the upstream propagating mode is readily apparent by noting that the value of Vg is negative in the bottom right frames of Figures 4.4 and 4.5. The interesting feature of the upstream propagating mode is that both αr and αi are negative for the Blasius boundary layer, with the mode P4 in the left half plane, indicating the propagation of this mode to be in the upstream direction. For such propagation direction, negative αi indicates it to be damped. For this mode, the corresponding variation of the phase speed and group velocity with ω ¯ 0 are calculated; it travels upstream, both in terms of ability to transmit energy packet and phase variation. One also notices that the upstream propagating mode, upon increasing its frequency above the critical value in Table 4.1, continue to stay along the dotted line. If investigated for any value of ω ¯ 0 greater than the critical value, the grid search method will show all the four modes. Starting from a value of circular frequency below this critical value, and increasing it above the critical value, using the Newton– Raphson search method for the exercise does not convert the upstream mode to the

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˜ the Blasius boundary layer excited at postdownstream mode, P1. For a given Re, critical frequency will show the presence of both types of modes. The upstream propagating mode shows the property that the amplitude increases with y˜ with this mode able to support any excitation at the free stream. Hence, one distinguishes these two classes as the wall and the free stream modes. There are other unique properties of ˜ which are shown in Figure 3.27 of [412], the upstream propagating modes with Re, with αr , αi , c ph , and Vg plotted as a function of ω ¯ 0 . The results established that the upstream propagating mode is damped for the Blasius boundary layer and would not be responsible for linear instability.

5.3.1 Low frequency free stream excitation: Klebanoff mode It is noted that the Blasius boundary layer excited by two-dimensional moderate frequency at the wall creates TS wave-packets. As mentioned earlier, for a very low frequency wall excitation TS wave-packets are absent [145]. This experiment re-investigated the flow that was reported earlier by Taylor [514], who vibrated a diaphragm on a zero pressure gradient boundary layer to look for TS waves without success. A similar low frequency experiment was performed by Klebanoff [236], who called the boundary layer response the breathing mode of motion. For very low frequency wall excitations, corresponding two-dimensional linear instability studies do not reveal TS waves. Instead in the experiments for low frequency excitation, one notices the whole boundary layer to execute a heaving motion, which is now termed as the Klebanoff mode of motion. In a similar low frequency excitation of a diaphragm vibrated at 2 Hz [145], the authors could not explain the Klebanoff mode of motion, and noted that a proper mathematical account of low frequency disturbances is required. This has been resolved since then and explanation of the experiment in [448] provides the main observation that the response field is three-dimensional for this two-dimensional zero pressure gradient mean flow. However, the response field propagates predominantly in the streamwise direction, as the spanwise component of group velocity is almost zero. To explain the features of the breathing mode better, it is necessary to understand the setup, dimensions and parameters of the experiments in [145], that repeated the pioneering experiment of Taylor. In the low noise wind tunnel, a flat plate was mounted in the test section, which is 3.5m long and having a cross section of (0.91 m ×0.91 m) in dimension. A circular vibrating bump of 20 mm diameter, which is located 400 mm from the leading edge of the plate, where the undisturbed boundary layer has a physical thickness of δ∗ = 0.99 mm, created a three-dimensional disturbance field. Based on the boundary layer data and the free stream speed of 18.10 m/s, the Reynolds number is 1196 at the location of the bump. The nondimensional circular frequency of bump vibration was ω ¯ 0 = 6.248 × 10−4 . The span of the tunnel ∗ test section was 920 times δ , which fixes the maximum spanwise wavelength λz as twice the spanwise width. This provides the fundamental spanwise wavenumber of

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β0 = 3.41777 × 10−3 . It has been reported in [145] that most of the disturbance energy is carried by the first ten modes corresponding to βn = 0.7854. To understand the Klebanoff mode, the authors in [448] investigated the disturbance field for very low ω ¯ 0 excitation. Both Klebanoff and upstream propagating modes were found, while studying receptivity of the Blasius boundary layer by varying ω ¯ 0 from moderate to very low values. The phase speed is obtained as c ph = ω ¯ 0 /αr , while the group velocity is obtained from its definition, Vg = ddαω¯ r0 . Blasius boundary layer excited by very low frequency source at the wall behaves qualitatively differently in experiments [145, 514], with the boundary layer perturbed by the shallow oscillating wall-bump. The oscillation frequency was only 2 Hz [145] and inside the shear layer no waves were seen; instead the whole boundary layer executed a heaving motion. The experiment [145] is for the mean flow represented by the Blasius profile and for the very low frequency excitation, two-dimensional spatial instability analysis does not reveal any eigen solutions that decay with height. The response field is found to be three-dimensional, but the disturbances propagate in the streamwise direction. As the disturbance field has very large wavelength, the experimental setup cannot even accommodate a single wavelength. A small part of the wavelength instead indicates a heaving motion in the boundary layer. This is true for moderate length test sections, as in the experiments in [145, 514]. Measurement stations in [145] were located at 70δ∗ and 105δ∗ aft of the shallow bump. The explanation of this experiment has been provided in [448], recounted next. As the flow field excited by very low frequency excitation does not support twodimensional disturbances, it is natural to investigate whether the boundary layer would support three-dimensional disturbances. For ease of analysis, the signal problem assumption is adopted. In performing such an analysis, the wall-normal component of the disturbance velocity is expressed as v ( x˜, y˜ , z˜, t˜) = 0

1 4π2

Z Z

φ(α, y˜ , β, ω ¯ 0 ) ei(α x˜+β˜z−ω¯ 0 t˜) dα dβ

(5.48)

Br

The associated signal problem with the excitation frequency ω ¯ 0 , is as indicated above for the response field, with inverse transform to be performed in the streamwise and spanwise wavenumber planes. For experiments conducted in closed tunnels, β will have a lower cut-off (β0 ), fixed by the spanwise extent of the tunnel, i.e., β0 = 2π/λz , where λz is twice the tunnel width. It is then possible to convert the bilateral Fourier– Laplace transform in the spanwise direction to the Fourier series in the spanwise direction, considering the flow to be spanwise periodic. Therefore, one rewrites the disturbance field as ∞

1 X v ( x˜, y˜ , z˜, t˜) = 2π n=1

Z

φ(α, y˜ , β0 , ω ¯ 0 ) ei(α x˜+nβ0 z˜−ω¯ 0 t˜) dα

0

(5.49)

Br

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The aforementioned representation can be used in the three-dimensional linearized Navier–Stokes equation and by invoking parallel flow approximation, one obtains the following Orr–Sommerfeld equation for φ as φiv − 2(α2 + n2 β20 ) φ00 + (α2 + n2 β20 )2 φ ˜ = iRe{(αU + nβ0 W − ω ¯ 0 )[φ00 − (α2 + n2 β20 ) φ] − [αU 00 + nβ0 W 00 ] φ}

(5.50)

In this Orr–Sommerfeld equation, Eq. (5.50), the parallel mean flow components are given by U(˜y) and W(˜y) in the streamwise and spanwise directions. In this equation, ˜ is based on the displacement primes indicate differentiation with respect to y˜ . The Re thickness of the boundary layer at the central location of the bump, and the free stream speed. The experimental conditions in [145] correspond to when the two-dimensional modes disappear, as the value of ω ¯ 0 is lower than the critical value (ω ¯ 0 = 6.248 × 10−4 ) for Re = 1196 as reported in [448]. The analysis based on Eq. (5.50) have been used to check if three-dimensional modes are indeed present, when two-dimensional modes were noted to be absent in this reference. The spatial eigenvalues can be again located by the grid-search method [284, 412], as explained in Chapter 4. Once the eigenvalues are located, streamwise and spanwise components of group velocity can be obtained numerically from   ~ g = ∂ω0 , ∂ω0 V ∂αr ∂βr

(5.51)

It has been noted in [412, 448] that with all the modes as stable, the least stable modes will help describe the response field (these are the fourth and fifth modes in the experiment of [145]). One also notes that the least stable modes have streamwise wavelengths that are thousands times greater than δ∗ . To detect such large wavelength disturbances experimentally, test section of any tunnel used in the experiment has to be long enough to accommodate at least a few wavelengths. In the experiment [145], measurements were made only up to 500δ∗ downstream of the exciter, which gave an appearance of the full boundary layer to heave. The direction of propagation of the fourth and fifth modes is predominantly in the x˜-direction, since the z˜-component of the group velocity is vanishingly small as compared to the x˜-component. For a general flow field, not only is background disturbance field all-pervasive, its spectral content can also be wide. For wide-band input disturbances, the low frequency components will make the shear layer execute heaving motion, which is called the Klebanoff mode of motion [145, 448]. In many experiments involving free stream turbulence, researchers have reported this heaving motion [236]. This has also unfortunately led researchers to identify response to free stream turbulence with the Klebanoff mode, whereas it is just one component of the response field. Essentially, the Klebanoff mode here has been identified as the response due to very low frequency

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wall excitation. How does it relate to the free stream excitation cases? The answer to this issue lies in explaining how the fundamental modes, classified as wall and free stream modes, are related and how the causation of one triggers the other.

5.3.2 Coupling between wall and free stream modes The solution of the Orr–Sommerfeld equation with four fundamental modes is given in Eq. (4.27) and is rewritten as φ = a1 φ1 + a2 φ2 + a3 φ3 + a4 φ4

(5.52)

These modes in the free stream have asymptotic values given by: φ1∞ ∼ e−α˜y , φ2∞ ∼ eα˜y , φ3∞ ∼ e−Q˜y , φ4∞ ∼ eQ˜y , where |Q| = [α2 + iαRe(1 − c)]1/2 . The first and the third modes decay, whereas p the second and the fourth modes increase with y˜ whenever the real part of α and Q2 are positive. The decaying modes are for wall excitation—the wall mode, and clubbed together as ΦI = a1 φ1 + a3 φ3

(5.53)

Similarly, the free stream mode is defined as ΦII = a2 φ2 + a4 φ4

(5.54)

The mode ΦII grows with y˜ to match the applied disturbance at the free stream for a receptivity problem. One can fix the values of a2 and a4 by matching (u∞ , v∞ ) or (φ∞ , φ0∞ ). The far field boundary for the problem is considered so far out where φ1∞ and φ3∞ are vanishingly small and the excitation is supported by the dominant free stream modes, φ2 and φ4 . Having satisfied the free stream boundary condition, one must satisfy the homogeneous boundary conditions at the wall. Two qualitatively different possibilities exist, depending on the convection speed of free stream disturbance. If it moves at the free stream speed, then we term it as the pure convection problem, described here. Pure Convection Problem: If the convected disturbances move at a height Y˜ with U∞ , then the fundamental solutions coalesce, i.e., φ1∞ = φ3∞ and φ2∞ = φ4∞ . To satisfy the free stream boundary conditions, a2 and a4 are fixed from ˜ αY ΦII∞ = φ∞ = a2 eαY + a4 Ye

(5.55)

˜ ˜ eαY˜ Φ0II∞ = φ0∞ = αa2 eαY + a4 (1 + αY)

(5.56)

˜

˜

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One uses the conditions, φI∞ = φ0I∞ ≡ 0, for Y˜ → ∞, to obtain the solution as ˜

˜ ∞ − Yφ0∞ ] e−αY a2 = [(1 + αY)φ a4 = [φ0∞ − αφ∞ ] e−αY To satisfy the homogeneous wall boundary conditions, one must have, φ(˜y = 0) = 0 = ΦI0 + ΦII0 This fixes the wall boundary condition for the wall mode as φPC given by ˜ ˜ − Yφ ˜ 0∞ }φ¯ 20 φPC = ΦI0 = −ΦII0 = − e−αY [{φ∞ (1 + αY) + {φ0∞ − αφ∞ }φ¯ 40 ]

(5.57)

The fundamental solutions for pure convection have been written with an overbar. Similarly, one can write an expression for Φ0I0 , providing two equations to solve for a1 and a3 from a1 φ¯ 10 + a3 φ¯ 30 = −(a2 φ¯ 20 + a4 φ¯ 40 ) a1 φ¯0 10 + a3 φ¯0 30 = −(a2 φ¯0 20 + a4 φ¯0 40 ) Solution of which are given by, a1 = (r1 φ¯0 10 − r2 φ¯ 10 )/D a3 = (r2 φ¯ 30 − r1 φ¯0 30 )/D where D = (φ¯0 10 φ¯ 30 − φ¯ 10 φ¯0 30 ) r1 = e−αY [φ¯ 20 {φ0∞ Y − (1 + αY)φ∞ } + φ¯ 40 (αφ∞ − φ0∞ )] r2 = e−αY [φ¯0 20 {φ0∞ Y − (1 + αY)φ∞ } + φ¯0 40 (αφ∞ − φ0∞ )] The non-zero values of a1 and a3 obtained in terms of a2 and a4 provide the coupling for this case.

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Bypass Problem: If the convected vortices do not move at the free stream speed, then the modes are distinct, unlike the previous case. One can simply calculate the wall boundary condition for the wall mode, called as φBP , given by φBP = ΦI0 =

1 ˜ ˜ [e−αY (φ0∞ − pφ∞ )φ20 − e−pY (φ0∞ − αφ∞ )φ40 ] p−α

(5.58)

p where p = Real ( Q2 ). Thus, Eqs. (5.57) and (5.58) represent equivalent wall mode amplitudes at the wall, for free stream excitation in pure convection and bypass problems, where the real part of α and p are positive. For instability problems, usually |p| >> |α|. Moreover, note for the case of free stream vortical disturbances, φ∞ > φ0∞ , and then  φ¯ [φ (1 + αY) ˜ − Yφ ˜ 0∞ ] + φ¯ 40 [φ0∞ − αφ∞ ]  φPC 20 ∞ = −(p − α) 0 φBP (φ∞ − pφ∞ )φ20

(5.59)

This can be further simplified to φ0 φPC  φ¯ 20  = [1 + αY˜ − Y˜ ∞ ] φBP φ20 φ∞

(5.60)

Hence, for the same level of free stream excitation, the aforementioned ratio indicates that the pure convection of vortices is a far weaker mechanism to create disturbances inside the shear layer as compared to the case when c = ω ¯ 0 /α , 1. The aforementioned analysis is for the spatial stability problem associated for free stream excitation with an imposed time scale. The physical difference between the pure convection and bypass problem lies simply with the different convection speed. According to Kelvin’s vortex theorem, the vortices move with local speed. This local speed will be equal to the free stream speed (for pure convection), if the convecting vortex is moving in an isolated manner, without being affected by any vorticity distribution or vortices. Such an instability problem was solved in [374] for free stream convecting vortex, and no receptivity was observed. By contrast, even when a single vortex convects in the free stream above a flat plate, there is going to be a Biot–Savart interaction between the vortex and vorticity distribution of the flat plate boundary layer. As a result the convecting vortex in the free stream will travel at a speed that is different from the free stream speed. This is the physical essence of the bypass problem and is more physical in nature in deciding the receptivity by free stream excitation.

5.3.3 Exciting TS waves by pulsating free stream vortex: Linear receptivity It has been experimentally noted in [109] that the TS mode can be excited in a boundary layer, if a disturbance is imposed from the free stream with a fixed time scale. This idea motivated researchers [478] to investigate the receptivity to stationary

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pulsating vortex in the inviscid part of the flow over a semi-infinite flat plate from the solution of two-dimensional Navier–Stokes equation, with the pulsating frequency as the imposed time scale. In the computational effort [478], the authors noted the creation of a spatio-temporal wave front (STWF) for a pulsating vortex located, where ˜ is 1000. There is also the presence of the local solution and the TS wave-packet. Re In the following, one notes from linear receptivity of a semi-infinite flat plate, as shown in [478] that the response field scales with the amplitude of the pulsating vortex, when it was increased ten thousand times. This pulsating vortex is given by Γ = Γ0 sin ω ¯ 0 t˜, where ω ¯ 0 is the imposed time scale, as in the experiment with a vibrating ribbon in the free stream creating TS wave-packets [109]. The schematic of the receptivity to stationary pulsating vortex problem is shown in Figure 5.5. In this receptivity problem, explicit time scale is introduced by the pulsating vortex and the signal problem is solved for the case reported here, corresponding to the solution of the Navier–Stokes equation [478]. Γ0e–iω t 0

Oscillating Point Vortex

20 δ *

U∞ y

δ* x Flat Plate

Figure 5.5 Schematic of the receptivity to pulsating vortex in the free stream problem, which imposes the time scale ω ¯ 0 for the Blasius boundary layer. This is solved as the signal problem, with the origin defined on the plate at the location of the exciter. The Orr–Sommerfeld equation is solved in the α-plane with 5735 Chebyshev collocation points using the generalized differential quadrature method [487], in the range, (αr )min,max = ±12.867. The streamwise stretch of ( x˜) is ±700, with the pulsating ˜ Here, the Bromwich contour has been taken vortex located at x˜ = 0 and y˜ = Y. ˜ = 1000. In Figure 5.6, the disturbance stream function along αi = −0.01, for Re

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has been shown as a function of x˜, for the height, y˜ = 1.20577. The solution away from the exciter shows TS waves, which would be obtained if the exciter is placed at the wall and vibrated with ω ¯ 0 = 0.1. In the framework of parallel flow, signal ˜ cases have to be constructed problem assumption, different solutions for different Re ˜ based on δ∗ with the origin located at the station under consideration, for which Re ˜ shows are as indicated in Figure 5.6. The variations shown in the figure for each Re the typical local and asymptotic part of the solution. For the wall excitation case, ˜ considered, and hence, the local solution has been noted to be independent of the Re ˜ they collapse on each other in the immediate neighborhood of the exciter for the Re considered. However, for the free stream excitation case, the local solutions do not collapse on each other immediately on downstream side of the exciter location, but are virtually identical on the upstream side of the exciter.

Figure 5.6 Disturbance stream function variation with x for the Blasius boundary layer, for the indicated Re cases with results at the outer maximum of the TS mode (y˜ = 1.20577) with the excitation frequency for the problem given by ω ¯ 0 = 0.1.

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The maximum values of disturbance stream function for different height, for wall and free stream excitation cases, are shown in Figure 5.7, to compare the efficacy of exciting TS modes by these two types of excitation. Whereas the free stream vortex is placed at Y˜ = 20, the boundary condition is implemented at 0.9Y˜ in solving the free stream receptivity problem. One important observation is the behavior of the solution as one approaches the wall (˜y → 0) for the wall excitation, i.e. the solution goes to the applied Dirac delta function. For the free stream excitation cases, the solution approaches a vanishing value due to satisfaction of no-slip condition.

Figure 5.7 Maximum disturbance stream function as a function of the indicated height range, with the pulsating vortex placed at a height Y˜ = 20. Also shown is the disturbance stream function variation with height for the wall excitation case.

5.3.4 Instability control by wave cancellation with wall and free stream excitation Next, the instability control by simultaneous application of wall and free stream excitation is explored. The receptivity to free stream pulsating vortex excitation is ˜ cases. In conformity with the parallel flow, signal shown in Figure 5.6 for different Re

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problem assumption, all results are presented with respect to the same origin. To control the TS wave created by free stream excitation, one needs to not only position the wall exciter at an appropriate streamwise location, but the amplitude of wall exciter also has to be chosen correctly, with respect to the free stream excitation amplitude. These two aspects can be addressed by noting from previous discussion that the wall excitation has stronger receptivity as compared to free stream excitation, described in the coupling between the wall and free stream modes, and graphically seen in Figure 5.7. This shows the maximum disturbance stream function for the perturbation field by both types of excitation. There have been attempts to control flow instability by the wave cancellation principle [203, 265]. However in attempting to do so, both the sources of excitation have been placed inside the shear layer. In these experimental approaches, the downstream exciter’s amplitude and phase with respect to upstream exciter have been adjusted, without the guidance of any theoretical results. Here, this is shown using theoretical results based on the linear receptivity approach described in this chapter. The attempt here is to control instability created by a free stream exciter at x˜ = 0, by a wall exciter, with imposed wall excitation given by ψw = aδ( x˜ − x0 )e−iω¯ 0 t˜ One notes that apart from an amplitude control factor a, the Dirac delta function is shifted at the wall to x˜ = x0 . These two factors are different for different free stream exciter locations, determining the response field by parallel flow approximation. ˜ = 1000 and ω For example, in Figure 5.8, the case for Re ¯ 0 = 0.1 is shown. The ˜ is 1000, whereas the free stream exciter is located at a streamwise location where Re controller wall exciter is located downstream, with an amplitude obtained by multiple simulations to determine factor a. The details of this is given in [85]. For the results shown in Figure 5.8, the best control is obtained for a = 8.04 × 10−3 and x0 = 0.488. Thus, the wall exciter amplitude needed is very minimal and is shifted downstream by a small amount. The results are obtained with the sole purpose of controlling the asymptotic part of the solution. As the present exercise is performed by signal problem assumption, one only notices the local and asymptotic components of the solution, with the objective of virtual elimination of TS waves achieved. As there is no time shift between the wall and free stream excitation, the local solutions are additive and do not disappear in the controlled response. Moreover, if one removes the signal problem assumption, one would obtain the STWF, which can cause the eventual transition, as has been noted using both linear and nonlinear approaches [419, 452]. Computing such a flow field and control of transition solving the Navier– Stokes equation has not been attempted before.

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Figure 5.8 Instability control by wall and free-stream excitation for the Blasius boundary layer ˜ = 1000 with results shown at the outer maximum of the TS mode (y˜ = for Re 1.20577) with the excitation frequency for the signal problem given by ω ¯ 0 = 0.1. Apparently, the wall and free stream modes cancel each other leaving only the local solution present only.

5.4 Linear Receptivity Analysis: Spatio-Temporal Wave Front In this section, the linear receptivity of a semi-infinite flat plate excited by a fixed frequency exciter at the wall is discussed. This is how the spatio-temporal analysis by Bromwich contour integral method (BCIM) started for the Blasius boundary layer in [418] for a case, in which the spatial instability theory indicates a growing TS wave. The similar experimental results are given in [405]. The schematic of the experimental set up, as modeled in the simulation is shown in Figure 4.1, for which the governing Orr–Sommerfeld equation is given by Eq. (5.1), and the applied wall excitation is given by Eq. (5.2) in the physical plane. Note the presence of the Heaviside function, H1 (t) in Eq. (5.2). In the transformed circular frequency plane, its transform is given by

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H1 (t˜) ⇔ π δ(ω0 ) +

1 iω0

As a consequence, the boundary conditions in complex α and ω0 plane is given by Eq. (5.3). For this wall excitation problem, as the perturbations decay in the free stream, one writes the solution of the Orr–Sommerfeld equation in terms of φ1 and φ3 and one fixes the values of a1 and a3 in Eq. (5.4) from Eq. (5.5). From Eq. (5.5), one obtains the general expression of φ(˜y) from Eq. (5.4) as given in Eq. (5.6). After obtaining φ(˜y) from Eq. (5.6), one can get back the perturbation velocity components by the inverse Fourier–Laplace transform from Eqs. (5.7) and (5.8). The integrations are performed in Eqs. (5.7) and (5.8), along the Bromwich contours in complex α- and ω0 -planes, as shown in Figure 5.1. The demonstration of the BCIM is ˜ = 1500 and indicated circular frequencies made for the Blasius boundary layer, for Re (ω ¯ 0 ) in Figures 5.9 and 5.10. The results shown here have been obtained in [29] first by solving the Blasius boundary using 2400 uniformly distributed points in the range, 0 ≤ η ≤ 12. Following the sketch in Figure 4.1, the exciter is located at a place where ˜ = 1500. Thereafter, the governing equations for CMM have been solved using 1200 Re points. All the solutions of the Blasius boundary layer; CMM equations for second compounds and the solution for φ are obtained using four stage, fourth order RungeKutta method directly. Thus, the solution for φ are obtained using equidistant 600 points. In Figure 5.9, results are shown for a non-dimensional circular frequency of ω ¯0 = 0.025, which corresponds to a physical frequency of 3.6Hz. According to the spatial ˜ and ω theory, this combination of Re ¯ 0 produce the least stable mode, which happens to be stable. As noted before, the origin is fixed at the location of the exciter, where one notes the near-field or the local solution. This is followed by the asymptotic part of the solution that corresponds to the leading TS wave (which in this case is clearly stable with respect to x˜). These two elements are common with that noted for the signal problem before. However, in these frames, one can clearly note the presence of the third element labeled as the spatio-temporal wave front (STWF). This is in accordance with the original terminology for such a component of the solution by Brillouin [67] for electromagnetic wave propagation. In fluid dynamics, this was noted for the first time in [418, 451, 452]. The solution is shown at the indicated times as a function of streamwise distance, x˜, at a height of y˜ = 0.278. As this height is slightly above the wall, the disturbance signal is very prominent. The existence of STWF has been anticipated without understanding the underlying mathematical physics, some researchers called it simply as the nonmodal growth [398] in the linear framework for quite some time. However, from the various transform techniques applied to dynamical system theory, ever since the appearance of the works of Fourier and Laplace, this is a natural extension of the response field. In Figure 5.10, results are shown for a non-dimensional circular frequency of ω ¯ 0 = 0.10, which corresponds to a physical frequency of 14.4 Hz. According to the

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_ ~ Re =1500, ω0 = 0.025

0.002

~ (a) t = 160.22

0.001 u’

0 –0.001 –0.002

0

200

~ x

400

600 ~ (b) t = 320.44

0

200

~ x

400

600

0.002

u’

0.001 0 –0.001 –0.002

~ (c) t = 480.66

0.002

u’

0.001 0 –0.001 –0.002

0

200

~ x

400

600 ~ (d) t = 640.88

0.002

u’

0.001 0 –0.001 –0.002

0

200 Local sol

0.002

~ x

400

600 (e) ~t = 801.11

n

u’

0.001 0 –0.001

Asymptotic sol

–0.002

0

Spatio-temporal wave-front

n

200

~ x

400

600

Figure 5.9 Streamwise disturbance velocity plotted as a function of x˜ at y˜ = 0.278 for monochromatic wall excitation given by ω ¯ 0 = 0.025. The exciter is located where ˜ = 1500 in the timethe Reynolds number based on displacement thickness is Re frames.

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˜ and ω spatial theory, this combination of Re ¯ 0 produces an unstable TS mode. With the origin at the location of the exciter, the streamwise velocity shown at different time instants display three-components of (i) the near-field or the local solution; (ii) the asymptotic solution corresponding to the TS wave, which locally grows with x˜; and (iii) the STWF which remains fused with the TS wave-packet. For the actual boundary layer developing downstream, δ∗ keeps growing with x˜, and the disturbance field adjusts itself accordingly. While this is not solved analytically, the corresponding solution of the Navier–Stokes equation supports the conjecture of flow continually adjusting to local conditions. As a consequence, the TS wave stops growing as the signal emerges out of the neutral curve, for the constant frequency lines shown in Figure 4.8 for the zero pressure gradient boundary layer. Note that the nondimensional ω ¯ 0 is based on the length scale δ∗ and velocity scale Ue , which can be alternately expressed as ω ¯ 0 = 2π fex

δ∗  2πν fex  Ue δ∗ ˜ = = F Re Ue ν Ue2

(5.61)

In this equation, F is the non-dimensional physical frequency of excitation. Thus, a ˜ ω constant physical frequency line in the (Re, ¯ 0 )-plane is a straight line passing through the origin; different constant-frequency lines have slopes that correspond to different values of F. Different physical frequency of excitation lines are indicated in Figure 4.8. Due to these growth and attenuation patterns of constant frequency disturbance field, one notices the TS wave as a packet in actual flow, whereas it will appear as continually growing in signal problem due to the parallel flow assumption, without requiring the disturbance field to adjust to local conditions. This was a tentative explanation advanced by Arnal et al. [12, 13] in making use of local analysis with global dynamics of the disturbance field. However, the most important aspect of the STWF is its distinct dynamics which is clearly noted in Figure 5.9, whereas it is not so clear in Figure 5.10. Here, only in the last frame of the latter at t˜ = 801.11, can one notice the emergence of the STWF out of the growing TS wave-packet. Similar such exercise ˜ = 1000 and ω was undertaken in [418] for Re ¯ 0 = 0.10 and clear emergence of STWF was not noted. However, in [451] when the same exercise was repeated for combinations ˜ and ω of Re ¯ 0 , clear emergence of STWF was noted for the semi-infinite flat plate. It is emphasized that STWF is a consequence of the dynamics of disturbance in a flow which is triggered by wall excitation here. However, one will note later that STWF is also created by free stream excitation in boundary layer in Chapter 9. Although, lot of efforts have been spent by the research community to show theoretically and experimentally the so-called, nonmodal growth or hydrodynamic instability without eigenvalues in the linear framework, the existence of the STWF was not emphasized or even understood, simply due to the inability to look for the spatio-temporal growth before.

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~ (a) t = 160.22

_ ~ Re =1500, ω0 = 0.1

u’

0.01 0 –0.01 0

200

~ x

400

~ (b) t = 320.44

0.01 u’

600

0 –0.01 0

200

~ x

400

~ (c) t = 480.66

0.01 u’

600

0 –0.01 0

200

~ x

400

~ (d) t = 640.88

0.01 u’

600

0 –0.01 0

200

~ x

400

600 (e) ~t = 801.11

u’

0.01 0 –0.01 0

200

x~

400

600

Figure 5.10 Streamwise disturbance velocity plotted as a function of x˜ at y˜ = 0.278 for monochromatic wall excitation given by ω ¯ 0 = 0.10. The exciter is located where ˜ = 1500 in the timethe Reynolds number based on displacement thickness is Re frames.

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5.5 Closing Remarks In this chapter, we mainly discussed linear theory of receptivity, and its connection to experiments that tends to support various instability theories. In the process, there have been misinterpretations about the difference between instability and receptivity. Whereas linear spatial theory requires a fixed frequency excitation, experiments also followed suit, as tracking natural transition would require the preference of the dynamical system to pick up a particular frequency from the frequency spectrum of the input. Finding such systemic preferences itself would require another theory! Hence,, the early corroboration of the spatial viscous theory came from experiments, with background disturbances minimized to very low values; then the controlled deterministic disturbances was introduced to destabilize the flow. In reality, the modest goal reported in [405] was to show the creation of TS waves. The experiments performed in the NBS wind tunnel needed specially fitted damping screens to reduce the turbulence intensity by increase in solidity ratio and increasing the number of damping screens. In the natural transition experiments, the transition location could be shifted downstream by reducing turbulence intensity to 0.08 percent. Below this level, no further benefit was recorded. The natural transition observed without explicit excitation at a flow speed of 53 ft/s was recorded in film by photographing the oscillograph screen with a moving film camera. By contrast, the deterministic vortical excitation inside the boundary caused transition, whereas free stream acoustic excitation did not, thereby necessitating the coining of the term “receptivity,” without providing a rigorous definition. Hopefully, the present chapter bridged that gap, by showing the missing link between instability and receptivity, through the concept of dynamical systems. We have seen that the eigenvalues of the instability problem are the poles of the receptivity problem for the transfer function of the dynamical system, which define the receptivity of the flow. Of course, the input types will also affect the receptivity. The rudimentary concepts of receptivity of fluid dynamical systems are adequate to explain receptivity, and show its link with instability. It is shown that the artifact of the theory and experiment requiring fixed frequency excitation to create the TS wave of spatial theory got intertwined into the claim of TS wave as the sole precursor of transition to turbulence. The fact is that this conversion of TS wave to turbulence is purely conjectural without being shown systematically even today in a causal sequence. Equally unfortunate is the adoption of the signal problem assumption, because of which the theory and experiment supported each other’s claim. This led to the intuitive leap or assumption that the response of the fluid dynamical system must be at the frequency of excitation used in the spatial theory and corresponding experiment. Although this may seem a natural fit for spatial theory, it foreclosed the interpretation that experiments of an unstable flow should be analyzed by a more generic spatiotemporal theory. This situation pervaded for almost half a century, till the time the author’s group explained that a finite start-up time invariably excites a wide spectrum of time scales, and thereby allows the system’s response to be decided not only by the

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discrete eigenvalues, but also by the group-action of the transfer function, which allows formation of continuous spectrum and the nonmodal part of the solution naturally from the Orr–Sommerfeld equation, even in the linear framework. This has been attempted in recent decades, and given the sobriquet of nonmodal growth or growth of disturbances without any eigenvalue etc. Unfortunately, this came through with the artifact of linear algebra used in a specific type of studies, which are not at all generic. The actual route to such nonmodal growth has been shown to be developing a new method of studying the semi-infinite flat plate by the Bromwich contour integral method. Although the method was first demonstrated for the signal problem, soon it was used to develop the spatio-temporal growth of the disturbance field, showing the elusive element of the spatio-temporal wavefront. This chapter is an introduction to this spatio-temporal wave front. En-route to describing this nonmodal element of disturbance growth for wall excitation, we also look at the complementary topic of free stream excitation and the coupling between wall and free stream excitations. Although both wall and free stream excitations are tackled in this chapter, it also helps to look at downstream and upstream propagating modes. Having depended solely on one kind of experimental route (which was mistakenly thought to be the frequency response) of the fluid dynamical system, it is time to consider more fundamental types of excitations of dynamical system, namely, the study of impulse response, as studied in the next chapter.

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Dynamical System Theory of Linear Receptivity

6.1 Introduction One of the principal tenets in developing a dynamical system theory is to study the relationship between cause and effects. This is true for a fluid dynamical system characterized by a large number of degrees of freedom, as compared to other dissipative dynamical systems in many fields of physics. Experimental verification of any theory is imperative, and in this respect, theories of instabilities are difficult propositions. This is because instability theories rely on omnipresent imperceptible ambient disturbances as input to produce response, specifically in the limit of vanishingly small input that is needed in the dynamical system approach. Mathematically, the instability problem involves seeking the output of a system governed by a homogeneous differential equation, subject to a homogeneous boundary and initial conditions. Implicit in this is the requirement of an equilibrium state whose instability is studied, and for which imperceptible omnipresent disturbance resides and draws energy for its growth. For example, flow past a circular cylinder displays unsteadiness above a critical Reynolds number (based on oncoming flow speed and diameter of the cylinder), even when one is considering uniform flow over a perfectly smooth cylinder. Whereas this can be rationalized for experimental investigation where the presence of background disturbances cannot be ruled out, the situation is far from straightforward for computational efforts. Roles of various numerical sources of error triggering instability for uniform flow past a smooth circular cylinder is complicated. This issue has been dealt with in [469]. Inability to compute the equilibrium flow past a circular cylinder at relatively high Reynolds numbers is due to the presence of adverse pressure gradient experienced

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by the flow on the lee side of the cylinder. The situation is equally difficult for the flow over a very long flat plate. As the equilibrium flow is obtained with significant precision, it is possible to study the flow past a flat plate as a receptivity problem, as has been done experimentally to study the existence of TS waves by Schubauer and Skramstad [405], where the disturbances were created by a vibrating ribbon inside the boundary layer. We have already identified a few drawbacks of the linear instability theory formulated by a homogeneous governing equation with homogeneous boundary conditions, in search of eigenvalues to explain growth of disturbances. These have been noted by Trefethen et al. [532] and mentioned in the previous two chapters. The main difficulty for the instability of boundary layer is in classifying instability in terms of either temporal or spatial growth of disturbances. When inviscid temporal theory failed for zero pressure gradient (ZPG) flow past a flat plate, using Rayleigh’s equation and theorem [351, 356], viscous effects were included in the spatial theory by the Orr–Sommerfeld equation [321, 495], which predicted TS wave as a new element of the disturbance field [393, 527]; this was not verified experimentally immediately. Experimental verification of an instability theory is not straightforward, as by definition it requires excitation by unquantified background disturbances, whose mode of application is ambiguous. Unfortunately, dependence on background noise makes such experiments non-repeatable. This situation prevailed till the classic experiment of Schubauer and Skramstad [405], when the boundary layer was excited by a vibrating ribbon inside the boundary layer to obtain spatially growing TS waves. This does not conflict with the requirements of spatial instability, which requires keeping the frequency of excitation constant to excite the TS waves. In a recent work [509], the authors reported the importance of finding the true role of TS waves in natural transition. The linear theory gained in importance via theoretical and experimental studies of instability, where TS waves and their control seemed to play a major role, as in [118, 138, 183, 245, 346]. Experimental work relating to interaction between TS waves and free stream turbulence is given in [46], whereas some numerical results are available in [50, 271, 306]. Authors in [509] noted that “the problems of instability in the presence of adverse pressure gradient have been reported in [23, 192]. Given the overwhelming importance of TS waves, the present investigation is about finding the true role of TS waves in actual transition to turbulence for the ZPG boundary layer.” How does one interpret the experiment of [405] in a theoretical/ computational framework? This has been variously attempted in [14, 141]. But among many other references including [321], authors have questioned the adequacy of normal mode eigenvalue analysis [26, 80, 100, 128, 131, 251, 375]. Despite such critique of modal analysis in [74, 166, 179, 362, 398, 532], some researchers used modal analysis to point out that such modes may not be always orthogonal to each other, as a result of which interactions among modes can give rise to very high transient growths for threedimensional flows. Although Trefethen et al. [532] noted that the “growing attention to 3D linear, nonmodal phenomena represents a significant change in the traditional

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conception of problems of hydrodynamic stability,” they also acknowledged that “when only 2D perturbations are considered, some amplification can still occur, but it is far weaker!” The conflicting issue appears to be far from settled, as Kerswell [229] has noted: “Interestingly, this new formulation revolved around considering the evolution of a mixture of normal modes, rather than just one in isolation, and revealed how the nonorthogonality of these modes (present when the linearized operator is nonnormal) can lead to short-term energy growth.” In justifying the need to introduce new approaches for short-term behavior of the linearized initial value problem, these have been variously called as the transient growth [361], optimal perturbation theory [74], or nonmodal theory [398]. The authors have even questioned the validity of the use of the Fourier–Laplace transform in linear theory for early transient times. The shortcomings of such an idea has already been discussed in Sub-section 4.6.1, to establish the adequacy of the Fourier–Laplace transform as the complete tool, with the help of Abel and Tauber theorems. We have demonstrated the efficacy of the Fourier–Laplace transform in explaining the near-field structure of the disturbance field for localized wall excitations in Sub-section 5.2.2. Accepting the adequacy of linear theory to explain near-field/ early-time disturbance fields, along with the farfield with the same formalism, it is needed to explain the motivation and implication of the experiment reported in [405]. Concomitant with the development of spatial instability studies by solving the Orr–Sommerfeld equation [119, 412], the boundary layer excited by a monochromatic time-harmonic localized source in the experiment prompted Gaster [141], and Ashpis and Reshotko [14] to view the experiment [405] as the frequency response of the boundary layer. The implicit assumption in these conjectural works is that the time dependent response of the disturbance field is also at the frequency of excitation; an assumption often used as standard practice in dynamical system theory. Whereas this may appear to be true for stable dynamical systems (as often used in theoretical and experimental characterizations), this assumption is also adopted in linear stability theory, and termed as the signal problem assumption. Interestingly, Schmid and Brandt [399] made the following justification: “Due to the linearity of the governing equations, the output ... responds with the same frequency.” However, the critique of this “signal problem” has been provided in the previous chapter for any dynamical system that is strictly not stable in the physical sense (and not merely by the presence or absence of unstable eigenvalues noted in [399]). The dynamics will be dominated by any time scale for which the system has inherent dominant instability, noted from its transfer function. This is explained in Section 5.4, showing the link between instability and receptivity for the wall excitation problem. All the time scales are excited in the experiment reported in [405], despite the fact that the excitation imposed is at a constant frequency. This is explained as due to the finite start-up time of any experiment, and for the wall-normal velocity excitation given by Eq. (5.2) is multiplied by the Heaviside function, for which the boundary condition is given in Eq. (5.3) as 1 + πδ(ω0 − ω ¯ 0) φ(0) = i(ω0 − ω ¯ 0)

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The first term on the right-hand side is responsible for excitation at all frequencies, whereas the second term on the right-hand side is due to the excitation created by the imposed frequency. Thus, in the study of flow instability, there is no such thing as pure frequency response. For this reason, there is no sanctity of the signal problem assumption, and one should study problem of instability always in the spatio-temporal framework, be it in linear or nonlinear receptivity. It is interesting to note that whereas Gaster [141] advocated using the signal problem, which implies following the frequency response of the flow instability problem, Gaster and Grant [144] reported an experiment where the ZPG boundary layer is subjected to a localized impulse excitation. As the existence of TS waves could not be demonstrated outside the confines of laboratories, Gaster and Grant [144] experimentally studied ZPG boundary layer excited by an impulse, which is localized in space. Mathematically, this is equivalent to using an input, which is a delta function in space and time, and the results provide the impulse response of the dynamical system. Such an excitation theoretically should excite all space and time scales. The results displayed that this impulse response of the dynamical system created a wave packet; however the authors [144] tried to explain this wave packet as an ensemble of TS waves obtained from the linear spatial instability theory. It has been shown using linear theory, and from the DNS of the Navier–Stokes equation in [34] that the response of ZPG boundary layer to localized pulse indeed creates a spatio-temporal wave front (STWF). The impulse response in the linear framework, obtained using the BCIM creates the STWF. The achievement of the BCIM in the creation of STWF, by solving the Orr– Sommerfeld equation following both the experimental approaches in [144, 405], is a major advancement in the field with the publication of the results in [451, 509] for both the experimental approaches for impulse and “frequency” response. The authors in [34] have identified the impulse response with the STWF as the building block that explains diverse physical and geophysical events, such as transition to turbulence, rogue waves and tsunamis. The role of STWF in creating transition to turbulence via the so-called frequency response route has been conclusively established in [32, 419, 422]. It is noted that the search for STWF was sought in other branches of physics, with early efforts recounted in [67] for electromagnetic wave propagation and by Bers [27] for plasma physics.

6.2 Case Studies of Spatio-Temporal Growth It is important to understand the theoretical aspects of impulse, and the so-called frequency response of a fluid dynamical system experiencing instability. Here, the results for the ZPG boundary layer are used to theoretically explain the common elements of these responses. In the context of flow instability, the differences between the two responses continue to baffle researchers. The primary goal here is to explain these two responses with the solution of the Orr–Sommerfeld equation as to how STWF is created, and its ubiquitous role in manifesting unsteady effects, even when

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the excitation is imposed impulsively only once, whereas the disturbance continues to grow indefinitely for the eventual transition to turbulence from initial laminar state. Moreover, it is explained here how STWF is created for different start-up conditions using BCIM in solving the Orr–Sommerfeld equation for a two-dimensional response field.

a)

~ y

Blasius profile

Edge of shear layer

U∞ Equivalent parallel flow U(y)

Exciter

O

~ x

~

Re 1

b)

H1(t)

U1(t–t0)

0.8 Heaviside (H1(t)) U1(t–t0) : αE = 100

0.6

_

U1(U(t) t–t0) = 0.5 1 + erf

y

t–t0

2 παE

0.4

0.2

R(t–t0) t0 = 150

0 –500

0

time ( ~t )

500

1000

Figure 6.1 (a) Schematic of the domain for different wall excitations for a parallel flow given by Blasius profile at a location marked and b) the envelope of the time-dependent excitations given by the Heaviside function (H1 (t)) and an error function (U1 (t)), triggered at t = 0 for H1 (t), and t = t0 for U1 (t − t0 ).

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6.3 Formulation of Impulse and Frequency Responses The schematic of the problem is shown in Figure 6.1(a) in the physical plane, whereas it is solved in the spectral plane, involving complex α and ω0 . For the two-dimensional problem, the response is calculated for the linearized field following the governing Orr–Sommerfeld equation given by,   ˜ (αU(˜y) − ω0 )D21 φ − αU 00 (˜y)φ D41 φ = iRe (6.1) ˜ = Ue δ . The displacement thickness at the exciter location is where D21 = ddy˜2 − α2 and Re ν used as the length scale and the free stream speed is the velocity scale used to obtain the Orr–Sommerfeld equation in non-dimensional form. The time scale is derived with the help of these length and velocity scales. The disturbance stream function is given by its spectral transform as Z Z ¯ ψ= φ(α, y˜ , ω0 ) ei(α x˜−ω0 t) dα dω0 (6.2) ∗

2

Brα

Brω

which is solved for both the signal problem and the spatio-temporal problem using BCIM. The difference between these two methods lies in choosing the Bromwich contours in the spectral planes, which should be in the respective strips of convergence of the Bromwich contours [324, 412, 537]. Here, the response to wall excitation is studied for three types of excitation fields: (i) Where the input is simply a product of delta functions in space and time; this represents the pure impulse response. The other cases are shown in Figure 6.1(b), with the impulsive-start represented by the Heaviside function (H1 (t)) and the other represents a non-impulsive start given by   t − t  0 (6.3) U1 (t − t0 ) = 0.5 1 + er f √ 2 παE which is given in terms of the error function. One can study the impulse and socalled frequency response cases, where H1 (t) and U1 (t) represent the envelope for the ¯ Other cases have also been studied, amplitude of input disturbance stream function, ψ. where the dynamical system is excited by inputs shown in Figure 6.1(b) without any imposed time scale; these are termed as non-oscillatory transient cases. Additionally, a case of ramp start is also studied, which is non-oscillatory and shown as R(t − t0 ). Note that when αE approaches zero in Eq. (6.3), one recovers the Heaviside function, H1 (t − t0 ). Moreover, in Figure 6.1(b), the non-impulsive case U1 (t − t0 ) becomes nonzero from t = 0 onwards, while it is centered around t0 . For this reason, the Fourier transform has been calculated using time-scaling, frequency and time shift theorems [339, 412]. This is to be highlighted that for the frequency response case: the finite start-up with Heaviside function introduces all possible circular frequencies as excitation. Even in the case where U1 (t) is characterized by αE taking a very small value, one again excites a wide range of frequencies, apart from ω ¯ 0 , for the so-called frequency response

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case. Thus, for the study of instability, there is hardly any difference between impulse and “frequency” responses, as both the cases are excited by wide-band input with finite start-up. The differences are in terms of distribution of amplitude as function of ω0 . ω

αi Downstream modes

*

Br

ω

* αi

ω

0i

αr

0i

* *ω

0r

*

αBr

* Upstream mode

Bromwich contour

Figure 6.2 Bromwich contours used here have been shown in the complex α and ω0 -planes for the Bromwich contour integral method approach. For signal problems, only the contour in the α-plane is used. In the signal problem, it is assumed that the response is solely at the frequency of excitation, ω ¯ 0 , and as a consequence, the frequency is fixed, i.e., ω0 = ω ¯ 0 , and one solves Eq. (6.1) along the Bromwich contour in α-plane (= αr + iα¯ i ) only. Choice of constant α¯ i facilitates the use of Discrete Fast Fourier Transform (DFFT) for the inverse transform; otherwise any deformed contour inside the strip of convergence should suffice. The contours used in the BCIM are shown in Figure 6.2, with the choice of contours dictated by the position of various marked eigenvalues in the complex-α and -ω0 planes, with details explained in [413, 418]. Results presented in the following are from [509], with the indicated physical and numerical parameters as reported therein. The formulation of linear receptivity analysis for impulse, “frequency” and other forms of response is given in Eq. (6.1). Considering infinitesimally small perturbation (), the Orr–Sommerfeld equation is ˜ = 1000 by the compound matrix method along the Bromwich contours in solved for Re the α-plane with 213 points, and in the ω0 -plane with 211 points. The wall perturbations are given in Table 6.1; vw =  f ( x˜, t˜) is shown in the last column with  = 0.002. Cases 1 and 3 are for impulse [144] and “frequency” responses [405], respectively. In Table 6.1, case 3 is for the impulsive start of wall excitation at a fixed frequency (with ω ¯ 0 = 0.16), for which the linear theory predicts spatial stability for this higher ˜ = 1000). Cases 2a to 2c have been proposed by frequency and Reynolds number (Re the authors in [509], as alternative routes to perform transition experiments. For cases ¯ x˜)) with different start-ups are imposed for these 2a to 2c, a localized perturbation (δ( cases. For case 2a the Heaviside function (H1 (t˜−t0 )) implies a constant in time localized excitation that is started impulsively at t˜ = t0 . For the case 2b, one executes a smooth

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Table 6.1 Different wall perturbation forms studied. [Reproduced from Is Tollmien-Schlichting wave necessary for transition of zero pressure gradient boundary layer flow? - P. Sundaram, T. K. Sengupta , and S. Sengupta, Phys. Fluids, vol. 31, pp 031701 (2019), with the permission of AIP Publishing.] Case

Type

1

Impulse response

2a

Impulsive start-up

2b

Smooth start-up

2c

Ramp start-up

3

Frequency response

f ( x˜, t˜) ¯ x˜)δ(t˜) δ( ¯ x˜)H1 (t˜) δ( ¯ x˜)U1 (t˜ − t0 ) δ( ¯ x˜)R(t˜ − t0 ) δ( ¯ x˜)H1 (t˜)eiω¯ 0 t˜ δ(

start-up following the error function (U1 (t˜ − t0 )) given in Eq. (6.3) and a linear ramp start-up (R(t˜ − t0 )) has been proposed for case 2c. The wall-normal velocity excitation for case 2b is written as vw (t˜) = 

 t˜ − t  ¯ x˜)  δ( 0 1 + er f √ 2 2 παE

(6.4)

with αE = 100 and t0 the central location of the error function. For the limit of αE → 0, one recovers the impulsive start indicated for case 2a. The ramp start-up excitation for case 2c is given by ¯ x˜) t − t1 , vw (t˜) =  δ( t1 − t2

for t1 ≤ t ≤ t2

(6.5)

For finite acceleration cases shown in Figure 6.1(b), the excitations for cases 2a and 3 are started at t = 0, while for cases 2b and 2c, the excitation start-up is centered at t = t˜0 = 150. The ramp-start in case 2c is during the interval 0 ≤ t˜ ≤ 300, i.e., in Eq. (6.5), the choice of time is from t1 = 0 to t2 = 300.

6.4 Impulse, Frequency, and other Responses To demonstrate linear response cases, the governing equation is given by the Orr– Sommerfeld equation; the equilibrium flow is given by the Blasius boundary layer in [509]. The response of this ZPG boundary layer is reported for infinitesimally small wall perturbations. The parallel flow assumption requires fixing the exciter location, ˜ = 1000. where the Reynolds number based on displacement thickness is given by Re Along the chosen Bromwich contour in the α-plane (Brα ), 8192 points are taken in the range −4π ≤ αr ≤ 4π, for which Eq. (6.1) is solved along α¯ i = −0.01. In the ω0 -plane, the Bromwich contour (Brω ) is located at ω0i = +0.012 line having 211 points in the range (−π/2, +π/2). It has been reasoned already that the signal problem is inconsistent

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for instability studies; instead, one should solve any problem of instability as a spatio-temporal growth problem, as performed in [412, 418, 451] to study “frequency” response. Thus, one should solve Eq. (6.1) for any type of excitation implied, along the Bromwich contours shown in Figure 6.2. Finally, double inverse Fourier transform ¯ as given in Eq. (6.2). It is readily noted that in the BCIM is performed to obtain ψ, one needs to solve an equivalent signal problem for each and every point along the Bromwich contour, Brω in the complex ω0 -plane. .075

~t = 1250

ud

~t = 1250

case-2a

10x

0 –.075 .075

case-1

~t = 1250

case-2b

1x

~t = 1500

~t = 1500

~t = 1750

~t = 1750

~t = 1250

case-2c

20x

~t = 1500

20x

~t = 1500

~t = 1250

case-3 1x

~t = 1500

0

–.075 .075

~t = 1750

~t = 1750

~t = 1750

0 –.075 –250 0 250 500 750 –250 0 250 500 750 –250 0 250 500 750 –250 0 250 500 750 –250 0 250 500 750

~ x

Figure 6.3 Evolution of ud from the solution of the Orr–Sommerfeld equation for different wall excitation at height y˜ = 0.278. The vertical dotted lines in case 3 show the region of instability predicted by the linear spatial theory. [Reproduced from “Is TollmienSchlichting wave necessary for transition of zero pressure gradient boundary layer flow?”, P. Sundaram, T. K. Sengupta, and S. Sengupta, Phys. Fluids, vol. 31, pp 031701 (2019), with the permission of AIP Publishing.] The solution of the Orr–Sommerfeld equation is obtained along the Bromwich contours, for which a second order filter is used with a filter coefficient of α f = 0.4 for the α-plane data to avoid effects of Gibbs’ phenomenon [439]. Blasius profile has been used as the equilibrium flow to solve the Orr–Sommerfeld equation (Eq. (6.1)) with the already specified number of points along the Bromwich contours. From the solution for φ0 along the Bromwich contours, one can perform double inverse transform to obtain the streamwise component of disturbance velocity (ud ) at the selected height y˜ = 0.278. Results obtained for the different wall excitation cases in Table 6.1 are shown as a function of time in Figure 6.3 [509]. Responses for case 1 to case 2c are magnified by the factors given in the frames. All the cases except case 2c display evolution of single STWF. For the ramp start case 2c, due to two discontinuities, one notices two STWFs, one following the other due to the presence of slope discontinuity for the wall excitation.

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The response for case 1 has a very clear appearance of STWF [144], which the authors mistakenly tried to represent by an empirical weighted sum of TS waves for an ensemble of frequencies. It is clearly evident that among all the cases listed in Table 6.1, case 1 imparts the least amount of energy through the wall-normal velocity boundary condition, even though the value of  = 0.002 is same for all the cases. For this reason, in all the three frames for case 1 in Figure 6.3, the response has been shown with ten-times magnification for the streamwise disturbance velocity. This same case of excitation has been studied by solving the three-dimensional Navier– Stokes equations in [34, 508], to show unequivocally that the experiment in [144] provides the best possible demonstration of transition and turbulence. This will be shown with computational results of the Navier–Stokes equation in the next chapter. Thus, the experimental and numerical evidence of the flow transition is established only after the unambiguous role of the TS waves in causing flow transition was shown by the results in [509]. The clue to this has been provided by demonstrating, with the help of the solution for the Orr-Sommerfeld equation that TS wave is not necessary to cause transition for ZPG boundary layer. In case 3 of Figure 6.3, it has been noted in [412] that the corresponding response field comprises three basic elements: (a) the local solution or the near-field solution, as has been theoretically explained in Chapter 4, with the help of Tauber theorem [537]. This solution is noted in the immediate neighborhood of the exciter, and is explained for the signal problem in [412, 420]; (b) Following the local solution in the downstream direction, one notices the asymptotic part, which consists of the TS waves. This also follows from the Abel theorem given in [412, 537] and (c) the final and the third element is the STWF, which grows spatiotemporally convecting downstream, unlike the TS wave that appears to be stationary due to its growth and attenuation properties. It has been established that STWF leads to eventual transition to turbulence in [32, 419, 483], and is explained in further detail in the following chapters. The distinctive feature of case 3, for this moderate frequency of excitation, has been noted as due to absence of any interactions between the TS wave and the STWF. This presence and absence of interactions is another basis of classifying transition in [36], where it is noted from the solution of the Navier– Stokes equation for lower frequency of excitations that the TS wave and STWF interact with each other continuously, which has prompted earlier researchers to completely assigning the role of the TS wave in causing transition. In comparison to the pure impulse response of case 1, the “frequency” response of case-3 is completely an artifact of spatial stability theory, which demands the excitation to be at constant frequency, making such an experiment to be confined in the controlled ambience of experimental facilities and very difficult to identify with natural transition. Moreover in such a case, there is a constant need to pump in disturbance energy via the wall-normal velocity boundary condition at the wall; for this reason, in the last column of Figure 6.3, the disturbance streamwise velocity signal is ten times stronger, as compared to the signal for case 1. As cases 1 and 3 can be identified as the impulse and “frequency” responses of the dynamical system, the authors in [509] were encouraged to investigate the cases 2a

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to 2c to suggest other methods by which the STWF can be created, which will grow significantly leading to flow transition. These cases show the presence of only two elements of the response field with the local solution (as in case 3) and the growing STWF. The distinctive feature of cases 2a to 2c is the absence of any TS wave. In that respect, the response field for these cases are similar to that observed for case-1. It now also becomes apparent why the cases are numbered as such, with case 1 having only one element of STWF, while cases 2 have two elements of local solution and the STWF, whereas case 3 has all the three elements. Thus, in all the cases in Figure 6.3, the STWF is the common element which is the only true precursor of transition. For case 1 and cases 2a to 2c, one does not have TS waves at all, and yet the generated STWF is capable of causing sustained disturbance growth by the linear mechanism. It is noteworthy that cases 1 and 2a are similar for the localized wall excitation. Whereas in case 1, only an instantaneous pulse causes the disturbance that grows with time; in case 2a, the exciter remains localized, whereas the subsequent wall excitation is constant without time variation. Case 2b is similar to case 2a, but the excitation startup is not impulsive. Instead for this case, the wall excitation is increased gradually following the error function given by Eq. (6.4). This case has been undertaken to show that one does not require an impulsive start to create STWF and answers the legitimate question: Whether the STWF arises due to impulsive start or not? This has been shown earlier with the solution of the two-dimensional Navier–Stokes equation in [36]. The same has been shown in [509] from the solution of the Orr–Sommerfeld equation, as the linearized analysis. To emphasize the same issue of creating STWF by nonimpulsive start-up, the case 2c has been considered, with the localized excitation given in Eq. (6.5), so that the final approach of vw to  is achieved via a slower acceleration, as can be seen in Figure 6.1(b). For the ramp start, one notes the presence of two STWFs, which are caused by the discontinuous rate of change of wall excitation at the beginning and end of the ramp. The responses for cases 2b and 2c build up slower, and for that reason, one needs to see the response fields magnified twenty times. In Table 6.1, there are cases with start-up phase when vw increases with weaker acceleration, yet one can see the presence of ever-growing STWF in all the cases. This is the main feature for all the cases considered in Figure 6.3. For case 3 in Figure 6.3, a region of instability predicted by the linear spatial theory for monochromatic perturbation that is marked by vertical dotted lines. The maximum amplitude of STWFs indicated by udm is shown in Figure 6.4(a) for all the cases shown in Table 6.1. In plotting the frames of Figure 6.4, the local solution is excluded, so that one essentially focuses only on the asymptotic part of the solution along with the STWF. The solutions are displayed in the figure for x˜ ≥ 38. The ordinate is shown in logarithmic scale and one notices that cases 2a and 3 show the maximum amplitude with almost identical signal levels, as input energy is continuously pumped. By contrast, the other three cases show lower values of udm . However, the main point to note is that the temporal growth rate is log-linear, having the same slope. This frame also indicates a similar trend that depends on the input amplitude, whereas the growth trajectory shows the same temporal rate. The

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100

(a)

10–2 10–4 udm

case-2a case-3 case-1 case-2b case-2c

10–6 10–8 10–10 10–12 0

500

~t

1000

1500

2000

1500

2000

(b)

1000 xudm 500 0

0

~t

500

(c)

1000

~ t = 1750

0.005 ud

0 –0.005 –200

–100

~ x

0

100

Figure 6.4 (a) Temporal evolution of the maximum amplitude of the STWFs. (b) Corresponding streamwise location of this maximum of STWF plotted as function of time. (c) Streamwise disturbance velocity ud for all the cases showing near-field of the exciter at t˜ = 1750; vertical dotted lines are for case 3 to show the region of instability as predicted by the linear spatial theory. All the results are for the height, y˜ = 0.278. [Reproduced from “Is Tollmien-Schlichting wave necessary for transition of zero pressure gradient boundary layer flow?”, P. Sundaram, T. K. Sengupta, and S. Sengupta, Phys. Fluids, vol. 31, pp 031701 (2019), with the permission of AIP Publishing.] corresponding spatial location of udm is plotted in Figure 6.4(b). Before the STWF appears, one notices a very rapid growth rate in Figure 6.4(a) during the transient stage, which also relates to the nonmodal growth obtained from the solution of the Orr–Sommerfeld equation. This also indicates that the use of Bromwich contour integral with the Fourier–Laplace transform is perfectly capable of picking up the transient growth of nonmodal components, despite unsubstantiated claims by some

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[399, 400] against the use of the Fourier–Laplace transform for transient growth. This has spawned other models which propose transient growth by empirical models [251, 361, 400, 401, 532]. The rapid early growth in Figure 6.4(a) is also related to the growth of STWF, which is obtained from the same governing equation without additional models for transient growth, as has been reported by various researchers in [23, 398, 532]. Moreover, the depicted time growth rate is not the same as that is reported in [23]. The early higher growth rate noted specially for case 2b is explained as relating to the growth of the STWF at the onset. In Figure 6.4(a), the growth of the STWF is log-linear at later stages, which implies that the growth is exponential, and thus is inherent for the Fourier–Laplace transform in formulating the linear spatiotemporal receptivity by the Orr–Sommerfeld equation.

6.4.1 Time scales and meaning of transient growth It is ambiguous to talk about transient growth without noting the time scales used in different solution procedures. Note that the solution of the Orr–Sommerfeld equation uses the time scale δ∗ /Ue , whereas the solution of the Navier–Stokes equation uses the convection time scale, and the ratio between the two is different by two to three orders of magnitude, as one notes by comparing the presented results in [509] with those reported in [32, 34, 36]. Thus, the single digit time instants of the Navier– Stokes solution can be in hundreds for the corresponding time of solution obtained from the solution of the Orr–Sommerfeld equation. It is emphasized that the Orr– Sommerfeld solutions are therefore obtained with extreme precision, which is not achievable by present day solution methodologies of the Navier–Stokes equation. The corresponding spatial locations of udm are shown in Figure 6.4(b); it also shows that despite the different nature of excitations in Table 6.1, all the STWFs convect with the same constant speed, as indicated by the identical slopes for all the excitation types. The solutions for ud close to the location of exciter at the later time of t˜ = 1750 are shown for all the excitation cases in Figure 6.4(c). For case 3, the region of spatial linear instability is marked by the dotted vertical lines. As the exciter is placed downstream ˜ = 1000 and ω of this spatially unstable zone for the computed case, for Re ¯ 0 = 0.16, the TS wave for case 3 decays in Figure 6.4(c). One can also note clearly that all the other four cases (cases 1 and 2) do not exhibit the TS wave, establishing the fact that TS wave is not necessary for linear spatio-temporal instability. With respect to the results that will be shown in Chapter 7 for the Navier–Stokes equation, the time range shown in Figure 6.4 will be viewed as the early transient period, with a scale factor of hundred for converting the Orr–Sommerfeld equation solution time to the solution time for the Navier–Stokes equation.

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ud

~t = 55

case-2b

ud 10–6

10–10 10–12

~ x = 38 100

10–8 200

x

10–4

300

100

400

~t = 155

ud 10–6

10–8

10–8 100

200

x

10–4

300

200

x

10–4

ud 10–6

ud

~t = 105

10–4

10

–8

400

100

~t = 255

200

x

10–4

ud

10

–6

300

400

~ t = 205

300

400

~t = 305

10–6 10–8

10–8 100

200

x

300

400

100

200

x

300

400

Figure 6.5 (a) The near-field solution shown for case-2b, at the indicated time instants, with the identified range of x˜ ≥ 38 for which the solution is scanned for udm which is plotted in Figure 6.4. All the results are for the height, y˜ = 0.278, with the filled circles for udm for x˜ ≥ 38. It is noted that for these early times, the growth of STWF is very rapid, as compared to that observed at later times.

6.4.2 Transient growth for smooth start-up Rapid growth of udm early for case 2b is due to the emergence of the STWF from the site of the local solution, as shown in Figure 6.5. The maximum for STWF in this figure is identified in a manner to avoid confusion with the local solution, by looking at the solution for x˜ ≥ 38; the frames capture udm only during the time range 55 ≤ t˜ ≤ 305, during which time, case 2b shows growth of udm from the level of machine zero to the level of 10−4 , a very large dynamic range caused by the wall excitation changing smoothly from zero acceleration to the maximum acceleration achieved at t˜ = 150. This case, with single STWF, evolves slowly with least acceleration during this time range; the solution is shown by its absolute value in all the frames of Figure 6.5. This figure explains the rapid growth of ud for case 2b to be solely due to growth of the STWF, and the maximum is associated with it only, as there is no TS wave created for this excitation pattern without any imposed time scale.

6.5 Non-Oscillatory Start-Up Having noted the distinction between the impulse and the “frequency” responses, one notes the absence of TS wave for the former. At the same time, both the cases have STWF. Thus, STWF is the only common element between the solutions for the impulse and the “frequency” response cases. The ever-growing STWF for both the impulse and “frequency” response cases show that it is not necessary to impose any specific time-scale to cause transition. However, imposition of time-scale helps

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in creating the TS wave, which helps in transition for the lower frequencies—those closer to Branch-I of the neutral curve, where STWF is constantly fed by TS wave, and which does not remain stationary. These cases have been termed as interacting or I-type transition cases in [36]. Keeping in view that TS wave is not necessary for transition, the responses of Blasius boundary layer to wall excitations are reported; these responses are associated with a sudden jump used as the input, without any oscillation frequency associated with the input. They explain the nonmodal route of instability.

6.5.1 Non-impulsive wall excitation: Ramp and error function start-up In these cases, the input disturbance stream function is from Eq. (6.4) for the error function with αE = 100 and t0 = 112.5, and the ramp function increases linearly from zero at t = 0 to unit value at t = 225. To solve the Orr–Sommerfeld equation, the boundary conditions are obtained using DFFT of the time signal at the exciter. The results are obtained by solving the Orr–Sommerfeld equation along the Bromwich contours chosen as before. In Figure 6.6, the streamwise component of disturbance velocity is shown as a function of streamwise distance for the two cases at the indicated times. One observes that these cases produce response fields which are one order of magnitude lower as compared to the case shown in Figure 6.3 for the impulsive start case 2a. Due to faster growth rate of the error function excitation case as compared to the linear ramp start-up case, the response field amplitude is higher for the former. However small is the approach of the input disturbance field to the same final value, one notes the creation of the STWF, implying the ubiquitous nature of the STWF. Given sufficient length and presence of wall shear, the STWF will grow eventually to cause transition to turbulence.

6.5.2 Mechanism for the formation of STWF In Figure 6.7, although the response for the non-oscillatory Heaviside function is one order of magnitude higher than the other two cases, the spectra of the response fields indicate that the scales of the STWF for all the three cases are similar with difference in relative amplitude [463]. It is to be noted that the STWF starts forming due to constructive interference in a group. The most effective interference occurs, when the interacting elements have same amplitude and these are essentially neighbors, as has been explained in the Appendix of Chapter 1. In other words, constructive interference occurs at the optimum of the spectrum. Using the BCIM, we note that the spectrum is a continuous function of its argument, even in the neighborhood of the eigenmodes. Hence, in Figure 6.7, one clearly notes that the occurrence of a wavepacket forming a group takes place at the maximum of the spectrum. For all the three wall excitations shown in this figure, the formation of the STWF is thus noted for the same value of α, indicated by the vertical dashed line. Moreover, as one obtained the

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t = 450.00

0.005 ud

Erf: αE = 100 Ramp

Re = 1000

0

–0.005

t = 550.00

0.005 ud

0

–0.005

t = 650.00

0.005 ud

0

–0.005

0.005 ud

t = 750.00

0

–0.005

t = 850.00

0.005 ud

0

–0.005 0

100

~ x

200

300

400

Figure 6.6 Response of Blasius boundary layer to excitation given by error function and ramp ˜ = 1000 during the time interval of 0 ≤ t ≤ 225, with results shown function for Re for y˜ = 0.278. The input is given in the form of a constant disturbance stream function at the exciter location. spectrum along the Bromwich contour, it will not pass through any eigenmodes, and if there are any in the close proximity of the Bromwich contour, then there would be secondary optima. However, the STWF will correspond to a local maximum of the spectrum that will keep growing with time, while the local maximum corresponding to the eigenmode will have the same amplitude at all time. In Figure 6.7, for the three cases shown, absence of any local maximum implies absence of any TS wave.

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t = 450, ~ Re = 1000

10–1

Heaviside Erf: αE = 100 Ramp

10–3

10–5 Ud(α) 10–7 α = 0.273 10–9 10–11 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

α

Figure 6.7 Fourier transform of ud for the three non-oscillatory input cases, whose responses are shown in Figs. 6.3 and 6.6 at the indicated time, t = 450. Thus, the wide-band phenomenon centered around αr ≈ 0.273 for the displayed time up to t = 450, show the formation of the group around the maximum of the nonmodal spectrum in this figure. Subsequently, all the three cases amplify, which is the universal feature of the STWF with a fixed group velocity. This value of group velocity, taking a value of 0.273 for the zero pressure gradient boundary layer, is used for the receptivity study for convecting vortex over the boundary layer.

6.6 Closing Remarks In the previous chapter, we have described how the Bromwich contour integral method was developed first for the signal problem of spatial theory, and then for the full space–time dependent problem of spatio-temporal theory. The connection of the linear receptivity theory with experiments performed for the instability theory was highlighted, and the role of STWF was established. Using the Bromwich contour integral method in solving the Orr–Sommerfeld equation, here the role of spatiotemporal analysis is re-emphasized by performing various case studies. On one side is the case of the experiment described in [405], that has been described as the “frequency” response of the dynamical system, and on the other extreme, lies the case of the experiment reported in [144], which is truly the case of impulse response. Whereas these two are the standard tools of analysis in dynamical system theory, in this chapter the correct perspectives of these flow receptivity experiments

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are described. There are many differences between these two approaches: as impulse response excites only the STWF, the so-called frequency response has three components. Apart from the STWF, one also notices the local solution and the TS wave for the experimental arrangement in [405]. To elucidate the receptivity of the linear framework even better, authors in [509] performed a few other case studies which are given together in Table 6.1, with three new cases that create only the local solution and the STWF. Thus, using the linear receptivity theory, it is now established that the STWF is the common denominator of flow instability and receptivity. The main result that stands out is the lesser significance of the TS wave, as one can now perceive flow transition in the zero-pressure gradient boundary layer by STWF alone, without the TS wave. It is shown that the STWF is the nonmodal disturbance element that causes transition; it does not require any new empirical model, as have been reported by different researchers [398, 400, 532]. In the following chapters, the ramifications of this observation will be discussed by moving from the local to a global analysis via solving the Navier–Stokes equation, in the linear and nonlinear framework to further establish the STWF as the precursor of transition to turbulence; although STWF originates from the linear mechanism governed by the Orr–Sommerfeld equation, the nonlinearity is crucial for its evolution. This point of view perfectly blends with observed transition to turbulence for not only wall-bounded flows, but also for internal flows and free shear layers.

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Nonlinear, Nonparallel Effects on Receptivity, Instability, and Transition

7.1 Introduction Experimental study of flow transition began with the famous pipe flow experiment [365], in which Reynolds took pipes of different diameters and fitted them carefully with bell-mouth shaped entry sections. In Chapter 4, one noted that favorable pressure gradient delays transition by attenuating disturbances. The bell-mouth accelerates the flow, and the resultant favorable pressure gradient stabilizes the flow. Apart from this, Reynolds took other precautions to note that the transition in the pipe flow depends upon the non-dimensional parameter, now known as the Reynolds number, Re = Vd/ν, where V is the centreline velocity and d is the diameter of the pipe. He found that with all extra precautions taken against disturbance growth, the flow can be kept laminar up to Re = 12, 830. It was also noted by Reynolds that this critical value is very sensitive to disturbances in the oncoming flow, before it enters the pipe. Although this might also indicate receptivity of the flow, Reynolds remarked that “this at once suggested the idea that the condition might be one of instability for disturbance of certain magnitude and stable for smaller disturbances.” The relation between input and output amplitudes during disturbance growth is a typical attribute of nonlinear instability. There are other flows, e.g. the Couette flow, which are found to be linearly stable for all Reynolds numbers. This prompted researchers [282, 298] to suggest nonlinear routes of instabilities for such flows. However, it is interesting to note that some authors [399] have stated without proof that “it is easy to verify that the nonlinear terms of the incompressible Navier–Stokes equations are energy preserving: the role of the nonlinear terms is the distribution, scattering and transfer

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of energy, but this reorganization is accomplished in a conservative manner. Energy growth or decay can only come from linear processes.” The presented explanation in this book is contrary to this point of view, with enough evidences provided to show the central and important role of nonlinearity in causing transition [471]. One of the constraints of classical linear instability theories performing local analysis is the adoption of the parallel flow assumption of the equilibrium flow. This has been addressed in review articles by Chomaz [93] and Theofilis [519], with the main emphasis on giving up on local analysis in favor of a global analysis. Whereas linear global instability is highlighted in [519], the role of nonlinearity and nonnormality has been emphasized for specific flow in [93]. Many diverse attempts have been made to overcome these two limitations related to linearity and local analysis. Using weakly nonparallel theories in [53, 142], the authors tried to modify local modal stability theories. However, one may see pages 166–170 for the critique of these methods in [412]. In [191], the authors studied absolute/ convective instabilities for parallel and weakly nonparallel flows following the definition of global modes by Chomaz et al. [94]. One of the original attempts to study flow instability by solving the Navier–Stokes equation began with a number of studies reported by Fasel et al. in [132, 133, 134, 363, 367]. Whereas the early emphasis by the authors was on capturing Tollmien-Schlichting (TS) waves for strictly zero pressure gradient boundary layers in two-dimensions, the later works reported three-dimensional flow transition and capturing coherent structures. In [134], the authors discussed briefly structures that appear to be spatio-temporal wavefronts (STWF), but there were no results or explanations provided for the same in any subsequent publications by this team. A major shortcoming of the work reported by these authors is their continuous use of the Blasius boundary layer as the equilibrium flow, even when the authors seem to have claimed that they were investigating nonparallel effects. Moreover, there have been no efforts by these authors to specifically discuss nonmodal routes of transition. By contrast, the authors in [451] solved the Navier–Stokes equation to provide supporting results for the theoretical explanation of STWF provided by solving the linear and nonlinear Navier–Stokes equations in the physical and spectral plane. The direct presence of modal and nonmodal elements of the response field arising due to flow receptivity and instability was given in a series of publications [418, 452], with the solution of the Navier–Stokes equation used to explain the nonlinear, nonparallel effects reported in [424]; a comprehensive study of the complete route from receptivity to fully developed turbulent stage of two-dimensional flow has been presented for the first time in [422]. The authors have subsequently concluded STWF as the main precursor of transition in [32, 419] for both the two-dimensional and three-dimensional routes to turbulence. The main distinguishing feature of the studies in [32, 34, 419, 422, 424, 508] has been the use of the solution of the Navier–Stokes equation past a semi-infinite flat plate as the equilibrium flow. As a consequence, the equilibrium flow truly embeds the nonlinear, nonparallel effects for the study of receptivity and instability. It may be noted that the authors in [387, 388, 560] have also used Blasius boundary layer as the mean flow in their studies of transition routes. All

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the results presented in this and subsequent chapters have been obtained by using the solution of the Navier–Stokes equation past a semi-infinite flat plate as the equilibrium flow. In this chapter, the focus is only on receptivity and instability, with nonlinear and nonparallel effects incorporated in the true sense, as discussed above. There are other nonlinear, nonparallel studies performed from the various perspectives of basic hydrodynamics, geophysics and capturing coherent structures, which will be described later. A recent review of nonlinear, nonmodal stability theory using the adjoint equation approach is given in [229]. Another set of studies using the solution of the Navier–Stokes equation from the dynamical system theory is given in [125, 221], which is one among many other efforts on studying the transition problem for finding optimal perturbation [89, 91]. All these nonlinear, nonparallel studies are for steady equilibrium flows, whereas some rudimentary discussion is given in [229] with the observation that how this analysis can be modified to probe the nonlinear stability of a time-dependent reference state remains an open question. According to this author, there is little work done on this topic, and attention is drawn to two new theories proposed. These theories are based on the time-dependent Navier–Stokes equation for incompressible flows and which report such instabilities using the concept of growth of disturbance in mechanical energy and disturbance ienstrophy transport equations [431, 452, 465, 466, 474, 475].

O xin = –0.05

Far-field boundary

SBS strip at xex = 3.3771 (Re = 1000)

Edge of shear layer

Outflow boundary

Inflow boundary

U∞

ymax = 1.5

y

x

Flat plate xout = 120 (Re = 5961)

Figure 7.1 Schematic of the flow domain used to compute two-dimensional Navier–Stokes equation for receptivity to simultaneous blowing-suction (SBS) strip at the wall.

7.2 Nonparallel and Nonlinear Effects on Receptivity and Instability In this section, the nonlinear and nonparallel receptivity of the boundary layer formed over a semi-infinite flat plate is explained using the solution of the two-dimensional Navier–Stokes equation for wall excitation at fixed frequency. The excitation is

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triggered by a simultaneous blowing-suction (SBS) strip located near the leading edge of the plate, as shown in Figure 7.1. Specifically, wall-excitation for low Re and high ω ¯ 0 combinations are traced, which are spatially stable in a linear framework; the neutral curve in Figure 4.8 shows such combinations of Re and ω ¯ 0 that are able to produce spatially stable TS wave by linear analysis, as for case-3 of Table 6.1. The original experimental results in [405] show a mismatch with the modal approach of solving the Orr–Sommerfeld equation for the neutral curve. This has led to the proposal of weakly nonparallel theories [53, 142], all of which have subsequently been shown to be inadequate [412], when viewed from a receptivity framework. Early receptivity studies based on low accuracy methods to solve the Navier–Stokes equation were entirely devoted to obtaining TS waves, and as a consequence, the domain considered has been not only severely restricted in the streamwise direction [133], but the authors used Blasius boundary layer as the equilibrium flow. In recent times, high accuracy schemes for solving the Navier–Stokes equation have produced many unknown aspects of instability and receptivity of the boundary layer formed over a semi-infinite flat plate. The results presented here show the STWF (apart from TS waves) that are actually responsible for transition following both the threedimensional and two-dimensional routes shown in [32, 419], respectively. The role of the leading edge of the semi-infinite flat plate cannot be over-emphasized, as the nonparallel effects in this region is seen to be vital. A B C

0.2

Schubauer and Skramstad Linear parallel theory

0.175

R( f 0 )

0.15 P2

0.125 _

ω0

Stable

(Ff)A = 3.5 × 10–4 (Ff)B = 3.0 × 10–4 (Ff)C = 2.5 × 10–4

0.1 P1

0.075

Unstable

0.05 0.025 0

Stable

O 0

Q1 1000

Q2 ~ Re

2000

3000

˜ ω Figure 7.2 Neutral curve in the (Re, ¯ 0 )-plane as obtained by linear spatial theory with parallel flow assumption, compared with experimental results [405]. Identified three frequencies corresponding to OA, OB and OC are stable for TS mode according to this linear theory, which, however, the experiment identified as unstable. The classical linear stability theory depends on: (a) ambient disturbances to be essentially so small that the governing equations can be linearized and, (b) further making the parallel flow assumption, the linearized Navier–Stokes equation can be simplified to the Orr–Sommerfeld equation, (c) which can be then solved in the spatial

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stability framework with the signal problem assumption. One can also solve the Orr– Sommerfeld equation in the temporal framework by specifying the spatial scale as input excitation or in the spatio-temporal framework as already discussed in detail in Chapter 6. In the spatial framework, one obtains the TS wave as the eigen solution. Thus, the TS wave is an artifact of theoretical formulation only; its experimental verification requires appropriate arrangement in the design of the experiment [405]. Experimental verification of TS waves, and thereby the neutral curve was reported first by Schubauer and Skramstad [405]. The experimental points of the neutral curve are identified by discrete symbols in Figure 7.2, enclosing the region where the zero pressure gradient boundary layer supports unstable TS wave for a range of moderate frequencies. The neutral curve obtained by linear theory is shown as a continuous line in the (Re, ω ¯ 0 )-plane in this figure that shows good agreement with experimental results for moderate frequency and moderate Reynolds number combinations. The analysis with parallel flow model assumes implicit instantaneous adjustment of propagating disturbances in an actual growing boundary layer. This model has been proposed in [12, 13], and remained as the standard procedure in the aerospace industry to interpret linear stability for quite some time. A constant physical frequency excitation follows a ray starting from the origin, as explained in Eq. (5.61). For example, if one focuses attention on the single frequency f0 and the excitation source is placed anywhere between O and Q1 , the disturbance travels along OR, decaying up to Q1 (until the ray meets the point P1 ). Between Q1 and Q2 , the disturbance grows in space. In Figure 7.2, the experimental results reveal discrepancies for low-Re, high-ω ¯ 0 combinations. The experimental critical Re is not definitive and is significantly lower than the value obtained by linear instability theory, which is given by Recr = 520. It is to be noted that the experimental details presented in [405] clearly mentions the location of the exciter and the corresponding measurement stations. Thus, the experimental results still remain a source of speculation among researchers. By contrast, the computational receptivity calculations can provide enough details of the experimental results as compared to stability analysis. However, in computations, there is a lack of rigorous tests of the numerical methods used to solve the Navier–Stokes equation. Merely stating grid independence of results is not adequate. Following the linear spatial stability theory, one can track the frequencies corresponding to OA, OB and OC in the figure, which do not create unstable TS waves. However, the experimental results clearly show a finite streamwise extent where wall excitation can create unstable TS waves over a very short streamwise stretch. It is not known if such a small stretch of unstable TS waves will have a lasting effect on flow transition. Thus, this issue has been treated in the past as a pedagogic problem based on nonparallel, nonlinear instability theory. To accommodate the nonparallel, nonlinear growth of distrubances near the leading edge of the semi-infinite flat plate, it is possible to solve the nonlinear receptivity problem by solving the Navier–Stokes equation following the set up of the experiment in [405]. In Figure 7.3 a snapshot for a localized wall excitation at a fixed

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frequency is shown at t = 15, with the solution computed from the Navier–Stokes equation for the excitation frequency corresponding to OB in Figure 7.2. The details of the computations in solving the governing two-dimensional Navier–Stokes equation in stream function–vorticity (ψ, ω)-formulation are as follows. A well-resolved nonuniform grid is used in the physical plane, whereas the equations are solved in a transformed (ξ, η)-plane, using high accuracy compact scheme [413]. The grid transformations are written as x = x(ξ, η) and y = y(ξ, η), which are usually prescribed analytically for the flow over a semi-infinite flat plate having a sharp leading edge. The computational domain used 4000 points in the streamwise (ξ) and 350 grid points in the wall-normal (η) directions. The governing equations for the stream function and vorticity, as explained in Chapter 3, are given by [411, 413] ∂  h2 ∂ψ  ∂  h1 ∂ψ  + = −h1 h2 ω ∂ξ h1 ∂ξ ∂η h2 ∂η

(7.1)

" # ∂ω ∂ω ∂ω 1 ∂  h2 ∂ω  ∂  h1 ∂ω  h1 h2 + h2 u + h1 v = + ∂t ∂ξ ∂η Re ∂ξ h1 ∂ξ ∂η h2 ∂η

(7.2)

where h1 and h2 are the scale factors of transformation given by h21 = xξ2 + y2ξ and h22 = xη2 +y2η . These equations are non-dimensionalized with L as the length scale and U∞ as the velocity scale. The length scale L is so chosen that one can construct a Reynolds number based on it as Re = U∞ν L = 105 . The domain in the streamwise direction is given by −0.1 ≤ x ≤ 50; in the wall-normal direction, the height is equal to L in dimensional units. The details of formulation, numerical methods, boundary and initial conditions are available in [419, 422]. All the distances are referred to with respect to L, which is roughly given as L ≡ 60δ∗D , where δ∗D is the displacement thickness at the outflow of the computational domain. The domain length used is five times longer than those used for two- and three-dimensional simulations in [132, 133, 388]. The computation is set up to mimic the experiment reported in [405], and thus, this is a case of receptivity to time-harmonic wall excitation, with wall-normal velocity given by [133, 419, 424] vwall (x) = Am (x) sin(ω ¯ 0 t),

(7.3)

for x1 ≤ x ≤ x2

where x1 and x2 represent the beginning and the end of the streamwise extent of the SBS strip placed on the wall, as shown in Figure 7.1. On the rest of the plate, zero wall-normal transpiration velocity is imposed, along with no-slip conditions. The amplitude function Am (x) is defined in terms of the amplitude used for the SBS strip [133] as given by x − x1 AFK (x) = a x st − x1

!5

x − x1 −b x st − x1

!4

x − x1 +c x st − x1

!3 (7.4)

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for x1 ≤ x ≤ x st , and x2 − x AFK (x) = −a x2 − x st

!5

x2 − x +b x2 − x st

!4

x2 − x −c x2 − x st

!3 (7.5)

for x st ≤ x ≤ x2 with x st = (x1 + x2 )/2. Here, a = 15.1875, b = 35.4375 and c = 20.25 are used [133] such that the maximum of AFK (x) is unity. Thus, one notes that despite the authors’ claim of using a small wall-perturbation [133], this is a massive wallnormal velocity excitation, which is equal to the free stream speed in magnitude. In comparison, the authors in [419, 424] have truly used a small fraction of AFK (x), with as low a value as 0.2 percent of the free stream speed. The high accuracy compact scheme based methods used for these simulations allow one to take a small fraction of AFK , i.e. Am = α1 AFK . In Figure 7.3, results are shown for the case where the SBS strip is defined by the amplitude in Eq. (7.3) with Am = 0.1AFK . One of the conditions of any receptivity computation to test any instability theory demands that the applied deterministic excitation must be as small as possible. For example, the wall-normal excitation used in [419] is a very low value of Am = 0.002AFK (x), i.e., the wall-normal excitation velocity is 0.2 percent of the free stream speed. Moreover, one uses Eq. (5.61) to relate the circular frequency, ω ¯ 0 with the non-dimensional physical frequency ( f ) as Ff =

ω ¯0 2π f ν = 2 ReL U∞

Here, t = 0 corresponds to the onset of the wall excitation. The solution should be completely damped according to the linear theory for the choice of excitation frequency, F f B = 3.0 × 10−4 . The SBS strip is placed near the leading edge in the range, 0.2915 ≤ x ≤ 0.3815. The corresponding Re is 315.5 based on δ∗ (considering Blasius boundary layer profile), which is lower than Recr ' 520. The excitation starts clearly in the stable range (with respect to TS wave), as seen in Figure 7.2. Solution of the Navier–Stokes equation in Figure 7.3 indicates a local solution, followed by the TS wave, and both of these are preceded by the STWF. The ud shown for y = 0.0028155 is not expected from the modal solution of linear stability theory, according to which only a highly damped TS wave is expected. The numerical solution indicates the presence of multiple wavenumbers and more than one frequency. The local solution and the STWF are expected from linear receptivity theory [451] as described in Chapter 5. However, the solution in Figure 7.3 has features that display nonlinear and nonparallel effects for the TS wave. In Figure 7.4, vorticity time series and their fast Fourier transform (FFT) are shown for the case of Figure 7.3, sampled at a streamwise station given by x = 2, for the three indicated heights. Thus, the sampling points are where the TS wave is downstream of the mean exciter location (x st = 0.3365). Moreover, the chosen heights are very close to the plate, with the first height located at y = 0.0028155. The time series on the

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Nonlinear, Nonparallel Effects on Receptivity, Instability, and Transition 0.4

(Ff)B = 3.0 × 10–4

Local solution

Am = 0.1AFK

0.3 Asymptotic soln.

Xst = 0.3815

wave front

y = 0.0028155, t = 15

0.2 0.1 ud

0

–0.1 –0.2 –0.3 –0.4

0

2

4

x

6

8

10

Figure 7.3 Streamwise disturbance velocity as receptivity solution at t = 15 for the height y = 0.0028155 showing the local, asymptotic and STWF components caused by SBS strip excitation at the plate. The frequency of excitation of the SBS strip is (F f )B = 3.0 × 10−4 . left column shown for lowest height clearly indicates that the time variation is multiperiodic, as noted also from the Fourier transforms on the right of Figure 7.4; hence, the first super-harmonic of the excitation frequency has almost the same amplitude for the lowest height shown. A weak second super-harmonic is also noted for this lowest height. However, for the other two heights, the super-harmonics are significantly subdominant as compared to the fundamental. The disturbance vorticity time series is shown for another case with the exciter located closer to the leading edge of the plate at x st = 0.27 (center of the SBS strip) on the wall in Figure 7.5. All the excitation parameters for this case is identical, including the frequency, with those used for the case shown in Figure 7.4. The vorticity is sampled at identical points, as in case of Figure 7.4. In this case, the exciter being located closer to the leading edge, one would expect to notice higher nonparallel effects. This is clearly evident from the time series and the Fourier transform for y = 0.0028155. One notes the special feature for this height, where the first superharmonic has more than two times higher amplitude as compared to the fundamental. The amplitude for the fundamental frequency remains the same in Figures 7.4 and 7.5. The nonparallel effects are also noticeable for the second height at y = 0.0057698,

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20 15 10 5 ωd 0 –5 –10 –15 –20

Amplitude

1500 1000 500

16

t 18

20

5

10 Frequency

15 x = 2, y = 0.0057698

4000 Amplitude

20 15 10 5 ωd 0 –5 –10 –15 –20

Am = 0.1AFK (Ff)B = 3.0 × 10–4 xex = 0.2915 to 0.3815 x = 2, y = 0.0028155

2000

3000 2000 1000

16

t

18

20

5

Amplitude

20 15 10 5 ωd 0 –5 –10 –15 –20

16

t

18

20

7000 6000 5000 4000 3000 2000 1000 0

10 Frequency

15

x = 2, y = 0.0088695

5

10 Frequency

15

Figure 7.4 Disturbance vorticity time series at x = 2.0 (left) for the case shown in Figure 7.3 are shown here for the three indicated heights. The corresponding Fourier transforms are shown on the right column. shown in Figure 7.5. The time variation is characterized by the presence of superharmonics and many other incommensurate frequencies. These results also clearly show that the assumption of treating it as a signal problem is incorrect. One notices many other frequencies apart from the input frequency of excitation. This assumption is thus shown to be inapplicable due to the nonlinear and nonparallel effects in addition to the reasons provided in Chapters 5 and 6. Presented results also indicate the effects due to shear layer growth, which are visible near the exciter due to the local solution. The signal problem assumption has been removed in the linear spatio-temporal receptivity analysis (the details are provided in Chapter 5). The original developments have been provided in [418, 451] for the spatio-temporal dynamics showing the presence of all the three essential elements of local solution, asymptotic solution and the STWF, as is shown by the Navier–Stokes solution in Figure 7.3. The amplitude of the SBS strip used in the case displayed in Figures 7.4 and 7.5 is one-tenth of that was used in [133], who reported only a moderate frequency excitation case. Here, high frequency excitations at lower Re are considered for those cases that

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Amplitude

40 20

ωd

0 –20 –40 10

12

t

Amplitude

0 –20

10 Frequency

15

20

x=2 y = 0.0057698

4000 3000 2000 1000

12

t

14

5

10 Frequency

8000 Amplitude

50 40 30 20 10 ωd 0 –10 –20 –30 –40 10

x=2 y = 0.002815

5000

20

–40 10

(Ff)B = 3.0 × 10–4 xex = 0.225 to 0.315 Am = 0.1AFK

5

14

40

ωd

6000 5000 4000 3000 2000 1000

12

t

14

15

20

x=2 y = 0.0088695

6000 4000 2000 5

10

Frequency

15

20

Figure 7.5 Disturbance vorticity time series at x = 2.0 (left) for the indicated amplitude and frequency of excitation are shown for the three indicated heights. The corresponding Fourier transforms are shown on the right column. The exciter is centered at x st = 0.27 with a width of 0.09 as in Figure 7.4. show discrepancy between linear stability theory and experiment [405] in Figure 7.2. For the small computational domain considered in [133], only TS waves have been shown from the solution of the Navier–Stokes equation, without showing the local solution and the STWF. The presented results in [424], using high accuracy methods [413] were obtained in a small domain, showing early time results, demonstrating the local solution and TS waves. In comparison, results are presented here in a longer domain, which also show the evolution of the local solution and the STWF. Accuracy of the present method is higher compared to other methods, and one can compute receptivity cases with amplitude of excitation reduced even further by a factor of ten than those shown in Figures 7.3 to 7.5. Detailed longer domain results obtained by solving the Navier–Stokes equation up to the fully developed turbulent stage have been reported with a value of Am = 0.002AFK using a higher accuracy method in [419]. To show the presence of the STWF for the cases considered in Figures 7.4 and 7.5, as they are not due to nonlinearity, another case is computed with Am = 0.01AFK , while keeping the exciter at the same location. In Figure 7.6, the disturbance streamwise velocity and vorticity are compared at t = 50 for y = 0.0028155, as a function of x for the three frequencies noted in Figure 7.2 as OA, OB and OC.

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All the three frequency cases in Figure 7.6 show the three elements of the local solution, asymptotic TS wave and the STWF, as similarly identified in Figure 7.3 for OB, i.e., F f = 3 × 10−4 . In this figure, the disturbance velocity (ud ) is plotted on the left-hand side, while the disturbance vorticity (ωd ) is plotted on the right-hand side. All the cases show no interaction between the TS wave and the STWF, and according to the classification in [36], all these belong to class-N or non-interacting type transition cases. Comparing the STWF for the three F f cases, it is clearly evident that the amplitude is lower for higher frequencies, for both ud and ωd . One also notices that despite the TS wave, both ud and ωd are decaying functions of x, while the STWF keeps growing with space and time. This may seemingly reconcile the discrepancy in stability reported by linear theory and the disturbance growth reported in experiment [405]. Thus, there is a definitive case to re-investigate experimentally the low-Re, highfrequency cases for boundary layers forming over a semi-infinite flat plate, with the excitation source near the leading edge of the plate. 0.08 Am = 0.01AFK , (Ff)C = 2.5 × 10–4 0.06 t = 50, y = 0.0028155 ud = 0.02346 0.04 xex = 0.225 to 0.315 0.02 ud 0 –0.02 –0.04 –0.06 –0.08 0 5 10 x 15 20 25

20 15 10 5 ωd 0 –5 –10 –15

0.08 0.06 0.04 0.02 ud 0 –0.02 –0.04 –0.06 –0.08

20 15 10 5 ωd 0 –5 –10 –15

0.08 0.06 0.04 0.02 ud 0 –0.02 –0.04 –0.06 –0.08

(Ff)B = 3.0 × 10–4 ud = 0.01086

0

5

10

x

15

20

25

(Ff)A = 3.5 × 10–4 ud = 0.01086

0

5

10

x

15

20

25

20 15 10 ωd 5 0 –5 –10 –15

ωd = 4.71661

0

5

10

x

15

20

25

ωd = 1.9389

0

5

10

x

15

20

25

ωd = 0.71376

0

5

10

x

15

20

25

Figure 7.6 Disturbance streamwise velocity (left) and vorticity (right) plotted against streamwise distance at t = 50 for y = 0.0028155. The exciter is located at xex = 0.27 with a width of 0.09. The three indicated frequency cases have amplitude of excitation given by Am = 0.01AFK .

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Nonlinear, Nonparallel Effects on Receptivity, Instability, and Transition

An interesting scenario is noted at a higher height, y = 0.0088695 in Figure 7.7, as compared to the results shown in Figure 7.6. At this height, the STWF for ud has higher amplitude, while ωd has lower amplitude for the STWF. Note the distinct feature of local solution for the disturbance vorticity in Figure 7.7. At the same time, the disturbance velocity shows higher growth, as compared to the corresponding frames in Figure 7.6. The decay for higher frequencies are also noted for this higher height. 0.03 0.02 0.01 0 ud –0.01 –0.02 –0.03 0.03 0.02 0.01 ud 0 –0.01 –0.02 –0.03

ud = 0.027265 Am = 0.01AFK, (Ff)C = 2.5 × 10–4 t = 50, y = 0.0088695 xex = 0.225 to 0.315

6 ωd = 1.2252

4

ωd

2 0 –2 –4

0

5

10

x

–6

15 20 25 (Ff)B = 3.0 × 10–4 ud = 0.01264

0

5

10

x

15

20

25

6 4

ωd = 0.41789

2

ωd

0 –2 –4

0

0.03 0.02 0.01 ud 0 –0.01 –0.02 –0.03 0

5

10

x

15

20

–6

25

(Ff)A = 3.5 × 10

0

5

10

x

15

20

25

6

–4

4

ud = 0.004997

ωd = 0.09677

2

ωd

0 –2 –4

5

10

x

15

20

25

–6

0

5

10

x

15

20

25

Figure 7.7 Disturbance streamwise velocity (left) and vorticity (right) plotted against streamwise distance at t = 50 for a higher height, y = 0.00886952 as compared to that shown in Figure 7.6.

7.3 STWF Created by Impulsive Start In [419], the authors changed the classical view of flow over a zero pressure gradient boundary layer undergoing transition to turbulence. This classic view emphasized the main role played by secondary and nonlinear instability of TS wave that is formed by a primary mechanism from the eigenvalue formulation governed by the Orr–Sommerfeld equation. However, the zero pressure gradient boundary layer has also been shown to display the onset of STWF by a linear mechanism in [451]. Subsequently, the STWF grows by the linear mechanism also governed by the Orr–

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Sommerfeld equation formulated in the parallel flow framework. The authors in [419] were the first to demonstrate that deterministic time-harmonic excitation creates the STWF by solving the two-dimensional Navier–Stokes equation without any limiting assumptions for flow past a semi-infinite flat plate. This STWF upon further tracking leads to the two-dimensional turbulent state. This is one of the first theoretical and numerical demonstration of fully developed turbulence being created by deterministic excitation from the solution of the Navier–Stokes equation. Transition from laminar to turbulent flow can take many routes depending on the associated physical parameters of the equilibrium flow and the excitation pattern. Classical linear viscous instability theories use the local parallel flow approach for spatially growing TS waves inside the neutral curve, in the unstable region shown in Figure 7.2. The linear stability theory has some interesting features. For example, the neutral curve is the same for both spatial and temporal analyses, as can be seen from linear modal analysis based on parallel flow approximation. For mathematical convenience, spatial and temporal growths are often interchanged in normal mode approach where the single dominant TS wave is considered responsible for transition. In the previous chapters, it has been shown that this dependence on normal modes is overcome using linear receptivity analysis by the Bromwich contour integral method. The linear nonmodal approach in [532] assumes a bypass transition route to cause transient/ algebraic growth of disturbances. Despite this, there are other global linear and nonlinear receptivity approaches, as cited in [471]. Historically, the very limited success of linear theory in predicting sub-critical transition has prompted researchers to look for alternate nonlinear theories [254], secondary instabilities [22], non-normal eigenvectors causing large transient energy growth for stable systems [398]. However, it has been pointed out in [532] that such theories can only show some growth for three-dimensional perturbation. Most importantly, all these routes of energy growth and mechanisms are yet to be demonstrated unequivocally by appropriate experiments; those proposing transition by the STWF route reproducing the experimental routes followed in [144, 405] are the exceptions. The results shown in this section are for the actual transition to turbulence, with nonmodal growth dominant from onset by the linear mechanism, and thereafter through the nonlinear growth of STWF. The Navier–Stokes equation has been solved in [419], as shown in the schematic of Figure 7.1, excited by the time-harmonic SBS strip described already. Success in showing STWF growing into a fully developed two-dimensional turbulence following the growth of STWF can be attributed to the accuracy and dispersion relation preservation (DRP) property of the numerical method used, with computations performed for longer times in a longer domain. Solutions from the two-dimensional Navier–Stokes equation is more generic, unlike the collections of all such proposed transient energy growth models, which are limited to three-dimensional flows only [532]. Transition to turbulence was considered as one of the unsolved problems of classical physics without definitive explanation showing its causality. Thus, the study in [419] is a pioneering effort (even the events during two-dimensional turbulence

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was later shown also for the three-dimensional route in [32]) which explained creation of turbulence from receptivity of laminar flow as the response to deterministic wallexcitation. The two-dimensional Navier–Stokes equation is solved using (ψ, ω)formulation due to its inherent accuracy, with wall excitation by an imposed timedependent normal velocity given by vwall (x) = Am (x) sin (ω ¯ 0 t), for x1 ≤ x ≤ x2 , as detailed before. Using the high accuracy DRP method [413], a small fraction of AFK is used for wall excitation, i.e., using only Am = α1 AFK . Here, three amplitudes given by ˜ 1 ) = 656 and α1 = 0.002, 0.01 and 0.05 have been used, with SBS strip defined by Re(x ∗ ˜ 2 ) = 676 [419]. The length scale used is given by L = 16.78δout , with displacement Re(x thickness at the outflow of the domain given by δ∗out . The Reynolds number based on displacement thickness at outflow is 5958 [419]. The computational domain spanned over a very long stretch given by −0.05L ≤ x ≤ 120L, so that the growth of the boundary layer is accounted right from the leading edge, with equilibrium solution obtained from the solution of the Navier–Stokes equation, which match with the Blasius profile everywhere, except near the leading edge. It is noted that the authors in [32, 36, 34, 419, 424, 483, 508] have always reported receptivity calculations with leading edge of the flat plate included inside the computational domain. Thus, these results are novel and unique, incorporating the varying pressure gradient near the leading edge, even if the equilibrium flow eventually settles down to zero pressure gradient Blasius boundary layer, as has been shown in [482] and in Figure 7.8. Transition of fluid flow has been classically studied as an eigenvalue instability problem, which is often referred to as the modal approach [119, 412]. Flow over a semi-infinite flat plate is represented in this framework by the Blasius boundary layer; however, many researchers have highlighted the nonmodal approach to represent a wider class of flow fields (including flows which have been noted to be stable by the linear modal studies) by invoking concepts of algebraic growth [128, 251] and transient nonmodal growth in the linear [398] and nonlinear framework [125, 221, 229]. In addition, some researchers have studied the effects of equilibrium flow distortion, specifically for those flows whose ideal representation (sometimes referred to as the exact solution) marginally departs from the non-ideal situation in theory and experiments [52]. The case in point is the classical approach of studying the instability of zero pressure gradient boundary layer, idealized by the Blasius velocity profile as a solution obtained by applying similarity transformation to the governing Navier– Stokes equation. It is to be noted that such a transformation is valid far away from the leading edge of the semi-infinite flat plate, as shown in Figure 7.8 using a nondimensional co-ordinate system with a length scale (L), such that the Reynolds number based on free-stream speed (U∞ ) and L is given by ReL = 105 . In this figure, the contour plot displays the mean streamwise pressure gradient obtained by solving the timedependent Navier–Stokes equation, as detailed in [36, 419]. In these references, twodimensional transition to turbulence have been shown for flow past a semi-infinite flat plate, whose onset has been created by wall excitation in the form of a harmonically

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excited SBS strip [133, 412]. It is readily noted from Figure 7.8, that the imposed pressure gradient is non-zero in the neighborhood of the leading edge, with adverse pressure gradient noted upstream, while the flow accelerates near the immediate downstream of the leading edge. This is followed by another stage where the flow experiences adverse pressure gradient, which is also non-negligible. It is only after a location given by Re x ≈ 5.2 × 105 , where the streamwise pressure gradient becomes lower than ddxp = 10−4 , displaying the range of validity for the Blasius boundary layer used. In the studies on flow transition by wall excitation in [133, 388], the computational domain is noted to be in the range 30, 393 ≤ Re x ≤ 408, 624, for which the authors used Blasius boundary layer as the equilibrium flow. This provides the justification for solving the Navier–Stokes equation to obtain the equilibrium flow in [36, 419, 424, 466, 471, 475, 509], for which the leading edge of the plate is always inside the computational domain, and which extends up to Re x = 5 × 106 for threedimensional transition routes and Re x = 12 × 106 for two-dimensional transitional studies. 1.5

0.0001

–0.0001

dp dx

1E-05

1 0.1 0.01 0.001 0.0001 1E-05 0 –1E-05 –0.0001 –0.001 –0.01 –0.1 –1

1 y

–0.001

0.5

0

0.0001

0.001

0

5

x

10

15

20

Figure 7.8 Streamwise pressure gradient contours for flow past a semi-infinite flat plate considered to study zero pressure gradient boundary layer. Near the leading edge dp ( x = 0), one notices ref-id variation of dx . The STWF shown in [419] for ud is reproduced for the amplitude control parameter of α1 = 0.01 in Figure 7.9. The non-dimensional frequency of wall excitation (F f ) investigated is 1.0 × 10−4 in [419]. A TS wave is created and it remains rooted at almost the same location in all the three time frames, at t = 56, 179 and 254. The data are sampled at a height located close to the wall at y = 5.7698 × 10−3 . At the early time of t = 56, apart from the TS wave, one notes the already developed STWF (shown as A), whose peak-to-peak amplitude is distinctly more than the TS wave. At later times, not only does this STWF increase first by a combinations of linear and nonlinear mechanisms it also displays a nonlinear saturation later, while at the same time, it induces creation of multiple STWFs. Following the cases described in Table 6.1 of Chapter 6, it is apparent that any time varying event inside the boundary layer is

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Nonlinear, Nonparallel Effects on Receptivity, Instability, and Transition 0.5

α1 = 0.01, F = 1.0 × 10–4

ud / U∞

TS

A

0 (i) t = 56

–0.5 0

20

40

x

60

80

0.5

A B

TS ud / U∞

100

0 (ii) t = 179

–0.5 0

20

40

x

60

80

ud / U∞

0.5

100 A

C

TS 0 (iii) t = 254 –0.5 0

20

40

x 60

80

100

Figure 7.9 Streamwise disturbance velocity (ud ) plotted for the amplitude control factor case of α1 = 0.01 at the indicated times. Note that the STWF at A spawns the second STWF at B, and which in turn creates C upstream of it. The STWF upon reaching nonlinear stage creates turbulent spots, which merges together. [Reproduced from “Onset of turbulence from the receptivity stage of fluid flows”, T. K. Sengupta and S. Bhaumik, Phys. Rev. L., vol. 107(15), 154501 (2011), with the permission of APS Physics.] capable of spawning another STWF, as one noted that even the ramp start-up created two STWFs, following the basic physical process involved in the impulse response excitation shown in the experiments of [144]. While the leading STWF convects ˜ is prone to formation of another STWF downstream, the flow behind it at high Re behind the leading one. These sequence will continue till a steady state is reached and a fixed transition region is established in the boundary layer, where the turbulent flow

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reaches a statistical steady state. It is obviously apparent that during this evolution process, the disturbance propagates upstream. It is for this reason that one must view the parabolized stability equation (PSE) approach for transition prediction purely as a modeling activity. This topic of PSE is not discussed further for the same reason. The growth and propagation of the STWF severely distorts the upstream mean flow, causing unsteady separation bubbles upstream, which can be termed as secondary instabilities in Figure 7.9. In the second frame at t = 179, the first STWF, A is dispersed over a long streamwise stretch, that induces the second STWF noted as B. Continuous feeding of energy by the exciter into the TS wave and an induced upwelling of the downstream separation bubbles are responsible for the appearance of further STWFs. In the bottom frame at t = 254, apart from A and B, a third STWF is noted to grow as C. This sequence of successive creation and growth of STWF is symptomatic of the regeneration mechanism of turbulence. α1 = 0.01, F = 1.0 × 10–4, Xex = 1.5 x = 60

x = 40

0.4

B

A A 0.2 B ud / U∞

C TS

0

–0.2

–0.4

0

50

100

Time

150

200

250

Figure 7.10 Streamwise disturbance velocity (ud ) plotted as a function of time for the amplitude control factor case of α1 = 0.01 at the indicated x-locations and y = 5.7698 × 10−3 . [Reproduced from “Onset of turbulence from the receptivity stage of fluid flows”, T. K. Sengupta and S. Bhaumik, Phys. Rev. L., vol. 107(15), 154501 (2011), with the permission of APS Physics.] For the studied flow field, the region from the receptivity to fully developed turbulent flow stage, would also include the transitional flow region. An observer located at a fixed station in this region will note the flow as intermittent, as shown in Figure 7.10 for ud at two streamwise indicated stations at x = 40 and 60 for the same

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Nonlinear, Nonparallel Effects on Receptivity, Instability, and Transition

height. In keeping with the notations of marking the STWF in Figure 7.9, the time variation of ud due to successive STWF are also marked as A, B and C in this figure. From such a time series, one can calculate the intermittency factor of the flow as a function of space using the DNS data.

7.3.1 Effects of excitation amplitude of wall-normal velocity The effects of the amplitude of wall excitation on the flow evolution from primary instability to the turbulent spot stage is shown by ud -variation in the streamwise direction at t = 192 in Figure 7.11, for the three chosen value of amplitude control factor, α1 = 0.002, 0.01 and 0.05, with successive input amplitude five times higher than the predecessor. For this reason, the scale of the ordinate is different in the top (i) α1 = 0.002

ud / U∞

0.02

0

–0.02

0

20

ud / U∞

x

60

80

100

(ii) α1 = 0.01

0.5

0

–0.5

0

20

0.5 ud / U∞

40

40

x

60

80

100

60

80

100

(iii) α1 = 0.05

0

–0.5

0

20

40

x

Figure 7.11 Streamwise disturbance velocity (ud ) plotted for (i) α1 = 0.002, (ii) α1 = 0.01 and (iii) α1 = 0.05 at t = 192 to show the effects of input amplitude, as indicated by the value of α1 . [Reproduced from “Onset of turbulence from the receptivity stage of fluid flows”, T. K. Sengupta and S. Bhaumik, Phys. Rev. L., vol. 107(15), 154501 (2011), with the permission of APS Physics.] from the bottom two frames shown in Figure 7.11. The TS wave remains rooted about the same mean location, while the width of the TS wave-packet increases with α1 .

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Moreover, due to such wide variation of input amplitude, the TS wave structures are also dissimilar in these frames. It is important to note that due to the use of high accuracy computing methods, the highest amplitude is merely 5% of the amplitude used in [133]. This aspect is desirable for any receptivity study, with input amplitude as small as possible. This was also noted in the linear receptivity studies reported in [509].

α1 = 0.05

Amplitude

0.4

α1 = 0.002

α1 = 0.01

0.2

0 0

50

100

150 Time

200

250

300

Figure 7.12 Maximum amplitude of Streamwise disturbance velocity (ud ) plotted for (i) α1 = 0.002, (ii) α1 = 0.01 and (iii) α1 = 0.05 as function of time, for the three values of α1 . [Reproduced from “Onset of turbulence from the receptivity stage of fluid flows”, T. K. Sengupta and S. Bhaumik, Phys. Rev. L., vol. 107(15), 154501 (2011), with the permission of APS Physics.] Here, the streamwise disturbance velocity (ud ) does not scale with input amplitude, and thus, it indicates nonlinear, nonparallel effects on receptivity and instability of the boundary layer over the semi-infinite flat plate. It is important to note that for the case of α1 = 0.05, nearly 50 % reduction of peak amplitude of TS wave is with respect to the scaled response of the lowest amplitude case. The turbulent spotlike behavior is the signature of unsteady separation bubbles that form in the late transitional flow due to nonlinear saturation of STWF for the two higher amplitude cases, with rapid intermittent variation of ud . The STWF appears earlier for the higher excitation amplitude cases; it also evolves faster for these higher amplitude cases. The bottom two frames show the larger number of saturated STWFs for the higher amplitude cases, which as a consequence affects the longer upstream stretch. This does not imply any qualitative differences for the long time dynamics for the lowest amplitude, α1 = 0.002 case, as compared to the other two cases.

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Nonlinear, Nonparallel Effects on Receptivity, Instability, and Transition

10–2

10–2

(i) t = 19

10–3 Ud(k)

Ud(k)

10–3 10–4 10–5 k

10–6

50 100

50 100 (iv) t = 94

10–2

10–3 10–4

10–3 10–4

10–5 10–6

k

10–5 (v) t = 194 Ud(k)

101 100 10–1 10–2 10–3 10–4 10–5 10–6

k

50 100 (vii) t = 264

k–3 k

k

50 100

Ud(k)

Ud(k)

k

10–1

(iii) t = 34 Ud(k)

Ud(k)

10–2

Ud(k)

10–4 10–5

10–6

102 101 100 10–1 10–2 10–3 10–4

(ii) t = 24

50 100

102 101 100 10–1 10–2 10–3 10–4 103 102 101 100 10–1 10–2 10–3 10–4

50

100

(vi) t = 234

k–3 k

50 100 (viii) t = 284 k–5/3

k–3 k

50 100

Figure 7.13 Fourier transform of the STWF associated with ud is shown at the indicated times for the lowest amplitude of excitation, α = 0.002. [Reproduced from “Onset of turbulence from the receptivity stage of fluid flows”, T. K. Sengupta and S. Bhaumik, Phys. Rev. L., vol. 107(15), 154501 (2011), with the permission of APS Physics.] Variations of peak-to-peak maximum amplitude of ud are plotted in Figure 7.12 as a function of time for the three different α1 values. Early exponential growth is prominently noted for the lower excitation amplitude cases of α1 = 0.002 and 0.01. This is expected for the validity of the linear receptivity analysis [451]. However at later

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times, even the lowest amplitude case of α1 = 0.002 shows the unsteady separation at around t = 260, indicating unsteady separation associated with the secondary stage of transitional to fully developed turbulent flows. Thus, the nonlinear saturation following the different early time behavior indicates some form of universality for all the input amplitude cases. However, when the excitation amplitude is increased to α1 = 0.05, the early exponential growth during linear stage is absent, and the maximum ud begins with a significantly larger amplitude due to separation bubbles induced immediately upon the onset of wall excitation. It will be shown in Chapter 9 that this type of unsteady bubble formation is quite common during transition following a bypass route caused by free stream convecting vortex [431, 474, 475]. Once the unsteady separation leading to bubble formation occurs, the value of ud saturates to a high fraction of the free stream speed for all the excitation amplitude cases. To explain the scale selection, as indicated by the spectrum of ud , the lowest amplitude case of α1 = 0.002 is chosen. For such a case, one notes a long time evolution of the flow field, starting from the linear receptivity to a fully developed turbulent flow stage. This is shown in the log–log plot in Figure 7.13, which displays the Fourier transform of ud plotted at different t. For the first frame at t = 19, the spectrum shows only two dominant peaks, flanked by weaker side-lobes. The STWF during the linearized growth stage cleanly detaches from the TS wave, and is unhindered in the front allowing it to grow over a longer stretch. Such long length scale events are responsible for the peak at lower k values. Thereafter, the amplitude decreases further due to redistribution of energy to many other scales, i.e., the dispersion is noted up to t = 74, with the spectrum becoming wider in the front, shifting wavenumbers of the packet to lower values. After the dispersion reduces, one notices the linearized growth showing the exponential growth of peak-to-peak amplitude, which can be noted in the frames at t = 34 and 94. This growth continues till t = 200 (as noted in Figure 7.12, without any significant flow distortion till t = 220). Following this, the total flow is distorted by wave-induced stresses [408, 528], with the appearance of unsteady separation bubbles. This shows that the linear receptivity analysis holds up to t = 220 for this amplitude value of excitation. From t = 234 onwards, the spectrum shows a k−3 variation of Ud (k), which is typical of two-dimensional turbulence enstrophy cascades [21, 247]. In the last frame at t = 284, one continues to see the two-dimensional spectrum, while a line for three-dimensional spectrum line given by k−5/3 is shown for the sake of comparison. This is often attributed to inverse energy cascades of twodimensional turbulence. Results show that the flow changes from the laminar to a fully developed turbulent stage, indicating how the spectral peaks emerge, superpose and eventually show the wide-band structure, along with appearance of some low wavenumber peaks. This latter phenomenon has often been termed as inverse cascade in the literature.

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7.4 STWF Created via Non-Impulsive Start In the previous chapter, it is noted that there are various ways of causing instability for the Blasius boundary layer, without even creating TS waves by infinitesimal wallnormal perturbation at the wall. These counter-intuitive examples are noteworthy as the transition research community were convinced that the TS wave is the precursor of turbulence, which is obtained by the linear spatial theory and was verified by the experiments in [405]. The results reported in [509] and reproduction of the linear receptivity results in Chapter 6 show conclusively that the TS wave is certainly not an essential element for causing transition with infinitesimal perturbation for the zero pressure gradient boundary layer. In this chapter, we discuss the creation and growth of STWF using the solutions of the Navier–Stokes equation to further support the observations of the previous chapter. The creation and growth of STWF occur for a varying range of non-impulsive wall excitation, as can be seen from the solution of the Navier–Stokes equation [36]. This should dispel any remaining doubts about the role of STWF in transition, as it includes all nonparallel and nonlinear effects. The remaining problem to be considered is if STWFs can be created by wall excitation started non-impulsively. The start-up effects have also been explained with the results in [36], by showing efficacy of creation of the SWTF by both impulsive and non-impulsive start for two-dimensional routes. Moreover, the distinction between high and low frequency excitations is shown, with each following different routes of transition. The start of the excitation is considered to be with finite time rate for non-impulsive start-up. When a time-harmonic excitation is used, the onset of STWF is due to the finite time start of this harmonic wall excitation that eventually takes the flow to the fully developed turbulent state. The creation of STWF has already been verified from the solution of the Navier– Stokes equation for which the time-harmonic wall excitation is started impulsively in the studies reported in [419, 422]. It has been shown that the transition follows mainly two routes for low amplitude, time-harmonic wall excitation: (a) With moderate to high-frequency excitation, the STWF is responsible for the eventual transition, in which the TS wave-packet remains spatially located close to the exciter. This is termed as the non-interacting or class-N type transition and (b) With lower frequency wall excitation, the STWF and TS wave-packet continually interacts to cause interacting or class-I type transition [36]. Although these have been created by impulsive timeharmonic wall excitation in [419, 422], it is also necessary to present results when the start-up of harmonic excitation is not impulsive, as has been shown for these two routes of transitions in [36]. For the receptivity problem with wall excitation started non-impulsively, a timedependent wall-normal velocity imposed by the SBS strip is prescribed in [36] vw = α1 A(t − t0 ) Am (x) sin (ω ¯ 0 t)

(7.6)

The non-dimensional circular frequency (ω ¯ 0 ) is related to physical frequency ( f ) by 2 F f = 2πν f /U∞ = ReL ω ¯ 0 ; Am (x) is as defined before for the SBS strip, and A(t − t0 ) is the

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time-dependent amplitude function corresponding to impulsive and non-impulsive start-up processes. The quantity α1 acts as an additional amplitude control parameter. For impulsive start at t = t0 , A(t − t0 ) is the Heaviside function H1 (t − t0 ). For nonimpulsive smooth start-up, the following expression for A(t − t0 ) has been used [36]  t − t  1 0 A(t − t0 ) = 1 + erf √ (7.7) 2 2 παE where αE controls the smoothness of the excitation onset. A larger value of αE implies more gradual onset of the excitation, whereas at the other extreme, for αE → 0, one obtains A(t − t0 ) → H1 (t − t0 ), the Heaviside function. The utilized computational domain for flow over the semi-infinite flat plate has the outflow placed at 120L, with the origin located at the leading edge of the plate. The height of the computational domain is located at y∗max = 1.5L. Based on the displacement thickness at the outflow δ∗ , the Reynolds number at the outflow is noted to be Reδ∗ = 5958. Based on the length scale L, the Reynolds number is ReL = 105 . The boundary layer, with or without the excitation, has to be well resolved with clustered grid points in the wall-normal direction by an analytic tangent hyperbolic function [413]. Similar clustering of grid points are also used in the streamwise direction around the leading edge of the plate (details of the numerical methods involved are discussed in [36]). Along the x direction, grid points are stretched up to x∗ = 10L, and uniform distribution of points are prescribed thereafter. The computed results have been obtained using 12,001 points in the streamwise direction and 401 points in the wall-normal direction. The non-impulsive and impulsive start-up cases are compared for time-harmonic wall excitation; the results for the impulsive start-up excitation case is shown in Figure 7.14, with two distinct receptivity routes. The amplitude control parameter for all the cases considered in [36] is α1 = 0.002. For a moderate frequency of F f = 7.5 × 10−5 , the STWF and the TS wave-packet do not interact with each other, as shown in the left columns of Figure 7.14. For this case, the STWF amplifies, leading to an eventual flow transition, with the TS wave-packet remaining unaffected, appearing almost stationary for this class-N or the non-interacting route of transition. When the excitation frequency is halved to F f = 3.75 × 10−5 , one notices interaction between the STWF and the TS wave continually, leading to an evolving spatio-temporal, elongated wave-packet, whose leading edge suffers nonlinear saturation due to formation of unsteady separation bubble. Thus, this is the class-I or the interacting route of transition. The results of class-N route of receptivity is shown in Figure 7.14(a)–(d) for F f = 7.5 × 10−5 , when the exciter is located at x st = 3.5 (Reδ∗ = 1018 at x = x st ). Results show ud as a function of x, for a height of y = 0.00572, at the indicated times. Both STWF and TS waves are noted in the first frame of Figure 7.14(a). The TS wave and the STWF separate early, and these remain distinct with no visual interactions. The STWF initially grows and disperses via linear mechanism in frames (a) and (b). This is followed by growth via secondary/ nonlinear instabilities. Finally, nonlinear

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Nonlinear, Nonparallel Effects on Receptivity, Instability, and Transition

(a) t = 30 Ff = 7.5 × 10–5

0.04

ud

xst = 3.5 0

STWF

–0.04

TS wave-packet 0

20

–0.01 0

–0.1

–0.1 20

40 (c) t = 60

x

60

0

ud 0

–0.4

–0.4 20

40

x

60

(d) t = 117

0.4

0

ud 0

–0.4

–0.4 20

40

60

x

60

0

40

20

40 x (g) t = 60

60

20

40 x (h) t = 105

60

20

40

60

(f) t = 45

0.4

0

0

x

0.4

0

0

20

0.1

ud 0

0.4

ud

60

xst = 2.25

ud 0

0

0

ud

x

(b) t = 42

0.1

ud

40

(e) t = 30 Ff = 3.75 × 10–5

0.01

x

Figure 7.14 (a)-(d): Class-N route of transition for F f = 7.5 × 10−5 and exciter located at xex = 3.5. (e)-(h): Class-I route of transition for F f = 3.75×10−5 and x st = 2.25. For both the cases, the amplitude control parameter is α1 = 0.002. saturation of the disturbance is noted, when ud in the boundary layer is of the order of free-stream speed (as seen at t = 60 and t = 117 in frames (c) and (d) of Figure 7.14). As noted before, the STWF during the nonlinear stage spawns secondary STWFs owing to flow distortion by the primary instability, and then the disturbances grow at a higher rate in the presence of other saturated STWFs. Regeneration and mergers of new secondary STWFs expands the streamwise stretch of the highly unsteady zone in Figure 7.14(d). Thus, for this route of transition, the STWF plays the dominant role causing flow transition.

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Evolution of disturbances following the Class-I route of receptivity are shown in frames (e) to (h) of Figure 7.14 for the time-harmonic wall excitation with frequency of F f = 3.75 × 10−5 and its location given by x st = 2.25. The extent of linear spatial instability is noted by two vertical dashed lines in these frames. Here, one cannot distinguish between the TS wave and STWF in frames (e) and (f), as these elements interact from their onset. Importantly, the TS wave-packet also exhibits spatiotemporal variation; hence, the bounds of linear instability has no correlation with the results obtained by solving the time-dependent full Navier–Stokes equation. At later times, the combined disturbance packet forms a single unit, which has been termed as the spatio-temporal wave-packet (STWP) in [36], to distinguish it from STWF. This STWP continually changes with time, with its trailing edge moving out of the linearly unstable region. The leading edge of the STWP moves relatively faster than the trailing edge, stretching the unstable region noted in frames (g) and (h). Thus, the combined TS wave and STWF in frames (e)–(h) show that for Class-I route of transition both the elements have dominant roles from the onset, with the STWF dominant from early times. Next, a case is considered when the time-harmonic excitation is switched on smoothly in a non-impulsive manner. The dynamics of STWF for the receptivity route is shown in Figure 7.15 for excitation parameters given by F f = 7.5 × 10−5 , x st = 3.5, and α1 = 0.002. This is a case of excitation with αE = 0.1 and t0 = 1.5. A primary STWF smoothly comes out of the TS wave at an early time. In frame (b) of Figure 7.15 at t = 165, the amplitude of the STWF is much larger than the TS wave. In these two frames (a) and (b), the TS wave amplitude remains virtually the same, whereas STWF amplitude increases by an order of magnitude. Comparing Figure 7.14(a) and Figure 7.15(a), the TS wave amplitude is noted to be almost similar for these two cases. However, the amplitude of the STWF after the onset is noted at t = 30 in Figure 7.14(a) to be significantly lower, with the smooth start-up of excitation. This is orders of magnitude higher compared to the case in Figure 7.15(a). Subsequently, the STWF in Figure 7.14 shows earlier rapid nonlinear growth, as can be seen after the time noted in frame (b). The initial amplitude of the STWF is lower in Figure 7.15. The eventual flow transition in this case follows the Class-N route of disturbance growth, which is responsible for nonlinear breakdown. At t = 237 in Figure 7.15(d), amplitude of the disturbance velocity inside the boundary layer is of the order of free stream velocity, whereas the TS wave displays very negligible amplification. The authors in [36] noted that other simulations performed for lower αE cases (i.e., the start-up approaching impulsive start case) show that by lowering αE , the amplitude of the STWF increases; one also notices earlier onset of nonlinear growth-stage. For all lower αE cases, flow transition follows Class-N route for moderate to high frequency of wall excitation. Further investigation of a non-impulsive case is performed, and its results are shown in Figure 7.16, for a case with the excitation amplitude same as in the previous case in Figure 7.15, i.e., for α1 = 0.002. The only difference is the placement of the exciter and reducing the frequency of excitation by a factor of two to F f = 3.75 × 10−5 . In this case, one notes that ud follows the Class-I type route, as noted in Figures 7.14(e)

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Nonlinear, Nonparallel Effects on Receptivity, Instability, and Transition (b)

(a)

0.4 0.2

Ff = 7.5 × 10–5

t = 132 0.4

xst = 3.5 αE = 0.1

ud

ud

–0.4 0

TS wave-packet

50 x

(c)

0.4

0

–0.2

100

0

STWF

0

ud

0

100

t = 237 STWF

0.4 0.2

ud

50 x

(d)

t = 195

0.2

–0.2

–0.2

–0.4 0

t = 165

0.2

STWF

0

–0.2

STWF

–0.4 50 x

100

0

50 x

100

Figure 7.15 Disturbance evolution for non-impulsive start-up with αE = 0.1 for, F f = 7.5 × 10−5 , x st = 3.5, t0 = 1.5 and α1 = 0.002. All computations are done with 12001 points in the range −0.05L ≤ x ≤ 120L and 401 points in the y-direction clustered near the wall. to (h), for the impulsive excitation case. Here, the non-impulsive start is characterized by αE = 0.1 and t0 = 1.5. For this case, the primary STWF interacts with the TS wave, which remains trapped and the combined STWP displays significant spatiotemporal variation. The STWP amplifies faster here, showing secondary/ nonlinear stage of STWP evolution with a sharp leading edge, as noted in Figure 7.16(b). The growth of the disturbance front is identical with that noted for the impulsive startup case shown in Figure 7.14(e)–(h) for α1 = 0.002. Thus, whether transition follows the Class-N or Class-I route will depend upon the frequency of wall-excitation. The comparison between results shown in Figures 7.14 and 7.16 are for the same frequency, although the latter has been started non-impulsively. One notes that for Figure 7.14, the harmonic excitation is started impulsively, whereas for the case of Figure 7.16, the harmonic excitation is started non-impulsively, with significantly lower initial acceleration, characterized by αE = 0.1.

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(a)

0.4

xst = 2.25 αE = 0.1

0.4 t = 66

t = 57 0.2

0

ud

ud

0.2

Ff = 3.75 × 10–5

–0.2

–0.4 0

–0.2

20

x

40

60

–0.4 0

80

t = 75

x

40

60

80

t = 114

0.4

0.2

0.2

0

ud

ud

20

(d)

(c)

0.4

–0.2

–0.4 0

0

0 –0.2

20

x

40

60

80

–0.4 0

20

40

60

80

x

Figure 7.16 Disturbance evolution shown for non-impulsive onset of excitation with αE = 0.1. Here, for this same amplitude of excitation, F f = 3.75 × 10−5 , xex = 2.25, t0 = 1.5 and α1 = 0.002, with the other two parameters remaining the same as in Figure 7.15.

7.4.1 Receptivity to high amplitude wall excitation Another popular nomenclature of bypass transition is for one of the routes which is not associated with TS waves at all [298, 412]. It is implied in this classification that for very low amplitude of excitations, one observes TS waves, which is absent for higher amplitude of excitations. However, there are many examples of instabilities, e.g., flow past bluff bodies, Couette and pipes flows, leading edge contamination on swept wings, two-dimensional and three-dimensional roughness effects on flows, all of which do not exhibit TS waves and could be viewed as bypass transitions. However, it has been well established in Chapter 6 that even for zero pressure gradient boundary layers, one can observe transition without TS waves for infinitesimal disturbance amplitude. The classification is qualitative and there are no definitive

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mechanisms identified in [298] for any of these examples of bypass transition. Probable mechanisms have been conjectured to be related to nonlinear, nonparallel or some unknown physical, mathematical reasons. Here, the known TS wave creation mechanism for small amplitude excitation is shown to be modified under the actions of nonlinearity and nonparallel effects. To understand the effects of nonlinearity due to high amplitude wall excitation created by the SBS strip, results for the case of Am = 0.10AFK are shown next, identifying a physical mechanism. This is related to unsteady separation near the exciter on the plate; the resultant separation bubbles are confined to a small height near the wall, as shown in the ψ contours in Figure 7.17 at t = 20. Only few representative contours are shown in this figure to clearly identify the micro-bubbles near the exciter up to x ≤ 4. 0.005

0.004

ψ - contours Ff = 3 × 10–4, Am = 0.1AFK xst = 0.22 to 0.264 t = 20 ψ = 10.000407

0.003 y

Bubbles

0.001

0

0

1

2 x

3

ψ = 10.000043 ψ = 10.000021 ψ = 10.000039 4

Figure 7.17 Streamline contours shown at t = 20 for the case of Am = 0.1AFK , F f = 3 × 10−4 with the exciter located between 0.22 and 0.264. Note the micro-bubbles near the exciter on the plate. The evolution of disturbances for the higher amplitude case is understood from the snapshots shown at t = 20 for ud varying with x for the three indicated heights in Figure 7.18. The Fourier transform of these disturbances at the three heights are shown in the right column of the figure. Various peaks in the transform help one identify length scales corresponding to different components of the response field.

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The top two frames for the lower heights indicate that the TS wave has higher peak for a higher value of k = k2 . 0.2 0.15 0.1 0.05

ud

0 –0.05 –0.1 –0.15 –0.2

y = 0.0028155, t = 20 xex = 0.22 to 0.264

Ud(k)

0

2

STWF 4

x6

8

10

0.2 0.15 0.1 0.05

ud

0 –0.05 –0.1 –0.15 –0.2

0 –0.05 –0.1 –0.15 –0.2

k1

5 0

5

10

k

15

20

25

y = 0.0057698, t = 20

10

Ud(k) 5

0

2

4

x6

8

10

0

5

10

k

15

20

25

10

0.2 0.15 0.1 0.05

ud

Ff = 3 × 10–4,

Am = 0.1AFK 10

TS Wave

k2

15

y = 0.0088695, t = 20

8 6

Ud(k)

4

2 0

2

4

x6

8

10

0

5

10

k

15

20

25

Figure 7.18 (a) Disturbance streamwise velocity component plotted for the heights y = 0.0028155, 0.0057698 and 0.0088695, from the top to bottom at t = 20, for the case of Am = 0.1AFK , F f = 3 × 10−4 with the exciter located between 0.22 and 0.264. (b) Fourier transform of the signals shown in (a). The top two frames of Figure 7.18 for ud shown at heights y = 0.0028155 and 0.0057698 clearly display that at this time, the TS wave is dominant over the STWF. However, for the height corresponding to the bottom frame, one notices that the TS wave is weaker, whereas the STWF is more or less of the same magnitude. As a consequence, the spectral peak for the STWF is higher than that for the TS wave. The manifestation of micro-bubbles in Figure 7.18 is for the total flow field. For points in the near vicinity of the exciter, and at points very close to the wall, the disturbance flow is larger in magnitude, as compared to the mean flow. When the flow is excited harmonically by the SBS strip at higher amplitude, then during the

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0.2

80

x = 0.3, y = 0.0028155 Ff = 3 × 10–4, Am = 0.1AFK

0.1

60

ud (Feq.)

xex = 0.22 to 0.264

0.05

0.15

ud

40

0

–0.05 20

–0.1 –0.15 10

11

12

t

13

14

15

0

2

4

6

20

0.02 0

15

–0.02

ud (Feq.)

ud

8

10

Frequency

12

14

x = 0.3, y = 0.0057698

10

–0.04

–0.06

5

–0.08 10

11

12

t

13

14

15

0

2

4

6

0.04

10

12

14

x = 0.3, y = 0.0088695

15

0.03

8

Frequency

0.02 10

ud 0.01 0

ud (Feq.)

–0.01

5

–0.02 –0.03 10

11

12

t

13

14

15

2

4

6

8

10

Frequency

12

14

Figure 7.19 Normalized ud time series plotted at x = 0.3 from the leading edge at the heights 0.0028155, 0.0057698 and 0.0088695 for the SBS exciter with Am = 0.1AFK , F f = 3 × 10−4 with the exciter located between 0.22 and 0.264. (b) Fourier transform of the time series shown in (a). blowing phase, flow separates locally in an unsteady manner. This separation bubble pulsates and becomes extinct during the suction phase of excitation. For increased width of the exciter and/or increased amplitude of excitation, separation bubbles are more pronounced, which interferes destructively with TS waves. There is another difference for the higher amplitude excitation case. For the case of Am = 0.10AFK , results shown for y = 0.0088695 distinctly indicate a streamwise stretch over which the TS wave actually grows. Presence of TS waves is noted for the Am = 0.1AFK case in Figure 7.19, from the time series of ud shown at x = 0.3 (in the immediate neighborhood of the exciter) for the three indicated heights. These time series are weakly multi-periodic as seen from the Fourier transform shown in the right frames of Figure 7.19. The presence of

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second and third harmonics are noticeable in the bottom frame only, whereas these are negligibly small at the other two heights. Thus, even when micro-bubbles are created, sub-dominant TS waves are present, negating the definition given in [298] for bypass transition event as one in which TS waves are absent. Another case is considered next, for which the streamwise extent of the exciter is increased with the SBS strip from x1 = 0.2 to x2 = 0.29. Moreover, the exciter begins from a location which is closer to the leading edge of the plate. Thus, one would expect to note higher nonparallel effects. Simultaneous extension in the downstream direction implies that for the same amplitude of the wall excitation, more disturbance energy is fed to the flow inside the boundary layer. In Figure 7.20, stream function and vorticity contours are compared for the two cases having different length of exciters. For these two cases, the amplitude of excitation is taken as Am = 0.1AFK , with the frequency of excitation given as F f = 3.0 × 10−4 . The stream function (ψ)-contours shown in Figure 7.20, indicate the presence of small bubbles immediately downstream of the exciter. These are spaced by a wavelength of the TS wave, which is created at the frequency of excitation, ω ¯ 0. The wall-normal co-ordinate is stretched in all the frames in the figure, for ease of identification of the ensuing physical events. The vorticity contours display twotier structures, with the lower layer showing an upstream cant, and the upper deck bends slightly forward. For the longer excitation strip case shown in Figure 7.20, the ψ-contours show bubbles over the streamwise stretch up to displayed range, as compared to the case of the shorter length exciter. The smaller length case shows smaller undulations. The bubbles are of higher dimension in the wall-normal direction, for the longer exciter. Despite this, the physical mechanism of disturbance generation is the same, as is the case in creating TS waves by harmonic excitation shown [418], establishing once again the appearance of convecting separation bubbles for the higher amplitude case of Am = 0.1AFK . Further increase in excitation amplitude of harmonic wall excitation would cause secondary and higher order instabilities leading to premature transition by nonlinear growth of STWF and/ or STWP.

7.5 Bypass Transition by Free Stream Excitation: By Pulsating Stationary Vortex The schematic of this problem is already shown in Figure 5.6, where the pulsating vortex has an instantaneous value given by Γ(t) = Γ0 sin (ω ¯ c t), with the chosen value of Γ0 = 0.1, ω ¯ c = 3π (based on the convection length scale, which corresponds to viscous scale frequency given by F f = 3π × 10−5 ). One can evaluate the time period of pulsation, which is given by τ = 2/3. This vortex is placed in the free stream at ˜ = 1000. x = 3.38 and y = 1.6, which corresponds to the streamwise location, where Re The domain is defined by −0.05 ≤ x ≤ 120 with 12,000 uniformly distributed points and 0 ≤ y ≤ 1.5 with 400 points distributed following a tangent hyperbolic function [413]. Here the pulsation frequency imposes the disturbance time scale; the equivalent

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0.025 0.025 ψ--contours contours ψ 0.22 to 0.264 xXexex==0.22 to 0.264 0.02 0.02 y y

t = 20

10-4–4 Ff F= 3=×3x10 f

0.015 0.015 0.01 0.01 0.005 0.005

00

00

x

0.5 0.5

11

xx

1.5 1.5

22

2.5 2.5

1.5

1.5

2

2

2.5

1.5 1.5

22

2.5 2.5

1.5 1.5

2

2.5 2.5

0.025 0.025 ω ω- -contours contours = 0.22 toto0.264 x X = 0.22 0.264 ex 0.02 ex 0.02

y y

0.015 0.015 0.01 0.01 0.005 0.005

00

0

0.5

11

x

00

0.5 0.5

11

xx

11

xx

0

0.5 ψ -cocontours ψ ntours 0.025 0.025 Xxexex==0.20.2 to 0.29 to 0.29

x

2.5

0.02 0.02

y y

0.015 0.015 0.01 0.01 0.005 0.005

00

0.025 0.025 ω- -contours contours ω Xex= 0.2 to 0.29 = 0.2 to 0.29 x 0.02 0.02 ex

yy

0.015 0.015 0.01 0.01 0.005 0.005

00

00

0.5

0.5

2

Figure 7.20 Stream function and vorticity contours for the two different strip exciter cases having different widths. (a) and (b) correspond to exciters located between 0.22 and 0.264; (c) and (d) correspond to exciters located between 0.2 and 0.29.

receptivity mechanism should have resemblance to the same problem solved for the Orr–Sommerfeld equation in Chapter 5. However, the essential difference here is that the parallel flow assumption is not made, and all nonlinear effects are captured from the solution of the Navier–Stokes equation. After the equilibrium flow is obtained,

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the pulsation of the stationary, free stream vortex is initiated at t = 0. Some typical solutions are shown in Figure 7.21, displaying ωd variation with x for data collected at y = 0.0028, for the indicated times to show the presence of growing STWFs. The frames 1

1

time = 120

time = 120

ω d ωd

0

Γ = 0.1 y = 0.0028

0 Γ0 = 0.1 y = 0.0028

0 –1

–1 -1

30 30

40 40

1

50

60 xx

70

80

70

80

70

80

70 70

80 80

time = 150

time = 150

ω d ωd

0 –1 -1

30 30

40 40

60 xx

time = 180

11

ω d ωd

50

time = 150

00 –1 -1

30 30

40 40

xx

60

time = 210

22

ω dωd

50

time = 150

00 –2 -2

30

40 40

50 50

60 60 xx

Figure 7.21 Disturbance vorticity plotted at the height, y = 0.0028, for the indicated times to track the growth of STWFs. in Figure 7.21 are shown for a near wall height of y = 0.0028; from t = 120 itself, one can see the leading STWF, ahead of very small amplitude TS waves that can be seen

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for this case upon zooming. For the excitation circular frequency given in the viscous ˜ = 1000, the spatial theory indicates an unstable complex scale as ω ¯ 0 = 0.1 and for Re wavenumber, α = (0.26610073 − i0.0074645), with the phase speed of c ph = 0.35390236 and the group velocity as Vg = 0.41795486. As a consequence, the trailing TS wave interacts with the leading STWF, which is noted at t = 150, to form a second STWF. This is a consequence of continuous pumping of energy via the coupling of free stream and wall modes. It is already established that the formation of STWF is also via a linear mechanism and an interaction between TS wave and STWF has been termed to cause Class-I type of transition in [36]. It is noted that the second STWF, which is clearly visible at t = 150, keeps on interacting with the following STWF upstream to receive energy at the rear, whereas the leading STWF also keeps increasing in its amplitude, as noted in the frame at t = 180. At this time, one can note the presence of four distinct STWFs, with the rear ones possessing very small amplitude. In the last frame shown at t = 210, the amplitude of the leading STWFs are significant and for that reason, the ordinate scale has been increased by more than double. In this zoomed-out view, one can spot the first four leading STWFs as distinct structures, while there are other trailing nascent STWFs in the process of growing. In Figure 7.22, vorticity contours are shown plotted in the computational domain at the indicated times. The vertical streaks near the outer part of the boundary layer are noted, where the STWFs have been noted in Figure 7.21. During the interval shown in the figure, one notices the induced vorticity layers near the surface of the plate in the form of two decks. However, we have noted that despite the presence of the stationary pulsating vortical structures, it is the STWFs which show growth, as the STWF approaches the outflow of the domain. For example, at t = 210, one can clearly see four sites of STWFs in the form of four humps. The change of signs of the vortical structures originating from the inflow of the domain is directly related to the pulsation of the free stream source. At later times, such distinctions among the STWFs blur, whereas the leading STWF becomes more and more stronger, as compared to trailing STWFs. However, one notes that the range of vorticity contour values plotted are lower, which actually grows more from t = 230 onwards, associated with vortical eruptions piercing through the boundary layer. As noted in Figure 7.21, the spatio-temporal growth of the STWF can be viewed by plotting peak-to-peak maximum amplitude as a function of time in Figure 7.23. One clearly notices two different growth rates, one up to t = 200, which is essentially lower compared to the steeper growth rate noted for the later time instants, and one beyond t = 250, where one can note the nonlinear saturation that is typical of tertiary stage of the transition to turbulence.

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0.2 0.15 y

max:3.1175E+00 min:–1.2952E+01

time = 190

0.075 –0.025 –0.1

0.1 0.05 0 0.2

40

50

55

max:6.6733E+00 min:–1.2936E+01

0.15 y

45

60 x

65

70

75

80

85

time = 210

0.075 –0.025 –0.1

0.1 0.15 0 40 0.2

y

45

50

55

60 x 65

70

75

80

max:2.2728E+01 min:–2.5905E+01

85 90 time = 230

0.15 0.075 –0.025 –0.1

0.1 0.15 0 40

50

60 x

80

70

85 time = 250

0.3 max:1.9016E+02 min:–2.2443E+02 0.2

0.075 –0.025 –0.1

y 0.1 0 40

50

60

80

x 70

90

0.4 max:2.3671E+02 min:–2.3759E+02 0.3 y

100 time = 270

0.075 –0.025 –0.1

0.2 0.1 0 40

50

60

70 x

80

90

100

110

Figure 7.22 Disturbance vorticity contours are shown for the indicated times for the case of free stream pulsation with Γ0 = 0.1 starting at t = 190.

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100

Γ0 = 0.1 y = 0.0028

80 ωdm

i

60 40 20 0

0

50

100

t

150

200

250

Figure 7.23 Peak-to-peak maximum amplitude of disturbance vorticity variation with time at the height y = 0.0028.

7.6 Closing Remarks • Nonlinear, nonparallel effects are described for transition by wall excitation. By nonparallel effects, one avoids the simplistic connotation of streamline deviating from the zero pressure gradient boundary layer. Instead one should use ddxp vs X, in the neighborhood of the leading edge of the flat plate in Figure 7.2. • One must also pay attention to excitation amplitude vis-´a-vis the value(s) used by other research groups (at Stuttgart, CTR) where vw ∼ U∞ is too large. Comparatively, researchers at HPCL have used really small amplitude excitation for the linearized and nonlinear Navier–Stokes equation, given in Eqs. (7.3) to (7.5). • There is a distinct need for well-catalogued experimental results for receptivity with respect to the experiment given in [405], with neutral curve shown in Figure 7.2, redrawn very carefully by locating the exciter and the location where response is noted. • It has been found that STWF is the main result for the linearized Navier– Stokes equation with parallel flow approximation over the last three decades, first obtained with Bromwich contour integral method, and later by using high accuracy numerical methods for the complete time-dependent Navier–Stokes equation. One should compare results obtained by solving the linearized and nonlinearized Navier–Stokes equation for wall excitation, as an exercise in global stability studies for wall excitation. Such an exercise will be detailed for free stream excitation in Chapter 14; it has been reported only recently in [471]. • It is necessary to ask if bypass transition is relevant in view of the results given in Chapter 6 with results drawn from [509]. In the same way, it is relevant to state what constitutes natural transition.

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8.1 Introduction By now what has been established is: (a) The onset of instability is due to the growth of imperceptible background disturbances for the canonical zero pressure gradient boundary layer or the boundary layer formed over a semi-infinite flat plate. Here, the connotation of zero pressure gradient has to be understood in the context of the discussions in Section 7.3; that the streamwise pressure gradient on the semi-infinite flat plate is variable in nature, as shown in Figure 7.8. The transition that the semi-infinite flat plate undergoes will be a strong function of the flow receptivity depending on the location of the exciter with respect to the leading edge of the plate. For the same reason, results reported in literature or books that employ the Blasius boundary layer should be viewed as pedagogic exercises primarily [133, 388] with poor reliability. (b) For experimental verification of instability theories, one has to follow a receptivity approach, with respect to deterministic excitation, for this allows replicability of the experiment. (c) Such experiments have to be for the conditions of the theory, just as it has been for the spatial instability theory, which requires that time periodic excitation be inside the boundary layer, even though the real excitation is completely different from such monochromatic excitation. This has led to the incorrect identification of TS waves as the precursors of natural transition to turbulence, and misdirecting researchers right from the beginning of the twentieth century. (d) Even when conducting experiments based on time aperiodic excitation [96, 144, 290], researchers and experimenters always tried to locate TS waves to explain their experiments. (e) However, when researchers investigated further for more generic spatio-temporal route of transition using linear theory, the concept of

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spatio-temporal wave front (STWF) emerged in [418, 451], which was later identified by solving the Navier–Stokes equation without linearization [419, 422]. Variety of researchers have tried to explain such events as due to transient growth and/or bypass transition [299, 364], or by nonmodal growth [229, 398, 532] and so on. The generic nature of STWF for seemingly different routes of transition via alternate routes for two-dimensional transition has been explained in the previous two chapters and is based upon research by Sundaram et al. [509]. The researchers no longer have to go looking for TS waves for every episode of transition. However, the question still remains, whether this point of view of ascribing the central role to STWF also holds true for three-dimensional transition route(s) for wall bounded flows? Can one take the same canonical boundary layer solution from the Navier– Stokes equation for flow past a semi-infinite flat plate, and replicate the experimental scenario followed in Klebanoff et al. [237]? This then, becomes the focus of discussion in the present chapter. The absence of a complete theory of turbulence will keep the understanding of it limited to merely explaining a physical problem. Meanwhile one has to acknowledge that there can be useful tools for practical applications based mainly on phenomenology and empiricism for turbulent flows. This has led Feynman [135] to note turbulence as a physical problem that is common to many fields, that is very old, and that has not been solved. The intractable nature of the Navier– Stokes equation has finally become easier to solve due to powerful computers in terms of speed of computation, using extremely elegant physics-based high accuracy numerical methods (DRP schemes), some of which have been introduced already in this book and which are further available in [29, 33, 413, 420]. The materials described in this chapter on the three-dimensional routes of transition are based on the reported works in [29, 32, 33, 34, 35, 482, 483, 508], supplemented by other results in [214, 215], and by Fasel et al. [134, 367] and few others [387, 388, 560]. Experiments on transition as reported in [209, 210, 237, 405], create disturbances deterministically in wind tunnels having extremely low background disturbances, while a ribbon is vibrated inside the boundary layer at a fixed frequency to cause transition. This conforms to the familiar spatial instability theory, which requires a fixed frequency excitation inside the boundary layer. Such an experiment can also be computationally studied, provided adequate care is exercised in choosing the grid, domain, formulation and discretization techniques. However, early DNS reported in [54, 493] are not suitable, as these modify the governing equation (described in appendix A of [32]). These approaches also employ additional numerical artifacts, such as hyper- and hypo-viscosity terms; distributed body forces, and random noise at the inflow of the computational domain in solving the Navier–Stokes equation. In other references [134, 388, 565], the complete Navier–Stokes equation is solved to simulate the experiments in [237, 405], but only with partial success. The difference between the unsuccessful simulations in [134, 388, 565], and the successful ones in [32, 33, 419, 422] to simulate transition experiments [209, 210, 237], is due to the use of a very small domain in the former, which did not even allow detection of the STWF itself. In [132, 134], the simulations report results which required wall-normal

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velocity perturbation that is as high as the free stream speed, and perhaps indicative of troubling numerical methods with excessive diffusion. In contrast, the authors in [32, 34, 419, 422, 508, 509] have used truly small wall perturbation to create the generic components of the disturbance field, and identifying the central role of STWF for wall excitation cases. Two-dimensional flow can be very well resolved, especially using high accuracy computing methods, and the topic completely explored in the previous chapters for wall excitation. The reasons for limited success or absence of tracking transition routes in [388] is due to (i) complications created by the use of implicit and explicit time integration methods with an overlap region; (ii) the use of a fourth order, lower accuracy spatial discretization scheme (compare the correct report in [387] with the incorrect reference of using a compact scheme in [388]) and (iii) use of random noise at the inflow. In a small domain used in [134, 388], a STWF will not appear for the frequency of excitation used. Even if it is in nascent stage, it will emerge out of the computational domain by convection, without affecting subsequent events. In the overlap region between the implicit near-wall part and the explicit outer part of the boundary layer; which is very close to the wall there will be the formation of spurious error-packets, acting as vortical excitation which will alter the physical flow drastically. It is a situation so adverse that the authors did not even notice any TS wave, quite unlike that in [32, 33, 35]. The issue has been discussed in detail in appendix B of [32] and some results are presented here. The domain in [419] is twelve times longer than that has been used in [388], which is one of the reasons that the authors [419] could depict STWF dynamics in two-dimensions, while others [388] could not do so. Even for the three-dimensional results presented in [32] and here, the domain is five times longer than that used in [388]. It is to be noted that in the context of spatio-temporal dynamics, a three-dimensional field is not a mere transformation of two-dimensional results. Some new results on the nature of STWF are presented in Chapter 14 showing the dominant role played by a two-dimensional disturbance field in the form of STWF, as compared to three-dimensional disturbance field. When the vibrating ribbon problem is solved as an excitation problem by the Bromwich contour integral method [418], three elements of a disturbance field emerge, as previously explained [451]. First, the STWF is not a mere transient, (as explained in Chapter 6); next, that this is caused by the spatio-temporal growth of mutually interfering neighbors in wavenumber plane for the Fourier-Laplace transform of the disturbance field. This is an elementary mathematical physics concept clearly explained by Rayleigh [266, 351]. While the constructive interference is noted whenever the STWF is formed, an interesting demonstration case has been presented in [452], where the authors have shown on the other hand, the destructive interference among a band of TS waves, all of which have been noted to be unstable according to spatial instability theory. These reinforce the necessity of considering interference among every component of the spectrum, even if it is not due to interference among

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discrete peaks (as with the eigenvalues in Craik’s triad interaction theory [99]) or discrete peaks interfering with smooth part of the spectrum in the vicinity of the discrete peaks. The STWF is created by a linear mechanism, which grows continually till dispersion and nonlinearity gains dominance. Thus, if one solves the linearized Navier–Stokes equation by employing global stability analysis [519], then one would see unlimited growth of the STWF, till it numerically overflows. In contrast, solving the fully nonlinear Navier–Stokes equation, the onset of STWF is by a linear mechanism, but its accelerated growth by nonlinearity will create nonlinear saturation, and cause separation upstream of STWF due to an adverse pressure gradient, as has been reported for two-dimensional cases [419]. This leads upstream propagation of disturbances and automatically invalidates parabolized stability equation (PSE) based approaches. Such adverse pressure gradients create new STWFs in turn; all of which saturate nonlinearly. The self-regeneration property of STWFs explains how irreversible the process of transition can be. Once a STWF is set in motion, transition to turbulence is inevitable, as has been shown for two-dimensional transition in [419, 422] while solving the transition problem over a long computational domain for the semi-infinite flat plate. It is further shown in this chapter that for a three-dimensional disturbance field as well, the disturbance field evolves into turbulence following the creation of STWF. A typical schematic of the computational domain with the exciter depicted shown as a circular patch or a rectangular strip is presented in Figure 8.1. In this figure the Gaussian circular patch (GCP) or Gaussian bump exciter will provide a three-dimensional response originating from the center of the GCP. In contrast, the spanwise modulated (SM) exciter will also create a threedimensional disturbance field, which will propagate as a plane wave front.

8.2 Different Routes of Three-Dimensional Transition Experiments on three-dimensional transition routes show vortical structures in late stages to be different in [210, 237, 384]. In these experiments, a rectangular ribbon with spanwise spacers is vibrated at a single frequency near a long flat plate, where disturbances evolve into Λ-vortices. When these vortices are aligned during the late stages of transition, it is called the K-type transition, as it was originally reported in [237] with frequency of excitation of the vibrating ribbon as 1489Hz and above. In another experiment reported in [210], created disturbance vortices during transition are noted as staggered, for the lower frequency of excitation of 120Hz. This is attributed to H-type transition, as it has common attributes of secondary route of transition proposed by Herbert [180]. The essential differences between the experiments in [237] and [210] are solely due to variation of excitation frequency. The computational and theoretical approaches which employed to explain these routes did not use the frequency data. Various artifacts have been used instead to present H-type breakdown as occurring due to resonance between a two-dimensional disturbance

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y ma

x

Far-field boundary y

Outflow boundary

Inflow

x ed ulat mod e s i nw Spa ter exci

z U∞

x out e

Plat

x ex

Gaussian bump exciter

z m ax

Plate leading edge

x in

Figure 8.1 Schematic diagram of a computational domain shown for Gaussian circular patch (GCP) and spanwise modulated (SM) exciters with center located at xex . For both the cases xex = 1.5 and streamwise extent of the domain is xin ≤ x ≤ xout ; wall-normal extent is 0 ≤ y ≤ ymax and spanwise extent is zmax /2 ≤ z ≤ + zmax /2. A length scale L is chosen such that Reynolds number based on free stream speed and L is ReL = 105 . Here the cases are computed with xin = −0.05L, xout = 50L, ymax = L and zmax = L. [Reproduced from “A new velocity-vorticity formulation for direct numerical simulation of 3D transitional and turbulent flows”, S. Bhaumik and T. K. Sengupta, J. Comput. Phys., vol. 284, pp 230-260 (2015), with the permission of Elsevier.] wave interacting with two oblique waves with half the excitation frequency of the two-dimensional wave, and is attributed to Craik’s triad interaction [99]. However, it has been shown in [32, 35, 417] that such complications of input excitation are not truly involved in H-type transition. In [134, 388, 565], the authors in attempting to detect H-type transition computationally, considered an exciter vibrating at both a fundamental and its sub-harmonic frequencies simultaneously by wall-normal perturbation, which is not the case in actual experiments. Similarly, authors in [388] added unphysical white noise along with wall excitation to “encourage flow randomization and asymmetry in post-transitional turbulent region.” Characteristic difference between K- and H-type transition routes is noted as a feature of Λ-vortices. While to begin with, H-type transition onset is delayed, but once established it builds up exponentially. In computational simulation, the role of TS waves dominates with barely any development of STWF [134, 367, 388, 565]. It has been shown in [32, 482] that STWF is not only created by three-dimensional excitation, but also supports both these routes of transition, simply because it is present.

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8.3 Three-Dimensional Receptivity of Boundary Layer to Wall Excitation Study of the role of STWF in disturbance fields during three-dimensional transition is conducted by solving the Navier–Stokes equation using dispersion relation preserving (DRP) space–time discretizing schemes [33, 413, 420]. The schematic of the domain used is shown in Figure 8.1, for the boundary layer that forms over a semi-infinite flat plate excited by a GCP or a SM exciter; which is vibrated time-harmonically. Results for cases with different spanwise periods are used to learn about the different effects due to frequency and spanwise wavenumber [482]. The spanwise wavenumbers are fixed subject to periodic boundary conditions at the extreme spanwise limits of the domain. For example, one can consider the extremes as the locations of the nodes, i.e. the spanwise width (zmax ) is half the spanwise wavelength. Also, the exciter geometry decides whether one generates planar and oblique three-dimensional TS waves for SM and GCP exciters, respectively. It is reiterated that the equilibrium flow is once again obtained over the semi-infinite flat plate by solving the Navier–Stokes equation, and this is completely unlike the Blasius boundary layer used in [388, 560].

8.3.1 Governing equation The computational domain has been described above in the physical plane. However, the simulations have been performed in the transformed (ξ, η, ζ)-plane, with independent separable transformations given by, x = x(ξ), y = y(η) and z = z(ζ). Primitive variables are not used in computation; instead derived variables are used by using velocity and vorticity as the dependent variables. The use of this derived variable formulation is preferred due to the reasons provided in [147, 148, 149, 163, 291, 308, 413] for higher accuracy. The recent updates of various types of velocity-vorticity formulations are given in [33, 420], and among many variants, the rotational form of → − − the ( V ,→ ω )-formulation is found to be preferable in [32, 483, 508] for the incompressible Navier–Stokes equation given by − ∂→ ω → − +∇× R =0 (8.1) ∂t → − → − − − where R = → ω × V + (1/ReL )∇ × → ω , and the respective components are indicated by the corresponding transformed coordinates. Here ReL is the reference Reynolds number defined in terms of the reference length (L) and the free stream speed (U∞ ). Equation (8.1) in transformed coordinate can be written as ∂ωξ  1 ∂Rζ 1 ∂Rη  + − =0 (8.2) ∂t h2 ∂η h3 ∂ζ

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∂ωη  1 ∂Rξ 1 ∂Rζ  =0 + − ∂t h3 ∂ζ h1 ∂ξ

(8.3)

∂ωζ  1 ∂Rη 1 ∂Rξ  =0 + − ∂t h1 ∂ξ h2 ∂η

(8.4)

→ − − ω = bωξ , ωη , ωζ cT and V = bu, v, wcT . The scale factors of orthogonal where, → ∂y ∂z transformation, h1 , h2 and h3 are given as h1 = ∂x ∂ξ , h2 = ∂η and h3 = ∂ζ . In Eqs. (8.2) to (8.4), the terms Rξ , Rη and Rζ are obtained from 1  1 ∂ωζ 1 ∂ωη  Rξ = (wωη − vωζ ) + (8.5) − ReL h2 ∂η h3 ∂ζ Rη = (uωζ − wωξ ) +

1  1 ∂ωξ 1 ∂ωζ  − ReL h3 ∂ζ h1 ∂ξ

(8.6)

Rζ = (vωξ − uωη ) +

1  1 ∂ωη 1 ∂ωξ  − ReL h1 ∂ξ h2 ∂η

(8.7)

→ − − ω )-formulation, the velocity vector is governed by the velocity Poisson In the ( V ,→ → − − equation, ∇2 V = −∇ × → ω , which in the transformed (ξ, η, ζ)-plane are given by  ∂ωη ∂ωζ  ∇2ξηζ u = h1 h2 − h3 h1 (8.8) ∂ζ ∂η  ∂ωζ ∂ωξ  ∇2ξηζ v = h2 h3 − h1 h2 ∂ξ ∂ζ

(8.9)

 ∂ωξ ∂ωη  ∇2ξηζ w = h3 h1 − h2 h3 ∂η ∂ξ

(8.10)

where the operator ∇2ξηζ is given as h1 h2 h3 ∇2ξηζ =

∂  h2 h3 ∂  ∂  h3 h1 ∂  ∂  h1 h2 ∂  + + ∂ξ h1 ∂ξ ∂η h2 ∂η ∂ζ h3 ∂ζ

→ − The velocity field must satisfy the solenoidality condition Dv = ∇ · V = 0, which in the transformed (ξ, η, ζ)-plane is given by Dv =

1 ∂u 1 ∂v 1 ∂w + + =0 h1 ∂ξ h2 ∂η h3 ∂ζ

(8.11)

The three-dimensional receptivity to wall excitation has been solved in [32, 34, 482, 483, 508] using the same formulation and methodologies reproduced here. The velocity field is obtained first, from the vorticity field given at any time by the Poisson equations. For the components which are in streamwise and spanwise directions (uand w-components) given in Eqs. (8.8) and (8.10) are solved first. The wall-normal,

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v-component of velocity is obtained by enforcing the divergence-free condition for the velocity field by integrating Eq. (8.11) from the wall (η = 0) to any height (η) given by Z η h2 ∂u h2 ∂w  v(ξ, η, ζ) = v(ξ, 0, ζ) − dη (8.12) + h3 ∂ζ 0 h1 ∂ξ Use of Eq. (8.12) satisfies the solenoidality condition for velocity, and avoids imposing any soft boundary condition for the wall-normal velocity at the far-field boundary shown in Figure 8.1.

8.3.2 Boundary conditions Solution of the boundary value problem for the Poisson equation for velocity, and transport equation for the vorticity components would require prescription of boundary conditions on different boundary segments. For example, at the inflow of the domain shown in Figure 8.1, the boundary condition for the components of → − − velocity V , and vorticity → ω , are given as u = 1 and

∂v ∂w ∂ωξ = = = ωη = ωζ = 0 ∂ξ ∂ξ ∂ξ

(8.13)

Similarly, for the far-field of the computational domain (located at η = 1), one obtains the boundary condition indirectly for v given by Eq. (8.12). Other velocity components and vorticity components require boundary conditions given by the following homogeneous and symmetry conditions u = 1 and w = ωξ =

∂ωη = ωζ = 0 ∂η

(8.14)

Due to limited computational resources, at the present time a periodic condition is used in the spanwise direction for all the three components of velocity and vorticity. On the solid bottom wall (η = 0), there must be the provision for exciting the boundary layer. This is provided by time-varying v-velocity component (vw (ξ, ζ)) given by the excitation as prescribed. Additionally, no-slip boundary conditions are imposed for the u- and w-components of velocity. These, lead to the following boundary conditions for the variables given for η = 0 by u = w = 0, v = vw (ξ, ζ, t), ωη = 0,   1 ∂(h3 w) ∂(h2 vw ) 1  ∂(h2 vw ) ∂(h1 u)  ωξ = − and ωζ = − h2 h3 ∂η ∂ζ h1 h2 ∂ξ ∂η

(8.15)

A special feature of the results reported in [32, 34, 36, 419, 431, 474, 483, 508] are in solving the Navier–Stokes equation for flow over a semi-infinite flat plate, while including all possible physical effects. The solutions in these references differ from other methods by obtaining solutions for the equilibrium flow by solving the Navier– Stokes equation in a computational domain, while taking into account the sharp leading edge of the semi-infinite flat plate, which is the site of the stagnation point.

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Hence, at the plane ahead of the leading edge shown in Figure 8.1, one employs symmetry conditions for all the variables. This fixes the boundary conditions on different variables [32, 33] as follows ∂ωη ∂w ∂u =v= = ωξ = = ωζ = 0 ∂η ∂η ∂η

(8.16)

The Sommerfeld boundary conditions used at the outflow boundary for the variables u, ωη and ωζ are ∂u ∂u + Uc =0 ∂t ∂ξ

(8.17)

∂ωη ∂ωη + Uc =0 ∂t ∂ξ

(8.18)

∂ωζ ∂ωζ + Uc =0 ∂t ∂x

(8.19)

The boundary condition for ωξ at the outflow boundary is obtained by using the solenoidality condition on vorticity as  h ∂ω ∂ωξ h1 ∂ωζ  η 1 =− + (8.20) ∂ξ h2 ∂η h3 ∂ζ The boundary conditions for v- and w-components of velocity at the outflow boundary are derived from the definition of the vorticity components ωη and ωζ as ∂(h2 v) ∂(h1 u) = + h1 h2 ωζ ∂ξ ∂η

(8.21)

∂(h3 w) ∂(h1 u) = − h1 h3 ωη ∂ξ ∂ζ

(8.22)

8.3.3 Initial condition for disturbance field To evaluate the proper three-dimensional equilibrium flow, one needs to start from the two-dimensional equilibrium flow, for which ωξ , ωη and w are identically zero. Thus, for the two-dimensional equilibrium flow, one needs to solve the vorticity transport equation for ωζ , and u- and v-components of velocity obtained by solving one of the Poisson equations, supplemented by the solution from the continuity equation for the other component of velocity. This flow, in turn, is simulated with the initial condition of an impulsive start u = 1, v = 0 and ωζ = 0

(8.23)

After having obtained the two-dimensional equilibrium flow, the three-dimensional flow computation is started with u, v and ωζ prescribed at all the spanwise stations, and forcing other variables, w, ωξ and ωη to assume a value of zero at all spanwise

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stations. These initial conditions are used in three-dimensional solvers for a few thousand iterations, till the flow adjusts itself in the full domain, satisfying all the boundary conditions. This equilibrium flow is ready to be subjected to the time periodic excitation for the receptivity calculation by solving Eqs. (8.1) to (8.11), subject to the boundary conditions described in Sub-section 8.3.2. The GCP type of excitation has been used in the experiments reported in [96, 144, 290], while the SM type of excitation has been used in [210, 237, 384]. Imposed wallnormal velocity created by the exciter is given by vwall (x, z) = α1 Am (x, z) H1 (t) eiω0 t , where α1 and ω0 denote amplitude and frequency of excitation; H1 (t) is the Heaviside function required to simulate an impulsive start of the exciter and max (|Am (x, z)|) = 1. Amplitude of the GCP exciter varies as Am = [1 + cos(π r/rmax )]/2, where rmax is the radius of the circular patch, and for the SM exciter case the amplitude variation is given by, Am = A x (x) sin (2π z/zmax ), with A x (x) has been given before for twodimensional cases in [419, 422], as described in Chapter 7. Spanwise periodicity indicate responses repeating itself at an interval of zmax .

8.4 Grid System The computational domain taken in [32, 33] is similar to [367, 388, 565], but the streamwise extent is at least five times longer to properly gauge the onset and the growth of any STWF properly. The domain shown in Figure 8.1, in the streamwise direction is given as xin ≤ x∗ ≤ xout , with x∗ = 0 denoting the leading edge; 0 ≤ y∗ ≤ ymax along the wall-normal direction and −zmax /2 ≤ z∗ ≤ zmax /2, along the spanwise direction. Variables marked with asterisk indicate dimensional values, and those without asterisk are non-dimensional. The origin of the coordinate system is placed at the center of the leading edge of the plate. In [419, 422], the length of the computational domain is such that xout = 120L for simulating two-dimensional turbulence. In [32, 33], it has been taken as xout = 50L, as compared to the domain length in [388] of 10L. The smaller domain length is inadequate for the creation of STWF. The grid is created by clustering points following a tangential hyperbolic distribution [413, 420] near the leading edge, and which is uniform after x = 5. In the wall-normal direction also, the grid is clustered by similar tangential hyperbolic distribution, which has been shown to reduce aliasing error in [411]. The grid in the spanwise direction has been chosen to be uniform. Thus, the computations performed in the transformed (ξ, η, ζ)-plane use the independent transformation given by, x = x(ξ), y = y(η) and z = z(ζ). The grid transformation in the x-direction is given by two stages, such that for, ¯ xin ≤ x ≤ xex (0 ≤ ξ ≤ ξ),  tanh[∆ x (1 − ξ)]  (8.24) x(ξ) = xin + (xex − xin ) 1 − tanh ∆ x

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while in the second stage for xex ≤ x(ξ) ≤ xout (ξ¯ ≤ ξ ≤ 1), one uses the following  ∆  ξ − ξ¯  x x(ξ) = xex + (xex − xin ) (8.25) tanh ∆ x ξ¯    xout −xex tanh ∆ x 1 ¯ . Similarly, the grid-transformation in the where ξ = 1+A0 and A0 = xex −xin ∆x wall-normal direction is given by,  tanh[∆y (1 − η)]  y(η) = ymax 1 − tanh ∆y

(8.26)

where 0 ≤ η ≤ 1, and ∆ x and ∆y are the clustering parameters controlling grid stretching in the streamwise and wall-normal directions, respectively. For all these cases, the value of ∆ x = 2 and ∆y = 2 have been used. A total of 2500 points in the streamwise direction and 350 points in the wall-normal direction have been used. A total of 100 points are taken in the spanwise direction. In addition, the computations in [32, 33] did not use random wide-band noise, as that has been used in [388]. More importantly, use of two-level time integration method ensures only the simulation of physical modes, without creating any spurious numerical modes often noted. This is clearly explained in [413, 416].

8.5 Numerical Methods In Chapter 3, we have discussed about analysis techniques of numerical schemes to check the suitability of the method with appropriate model equations. For example, the extremely accurate combined compact difference method for spatial discretization in conjunction with four stage, fourth order Runge-Kutta scheme for time integration is known to be capable of performing DNS. Having detailed the velocity-vorticity formulation in this chapter for three-dimensional transition routes to fully developed turbulent flow, it is time to proceed to a novel method introduced in [29, 32, 33] using a staggered arrangement of variables for better accuracy [190]. Velocity components (u, v, w) are defined at the center of the cell-surface perpendicular to it, and the vorticity components (ωξ , ωη , ωζ ) are defined at the twelve edges of the cell which are parallel to the components for a staggered arrangement. In this arrangement, an optimized version of staggered compact schemes reported and used in [30, 32, 33, 482, 483] has been used. It requires mid-point interpolation, following the evaluation of first derivatives at mid-point locations. The basic scheme is a variant of a sixth order compact scheme introduced in [305]. The basic stencil of this mid-point interpolation scheme is given as γI φˆ j−1 + φˆ j + γI φˆ j+1 =

aI bI (φ j− 12 + φ j+ 21 ) + (φ j− 32 + φ j+ 32 ) 2 2

(8.27)

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The basic staggered compact scheme for the first derivative at the mid-point used is similarly given by γII φ0j−1 + φ0j + γII φ0j+1 =

bII aII (φ j+ 21 − φ j− 12 ) + (φ 3 − φ j− 32 ) h 3h j+ 2

(8.28)

0

Here, φˆ j s and φ j s are the mid-point values of the function being interpolated, and of the first derivative of the function at the jth-location, respectively, which are evaluated from the given φ j±1/2 s at the locations in the subscript. In Eq. (8.28), h is the uniform grid spacing of the transformed plane. The fourth order accuracy can be achieved, if the interpolation scheme given by Eq. (8.27) is fixed by choosing aI = 81 (9 + 10γI ) and bI = 18 (6γI − 1), with γI as the parameter. The accuracy is further increased to the sixth 3 order in [305] by additionally fixing γI = 10 . A fourth order formal accuracy for first derivatives is similarly given in Eq. (8.28), for aII = (9 − 6 γII )/8 and bII = (22 γII − 1)/8, with γII as the parameter. Equating the next higher order terms of the Taylor series expansion on both sides, one obtains the sixth order scheme given by, γII = 9/62. These schemes from [305] have been further optimized, as a fourth order version of the original concept by optimization in the spectral plane for better DRP properties, as given in [420]. The optimization yielded γI = 0.41 and γII = 0.216 for Eqs. (8.27) and (8.28), respectively [29, 30]. These have been termed as Optimized Staggered Compact (OSC) scheme in [29, 32, 33, 482, 483]. The resulting schemes being of fourth order formal accuracy, show superior resolution and DRP properties, as compared to the sixth order schemes in [305], as has been shown in [29]. The OSC scheme has been used for second and mixed derivatives appearing in the Navier–Stokes equation by repeated applications of Eq. (8.28). Central difference of second order accuracy is used in discretizing the velocity Poisson equations, while a trapezoidal method is used for numerical integration of Eq. (8.12) in calculating the v-component of velocity. The time integration of the vorticity transport equations is performed using the optimized three stage Runge-Kutta scheme, or ORK3 developed in [450]. The sixth order compact spatial discretization schemes have been used for receptivity problems in [291, 565], where authors have used the four stage, fourth order Runge-Kutta (RK4 ) scheme for time integration. In [388], the authors have used a 4th order central difference (CD4 ) scheme, as given in Eqs. (2.13) and (2.14) in [387], which has been used with three stage, Runge-Kutta (RK3 ) scheme. The explicit, three stage optimized Runge-Kutta method, ORK3 developed for good DRP properties is used with the combined space–time discretization method which is called the OSC-ORK3 scheme in [32, 33, 482, 483] and for the results shown here. In Figure 8.2, the numerical properties of the present OSC-ORK3 scheme is compared with the CD4-RK3 scheme of [388] for the model convection equation ∂u ∂u +c =0 ∂t ∂x The usefulness of this model equation is well-known for the reason of developing methods for convection-dominated flow problems in [413]. For this non-dissipative,

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Figure 8.2 Numerical properties of the CD4-RK4 method are compared with the present OSCORK3 method. These are shown for (a, d) Numerical amplification factor (|G|); (b,e) Normalized phase speed (cN /c) and (c,f) normalized group velocity (VgN /c) in (Nc , kh)-plane. [Reproduced from “Precursor of transition to turbulence: Spatiotemporal wave front”, Swagata Bhaumik and Tapan K. Sengupta, Phys. Rev. E, 89(4), 043018 (2014), with the permission of APS Physics.] non-dispersive equation, the relevant parameters are the numerical amplification rate (|G|), normalized numerical phase speed (cN /c) and normalized group velocity (VgN /c),

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as shown in Figure 8.2. This equation being neutrally stable, the numerical method must also have the same attribute. This must be checked for both CD4-RK3 and OSCORK3 methods for identical space and time steps. Spatial resolution in the spectral plane is given by the non-dimensional wavenumber (kh), where h is the uniform spacing and time resolution as measured by the CFL number given by Nc = c∆t/h. For the non-dispersive dynamics of the model equation, one expects that the normalized phase speed and group velocity must take a unit value in order to minimise dispersion error. For these metrics, the CD4-RK3 method is deficient, as compared to the OSCORK3 method. When one looks at extreme dispersion error given by the q-waves for which VgN /c < 0 [413, 427], one obtains significantly better properties for the OSCORK3 scheme, as compared to the CD4-RK3 scheme. There is a serious concern that the methods used in [388], relating to the use of explicit and implicit time integration methods together, also known as the IMEX method, may not be adequate. In this case, a very thin layer close to the wall is considered where an implicit Crank-Nicolson scheme is applied close to the wall to overcome the severe restriction on numerical time-steps posed by the use of explicit methods due to Peclet and CFL numbers, as explained in Chapter 3. Explicit time integration in the form of a three stage, Runge-Kutta method is used above the bottom layer using such an implicit method. As explicit methods are less error-prone compared to implicit methods, it is desirable that the layer employing implicit method should be as thin as possible. An overlapping region of six points have been taken in [388] for the switch-over from the implicit to the explicit method of time integration to reduce error arising out of abrupt switch of methods. Otherwise, only a twopoints overlap is required between the blending of the two schemes. Next comes the testing of the consequences of such blending with the methods used in [32, 33] that uses a compact scheme for spatial discretization with a ten point overlap in the implementation of the parallel algorithm used. It has been noted already [32] that applications of a parallel scheme with compact spatial discretization may require overlap points. One such method using Schwarz domain decomposition in [435], has shown that even solving one-dimensional convection equation requires at least six overlap points. In reporting the results in [32], the authors have used ten overlap points. To show the importance of using the correct method for computing transitional flow as reported in [388] and [32], the IMEX method used in [388], is compared with the OSC-ORK3 method to compute two-dimensional convection equation with a wave-packet as the initial condition moving at an angle of 45o to the x-axis. The initial condition is given by u(x, y, t = 0) = exp[−500s] cos s where s2 = (x − x0 )2 + (y − y0 )2 . The solutions are obtained inside the domain 0 ≤ x ≤ 5; 0 ≤ y ≤ 5, with the wave-packet located initially at x0 = y0 = 1.6, while the phase speed components are given by c x = 0.2 and cy = 0.2 in x- and y-directions,

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respectively. For these two methods of [388] and [32], a grid with 500 × 500 points has been used. The time step is fixed by choosing CFL number as 0.1. With respect to the exact solution, errors are plotted as a function of time in Figure 8.3 on the left for the IMEX method [388] and on the right for the OSC-ORK3 method. As the errors differ by order of magnitudes, these have been designated by the exponent of the error. Most methods have dispersion, and the error is of the order of the signal in the neighborhood of the wave-packet. This causes the maximum exponent to keep growing with time. Additionally, when the wave-packet crosses the overlap region, another source of error creeps in as an error packet, which moves upstream due to reflection from the overlap region. Such an error is higher for IMEX method. Similar error packets also occur for the compact scheme with the overlap region at sub-domain boundaries, as seen in Figure 8.3. However, the error in the case of the compact scheme is significantly lower, as compared to the CD4-RK3 with IMEX method. Furthermore, the error attaches itself behind the advancing wave-packet, and grows larger with higher strength. Again this is greater in the case of the CD4-RK3 and IMEX method. These two sources of error contaminate the computed flow field, leading to physically wrong results in [388]. These are so incorrect that the method failed to show the TS waves, and its large numerical error interferes with formation of STWF. Also, the use of implicit time integration itself is not suitable for the transitional flow simulation shown in [454]. It is important to note that continuous excitation cases when simulated, exhibit recurring error, and this accumulates over the full domain. In [509], the authors have shown that switching the exciter on and off is the source of STWF for every such transient. All STWF cause adverse pressure gradients upstream, and in turn creat additional STWFs. Thus, for computation performed in a small domain [388], any STWF created will convect out of the computational domain without regeneration of STWF, as noted in the previous chapter for two-dimensional routes [419, 422]. While transition is not seen in [388] due to formation of STWF, the situation is further complicated by adding random excitation, along with the time-harmonic wall excitation. Such additional random excitation will create innumerable nascent STWFs along with local solutions, which will contaminate the flow although one will not notice distinct and coherent STWFs. Such synthetic random excitation is unphysical, as compared to disturbances in experiments with free stream turbulence present. This unrealistic wall excitation and multitudes of numerical error packets inside the shear layer are the reasons for the observation of so-called bypass transition in [388, 560] for a very small domain (0 ≤ x ≤ 10). Error packets will convect downstream, similar to the free stream excitation causing bypass transition shown in [123, 431, 474] for vortexinduced instability, but, with qualitative difference. The error packet in [388] moves very close to the wall, while in the vortex-induced instability reported in [123, 431, 474] is due to a convecting free stream vortex which moves far removed from the wall. Kleiser and Zang [239], have similarly pointed out that the used “splitting methods are susceptible to errors that may be particularly large near the boundaries and may even prevent the numerical solution from converging to the correct result.” Presence

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log(error)-contours plotted from -10 to -1.

log(error)-contours plotted from -10 to -1. (d) t = 1

(a) t = 1

4

4 CD4-CN

-10

3

y

y

3 2

-2

OSCS-ORK3

2

CD4-RK3

1 0 0

OSCS-ORK3 -3.5

1 1

2

x

3

0 0

4

1

2

x

3

4

(e) t = 7

(b) t = 7

4

4 CD4-CN

-5

-2.16

-6

3

-1.6

y

y

3

CD4-RK3

2

2 OSCS-ORK3

OSCS-ORK3

1 0 0

1 1

2

x

3

0 0

4

1

2

x

3

(f) t = 11

(c) t = 11

4

CD4-CN

4

-1.3 -3.8

-1.8

-4.81

3

y

y

3

4

2

2

OSCS-ORK3

OSCS-ORK3

CD4-RK3

1 0 0

1 1

2

x

3

4

0 0

1

2

x

3

4

Figure 8.3 Numerical solution of one-dimensional convection equation by (a-c) the IMEX method using a combination of Crank-Nicolson and CD4 schemes used on the left of the overlap region and CD4-RK3 method on the right, as used in [388]. Results of IMEX method are compared with the results of OSCS-ORK3 method shown in frames (d-f). In the IMEX method six point overlap is used, while ten point overlap is used for the compact scheme. [Reproduced from “Precursor of transition to turbulence: Spatiotemporal wave front”, Swagata Bhaumik and Tapan K. Sengupta, Phys. Rev. E, 89(4), 043018 (2014), with the permission of APS Physics.]

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of such large sources of error prevents formation of STWF, which have been shown to be created for much lower amplitudes of wall excitation in [32, 419].

8.5.1 Three-dimensional route of transition by harmonic wall excitation Receptivity of the boundary layer forming over a semi-infinite flat plate, as obtained by the three-dimensional Navier–Stokes equation is studied for time-harmonic threedimensional wall excitation. In [33], the case of time-harmonic GCP wall excitation has been studied. The imposed wall-normal velocity in the physical plane vw (x, z) on the circular patch is given by vw (x, z) = α1 Am (x, z) sin(ω ¯ 0 t)

(8.29)

with α1 as the amplitude parameter; Am (x, z) as the amplitude distribution on the plate for wall-normal velocity, with absolute value varying between zero and one. The quantity, ω ¯ 0 is the non-dimensional circular frequency given as, ω ¯ 0 = F f ReL . The amplitude function for the GCP type excitation is represented as Am (r) and given by  πr ! 1 1 + cos (8.30) Am (r) = 2 rmax p for r ≤ rmax and Am (r) = 0 for r > rmax , where r = (x − x0 )2 + (z − z0 )2 and (x0 , z0 ) indicates the center of the circular patch. The results shown in [33] are for rmax = 0.09, x0 = 1.5 (corresponding Reδ∗ = 666) and z0 = 0. In Figure 8.4(a), the amplitude function of the GCP exciter, Am (r) is shown plotted as a function of non-dimensional radial distance, r/rmax . For the SM exciter the amplitude function Am (x, z) is given as Am (x, z) = Aex (x) sin (8πz/zmax )

(8.31)

The amplitude of exciter is obtained as Aex (x) = α1 A1 (x). For an exciter located between x1 and x2 , the amplitude distribution for the exciter is given by A1 (x): For x1 ≤ x ≤ xex x − x1 A1 = 15.1875 xex − x1

!5

x − x1 − 35.4375 xex − x1

!4

x − x1 + 20.25 xex − x1

!3 (8.32)

and for xex ≤ x ≤ x2 x2 − x A1 = −15.1875 x2 − xex

!5

x2 − x + 35.4375 x2 − xex

!4

x2 − x − 20.25 x2 − xex

!3 (8.33)

2 where xex = x1 +x is the center-point of the SBS strip, which is also used for two2 dimensional studies in [133, 412]. Variation of A1 with (x − x1 )/(x2 − x1 ) is plotted in Figure 8.4(b). Presented results are obtained [33] for x1 = 1.455 and x2 = 1.545, so that xex = 1.5, where the Reynolds number based on displacement thickness is Re∗δ = 666. In

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(a) 1

Am

0.8

0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

0.8

1

r/rmax

(b) 1

A1

0.5

0

0.5

1 0

0.2

0.4

0.6

(x − x1)/(x2 − x1 )

Figure 8.4 Amplitude function of the (a) Gaussian circular patch (GCP) and (b) spanwise modulated (SM) time-harmonic wall exciter. [Reproduced from “A new velocity–vorticity formulation for direct numerical simulation of 3D transitional and turbulent flows”, S. Bhaumik and T. K. Sengupta, J. Comput. Phys., vol. 284, pp 230-260 (2015), with the permission of Elsevier.]

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the results given next for the three-dimensional transition route, the amplitude control parameter is taken as α1 = 0.01, i.e. the maximum amplitude of the input perturbation is 1% of the free stream velocity. In Figure 8.5, perspective views of streamwise disturbance velocity (ud ) are shown in (x, z)-plane for y = 0.00442 at t = 15, 20 and 40, with GCP excitation at the wall. For this case, zmax = 1 and the GCP exciter is tuned to the non-dimensional 2 frequency of F = 5.0 × 10−5 , where F = 2πν f /U∞ ; and f is the frequency in Hertz. In all the frames, one can identify the three elements of response, which are typical of fixed frequency wall excitation cases, as used in the experiments in [237, 405] and noted in the computational results in [32, 33, 419, 422, 451] for two- and threedimensional routes of transition to turbulence. For the GCP excitation, one notes the local solution at the location of the GCP patch, and following this is a pair of oblique TS wave-packets that terminate on the spanwise edges of the domain and are reflected from there. The leading structure of the disturbance field is the STWF, which grows spatio-temporally in a more violent manner than what has been observed for two-dimensional disturbance fields in [419, 422], due to the spanwise periodic boundary condition. Additional growth in this three-dimensional route is also due to the presence of the vortex stretching mechanism present in the three-dimensional flow. By t = 40, one can clearly see the formation of turbulence spots. Spatio-temporal growth of the disturbance field is distinctly observable in the planview shown in Figure 8.6 for ud at the non-dimensional height, y = 0.00215. One notes higher wavenumber fluctuations (turbulent spots) at selected locations near the spanwise boundaries due to multiple interactions among STWFs, which have disturbance levels of the order of U∞ . Two such zones (S 1 and S 2 ) identifying turbulent spots (in frames (b) and (c)) are shown at t = 26 and 29 at S 1 and S 2 . These spots while spreading in the streamwise direction, as well as in the spanwise directions, spawn newer spots upstream (marked as S 3 and S 4 ). This regeneration and self-induction mechanism establishes a fully turbulent zone extending from x ' 13 to 25 at t = 36 in Figure 8.6(d). In Figure 8.7, the perspective plot of ud is shown at the height, y = 0.00442, for the case of wall excitation created by the SM exciter with two full wavelengths in the spanwise direction. Although the local solutions for SM exciter are qualitatively different from the GCP case in the form of honeycomb cells near the exciter, the main difference is also related to the TS waves which are aligned parallel in the streamwise direction for SM exciter case. The STWF and the TS wave are seen to be contiguous, even at t = 15. The growth rate of the STWF takes it above 15% of the free stream speed even at such an early time. By t = 18, this amplitude grows significantly higher to the level of 60% of the free stream speed. In frame (c) of Figure 8.7, one can discern the nonlinear saturation of ud and turbulent spot-like features are clearly noted. This aspect can be noted better in the plan view shown in Figure 8.8, which shows the flow field up to t = 27. The plan view shown in Figure 8.8 for the SM excitation case shows the typical growth of the disturbance field. The frame (a) at t = 15 shows two spanwise waves for

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a) GCP: t = 15 Wave-fronts

0.04

0.02

ud

z 0

0

2

4

TS wave-packets −0.2 b) GCP: t = 20

0.2

8

x

Local solution

ud

z 0 2

0

4 TS wave-packets

−0.2 c) GCP: t = 40

0.2

8

x

ud

z

0

5 15

−0.2

x

Figure 8.5 Perspective view of streamwise disturbance velocity (ud ) plotted in (x, z)-plane at y = 0.00442 for the flat plate boundary layer excited by a Gaussian circular patch (GCP) wall exciter, with non-dimensional excitation frequency is F f = 5 × 10−5 . Results are shown at (a) t = 15, (b) t = 20 and (c) t = 40, with the three elements identified as the local solution in the neighborhood of the circular patch, the oblique TS waves originating from the local solution and leading these is the spatio-temporally growing STWF. [Reproduced from “A new velocity–vorticity formulation for direct numerical simulation of 3D transitional and turbulent flows”, S. Bhaumik and T. K. Sengupta, J. Comput. Phys., vol. 284, pp 230-260 (2015), with the permission of Elsevier.]

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a)

−0.1 −0.005 0.005

0.1

0.25

b) t = 26 0.5

0.4

0.5 t = 20; y = 0.00215

N0

N0 −0.25

−0.25

−0.5

x 0

5

10

c) t = 29 0.5

15 S3

20

25

0.25

N0

N0

−0.25

−0.25 0

5

x

10

S4 15 S2

0

5

x

10

S2 15

20

25

5

x

10

15

20

25

d) t = 36 0.5

S1

0.25

−0.5

S1

0.25

0.25

−0.5

i

20

25

−0.5

0

Figure 8.6 Plan-view of streamwise disturbance velocity (ud ) plotted in (x, z)-plane at y = 0.00442 for the flat plate boundary layer excited by a GCP wall exciter, excited by the non-dimensional frequency of F f = 5 × 10−5 . Results are shown at (a) t = 25, (b) t = 26, (c) t = 29 and (d) t = 36, with the flow structures marked with the faint impression of local solution, equally weak oblique TS waves originating from the local solution and the very well formed spatio-temporally growing STWF with marked turbulent spots. [Reproduced from “Precursor of transition to turbulence: Spatiotemporal wave front”, Swagata Bhaumik and Tapan K. Sengupta, Phys. Rev. E, 89(4), 043018 (2014), with the permission of APS Physics.] the local solution. The TS waves follow each of the half-waves of the local solution and indicate streamwise growth. However, such growth is impeded by the presence of the STWF which is large in structure at the leading edge. In frame (b) of Figure 8.8 at t = 22, one notices weakening of the TS wave amplitude in the spanwise direction, while the length scale associated with the STWF is seen to decrease along with steepening of the amplitude of the STWF near the front. One also notices some amount of dilatation of STWF at the leading edge, while near the rear of the STWF the structures merge together. Such dynamics are also noted in frame (c) of the figure with the leading edge of STWF approaching x = 20, while the trailing edge moves slower, thereby affecting a larger streamwise stretch of the domain. Upstream regeneration of STWF is also noted for this case of excitation, and the sequence of events depicted in Figure 8.8 takes the flow from the laminar to the turbulent stage. Time averaged skin friction coefficient (< C f >) and the shape factor, H = δ∗ /θ are shown in Figure 8.9 for the data along midspan-line (z = 0) for the collected

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Three-Dimensional Routes of Transition to Turbulence a) SM exciter at t = 15 0.2

Local solution

Wave-fronts

U∞ 0.15

0.1

ud 0

0.05

2 4

0 8

x

6 TS wave-packets

b) SM exciter at t = 18 0.6

Wave-fronts

Local solution

0.5 0.4 0.3

ud

0.2 0 2

0.1 4

0 6 −0.1 c) SM exciter at t = 20 0.6

8

x

TS wave-packets

Wave-fronts Local solution

0.5 0.4 0.3

ud

0.2 0 0.1

2 4

0 6 x

−0.1

TS wave-packets

8

Figure 8.7 Perspective view of streamwise disturbance velocity (ud ) plotted in (x, z)-plane at y = 0.00442 for the flat plate boundary layer excited by a SM wall exciter, excited by non-dimensional frequency, F f = 5 × 10−5 . Results are shown at (a) t = 15, (b) t = 18 and (c) t = 20, with the three elements identified as the local solution near the SM exciter, the spanwise parallel TS waves originating from the local solution and leading these is the spatio-temporally growing STWF, which is well formed by t = 20. [Reproduced from “A new velocity–vorticity formulation for direct numerical simulation of 3D transitional and turbulent flows”, S. Bhaumik and T. K. Sengupta, J. Comput. Phys., vol. 284, pp 230-260 (2015), with the permission of Elsevier.]

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a) t = 15, F = 0.5 × 10

0.25 0.15 0.05 0.001 −0.001 −0.01 −0.1 −0.2

0.2 0.1 N0 −0.1 −0.2 0 b) t = 22

5

10

x 15

20

5

10

x 15

20

5

10

x 15

20

0.2 0.1 N0 −0.1 −0.2 0 c) t = 27 0.2 0.1 N0 −0.1 −0.2 0

Figure 8.8 Plan view of streamwise disturbance velocity (ud ) plotted in (x, z)-plane at y = 0.00442 for the flat plate boundary layer excited by a SM wall exciter, excited by non-dimensional frequency F f = 5 × 10−5 . Results are shown at (a) t = 15, (b) t = 22 and (c) t = 27. [Reproduced from “A new velocity–vorticity formulation for direct numerical simulation of 3D transitional and turbulent flows”, S. Bhaumik and T. K. Sengupta, J. Comput. Phys., vol. 284, pp 230-260 (2015), with the permission of Elsevier.] time-series between t = 40 to 50. The abscissa is converted to the Reynolds number based on current length from the leading edge, as Re x . It is well known that for laminar zero pressure gradient boundary layer, the steady skin friction is given by C f = 0.664 × Re−1/2 and this is indicated in Figure 8.9(a) by a dashed line. Also, for x the fully developed zero pressure gradient turbulent boundary layer, an estimate for

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= 0.074 × Re−1/5 x

a) Time averaged between t = 40 to 50.

CfL = 0.664 × Re−1/2 x Rex × 10−5 HL = 2.59

H

HL ~ 1.4

−5

Rex × 10

Figure 8.9 (a) Time-averaged skin friction coefficient < C f > and (b) the shape factor (H ) along midspan, z = 0 plotted as function of Re x , respectively. Theoretical variation of < C f > and H for laminar and turbulent flows are also marked in frames (a), (b), respectively. [Reproduced from “A new velocity–vorticity formulation for direct numerical simulation of 3D transitional and turbulent flows”, S. Bhaumik and T. K. Sengupta, J. Comput. Phys., vol. 284, pp 230-260 (2015), with the permission of Elsevier.] time-averaged skin friction is given in [335, 518] as, < C f >= 0.74 × Re−1/5 , depicted by x the dashed-dotted line above the curve for the laminar flow. While the quiet laminar flow skin friction matches with the derived analytical expression given above, the region for the turbulent spot between x = 17 to 24 shows the tendency to match the fully developed turbulent flow trend. In the other part of the perturbed flow, the time averaged skin friction lies between laminar and turbulent flows, as one would expect. In Figure 8.9(b), plotted shape factor is seen to be between the line for laminar flow (HBlasius = 2.59) and fully developed turbulent flow value (HT ∼ 1.4). It is observed that the transitional flow has higher skin friction (and hence higher momentum thickness, even though the displacement thickness is also higher), as compared to the fully turbulent value noted experimentally.

8.5.2 Spectrum of inhomogeneous turbulent flow over flat plate In Figure 8.10, compensated streamwise spectral densities [380] (E11 , E22 and E33 ) are shown for the spanwise station located at z = 0.38, which are the square of the Fourier-Laplace transform of ud , vd and wd , respectively. From this figure one notes the existence of an intermediate wavenumber range, where spectral densities vary as k1−5/3 – a variation in the inertial subrange predicted for three-dimensional isotropic homogeneous turbulence [242]. More importantly, displayed spectra show

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(a)

Time averaged spectral density

2

10

5/3

103

1

10

z=0 z = 0.38 z = 0.48

100

10

0

1

2

k1

10

10

k1

101

102

(b)

10

1

10

2

z=0 z = 0.38 z = 0.48

5/3

100

103

100

(c) 2

5/3

10

1

10

100 10–1 10

–2

10

–3

z=0 z = 0.38 z = 0.48

100

k1

101

102

Figure 8.10 Compensated streamwise spectral density plotted as a function of streamwise wavenumber (k1 ) for z = 0.38 at t = 36.0, for the case shown in Figure 8.7. similarity with the experimental data for inhomogeneous flow past a flat plate [380]. Thus, the STWF is observed to take the three-dimensional transition route to fully developed turbulent flow all the way from receptivity to turbulent stages, similar to two-dimensional cases reported in the previous chapter and in [419].

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8.6 Effects of Frequency and Wavenumber on Three-Dimensional Transition So far in this chapter, the discussion has been related to simulating experimental cases reported in [209, 237], which relates mostly to the SM type of wall excitation. Although the computational results in [32] was primarily for GCP type of wall excitation, results in [33] are for both types of excitation. The authors in [32] have noted that during the later instability stages of three-dimensional transition, different arrangements of vortices are seen. Based on classification of routes during three-dimensional transition to turbulence, the processes follow either K- or H-types of transition. Sharma and Sengupta [482] have noted that “two 3D routes of boundary-layer transition are referred to in the literature: a) the K-type route (after Klebanoff et al. [237]) and b) the H-type route [209], which is also known as N-type transition. The K-type route shows a fundamental wave with amplitude modulated in spanwise direction to form ‘peaks’ and ‘valleys’, such that one notices aligned Λ-vortices [386]. Additionally, at a later stage there appear ring-vortices at the tip of the Λ-vortices for the K-type route [49, 172, 209, 368]. However, the H-type of transition is characterized by the presence of a staggered pattern of Λ-vortices. Theoretical and experimental explanations have been attempted for both K- and H-type transition routes, following resonant triad interaction of waves [99, 210]. This topic has been resolved for the three-dimensional routes of transition, as affected by the frequency of monochromatic wall excitation started impulsively by DNS in [482].” The computed results for the two frequencies of excitation reported in [482] resemble the experimental set up of [237], where two-dimensional boundary layer is excited by the SM-type of wall excitation. The monochromatic wall excitation creates three-components disturbance field: a near-field followed by the TS wave and the STWF, which is the precursor of eventual transition. It is shown that for moderate frequency of excitation, the near-field does not interact with the STWF. More importantly, another route of transition that has been reported computationally, is caused by a lower frequency of wall excitation. This case displays an interacting nearfield solution with the STWF. Both the frequencies of excitation studied in [482] can cause transition for moderate spanwise wavelength (or λz that is half the width of domain in the spanwise direction) disturbances, and dependence of transition on λz is also reported in this reference. It is shown that by halving λz , the STWF disappears without causing flow transition. This has immense implication on the role of twoversus three-dimensional disturbances on spatio-temporal growth of disturbances, and appears to follow the Squire’s theorem (discussed in [400, 412] for temporal instability). One observes that the squire theorem has been shown to be valid for temporal instability only. Thus, this result has far reaching consequences, which is in opposition to concepts proposed by various researchers about the centrality of three-dimensional transition, as against the study of two-dimensional instabilities. Interestingly, the three-dimensional study in [482] highlights the greater importance of two-dimensional wall excitation.

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Many researchers have studied the alterations in receptivity of boundary layers that have been exposed to different types of disturbances to develop better understanding of active mechanisms in different routes of transition. For example, presence of three-dimensional surface indentations interfere with two-dimensional TS wave via linear and nonlinear analysis for the instability of a boundary layer, as in [566]. The receptivity of swept-wing boundary layers in the presence of imposed steady vortical disturbances has been studied by DNS in [249] for highspeed subsonic flows. The receptivity is observed to be lower for higher Reynolds and Mach numbers. Receptivity of laminar boundary layers deterministically excited by localized disturbance upstream of a flat plate has been studied experimentally in [550]. The spanwise gradient of v and ωξ are observed to play major role on transfer of disturbance energy inside the boundary layer from outside. Another mechanism that has been suggested for three-dimensional transition in literature [251, 279], is called the lift-up mechanism, and attempts to explain algebraic growth of disturbance with time [60]. This is an inviscid analysis following the concept in [128], and is based on the assumptions regarding: Streamwise independence of streamwise velocity and vorticity; and for the wall-normal velocity component to be time-independent. But, it has been shown that these assumptions are not tenable in results using DNS in [483]. Once again, the velocity-vorticity formulation that has reported STWF in a long domain using high accuracy methods in [32, 33, 483], is employed in simulating the vibrating ribbon experiments performed for the frequency response in [209, 237]. In this formulation, vorticity transport equation and velocity Poisson equations are solved with high accuracy methods by making the dependent variables divergencefree [190]. Also to minimize aliasing error, the staggered grid is adopted, as in [32, 33], with velocity components evaluated at face centers and the vorticity components at the edge centers. The optimized staggered compact scheme, and an optimized DRP, three-stage, Runge-Kutta method is used for time integration

8.6.1 Computational domain The computational domain in Figure 8.1 has a non-dimensional length 50.05 in the x-direction; unit height in the y-direction and spanwise width from −0.25 to 0.25 is taken in z-direction. The grid has 2501 × 351 × 49 points, in the x-, y- and z-directions, respectively. The leading-edge is accommodated by taking a small section ahead of the flat plate at xin = −0.05. The lengths are non-dimensionalized with a reference length scale L, such that ReL = 105 . The grid clustering used has been described in Sub-section 8.4.

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8.6.2 Grid parameters The stretching parameters of the grid used in the streamwise and wall-normal directions, are the same. The grid resolutions in three spatial directions are given by [482]: i)

In the wall-normal direction: (∆y)min = 4.21 × 10−4 and (∆y)max = 6.0 × 10−3 .

ii)

Along streamwise direction: (∆x)min = 1.575 × 10−3 and (∆x)max = 2.22 × 10−2 .

iii)

Along spanwise direction: ∆z = 1.04 × 10−2 .

A non-dimensional time step of ∆t = 8.0 × 10−5 is used for time integration by ORK3 method.

8.6.3 Boundary conditions At the inlet of Figure 8.1, uniform free stream velocity is used. At the outflow, the Sommerfeld boundary condition is used for velocity and vorticity components. Periodic conditions apply on boundaries in z-direction. Far-field outflow boundary conditions are imposed on the top plane with u = 1, w = 0 and v-component of velocity evaluated using Eq. (8.12). Boundary conditions imposed on segment ahead of the ∂ωη ∂w leading edge for y = 0 are ∂u ∂y = ∂y = ∂y = 0 and v = ωζ = ωξ = 0.

8.6.4 Spanwise modulated wall excitation The SM time-harmonic input (spanwise punctuated suction-blowing strip) are applied near the leading edge of the plate. Based on displacement thickness, the Reynolds √ number at the exciter location (x = 1.5) is given by Reδ∗ = 1.72 (x∗ /L)ReL ≈ 666, where x∗ is the dimensional length [482]. The variation of wall excitation amplitude in x- and z-directions are given as A x = AF (1.0 + cos(2πxex ))/2 # " 2π z Az = sin λz such that

(8.34) (8.35) A(x, z) = A x Az

(8.36)

where the amplitude is AF = 0.01, with the exciter centered at xexciter = 1.5 and stretch of exciter from x1 = 1.455 to x2 = 1.545, as shown in Figure 8.1. Thus, in Eq. (8.34): xex = (x − xexciter )/(x2 − x1 ) for x1 ≤ x ≤ x2 . Two cases of spanwise variation are reported in [482] with two and four periods of a sinusoid in the range z = ±0.25, with non-dimensional wavelengths (λz ) as 0.25 and 0.125, respectively. The nondimensional spanwise wavenumbers (kz ) for these two cases therefore are 25 and 50. The investigated cases are given in Table 8.1. The spanwise variations of the exciter amplitude with two- and four-spanwise waves are also shown in Figure 8.12.

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A

A

A

A

A

0.01 Y

Z N

N

N

N

N

N

N

N

N

0 0.2

0.1

0

A

A A = Antinode N = Node

A

0.1

0.2

A

A

0.01

Figure 8.11 The spanwise variation of the wall excitation shown with solid line having two spanwise waves and dashed line with four spanwise waves. [Reproduced from “Effect of frequency and wavenumber on the three-dimensional routes of transition by wall excitation”, Pushpender K. Sharma and Tapan K. Sengupta, Phys. Fluids, vol. 31, pp 064107 (2019), with the permission of AIP Publishing.] The wall excitation frequency is F f and is started impulsively, so that the wallnormal velocity is given as vex = A × H1 (t) sin(F f ReL t)

(8.37) ω ¯0 ReL , given in the viscous scale. The same ω∗ L convective scales is given by, ω ¯ 0 = (2πU∞f )L = U0∞ ,

where the non-dimensional frequency is, F f =

non-dimensional circular frequency in with ω∗0 = 2π f as the dimensional frequency. In Eq. (8.37), H1 (t) is the Heaviside function for the impulsive start, i.e. vex is activated for t ≥ 0. Two F f values have been considered in [482], one with moderately high frequency of F1 = 1.0×10−4 , and a lower frequency of F2 = 5 × 10−5 . For these two frequencies with the given exciter location, there are finite intermediate spatial ranges, where the respective TS waves will show growth. Table 8.1 Three-dimensional wall excitation cases considered. Case No. 1

Ff −4

1.0 × 10

λz and kz

No. of spanwise waves

0.5, 25.13

4

0.25, 50.26

8

−4

2

0.5 × 10

3

1.0 × 10−4

4

0.5 × 10−4

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8.7 Effects of Frequency and Wavenumbers on Three-Dimensional Transition A GCP exciter has been used to excite the boundary layer in a spanwise periodic domain, with maximum extent of zmax = 1 used in [32], which creates oblique threedimensional TS waves, apart from the STWF. This STWF evolves into turbulent spots, which merge together to form fully developed turbulence for this moderate frequency excitation case. The two-dimensional [36, 509] and three-dimensional [483] cases of excitation have established that the STWF is created by non-impulsive start also, once again confirming the STWF as the generic precursor of transition to turbulence. For two-dimensional transition, excitation with moderately high frequency causes non-interacting Class-N type of transition, whereas lower frequency excitation creates Class-I type of transition, which is based on classification scheme in [36, 419]. Similar types of transition for moderate and lower frequency excitations are noted for threedimensional route also, with STWF always playing the main role. The study in [482] used SM wall excitation to simulate the experiments in [209, 237].

8.7.1 Different frequency and wavenumber cases of evolving STWF For normal mode analysis, one can easily see as pointed out in [412] that spatial instability is more complicated as compared to temporal instability. This is due to the fact that for such TS waves, the wave vector points in one direction, while the spatial growth will occur in other directions. Additionaly, for three-dimensional transition routes, frequency of excitation is a major determinant. Two different frequency cases (cases 1 and 2 in Table 8.1 with the exciter having two spanwise waves) are compared in Figure 8.13 showing ωηd . For cases 3 and 4 with the exciter having four spanwise waves, a similar comparison is shown in Figure 8.14. In these figures, a three-dimensional isometric view for ωηd is shown for a near-wall plane located at y = 0.00215. Results for this height are shown for ωηd , which is known as the squire mode and is considered important for parametric resonance and instability [400]. To show the effects of input frequency in Figure 8.12; ωηd is plotted on the left for F1 = 1 × 10−4 , and on the right column for F2 = 5 × 10−5 . In Figure 8.12, early time results can be noted at t = 5 with results shown for a domain up to x = 5, that displays qualitatively different near-field solutions for F2 ; “with streamwise streak for F1 and dimpled cellular structure for F2 ” [482]. At the later stage of t = 20, ωηd is plotted over an extended range up to x = 25 for clarity of view for the growth of STWF, that is more rapid for F2 . For F1 , the STWF appears later (not shown, but occurs at t = 15), with the three components of solution visible. One distinct feature for the case of F1 is that the STWF separates clearly from the exciter, while for the case of F2 , one observes interaction between the TS wave and the STWF. According to stated convention, the F1 -case follows the Class-N type of transition, and the F2 -case follows the Class-I type of transition [36].

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For both the cases in Figure 8.12, the nonlinear stage displays growth as high wavenumber fluctuations, are noted at t = 20 for the F1 -case and at t = 15 for the F2 -case (which is not shown). For both the cases, growth of STWF is responsible for the transition mechanism. The times for the secondary and nonlinear growth stages happen earlier for the F2 -case as a consequence of continued interactions by which energy is pumped to the STWF from the exciter via the TS wave. Thus, this growth route is different as compared to the F1 -case. Also, as a result of this, the secondary and nonlinear growth stages occur upstream for the F2 -case, as compared to the F1 -case. Thus, once again, one can conclude that the lower frequency cases experience transition, quicker and with more acceleration, compared to moderate/ higher frequency cases. Once the nonlinear stage begins, faster disturbance growth is noted in the frames at t = 40 and t = 55 for the F1 - and F2 -cases, with the turbulent spot forming over the full spanwise width, even at early stages of nonlinear growth. The F2 -case is distinguished by disturbance spread over a longer streamwise region, compared to the F1 -case. In the case of F1 , a fixed location of the trailing edge of the disturbance field is noted near x = 13.2, while for the F2 -case, the trailing part of the disturbance field is observed to end near x = 11. For cases-3 and -4 in Table 8.1 (with the exciter having four spanwise waves), the variation of ωηd is shown in Figure 8.13. In this figure, one can observe the near-field in the immediate vicinity of the exciter at early time stages of t = 0.5 and 5. Thereafter, for both the frequencies (F1 and F2 ), one does not notice the creation of the STWF, quite unlike the cases-1 and -2. Furthermore, the near-field is seen to decay with time, and the disturbance field becomes progressively smaller as it convects downstream, eventually moving out of the domain. Therefore, increasing the input disturbance to a higher spanwise wavenumber by doubling it, does not lead to an even (or steady) growth of disturbances. Thus, eventual transition to turbulence is impossible for such wall excitation with higher spanwise wavenumbers for the flow over the semi-infinite flat plate.

8.7.2 Evolution of disturbances from laminar to turbulent flow Time variation of the disturbance streamwise velocity (ud ) is shown at the indicated spanwise locations at the nodes and anti-nodes in Figure 8.11 of the exciter, in Figures 8.14 and 8.15, for cases-1 and -2 of Table 8.1. These are the cases where the exciter has two spanwise waves. The results are presented as semi-log plots with the ordinate in log scale. The spanwise stations at z = 0 and −0.125 are at the nodes, whereas z = −0.0625 and −0.1875 are at the anti-nodes, as marked in Figure 8.12. The corresponding case for exciter with four spanwise waves are not important, as those excitation cases do not show growth in disturbance. Figure 8.14 shows time variation for ud for the frequency value of F1 , with the top frame for t = 15, and up to t = 70, when the leading front of disturbances reaches the outflow. In frame a), the STWF is near x = 10 at t = 15, which suffers secondary and nonlinear growth stages between t = 20 and 30, with the front steepening. Such

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t=5

t=5

t = 20

t = 20

t = 40

t = 40

t = 55

t = 55

Figure 8.12 Three-Dimensional isometric view of the disturbance wall-normal vorticity (ωηd ) shown in ( x, z)-plane for y = 0.00215. Spanwise wall exciter has two spanwise waves for case-1 and case-2 of the table with frequency for case-2 at half of what it is for case-1. [Reproduced from “Effect of frequency and wavenumber on the three-dimensional routes of transition by wall excitation”, Pushpender K. Sharma and Tapan K. Sengupta, Phys. Fluids, vol. 31, pp 064107 (2019), with the permission of AIP Publishing.]

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Figure 8.13 (a) Three-dimensional isometric view of the disturbance wall-normal vorticity (ωηd ) in ( x, z)-plane, showing comparison for two frequencies, at early times indicated in the figure. Spanwise exciter has four spanwise waves for case-3 and case-4. The later time evolution is shown in Figure 8.14. [Reproduced from “Effect of frequency and wavenumber on the three-dimensional routes of transition by wall excitation”, Pushpender K. Sharma and Tapan K. Sengupta, Phys. Fluids, vol. 31, pp 064107 (2019), with the permission of AIP Publishing.]

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discontinuous fronts excite wider range of wavenumbers for ud . As the leading edge of the STWF convects faster, in comparison to the trailing edge, the disturbance front widens with time, and in frame f ) for t = 70, the leading front has exited the outflow. The dashed line at x = 13.2 approximately marks the transition location, where the trailing edge of STWF settles. The amplitude of ud saturates eventually to remain below 0.6U∞ . Clear demarcation between the TS wave and STWF is noted in all the frames, indicating the non-interacting type of transition. Figure 8.15 shows the evolution of ud for the lower frequency (F2 ) case, with the frames starting from t = 15 till 65. Due to continuous interaction between TS wave and STWF, the latter suffers secondary and nonlinear growth at upstream locations, as compared to the higher frequency (F1 ) case, with such growths also occurring at an earlier time. For this lower frequency case, the trailing edge of the STWF is noted at x ≈ 11, which is at an upstream transition location, as compared to the case of higher frequency (F1 ).

8.7.3 Energy propagation speed and group velocity From Figures 8.14 and 8.15, it is noted that the STWF convects downstream by first going through a linear growth stage, followed by rapid secondary and nonlinear stages of growth. The streamwise convection speed of the STWF is the group velocity, which can be calculated by plotting ud in the (x, t)-plane, as shown in Figure 8.16. The ud variation with x at various times is extracted at the anti-nodal location z = −0.1875 for the height, y = 0.00215 in this figure. The position of the leading edge of STWF is noted as function of time, indicated by the long-dashed line in Figure 8.16, for both the frequencies, F1 - and F2 -cases. The slope of these lines in these two frames provide the group velocity, i.e. Vg = dx/dt and the values are noted approximately as Vg = 0.68 for the frequency F1 and Vg = 0.65 for the frequency F2 . Thus, there is hardly any difference for the group velocity between the Class-N and Class-I types of three-dimensional routes of transition. There is marginal difference for the onset of transition location.

8.7.4 Maximum disturbance amplitude growth for high and low frequencies Transition to turbulence occurs following the nonlinear growth of STWF. For all the cases in Table 8.1, this happens for cases-1 and -2 only, with the exciter having two spanwise waves, λz = 0.25 and kz = 25.13. However, for cases-3 and -4, one does not observe disturbance growth. The growth of maximum value of disturbance streamwise velocity, indicated by udm , is therefore extracted over the full domain for cases-1 and -2 only. The time evolution of udm for these cases are shown in Figure 8.17. Results are presented in a semi-log plot, with the ordinate in log scale for the same spanwise locations chosen to trace the growth at nodes and anti-nodes of the exciter. The left column of Figure 8.17 shows the variation along anti-nodes, while the right column shows the variation along the nodes. For the case of frequency F1 , the primary

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Figure 8.14 Disturbance streamwise velocity (ud ) plotted as function of x for F1 = 1 × 10−4 , with frames a), b), c), d), e), and f ) showing variation at t = 15, 20, 30, 40, 50, and 70, respectively. [Reproduced from “Effect of frequency and wavenumber on the three-dimensional routes of transition by wall excitation”, Pushpender K. Sharma and Tapan K. Sengupta, Phys. Fluids, vol. 31, pp 064107 (2019), with the permission of AIP Publishing.]

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Figure 8.15 Disturbance streamwise velocity (ud ) plotted as function of x for F2 = 5 × 10−5 , with frames a), b), c), d), e), and f ) showing the variation at t = 15, 20, 30, 40, 50, and 65, respectively. [Reproduced from “Effect of frequency and wavenumber on the three-dimensional routes of transition by wall excitation”, Pushpender K. Sharma and Tapan K. Sengupta, Phys. Fluids, vol. 31, pp 064107 (2019), with the permission of AIP Publishing.] growth is noted at about t = 10, while for the frequency F2 this is at about t = 3. The STWF appears earlier for frequency F2 , as compared to the case of frequency F1 , as

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Figure 8.16 The ud contour plot in ( x,t)-plane for the frequency of excitation cases, F1 and F2 . The position of the leading edge of STWF is marked with a long-dashed line. Slope of this line provides the group velocity value of the STWF, which is Vg ≈ 0.68 for F1 and Vg ≈ 0.65 for F2 . [Reproduced from “Effect of frequency and wavenumber on the three-dimensional routes of transition by wall excitation”, Pushpender K. Sharma and Tapan K. Sengupta, Phys. Fluids, vol. 31, pp 064107 (2019), with the permission of AIP Publishing.] noted before. The time when udm reach 1 % of U∞ is also marked in the frames for the cases shown. The frames indicate that at both the nodes and anti-nodes, udm reaches 0.01U∞ earlier for the case of F2 , as compared to the case of F1 . Furthermore, the time to reach this level is earlier for the anti-node, as shown in frames a1) and a2) at t ≈ 7.7, which can be compared to the time for the node shown in frames b1) and b2) occurring at t ≈ 10.5. The maximum disturbances for streamwise velocity saturate in all cases when transition occurs, and remain bound to within 0.6U∞ .

8.7.5 Near-field solution and receptivity routes Near-field solutions for the cases in Table 8.1 are shown in Figure 8.18 for the plane at a height y = 0.00215, with frames i) and iii) showing ud for cases-1 and -3, respectively, for frequency F1 . Frames ii) and iv) show ud for cases-2 and -4, respectively, for frequency F2 . The feature to note for the F1 -case in frame i) is that downstream of the exciter between x = 1.5 and 3, disturbances align in such a way that crests of the response remain aligned with the nodes of the exciter. Similarly, the troughs of the response remain along the anti-nodes of the exciter, as seen in frame i) of Figure 8.18. One also notices that the spanwise variation of ud occurs between x = 2 and x = 5, which is half of the spanwise wavelength of the exciter, i.e. there are twice the

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100

10 0 10 2

F1 F2

10 4 u dm 10 6

104 u dm 106

t = 15.8

Z = 0.1875 Antinode

8

10

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10

t = 10.5

20

a1)

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0

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b1)

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2

10

4

10 u dm

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0

2

10 10 u dm

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10

Z = 0.0625 Antinode

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10 t

40

t = 18.8

Z=0 Node

8

t = 7.7

20

4

6

t = 17.5

10

a2)

Z = 0.125 Node

8

t = 7.7

10

t = 18.8

60

b2)

t = 10.5

20

t

40

60

Figure 8.17 Maximum disturbance streamwise velocity (udm ) associated with STWF compared at four z-locations for the two frequency cases. Frames a1), a2), b1), and b2) show the udm variation along z = −0.1875, z = −0.0625, z = −0.125, and z = 0, respectively. The time when the disturbance achieves 1% of U∞ ; is marked in the frames, which occur at different x locations. [Reproduced from “Effect of frequency and wavenumber on the three-dimensional routes of transition by wall excitation”, Pushpender K. Sharma and Tapan K. Sengupta, Phys. Fluids, vol. 31, pp 064107 (2019), with the permission of AIP Publishing.] numbers of spanwise waves, as there are spanwise wave at the exciter. At x = 2.34 and x = 3.65, one notices the signal to have the anti-node for ud . For the case of frequency F2 shown in frame ii) of Figure 8.18, the near-field shows cellular patterns. One observes oscillation of ud with time, along the spanwise antinodes between x = 1.5 and x = 3, with sharp ridges forming along the spanwise node. This is unlike the case of frequency F1 , with spatial distribution of ud remaining invariant with time. The STWF for frequency F2 is seen to grow and convect beyond x = 3. The created STWFs emerge continuously, while interacting for this case. The spanwise wavelength associated with the STWFs is the same as that of the exciter for frequency F2 case.

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Figure 8.18 Zoomed view of ud at y = 0.00215, showing the near-field solution. Frames i) and iii) are plotted for exciter with two and four spanwise waves, respectively, for F1 = 1.0 × 10−4 . Frames ii) and iv) are plotted for exciter with two and four spanwise waves, respectively, for F2 = 0.5 × 10−4 . [Reproduced from “Effect of frequency and wavenumber on the three-dimensional routes of transition by wall excitation”, Pushpender K. Sharma and Tapan K. Sengupta, Phys. Fluids, vol. 31, pp 064107 (2019), with the permission of AIP Publishing.] In frame iii) of Figure 8.18, one observes diamond shaped structures form between x = 1.5 and x = 3, which are completely different from the case shown in frame i). These vanish beyond x = 3 for this case. The structures in frame iv) have similarity with that in frame ii), occurring between x = 1.5 and x = 3, but these also attenuate beyond x = 3.

8.8 Closing Remarks In this chapter a number of different three-dimensional routes of transition have been examined. Classically, these routes are classified as K-type and H-type, depending upon the alignment of Λ-vortices noted in late-transitional flows. In the earliest experiment reported in [237], these vortices were observed to be aligned for the high value of time-harmonic SBS-type spanwise modulated (SM) wall excitation. When the SM excitation was repeated in [210] with excitation frequency significantly lower, then one saw the H-type of vortical structures and this was thought to be caused by resonant interaction. One such earlier theory of resonant triad interaction had been

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proposed by Craik [99]. This modal theory was based on how TS waves interact. However, as pointed out in Chapter 7, there are two types of two-dimensional routes of transition based on interactions between TS waves and the STWF, with STWF as the nonmodal component of the solution. There is also the non-interacting or Class-N type of transition, where TS waves and STWF do not interact, and which is noted for moderate to high frequency of wall excitation. In the second route, for the interacting or Class-I type of transition, the modal (TS wave) and nonmodal (STWF) components are seen to continually interact for the low frequency excitation of the boundary layer. For the three-dimensional routes of transition, parallels exist with corresponding two-dimensional routes, with Class-N type similar to K-type transition and Class-I type transition pairing with H-type transition. This provides the great synergy of transition routes for both two- and three-dimensional wall excitation and is the significant development in the subject in the last decade. We have noted that transition experiments are specially designed and their validation by specific theoretical formulations are mutual vindication of both theory and experiment. However, how does one progress from experimentally designed routes of transition to natural transition? Attempts have been made from the early era of transition research. One of the pioneering efforts was by GI Taylor [510], which was endorsed by Monin and Yaglom [295] as a valid conjecture. Some additional experiments have been performed by Kendall [225, 226], Lim et al. [267] in the last four decades and definitive progress have been made with the solution of the Navier–Stokes equation in [431, 474, 475], along with two new theoretical concepts of disturbance mechanical energy and disturbance enstrophy transport energy. This aspect of transition is the subject matter of Chapters 9 and 13.

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Receptivity to Free Stream Excitation: Theory, Computations, and Experiments

9.1 Introduction to Free Stream Excitation The receptivity to free stream disturbances was introduced in Chapter 5 from the perspective of its role in creating TS waves. This problem is addressed by few researchers due to the failure reported in the experiments of [405]. Its authors could not create TS waves by acoustic excitation from the free stream, as was theoretically predicted earlier by Tollmien and Schlichting using linear spatial instability theory. Reasons for these are many-fold: first, an acoustic wave is three-dimensional and thus does not excite two-dimensional TS waves. Free stream excitation is not monochromatic, and the linear spatial theory demands monochromatic excitation. It is shown in Sub-section 5.3.2 that receptivity of the laminar boundary layer to free stream disturbance convecting with free stream speed shows very weak coupling, unless one follows the bypass route (convection speed is significantly lower than the free stream speed). Experimental work also started with that reported in [510]; to estimate the dependence of critical Reynolds number on free stream turbulence (FST). While discussing Taylor’s work in [295], the authors (of this treatise) conjectured that FST consisting of convected vortices is responsible for creating adverse pressure gradients locally, which gives rise to unsteady separation. Such separations trigger the rapid vortex-induced instability leading to transition. The assumption in [295] is that the “effect is connected with the generation of fluctuations of longitudinal pressure gradient by these disturbances, leading to the random formation of individual spots

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of unstable S-shaped velocity profile.” Exciting a shear layer by sources outside it has been experimentally investigated later in [109, 225, 226, 267]. Dietz [109] created disturbances inside the boundary layer by vibrating a ribbon in the free stream at a single frequency. This is supposed to supplement the experiment in [405] to create TS waves by free stream excitation at a monochromatic frequency. Dietz [109] actually did not demonstrate progressive TS wave, and instead plotted velocity profiles which looked like the TS wave eigenfunctions, shown in [412]. It is pertinent to note that even the STWF displays a velocity profile in [29], which has the same wall-normal distribution of streamwise disturbance velocity. So the demonstration in [109] is qualitative only. In Sub-section 5.3.2, the coupling mechanism between wall and free stream excitation is explained. This is despite the shear sheltering effect shown in [194], which states that disturbances in the free stream remain sheltered from penetrating inside the boundary layer. Physically, this effect is further corroborated in [431, 474] from direct simulation of the Navier–Stokes equation. Thus, the work in [109] reinforced the idea that free stream excitation does also create TS waves, originally shown in [405] for wall excitation (although the authors failed to do so by free stream acoustic excitation). This apparent demonstration of a TS wave by free stream monochromatic excitation in [109] supported also the belief of the research community about TS waves being the central factor in causing flow instability and transition to turbulence. In contrast, Kendall reported two sets of experiments in [225, 226] for free stream excitation to bring in practical considerations of transition. Thus in [226], jet-induced FST (which is polychromatic in nature) results were purported to show evidence of TS wave-packets forming on a nominally flat plate boundary layer. The interpretation of such a result was also not proper, as discussed in [412]; that the displayed structures are actually a train of local solutions and not a TS wave-packet. Here, unlike the experiment of [405], instead of a TS wave, the author referred to the creation of TS wave-packets in [226]. Also, it went unnoticed that such packets actually convected with the flow, and did not appear stationary as shown in Chapters 7 and 8 for the frequency response experiments in two- and three-dimensional routes of transition. Morkovin [298] attempted to classify all transitions in fluid flows into two simple categories. The first one, was termed as natural transition, which displays a TS wave in the formative stage. All other routes have been given the generic term of bypass transition. The readers may find this too simplistic based on the contents of previous chapters, specifically the knowledge gained in Chapter 6, where the results from [509] have been used to show how the STWF controls transition, even in the so called natural transition cases per the results presented in Table 6.1; and associated discussion which is outlined in Section 6.4. It is appropriate to note that in [298], it is a mere conjecture whether bypass transition is due to nonlinear, nonparallel effect(s) and or some altogether unknown mechanisms. With the present state of knowledge we can identify the unknown mechanism as the action of STWF which is omnipresent, even for the so-called natural transition.

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However, it is pertinent to examine developments with free stream excitation from the perspective of so-called bypass transition. By now, readers are familiar that without considering STWF, any classification scheme is going to be noninclusive and incomplete. We quote from [474] “that detected TS waves in the experiments are always for a ZPG boundary layer created by a monochromatic vibrating source. Considering that natural transition is caused by imperceptible, polychromatic excitations, it is next to impossible to trace TS waves without a properly designed experiment using a monochromatic source in a ultra-quiet wind tunnel, as in [405].” In trying to create natural transition, Gaster and Grant [144] performed a pure impulse response experiment, by wall excitation creating a localized pulse (but they mistakenly thought that the response field was an ensemble of TS waves). It has been correctly inferred in [34] and for case 1 of Table 6.1 that such excitations lead to transition without the attributes of frequency response, i.e. the local solution and the TS wave-packet. It is simply impossible to have classification of transition, as proposed by Morkovin [298], even for a zero pressure gradient boundary layer. Furthermore in [64, 412], the authors identify flow past bluff bodies; Couette and pipe flows; leading edge contamination on swept wings [319, 434] as examples of bypass transition. There are also reports of this route in flows with adverse pressure gradient, caused by two- and three-dimensional roughness effects [276, 331]. The instability by roughness element is via temporal growth of disturbances by an inflectional velocity profile. This leads to added confusion that temporal instability should be called as bypass transition. Similar other examples of bypass transition are given in [400]. Another definition of the bypass route is given in [561] as “similar to the secondary instability and breakdown process in boundary layer natural transition.” It is noted in [267] that all cases of bypass transition in [298] are triggered by vortical disturbances affecting the boundary layer from outside. Kendall [225] reported experimental results, obtained by moving a pair of circular cylinders in a circular trajectory outside a zero pressure gradient boundary layer, to create a train of periodic disturbances, which were identified as pressure pulses. Kendall reported that the boundary layer is highly receptive when the free stream disturbances moved at a speed around c∗ = 0.3U∞ (where c∗ is the dimensional translation speed of the free stream vortex and U∞ is the oncoming flow velocity). This is explained in [429, 474] with the solution of the Navier–Stokes and the Orr–Sommerfeld equations. The linearized disturbance in this case, follows a route of amplification along c = constant lines in the (Reynolds number, circular frequency)-plane, as opposed to following the constant frequency route in wall excitation cases. A similar experiment has been reported in [270], where periodic disturbances have been directed towards the wall of a boundary layer, with high wall-normal velocity to simulate events in turbomachinery. Wu et al. [559] performed DNS for this experiment [270] to validate features of the experiment qualitatively. Researchers have attempted to find out the physical mechanisms responsible for various routes of transition which are called as bypass transition. These are classified into categories of streak instabilities [61, 62, 63, 296, 390], and those originating due

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to interaction between continuous and discrete modes [124, 202, 271, 569, 570]. Note that all these still refer to linear analysis, while discussing discrete modes. However we have already learnt in Chapter 7, that STWF has linear mechanism at onset, that is not related to discrete eigenvalues, either obtained by spatial or temporal instability route. In the streak instability, bypass transition is associated with the appearance of streamwise elongated near-wall streaky structures of high and low streamwise velocity. The proponents of this point of view interpret the non-normality of linearized operator of the Navier–Stokes equation which cause disturbances to grow significantly by a mechanism called transient energy growth. For boundary layers, such streaks arise due to counter-rotating streamwise vortices. Such vortices lift low-momentum fluid from the wall on one side and push high-momentum fluid from the outer parts towards the plate on the opposite side. This creates regions of accelerated and decelerated streaks, on either side of the vortex. This has been also noted in the literature as lift-up effect. However, in the original concept the mechanism involved in the lift-up effect the focus was to look at it as an inviscid effect. Thus, there appears some ambiguity in discussions on embedded vortices inside the boundary layer which cause disturbance growth by an inviscid mechanism. After the formation of such streaks, the boundary layer experiences secondary instabilities with the response field displaying very high-frequency components. This is a signature of instabilities formed due to the wall-normal and spanwise velocity components both displaying inflectional velocity profiles, as in cross-flow instability noted over sweptback wings [434]. The secondary instabilities with high frequency, symmetric and antisymmetric oscillating streaks are readily prone to formation of turbulent spots, which merge to form fully developed turbulent flow. However, the mechanism for the formation of streamwise vortices is not stated. Also, this implies that transition can only occur following a three-dimensional route. We have noted in Chapter 5 that the low frequency contents of free stream excitation can create the Klebanoff mode which can be triggered for the streamwise vortices. This is actually relevant here for forming streamwise elongated streaks. Brandt et al. [61] studied secondary instability due to finite-amplitude streaks in flat plate boundary layer with high levels of FST, which was termed as noise, that led the authors to conclude the bypass transition to be noise-driven. In [63], effects of high levels of FST on bypass transition for a Blasius boundary layer have been studied by DNS. The authors used synthetic turbulent inflow by superposing “continuous modes” of the Orr–Sommerfeld and Squire operators to constitute an optimal set for the onset of instability. The Squire operator refers to the governing vorticity transport equation for the wall-normal component. The perturbations which are created become unstable via sinuous/ varicose breakdown for the underlying mechanism to trigger bypass transition. Furthermore, two physical mechanisms have been identified, based on the energy content of the imposed perturbation: 1) a linear mechanism for FST with low frequency disturbances, and 2) a nonlinear process if the FST predominantly

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contains high-frequency disturbances. This is also relevant for a three-dimensional perturbation field. In [390], the role of streak instabilities during bypass transition by FST is discussed. Although the authors state that the leading edge is important for the FST to enter the boundary layer, it is not apparent how this is accounted for in the study. In [62], the authors show turbulent breakdown of the flat plate boundary layer due to streamwise velocity streaks of finite amplitude. The unsteadiness and interactions among streaks are important triggers for the breakdown. In [202], DNS of bypass transition of a zero pressure gradient boundary layer forced by FST has been studied by considering the inflow to consist of the Orr–Sommerfeld continuous modes. In [569], the manner by which external vortical disturbances are internalized inside the laminar boundary layer to induce transition is studied. The authors showed that the entire transition process is realized by the interactions of only two Orr–Sommerfeld continuous mode eigenfunctions: one with low frequency that generates Klebanoff streaks, and the other a high frequency mode. The former is not affected by shear sheltering and is able to penetrate the boundary layer. As explained in Chapter 5 free stream excitation supports upstream propagating Klebanoff modes and thus can explain the backward perturbation jets, while the induced wall mode by the coupling mechanism can account for the forward moving jets. The highfrequency disturbance, according to these authors, is prevented by the shear sheltering effect, and such disturbances can only exist in the free stream. This high frequency component, constrained to stay out of the shear layer, can interact with the ejected backward jets at the top of the boundary layer, giving rise to an inflectional velocity profile responsible for temporal instability. Such a high frequency instability is more intense in the downstream direction, and while breaking down into a turbulent spot. However, one of the important aspects in this point of view is identifying the continuous mode of the Orr–Sommerfeld equation, which these authors take along the imaginary axis of the streamwise wave number (α). As explained in the properties of the Orr–Sommerfeld equation, these are essentially the Stokes line described in Subsection 4.6.3 in Chapter 4. However, these authors do not discuss the other Stokes line for p˜ = 0 in Figure 4.6. In [570], the route of bypass transition that has just been discussed, through interactions of continuous mode of the Orr–Sommerfeld equation, is considered for an adverse pressure gradient boundary layer. According to such proponents, the influence of pressure gradient is not due to the inflectional mean velocity profile, but instead is due to the coupling of the free stream mode with the boundary-layer shear, which determines the intensity of streamwise streaks. It is equally interesting that they state that there are no leading-edge effects due to insensitivity of bypass transition to it. In [271], the interaction of TS waves and boundary-layer streaks is studied by numerical simulation. However all of these studies which invoke adverse pressure gradients, use local instability aspect of the Orr–Sommerfeld equation, and thus suffer due to limitations of the parallel mean flow assumption.

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The authors in [557, 560, 562], discuss bypass transition caused by patches of isotropic turbulence introduced periodically at the inflow. The authors claim that the mechanism has similarity with the secondary instabilities noted in boundary layer H-type transition [181]. It is concluded that “a well-controlled bypass transition will proceed along a path broadly resembling one of the secondary instabilities in natural transition.” While non-physical streamwise periodicity is not considered nor are any terms being added to the Navier–Stokes equations, time scales are yet added indirectly via the periodical insertion of inflow turbulence patch. This simple artifice adds a whole range of time scales that are not strictly physical.

9.2 Experiments on Transition without TS Waves The experiments reported in [109, 226] have been described as ones which emphasize the role of TS waves for transition. One notes that the experiment in [226] is not related to TS wave dictating transition by free stream jet, rather it is still the STWF which is responsible for transition. The other experiment reported in [225] is more interesting. Its importance was noted by the authors in [267], in designing a more controlled experiment, which can be called as the kernel experiment due to its importance as highlighted in [492]. A very good account of the experiments in [267] and their significance have been reported in [474]. The above discussion on different mechanisms for bypass transition, which use a mixture of spatial theory (to explain the creation of streamwise streaks) with the temporal growth route during the secondary and higher order instability stages, are without theoretical rigor. These are heuristics at best to explain the computed results. One notes that the role of STWF has completely eluded some. It can be considered as the primary instability itself, and thus the STWF is the building block of transition to turbulence. In this context, careful inspection of experiments is mandatory. In discussions about free stream vortices interacting with boundary layer vorticity, one misses out the vital factor of action at a distance. This is important, in view of the shear sheltering effects described in [194]. Such an interaction can take place by BiotSavart interaction [414], and this has been addressed in the experiments reported in [225, 226, 267]. A pertinent excerpt from [474] examines requirements of receptivity experiments. Experiments on receptivity have to be designed as in [405], where existence of TS waves was established as part of verifying spatial theory. Thus, one excites the boundary layer at a fixed single frequency to create TS waves. However in natural transition by wall excitation, the background disturbances are never at a single frequency, and it is next to impossible to clearly identify TS waves in natural transition. The same equilibrium flow in a similar experimental set-up was used for the threedimensional receptivity experiment in [237]. The zero pressure gradient boundary layer is neutrally stable over a finite plate, if the background disturbances are kept very small, which was the motivation for designing a low-noise wind tunnel. The use of specific equilibrium flow is necessary to study transitional and turbulent flows. This

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is the reason behind receptivity experiments and DNS of transitional flows selecting zero pressure gradient boundary layer as the equilibrium state. It is also used in the experiment in [270] to simulate transition in a turbomachine, which was later also simulated in [559]. Monin and Yaglom [295] observed that FST can cause “separation of the boundary layer under the action of a negative longitudinal pressure gradient” to explain the effect of disturbances in the ambient flow and which is related to the creation of fluctuations of the longitudinal pressure gradient. These can lead to the random formation of individual spots of unstable S-shaped velocity profiles causing separation and transition of the boundary layer. On the basis of this hypothesis, G. I. Taylor [510] tried to estimate theoretically the dependence of the critical Reynolds number on the turbulence intensity of the oncoming flow. Taylor proposed that critical Reynolds δ2 ∂p number is fixed by the modified Pohlhausen parameter Λ = − µU ∂x , where δ is the boundary layer thickness. One notes that in a semi-infinite flat plate boundary layer there can be fluctuations in pressure, as noted in Figure 7.8. And so, Taylor further proposed that the character of the motion in a fixed section is determined here by the δ2 δp0 0 parameter Λ = − µU δx (where p is the pressure fluctuation and δ/δx signifies a typical value of the derivative ∂/∂x). According to Taylor, the point of transition from laminar to turbulent flow is determined by the parameter Λ attaining some critical value [295]. This provided justification to study effects of FST on a semi-infinite flat plate boundary layer by creating controlled pressure fluctuations experimentally, as in [225, 267], and have been studied theoretically and computationally in [431, 474, 475]. In the experiments of [225], the boundary layer is excited by periodic free stream disturbances introduced outside the boundary layer. The schematic of the experiment and the summary of results in [225] are shown in Figure 9.1, as reproduced from [474]. In the top panel, the test configuration shows a rotor with two cylinders placed at a distance R from the axis of rotation, rotating at a rate Ω0 in an anti-clockwise direction. The translation speed of the periodic pulse is then given by c ' Ω0 R. This creates input disturbances that remain outside the boundary layer, while the response is shown in the bottom frame of Figure 9.1. Evidently, the induced streamwise fluctuations display receptivity for speeds c in the range 0.2 to 0.6, with a maximum receptivity for c = 0.35, that falls off rapidly on either side of this peak value. For a periodic input, the induced pressure field inside the boundary layer that is ahead of the rotor, creates an adverse pressure gradient. In the computations of [429], for periodic free stream excitation an identical trend was noted. For vortex convection speed of c = 0.3, a rapid growth of disturbances was reported (Figure 10 in [429]), whereas for the free stream speed of c = 1, the flow was not receptive (as in Figure 9 of [429]). The linear stability diagram in Figure 6 of [429] shows disturbance growth to be in the phase speed range of 0.26 to 0.39. In Figure 11 of [225], it is observed that when the rotor rotated in the clockwise direction (thereby creating a favorable pressure gradient ahead of the rotor), the flow was not receptive.

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Figure 9.1 Schematic of the experimental setup used by Kendall [225] for the periodic free stream vortex excitation problem over a zero pressure gradient boundary layer (top); Peak response of streamwise disturbance velocity (u0 ) measured for different rotor speeds as obtained by Kendall [225]. [Reproduced from “Direct numerical simulation of vortex-induced instability for zero pressure gradient boundary layer”, A. Sengupta, V. K. Suman, and T. K. Sengupta, Phys. Rev. E, vol. 100, 033118 (2019), with the permission of APS Physics.] The effects of FST resemble those that are created by a free stream convecting vortex. This is explained in [474] that if “a free stream vortex is far away from any wall, then it will convect at the local speed dictated by the vorticity transport equation. However, when the vortex is placed above a flat plate, it will not do so due to Biot-Savart interactions among the free stream vortex, its image system and the boundary layer forming over the flat plate. For the case of FST, this situation is further complicated as FST is a collection of many vortices, which interact continually; causing each vortex in the collection to travel at changing local speed, which is distinctly

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different from the free stream speed. One of the goals of the present study has been to find out the speed of translating vortex with higher receptivity.” This was the motivation behind the conceptualization of the experiments in [267], where total absolute control was achieved over the fluctuating pressure by controlling the speed, which in turn kept the strength and sign of the circulation of the vortex fixed by using a translating and rotating cylinder, moving at a fixed height with constant rotation rate, and a constant translational speed. The rotating cylinder with a high rotation rate, creates a compact and stable captive vortex in the free stream, that is constrained to move at a fixed speed and height when it is translated in the streamwise direction. The use of a captive vortex created by a rotating cylinder is supported by experimental and computational work in [442, 552] for super-critical rotation rates. In that scenario, the Karman vortex street is absent for moderate Reynolds numbers. Any shed vortices behind the rotating cylinder, will not affect the boundary layer, due to the weak strength of such vortices, as compared to the captive vortex strength associated with the rotating cylinder. If the cylinder is not rotated and simply translated in the experiment, then the boundary layer is not perturbed, as has been shown in [267]. The experimental investigation in [267] relates to the excitation of the semi-infinite flat plate boundary layer by an aperiodic free stream vortical disturbance. The experimental and computational set-ups of the problem in [474] are shown in Figure 9.2. Results shown in [267] are for those parameters which perturb the equilibrium flow effectively by this translating vortex. In [474], the authors explain the strong receptivity route shown in this experiment to be similar with research trends of determining optimal forcing by various models of transient growth, nonmodal growth etc. Thus, apart from the need to perform such kernel experiments, one is motivated by the physical insights provided in [295, 510]. The convecting free stream vortex is a representative vortex in the ensemble of vortices associated with FST, which can create an adverse pressure gradient to destabilize the boundary layer. While the experiment identifies the parameter values for strong receptivity, the question related to convection speed has to be addressed, as there seems to be misconception among researchers in presuming that all vortices in FST convect at the free stream speed. At the same time, the investigations reported in [328, 329, 428] have clearly established the role played by Biot-Savart interaction in deciding the local speed for any vortex in the ensemble. The experiment in [267] was designed to test the hypothesis that a distant vortex can induce a streamwise adverse pressure gradient to destabilize a boundary layer, the idea proposed to study dependence of critical Reynolds number on FST in [510]. Monin and Yaglom [295] following up on this idea, observed that such change in critical Reynolds number due to streamwise fluctuating adverse pressure gradient is due to a sequence of unsteady separations, created by vortices in the FST. The implicit assumption here is that the effect is a consequence of fluctuating longitudinal pressure gradient by created FST, leading to formation of individual spots [295]. The experiment [267] and computations [431, 474] point out the role of FST in causing transition, triggered by vortex-induced instability without TS waves. Hence,

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(a)

R4

u,v)

v v Far- eld boundary

a y

G

Q4 P4

c

P3

xout

y

te Pla

x

In ow u , v U v v Q1

ow

w O u tflo

R1

Q3 z

R2

x

x out

P1

Plate x

z ma

zmax (b)

R3 ymax Ou

P2 xxinin

x

Plate leading Plateedge leading edge z Rotating and translating cylinder y

Dyelines U∞

c w Ωr

x Leading edge of theωplate (L.E.)

Dye port

z Hc

Ωr

Flat plate Flap

U∞ c

(c)

25

Figure 9.2 (a) Schematic of the computational domain used to study vortex-induced instability by a translating vortex (speed, c and circulation, Γ) in the free stream over a zero pressure gradient boundary layer. (b) The plan and side view of the experimental arrangement, with the flat plate fitted with a flap; a rotating and translating circular cylinder causing the free stream vortex and arrangement for dye visualization are shown. (c) Photographic experimental set-up in the water tunnel shown [474]. [Reproduced from “Direct numerical simulation of vortex-induced instability for zero pressure gradient boundary layer”, A. Sengupta, V. K. Suman, and T. K. Sengupta, Phys. Rev. E, vol. 100, 033118 (2019), with the permission of APS Physics.]

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this has been referred to as bypass transition in [267, 431]. The interaction of the shear layer with finite-core vortex convecting outside the boundary layer will lead to unsteady separation on the wall, which is a new physical mechanism suggested in [295, 510] and verified in [267, 431]. The vortex-induced instability has been noted as “one of the most important unsolved problems of fluid dynamics” in [115]. This problem can have many variations, but one can consider only the case where the primary instability is triggered by a convecting vortex over the boundary layer forming on a flat plate with sharp leading edge. The same mechanism governs unsteady flow separation [108], near-wall eddy formation in turbulent boundary layers [68], and has also been thought to form hairpin vortices in the near-wall region of turbulent flows [372, 492]. In reporting the results in [474], all dimensional quantities are denoted with an asterisk, while the corresponding non-dimensional quantities are presented without ∗ such. To non-dimensionalize quantities, incoming free stream speed (U∞ ) is taken as ∗ the velocity scale, and L as the length scale, so that the Reynolds number is defined ∗ ∗ ∗ L /ν , with ν∗ as the kinematic viscosity. The time scale is derived from by ReL = U∞ the chosen length and velocity scales. The two-dimensional simulation for one of the cases has been reported in [431], reporting bypass transition to occur ahead of the vortex of strength, Γ = 9.1, translating at a speed of c = 0.154 which is compared with the experimental results in [267]. The experiment is performed for sub-critical instability with respect to linear theory following the Orr–Sommerfeld equation. The vortexinduced instability onset is due to unsteady separation, and hence linear theories to study it do not exist, and a nonlinear receptivity theory for incompressible flow has been developed using disturbance mechanical energy [431]. Another nonlinear theory for receptivity has been developed using disturbance enstrophy transport equation in [474, 475].

9.3 Direct Numerical Simulation of Three-Dimensional Vortex-Induced Instability Computations reported in [474] tried to reproduce the experimental observations in [267] to provide data in accounting for the effects of translating vortex in the free stream. The free stream vortex is modeled to mimic experimental arrangements in [267]. Ideally, receptivity experiments should use an amplitude as small as possible, to meet the expected definition of instability. For the two-dimensional simulation in [431], the free stream vortex strength was Γ = 9.1, while in the three-dimensional simulations in [474], the magnitude of Γ has been reduced to 0.5 and 2. Two-dimensional computations have been shown in [431, 475] and threedimensional computational results are shown in [474] to explain the physical mechanism during transition caused by vortex-induced instability. It is necessary to describe how the initial two-dimensional response evolves into a three-dimensional field for the counter-clockwise vortex cases. Also, a case for a higher speed of c = 0.77

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is computed, when the free stream vortex rotate in counter-clockwise direction. This is followed by the case of clockwise rotating vortex translating at c = 0.19, as shown in the experiment of [267]. The numerical methods use high accuracy compact schemes for solving the Navier–Stokes equation in velocity–vorticity formulation with a staggered grid that has been described before in [32], to explain transition from receptivity to fully developed turbulent flow for wall excitation, by comparing with experimental results in [237, 405]. The impulse response experiment reported in [144], has also been validated in [34] by three-dimensional DNS using the same formulation and methods. The vortex-induced instability in the experiment [267] shows the response field to remain two-dimensional for long time after the onset of instability, as noted in [475] using the same formulation during the primary instability. The receptivity of the equilibrium flow for semi-infinite flat plate boundary layers to translated vortex in the free stream is due to the time-dependent boundary forcing. The origin of the co-ordinate system is at the mid-point of the leading edge, while the computational domain begins slightly ahead of it.

9.3.1 Problem set-up and numerical methods The Reynolds number based on L∗ is used as ReL = 105 for all the simulations reported in [474], with the schematic already shown in Figure 9.2(a). The experimental set-up in [267] is shown in frames (b) and (c) of the figure. The experiment was performed in a water tunnel with a flat plate held vertically and the cylinder moving parallel to the plate. The cylinder is rotated and translated by two motors to create the free stream translating vortex. The extent of computational domain is given by xin ≤ x ≤ xout along the streamwise direction; 0 ≤ y ≤ ymax along the wall-normal direction and −zmax /2 ≤ z ≤ zmax /2 along the spanwise direction. The free stream vortex translates in the x-direction at a constant speed c and at a constant height Hc from the plate. The inlet (for computational purposes) is upstream of the leading edge, and the flow field is periodic in the spanwise direction. The free stream vortex has constant circulation given by Γ per unit length, with its axis along the spanwise direction. The sign of the vortex is determined by the direction of rotation and is positive for the counterclockwise case. ~ = bωξ ωη ωζ cT and The vorticity and velocity components are denoted as ω ~ = bu v wcT , in streamwise, wall-normal and spanwise directions, respectively. The V governing equations along with boundary conditions are described in Section 8.3, and are also used in [475] for this problem. This solution strategy requires a staggered grid formulation, with the location of the variables as indicated in Figure 9.3. The vorticity components are defined at the center of each edge of the cell, while the velocity components are placed at the center of the faces of the computational cell in the transformed plane. To evaluate first derivative and interpolate, an optimized staggered compact scheme (OSCS) is used [32]. Interpolation is required for staggering, and second and mixed derivatives

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are obtained by using the OSCS for first derivatives repeatedly. An optimized three stage Runge-Kutta ORK3 -scheme is used for time integration of the vorticity transport equation, with a time-step of ∆t = 8 × 10−5 or less. Control of numerical instabilities and aliasing error are performed by using a two-dimensional filter in (ξ,η)-plane with a filter coefficient of 0.18 ≤ α2D ≤ 0.22 with minimal frequency. Schwarz domain decomposition technique using MPI parallelization framework is used for developing a parallel algorithm [435]. The computations use compact schemes in the (ξ, η, ζ)plane, where x = x(ξ), y = y(η) and z = z(ζ). With respect to Figure 9.2(a), the computational domain and parameters used are given as: the streamwise extent is, xin = −0.05 and xout = 20; the wall-normal extent is defined by, ymax = 0.75 (for c = 0.3, 0.19 and 0.77), and 1.5 (for c = 0.386), and the spanwise extent is defined by, zmax = 1.6. The free stream vortex is released from Hc = 2 and x0 = −1. A grid with (1001 × 301 × 129) points is used in x-, y- and z-directions, respectively, for the cases of c = 0.3, 0.19 and 0.77. The minimum and maximum grid spacing + + = 108.364 in the x-direction, ∆y+min = in terms of wall units are ∆xmin = 46.742, ∆xmax 1.8476, ∆y+max = 25.9838 in the y-direction and ∆z+ = 62.6237 in the z-direction. A grid with 1001 × 351 × 129 points is used for the c = 0.386 case. For this case, in terms + + = 108.364 in the x-direction, of wall units the grid spacing is ∆xmin = 46.742, ∆xmax + + ∆ymin = 1.9986, ∆ymax = 50.399 in the y-direction and ∆z+ = 62.6237 in the z-direction.

9.3.2 Three-dimensional equilibrium flow: Initial condition The equilibrium flow is obtained without excitation, first in two-dimensions and the solution is stacked along the third dimension. This is run with the three-dimensional code for some time to obtain the three-dimensional equilibrium flow. Subsequently, two-dimensional free stream excitation is initiated by the translating vortex to study the onset of the three-dimensional receptivity problem.

9.3.3 Imposed free stream excitation The receptivity to translating free stream vortex is understood from the expression of induced stream function created by the finite core translating potential vortex in the free stream given as  (y − H )(d/2)2 (y + H )(d/2)2  c c + 2 ψ∞ = y − (1 − c) 2 x¯ + (y − Hc )2 x¯ + (y + Hc )2 Γ x¯2 + (y + Hc )2 + Ln 2 4π x¯ + (y − Hc )2

(9.1)

where d is the diameter of the vortex core; the vortex strength is Γ; whose translation speed is c and which is moving at the constant height Hc . The displacement effect of the finite core vortex depends on the relative speed, (1 − c), given by the second term on the right-hand side, and the circulation effect is given by the last term on the

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Receptivity to Free Stream Excitation y v ωξ

(i, j+1, k) ωζ

ωζ ωη

ωξ

u

ωη (i, j, k)

ωη w

ωξ ωη

(i+1, j, k)

x

ωζ

ωζ ωξ

(i, j, k+1) z

Figure 9.3 Staggered grid for velocity and vorticity components used for the three-dimensional formulation [474]. [Reproduced from “Direct numerical simulation of vortexinduced instability for zero pressure gradient boundary layer”, A. Sengupta, V. K. Suman, and T. K. Sengupta, Phys. Rev. E, vol. 100, 033118 (2019), with the permission of APS Physics.] right-hand side of Eq. (9.1). As the vortex translates with speed c, the instantaneous streamwise location x¯, in the example given above depends on time, with x¯ = x − (x0 − ct), with x0 as the initial location of the free stream vortex. This equation makes use of the method of images in potential flow, accounting for the presence of the flat plate. The displacement effect is seen to be directly reduced for higher values of c. It is noted that Γ is indirectly related to the effective speed, i.e. on (1 − c), along with the rotation rate of the cylinder, through the Reynolds number. However, the circulation effect was experimentally observed to be more important in [267], such as when the cylinder is translated without rotation at the speed of c = 0.39, there was hardly any receptivity. It may be concluded from the expression of the total stream function that the relative speed plays a stronger role in determining the circulation effects, and is the main factor of receptivity of the boundary layer to convecting vortex. The streamlines for the ideal inviscid cases for counter-clockwise and clockwise vortex in the free stream are shown at the bottom in Figure 9.4. These are schematics of streamlines sketched from the expression of imposed stream function given by Eq.

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250

(a) (a)

250

200

200

150

150

100

100

50

50

0

–1

–0.5

α 0

0.5

0 –1

1

(b) (b)

–0.5

α

0

0.5

1

Adverse pressure gradient

Figure 9.4 Schematic of streamlines for the potential flow created by a translating and rotating cylinder given by Eq. (9.1) are shown in the bottom. (a) Depiction of the clockwise rotating vortex; above it is the spectrum of the induced flow field. (b) Streamline and spectrum depicted for a counter-clockwise vortex [474]. The region of imposed adverse pressure gradient is marked by slanted dashed lines. [Reproduced from “Direct numerical simulation of vortex-induced instability for zero pressure gradient boundary layer”, A. Sengupta, V. K. Suman, and T. K. Sengupta, Phys. Rev. E, vol. 100, 033118 (2019), with the permission of APS Physics.] (9.1). These are ideal streamlines in the inviscid part of the potential flow, created by the rotating and translating cylinder. The bottom horizontal line depicts the flat plate, and in frame (a), one notices the stagnation points located below the mid-plane for the clockwise rotating vortex case. The flow enclosed between the stagnation streamline and the flat plate helps one understand qualitatively that the imposed pressure gradient is favorable, ahead of the finite core vortex. Meanwhile the flow behind the vortex experiences a milder adverse pressure gradient, where the limiting streamline diverges the most behind the vortex. These regions with adverse pressure gradients based on streamline divergence are marked in both the frames with slanted dashed lines. One can perform the Fourier transform of the wall-normal velocity, and the spectrum is shown on top of the streamline plots. For the clockwise rotating case in frame (a), two regions of adverse pressure gradients are indicated: one behind the cylinder and the other region is just below the cylinder and towards its front part. For the counter-clockwise rotation case, an adverse pressure gradient is noted in

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the front half of the region below the cylinder. For this case as depicted in Figure 9.4(b), the streamline divergence downstream of the cylinder is more pronounced, than that of the clockwise rotating vortex case in frame (a). Thus, the imposed adverse pressure gradient downstream of the counter-clockwise rotating vortex creates a significantly stronger effect that causes higher disturbance growth leading to vortexinduced instability. This schematic is in accordance with the two-dimensional Navier– Stokes computational results in [431] for a low value of c. The Fourier spectrum of wall-normal velocity on top of frame (b) corresponding to counter-clockwise vortex case is broadband, as compared to the clockwise rotating case in frame (a). In both the cases, the zero wavenumber component is almost same, and the lower wavenumbers of the counter-clockwise case impose a stronger adverse pressure gradient, causing vortex-induced instability. These adverse pressure gradient effects by free stream vortex are qualitative, as mutual interactions between the boundary layer vorticity and free stream vortex can be complex, as is shown in [474] for the rapidly translating vortex (c = 0.77) case. The information from Figure 9.4 should be interpreted for the imposed pressure gradient in the inviscid part of the flow. However, once unsteady separation is initiated near the wall, there will be secondary pressure gradients created by the unsteady separation bubble, which will convect with the flow.

9.4 Validation of Direct Numerical Simulation for Vortex-Induced Instability The DNS results in [474] have been compared with the experimental results in [267]. Three experimental cases have been identified for validation in [474] which are for (i) strong vortex-induced instability case of a counter-clockwise free stream vortex with c = 0.386, Γ = 2; (ii) weaker counter-clockwise free stream vortex with c = 0.77, Γ = 0.5 and (iii) another weak case of clockwise free stream vortex with c = 0.19, Γ = −0.5. The frames for validating computations have been obtained from videos and these frames are complementary to those noted in [267].

9.4.1 Strong vortex-induced instability caused by counter-clockwise free stream vortex This case from [267] is for c = 0.386 that shows strong receptivity. For this case the cylinder rotates in the counter-clockwise direction, which imposes an adverse pressure gradient ahead of the vortex in the inviscid part to thicken the boundary layer ahead of the vortex. The experimental figures are shown in Figure 9.5(a), with the rotating cylinder translating in the free stream from left to right at the relative speed of (1 − c). The flat plate boundary layer is seen in the top frame with a nascent disturbance in wall-normal direction, immediately downstream of the translating vortex. The dye streaks are more perturbed in the middle frame of Figure 9.5(a). It should be noted that the counter-clockwise free stream vortex stabilizes the boundary layer upstream of

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(a)

Experimental visualization (plan view) (b) Computational Results (side view) c = 0.386 t = 25 c = 0.77, G = 2 Wr ωη: –18.1339

0

16.0874

5

U∞

x cv = 8.6

C

8

10

12

x 14

16

18

x 14

16

18

16

18

t = 35.6 Wr

U∞

xcv = 12.746 C

8

10

12

t = 38.2 Wr

U∞

xcv = 13.7452 C

8

10

12

x 14

Figure 9.5 (a) Snapshots of the receptivity experiment for c = 0.386 shown in the left column, with experimental parameters described in the text. (b) Corresponding computational results in the right column for counter-clockwise rotating vortex case shows strong receptivity for a boundary layer forming over a flat plate with sharp leading edge. the vortex, while it destabilizes the boundary layer downstream. This is visible in the computed results in Figure 9.5(b), showing very good correlation with experimental visualization. The instantaneous streamwise location of the free stream vortex is indicated as xcv in all the frames of Figure 9.5(b). The rectangle with arrowheads on top shows the location of the free stream vortex. The strong interaction reflects scouring of the boundary layer by the imposed adverse pressure gradient, as noted in the bottom frames of Figure 9.5. The top frame shows the perturbation field to have predominantly spanwise vorticity (ωζ ) with no spanwise variation, while the other two components of vorticity (ωξ , ωη ) slowly increase from negligible values by vortex-stretching in the enstrophy transport equation [348, 468]. Evidently, the strong three-dimensional nature of the flow is revealed at later times, as noted in the experiments [267]. In visualization pictures, three dimensionality is visible with dye streaks spreading by strong eddy diffusivity due to vortex-induced instability. This is despite the boundary layer on the flat plate being sub-critical [267]. Thus, vortex-induced instability is an example of

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bypass transition, as growing TS waves are not created in the experiments and the noted events are reflected in computations.

9.4.2 Weak vortex-induced instability for counter-clockwise free stream vortex For vortex-induced instability caused by the free stream vortex, reduced receptivity is noted with increased c. A case was reported in [267] with c = 0.77, which is twice the value of c for the case in Figure 9.5. The experimental snapshots for this case are shown in Figure 9.6(a). It is seen that the boundary layer over a semi-infinite flat plate (a)

Experimental visualization (plan view) c = 0.77

(b)

Computational Results (side view) c = 0.77, G = 0.5

t = 15

ωη: –0.0915812

0.09

xcv = 10.5

04

6

8

ωζ = –24 10 12 14x 16

20

iso-surface of ωζ = –13, –24

t = 20 Wr

xcv = 14.4 ωζ = –24

U∞

18

C

04

6

8

10

12

14 16 x

18

20

18

20

t = 26 Wr

xcv = 19.02 ωζ = –24

U∞

C

04

6

8

10

12

14 16 x

Figure 9.6 (a) Snapshots of the receptivity experiment for c = 0.77, with parameters given in the text. For this counter-clockwise rotating vortex case, weak receptivity is noted for the boundary layer forming over a flat plate with sharp leading edge. (b) Computed results for c = 0.77 and Γ = 0.5 are shown for ωζ contours at the noted times, colored with ωη . shows no receptivity in the experiment. In Figure 9.6(b), corresponding numerical results are shown at noted times, obtained for the parameters: x0 = −1.0; c = 0.77 and Γ = 0.5. The edgewise view shows the computed results at t = 15, 20 and 26 for two values of ωζ = −13 and −24, illustrating very mild perturbations induced in this case. The higher contour value spanwise vorticity remains confined close to the wall, while

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the lower contour value of -13 is deflected away from the shear layer, as is noted in all the three frames of Figure 9.6(b). Such splitting of dye streaks have been also noted in the experimental results in [267]. It is clearly observed that the flow remains twodimensional for this case up to t = 26, beyond which the vortex convects out of the computational domain.

9.4.3 Weak induced perturbation by clockwise free stream vortex In Figure 9.7, results for the clockwise rotating vortex case are shown for the translating speed of c = 0.19. This case does not exhibit violent instability, as noted for (a)

Experimental visualization (plan view) c = 0.19 Wr

(b)

Computational Results (side view) c = 0.19, G = –0.5 t = 60 xcv = 10.4 A

U∞

6

C

x

18

16

14

iso-surface of ωζ = –20, –25, –30

t = 80

Wr

12

10

8

xcv = 14.2 A

U∞

C

8

10

12

14

x

18

16

20

t = 104 xcv = 18.76

Wr A U∞

C

8

10

12

14

x

16

18

20

Figure 9.7 (a) Snapshots of the receptivity experiment for the clockwise rotating vortex case for c = 0.19. For this case weaker receptivity is observed, with induced disturbance visible upstream of the free stream vortex. (b) Computed results are shown for this case with spanwise vorticity, ωζ taking three values, at the noted times with indicated contour levels [474]. the counter-clockwise rotating vortex cases with similar value of c in [267, 475]. The perturbed shear layer for this case appears as a bulge, upstream of the translating vortex. For the counter-clockwise vortex, onset of instability is observed with the

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appearance of a separation bubble on the wall. For the clockwise rotating vortex case, no separation bubble is seen on the wall for the less intense adverse pressure gradient created upstream of the vortex. Instead, one notices the following in the experimental results shown in Figure 9.7(a): (i) the disturbances trail behind the translating vortex and (ii) the growth of the disturbance is insufficient to cause transition to turbulence. The reason for these is due to the clockwise circulation creating weaker adverse pressure gradients upstream of the vortex. In the downstream, induced pressure gradient is favorable. Also, there is negligible spanwise mixing. These are graphically shown from the three-dimensional simulation of the flow field in Figure 9.7(b) for three values of ωζ = −20, −25 and −30. In the three frames, it is clearly observed that the vorticity field is demarcated into two zones about the marked point A, with the upstream part showing a bulge, and downstream part showing instead a compressed vortical layer, in common with the experimental frames.

9.5 Vorticity Dynamics during Vortex-Induced Instability Three-dimensional computed results are shown in [474] for various cases with different parameters of free stream vortical excitation, with corresponding experimental results given in [267]. The response field becomes three-dimensional as time progresses due to increased spanwise variation during vortex-induced instability. This is evident from the vortex stretching contribution. It should be noted that the initial seed for three-dimensionality arises due to numerical errors as the computational framework is three-dimensional, but the excitation is two-dimensional. This aspect of numerical error is shown by single and double precision computations for the case of c = 0.3 for the case of counter-clockwise rotating and translating vortex.

9.5.1 Evolution of three-dimensional response during vortex-induced instability In Figure 9.8, the absolute maximum value of vorticity components obtained by solving the Navier–Stokes equation for the cases of c = 0.3, 0.386 and 0.77 with Γ = 2, are compared. Similar to periodic free stream vortical excitations, one notes earlier receptivity for c = 0.386, as compared to c = 0.3 case. This is heralded by an early onset of three-dimensional nature for c = 0.386, as compared to c = 0.3. It is natural that the application of two-dimensional excitation will cause two-dimensional response field at earlier times, with ωξ and ωη seen to have values which are eleven orders of magnitude smaller than ωζ . This is noted till t ≈ 18 for the case of c = 0.386. It is to be noted that even when ωξ and ωη grow rapidly during primary instability (after t = 18 for c = 0.386 and t = 28 for c = 0.3), the maximum ωζ retains the same value for both the cases. Thus, the evolution from two- to three-dimensional disturbance field is seen with respect to ωξ , ωη becoming non-negligible. Noting the time variation of ωξ , ωη

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in Figure 9.8, one observes the growth to occur in stages following the primary and multiple secondary instabilities. 103 ωξ ,[c = 0.386] ωη ,[c = 0.386] ωζ ,[c = 0.386] ωξ ,[c = 0.3] ωη ,[c = 0.3] ωζ ,[c = 0.3] ωξ ,[c = 0.77] ωη ,[c = 0.77] ωζ ,[c = 0.77]

1

10

Max (ωξ , ωη,ωζ)

10–1 10–3

Primary instability phase

10–5 10–7 10–9

10–11

0

5

10

15

20

t

25

30

35

40

45

Figure 9.8 Maximum vorticity components in the flow domain shown as function of time, for the cases of c = 0.3, 0.386 and 0.77, for counter-clockwise vortex with strength given by Γ = 2. For the primary instability of both the cases (c = 0.3 and 0.386), ωξ and ωη grow more than million times in a short time interval during the primary instability. This is followed by a further growth at a lower rate during the phase of threedimensionalization, with ωξ , ωη to further grow by three to four orders of magnitude during this secondary instability. For the higher translational speed of c = 0.77 for the free stream vortex the computation is limited to t = 26 (after which the vortex convects outside the domain for this higher convection speed). This is also the time up to which the flow essentially remains two-dimensional. Three-dimensionalization is shown in perspective plots of ωζd in Figures 9.9 to 9.13, for c = 0.386 and 0.3 cases, by a single iso-contour of the disturbance component of ωζ , i.e. ωζd = −11, at indicated times. The disturbance component is obtained by subtracting the equilibrium value from the instantaneous value computed by DNS. To emphasize the three-dimensionalization, this iso-contour is colored with ωηd . In Figure 9.9, the onset of primary instability for c = 0.386 case is noted at t = 16, when the free stream translating vortex is located at xcv = 5.176. The ωζd = −11 isocontour is colored with ωηd in the range of −18.1339 to +18.1339. We note that although the minimum and maximum values of ωηd vary in individual frames in Figures 9.9 and

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9.10, particular choice of ranges brings out the details of the vortical structures better during bypass transition. In each figure, for ease of comparison among the frames, an identical range is used. The ranges do not reflect the maximum and minimum values of ωη , which have been already indicated in Figure 9.8. We note that the response field is two-dimensional without any spanwise variation, and a bubble-like structure is noted on the wall, between x = 6 and 8 in Figure 9.9. Absence of spanwise variations at t = 24 and 25 highlight the two-dimensional nature of the perturbation field. The iso-contour of ωζd shows the evolution of the flow at early times in this figure. We note that the maximum value of ωζ remains invariant in Figure 9.8 for a long time, while the other two components start growing from negligible to significant values. In Figure 9.10, the frame at t = 28.4 shows the two-dimensional nature of the flow retained up to this point of time. Although mild streamwise streaks are seen across the spanwise direction for the ωζd contours, these are lifted from the boundary layer in the form of recirculation bubbles. In the time frame at t = 34.4, one notices the growth of disturbances by the primary instability in the form of bubbles attached to the wall. Also, a secondary instability is seen with varying strength in the streamwise location from x = 12 to 15. However, the leading wave front that is lifted off the plate displays three-dimensionality, but this structure weakens also. In comparison, the primary wave front becomes larger and stronger. Thus, one notices the evolution of distinct sites over which vortex-induced instability is observed, which are centered at the streamwise location of x = 13.3 and x = 14.6 at t = 34.4. In the bottom frame at t = 37.8 in Figure 9.10, one notices interaction and amalgamation of these fronts into a single stronger and longer perturbed region. The effects of the evolving perturbation field moves upstream of the location of the free stream vortex at x = 13.668. One then observes that during t = 35 to 38, the maximum ωζ is about 30 times larger as compared to the maximum ωξ at t = 35. By t = 37, this ratio reduces to about 12 times in Figure 9.8. The flow is still transitional, as depicted in instantaneous and span-averaged skin friction C f plots shown at mid-span, i.e. at (z = 0)-plane in Figure 9.11, at the indicated times of t = 36.5, 37.5 and 38.5. Instantaneous C f is shown with solid lines, while dotted and dash-dotted lines represent the laminar and turbulent correlations, respectively, for flow over a flat plate at zero incidence. Subsequently, the flow progresses towards homogenization by a vortex stretching mechanism, along with diffusion and dissipation by breaking the front into smaller vortices. In all the frames of Figure 9.11, spanwise-averaged C f shows that there are very minimal spanwise variations of the flow, even at these later stages of secondary instabilities. The other case computed for a reduced translation speed of c = 0.3 for the free stream vortex with Γ = 2, shows similarities with the vortex-induced instabilities in Figures 9.9 and 9.10. In Figures 9.12 and 9.13 the iso-contour of ωζd = −11 is plotted for primary and secondary instabilities for the case. In Figure 9.12, the primary instability is noted, with the instantaneous streamwise location of the free stream translating vortex marked as xcv in the frames. The disturbance field remains two-dimensional without spanwise variation, during the primary instability in all the frames shown in Figure 9.12. At t = 26, one notices an elongated primary vortex attached to the

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Iso-surface of ωζd = –11

c = 0.386, G = 2

t = 16

ωηd: –18.1339

0

18.1339

xcv = 5.176 0

1

5

4

3

2

6

7

x

8

9

10

12

11

t = 24

xcv = 8.264 2

3

4

5

6

7

8

x

9

10

11

12

9

10

11

12

t = 25

xcv = 8.65 2

3

4

5

6

7

8

x

Figure 9.9 Iso-contours of spanwise disturbance vorticity component shown at t = 16, 24 and 25 for the counter-clockwise vortex case, with c = 0.386 and Γ = 2. wall, while another vortex sheet originating from the leading edge remains outside the shear layer. Such separation of the free stream disturbance vortices from the boundary layer is due to the shear sheltering effect described in [194]. While this is always noted in experiments, to show the same effect numerically, one must take a computational domain which includes the leading edge of the plate. This has been emphasized earlier also for wall excitation cases, and for the two-dimensional vortex-induced instability case in [431, 475]. The ωζd iso-contour is colored with ωηd in the range of −18.1339 to

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Receptivity to Free Stream Excitation Iso-surface of ωζd = –11

c = 0.386, G = 2

t = 28.4

ωηd: –60

–4.4454

9.24313

xcv = 9.9624 8

12

11

10

9

13

14 x

15

18

17

16

t = 34.4

xcv = 12.2784 15

14

13

12

11

10

x

t = 37.8

xcv = 13.5908

10

11

12

13

14

16

15

17

18

x

Figure 9.10 Iso-contours of spanwise disturbance vorticity component shown at t = 28.4, 34.4 and 37.8 for the counter-clockwise vortex case, with c = 0.386 and Γ = 2. +16.0874 in the displayed frames of Figures 9.12 and 9.13. As noted before, this choice of range brings out the details of the vortical structures during bypass transition. With time, the iso-contour of ωζd develops folds through stretching, along with induced separation bubbles forming on the plate. In the bottom frame, one notices two clean vortical rolls lifted off the shear layer in Figure 9.12.

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Laminar Turbulent Cf (z = 0) Cf (spanwise average)

(a) t = 36.5 0.008

Cf

0.006 0.004 0.002 5

10 x

15

20

5

10 x

15

20

5

10 x

15

20

(b) t = 37.5 0.008

Cf

0.006 0.004 0.002

(c) t = 38.5 0.008

Cf

0.006 0.004 0.002

Figure 9.11 Instantaneous and spanwise-averaged skin friction contours C f shown at the indicated times for the counter-clockwise vortex case with c = 0.386 and Γ = 2. In the top two frames of Figure 9.13 at t = 39, one still notes the quasi twodimensional nature of the flow, even though mild spanwise variation is visible.

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t = 26

Iso-surface of ωζd = –11

xcv = 6.8 10

8

6

4

2

0

x

t = 28

xcv = 7.4

0

8

6

4

2

14

12

10 x

t = 31 ωηd: –18.1339

0

16.0874

xcv = 8.3

5

6

7

8

9

11

10

12

13

14

15

x

Figure 9.12 Iso-contours of spanwise disturbance vorticity component shown at t = 26, 28 and 31 for the counter-clockwise vortex case for c = 0.3 and Γ = 2. This is observed in the form of streamwise streaks on the vortex sheet originating from the leading edge, along with coherent ωζd contours, which are lifted off the boundary layer. Also the leading front of vortical rolls, which are distinctly above

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c = 0.3, G = 2

t = 39

ωηd: –18.1339

0

16.0874

xcv = 10.7 11

10

12

13

12

13

x

14

15

14

15

t = 41.45

xcv = 11.435 11

10

x

t = 43

xcv = 11.9 10

11

12

13

14

x

15

16

17

Figure 9.13 Iso-contours of spanwise disturbance vorticity component shown at t = 39, 41.45 and 43 for the counter-clockwise vortex case for c = 0.3 and Γ = 2. the boundary layer, display spanwise modulation. For the frame at t = 41.45, one can also see the growth of two rolls, which were noted at t = 31 and 39, as the leading three-dimensional disturbance structures. Also, the second trailing three-

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0.01

c= 0.3, C plot c = 0.3,f C plot

a) t = 38

f

Laminar La minar Turbulent Turbulent (z = 0) Cff (z=0) C C ff spanwise spanwiseavg avg

0.008

Cf

0.006 0.004 0.002 0

0.01

5

10

5

10

5

10

x

15

20

15

20

15

20

b) t = 41.7

0.008

Cf

0.006 0.004 0.002 0

0.01

x

c) t = 42.15

0.008

Cf

0.006 0.004 0.002 0

x

Figure 9.14 Instantaneous and spanwise-averaged skin friction contours C f shown at the indicated times for the counter-clockwise vortex case for c = 0.3 and Γ = 2. dimensional packet has its upstream end extending beyond the location of the free stream vortex. Thus, one sees the evolution of two distinct sites of vortex-induced

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instability at this time, one centered at x = 13 and the other at x = 14 from the frame at t = 41.45. Such primary and secondary instabilities are also noted in experimental visualization of Figure 9.5 for c = 0.386. However, one notices distinct turbulent spot-like packets in this case, as compared to the case shown in Figure 9.10, where the vortical packets merge. The effects of an evolving perturbation field can also be seen to move upstream of the location of the free stream vortex at t = 43. Similar to the previous case, this flow too is in a transitional stage, as may be noted from the instantaneous skin friction C f plot at the mid-span, (z = 0)-plane, in Figure 9.14. Plotted spanwise-averaged C f in all the frames, indicate some spanwise variations in the bottom two frames, at the late stages of secondary instabilities of bypass transition, where the incipient turbulent spots are forming at t = 41.7 and 42.15. Another noteworthy aspect emerges by comparing the C f in this figure at the mid-span with the spanwise averaged value. In the bottom two frames, the C f is seen to be more spiky, as compared to the spanwise averaged value, where secondary instability is perceptible. This implies that the two-dimensional case will be much more stronger (indicated by the mid-span station properties) as compared to the three-dimensional case, where the perturbation field has more degrees of freedom to distribute disturbance energy in the spanwise directions. This behavior is specifically noted for the highest peak value of C f observed in these frames.

9.5.2 Evolution of three-dimensionality during vortex-induced instability In Figure 9.8, time variation of maximum values of the three vorticity components have been shown to explain the three-dimensional nature of the unsteady flow caused by vortex-induced instability. It does not pin-point where and how the flow becomes three-dimensional. It is noted from the experimental results in [267] and computational framework of vortex-induced instability in [431, 475], that the equilibrium flow is steady and two-dimensional. This is specially true in [474, 475] wherein the computed results show two-dimensionality, even when this is obtained ~ ω ~ )from the time-dependent three-dimensional Navier–Stokes equation by using (V, formulation with a periodicity condition in the spanwise direction. The imposed excitation applied through the time dependent boundary condition created by the free stream vortex is also two-dimensional, i.e., the conditions do not change in the spanwise direction. Despite this, the flow eventually becomes three-dimensional due to vortex-induced instability. The onset and evolution of three-dimensional nature is investigated first by looking at the computed vorticity field directly, along a specific streamwise plane. This provides a view of the global dynamics enabling a nominally two-dimensional equilibrium flow (computed using the three-dimensional Navier–Stokes equation) under the action of two-dimensional free stream excitation to become threedimensional eventually. In two-dimensional flows, one has only the spanwise component of vorticity (ωζ ). Thus, it is of great interest to systematically study how the other two components of vorticity, ωξ and ωη , develop from negligibly small

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values proportional to machine zero, to values those are comparable to ωζ . The flow is considered spanwise periodic, and hence one would expect symmetry or antisymmetry of the computed flow field about the mid-spanwise plane (z = 0).

Figure 9.15 Contours plotted in mid-span (z = 0) plane for ωξ (left) and ωη (right) at t = 25, 34 and 38 for the case of Γ = 2 and c = 0.386. The location of the free stream vortex is indicated by the vertical arrowhead and the point is marked by xcv . Also, indicated are minima and maxima values for each frame for both the components of vorticity.

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In Figure 9.15, the contour plots of ωξ and ωη are shown for the mid-span, (z = 0)plane at the indicated times, for the case of Γ = 2 and c = 0.386. The x-location of the free stream vortex is marked by a vertical arrow in the frames with the maximum and minimum values of the plotted quantities marked in the figure. Also, the contours for (z = ±0.4)-planes are checked for symmetry or antisymmetry of the flow in the spanwise direction. From Figure 9.15, the following features of three-dimensional nature are noted [474]: (i) The primary instability occurs at early times, which is strictly two-dimensional. This leads to unsteady separations at the wall which gathers into two rolls, as shown in Figures 9.9 and 9.10. These two rolls at later times in the advanced stages appear as the STWF, which have been reported in the previous chapters for the wall excitation cases; (ii) There is asymmetry of the flow in the spanwise direction, only in the secondary stage of instabilities shown in Figures 9.9 and 9.10. With the use of periodic boundary conditions in the ζ-direction, such asymmetry arises due to nonlinearity noted during secondary instabilities; (iii) In Figure 9.8, the streamwise component of vorticity (ωξ ) is seen to be more than the wallnormal (ωη ) component; (iv) The locations where three-dimensionality is noted with significant values are highly localized, which lead to the origin of the two primary unstable rolls described before; (v) The unstable regions formed ahead of the free stream vortex for this counter-clockwise case, return to an undisturbed state once the free stream vortex has passed over; (vi) The two sites which lead to formation of STWF are caused by the vortex-induced instability. Despite vortical eruptions, the locations where ωξ and ωη are minimum and maximum are always close to the wall. For the time instants shown in Figure 9.15, the minima and maxima are given in Table 9.1. The reason why a two-dimensional flow becomes three-dimensional is further investigated by computing two cases for c = 0.3 and Γ = 2. In the first computation, double precision arithmetic is used, whereas for the second case, single precision arithmetic is used. Similar to Figure 9.8, the maximum vorticity components in the domain are plotted and compared for these two computations and shown in Figure 9.16. One notes that ωξ and ωη values are orders of magnitude higher (≈ 108 times) for the single precision case, as compared to the double precision case. This shows that the flow becomes three-dimensional earlier for single precision computations. The results are explained by the fact that single precision cases have higher round-off errors to trigger higher seeding and growth. Thus, the numerical noise is an additional source of excitation to flows which are receptive to such sources. This exercise shows that the seed for symmetry breaking about the mid-span for the two-dimensional flow is the numerical noise, which is accentuated in single precision computation. Thus, in actual computations of transitional flows, one must perform double or higher precision arithmetic for computing. In Table 9.1, the streamwise and wall-normal locations are given for the minimum and maximum values of ωξ and ωη , for different times. One notes that the minima and maxima are usually located very close to the plate. Also for t = 25, the maxima and the minima are both downstream of the instantaneous streamwise location of the free stream vortex. These can be seen during primary instability in Figure 9.9. The vortex

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c = 0.3; G = 2 107

Max. Max. Max. Max. Max. Max.

|ωξ|, double precision |ωη|, double precision |ωζ|, double precision |ωξ|, single precision |ωη|, single precision |ωζ|, single precision

4

5

log(|ω|)

102

10–3

10–8

10–13

0

1

2

t

3

Figure 9.16 Maximum vorticity components in the flow domain shown as function of time, for the case of c = 0.3 computed using single and double-precision arithmetic for the counter-clockwise vortex, with strength given by Γ = 2. Table 9.1 The location of maximum and minimum values of ωξ and ωη with time. ωξ

Time

ωη

Minimum

Maximum

Minimum

Maximum

25

x y

10.2527 0.00580

10.2229 8.822 × 10−5

10.26600 0.048688

10.26612 0.01393

30

x y

12.084 2.78 × 10−5

11.996 8.39 × 10−6

12.722 0.0335

12.0614 0.037762

34

x y

13.05615 6.4718 × 10−6

12.87140 1.2142 × 10−5

13.2086 0.03874

12.841366 0.005945

38

x y

13.8355005 0.050906363

13.77099 0.0544398

13.7302 0.04946

13.68795 0.02206

roll-up seen in Figure 9.10 at t = 25 corresponds to the location of minimum and maximum. However at later times (t = 34 and 38) during the secondary instability stage, one notes that the location of minima and maxima relative to instantaneous streamwise vortex location is not as far downstream as those seen at earlier times.

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9.6 Nonlinear Instability Theory Based on Disturbance Energy and Enstrophy We have seen so far that the vortex-induced instability is caused due to a sequence of unsteady separations, and one does not notice the formation of TS waves. This instability cannot be investigated by spatial linear theory using the Orr–Sommerfeld equation. Also, the traditional Orr–Sommerfeld analysis is based on disturbances which decay with wall-normal distance. Even in developing the compound matrix method, we have noted this aspect and stated that for problems which involve free stream excitation, it is not a straightforward matter to describe the disturbance field in terms of wall modes. One of the reasons for solving the Navier–Stokes equation to study free stream convecting vortex in [412], is due to the absence of any tangible tools to analyze bypass transition due to free stream convecting vortex. This has been partly addressed by the authors in [431, 452] by developing and using a tool for disturbance mechanical energy (DME), which originates from the incompressible Navier–Stokes equation, which is described in the following sub-section. One of the disadvantages of the DME equation based method is that the equation is obtained by taking divergence of the Navier–Stokes equation. As a consequence, the unsteady and viscous terms do not contribute directly. The indirect effects come through the vorticity field. This issue is circumvented by developing another equation based on disturbance enstrophy transport equation (DETE) in [465, 475]. As noted for enstrophy transport equation in Chapter 3, one can develop another equation for the DETE budget, which consists of contributions from the vortexstretching, unsteady and viscous diffusion terms. At early stages of vortex-induced instability, the flow is dominated by enstrophy diffusion and vorticity gradient terms that are seen to be relevant downstream of the instantaneous vortex location. As both the methods of DME and DETE are based on the Navier–Stokes equation without using any approximation, these are viewed as tools for nonlinear instability studies, and are described next.

9.6.1 Nonlinear instability study by disturbance mechanical energy The authors in [252] emphasized their belief that to understand turbulence one should look for a mechanism which accounts for growth of total mechanical energy, not merely for the growth of turbulent kinetic energy, and furthermore stated that “it is possible to understand such behavior by studying the redistribution of the total mechanical energy of the flow.” Instead of characterizing turbulence by kinetic energy, one must include the fluctuating pressure also, as it has been explained following the works in [295, 510] that during bypass transition, the unsteadiness associated with unsteady separation is due to fluctuating pressure, which is also termed as shear noise in the Poisson equation for static pressure by Morkovin [298]. This prompted Sengupta et al. [431] to develop an equation for the mechanical energy starting from the incompressible Navier–Stokes equation, for both the equilibrium and the

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disturbance flow fields. The derivation that follows is for mechanical energy of incompressible fluid flows. In the Navier–Stokes equation one uses the vector identity for the convective acceleration term given by → − → − → − → − 1 → − ( V · ∇) V = ∇(| V |2 ) − V × (∇ × V ) 2 Thus the Navier–Stokes equation is rewritten as → − ∂V 1 → ∇p − → − − → − + ∇(| V |2 ) − V × → ω=− + ν∇2 V ∂t 2 ρ

(9.2)

Further, one alters the diffusion term with another vector identity given by → − → − → − ∇2 V = ∇(∇ · V ) − ∇ × (∇ × V ), These two modifications enable one to obtain a rotational form of the Navier–Stokes equation as → − − 2  p |→ V| ∂V → − − − ω) − V ×→ ω = −∇ + − ν(∇ × → ∂t ρ 2

(9.3)

Incorporating the gradient term, one can define the total mechanical energy as E=

→ − p | V |2 + ρ 2

so that the Navier–Stokes equation with E becomes → − ∂V → → − − − ω = −∇E + ν∇2 V − V ×→ ∂t

(9.4)

If one takes the divergence of Eq. (9.4) and assumes incompressibility, one arrives at the governing Poisson equation for E as ~ ×ω ~) ∇2 E = ∇ · (V

(9.5)

Further using the identity ~ ×ω ~ −V ~ · (∇ × ω ~) = ω ~ · (∇ × V) ~) ∇ · (V One finally derives gets the governing equation for the total mechanical energy as ~ · (∇ × ω ~ ·ω ~ −V ~) ∇2 E = ω

(9.6)

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It is to be noted that the first term on the right-hand side is the enstrophy, and the ~ = ∇× ω ~ ). second term is the dot product of the velocity vector with the palinstrophy (Π This definition of palinstrophy follows the notation used in [474], and is different from what is conventionally used for two-dimensional turbulence. To account for conservative body force; it may be written as a gradient term and one can augment the expression of E accordingly. In defining the total mechanical energy, if one retains the body force caused due to gravitational potential, then the redfined E is nothing but the Bernoulli’s head used for inviscid, irrotational flows. The Bernoulii’s equation is often treated as the energy equation, even though it is derived from a momentum conservation equation. The present analysis for deriving the Poisson equation for E is a generalization of the Bernoulli’s equation, derived for viscous flow from the Navier– Stokes equation. This requires the simplifying assumption for the incompressible flow (requiring the velocity field to be divergence-free), and one notes that the contribution to E comes from the convective acceleration and pressure gradient terms. The local acceleration and the viscous diffusion terms do not contribute in deriving the DME equation. The spatial distribution of E is determined by enstrophy and velocity field interacting with palinstrophy. The viscous term creates rotationality, as has been shown in deriving the enstrophy transport equation [468] or vorticity can be created by shock waves via the entropy gradient (as given by Crocco’s theorem relating vorticity with entropy gradient [426, 479]) and such sources will affect the distribution of E by the forcing term on the right-hand side terms in Eq. (9.6). The distribution of E can be obtained by solving the Poisson equation. However, one can qualitatively interpret the solution of Eq. (9.6), without solving it by looking at the sign of the right-hand side of the Poisson equation [494]. This is indicated by the presence of a source or sink of E, depending on whether the right-hand side term has a negative or positive sign, respectively. In [452], the DME equation has been used to show the presence of a wavy solution, which looks identical to TS waves, except that such DME waves appear smoother. This is due to absence of the viscous and unsteady terms contributing to the governing Poisson equation. However, one must realize that the effects of viscous and unsteady terms are present through the dependence of vorticity on these via the transport equation, or the enstrophy transport equation derived in Sub-section 3.7.1. As vorticity is directly related to rotationality of flow, it is preferable to identify it by diagnostic tools, which do not remove high wavenumber fluctuations. One also notes in Eq. (3.100) that there is a confusion regarding the roles of enstrophy, including growth and decay of disturbances [114], where for periodic two-dimensional flows the integrated enstrophy over the domain (||ω||22 ) is shown as a decaying function of time. The implication of Eq. (3.100) developed for homogeneous two-dimensional flow has been extrapolated to inhomogeneous three-dimensional flows, and it has been claimed that the primary role of enstrophy is related to viscous dissipation, as the right-hand side originates from a diffusion term and its negative sign indicates dissipation of total enstrophy [116, 117, 227, 567]. However, it has been shown using the enstrophy

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transport equation that enstrophy can grow for inhomogeneous flows as a point function [468]. Thus, it is now clear that the enstrophy itself can grow, and hence can destabilize E in Eq. (9.6). For nonlinear evolution of E, destabilization of the equilibrium state can be studied, if one expresses E as E = Em + Ed , with subscripts m and d signifying the equilibrium and disturbance components, respectively. If one is interested in growth of small perturbations applied to the equilibrium state, then the parameter  represents a small value, and one can apply the same splitting scheme for all the quantities appearing in Eq. (9.6). The equation for disturbance energy, Ed , is obtained by subtracting the equation for equilibrium state from the equation for the instantaneous state as ~d −V ~ m − V ~d ~m · Π ~d · Π ~d · Π ~d · ω ~d − V ∇2 Ed = Ωd +  ω

(9.7)

~ d , is the linearized disturbance enstrophy. where Ωd = 2~ ωm · ω Existence of waves for Ed caused by spatially localized excitation for a boundary layer has been reported in [452] by taking the equilibrium state as the Blasius profile and solving the linearized version of Eq. (9.7). In this approach, growth of primary disturbances is due to interactions of velocity and vorticity fields acting as source terms on the right-hand side of Eqs. (9.6) and (9.7). Thus, the major issue is about how the energy is initially exchanged from the equilibrium to the disturbance field, and this is clearly brought out by the right-hand side of Eq. (9.7), which shows interaction between equilibrium and disturbance vorticity fields. For the vortex-induced instability problem, some results are presented next using results of two-dimensional simulations in [475]. This is the case for Γ = 2 and the translation speed for the free stream vortex is given by, c = 0.30. In Figure 9.17, for all the frames shown, the same stream function contours are plotted at indicated times to indicate the onset and growth of the disturbance field. For the frame at t = 15, the free stream vortex is upstream of the leading edge and no imposed disturbance is visible inside the boundary layer, and Biot-Savart interaction is not strong enough to create the onset of instability. For t = 25, the free stream vortex is downstream of the leading edge at x = 2.5, and anti-clockwise circulation scours the boundary layer ahead for x ≥ 5. Upstream of the vortex, due to the favorable pressure gradient, the boundary layer shows a thinning tendency in the stream line contours. While downstream of the vortex, the induced adverse pressure gradient creates unsteady separation bubbles. For farther downstream locations of the vortex, the boundary layer ahead of it experiences sustained adverse pressure gradient, and in Figure 9.17(c) at t = 30, unsteady separation bubbles are noted over an extended location starting from x = 7 onwards. The flow over the flat plate experiences additional adverse pressure gradient due to the formation of new bubbles. This is readily seen in frames (d) and (e) at t = 35 and 50, respectively, with increasing number of unsteady separation bubbles convecting downstream. Such unsteady separation bubbles during bypass transition have been conjectured to be a result of constant buffeting of the boundary layer by

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(a) t = 15

y

0.2 0.1 00

10

5

10

5

x

15

20

x

15

20

10

x

15

20

5

10

x

15

20

5

10

x

15

20

(b) t = 25

y

0.2

5

0.1 00

(c) t = 30

y

0.2 0.1 00

(d) t = 35

y

0.2 0.1 00

y

0.2

(e) t = 50

0.1 00

Figure 9.17 Stream function contours plotted in the computational domain at the indicated times. Arrowheads at the top show the streamwise location of the convecting free stream vortex. FST vortices [295]. Here without any model, the vortex-induced instability is shown from the solution of the Navier–Stokes equation. In Obabko and Cassel [318] a model is proposed, where the primary vortex is a Batchelor vortex which is made to move at a constant speed along the wall to study unsteady flow evolution.

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The unsteady separations and vortical structures on the wall during the vortexinduced instability are shown in Figure 9.18, at the same time instants for which the stream function contours are shown in Figure 9.17. Here, the strong receptivity of the zero pressure gradient boundary layer to free stream inviscid vortical excitation

max = 172.94 min = –4006.03

(a) t = 15

y

0.2

0.1

0

–1.12

–50

0

5

10

x

(b) t = 25

15

20

max = 544.47 min = –2928.20

y

0.2

0.1

0

0

5

10

x

(c) t = 30

20

max = 524.45 min = –2833.15

y

0.2

15

0.1

0

0

5

10

x

(d) t = 35

20

max = 537.08 min = –2790.65

y

0.2

15

0.1

0

0

5

10

x

15

20

10

x

15

20

(e) t = 50

0.2

y

max = 541.23 min = –2747.19

0.1

0

0

5

Figure 9.18 Vorticity contours plotted in the computational domain at the indicated times, as in Figure 9.17. Same contour values are plotted in all the frames. Arrowheads at the top show the streamwise location of the free stream convecting vortex.

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is noted, as reported experimentally in [267, 431]. The onset and growth of the primary bubble and secondary separation bubbles are consequences of vortexinduced instability, as noted in this figure. At the latter stages in frames (c) to (e), one notices that some of the secondary bubbles are ejected out of the unperturbed boundary layer. The mechanics of such motion is next discussed with the help of DME. The role of various parameters responsible for vortex-induced instability is characterized by the redistribution of Em to Ed in the flow, as given by Eq. (9.7), which shows the distribution at any time instant to depend on the enstrophy and interaction between the velocity field and the palinstrophy. One notices that the evolution of Ed given in Eq. (9.7), for which the lead term on the right-hand side is given by Ωd , and this can be of either sign. Thus, Ed can display growth or decay of DME, depending on sign of Ωd . Growth is indicated when the equilibrium and disturbance vorticities are of opposite sign, transferring energy from equilibrium to disturbance flow. Equation (9.7) is used to explain the complete nonlinear evolution of DME. In Figure 9.19, distributed sources of Ed are plotted as negative contours, and at t = 15 one observes two sites of instability, one from the leading edge of the plate and the other on the plate itself, originating downstream of the leading edge. The disturbances originating at the leading edge site remain outside the boundary layer due to the shear-sheltering effect [194]. Despite its extent in the streamwise direction, it cannot directly affect the instability inside the shear layer, which originates around x = 2.4, as also seen in Figure 9.18. Disturbance energy structures from these two sites remain distinct till t = 25, and at this time the instability structure inside the shear layer is seen to erupt outside, merging with the outer structure. In the figure, these two sources of Ed interact and are visible from the bottom three frames in Figure 9.19. The spikes forming at the downstream site of the wall interact with the vortical structures outside, which originate from the leading edge, while remaining outside the shear layer. Therefore it is important to include the leading edge in computations of the Navier– Stokes equation for this flow field. Otherwise the computed unimpeded spikes, as in [318, 492], will be unphysical. Also the presence of the outer structure moderates the instability originating from the plate inside the boundary layer. This type of analysis based on DME reveals the physical mechanism of the instability, which can provide the nonlinear mechanism of disturbance growth. In DME, the unsteady and viscous terms do not participate. These can be explicitly included by studying the growth of disturbances with the help of disturbance enstrophy. Thus, to study the growth of disturbances over an equilibrium state given by, Ωm , one needs to investigate the evolution of disturbance enstrophy (Ωd ), where Ω1 = Ωm + ~ d , respectively), 1 Ωd . In terms of equilibrium and disturbance vorticity fields (~ ωm and ω one can define: ~m · ω ~ m and Ωd = 2~ ~d Ωm = ω ωm · ω

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y

max = 896666.55 min = –1790575.42

–10

–5.6

–1.2

3.6

8

0.1 0

5

0

10

x

15

20

max = 354567.47 min = –629854.20

(b) t = 25

y

0.2 0.1 0

5

0

10

x

15

20

max = 172845.97 min = –294857.11

(c) t = 30

y

0.2 0.1 0

5

0

10

x

(d) t = 35

15

20

max = 153847.69 min = –166281.06

y

0.2 0.1 0

5

0

10

x

15

20

10

x

15

20

(e) t = 50 max = 220045.57 min = –150417.24

y

0.2 0.1 0

0

5

Figure 9.19 Contours of the right-hand side of the equation for DME, Eq. (9.7). The negative values indicating disturbance energy sources are plotted as dark contours. ~d · ω ~ d. Additionally, Ωd can have higher order positive contribution coming from  ω Thus, to study the nonlinear instability of fluid flow, there is a specific need to develop an evolution equation for Ωd , in addition to that provided by the equation for DME. Such a study for the evolution of Ωd has been performed in [465, 475].

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A detailed analysis is made to identify criteria for growth of Ωd and quantify the growth rate for an inhomogeneous flow experiencing vortex-induced instability, by developing an evolution equation for Ωd . To develop this equation in [475], attention has been focused mainly on two-dimensional flow. For three-dimensional flows, vortex stretching can additionally contribute to growth of enstrophy [348, 468], and hence a two-dimensional flow provides a more critical limit. However in the following section, the three-dimensional vortex-induced instability problem is examined with the help of the developed disturbance enstrophy transport equation.

9.7 Disturbance Enstrophy Transport Equation: Three-Dimensional Vortex-Induced Instability There are two specific aspects of the works in [267, 474, 475] which are noteworthy. First, the experiment was designed in [267] to keep the disturbance field twodimensional during primary instability, yet the computations are performed for threedimensional flow to note the evolution of three-dimensionality, during transitional and turbulent stages of disturbance growth. In [474], the three-dimensional Navier– ~ ω ~ )-formulation has been solved for different cases with Stokes equation using (V, opposite signs of free stream vortex, translating at different speeds, to explain the physical mechanisms of vortex-induced instability. The equilibrium flow is obtained as a steady state of the three-dimensional Navier–Stokes equation in a domain that includes the leading edge of the flat plate, as shown in Figure 9.2. With spanwise periodicity imposed, the equilibrium flow is a steady, two-dimensional field. The free stream excitation is also two-dimensional, and yet the flow eventually becomes three-dimensional in the experiment [267]. In the computations, the origin for three-dimensionalization is due to numerical errors, which trigger the growth of streamwise and wall-normal components of vorticity in the flow. The growth increases with time due to physical instability, causing the flow to become three-dimensional. With the use of enstrophy transport equation [348, 468] and disturbance enstrophy transport equation (DETE) [475], the roles of vortex-stretching and diffusion terms in leading the flow to three-dimensionality can be compared. Secondly, developed nonlinear methods using DME equation and DETE are applicable for any flow to trace disturbances evolving in incompressible flows. The presence of reliable experimental results in [267] along with computational results using high accuracy methods for ~ ω ~ )-formulation [32] enables one to understand the physical three-dimensional (V, mechanisms governing the vortex-induced instability. From a quantitative description of three-dimensionalization from the computed three-dimensional Navier–Stokes equation, it is possible to show this through study of the enstrophy transport equation [468, 475].

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It is pertinent to inquire why enstrophy is important in fluid flow. This is because although the governing fluid dynamical equations are based on conservation of mass, momentum and energy, yet the conservation of angular momentum is never invoked. This is due to the fact that translation momentum can be easily conserved, as the inertia parameter, i.e. mass remains the same with time for any control volume. However, the inertia property does not remain the same for angular momentum, with the moment of inertia tensor becoming a complex time dependent quantity, which cannot be predicted a priori. This lack of conservation of angular momentum is weakly addressed via the invocation of vorticity, as a measure of rotationality. As vorticity can take either sign, another alternative is enstrophy as the measure of rotationality. Thus the enstrophy, Ω1 , in flows represents a measure of the rotational energy of the flow. It shows the importance of rotationality during transition and turbulence. It is important to understand the rotational energy budget during flow evolution. The enstrophy transport equation developed in [468, 475] helps in doing so, as explained in Chapter 3. The budget is traced to study effects of all the terms of the enstrophy transport equation. The creation of different smaller scales in a general incompressible flow (by forward and inverse cascade) has already been explained. Flow instabilities take an equilibrium flow to another state or to turbulent states by processes which redistribute the energy of the system into various rotational and translational degrees of freedom. Such instabilities are explained in [465, 475], with enstrophy transport equation derived from the non-dimensional vorticity transport equation. The enstrophy transport equation (for Ω1 ) is given for three-dimensional flow as ∂ui ∂Ω1 1 ∂2 Ω1 2 ∂ωi ∂Ω1 = 2ωi ω j + − + uj ∂t ∂x j ∂x j Re ∂x j ∂x j Re ∂x j

!

∂ωi ∂x j

! (9.8)

The first term on the right-hand side reflects vortex-stretching. In a two-dimensional flow this is absent. Viscous diffusion contributions to enstrophy are represented by the second term on the right-hand side in Eq. (9.8). The last term on the right-hand side represents a loss or dissipation of Ω1 . The quantity Ω1 is a point property, and is different from enstrophy defined in [114], which is integrated over the domain for homogeneous and periodic flows. For two-dimensional flows with ω, as vorticity, the enstrophy transport equation is written as [468] " # DΩ1 2 1 2 2 = ∇ Ω1 − (∇ω) Dt Re 2

(9.9)

The first term on the right-hand side of this equation is absent for homogeneous and periodic flows when integrated over the domain. For inhomogeneous flows, this term is non zero and can be of either sign. Thus, the term originating due to diffusion can increase or decrease rotationality depending on the sign of the right-hand side in

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Eq. (9.9). As noted in [475], “positive RHS indicate the diffusion to cause instability. ... This provides a mechanism of creating rotationality at different scales by diffusion and is distinctly different form the concept of creating smaller scales by vortex stretching, as the dominant mechanism of generating small eddies for 3D flows. We note that the role of diffusion in creating new length scales is ubiquitous for both 2D and 3D flows.” It is further emphasized in [468] that the diffusion for inhomogeneous flows is not strictly dissipative, as it is for homogeneous turbulence. In [475], a regular perturbation analysis is done by splitting Ω1 in terms of an equilibrium and a ~m · ω ~ m and Ωd = 2~ disturbance component by, Ω1 = Ωm +  Ωd , so that Ωm = ω ωm · ~d +  ω ~d · ω ~ d . Substituting this in Eq. (9.8), one obtains a transport equation for Ωd ω which has been called the DETE and is given by Term1

∂umi  ∂ui DΩd = 2ωi ω j − 2ωmi ωm j Dt ∂x j ∂x j 

Term2

 1 ∂2 Ω 1 ∂2 Ωm  1 + − Re ∂x j ∂x j Re ∂x j ∂x j Term3



+ −

2 ∂ωmi ∂ωmi  2 ∂ωi ∂ωi + Re ∂x j ∂x j Re ∂x j ∂x j

(9.10)

Marked terms in this equation are contributed by stretching, diffusion and vorticity gradients, respectively. In Eq. (9.8), one could strictly associate the second and last terms on right-hand side as diffusive and dissipative, but one cannot strictly call Term2 of Eq. (9.10) as dissipative, as this can be of either sign. The DETE is derived from the Navier–Stokes equation without approximations. This is unlike the stability theories developed with assumptions of linearity, parallel mean flow and so on. DETE traces instabilities and creation of rotationality for the disturbance field. Moreover, this can be used for turbulent flows to identify patterns and coherent structures [466], a topic to be taken up in Chapter 13. Unlike Ωm , which is a strictly positive definite quantity, Ωd can be of either sign. For laminar unperturbed or perturbed periodic homogeneous flows, Ω1 or Ωm is usually associated with dissipation [335, 567]. In that respect, Eq. (9.10) is special, as Ωd can be of either sign, and for both the signs, the associated quantity can grow in magnitude. For example, if ωm and ωd are of same sign, then d Ωd is strictly positive and will grow with time, if DΩ Dt > 0. On the other hand, if ωm and ωd are of opposite signs, then Ωd is negative and the magnitude of Ωd will grow d with time, if DΩ Dt < 0. Both these conditions of disturbance enstrophy growth given by DETE is written as a single inequality as DΩd ≶ 0 for Ωd ≶ 0 Dt

(9.11)

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Although the DETE is developed to track nonlinear instability, this can also be used to detect flow structures exhibiting three-dimensionality, and is used as such in [474] during vortex-induced instability. Before one uses DETE in trying to explain evolution of three-dimensionality, one should first use the computed flow field to identify stages of development for the vorticity field. In Figure 9.20, the DETE budget is shown at t = 25, as given by Eq. (9.10), for contributions to DΩd /Dt. The top frame shows contribution from the vortex-stretching term (Term1) for the marked contour value; the second frame from the top shows contribution of the diffusion term (Term2), again for the marked contour value; the third frame from the top shows the contribution from the vorticity gradient term (Term3) and the bottom frame shows the growth of Ωd . The dark color indicates negative Ωd value in each frame. The light (gray) contours similarly indicate positive value. Evidently, the bottom frame indicates that the flow is unstable as a twodimensional quantity, without any spanwise variation. The middle two frames (Term2 and Term3 of Eq. (9.10)) taken together indicate the instability to be contributed by viscous diffusion, as the Term1 (due to stretching) is absent for two-dimensional flow. One notes that the viscous diffusion term in the Navier–Stokes equation contributed by Term2 and the term due to vorticity gradient (Term3), can be of either sign, indicating that this can either be dissipative or it can contribute to instability. Thus, the primary instability (indicated in this figure) is due to the two-dimensional mechanism for vortex-induced instability. Figure 9.21 shows the DETE budget at t = 30, with growing instability indicated by increasing magnitude of Ωd . Due to spanwise periodicity of the flow in the domain, disturbance growth due to vortex stretching originates from the domain boundary in the spanwise direction. However, Term2 and Term3 originating from enstrophy diffusion and vorticity gradient show two-dimensional structures. This two-dimensional mechanism predominantly lifts the disturbances above the plate surface, while the disturbance growth due to stretching at this time is seen to be located primarily near the surface, with very little contribution for the chosen level of 95. In contrast, the vorticity gradient is spanwise invariant, but present everywhere with the chosen lower contour value. These two complementary natures of growth of Ωd are shown in the bottom frame near x > 11. Downstream of this location, one can also see the advancing two-dimensional rolls due to primary instability at this time. As compared to the events at t = 25, the downstream primary instability is weaker, although the stretching contribution is growing with time. The evolving three-dimensionality in the flow is prominent in Figure 9.22, from the DETE budget shown at t = 34. In comparison to events noted at t = 30, the vortex stretching term here, is perceptibly dominant and it is to be noted as clustered around x = 13 and 14 along the streamwise direction. These disturbance structures are at a lifted location with respect to the flat plate, while the leading edge of the advancing front is noted beyond x > 15. This part of the disturbance front remains attached to the wall, as shown in the top frame. Due to three-dimensionalization of the flow becoming evident, the corresponding contributions given by Term2 and Term3

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Figure 9.20 Perspective plots for the DETE budget in Eq. (9.10) at t = 25; the top frame shows vortex-stretching term (Term1); the second frame from the top shows contribution from enstrophy diffusion (Term2) and the third-from-top frame shows vorticity gradient contribution (Term3). In the bottom frame, dark regions are of negative Ωd becoming more negative and light (gray) regions are of positive Ωd becoming more positive due to conditions in Eq. (9.11), for free stream vortex with Γ = 2 and c = 0.386 located at xcv .

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Figure 9.21 Perspective plots shown for the DETE budget in Eq. (9.10) at t = 30; the top frame shows vortex-stretching term (Term1); the second frame from the top shows enstrophy diffusion (Term2) and the third-from-top frame shows vorticity gradient contribution (Term3). In the bottom frame, dark regions are of negative Ωd becoming more negative and light (gray) regions are of positive Ωd becoming more positive due to conditions in Eq. (9.11), for excitation by free stream vortex with Γ = 2 and c = 0.386 located at xcv . of Eq. (9.10) contribute to three-dimensionality around these same two locations, as also to be noted for Term1. The spanwise scales at these two sites are finer for Term2 and Term3, while the stretching term shows relatively longer spanwise length scales. Consequently in the bottom frame, the total contribution to the instability patterns are noted to indicate the evolving three-dimensionality in the instability pattern present in the trailing part of the disturbance field, while the leading front of the instability pattern indicates the dominance of the two-dimensional mechanism.

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Figure 9.22 Perspective plots shown for the three terms in DETE budget, Eq. (9.10) at t = 34, with the top frame showing vortex-stretching contribution (Term1); the second frame from the top shows enstrophy diffusion contribution (Term2) and the third frame from the top shows vorticity gradient contribution (Term3). In the bottom frame, the dark regions are where negative Ωd becomes more negative and light (gray) regions are where positive Ωd becomes more positive due to condition given in Eq. (9.11), for free stream vortex with Γ = 2 and c = 0.386 located at xcv . The DETE budget at t = 38 is shown in Figure 9.23, for which one notices even stronger contributions coming from vortex-stretching term, as compared to that was noted at t = 34. Also, the advancing front is no more quasi-two dimensional, due to total three-dimensionalization of the flow field. The leading front also appears to be lifted off the plate near the advancing front, while the trailing part shows streaky structures. One notes the trailing edge location of this contribution is almost at the same place, despite the fact that the free stream vortex has moved downstream. This has the distinct streaky structure, that was called the Klebanoff mode in Chapter 5. The

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Figure 9.23 Perspective of the three contributing terms are shown for the DETE budget given by Eq. (9.10) at t = 38, with top frame is for the vortex-stretching (Term1); the second frame from the top shows enstrophy diffusion (Term2) and the third frame from the top shows contribution from vorticity gradient (Term3). In the bottom frame, the dark regions are where negative Ωd becomes more negative and light (gray) regions are where positive Ωd becomes more positive by instability given in Eq. (9.11), for free stream vortex with Γ = 2 and c = 0.386 located at xcv . excitation is from the free stream, and therefore the disturbance field has the tendency to show upstream propagation, with respect to the source of perturbation. Once threedimensionality is evident beyond t ≥ 34, the Term2 and Term3 mirror similar behavior and so seen in Figure 9.23, as compared to Figure 9.22. The proliferation of threedimensionalization at the later stages of the instability is also noted in the growth rate of Ωd shown in the bottom frame. This indicates how vortex stretching plays a major role in augmenting enstrophy diffusion and vorticity gradient terms in furthering three-dimensionality and the instability. The spanwise length scales associated with

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the stretching term are distinctly different near the leading front, as compared to finer scales at upstream locations, associated with the streak formation. A complete portrait is presented in the bottom frame showing the growth rate of Ωd .

9.8 Dependence of Instability on Speed and Strength of Free Stream Vortex So far, the emphasis on study of vortex-induced instability has been focused on receptivity aspects of the speed and sign of rotation of the translating vortex, as reported in experiments [267, 431] and computations [431, 474, 475]. The flow dynamics can strongly depend upon other parameters, such as the height and strength of the vortex, and this is reported in [474]. One can notice the effects of Γ directly from Eq. (9.1) which helps one understand how this instability depends on the imposed fluctuating streamwise pressure gradient. Applying the steady boundary layer theory [551], leads to a major simplification of the governing equation, brought about by the imposed pressure gradient being transmitted on the plate in an unaltered fashion from the inviscid part of the flow. This is used for both laminar and turbulent flows, and while it is so for laminar flow, such a condition is less valid for turbulent flow, where the wall-normal pressure gradient is seen to depend on the variation of Reynolds stress as well. So if the imposed pressure gradient is qualitatively the same, then according to [295, 510] the flow should show a similar response for the initial laminar flow. For the case of the convection speed c = 0.77, one observes milder receptivity without any instability, as in Figure 9.6 for Γ = 0.5, and as was also shown in the experimental results in [267]. The counter-clockwise free stream vortex case shows vortex-induced instability for the translation speed of c = 0.386, while the receptivity has been distinctly milder when the convection speed was raised to c = 0.77 [267]. For another experimental case for counter-clockwise vortex shown for c = 0.154 in [431], corresponding computational results for the two-dimensional route in [475] indicates a primary instability, which can act as a trigger for bypass transition. In [431], the strength of the free stream vortex is used as Γ = 9.1. It is to be noted that experimentally measuring circulation caused by a rotating and translating cylinder is not possible, and an indirect method has been suggested for experimenting in [526]. As required for instability studies, in [475] computed results are for c = 0.3, with the vortex strength reduced to Γ = 2 for the two-dimensional case. While the experimental case of clockwise free stream vortex with c = 0.19 has been validated in [474], in all the cases the strength of the free stream vortex is taken as Γ = 2, except for c = 0.77 in [474], for a definite reason. This makes it possible for detailed explanation about the roles played by the strength and translation speeds of the free stream vortex. It has been observed in [267, 431, 475] that the free stream vortex with positive Γ tries to scour the boundary layer ahead of it. However, the investigation in [474] for c = 0.77 and Γ = 2, does not exhibit flow physics following such a general rule. This is due

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c = 0.77, G = 0.5

(a) t = 24 1

xcv = 17.48

0.8 x = 17.188 x = 16.9717 x = 15.6738

y

0.6 0.4 0.2 0

0.5

1

u

min = –2727.60 max = 193.47 7 963 3. 09 –2 2 .

36

429 9 – 2 1 .8 75

– 22.9096

9 –2 2 .4 2

9

–23.0 541

–2 5.73 52

–2 7 .4 1 5 8

–2 8 .6 0 2 5

y

–22.9096 –22.4299 –22.4299

–2

12

–2 4.99 06

–2 7 .0 1

–3 0 .0 7 4 7 –31 .1 6 46

y (x 10–3)

88

–2 3 .3 8 8

– 2 3 .3

-24.4 327

314

1

0.008

0.004

–23.6115

-2 9 .4 2

xcv = 18.25

– 3 0 .0 7 4 7

– 31 .4 45 8

0.0005

t = 25

– 2 8.1 71 2 – 2 9 .1 6 0 5

xcv = 17.48

t = 24

-2 7 .1

–29.027 –30.0747

–2 3 .

–28.0857

–25.0441

–2 3 .4 0 2 42 3 .8 7 8 9

xcv = 16.71

–24.1 147 –2 3 .6 8 25 –2 3 .0 12

t = 23

-25.77 26

(c) ωζ-contours

(b) streamtraces

– 23.4 024

0

–21. 50 27 0

0

5

10 x

15

20

0 12

14

16 x

18

20

Figure 9.24 Flow field for Γ = 0.5 and c = 0.77 case is shown in: (a) The velocity profiles for x = 15.6738, 16.9717 and 17.188 at t = 24 with the free stream vortex at xcv = 17.48; (b) Stream-traces at the mid-span (z = 0) at t = 23, 24 and 25, with xcv = 16.71, 17.48 and 18.25, respectively; (c) Spanwise vorticity contours in z = 0 plane.

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to variation of streamwise pressure gradient at different heights for the vortex-induced instability occurring with different combinations of Γ and c. Taylor [510] was the first to conjecture that a distant vortex can create a small longitudinal pressure gradient, and once created would be adequate to destabilize the underlying boundary layer. This has been considered as the dominant effect of free stream turbulence. Monin and Yaglom [295] in supporting this, conjectured that the change in critical Reynolds number due to free stream turbulence can be caused by small longitudinal adverse pressure gradients leading to unsteady separation, which is the precursor of bypass transition. In Figure 9.6, the computed results with Γ = 0.5 for c = 0.77 have been validated with experimental results in [267], showing the experimental visualized results to match satisfactorily. In Figure 9.24, various flow quantities for this case are validated again, with velocity profiles shown for the indicated streamwise stations (x = 15.6738, 16.9717 and 17.188), at t = 24 in the top frame [474]. The bottom three frames on the left in Figure 9.24, show the stream-traces at the indicated times of t = 23, 24 and 25 in the mid-span plane (z = 0) along with a vertical arrowhead indicating the streamwise location of the vortex at that instant. The presence of the free stream vortex causes the streamlines to show a depression, as a consequence of which the flow behind the vortex accelerates, while ahead of the vortex an adverse pressure gradient is indicated by the streamline divergence. The corresponding spanwise component of vorticity (ωζ ) is shown for the same times, in the three frames on the right-hand side marked as (c). As the imposed excitation is for Γ > 0, the instability is noted ahead of the translating vortex location. This is consistent with the idea [267, 431] that the action of the translating counter-clockwise vortex is to impose an adverse pressure gradient ahead of it in the free stream. This is also consistent with the imposed disturbance field given by Eq. (9.1). The same case of c = 0.77 is again taken up, but this time with the strength of the vortex increased to Γ = 2. Now the vortex-induced instability is qualitatively different for this higher Γ case, for the same imposed pressure gradient outside the boundary layer. If the steady boundary layer theory [551] holds, then this flow should show similar behavior, for qualitatively same imposed pressure gradient in the streamwise direction. The computed results are shown in Figure 9.25 for Γ = 2 with c = 0.77 case. Evidently, the figure for this higher Γ case shows earlier instability at t = 22. Now the instability is noted ahead of the free stream vortex, demonstrating the instability to be not only stronger due to increased Γ, but to be qualitatively different too. To decipher the event, streamwise velocity profiles are shown for the streamwise locations at t = 24 in the top left frame of Figure 9.25. As compared to the monotonically growing velocity profile shown in Figure 9.24 for lower Γ case, here the velocity profile for x = 16.9717 shows an overshoot. The other two displayed streamwise instantaneous velocity profiles are monotonic, which are ahead and behind the velocity profile with overshoot. The large overshoot clearly is an indicator of the presence of an inflection point inside the boundary layer. This can be viewed as the conjectured S-shaped profile in [295], and is the direct consequence of the free stream turbulence. According

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Receptivity to Free Stream Excitation c = 0.77, G = 2

(a) t = 24 0.8

t = 23 t = 24 t = 25

(b) x = 16.9717 0.6

xcv = 17.48 0.6

0.02

0.4

x = 17.188 x = 16.9717 x = 15.6738

y

y

0.01

y

0.4

0

0.2

0.2

–0.01 – 0.1

0

u

0.2

0.1

0 0

0.5

u

0

1 t = 21

(c) streamtraces

0

0.5

(d) ωζ-contours xcv = 15.17

xcv = 15.17

u

1 min = –2738.14 max = 194.01

t = 24 min = –2734.32 max = 193.82

xcv = 17.48

xcv = 17.48

x

t = 25 min = –2733.37 max = 193.77

xcv = 18.25

xcv = 18.25

y

0.5

0 0

x

10

20

Figure 9.25 The flow field for Γ = 2 and c = 0.77 case is shown with: (a) The velocity profiles for x = 15.6738, 16.9717 and 17.188 at t = 24 with the free stream vortex at xcv = 17.48; (b) Velocity profile shown at t = 23, 24, 25 for x = 16.9717; (c) Stream-traces at the mid-span plane (z = 0) at t = 21, 24 and 25, with xcv = 15.17, 17.48 and 19.02, respectively; (d) Spanwise vorticity contours in the (z = 0)-plane.

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to Rayleigh’s theorem [119, 412], such a velocity profile is also a necessary condition for temporal instability locally. Such an instability gives rise to unsteady separation, that would lead to secondary bubble formations. This is evident from the ωζ contours at t = 21, 24 and 25 in this figure, which clearly show strong unsteady separations on the wall. Similar strong unsteady separations were also noted for two-dimensional simulations in [431], which was termed as bypass transition, due to absence of TS waves. However, we know that based on the works reported in [509] that this same equilibrium flow can experience transition due to the nonmodal element of STWF. It is also important to distinguish between instability types noted in Figures 9.24 and 9.25. For the lower strength case of Γ = 0.5, the receptivity of the boundary layer is essentially via a two-dimensional mechanism in which the disturbance hardly grows and does not cause transition. It may be viewed as mild spatio-temporal growth. However, for the higher strength case of Γ = 2, one notices stronger perturbations which spawn a series of unsteady separations, leading to strong spatio-temporal instability, with stronger unsteady separations for c = 0.77, noted in the stream-traces shown in Figure 9.25. The nature of the temporal instability can be further understood by looking at the close up of the velocity profile shown in Figure 9.25(b) at t = 23, 24 and 25 for x = 16.9717. At t = 24, one notices a reverse flow at the wall for this station, which was monotonic in the same wall-normal range at t = 23. This is the reason which supports the stated conjecture due to Monin and Yaglom [295] for the presence of S-shaped profile at t = 25. One interesting observation is about the speed of propagation of the free stream vortex and the speed with which the disturbance field inside the shear layer moves. For Γ = 0.5 case, the disturbance field moves slowly as compared to the speed of the free stream vortex (c = 0.77). This is seen in the ωζ contours in Figure 9.24. Such relative position of the free stream vortex with respect to separated flow near the wall, can intensify the pressure gradient, or weaken it. For example, for the weaker free stream vortex case, the free stream vortex not only catches up with the wall perturbations at t = 25, but in the process of overtaking it, attenuates the disturbance field. For the higher strength free stream vortex, the perturbation field shown in Figure 9.25 is noted to be centered around the location of the free stream vortex, with disturbances noted both upstream and downstream of it. As depicted in Figure 9.25(b), the velocity profiles show not only micro-bubbles close to the plate (at t = 24), but one can also notice another inflection point in the velocity profile at t = 25. This is shown for x = 16.9717 for which the velocity profile shows inflection point(s) during the interval, t = 23 to 26. For t = 24, there are two inflection points, which causes micro-bubbles in the ωζ contours shown at t = 25. Rapid spatial fluctuations at heights corresponding to the outer inflection point are responsible for high frequency disturbances in the response field due to temporal instabilities. The absence of three-dimensional nature of the flow for c = 0.77 and Γ = 2 case has been demonstrated in [474]. Here, the same absence is highlighted with the help of the perspective plots in Figure 9.26, which complement the vorticity contours plotted at multiple spanwise stations. In these perspective plots, one can

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t = 22

Iso-surface of ωζd = –20, –30, –40

xcv = 15.94

12

10

8

6

4

x

18

16

14

t = 24

xcv = 17.48

12

10

8

6

4

x

18

16

14

t = 26

ωζd : –8.14E-08 –3.04E-08

4

6

8

2.06E-08

7.16E-08

10

12

xcv = 19.02

x

14

16

18

20

Figure 9.26 The perspective plots of spanwise vorticity ωζ for Γ = 2 and c = 0.77 case at the indicated times, showing the absence of three-dimensionality during the violent vortex-induced instability. see the disturbance field as completely two-dimensional for the excitation parameters, even in the presence of such strong temporal instability due to severe unsteadiness caused by the free stream excitation. The vorticity contours show the presence of small scale separation bubbles and their interactions [328, 329]. These are features of another mechanism of transition studied in this chapter for which the unsteadiness is a consequence of fluctuating pressure gradient. This mechanism has been reported in depth in [474, 475], while the experiment has been reported earlier in [267, 431]. This mechanism was conjectured earlier in [295, 510], and now it is well established that whether it is due to wall excitation or free stream excitation, the actual mechanism of transition is through the creation of nonmodal STWF, whose eventual evolution leads to fully developed turbulence.

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9.9 Closing Remarks In this chapter, a completely different route of transition than that caused by vortical disturbances imposed on the wall is shown. Originally, this mode of transition was conjectured by Taylor [510] and supported in [295]. However, a major observation was made experimentally by Kendall [225, 226], which has been furthermore explained in [429, 431, 474, 475] for the bypass transition caused by convecting free stream vortex. Even though the vortex convects in the inviscid part of the flow, it is the speed of convection of the vortex, its sign and height over the plate, which are the major parameters determining the receptivity. It is shown that a counter-clockwise vortex scours the boundary layer ahead of it to trigger the instability. In comparison, a clockwise vortex mildly perturbs the flow upstream of it. One of the major observations regarding the nature of the vortex-induced instability in this chapter, is that for a two-dimensional mean flow, acted upon by purely two-dimensional excitation shows the disturbance field becoming three-dimensional. The evolution of such a disturbance field without the presence of TS waves has been explained by the developed disturbance mechanical energy theory and disturbance enstrophy transport equation. A special case with vortex having high amplitude excitation and moving at a relatively high speed, shows the presence of an overshoot in the velocity profile. Such a velocity profile with inflection points remains two-dimensional and demonstrates temporal instability, which can be explained by Rayleigh’s theorem.

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Nonlinear Receptivity Theories: Hopf Bifurcations and Proper Orthogonal Decomposition for Instability Studies

10.1 Introduction It has already been explained how linear theories display unconditional stability for flow in a pipe and plane Couette flows. Even for plane Poiseuille flow, the computed critical Reynolds number obtained by linear theory is about almost six times higher than that has been observed experimentally by Davies and White [106]. Also, of interest is the classical experiment of Reynolds [203] for the pipe flow, in which the critical Reynolds number was raised to a value of 12,830 after careful control of the experimental conditions. Monin and Yaglom [295] noted that “in the case of a tube with sharp entrance, pushed through the plane wall of the reservoir, the end of the tube will create considerable disturbances, and Recr will equal approximately 2800. Conversely, if the degree of disturbance at the intake into the tube is decreased strongly by some means or other, we can delay the transition from laminar to turbulent flow until very high Reynolds numbers.” It has been noted that Pfenninger [332] could delay transition in pipe flow for up to Re = 100, 000, by arranging twelve special screens to damp disturbances in the flow approaching the inlet of the tube. Interestingly, Reynolds [203] noted the events in his experiments “at once suggested the idea that the condition might be one of instability for disturbance of a certain magnitude and stable for smaller disturbances.” While the relation between response

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with amplitude of input is an attribute of nonlinear instability, authors in [295] interpret the situation somewhat differently by concluding that “Reynolds number in itself is not a unique criterion for transition to turbulence; for a flow in a tube it is apparently impossible to find a universal critical value of Recr such that for Re ≥ Recr the flow regime is bound to be turbulent. To establish the upper value of Re for laminar flows in tubes it is necessary to have some knowledge of the level of inlet turbulence of the laminar flows considered.” To support this conjecture, a figure showing the variation √of Reynolds number at transition is plotted against turbulence 02 02 +w02 ) in Figure 10.1 for flow over a zero pressure gradient intensity (T u = 100 × u √+v 3U∞ boundary layer. 3

[406]

Ret × 10–6

2

[119] [167] [556]

1

0

1

2

3

Figure 10.1 Effect of turbulence intensity on transitional Reynolds number for flow over a flat plate. Circles are the data from Schubauer and Skramstad [406]; rectangles are the data due to Dryden [120]; triangles are data from Hall and Hislop [168] and inverted triangles data are from Wright and Bailey [556]. This figure is adopted from Dryden [120], with the difference that in this figure Ret refers to Reynolds number at transition, while in [295] this is incorrectly attributed to critical Reynolds number. Such detailed data are given in Figures 3.14 and 3.15 in [412] also. The point that is made with Figure 10.1 is that the transition Reynolds number seems to level off with decrease in T u (plotted along the abscissa) to the value of three million. More detailed results for lower values of T u are given in [405], and are shown in Figure 3.14 of [412]. While this is the case for flow over a flat plate, for pipe flows there are no records of such limiting behavior with reduction in T u apart from the observation in [295]. In the chapters for transition of flow over flat plates, one noted the modal and nonmodal aspects of disturbance generation by vortical and/ or acoustic excitation. In the last chapter (Section 9.2), one noted how free stream excitation can cause instability

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and transition by the sheer presence of a fluctuating pressure gradient, which led to creation of spatio-temporal wave front (STWF) [474, 475], as shown for wall excitation in [509]. Going by the simple classifications in [299], the free stream excitation can be viewed as an example of bypass transition. While both the wall and free stream excitations create STWF, the vortex-induced instability [474] shows predominantly temporal growth. In contrast, the wall excitation cases in [32, 36, 509] show spatiotemporal growth. There are other examples of internal and external vortex-dominated flows which display linearized temporal growth of infinitesimal omnipresent disturbances. When such disturbances grow and a threshold amplitude is crossed, then the nonlinearity can work definitively by taking the system from one equilibrium state to another, without going through the full cycle of transition to turbulence. An appropriate example is that of vortex shedding behind a bluff body, for which flow instability initiates through the temporal growth of disturbances, and the passage to another equilibrium state is by nonlinear action leading to another saturated state. This flow has been treated as a nonlinear dynamical system representing instabilities and vortex shedding for external flows in [341, 469, 499]. Such flows are also termed as being oscillator-type due to the demonstration of fixed time-scale response field characterized by the Strouhal number. Another aspect is from a misplaced emphasis spurred by numerical observations of near-independence of the response from the initial condition for flow past a circular cylinder. This idea was aided by the heuristic development of Landau’s theory on nonlinear aspect of instability proposed in [253, 254, 255]. This idea of independence from initial flow condition can be developed with the help of the dynamical system as proposed in [469], in which the transfer function is central with only a single dominant mode, with the input spectrum playing only a minor role for the new equilibrium state following the instability of the steady equilibrium laminar flow. This has been referred to as hydrodynamic oscillators for post-critical Reynolds number. The critical condition refers to primary temporal instability, a modal feature of the flow instability, and in the literature it is also described as a Hopf bifurcation in the parameter space [160]. The characteristic of bluff body flow is the presence of large scale vortices in its wake. Another interesting example of vortex-dominated flow that assists one to study such oscillator type of flow is seen for the flow inside a lid-driven cavity (LDC). The LDC flow has the attractive feature of a canonical status of incompressible flow due to its simple geometry and unambiguous boundary conditions. The oscillator-type feature of the flow arises due to the fact that the flow is shear driven, and in the constant presence of corner singularities. Here the period of oscillation for LDC will not be determined by the forcing frequency, as this forcing frequency is indeterminate. Quite like the case of flow over the flat plate excited at the wall by a vibrating ribbon, the response will also be given by spatio-temporal dynamics for LDC. Of course, for separated flows with recirculating vortices the Orr–Sommerfeld equation is not going to be the governing linear perturbation equation. In trying to study some of these

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flows for disturbance growth, one needs to first understand the primary temporal instability, followed by inclusion of nonlinear effects on the primary instability. It has been noted that these flows, including flow past a cylinder [469] present intrinsic dynamics, due to the so-called success of a few numerical studies of the flow in [19, 201, 300, 571], which all predicted a critical Reynolds number for the onset of vortex shedding in the range 45 ≤ Recr ≤ 47. These simulations by different formulations and methods are for a uniform flow over a smooth cylinder. However, it is to be highlighted here, supported by further high accuracy numerical simulations, that even though the vortex shedding frequency may remain the same, the critical Reynolds number and loads experienced by the cylinder is a strong function of spatial discretization used. This insistence that the final limit cycle following the secondary instability is independent of initial condition, and existence of a universal feature is due to the proposed equation by Landau [253], which is described next.

10.2 Nonlinear Effects and Stuart–Landau Equation In the previous chapters, it has been pointed out that in recent decades the issue of non-orthogonality of eigenfunctions for some of the flows, and how it affects the nonmodal transient growth, have been discussed at length. One has concluded that the STWF is the nonmodal contributor to transition for boundary layer over a semiinfinite flat plate, even if one finds many eigenvalues and eigenfunctions. For plane Couette and Poiseuille flows, Grohne [162] reported an infinite sequence of a number of strongly damped modes, and few nonmodal transient growth theories have relied upon the non-orthogonality of these modes. Perhaps implicit in this point of view is the limited use of normal mode analysis for such flows, while the interactions among non-orthogonal modes may explain large finite disturbances created during the transient state for three-dimensional flows. In contrast, for the zero pressure gradient boundary layer analyzed by the Orr–Sommerfeld equation it has been shown [418, 451, 452] that the number of modes are far fewer, and these disappear with decrease in Reynolds number, irrespective of the cases for stable or unstable TS modes. In effect, the debate about non-orthogonality of modes is pointless, as any disturbance field undergoing spatio-temporal variations can be expressed by Bubnov– Galerkin type expansion. This formalism is often used for computing in finite difference and finite element frameworks. For flow instability this has been used in [119, 127, 469, 473, 477, 505]. This Galerkin-type expansion is also the basis for proper orthogonal decomposition (POD) pioneered by Karhunen [216], Kosambi [244] and Loeve [275]. Here, the main interest is to study primary and secondary instabilities by POD. This eigenfunction-type expansion is not to be confused with mere separation of variables. If one starts by considering a steady equilibrium flow, as may be the case for flow past a circular cylinder, then the theory of stability of a steady basic flow generally gives a

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spectrum of independent modes with perturbation of the form is given for vorticity field by ~ t) = ω0 (X,

∞ X ~ + A∗j (t) g∗j (X)] ~ [A j (t) g j (X)

(10.1)

j=1

where A j s represent time-dependent amplitude functions of the disturbance field, with the corresponding g j s belonging to a complete set of space-dependent functions ~ t) is the instantaneous vorticity and ωm (X, ~ t) satisfying boundary conditions. If ω(X, 0 is its equilibrium value, then the disturbance vorticity is denoted as, ω = ω − ωm . It will be shown how to obtain these stability modes from the POD modes following the method in [421, 469] using Galerkin representation. In Eq. (10.1), the quantities marked with asterisks are the corresponding complex conjugates, and the Galerkin representation is for the instability modes. First, the origin of the linear instability is explained, followed by the development of Landau equation and incorporating Stuart’s derivation of a formal equation, which has been called here as the Stuart– Landau equation [469]. It is noted that the POD modes are orthogonal, while we have observed that the modes obtained from the Orr–Sommerfeld equation are not always infinite in number and are not necessarily orthogonal. There appears to be a confusion in the literature about this, as already referred with respect to the work of [162] for plane Poiseuille flow, and the zero pressure gradient boundary layer in [418, 451, 452]. In the former, there are infinite modes, while in the latter, there are very few Orr– Sommerfeld modes, and even these disappear as the Reynolds number or excitation frequency is reduced. This discussion is important, as in a subsequent improvement of Stuart–Landau nonlinear theory by Eckhaus [127], an eigenfunction expansion approach has been proposed, with the help of Orr–Sommerfeld eigenfunctions. Evidently, such an approach will not work for zero pressure gradient boundary layers. But the approach proposed here based on POD modes converted to the instability modes will work for any flow. This is the basis for much of the material in this chapter. For linear stability studies if the complex amplitude is given by A j (t) = constant eω0 j t

(10.2)

then during the linear growth stage, the individual modes are governed by the amplitude growth rate given by dA j = ω0 j A j dt If one multiplies this expression by A∗j (t) on both sides, and noting that |A j |2 = A j (t) A∗j (t) is the instantaneous amplitude squared, then one can write the variation of this by d|A j |2 = 2(ω0 j )real |A j |2 dt

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where (ω0 j )real , and (ω0 j )imag are the real and imaginary parts of ω0 j , the jth mode. This equation is often used and modified to incorporate nonlinear effects. In Figure 10.2, the time evolution of disturbance vorticity (ω0 ) for flow past a cylinder (of diameter D) and flow inside a square LDC are shown, when the flow evolves to long term sustained unsteadiness. For the cylinder, the sampling point is placed along the the wake center-line at x/D = 0.504, with the center of the cylinder as the origin of the axis system being considered. It is noted that ω0 , along the wake center-line has this typical behavior with respect to time, starting from a quiescent condition, it then displays exponential growth in time, which is typical of linear instability given by Eq. (10.2). One also notices that following the linear growth stage, the amplitude of time variation saturates to a constant value, indicated as Ae , known as the equilibrium amplitude, while the time variation corresponds to the shedding frequency of vortices in the wake, given by the Strouhal number (S t = f D/U∞ , and f as the characteristic frequency of vortex shedding). This flow past the circular cylinder (a)

2 1.5 1 ω'

0.5 0 –0.5 –1.5 –2

2Ae

0

100

200 (b)

–5

t

300

400

–6 ω'

–7 –8 –9 500

1000

t

1500

2000

Figure 10.2 Time variation of disturbance vorticity shown in (a) flow past a circular cylinder for point on the wake center-line at x/D = 0.504 for Re = 80 (based on diameter and free stream speed), with the nonlinear saturated equilibrium amplitude shown as Ae . (b) The evolution of vorticity inside a lid-driven square cavity at a location which is near the top right corner, with the top-lid driven from left to right for the flow with Re = 9500 (based on constant lid velocity and the side of the cavity).

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starts impulsively, with the free stream speed given by U∞ . In comparison to the flow past the cylinder, the flow inside the LDC (of side L) starts from the quiescent condition and ω0 at the sampling point of x = 0.95L, y = 0.95L, is very close to the top right corner of the LDC. The origin is fixed at bottom left corner of the LDC. The flow evolves, as the lid is given a constant velocity from left to right, causing the entrainment of the fluid with the lid, and the time variation at the sampling point shows very strong transient. However, the high frequency oscillation at the sampling point gives way to a different time variation with bounded limit, which attenuate in the intermediate interval. Eventually, one notices an equivalent exponential time variation once a limit cycle is reached. The calculated results are for Re = 9500, which is defined with L and U as length and velocity scales for the LDC problem. It is interesting to note that the first limit cycle shown in Figure 10.2(b), becomes unstable and evolves into a second limit cycle beyond t = 1500. Such secondary instability has been noted and reported for LDC in [277, 507].

10.2.1 Nonlinear stability: Stuart–Landau equation Beyond the linear instability stage in Figure 10.2, the disturbance field saturates due to nonlinearity and extending Eq. (10.2) one can express the following equation used in [119, 469] as X dA j = ω0 j A j + N j (Ak ) dt k

(10.3)

P where the second term on the right-hand side, denoted as k N j (Ak ), accounts for all nonlinear interactions among the instability modes. This also includes the selfinteraction that was first proposed without accompanying justification in [253]. This has been provided in [505], following a procedure introduced in [547], starting from the Navier–Stokes equation for plane Poisseuille flow. It is important to note that Landau in proposing his theory for amplitude of the disturbance field [253], emphatically stated that the phase cannot be predicted in the nonlinear framework, which remains non-deterministic, i.e. he did not see how the time scale of variation, such as the Strouhal number of vortex shedding can be evaluated from his proposed nonlinear theory. Depending on the form and sign of the nonlinear interaction terms, a nonlinear model with a single dominant mode has been used in [119] to explain the role of nonlinearity in promoting sub-critical instability and causing supercritical stability. To begin with, it is necessary to examine the phase in the nonlinear framework, by considering the specific self-interaction for the nonlinear term in Eq. (10.3). Consider only a single dominant mode, with ω0 = ωr0 + iωi0 representing its linear complex temporal growth exponent. With a single mode (A j = A), Landau and Stuart considered the self-interaction term only in Eq. (10.3), in the form of a cubic nonlinearity as N j = N = − 21 La A|A|2 , with Landau inserting the complex coefficient,

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La = Lr + iLi . This was not known a priori, and is the Landau coefficient. Writing A = |A| eiθ , one can split the complex Eq. (10.3) for the amplitude (|A|) and phase (θ) of the dominant mode as d|A|2 = 2ωr0 |A|2 − Lr |A|4 dt

(10.4)

dθ Li = ωi0 − |A|2 dt 2

(10.5)

The steady state or limit cycle solution can be obtained from Eqs. (10.4) and (10.5) by equating the time to obtain the equilibrium √ derivatives to zero for the former equation Li state as, Ae = 2ωr0 /Lr and from the latter as ωie = ωi0 − 2 |Ae |2 . These estimates are useful, provided one has knowledge of the real and imaginary parts of the Landau coefficient. Needless to note that such an approximation is valid provided the StaurtLandau model is viable. Another aspect is that despite Eq. (10.4) being nonlinear, it admits the exact solution given by |A|2 =

A20 (A0 /Ae )2 + [1 − (A0 /Ae )2 ] e−2ωr0 t

(10.6)

where A0 is initial amplitude, i.e. A0 = A(0). However, this solution in Eq. (10.6) shows the property that as t → ∞, A approaches Ae , completely independent of A0 . This prompted many computational results to also claim that for flow past a circular cylinder, the eventual vortex shedding does not depend on the initial condition, and such a solution is universal. In the absence of any knowledge about the Landau coefficient, such unproven conjecture was the basis of many such results declared in [19, 201, 300, 571]. The insistence of having a universal critical Reynolds number, despite having well-documented and featured experimental results due to Homann [187] in [20, 396] is a mystery. The appearance of very high accuracy solutions in [469] that explained the receptivity aspect of the flow past circular cylinder by matching these with experimental results [187], and there is now the awareness that there is no universality of flow past bluff bodies like a circular cylinder. Furthermore, the universality of instability modes created from the POD modes has been established in [477]. Some of the early discussions are also provided in [412], while the recent developments highlighting the flow inside LDC are emphasized next. Different signs of real and imaginary parts of La , enables one to qualitatively describe nonlinear effects through Eqs. (10.4) and (10.5) as given in [119]. The present interest is focused on flows which exhibit linear temporal instability at onset for the unsteady nature, and subsequent nonlinear saturation, as evidenced experimentally. Such cases can be explained qualitatively by Eq. (10.4), only when Lr > 0 and Re > Recr , which takes the temporally growing flow to a time periodic neutral state, as an event of super-critical stability. For super-critical Re: ωr0 ∼ (Re − Recr ) and thus

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a plot of Ae versus Re should represent a quadratic variation between Ae with Re, √ provided the Stuart–Landau model is valid near Recr , giving Ae = 2ωr0 /Lr . Similarly, one can also obtain the equilibrium Strouhal number beyond nonlinear saturation from Eq. (10.5) as ωie = ωi0 −

Li Li |Ae |2 = ωi0 − ωr0 2 Lr

(10.7)

Thus, Hopf bifurcation is defined as the passage of a dynamical system from a steady state to a periodic state, as a typical bifurcation parameter is varied (such as the Reynolds number [160]). One of the highlights of the works reported in recent times is that there are in effect multiple such bifurcations. This makes flow control plausible, as shown in the experiments of [504] and computed results in [112]. This dependence of external bluff body and vortex-dominated internal flows brings forth the receptivity aspects of the dynamical fluid system, as reported in [469].

10.3 Receptivity of Vortex-Dominated Flows We have noted that such flows become unsteady initially due to linear temporal instability, and finally saturate due to nonlinearity, as shown in Figure 10.2. Thus, there is every possibility for such flows to demonstrate strong receptivity, just the opposite of the universality of unsteadiness, i.e. independence of equilibrium solution on initial amplitude of disturbances as suggested by the Stuart–Landau equation in Eq. (10.6). Apart from the experimental demonstration of flow past circular cylinder in [187] for high Recr ≈ 65, it has been described in [430, 469] that the reported Recr from different experimental facilities are scattered. Even in the same experimental facility, there are reports in [341] (Figure 3) and in [499] (Figure 6) of different Recr values for flow past circular cylinder having different length to diameter ratios. Similar variations are also noted in [314], and all these authors attributed this to the three-dimensional nature of the flow field. However, according to Williamson [555] such flows at low Re ought to be two-dimensional, as many believe that a threshold of three-dimensionality exists only for Re > 180. Three-dimensional effects have been computationally however, noted by Henderson [178] for Re ≥ 250 by spectral element calculations. Such variations of the flow past bluff bodies in the same experimental facility has led to, in [430, 469], designing experiments by taking two different diameter cylinders, and exposing those to different oncoming flow speeds to keep the flow for the same Re = 53. Such flows in the tunnel were observed to be displaying different free stream turbulence (FST) at different operating speeds to result in dramatically different flows. While there is nothing wrong with dimensional analysis, such analysis must include all the relevant physical effects. For example, to perform dimensional analysis, one must include the FST (T u), as one of the factors affecting the flow. Adding

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the effects of FST for the flow past a circular cylinder has been reported in [430, 469] and is also described in [412].

10.4 Multimodal Dynamics of Vortex-Dominated Flows To show that there is more than one dominant mode present for the flow past a circular cylinder, disturbance vorticity for a post-critical Reynolds number of Re = 60 is shown in Figure 10.3(a), as obtained by solving the Navier–Stokes equation using the stream function and vorticity formulation. If the Stuart–Landau model of single dominant mode has to be valid, then the amplitude envelope would have to be smooth. In frame (b) of the same figure, the results from [503] are shown for the flow past a circular cylinder for Re = 49 obtained in a wind tunnel. The fact that the experimental figure also shows a jagged envelope testifies to the fact that there are more than one mode active for such a low post-critical Reynolds number. It is noted that the experimental time is given in seconds, while the computational times are non-dimensional and for the experimental data in [503], one second is equivalent to 378 units of non-dimensional time for Re = 49. Thus, computational events are seen to be faster for the higher Reynolds number of Re = 60, which may be expected, as the temporal growth rate is proportional to (Re − Recr ), even though the critical Reynolds numbers are not the same for the computation and experiment. Having noted the presence of more than one dominant mode for this flow, it is natural that one would like to develop a model, which can account for the presence of multiple modes. This has been achieved in [469, 477] with the help of POD modes, and is introduced next.

10.5 Proper Orthogonal Decomposition and Nonlinear Instability Kosambi [244] was a pioneering researcher on the subject of proper orthogonal decomposition (POD) who laid the foundation for studying such complex stochastic dynamics of any spatio-temporally varying dynamical system, as it is done for finite element method using Bubnov–Galerkin techniques that are part of finite elements methods [5, 360]. For the stochastic field given by vi (~r, t), POD projects the random field onto a set of deterministic vectors (φi (~r)), such that < |vi φi |2 > is maximized, subject to initial and boundary conditions with angular brackets representing time or ensemble averaging of the random field. In recent times, POD has been developed further to explain complex spatio-temporal dynamics of fluid flow representing transitional and turbulent flows in [153, 421]. This problem of maximization of the projection onto a deterministic field turns out to be the problem of variational calculus, where the choice of φi s is given by the following integral equation, Z

Ri, j (r, r0 ) φ j (r0 ) dr0 = λi φi

(10.8)

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Figure 10.3 Time variation of disturbance vorticity shown in flow past a circular cylinder (a) for a point in the wake center-line at x/D = 0.5044 for Re = 60, with multimodal nature of the time series shown in this figure. (b) The experimental data shown in [503] for fluctuating streamwise speed shown for Re = 49. The velocity and diameter of the cylinder is such that the unit dimensional time is equal to 378 times in the non-dimensional units. where λi is the ith eigenvalue of this integral equation for the non-trivial eigenfunction φi , as the ith POD mode. The kernel of the above integral equation is the twopoint correlation function given by Ri j =< vi (~r, t)v j (~r0 , t) > of the spatio-temporal field. For flows with finite energy, classical Hilbert–Schmidt theory is applicable and gives denumerable POD modes, for both transitional and turbulent flows, as energy conservation is the cornerstone of the governing equation for any flow. Specifically,

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for transitional or unstable flows, the disturbance energy is driven by supply of energy from the equilibrium flow. And, it has been shown by DNS [32, 34, 507] that the final turbulent state occurs after the nonlinear saturation, following the primary, secondary and tertiary instabilities. The solution for the turbulent flow was considered not directly evident by the authors of [153, 281], due to uncertainty about the route(s) of transition, e.g. if it occurs due to chaotic dynamics, then the finite energy supplied cannot be determined with surety. However in recent times, the nonlinear dynamical system approach has been applied using DNS to study the chaotic route to turbulence [125, 221, 229]. But, theoretical analysis is still not available that by following this route, a finite source of energy supply can be identified along the chaotic route to turbulence. Such a problem does not arise, when one shows the existence of turbulent flow starting from its receptivity stage following deterministic excitation as shown for flow over a flat plate for two- and three-dimensional routes of transition to turbulence in [32, 34, 36, 419, 508, 509]. One of the strongest attributes of POD modes is that these are orthogonal to each other, simplifying two essential theoretical problems: (i) There is no need to establish a complete basis of such modes, and (ii) these modes are orthogonal like those used in Sine and Cosine transforms, one can establish the simultaneous existence of modal and nonmodal contributions for transition to turbulence, without proposing ad hoc theories of linear transient growth, as has been done in [251, 398]. No restrictions for using POD modes to explain routes of transition to turbulence apply, as the disturbance field satisfies the Hilbert–Schmidt theorem. Early contributions to POD [216, 244, 275] were generically for stochastic dynamical systems. Related concepts in linear algebra for singular value decomposition have been noted earlier by Pearson [326]. POD for fluid flow was formally initiated by Lumley and co-authors [281] in an attempt to conceptualize coherent structures for fully developed turbulent flows [186], using the method of snapshots developed by Sirovich [489, 490]. POD has been used as a reduced order model for laminar unsteady flow, or undergoing transition in [98, 469], with the intention of converting the governing partial differential equations to ordinary differential equations. In the latter, enstrophy based POD has been developed with an intention of using the multimodal nature of POD in trying to enlarge the scope of the Stuart–Landau equation, as has been attempted by Eckhaus [127] by incorporating the eigenfunctions of the Orr–Sommerfeld equation. However, a massively separated flow in the lee-side of the circular cylinder does not allow incorporation of the disturbance growth by the Orr–Sommerfeld equation, and a novel approach was introduced in [469], with the capability of converting the POD modes to instability modes, allowing use of a similar extension for multimodal expansion. Despite the major difference of replacing Orr–Sommerfeld eigenfunction by nonlinear eigenfunctions for the instability modes obtained via the POD modes, the authors consider this approach as valid and call it the Stuart–Landau–Eckhaus equation based method [469]. This upgraded version of the eigenfunction expansion approach as given in [437] is described here.

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Another important aspect of POD analysis is: The method is based on optimizing the projection of the stochastic field onto a deterministic field by the method of variational calculus by working with quantities that are time-averaged or ensembleaveraged. This enables choice of different time horizons to study transient growth as well, from the DNS results. Finally, one notes that initially, POD analysis was used with the primitive variables, with velocity as a dependent variable, so that the cumulative sum of the eigenvalues provided a measure of the kinetic energy. In doing so, in most such POD analyses (with very few exceptions), the pressure gradient term from the Navier–Stokes equation is omitted, and this becomes a major source of error. To avoid such error, the vorticity field is often also used, so that the eigenvalues provide a measure of enstrophy of the investigated flow field [278, 421, 431, 437, 440, 469]. Vorticity being a primary flow quantity of interest, this is a preferable approach and further, for two-dimensional flows, solutions so obtained using stream function and vorticity provides more accurate results. Henceforth, references to such analysis will be restricted to enstrophy-based POD analysis only. Meanwhile, the disturbance vorticity field is defined as 0

ω (~r, t) =

M X

am (t) φm (~r)

(10.9)

m=1

Here, the φm s are theR eigenfunctions of the covariance matrix, whose elements are R 0 0 given by, Ri j = (1/M) ω (~r, ti ) ω (~r, t j ) d2~r, with i, j = 1, 2, ..., M defined over all the collocation points in the domain. This is essentially following the method of snapshots [489]. Therefore in analogous terms, the eigenvalues in Eq. (10.8) indicate the sum providing the enstrophy of the disturbance flow, while the partial sums provide the cumulative enstrophy of the system 0

Ω =

M X

~ tm ) ω02 (X,

m=1

An alternative to the method of snapshots is used in [432], by evaluating eigenvalues and eigenfunctions of the covariance matrix using Lanczos iteration. While this method can be easily parallelized, but the Lanczos method can become very expensive when implemented by a sequential algorithm, even though the accuracy and conceptual validity has been shown to be better in [432]. Sirovich [489] proposed the method of snapshots to circumvent the problems of handling very large data sets for multidimensional problems needed in methods like that is given in [432]. In such a method of snapshot, any element of eigenfunction set, φm (~r) is considered to be a weighted snapshots of the flow field taken at discrete times ti s given as, N qmi ω0 (~r, ti ) φm (~r) = Σi=1

(10.10)

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with the limit of sum on the right-hand side given by N is the number of snapshots taken for the method, and qmi are the corresponding weights. This equation, Eq. (10.10) used in the expression for the covariance matrix element Ri j , helps to rewrite Eq. (10.8) as an algebraic eigenvalue problem [C]{q} = λ{q}

(10.11)

where the discrete elements of covariance matrix is given by 1 Ci j = N

" ω0 (~r, ti ) ω0 (~r, t j ) d2~r

(10.12)

with the indices of covariance matrix element span over, i, j = 1, 2 . . . N to incorporate all the snapshots of the flow which have been stored. The amplitude function am (t)s in Eq. (10.9) are obtained from the implicit defining property of orthogonality for the POD eigenfunction given by ! am (t) =

ω0 (~r, t) φm (~r) d2~r ! φ2m (~r) d2~r

The POD analysis by method of snapshot has been performed in [437], for few Reynolds numbers with the large number of 2000 snapshots taken in the time interval of t = 0 to 400 for flow past a circular cylinder. Having obtained the amplitude function am (t) and the corresponding eigenfunction φm (~r) for Re = 100, these are shown respectively in Figures 10.4 and 10.5. The amplitude functions shown in Figure 10.4, display the first two modes to form an orthogonal pair. This type of amplitude variation are referred to as for the regular or R-modes in [421, 437, 469, 477]. By convention, the POD modes are always numbered in a decreasing sequence with respect to their enstrophy contents. By the same convention, one notes missing numbers (such as 4, 10, 14 and 16 in Figure 10.4) which represent a missing member of the pair, with the present amplitude function in isolation. For example, in this figure, the third POD mode appears in isolation and to indicate this, the mode-4 is left blank. In [315], only the first isolated POD mode was identified and termed as the shift mode. According to the present adopted notation following [421, 437, 469, 477], this will be termed as the T 1 - or anomalous mode of the first kind. In [315], only one shift mode was reported from the RANS solution. Here, in Figure 10.4, one can identify multiple T 1 -modes obtained from the DNS solution. By performing Fourier transform of the amplitude functions in Figure 10.4 one can identify the time scales of the various POD modes. For example, one observes the third mode to be at twice the Strouhal frequency, while the first two POD modes are at the Strouhal frequency. The time variation of the third mode is physically relevant, as this mode arises from the Reynolds stress-like term in RANS-like approach, that originate from the product of two fluctuating quantities, as reported in [469]. If the individual

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POD mode 3

POD mode 1 and 2 0.03

0.04

0.02

0.02

0.01 a___ (t) 1 0 a2(t) –0.01

0 a3(t) –0.02

–0.02

–0.04

–0.03

–0.06

0

50

t

100

150

200

0

50

t

100

POD mode 5 and 6

200

POD mode 7 and 8

0.03

0.03

0.02

0.02

0.01 a___ (t) 5 0 a6(t) –0.01

0.01 a___ (t) 7 0 a8(t) –0.01

–0.02

–0.02 –0.03

–0.03 0

150

50

t

100

150

200

0

POD mode 9

0.1

50

t

100

150

200

POD mode 11 and 12

0.15 0.1

0.05 a9(t)

a___ (t) 11 a12(t)

0

0.05 0

–0.05 –0.1

–0.05 0

50

t

100

150

200

0

50

t

100

150

200

POD mode 15

POD mode 13 0.2 a9(t) 0.2

a9(t) 0.1

0.1 0 0 –0.1 0

50

t

100

150

200

0

50

t

100

150

200

Figure 10.4 The first sixteen amplitude functions for the POD representation of flow past a circular cylinder for Re = 100 showing disturbance vorticity. Regular modes are shown together, while the anomalous modes are followed by a blank POD mode, as explained in the text. [Reproduced from “Enstrophy-based proper orthogonal decomposition for reduced-order modeling of flow past a cylinder”, T. K. Sengupta, S. I. Haider, Parvathi M. K. and Pallavi G., Phys. Rev. E, vol. 91(4), pp 043303 (2015), with the permission of APS Physics.]

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perturbation quantities are seen to vary with the time scale given by the Strouhal frequency, then the product of the velocity perturbations will have time variation at twice the Strouhal frequency. The product would also have give a time-independent component, which for the sake of consistency should be accounted for in the timeaveraged mean field, as it has been shown in another context in [408, 445]. The role of an anomalous mode, such as φ3 , is explained next for the dynamics at a post-critical Reynolds number by looking at the action of disturbance velocity components of a two-dimensional flow given by ud (x, y, t) =

Z Z

vd (x, y, t) =

Z Z

[ˆu ei(αx−βt) + uˆ ∗ e−i(αx−βt) ] dα dβ

(10.13)

[ˆv ei(αx−βt) + vˆ ∗ e−i(αx−βt) ] dα dβ

(10.14)

where x is the homogeneous streamwise direction and quantities with asterisks denote complex conjugates. If one performs Reynolds’ decomposition for physical variables to be a sum of the mean plus a disturbance component, and substitutes these in the Navier–Stokes equation, then one obtains the governing RANS equation indicated by angular brackets for the mean velocity, with gradient transport terms, containing a Reynolds stress-like terms given by < ud (x, y, t)vd (x, y, t) >=


(10.15)

Note that in Eq. (10.15), the first contribution on the right-hand side is phase independent, and thus included in the mean-field equation, irrespective of whether a time or ensemble averaging has been performed to obtain this equation. In [408, 445], this has been called the wave-induced stress, where turbulent flow past a compliant wall was studied. Thus, the unsteadiness causes a steady-streaming in the Navier– Stokes equation for any flow field in the presence of disturbances. It is thus evident that there can be many such contributions from different anomalous modes. However, as the enstrophy contents keep decreasing with the number, the latter T 1 -modes will progressively contribute lower wave-induced stress to the mean field. The wavenumber and the circular frequency are related by the dispersion relation, which is not easy to determine for nonlinear equations. This wave-induced stress term does not arise with R-mode, which consists of orthogonal pairs. There are no restrictions on the number of anomalous modes. In

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Figures 10.4 and 10.5, the third, ninth and fifteenth are the T 1 -modes and the (11, 12)modes are termed as the T 2 -mode, which appears like a wave-packet. The third mode is an anomalous one, and thus by the adopted convention, one will set a4 ≡ 0 and φ4 ≡ 0. The subsequent two modes, mode-5 and mode-6 are a pair of R-modes, with their phase varying at twice the Strouhal frequency. For the governing nonlinear Navier–Stokes equation, the response field expressed by POD modes will demonstrate higher super- and sub-harmonics. Such a situation is also observed with the 7th and 8th modes, which constitute the third pair of R-modes, with its time variation at thrice the Strouhal frequency [469]. This is a demonstration of POD analysis indicating the nonlinear aspects of the flow field. The dynamics of the flow system is multimodal, also showing the presence of limit cycles and multiple bifurcations. In [421, 469, 477], it has been shown that POD can be used to develop a nonlinear global instability analysis, to show nonlinear effects. Once again, the 9th POD mode can be viewed as a T 1 -mode, with time variation at twice the Strouhal frequency, which is similar to the 3rd mode. The shift mode [315] and more generally the T 1 -modes provide a way to split the Navier–Stokes equation into a time-averaged mean field, which will be affected by a time-dependent perturbation field through wave-induced stress, as explained above in Eq. (10.15) for the RANS approach that was adopted in [315]. With available DNS results, one can comprehend the dynamical system aspect of the flow with more clarity than that provided by the RANS type approach. The following two modes display another interesting aspect of POD analysis. These two (11, 12)-modes are paired, but their time variation is different from the Rmodes. These modes have been identified in [421, 469] as the anomalous modes of second kind or T 2 -mode. These behave like a wave-packet, showing initial growth followed by decay, following which the time series settles down onto constant amplitude oscillation at the Strouhal frequency, as can be clearly seen in Figure 10.4. In the discussion here about external flow past a circular cylinder, this has been extended in [421, 477] showing that there is a universality for the classification of amplitude functions of POD modes, applicable for both internal and external vortex-dominated flows. It has also been shown in [421, 477] that internal flows display many T 2 -modes, as compared to external flows. However, given that for flow past a semi-infinite flat plate, the STWF matures into fully developed turbulence and at the onset, it may remain sub-dominant to the modal component of disturbance field. In Figure 10.4, the following two modes, namely the 13th - and 15th -modes are again anomalous modes with asymptotic phase variation at Strouhal frequency. Although the (5,6)-modes form a pair and appear as though these are R-modes, there is clear departure from R-mode characteristics. For example, during the interval t = 0 to 80, the mode-6 oscillates about a non-zero mean in the beginning, but the mean eventually settles to zero later. Despite such minor differences, external flows can be classified into these three classes of POD modes (R-, T 1 - and T 2 -modes) as advocated in [421, 469, 477].

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1

1

0

0

–1

–1

–2

–2

0

2

4 X

6

POD mode2

2

Y

Y

2

8

0

2

4 X

POD mode3 1

0

0

Y

Y

2

1

–1

–1

–2

–2

0

2

4 X

1

1 0

–1

–1

–2

–2

0

2

4 X

6

8

POD mode8

1 0

Y

1

–1

–1

–2

–2

2

4 X

6

8

POD mode11

2

1 0

Y

1

–1

–1

–2

–2

2

4 X

6

8

POD mode13

2

0

Y

1

0 –1

–1

–2

–2

2

4 X

6

8

6

8

POD mode7

0

2

4 X

6

8

POD mode9

0

2

4 X

6

8

POD mode12

0

2

4 X

6

8

POD mode15

2

1

0

4 X

2

0

0

2

2

0

0

0

2

Y

Y

8

0

2

Y

6

POD mode6

2

Y

8

POD mode5

2

Y

6

0

2

4 X

6

8

Figure 10.5 The first sixteen eigenfunctions for the POD representation of flow past a circular cylinder for Re = 100 showing disturbance vorticity. Regular modes are shown together, while the anomalous modes are followed by a blank POD mode, as explained in the text. [Reproduced from “Enstrophy-based proper orthogonal decomposition for reduced-order modeling of flow a past cylinder”, T. K. Sengupta, S. I. Haider, Parvathi M. K. and Pallavi G., Phys. Rev. E, vol. 91(4), pp 043303 (2015), with the permission of APS Physics.]

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While the POD amplitude functions can be classified into distinct groups, corresponding eigenfunctions are somewhat more difficult to identify. For example, for the flow past a circular cylinder for Re = 100 the POD eigenfunctions φm (~r)s are shown in Figure 10.5. This may also help in the verification of the classification scheme explained for the amplitude functions in Figure 10.4. The first pair of R-modes show spatial vortical structures with signs flipped at similar locations in the near-wake, implying the phase shift of quarter cycle. This is of course due to orthogonality between these two modes of the pair. The first T 1 -mode has been noted as isolated without any pairing, based on the amplitude function property. Looking at the eigenfunction φ3 , one observes an anti-symmetry of the spatial structure about the wake center-line. This is also noted as being the same for mode-9, the second T 1 -mode among the POD modes. However, the mode-13 and mode-15 (also being T 1 -modes) do not show such specific patterns as noted for mode-3 and mode-9. In the same way, the second and third R-mode pairs also show different symmetries and antisymmetries in the near wake. As noted earlier, POD eigenfunctions are not amenable for easy classification, such as with the amplitude function. Looking at the literature, one notices that early uses of POD analysis rested more on POD eigenfunction, and apart from the reported studies in [421, 437, 469, 477] there are hardly any worthwhile POD analysis that depends on the amplitude function.

10.5.1 Instability modes and multimodal Stuart–Landau equation One of the early goals of POD analysis with primitive variable formulation was essentially to reduce the partial differential equation (the Navier–Stokes equation) to set of ordinary differential equations for the POD amplitude functions, am (t)’s [186, 283, 315, 363, 489] instead. In [469], the time variation of am (t)s have been derived from DNS data using stream function-vorticity formulation, as given by Eqs. (7.1) and (7.2). The disturbance vorticity in Eq. (10.9) can also be expressed in terms of Galerkin expansion of instability modes, as given in Eq. (10.1). The classification of POD modes helped, as these can be related to instability modes. The instability modes are related to the regular or R-modes by the relationship in the following manner for the amplitude and eigenfunctions for the former [421, 469, 477]: √

 j (a2 j−1 (t) + ia2 j (t))

(10.16)

1 g j (~r) = √ (φ2 j−1 (~r) − iφ2 j (~r)) j

(10.17)

A j (t) =

N where  j = (λ2 j−1 + λ2 j )/Σk=1 (λk ) and this normalization factor has been introduced to make the dimension of A j (t) the same as that of vorticity, as  j is a measure of the point property of enstrophy. The above relations between POD and instability modes

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do not hold for anomalous modes, and have been introduced in [437] to explain a reduced order model (ROM) developed from POD modes. The relation between POD and instability has been further investigated using a refined grid calculation in [437], as compared to the results obtained in [469, 477]. This helps with further improving the relation between POD and instability modes, as the POD modes and amplitude functions are obtained by singular value decomposition of DNS data obtained with the refined grid. This is due to the fact that there exists a connection between numerical errors and vortex shedding behind a cylinder for super-critical Re. The simulation results given in [437] used 1001 points in the azimuthal direction, and 401 point in the wall-normal direction. This is finer than the (153 × 401)-grid used in [469]. The case for the higher Reynolds number of Re = 100 is demonstrated in [437]. It is noted for this flow past a stationary circular cylinder that φ1 and φ2 are phase shifted by quarter of a cycle, i.e. by π/2, so that these two modes are perfectly orthogonal. Thus, these can be expected to compose of the real and imaginary parts of a dominant instability mode, A1 (t). In [437, 477], it is shown that flow past a circular cylinder and flow inside a square lid-driven cavity exhibit similar POD modes, with first two POD modes forming a conjugate pair. Such pair of modes, phase shifted by π/2, are the regular or R-modes. In [315], it is shown that following this pair an isolated mode exists, which alters the mean flow, and is termed the shift mode, obtained from an RANS solution. Thus, its phase variation could not be reported. But, based on the time variations of am s in [421, 437, 469, 477], POD modes are classified into regular and anomalous modes. Shift mode of [315] is called the anomalous mode of the first kind or T 1 -mode in this scheme. Another class of modes occurring in pair, but having different time variations in the form of a wave-packet, have been called the anomalous mode of second kind or T 2 -mode in [421, 437, 469, 477]. Such classification of modes was done for the first time, based on simulation of the time-accurate Navier– Stokes equation. Classifying modes such as these, reported for bluff body flows, also highlights the needless emphasis placed on orthogonality of instability modes for various flows described in the previous chapters for Poiseuille and Couette flows. Why the equilibrium amplitude is not given by the Stuart–Landau equation can be explained with Figure 10.6(a), where Ae of streamwise fluctuating velocity (rms) variation is plotted against Re, with experimental data from [503] and computations in [469]. The time variation is not strictly parabolic in the figure, as one would expect from Stuart–Landau equation, Eq. (10.6). The reported measured disturbance quantity in [503] is the root mean squared value of the azimuthal component of velocity (ud (rms)). Numerical results are obtained by solving Eq. (7.2), with Ae is the equilibrium value of ud in the limit cycle stage. In Figure 10.6(b), time variation of disturbance vorticity (ω0 ) is shown at the point in the wake centerline (at x/D = 0.504), which shows saturation of ω0 with time, indicated as Ae (ω0 ). In Figure 10.6(c), computed values of Ae (ω0 ) at the wake centerline (x/D = 0.504) show its variation with Re. The composite solid line indicates presence of five intersecting parabolas, with each parabola representing stages of growth of Ae with Reynolds numbers, starting from a virtual origin at different Re. Thus each parabola indicates one bifurcation,

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Figure 10.6 (a) Comparison of experimental and DNS results for streamwise disturbance velocity, as reported in [469]. (b) The temporal variation of disturbance vorticity at the wake centerline ( x/D = 0.5044). (c) The composite bifurcation diagram as obtained computationally, showing multiple Hopf bifurcations, as described in the text. [Reproduced from “Enstrophy-based proper orthogonal decomposition for reduced-order modeling of flow a past cylinder”, T. K. Sengupta, S. I. Haider, Parvathi M. K. and Pallavi G., Phys. Rev. E, vol. 91(4), pp 043303 (2015), with the permission of APS Physics.] which was also identified in [469] for a less refined grid calculation for the flow past a circular cylinder. This idea of multiple Hopf bifurcations has been supported by numerical data in [437, 469] and experimental results in [187], with different modes responsible for vortex shedding at different Reynolds numbers. Each bifurcation is determined by various sources of disturbances active in different parts of the flow. Thus, it is possible to suppress the primary instability, as in [187] by the fluid property, and in the vortex suppression experiment in [503, 504] by another control cylinder. In [469], the first bifurcation was noted for Re = 53.2907; the second one at Re = 62.5326 and the third one at Re = 78.2071.

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The observed bifurcations during computation of flow past a circular cylinder are due to disturbances owing to numerical errors. An accurate dispersion relation preserving method will control dispersion error and delay onset of instabilities. In the experiments reported in [187], use of highly viscous oil attenuates disturbances. This enables the flow to bypass the first instability, as the visualization results in Batchelor [20] and Schlichting [396], testifies. Similar evidence for missed bifurcation for flow past a cylinder has been reported by flow control achieved by placing a smaller control-cylinder in the near wake [503, 504]. These flow control exercises have been checked numerically in [112]. The observed computational instabilities depend on accuracy of the method and errors due to truncation and round-off errors. In [437, 469], high accuracy compact schemes are used to discretize the convective acceleration terms. In [437], a finer grid with 1001 points in the azimuthal direction and 401 points in the radial direction has been used showing five critical Re. With vorticity recorded on the wake center line for Re = (55, 58, 60), the first critical Reynolds number Recr1 is obtained as 47.16. The second critical Reynolds number (Recr2 ) has been obtained as 53.72 using the data set of Re = (70, 72, 76), while using the data set of Re = (80, 83, 90), the third critical Reynolds number (Recr3 ) has been obtained as 58.31. The fourth and fifth bifurcations are noted for critical Reynolds numbers of Recr4 = 64.949 and Recr5 = 77.64, using the data sets of Re = (92, 95, 100) and Re = (100, 104, 110), respectively. The use of refined grid spacing in the azimuthal direction by a factor of more than six times (while keeping grid spacing same in the wall-normal direction), reduced the critical Reynolds numbers with respect to coarse grid simulation. In [469], use of (153×401) grid predicted the first critical Reynolds number as 53.2907, while the finer grid computations using (1001 × 401) points provide a critical Reynolds number around Re = 53.72, and with the finer grid, an additional instability is noted earlier for Recr1 = 47.16. There remains an open question of reducing the first critical Reynolds number by refining the grid in the azimuthal direction. Multiple instabilities also indicate the existence of more than one bifurcations, as has been noted in [469]. The multiple Hopf bifurcations was also noted for internal and external flows in [477]. This shows the limitation of the Stuart–Landau equation, which presumes single bifurcation for a single instability mode. To explain multiple Hopf bifurcations, one must have more than one instability mode, leading to development of multimodal Stuart–Landau equations [469, 477] which resort to the eigenfunction expansion method, using the POD modes and not the Orr–Sommerfeld modes used in [127]. In this approach, the instability modes are governed by dA j M = α j A j + Σk=1 N j (Ak , A j ) dt

(10.18)

The last term on the right-hand side accounts for all non-linear interactions among the M modes, including self-interaction. Eckhaus [127] suggested taking the non-linear term as β jk A j |Ak |2 , and the M-mode Stuart–Landau equations are written as follows

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dA j M = α j A j + Σk=1 β jk A j |Ak |2 dt

(10.19)

The amplitude and phase equations are obtained from Eq. (10.19) and written as d|A j | M = (α j )Real |A j | + Σk=1 (β jk )Real |A j ||Ak |2 dt

(10.20)

dθ j M = (α j )Imag + Σk=1 (β jk )Imag |Ak |2 dt

(10.21)

and

where α j = (α j )Real +i(α j )Imag and β jk = (β jk )Real +i(β jk )Imag are the complex parameters in Eq. (10.19). Following the definition of instability modes, the POD modes are paired to construct the instability amplitude functions for Re = 100. The POD modes-(a1 , a2 ) make up the first instability mode, A1 ; the anomalous T 1 -mode a3 gives rise to the instability mode A2 ; the regular modes-(a5 , a6 ) makes the third instability mode A3 , and the regular pair given by modes-(a7 , a8 ) gives rise to the fourth instability mode, A4 and so on. From Eqs. (10.16) and (10.17), instability eigenfunctions (g j (~r)) and amplitude functions (A j (t)) are obtained from the POD data of DNS performed for the flow. The imaginary part is plotted against the real part of the instability modes, with time as the parameter in the phasor plots shown in Figure 10.7 for Re = 100. Instability mode A5 is formed with the T 1 -mode a9 , while A6 is formed using T 2 -mode pair (a11 , a12 ). Thus, A2 and A5 are straight lines along Im(A j ) = 0. The limit cycles of the regular instability modes are annulus of finite thickness in Figure 10.7. The thickness of the ring provides a measure of the basin of attractor for the saturation amplitude (A je ). From DNS data, one constructs am (t) and φm (~r) using Eqs. (10.9) and (10.10). From the linear algebraic equation, the POD amplitudes and eigenfunctions are obtained for A j (t) and g j (~r), using Eqs. (10.16) and (10.17). Thus, the amplitude of the instability mode is obtained as R R g j (~r) ω0 (~r, t) d2~r A j (t) = R R g j (~r)g j (~r) d2~r Using the POD modes, one constructs the instability modes, using Eqs. (10.16) and (10.17). Amplitude of instability modes are shown for Re = 100 in Figure 10.8. For the first instability mode, the time variation is at Strouhal frequency, with the final amplitude settling to a non-zero value. The constituent POD modes, a1 (t) and a2 (t) have a zero mean in Figure 10.4, and yet the instability mode settles to a non-zero value due to the phase shift between the constituent modes. The second instability mode amplitude (A2 ) is due to the anomalous POD mode a3 (t), whose time variation is at twice the Strouhal number (as noted from the FFT of data shown in Figure 10.4).

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Re = 100

Re = 100 0.03 0.02

0.02 t

0.005 t

0

–0.01

Im(A3)

0.01 Im(A2)

Im(A1)

0.01

0

–0.01

–0.02 –0.03

Re = 100

0.03

–0.005

–0.02 t=0 –0.02

0

Re (A1)

0.02

–0.03

0

t=0 –0.02

0

Re (A2)

0.02

–0.005

Re = 100

Re = 100

Re (A3)

0.005 Re = 100

0.03 0.02 t

0.01

0

Im(A6)

0.01 Im(A5)

Im(A4)

0.005

0

–0.01

–0.005

0 t

–0.02 t=0 –0.005

0

Re (A4)

0.005

–0.03

–0.02

Re (A5)

0.02

–0.01 –0.01 –0.005

t=0

0 0.005 0.01 Re (A6)

Figure 10.7 The phasor plots for the six instability modes for flow past a circular cylinder for Re = 100 showing disturbance vorticity. Regular modes show finite width basin of attraction, while the anomalous modes are straight lines, as explained in the text. [Reproduced from “Enstrophy-based proper orthogonal decomposition for reduced-order modeling of flow a past cylinder”, T. K. Sengupta, S. I. Haider, Parvathi M. K. and Pallavi G., Phys. Rev. E, vol. 91(4), pp 043303 (2015), with the permission of APS Physics.] Quite unlike the first instability mode amplitude (A1 ), the second instability mode (A2 ) displays a time periodic oscillatory state. In the time-averaged framework, the only effect of shift or anomalous mode (as noted in [315, 316, 488]) can at most alter the mean field, and will be unable to indicate any time variation. To capture the time variation of A2 (t), a DNS based approach will be effective. In Figure 10.8, the time variation of third and fourth instability mode amplitude (|A3 | and |A4 |) are due to the two regular mode pairs, which also show mode-switching as Reynolds number is increased. This has been shown in [437], for Re = 100 and 150. In Figure 10.8, mode-A3 depicts higher fluctuation about the mean after the periodic equilibrium is reached, while mode-A4 displays much lower level fluctuations. It has been noted in [437] for Re = 150, the roles of A3 - and A4 -modes are switched with fluctuations of A3 significantly lower compared to A4 . Also in Figure 10.8, |A3 | shows phase variation at twice the Strouhal frequency, while |A4 | shows this variation at thrice the Strouhal frequency.

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0.03

Re=100

Re=100

0.015

0.01

0.008

0.01

|A3|

|A2|

|A1|

0.02

0.005 100

200 t

300

0.006 0.004 0.002

100

Re=100

200 t

300

100

Re=100

200 t

300

200 t

300

Re=100

0.01 0.006 0.004

0.01

0.006

|A6|

|A5|

|A4|

0.008 0.004

0.002

0.005

0.002 100

200 t

300

100

200 t

300

100

Figure 10.8 The time variation of instability amplitude for flow past a circular cylinder with Re = 100 for the disturbance vorticity. [Reproduced from “Enstrophy-based proper orthogonal decomposition for reduced-order modeling of flow a past cylinder”, T. K. Sengupta, S. I. Haider, Parvathi M. K. and Pallavi G., Phys. Rev. E, vol. 91(4), pp 043303 (2015), with the permission of APS Physics.] However, an important feature of Figure 10.8 is how A2 shows significant phase variation at later times, while the phasor plot in Figure 10.7 does not show any phase variation due to the construction of A2 (t) by the method given in Eqs. (10.16) and (10.17). This is addressed and rectified in [437] by another model to be described next. The instability modes |A5 | and |A6 |, have residual effects at later times and their phase variations are at twice and at the Strouhal frequency, respectively. In the following, a model developed in [437], by obtaining a formulation through the least square formalism for the multimodal Stuart–Landau equations is described.

10.5.2 Formulation and modeling of a multimodal Stuart–Landau equation Using Eqs. (10.20) and (10.21) with the DNS data, the instability modes with characteristic amplitude and phase variations can be obtained. These involve fixing the coefficients in these equations governing the temporal growth of the instability modes. As a first attempt, these coefficients were obtained by using the temporal rates at many time instants, each corresponding to an unknown, in [469]. The coefficients obtained in this way show extreme sensitivity based on the chosen times. In [437] the sensitive dependence of coefficients, as noted in [469], have been circumvented. The values of (α)Real , (α)Imag , (β jk )Real and (β jk )Imag , have been evaluated by a least squares formulation using the data from the complete time interval, which also incorporates the transient stage. The passage from the quiescent state to the second equilibrium state does not occur as the system settles to the final constant

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amplitude of the limit cycle. This is indicated from the thickness of the limit cycles in the phasor plots, which can be viewed as the basin of attractors in dynamical system terminology. The attractors of A3 - and A4 -modes are wider, than that of the A1 -mode. During the early transient, the variations are very pronounced, so that a lower order model would be erroneous, if one attempts a deterministic evaluation of constants in the multimodal Stuart–Landau model. This provides reason for determining the model constants by a least square framework which minimizes modeling error. For a M-mode multimodal Stuart–Landau model for |A j |, one requires (M + 1) unknowns, as (α j )Real and (β jk )Real , where k varies from 1 to M. Identical procedure is conducted for the phase equation for θ j . There are N >> M linear equations available, each for a time, where N is the number of snapshots used for POD. These equations for |A j | and θ j , are given at t = ti by d|A j (ti )| M = (α j )Real |A j (ti )| + Σk=1 (β jk )Real |A j (ti )||Ak (ti )|2 dt

(10.22)

dθ j (ti ) M = (α j )Imag + Σk=1 (β jk )Imag |Ak (ti )|2 dt

(10.23)

where i = 1 to N, correspond to each snapshot. Least square approach gives one a solution that minimizes the error-norm in the set of N equations given by Eqs. (10.22) and (10.23). Written in a single symbolic form, these equations are M U j = δ j V1, j + Σk=1 γ jk Vk+1, j

(10.24)

For the instability amplitude mode the notations imply: Uj =

d|A j | and V1, j = |A j | dt

Vk+1, j = |A j ||Ak |2 δ j = (α j )Real and γ jk = (β jk )Real Similarly for the phase variation of instability mode, the following hold: Uj =

dθ j and V1, j = 1 dt

Vk+1, j = |Ak |2

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δ j = (α j )Imag and γ jk = (β jk )Imag The square of the error associated with Eq. (10.24) is given by the norm N M E j = Σm=1 [U j (tm ) − {δ j V1, j (tm ) + Σk=1 γ jk Vk+1, j (tm )}]2

(10.25)

Method of least squares can also be used to find the unknowns, δ j and γ jk by minimizing the above error-norm in Eq. (10.25). From the (M + 1) equations, minimizing the error-norm for the unknowns yields the following ∂E j ∂E j = 0 and =0 ∂δ j ∂γ jk By substituting for E j in these equations, one gets ∂E j N M = Σm=1 [U j (tm ) − {δ j V1, j (tm ) + Σk=1 γ jk Vk+1, j (tm )}]V1, j (tm ) = 0 ∂δ j

(10.26)

∂E j N M = Σm=1 [U j (tm ) − {δ j V1, j (tm ) + Σk=1 γ jk Vl+1, j (tm )}]Vk+1, j (tm ) = 0 ∂γ jk

(10.27)

The above set of (M + 1) linear algebraic equations are written notationally as ¯ [P]{x} = {q} ¯ is a (M + 1) × (M + 1)-matrix, {x} denotes the (M + 1) unknown coefficients. where [P] The solution to this, gives the requisite coefficients of the multimodal Stuart–Landau equations with the unknowns, δ j and γ jk . In the following, the case of Re = 100 is treated to obtain the corresponding coefficients.

10.5.3 Reconstruction of instability modes using multimodal Stuart–Landau equations Results of reconstruction using the least square formulation given in the previous subsection are provided next. The coefficients of amplitude and phase equations (Eqs. (10.20) and (10.22)) for a 3-mode, (A1 , A2 , A3 )-Stuart–Landau equations are provided in Tables 10.1 and 10.2. Before one solves Eqs. (10.22) and (10.23), one needs initial conditions in developing a reduced order model (ROM), that will replace the intensive solution of the Navier–Stokes equation with different boundary and initial conditions. Equations (10.22) and (10.23) apart from being nonlinear are also stiff, as the original Navier– Stokes equation for super-critical Re is stiff too. The characteristic of a stiff differential equation is its sensitive dependence on auxiliary conditions. One therefore looks for

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Table 10.1 Coefficients of Eq. (10.22) for Re = 100. [Reproduced from “Enstrophybased proper orthogonal decomposition for reduced-order modeling of flow a past cylinder”, T. K. Sengupta, S. I. Haider, Parvathi M. K. and Pallavi G., Phys. Rev. E, vol. 91(4), pp 043303 (2015), with the permission of APS Physics.] j

αj

β j1

β j2

β j3

1

8.941 × 10−2

−8.412 × 101

1.641 × 102

−1.504 × 102

2

−2.429 × 10−3

−1.967 × 102

−9.148 × 101

2.005 × 103

3

5.883 × 10−4

−3.604 × 102

−1.231 × 102

3.122 × 103

Table 10.2 Coefficients of Eq. (10.23) for Re = 100. [Reproduced from “Enstrophybased proper orthogonal decomposition for reduced-order modeling of flow a past cylinder”, T. K. Sengupta, S. I. Haider, Parvathi M. K. and Pallavi G., Phys. Rev. E, vol. 91(4), pp 043303 (2015), with the permission of APS Physics.] j 1 2 3

αj

β j1 −1

1.490 × 10 0.00

β j2 2

8.849 × 10 0.00

−3.182 × 10

−1

−2.227 × 10

3

β j3 3

2.660 × 10

8.706 × 102

0.00

0.00

1.126 × 10

3

3.277 × 103

optimum initial conditions for Eqs. (10.22) and (10.23) for any choice of Re. For the demonstrated case in [437] for Re = 100, coefficients are given in Tables 10.1 and 10.2 for Eqs. (10.22) and (10.23). To solve these equations, one must locate optimum initial conditions given by A10 , A20 , A30 for the 3-mode multimodal Stuart-Landau equation reconstruction. The ordinary differential equations are solved by the implicit second order, Runge–Kutta (IRK2 ) method [31] for these stiff differential equations, with a time step of 0.001. The main idea is to obtain the optimal initial conditions, for which the integrated error-norm in Eq. (10.25) is minimum. The reason behind a 3-mode multimodal Stuart-Landau equation model is due to the fact that this is a minimal set for ROM of this flow. In what follows, optimal initial conditions are computed which minimize the departure between the DNS and the model in solving Eqs. (10.22) and (10.23). To minimize error in Eq. (10.25), the full time range is used to train the model with known DNS data for different Reynolds numbers. It is highlighted that replacing the problem of Navier–Stokes equation by a set of initial value problems, requires information about the unused boundary condition. This is why an optimal set of initial conditions are needed to solve Eqs. (10.22) and (10.23) with coefficients given in Tables 10.1 and 10.2. Any arbitrary initial condition will not do.

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Minimum Error = 0.498

A10 = 1 × 10–5

0.006

0.007

0.005

2

0.008

48

83

0.004

0.4 97 91 8

|A30|

0.

49

0 .5

0.005

0

6 .4

18

9

0 .5 0 3 1 8

|A30|

0.006

Minimum Error = 0.421

A10 = 2 × 10–5

0.004

4 0.

44

11

3

0.003

0.

4 42

14

2

0.002

0.003 0.002 0.01

0.012

0.014

A10 = 3 × 10–5

0.016

|A20|

0.018

0.001

0.02

0.01

0.02

|A20|

Minimum Error = 0.408

A10 = 4 × 10–5

0.003

Minimum Error = 0.420

0.003

062 0 .4 3 0.002

3

0.444113 0.444113

0 .424 142 0.002

0 .40940 3

|A30|

|A30|

0.420979

0.001

0.001

0.006

0.008

0.01

|A20|

0.012

0.014

0.006

0.008

0.01

0.012

0.014

|A20|

Figure 10.9 The integrated error contours for 0 ≤ t ≤ 396 are plotted in (A20 , A30 )-plane for the indicated choice of A10 , for the case of Re = 100. [Reproduced from “Enstrophy-based proper orthogonal decomposition for reduced-order modeling of flow a past cylinder”, T. K. Sengupta, S. I. Haider, Parvathi M. K. and Pallavi G., Phys. Rev. E, vol. 91(4), pp 043303 (2015), with the permission of APS Physics.]

For the proposed ROM in [437], one requires the global minima for A10 , A20 , A30 with error being least between the computed mutlimodal Stuart–Landau equations and the corresponding DNS data. For this, one fixes A10 , and sets of A20 and A30 values are chosen in the (A20 , A30 )-plane. Time integrated errors obtained from t = 0 to 396 are next plotted as contour plots in (A20 , A30 )-plane, as shown in Figure 10.9 for the choice of A10 . In each frame, the minimum error is marked and corresponding location noted for different A10 in (A20 , A30 )-plane. The minimum obtained in (A20 ,A30 )-plane for each choice of A10 are plotted as a function of A10 in Figure 10.10. This helps to identify the global minimum error and the desired initial conditions. This global minimum error of 3-mode multimodal Stuart–Landau model reconstruction is given for Re = 100, with the following initial conditions: A1 (0) = 3.0 × 10−5 , A2 (0) = 1.05 × 10−2 and A3 (0) = 1.6 × 10−5

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0.5

Local minimum in A20 – A30 plane

0.49 0.48 0.47 0.46 0.45 0.44 Global minimum

0.43 0.42 0.41 0.4 1.0

2.0

3.0 A10

4.0

5.0

×10–5

Figure 10.10 The global minimum is located by plotting local minimum error plotted as a function of A10 for the flow past a circular cylinder for Re = 100. [Reproduced from “Enstrophy-based proper orthogonal decomposition for reduced-order modeling of flow a past cylinder”, T. K. Sengupta, S. I. Haider, Parvathi M. K. and Pallavi G., Phys. Rev. E, vol. 91(4), pp 043303 (2015), with the permission of APS Physics.] With these optimal initial conditions, Eqs. (10.22) and (10.23) are solved using the coefficients in Tables 10.1 and 10.2. The results of the reconstruction given in Figure 10 of [437] show that the reconstructed amplitude variation of the regular instability modes (with A1 and A3 ), match well with DNS results. The reconstruction of the first T 1 -mode (A2 ) does not work for the reason that the procedure is incapable of capturing oscillatory behavior. The POD of DNS data possesses this information, and modeling by a multimodal version of the Stuart–Landau equation needs to incorporate the oscillatory T 1 -modes introduced, as it has been done in [437]. In similarity with the amplitude of instability modes, the phase equations also require appropriate initial conditions for time integration. The phase is simply shifted by initial conditions, as can be seen from Eq. (10.23). In view of this, a new methodology has been adopted for the initial conditions of phase in [437]. With all phases initialized to zero, time integration is carried out to compare phase variation from DNS to note the difference between the two (= θDNS (t) − θreconstructed (t)). The time average of this is set as the initial phase. The initial phase values obtained for Re = 100 are: θ1 (0) = −7.637 rad, θ2 (0) = 0 rad and θ3 (0) = 7.766 rad,

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10.5.4 Modeling the anomalous mode of the first kind (T 1 -mode) In constructing the instability mode associated with T 1 -modes, one uses information from Figure 10.4 for the first T 1 -mode (POD mode a3 ) which shows oscillatory behavior at saturation with time varying phase. Thus, there is a need to have a model for the T 1 -mode. The anomalous modes obtained from the POD analysis show different time-periodic behavior during and after saturation, implying presence of two disparate time scales. A slowly varying component alters the mean field (as by the shift mode in [315] for RANS), and a rapidly varying time-periodic component that has to be explicitly modeled.

T1s(t)

0.05 T1s at saturation = –0.067

0 0

100

200 t

300

100

200 t

300

T1r(t)

0.01 0

–0.01

Centre of the limit cycle at (–0067, 0) Im(A2)

0.005 t 0

–0.005

t= 0 –0.01

0

0.01 Re(A2)

0.02

0.03

0.04

Figure 10.11 The slow and fast varying components of T 1 -mode (POD mode a3 ) for Re = 150 shown in top two frames. The bottom frame shows the phasor plot, with data from [437]. [Reproduced from “Enstrophy-based proper orthogonal decomposition for reduced-order modeling of flow a past cylinder”, T. K. Sengupta, S. I. Haider, Parvathi M. K. and Pallavi G., Phys. Rev. E, vol. 91(4), pp 043303 (2015), with the permission of APS Physics.]

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One needs to represent the T 1 -mode to account for the phase variation at fast scales. To achieve this one splits the T 1 -mode into slow and rapid components at two time scales. The slow variation is obtained by smoothing the data, removing the fluctuating component, to obtain T 1s (t). One uses a low-pass filter on the data during post-processing. It is shown in the top frame of Figure 10.11 for Re = 150. The rapidly varying component T 1r can then be found by subtracting the slow variation from the original T 1 -mode given by T 1r (t) = aT1 (t) − T 1s (t)

(10.28)

For Re = 150, T 1r is shown as a function of time in the middle frame of Figure 10.11, and FFT of T 1r (t) allows one to identify the frequency, ω ¯ d of the fast scale. The amplitude envelope T ra (t) of T 1r (t) is directly obtained using a cubic Hermite interpolation of the envelope of maxima and minima to get the amplitude variation as function of time. Using the frequency and the amplitude of the rapidly time-varying component, it is written as T 1r (t) = T ra (t) cos(ω ¯ d t) Using this, the instability mode A2 (t) is modeled as A2 (t) = T 1s (t) +

T ra (t) iω¯ d t e 2

(10.29)

The model for this T 1 -mode is to split A2 (t) into A2m (t) (for slow variation) for the mean and A2d (t) for the fast scale are given by: A2m (t) = T 1s (t)

A2d (t) =

T ra (t) iω¯ d t e 2

(10.30)

(10.31)

10.6 Reduced Order Model Using Multimodal Reconstruction by POD Analysis The minimal instability mode based model is now presented as shown below, using equations for (A1 , A2m , A2d , A3 ) in this unique model presented in [437]. The multimodal Stuart-Landau equations are governed by the following for the amplitude as d|A1 | = α1 |A1 | + β11 |A1 ||A1 |2 + β12m |A1 ||A2m |2 + β12d |A1 ||A2d |2 + β13 |A1 ||A3 |2 dt

(10.32)

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d|A2m | = α2m |A2m |+β2m1 |A2m ||A1 |2 +β2m2m |A2m ||A2m |2 +β2m2d |A2m ||A2d |2 +β2m3 |A2m ||A3 |2 (10.33) dt d|A2d | = α2d |A2d | + β2d1 |A2d ||A1 |2 + β2d2m |A2d ||A2m |2 + β2d2d |A2d ||A2d |2 + β2d3 |A2d ||A3 |2 (10.34) dt d|A3 | = α3 |A3 | + β31 |A3 ||A1 |2 + β32m |A3 ||A2m |2 + β32d |A3 ||A2d |2 + β33 |A3 ||A3 |2 dt

(10.35)

The coefficients of the amplitude equations obtained using the least square approach for different Re are plotted in Figures 13 to 16 of [437], and can be consulted.

Table 10.3 Initial conditions for amplitude equations. [Reproduced from “Enstrophy-based proper orthogonal decomposition for reduced-order modeling of flow a past cylinder”, T. K. Sengupta, S. I. Haider, Parvathi M. K. and Pallavi G., Phys. Rev. E, vol. 91(4), pp 043303 (2015), with the permission of APS Physics.] Re

|A10 |

|A2m0 |

|A2d0 |

|A30 |

100

8.1 × 10−5

3.26 × 10−3

8.0 × 10−7

1.9 × 10−3

130

3.46 × 10−5

5.0 × 10−3

1.69 × 10−6

2.31 × 10−3

150

2.5 × 10−5

2.5 × 10−3

2.1 × 10−6

1.0 × 10−8

The optimal initial conditions for the amplitudes are found through a timeintegrated error analysis in the (A10 , A2m0 , A2d0 , A30 )-hyperspace, similar to the approach stated before, with local minima found in (A2m0 , A2d0 , A30 )-space for a choice of A10 , from which one obtains the global minimum. The final global optimal initial conditions obtained by this process are given in Table 10.3. The governing multimodal Stuart–Landau equations for the phase of the 3-modes (A1 , A2m , A2d , A3 ) are given in [437] as dθ1 = α1 + β11 |A1 |2 + β12m |A2m |2 + β12d |A2d |2 + β13 |A3 |2 dt

(10.36)

dθ2m =0 dt

(10.37)

dθ2d = α2d + β2d1 |A1 |2 + β2d2m |A2m |2 + β2d2d |A2d |2 + β2d3 |A3 |2 dt

(10.38)

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dθ3 = α3 + β31 |A1 |2 + β32m |A2m |2 + β32d |A2d |2 + β33 |A3 |2 dt

(10.39)

Table 10.4 Initial conditions for phase equations (in radians). [Reproduced from “Enstrophybased proper orthogonal decomposition for reduced-order modeling of flow a past cylinder”, T. K. Sengupta, S. I. Haider, Parvathi M. K. and Pallavi G., Phys. Rev. E, vol. 91(4), pp 043303 (2015), with the permission of APS Physics.] Re

θ10

θ2m0

θ2d0

θ30

100

−9.409

0

−9.685

12.410

130

1.257

0

8.45

0.944

150

1.606

0

11.210

−6.147

The coefficients of phase equations obtained using the least square approach are plotted in [437], for different Re values. The procedure to obtain optimal initial conditions for phase angles are less complicated and such data is generated by averaging the difference of the reconstructed and actual phase from DNS. In starting this exercise, all phases are initialized as zero. The optimal initial conditions for phase angles are given in Table 10.4 following the details given in [437]. The amplitudes of instability modes are thereafter obtained by solving multimodal Stuart–Landau equation and it is noted [437] that the ROM from the POD data matches even better with increase in Reynolds number. One of the successes of this effort stems from the reconstructing and using the A2 -mode, contributed by the first T 1 -mode. The phase matches rather well for all the Reynolds numbers, as reported in [437].

10.7 Universality of POD Modes It is noted that there are more than one dominant instability mode present for flow past a cylinder. These have been related to POD modes in Eqs. (10.16) and (10.17). The original goal of using POD in fluid mechanics has been to analyze turbulent flows. However in [107, 283], flow past a circular cylinder was studied using POD to explain vortex shedding. Flows past a circular cylinder have also been studied in [315, 488] in a time-averaged manner and portrays a dynamical system. In this chapter, instability modes are constructed from POD modes. In early applications of POD, it was performed with the assumption that the dynamical system is characterized by limit cycle oscillation, and hence the resultant spatial modes are appropriate only for the limit cycle. Siegel et al. [488] concluded that “POD in its original form is not well suited to describing transient data sets.” In calculating POD modes for a cavity flow in [208], the authors used a method called the sequential POD to process data from different flow states, by ensuring that the modes are orthogonal. The success of this and the results in [437, 469, 477] show that POD modes are perfectly capable of

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capturing transient dynamics for flow past a cylinder. In [477], DNS results for twodimensional flow past a circular cylinder, and flow inside a lid-driven cavity (LDC) have been used to show the universality of POD modes, despite the flows appearing so dissimilar. In what follows, the dynamics, and bifurcation sequences for flow inside a LDC are described further, with focus on instability sequences affected by numerical seeding of the events. y 1

E0

U P

(0.02, 0.98)

(0.95, 0.95)

(a)

0 –6.0

1

x

Vorticity time series at P(0.95, 0.95), Re=9800

(b)

Vorticity

–6.5 –7.0

2 Ae

–7.5 –8.0 –8.5 –9.0

0

Primary instability

500

Secondary instability

1000 time

1500

2000

Figure 10.12 (a) The schematic of flow inside the LDC, with the point P marked where the data is sampled and the flow can be excited at E0 for selected cases. (b) The vorticity time series at the sampling point for Re = 9800 is shown to identify primary and secondary instabilities and equilibrium amplitude Ae of the eventual limit cycle oscillation. [Reproduced from “Grid sensitivity and role of error in computing a lid-driven cavity problem”, V. K. Suman, Siva Viknesh S., Mohit K. Tekriwal, S. Bhaumik, and T. K. Sengupta, Phys. Rev. E, vol. 99, pp 013305 (2019), with the permission of APS Physics.] The two-dimensional flow in a square LDC has been often used as a canonical problem to test numerical methods for the incompressible Navier–Stokes equation. The lid is given a constant-translation speed (U), which creates corner singularities on it, schematically depicted in the top frame of Figure 10.12. In fact, the corner singularity is avoided by prescribing the wall vorticity to be zero exactly at the corner. In actual computational framework, the singularity is transferred to the next point

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on all the walls in the form of a discrete jump in the value of wall vorticity, which in turn gives rise to the Gibbs’ phenomenon. This is observed most acutely on the top lid, and has been reported by pseudo-spectral computation of the Navier–Stokes equation [15, 51], and also additionally by aliasing error for under-resolved grids in [443, 476]. This is somewhat paradoxical, as computing flow in LDC by low order methods [73, 154, 404], do not face the problems due to high implicit filtering by the spatial discretization [447]. It is only in high accuracy methods such as pseudospectral methods and compact schemes, that one becomes aware of these sources of errors. However, low order methods are poor in computing unsteady flows for high Reynolds numbers (Re = UL/ν). For example in [154], results for a wide range of Re up to 10000 are presented, but the flow has been reported to be steady for Re = 10000. Numerical results obtained by high accuracy combined compact difference (CCD) schemes in [277, 443, 476, 507] display the creation of a transient polygonal core vortex, with permanent satellite vortices moving around it for a case of Re = 10000, that eventually leads to limit cycle oscillation. In [277, 443, 476, 507], formally sixth order CCD scheme has been used to discretize both the convection and diffusion terms of the vorticity transport equation. It is well known [413, 438] that compact schemes used for spatial discretization filters minimally, and along with carefully chosen boundary closure schemes largely control the Gibbs’ phenomenon for the LDC problem by smoothing the solution near the Nyquist limit for the wavenumber. Aliasing error, on the other hand can provide distributed numerical seeds for triggering unsteadiness in the flow, if the grid resolution is slightly inadequate. For lower Re, researchers in [15, 16, 51] have tried to remove the effects of the corner singularity by subtracting the contribution due to singularity (divergence of pressure and vorticity at the top corners) to obtain a steady solution by pseudospectral methods. The singularity diverges as 1/r, with r as the radial distance from the corner. This method has not been used for Re exceeding 1000; instead for higher Re LDC flows, such singularity is removed by altering the lid velocity smoothly to zero at the corners, with the process called regularization [263]. As explained above, the flow field is essentially excited by the Gibbs’ phenomenon and aliasing error, even when the so-called corner singularities are subtracted. There have been attempts to obtain unsteady LDC flow by studying the linear temporal instability of steady solutions obtained numerically, in an attempt to explain the bifurcation problem [48, 137, 334]. However, less accurate equilibrium flow data contaminates eigenvalues. Simulation of the full time-dependent Navier–Stokes equation [156, 164, 323, 443, 476] can also capture instability through the Hopf bifurcation, with increasing Re. Critical Re and frequencies obtained in this manner and eigenvalue analysis show mismatch due to inherent differences between linear and nonlinear instability [471]. It is noted in [476, 477] that the first critical Reynolds number Recr1 is dependent on the accuracy of methods and the way the flow is started. Impulsive start of the flow is a standard theoretical way to study the onset, as this triggers all the frequencies at t = 0 [277, 443, 476, 507]. This type of multimodal global analysis is superior to linear

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normal mode analysis, as it has been already explained for the flow over zero pressure gradient boundary layer or flow over semi-infinite flat plate [471]. It has been shown [507] that trying to obtain the limit cycle oscillation at a higher Re from the solution obtained at a lower Re is not appropriate. This may speed up computations [323], but produces an incorrect Recr1 , as compared to impulsive start results [277, 476]. The role of aliasing error has been highlighted in [476], and re-emphasized again in [507]. Also, parallel computing itself is an additional source of error for all high accuracy methods, and this is highly sensitive, so much so that early high accuracy solutions in [443, 476, 277] have been obtained by sequential computing. The nature of error and its solution for parallel computing has been solved only recently in [472, 507]. Unsteadiness in flows with varying parameters studied in bifurcation theory [389] is an attribute of flow instabilities [412, 477]. Linear instability of equilibrium flow and direct simulation have been used to trace onset of unsteadiness, i.e. to obtain Recr1 for LDC. Both these approaches report widely scattered values for Recr1 . For example, Recr1 = 8018 is reported in [16] and as 8031.93 in [382]. In [81], authors have reported Recr1 for 7972 using a finite volume method. Sengupta et al. [477] have reported multiple Hopf bifurcations, showing the first one at 7933 and the second at 8187, using a uniform (257 × 257) grid. Osada and Iwatsu [323] have identified this value at 7987 ± 2%, a value obtained using the compact scheme on non-uniform (128 × 128) and (257 × 257) grids. Shen [486] reported Recr1 in the range of 10000 to 10500 by using regularization for the top lid velocity; Peng et al. [327] reported a value of Recr1 = 7402 ± 4% using the finite difference, marker and cell (MAC) method. A typical high accuracy method used in [277, 278] produces time accurate results, as shown in Figure 10.12(b) for Re = 9800 obtained using a (257×257) uniform grid for the CCD scheme. This frame demonstrates a highly active point near the top right corner of the lid, which moves from left to right. After the initial transient dies down, the primary instability causes multiple harmonics to be present (as shown by the highly modulated time series), which leads to a limit cycle beyond t ≈ 320, with amplitude increasing very slowly. At around t ≈ 900, one again starts noticing a modulated amplitude envelope, indicating the onset of a secondary instability, which leads to the final limit cycle with steady envelope amplitude, as marked in the figure by its amplitude Ae .

10.7.1 Role of disturbances in triggering instabilities The vorticity dynamics inside the LDC is driven by the creation of vorticity on the wall segments. Of specific interest is the way the wall vorticity is created on the top wall, as described next for the used stream function-vorticity formulation, whose governing equation is given by Eqs. (7.1) and (7.2). These equations are solved using a uniform grid of a Cartesian frame, with the origin at the bottom left corner. These are solved subject to the following boundary conditions. The wall vorticity is given 2 exactly as, ωb = − ∂∂nψ2 , with n as the wall-normal co-ordinate chosen for the wall

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∂ψ segments. The lead truncation error in calculating ωb is ∆n 3 ∂n3 . On the walls of LDC, for the streamfunction, ψ = constant is prescribed to satisfy the no-slip condition, and to evaluate wall vorticity. The wall vorticity can be obtained from the Taylor series expansion at the second point from the walls in terms of quantities evaluated at the wall. This Taylor series is given for the top wall by 3

ψ(x, L − dy) = ψ(x, L) − dy Since, U = as

∂ψ ∂y

ωb (x) =

∂ψ dy2 ∂2 ψ + − ..... ∂y 2 ∂y2

at the top wall, the wall vorticity can be written in truncated series form  2  ψ(x, L) − ψ(x, L − δy) − δy δy2

(10.40)

On the right-hand side of Eq. (10.40), the last term is due to the constant top lid motion with the value U, which in non-dimensional form is unity. One can similarly obtain the expression for the wall vorticity at other wall-segments, where one has to use ∂ψ ∂n ≡ 0. In Figure 10.13, variation of ωb with x for the top lid shows the maximum amplitude of wall vorticity to scale inversely with δy. The convection and diffusion terms of the vorticity transport equation are discretized using the CCD scheme [277, 443, 476] for the first and second derivatives, simultaneously. For time advancement of the vorticity transport equation, the four-stage, fourth-order Runge–Kutta (RK4) method is used. The CCD scheme with RK4 method has been examined in detail for solving convection-diffusion equations in [472, 506] and highlight the role of numerical Peclet number, which essentially depends on the time-step for the same grid. A time step of ∆t = 0.001 is used with the lower resolution grids, while the (1025 × 1025)-grid used ∆t = 0.0005 in Figure 10.13. Some properties of the wall vorticity shown in this figure are noteworthy. In the finer grids, the maximum wall vorticity is noted for the second and second-last grid points, which also shows distinct asymmetry of the values due to the preferred direction of the lid motion. Presence of these maxima are the sources of excitation via the Gibbs’ phenomenon, and grid refinement causes this to appear more aggravated near the top left. In contrast, the flow displays presence of higher wavenumber phenomena near the top right corner. This manifests as the aliasing error (grid-scale oscillations) for the coarsest grid with (257 × 257) points. The inset of Figure 10.13 clearly demonstrates that the aliasing error reduces significantly for the (513 × 513) grid and aliasing error is invisible in the (1025 × 1025) grid, while the jump discontinuity is almost double for the most refined grid in this figure, as compared to that of the (513 × 513) grid. In perspective, the compact schemes cause significantly lower discretization error. Requirement of lesser number of points for compact schemes, causes lower round-off error too. However, not all compact schemes are equal in countering aliasing error. For example, this is a serious source of error for the sixth order scheme given in [261] for the LDC problem. Using the (257 × 257) grid and employing Lele’s method, it is found that

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t = 1750

–500 257×257 513×513 1025×1025

ωb

–1000

–1500

–2000

–2500 0

0.2

0.4

x

0.6

0.8

1

Figure 10.13 Comparison of computed wall vorticity distribution ωb , at the top wall for the LDC flow for Re = 9600 at t = 1750 using (257 × 257), (513 × 513) and (1025 × 1025) grids [507]. [Reproduced from “Grid sensitivity and role of error in computing a lid-driven cavity problem”, V. K. Suman, Siva Viknesh S., Mohit K. Tekriwal, S. Bhaumik, and T. K. Sengupta, Phys. Rev. E, vol. 99, pp 013305 (2019), with the permission of APS Physics.]

solution breakdown occurrs due to numerical instability by aliasing error, as reported in [413]. But, the same method works in [507] while using a (1025 × 1025) grid. It is therefore apparent that different sources of error trigger the flow in LDC for high accuracy methods because of the Gibbs’ phenomenon, while aliasing can destabilize the computations or produce results which are not correct, for some high accuracy methods [261]. Aliasing error even when present, may not cause solution breakdown, as seen in [277, 443, 476], and shown in Figure 10.13. Upon regularizing the top lid velocity boundary condition, both the problems associated with Gibbs’ phenomenon and aliasing error are removed, as shown in the results of [486], where Recr1 has been reported to be delayed, and observed to be between 10000 and 10500 for the used compact scheme. The authors in [482] have noted that the onset of flow instability is due to distributed aliasing error causing grid-scale oscillations for relatively coarse grids, near the top lid. Additionally, the Gibbs’ phenomenon is triggered near the corners of LDC, for which the discontinuity is similar to the physical flow. The corner points have zero vorticity, while the neighboring points have higher wall vorticity values. These jumps near the corner points are especially strong near the top corners.

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10.8 Forced LDC Flow and Sub-Critical Instability Solutions of the Navier–Stokes equation for LDC in [277] show the flow as steady for Re < Recr1 . This critical value of Re for flow becoming unsteady without additional excitation, depends on the choice of grid. In [277], sub-critical instability is reported for (257 × 257) and (513 × 513) grids, with flow becoming unsteady with external imposed excitation caused by a pulsating vortex given by, ωex = Aex sin(2π fo t), applied at a location E0 with the co-ordinates (0.02, 0.98), as shown in Figure 10.12. It is noted that for certain amplitudes of excitation [277] the flow persists as unsteady, even when the excitation is switched off. This is referred to as the case of selfsustained unsteadiness. High accuracy computing methods will cause the flow in LDC to be unsteady, as it is excited either explicitly or implicitly. For a neutrally stable numerical method calibrated with one-dimensional convection equation [413], some trans-critical Reynolds numbers will display steady flow, despite the presence of implicit excitation by numerical errors that turn out to have negligible effect. For such cases, explicit excitation is needed to make the flow unsteady for some sub-critical Re values near Recr1 . Existence of a Recr1 for a flow implies that the existing numerical errors (including the Gibbs’ phenomenon and aliasing error) for a sub-critical Re < Recr1 is not adequate to excite physical instability. The nature of sub-critical instability can be understood by observing Figures 13 and 14 of [277], where it has been shown that unsteadiness is self-sustained only when the explicit excitation is above a certain amplitude level indicated by Aex . This is the threshold value of excitation amplitude, above which it can be established that the instability is self-sustaining. That this threshold amplitude Aex , is a primary variable, and the frequency of excitation ( f0 ) is not, has been established in [507] by choosing three widely different values of f0 , which are in turn significantly different from the natural frequency of excitation for the flow inside LDC.

10.8.1 Threshold amplitude for primary instability Next, the threshold amplitude curve for sub-critical Reynolds numbers for LDC is determined. The computations have been performed using a (1025 × 1025)-grid using the CCD method. As compared to earlier reports for flow inside a LDC by the same numerical method which used a (257 × 257)-grid in [277, 443, 476], this grid is significantly refined. As illustrated in Figure 10.13 the refined grid does not display any aliasing error in the computational domain. Thus, the results that accrue by using this grid can be considered highly accurate. The threshold amplitude is determined by following an approach of finding a bracket of values, above which the unsteadiness is due to implicit numerical error and self-sustaining, while below this interval one requires explicit excitation to create unsteadiness that disappears when the explicit excitation is removed. The width of this bracket successively narrows down with increase in Reynolds number. For a choice of sub-critical Re, a steady solution is obtained by solving the Navier–Stokes equation. This flow is excited by a vortical

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Vorticity

–6.5 0

500

1000

Vorticity

–5.5

1500 Time 2000

Exciter On

2500

3000

3500

Re = 9000, Aex = 0.135 Exciter Off

–6 –6.5 –7 0

500

–5.5 Vorticity

Exciter Off

Exciter On

–6

–7

1500 Time 2000

1000

Exciter On

2500

3500

3000

Re = 9300, Aex = 0.08 Exciter Off

–6 –6.5 –7 0

500

1000

1500 Time 2000 2500 3000 Re = 9500, Aex = 0.058 Exciter On Exciter Off

3500

0

500

1000

1500 Time 2000

3500

–5.5 Vorticity

1025×1025

Re = 8800, Aex = 0.21

–5.5

–6 –6.5 –7 2500

3000

Figure 10.14 Vorticity time series at (0.95, 0.95) is plotted for vortical excitation imposed at (0.02, 0.98) for sub-critical Reynolds numbers. The results are for (1025×1025)grid and the amplitude of excitation is indicated by Aex [507]. [Reproduced from “Grid sensitivity and role of error in computing a lid-driven cavity problem”, V. K. Suman, Siva Viknesh S., Mohit K. Tekriwal, S. Bhaumik, and T. K. Sengupta, Phys. Rev. E, vol. 99, pp 013305 (2019), with the permission of APS Physics.] excitation ωex with the amplitude Aex . The flow field is monitored by sampling the vorticity at the location (0.95, 0.95). Each case is executed till the vorticity shows a clear limit cycle at the sampling point. Thereafter the excitation is switched off. If the vorticity time series at the sampling point continues on the limit cycle, then it is a case of self-sustained excitation. The explicit amplitude of excitation (Aex ) is reduced by repeating the exercise to test for self-sustenance. If the limit cycle continues after

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removal of excitation, then Aex is further reduced. This is continued till the selfsustenance test fails, then that point becomes the threshold amplitude of excitation i.e. Aex,u , for that sub-critical Re value. Proceed with another set of computations with a very small initial amplitude of Aex , such that the self-sustenance test fails; the amplitude of excitation is increased incrementally, till the point where self-sustenance test is successful. This provides the second threshold amplitude, which is called Aex,l , above which all amplitudes will exhibit self-sustained limit cycle oscillation at the sampling point. Thus, for every sub-critical Re near Recr1 , one obtains the bracket given by (Aex,l , Aex,u ). The threshold amplitude is shown in the center frame of Figure 10.15 for sub-critical Re, for a constant frequency of excitation. In Figure 10.15, the sub-critical threshold amplitude is explained for Re = 9400. The top frame is for a value of excitation Aex,u = 0.065, above which self-sustenance is found [507]. For this excitation amplitude, the flow field remains unsteady indefinitely when the excitation is removed, as shown by dark lines in the figure. The bottom frame in the figure shows a case for which the excitation amplitude is slightly lower than Aex,l . In the middle frame, these two curves are shown together, with the upper curve being Aex,u , and lower one Aex,l . For the excitation amplitude Aex = 0.05, the LDC shows unsteadiness with the excitation kept switched on, with a stable limit cycle oscillation achieved after t ≈ 1500. At t = 1650, the excitation is then switched off. It is then seen that the vorticity fluctuations quickly decay to zero in the bottom frame of Figure 10.15. Two features are distinct from the curves shown in Figure 10.15: i) The instability is self-sustaining, at and above the upper threshold amplitude curve, providing the value of Aex,u . The lower curve shows Aex,l , below which the excitation is not self-sustained, and ii) The upper and lower bracket amplitudes are decaying functions of increasing Re. The threshold amplitude decreases with increasing Re due to higher receptivity of the flow at higher Re. For external excitation superposed with implicit numerical disturbances, the flow is even more receptive to become unsteady because of instability at lower imposed excitation amplitude. The effect of grid resolution on Aex,u is noted in Figure 10.16, with values plotted for the three grids shown. One notes that for same Re, the excitation amplitude required is least for the coarsest (257 × 257)-grid, while it is most for the finer (1025 × 1025)grid. This reinforces earlier observation about a finer grid being associated with lesser numerical error in determining the trigger point of instability. Hence Recr1 is higher for finer grid, and the threshold amplitude of imposed excitation is zero, i.e. Recr1 = 9580 for the (1025 × 1025)-grid. While demonstrating the receptivity aspect of the flow, some numerical finesse is required whereby one can gradually increase Recr1 , as has been shown in [486] using the compact scheme and regularization of top lid boundary condition, with Recr1 found to be between 10000 and 10500.

10.8.2 Frequency spectrum of sub-critical excitation So far the issue of the frequency of imposed excitation, in the context of sub-critical Re cases studied above in Figures 10.14 to 10.16, has not really been examined apart from

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Re = 9400, A0 = 0.065 Exciter on

Vorticity

–5.5

1025×1025

Exciter off

–6 –6.5

0

500

1000

0.4

1500 t

2000

2500

3000

Lower bracket

Amplitude, A0

Upper Bracket 0.3 Limit cycle oscillation

0.2 0.1

Trivial solution

0 8400

8600

8800

9000 Re

9200

9400

9600

Re = 9400, A0 = 0.05

Vorticity

–6.01

Exciter on

Exciter off

–6.015 –6.02 –6.025

1550

1600

1650 t

1700

1750

Figure 10.15 Vorticity time series at the sampling point (0.95,0.95), noted for vortical excitation applied at (0.02,0.98) for the sub-critical Re = 9400 at the top. Results are obtained using (1025 × 1025)-grid, and the amplitude of excitation is indicated by A0 along the ordinate [507] in the middle frame. [Reproduced from “Grid sensitivity and role of error in computing a lid-driven cavity problem”, V. K. Suman, Siva Viknesh S., Mohit K. Tekriwal, S. Bhaumik, and T. K. Sengupta, Phys. Rev. E, vol. 99, pp 013305 (2019), with the permission of APS Physics.] stating that the frequency is not a very important parameter. It has been noted in [507] that for very low frequencies with fo ≤ 0.01) the LDC flow is not receptive to pulsating vortical excitation. For fo > 0.02, the flow exhibits a stable limit cycle oscillation (LCO) following the primary and secondary instabilities. The LDC flow’s response and its frequency spectrum, with and without excitation, are shown in Figure 10.17. The displayed flows for Re = 8500 have been excited at three different frequencies ( fo = 0.173, 0.41 and 0.6133) and the time series at the

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0.8

A

257×257 1025×1025 513×513

0.7 0.6

Amplitude, Aex

0.5 0.4

A

0.3

A A A

0.2

A A

0.1 0 7500

8000

8500 Re

9000

AAA A

A

A A A

9500

Figure 10.16 Upper threshold amplitude curve of pulsating vortical excitation needed for the three grids used with (257 × 257)-, (513 × 513)- and (1025 × 1025)-grid points [507]. sampling point are shown in Figure 10.17. The excitation amplitude used is Aex = 1 for all the three cases. Results in the form of time series are shown for the (257×257)-grid and the corresponding frequency spectrum of the vorticity time series are shown in the right frames, showing the exciter-on and exciter-off cases. In the time series, lighter parts are for the results with the exciter switched on, while the darker time series are for when the exciter is switched off. While the switched off time series show the same peak to peak oscillations, the switched on parts have different amplitudes for the three different frequencies. The most important aspect of the response for different excitation frequency in LDC flow is revealed in Figure 10.17 on the right-hand side frames shown for the three frequencies of input excitation, which are different from the fundamental frequency of LDC ( fN = 0.44) for the coarsest grid [277]. It has already been noted that the natural frequency is not going to be different for different grid points. Thus the information gleaned from this figure, will have its utility for all grids, including finer grids excited for sub-critical Reynolds number. For the time series shown in Figure 10.17, corresponding frequency spectra exhibit an interesting feature. For different frequencies of excitation, the spectra for f0 = 0.173 and 0.6133 are different from each other and also different from the no-excitation case reported in [277]. However,

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–6.5 –7 0

103

Excitation Off Amplitude

Vorticity

–6

2000

4000

6000 Time f0 = 0.41

10

1.3133

100 0.5

Amplitude

Vorticity

0.88

1

103

–7

1 Frequency

1.5

0.44

102

0.88

101

1.3133

10

0

10–1 2000

4000

6000 Time f0 = 0.6133

8000

0.5 103 Amplitude

–6 Vorticity

102

8000

–6.5

–6.5 –7 0

Excitation On Excitation Off

0.44

10–1

–6

0

0.173

2000

4000

6000 Time

8000

10

2

0.173

0.44

1 1.5 Frequency 0.6133 0.88 1.3133

101 100

10–1 0.5

1 Frequency

1.5

Figure 10.17 Fast Fourier transform of the vorticity time series for the different excitation frequencies: (a) fo = 0.173, (b) fo = 0.41, (c) fo = 0.6133, for the amplitude of excitation given by Aex = 1 [507]. [Reproduced from “Grid sensitivity and role of error in computing a lid-driven cavity problem”, V. K. Suman, Siva Viknesh S., Mohit K. Tekriwal, S. Bhaumik, and T. K. Sengupta, Phys. Rev. E, vol. 99, pp 013305 (2019), with the permission of APS Physics.] for the frequency of excitation of f0 = 0.41, the spectrum appears similar for lower frequencies. The most interesting aspect is the frequency spectra for all the three excitation cases obtained when the excitation is switched off. All the spectra clearly display the first three spectral peaks, which match with no-excitation cases. Thus, the excitation cases are a powerful means of finding the natural frequency ( fN ) for this type of flow, irrespective of amplitude and frequency of excitation. This has potent application for the cases of sub-critical Re for any grids, preferably using a relatively coarse grid. Figure 10.17 shows three cases of excitation frequencies, and one

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observes the spectrum to be rich with higher harmonics of fN and f0 , which is typical of a nonlinear dynamical system with multiple modes interacting. With increase in exciting frequency, the response field displays larger number of peaks. When the excitation is withdrawn, irrespective of different vortical exciting frequencies, the dynamical system attains stable LCO with fN = 0.44 and its higher harmonics at 2 fN , 3 fN and so on. The same behavior is exhibited in the finer grid using (1025 × 1025) points, with the spectrum recorded for different Re cases in Table 10.5. One can see how the natural frequency and its super-harmonics are obtained starting from the lowest possible subcritical Re for this choice of grid. For example, choosing a coarser (513 × 513)-grid, will allow one to reach lower sub-critical Re values than 8500. In Table-10.5, apart from the frequency, the amplitudes are provided in the parenthesis.

10.9 Multiple Hopf Bifurcations for Different Grids In Figure 10.18, various bifurcation diagrams are shown for the three grids using (257 × 257), (513 × 513) and (1025 × 1025) points. This figure is a compilation of

0.8

Excited - 257×257 Unexcited - 257×257 Excited - 513×513 Unexcited - 513×513 Excited - 1025×1025 Unecited - 1025×1025

Amplitude

0.6

0.4

0.2

0 8000

8500

9000 Re

9500

10000

Figure 10.18 Bifurcation diagrams for the three grids with (257 × 257), (513 × 513) and (1025 × 1025) points, shown for with (solid symbols) and without excitation cases (open symbols) [507].

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Table 10.5

Natural frequencies and amplitudes (in bracket) for the (1025 × 1025)-grid for LDC flow. [Reproduced from “Grid sensitivity and role of error in computing a lid-driven cavity problem”, V. K. Suman, Siva Viknesh S., Mohit K. Tekriwal, S. Bhaumik, and T. K. Sengupta, Phys. Rev. E, vol. 99, pp 013305 (2019), with the permission of APS Physics.]

Re

F1

F2

F3

F4

8500

0.4466 (792.41)

0.8909 (112.76)

1.3298 (6.32)

1.7662 (1.1797)

8600

0.4488 (818.24)

0.8912 (89.22)

1.3399 (7.12)

1.7709 (0.1791)

8700

0.4489 (558.21)

0.8908 (78.66)

1.3397 (9.58)

1.7863 (6.6108)

8800

0.4466 (1012.83)

0.8933 (132.01)

1.3410 (9.84)

1.784 (0.71755)

8900

0.4467 (1045.04)

0.8934 (128.67)

1.3330 (15.81)

1.7836 (2.0636)

9000

0.4466 (1040.57)

0.8867 (144.76)

1.3330 (21.98)

1.7865 (2.9502)

9100

0.4466 (1002.79)

0.8867 (199.46)

1.3333 (24.70)

1.7733 (1.9632)

9130

0.6199 (693.27)

1.2330 (30.23)

1.8533 (5.67)

2.4735 (0.6432)

9180

0.6199 (690.33)

1.2330 (35.98)

1.8533 (5.81)

2.4665 (1.5347)

9200

0.6199 (685.31)

1.2330 (37.01)

1.8534 (6.29)

2.4667 (1.5134)

9300

0.6133 (761.60)

1.2333 (37.66)

1.8466 (5.54)

2.4666 (1.5297)

9400

0.6133 (938.01)

1.2334 (45.07)

1.8467 (10.02)

2.4678 (1.9155)

9500

0.6136 (111.35)

1.2334 (35.13)

1.8467 (14.34)

2.461 (2.183)

9530

0.6133 (1164.22)

1.2266 (39.81)

1.8467 (11.01)

2.46 (0.592)

9580

0.4466 (1290.87)

0.8866 (235.35)

1.3267 (33.41)

1.7667 (6.0165)

9600

0.4467 (1042.69)

0.8867 (207.42)

1.3333 (30.69)

1.7733 (2.5189)

9650

0.4466 (1355.25)

0.7133 (273.16)

0.8867 (214.50)

0.273 (190.895)

9700

0.4415 (1402.38)

0.7166 (318.22)

0.8845 (253.11)

0.9865 (58.42)

9800

0.4412 (1396.35)

0.7134 (637.57)

0.8811 (260.11)

0.9865 (58.42)

9900

0.4401 (1334.84)

0.7133 (774.45)

0.8810 (228.92)

1.1533 (58.8711)

10000

0.4400 (1251.31)

0.7133 (835.603)

0.88 (181.341)

0.5467 (172.383)

all the information, for all the cases reportedly obtained by the high accuracy CCD scheme, with and without any external excitation to show similarities and differences of the presented results, with those shown earlier for the relatively less finer grids

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used in [277]. The main features of these diagrams are as follows: (a) The flow with (257 × 257)-grid with sub-critical excitation shows first bifurcation for Recr1 = 8030, Recr2 ≈ 9400 and Recr3 ≈ 10, 500 (obtained by fitting appropriate quadratic functions, as shown for the circular cylinder case in Figure 10.6); for the relatively finer grid with (513 × 513) points, the bifurcation onset is at Recr1 = 8300 with sub-critical excitation, with a second bifurcation at Recr2 = 9450, and the third bifurcation at Recr3 ≈ 9700. The refined grid with (1025 × 1025) points with the help of sub-critical excitation shows Recr1 = 8500, the second at Recr2 ≈ 9130, and the third one at Recr3 ≈ 9580 (without excitation). (b) The reasons for obtaining different bifurcation sequences are due to basic differences of the equilibrium flows, and the implicit errors (comprising of the Gibbs’ phenomenon, aliasing error (if any), and all other sources of numerical error), which determine flow receptivity. (c) A new branch is noted for the equilibrium amplitudes from Re ≈ 9100 to 9530 with lower amplitudes, which was not completely established in [277] for the unexcited case using a (513×513) grid. With the refined grid using (1025 × 1025) points, this is noted clearly with excitation, as a sub-critical branch. (d) With sub-critical excitation, all the three grids show nearly same amplitudes of the LCO for the range: 8700 ≤ Re ≤ 9100, establishing the important role of subcritical excitation. (e) Finally, for none of the grids does bifurcation start smoothly from vanishingly small equilibrium amplitude (Ae ∼ 0), despite showing sub-critical excitation as an additional means of initiating instability.

10.10 Towards Grid-Independence and Universality The study of LDC flows has seen a resurgence in recent times, mostly using the CCD scheme [277, 278, 443, 476, 477, 507], reporting the discovery of a polygonal core vortex as transient and permanent gyrating vortices around the core. Presence of the polygonal core vortex and specifically triangular shaped vortices have been reported in lab-scale experiments in [24, 77]. Interestingly, polygonal vortical structures have been noted in atmospheric dynamics on a planetary scale, e.g. above the north pole of Saturn in [79], demonstrating that shear-driven confined flows are capable of exhibiting such polygonal structures. Such flow structures in LDC been reported mostly using the CCD scheme, with very few researchers using different methods have reporting any triangular core vortex, as in [41, 481]. The polygonal vortex can be captured by other high accuracy methods too, and one such examples is provided in [507] using Lele’s compact scheme [261] for Re = 10000. As noted before, Lele’s scheme [261] failed to avoid numerical instability using a (257 × 257)-grid, as reported in [413]. As such, Lele’s scheme is given for interior points only, without any boundary closure schemes. When implicit closure schemes of same order are used, then the full scheme displays anti-diffusion at multiple points, as noted by global analysis in [413, 438]. Anti-diffusion causes catastrophic solution blow-up, specifically noted for internal flows. This problem of anti-diffusion leading to numerical instability is noted for compact schemes, and a solution to this has

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been also provided in [413, 438]. Even after this fix, the solution blew up due to excessive aliasing error for flow inside LDC, as reported in [413]. One of the easiest ways of circumventing aliasing error is to increase the number of points, (just as the authors did in [507] when they used a refined grid with (1025 × 1025) points to use Lele’s scheme), with well proven boundary closure schemes. Also to explore gridindependence, another case has been reported using CCD scheme with (2049 × 2049) points in [507]. These additional cases have been run for significantly long times, with wall vorticity compared at t = 1800 in Figure 10.19 for Re = 10000. The plotted wall vorticity in Figure 10.19, for the three test cases display the vorticity variation for the interior points on the top lid to be identical. We have already noted that for LDC, symmetry of the geometry and the ninety degree bend at the corners demands that the vorticity at these corner-points is identically zero, which follows from the definition of vorticity, with the stream function being constant on the wall. The points near the corner on the wall

Re = 10000, t = 1800 0 −500 −1000 NCCD 1025X1025 Lele 1025X1025 NCCD 2049X2049

−1500 −2000

ωβ

−2500 −3000 −3500 −4000 −4500 −5000 0

0.2

0.4

0.6

0.8

1

x

Figure 10.19 Wall vorticity at the top lid of LDC flow compared for Re = 10000 at t = 1800, using (1025 × 1025)- and (2049 × 2049)-grid by using CCD [443] and Lele’s scheme [261].

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will have non-zero wall vorticity. From Eq. (10.40), one notices that ωb is dependent on grid spacing at the wall, and one expects this to be the same for Lele’s and CCD scheme using identical grids. Also, ωb is not symmetric about the mid-plane due to top lid motion, which can be seen in Figure 10.19. One notes that the grid using (2049 × 2049) points will have highest value of ωb at the second, and the second-last points, next to left and right vertical walls. This ωb takes large values with grid refinement due to its impulsive start, for which there are no

t = 100

1

t = 400

1 0.8

0.8

0.6

0.6

0.6

y

y

y

0.8

0.4

0.4

0.4

0.2

0.2

0.2

0 0

0.2

0.4 x 0.6 0.8

1

t = 850

1

0 0

0.2

0 0 1 0.2 0.4 x 0.6 0.8 Re = 10000 − 1025 × 1025 − NCCD

1

0.4 x 0.6 0.8

Vorticity

−5 −6 −7 0

500 t = 1000

1

Time 1000 t = 1400

1

0.8

0.6

0.6

0.6

y

y

y

0.8

0.4

0.4

0.4

0.2

0.2

0.2

0.2

0.4 x 0.6 0.8

1

t = 1750

1

0.8

0 0

2000

1500

0 0

0.2

0.4 x 0.6 0.8

1

0 0

0.2

0.4 x 0.6 0.8

1

Figure 10.20a Vorticity time series and contours showing the dynamics of flow inside LDC at different times for Re=10000 for (a) CCD scheme with (1025 × 1025) points, (b) Lele’s scheme using (1025 × 1025) points and (c) CCD scheme with (2049 × 2049) grid points [482] (Cont.). [Reproduced from “Grid sensitivity and role of error in computing a lid-driven cavity problem”, V. K. Suman, Siva Viknesh S., Mohit K. Tekriwal, S. Bhaumik, and T. K. Sengupta, Phys. Rev. E, vol. 99, pp 013305 (2019), with the permission of APS Physics.]

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theoretical upper limits at t = 0. These large values near the corner points for ωb give rise to Gibbs’ phenomenon becoming stronger and stronger with grid refinement, as is also evident from Figure 10.19 with the ωb value largest for the (2049 × 2049)-grid. Thus grid refinement reduces aliasing error, while at the same time it increases the adverse effects of Gibbs’ phenomenon. The Gibbs’ phenomenon has been described for high accuracy compact schemes in capturing shocks and discontinuity [439]. t = 100

1

t = 400

1 0.8

0.8

0.6

0.6

0.6

y

y

y

0.8

0.4

0.4

0.4

0.2

0.2

0.2

0 0

0.2

0.4 x 0.6 0.8

1

t = 850

1

0 0

0.2

0 0 1 0.2 0.4 x 0.6 0.8 Re = 10000 − 1025 × 1025 − NCCD

1

0.4 x 0.6 0.8

Vorticity

−5 −6 −7 0

500 t = 1000

1

Time 1000 t = 1400

1

0.8

0.6

0.6

0.6

y

y

y

0.8

0.4

0.4

0.4

0.2

0.2

0.2

0 0

0 0

0.4 x 0.6 0.8

1

t = 1750

1

0.8

0.2

2000

1500

0.2

0.4 x 0.6 0.8

1

0 0

0.2

0.4 x 0.6 0.8

1

Figure 10.20b Vorticity time series and contours showing the dynamics of flow inside LDC at different times for Re=10000 for (a) CCD scheme with (1025 × 1025) points, (b) Lele’s scheme using (1025 × 1025) points and (c) CCD scheme with (2049 × 2049) grid points [482] (Cont.). [Reproduced from “Grid sensitivity and role of error in computing a lid-driven cavity problem”, V. K. Suman, Siva Viknesh S., Mohit K. Tekriwal, S. Bhaumik, and T. K. Sengupta, Phys. Rev. E, vol. 99, pp 013305 (2019), with the permission of APS Physics.]

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Figure 10.20c Vorticity time series and contours showing the dynamics of flow inside LDC at different times for Re = 10000 for (a) CCD scheme with (1025 × 1025) points, (b) Lele’s scheme using (1025 × 1025) points and (c) CCD scheme with (2049 × 2049) grid points [482]. [Reproduced from “Grid sensitivity and role of error in computing a lid-driven cavity problem”, V. K. Suman, Siva Viknesh S., Mohit K. Tekriwal, S. Bhaumik, and T. K. Sengupta, Phys. Rev. E, vol. 99, pp 013305 (2019), with the permission of APS Physics.] In Figures 10.20a to 10.20c, vorticity contours are shown for Re = 10000 for the different schemes shown at the indicated times. The vorticity time series at the sampling point are placed at the center of the contour plots shown for typical time instants for each figure. As with coarser grids, one notes that the primary instability leads to formation of a polygonal core vortex, which in this case is a pentagonal vortex for the refined grids at t ≈ 450. This pentagonal vortex transforms into a triangular vortex, when the first limit cycle oscillation suffers the secondary instability, seen here in the frame at t = 850. The contour plots and the time series with the presence of the polygonal core vortex for the

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refined grids, give an indication of grid-independence of the flow. There appear to be minor differences for different times, which can be ascribed as dispersion effects causing apparent time-shift with respect to each other. The fact that the (1025 × 1025)- and the (2049 × 2049)-grids are more than adequate to resolve the flow field can be noted quantitatively. The spectrum of the time series shown in Figures 10.20a to 10.20c are displayed in Figure 10.21, showing a very strong similarity for the CCD and Lele’s scheme using identical grids. As one would expect, that the most refined grid using (2049 × 2049) points would resolve finer scales, which is visually noted in Figure 10.21. However, when one tabulates the five most dominant modes, as given in Table 10.6, one again notices that the tabulated values of the frequencies are almost the same, matching up to the second and third decimal places. The apparent differences in the strength of the peaks that are seen to be higher for the most refined grid is also not unexpected, and are given in the parenthesis in the table. As the time step has been reduced by a factor of five from ∆t = 0.0005 to 0.0001, moving from the (1025 × 1025)-grid to (2049 × 2049)-grid, the sampling rate has been increased by the same proportion, from 0.025 (for the (1025 × 1025)grid) to 0.005 (for the (2049 × 2049)-grid). This is noted in the amplitudes given in Table 10.6; that the ratio of the amplitudes are almost between 4.5 to 4.9 times. The appearance of additional peaks in the most refined grid case get their supply of energy from the leading energy carriers, according to the well-known Parseval’s theorem in operational calculus [324, 339] for the Fourier transform. However, as is typical of any other bluff body flows, universality is defined in terms Strouhal number [469, 477], which is clearly shown in Figure 10.21 and Table 10.6. Interestingly, the Strouhal number in this case is given by the frequency value of 0.440, whose first super-harmonic is noted as the fifth dominant mode. Table 10.6 Leading frequencies and amplitudes for the time series of vorticity recorded at the sampling point obtained using two compact schemes using the displayed grids. [Reproduced from “Grid sensitivity and role of error in computing a lid-driven cavity problem”, V. K. Suman, Siva Viknesh S., Mohit K. Tekriwal, S. Bhaumik, and T. K. Sengupta, Phys. Rev. E, vol. 99, pp 013305 (2019), with the permission of APS Physics.] Frequency

NCCD

Lele

NCCD

(1025 × 1025)

(1025 × 1025)

(2049 × 2049)

F1

0.440 (1619.60)

0.440 (1631.39)

0.440 (1581.75)

F2

0.710 (1023.16)

0.709 (1033.6)

0.715 (848.07)

F3

0.275 (378.13)

0.275 (383.80)

0.270 (343.85)

F4

0.545 (259.09)

0.545 (249.47)

0.545 (256.04)

F5

0.880 (231.16)

0.880 (221.38)

0.880 (245.44)

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Amplitude

10 3 10 2 NCCD Scheme 1025X1025 10 1 10 0 0

1

2 Frequency

3

4

5

4

5

4

5

10 4

Amplitude

10 3 10 2 Lele Scheme 1025X1025

10 1 10 0 0

1

2 Frequency

3

10 4 10 3

Amplitude

10 2

NCCD Scheme 2049X2049

10 1 10 0

0

1

2 Frequency

3

Figure 10.21 Fourier transform of vorticity time series for the LCD flows for Re = 10000 obtained by solving the Navier–Stokes equation using two uniformly spaced grids: (1025 × 1025) and (2049 × 2049) points using CCD and Lele’s schemes. [Reproduced from “Grid sensitivity and role of error in computing a lid-driven cavity problem”, V. K. Suman, Siva Viknesh S., Mohit K. Tekriwal, S. Bhaumik, and T. K. Sengupta, Phys. Rev. E, vol. 99, pp 013305 (2019), with the permission of APS Physics.]

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10.11 Closing Remarks The major observations of the present chapter are summarized here for two apparently dissimilar flows, one external and the other internal, that flow past a cylinder and flow inside the lid-driven cavity (LDC). These two have been termed as oscillator-type flows, as both the flows display presence of a unique time scale, the Strouhal number. While the flow past a circular cylinder has been widely studied from theoretical and experimental perspectives, the emphasis has been to describe the appearance of unsteadiness on this time scale. Interestingly, however most of the theoretical efforts have been spent on developing an amplitude equation. These two flows have been studied with different emphasis, with more focus on flow past the cylinder due to its versatile applications in engineering. In contrast, flow inside the LDC has been pedagogic, with interest in studying instability and bifurcation. The authors in [477], have shown the instability and bifurcation of these two flows using POD to display some universal features. Once the POD modes have been used to represent the instability modes, further developments have taken place and presented in [437, 469, 477] by treating the flow as a dynamical system. This is in contrast to prevalent Stuart–Landau equation approach which seeks to show the limit cycle oscillation at Strouhal number, whose amplitude is independent of the initial condition. Further, the approach presented here, of the dynamical system establishes multimodal dynamics with the help of POD. This has been extended to classify POD modes as regular and anomalous modes of two kinds [469, 477]. The importance of, and a model for, anomalous modes have been presented. This has been used to develop a robust reduced order model for flow past a circular cylinder. The comparative study performed here displays the basic fact that the bifurcation sequences are universal with respect to the dominant frequency, as is usual for vortexdominated flows [469, 477]. Both the flows show the central importance of receptivity to disturbances in fixing the Hopf bifurcation and primary instability onset. The analysis by solving the Navier–Stokes equation is specifically sensitive to different sources of numerical error, while using high accuracy computing methods. This has been described in details for flow in LDC, highlighting the roles of the aliasing error [477] and error introduced due to Gibbs’ phenomenon [439]. For high accuracy methods the preference should be for employing a large number of grid points, which have lower truncation and aliasing errors. The reported computations are for extrarefined grids using (1025 × 1025) and (2049 × 2049) points using parallel computing (which by itself introduces additional error, if special care is not exercised), as reported in [507]. Finer grids experience stronger Gibbs’ phenomenon and more accumulation of round-off error, while alleviating aliasing error. For flow in LDC, one can regularize top lid velocity at the corners to significantly raise Recr1 = 10000 to 10500, as reported in [486], which control both aliasing error and the Gibbs’ phenomenon. For the LDC, the primary instability can be initiated at lower Re values by imposed monochromatic excitation inside the domain. Such excitation does not disappear when the excitation is switched off. The effects of grid sensitivity for different

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refinements is reflected in the requirement of different threshold amplitudes of subcritical excitation for self-sustained unsteadiness. The resulting dynamics is noted to be independent of frequency of excitation for a given grid. The sub-critical excitation also provides the way of finding the natural frequency of the fluid dynamical system, by exciting the system at a frequency with sufficient amplitude and then removing the excitation after the limit cycle oscillation attains a steady state.

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Chapter

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11

Mixed Convection Flow

11.1 Instability of Mixed Convection Flows At this point it would be relevant to examine heat transfer effects for incompressible fluid flows governed by the Navier–Stokes equation, along with the mass conservation equation. Any reckoning of heat transfer, demands consideration of energy conservation, taking into account possible heat transfer in the flow, originating from the boundary and initial conditions. In forced convection, the flow is caused by an external agency, such as by a fan, a pump or by atmospheric wind. In contrast, free convection is caused by buoyancy force that exists only in the fluid. Mixed convection flow is a combination of free or natural convection (without any background flow of the medium) and forced convection [496], and in the present context, the examination about very low speed flow, with heat transfer taking place because of a small temperature gradient. Such heat transfer can be modeled by Boussinesq approximation. This approximation is limited however to small temperature gradients. For the small temperature gradients that are responsible for heat transfer effects from the boundaries, it is to be investigated how the associated buoyancy force comes into play. This can be attempted through the Boussinesq approximation, on the premise that the buoyancy force is the result of change in density. Otherwise, the density is treated as constant, as representative of a reference temperature (call it T 0 ), and the temperature differential as δT = (T − T 0 ). The buoyancy force is then inserted in the momentum equation, in an appropriate direction, given by δρg in the direction of gravity (say, along the y- axis). Introducing the volumetric expansion coefficient as βT = −(dρ/dT )ρ−1 , the buoyancy force is given by, Fb = −ρgβT (T − T 0 ) ˆj. As further explained in the next section, the reason is that there is an added variable to deal with for the mixed convection flows. And, energy equation is intrinsic to all study of flows

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that involve heat transfer. Thus, study of corresponding perturbation fields would require consideration of entropic disturbances. Mixed convection flows are found in many natural and engineering devices, as in geophysical fluid dynamics and various engineering applications that are affected by mixed convection flow instabilities. Studies of these flows in the presence of different disturbance sources are important. It has been shown [412, 420] that vortical excitation of small amplitude can couple to create thermal fluctuations, implying possible flow control in many engineering devices. Another complication arises in the study of mixed convection flows, which is due to the nature of the disturbance field, with an added degree of freedom for heat exchange. However, a typical problematic issue in the study of mixed convection flow instability is the lack of any standard canonical equilibrium flow, as opposed to that for hydrodynamic instability studies, where the fundamental physical processes can be understood with the Blasius’ profile for zero pressure gradient flow, and the Falkner– Skan profile to study the effects of pressure gradient. For mixed convection flows, there is the Schneider’s profile for flow past a horizontal plate, with outer flow taking a constant velocity [402]. The theoretical construct of this profile is such that there is no heat transfer anywhere on the plate except at the leading edge, where the heat transfer occurs singularly, as the wall temperature scales as x−1/2 , and thus the leading edge has infinite temperature. This implies that the equilibrium flow is given by a similarity solution, requiring the wall temperature (T w (x)) to vary as inverse square root of the distance from the leading edge, for the boundary layer edge velocity to remain constant. It will be shown that heat transfer induces an additional pressure gradient, which alters not only the equilibrium flow, but also modifies the instability properties significantly, as compared to isothermal flows. In the context of mixed convection flow instability, the similarity profile given in [402] is important. For edge velocity varying as xm , generalization is possible for this equilibrium flow, as given in [303], and has been reported in [420]. Flow visualizations have shown transition to turbulence for the natural convection problem in [126] through unstable growth of small disturbances. Natural convection flow past inclined plates has been studied, and it has been noted that an array of longitudinal vortices have been created in [497]. Also, experimental studies on instability of natural convection flow past an inclined plate [272, 579] have shown the presence of two types of flow instabilities depending on flow inclination. For angles lower than 14o with respect to the vertical, one notices a wave-like instability, which has been also studied in [86]. For inclination angle greater than 17o , one notices dominance of vortices, and so this is called vortex instability. This is completely different from the vortex-induced instability studied in Chapter 9. This mixed convection flow instability is present for both horizontal and inclined plates [155, 167, 301, 485, 544, 558]. Instability of forced-convection boundary layers over horizontal heated plate is classified in [169], by two prototypical instabilities: the Rayleigh–Benard type, as observed for a closed system heated from below, and the

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Tollmien–Schlichting type, which is observed for isothermal open flows, as in pure hydrodynamics. For mixed convection flows, linear temporal analysis has been conducted in [302] for isothermal vertical flat plate, with equilibrium flow obtained by a local nonsimilar method. For assisting flows, the effect of buoyancy is observed to stabilize the flow. However, description of non-similar flow for instability studies have been critiqued [65]. For mixed convection flow over inclined and horizontal isothermal plates, instabilities have been studied in [86] and in [87], respectively. Reported results indicated the flow along vertical and inclined plates to be more stable, if the buoyancy force aids external convection, while stability decreases for inclination angle approaching the horizontal. For horizontal plates there is a tendency for the flow to become more unstable, with buoyancy force always acting in a direction away from the plate surface. There are some important aspects of instability and receptivity of mixed convection flows. The first aspect is the importance of the boundary layer, as compared to the Navier–Stokes equation noted in some experiments. As will be taken up in the next section, the Navier–Stokes equation is considered, with the non-dimensional parameter involving buoyancy force given by the Boussinesq approximation, appears in terms of Ri = Gr/Re2 , where Gr is the Grashof number and Re is the Reynolds number defined in terms of appropriate length, velocity and temperature scales. The parameter Ri is the Richardson number. The authors in [259, 498], have shown that for boundary layers, another combination of these nondimensional parameters appear in the governing equation given by, K = Gr/Re5/2 . It is noted in experiments reported in [155, 544] that the onset of instability occurs at the same value of K, highlighting the importance of K and not Gr/Re2 . The two similarity profiles given in [303, 402] also depend on K alone. The main aim is to identify a critical value of K = Kcr , above which the heat transfer properties will change qualitatively for mixed convection flow. For example in [88], the authors have noted strong buoyancy effects for K x ≥ 0.05 and K x ≤ −0.03, for aiding and opposing flows past a horizontal plate, where K x = Gr x /Re5/2 x . The second important aspect of concern is to find out the relative importance of spatial and temporal growth of disturbances, specially if the mixed convection flow displays both of these routes to be unstable. Is it possible to invoke the full Navier–Stokes equation using compressible flow formulation (so that the Boussinesq assumption is not needed), in such a scenario? Discussion follows.

11.2 Mixed Convection Flow: Governing Equations To develop the governing equation for the study of flow receptivity and instability, in the following Navier–Stokes equation, with the body force modeled by Boussinesq approximation, and the corresponding boundary layer equation are stated first. The detailed derivation for these governing equations for the velocity and temperature fields can be noted from [152], with the spatial stability equations as given in [538].

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For the sake of demonstration, the vector equations for an external flow are first stated, with the free stream velocity and temperature denoted by U∞ and T ∞ , respectively. Consider laminar flow past a horizontal hot plate, with temperature on the top of the plate being given by T w (x) > T ∞ , with the leading edge of the plate considered fixed as the stagnation point. The governing equation in dimensional form, indicated by the quantities with asterisks, is given in vector notation for mass conservation as ~∗ = 0 ∇∗ · V

(11.1)

The momentum conservation equation along with the Boussinesq approximation for the body force is given by ~∗ 1 DV ~∗ = ~g j βT (T ∗ − T ∞ ) − ∇∗ pˆ ∗ + ν ∇∗2 V ∗ Dt ρ

(11.2)

Here DtD∗ represents the substantial derivative, g j = [0, g, 0]T represents the gravity vector contributing to the body force derived from Boussinesq approximation in the y-momentum equation only, as ˆj is the unit vector in the wall-normal direction for the plate in horizontal position. We note that the pressure term ( pˆ ∗ ) used in the above equation is the gauge pressure (= p∗ − p∞ ) for the following reason. The incompressible flows with constant property can be written as ρ

~∗ DV ~∗ = −ρ∗~g j − ∇∗ p∗ + µ ∇∗2 V Dt∗

The body force is due to gravitational force (taken as acting downwards). The buoyancy term can be simplified if the temperature approaches a uniform value, as one goes further away from the body. Then the convection terms will drop out, for natural convection, so that, 0 = −ρ∞~g j − ∇∗ p∞ . Using this in the above equation, it can be derived that ρ

~∗ DV ~∗ = −(ρ∗ − ρ∞ )~g j − ∇∗ (p∗ − p∞ ) + µ ∇∗2 V Dt∗

where (ρ∗ − ρ∞ ) = βT (T ∗ − T ∞ ) and pˆ ∗ = (p∗ − p∞ ), and so arriving at Eq. (11.2). The energy equation is given in terms of temperature field as DT ∗ ν qˆ = αˆ ∇∗2 T ∗ + Lviscous + Dt∗ Cp ρCv

(11.3)

where αˆ is the thermal diffusivity, ν as the kinematic viscosity, C p and Cv are the specific heat at constant pressure and constant volume, respectively. In the presented analysis, the viscous dissipation (Lviscous ) and heat source terms (q) ˆ will not be considered in the

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energy equation. These are the equations for the evolution of velocity and temperature fields with space and time. The above equations are non-dimensionalized with a length scale (L), a velocity 2 ). The scale (U∞ ), a temperature scale (∆T s = T w (L) − T ∞ ) and a pressure scale (ρU∞ 5 length scale L is such that the Reynolds number based on it is 10 for all the presented results in this chapter. The non-dimensional equations are given by ~ =0 ∇·V

(11.4)

~ DV Gr ˆ 1 2~ = ∇V θ j − ∇ pˆ ∗ + Dt Re Re2

(11.5)

Dθ 1 = ∇2 θ Dt RePr

(11.6)

~ = where V

~∗ V U∞ ,

T ∗ −T ∞ ∆T s . The Grashof, Reynolds and Prandtl numbers, respectively, 3 sL , Re = U∞ν L , Pr = ανˆ . The Grashof number (Gr) is the = gβT ∆T ν2

θ=

are given by, Gr ratio of buoyancy to viscous force in the flow and the Richardson number (Ri) or the Gr Archimedes number is given by, Re 2 , that appears in the governing Navier–Stokes equation, as noted in Eq. (11.5). Positive and negative signs of Ri correspond to assisting and opposing flows. If Gr > Re2 , then one has natural convection. In mixed convection regimes, Ri is of order one, i.e. Gr ≈ Re2 , which imply that natural and forced convection effects are of equal importance. The Prandtl number (Pr) defines the ratio of momentum and heat diffusion, and if only flow of air is considered then this number is 0.71, a value for air as the working medium. In earlier investigations on mixed convection flow instabilities, further simplification to the Navier–Stokes equation was brought about as has been sought by making equivalent boundary layer equations. This can then be used as the equilibrium flow for such studies.

11.3 Mixed Convection Boundary Layer Flows Similar to isothermal flows, where momentum conservation equations have been simplified by postulating the existence of a thin shear layer close to the body, one can simplify mixed convection flows as well. In isothermal flows, this gave rise to a velocity boundary layer, but in the case of mixed convection flows, it becomes an additional necessity to determine a thermal boundary layer, as shown in Figure 11.1 for flow over a heated flat plate. For high Reynolds number mixed convection flows, the transport of mass, momentum and energy is due to occurrence of gradients, and flow quantities vary

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Transition to Turbulence Velocity boundary layer Ue

Te Thermal boundary layer

Tw

Figure 11.1 Velocity and thermal boundary layer for mixed convection flow past a heated flat plate. rapidly in the narrow region inside the boundary layer. This enables one to introduce −1/2 a new length scale in the wall-normal direction (which varies ), with which √ as Re ∗ ∗ one can non-dimensionalize variables as, X = x /L, Y = y Re/L, U = u∗ /U∞ , V = √ T ∗ −T ∞ ∗ ∗ 2 v Re/U∞ , θ = ∆T s and P = pˆ /(ρ∞ U∞ ). Thus, the boundary layer approximation applied to the Navier–Stokes equation provides the following equations for twodimensional incompressible flow in transformed Cartesian coordinates as in [402] ∂U ∂V + =0 ∂X ∂Y

U

∂U ∂P ∂2 U ∂U +V =− + ∂X ∂Y ∂X ∂Y 2

0 = Kθ −

U

(11.7)

(11.8)

∂P ∂Y

(11.9)

∂θ ∂θ 1 ∂2 θ +V = ∂X ∂Y Pr ∂Y 2

(11.10)

Thus the instability of boundary layer is also dependent upon the buoyancy parameter present in Eq. (11.9) and is defined as K=

Gr Re

5 2

1

−5

= g βT [∆T s ] (Lν) 2 U∞2

(11.11)

This is alternately defined in terms of Ri by K = √RiRe . The parameter K = 0 is for a flow over flat plate without heat transfer. Upon introduction of the similarity

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transform, it will be shown that K = 0 (no heat transfer situation) also transforms the boundary layer to the Blasius boundary layer over a flat plate. The condition K > 0, is for assisting flows, for which the flow considered is above a heated flat plate, and similarly, K < 0 refers to opposing flows over a cooled flat plate. These conditions on K switch signs, when studying the flow on the bottom side of the flat plate. The boundary layer equations Eqs. (11.7) to (11.10), have to be solved using the following boundary conditions: i) at the wall (Y = 0 and X > 0): U = V = 0; and ii) at the free stream (Y → ∞): U = 1, p∗ = p∞ , i.e. P = 0 and θ = 0. (ii)

1

0.8

4E-10

–0.06

2E-10

f′

–0.08

0.4 0.2

0

–0.1

–2E-10

–0.12

–4E-10 11

–0.14 0

2

4

6 η

8

10

12

(iii) 1

(iv)

11.2 11.4 11.6 11.8 12 η

0

2

4

6 η

8

10

12

0

2

4

6 η

8

10

12

0

–0.05

0.8

–0.1

0.6 Θ

Θ′

–0.15

0.4

–0.2

–0.25

0.2 0

–0.04 f′′′

0.6

0

0

–0.02

f′′′

(i)

–0.3 0

2

4

6 η

8

10

12

–0.35

Figure 11.2 Variation of equilibrium flow quantities obtained for K = 0, i.e. for Blasius profile. The buoyancy effects are explicitly observed through the parameter K, which appears directly in the Y-momentum equation, accounting for the wall-normal pressure gradient. Its effects may be reflected in the X-momentum equation by integrating the Y-momentum equation with respect to Y, by using the far field boundary condition at Y → ∞ by P =R 0. One can differentiate the expression for ∞ P, with respect to X, to get ∂P/∂X = −K Y θX dY, where the subscript of θ is the partial

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1

f′ 0.4

2E-07

–0.08 –0.1

0

2

4

6 η

8

10

12 (iv)

η

11

12

0

2

4

6 η

8

10

12

0

2

4

6 η

8

10

12

0

Θ′

Θ

–0.15

0.4

–0.2

–0.25

0.2

–0.3 0

2

4

6 η

8

10

–0.35

12 (b)

1

K = 1 × 10–5 (ii) 0 –0.02

0.8

–0.04

4E-10

–0.06

2E-10

f′

f′′′

0.6

–0.08

0.4

–0.1

–4E-10 11 11.2 11.4 11.6 11.8 12 η

–0.14 0

2

4

6 η

8

10

12

(iii) 1

(iv)

0

–2E-10

–0.12

0.2

0

2

4

6 η

8

10

12

0

2

4

6 η

8

10

12

0

–0.05

0.8

–0.1

0.6 Θ

Θ′

–0.15

0.4

–0.2

–0.25

0.2 0

10

–0.1

0.6

0

–2E-07 –4E-07 9

–0.05

0.8

0

0

–0.14

(iii) 1

(i)

–0.06

–0.12

0.2 0

4E-07

f′′′

0.6

–0.04

f′′′

–0.02

0.8

f’’’

(i)

(a) K = 1 × 10–6 (ii) 0

–0.3 0

2

4

6 η

8

10

12

–0.35

Figure 11.3 Variation of equilibrium flow variables for Schneider’s profile with (a) K = 1×10−6 and (b) K = 1 × 10−5 .

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derivative with respect to X. Thus, depending on the sign of K, one can create either a retarded or accelerated streamwise pressure gradient in the boundary layer. It is incorrect to call these as adverse or favorable pressure gradients, as customarily done for flows without heat transfer, where the nomenclature is based on the property of linear stability. For a mixed convection flow, that can only be done after stability analysis. For flow over a heated flat plate (K > 0), with the buoyancy effect imposes a pressure gradient that will accelerate the flow in the streamwise direction. Such dependence of flow field on streamwise coordinates makes the flow non-similar.

11.3.1 Similarity transform of boundary layer flow The boundary layer equation developed for mixed convection flows can be further simplified by invoking a similarity concept as given by Schneider [402]. A similarity solution for the mixed convection boundary layer equation is derived by using the stream function (Ψ) first. This ensures satisfaction of the mass conservation equation. Next, using the expression for ∂P/∂X in the X-momentum and the energy equation, the following may be derived ΨY ΨXY − ΨX ΨYY − K

∞

Z

θX dY = ΨYYY

(11.12)

Y

ΨY θX − ΨX θY =

1 θYY Pr

(11.13)

with subscripts denoting partial derivatives. These equations admit a similarity solution, provided the wall temperature distribution is given by θw ∝ X −1/2 , where ∗ ∞ . This similarity transform will convert the partial differential equations θw = Tw (x∆T)−T s in X and Y for the boundary layer into an ordinary differential equation, with the similarity variable defined by, η = Y X −1/2 [402]. Transforming the dependent variables by, Ψ = X 1/2 f (η) and θ = θw Θ(η), will provide the following governing equations 2 f 000 + f f 00 + KηΘ = 0

(11.14)

2 00 Θ + f Θ0 + f 0 Θ = 0 Pr

(11.15)

Here, a prime indicates derivative with respect to η. In Eq. (11.14), one readily observes that K = 0 makes the equation identical to the governing Blasius equation for zero pressure gradient isothermal flow. Hence, the coupling between the two dependent variables is related to the added pressure gradient effects brought about by heat transfer. These coupled equations are solved subject to the boundary conditions, at η = 0: f = f 0 = 0 and Θ = 1, and as η → ∞: f 0 = 1 and Θ → 0 (from the definition

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of θ in Eqs. (11.5) and (11.6)). Equation (11.15) can be integrated analytically once to obtain the following equation 2 0 Θ + fΘ = 0 (11.16) Pr One notes that the similarity profile explicitly depends on K. There is an implicit dependence of the flow field on Re, which appears through the definitions of Y and η. From Eq. (11.16), along with the boundary condition that f (η = 0) = 0, one observes an adiabatic condition (Θ0 = 0) to exist all over the plate, irrespective of the value of K. Heat transfer occurs for this similarity profile singularly at the leading edge only, due to T w ∝ X −1/2 . This is one of the special feature of the Schneider’s profile that is difficult to replicate in experiments or set computationally to determine the existence of this similarity. The similarity solutions for the equilibrium flow problem are given for f 0 , f 00 , Θ and Θ0 as functions of η for different values of the buoyancy parameter K. These solutions are shown in Figures 11.2 to 11.6, with the velocity profile f 0 , shown in frame (i); the wall-normal second derivative of the streamwise velocity component shown in frame (ii); non-dimensional temperature Θ plotted against η in frame (iii) and the wallnormal derivative of the non-dimensional temperature plotted versus η in frame (iv) of all these figures. In the frame (ii), an inset zooms into the part of the curve where there is the possibility of the ordinate taking on a zero value. In frame (i), the velocity profile monotonically grows following the boundary condition at η = 0 to η = ηmax , where it tends to unity. Similarly in frame (iii), the temperature profile reaches the free stream value, where it is zero by the definition. In Figure 11.2, the velocity and temperature profiles are for the case where K = 0, corresponding to the Blasius profile for the velocity field, which is independent of the temperature profile. All the quantities shown in the frames show a smooth approach to free stream boundary conditions, without any inflection point. In Figures 11.3(a) and (b), the cases for K = 1 × 10−6 and 1 × 10−5 are shown. For these low values of K, the velocity and temperature profiles remain similar to that observed for K = 0, as shown in Figure 11.2. In frames (i) to (iv) of Figure 11.4, two higher values of the buoyancy parameter are compared for K = 1×10−4 and 1×10−3 . While it is not visible to distinguish the velocity and temperature profiles for these cases with the previous ones, in frame (ii) however, one can notice the appearance of inflection points in the zoomed second derivative plot near the edge of the boundary layer. With increase in the value of K in sub-figure (b), the overshoot also increases and the point of inflection moves closer towards the plate. There are no qualitative changes in the temperature profile, as in all the cases the temperature differences are small, which fulfills the requirement for the applicability of Boussinesq approximation. Sub-figures (a) and (b) of Figure 11.5 show the velocity and temperature profiles for K = 3 × 10−3 and 0.01, respectively. In frame (ii), one notices an inflection

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1

K = 1 × 10–4 (ii) 0 –0.02

0.8

3E-10

–0.04

2E-10

–0.06

1E-10 f′′′

f′

f′′′

0.6

–0.08 0.4

0

–1E-10

–0.1

–2E-10

–0.12

0.2

–3E-10 11

11.2 11.4 11.6 11.8 12

η

–0.14 0

0

2

4

6 η

8

10

12

(iii) 1

(iv)

0

2

4

6 η

8

10

12

0

2

4

6 η

8

10

12

0

–0.05 0.8 –0.1 0.6 Θ

Θ′

–0.15

0.4

–0.2

–0.25 0.2

–0.3 –0.35 0

2

4

6 η

8

10

12 (b)

(i)

1

–0.02

0.8

4E-08

–0.06

2E-08

f′

–0.08

0.4

0

–2E-08

–0.1

–4E-08

–0.12

0.2

9

10

η

11

12

–0.14 0

2

4

6 η

8

10

12

(iii) 1

(iv)

0

2

4

6 η

8

10

12

0

2

4

6 η

8

10

12

0

–0.05

0.8

–0.1

0.6 Θ

Θ′

–0.15

0.4

–0.2

–0.25

0.2 0

–0.04 f′′′

0.6

0

K = 1 × 10–3 (ii) 0

f′′′

0

–0.3 0

2

4

6 η

8

10

12

–0.35

Figure 11.4 Variation of equilibrium flow for Schneider’s profile with (a) K = 1 × 10−4 and (b) K = 1 × 10−3 .

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Transition to Turbulence

1

–0.02

0.8

1E-06

–0.06

5E-07

f′

f′′′

0.6

–0.04

0.2 0

2

4

6 η

8

10

–5E-07

–0.12

–1E-06

12

(iii) 1

0 (iv)

10 η

11

12

2

4

6 η

8

10

12

2

4

6 η

8

10

12

0

Θ′

Θ

–0.15

0.4

–0.2

–0.25

0.2

–0.3 0

2

4

6 η

8

10

12

–0.35

0

(b) K = 1 × 10–2 (ii) 0

1

–0.02

0.8

0.0002

–0.04

f′

f′′′

0.6 0.4

0.00015 0.0001 5E-05

–0.06 –0.08

0 –5E-05

–0.1

–0.0001 –0.00015 –0.0002 6

–0.12

0.2

7

–0.14 0

2

4

6 η

8

10

12

(iii) 1

(iv)

8

η

9

10 11 12

0

2

4

6 η

8

10

12

0

2

4

6 η

8

10

12

0

–0.05

0.8

–0.1

0.6

Θ′

Θ

–0.15

0.4

–0.2

–0.25

0.2 0

9

–0.1

0.6

0

8

–0.05

0.8

(i)

0

–0.1 –0.14

0

0

f′′′

–0.08

0.4

f′′′

(i)

(a) K = 1 × 10–3 (ii) 0

–0.3 0

2

4

6 η

8

10

12

–0.35

Figure 11.5 Variation of equilibrium flow for Schneider’s profile with (a) K = 3 × 10−3 and (b) K = 1 × 10−2 .

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Mixed Convection Flow

point near η ≈ 8.6 for the former, while it is found near η ≈ 7.2 for the latter case. The velocity profile smoothly approaches the free stream value for both the cases. Similar features are observed in Figure 11.6 for K = 0.09 and 0.10 in sub-figures (a) and (b), respectively. Both these are for heat transfer cases modeled for buoyancy effects. For these higher heat transfer cases, it can be additionally noted that an overshoot exists for the velocity profile in frame (i), along with more pronounced inflection points for these values of K. Such overshoot was noted in Schneider [402], for higher values of K within the boundary layer. The application of low temperature gradients allows for modeling the heat transfer by Boussinesq approximation (as indicated by the Richardson number). Despite this the second derivative with respect to η of streamwise velocity remains zero at the wall, and at the free stream that follow from the governing equations and the boundary conditions. For cases of low heat transfer from the plate, an inflection point is observed in the interior for K ≥ 10−4 . Furthermore, it can be seen from the figures that the streamwise velocity varies over a lower range of η, as the value of K increases. Thus, the flow and temperature fields and corresponding wall-normal gradients achieve the free stream conditions at lower values of η. This implies an increasing temperature difference between the hot plate and the free stream, with the boundary layer thickness coming down. As a consequence, the inflection point will, progressively, be closer to the plate. Also observed is an overshoot in the velocity profile for higher K values, within the boundary layer reported in [303] for wedge flow, for which the temperature on the wall is held fixed. It is also evident from Eq. (11.9) that heat transfer causes a wall-normal pressure gradient inside the boundary layer, which is in stark contrast to pure hydrodynamic flows without heat transfer. In flows without heat transfer, the pressure is impressed upon the boundary layer by the outer inviscid flow, i.e. the pressure does not change with height inside the boundary layer. This makes mixed convection boundary layers significantly different from boundary layers without any heat transfer. Also, as a consequence, the wall-normal variation of pressure introduces a streamwise pressure gradient in the mixed convection boundary layer that can be estimated exactly. Thus, in the X-momentum equation the pressure gradient term is given by ∞

Z −K Y

∂θ dY ∂X

For the Schneider’s profile, the following holds good for dependent and independent variables, respectively, θ = θw Θ = X −1/2 Θ,

and η = Y X −1/2

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Transition to Turbulence K = 0.09

(ii) 0

0.8

–0.05

0.6

–0.1

f′

f′′′

1

0.01 0.005 f′′′

(a)

(i)

–0.15

0.4

0

–0.005

0.2 0

–0.2 0

2

4

6 η

8

10

12

(iii) 1

–0.25 (iv)

10

12

0

2

4

6 η

8

10

12

0

2

4

6 η

8

10

12

0

Θ

Θ′

–0.15 –0.2

0.4

–0.25

0.2

–0.3 0

2

4

6 η

8

10

12

–0.35

(b) K = 0.10 (ii) 0

(i) 1

–0.05

0.8

0.01

–0.1

0.4

–0.15

0.2

–0.2 0

2

4

6 η

8

10

12

–0.25

0.005 f′′′

f′

f′′′

0.6

0

–0.005 –0.01 4

6

8 η

10

12

0

2

4

6 η

8

10

12

0

2

4

6 η

8

10

12

(iv)

(iii) 1

0 –0.05

0.8

–0.1

0.6

Θ′

Θ

–0.15

0.4

–0.2

–0.25

0.2 0

8 η

–0.1

0.6

0

6

–0.05

0.8

0

–0.01 4

–0.3 0

2

4

6 η

8

10

12

–0.35

Figure 11.6 Variation of equilibrium flow for Schneider’s profile with (a) K = 0.09 and (b) K = 0.1.

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Mixed Convection Flow

Thus, the temperature gradient in X direction is given by  ∂Θ ∂η 1 ∂  −1/2 X Θ(η) = − X −3/2 Θ + X −1/2 ∂X 2 ∂η ∂X

With

∂η ∂X

η = − 2X

The pressure gradient is given with Y held fixed by ∂θ 1 = − 3/2 [Θ + η Θ0 ] ∂X Y 2X Therefore, the induced streamwise pressure gradient in the boundary layer equation is given by ∞

Z

θX dY = −

−K Y

K  2X 3/2

∞

Z η

 (Θ + η Θ0 )X 1/2 dη

Integrating this pressure gradient term, the contribution from the streamwise pressure gradient term becomes ∞

Z

θX dY = −

−K Y

K  ∞ K ηΘ = − ηΘ 2X 2X η

(11.17)

In arriving at this relation, the far field (for η → ∞) boundary condition for the non-dimensional temperature as Θ → 0, has been used above for the streamwise pressure gradient given by, Eq. (11.17). This equation indicates that the buoyancy effects do not induce a streamwise pressure gradient at the wall (η = 0), and at the far field. However, at the intermediate heights inside the boundary layer, there is the η-dependent streamwise pressure gradient present. Considering mixed convection boundary layer developing over a heated plate, the buoyancy parameter satisfies K > 0; therefore the streamwise pressure gradient inside the boundary layer is negative, i.e. the buoyancy effect in such cases will accelerate the flow. Note that premature conclusions should not be made that such heating will delay transition due to induced effects of pressure gradient, as it happens for the case of flows without heat transfer. This aspect is supported with the help of linear instability theory and DNS of receptivity cases for mixed convection flows in this chapter. Noting that (KηΘ) fixes this pressure gradient magnitude for fixed streamwise locations, it is possible to plot this value as a function of η for different K, as shown in Figure 11.7, with the following features that emerge from this figure. It is noted that increasing buoyancy effects by increasing K makes the boundary layer thinner, hence the location of the maximum accelerating pressure gradient inside the boundary layer moves towards the plate, as K increases. The magnitude of the maximum negative pressure gradient is noted to be of the same order, that is, corresponding to the magnitude of K. This maximum modulus of negative pressure gradient, and the location where it occurs (ηacc ) are given in Table 11.1.

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K = 1 × 10–6 K = 1 × 10–4 K = 1 × 10–3 K = 3 × 10–3 K = 1 × 10–2 K = 9 × 10–2 K = 1 × 10–1

0.15

KηΘ(η)

0.1

0.05

0 0

2

4

6 η

8

10

12

Figure 11.7 Magnitude of buoyancy-induced streamwise pressure gradient given by KηΘ plotted as a function of η. It is thus observed above that for boundary layers that do not involve heat transfer, a negative pressure gradient makes the flow less unstable, and also that the impressed pressure gradient acting on the boundary layer is transmitted unaltered from the outer inviscid flow. For the Schneider’s profile under consideration, there is a varying streamwise pressure gradient according to its height inside the boundary layer. Effects of this negative pressure gradient accelerates the flow differently at different heights. This can result in “velocity overshooting” inside the boundary layer. Existence of such velocity overshoot causes the profile to have an inflection point. The location and the extent of overshoot strongly depend on the value of the buoyancy parameter K. The higher the value of K, the greater will be the extent of the velocity overshoot. As noted before, from Figure 11.7 and Table 11.1 it is evident that for higher values of K, the maximum of the streamwise pressure gradient moves toward the plate. Contours of pressure gradient are shown in Figure 11.8, calculated using Eq. (11.17) for values of K = 1 × 10−6 and 1 × 105 in the physical (x, y)-plane (with the nondimensional co-ordinates given by x = x∗ /L and y = y∗ /L). The value of η at which the maximum pressure gradient occurs are joined by a dashed line in each of the sub-

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Mixed Convection Flow

Table 11.1 Buoyancy induced streamwise pressure gradient effect over a hot flat plate for different K . Case

K

1

0

ηacc

Maximum Pressure Gradient Parameter (KηΘ)

-

-

2

1 × 10

−6

2.62199

1.86447 × 10−6

3

1 × 10−5

2.62199

1.86445 × 10−5

4

1 × 10−4

2.62199

1.86418 × 10−4

5

1 × 10−3

2.61799

1.86149 × 10−3

6

3 × 10−3

2.61099

5.56688 × 10−3

7

1 × 10−2

2.58499

1.83618 × 10−2

8

9 × 10−2

2.38399

1.51604 × 10−1

9

1 × 10−1

2.36499

1.67095 × 10−1

figures of Figure 11.8. For K = 1 × 10−6 and K = 1 × 10−5 , and, this value is noted at the identical height of η = 2.62199. In this figure, with increasing value of K the isocontours of the pressure gradients spread outwards over a larger region in the physical plane. In Figure 11.9, the iso-contours of −(KηΘ/2X) are once again plotted in (x, y)-plane for the buoyancy parameter values of K = 1 × 10−4 and 1 × 10−3 in frames (a) and (b), respectively. Between the two frames, one notices that the maximum value changes in the second decimal place for the case of K = 1 × 10−3 after three orders of magnitude increase in the value of K. The trend of iso-contours expanding outwards with the increase of K is noted as before for the lower values of K. In Figure 11.10, the iso-contours of the pressure gradient given by −(KηΘ/2X) are plotted in the (x, y)-plane for higher values of K, as indicated in the frames. The trend of increase in the value of the maximum streamwise pressure gradient is seen to continue, and so are the iso-contours that expand outwards. A tenfold increase in the value of K between the two frames again brings in an increase of the maximum streamwise pressure gradient only in the second decimal place, while the iso-contours expand further outwards from the origin. Finally in Figure 11.11, the iso-contours of the pressure gradient given by −(KηΘ/2X) are shown in the (x, y)-plane for higher values of K increased nine and ten fold of that which is shown in sub-figure (b) of Figure 11.10. The trend of increase in the value of the maximum streamwise pressure gradient continues and its locus is seen at lower distances from the plate. The iso-contours of pressure gradients expand outwards as before. For hydrodynamic flows, i.e. isothermal flows without heat transfer, the existence of an inflection point for the equilibrium flow provides a necessary condition for

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Transition to Turbulence (a) K = 1 × 10–6 0.3 14

–5 E

0.25

0.2

–4 E

-1 2

0

y

–5 E -1

0.15

08

2.6 21 99

E-

2.62 19 9

-0 –1.2 E

–2 .5

–5 E

-5E -10

0

E -5

-0 9 –5 EEE--0088 2 –-1..4

08 E-

0.05

.8 –1

-1 4

0.1

8 -0 -9E-08

8

8 –1.4 E -0 -5E-10

–5E -09

20

40 x

60

80

(b) K = 1 × 10–5

0.3

0.25

–5 E

–5 E

0.2

11

0

y

-1 –5 E

-1 3

0.15 –5 E

-0 9

0.1

-1 .2-5 E-E07 -0

8

2.62 19

0

–5 E

9

7

2.6 21 99 E –1 .2

-0 7

0.05

–1.4 E -0

7

–2 .5 E

–1 .8 E -0

-0 7

-0 7

–5E-08

20

40 x

-5E-09

60

80

Figure 11.8 The contours of −(KηΘ/2X) are plotted in the ( x, y)-plane for (a) K = 1 × 10−6 and (b) K = 1 × 10−5 .

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Mixed Convection Flow (a) K = 1 × 10–4

0.3

0.25

–5 E

–5 E

-1 2

11

-1 0 –5 E

y

0.2

09

–5 E

0.15

–5 E -0

0.1

0.05

6 2 1 -5 E -1–5 9 9 0E -1 1

E –5

0

-0

7

–1E-06

6 –1 .2 E -0

8

2.6 21 99

–3 E

-0 6 2.62 19 9

–5E -0 6

–2 E

-0 6

–1 .2 E -0

6

–5E -07 -1E-0 6

20

40 x

60

80

(b) K = 1 × 10–3 0.3

0.25

0.2 y

–5 E

-0 9

0.15 –5

E11

07 -5 E 2.61799

0.1 –1 .2 E -0

5

2.6 17 99

6-3E 0 . -5 E 1 7 9--06 9 0 0 -0 8 01

0.05

0-3E-0 5

–2 E

-0 5

2.61 79 9

20

–1 .2 E-05 -5E-07

E –1 .5

40 x

-0 5

60

80

Figure 11.9 The iso-contours of −(KηΘ/2X) are plotted in the ( x, y)-plane for (a) K = 1×10−4 and (b) K = 1 × 10−3 .

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Transition to Turbulence (a)

K = 3 × 10–3 0.3

–1E

0.25

-1 0

–5 E

09

0.2 y

08

0.15

–1

–5 E

09 E–5

10 E-

–5 E

07

–5 E -0

–2 E-05

0.1

-5 E -5 E -0 7-0 8 -5E -09

–5E -05

–4 E -0 5–6E -05 –8E-0 5

06 E–0 .0 00 1 -5-0 5 E -2 2 0 0 99 .0-0.0003 –0 2 .6 1 0

5 – 6 E -0 – 4E -0 5

20

– 4 E -0 5

0.05

0

6

2.61 09 9

5 –5 E -0 –2E -05 -5E-07

–5E-06

40

60

x

80

(b) K = 1 × 10–2

0.3

0.25

–5–E

–5 E

0.2 y

–5 E

0.1

0 0 1 -5-E5 E 5 - 0-91 0

–5

-0 .0

0.05

0

-0 .

2

00 919 .5 8 4

-0.000 2

08 E–5 -0 6 7 0 –5 E E-

1 .0 0 0 -0 5 – 0 0 2 –5 E 0.00 – 025 –0 .0 0 –0. 00 05

–5E -0

7

08

–5E -0

6

–5 E -0 5 –0.0 001

–0 .00 01 5

5 002

2.5 84 99

2.58 49 9 2 000 –0 .

– 0 .0 00 01 -0 .0 00 15 -0. -5E-0 5

20

8

-0 9

07

–5 E

10 E–5 -0 9 E –5

0.15

-0 –5 E

-1 0

–0.

00

015 – 0 .0 0 0

1

–5E -05

40 x

-5E-06

60

80

Figure 11.10 The iso-contours of −(KηΘ/2X) are plotted in the ( x, y)-plane for (a) K = 3 × 10−3 and (b) K = 1 × 10−2 .

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Mixed Convection Flow (a) K = 9 × 10–2 –5E

0.3

–5 E

0.25 –5

–5 E

y

09 E–5 8 -0 E –5

0.15

E –5

09

-0 8 –5 E

0 E -1

0.2

-1

-0 7 –5 E -0

6

–5 E -0 5

0 -1

07 E–5

–5E

-0 6

–5 E -0

5 –0 .00 05

0.1 -5 E -05 8E 0

9

–-0.001 00 5 –0.0–0 .0 0 1 –0 .00 25

06

2 .0 0

99

0054.0 0 3 .0 0.0 .002 5 --0 -0 -0 -0.000 5

.0 –0

0-0.00 2

–0 .0 0

5 –0 .0 0 1 –0 .00 1

1

-5 E -

2 .3 8 3

2.3 83 99

–0

05 E-5 53 2 1 00 0 ..000.0 --00-

0.05

–0.0 015

20

–0.000 5 -5E-05

40 x

1

60

80

(b) K = 1 × 10–1

0.3

–5 E

0.25 –5E

0.2 y

E –5

0.15

E –5

E –5

0.1

-0

-1 0 -0-.0 0-0.00-02 -5-5 .00.500 8EE- -0 1 0 0-659E - – 2 08 5E

0.05

00

–0 15

.0 0

05

08 9 -0 –5 E -

–5

7

–5 E

–0 .00 3

07

-0 8

0 –5 E-

– 5 E-0

6

5

–0.0 005 –0 .0 00 5

–0 .0 00 8 2 –0.0015 –0 .0 01 –0 .00 2 –0 .0 0 25 2.36 49 9

20

–5 E

6 E -0

-0 5

4 05 .0 0 .0 0 –0 – 02 .0 .0 03 012-0 008 -0.0-0.0 015 -0.0-0 0-0.0 025

. -0

–5 E

0 -1

-09

-1 0

– 0 .0 0 -5E-05

–0.0008 –0.001 2 2.3 64 99

15

2 –0.0 0 1 –0 .00 08

–0.000 5

–0.0 005

40 x

60

–5E-05

80

Figure 11.11 The iso-contours of −(KηΘ/2X) are plotted in the ( x, y)-plane for (a) K = 9 × 10−2 and (b) K = 1 × 10−1 .

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Transition to Turbulence

inviscid temporal instability, which is given by the Rayleigh–Fj∅rtoft theorem [119, 412], which is a consequence of the Rayleigh equation given by Eq. (4.10). It is therefore natural to ask if any equilibrium flow with inflection point shows an inherent tendency of inviscid temporal instability? More specifically, one may ask whether the Rayleigh equation remains valid for mixed convection flow? This has been unambiguously answered in [423]; that this is not necessarily the case, and instead, there is a need to develop the governing inviscid instability equation from the first principle, without resorting to the Rayleigh equation. It has been established for the problem of hydrodynamic instability that the STWF is the precursor for transition to turbulence, with disturbance growing spatio-temporally along a nonmodal path. However, the existence of inviscid temporal growth of eigenmodes is a stronger route to transition, and there is a strong motivation for deriving a new equation in [423], which is much more general and Rayleigh’s equation for pure hydrodynamics without heat transfer is a special case. Thus, one need not presuppose any disturbance growth route, whether it is modal or nonmodal, or it is temporal or spatio-temporal.

14

12

ηip

10

8

6

4

2

10–6

10–5

10–4

K

10–3

10–2

10–1

100

Figure 11.12 Location of inflection point (ηip ) plotted as a function of K . In Figures 11.7 to 11.11, the effect of heat transfer (as modeled by Boussinesq approximation) is to induce a height-dependent streamwise pressure gradient. In consequence, for relatively larger values of K for the Schneider’s similarity profile,

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in the interior of the boundary layer, a point of inflection can occur. For higher K, one may also encounter the presence of velocity overshoot for the mixed convection boundary layer. In Figure 11.12, variation of the height at which the inflection point exists is shown for different values of K. This figure shows that the inflection point approaches the wall, as the value of K increases. It suggests that the mean flow becomes progressively unstable for the mixed convection boundary layer with heat added from the wall, as the present inflection point is nearer to the wall [535]. This heuristic approach was abandoned in [423] to quantitatively assess the effects of heat transfer from the first principle, without using Rayleigh equation.

11.4 Equilibrium Solution for Isothermal Wedge Flows The previous section establishes the fact that even if boundary layers in mixed convection cases are cast as ordinary differential equations, in terms of a similarity transformation by Schneider’s formulation [402], as given by Eqs. (11.14) and (11.16), the flow does not truly represent a similarity profile. This is due to the fact that heat transfer causes a pressure gradient to build inside the boundary layer, which is both a function of X and Y. This is a unique property of mixed convection flows, quite unlike the case for pure hydrodynamics, for which the Blasius profile provides the canonical state, with respect to which all receptivity and instability analysis can be performed by local analysis. The additional problem with Schneider’s solution is that the wall temperature is given by θw ∝ X −1/2 , implying a singularity at the leading edge. Additionally, applying the boundary conditions in Eq. (11.16) it is observed that there is no heat transfer over the complete wall (η = 0), except at the leading edge. Thus, the flow is essentially adiabatic, with singular heat transfer from the leading edge of the plate. This is an artifact of the transform used in [402]. In [303], the authors have introduced a new formulation, which is a generalization of the formulation in [402], for isothermal wall condition of mixed convection flow past a wedge. This generalization avoids displaying any singular heat transfer at the leading edge, as was the case with Schnieder’s solution. One attempts to transform the governing partial differential equation again to an ordinary differential equation for which the external flow is given as Ue ' U∞ X m¯ , along with the independent ¯ variable given by, η = Y X (m−1)/2 . Additionally, the wall temperature is distributed (5m−1)/2 ¯ as θw = X . One immediately notices that the choice of m ¯ = 1/5 will create a wall-temperature distribution independent of X, which can be shown to correspond to a wedge with an included angle of 60◦ [423]. As the driving inviscid flow for the boundary layer in [303] continues to be a function of X, the resultant velocity profiles will not represent a truly similarity solution. In [303], for a general equilibrium flow a buoyancy parameter is introduced as G0 , which was previously denoted as K, following the notations used in [402]. Henceforth,

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Transition to Turbulence

the following will be used for the cases indicating isothermal wall condition for the general wedge: 1 Gr −5 K¯ = = g βT [T w (x∗ ) − T ∞ ] (x∗ ν) 2 Ue 2 5 Re 2

It has been already noted that K¯ = 0 corresponds to flows given by Blasius profile with no heat transfer.

11.4.1 Boundary layer equation for flow over isothermal wall The two-dimensional mixed convection boundary layer is obtained from the steady momentum and energy equations for incompressible mixed convection flow with ∗ −T ∞ . The Boussinesq approximation, as given in Eqs. (11.7) to (11.10) with θ = TwT(L)−T ∞ leading term on the right-hand side of Eq. (11.8) arises due to the pressure gradient in the streamwise direction in the boundary layer given by the edge velocity Ue (X). The terms on the right-hand side of Eq. (11.9) show that the wall-normal pressure gradient arises due to buoyancy-induced heat transfer. It has been already observed that the sign of K determines whether the flow inside the boundary layer is accelerated or decelerated. Equation (11.9) shows the flow R ∞ to experience wall-normal dependence of streamwise pressure gradient [412] by K Y θX dY. For flows with K < 0, one notices induced accelerated velocity profiles. Similar conditions hold for isothermal wedge ¯ flows, with different signs of K. For the wedge flow, Eqs. (11.7) to (11.10) are solved subject to the following boundary conditions for the schematic shown in Figure 11.13: The wall condition is given by (Y = 0 and X > 0) : U = V = 0, At the free stream (Y → ∞) : U = Ue (X) and θ = 0. One proceeds further by transforming the partial differential equations to ordinary ¯ differential equations using the independent variable, η = Y X (m−1)/2 . The equilibrium solution is obtained following [303, 402], with the wall temperature already given by, ¯ θw = X (5m−1)/2 . New dependent variables for velocity, temperature and pressure inside the boundary layer are given as ¯ ¯ U = Ue f 0 (η), θ = X (5m−1)/2 Θ(η), P = X 2m¯ P(η)

(11.18)

Presence of stream function ( f (η)), ensures that Eq. (11.7) is automatically satisfied. Equations (11.8) to (11.10) are transformed to ordinary differential equations given in terms of the introduced non-dimensional dependent variables as 1 1 f 000 = m( ¯ f 02 − 1) − (m ¯ + 1) f f 00 + 2m ¯ K¯ P¯ + (m ¯ − 1)K¯ η P¯0 2 2

(11.19)

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Mixed Convection Flow

P¯ 0 = Θ

Θ00 =

(11.20)

1 Pr[(5m ¯ − 1)Θ f 0 − (m ¯ + 1) f Θ0 ] 2

(11.21)

which are solved subject to the boundary conditions: For η = 0 : f = f 0 = 0 and Θ = 1 And as η → ∞ : f 0 = 1 and P¯ = Θ = 0 These equations are for mixed convection boundary layer flow over a wedge, with the wedge angle: 2¯γ = 2mπ/( ¯ m ¯ + 1). Derivation of these equations and boundary conditions are given in Appendix A of [420], and not reproduced here. This reduces to Schneider’s case [402] by taking m ¯ = 0, to study mixed convection flow over a horizontal plate, with the attendant singularity of heat transfer at the leading edge once again given by the wall temperature θw ∼ X −1/2 . y

_

_

Ue = U∞ (cos γ + x m )

U∞, θ∞ _ γ

Momentum boundary layer thickness

SBS Strip

_

x inflow

Symmetry condn.

Far-field boundary

θw ~ x (5m–1)/ 2 x1

x2

xm

wall

y = ymax Thermal boundary layer thickness

δ

δΤ xoutflow

x

Figure 11.13 Schematic diagram for mixed convection flow over a wedge. Identical computational domains are used for computing equilibrium flow and its receptivity to wall–excitation. [Reproduced from “Direct numerical simulation of transitional mixed convection flows: Viscous and inviscid instability mechanism”, Tapan K. Sengupta, Swagata Bhaumik and Rikhi Bose, Physics of Fluids, 25, 094102 (2013), with the permission of AIP Publishing.] Finally, one notes that the wall boundary condition given in Eq. (11.18) with m ¯ = 0.2, makes θw independent of X, i.e. one has a mixed convection flow over an isothermal wedge with the half-wedge angle (¯γ) of 30◦ , as shown in Figure 11.13. For m ¯ = 0, one obtains the mixed convection flow over a flat plate, which upon integrating Eq. (11.21), one gets, Θ0 = − Pr 2 f Θ. With the wall boundary condition, gives one notes ¯ as f = 0 for η = 0. that Θ0w = 0 for any X and K,

11.4.2 Governing equations and boundary conditions for DNS of mixed convection flows Just as in the studies of pure hydrodynamic flow receptivity, an identical route is followed here to investigate nonlinear receptivity of mixed convection flows. This

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starts with first obtaining the equilibrium flow, whose receptivity or instability has to be studied. It is shown for two-dimensional problems by solving the Navier–Stokes equation in (ψ, ω)-formulation, along with Boussinesq approximation for modeling heat transfer. The physical plane is defined by non-dimensional Cartesian co-ordinates (x, y), for which the same reference length (L) that was used earlier to obtain X√in Eqs. (11.7) to (11.10). But, Y in the boundary layer equations is related to y by Y = y Re. As stated before, both the equilibrium and the disturbance fields are obtained by direct simulation of the Navier–Stokes and the energy equations in the transformed (ξ1 , η1 )plane. The schematic of the problem is shown in Figure 11.13. In the transformed plane, vorticity transport, stream-function and energy equations are obtained as [423]: ∂ω ∂ω 1  ∂  h2 ∂ω  ∂  h1 ∂ω  ∂ω + h2 u + h1 v = + h1 h2 ∂t ∂ξ1 ∂η1 ReL ∂ξ1 h1 ∂ξ1 ∂η1 h2 ∂η1 p ∂ (h2 θ) (11.22) +K ReL ∂ξ1 ∂ h2 ∂ψ ∂ h1 ∂ψ ( )+ ( ) = −h1 h2 ω ∂ξ1 h1 ∂ξ1 ∂η1 h2 ∂η1

h1 h2

∂θ ∂θ 1  ∂  h2 ∂θ  ∂θ ∂  h1 ∂θ  + h1 v = + h2 u + ∂t ∂ξ1 ∂η1 ReL Pr ∂ξ1 h1 ∂ξ1 ∂η1 h2 ∂η1

(11.23)

(11.24)

The contra-variant components of the velocity in the transformed plane are given as u=

1 ∂ψ 1 ∂ψ and v = − h2 ∂η1 h1 ∂ξ1 1

with the scale factors of orthogonal transformation are defined as, h1 = (xξ21 + y2ξ1 ) 2 and 1

h2 = (xη21 + y2η1 ) 2 . In the transformed plane, ξ1 is in the direction of the wall and η1 is in the wall-normal direction. Thus, if the uni-directional transformations are given as, x = x(ξ1 ) and y = y(η1 ), then the scale factors are simplified as, h1 = xξ1 and h2 = yη1 . The presented results have been obtained with the parameters, ReL and Pr (for air) as 105 and 0.71, respectively [423].

11.4.3 Boundary and initial conditions The receptivity of mixed convection flows to wall excitation is studied next by solving Eqs. (11.22) to (11.24) sequentially. First, an equilibrium flow is obtained by solving these equations without any excitation, and thereafter the response due to imposed disturbances is followed in receptivity studies. As practiced in isothermal flows without heat transfer, here also the dependent variables are combinations of the equilibrium flow variables (indicated by overbars) and the disturbance quantities (which will be indicated by the subscript d). Thus, the

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vorticity and temperature are given by ω = ω ¯ + ωd and θ = θ¯ + θd respectively. For the computation of equilibrium flow, free stream conditions are prescribed at the top of the domain for all ξ1 as ∂ψ¯ = Ue h2 , ω ¯ = 0 and θ¯ = 0 ∂η1

(11.25)

with the boundary layer edge velocity given by Ue = U∞ , ahead of the wedge (for x ≤ 0) and after the leading edge, Ue = U∞ (cos γ¯ + xm¯ ) for x > 0, with γ¯ , as the semiwedge angle. As noted in [423], the “no-slip and impervious conditions are used at the wall as the boundary conditions, which provides ψ and ω for the equilibrium flow. Thus, for x ≥ 0, the boundary conditions at y = 0 are” ψ¯ w = ψ¯O = constant

ω ¯w = −

1 ∂2 ψ¯ h22 ∂η1 2

¯ θ¯w = x(5m−1)/2

(11.26)

(11.27)

(11.28)

Ahead of the wedge (x < 0), Eq. (11.26) is modified to ψ¯ = ψ¯O + U∞ x sin γ¯

(11.29)

The wall vorticity is obtained from kinematic definition relating ψ with ω, by noting the stream function as a function of η1 , and the no-slip condition is used to evaluate the derivative in Eq. (11.27). At the outflow, the boundary conditions for equilibrium and disturbance fields are: ∂v =0 ∂x

(11.30)

∂ω ∂ω + UO =0 ∂t ∂x

(11.31)

∂θ ∂θ + UO =0 ∂t ∂x

(11.32)

The last two outflow conditions for vorticity and temperature are due to Sommerfeld [413], and is commonly used in computational transition and turbulence studies, as can be noted in [133, 419, 422]. This enables disturbances to convect out of the

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computational domain smoothly with the velocity, UO , which is usually greater, or at least equal to the free stream speed [344, 419, 474, 508]. The computation of equilibrium flow is initialized in the domain with a potential flow solution, which for the simple domain provides the initial conditions given in Eq. (11.25). The computational domain used in [423] is given as, −0.05 ≤ x ≤ 80; 0 ≤ y ≤ 2. In the wall-normal direction, a stretched tangent hyperbolic grid is used with ymax = 2 and βy = 2. This will cluster the grid lines close to the wall with a total of 501 points taken in this direction. The jth line above the wedge is obtained from the following equation as  tanh[βy (1 − η1 j )]  (11.33) y j = ymax 1 − tanh βy Apart from clustering the grid lines, the tangent hyperbolic stretching also helps in controlling aliasing error, as explained in [411]. In the streamwise direction, the grid points are distributed in two segments in [423]. In the first part from xin = −0.05 to xm = 10, a stretched grid is created using a tangent hyperbolic function, such that the ith point can be expressed as  tanh[β x (1 − ξ1i )]  xi = xin + (xm − xin ) 1 − tanh β x In the second part from xm = 10 to xout = 80 along the streamwise direction, the grid points are distributed uniformly. A total of 4501 points has been used altogether, in the streamwise direction. The grid resolution at the wall is given by, ∆yw = 4.36 × 10−4 . The grid spacing in the x-direction is such that at the exciter location it is given by, ∆xexciter = 0.002, that is adequate for the used compact schemes used to discretize the convection terms. The Laplacian operator (∇2 ) in the stream function equation in Eq. (11.23) is discretized by second order central differencing and the equation is solved by a BiCGSTAB algorithm [572]. To discretize various terms in Eqs. (11.22) to (11.24), the following well proven methods have been used. The convective acceleration terms in the vorticity transport equation, and the gradient transport terms in the energy equation are discretized using the OUCS3 scheme [438] and optimized threestage, Runge–Kutta method (ORK3) [450] is used for time advancement in [423]. A thin buffer zone has been used near the outflow at (79.5 ≤ x ≤ 80), where a onedimensional, second order filter is used in both the x- and y-directions, to prevent reflections from the outflow boundary [40, 413]. Use of a buffer domain for wave propagation is quite prevalent and such explicit domains have been used and reported

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by many researchers in [59, 269, 429], particularly for solving the Navier–Stokes equation. In Figure 11.14, solutions of Eqs. (11.19) to (11.21) are shown for K = −0.01 and −0.001, for the velocity profile ( f 0 ), second derivative of the velocity profile ( f 000 ), temperature (Θ) and its gradient (Θ0 ). For these cold plate cases, solution for the Navier–Stokes equation are also obtained by solving Eqs. (11.22) to (11.24), with appropriate boundary and initial conditions. Results from these DNS are compared with the Schneider’s similarity solution in Figure 11.15, for different x stations. Results of the Navier–Stokes equation show a decent match with Schneider’s profile for velocity profiles for both the buoyancy parameters. Temperature profiles show mismatch due to the requirement of the Schneider’s formulation which requires

1

(i)

(ii)

0

0.8

−2

f¢

K = −10 −3 K = −10

0.4

f¢¢¢

–0.05

0.6

−2

K = −10 −3 K = −10

–0.1

0.2 0 0 1

1

2 η 3

4

5

6

0

(iii)

0

−2

K = −10 −3 K = −10

0.8

1

2

3η 4

5

7

8

(iv)

–0.1

θ

θ¢

0.6 0.4

–0.2

−2

K = −10 −3 K = −10

0.2 0

6

0

1

2

3 η 4

5

6

7

–0.3 0

2

4 η

6

8

10

Figure 11.14 Streamwise velocity and temperature simulated for flow past an adiabatically cooled plate for K = −0.001 and −0.01, and plotted as a function of the boundary layer similarity variable η following Schneider’s profile [402]. [Reproduced from “Direct numerical simulation of transitional mixed convection flows: Viscous and inviscid instability mechanism”, Tapan K. Sengupta, Swagata Bhaumik and Rikhi Bose, Physics of Fluids, 25, 094102 (2013), with the permission of AIP Publishing.]

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(a) K = −10–3; t = 220 1

f′; θ

0.8 Schneider’s solution x = 9.27 x = 29.47 x = 49.65 x = 79.89

f’

θ

0.6 0.4 0.2 0

0

1

2

3

4

η

5

6

7

8

(b) K = −10–2; t = 250 1 0.8

θ

f′; θ

0.6

Schneider’s solution x = 9.27 x = 29.47 x = 49.65 x = 79.89

f’

0.4 0.2 0

0

1

2

3

η

4

5

6

7

8

Figure 11.15 Streamwise velocity and temperature simulated for flow past an adiabatically cooled plate for K = −0.001 and −0.01 by DNS are compared with Schneider’s similarity solution as a function of the similarity variable η for the indicated streamwise stations. [Reproduced from “Direct numerical simulation of transitional mixed convection flows: Viscous and inviscid instability mechanism”, Tapan K. Sengupta, Swagata Bhaumik and Rikhi Bose, Physics of Fluids, 25, 094102 (2013), with the permission of AIP Publishing.] it to be singular at the plate leading edge (x = 0) for wall temperature [402]. This mismatch is seen for all the streamwise stations seen in the figure, whose origin is, of course, at the leading edge. However, hot plate cases have shown a better match between DNS and Schneider’s solution in [412]. The situation for cold plate cases worsen for higher degree of cooling

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cases, so much so that the DNS does not even produce the equilibrium solution. One such case is considered next, for which the Schneider’s solution for adiabatic cold plate is shown in Figure 11.16 for K = −0.05 for the solution of Eqs. (11.19) to (11.21). In frame (ii) of Figure 11.16, one notes an inflection point inside the mixed convection boundary layer near η ≈ 2.0. Drawing an analogy with flow without

(i)

(ii)

1 0

0.8

3E-06

–0.02

2E-06 1E-06

f

f′

f′′′

0.6 Adiabatic wall condition _ with K = –0.05

0.4

–0.04

0.2 0 0

1

0

-1E-06 0

0.01 η 0.02

0.03

–0.06

2

4

η

6

8

10

0

(iii)

2

4

η

6

8

10

6

8

10

(iv) 0

0.8 –0.1

Θ

Θ′

0.6 0.4

–0.2 0.2 0 0

2

4

η

6

8

10

–0.3 0

2

4

η

Figure 11.16 Streamwise velocity and temperature simulated for flow past an adiabatically cooled plate for K = −0.05 plotted as a function of the boundary layer similarity variable η following Schneider’s profile [402]. [Reproduced from “Direct numerical simulation of transitional mixed convection flows: Viscous and inviscid instability mechanism”, Tapan K. Sengupta, Swagata Bhaumik and Rikhi Bose, Physics of Fluids, 25, 094102 (2013), with the permission of AIP Publishing.]

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heat transfer in [412], the existence of an inflection point is a necessary condition for inviscid temporal instability following Rayleigh’s theorem [119, 412] for flow without heat transfer. The velocity and temperature profiles show monotonic growth of the plotted quantities from the wall to the free stream values. Another case of heated isothermal plate results is shown in Figure 11.17 for K¯ = 0.04, obtained by solving Eqs. (11.19) to (11.21). One notes an inflection point distinctly in the inviscid part of the mixed convection boundary layer flow (η ≈ 6.3), as given by the formulation of Denier and Mureithi [303]. The cases shown in Figures 11.16 and 11.17 are for high cooling and high heating of a flat plate and an isothermal wedge, respectively. While the formulation given in [402] for the cooled flat plate indicates existence of an equilibrium flow; high heating of isothermal wedge will similarly provide an equilibrium state. However, these are not similarity solutions, despite the fact that the governing equations are ordinary differential equations. It is necessary to investigate whether one can obtain such solutions from the Navier–Stokes equation itself by DNS. The equilibrium solution of Eqs. (11.22) to (11.24) are shown in Figure 11.18, for the indicated times corresponding to the case shown in Figure 11.16 for the case of K = −0.05. A negative K implies, by definition, that ∆T s = (T w (L) − T ∞ ) is also negative. In Figure 11.18, vorticity contours are shown up to t = 90 obtained by DNS, where one notices growing fluctuations originating from the vicinity of the leading edge. Such disturbances occupy longer streamwise stretches with time. It is natural to conjecture that disturbance growth corresponds to absolute instability. Also, such disturbances proliferate along the edge of the boundary layer. At later times, fluctuations are noted to build up progressively on the plate and inside the boundary layer simultaneously. Despite the inflection point being deep inside the boundary layer (as noted in frame (ii) of Figure 11.16 near η ≈ 2.0), the fluctuations appear near the edge of boundary layer. Such behavior is reminiscent of disturbance energy growth mechanism explained in [431, 474] for bypass transition via vortex-induced instability in Chapter 9. In this bypass transition scenario, disturbance originate from the leading edge of the semi-infinite flat plate boundary layer without any heat transfer. For t = 90, one notices significant fluctuations inside the boundary layer, induced by outer disturbances, including some oscillations on wall vorticity as well. These two sets of vortical structures further interact with each other via Biot-Savart interactions and are the source of selfperpetuating unsteadiness that prevents the equilibrium flow to be calculated by DNS. It is apparent that the signature of temporal instability is clearly visible in all the frames of the figure, which has been explained satisfactorily in [423], and is further explained after investigating linear instability of the velocity profile obtained by the formulation in [402]. After obtaining the boundary layer solution for the isothermal hot wedge case of Figure 11.17 with K¯ = 0.04; the formulation in [303] is investigated by solving the Navier–Stokes equation and the vorticity contours are shown in Figure 11.19. This case was also not reproduced in [423]. For this case, one notices unsteady fluctuations near the outer edges of the momentum boundary layer in Figure 11.19. However,

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(i)

(ii)

1

0

0.8

–0.05 4E-05

0.6

–0.1

0.4 0.2

1

–0.15

Isothermal_heated wedge case with K = 0.04

0 0

2

η

4

6

00

ff′′′″′

f′

f ′′′

2E-05 2E-05 1E-05

-1E-05 –2E-05 -2E-05

–0.2

8

–0.25 0

(iii)

5.5

5

6

η

6

6.5 7 η 8

7

7.5 9

8 10

4

6

8

4

6

8

(iv)

0

0.8

2

–4E-05 5

–0.1

Θ

Θ′

0.6 –0.2

0.4 –0.3

0.2 0 0

2

η

4

6

8

0

2

η

Figure 11.17 Wall-normal variation of equilibrium flow obtained from a similarity solution for the case of flow over isothermal wedge with buoyancy parameter K¯ = 0.04. [Reproduced from “Direct numerical simulation of transitional mixed convection flows: Viscous and inviscid instability mechanism”, Tapan K. Sengupta, Swagata Bhaumik and Rikhi Bose, Physics of Fluids, 25, 094102 (2013), with the permission of AIP Publishing.] unlike what is noted in Figure 11.18, no fluctuations inside the boundary layer are seen. We note the observation by the authors in [423] that there exist other qualitative differences between the results depicted in Figures 11.18 and 11.19. For the hot isothermal wedge, disturbances do not originate from the leading edge, as was the case for cold plate with adiabatic boundary condition applied at the plate surface. But, the disturbances are noted at the edge of the shear layer for both the cases.

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0.15

(i) t = 30

K = –0.05

y

0.1 0.05 00

–30

–10

–20

20

40

x

–2

–5 60

80

(ii) t = 60

0.15

y

0.1 0.05 00

–20

20

–10

–20

40

x

–5

–10

60

80

(iii) t = 70 0.1

y

–2

0.05 00

–5

–10 –20

20

40

x

0.15

60

80

(iv) t = 80

0.1 y

–2

0.05 00

–20

20

40

x

0.15

60

80

(v) t = 90

0.1 y

–2

0.05 00

–5

–10

–10

–5

–20

20

40

x

60

80

Figure 11.18 Vorticity (ω) contours plotted in ( x, y)-plane at indicated times for equilibrium flow, which is simplified in [402] by a similarity transform for mixed convection boundary layer with buoyancy parameter K = −0.05 for cold plate with respect to the free stream. [Reproduced from “Direct numerical simulation of transitional mixed convection flows: Viscous and inviscid instability mechanism”, Tapan K. Sengupta, Swagata Bhaumik and Rikhi Bose, Physics of Fluids, 25, 094102 (2013), with the permission of AIP Publishing.]

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0.2

_

K = 0.04

(i) t = 15

y

0.15 0.1 0.05 00 0.2

–0.05

–0.4

20

–4

x

–50

40

–4 –100

60

80

(ii) t = 20

y

0.15 0.1 –0.05

0.05

–4 –50

00 0.2

20

x

–0.4

–15

40

60

80

(iii) t = 25

y

0.15 0.1 –0.4

0.05 00

20

x

–0.05 –0.4 –15 –50

40

60

80

40

60

80

40

60

80

(iv) t = 35

y

0.3 0.2 0.1 –4 –0.4 –15

00 1.5

–50

20

x

–4

–15-0.4 –50

(v) t = 45

y

1 0.5 00

-100

20

x

Figure 11.19 Vorticity (ω) contours plotted in ( x, y)-plane at indicated times while computing equilibrium flow of a heated isothermal wedge with buoyancy parameter K¯ = 0.04. The dotted line in the top frame indicates where IF (Eq. (11.68)) is negative above the line, indicating possibility of inviscid instability following Theorem-II described later. Furthermore, fluctuations for the isothermal wedge case reveal significantly taller vortical disturbance structures with lower fluctuation values. These similarities between the two cases are perplexing, as the cold adiabatic plate with higher cooling

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from the leading edge induces an accelerated velocity profile, as compared to the hot wedge case. Despite this, the high wavenumber fluctuations do not allow computing the equilibrium flows for both the cases, as obtained by the boundary layer formulations given in [303, 402]. In trying to calculate these equilibrium flows, no external explicit excitation is applied in either case. Thus, the fluctuations noted in the study [423], are due to amplification of numerical errors triggered due to truncation and round off errors. Having performed all the computations in double precision indicate that there is a need to investigate linearized flow instabilities for which the response will be for very low amplitude small scale disturbances. Such instabilities have been noted in Kaiktsis et al. [211]. The numerical methods used in [423] are the same ones, which have successfully solved the receptivity study of the semi-infinite flat plate boundary layer in [32, 419]. These methods have also been used to study instability of the adiabatic hot plate without any numerical manifestation of physical instability in [412, 535]. In these cases, equilibrium flows have been obtained by DNS first, and the receptivity to vortical wall excitation has been studied. These counterintuitive responses observed for the solution of the Navier–Stokes equation have been investigated in [423] for the linear viscous stability of mixed convection boundary layer for spatial and temporal growths.

11.4.4 Linear modal theory for viscous instability: Spatial and temporal routes In Chapter 2, the issues of obtaining the equilibrium flows for pure hydrodynamic problems have been discussed. Some of these equilibrium flows have been used to study the receptivity and instability in Chapter 4 with the help of the compound matrix method (CMM). Here, one investigates the linear receptivity and instability of mixed convection problems from the first principle, highlighting the fact that the equilibrium flows obtained in [303, 402] are from the solution of ordinary differential equation obtained by special transformations. Yet these are not termed as similarity profile due to varying pressure gradients in both directions for the two-dimensional flow under discussion. Linear viscous theory showing spatial and temporal growths of the equilibrium flow obtained in [303, 402] for the mixed convection flows are developed as follows. To derive the equations for linear analysis, one starts with the nondimensional Navier– Stokes equation and the energy equation with Boussinesq approximation. These equations are nondimensionalized by the boundary layer displacement thickness (δ∗ ) as the length scale, boundary layer edge velocity (Ue ) as the velocity scale and the temperature scale is given by ∆T (x∗ ) (difference between local plate and free stream temperatures, i.e., ∆T (x∗ ) = T w (x∗ ) − T ∞ ), with the nondimensional quantities represented by the hat symbol. These equations are given as: ∂ˆu ∂ˆv + =0 ∂ xˆ ∂ˆy

(11.34)

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Mixed Convection Flow

∂ˆu ∂ˆu ∂ˆu ∂ pˆ 1  ∂2 uˆ ∂2 uˆ  + uˆ + vˆ =− + + ˆ ∂ xˆ2 ∂ˆy2 ∂ xˆ ∂ˆy ∂ xˆ Re ∂tˆ

(11.35)

∂ˆv ∂ˆv ∂ˆv ∂ pˆ 1  ∂2 vˆ ∂2 vˆ  + uˆ + vˆ = Ri θˆ − + + ˆ ∂ xˆ2 ∂ˆy2 ∂ xˆ ∂ˆy ∂ˆy Re ∂tˆ

(11.36)

∂θˆ ∂θˆ ∂θˆ 1  ∂2 θˆ ∂2 θˆ  + uˆ + vˆ = + ˆ Pr ∂ xˆ2 ∂ˆy2 ∂ xˆ ∂ˆy Re ∂tˆ

(11.37)

ˆ are Richardson and Reynolds numbers, respectively. In performing where Ri and Re linear stability analysis, flow variables like velocity, pressure and temperature are shown as a combination of time-independent part and a fluctuating part, with a generic variable given by zˆ ( xˆ, yˆ , tˆ) = Z( xˆ, yˆ ) +  z¯ ( xˆ, yˆ , tˆ)

(11.38)

ˆ T represents the full quantity; Z = [U, V, P, T ]T is the mean and where, zˆ = [ˆu, vˆ , p, ˆ θ] T ¯ are the fluctuating components. Mean components can be obtained, as z¯ = [¯u, v¯ , p, ¯ θ] given in [303, 402] via the transformations of the boundary layer equations described. Two-dimensional local linear receptivity study requires assumption of parallel flow as, U = U(ˆy), V = 0, P = P(ˆy) and T = T (ˆy). The non-dimensional numbers in ˆ = Ue δ∗ /ν and Ri = Gr/ ˆ 2 , where the Grashof ˆ Re Eqs. (11.34) to (11.37) are given as, Re ˆ = gβT ∆T (x∗ )δ∗3 /ν2 . Linear analysis is performed using Fourier-Laplace number is Gr transforms for near- and far-field, with the disturbance quantities given as " ¯ = [¯u, v¯ , p, ¯ θ]

[ f1 (ˆy), φ(ˆy), π(ˆy), h(ˆy)] ei(α xˆ−βtˆ) dαdβ

(11.39)

Here, α = αreal + iαimag and β = β0 + iβi , are the complex wavenumber and circular frequency, respectively. For disturbance growing only in space, one considers wavenumber α as complex, while β is real. Spatial growth indicated when αimag is negative. For pure temporal growth of disturbances, the wavenumber α is real (α = αreal ), while the circular frequency β = βreal + i βi is complex. Positive values of βi indicate temporal growth of disturbances. Substituting Eqs. (11.38) and (11.39) in Eqs. (11.34) to (11.37) and linearizing by retaining O() terms, one obtains linearized equations for disturbance quantities. Using the notation in Eq. (11.39) in these linearized equations, leads to ordinary differential equations, which govern the evolution of disturbance amplitude functions (as functions of yˆ ). These equations can be further simplified into a fourth and second order ordinary differential equations as [423],

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i(αU − β)(α2 φ − φ00 ) + ikU 00 φ = Ri α2 h −

i(αU − β)h + T 0 φ =

1 iv 00 (φ − 2α2 φ + α4 φ) ˆ Re

1 (h00 − α2 h) ˆ RePr

(11.40)

(11.41)

where a prime indicates a derivative with respect to yˆ . These two equations can be coupled to give a sixth order Orr–Sommerfeld equation valid for mixed convection boundary layer. The system given by Eqs. (11.40) and (11.41) can be solved with six boundary conditions as at and as

yˆ = 0 :

φ, φ0 = 0;

yˆ → ∞ :

h=0

(11.42)

φ, φ , h → 0

(11.43)

0

A general solution for these ordinary differential equations can be written in terms of fundamental modes as φ = a1 φ1 + a2 φ2 + a3 φ3 + a4 φ4 + a5 φ5 + a6 φ6

(11.44)

h = a1 h1 + a2 h2 + a3 h3 + a4 h4 + a5 h5 + a6 h6

(11.45)

In traditional instability analysis, one notes the use of boundary conditions which imply that the disturbances decay in the far stream (ˆy → ∞), i.e. only the wall mode is studied. One also notes that in the free stream, the governing equation for h decouples from the governing equation for φ; since U ≈ 1 and U 00 , T 0 ≈ 0. Thus in the free stream, the energy equation given by Eq. (11.41) simplifies to ˆ h00 − [iRePr(α − β) + α2 ]h = 0

(11.46)

which is solved exactly in the free stream in terms of the characteristic modes by h∞ = a5 e−S y˜ + a6 eS y˜

(11.47)

q ˆ where S = α2 + iRePr(α − β). In the study of wall modes, one needs to satisfy the vanishing free stream boundary condition, and one must have a6 = 0, for real(S ) > 0. It is seen that as yˆ → ∞, h1∞ = h3∞ = 0. Approaching the free stream one has U = 1, and all derivatives of equilibrium flow quantities are zero. This simplifies the linearized momentum equation to 00

ˆ ˆ ˆ α2 h φiv − [2α2 + iRe(α − β)]φ + [α4 + iRe(α − β)α2 ]φ = Ri Re

(11.48)

In the free stream, the momentum equation is coupled with the thermal field, with the latter, as given by Eq. (11.47), providing the forcing for the momentum equation. Thus, the solution of Eq. (11.48) is the sum of a homogeneous solution and a particular

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integral, which after removing exponentially growing terms with respect to yˆ (i.e. by enforcing a2 = a4 = 0) gives ˆ −S yˆ φ∞ = a1 e−αˆy + a3 e−Qˆy + a5 Γe

(11.49)

q ˆ ˆ α2 /[S 4 − (α2 + Q2 )S 2 + α2 Q2 ], when the real − β) and Γˆ = Ri Re where Q = α2 + iRe(α parts of α, Q and S are positive. Hence, the general solutions of the coupled sixth order Orr–Sommerfeld equations satisfying prescribed homogeneous boundary conditions at free stream per Eq. (11.43) are φ = a1 φ1 + a3 φ3 + a5 φ5

(11.50)

h = a1 h1 + a3 h3 + a5 h5

(11.51)

ˆ −S yˆ . Three with h1∞ = h3∞ = 0, h5∞ = e−S yˆ , and φ1∞ = e−αˆy , φ3∞ = e−Qˆy and φ5∞ = Γe fundamental solutions are retained and these decay exponentially at three widely different rates, which is the reason behind Eqs. (11.40) and (11.41) being stiff ordinary differential equations. Stiff solvers like the CMM [3, 313, 412] are required to solve such equations. In CMM, stiff equations are replaced by a set of auxiliary equations, in terms of second-compound variables, which are well-defined combinations of φ j , h j and their higher derivatives. These new auxiliary second compounds grow/ decay exponentially at comparable rates, as shown in [412]. The sixth order Orr–Sommerfeld equation (with six boundary conditions) is converted into an initial value problem in CMM with twenty (20) second-compounds. The reason that the boundary value problem can be converted to an initial value problem is due to the nature of the disturbance field which decays with height. The number of equations are determined by the fact that out of six fundamental solutions, only three are admissible and the number of linearly independent unknown variables is given by 6C3 = 20. The eigenvalues of Eqs. (11.40) and (11.41) are obtained by solving the CMM equations from the free stream to the wall, with initial conditions described in [412, 535] and satisfying the dispersion relation, which are the homogeneous conditions at the wall given by Eq. (11.42). For spatial stability problem, the eigenvalues are the complex wavenumber (α), for the fixed combination ˆ β = β0 and K¯ (or K). For temporal stability analysis, one looks for the complex of Re, ˆ αreal and K¯ (or K). circular frequency β, for a fixed combination of Re, The dispersion relation is obtained in terms of CMM variables from the homogeneous wall conditions in the following manner. For the disturbance values of temperature and velocity at the wall given by Eq. (11.43); these are written in terms of fundamental solutions given as a1 φ1 + a3 φ3 + a5 φ5 = 0

(11.52)

a1 φ01

(11.53)

+

a3 φ03

+

a5 φ05

=0

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a1 h1 + a3 h3 + a5 h5 = 0

(11.54)

For non-trivial solutions for a1 , a3 and a5 , the characteristic determinant of the above linear algebraic system must be equated to zero. Using the definition of secondcompounds in [412] provides this dispersion relation as Dr + iDi = y3 = 0 at yˆ = 0 where the second compound y3 is given by [412] φ1 φ3 φ5 y3 = φ01 φ03 φ05 . h h h 1 3 5

(11.55)

(11.56)

11.4.5 Linear spatial and temporal theories for mixed convection boundary layer To study linear viscous instability for mixed convection flows, adopting the classical approaches is adequate. For example, in the study of linear temporal viscous instability, the wavenumber α could be considered as real (α = αreal ), while looking for the complex frequency β = β0 + iβi . Due to temporal instability, one would expect the disturbance to grow with time, only when βi is positive. Thus, one would solve ˆ αreal )an eigenvalue problem for complex β in the parameter space defining the (Re, plane. Overall, the eigenvalue problem is solved for a prescribed buoyancy parameter, ¯ Similarly, in linear spatial viscous instability studies, the wavenumber α is K and K. considered as complex (α = αreal + αimag ), for a fixed real circular frequency β = β0 . Spatial growth of disturbances in the downstream direction is indicated, for αimag to ˆ β0 ) be negative. Thus, the task is to find eigenvalues as complex α, for the choice of (Re, ¯ with prescribed buoyancy parameter, K and K. Linear spatial viscous instability of mixed convection boundary layer over an adiabatic cold plate [402] and an isothermal heated wedge [303] are reported with neutral curves shown in Figure 11.20 (a) for αimag = 0, obtained from spatial theory and (b) for βi = 0 obtained from temporal linear theory. From these figures it is noted that the critical Reynolds number increases as the plate is cooled more, and which indicates the same value from both the theories, for the same value of K. One also notices that unstable frequency ranges shrink as the value of the highest unstable frequency comes down with increased cooling. In contrast, with higher cooling, the maximum unstable wavenumber increases, i.e. the unstable waves become smaller. The ray OA in Figure 11.20(a) corresponds to the constant non-dimensional physical frequency defined by F = β0 /Reδ∗ = 2πν f˜/Ue2 where f˜ is the physical excitation frequency at the wall in Hertz. For the flow excited monochromatically, one would track the disturbance following OA, along which F =

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Mixed Convection Flow (a) Neutral curves for flow over adiabatic cold plate case from spatial approach A K = –0.01 0.1

P2

0.4 0.3

F = Const. ray

β0

(b) Neutral curves for adiabatic cold plate case from linear temporal viscous analysis

K = –0.05

P1

K = –0.01

α real

0.15

0.2

K = –0.05

0.05 0.1 K = –0.001

00 O

500

1000

1500

2000 ^ Re

2500

3000

00

3500

K = –0.001

500

1000

1500

2000 ^ Re

2500

3000

3500

ˆ β0 )-plane Figure 11.20 (a) Neutral curve obtained from linear spatial analysis shown in (Re, for the indicated buoyancy parameter (K ) values for the cold plate case from [402]. (b) The same neutral curves for the flow over the cold plate cases shown ˆ αreal )-plane. The different appearances of the curves are in (a), plotted in (Re, due to change in the quantities plotted along the y-axis in these two frames. constant for the spatial instability theory. The ray enters the neutral curve for K = −0.01 at P1 in Figure 11.20(a), and exits the neutral curve through P2 . (a) Neutral curves for the case of flow over heated isothermal wedge from spatial approach

0.15

(b) Neutral curves for heated isothermal wedge from linear temporal viscous analysis

0.4

_

K = 0.04 B

_

β0

K = 0.025

Q2

K = 0.04

0.2

K=0

_

_ K = 0.001

_

0.05

_

0.3 αreal

0.1

K=

0.1

.01

_

0.0 25

K=0 .01

_

K = 0.0 01

Q1

0

O

0

1500

3000

^ Re

4500

6000

7500

0

0

1500

3000

^ Re

4500

6000

7500

ˆ β0 )-plane for Figure 11.21 (a) Neutral curve obtained from linear spatial analysis shown in (Re, ¯ the indicated buoyancy parameter (K ) values for the heated isothermal wedge flow case from [303]. (b) The same neutral curves for the flow over the hot ˆ αreal )-plane. The different appearances plate cases shown in (a), plotted in (Re, of the curves are due to change in the quantities plotted along the y-axis in these two frames. For mixed convection flow over heated isothermal wedges with different buoyancy ¯ neutral curves (given by αimag = 0 contour) are shown in parameter values (K), ˆ β0 )-plane in Figure 11.21(a), obtained by linear spatial theory. The neutral curves (Re,

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are shown for the similarity solution given by Mureithi and Denier in [303]. The flow supports spatially growing disturbances for the parameters inside the neutral ˆ cr decreases, while curve. For this similarity profile, the critical Reynolds number (Re) range of unstable circular frequency increases, when the heat transfer increases with ¯ implying that heating the plate will destabilize the flow even increasing value of K, more when following this linearized viscous spatial theory route. In Figure 11.21(b), neutral curves are obtained using linear temporal theory for the flow over the same ˆ cr heated plate cases. Here, the trends are similar, with critical Reynolds number (Re) ¯ decreasing with heating, while the range of unstable values of αreal increases with K.

0

0.015

0.008

-0.01

-0.15

1000 ^ 2000 Re

0.3

αreal -0.01 -0.1

0 0

0

-0.1

1000 ^ 2000 Re

6000

3000

-0.1

-0.1

4000

(cimag)-contours Min = -0.58 Max = 0.003 -0.025 -0.015 -0.01 -0.005

0.2

0

-0.05

0.025

2000 ^ 4000 Re

-0.1

-0.025

0.1

0.015

-0.01

-0.02 -0.03

-0.15

(d) K = 0.001 0.4

0.02

0.02

0

-0.01

-0.03

0.3

0.01 0.015

0.005

-0.00 3

-0.0 1

- 0 .0 5

_

-0.01 0

0 0

4000

Min = -0.64 Max = 0.0255

-0.01

0.1

3000

(cimag)-contours

_

3 0.00

0

-0.05

-0.05

-0.1

(c) K = 0.04 0.4

0.2

0.1

0

-0.01

0

-0.0 2

-0.1

0.1

.0 5

0.2

0.022

0.02

1 .0 -0

-0

αreal

αreal

0

-0.05

0

0.3

Min = -0.71 Max = 0.0058

03 -0.0

-0.03

0.008

0.2

0

-0.05

-0.01

(cimag)-contours

(b) K = -0.05 0.4 -0.1

0.3

Min = -0.71 Max = 0.022

2 -0.0

-0.01

(cimag)-contours

αreal

(a) K = -0.001 0.4

0 0

-0.025

-0.015

-0.1

0.003 -0.015 -0.005

-0.05 -0.1

2000 ^

Re

4000

6000

ˆ αreal )Figure 11.22 Temporal growth rate contours of cimag = βimag /αreal are shown in (Re, plane for indicated mixed convection boundary layer cases.

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11.4.6 Temporal viscous growth rates for mixed convection boundary layer flows Temporal growth rates are often indicated by cimag (= βi /αreal ), and the growth rates ˆ αreal )-plane in Figure are shown by plotting the contours of this quantity in the (Re, 11.22. The maximum and minimum values of cimag are indicated in each frame of the figure, for the indicated cases of heating and cooling. The adiabatic cold plate case with K = −0.05 in frame (b) displays a lower temporal growth rate, as compared to the case of K = −0.001 in frame (a). Thus, according to linear temporal instability theory, cooling reduces instability by increasing the critical Reynolds number in frame (b), as compared to that shown in frame (a). For the isothermal heated wedge case of [303], it is seen that heating increases the temporal growth rate, which are depicted in frames (c) and (d). Thus, for mixed convection flows studied for linear growth mechanisms, the viscous temporal theory applied for flow over an adiabatic cold plate and an isothermal heated wedge show identical trends, and thus seen to be following linear spatial theory. But, it is apparent that both the viscous linear theories are incapable of explaining the instabilities noted in Figures 11.18 and 11.19, obtained by DNS and without requiring any explicit excitation. It is noted from DNS that instabilities in mixed convection flows intensify for both high heating and high cooling. The conflicting results from DNS and linear theory can tempt one to conclude that the mixed convection flow instabilities are not explained by eigenvalue analysis, and instead, one should perform global linear analysis, which shows compatability with DNS results. Already for hydrodynamic instability studies, the shortcoming of local linear viscous theories have been identified in previous chapters. It has been noted that linear eigenvalue analysis is not correct for flow over a semi-infinite flat plate, where nonmodal analysis shows STWF as the precursor of transition in [32, 34, 419, 422, 471]. Additionally, linear theories using parallel flow assumption fail due to the nature of flow dynamics near the leading edge of the plate. Results in Figure 11.18, clearly show onset of disturbance near the proximity of the leading edge. Linear analysis that views instability as either a spatial or a temporal growth problem is inadequate for semi-infinite flat plates, as has been pointed out in Chapters 5 to 7. To incorporate the streamwise variation of the flow, one can perform either global linear studies [519], or use fully nonlinear formulation [419, 422]. The latter provides results including nonlinear effects, which can help in relating these with those obtained by local analysis in some specific cases. In a very recent research, it has been categorically shown that global nonlinear, nonmodal analysis is the all-inclusive approach, as compared to global linear studies for a test case of excitation from the free stream by a convecting vortex [471]. Results in [423] have shown ambiguities of linear theories with respect to Figures 11.18 and 11.19 for mixed convection type of boundary layer. In earlier studies of mixed convection, the inviscid instability mechanisms has been completely overlooked, with the exception of that reported in [423]. This reference studied the spatio-temporal receptivity of mixed convection boundary layers to understand instability mechanisms by solving the Navier–Stokes equation, as explained in the following.

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Figures 11.18 and 11.19 show that for the investigated mixed convection flow cases with high heat transfer parameters, equilibrium flows cannot be obtained by solving the Navier–Stokes equation, even though there are steady similarity solutions reported in [303, 402]. DNS can produce steady equilibrium flows for lower rate of heat transfer cases. In [420], a few cases have been reported, with equilibrium flows obtained by solving the Navier–Stokes equation and receptivity to wall excitation of such flows are obtained for (i) cold plate with K = −0.01 and −0.001 and (ii) the isothermal hot wedge with K¯ = 0.025.

11.5 Strict Spatial and Temporal Linear Instability Theories: Re-evaluating Mixed Convection Flows One of the key features noted from Figures 11.18 and 11.19 is that for the mixed convection flows with heat transfer modeled by Boussinesq approximation, the equilibrium flows themselves are not computed for higher heat transfer rates in [423]. This is the case for the equilibrium flows: (a) For a highly cooled flat plate with heat transfer only from the leading edge and (b) A highly heated isothermal wedge flow with the wedge angle of 60◦ . The cold plate case is particularly perplexing, because linear spatial theory indicates the flow to become more stable with increased cooling, while the DNS results indicate such mixed convection flow to become unstable with significantly higher magnitude cooling. According to the DNS solution, the disturbance grows outside the shear layer. In [423], various mechanisms of instability have been investigated for mixed convection flows to explain viscous and inviscid mechanisms for strong heat transfer effects. The question still lingers about what happens to a flow, when both the temporal and spatial theories indicate instability. This can be addressed by spatio-temporal receptivity analysis and nonlinear, nonmodal global analysis by DNS. Figures 11.18 and 11.19 are the partial demonstration of the latter nonlinear approach. But even this fails to answer whether the disturbance growth is due to modal or nonmodal growth routes. DNS has to be performed using a highly accurate scheme, to properly include nonparallel and nonlinear effects. However, very interesting research results have been presented along with new theorems for inviscid instability in [423], which are more generic than Rayleigh and Fjφrtoft theorems for hydrodynamic instability. This is a significant development in instability studies for mixed convection flow, with implications for atmospheric flows also [119, 412]. Receptivity studies are usually for steady equilibrium flows, preferably defined by similarity profile(s). For example, the Blasius profile is used to study zero pressure gradient boundary layer, whose receptivity has been studied in [133, 391]. Use of such a Blasius profile is incorrect for instability studies, as it excludes the leading edge zone. To include that zone, one must obtain an equilibrium flow by solving the Navier– Stokes equation including the leading edge. It has been shown in Chapters 6, 7 and 9 that the leading edge is a primary site of disturbance energy, creating disturbances

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which remain outside the boundary layer by a shear sheltering effect, and still affect the flow inside the boundary layer. Another factor to be accounted for is the rapid growth of the boundary layer near the leading edge. Thus, for mixed convection flows, the equilibrium flow must be obtained by solving the Navier–Stokes equation inclusive of nonlinear, nonparallel effects.

11.5.1 DNS of mixed convection flow instability: New theorems of instability Contradictory results are obtained for mixed convection flow instabilities for higher heating and cooling from the wall by solving linear and nonlinear sets of equations. For flow past a horizontal cold plate increased stabilization is noted with cooling in Figure 11.20 by linear viscous thoeries for the equilibrium flow obtained from [402]. On the other hand, performing DNS with intense cooling, even the equilibrium flow is not achieved for K = −0.05. Lower rate of cooling cases allow achieving equilibrium flow and study its receptivity, without any contradiction between linear viscous theories and DNS approaches. The same holds true for equilibrium and disturbance flows for lower rate of heating cases for an isothermal wedge. However, when heating is increased with K¯ = 0.04, DNS is unable to solve for equilibrium flow. These suggest the existence of other mechanisms for mixed convection flow instabilities. The governing equations may not be the Orr–Sommerfeld equation, or the nonmodal routes of spatio-temporal growth. Results of DNS in Figures 11.18 and 11.19 show appearance of disturbances without excitation, outside the boundary layer. This implies the presence of stronger instability mechanisms, which amplify numerical errors naturally and those show temporal growth without significant convection. Subsequently, disturbances occupy larger streamwise stretches, but stay outside the boundary layer. Presence of disturbances in the inviscid part of the flow suggests some inviscid mechanism. For flows without heat transfer, Rayleigh’s stability equation and theorems provide necessary conditions for disturbance growth [119, 412]. Corresponding analysis to study inviscid temporal instabilities for mixed convection flow have been reported for the first time in [423]. The governing equations for inviscid disturbances for mixed convection flows with heat transfer modeled by Boussinesq approximation are given in non-dimensional form as: ∂ˆu ∂ˆv + = 0, ∂ xˆ ∂ˆy

(11.57)

∂ˆu ∂ˆu ∂ˆu ∂ pˆ + uˆ + vˆ =− , ∂ xˆ ∂ˆy ∂ xˆ ∂tˆ

(11.58)

∂ˆv ∂ˆv ∂ pˆ ∂ˆv + uˆ + vˆ = Ri θˆ − , ∂ xˆ ∂ˆy ∂ˆy ∂tˆ

(11.59)

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∂θˆ ∂θˆ ∂θˆ + vˆ = 0. + uˆ ∂ xˆ ∂ˆy ∂tˆ

(11.60) ˜

For the linearized analysis, Richardson number Ri = Gr is retained in Eq. (11.59) for ˜2 Re an inviscid mechanism. This is due to the fact that the buoyancy effects will affect the mean flow as well and the associated terms are not a lower order perturbation. For mixed convection flow, the momentum boundary layer grows as O(Re−1/2 ). To have equal effects of free and forced convection in the mixed convection regime, one must have K¯ = Ri/Re1/2 ≈ O(1), and this is the reason in retaining buoyancy effects in the governing equation. If parallel flow is assumed for the equilibrium flow, and perturbation quantities are expressed by the Fourier-Laplace transform given in Eq. (11.38), then using these relations in Eqs. (11.57) to (11.60), and after algebraic manipulations one gets a simplified single equation for φ as  Ri dT d2 U  d2 φ dyˆ 2 (U − c) + φ α (c − U) + =0 − (U − c) dyˆ 2 dyˆ 2

(11.61)

where, c = β/α is the complex phase speed that fixes temporal growth rate. This is the linearized inviscid equation for temporal instability of mixed convection flows. In form, this is the same as that is obtained for hydrodynamic instability by Rayleigh’s stability equation. For temporal instability, α is real, i.e. (α = αreal ) with β being complex. Also, c = creal + i cimag , where creal and cimag are the physical phase speed and the temporal growth (or decay) rate of the disturbance. In hydrodynamic instability, Rayleigh’s and Fjφrtoft’s theorems provide necessary conditions for temporal inviscid instability, obtained by multiplying Eq. (11.61) with φ∗ (the complex conjugate of φ) and integrating over full possible limits (0, ∞) for the independent variable to yield Z 0

∞

d2 U   d2 φ  Ri dT dyˆ dyˆ 2 ∗ 2 2 φ + |φ| −α + − dyˆ = 0 2 2 (U − c) dyˆ (U − c)

(11.62)

This can be further simplified by integrating the above by parts, and using boundary conditions: φ(0) = 0 and φ → 0, as yˆ → ∞ to get Z  Z   Ri dT |φ|2  d2 U dyˆ dφ 2 + α2 |φ|2 dyˆ + ∗ ∗ 2 (U − c ) − (U − c ) dyˆ = 0 dyˆ |U − c|2 dyˆ 2 |U − c|2

(11.63)

where c∗ is the complex conjugate of c, i.e. c∗ = creal − icimag . The imaginary part of Eq. (11.63) is Z

dT cimag |φ|2  −2Ri dyˆ (U − creal ) d2 U  + 2 dyˆ = 0 |U − c|2 |U − c|2 dyˆ

(11.64)

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For inviscid temporal instability to be indicated, one requires cimag , 0. Therefore, the integral given above in Eq. (11.64) will be zero, only if the term inside the square brackets changes sign, within the flow domain. This quantity is defined as IR =

 −2Ri

dT dyˆ

(U − creal )

|U − c|2

+

d2 U  dyˆ 2

(11.65)

Thus, a necessary condition for inviscid temporal instability is that IR must be zero in the interior of the domain, 0 ≤ yˆ ≤ ∞. This becomes the Rayleigh’s inflection point 2 theorem for flows without heat transfer (Ri = 0), with IR ≡ ddyˆU2 . This is a logical extension of Rayleigh’s theorem propounded in [423]: Mixed Convection Instability Theorem-I: - The necessary condition for inviscid instability of a mixed convection parallel boundary layer described by the velocity and temperature profiles, U(ˆy) and T (ˆy), the integrand IR given in Eq. (11.65) must vanish in the interior of the domain. The advantage of having the Mixed Convection Instability Theorem-I for inviscid temporal growth is: (i) If the equilibrium flow can be defined by the parallel profiles for the velocity and temperature fields, then all the viscous information of the flow is embedded therein, and one can use DNS results, provided a local parallel flow approximation holds; (ii) This is significantly unlike flows without heat transfer, where the existence of an inflection point of velocity profile alone is sufficient to define this instability. It is often noted that researchers try to explain instability in compressible flows by merely noting the presence or absence of inflection point alone. For compressible flows, heat transfer is definitely involved and the Mixed Convection Instability Theorems are more appropriate than the Rayleigh’s theorem. Existence of the vanishing point, where IR changes sign is a function of the complex phase speed c. This demonstration is not straight forward and was shown in [423], using a parametric analysis. Noting that IR is bounded for cimag , 0, it is possible to locate a critical layer for such a mixed convection problem. However, with the knowledge of the existence of multiple modes in instability studies, it is not necessary that the location of IR = 0 for all modes to happen together. This is a misconception associated with normal mode analysis. For such normal mode analysis, neutral disturbances are associated with U(ˆycr ) = creal at the so-called critical layer. Mixed Convection Instability Theorem-I is shown from the imaginary part of Eq. (11.63), following the procedure used in Rayleigh’s theorem for inviscid temporal theory of flows without heat transfer. The same has been subsequently modified by Fjφrtoft [119, 412], who then used the real part of Eq. (11.63) to derive a second necessary condition. This condition utilizes the real part of Eq. (11.63) to study

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(i) cimag = 0.01

Rayleigh integrand IR for adiabatic cold plate, K = -0.05 Edge of the momentum boundary layer

0.02 0.5 0.01 0

IR Φ

0.2

0 -0.5

0

-1 -0.01

IR

0

0.1 0.10.2

^0.30.2 0.4 yy

0.5 0.3

-0.2

-0.4 creal = 0.001 creal = 0.01 creal = 0.1

-0.6 0

1

2

3

y^

4

(ii) cimag = 0.1 creal = 0.001 creal = 0.01 creal = 0.1

0

0.0001 Edge

Φ

-0.1

IR

-0.2

of the momentum 0 boundary layer

-0.0001 5

-0.3

5.5

6

y

6.5

7

-0.4 -0.5 -0.6 0

2

y^

4

Figure 11.23 The integrand IR indicating temporal inviscid instability for flow over a cold plate, with buoyancy parameter K = −0.05 for the growth rates given by (i) cimag = 0.01 and (ii) cimag = 0.1. [Reproduced from “Direct numerical simulation of transitional mixed convection flows: Viscous and inviscid instability mechanism”, Tapan K. Sengupta, Swagata Bhaumik and Rikhi Bose, Physics of Fluids, 25, 094102 (2013), with the permission of AIP Publishing.]

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Mixed Convection Flow

inviscid temporal instability of the mixed convection flows, which for the present analysis is Z

|φ|2  −Ri |U − c|2

dT dyˆ

{(U − creal )2 − c2imag } |U − c|2

 d2 U (U − c ) dyˆ real dyˆ 2 Z   dφ 2 + α2 |φ|2 dyˆ = 0 + dyˆ +

(11.66)

In the expression given above, the second integral is always positive. Hence, the integrand of the first integral has to be negative inside the domain in order for the total to vanish. Indicating a height yˆ = yˆ s , where IR (yˆs ) = 0 and U(ˆy s ) = U s , then multiplying Eq. (11.64) by (creal − U s ) and then adding it with the first integral of Eq. (11.66) and simplifying, one gets Z 2

|φ|

 Ri dT dyˆ

{(U s − creal )2 + c2imag − (U − U s )2 } + 2

|U − c|

 d2 U (U − U ) dyˆ < 0 s dyˆ 2

(11.67)

One can define the integrand within the square bracket as IF =

 Ri dT dyˆ |U − c|

{(U s − creal )2 + c2imag − (U − U s )2 } + 2

 d2 U (U − U ) s dyˆ 2

(11.68)

Calling IF as the second integrand, one can state the following theorem for inviscid temporal instability of mixed convection flow as [423]: Mixed Convection Instability Theorem-II: The necessary condition for the inviscid instability for a mixed convection parallel flow described by velocity and temperature profiles U(ˆy) and T (ˆy) is that the second integrand IF (in Eq. (11.68)) must be negative in the interior of the domain. This theorem is similar to Fjφrtoft’s theorem developed for hydrodynamic instability, and is more useful than Rayleigh’s theorem for inviscid temporal instability of mixed convection flows. It is noted that like IR , the second criterion given in terms of IF is also the function of complex c, enabling one to investigate the occurence of inviscid instability by tracing different temporal growth rates (cimag ) and associated time scales (creal ). The usefulness of Mixed Convection Flow Theorems-I and -II is to explain why in certain high heat transfer cases, the equilibrium flows given in [303, 402] are not obtained from time-accurate solutions of the Navier–Stokes equation. These theorems serve in explaining the cases shown in (i) Figures 11.16 and 11.18 for the cold plate case with buoyancy parameter value of K = −0.05 and (ii) the results shown in Figures 11.17 and 11.19 for the hot isothermal wedge flow case with the buoyancy parameter value of K¯ = 0.04. The inviscid instability is predicted by Mixed Convection Instability

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Theorems-I and -II, when IR = 0 and IF < 0 conditions are satisfied in the interior of the flow. The necessary condition involving IR is plotted as a function of yˆ in Figure 11.23, for flow over a cold plate with K = −0.05; these results were first presented in [423]. The profiles obtained as functions ofR the similarity variable η have been converted to ˆ where Cˆ = ∞ (1 − f 0 ) dη. Variation of IR with η, for indicated functions of yˆ as yˆ = η/C, 0 different values of creal are shown in frame (i), for the growth rate value of cimag = 0.01. The three curves are for the values of creal = 0.001, 0.01 and 0.1. One notes that for the lower values of creal , the zero crossing of IR occurs near the wall, shown in the inset of the frame. For the frequency creal = 0.1, one notes that IR changes sign discontinuously for a value of η. In Figure 11.23(ii), similar variation is shown for the case of cimag = 0.1. Zero-crossing is noted for all the values of creal , implying that all the three time scales separated by factor of hundred are unstable. Thus, for both the temporal growth rate cases (cimag = 0.01 and 0.1), the mixed convection flow shows linear inviscid temporal instability. This growth originates at the outer part of the shear layer near the plate leading edge in Figure 11.18. This clearly provides the explanation why equilibrium flow cannot be computed for K = −0.05. This inviscid temporal instability for the highly cooled plate shows absolute instability of the flow, near the leading edge. Similarly, IF plotted as a function of yˆ in Figure 11.24, depicts the case of cold plate for K = −0.05. The condition of inviscid instability given by Mixed Convection Instability Theorem-II is obtained from IF that has to be negative in the domain. The cases in Figure 11.24 are for cimag = 0.01 and 0.1; with both cases exhibiting inviscid temporal instability. Satisfaction of both the necessary conditions of Theorems-I and -II, help explain why an equilibrium flow is not computed. Also, it should be noted that if a flow is found to be both temporally and spatially unstable, then the stronger mechanism will dominate. As the present flow shows weaker spatial instability, the flow will show the dominant temporal instability, and this is supported by the results of DNS shown in Figure 11.18. The first integrand, IR is shown for flow over an isothermal heated wedge with K¯ = 0.04 in Figure 11.25, for cimag = 0.01 and 0.1. The three curves shown in each frame corresponds to creal = 0.001; 0.01 and 0.1. Existence of more than one zerocrossing point, as indicated by IR = 0 is seen in the top frame of Figure 11.25. It may also be noted from the top frame that as creal decreases, the first zero-crossing point moves towards the wall. But for all the values of creal shown here, there is also a zero-crossing point around yˆ ' 5.2, which is nearer to the edge of the momentum boundary layer (approximately at yˆ = 4.54). Therefore, the inner zero-crossing point corresponds to temporal instability in the interior of the shear layer, very close to the wall. In the same way, the outer zero-crossing point is located in the inviscid part of the flow and indicates the inviscid mechanism of temporal instability. These zerocrossings responsible for the inviscid instability are clearly seen in the inset of top frame in Figure 11.25. For the higher value of cimag = 0.1 in the bottom frame of Figure 11.25, there is only an outer zero-crossing point near yˆ ' 5.5. This indicates stronger temporal instability of mixed convection flows outside the shear layer for all

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(i) cimag = 0.01

i

Fjfrtoft integrand IF for adiabatic plate case, K = -0.05 creal = 0.001 creal = 0.01 creal = 0.1

0

1 0.5

α

-0.1

0

Edge of the

IF

-0.5 0

-0.2

0.2 momentum 0.3 0.4

0.1

y boundary layer

0.004

α

-0.3

0

-0.004

-0.4

3

0

(ii) cimag = 0.1 0

2

y4

5

4

y^

creal = 0.001 creal = 0.01 creal = 0.1

IF

-0.1

Edge of the momentum boundary layer

0.004

α

-0.2

0

-0.004 3

y4

-0.3 0

2

y^

5

4

Figure 11.24 The integrand IF indicating the second necessary condition for temporal inviscid instability for flow over a cold plate, with buoyancy parameter K = −0.05 for the growth rates given by (i) cimag = 0.01 and (ii) cimag = 0.1. [Reproduced from “Direct numerical simulation of transitional mixed convection flows: Viscous and inviscid instability mechanism”, Tapan K. Sengupta, Swagata Bhaumik and Rikhi Bose, Physics of Fluids, 25, 094102 (2013), with the permission of AIP Publishing.]

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Transition to Turbulence _ Rayleigh integrand IR for flow over isothermal wedge, K = 0.04

(i) cimag = 0.01 1.5 1

Edge of momentum boundary layer

creal = 0.001 creal = 0.01 creal = 0.1

0.5

IR

0 -0.5

1 0.5 0.0001

IR

IR Φ

-1

00

-0.5

0

-1

-1.5

-1

-0.0001 4.5

-2 0

5

5.5

y^ 6 2

6.5

-2 0

7

y^

0.1

0.2 y^ 0.3 y

0.4

4

0.5

6

(ii) cimag = 0.1

0 0.0001

IR

-0.1

0

-0.0001

-0.2

IR

5

-0.3

5.5

y^

6

6.5

7

creal = 0.001 creal = 0.01 creal = 0.1

-0.4 0

2

y^

4

6

Figure 11.25 The integrand IR indicating temporal inviscid instability for flow over a isothermal hot wedge, with buoyancy parameter K¯ = 0.04 for the growth rates given by (i) cimag = 0.01 and (ii) cimag = 0.1. [Reproduced from “Direct numerical simulation of transitional mixed convection flows: Viscous and inviscid instability mechanism”, Tapan K. Sengupta, Swagata Bhaumik and Rikhi Bose, Physics of Fluids, 25, 094102 (2013), with the permission of AIP Publishing.]

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frequencies corresponding to creal < 1, while the lower growth rate case also indicates additional temporal inviscid instability nearer to the wall. The second integrand IF of Eq. (11.68) is plotted in Figure 11.26, as a function of yˆ for the heated isothermal wedge case with K¯ = 0.04. To calculate IF in Eq. (11.68), the

(i) cimag = 0.01

_

Fjfrtoft integrand IF for flow over isothermal wedge, K = 0.04

1.5

2

creal = 0.001 creal = 0.01 creal = 0.1

1

IF

1

IF

0.5

0

-1 0

0.1

0.2

y^

0.3

4

^y

5

0

0.4

0.004

IF

-0.5

0

-1 -0.004 3

-1.5

0

1

2 y^

3

4

(ii) cimag = 0.1 creal = 0.001 creal = 0.01 creal = 0.1

0.4

0.2

0.004

IF

IF

0.3

0.1

0

-0.004

3

4

^y

5

0 0

1

2 ^ y

3

4

Figure 11.26 The integrand IF indicating the second necessary condition for temporal inviscid instability for flow over hot isothermal wedge, with buoyancy parameter K = 0.04 for the growth rates given by (i) cimag = 0.01 and (ii) cimag = 0.1. [Reproduced from “Direct numerical simulation of transitional mixed convection flows: Viscous and inviscid instability mechanism”, Tapan K. Sengupta, Swagata Bhaumik and Rikhi Bose, Physics of Fluids, 25, 094102 (2013), with the permission of AIP Publishing.]

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mean speed U s corresponding to the outer zero-crossing point for IR of Figure 11.25 is used. For creal = 0.01, one notices that IF < 0 occurs in two ranges, with the first one located closer to the wall, while the other range is near the edge of the boundary layer. Also, for the higher value of creal = 0.1, the negative second integrand (IF < 0) occurs away from the wall, as compared to the lower values of creal . In Figure 11.23, only one zero-crossing point exists for the three creal ’s shown in the frames for both the temporal growth rates (cimag ), unlike the case of the isothermal heated wedge in Figure 11.25 where one observes multiple zero-crossing points. Hence for the cold plate case (K = −0.05), the temporal instabilities occur within the shear layer by the same inviscid temporal growth mechanism, and the viscous effects are embedded in the equilibrium velocity and temperature profiles obtained by the solution of the Navier–Stokes equation. One notes the negative second integrand (IF < 0) occurs over an extended range of yˆ inside the shear layer, for all combinations of creal and cimag in Figure 11.24. The necessary condition of IF < 0 predicts temporally unstable disturbances to occur inside the boundary layer. One expects that the second necessary condition of IF < 0 will have a stronger indication of instability, as compared to the first condition given by IR = 0, since the second condition with IF constitutes a necessary condition by using both the real and imaginary parts of Eq. (11.62). For any combinations of K¯ or K and cimag , there exists a critical value (creal )crit , such that for creal > (creal )crit , no zero-crossing point of IR exists in the flow. For, a fixed frequency wall excitation, creal is directly proportional to length scale, obtained from the dispersion relation. Therefore, every critical (creal )crit represents a critical lengthscale of disturbances lcrit = 2π(cβreal0 )crit , for a given real part of the circular frequency, β0 . All scales larger than lcrit , do not suffer from temporal instability. Thus, there exists cimag > 0 for the disturbance field for creal < (creal )crit , which satisfies the necessary conditions for inviscid temporal instability. These two inviscid instability theorems explain the instabilities noted in the results shown in Figures 11.18 and 11.19, obtained from the simulations of the Navier– Stokes equation. Observed instability in Figure 11.18 for the cold plate displays high wavenumber (frequency) instability at the edge of the shear layer, which originates from the leading edge. Additional lower wavenumber (frequency) disturbances are also noted inside the boundary layer, all of which originates from the leading edge of the flat plate. These are noted from Figures 11.23 and 11.24, showing the presence of IR = 0 (according to Theorem-I) closer to the wall for higher creal . As creal = β0 /α, this implies that disturbances with lower values of α are seen near the wall, as noted in Figure 11.18. Also from Figure 11.24, one notes IF < 0 to occur for a larger range of heights, with maximum negative value near yˆ ' 2. This implies that the maximum inviscid temporal instability occurs at the edge of the shear layer for all α, as seen in Figure 11.18. One justifies the instability and corresponding length and time scales for the case of the isothermal heated wedge to occur primarily at the edge of the shear layer. Also, in Figure 11.19, a dotted line is drawn, above which IF is negative for high wavenumbers, as shown in Figure 11.26.

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¯ cimag )- and (K, cimag )-planes, In Figure 11.27 (a) and (b), (creal )crit is plotted in (K, respectively. These two frames allow one to distinguish the instabilities. For example, for the case of mixed convection flow over an isothermal wedge shown in frame (a), the line AB represents the (creal )crit values including and above which, all creal are temporally unstable by the discussed inviscid mechanism. In the same way, one can reason that the region below AB indicates instability for all values of creal = 1. This in ¯ one will get higher growth rates for larger essence implies that for higher values of K, ranges of length scales. It is also noted that for the region above CD, (creal )crit = 0.0001 everywhere, and so only very small length scales experience instability, but with larger

(a) Isothermal wedge case

D

0.015 Existence of IR = 0

0.1

for (creal)crit ~ 0.0001

B

0.25

cimag

0.01

= (c real) crit

0.5 0.75

0.005

9 0.999

Existence of IR = 0 for (creal)crit ~ 1

C A

0.002

_

0.004

0.006

K

0.008

0.01

(b) Adiabatic cold plate case

0.02 Existence of IR = 0 for (creal)crit ≤ 1

0.75

Existence of IR = 0 for (creal)crit < 0.00001

0.01

0.005

07 0.0 05 0.0 0.0045

-0.001

-0.0008

0 0.0

6

0.001

0.003

0.0015 0.0035 0.004

-0.0006

0.0025

K

0.002

-0.0004

1E-05

cimag

0.015

-0.0002

Figure 11.27 Critical value (creal )crit plotted for cases of (a) flow over an isothermal wedge ¯ cimag )-plane and (b) for flow over a cold plate in (K, cimag )-plane. Here, in (K, (creal )crit represents the critical phase speed such that for all creal > (creal )crit , no zero-crossing point of IR exists in the flow.

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growth rates, due to the fact that the corresponding cimag values are larger above CD. The (creal )crit contours in frame (b) are qualitatively different for the cold plate case, and with increased cooling of the plate, i.e. approaching the origin along the x-axis, one notices growth rates for all the plotted range along the y-axis for (creal )crit ≥ 0.005. Similarly, other conclusions can be drawn for very small length scales, as marked by the range on the right-hand side of the frame for (creal )crit < 0.00001. Overall, one can conclude that for cold plate case shown in Figure 11.27(b), enhanced temporal instability is noted when cooling rate is increased. The second necessary condition for linear temporal inviscid instability is given by |φ|2 IF dyˆ < 0, which is a consequence of the Mixed Convection Instability TheoremII. Its utility is demonstrated here in the investigation of the cold plate case. As |φ|2 is strictly non-negative and less than one, another condition can R be obtained by reducing the above to a more conservative inequality given by IΓ = IF dyˆ ≤ 0. In Figure 11.28, the contours of IΓ , in the (creal , cimag )-plane for the indicated values of K, are shown. For R

(a) K = -0.05 ; max = 0.38 & min = -2.45

(b) K = -0.03 ; max = 0.44 & min = -1.83

0.012

0.012

cimag

0.014

cimag

0.014

0.01

0.008

-2.2

-1.8

0 -1.4

0.01

-0.7

0.008

-1 -0.7 -0.22

0.006

0.006

0.004

0.004

0.002

-0.22 -1.4

0

-1

0.002 0.2

0.4

0.6 real

c

0.8

1

0.2

0.4

creal0.6

0.8

1

(c) K = -0.01 ; max = 0.39 & min = -1.45 0.014

0.01

0.008

-1

-0.22

cimag

0.012

-0.7

0

0.006-1.4 0.004 0.002 0.2

0.4

creal0.6

0.8

1

R Figure 11.28 Contours of IΓ = IF dyˆ plotted in (creal , cimag )-plane for flow past a cold plate with indicated values of buoyancy parameter K .

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K = −0.05, the onset of temporal instability is indicated with the help of this criterion for creal ≤ 0.892, which is marked in the figure. For a fixed creal , all cimag ’s are indicated as possible in the plotted range. The intensity of instability is marked by the minimum value of IΓ in the flow. From all the frames, the maximum instability is noted for creal → 0, i.e. for vanishing length scales. Such instabilities are seen in Figure 11.18 with disturbances having very small length scales near the edge of the shear layer, closer to the leading edge. This criterion depends on the value of U s corresponding to Theorem-I. It is necessary to use actual velocity profiles obtained from DNS. Snapshots in Figure 11.18 indicate disturbances in the outer layer, which spread in the streamwise direction. Both the Theorems-I and -II are important in explaining instabilities which do not allow obtaining an equilibrium flow from DNS. This also indicates that both strong heating and cooling lead to flow instability and these are obtained from DNS. This predominant role of the inviscid instability mechanism to destabilize the flow is emphasized by solving Eq. (11.61) for flows past a cold plate for K = −0.001, 0.05 and for isothermal heated wedge cases with K¯ = 0.001 and 0.05. In solving Eq. (11.61), a fourth-order Runge–Kutta method is used for integrating from yˆ max = 12 to wall, for a particular choice of α. Initial conditions for the integration of Eq. (11.61) is given by specifying φ∞ = e−αˆy and φ0∞ = −αe−αˆy at yˆ = yˆ max for the studied wall mode. Rayleigh’s equation is a regular ordinary differential equation, and hence direct integration of Eq. (11.61) is possible, quite unlike the Orr–Sommerfeld equation which is known for its stiffness. In integrating Eq. (11.61), one can avoid considering neutral disturbances (as is done in [423]), otherwise the solution blows up near the critical layer. An eigenvalue satisfies the homogeneous boundary condition at the wall, i.e. when φreal + iφimag = 0 at yˆ = 0. Thus in Figure 11.29, the contours of φreal = 0 (solid lines) and φimag = 0 (broken lines) are plotted for yˆ = 0, in (creal , cimag )-plane, for cold plate cases, with indicated values of K. Only two values are chosen: α = 0.2 (frames (a) and (b)) and α = 0.05 (frames (c) and (d)). In this figure, plotted ranges are for 0.38 ≤ creal ≤ 0.45, for the sake of illustration. Intersection of these two sets of curves occur at eigenvalues for the inviscid instability analysis. One notes from Figure 11.29 that as one reduces the cooling rate, the number of eigenvalues reduces. The symmetry of the contours in Figure 11.29, is due to the fact that if c = creal + icimag is an eigenvalue of Eq. (11.61) with eigenfunction φ, then c∗ = creal − icimag is also an eigenvalue of Eq. (11.61) with the complex conjugate eigenfunction φ∗ . Only the first ten leading unstable eigenvalues are listed in Table 11.2. One notices from this table that for K = −0.05, lower α disturbances are more unstable than higher α disturbances. However, for the lower cooling rate of K = −0.001, the higher α disturbances exhibit pronounced instability. Figure 11.29 together with Table 11.2 establish that cooling destabilizes the flow via the inviscid instability mechanism, and this cannot be explained by the Orr–Sommerfeld equation. In Figure 11.30, the real (φreal = 0) and imaginary parts (φimag = 0) of eigenfunction contours are plotted at the wall (ˆy = 0) for the flow past isothermal heated wedge, with buoyancy parameter values of K¯ = 0.05 and K¯ = 0.001. It is noted by the direct simulation of the Navier–Stokes equation that inviscid instability mechanism shows

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(a) K = -0.05, α = 0.2 0.0004

φ real = 0 φ imag = 0

0.0004

0

0

-0.0002

-0.0002

-0.0004

-0.0004

0.38 0.39 0.4 0.41 0.42 0.43 0.44 0.45

0.38 0.39 0.4 0.41 0.42 0.43 0.44 0.45

(c) K = -0.05, α = 0.05

(d) K = -0.001, α = 0.05

creal

φ real = 0 φ imag = 0

0.001

0.0005

creal

0.0004

φ real = 0 φ imag = 0

0.0002

cimag

cimag

φ real = 0 φ imag = 0

0.0002

cimag

cimag

0.0002

(b) K = -0.001, α = 0.2

0

0

-0.0002

-0.0005

-0.0004 -0.001 0.38 0.39 0.4 0.41 0.42 0.43 0.44 0.45

creal

0.38 0.39 0.4 0.41 0.42 0.43 0.44 0.45

creal

Figure 11.29 Contours of φreal = 0 (solid lines) and φimg = 0 (broken lines) plotted at the wall in (creal , cimag )-plane for flows past an adiabatic cold plate with indicated buoyancy parameter K and real disturbance wavenumber α. that heating destabilizes the wedge flow, as shown in Figure 11.30. The results are shown in three different ranges of creal values. Therefore, for mixed convection flow past a hot plate, destabilization of flow is indicated simultaneously through viscous and inviscid mechanisms, with the latter dominating over the former.

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Mixed Convection Flow _

(a) K = 0.05 and α = 0.2 0.004

_

φ real = 0 φ img = 0

_

φ real = 0 φ img = 0

0

0

-0.002

_

0.2

creal

0.22 0.24 0.26

(d) K = 0.001 and α = 0.2 0.004

0.3

0.35

0.4

creal

0.45

(e) K = 0.001 and α = 0.2 0.004

0.5

0.55

-0.002

φ real = 0 φ img = 0

0.2

0.22 0.24 0.26

0

-0.004 0.25

0.68 0.7 0.72 0.74 0.76 0.78 0.8 0.82

creal

(f) K = 0.001 and α = 0.2 0.004

φ real = 0 φ img = 0

0.002

-0.002

creal

0

_

0.002

0

-0.004 0.14 0.16 0.18

-0.004 0.25

cimag

0.002

φ real = 0 φ img = 0

-0.002

_

φ real = 0 φ img = 0

0.002

cimag

-0.004 0.14 0.16 0.18

(c) K = 0.05 and α = 0.2

cimag

0.002

-0.002

cimag

0.004

cimag

cimag

0.002

(b) K = 0.05 and α = 0.2

0

-0.002

0.3

0.35

0.4

creal

0.45

0.5

0.55

-0.004

0.7

0.75

creal

0.8

Figure 11.30 Contours of φreal = 0 (solid lines) and φimag = 0 (broken lines) plotted at the wall in (creal , cimag )-plane for flows past an isothermal hot wedge for the indicated buoyancy parameter K¯ = 0.05 and 0.001. A fixed disturbance wavenumber (α = 0.2) has been chosen and results shown in three segments of creal in different frames.

11.6 Closing Remarks In this chapter, the effects of heat transfer on flow instability is studied, with the help of Boussinesq assumption to account for the heat transfer effect. Thus, the heat transfer is affected via the buoyancy effect in the conservation of momentum equation in the direction along which gravitational forces act. It is also noted that for mixed convection flows, one can use canonical solution of flows past a horizontal plate and a wedge using the formulations given in [402] and [303], respectively. It is seen that heat transfer induces a pressure gradient, with consequent effects in both the streamwise and wall-normal directions for the boundary layer. As a result, such flows are prone to having velocity profiles with inflection points and also velocity overshoots. In such scenarios one may be tempted to use Rayleigh and Fjφrtoft theorems which contain necessary conditions for temporal instability for a parallel mean flow without heat transfer. However, if the heat transfer quantum is increased (both heating and cooling), then one notices that the solutions obtained following the boundary layer formulations of Schneider [402] and that those due to Mureithi and Denier [303] cannot

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Table 11.2 First ten most dominant unstable modes from inviscid instability analysis tabulated for flows past adiabatic cold plate with K = −0.05 and K = −0.001. We have tabulated only for disturbance wavenumber of α = 0.2 and 0.05. [Reproduced from “Direct numerical simulation of transitional mixed convection flows: Viscous and inviscid instability mechanism”, Tapan K. Sengupta, Swagata Bhaumik and Rikhi Bose, Physics of Fluids, 25, 094102 (2013), with the permission of AIP Publishing.] K = −0.05

K = −0.001

Mode No.

α = 0.2

α = 0.05

α = 0.2

α = 0.05

1

(0.192, 4.19e-4)

(0.332, 4.24e-4)

(0.263, 3.41e-4)

(0.595 1.51e-4)

2

(0.223, 3.84e-4)

(0.337, 4.23e-4)

(0.289, 2.96e-4)

3

(0.246, 3.57e-4)

(0.342, 4.18e-4)

(0.411, 2.56e-4)

4

(0.265, 3.29e-4)

(0.321, 4.18e-4)

(0.421, 2.52e-4)

Other modes

5

(0.284, 2.94e-4)

(0.306, 4.18e-4)

(0.312, 2.49e-4)

are not

6

(0.459, 2.58e-4)

(0.316, 4.16e-4)

(0.362, 2.31e-4)

clearly

7

(0.489, 2.58e-4)

(0.348, 4.16e-4)

(0.331, 2.23e-4)

distinguishable.

8

(0.505, 2.55e-4)

(0.327, 4.12e-4)

(0.348, 2.23e-4)

9

(0.526, 2.48e-4)

(0.295, 4.12e-4)

(0.348, 2.23e-4)

10

(0.445, 2.45e-4)

(0.311, 4.06e-4)

(0.377, 2.16e-4)

be reproduced by solving the Navier–Stokes equation. Instead, inherently unsteady flow is seen in the solution of the Navier–Stokes equation. Such inability led to using the Orr–Sommerfeld equations to probe instability of the boundary layer as given in [303, 402] by similarity transforms. This technique of analysis revealed that cooling stabilizes, while heating destabilizes such flows. It is also observed that such equilibrium flows support both temporal and spatial instabilities. While conjecturing about the actual nature of instability by DNS, the physical mechanism may appear elusive, and to resolve that a fresh inviscid instability mechanism is sought. This search culminates in developing an appropriate governing equation and a couple of new theorems, similar to Rayleigh’s and Fjφrtoft’s theorem in hydrodynamics. Not only are the necessary conditions stated for these new theorems for mixed convection flows, but these also provide detailed estimates about the length and time scales during the instability of the flow. It may appear that such a study depends significantly on the Boussinesq approximation for heat transfer effects, and so, one may like to develop further instability theories without such limiting assumption. This is the subject of study in the following chapter on barotropic instability, in which one studies instabilities using the compressible Navier–Stokes equation.

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Baroclinic Instability: Rayleigh–Taylor Instability

12.1 Baroclinic Instability Baroclinic instability of flow is of interest in multiple branches of engineering, geophysics and astrophysics. It is considered very important in explaining motion in earth’s atmosphere and in the ocean. Rayleigh–Taylor instability is an example of a baroclinic instability introduced in Chapter 1, as a special case of the inviscid Kelvin– Helmholtz instability. Rayleigh–Taylor instability, in its simple connotation, arises when a heavier fluid is made to rest on top of a lighter one, as the initial equilibrium state. If the heavier and lighter fluids are perfectly aligned with the direction of gravity, then such an equilibrium state would remain stable. If an instability misaligns this configuration, then the ensuing turning moment will unabatedly destabilize the initial state. However, if the opposite initial configuration of a lighter fluid resting on top of heavier fluid is perturbed, then the turning moment will restore the equilibrium state. This qualitative explanation of Rayleigh–Taylor instability can be given more meaning, physically and mathematically, by considering the Euler equation for a compressible flow. Taking the curl of the momentum conservation equation in its primitive variable form, one can depict the corresponding inviscid vorticity transport equation as      D~ ω ∂~ ω ~ ~ −ω ~ + 1 ∇ρ × ∇p ~ = ω ~ ·∇ V ~ ∇·V ≡ + V ·∇ ω Dt ∂t ρ2

(12.1)

~ and ω ~ are the velocity and vorticity vectors, and the last term on the where V right-hand side ( ρ12 ∇ρ × ∇p) is the baroclinic contribution to vorticity generation by

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the misalignment of the pressure gradient from the density gradient. Thus, for baroclinic flows, pressure is not dependent on density alone, but also depends on the temperature. In contrast, an atmospheric flow is called barotropic, for which the pressure depends only on density, and  thus,  the cross product is identically zero. It ~ , is due to compressibility and prediction ~ ∇·V should also be noted that the term, ω of such instabilities should justifiably depend upon bulk viscosity for the constitutive relation between stress and rates of strain tensors. In atmospheric motion, baroclinic instability is the main source for the formation of cyclones and anti-cyclones, occurring mainly in temperate latitudes. At the tropics, the atmospheric flow is barotropic. The baroclinic instability is observed to contribute to the formation of eddies in the mesoscale. However, such instabilities are for a rapidly rotating fluid, which also displays strong density stratification. The nondimensional parameter, Richardson number, is an indicator of the strength of stratification, with large values indicative of stable stratification. It has been noted [462] that for strong rotation (corresponding Rossby number being very small) the flow demonstrates twodimensionality, according to the Taylor–Proudman theorem. It is observed that the Rossby number must be small for baroclinic instability to be present. Thus, it appears that baroclinic instability would be more predominant for two-dimensional flows and many of the studies presented here focus more on two-dimensional Rayleigh– Taylor instability. Rudimentary analysis of Kelvin–Helmholtz instability in Chapter 1 indicates that for effects of shear, when both the domains across the interface contain the same fluids, two-dimensional disturbances are stronger promoters of disturbance growth by modal amplification, as compared to three-dimensional disturbances. One can follow the discussion after Eq. (1.23), which shows that two-dimensional growth rates are higher than the growth rates for three-dimensional disturbances. The prototypical Rayleigh–Taylor instability is studied where fluids of different densities at rest are used, with heavier fluid on top of the lighter one. Such an unstable arrangement of two miscible fluids will result in interfacial instability with associated baroclinic vorticity generation at the interface. To focus more on the fundamental physical aspects of Rayleigh–Taylor instability, the case reported in [459] is discussed, which emphasizes more on the physical aspects of the instability by considering two air masses initially having a large temperature difference. The large discontinuous difference of temperature does not allow one to apply the Boussinesq approximation [403], which is regularly used in problems of heat transfer for incompressible flows. And, it has been pointed out by Mikaelian [293] that Boussinesq assumption leads to about 40 % error on quantities calculated using incompressible flows. To circumvent this problem, the authors in [7, 18, 151, 359, 459, 460, 461] have used compressible flow formulation without the Boussinesq approximation. In [151], compressible Newtonian flows have been studied for the Rayleigh–Taylor instability by pseudo-spectral formulation of the Navier–Stokes equation starting from a strongly stratified initial condition. The problem studied is for the case where the physical variables are periodic in the horizontal planes parallel to the interface, and

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so, using Fourier series in these homogeneous directions is natural, while Chebyshev expansion is used in the perpendicular direction to the interface. The temporal growth of disturbance associated with Rayleigh-Taylor instability has been studied via an inviscid mechanism in [119, 352, 516, 568], while it has been studied as viscous instability in [75, 84, 97]. O

R (Tu,ρu) (u,v) = 0; ∂T = 0; ∂p = 0; ∂n ∂n

Initial interface positon (Tl,ρl) Tl > Tu; ρl < ρu

y

x

Q

P O

R ρ=ρ u

Initial interface positon

Mixed region

H

hm

ρ=ρ l y P

Q x

Figure 12.1 Schematic of Rayleigh–Taylor instability in a section shown by the rectangular box, which could be a two- or three-dimensional problem described in this chapter. Apart from applications in geophysics, Rayleigh–Taylor instability is also central to understanding “inertial confinement fusion where the fuel capsule is imploded by the lasers to yield enough power to sustain another fusion reaction,” as noted in [460]. According to Gauthier [151], this instability also “plays a prominent role in supernova explosions in astrophysics” as also as discussed in [75, 577]. The rudimentary study of Rayleigh–Taylor instability can be performed by a schematic arrangement shown in Figure 12.1, with the initial interface position located halfway along the vertical edges of a box as investigated in [84, 347, 352, 357, 484, 516]. This could be the domain of interest for a two-dimensional study [6, 357, 459, 460, 461], or it can be

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considered as a plane of interest for a three-dimensional study [357]. The difference of density between the two fluids across  the initial interface is parameterized by the non-dimensional Atwood number, A =

ρ1 −ρ2 ρ1 +ρ2

, where ρ2 and ρ1 are densities of upper

and lower fluids shown in Figure 12.1. If the density gradient between the two fluids is large compared to the temperature differential, then using the Boussinesq approximation is necessary. This is due to the fact that in this approximation, the density variation is small and treated as constant, except that is often used to describe the applied gravitational force acting as the body force. This was the reason that Mikaelian noted that “direct numerical simulations are the best way to assess the validity” of Boussinesq approximation, while investigating its effects on Rayleigh– Taylor and Richtmeyer–Meshkov instabilities. It has been also noted by Wei and Livescu [548] that higher order effects are missed by the Boussinesq approximation for lower Atwood numbers. Wang and Robertson [545] are the early investigators in studying Rayleigh–Taylor instability using a compressible formulation to model accreting X-ray in neutron stars by considering identical media, differing from each other by temperature discontinuity, with and without a magnetic field. However, the instability was initiated by single mode and random amplitude perturbations as this and most other studies formulate the instability problem by considering periodic boundary conditions in the horizontal direction [151, 207]. A completely different approach was adopted in Bhole et al. [38] and Sengupta et al. [459, 467], where the authors used a very high accuracy dispersion relation preserving numerical method to solve the non-periodic Rayleigh–Taylor instability problem in two-dimensional DNS framework by solving the compressible Navier– Stokes equation. Near-spectral accuracy compact scheme [413, 439] with appropriate boundary closure helped solve the non-periodic instability problem and matching the experimental results in [357]. It has been noted in [459] that one can perform high resolution two-dimensional analysis for Rayleigh–Taylor instability, as compared to poorly resolved three-dimensional computations. It has been further pointed out that mere use of a compact scheme is not a guarantee for success, as earlier on, some researchers [75] observed; saying that “the availability of even more powerful computers has led to a somewhat ironic state of affairs, in that agreement between simulations and experiments is worse today than it was several decades ago.” This has nothing to do with hardware or the use of compact schemes, as the accuracy reported in [38, 456, 459, 460, 461, 467] is quite spectacular in matching the experiment in [357]. The reason for poor results in [75] have been since diagnosed [459] as due to: (a) use of the compact scheme with poor numerical dispersion and dissipation properties; (b) use of the periodicity condition in horizontal directions and (c) the use of incompressible formulation with Boussinesq approximation. In fact, the numerical diffusion was so excessive that a long wavelength single mode forcing was required to trigger onset of the instability. Like other similar methods, this approach of single mode excitation depends on a forward energy cascade, while the experiments [6, 357]

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clearly reveal the onset to be due to a very high wavenumber event originating from the junction of the interface with the vertical wall, which has been replicated in [38, 456, 459, 460, 461, 467]. In [459, 467], results have been presented for A = 0.105583 and a temperature differential of 70K, for which Boussinesq approximation will be totally inadequate. The numerical method has been thoroughly analyzed [467, 456] to show that the method used is neutrally stable for the convection operator, with very good dispersion relation preservation property. Despite the good properties, it is noted in [456] that the results obtained using CFL numbers of 0.09 and 0.009 showed drastically different onset, even when the terminal mixed stage appeared similar. This only highlights the extreme sensitivity of Rayleigh–Taylor instability to residual background disturbances arising due to truncation and round-off error, as we have also noted for lid-driven cavity flow in Chapter 10. There have been few other experiments performed for Rayleigh–Taylor instability. Ratafia [347] experimentally demonstrated the instability for alcohol and water mixture with A = 0.095, when the interface is perturbed by a single sinusoid. The perturbation was created by a downward acceleration which is greater than acceleration due to gravity, leading to formation of structures called the bubbles and spikes. A computed case is shown in Figure 12.2 with these structures marked. As noted already, the onset of instability is due to the baroclinic vorticity generation evidenced by creation and evolution of these bubbles and spikes. These arise when lighter fluid penetrates heavier fluid and vice versa. These events are associated with incipient pressure waves. Lawrie [257] explains the onset as due to small initial perturbations caused by baroclinically generated vorticity, which organize into “selfadvected structures known as bubbles, if they contain relatively less dense fluid and are rising upwards, and spikes if they are more dense..... While the bubbles and spikes remain coherent structures, the growth rate of the instability decreases” as a result of dynamic balance between buoyancy and drag. The density contours in Figure 12.2 show migration of bubbles upwards via a leapfrog motion, with new bubbles

spatio-temporal wavefront bubbles

rho 1.00761 1.00244 1.00085 0.998308 0.977895 0.959519 0.924066 1.86193 0.841643 0.830774 0.811957

spike

Figure 12.2 The mechanism of generation and propagation of bubbles and spikes as fronts shown with density contours.

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appearing above the spike, while older bubbles weaken and disappear. In a similar action of balance between drag and buoyancy, the denser fluid sinks into lighter fluid, assisted by gravity to create the spikes. Thus, it is observed that the coherent vortical structures in the form of spikes and bubbles are caused by pressure fronts at the onset stage. Growth of Rayleigh–Taylor instability has been investigated experimentally [357] by generating constant acceleration up to 150 m/sec2 with the help of solid propellant rocket motor. Read [357] was able to create Rayleigh–Taylor instability without any initial perturbation, which is relevant for the accurate simulations reported in [459, 460, 461, 467] without any artificial periodicity imposed in the horizontal direction. Also, Read [357] reported some experiments which used a tank with square cross section, so that the instability is caused by a three-dimensional mechanism. Other experiments used a tank with depth being one sixth of the width, so that the resultant flow can be treated as two-dimensional, and these experimental results were used for comparison in [459, 460, 461]. Andrews and Spalding [6] reported effects of tilt on Rayleigh–Taylor instability and observed bowing of the mixing region, apart from billowing from the side wall, as has been previously shown in [357]. Effects of the time rate of acceleration on mixing and amplitude of bubble growth have been reported in [110]. An experimental investigation has been specifically carried out by Roberts and Jacobs [373] inducing Rayleigh–Taylor instability by forcing at small scale with finite bandwidth perturbations. Such forcing accelerates creation of turbulence. Despite the widespread use of the Stokes’ hypothesis [501] in simulating the compressible Navier–Stokes equation, many researchers in fluid mechanics and thermodynamics have raised questions about the correctness of adopting it [71, 101, 129, 140, 161, 219, 264, 343, 376]. The Lame coefficients µ and λ used in the constitutive relation between stress and rates of strain are relevant for compressible flows [255]. Stokes hypothesis postulates the bulk viscosity (κ = λ + 23 µ) to disappear in the flow field. While this is not relevant for incompressible flow, it amounts to treating thermodynamic pressure and mechanical pressure as one and the same in compressible flows. It is noted in [461] that λ is independent of µ and can be orders of magnitude higher in amplitude, with the sign reversed [219, 264, 376]. Stokes’ hypothesis is questionable such as during spacecraft making their re-entry into the earth’s atmosphere [129, 140]. Without Stokes’ hypothesis, the second coefficient of viscosity is given as, λ = (− 23 + µκ )µ, with a regression model developed in [460] based on acoustic absorption data in [10, 11]. In this chapter, results will be presented with and without Stokes’ hypothesis, by clearly identifying this in each figure.

12.2 Numerical Simulation of Rayleigh–Taylor Instability The numerical computational domain used here is shown in Figure 12.1, with the initial schematic shown in frame (a). This involves taking an insulated box, and

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making two chambers initially by an insulating partition. These chambers are filled with air at t = 0 kept at constant temperatures. To initiate the instability at t = 0, this partition is removed impulsively, allowing the fluids with density and temperature discontinuities to mix. Thus, the flow properties can be represented by a Heaviside function as one crosses the initial interface. Similar variations are noted along the xdirection, at the sidewalls of the box. The time variation of physical variables also displays similar jump discontinuity. Thus, mathematically, such initial perturbation excites all wavenumbers and circular frequencies. The physical process is driven by the destabilizing potential energy, as the heavier fluid is atop the lighter fluid. Physically, the range of wavenumbers and circular frequencies excited are functions of this destabilizing potential energy, and the relation between spatial and temporal scales is dictated by the dispersion relation for the instability. The aim of the investigation from the perspective of thermodynamics in [459, 460, 461], is to consider the isolated system at all times, and follow the physical process by a consistent formulation (considering compressible flow without Boussinesq approximation, and comparing the flow behaviors, with and without the Stokes’ hypothesis) and highly accurate numerical methods suitable for the non-periodic problem. The accuracy of the method is ensured by monitoring conservation laws and tracking the transient variation of the entropy for the isolated system. This entropy variation can be associated with vorticity generation at no-slip wall and baroclinic vorticity generation at the interface during the Rayleigh–Taylor instability. For the two-dimensional flow considered [459, 460, 461], the governing equations are given by the unsteady Navier–Stokes equation involving conservation of mass, momentum and energy. The four nonlinear partial differential equations for the two-dimensional flow are given in conservation form, expressed for the fluxes as follows [184] ∂F ∂Ev ∂Fv ∂Q ∂E + + = ∗ + ∗ ∂t∗ ∂x∗ ∂y∗ ∂x ∂y

(12.2)

with the dimensional variables given by the quantities marked with asterisks  ∗  ρ  ρ∗ u∗ Q =  ∗ ∗  ρ v ρ∗ e∗t

     

(12.3)

The convective inviscid fluxes E and F are given as  ρ∗ u∗   ρ∗ u∗2 + p∗ + (ρ∗ − ρ )g x s x E =  ρ∗ u∗ v∗  (ρ∗ e∗t + p∗ )u∗

     

(12.4)

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 ρ∗ v∗   ρ∗ u∗ v∗ F =  ∗ ∗2  ρ v + p∗ + (ρ∗ − ρ s )gy y (ρ∗ e∗t + p∗ )v∗

     

(12.5)

and the viscous fluxes Ev and Fv are    Ev =  

0 τ∗xx τ∗xy ∗ ∗ u τ xx + v∗ τ∗xy − q∗x

     

(12.6)

    Fv =   

0 τ∗yx τ∗yy u∗ τ∗yx + v∗ τ∗yy − q∗y

     

(12.7)

In Eq. (12.2), the variables ρ∗ , p∗ , u∗ , v∗ , T ∗ and e∗t represent the density, the fluid pressure, Cartesian components of fluid velocity, the absolute temperature and the specific internal energy of the fluid, respectively, and τ∗xx , τ∗xy , τ∗yx , τ∗yy are the components of the stress tensor which are related to velocity gradients as τ∗xx

∂u∗ ~∗ = 2µ ∗ + λ∇∗ · V ∂x

τ∗yy

∂v∗ ~∗ = 2µ ∗ + λ∇∗ · V ∂y

τ∗xy

=

τ∗yx

!

!

∂u∗ ∂v∗ =µ + ∂y∗ ∂x∗

!

The thermal conductivity (¯κ) has been held constant in the reported results in [459, 460, 461]. Use of the Stokes’ hypothesis, relates µ with λ by the expression, λ = −2µ/3. The heat conduction terms are given as q∗x = −¯κ

∂T ∗ ∂T ∗ ∗ and q = −¯ κ y ∂x∗ ∂y∗

The governing equations are closed by the equation of state for perfect gas, given by p∗ = ρ∗ R∗ T ∗

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and the perfect gas law is used to define specific energy by e∗t = Cv T ∗ +

(u∗2 + v∗2 ) 2

The same governing equations and numerical framework have been developed for the study of transonic flows past aerofoils in [37, 426]. The governing equations are nondimensionalized for all variables with appropriate scales shown here with subscript s as: t=

t∗ U s , L

ρ=

ρ∗ , ρs

x=

θ=

x∗ , L

y=

T ∗ − Ts , ∆T

y∗ , L

u=

p=

p∗ − p s , ρ s U s2

u∗ , Us

v=

et =

v∗ Us

e∗t − ets U s2

L, U s , ∆T , and ρ s are the characteristic length, velocity, temperature difference and density, respectively. The standard conditions correspond to air at normal temperature (T s = 298K), pressure (p s = 101 kN/m2 ) and density (ρ s = 1.225 kg/m3 ). Five characteristic numbers emerge from nondimensionalization and these are: Reynolds number (Re), Prandtl number (Pr), Froude number (Fr), Eckert number (Ec) and GayLussac number (Ga) given by Re =

ρs U s L , µs

Pr =

µ sC p , κ¯

Fr =

U s2 , gL

Ec =

U s2 , C p ∆T

Ga =

∆T Ts

where Froude and Eckert numbers represent the ratio of inertial to gravitational force and enthalpy difference, respectively. For the results reported here are those have been presented in [459, 460, 461]. The Gay–Lussac or thermal expansion number relates the temperature difference from the reference temperature which is 298K. The reference velocity is taken as 26.52357 m/sec and the length scale used is 5.5262m, so that Re = 107 [459]. Other reference values needed are for µ s , C p and κ¯ , which are obtained by fixing the value of Ec = 0.01, and Pr = 0.7 (for air). Similarly, the reference Mach number is calculated from ρ s and ∆T . The nondimensionalized governing equations are obtained in similar form (without any asterisk) with the fluxes indicated with hat as ∂Qˆ ∂Eˆ ∂Fˆ ∂Eˆv ∂Fˆv + + = + ∂t ∂x ∂y ∂x ∂y

(12.8)

The normal and shear stress components with Stokes hypothesis are obtained as τ xx =

" # 1 4 ∂u 2 ∂v − Re 3 ∂x 3 ∂y

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" # 1 4 ∂v 2 ∂u τyy = − Re 3 ∂y 3 ∂x " # 1 ∂u ∂v τ xy = τyx = + Re ∂y ∂x

(12.9)

The nondimensional heat conduction terms are simplified as qx = −

∂θ 1 ∂θ 1 and qy = − RePrEc ∂x RePrEc ∂y

(12.10)

R The equation of state transforms to, p = ρRθ − Ga (1 − ρ), where R = R∗ ∆T/U s2 . The presented results in [459, 460, 461] are obtained using the nondimensional physical Cartesian grid, which provides the benefit of avoiding aliasing error faced with transformed plane equations, even for linear terms.

12.2.1 Physical domain and auxiliary conditions The flow computed and reported in [459, 460, 461] are for a domain: 0 ≤ x ≤ 60 and 0 ≤ y ≤ 30, discretized by uniformly spaced grid lines having 3000 and 1800 points in the x- and y-directions, respectively. When energy travels from large to small scales in a flow, then it is termed as a direct cascade. For Rayleigh–Taylor instability, one observes an inverse cascade following the onset of instability through the creation of scales starting from small to large. The exact onset location is at the junction of the interface with the side-walls and energy transfer is via a billowing motion. It is followed by creation of larger scales by the bowing motion, which has been reported experimentally in [6, 357]. The success of the computing Rayleigh–Taylor instability rests on the use of compact OUCS3 scheme for discretizing convective acceleration terms, along with two-dimensional compact filters (to control numerical instability at very high wavenumbers) that are used for spatial discretization and time integration is by a four-time level OCRK3 method. Initially a horizontal interface separates the same fluid of different densities exactly at the middle height of the box (OPQR) shown in Figure 12.1. The boundary conditions are applied on the variables u, v, p, θ (or T ), as deduced from the components of state vector Q in Eq. (12.3), at all times. Along the solid walls (OP, PQ, QR, RO) for x ≥ 0 and y ≥ 0, no-slip (u, v ≡ 0) and adiabatic conditions ( ∂T ∂n = 0) are imposed. These render the thermodynamic system completely isolated from the surrounding. Therefore, it can be treated as a “thermodynamic universe consisting of both the system and the surrounding” [459, 460]. Results have been presented by solving the compressible formulation of the Navier–Stokes equation, with and without Stokes’ assumption. This study allows a deeper understanding of Rayleigh– Taylor instability from flow instability and thermodynamics perspectives. For fluid dynamics, one studies Rayleigh–Taylor instability from linear to nonlinear stages,

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as well as estimating rotationality created by the baroclinic mechanism. From the thermodynamics aspect, one studies it in order to know how entropy is generated for a non-equilibrium thermodynamic system having a finite mass and fixed identity. Reported entropy variation in [459, 460, 461] shows non-monotonic growth of entropy viewed as fluctuations and justified in [286], using Rayleigh–Taylor instability as the test case for a finite-sized evolving dissipative system by considering a fluctuation theorem and the principle of least action. In [459], the flow is simulated with the heavier (colder) air on the top chamber at t = 0 with temperature given by T = T s (298K), so that θ = 0; and θ = 1 for the warmer air in the bottom chamber. The reference density is ρ s and the temperature on the bottom of the interface at t = 0 is T l = 368K. The densities of air, above and below the interface, are obtained at t = 0, in terms of reference density ρ s as, ρu = ρ s and ρl = ρ s TTul . The boundary condition ∂T ∂n = 0 implies zero heat transfer with the surrounding, while the pressure boundary condition ( ∂p ∂n = 0) is somewhat subtle. Here, the equilibrium state is quiescent and close to the surrounding wall convection is negligible. Thus, in the proximity of the wall, one can consider it as a creeping flow, with the pressure gradient balanced by viscous diffusion. For example, on the wall segment PQ, the wall-normal pressure gradient will be balanced by ∇2 v. However, the latter being negligibly small, it justifies the use of the pressure boundary condition.

12.3 Numerical Methods and Validation The numerical methods to solve the Navier–Stokes equation are the same as those used in [38, 426]. The optimized upwind compact scheme developed in [438], as the OUCS 3 scheme, is used to discretize inviscid flux derivatives; the classical second order central difference (CD2) scheme is used to discretize viscous diffusion terms and optimized dispersion relation preserving OCRK3 scheme; and a variant of four-stage Runge–Kutta method is used [450] in all simulations reported on Rayleigh–Taylor instability in [459, 460, 461].

12.3.1 Validation of numerical results for Rayleigh–Taylor instability The numerical methods used for simulation of Rayleigh–Taylor instability are validated with available experimental data. In [38, 456, 459], such a validation of numerical simulations has been performed with the visualization pictures shown in [357]. Despite so many computed results available in the literature, very few report such a comparison, except the acknowledgment in [75] about one such failed attempt using a so-called high order compact scheme of Lele [261]. The reasons for this failure have been stated in Section 12.1, as given in [459]. To that one can add the fact that this instability onset is due to high wavenumber components originating from the junctions between the interface and the side walls. Subsequent events via billowing and bowing imply creation of disturbances at smaller wavenumbers followed by

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transfer to even smaller wavenumbers. Such inverse cascade of energy associated with disturbance evolution requires numerical methods which strictly abide by the physical dispersion relation of the instability. This is the reason that simulations in [459, 460, 461] are successful and others are not. This successful comparison is evident in the computed density contours shown in Figure 12.3 compared with the visualization results taken from Read [357]. The experimental results shown in the right frames indicate the onset with the first frame setting the datum marked as t = 0 ms, which is characterized by small scale structures at the junction of the interface with the sidewalls. Computations have been performed using a uniform grid with 3000 points in the horizontal direction and 1800 points in the vertical direction and the results shown in the left column are shown at the indicated non-dimensional time for the non-dimensional density contours. In the subsequent frames one can see the inception of disturbance fronts from the sidewalls in the experimental and computational frames. While the experimental results reported in [357] are for both two-dimensional and three-dimensional set-ups, the comparison is made here with the former set of experiments. The onset of Rayleigh–Taylor instability, characterized by billowing along the interface, starts from the junction and progresses inwards. The experimental frames from [357] are shown for qualitative comparison of simulation of the compressible Navier–Stokes equation. In the top frames of Figure 12.3, computed results for t = 3 and the experimental results at t = 0 ms, clearly show the disturbances originate from the junctions at very small length scales. In the following frames one notices similar small scale disturbances all along the interface triggered by baroclinic vorticity instability, seen in the experimental result at t = 22.1 ms. Comparisons of subsequent frames show good qualitative match between sequence of events in the computation and the two-dimensional experiment. The computational time scale is about 200ms, which is slower compared to the experimental value. This good qualitative match suggests that solving it as a non-periodic problem is the correct approach, rather than solving it as a periodic problem [75]. In later time-frames of Figure 12.3, further billowing of disturbances from the interface junction with the side-walls are noted. At the same time, prevalent disturbances in the interior evolve into spikes and bubbles.

12.3.2 How sensitive is baroclinic instability? It is known thus far that baroclinic instability has a strong disturbance growth mechanism, and yet many past numerical simulations used either dissipative numerical methods or used methods which inherently filter out high wavenumbers due to the spectral properties of spatial discretization. Thus, these required an explicit excitation to trigger the instability [75, 97, 103, 345, 358, 548, 568]. Some of these efforts needed single wavelength excitation (which is equivalent to studying temporal instability with real wavenumber specified for the Rayleigh–Taylor instability), as in [358]. The simulation has been performed in an infinite domain to avoid reflection of pressure waves from the lateral boundaries. In that respect, the simulations reported

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Baroclinic Instability t = 16.00

(a)

Min = 0.7774; Max = 1.0353

y

4

2 0

0

2

(b)

4

t = 18.00

x 6

8

10

1.02 0.98 0.96 0.94 0.92 0.9 0.88 0.86 0.84 0.82 0.8

y 2

0

2

(c)

0ms

Min = 0.7810; Max = 1.0605

4

4

t = 22.00

x 6

8

10

16.6ms

Min = 0.7968; Max = 1.0361 1.02 0.98 0.96 0.94 0.92 0.9 0.88 0.86 0.84 0.82 0.8

y

4

2 0

1.02 0.98 0.96 0.94 0.92 0.9 0.88 0.86 0.84 0.82 0.8

0

2

(d)

4

t = 30.00

x 6

8

10

Min = 0.7826; Max = 1.027

y

4

2 0

0

(e)

2

4

t = 18.00

x 6

8

10

27.6ms

1.02 0.98 0.96 0.94 0.92 0.9 0.88 0.86 0.84 0.82 0.8

y 2

0

2

4

t = 50.00

(f)

x 6

8

10

Min = 0.7884; Max = 1.0208

y

4

2 0

1.02 0.98 0.96 0.94 0.92 0.9 0.88 0.86 0.84 0.82 0.8

Min = 0.7810; Max = 1.0605

4

0

22.1ms

0

2

4

x 6

8

10

44.1ms

1.02 0.98 0.96 0.94 0.92 0.9 0.88 0.86 0.84 0.82 0.8

60.8ms

Figure 12.3 Numerical simulations compared with Read’s experiment [357] with simulation results showing non-dimensional density at non-dimensional times: (a) t = 3; (b) t = 5.5; (c) t = 8; (d) t = 12; (e) t = 21; and (f) t = 34 shown in the left frames. On the right frames are the experimental results at the indicated dimensional times in milliseconds.

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in [459, 460, 461] not only captured the onset of Rayleigh–Taylor instability, as in the experiment [357], but the method did not require any added excitation at all in depicting the subsequent spikes and bubbles. The efficacy of the numerical method employed in these simulations was analyzed by Sengupta et al. [456]. It has been noted in Eq. (12.1) that the baroclinic instability term originates from the vorticity transport equation obtained from the Euler equation, and hence the numerical method has been calibrated with the help of a one-dimensional wave equation in [456]. It should be understood that to simulate physical instabilities, adopted methods must display neutral numerical instability, as has been explained in [413, 420, 456]. In [456], computations are compared for the CFL numbers of Nc = 0.009 and 0.09, to exhibit that the onset of Rayleigh–Taylor instability are completely different, even though the terminal states have similar nonlinear, fully developed, evolved mixed stages. The difference was traced to almost an insignificant variation in numerical amplification factors in the seventh decimal place between the two values of CFL numbers in [456]. The evidence of different structures of spikes and bubbles for these two CFL numbers are shown in Figure 12.4, with a small section of the interface zoomed at t = 6.599 for Nc = 0.09 and at t = 9.269 for Nc = 0.009. These spikes and bubbles are indicators for the formation and growth of the mixing region. As described before, the spikes are packets of heavier fluid parcels going below the original interface. The bubbles are parcels of fluid with lower density rising into the heavier fluid which was initially above the interface. (a)

t = 009.269

(b)

Nc = 0.009

t = 006.599 2.9

2.8

2.8

2.7

2.7

rho

1.00761 1.00244 1.00085 0.998308 0.977895 0.959519 0.924066 0.86193 0.841643 0.830774 0.811957

y

y

2.9

Nc = 0.09

2.6

2.6

2.5

2.5

2.4

2.4

4.4

4.6

4.8 x

5

5.2

4.4

4.6

4.8 x

5

5.2

Figure 12.4 Onset of Rayleigh–Taylor instability shown by numerical simulation using a (3000 × 1800)-grid with different time steps, for non-dimensional density at the indicated times for: (a) t = 9.269 with CFL number of 0.009; (b) t = 6.59 with CFL number of 0.09. One of the striking features of the spikes and bubbles at the onset of instability is the geometrical symmetry displayed by these. Lawrie [257] has noted that the “bubble

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Baroclinic Instability

and spike structures eventually overturn due to their own vorticity and acquire a mushroom-like appearance,” which is clearly noted in Figure 12.4(b). However, one notices distinct differences between the flows shown in frames (a) and (b). While one can seek reasons for these from numerical analysis, the physical origin of the differences can be viewed from the perspective that the case with lower CFL number is closer to numerical symmetry with growth rate almost absent and hence tries to retain its equilibrium state for a longer duration. Also in Figure 12.4, one clearly notes the formation of a front above the bubbles in both the frames, where the fluid density is higher than what was present in the top half of the box at the onset. This indicates a compression front preceding the rows of bubbles. For instabilities caused by wall and free stream excitation noted for boundary layer forming over a semi-infinite flat plate, one has noted such fronts growing in space and time as the STWF, which cause eventual transition, as described in previous chapters. This again highlights that receptivity of fluid flow, with the case shown in Figure 12.4(b), experiences higher disturbance growth by larger departure of numerical amplification rate from neutral numerical stability (noted in the seventh decimal place). This is determined by the error dynamics in computing [436].

12.3.3 Numerical amplification rate and error dynamics One can explain the reasons for the differences noted in Figure 12.4 between the frames (a) and (b) by reproducing the salient features of error dynamics explained with the help of the one-dimensional convection equation given in [413] as ∂u ∂u +c = 0; ∂t ∂x

for c > 0

(12.11)

To analyze any numerical method, the dependent variable here, is represented by a R hybrid notation, u(xm , tn ) = U(kh, tn ) eikxm dk. The physical plane is transformed to the spectral wavenumber (k)-plane, with the physical time remaining unchanged. One of the vital numerical parameters in characterizing any numerical method is the n +∆t) amplification factor which is defined by, G(∆t, kh) = U(kh,t U(kh,tn ) , introduced already in Chapter 3 for the convection-diffusion equation. G is a complex quantity, as is U(kh, t). It is obvious that in the continuum limit ∆t → 0, one must obtain, |G| ≡ 1, even though the real and imaginary parts can define different phase relations for different methods. This gives rise to dispersion error and is briefly explained in the following. If the initial condition for Eq. (12.11) is given as u(x j , t = 0) = u0j =

Z U0 (k) eikx j dk

(12.12)

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then the solution at any arbitrary time is written as unj =

Z U0 (k) [|G j |]n ei(kx j −nφ j ) dk

(12.13)

where |G j | = (G2r j + G2i j )1/2 and tan φ j = −Gi j /Gr j , with Gr j and Gi j being the real and imaginary parts of G j , respectively. The numerical phase speed (cN ) is determined from nφ j = kcN t, while, the physical phase speed is constant for all k. This implies that the numerical solution is dispersive, in contrast to the nondispersive R physical solution. The numerical solution of Eq. (12.11) is thus, denoted as, u¯ N = A0 [|G|]t/∆t eik(x−cN t) dk. The numerical dispersion relation is, ωN = cN k, as compared to the physical dispersion relation: ω ¯ = ck, implying that the non-dimensional phase speed and group velocity at the jth node is expressed as [413] c  N

c V

gN

c

j

=

 j

φj ω∆t ¯

=

(12.14)

1 dφ j hNc dk

(12.15)

Next, the numerical error dynamics is elucidated in terms of these metrics, defining the space–time discretization used in the numerical method. The computational error is defined as, e(x, t) = u(x, t) − u¯ N (x, t), for the governing equation, Eq. (12.11), and its dynamics is governed by [436]  ∂e ∂e cN  ∂¯uN +c = −c 1 − − ∂t ∂x c ∂x Z −

Z

Z  dcN  0 ik0 A0 [|G|]t/∆t eik (x−cN t) dk0 dk dk

Ln |G| A0 [|G|]t/∆t eik(x−cN t) dk ∆t

(12.16)

This is the correct error propagation equation, which is dependent on the numerical amplification factor, numerical phase speed and numerical group velocity as functions of nondimensional wavenumber kh, with h as the uniform grid spacing (as is usually the case in transformed plane computations). It is readily observed that the right-hand side terms contribute to forcing the spatio-temporal growth of the error. The third term on the right-hand side clearly shows why neutral stability is mandatory for the wave equation. To depict baroclinic stability, it is to be noted that physical instability is governed by the Euler equation as given by the misalignment of the pressure gradient with the density gradient. Also to be noted is that a so-called numerically stable method will also have non-zero contribution and contribute to error, which in turn

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Baroclinic Instability

will trigger physical instability. The least error occurs when cN  c and VgN  c, i.e. there is no dispersion error (as explained in Chapter 3). Such low error is obtained near the origin in the (kh, Nc )-plane, for the numerical properties plotted in the parametric plane. The properties |G| − 1, 1 − (cN /c) and VgN /c are shown in Figure 12.5 for Nc = 0.001 to 0.009, with the plotting done using eight digit precision, while all the computations are done using double precision arithmetic. In frame (a), i.e. plotted values of (|G| − 1)

(a) Effect of |G| regarding the initial instability used to trigger RTI Nc = 0.001 Nc = 0.002 Nc = 0.003 Nc = 0.004 Nc = 0.005 Nc = 0.006 Nc = 0.007 Nc = 0.008 Nc = 0.009

1E-07

|G|-1

8E-08 6E-08 4E-08 2E-08 0

0.5

1

1.5 kh

2

2.5

3

(b) 1 1

0 −1

0.6 Nc = 0.001 Nc = 0.002 Nc = 0.003 Nc = 0.008 Nc = 0.009

0.4

VgN /c

1-(cN /c)

0.8

−2 Nc = 0.001 Nc = 0.002 Nc = 0.003 Nc = 0.008 Nc = 0.009

−3 −4

0.2

−5 0 1

kh

2

3

0.5

1

1.5 kh

2

2.5

3

Figure 12.5 Numerical properties of the method used [456, 459, 460, 461] for Rayleigh–Taylor instability calibrated with a one-dimensional convection equation with the help of (a) (|G| − 1); (b) (1 − cN /c) and (c) VgN /c for the CFL number range of 0.001 to 0.009.

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shown for the range of Nc , it may be observed that the quantity retains numerically the machine-zero value for Nc ≤ 0.007, while for 0.008 and 0.009 the numerical amplification factor is marginally greater than one in the eighth decimal place for a limited range of higher wavenumbers. This provides the necessary amplification of the seed provided by numerical error (truncation and/or round-off error) for the physical baroclinic instability in the cases with Nc = 0.009 and the higher value of 0.09 in Figure 12.4. For the higher Nc value, error amplifies faster and one notices the earlier formation of the spikes and bubbles into distinct structures. For the lower Nc value, the evolution of interface instability occurs as a sequence of pressure waves on the bubble-side, while the spike formation is aided by buoyancy as discrete structures. Thus, a near-neutral method, with plotted |G| against kh helps one identify the scale ranges. It is also observed that with increased Nc value, excitation is evident over a wider range of excited wavenumbers. In frames (b) and (c) of Figure 12.5, one does not note any difference in the dispersion error patterns for the plotted range of Nc values. The dispersion error becomes significant for kh  2.0 onwards, both in terms of numerical phase speed and numerical group velocity. Thus, the difference in the spikes and bubbles shown in Figure 12.4 are essentially due to difference in the values of |G| for the two CFL values. One notes that the value of Nc used in [75] is significantly larger than 0.09, despite which the authors used long length-scale forcing, as the preferred numerical method along with the filter attenuated disturbances due to excessive numerical stabilization. In Figure 12.4, one saw the different spikes and bubbles for Nc = 0.009 and 0.09 cases. Explanation for these has been provided, with the help of numerical properties in Figure 12.5. It is relevant to note that despite the difference in the onset process of Rayleigh–Taylor instability for these CFL numbers differing by a factor of ten, the eventual mixing process is not different. This can also be seen from the frames of Figure 12.6 for two dissimilar CFL numbers. The baroclinic instability shows slower appearance for the lower CFL number and its evolution is also slower with delayed appearance of the spikes and bubbles. These spikes and bubbles are associated with the formation of vortices which assist in mixing of fluids near the interface. From the bottom frames of Figure 12.6, one can note that the width of mixing region evolves at slightly different rates, but despite such differences the topology of the mixing region is driven by the same physical mechanism of baroclinic instability which originates at high wavenumbers and then transfers energy to lower wavenumbers. This is the inverse energy cascade and in literature, this is attributed to two-dimensional turbulent flows. However, for Kelvin–Helmholtz and Rayleigh–Taylor instabilities, two-dimensional disturbances grow more than the three-dimensional disturbance field for the inviscid model described in Chapter 1. As has been variously reported, from experiments and numerical simulation, baroclinic instability begins at high wavenumbers from the sidewalls, following the real flow governed by the compressible Navier–Stokes equation.

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Baroclinic Instability Without Refined Time-step

0.993529 0.887099 0.81081 0.805494

2

x6

8

10

t = 014.000

© 4

0.993529 0.887099 0.81081 0.805494

y

3 2

2

(e)

4

x6

8

0.993529 0.887099 0.81081 0.805494

y

3 2

2

4

(g)

x6

8

t = 018.000

0.993529 0.887099 0.81081 0.805494

y

3 2 1 00

2

(iiI)

4

x6

8

t = 020.000

y

0.993529 0.887099 0.81081 0.805494

3 2 1 2

4

x6

0.993529 0.902673 0.819785 0.805494 2

8

10

4

x6

8

10

t = 020.000

4

0.993529 0.887099 0.81081 0.805494

3 2 1 0 0

2

4

x6

8

10

t = 022.000

4

0.993529 0.887099 0.81081 0.805494

3 2 1 00

2

4

x6

8

10

t = 023.000

(h) 5

4

00

1 0 0

10

5

10

3

(h) 5

4

8

2

10

5

x6

t = 018.009

(f) 5

5

4

4

10

t = 016.000

4

1 0 0

2

(d) 5

5

1 0 0

1 0 0

y

4

0.993529 0.887099 0.81081 0.805494

3 2

y

2

Nc = 0.09

t = 016.000

4

y

y

3

(b) 5

y

4

1 0 0

Without Refined Time-step Nc = 0.09

t = 012.000

4 y

(a) 5

0.993529 0.887099 0.81081 0.805494

3 2 1 00

2

4

x6

8

10

Figure 12.6 Density contours as function of time for Nc = 0.09 (on the left) and Nc = 0.009 (on the right) cases to illustrate the flow to be identical at later times, despite the onset being dissimilar.

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12.3.4 Creation of pressure and density fronts during primary instability Figure 12.7 shows the magnified view of the interface in the range 4.4 ≤ x ≤ 5.2 to explain events during primary instability with the help of density contours. The calculation is done with the CFL number of 0.09, and the frame at t = 5.019 displays the early stage of primary instability. One notes the inclined contours in the shape of fingers at intermediate length scales, and the alignment of the pressure gradient with respect to the mean interface is indicated by the slant angle of these structures, which are in the opposite directions with respect to the middle of the interface located at x = 4.8. The interface evolves into two distinct symmetric structures at t = 5.499, noted on either side of the central point featuring the leading spike. Bubbles are observed to form on top of the mixing region in a layered sequence in Figure 12.7, with the most two prominent bubbles located near x = 4.57 and 5.12. The other aspect of the onset of primary instability is in the formation of two density fronts featuring density differences in ambient fluids. In the top left frame, a density front at y = 2.76 is seen and this is of higher density than the maximum density at the beginning (t = 0). Just below this density-front, above the finger-like structures, one observes even higher density, implying that the interface is the source of successive density-fronts. This successive shockwave-like feature above the interface with increased density and pressure is accompanied by a complementary rarefaction-front below the interface at y = 2.6. This rarefaction-front has a density more than the density of the lighter fluid at t = 0. In the early stages, the evolution of the instability occurs in a quasi-steady fashion and a rarefaction front is expected to lower entropy. This is not precluded for the system with finite mass and energy undergoing Rayleigh–Taylor instability, which makes the fluid system to deviate significantly from the equilibrium stage. The rarefaction-front is not continuous during the primary instability, which is punctuated by connecting links with viscous fingers in the center, as seen at t = 5.019. With time, the spikes and bubbles grow and the associated fronts about the mean interface conform to the growing shapes of bubbles and spikes, while maintaining symmetry about the centre of the interface. The finite size of the box on all sides affects the primary structures by the no-slip walls on all sides, which eventually leads to breakdown of symmetries, as noted in frames of Figure 12.7 at later times.

12.3.5 Baroclinic vorticity generation during primary instability The onset of Rayleigh–Taylor instability is caused by the omnipresent background disturbances in the quiescent equilibrium flow by the destabilizing moment caused by the baroclinic term in Eq. (12.1). This further deforms the interface by the baroclinic generation of vorticity as a consequence of progressive misalignment of pressure and density gradients. This onset of vorticity generation during the instability is concentrated at the junction of side-walls with the interface of the fluid. These vortices advect causing the interface to progressively stratify in a slanted manner. In Figure 12.8, the vorticity contours are plotted for early times in a region spanned

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Baroclinic Instability t = 005.499

t = 005.019 2.9

2.9

y

2.7 2.6 2.5

2.7 2.6 2.5

2.4

2.4 4.6

4.8 5 x t = 005.019

4.4

5.2

2.9 2.8 y

2.7 2.6 2.5 4.6

4.8 5 x t = 007.019

2.7 2.6 2.5 4.4

5.2

y

2.7 2.6 2.5

4.8 5 x t = 007.499

5.2

rho 1.00301 1.0085 0.981972 0.966493 0.924066 0.851419 0.834919 0.811957

2.8 2.7 y

1.00301 1.0085 0.981972 0.966493 0.924066 0.851419 0.834919 0.811957

4.6

2.9

rho

2.8

2.6 2.5

2.4

2.4 4.6

4.8 5 x t = 005.019

4.4

5.2

2.9

2.5

4.8 5 x t = 005.019

5.2

rho

2.8

1.00301 1.0085 0.981972 0.966493 0.924066 0.851419 0.834919 0.811957

2.7 y

y

2.7 2.6

4.6

2.9

rho 1.00301 1.0085 0.981972 0.966493 0.924066 0.851419 0.834919 0.811957

2.8

2.6 2.5 2.4

2.4 4.4

rho 1.00301 1.00085 0.981972 0.966493 0.924066 0.851419 0.834919 0.811957

2.4

2.9

4.4

5.2

2.8

2.4 4.4

4.8 5 x t = 005.019

2.9

rho 1.00301 1.0085 0.981972 0.966493 0.924066 0.851419 0.834919 0.811957

4.6

y

4.4

1.00301 1.0085 0.981972 0.966493 0.924066 0.851419 0.834919 0.811957

2.8 y

1.00301 1.0085 0.981972 0.966493 0.924066 0.851419 0.834919 0.811957

2.8

4.6

4.8 x

5

5.2

4.4

4.6

4.8 x

5

5.2

Figure 12.7 Events during Rayleigh–Taylor instability from viscous finger formation to mushroom-shaped bubbles and spikes and eventual breakdown of symmetry, shown with non-dimensional density contours at non-dimensional times: (a) t = 5.019, (b) t = 5.499, (c) t = 6.019, (d) t = 6.499, (e) t = 7.019, (f) t = 7.499, (g) t = 8.019, (h) t = 8.499. [Reproduced from “Non-equilibrium thermodynamics of Rayleigh–Taylor instability”, T. K. Sengupta et al., Int. J. Thermophysics, vol. 37(4), pp 1-25 (2016), with the permission of Springer Nature.]

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t = 004.700 omega: -23.1368

t = 004.850 omega: -23.1368

6.10829

2.8

y

y

2.8

2.6

2.6

2.4

3.2

3.4

3.6

x

3.8

2.4

3.2

t = 005.019 omega: -23.1368

3.6

3.8

3.6

3.8

3.6

3.8

3.6

3.8

6.10829

2.8

y

y

2.6

3.2

3.4

3.6

x

3.8

2.4

3.2

t = 006.019 omega: -23.1368

3.4

x

t = 006.499 omega: -23.1368

6.10829

2.8

6.10829

y

y

2.8

2.6

2.6

3.2

3.4 x

3.6

3.8

2.4

3.2

t = 007.019 omega: -23.1368

3.4 x

t = 007.499 omega: -23.1368

6.10829

6.10829

2.8

y

y

2.8

2.6

2.6

2.4

x

omega: -23.1368

6.10829

2.6

2.4

3.4

t = 005.499

2.8

2.4

6.10829

3.2

3.4

x

3.6

3.8

2.4

3.2

3.4

x

Figure 12.8 Generation and growth of vorticity shown by vorticity contours at: (a) t = 4.7, (b) t = 4.85, (c) t = 5.019, (d) t = 5.499, (e) t = 6.019, (f) t = 6.499, (g) t = 7.019, (h) t = 7.499 to characterize the onset of primary instability, narrowing strands of vortices into discrete vortices as a secondary instability which develops into spikes and bubbles. [Reproduced from “Non-equilibrium thermodynamics of Rayleigh–Taylor instability”, T. K. Sengupta et al., Int. J. Thermophysics, vol. 37(4), pp 1-25 (2016), with the permission of Springer Nature.]

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by 3.05 ≤ x ≤ 3.85 and 2.4 ≤ y ≤ 2.9. In the first frame at t = 4.7, alternately slanted and elongated rolls are noted in alternate directions. In each roll, long strips of vortices of opposite signs are seen, which are stacked side by side. Such symmetry is retained during early times, shown till t = 5.019 in Figure 12.8. Furthermore, the slanted vortices which are initially straight and narrow, undergo an oblique instability, with the straight elements becoming wavy, displaying undulations for this secondary instability progressing along the vortices. Each of the vortex elements stretch in the respective aligned directions leading to the formation of a neck at the center. Simultaneously, the ends of the straight elements grow in size and strength. This is noted for both signs of vortices. Eventually these two ends detach forming rounded vortices. Further growth of waviness of these elements causes the roll-up into distinct vortices, which can be noted in frame at t = 5.499 in Figure 12.8, which is in an intermediate stage.

12.4 Entropy and Vorticity Creation during Rayleigh–Taylor Instability A distinct feature of flow fields in a closed space is the over-riding effect of rotational motion over rectilinear motion. In Rayleigh–Taylor instability, the onset occurs by the destabilizing potential energy of a heavier fluid over a lighter one. For the configuration computed, it is determined as 439005.84 Joules. Translational motion is absent for the equilibrium state. In contrast, quantifying angular momentum and rotational energy for fluid flows is difficult, as there is no easy way to identify the inertia property of fluid parcels having distinct vorticity. Unlike mass as the inertia property for kinetic energy, there is the nine-component moment of inertia tensor for rotational motion. Additionally, deforming fluid flow will have an inertia tensor which is varying with time. To avoid such problems of quantification, enstrophy is used as a measure for specific rotational energy and its time evolution is of central interest. It has been shown [459] that the rotational motion during Rayleigh–Taylor instability accounts for more energy than rectilinear motion. During Rayleigh–Taylor instability, the mixing of two fluids with dissimilar density and temperature takes place which have ordered states at the onset. The mixing associated with instability causes the isolated system to become disorderly, with change in entropy of the system. Air is the operating fluid which can be treated as a perfect gas and removal of the barrier results in mass and heat diffusion across the air masses. For a computational time step (∆t), with temperature-dependent specific heats (as given in [379] by a cubic polynomial for c p ), one defines the change of entropy for a discrete cell by ∆smn = s(xm , yn , t + dt) − s(xm , yn , t) as    T (x , y , t + ∆t)  m n + b T (xm , yn , t + ∆t) − T (xm , yn , t) + ∆smn (t) = (a − R) Ln T (xm , yn , t)

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 d  c 2 T (xm , yn , t + ∆t) − T 2 (xm , yn , t) + T 3 (xm , yn , t + ∆t) − T 3 (xm , yn , t) 2 3

−R Ln

ρ(xm , yn , t + ∆t) ρ(xm , yn , t)

(12.17)

With R as the specific gas constant for dry air (287 J/kgK), one evaluates cv also a function of time. In [379], the coefficients of the cubic polynomial for dry air are given as, a = 970.6491713J/(kgK), b = 0.06792127J/(kgK), c = 1.65814917 × 10−4 J/(kgK), d = −6.7886740 × 10−8 J/(kgK), in Appendix A10 of the reference.

12.4.1 Relationship between entropy and vorticity The specific entropy change in a cell ( s¯mn ) identified by its co-ordinate as (xm , yn ), can be estimated from Eq. (12.17), with respect to the datum. For compressible flows, according to Crocco’s theorem, generated entropy is related to vorticity generation by [479]   ~ ~ − 1 ∇ λ(∇ · V) ~ ×V T ∇S = ∇h0 + ω ρ

(12.18)

Figure 12.9 shows the entropy (left frames) and vorticity contours (right frames) during the instability. With respect to datum state given by the temperature T s and pressure p s , the instantaneous entropy change of the system at the cell (xm , yn ) is given by s¯mn (t) = c p Ln

p(xm , yn , t) T (xm , yn , t) − R Ln Ts ps

(12.19)

Vorticity is generated during the formation of spikes and bubbles, and so also responsible for increase and decrease of entropy for the closed isolated system. Although it may appear counter-intuitive, the decrease in entropy is associated with formation of coherent vortices, which can be viewed as an enhanced order in the system which explains the fluctuating entropy. This is in variance to what is understood for equilibrium thermodynamics with associated monotonic increase in entropy. One expects such a monotonic rise in entropy for an irreversible process with infinite mass and no instability, according to the second law of thermodynamics, when the process can be construed to be represented by infinitesimal departure from equilibrium state. However, the observed variation of entropy in Figure 12.9 is due to non-equilibrium thermodynamic processes for the isolated system affected by physical instability. Prigogine [340] has stated that “if the system is perturbed, the entropy production will increase, but the system reacts by coming back to the minimum value of entropy production,” further adding that “non-equilibrium may be a source of order.” This is further observed by calculating the total entropy of the

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Baroclinic Instability Entropy Contours at t = 005.019 5

Vorticity Contours at t = 005.019 5

Entropy

4

y

3 2

4 3 2

1 0

1 2

4 0 x 6

8

0

10

Entropy Contours at t = 010.019 5

10

omega

0.015 0.01 0.005 0.001 0.0001 -0.0001 -0.001 -0.01 -0.015

4 3 2

1

1 2

4 0 x 6

8

0

10

Entropy Contours at t = 024.999

2

4 0 x 6

8

10

Vorticity Contours at t = 024.999

5

5

Entropy

y

3 2

omega

0.015 0.01 0.005 0.001 0.0001 -0.0001 -0.001 -0.01 -0.015

4 3

y

0.015 0.01 0.005 0.001 0.0001 -0.0001 -0.001 -0.01 -0.015

4

2 1

1 0

8

y

y 2

4 0 x 6

5

Entropy

3

2

Vorticity Contours at t = 010.019 0.015 0.01 0.005 0.001 0.0001 -0.0001 -0.001 -0.01 -0.015

4

0

omega

0.015 0.01 0.005 0.001 0.0001 -0.0001 -0.001 -0.01 -0.015

y

0.015 0.01 0.005 0.001 0.0001 -0.0001 -0.001 -0.01 -0.015

2

4 0 x 6

8

10

0

2

4 0 x 6

8

10

Figure 12.9 Crocco’s Theorem [479] provides a relation between entropy (left, in J/(kg-K)) and vorticity (right), and is shown at times: (a) t = 5.019, (b) t = 10.019, (c) t = 24.999. [Reproduced from “Non-equilibrium thermodynamics of Rayleigh– Taylor instability”, T. K. Sengupta et al., Int. J. Thermophysics, vol. 37(4), pp 1-25 (2016), with the permission of Springer Nature.]

isolated system as a function of time, as shown in Figure 12.10, and explained in the following.

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Domain size: 60 × 30 Domain size: 60 × 30

12 10

Entropy (∆S)

8 6 4 2 0 -2 0

1

2

3

time

4

5

6

7

Figure 12.10 Total entropy in J/K for rectangular and square (boxes) domain plotted as function of non-dimensional time. The interface instability causes spikes and bubbles forming during mixing. The discontinuous variations are due to dominance of pressure waves, while the mixing is due to diffusion with increasing entropy. [Reproduced from “Non-equilibrium thermodynamics of Rayleigh– Taylor instability”, T. K. Sengupta et al., Int. J. Thermophysics, vol. 37(4), pp 1-25 (2016), with the permission of Springer Nature.] In the previous figures, simulation results have been for a rectangular box of dimension (60 × 30). An additional set of calculation have also been reported in [459], for a square box of dimension (60 × 60). Thus, the square domain is deeper as compared to the rectangular box. The computed entropy for these two domains are shown in Figure 12.10 for elucidating the physical processes during the ensuing instability through comparison of total entropy of the isolated system for the two boxes. The solid line in the figure with diamond symbols represents the total entropy for the rectangular box, while the dashed line is for the square box. There is no exchange of energy of the system with the surrounding, and during the primary instability the total entropy shows continual growth in the mean. The effect of the depth, i.e. the distance of the walls from the interface, shows up as the entropy variation not being monotonic for the total entropy of the system. Fluctuations in the form of letter ‘M’ are observed for total entropy for the rectangular

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and the square boxes. The total entropy for the deeper square box shows distortion in the fluctuations. The rate of increase of total entropy is greater for the deeper box, and is roughly double of that in the rectangular box. This is due to the fact that mixing occurs early for the square domain, and onset is at smaller scales, as compared to the rectangular box. Pressure pulses are created after the removal of the partition and these propagate from the interface to the top and bottom of the box and upon reflection propagate back towards the interface. These multiple pulses interfere with each other in various stages of propagation, but the reflection from the solid walls are responsible for the formation of the atypical M-shape fluctuations in Figure 12.10. The increase in fluctuation in the square domain is due to the pressure pulses taking longer to reach the top and reflect back in the opposite direction resulting in rarefaction of waves in each cycle. Entropy fluctuations with time in finite-sized time-evolving dissipative systems is investigated theoretically in [286] to decipher the role of these fluctuations on the averaged entropy generation rate, over a long time interval. From the perspective of non-equilibrium thermodynamics, the fluctuation theorem and principle of least action are used by the authors [286] for the dissipative dynamical systems. The principle of maximum entropy production is conceptually the principle of least action for dissipative systems. A Thermodynamic Fusion Theorem including the fluctuation theorem and maximum entropy principle together is proposed to address the role of fluctuations in entropy production. “It identifies “entropy fluctuations” as the “least action path” for maximizing the time-averaged entropy production in a dissipative system. This theorem is explained with the entropy fluctuations in Rayleigh–Taylor instability” [286].

12.5 Role of Bulk Viscosity on Rayleigh–Taylor Instability In relating deviatoric stresses with rates of strain, one requires in principle 34 elements of a fourth ranked tensor, which is the defining relation between these tensors of rank two. However, postulating various symmetries, isotropy and homogeneity of the fluid medium, the number of unknown coefficients can be brought down to two, namely the dynamic viscosity (µ) and the second coefficient of viscosity (λ). These two are the least numbers of unknown coefficients one requires for compressible flows (see [396] for more details). However, Stokes further simplified this relation by proposing an often-used hypothesis. This hypothesis relates λ with µ, via the coefficient of bulk viscosity (κ) which is given by λ + 23 µ = κ. Stokes’ hypothesis neglects any loss of energy when a fluid parcel is reversibly compressed and dilated, as it occurs during sound propagation by longitudinal waves. Thus, κ is set to zero with Stokes’ hypothesis, which is used in many studies of Rayleigh–Taylor instability [75, 320, 358]. Computations without Stokes’ hypothesis have been reported for Rayleigh–Taylor instability in [460, 461] and in [37] for transonic flow past an aerofoil. There are some discussion on the roles of bulk viscosity, for flow of gases, as viewed from

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a gas dynamics perspective, where the continuum is explained as a consequence of reaching reversible thermodynamic equilibrium with a mono-atomic gas model making a statistically large number of collisions without loosing energy for the elastic collision model adopted. However, transfer of energy during inelastic collisions between poly-atomic molecules of gases changes the translational energy. This results in the change of the normal stresses [129] to make κ , 0. The bulk viscosity is irrelevant with incompressible flow assumption, even if the flow speeds are extremely low [459, 460, 461]. Rajagopal [343] has noted the need for a new way of writing the Navier–Stokes equation without neglecting the bulk viscosity.

12.5.1 Ash et al.’s model for bulk viscosity Ash et al. [11] have proposed a unified framework capable of handling both acoustics and fluid mechanics together by conceptualizing a viable model which will operate with non-zero bulk viscosity. This concept hinges on Zuckerwar’s [578] measured accoustic signal attenuation data to quantify bulk viscosity for air [11]. The experimental variation of γII with temperature are shown for dry and saturated air in Figure 12.11, where γII is the non-dimensional value of κ (= µγII ). Using linear 106 Marcy et al. Ash et al.

γ II

105

Dry Air

9.7

104

7.1

9.4 13.7 20.4 24.2 16.5 38.3 25.7 35.5 46.0 32.9 45.8 65.3 52.1 47.9 73.5 91.2 65.5 7.9

28.3

10

6.2 17.4 28.8

3

Saturated Air

10

20

30 T(0C)

40

50

60

Figure 12.11 Variation of nondimensional second coefficient of viscosity (γII = µλ ) at standard atmospheric pressure with temperature for air and relative humidity readings being taken at discrete points from [11], with the regression curve obtained from these data points [461]. [Reproduced from “Roles of bulk viscosity on Rayleigh–Taylor instability: Non-equilibrium thermodynamics due to spatiotemporal pressure fronts”, T. K. Sengupta et al., Phys. Fluids, 28, 094102 (2016), with the permission of AIP Publishing.]

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regression analysis of the data given in [11], a line starting from the origin provides the following relation between κ and T ∗ (in K) as in [461] κ = −7.383 × 10−4 + 3.381 × 10−4 T ∗

(12.20)

Thus, the viscous normal stress components with non-zero bulk viscosity are given by " # 1  4 κ  ∂u  2 κ  ∂v + + − + Re 3 µ ∂x 3 µ ∂y

(12.21)

" # 1  4 κ  ∂v  2 κ  ∂u τyy = + + − + Re 3 µ ∂y 3 µ ∂x

(12.22)

τ xx =

In Figure 12.11, variation of γII with T ∗ is shown with the ordinate in log scale. This is the reason that Eq. (12.20) shows a linear relation between κ and T ∗ , and the variation in Figure 12.11 for γII with T ∗ appears as nonlinear. The mean of the data for dry and saturated air is also shown. The dynamic viscosity follows Sutherland’s relation, and hence µ is not constant with T ∗ .

12.5.2 Entropy of the system: Role of bulk viscosity The bulk viscosity changes the viscous normal stresses, allowing incorporation of additional loss due to bulk action in a fluid flow. Furthermore, the altered stress system can have an effect on the onset and propagation of Rayleigh–Taylor instability. A priori, one cannot state how the presence of bulk viscosity will affect the irreversibility of the system through its effect on entropy. The resultant change in entropy of the system as a function of time is shown in Figure 12.12 to compare the results obtained using this model of Ash et al. that incorporates the bulk viscosity against results obtained by using the Stokes’ hypothesis. The computed results are based on using an identical grid and time-step (∆t = 3×10−5 ) in [459, 461]. As expected, the computed entropy is higher without Stokes’ hypothesis. This is on account of the presence of higher viscous normal stress contributing to higher losses. But, it can also be seen that time variation of entropy is almost same for both results i.e. computed with; and without Stokes’ hypothesis; till t ≈ 4, a duration within which the convection in the bulk is negligible. Following t = 4, the computed entropy is higher without the Stokes’ hypothesis. It has been shown in [460] that the increased entropy is due to enhanced instability by higher numerical amplification factors for higher CFL numbers, and this has also been observed during the onset of Rayleigh–Taylor instability in [456]. The lower seeding of physical instability during computations with Stokes’ hypothesis causes lower flux transport, and hence the entropy production is lower. As is evident, this will also reduce the slope of the total entropy curve for computations with Stokes’ hypothesis.

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( t = 3 × 10-5) 12

Asht et al.’s model With stokes’ hypothesis

10 8

STotal

6 4 2 0 -2 -4 0

2

4

6 time

8

10

12

Figure 12.12 Entropy evolution for Rayleigh–Taylor instability in an isolated box computed with Ash et al.’s model using ∆t = 3 × 10−5 . [Reproduced from “Roles of bulk viscosity on Rayleigh–Taylor instability: Non-equilibrium thermodynamics due to spatio-temporal pressure fronts”, T. K. Sengupta et al., Phys. Fluids, 28, 094102 (2016), with the permission of AIP Publishing.]

The reason for higher entropy for the case with bulk viscosity modeling has been attributed in [460] to the explanation provided in [71] as “... if we renounce to any form of Stokes’ hypothesis, it is easily seen that a further term equal to κ (divV)2 would appear in the internal energy balance,” which is the unavailable energy in the form of increased entropy. Thus, for Rayleigh–Taylor instability with the flow far from equilibrium, the contribution from κ should not be ignored. One also notices in Figure 12.12 that the fluctuation in entropy by way of M-shaped variations is present from the onset of instability, while the total entropy increases in the mean.

12.6 Pressure Waves: Outcome of Non-Periodic Boundary Conditions and Compressible Formulation The computational strategy adopted for computing the entropy of the system is the following. At each node, the entropy is evaluated using Eq. (12.19) and all such contributions are summed over the full domain, defining the thermodynamic system.

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When the partition is removed at t = 0, pressure will change immediately for very low speed (it is an incompressible flow with speed of sound being very high and so creating acoustic pressure), with temperature remaining the same due to high initial impedance. Consequently, the first term in Eq. (12.19) is virtually negligible, while the second term with the negative sign causes entropy to drop, as seen in the top frame in Figure 12.13 up to the point A. This figure shows early time variation of entropy (during t = 0 to 1) for the bulk viscosity model of Ash et al. compared with results obtained with Stokes’ hypothesis. Above t = 1 heat transfer will rapidly cause greater mixing of the two air-masses. This will also result in entropy increasing very rapidly. In the beginning, pressure will travel as pulses from the interface moving toward the top and bottom of the box, as explained in [460]. To explain the rapid decrease in entropy, between F and G in the top frame, it is necessary to plot pressure versus y in other frames of Figure 12.13. The higher value of entropy at F is due to a sharp discontinuous jump in pressure shown in the top left frame of the figure. With time progressing from F to G, the pressure pulses approach each other with opposite phase. Thus, in the process of superposition, these pulses cancel the discontinuity to a small value, resulting in a very low pressure jump, and the entropy achieves a lower value in the displayed cycle. Thus, the entropy variation is created by pressure pulses either by discrete pressure jumps, or superposition of pulses approaching each other and canceling the pressure jump, which is the characteristic of non-equilibrium thermodynamic processes. It has been already pointed out that at early times, pressure terms contribute more over temperature terms in changing the entropy. Also, convection being negligible ~ on normal viscous stresses is also marginal. at early times, contribution from κ∇ · V Thereafter, convection increases by induced contribution created by discrete vortices which form with the spikes and bubbles. One notes differences between entropy obtained with and without Stokes’ hypothesis, as shown in the top frame of Figure 12.14. Other frames show pressure variation with y for the middle of the domain at distinctive points. Pressure variation with y for the case of computations with Stokes’ hypothesis is marked with subscript 2, which is lower than the computed results obtained with models of bulk viscosity such as of Ash et al. Higher diffusion at later ~ 2 to times causes smoother time variation of entropy, while the contribution of κ(∇ · V) internal energy results in higher entropy for the model with non-zero bulk viscosity.

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S total

0 -2

1

0.1

0.05

C

0

0.2

0.4 time

0

-0.05 -0.10

3

1

Y 3

4

-0.05

0

-0.10

4

1

2

Y 3

4

5

time = 000.91199 with Stokes hypothesis without Stokes hypothesis

0.1

0.05 0

-0.05 1

2

Y 3

4

0.05

-0.10

5

time = 000.91598 with Stokes hypothesis without Stokes hypothesis

0.1

0

6

0.1

-0.05

1

G

2

Y 3

4

5

time = 000.92000 with Stokes hypothesis without Stokes hypothesis

0.05

p

p

0.05

5

0

-0.10

1

time = 000.90398 with Stokes hypothesis without Stokes hypothesis

0.1

p

p

2

time = 000.90800 with Stokes hypothesis without Stokes hypothesis

0.05

5

2

0.8

-0.05

0.1

-0.10

G 0.6

p

p

E

A

time = 000.8999 with Stokes hypothesis without Stokes hypothesis

F

F Without Stokes hypothesis

B

-1 -3

With Stokes’ hypothesis

D

1

0

-0.05 1

2

Y 3

4

5

-0.10

1

2

Y 3

4

5

Figure 12.13 Pressure distribution across the height of the box, for different times during a steep jump of entropy between points F and G shown in the top frame at times: (1) t = 0.8999, (2) t = 0.90398, (3) t = 0.90800, (4) t = 0.91199, (5) t = 0.91598, (6) t = 0.92000. The top frame depicts the distinct M pattern observed in the total entropy plot in Figure 12.12, which is explained in [461]. [Reproduced from “Roles of bulk viscosity on Rayleigh–Taylor instability: Non-equilibrium thermodynamics due to spatio-temporal pressure fronts”, T. K. Sengupta et al., Phys. Fluids, 28, 094102 (2016), with the permission of AIP Publishing.]

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Without Stokes Hypothesis

4

E1

Entropy (φ S)

3

C1

E2

2

D1

C2

1

A

0

D2

-1

F1

B1

-2

F2

B2

-3

4

4.2

4.4

4.6

time

A) t = 3.90 0.05

0.04

0.04 without Stokes

0.02

0.01

0.01

0

0

-0.01

-0.01

-0.02

-0.02

-0.03

-0.03

-0.04

without Stokes

0.03

0.02

p

p

0.03

-0.04 0

1

2

y

3

4

5

-0.05

0

1

B1,B2) t = 4.08

4

5

0.02

0.01

0.01

0

0

-0.01

-0.01

-0.02

-0.02

-0.03

-0.03

-0.04

without Stokes

0.03

0.02

p

p

3

0.04 without Stokes

0.03

-0.04 0

1

2

y

3

4

5

-0.05

0

1

2

y

3

4

5

F1,F2) t = 4.66

C1,C2) t = 4.20 0.05

0.05

0.04

0.04 without Stokes

0.03

0.02

0.01

0.01

0

0

-0.01

-0.01

-0.02

-0.02

-0.03

-0.03

-0.04

without Stokes

0.03

0.02

p

p

y

0.05

0.04

-0.05

2

E1,E2) t = 4.56

0.05

-0.05

4.8

D1,D2) t = 4.38

0.05

-0.05

i

-0.04 0

1

2

y

3

4

5

-0.05

0

1

2

y

3

4

5

Figure 12.14 The computed variation of entropy with and without Stokes’ hypothesis, explained by pressure distribution across the height at (A) t = 3.90, (B1, B2) t = 4.08, (C1, C2) t = 4.20, (D1, D2) t = 4.38, (E1, E2) t = 4.56, (F1, F2) t = 4.66. [Reproduced from “Roles of bulk viscosity on Rayleigh–Taylor instability: Non-equilibrium thermodynamics due to spatio-temporal pressure fronts”, T. K. Sengupta et al., Phys. Fluids, 28, 094102 (2016), with the permission of AIP Publishing.]

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12.6.1 Compression and rarefaction fronts during Rayleigh–Taylor instability In Figure 12.2, the mechanism of generation and propagation of bubbles and spikes were fronts or STWFs, shown at early times with density contours, and these STWFs emanated from the interface into both the lighter and heavier fluids. The same phenomenon is illustrated in Figures 12.15a and 12.15b with the help of pressure contours at different times. The importance of pressure fronts for visualization becomes more relevant, when one compares the pressure fronts in two domains with different depths. The rectangular domain corresponds to the shallower domain, while the square domain has double the depth. If the events during Rayleigh–Taylor instability are dominated by vortices forming at the interface alone, then both the domains should exhibit identical behavior. However, if pressure is also created at the interface for very low convection speeds, then such pressure pulses would travel very fast, yet the distances traveled are so dissimilar between rectangular and square domains, that one could expect to see significantly different instability sequences for different depths. Rectangular domain

ii

Max = 0.5610

t = 2.2

y

4 2 0

0

2

4

x6

8

10

Min = -1.0100

0.026 0.019 0.002 -0.012 -0.015 -0.022 -0.031 -0.044 -0.054

Rectangular domain

iii

Max = 0.8090

t = 5.0

y

4 2 0

2

4

x6

8

10

p 0.031 0.020 0.007 -0.012 -0.015 -0.022 -0.031 -0.044 -0.054

Square domain t = 2.2

Max = 0.9238

p

8 6 4 2 0 0

5 x

10

0.600 0.052 0.003 -0.014 -0.030 -0.044 -0.102 -0.900

Square domain

iv

Min = -0.6305

0

10

p

y

Min = -0.3956

Min = -1.6979

Max = 1.1093

t = 5.0

p

10 8 y

i

6 4 2 00

5 x

10

0.400 0.036 0.000 -0.014 -0.030 -0.044 -0.102 -0.900

Figure 12.15a Pressure contours for the rectangular domain in frames (i) and (iii) and for the square domain in frames (ii) and (iv) at t = 2.2 and t = 5.0. Reflection and rarefaction of waves are visible in the frames, with pockets of shocks also observed. [Reproduced from “Roles of bulk viscosity on Rayleigh–Taylor instability: Non-equilibrium thermodynamics due to spatio-temporal pressure fronts”, T. K. Sengupta et al., Phys. Fluids, 28, 094102 (2016), with the permission of AIP Publishing.]

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Baroclinic Instability Rectangular domain

vi

Max = 1.1282

t = 15

y

4 2 0

0

5 x

10

p 0.031 0.020 0.007 -0.012 -0.015 -0.022 -0.031 -0.044 -0.054

Rectangular domain

vii

Max = 0.6181

t = 25

y

4 2 0

5 x

10

p 0.031 0.020 0.007 -0.012 -0.015 -0.022 -0.031 -0.044 -0.054

Square domain t = 15

Max = 1.20975

10

5

00 viii

Min = -0.7781

0

Min = -2.1133

y

Min = -0.8404

Min = -1.2100

5 x

10

Square domain t = 25

5

0

0

5 x

p 0.900 0.071 0.004 -0.011 -0.030 -0.044 -0.102 -0.900 Max = 0.7487

10

y

v

10

p 0.900 0.071 0.004 -0.011 -0.030 -0.044 -0.102 -0.900

Figure 12.15b Pressure contours for the rectangular domain in frames (v) and (vii) and for the square domain in frames (vi) and (viii) at t = 15 and t = 25. Reflection and rarefaction of waves are visible in the frames, with spikes and bubbles marked. [Reproduced from “Roles of bulk viscosity on Rayleigh–Taylor instability: Non-equilibrium thermodynamics due to spatio-temporal pressure fronts”, T. K. Sengupta et al., Phys. Fluids, 28, 094102 (2016), with the permission of AIP Publishing.] In Figure 12.15a, pressure contours are shown for early times of t = 2.2 and 5, with the frames on the left representing the rectangular domain and the frames on the right representing the square domain. One notes pressure waves in Figure 12.15a, which move normal to the initial interface. The frames at t = 2.2 show oblique waves moving from the interface to the top and bottom of the domain, and then reflect from the top, bottom and side walls. The pressure contours are completely different for the rectangular and square domains. The interactions among previous and nascent waves from the junction of the side walls and the interface create complex dynamics. Shocks are created when waves intersect, as noted in the top frames. The frames at t = 5.0, show further reflections of pressure waves, after the onset of Rayleigh–Taylor instability. The start of plumes of fluid at the junction points of the interface with the side-wall are also visible in these frames. One also notices formation of vortices at the junction where interface meets the side-walls [461], as reported in the experiments [6, 110, 357]. Afterwards, mixing is noted at t = 15.0 in Figure 12.15b. It is stated in [461] that the “wave-packets of high pressure zones are evident in the top compartment of the domain. The high pressure zones or wave-packets depicted in the figure by structures resembling spikes and bubbles are similar to the shocklets described in the large-eddy

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compressible simulations of Olson & Cook [320].” It has been emphasized [461] that the pressure differential over the static pressure remains small and the presence of such “shocklets” are noted in Mach contours. One notes coalescing bubbles rising at t = 15, that is more prominent for the rectangular domain, compared to the square domain. Such differences persist also at t = 25 in Figure 12.15b. At these later times, mixing is pronounced giving rise to higher pressure zones in the lower compartment of the domain. Mach number variation near the interface is shown in Figure 12.16a, for t = 4.99, 14.99 and 24.99, for the case of rectangular domain. Fluctuations at t = 4.99 show slanted elongated structures, made visible in the density and pressure contours. This indicates the onset of convection by Rayleigh–Taylor instability. These structures during the primary instability are associated with fluctuating Mach numbers, which are extremely small. However, the accuracy of the compressible formulation allows one to capture even small fluctuations during Rayleigh–Taylor instability. At t = 14.99 in Figure 12.16a, fluctuating Mach numbers are diffused over a wider region for the same range of contours. The maximum value of Mach number reaches to M = 0.076 at this time. The fluctuations become smoother with time, noted by comparing frames at t = 4.99 and 14.99. At t = 24.99, one notices the maximum value of M to remain of the same order, while diffusion widens the fluctuating region, with larger values of maximum M remaining near the interface. M: 0.01 0.02 0.03 0.04 0.05 0.07 0.08 Time = 14.99 Time = 4.99 Max = 0.76 Min = 0.000 Max = 0.047 Min = 0.000 60×30 60×30 5 3.2 M:

0.01 0.01 0.02 0.03 0.03 0.04 0.05

4.5 3

M: 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Time = 24.99 Min = 0.000 Max = 0.76 4 60×30 3.5

4 3.5

2.8

3

y

y

y

3 2.5

2.5

2.6

2 2.4

2

1.5 1

2.2

1.5

0.5 2 4.2

4.4

4.6 x

4.8

0

4.5

5

x

5.5

6

5

6

x

7

8

Figure 12.16a Mach contours for the rectangular domain at indicated times of t = 4.99, 14.99 and 24.99. [Reproduced from “Roles of bulk viscosity on Rayleigh–Taylor instability: Non-equilibrium thermodynamics due to spatio-temporal pressure fronts”, T. K. Sengupta et al., Phys. Fluids, 28, 094102 (2016), with the permission of AIP Publishing.]

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Baroclinic Instability

0.01 0.02 0.03 0.04 0.05 0.06 0.07

M:

Min = 0.000

M:

Min = 0.105

0.02 0.03 0.05 0.06 0.08 0.09 0.11

Time = 24.99 Min = 0.000

8

60×60

Min = 0.110 60×60

7.5

6.5

6.5

6

6

5.5

5.5

7 6.5 6 5.5

y

y

5.5 y

0.02 0.03 0.05 0.06 0.08 0.09 0.11

Time = 15.00

Min = 0.000 Min = 0.066 Min = 0.000 Min = 0.091 60×60 60×60

6

M:

0.01 0.03 0.04 0.05 0.06 0.08 0.09

Time = 9.99

Time = 5.00

5

y

M:

5

5

4.5

4.5

3.5

4

5 4.5

3 4

4 4.5

4

4.5 x

5

4

4.5 x

5

2.5 4

4.5 x

5

2 2

4

x

6

8

Figure 12.16b Mach number contours for the square domain at indicated times of t = 5, 9.99, 15 and 24.99. [Reproduced from “Roles of bulk viscosity on Rayleigh–Taylor instability: Non-equilibrium thermodynamics due to spatio-temporal pressure fronts”, T. K. Sengupta et al., Phys. Fluids, 28, 094102 (2016), with the permission of AIP Publishing.] Figure 12.16b shows the Mach number fluctuations for the square domain, near the interface at t = 5, 9.99, 15 and 24.99, to compare the effects of compressibility for the higher depth case of the square domain, as compared to the shallower rectangular domain. The fluctuating values of M at t = 5 show the same elongated, slanted small scale structures, that are seen for the rectangular domain also. At this time, one can observe low values of fluctuating M indicating early signs of primary Rayleigh–Taylor instability. Furthermore, the fluctuating M does not attenuate – unlike the case for the rectangular domain. Also, the maximum Mach number increases, as compared to that of the rectangular domain, due to the higher size of the domain allowing the flow to evolve without obstruction from the top and bottom walls. With time, fluctuating M-contours diffuse over larger mixed region. Thus, one notices increasing value of maximum Mach number in the other frames at t = 9.99 and 15. In the last frame at t = 24.99, the maximum M to reaches all the way up to M = 0.11, with the fluctuations smoother at later times, as compared to early times, with higher M values noted over larger parts of the domain.

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12.7 Closing Remarks A completely different type of instability, originating from the different physical mechanisms may be understood by studying the Rayleigh–Taylor instability. This baroclinic instability arises due to compressibility effects, when the pressure and density gradients are not aligned. This in turn, creates a destabilizing turning moment which acts more like a static instability, as the perturbed state further deviates into instability. To capture the stages of, and, understand this instability, one must use very accurate formulation and numerical methods, as reported here. The onset of instability is from the singular point of the interface, with the smallest length scales triggering the instability. Thereafter, a billowing and a bowing motion propels a disturbance front along the interface. Simultaneously, pressure fronts are created which move normal to the interface. As a consequence of creation of the instability at the smallest scale, one should not try to compute the flow using periodic conditions in the horizontal directions. This has been explained by the success that many computations achieved which use high accuracy compact schemes. At the same time, use of a compact scheme in conjunction with any periodic conditions in the horizontal plane leads to poorer results. Unlike the other instability mechanisms described in previous chapters, presence of an inverse cascade and its accurate depiction by an appropriate method is a special feature of Rayleigh–Taylor instability. The results presented for a finite mass, finite energy system also show the presence of spatio-temporal pressure fronts or STWFs, which properly depict non-monotonic growth of entropy, due to presence of compression and rarefaction fronts.

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13

Coherent Structure Tracking in Transitional and Turbulent Flows

13.1 Introduction In Chapter 9, two nonlinear theories have been described to predict the onset and growth of disturbance fields. For incompressible flows, these are derived from the Navier–Stokes equation without resorting to any assumptions, based on the disturbance mechanical energy (DME) and the disturbance enstrophy transport equation (DETE), and are given in Eqs. (9.7) to (9.11). In studying transitional and turbulent flows, one begins with the details of the receptivity of the equilibrium flow to imposed perturbations. These are ingested in the flow and evolve via various instability stages. The onset or the receptivity stage strongly depends on the way the equilibrium flow is excited. It has been clearly described in Chapters 5, 6, 8 and 9 that there are two prototypical routes of causing transition for the flow over the canonical semi-infinite flat plate [298]: (i) where the boundary layer is excited at the wall [32, 34, 134, 509], experimentally investigated in [236, 405, 565], and (ii) where the flow transition is triggered by free stream excitations, studied theoretically and experimentally in [225, 267, 431, 559]. The second route of excitation has been originally conceptualized by Taylor [510] in trying to quantify the effects of free stream turbulence. This has been endorsed by Monin and Yaglom [295] subsequently. In this chapter, the methods which trace disturbances from the onset to fully developed turbulent stages are discussed. Thus, it is essential that one understands the genesis and growth of disturbances in the first place. That there is multiplicity of point of views about the receptivity or onset stage itself, is well known. Even for the canonical flow over a semi-infinite flat plate, various aspects of flow transition have been emphasized by different researchers. For example, Saric et al.

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[385] provided a roadmap of transition, by highlighting the role of amplitude of the imposed perturbation, noting that weak disturbances inside the boundary layer can cause instabilities that can be described by the Orr–Sommerfeld equation. With higher amplitude of imposed perturbation, nonlinear interactions can directly occur in the form of secondary instabilities, bypassing the primary linear instability [299] with turbulent spots appearing directly. Such a bypass route is also noted for high free stream turbulence in [364]. As a critique of unmitigated importance given to amplitude of input excitation, authors in [399] have noted the importance of linear mechanisms by stating that the “energy growth or decay can only come from linear processes,” indicating an important role of linear growth of disturbance right from the onset. The precise statement in [399] relates to the energy conservation property of the convective acceleration terms in the Navier–Stokes equation which has been critiqued in this book. In the classical point of view, the onset of the disturbance has been conceptualized to be related to normal mode or eigenvalue analysis by some researchers. There are two impediments for such an approach. The process of growth of disturbances turns out to be dependent on the ansatz adopted for instability. For example, in deriving Rayleigh’s equation, one assumes that energy grows in time, as noted in Chapters 4 and 6. The complementary development of the Orr–Sommerfeld equation for flow over semi-infinite flat plate has taken the spatial growth route, which led researchers to adopt the so called frequency response approach with a signal problem assumption (see the discussion in Chapter 6). In Chapter 11, one noted the difficulty accurately estimating mixed convection flows for which both temporal and spatial instability is present, as found by the eigenvalue approach. According to authors in [532], eigenvalue analysis produces results which can be verified experimentally for flows driven by forces arising due to buoyancy and centrifugal actions, as in Taylor–Couette flow between a stationary outer cylinder with a rotating inner cylinder and in Rayleigh– B´enard convection. However, for the cases of pipe and Couette flows, unconditional stability is indicated by the eigenvalue approach. Also, plane channel or Poiseuille flows display linear instability at significantly higher Reynolds number, than that noted experimentally [106]. According to Trefethen et al. [532]: “Other examples for which eigenvalue analysis fails include pipe Poiseuille flow (in a cylindrical pipe) and, to a lesser degree, Blasius boundary layer flow (near a flat wall).” This prompted researchers [45, 74, 131, 166, 361, 362, 398] to look for disturbance growth mechanisms other than those dictated by eigenvalue or modal analysis. The idea behind transient growth, as the primary route of energy growth in a finite time, has also some unexplained aspects which rest on non-normal/ non-orthogonal modes studied without knowledge of the weightage of each of these discrete modes and their phase relationship [362, 532]. The major problem arises due to adoption of the signal problem assumption, fixing the time scale of the response field, which is mandatory for spatial analysis based on this assumption [141]. The basic mistake of fixing the imposed time scale with such an assumption has already been pointed out in Chapter 4, is

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negated by the finite start-up of a time-harmonic excitation. This start-up implies that the excitation boundary condition must be associated with a Heaviside function, (given in Eq. (4.15)). The issue of signal problem assumption also created additional conceptual difficulties. One of which is clear, if one looks at the schematic of the flow past the semi-infinite flat plate in Schmid and Henningson [400]. The lack of knowledge of time scales other than the imposed frequency may have also contributed to not showing any response field in the immediate neighborhood of the exciter, which has been termed in this book as the local solution. In [409], the concept of local solution is explained, even for the signal problem assumption by using the Bromwich contour integral method (BCIM), as explained in Section 5.2. Instead, some researchers in attempting to explain early transients used other heuristic explanations, as described next. Gaster and Grant [144] in an attempt to circumvent the signal problem assumption, reported additional experimental results, where transition is caused by impulsively exciting the boundary layer forming over a semi-infinite flat plate with a localized pulse. This has been described in Section 6.3 as the impulse response. However, the authors in [144] misinterpreted the response to be due to an ensemble of a large number of excitations over a very wide band of frequencies, each of which constitute the signal problem with individual response of each frequency added together with an empirical weight. Ironically, the authors in [144] were trying to circumvent the objection of the proponents of transient growth against the non-orthogonality of eigenvalues obtained by modal analysis. While there is nothing wrong with the nonorthogonality of discrete modes, no attempt was made to develop the equivalent of BCIM for spatial and spatio-temporal analysis which have been reported for the first time in [409] and [418], respectively. As explained in Chapter 6, the experiment of impulse response [144] has been reproduced theoretically in the linear framework of the Orr–Sommerfeld equation by BCIM recently in [509]. This explains a variety of response fields which can be realized by following the route of transition to turbulence. Algebraic or transient growth has been suggested for a long time [26, 128, 193, 251] only for three-dimensional disturbance fields caused by an inviscid mechanism, and these can also be found in reviews [364, 399, 400]. Attempts to explain such growth involve both linear and nonlinear nonmodal approaches [229, 398]. The linear approach studies three-dimensional disturbance as an initial value problem with coupled governing equations for wall-normal velocity and wall-normal vorticity. Nonlinear dynamical systems approaches (as in [542] to identify coherent structures in shear flows) have been reviewed in [125, 221] by invoking solutions of the Navier– Stokes equation. All the approaches for algebraic or transient growth have arisen due to the misconception that disturbance quantification cannot be explained by Fourier–Laplace transform for vanishingly small times, “as it only describes the asymptotic fate of the perturbation and fails to capture short-term characteristics ”[398, 400]. This is largely due to oversight by several researchers about the well-founded Abel and Tauber

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theorems, which relate transient growth or the local solution in the physical plane to the image at infinity in the spectral plane [412, 537], thereby validating the use of Fourier–Laplace transform applied along the Bromwich contours [418, 451]. Thus there is no need to separately perform “short-term stability analysis,” as advocated by Schmid [398]. Modal analysis is validated by the successful experimental detection of the Tollmien–Schlichting wave [405], following the theoretical prediction of the same in [392, 528]. The authors in [418, 451, 509] have successfully demonstrated the presence of modal and nonmodal components of the response field by performing BCIM. These analyses show the existence of the nonmodal component of the disturbance field, known as the spatio-temporal wave front (STWF) from the solution of the Orr– Sommerfeld equation. The existence of STWF is validated with experiments [144, 237, 405] and extended to nonlinear analysis of the Navier–Stokes equation viewing transition as spatio-temporal growth phenomenon in [32, 34, 451, 483, 508, 509]. One notes the STWF only when the disturbance growth is studied in the general spatio-temporal framework by BICM [418, 451]. STWF growth is not related to any eigenvalue growth, but rather, conjoining both the modal and nonmodal contribution from the Fourier–Laplace transform. It is worth recalling that the Bromwich contour is chosen such that it does not pass through any eigenvalue and the nonmodal growth is noted by constructive interference of neighboring points of the spectrum with identical amplitude, in the linear and nonlinear framework for wall excitation cases [32, 34, 509]. The aim of the work in [466] is to trace disturbances from receptivity to fully developed turbulent flow stage by DME and DETE methods [431, 474, 475]. Transition caused by FST has been studied by nonlinear, nonmodal approach in [90], and proposed as bypass transition for high FST levels which directly creates coherent structures. Moderate levels of FST in experiments have also shown streaky growth [288] for flow over semi-infinite flat plate with FST intensity in the range of 1 to 6%. Authors in [549] have experimentally demonstrated modification of the boundary layer over a semi-infinite flat plate exposed to near-isotropic FST. Fluctuations are noted inside the boundary layer, predominantly with streaky structures, growing in the streamwise direction with increasing amplitude and length scale. Schlatter et al. [390] similarly identified transition via primary instability in the form of long streaks; algebraic/ transient growth as the secondary instability. During such transition, coherent structures associated with pressure minima are not seen, and therefore classical means of tracing techniques for disturbance evolution fail. The techniques following DME and DETE are found to be useful for such cases [466]. Not identifying the streaky structures as Klebanoff modes is a shortcoming, which has been adequately described in Section 5.3 following the work in [448]. Transition caused by free stream excitation for different flows [225, 267, 412, 559] can be studied by considering vortex-induced instability as the unit process. In this, a vortex convects in the free stream over a semi-infinite flat plate at speeds significantly lower than the free stream speed, and thereby destabilizes the boundary layer [431, 474, 475], as described in Chapter 9. Thus, distinguishing between excitation applied

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at the wall and that applied from the free stream, is key to understanding transition to turbulence for wall bounded flows. Apart from comparing these two excitations with the help of vorticity dynamics, one can trace disturbances with the help of DME [431] and DETE [475], as described in Chapter 9. One must distinguish between wall and free stream excitation. G.I. Taylor was one of the pioneers interested in experimentally and theoretically studying the transition to turbulence. The successful capture of TS waves by wall excitation experimentally in [405], was preceded by Taylor trying to detect TS waves by vibrating a diaphragm at the wall over which a nominally zero pressure gradient boundary layer was investigated. Lack of success by Taylor has been explained in Chapter 5, as due to the very low frequency of excitation of the diaphragm. But, he is credited in [448] with demonstrating the breathing mode instead in this experiment. Before this vibrating diaphragm experiment, Taylor tried to decipher the transition phenomenon in real flows by studying effects of FST. He tried to estimate a theoretical dependence of the critical Reynolds number at which transition occurs on the FST of oncoming flow [510] by employing the modified Pohlhausen parameter Λ = −(δ2 /µU) δp δx , where δ is the boundary layer thickness, δp is the fluctuating pressure of the flow in the free stream, and U is the free stream speed. For semi-infinite flat plates, one resorts to the Blasius profile, for which pressure gradient is absent. Monin and Yaglom [295] noted that free stream excitation can cause “separation of the boundary layer under the action of a negative longitudinal pressure gradient,” which “is connected with the generation of fluctuations of the longitudinal pressure gradient by these disturbances, leading to the random formation of individual spots of unstable S-shaped velocity profiles .... and hence, to separation and transition of the boundary layer” which is predicated to the observation that even the so-called zero pressure gradient boundary layer supports fluctuations in pressure, as “Taylor proposed that the character of the motion in a fixed 0 0 section is determined here by the parameter Λ = −(δ2 /µU) δp δx , where δp is the pressure fluctuation. In other words, according to Taylor [510], the point of transition from laminar to turbulent flow is determined by the parameter Λ attaining some critical value” [295]. The study of free stream disturbances on a finite flat plate boundary layer, can support controlled pressure fluctuations, as demonstrated in [225, 267] following the vortex-induced instability. These fluctuations drive the primary mechanism of disturbance growth through unsteady separation for transition. While the above discussion about the coherent disturbance sources for transitional flows is relevant, it is also well-known that coherent structures are present in turbulent flows, as reviewed in [206]. Early works on coherent structures have been presented for different flows in [69, 240]. Tracking coherent structures from numerical and experimental results have been reported in [197, 198]. In [466], the authors have shown that growing vortical structures during late-transition stage evolve into coherent structures, which are also associated with fully developed turbulent flows. Such structures in fully developed turbulent stage can be detected by either Eulerian or Lagrangian methods. The former is popular for its ease and efficiency. Two such Eulerian schemes are the λ2 - and the Q-criteria [196, 204], and structures detected using

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~ from the observation that these methods are from the velocity gradient tensor (∇V), pressure attains a minimum at the vortex core associated with any coherent structures. In the other methods, one traces Lagrangian coherent structures by studying the stability of tracer fluid particles’ trajectories, by quantifying material transport in predicting large-scale flow features and mixing events [171]. It has been noted that coherent structures are regions of high vorticity created during interactions of vortex filaments [83, 198, 289]. The Eulerian approaches are computationally easier due to, (i) the local nature of such schemes (i.e. the schemes can be applied at any point in the domain), (ii) one can operate the instantaneous velocity field, and (iii) define a function in the domain for all points being either inside or outside a vortex, and its use is based on a criterion for choosing the function values [83]. It has been shown [466] that pre- and post-transitional flows are dominated by vortices. In fully developed turbulent flows, such structures are not visible directly, being inside chaotic and random data. The vortices in fully developed turbulent flows have been studied [372, 492], motivating others in search of coherent structures in [92, 196, 204, 573] with varied success. While these are effective in the presence of vortices during later stages of transition and in fully developed turbulent flows, in some flows like that in a channel, there are false positive and negative indications of coherent vortices. In the following section, various definitions for a vortex and coherent structures and methods for their identification are provided, highlighting their merits and disadvantages.

13.2 Definition of a Vortex and Coherent Structures Late-transitional and fully developed turbulent flows are considered as entanglements of multiple vortices, with various sizes, scales and orientations. An understanding of their dynamics helps in understanding the flows [195, 518]. Loosely stated, spatial coherence and regularly evolving temporal structures constitute turbulent shear flows, and are referred to as coherent structures [76, 372]. A number of definitions exist for these coherent structures one of which is stated in [372] as: “a three-dimensional region of the flow over which at least one fundamental flow variable (velocity component, density, temperature, or other) exhibits significant correlation with itself or with another variable over a range of space and/or time that is significantly larger than the smallest local scales of the flow.” Researchers are in constant search for better definitions of a vortex [130, 170]. The detection and analysis of coherent structures help: i) one understand basics of turbulence associated with mixing and transfer of mass, momentum, and energy; ii) device flow control for drag reduction and noise mitigation and iii) develop analytical and computational aids for turbulence modeling [200]. As the coherent structure is postulated to contain a vortex with a core [204], it should possess the following

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properties: (a) The net vorticity, and therefore net circulation, inside the vortex core must be non-zero and (b) The vortex core should be a Galilean invariant. A vortex is a region of fluid rotating about an axis [280], and the following intuitive definitions of vortex are as given in [204]: 1.

a vortex should display closed circular streamlines and path-lines in an appropriate co-ordinate system.

2.

a vortex must be characterized by a magnitude given by |~ ω|, and

3.

the rotating region must be associated with a pressure minimum.

13.2.1 Circular streamlines and path-lines Vortices are associated with spiraling and/or closed path-lines, as proposed by Lugt [280]. The fluid particles in a vortex rotate about a common center or axis. It is to be appreciated that this has a major drawback, as fluid particles need not complete a full circular motion about the axis of rotation; instead, they may define a helical trajectory without forming a closed path-line around the axis. Before completing a full revolution, the vortex may breakdown, as it happens in flow over a delta wing at high angles of attack. The path-lines may not satisfy the Galilean invariance condition, which is a required property for defining a vortex. The streamline patterns from different reference frames look different compared to a closed or spiraling pattern defining a vortex. Thus, a definition based on streamlines has been adopted by Robinson [370]: “A vortex exists when instantaneous streamlines mapped onto a plane normal to the vortex core exhibit a roughly circular or spiral pattern, when viewed from a reference frame moving with the center of the vortex core.” The relevance of Galilean invariance and singular points are discussed with respect to the Lamb–Oseen vortex in Figure 13.1, also described in [204]. In frame (a) of this figure, the vortex is shown in the inertial frame of reference (for which the streamlines are azimuthal), by the radial and azimuthal components of velocity, vr = 0 and vθ = Γ 2πr (1

r2

− e− 4νt ), where r is the radial distance from the origin (P0 ) and t is the time. The Lamb–Oseen vortex has strength indicated by Γ and ν is the kinematic viscosity of the fluid. The displayed closed contours are drawn for Γ = 1 and νt = 0.5, while the streamwise velocity profile is plotted along x = 0 with y as the independent variable. One can readily note the singular point at (x, y) = (0, 0), with two more points at P1 and P2 marked along x = 0 with y-coordinates being -0.73 and -3.15, both having same x-component of velocity, u = 0.05. Now, if one moves with this velocity in the x-direction, then the streamlines are depicted in frame (c), with two singular points at P1 and P2 , in this frame of reference. If one moves with a velocity given by, u = 0.1, then the streamlines would be as indicated in frame (d), without showing any closed contours. This is apparent from frame (b) as well, as u does not exceed u = 0.075, and hence no singular points are noted in this frame of reference.

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b)

4

4

2

2

y 0

P0 :(0,0)

y 0

P0

P1 :(0,–0.73)

–2

–2

P2 :(0,–3.15) –4

–4 –4

–2

0 x

2

4

–0.1

–0.05

0 u(x = 0, y)

0.05

0.1

d)

c) 4

4

2

2

y2

y1 0 P1

–2

0 –2

P2 –4

–4 –4

–2

0 x1

2

4

–4

–2

0 x2

2

4

Figure 13.1 a) Streamlines of Lamb–Oseen vortex are plotted in an inertial frame. b) The xcomponent of velocity profile with y is plotted at the station, x = 0 is plotted, with typical points marked. c,d) Streamlines of the Lamb–Oseen vortex viewed from the moving frames of reference, at constant velocity 0.05 and 0.1, respectively, are shown.

13.2.2 Vorticity magnitude A vortex has lumped vorticity in the absence of background shear. Originally, an instantaneous vorticity field has been thought to be absolutely necessary to detect vortical structures [44, 199]. However, Robinson [372] has shown that the vorticity field is inadequate to detect coherent structures in a turbulent boundary layer, while it is adequate for free shear layers. Jeong and Hussain [204] have noted that if a vortex core is in a shear flow, then the magnitude |~ ω| does not identify the cores unambiguously. Especially if the background shear is equal to the vorticity

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magnitude. Of course, in a boundary layer, one has vorticity which is not sufficient to guarantee the presence of vortices.

13.2.3 Pressure minimum It is conjectured that the location of minimum pressure is at the core of the vortex. This is due to fluid particles closer to the axis moving faster and experiencing falling pressure, as all fluid particles in the core have the same mechanical energy. Similar conditions are prevalent for steady inviscid vortices having equivalent Bernoulli head, which experience lower pressures on inner circuits, as the azimuthal velocity increases towards the core of the vortex. But this will not be true for all scenario involving high unsteadiness, and in such cases, the condition of a pressure minimum is neither sufficient nor necessary for the existence of a vortex core. There could be locations with pressure minimum, even without the presence of any vorticity. Of course, the necessary condition for the existence of a vortex without vorticity is meaningless. Examples of this include planar irrotational source/sink flows, which have a pressure minimum at the origin. Similar possibility occurs, when centrifugal force is balanced by viscous action and not by a pressure gradient term, as in the Karman viscous pump [218] and Stokes or creeping flow (at very low Reynolds number). Unfortunately, the condition of a pressure minimum is commonly adopted for identifying vortical coherent structure. This is the basis for various popular criteria, such as Q- [196] and λ2 -criteria [204], which is also supported in [371] for turbulent boundary layers, with vortical structures forming very close to the wall in the lowpressure region. The major difficulty is in specifying a single minimum that can account for all vortical structures. Most importantly, pressure for incompressible flow is related to velocity by the Poisson equation, ∇2 p = −ρui, j u j,i , with comma in the subscript implying partial differential with respect to that index, and repeated index implies summation of that index.

13.3 Definitions of Coherent Structures Based on Velocity Gradient and Its Invariants It has been noted in the previous section that Galilean invariance is a weak requirement for any detected coherent structure. This is immediately understood, when one attempts to view vortices in the wake of a stationary cylinder in a moving frame of reference, instead of the traditional Eulerian framework. The vortices are not apparent and streamlines appear to intersect the cylinder at multiple points on the surface. Despite this, Galilean invariance is considered as an important requirement to define a vortex, and many attempts have been made to detect vortices based on the ~ in [92, 196, 294]. The velocity gradient invariants of the velocity gradient tensor (∇V) is a tensor of rank two, containing information on rotational and shear components

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~ while in it. In the following subsections, we introduce some of the invariants of ∇V, defining some of the popular criteria. This is a point property of the flow field, defined by the nine components of the tensor or the matrix defined using the dimensions of the problem as the independent variable. For a three-dimensional flow field, one has nine components of the velocity gradient matrix.

13.3.1 ∆-Criterion This method is based on defining a core region of the vortex in terms of the complex ~ [92]. The complex eigenvalues of the velocity gradient indicate eigenvalues of ∇V helical or closed streamlines about a point in the flow, when the flow field is viewed following a reference frame which moves with the velocity of that point. These points can be viewed using critical-point theory, with critical or singular points in the flow field identified as those for which all the velocity components vanish and the derivatives of the streamline are undefined [92]. ~ can be written as follows The characteristic equation for the eigenvalues (Φ) of ∇V Φ3 − PΦ2 + QΦ − R = 0 (13.1)   ~ are the three invariants of where, P = ui,i , Q = 12 u2i,i − ui, j u j,i , and R = Det(ui, j ) = |∇V|; ~ For incompressible flows, ui,i = 0, which simplifies the following invariants as: ∇V.   P = 0 and Q = − 12 ui, j u j,i . Two complex conjugate eigenvalues exist when the discriminant (∆) of Eq. (13.1) for incompressible flow (i.e. P = 0) is positive, which is given as ∆=

 Q 3 3

+

 R 2 2

>0

(13.2)

This condition identifies helical or closed streamlines about the center of a convecting vortex, when the frame of reference moves with the particle at the center of the vortex (which is a critical point). It can also be noted from Eq. (13.2) that for ∆ > 0; the additional condition of Q > 0 is more restrictive, and is known as the Q-criterion. Another variant of the ∆-criterion, is called the λ2ci -criterion [573], which uses the ~ to identify vortices. imaginary part of the complex conjugate eigenvalues of ∇V

13.3.2 Q-Criterion ~ is the well-known QAnother method that is based on the second invariant of ∇V criterion proposed by Hunt et al. [196] to identify vortex or “eddy” as a region ~ i.e. Q of Eq. (13.1), is positive. This criterion where the second invariant of ∇V, additionally requires a condition that the pressure in this region is lower than the value surrounding it. For incompressible flows, Q is given by

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Q=

  1 1 ui,i ui,i − ui, j u j,i = − ui, j u j,i 2 2

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(13.3)

~ or ∂ui , can be decomposed into a symmetric (S or S i j ) The velocity gradient tensor, ∇V ∂x j ¯ or Ω ), such that ∇V ¯ or in tensor notation, ~ = S + Ω, and an antisymmetric part (Ω ij ∂ui ¯ represent the parts of ∇V ~ due to shear strain-rate and = S i j + Ωi j . Here, S and Ω ∂x j

rotation rate, respectively, and are shown as S=

i 1h ~ ~ T ∇V + (∇V) 2

or S i j =

1 ∂ui ∂u j + 2 ∂x j ∂xi

! (13.4)

And, h i ¯ = 1 ∇V ~ − (∇V) ~ T Ω 2

1 ∂ui ∂u j or Ωi j = − 2 ∂x j ∂xi

! (13.5)

¯ 2 ) = u u . Thus, Q can be Using these notations, it is easy to see that trace(S2 + Ω i, j j,i ~ as written in terms of the symmetric and antisymmetric parts of ∇V Q=−

 1 1 ¯ 2) ui, j u j,i = − Trace(S2 + Ω 2 2

(13.6)

An alternative variant of Q is written as, Q=

1 ¯Ω ¯ t − SSt ) Trace(Ω 2

(13.7)

¯ t = −Ω, ¯ and symmetry, St = S properties have been used. For Both antisymmetry, Ω incompressible flows, positive Q shows that there is excess rotation over the strain rate. The static pressure is related to the velocity field through the Poisson equation for incompressible flows, obtained by taking divergence of the Navier–Stokes equation [335] as ∇2 p = −ρui, j u j,i

(13.8)

Now, using Eq. (13.6) in Eq. (13.8), the pressure Poisson equation can also be written as ∇2 p = 2ρQ

(13.9)

The properties of the Poisson equation given in [494], indicate that the sign of Q determines whether this is a source or a sink of pressure. A positive value of Q implies a sink of pressure, i.e. a minimum, while a negative value implies a source

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of pressure. This is the essence of the Q-criterion. It may be observed from Eq. (13.9) that the Q-criterion does not involve the effects of unsteady and viscous terms of the Navier–Stokes equation. It has been noted in [122] while comparing four detection methods for coherent structures based on low pressure, high vorticity, negative λ2 and positive Q-criterion, that the Q-criterion performs well for isotropic turbulence, channel flow, mixing layer and for flow over backward-facing step. However, it has been observed in [466] that Eq. (13.9) is an exact equation for incompressible flows, which is quite contrary to common imprecise perceptions that the Q-criterion is a kinematic condition derived from a velocity gradient tensor, and hence can also be applied for compressible flows.

13.3.3 λ2 -Criterion As discussed above, a pressure minimum is not the only definitive detection criterion for coherent structures. But, it is still used as the building block in developing the 2 λ2 -criterion. Jeong & Hussain [204], used the pressure Hessian ( ∂x∂i ∂xp j or p,i j in tensor notation) to detect pressure extrema in transitional and turbulent flows. A pressure Hessian is defined as the square matrix consisting of possible second-order derivatives (including the mixed ones) of pressure, which is used to describe the local curvature of the static pressure at any point in the domain p(x, y, z). Thus, one must construct the pressure Hessian (p,i j ), obtained from the gradient operation on the incompressible Navier–Stokes equation. Using the Navier–Stokes equation for the substantive derivative of ui in the following ai =

1 ∂p ∂2 ui ∂ui ∂ui =− +ν + uk ∂t ∂xk ρ ∂xi ∂xk ∂xk

(13.10)

one obtains the pressure Hessian (p,i j ) by differentiating ui with respect to x j for an expression of p,i j . This is equivalent to taking the gradient of the Navier–Stokes equation given in Eq. (13.10) as ! ! ∂ ∂ui ∂ui ∂ 1 ∂p ∂2 ui = − +ν ai, j = + uk ∂x j ∂t ∂xk ∂x j ρ ∂xi ∂xk ∂xk ! ! 2 ∂ ∂ui ∂ ∂ui 1 ∂ p ∂3 ui +ν ai, j = + uk =− ∂t ∂x j ∂x j ∂xk ρ ∂x j ∂xi ∂x j ∂xk ∂xk

(13.11)

~ As the velocity gradient tensor ∇ tensor, S =  V is written as a sum of strain rate h i h i ∂u ∂u j 1 1 1 ¯ T i ~ ~ ~ − (∇V) ~ T or or S i j = 2 ∂x j + ∂xi and the rotation rate tensor, Ω = 2 ∇V 2 ∇V + (∇V)   ∂u j ∂ui ¯ ~ Ωi j = 12 ∂x − ∂xi , then in vector calculus notation, ∇V = S + Ω, or in tensor notation, j ∂ui ∂x j

= S i j + Ωi j . Thus, one can represent ai, j into symmetric and antisymmetric parts as

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follows Term1

Term2

∂ ∂ui ∂ ∂ui + uk ∂t ∂x j ∂x j ∂xk !

ai, j =

! (13.12)

where the following terms are obtained as Term1: !  ∂S i j ∂Ωi j ∂ ∂ui ∂  S i j + Ωi j = + = ∂t ∂x j ∂t ∂t ∂t

(13.13)

Term2: ! ∂ ∂ui ∂uk ∂ui ∂2 ui uk = uk + ∂x j ∂xk ∂x j ∂xk ∂xk ∂x j ! ∂ui ∂uk ∂ ∂ui + = uk ∂xk ∂x j ∂xk ∂x j i ∂ h S i j + Ωi j + (S ik + Ωik )(S k j + Ωk j ) = uk ∂xk  ∂S i j ∂Ωi j  + uk + S ik S k j + Ωik Ωk j + S ik Ωk j + Ωik S k j (13.14) = uk ∂xk ∂xk Collating the terms in Eq. (13.12) gives, ! !   ∂S i j ∂S i j ∂Ωi j ∂Ωi j ai, j = + + S ik S k j + Ωik Ωk j + S ik Ωk j + Ωik S k j + uk + uk ∂t ∂xk ∂t ∂xk ! !     DS i j DΩi j = + + S ik S k j + Ωik Ωk j + S ik Ωk j + Ωik S k j Dt Dt ! ! DS i j DΩi j = + S ik S k j + Ωik Ωk j + + S ik Ωk j + Ωik S k j (13.15) Dt Dt Symmetric

Antisymmetric

Also, the right-hand side of Eq. (13.11) can be modified as, 1 ∂2 p ∂3 ui +ν ρ ∂x j ∂xi ∂x j ∂xk ∂xk 1 − p,i j + νui, jkk ρ ! 1 ∂2 ∂ui − p,i j + ν ρ ∂xk ∂xk ∂x j  1 ∂2  − p,i j + ν S i j + Ωi j ρ ∂xk ∂xk ∂2 S i j ∂2 Ωi j 1 − p,i j + ν +ν ρ ∂xk ∂xk ∂xk ∂xk 1 − p,i j + νS i j,kk + νΩi j,kk ρ

ai, j = − = = = = =

(13.16)

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Here, p,i j is the pressure Hessian. Bringing the symmetric and antisymmetric parts from Eqs. (13.15) and (13.16) together, one gets ! ! DS i j DΩi j 1 + S ik S k j + Ωik Ωk j − νS i j,kk + p,i j + + S ik Ωk j + Ωik S k j − νΩi j,kk = 0 Dt ρ Dt Symmetric

Antisymmetric

(13.17) The antisymmetric part of Eq. (13.17) is identically zero, as it is the vorticity transport equation. One is therefore left with the following symmetric part of the equation as ! DS i j 1 + S ik S k j + Ωik Ωk j − νS i j,kk + p,i j = 0 (13.18) Dt ρ After rearrangement, one can rewrite the equation for the pressure Hessian (p,i j ) as ! DS i j 1 (13.19) − νS i j,kk + S ik S k j + Ωik Ωk j = − p,i j Dt ρ It is noted in [204] that the local pressure minimum requires two of the eigenvalues to be positive. Without much discussion, the authors in [204] dropped the unsteady  DS term Dti j , being of the opinion that this is unsteady irrotational straining, and creates a pressure minimum without the presence of a vortical or swirling motion. The second   viscous term νS i j,kk of Eq. (13.19) was also discarded for it could lead to elimination of a pressure minimum in a vortical motion. Thus, the final expression for p,i j can be written as   1 (13.20) p,i j = − S ik S k j + Ωik Ωk j ρ The Hessian matrix requires two positive eigenvalues of p,i j , for the occurrence of local pressure minimum in a plane. Due to the negative sign ahead of S ik S k j + Ωik Ωk j on the right-hand side, this quantity must have two negative eigenvalues. This reduces the search for the pressure minimum to simply looking at S ik S k j + Ωik Ωk j to correlate local pressure minimum to a coherent structure defined as “a connected region with two negative eigenvalues of S ik S k j + Ωik Ωk j ” [204]. Furthermore, as S ik S k j + Ωik Ωk j is symmetric, the associated eigenvalues are always real. As a consequence, the three eigenvalues (namely λ1 , λ2 and λ3 ) can be arranged as follows, λ1 ≥ λ2 ≥ λ3 . The requirement of two negative eigenvalues are ensured for the λ2 -criterion by alternatively stating λ2 < 0 as a criterion for detecting coherent structure.

13.3.4 Relation between Q- and λ2 -criteria It is easy to obtain the Poisson equation, Eq. (13.9), by taking the trace of Eq. (13.19). First, one notes that the first (unsteady) and second (viscous) terms on the left-hand

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side of Eq. (13.19) disappear due to the incompressibility condition, as Trace(S i j ) = S ii = 0), and S ii,kk ≡ 0. Hence, Eq. (13.19) is written as ! 1 Trace S ik S k j + Ωik Ωk j + p,i j = 0 ρ 1 Trace(S ik S k j + Ωik Ωk j ) = − Trace(p,i j ) ρ 1 S ik S ki + Ωik Ωki = − p,ii ρ 2 ¯ ) = − 1 ∇2 p or in vector notation; Trace(S2 + Ω (13.21) ρ From Eq. (13.6), one can rewrite Eq. (13.21) to obtain Eq. (13.9) ∇2 p = 2ρQ ¯ 2 ) as One can also note that, Q can be written in terms of the eigenvalues of (S2 + Ω ¯ 2 ) = − 1 (λ + λ + λ ). Q = − 21 Trace(S2 + Ω 2 3 2 1 Despite the fact that we have derived the Q-criterion from the λ2 -criterion in what is given just above, it is appropriate to mention that the latter is an approximation, while the Q-criterion is directly obtained from incompressible Navier–Stokes equation without any approximation. The fact is that in deriving Q-criterion, the unsteady and viscous terms identically drop out due to the incompressibility condition. Therefore, in λ2 - and Q-criteria, the direct effects of unsteadiness and viscous terms are absent. As a consequence, structures detected by these two methods are apparently devoid of high frequency and high wavenumber effects at the small scales. This lends artificial coherence to the detected structure, which is not necessarily physical. Thus, the authors in [205] have noted that low-pressure regions and λ2 iso-contours near the wall of a turbulent channel flow show serious shortcomings. Furthermore, in several regions, pressure minima do not coincide with negative λ2 values. These make both Q- and λ2 -criteria dependent on the choice of threshold values, that one chooses to allocate to the ‘structures’. Such threshold values for Q and λ2 , must be correlated to physical quantities, such as the vorticity for the transitional and turbulent flows. In [466], threshold values of Q and λ2 are correlated with new generic methods in tracing disturbances for flows governed by the incompressible Navier– Stokes equation, with the concepts of DME [431] and DETE [465, 474, 475]. In a critique about structure detection methods based on pressure minimum, Kolar [241] noted that the “existence of a local pressure minimum is neither a sufficient nor a necessary condition for the presence of a vortex in general” and the terms removed in developing λ2 -method are found to be the main cause of this inaccuracy. Wu et al. [559] analytically examined Q- and λ2 -criteria using a Burgers and Sullivan vortex to note that the connected vortex detected by these methods fragments into different parts due to strong axial stretching. This provides reason to further explore other disturbance tracking methods, such as the methods based on DETE [465, 474, 475]

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and the DME [431]. A detailed investigation has been presented in [466] showing the relationship among these disturbance tracking methods and the prevalent coherent structure identification criteria.

13.4 Alternative Methods to Detect Coherent Structures in Transitional and Turbulent Flows ~ primarily used to identify The vortex detection methods based on the invariant of ∇V coherent vortical structures in turbulent flows, have in general, been popular in the absence of something better. As these methods require only a kinematic description of coherent structures for turbulent flows, it has often been used for all kinds of flows. Caution is advised however, because, as pointed out in the previous section, both the Q- and λ2 -criteria are developed from the Navier–Stokes equation strictly for incompressible flows. The fact that these methods do not involve kinetics of the flow is due to either incompressibility for the Q-criterion (where viscous terms drop out automatically) or by neglecting the viscous term without sufficient physical justification in the λ2 -criterion. As these are also developed with pressure minima located at the core of coherent structures, these methods cannot be used for the complete evolution of disturbance structures from the onset of disturbance field. This is true even if the disturbance field shows vorticity evolution. To trace disturbances from early stages of their inception to the fully-developed turbulent stage, one must use more general methods such as those based on DME and DETE. Thus any method originally designed for tracking instability in a flow governed by the Navier–Stokes equation or the vorticity transport equation, can also work for tracing coherent vortical structures from the inception to the fully developed turbulent flow stage. In Sections 9.6 and 9.7, we have already introduced two new methods based on DME and DETE to track disturbances from the receptivity stage itself to the fully developed turbulent flow stage. The first method based on DME [431, 452] is obtained from the primitive variable formulation of the Navier–Stokes equation by taking its divergence, with detailed derivations given in Sub-section 9.6.1. The second method based on the evolution equation for disturbance enstrophy (i.e. DETE) [465, 475], is obtained from the vorticity transport equation, after taking its inner product with the vorticity vector to obtain the enstrophy transport equation in Eq. (9.8). These are based on the complete nonlinear Navier–Stokes equation, developed to track disturbances from the receptivity stage. It has been noted in [466] that the earlier structure detection methods help identify coherent vortices in turbulent flows with static pressure minima. The detection mechanisms are based on the right-hand side of pressure Poisson equation in Qcriterion, whereas λ2 -criterion is based on eigenvalue of the right-hand side of the pressure Hessian equation. Alternative search started with this in mind to find methods based on DME and disturbance enstrophy. DME measures disturbance

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mechanical energy, with similar basis as that of Q-criterion, looking at the Poisson equation for total pressure, which for an incompressible flow is the measure to total mechanical energy. ~ 2 . If one The total mechanical energy in a flow is defined [431] as E = ρp + 21 |V| uses subscripts m and d to represent the equilibrium and disturbance quantities, respectively, then the disturbance component of mechanical energy represented by Ed would be obtained by subtracting the mean from the instantaneous energy as, Ed = E − Em . Retaining the full nonlinear contributions, the equation for Ed is given as follows [412, 431, 474, 475] h i h i ~ ×ω ~m × ω ~ ) − ∇ · (V ~ m) ∇2 Ed = ∇ · (V (13.22) ~ =V ~m +V ~ d . For this Poisson equation, the negative right-hand ~ =ω ~ m +ω ~ d and V where, ω side of Eq. (13.22) represents a source of Ed , while a positive value indicates a sink of Ed . Holmen [185] defined a coherent structure as a region in a flow with less mixing and movement as compared to local convection, so that the lump of fluid retains its identity with coherence. But, it has been noted that “vortices are coherent structures, and while the inverse is generally true, it is not necessarily so.” Coherent structures are supposed to originate during instability as vortical structures, and as solutions to the vorticity transport equation. As noted in [466] that a “measure for vorticity is enstrophy, and to follow transition to turbulence in both qualitative and quantitative manner, a new method uses the DETE [465, 475] for tracking disturbance growth into vortices in transitional and turbulent flows. Both the methods using DME equation and DETE are nonlinear tools, but both are equally capable of tracing linear as well as nonlinear routes of disturbance evolution.” ~ ·~ Enstrophy (Ω1 ) as a point property is defined as the square of vorticity by Ω1 = ω ω. ~m · ω ~ m ). An important way One obtains the equilibrium state enstrophy as, Ωm (= ω adopted to describe a disturbance enstrophy, is in the same way one defined Ed and thus, Ωd = Ω1 − Ωm . It is important to note that Ωm is strictly positive by definition. The same convention does not hold for Ωd , which can be both positive and negative. It is understood better by noting the linearized version of the disturbance enstrophy ~ d ), while the nonlinear version is Ωd = 2~ ~d +ω ~d ·ω ~ d . The evolution as, Ωdl (= 2~ ωm · ω ωm · ω equation for disturbance enstrophy, i.e. DETE is given as follows, with the complete derivation provided in Section 9.7 Term2

Term1

∂ui ∂uim 1 ∂2 Ω1 1 ∂2 Ωm DΩd = 2ωi ω j − 2ωim ω jm + − Dt ∂x j ∂x j Re ∂x j ∂x j Re ∂x j ∂x j !

!

Term3

2 ∂ωi ∂ωi 2 ∂ωim ∂ωim + − + Re ∂x j ∂x j Re ∂x j ∂x j

! (13.23)

The first and second terms in the right-hand side of Eq. (13.23) are due to stretching (Term1) and diffusion (Term2). In the above, the third term (Term3) originates from

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the diffusion term, yet it is not strictly dissipative. One can use the full nonlinear Navier–Stokes equation or its linearized equation for analysis [475]. The disturbance enstrophy equation is developed to show that Ωd provides a way of tracking instability and rotationality. It is not necessary to associate Ωd with dissipation only as a point property, as has been customarily done with total enstrophy [335, 567]. Conditions of growth for Ωd have been shown in Eq. (9.11) and are repeated below as DΩd ≶ 0 for Ωd ≶ 0 Dt

(13.24)

d That is the growth of positive Ωd is noted when DΩ Dt > 0, and for negative Ωd , the d growth occurs when DΩ Dt < 0. Negative-Ωd has been introduced to study vortexinduced instability in [475], while all possibilities have been considered in [466]. In reference [466], both signs of Ωd and its growths are compared quantitatively, for the same equilibrium flow excited by a localized monochromatic wall excitation [32, 483] and free-stream excitation [431, 474, 475]. The three-dimensional transitional and turbulent flows have been simulated and the DNS results are analyzed by tools of DETE and DME to identify the disturbance structures. The structures identified using DETE are compared with those obtained using the λ2 -criterion, since these two methods (in unaltered form) have both the unsteady and viscous terms. However, in applications using λ2 -criterion, the viscous and unsteady terms are removed following the observations in [204]. Similarly the structures identified by DME are compared with those obtained with Q-criterion, because, while deriving the governing equations for these methods from the Navier– Stokes equation, these simplify due to the incompressibility condition. Both of these methods can be reduced to providing the Poisson equation. In Q-criterion this is for the static pressure and in the DME method, this is for the total pressure or disturbance mechanical energy. The comparisons between the pairs of DETE/λ2 -criterion and DME/Q-criterion are shown in the following sections for two prototypical flow systems highlighted in Chapters 8 and 9, for wall and free stream excitations.

13.5 Wall and Free Stream Excitation for DNS Disturbance tracking is demonstrated for two prototypical excitation cases of incompressible flow over a semi-infinite flat plate, which excite the flow from inside and outside the boundary layer. The three-dimensional flow field excited at the wall and the free stream requires high accuracy solution methods, for which the velocity-vorticity formulation of the Navier–Stokes equation has been used for direct simulation from first principle [32, 33, 420]. In this approach, the vorticity is obtained by solving its transport equation in the non-dimensionalized rotational form as ∂~ ω ~v = 0 +∇×H ∂t

(13.25)

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~ v = (~ ~ + 1 ∇×ω ~ ). The Reynolds number, ReL = U∞ L/ν, is defined using where, H ω×V ReL the free stream velocity (U∞ ) and the length scale L, used to non-dimensionalize the Navier–Stokes equation. The velocity is obtained by solving vector Poisson equations ~ = −∇ × ω ~ ∇2 V The velocity and vorticity vectors are called solenoidal, when these satisfy divergencefree condition, and lends accuracy to the simulations performed using a staggered grid [32, 33]. As noted before, grid staggering improves accuracy by removing aliasing error. Furthermore, an optimized dispersion relation preserving, four-stage Runge–Kutta method is used for time integration [33]. The u- and w-components of velocity are obtained by solving the velocity Poisson equations, and the v-component of the velocity is calculated by integrating the continuity equation, thereby nominally ensuring a divergence-free velocity field at all times. The equilibrium flow over a semi-infinite flat plate is used to study wall and free stream excitation cases, which is obtained by solving the Navier–Stokes equation in the absence of any excitation in [466]. The schematics of the flows are as given in Figure 13.2. The wall excitation cases have been reported in [32, 483] for the transition experiments in [236, 405]. The free stream excitation case in Figure 13.2(b) has been studied experimentally [267] by a translating and rotating circular cylinder for different heights, convection speeds, circulations of the vortex and its sign. All of these determine the receptivity of the equilibrium flow to free stream excitation. Earlier experiments by Kendall [225] helped identify the range of convection speed of the periodic free stream excitation sources. This also identifies the optimal case of transition to turbulence by free stream convecting vortices. Other important parameters can be observed for a single convecting vortex in [267] with constant strength circulation, sign and height above the plate. The experimental results have been used for validation of two- and three-dimensional simulations presented in [431, 474, 475], in Chapter 9. Figure 13.2(a) and (b) show the schematics for the wall and free stream excitation problems to provide a systematic study of the tracking methods. The equilibrium flow obtained for the semi-infinite flat plate by solving the time dependent Navier– Stokes equation is used as the initial condition for both the excitation cases, with ReL = 105 . The equilibrium flow is obtained by solving Eq. (13.25) without any excitation, ω −6 while the simulation is continued up to a time when ∂~ ∂t reaches a value below 10 everywhere in the domain. This can be treated as a steady state for flow past a semiinfinite flat plate with the leading edge of the plate included inside the computational domain. One of the features of detecting coherent structures is the magnitude of vorticity, which is described in Chapters 8 and 9 for wall and free stream excitation. In the following, brief descriptions of these two excitation cases are provided with computational details.

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Far

a)

B4

R3

Ou

x

C1 B3

P4

x out

P3

E1 D1 B1

Semi-in

x2 x1 B2

x in z

D2

E2

e plate

U∞ + uv , vv

c R1

Q3

C2

A3

z max

G

Q4

A4

A1

R4

Ou

y

U∞

(uv , vv)

Far

b)

C3

x

R2 out

y

x

e Plat

Q1

P1

exciter strip

Plate leading edge

z max

P2

xin

Q2

z

Plate leading edge

Figure 13.2 Schematics of computational domains for transitional flow triggered by a) wallexcitation, and b) free stream convecting vortex. [Reproduced from “Tracking Disturbances in Transitional and Turbulent Flows: Coherent Structures”, T. K. Sengupta, P. K. Sharma, A. Sengupta, and V. K. Suman, Phys. Fluids, vol. 31, p 124106 (2019), with the permission of AIP Publishing.]

13.5.1 Wall excitation case For the wall excitation case, a computational domain as depicted in Figure 13.2(a) has been used with range defined by: −0.05L ≤ x ≤ 50L; 0 ≤ y ≤ L and −0.25L ≤ z ≤ 0.25L, such that the number of points used in x-, y- and z-directions are 2501 × 351 × 49. At the inflow of the domain uniform velocity is used, and at the outflow, Sommerfeld boundary conditions on velocity and vorticity are used. Periodic boundary conditions in the z-direction have been imposed. The non-impulsively started, harmonic wall-excitation is activated over a spanwise-punctuated suction-blowing strip. The wall excitation in the form of wallnormal velocity is given by, v(x, y = 0, z) = Am (x, z)A(t), where Am = 0.005(1 + t−4 cos 2πxex ) sin(8πz), with xex = 11.111 × (x − 1.5) and A(t) = (1 + er f ( 0.25066 )) sin 10t. The exciter is centered at xexciter = 1.5, located between x1 = 1.455 to x2 = 1.545. The nonimpulsive start-up is centered around t = 4 for the applied excitation. The exciter ω ¯0 has the non-dimensional frequency F, given by F = 2πU 2f ν = Re . Two values of F are L ∞ considered for the wall excitation, one with a moderate value of F1 = 1.0 × 10−4 , and the second is at a lower frequency of F2 = 0.5 × 10−4 .

13.5.2 Free stream convected vortex case For this case, the excitation source is a convecting vortex of strength Γ which moves over the semi-infinite flat-plate at a constant height, Hc , with a fixed velocity (c), as described in [474, 475], and shown here in Figure 13.2(b). The receptivity to excitation

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by a free stream vortex with finite core of diameter, d, moving in the inviscid part of the flow is given by the disturbance stream function as  (y − H )(d/2)2 (y + H )(d/2)2  Γ x¯2 + (y + Hc )2 c c ψ∞ = U∞ y − (U∞ − c) 2 + Ln + (13.26) 4π x¯2 + (y − Hc )2 x¯ + (y − Hc )2 x¯2 + (y + Hc )2 The disturbance stream function is given in terms of (i) the displacement effect due to the finite core vortex, which depends on the relative speed, (U∞ − c) given by the doublet in the second term of the right-hand side of Eq. (13.26), and (ii) the circulation effects arising from the vortex in the last term of the right-hand side. The dependence on time for the disturbance stream function is implicit through, x¯ = x0 − ct, where x0 is the initial location of the vortex. The vortex is idealized as inviscid which is strictly placed far outside the boundary layer. The method of images is used to ensure zero normal velocity on the wall. For the potential flow, this is accounted for by taking the images of the doublet and the vortex below the plate. It has been noted in [267] that a rotating and translating cylinder in the inviscid part of the flow, creates a perturbation field given in Eq. (13.26). With increasing c, one notices that both the displacement and circulation effects reduce. In this approach, one does not know the value of Γ a priori, without solving the Navier–Stokes equation. Thus, the value of Γ has to be parameterized, as it is also related to (U∞ − c), along with the rotation rate of the cylinder. The boundary conditions at the inflow and the far-field in Figure 13.2(b) are obtained using Eq. (13.26). Periodic boundary conditions apply on the spanwise boundaries for all variables. Two different domains having same extent in the x- and zdirections, with different extents in the y-direction have been used in [466]. For c = 0.3 the domain is given by, −0.05L ≤ x ≤ 20L; 0 ≤ y ≤ 0.75L, and −0.8L ≤ z ≤ 0.8L. The domain for c = 0.386 case has an extended y-range of: 0 ≤ y ≤ 1.5L. The number of points used in x-, y- and z-directions are 1001 × 301 × 129 for c = 0.3 and 1001 × 351 × 129 for c = 0.386. Kendall [225] noted experimentally that there is an optimum convection speed of the vortex for receptivity to be at around c = 0.3. The experimental optimum convection speed referred with respect to experiments of [225] is often misunderstood. It is thought that the convecting vortex in the free stream must convect with the free stream speed. This is following the inviscid vorticity transport equation, i.e. the Euler equation which indicates that the vorticity convects with local convection speed for two-dimensional flow. However, this is only true if a free vortex were to convect in an infinite fluid expanse, unaffected by any wall effects. But, an ensemble of vortices in free stream can have mutual interactions due to Biot–Savart law, affecting the individual vortex trajectory. In the presence of a wall, any free stream vortex is further affected by Biot–Savart interactions with the image system, as well as with the vorticity associated with the boundary layer forming over the geometry. It is equally relevant to note the results in [328, 329, 428], where a vortex is placed above a plate in quiet fluid, i.e. U∞ = 0. If the logic of convection by local speed is to hold, then the vortex would remain stationary. In [428], the Navier–Stokes equation has been solved for the unsteady flow field, and the results indicate that

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the vortex move with non-constant variable speed. It has been physically explained that there are multiple interactions among the singularities responsible for the flow field and their images to satisfy zero-normal velocity at the wall, and Biot–Savart interaction of the vortex in the free stream with the boundary layer forming over the solid geometry. One may also note that for three-dimensional inviscid vorticity transport equation, there would be forcing due to the vortex stretching term. For free stream turbulence causing transition, knowledge of receptivity is important for determining the optimal perturbation, even when it is imposed in a non-deterministic manner. The physics of such optimal perturbation parameters can be obtained from controlled experiment, as in [225, 267] or from computations in [431, 474, 475].

13.6 Vorticity Dynamics of Flows Caused by Wall and Free Stream Excitations Routes of transition experienced by the boundary layer forming over a semi-infinite flat plate by wall and free stream excitations differ due to different vorticity dynamics shown in Figures 13.3 and 13.4. It is noted in [412, 429] that the disturbance field is governed by the Orr–Sommerfeld equation for vanishingly small perturbations. However, these cases demonstrate different responses for wall and free stream excitations. Even though a free stream excitation can equivalently be represented by wall excitation, the coupling between the two mechanism is very weak. In Figure 13.3, the |~ ω| = 2 iso-surfaces (top) and two-dimensional contour plots are shown for the wall excitation frequencies given by F1 = 1 × 10−4 and F2 = 0.5 × 10−4 . The iso-surface perspective plot of |~ ω| is colored with wall-normal vorticity. As noted in Chapter 8, the higher frequency case shows a non-interacting type of transition [36, 483], as the TS wave-packet and the STWF remain distinct. The lower frequency case shows continuous interactions between TS wave-packet and STWF [422, 483], and also early and vigorous wall-normal eruptions. Also, the non-interacting case in the left spawns a turbulent spot, which is narrower compared to the lower frequency case. The twodimensional contour plots in the bottom frames are shown for a spanwise station at z = 0.0625, where the wall-normal velocity is maximum. Variation of results at nodal and anti-nodal spanwise stations have been described in Chapter 8. For the case of convecting counter-clockwise vortex in the free stream, with translation speed c = 0.386 and strength Γ = 2, some results are shown in Figure 13.4, with a perspective plot of spanwise vorticity colored by wall-normal vorticity component in the bottom frame, while the two-dimensional line contours of spanwise vorticity are shown above it. Comparing iso-surfaces in Figure 13.3 (for wall excitation) and Figure 13.4 (for free stream excitation), one notes the presence of Λ-vortices in the transitional region for the wall excitation case. No such vortices are present for the free stream excitation case. The vortex-induced instability in Figure 13.4, starts as two-dimensional spanwise rolls which evolve spatio-temporally, exhibiting slow spanwise variations, and eventually the vortical structures break

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0.1

0.2 0 24 0.2 20 22 14 16 18 12 0 x 8 10 z 0.2 0 ωy

6

4

2

0.4 y

-20 -16 -12 -8

t=20

-4 0

4

8

0.1

00 0.1

2

50

30

10

10

12 14 x

16 18

20 22

t=25

00 0.1

0 1 2 3 4 5 6 7 8 9 10

0.05

00 0.1

2

50

4

6

10

12 14 x

16 18

20 22

50

4

6

16 18 20 22

12 14 x

16 18 20 22

12 14 x

16 18 20 22

1

00

2

50

4

6

20

8

10

t=30

0.05

1

10

20

10

12 14 x

16 18

20 22

00

2

50

4

20

30

40

8

12 14 x

t=25

30

40

10

0 1 2 3 4 5 6 7 8 9 10

1

10

8

10

0.1

0.05

6

4

40

t=30

2

50

0.05

8

12 16 20

20

20

0 1 2 3 4 5 6 7 8 9 10

00

2

30

40

8

0 1 2 3 4 5 6 7 8 9 10

1

10

4

z = 0.0625

50

8

-4 0

1

30

6

4

-20 -16 -12 -8

t=20

0.05

1

40

ωy

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

10

24 18 20 22 12 x 14 16 8 10

6

4

2

12 16 20

z = 0.0625

0.05

t =35 ω =2 -4 F2 = 0.5×10

10

0 z 0.2 0

0.6

30

0.2 0 0.2

Æ

Æ

ω =2

t =35 -4 F1 = 1.0×10 0.4 y 0.6

6

8

10

Figure 13.3 Iso-surface of |~ ω| colored with ωy (top) and two-dimensional contours of |~ ω| at a spanwise plane for z = 0.0625 for wall excitation cases with imposed frequencies, F1 = 1 × 10−4 (left) and F2 = 0.5 × 10−4 (right), at the indicated times. [Reproduced from “Tracking Disturbances in Transitional and Turbulent Flows: Coherent Structures”, T. K. Sengupta, P. K. Sharma, A. Sengupta, and V. K. Suman, Phys. Fluids, vol. 31, p 124106 (2019), with the permission of AIP Publishing.] down to small-scales. The primary events occur downstream of the instantaneous streamwise location of the free stream vortex. Subsequently, the flow evolves threedimensionally, with stronger streamwise streaks upstream of the convecting vortex. For this instability, the wall-normal eruptions achieve larger heights, compared to wall excitation cases. For the wall excitation case, during the onset of STWF and its growth with time, the nonlinear effects become prominent in the turbulent region, noted in the frames of Figure 13.3 at t = 30 for F1 = 1 × 10−4 , and at t = 25 for F2 = 0.5 × 10−4 . This turbulent region grows with time in the downstream direction, while the upstream region shows

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0.4

ωz- contours

Min = -2684.1117 Min = 403.32564

0.3 Y

c = 0.386, G = 2

t = 31

0.2

-3

.5

-0.5

-14.5

0

0.4

-25

-1

5

10 X

20

ωz- contours

Min = -2496.109 Min = 328.45756 -3.5

-0.5

0

0

10 X

-3.5

-5

-15

-0.5

-14.5

-14.5

0.1

Y

-20

15

0.2

0

-3.5

-15

t = 33

0.3 Y

-1

-7

-7

0

-5

0.1

-7

-20

15

20

ωz- contours

0.4

t = 35

0.3

Min = -1459.8818 Min = 52.37707

0.2

-1 5 -3.

0.1 0

-0.5

-15

0

0

0 -2

10 X

-7

5 -2

-7

-20

15

20 Iso-surface of ω z = -10

t = 38.2

ω y : -60

xcv = 13.7452

10

11

12

13

-5

14

15 X

16

17

18

60

19

Figure 13.4 Two-dimensional contours of |ωz | for spanwise mid-plane, at indicated times, and iso-surface of ωz colored with ωy (bottom) for the free-stream excitation case with c = 0.386 and Γ = 2. The instantaneous streamwise location of the vortex is indicated by x¯ = xCV . [Reproduced from “Tracking Disturbances in Transitional and Turbulent Flows: Coherent Structures”, T. K. Sengupta, P. K. Sharma, A. Sengupta, and V. K. Suman, Phys. Fluids, vol. 31, p 124106 (2019), with the permission of AIP Publishing.]

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limited variation. Even though the TS wave-packet is present for wall excitation case with higher frequency, it does not play a role in transition, as shown in Figure 13.3; whereas for the vortex-induced instability in Figure 13.4, TS waves are not present. For both the excitation cases, unsteady separation and three-dimensionality are noted with ω x dominating over ωy during pre- and post-transitional stages.

13.7 Disturbance Tracking Using DME In Figure 13.5, iso-contours of the right-hand side of DME with both signs are shown for the wall excitation case with F1 = 1 × 10−4 . The exciter is centered at x = 1.5, the evolving disturbances are seen from the sources (right) and sinks (left) of DME shown in the figure. The DME equation has no contribution from unsteady and viscous terms of the Navier–Stokes equation due to divergence-free condition for the velocity field due to incompressibility. This makes the disturbance structures large and smoother, something akin to the case of Q-criterion. The physical appearances of the disturbance sources/ sinks are complementary, with distinct eruptions from the STWF visible in the bottom frames of Figure 13.5. The presence of a disturbance source helps its growth. But, the presence of sinks adds secondary motion from source to sink to enhance unsteadiness. This secondary motion is responsible for unsteadiness for incompressible flows, showing the role of DME in causing eruptions, as noted at t = 30 in the form of a coherent structure near x ≈ 16.5. This becomes more prominent at t = 35. The iso-contours of the right-hand side of DME equation (Eq. (13.22)), are plotted for the free stream vortex excitation case with c = 0.3 and Γ = 2 in Figure 13.6. For the free stream excitation, two sites of disturbances are seen to form: the first starting from the leading edge, and staying outside the shear layer while the other is the near-wall structure inside the shear layer. This separation of disturbances inside and outside the shear layer is due to the shear-sheltering effect noted in [194, 431]. The structure outside the shear layer can be associated with the free stream mode of the Orr–Sommerfeld equation (see the discussion in Section 4.4), while the second site inside the shear layer can be termed as due to the wall mode [429]. While the wall mode is noted near the leading edge at t = 34, the spatial location is at x ≈ 4. This highlights the necessity for including non-parallel effects on disturbance growth, which in turn requires that the equilibrium flow be obtained from the solution of the Navier–Stokes equation in a domain including the leading edge. Thus, it is mandatory to include the leading edge in computing to capture the shear sheltering effect, as this significantly affects the later-time interactions between the wall and free stream modes. The eruptions become milder due to interactions between wall and free stream modes, where the wall mode structures are lifted off making the downstream beyond the point of eruption quieter. For the case shown in Figure 13.6, the small scale remnant wall mode structures are noted to persist.

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Min = -3.90e+03 t = 20

RHS of DME(+ve)

t = 20

0.1

RHS of DME(-ve)

0.1

Y

1 10

1 10

0.05

1 10

Y

100

0.05

100

10

100

0

0

2

4

6

8

Max = 6.92e+03

10 X 12

14

16

1 10

0

2

4

6

10 X 12

14

16

10 X 12

14

16

18

0

2

-1 -10 -100

-1 -10

-1 -10 -100

0

2

4

6

8

10 X 12

14

16

Min = -1.00e+05 t = 30

0.05

0.05

1

2

-1 -10 -100

Y

Y

18 -1 -10 -100

0.1 1 10

2

18

-100

0.05

t = 30

0.1

8

-1 -10

1

8

6

Y

100

1

Max = 9.02e+04

4

10

100

10

100

2

0.1

1 1

Y

0

Min = -8.97e+03 t = 25

10

0.05

0

2

t = 25

0.1

0

18

-1 -10 -100

10

-1 -10 -100

100

0

0

4

6

8

RHS of DME = 10

0.6 0.4

2

10 X 12

14

16

18

t = 35

0.6

Y

0.4

0.2

0

8

10 X 12

RHS of DME = -10

2

4

6

t = 35

14

16

18

2

Y

0.2

0 0.2

Z

0

2

0 0.2 0

0

4

6

8

10

12

X

14

16

18

20

22

24

0 0.2

Z

0 0.2 0

0

4

6

8

10

12 X14

16

18

20

22

24

Figure 13.5 Two-dimensional contours of DME shown for the wall excitation case, with F = 1 × 10−4 for a spanwise plane at z = 0.0625 for the indicated times (top) and three-dimensional perspective plots of DME at t = 35 are shown in the bottom. The left and right panels show sinks and sources of DME, respectively. The start-up of excitation is in a smooth, non-impulsive manner [483]. [Reproduced from “Tracking Disturbances in Transitional and Turbulent Flows: Coherent Structures”, T. K. Sengupta, P. K. Sharma, A. Sengupta, and V. K. Suman, Phys. Fluids, vol. 31, p 124106 (2019), with the permission of AIP Publishing.]

13.8 Disturbance Tracking by DETE Disturbance tracking using DETE [475] is shown next for the cases depicted in Figures 13.5 and 13.6. The methods of DME and DETE are rooted to the Navier–Stokes equation and the development of the equations do not require any assumption, yet these have major differences due to the operator involved in deriving these equations. One notes that DME does not include unsteady and viscous terms, rendering smoother detected structures in Figures 13.5 and 13.6, which are prominent

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Figure 13.6 Two-dimensional contours of the right-hand side of the DME equation for the free stream vortex excitation case with c = 0.3 and Γ = 2 for the spanwise mid-plane at the indicated times shown in the top, and a three-dimensional perspective plot of the right-hand side of the DME equation at t = 45 is shown in the bottom. The left and right panels show sinks and sources of DME, respectively. [Reproduced from “Tracking Disturbances in Transitional and Turbulent Flows: Coherent Structures”, T. K. Sengupta, P. K. Sharma, A. Sengupta, and V. K. Suman, Phys. Fluids, vol. 31, p 124106 (2019), with the permission of AIP Publishing.] outside the shear layer. On the contrary, the unsteady and viscous terms remain after curl operation followed by taking a scalar product with vorticity to obtain the enstrophy transport equation. This equation is further split by regular perturbation to derive the DETE, and so, the method will capture smaller scale events inside the shear layer. As a consequence the detected structure will display very high wavenumbers and frequencies.

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(i) The regions where Ωd and DΩd /Dt are both positive, i.e. Ωd > 0 & DΩd /Dt > 0, we find DΩd /Dt as equal to 0.5(Ωd + |Ωd |) × 0.5

 DΩ

d

Dt



d + | DΩ Dt |

(13.27)

0.5 (Ωd + |Ωd |) + 

(ii) The regions where Ωd and DΩd /Dt are both negative, i.e. Ωd < 0 & DΩd /Dt < 0, we find DΩd /Dt as equal to 0.5(Ωd − |Ωd |) × 0.5

 DΩ

d

Dt



d − | DΩ Dt |

0.5 (Ωd − |Ωd |) + 

(13.28)

here  is a very small number of the order of 10−15 to avoid division by zero. In Figure 13.7, growth of ±Ωd is shown for the same wall excitation case in Figure 13.5, by disturbance tracking with the DME method. For the equilibrium flow over ˆ inside the boundary a semi-infinite flat plate, the vorticity is negative (~ ωm = −|~ ωm |k) ~ d = ωdxˆi + layer. If the disturbance vorticity is written in Cartesian co-ordinates as, ω ˆ then the positive (or negative) Ωdl must have negative (or positive) ωdz . It ωdy ˆj + ωdz k, is also seen that positive Ωd is predominant in the outer part, while the negative Ωd are present closer to the wall for the transitional flow. The streamwise extent for Ωd < 0 is longer, as compared to where Ωd > 0. This indicates the role of wall mode for Ωd < 0, while Ωd > 0, is caused by free stream mode near the leading edge. The effects of wall and free stream modes explain the multi-deck disturbance structure. The perspective plot at t = 35 shows coherent structures associated with Ωd > 0. The structures of Ωd < 0 at t = 35 are noted close to the plate having smaller scales. These finer scales (near-wall structures) are not captured by other methods – those based on originating equations for which viscous and unsteady terms are either neglected (in λ2 -criterion) or which drop out for incompressible flows (as in DME method and Q-criterion). Figure 13.8 shows the growth of positive and negative values of Ωd for the case of free stream excitation, whose DME counterpart has been shown in Figure 13.6. For free stream excitations, it is distinctly observed that the free stream mode is sheltered from the wall mode, for all frames with positive and negative values of Ωd . As in case of wall excitation, here also, negative ωdz (i.e., positive Ωdl ), erupts as the free stream mode, while positive ωdz structures are seen close to the wall. Such distinctions between disturbance structures such as the wall (positive ωdz ) and free stream modes (for negative ωdz ) are evident in the perspective plot of DETE structures at t = 45 in the bottom of Figure 13.8. There is a special feature shared by DME and DETE methods, i.e. the disturbances are seen as continuous from the leading edge to the STWF. There are the hairpin and Λ-vortices for the wall excitation case, while there are ramp-like and streaky structures noted for the free stream excitation case. Either of the excitation cases shows small scale structures near the plate due to the presence of viscous and unsteady terms in DETE method only [466].

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Coherent Structure Tracking in Transitional and Turbulent Flows DWd/Dt > 0 & Wd > 0

t = 20

Max = 1.41e+05 0.1 Y

Y

0.05

0

0.05

2

0

4

6

8

10

12

14

16

0

18 20

0

t = 25

Max = 1.59e+05 0.1 Y

10 8 t = 25

12

14

16

18 20

10 8 t = 20

12

14

16

18 20

10 8 t = 35

12

14

16

18 20 Y

Y 0.05

2

0

4

6

Max = 6.28e+05

10 8 t = 30

12

14

16

0

18 20

0

2 4 6 Max = -3.80e+05

0.1

0.1

Y

Y 0.05

0.05

0

2 4 6 Max = -3.31e+04

0.1

0.05

0

DWd/Dt < 0 & Wd < 0

t = 20

Max = -2.51e+04 0.1

0

0.6

2 4 6 DWd/Dt = 1 & Wd > 0

10 8 t = 35

12

14

16

X

0.4 Y 0.2

0

0.6

2

4

6

DWd/Dt = 1 & Wd > 0

X

0.4 Y 0.2

0 0.2

0 Z 0.2 0

0

18 20 Y

2

4

6

8

22 24 16 18 20 10 12 X14

0 0.2

0 Z 0.2 0

2

4

6

8

22 24 16 18 20 10 12 X14

−4 d Figure 13.7 Two-dimensional contours of DΩ Dt for the wall excitation case with F 1 = 1 × 10 at z = 0.0625, an anti-node plane at indicated times for indicated signs, and a d three-dimensional perspective plots of | DΩ Dt | = 1 shown at t = 35 in the bottom. The left and right panels are for the growth of Ωd > 0 and Ωd < 0, respectively. [Reproduced from “Tracking Disturbances in Transitional and Turbulent Flows: Coherent Structures”, T. K. Sengupta, P. K. Sharma, A. Sengupta, and V. K. Suman, Phys. Fluids, vol. 31, p 124106 (2019), with the permission of AIP Publishing.]

13.9 Comparing Coherent Structure Detection Methods The correlations among the various coherent structure and vortex identification methods are shown with magnitude of vorticity. There is similarity between the DETE method and the λ2 -criterion to trace vortical structures, as both have the ability to include unsteady and viscous effects for incompressible flows. As coherent structures

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t = 34

0.15

min = 0 max = 5.97e+07

DW d/Dt > 0 & Wd > 0

0.15

t = 34

min = -1.87e+07 max = 0

DW d/Dt > 0 & Wd > 0

0.1

y

y

0.1

0.05

0.05 00

10x

5

15

min = 0 max = 5.98e+07 0.15

t = 39.6

0.15

00

20

0.1

10x

5

15

20

min = -1.90e+07 max = 0

t = 39.6

y

y

0.1

0.05

0.05

00

y

0.1

15

t = 39.6 min = 0 max = 9.98e+07

20

t = 43

0.1

10 x

5

0.6

15

00

20

t = 45

Y

0

2

4

6

8

10

12

14

16

DWd/Dt = 0.1 & Wd > 0

0 18 -0.5

15

20

t = 45

Y

Y

0.2

0 -0.5

10 x

5

0.6 0.4

Y

0 Z 0.5

15

min = -4.55e+07 max = 0

0.05

00

0.2

10 x

5

0

0.15

0.05

0.4

0

20

y

0.15

10x

5

0

Z 0.5

0

2

4

6

8

10

12

14

16

18

DWd/Dt = 0.1 & Wd < 0

d Figure 13.8 Two-dimensional contours of DΩ Dt are shown for the free stream excitation case for c = 0.3 and Γ = +2 at the mid-span plane at the indicated times in the d top, and three-dimensional perspective plots of | DΩ Dt | = 0.1 shown at t = 45 in the bottom frame. The left and right panels are for the growth of Ωd > 0 and Ωd < 0, respectively. [Reproduced from “Tracking Disturbances in Transitional and Turbulent Flows: Coherent Structures”, T. K. Sengupta, P. K. Sharma, A. Sengupta, and V. K. Suman, Phys. Fluids, vol. 31, p 124106 (2019), with the permission of AIP Publishing.]

consist of coherent vortices, there is a possibility to compare between DETE method with λ2 -criterion, taking vorticity-magnitude as the reference. In the same way, the DME method and Q-criterion do not have contributions from unsteady and viscous terms due to solenoidality of velocity. The Q-criterion tracks pressure minima for vortical cores while the DME method uses the total pressure to track disturbance structures, and thus a correlation is provided between these two methods. The first step in correlating different methods is to identify a region in space (which can be the full domain also, as is the case for the wall excitation), where prominent

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Coherent Structure Tracking in Transitional and Turbulent Flows 103 102

d ( DW Dt )(Mean)

101

t = 20 t = 25 t = 30 t = 35

(W d > 0)

100 10–1 5 6000 4000

15

10

20

d ( DW Dt )(Mean + Std. Dev.)

25 |ω | 30 t = 20 t = 25 t = 30 t = 35

35

40

45

40

45

40

45

35

40

45

35

40

45

35

40

45

(W d > 0)

2000 0

5

15

10

20

0.5

25

|ω| 30

35

λ 2 – (Mean)

0 –0.5 –1 –1.5

A(11, –0.51) C(3.8, 0.064) D(9.4, –0.5257)

5

10

B(41.8, –0.168)

15

20

25

|ω| 30

35

–5 –10 –15

λ 2 – (Mean – Std. Dev.) 5

10

15

20

25

|ω| 30

0 -50 –100

d ( DW Dt )(Mean)

5

10

(W d > 0) 15

20

25

| ω | 30

0 –1000 –2000

d ( DW Dt )(Mean – std. Dev.)

5

10

(W d > 0) 15

20

25

|ω| 30

Figure 13.9 Correlation curves of growth rates of positive and negative Ωd and λ2 -values, with |~ ω|, at the indicated times for wall excitation case with F1 = 1 × 10−4 . The spatial mean and standard deviation are evaluated over the full domain. [Reproduced from “Tracking Disturbances in Transitional and Turbulent Flows: Coherent Structures”, T. K. Sengupta, P. K. Sharma, A. Sengupta, and V. K. Suman, Phys. Fluids, vol. 31, p 124106 (2019), with the permission of AIP Publishing.]

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disturbance structures are observed. Identify a fixed reference value of |~ ω| with a negligibly small tolerance. One must extract the values of the dependent variable of the criterion under consideration at all the grid points of the chosen region. With this data spread in the identified zone, the mean and variance of the dependent variables of all the criteria are calculated, for some identified time. The mean and variance of the spread of the dependent variable provides the spatial distribution of the data to be correlated within the chosen region in space, as an interrogation box. This procedure is repeated for each reference value of |~ ω| for the dependent variables used in the criteria. In the following figures, different criteria can be compared among themselves with respect to spatial mean value, with and without standard deviation, for that particular criterion by plotting iso-surfaces of these correlation methods against the reference value of |~ ω| chosen.

13.9.1 Correlation for wall excitation case Figure 13.9 shows the correlations obtained between |~ ω| and disturbance tracking methods based on DETE and λ2 -criterion for the wall excitation case with F1 = 1×10−4 . A tolerance limit of ±0.1 is used to calculate the spatial mean and standard deviation for |~ ω| considering the full computational domain. The top frame depicts the spatial mean of positive growth rate of Ωd with |~ ω|, which increases with increasing |~ ω| for the indicated times. For values of |~ ω| ≤ 21, the growth rates are very small, as is noted by the logarithmic ordinate scale. In the frame just below, the sum of the spatial mean and the standard deviation for the same case is shown as the ordinate, for which the displayed time instants show an increasing trend with |~ ω| during late transition stages for Ωd > 0 at t = 30 and 35, with higher values of |~ ω| dominating the flow. In the third frame from the top, mean value for the λ2 -criterion is shown. For |~ ω| > 8, one notices this value to be negative. Thus, in the fourth frame from the top, the standard deviation of the quantity is subtracted from the mean for λ2 and correlated with |~ ω|. It is noted in the third frame that the lower mean values of |~ ω| have positive λ2 , which according to [204] implies the absence of pressure minima for such vortical structures, and thus would not be identified as coherent structures. Also, such lower positive values of λ2 have larger standard deviation, so that in the fourth frame, the sum of the magnitude of mean and standard deviation display the desired negative value of λ2 . This implies a suggested augmentation for the λ2 -criterion, without discarding the unsteady and viscous term, by using it the way it is done in the fourth frame (by combining the mean and variance of the data). The bottom two frames in Figure 13.9 show the case for growth of disturbance enstrophy for negative Ωd , which show that an intermediate range of |~ ω| will have the highest growth rate. As time progresses, this peak becomes higher, and shifts to lower values of |~ ω|. In Figure 13.9, the standard deviation for all the growth quantities are noted to be significantly higher at a later time, due to dominance of nonlinear actions. The Q-criterion and DME method are correlated with |~ ω| next, for the same case of wall excitation in Figure 13.10. For the Q-criterion, coherent structures require the

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presence of positive Q, so that the static pressure is a minimum corresponding to the maximum value of Q. For the DME method, disturbance sources are associated with negative right-hand side of Eq. (13.22). The DME method and Q-criterion shown in Figure 13.10, also indicate higher growth rates for higher values of |~ ω|. Similarly, the standard deviation for each of the methods displays higher fluctuations and add to the growth of disturbances. As in λ2 -criterion for small values of |~ ω|, the Q-criterion also fails to satisfy the sign requirement for the pressure minima. Overall, one notices

Mean of RHS of DME

200 0 –200 –400

5

10

15

20

25

|ω | 30

35

40

45

20

25

|ω | 30

35

40

45

35

40

45

40

45

–1000 –2000 –3000

(Mean – Std. Dev.) of RHS of DME 5

15

10

15

Q - (Mean)

E(10, 2.138)

10 5

F(5.8, 0.5445)

0 –5

5

10

15

20

25

|ω | 30

Q - (Mean + Std. Dev.)

150 100 50 0

5

10

15

20

25

|ω| 30

35

Figure 13.10 Correlation curves showing Q- and RHS of DME criteria correlated with |~ ω|, at the indicated times for wall excitation case with F1 = 1 × 10−4 . The spatial mean and standard deviation are evaluated over the full domain. [Reproduced from “Tracking Disturbances in Transitional and Turbulent Flows: Coherent Structures”, T. K. Sengupta, P. K. Sharma, A. Sengupta, and V. K. Suman, Phys. Fluids, vol. 31, p 124106 (2019), with the permission of AIP Publishing.]

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from Figures 13.9 and 13.10 that higher |~ ω| show larger growth of disturbance by both DETE and DME methods indicating presence of coherent structures, dictated by λ2 and Q-criteria.

13.9.2 Correlation for free stream excitation case Comparing the effectiveness of the four methods for the wall excitation case, the free stream excitation case is correlated next for the same methods in Figures 13.11 and 13.12. In finding the spatial mean and standard deviation of the properties in these figures, domains of interrogation are chosen at different times, unlike the case of wall excitation where the full computational domain has been used. For the free stream excitation case shown in Figure 13.11 and 13.12, the complete y- and z-ranges of the computational domain are considered. For the data of t = 34 and 39.6, the streamwise range chosen is given by 9 ≤ x ≤ 12 and for t = 43 and 45, the enlarged streamwise range is given by, 11 ≤ x ≤ 16. Figure 13.11 correlates the DETE method with λ2 -criterion, for the free stream excitation case with free stream vortex strength of Γ = 2, and speed of c = 0.3. Once again |~ ω| is used as the currency for comparison of different methods. In Figure 13.11, results are shown for the primary instability at t = 34 and beyond. Here also, one notices significant disturbance growth for Ωd > 0 for larger values of |~ ω| at t = 34, with a discrete jump of the growth rate for |~ ω| ≥ 32.0. Subsequently, all the values of |~ ω| keep increasing with time, and with higher rates. In contrast, the growth rates for Ωd < 0 are smaller by at least one order of magnitude at all times. For the λ2 -criterion, one notes a significantly large negative value of λ2 for |~ ω| ∼ 20 at t = 34. Above this value of |~ ω|, the eigenvalue becomes positive and thus these vorticity values cannot be associated with λ2 -criterion. At t = 39.6 also, one notices a negative λ2 -value for |~ ω| = 20. For t = 43 and 45, one notices negative values of λ2 across all |~ ω|, except for very small values of vorticity. Figure 13.12 shows the Q-criterion to have a large positive value of Q for |~ ω| ∼ 20 and some lower values at t = 34 and 39.6. Higher values of |~ ω| show negative values of Q, and thus, those values cannot be associated with coherent structures at t = 34 and 39.6. At later times of t = 43 and 45, the higher values also have positive values of Q, so one can relate these to coherent structures. In comparison, the right-hand side of the DME method alternates in sign at t = 34, implying strong unsteadiness and therefore growth of disturbances, without any preferential value of |~ ω|. At later times, the fluctuations show a reduction in values to lower or moderate levels, but negative values are seen for |~ ω| ∼ 25. The higher values of |~ ω| show stronger switching of signs implying very high unsteadiness.

13.9.3 Comparison of DETE method and λ2 -criterion: Wall excitation case Figures 13.9 to 13.12 display correlation curves for different Eulerian methods in tracking disturbance structures. With the correlation curves, specific cases for distinct values of |~ ω| can be presented. In Figures 13.13 and 13.14, the wall excitation case is

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Coherent Structure Tracking in Transitional and Turbulent Flows

c = 0.3, G = 2

103

t = 43

t = 45

DWd /Dt

102 t = 39.6

101 100

t = 34

10–1 0

5

10

15

10

20 |ω |

25

30

35

40

t = 34 A1

0 λ2

t = 45 A2

–10

B1

B2

B3

t = 39.6

A3

t = 43

–20 0

5

10

15

20 |ω |

25

30

35

40

103 t = 39.6 DWd /Dt

102 t = 43

101 100 10–1

t = 34

0

5

10

15

|ω|

20

25

t = 45 30

Figure 13.11 Correlation curve with |~ ω| for DETE methods for positive and negative Ωd and λ2 criterion, for free stream excitation case with c = 0.3 and Γ = 2. [Reproduced from “Tracking Disturbances in Transitional and Turbulent Flows: Coherent Structures”, T. K. Sengupta, P. K. Sharma, A. Sengupta, and V. K. Suman, Phys. Fluids, vol. 31, p 124106 (2019), with the permission of AIP Publishing.]

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c = 0.3, G = 2

1000

t = 34

RHS of DME

500 t = 45 0

–500

t = 43

–1000

t = 39.6 0

5

10

15

20 | ω|

25

30

40

C1

200 C2 150

t = 43 t = 39.6

C3

D2

100 Q

35

D1

50 t = 45 0 t = 34

–50 0

5

10

15

20 | ω|

25

30

35

40

Figure 13.12 Correlation curve showing Q-criterion and DME-method with |~ ω|, for free stream excitation case with c = 0.3 and Γ = 2. [Reproduced from “Tracking Disturbances in Transitional and Turbulent Flows: Coherent Structures”, T. K. Sengupta, P. K. Sharma, A. Sengupta, and V. K. Suman, Phys. Fluids, vol. 31, p 124106 (2019), with the permission of AIP Publishing.]

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considered, correlation curves for which have been shown in Figures 13.9 and 13.10 at t = 30. Figure 13.13 displays two such values of |~ ω| = 11 and 41.8, with the purpose of comparing the λ2 -criterion and DETE method for positive Ωd . For |~ ω| = 11, this is |DWd/Dt| = 18.2 DWd/Dt > 0 for (Wd > 0)

i) 0.3

0.3

0

2

4

6

8 X 10

12

14

16

18 0 Z

0

2

4

6

u: 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

λ2 = –0.51

ii) Y

Y

Z

0

2

4

8 X 10

12

14

16

18 0 Z

0

2

4

6

u: 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

iii) 0.3

18

8 X 10

12

14

16

18

Y

0 Z

16

Æ |ω| = 41.8

0.3

Y

14

u: 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

vi)

Æ |ω| = 11

12

u: 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.3

6

8 X 10

λ2 = –1.68

v)

0.3

0

t = 30

Y

Z

0

|DWd/Dt| = 583.6 DWd/Dt > 0 for (Wd > 0)

iv)

X

Y

Z

t = 30

Y

0

2

4

6

8 X 10

12

14

16

18 0

u: 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Z

0

2

4

6

8 X 10

12

14

16

18

u: 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 13.13 Iso-surfaces of (i,iv) positive indicated growth rate of Ωd values; (ii,v) indicated negative λ2 values, those calibrated with (iii,vi) vorticity magnitudes, |~ ω| for the wall excitation case with F1 = 1 × 10−4 at t = 30. [Reproduced from “Tracking Disturbances in Transitional and Turbulent Flows: Coherent Structures”, T. K. Sengupta, P. K. Sharma, A. Sengupta, and V. K. Suman, Phys. Fluids, vol. 31, p 124106 (2019), with the permission of AIP Publishing.] marked as point A in the third frame from the top of Figure 13.9, with the co-ordinate (λ2 = −0.51, |~ ω|=11) and DΩd /Dt = 18.2, |~ ω| = 11 in the top frame of Figure 13.9. The comparison among these three results are shown in the left frames of Figure 13.13. Evidently, these methods help identifying the STWF, with the DETE method showing better match with |~ ω|-contours, compared to the λ2 -contour. In the right frames of

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i) 0.7

|DWd/Dt| = 3.5 Y DWd/Dt > 0 for (Wd > 0)

0.7

0

5

10

X

15

20

25 0 Z

0

5

10

u: 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

λ2 = 0.064

ii)

0.7

Y

Y

0

5

10

X

15

20

18 0 25 Z

0

5

10

u: 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

iii)

20

25

Y

0 Z

X

15

Æ |ω| = 9.4

0.7

Y

25

u: 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

vi)

Æ |ω| = 3.8

0.7

20

u: 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.7

Z

X

15

λ2 = –0.5257

v)

0

t = 35

Y

Z

0

|DWd/Dt| = 10.11 DWd/Dt > 0 for (Wd > 0)

iv)

X

Y

Z

t = 35

0

5

10

X

15

20

25 0

u: 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Z

0

5

10

X

15

20

25

u: 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 13.14 Iso-surfaces of (i,iv) positive growth rate of Ωd , (ii,v) λ2 value, calibrated with (iii,vi) indicated vorticity magnitudes, |~ ω| for the wall excitation case with F1 = 1 × 10−4 at t = 35. These points are marked as C and D in Figure 13.9. [Reproduced from “Tracking Disturbances in Transitional and Turbulent Flows: Coherent Structures”, T. K. Sengupta, P. K. Sharma, A. Sengupta, and V. K. Suman, Phys. Fluids, vol. 31, p 124106 (2019), with the permission of AIP Publishing.] Figure 13.13, the comparison is made for |~ ω| = 41.8 for positive Ωd method and λ2 criterion. In the third frame from the top of Figure 13.9, this point is marked as B, for the correlation with λ2 -criterion. In this higher vorticity amplitude case, one notices that the λ2 -criterion displays exaggerated coherent structures near the STWF, while missing the structures near the leading edge of the plate. This is despite the fact that λ2 shows a distinct negative peak in Figure 13.9. Once again, the superiority of the DETE method over λ2 -criterion is established.

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In Figure 13.14, the DETE method and λ2 -criterion are compared at the later time of t = 35 with smaller values of |~ ω| = 3.8 and 9.4 for the wall excitation case. Both the methods show larger STWF structure as compared to |~ ω| structures, with λ2 criterion faring poorly, completely missing the structures downstream of the STWF. In comparison, the DETE method performs much better, only missing the continuous sheet of vorticity originating from the leading edge that stays outside the shear layer. It has been noted that such structures are easily detected by the DETE method for Ωd < 0.

13.9.4 Free stream excitation case: Comparing DETE method and λ2 -criterion Figures 13.15 and 13.16, show a comparison between the λ2 -criterion and DETE method for Ωd > 0, for free stream excitation case as reported in [466]. The corresponding correlations of these methods are shown in Figures 13.11 and 13.12. In Figure 13.15, comparison is made between DETE method and λ2 -criterion at t = 34 for |~ ω| = 14.6, 15.8 and 17.4. The corresponding parameters are marked in Figure 13.11 as A1, A2 and A3 in the middle frame, for those cases with λ2 < 0. The |~ ω| contours are shown in the bottom frame of Figure 13.15, with the frame for the λ2 -criterion just above. For the DETE method the growth rates are shown on top for ±Ωd . The complementarity of ±Ωd is evident, with the structures identified by DETE method matching extremely well with |~ ω| contours. The λ2 -criterion does not perform as well as it was for the wall excitation case. Here, one sees much bigger structures, which appear smoother and larger, as rolled-up structures. In Figure 13.16, the methods are compared at t = 43 for the |~ ω| values noted as B1, B2 and B3 in the middle frame of Figure 13.11. For this time, the STWF is much smaller, as seen in |~ ω|-contours. This is detected by both the DETE method and λ2 -criterion, with the bigger structures in the outer part of the boundary layer (with Ωd > 0). The small scale, near-wall structures are captured only by DETE method (with Ωd < 0).

13.9.5 Comparing DME method and Q-criterion: excitation

Wall and free stream

In Figure 13.17, the DME method is compared with Q-criterion for the points marked as E and F in Figure 13.10. The point F has the value of Q = 0.5445, while the point E has a higher value of Q = 2.138, both being positive - as required by the Qcriterion. For this comparison one is looking for vorticity contours with magnitude, |~ ω| = 10 and 5.8, on the left and right frames of Figure 13.17. At both the time instants, the vorticity contours detected by the Q-criterion shows exaggerated STWF, failing to capture the other structures at all. In comparison, the structures captured by the DME method have a decent match with the vorticity contours. This helps with the observation that vortical structures in transitional and turbulent flows are not always due to static pressure achieving minimum value. In contrast, methods using physical disturbance quantities like DME and DETE, which in turn, are based on the Navier–

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i)

t = 34

DWd/Dt at (A1, A2, A3) for (Wd < 0)

ii) 0.2

0.2 Y

Y

0 Z

10

8

6

X

12

14

0 Z

6

10

8

u: 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

14

12

u: 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

λ2 (A1, A2, A3)

iii) Y

0.2

X

X

Y

14

Z

12

0

10 8

Z

X

6

u: 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

iv)

Æ

|ω| = 14.6, 15.8, 17.4

0.2 Y

14 12

0

10 Z

8 6

X u: 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

DΩd d Figure 13.15 Iso-surfaces of (i) DΩ Dt > 0, (ii) Dt < 0, and (iii) λ2 values calibrated with (iv) the indicated vorticity magnitude, |~ ω|, for the free stream excitation case with c = 0.3 and Γ = 2 at t = 34. The points A1, A2 and A3 are as marked in Figure 13.11. [Reproduced from “Tracking Disturbances in Transitional and Turbulent Flows: Coherent Structures”, T. K. Sengupta, P. K. Sharma, A. Sengupta, and V. K. Suman, Phys. Fluids, vol. 31, p 124106 (2019), with the permission of AIP Publishing.]

Stokes equation detect disturbance structures better. Even though the DME method and Q-criterion have no viscous and unsteady terms, the DME method still performs

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Coherent Structure Tracking in Transitional and Turbulent Flows i) DWd/Dt at (B1, B2, B3) for (Wd > 0)

t = 43

0.2

0.2

Y 0 Z

DWd/Dt at (B1, B2, B3) for (Wd < 0)

ii)

Y 14

12

10

X

18

16

0 Z

10

14

12

u: 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

X

18

16

u: 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

λ2 (B1, B2, B3)

iii) Y 0.2 Y

X 18

Z

X

0

16 14

12

Z

X

10

u: 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

iv)

Æ

|ω| = 21.8, 26.6, 28.4 0.2

Y 18 16

0 Z

14 12 10

X u: 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

DΩd d Figure 13.16 Iso-surfaces of (i) DΩ Dt > 0, (ii) Dt < 0 and (iii) λ2 values, calibrated with (iv) the indicated vorticity magnitude, |~ ω| for the free stream excitation case with c = 0.3 and Γ = 2 at t = 43. The points B1, B2 and B3 are as marked in Figure 13.11. [Reproduced from “Tracking Disturbances in Transitional and Turbulent Flows: Coherent Structures”, T. K. Sengupta, P. K. Sharma, A. Sengupta, and V. K. Suman, Phys. Fluids, vol. 31, p 124106 (2019), with the permission of AIP Publishing.]

better. This has been also observed in Chapter 9 - that DME is based on physical quantities, enabling one to capture disturbance structures better. In Figure 13.18, these two methods are again compared for a free stream excitation case at t = 34 and 43, to track structures for (C1, C2, C3) and (D1, D2), as marked in Figure 13.12. At both these instants Q-criterion fares poorly, as compared to the DME method, specially at the earlier stage. This is surely due to the reason that Q-criterion

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t = 30

RHS of DME = –49.6 Y

(iv)

Y

0.5

Y

X

X

Y Z

Z 0 Z

0

10

5

15

0

X

Z

0

5

20

25

X

u: 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

(v)

Q = 2.138

15

10

u: 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

(ii) 0.5

t = 35

RHS of DME = –46.9

Q = 0.5445

0.5 Y

Y 0 Z 0

10

5

15

0 Z

X

0

5

(vi)

Æ

|ω| = 10

25

Æ

|ω| = 5.8

0.5

Y

Y

0 Z

20 X

u: 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

u: 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

(iii) 0.5

15

10

0

5

10

15 X

u: 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0 Z

0

5

10

15

20

25

X

u: 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 13.17 Iso-surfaces of (i,iv) RHS of DME Equation and (ii,v) Q-criterion calibrated with (iii,vi) the indicated vorticity magnitudes, |~ ω| for the wall excitation case with F1 = 1 × 10−4 at t = 30 and 35. [Reproduced from “Tracking Disturbances in Transitional and Turbulent Flows: Coherent Structures”, T. K. Sengupta, P. K. Sharma, A. Sengupta, and V. K. Suman, Phys. Fluids, vol. 31, p 124106 (2019), with the permission of AIP Publishing.] is based on pressure minimum, which are absent early on, and disturbances are purely vortical in nature.

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Coherent Structure Tracking in Transitional and Turbulent Flows

i) 0.2

Y

RHS of DME at (C1, C2, C3)

t = 34

0 Z

Y

Z

6

10

8

X

12

14

0 Z

10

14

12

u: 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Q (C1, C2, C3)

ii) 0.2

16

18

Q (D1, D2)

0.2 Y

6

10

8

X

12

14

0 Z 10

14

12

u: 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

iii)

|ω| = 17.4, 18.4, 19.6

X

16

18

u: 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

vi)

Æ

0.2

Æ

|ω| = 27.6, 29.8

0.2

Y 0 Z

X

u: 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

v)

Y 0 Z

t = 43

0.2

X Y

RHS of DME at (D1, D2)

iv)

Y

6

8

10

X

12

14

u: 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0 Z

10

12

14

X

16

18

u: 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 13.18 Iso-surfaces of (i,iv) RHS of DME equation and (ii,v) Q-criterion, calibrated with (iii,vi) the indicated vorticity magnitudes, |~ ω|, for free stream excitation case with c = 0.3 and Γ = 2, at t = 34 and 43. [Reproduced from “Tracking Disturbances in Transitional and Turbulent Flows: Coherent Structures”, T. K. Sengupta, P. K. Sharma, A. Sengupta, and V. K. Suman, Phys. Fluids, vol. 31, p 124106 (2019), with the permission of AIP Publishing.]

13.10 Closing Remarks In this chapter qualitative and quantitative comparisons of methods developed for tracking disturbances (based on physical quantities, such as the DME and the DETE methods), has been conducted. The DME method is based on the Poisson equation connected to the DME [431] equation, while the DETE method is based on the evolution equation for disturbance enstrophy [475]. With this objective, calibration of these two methods are performed with two well-known methods for vortex identification, the Q- and λ2 -criteria. The DME and DETE methods have been developed to trace disturbances from onset to nonlinear stages. In this chapter, the capabilities of these methods in identifying the coherent vortical structures from their early stages to the late transitional stages have been examined. The Q- and λ2 -criteria

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are based on pressure minima at the vortex core to identify such coherent structures, and both work satisfactorily for fully-developed turbulent flow, even though there are failures reported in Jeong et al. [205]. In contrast, DME and DETE methods are applicable for all regimes of disturbance growth, from incipient to fully-developed stages. Thus, the main objective is to show that DME and DETE methods are capable of tracing incipient disturbances and the coherent vortical structures simultaneously, from receptivity to the fully-developed turbulent stage. The effectiveness of Q- and λ2 -criteria has been demonstrated already for identifying coherent structures, as they reveal even the Λ- and hairpin-vortices with aligned and staggered patterns. However, the operational range for these methods is mainly in the late transitional and turbulent stages. The DME and DETE methods have the added advantage of capturing evolving disturbance structures, which later turn into coherent vortical structures. These methods are derived from the Navier–Stokes equation without any assumptions and have physical meaning associated with the identified structures, as these methods are based on physical terms, such as mechanical energy and enstrophy. The Q- and λ2 criteria, on the other hand, try to do the same by identifying the pressure minima indirectly. In DME method, the mechanical energy is the total pressure for the incompressible flows, and incorporates the Q-criterion and the velocity fluctuations, as given by ∇2 Ed = 2Q+∇2 [ 12 (V 2 −Vm2 )]. This enables the DME method to not only detect the vortical coherence through Q, but also detect near-wall streak coherence through the velocity fluctuations. This highlights the distinction between the DME and Q-criterion, even though both are based on the Poisson equation for total mechanical energy and static pressure, respectively. In the λ2 -criterion, the pressure Hessian is obtained and then the unsteady and viscous terms are dropped. The trace of pressure Hessian reproduces the Q-criterion, without the need of neglecting unsteady and viscous terms. The Qand λ2 -criteria are for incompressible flow, and depend on velocity gradient terms only. This may lead researchers to erroneously view it as a kinematic tool, and extend its application to compressible flows. In any case, removal of the unsteady and viscous terms cause inaccuracy for the λ2 -criterion. The DETE method and the λ2 -criterion are grouped together and the DME method and the Q-criterion are placed into another set, due to the similarity of their links with the Navier–Stokes equation and retained operators. All such methods must be correlated with the vorticity magnitude (|~ ω|) for tracking vortical coherent structures, if these are developed to trace coherent vortices primarily for fully developed turbulent flows. Such correlations are necessary, if one wants to link these methods. These correlation plots, have assisted in explaining the varying degree of success achieved by the methods described in this chapter. However, the DME and DETE methods show major vortical structures, while capturing other evolving weaker structures likely to be missed by the Q- and λ2 -criteria.

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The Route of Transition to Turbulence: Solution of Global Nonlinear Navier–Stokes Equation

14.1 Introduction Turbulence continues to be a largely unsolved problem of physics, despite accurate numerical results available for some canonical problems. One of the dominant approaches in studying turbulence is nonlinear dynamics, sharing certain universal properties of fully developed turbulence. The other approaches include studies where turbulence is traced as a receptivity problem starting from the excitation of an equilibrium flow by input disturbances and the disturbances propagate via multiple instabilities accounting for the overall growth processes. This latter approach has been the one followed in this book so far. In Figure 2.1, a schematic of flow transition indicated the dynamical system approach as a possible route. Two other such roadmaps are now presented in Figure 14.1, and these are from [90] and [385], both of which classify transition routes based on the amplitude of excitation only. According to Saric et al. [385] the amplitude of input excitation increases for routes followed along A to E in Figure 14.1. In the other road-map, Cherubini et al. [90] also cites the primary instability associated with TS waves as due to low amplitude excitation, as in the path A due to [385] with routes are somewhat similar in these maps. In explaining the relation between instability experiments and receptivity analysis in Chapter 5, it is now clear that TS wave or wave-packet is strictly an artifact of experiments created to validate spatial instability theory. Discussion in Chapter 6 also establishes that transition can be initiated in many ways, with harmonic wall excitation (as in [405]) as just one of the many routes described in Chapter 6. The classification of a route

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as bypass transition is therefore an anachronism, as the original connotation of it in [298, 364], was absence of TS wave or wave-packet in any route being the rule (and not exceptional cases) for the canonical flow past zero pressure gradient boundary layer. The same can be said about the transient growth processes, which are marked as routes B, C and D in Figure 14.1(a) due to spanwise modulation, mean flow distortion or due to some bypass route - as one of the many possibilities whose generic route happens to be the spatio-temporal route espoused correctly since the necessary approaches developed in [418] and demonstrated in [34, 451, 452, 508, 509] for both two- and three-dimensional transition routes for wall excitation. In all the cases, the authors unequivocally make the point that spatio-temporal wave front (STWF) is the true precursor of transition. Similarly, for free stream excitation, the role of STWF as the precursor of transition has been shown in [431, 475, 474]. These have been described in Chapters 4 to 9 for two- and three-dimensional wall and free stream excitation cases using both linear and nonlinear approaches, including solving the full Navier–Stokes equation. To show the power of the solution methods for the Navier–Stokes equation, an illustration of the concept of STWF is presented. Its relevance lies in its creation of tsunami-like disturbances due to different boundary motions. Distinction is made between tangential wall motion, with the boundary motion in the wall-normal direction in [508], establishing the latter to be more effective than the former in creating tsunami-like disturbances. Also, having established identical routes of transition between the three- and two-dimensional disturbances through the growth of STWF, it is necessary to establish the effectiveness of one over the other. Traditionally, following the temporal growth route, Squire’s theorem [119] is used to show the primacy of the two-dimensional route over the three-dimensional route. With respect to STWF, results on this aspect are needed, and are presented in this chapter. It is noted that for wall excitation cases, two-dimensional disturbance fields are stronger than threedimensional fields for the creation and sustenance of STWF. The linear analysis of wall excitation cases allows one to explore the modal and nonmodal components of the response field, following the most generic spatiotemporal approach. Such approaches are not readily available for free stream excitation or application of simultaneous wall and free stream excitations. This is ultimately due to stiffness of the governing Orr–Sommerfeld equation. In [471], the onset and growth of disturbances leading to two- and three-dimensional turbulence have been theoretically and computationally shown for wall excitation. The response field is found to have both modal (TS wave-packet) and nonmodal (STWF) components of the spectrum. The STWF is not only the dominant mechanism, but it also has the self-regenerative property following the initial trigger, which eventually perpetuates the process of transition to turbulence. The case of free stream excitation for the same equilibrium flow over a semiinfinite flat plate is presented by solving linearized and nonlinear two-dimensional Navier–Stokes equations, which also show that nonmodal growth leads the process of transition to turbulence. It is also firmly established that transition to turbulence

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The Route of Transition to Turbulence Forcing Environmental Disturbances amplitude

Receptivity

A

Transient Growth B

E

D

C

Bypass

Primary Modes Secondary Mechanisms

Breakdown

Turbulence

Environmental di sturbance amplitude

(a) Routes according to Saric et al. (2012)

High

Non-linear coherent structures

Turbulence

Receptivity Medium

to ambient

Transient growth (streaks)

Secondary instability

Turbulence

perturbations Low

TS Waves

Secondary instability

Turbulence

Sketch of the different scenarios of transition in a boundary layer (b) Routes due to Cherubini et al. (2012)

Figure 14.1 The road-maps of transition as given by (a) Saric et al. [385] and (b) Cherubini et al. [90]. definitely requires nonlinearity, even though the onset process is similar in linear mechanism. The comparison between linear and nonlinear process has been made by global receptivity analysis in [471]. This would lead one to conclude that transition to turbulence for wall-bounded flows occurs by a nonmodal, nonlinear mechanism for any disturbance.

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14.2 Three-Dimensional Impulse Response of the Boundary Layer over Semi-Infinite Flat Plate by Different Wall Excitations Having distinguished between wall and free stream excitations with the help of the Orr–Sommerfeld equation, one notes the greater effectiveness of the former in triggering disturbances, as it affects the boundary layer directly. However, a further classification has been presented in [508] for the wall excitation imposed by either tangential or wall-normal action. The physical mechanism of transition, in the framework of impulse response for flow over a semi-infinite flat plate, has been presented recently in [34], which explains the experimental observations in [144]. In this approach, generation of the STWF is noted as the precursor of turbulence, caused by localized wall-excitation in the form of pulsed delta function. This follows the frequency response study of the same boundary layer in [32, 483] involving time-harmonic wall-normal velocity perturbations, started both impulsively and non-impulsively. In experimental transition research, monochromatic, wall-normal velocity perturbation has been used in [209, 237, 405]. These experiments followed the theoretical developments in spatial instability, leading to the postulation of TS waves and their detection as that is required for harmonic wall excitation in the experiments. The impulse response approach has been studied in [508], having common features with some geophysical events like tsunami and/or rogue waves in ocean. A tsunami is caused due to surface waves triggered by earthquake along the fault-lines in the ocean bed or continental shelf. However, not all earthquakes in the ocean bed cause a tsunami. For example, strong tsunamis are usually noted for earthquakes with order of magnitudes of 8.0 Mw. Here, Mw is the moment magnitude scale, as given in Hank and Kanamori [173], to characterize earthquakes’ potential in causing tsunami. Direct numerical simulation of tsunami is a massive, near-impossible undertaking at this point in time, demanding inclusion of all relevant physical geophysical parameters and the bottom topography due to present day limited computational power and memory. It is a hugely arduous task to correlate seismological motions with ocean dynamics leading up to tsunami. One such unit process has been studied in [508] correlating subduction motion with receptivity of the boundary layer forming over a semi-infinite flat plate, noting fully well that significant differences exist between the Ekman layer or benthic (close to the sea-floor) layer with the semi-infinite flat plate boundary layer. The essence of this study [508] is to mimic the unit process of subduction. Subduction motion caused by an earthquake results in either a vertical displacement (dip-slip event), or a horizontal displacement (strike-slip event), or a combination of the two, as shown in Figure 14.2 [500]. According to the authors in [508], earthquakes are classified based on an effective moment, which is a seismic moment of a virtual step function dislocation, whose zero-frequency component is noted to cross a threshold to create a tsunami; as postulated by Kanamori [213]. The author in [213] used this criterion to explain massive tsunamis caused by moderate earthquakes, implying that the earthquake magnitude is not the sole determinant of

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Figure 14.2 The schematic of subduction motions caused by: (a) strike-slip fault and (b) dip-slip fault. [Reproduced from “The three-dimensional impulse response of a boundary layer to different types of wall excitation”, Prasannabalaji Sundaram, T. K. Sengupta, and S. Bhaumik , Phys. Fluids, vol. 30, pp 124103 (2018), with the permission of AIP Publishing.] a tsunami. This point of view also implicitly assumes tsunami to be related with the receptivity problem, and not merely as the forced excitation by the large perturbation of the earthquake. A benthic layer occurring near the ocean-bed is controlled by bottom friction. Near the ocean surface, it is subjected to wind stress [102, 189], where the horizontal stress is balanced by the Coriolis force due to earth’s rotation. None of these are considered for the semi-infinite flat plate boundary layer, yet one expects that the impulse response studied will qualitatively resemble the creation and sustenance mechanism of the process during its early stage caused by subduction motion. In [34], a localized excitation is applied at the boundary by vertical velocity in the form of a pulseexcitation with respect to time, categorized as dip-slip subduction motion in Figure 14.2. In the strike-slip motion, the vertical displacement is indirect and small, with the shearing motion disturbing the flow in the tangential direction.

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14.3 Governing Equations, Boundary Conditions, and Numerical Methods for Subduction Motion The complete three-dimensional Navier–Stokes equation in velocity-vorticity formulation is used, as has been described in Section 8.3. The vertical component of velocity is obtained by solving the mass conservation equation. The equations are nondimensionalized by the free stream velocity and a length scale (L) obtained from the Reynolds number ReL = 105 in [508]. If δ∗out is the displacement thickness at the outflow of the domain, then L ≈ 41.11δ∗out . The computational domain is from x = −0.05 to 20 in the streamwise direction, y = 0 to 0.75 in the wall-normal direction and z = −2 to 2 in the spanwise direction. A non-parallel equilibrium flow is used in [34], to study the impulse response by dip-slip boundary perturbation. For both the subduction cases, the flow condition at the inflow and far-field boundaries are as indicated in Figure 14.3. The Sommerfeld boundary condition at the outflow convects disturbances y ma x

Far-field boundary y Inflow Outflow

X

z U∞

Xout Plate Exciter ax

Zm Xin

Xout Plate leading edge

Figure 14.3 Schematic of computational domain, showing a strike-slip subduction motion. [Reproduced from “The three-dimensional impulse response of a boundary layer to different types of wall excitation”, Prasannabalaji Sundaram, T. K. Sengupta, and S. Bhaumik, Phys. Fluids, vol. 30, pp 124103 (2018), with the permission of AIP Publishing.] out of the computational domain, allowing one to keep the length of computational domain finite. The localized pure impulsive strike-slip event is modeled by u0w given as in [508] ¯ − t0 ) Aamp (x, z) u0w (x, y = 0, z, t) = α1 δ(t

(14.1)

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¯ − t0 ) = sin(β0 (t − where, the time-shifted dirac delta function is approximated by, δ(t t0 ))/(t − t0 ) with β0 = 100 and t0 = 0.01 [537]. The amplitude control parameter α1 is used as 0.002, i.e. the amplitude of excitation is limited to 0.2 percent of the free stream speed. The excitation for the strike-slip event is centered at xex = 1.5 with its spatial variation is given by the amplitude function as    1 for xex − 0.045 ≤ x ≤ xex + 0.045 and 0 < z ≤ 0.045    −1 for xex − 0.045 ≤ x ≤ xex + 0.045 and − 0.045 ≤ z < 0 Aamp (x, z) =      0 otherwise For the dip-slip event the wall boundary condition is given by ¯ − t0 ) Adip (x, z) v0w (x, y = 0, z, t) = α1 δ(t

(14.2)

where the amplitude of the dip-slip excitation is given by Adip (x, z) = 0.5[1 + cos(πr/r0 )] for 0 ≤ r ≤ r0

(14.3)

For the dip-slip excitation, wall-normal velocity is set over a circular patch of radius r0 = 0.09 as a Gaussian function, as reported in [34]. The strike-slip on the boundary is prescribed as a shear action along the middle of the rectangular patch shown in Figure 14.3, with the movement in the x-direction. Thus, the input energy to the flow by the strike-slip action is the same (for identical linear dimension), irrespective of the earthquake strength. The wall-normal motion caused by dip-slip action, in contrast, is directly proportional to the strength of the earthquake. In the presented results for the strike-slip event the energy is imparted over a square patch of area of 8.1 × 10−3 with all points moving with the same amplitude. In the case of dip-slip event the energy is imparted over an area of 25.44 × 10−3 , while the movement of all points are given by the Gaussian distribution having maximum displacement at the center of the circular patch and edges decaying to zero. Strike-slip excitation can also be prescribed by the w-component of perturbation velocity, instead of the u-component. It is reasoned that this would create a benign disturbance compared to the reported excitation in [508]. The fundamental disturbance quantities are the perturbation wall-normal velocity and wall-normal 0 ∂w0 component of vorticity given by, ω0y = ( ∂u ∂z − ∂x ). Therefore, the basic idea in the strike-slip excitation is to prescribe ω0y at the wall. As u0 is aligned in the primary flow direction, it will create stronger receptivity, compared to w0 prescribed on the wall. The grid points are clustered near the wall, and at locations with large streamwise gradient near x = 0, to accurately capture the initial growth of disturbances, as in [508]. The minimum and maximum grid-spacing along x- and y-directions are ∆xmin = 9.106 × 10−3 , ∆xmax = 2.16 × 10−2 , ∆ymin = 3.69 × 10−4 and ∆ymax = 5.186 × 10−3 with uniform spacing, ∆z = 1.25 × 10−2 , maintained in the z-direction, for the simulations reported in [508]. Vorticity is time-advanced explicitly by the optimized four stage Runge–Kutta scheme [344], with a time step of ∆t = 8 × 10−5 .

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14.4 Direct Numerical Simulation of Dip-Slip and Strike-Slip Events The DNS of the Navier–Stokes equation with dip-slip and strike-slip spatial boundary actions have been obtained for temporal pulse, starting from the identical equilibrium flow, which is obtained as the solution of the Navier–Stokes equation performed till the unsteady terms go below a prescribed tolerance. In Figure 14.4, streamwise component of disturbance velocity ud (x, y, z, t), on the (y = 0.00189)-plane is compared at indicated time instants for the two types of wall excitations. The disturbance quantities are obtained by subtracting the equilibrium from the instantaneous velocity component. The created wave-packet grows spatio-temporally as it travels downstream, with higher wavenumbers emerging slowly, as noted in frame (b) for the dip-slip excitation at t = 15. Appearance of higher wavenumbers in the spectrum indicate formation of turbulent spots. The evolution of STWF during nascent stages of tsunami are noted near the coast, with very few crests and troughs, with relatively lower wavenumber components. t = 10.0

(a)

U∞

Dip-slip event

Max : 0.02792

0.1

0

0.075 ud 0.05

5

0.1

0

x

-2

15

-1

0

1

2

0 5

0.05 ud 0.025

10

x

0 15 1

2

x

-1

U∞

0.075

z0

10

0

20

(b) t = 15.0 Max : 0.04333 0.1

-1

5

0.05 ud 0.025

10

0

z

-2

U∞

Max : 0.02792

0.075

0.025 -2

Strike-slip event

(c) t = 10.0

20

0.1

15

z0

1

2

20

U∞

(d) t = 15.0 Max : 0.02210

0

0.075 5

0.05 ud 0.025 -2

10

x

0 -1

z

15 0

1

2

20

Figure 14.4 Evolution of streamwise disturbance velocity for the response of dip-slip event (left) and strike-slip event (right) in plane y¯ = y/δ∗ = 0.2837 at different time instants. Here, δ∗ is the local displacement thickness of the boundary layer. For both the cases, onset of excitation causes generation of a downstream propagating wave-packet (in the form of an arrowhead), seen clearly in the planview for the dip-slip excitation shown in Figure 14.5. The generated wave-packet and its subsequent evolution is predominantly along the streamwise direction, as the

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group velocity components obtained from the dispersion relation are mainly in the xdirection. The plan-view in Figure 14.5 also shows a qualitative difference between the dip-slip and strike-slip events. For the dip-slip event, the excitation is geometrically symmetric in the circular patch, which triggers a symmetric disturbance about the mid-plane (z = 0) for the response in both the time-frames. In the case of strike-slip event, the wall excitation is equivalent to creating a wall-normal vorticity component, due to which the response shows an anti-symmetry at early times, about (z = 0)-plane. This is clearly evident in both the perspective and plan-view plots in Figures 14.4 and 14.5. This qualitative difference of excitation is noted for all subsequent times between dip-slip and strike-slip events, and clearly shown in a video in [581]. (a) t = 10.0

(c) t = 10.0

Dip-slip event

2

Strike-slip event

2

1

1

z 0

z 0

-1

-1

0.008 0.0015 0.000516281 -5.36355E-05 -0.001 -0.002

Max : 0.02792 -2

0

2

4

6 x

8

10

Max : 0.02792 -2

(b) t = 20.0

0

2

4

6 x

10

(d) t = 20.0

2

2

1

1

z 0

z0

-1

-1 Max : 0.02214

Max : 0.10822 -2 5

8

10

x

15

-2

5

10

x

15

Figure 14.5 Plan-view of evolving streamwise disturbance velocity for the response of dip-slip event (left) and strike-slip event (right) in plane y¯ = y/δ∗ = 0.2837 at different time instants. Here, δ∗ is the local displacement thickness of the boundary layer.

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Two-dimensional Fourier transforms of the streamwise velocity perturbation in Figure 14.4, are shown in Figure 14.6, with the dip-slip spectrum represented on the left, and strike-slip spectrum on the right frames. The maximum amplitude of the two-dimensional spectrum is noted at the origin in the spectral plane, consisting of streamwise and spanwise wavenumbers, in (k x , kz )-plane. At t = 10, apart from the spanwise modes (noted along k x = 0), one also observes oblique modes for both the excitation cases. The amplitudes for the strike-slip case are lower as compared to the dip-slip events. For the frames at t = 15, both the cases display larger bandwidth of wavenumbers. However, the dip-slip case displays richer spectrum, compared to the strike-slip case. (a) 50

time = 10 Max : 40.3915

(c)

Dip-slip event 50

kz 0

kz 0

-50

-50 -5

kx 0

5

-5

10

(b) time = 15 50 Max : 149.0763

(d) 50

kz 0

kz 0

-50

-50 -5

kx 0

time = 10 Max : 17.2757

5

10

Strike-slip event

kx 0

5

10

5

10

time = 15 Max : 36.8054

-5

kx 0

Figure 14.6 Two-dimensional spatial Fourier transform of ud data at the indicated times, comparing dip-slip (left) and strike-slip (right) events, with the physical plane data shown in Figure 14.4. Results are shown for a height of y¯ = y/δ∗ = 0.2837. Here, δ∗ is the local displacement thickness of the boundary layer. Comparing the evolution of ud for dip-slip and strike-slip events establishes the stronger amplification and dispersion of the STWF for dip-slip action. Apart from the quantitative differences, there are also qualitative differences between dip-slip and

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strike-slip events which magnify later. To demonstrate this clearly, ud contours are plotted at t = 20 in the (x, z)-plane, for y¯ = 0.2837 in Figure 14.7.

U∞

(a) t = 20.0 Max ; 0.10822

0.1

0

0.075 ud 0.05

5

0.025 -2

10 x

0 15

-1

0 z

(b)

1

2

20

0.01 0.00183333 0.001 0.000333333 7.01987E-08 -1.03825E-05 -0.000333333 -0.00116667 -0.002

t = 20.0 Max : 0.02214

U∞

0.1 0

0.075 ud 0.05

5

0.025 -2

10 x

0 15

-1

0 z

1

2

20

Figure 14.7 The streamwise disturbance velocity for the response of (a) dip-slip and (b) strikeslip events in the plane y¯ = y/δ∗ = 0.2837 at t = 20, showing the later nonlinear stage of disturbance growth. Note the asymmetry of the disturbance field in both the cases, with strike-slip event showing larger asymmetry. Both the dip-slip and strike-slip events show significant growth of the nonmodal part of the response via the growth of STWF. As noted before, the dip-slip event is associated with symmetric disturbance and earlier onset caused by the Gaussian circular patch excitation about (z = 0)-line, aided by spanwise periodicity. This

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symmetry is maintained at the advanced time of, t = 20 for dip-slip event where significant nonlinear evolution of the disturbance field is noted, compared to the strike-slip excitation case in Figure 14.8. Along the mid-plane, one can notice the significant high wavenumber growth for the dip-slip event, heralding the earlier formation of turbulent spot. The strike-slip event, in contrast, creates asymmetric disturbance field due to imposition of wall-normal component of vorticity. It is noted in Chapter 9, the presence of streamwise and wall-normal components of vorticity indicate the onset of three-dimensionality in the disturbance field. Such lack of symmetry and emerging three-dimensionality is clearly evident in frame (b) of Figure 14.7, with its attendant lower peak disturbance amplitude. In both the cases of excitation, the onset of three-dimensionality in the disturbance field is accentuated during the late nonlinear stage of disturbance growth due to vortex stretching mechanism. The maximum amplitude of response due to dip-slip excitation is more than four times the corresponding amplitude for the strike-slip excitation, at t = 20. Of specific interest is the presence of streamwise streaks in the disturbance field, for both the cases shown in Figures 14.4 and 14.7. This is nothing but the wellknown Klebanoff mode, explanation of which is given in [412, 448] as occuring due to very low frequency excitation for a flat plate boundary layer. It was discovered experimentally by Taylor [510], with the excitation of a circular bump at a very low frequency of 2Hz and he called it the breathing mode. As has been theoretically explained [448] later that this is caused by very low frequency excitation, which does not support two-dimensional modes. But, the excited three-dimensional modes specifically feature very large streamwise wavelengths, while the spanwise component of group velocity is almost zero. Thus, the three-dimensional disturbance field moves in the streamwise direction, as noted in Figures 14.4 to 14.7. Klebanoff also experimented with similar phenomenon triggered by the low frequency component of free stream turbulence, and so, the commonly held perception in literature that he was first to report such streamwise streaky structures. These are hence known as the Klebanoff mode. It is incorrect however, to say that the Klebanoff mode is only related to free stream turbulence [90, 385]. Rather, it is a feature of wide-band excitation of boundary layer (irrespective of wall or free stream excitation), with lower frequency components contribute to Klebanoff mode. This is explained in [448] for a zero pressure gradient boundary layer. In [34, 508], use of delta function excitation in time has the potential of dealing with extremely low frequency excitation components, which can create the Klebanoff mode. The flow field and associated dynamics shown in Figure 14.7 reveal higher receptivity to dip-slip events, rather than to strike-slip events. The solution at t = 20 shows an earlier nonlinear evolution for the dip-slip event. Strike-slip event shows anti-symmetry in the solution, indicated in the spectrum in Figure 14.6 at t = 15. In Figure 14.8 the spectrum of dip-slip event is compared with that for the strikeslip event at t = 20. At this later time (t = 20), the dip-slip case displays a stronger multimodal transform with an enriched spanwise component, while the

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(a) time = 20 Max : 211.7050 50

kz

0 96.25 85 73.75 58.75 47.5 32.5 21.25 10

-50 -5

0

kx

5

10

5

10

(b) time = 20 Max : 59.4551 50

kz

0

-50 -5

0

kx

Figure 14.8 Two-dimensional spatial Fourier transform of ud data at t = 20, comparing (a) dipslip with (b) strike-slip event, with the physical plane data shown in Figure 14.7. Results are shown for the height of y¯ = y/δ∗ = 0.2837.

streamwise bandwidth remains the same. The motion remains symmetric in the spanwise direction. For the strike-slip case, the amplitude remains more than three times smaller, while asymmetry is not only retained, but tendency towards threedimensionalization is more for this event and transition to turbulence will be more inhomogeneous. The difference between dip-slip and strike-slip events is quantified further by tracking the evolution of maximum disturbances, ud max , as a function of time, for

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0.15

0.15

0.1 u d max

0.1 u d max

0.05

0.05

0

5

t

10

15

Strike-slip event

(b)

Dip-slip event

y = 0.00189 y = 0.0039 y = 0.0061

0

5

t

10

15

Figure 14.9 Evolution of maximum streamwise disturbance velocity for the response of dip-slip event and strike-slip event at different indicated heights. [Reproduced from “The three-dimensional impulse response of a boundary layer to different types of wall excitation”, Prasannabalaji Sundaram, T. K. Sengupta, and S. Bhaumik, Phys. Fluids, vol. 30, pp 124103 (2018), with the permission of AIP Publishing.]

the indicated heights in Figure 14.9. For both the excitation events, decay of initial amplitude is due to dispersion, both streamwise and spanwise, of the disturbance. However, the growth of the nonmodal component of disturbance follows an initial linear route with subsequent secondary growth because of nonlinear interactions, signaled by the growth of ud max . A similar growth pattern is exhibited by the disturbance amplitude for the strike-slip event. The primary and secondary growth rates are strong functions of height. Significantly lower primary growth is noted, at early times, for heights close to the wall. It is noted that for the strike-slip event, the amplitude starts to fall smoothly following the secondary and nonlinear growth becoming dominant at t ≈ 12, which is not the case for dip-slip event. Overall, disturbance amplitudes are higher for the dip-slip event after the nonlinear saturation stage, noted in Figures 14.7 and 14.9. This is again indicative of threedimensionalization of the disturbance field by the strike-slip event, initiated by wallnormal component of vorticity. In contrast, for dip-slip event, the disturbance remains symmetric about the mid-span. In Figure 14.10, ud as a function of z are compared at four different x-stations for the indicated heights. The disturbance for t = 20, highlights the symmetric structure of the STWF for the dip-slip event, at all streamwise stations. The STWF for the strike-slip event displays symmetry in its aft part, which is progressively lost near the leading part of the STWF due to the inherent excitation by streamwise and

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(a1)

Dip-slip event

x = 9.0548 y = 0.00189 y = 0.0039 y = 0.0061

0.2

ud

0.1

(a1)

ud

0 -2

-1.5

-1

-0.5 Z 0

0.5

1

1.5

2

-2

x = 10.0282

-1.5

-1

-0.5 Z 0

0.5

1

0.2

ud

0.1 0

1.5

2

-2

-1.5

-1

-0.5 Z 0

0.5

1

1.5

2

-2

x = 11.0016

-1.5

-1

-0.5 Z 0

0.5

1

0.2

ud

0.1 0

1.5

2

x = 11.0016

0.1 0

-1.5

-1

(a4)

-0.5 Z 0

0.5

1

1.5

2

-2

-1.5

-1

(b4) x = 12.0831

0.2

-0.5 Z 0

0.5

1

1.5

2

x = 12.0831

0.2

ud

0.1

0.1 0

0 -2

0.1

(b3)

0.2

-2

x = 10.0282

0

(a3)

ud

0.1

(b2)

0.2

ud

x = 9.0548

0

(a2)

ud

Strike-slip event

0.2

-1.5

-1

-0.5 Z 0

0.5

1

1.5

2

-2

-1.5

-1

-0.5 Z 0

0.5

1

1.5

2

Figure 14.10 Cross-section of wavefront in different x− planes at different heights for t = 20 for the indicated streamwise stations for the dip-slip and strike-slip excitation. [Reproduced from “The three-dimensional impulse response of a boundary layer to different types of wall excitation”, Prasannabalaji Sundaram, T. K. Sengupta, and S. Bhaumik, Phys. Fluids, vol. 30, pp 124103 (2018), with the permission of AIP Publishing.] wall-normal components of vorticity. The dip-slip event shows a steep increase in amplitude towards the front end of the STWF, which is not the case for the strike-slip event. This supports the earlier observation that the evolution and amplification of the STWF is milder for the strike-slip, than for the dip-slip excitation. Also for the dip-slip event, the motion is symmetric in the spanwise direction, while the strike-slip case shows a bias towards negative z-direction. This is consistent with the FFT data shown in Figures 14.6 and 14.8. Results indicate stronger receptivity of the boundary layer to dip-slip than to strike-slip excitations. If the subduction in the sea-bed by earthquake is similar to a dip-slip event occuring, then such an event is more likely to create a tsunami. However, if the earthquake delivers the same energy, but it is similar to a strike-slip event, then the occurrence of a tsunami will be less likely.

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14.5 Nonmodal Disturbance Growth and Squire’s Theorem Squire’s theorem [119, 412] relates the two- and three-dimensional disturbance fields for a temporal instability with parallel equilibrium flow, studied by normal mode analysis. First a short account of it is provided, followed by its extension to other aspects of the theorem, and specifically probing its utility for the alternatives proposed as transition routes which are different from modal eigenvalue analysis. If φ(˜y; α, β, ω0 ) is the Fourier–Laplace amplitude of the wall-normal velocity component, then for three-dimensional mean and disturbance fields, the linearized governing Orr–Sommerfeld equation is given by  00 ˜ (αU + βW − ω0 )[φ00 φiv − 2(α2 + β2 )φ + (α2 + β2 )2 φ = iRe  00 00 −(α2 + β2 )φ] − (αU + βW )φ (14.4) Now for a two-dimensional mean and disturbance fields, the Orr–Sommerfeld equation simplifies to 00 00 00 ˜ φiv − 2α2 φ + α4 φ = iRe[(αU − ω0 )[φ − α2 φ] − αU φ]

(14.5)

In Chapter 4, resultant wavenumber has been introduced for three-dimensional disturbance field by, γ2 = α2 + β2 . Additionally one defines ˜ α = Re ˆ γ ω0 γ = ω ˆ 0 α and Re Thus, for a two-dimensional mean flow, the three-dimensional disturbance field is governed by 00 ˆ [(γU − ω0 )[φ00 − α2 φ] − γU 00 φ] φiv − 2γ2 φ + γ4 φ = iRe

(14.6)

Comparing Eqs. (14.5) and (14.6), one can appreciate how similar these two governing ˆ < Re, ˜ and the mean flow remains unchanged, equations actually are. However, as Re ˜ is replaced by an equivalent twothe three-dimensional instability problem at Re ˆ dimensional instability problem at the lower Reynolds number of Re. Such an equivalence shows the stronger instability in the two-dimensional disturbance field, compared to the corresponding three-dimensional disturbance field. This is a classic result known as the Squire’s theorem, stated as: In a two-dimensional boundary layer, with real wavenumbers, instability appears first for two-dimensional disturbances. This theorem proposed by normal mode analysis is valid only for temporal instability, with real wavenumber. It is not relevant for complex α and β or if the mean flow is three-dimensional. Using new results presented here, this theorem is linked with the nonmodal growth of disturbance fields, which has been the central theme of

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this treatise. A question raised by the discerning is regarding the implicit assumption in Squire’s theorem, which states that the temporal eigenvalue simply scales linearly by maintaining the governing Orr–Sommerfeld equation at, ω ˆ 0 = ω0 γ/α. The answer is that eigenvalues do not simply depend on the governing disturbance equation, but more so on the boundary conditions, either at the wall or at the far-field, and thus would not scale by ω ˆ 0. So far, the results and connected discussions in this chapter and in Chapters 8 and 9 indicate the stronger attributes of the two-dimensional route of transition, compared to the three-dimensional route of transition. Using the numerical framework of threedimensional route of transition presented in Chapter 8 with respect to the experiment in [209, 237], the problem is investigated here in a methodical manner, with further details can be noted from [482] and the thesis [580]. R4

(a)

R3 y P4

Inflow P1

Out-flow R1

Q3

P3

U

Far-field

Q4

Q1

zmax

Xin

R2 xcut Plate

ST1 1 O P2

X1 Q2

X2 T2 S2 x ex z

(b)

0.01 y

-0.25

-0.1875

-0.125

-0.0625

0.0625

0

0.125

0.1875

0.25 Z

Node Anti-node -0.01

Figure 14.11 (a) Schematic of the computational domain for spanwise punctuated wall exciter, which is periodic in spanwise direction. (b) Spanwise variation of exciter with different spanwise wavenumber with nodes and antinodes marked. In Figure 14.11(a), spanwise modulated time-harmonic disturbance is imposed by following the arrangement in the experiments of [237], in the form of blowing-suction strip on the wall of the semi-infinite flat plate located at, x1 ≤ x ≤ x2 . The amplitude distribution in space is given by A(x, z) = A x Az =

 2π  Af [1 + cos(2πxex )] sin z 2 λz

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Table 14.1 Three-dimensional DNS cases considered. Case 1a 2a 2b 3a 3b 3c

Frequency F1 = 1.0 × 10−4 F1 = 1.0 × 10−4 F2 = 0.5 × 10−4 F1 = 1.0 × 10−4 F2 = 0.5 × 10−4 F3 = 0.15 × 10−4

λz / kz 1.0/ 6.283 0.25/ 25.13

No. of spanwise waves 1/2 2

0.125/ 50.26

4

where A f is the amplitude control parameter, for which a value of 0.01 is used. The streamwise extent of the exciter is fixed by x1 = 1.455 and x2 = 1.545. The timeharmonic wall-normal excitation is started impulsively by, vw = A(x, z) H1 (t) sin ω0 t, 2 with ω0 = FReL and F = 2π f ν/U∞ , where f is the frequency in Hz. Here, H1 (t) is the Heaviside function, indicating the impulsive start. Three frequencies of excitation: F1 (a moderate value of 10−4 ) and lower frequencies, F2 = 5×10−5 and F3 = 1.5×10−5 have been used to demonstrate the properties of the response fields for different inputs. Also, three different spanwise wavenumbers have been considered, with number of waves along the spanwise extent of the domain as: (a) half, (b) two and (c) four complete waves. All these cases are collated together in Table 14.1. The spanwise distribution of waves for the three spanwise cases are shown in Figure 14.11(b). In Figure 14.12, these three different spanwise wavenumber cases of Table 14.1 are shown for the frequency of input excitation, F1 = 10−4 , to display different responses with the spanwise wavenumber of wall excitation. The response fields are characterized by the presence of the local solution, the TS wave-packet and the STWF (if this exists) till t = 20. The Case-1a graphically demonstrates evidence of all the three components, with the TS waves noted as a plane wave-front, ahead of the local solution showing half the wavelength. It is only beyond x ≈ 4, that one can discern the STWF. For the Case-2a, the local solution displays two crests corresponding to the two spanwise waves excited by the blowing-suction strip. The TS waves following the local solution, display four distinct wave-packets, up to x ≈ 3.6, following which the STWF is seen to be present with distinctly reduced amplitude, compared to Case-1a. For the case with four spanwise wave excitations, while the local solution always has the requisite numbers of peaks, the TS waves display a checkered pattern, following which no STWF is noted. Thus in this overview, one clearly observes that as the spanwise wavenumber is increased, the STWF reduces in amplitude. As Case-1a of Table 14.1 is seen to support all the three elements of frequency response, a detailed snap-shot of the flow field is shown in Figure 14.13 for a plane at the height which is quite close to the plate at y = 0.00215 for t = 15. It is clearly noted that the streamwise wavenumber of the TS wave-packet is significantly higher than the wavenumber of STWF. While the growth and decay of TS wave-packet is

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ud -0.05 0.05 0.15 0.25

ii) ( 0.2

(ii)

Local solution

TS waves

0.2

0.15

0.15

ud 0.1 0.05 -0.2

ud 0.1

Local solution

TS waves

0.05 -0.2 z 0 0.2

Case- 1a

5

3

4

2 x

z 0 0.2

5

Case- 2a (iii) 0.2

4

3

2 x

Local solution

TS waves

0.15 ud 0.1 0.05 -0.2 z

0 0.2

Case- 3a

5

4

3

x

Figure 14.12 Streamwise disturbance velocity, ud -contours are plotted for the Cases-1a (half spanwise wavelength), -2a (two spanwise wavelengths) and -3a (four spanwise wavelengths) at the later time of t = 20 for the nondimensionalized height indicated as y = 0.00215, with the frequency of wall excitation given by F1 = 10−4 . dependent on local Reynolds number and displacement thickness, the STWF is noted to grow over the domain. While the input excitation has a wavenumber of 2π, the STWF has twice that for the maximum amplitude of the response. Wall-normal variation of the response field (ud ) is shown in Figure 14.14, (for the Case-1a shown in Figure 14.13), for different heights over the plate. With such wide variations of height, one notices interaction between the TS wave-packet and STWF at the lower heights, in the form of spanwise undulations. The increased height of the TS wave-packet, at two spanwise stations, is noted up to y = 0.0148. Above this, undulation is absent, as noted for y = 0.0278. The last height of y = 0.0753 does not show any perceptible components of the disturbance field. While the perspective plots provide graphic overview of the disturbance field, one would also be interested in quantitative data about the disturbance field for the

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ud 0.1

-0.06 -0.02 0.02 0.06 0.1

t = 15 -4 F1 = 1 × 10

STWF Local solution

0.05 ud 0 aves TS w

-0.05 -0.2

U∞ z 0 0.2

10

0

5

x

Figure 14.13 Perspective plot of streamwise disturbance velocity, ud -contours are plotted for the Case-1a (half spanwise wavelength) at t = 15 for the nondimensionalized height indicated as y = 0.00215, with the frequency of wall excitation given by F1 = 10−4 . various cases of Table 14.1. In Figures 14.15 and 14.16, the cases of half and two spanwise wavelength excitations are shown for the mid-span and anti-node locations, respectively. The results are shown till t = 30 and the x-scales are adjusted accordingly. Two vertical lines plotted in these frames, corresponding to the ingress and exit points of TS wave disturbances, to indicate the spatially unstable region. Between t = 5 and 10, one can see the growing STWF for both the cases. By t = 20, one notices the onset of nonlinear growth of the STWF. At a later time of t = 30, one notes the formation of not only the leading turbulent spot, but also the dispersion of the STWF appears for Case-1a in Figure 14.15. For Case-2a, the second STWF shows its appearance by t = 30 near x ≈ 13, but the amplitude is significantly smaller, compared to Case-1a. Also, for Case-1a there are interactions between the TS wave-packet and the STWF, which are absent for Case-2a. In Figure 14.17, ud is plotted as a function of x for the indicated heights for Case-2b, plotted along the anti-node line of z = −0.0625, for the lower excitation frequency of F2 = 5 × 10−5 . Case-2b displays stronger disturbance growth compared to Case-2a shown in Figure 14.16. By comparing the time frames at t = 30, it is evident that a longer stretch of the turbulent spot is noted for Case-2b. The vertical lines demarcate the region of instability obtained by linear theory, and here one notices the onset and growth of STWF from inside this range at t = 10. Nonlinear growth is noted inside this range of linear growth at t = 20. It is only at t = 30, that one notices the turbulent spots emerging out of the linearly unstable region.

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The Route of Transition to Turbulence y = 0.00215

0.4

ud

y -0.1 -0.06 -0.02 0.02 0.06

x

0.1

y = 0.0148

0.4

z

0.2

0.2 0

x

10

0

0

0 5

x

y = 0.00442

0.4

y = 0.02780.4

0.2

0.2

ud

0

0

0

x

10

0

5

x

y = 0.00936

5

5

y = 0.0753

0.4

0.4

0.2

0.2

0

x

10

0

0

0 5

x

10

5

Figure 14.14 Streamwise disturbance velocity, ud -contours plotted for the Case-1a (half spanwise wavelength), at t = 15 for different indicated heights for frequency of wall excitation given by F1 = 10−4 . 0.04

ud

z=0

t=5

0

0.04 0.08

x in = 1.796 2

0

0.2

x out = 4.97

4

x

6

y = 0.00442 y = 0.00936 y = 0.0148 y = 0.0278 y = 0.0753

8

t = 10

ud 0 0.2 0

2

4

6

8

10

x

12

x

20

t = 20

ud 0 0 0.8

5

10

15

t = 30

0.4

ud 0 0.4 0

5

10

15

20

x

25

Figure 14.15 Streamwise disturbance velocity, ud plotted as function of x for Case-1a (half spanwise wavelength) at indicated times, for the indicated heights, for the frequency of wall excitation given by F1 = 10−4 plotted along z = 0.

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0.04

x in = 1.866

t=5

ud0

x out = 4.45 y = 0.00442 y = 0.00936 y = 0.0148 y = 0.0278 y = 0.0753

z = 0.0625

0.04 0.08 0 0.06

1

2

3

4

5

6

x

7

t = 10

ud 0 0.06 0 0.3

5

10

x

t = 20

ud 0 0.3 0 0.5

5

10

15

x

20

x

25

t = 30

ud 0 0.5

0

5

10

15

20

Figure 14.16 Streamwise disturbance velocity, ud plotted as function of x for the Case-2a (two spanwise wavelengths), at indicated times for the indicated heights for the frequency of wall excitation given by F1 = 10−4 shown for z = −0625. 0.05

z = 0.0625

t=5

ud 0 0.05 0.1

x in = 3.593 0

0.2

5

10

5

10

y = 0.00442 y = 0.00936 y = 0.0148 y = 0.0278 y = 0.0753

x out = 12.54

x

15

x

15

x

20

t = 10

ud 0 0.2 0 0.5

t = 20

ud 0 0.5

0

0.8

5

10

15

t = 30

0.4

ud

0

0.4

0

3

6

9

12

15

18

21

24

x

27

Figure 14.17 Streamwise velocity, ud plotted as function of x for the Case-2b (two spanwise wavelengths), at indicated times for the indicated heights for the frequency of wall excitation given by F2 = 5 × 10−5 , with results shown for z = −0625.

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The top view of ud are plotted in the (x, z)-plane in Figure 14.18, to distinguish between Case-1a and Case-2a. For these two cases, the local solution is distinctly different. The STWF has significantly higher amplitude for Case-1a, compared to Case-2a. The advent of nonlinearity is seen at t = 15 for Case-1a, while it is noted at t = 20 for Case-2a. Also, the onset of receptivity is noted at an upstream location

t=5 xin

(a1)

casr-2a ud

xout

-0.04 0.02 0.08

t=5 xin

xout

x 10

15 0 (a2) t = 10

5

x 10

15 (b2)

5

x 10

15 0 (a3) t = 15

5

x 10

15 (b3)

5

x 10

15 0 (a4) t = 20

5

x 10

15 (b4)

5

x 10

5

x 10

15

t = 15

0 t = 20

0

-0.04 0.02 0.08

5

t = 10

0

ud xin = 1.796 xout = 4.97

xin = 1.865 xout = 4.45 0

(b1)

case-1a

15 0

Figure 14.18 Plan view of streamwise disturbance velocity, ud compared between Case-1a with Case-2a (two spanwise wavelengths), at the indicated times for the frequency of wall excitation given by F1 = 10−4 .

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for Case-1a. Thus, the instability appears earlier in time and at an upstream location for Case-1a (half spanwise wave), as compared to Case-2a (two spanwise wave). When these two cases are compared with the four spanwise wave case (Case-3a), one concludes that lower spanwise wavenumber cases are more receptive to the instability for the boundary layer forming over a semi-infinite flat plate. This is equivalent to stating that two-dimensional disturbance sources destabilize the equilibrium flow more effectively, compared to disturbances which are more three-dimensional. It is furthermore noted that a similar sequence of events is noted when the frequency is lowered, as has been reported in [482].

10

1

10

2

10

3

10

4

10

5

10

6

10

7

10

8

10

9

u dm

Case1a Case2a Case2b Case3a Case3b Case3c 0

5

10

t

15

20

25

30

Figure 14.19 Maximum streamwise disturbance velocity, udm plotted as function of t for all Cases at the height, y = 0.00215 shown for z = −0625 for Case-2a and -2b; along z = −0.03125 for Case-3c and z = 0 for Case-1a. For Cases-3a to 3c, transitions to turbulence do not occur for the spanwise wavenumber and frequency combinations. This has been further investigated by studying Case-3a, and it is noted that there is no distinct, complete transition occurring for this case with higher spanwise wavenumber excitation, even though amplitude of wall excitation is kept the same. In Figure 14.19, maximum amplitude of streamwise disturbance velocity (udm ) is plotted for all the cases studied. As noted, the four spanwise wave excitations for Case3a to -3c show only linearized growth of the disturbance field, with secondary and nonlinear growth stages missing. Instead, one notices a steady mild decrease in the udm with time. Among these three cases (Case-3a to -3c), the trend of increasing disturbance level with decreasing frequency is clearly noted in Figure 14.19, so much so that one can find a critical maximum frequency, below which transition will occur. For Case-2a, the amplitude of disturbance is insignificant till t ≈ 10,

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following which the growth of disturbance builds up with almost identical rate in the primary growth regime. Secondary growth is noted in spurts for Case-2a, following which one notices nonlinear saturation to turbulent state. Thus, there appear some qualitative differences in the growth of disturbances for the three spanwise wavenumber excitation cases reported. However, the basic trend remains that two-dimensional wall excitation is more efficient in causing transition, compared to disturbances having higher spanwise wavenumber components (more threedimensionality). These seemingly subtle differences among various wall excitation cases with different degree of three-dimensionality can also be understood better by using linear stability analysis, as differences in disturbance growth during the primary stage in Figure 14.19, are captured effectively.

14.6 Linear Stability Analysis Caused by Different Spanwise Excitation A general three-dimensional framework of disturbance growth is sought to be developed here for the study of linearized growth rate in a well designed experimental set up. This has already been done in Section 5.3, while studying the Klebanoff mode induced by extremely low frequency of excitation. For a two-dimensional equilibrium flow supporting a three-dimensional disturbance field with signal problem assumption, one can present the wall-normal component of disturbance velocity as in Eq. (5.48). As explained in Sub-section 5.3.1 for describing creation of the Klebanoff mode, one can introduce a cut-off spanwise wavenumber given by β0 , which is determined by the spanwise extent of the experimental or numerical domain as, β0 = 2π/λz , where λz is twice the size of the spanwise domain. This ansatz also allows one to cast the unknown disturbance field as in Eq. (5.49), with the governing Orr–Sommerfeld equation given in Eq. (5.50), by setting W = 0 for the two-dimensional equilibrium flow. This equation is rewritten here for the sake of convenience as   00 00 00 φiv − 2(α2 + n2 β20 ) φ + (α2 + n2 β20 )2 φ = iReδ∗ (αU − ω ˜ 0 )[φ − (α2 + n2 β20 ) φ] − αU φ (14.7) where primes indicate differentiation with respect to the nondimensional wall-normal co-ordinate (˜y). Equation (14.7) can be solved as an eigenvalue problem. At the free 00 stream (˜y → ∞), the equilibrium flow simplifies as U(˜y) = 1 and U (˜y) = 0, and Eq. (14.7) simplifies to the constant coefficient ordinary differential equation given by   00 00 φiv − 2(α2 + n2 β20 ) φ + (α2 + n2 β20 )2 φ = iReδ∗ (α − ω ˜ 0 )[φ − (α2 + n2 β20 ) φ]

(14.8)

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λ˜y Solution of this equation can p be written in the form, φ ∼ e , so that the characteristic 2 2 roots are given by λ˜ 1,2 = ∓ α + (nβ0 ) and λ˜ 3,4 = ∓Q, where Q2 = (α2 + n2 β20 ) + iReδ∗ (α + nβ0 − ω ˜ 0 ). The general solution of Eq. (14.7) can be written as ˜

φ = a1 φ1 + a2 φ2 + a3 φ3 + a4 φ4 For real(α, Q) > 0, to satisfy boundary conditions at y˜ → ∞, one requires a2 = a4 = 0, so that the general solution is written in terms of the admissible fundamental solution modes as φ = a1 φ1 + a3 φ3

(14.9) 0

The wall boundary conditions (φ, φ = 0) result into the dispersion relation as derived in Eq. (5.6) 0

0

(φ1 φ3 − φ1 φ3 )y˜=0 = 0

(14.10)

The Orr–Sommerfeld equation given by Eq. (14.7) is a stiff differential equation, with each fundamental solution growing or decaying at very different rates. The compound matrix method introduced in Sub-section 4.4.1 has been used to find the numerical solution, for both the problems of stability and receptivity. As explained in Chapter 4, one works with the second compounds, which are combinations of the fundamental solutions φ1 and φ3 . In the stability analysis for flow past a semi-infinite flat plate, one makes the idealization of Blasius velocity profile for providing qualitative analogy with the DNS results provided so far. Furthermore, the introduction of β0 allows another improvisation. By fixing the imposed time scale ω0 , and the spanwise wavenumber, the linearized problem involves solving a twodimensional Orr–Sommerfeld equation, as has been also followed for the study of Klebanoff mode. Further, the stability problem appears like a spatial stability analysis, and yet readers should note that instead of looking at spatial growth rates and such, the entire focus should instead be directed on the neutral curve for stability and spatiotemporal analysis by the Bromwich contour integral method (BCIM) for receptivity analysis. These approaches are compatible, as the neutral curve is not the sole criterion of either spatial or temporal instability theory. As both these theories produce the same neutral curve, it implies that the neutral curve truly belongs to spatio-temporal analysis.

14.6.1 Results and discussion for linear stability analysis The Reynolds number based on displacement thickness (Reδ∗ ) is related to Reynolds number based on convection scale (ReL ) by, Reδ∗ = 1.72(xReL )1/2 for a Blasius boundary layer. Therefore, the Reynolds number at the non-dimensional exciter location, xex = x∗ /L=1.5 is given as Reδ∗ |ex = 666.153 for ReL = 105 used in the convection scale. Thus,

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the displacement thickness at the exciter location is δ∗ex /L = 6.66 × 10−3 , where Reδ∗ |ex = U∞ δ∗ex /ν. Hence, the relationship between viscous scales and convection scales is given as [ x˜, y˜ , z˜] = (x, y, z) × (L/δ∗ex ). The non-dimensional spanwise wavenumber in viscous scale is going to be, β = kz (δ∗ex /L), and for a single spanwise wave in the full spanwise domain length (i.e. zmax = 0.5), the non-dimensional wavelength and wavenumber are λz0 = 0.5 and kz0 = 2π/0.5 = 12.56, respectively. Therefore, the wavenumber in viscous scale for a single spanwise wave is β0 = 12.56 × (6.66 × 10−3 ) = 0.0837.

–4

DNS does not follow either purely temporal, or, purely spatial growth routes, but instead it examines spatio-temporal growth of disturbances holistically. The neutral curve is one of the elements of such spatio-temporal growth, and has been commonly sought after by researchers in [229, 398, 532], for the nonmodal growth of disturbance to replace normal mode theories. Both the linear and nonlinear aspects of spatiotemporal growth have been advanced here based on the results in [32, 482, 483], with the neutral curve providing the set of parameters for which one notes disturbances displaying no growth in space and time. Thus, knowledge of the neutral curve is vital in discussing spatio-temporal growth, with the sole limitation being its dependence upon parallel flow approximation.

F

2

=

0.

5

F

1

×

=

10

–4

1×

10

0.05

β 0.5β0 β0 2β0 3β0 4β0

Reδ*lcr 524 545 618 830 2079

0.1 0.5β0 ω~0

β0 2β0 3β0

0.05

4β0 Reδ* lex = 666.153 0

0

1000

2000

Reδ*

3000

4000

Figure 14.20 Neutral curves for different spanwise wavenumber excitation cases, as indicated in the figure from 0.5β0 to 4β0 . Note the critical Reynolds numbers are given in the inset for different β values.

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0.0002

Case2a

~

0.0002

~ t = 750

t = 750

~ u

d

u~d 0

0

0.0002

0

0.0002

~ x

1500

0

~x

1500

~ t = 1650

~ ud 0

0

0.0002

~x

1500

0.0002

~u d

0

~ x

0

0.0002

~ t = 2450

~ ud 0

0.0002

0.0002 0.0002

~ t = 1650

~u 0 d

0.0002

Case2b

1500

0.0002

~ x

1500

~ t = 2450

0

0

~ x

1500

Figure 14.21 Disturbance u-component of velocity plotted versus x˜ at the indicated times, obtained by the Bromwich contour integral method. Results are shown for ¯˜ 0 = 0.033 and 0.066, Cases-2a and -2b. Left and right frames correspond to ω respectively. The results are shown for the height, y˜ = 0.278. Figure 14.20 shows neutral curves for different spanwise sub- and super-harmonics of the fundamental spanwise wavenumber (β0 ), i.e. for β = 0.5β0 , β0 , 2β0 , 3β0 , 4β0 in which 0.5β0 , 2β0 , 4β0 have already been identified as Cases-1, -2 and -3, in Table 14.1. The frequencies F1 and F2 are drawn as constant-slope lines passing through origin in (Reδ∗ , ω ˜ 0 )-plane, with the intersection of the exciter location marked on these lines at Reδ∗ = 666.153. One observes that with increase in number of spanwise waves, the critical Reynolds number (Reδ∗ |cr ) keeps increasing. One can clearly see that the frequencies F1 and F2 are passing through the neutral curve for β = 2β0 , whereas these constant frequency lines lie outside the neutral curve in the stable region for β = 4β0 . This means that the disturbances will grow spatio-temporally for lower spanwise wavenumber cases, whereas these would not grow for higher wavenumber cases. This explains why flow transition was not captured at higher wavenumber cases by DNS. For Cases-2a and -2b, the constant frequency rays cut through the neutral curve and pass through the unstable region. This is not the case for Cases-3a and -3b.

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It has been shown that the STWF is an outcome of the spatio-temporal analysis of the linearized governing equations [418, 451]. For its analysis with the time-harmonic excitation, BCIM is performed in α- and ω ˜ 0 -planes, with boundary conditions at y˜ = 0 as u¯˜ d v˜¯ d

= 0 ¯ x˜)H1 (t˜)e−iω¯˜ 0 t˜ = δ(

(14.11)

¯ x˜) is the delta function in space, ω ¯˜ 0 is the angular and for y˜ → ∞; u˜ d , v˜ d → 0. Here, δ( frequency of the exciter, and H1 (t˜) is the Heaviside function in time. The Bromwich contour in α-plane is below the real axis at αi = −0.001 and the Bromwich contour in ω ˜ 0 -plane lies above the real axis at ω ˜ 0i = 0.01, such that all the downstream and upstream propagating modes are captured with correct directionality, and without violating the causality principle. The domain is traversed with a total number of points along the Bromwich contour in α and ω ˜ 0 planes as Nα = 16384 and Nω˜ 0 = 2048, respectively, with corresponding spectral range as −4π < αr < 4π and −π/4 < ω ˜ 0 < π/4. The physical domain extends from {− x˜max , x˜max } in streamwise direction, and up to ˜ 0 max : The spatial t˜max in time, where x˜max = (Nα − 1)π/2αmax and t˜max = (Nω˜ 0 − 1)π/2ω ˜ and temporal step sizes are ∆ x˜ = π/αmax and ∆t = π/ω ˜ 0 max , respectively. Discrete fast Fourier transform (DFFT) algorithm is employed here, as explained in [34]. Implicit second order filter with α f = 0.44 has been used for windowing to remove high wavenumbers and high frequency components which can induce Gibbs’ phenomenon [413]. To compare the results obtained from the receptivity analysis with DNS results, BCIM has been used for 2β0 and 4β0 cases, as listed in Table 14.1. For these cases, the Reynolds number at exciter location is Reδ∗ |ex = 666.153, hence the circular frequency ¯˜ 0 ) is given as ω ¯˜ 0 = FReδ∗ |ex , and results are shown in Figure 14.21. at exciter location (ω ¯˜ 0 = 0.066) and CaseOne can observe from this figure that for both Case-2a (2β0 and ω ¯ 2b (2β0 and ω ˜ 0 = 0.033), the STWF is growing in time. The TS waves decay for both these cases, with faster asymptotic decay for Case-2a. This is because the exciter for both these cases is placed outside the neutral curve where αi > 0. However, the exciter location for F2 is farther away from the neutral curve compared to F1 (in Figure 14.20), and hence suffers more attenuation.

14.7 Nonmodal, Nonlinear Global Route for Free Stream Excitation Thus far in this chapter, discussion has hinged on two-dimensional disturbances being more effective in triggering transition for the canonical flow past a semi-infinite flat plate, as compared to three-dimensional routes of transition for wall excitation. It has also been noted that for wall excitation, with fixed amplitude, frequency and location of the exciter, the onset of STWF following linear and nonlinear routes, is from sites very close to each other. The transition caused by the STWF also displays a self-regeneration mechanism, and thus, is qualitatively similar for both two- and

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three-dimensional transition routes. However, because two-dimensional routes are more effective in causing transition they deserve greater attention for determining their critical parameters for transition. Such is the reasoning behind the study of twodimensional routes of transition caused by free stream excitation as well, as it has been studied in [431, 475] by solving the Navier–Stokes equation. There are other reasons for which study of two-dimensional routes of transition is vital. Absence of a complete theory still eludes the physics of turbulence and thus complete understanding of the subject continues to be distant. This is despite the Navier–Stokes equation being widely accepted as the formulation for describing turbulence. [20, 114]. While the nonlinearity of the Navier–Stokes equation prevents one from obtaining an analytical solution, the linearized theories, started with the original works in [321, 351, 495], for the onset of instability have been extended for different flows and processes. Originally, this was the modal approach used to study individual normal modes one at a time. This was the thrust area, after viscous effects were considered essential in describing the disturbance field. The governing Orr–Sommerfeld equation was first solved by Heisenberg [176] and subsequently by Tollmien and Schlichting [393, 528] by formulating disturbance growth in space, for a fixed moderate frequency excitation. Tollmien and Schlichting used linear spatial theory to report the existence of TS waves for zero pressure gradient boundary layer. The verification of TS waves followed after careful design of experiments in [405] for the two-dimensional excitation. This experiment led scientific opinion to the observation that TS waves only decides transition to turbulence for boundary layers. The actual role of the TS wave in causing transition has not been traced, as noted in the critique of linear instability theory [285] that it “tells nothing about turbulence, or about the details of its initial appearance, but it does explain why the original laminar flow can no longer exist,” that too under the specific condition of excitation by specific monochromatic timeharmonic source placed inside the boundary layer. The other strong criticism against the analysis using the Orr–Sommerfeld equation with parallel flow approximation is due to the use of assumptions that make such analysis local in nature. The main problem of linear spatial theory is the signal problem assumption that forces the response to be strictly at the excitation frequency. When signal problem assumption is revoked in favor of spatio-temporal growth of disturbances and the receptivity problem is studied by solving the Orr–Sommerfeld equation the response field reveals a three-component structure [451] in Chapters 6 and 7. The leading STWF has been termed as the forerunner in [67] for electromagnetic wave propagation. From the material in previous chapters, it is apparent that the presence of STWF has features of “hydrodynamic instability without eigenvalues” [532], transient growth by non-orthogonal eigenvectors in the linear framework [398] and transition by nonlinear, nonmodal route of transition [125, 221, 229]. There is also enough justification to use a Fourier–Laplace transform for the governing Orr–Sommerfeld equation in linear theory without the need to perform normal mode analysis. The correct approach has been identified in [412, 451, 539] to identify the role of Abel

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and Tauber theorems [537] for the use of Fourier–Laplace transform to be natural, providing self-consistent formulation without any simplification. These theorems allow for determining (i) near- and far-field response and (ii) transient and asymptotic response to imposed perturbation by considering different parts of the spectrum. Near-field for the fixed frequency excitation is obtained by the essential singularity for the wavenumber, α → ∞ [412]. The STWF propagates with group velocity [354, 355], per the rudimentary explanation provided in Appendix, Chapter 1 using the nonmodal part of the spectrum. For the zero pressure gradient boundary layer, such a spectrum has been depicted in Figure 6.7, with the STWF arising from the nonmodal flat part of spectrum of the disturbance field, which corresponds to a maximum around α ≈ 0.3. Researchers [412, 419] have shown the existence of STWF from the solution of the two-dimensional Navier–Stokes equation and its role in two-dimensional turbulence [21, 247] shown by typical dependence of energy spectrum on the wavenumber as, E(α) ∼ α−3 . These computations require high accuracy, dispersion relation preserving (DRP) numerical methods [413], described briefly in Chapter 3, using stream functionvorticity formulation. Similar DRP methods have been used for three-dimensional routes of transition in [32, 34], with velocity-vorticity formulation. Two-dimensional turbulence simulation is distinctly possible, as it can be performed by using finer resolution grids. Also, typical spectrum of atmospheric data in [310], shows the energy spectrum depicting zonal and meridional wind with the two-dimensional trend alluded to above, for the energy spectrum to follow E(α) ∼ α−3 . This part of the spectrum accounts for more than 98% of the total energy. Only a small fraction follows the other well-known α−5/3 spectrum that is typical of three-dimensional homogeneous isotropic turbulence. The above mentioned shortcomings, which are inherent with the signal problem assumption used in linear spatial theory, have prompted some researchers to look at alternative concepts to explain the near-field solutions at early moments, by invoking mechanisms such as “lift up” [128, 251], nonmodal linear transient growth due to nonnormality of eigenmodes of Orr–Sommerfeld equation [398, 399, 400] and so on. These authors have noted that “linear effects play a central role in hydrodynamic instability” and reconciliation has been presented “based on the “pseudospectra” of the linearized problem, which imply that small perturbations to the smooth flow may be amplified by factors of the order of 105 by a linear mechanism even though all the eigenmodes decay monotonically. The methods suggested here apply also to other problems in the mathematical sciences that involve non-orthogonal eigenfunctions” [532]. However, the authors in [532] state that an “essential feature of this nonmodal amplification is that it applies to three dimensional perturbations” and asserted that their “new results indicate that this emphasis on 2D perturbations has been misplaced. When only 2D perturbations are considered, some amplification can still occur, but it is far weaker.” This is in sharp contradiction to results presented in the previous section of this chapter showing stronger receptivity of two-dimensional disturbance fields, compared to three-dimensional disturbance fields.

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The moot point is that the dispersion relation for instability problems arise not only from the governing differential equation, but also depends on auxiliary conditions of the problem. In that context, any theory that depends on dimensionality of the problem must be deficient in its scope and cannot be strictly correct. It should be viewed as an empirical model only. Proponents of nonmodal theories critique the lack of orthogonality of eigen-functions as their inappropriateness for instability study, while at the same time they continue to use such non-orthogonal eigen-functions to explain transient growth. This is pointed out by Kerswell [229], who instead, suggested the inception and growth of disturbances to be essentially governed by nonlinear mechanisms. This misunderstanding originates from the fact that the receptivity framework built around Fourier–Laplace transform is for any arbitrary auxiliary conditions. For this the constituents of the response field need not be orthogonal and mutual interactions depend on the phase of the different components, which is clearly accommodated in the BCIM in the linear framework of receptivity as shown in Section 5.4. In retrospect, researchers have migrated from temporal instability theory to spatial theory, as the former was found inadequate to explain transition over zero pressure gradient boundary layers. When linear spatial stability theory pronounced pipe and Couette flows to be unconditionally stable for all Reynolds numbers, the switch back to linear nonmodal theory by some [74, 361, 362, 398, 399], was justifiably criticized. At the same time, other researchers adopted global theory to study instability, using linearized Navier–Stokes equation [519, 520]. According to Theofilis [519], any analysis based on the solution of an eigenvalue problem or an initial value problem, with two- or three-dimensional equilibrium flow constitutes a global instability theory. This approach is the median of many such approaches governed by the complete Navier–Stokes equation and linearized instability theory based on local approach of a parallel mean flow with signal problem assumption. Now, this approach is investigated by comparing linear and nonlinear global receptivity analysis. In recent times, an alternative, fully nonlinear dynamical system approach has been developed by some researchers [125, 221, 229] which looks for nonlinear theories for the evolution from a laminar to a turbulent state. In this book, the dynamical system approach has been followed to a large extent in the form of receptivity analysis, correlating the cause with its effect. Another complementary approach has been examined where researchers discovered finite-amplitude solutions disconnected from the equilibrium state in [125, 221, 230]. This is a complete departure from the classical instability theory, where equilibrium flow plays the central role, while these dynamical system theory based approaches refer to transition which is totally disconnected from the equilibrium solution, with complete emphasis on the governing equation. Here, the flow displayed by the dynamical system evolves in the phase space “populated by various exact solutions and their stable and unstable manifolds” [229]. Nonlinear instability is stated to occur under the action of finite disturbance which takes the dynamical system beyond the “basin of attraction” of the equilibrium state. Linear nonmodal theories that work within the “basin of attraction” have been identified, as

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well as those nonlinear nonmodal theories that are better suited to basin boundaries and beyond. According to Kerswell [229], nonlinear nonmodal theory should not be viewed merely as a higher input amplitude version of linear nonmodal theory, as in the former the problem of seeking maximum energy growth changes to a fully nonlinear, nonconvex optimization from the convex optimization problem that is solved in linear nonmodal theory. Readers’ attention is drawn to Figure 6.7, which shows the Fourier– Laplace transform of the linear nonmodal response in the streamwise disturbance velocity, for three different types of input function. The associated discussion in Section 6.5 explains how creation of the STWF is related to an optimization problem, with respect to solution of the Orr–Sommerfeld equation.

14.8 Comparing Linear and Nonlinear Global Instability Problem Having noted the similarities and differences of different analysis techniques applied to studying an equilibrium flow, it has been established that two-dimensional disturbances are more receptive to cause transition for wall excitation. The only study that has not been reported so far is the comparison between linear and nonlinear global instability studies. One has noted that wall excitation experiments are mostly designed to test various version of receptivity and instability studies. In real flows, disturbances are omnipresent with incoming free stream turbulence, and a complementary research on transition was initiated by Taylor [510] to estimate critical Reynolds number for the free stream excitation problem, which led to the subject of vortex-induced instability described in Chapter 9 showing both two- and threedimensional routes.

Figure 14.22 The computational domain for the receptivity created by convecting vortex in the free-stream over a zero pressure gradient boundary layer on a flat plate. [Reproduced from “Nonmodal nonlinear route of transition to two-dimensional turbulence”, A. Sengupta, Prasannabalaji Sundaram, and Tapan K. Sengupta, Phys. Rev. Res. 2, 012033(R) (2020).]

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Two-dimensional routes exhibit stronger transition, and a similar physical problem has been studied in [471], which is depicted in Figure 14.22, for a vortex convecting over the boundary layer above the semi-infinite flat plate as the disturbance source. As explained in Chapter 9, this vortex induces an adverse pressure gradient to cause unsteady separation of the boundary layer with recirculating structures moving downstream. Unsteady separation is an essential element of the transition caused by free stream excitation. Events are noted in experiments with turbulent flow created over boundary layers for a viable transition route created by pressure fluctuations in the oncoming flow [295]. For the reported two-dimensional studies, the equilibrium flow is again obtained by solving the Navier–Stokes equation in the absence of free stream convecting vortex. Transition is initiated by a free stream convecting vortex with counter-clockwise circulation affects the boundary layer over the semi-infinite flat plate globally, which creates a time-dependent adverse pressure gradient ahead of the free stream vortex leading to unsteady separation. Additionally, such excitation in the free stream indirectly creates an equivalent wall excitation, as explained in Sub-section 5.3.2. Due to this reason, all the four fundamental solutions of the Orr–Sommerfeld equation needs to be retained, and one cannot use the compound matrix method, as has been performed for pure wall excitation cases. For this reason in [429], the linearized Navier–Stokes equation has been solved for a free stream excitation problem, instead of the Orr–Sommerfeld equation. The alternative is to solve the complete Navier–Stokes equation, as has been suggested to study transition by global nonlinear instability analysis in [519]. The problem shown in Figure 14.22 has been experimentally [267] and numerically [431] studied, and described in Chapter 9. Next, a complete description of the twodimensional instability and transition to turbulence is provided by global analysis using linear and nonlinear frameworks. This will bridge the existing knowledge gap for flow transition over a flat plate excited from the free stream. With results obtained from linear and nonlinear frameworks using identical numerical formulation and grids, one can understand the role played by nonlinearity from the onset to fully developed turbulent flow stage. This is also a dynamical system approach relating cause and effect, and is distinctly different from the nonlinear, nonmodal approach [229]. These aspects of disturbance growth following linear and nonlinear routes, as obtained from the spatio-temporal response field over the whole domain to account for both modal and nonmodal contributions have been reported recently in [471]. It is shown in [509] that transition can occur without the modal component for wall excitation; for free stream excitation, just as it has been shown that transition occurs without TS waves in Chapter 9. This is achieved by high accuracy computing, while treating this as a global receptivity problem for convecting free stream vortex over a semi-infinite flat plate. In [474], its authors explain how an imposed two-dimensional free stream vortex displays strong receptivity for a convection speed around a value of 0.3U∞ . Using this

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information, very high accuracy simulation results have been presented in [471], to compare the role of a nonlinear mechanism with the linear mechanism for the first time, for the free stream excitation case. Both linearized and nonlinear Navier–Stokes equation are solved for the domain in Figure 14.22, solving the vorticity transport equation and the stream function equation, as this formulation provides the most accurate solution. With respect to the equilibrium solution, disturbance vorticity is evaluated from the solutions of the linearized and nonlinear Navier–Stokes equation by subtracting the equilibrium flow from the instantaneous solutions. With instantaneous velocity and vorticity given by ~ and ω for the two-dimensional flow field, the corresponding equilibrium quantities V ~ b and ωb . Similarly, the disturbance component of velocity and are represented as, V vorticity are represented as ~vd and ωd . The nonlinear Navier–Stokes equation (NNSE) is given by the stream function and vorticity formulation by ∇2 ψ = −ω

∂ω ~ · ∇)ω = Re−1 ∇2 ω + (V ∂t

(14.12)

here ψ is the stream function and the Reynolds number (Re) is defined with the length (L) and velocity scales (U∞ ), which have been used to nondimensionalize all variables including the time. The length L is fixed by considering the Reynolds number based on it as Re = 105 . This is the identical scale used in [32, 34, 419, 474] and in previous chapters. The linearized Navier–Stokes equation (LNSE) is similarly written as ∇2 ψd

= −ωd

∂ωd ~ b · ∇)ωd + (~vd · ∇)ωb = Re−1 ∇2 ωd + (V ∂t

(14.13)

As before in the cases shown in Chapter 9, here also the solutions of the LNSE and NNSE are obtained using nonuniform mesh in the x- and y-directions. Computational domain for the semi-infinite flat plate is restricted from −0.05 to 120 in x-direction, as has been pointed out before, to capture the non-parallel effects, associated with mean flow distortion near the leading edge from the Blasius solution. As compared to the results shown in Chapter 9, the domain in the y-direction is from 0 to 1.5, a significantly higher domain for the sake of producing more accurate results. The results presented here are significantly different from other studies, which do not include the leading edge, and also use the similarity profile of the Blasius boundary layer as the equilibrium solution. Though the intention was to study a zero pressure gradient boundary layer, with the leading edge stagnation point, true global effects are observed in the present study and indicate a finite, non-trivial variable pressure gradient. This requires a higher domain size in the y-direction, for the study performed here, as compared to studies using zero pressure gradient Blasius boundary layer as the equilibrium flow. The optimized upwind compact scheme (OUCS3) as introduced in Chapter 3 has been used for spatial discretization of convection terms and the optimal three-stage

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Runge–Kutta scheme [413] has been used for time integration with a time-step of ∆t = 2.5 × 10−5 in [471]. This time step is smaller than that has been used earlier for the cases of vortex-induced instability in Chapter 9. The special feature of these spatial and temporal discretization schemes is the DRP property explained in the literature [32, 34, 413] for high accuracy solutions to capture onset and propagation of the STWF. For transitional flow, the group velocity [67] of the STWF is an important parameter, as it transports energy in the system for all scales, and a short explanation has been provided in Chapter 1, Appendix. a) 12

c)

2400

6 0 t = 110 t = 140 t = 120 t = 100

6

1600

24

28

32

t = 124

t = 108

t = 100

0

36

40

44

48

52

2

d)

18

4

(II)

t = 60

t = 68

W d(α)

t = 56

6 0

1600 (I)

8

10

12

14

t = 60 t = 72 t = 68 t = 64 t = 56 t = 52

(III)

800

6

6 t = 76

2400

12

ωd

t = 116

0

20

t = 140 t = 132

800

12

b)

α

(I)

W d(α)

ωd

t = 52

0 8

10

12

14

16

x

18

20

22

24

26

0

4

8

12

16

20

24

28

Figure 14.23 Disturbance vorticity (ωd ) and its spectrum at indicated times for y = 0.0058, evaluated from (a,c) LNSE and (b,d) NNSE. While the LNSE indicates later onset of STWF, the NNSE displays strong effects of nonlinearity in shorter span of time, as noted at t = 68. In frame (d), the spectrum shows the interaction between local solution and the STWF. The physical problem here is for the free stream vortex convecting with a speed of c = 0.3, so that the input shows very strong receptivity, as noted in the experiments [225, 267]. The evolving disturbance vorticity caused by the imposed free stream excitation is shown in the supplementary video [582], over the full domain using NNSE, corresponding to frame (b). The equilibrium solution is computed as before, without excitation in the computational domain, and thereafter the free stream vortex is released from x = −1, y = 2 starting at t = 0. The choice of Re = 105 , corresponds to the Reynolds number based on the displacement thickness at the outflow to be of a very large value of 5960, significantly larger compared to wall excitation cases reported in [133, 388]. One can note the onset and propagation of the STWF from the LNSE and NNSE solution shown in Figure 14.23, along the top and bottom rows, respectively. The

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STWF is marked as (II) in the spectral plane in frames (c) and (d). The spectrum of the solutions obtained by solving LNSE and NNSE is demonstrated in the supplementary video [583]. For the LNSE solution early indication of STWF is noted in frame (a) at t = 100. For the free stream excitation also, one observes the STWF emerging from out of local solution slowly, with its location noted just below the convecting free stream vortex. Thereafter, one observes an exponential growth of STWF in space and time, in accordance with linear instability theory. For this linearized solution of LNSE, it is observed that by t = 140, the STWF, as measured by peak-to-peak amplitude, larger than the local solution. The local solution of LNSE remains time-invariant in the spectral plane. In comparison, for NNSE solution shown in Figure 14.23(b), the STWF appears significantly earlier at t = 52. Unlike the LNSE solution, the local solution noted for NNSE shows that its Fourier–Laplace amplitude decreases with time. Also by t = 68, additional peaks make their appearance in its vicinity due to the interactions between the STWF and the local solution, with the appearance of multiple local peaks indicated as (III) in frame (d). The growth of the STWF given by NNSE indicates a similar onset mechanism to that of the LNSE solution. Also, the formulations and identical grids for the numerical solution indicate that both of these suffer from the same nonparallel effects of the identical equilibrium flow. However, the nonlinear effects are significantly different between the linear and nonlinear solutions. The formation of STWF is by the same mechanism for wall-excitation cases reported in [32, 34, 419, 451]. In Figure 14.23(b) at t = 68, the nonlinear stage of the growth of the STWF is manifested by the appearance of spikes. Along with earlier onset, saturation of amplitude of the STWF is also rapid for NNSE, while LNSE shows later onset and no saturation. The nonlinear saturation for NNSE indicates multiple length scales in Figure 14.23(d), while the LNSE shows a constant length scale, as noted in frame (c). Two consequences result from this difference. If the LNSE computations are continued in time, then the STWF amplitude will continue its exponential growth till there is numerical overflow of the solution. Surprisingly in literature, such an event has not been reported. In contrast, for NNSE solution one always sees saturation of the disturbance field to its turbulent state. This makes it apparent that flow transition must be studied by NNSE formulation and not by LNSE. The STWF is due to constructive interference in the nonmodal part of the spectrum, as has been explained in Chapter 6 with the help of Figure 6.7. The maximum wavenumber of the peak is traced by a dotted line, which identifies the STWF as (II). In Figure 14.23(d), one can see at t = 64 that in addition to the highest peak for the central wavenumber (indicated by a dashed-dotted line), there are secondary peaks at higher α values. These additional peaks attain significant amplitude with time. As has been noted in [229] for nonlinear nonmodal components, the problem of finding the highest disturbance growth is not just one of finding the maximum of a convex optimization problem, as there are multiple maxima noted during transition. The presence and interactions of multiple peaks make the bandwidth of the spectra of solutions obtained by the NNSE, wider. In Figure 14.23(d), from t = 72 onwards, additional peaks in the

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spectrum near the local solution component is due to interaction with the STWF, which is marked as (III). These interactions have significant global effects, as according to Abel’s theorem [537], the far-field in the physical plane is affected by events occurring in the vicinity of origin in the spectral plane. 0.15 NNSE: t = 60, x = 18.5 LNSE: t = 120, x = 38.7

0.1

y 0.05

y = 0.00542 0 4

2

0

2

4

ωd

Figure 14.24 Wall-normal variation of ωd from the solution of the LNSE (at t = 120, for a station, x = 38.7) and NNSE (at t = 60, for a station, x = 18.5) showing the inner maximum at y = 0.0058. In Figure 14.24, the wall-normal variation of ωd , as obtained from the solutions of LNSE and NNSE, are plotted for t = 120 at a station located near x = 38.7, and at t = 60 for a station located at x = 18.5, respectively. At these time instances, one notes the linearized growth stage of the STWF for both the cases of LNSE and NNSE, with the STWFs centered around these x-stations. The disturbance vorticity, ωd decays with height and an inner maximum is noted near y = 0.0058, where the data has been logged for plotting Figure 14.23. The height of the inner maximum also does not change appreciably with time. For this reason, all the results are presented for this height. After the NNSE solution evolves to the nonlinear saturated state of the STWF, the disturbance vorticity and its spectrum are shown in Figure 14.25. One clearly notes the wide-band spectrum for the two-dimensional turbulence caused by free stream excitation in the frame (a) to (c) of this figure. Such wide-band spectrum have also been noted for wall excitation in [419]. The time instances are specifically chosen

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a) t = 97

10

4

10

2

10

0

d)

Wd(α)

ωd

40 0 40 2

10

10 4

b) t = 98

e)

40 Wd(α)

ωd

10 2 0

10 0

40 10 2 xo = 38.75

c) t = 99

4

f)

10

40 Wd(α)

ωd

2

0

10

0

10 40

2

10 24

28

32

36 x

40

44

0

60

120 α

180

240

Figure 14.25 (a-c) Disturbance vorticity shown as function of x, at the indicated time frames; and in frames (d-f) shown are the corresponding Fourier transforms (Ωd (α)), as obtained from NNSE for the data at the height of y = 0.0058. Wide-band shift in spectrum in (e) is observed at all wavenumbers due to the intermittent formation of spikes, as the STWF evolves (shown in frame (b) at xo = 38.75). [Reproduced from “Nonmodal nonlinear route of transition to two-dimensional turbulence”, A. Sengupta, Prasannabalaji Sundaram, and Tapan K. Sengupta, Phys. Rev. Res.2, 012033(R) (2020).] in Figure 14.25 to show the occurrence of novel events where intermittent spikes are seen in the physical plane. These are associated with the formation of vortexdoublets, as evident in frame (b) at the location of xo = 38.75. Formation of such localized singularity due to vorticity dynamics of the problem at any time causes the corresponding spectrum shown in frame (e), which is qualitatively different from the spectrum for other times when such singularities are absent.

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This is evident by noticing that at t = 99, with the disappearance of the vortexdoublet, the spectrum in frame (f) is similar to that in frame (d) of Figure 14.25. Thus, the sudden appearance of a spike adds a component in the form of eiαxo at all wavenumbers (which can be related by an equivalent time shift property of Fourier– Laplace transform [324, 412]), which has been highlighted for the first time in [471]. Events altering the wide-band signal are very special effects of nonlinearity for fluid dynamical system, only noted at high wavenumbers of the spectrum in frame (e). Such events repeatedly occur, whenever secondary separation bubbles are seen to form on the wall. This has been identified for the first time and termed as twinkling of the spectrum in [471]. These attributes of nonlinearity are further explained in Figure 14.26, with the spectra shown in the wavenumber-frequency plane for indicated time ranges, in order to get a glimpse of the dispersion relation for the nonlinear dynamics. In the supplementary video [584], the spectrum of disturbance vorticity for LNSE and NNSE are compared as a function of time. The spectrum of disturbance vorticity, ωd obtained as the solution of the LNSE and NNSE are shown in Figure 14.26. The energy propagation speed given by the group velocity of the STWF in its early stage is calculated from these two-dimensional spectra shown in frames (a) to (e), by evaluating dωo /dα at the peak amplitude, where constructive interference gives rise to formation of the STWF. This is the mechanism also depicted in Figure 6.7, as the nonmodal component of disturbance spectrum. In the spectrum of Figure 14.26(a) for the solution of LNSE, one observes that the local solution dominates over the STWF. Figure 14.26(b) at a subsequent time for LNSE shows the spectrum with the local solution and the STWF retaining their distinct identities. As the local solution of LNSE retains its amplitude with time (shown in Figure 14.23(a) and (c)), and the STWF grows exponentially, this feature may be clearly observed in Figure 14.26(b). As identified in frame (b) for LNSE, the wavenumber and frequency of the STWF change slowly with time. In contrast, for the NNSE in Figure 14.26(c), the onset of STWF occurs earlier, with the nonlinear effect accentuating the growth rate dominantly. Also, in the spectrum of the NNSE case shown in Figure 14.26(d) and (e), one observes the stronger nonlinear effects to cause significant dispersion over a wider ranges of wavenumbers and frequencies with time. Due to interactions between local solution and the STWF, the spectrum of the local solution changes significantly with time, noted by the appearance of higher amplitude and additional peaks induced by the nonlinear interactions. For external two-dimensional fully developed turbulent flows, one notes a direct cascade of energy, which is dictated by the enstrophy. This energy spectrum is given by E(α) ∼ α−3 , as given in various references [21, 114, 247, 412, 451]. Such a spectrum obtained from the solution of NNSE is shown in Figure 14.27. Additionally, in twodimensional turbulence, one observes an inverse cascade given by, E(α) ∼ α−5/3 , also marked in Figure 14.27. In the literature [114], the direct cascade is attributed to enstrophy and the inverse cascade is termed as the energy cascade. This showcases the importance of working with a nonlinear, spatio-temporal framework to account

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a)

i

LNSE: t = 0 to t = 130

6

27

198 374 751 13293

ω0

4 STWF

2

max = 1167.8

0 LNSE: t = 0 to t = 220

b) 6 ω0

4 2

STWF

max = 21578.6

0 c)

NNSE: t = 0 to t = 65

6

ω0

4 STWF

2

max = 805.0

0 NNSE: t = 0 to t = 85

d) 6 ω0

4 STWF max = 841.1

2 0 e)

NNSE: t = 0 to t = 166

6

ω0

4 2 0

max = 8483.4 0

2

4

α

6

8

10

12

Figure 14.26 Fourier–Laplace transform of ωd shown for the height, y = 0.0058 in (α, ωo )-plane indicating the local solution (near the origin of the spectral plane) and the STWF from (a,b) LNSE and (c-e) NNSE for the identified time ranges. The evolution of STWF for NNSE shows the single peak in (c) transforming to dispersed spectrum in frames (d) and (e). [Reproduced from “Nonmodal nonlinear route of transition to two-dimensional turbulence”, A. Sengupta, Prasannabalaji Sundaram, and Tapan K. Sengupta, Phys. Rev. Res.2, 012033(R) (2020).]

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2

10

10

1

α

3

0

E(α) 10

α

5/3

1

10

10

2

10

0

1

10 α

10

2

Figure 14.27 Energy spectrum of solution of NNSE at t = 165 and y = 0.0058 showing −5/3 and −3 type variation typical of 2D turbulence [114]. [Reproduced from “Nonmodal nonlinear route of transition to two-dimensional turbulence”, A. Sengupta, Prasannabalaji Sundaram, and Tapan K. Sengupta, Phys. Rev. Res.2, 012033(R) (2020).] for all stages of the receptivity problem considered for global, nonmodal, nonlinear analysis.

14.9 Closing Remarks The routes of transition to turbulence for flow past a semi-infinite flat plate excited at the wall have been shown to be due to the STWF in [32, 34, 419, 451]. Furthermore, the role of TS waves in causing transition is shown to be exaggerated in [509]. Instead, STWFs are the definitive cause for transition to turbulence. In this chapter, it has been conclusively shown that the transition is strongly determined by the type of supply of input energy that is causing transition. Unfortunately, the classical eigenvalue analysis fails to do so. Here, an example is shown for the three-dimensional route of transition, which can even explain the physical mechanism of tsunamis; that they are more effectively created by dip-slip events that provide large wall-normal excitation. We

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have explored the three-dimensional DNS route in this study to explain the stronger receptivity in the case of dip-slip events, as compared to strike-slip events, due to subduction motion. Another special feature of transition routes is highlighted by showing that twodimensional input perturbation is more efficient in causing transition, as compared to three-dimensional routes. This quantitative demonstration should convince the reader that though the transition is caused primarily by STWF, it is the two-dimensional disturbance field that is dominant over the three-dimensional route even for the creation of STWF. Next, the global receptivity analysis conducted, shows the effects of free stream excitation. The same equilibrium flow that was conjectured to undergo transition in [295, 510] by free stream turbulence, has now been shown to encounter transition because of unsteady separation, as given in Chapter 9. This is triggered by an unsteady pressure gradient causing unsteady separation. The same transition mechanism is proven here by solving the Navier–Stokes equation in two-dimension to be one of nonlinear, nonmodal mechanism caused by STWF, which is now demonstrated for both two- and three-dimensional routes for wall and free-stream excitation. Here, the element of receptivity for this canonical flow excited by a convecting vortex in the free stream has been studied, using both linear and nonlinear formulations with identical numerical parameters. It is noted that the onset of STWF is captured by both the formulations, but at different times and different locations. This should set the tone for discussion that the nonlinear, nonmodal approach has to be the preferred route of study, rather than the linear, nonmodal route. This is due to the fact that the evolution of STWF is completely different following these formulations, with the nonlinear route capable of capturing transition to turbulence more realistically. Also, the later stages of transition are determined by secondary and nonlinear effects, causing unsteady separation which eventually become the coherent structures, as shown in Chapter 13. The first STWF forms a turbulent spot, which has the regeneration mechanism to create subsequent STWFs, all or some of which can merge downstream to create the turbulent flow. The free-stream excitation case considered here is distinctly different from the wall excitation case [509], as the local solution and the STWF continually interact for the former, while for the latter, these two components do not interact after the onset of STWF. As a consequence, the linear and nonlinear solutions are completely different for free-stream excitation case. The extension to three-dimensional route driven by free stream excitation is straightforward, with initial results presented in [474, 471], where a parametric study of speed, strength and sign of rotation of the vortex was conducted. This was done with the aim of corroborating the results with experiments in [267]. Based on an optimal convection speed of the vortex (obtained from [225, 474]), highly resolved two-dimensional simulations have been reported in [471], that distinguish between the linear and nonlinear operators of the Navier–Stokes equation for global receptivity to free stream excitation. Earlier studies on wall excitation [32, 34, 419, 451] have shown the response field to consist of modal and nonmodal

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components, with the nonmodal STWF as the precursor to transition. Finally, the same STWF is shown to play the primary role for free stream excitation as well. While the receptivity to wall excitation is essentially a carefully designed laboratory exercise, the free-stream excitation case is a real practical flow. Thus, this kind of detailed global receptivity study essentially provides the final answer about route of transition, bridging multiple concepts, which at first sight may appear dissonant. With this description of a recent work providing a complete picture for wall-bounded turbulence, by showing the common elements of wall and free-stream excitations, between global linear and nonlinear receptivity, one hopes that further progress will rapidly materialize in the final quest in our understanding of turbulence.

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