Transition and Turbulence Control

TRANSITION AND TURBULENCE CONTROL This page intentionally left blank TRANSITION AND TURBULENCE CONTROL Mohamed Bad...

Author: Mohamed Gad-El-Hak; Her Mann Tsai

85 downloads 1065 Views 9MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

TRANSITION AND TURBULENCE CONTROL

This page intentionally left blank

TRANSITION AND TURBULENCE CONTROL

Mohamed Bad-al-Hak Her Mann Taai

World Scientific

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Transition and turbulence control / editors, Mohamed Gad-el-Hak, Her Mann Tsai. p. cm. Includes bibliographical references and index. ISBN 981-256-470-5 -- ISBN 981-256-594-9 (pbk.) 1. Turbulence--Congresses. 2. Fluid dynamics--Congresses. 3. Transition flow--Congresses. I. Gad-el-Hak, Mohamed, 1945– II. Tsai, Her Mann. TA357.5.T87T72 2006 620.1'064--dc22 2005055159

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Cover Photograph: Slow-moving, large-amplitude, highly asymmetric static-divergence waves forming on the surface of a viscoelastic coating subjected to a turbulent boundary layer (Fig. 9, Chap. 12). Divergence instabilities are difficult to excite in a laminar flow environment (Fig. 11, Chap. 12). Blue Strip on the Back Cover: A boundary layer, seen in plan view from above, breaks into turbulence. The flow in this computer simulation is from top to bottom. Contours of velocity provide the visualization. Transition between laminar flow at the top and continuously turbulent flow at the bottom is maintained by intermittent patches of turbulence in between (Fig. 9, Chap. 3).

Copyright © 2006 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

Typeset by Stallion Press Email: [email protected]

Printed in Singapore.

CONTENTS

Foreword

vii

Preface

ix

About the Editors

xiii

Contributors

xv

1. Modeling Transition: New Scenarios, System Sensitivity and Feedback Control John A. Burns and John R. Singler 2. Dynamics of Transitional Boundary Layers Cunbiao Lee and Shiyi Chen

1

39

3. Continuous Mode Transition Paul Durbin and Tamer Zaki

87

4. Transition in Wall-Bounded Shear Flows: The Role of Modern Stability Theory Peter J. Schmid

107

5. A Framework for Control of Fluid Flow Alan Guégan, Peter J. Schmid and Patrick Huerre

141

6. Instabilities Near the Attachment-Line of Swept Wings J¨ orn Sesterhenn and Rainer Friedrich

167

v

vi

Contents

7. Experimental Study of Wall Turbulence: Implications for Control Ivan Marusic and Nicholas Hutchins 207 8. Turbulent Boundary Layers and Their Control: Quantitative Flow Visualization Results Michele Onorato, Gaetano M. Di Cicca, Gaetano Iuso, Pier G. Spazzini and Riccardo Malvano

247

9. Mean-Momentum Balance: Implications for Wall-Turbulence Control Joe Klewicki

283

10. The FIK Identity and Its Implication for Turbulent Skin Friction Control Nobuhide Kasagi and Koji Fukagata

297

11. Control of Turbulent Flows Using Lorentz Force Actuation Kenneth S. Breuer

325

12. Compliant Coatings: The Simpler Alternative Mohamed Gad-el-Hak

357

13. Noise Suppression and Mixing Enhancement of Compressible Turbulent Jets Dimitri Papamoschou Index

405 423

FOREWORD

The Institute for Mathematical Sciences at the National University of Singapore was established on 1 July 2000 with startup funding from the Ministry of Education and the University. Its mission is to provide an international center of excellence in mathematical research and, in particular, to promote within Singapore and the region active research in the mathematical sciences and their applications. It seeks to serve as a focal point for scientists of diverse backgrounds to interact and collaborate in research through tutorials, workshops, seminars and informal discussions. The Institute organizes thematic programs of duration ranging from one to six months. The theme or themes of each program will be in accordance with the developing trends of the mathematical sciences and the needs and interests of the local scientific community. Generally, for each program there will be tutorial lectures on background material followed by workshops at the research level. As the tutorial lectures form a core component of a program, the lecture notes are usually made available to the participants for their immediate benefit during the period of the tutorial. The main objective of the Institute’s Lecture Notes Series is to bring these lectures to a wider audience. Occasionally, the Series may also include the proceedings of workshops and expository lectures organized by the Institute. The World Scientific Publishing Company and the Singapore University Press have kindly agreed to publish jointly the Lecture Notes Series. This volume, “Transition and Turbulence Control”, is the eighth of this Series. We hope that through regular publication of lecture notes the Institute will achieve, in part, its objective of promoting research in the mathematical sciences and their applications. November 2005

Louis H. Y. Chen Denny Leung Series Editors

vii


PREFACE

In the Nevada desert, an experiment has gone horribly wrong. A cloud of nanoparticles — micro-robots — has escaped from the laboratory. This cloud is self-sustaining and self-reproducing. It is intelligent and learns from experience. For all practical purposes, it is alive. It has been programmed as a predator. It is evolving swiftly, becoming more deadly with each passing hour. Every attempt to destroy it has failed. And we are the prey. (From Michael Crichton’s techno-thriller “Prey”, HarperCollins Publishers, 2002) The ability to actively or passively manipulate a flow field to obtain a desired change is of immense technological importance, and this undoubtedly account for the fact that the subject is hotly pursued at present by more scientists and engineers than any other topic in fluid mechanics. The art of flow control is as old as prehistoric man, whose sheer perseverance resulted in the invention of streamlined spears, sickle-shaped boomerangs and finstabilized arrows. The German engineer Ludwig Prandtl pioneered the science of flow control at the beginning of the twentieth century. The potential benefits of realizing efficient flow control systems range from saving billions of dollars in fuel cost for land, air and sea vehicles to achieve more economically/environmentally competitive industrial processes involving fluid flows. The purpose of this book is to provide an up-to-date view of the fundamentals of a few basic flows and control practices that can be employed to achieve transition and turbulence control. The understanding of some basic mechanisms in free and wall-bounded turbulence has increased substantially in the last few years. This understanding suggests that the taming of turbulence — the quintessential challenge in the field of flow control — is possible so as to eliminate some of its deleterious effects while enhancing its useful traits. The control of turbulent flow remains a very challenging ix

x

Preface

problem as compared to laminar flow control or separation prevention. Flow instabilities magnify quickly near critical flow regimes, and therefore delaying transition or separation is relatively an easy task. In contrast, traditional control strategies, whether passive or active, are often ineffective for turbulent flows. There are new ideas to achieve for turbulent-flow control, for example, skin-friction drag reduction focus on the direct onslaught on coherent structures. Spurred by the recent developments in chaos control, microfabrication and soft computing tools, reactive control of turbulent flows — where wall sensors detect oncoming coherent structures — and wall actuators attempt to favorably modulate those quasi-periodic events, are now in the realm of the possible for future practical devices. In particular, the current availability of inexpensive, low-energy-consumption microelectromechanical systems (MEMS), e.g., distributed microsensors/microactuators with control logic, opens new opportunities for the effective taming of turbulence. While surely benefiting future humankind, nanotechnology does not have to be as terrifying as the imaginative novelist Crichton has foretold, but rather enabling new frontiers in flow control, e.g., by utilizing adaptive wings and other smart structures. This book contains articles based on lectures given at the Workshop on Transition and Turbulence Control, hosted by the Institute for Mathematical Sciences, National University of Singapore, Singapore, 8–10 December 2004. Lecturing at the workshop were thirteen of the world’s foremost experts in the control of transitioning and turbulent flows. Attending the lectures were faculty and students from universities in Singapore as well as from neighboring countries. The chapters in this book cover a limited number of subjects in the broad area of flow control, and the book should prove useful to researchers working in this area in academia, industry and government laboratories. Topics covered herein include control theory, passive, active and reactive methods for controlling transitional and turbulent wallbounded flows, noise suppression and mixing enhancement of supersonic turbulent jets, passive compliant coatings, modern flow diagnostic systems and swept-wing instabilities. The chapters provide lucid treatment of modern subjects in flow control, and this should be useful as a reference book to scientists and engineers already experienced in the field or as a primer to researchers and graduate students just getting started in the art and science of flow control. The editors are very grateful to all the contributing authors for their dedication to this endeavor, to the Institute for Mathematical Sciences for hosting the

Preface

xi

meeting and generously supporting the travel of all speakers, and to the faculty and staff of the National University of Singapore for providing an efficient, welcoming ambiance to all attendees. While traveling to Singapore was a grueling two-day trip for most lecturers, the warm welcome and hospitality of the “locals” made it all worthwhile. The menacing cloud of nanoparticles never muscled itself into the workshop even for a microsecond. Mohamed Gad-el-Hak Richmond, Virginia, USA 1 May 2005


ABOUT THE EDITORS

Mohamed Gad-el-Hak received his B.Sc. (summa cum laude) in mechanical engineering from Ain Shams University in 1966 and his Ph.D. in fluid mechanics from the Johns Hopkins University in 1973. Gad-el-Hak has since taught and conducted research at the University of Southern California, University of Virginia, University of Notre Dame, Institut National Polytechnique de Grenoble, Université de Poitiers, Friedrich-AlexanderUniversit¨ at Erlangen-N¨ urnberg, Technische Universit¨ at M¨ unchen and Technische Universit¨ at Berlin, and has lectured extensively at seminars in the United States and overseas. Dr. Gad-el-Hak is currently the Inez Caudill Eminent Professor of Biomedical Engineering and Chair of Mechanical Engineering at Virginia Commonwealth University. Dr. Gad-el-Hak has published over 430 articles, authored/edited 14 books and conference proceedings and presented 250 invited lectures. He is the author of the book Flow Control: Passive, Active, and Reactive Flow Management and editor of the books Frontiers in Experimental Fluid Mechanics, Advances in Fluid Mechanics Measurements, Flow Control: Fundamentals and Practices and The MEMS Handbook. Professor Gad-el-Hak is a fellow of the American Physical Society, the American Society of Mechanical Engineers and the American Academy of Mechanics. In 1998, Professor Gad-el-Hak was named the Fourteenth ASME Freeman Scholar. In 1999, Gad-el-Hak was awarded the prestigious Alexander von Humboldt Prize, Germany’s highest research award for senior U.S. scientists and scholars in all disciplines. In 2002, Gad-el-Hak was named ASME Distinguished Lecturer, as well as inducted into the Johns Hopkins University Society of Scholars.

xiii

xiv

About the Editors

Her Mann Tsai received his B.Sc. in aeronautical engineering in 1978 from Imperial College of Science & Technology, London. Upon graduation, he did his Ph.D. in the same college in experimental fluid mechanics. Dr. Tsai has conducted research in Queen Mary College, London, in direct and large-eddy numerical simulations of turbulent flows. He has worked for DSO National Laboratories, Singapore, where he researched and developed major codes for flow analysis and conducted studies in applied aerodynamics. Currently, Dr. Tsai is a Principal Research Scientist in Temasek Laboratories, National University of Singapore, where he leads a team of researchers in a range of aeronautical problems from flow control to flow computations and analysis for design, and optimization of aerodynamic devices. He also teaches as an adjunct staff in the Mechanical Engineering Department, National University of Singapore. Dr. Tsai has published over 80 articles.

CONTRIBUTORS

Kenneth S. Breuer Division of Engineering Brown University Providence, RI 02912 U.S.A. E-mail: [email protected]

Paul Durbin Department of Mechanical Engineering Stanford University Stanford, CA 94035-3030 U.S.A. E-Mail: [email protected]

John A. Burns Interdisciplinary Center for Applied Mathematics Virginia Tech Blacksburg, VA 24061-0531 U.S.A. E-mail: [email protected]

Rainer Friedrich Fachgebiet Str¨ omungsmechanik Technische Universität M¨ unchen 85747 Garching GERMANY E-Mail: [email protected]

Shiyi Chen Department of Mechanical Engineering Johns Hopkins University Baltimore, MD 21218 U.S.A. E-Mail: [email protected]

Koji Fukagata Department of Mechanical Engineering The University of Tokyo Tokyo 113-8656 JAPAN E-Mail: [email protected]

Gaetano M. Di Cicca DIASP Politecnico di Torino 10129 Torino ITALY E-Mail: [email protected]

Mohamed Gad-el-Hak Department of Mechanical Engineering Virginia Commonwealth University Richmond, VA 23284-3015 U.S.A. E-Mail: [email protected] xv

xvi

Contributors

Alan Guegan Laboratoire d’Hydrodynamique (LadHyX) ´ CNRS-Ecole Polytechnique F-91128 Palaiseau FRANCE E-Mail: alan.guegan@ladhyx. polytechnique.fr

Joe Klewicki Department of Mechanical Engineering University of Utah Salt Lake City, UT 84112-9208 U.S.A. E-Mail: [email protected]

Patrick Huerre Laboratoire d’Hydrodynamique (LadHyX) ´ CNRS-Ecole Polytechnique F-91128 Palaiseau FRANCE E-Mail: patrick.huerre@ladhyx. polytechnique.fr

Cunbiao Lee State Key Laboratory for Turbulence Research and Complex Systems Peking University Beijing 100871 CHINA E-Mail: [email protected]

Nicholas Hutchins Department of Aerospace Engineering and Mechanics University of Minnesota Minneapolis, MN 55455-0153 U.S.A. E-Mail: [email protected] Gaetano Iuso DIASP Politecnico di Torino 10129 Torino ITALY E-Mail: [email protected] Nobuhide Kasagi Department of Mechanical Engineering The University of Tokyo Tokyo 113-8656 JAPAN E-Mail: [email protected]

Ivan Marusic Department of Aerospace Engineering and Mechanics University of Minnesota Minneapolis, MN 55455-0153 U.S.A. E-Mail: [email protected] Riccardo Malvano DIASP Politecnico di Torino 10129 Torino ITALY E-Mail: [email protected] Michele Onorato DIASP Politecnico di Torino 10129 Torino ITALY E-Mail: [email protected]

Contributors

Dimitri Papamoschou Department of Mechanical and Aeropsace Engineering University of California, Irvine Irvine, CA 92697-2700 U.S.A. E-Mail: [email protected]

Peter J. Schmid Laboratoire d’Hydrodynamique (LadHyX) ´ CNRS-Ecole Polytechnique F-91128 Palaiseau FRANCE E-Mail: peter@ladhyx. polytechnique.fr

J¨ orn Sesterhenn Fachgebiet Str¨ omungsmechanik Technische Universität M¨ unchen 85747 Garching GERMANY E-Mail: [email protected]

xvii

John R. Singler Department of Mathematics Virginia Tech Blacksburg, VA 24061-0531 U.S.A. E-mail: [email protected] Pier G. Spazzini DIASP Politecnico di Torino 10129 Torino ITALY E-Mail: piergiorgio. [email protected] Her Mann Tsai Temasek Laboratories National University of Singapore Singapore 119260 E-Mail: [email protected] Tamer A. N. Zaki Department of Mechanical Engineering Stanford University Stanford, CA 94035-3030 U.S.A. E-Mail: [email protected]


MODELING TRANSITION: NEW SCENARIOS, SYSTEM SENSITIVITY AND FEEDBACK CONTROL

John A. Burns∗ and John R. Singler† Center for Optimal Design and Control Interdisciplinary Center for Applied Mathematics Virginia Polytechnic Institute and State University Blacksburg, VA 24061-0531, USA E-mails: ∗[email protected] † [email protected] The problem of controlling or delaying transition to turbulence in shear flows has been the subject of numerous papers over the past twenty years. Although there is no single mathematical framework that describes transition for all possible flows, new approaches to (non-classical) linear hydrodynamic stability theory have provided tremendous improvements in the fundamental understanding of this process. In particular, ideas from robust control theory have been used to develop new linear theories in an attempt to explain some of the failures of classical linear hydrodynamic stability theory. This mostly linear theory has produced some new scenarios that may be exploited in control design and analysis. In addition, these theories have been tested on low-dimensional model problems with mixed success. In this paper, we review some of these linear theories and discuss the roles of uncertainty, system sensitivity and modern feedback control in the transition problem. A boundary control problem defined by Burgers’ equation is employed to illustrate how the distributed parameter control theory can be used as a framework for computing feedback functional gains that provide practical guidance in sensor/actuator placement. Low-dimensional models are employed to explain the basic ideas and to illustrate how one can employ bifurcation analysis to predict transition. These examples are also used to show how feedback controllers can delay transition and alter the global dynamics of such systems. Contents 1

Introduction and Motivating Problem

2

1.1

4

A state space formulation 1

2

J. A. Burns and J. R. Singler

1.2 A feedback control problem 1.3 Hydrodynamic stability and feedback control 2 A Mathematical Framework 2.1 The LQR control problem 2.2 Control of Burgers’ equation 2.3 Numerical examples 3 Low Order Model Problems 3.1 The two-dimensional model 3.2 The three-dimensional model 4 Summary and Conclusions Acknowledgment References

6 9 12 13 17 20 25 27 29 31 33 33

1. Introduction and Motivating Problem Designing feedback controllers for active control of fluid flows has received considerable attention from the research community (Gad-el-Hak, 1996, 2000; Gunzburger, 1995; Gunzburger et al., 1999; Sritharan, 1995, 1998 and the references therein). Although the basic problem has been the subject of many experimental and computational studies, much work remains to be done on the development of a “practical theory” (and the corresponding computational tools) that can be used to attack realistic 3D problems at high Reynolds numbers. However, recent advances in hydrodynamic stability theory combined with new mathematical and computational tools offers the potential for breakthroughs on this problem. In this paper, we discuss a portion of this work and indicate some of the mathematical and computational challenges that remain to be addressed. The problem of controlling an incompressible viscous fluid in a given domain Ω ⊂ Rn (n = 2 or 3) by forcing on the boundary ∂Ω = Γ may be modeled by the Navier–Stokes equations. In many important flow control problems, the domain Ω is unbounded (channel and external flows). However, in order to keep the discussion as simple as possible, we focus on regular bounded domains and only briefly indicate how the unbounded 1 >0 domain problem may differ. In particular, let w(t, ˜ x), p(t, x) and ν = Re denote the velocity field, pressure field and kinematic viscosity, respectively. We consider the controlled Navier–Stokes equations given by ˜ x) · ∇)w(t, ˜ x) = ν∆w(t, ˜ x) − ∇p(t, x), x ∈ Ω, t > 0, (1.1) w ˜ t (t, x) + (w(t, ∇ · w(t, ˜ x) = 0,

x ∈ Ω,

t > 0,

(1.2)

3

Modeling Transition

with initial data w(0, ˜ x) = w ˜ 0 (x),

x ∈ Ω,

(1.3)

and “controlled” boundary condition w(t, ˜ x) = u ˜ (t, x),

x ∈ ∂Ω,

t > 0.

(1.4)

Conservation of mass requires that the boundary control u ˜ (t, x) satisfy u ˜(t, x) · n ˜(x) dΓ = 0, (1.5) Γ

where n ˜(x) is the unit outward normal to the boundary Γ = ∂Ω. In many practical situations, the control has the form u ˜ (t, x) =

m

ui (t)˜ gi (x),

(1.6)

i=1

where for each i = 1, 2, . . . , m, gi (x) · n ˜ ˜(x) dΓ = 0 Γ

and hence the control constraint (1.5) is satisfied automatically. ˜ Let W(x) and P (x) denote a steady state (equilibrium) solution to the uncontrolled problem (w(x) ˜ · ∇)w(x) ˜ = ν∆w(x) ˜ − ∇p(x), ∇ · w(x) ˜ = 0,

w(x) ˜ = 0,

x ∈ Ω,

x ∈ Ω,

(1.7)

x ∈ ∂Ω

and define the velocity and pressure perturbation fields by v ˜(t, x) = ˜ W(x) − w(t, ˜ x) and q(t, x) = P (x) − p(t, x), respectively. The controlled perturbation equations become ˜ · ∇)˜ ˜ − ∇q, v · ∇)˜ v = ν∆˜ v − ((W v + (˜ v · ∇)W) v ˜t + (˜

x ∈ Ω,

t > 0, (1.8)

where ∇·v ˜(t, x) = 0,

x ∈ Ω,

v ˜(0, x) = v ˜0 (x),

t > 0,

x ∈ Ω,

(1.9) (1.10)

and v ˜(t, x) = u ˜(t, x),

x ∈ ∂Ω,

t > 0.

(1.11)

4


1.1. A state space formulation Here we briefly outline the steps one takes to construct a (rigorous) state space model for the system of partial differential equations from above. Details may be found in the references Barbu and Triggiani (2004), Barbu et al. (2005), Fursikov (2004), Sritharan (1990, 1998), Temam (1984, 1988). The important function spaces are defined by ˜ = 0, v ˜·n ˜ = 0, x ∈ Ω H= v ˜ ∈ L2 (Ω; Rn ) : ∇ · v and

Here,

V= v ˜ ∈ H 1 (Ω; Rn ) : ∇ · v ˜ = 0, v ˜ = 0, x ∈ Ω . 2 n L (Ω; R ) = v ˜:Ω→R : ˜ v(x)Rn dx < +∞ 2

n

Ω

is the usual Lebesgue space of square integrable vector functions and for m > 0, H m (Ω; Rn ) is the Sobolev space of vector functions whose distributional derivatives of order up to m belong to L2 (Ω; Rn ). If V = {˜ v : Ω → Rn : v ˜ ∈ C0∞ (Ω), ∇ · v ˜ = 0} , then one can show (Sritharan, 1990; Temam, 1984) that H is the closure of V in L2 (Ω; Rn ) and V is the closure of V in H01 (Ω; Rn ). Let PH : L2 (Ω; Rn ) → H be the orthogonal projection onto H (the Leray projection) and define the Stokes operator AS : D(AS ) ⊂ H → H on the domain D(AS ) = H 2 (Ω; Rn ) ∩ V by AS v ˜ = PH ∆˜ v,

v ˜ ∈ D(AS ).

It follows (see p. 14 in Sritharan, 1990) that the dense and continuous embeddings D(AS ) ⊂ V ⊂ H = H′ ⊂ V′ ⊂ [D(AS )]′ are compact, where the prime denotes the dual space. Let A0 : V → V′ be the lifting of AS : D(AS ) ⊂ H → H to V and define the linear operator R : V → V′ by ˜ · ∇)˜ ˜ R˜ v = −PH ((W v + (˜ v · ∇)W)

5

Modeling Transition

and hence the Oseen operator A(ν) : V → V′ given by A = A(ν) = νA0 + R is the linearized operator corresponding to the steady state solution given by (1.7). The nonlinear operator F : V → V′ is defined by n ∂vj [F (˜ v)](ϕ) ˜ = vi ϕj . ∂x i i,j=1 Ω In order to complete the model, one must define the control input operator. The basic idea is to use the Lions structure (Lions, 1969) (see also Barbu et al., 2005; Bensoussan et al., 1992a). In particular, one “lifts” A : V → V′ to A1 : H → [D(A∗ )]′ by ˜=w ˜ A1 v

if and only if w( ˜ ϕ) ˜ = ˜ v, A∗ ϕ ˜H

and defines B : L2 (Γ; Rn ) → [D(A∗ )]′ by B˜ g = −A1 D˜ g where D is the Dirichlét map D : L2 (Γ; Rn ) → H 1/2 (Ω; Rn ) ∩ H. The state space model for the boundary control problem (1.8)–(1.11) now has the (very) weak formulation d v ˜(t) = (νA0 + R)˜ v(t) + F (˜ v(t)) + B˜ u(t) ∈ [D(A∗ )]′ . dt

(1.12)

The system (1.12) has some important features that are typical of many flow control problems. Note that although the Stokes operator νA0 is selfadjoint, the Oseen operator A = νA0 + R can be highly non-normal if R=

0. Therefore, the linearized control system d v ˜(t) = (νA0 + R)˜ v (t) + B˜ u(t), dt

v ˜(0) = v ˜0

(1.13)

can be highly sensitive to parameter variations, initial data and inputs u(·). Computational algorithms for linear control problems with a non-normal A operator require special effort. In addition, the nonlinear operator is conservative in the sense that F (˜ v), v ˜ V = 0

(1.14)

for all v ˜ ∈ V and this special structure tends to be highly sensitive to disturbances (Burns, 2003). Sensitivity and non-normality of the linear system lead to many difficulties in the corresponding control problem.

6


1.2. A feedback control problem We consider the feedback control problems for the Navier–Stokes system d v ˜(t) = (νA0 + R)˜ v(t) + F (˜ v(t)) + B˜ u(t) dt and the corresponding linearized system d v ˜(t) = (νA0 + R)˜ v(t) + B˜ u(t), dt

(1.15)

(1.16)

1 where again ν = Re . It is important to note that there are many “feedback control problems” (linear feedback, nonlinear feedback, LQR, LQG, H ∞ , Min-Max, etc.) and several technical approaches to each of these problems. Obviously, one cannot cover all these areas in a single paper, so we focus on a feedback stabilization problem that has a direct connection to transition control. The basic scenario is that there is a critical Reynolds number, Recrit , such that if Re < Recrit, the open-loop linearized operator A(Re) = (νA0 + R) generates an exponentially stable C0 -semigroup S(t, Re) = eA(Re)t satisfying

S(t, Re)H ≤ M e−γt where M = M (Re) ≥ 1 and γ = γ(Re) > 0. If Re > Recrit , then there is an initial condition v ˜(0) = v ˜0 such that lim S(t, Re)˜ v0 H = +∞,

t→+∞

i.e. S(t, Re) is unstable. In this case, this initial condition may produce transition in the full nonlinear equation. It is well known that the situation is much more complex and transition can occur for values much lower than Recrit . In fact, if one considers the A(Re) = (νA0 + R) linearization about the plane Couette flow, then Recrit = +∞ and the linearized operator is always stable. Attempts to understand and explain this phenomena has motivated research in classical linear hydrodynamic stability theory for more than a century (see the excellent summary in Drazin, 2002). Modern hydrodynamic stability theory based on robustness, pseudospectrum and sensitivity analysis has provided a much better understanding of this process and generated new scenarios to explain transition (Drazin, 2002; Schmid and Henningson, 2001). A fundamental new idea in these approaches is that because A(Re) = (νA0 + R) is non-normal and M = M (Re) grows like [Re]θ where θ > 1, a small initial data can produce large transient growth due entirely to the linear part of the equation.

Modeling Transition

7

Therefore, even when A(Re) = (νA0 + R) is stable, once this transient growth becomes “large enough” the nonlinear terms become important and nonlinear “mixing” leads to transition. Clearly, there are still gaps in building a complete theory for such scenarios, but the basic idea helps understand why and how feedback control might be used to delay or prevent transition. For example, assume that one can find a linear feedback gain operator K : D(K) ⊆ H → U = L2 (Γ; Rn )

(1.17)

such that the closed-loop operator ACL (Re) A(Re) − BK = (νA0 + R − BK)

(1.18)

generates a closed-loop semigroup SCL (t, Re) = eACL (Re)t satisfying SCL (t, Re)H ≤ e−ˆγ t with γˆ > 0. In this case, the corresponding closed-loop nonlinear Navier– Stokes equations d v ˜(t) = (νA0 + R − BK)˜ v(t) + F (˜ v(t)) (1.19) dt would be “monotonically stable” as defined on p. 5 in Schmid and Henningson (2001). It is important to note that this does not automatically imply asymptotic stability of the base flow. One must also establish that the zero equilibrium v ˜0 = 0 for the nonlinear closed-loop system (1.19) is stable. In particular, one needs to show that for ε > 0, there is a δ > 0 such that ˜ v0 H < δ implies that: 1. there exists a unique solution v ˜(t, v ˜0 ) to (1.19) with v ˜(0, v ˜0 ) = v ˜0 defined for all t > 0, 2. ˜ v(t, v ˜0 ) ≤ ε for all t > 0. Producing a monotonically stable closed-loop system may prove difficult, but it is not impossible for certain flows (Kang and Ito, 1994) and there are recent results that imply one can stabilize a 3D nonlinear flow with linear feedback even when Re > Recrit. Fursikov (2004) has some very interesting results along this line. Also, Barbu, Lasiecka and Triggiani (see Theorem 6.1 in Barbu et al., 2005) have proven the existence of a linear feedback operator K : H 1/2+ε (Ω; Rn ) ∩ H → U = L2 (Γ; Rn ) such that the closed-loop Navier–Stokes equation (1.19) is exponentially asymptotically stable. Moreover, this feedback gain operator can be “computed” by solving an abstract Riccati operator equation. Although the results in Barbu et al.

8


(2005) only apply to problems on bounded domains, these results provide insight and some promise that one can deal with certain exterior flows. In addition, there is considerable numerical and experimental evidence that the same is true for channel flows (Choi et al., 1994; Cortelezzi, et al., 2001; Wang et al., 1992; Wiltse and Glezer, 1993). One benefit of this form of a feedback law is that it can generate useful spatial information about sensor and actuator placement (Burns and King, 1994, Burns et al., 1995, 1998). Under suitable conditions, it is possible to represent the gain operator as an integral of the form k(x, y)˜ v(t, y) dy, (1.20) u ˜(t, x) = −K v ˜(t, ·) = − Ω

where the kernel is called the functional gain. Moreover, there are many practical cases where the functional gain has highly localized support. In some cases (LQR boundary control), the functional gain may have local support so that practical information comes from being able to compute k(x, y). This will be illustrated by the simple Burgers’ equation below and has been applied to a wide variety of distributed parameter control problems in Bewley (2001), Burns and King (1994), and Burns et al. (1995, 1998). In view of these results, we focus on the use the linearized control system (1.16) to design feedback controllers and then apply this linear control to the full nonlinear system (1.15). This is a standard approach and a good “first step” in any design process. It is desirable to have some basic knowledge about the open-loop uncontrolled dynamics for both the nonlinear and linear systems. When Ω is smooth and bounded, much is known about the spectrum of the linearized Oseen operator A = νA0 + R and the stability of the linearized problem. In particular, one has the following lemma (see p. 1448 in Barbu and Triggiani, 2004 and Theorem 3.6 in Sritharan, 1990). Lemma 1: If Ω is bounded with smooth boundary, then the Oseen operator A = νA0 +R has compact resolvent and generates an analytic semigroup on H. The spectrum of A is only point spectrum (i.e. σ(A) = σp (A)), all the eigenvalues have finite geometric multiplicity, accumulate only at −∞ and there are at most a finite number of eigenvalues λi , i = 1, 2, . . . , m satisfying Re(λi ) ≥ 0. Lemma 1 implies that in order to stabilize the linearized flow, one needs only to “move” a finite number of eigenvalues to the left hand complex

Modeling Transition

9

plane. This fact is the basis of much of the work in the paper of Barbu and Triggiani (2004) and has been extended to boundary control in Barbu et al. (2005). However, for unbounded domains, A = νAS + R can have a nonempty essential spectrum. For example, for certain exterior domains the spectrum of A = νAS + R has the form σ(A) = σp (A) ∪ Λ(Re) where again there are at most a finite number of eigenvalues λi , i = 1, 2, . . . , m satisfying Re(λi ) ≥ 0 and the essential spectrum lies inside the parabolic region Λ(Re) = λ = α + βi : α ≤ 0, β 2 ≤ −Reα

(see Theorem 3.11 in Sritharan, 1990). Observe that the parabolic region Λ(Re) opens up as Re → +∞ and hence the non-zero essential spectrum can move closer to the imaginary axis. This can impact sensitivity and control design. The case of channel flows is again different because the boundary is not compact. 1.3. Hydrodynamic stability and feedback control We close this section by noting some recent work that has considerable impact on the control problem. Also, we point out some important technical issues that need to be addressed when one considers control problems governed by highly sensitive nonlinear systems with non-normal linearizations. A detailed discussion of these issues along with several illustrative examples may be found in Burns (2003). In 1880, Lord Rayleigh (Rayleigh, 1880) wrote a fundamental paper on the stability of fluid motions and since then the field of linear hydrodynamic stability theory has been a centerpiece of classical fluid dynamics. One hundred years later, beginning in the late 1980’s and early 1990’s, Henningson, Reddy, Schmid, Trefethen and co-workers begin to develop a new approach to hydrodynamic stability. This modern approach is still based on a linear theory, but differs from classical linear hydrodynamic stability in that singular values and pseudo-spectrum play the key role in their work. The observation that linearization about a non-trivial laminar flow leads to a non-normal linear problem is the key to this theory. The references Baggett et al. (1995), Henningson (1987), Henningson et al. (1993), Henningson and Reddy (1994), Reddy and Henningson (1993), Reddy et al. (1993), Schmid and Henningson (1994), Schmid et al. (1996), Schmid (2000), Trefethen et al. (1993) provide the foundations for this work and the recent book by Schmid and Henningson (2001) provides an excellent and modern treatment of this area. Much of this work (certainly not all) focuses on the idea that small (but very specific) “initial” data can produce large transient

10


growth due to the non-normality of linear operator until the nonlinear terms become “important” and produce transition. Considerable effort has been devoted to the problem of identifying the specific initial data (Tollmien– Schlichting waves, oblique waves, etc.) and the corresponding threshold amplitudes that generate this initial large transient growth. In the mid 1990’s, a group of researchers including Bamieh, Dahleh, Farrell, Ioannou and co-workers developed a similar linear theory based on ideas from robust control theory. In addition to identifying amplitude thresholds for specific initial data, this effort focused on possible input disturbances that also get magnified due to the non-normality of the operator A = νA0 + R (Bamieh and Dahleh, 1998, 2001; Butler and Farrell, 1992; Farrell and Ioannou, 1993; Trefethen, 1997). Bamieh and Dahleh suggested that for channel flows, an unmodeled disturbance could come from extremely small wall-roughness or forced boundary conditions and this disturbance could be amplified by the non-normal linear system leading to transition. Moreover, this observation also suggests that boundary control has the potential to significantly delay or eliminate transition in a wide variety of shear flows (Choi et al., 1994; Cortelezzi and Speyer, 1998; Cortelezzi et al., 2001; Joshi et al., 1997). Almost all of this work focuses on linear input-output theory and the role that the nonlinear term plays in this scenario is not fully understood. Although the basis for both scenarios is linear stability analysis, the ideas put forth by these groups have proven to be very useful and have been used to more accurately predict critical transition numbers. However, as noted in Schmid and Henningson (2001), in order to provide a complete description of the total transition process, one must develop a framework that can be used to analyze the precise role that the nonlinear term plays in the transition mechanism. At this time, it is probably not possible to develop a theory directly applicable to the 3D Navier–Stokes equations. However, even simple 1D partial differential equations can exhibit extreme sensitivity to boundary disturbances. For example, Burgers’ equation is known to be supersensitive to changes in Dirichlét boundary conditions (Garbey and Kaper, 2000; LaForgue and O’Malley, 1993) and hypersensitive to changes in the Neumann boundary conditions (Allen et al., 2002). In Allen et al., (2002), it is shown that the specific nonlinearity, combined with hyper-sensitive boundary conditions can produce an unexpected transition from a small initial state to a large steady state solution of a “nearby” problem. This problem can be analyzed completely and it is possible to see the exact cause of the breakdown. This extreme sensitivity to

Modeling Transition

11

boundary disturbances makes Burgers’ equation a good infinite-dimensional model for testing transition scenarios. In addition, we shall use Burgers’ equation to illustrate the computation of functional gains. We also present some bifurcation analysis and control results for low order finite-dimensional models found in the literature (Henningson et al., 1993; Henningson and Reddy, 1994; Henningson, 1996; Schmid and Henningson, 2001; Trefethen et al., 1993; Waleffe, 1995a). These “simple” models can provide insight into possible transition scenarios and provide useful cases to illustrate the power of linear feedback control. There are numerous control-related issues that need to be addressed before one can confidently attack realistic flow control problems. Two issues are specific to problems with the structure described above. 1. Small variations in the non-normal operator (νA0 + R) can produce dramatic changes in the stability of the linear system and radical changes in the (global) dynamics can be produced by small perturbations of the nonlinear term (Allen et al., 2002; Burns, 2003; Burns and Singler, 2005). This high sensitivity and lack of robustness can be used to explain certain routes to transition (Henningson et al., 1993; Henningson and Reddy, 1994; Henningson, 1996; Waleffe et al., 1993; Waleffe, 1995a). In terms of control, the open-loop system is not robust with respect to uncertainties and even a stable system can proceed to transition when subjected to a small change. Therefore, one objective of the feedback control might be to robustly stabilize the system and it is not always clear how best to approach this problem even for the simple finite-dimensional models. 2. Since A is not normal, then one must be careful when developing approximations for feedback control or optimization of such systems in order to ensure convergence of the design. In addition, there are other important computations (e.g., power density functions and pseudo-spectra) that require careful approximations of the adjoint A∗ and this is a non-trivial problem which is often ignored (Banks and Burns, 1978; Banks and Kunisch, 1982; Banks and Ito, 1997; Burns et al., 1988; Burns, 2003; Burns et al., 2003). It is important to develop accurate and rigorous algorithms so that the numerical control laws can be used to address practical design questions such as where to locate sensor/actuator pairs (Banks and Ito, 1997; Burns and King, 1994; Burns et al., 1995, 1998). The uncontrolled system (1.12) with conservative nonlinearity satisfying (1.14) falls into a general class of distributed parameter systems

12


considered by Ghidaglia (1984) and Temam (1988). This framework provides a powerful tool for investigating existence, uniqueness and regularity results for the nonlinear Navier–Stokes equations. In addition, this framework can be used to investigate control design and sensitivity for such systems (Burns, 2003). However, several theoretical questions remain open and certain “gaps” exist when the boundary is unbounded or the problem is 3D (i.e. when n = 3). We turn now to a general distributed parameter control problem that is motivated by the structure of the flow control problem above. 2. A Mathematical Framework Navier–Stokes equations, Burgers’ equation and most of the proposed loworder finite-dimensional models of transition fall into a general framework first developed by Ghidaglia (1984). The basic structure is summarized as follows. Let W , V and Z be separable Hilbert spaces satisfying W ⊂ V ⊂ Z = Z′ ⊂ V ′ ⊂ W ′,

where W ′ , V ′ and Z ′ are the dual spaces of W , V and Z, respectively. We assume that the injections are continuous and each space is dense in the following one. Let a(·, ·) be a symmetric bilinear form on V satisfying a(v, v) ≥ γ vV 1 Re

(2.1)

for some γ > 0. Let ν = > 0 and define the associated isomorphism A0 : V → V ′ by [A0 z]v = −a(z, v). The (unbounded) selfadjoint restriction operator AS : D(AS ) ⊆ V ⊂ Z → Z is defined on D(AS ) = {z ∈ V : A0 z ∈ Z} by AS z = A0 z for all z ∈ D(AS ). Note that for z ∈ D(AS ) and v ∈ Z, a(z, v) = − AS z, v . We assume that the self-adjoint linear operator νAS generates a C0 -semigroup on Z. The input operator B : U → W ′ is a bounded linear operator mapping the control space U into a space W ′ containing Z. Although there are considerably technical issues to be addressed, roughly speaking the Dirichlét boundary control problem requires that W = D(A∗S ) and for the Neumann boundary control problem, one sets W = V . When control is applied through internal “body forces”, then W is not required and B : U → Z is a bounded linear operator with range in Z. Let R : V → V ′ be a bounded linear operator which maps D(AS ) into Z. Thus, we can define the (possibly unbounded) linear operator RS : D(RS ) = D(AS ) → Z to be the restriction of R to D(AS ). In addition, we assume that [νAS + RS ] generates a C0 -semigroup S(t) : Z → Z on Z and

13

Modeling Transition

note that A = [νA0 + R] is the standard extension of [νAS + RS ] to V . Thus, we follow the standard abuse of notation and say that A = [νA0 + R] generates S(t) on Z. The nonlinear operator is defined by a trilinear form. Therefore, let f : V × V → V ′ be a continuous bilinear operator with the property that f maps D(AS ) × D(AS ) into Z. The nonlinear operator F : V → V ′ is defined by F (v) = f (v, v). Given the framework above, we consider the abstract control system z(t) ˙ = [νA0 + R]z(t) + F (z(t)) + Bu(t),

(2.2)

z(0) = z0 ∈ Z.

(2.3)

with initial data Although this system is quite general, there are reasonable control systems that are not covered by this framework and much remains to be done to complete the theory. The important point for this paper is that nonlinear term F is often conservative. In particular, if ·, · : V ′ × V → C denotes the duality map defined by u, v = u(v), and F (v), v = [F (v)]v = [f (v, v)]v = f (v, v), v = 0,

(2.4)

for all v ∈ V , then we say that F is conservative. When the linear part of the system is stable but near an unstable operator and the corresponding nonlinear term is conservative, then the nonlinear system can be hypersensitive to small disturbances. This plays a central role in a recent scenario presented in Burns and Singler (2005). Before discussing possible transition mechanisms, we present some basic results on linear feedback control. 2.1. The LQR control problem We now consider specific feedback control problems for the distributed parameter system z(t) ˙ = [νA0 + R]z(t) + F (z(t)) + Bu(t),

z(0) = z0 ,

(2.5)

and the corresponding linearized system z(t) ˙ = [νA0 + R]z(t) + Bu(t),

z(0) = z0 .

(2.6)

In order to keep the discussion as simple as possible, we limit ourselves to the case with bounded input operator B : U → Z. We use the linearized system (2.6) to design a feedback controller and then apply this linear control to the full nonlinear system (2.5).

14


The linear system (2.6) is exponentially stabilizable if there is a bounded linear operator K : Z → U such that the closed-loop operator [νA0 + R] − BK generates an exponentially stable C0 -semigroup SCL (t) (for the basic definitions, see Curtain and Zwart, 1995; Kato, 1976; Lions, 1969). Let Q : Z → Z be a self-adjoint bounded linear operator, r > 0 and define the cost function +∞ {Qz(t), z(t) Z + r u(t), u(t) } dt, (2.7) J(u, z0 ) = 0

where z(t) is the mild solution of (2.6) defined by t z(t) = S(t)z0 + S(t − s)Bu(s) ds,

(2.8)

0

and S(t) is the semigroup generated by νA0 + R. The LQR problem is to find a u(·) ∈ L2 (0, +∞; U ) that minimizes J. It follows from standard distributed parameter control theory (Bensoussan et al., 1992a; Curtain and Zwart, 1995; Lions, 1969) that if (2.6) is exponentially stabilizable, then the minimizer of (2.7) exists and is given by state feedback u(t) = −Klqr z(t).

The feedback gain operator Klqr : Z → U is given by

1 ∗ B Π r where Π = Π∗ is the self-adjoint linear operator Π : Z → Z that solves the operator Riccati equation Klqr =

A˜∗ Π + ΠA˜ − rΠBB ∗ Π + Q = 0,

(2.9)

where A˜ = [νAS + RS ]. Moreover, the gain operator produces an exponentially stable linear closed-loop system. We use this result to compute the functional gains that define the gain operator Klqr . This controller is then applied to the nonlinear system which yields the full closed-loop system z(t) ˙ = [νA0 + R − BK]z(t) + F (z(t)),

z(0) = z0 ∈ Z.

(2.10)

In order to make this approach practical, we need computational algorithms to solve the operator Riccati equation (2.9) and mathematical tools to analyze the resulting closed-loop nonlinear system. Again, we emphasize the point that it is not necessary to have a complete existence theory for the nonlinear open-loop system in order to proceed with this method. In fact, there is no such theory for the 3D Navier–Stokes equations yet the results

Modeling Transition

15

in Barbu and Triggiani (2004) and Barbu et al., (2005) are valid for this 3D problem. As the simple example below illustrates, the closed-loop system (2.10) can be stabilized by linear feedback even when the open-loop problem has the property that every neighborhood of zero contains a solution with finite blowup time. Example 1: Consider the controlled ordinary differential equation d x(t) 1 0 x(t) −0.5 1 u(t) (2.11) + + = 1 [y(t)]2 y(t) 0 0 dt y(t) with initial condition

x(0) x0 . = y0 y(0)

(2.12)

First note that (2.11) and (2.12) is well-posed on R2 , i.e. given any initial

T there exist a unique solution data z0 = x0 y0 x(t, z0 ) z(t, z0 ) = y(t, z0 ) defined on a finite interval (0, T ) where T = T (z0 ) > 0. If y0 > 0 and

T y0 z0 = 0 y0 , then y(t, z0 ) = 1−ty so that the solution always has finite 0 1 blowup time T = T (z0 ) = y0 . Note that the open-loop system has only one

T equilibrium ze = 0 0 = 0 and ze is certainly not stable. We fix r = 1 and solve two LQR problems with different Q matrices. In particular, if 0 0 0.5 0 Q1 = and Q2 = , 0 1 0 0.5

then K1 = 0 1 and K2 = 0.3985 0.7841 , respectively. The corresponding closed-loop linear operators are given by −0.8985 0.2159 −0.5 0 ACL1 = , and ACL2 = −0.3985 −0.7841 0 −1 and when the feedback controllers are applied to the nonlinear system both controllers exponentially stabilize z0 = 0. Moreover, K1 = 1 and K2 = 0.8796, so the control energies required to stabilize the system are about the same. However, there are major differences between the controllers. First, the controller defined by K1 is “local” in that it makes use of only one state y(t). Although the control defined by K2 requires both states (not local), it is more robust in the sense that the closed-loop operator ACL2 has a stability radius of 0.7879 while the closed-loop operator ACL1 has a

16


stability radius of 0.5. The radius of stability for the closed-loop nonlinear system with feedback gain K1 is equal to 1. If y0 < 1, then any initial data

T of the form z0 = x0 y0 produces a stable response. The half-plane Σ1 = {[ x0 y0 ]T : y0 < 1} is an invariant set with attractor M = {0}. The radius of stability for the closed-loop nonlinear system with feedback gain K2 is less than 1. Moreover, there is an invariant set Σ2 (the shaded region in Fig. 1) defined by the set of all initial conditions below the stable manifold Υ of the unstable hyperbolic equilibrium. Observe that this closedloop system allows for large values of y0 if x0 is also large. As noted above, it has recently been shown in Barbu et al. (2005) that the linear LQR boundary feedback controller designed by the Riccati equation will (locally) stabilize the full 3D Navier–Stokes equation. On one hand this is a very strong theoretical result, but this control law has two possible drawbacks. The LQR controller is infinite dimensional in that it must be applied to the entire boundary and nothing is known about the support of the functional gain. It is not obvious that LQR type design will prove practical and it is not clear that the LQR controllers will work in channel or other shear flows. This requires additional work. Also, in order to use this 2 1.5 1

y

0.5 0 −0.5

Σ2

−1 −1.5 −2 −2

Fig. 1.

−1

0 x

1

2

Invariant set for the nonlinear closed-loop system, K2 .

17

Modeling Transition

type of design, one must be able to compute the feedback gains. Considerable progress has been made in the area, but new computational algorithms need to be developed to handle fully 3D flows. The simple example above illustrates several important issues. However, in order to illustrate the potential practical application of the distributed parameter control theory we turn to a control problem for Burgers’ equation. 2.2. Control of Burgers’ equation Now we turn to an infinite-dimensional system that falls into the above framework so that we can illustrate some practical benefits of distributed parameter control theory. We use Burgers’ equation as an example because it is known to be highly sensitive to “small disturbances” in the boundary conditions and it provides an infinite-dimensional problem where we can illustrate the calculations of functional gains. Therefore, we consider Burgers’ equation with a convective term zt (t, x) + z(t, x)zx (t, x) = νzxx (t, x) + κzx (t, x),

0<x 0, u(t) ∈ L2 (0, +∞) and ϕ(·) ∈ L2 (0, 1). Let Z = L2 (0, 1), D(AS ) = H01 (0, 1) ∩ H 2 (0, 1), and define the differential operator AS : D(AS ) → Z by AS w(·) = wxx (·).

(2.15)

Let V = H01 (0, 1) and define a(·, ·) to be the symmetric bilinear form on V given by 1 wx (x)vx (x) dx. (2.16) a(w, v) = 0

It follows that V ⊂ Z = Z ′ ⊂ V ′,

(2.17)

18


where the injections are continuous and each space is dense in the following one. Moreover, a(v(·), v(·)) satisfies a(v(·), v(·)) ≥ α v(·)V

(2.18)

for some α > 0 and a(v(·), v(·)) defines an associated isomorphism A0 : V → V ′ by [A0 z(·)]v(·) = −a(z(·), v(·)). Let R : V → V ′ be defined by [Rw(·)]v(·) = κ

(2.19)

1

wx (x)v(x) dx

(2.20)

0

and note that R maps D(AS ) into V ⊆ Z so that RS is well defined by restricting R to D(AS ). In addition, it is easy to show that [νAS + RS ] generates a C0 -semigroup S(t) : Z → Z on Z. Define the continuous bilinear mapping f : V × V → V ′ by [f (z(·), w(·))]v(·) = −

1

z(x)wx (x)v(x) dx

(2.21)

0

and observe that f maps D(AS )×D(AS ) into H 1 (0, 1) ⊂ Z. In particular, if z(·) and w(·) belong to D(AS ), then f (z(·), w(·)) = −z(x)wx (x) ∈ H 1 (0, 1). Let F : V → V ′ denote the operator given by F (z(·)) = f (z(·), z(·))

(2.22)

and note that

[F (z(·))]v(·) −

1

z(x)zx (x)v(x) dx

(2.23)

0

maps D(AS ) into Z. For each z(·) ∈ V , F satisfies F (z(·)), z(·) = −

1

z(x)zx (x)z(x) dx = 0,

(2.24)

0

and hence the nonlinear term in conservative. As noted above, the B operator will not map into Z ⊂ V ′ . One approach that works for this simple (essentially self-adjoint) problem is to define the space W by the domain of AS , extend the state space and consider a very

19

Modeling Transition

weak form of the problem. Define W to be the space D(AS ) = D([AS ]∗ ) with graph norm z(·)W = [AS ]∗ z(·)Z + z(·)Z .

(2.25)

It follows that the injections W ⊂ V ⊂ Z = Z′ ⊂ V ′ ⊂ W ′

(2.26)

are all continuous and dense. One now lifts the operator A0 : V → V ′ defined by [A0 z(·)]v(·) = −a(z(·), v(·)) to an operator A1 : Z → W ′ . The basic idea is to integrate by parts twice and define A1 : Z → W ′ by 1 ∗ [A1 z] w = z, [AS ] w Z = z, [AS ]w Z = z(x)wxx (x) dx (2.27) 0

′ for all w(·) ∈ W ′ = H01 (0, 1) ∩ H2 (0, 1) . Let D : R1 → L2 (0, 1) = Z be the Dirichlét map [Du](x) = xu,

(2.28)

B = −A1 D.

(2.29)

and define B : R1 → W ′ by

It is easy to see that for w(·) ∈ W 1 [xu]wxx (x) dx [Bu]w(·) = 0

x=1

= [xu]wx (x)|x=0 − = uwx (1) −

1

[u]wx (x) dx

0 x=1 [u]w(x)|x=0

= u[δ1′ ](w(·)),

where δ1′ is the (distributional) derivative of the delta function at x = 1. It follows (Bensoussan et al., 1992a; Curtain and Zwart, 1995; Lions, 1969) that the linearized system may be formulated as the well-posed control system in W ′ z(t) ˙ = [νAS + RS ]z(t) + Bu(t) ∈ W ′ .

(2.30)

This is the linear system we use for control design. Note that we do not consider the nonlinear system until we close the loop. The nonlinear

20


closed-loop system will have the form z(t) ˙ = [νA0 + R − BK]z(t) + F (z(t)),

z(0) = z0

(2.31)

and one makes use of regularity results to show that this nonlinear closedloop system is stable (Bensoussan et al., 1992a; Burns and Kang, 1991; Burns et al., 1998). If one applies LQR theory (or LQG, Min-Max, etc.), then the optimal controllers have the form 1 k(x)w(t, x) dx, uopt (t) = −Kw(t, ·) = − 0

where k(x) is the functional gain. For LQR problems where the control is applied through a Dirichlét boundary term, these gains tend to become singular near the controlled boundary (Burns et al., 2002a). We focus on the accurate computation of these functional gains and discuss how the choice of a LQR problem impacts the support and singularity of the functional gains. We do make use of the nonlinear system (2.31) to develop finite element approximation schemes. For convection dominated flows, one needs to use upwinding or some form of stabilized finite element scheme. The approximation theory is complete for the linear equation (2.30). However, the implementation of these schemes in more than one space dimension can be complicated. 2.3. Numerical examples √ Let 0 < ν < 1 and bL = 1 − ν . Define q(·) : [0, 1] → R by q(x) =

qL , bL ≤ x ≤ 1 , qS , 0 ≤ x < bL

(2.32)

where qS and qL are positive numbers. For r > 0 and α ≥ 0, define the cost function by Jα (u(·)) =

0

∞

eαt

1 0

2 q(x)|w(t, x)|2 dx + r |u(t)| dt.

(2.33)

Note that if 0 < qS ≪ qL , then the cost function places a large penalty on the solution in the “boundary layer”, bL ≤ x ≤ 1. Also, when α > 0 there is an additional performance requirement. The boundary control problem

Modeling Transition

21

is to minimize Jα (u(·)) defined by (2.33) subject to (2.30). Standard representation theory implies that the optimal controller has the form 1 kµ,α (x)w(t, x) dx, (2.34) uopt (t) = −Kµ,α w(t, ·) = − 0

where the functional gain kµ,α (·) ∈ L2 (0, 1) (Burns and Kang, 1991; Burns and King, 1994). Note that this functional gain depends on the choice of µ, α and the weight q(x). We compute approximations [kµ,α ]N (·) of kµ,α (·) for α ≥ 0 and 0 < µ ≪ 1 by using finite element methods with mesh refinement schemes given in references Burns et al. (2002) and Burns et al. (2002a). In Run 1, we show the functional gains for α = 0, α = 0.1 and α = 0.2 to illustrate how the choice of weighting can impact the support of the optimal functional gains. In Run 2, we consider a convection dominated problem to illustrate some of the numerical and convergence issues connected to solving the Riccati equation. 1 , κ = 0, r = 0.25 qS = 1 and qL = Run 1. √ For the first run we set ν = 120 50 √ = 50 120 = 547.723. Here, the selection of the weights places a heavy ν penalty on the boundary layer near the control boundary. In particular, we √ 1 = 0.0913. focus on the region bL < x < 1 which has thickness ν = √120 To emphasize the role that α plays in the problem, we consider three cases corresponding to α = 0, α = 0.1 and α = 0.2 and use a uniform mesh to compute the functional gains. In Fig. 2 we see that the functional gains for α > 0 has global support over the entire interval [0, 1] and the gains become more significant on the interior of the domain as α increases. However, in all the cases above, the gains are singular near the boundary.

Run 2. This run illustrates the importance of developing good approximation schemes for convection dominated flow when the Peclet number P e κRe is large. This points to the need for the development of special numerical methods to solve the forward problem and the Riccati equation. 1 , κ = 1, r = 0.25, but fix the weightWe consider the case α = 0, ν = 10,000 ing function to be q(x) ≡ 1 for all x. Note that Peclet number is given by P e = κRe = 10, 000. Also, we use a distributed control so that [Bu](x) = b(x)u, where b(x) = x10 . This simple problem is sufficient to demonstrate the need for good algorithms. We solve the Riccati equation by using two finite element schemes. Scheme one is the standard Galerkin finite element method

22


800 700 600 500 400 300 200 100 0 0

N=320 Uniform mesh Alpha=0

0.1 Fig. 2.

0.2

N=320 N=320 Uniform mesh Uniform mesh Alpha=0.2 Alpha=0.1

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Functional gains for α = 0, α = 0.1 and α = 0.2.

and scheme two is a stabilized finite element (upwind) scheme. It can be shown that both methods produce strongly convergent functional gains. Moreover, the optimal functional gain is bounded and smooth on the interior of (0, 1). Again we see that the LQR controller produces functional gains with support near the control boundary at x = 1. However, Fig. 3 illustrates that the functional gains computed by standard Galerkin finite elements become oscillatory near the x = 1. These numerical oscillations can be reduced if one uses a stabilized finite element scheme. Also, Figs. 4 and 5 show that both schemes converge, but the upwind scheme produced almost no oscillations and had converged by N = 64. The finite-dimensional model problem in Example 1 with finite blowup time and the controlled Burgers’ equation serve to illustrate the following important issues. 1. Linear feedback can stabilize highly nonlinear systems. Even if the openloop model is not known to be well-posed, it may still be possible to use the linearization to compute a feedback controller. Therefore, one does not need to prove the existence of global solutions to the open-loop

23

Modeling Transition

1.8 N = 64

1.6 1.4 1.2 WITHOUT UPWINDING 1 0.8 0.6 0.4 0.2 0 -0.2 0

0.1

Fig. 3.

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Functional gains when P e = 10, 000 and N = 64 elements.

0.6 N = 128 0.5

0.4 WITHOUT UPWINDING 0.3

0.2

0.1

0

-0.1 0 Fig. 4.

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9


1

24


0.4 N = 256 0.35 0.3

WITHOUT UPWINDING

0.25 0.2 0.15 0.1 0.05 0 -0.05 0 Fig. 5.

2.

3.

4.

5.

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


equations to design a control law. This is important since the 3D Navier– Stokes are not known to be globally well-posed. Linear feedback is sufficient to dramatically alter the closed-loop dynamical system. In addition, different design methodologies generally produce distinct closed-loop systems. Although many methods such as LQR produce feedback laws that depend on knowledge of the entire state, some designs lead to controllers with local support. Therefore, this information helps give practical guidance about where to place sensors and what spatial information one may need to implement the feedback law. This type of information can help build robust low order controllers. In order to take advantage of these observations one must be able to accurately compute the functional gains. This is a difficult problem and requires special numerical methods that address dual convergence and other approximation issues. However, considerable progress has been made in understanding the important issues and computational tools are under development. All “general” methods should be viewed as a first step in the total design process. In order to build a practical controller, one must have

Modeling Transition

25

a mathematical framework to analyze and compare various controllers. The framework above has been helpful in this respect. However, a better understanding of the nonlinear open-loop dynamics is always helpful. For example, if one understands the mechanism that triggers transition, then the control problem should be formulated to yield a closed-loop system that avoids this mechanism. In the next section, we present some numerical results that offer some insight into a possible mechanism for transition. We make use of the lowdimensional models found in Baggett and Trefethen (1997). 3. Low Order Model Problems As noted above, during the past few years, several low-dimensional model problems have been suggested in an attempt to describe specific aspects of transition. We consider 2D and 3D systems that are typical of the those found in the papers Baggett and Trefethen (1997), Henningson (1996), Waleffe et al. (1993) and Waleffe (1995a). However, we focus on the role that small constant uncertainties play in transition and illustrate how feedback can delay or eliminate transition in these cases. Both systems have the form z(t) ˙ = A(R)z(t) + z(t)Sz(t) + Bu(t) + Gε,

(3.1)

1 A0 + R], A0 < 0 is diagonal and S = −S ∗ is skewwhere A(R) = [ R adjoint. The system (3.1) is sensitive to initial data and inputs. The impact of the nonlinear term can be very complex and extremely sensitive to small perturbations. These low order models are constructed to mimic the main features of the Navier–Stokes equations and they are typical of many special flow problems (Baggett and Trefethen 1997; Schmid and Henningson, 2001; Waleffe, 1995). It is interesting to note that similar model problems have been used by the control theory community to test various approaches to robustness, to evaluate approximation schemes and as examples to illustrate ill-conditioning, sensitivity and non-convergence of computational control methods (Burns et al., 1988; Burns, 2003; Burns et al., 2003; Datta, 2004). Although there are several models of this type, Baggett and Trefethen (1997) have shown that all these low-dimensional models have many common features. We will discuss two such models and show how bifurcation analysis under uncertainty can describe a possible route to transition. The idea is to view the disturbance as a perturbation of the conservative nonlinearity.

26


If in addition the linear operator is highly non-normal, then the dynamical system can become extremely sensitive to small disturbances and transition occurs even when the linearized system is stable. In particular, the two dimensional system is defined by −α/R 1 0 −1 A(R) = , S= (3.2) 0 −β/R 1 0 and 0 B= , 1

1 G= . 1

(3.3)

The three-dimensional system is defined by     −α/R 1 0 0 −1 −1/2 A(R) =  0 −β/R 1 , S =  1 0 1/4  (3.4) 0 0 −γ/R 1/2 −1/4 0 and   0  B = 0, 1

  1  G = 1, 1

(3.5)

where all constants are positive. Both models have the property that the linear operator A(R) is stable for all R > 0 and the two-dimensional nonlinear model is also dissipative. In particular, the nonlinear two-dimensional system defined by (3.2) and (3.3) has a compact global attractor. The nonlinear three-dimensional system defined by (3.4) and (3.5) is more complex, but exhibits features very similar to those one finds in plane Couette flows. As noted above, the problem with classical linear analysis is that it fails to predict the correct critical Reynolds number that yields transition. For plane Couette flows, the linearized equations are always stable and theoretically, one should not see transition if the initial flow state is sufficiently close to the plane Couette flow. However, if one views a “small” constant disturbance as a perturbation of the conservative nonlinear term, then standard bifurcation theory under uncertainty yields a transition scenario which matches many flow cases. Understanding this mechanism is crucial to the development of feedback control laws. The following simple models are sufficient to illustrate the basic ideas and to demonstrate how feedback can be useful in the delaying of transition.

27

Modeling Transition

3.1. The two-dimensional model In this case, we set α = 1.2 and β = 1.4. We call the eigenvector zT S = [ 1 0 ]T corresponding to the smallest eigenvalue −α/R the Tollmien– Schlichting initial state because of the similarity to the Tollmien–Schlichting waves in plane Poiseuille flows. We refer to the vector zOB = [ 1 1 ]T as the oblique state. Observe that A(R) is stable for all R > 0. In addition, one can show that this two-dimensional system has a compact global attractor. In Fig. 6, the light lines are the stable manifold and the dark lines are the unstable manifolds for the hyperbolic critical points. The union of the five equilibrium and unstable manifolds is the global attractor. The basin of attraction for the zero equilibrium lies between the stable manifolds. If ε = 0, then the zero (z0 = 0) equilibrium is locally asymptotically stable for all R. However, the radius δ(R) of the largest ball about z0 that lies in the domain of attraction converges to 0 and is approximately given by δ(R) = O(R−2 ). When one adds a small “uncertainty” such as an ε = 0.0001 perturbation to the nonlinear term, there is a subcritical bifurcation near R = 6 as illustrated in Fig. 7. The light lines are the stable manifold and the dark lines are the unstable manifolds for the single hyperbolic critical point. The union of the three equilibrium and the unstable manifolds is the global attractor. In this case, all initial states near z0 = 0 transition.

1 0.8 0.6 0.4

y

0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1

Fig. 6.

-0.8

-0.6

-0.4

-0.2

0 x

0.2

0.4

0.6

0.8

1

Phase portrait without disturbance (α = 1.2, β = 1.4, R = 4, ε = 0).

28


1 0.8 0.6 0.4

y

0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1

Fig. 7.

-0.8

-0.6

-0.4

-0.2

0 x

0.2

0.4

0.6

0.8

1

Phase portrait with disturbance (α = 1.2, β = 1.4, R = 6, ε = 0.0001).

0.1 0.08 0.06

OBLIQUE IC

0.04

y

0.02 0

TS IC

-0.02 -0.04 -0.06 -0.08 -0.1 -0.1

Fig. 8.

-0.08 -0.06 -0.04 -0.02

0 x

0.02

0.04

0.06

0.08

0.1

Phase portrait with disturbance (α = 1.2, β = 1.4, R = 6, ε = 0.0001).

In Fig. 8, we zoom in near zero. The dashed lines are the nullclines and the disturbance of size ε = 0.0001 produces a subcritical bifurcation. There are only three critical points. The light lines are the stable manifold and the dark lines are the unstable manifolds for the single hyperbolic critical

29

Modeling Transition

0.8

CLOSED-LOOP SYSTEM

0.6 0.4

y

0.2 0 TS IC -0.2 -0.4 -0.6 -0.8 -0.6

-0.4

-0.2

0 x

0.2

0.4

0.6

Fig. 9. Phase portrait with disturbance (α = 1.2, β = 1.4, R = 6, ε = 0.0001) with a LQR feedback controller.

point. The union of the three equilibrium and the unstable manifolds is the global attractor. However, all initial states above the stable manifold go into transition to the distant stable equilibrium. However, this simple example illustrates how and why the oblique initial state goes into transition before the Tollmien–Schlichting initial state as observed in Schmid and Henningson (2001). Finally, Fig. 9 shows that if one applies a LQR feedback control to this system, then the closed-loop system looks much like the R = 4 openloop system and hence feedback delays transition. The LQR control was computed with weighting matrices Q = I2 and r = 25. The disturbance of size ε = 0.0001 no longer produces a subcritical bifurcation and there are five critical points. Again, the basin of attraction for the zero equilibrium lies between the stable manifolds and is much greater than the open-loop system with no disturbance. 3.2. The three-dimensional model We present this three-dimensional system to illustrate how one might use feedback in a fully developed chaotic flow. Because we are no longer restricted by dimension, this system is more complex and, for various values of the parameter R > 1, it exhibits periodic, quasi-periodic and chaotic

30


x 10-4

1.2

TOLLMIEN–SCHLICHTING INITIAL DATA

1 0.8 0.6 0.4 0.2 0

0

50

100

150

200

250

300

350

400

450

500

350

400

450

500

OBLIQUE INITIAL DATA

2 1.5 1 0.5 0 0

50

Fig. 10.

100

150

200

250

300

Open-loop energies with initial data of norm z0 = 10−4 .

attractors. For the runs presented below, we set α = 0.5, β = 0.75 and γ = 1.0. Again, the eigenvector zT S = [ 1 0 0 ]T corresponding to the smallest eigenvalue −α/R is called the Tollmien–Schlichting state. The vector zOB = [ 1 1 1 ]T is called the oblique state. If 9.5 < R < 23, then there is a chaotic (local) attractor. The results presented below are based on R = 10 and initial states z¯ satisfying ¯ z = 10−4 . In Fig. 10, we plot the energies of the open-loop responses to the Tollmien–Schlichting and oblique initial states, respectively. The Tollmien–Schlichting initial state returns to zero state, but the oblique initial state goes into transition to the chaotic attractor with a transition time of approximately 50 s. As observed in the two-dimensional example, if one sets ε = 10−6 , then the Tollmien– Schlichting initial state also goes into transition to the chaotic attractor and the transition time increases to approximately 100 s. In order to test the feedback control, we computed a LQR controller and used a “capturing” algorithm that turns on the control only if t > 150 and the trajectory “wanders” into the domain of attraction for the closed-loop system. A version of this method was suggested by Yorke and co-workers in the papers of Shinbrot et al. (1992; 1992a). For the case here, we wait

31

Modeling Transition

OPEN–LOOP and CLOSED–LOOP ENERGIES 2

1.5

1

0.5

0

-0.5

-1 0

50 Fig. 11.

100

150

200

250

300

350

400

450

500

Energies of the open-loop and closed-loop 3D system.

until the flow is fully chaotic (t > 150 for both initial states) and then only turn on the feedback control law when z(t) < 1. The weighting matrices for the LQR problem were Q = I3 and r = 1. Figure 11 shows the openloop and closed-loop energies for the oblique initial state. The capturing feedback control law is turned on at t = 150 and the fully developed flow is stabilized by t = 190 s. This example illustrates the power of feedback to change the global nature of the nonlinear dynamics. 4. Summary and Conclusions The two models considered in the previous section have many of the mathematical features common to flow control problems. Also, all the examples above clearly show that it might be possible to develop a rigorous theoretical framework to explain some transition scenarios as a subcritical “bifurcation under uncertainty”. The linear part of such non-normal systems is extremely important in understanding sensitivity and control design. However, it is the perturbation of the conservative nonlinear term that might explain a transition mechanism. Even a small perturbation to the condition F (z), z = 0

32


AMPLITUDE OF PERTURBATION

HEURISTIC BIFURICATION DIAGRAM

TURBULENT FLOW

1

0 LAMINAR FLOW R 0

2

4

6

8

10

Fig. 12. Heuristic bifurcation diagram for low-dimensional models. The laminar flow is stable for all R > 0 but the stability radius decays to 0 as R → +∞. An initial state must be above the dashed blue line to transition.

can produce bifurcation diagrams such as shown in Figs. 12 and 13. In particular, if ε = 0 then the perturbed nonlinear term becomes Fε (z) = F (z) + Gε, so that Fε (z), z = F (z), z + Gε, z = Gε, z = ε(u + v + w) is no longer conservative and not definite. It is the perturbation of the conservative nonlinear term that provides the transition mechanism. In addition to providing a framework to help with the fundamental understanding of transition, the abstract formulation above can be used to quantify sensitivity and uncertainty. Moreover, distributed parameter control theory combined with numerical analysis provides a basis for developing control laws and computational algorithms. However, much remains to be done in all of these areas.

33

Modeling Transition

AMPLITUDE OF PERTURBATION

BIFURCATION UNDER UNCERTAINTY

TURBULENT FLOW 1

0 LAMINAR FLOW R 0

2

4

6

8

10

Fig. 13. A bifurcation under uncertainty. The small constant disturbance produces a non-conservative nonlinear term which leads to a subcritical bifurcation. The laminar flow state is no longer an equilibrium for R > Rcrit and transition occurs.

Acknowledgment This research was supported in part by the Air Force Office of Scientific Research under grant F49620-03-1-0243 and by the DARPA Special Projects Office. References 1. E. Allen, J. A. Burns, D. S. Gilliam, J. Hill and V. I. Shubov, The impact of finite precision arithmetic and sensitivity on the numerical solution of partial differential equations, J. Math. and Computer Modelling 35 (2002) 1165–1195. 2. J. S. Baggett, T. A. Driscoll and L. N. Trefethen, A mostly linear model of transition to turbulence, Phys. Fluids A7 (1995) 833–838. 3. J. S. Baggett and L. N. Trefethen, Low-dimensional models of subcritical transition to turbulence, Phys. Fluids 9 (1997) 1043–1053. 4. B. Bamieh and M. Dahleh, Disturbance energy amplification in threedimensional channel flows, 1998 Amer. Control Conf. (1998) 4532–4537. 5. B. Bamieh and M. Dahleh, Energy amplification in channel flows with stochastic excitation, Phys. Fluids 13 (2001) 3258–3269.

34


6. H. T. Banks and J. A. Burns, Hereditary control problems: numerical methods based on averaging approximations, SIAM J. Control Optim. 16 (1978) 169–208. 7. H. T. Banks and K. Kunisch, An approximation theory for nonlinear partial differential equations with applications to identication and control, SIAM J. Control Optim. 20 (1982) 815–849. 8. H. T. Banks and K. Ito, Approximations in LQR problems for infinitedimensional systems with unbounded input operators, J. Math. Sys., Estimation and Control 7 (1997) 1–34. 9. V. Barbu and R. Triggiani, Internal stabilization of Navier–Stokes equations with finite-dimensional controllers, Indiana Univ. Math. J. 7 (2004) 1443– 1494. 10. V. Barbu, I. Lasiecka and R. Triggiani, Boundary stabilization of Navier– Stokes equations, to appear in Memoires of the AMS. 11. A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems: Volume I (Birkh¨ auser, Boston, 1992). 12. A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems: Volume II (Birkh¨ auser, Boston, 1992a). 13. T. R. Bewley, Flow control: new challenges for a new renaissance, Progress in Aerospace Sciences 37 (2001) 21–58. 14. J. A. Burns, K. Ito and G. Propst, On non-convergence of adjoint semigroups for control systems with delays, SIAM J. Control Optim. 26 (1988) 1442– 1454. 15. J. A. Burns and S. Kang, A control problem for Burgers’ equation with unbounded control and observation, Control and Estimation of Distributed Parameter Systems, eds. F. Kappel, K. Kunisch and W. Schappacher (Birkh¨ auser Verlag 100, p. 51–72). 16. J. A. Burns and B. B. King, Optimal sensor location for robust control of distributed parameter systems, Proc. 33rd IEEE Conf. on Decision and Control (1994) 3967–3972. 17. J. A. Burns, B. B. King and Y. R. Ou, A computational approach to sensor/actuator location for feedback control of fluid flow systems, Sensing, Actuation, and Control in Aeropropulsion, ed. J. D. Paduano, SPIE Proceedings Series 2494 (1995), pp. 60–69. 18. J. A. Burns, B. B. King and D. Rubio, Feedback control of a thermal fluid using state estimation, Int. J. Comput. Fluid Dyn. (1998) 1–20. 19. J. A. Burns, B. B. King and L. Zietsman, Functional gain computations for a 1D parabolic equation using non-uniform meshes, Proc. 2002 MTNS, Paper 23323–5 (2002). 20. J. A. Burns, B. B. King and L. Zietsman, On the computation of singular functional gains for linear quadratic optimal control, 2002 AIAA Flow Control Conference, St. Louis, MO, (2002a) Paper AIAA 2002–3074. 21. J. A. Burns, Nonlinear distributed parameter control systems with nonnormal linearizations: applications and approximations, Research Directions

Modeling Transition

22.

23. 24. 25. 26. 27.

28.

29. 30. 31. 32. 33. 34. 35. 36. 37.

38. 39.

40.

35

in Distributed Parameter Systems, eds. R. C. Smith and M. A. Demetriou, (SIAM Publications, Philadelphia, 2003), pp. 17–53. J. A. Burns, B. B. King, D. Krueger and L. Zietsman, Computation of feedback operators for distributed parameter systems with non-normal linearizations, Proc. 2003 Amer. Control Conf., Paper TA04–2. J. A. Burns and J. Singler, Feedback control of low dimensional models of transition to turbulence, submitted (2005). K. M. Butler and B. F. Farrell, Three dimensional optimal perturbations in viscous shear flow, Phys. Fluids A4 (1992) 1637–1650. H. Choi, P. Moin and J. Kim, Active turbulence control for drag reduction in wall-bounded flows, J. Fluid Mech. 262 (1994) 75–110. L. Cortelezzi and J. L. Speyer, Robust reduced-order controller of laminar boundary layer transitions, Phys. Rev. E58 (1998) 1906–1910. L. Cortelezzi, K. H. Lee and J. L. Speyer, Skin-friction drag reduction via robust reduced-order linear feedback control, Int. J. Comput. Fluid Dyn. 11 (2001) 79–92. H. F. Creveling, J. F. De Paz, J. Y. Baladi and R. J. Schoenhals, Stability characteristics of a single-phase free convection loop, J. Fluid Mech. 67 (1975) 65–84. R. F. Curtain and H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems (Springer-Verlag, New York, 1995). B. N. Datta, Numerical Methods for Linear Control Systems (Elsevier Academic Press, Amsterdam, 2004). P. G. Drazin, Introduction to Hydrodynamic Stability Theory (Cambridge University Press, Cambridge, 2002). B. F. Farrell and P. Ioannou, Stochastic forcing of the linearized Navier– Stokes equation, Phys. Fluids A5 (1993) 2600–2609. A. V. Fursikov, Stabilization for the 3D Navier–Stokes system by feedback boundary control, Discrete Contin. Dyn. Sys. 10 (2004) 289–314. M. Gad-el-Hak, Modern developments in flow control, Appl. Mech. Rev. 49 (1996) 365–379. M. Gad-el-Hak, Flow Control: Passive, Active, and Reactive Flow Management (Cambridge University Press, 2000). M. Garbey and H. Kaper, Asymptotic-numerical study of supersensitivity for generalized Burgers’ equation, SIAM J. Sci. Comput. 22 (2000) 368–385. J. M. Ghidaglia, Etude d’´ ecoulements de fluides visquex incompressibles: Comportement pour les grands temp et applications aux attracteurs (Thèse de 3e Cycle, Universite Paris Sud, Orsay, 1984). M. D. Gunzburger, Flow Control: IMA Volume 68 (Springer-Verlag, New York, 1995). M. D. Gunzburger, A. Fursikov and L. Hou, Optimal boundary control of the Navier–Stokes equations with bounds on the control, Proc. Korean Adv. Inst. Sci. and Tech. Math. Workshop on Finite Elements (1999) 41–60. D. S. Henningson, Stability of parallel inviscid shear flow with mean spanwise variation, FFA-TN 1987-57 (Aeronautical Research Institute of Sweden, Bromma, 1987).

36


41. D. S. Henningson, A. J. Lundbladh and A. V. Johansson, A mechanism for bypass transition from localized disturbances in wall bounded shear flows, J. Fluid Mech. 250 (1993) 169–207. 42. D. S. Henningson and S. C. Reddy, On the role of linear mechanisms in transition to turbulence, Phys. Fluids A6 (1994) 1396–1398. 43. D. S. Henningson, Comments on transitions in shear flows. Nonlinear normality versus non-normal linearity, Phys. Fluids A8 (1996) 2257–2258. 44. S. S. Joshi, J. L. Speyer and J. Kim, A systems theory approach to the feedback stabilization of infinitesimal and finite-amplitude disturbances in plane Poiseuille flow, J. Fluid Mech. 332 (1997) 157–184. 45. S. Kang and K. Ito, A dissipative feedback control synthesis for systems arising in fluid dynamics, SIAM J. Control Optim. 32 (1994) 831–854. 46. T. Kato, Perturbation Theory for Linear Operators (Springer-Verlag, New York, 1976). 47. J. G. LaForgue and R. E. O’Malley, Supersensitive boundary value problems, in Asymptotic and Numerical Methods for Partial Differential Equations with Critical Parameters, eds. H. G. Kaper and R. E. O’Mally (Kluwer Academic Press, Dordrecht, The Netherlands, 1993), pp. 215–223. ´ 48. J. L. Lions, Controle Optimal des Systèmes Gouvernés par des Equations aux D´ erivées Partielles (Dunod, Paris, 1969) (English translation SpringerVerlag, New York, 1971). 49. L. Rayleigh, On the stability of certain fluid motions, Proc. Math. Soc. Lond. 11 (1880) 57–70. 50. S. C. Reddy and D. S. Henningson, Energy growth in viscous channel flows, J. Fluid Mech. 252 (1993) 209–238. 51. S. C. Reddy, P. J. Schmid and D. S. Henningson, Pseudospectra of the Orr– Sommerfeld operator, SIAM J. Appl. Math. 53 (1993) 15–47. 52. S. C. Reddy and L. N. Trefethen, Pseudospectra of the convection-diffussion operator, SIAM J. Appl. Math. 54 (1994) 1634–1649. 53. P. J. Schmid and D. S. Henningson, Optimal energy density growth in Hagen– Poiseuille flows, J. Fluid Mech. 277 (1994) 197–225. 54. P. J. Schmid, S. C. Reddy and D. S. Henningson, Transition thresholds in boundary layer and channel flows, Advances in Turbulence VI, eds. S. Gavrilakis, L. Machiels and P. A. Monkewitz (Kluwar Academic Publishers, 1996), pp. 381–384. 55. P. J. Schmid, Linear stability theory and bypass transition in shear flows, Phys. Plasmas 7 (2000) 1788–1794. 56. P. J. Schmid and D. S. Henningson, Stability and Transition in Shear Flows (Springer-Verlag, New York, 2001). 57. T. Shinbrot, C. Grebogi, E. Ott and J. A. Yorke, Using chaos to target stationary states of flows, Phys. Lett. A169 (1992) 349–354. 58. T. Shinbrot, W. Ditto, C. Grebogi, E. Ott, M. Spano and J. A. Yorke, Using the sensitive dependence of chaos (the butterfly effect) to direct orbits to targets in an experimental chaotic system, Phys. Rev. Lett. 68 (1992a) 2863– 2866.

Modeling Transition

37

59. S. S. Sritharan, Invariant Manifold Theory for Hydrodynamic Transition (Wiley, New York, 1990). 60. S. S. Sritharan, Optimal feedback control of hydrodynamics: A progress report, Flow Control, ed. M. D. Gunzburger (Springer-Verlag, New York, 1995), pp. 257–274. 61. S. S. Sritharan, Optimal Control of Viscous Flows (SIAM Publications, Philadelphia, 1998). 62. R. Temam, Navier–Stokes Equations (Elsevier Science Publishers, New York, 1984). 63. R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics (Springer-Verlag, New York, 1998). 64. L. N. Trefethen, A. E. Trefethen, S. C. Reddy and T. A. Driscoll, Hydrodynamic stability without eigenvalues, Science 261 (1993) 578–584. 65. L. N. Trefethen, Pseudospectra of linear operators, SIAM Review 39 (1997) 383–406. 66. F. Waleffe, J. Kim and J. Hamilton, On the origin of streaks in turbulent flows, in Turbulent Shear Flows 8: Selected Papers from the Eighth International Symposium on Turbulent Shear Flows, eds. F. Durst, R. Friedrich, B. E. Launder, F. W. Schmidt, U. Schumann and J. H. Whitelaw (SpringerVerlag, Berlin, 1993), pp. 37–49. 67. F. Waleffe, Hydrodynamic stability and turbulence: Beyond transients to a self-substaining process, Stud. Appl. Math. 95 (1995) 319–343. 68. F. Waleffe, Transitions in shear flows, Nonlinear normality versus Non-normal linearity, Phys. Fluids 9 (1995a) 1043–1053. 69. Y. Wang, J. Singer and H. Bau, Controlling chaos in a thermal convection loop, J. Fluid Mech. 237 (1992) 479–498. 70. J. M. Wiltse and A. Glezer, Manipulation of free shear flows using piezoelectric actuators, J. Fluid Mech. 249 (1993) 261–285.


DYNAMICS OF TRANSITIONAL BOUNDARY LAYERS

Cunbiao Lee∗,† and Shiyi Chen∗,‡ ∗

State Key Laboratory for Turbulence Research and Complex Systems Peking University, Beijing 100871, China † E-mail: [email protected] ‡

Department of Mechanical Engineering Johns Hopkins University, Baltimore, MD 21218, USA E-mail: [email protected] This chapter presents direct comparisons of related visualizations and hot film measurement studies of the nonlinear, late stages of transition in a boundary layer. The earlier nonlinear stages of the transition process are well-documented in previous studies. The visualizations and measurements were both performed with controlled disturbance conditions excited by an instability wave in a flat-plate boundary layer. The threedimensional wave packets called soliton-like coherent structures (SCS), the Λ-vortex, the secondary closed vortex and the chain of ring vortices are postulated to be the basic flow structures of the transitional boundary layer. New mechanisms for their formation are also given. Despite somewhat different initial disturbance conditions used in these and previous experiments, the flow structures were found to be practically the same. The experimental results show new dynamic processes and new flow structures, i.e. the secondary closed vortex, in a transitional boundary layer. Contents 1 2 3 4

Introduction Experimental Methods Experimental Results Formation of the SCS and the Λ-vortex 4.1 Main features of the SCS 4.2 Physical mechanism for the SCS formation 4.3 SCS and turbulent spots 39

40 43 45 49 49 50 54

40

C. Lee and S. Chen

5

Secondary Closed Vortex

55

6

Formation of the Chain of Ring Vortices

58

7

Breakdown of the Chain of Ring Vortices

64

8

Formation of the Streamwise Vortices

67

9

Breakdown of the SCS

10 Discussion

70 72

10.1 Description of turbulent bursting in a transitional boundary layer

72

10.2 Long streak breakdown mechanism

73

11 Conclusion

79

References

80

1. Introduction In 1883, Osborne Reynolds published the outcome of his painstaking flow visualizations at Manchester in the Philosophical Transaction of Royal Society. These showed that the flow transition in a pipe from direct to sinuous (nowadays we would say laminar to turbulent) depended on the Reynolds number. Transition from laminar to turbulent flow is still an important problem in fluid mechanics which has attracted the interest of investigators for more than 100 years. The particular case of boundary-layer flow has received the most attention and has been more successfully treated than any other flows. However, despite the success of linearized theories in revealing the nature of the initial stages of boundary-layer instabilities, there remains a deep void in the understanding of the subsequent nonlinear behavior and the actual breakdown of the laminar boundary layer. The present state of affairs is such that one must depend on experiments to bridge the gap. The classic investigations into the mechanics of transition called the K-regime were undertaken at the U.S. National Bureau of Standards by Schubauer (1957) and, Klebanoff and Tidstrom (1957, 1962). The work of Schubauer and Klebanoff (1956) showed transition to be a process involving the formation of turbulent spots, as had been postulated earlier by Emmons (1951). Experimental investigations by Hama and Long (1957), Kovasznay, Komoda and Vasudeva (1962), and Hama and Nautant (1963) utilizing dye techniques in water demonstrated the occurrence of characteristic three-dimensional dye configurations before transition occurs. The three main stages were identified both experimentally and theoretically as (a) receptivity, (b) linear stability, and (c) nonlinear breakdown.

Dynamics of Transitional Boundary Layers

41

The receptivity problem was clearly formulated for the first time by Morkovin (1969). The idea that the Reynolds number of the pipe-flow transition has to increase when the amplitudes of the disturbances in the incoming flow are attenuated had been suggested by Reynolds more than a century ago and corroborated later in 1905. Loehrke (1975) reviewed the importance of the receptivity problem for understanding transition. The effect of a model vibration in an acoustically excited boundary layer was used experimentally to illustrate the receptivity by Kachanov et al. (1974a, 1974b). The recent review by Saric (2002) gives more detailed information about the receptivity. Theoretical studies of the receptivity problem focused on investigating the boundary-layer receptivity to free stream vortices (Rogler, 1977; Rogler and Reshotko, 1975) and to acoustic waves (Manger, 1977). Wu (1996) proposed a new mechanism for the generation of Tollmien–Schlichting waves by freestream turbulence. The mechanism is described using triple-deck formalism to preserve the definiteness and self-consistency. The freestream turbulence is represented by converting gusts consisting of the so-called vortical and entropy waves of small amplitude. Wu et al. showed that suitable converting gusts can interact with sound waves in the freestream to produce a forcing that has the same time and length scales as those of the T–S waves. Wu and Lu (2001) and, Wu and Choudhari (2001) also showed that the instability of the incompressible Blasius boundary layer can be significantly modified, and even fundamentally altered, by certain small-amplitude distortions which feature low-speed streaks. The instability of the perturbed flow was shown to be governed by a remarkably simple system described by a Schrödinger-like equation with a purely imaginary potential. The boundary-layer instabilities with three-dimensional disturbances were experimentally investigated in the 1960s by Vasudeva (1967) who used a localized source of instability waves. Later, Gaster and Grant (1975) obtained extensive information about the development of three-dimensional packets of instability waves in a Blasius boundary layer. Kachanov (1994) made a direct quantitative comparison of theoretical and experimental data for the dispersion and stability characteristics of three-dimensional instability waves propagating in a flat plate boundary layer. Direct numerical simulation (DNS) results quantitatively modeled the K-regime of transition experiments using a DNS scheme based on the spatial simulation model by Bake et al. (2002) and Borodulin et al. (2002).

42

C. Lee and S. Chen

Although the region of nonlinear breakdown has been studied for more than forty years, many aspects remain a mystery. Several new structures have been found in recent years by Kachanov (1994), Bake et al. (2002) and Borodulin et al. (2002), Rist and Fasel (1995), and Lee (2000, 2001a, 2001b) with several new scenarios to describe transition proposed by Kachanov (1994), Reddy et al. (1998), Thefethen et al. (1993), Waleffe (1995, 1998) and Lee (2000, 2001b). The structural similarity between transitional and developed boundary layers was first briefly discussed by Blackwelder (1983). Kachanov (1994) discussed the connection between the K-breakdown and the developed turbulence. The flow structures in developed turbulence described by Fukunishi, Sato and Inous (1987) and, Thomas and Saric (1981) were very similar to those described by Borodulin and Kachanov (1994). The seminal experimental studies of Kline et al. (1967) on the turbulent boundarylayer structure inspired much experimental work (as well as direct numerical simulations of turbulence), some of which are discussed by Walker et al. (1989) and Robinson (1990, 1991a, 1991b). The general character of the readily observable features of boundary-layer flows is well established, although an understanding of the cause-and-effect relationships has proven elusive. There are two main aspects that dominate the near-wall flow, namely the “low speed streaks” and the “bursting” phenomenon. Along a given area of the wall, streaks may be readily observed during most of any observation period when a visualization medium, such as dye or hydrogen bubble, is introduced into the flow near the surface. The streaks delineate regions where the cross-stream motion converges and the streamwise velocity is in deficit relative to the local mean velocity, with high-speed flow where the streamwise velocity exceeds the local mean sandwiched between the low-speed regions (Lian 1990). Falco (1977, 1991) found other bursting phenomena called outer layer bursting with the formation of “typical eddies”. Falco (1991) and Smith et al. (1991) suggested the existence of several secondary hairpin vortices. They each established structural models based on these observations. Several similar structures have been found in a transitional boundary layer. These flows are compared in Table 1. Combined visual and quantitative techniques have further revealed a number of significant features of transitional boundary layers. The physical processes for the formation of the SCS, the Λ-vortex, the secondary closed vortex, the streamwise vortex and the chain of ring vortices are described briefly in this paper. A new model similar to that suggested by Smith et al.


43

Table 1. Comparison of flow structures in transitional and developed turbulent boundary layers. Turbulent boundary layer

Transitional boundary layer

Hairpin vortex (Theodorsen, 1952) Secondary hairpin vortex (Falco, 1991, Smith et al., 1991) Long streak or low-speed streak (Kline et al., 1967) Typical eddies (Falco, 1977; Adrian, 2000) Streamwise vortex

First closed vortex (Λ-vortex) Secondary closed vortex Long streak (SCS) Chain of ring vortices Streamwise vortex

(1991), Falco (1991) and Robinson (1990, 1991a, 1991b) is given based on these observations instead of the hairpin vortex mechanism.

2. Experimental Methods The experiments were performed in an open surface recirculating water channel, shown schematically in Fig. 1. The low-turbulence level water channel in the State Key Laboratory for Turbulence Research and Complex Systems (LTCS) in Peking University had a freestream velocity U∞ = 20 cm/s with a turbulence level of around 0.1%. The cross-section was 600 mm × 400 mm, and the test section was about 6000 mm long. A flat plate with a chord length of 1.8 m, a span of 0.8 m and a thickness of 15 mm was mounted vertically. Part of the flat plate was above the open water surface. The leading edge had two 90◦ arcs with different radii. The plate was mounted in the test section at zero angle of attack. The streamwise and spanwise pressure gradients were nearly zero far from the leading edge. A downstream flap was used to make the flow more uniform. The disturbance generator (T–S wave generator) was a spanwise slit in the plate of length 150 mm and width 1 mm on the working side mounted at a distance x = 200 mm from the leading edge of the plate. Water was periodically pumped in and out of the slit at a frequency of 2 Hz. A water tank was connected to both the slit and two tubes on opposite sides of the plate. A loudspeaker was set on top of a round barrel with the two tubes mounted on the outer edge of the barrel’s bottom. The instability waves had a frequency of 2 Hz and amplitude of 1.8% of the freestream velocity, U0 , as set by the voltage input to the loudspeaker. The development of the disturbances in the boundary layer and the structure of the mean flow were investigated with a hot wire anemometer made by Kanomax Company. The

44

C. Lee and S. Chen

Fig. 1.

Experimental set-up.

hot films were made by TSI. The sensitive part of the probe was less than 2 mm long. Experimental data was acquired from x = 250 mm to 700 mm. At each measured point, three characteristics were measured: the mean value of the streamwise velocity, U0 , and the amplitude and phase of the streamwise disturbance velocity, u, filtered at the fundamental frequency. The distributions of these characteristics were measured along the streamwise direction (x), normal to the plate (y) and along the spanwise direction (z). Along with these measurements, an improved hydrogen bubble technique was used to carefully visualize the flow structures. Complete visualization of the flow structures was accomplished by placing the hydrogen bubble wire at positions from x = 250 mm to 700 mm in steps of 50 mm and from y = 0.25 mm to 6 mm in steps of 0.25 mm. This technique made it possible to clearly visualize the spatial flow structures. As shown in Fig. 2, different sections in the plan view were obtained by placing the electrode wire at different y-positions. If the flow was laminar, several hydrogen bubble planes were obtained (Fig. 2). Continuous plan views of the hydrogen bubbles were produced to visualize the flow structures. The water temperature during the experiments was about 20◦ C. Constant water temperature was obtained by starting the water channel more than 9 hours before each test. The kinematic viscosity was 1.01 × 10−6 m2 /s and the Reynolds number based on the length was 2.0 × 105 .

45


Fig. 2. Table 2.

Visualization techniques.

Operating conditions and flow parameters changes.

Free stream velocity Tollmien–Schlichting wave amplitude T–S wave frequency Formation of the SCS Breakdown of Λ-vortex and formation of chain of ring vortices Breakdown of the long streak

Present

Previous (Lee, 2000)

20 cm/s 1.8% U0 2 Hz x = 245 mm x = 390 mm

17 cm/s 1.6% U0 2 Hz x = 267 mm x = 420 mm

x = 600 mm

x = 675 mm

Some changes made from our previous experiments (Lee, 2000) are as listed in Table 2. 3. Experimental Results Figure 3 shows the spanwise distributions of the turbulent intensity at different y-positions from the surface and at 250 mm, 300 mm, 350 mm and 400 mm downstream from the leading edge. The distributions shown were obtained with fixed source amplitude. The intensity of U decreased with increasing spanwise distance z from the “peak position” (Bake et al., 2002; Borodulin et al., 2002) and became very small for z > 10 mm. The spanwise wavelength was about 28 mm. Attention was focused on to the region from z = −14 mm to 14 mm. Figure 4 shows the growth in the wave intensity. The measurements were made along a line corresponding to the “peak position” (z = 0). The intensity is plotted relative to U∞ . The peak and the valley maintained a fixed spanwise position as they intensified in the downstream direction. In contrast to the intensity distribution in the valley, the intensity in the peak

46

C. Lee and S. Chen

Fig. 3. Distribution of the intensity of the u fluctuations across the boundary layer for U∞ = 20 cm/s. (a) x = 250 mm; (b) x = 300 mm; (c) x = 350 mm; (d) x = 400 mm.

increased very rapidly from a minimum value of about 4% at x = 250 mm to a maximum value of about 16% at x = 550 mm. The position of the maximum intensity distributions agrees well with the previous results by Klebanoff (1962). Figure 5 shows the T–S wave (TS↑) and its evolution visualized using the hydrogen bubble visualization technique. In these pictures, the wire


Fig. 4.

47

Intensity of u fluctuations in the peak for larger wave amplitude.

Fig. 5. Plan-view of the T–S wave (TS↑) and a wavy structure at the two peak positions (PP↑). Two turbulent production positions were generated in this experiment.

was located parallel to the plate and normal to the direction of the flow with the flow from left to right in the pictures. The wire position was moved ∆y = 0.25 mm for each successive picture. The flow is both threedimensional and unsteady at the peak position (PP↑). The collection of bubbles at the peak position due to fluctuations of both the streamwise velocity components and a spanwise velocity component is the noticeable feature of this region. The streaks waver and oscillate much like a spring carried by a moving train with constant speed. Figure 6 shows a typical T–S wave and its evolution. At the peak position, additional flow structures are observed which was previously called SCS (CSS↑) (Lee 1998, 2000, 2001a, 2001b). Figures 6(a) and 6(b) show the formation of a typical Λ-vortex.

48

Fig. 6.

C. Lee and S. Chen

T–S wave and its growth. The wire was placed at x = 250 mm and y = 0.75 mm.

Fig. 7. Oscilloscope traces of velocity disturbances from the wall at x = 250 mm and z = 1 mm.

Two peak positions occurred in this figure. Since these two “peak positions” seemed to be very similar, attention was mainly focused on the upper “peak position”. A typical set of oscilloscope traces measured at various distances from the wall is shown in Fig. 7. The traces show that additional kinks (indicated by the arrows) in the time traces are periodic and that the phases of the additional kinks (arrows A) at different y-positions are equal.


49

4. Formation of the SCS and the Λ-vortex 4.1. Main features of the SCS Figure 6 clearly shows the SCS formation process (Lee 1998, 2000) and the so-called Λ-vortex. The SCS forms before the formation of the Λ-vortex. A widely accepted mechanism for the formation of the Λ-vortex was summarized by Hinze (1975) before detailed visualizations such as in Fig. 6. The corresponding measured results are shown in Fig. 7. A new mechanism has been suggested based on new experimental observations by Lee (2000). Figure 8 shows the formation of the SCS and the Λ-vortex. The successive pictures show that a kink-like structure appears first which is the initial form of the SCS (CSS↑) and which eventually becomes a rhombuslike structure (B↑).

Fig. 8. Visualization of the formation of the SCS and the well-known Λ-vortex. In these pictures, the wire was located parallel to the plate and normal to the flow direction with the flow from left to right in the pictures. The wire was positioned at y = 0.75 mm. The time interval between successive pictures was 1/12 s. (a) to (f) show the SCS (CSS↑). The formation of the Λ-vortex is clearly visualized in (e) to (h) (Λ↑).

50

C. Lee and S. Chen

Fig. 9.

Amplitude distributions of the SCS in the x-direction.

Figure 9 shows the amplitude variation of the SCS (Lee, 2000) at three y-positions. The measurements were made along a line corresponding to a peak position in the near-wall region. The low-amplitude wave which was initially amplified and then damped obeys linear theory. At larger amplitudes, the wave at the peak first grows as predicted by linear theory, but then exhibits the different characteristic behavior associated with the region of finite amplitude, namely very rapid growth at the peak with initial growth in the valley that is less that predicted by linear theory. Positions A, B, C and D in the figure are different downstream positions at which detailed observations were made, with position A corresponding to the position of the departure from linear theory, position B to the end of the nonlinear interaction period and position D to the breakdown of the SCS (Lee 2000). The region from departure to the breakdown of the SCS and the region from breakdown to the so-called fully developed turbulent flow are of principal interest. The measured instantaneous velocity profiles during one period, 0.5 s, are shown in Fig. 10. With some imagination, the occurrence of the SCS and the Λ-vortex in the U oscillograph traces, can be deduced from the instantaneous velocity profiles. Note that at locations where the U -component has a kink in its distribution, the U -component becomes appreciable. At z = −2 mm, the kinks are small. The experimental technique used to draw the time dependent profiles is similar to that used by Kachanov (1994).

4.2. Physical mechanism for the SCS formation Receptivity is defined as the mechanism by which disturbances enter the boundary layer and create the initial conditions for unstable waves.


51

Fig. 10. Low-frequency instantaneous velocity profiles observed at the “peak position” (z = 0) and other positions at an early stage (x = 250 mm). (a) z = 0; (b) z = −2 mm.

However, only certain kinds of unstable waves can generate the rhombuslike SCS seen in plan view. The wave resonant (WR) concept was suggested (Kachanov 1994) on the basis of a detailed analysis of experiments by Kachanov and Levchenko (1984), as well as the theoretical results by Craik (1971, 1982). The results obtained by Borodulin and Kachanov (1994) showed that the system of parametric subharmonic resonances, postulated within the framework of the WR concept, was actually observed in the K-regime of breakdown at the stage of spike formation. Simulations of oblique wave interactions in a Blasius boundary layer performed by Joslin et al. (1993) showed that the oblique wave interaction generates a strong spanwise-dependent mean-flow distortion. No two-dimensional (2D) T–S waves take part in this process but the SCS can generate this kind of strong spanwise-dependent meanflow distortion. One possible physical reason for the SCS is that they are generated by the interaction between two oblique waves. The effect of twodimensional T–S waves on the formation of the SCS was investigated using a theoretical approach called the Phase-locked method which showed that the

52

C. Lee and S. Chen

two-dimensional T–S wave has a catalytic effect on the three-dimensional structure formation. Recently, a self-consistent asymptotic theory developed by Wu et al. (2001a, 2001b) showed that the instability of the perturbed flow is governed by a remarkably simple system represented by the Schr¨ odinger wave equation. The instability modes can be viewed as a kind of modified oblique T–S wave, because these modified T–S waves are governed by the Schr¨ odinger wave equation, the solution is a soliton-like structure. In fact, the oblique waves may exist anywhere because they are induced by any three-dimensional disturbance. Figure 11 shows the oscilloscope traces of the velocity disturbances at various distances from the wall for different z at the same x (x = 300 mm). An additional flow structure was generated periodically as clearly shown in Figs. 11(a) and 11(b). Note the velocity fluctuations at y = 1.22 mm and 1.42 mm. The SCS evolving from a basic wave was measured at x = 250 mm, with a new structure which lags behind the basic wave as shown in Fig. 7. The same new structure was most apparent at x = 300 mm, z = 4 mm, from y = 1.02 mm to y = 2.22 mm. The structure was created in the valleys of the SCS in the near-wall region, with the amplitudes of the additional kinks varying from a minimum at y = 1.02 mm to a maximum at y = 2.82 mm, while the basic wave varies from a maximum at y = 1.02 mm to a minimum at y = 2.82 mm. This detail has never been observed before. At x = 300 mm, the visualization results in Figs. 6 and 8 show that two kinds of flow structures, SCS (arrows B) and Λ-vortices, exist. The additional kinks (arrows A) were generated by the Λ-vortices. The occurrence of a packet vortex in a turbulent boundary layer was suggested by Falco (1977, 1991) and Smith et al. (1991). A packet can form as the result of the interaction of a typical eddy passing along the wall and the sublayer, which results in the lift-up and formation of the hairpin vortices and by the induction of fluid that is initially away from the wall towards the wall by strong streamwise vortices. Moin et al. (1986) showed that a sufficient concentration of markers covering an area on the wall is needed to detect a packet vortex, and that hydrogen bubble visualization does not provide enough markers, in general, to observe them. The packet vortex becomes highly stretched because it induces itself to remain within the boundary layer. Furthermore, because it is so close to the wall the impermeability condition is important which adds to the stretching. Markers will build up on the outer sides of the packet vortex, where fluid is moving away from the wall forming a pair of streaks along the sides of the packet. The


53

Fig. 11. Oscilloscope traces of velocity disturbances at various distances from the wall. (a) x = 300 mm, z = 1 mm; (b) x = 300 mm, z = 4 mm.

mechanism for this lift-up of markers was well described by Doligalski and Walker (1984). The elongation of the packet vortex, from the moment of its formation, continually reforms the upstream and lateral boundaries of the packet. This was clearly observed in the high-speed movies of Falco (1991). Smith et al. (1991) also noted that the upstream kinks of bent vortex tubes would have the “most active” viscous response. A further consequence of this stretching is that the packet vortex is rapidly dissipated leaving behind a pair of streaks inside the long streak pair.

54

C. Lee and S. Chen

4.3. SCS and turbulent spots A turbulent spot was first found by Emmons (1951). A nice picture of a growing turbulent spot in plan view was obtained by Elder (1960). Note that the flow conditions upstream of the turbulent spot, after the spot passes, are again laminar. Earlier, other investigators observed the existence of turbulence spots during the transition process (Schubauer et al. 1956; Mitchner 1954). Landahl (1977) analyzed the effects of initial “lift-ups” i.e. localized three-dimensional up-down movements in the high ∂u/∂y regions of a laminar boundary layer. Inviscid linear theory indicates algebraic growth due to the straining action. If in real flows, the growing disturbance exceeds a nonlinear threshold before its viscous decay and it could open up a relatively low-level bypass to a small turbulent spot. Numerical inviscid analyzes of the evolution of disturbances in a Blasius layer, experiments on weak (nonbypass) and stronger (bypass) “lift-ups” and Navier–Stokes simulations of the stronger lift-ups (besides the Gaster–Grant 1975 Laminar T–S spots) have shown rather similar patterns of intensifying shear over the center of the disturbance and elongating flow structures on the sides. For stronger disturbances, the breakdown seems to be associated with long strips of high-speed fluid surrounding a low-speed region and with rapid inflectional instabilities of the distorted mean profile. Kachanov (1994) suggested a new transition called “without turbulent spot transition”. He believed that the only condition that must be satisfied to generate turbulent spots is that the initial instability wave must have sufficiently strong temporal modulation in its initial amplitude. The socalled microscopic have also been suggested for the onset of turbulent spots. Lee (2000) discussed the differences between SCS and the turbulent spots. The main conclusions are listed in Table 3. Previous studies must be carefully analyzed to distinguish between a real SCS and a turbulent spot. As described by Theodorsen (1952), turbulent bursts may be identified with turbulent spots, i.e. the appearance of turbulent spots is associated with localized turbulence. In fact, only one or two turbulent production

Table 3.

Differences between a SCS and a real turbulent spot. SCS

Location Structure Scales

appears at early stage simple wave packet single scale

Turbulent spot appears further downstream several SCS and their bounded vortices multiple scales


55

channels were generated under controlled conditions such as those used by Kachanov (1994). The basic differences between the transition with or without turbulent spot have not yet been found. It is possible that the SCS can not be distinguished from turbulent spots by using only hot wire data as in Kachanov (1994).

5. Secondary Closed Vortex Figure 12 shows a new closed structure (SCV↑) separated from the SCS (arrows B) that has never been reported in transitional boundary layers.

Fig. 12. Secondary closed-vortex (SCV↑) formation. From (a) to (d), part of the first closed vortex, i.e. the Λ-vortex and a SCS appear (CSS↑). From (e) to (h), the righthand side of the secondary closed vortex appears and is then separated from the SCS. At the same time, the Λ-vortex is stretched (FCV↑). The time interval between successive pictures is 1/12 s. The hydrogen bubble wire was located parallel to the plate and normal to the flow direction. The wire position was x = 300 mm from the leading edge and y = 1.25 mm from the wall.

56

C. Lee and S. Chen

Falco (1991) and Smith et al. (1991) observed the so-called secondary closed hairpin vortex in a turbulent boundary layer with adverse pressure gradient. As sketched by Smith (1991), a new mechanism called surface layer separation can be provoked by a symmetric hairpin vortex at three locations, behind the head and immediately inboard of each leg. Figure 13 shows the flow structures from the near-wall region to the outer layer including the secondary closed votex. The secondary closed vortex (SV↑) has not been reported in a transitional boundary layer before. The wire was moved ∆y = 0.25 mm for each step shown in the picture series. Figure 13(a) shows a long streak (L↑) and its evolution at different times. The rotation of this long streak has not been observed indicating that the long streak behavior differs from that of the streamwise vortices. However, two streamwise vortices exist on the two sides of the long streak. The long streak (L↑) is also present in Fig. 13(b). The structure is modulated in time with the fundamental frequency. The long streak in a transitional boundary layer differs from that in a developed turbulent boundary layer in several aspects. First, the long streaks in a transitional boundary layer appear in the near-wall region in the “peak positions”. In contrast, the long streaks called streamwise vortices in a developed turbulent boundary layer appear both at the interface between the high-speed streaks and the low-speed streaks and in the near-wall region. Secondly, the long streaks in a transitional boundary layer represent the features of the SCS (CSS↑). A complete long streak is composed of several SCS as clearly shown in Fig. 13(b). In Fig. 13(c), the two middle parts of the Λ-vortex on the two sides of the long streak rotate at a much higher speed. Therefore, two hornlike vortex tubes (VT↑ and VT↓) are formed on the two sides of the long streak, which causes interaction between the secondary closed vortex and the Λ-vortex. The left-hand side of the secondary closed vortex can be seen in Fig. 13(c) (SV↑). The two vortex tubes are formed on the two sides of the “peak position” in Fig. 13(c). They then develop downstream and become very strong. The left part of the secondary closed vortex is also present (the “peak position” in Fig. 13), an heir separates from an SCS (Lee, 1998). A long streak and two vortex tubes are seen in Fig. 13(d). The tubes look similar to a sheep’s horn. One or two vortices in a chain of ring vortices (C↑) also appear. The interaction between the long streak and the chain of ring vortices is also shown here. A perfect secondary closed vortex is shown in Figs. 13(e) and 13(f) (SV↑) where it is completely closed instead of in the Λ-like form. Figure 13(f) shows the formation of the secondary closed vortex (SV↑). Figure 13(g) shows both the streamwise vortices (SW↑) and the


57

Fig. 13. Flow structures at different y-positions. The hydrogen bubble wires were located at x = 350 mm and y = 0.25 mm, 0.5 mm, 0.75 mm, 1.0 mm, 1.25 mm, 1.5 mm and 1.75 mm for each of the respective photos.

58

C. Lee and S. Chen

left part of the secondary closed vortex (arrow A). The existence of this left part of the vortex shows that the secondary closed vortex is different from the form of the Λ-vortex. The pictures illustrate the interaction between the secondary closed vortex and the side part of the Λ-vortex. Besides the two vortex tubes, the filaments from the Λ-vortex and the secondary closed vortex are twisted together. The pictures in Figs. 12 and 13 show several new features. 1. The secondary vortex is closed instead of a hairpin vortex. 2. The additional hairpin vortices on the two legs of the main hairpin vortex were not found. 3. The development of the instability along the borders of the SCS is the necessary condition for the separation of the secondary closed vortex from the SCS (Lee, 1998). 4. These results do not suggest that the original work by Smith et al. was in error, but just incomplete. A different mechanism is suggested for the different vortex form which is similar to an earlier proposal by Lee (2000, 2001a, 2001b) had been suggested. The quantitative results are present but do not clearly show which kinks were generated by which structures (Fig. 14). At this stage, four kinds of flow structures have been identified which strongly affect the time traces, the SCS, the Λ-vortices, the secondary closed vortices and the first ring vortices that will be discussed later. After extensive analysis of the figures, the kinks indicated by arrows A, B, C and D were related to the SCS, the Λ-vortices, the first ring vortices and the secondary closed vortices, respectively. 6. Formation of the Chain of Ring Vortices The typical eddies in the transition boundary layer was first noticed by Falco (1977, 1990). As described by Falco, the typical eddies are local compact regions of vorticity concentration that have distorted vortex ring configurations and behavior. The measured frequency of occurrence of the typical eddies, when normalized by the freestream velocity and boundary layer thickness, f δ/U∝ , increase from 0.78 to 1.16 as Re θ increases from 753 to 3853. Falco’s overall sense obtained from several thousand feet of visual data taken with high-speed cameras, as well as time observations of lower-speed boundary layers, and cross-light sheet experiments is that the


59

Fig. 14. Oscilloscope traces of the velocity disturbances at various distances from the wall at x = 400 mm for z = 0.

typical eddies have a vortex ring configuration and evolution, as opposed to an attached hairpin eddy configuration and evolution. At low and moderate Reynolds numbers, the typical eddies appear to be laminar vortex rings, and thus are viscous controlled. At higher Reynolds numbers, the typical eddies have been observed to appear as wavy-cored vortex rings. At still higher Reynolds numbers (such as characteristic of atmospheric boundary layers), they have been observed as completely typical eddy motions. The formation and evolution of typical eddies in a highly perturbed vortex

60

C. Lee and S. Chen

environment strongly governs the shape of these eddies. It is unrealistic to expect that an observer will often see an idealized picture of a developing or developed typical eddy. Therefore, understanding of the formation process is not yet complete, but there are indications that at least two mechanisms are involved. Formation could take place by pinch-off and reconnection of hairpin eddies observed to lift-up from the wall (Melander and Zabuski, 1988). Falco (1977) presented data that show evolution by this process. The experiments of Chu and Falco (1988) show the reconnection of hairpin vortices, as do the numerical simulations of Moin et al. (1986). The other likely mechanism is the formation of typical eddies from instability in a local region of high vorticity fluid exposed to an applied force, as described by the solutions of Cantwell (1986). These solutions describe the formation of the vortex rings in an infinite fluid through the action of an impulsive force on a vortical region. As with Falco, Borodulin et al. (2002) and Bake et al. (2002) suggested a simple formation process for the high frequency vortex ring formation in a transitional boundary layer called self-induction of the two legs of the Λ-vortex. Lee (2000, 2001a, 2001b) suggested a new process based on experimental observations which describes how continuous separations along the border of the SCS can generate high frequency vortices. The chain of ring vortices (CRV) shown in Fig. 15 were found to be periodic instead of random as described earlier (Lee, 2000, 2001a, 2001b; Borodulin et al., 2002). A secondary closed vortex (arrows SV in Fig. 13) appears after the formation of the Λ-vortex; however, the quantitative results do not clearly show which kinks were generated by which structures (Fig. 14). At this stage, there exist four kinds of flow structures which strongly affect the time traces, the SCS, the Λ-vortices, the secondary closed vortices and the first ring vortices which will be discussed later. After extensive analysis with the figures, the kinks indicated by arrows A, B, C and D were related to the SCS, the Λ-vortices, the first ring vortex in a chain of ring vortices and the secondary closed vortices, respectively. Lee (2001a) earlier suggested the possibility of the existence of this kind of closed vortex based only on visual observations. The direction of the rotation velocity in this vortex is the same as in a Λ-vortex. The angle between the plane with the closed vortex and the flat plate is less than that between the plane with the Λ-vortex and the flat plate, but the y-position of the closed vortex is higher than that of the Λ-vortex at the same x-position. Since the closed vortex has a


61

Fig. 15. Plan view of the chain of ring vortices (↑) associated with a Λ-vortex (A↑) (the wire was at x = 360 mm and y = 1.75 mm) (Lee, 2001a). The first ring vortex propagated downstream (1↑) while the other three (2↑, 3↑ and 4↑) appeared at nearly the same time. The time interval between successive pictures was 1/24 s. (a) shows the first ring vortex. (b) to (e) show the other three ring vortices.

Fig. 16. Generation of the first ring vortex (1↑) by the interaction between the secondary closed vortex (SCV↑) and the Λ-vortex (Λ↑) (Lee, 2001a). The hydrogen bubble wire is at x = 250 mm and y = 1 mm from the leading edge of the flat plate and the time interval between successive pictures was 1/12 s.

higher downstream convection velocity compared with the Λ-vortex, they will eventually meet each other. Figure 16 shows the different stages of the secondary closed vortex interacting with the two legs of the Λ-vortex. Figure 16(a) shows the closed vortex close to the two legs of the Λ-vortex as it catches up to the Λ-vortex.

62

C. Lee and S. Chen

Figure 16(b) shows that when the two vortices are close to one another, the interaction increases the rotational velocities of both vortices. The Λ-vortex is divided into three sections: the head part, the interaction part and the upstream part. The filaments connecting the head part where an Ω-shaped vortex has formed and the left part of the Λ-vortex become much more slender and will break when their rotational velocity is sufficiently large as shown in Figs. 16(c) and 16(d). The Ω-shaped vortex was found many years ago (Hama et al., 1963; Knapp et al., 1968) but was more recently found to be the first ring vortex in the chain of ring vortices (Lee, 2000; Borodulin et al., 2002; Bake et al., 2002). Few studies after the works by Crow (1970) and Moin et al. (1986) have analyzed the physical mechanism for the formation of the Ω-shaped vortex because researchers such as Bake et al. (2002) believed that it is induced by vortex stretching and self-induction. A different mechanism was suggested for the different vortex form which is similar to an earlier proposal by Lee (2000, 2001a, 2001b). In general, the first ring vortex forms due to vortex stretching, induction by the secondary closed vortex and axial instabilities of the vortex filaments. Vortex stretching is a well-known effect so it is not repeated here. The induction of the secondary closed vortex which bring the two legs of the Λ-vortex close to each other is shown in Fig. 16. This effect, which is needed to describe the axial instability of the vortex filament, has not been previously suggested in boundary-layer flow. Several later stages of the interaction between the Λ-vortex and the secondary closed vortex are clearly seen in Figs. 17(a)–17(l). Two symmetric filaments appearing on the inside-face of the Λ-vortex in the interaction zone are observed to move into the center of the secondary closed vortex. The middle parts of the two filaments look like two half-ring vortices appearing in symmetry in the center of the secondary closed vortex. At the same time, the parts of the two filaments in the head part move from the inside out due to the secondary closed vortex because of a positive angle between the Λ-vortex and the flat plate. For the same reason, the filaments in the upstream parts then go into the secondary closed vortex so that two narrow necks of these two symmetric filaments exist inside the secondary closed vortex. After the two symmetric filaments meet each other where their two narrow necks come together, they break and reconnect into three small ring vortices. The filaments of the third ring vortex come from the two symmetric filaments. The second and fourth ring vortices are the direct results of the breaking and reconnecting of the filaments of the Λ-vortex and the secondary closed vortex. The filaments on the right-hand side of the fourth


63

Fig. 17. Formation of the second (2↑), third (3↑) and fourth (4↑) ring vortices in a chain of ring vortices (Lee, 2001a). The hydrogen bubble wire was at x = 350 mm and y = 1.5 mm. The time interval between successive pictures was 1/24 s. (a) shows the filaments of the Λ-vortex moving into the center of the secondary closed vortex (SCV↑) and the two symmetric filaments with two narrow necks formed inside the center of the secondary closed vortex. (c) shows the breaking and reconnection of the two symmetric filaments at their narrow necks inside the secondary closed vortex and the third ring vortex is clearly seen. In (e) to (f), the fourth ring vortex appears. The filament on the left-hand side of the vortex comes from the secondary closed vortex and the filament on the right-hand side of the vortex from the symmetric vortices. (h) to (k) show the formation of the second ring vortex. The filaments of this vortex come from both the secondary closed vortex and the two symmetric filaments. All three ring vortices appear clearly in (i) to (k). The two stream-wise vortices (SW↑) on the two sides of these three vortices are the well-known streamwise vortices, also known as long streaks. The filaments of the streamwise vortex come from both the secondary closed vortex and the Λ-vortex.

64

Fig. 18. z = 0.

C. Lee and S. Chen

Oscilloscope traces of the velocity disturbances at x = 500 mm, y = 1.5 mm,

vortex come from the Λ-vortex while those on the left-hand side come from the secondary closed vortex, the filaments on the left-hand side of the second vortex come from the Λ-vortex while those on the right-hand side come from the secondary closed vortex. The second, third and fourth ring vortices then move from the inside to the outside of the secondary closed vortex, are lifted up to higher y-positions and then propagate downstream at a higher convection velocity relative to the Λ-vortex because a positive induction velocity field exists in the y-direction at the center of the secondary closed vortex. When two vortex filaments are brought into contact by an induced velocity field, viscous diffusion in the contact region causes annihilation of vorticity. The annihilation of vorticity effectively “severs” the filaments and, due to the kinematic constraint that vortex lines cannot end inside a field, they reconnect on either side of the contact region. Various views on vortex breaking and reconnection were given by Saffman (1972, 1990), Zawadzki and Aref (1991), Melander and Hussain (1989), Kokshaysky (1979), Rott (1956) and Sears (1956). Figure 18 shows the measured data generated by the chain of ring vortices. The number of spikes in a period equals the numbers of vortices in a chain. Both the measured data and the visual results show the existence of the interactions between the Λ-vortex and the secondary closed vortices. A new mechanism for the formation of the chain of ring vortices is suggested based on the previous visual study (Figs. 15, 16 and 17) and the present quantitative and visual results (Figs. 14, 15 and 18) that differs from that of Borodulin et al. (2002) and Bake et al. (2002). 7. Breakdown of the Chain of Ring Vortices The regular breakdown of the chain of ring vortices is shown during boundary-layer transition in Fig. 19. At first, the filaments of the ring


65

Fig. 19. Breakdown of the first ring vortex in a transitional boundary layer. (A) to (E) show breakdown of the near-wall part of the vortex. In (A), the first ring vortex appears. Arrows indicate the position of the breakdown in the near-wall region of the boundary layer. In (a) to (e), there is a breakdown of the first ring vortex in the outer region of the boundary layer. In (a), the vortex in the outer layer starts to break (1↑) and the vortex in the near-wall region has already broken (W↑). In (c), (d) and (e), the arrows show the points of the breakdown. (e) shows the first ring vortex and with other ring vortices in a chain (1↑, 2↑, 3↑ and 4↑).

66

C. Lee and S. Chen

Fig. 20. Time traces starting from the 4-spike stage to the multiple spike stage and their corresponding spectra. The Tollmien–Schlichting wave frequency was 2 Hz. Lines 1, 2, 3 and 4 on the right side of the figure are the spectra at x = 500 mm, 600 mm, 650 mm and 700 mm.

vortex are twisted back and forth in a regular pattern. Then the filaments are broken on the near-wall side (arrows) and finally broken completely. The breakdown of the other three ring vortices in the chain occurs in a similar manner as with the first ring vortex, as seen on a video of the whole process. Typical velocity fluctuations and frequency spectrums of the chain of ring vortices after its breakdown are illustrated in Fig. 20. At the breakdown stage of the chain of ring vortices, the vortices produce a wide spectrum of strongly coupled frequencies. The videos showed that the breakdown was nearly periodic. Observation of the process showed a solid bridge connecting the high frequency vortex (4-spike stages) to the much higher frequency velocity fluctuations in Fig. 20. Direct comparison of the measured and visualized results presented here demonstrates that the breakdown of the chain of ring vortices plays an important role in the later stages of transition. They excite the near-wall region around the peak position producing very intense vortex fluctuations in the boundary layer. But obvious contributions to the flow randomization process by the breakdown have not been found. The frequency spectrum due


67

to the chain of ring vortices and after their breakdown, Figs. 19 and 20, describe the deterministic features of the breakdown near the last stage of transition which have not been previously reported. These results will facilitate understanding of the later stage of transition.

8. Formation of the Streamwise Vortices A streamwise vortex is a typical coherent structure in both transitional and developed turbulent boundary layers. Streamwise vortices were observed early in the history of turbulent structure research. Kline et al. (1967) and Kim et al. (1970) noted the common appearance of quasi-streamwise vortices in conjunction with the oscillation phase in the turbulence-generating bursting process. Clark and Markland (1971) made careful observations of relatively long quasi-streamwise vortices with 3 to 7 degrees upward tilt in the wall region of a turbulent water channel. Perhaps the most extensive direct information concerning quasistreamwise vortices came from the end view hydrogen bubble visualization studies of Smith et al. (1991). These studies confirmed the common occurrence of quasi-streamwise vortices in the near-wall region, including frequent observation of counter-rotating pairs. In the simultaneous top and end views by Smith and Schwartz (1983), counter-rotating pairs in the near-wall region are always associated in space and time with low-speed streak formation. The study by Kasagi (1988) suggested that quasi-streamwise vortices are more common than vortex pairs in the near-wall region, and that the vortical structures are not as long as the near-wall low-speed streaks. Lee (2000, 2001a, 2001b) analyzed almost all of the structures referred to as streamwise vortices and found that there are two different kinds of flow structures which have been referred to as streamwise vortices. One is the real streamwise vortex such as that found by Clack and Markland (1971) and Lian (1990). The other is the long streak containing several SCS, called a solitary quasi-streamwise vortex by Kasagi (1988). For example, the long streak very near the wall found by Kline et al. (1967) was actually a solitary quasi-streamwise streak instead of a streamwise vortex. Figure 21 shows both the primary vortex called the Λ-vortex and the secondary closed vortex. A long streak containing several SCS appears in the very near-wall region. The middle layer has two real streamwise vortices which are the relatively long quasi-streamwise vortices (often counterrotating pairs).

68

C. Lee and S. Chen

Fig. 21. Flow structures at x = 400 mm. The wire was at y = 0.75 mm. (b) and (c) show that the left-hand side of the first closed vortex appears (FCV↑). The complete secondary closed vortex appears in (d), (e) and (f) (SCV↑). A long streak associated with two streamwise vortices is always present in the “peak position”.


69

Fig. 22. Formation of the streamwise vortices. (a) shows the start of interaction between the primary vortex and the secondary closed vortices (SCV↑). (b) to (f) show the filaments of both vortices are rolled up together (↑). The hydrogen bubble wire was at x = 350 mm and y = 1.5 mm. The time interval between successive pictures was 1/12 s.

Figure 22 presents a sequence of photographs showing the interaction of a Λ-vortex and a secondary closed vortex. The leg of the Λ-vortex is divided into three sections: the upstream part, the interaction part and the downstream part. Attention is focused here on the interaction part. In general, the velocity fields of both vortices will cause their diameters to increase. Parts of the Λ-vortex filaments are induced into the secondary closed vortex, as discussed in the previous section. The other parts of the filaments of the Λ-vortex and the secondary closed vortex are rolled up together. Since the rolling velocity at their outer edge is much higher than that in the other two sections, the structure will break at two points, one connecting the interaction section and the upstream section and the other connecting the downstream and interaction sections. The videos showed that these two bullhorn-like vortices have higher rotational speeds than the other two sections. This process, reported here for the first time in a boundary-layer flow, is a typical vortex axial instability. Klebanoff et al. (1962) showed that streamwise vortices are generated from the three-dimensional development of the fundamental “peak-valley”

70

C. Lee and S. Chen

spanwise structures. The present visual results show a new instability inducing the formation of the streamwise vortices. 9. Breakdown of the SCS Streaky structures elongated in the streamwise direction play a fundamental role in sustaining turbulence in wall-bounded shear flows (Kline et al., 1967; Kasagi, 1988; Lian, 1990), with much work done in investigating the origin and breakdown of these structures in the turbulent regime. In this model, streak breakdown is one phase of self-sustaining cycles in turbulent flows which include streak formation, streak breakdown and streamwise vortex regeneration from the nonlinear interaction of the streak instability eigenmode. Figure 23 shows the regular breakdown of the long streak (the same as the low speed streak in Asai et al. (2002)) at x = 600 mm. A chain of ring vortices and the long streak can be seen. The chain of ring vortices was observed from their formation to the position of the long streak breakdown. Three stages of their formation and evolution were found (Lee, 1998, 2000). First, they form along the borders of the SCS and then separate from the SCS one by one. Then, they meet the head of the Λ-vortex which increases their rotational speed. Because the convection velocity of the chain of ring vortices is higher than that of the closed vortex, the ring vortices separate from the Λ-vortex one by one. Next, the angle between the plane containing the vortices and the flat plate changes quickly to nearly 90◦ and the vortices propagate downstream at a speed of about 0.9 times the freestream velocity. They are visible even in the very late stage when the boundary layer becomes turbulence. They strongly affect not only the outer layer but also the near-wall region. Besides the main low-frequency fluctuations, high amplitude fluctuations appeared in the near-wall regions which were generated by the SCS as shown in Fig. 8 with additional spikes on the time traces generated by some of the chain of ring vortices. The breakdown occurs at x = 600 mm to 640 mm. The streak amplitude increase could not be clearly related to the streak breakdown which is considered to be the necessary condition for streak breakdown suggested by Waleffe (1995, 1998). The term breakdown used by Klebanoff et al. (1962) and other researchers describe what appears to be an abrupt change in the character of the wave motion at the peak. The breakdown process is characterized by intense fluctuations in the direction of the lower velocities which occur for


71

Fig. 23. Breakdown of a long streak at x = 600 mm (L↑). In (a) to (c), a regular long streak. In (d) to (f), a wavy long streak. In (g), start of the breakdown of a long streak. From (h) to (l), breakdown process associated with the appearance of the chain of ring vortices (HFV↑) in the “peak position”. In (l), the arrow points to the breakdown of the long streak. The hydrogen bubble wire was at x = 550 mm and y = 0.75 mm. The time interval between successive pictures was 1/12 s.

72

C. Lee and S. Chen

each cycle of the primary wave. The spatial extent of the initial breakdown as well as its physical significance can be determined by analyzing the manner in which the high frequency fluctuations characteristic of breakdown vary in the y- and z-directions. Although these characteristics are used to clearly define the breakdown, confusion often occurs. Our new definition for the breakdown of the Λ-vortex describes the formation of the high frequency vortices, i.e. the chain of ring vortices. The breakdown of the long streak describes the strong effects of the chain of ring vortices on the long streak with the harmonics found on the time traces in the near-wall region. 10. Discussion 10.1. Description of turbulent bursting in a transitional boundary layer Early investigations of the near-wall streaks identified them as streamwise regions of low-speed fluid (relative to the mean), and revealed their participation in the “bursting cycle”. A turbulent burst begins when a low-speed streak is perturbed and begins to oscillate. The sinuous oscillations increase in magnitude and the streak gradually lifts up, away from the wall. A portion (or portions) of the streak is then rapidly “ejected” from the near-wall region into the outer portion of the flow. The ejection is a strong contributor to the Reynolds stress with the low-speed fluid (−u′ ) moving away from the wall (+v ′ ) creating a positive product, −u′ v ′ . From continuity considerations, the ejection of near-wall fluid is followed by an inrush of highspeed fluid from farther out in the boundary layer. This “sweep” motion constitutes another positive contribution to the Reynolds stress (+u′ , −v ′ ) (Corino and Brodkey, 1969). Taken together, the ejection/sweep sequence is termed a turbulent burst (this definition will be used here although other definitions of bursting have been employed elsewhere). Estimates of the contribution of the bursting process to the production of turbulent stresses vary, but it is generally considered to be responsible for 60–80% of the total (Grass, 1971; Kim et al., 1971). The source of the initial perturbation which triggers the oscillation of the near-wall streaks is an open question. Brown and Thomas (1977) showed that the passage of large-scale structures in the outer layer can create conditions in which streamlines in the near-wall region become curved, so that a Taylor–G¨ ortler type instability is possible. Similar explanations of the instability based on the influence of outer-layer structures through an imposed pressure field have been proposed by, for example, Kovasznay et al. (1970),


73

and Blackwelder (1978). In contrast, Offen and Kline (1974) and, Offen and Kline (1975) suggest that the instability is imposed by localized sweep motions from the logarithmic region near the wall, which are caused by earlier bursting events upstream. There have been many attempts to link the near-wall bursting cycle to structures in the outer portion of the boundary layer. Consequently, the large-scale structures in that region have been the subject of numerous investigations. The discovery of the SCS has helped clarify the complex burst phenomena. The processes associated with the burst phenomena are consistent with the SCS: (a) the vertical movement of the SCS generates both the Λ-vortex and the closed vortex and (b) the interaction between the secondary closed vortex and the Λ-vortex produces the chain of ring vortices (i.e. high frequency vortices) (Lee 2001a, 2001b). The SCS has very large velocity in the y-direction before its breakdown (Figs. 24 and 25). The burst appears in the region where the secondary closed vortex and the high frequency chain of ring vortices are formed. The SCS carries the flow from the near-wall region to the outer layer while the secondary closed vortex enables sweeping of the flow down to the near-wall region along the border of the SCS. The production of the chain of ring vortices is shown to be a periodic process instead of a random process because this very quick process cannot be clearly seen in a turbulent boundary layer (Lee, 2001a). Because all these dynamic processes including the formation of the Λ-vortex, the secondary close vortex and the chain of ring vortices are periodic, the bursting process can also be observed under controlled conditions.

10.2. Long streak breakdown mechanism The evolutions from the SCS to a long streak (the same as the low-speed streak) as well as the disappearance of the long streak are summarized in Fig. 26. Besides the main low-frequency, high-amplitude fluctuations in the near-wall region generated by the SCS shown in Fig. 27, additional spikes (indicated by arrows) are also present on the time traces which are generated by some of the chain of ring-like vortices. The breakdown occurs at x = 675 mm. The measurements did not show any obvious increase of the streak amplitude at this time, which has been considered to be the necessary condition for its breakdown. The near-wall streaky structure with high- and low-speed regions aligned in the streamwise direction was first visualized in the developed turbulent boundary layer by Kline et al. (1967). They suggested that the

74

C. Lee and S. Chen

Fig. 24. Visualization of the SCS. The hydrogen bubble wire was located parallel to the plate and normal to the flow direction at x = 450 mm and y = 0.25 mm with the flow from left to right in the pictures. A DC voltage was applied to the wire to continuously form hydrogen bubbles. The light sheet was transmitted from the bottom to the top of the pictures with the camera set to normal to the light sheet. The time interval between two successive pictures was 1/12 s. Pictures show both a SCS and its shadow (S↑). The shadows appeared periodically at the frequency of the T–S waves (2 Hz). The shadow of the SCS was also seen when the wire was set at two other positions (x = 350 mm and y = 0.25 mm; x = 400 mm and y = 0.25 mm). SCS periodically appeared early in the transitional flow (x < 250 mm) and were transported downstream to the later stages (x > 550 mm). The SCS shadow was not seen earlier because previous visualizations were taken with bubbles at y larger than 0.5 mm where the density of the hydrogen bubbles organized by the flow field was not sufficient to reflect the light.


Fig. 24.

75

(Continued)

Fig. 25. Oscilloscope traces of velocity disturbances in both the x- and y-directions at various distances from the wall at x = 450 mm. T represents the period of the T–S waves (0.5 s). (a) Velocity fluctuations in x-direction; (b) Velocity fluctuations in y-direction.

near-wall low-speed streaks with an average spanwise spacing of about 100 wall units play an important role in generating the turbulent energy through a sequence of bursting events. Low-speed streaks and a nearwall activity similar to bursting were also observed in the final stages of laminar–turbulent transition initially controlled by the convective growth of Tollmien–Schlichting waves. In the so-called ribbon-induced transition where periodic Tollmien–Schlichting waves are introduced by means of a

76

C. Lee and S. Chen

Fig. 26. (a) T–S wave and CSS. The hydrogen bubble wire was at x = 250 mm and y = 0.75 mm; (b) The Λ-vortex structure. The wire was positioned at x = 300 mm and y = 0.75 mm; (c) Development of the Λ-vortex and the long streak. The wire was at x = 350 mm and y = 0.75 mm; (d) The long streak is composed of several CS-solitons and the wire was at x = 450 mm and y = 0.5 mm; (e) Breakdown of a long streak. The wire was at x = 550 mm and y = 0.75 mm.

vibrating ribbon, the phenomenon of wall turbulence generation occurs in the later stage of the high-frequency secondary instability, i.e. the breakdown of thin internal shear layers (formed away from the wall) into high-frequency hairpin vortices. Through a series of ribbon-induced transition experiments of plane Poiseuille flow, Nishioka et al. (1981) and,


77

Fig. 27. Corresponding velocity fluctuations generated mainly by SCS at different stages of transition.

Nishioka and Asai (1984) were the first to show this near-wall phenomenon similar to a bursting event. That is, after the passage of the hairpin vortices resulting from the high-frequency secondary instability, the near-wall fluid is lifted up to develop a low-speed streak in between neighboring highspeed regions. The uplifted wall shear layer along the low-speed region soon evolves (or breaks down) into near-wall vortices. Sandham and Kleiser (1992) and, Rist and Fasel (1995) used numerical simulations to reproduce the disturbance development observed in the ribbon-induced transition experiments. In these transition studies, hairpin vortices were identified as the key elements directly triggering the occurrence of the wall turbulence structure, though the details of the near-wall bursting events were not sufficiently clarified. In order to do so, Asai and Nishioka (1990, 1995a, 1995b) further examined the development of the wall turbulence structure through the observation of a subcritical boundary-layer transition triggered by hairpin vortices. In the lateral growth of turbulent spots or turbulent wedges, low-speed streaks appear in succession in the ambient laminar flow region on both sides of the spot or wedge, as visualized, for instance, by Perry et al. (1981) and others. As far as these low-Reynolds number visualization

78

C. Lee and S. Chen

studies are concerned, hairpin-like vortices are generated from near-wall low-speed streaks through their breakdown. Recent experiments (Lee, 1998, 2000) have shown that the Λ-vortices (the same as the hairpin-like vortices) are generated from the SCS and the low-speed streak is composed of several SCS. Streak instability is also thought to govern the secondary instability process occurring beyond the growth of G¨ ortler vortices on a concave wall. In G¨ ortler flows, as observed by Swearingen and Blackwelder (1987), low-speed streaks caused by each pair of counter-rotating streamwise vortices undergo a secondary instability to generate hairpin-shaped (or arch-like) vortices or wavy meandering motions of G¨ ortler vortices. The review by Saric (1994) further describes these structures. The secondary instability is no doubt caused by the associated inflectional velocity profile both in the normal-to-wall and spanwise directions, which can excite both varicose and sinuous instability modes. Hall and Horseman (1991) and, Park and Huerre (1994) theoretically studied the mechanism of the secondary inflectional instability of G¨ ortler vortices leading to the development of sinuous and varicose modes on the basis of three-dimensional stability equations. Similar instability phenomena have been observed in by-pass boundary-layer transition caused by high-intensity free-stream turbulence (Morkovin and Reshotko, 1990). The bypass transition results from the algebraic growth of non-modal disturbances first described as transient growth by Landahl (1980). Matsubara et al. (2000) demonstrated experimentally that the growth of near-wall streaks is a key phenomenon triggering the bypass transition and that when the near-wall streaks are intensified, a secondary instability occurs which leads to a time-dependent oscillation of the streaks and subsequent breakdown into new turbulent spots. As already stated, the streak instability must be studied under well-controlled flow conditions to examine if and how the streak instability can operate as a dominant mechanism for generating and sustaining coherent structures (Asai et al., 2002). In the present experimental study, a single low-speed streak was produced in a laminar boundary layer. The life time of the coherent structures was then obtained under the controlled conditions. Table 4 lists the life time and positions of the various coherent structures found in a transitional boundary layer. At x = 550 mm (Figs. 9 and 23), the fact that the disturbance amplitude of the SCS is very small shows that the SCS at this stage is very weak. Otherwise the chain of ring vortices would strongly affect the other structures such as the SCS (Fig. 23). The time traces at x = 600 mm in Fig. 27 (also Fig. 23) show that high amplitude spikes are generated by the chain of ring vortices.

79

Dynamics of Transitional Boundary Layers Table 4.

Life time of various coherent structures.

Streamwise position (mm)

Observed distance in the streamwise direction (mm)

Existence time (s)

Flow structures

Appearance

Disappearance

SCS Λ-Vortices Streamwise vortex Secondary closed vortices Chain of ring-like vortices

255 280 390 420

620 460 580 510

385 180 190 90

5.65 3.5 3.7 1.8

500

675

175

2.5

11. Conclusion Detailed flow structures have been presented describing the fluid dynamic processes in transitional boundary-layer flows. The key element in the model is the SCS, which manifests the physics necessary to explain both the regeneration of vortices and the observed growth to larger scales farther from the wall. The SCS are proposed to be the direct results of oblique wave interaction with the T–S waves catalyzing their formation based on the theory presented by Wu (1996) and Wu et al. (2001a, 2001b). The sequential process describing the interaction between the Λ-vortex and the secondary closed vortex controls the manner in which the chain of ring vortices is periodically introduced from the wall region into the outer region of the boundary layer. There are several proposals to explain the generation of the high frequency vortices (Borodulin et al., 2002; Bake et al., 2002 and Lee 2000, 2001a, 2001b) which is one of the key problems for understanding both transitional and developed turbulent boundary layers as well as other flows. The observation of the secondary closed vortex makes it possible to establish a real physical process for the formation of the high frequency vortices, i.e. the chain of ring vortices. The secondary closed vortex suggested here and the secondary hairpin vortex must be understood to explain the entire process. The present result suggests that the secondary hairpin vortex is just part of the closed vortex. If the secondary closed vortex is not considered, the dynamic processes and flow structures in a transitional boundary layer are very different from those discussed in previous studies (Borodulin et al., 2002; Bake et al., 2002 and Lee, 2000, 2001a, 2001b). The present result also shows that both the breakdown of the chain of ring vortices and the breakdown of the long streak are roughly periodic,

80

C. Lee and S. Chen

which has not been reported previously even in the very recent studies by Bake et al. (2002).

References 1. R. J. Adrian, C. D. Meinhart and C. D. Tomkins, Vortex organization in the outer region of the turbulent boundary layer, J. Fluid Mech. 422 (2000) 1–53. 2. M. Asai and M. Nishioka, Development of wall turbulence in Blasius flow, in Laminar–Turbulent Transition, eds. D. Arnal and R. Michel (Springer, 1990), pp. 215–222. 3. M. Asai and M. Nishioka, Boundary-layer transition triggered by hairpin eddies at subcritical Reynolds numbers, J. Fluid Mech. 297 (1995a) 101–122. 4. M. Asai and M. Nishioka, Subcritical disturbance growth caused by hairpin eddies, in Laminar–Turbulent Transition, ed. R. Kobayashi (Springer, 1995b), pp. 111–118. 5. M. Asai, M. Minagawa and M. Nishioka, The instability and breakdown of a near-wall low-speed streak, J. Fluid Mech. 455 (2002) 289–314. 6. S. Bake, D. G. W. Meyer and U. Rist, Turbulence mechanism in Klebanoff transition: A quantitative comparison of experiment and direct numerical simulation, J. Fluid Mech. 459 (2002) 217–243. 7. R. F. Blackwelder, The bursting process in turbulent boundary layers, in Coherent Structures of Turbulent Boundary Layers, eds. C. R. Smith and D. E. Abbott (AFOSR/Lehigh University, Bethlehem, PA, 1978). 8. R. F. Blackwelder, Analogies between transitional and turbulent boundary layers, Phys. Fluids 26 (1983) 2807–2815. 9. V. I. Borodulin, V. R. Gaponenko, Y. S. Kachanov, Q. X. Lian, Y. H. Qin, H. Guo and C. B. Lee, Experimental study of mechanisms of flow randomization at late stages of boundary layer transition, in Intl Symp. on Actual Problems of Physical Hydroaerodynamics, Trans. Part II, Novosibirsk: Inst. Thermophysics (1999), pp. 25–26. 10. G. L. Brown and A. A. Thomas, Large structure in a turbulent boundary layer, Phys. Fluids 20 (1977) S243–S252. 11. B. J. Cantwell, Viscous starting jets, J. Fluid Mech. 173 (1986) 159–189. 12. C. C. Chu and R. E. Falco, Vortex ring/viscous wall layer interaction model of the turbulence production process near walls, Experiments in Fluids 6 (1988) 305–315. 13. J. A. Clark and E. Markland, Flow visualization in turbulent boundary layer, J. Hydraul Div. ASCE 97 (1971) 1653–1664. 14. E. R. Corino and R. S. Brodkey, A visual investigation of the wall region in turbulent flow, J. Fluid Mech. 37 (1969) 1–30. 15. A. D. D. Craik, Wave Interactions and Fluid Flows (Cambridge University Press, 1985). 16. A. D. D. Craik, Nonlinear resonant instability in boundary layers, J. Fluid Mech. 50 (1971) 393–413.


81

17. S. C. Crow, Stability theory for a pair of trailing vortices, AIAA J. 8 (1970) 2172–2179. 18. T. L. Doligalski, C. R. Smith and J. D. A. Walker, Vortex interactions with walls, Annu. Rev. Fluid Mech. 26 (1994) 573–616. 19. J. W. Elder, An experimental investigation of turbulent spots and breakdown to turbulent, J. Fluid Mech. 9 (1960) 235–246. 20. H. W. Emmons, The laminar–turbulent transition in a boundary layer, part 1, J. Aero. Sci. 18 (1951) 490–498. 21. R. E. Falco, Coherent motions in the outer region of turbulent boundary layers, Phys. Fluids 20 (1977) S124–S132. 22. R. E. Falco, A coherent structure model of the turbulent boundary layer and its ability to predict Reynolds numbers, Phil. Trans. R. Soc. Lond. A 336 (1991) 103–129. 23. H. F. Fasel, U. Rist and U. Konzelmann, Numerical investigation of the threedimensional development in boundary-layer transition, AIAA J. 28 (1990) 29–37. 24. Y. Fukunishi, H. Sato and O. Inous, Study of developing process of coherent structures in turbulent boundary layer, AIAA Pap. 87 (1987) 1253–1265. 25. M. Gaster and T. Grant, An experimental investigation of the formation and development of a wave packet in a laminar boundary layer, Proc. R. Soc. London Ser. A 347 (1975) 253–269 26. A. J. Grass, Structural features of turbulent flow over smooth and rough boundaries, J. Fluid Mech. 50 (1971) 233–255. 27. P. Hall and N. J. Horseman, The linear inviscid secondary instability of longitudinal vortex structures in boundary layers, J. Fluid Mech. 232 (1991) 357–375 28. F. R. Hama, J. D. Long and J. C. Hegarty, On transition from laminar to turbulent flow, J. Appl. Phys. 28 (1957) 388–394. 29. F. R. Hama and J. Nutant, Detailed flow field observations in the transition process in a thick boundary layer, in Proc. Heat Transfer and Fluid Mech. Inst. (Stanford University Press, 1963), pp. 77–93. 30. J. O. Hinze, Turbulence (McGraw-Hill Company, New York, 1975). 31. R. D. Joslin, C. R. Street and C. L. Chang, Spatial direct numerical simulation of boundary-layer transition mechanism: Validation of PSE theory, Theor. Comput. Fluid Dyn. 4 (1993) 271–288. 32. N. Kasagi, Structural study of near-wall turbulence and its heat transfer mechanism, in Near Wall Turbulence, Zaric Memorial Conf. eds. S. J. Kline and N. H. Afgan (Hemispere, New York, 1988). 33. Y. S. Kachanov, Physical mechanisms of laminar boundary-layer transition, Annu. Rev. Fluid Mech. 26 (1994) 411–482. 34. Y. S. Kachanov, The resonant interaction of disturbance at laminarturbulence transition in a boundary layer, J. Fluid Mech. 138 (1984) 209–247. 35. Y. S. Kachanov, V. V. Koznov and V. Y. Levchenko, Experimental investigation of the influence of cooling on the stability of laminar boundary laye, Izv. Sib. Otd. Akad. Nauk SSSR, Ser. Tekh. Nauk. 8 (1974a) 75–79.

82

C. Lee and S. Chen

36. Y. S. Kachanov, V. V. Koznov and V. Y. Levchenko, Experimental study of laminar-boundary-layer stability on a wavy surface, Izv. Sib. Otd. Akad. Nauk SSSR, Ser. Tekh. Nauk. 13 (1974b) 2–6. 37. H. Kim, S. Kline and W. Reynolds, The production of turbulence near a smooth wall in a turbulent boundary layer, J. Fluid Mech. 50 (1971) 133–160. 38. P. S. Klebanoff and K. D. Tidstrom, Evolution of amplified waves leading to transition in a boundary layer with zero pressure gradient, NACA Tech. Note (1959) D-195. 39. P. Klebanoff, K. D. Tidstrom and L. M. Sargent, The three-dimensional nature of boundary layer transition, J. Fluid Mech. 12 (1962) 1–34. 40. S. J. Kline, W. C. Reynolds, F. A. Schraub and P. W. Runstadler, The structure of turbulent boundary layers, J. Fluid Mech. 30 (1967) 741–773. 41. C. R. Knapp and P. J. Roache, A combined visual and hot-wire anemometer investigation of boundary-layer transition, AIAA J. 6 (1968) 29–36. 42. N. V. Kokshaysky, Tracing the wake of a flying bird, Nature 279 (1979) 146–148. 43. L. S. G. Kovasznay, H. Komoda and B. R. Vasudeva, Detailed flow field in transition, in Proc. Heat Transfer and Fluid Mech. Inst. (Stanford University Press, 1962), pp. 1–26. 44. L. S. G. Kovasznnay, V. Kibens and R. F. Blackwelder, Large-scale motion in the intermittent region of a turbulent boundary layer, J. Fluid Mech. 41 (1970) 283–325. 45. M. T. Landahl, A note on an algebraic instability of inviscid parallel shear flows, J. Fluid Mech. 98 (1980) 243–251. 46. C. B. Lee, New feathers of CS-solitons and the formation of vortices, Phys. Lett. A 247 (1998) 397–402. 47. C. B. Lee, Possible universal transitional scenario in a flat plate boundary layer: measurement and visualization, Phys. Rev. E 62 (2000) 3659–3670. 48. C. B. Lee and S. Fu, On the formation of the chain of ring-like vortices in a transitional boundary layer, Exp. Fluids 30 (2001) 354–357. 49. C. B. Lee, On the breakdown of the chain of ring-like vortices in a transitional boundary layer, in Proc. Heat Transfer and Fluid Mech. Inst. (California State University Press, 2001), pp. 61–80. 50. Q. X. Lian, A visual study of the coherent structure of the turbulent boundary in flow with adverse pressure gradient, J. Fluid Mech. 215 (1990) 101–124. 51. R. I. Loehrke and M. B. Morkivin, Transition in nonreversing oscillating boundary layers, J. Fluid Eng., Trans. ASME 97 (1975) 534–550. 52. C. J. Mangur, On the sensitivity of shear layers to sound, AIAA pap. (1977) 77–1369. 53. M. V. Melander and N. J. Zabusky, Interaction and “apparent” reconnection of 3D vortex tubes via direct numerical simulations, Fluid Dynamics Res. 3 (1988) 247–250. 54. M. V. Melander and F. Hussain, Cross-linking of two antiparallel vortex tubes, Phys. Fluids A 1 (1989) 633–636.


83

55. M. J. Mitchner, Propagation of turbulence from an instantaneous point disturbance, J. Aeronaut Sci. 29 (1954) 350–351. 56. P. Moin, A. Leonard and J. Kim, Evolution of a curved vortex filament into a vortex ring, Phys. Fluids 29 (1986) 955–963. 57. M. V. Morkovin, On the many faces of transition, in Viscous Drag Reduction, ed. C. Wells (Plenum, 1969), pp. 1–31. 58. M. V. Morkovin and E. Reshotko, Dialogue on progress and issues in instability and transition research, in Laminar–Turbulent Transition, eds. D. Arnal and R. Michel (Springer, 1990), pp. 3–39. 59. M. Nishioka, M. Asai and S. Iida, An experimental investigation of the secondary instability, in IUTAM Sym. on Laminar–Turbulent Transition, eds. R. Eppler and H. Fasel (Springer, 1980), pp. 37–46. 60. M. Nishioka and M. Asai, Evolution of Tollmien–Schlichting waves into wall turbulence, in Turbulence and Chaotic Phenomena in Fluids, ed. T. Tatsumi (North Holland, 1984), pp. 87–92. 61. M. Nishioka, M. Asai and S. Iida, Wall phenomena in the final stage of transition to turbulence, in Transition and Turbulence, ed. R. Meyer (Academic, 1981), pp. 113–126. 62. M. Nishioka, S. Iida and Y. Ichikawa, An experimental investigation of the stability of plane Poiseuille flow, J. Fluid Mech. 72 (1975) 731–751. 63. D. W. Moore and P. G. Saffman, The motion of a vortex filament with axial flow, Trans. R. Soc. London Ser. A 272 (1972) 403–429. 64. D. S. Park and P. Huerre, Primary and secondary instabilities of the asymptotic suction boundary layer on a curved plate, J. Fluid Mech. 283 (1994) 249–272. 65. G. R. Offen and S. J. Klein, Combined dye-streak and hydrogen-bubble visual observations of a turbulent boundary layer, J. Fluid Mech. 62 (1974) 223–239. 66. G. R. Offen and S. J. Klein, A proposed model of the bursting process in turbulent boundary layers, J. Fluid Mech. 70 (1975) 209–228. 67. A. E. Perry, T. T. Lim and E. W. Teh, A visual study of turbulent spots, J. Fluid Mech. 104 (1981) 387–405. 68. S. C. Reddy, P. J. Schmid, J. S. Baggett and D. S. Henningson, On stability of streamwise streaks and transition thresholds in plane channel flows, J. Fluid Mech. 365 (1998) 269–303. 69. U. Rist and H. Fasel, Direct numerical simulation of controlled transition in a flat plate boundary layer, J. Fluid Mech. 298 (1995) 211–248. 70. S. K. Robinson, A perspective on coherent structures and conceptual models for turbulent boundary layer physics, AIAA pap. (1990) 90–1638. 71. S. K. Robinson, Coherent motions in the turbulent boundary layer, Annu. Rev. Fluid Mech. 23 (1991) 601–639. 72. S. K. Robinson, The kinematics of turbulent boundary layer structure, NASA Tech. Memo. (1991) 103859. 73. H. L. Rogler, The coupling between freestream disturbances driver oscillations, forced oscillations and stability waves in a spatial analysis

84

74. 75. 76. 77. 78. 79. 80. 81. 82. 83.

84. 85.

86. 87. 88. 89. 90. 91. 92.

93.

94.

C. Lee and S. Chen

of a boundary layer, AGARD Symp. Laminar–Turbulent Transition 16 (Copenhagen, 1977) AGARD-CP-244. H. L. Rogler and E. Reshotko, Disturbances in a boundary layer introduced by a low intensity array of vortices, SIAM J. Appl. Mech. 28 (1975) 431–462. W. Rott, Unsteady viscous flows in the vicinity of a separation point, Q. Apply Math 13 (1956) 444–451. P. G. Saffman, A model of vortex reconnection, J. Fluid Mech. 212 (1990) 395–402. W. S. Saric, G¨ ortler vortices, Annu. Rev. Fluid Mech. 26 (1994) 379–409. W. S. Saric, H. L. Reed and E. J. Kerschen, Boundary-layer receptivity to freestream disturbances, Annu. Rev. Fluid Mech. 34 (2002) 291–319. W. R. Sears, Some recent developments in airfoil theory, J. Aeronaut. Sci. 23 (1956) 490–499. H. Schlichting, Boundary Layer Theory (McGraw-Hill, New York, 1979). G. B. Schubauer, Boundary Layer Research (Springer-Verlag, 1957). G. B. Schubauer and P. S. Klebanoff, Contributions on the mechanics of boundary-layer transition, NACA Rep. (1956) 1289. C. R. Smith and S. P. Schwartz, Observation of streamwise rotation in the near-wall region of a turbulent boundary layer, Phys. Fluids. 26 (1983) 641–652. C. R. Smith, J. D. A. Walker, A. H. Haidari and U. Sobrun, On the dynamics of near-wall turbulence, Phil. Trans. R. Soc. Lond. A 336 (1991) 131–175. J. D. Swearingen and R. F. Blackwelder, The growth and breakdown of streamwise vortices in the presence of a wall, J. Fluid Mech. 182 (1987) 255–290. L. N. Trefethen, A. E. Trefethen, S. C. Reddy and T. A. Driscoll, Hydrodynamic stability without eigenvalues, Science 261 (1993) 578–584. T. Theodorsen, Mechanism of turbulence, Proc. 2nd Midwestern Conf. Fluid Mech., Ohio State University (1952). A. S. W. Thomas and W. S. Saric, Harmonic and subharmonic waves during boundary-layer transition, Bull. Amer. Phys. Soc. 26 (1981) 1252. B. R. Vasudeva, Boundary-layer instability experiment with localized disturbance, J. Fluid Mech. 29 (1967) 749–763. F. Waleffe, On a self-sustaining process in shear flows, Phys. Fluids. 9 (1995) 883–900. F. Waleffe, Three-dimensional coherent states in plane shear flows, Phys. Rev. lett. 81 (1998) 4140–4143. J. D. A. Walker, D. E. Abbott, R. K. Scharnhorst and G. G. Weigand, Walllayer model for the velocity profile in turbulent flows, AIAA J. 27 (1989) 140–149. D. R. Williams, H. Fasel and F. R. Hama, Experimental determination of the three-dimensional vorticity field in the boundary-layer transition process, J. Fluid Mech. 149 (1984) 179–203. X. Wu, Generation of Tollmien–Schlichting waves by convecting gusts interacting with sound, J. Fluid Mech. 397 (1996) 285–316.


85

95. X. Wu and J. Luo, Instability of Blasius boundary layer in the presence of steady streak, Annual Research Briefs of Center for Turbulence Research at Stanford University, Stanford University (2001). 96. X. Wu and M. Choudhari, Effects of long-wavelength Klebanoff modes on boundary-layer instability, Annual Research Briefs of Center for Turbulence Research at Stanford University, Stanford University (2001). 97. I. Zawadzki and H. Aref, Mixing during vortex ring collision, Phys. Fluids A 3 (1991) 1405–1410.


CONTINUOUS MODE TRANSITION

Paul Durbin* and Tamer Zaki† Department of Mechanical Engineering, Stanford University Stanford, CA 94035-3030, USA E-mails: ∗[email protected] † [email protected] Some aspects of transition induced by freestream turbulence can be understood in terms of the continuous spectrum of the Orr–Sommerfeld and Squire equations. Instead of discrete mode (Tollmien–Schlichting wave) precursors, transition can be studied, starting from the continuous mode precursors. An examination of mode shapes leads to a theory of how freestream disturbances penetrate the boundary layer and, ultimately, provoke transition. Basically, low-frequency modes penetrate the boundary layer, while high frequencies are expelled — a result referred to as shear sheltering. Low-frequency penetration can be characterized by a “coupling coefficient”. This only describes the initial route into the boundary layer. Transition subsequently involves an interaction between low- and high-frequency modes, to produce breakdown near the top of the boundary layer. Continuous mode transition is illustrated by numerical simulations of mode interaction. Either one or two modes are prescribed at the inlet to the computational domain. One low-frequency mode will generate perturbation jets in the boundary layer. Transition does not occur. One low-frequency mode and one high-frequency mode will suffice to induce transition. These studies of mode interactions provide a fundamental perspective on the transition mechanism seen in full simulations with turbulent inflow. DNS of transition induced by grid turbulence and by swept wakes are also reviewed. Contents 1

Introduction

88

2

The Continuous Spectrum and Shear Sheltering

88

87

88

P. Durbin and T. Zaki

3 Computer Simulations References

95 105

1. Introduction This article is based on a talk presented at the NUS–IMS workshop on Transition and Turbulence Control. It reviews our recent work on transition theory and simulation. It is not intended as a literature survey. The process by which freestream, vortical disturbances induce transition to turbulence in an underlying boundary layer, without the intervention of instability waves, is an instance of what has been called a “bypass transition” (Morkovin, 1969). That terminology is all-encompassing; it is defined by what does not occur, i.e. Tollmien–Schlichting waves are bypassed. Laboratory experiments and DNS have shed considerable light on what does occur. Our starting point is the shapes of Orr–Sommerfeld continuous spectrum modes and the notion of shear sheltering. The title of this article is borrowed from the former. A number of aspects of bypass transition can be understood in terms of properties of the continuous spectrum. Computer simulations can be initialized with continuous spectrum modes. Hence, the title Continuous Mode Transition properly cites the perspective developed herein. 2. The Continuous Spectrum and Shear Sheltering Rapid distortion theory (RDT) shows how the energy of a sheared, broadband disturbance grows linearly in time. Moffatt (1967) showed that at long times, Deissler’s (1961) solution for inviscid, homogeneous shear flow approaches u2 ∝ t. A succinct physical explanation was provided by Phillips (1969): algebraic growth occurs in consequence of displacement of mean momentum. Momentum is not conserved by fluid elements due to acceleration by pressure gradients. However, Moffatt (1967) found that disturbances with kx → 0 dominate at large times. For these components, the x-pressure gradient is small. Hence, momentum is approximately conserved. Such perturbations can continue to grow in consequence of persistent displacement. If the mean velocity is in the u-direction, the lift-up of a fluid element, by conserving its momentum, produces a negative u-component in the perturbation field. This is quite similar to the Prandtl mixing length argument.

Continuous Mode Transition

89

In a homogeneous mean shear, the RDT equations for a Fourier component are dt uˆ1 = S uˆ2 2e21 − 1 , dt uˆ2 = 2S uˆ2 e1 e2 ,

(2.1)

dt uˆ3 = 2S uˆ2 e1 e3 , in which ei = κi /|κ| with κ2 = k2 − Stk1 ,

κ1 = k1 ,

κ3 = k3 ,

where k is the Fourier wavenumber vector and S is the rate of shear, dU/dy. The mechanism of streak formation is quite simple. For components with k1 ≈ 0, uˆ2 = uˆ02 and dt uˆ1 = −S uˆ02 according to (2.1). This equation for uˆ1 simply equates its evolution to vertical displacement of mean momentum by the u2 -component of velocity. For these low wavenumber components, the growth of u21 is simply due to displacement of mean momentum by cross-stream velocity fluctuations; the pressure gradient is negligible. In the large St limit, the full Deissler–Moffatt analysis gives u2 → u20 St log 8 for sheared, initially isotropic turbulence. Through this reasoning, and the RDT analysis, it is seen that the amplifying portion of the broadband disturbance is elongated in the x-direction, and also that the u velocity predominates. In short, the disturbance is like a jet in the perturbation field. These perturbation jets are commonly seen in sheared broadband disturbances — often cited as “streaks”. Of course, in linear theory the displacement of fluid elements is infinitesimal. Computer simulations, surveyed in the next section, provide a fully nonlinear picture of how the lift-up evolves. Amplification of broadband disturbances by shear suggests a mechanism by which freestream turbulence can have a disproportionate effect on an underlying boundary layer: it induces a perturbation in the shear layer, which then grows in accord with the RDT mechanism. Some time ago (Hunt, 1977), we performed a rapid distortion analysis of this problem and came to a disconcerting conclusion. The configuration was a piecewise linear velocity profile, U = 1 in the freestream, y > 0 and U = 1 + y in the shear layer, y < 0. The solution consisted of two parts. The particular solution, which was forced by the freestream turbulence, vanished inside the shear layer — it did not produce an amplifying, broadband disturbance. An eigensolution could be added to this, which corresponded to vorticity waves on the interface. The vorticity waves did induce a disturbance in the

90


shear layer, that amplified in time, approaching linear growth asymptotically. However, the initial amplitude of the eigensolution was arbitrary, not obviously connected with the freestream turbulence. The first part of the analysis is simply a solution to the Rayleigh stability equation. For a piecewise linear profile, Rayleigh’s equation reduces to D 2 ∇ v = 0. Dt

(2.2)

∇2 v = F (x − U t, y, z).

(2.3)

Thus

If the disturbance is initially confined to the freestream, the right-hand side vanishes in the shear layer. The other part of the classical analysis of disturbances on a piecewise linear shear is the matching condition. Both displacement and pressure must be continuous at y = 0. Continuity of pressure applied to the x-momentum equation gives Dt [u] = v− S, where S = dU/dy is the linear shear and v− is velocity on the lower side of the interface, limy↑0 v. The brackets in [u] denote jump across y = 0; hence [u] measures the strength of a vortex sheet, at the interface. In the absence of a vortex sheet, the matching condition is v− = 0 where the freestream vorticity does not induce a vertical velocity in the shear layer; the sheared region can only be perturbed by the velocity induced by a vortex sheet at the interface. Thus, the RDT solution for this configuration was written as a sum of the particular solution to (2.3), plus a vortex sheet. The particular solution has v ≡ 0 for y < 0. The other components of velocity also vanish in the shear layer. For instance, if F ∝ ei(kx+ly+mz) the particular solution is proportional to eily − e−

√ m2 +k2 y

for y > 0 and vanishes for y ≤ 0. The vortex sheet is a homogeneous solution to (2.3). It can be added to the particular solution, with an arbitrary amplitude, and independent of the freestream, vortical perturbation. It has a discontinuous derivative at y = 0, for instance, it can be proportional to e−

√ m2 +k2 |y|

.

The corresponding u-component is discontinuous, representing a vorticity wave on the interface. Let us ignore this homogeneous solution, since it is

91


5

5

4.5

4.5

4

4

3.5

3.5

3

y /δ 99

y /δ 99

not directly connected with the freestream turbulence. We are left with no perturbation in the shear layer and the solution is as if an inviscid wall were inserted at y = 0. Although the analysis alluded to was done some time ago, in connection with rapid distortion theory, it was resurrected as an embodiment of the shear sheltering phenomenon in Hunt and Durbin (1999). They proposed that in some circumstances, freestream disturbances are unable to perturb the boundary layer. One might also attribute this to the absence of pressure fluctuations when a vortical disturbance is convected at a uniform speed. It derives from a linear, inviscid analysis of a piecewise linear flow, but is it anomalous? Or can shear sheltering be derived from a more realistic model? In fact, the phenomenon has been known since Grosch and Salwen (1978). Mode shapes of the continuous spectrum of the Orr–Sommerfeld equation illustrate this shear sheltering. Figure 1 shows mode shapes obtained by solving the Orr–Sommerfeld and Squire equations. When frequencies and wavelengths are of order one, the eigenfunctions become very small inside the boundary layer. The horizontal line indicates the 99% thickness. The shear is quite mild at that height. The shear increases below δ99 , and the modal amplitude falls rapidly; it is nearly zero in the lower part of the boundary layer. Hence, shear sheltering is seen with finite viscosity and a smooth, Blasius profile; it is not an artifice of the piecewise linear profile. The continuous spectrum is analogous to the particular solution of (2.3). In that sense, the continuous spectrum might be regarded as a forced solution,

2.5

3 2.5

2

2

1.5

1.5

1

1

0.5

0.5

0 −1.5

−1

−0.5

0

Real(φ)

0.5

1

1.5

0 −1.5

−1

−0.5

0

0.5

1

1.5

Real(χ)

Fig. 1. Sheltered modes of the continuous spectrum of Orr–Sommerfeld and Squire equations. ω = π.

92


5

5

4.5

4.5

4

4

3.5

3.5

3

y /δ 99

y /δ 99

instead of as a set of eigenfunctions. However, from the perspective of completeness, the general solution to an initial/boundary value problem will contain the continuous spectrum. The numerical method that Grosch and Salwen (1978) used to compute mode shapes is effective for moderate and high frequencies, but it fails for lower frequencies. Jacobs and Durbin (1998) developed an alternative algorithm that is able to compute all frequencies. They recovered the shapes seen by Grosch and Salwen (1978) at high frequencies, but they found that low-frequency modes can penetrate the boundary layer. Thus, shear sheltering becomes shear filtering. High-frequency modes are prohibited from the boundary layer, but low frequencies are admitted. In Fig. 2, mode shapes are plotted for two low frequencies. The lowest, ω = π/100, penetrates to the wall. Frequency is non-dimensionalized by boundary-layer thickness and freestream velocity. It is noted in Zaki and Durbin (2005) that modes also penetrate when ky is large and when the Reynolds number is small. By Squire’s transformation, the latter includes highly oblique modes. These cases of penetration are not interesting because they are disturbances that decay rapidly. The nondimensional dispersion relation for temporal Orr–Sommerfeld continuous spectrum modes is i 2 (2.4) kx + kz2 + ky2 . ω = kx − R The imaginary part is the decay rate. Modes with large ky or low R have high decay rate.

2.5

3 2.5

2

2

1.5

1.5

1

1

0.5

0.5

0 −1.5

−1

−0.5

0

Real(φ)

0.5

1

1.5

0 −1.5

−1

−0.5

0

0.5

1

1.5

Real(χ)

Fig. 2. Penetrating modes of the continuous spectrum. —— refers to ω = π/10; — · — refers to ω = π/100.

93


The spatial dispersion relation is −1/2 2 4ω r kx = ω M2 + 2 + M ; R

kxi =

R ω − 1 2 kxr

(2.5)

with 1 2(ky2 + kz2 ) . + 2 R2 The only penetrating modes of interest are those with low frequency and high Reynolds number. If R is large, the spatial dispersion relation is approximately M=

ky2 + kz2 + ω 2 . R The first indicates that the mode propagates with the freestream speed, which is normalized to unity. The second indicates the rapid decay of short wavelengths in y or z. The full three-dimensional eigensolution consists of an Orr–Sommerfeld mode for the vertical velocity and a Squire mode for the vertical vorticity. The linear equations are (Schmid and Henningson, 2001) kxr ≈ ω;

kxi ≈

(L + iω)φ(y) = 0, (S + iω)χ(y) = Cφ(y),

(2.6)

with ∆ + ∆−1 {ikx U ′′ − ikx U ∆}, R ∆ − ikx U , S= R

L=

C = ikz U ′ .

where ∆ is the Laplacian operator, and ∆−1 is its formal inverse. The second equation of (2.6) is Squire’s equation. The Orr–Sommerfeld eigenfunction provides forcing on its right-hand side. With the right-hand side equated to zero, the equation becomes the Squire’s eigenvalue problem. The dispersion relation for the continuous spectrum is (2.4) for either Squire or Orr–Sommerfeld modes. Hence, the forcing on the right-hand side of the Squire equation (2.6) is exactly resonant. Klebanoff (1971) observed characteristic, low-frequency disturbances in the laminar boundary layer, perturbed by grid turbulence. They consisted of streamwise elongated motions, dominated by the u-component of fluctuating velocity, that is, they are jets in the perturbation field. Klebanoff used

94


the term “breathing mode” to identify his observations. This name was suggested by the idea that they were associated with thickening and thinning of the boundary layer. The notion that the perturbation field could be characterized by oscillations of the boundary-layer thickness had been suggested in earlier work by Bradshaw (1965) (see Goldstein and Wundrow, 1998). This simple idea predicts that the rms velocity fluctuation is proportional to y∂y U . It is in excellent agreement with measured profiles of u2 . Numerous alternative perspectives on “breathing modes” have been developed. That they are streamwise elongated disturbances, with the u-component being dominant is in accord with the Deissler–Moffatt rapid distortion analysis. When the disturbance consists of broadband freestream turbulence, linear growth is observed, analogous to Moffatt’s large time RDT result. The prolonged lift-up which produces algebraic growth is similar to the notion of boundary-layer thickness oscillations. Certain aspects of the theory of optimal disturbances are also consistent with properties of these “modes” (Schmid and Henningson, 2001). A further perspective, which draws a link to the present notions of mode penetration, is that breathing modes can be expanded as a superposition of Squire modes. The forcing on the right-hand side of Eq. (2.6) generates the Squire modes, and hence is a route for freestream turbulence to enter the shear layer. The algebraic growth can be attributed to the exact resonance between Squire and Orr–Sommerfeld modes. Zaki and Durbin (2005) invoke that perspective in order to characterize modal penetration. It is measured in proportion to the resonant forcing term. In particular, the inner product of the right-hand side of (2.6) with the adjoint eigenfunction of Squire’s equation gives the amplitude of the resonant forcing. The tendency to generate low-frequency disturbances is inversely related to the modal decay rate; alternatively, the integrated effect of the forcing is proportional to 1/ωi . Thus, the coupling coefficient, Θ, proposed in Zaki and Durbin (2005) is † χ , Cφ χ† , U ′ φ . Θ≡ = kz ω i i ωos os

(2.7)

This is used only to characterize continuous spectrum modes. The factor of U ′ causes Θ to increase as modes penetrate more deeply into the boundary layer. The factor of kz shows that only three-dimensional modes will generate perturbation jets. The factor of 1/ωi is included so that rapidly decaying modes are characterized as weakly coupled to boundary-layer disturbances.


95

Indeed, there is a good deal of linear theory that is consistent with Klebanoff’s observations. But the relation between the long, jet-like disturbances to the laminar boundary layer and subsequent transition to turbulence has little theory to explain it. What has been provided in the literature is largely speculations about the next step in the process. They all involve an inflectional instability of the perturbation jets, in one way or another. Rayleigh’s inflection point theorem was invoked to suggest that inflections in either the vertical or horizontal profiles of U will lead to instability. One way or another, such instabilities are expected to be precursors to the breakdown into turbulence. Recent computer simulations by our group and others have provided a detailed, empirical picture. While transition is seen to originate in an inflectional instability, the mechanism differs from all these speculations. The next stage is not an instability of the perturbation jets.

3. Computer Simulations Linear theory provides a theoretical framework. The Orr–Sommerfeld modes that precede orderly transition become replaced by continuous spectrum modes. Thus we have a starting point for studies of bypass transition. That is the motive for our title Continuous Mode Transition. Basic studies via direct numerical simulations can be conducted by prescribing one, two or a few modes at the inlet. Grid turbulence can be modeled by a broadband spectrum of modes. In the freestream, the continuous modes are sinusoidal. A Fourier representation of homogeneous, isotropic turbulence defines the modal amplitudes and phases for grid turbulence simulations. In each case, linear theory was invoked only when specifying the inlet boundary condition. Inside the domain, the full Navier–Stokes equations are computed. Simulations with a single mode at the inlet produce an evolution that agrees with linear theory. The perturbation decays by viscous action. This serves as a code validation (Jacobs and Durbin, 2000) and is illustrated by Fig. 3. Two modes at the inlet interact within the computational domain. If both are of high frequency — more correctly, are non-penetrating — they also decay, with little effect on the boundary layer. The case of two 2D modes is illustrated by Fig. 4. One penetrating and one non-penetrating mode are chosen. They superpose in the freestream. In the boundary layer, only the lower frequency mode is seen. This illustrates filtering by the shear.

96

Wall Normal (y)


δ99 Streamwise (x) One non-penetrating mode in the freestream.

Wall Normal (y)

Fig. 3.

δ99 Streamwise (x)

Fig. 4. layer.

Two modes in the freestream; only the low frequency penetrates the boundary

Fig. 5. Instantaneous u, contoured between −0.3 and 0.3. The plane is at y/δ = 0.8. The spanwise coordinate is enlarged by a factor of 3.

Vertical vorticity, or Squire modes in linear theory, are not present because these are two-dimensional disturbances. Figure 5 is a plan view of a boundary layer perturbed by two penetrating, three-dimensional modes. Now there is a Squire equation forcing term. Contours of u show that elongated disturbances are produced. They can be described as jets in the perturbation field since u contours and elongates in the x-direction — hence, jets. The behavior of two low-frequency modes is still innocuous. They generate boundary-layer disturbances, but by the end of the domain they have decayed. But a simulation with one penetrating and one non-penetrating modes tells a different story. Now, in Fig. 6, transition occurs near the end of the domain. The penetrating mode generates perturbation jets, which seem to start decaying, but suddenly a patch of turbulence appears in the

97


Fig. 6. Instantaneous v contours. The plane is y/δ0 = 0.5. The boundary layer is perturbed by a high- and a low-frequency mode.

0.015

Cf

0.01

0.005

0

0.5

1

1.5

2

2.5

Rex

3

3.5

4

4.5 5

x 10

Fig. 7. Instantaneous (——) and time averaged (− − −−) skin friction profiles. Inlet modes: ω = 0.032, ky = 1.87, kz = 2π/3.2, Θ = 14.3 and ω = 0.64, ky = π/3, kz = π, Θ = 0.17.

middle of the domain. This is a turbulent spot. It does not look like a classic, Emmons spot. Computer simulations and experiments with freestream turbulence, consistently show that the patches of turbulence in the transitional region do not look like classic spots. Those are produced by forcing within the boundary layer whereas forcing by freestream disturbances has a different signature. Nevertheless, the term “turbulent spot” is still used. Below the contours, in Fig. 7, the skin friction shows the transition from a smooth, long wavelength variation, to a more erratic x-dependence. With just two modes at the inlet, the flow is able to transition. The sudden appearance of a turbulent spot is characteristic of bypass transition. No evidence of instability is seen in this figure prior to the formation of the spot. The apparent waviness just upstream of the spot is simply a consequence of

98


contour plotting the disturbed field. An instability would show a wavelength comparable to the size of the spot. That instability can be discovered, as will be shown later, but it is not apparent here. First, we present figures from the simulation by Jacobs and Durbin (2000) of transition induced by freestream grid turbulence. Again, the inflow is disturbed by a superposition of continuous spectrum modes, but now a large number of three-dimensional modes are summed, with amplitudes and phases selected in accord with an isotropic turbulent spectrum. Three plan views in Fig. 8 show the freestream turbulence, the turbulence at the top of the boundary layer and the perturbation within the shear layer. The uppermost view shows the decaying freestream turbulence. The lowermost is not at all reminiscent of the freestream. Due to the shear filtering, only low-frequency disturbances enter the boundary layer. They grow by displacement of the mean momentum, producing long jets in the perturbation field. It is quite remarkable that grid turbulence, looking like the uppermost panel, produces a disturbance deep in the boundary layer that looks like the lowest panel. In the midst of the jets, a turbulent spot appears. Again, it seems to arise spontaneously without being preceded by instability of the streaks. Figure 5, (Matsubara and Alfredsson, 2001) shows a very similar behavior in experimental smoke visualizations. Straight smoke streams are seen to suddenly develop into a spot of turbulence, with no prior instability.

Fig. 8. Horizontal sections through the freestream and the boundary layer. The sections from top to bottom are in the freestream, at y ∼ δ and at y ∼ 13 δ where δ is the 99% thickness at Rθ = 250.

99


Fig. 9.

Contours of v component.

0.6

u / U∞

0.4 0.2 0 -0.2 -0.4 0

1∗10 5

2∗10 5

3∗10 5

4∗10 5

5∗10 5

Rex

u (m/s)

8 7 6 5 0.0

0.1

0.2

0.3

0.4

0.5

t (s) Fig. 10. Instantaneous trace versus x at various heights from the DNS of Jacobs and Durbin (2000) and versus t from the experiments of Matsubara (1990).

The v-component of velocity is contoured in Fig. 9. Streaks are not seen in the laminar boundary layer. That is because they are jets, seen in the u-velocity, but not apparent in v. The patch of turbulence appears even more abruptly now; the lack of an instability prior to its emergence is even more obvious. In Fig. 10, the graph plots of u versus x at an instant of the DNS are compared to traces of u versus time obtained by Matsubara with a hot wire. The contrast between freestream and boundary layer is much the same in experiment and simulation. In the freestream, the signal consists of lowamplitute, high-frequency noise. Inside the boundary layer this becomes a

100


larger amplitude, low-frequency unsteadiness. These and other quantitative and qualitative comparisons between DNS and the experiment, made by Wu et al. (1999) and by Jacobs and Durbin (2000), provide a strong case that the simulations produce the same phenomenology as seen in laboratory experiments. The perplexing question of what leads to the apparently spontaneous appearance of a turbulent spot is answered by examining the DNS flow field in three dimensions. The plan view is a two-dimensional section. The absence of an instability preceding the appearance of the spot is a misleading consequence of this perspective. Some of the detective work involved in uncovering the origins of breakdown is described in Wu et al. (1999). A primary factor is the ability to restart the DNS at a time before the spot emerges. Then data sets are generated at the right times and the right places to see the spot precursors. In short, it is discovered that the disturbance seen near the surface as a turbulent spot had its origins at a higher elevation, near the top of the boundary layer. Figure 11 is from the two-mode simulation of Fig. 6.

8 6 4 2 0 140

160

180

200

220

Fig. 11. Instability of elevated backward jets as seen in a time sequence. v-contours with velocity vectors of the perturbation field superposed on the lower views.

101


Vertical sections are shown, through spanwise locations where the spot was seen to emerge. An instability of Kelvin–Helmoltz appearance is seen to begin at the left side of the top figure. The line across the figure shows the 99% boundary-layer thickness. A cat’s eye pattern is seen in the velocity contours just near that height. The lower views are of this instability at later times. At the latest time, breakdown has filled the boundary layer. Then a turbulent spot would be detected in plan views near the wall. This same process of instability near the top of the boundary layer was seen in the simulations by Jacobs and Durbin (2000) with freestream grid turbulence and by Wu et al. (1999) of transition by incident wakes. Figures 12–14 are from the latter study. Figure 12 presents the velocity with the phase average removed. This is the fluctuation field produced by wakes swept across the inlet. With time, the wakes translate from left to right. The boundary layer transitions shortly at the back of the wake. Since the freestream disturbance is intermittent, very long streaks are not seen in the initial, laminar boundary layer. However, elongated velocity contours are seen; so perturbation jets are found again though not as

Fig. 12. Transition induced by passing wakes. Fluctuation velocity contours showing the incident wakes.

0.10 y 0.08 0.06 0.04 0.02 x 0.75

0.875

1.0

1.125

u′

Fig. 13. Localized instability seen in contours where 0.02 ≤ lines are negative values and dashed are positive.

1.25

|u′ |

1.375

≤ 0.34 Uref . Solid

102


Fig. 14. Shorter jets are seen prior to wake-induced transition. Laboratory visualizations from Zhong et al. (1998) and u contours from DNS by Wu et al. (1999).

long as beneath freestream turbulence. The velocity contours in Fig. 14 are representative of those seen prior to wake-induced transition. Laboratory visualizations from Zhong et al. (1998) show the same features observed in DNS. Elongated u-contours occur in the region behind the wake. Time histories of the three patches seen in this figure show that the upper two patches develop into turbulent spots and the lowest patch decays. Hence, these are not necessarily precursors to transition. The instability induced by incident wakes is shown in Fig. 13. The solid contours show negative fluctuating velocity. Here, and in the other simulations, the instability originates near the top of the boundary layer, on a negative perturbation jet. The jet is produced by lift-up of the mean momentum — as in the linear rapid distortion theory. Nonlinearity primarily implies that negative perturbation jets will be seen high in the boundary layers, while positive jets occur close to the wall. That is, the lift-up of lowspeed fluid creates a negative perturbation. As explained by Phillips (1969),


103

streamwise elongated disturbances are subject to prolonged lift-up, without a counteracting pressure. Hence, jet-like features become pronounced near the top of the boundary layer. The nonlinear phase of lift-up precedes instability and breakdown. That is why speculations that have appeared in the literature about instability of the perturbation jets have been incorrect. They are based on the jets that are first seen near the wall. At that stage, we do not observe instabilities; rather the negative jets are seen to remain coherent as they lift away from the wall. In the upper part of the boundary layer, they are perturbed by the higher frequency, freestream turbulence. That triggers a localized instability. The subsequent breakdown appears as a turbulent spot. That process has been seen in simulations of transition under grid turbulence, in transition induced by passing wakes and in two mode simulations. Velocity vectors in Fig. 15 provide another view of the breakdown. This is the case of period, passing wakes. The cross-section was selected to correspond to the center of a turbulent spot that was observed in plan view sections. Three instants leading up to the appearance of a turbulent spot are shown: at first, a jet is seen to form away from the wall; then an oscillatory instability is triggered on the jet; the instability grows and breaks down into a local patch of turbulence. These views are from the simulation by Wu et al. (1999). Vectors of the perturbation field are plotted in a plan view section in Fig. 16. This is from the simulation by Jacobs and Durbin (2000) of transition beneath freestream turbulence. It shows the long, backward jets. Figure 16 is a plane in the upper part of the boundary layer, which is why jets with negative velocity predominate. Upward velocities transport low velocity fluid into the upper portion of the boundary layer. Persistent upward velocity is associated with perturbations having small kx ; hence, they generate the jets seen in this figure. In this article, our own computer simulations have been reviewed. A theoretical framework that motivated the work, and which has been invoked to explain the very early stages has been cited. A number of questions can be asked. Would a leading edge play a role in the initial coupling? Evidence from simulations of geometries that have a leading edge suggest not — perhaps, unless it is very blunt compared to the incident turbulence. Can the later stages be analyzed, beyond suggesting that they start from instability of the lifted jets? Aside from the inflectional instability, the

104


Fig. 15. Breakdown in wake-induced transition. Instability appears on the backward jet in the perturbation field, seen in the top view. These are successive times, shown in a frame that moves downstream with the incident wake.

coupling to higher frequency, freestream disturbances must be an element of any such analysis. In light of the other contributions to the volume, is there any prospect for controlling continuous mode transition? It might involve preventing the perturbation jets from lifting from the wall, or contouring the velocity profile to reduce coupling to the higher frequency


Fig. 16.

105

Vectors in upper part of boundary layer, showing long backward jets.

perturbations. The prospect for micro-control of the disturbances is not promising. References 1. R. G. Deissler, Effects of inhomogeneity and of shear flow in weak turbulent fields, Phys. Fluids 4 (1961) 1187–1198. 2. M. E. Goldstein and D. W. Wundrow, On the environmental realizability of algebraically growing disturbances and their relation to Klebanoff modes, Theoret. Comp. Fluid Dynamics 10 (1998) 171–186. 3. C. E. Grosch and H. Salwen, The continuous spectrum of the Orr–Sommerfeld equation, Part 1; The spectrum and the eigenfunctions, J. Fluid Mech. 68 (1978) 33–54. 4. J. C. R. Hunt, A review of rapidly distorted turbulent flows and its applications, Fluid Dyn. Trans. 9 (1977) 121–152. 5. J. C. R. Hunt and P. A. Durbin, Perturbed shear layers, Fluid Dyn. Res. 24 (1999) 375–404. 6. R. G. Jacobs and P. A. Durbin, Simulations of bypass transition, J. Fluid Mech. 428 (2000) 185–212. 7. R. G. Jacobs and P. A. Durbin, Shear sheltering and the continuous spectrum of the Orr–Sommerfeld equation, Phys. Fluids 10 (1998) 2006–2011. 8. J. M. Kendall, Studies on laminar boundary-layer receptivity to freestream turbulence near a leading edge, Boundary Layer Stability and Transition to Turbulence, eds. Reda et al., ASME-FED 114 (1991) 23–30. 9. P. S. Klebanoff, Effect of freestream turbulence on a laminar boundary layer, Bull. Am. Phys. Soc. 16 (1971). 10. M. Matsubara and P. H. Alfredsson, Disturbance growth in boundary layers subjected to freestream turbulence, J. Fluid Mech. 430 (2001) 149–268.

106


11. H. K. Moffatt, The interaction of turbulence with a strong wind shear, Proc. URSI-IUGG Int. Colloq. on Atmos. Turbul. and Radio Wave Propag., Moscow (1967), pp. 139–154. 12. M. V. Morkovin, On the many faces of transition, Viscous Drag Reduction, ed. C. S. Wells (Plenum Press, New York, 1969). 13. O. M. Phillips, Shear-flow turbulence, Ann. Rev. Fluid Mech. 1 (1969) 245–264. 14. M. Rosenfeld, D. Kwak and M. Vinokur, A fractional step solution method for the unsteady incompressible Navier–Stokes equations in generalized coordinate systems. J. Comp. Phys. 94 (1991) 102–137. 15. P. J. Schmid and D. S. Henningson, Stability and Transition in Shear Flows (Springer, New York, 2001). 16. X. Wu, R. G. Jacobs, J. C. R. Hunt and P. A. Durbin, Simulation of boundary-layer transition induced by periodically passing wakes, J. Fluid Mech. 398 (1999) 109–153. 17. T. Zaki and P. A. Durbin, Mode interaction and the bypass route to transition, J. Fluid Mech. (2005) to appear. 18. S. Zhong, C. Kittichaikarn, H. P. Hodson and P. T. Ireland, Visualization of turbulent spots and under the influence of adverse pressure gradients, in Proceedings of the 8th Int. Conf. Flow Visualization, Italy (1998).

TRANSITION IN WALL-BOUNDED SHEAR FLOWS: THE ROLE OF MODERN STABILITY THEORY

Peter J. Schmid Laboratoire d’Hydrodynamique (LadHyX) ´ CNRS-Ecole Polytechnique F-91128, Palaiseau, France E-mail: [email protected] Instabilities are often the precursors of transition to turbulent fluid motion in wall-bounded shear flows. This review motivates and presents mathematical tools for detecting and quantifying instability mechanisms — of both modal and non-modal type — in shear flows. The role of these instabilities during the transition process is illustrated, and extensions of the tools to capture unsteady effects, spatially evolving flows and inhomogeneous coordinate directions are given. Contents 1 2

3

Introduction Stability Theory 2.1 A motivational example 2.2 Modal versus non-modal analysis 2.3 Choice of norm 2.4 Spectral representations 2.4.1 The numerical range 2.4.2 The resolvent norm 2.5 Rayleigh–Benard convection versus Poiseuille flow — revisited 2.6 Link between normality/non-normality and sub/supercritical behavior 2.7 Application to plane Poiseuille flow Transition to Turbulence 3.1 Pattern selection 3.2 Transition thresholds 107

108 109 109 111 113 113 114 116 117 118 119 120 121 123

108

4

P. J. Schmid

Extensions 4.1 Time-dependent flows 4.1.1 A model problem 4.1.2 Arbitrary time-dependence 4.1.3 Energy growth rate and Lyapunov exponent 4.2 Spatial theory 4.3 Inhomogeneous directions 4.3.1 Self-sustained oscillations of a liquid curtain 4.3.2 Pseudo-wavepacket solutions 5 Summary and Conclusions References

125 125 127 128 130 131 133 134 138 138 139

1. Introduction Hydrodynamic stability theory is one of the central fields of fluid mechanics. It concerns the evaluation and quantification of the response behavior of laminar flows to small perturbations. If the slightly perturbed flow returns to its laminar state, the mean flow is labeled as stable; on the other hand, if the small perturbation grows unboundedly, the mean flow is deemed unstable. Various alternatives or more general definitions of stability are also common. Instabilities are often thought of as precursors to the transition process from laminar to turbulent flows. However, experimental and numerical studies of the transition process have uncovered routes to turbulent fluid motion that exhibit rather complex behavior but lack a satisfactory theoretical explanation. Despite this complexity, the categorization depicted in Fig. 1 presents a rough outline of the mechanisms involved at various stages of the transition process (Reshotko et al., 1994). Under realistic conditions, transition to turbulence is initiated through an external disturbance environment, which, e.g., can take the form of wall roughness or freestream turbulence. This disturbance environment then forces perturbations inside the shear layer through a filtering mechanism known as receptivity. Only perturbations that are receptive, in scale and frequency, to the external forcing will be seeded. Primary instabilities, of both exponential and transient types, then yield finite-amplitude disturbances that often modify the underlying laminar profile which subsequently becomes susceptible again to infinitesimal perturbations via secondary instabilities. At this stage, the fluid behavior is rather complicated and often breakdown into turbulence occurs. As Fig. 1 suggests, instabilities figure prominently during various stages of the transition process.

Transition in Wall-Bounded Shear Flows

109

Forcing Environmental Disturbances Receptivity Mechanisms Transient Growth Eigenmode Growth Parametric Instabilities Bypass Mechanisms & Mode Interactions Breakdown Turbulence Fig. 1.

Outline of transition to turbulence. From Reshotko et al. (1994).

In this article, we will introduce and review methods of hydrodynamic stability theory that will aid in the quantitative description of mechanisms present, at various stages, during the transition from laminar to turbulent fluid motion. 2. Stability Theory In this section, we will motivate and develop a set of tools to analyze the stability of wall-bounded shear flows. Since the early days of fluid dynamics, fluid dynamicists have approached the issue of stability in two distinct ways. The first method, referred to here as traditional stability analysis, determines the lowest possible critical parameter above which the mean state becomes unstable to infinitesimal perturbations. Mathematically, this type of analysis leads to an eigenvalue problem based on the linearized governing equations. The second method, referred to here as energy stability analysis, determines the largest possible critical parameter below which all perturbations monotonically decay in energy. Mathematically, this type of analysis is based on a variational principle applied to the energy transport (or Reynolds–Orr) equation. Both techniques will be applied to two prototypical flow configurations: Rayleigh–Benard convection and plane Poiseuille flow. 2.1. A motivational example As a first example, we are interested in determining the critical Rayleigh number (non-dimensional temperature gradient) for the onset of convective

110

P. J. Schmid

T2 < T1

U(y)

T1 (a) Fig. 2.

(b)

(a) Sketch of Rayleigh–Benard convection; (b) Sketch of plane Poiseuille flow.

motion when a fluid between two parallel plates is heated from below and cooled from above. At this critical Rayleigh number, the conductive state becomes unstable to infinitesimal perturbations and convection rolls appear, transporting heat between the two plates (Fig. 2(a)). Applied to the Rayleigh–Benard problem, traditional stability analysis predicts the onset of convection at a Rayleigh number of Ra = 1708; whereas energy stability analysis predicts that below a Rayleigh number of Ra = 1708 all disturbances decay monotonically. Additionally, experiments determine the onset of convection rolls at Ra = 1710 ± 10. Our second example is flow between parallel plates, i.e. plane Poiseuille flow (Fig. 2(b)). Again, we are interested in the stability characteristics of this flow. The critical parameter is the Reynolds number, Re. Traditional stability theory predicts an instability of the parabolic mean flow to infinitesimal disturbances at a Reynolds number of Re = 5772. On the other hand, energy stability theory states that any disturbance superimposed on the mean flow will decay monotonically for a Reynolds number below Re = 49.6. Experiments of plane Poiseuille flow have established a critical Reynolds number of Re ≈ 1000 for the onset of instabilities. For the above examples, we have seen both success and failure of traditional stability theory and energy stability theory (Table 1), and the Table 1. Results of stability analysis and experiments for Rayleigh–Benard convection and plane Poiseuille flow.

Rayleigh–Benard plane Poiseuille

Traditional stability theory

Energy stability theory

Experiment

1708 5772

1708 49.6

1710 1000


111

question that arises is: why did traditional and energy stability theory succeed in predicting the onset of convection, but failed in providing a critical Reynolds number for the onset of instabilities in the plane poiseuille flow? Before answering this questions, another paradox shall be mentioned. It is well-known that the nonlinear terms of the Navier–Stokes equations conserve energy. During the transition process, experiments as well as numerical simulations show a significant gain in kinetic perturbation energy. It is correct to conclude that only a linear mechanism, extracting energy from the laminar mean flow, can be responsible for this gain. At subcritical conditions, i.e. at Reynolds numbers below the critical one, the spectrum of the linearized Navier–Stokes equations predicts only stable modal solutions. To explain the energy gain during transition, it seems that we need perturbation energy growth without relying on unstable modes. 2.2. Modal versus non-modal analysis When computing the spectrum of a linear stability operator, the operator is diagonalized by a similarity transformation. Many linear stability operators for wall-bounded shear flows are non-normal, i.e. do not commute with their adjoint. As a consequence, they can be diagonalized (if non-defective) only by a non-unitary transformation. In other word, non-normal operators have a set of non-orthogonal eigenfunctions (Trefethen and Embree, 2005). Formally, we can write the solution of the linear evolution equationa d q = Lq, dt

q(t = 0) = q0 ,

(2.1)

in the form q = exp(tL)q0 = S exp(tΛ)S−1 q0

(2.2)

using the matrix exponential and a similarity transformation to diagonalize the matrix L according to L = SΛS−1 with Λ as a diagonal matrix containing the eigenvalues of L and the eigenvectors of L forming the columns of the matrix S. It becomes immediately apparent from (2.2) that considering the spectrum of L and deducing from it, the dynamics of disturbances neglect the influence of the similarity transformation involving the eigenvectors of L. If one of the eigenvalues lies in the unstable domain, we expect exponential a For

simplicity, only finite-dimensional operators (matrices) are considered.

112

P. J. Schmid

growth, and the smallest Reynolds number for which an eigenvalue crosses into the unstable half-plane yields the critical Reynolds number. Conversely, if all the eigenvalues lie in the stable half-plane, it is customary to conclude that the base flow is stable. Conclusions of this sort are independent of the eigenvector structure contained in S. For non-normal L, i.e. when S is non-unitary, there exist initial conditions that can exhibit short-term growth, even though all eigenvalues are confined to the stable half-plane (Butler and Farrell, 1992; Trefethen et al., 1993; Reddy and Henningson, 1993). A small geometric model in two dimensions will help understand the evolution of initial conditions by a non-orthogonal modal representation (Schmid and Henningson, 2001). Let us assume that we expand an initial condition q of unit length in a non-orthogonal (two-dimensional) basis Φ1 , Φ2 as shown in Fig. 3. We notice that the expansion coefficients are not of order one — a characteristic of non-orthogonal expansions. Let us further assume that the eigendirections are associated with decaying modes, but that the component in the first direction decays more rapidly than the component in the second. After a short time, the subtle cancellation of the non-orthogonal vectors at time t = 0 ceases to exist which gives rise to transient growth (measured by the length of the vector q). Since we assumed exponentially decaying solutions, the length of q will eventually decay to zero.

t=0 q Φ2

Φ1

t>0

Φ2

Φ1

Φ2 Φ1 Fig. 3. Sketch illustrating transient growth due to non-orthogonal superposition of two vectors that decay at different rates as time evolves.


113

This simple example demonstrates the following facts that also carry over to higher-dimensional linear operators: (i) the non-orthogonal superposition of exponentially decaying solutions can give rise to short-term transient growth measured in a suitable norm, (ii) eigenvalues alone only describe the asymptotic fate of the disturbance, but fail to capture transient effects, and (iii) the “source” of the transient amplification of the initial condition lies in the non-orthogonality of the eigenfunction basis. A more appropriate way to analyze the behavior of disturbances governed by exp(tL) is to compute its potential to amplify a given disturbance over time. We will measure the size of the disturbance by an appropriate norm and define as the maximum amplification the ratio of disturbance size to its initial size optimized over all possible initial conditions. We have exp(tL)q0 q = max = exp(tL) ≡ G(t). max q0 q0 q0 q0

The quantity G(t) represents the maximum possible amplification of unitnorm initial conditions over a time period [0, t]. 2.3. Choice of norm

At the core of the analysis of non-normal operators lies the nonorthogonality of the eigenfunctions. The angle between various eigenfunctions is computed using an inner product. This same inner product provides a norm to measure the size of our state variables. This choice of inner product will quantitatively influence the maximum amplification potential of the underlying operator. Therefore, the norm and inner product have to be chosen carefully. For the temporal evolution of disturbances in shear flows, we choose the kinetic energy of the perturbations. For more complex cases, e.g. for compressible flow or flows with additional physical effects (e.g., surface tension, Marangoni stresses, non-Newtonian behavior, etc.), it is not as obvious to construct a meaningful inner product. In these cases, the choice of inner product and norm is unique; it has to be motivated or rationalized by arguments involving the specifics of the problem at hand. 2.4. Spectral representations Computing the norm of the matrix exponential exp(tL) is rather costly, and the question presents itself whether there is a simpler set of measures that allows us to probe a non-normal operator as to its potential to transiently amplify initial energy.

114

P. J. Schmid

We concluded in the previous section that the spectrum of non-normal operators describes their behavior in the limit of large times. It is instructive to determine another set in the complex plane which describes the behavior of exp(tL) in the limit of small times. 2.4.1. The numerical range Starting with an expression for the time rate of change of energy

d d d 2 q = q, q + q, q dt dt dt = (Lq, q) + (q, Lq) = 2 Real{(Lq, q)},

numerical range

the numerical range of L emerges which is defined as the set of all Rayleigh quotients F (L) ≡ {z : z = (Lq, q)/(q, q)}.

The Rayleigh quotients of L are, in general, complex numbers. A simple numerical algorithm (Horn and Johnson, 1991) computes the boundary of a region in the complex plane that contains all the Rayleigh quotients of L. An expansion for small times yields the following result: E(t) = qH q, = (exp(tL)q0 )H (exp(tL)q0 ) ≈ q0 (I + tL)H (I + tL)q0

H H ≈ qH 0 q0 + q0 (L + L)tq0 ,

and the energy growth rate at t = 0+ is given as qH (LH + L)q0 1 dE = 0 . E dt qH q t=0

0

0

This expression represents the Rayleigh quotient for the composite matrix LH + L, and the maximum energy growth rate at t = 0 is given by the largest eigenvalue of this (Hermitian) matrix. We have (Farrell and Ioannou, 1996a; Schmid and Henningson, 2001; Trefethen and Embree, 2005): 1 dE = λmax (LH + L) E dt t=0

which coincides with the real part of the numerical range boundary for θ = 0 (Horn and Johnson, 1991). The initial condition that achieves the

115


largest initial energy growth rate can easily be determined from the principal eigenvector of LH + L. The numerical range of a matrix L has many properties (Horn and Johnson, 1991), three of which are important for hydrodynamic stability applications: (i) The boundary of the numerical range is convex. This means that, connecting any two points in the numerical range by a straight line, all points on the straight line lie inside the numerical range. (ii) The numerical range of L contains the spectrum of L. This readily follows from the definition of the numerical range as a set of Rayleigh quotients. By choosing eigenfunctions of L as test functions, we obtain the corresponding eigenvalues as the Rayleigh quotients. Thus, eigenvalues are special Rayleigh quotients, and since the numerical range is convex it must contain the entire spectrum of L. (iii) If L is a normal operator, its numerical range is the convex hull of the spectrum. These three properties are demonstrated below. We wish to compute the numerical range of the following two matrices.     −5 4 4 −5 0 0 , A2 =  0 −2 − 2i . A1 =  0 −2 − 2i 4 0 0 0 −0.3 + i 0 0 −0.3 + i

Clearly, both matrices have the same set of eigenvalues, but whereas matrix A2 is normal, matrix A1 is non-normal. Figure 4 shows the numerical range for each matrix as the shaded area. 6

6

4

4

2

2

0

0

−2

−2

−4

−4

−6 −8

stable −6

−4

unstable −2

(a)

0

2

4

−6 −8

unstable

stable −6

−4

−2

0

2

4

(b)

Fig. 4. Numerical range demonstration. (a) For the normal matrix A2 ; (b) For the non-normal matrix A1 .

116

P. J. Schmid

For the normal matrix A2 we have a numerical range that consists of the convex hull of the spectrum — as stated above. For the non-normal matrix A1 the numerical range is also convex and contains the eigenvalues, but is significantly larger than the convex hull of the spectrum. In addition, the numerical range protrudes into the unstable half-plane, thus indicating a positive energy growth rate (1/E)dE/dt at t = 0+ . Transient growth is to be expected if the matrix A2 were used as an evolution operator. 2.4.2. The resolvent norm Having estabished two sets in the complex plane, the eigenvalues and the numerical range, to describe the behavior of non-normal matrices for large and small times, respectively, we now introduce another set in the complex plane that can be used in the analysis of maximum energy amplification. This set is given by the resolvent norm (Trefethen et al., 1993) defined as R(z) = (zI − L)−1 ,

z ∈ C.

The resolvent norm is defined in the complex plane and exhibits singularites at locations z where the inversion fails, i.e. at the eigenvalues of L. The resolvent of L can be used to bound the amount of maximum transient growth Gmax = maxt≥0 G(t). Experience shows that for applications in hydrodynamic stability theory, both these bounds are accurate within a factor of about two. A lower bound of the maximum energy amplification Gmax is based on the Kreiss matrix theorem. We start by taking the Laplace transform of the solution of the initial value problem (2.1). ∞ ˜ = (L − sI)−1 q0 = e−st etL q0 dt. q = exp(tL)q0 → q 0

Taking bounds on each side results in ∞ (L − sI)−1 ≤ etL |e−st | dt 0

≤ ≤

∞

0

e−Real{s}t dt max etL t≥0

1 max etL , Real{s} t≥0

which leads to a lower bound for Gmax given by Gmax = max etL ≥ max Real{s}(L − sI)−1 . t≥0

Real{s}


117

This last expression measures how far the resolvent contours reach into the unstable half-plane. An upper bound can be derived in a fairly straightforward manner, again using the resolvent norm. We start with Cauchy’s integral formula applied to analytical matrix functions (in our case the matrix exponential); we get the expression " 1 etz (zI − L)−1 dz, etL = 2πi Γ where the closed contour Γ has to include the singularities of (zI − L)−1 , i.e. the spectrum (eigenvalues) of L. Applying norms on both sides results in " 1 (zI − L)−1 |dz|. etL ≤ 2π Λ We have thus reduced the norm of the matrix exponential to a pathintegral in the complex plane that includes the spectrum of L. To summarize so far, we have learned that analyzing and quantifying the behavior of non-normal operators requires somewhat more care than the equivalent analysis of normal operators. Whereas the behavior of normal operators is entirely given by its spectrum, the three tools (or, sets in the complex plane) that allow an accurate or approximate description of non-normal behavior are: (i) the numerical range, which governs the behavior in the limit t → 0+ , (ii) the resolvent, which allows upper and lower bounds on the maximum transient amplification, and (iii) the spectrum, which determines the behavior in the limit t → ∞. 2.5. Rayleigh–Benard convection versus Poiseuille flow — revisited The connection between initial energy growth and the maximum real part of the numerical range allows for an interesting observation that sheds light on the success of traditional and energy stability theory for the Rayleigh– Benard convection and on the failure of traditional and energy stability theory for the plane Poiseuille flow. The critical Reynolds number based on traditional linear stability theory is determined by an eigenvalue analysis: the lowest Reynolds number at which an eigenvalue crosses into the unstable half-plane is the critical Reynolds number.

118

P. J. Schmid

On the other hand, the critical Reynolds number based on the energy stability theory is determined by a numerical range analysis: the lowest Reynolds number at which the numerical range crosses into the unstable half-plane (and thus causes a positive initial energy growth rate) determines the critical energy Reynolds number in this case. The stability operator for Rayleigh–Benard convection is normal. From this fact we conclude that the numerical range is the convex hull of the spectrum. Consequently, the numerical range and the spectrum cross into the unstable half-plane at the same Rayleigh number (since the numerical range is “attached” to the spectrum). For this reason, the critical Rayleigh numbers based on linear and energy stability theory are identical for Rayleigh– Benard convection (Ra = 1708). The stability operator for plane Poiseuille flow is non-normal. As we have seen from the simple matrix example above, the numerical range is a much larger set than the convex hull of the spectrum; i.e. the boundary of the numerical range is “detached” from the spectrum. For this reason, the numerical range crosses into the unstable half-plane (causing positive initial energy growth) at Reynolds numbers where the spectrum is still confined to the stable half-plane (causing ultimate decay of energy for large times). This explains the wide discrepancy between the critical energy Reynolds number (Re = 49.6) and the critical linear Reynolds number (Re = 5772).

2.6. Link between normality/non-normality and sub/supercritical behavior The above observation can be further advanced by establishing a connection between the nature of the underlying linear stability behavior and the bifurcation behavior of the full nonlinear system. As a cautionary note, it has to be stressed that this argument is only valid for systems with energypreserving nonlinearities (as is the case for Rayleigh–Benard convection and plane channel flow) and for bifurcations from the linear state. In the presence of energy-preserving nonlinearities, we can achieve positive energy growth only by a numerical range that protrudes into the unstable half-plane. For normal operators, the numerical range is the convex hull of the spectrum, and a numerical range that protrudes into the unstable half-plane implies an unstable eigenvalue. In fact, we obtain positive energy growth (and thus a finite-amplitude state) at the same time as the (E = 0)state becomes unstable to infinitesimal disturbances. This is the definition

119


of supercritical bifurcation behavior. We therefore conclude that a normal linear operator behaves supercritically. On the other hand, subcritical bifurcation behavior is characterized by the presence of finite-amplitude states before the infinitesimal state becomes unstable. We thus have to require an energy-producing linear mechanism that acts at Reynolds numbers below the critical linear Reynolds number. In other words, the numerical range has to protrude into the unstable domain, at the same time as the spectrum is still confined to the stable half-plane. Clearly, the numerical range has to be “detached” from the spectrum — the defining characteristic of a non-normal operator. We therefore conclude that subcritical behavior requires a non-normal linear operator. 2.7. Application to plane Poiseuille flow In the following figures, we apply the tools introduced above to analyze the stability behavior of plane Poiseuille flow. Figure 5 shows the spectrum and resolvent contours for plane Poiseuille flow for streamwise and spanwise wavenumbers of α = 1, β = 0, respectively, and a Reynolds number of Re = 1000. We observe that the resolvent contours protrude into the unstable (shaded) half-plane which indicates the potential for transient amplification of initial energy. This is confirmed 7 0.2

1

6

1.5 0

5 -0.2

4

ci 3

-1 0

3

2.5

-0.6 -0.8

G(t )

3.5 -0.4

2 2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 cr

(a)

1

1

0

0

5

10

15

20 t

25

30

35

40

(b)

Fig. 5. (a) Spectrum and resolvent contours for plane Poiseuille flow with streamwise and spanwise wavenumber α = 1, β = 0 and Reynolds number Re = 1000; (b) Energy amplification G(t) versus time.

120

P. J. Schmid

200 0.2

1

180 160

0 1.5

2

140

-0.2 120

G(t )

ωi -0.4

100 80

-0.6

60 40

-0.8

20 -1 -0.5 -0.4 -0.3 -0.2 -0.1 0 ωr

0.1 0.2 0.3 0.4 0.5

(a)

0

0

50

t

100

150

(b)

Fig. 6. (a) Spectrum and resolvent contours for plane Poiseuille flow with streamwise and spanwise wavenumbers of α = 0, β = 2, respectively and Reynolds number of Re = 1000; (b) Energy amplification G(t) versus time.

in Fig. 5(b) where the computation of G(t) demonstrates that an energy amplification of nearly seven times is achievable at these parameter settings. Figure 6 shows the spectrum and resolvent contours for plane Poiseuille flow for streamwise and spanwise wavenumbers of α = 0, β = 2, respectively, and a Reynolds number of Re = 1000. The spectrum and the resolvent contours look markedly different in this case. We notice that the resolvent contours again protude far into the unstable half-plane. Figure 5(b) displaying the energy amplification G(t) versus time illustrates that energy growth of nearly 200 times is feasible for the given parameters. The significantly higher transient energy amplification of streamwise elongated perturbations (α ≈ 0) will be further discussed in the next section. 3. Transition to Turbulence In this section, we will try to suggest or establish a link between the route to turbulence known as bypass transition and the transient amplification of energy analyzed in the previous section. The very definition of bypass transition, i.e. transition that does not rely on or involve exponential instabilities, requires a linear process for the amplification of energy that operates at subcritical conditions. Based on the analysis in the previous section, we thus hypothesize that transient

121


amplification, due to the non-normal nature of the linearized Navier–Stokes equations for wall-bounded shear flows, plays an important role during the bypass transition process. It seems fitting then to investigate the transient growth mechanism as to its favoring of specific perturbation characteristics. We have already seen in Figs. 5 and 6 that perturbations with a zero streamwise wavenumber show a more significant amount of energy growth than disturbances with zero spanwise wavenumbers. 3.1. Pattern selection To make this observation more concrete, the maximum attainable energy amplification as a function of streamwise and spanwise wavenumbers has been computed. The results are shown in Fig. 7. The plot has been obtained for Blasius boundary-layer flow, but is qualitatively equivalent to, e.g., plane Poiseuille flow. As is apparent from Fig. 7, the maximum energy amplification is strongly slanted toward perturbations with zero or small streamwise 1 1000

0.9

800

1200 0.8 1400 600

0.7

β

0.6 0.5 400 0.4 0.3

200

0.2 unstable 0.1 0

0.05

0.1

0.15 α

0.2

0.25

0.3

Fig. 7. Contours of maximum amplification of energy for Blasius boundary-layer flow as a function of streamwise (α) and spanwise (β) wavenumbers.

122

P. J. Schmid

wavenumbers. An amplification maximum of more than 1400 times the initial energy is attained for perturbations independent of the streamwise direction (streamwise wavenumber α = 0) and a spanwise wavenumber β ≈ 0.7. Assuming an equidistribution of energy in the (α, β)-wavenumber plane by nonlinear terms, the linear mechanisms should strongly favor and amplify structures elongated in the streamwise direction. Moreover, due to their substantial amplification potential, the rise of these elongated structures should occur naturally without any high-energetic forcing. Computations of this type have motivated experiments to confirm the natural occurrence of streamwise elongated structures that best exploit the transient growth mechanism. Elofsson (1998) has conducted an experiment forcing a boundary layer with a pair of low-energetic oblique waves. Due to nonlinear interactions, energy is transferred into streamwise elongated structures, spanwise elongated structures (Tollmien–Schlichting waves) and higher-harmonic oblique waves. However, only the streamwise elongated structures exhibit a strong transient amplification (according to Fig. 7) which is reflected in the prevalence of streaks in the smoke visualization of Fig. 8. The spacing of the streaks does not coincide with the theoretical calculations, but it is difficult to deny that the occurrence and persistence of streamwise elongated structures are based on the strong transient amplification of perturbations with small and vanishing spanwise wavenumbers (Fig. 7).

Fig. 8. Smoke visualization of streak formation from a pair of oblique waves in a boundary layer. From Elofsson (1998).


123

Non-normal effects and transient energy growth also play an important role in the receptivity process. In other words, streamwise elongated structures are also favored when wall-bounded shear flows are forced by freestream turbulence or surface roughness. In this case, the linear analysis rests on the resolvent norm (rather than the norm of the matrix exponential) as a measure of maximum response to external forcing. Using simple scaling assumptions, it is straightforward to show that the resolvent norm, and thus the strongest response, is achievable for streamwise elongated structures. Consequently, by forcing a boundary in the freestream one should observe the rise of streaky structures in the boundary layer. Jacobs and Durbin (2001) have conducted an interesting numerical experiment to confirm this conclusion. In their simulation (Fig. 9), random fluctuations were seeded in the freestream (top panel) which decayed in the downstream direction. In the boundary layer (bottom panel), however, the rise of elongated structures can be observed. Further downstream, the streaks break down and form turbulent spots, which, even further downstream, led to fully-developed turbulent flow. A similar setup has been investigated in a physical experiment by Alfredsson and Matsubara (2000). Again, freestream disturbances have been introduced which swiftly generated streaky structures in the, otherwise unforced, boundary layer (Fig. 10). The agreement between the numerical simulation (Fig. 9) and experiment (Fig. 10) — including streak formation, turbulent spots and breakdown — is striking. 3.2. Transition thresholds From the previous section, it becomes clear that streak-like structures play a significant role in the bypass transition to turbulence. Due to the large amplification of this type of perturbations, a rather low initial energy is necessary to generate and maintain these structures. Once they reach

Fig. 9. Numerical simulation of the receptivity of a boundary layer to freestream disturbances. From Jacobs and Durbin (2001).

124

P. J. Schmid

Fig. 10. Experiment of boundary-layer receptivity to freestream disturbances. From Alfredsson and Matsubara (2000).

saturation at finite amplitudes, these elongated structures become susceptible to secondary instabilities. For large enough streak amplitudes, positive secondary growth rates occur, that quickly lead to breakdown and turbulent fluid motion. A secondary stability analysis for plane channel flow (Reddy et al., 1998) based on Tollmien–Schlichting waves optimal streamwise vortices or optimal oblique waves, showed that the two transition scenarios based on transient growth (optimal streamwise vortices and optimal oblique waves) required substantially lower energy to lead to transition than the scenario starting with Tollmien–Schlichting waves. Even by starting with random noise, transition to turbulence could be triggered with lower initial energies than by starting with Tollmien–Schlichting waves. The threshold amplitudes for various scenarios are shown in Fig. 11. To conclude, linear transient growth is an important component of the bypass transition process yielding a viable and low-energetic path to turbulence. A three-component transition model, based on transient amplification, streak breakdown and streamwise vortex regeneration, has been proposed by Hamilton et al. (1995) that was extracted from numerical simulations and condensed into low-dimensional dynamical systems (Waleffe, 1997). Alternatively, a breakdown process based on reverse jets observed in numerical simulations of boundary-layer flow has been proposed by Jacobs

125


Threshold for Transition in Poiseuille Flow

−1

10

−2

10

−3

(T–S) (2DOPT)

initial energy density

10

(N) −4

10

(SV) −5

10

(OW)

−6

10

−7

10

−8

10

3

4

10

10 Reynolds number

Fig. 11. Threshold energy density for transition in Poiseuille flow for various transition scenarios: (T–S) Tollmien–Schlichting waves, (2DOPT) two-dimensional optimal, (N) three-dimensional random noise, (SV) streamwise vortex and (OW) oblique waves. The circles correspond to data from simulations. The lines are fits to the data. From Reddy et al. (1998).

and Durbin (2001) as the mechanism responsible for the formation of spots from streamwise elongated structures. 4. Extensions Only the most generic or academic configurations fall within restrictions that allow the direct application of the above analysis tools. For more realistic configurations, adjustments have to be made. In the following sections, we will briefly discuss various extensions and touch upon techniques necessary to incorporate additional effects. 4.1. Time-dependent flows In the previous section, we have developed a mathematical framework for the analysis of the stability characteristics of time-steady flows. We

126

P. J. Schmid

especially concentrated on the additional techniques that are necessary for non-normal linear operators (Schmid and Henningson, 2001). In the following sections, we will relax the condition of time-steadiness and allow for time-periodic flow and for arbitrary time-dependence. Again, the focus will be on developing general tools for the analysis of non-normal stability operators. We will first consider evolution problems of the form d q = L(t)q, L(t + T ) = L(t), dt with T -periodic operator L. Again, we can formally write the solution in the form q = X(t)q0 with X(t) as the fundamental solution operator which propagates the initial condition q0 forward in time. Periodicity requires that X(t + T ) = X(t)C, where C is called the monodromy matrix. Floquet’s theorem states that the fundamental solution operator X(t) can be decomposed into a T -periodic part and an exponential part. We get X(t) = P(t)etB . The monodromy operator C governs the amplification or damping of solutions to the initial value problem over one period (of length T ). Generally, we compute the eigenvalues of C which we refer to as the Floquet multipliers. The reasoning about stability or instability then goes: if all eigenvalues of C fall inside the unit disk, the solutions are stable and the initial value problem is referred to as contractive. For one or more eigenvalues of C outside the unit disk, we experience instability. Following the argument for time-steady stability analysis, we reason that, unless the operator C is normal, looking at the eigenvalues of C may not capture short-term phenomena but only the long-term behavior. In mathematical terms, the mapping from one period to the next is formulated as qn = Cqn−1 = Cn q0 , where the index n denotes the nth period and q0 stands for the initial condition. Growth or decay of the norm of q is related to the power boundedness of the operator C (Trefethen and Embree, 2005). Following an earlier


127

derivation, we have the following bounds on Cn , ρn ≤ Cn ≤ SS−1 ρn , where ρ is the largest (in modulus) Floquet multiplier of L, or eigenvalue of C, and S is composed of the eigenfunctions of C. The condition |ρ| < 1 is a necessary and sufficient condition for asymptotic stability, i.e. stability in the limit n → ∞. For highly non-normal monodromy matrices C, the factor S S−1 is large, and substantial transient amplification may occur before the asymptotic behavior, governed by |ρ|, is observed. For the stable case we can estimate the maximum transient amplification using Cauchy’s integral formula. We get " 1 (zI − C)−1 |dz| Cn ≤ 2π C

with C as the unit circle. Again, the complete behavior of the resolvent formed with the monodromy operator captures transient effects; the singularities of the resolvent, i.e. the eigenvalues of C are insufficient to account for short-term phenomena. 4.1.1. A model problem To motivate and demonstrate the application of non-modal stability analysis to oscillatory flows, we consider Mathieu’s equation d2 y + (δ + ǫ cos t)y = 0. dt2 We proceed by invoking Floquet’s theorem that states that the solution to a linear equation with time-periodic coefficients can be decomposed into an exponential and a time-periodic part according to ak eikt . y(t) = eλt k

Clearly, the sign of λ determines the stability of the solution. Substituting this ansatz into Mathieu’s equation and reducing the second-order equation to a system of first-order equations, we arrive at a coupled eigenvalue problem for λ,

Ak ak = λ , (k 2 − δ)ak − 2ikAk − 12 ǫ(ak−1 + ak+1 ) Ak

which is solved by truncating at sufficiently high k. Figure 12(a) displays the familiar stability diagram for Mathieu’s equation in the (δ, ǫ)-parameter plane. For values of δ and ǫ that fall within

128

P. J. Schmid

2

104

1.8 1.6

103

GB (t )

1.4 1.2 ω

102

1 0.8 0.6

101

0.4 0.2 0 0.5

0

0.5

1

1.5

δ

(a)

2

100 0

50 100 150 200 250 300 350 400 450 500 n

(b)

Fig. 12. (a) Stability diagram for Mathieu’s equation; (b) Transient growth for Mathieu’s equation. ǫ = 1.47585, 1.4759, δ = 1.5. The dashed line represents the modal solution for the unstable parameter setting. From Schmid and Henningson (2001).

the unstable region, we expect exponentially growing solutions; whereas for δ and ǫ chosen from the stable region, purely oscillatory behavior prevails. Applying the transient growth analysis based on the algorithm given above yields the results shown in Fig. 12(b). We have chosen two cases. The first case (δ = 1.5, ǫ = 1.47585) is stable based on traditional Floquet analysis. The plot of GB (t) shows oscillatory behavior but also reveals an amplification of two orders of magnitude due to non-normal effects. Even for the (second) unstable case with δ = 1.5 and ǫ = 1.4759, we noticed a marked amplification, again about two orders of magnitude, before the exponential growth stemming from the positive Floquet exponent dominates for large times. The dashed line represents the modal solution in this case, and the difference between the solid line and dashed line can solely be attributed to non-modal effects.

4.1.2. Arbitrary time-dependence For arbitrary time-dependence, no simplifying decomposition (such as the one given by Floquet’s theorem) is known. To analyze the stability of unsteady flows, two methods are conceivable. The first method is the natural extension of the steady-flow analysis using the fundamental solution matrix rather than the matrix exponential.


129

Writing the formal solution of the time-dependent problem as q(t) = X(t)q0 , we can formulate the maximum growth equivalently as G(t) = max q0

X(t)q0 = X(t). q0

Correspondingly, the optimal disturbance at a given time tspec satisfies X(tspec )q0 = X(tspec )q(tspec ) which can be solved for q0 using the singular value decomposition (SVD). Identical to the time-steady case, we obtain the optimal initial condition as the right principal singular vector of X(tspec ). The above procedure requires the computation of the fundamental solution operator and therefore relies on an efficient algorithm to obtain it. Recently, Iserles et al. (2000) suggested a family of methods based on Magnus’ technique and Lie-brackets. A fourth-order accurate method for computing X(t) reads X(t → t + ∆t) = eσ ,

√ 3 2 1 ∆t [L2 , L1 ], σ = ∆t(L1 + L2 ) + 2 12 √

1 3 − L1 = L t + ∆t , 2 6 √

1 3 + L2 = L t + ∆t , 2 6

with [., .] as the commutator. Other schemes within the same family are given in Iserles et al. (2000). An alternative and elegant method of computing time-dependent stability results uses adjoint operators. Starting with the definition of maximum (energy) amplification, and using the definition of the adjoint operator, we obtain (Xq0 , Xq0 ) G2 (t) = max q0 (q0 , q0 ) = max q0

+

(X+ Xq0 , q0 ) , (q0 , q0 )

where X denotes the operator adjoint to X. We identify the last expression above as the Rayleigh quotient of the composite operator X+ X. In addition, we notice that this composite operator is self-adjoint. The maximum of G2 (t) over all non-zero q0 is achieved

130

P. J. Schmid

for the principal eigenvector of X+ X. For this reason, we can reformulate the above Rayleigh quotient into the equivalent eigenvalue problem X+ Xq0 = λq0 , and G2 (t) is given by the largest eigenvalue λmax . Since it is only the largest eigenvalue of X+ X we are interested in, we can use the familiar power iteration method to extract it. Power iteration, is an iterative procedure that successively applies the operator X+ X to a given initial condition; after many iterations, the eigenfunction corresponding to the largest eigenvalue (in magnitude) will emerge. One iteration step in this procedure can be written as (n+1)

q0

(n)

= ρ(n) X+ Xq0 ,

with (n) as the iteration count. The scaling factor ρ has been introduced to avoid excessively large or small amplitudes. We break one iteration into two steps: 1. Given the initial condition q0 , we apply the fundamental solution operator X to obtain the result q(T ) at a given time T. 2. We then apply the adjoint operator X+ to the outcome of step (l) to arrive at the new input q0 (appropriately scaled) for the next iteration. Computationally, we do not explicitly form the fundamental solution operator X(t), nor its adjoint X+ (t). Instead, we recognize the fact that applying X(T ) to an initial condition is equivalent to solving the initial value problem (forward in time) up to time T starting with q0 . Analogously, applying X+ (0) to a terminal condition q(T ) is equivalent to solving the adjoint initial value problem (backward in time) up to time t = 0 starting with q(T ). 4.1.3. Energy growth rate and Lyapunov exponent In the presence of arbitrary time-dependence, diagnostic measures of energy amplification have to be modified from the ones used for steady flows (Farrell and Ioannou 1996b; Schmid and Henningson, 2001). The instantaneous energy growth rate is defined as the logarithmic derivative of the energy with respect to time. We have

E qH (L+ + L)q d 1 dE ln = . = dt E0 E dt qH q In the last expression, we recognize the Rayleigh quotient for L+ + L from which we deduce that the maximum energy growth rate is given by using


131

the principal eigenfunction of L+ + L whereas the minimum energy growth rate is determined by using the eigenfunction associated with the smallest eigenvalue. Consequently, we obtain the following time-dependent bounds on the energy growth rate:

E d ln ≤ λmax (t). λmin (t) ≤ dt E0 Another familiar quantity in the study of evolution problems is the ¯ which is defined as Lyapunov exponent Λ ¯ = lim sup lnX Λ t→∞ t with X as the fundamental solution operator. Relating the Lyapunov exponent to the previous expression for energy growth rate we get

t t E ≤ λmax (s) ds, λmin (s) ds ≤ ln E0 0 0 and thus the following bounds on the Lyapunov exponent #t #t 0 λmin (s) ds ¯ ≤ lim sup 0 λmax (s) ds . lim sup ≤Λ t→∞ t→∞ 2t 2t For the special case of a time-periodic system, one obtains ¯ = lim sup lnX Λ t→∞ t = lim sup t→∞

lnP(t)etB t

lnP(tmax )etB t→∞ t = λmax (B),

= lim

where tmax is the time at which P(t) reaches its maximum. 4.2. Spatial theory Up to this point, our analysis of the stability of shear flows has been temporal, i.e. we described and quantified the evolution of a perturbation measure in time. In many situations (encountered in physical experiments), the evolution of a perturbation is more appropriately described in a spatial framework. Disturbances introduced by a vibrating ribbon, shedding off a roughness element or developing from freestream perturbations, are classical examples where a spatial framework is preferred over a temporal one.

132

P. J. Schmid

Mathematically, the spatial framework is more complicated as it requires the reformulation of the governing equations as an evolution equation in one spatial coordinate. Referring back to the section on arbitrary timedependence, we will look at a spatial evolution problem of the form ∂ q = A(x)q, ∂x where we assume a known dependence of the spatial evolution operator A on x. The equations governing the evolution of steady disturbances in a growing boundary layer (exploiting the boundary layer approximation) fall into this category. They are given as ux + vy + wz = 0, (U u)x + V uy + Uy v = uyy − β 2 u,

(V u + U v)x + (2V v)y + βV w + py = vyy − β 2 v,

(U w)x + (V w)y − βp = wyy − β 2 w,

with β as the spanwise wavenumber. The boundary layer approximation ensures that the above equations are well-posed as an evolution problem in the streamwise direction x. We choose the limit of large Reynolds numbers and adopt an input-output formulation (in primitive variables) by choosing the input and output vectors of the form uin = (0, v0 , w0 ),

uout = (u1 , 0, 0),

i.e. the initial disturbance consists of the normal and spanwise velocity components only, whereas the output perturbation has significant contributions only in the streamwise direction. An evolution operator X will map the input vector onto the output vector (solving the boundary layer equations above). We can write

v , u1 = Bq, q = w and the spatial amplification then follows the same steps as the temporal amplification derived earlier. We have u1 (X q, X q) (X + X q, q) G(x) = max = max = max q q q Re q (q, q) (q, q)

with X + as the operator adjoint to X . The extra factor of 1/Re stems from the boundary layer scaling of the streamwise velocity. Due to the explicit appearance of the Reynolds number, the evaluation of the right-hand side above becomes independent of Re.

133


3 4 x10

3.5 3

G/Re

2.5 2 1.5 1 0.5 0

0

1

2

3

4

5

6

7

8

9

10

11

xf Fig. 13. Maximum spatial transient growth versus streamwise distance. From Andersson et al. (1999).

As in the temporal case for arbitrary time dependence, we do not form the operators X (x) and X + (x) explicitly. We rather solve the spatial initial value problem forward in space from the starting location x0 to a final location xf . The output of this forward integration is then used to determine the input for the backward integration of the adjoint initial value problem from xf to x0 . Another mapping of the output of the adjoint equations to the input of the boundary layer equations is employed before the next iteration commences. Applying the iterative procedure, using the direct and adjoint boundary layer equations, yields the results displayed in Fig. 13 (Andersson et al., 1999). As in the temporal case, we notice strong transient growth in the spatial direction followed by exponential decay farther downstream. 4.3. Inhomogeneous directions Despite their value as test beds for developing analysis tools and gaining experience, generic flows in simplified geometries (such as plane channels, pipes, boundary layers, etc.) are far removed from the complexities encountered in realistic flow configurations. Flows with, e.g., weakly parallel mean velocity profiles can successfully be treated by a quasi-local

134

P. J. Schmid

perturbation analysis. For stronger inhomogeneities, different methods have to be brought to bear. For flows in complex domains, a global stability analysis is necessary where the inhomogeneous directions become “eigendirections”. The resulting eigenfunctions (global modes) are entire flow fields that capture the dynamics of perturbations superposed on an inhomogeneous mean field. In general, global stability analysis requires substantially larger computational resources, and the diagonalization of the linearized equations based on direct methods (such as the QR-method) quickly becomes prohibitively expensive and inefficient. Iterative schemes are favored for a global stability analysis. Just as for simplified geometries, flows in complex geometries also have to be investigated as they may support a perturbation dynamics that is not captured by eigenvalue analysis alone. In analogy with previous sections, we are interested in the dynamics of a superposition of global modes rather than the dynamics of individual global modes. In an early paper, Cossu and Chomaz (1997) have shown that localized convective instabilities are possible by a linear combination of stable global modes. We will illustrate this finding by analyzing the stability of a simple model describing the oscillatory motion of a falling liquid sheet (Schmid and Henningson, 2002). 4.3.1. Self-sustained oscillations of a liquid curtain A thin two-dimensional liquid curtain falling under gravity encloses an air cushion on one side that is assumed to be weakly compressible (see Fig. 14 for a sketch of the geometry). It is well known that this flow configuration exhibits self-sustaining oscillations which can lead to significant noise levels and/or structural damage. The source of these oscillations can be determined as the coupling of a fluid instability to a feedback mechanism due to the pressure variations in the air pocket. A simplified model, capturing the main mechanisms, can be formulated in terms of an integro-differential initial-boundary value problem of the form ∂v u ¯(x) 1 ∂v + u¯(x) = −κ f (x)dx, ∂t ∂x U 0 ∂f ∂f +u ¯(x) =v ∂t ∂x with v and f as the lateral velocity and displacement of the sheet, and u¯(x) as the vertical mean velocity.

135


U H y B h (x, t )

g L

f (x, t ) v u air cushion

x Fig. 14. (2002).

Sketch of the geometry for a falling liquid sheet. From Schmid and Henningson

Applying the standard modal approach, we obtain an integro-differential eigenvalue problem. The spectrum for a representative parameter combination is displayed in Fig. 15. The (global) eigenfunctions for the first, second, third, fifth and tenth eigenvalues are depicted in Fig. 16 showing the familiar structure of increasing zero-crossings. For the chosen configuration, the frequency of the least stable mode can be computed as f = 37.3 Hz. Experimental efforts for the same parameter values have reported a robust frequency selection mechanism at a value of 4.1 Hz, nearly one order of magnitude lower than the theoretical value. This discrepancy between stability theory and measurements has been observed over a wide range of parameters. Based on the first 44 global eigenfunctions, a superposition of modes has been designed to optimize the transient growth of the sheet’s kinetic energy and its compression work exerted on the air cushion. Based on

136

0.05 0 −0.05 −0.1 −0.15 −0.2

P. J. Schmid

−150

−100

−50

0 ωr

50

150

100

0

0

0.1

0.1

0.1

0.1

0.1

0.2

0.2

0.2

0.2

0.2

0.3

0.3

0.3

0.3

0.3

0.4

0.4

0.4

0.4

0.4

0.5

0.5

0.5

0.5

0.5

x

0

x

0

x

0

x

x

Fig. 15. Spectrum for a falling liquid sheet for κ = 5 × 104 and U = 0.4. From Schmid and Henningson (2002).

0.6

0.6

0.6

0.6

0.6

0.7

0.7

0.7

0.7

0.7

0.8

0.8

0.8

0.8

0.8

0.9

0.9

0.9

0.9

0.9

1

1

1

1

1

Fig. 16. Selected eigenfunctions for liquid curtain flow with κ = 5 × 104 and U = 0.4. From left to right: 1st, 2nd, 3rd, 5th and 10th eigenfunctions (sorted by ascending positive phase velocity). From Schmid and Henningson (2002).

this optimization, a simple numerical simulation revealed a perturbation dynamics that is significantly different from the one predicted by a global eigenvalue analysis. Figure 17(a) shows the energy amplification as a function of time. The corresponding pressure signal in the air cushion (Fig. 17(b)) shows strong peaks at a distinct frequency. This frequency has been determined as 4.2 Hz, in close agreement with the experimental value (4.1 Hz). The numerical simulation shown in Fig. 18 reveals a localized (wavepacket) solution propagating down the liquid sheet. As this localized disturbance reaches the bottom of the sheet, a global deformation of the curtain is observed which in turn triggers the next wavepacket. It is very interesting

137


2 103 1.5 1

102

p

G

0.5 101

0

−0.5 −1

100

−1.5 10−1

−2 0

0.5

1 1.5 t / Tfall

2

2.5

(a)

0

0.5

1

1.5 t / Tfall

2

2.5

3

(b)

Fig. 17. Multimodal stability analysis of a falling liquid sheet for κ = 5 × 104 and U = 0.4. (a) Optimal energy amplification G(t) versus time; (b) Pressure difference versus time, obtained from the linear evolution of the optimal initial condition. From Schmid and Henningson (2002).

t / Tfall

Fig. 18. Curtain shape versus time for κ = 5 × 104 and U = 0.4, starting with the optimal initial condition. From Schmid and Henningson (2002).

138

P. J. Schmid

and instructive to realize that the optimal perturbation does not resemble, in any way, the global eigenfunction structure. Rather, the relevant dynamics is governed by a multimode phenomena — a wavepacket formed by the superposition of many global eigenfunctions. As before, the behavior of linear perturbations is not captured by the (global) spectrum alone.

4.3.2. Pseudo-wavepacket solutions A conclusion to draw from the work of Cossu and Chomaz (1997), and the liquid curtain model above is that the observed perturbation dynamics can be significantly different from the exponential behavior predicted by individual global eigenmodes. A superposition of global modes, often resulting in localized structures, is a more appropriate representation of the disturbance dynamics. Chapman (2002) and Trefethen (2004) have recently shown that linear stability operators, or, more generally, variable-coefficient differential operators, can support localized wavepacket solutions that satisfy the differential equation and/or the boundary conditions up to an exponentially small error ε. These ε-pseudo-wavepacket modes describe the perturbation behavior more efficiently than global modes and can be computed using a WKB-approach (Chapman, 2002) or arise as a result of a winding number argument based on the operator’s symbol (Trefethen, 2004). If appropriate conditions are satisfied, regions in the complex plane can be identified where the resolvent norm is exponentially large; in these regions, ε-pseudowavepacket modes exist. Many shear flows with inhomogeneous coordinate directions await an analysis in terms of ε-pseudo-wavepacket modes; new insight into the dynamics of localized disturbances is expected from such an analysis.

5. Summary and Conclusions Non-normality is prevalent in many wall-bounded shear flows and must not be ignored in a stability analysis of such flows. The presence of nonnormality requires an extended spectral analysis that involves not only the spectrum, but also the resolvent norm and the numerical range of the stability operator. A significantly different short-term perturbation dynamics is often observed that cannot be described by a normal mode analysis; rather a non-modal or multimodal approach has to be taken to capture the short-time behavior of the linearized stability operator.


139

Even for more complex flows, e.g., spatially evolving flows, unsteady flows or highly non-parallel flows, a non-modal view point captures more accurately the dynamics of infinitesimal perturbations. Pseudomodes — nearly solutions, but not near solutions (Domokes and Holmes, 2003) — are just as important as exact solutions to the dispersion relation; in fact, many times an inexact analysis of hydrodynamic stability problems reveals more about the perturbation dynamics than an exact eigenvalue calculation. It is hoped that the examples and tools discussed in this article have demonstrated this fact and given the reader the means and motivation to assess and quantify the stability behavior of wall-bounded shear flows. The work was partially done while the author was visiting the Institute for Mathematical Sciences, National University of Singapore in 2004. The visit was supported by the institute. The author wishes to thank Her Mann Tsai and Mohamed Gad-el-Hak for their invitation, support and hospitality.

References 1. P. Alfredsson and M. Matsubara, Freestream turbulence, streaky structures and transition in boundary layer flows, AIAA pap. (2000) 2000–2534. 2. P. Andersson, M. Berggren and D. Henningson, Optimal disturbances and bypass transition in boundary layers, Phys. Fluids 11 (1999) 134–150. 3. K. Butler and B. Farrell, Three-dimensional optimal perturbations in viscous shear flows, Phys. Fluids A 4 (1992) 1637–1650. 4. S. Chapman, Subcritical transition in channel flows, J. Fluid Mech. 451 (2002) 35–97. 5. C. Cossu and J.-M. Chomaz, Global measures of local convective instabilities, Phys. Rev. Lett. 78(23) (1997) 4387–4390. 6. G. Domokes and P. Holmes, On nonlinear boundary-value problems: ghosts, parasites and discretizations, Proc. Roy. Soc. Lond. A 459 (2003) 1535–1561. 7. P. Elofsson, Experiments on oblique transition in wall-bounded shear flows, PhD thesis, Royal Institute of Technology (KTH), Stockholm, Sweden (1998). 8. B. Farrell and P. Ioannou, Generalized stability theory. I. Autonomous operators, J. Atm. Sci. 53 (1996a) 2025–2040. 9. B. Farrell and P. Ioannou, Generalized stability theory. II. Nonautonomous operators, J. Atm. Sci. 53 (1996b) 2041–2053. 10. J. Hamilton, J. Kim and F. Waleffe, Regeneration of mechanisms of near-wall turbulence structures, J. Fluid Mech. 287 (1995) 317–348. 11. R. Horn and J. Johnson, Topics in Matrix Analysis (Cambridge University Press, 1991). 12. A. Iserles, H. Munthe-Kaas, S. Norsett and A. Zanna, Lie-group methods, Acta Numerica (2000) 215–365. 13. R. Jacobs and P. Durbin, Simulations of bypass transition, J. Fluid Mech. 428 (2001) 185–212.

140

P. J. Schmid

14. S. Reddy and D. Henningson, Energy growth in viscous channel flows, J. Fluid Mech. 252 (1993) 209–238. 15. S. Reddy, P. Schmid, J. Baggett and D. Henningson, On stability of streamwise streaks and transition thresholds in plane channel flows, J. Fluid Mech. 365 (1998) 269–303. 16. E. Reshotko, M. Morkovin and T. Herbert, Transition in open flow system: a reassessment, Bull. Amer. Phys. Soc. 39 (1994) 1882. 17. P. Schmid and D. Henningson, Stability and Transition in Shear Flows (Springer-Verlag, 2001). 18. P. Schmid and D. Henningson, On the stability of a falling liquid curtain, J. Fluid Mech. 463 (2002) 163–171. 19. L. Trefethen, Wavepacket pseudomodes of variable coefficient differential operators, Lond. Math. Soc., submitted (2004). 20. L. Trefethen and M. Embree, Spectra and Pseudospectra: The Behavior of Non-Normal Matrices and Operators (Princeton University Press, 2005). 21. L. Trefethen, A. Trefethen, S. Reddy and T. Driscoll, Hydrodynamic stability without eigenvalues, Science 261 (1993) 578–584. 22. F. Waleffe, On a self-sustaining process in shear flows, Phys. Fluids 9 (1997) 883–900.

A FRAMEWORK FOR CONTROL OF FLUID FLOW

Alan Guégan∗ , Peter J. Schmid† and Patrick Huerre‡ Laboratoire d’Hydrodynamique (LadHyX) ´ CNRS-Ecole Polytechnique F-91128, Palaiseau, France E-mails: ∗[email protected] † [email protected] ‡ [email protected] A mathematical framework is presented that allows the efficient calculation of stability properties and control strategies for fluid flows. The framework is based on an optimization scheme using a variational formulation. Adjoint equations and optimality conditions are used to compute the necessary gradient information and to advance the cost objective to an extremum. The proposed scheme is then applied to swept attachment-line boundary layers, and disturbances favored by the nonhomogeneous mean flow are determined together with their temporal evolution. The same optimization scheme is then used to apply an optimal blowing/suction strategy to minimize the rise of previously identified instabilities. Contents 1 2

3

Introduction A Mathematical Framework for Flow Control 2.1 Mathematical preliminaries and notation 2.2 The Lagrangian: From constrained to unconstrained optimization 2.3 Control objectives 2.4 Finding the stationary points of the Lagrangian 2.5 Optimization procedure Application to Swept Hiemenz Flow 3.1 Numerical methods 3.2 Optimal perturbations 141

142 142 143 145 145 146 150 151 155 156

142

A. Gu´ egan, P. J. Schmid and P. Huerre

3.2.1 Optimal energy growth 3.2.2 Catalytic role of the chordwise velocity component 3.3 Optimal control 3.3.1 Control of optimal perturbations: Effect of control time Tw 3.3.2 Control of optimal perturbations: Effect of the control penalty 3.3.3 Control of linearly unstable flow 4 Conclusions References

156 158 159 159 161 163 163 164

1. Introduction Recent advances in flow control theory have demonstrated that it is possible to enforce specific properties on flows of engineering interest, with very little energy input, by blowing/suction at the wall or other localized controlled perturbations. For instance, relaminarization has been achieved by Bewley et al. (2001) in parallel wall-bounded flows thereby inducing a significant drag reduction. These results show great promise for drag reduction, energy savings, noise suppression or alleviation of structural fatigue on airplane wings. The general idea of control is to design a cost or objective functional I expressing the control objectives, such as energy or drag reduction, and then search for the extrema of this functional. In this article, we will use the energy growth of initial perturbations over a given finite time interval as the cost functional. The optimal perturbation problem consists of finding the maxima of I, whereas the optimal control is achieved when a minimum of I is attained. Several optimization formulations are available, which are reviewed in Bewley (2001). The procedure applied here is referred to as the adjointbased method: starting from a guess value of the optimal perturbation (or optimal control strategy), one computes the gradient of the cost functional with respect to the initial perturbation (or the control variable) which is accomplished using adjoint fields. This gradient is subsequently used in an optimization algorithm and an iterative technique is employed until an optimum is reached within a prescribed error tolerance. 2. A Mathematical Framework for Flow Control In this section, we will introduce a mathematical framework for computing the optimal energy amplification and optimal control strategy for wallbounded shear flows. The goal is either to find the initial perturbation that exhibits the maximum energy growth over a given time interval [0, T0 ], or

A Framework for Control of Fluid Flow

143

minimize the energy growth of a given initial perturbation over a given time interval [0, Tw ] by applying wall-normal blowing or suction. 2.1. Mathematical preliminaries and notation Our notation follows a general pattern where the subscript w stands for functions which are defined at the wall, i.e. y = 0, over the time interval [0, T ]. The subscripts 0 or W designate functions of y defined at a fixed time (0 or T ). Functions with no subscripts are functions of space and time, defined on the entire space and time intervals. The variable q stands for the state vector, i.e. a vector fully describing the dynamics of the flow at a given point and time. For instance, in standard primitive variables, we have q = (u, v, w, p) for incompressible flow. The adjoint functions and operators are identified by a tilde ( ˜ ), while an asterisk ( ∗ ) denotes the conjugate transpose for both vectors and matrices. The prefix δ is used for test functions. It is tacitly assumed that all functions are sufficiently differentiable and satisfy the boundary conditions at infinity. Furthermore, the boundary conditions at the wall, when not part of the control variables, are assumed to be satisfied. Suitable scalar products have to be introduced for the optimization procedure. We define q1∗ · q2 dy dt + c.c., (q1 , q2 ) = L

[q1 , q2 ] = q1 , q2 = [[q1 , q2 ]] = q1 , q2 =

T

L

q1∗ · q2 dy + c.c.,

T

q1∗ · q2 dt + c.c.,

L

q1∗ · M · q2 dy + c.c.,

T

q1∗ · M♦ · q2 dt + c.c.

The first scalar product applies to functions of space and time; the second and third inner products apply, respectively, to functions of space and time only. The second inner product is closely related to the standard inner product for functions q1 · q2 = (u∗1 u2 + v1∗ v2 + w1∗ w2 ) dy + c.c. L

For the case that the components of q1 and q2 satisfying the continuity equation given (here for swept Hiemenz flow) as u + ∂v/∂y + ikw = 0, the

144


above expression reads 1 ∗ ∗ ∗ u1 u2 + v1 v2 + 2 (u1 + ∂y v1 ) (u2 + ∂y v2 ) dy + c.c. q1 · q2 = k L Provided that the no-slip boundary condition is satisfied at the wall, integration by parts yields q1 · q2 = [[q1 , q2 ]]. In order to have [[q, q]] represent the kinetic energy of the flow, M has to be defined as   ∂ ∆ 1  ∂y  , M = − 2  ∂  k −1 − k 2 − ∂y

where ∆ stands for the Laplacian operator ∂ 2 /∂y 2 − k 2 . The . , . inner product is used to express the energy of the control in the cost function. The operator M♦ has been introduced to allow weighing in time; for example, significant weight can be applied during the first few time steps so as to penalize a too abrupt initiation of control efforts. In addition, this commonly used technique (Corbett and Bottaro, 2001a; Corbett and Bottaro, 2001b) also avoids numerical difficulties. Let us introduce yet another scalar product {. , .} as

1 2 , qw {Q1 , Q2 } = (q 1 , q 2 ) + q01 , q02 + qw

1 2 1 2 1 2 + (˜ q , q˜ ) + q˜0 , q˜0 + ˜ . qw , q˜w As we will see below, this composite inner product is essential for the computation of the gradient of the Lagrangian. A crucial step in the formulation of the optimization problem involves the derivative of the Lagrangian functional with respect to the direct and adjoint variables, and care has to be exercised when problems are to be avoided at this step. Optimization theory is mainly based on the concept of differentiation in the sense of Frechet. In this article, however, we use the more restrictive Gateaux differentiation as the basis of our optimization scheme. A function f (x) is differentiable in the sense of Gateaux at a point x if there exists a linear form df |x such that for all h, we have lim

ε→0+

f (x + εh) − f (x) = df |x (h). ε

The linear form df |x is called the differential of f at point x. With (. , .) as the scalar product, a unique vector ∇(x) exists such that df |x (h) = (∇(x), h). This vector is referred to as the gradient of f at point x.


145

2.2. The Lagrangian: From constrained to unconstrained optimization With scalar product and differentiation defined, we can now formulate the optimization problem and establish a link between optimal solutions and the stationary points of the Lagrangian (Abergel and Temam, 1990; Gunzburger, 1997). It is convenient to write the equations governing the state vector q in the form of:

∂j q ∂ iq ∂ Ai i + Bj j = 0, Linearized Navier–Stokes F (q) = ∂t ∂y ∂y equations (2.1) Initial conditions Boundary conditions

G(q, q0 ) = q(y, 0) − q0 (y) = 0,

H(q, qw ) = q(0, t) − qw (t) = 0.

(2.2) (2.3)

In view of the application of the above formulation to the attachmentline boundary layer flow, it is assumed that the boundary conditions at infinity (y → ∞) are satisfied by the functions under consideration. The constrained problem can then be recast into an unconstrained optimization problem by introducing the Lagrangian q , F (q)) − [˜ q0 , G(q, q0 )] L(q, q0 , qw , q˜, q˜0 , q˜w ) = I − (˜ − ˜ qw , H(q, qw )

(2.4)

with I as the cost functional. The Lagrangian represents a reformulation of the optimization problem where the constraints are not imposed, rather, they are simply penalized. The Lagrangian L has to be regarded as a functional of Q = (q, q0 , qw , q˜, q˜0 , q˜w ), and the quantities q, q0 , qw , q˜, q˜0 , q˜w are assumed to be independent of each other, i.e. they may not satisfy the governing equations. The solutions to the optimization problem may be found at the stationary points of the Lagrangian. Adjoint-based optimization techniques are commonly designed with the intention of finding these points using efficient iterative algorithms. 2.3. Control objectives Before finding the stationary points of the Lagrangian L, we need to decide on the specific form of the cost functional I. We will use the following cost functional E(T ) l2 + qw , qw . (2.5) I= E(0) 2

146


The operators M and M♦ are chosen according to the quantities one seeks to optimize: if q(y, t) = (u(y, t), v(y, t), w(y, t)), setting M to the identity operator means that [[q0 , q0 ]] is the energy of the initial perturbations. If M♦ q = (0, v, 0), then qw , qw is the energy exerted in wall-normal blowing/suction during the optimization time interval [0, Tw ]. The term E(T )/E(0) in the cost functional stands for the energy amplification over the optimization time interval [0, T ]. This term alone constitutes a valid cost functional and is commonly used to calculate optimal perturbations. For control problems, on the other hand, the control energy as well as the perturbation energy should be minimized, which is accomplished by including the last two terms in the cost functional. Thus, the second term in I represents the mean perturbation energy and the third term measures the control effort. The quantity l is a parameter that accounts for the control energy penalty. A large value of l penalizes the control and leads to a parsimonious control strategy. 2.4. Finding the stationary points of the Lagrangian Under sufficient regularity assumptions, differentiating the Lagrangian L in the sense of Gateaux consists of computing the effect of a small change in the vector Q on L(Q). This operation can be interpreted as finding the tangent linear form dL at point Q by computing its action on test vectors δQ. The Gateaux differential of L at point Q reads

L(Q + εδQ) − L(Q) . (2.6) ε By definition, at the stationary points Qs of the Lagrangian, dL(δQ) is equal to zero, independent of the choice of δQ. From the above expression, it is possible to deduce gradient information that indicates, starting from a given guess for the control variables, the direction in which one is likely to find stationary points of the underlying Lagrangian. We will follow Gunzburger (1997) and present an intuitive explanation why the solutions of the constrained optimization problem are equivalent to the stationary points of the Lagrangian. At the stationary points, a small variation of the adjoint variables has no effect on the Lagrangian to first order which is equivalent to stating that the functionals F , G, H are zero over the space (and/or time) interval. Hence the equations of the direct problem are recovered. dL(δQ) = lim

ε→0


147

Suppose that the Lagrangian can be evaluated at the point Q + δQ given by Q + δQ = (q + δq, q0 + δq0 , qw + δqw , q˜ + δ q˜, q˜0 + δ q˜0 , q˜w + δ q˜w ). Recalling the definition of the Lagrangian, we obtain δI = δL + δ(˜ q , F (q)) + δ[˜ q0 , G(q, q0 )] + δ˜ qw , H(q, qw ) which further reduces to δI = {∇q0 L, δq0 } + {∇q L, δq} + {∇qw L, δqw }

+ {∇q˜L, δ q˜} + {∇q˜0 L, δ q˜0 } + {∇q˜w L, δ q˜w }

+ (δ q˜, F ) + [δ q˜0 , G] + δ q˜w , H + (˜ q , (∇q F, δq))

+ [˜ q0 , [∇q0 G, δq0 ] + [∇q G, δq]] + ˜ qw , ∇q H, δq + ∇qw H, δqw .

(2.7)

With (q, q0 , qw ) as a solution of the linearized Navier–Stokes equations and associated boundary conditions, the three functions F , G, H are equal to zero and the third line on the right-hand side of (2.7) vanishes, and, as can be inferred from the second line, the Lagrangian is stationary with respect to the three adjoint variables q˜, q˜0 , q˜w . If (δq, δq0 , δqw ) are chosen such that (q + δq, q0 + δq0 , qw + δqw ) also satisfy the linear system of equations, the last three lines on the right-hand side will vanish and (2.7) will reduce to δI = {∇q0 L, δq0 } + {∇qw L, δqw },

(2.8)

or, in gradient notation, {∇q0 I, δq0 } + {∇qw I, δqw } = {∇q0 L, δq0 } + {∇qw L, δqw }.

(2.9)

Solutions to the optimization problem are extrema of the cost functional I with respect to the control variables (δq0 , δqw ). Equation (2.9) shows that these solutions lie at stationary points of the Lagrangian. In more practical terms, at points where the linear system and the associated boundary conditions are satisfied, the gradients of the cost functional with respect to the control variables coincide with the gradients of the Lagrangian. Contrary to the gradients of I, the latter gradients are easily accessible through differentiation in the unconstrained space in which L has been defined.

148


Differentiating L with respect to the adjoint variables, one obtains dL(δ q˜) = −(δ q˜, F (q)),

dL(δ q˜0 ) = −[δ q˜0 , G(q, q0 )],

dL(δ q˜w ) = −δ q˜w , H(q, qw ) . A standard linear solver can be used to ensure that the Lagrangian is stationary with respect to any of the adjoint variables, by setting F , G, H to be identically zero, thus recovering the equations governing the direct problem. The derivative of the Lagrangian with respect to the control variable qw produces the following expression dL(δqw ) = l2 qw , δqw + ˜ qw , δqw .

(2.10)

According to the definitions of the gradient ∇qw and of the {. , .} inner product, Eq. (2.10) is equivalent to qw , δqw . ∇qw , δqw = l2 qw , δqw + ˜ When the blowing at the wall vanishes at t = 0, the single-bracket product Tw may be converted into a double-bracket product since (−1) (−1) qw , δqw = qw , M♦ M♦ δqw = qw , M♦ δqw . The operator M♦ can easily be chosen to be self-adjoint. Consequently, the single-bracket term in (2.10) is equal to M♦−1 q˜w , δqw , and (2.10) is equivalent to ∇qw , δqw = l2 qw , δqw + M♦−1 q˜w , δqw . It follows that the gradient of the Lagrangian with respect to the control qw at point (q0 , qw ) reads (−1)

∇qw = l2 qw + M♦

q˜w .

(2.11)

With the gradient ∇qw at point Q determined, one may use a gradientbased optimization algorithm to find a stationary point Qs , starting from an arbitrary point Q. At the stationary points, the gradient vanishes resulting in the first optimality condition (−1)

l2 qw + M♦

q˜w = 0.

Differentiation with respect to the initial perturbation q0 yields ET q0 , δq0 ], dL(δq0 ) = −2 2 [[q0 , δq0 ]] + [˜ E0 which is equivalent to ET (−1) [[∇q0 , δq0 ]] = −2 2 [[q0 , δq0 ]] + [[M q˜0 , δq0 ]]. E0

(2.12)

(2.13)

149


The explicit expression for the gradient of the Lagrangian with respect to the initial perturbation (2.14) thus reads (−1)

∇q0 = M

q˜0 .

(2.14)

The derivative with respect to the direct state variable q remains to be computed. The procedure is rather tedious but may be summarized in the following way: when differentiating the (˜ q , F (q)) term in the Lagrangian, spatial and temporal derivatives of the test function δq result. Repeated integration by parts then transforms the derivatives of δq onto the adjoint variable q˜. Finally, invoking the fact that the derivative of L must be zero, one finds the equation for the adjoint variables (−1)i+1

∂ j (Bj∗ q˜) ∂ i+1 (A∗i q˜) j + (−1) = 0. ∂y i ∂t ∂y j

(2.15)

The adjoint boundary conditions are found at the same time. The adjoint equation together with the adjoint boundary conditions make up the adjoint system. It is worth pointing out that the adjoint equation has to be solved backward in time using the adjoint terminal condition (−1)i

1 ∂(A∗i q˜) (T ) = M q(T ). ∂y i E0

(2.16)

Expressions for q˜0 and q˜w as functions of q˜ can also be determined by differentiation with respect to q, which yields the gradients of the Lagrangian with respect to the initial perturbation and control i ∗ E(T ) ˜) −1 ∂ (Ai q i −2 q0 , (2.17) ∇q0 = (−1) M ∂y i t=0 E(0)2 ' & i ∗ ∂ j−1 (Bj∗ q˜) ˜) −1 i ∂ (Ai q 2 j−1 . (2.18) ∇qw = l qw + M♦ (−1) + (−1) ∂y j−1 ∂y i−1 ∂t y=0

y=0

The above gradients give information on the local shape of the Lagrangian. As demonstrated above, in the vector space where the direct equations (3.5)–(3.9) are satisfied, the gradients ∇q0 and ∇qw point in the directions along which the Lagrangian increases most rapidly with respect to the initial perturbation q0 and the control qw , respectively. In closing, it should be noted that the two optimization procedures (minimization of energy amplification by an optimal control strategy and maximization of energy amplification by well-chosen initial conditions) can be applied simultaneously. This type of problem is referred to as a robust control problem where an optimal control strategy for the worst-possible initial condition is determined. Mathematically, this corresponds to a saddle-point problem.

150


2.5. Optimization procedure In this section, the initial disturbance q0 is fixed. The procedure to find the optimal control of this perturbation is presented. The control is applied during the time interval [0, Tw ]. As the initial perturbation q0 is fixed, the term [[q0 , q0 ]] is constant and may be removed from the cost functional (2.5) which then becomes E(Tw ) l2 + qw , qw . (2.19) E(0) 2 The control penalty l weighs the control energy compared to the energy amplification E(Tw )/E(0). The procedure to find the stationary points of the Lagrangian is based on the gradient information obtained from the adjoint fields. Starting from an initial guess, the optimization algorithm iteratively improves the control k at the kth iteration, proceeding “downhill” along the direction strategy qw of steepest descent of the Lagrangian. The specific steps of this algorithm are detailed below. I=

0 (t) for the wall-blowing Step 1. Start with an initial guess value qw strategy 0 (t) = 0, is an acceptable guess value for the control as “No blowing” i.e. qw it satisfies the wall-normal boundary condition of the initial perturbation.

Step 2. Solve the direct problem (3.5)–(3.10) The direct problem is solved forward in time from t = 0 to t = Tw by standard numerical techniques, with initial condition q0 (y) and boundary 0 k condition qw (t) at the kth optimiza(t) at the first optimization step, qw tion step. The computed spatio-temporal field q ensures that F , G, H are equal to zero: at each step of the optimization scheme, the Lagrangian is rendered stationary with respect to the state variable q and to any adjoint variable. Step 3. Compute the terminal condition q˜(Tw ) for the adjoint field via (2.16) Solving (2.16) requires integration of a partial differential equation using the boundary conditions for the adjoint field. Step 4. Solve the adjoint problem (2.15) The adjoint problem has to be solved backward in time from t = Tw to t = 0 starting with the terminal value q˜(Tw ) from the previous step. The adjoint field q˜ contains the gradient information which is necessary for the optimization.


151

Step 5. Compute the gradient of the Lagrangian with respect to the control via (2.18) Solving Eq. (2.18) implies integrating the operator M♦ . In the present case, this simply means dividing by the time-dependent scalar weighting function S♦ (t) introduced in Eq. (3.19). Step 6. Compute the descent direction The direction of descent ∇desc is based on the gradient of the Lagrangian computed in step 5. The simplest choice is to go along the gradient of the Lagrangian by takinga ∇desc = −∇qw . This direction should lead, at least locally, to the largest decrease in the Lagrangian. This method, referred to as the steepest descent method, uses only local information and disregards the global shape of the Lagrangian that emerges after many iterations. A more elaborate technique to determine the descent direction is based on the conjugate gradient method (see, Allaire, 2003 for details). Conjugate gradient techniques often increase the convergence rate of the algorithm at a rather low additional computational cost (Bewley et al., 2001) and are often essential to ensure convergence. k k+1 k to qw = qw + αk ∇desc Step 7. Change qw k+1 The new control strategy qw (t) is determined by adding a multiple of the gradient direction computed in step 6 to the previous control strategy k (t). The parameter αk is computed using a line search algorithm which qw k + αk ∇desc ) reaches a determines αk > 0 such that the Lagrangian L(qw minimum.

Step 8. Return to step 2 The algorithm for finding the optimal perturbation is very similar, except that the optimization is performed with respect to the perturbation q0 , with the control qw being fixed; also, the minimization is replaced by a maximization in the line search algorithm and the optimization time is T0 instead of Tw . 3. Application to Swept Hiemenz Flow In the previous section, a general framework has been outlined for the optimization of a cost functional subject to constraints. This framework will now be used to compute optimal perturbations and optimal control strategies for the flow near the attachment-line of a swept wing. The geometry of a The

minus sign represents the “downhill direction”.

152


x z

y

Fig. 1.

Sketch of a leading edge boundary layer along a swept wing.

this configuration is sketched in Fig. 1. A uniform flow impinges on the leading edge of a swept wing dividing equally over its upper and lower surface. In the immediate neighborhood of the stagnation-line, curvature effects of the leading edge can be neglected and the wall can be modeled as being flat. Figure 1 also introduces the coordinate directions with x denoting the chordwise direction, y denoting the normal direction and z denoting the spanwise direction. The mean flow near the stagnation-line is adequately described by swept Hiemenz flow (Hiemenz, 1911), an exact solution to the Navier–Stokes equations. Following Hiemenz, we assume the mean flow in the form U = xRe−1 F ′ (y), V = −Re−1 F (y),

W = W (y),

with (′ ) denoting differentiation with respect to y. The Reynolds number Re is based on the freestream sweep velocity W∞ , the kinematic viscosity and the strain rate of the irrotational outer flow. After non-dimensionalization, one arrives at the following system of ordinary differential equations for F and W : F ′′′ − (F ′ )2 + F F ′′ + 1 = 0, ′

′′

F W + ReW = 0,

(3.1) (3.2)


153

which has to be solved subject to the boundary conditions F (0) = F ′ (0) = W (0) = 0,

(3.3)

′

(3.4)

F (∞) = Re,

W (∞) = 1.

As mentioned before, this flow model is valid only close to the attachment-line, but Gaster (1967) and Poll (1979) have found good agreement with experimental data. The mean flow is then perturbed by introducing the small disturbances (ˆ u, vˆ, w, ˆ pˆ). As a further simplification, we assume that the perturbations exhibit the same chordwise structure as the mean flow, i.e. u ˆ scales linearly in the chordwise direction x. This hypothesis was first proposed by G¨ ortler (1955) and H¨ ammerlin (1955), and is thus referred to as the G¨ ortler–H¨ ammerlin (GH) assumption. Substituting into the Navier–Stokes equations, it is straightforward to derive linear equations governing the evolution of infinitesimal disturbances. We obtain

∂ ∂ ∂ ′ −F − ∆ + ReW + 2F uˆ + F ′′ vˆ = 0, ∂t ∂y ∂z

∂ ∂ ∂ ∂ pˆ ′ −F − ∆ + ReW − F vˆ + = 0, ∂t ∂y ∂z ∂y

∂ ∂ ∂ ∂ pˆ −F − ∆ + ReW = 0, w ˆ + ReW ′ vˆ + ∂t ∂y ∂z ∂z ∂w ˆ ∂ˆ v + = 0, u ˆ+ ∂y ∂z with ∆ = ∂ 2 /∂y 2 + ∂ 2 /∂z 2 . Assuming periodicity in the spanwise direction allows us to introduce a spanwise wavenumber k via Fourier transformation in the z-direction. Furthermore, the spanwise velocity component can be eliminated, yielding a system of governing equations of the form ∂ v v A +B = 0, (3.5) u u ∂t with the boundary conditions ∂v = 0 at y = 0, ∂y v = vw at y = 0, ∂v = 0 at y = ∞. u=v= ∂y u=

(3.6) (3.7) (3.8)

154


The operators A and B read ∆ 0 A= , 0 I  ∂ ′ − ∆ + ikReW ∆ − F −F  ∂y   ∂  − ikReW ′′ −F ′′′ − F ′′ B= ∂y     F ′′

(3.9) −2F ′′ − 2F ′

−F

∂ ∂y



    .    

∂ − ∆ + ikReW + 2F ′ ∂y (3.10)

The above system governs the linear evolution of initial perturbations of the mean flow. The stability properties of this system of equations have been studied by Obrist and Schmid (2003a) for the case of asymptotic stability while the potential to exhibit transient growth has been investigated in Obrist and Schmid (2003b). Following the outline of the first section of this article, we derive the adjoint set of equations from the above system by repeated integration by parts. After some tedious, but straightforward algebra, we arrive at ∂ v˜ ˜ v˜ = 0, +B (3.11) A˜ u ˜ ∂t u˜ ˜ are where the adjoint operators A˜ and B ∆ 0 A˜ = , (3.12) 0 I   ∂ ′′ + ∆ + ikReW ∆ −F −F   ∂y     2  −2F ′ ∂ + (2ikReW ′ − F ′′ ) ∂ . ˜  B= ∂y 2 ∂y        ∂ ′ ′ ∂ −F + ∆ + ikReW − 3F −2F ∂y ∂y (3.13) The boundary conditions for the adjoint variables as y → ∞ are given as u˜ = v˜ =

∂˜ v =0 ∂y

at y = 0, ∞

(3.14)


155

while the terminal condition for the adjoint system at t = T reads q˜(y, T ) =

1 (−1) A M q(T ), E0

(3.15)

with q˜ = (˜ v, u ˜)t and the operator M given as before. We notice that the terminal condition (3.15) for the adjoint requires the inversion of the operator A using the boundary conditions (3.14). The gradients of the Lagrangian with respect to the initial perturbations ∇q0 and with respect to the control ∇qw defined in (2.14) and (2.11) become, respectively, ET A˜ q (y, t = 0) − 2 q0 , E0

3 ∂ ∂2 (−1) q˜(y = 0, t), − F = l2 qw + M♦ ∂y 3 ∂y 2 (−1)

∇q0 = M

(3.16)

∇qw

(3.17)

where the operator M♦ is given by 1/S♦ (t) M♦ = 0

0 0

(3.18)

with S♦ (t) as the “switch function” which allows for a time-dependent weighing of the control energy in the cost function (2.5) (Corbett and Bottaro, 2001a; Corbett and Bottaro, 2001b). 3.1. Numerical methods The direct and adjoint problems were solved using standard numerical techniques and an optimization program has been added to this program. Steps 2 and 4 of the optimization loop presented in previous section require the computation of the direct and adjoint problems, which is accomplished using a Chebyshev collocation method in space and a second-order BDF scheme in time. Steps 3 and 5 of the optimization loop require the inversion of the differential operators M and M♦ which are discretized using a Chebyshev collocation method. In step 6 of the optimization loop, a conjugate gradient method of Polak–Ribière type has been implemented to determine the new search direction. In step 7 of the optimization scheme, a line search algorithm (Brent’s algorithm) has been used to ensure the robust convergence of the optimization algorithm (Bewley, 2001).

156


To summarize, one optimization step for computing the optimal initial condition proceeds as follows: starting with an initial condition q0k , the direct problem is advanced in time until t = T0 where the terminal adjoint condition is determined; the adjoint problem is then solved backward in time until t = 0 at which point the gradient with respect to the initial disturbance is computed. A line-search then determines the new iterate q0k+1 . Analogously, one optimization step for computing the optimal control strategy proceeds similarly, except that the gradient with respect to the control variable has to be computed and stored at each time step of the k+1 has to be imposed at adjoint problem. The updated control strategy qw each time step of the direct problem. In what follows, we used N = 150 collocation points in the wall-normal direction and a time step of ∆t = 0.1. In general, it took less than five iterations to converge to an optimum in both the optimal control and optimal perturbation problems. 3.2. Optimal perturbations In their study of non-modal effects in swept Hiemenz flow, Obrist and Schmid (2003b) present several computations, at a Reynolds number of 550 and spanwise wavenumber k = 0.25, that will serve as a reference to compare with the present computations. With these parameter settings, the flow is susceptible to transient energy growth but is asymptotically stable. For the computations in this section, no control is applied (qw is set equal to zero). 3.2.1. Optimal energy growth Optimizing the cost functional I by manipulating the initial condition q0 yields the results displayed in Fig. 2. Most of the initial perturbation energy is located in the chordwise velocity component u with the remaining two components contributing significantly less to the overall initial energy. The optimal initial condition is located within the boundary layer (which extends to a value of y ≈ 3) as evident from the exponential decay of the velocity amplitudes in the freestream. The initial condition depicted in Fig. 2 achieves its maximum energy amplification at a time T0 = 14 after which it decays to zero. The energy amplification of this optimal perturbation versus time is displayed in Fig. 3. We observe a strong transient amplification that results in a disturbance energy more than 120 times its initial value. The decay

157


1

U

0

−1 0 1

V

4

6

8

10

2

4

6

8

10

2

4

6

8

10

0

−1 0 1

W

2

0

−1 0

y Fig. 2. Velocity profiles of the optimal perturbation, chordwise (u), wall-normal (v), and spanwise (w) components on a linear plot. Dots represent the real (gray) and imaginary (black) part of the optimal perturbations, and the solid line its amplitude. Parameter settings: T0 = 14, Re = 550, k = 0.25.

140

energy of the perturbations

120 100 80 60 40 20 0 0

10

20

30 time

40

50

60

Fig. 3. Energy growth of the optimal perturbation versus time, between t = 0 and t = 4T0 . The solid line represents the energy growth during the optimization interval [0, T0 ]. Parameter settings: T0 = 14, Re = 550, k = 0.25.

158


rate of the energy for large times is given by the least stable eigenvalue, a fact that is easily verified for the parameter settings in Fig. 3.

3.2.2. Catalytic role of the chordwise velocity component

3 2.5 2 1.5 1 0.5 0 0

5

10 y

(a)

15

20

amplitude of wall-normal perturbation velocity

amplitude of chordwise perturbation velocity

We observe in Fig. 2 that the initial condition consists mostly of the chordwise velocity component u. Following this initial perturbation in time (Fig. 4), however, reveals that the wall-normal velocity quickly rises to become the dominant component of the perturbation energy. The initial wall-normal velocity component is amplified by more than one order of magnitude before decaying to zero, whereas the chordwise velocity u, though dominant at t = 0, shows monotonic decay. This phenomenon has been observed over a wide range of parameter values. Careful analysis shows that the spanwise velocity component w also shows significant growth. Due to the reduced state vector representation, q = (v, u), any growth in w is reflected in the wall-normal velocity component v. The above observations suggest the conclusion that the velocity components in the y-z-plane are susceptible to transient growth which may be enhanced by the presence of a finite chordwise velocity component u in the initial disturbance. In a sense, the chordwise velocity component, though not exhibiting transient growth on its own, aids in the amplification of initial

3 2.5 2 1.5 1 0.5 0 0

5

10 y

15

20

(b)

Fig. 4. Perturbation velocity component amplitude at initial time t = 0 (solid line), final time t = T0 (dotted line) and half-time t = T0 /2 (dashed line). The wall-normal velocity component v displayed in (b) is amplified by more than one order of magnitude while the chordwise velocity component u displayed in (a), initially larger than the v-component, is damped. Parameter settings: T0 = 14, Re = 550, k = 0.25.


159

perturbation energy (mostly in the spanwise and wall-normal component). Its role can be aptly described as catalytic. A similar behavior has been observed by Obrist and Schmid (2003b) who showed that the discrete modes of the swept Hiemenz spectrum associated with G¨ ortler–H¨ ammerlin perturbations are susceptible to transient growth which may be further strengthened by adding continuous modes to the initial disturbance. Realizing that the discrete G¨ ortler–H¨ ammerlin modes are dominated by the wall-normal velocity while the continuous modes consist mostly of the chordwise velocity component, their results support the observations above.

3.3. Optimal control After the optimal perturbation has been determined, we are now interested in controlling its growth by applying appropriate blowing/suction boundary conditions during the evolution of the optimal initial condition.

3.3.1. Control of optimal perturbations: Effect of control time Tw As a first attempt, we will not penalize the cost of the control effort by setting the parameter l to zero in the cost functional I. This allows, in theory, to apply an infinite amount of control energy to achieve our goal. Furthermore, we recall that only wall-normal blowing and suction will be considered, as is reflected in the form of the operator M♦ . The switch function S♦ (t) allows us to turn on the control smoothly and to cut the control a few time steps before the end of the control interval. Such a mollified control effort constitutes the type of regularization opportunities pointed out by Protas et al. (2002). In our case, the switch function is essential: if the wall-blowing does not vanish at the final time t = Tw , the terminal condition for the adjoint problem does not satisfy the boundary condition v˜(y = 0, Tw ) = 0. In a similar manner, as no boundary condition at the wall is imposed on the higher derivatives of the adjoint field v˜, the k + αk ∇kdesc may not satisfy gradient ∇kqw and the modified guess value qw the homogeneous boundary conditions at the wall, if no switch function is introduced. For our case, the switch function, depicted in Fig. 5(a), takes the general form S♦ = 1 − exp[−(t/τ )2 ] − exp[(1 − t/τ )2 ]. It approaches zero at the two ends of the optimization interval and is nearly one during more than 80% of the time interval.

160


1

wall-normal blowing

Switch function

1 0.8 0.6 0.4

0.5

0

−0.5

0.2 0 0

2

4

6

time

(a)

8

10

12

14

−1 0

2

4

6

8

10

12

14

time

(b)

Fig. 5. (a) “Switch function” S♦ (t) modulating the blowing velocity at the wall: at the beginning and at the end of the control time interval [0, Tw ], the control vanishes whereas its intensity is nearly uniformly weighed over about 80% of the time interval; (b) Wall-blowing time sequence for optimal control: real (dashed line) and imaginary (dotted line) part and amplitude (solid line) versus time. As expected, the wall-blowing vanishes at both ends of the time interval. Parameter settings: l = 0, Re = 550, k = 0.25.

Applying optimal control, it was possible to decrease the maximum energy amplification by almost 80%, from 123 to 26 times the initial energy. Varying the extent of the time interval over which control was applied, the best result was obtained when the control objective was to minimize the energy amplification at Tw = 14, i.e. when Tw coincides with the time at which the uncontrolled initial perturbation reaches its maximum energy. The optimal control strategy, i.e. the temporal evolution of the blowing velocity, is displayed in Fig. 5(b). As expected, the control velocity vanishes at the two ends of the optimization interval; the blowing amplitude is almost constant during most of the time interval. As the control time interval is increased, the control becomes increasingly effective — the controlled quantity (the energy amplification at time Tw ) almost vanishes (Fig. 6). However, even in this case, an energy peak still occurs between t = 0 and t = Tw , and the fact that, despite the long control interval (Tw = 28), the maximum energy amplification remains high (about 70) is rather disappointing. In order to eliminate this energy peak, one would have to redesign the cost functional I. In its present form it does not take into account any information during the control interval, and the disappointing performance of long-term control should not come as a surprise. For examples of alternative cost functionals addressing this issue, the reader is referred to Corbett and Bottaro (2001a), Corbett and Bottaro (2001b) and Bewley and Liu (1998).

161



150

100

50

0 0

5

10

15 time

20

25

Fig. 6. Energy amplification of the optimal perturbation versus time when no control is applied (thick solid line), and when the control interval is [0, Tw ] with Tw = 7 (dashed line), Tw = 14 (thin solid line) and Tw = 28 (dotted line). Parameter settings: l = 0, Re = 550, k = 0.25.

Alternatively, computing suboptimal control sequences could be applied to eliminate energy peaks; controlling the disturbances “step by step” has been shown to produce very efficient control strategies (Bewley et al., 2001). 3.3.2. Control of optimal perturbations: Effect of the control penalty Allowing unrestricted control energy is rather unrealistic. For this reason, we will fix the control interval to [0, Tw = 14] and investigate the cost of control, represented by the parameter l in the cost functional. The lower the value of l, the more energy can be applied to control without significantly influencing the value of the cost functional. In general, this will lead to efficient damping of the perturbation energy. High values of l, on the other hand, penalize any control effort quite significantly. In this case, only a modest reduction in energy amplification should be expected. Figure 7 plots the value of the cost functional versus the control cost parameter l. Two asymptotic limits can be observed. If l > 10, the control is ineffective in reducing energy amplification; the value of the cost functional remains rather high and close to the value for no control. For 1 < l < 10, control efforts are very effective; the energy amplification can be decreased from 123 to 12. Decreasing the value of l below one, however,

162


value of the cost function

102

101 10−2

100 control penalty l

102

Fig. 7. Cost function I versus control penalty l. Parameter settings: Tw = 14, Re = 550, k = 0.25.


102

l=16 l=8 l =4

101 l=0

100

0

10

20

30 time

40

50

60

Fig. 8. Evolution of the energy of the perturbations when the control penalty l is 0, 4, 8, 16. When l is set equal to 16 and above, the control is inefficient and the energy amplification of the optimal perturbations without control is recovered. Parameter settings: Tw = 14, Re = 550, k = 0.25.

does not yield any improvement. In this case, the most successful reduction of energy growth is accomplished for values of l near one. The effect of the control penalty parameter l is also observed in the time evolution of the perturbation energy, displayed in Fig. 8. For all values


163

of l, the energy decays at the rate of the least stable mode as t tends to infinity. However, the long-term value of the perturbation energy, when control efforts are not penalized, i.e. l = 0, is one order of magnitude lower than the energy when the control is rather expensive, l = 16. Hence, finitetime control yields long-term benefits. 3.3.3. Control of linearly unstable flow The critical Reynolds number above which swept Hiemenz flow is linearly unstable is Rec = 583.1 (Hall et al., 1984). Consequently, controlling the transient growth of linearly stable swept Hiemenz flow may not be sufficient for industrial applications, such as the control of the leading edge boundary layer on the wings of civil aircrafts. The above optimization framework also yields effective control strategies that show significant improvement for linearly unstable flows. As the mean flow is linearly unstable, the evolution of the energy of the optimal perturbations at very large times is governed by the amplification rate of the most unstable mode. When the control is applied, the energy can be driven significantly lower when compared to the uncontrolled flow. After the control is switched off, the energy resumes to increase exponentially at the rate of the most unstable mode. For further results on controlling linearly unstable flows, and on the relation between optimal and opposition controls in this case, we refer the reader to Guégan et al. (2005). 4. Conclusions A general framework using optimization techniques for unconstrained problems has been presented. It allows the efficient calculation of optimal perturbations and optimal control strategies for fluid flows that can be described by linearized equations. This primal-dual formulation, based on the Navier– Stokes equations and their adjoint set, has then been applied to swept Hiemenz flow, a model for the flow near the attachment-line of swept wings. It has succeeded in identifying the initial perturbations leading to the largest amplification of energy over a finite specified time interval. Furthermore, optimal control strategies for suppressing this transient energy growth have been determined within the same framework. The linear swept Hiemenz flow model has been shown to support strong transient energy growth in the linearly stable Reynolds/spanwise wavenumber domain. At a Reynolds number of 550 and at a spanwise wavenumber of k = 0.25, initial perturbations may undergo an energy

164


amplification as high as 123, before asymptotically decaying to zero. The energy growth is mainly driven by the wall-normal velocity component, while the chordwise velocity component plays a strong catalytic role in this transient growth. Optimal wall-normal blowing/suction succeeded in reducing the maximum energy amplification by 80%. More modest reductions have been accomplished as the control efforts were increasingly penalized. In this study, only the energy amplification at the final time has been taken into account in the cost functional I resulting in energy peaks within the time interval. An improved cost functional including a time-averaged energy term has been shown to eliminate similar energy peaks in twodimensional boundary layers (Corbett and Bottaro, 2000). The mathematical framework is readily capable of dealing with this improved cost functional. References 1. E. Abergel and R. Temam, On some control problems in fluid mechanics, Theor. Comp. Fluid Dyn. 1 (1990) 303–325. 2. G. Allaire, Optimisation et analyse num´ erique, Une introduction a ` la mod´ elisation math´ ematique et a ` la simulation num´ erique, Notes de Cours Ecole Polytechnique (2003). 3. T. Bewley, Flow control: New challenges for a new Renaissance, Progr. Aerosp. Sci. 37 (2001) 21–58. 4. T. Bewley and S. Liu, Optimal and robust control and estimation of linear paths to transition, J. Fluid Mech. 365 (1998) 305–349. 5. T. Bewley, P. Moin and R. Temam, DNS-based predictive control of turbulence: an optimal benchmark for feedback algorithms, J. Fluid Mech. 447 (2001) 179–225. 6. P. Corbett and A. Bottaro, Optimal perturbations for boundary layers subject to streamwise pressure gradient, Phys. Fluids 12 (2000) 120–130. 7. P. Corbett and A. Bottaro, Optimal control of non-modal disturbances in boundary layers, Theor. Comp. Fluid Dyn. 15 (2001a) 65–81. 8. P. Corbett and A. Bottaro, Optimal linear growth in swept boundary layers, J. Fluid Mech. 435 (2001b) 1–23. 9. M. Gaster, On the flow along swept leading edges, Aero. Q. 18 (1967) 165–184. 10. H. G¨ ortler, Dreidimensionale Instabilität der ebenen Staupunktsstr¨ omung gegen¨ uber wirbelartigen St¨ orungen, in 50 Jahre Grenzschichtforschung (G¨ ortler & Tollmien, Vieweg, Braunschweig, 1955). 11. A. Guégan, P. Schmid and P. Huerre, Optimal energy growth and optimal control of swept attachment-line boundary layers, J. Fluid Mech., submitted (2005). 12. M. Gunzburger, Inverse design and optimisation methods, in Lecture Series 1997–2005 , Von Karman Institute for Fluid Dynamics (1997).


165

13. P. Hall, M. Malik and D. Poll, On the stability of an infinite swept attachment line boundary layer, Proc. Roy. Soc. Lond. A395 (1984) 229–245. 14. G. H¨ ammerlin, Zur Instabilitätstheorie der ebenen Staupunktsstr¨ omung, in 50 Jahre Grenzschichtforschung (G¨ ortler & Tollmien, Vieweg, Braunschweig, 1955). 15. K. Hiemenz, Die Grenzschichten an einem in den gleichförmigen Fl¨ ussigkeitsstrom eingetauchten geraden Kreiszylinder, PhD thesis, G¨ ottingen (1911). 16. D. Obrist and P. Schmid, On the linear stability of swept attachment-line boundary layer flow. Part 1. Spectrum and asymptotic behaviour, J. Fluid Mech. 493 (2003a) 1–29. 17. D. Obrist and P. Schmid, On the linear stability of swept attachment-line boundary layer flow. Part 2. Non-modal effects and receptivity, J. Fluid Mech. 493 (2003b) 31–58. 18. D. Poll, Transition in the infinite swept attachment line boundary layer, Aero. Q. 30 (1979) 607–629. 19. B. Protas, T. Bewley and G. Hagen, A comprehensive framework for the regularization of adjoint analysis in multiscale PDE systems, J. Comp. Phys. 195 (2002) 49–89.


INSTABILITIES NEAR THE ATTACHMENT-LINE OF SWEPT WINGS

J¨ orn Sesterhenn∗ and Rainer Friedrich† Fachgebiet Str¨ omungsmechanik Technische Universit¨ at M¨ unchen 85747, Gaching, Germany E-mails: ∗[email protected] † [email protected] In the vicinity of the leading edge of a swept wing, two classes of primary fluidmechanical instabilities are observed: the attachment-line instability and the cross-flow instability. The further it is at the leading edge, the latter is observed at several viscous length scales downstream. Both the attachment-line and the cross-flow instability exist in a large parameter range and are usually investigated in isolation and in several degrees of approximations. The present approach incorporates the geometrically complex of a three-dimensional nature of the flow situation, at the leading edge of a swept wing. In this situation, the different instabilities should more properly be called different (unstable) eigenmodes of the flow. The aim is to understand this situation and prepare for a global stability analysis of the full problem. Three situations are investigated here: (i) the compressible attachment-line flow on a flat plate in subsonic and supersonic flow, (ii) the same flow along a parabolic leading edge and (iii) the supersonic flow along the same body with a detached bow shock. We use a numerical method which is fifth order in space and fourth order in time on a curvilinear, time-dependent body-fitted grid (Sesterhenn, 2001). In the supersonic case, a bow shock develops which is described using the Rankine–Hugoniot conditions as an inlet condition (Fabre et al., 2001). In a Reynolds number range where the attachment-line instability and cross-flow instability are expected, several base flows are computed. In the spanwise direction, periodicity is assumed. Therefore, the base flow can be computed in two dimensions and extended

167

168

J. Sesterhenn and R. Friedrich

into different depths in the third direction, corresponding to different wavelengths. This flow is confronted with random and coherent disturbances. Contents 1

2

3 4

5

Physical Situation 1.1 General remarks concerning the attachment-line 1.2 Structure of the boundary layer in the vicinity of the leading edge 1.3 Synopsis of instabilities Characterization of the Flow 2.1 Non-dimensional parameters 2.2 Length scales and timescales 2.2.1 Length scales 2.2.2 Timescales 2.3 Brief literature survey 2.3.1 Incompressible flow on a flat plate 2.3.2 Nose radius 2.3.3 Subcritical transition 2.3.4 Compressibility 2.3.5 Wall temperature effects 2.3.6 Hypersonic flow 2.4 Summary Computation of the Base Flow Configurations 4.1 Subsonic flow on a flat plate 4.1.1 The swept Hiemenz solution 4.1.2 Compressibility correction 4.2 Subsonic flow on a parabolic leading edge 4.3 Supersonic flow on a parabolic leading edge Perturbation 5.1 Subsonic flows 5.1.1 Random perturbation 5.1.2 Attachment-line instability 5.1.3 Changes in the structure of the instability 5.1.4 Influence of wall boundary condition 5.2 Supersonic flow 5.2.1 Random perturbation 5.2.1.1 Perturbance growth depending on Reynolds number and wavenumber

169 169 170 170 172 172 174 174 175 175 175 177 177 178 178 178 179 179 180 180 180 182 182 183 187 187 187 187 189 191 191 191 192

Instabilities Near the Attachment-Line of Swept Wings

5.2.2 Coherent disturbances

195

5.2.2.1 Perturbation growth 6

169

196

Conclusion

202

A Definitions

203

A.1 Kinetic energy

203

A.2 Modal energy

203

A.3 Growth rates

203

A.4 Q-criterium

204

References

204

1. Physical Situation 1.1. General remarks concerning the attachment-line At the attachment-line of a swept wing, a stagnation line develops. The situation is depicted in Fig. 1. The origin of the coordinate system lies on the attachment-line with the z-axis oriented in the direction of the attachmentline, and the x-axis in the direction of the normal on the body surface. The flow direction together with the normal on the body surface at the attachment-line forms the sweep angle Λ. The flow velocity is described by the normal velocity u∞ and the sweep velocity w∞ . The angle of attack is always kept at zero. If the normal velocity is supersonic, a detached bow shock forms.

Body

Bow Shock Fig. 1.

Physical situation.

170


The flow along the attachment-line is of crucial importance for keeping the flow laminar. If the flow is turbulent here, little hope exists in keeping the downstream flow laminar. One of the principal aims of transition research is to delay the locus of laminar/turbulent transition. There are two reasons that coexist: in civil transportation, applications on energy consumption is the main focus whereas in re-entry vehicles, heat transfer and thermal loads are of interest. 1.2. Structure of the boundary layer in the vicinity of the leading edge The boundary layer in the vicinity of the leading edge is three-dimensional. That means that the local velocity vector is not confined to the plane spanned by the surface normal and the freestream velocity (cf. Fig. 9). The component of the velocity perpendicular to this plane is called cross-flow velocity and it is responsible for the complexity of the flow. 1.3. Synopsis of instabilities The possible instabilities which finally lead to laminar/turbulent transition are described by Bippes (1999, Fig. 2), for example. Those are the attachment-line instability, the Tollmien–Schlichting instability, the crossflow instability and the centrifugal instability. At high Mach numbers, Mackmodes may occur. The constitutive features of the instabilities relevant in our case shall be briefly recollected here: Attachment-line instability The attachment-line instability is a viscous, linear instability. Its base flow is a swept Hiemenz-flow. This is a stagnation point flow with a superimposed spanwise component. It is named after Hiemenz, a student of Prandtl who found this exact solution to the incompressible Navier–Stokes equations (Hiemenz, 1911). The adjective linear indicates the fact that its behavior is described by the linearized Navier–Stokes equations. The principal result of a linear stability analysis is a critical Reynolds number, above which the flow is unstable against all disturbances, however small. Additionally, information about wavelengths and growth rates of disturbances is found. Alas, important information is destroyed by the linearization, namely about the amplitudes of the disturbances, its influence on the base flow and the nonlinear interaction of the modes.


171

No wonder that even below the critical Reynolds number, amplification of disturbances can be observed, provided the initial disturbance is sufficiently large. These instabilities, so they exist, are called subcritical ones. We refrain from further subdivisions into subtleties of transient growth mechanisms, not referred to in the remainder of the chapter. Common control practice is to keep the Reynolds number ( q∞ R sin2 Λ , (1.1) Reθ = 0.404 (1 + ǫ) cos Λ which is based on the approximate displacement thickness, less than Reθ = 100. The value is less than half of the critical one and takes care of subcritical instabilities. This is achieved by holding the nose radius small (Saric and Reed, 2003). For future large, long-range aircrafts, this is no longer feasible. Suction is employed as well. Cross-flow instability The cross-flow instability is an inviscid, linear instability. Its base flow is a three-dimensional boundary layer. The instability is due to an inflection point in the velocity profile. In the compressible case, a generalized inflection point criterion holds (Mielke, 1999):

∂u ∂ ̺ = 0. (1.2) ∂y ∂y A proposed control strategy (Saric and Reed, 2003) consists in distributing roughness elements along the wing surface which favor slower growing modes and thereby hamper the most unstable ones. In this way, the transition location is postponed downstream. Streamline curvature instability Itoh (1994) discovered a further instability in the vicinity of the stagnation line. It occurs at a sufficiently large streamline curvature due to a centrifugal term in the linearized equations. The mechanism is similar to the one of the cross-flow instability and hard to distinguish from the latter as the eigenfunctions have the same structure. However, two differences are known. The phase relationship of the cross-flow velocity component with distance from the wall is different as well as the orientation with respect to the freestream streamline. This instability is still poorly understood and not a target of control strategies yet.

172


Secondary instabilities Beyond the field of these primary instabilities, the vast field of secondary instabilities opens, describing the decay of the primary ones. When they in turn decay due to further instabilities, numbering ceases and one talks about the transition to turbulence. Here we do not discuss those instabilities, but rather, we try to understand the relationship of the above primary instabilities. 2. Characterization of the Flow 2.1. Non-dimensional parameters Even a simplified representation of the attachment-line problem is characterized by a wealth of parameters. Obvious dimensional quantities are as follow: Flow velocity Sweep angle Nose radius Wall temperature Pressure Speed of sound Dynamic viscosity

q∞ Λ R Tw p∞ c∞ µ∞

However, this enumeration is not complete. Think of an angle of attack, wall roughness or suction parameters. In the following, we characterize the flow by the non-dimensional numbers given below: Reynolds number

Nose radius

Re = ) R = δ

Temperature ratio

2u∞ R νr

2Ru∞ 2Re = νr tan Λ

τ= Mach number

w∞

(2.1)

(2.2)

Tw − Tr To − T∞

(2.3)

w∞ c∞

(2.4)

M=

173


In the case of a detached bow shock, the quantities u∞ , c∞ and Λ are replaced by those behind the shock. The kinematic viscosity νr is taken at recovery temperature. The flat wall is recovered for R → ∞ when 2uR∞ is kept constant. These parameters are chosen such that they may be computed a priori, i.e. without prior computation of the base flow. They naturally degenerate to the incompressible flat plate case known from the literature of Spalart (1988), Theofilis (1998), Joslin (1996), Obrist (2000) and, Obrist and Schmid (2003). In this case, only one parameter exists, commonly given as the following Reynolds number: ) ∞ w∞ ν/ ∂u ∂x . (2.5) Re = ν This is not a lucky choice in the case of finite nose radii since at infinity. We use

∂u u∞ , =2 ∂x wall R

∂u∞ ∂x

vanishes

which is the derivative of the wall-normal velocity about a circular cylinder, 2 2 2 y −x , (2.6) u(x, y) = u∞ 1 + R (x2 + y 2 )2 as given by the potential flow solution, in the stagnation point (x, y) = (−1, 0). In the compressible case, besides the specific heats and the Prandtl number that are kept constant in what follows, Mach number and wall temperature are new parameters. Additionally, the temperature at which the reference viscosity is evaluated has to be given. We choose the recovery temperature which can be computed following Reshotko and Beckwith Reshotko and Beckwith (1958) using a table for the recovery factor R :=

Tr − T∞ = 1 − (1 − ζw ) sin2 Λ To − T∞

(2.7)

for an infinite, swept cylinder. ζw is called the local recovery factor and tabulated as ζw (κ) with κ :=

γ−1 2 2 M∞ γ−1 2 2 2 M∞ cos

1+ 1+

Λ

(2.8)

174


for P r = 0.7 as follows: Table 1. Local recovery factor according to Reshotko and Beckwith (1958). κ

ζw

1.0 1.6 3.0 6.5

0.8485 0.8518 0.8567 0.8627

The results for the recovery temperature Tr obtained using this table differ from the a posteriori values from our computations within an relative error of less than 10−3 . Equations (2.6) and (2.7) show that for given To and T∞ , the temperature on the stagnation line depends on M∞ and Λ. Λ may be replaced by M = M∞ cos Λ. T∞ /To is, for constant γ, a function of the Mach number M∞ . Thus, the temperature on the stagnation line is dependent on M∞ and M . It follows that for) reference temperatures other than the recovery

temperature in Re = w∞ / 2uR∞ νr additional parameters have to be specified. E.g., besides M (or Λ), M∞ has to be specified. The reference pressure in the stagnation point is determined from the inviscid flow. Reshotko and Beckwith give the following relationship: γ 1

γ−1

γ−1 γ+1 γ+1 ps 2 (M∞ cos Λ) = . (2.9) p∞ 2 2γ(M∞ cos Λ)2 − (γ − 1) In the shock-free case, we have γ

γ−1 γ−1 ps 2 (M∞ cos Λ) = 1+ . (2.10) p∞ 2 The chosen Reynolds number implies a viscous length scale * +

+ ∂u∞ , δ = νr . (2.11) ∂x wall In the curved case, the nose radius is non-dimensionalized by this length. By introducing a finite nose radius, the sweep angle is introduced as well, as seen from Eq. (2.1). 2.2. Length scales and timescales 2.2.1. Length scales

Nose radius R and the viscous length scale δ are the natural length scales of the flow. A non-dimensional number δ/R is a function of the sweep


175

angle and Reynolds number. A correction which allows for a displacement u∞ ) = 2 R+δ might be used. The Taylor series thickness δ at the wall in ( ∂u . ∂x wall / 2 Λ tan Λ for δ/R is Rδ = tan 2Re 1 + 32Re2 + · · · . For sweep-angles which are not extreme, the correction is negligible. In the literature, one finds the momentum thickness θ used as length scale. This choice is motivated by a similarity solution for the flat plate flow (Rosenhead (1963, p. 471) for β = 1.0). One finds Reθ = 0.4044 Re. In this case, the following relationships hold: δ99 = 3.055δ, δ1 = 1.0646δ and θ = 0.4044δ. Unlike common practice, we do not use this nomenclature since the results are not valid for the curved surface and differ significantly. In the case with a shock, the stand off distance ∆ of the shock is an additional length scale. For very high Mach numbers, ∆ is comparable to δ.

2.2.2. Timescales ∞ −1 exists. In the In the incompressible case, only the timescale τ = ( ∂u ∂x ) compressible case, it corresponds to the timescale built with the stand off distance, τ∆ = u∆2 , since the velocity distribution from the shock into the stagnation line is close to linear. A particle thus travels 2τ from the shock to the boundary layer. Acoustic disturbances exhibit a timescale τc = c∆2 , and so the ratio γ−1 τc τ∆ = M. The limit for very high Mach numbers M1 → ∞ is 2γ . For τc γ = 1.4, the ratio reaches τ∆ = 1/7 at maximum. 2∆ M Acoustic waves travel the time u22∆ +c2 = u2 M+1 . In the limit case of M1 → ∞, this is about τ /4. This means that timescales for the flow and the acoustics are comparable. Acoustic waves may couple shock and boundary layer. Far field boundary conditions replacing a shock might thus not be applicable.

2.3. Brief literature survey 2.3.1. Incompressible flow on a flat plate Three or more dimensional parameters govern the problem. Based on a formal dimensional analysis for the incompressible flow on a flat plate, Cumpsty and Head (1967) suggested

Π=

2 ̺∞ w∞ ∂u∞ µ∞ ∂x

(2.12)

176


as a unique non-dimensional parameter. More recent investigations use the (sweep-)Reynolds number √

Π = Re =

̺∞ w∞ δ . µ∞

(2.13)

Linear stability analysis by Hall, Malik and Poll (1984) exhibited a critical Reynolds number Re = 583 for a wavenumber α = 2πδ λ = 0.288 in spanwise direction. The corresponding wavelength is λ ≈ 22δ. They assumed a linear dependence of the perturbation with chord length. This assumption was used by the pioneering work of G¨ ortler (1955) and H¨ aemmerlin (1955) and was proved by Lin and Malik (1996) as well as by numerical simulations (e.g., Spalart (1988)) to exhibit the most unstable mode indeed. Obrist and Schmid (2003) provide the stability diagram in Fig. 2. Experiments for the incompressible flow on a flat plate do not exist but several experiments are known since the 1960s performed by Pfenninger (1965) on airfoils and later on cylindrical bodies by Cumpsty and Head (1969) and Poll (1985). They used (one and the same) cylindrical body with circular nose and a wedge in the rear and they found a critical Reynolds number Re ≈ 570.

0.4

0.3 α 0.2

0.1

0

500

1000

1500

2000

2500

3000

Re Fig. 2. Stability diagram of the swept Hiemenz-flow for an incompressible flow over a flat plate. From Obrist and Schmid (2003).


177

2.3.2. Nose radius When using cylinders, the flat plate approximation for theory has to be questioned. The ratio of viscous length scale to nose radius in the experiments by Poll ranges for the tests in the vicinity of the critical Reynolds number from about δ/R ≈ 1.3 × 10−3 to 2.4 × 10−3 , corresponding to a range of 1/770 to 1/417. For stability purposes, this is not necessarily small. For example, the instability between two rotating coaxial cylinders differ from the one between two horizontally-moving flat plates unless the gap to curvature radius is less than d/R = 2.5 × 10−5 = 1/40,000. Chandrasekhar (1981, p. 350). Lin and Malik (1997) found in a linear stability analysis that the convex curvature has stabilizing influence. The critical Reynolds number is reported to rise from Re = 570 to Re = 637 when the nose ratio δ/R varies within δ 1 1 ≤ ≤ . 1430 R 143 This result raises doubts about whether the experimental findings quoted above are comparable to a flat plate, which is recovered only for R 1500δ.

2.3.3. Subcritical transition For sufficiently large disturbance amplitudes, the flow was found to become turbulent for substantially lower values than Recrit = 583. The lowest value observed was Re ≈ 250.a This was first reported by Pfenninger and Bacon (1969) (quoted after Hall and Malik (1986)) and described by Poll (1984) as follows: depending on a trip-wire thickness d three ranges exist; for (d/δ) < 0.8, no subcritical transition is observed. In 0.8 < (d/δ) < 1.6, the subcritical Reynolds number falls from 570 to 400, with transition downstream of the wire. For 1.6 < (d/δ) < 2.0, turbulence occurs right at the wire and the subcritical Reynolds number becomes as low as 250. No transition is observed below this threshold value. This behavior was confirmed by investigations by Hall and Malik (1986) and Joslin (1996) by direct numerical simulation of a flat plate flow.

a The

value Re = 250 corresponds with Reθ = 100 from Eq. (1.1), see Sec. 1.3.

178


2.3.4. Compressibility Further parameters are introduced by compressibility. Entropy convection and acoustic waves augment the picture. At high Mach numbers, acoustic modes in boundary layers are known. Shocks and wall temperature effects may occur. In the very vicinity of the attachment-line, there are two Mach numbers that have to be distinguished. The Mach number normal to the wall M ⊥ = uc is always small. No compressibility effects are expected from this side. ∞ , The Mach number of the flow parallel to the leading edge, M = wc∞ may acquire high values. From the stability theory, qualitative changes in the stability characteristics are known for M > 4. For values M = 1.7 up to M = 3.3, experimental investigations exist (Creel et al., 1987; Murakami et al., 1996), as well as a re-evaluation of older data by Bushnell and Huffman (1967). These results indicate a slight stabilization (Murakami et al., 1996); for this comparison, the reference temperature is chosen according to a suggestion by Poll (1985). The contrary is claimed by Lin and Malik (1995). They predict a drastic destabilization indicated by a change of the critical Reynolds number from 570 to 349 for M ≈ 3. This result is questionable from the view of the experiments and has to be investigated further. 2.3.5. Wall temperature effects Poll (1983) was able to reconcile the influence of different wall temperatures from data by Topham (1965) and, Bushnell and Huffman (1967) empirically with the known criterion provided that Re was evaluated at a reference temperature T ∗ = Te + 0.10(Tw − Te ) + 0.60(Tr − Te ).

(2.14)

Le Duc (2001) observed for adiabatical and isothermal boundary conditions at the recovery temperature of different growth rates. Malik and Beckwith (1988) and Kazakov (1990) report stabilization by wall cooling. 2.3.6. Hypersonic flow For hypersonic flow, a strong detached bow shock forms. No investigations of the influence of this shock on the stability of the flow are known. Even when dealing with perfect gas flow only, three points have to be


179

considered: 1. Reshotko and Beckwith (1958) showed theoretically that heat transfer is strongly dependent on the freestream Mach number, 2. Mack-modes may come into play, 3. interactions of the shock with the stability behavior are likely due to the small shock distance. 2.4. Summary In a relatively simple configuration, five parameters govern the swept, compressible flow around a curved leading edge. At cruise conditions of an aircraft, they may roughly be enumerated according to their importance as follows: Tripwire-diameter d/δ: dominating for d/δ > 0.8 and Re > 250. Reynolds number Re = nose radius.

̺w∞ δ µ :

critical within 570–637, according to

Nose radius R/δ: stabilizing for smaller radii. Mach number M = not known.

w∞ c :

up to M ≈ 3 stabilizing, “Mack-modes” are

w : detailed effects are not known. An empirical Wall temperature TTrec temperature criterion exists.

3. Computation of the Base Flow The stability of the flow described above is investigated using a numerical solver of the compressible Navier–Stokes equations. First, a base flow is computed for each Reynolds number. In a next step, this flow is extended in spanwise direction to accommodate the desired wavelength. The small amplitude disturbances are introduced and the initial value integrated for later times. The medium is described as a perfect gas together with the Sutherlandlaw for the temperature-dependency of the viscosity. The Prandtl number was fixed to be P r = 0.71. The ratio of specific heats was specified as γ = 1.4. Bulk viscosity µd is neglected. When computing the base flow, we make use of the fact that the equations allow for a laminar solution with no dependency on the spanwise direction. Therefore it is sufficient to compute the solution, including the reduced spanwise momentum equation, in the xy-plane. This implies the assumption that spanwise growth of the disturbances is negligible within a few wavelengths. This approach is called a “temporal simulation”.

180


The following computations aim at the resolution of a few modes. Demands on the numerical resolution are thus moderate in the spanwise direction. For the computation of the first mode, 8 points were used. This was backed up with control computations using 16, 32 and 64 points (see Fig. 32). For the initial development of the computation no difference was observed except in the case of the lowest Mach number where more than 8 points were used. The reason is that the modified wavenumber of the space discretization depends on the condition of the problem which, for small Mach numbers, behaves as 1/M. 4. Configurations The following cases were considered: α: subsonic flow on a flat plate, β: subsonic flow on a parabolic leading edge, γ: supersonic flow on a parabolic leading edge. The flow case is denoted by a signature containing a Reynolds number, a nose radius measured in the length scale δ from (2.10), wall temperature ratio of the base flow or “a” to designate an adiabatic wall and finally the wall boundary condition employed for the perturbation, namely an adiabatic or isothermal wall, e.g., (γ/642/409/25/a/a). Figure 3 shows the base flow profile normal to the body surface at the attachment-line. The symbols designate the incompressible solution following Rosenhead (1963). The cases α and β coincide within the visibility of the plot and are not shown. The supersonic case γ is shown by the full lines. They do not coincide since they depend on additional parameters. Similarity solutions for this flow were given by Reshotko and Beckwith (1958) and, Saljnikov and Dallmann (1989), they were not used for reasons discussed below. 4.1. Subsonic flow on a flat plate This flow case (Le Duc, 2001) was computed for one Reynolds number Re = 644 and wavenumber α = 0.2787 for several Mach numbers from M = 0.1 to M = 1.3. 4.1.1. The swept Hiemenz solution As an initial condition, we used the Hiemenz solution for the incompressible stagnation point flow (Hiemenz, 1911), cf. Schlichting (1982, p. 97). It is

181


5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 -0.5

0

1

2

3

4

5

Fig. 3. Base flow on the stagnation line (γ/642/409/ − /a/−) (full lines) and the similarity solution by Rosenhead (1963) (symbols).

obtained by modifying the potential solution for the stagnation point flow close to the wall. The stream function of this solution is

∂u∞ Ψ= xy (4.1) ∂x ∞ with a prescribed constant velocity gradient ∂u ∂x . The velocity components are

∂u∞ x, −Ψy = u = − ∂x

∂u∞ y. Ψx = v = ∂x To fulfil the boundary conditions, the equation is modified in the viscous flow as

∂u∞ f (x)y, (4.2) Ψ= ∂x

obtaining

u ∼ −f (x),

v ∼ f ′ (x)y.

182


This ansatz is introduced in the Navier–Stokes equations and leads with the boundary conditions f (0) = f ′ (0) = 0 and f (∞) = 1 to an ordinary differential equation for f . The solution is given by Schlichting and Rosenhead. For the swept case, the solution is obtained by integrating the momentum equations in the spanwise direction.

4.1.2. Compressibility correction The influence of the Mach number is approximated for the adiabatic wall by an ansatz of the form ϕ = ϕo + M 2 ϕ2 + · · ·

(4.3)

for all flow quantities ϕ = (p, u, v, w, s) (Rayleigh, 1916; Le Duc, 2001). This solution is used as an initial condition as well as condition at the lateral boundaries. Using this approximation, the incoming charateristic information

1 ∂p ∂v ± (4.4) Y ± = (u ± c) ̺c ∂y ∂y was specified and kept constant. With this information, a steady state solution was computed. Local time stepping was used to accelerate the convergence.

4.2. Subsonic flow on a parabolic leading edge For the subsonic flow on a parabolic leading edge, similarity solutions exist as well. They were given by Reshotko and Beckwith (1958) and more generally by Saljnikov and Dallmann (1989). Both methods help to understand the physics of the flow but were not used as initial conditions for the base flow computation since little time is saved by having a better initial guess and programming the methods is tedious. Instead of the similarity solutions, we used a potential flow solution which is mapped by a conformal mapping onto the parabolic leading edge. The no-slip boundary condition is imposed by a power law in the wallnormal direction in the form (ξ/δ)n , prescribing an approximate boundarylayer thickness. This method was used for the outflow boundary conditions as well. The quality was checked by comparing solutions with different domain lengths (Schwertfirm, 2002).


183

4.3. Supersonic flow on a parabolic leading edge The supersonic case differs from the subsonic case by the entropy gradient which is produced by the shock, implying vortical flow as well as the possibility of the influence of the bounded domain. In this case, the initial shock location is prescribed as a parabola and the stand off distance ∆ is approximated by the empirical relation of Billig (1967): 2 ∆ = 0.386e4.67/M∞ . (4.5) R We also prescribed the shock form as given by Billig but this method is less accurate since the initial grid cannot be computed using a conformal mapping. Velocities and temperature were adjusted using the power law method as described above. For a given Mach number, the stand off distance of the shock is almost independent of the Reynolds number ∆ = 0.53 R. For Re = 642 to Re = 800, this corresponds to ∆/δ varying between 217 and 267. The numerical resolution was about 320 to 400 points # x in this direction. The arclength of the bodyb is 0 1 + y ′2 dx/δ ≈ 2000. Since the structures in this direction are much longer, about 150 points were sufficient. In Figs. 4–8, the base flow is depicted. All quantities are normalized with the values behind the shock. The full domain is displayed. Figure 9 shows the velocity profile in streamline coordinates. The x-axis points in direction of the potential streamline. u denotes the velocity

1.2 1 0.8 0.6 0.4 0.2 -1

0

1

2

3

4 -6

-4

-2

0

2

4

6

Fig. 4. Cut of the base flow in the xy-plane: Pressure, (γ/800/504/ − /a/−). Only every fourth grid line is displayed. b cf.

footnote c on p. 199

184


1.08 1.04 1 0.96

-1

0

1

2

3

4

-6

-4

-2

0

2

4

6

Fig. 5. Cut of the base flow in the xy-plane: Entropy, (γ/800/504/ − /a/−). Only every fourth grid line is displayed.

4 3 2 1 0 -1

0

1

2

3

4 -6

-4

-2

0

2

4

6

Fig. 6. Cut of the base flow in the xy-plane: Velocity component u, (γ/800/504/ − /a/−). Only every fourth grid line is displayed.

2 1 0 -1 -2

-1

0

1

2

3

4

-6

-4

-2

0

2

4

6

Fig. 7. Cut of the base flow in the xy-plane: Velocity component v, (γ/800/504/ − /a/−). Only every fourth grid line is displayed.

185


1 0.8 0.6 0.4 0.2 0 -1

0

1

2

3

4

-6

-4

-2

0

2

4

6

Fig. 8. Cut of the base flow in the xy-plane: Velocity component w, (γ/800/504/ − /a/−). Only every fourth grid line is displayed.

1.2 1 0.8 0.6 0.4 0.2 0 -0.2

0

0.5

1

1.5

2

2.5

3

Fig. 9. Velocity profile in streamline coordinates at (x/δ, y/δ) = (55,176). (γ/700/ 446/ − /a/−).

in this direction. The y-coordinate with velocity component v is normal to the potential streamline and coplanar with the local tangential plane to the body surface. The streamline curvature K will be of interest later. It is given by K2 =

r˙ 2 r¨2 − (r¨ ˙ r2 ) 3

(r˙ 2 )

,

(4.6)

where r is the coordinate vector of a “fluid particle” in a Lagrangian description. The dot indicates the temporal derivative. Contour lines of K are shown in Fig. 10. Figure 10(a) shows the streamline curvature. It

186


varies from very small values at the stagnation point to values as high as KR = 0.7. The streamlines have no inflection point. An inflection point occurs only at very small sweep angles. x and y are non-dimensionalized with the viscous length scale δ. In Fig. 10(b), the cross-flow instability is shown as well. It occurs at the locus of highest streamline curvature. It will be discussed in the following.

(a) Streamline curvature in the xy-plane. Blue indicates the small curvature and red, the large curvature.

200 100 0 -100 -200 -300 -1000

-500

0

500

-400 1000

(b) Contours of the streamline curvature as in (a). The cross-flow instability is depicted simultaneously. It is visible as the two thin symmetrical strips. The cross-flow instability appears first at the locus of the highest streamline curvature. Fig. 10.

Streamline curvature in the xy-plane, (γ/642/409/ − /a/−).

187


5. Perturbation 5.1. Subsonic flows 5.1.1. Random perturbation In both subsonic cases, we superimposed harmonic perturbations with random phase and amplitude to the base flow. These perturbations were divergence-free. The form of the perturbations was u = A(y − yo )E/ro , v = A(xo − x)E/ro , w = A (−kx (y − yo ) + ky (x − xo )) E/kz ro .

(5.1)

It is E = exp(i(kx x+ky y +kz z +Φx +Φy )) exp(−r2 ln 2/ro2 ). The amplitude A is a random number within [−1, 1] and the phase Φ is chosen within [0, 2π]. ro is the length scale of the perturbation. 5.1.2. Attachment-line instability When the initial perturbation is introduced, a decay in perturbation energy (A.1) is observed (cf. Fig. 11). In a projection of the initial perturbation onto the eigenmodes of the flow, this corresponds to a decay of all energy -2

10

w u v ρ p T s

-6

10

10

-10

10

-14

fL2 (t )

10-18

10

-22

0

20

40 t

60

80

Fig. 11. Growth rate of several flow quantities over time. (α/644/∞/22/a/a) at M = 0.3. The growth rate is w1 = 0.187.

188


in the non-amplified modes. Later, a growth of the instable mode over six decades is observed until nonlinear effects lead to a saturation. The eigenmodes in the weakly compressible case show, as expected, a similar structure as the incompressible swept Görtler–H¨ ammerlin mode. All flow quantities except the velocity in spanwise direction are independent of y. The velocity v shows a linear dependency (Fig. 12). This is also visible in the three-dimensional plot of the rms-value of v in spanwise direction, shown in Fig. 13. Figure 14 shows a contour plot of the second invariant of the velocity gradient tensor Q = − 21 ui,j uj,i in the zx-plane. The center of the vortices is observed at x/δ ≈ 1.5. The highest strain is observed close to the wall. As in boundary layers, an increase in Mach number is expected to damp the instabilities. We performed computations at M = 0.1, 0.3, 0.7 and M = 1.3 which confirm this expectation. The Mach number scales within this region as M 2 . The growth rates are displayed in Fig. 15. The arrow at the left margin documents the growth rate predicted for the incompressible case (Obrist and Schmid, 2003). Results by Heeg (1998, Figs. 5–9) do not show this dependency, but Heeg defines the Reynolds number with the viscosity at infinity using T∞ .

1.2

1 w u v ρ p θ s

frms / wrms(y=0)

0.8

0.6

0.4

0.2

0 -100

-80

-60

-40

-20

0

20

40

60

80

100

y Fig. 12. Maxima of the disturbances in wall-normal direction in dependence of span. (α/644/∞/22/a/a) at M = 0.3.

189


vrms 2 10

-7

1 10

-7

0 100

-80 -40 0 40 80

0

-2

-4

-6

-8

-10

Fig. 13. The rms-value of the perturbation in v along the spanwise direction. (α/644/∞/22/a/a) at M = 0.3.

Fig. 14. Contour lines of Q′ = − 12 u′i,j u′j,i . The full lines are positive values of Q′ . (α/644/∞/22/a/a) at M = 0.3.

His results are given for Re = 1500. As shown in Sec. 2.1, this information does not uniquely characterize the flow.

5.1.3. Changes in the structure of the instability Different behavior of the instability for subsonic and supersonic flow is documented by the amplitudes of the fundamental mode (for u, w and w) shown versus distance to the wall in Fig. 16, for the swept Hiemenz flow.

190


0.4

extrapolated growth rate for linear least square t

0.2 -0 -0.2 -0.4 -0.6 0

Fig. 15.

(a)

0.2

0.4

0.6

0.8

1.2

1

1.4

1.6

1.8

Growth rate depending on Mach number (α/644/∞/22/a/a&i).

1 0.8 0.6 0.4 0.2

(b)

0 -10

-8

-6

-4

-2

0

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -10

-8

-6

-4

-2

0

Fig. 16. Normalized amplitudes of the fundamental mode of the velocities as a function of the distance to the wall, for an initially-centered vorticity perturbation and an adiabatic wall. (a) The Mach numbers are M = 0.1; (b) M = 1.3.


191

For high Mach numbers, we observed a significantly slower decay of the eigenfunction with wall-normal distance x/δ. Especially when the stand off distance of the bow shock is comparable to δ, interaction of the eigenmode will take place. 5.1.4. Influence of wall boundary condition On the flat plate, we also changed the boundary condition for temperature. We fixed the wall temperature at recovery temperature. In the undisturbed base flow, no heat transfer takes place, but for the perturbation it does. The growth rate is slightly higher than in the adiabatic case. 5.2. Supersonic flow Several base flows with Reynolds numbers from Re = 642 to Re = 800 were computed. The nose radii varied between R/δ = 409−500 and thus we expect situations in which the attachment-line mode is unstable according to Lin and Malik’s (1991) results for the incompressible curved surface. From the two-dimensional base flow, a three-dimensional flow field was generated with a depth of 20δ–30δ. The critical Reynolds number in the incompressible, flat plate case corresponds to α ≈ 2π 22 ≈ 0.288. 5.2.1. Random perturbation The base flow was disturbed with random and coherent perturbations. The 2 2 random perturbation was modulated by an amplitude factor e−(ξ−ξo ) /B and was set to zero at the wall and at the shock location. If r(x, y, z) is a random field with entries within − 12 and + 21 , then the flow field was perturbed for all flow quantities ϕ = (p, u, v, w, s) as ϕ := ϕ + ǫϕo r.

(5.2)

An amplitude of ǫ = 10−8 was found to guarantee that the response is linear. This was tested using several amplitudes and plotting the energy growth normalized with this value. The most important runs are reported in Table 3. The Reynolds number varies between Re = 600 and Re = 800. The wavenumber was within 2π α ≈ 2π 20 · · · 30 ≈ 0.3 · · · 0.2. The Mach number and sweep angle were kept constant. Thus the nose radius varies according to Eq. (2.1). This corresponds to the design problem for an airplane wing. For constant Mach number M = 1.25 and constant

192

J. Sesterhenn and R. Friedrich Table 2. Reynolds number and nose radius for M = 1.25 and Λ = 30◦ . Re R/δ

600 377

642 409

700 446

750 472

800 504

0.01 0.0001 1e-06 1e-08 1e-10 1e-12 1e-14 1e-16 0

1

2

3

4

5

6

7

8

9

10

t Fig. 17.

Modal energy growth for (γ/700/509/20, 22, 30/a/a).

sweep angle Λ = 30◦ , we find the values in Table 2 for Reynolds number and nose radius: 5.2.1.1. Perturbance growth depending on Reynolds number and wavenumber Full physical domain Figure 17 depicts the kinetic energy of the disturbances for the first harmonic in the full computational domain. The case with 30δ is expected to be stable with respect to the attachment-line instability (compare with Fig. 2). In all cases only the cross-flow instability was visible. Measurements in the vicinity of the stagnation line Table 3 reports the growth rates of the perturbance energy measure in the vicinity of the stagnation line only. In restricting the measurement domain to the vicinity of the leading edge, we wanted to exclude the cross-flow instability. This is not fully possible and Table 3 is only a rough guess of the growth rates of

193

Instabilities Near the Attachment-Line of Swept Wings Table 3. Growth rates of the perturbance energy measure in the vicinity of the stagnation line, dependent on Reynolds number and wavenumber. α/Re

600

642

700

750

800

2π/20 2π/22 2π/24 2π/26 2π/28 2π/30

−1.0672 −0.6381 −0.1085 −1.3554 −1.3447 −1.4227

−1.201 −0.580 −0.2501 −0.1333 0.2719 −0.3572

−1.3842 −0.702 −0.880 0.6291 0.562 0.4062

−1.694 −0.656 0.0804 0.5018 0.7268 0.5409

−1.4605 −0.7515 0.3103 0.7563 0.9164 0.8382

600

Fig. 18.

650

700

750

0.32 0.3 0.28 0.26 0.24 0.22 0.2 800

Tentative stability diagram according to Table 3.

the instability. The contour plot of this table is depicted in Fig. 18. Full lines indicate the unstable region. The distance of the contour lines is ∆w1 = 0.5. The amplitudes of the attachment-line instability are small compared to the other instability. This and its structure can be seen in Figs. 19 and 20 for Re = 800. Measurement at the locus of maximal streamline curvature Table 4 shows the corresponding values evaluated at the locus of maximal streamline curvature. At this location, the cross-flow instability develops first and spreads in both directions. The development will be discussed later. As expected, the growth rates are positive everywhere and increase with increasing Reynolds number and wavenumber. Structure of the modes Figures 19 and 20 show in the upper part, the v-velocity in the vicinity of the stagnation line and in the lower part, at the maximum of the cross-flow instability. The cases (γ/800/504/20/a/a) and (γ/800/504/26/a/a) are reported.

194

J. Sesterhenn and R. Friedrich Table 4. Growth rates of the perturbance energy measure in the vicinity of the locus of maximal streamline curvature, dependent on Reynolds number and wavenumber. α/Re

600

642

700

750

800

2π/20 2π/22 2π/24 2π/26 2π/28 2π/30

2.673 3.857 4.554 4.909 5.017 4.958

2.901 3.566 4.676 4.655 4.495 4.903

2.631 4.064 4.096 5.269 5.595 5.537

2.168 3.699 4.742 5.431 5.829 6.019

1.998 3.665 4.739 5.484 5.956 6.029

2 1.5 1 0.5

0

2

4

6

8

0 10 12 14 16 18

2 1.5 1 0.5

0

2

4

6

8

0 10 12 14 16 18

Fig. 19. Contour lines of the v-velocity. (γ /800/504/20/a/a). Maxima of the amplitudes are 2 × 10−9 in the upper plot and 5 × 10−5 in the lower one.

Table 3 shows that in the first case, the flow at the stagnation line is stable and in the second case, instable. The other instability grows in both cases. Flow visualizations show that the damped structure is of random nature whereas the others represent large-scale vortical structures. They are parallel to the y-axis close to the stagnation line and inclined as the freestream streamline.


195

3 2.5 2 1.5 1 0.5 0 0

5

10

15

20

25 3 2.5 2 1.5 1 0.5 0

0

5

10

15

20

25

Fig. 20. Contour lines of the v-velocity. (γ/800/504/26/a/a). Maxima of the amplitudes are 5 × 10−8 in the upper plot and 2.5 × 10−5 in the lower one.

The phase of the cross-flow instability is depicted for the v-velocity in Fig. 21. With increasing distance from the wall, the phase is running ahead. This is comparable with (Itoh, 1994, Fig. 3) and would indicate a cross-flow instability. Summary The two instabilities are difficult to distinguish with the present tools. The cross-flow instability was observed in all flow cases while the attachment-line instability was observed only in a few cases. The stabilizing influence of nose curvature and Mach number are confirmed. The structures of both instabilities are similar. This indicates a connection found by Bertolotti (1999). 5.2.2. Coherent disturbances We tried to favor the attachment-line instability by tailored disturbances which impinge on the body. The results were discouraging, but gave some insight in the growth of the cross-flow instability.

196


5 Phase of 4 3 2 1 0 -150

Fig. 21.

-100

-50

0 Angle

50

100

150

Phase of the v-velocity. (γ/800/504/26/a/a) at (x/δ, y/δ) = (55,176).

Nature of the disturbance Ahead of the shock, an analytical disturbance in entropy was prescribed. It had the form of „ «

2 2 o) − (x−xo ) δ+(y−y 2πz 2 e . (5.3) s = s∞ + ǫcv 1 + cos Lz This corresponds to a vehicle flying at supersonic speed through a spot of hot air. The interaction of the entropy disturbance with the shock creates vortical, entropy and acoustic disturbances behind the shock. The investigation range was 642 to 700 and 20δ to 40δ. The amplitude −6 v · · · 10−2 . The other quantities were chosen as in was chosen as ǫc s∞ = 10 Sec. 5.2.1 and Table 2. 5.2.2.1. Perturbation growth Kinetic energy Figure 22 shows the kinetic energy over time. Two different integration domains were used. One of them comprises the full physical domain, the other only the boundary layer. One can observe that all perturbation energy is confined to the boundary layer for times larger than t ≈ 3.5. In the following, we describe the perturbation development over time. The numbers correspond to those in Fig. 22. 1. External perturbation hits the shock The external perturbation is given analytically and has exponential shape. Its initial location (xo , yo ) was chosen such that the exponential tails are smaller than machine accuracy at the shock location. At the instant in time, that is denoted by 1 in

197


1e-5 1e-7

3

1

6

2

1e-9 5 4

1e-11 1e-13 1e-15 1e-17 1e-19

boundary layer full comp. dom.

1e-21 0

Fig. 22.

1

2

3

4

5

6

7

Perturbation growth. The numbers correspond to the paragraphs below.

the diagram, it has passed the shock and generates vortical, entropy and acoustic disturbances behind the shock (Fabre et al., 2001). Figure 23 displays isolines of pressure and Q = 21 u′i,j u′j,i at time t = 0.3 in a xy-cross-section for z = 0. At the left boundary, the shock is visible. The body is not displayed since the stagnation point is at (0, 0). The pressure was normalized with the pressure behind the shock p2 and the isolines scale from −6 × 10−3 to 4 × 10−3 with an increment of 10−4 . The picture shows a pressure peak followed by a depression. ∞ 2 and the values of the isolines Q was made non-dimensional with ∂u ∂x −8 −7 vary between 2 × 10 and 1.8 × 10 . 2. Acoustic wave impinges on the body At time t ≈ 0.5, the acoustic wave hits the boundary layer which is visible in the kinetic energy. It is reflected at the body (cf. Fig. 24) and leaves the boundary layer again. Perturbations are generally damped until time t = 1. Figure 32 shows that modal energy is excited but decays. 3. Acoustic wave reflects at the shock At time t = 1, the reflected acoustic wave hits the shock again. The kinetic energy was constant until now and now leaves the domain only via the shock up to time t = 4 (Figs. 25 and 26).

198


200

200

100

100

0

0

-100

-100

-200 -200

-100

0

-200 -200

-100

0

Fig. 23. Isolines of pressure and Q = 12 u′i,j u′j,i at time t = 0.3. On the left, the shock is visible. The stagnation point is located at (0, 0).

200

200

100

100

0

0

-100

-100 -200

-200 -200

-100

0

-200

1 ′ u u′ 2 i,j aj,i

Fig. 24. Isolines of pressure and Q = acoustic wave at the body.

-100

0

at time t = 0.57. After reflection of the

200

200

100

100

0

0

-100

-100 -200

-200 -200

-100

0

Fig. 25. Isolines of pressure and Q = at the shock.

1 ′ u u′ 2 i,j j,i

-200

-100

0

at time t = 0.85. Prior to the reflection


200

200

100

100

0

0

-100

-100 -200

-200 -200 Fig. 26. shock.

-100

0

Isolines of pressure and Q =

199

-200 1 ′ u u′ 2 i,j j,i

-100

0

at time t = 1.2. After reflection at the

4. Acoustic wave impinges on the body for the second time The wave, reflected for the second time, now becomes a plane wave but is very weak. Despite this, it triggers modal energy growth in the boundary layer at t ≈ 1.5 as seen in Fig. 32. 5. Instability growth The convection of the generated vortices along the body surface is seen in Fig. 27. In this plot, the first harmonic of the v-velocity F1 {v} is depicted along the arclengthsc s for three subsequent times. One observes the vortices coming from the stagnation point. These are the three maxima at s ≈ 800δ, 1100δ and 1600δ, with decaying amplitude. The vortices are outside of the boundary layer which is visible in Fig. 28. Here F1 {v} is depicted along the normal on the body surface n. This cut was made at arclength s = 177δ. Within the boundary layer, an enormous growth of the disturbance is triggered. Its maximum is at about n = −δ. The location of the first instability growth within the boundary layer is at s ≈ 80δ. Its maximum develops further downstream. In this region, the streamline curvature is maximal. 6. Occurence of a second maximum and further growth Subsequently, a second maximum develops at s ≈ 350δ. It grows faster than the first one. At t ≈ 7.2 both vortices merge. Figure 29 shows that the vortices grow within a very short time interval from an amplitude of 0.1% (cf. Fig. 28) to 40%. c s(x)

=

Rxp 0

1 + y ′2 dx =

√ p x x + R/2 +

R 2

log

√ √ x+ x+R/2 √

R/2

.

200


0.14

t =4.3 t =5.2 t =6.0

0.12 0.1 0.08 0.06 0.04 0.02 0 0

Fig. 27.

0.009 0.008

500

1000

1500

2000

Convection of the vortices along the body.

t =4.3 t =5.2 t =6.0

0.007 0.006 0.005 0.004 0.003 0.002 0.001 0 -10

Fig. 28.

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

Energy of the first mode F1 {v} along the body normal n. At s = 177δ.

The structure of the mode is like a cross-flow, see Fig. 31. A cartoon of the situation is displayed in Mielke (1999, Fig. 1.5), a similar plot is found in Schmid and Henningson (2001, Fig. 9.48a). This vortex is spatially confined, as is to be seen in Fig. 30. On the left, an isosurface of Q = 12 ui,j uj,i with

201


0.4

t =4.3 t =5.2 t =6.0 t =6.7 t =7.1 t =7.4

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0

Fig. 29.

Fig. 30.

100

200

300

400

500

600

700

Growth of the second vortex, merging of both.

Isosurface of Q =

1 u u 2 i,j j,i

and Q′ =

1 ′ u u′ 2 i,j j,i

at time t ≈ 7.

a positive Q is depicted, while on the right we show Q′ = 12 u′i,j u′j,i , both at time t ≈ 7. The gray background is the body surface. This plot shows a periodical continuation of the domain. The flow is from left to right and the coordinate system lies in the stagnation line. The figure on the right-hand side corresponds to Fig. 29. Saturation At times greater than t ≈ 7.0, the vortices saturate and stop developing. Further decay cannot be computed since the resolution is not

202


3 2.5 2 1.5 1 0.5 0 0

5 10 15 20 25 30 35 40 45 50

Fig. 31. Contour lines of the v-velocity at time t ≈ 7.4, at s = 189δ. The distance of the isolines is ∆v = 0.1 u2 .

0

1

2

3

4

5

6

Fig. 32. Energy of the first Fourier modes in the boundary layer (γ/656/418/22/A/A) for ∆s/s∞ = 10−5 .

sufficient. Further computations which follow the later development are currently performed. Modal energy growth The time evolution of the modal energy is displayed in Fig. 32 for eight modes at a resolution of 16 gridpoints in spanwise direction. The disturbance was s/s∞ = 10−5 . The second mode is well below the first and the others are unimportant. This shows that for the initial phase of the computation 8 points are sufficient. 6. Conclusion The attachment-line instability in compressible flow was detected and is in concordance with the known results from the incompressible case. In


203

the range of M = 0 to M = 1.3, the flow stabilizes according to M 2 . Curvature stabilizes the flow as well, but in this case another instability, likely a streamline curvature instability, appears and which grows with a significantly higher growth rate. The main difficulty in the present approach, solving an initial value problem by direct numerical simulation, is the identification of evolving instabilities. Constructing a stability diagram is therefore difficult and even if the identification problem were solved, it is prohibitively expensive. Therefore, we presently work on a global stability analysis of this flow.

A. Definitions A.1. Kinetic energy The kinetic energy was computed as

E=

Ω

1 (ui − Ui )2 dΩ ≡ 2

Ω

1 ′ ′ (u u ) dΩ. 2 i i

(A.1)

Ω is either the full computational domain or a part of it.

A.2. Modal energy The modal energy was defined as Em =

(xy)

1 Fm {(ui − Ui )2 )} dx dy ≡ 2

(xy)

1 Fm {(u′i u′i )} dx dy. 2

(A.2)

The Fourier transform was performed in spanwise direction and is named Fm {·} for the mth mode. The result was integrated in the xy-plane. A.3. Growth rates The growth rate is computed using the above energy measures as wm =

1 log Em (t + ∆t) − log Em (∆t) . 2 ∆t

(A.3)

It has the dimension of an inverse time and was non-dimensionalized ∞ with ∂u ∂x .

204


A.4. Q-criterium Vortices are identified using the Q-criterium. A discussion of several possibilities for detecting vortices is given in (Mielke, 1999). The velocity gradi∂u ent tensor ∂xpq has the characteristic equation (A.4) λ3 + P λ2 + Qλ + R = 0, ∂u ∂u ∂u ∂u with P = − ∂xpp , Q = − 12 ∂xpq ∂xpq and R = ∂xpq . Its second invariant Q may be expressed using the rotation tensor rij and the deformation tensor sij as Q=−

1 ∂up ∂uq 1 = (rkk − skk ) . 2 ∂xq ∂xp 2

(A.5)

Positive values of Q therefore indicate regions where rotation is dominant. If a pressure minimum is observed at the same place, then we are talking of a vortex. References 1. F. P. Bertolotti, The connection between cross-flow vortices and attachmentline instabilities, in IUTAM Symposium on Laminar–Turbulent Transition, Sedona, USA (September 1999), pp. 625–630. 2. F. S. Billig, Shock-wave shapes around sperical- and cylindrical-nosed bodies, J. Spacecraft and Rockets 4(6) (1967) 822–823. 3. H. Bippes, Basic experiments on transition in three-dimensional boundary layers dominated by cross-flow instability, Progr. Aerospace Sciences 35 (1999) 363–412. 4. D. M. Bushnell and J. K. Huffman, Investigation of heat transfer to a leading edge of a 76◦ swept fin with and without cordwise slots and correlations of swept leading edge transition data for Mach 2 to 8, Technical Report TMX1475, NASA (1967). 5. S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, republication of the 3rd printing of 1970 edition (Dover Publications, Incorporated, 1981). 6. T. T. Creel, I. E. Beckwith and F. J. Chen, Transition on swept leading edges at Mach 3.5, J. Aircraft 24(10) (1987) 710–717. 7. N. A. Cumpsty and M. R. Head, The calculation of three-dimensional turbulent boundary layers. Part II. Attachment-line flow on an infinite swept wing, The Aeronautical Quarterly, XVIII (1967) 150–164. 8. N. A. Cumpsty and M. R. Head, The calculation of the three-dimensional turbulent boundary layer. Part III. Comparison of attachment line calculations with experiment, The Aeronautical Quarterly 20 (1969) 99–113. 9. D. Fabre, L. Jacquin and J. Sesterhenn, Linear interaction of a cylindrical entropy spot with a shock, Phys. Fluids 13(8) (2001) 2403–2422.


205

10. H. G¨ ortler, 50 Jahre Grenzschichtforschung, Chapter Dreidimensionale Instabilit¨ at der ebenen Staupunktstr¨ omung gegen¨ uber wirbelartigen St¨ orungen (Vieweg, Braunschweig, 1955), pp. 304–314. 11. P. Hall, M. Malik and D. I. Poll, The stability of an infinite swept attachment-line boundary layer, Proc. R. Soc. Lond. A395 (1984) 229–245. 12. P. Hall and M. Malik, The instability of a three-dimensional attachment-line boundary layer: Weakly nonlinear theory and a numerical approach, J. Fluid Mech. 163 (1986) 257–288. 13. R. Heeg, Stability and Transition of Attachment-line Flow, PhD thesis, University of Twente (1998). 14. K. Hiemenz, Die Grenzschicht an Einem in den Gleichförmigen Fl¨ ussigkeitsstrom Eingetauchten, Geraden Kreiszylinder, PhD thesis, Göttingen, Dingl. Polytechn. J. 326 (1911) 321. 15. G. H¨ ammerlin, 50 Jahre Grenzschichtforschung, Chapter Zur Instabilit¨ atstheorie der ebenen Staupunktstr¨ omung (Vieweg, Braunschweig, 1955), pp. 315–327. 16. N. Itoh, Instability of three-dimensional boundary layers due to streamline curvature, Fluid Dyn. Res. 14 (1994) 353–366. 17. R. D. Joslin, Simulation of nonlinear instabilities in an attachment-line boundary layer, Fluid Dynam. Res. 18 (1996) 81–97. 18. A. V. Kazakov, Effect of surface temperature on the stability of swept attachment-line boundary layer, Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkostu i Gaza (1990) 78–82. 19. A. Le Duc, Simulation Numerique Directe des Instabilites dans l’Ecoulement Tridimensionel Compressible Hiemenz, PhD thesis, Laboratoire de Mecanique des Fluides, EC Lyon & Fachgebiet Strömungsmechanik, TU M¨ unchen (2001). 20. R.-S. Lin and M. R. Malik, The stability of attachment-line boundary layers. Part 1. The incompressible swept Hiemez flow, J. Fluid Mech. 311 (1996) 239–255. 21. R.-S. Lin and M. R. Malik, The stability of attachment-line boundary layers. Part 2. The effect of leading edge curvature, J. Fluid Mech. 333 (1997) 125–137. 22. R.-S. Lin and M. R. Malik, Stability and transition in compressible attachment-line boundary layer flow, Technical Report 952041, SAE (1995). 23. M. R. Malik and I. E. Beckwith, Stability of a supersonic boundary layer along a swept leading edge, in AGARD Conf. Proc. 438 (1988). 24. C. Mielke, Numerische Untersuchungen zur Turbulenzentstehung in dreidimensionalen kompressiblen Grenzschichtströmungen, PhD thesis, ETH Z¨ urich (1999). 25. A. Murakami, E. Stanewsky and P. Krogmann, Boundary-layer transition on swept cylinders at hypersonic speeds, AIAA J. 34(4) (1996) 649–654. 26. D. Obrist, The stability of the swept leading-edge boundary layer, PhD thesis, University of Washington, Seattle (2000).

206


27. D. Obrist and P. Schmid, The linear stability of swept attachment-line boundary-layer flow. Part 1. Spectrum and asymptotic behavior, J. Fluid Mech. 493 (2003) 1–29. 28. W. Pfenninger, Flow phenomena at the leading edge of swept wings, Technical Report AGARDoraph 97, AGARD (1965). 29. W. Pfenninger and J. W. Bacon, Amplified laminar boundary layer oscillations and transition at the front attachment-line of a 45 degree swept flat-nosed wing with and without boundary layer suction, in Viscous Drag Reduction (Plenum Press, 1969). 30. D. I. A. Poll, The development of intermittent turbulence on a swept attachment-line including the effects of compressibility, The Aeronautical Quarterly 34 (1983) 1–23. 31. D. I. A. Poll, Transition description and prediction in three-dimensional flows, in Special Course on Stability and Transition of Laminar Flow, AGARD (1984) 5/1–5/23. 32. D. I. A. Poll, Some observations of the transition process on the windward face of a long yawed cylinder, J. Fluid Mech. 150 (1985) 329–356. 33. L. Rayleigh, The flow of compressible fluid past an obstacle, Phil. Mag. S. 6 32(187) (1916). 34. E. Reshotko and I. E. Beckwith, Compressible laminar boundary layer over a yawed infinite cylinder with heat transfer and arbitrary Prandtl number, Technical Report 1379, National Advisory Committee for Aeronautics (1958). 35. L. Rosenhead (ed.), Laminar Boundary Layers (Oxford at the Clarendon Press, 1963). ¨ 36. V. Saljnikov and U. Dallmann, Verallgemeinerte Ahnlichkeitsl¨ osungen f¨ ur dreidimensionale, laminare, stationäre, kompressible Grenzschichtstr¨ omungen an schiebenden profilierten Zylindern, Technical Report DLR-FB 89–34, DLR, G¨ ottingen (1989). 37. W. S. Saric and H. L. Reed, Cross-flow instabilities — Theory and technology, in AIAA Pap. (2003) 2003–0771. 38. H. Schlichting, Grenzschicht-Theorie (Verlag G. Braun, Karlsruhe, 1982). 39. P. Schmid and D. Henningson, Stability and Transition in Shear Flows (Springer, New York, 2001). 40. F. Schwertfirm, DNS von Instabilit¨ aten in der dreidimensionalen kompressiblen Str¨ omung entlang der Vorderkante eines gepfeilten Tragfl¨ ugels, Master’s thesis, TU-M¨ unchen, Fachgebiet Str¨ omungsmechanik (2002). 41. J. Sesterhenn, A characteristic-type formulation of the Navier–Stokes equations for high order upwind schemes, CAF 30(1) (2001) 37–67. 42. P. R. Spalart, Direct numerical study of leading edge contamination, Technical Report CP-438, Fluid Dyn. of 3D Turb. Shear Flows and Transition, AGARD (1988). 43. V. Theofilis, Linear and nonlinear instability of the incompressible swept attachment-line boundary layer, J. Fluid Mech. 355 (1998) 193–227. 44. D. R. Topham, A correlation of leading edge transition and heat transfer on swept cylinders in supersonic flow, J. Roy. Aeronaut. Soc. 69 (1965) 49–52.

EXPERIMENTAL STUDY OF WALL TURBULENCE: IMPLICATIONS FOR CONTROL

Ivan Marusic∗ and Nicholas Hutchins† Department of Aerospace Engineering and Mechanics University of Minnesota Minneapolis, MN 55455-0153, USA E-mails: ∗[email protected] † [email protected] A review is presented of recent multi-plane PIV measurements undertaken to investigate coherent structures in the log and wake regions of the turbulent boundary layer. The results show that the wall-parallel plane measurements are dominated by long meandering stripes of positive and negative u fluctuation, alternating in the spanwise direction and carrying a large proportion of the Reynolds shear stress in the outer regions. The size of these features appears to scale well with outer variables, and there are signs of preferred spanwise spacing modes (a notable repetition or periodicity of large-scale u fluctuations in the spanwise direction). A pronounced vortical structure is noted to be clustered about the elongated regions of momentum deficit in arrangements that are consistent with the hairpin packet paradigm. These large-scale structures maintain a presence in the near-wall region, imposing low wavenumber energy onto the buffer region. Through complementary hot-wire measurements, it is found that these structures can be extremely long in the streamwise direction (often exceeding 20δ in length). The consequent implications for control of wall turbulence are briefly discussed. Contents 1

Introduction

208

2

Experimental Details

209

2.1

Facility

209

2.1.1 Wall-parallel plane PIV

210

2.1.2 Inclined-plane cross-stream PIV

211

2.1.3 Combined-plane PIV

213 207

208

I. Marusic and N. Hutchins

3

Resolved Structure in the Log and Wake Regions of Turbulent Boundary Layers 3.1 Large-scale stripiness in the log region 3.1.1 Example velocity fields 3.1.2 Two-point correlations 3.2 An associated vortical structure 3.2.1 Vortex identification 3.2.2 Example swirl fields 3.2.3 The hairpin packet model 3.2.4 Statistical evidence 3.2.5 Alternative explanations 3.3 Reynolds number scaling 3.4 Influence at the wall 3.5 Extremely long streamwise modes 3.6 Associated Reynolds stress 3.7 Spanwise repetition 4 Summary 5 Implications to Flow Control Acknowledgments References

214 214 214 218 221 221 222 226 228 231 232 233 233 236 238 242 243 244 244

1. Introduction A great advantage of the Particle Image Velocimetry (PIV) is its ability to provide an instantaneous quantitative snapshot of spatial coherence. Of the other available techniques for boundary layer study, only Direct Numerical Simulations (DNS) can provide comparable spatial views. Despite the obvious successes of DNS in resolving the near-wall regenerative cycle, the larger scales that inhabit the log and wake region have, until quite recently, been inadequately covered by simulation owing to low Reynolds numbers and limited computational box sizes. Furthermore, most experimental studies have tended to focus on the near-wall viscous dominated region of the flow. A review of such studies, as well as a comprehensive overview of PIV in boundary layers, is given in the accompanying chapter by Onorato et al. For the present chapter, we focus on those structures that inhabit the outer regions of the flow (here we define the outer region as z + ≥ O(100)).

Experimental Study of Wall Turbulence

209

2. Experimental Details Numerous PIV experiments have been undertaken at the University of Minnesota with the aim of learning more about the coherent structure in the log and wake regions of turbulent boundary layers. A condensed review of these results will be attempted over the following chapter. Prior to this, a brief overview of the experimental facility will be given, along with details of any experimental issues and challenges particular to each configuration. 2.1. Facility Experiments are conducted in an open return suction wind tunnel (Eiffeltype), with working section 4.7 m × 1.2 m × 0.3 m. The measurement station is 3.3 m downstream of a 1.5 mm diameter trip wire. Optical access is through 0.5 in. float glass on the floor and sidewalls. Flows are seeded with olive oil droplets from an array of eight Laskin nozzles located upstream of the tunnel inlet. The seeded flow is illuminated by pulsed sheets from two Nd:YAG lasers (either a Big Sky CFR200 or Spectra Physics Quanta-Ray). For imaging, we use a pair of TSI Powerview 2048 × 2048 pixel resolution digital cameras. The combined-plane experiments use an additional pair of Kodak Megaplus CCD 1024 × 1024 cameras to image the streamwise/wallnormal plane. All PIV measurements discussed here are stereoscopic, meaning that all three velocity components are obtained on the imaging plane. A more detailed description of the tunnel facility and PIV system can be found in Ganapathisubramani et al. (2003). Details of all the experimental conditions are given in Table 1. The boundary-layer thickness δ used to calculate Reτ (= δUτ /ν) was determined from a Coles law of the wall/wake fit to mean velocity profiles obtained using a Pitot-static tube. The friction velocity (Uτ ) is obtained from a Clauser chart fit (with log law constants κ = 0.41 and A = 5.0). U∞ is the freestream velocity and z + (= zUτ /ν) and z/δ are the respective inner and outer scaled wall-normal heights. For the most part, the current discussion will focus on results from experiments made at Karman numbers of approximately Reτ ≈ 1000. However, the inclined-plane measurements were repeated across a range of Reynolds numbers to analyze scaling of spanwise length scales, and these results will be touched upon briefly. Throughout this paper, the axis system x, y and z refer to the streamwise, spanwise and wall-normal directions, with u, v and w describing the

210

I. Marusic and N. Hutchins Table 1. Experimental parameters for PIV experiments. Parentheses for the inclined-plane experiments indicate 135◦ case. Experiment

Reτ

Reθ

U∞ m/s

Uτ m/s

Wall-parallel

1060

2500

5.9

0.244

Inclined-plane

690 1010 1840 2800 1100

1430 2680 4740 7440 2600

Combined-plane

3.05 5.33 — 17.23

(3.13) (5.25) (10.12) (17.28) 6.00

0.138 0.212 — 0.607

(0.138) (0.208) (0.376) (0.609) 0.25

z+

z/δ

92 150 198 530

0.087 0.141 0.187 0.500

— — — — 98

— — — — 0.089

respective fluctuating velocity components. Capitalised velocities (e.g., U ) and overbars indicate time-averaged values whilst the prime symbol for velocities (u′ ) denotes RMS values. 2.1.1. Wall-parallel plane PIV For these experiments, the seeded boundary layer is illuminated by a laser sheet parallel to the glass wall. A stereo pair of cameras view from below to image a streamwise/spanwise measurement domain (Fig. 1). Four data sets are obtained with the wall-normal height of the laser sheet (and hence, the image plane) ranging from z/δ = 0.087–0.5 (Table 1). In this way, comparative views of the log and wake regions are obtained. Full details are given in Ganapathisubramani et al. (2003, 2005). z

idealized structure

U∞ x y

light-sheet

2K× 2K cameras

Fig. 1. Basic setup for streamwise/spanwise plane PIV, showing viewing window, laser sheet and camera location.


211

2.1.2. Inclined-plane cross-stream PIV These experiments are essentially a reprise of the well-known visualizations by Head and Bandyopadhyay (1981). The laser light-sheet and image plane are arranged in a cross-stream orientation inclined at either 45◦ or 135◦ to the x-axis. Figures 2(a) and 2(b) show the experimental configuration and transformed axis system for both inclinations. In each case, the out-of-plane direction is referred to as x′ , spanwise ordinates are unchanged (y ′ = y) and the third ordinate z ′ is defined appropriately by the right-hand rule. The velocity components along the x′ -, y ′ - and z ′ -axes are denoted as uθ , vθ and wθ respectively, where θ is the image-plane inclination angle (45◦ or

inclined-plane laser sheet (45° )

z′ (a)

y′ z

(a) x′

x

idealized structure

U∞

y

2K × 2K cameras (vp)

(b)

x′

z′ inclined-plane laser sheet (135° )

y′ z idealized structure

x

y

U∞ 2K × 2K cameras (vp)

aluminium separator

Fig. 2. Basic PIV setup showing viewing window, laser sheet, camera location and representative idealized structure. (a) For 45◦ case; (b) For 135◦ case.

212


135◦ ). It is these components that are measured during the experiment. Streamwise, spanwise and wall-normal velocity components are recovered from simple trigonometric conversion. w45 − u45 u45 + w45 √ √ , v = v45 , w = . (2.1) u= 2 2 u135 − w135 w135 + u135 √ √ , v = v135 , w = . (2.2) 2 2 Successful PIV measurements are based on the intrinsic assumption that the particles imaged by the first laser pulse (frame A) will remain within the light-sheet for the duration of the time delay (such that they are imaged onto frame B by the second laser pulse). For cross-plane measurements, this assumption is undermined by the large component of the mean velocity vector that acts in the out-of-plane direction. The approximate rule that out-of-plane particle displacement should not exceed 0.25–0.30 of the lightsheet thickness (Adrian, 1991) imposes a maximum time delay between image pairs for each given freestream velocity. For a fixed light-sheet thickness and camera resolution (2k × 2k), this leads to a necessary compromise between mean particle displacements (and dynamic range), spatial resolution and overall viewing area. It is not desirable to thicken the light-sheet beyond approximately 1.5 mm (this limit yields an interrogation cube for a 32 × 32 pixel spot-size), meaning that the only way remaining to increase particle displacements is to increase magnification, which though beneficial in terms of increased spatial resolution, incurs the penalty of a reduced field-of-view. A primary focus of this experiment is to study the largescale features in the log and wake regions. However, it is also desirable to maintain sufficient resolution to enable analysis of smaller-scale swirling motions. A careful balance must be struck to meet the overall goals of the experiment. The optimum compromise for these conflicting requirements was a field-of-view that spanned 1.2δ in the spanwise direction and 0.8δ in the wall-normal, such that all of the log region and most of the wake region of the boundary layer is imaged. Using a 32 × 32 pixel interrogation window, this yields a spatial resolution of 24 viscous wall units (i+ = 24) at Reτ = 1010 (most analysis is conducted at this Reynolds number). In meeting these conditions, the pixel displacements are necessarily small. It has been shown that data with a small dynamic range can be especially susceptible to errors caused by pixel locking. Christensen (2004) has shown that in cases where the dynamic range is less than 0.5 pixels (which is the case here), the influence of peak locking can lead to an underestimation of the turbulent fluctuations, especially noticeable in the mean Reynolds u=


213

vertical plane (vp)

z horizontal plane (hp)

U∞ x y

1K × 1K cameras (hp) 2K × 2K cameras (vp)

Fig. 3. Basic PIV setup for the combined-plane experiments showing vertical and horizontal laser sheets with associated stereo camera pairs.

stress profile. In spite of these challenges, the mean statistics reported in Hutchins et al. (2005b) (where a full error analysis is given) show that with careful attention to the above details, the method of inclined-plane PIV has successfully resolved the salient features of the turbulent boundary layer. Other specific issues about the experimental design are discussed in Hutchins et al. (2005b). For this chapter, we will also consider an earlier inclined-plane data set with a wider field-of-view and Reτ = 1142 (as detailed in Ganapathisubramani et al., 2005). This data set predates the results reported in Hutchins et al. (2005b). The increased field-of-view is achieved at the expense of spatial resolution, with the interrogation window resized at i+ = 36. In recognition of the small-scale filtering effect due to the larger i+ , this data set will only be used to analyze the largest energetic modes (which are on average an order of magnitude larger than i+ ). 2.1.3. Combined-plane PIV For these experiments, stereoscopic planar measurements are made simultaneously on two mutually perpendicular planes (both streamwise/wallnormal and streamwise/spanwise) as shown schematically in Fig. 3. These planes will be respectively referred to here as the vertical and horizontal planes (abbreviated to vp and hp). The two light-sheets are formed by splitting the 400 mJ pulse from the Spectra Physics laser into two separate orthogonally-polarized beams. Each of these is sent through its own set of spherical and cylindrical lenses. The seeding particles illuminated by the light-sheets are assumed to be small enough such that the original lightsheet polarization is retained by the scattered light, and thus polarizing filters on the camera lenses can be used to filter out illuminated particles

214


that are not in the sheet of interest. For a description of this technique, see Christensen (2001). Due to the imperfect nature of the polarizing filters and optics, some contamination occurs between the images, and this is most noticeable at the intersection of the sheets, producing a background glow. Such stationary features cause errors in computed particle displacements, such that fluctuations at the planar intersection are under-resolved (this error is clearly visible in similar experiments by Kim and Kwon, 2004). A simple image pre-conditioning procedure has been implemented to minimize this error. A mean intensity field, created from the average of all experimental captures, is subtracted from each image prior to cross-correlation, equalizing particle intensities across the field-of-view and removing any consistent bright spots. Further details of these experiments are given in Hambleton et al. (2005), where it is shown that this technique of background subtraction can significantly improve the mean velocity, turbulent intensity and Reynolds shear stress profiles. 3. Resolved Structure in the Log and Wake Regions of Turbulent Boundary Layers 3.1. Large-scale stripiness in the log region While statistical analyses of the PIV data are extremely useful, much can also be learned from simply viewing individual frames. During the course of this study, we have viewed many thousands of PIV acquired flow fields (over twelve thousand frames are being considered in all for this chapter). By acquainting ourselves to this extent with the raw data, some obvious recurrent features emerge. We will start this discussion of the data with some examples of instantaneous flow fields from the different experiments, each of which is chosen as being representative of the large-scale features we have noticed in the log and wake regions of the turbulent boundary layer. Throughout the subsequent analysis, we will build on these examples, using more rigorous statistical techniques to cement our initial observations but always returning to instantaneous flows in an attempt to explain the finer details of the time-averaged statistics. 3.1.1. Example velocity fields A strong tendency for long streamwise regions of positive and negative u fluctuation (alternating in the spanwise direction) is immediately noticeable from the instantaneous PIV data. As an example, Figs. 4(a) and 4(b)


215

Fig. 4. Example of instantaneous negative u fluctuations. (a) From wall-parallel PIV; (b) From 45◦ inclined-plane PIV. Darker shading shows larger negative fluctuation. All positive fluctuation is shaded in white (see gray scale). Arrows show mean flow direction.

give instantaneous snapshots of the negative streamwise velocity fluctuation in both the wall-parallel and inclined-planes respectively. In both cases, the darker gray shading shows negative u fluctuation and all positive fluctuations are shaded white. Figure 4(a) is characterized by long low-speed regions, the length of which seems to exceed the measurement domain. The regions between these low-speed features are typically filled by high-momentum fluid. Together, these features give rise to an over-riding impression of “spanwise stripiness” when viewing wall-parallel plane velocity fields. The inclined-plane data shown in Fig. 4(b) gives some clues to the

216


typical wall-normal extent of these features. It would seem that in crosssection, the low-speed stripes of plot (a) appear as “plumes” of low-speed fluid, which in this case span from the near-wall buffer region well into the wake region (beyond 0.5δ). Viewed in this way, these features very much resemble the “superbursts” discussed by Na et al. (2001). It is important to note that the two frames in Figs. 4(a) and 4(b) were not acquired simultaneously. Each are from separate experiments, performed at different times. Thus, although the spanwise repeating plumes in the inclined-planes (see plot (b)) would seem to be the only features that can satisfactorily account for the spanwise stripiness noted of plot (a), the evidence in Fig. 4 could still be considered somewhat circumstantial. Such issues are addressed by the combined-plane data set, where a simultaneous view of the spanwise stripiness and its associated wall-normal extent is obtained. Figure 5 shows an example flow field pair from the combined-plane data set. The horizontal plane (shown by the plan view of plot (a)) exhibits the spanwise stripiness previously noted of Fig. 4(a). This particular frame set is chosen such

Fig. 5. Example of instantaneous negative u fluctuations from the combine-plane PIV. (a) Plan view showing horizontal plane; (b) Side view showing vertical plane; (c) Orthoganal projection showing both planes. Gray scaling as Fig. 4. Thick horizontal lines on plots (a) and (b) show plane intersects.


217

that a low-speed stripe lies approximately on the intersection between the vertical and horizontal planes (shown by the bold black line in Fig. 5(a)). The side view (plot (b)) shows the vertical plane. The intersection of the horizontal plane is marked by the bold line at z/δ = 0.087. An analysis of these simultaneous perpendicular flow fields offers clear proof that the spanwise stripiness noted in the wall-parallel plane is indeed associated with a much taller low-momentum region. This scenario is clarified by the threedimensional projection of Fig. 5(c). The low-speed region, first identified as stripiness in the horizontal plane, can occasionally extend well beyond 0.5δ in the wall-normal direction. For clarity, only the negative u fluctuations have been shown on these monochrome plots. However, it is clear from viewing many thousands of frames that the areas between the elongated low-momentum zones (shaded gray on Figs. 4 and 5) are commonly filled with accelerated u regions. As clarification of this phenomena, Fig. 6 shows a different realization from the combined-plane experiments, which though exhibiting the spanwise stripiness in the horizontal plane, is selected such that the intersection of the two planes lies between adjacent low-momentum stripes. In this case the

Fig. 6.

See Fig. 5 for caption.

218


vertical plane, visible in plots (a) and (c), is almost fully white. There is little sign of the negative fluctuations that dominated the vertical plane in Fig. 5. Indeed for the vertical plane shown, almost the entire bottom half of the boundary layer (z < 0.5δ) is predominated by strongly positive u fluctuations. 3.1.2. Two-point correlations In an attempt to anchor these observations within a more statistical framework, the two-point correlation tensor of the streamwise velocity component has been calculated at a common zref for the three experimental configurations (at comparable Reτ ). For these purposes, the wall-parallel plane is considered homogenous in x and y such that the two-point correlation coefficient between any two fluctuating quantities a and b can be defined as, a(x, y)b(x + ∆x, y + ∆y) , (3.1) σa σb where σa and σb are the standard deviations of a and b and, ∆x and ∆y are the in-plane separations between these two fluctuating components. The overline notation for Eq. (3.1), denotes an ensemble average over all x and y locations and for all realizations. For the inclined-plane results and for the vertical combined-plane, there is an inhomogeneity in z. Thus the respective two-point correlations must be calculated at a specific wallnormal position, as given by Eqs. (3.2) and (3.3). For a given number of frames, the assumed homogeneity in the wall-parallel plane will yield a greater number of ensembles, and therefore a smoother more converged twopoint correlation than those planes that involve a wall-normal component. Rab (∆x, ∆y) =

Rab (zref , ∆y, ∆z) =

a(y, zref )b(y + ∆y, zref + ∆z) , σa (zref )σb (z)

(3.2)

Rab (zref , ∆x, ∆z) =

a(x, zref )b(x + ∆x, zref + ∆z) . σa (zref )σb (z)

(3.3)

Figure 7 shows the two-point correlation of the streamwise velocity fluctuations at zref /δ ≈ 0.087 (chosen to correspond with the height of the z + = 92 wall-parallel data set). Projections from all three experiments are shown (the Reτ = 1010 inclined-plane data is used). Considering first the wall-parallel result of Fig. 7(a), a prominent region of positive correlation is flanked in the spanwise direction by anti-correlated behavior which peaks at approximately ∆y/δ ≈ ±0.35. These correlated and anticorrelated regions are highly elongated in the streamwise direction, such

219


1 0.8

135°

(b)

(a)

45°

(c)

0.6 0.4

∆y / δ

0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1

−0.5

0 ∆x /δ

0.5

1 0

−0.5

0 ∆x /δ

0.5

1

0.5 z /δ

1

0

0.5 z /δ

1

1

(d)

∆z / δ

0.8 0.6 0.4 0.2 0 −1

Fig. 7. Two-point correlations of the streamwise velocity fluctuation Ruu calculated at zref /δ = 0.087. (a) For wall-parallel plane; (b) For 135◦ inclined-plane; (c) For 45◦ inclined-plane; (d) For vertical-plane from combined experiments. Contour levels are from Ruu = −0.12 to 0.96 in increments of 0.06. Solid lines show positive contours and dashed lines show negative contours.

that the Ruu = 0.05 contour has an overall streamwise length of almost 4δ. The spanwise width of the positive and negative correlation regions seems to be approximately 0.35δ. This pattern of anti-correlated regions flanking positive correlation reaffirms the stripiness that was previously noted of the example wall-parallel planes in Figs. 4–6 (where elongated high- and lowspeed regions were observed to alternate in the spanwise direction). From the remaining projections of Figs. 7(b) to 7(d), we get a clear sense that the stripiness at z/δ = 0.087 is actually a slice through a far wider structure. The inclined-plane plots (b) and (c) show clearly that the correlated and anti-correlated regions exhibit a considerable coherence in the wallnormal direction. This reflects the initial observations from Figs. 4 and 5 that the low- and high-momentum regions can extend well into the wake region (frequently extending beyond z/δ = 0.5). The disparity between the wall-normal extent of the correlation contours in the 135◦ and the 45◦

220


planes indicate that the correlated (and anti-correlated) regions have a pronounced streamwise inclination. This becomes increasingly apparent from the vertical plane correlation contours of Fig. 7(d), where the dash-dot lines show the 45◦ and 135◦ planes. In this view, we see an obvious inclination angle producing Ruu contours that extend further in the 45◦ direction than for the 135◦ plane (which explains the disparity in plots (b) and (c). Such inclined tendency in Ruu correlations has been studied in detail for these data sets by Ganapathisubramani et al. (2005), and has also been noted on numerous previous occasions (e.g., Kovasznay et al., 1970; Christensen, 2001; Christensen and Adrian, 2001). This streamwise asymmetry of the correlation contours suggests that some instantaneous inclination should manifest in the elongated low and high u momentum regions. Signs of this tendency are evident in the example combined-plane flow fields (see Figs. 5 and 9). The correlation construct assembled for Fig. 7 is derived from nonsimultaneous acquisitions. It is perfectly correct (and mathematically rigorous) to infer the three-dimensional structure of correlations in this way, since the condition points for all three experimental sets lie in the same homogenous plane. However, with the simultaneous acquisitions from the combined-plane experiments, we can also calculate the Ruu correlation for condition points that do not lie on the planar intersect. As an example of this, Fig. 8 shows Ruu contours where the condition point is located in the vertical plane some distance above the intersection of the two planes (just beyond the edge of the log region at zref /δ = 0.17). The streamwise velocity fluctuations are correlated everywhere in the horizontal plane with the u fluctuations at the reference height in the vertical plane, as given by, Ruvp uhp (zref , ∆x, ∆y) =

uvp (x, zref )uhp (x + ∆x, y + ∆y) , σuvp (zref )σuhp

(3.4)

where (hp) and (vp) denote velocity fluctuations from the vertical and horizontal planes respectively. Figure 8 shows the resulting cross-correlation between uvp and uhp . Clearly the velocity fluctuations in the vertical plane are strongly and simultaneously correlated with those in the horizontal (wall-parallel) plane, demonstrating that events some distance above the intersect (at z/δ = 0.17) leave a clear footprint on the horizontal plane (at z/δ = 0.087). The quality of the match between correlation contours (from hp and vp) at the intersect is taken as further evidence of the accuracy of the combined-plane experiments.


221

Fig. 8. Two-point correlations calculated on the u fluctuations in the vertical plane of the combined-plane experiments (zref /δ = 0.17). Contour levels are from Ruu = −0.12 to 0.96 in increments of 0.06. Solid lines show positive contours and dashed lines show negative contours.

The two-point correlations of Figs. 7 and 8 have reinforced all of the initial observations made from the example flow fields. The spanwise alternating sign and all of the length scale information contained within the Ruu plots are entirely consistent with the structural scales we have previously witnessed in the instantaneous examples of Figs. 4–6. The large-scale instantaneous u fluctuations, so clearly evident in the example frames, have left a clear statistical imprint on the two-point correlations, to such an extent where the latter must be considered as a time-averaged view of the former. 3.2. An associated vortical structure Up to now, this discussion has concentrated exclusively on streamwise velocity fluctuations. In reality, however, the resolved large-scale u fluctuations are strongly associated with a wider (and more complex) vortical structure. 3.2.1. Vortex identification Here we use the two-dimensional swirl approximation λci as an identifier of vortex cores, as described in Adrian et al. (2000a, 2000b). In the standard form, λci represents swirling magnitude and contains no

222


directional information. Sign is recovered by multiplying λci with the sign of the local out-of-plane vorticity component. For example, in the wall-parallel plane, signed 2D swirl will be defined as,

ωz λs = λci , (3.5) |ωz |

where ωz is the wall-normal vorticity component. When calculating λs in the vertical plane, the spanwise vorticity ωy is used to determine sign. Likewise, for the inclined-planes, ωx′ provides the sign (see transformed axis system in Fig. 2). 3.2.2. Example swirl fields

Figure 9(a) shows the streamwise velocity fluctuations in an example horizontal measurement plane (from the combined-plane experiments). The frame is selected such that an elongated low-momentum streak lies on the intersection of the vertical and horizontal planes. Figure 9(b) shows the associated magnitude of the signed swirl λs . In the absence of vortical activity, swirl is everywhere zero (uniform mid-gray shading). Darker shaded patches show regions of positive signed swirl (counterclockwise), whilst the whiter areas indicate negative rotation (clockwise). As a first impression, the arrangement of swirl patches on Fig. 9(b) can appear almost random, and certainly serves as a reminder of the complexity intrinsic to turbulent boundary layers. However, when we superimpose the borders of the negative u fluctuations (shown by the solid contours at u = −Uτ ), a clear pattern emerges. The swirl events of Fig. 9(b) are strongly associated with the flanks of the low-speed regions (the patches are predominantly arranged on the contours). A closer inspection reveals that the patches are arranged such that positive swirls tend to be situated on the −y flank of the low-speed region, whilst negative swirls tend to align on the +y side. Two-dimensional swirl essentially marks vortex cores that have pierced the measurement plane (at some oblique angle). The implication from Fig. 9(b), is that the previously observed large-scale stripes are associated with complex (yet quite well defined) arrangements of smaller-scale vortical structures. Such arrangements have been previously reported from wall-parallel plane measurements. Tomkins and Adrian (2003) found the same instantaneous vortex pattern associated with the elongated low-momentum regions. Using conditional averaging techniques (with a negative u fluctuation as the condition event), they have shown that the time-averaged low-speed regions have an associated counter-rotating vortex pairing with a sign


223

Fig. 9. Example frame-set from combined-plane experiment. Left-hand plots show negative streamwise velocity fluctuations in (a) the horizontal and (c) the vertical planes. Gray shading shows u < 0.5Uτ (see gray-level scale). Right-hand side shows signed swirl for (b) horizontal and (d) vertical planes. Gray shading shows positive and negative + swirl, (dark) λ+ s > 0, (light) λs < 0 (see gray scale). Solid contours show negative u fluctuations (u+ = −1).

and arrangement consistent with the instantaneously observed structure. Ganapathisubramani et al. (2003) employed feature extraction algorithms to show that these same events contain a large proportion of the Reynolds shear stress in the log region. Figure 9(c) shows the simultaneous velocity fluctuations in the vertical plane. This side view slice reveals that the elongated low-momentum region on the intersect of plot (a) extends well into the wake region. Furthermore, the wall-normal extent of the low-speed region exhibits a clear inclination angle (consistent with the Ruu maps of Figs. 7 and 8). The corresponding swirl pattern for this vertical plane is shown on Fig. 9(d). In this view, almost all of the swirl is positive, which of course reflects the mean spanwise

224


vorticity due to the shear layer (in side view, positive swirl is indicative of clockwise rotation, since the y-axis acts into the page). These swirl patches mark vortex cores that are primarily spanwise aligned. Though the structure is clearly complex, we again see signs that vortex cores are aligned along the edge of low-speed regions. There appears to be an array of spanwise vortices arranged along the inclined back of the previously observed low-momentum region. It was such an arrangement of vortices, interpreted as the heads of hairpin vortices, that originally led Adrian et al. (2000b) to propose the hairpin packet model. This model will be elaborated upon below. However, before doing so, we will first consider the in-plane swirling motions in the 45◦ and 135◦ inclined-planes. Figure 10 shows an example of instantaneous flow fields for both inclined planes (at Reτ = 1010). The upper two plots show the 45◦ case and the lower two show 135◦ data. It must be noted that these two inclinations represent separate experiments (non-simultaneous measurements). The two example frames are chosen on the basis that each contains similar evidence of large plumes of low u momentum erupting from the wall. Such features are typical of many instantaneous observations. In each plane, the tall low-speed regions are actually strong ejection events (−u, +w) and they are typically flanked in the spanwise direction by high-speed sweep events (+u, −w). Together these sweep and ejection regions form a large positive contribution to the Reynolds shear stress (for clarity, only the negative u fluctuations are shaded in plots (a) and (c)). Figures 10(b) and 10(d) show the corresponding in-plane swirl patterns for both inclined-planes (with the same gray-level scale as the previous figure). From this viewpoint (viewed along the negative x′ direction), positive swirl implies a counterclockwise rotating vector field. It is immediately noticeable from a comparison of the right-hand plots of Figs. 10(b) and 10(d) that the in-plane swirl is much more prevalent in the 135◦ plane than the 45◦ . This is consistent with the observations of Head and Bandyopadhyay (1981) who noted considerable in-plane vortical activity in their 135◦ flow visualizations as opposed to the corresponding 45◦ images, which exhibited mainly elongated hoop-like patterns. Such patterns are entirely consistent with the inclined structures proposed by the hairpin vortex and packet models. This disparity in swirling motions can be highlighted by calculating the respective mean swirl magnitudes at all wall-normal locations across the boundary layer, as shown in Fig. 11. Averaging across the data set in this way clarifies our initial observation that there is substantially less in-plane swirling activity in the 45◦ vector fields. The plots reveal that this disparity is greatest near the wall.


225

Fig. 10. Example inclined planes; (a and c) 45◦ and 135◦ negative u fluctuations. Gray shading shows u < 0.5Uτ (see gray scale); (b and d) 45◦ and 135◦ signed swirl. Gray + shading shows positive and negative swirl, (dark) λ+ s > 0, (light) λs < 0 (see gray scale). Solid contours show negative u fluctuations (u+ = −1).

As was noted for the vertical and horizontal planes of Fig. 9, the swirl patches in the 135◦ plane of Fig. 10(d) are clearly associated with the borders of the low-momentum region. The tall plume of negative u fluctuation labeled (I), is clearly surrounded by swirling motions, with the +y flank bordered by negative-signed swirls, and the −y side bordered by positive swirls. In this view, we get some wider sense of the structural geometry. The arrangements of counter-rotating vortex cores, on either side of the

226


Fig. 11. Wall-normal variation of outer-scaled mean swirl for (open) 45◦ and (closed) 135◦ cases at Reτ = 1010.

low-speed region, appear to be stacked above one another in the wall-normal direction. 3.2.3. The hairpin packet model The observed arrangements of vortices about negative u fluctuations (as noted in Figs. 9 and 10) can be largely explained by the hairpin packet paradigm. Figure 12 shows sketches of an idealized hairpin packet along with representations of the various light-sheet orientations used in the different experiments (light-sheets are shown as green planes). In all instances, the darker green shading in the interior of the packet structure shows regions of predicted negative u fluctuation (this is due to a mutual backwards flow induced by the hairpin vortices). Regions where positive and negative twodimensional swirls occur are colored red and blue respectively (i.e. regions where the measurement plane makes an approximately perpendicular cut through a vortex core). For all three cases shown, the shape of the lowmomentum regions and the arrangements of the swirl patches about these, match exactly the observations from the instantaneous examples shown in Figs. 9 and 10. In the wall-parallel plane (Fig. 12(a)), the packet structure imposes an elongated low-speed streak onto the measurement plane. The cut made through the necks of the vortices will lead to a series of positive-signed


227

Fig. 12. Idealized packet of inclined hairpin eddies imaged by (a) wall-parallel lightsheet; (b) vertical (streamwise-spanwise) light-sheet; (c) 45◦ and 135◦ inclined planes. • Red patches show positive swirl. • Blue patches show negative. Darker green shadow on light-sheet indicates predicted region of negative u fluctuation.

228


swirls aligned on the −y side of the streak (with negative-signed vortices populating the opposite flank). This is the pattern observed in Fig. 9(b).a For the vertical plane (Fig. 12(b)), the packet signature will be an inclined low-speed region with positive-signed swirl events aligned along the inclined back. In this plane, there is no negative-signed swirl attributed to the packet structure. This matches closely the observations from Figs. 9(c) and (d). Figure 11 has shown clearly that the inclined-plane swirl is more prevalent in the 135◦ than the 45◦ planes. The individual hairpin structures in Fig. 12 are depicted at an inclination angle that is close to 45◦ (but varying with wall-normal distance). The two inclined-planes bisecting this packet in Fig. 12(c) show that if such an arrangement of structures were typical, we would indeed expect to find far greater evidence of in-plane swirling activity in the 135◦ plane than the 45◦ case (in fact, if all structures were ideal and aligned at 45◦ to the streamwise axis, there would be no 2D swirling activity evident in the 45◦ plane). Furthermore, the packet arrangement of Fig. 12(c) would seem to imply the occasional occurrence in the 135◦ plane of well-ordered arrangements of positive and negative swirl, stacked above one another and flanking a region of reduced streamwise momentum. The negative u region labelled (I) in Figs. 10(c) and 10(d), is a clear example of such a packet signature. Certainly this region is qualitatively similar to the flow pattern shown in the 135◦ plane of Fig. 12, with predominantly positive swirling motions to the left of region (I) and negative swirling motions to the right. 3.2.4. Statistical evidence Previously, most statistical evidence for hairpin-like vortex organizations has been provided by Adrian and co-workers. Tomkins and Adrian (2003) conditionally averaged wall-parallel vector fields on the occurrence of a negative u event. The resultant ensemble average revealed a counter-rotating vortex pair arranged about the low-speed condition event, with a sign that a In keeping with the rest of the discussion, where we have tended to concentrate on negative u fluctuations, regions of expected u acceleration are not shown on Fig. 12. However it is noted that, owing to the sign of rotation of the constituent eddies, forwardinduced flows would be predicted everywhere on the packet exterior. For the wall-parallel plane of Fig. 12(a), therefore, the model predicts that the marked low-speed event will be flanked on either side in y with corresponding high-speed regions. This can account for the over-riding impression of stripiness when viewing u fluctuations in the wall-parallel planes.


229

is consistent with a proposed hairpin eddy. They also computed swirlswirl correlations in this plane (Rλci λci ), uncovering a weak secondary peak located upstream and downstream of the primary correlation. This is interpreted as an artifact of streamwise-aligned same-sign vortex arrangements (such as those depicted in Fig. 12(a)). Christensen and Adrian (2001) employed linear stochastic estimation (LSE) to obtain the conditional flow field on the occurrence of swirl in the vertical plane. They found secondary swirling motions located upstream and downstream of the condition event and aligned on an inclined shear layer (with inclination angle of approximately 14◦ ). This scenario is clearly evident in the instantaneous example of Figs. 9(c) and 9(d), and is predicted by the packet model (Fig. 12(b)). Hutchins et al. (2005b) performed a conditional average on the occurrence of a positive-signed swirl in the 135◦ inclined-plane data set. The resulting ensemble average is asymmetric, exhibiting a preference for counter-rotating vortex pairs occurring about a low-speed region and with an ejection at their confluence. A boundary layer that consisted of hairpin vortices would exhibit just such a pairing tendency (see, for example, the counter-rotating vortex pairs in the 135◦ plane of Fig. 12). We can make use of the simultaneous vertical and horizontal views provided by the combined-plane experiments to add to this statistical evidence. In the manner of Christensen and Adrian (2001), an LSE is performed, with the occurrence of positive-signed swirl in the vertical plane as the condition event. In the vertical plane, such an estimate is essentially a function of the two-point correlation tensors Rλs u , Rλs v and Rλs w . The horizontal estimate is based on cross-correlations between λs in the vertical plane and velocity fluctuations in the horizontal plane (of the form given by Eq. (3.4)). The condition point is just above the upper limit of the log region at z/δ ≈ 0.19. The resulting conditional event is shown in Fig. 13 (the vector magnitude has been everywhere set to unity, such that these plots show just the direction field). Plot (b) shows the side view of the LSE event with the (×) symbol marking the condition point. Since we have conditioned on positive λs , there is a primary clockwise swirling event occurring about this point. Upstream and downstream of this point we see an inclined shear layer (at approximately 13◦ ), along which there are occasional signs of secondary swirling motion. This reiterates the results of Christensen and Adrian (2001). The unique feature of the current result is provided by the simultaneous view given by the horizontal plane of Fig. 13(a). The condition event in the vertical plane is accompanied by two pronounced swirling motions that occur just upstream of the condition point at ∆y ≈ ±0.15δ (marked by

230


Fig. 13. LSE conditioned on a positive-signed swirl event at z/δ = 0.15. (a) For horizontal plane; (b) For vertical plane. Vectors show in-plane velocity components (magnitude set everywhere to unity). The (×) symbol shows the condition point, and the (◦) marks the approximate locations of the primary-conditioned vortices.

the white circles). This would seem to suggest that the time-averaged conditional event is an inclined hairpin structure, with an inclination angle of approximately 45◦ (the line drawn between vortex cores in Fig. 13(a) shows this inclination angle). Between the legs of this conditional eddy, there is a pronounced elongated low-speed region extending some distance up and downstream of the condition point. This structure seems to widen with increasing ∆x and is flanked in y by accelerated regions. Some evidence of secondary swirling activity is noted at the shear layer between these lowand high-speed regions. A further interesting feature in the horizontal plane of Fig. 13(a) is the saddle-point that seems to occur at ∆x ≈ −0.65δ. This feature corresponds to the point in the side view (plot (b)), where the inclined shear layer crosses the horizontal measurement plane (z/δ ≈ 0.89). Certainly such saddle points would be predicted by the inclined packet scenarios of Fig. 12(a) if the heads of the smallest hairpin structures (at the upstream end of the packet) were below the wall-parallel measurement plane.


231

3.2.5. Alternative explanations Variations on the hairpin packet model have on occasion been invoked to explain the large-scale structural features in the log and wake region. Though more an observation than a model, Na et al. (2001) proposed the notion of the “superburst” to explain the occurrence of large ejecting lowspeed regions. Liu, Adrian and Hanratty (2001) conducted proper orthogonal decomposition (POD) on PIV data, finding that large-scale motions (captured by the first few eigenmodes) contained a large fraction of the total kinetic energy due to the streamwise velocity fluctuations (hardly surprising since the observed stripiness and associated three-dimensional structure are clearly the most energetic features in the PIV results). Further analysis of DNS data showed these large-scale motions to be “plumes” of low-speed fluid erupting from the viscous wall-layer and extending a considerable distance towards the channel centerline. The “plumes” that we have observed in the inclined-planes are very reminiscent of such structures. Our own instantaneous measurements on inclined 135◦ planes indicate that swirl events commonly flank such “plumes”. This suggests that the hairpin packet model may provide a basis for such events, with ejecting low-speed regions occurring within the packet (certainly spanwise vortices are clearly visible crowning the streamwise/wall-normal plane “superburst” examples shown in Na et al., 2001). ´ More recent analysis of high Reynolds number DNS data (see del Alamo ´ et al., 2004, for simulation details) has led Jiménez and del Alamo (2004) to attribute the elongated negative u fluctuations to the formation of “passive wakes” from smaller-attached clusters of vortices that have ejected from the buffer region. We have used the same DNS data set to analyze the isosurfaces of three-dimensional swirl occurring about one of these elongated low-speed regions. This has uncovered a complex clustering of vortex cores occurring around the fringes of the low-speed region. Regardless of the precise mechanisms by which the u stripiness is formed, there appears to be a clear association between the vortex tubes and the large-scale u fluctuations. Instantaneously, these arrangements rarely resemble the simplified hairpin packet sketched in Fig. 12. However, in a time-averaged sense (and indeed in instantaneous planar views), it would appear that this association is well-represented by the hairpin packet paradigm. The principle difference between the superburst or passive wake explanations and the hairpin packet model lies in the significance placed on given flow features. The notion of superbursts and passive wakes tends

232


to focus attention (and dynamic significance) on the elongated regions of low-momentum (ejecting) fluid, whilst the packet scenario places greater emphasis on the vortical structure. There is a subtle cause and effect issue inherent in these two views. The question of whether hairpin vortices cause the regions identified by Hanratty and co-workers as “superbursts”, or whether hairpins form over pre-existing lifted low-momentum regionsb (plumes or passive wakes), cannot be answered definitively by the current data set and almost certainly requires a time-resolved view. For this reason, we will limit discussion here to simply a report of the existence and form of the large-scale structure in the log and wake regions. 3.3. Reynolds number scaling Thus far, all discussions have been limited to measurements obtained at or close to Reτ = 1100. It is worth pausing briefly to consider the Reynolds number scaling of the largest-scale u fluctuations in the log and wake regions. From correlation results, Wark, Naguib and Robinson (1991) noted that “. . . the structures in the log region seem to scale with outer variables over the entire range of spanwise scales”. Similarly, Mclean (1990) concluded, for y + 40, that the spanwise and streamwise correlations (and hence length scales) of Ruu scaled with outer variables. More recently, Hutchins et al. (2005b) have shown, from a Reynolds number comparison of the inclined-plane data, that the spanwise and wall-normal extent of the u, v and w two-point correlations all scale with outer variables. As an example of this, Fig. 14 shows both inner and outer-scaled views of a single Ruu contour (Ruu = 0.2) across a range of Reynolds numbers (690 < Reτ < 2800). It is clear from a comparison of plots (a) and (b) that the correlation contour scales on the boundary-layer thickness. At such heights from the wall (zref = 0.15δ), the two-point correlations essentially reflect the large-scale instantaneous stripiness that we have noted in the example PIV frames. Hence, the implication arising from outer-scaling of two-point correlations is clear. The largest scales in the log and wake regions (namely the “packet structure”, “superburst” or “large-scale passive wakes”) seem to scale with the boundary-layer thickness (at least over the Reynolds number range considered thus far). b Certainly

a low-speed region of the approximate shape given by the two-point correlations of Figs. 7 and 8 would be bounded by a shear layer that was primed for roll-up into hairpin packet-like structures.

233


800

600

0.5 (a)

increasing Re

0.4

(b)

z/δ

z+

0.3 400

0.2 200

0

0.1

−500

0 + ∆y

500

0 −0.5

0 ∆ y /δ

0.5

+ Fig. 14. The Ruu = 0.2 contour. (a) For inner scaling at zref = 150; (b) For outer scaling at zref /δ = 0.15. Contours are shown for Reτ = 690, 1010, 1840 and 2800 for the 135◦ inclined-plane data set.

3.4. Influence at the wall Hutchins et al. (2005b) have shown that the large-scale log and wake region events retain a measurable correlation with the near-wall buffer region. When looking at the inclined-plane correlation contours, a clear asymmetry is noticeable about the reference point, such that the contours can appear curtailed by, or splatted onto, the wall. Figure 14(b) can give some indication of this asymmetry. Clearly the reference point (marked with the + symbol) is not at the wall-normal mid-point of the correlation contour. Beyond a given height, signs of this “splatting” cease and the contours appear to lift off from the wall, giving rise to the “attached” and “detached” correlation regimes as proposed by Hutchins et al. (2005b). Such correlation behavior indicates that the large-scale log and wake region structures have a well-defined footprint in the buffer region. In other words, events in the buffer region are “aware” of the these large-scale structures as they pass overhead. By such mechanisms, very low wavenumber energy leaches into the near-wall region. This can explain the Reynolds number dependence of the streamwise energy in the sub-layer and buffer region (Jiménez et al., 2004; Metzger and Klewicki, 2001; DeGraaf and Eaton, 2000; Marusic and Kunkel, 2003). An instantaneous example of this correlation behavior is given by Fig. 16, where u fluctuations in the log region (plot (a)) are compared with those in the sub-layer (plot (c)). 3.5. Extremely long streamwise modes In all of the example frames referred to so far, the streamwise length of lowspeed regions have exceeded the PIV measurement domain (x ≈ ±δ, see

234


Figs. 4, 5, 6 and 9). To redress this situation, we have made measurements with a spanwise rake of 10 hot-wire sensors. Preliminary details of these experiments are given in Hutchins et al. (2004, 2005a). The idea here is to use the fluctuating signals from the 10 hot-wire probes to reconstruct the instantaneous spanwise profile of the u velocity fluctuation. By projecting this signal in time and using Taylors Hypothesis (frozen convection), a view of the long high- and low-speed streaks can be constructed that covers a much larger streamwise domain than that available with PIV. An example section of the reconstructed field is shown in Fig. 15(a). The gray shading shows only the negative fluctuations. A typical size PIV vector field for this Reynolds number is shown in plot (b). Clearly there are some very long features in the flow that the PIV data will fail to adequately capture. A long, meandering low-speed region wanders through the measurement domain for the entire 14δ shown. Indeed when we run movies of the frozen turbulence as it advects past the probe array, there are many instances where the length of the streaks exceeds 20δ. In the log region, the hot-wire measured u fluctuations exhibit an autocorrelation curve that falls to zero, becoming slightly negative for signal shifts ∆x 4δ. The broad peak in the pre-multiplied streamwise spectra kx Φuu occurs for similar length scales (≈ 6δ). Such features in classical single point statistics have previously informed our view of the largest energetic scales in turbulent boundary layers. However, as indicated by Fig. 15,

Fig. 15. (a) Example signal section from 10 sensor hot-wire rake at z/δ = 0.14. Spatial view is reconstructed using local mean velocity (x = −U∞ t). Shading shows negative u fluctuations only (see gray scale); (b) Comparison with typical PIV frame.


235

the rake data demonstrate that much larger scales inhabit the flow. It is proposed that these length scales are not resolved by classical single point techniques due to a spanwise wandering or meandering in the streaks (a measure of which is clearly evident in Fig. 15(a)). Indeed, the Ruu maps from the spatially reconstructed hot-wire signals show some unusual correlation behavior for ∆x > ±4δ which may be an artifact of this meandering. Hutchins et al. (2004) modeled the log region with a fake flow comprised entirely of meandering low- and high-speed zones, successfully recreating these distinctive patterns in the Ruu map. Based on these results, our preliminary conclusions are that the stripiness we have reported in the log region is, in reality, extremely persistent in the streamwise direction. It is not uncommon for the elongated lowmomentum regions to exceed 20 boundary-layer thicknesses in length, and these meander appreciably (with very long meandering wavelengths) as they advect downstream. ´ Kim and Adrian (1999) and, def Alamo and Jiménez (2003) found streamwise energy residing at comparable length scales for pipe and DNS channel flows. Figure 16(a) shows streamwise fluctuations from the chan´ et al., nel flow simulations at Reτ = 940 and z/h = 0.164 (see del Alamo 2004, for simulation details). Clearly the numerical simulations exhibit a similar spanwise stripiness in the u fluctuations, with evidence of very long streamwise features (and spanwise wandering). Figure 16(c) shows the corresponding negative velocity fluctuations at z + = 15 (z/h = 0.016). If we look through the small-scale fluctuations, we can see obvious proof that the large-scale fluctuations in the log region have a pronounced footprint in the near-wall. This is clear evidence of an outer-scaled log region event imposing low wavenumber energy onto the near-wall u fluctuations (as was suggested by the “attached” correlation regime of Hutchins et al., 2005b). Our own hot-wire rake experiments were recently repeated in a much larger high Reynolds number boundary layer facility at the University of Melbourne (up to Reτ = 20,000, see Hafez et al., 2004, for details of facility). Data from these studies are still under analysis. However, it is clear from initial results that the same long features, with a streamwise length that commonly exceeds 15δ, populate the log region of these higher Reynolds number flows.c c In

the Melbourne wind tunnel, 15δ equates to a physical length scale of approximately 5 m for the largest u fluctuations.

236


Fig. 16. Example of instantaneous flow fields from the Reτ = 950 channel flow simu´ lations of de Alamo et al. (2004); (a) Negative u fluctuations at z + = 150, z/h = 0.16, gray scale as Fig. 15; (b) Q2 ejection events at z + = 150, z/h = 0.16, gray scale as Fig. 17; (c) Negative u fluctuations at z + = 15, z/h = 0.016, gray scale as Fig. 15.

3.6. Associated Reynolds stress Ganapathisubramani et al. (2003) have shown that a considerable proportion of the Reynolds shear stress in the log and wake region is associated with the large-scale features under discussion here. Figure 17 shows the wall-normal and Reynolds shear stress fluctuations for the same

237


0.6

(a)

(b)

(c)

(d)

0.4 0.2

y/δ

0 −0.2 −0.4 −0.6 −0.8 −1 −1.2

1

z/δ

0.8 0.6 0.4 0.2 0

−1 −0.8 −0.6 −0.4 −0.2 0 x /δ 0

1

0.2 0.4 0.6 0.8

2 +

w

3

−1 −0.8 −0.6 −0.4 −0.2 0 x /δ 0

2

4

0.2 0.4 0.6 0.8

6 8 −u +w+

10

12

Fig. 17. Example frame set from combined-plane experiment. Left-hand plots show positive wall-normal velocity fluctuations in (a) the horizontal and (c) the vertical planes. Gray shading shows w > 0.5Uτ (see gray scale). Right-hand side shows Reynolds shear stress for (b) horizontal and (d) vertical planes. Gray shading shows −u+ w + > 0 (see gray scale). Solid contours show negative u fluctuations (u+ = −1).

combined-plane frame set originally studied in Fig. 9. The left-hand plots show the wall-normal velocity fluctuations. Only the positive w fluctuations are shown (shaded grey) with contours of negative u fluctuation superimposed over the top (at u+ = −1). Clearly, the elongated low-speed regions are predominantly characterized by positive wall-normal fluctuations. These ejection events are strong positive contributors to Reynolds shear stress. This is confirmed by the right-hand plots (b) and (d) which show the fluctuating Reynolds shear stress −uw for the horizontal and vertical planes respectively. The localized concentrations of −uw primarily occupy the interior of the low-speed contour. Such association between the Reynolds stress and the long low-momentum regions is further highlighted

238


by Fig. 16(b) which shows the magnitude of Q2 events in the DNS channel at z/h = 0.16. Concentrations of Reynolds shear stress are aligned all the way along the 20h long low-speed region that runs across the bottom of Fig. 16(a) (between 0 < y/h < 2). It is noted from Figs. 16 and 17 that the patches of Reynolds shear stress concentration are highly localized. This is presumably because the Q2 and Q4 events are associated with individual vortex cores that are themselves clustered about the long low-speed regions. Certainly a comparison of Figs. 9 and 17 would support this scenario, with Reynolds stress concentrations tending to coincide with nearby swirl events. This spatial compactness is reflected in single point statistics such as the energy spectra of the Reynolds shear stress fluctuations. However, the important point that could be missed from such statistics is that, though the Reynolds stress fluctuation itself does not contain energy at very low wavenumbers, the individual spatially compact concentrations exhibit a strong alignment with the largest u modes (and these modes are extraordinarily long in the streamwise direction).

3.7. Spanwise repetition Almost all of the example instantaneous frames included in this study have contained evidence of spanwise repetition. This is immediately apparent in the wall-parallel plane results of Figs. 4, 5, 6 and 9, where the spanwise arrangements of elongated positive and negative u fluctuation show obvious signs of periodicity in the spanwise direction. Indeed the initial impression of stripiness arises precisely because of this repetition (i.e. it is often the case that more than one stripe is evident in each PIV frame). It is perhaps surprising that such stripiness does not seem to manifest in any of the time-averaged statistics. The pre-multiplied spectra of u fluctuations in the spanwise direction does exhibit a peak (occurring at a spanwise length scale Ly ≈ 0.7δ). However, this is not really a true signifier of spanwise repetition since it can be readily shown that a single low-speed streak of width 0.35δ, flanked by similar-sized high-speed regions and in the absence of any spanwise repetition of this pattern, could lead to such an energy peak. Indeed the peak in ky Φuu really just reflects the length scale information contained in the spanwise Ruu correlation plots such as those shown in Figs. 7(a) to 7(c). The most obvious and convincing sign of spanwise repetition would be a spanwise ringing of the primary positive correlation regions, and such behavior is notably absent in these

239


figures. This raises the obvious question of why the spanwise repetition, which is so obvious in the instantaneous flow fields, does not manifest in the time-averaged statistics. The answer lies most likely in the multitude of scales that reside in turbulent flows. The resultant statistical smearing due to a superposition of scales masks any artifact of spanwise repetition. To overcome these issues, a simple method of sorting the PIV data has been proposed (Hutchins et al., 2004, 2005a). Figure 18 illustrates the sorting process for the wider field-of-view (Reτ = 1142) 45◦ inclined-plane data set. A spanwise trace of the u fluctuation is extracted at a given reference height. For the example given in Fig. 18(a), zref = 0.14δ and the signal is extracted from the example frame originally presented in Fig. 4(b). A Fourier analysis of this signal reveals the dominant spanwise mode Ly . The PIV frame is then sorted or “binned” according to this dominant mode. For the case shown in Fig. 18(a), the dominant mode has a wavelength of 0.6δ (shown by the dashed sinusoid), and hence is placed in the 0.5 < Ly < 0.75 bin. The probability distribution for the sorted frames is shown by Fig. 18(b). For the relatively broad bin sizes used here (0.25δ increments from 0 to 1.5), it is found that most of the energy resides in four modes, with 0.5 < Ly /δ < 0.75 being the most 6 4

Ly / δ = 0.6

(a)

u+

2 0 −2 −4 −6 −1

−0.5

0 y /δ

0.5

1

0.4 (b)

probability

0.3 0.2 0.1 0 0

0.25

0.5

0.75

1 Ly / δ

1.25

1.5

1.75

2

Fig. 18. (a) Spanwise trace of u fluctuation extracted at zref = 0.14δ (from example of 45◦ plane shown in Fig. 4(b)). Dashed line shows dominant spanwise Fourier mode; (b) Probability distribution of dominant spanwise wavelengths (Ly /δ).

240


populated bin. Of the entire data set, 37% of all frames exhibit a dominant spanwise u fluctuation of this wavelength. Having sorted the PIV frames in this manner, we can now compute standard statistics on the “binned” data. The two-point correlations as calculated on these four dominant modes are shown in Fig. 19 plots (a)

Fig. 19. Ruu calculated at zref = 0.14δ for each of the four dominant modes. (a) 0.25 < Ly /δ < 0.5; (b) 0.5 < Ly /δ < 0.75; (c) 0.75 < Ly /δ < 1.0; (d) 1.0 < Ly /δ < 1.25; (e) shows the sum of these four modes as compared to (f) the standard Ruu map. Contours levels are from Ruu = −0.12 to 0.96 in increments of 0.06. Solid lines show positive contours and dashed lines show negative; (g) Ruu calculated for the mode 0.5 < Ly /δ < 0.75 for both wall-parallel and 45◦ inclined-plane data.


241

to (d). Note that these modes recover 93% of the total energy (93% of all PIV frames exhibit these spacing modes). With the data sorted in this way, there is clear evidence of ringing in the “binned” two-point correlations indicating an underlying spanwise periodicity. This is expected along the line z = zref (due to the sorting condition), but the fact that these spanwise modes extend a considerable distance in the wall-normal direction (>0.5δ) implies a far wider coherence. Note that these data have not been filtered in any way (apart from being sorted). Yet 93% of all frames are well described by a single sinusoidal mode for all of the log region and slightly beyond (up to 0.5δ). Figure 19(e) shows the sum of the four dominant modes in plots (a) to (d). Despite the fact that repeating modes characterize the majority of the data, the superposition of the individual modes leads to an Ruu plot that exhibits no obvious sign of spanwise periodicity. The sum of the four dominant modes shown in plot (e) almost completely recovers the standard Ruu profile (included for comparison as plot (f)). This is a good illustration of the caution that should be exercised when interpreting statistical quantities such as Ruu , where multiple scale interactions and superposition can mask underlying organization. The same de-jittering process has been applied to the wall parallel data set revealing the streamwise extent of these dominant modes. Figure 19(g) shows the Ruu construct from the most populated bin (0.5 < Ly /δ < 0.75) for both the wall-parallel and inclined-plane PIV data sets. Clearly the previously observed spanwise modes actually persist for a substantial distance in the streamwise direction, extending well beyond the x = ±δ view afforded by the PIV data. The implications of the correlation plots in Fig. 19 are clear. The very long meandering stripes of low- and high-speed u fluctuation exhibit a measurable trace of dominant spanwise spacing modes. Hence to some extent, we would expect the wider associated vortical structure, Reynolds shear stress fluctuations and the wall footprint to exhibit similar modes. At the present time, the precise dynamics that leads to this spanwise periodicity is unclear. However, the existence of such preferential spacings lends an additional degree of spatial predictability to the log and wake regions, and would seem to be of obvious interest to those who seek to model or control turbulent boundary layers. Some implications for flow control strategies will be discussed below. However, first it would seem worthwhile to present a brief summary of the major findings described in this chapter.

242


4. Summary Multi-plane stereoscopic PIV measurements have enabled a detailed characterization of the larger-scale coherent structure in the logarithmic and wake regions of a turbulent boundary layer. The principle findings arising from these studies are as follows: • Stripiness In the wall-parallel plane, the largest scale u fluctuations are characterized by a pronounced spanwise stripiness of alternating highand low-momentum. Views from additional PIV planes confirm that such stripiness extends a considerable distance in the wall-normal direction (and is a slice through an elongated three-dimensional ridge of low- and high-speed fluid). • Associated vortical structure Multiple swirling motions appear to be clustered around the stripes of momentum deficit. The precise arrangement and sign of these associated swirl patches would suggest that the hairpin packet paradigm provides a good descriptive model for this clustering. • Scaling on boundary layer thickness The large-scale stripiness observed in the log and wake regions appears to scale on outer variables (certainly in terms of u, v and w fluctuations). Scaling of the associated swirling regions (or clustered vortical structure) remains to be properly established. • A wall footprint These large-scale outer-region structures impose low wavenumber energy onto the buffer region. • Extremely long streamwise modes Hot-wire measurements and analysis of DNS results in the log region have revealed that the structure can be extremely persistent in the streamwise direction (sometimes exceeding 20δ in length). • Meandering A noted spanwise meandering will tend to mask the true length of these features from single-point statistics. • Associated Reynolds stress Spatially compact Reynolds shear stress concentrations are aligned along these elongated features (to the extent that these long events are the main shear stress producing events in the outer regions). • Spanwise repetition The entire resolved structure exhibits signs of spanwise organization with inferred preferential spacings.


243

5. Implications to Flow Control It is not surprising that most flow control strategies have tended to concentrate on modifications to the near-wall structure. In this region a well-defined stress-producing cycle leads to the large peak in the turbulence production (occurring at z + ≈ 15). DNS results have proven remarkably persuasive in cementing a view of this region, providing a unique spatiotemporal view of the developing near-wall events. Up until quite recently, such simulations have been limited to fairly low Reynolds numbers and small computational boxes. By virtue of this, DNS has necessarily tended to concentrate attention on the near-wall structure. PIV, on the other hand, is well suited to surveying the turbulent structure in the log and wake regions of higher Reynolds number flows and has, since its inception, been instrumental in resolving the larger scales that inhabit these regions (along with ´ recent high Reynolds number simulations such as those by del Alamo et al. 2004). Whilst the near-wall structure will continue to hold obvious attraction for skin friction reduction strategies (most of which, after all, involve a wall-based actuation), the emergent picture from the log and wake region would suggest that the large-scale structure should not be ignored. Indeed, as Reynolds number increases, the integrated turbulence production across the log region will become increasingly significant, eventually surpassing the production in the buffer region. A few obvious flow control strategies do appear to have successfully targeted the larger-scale structures (LEBU devices for example, see Anders, 1989). Other control strategies have been studied exclusively at low Reynolds numbers, where the separation between inner and outer scaling is not immediately apparent. As an example, the majority of wall oscillation studies (Choi, 2002; Park et al., 2002; Di Cicca et al., 2002) have been performed at Reynolds numbers in the range 200 < Reτ < 400, for which optimum oscillation periods of 100 < T + < 200 are commonly reported. Based on a conservative estimate of local convection velocity, this implies an oscillation wavelength of over 5δ, which could conceivably place such control schemes in the domain of larger-scale modification (indeed the results of Park et al., 2002, might suggest a Reynolds number dependency for the optimum oscillation period). Over the previous pages, the dominant structure in the log region has been fairly well-defined. The fact that this structure is found to carry so

244


much of the Reynolds stress in the log region, whilst maintaining an apparent dynamic link with the near-wall region, would seem to make them a viable target for control schemes. Though perhaps not so amenable to wallbased actuation, these structures do have the advantage that they are very large, they scale on boundary-layer thickness and also seem to obey preferred spanwise spacing modes. Thus for higher Reynolds numbers, they might provide a more realistic target for active control schemes (less demanding in terms of sensor/actuator coverage, response, etc). In the interim, it would certainly be interesting to study the passive modification of this structure (maybe even revisiting LEBU type devices in light of this resolved structure). As a final note to this effect, we should caution that these structures appear to be quite robust. Our own studies with hot-wire rakes have shown that the large-scale stripiness in the log region persists even over fully roughened walls (where in theory the standard near-wall structure is completely destroyed). This suggests that these structures may not need to originate from the near-wall cycle. Acknowledgments The authors gratefully acknowledge support from the National Science Foundation (Grant CTS-0324898) and, the David and Lucile Packard Foundation. The majority of the experimental work was undertaken jointly with B. Ganapathisubramani and W. Hambleton. The authors would also like to thank Professor R. D. Moser for kindly making the Reτ = 940 DNS database available, and Professor M. S. Chong and S. Hafez for hosting the Melbourne experiments. References 1. R. J. Adrian, Particle-imaging techniques for experimental fluid mechanics, Annu. Rev. Fluid Mech. 23 (1991) 261–304. 2. R. J. Adrian, K. T. Christensen and Z.-C. Lo, Analysis and interpretation of instantaneous turbulent velocity fields, Exp. Fluids 29 (2000a) 275–290. 3. R. J. Adrian, C. D. Meinhart and C. D. Tomkins, Vortex organization in the outer region of the turbulent boundary layer, J. Fluid Mech. 422 (2000b) 1–54. ´ 4. J. C. del Alamo and J. Jiménez, Spectra of the very large anisotropic scales in turbulent channels, Phys. Fluids 15 (2003) 41–44. ´ 5. J. C. del Alamo, J. Jiménez, P. Zandonade and R. D. Moser, Scaling of the energy spectra of turbulent channels, J. Fluid Mech. 500 (2004) 135–144. 6. J. B. Anders, Outer-layer manipulators for turbulent drag reduction, in Viscous Drag Reduction in Boundary Layers, eds. D. M. Bushnell and


7. 8.

9. 10. 11. 12.

13.

14.

15.

16. 17. 18.

19.

20.

21.

22. 23.

245

J. N. Hefner, Progress in Astronautics and Aeronautics, Vol. 123 (AIAA, Washington, D.C., 1989). K.-S. Choi, Near-wall structure of turbulent boundary layer with spanwise oscillation, Phys. Fluids 14 (2002). K. T. Christensen, Experimental investigation of acceleration and velocity fields in turbulent channel flow, PhD thesis, University of Illinois, USA (2001). K. T. Christensen, The influence of peak-locking errors on turbulence statistics computed from PIV ensembles, Exp. Fluids 36 (2004) 484–497. K. T. Christensen and R. J. Adrian, Statistical evidence of hairpin vortex packets in wall turbulence, J. Fluid Mech. 431 (2001) 433–443. D. B. DeGraaf and J. K. Eaton, Reynolds number scaling of the flat-plate turbulent boundary layer, J. Fluid Mech. 422 (2000) 319–346. G. M. Di Cicca, G. Iuso, P. G. Spazzini and M. Onorato, Particle image velocimetry investigation of a turbulent boundary layer manipulated by spanwise wall oscillations, J. Fluid Mech. 467 (2002) 41–56. B. Ganapathisubramani, N. Hutchins, W. T. Hambleton, E. K. Longmire and I. Marusic, Investigation of large-scale coherence in a turbulent boundary layer using two-point correlations, J. Fluid Mech. 524 (2005) 57–80. B. Ganapathisubramani, E. K. Longmire and I. Marusic, Characteristics of vortex packets in turbulent boundary layers, J. Fluid Mech. 478 (2003) 35–46. S. Hafez, M. S. Chong, I. Marusic and M. B. Jones, Observations on high Reynolds number turbulent boundary layer measurements, in Proc. 15th Australasian Fluid Mech. Conf., eds. M. Behnia, W. Lin and G. D. McBain (2004). W. T. Hambleton, N. Hutchins and I. Marusic, Multiple plane PIV measurements in a turbulent boundary layer, J. Fluid Mech., in preparation (2005). M. R. Head and P. Bandyopadhyay, New aspects of turbulent boundary-layer structure, J. Fluid Mech. 107 (1981) 297–337. N. Hutchins, B. Ganapathisubramani and I. Marusic, Dominant spanwise Fourier modes and the existence of very large scale coherence in turbulent boundary layers, in Proc. 15th Australasian Fluid Mech. Conf., eds. M. Behnia, W. Lin and G. D. McBain (2004). N. Hutchins, B. Ganapathisubramani and I. Marusic, Spanwise periodicity and the existence of very large scale coherence in turbulent boundary layers, in Proc. 4th Int. Symposium Turbulence and Shear Flow Phenomena (2005a). N. Hutchins, W. T. Hambleton and I. Marusic, Inclined cross-stream stereo PIV measurements in turbulent boundary layers, J. Fluid Mech., in press (2005b). ´ J. Jiménez and J. C. del Alamo, Computing turbulent channels at experimental Reynolds numbers, in Proc. 15th Australasian Fluid Mech. Conf., eds. M. Behnia, W. Lin and G. D. McBain (2004). ´ J. Jiménez, J. C. del Alamo and O. Flores, The large-scale dynamics of nearwall turbulence, J. Fluid Mech. 505 (2004) 179–199. K. C. Kim and R. J. Adrian, Very large scale motions in the outer layer, Phys. Fluids 11 (1999).

246


24. K. C. Kim and S. H. Kwon, Three-dimensional topology of hairpin packet structure in turbulent boundary layer, in Proc. 12th Int. Symposium Appl. Laser Techniques to Fluid Mech. (2004). 25. L. S. G. Kovasznay, V. Kibens and R. F. Blackwelder, Large-scale motion in the intermittent region of a turbulent boundary layer, J. Fluid Mech. 41 (1970) 283–326. 26. Z. Liu, R. J. Adrian and T. J. Hanratty, Large-scale modes of turbulent channel flow: Transport and structure, J. Fluid Mech. 448 (2001) 53–80. 27. I. Marusic and G. J. Kunkel, Streamwise turbulence intensity formulation for flat-plate boundary layers, Phys. Fluids 15 (2003) 2461–2464. 28. I. R. Mclean, The near-wall eddy structure in an equilibrium turbulent boundary layer, PhD thesis, University of Southern California, USA (1990). 29. M. M. Metzger and J. C. Klewicki, A comparative study of near-wall turbulence in high and low Reynolds number boundary layers, Phys. Fluids 13 (2001). 30. Y. Na, T. J. Hanratty and Z. Liu, The use of DNS to define stress producing events for turbulent flow over a smooth wall, Flow Turbul. Combust. 66 (2001) 495–512. 31. J. Park, C. Henoch and K. S. Breuer, Drag reduction in turbulent flows using Lorentz force actuation, in Proc. IUTAM Symposium on Scaling in Turbulent Flows, ed. A. Smits (2002). 32. C. D. Tomkins and R. J. Adrian, Spanwise structure and scale growth in turbulent boundary layers, J. Fluid Mech. 490 (2003) 37–74. 33. C. E. Wark, A. M. Naguib and S. K. Robinson, Scaling of spanwise length scales in a turbulent boundary layer, AIAA Pap. 91-0235 29th Aerospace Sciences Meeting, Reno, Nevada (1991).

TURBULENT BOUNDARY LAYERS AND THEIR CONTROL: QUANTITATIVE FLOW VISUALIZATION RESULTS Michele Onorato∗ , Gaetano M. Di Cicca† , Gaetano Iuso‡ , Pier G. Spazzini§ and Riccardo Malvano¶ DIASP Politecnico di Torino 10129, Torino, Italy E-mails: ∗[email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] Results obtained by applying PIV to canonical turbulent boundary layers and to wall flows under the action of external forcing produced by spanwise wall oscillations are shown and analyzed. In particular, results obtained at the Aerodynamic Laboratory of the Politecnico di Torino are discussed, with reference, where it is possible, to results produced in other laboratories. For both flows, PIV images in planes streamwise-normal and parallel to the wall are analyzed. Contents 1 2

Introduction Flat Plate Boundary-Layer Flow 2.1 Streamwise-spanwise planes 2.2 Streamwise-wall-normal plane 3 Turbulence Control by Wall Oscillation References

247 249 249 262 271 280

1. Introduction The strong development of PIV in the last few years has opened a new field for the experimental study of turbulent flows. One of the first application to 247

248

M. Onorato et al.

near-wall turbulence has been performed by Meinhart and Adrian (1995), showing that PIV is a suitable technique for turbulent boundary layer studies. The main importance of PIV with respect to classical point measurement techniques in the study of wall turbulence is the ability of producing instantaneous maps of the velocity field, giving information about organized motions and their relation to the skin-friction growth and turbulence production. It is fair to state that the bulk of our recent understanding of the structure of turbulent wall flows has been achieved by studies based on the analysis of data obtained by direct numerical simulations. Although the capabilities of current computers restrict such simulations to relatively low Reynolds numbers and relatively simple geometries, the results have been extremely illuminating because they provide full instantaneous fields of velocity, vorticity, rate-of-strain and pressure with spatial and temporal resolution. One of the goals of using PIV is to extend such knowledge to higher Reynolds numbers of greater interest for industrial and environmental applications as well as to more complex flows such as, e.g., turbulent boundary layers manipulated by external forces in order to control turbulence, skin friction, wall heat exchange and aeroacoustic emission. In the present paper, results obtained by applying PIV to canonical turbulent boundary layers and to wall flows under the action of external forcing are shown and analyzed. In particular, results obtained at the Modesto Panetti Aerodynamic Laboratory of the Politecnico di Torino are discussed, with reference, where it is possible, to results produced in other laboratories. Turbulent boundary layers manipulated by external forcing, in a way to reduce skin friction and near-wall turbulence activity, have been in recent years the object of studies at the Politecnico di Torino, mostly with reference to researches on basic fluidynamics rather than with reference to specific applications. These researches, that are still going on, are based on the belief that a powerful technique for investigating the physics of nearwall turbulence behavior and regeneration is to study the response of the flow to external perturbations and to try to explain the differences observed between the natural and manipulated cases through proposed turbulence regeneration scenarios. Obviously, a better knowledge of the wall turbulence regeneration mechanisms will allow the development of more efficient flow control techniques.

Turbulent Boundary Layers and Their Control

249

In addition to PIV results on the basic canonical zero-pressure-gradient turbulent boundary layer, data are shown here and discussed for the control mechanism consisting of a forced transversal oscillation of the wall. The reasons for the choice of this external forcing are that it is an example of large-scale manipulation and it is representative of manipulation consisting in forcing the boundary layer by means of transversal motions superimposed to the mean flow. All details about measurement procedures are omitted in the following and are referred to the appropriate bibliographic references. 2. Flat Plate Boundary-Layer Flow 2.1. Streamwise-spanwise planes In Figs. 1 to 3 (Gottero and Onorato, 2000), color maps of the instantaneous longitudinal component of the fluctuating velocity u, in plane (x, z) parallel to the wall, are reported. The streamwise direction of the flow is in the x+ -direction. The observed flow field covers about 700 viscous units in the longitudinal x-direction and 1000 viscous units in the spanwise z-direction. As expected in the buffer layer region at y + = 20 (Fig. 1), quasi-streamwise velocity streaks characterize the flow. The typical irregular wavy appearance of the streaky structures, producing steep streamwise gradients of u, as focused by theoretical models (Landahl, 1990), as well as by numerical simulations (Johansson et al., 1991), is clearly evident also in the present results. Figures 2 and 3 refer to planes in the logarithmic region, showing, as already found by Smith and Metzler (1983), by flow visualization, the increase of the streak spanwise spacing and the decrease of the streamwise scale in the logarithmic layer with the distance from the wall. At a distance of 130 viscous units from the wall, the streak visual identification becomes very uncertain. Instantaneous flow fields parallel to the wall allow furthermore the observation of swirling motions, as shown in Figs. 1 to 3, where instant signatures in the plane (x, z) of vortical structures are shown, superimposed to the streamwise velocity streaks. The observed swirling motions are the crosssections of quasi-streamwise vortices or legs of hairpin structures that are cut by the laser light sheet. Swirling motions have been identified by the Zhou et al. (1999) method, involving the analysis of the eigenvalues corresponding to the local velocity gradient tensor. Vortical motions are identified by plotting isoregions of λci > 0, where λci (swirling strength) is the

250

M. Onorato et al.

Fig. 1. Instantaneous flow field in a plane (x, z) at y + = 20. Reθ = 1010. Color map: Streamwise component of the fluctuating velocity. Isolines: Detected swirling motions. From Gottero and Onorato (2000).

imaginary part of the pair of complex conjugate eigenvalues, when the discriminant of the tensor characteristic equation is positive. Vortex signatures in Figs. 1 to 3 and in similar PIV images have elliptical shapes and almost circular shape far from the wall and identify vortical structures whose vertical inclination at that instant is large enough to display a swirling motion in the plane (x, z). The vortex rotation directions are detected by looking at the sign of the out-of-plane vorticity, not reported here, showing peak values in correspondence to the vortex detection regions. Observing the maps, the vortices appear on both sides of the low-speed streaks and with different rotation directions, clockwise on the left side of the streaks and counterclockwise on the right side (looking downstream). This association of low-speed streaks with vortices is found in every realization of the flow and it is a consequence of the genesis of a low-speed streak, due to the induced low-momentum flow on the side of the quasi longitudinal vortex or of one of the hairpin vortex leg, where the structure lifts up the fluid


251


from the wall. This concept is self-explained by the sketch in Fig. 4. The instantaneous flow fields in Figs. 1 to 3, as well as in similar images taken at different instants, show that several vortical structures may be associated with a low-speed streak and that even in the buffer layer, many vortices have an inclination angle presumably higher than 9◦ , which is the mean vortex inclination forecast by the near-wall structure conceptual model of Jeong et al. (1997). Moreover, a copious number of couples of counterrotating vortices nearly symmetrical in the spanwise direction with respect to a low-speed streak are observed; they can be interpreted as sections of hairpin legs. This is in agreement with PIV results obtained by Tomkins and Adrian (2003), as it will be commented later on. The streamwise extent and spanwise spacing of the velocity streaks have been the subjects of several investigations, both experimental and numerical. Several authors report estimations of streamwise scales varying from 500 to more than 1000 wall units, with widths ranging from 20 to 80 wall

252

M. Onorato et al.


Fig. 4.

Sketch of an inclined vortex and its cross-section in a plane parallel to the wall.

253


20 Natural Flow

%

10

L+ 0 9

44

80

116

151

187

222

258

293

Fig. 5. Histogram of low-speed streak width. y + = 20. Reθ = 1160. From Iuso et al. (2003).

units. The reported mean spacing is about 100 wall units, λ+ = 100, in the viscous sub-layer and in the inner part of the buffer layer, increasing to a value of about λ+ = 140 at the external edge (Smith and Metzler, 1983), essentially invariant with Reynolds number. Values of λ+ have generally been obtained by visual inspection of flow visualization or numerically simulated data and are confirmed by observing the present PIV images. PIV results allow the acquisition of more detailed informations, both about the scale distribution and the energy distribution between the different scales. In Figs. 5 and 6, histograms of spanwise low-speed streak width and spacing are reported, in a plane (x, z) at y + = 20 (Iuso et al., 2003). Both quantities show a continuous distribution with significant values going from 5 to 150 wall units for the width and from 40 to 300 wall units for the spacing. The peaks of the distributions are positioned at about L+ = 40 for the width and about λ+ = 100 for the spacing, confirming previous observations reported in the literature. The statistics has been extracted by applying a streak eduction procedure, described in Iuso et al. (2003), based on the one reported in Schoppa and Hussain (2002). The evaluation of the energetic content of each scale may be obtained by applying spectral analysis. PIV data allow quantitative evaluation of the spanwise mode of the streaky structures by two-dimensional Fourier analysis, as suggested by Liu et al. (1996). Figure 7 (Gottero and Onorato, 2000) displays the spatial Fourier spectrum versus spanwise wavenumbers kz along with streamwise wavenumbers kx as parameters. Suu is the

254

M. Onorato et al.

9 %

Natural Flow

6

3

0 36

80

125

169

213

258

302

λ + 347

Fig. 6. Histogram of low-speed streak spacing. y + = 20, Reθ = 1160. From Iuso et al. (2003).

Fig. 7. Two-dimensional Fourier spectrum of the streamwise velocity component. y + = 20, Reθ = 1010. From Gottero and Onorato (2000).

two-dimensional power spectrum of the u-velocity component normalized with respect to the external flow velocity Ue2 . The power spectrum Suu is multiplied by the spanwise wavenumber, and it represents the energy of modes in a wavenumber bandwidth kz+ wide, centered on kz+ . The spectrum reveals that the streaky structures have many spanwise modes and that the mode observed more frequently by flow visualization, λ+ = 100,


255

Fig. 8. Fourier spectra of the streamwise velocity component at different distances from the wall. kx+ = 0, Reθ = 1010. From Gottero and Onorato (2000).

is relatively weak compared with modes related to larger scales. This is in agreement with the results of Liu et al. (1996) in a turbulent channel flow, where it is suggested that whereas the λ+ = 100 mode is the most readily observed mode by flow visualization, it is also one of the least important with respect to total turbulent energy. The most energetic spanwise mode, here observed, occurs at λ+ = 2π/kz+ = 260. Information about spectra at different distances from the wall are given in Fig. 8, where power spectra are shown for three planes, for kx+ = 0. As a consequence of the increase of the streak spanwise spacing, the most energetic modes occur on the scales λ+ = 370 and λ+ = 490, for y + = 70 and y + = 130 respectively. An important finding of the past research work on near-wall turbulence, using hot wire or laser Doppler anemometry, was that the occurrence of VITA (Variable Interval Time Average) events (e.g., bursting events and internal shear layer events) is associated with instants of large turbulence activity. These events are detected when the local velocity signal variance is larger than the signal variance (Blackwelder and Kaplan, 1978). It was found by different authors that the frequency of occurrence of VITA events in the buffer layer f is about constant and independent of the Reynolds number, if scaled with respect to viscous units. It was found that f + ranges from 0.0035 to 0.004 (Blackwelder and Haritodinis, 1985). The pointwise nature of HW and LDA measurements did not allow us to

256

M. Onorato et al.

understand the spatial relationship between such bursting events and the organized structures of the flow. This, on the other hand, can be readily obtained through planar PIV measurements. In Fig. 9 (Gottero and Onorato, 2000), VISA (Variable Interval Space Average) events, the spatial counterpart of VITA events, represented by closed isolines, are superimposed to the vectorial field of the fluctuating component of the velocity and to the u-velocity component color map, at y + = 20. The local variance was averaged over a short integration distance of 115 viscous units in the streamwise direction. This length, assuming a propagation velocity of the internal shear layers of 10.6 times the friction velocity, refers to a short averaging time of 10 viscous units. This averaging time has been used in most studies to detect VITA events from hot wire probe signal. The number of events per unit time, deduced from 15 realizations as the one in Fig. 9, yields a non-dimensional frequency f + = f ν/u2τ = 3.2 × 10−3, in agreement with hot wire literature.

Fig. 9. Velocity field and VISA events. y + = 20, Reθ = 1010. Vectors: Planar instantaneous velocity field. Color map: Instantaneous u-velocity field. Isolines: VISA events. From Gottero and Onorato (2000).


257

Representations of the flow field as in Fig. 9 are suitable for observing the spatial relationship between low-speed streaks and internal shear layer motions. High levels of local variance are localized along the boundaries of the streaks; the strongest events, always with du/dx < 0, are found by the side of the streaks, often near the upstream end of the region where the streak seems to divide or to change the tilting direction. The characteristics of the streaks of appearing as quasi-segmented elements, alternatively tilted with respect to the mean flow, seem to be essential in the production of strong internal shear layers (Johansson et al., 1991; Gottero and Onorato, 2000). A conditional analysis has been applied in Gottero and Onorato (2000) to a series of 15 images, as the one in Fig. 9: detected internal shear layer events were ensemble-averaged centering individual realizations in both x- and z-directions. The VISA technique has been applied in order to retain asymmetry features of the streaks; individual events with different sign of du/dz have been ensemble-averaged separately. Only the socalled accelerated events (du/dx < 0) have been considered. In Fig. 10 (Gottero and Onorato, 2000), the results corresponding to du/dz < 0 are shown; the VISA detection point, corresponding to the peak of the local u-variance, is at x+ = z + = 0. Continuous and dotted contour lines highlight phases of averaged low-speed and high-speed bands respectively. In Fig. 10(a) the color map represents the conditionally-averaged u-velocity, while the vector field represents the conditionally-averaged fluctuating velocity parallel to the wall. The ensemble averaged structure retains most of the characterizing features of single realizations. In particular, the segmented character of the streaks is clearly shown in the isocontour plot by the two low-speed islands tilted in different directions. The position of the internal shear layer event (x+ = z + = 0), highlighted by the distribution of du/dx in Fig. 10(b), corresponds to the merging region of the two kinked low-speed segments, at the border between low- and highspeed bands. The structures in Fig. 10 bear a close resemblance to results obtained by Johansson et al. (1991) by applying VISA technique to the turbulent channel flow DNS data of Kim et al. (1987); a similar picture is presented by Jeong et al. (1997) as an outcome from the application of Jeong and Hussain (1995) vortical structures detection scheme to the same database. The conditionally-averaged fluctuating flow field in Fig. 10(a) appears as if it was generated by two jets having opposite direction, strongly interacting in the VISA detection region and producing a complex flow behavior. The complex vectorial phase-averaged flow field shown in

258

M. Onorato et al.

Fig. 10. Conditional ensemble-averaged flow fields. y + = 20, Reθ = 1010. (a) Vectors: Planar instantaneous velocity field. Color map: Instantaneous u-velocity field. Isolines: Low-speed streaks (continuous lines) and high-speed streaks (dotted lines) in mm/s; Contours = −32, 32 increment 8; (b) Color map: du/dx (s−1 ); (c) Color map: du/dz (s−1 ). From Gottero and Onorato (2000).

Fig. 10(a) gives evidence of the presence of strong spanwise u-velocity gradient (du/dz) at the high-speed low-speed interface, in proximity of the VISA detection region, as it can be seen also in single realizations. This derivative constitutes the main contributor to the wall-normal vorticity ωy . The color map in Fig. 10(c) represents VISA-event-conditioned wall-normal vorticity. The importance of the normal vorticity component in the turbulence production cycle has been stressed, among others, by Jimenez and Pinelli (1997) and, Schoppa and Hussain (2002), according to whom the instability of the vorticity sheet flanking the streaks is one of the basic regeneration mechanisms of streamwise vortices. In Gottero and Onorato (2000), it is finally observed that the scenario described in Fig. 10 appears to be consistent with the conceptual model of coherent structures of Jeong et al. (1997). As sketched in Fig. 11, the flow


Fig. 11.

259

Quasi-longitudinal vortical structure, low-speed streaks, VISA event.

structure obtained from the conditional analysis of PIV data may be speculatively interpreted as the flow field induced by two adjacent overlapping counter-rotating vortices, asymmetrically tilted due to their mutual induction and advected by the mean flow. The organization of the flow into lowand high-speed regions comes from the pumping of fluid away from and towards the wall respectively. According to the model, motions of opposite signs induced by the two counter-rotating vortices near the vortex overlapping region would be responsible for the creation of the very strong internal shear layer detected by the VISA technique.

260

M. Onorato et al.

Except the ones previously reported, Particle Image Velocimetry observations in planes parallel to the wall, both in the buffer layer and in the logarithmic layer, have been object of very few studies, as appears from the open literature, probably because of the peculiar and special accuracy needed in taking PIV measurements in the region very close to the wall. Nevertheless, the accurate and exhaustive PIV measurements performed at the University of Illinois, Urbana, by Adrian and co-workers, in both streamwise-spanwise planes and in the streamwise-wall-normal plane of zero-pressure-gradient turbulent boundary layers, at low and high Reynolds number, should be mentioned here. In Tomkins and Adrian (2003), detailed measurements are reported at Reθ = 1015 and Reθ = 7705 in plane (x, z), at different distances from the wall. Figure 12 (Tomkins and Adrian, 2003) shows instantaneous velocity vectors in the plane y + = 100, at Reθ = 7705. The velocity vectors are plotted with a constant convection velocity (Uc = 0.65 Ue ) subtracted, in order to visualize the signatures of vortical motions. The flow direction is from the bottom to the top, so that downward vectors represent low-speed fluid. On the flank of the elongated low-momentum region (low-speed streak), a series of plane (x, z) vortex signatures, closely spaced and aligned roughly streamwise, are pointed out by circles and identification letters. Several of these vortical structures appear to be counter-rotating vortex pairs, consistently with the presence of two-legged hairpin-type vortices; others are one-sided vortex signatures. As already noted for the instantaneous flow field shown in Figs. 1 to 3, the sign of the swirling motion is consistent with the hypothesis that the velocity streak is a consequence of vortex induction: counterclockwise vortices appear on the right of the low-speed streak and clockwise vortices appear on the left. The observed vortical motions at the border of the velocity streak offer to the authors strong support in favor of the vortex packet model (Adrian et al., 2000) which assumes a series of streamwise aligned hairpin-type vortices, dominating the logarithmic layer, inducing the low-momentum regions and responsible for all the events observed in the inner and outer layer of a turbulent boundary layer. It should be commented that the definition of a hairpin vortex is seen by the authors in a modern version, including quasi-streamwise vortices (considered as legs of hairpins), one-sided and two-sided symmetric and asymmetric hairpins. The results in Fig. 12 are found near the bottom of the logarithmic region. A similar flow structure is found at the top of the logarithmic layer, at y + = 440, with some differences: the percentage of vortices paired reduces further from the wall and the spanwise size of the structures is greater than


Fig. 12. (2003).

261

Instantaneous field at y + = 100 and Reθ = 7705. From Tomkins and Adrian

in the earlier case, at y + = 100. Measurements in Tomkins and Adrian (2003), coherently with the results shown in Figs. 1 to 3, reveal indeed that at y + = 20, velocity streaks are the dominant structures and that vortices with significant component of vertical rotation are observed, but less frequently than further from the wall. Vertically-oriented vortices start to be very common at y + = 46, where both single and counter-rotating pairs exist.

262

M. Onorato et al.

One of the major contributes from the analysis of the PIV results in Tomkins and Adrian (2003) is the study to understand the mechanisms by which scale growth occurs in the spanwise direction, coupled with scale growth in the vertical direction, as observed in their instantaneous results at different distance from the wall and as already shown in Figs. 1 to 3. In Tomkins and Adrian (2003), it is proposed that the mechanism of spanwise scale growth is the merging of the vortex packets and hence the merging of associated low-speed regions and it is shown that the merging of vortex packets appears to occur quite frequently in the approximate range 20 < y + < 100. In particular, the coalescence of low-speed streaks occurs more frequently in the region y + < 30. As a result of the merging process, it is finally observed that the mean value of the spanwise length scales vary linearly with the distance from the wall, whereas individual structures do not grow in a strict self-similar fashion over the course of their development.

2.2. Streamwise-wall-normal plane Trains of vortical structures are particularly evident in the streamwise plane normal to the wall. It should be said that the detection of such swirling motions in wall normal planar measurements presents a degree of ambiguity due to the large spanwise unorganized vorticity produced by the shear nature of the mean flow field. In Fig. 13 (Di Cicca et al., 2001), iso-regions of λci > 0 (the closed isolines) are reported in conjunction with the map of the vorticity field and local vectorial fields, representing local instant velocity fields after the subtraction of a local velocity (Galileian decomposition) nearly equal to the structure convection velocity. The detected swirling motions are signatures of quasi longitudinal vortices displaying a tilting deviation angle in the horizontal plane high enough to be clearly intersected by the (x, y) plane and possibly much larger than the mean tilting angle of ±4◦ claimed by Jeong et al. (1997) in their conceptual model of nearwall structure. Hairpin shaped vortices, on the other hand, are expected to be easily detected when the PIV image plane cuts their head. Swirling patterns A, B and C are aligned in a direction forming an angle of about 15◦ with respect to the mean velocity and their distance is of the order of 100–200 wall units. Similar configurations have been found in several other realizations. The correspondence between the vortical structure organization observed in Fig. 13 and the packets of hairpin vortices observed by Adrian and co-workers in the outer layer region is evident. They found,


263

Fig. 13. Streamwise wall-normal instant field. Reθ = 1160. The mean flow is from the left. Isolines: Loci of λci > 0. Vectors: Velocity field after Galileian decomposition. Color map: Vorticity field. From Di Cicca et al. (2000).

both in DNS results (Zhou et al., 1999) and PIV measurements (Adrian et al., 2000), aligned hairpin packets growing upwards in the streamwise direction at an average angle of approximately 12◦ , with hairpins spaced several hundred viscous lengths apart in the streamwise direction. Vortex D, in the lower part of the image (Fig. 13), is also consistent with the packet model described in Zhou et al. (1999). The vortex signatures observed in Fig. 13 are indicated in the next figure, Fig. 14 (Di Cicca et al., 2001), as open circles (on which the rotation direction is also reported) superimposed to the instantaneous fluctuating velocity field after Reynolds decomposition. In the figure, an occurring VISA event, the instant contribution to the turbulent kinetic

264

M. Onorato et al.

Fig. 14. Streamwise wall-normal instant field. Reθ = 1160. Vectors: Velocity field after Reynolds decomposition. Color map: Instant contribution to the turbulent kinetic energy production term. Q2 and Q4: Quadrant events. V: VISA event. From Di Cicca et al. (2001).

energy production (color map) and detected quadrant events are also reported. A quadrant event is detected (Alfredson and Johansson, 1984) when |uv| > H(u′ v ′ ), where H is the threshold constant and u < 0, v > 0 (Q2 event, ejection) or u > 0, v < 0 (Q4 event, sweep); u′ and v ′ are the actual rms values of the velocity fluctuations u and v. The signature of a low-speed streak, clearly generated by the vortex packet, is evident in the near-wall region, where reverse (fluctuating) flow is present. Below the signature of the swirling structures, mostly evident for vortices B and C, a strong ejection of low-momentum fluid is present as a result of the their collective pumping effect. These ejections of low-momentum fluid represent


265

Q2 (second quadrant) events whose occurrence and frequency have been the subject of several experimental studies conducted with pointwise measurements, hot wire and laser Doppler velocimetry. Planar PIV measurements offer the opportunity of directly observing and understanding these motions in conjunction with the causes determining their behaviour (Fig. 14). The low-momentum fluid (Q2 event) meets the high-momentum fluid (Q4 event) from the outer region, forming internal shear layers. These Q2 (ejections) and Q4 (sweeps) events produce strong contributions to Reynolds stress and are the source of turbulent kinetic energy. Moreover, the sweep events are responsible for the generation of turbulent wall shear-stress. The color map in Fig. 14 reports the quantity (uv) · dU/dy (uv = instant Reynolds stress, U = mean velocity) whose negative counterpart, when time-averaged, represents the production term in the turbulent kinetic energy equation. This instant image shows that the main contribution to the turbulence production comes from Q2 events below y + = 100 and that Q4 events do not provide significant production. Finally, it may be inferred that the strong internal shear layer, detected in Fig. 14 by VISA analysis, as for the case seen in the plane parallel to the wall, is associated with jets (in the Reynolds decomposed flow field) generated by the vortical structure of the flow; these jets have opposite direction and strongly interact in the detection region. In other words, VISA events occur at the stagnation point where Q2 flow encounters a Q4 flow. PIV data in streamwise planes normal to the wall, in a zero pressure gradient boundary layer, with a much wider field of view have been examined carefully by Adrian and co-workers in a few papers, most recently in Adrian et al. (2000), where experiments at three Reynolds numbers in the range 930 < Reθ < 6845 are commented with great details. Based on a suitable hairpin vortex signature defined in the 2D PIV vectorial fields, it is clearly shown that packets of hairpins, which propagate while keeping aligned in the streamwise direction, with small velocity dispersion, populate the turbulent boundary layer. Such pockets are most frequently seen between the wall and the top of the logarithmic layer. The hairpin signature pattern consists of a spanwise vortex core located above a region of strong Q2 fluctuations that occur on a locus inclined at 30◦ –60◦ to the wall. It is worthwhile here to insist on the “extended” definition of hairpins adopted by the authors, including all variations of a common basic streamwise elongated vortical structure at different stages of evolution or in different surrounding flow environments and with varying size, aspect ratio, symmetry and age. In Adrian et al. (2000), the observed frequency of

266

M. Onorato et al.

occurrence of hairpin vortex packets is fairly high; at least one packet was found in 85% of all PIV images examined. As already mentioned, individual packets are seen to grow upwards in the streamwise direction at a mean angle of approximately 12◦ . The number of hairpins present in a packet depends on the Reynolds number: at a low Reynolds number, the packets contain fewer hairpins than in higher Reynolds number flow. In agreement with the results of Di Cicca et al. (2001), a typical low Reynolds packet may contain only two or three vortices. Moreover in Adrian et al. (2000), it is shown that several instantaneous realizations display small, presumably young, packets lying close to the wall within larger, presumably older packets, which may exist within still larger and older packets. It is believed that the older, larger packets have been originated from upstream and, moving in faster zones, overrun the younger, more recently generated ones. As a consequence of the flow fields induced by the system of packet structures, large and irregularly-shaped instant regions of flow having relatively uniform values of the streamwise momentum, separated by thin regions of large ∂u/∂y, are observed, in which the streamwise component of the velocity usually changes significantly between zones, but remains roughly constant within a zone. These zones of uniform streamwise momentum were previously observed by Meinhart and Adrian (1995), who suggested that the long region of uniformly retarded flow in each zone is the backflow induced by several hairpins that are aligned in a coherent pattern in the streamwise direction. The flow features shortly described before are evident in Figs. 15(a) and 15(b) (Adrian et al., 2000), reported here as an example. In the figures, the uniform momentum zones are separated by hand drawn lines and labelled zones I, II and III. The lines pass through the centers of the detected vortical structures. In Fig. 15(a), the instantaneous velocity vector map is reported, as viewed in a convecting frame of reference Uc = 0.8 × U∞ and scaled with inner variables. The gray color map in Fig. 15(b) provides a more direct way of seeing the three zones. The coincidence between the “heads of the hairpin” and the boundaries between the zones of uniform momentum clearly demonstrates an association between hairpin packets and uniform momentum zones. Finally, it should be noted that the buffer layer streaks are a part of the low-momentum zones phenomenon. A more recent analysis of PIV data in the streamwise-wall-normal plane of a turbulent channel flow, in the range of Reτ = 566–1759, has been done by Christensen and Wu (2004). They also found that hairpin-like vortices, aligned coherently in the streamwise direction, creates large-scale organized


267

Fig. 15. Hairpin vortex signatures and uniform-momentum zones. Reθ = 2370. (a) Instantaneous velocity map viewed in a convecting frame of reference; (b) Gray level map of constant streamwise momentum. From Adrian et al. (2000).

268

M. Onorato et al.

motions (hairpin vortex packets), dominating the turbulence dynamics in the outer region. They examined both spanwise clockwise and counterclockwise vortices. The former, whose rotation is consistent with the mean shear layer, were believed to be the head of hairpin-like vortices while the latter, or at least some of them, were postulated to be created by the local shear generated between two or more consecutive clockwise vortices within a packet. Counterclockwise vortices were scarcely found in the very-nearwall region, but their presence was observed to steadily increase within the logarithmic layer. Moreover a detailed analysis of the PIV data showed that the diameters of both type of vortices scale with inner units and that the percentage of clockwise and counterclockwise vortices, as function of wallnormal position y collapse with Reynolds number when y is scaled in inner units within the logarithmic layer. The previously shown PIV results, obtained in laboratory experiments at low or moderately high Reynolds number, demonstrate unequivocally the existence of streamwise-oriented trains of vortical structures dominating the inner and outer layers of near-wall turbulent flows. This flow organization is expected to be present also at higher Reynolds numbers as suggested by the flow visualization work of Head and Bandyopadhyay (1981), in which it was concluded that for Reynolds numbers up to Reθ = 10,000, the turbulent boundary layer consists of vortex loops and hairpin structures that are inclined at a characteristic angle of 45◦ to the wall. The flow behavior at much higher Reynolds number (Reθ of the order of 106 ) has been observed very recently (the data were acquired on May, 2004) in an open field experiment by Slaboch et al. (2004) in the atmospheric boundary layer at the Surface Layer Turbulence and Environmental Test site, Utah, USA. In about 40% of the 1700 instantaneous realizations acquired, the authors found, as in the previously cited laboratory experiments, zones of uniform momentum, separated by thin layers of high shear, aligned at a mean angle of 18◦ ; coherent vortex motions were sometimes seen to be aligned along the shear layer. A sample of the PIV images acquired is shown in Fig. 16 (Slaboch et al., 2004) where a shear layer, separating a low-momentum region from a region of higher momentum, may be identified by observing the instantaneous realizations of streamlines with a convection velocity subtracted and the u-velocity map. The authors claim that the instantaneous realizations in which vortex motions were found are strikingly similar to those of Adrian and co-workers. However, several examples were found where an inclined shear layer exhibited no discrete vortex motions. The distinctive features of the data as compared to laboratory


269

Fig. 16. Instantaneous realization of streamlines and longitudinal velocity color map, in an open field turbulent boundary layer. Reθ ≈ 106 . From Slabock et al. (2004).

lower Reynolds number results are to be attributed to surface roughness present in the open field experiments, to the fact that the open field PIV images cover only 1% of the boundary layer thickness and to the spatial resolution of the experiments. Moreover a physical Reynolds number effect can not be excluded. Innovative PIV arrangements are applied to zero-pressure-gradient turbulent boundary layer studies by Marusic and co-workers at the University of Minnesota, using single-plane and simultaneous dual-plane stereoscopic PIV. Recent measurements in streamwise-wall-normal and streamwisespanwise planes, as well as in cross-stream planes inclined at 45◦ and 135◦ to the streamwise direction are reported in Ganapathisubramani et al. (2003, 2005) and have been shown at the Workshop on Transition and Turbulence Control, Singapore, (December 2004). Consistently with the scenario of spatially coherent packets of hairpins vortices in the logarithmic region of the flow, the Marusic and co-workers results show clear evidence of largescale organization with long streamwise low-momentum zones and indicate the occurrence of coherent structures that become increasingly decorrelated from the wall as they grow beyond the logarithmic region. The shown PIV data constitute the experimental evidence of the existence of small and large-scale organized motions in streamwise planes parallel and normal to the wall, at low and high Reynolds numbers. Nevertheless, fundamental aspects of the physical behavior of near-wall turbulence remain unsolved: first of all, the mechanisms of regeneration of the vortical structures, either by instability of the velocity streaks or by

270

M. Onorato et al.

strong shears generated by previous vortical structures or by other mechanisms not yet been explored. Moreover, besides this fundamental aspect, many other unsolved problems still need experimental (and numerical) work; examples among others are the mechanisms of vortex packets merging and the related coalescence of the low-speed streaks, the scale growth with the distance from the wall and its self-similar behavior, the implications of the near-wall eddy structure dynamics on the logarithmic law and on the universal von Karman constant, the influence of the physical state of the wall and of the external flow on the eddy structure. Further experimental contributions could be obtained from PIV by applying the 3D (volumetric) and the time-resolved versions of the technique. To show the potentiality of the time-resolved PIV, preliminary results from data recently collected at INSEAN Institute, Rome, will be displayed here. The PIV images were taken in a zero-pressure-gradient boundary layer, Reθ = 2086, at 2000 frames per sec (the wall viscous time was 2 ms). In Fig. 17 (Dolcini et al., 2004), a sequence of realizations is reported. The time interval ∆t between two consecutive images in the figure is equal to 1 ms. The time-resolved PIV images show the velocity field, after Reynolds decomposition, superimposed to the vorticity field (the color map), in the streamwise wall-normal plane. The wall is at the top of the image, the mean flow coming from the left. Each frame covers about 850 wall units in the streamwise direction and about 650 wall units in the wall-normal direction. The top of the logarithmic layer is situated about 15 mm from the wall. This sequence of images, as other sequences in different interval of times, is characterized by vorticity sheets inclined with respect to the wall at about 15◦ –30◦ . In frame (d), for instance, two of such structures are clearly formed in the logarithmic layer and convected downstream, while keeping about the same inclination with respect to the wall. At the head of both vorticity sheets, a spanwise vortical motion is visible in the Reynolds decomposed flow field. The rotation of this swirling motion is coherent with the sign of the near-wall mean vorticity. Speaking of the color representation of the vorticity field, it should be noted that the colors are clearly saturated, therefore the picture must be observed as a flow visualization. The upstream structure, the one on the left of frame (d), though not yet formed in the first frame (a), is seen to grow through the sequence and, at a later time, to be segmented in smaller structures in the last frame. Moreover, a counterrotating (blue color) vorticity sheet is seen to develop between the two and to grow up in time. According to the author, the phenomenology shown in the typical sequence of Fig. 17 represents a further experimental evidence of


271

Fig. 17. Sequence of time-resolved velocity end vorticity fields. Reθ = 2086. Viscous length = 0.046 mm. From Dolcini et al. (2004).

the vortical structure regeneration mechanisms postulated by Adrian and co-workers (Zhou et al., 1996; Adrian et al., 2000 and Tomkins et al., 2003). 3. Turbulence Control by Wall Oscillation As anticipated in the introduction, turbulent boundary layers forced by spanwise wall oscillations are representative of large-scale turbulence

272

M. Onorato et al.

control strategies where spanwise motions or oscillating pressure gradients are applied leading to copious reductions of skin-friction and near-wall turbulence. Besides the oscillating wall, other examples of such flows are the one forced by a spatially or temporally oscillating, spanwise oriented, Lorentz force in an electrically conducting fluid (Berger et al., 2000) and the one excited by transverse traveling waves (Du et al., 2002). The properties of such flows are discussed in a recent review by Karniadakis and Choi (2003). Jung et al. (1992) conducted a DNS study (Reh = 3000) of a turbulent channel flow subject either to an oscillatory spanwise cross flow or to spanwise oscillatory motions of one of the channel walls. Their results indicate a 40% reduction in friction drag when the period of oscillation T + , normalized with respect to inner units, was set to 100. The oscillations also gave rise to a 40% reduction in the streamwise component of the Reynolds stress, with no significant increase in the spanwise component. These friction drag and turbulence reduction results were experimentally confirmed by Laadhari et al. (1994) for a boundary layer flow at Reθ = 950. The effect of the wall-oscillation amplitude on the total energy balance was investigated by Baron and Quadrio (1996) using DNS. For an oscillation period of T + = 100, they found 10% of net energy saving with a suitable spanwise wall velocity oscillation amplitude. Experimental investigations into changes in the turbulent boundary layer structure with spanwise wall oscillations were carried out by Choi et al. (1998), Choi (2000) and more recently by Choi (2002). In agreement with previously mentioned DNS and laboratory experiments, a reduction of 45% in the skin friction coefficient was measured with wall oscillations. Despite the number of studies available in the literature, the mechanisms leading to this flow control are not yet completely explained. Jung et al. (1992) observed that the turbulence reduction occurs because of the decrease in the number and intensity of turbulent bursts in the oscillatory channel as compared to the unperturbed flow. Later, the same authors (Akhavan et al., 1993) attributed the suppression of turbulence to the continuous shifting of the longitudinal vortices to different positions relative to the wall velocity streaks, weakening the intensity of the streaks by injecting high-speed fluid into low-speed streaks and low-speed fluid into high-speed regions. A similar argument was reported by Laadhari et al. (1993). Choi and co-workers (Choi et al., 1998; Choi, 2000) related the drag reduction by spanwise wall oscillation to the spanwise vorticity generated by the periodic Stokes layer, which reduces the mean velocity gradient within the viscous


273

sub-layer and therefore, the skin friction. At the same time, the twisting action caused by the periodic Stokes layers on the longitudinal vortices reduces the streamwise component of the vorticity in the near-wall region, weakening the near-wall ejection and sweep activity (Choi et al., 1998). Dhanak and Si (1991) could demonstrate numerically that the interaction between evolving, axially-stretched, streamwise vortices and a Stokes layer on the oscillating surface beneath them leads to reductions in skin friction. More recently, Di Cicca et al. (2002) and Iuso et al. (2003) applied PIV to observe the behavior of the turbulent near-wall organized motions under the action of spanwise wall oscillations. In this section, results from the last two papers are reported. These results refer to a basic flat plate turbulent boundary layer with Reθ = 1160 and to a wall oscillation frequency corresponding to a period T + = 100 and to a peak-to-peak amplitude of the oscillating wall of 324 viscous units. In Fig. 18 (Di Cicca et al., 2002), mean velocity profiles are displayed in semi-log plots; the forced velocity profiles were normalized using the friction velocities of both the canonical and the forced cases. The marked upward shifting of the logarithmic velocity profile for the forced flow, when actual inner-scaling (i.e. the one performed with the inner variables corresponding to the case being discussed) is applied, confirms that the skin-friction drag

fixed wall osc. wall - canonical u τ osc. wall - actual u τ LDV data

Fig. 18.

Log-law plot of mean velocity profiles. Reθ = 1160. From Di Cicca et al. (2002).

274

M. Onorato et al.

is reduced by spanwise wall oscillation. This indication is typical of most drag-reducing flows and suggests the thickening of the viscous sub-layer. The increase in depth of the viscous sub-layer reflects a reorganization of the turbulent structures, including an increase in the smallest eddies scale and a displacement of turbulent events outwards from the wall (Choi et al., 1998). When quantities are normalized with respect to the friction velocity of the standard boundary layer, or more evidently when they are outerscaled in linear coordinates, a reduction of the mean velocity is observed throughout the region y + < 30 (y/δ < 0.04). Further from the wall, the mean velocity is slightly larger in order to conserve momentum. Velocity fluctuation results are compared in Fig. 19 (Di Cicca et al., 2002) where considerably large reductions in both components of the veloc2 2 ity variance, longitudinal u′ and wall normal v ′ , are shown within the inner 2 2 region. The maximum u′ and v ′ reductions, located in the buffer region, are of the order of 30% and 40% respectively. Turbulence reduction is still evident when mean fluctuating quantities are normalized with respect to the actual friction velocity. In order to observe the overall structure of the flow, Figs. 20(a) and 20(b) (Di Cicca et al., 2002) display the double spatial correlation function, Ruu , for the streamwise fluctuating velocity in the plane y + = 20 parallel to the

fixed wall osc. wall - canonical u τ osc. wall - actual u τ LDV data

Fig. 19. (2002).

Profiles of velocity components variance. Reθ = 1160. From Di Cicca et al.


275

Fig. 20. Double spatial correlation function Ruu in plane y + = 20. Reθ = 1160. (a) Natural bounday layer; (b) Forced boundary layer. From Di Cicca et al. (2002).

wall, respectively for the standard and manipulated boundary layers. Comparison between Figs. 20(a) and 20(b) shows similarities between the two cases. This implies that, in spite of the great reduction in turbulence due to wall oscillation, the global organization of the flow is not qualitatively affected by the manipulation. Both correlograms exhibit an elliptical shape, because of the strong anisotropy between streamwise and spanwise directions. Nevertheless, from the quantitative analysis of the correlation functions and in particular of the spanwise (∆x = 0) correlation reported in Fig. 21 (Di Cicca et al., 2002), a marked increase in the integral lateral scale (the area under the curves) produced by the wall oscillation, clearly appears. This presumably means that the flow control increases the mean spacing of the low-speed streaks. Similar result is reported in Choi et al. (1998), where infrared images show that several low-speed streaks coalesce into a single streak as the wall oscillates, leading the mean streak spacing to increase by about 45%, whereas the duration of the streaks is multiplied by a factor of 4 (Choi, 2000). In the previous section, Q2 motions were shown in plane (x, y) evidencing their implication in the production of turbulent kinetic energy and their physical and spatial relationship with vortex signatures and with internal shear layer motions (VISA events). It is then of interest to see how such motions are influenced by the wall oscillation. In Fig. 22 (Di Cicca et al., 2002), the number of Q2 events per unit time N at y + = 50, are displayed

276

M. Onorato et al.

Fig. 21. Spanwise correlation functions Ruu at y + = 20. Reθ = 1160. From Di Cicca et al. (2002).

Fig. 22.

Frequency of Q2 events at y + = 50. Reθ = 1160. From Di Cicca et al. (2002).

as a function of the threshold constant H. Both curves are approximately coincident and have a linear behavior as shown in Alfredson and Johansson (1984), where the analysis was performed also at y + = 50. Moreover, if the Q2 events were detected by using (for both the basic and the manipulated flow) the value of u′ v ′ corresponding to the natural flow, the resulting number of events per unit time N would be lower for the forced flow, being u′ v ′

277


reduced by about 25% (at y + = 50) by the wall oscillation. Furthermore in Di Cicca et al. (2002), it is shown that also the frequency of the VISA events, spatially associated with the Q2–Q4 motions, are reduced by the wall oscillation, for any value of the local integration region size and the constant threshold. An enlighten statistical analysis of the modification induced by the wall oscillations in the velocity streak structure is reported in Iuso et al. (2003). Histograms of spanwise low-speed streak spacing and width are shown in Figs. 23 and 24 (Iuso et al., 2003). The observation of the two histograms and the visual analysis of some instantaneous realizations suggest that a partial process of coalescence between adjacent streaks is under way as the wall oscillates. This process can explain the shift of both histograms towards higher values of the streak spacing and width for the case of the manipulated flow. In Fig. 23, the number of streaks spaced only a few viscous lengths apart appears to be considerably reduced, while the number of streaks with separation higher than the mean value increases. This observation also agrees with the result shown in Fig. 21 where the spanwise integral scales of the fluctuating streamwise velocity are compared for the two flows. In order to explain the increase in streak spacing when the wall is oscillating, in addition to the advective mechanism of coalescence, reference must be done to another mechanism, namely the enhancement of the streaks’ vorticity annihilation through viscous effects due to the alternating 9 Natural Flow Forced Flow

%

6

3

0 36

80

125

169

213

258

302

λ + 347

Fig. 23. Histograms of spanwise low-speed streak spacing. y + = 20. Reθ = 1160. From Iuso et al. (2003).

278

M. Onorato et al.

20 Natural Flow Forced Flow

%

10

L+ 0 9

44

80

116

151

187

222

258

293

Fig. 24. Histograms of spanwise low-speed streak width. y + = 20. Reθ = 1160. From Iuso et al. (2003).

8 Natural Flow Forced Flow

%

6

4

2 |ω+y | max 0 0.00

0.04

0.07

0.11

0.14

0.18

0.21

0.25

0.28

0.32

Fig. 25. Histograms of spanwise low-speed streak strength. y + = 20. Reθ = 1160. From Iuso et al. (2003).

Stokes layers: the weaker and smaller scale velocity streaks are selectively reduced by this viscous effect. An important behavior of the low-speed streaks under the wall oscillation, which is likely to affect the near-wall turbulence regeneration, is shown in Fig. 25 (Iuso et al., 2003) where histograms of the streak strength |ωy+ |max are reported. |ωy+ |max is the maximum value of the wall-normal vorticity flanking a low-speed streak. The histograms show a weakening


279

action of the wall oscillation on the streak strength. The forced flow distribution shows a much sharper and higher peak around its most probable value of |ωy+ |max = 0.123 which in turn is shifted towards lower values (reduction by about 10%) with respect to the natural case. In both flows, the distribution is approximately symmetric about the corresponding most probable value, which is essentially coincident with the average value. It is interesting to observe that, for values of |ωy+ |max higher than the natural flow most probable value, the percentage of low-speed streaks realizations present in the forced flow is considerably reduced (31% versus 56%). Moreover, the percentage of very weak streaks is substantially reduced by the manipulation, which could be a consequence of the discussed coalescence of streaks at small distances from each other. The direct association between the weakening of the low-speed streaks and the reduction of turbulence activity, demonstrated for the present flow, is not new and has already been the subject of various speculations. Jimenez and Pinelli (1999) showed by a numerical experiment that, through filtering out the high values of ωy flanking the velocity streaks, a turbulent flow can be brought to decay back to a laminar state. Schoppa and Hussain (2002) recently proposed a new mechanism for generation of nearwall quasi-longitudinal streamwise vortices using linear perturbation analysis and direct numerical simulation of a turbulent channel flow. According to their suggestion, the streak waviness in the plane parallel to the wall, caused by w(x) disturbance, grows mainly by linear transient instability, generating horizontal sheets of streamwise vorticity that collapse in organized streamwise vortices via a stretching process. Within the framework of this mechanism, the instability of the low-speed streaks would play a fundamental role in the production of near-wall turbulence. In the same paper, the authors suggest the possibility to control friction drag and heat transfer by reducing ωy flanking the velocity streaks, hence partly suppressing the velocity streak perturbation growth responsible for new longitudinal vortices generation. The results shown in this section are consistent with the previously commented numerical experiment of Jimenez and Pinelli (1999) and the turbulence regeneration mechanisms proposed by Schoppa and Hussain (2002). It may be then speculated that the key feature of the wall oscillation responsible for the turbulence decay, with respect to the natural boundary layer, is the presence of the alternate Stokes layers superimposed to the mean motion. These layers, in addition to viscous annihilation of ωy may reduce

280

M. Onorato et al.

the vorticity flanking the streaks by their convective action, decreasing the number of unstable streaks. The alternating Stokes layer may be also considered responsible for promoting the coalescence between adjacent low-speed streaks, contributing to increase their width and spacing. References 1. R. J. Adrian, C. D. Meinhart and C. D. Tomkins, Vortex organization in the outer region of the turbulent boundary layer, J. Fluid Mech. 422 (2000) 1. 2. R. Akhavan, W. G. Jung and N. Mangiavacchi, Turbulence control in wallbounded flows by spanwise oscillations, Appl. Sci. Res. 51 (1993) 299. 3. P. H. Alfredson and A. V. Johansson, On the detection of turbulencegenerating events, J. Fluid Mech. 139 (1984) 325. 4. T. W. Berger, J. Kim, C. Lee and J. Lim, Turbulent boundary layer control utilizing the Lorentz force, Phys. Fluids 12(3) (2000) 631. 5. A. Baron and M. Quadrio, Turbulent drag reduction by spanwise wall oscillations, Appl. Sci. Res. 55 (1996) 311. 6. R. F. Blackwelder and J. H. Haritodinis, Scaling of the bursting frequency in turbulent boundary layers, J. Fluid Mech. 132 (1983) 87. 7. R. F. Blackwelder and R. E. Kaplan, On the wall structure of turbulent boundary layers, J. Fluid Mech. 76 (1978) 89. 8. K. S. Choi, European drag reduction research — Recent developments and current status, Fluid Dyn. Res. 26(5) (2000) 325. 9. K. S. Choi, J. R. Debisschop and B. R. Clayton, Turbulent boundary layer control by means of spanwise wall oscillations, AIAA J. 36(7) (1998) 1157. 10. J. I. Choi, C. X. Xu and H. J. Sung, Drag reduction by spanwise wall oscillation in wall-bounded turbulent flows, AIAA J. 40(5) (2002) 842. 11. K. T. Christensen and Y. Wu, A population study of small-scale spanwise vortices in turbulent channel flow, 11th Int. Symp. on Flow Visualization, Notre Dame, Indiana, USA (2004). 12. M. B. Dhanak and C. Si, On reduction of turbulent wall friction through spanwise wall oscillations, J. Fluid Mech. 383 (1999) 175. 13. G. M. Di Cicca, G. Iuso, P.G. Spazzini and M. Onorato, Particle image velocimetry investigation of a turbulent boundary layer manipulated by spanwise wall oscillations, J. Fluid Mech. 467 (2002) 41. 14. G. M. Di Cicca, M. Onorato, G. Iuso and P. G. Spazzini, Application of particle image velocimetry to wall turbulence studies, Acc. Sc. Torino-Atti Sc. Fis. 135 (2001) 43. 15. A. Dolcini, F. Di Felice and G. P. Romano, Time-resolved PIV investigation of a turbulent boundary layer at high Reynolds numbers, INSEAN-PIVNET workshop, Rome (May 2004). 16. Y. Du, V. Symeonidis and R. Karniadakis, Drag reduction in wall-bounded turbulence via a transverse traveling wave, J. Fluid Mech. 457 (2002) 1.


281

17. B. Ganapathisubramani, E. Longmire and I. Marusic, Characteristics of vortex packets in turbulent boundary layers, J. Fluid Mech. 478 (2003) 35. 18. B. Ganapathisubramani, N. Hutchins, W. T. Hambleton, E. K. Longmire and I. Marusic, Investigation of large-scale coherence in a turbulent boundary layer using two-point correlations, J. Fluid Mech. 524 (2005) 57. 19. M. Gottero and M. Onorato, Low-speed streak and internal shear layer motions in a turbulent boundary layer, Eur. J. Mech. B-Fluids 19 (2000) 23. 20. M. R. Head and P. Bandyopadhyay, New aspects of turbulent boundary layer structure, J. Fluid Mech. 107 (1981) 297. 21. D. K. Heist and T. J. Hanratty, Observations of the formation of streamwise vortices by rotation of arch vortices, Phys. Fluids 12 (2000) 2965. 22. G. Iuso, M. Di Cicca, M. Onorato, P. G. Spazzini and R. Malvano, Velocity streak structure modifications induced by flow manipulation, Phys. Fluids, 15(9) (2003) 2602. 23. G. Karniadakis and K. S. Choi, Mechanisms on transverse motions in turbulent wall flows, Ann. Rev. Fluid Mech. 35 (2003) 45. 24. J. Jeong and F. Hussain, On the identification of a vortex, J. Fluid Mech. 285 (1995) 69. 25. J. Jeong, F. Hussain, W. Schoppa and J. Kim, Coherent structures near the wall in a turbulent channel flow, J. Fluid Mech. 332 (1997) 185. 26. J. Jimenez and A. Pinelli, Wall turbulence: How it works and how to damp it, 28th AIAA Fluidynamics Conference, Snowmass Village, Colorado, USA (1997) Paper 97-2112. 27. A. V. Johansson, P. H. Alfredsson and J. Kim, Evolution and dynamics of shear-layer structures in near-wall turbulence, J. Fluid Mech. 224 (1991) 579. 28. W. Jung, N. Mangiavacchi and R. Akhavan, Suppression of turbulence in wall-bounded flows by high-frequency spanwise oscillations, Phys. Fluids A4(8) (1992) 1605. 29. J. Kim, P. Moin and R. D. Moser, Turbulence statistics in fully developed channel flow at low Reynolds number, J. Fluid Mech. 177 (1987) 133. 30. F. Laadhari, L. Skandaji and R. Morel, Turbulence reduction in a boundary layer by a local spanwise oscillating surface, Phys. Fluids 6(10) (1994) 3218. 31. M. T. Landahl, On sublayer streaks, J. Fluid Mech. 212 (1990) 593. 32. Z. C. Liu, R. J. Adrian and T. J. Hanratty, A study of streaky structures in a turbulent channel flow with particle image velocimetry, Int. Symp. on the Application of Laser Technology to Fluid Mechanics, Lisbon (1996). 33. C. D. Meinhart and R. J. Adrian, On the existence of uniform momentum zones in a turbulent boundary layer, Phys. Fluids 7(4) (1995) 694. 34. S. H. Robinson, The kinematics of turbulent boundary layer structures, NASA TM-103859 (1993). 35. W. Schoppa and F. Hussain, Coherent structure generation in near-wall turbulence, J. Fluid Mech. 453 (2002) 57. 36. P. E. Slaboch, S. R. Stolpa and S. C. Morris, Spatially resolved near surface motions in the atmospheric boundary layer using PIV, 11th Int. Symp. on Flow Visualization, Notre Dame, Indiana, USA (2004).

282

M. Onorato et al.

37. C. R. Smith and S. P. Metzler, The characteristics of low-speed streaks in the near-wall region of turbulent boundary layer, J. Fluid Mech. 129 (1983) 27. 38. C. D. Tomkins and R. J. Adrian, Spanwise structure and scale growth in turbulent boundary layers, J. Fluid Mech. 490 (2003) 37. 39. J. Zhou, R. J. Adrian, S. Balachandar and T. M. Kendall, Mechanisms for generating coherent packets of hairpin vortices in channel flow, J. Fluid Mech. 387 (1999) 353.

MEAN-MOMENTUM BALANCE: IMPLICATIONS FOR WALL-TURBULENCE CONTROL

Joe Klewicki Physical Fluid Dynamics Laboratory Department of Mechanical Engineering University of Utah Salt Lake City, UT 84112-9208, USA E-mail: [email protected] Mean flow structure commonly constitutes the foundational origin of efforts seeking to describe the instantaneous mechanisms of wall turbulence. As it pertains to flow control, these mean flow-based interpretations are significant since they often suggest the nature of potentially promising control strategies, including the locations in the layer where, for example, imparting control actuations are likely to be most effective. Recent results relating to the structure of the mean-momentum balance in turbulent wall flows reveal a physical layer structure whose associated dynamics and Reynolds number scaling behaviors are quantitatively and qualitatively distinct from the existing predominant view. Furthermore, analytical explorations into the mathematical properties of the mean-momentum balance reveal behaviors that are also distinct from the well-accepted view. The present study explores the implications of these new physical and theoretical findings as they pertain to issues of turbulent wall flow control. Contents 1 2

3

Introduction Review of Results Relating to the Mean-Momentum Balance 2.1 Layer structure 2.2 Reynolds number scaling behaviors 2.3 Self-consistent physical model Flow Control Implications 3.1 Flow physics and flow control 3.2 Mean-momentum balance as a diagnostic 3.3 The Lamb vector and wall-flow control 283

284 285 286 286 289 290 290 292 293

284

4 Summary Acknowledgments References

J. C. Klewicki

294 294 295

1. Introduction The common depiction of turbulent wall flow structure is largely derived from the observed properties of the mean velocity profile and the relative magnitudes of the mean viscous stress and Reynolds shear stress (see, for example, Tennekes and Lumley, 1972; Pope, 2000). Together these can be used to motivate the familiar picture of wall flows as being composed of a viscous sub-layer, buffer layer, logarithmic (overlap) layer and wake layer. Perspectives that view the boundary layer as a dynamical machine are derived from this structural framework, and thus provide a basis for implementing strategies intending to modify mean dynamics. For example, independent of the Reynolds number, this picture naturally focuses the attention on the thin region immediately adjacent to the surface, commonly referred to as the viscous wall-layer. With increasing Reynolds number, a particular challenge associated with this focus is the diminishingly small fraction of the flow occupied by this viscous wall-layer (i.e. buffer layer and below). Recent data analyses and complementary theoretical developments are employed to ascertain flow structure as directly specified by the mean dynamical equation. Available high quality data reveal a four-layer description that is a considerable departure from the four-layer structure (discussed above) traditionally and nearly universally ascribed to turbulent wall flows (Wei et al., 2005). Each of the four new layers is well characterized relative to the contributions required to balance the governing equation, and thus the mean dynamics of these four layers are unambiguously defined. The inner normalized physical extent of three of the layers exhibit significant Reynolds number dependence. The scaling properties of these layer thicknesses are determined. Particular significance is attached to the viscous/Reynolds stress gradient balance layer. For example, contrary to the common view, the existence of this layer reveals that viscous effects remain as dynamically significant as turbulent inertia out to a wall-normal position beyond the peak in the Reynolds stress. Multi-scale analysis substantiates the four-layer structure in developed turbulent channel flow. In particular, the analysis verifies the existence of a third layer, with its own characteristic scaling, between the traditional inner and outer layers.

Mean-Momentum Balance: Implications for Wall-Turbulence Control

285

In what follows, these new results are briefly reviewed. Then, their implications are discussed relative to the problem of wall flow control. Attention is paid to the utility of the present framework for both implementing and assessing the effectiveness of candidate flow control strategies. In this context, a new model of the boundary layer (Klewicki et al., 2004, 2005) whose dynamical attributes are in accord with the properties of the meanmomentum balance, is presented and discussed. Finally, a brief discussion is provided that identifies connections between the present findings to previous efforts that have utilized the Lamb vector in assessing the effectiveness of strategies intending to modify wall flow dynamics (see, for example, Crawford and Karniadakis, 1997; Crawford et al., 1998). 2. Review of Results Relating to the Mean-Momentum Balance Consideration is given to the Reynolds Averaged Navier–Stokes (RANS) equation in its unintegrated form as applied to turbulent boundary layer flow over a planar surface located at y = 0, U+

+ ∂U + ∂2U + ∂T + + ∂U + V = + , ∂x+ ∂y + ∂y +2 ∂y +

(2.1)

and to a fully-developed flow in a planar channel of height 2δ, 0=

d2 U + dT + 1 + + , δ+ dy +2 dy +

(2.2)

where a superscript + denotes normalization by the friction velocity (uτ = τw /ρ, τw is the mean wall shear stress) and kinematic viscosity, ν. T is the so-called Reynolds shear stress, −ρuv. As is customary, x is the axial coordinate, y is the wall-normal coordinate, U and V are the velocity components in the x- and y-directions respectively. Upper case letters represent mean quantities, lower case letters denote fluctuating quantities, tilde denotes instantaneous quantities (i.e. u ˜ = U + u), an overbar denotes time averaging, and vorticity components are identified by their subscript. The lefthand side of Eq. (2.1) represents mean flow advection, while the right-hand side terms represent the viscous and Reynolds stress gradients respectively. These latter two forces are also present in channel flow (Eq. (2.2)), while a mean pressure gradient is present instead of mean advection. Since, for either the flat plate boundary layer or fully-developed channel flow, there are only three distinct dynamical effects, the ratio of any two determine the nature by which the equation is balanced.

286

J. C. Klewicki

2.1. Layer structure Wei et al. (2005) explored the structure of boundary layer, pipe and channel flows by examining the ratio of the last two terms in Eq. (2.1) and Eq. (2.2). Interpretation of this ratio for the boundary layer is as follows: 2

+

+

1. If | ∂∂yU+2 / ∂T ∂y + | ≫ 1, then the Reynolds stress gradient term is negligible and Eq. (2.1) sums to zero essentially through a balance of the mean advection and viscous stress gradient terms. + 2 + 2. If | ∂∂yU+2 / ∂T ∂y + | ≪ 1, then the mean viscous stress gradient term is negligible and Eq. (2.1) sums to zero essentially through a balance of the mean advection and Reynolds stress gradient terms. 2 + + 3. If | ∂∂yU+2 / ∂T ∂y + | ≃ 1, then the Reynolds stress and viscous stress gradients balance, and are either greater or of the same order of magnitude as the mean advection term. If channel flow is considered, then mean advection is replaced by the mean pressure gradient. Existing high quality data from the experiments of Zagarola and Smits (1997), DeGraaff and Eaton (2000), and the direct numerical simulations of Moser et al. (1999) were employed to examine the indicated ratio as a function of distance from the wall, y + = yuτ /ν, for a range of Reynolds numbers, δ + , where δ is either the boundary-layer thickness or the half channel height. The sketch of Fig. 1 depicts the behavior of the stress gradient ratio at any fixed δ + . As indicated, there exists a four-layer structure. Layer I nominally retains the traditionally held character of the viscous sub-layer, and in the boundary layer is a region where the viscous stress gradient nominally balances mean advection. In the channel, layer I is characterized by a balance between the viscous stress gradient and pressure gradient. In layer II, the magnitude of the ratio is very close to unity, and thus is called the stress gradient balance layer. As described by Wei et al. (2005) and Fife et al. (2005a), across the mesolayer (layer III), the Reynolds stress gradient changes sign and the terms in Eq. (2.1) undergo a process of balance breaking and exchange. The net result of this process is that from the outer edge of layer III to y = δ (i.e. layer IV). Equation (2.1) is characterized by a balance between mean advection and turbulent transport via the Reynolds stress gradient. 2.2. Reynolds number scaling behaviors The qualitative features of Fig. 1, depicted for any fixed Reynolds number, persist for the Reynolds number range currently accessible to inquiry.


287

Fig. 1. Sketch of the ratio of the viscous stress gradient to the Reynolds stress gradient (last two terms in Eqs. (2.1) and (2.2)) in turbulent boundary layer or channel flows, at any given Reynolds number. Layer I is characterized by a balance between mean advection and the viscous stress gradient; dashed line is for the boundary layer; solid line is for the channel. (Note that in a pipe, this balance is between the mean pressure gradient and the viscous stress gradient.) In layer II, the balance is between the viscous and Reynolds stress gradients. Layer III is a mesolayer in which all three terms in Eq. (2.1) are of the same order of magnitude, except that in a part of it, the Reynolds stress gradient is negligible. Layer IV is defined by a balance between mean advection and turbulent inertia. From Wei et al. (2005). Table 1. Inner-normalized scaling behaviors of the layer thicknesses and velocity increments. Note that the layer IV properties are asymptotically attained as δ+ → ∞. Layer I II III IV

∆y + increment O(1) √ O(√δ+ ) O( δ+ ) O(δ+ )

(≃ 3) √ (≃ 1.6√δ+ ) (≃ 1.0 δ+ ) (→ δ+ )

∆U + increment O(1) + O(U∞ ) O(1) + O(U∞ )

(≃ 3) + (≃ U∞ /2) (≃ 1) + (→ U∞ /2)

Quantitatively, however, this layer structure has been shown by Wei et al. (2005) to exhibit distinct Reynolds number dependencies relating to both the wall-normal extent of the layers and the velocity increment across each of the layers. Table 1 presents these scaling behaviors as normalized by

288

J. C. Klewicki

inner variables. As is evident, layers I and IV adhere (at least asymptotically) to the traditional inner and outer scalings respectively. On the other hand, layers II and III exhibit mixed scaling properties. The inner normalized thickness of layer II grows like the geometric mean of the Reynolds √ number defined as the ratio of outer to inner length scales (i.e. ∼ δ + ), while its velocity increment remains a fixed √ fraction of U∞ , independent of Reynolds number. Similarly, ∆III y + ∼ δ + , while its velocity increment is only about 1.0uτ , independent of δ + . As discussed in detail by Wei et al. (2005), these scaling behaviors differ considerably from the classical view of boundary layer structure, and are associated with the existence of a third fundamental length scale, νδ/uτ , that is intermediate to ν/uτ and δ. Layer II is called the stress gradient balance layer since the dominant dynamical mechanisms are the viscous and Reynolds stress gradient terms on the right-hand side of Eq. (2.1); underlying their ratio being −1 in Fig. 1. Contrary to the prevalent notion that boundary layer dynamics are inertially-dominated outside the buffer layer (independent of δ + ), momentum balance data incontrovertibly reveal that an equal competition between the mean viscous force and the time-averaged turbulent inertia persists to wall-normal distances beyond the peak in the Reynolds stress, Tmax .a Consistent with the mathematical scaling layer hierarchy revealed by Fife et al. (2005a), the physical model of Klewicki et al. (2004) postulates that this competition is associated (in a time mean sense) with the vortical motions forming and evolving from the intense near-wall vorticity field. It is significant to note, however, that the balance in layer II comes about via two nearly equal but opposite decreasing functions that lose their dominance over mean advection (or the mean pressure gradient) as layer II transitions into layer III. The scalings of Table 1 reveal that the layer thickness of II and III are coupled such that their velocity increments adhere to outer and inner scaling respectively. These properties underlie new interpretations relating to, for example, the nature of the inner/outer interaction in boundary layers. In this regard, it is relevant to note that major portions of layers II and IV, and all of layer III reside within the bounds of the region of the mean profile that exhibits a logarithmic-like variation with y + . The lower edge of layer II + ), while is fixed near the edge of the viscous sub-layer (independently of δ√ + the position of its outer edge extends to increasing y values like δ + such a Thus,

for example, at Reynolds number δ+ = 1 × 106 , this corresponds to a y + location of about 2000.


289

that ∆II U = U∞ /2. (Note that relative to outer scaling, the position of √ + the outer edge of layer II moves “inward” like 1/ δ .) Because of this positioning behavior for layer II, both end points of layer III vary with δ + . Thus, while the layer III thickness exhibits the same Reynolds number scaling behavior as layer II, its velocity increment is only ≃ 1uτ owing to the fact that with increasing δ + , its position is located at increasing y + locations in a region where U + ∼ log(y + ). 2.3. Self-consistent physical model Klewicki et al. (2004, 2005) have proposed a new physical model for the boundary layer and wall flows in general. Elements of this model are presented in the schematic of Fig. 2. In contrast to the physical picture promoted by the traditional (sub, buffer, log and wake layer) view, this model is consistent with the properties of the mean-momentum balance reviewed above. Primary aspects of the model depicted in Fig. 2 include the following: 1. The change in sign of the Reynolds stress gradient indicates that, relative to affecting a mean time rate of change of momentum, turbulent inertia acts as a source in the mean dynamical equation for y < ym and a sink for y > ym , where ym is the position of Tmax . 2. Consistent with the solenoidal condition on the vorticity field and existing vorticity field data, the predominant motions residing in the region y < ym (layer II and below) are viewed as attached eddies in that their vorticity filaments extend to the single-signed distribution of spanwise vorticity that exists in the near-wall region. The predominant resident motions in the region y > ym (layer IV) are viewed as detached eddies

Fig. 2. Schematic representation of some of the dynamical attributes of the proposed model for the turbulent boundary layer (from Klewicki et al. (2004)). Layer numbers are the same as those identified in Fig. 1.

290

J. C. Klewicki

in that they are spatially compact, highly three-dimensional, and locally · satisfy ∇ ω = 0. The representative attached eddy is hypothesized to be a hairpin-like vortex, and the representative detached eddy is hypothesized to be a ring-like motion. 3. The observed increments of mean-momentum (or circulation per unit length) across each layer add clarity to the notions that characteristic attributes of boundary layer development are associated with an outward transport of wall layer vorticity from layer II simultaneous with an inward transport of freestream momentum from layer IV. As δ + → ∞, these two processes approach a balance as reflected in the asymptotic equalization of the velocity increments across layers II and IV. 4. Vorticity field data and mean-momentum balance structure support the hypothesis that processes (involving attached and detached eddies) relating to the so-called inner/outer interaction occur in a zone centered (in a time-averaged sense) on layer III. An important property of these interactions is the aforementioned Reynolds number dependence in the position of the Tmax . 3. Flow Control Implications There is utility in viewing a turbulent wall flow as a complex mechanical system whose time-averaged dynamics are described by the differential statement of Newton’s second law (i.e. the mean-momentum balances of Eqs. (2.1) and (2.2)). Given this, one may relatively broadly describe flow control applications as having the purpose to modify the inherent dynamical mechanisms such that the momentum transport that would otherwise naturally occur is predictably modified. It is then rational to expect that any such dynamical modifications will be reflected in the structure of the meanmomentum balance. These considerations and their implications motivate the following discussion. 3.1. Flow physics and flow control A major outcome of the results described above relates to the significantly altered view of boundary layer physics. Relative to flow control, these provide new insights and notions pertaining to the nature of the flow field interactions to be modified via flow control, their relevant length and time scales, and the locations in the layer where control actuations might best be imparted. Of course, an important consideration implicit to all of these issues is the question of Reynolds number dependence.


291

Flow Field Interactions The new insights derived from the properties of the mean-momentum balance allow specific attributes to be associated with the attached/detached eddy structure. For example, under the proposed model, attached eddies form and evolve across layer II, and thus their dynamical signature is that they produce instantaneous contributions to positive −∂ uv/∂y. Similarly, the characteristic eddies of layer IV are detached. Therefore, their dynamical signature is that they produce negative −∂ uv/∂y. In the context of these dynamical signatures, it is useful to examine the equation, e.g., (Klewicki, 1989), −

∂ 2 ∂ uv = vω z − wω y + (v + w 2 − u2 ). ∂y ∂x

(3.1)

For turbulent channel flow, the last term is identically zero; while for boundary layers, this term is small, especially as δ + becomes large (Klewicki, 1989). Thus, to a very good approximation, the gradient of the Reynolds stress is established by the difference of the indicated velocity-vorticity correlations. Given this, the interpretation is that in layer II, the attached eddies interact with the velocity field to generate a net positive sum and in layer IV, the detached eddies generate a net negative sum. The dominant terms in Eq. (2.1) indicate that the dynamics underlying the evolution of attached eddies are characterized by a competition between viscous shear forces and turbulent advection. Similarly, detached eddy dynamics in layer IV are characterized by a competition between mean flow and turbulent advection. The velocity-vorticity field interactions underlying Eq. (3.1) in either layer II or layer IV have recently been shown to have significant contributions at selective intermediate and large scales (Priyadarshana and Klewicki, 2003; Priyadarshana et al., 2005). Physically, the reason for this is attributable to the fact that as δ + → ∞, velocity spectra peak at decreasingly low wavenumber, while vorticity spectra peak at increasingly high wavenumber (Priyadarshana et al., 2005). This description as to how the Reynolds stress distribution is established and maintained suggests a number of promising avenues regarding flow control.b Notable among these are to (i) modify the velocity-vorticity field interactions that individually lead to the capacity of the attached and detached b Especially

so, in light of the analysis of Fukagata et al. (2002) indicating that through the third integral of the mean momentum balance one may directly associate the surface drag with the area enclosed by the Reynolds stress profile.

292

J. C. Klewicki

eddies to generate positive and negative Reynolds stress gradients respectively, (ii) modify the relative proportion of attached and detached eddies, and (iii) modify the interaction between the attached and detached eddies. Length and Time Scales Important information required to implement any given control strategy relates to the length and time scales of the relevant dynamical motions. In this regard, the characteristic length and velocity data in Table 1 suggest a much richer set of possibilities than can be realized by the traditional inner and outer scales alone. Specifically, the entries in the table for layers II and III exhibit an intriguing mixture of inner, outer and intermediate scales. Actuation Locations An apparently unique feature of the physical model of Klewicki et al. (2004, 2005) is its direct connection with the mean dynamics. In particular, the dynamical signatures of the characteristic motions in each layer are defined such that they are consistent with the time-averaged dynamics. This self-consistency provides clarity regarding where to locate actuators having specific dynamical modification objectives. For example, a control strategy seeking to modify the dynamical interactions between mean advection and turbulent inertia would best be placed in layer IV. Similarly, strategies seeking to modify the inner/outer interaction (and thus interrupt the aforementioned wall-layer vorticity and freestream momentum fluxes) would, on average, be optimally placed in layer III. In this regard, it is important √ to reiterate that the wall-normal position of layer III scales like y/δ ∼ 1/ δ + . In contrast, the traditionally held view is that the primary inertial/viscous interaction (constituting the inner/outer interaction) occurs in the buffer layer for all Reynolds numbers. The buffer layer thickness scales like 1/δ + , and thus with increasing Reynolds number becomes inaccessible to physical actuators much more quickly than the center of layer III. 3.2. Mean-momentum balance as a diagnostic It is asserted that the variations induced in the mean-momentum balance constitute a straight-forward indication of the net dynamical effects imparted by any given flow management strategy. The notion here is simple and direct, and is akin to previous approaches that have utilized the variations in the turbulence kinetic energy balance in an analogous diagnostic fashion (see, for example, Rathnasingham and Breuer, 2003). In this regard, it is worth noting that while flow field energetics are important, cause and effect relationships are governed by the dynamics.


293

Fig. 3. Ratio of the viscous stress gradient to the Reynolds stress gradient in favorable pressure gradient turbulent boundary layers. Profiles are derived from the DNS of Spalart (1986).

Flow management devices intending to change dynamics must impart a force. Direct examination of the mean-momentum balance reveals how the baseline (canonical) structure depicted in Fig. 2 is modified. For example, Fig. 3 shows the stress gradient ratio for turbulent boundary layers under the influence of a favorable pressure gradient. In this case, a constant, positive force in the x-direction has been added to the canonical flow. As can be seen, the baseline four-layer structure exists, but in a modified form. Specifically, the growth of layer II appears to be largely arrested (albeit over the admittedly small range of Reynolds numbers). Multi-scale √ analyses by Metzger and Fife (2005) reveal that for this flow, the δ + dependence of the growth rate is retained for layer II, but the growth rate of δ itself is inhibited. Relative to determining the Reynolds number sensitivity of a control strategy, a broader point worth mentioning is that if the applied force distribution can be analytically characterized, then the problem may be amenable to the multi-scaling analysis framework developed for the mean-momentum balance (Fife et al., 2005b). 3.3. The Lamb vector and wall-flow control More broadly, the difference in velocity-vorticity correlations shown in Eq. (3.1) constitutes the axial component of the Lamb vector (see, for example, Lele, 1992; Wu et al., 1996). Previous studies, for example, by Crawford et al. (1998), have effectively argued and subsequently demonstrated that

294

J. C. Klewicki

the behavior of the Lamb vector can be rather directly correlated with mean reductions in surface drag. The empirical evidence, multi-scale analyses and dynamical model reviewed herein substantiate and extend these earlier flow control notions by clarifying the underlying physical mechanisms at their connections to the structure of the mean dynamical equation. With regard to these issues, it is relevant to note the recently developed theory of Marmanis (1998) that incorporates the transport equation for the Lamb vector in a system of equations analogous to Maxwell’s equations of electromagnetics. An important element of this theory is the concept of turbulent charge density, n. Physically, this quantity is a measure of the local concentration of the sum of static pressure and directed momentum along a streamline. Mathematically, n is given by the divergence of the Lamb vector, and in the flat plate boundary layer or fully-developed channel can be shown to be equal to ∂ 2 T /∂y 2. This is particularly relevant to the development of the mean-momentum field since Fife et al. (2005a) have, among other related results, shown analytically (no assumptions or approximations) that the constancy of ∂ 2 T /∂y 2 (appropriately normalized) is the explicit determinant of logarithmic mean profile dependence across an internal hierarchy of scaling sub-domains. 4. Summary A newly developed framework for interpreting the mean dynamics of turbulent wall flows was outlined herein. The flow control implications of this new theoretical and physical framework was then discussed. Primary among these are: 1. The physical insights provided via analysis and interpretation of the mean-momentum equation provide an array of new ideas and possibilities relating to flow control. 2. The momentum balance-based analysis provides critical information relating to the Reynolds number dependencies of the dynamically relevant lengths, velocities and forces within the flow. 3. The mean-momentum balance provides a useful (arguably, the natural) diagnostic for assessing the dynamical impact of any given flow management strategy. Acknowledgments Aspects of this work were supported by the National Science Foundation under grant CTS-0120061 (grant monitor, M. Plesniak), the Office of Naval


295

Research under grant N00014-00-1-0753 (grant monitor, R. Joslin), and the Department of Energy through the Center for the Simulation of Accidental Fires and Explosions under grant W-7405-ENG-48.

References 1. C. Crawford and G. Karniadakis, Reynolds stress analysis of EMHDcontrolled wall turbulence. Part I. Streamwise forcing, Phys. Fluids 9 (1997) 788–806. 2. C. Crawford, H. Marmanis and G. Karniadakis, The Lamb vector and its divergence in turbulent drag reduction, in Proc. Int. Sym. Seawater Drag Reduction, Newport, RI, USA (July, 1998), pp. 22–24. 3. D. B. DeGraaff and J. K. Eaton, Reynolds number scaling of the flat plate turbulent boundary layer, J. Fluid Mech. 422 (2000) 319–346. 4. P. Fife, T. Wei, J. Klewicki and P. McMurtry, Stress gradient balance layers and scale hierarchies in wall bounded turbulent flows, J. Fluid Mech. 532 (2005) 165–189. 5. P. Fife, J. Klewicki, P. McMurtry and T. Wei, Multiscaling in the presence of indeterminacy: Wall-induced turbulence, in press, Multiscale Modeling and Simulation (2005). 6. K. Fukagata, K. Iwamoto and N. Kasagi, Contribution of the Reynolds stress distribution to the skin friction in wall-bounded flows, Phys. Fluids 14 (2002) L73–L76. 7. J. Klewicki, Velocity-vorticity correlations related to the gradients of the Reynolds stress in parallel turbulent wall flows, Phys. Fluids A1 (1989) 1285– 1288. 8. J. Klewicki, P. McMurtry, P. Fife and T. Wei, A physical model of the turbulent boundary layer consonant with the dynamical structure of the mean-momentum balance, in Proc. 15th Australasian Fluid Mech. Conf., The University of Sydney, Sydney, Australia (December, 2004), pp. 13–17. 9. J. Klewicki, P. McMurtry, P. Fife and T. Wei, A physical model of turbulent wall flows consonant with the structure of the mean-momentum equation, in preparation (2005). 10. S. Lele, Vorticity form of the turbulence transport equations, Phys. Fluids 4 (1992) 1767–1772. 11. H. Marmanis, Analogy between the Navier–Stokes equations and Maxwell’s equations: Application to turbulence, Phys. Fluids 10 (1998) 1428–1437. 12. M. Metzger and P. Fife, Scaling properties of turbulent boundary layers with favorable pressure gradient, in preparation (2005). 13. R. D. Moser, J. Kim and N. N. Mansour, Direct numerical simulation of turbulent channel flow up to Rτ = 590, Phys. Fluids 11 (1999) 943–945. 14. S. B. Pope, Turbulent Flow (Cambridge University Press, 2000). 15. P. Priyadarshana and J. Klewicki, Reynolds number scaling of wall-layer velocity-vorticity products, in Reynolds Number Scaling in Turbulent Flow, ed. A. J. Smits (Kluwer Academic Publishers, 2003), pp. 117–122.

296

J. C. Klewicki

16. P. Priyadarshana, J. Klewicki, S. Treat and J. Foss, Statistical structure of turbulent boundary layer velocity-vorticity products at high and low Reynolds numbers, J. Fluid Mech., under review (2005). 17. R. Rathnasingham and K. Breuer, Active control of turbulent boundary layers, J. Fluid Mech. 495 (2003) 209–233. 18. P. Spalart, Numerical study of sink flow boundary layers, J. Fluid Mech. 172 (1986) 307–328. 19. H. Tennekes and J. Lumley, A First Course in Turbulence (MIT Press, 1972). 20. T. Wei, P. Fife, J. Klewicki and P. McMurtry, Properties of the meanmomentum balance in turbulent boundary layer, pipe and channel flows, J. Fluid Mech. 522 (2005) 303–327. 21. J. Wu, Y. Zhou and J. Wu, Reduced stress tensor and dissipation and the transport of the Lamb vector, Tech. Rep. 96–21, ICASE, NASA Langley Research Center, Hampton VA (1996). 22. M. V. Zagarola and A. J. Smits, Scaling of the mean velocity profile for turbulent pipe flow, Phys. Rev. Lett. 78 (1997) 239–242.

THE FIK IDENTITY AND ITS IMPLICATION FOR TURBULENT SKIN FRICTION CONTROL

Nobuhide Kasagi∗ and Koji Fukagata† Department of Mechanical Engineering The University of Tokyo Hongo 7-3-1, Bunkyo-ku, Tokyo 113-8656, Japan E-mails: ∗[email protected] † [email protected] The identity equation derived by Fukagata, Iwamoto and Kasagi (Phys. Fluids 14 (2002) L73–L76) leads to a general strategy for accomplishing turbulent skin-friction drag reduction. This is demonstrated by referring to several typical examples of recently studied control schemes including local blowing/suction and surfactant additives. Based on the FIK identity and numerical experiment of direct numerical simulation, the performance of active feedback control of wall turbulence, which has been paid much attention over the decade but only has been studied at low Reynolds numbers, is estimated at higher Reynolds numbers. A possible control scheme for enhancing heat transfer while keeping skin friction moderate is also discussed. Contents 1

Introduction

298

2

The FIK Identity

302

3

Analysis of Manipulated Flows

305

4

Development of Control Schemes

310

5

Control Feasibility at High Reynolds Numbers

313

6

Enhancement of Heat Transfer

317

7

Concluding Remarks

320

Acknowledgments

322

References

322

297

298

N. Kasagi and K. Fukagata

1. Introduction The modern turbulence research has a history of more than a hundred years since Osborne Reynolds’ pioneering work in the late 19th century. Its three major aims have been to understand highly nonlinear turbulence mechanics, develop predictive methods for turbulent flow phenomena and devise schemes of controlling them. It is this third target that we focus upon in this chapter and our efforts are directed toward innovating highly advanced control methodologies. It is well known that control of turbulent flows and the associated transport phenomena should be a key in many engineering practices such as energy saving, efficient production process, securing high quality products and resolving global environmental problems. Its impact on future technology and human life would be enormous through manipulation and modification of turbulent drag, noise, heat transfer, mixing as well as chemical reaction. Hereafter, we mainly pay attention to the skin-friction drag in wall turbulence. The skin-friction drag in a wall-bounded turbulent flow is usually much higher than that of a laminar flow at the same Reynolds number. From extensive research over the last several decades, we presently have a common understanding that large frictional drag in turbulent wall flows is attributed to the existence of near-wall vortical structures and the associated ejection/sweep events (Kline, Reynolds, Schraub and Runstadler, 1967; Robinson, 1991). As an example, the spatial relationship between the near-wall quasistreamwise vortex and the production, destruction and diffusion of the instantaneous Reynolds shear-stress is shown in Fig. 1 (Kasagi, Sumitani, Suzuki and Iida, 1995). A low-pressure region corresponds to the core of an inclined streamwise vortex near the wall. On the sweep side of the vortex, the high-pressure region near the wall is produced by the fluid impingement onto the wall that is induced by the vortex motion. On the ejection side of the vortex, low-speed fluid is lifted up and its collision against high-speed fluid from upstream forms a local stagnation region with high pressure. Instantaneous high production rate of the Reynolds shear-stress takes place on both sides of the vortex. The low- and high-pressure regions are regarded as high destruction (pressure-strain correlation) regions of the Reynolds stress. The turbulent diffusion transports the Reynolds shearstress from the high production regions to the regions between the highand low-pressure regions. The essential dynamical mechanism of near-wall turbulence appears spatially and temporally intermittent. Thus, the production of turbulent

The FIK Identity and Skin-Friction Control

299

Fig. 1. Relationship between a near-wall quasi-streamwise vortex and the production, pressure-strain and diffusion of −u′ v′ . Reprinted from Kasagi et al., Int. J. Heat Fluid Flow 16 (1995) 2–10.

kinetic energy and the wall skin friction could be effectively reduced through selective manipulation of near-wall vortices. Figure 2 shows the spatiotemporal scales of the streamwise vortices in various applications (Kasagi, Suzuki and Fukagata, 2003). The typical length scale of vortices is found to be 100 µm. Although the coherent structures have such small scales, recent development of MEMS technology has made it possible to fabricate flow sensors and mechanical actuators of such small-scale range (see, e.g., Ho and Tai, 1996; Gad-el-Hak, 2002). Various control algorithms have been proposed with the aid of direct numerical simulation (DNS), as reviewed, e.g., by Moin and Bewley (1994), Gad-el-Hak (1996), Kasagi (1998), Bewley (2000) and Kim (2003). Those rigorously based on the modern control theory, e.g., the optimal control theory, are potentially very effective (Bewley, Moin and Temam, 2001). However, much simpler control algorithms are preferable for practical use, because the amount of measurable flow information is limited and realtime data processing is essential. The above-mentioned knowledge on the near-wall coherent turbulence structures resulted in, for instance, dynamical argument-based control algorithms for drag reduction in turbulent wallbounded flows. Choi, Moin and Kim (1994) demonstrated in their DNS that about 25% drag reduction can be attained by a simple algorithm, in which local blowing/suction is applied at the wall so as to oppose the wall-normal

300


Fig. 2. Spatio-temporal scales of coherent structure in real applications. Redrawn based on Kasagi et al. (2003).

velocity at 10 wall units above the wall. Subsequently, several attempts were made to develop control laws using the quantities measurable at the wall. Lee, Kim, Babcock and Goodman (1997) used a neural network and found a control law, in which the control input is given as a weighted sum of the spanwise wall shear stresses measured around the actuator. A series of analytical control laws were derived by Lee, Kim and Choi (1998) in the framework of the suboptimal control. Their DNS of channel flow at Reτ = 110 showed 16% to 22% drag reduction achieved by using the spanwise wall shear stress or the wall pressure as a sensor signal; in the former case, the control law is quite similar to that obtained by the neural network of Lee et al. (1997). Apart from these studies, several other types of control schemes have been proposed and assessed by using DNS (e.g., Koumoutsakos, 1999; Lee, Cortelezzi, Kim and Speyer, 2001). The control input assumed in all of these DNS studies is blowing/suction, which is assumed continuously distributed over the wall surface. However, the control effectiveness was unknown in a realistic situation, where sensors and actuators of certain sizes are distributed discretely on the wall. For this problem, Endo, Kasagi and Suzuki (2000) carried out DNS of turbulent channel flow, in which arrayed discrete wall shear-stress sensors and walldeformation actuators were assumed, and demonstrated the effectiveness of feedback control under more realistic condition.


301

Fig. 3. Feedback control system for wall turbulence with 192 wall shear-stress sensors and 48 shell-deformation actuators. From Yoshino et al. (2003).

Following these theoretical and numerical studies, some attempts are being made to develop a hardware system for active feedback control of turbulence (Rathnasingham and Brewer, 2003; Yoshino, Suzuki and Kasagi, 2003; Yamagami, Suzuki and Kasagi, 2005). Figure 3 shows the control system developed at the University of Tokyo (Yoshino et al., 2003). It has four rows of micro hot-film sensors and three rows of miniature magnetic actuators in between. Each sensor row has 48 micro wall shear-stress sensors with 1 mm spacing, and each actuator row has 16 shell-deformation actuators with 3 mm spacing. The size and frequency response of these sensors and actuators are found to fulfil the spatio-temporal requirements in the wind tunnel experiment. Performance evaluation of this feedback control system was made in turbulent channel flow of air (Suzuki, Yoshino, Yamagami and Kasagi, 2005). The control system was placed at the bottom wall of the test section. The bulk mean velocity was set to be 3 m/s, which corresponds to the friction Reynolds number of 300. An optimal control scheme based on genetic algorithm (Morimoto, Iwamoto, Suzuki and Kasagi, 2002) was employed. The resultant drag reduction rate was 7 ± 3% (Suzuki et al., 2005). Despite the extensive research on wall-turbulence and its control introduced above, the quantitative relation between the statistical quantities of turbulence and the drag reduction effect has not been completely clear. Recently, we derived an identity equation that quantitatively relates the skin-friction coefficient and the Reynolds stress distribution for three

302


canonical wall-bounded flows, i.e. channel, pipe and plane boundary layer flows (Fukagata, Iwamoto and Kasagi, 2002). Although the derivation itself is simple and straightforward, the result is suggestive and useful for analyzing the effect of the Reynolds stress on the frictional drag, especially in controlled flows. In the present chapter, this identity equation (hereafter, referred to as the FIK identity) is introduced with several example analyses. In the next section, we introduce the mathematical derivation of the FIK identity and its implication for drag reduction control. Subsequently, a theoretical analysis is presented concerning the Reynolds number effect on control by an idealized near-wall layer manipulation. Based on the idea of the FIK identity, control strategy for heat transfer enhancement is also discussed. 2. The FIK Identity First, we show the derivation of the FIK identity, including the detailed calculations that were omitted for brevity in the original paper (Fukagata et al., 2002). Here, only the simplest case, i.e. a steady, fully-developed, isothermal, incompressible turbulent flow of a Newtonian fluid in a plane channel, as shown in Fig. 4, is considered. The Reynolds-averaged Navier–Stokes equation in the x-direction is given by d 1 d¯ u d¯ p ′ ′ + + (−u v ) , (2.1) 0=− dx dy Reb dy where the overbar denotes the average. In this section, all variables without superscript are those non-dimensionalized by the channel half-width δ ∗ and twice the bulk mean velocity 2Ub∗ whereas dimensional variables are denoted by the superscript of ∗. The bulk Reynolds number is defined as Reb = 2Ub∗ δ ∗ /ν ∗ , where ν ∗ is the kinematic viscosity. The pressure in Eq. (2.1) is normalized by the density.

δ

Fig. 4.

Flow geometry.

303


The flow rate is assumed to be always constant. The velocities on the walls are no-slip, but wall-transpiration is allowed given the net flux is zero, viz., v¯(x, 0, z, t) = 0 and v ′ (x, 0, z, t) = 0. This enables the use of the FIK identity to, for instance, to analyze a flow controlled by blowing/suction with zero net flux, which is widely used in numerical studies of active feedback control. Under the conditions above, integration of Eq. (2.1) over y gives the relation between the pressure gradient and the skin-friction coefficient, d¯ p τw∗ u 8 d¯ = −8 . (2.2) = Cf = 1 ∗ ∗ Reb dy y=0 dx ρ Ub 2 The relation for componential contributions of different dynamical effects to the local skin-friction coefficient can be obtained by applying triple integration to Eq. (2.1). The first integration gives the well-known linear relation for stresses, which is readily derived from Eqs. (2.1) and (2.2) as u Cf 1 d¯ + (−u′ v ′ ) = (1 − y). Reb dy 8

(2.3)

The further integration leads to the mean velocity profile, which reads

y Cf y2 ′ ′ (−u v ) dy . u ¯ = Reb (2.4) y− − 8 2 0 The final integration is akin to obtaining the flow rate from the velocity profile, i.e. 1 y Cf 1 = Reb − (−u′ v ′ ) dy dy , (2.5) 2 24 0 0 where the relation of the dimensionless bulk mean velocity Ub = 1/2 was used. The double integration in Eq. (2.5) can be transformed to single integration by applying the integration by parts, viz., 1 y 1 y ′ ′ ′ ′ (−u v ) dy dy (−u v ) dy dy = 1 0

0

0

0

= y

0

=

y

(−u′ v ′ ) dy

1 0

−

1

y(−u′ v ′ ) dy

0

1 0

(1 − y)(−u′ v ′ ) dy.

(2.6)

304


Thus, Eq. (2.5) can be rewritten as

Cf =

12 + 12 Reb

1

0

2(1 − y)(−u′ v ′ ) dy .

(2.7)

This identity equation for a fully-developed channel flow indicates that the skin-friction coefficient is decomposed into the laminar contribution 12/Reb which is identical to the well-known laminar solution, and the turbulent contribution (the second integral term), which is proportional to the weighted average of Reynolds stress. The weight linearly decreases with the distance from the wall. A similar relationship can be derived also for other canonical flows. The FIK identity for a fully-developed cylindrical pipe flow is expressed as 16 + 16 Cf = Reb

0

1

2r u′r u′z r dr,

(2.8)

where the length is non-dimensionalized by the pipe radius. The wall and the cylindrical axis are located at r = 1 and r = 0, respectively. The FIK identity for a zero-pressure-gradient boundary layer on a flat plate is Cf =

4(1 − δd ) +4 Reδ

1

(1 − y)(−u′ v ′ ) dy−2

0

0

1

(1 − y)2

∂ uu ∂ u ¯ v¯ + ∂x ∂y

dy,

(2.9)

where the non-dimensionization is based on the freestream velocity and the 99% boundary-layer thickness. The third term is the contribution from the spatial development, while δd in the first term is the dimensionless displacement thickness. For a laminar plane boundary layer, the first contribution is 4(1 − δd )/Reδ ≈ 2.6/Reδ and the third contribution can be computed as 2.6/Reδ by using the similar solution of Howarth (1938). The summation of these contributions is identical to the well-known relation, i.e. Cf ≈ 3.3/Reδ . A more general form of the FIK identity (e.g., for channel flows) can be expressed as 12 + 12 Cf = Reb

0

1

2(1 − y)(−u′ v ′ ) dy + (III) + (IV) + (V).

(2.10)


305

The third term is the contribution from the spatial and temporal development, which reads (III) = 12

0

1

′′ ′′ ∂(uv) 1 ∂ 2 u′′ ∂p′′ ∂u′′ ∂(uu) − + − − (1 − y)2 − dy, ∂x ∂x Reb ∂x ∂x ∂t (2.11)

where the double-prime denotes the deviation of mean quantity from the bulk mean quantity, i.e. f ′′ (x, y, t) = f¯(x, y, t) −

1

f¯(x, y, t) dy.

(2.12)

0

The fourth term is the contribution from a body force bx and additional a such as that by polymer/surfactant (Yu, Li and Kawaguchi, stress, τxy 2004; Li, Kawaguchi and Hishida, 2004; White, Somandepalli, Dubief and Mungal, 2005), which can be expressed as (IV) = 12

0

1

a dy. (1 − y) (1 − y) ¯bx + 2¯ τxy

(2.13)

The fifth term is the contribution from the boundary momentum flux, such as uniform blowing/suction, which reads (V) = −12 Vw

2

0

(1 − y)¯ u dy,

(2.14)

where Vw denotes the wall-normal velocity at the walls. In this case, the integration of the other terms should also be made from 0 to 2, because the flow is not anymore symmetric around the center plane.

3. Analysis of Manipulated Flows The merit of the FIK identity derived above is that one can quantitatively identify each dynamical contribution to the drag reduction/increase even for a manipulated flow. The first example is a fully-developed turbulent pipe flow manipulated by the opposition control scheme (Choi et al., 1994). Namely, timedependent, continuous blowing/suction velocity is applied as the boundary condition at the wall, so as to oppose the wall-normal velocity at the detection plane assumed at y = yd . The data were obtained by DNS made

306


by the energy-conservative finite difference method (Fukagata and Kasagi, 2002) at the Reynolds number of Reb = 5300 (i.e. Reτ = 180 for uncontrolled flow). The detection plane was set at yd+ = 15. Here, the superscript of + denotes a quantity non-dimensionalized by the friction velocity of the uncontrolled flow. Figure 5 shows the Reynolds shear-stress, u′r u′z , and the weighted Reynolds shear-stress appearing in the FIK identity for pipe flow, 2r2 u′r u′z . As noticed from Eq. (2.8), the contribution of Reynolds stress near the wall dominates both in uncontrolled and controlled cases. The difference in the areas covered by these two (controlled and uncontrolled) curves of the weighted Reynolds stress is directly proportional to the drag reduction by control. In the present case, the turbulent contribution is reduced by 35%, while the total drag reduction is 24%. The contribution of Reynolds stress near the wall can be more clearly illustrated by plotting a cumulative T (cum) , to the turbulent part, defined here as contribution, Cf 1−y T (cum) 2r u′r u′z r dr, (y) = 16 (3.1) Cf 0

where y = (1 − r) is the distance from the wall. As is shown in Fig. 6, the Reynolds stress within 80 wall units from the wall is responsible for 90% of the turbulent contribution to the skin friction in the case of uncontrolled flow. This fact makes the opposition control algorithm proposed by Choi et al. (1994) very successful. Namely, it works to suppress the Reynolds stress near the wall, and this results in considerable drag reduction at a low Reynolds number flow. 0.0012 uruz, no control uruz, controlled 2r2 uruz no control 2r2 uruz, controlled

uruz , 2r2 uruz

0.001 0.0008 0.0006 0.0004 0.0002 0 0

0.2

0.4

0.6

0.8

1

r Fig. 5. Reynolds shear-stress and weighted Reynolds shear-stress in pipe flow at Reτ = 180 under opposition control. Reprinted from Fukagata et al., Phys. Fluids 14 (2002) L73–L76.


0.007

100 %

0.006

90 %

0.005 CT(cum) f

307

0.004 0.003

50 %

0.002 no control controlled

0.001 0 0

20

40

60

80 100 120 140 160 180 y+

Fig. 6. Cumulative contribution to skin friction in pipe flow at Reτ = 180 under opposition control. Reprinted from Fukagata et al., Phys. Fluids 14 (2002) L73–L76.

A more interesting analysis can be made when the feedback control is applied only periodically and partially to the wall (Fukagata and Kasagi, 2003). The opposition control of Choi et al. (1994) is applied in the region of nL < z < nL + Lc and the other region of nL + Lc < z < (n + 1)L is uncontrolled (where L is a streamwise computational size). In this case, the spatial development term corresponding to Eq. (2.11) appears, but its contribution to the skin friction is shown to be relatively small. Namely, the streamwise distribution of the skin-friction drag is primarily determined by the turbulent contribution term. Figure 7 shows the radial distribution of the weighted Reynolds shear-stress at different streamwise locations. Before entering into the controlled region, i.e. z + = −200 in Fig. 7(a), the profile is essentially the same as that of the uncontrolled flow. At the beginning of controlled region (z + = 200), the profile near the wall drastically changes to one similar to that of the entire-wall control. Detailed inspection reveals that the most of such streamwise evolution occurs in the region of −10 < z + < 50. The far-wall distribution gradually changes following the quick change in the near-wall region and this is considered as an indirect effect. A similar variation is observed also in the uncontrolled region (Fig. 7(b)). Most of the change in the near-wall region occurs right downstream of the controlled region. After that, the profile recovers to that of the uncontrolled flow. Another example of controlled flow is a fully-developed channel flow with uniform blowing on one wall and suction on the other. Figure 8 shows the componential contributions computed from the DNS database of Sumitani and Kasagi (1995), where the blowing/suction velocity is

308


(a)

0.0014 upstream

0.0012

No control

2r2 ur uz

0.001 0.0008 0.0006

downstream

0.0004

z+ =−200 200 600 1000 1400

0.0002 Entire-wall control

0 -0.0002 0

(b)

10

20

0.0014

30 y+

40

60

downstream

0.0012 No control 0.001 2r2ur uz

50

Entire wall control

0.0008 0.0006

upstream

0.0004

(z −Lc)+ =−200 200 600 1000 1400

0.0002 0 -0.0002 0

10

20

30 y+

40

50

60

CT , CC

Fig. 7. Weighted Reynolds stress distribution at different streamwise locations. (a) Right downstream of the onset of control; (b) Right downstream of the controlled region. Reprinted from Fukagata and Kasagi, Int. J. Heat Fluid Flow 24 (2003) 480–490.

0.001 0.0008 0.0006 0.0004 0.0002 0 −0.0002 −0.0004 −0.0006

blowing side

suction side

CT CC C T+C C 0

0.5

1 y

1.5

2

Fig. 8. Contributions to friction drag in channel flow at Reτ = 150 with uniform blowing and suction. Reprinted from Fukagata et al., Phys. Fluids 14 (2002) L73–L76.


309

Vw = Vw∗ /(2Ub∗ ) = 0.00172. The key in the figure C T denotes the integrand of turbulent contribution C T = (1 − y)(−u′ v ′ ), and C C corresponds to that of the convective contribution C C = −Vw (1 − y)¯ u (see Eq. (2.14)). The dotted line in the figure represents C T in an ordinary channel flow (Vw = 0) at the same bulk Reynolds number (Reb = 4360), computed by the pseudospectral DNS code (Iwamoto, Suzuki and Kasagi, 2002). The weighted Reynolds shear-stress on the blowing side (defined here, for convenience, as 0 ≤ y ≤ 1) is larger than that in the case of Vw = 0, while it is close to zero on the suction side (1 ≤ y ≤ 2). The total turbulent contribution is slightly reduced from the ordinary channel flow. The integrand of convective contribution C C is negative on the blowing side and positive on the suction side. The total convective contribution is slightly positive. Since the total convective contribution exceeds the amount of reduction in the turbulent contribution, the total Cf results in a larger value than that of the ordinary channel flow. The last example is a surfactant-added channel flow (Yu et al., 2004). Direct numerical simulation is performed by assuming the Giesekus fluid model. The bulk Reynolds number is 12,000. The friction Weisenberg number, which represents the memory effect of the surfactant-added fluid, is 54, corresponding to 75 ppm CTAC surfactant solution. The fractional contribution to Cf is shown in Fig. 9, where the turbulent contribution drastically decreases with the addition of surfactant. The viscoelastic contribution (see Eq. (2.13)), however, works to largely increase the friction drag. As a result of these changes, the total friction drag is reduced by about 30%. A similar analysis is also reported for experimental data of polymer-added zero-pressure-gradient boundary layer (White et al., 2005). The changes in

Contribution to C f

0.008 0.006 0.004

Turbulent contribution

0.002 0

Viscoelastic contribution Turbulent

Laminar

Laminar

Newtonian fluid

Surfactantadded fluid

Fig. 9. Contributions to friction drag in surfactant-added channel flow. Redrawn based on Yu et al. (2004).

310


the different contributions are found qualitatively similar to the case of the surfactant-added flow mentioned above. Likewise, the FIK identity can be used for investigation of drag reduction mechanism by other additives, such as microbubbles (Murai, Oishi, Sasaki, Yamamoto and Kodama, 2005).

4. Development of Control Schemes The FIK identity suggests that suppression of the Reynolds shear-stress in the near-wall region is of primary importance in order to substantially reduce the skin-friction drag. Once the near-wall Reynolds shear-stress is suppressed, the stress far from the wall is also suppressed through the indirect effect (Fukagata and Kasagi, 2003). From this argument, a new suboptimal control law was derived by Fukagata and Kasagi (2004a). In that work, the cost functional for a channel flow was defined as follows: J(φ) =

ℓ 2A∆t

t

t+∆t

S

φ2 dS dt +

1 2A∆t

t+∆t

t

(−u′ v ′ )y=Y dS dt.

S

(4.1) Here, φ denotes the local blowing/suction velocity at the wall, A is the area of wall S, ∆t is the time-span for optimization and ℓ is the price for the control. By approximating the Reynolds shear-stress at y = Y using the firstorder Taylor expansion, viz., (−u′ v ′ )y=Y = −Y φ

∂u , ∂y w

(4.2)

the control input φ that minimizes the cost functional can be calculated analytically by the procedure proposed by Lee et al. (1998). The result is 0 ∂u α φˆ = , 1 − iαγkx /k ∂y

(4.3)

w

√ where the hat denotes the Fourier component i = −1 and k = kx2 + kz2 . There are two parameters: α = Y /(2ℓ) is the amplitude coefficient and γ = 2Reb /∆t can be interpreted as an inverse of influential length (see Fukagata and Kasagi (2004a) for details). A similar control law can be developed also for a pipe flow. Following the procedure by Xu, Choi and Sung (2002), an approximate control law is

311


derived as

1z ∂u α φˆ = , ′ (k ) ∂r 1 − iαγIm (kz )/Im z

(4.4)

w

where Im is an mth-order modified Bessel function of the first kind and ′ is its derivative. Although the expressions look different, the control Im laws for channel and pipe have essentially the same dynamical effect on the controlled flow (Fukagata and Kasagi, 2004a). The derived control law can be transformed to the physical space through the following inverse Fourier transform, similarly to Lee et al. (1998), to read ∂u (4.5) Wij φ(x, z, t) = (x + i∆x, z + j∆z, t), ∂y i

j

w

where ∆x and ∆z are the streamwise and spanwise grid spacings, respectively. The weight distribution in the physical space is shown in Fig. 10. The weights are symmetric in the spanwise direction and asymmetric in the streamwise direction. The product of parameters αγ determines the tail length in the streamwise direction. Performance of the proposed control algorithm was tested by DNS of turbulent pipe flow (Fukagata and Kasagi, 2004a). About 12% drag reduction is obtained when φ+ rms ≈ 0.1 and αγ = 73. The profile of the Reynolds shear-stress is shown in Fig. 11. As expected, the near-wall Reynolds stress is suppressed by the present control. Note that, the profile of the present Wij / W00 2 1.5 1 0.5 0 -0.5 -1

-4

-3

-2 -1 0 1 j 2

3

4

-5 -10

15 20 10 5 i 0

25

30

Fig. 10. Weight distributions of the Reynolds shear-stress-based suboptimal control law. Reprinted from Fukagata and Kasagi, Int. J. Heat Fluid Flow 25 (2004) 341–350.

312


0.8 0.7 0.6 (ur′uz′)+u

0.5 0.4 0.3 0.2

No control Present control v-control (yd+u =5) v-control (yd+u =15)

0.1 0 -0.1 0

5

10

15

20 y+

25

30

35

40

Fig. 11. Reynolds shear-stress in pipe flow at Reτ = 180 under the Reynolds shearstress-based suboptimal control. Reprinted from Fukagata and Kasagi, Int. J. Heat Fluid Flow 25 (2004) 341–350.

control is nearly the same as that of the opposition control (denoted as v-control) with yd+ = 5. Comparison is also made with the opposition control with yd+ = 15, in which the Reynolds stress around 5 < y + < 10 is suppressed to give a higher drag reduction rate of 25%. The direct suppression with the present control seems to occur merely in the region of 0 < y + < 5. This is due to the first-order Taylor expansion used for the approximation of cost functional, i.e. Eq. (4.2). If the streamwise velocity above the wall, say at y + = 15, can be more accurately estimated, a higher drag reduction can be made by this control strategy. In fact, in DNS, using the streamwise velocity above the wall as an idealized sensor signal, the drag reduction rate was about 25% (Fukagata and Kasagi, 2004b), which is comparable to the opposition control. The FIK identity further suggests that a drastic drag reduction can be achieved if the near-wall Reynolds shear-stress is more ideally reduced. When an ideal feedback body force (instead of blowing/suction) was applied to DNS, the near-wall Reynolds shear-stress became negative to yield a friction drag much lower than that of the same Reynolds number laminar flow (Fukagata, Kasagi and Sugiyama, 2005). In that case, however, the actuating power consumption became larger than the power saved by the drag reduction.


313

5. Control Feasibility at High Reynolds Numbers Up to now, various Reynolds number effects in wall turbulence have been reported. Zagarola and Smits (1998) suggest that the overlap region between the inner and outer scalings in wall-bounded turbulence may yield a log law rather than a power law at very high Reynolds numbers. Moser, Kim and Mansour (1999) have made a DNS of fully-developed turbulent channel flows at Reτ = 180 to Reτ = 590, and they conclude that the wall-limiting behavior of rms velocity fluctuations strongly depends on the Reynolds number, but obvious low-Reynolds-number effects are absent at Reτ = 395. It is well known that the near-wall streamwise vortices play an important role in the transport mechanism in wall turbulence, at least, at low Reynolds number flows (Robinson, 1991; Kravchenko, Choi and Moin, 1993; Kasagi et al., 1995). Those streamwise vortices and streaky structures, which are scaled with the viscous wall units (Kline et al., 1967), are closely associated with the regeneration mechanism (Hamilton, Kim and Waleffe, 1995). The Reynolds number assumed in most previous studies on active feedback control of wall-turbulence remains at Reτ = 100 to Reτ = 180, where significant low-Reynolds-number effects must exist. Iwamoto et al. (2002) showed in their DNS at Reτ < 642 that the effect of the suboptimal control (Lee et al., 1998) gradually deteriorated as the Reynolds number increased. In real applications, the Reynolds number is far beyond the values that DNS can handle. For a Boeing 747 aircraft, for example, the friction Reynolds number is roughly estimated to be Reτ ∼ 105 under a typical cruising condition. For such high Reynolds number flows, where highly complex turbulent structures exist with a very wide range of turbulent spectra, no quantitative knowledge is available for predicting the effectiveness of active feedback control. Figure 12 shows the profiles of weighted Reynolds shear-stress in uncontrolled flow at different Reynolds numbers, which are calculated by using a simple mixing length model with the van Driest damping function. At higher Reynolds numbers, the contribution of near-wall Reynolds shearstress to the friction drag drastically decreases and the far-wall contribution becomes dominant. However, as mentioned above, the Reynolds shearstress far from the wall is indirectly reduced by near-wall manipulation (Fukagata and Kasagi, 2003). Then, the question is whether the near-wall

314


0.8 0.6

0 100

101

102

103

00 1000 Re τ=

0.2

0 1000 Re τ= 1000 Re τ=

0.4

180 Re τ=

(1−y+/Reτ)(−u+v+)

1

104

105

y+ Fig. 12.

Weighted Reynolds shear-stress at different Reynolds numbers.

flow manipulation is effective to friction drag reduction even in practical applications at high Reynolds numbers. Very recently, we theoretically investigated the Reynolds number effect on the drag reduction rate achieved by an idealized near-wall layer manipulation (Iwamoto, Fukagata, Kasagi and Suzuki, 2005). An assumption is made that all velocity fluctuations in the near-wall layer of 0 < y < yd are perfectly damped. We also assume a fully-developed turbulent channel flow under a constant flow rate, and derived a theoretical relationship among the Reynolds number of the uncontrolled flow Reτ , the dimensionless damping layer thickness yd /δ and the drag reduction rate RD . It is given as

1 yd2 yd 1 yd ln Reτ + F = + (1 − RD ) Reτ 1− κ δ δ 3 δ2 2

3 3 1 yd 2 yd 2 1 + 1− ln 1− (1 − RD ) 2 δ κ δ 3 1

×(1 − RD ) 2 Reτ

+F .

(5.1)

The sole empirical formula used in the derivation above is the Dean’s formula (Dean, 1978) on the bulk mean velocity (the logarithmic law version), which reads 1 Ub = ln Reτ + F. (5.2) uτ κ Figure 13(a) shows the dependency of RD on Reτ for constant values of yd . As Reτ increases, RD decreases. The Reynolds number dependency of RD , however, is very mild. For yd+ = 10, for instance, the drag reduction rate RD is about 43% at Reτ = 103 , and about 35% even at Reτ = 105 .

315


(a) 0.8

RD

0.6 0.4 +

+

yd = 10

yd = 20

+

0.2

+

yd = 30

yd = 40

+

+

yd = 50

0.0 3

4 5 6

2

10 (b)

yd = 60 3 4 56

3

Reτ

2

10

10

100 6 4

RD = 0.1 RD = 0.4

3 4 5 6

4

5

+

RD = 0.2 RD = 0.6

yd ~ ln Reτ

yd

+

2

10 6 4 2

13

4 5 6

2

10

3 4 56

3

Reτ

2

10

4

3 4 5 6

10

5

Fig. 13. Reynolds number dependency of idealized near-wall manipulation. (a) Dependency of the drag reduction rate RD on the Reynolds number Reτ with the constant thickness of damping layer yd ; (b) Thickness of the damping layer yd required to achieve the same drag reduction rate RD . Reprinted from Iwamoto et al., Phys. Fluids 17 (2005) 011702.

The damping layer in the latter case is extremely thin as compared to the channel half-width, i.e. yd /δ = 0.01%. The Reynolds number dependency of yd required to achieve the same drag reduction rate RD is shown in Fig. 13(b). As Reτ increases, yd gradually increases. For high Reynolds numbers, where yd /δ ≪ 1 holds, Eq. (5.1) can reduce to yd+ ∼ ln Reτ , and this means the Reynolds number dependency is very weak. The asymptotic relation is in good agreement with Eq. (5.1) when Reτ > 4 × 103 , as shown in Fig. 13(b). Thus, large drag reduction can be obtained even at high Reynolds numbers if we can control and completely damp the near-wall velocity fluctuations. Figure 14 shows the flow field computed in the corresponding DNS. The friction Reynolds number is about 650 and the damped layer thickness is

316


(a)

(b)

Fig. 14. Cross-sectional view of an instantaneous streamwise velocity in channel flow at Reτ = 650. (a) Uncontrolled; (b) With damping in the near-wall layer. Reprinted from Iwamoto et al., Phys. Fluids 17 (2005) 011702.

yd+ = 60. The turbulence is drastically suppressed in the damping layer, and found considerably suppressed also in the undamped region. The change in the Reynolds shear-stress gives a clue to explain the large drag reduction through the FIK identity. As shown in Fig. 15, the drag reduction rate directly caused by the decrease of the Reynolds shear-stress in the damped layer is 18%, while that due to the accompanied decrease of the Reynolds shear-stress in the undamped region is 56%. For higher Reynolds numbers, the relative thickness of the damping layer yd /δ becomes negligibly small, so that the contribution away from the damped layers should be dominant. Thus, possible large drag reduction at high Reynolds numbers should be mainly attributed to the decrease of the Reynolds stress in the region away from the wall.

317

Reynolds stress, w/o damping Reynolds stress, with damping Weighted Reynolds stress, w/o damping Weighted Reynolds stress, with damping

2.5 2.0 1.5 1.0

+

−u' v'

+

,

+

+

3(1−y/δ ) (−u' v' )


0.5 0.0 0.0

0.2

0.4

0.6

0.8

1.0

y /δ Fig. 15. Reynolds shear-stress in channel flow at Reτ = 650 with damping in the nearwall layer. Reprinted from Iwamoto et al., Phys. Fluids 17 (2005) 011702.

The present theoretical analysis provides a favorable support for the existing control schemes. Namely, attenuation of turbulence in the near-wall layer is still effective at higher Reynolds numbers appearing in real applications.

6. Enhancement of Heat Transfer In many industrial applications, such as heat exchangers and piping systems, it is desirable to keep the skin-friction drag reasonably small. As for the heat transfer, either enhancement or suppression is preferred depending on the function of the equipment. Due to the similarity between momentum and heat transport, simultaneous reduction of skin friction and heat transfer is straightforward. For simultaneous achievement of skin-friction reduction and heat transfer enhancement, however, no control strategy has been established. Very recently, we made an attempt for simultaneous, but independent control of skin-friction reduction and heat transfer enhancement in a turbulent channel flow (Fukagata, Iwamoto and Kasagi, 2005), extending the idea of FIK identity. Consider heat transfer in a fully-developed turbulent channel flow between isothermal walls kept at different temperatures T ∗ |y=0 and T ∗ |y=2 . Namely, one of the walls is heated and the other is cooled, and the mean temperature profile is anti-symmetric with respect to the center plane. The temperature difference ∆T ∗ is defined as ∆T ∗ = T ∗ |y=1 − T ∗ |y=0 = T ∗ |y=2 − T ∗ |y=1 ,

(6.1)

318


and by using this given temperature difference, a dimensionless temperature θ is defined as θ(y) =

T ∗ (y) − T ∗ |y=1 ∆T ∗

.

(6.2)

The transport equation for the dimensionless mean temperature reads 0=−

d v ′ θ′ 1 d2 θ¯ + , dy Reb P r dy 2

(6.3)

where the Prandtl number is defined as Pr = ν ∗ /α∗ and α∗ is the thermal diffusivity. The wall boundary conditions are given by θ¯y=0 = −1,

The Nusselt number is defined as

θ¯y=2 = 1.

Dh∗ q ∗w 4δ ∗ dT¯∗ dθ¯ Nu = = , =4 ∆T ∗ λ∗ ∆T ∗ dy ∗ w dy w

(6.4)

(6.5)

where λ∗ is the heat conduction coefficient, Dh∗ is the hydraulic diameter and the subscript of w denotes the quantity on the wall. In this problem, ∗ is determined as a result of turbulent heat the amount of wall heat flux qw transport. Due to the anti-symmetric mean temperature profile, the mean heat flux is the same on the both walls, i.e. dθ¯ dθ¯ =4 . Nu = 4 dy y=0 dy y=2

(6.6)

By integrating Eq. (6.3) once from 0 to y, we obtain Nu 1 dθ¯ = −v ′ θ′ + , Reb P r Reb P r dy

(6.7)

where v ′ θ′ y=0 = 0 and Eq. (6.6) were used. This integration is equivalent to the heat flux balance. Equation (6.7) is integrated again from 0 to 1, and then the following equation is obtained: Nu = Reb P r

0

1

−v ′ θ′ dy +

1 . Reb P r

(6.8)


319

Here, the boundary condition of θ¯y=0 = −1 and the anti-symmetry con dition θ¯y=1 = 0 were used. Hence, 1 −v ′ θ′ dy. (6.9) N u = 4 + 4Reb P r 0

The derived relationship suggests that the Nusselt number can be decomposed into two parts. The first term on the right-hand side is the laminar contribution, which is heat conduction, and the second term is the turbulent contribution. The latter is a simple integral of the turbulent heat flux, which is in contrast to the FIK identity for the skin friction. The difference in the weighting factors for the Reynolds shear-stress and turbulent heat flux in the turbulent contribution terms, i.e. (1 − y) and 1, suggests that simultaneous achievement of skin-friction drag reduction and heat transfer augmentation is made possible by suppressing turbulence near the wall and enhancing turbulence in the central region of the channel. Note that this difference in the weighting factors originated from the dissimilarity in the boundary conditions for momentum and heat. The control strategy presented here is less effective, e.g., in the case of isoflux walls (Fukagata et al., 2005). The proposed strategy is examined by means of DNS of channel flow at Reb = 3220 (i.e. Reτ = 110 in uncontrolled flow) and Pr = 0.71. The opposition control scheme (Choi et al., 1994) is adopted for the suppression of near-wall Reynolds stress. In addition, a virtual body force, i.e. −βf (y)θ′ , is added to the wall-normal momentum equation in order to enhance the turbulent heat flux in the central region of the channel. Here, β is an amplitude coefficient and f (y) is an envelope function, which has a value of unity in the central region away from the wall and zero near the wall: f (y) = 1 for 0.5 < y < 1.5 and f (y) = 0 otherwise. Figure 16 show the time traces of the skin-friction coefficient Cf and the Nusselt number Nu in three cases: (1) without control, (2) with opposition control (denoted as v-control), and (3) with v-control and body force. With v-control only, both Cf and Nu decrease. With v-control and body force, Nu increases whilst Cf is kept at the original level. Namely, the heat transfer performance is improved by 50% with the present control. The profiles of Reynolds shear-stress and turbulent heat flux exhibit the expected changes, as shown in Fig. 17. The Reynolds shear-stress profile is roughly similar to that in the uncontrolled case. This is consistent with the observation that Cf takes approximately the same value as that in the uncontrolled case. The turbulent heat flux is suppressed near the wall, whereas largely enhanced

320


(a)

−3

10.0x10

9.5

Cf

9.0 8.5

w/o control w/ v-control

w/ v-control and body force

8.0 7.5 7.0 0

200

400

600

t (b)

800

1000

1200

+

24 22

Nu

20 w/o control w/ v-control

18

w/ v-control and body force

16 14 12 0

200

400

600

t

800

1000

1200

+

Fig. 16. Time traces of skin friction and heat transfer (Fukagata et al., 2005). (a) Skinfriction coefficient; (b) Nusselt number.

in the central region of the channel. This leads to the enhancement of heat transfer as indicated by Eq. (6.9).

7. Concluding Remarks The FIK identity (Fukagata et al., 2002) mathematically shows decomposed dynamical contributions to the turbulent skin friction. Its usefulness is demonstrated through example analyses of controlled wall-bounded flows. It is reconfirmed that, for friction drag reduction control, suppression of the Reynolds stress near the wall should be of primary importance. Based on this recognition, effective control schemes can be proposed. As an example, we introduced the suboptimal control with an alternative cost functional, which incorporates the near-wall Reynolds shear-stress distribution.

321


(a)

0.8 w/o control w/ v-control w/ v-control and body force

+

− u' v'

+

0.6 0.4 0.2 0.0 0.0

0.2

0.4

0.6

0.8

1.0

y /δ (b)

1.6

+

− v' θ'

+

1.2 0.8 w/o control w/ v-control w/ v-control and body force

0.4 0.0 0.0

0.2

0.4

0.6

0.8

1.0

y /δ Fig. 17. Effects of control on turbulent transport (Fukagata et al., 2005). (a) Reynolds shear-stress; (b) Turbulent heat flux.

For flows of complex fluids, such as polymer/surfactant solutions, the FIK identity gives a clue to understand the drag-reducing mechanism. We also discussed the Reynolds number effect on the drag reduction control through the near-wall flow manipulation. The formula that we have derived indicates that, under a constant flow rate, considerable drag reduction can be attained even at high Reynolds numbers by only suppressing the turbulence near the wall, viz., without any direct manipulation of large-scale structures arising away from the wall. The analysis using the FIK identity reveals that the contribution from far-wall Reynolds shear-stress is largely reduced by an indirect effect of the near-wall layer manipulation. Therefore, the basic strategy behind the existing control schemes, i.e. attenuation of the near-wall turbulence, is also valid at higher Reynolds numbers appearing in real applications.

322


Finally, we discussed the extension of the FIK identity to a turbulent heat transfer problem. The derived relationship between the Nusselt number and the turbulent heat flux distribution in a turbulent channel flow with isothermal walls kept at different temperatures suggests that the nearwall turbulence should be suppressed, while far-wall turbulence should be enhanced, in order to achieve simultaneously skin-friction reduction and heat transfer enhancement. Validity of this control strategy was confirmed by DNS with an idealized feedback control. Acknowledgments We thank Dr. Y. Suzuki and former students at the University of Tokyo, who have been involved in this project, particularly, Drs. K. Iwamoto and T. Yoshino, for their creative and fine job. This work was supported through the Project for Organized Research Combination System by the Ministry of Education, Culture, Sports and Technology of Japan (MEXT). References 1. T. R. Bewley, Flow control: New challenges for a new Renaissance, Prog. Aerospace Sci. 37 (2001) 21–58. 2. T. R. Bewley, P. Moin and R. Temam, DNS-based predictive control of turbulence: An optimal benchmark for feedback algorithms, J. Fluid Mech. 447 (2001) 179–225. 3. H. Choi, P. Moin and J. Kim, Active turbulence control for drag reduction in wall-bounded flows, J. Fluid Mech. 262 (1994) 75–110. 4. R. B. Dean, Reynolds number dependence of skin friction and other bulk flow variables in two-dimensional rectangular duct flow, Trans. ASME J. Fluids Eng. 100 (1978) 215–223. 5. T. Endo, N. Kasagi and Y. Suzuki, Feedback control of wall turbulence with wall deformation, Int. J. Heat Fluid Flow 21 (2000) 568–575. 6. K. Fukagata and N. Kasagi, Highly energy-conservative finite difference method for the cylindrical coordinate system, J. Comput. Phys. 181 (2002) 478–498. 7. K. Fukagata and N. Kasagi, Drag reduction in turbulent pipe flow with feedback control applied partially to wall, Int. J. Heat Fluid Flow 24 (2003) 480–490. 8. K. Fukagata and N. Kasagi, Suboptimal control for drag reduction via suppression of near-wall Reynolds shear-stress, Int. J. Heat Fluid Flow 25 (2004a) 341–350. 9. K. Fukagata and N. Kasagi, Feedback control of near-wall Reynolds shearstress in wall turbulence, Proc. 4th Int. Symp. Advanced Fluid Information and Transdisciplinary Fluid Integration, Sendai, 2004 Japan (November 2004), pp. 346–351.


323

10. K. Fukagata, K. Iwamoto and N. Kasagi, Contribution of Reynolds stress distribution to the skin friction in wall-bounded flows, Phys. Fluids 14 (2002) L73–L76. 11. K. Fukagata, K. Iwamoto and N. Kasagi, Novel turbulence control strategy for simultaneously achieving friction drag reduction and heat transfer augmentation, Proc. 4th Symp. Turbulence and Shear Flow Phenomena, Virginia, USA (June 2005), pp. 307–312. 12. K. Fukagata, N. Kasagi and K. Sugiyama, Feedback control achieving sublaminar friction drag, Proc. 6th Symp. Smart Control of Turbulence, Tokyo (March 2005), pp. 143–148. 13. M. Gad-el-Hak, Modern developments in flow control, Appl. Mech. Rev. 49 (1996) 365–379. 14. M. Gad-el-Hak, The MEMS Handbook (CRC Press, 2002). 15. J. M. Hamilton, J. Kim and F. Waleffe, Regeneration mechanisms of nearwall turbulence structures, J. Fluid Mech. 287 (1995) 317–348. 16. C.-M. Ho and Y.-C. Tai, Review: MEMS and its applications for flow control, Trans. ASME J. Fluids Eng. 118 (1996) 437–447. 17. L. Howarth, On the solution of the laminar boundary layer equations, Proc. Roy. Soc. London Ser. A164 (1938) 547–579. 18. K. Iwamoto, K. Fukagata, N. Kasagi and Y. Suzuki, Friction drag reduction achievable by near-wall turbulence manipulation at high Reynolds number, Phys. Fluids 17 (2005) 011702. 19. K. Iwamoto, Y. Suzuki and N. Kasagi, Reynolds number effect on wall turbulence: Toward effective feedback control, Int. J. Heat Fluid Flow 23 (2002) 678–689. 20. N. Kasagi, Progress in direct numerical simulation of turbulent transport and its control, Int. J. Heat Fluid Flow 19 (1998) 125–134. 21. N. Kasagi, Y. Sumitani, Y. Suzuki and O. Iida, Kinematics of the quasicoherent vortical structure in near-wall turbulence, Int. J. Heat Fluid Flow 16 (1995) 2–10. 22. N. Kasagi, Y. Suzuki and K. Fukagata, Control of turbulence, Parity 18(2) (2003) 20–26 (in Japanese). 23. J. Kim, Control of turbulent boundary layers, Phys. Fluids 15 (2003) 1093–1105. 24. S. J. Kline, W. C. Reynolds, F. A. Schraub and P. W. Runstadler, The structure of turbulent boundary layers, J. Fluid Mech. 30 (1967) 741–773. 25. P. Koumoutsakos, Vorticity flux control for a turbulent channel flow, Phys. Fluids 11 (1999) 248–250. 26. A. G. Kravchenko, H. Choi and P. Moin, On the relation of near-wall streamwise vortices to wall skin friction in turbulent boundary layers, Phys. Fluids A5 (1993) 3307–3309. 27. C. Lee, J. Kim, D. Babcock and R. Goodman, Application of neural networks to turbulence control for drag reduction, Phys. Fluids 9 (1997) 1740–1747. 28. C. Lee, J. Kim and H. Choi, Suboptimal control of turbulent channel flow for drag reduction, J. Fluid Mech. 358 (1998) 245–258.

324


29. K. H. Lee, L. Cortelezzi, J. Kim and J. Speyer, Application of reducedorder controller to turbulent flows for drag reduction, Phys. Fluids 13 (2001) 1321–1330. 30. F.-C. Li, Y. Kawaguchi and K. Hishida, Investigation on the characteristics and turbulent transport for momentum and heat in a drag-reducing surfactant solution flow, Phys. Fluids 16 (2004) 3281–3295. 31. P. Moin and T. Bewley, Feedback control of turbulence, Appl. Mech. Rev. 47 (1994) S3–S13. 32. K. Morimoto, K. Iwamoto, Y. Suzuki and N. Kasagi, Genetic algorithm-based optimization of feedback control scheme for wall turbulence, Proc. 3rd Symp. Smart Control of Turbulence, Tokyo (March 2002), pp. 107–113. 33. R. D. Moser, J. Kim and N. N. Mansour, Direct numerical simulation of turbulent channel flow up to Reτ = 590, Phys. Fluids 11 (1999) 943–945. 34. Y. Murai, Y. Oishi, T. Sasaki, F. Yamamoto and Y. Kodama, Turbulent shear-stress profile in a horizontal bubbly channel flow, Proc. 6th Symp. Smart Control of Turbulence, Tokyo (March 2005), pp. 289–295. 35. R. Rathnasingham and K. Breuer, Active control of turbulent boundary layers, J. Fluid Mech. 495 (2003) 209–233. 36. S. K. Robinson, Coherent motions in the turbulent boundary layer, Annu. Rev. Fluid Mech. 23 (1991) 601–639. 37. Y. Sumitani and N. Kasagi, Direct numerical simulation of turbulent transport with uniform wall injection and suction, AIAA J. 33 (1995) 1220–1228. 38. Y. Suzuki, T. Yoshino, T. Yamagami and N. Kasagi, Drag reduction in a turbulent channel flow by using a GA-based feedback control system, Proc. 6th Symp. Smart Control of Turbulence, Tokyo (March 2005), pp. 31–40. 39. C. M. White, V. S. R. Somandepalli, Y. Dubief and M. G. Mungal, Dynamic contributions to the skin friction in polymer drag-reduced wall-bounded turbulence, Phys. Fluids (2005), submitted. 40. T. Yamagami, Y. Suzuki and N. Kasagi, Development of feedback control system of wall turbulence using MEMS devices, Proc. 6th Symp. Smart Control of Turbulence, Tokyo (March 2005), pp. 135–141. 41. T. Yoshino, Y. Suzuki and N. Kasagi, Evaluation of GA-based feedback control system for drag reduction in wall turbulence, Proc. 3rd Int. Symp. Turbulence and Shear Flow Phenomena, Sendai, Japan (June 2003), pp. 179–184. 42. B. Yu, F. Li and Y. Kawaguchi, Numerical and experimental investigation of turbulent characteristics in a drag-reducing flow with surfactant additives, Int. J. Heat Fluid Flow 25 (2004) 961–974. 43. M. V. Zagarola and A. J. Smits, Mean flow scaling of turbulent pipe flow, J. Fluid Mech. 373 (1998) 33–79. 44. C.-X. Xu, J.-I. Choi and H. J. Sung, Suboptimal control for drag reduction in turbulent pipe flow, Fluid Dyn. Res. 30 (2002) 217–231.

CONTROL OF TURBULENT FLOWS USING LORENTZ FORCE ACTUATION

Kenneth S. Breuer Division of Engineering, Brown University Providence, RI 02912, USA E-mail: [email protected] We discuss the use of electro-hydrodynamic “Lorentz force” actuators to affect the near-wall flow of a low Reynolds number of a fully turbulent channel flow. The actuators induce fluid motion due to the interaction between a magnetic field and a current density. The force generates spanwise velocity profiles with a penetration depth into the flow of a few millimeters and maximum velocities of approximately 4 cm/s. Although the actuators are effective in generating disturbances in the flow, their efficiency is poor, making their practical use in engineering systems doubtful. The actuators are used in a fully turbulent low Reynolds number channel flow and their effect on the structure of the turbulent flow is measured using both a direct measurement of turbulent drag and Particle Image Velocimetry. Drag is shown to be reduced by approximately 10% at an optimal condition. Similarly, PIV measurements of the mean and fluctuating velocity profiles indicate that certain amplitude and frequency combinations are effective in suppressing the turbulent fluctuations and Reynolds stresses. At these conditions, a locally-accelerated velocity profile is generated. Two-point velocity correlations indicate that the effect of forcing is to reduce the streamwise scale of the nearwall coherent structures and to sharply reduce the frequency of highamplitude turbulence-producing “bursts”. Contents 1 2 3

Introduction 1.1 Overview of chapter Scaling of Oscillatory Flow Control Lorentz Force Actuators 3.1 Actuator performance 3.2 Flows induced by Lorentz forces 325

326 327 328 330 333 336

326

4

K. S. Breuer

Turbulent Flow Control 4.1 Flow facility and measurement techniques 4.2 Drag reduction measurements 4.3 Fluctuation statistics in the controlled flow 4.4 Integrated flow rate and production 4.5 Two-point correlations 4.6 Production probability 5 Conclusions References

339 339 341 342 345 348 353 354 355

1. Introduction Interest in small-scale turbulence control has grown over the past two decades thanks to predictions from numerical computations at low Reynolds number flow (Bewley 2001; Choi et al., 1994; Jung et al., 1992) which indicate that the turbulence and drag can be reduced by a variety of fluidic actuation. The most sophisticated of these control schemes uses distributed sensing and actuation at the wall (Choi et al., 1994; Bewley et al., 2001; Lee et al., 1997) and report reductions in turbulent shear stress of as much as 30%. Although the insight gained from these approaches is valuable, physical implementation is not feasible due to unrealistic density of sensors and actuators. Some progress in resolving these issues has been achieved, including numerical simulations of turbulent channel flow using discrete synthetic jet actuators responding to shear stress inputs (Lee and Goldstein, 2001) as well as the successful experimental demonstration of feedforward control over a very small patch of turbulent flow (Rathnasingham and Breuer, 1997; 2003). Nevertheless, simpler control approaches that do not require massive arrays of sensors and actuators are desirable. One of the simplest approach that yields significant reductions in turbulent skin friction is the spanwise oscillation of the wall in which the entire surface is moved in a sinusoidal manner with a characteristic frequency and amplitude. This was first proposed by Jung et al. (1992) in a numerical simulation and has been confirmed experimentally in a boundary-layer flow by Choi et al. (1998) and more recently, by Di Cicca et al. (2002). However, despite its simplicity, the difficulty of physically moving the entire wall has prohibited any widespread implementation of this control approach. Realization of effective turbulence control using spanwise oscillation was revived with the idea that physical motion of the wall could be replaced by a body force acting on the fluid, and that the body force could be generated

Control of Turbulent Flows

327

in a conducting fluid using the Lorentz force, which is generated from the interaction between a magnetic field, B and a current density, J: FL = J × B.

(1.1)

Numerical computations of Berger et al. (2000) showed that turbulent skin friction in a low Reynolds number channel flow could be substantially reduced using the Lorentz force generated by a series of interlaced magnets and electrodes. Similar computations by Du and Karniadakis (2002) confirmed these results, although their forcing was achieved using a generic body force similar to, but not identical to, one that could be generated by a realistic arrangement of magnets and electrodes. Du and Karniadakis (2002) also proposed the use of a traveling wave rather than a static oscillating force as an alternative means to achieve efficient flow control. Experimental demonstrations of the use of the Lorentz force for turbulent flow control has a checkered past with somewhat inconclusive and controversial results. Nosenchuck and Brown (1993) claimed significant drag reduction using a configuration of magnets and electrodes designed to generate vertical wall motion (similar in spirit to the opposition control of Choi et al. (1994)). However, those results were not widely accepted and have not been reproduced either experimentally or using numerical simulations. More recently, the spanwise oscillation of the flow using the Lorentz force has been demonstrated by Breuer et al. (2004), who presented extensive results on the performance of the actuator and confirmed effective drag reduction using a direct measurement of the turbulent skin friction. Pang and Choi (2004) also demonstrated control in a turbulent boundary layer, inferring drag reduction from measurements of the local boundary layer profile. 1.1. Overview of chapter This chapter reviews results obtained in our lab at Brown University on both drag reduction and modifications to the structure of the turbulent flow achieved using Lorentz force actuation. The results concerning the actuator performance and drag reduction have been published (Breuer et al., 2004), while the more recent results concerning the structure of turbulent flow are part of a second manuscript, currently in preparation for publication in the archival literature. The organization of the chapter is as follows: in the following section, we present some basic scaling arguments, followed by some discussion of the design and performance of the Lorentz force actuators used in our studies. Results concerning turbulent flow control are

328

K. S. Breuer

presented, focussing first on drag reduction using Lorentz force actuation, followed by extensive PIV measurements of the structure of the turbulent flow structure subject to forcing by the Lorentz force actuators. 2. Scaling of Oscillatory Flow Control Before discussing the experimental setup and results, it is useful to review some fundamental scaling which will help to interpret the results. Here we review a intuitive scaling argument based on the underlying physical principles and a fundamental assumption that the control of a wall-bounded turbulent flow will be achieved when the Lorentz forcing is sufficiently strong to induce a spanwise velocity that is large enough to disrupt the near-wall turbulent dynamics. We take as an axiom that classical turbulent wall scaling applies andthus, the only relevant scaling parameters are the friction velocity, uτ = τw /ρ, the density, ρ, and the kinematic viscosity, ν. We consider a single electrode/magnet-pair with characteristic pitch, a. The volume of fluid affected by this actuator is approximately a2 w (w is the cross-stream width of the actuator). Note that here, and throughout this analysis, we ignore constant factors of order one. The electrical resistance, Ω of this fluid scales approximately with the distance between electrode pairs and inversely proportional to its cross-sectional area and the fluid conductivity, σ: 1 . (2.1) wσ The Lorentz force, FL , generated by a current, I, passed through this volume of fluid was defined earlier (Eq. (1.1)), and in order for the control to be effective, this force must be strong enough to induce the fluid to move with a spanwise velocity of approximately uτ , oscillating back and forth with a period T . Thus the required force on the fluid, FF , is: Ω=

FF ≈

ρa2 wuτ (mass)(velocity) = . (period) T

(2.2)

Equating the Lorentz force generated, FL , to the fluid force required for control, FF , we can solve for the current: ρawuτ , (2.3) I= BT and a corresponding current density of J=

I ρuτ = . aw BT

(2.4)


329

The ratio of the Lorentz inertia, JBa, to the fluid inertia, ρu2τ is the definition of the Stuart number, and thus we obtain an expression for the Stuart number at which we expect Lorentz force control to be effective: St =

a JBa = . ρu2τ uτ T

(2.5)

Note that this definition differs slightly from that used in numerical simulations (Berger et al., 2000) which uses the channel half-height, h, as the length scale. The definition here is more closely related to the underlying physics of the problem, and is somewhat more appealing since the length scale used is related to the local generation of the Lorentz force, not a somewhat arbitrary global scale. If we define a and T in terms of turbulent wall units: a = a+ ν/uτ ; T = T + ν/u2τ , then the definition of the optimal Stuart number is simply: St =

a+ . T+

(2.6)

This result is similar to that derived by Berger et al. (2000), but in their case, it was derived from physical arguments rather than the governing equations of motion. Although this predicts the functional relationship between parameters, the range over which one can freely vary the parameters is clearly not infinite. We know from both basic physical intuition as well as previous numerical simulations on wall-bounded turbulence control (Choi et al., 1994; Berger et al., 2000; Du and Karniadakis, 2002) that the region where control is most effective is the near-wall region extending approximately 40 l∗ away from the wall (l∗ = ν/uτ ). We also expect the optimal Stuart number, as defined, to be of order one — representing a balance between the induced control velocity and the natural inertia of the near-wall region. Thus, we can predict that the optimal forcing period will be approximately a+ , or T + ≈ 40. Given the simplicity of this scaling argument, this prediction is surprisingly in good agreement with the value of T + ≈ 100 found in the DNS studies of spanwise oscillation flow control (Jung et al., 1992; Berger et al., 2000; Du and Karniadakis, 2002) and, as will be shown in the next section, the present results. Lastly, we can extend this scaling argument slightly further in service of a physical experiment in which the electrode spacing, a, has a fixed size but is nevertheless used in flows with different Reynolds numbers. Thus, its “apparent” size, a+ , increases as the Reynolds number of the flow increases.

330

K. S. Breuer

It is trivial to rewrite the definition of the Stuart number in Eq. (2.6) as a Re∗ , (2.7) h T+ where h is the channel half-height and Re∗ is the turbulent Reynolds number, Re∗ = uτ h/ν. Thus we see that, for a fixed value of a, we predict that the optimal value of the period will also rise linearly with the Reynolds number and that for a fixed non-dimensional period, the optimal Stuart number will also scale with Re∗ . These predictions will be confirmed (at least for two values of Re∗ ) by the experimental data presented below. From this analysis, we can also predict the control efficiency. The power (per unit area) required to operate the control system is St =

ρ2 au2 I 2Ω = 2 2τ . (2.8) wa B T σ The power saved will be some fraction, K, of the total turbulent drag: Pcontrol =

Psaved = Kρu2τ U

(2.9)

where U is the freestream velocity. Thus, the total system efficiency is given by

+2 KB 2 σ KB 2 σ T 2 U U 1 Psaved T = = . (2.10) η= + Pcontrol ρ a µ a uτ u2τ Since U/uτ is more or less constant (varying only by a factor of two over a very wide range of Reynolds numbers), this result indicates that not only is the efficiency poor, but that at higher speeds, it only gets worse. The main culprit is the fact that while the actuation requirements increase (the necessary actuation velocity increases with uτ ), the magnet strength, B, is fixed so that all of the extra actuation authority must come from the current density. The poor conductivity of saltwater makes this current very expensive to use. 3. Lorentz Force Actuators The actuator used in our laboratory is based on a Lorentz force actuator developed by Henoch and Stace (1995) which used interlaced magnets and electrodes to generate an approximately uniform Lorentz force. In the case of Henoch and Stace, the force was aligned with the streamwise direction, while the current experiment rotates the actuator system by ninety degrees so that the Lorentz force acts in the spanwise direction. The present system, illustrated in Fig. 1(a), consists of linear arrays of permanent magnets lined

331


y

FLOW Electric field lines

x

Permanent Magnets

Magnetic field lines

S

N

S

N

N

S

N

S

Back Iron Electrodes

x z

FLOW

S

N

S

N

Flow direction

Lorentz force directon

Magnets

Back iron

Electrode pair

Alignment pins (nylon)

Fig. 1. (a) Schematic; (b) Composite photograph of the Lorentz force actuator. From Breuer et al. (2004).

332

K. S. Breuer

up in the spanwise direction, z and aligned such that the surface exposed to the flow has alternating polarity: N − S − N − S, etc. Between each of these magnet rows are a series of electrodes, also energized with alternating polarity: −, +, −, +, etc. The bulk of the electric and magnetic field lines are aligned in the streamwise direction, and the two are always crossing each other such that the Lorentz force acts in the direction spanwise to the core flow. The direction and magnitude of the Lorentz force is controlled by varying the electric field strength and polarity. The control of the spanwise structure of the Lorentz force is achieved by fabricating the actuator with multiple phases in the spanwise direction. For the present discussion, we restrict ourselves to sinusoidal forcing voltages and, in a further restriction for simplicity’s sake, we have linked all the electrodes in a single streamwise column together so that there is no streamwise variation in the generated Lorentz force. With this subset of available forcing signals, the Lorentz force generated in the fluid is F (x, y, z, t) = f (x, y) sin(ωt − λz),

(3.1)

where ω is the forcing frequency and λ is the spanwise wavelength. The phase speed of the disturbance is given by c = ω/λ. The “structure” function, f (x, y), is determined solely by the geometry of the electric and magnetic fields. Actuating all of the spanwise phases in unison (i.e. with zero phase shift) generates a uniform Lorentz force, equivalent to the uniform spanwise oscillation simulated by Berger et al. (2000). Alternatively, operating the electrodes with the same amplitude and time-signal, but with a spanwise phase shift generates a traveling wave, as simulated by Du and Karniadakis (2000; 2002). For the current experiments, the actuator tiles were fabricated with 16 independent phases in the spanwise direction. Fabrication of the actuators was a complex multi-step process. The base of the actuators is a 305 × 305 × 0.0032 mm low-carbon steel plate. Plastic alignment pins were inserted into the plate to secure the magnet array. Neodymium–Boron–Iron (45 M–O) permanent magnets were dual coated with nickel and epoxy to help prevent degradation in the harsh saltwater environment. The magnets have a 3.2 mm square cross-section and a length of 50.8 mm. Longer lengths would have made assembly much easier; however, the strength of the magnetic field as well as the brittleness of the magnets were limiting factors. A composite photograph containing both the magnet-pin structure as well as the electrode layer is shown in


333

Fig. 1(b). The electrode layer consists of a commercially fabricated twolayer printed circuit board. The electrodes protrude slightly above the surface, but are hydrodynamically smooth (step less than 1 l∗ ). To reduce the parasitic Lorentz force between adjacent spanwise electrodes, each electrode is designed with a “cut back” between spanwise columns. The surface of the circuit board was screen printed with a polymer ink to reduce the electrolytic effects during actuation and to prevent corrosion. This treatment has proved invaluable and our experiments show no signs of bubble formation or corrosion during months of testing. Electrical connections are made using through-plated vias onto which wires are soldered. The sides and back of the actuator plate are encased in an epoxy which protects the system from corrosion during use.

3.1. Actuator performance The amplitude and frequency response of the polymer-coated electrodes are shown in Figs. 2 and 3. Here, the currents are normalized by the driving voltage and are phase-aligned with the drive voltage. These responses can be summarized by the normalized power averaged over the cycle period, T : P 2 Vpeak

=

1 2 T Vpeak

T

V (t)I(t) dt.

(3.2)

0

At low frequencies, the sinusoidal signal shows marked attenuation due (presumably) to the onset of DC effect such as electrolysis and bubble generation. As the frequency rises, however, the current response becomes more sinusoidal and the delivered power rises to a plateau. Note that there is no degradation at high frequencies, and that there is almost no phase lag between the driving signal and the resultant current. This confirms that the saltwater-electrode system is an almost pure resistive load with little energy storage. The amplitude response shows a slightly different characteristic. At low voltages, the electrode potential of the system is not reached and current flow is minimal. Note that this is confirmed at a driving voltage of 0.5 Vpp which shows a nonlinear increase in current at the peak of the driving signal. For 1 Vpp and above, the signal is close to sinusoidal, and does not change with driving voltage, confirming that the system has reached a regime characterized by a linear resistive response. Electrochemistry at the electrodes is a necessary by-product of this class of actuator, usually generating hydrogen and chlorine at the two electrodes.

334

K. S. Breuer

Current/peak voltage [1/Ohms]

0.5 0.2 V 0.5 V 1.0 V 5.0 V

0

-0.5

0

0.2

0.4

0.6

0.8

1

Time (normalized by period)

Normalized Power [W/V2]

1

0.1

0.01 0.1

1

10

Amplitude [V]

Fig. 2. (a) The normalized current (I/Vpeak ) measured due to sinusoidal forcing at different forcing amplitudes. The traces are all synchronized with the forcing voltage so that t/T = 0 always coincides with the start of the forcing cycle; (b) The integrated 2 ). From Breuer et al. (2004). power (normalized by Vpeak

This has been a pervasive problem with the Lorentz force actuators in previous experiments resulting in both bubble generation and electrode corrosion. The gold-plated electrodes tested here were no exception and it was observed that bubble formation started immediately after the voltage was applied. Although the bubble formation is problematic, more serious is the accompanying corrosion of the electrodes and it was found that even after

335


Current [A]

0.5 0.2 Hz 0.3 Hz 0.8 Hz 10.0 Hz 20.0 Hz 100 Hz 0

-0.5

0

0.2

0.4

0.6

0.8

1

Time (normalized by period)

Power [W]

1

0.1

0.01 0.1

1

10

100

Frequency [Hz]

Fig. 3. (a) Measured current flow during a complete forcing cycle between two parallel, polymer-coated electrodes. The different curves on the upper frame show the current measured due to sinusoidal forcing with Vpeak = 1V , and at different frequencies. The traces are synchronized to the forcing voltage so that t/T = 0 always coincides with the start of the forcing cycle; (b) The integrated power as a function of forcing frequency. From Breuer et al. (2004).

a short time, the corrosion was sufficiently serious to render the electrodes useless. In the case of the polymer-coated electrodes, however, no bubble formation was observed during sinusoidal operation above 0.2 Hz. Furthermore, minimal degradation of the electrode performance was observed over

336

K. S. Breuer

months of testing. Although we do not have a definitive explanation for this very encouraging result, we believe, as was commented earlier, that the polymer coat serves to slow down the kinetics of bubble formation and allow the process to remain reversible above a critical frequency so that microbubbles are formed, but then disappear without any net gas bubble generation or electrode degradation. 3.2. Flows induced by Lorentz forces In order to characterize the performance of the actuators in more detail, Particle Image Velocimetry (PIV) was used to measure the flows induced by the Lorentz forces. The actuator tile was placed in the center of a zeromean-flow plexiglas tank, which measured 0.6 m by 0.6 m by 0.2 m and was filled with saltwater (σ = 5 Siemens/m). The actuator was divided into eight independent spanwise phases, driven by the amplified output of a Digital Signal Processing (DSP) board. The velocity fields measured, using PIV, were integrated in the z-direction so that a single velocity profile, w(y), was obtained at phase during the forcing cycle (Fig. 4). The velocity reaches a maximum at approximately 1 mm above the surface and dies out at a distance of about 4 mm. These length scales are solely determined by the streamwise spacing of the magnet-electrode pattern shown in Fig. 1(a) which sets the field penetration depth. These profiles are

Distance from surface [mm]

6

4

2

0 -0.04

-0.02

0

0.02

0.04

Spanwise velocity [m/s] Fig. 4. Velocity profiles due to oscillatory spanwise forcing obtained by integrating all profiles over the spanwise extent of the electrode. 16 profiles are plotted at equally-spaced phases during the forcing cycle. From Breuer et al. (2004).


337

in excellent agreement with the velocity profiles derived by Berger et al. (2000) for the flow induced by a spanwise oscillating body force in a laminar flow. These measurements were repeated over a variety of forcing voltages and frequencies ranging from 0.3 Hz to 2 Hz and from 0.5 V. to 3.0 V. The results are assembled in Figs. 5(a) and 5(b) which show the maximum velocity, Wmax , measured as a function of the forcing voltage and the electricalto-mechanical efficiency of the Lorentz force actuator (plotted against the input electrical power). In the case of the velocities, we see that the induced velocity increases approximately linearly with the forcing voltage, but for a fixed voltage, decreases as the driving frequency increases. The actuator efficiency is defined as the ratio of the output flow power, crudely defined as Pflow ≈

ρ 2 (Area)δp Wmax , 2T

(3.3)

to the input electrical power: 1 V I. 2

(3.4)

Here δp and Wmax are the penetration depth and maximum spanwise velocity, measured from the PIV data. V and I are the peak voltage and current respectively. The factors of 1/2 come from the integration over the sinusoidal forcing cycle. When plotted this way, we see that the actuator is performing with relatively uniform efficiency over the entire operating space, although there is some falloff at low input power, as expected given the low-voltage behavior shown in the previous section (Fig. 2). It is clear that the actuator exhibits very poor electrical-to-mechanical efficiency — less than 10−4 . This is primarily due to two facts, namely the limited field strength available from the permanent magnets (approximately 0.3 Tesla effective field strength on the surface) and the fact that saltwater has a low conductivity (5 Siemens/m). As with any Lorentz force actuation, the force is generated by the current density, and since power varies as I 2 R, a high-resistivity material (such as most fluids of practical interest), will require a substantial power dissipation before any practical fluid motion can be generated. Du and Karniadakis (2000) proposed that a traveling Lorentz force might be as effective for turbulent drag reduction as a stationary oscillatory Lorentz force. The actuators developed here are segmented in the spanwise direction and allow for arbitrary control of the eight spanwise phases. The flow induced by a traveling wave was measured by Breuer

338

K. S. Breuer

Maximum fluid velocity [m/s]

0.07 0.3 Hz 0.5 Hz 0.7 Hz 1.0 Hz 2.0 Hz

0.06 0.05 0.04 0.03 0.02 0.01 0

0

0.5

1

1.5 2 2.5 Actuator voltage [V]

3

3.5

Actuator Efficiency

104

0.3 Hz 0.5 Hz 0.7 Hz 1.0 Hz 2.0 Hz 105

0

5

10 15 20 25 Input Electrical Power [W]

30

Fig. 5. (a) Maximum velocity generated by the Lorentz force actuator, plotted as a function of the forcing voltage. The velocities increase approximately linearly with driving voltage, although some saturation at high voltages is seen; (b) Lorentz force actuator efficiency, measured as the ratio of the total fluid power induced to the electrical power. From Breuer et al. (2004).

et al. (2004) and consists of regions of both positive and negative spanwise velocity, propagated at a proscribed wave speed. For an observer located at a fixed spanwise location, the flow is similar to a uniform stationary oscillation as long as the wavelength over which the wave is imposed, is large and consists of a compressional flow followed by a dilational flow.


339

4. Turbulent Flow Control 4.1. Flow facility and measurement techniques The flow facility (Fig. 6) at Brown University is a constant pressure channel that can support both saltwater and freshwater experiments, with flow rates between Re ∗ = 100 and Re ∗ = 800. The entire channel is constructed of clear acrylic for ease of PIV measurement as well as corrosion resistance and is assembled on an aluminum frame. The working fluid is pumped from the collection tank through a series of four pumps up to the accumulation tank. Water then drains through a contraction into the development section, passing through a honeycomb and over a strip of velcro that forces early transition to turbulence. The development section is 3 m long, 1 m wide and 24 mm high, and ensures fully-developed turbulent flow by the time the fluid enters the measurement section. A small heat exchanger was used in the collection tank to maintain a constant working temperature over the duration of the experiment. The floor of the test section is a removable table that allows installation and adjustment of multiple actuators. Snorkels in the roof of the channel as well as a “fence” along the side of the test section permit the wires to be fed out of the channel to the amplifiers without disturbing the main flow. Portholes in the roof give easy access for calibration and cleaning. Salt was added to a mass concentration of 3.5% giving a conductivity of 4.5 Siemens/m, closely matching that of seawater. Particle Image Velocimetry (PIV) was used to obtain velocity data. A 15 mW Nd:YAG laser was mounted on a traverse above the channel and,

Fig. 6. Schematic of flow facility, showing the back pressure and water supply tanks, development section, test area with actuator and return flow path. The configuration shown is for local measurements of turbulent velocity profiles. However, for drag measurements, the entire lower surface of the test section is comprised of Lorentz force actuators.

340

K. S. Breuer

a cylindrical and spherical lens pair were used to generate a focused laser sheet in the x-y plane. A CCD camera with 1360 by 1024 pixel resolution was mounted on a three-axis micrometer stage on the side of the channel, and a long distance lens (f = 350 mm) was used to acquire image pairs during the laser flashes. The camera was mounted with the long axis aligned with the y-direction (wall-normal) and with these, optics, the field of view was 10 mm by 14 mm. The measurement plane was located approximately 2 cm from the downstream end of the actuator plates, and at the center of the actuator plate in the spanwise z-direction. The water used in the facility was city tap water (with added salt), and was filtered continuously during operation with a 20 micron pore filter mounted on the facility pump. This choice of filter left a healthy distribution of natural seeding particles in the working fluid, and further seeding of the water was found to be unnecessary. During PIV processing, an interrogation area measuring 36 square pixels was typically used to determine the fluid velocity from the particle displacements, yielding 50 by 75 velocity vectors from each image, although smaller areas were used close to the wall where the local velocity was lower. The second image was also shifted with respect to the first to account for the high mean velocity and to improve the PIV vector determination. Time separation between the laser pulses for the lower Reynolds number was 300 µs, compared to 150 µs at the higher velocity flow. 200 sets of image pairs were usually acquired at each test condition. Prior to an experimental session, the facility was run for some time at its operating flow rate to ensure complete thermal equilibration, and to allow for any air bubbles that may have accumulated on the channel walls to be flushed from the channel. After the system had reached a steady state, an initial baseline set of image pairs was taken. These image pairs were immediately processed using the PIV software and examined to ensure that the laser settings and camera position were accurate and that the seeding was adequate. Barring any abnormalities in the flow, a test suite consisting of a variety of current-frequency combinations was then run. In between each actuator operating condition, the channel was given 30 s to allow the new forcing conditions to propagate through the entire length of the test section. Baseline cases with no control were taken throughout the course of the experiment to ensure that there were no abrupt changes in the operating conditions during the testing sequence. During a typical testing sequence 20–30 actuator operating conditions were tested, and the experiment typically lasted several hours.

341


4.2. Drag reduction measurements For drag reduction measurements, the entire test section floor was covered by Lorentz force actuators and the active surface was suspended by thin wires from the roof of the facility. The motion of the plate is measured in both the streamwise and spanwise directions by three optical displacement transducers. Typical displacements due to the mean shear force (O(0.5) N at Re ∗ = 300) were of approximately 300 microns. The system was calibrated using two methods. Firstly, with no flow, known forces were applied to the plate and the corresponding displacement measured. Following this, the plate displacement was measured due to the mean turbulent drag (with no control) at different centerline water speeds; the results compared very well with standard correlations for turbulent skin friction. The drag system was measured to have a sensitivity (to measure local changes in drag) better than 0.01 N (2% of the mean drag force). The effects of Lorentz control on the turbulent shear stress were tested at two Reynolds numbers: Re∗ = 289 and Re ∗ = 418, (centerline velocities of 0.234 m/s and 0.280 m/s respectively). At these conditions, the friction velocity, uτ , was 0.0134 m/s and 0.0194 m/s respectively, and the non-dimensional magnet electrode pitch was 40 l∗ and 58 l∗ respectively. Figure 7 shows the percentage change in the turbulent drag measured using the full-channel drag balance as a function of various combinations of the forcing amplitude and frequency, and Reynolds number. Two sweeps 12.0

Drag Reduction (%)

10.0 Re* = 400

8.0 Re* = 290

6.0 4.0 2.0 0.0 -2.0 -4.0 0.0

0.5

1.0

1.5

T+ St/R* Fig. 7. Percent drag reduction versus similarity parameter, T + St/Re∗ . Data from two different Reynolds numbers are shown and at each Reynolds number, both amplitude and frequency have been varied.

342

K. S. Breuer

were made. In the first, the forcing period was fixed at T + = 100 while the amplitude was varied. In the second, the amplitude was fixed (St = 1.8 at Re ∗ = 289 and St = 1.1 at Re ∗ = 418), while the frequency was changed. The parameters are combined into a single similarity variable, T + St/Re∗ , as suggested by the scaling analysis presented earlier. It is clear that significant drag reduction is achieved over a variety of forcing amplitudes, reaching a maximum of approximately 10% at a value of T + St/Re∗ ≈ 0.5 before beginning to decrease again. The voltages required to generate these Stuart numbers were in the 3 V range, meaning that the spanwise velocities induced by the Lorentz force actuators are about 4 cm/s or approximately 2uτ . This value is slightly smaller than the optimal amplitude found by Berger et al. (2000) “idealized” forcing, but very similar to the optimal amplitude for their “realistic” forcing case. The flow induced by the actuator is, of course, modified by the base flow fluctuations and the profiles are not likely to remain as clean as was measured in the quiescent flow tank. It is reassuring to see that the results at two different Reynolds numbers match reasonably well, particularly at low values of the forcing amplitude. However, there is not enough data, especially at the higher forcing amplitudes for the Re∗ = 418 case to definitively capture the location of the maximum in the drag reduction. The 10% reduction in drag is the reduction for the entire active surface, and one should keep in mind that this includes the edge of the channel (where corner flows are likely contaminate the flow) and the development region where the flow must adjust from its canonical state to the fully-controlled state. Thus, the local drag reduction is much likely higher than 10%.

4.3. Fluctuation statistics in the controlled flow For the measurement of the turbulence statistics, the hanging test surface used in the drag measurements was replaced by a fixed surface. Two actuator tiles were mounted one behind the other on the floor of the test section, flush with the rest of the channel wall. Velocity measurements were taken close to the downstream end of this assembly, allowing a development length of approximately 4000 l∗ (at Re ∗ = 167), and 1000 l∗ of fully-controlled flow on either side of the measurement plane. The data presented was taken during four sessions, each with a different flow rate. Figures 8 and 9 show profiles of the mean velocity, Reynolds stress and turbulence production terms as they vary under different actuation conditions. In each figure, either the forcing frequency or amplitude is held

343


18 16 14

u+

12 10 8

Baseline T+St/Re* = 6.7 T+St/Re* = 2.9 T+St/Re* = 1.1

6 4 2 0

20

40

60

80

v+ Fig. 8. Mean flow profiles for a fixed forcing amplitude with changing frequency at Re∗ = 167. The fixed amplitude that was chosen was the highest possible given the actuator design. Actuation both increases and decreases the fluctuations, with a local minimum around T + St/Re∗ = 2.9.

constant and profiles are shown for a sweep through the other parameter. In all figures presented here, circles represent the baseline (unactuated) case. Figure 8 shows a frequency sweep for fixed high-amplitude forcing (4.5 V, St = 12.9) at Re∗ = 167. One immediately sees that the Lorentz forcing has a dramatic effect on the mean velocity profile, and that at the lowest frequency, the mean velocity is reduced, while at the highest frequency, the mean velocity is increased and at an intermediate frequency, the velocity is sharply increased over that of the baseline. One should remember that in this configuration (unlike the direct drag measurement configuration) which is driven by a constant pressure head, the flow rate is not maintained. Thus, the overall skin friction in the facility (including both the actuated and un-actuated portions of the channel) remains constant since the driving pressure head is approximately constant. Thus, the interpretation of the change in mean velocity profiles must be done with care. Furthermore, measurement of the velocity gradient near the wall (and hence, the skin friction) are extremely difficult and inaccurate, sensitive to minute errors in the velocity profile. Nevertheless, we can clearly see that over the controlled region of the channel, the Lorentz force actuators has a clear effect on the local flow rate, increasing for some range of parameters and decreasing at other combinations of amplitude and frequency.

344

K. S. Breuer

0.8

−u’v’

0.6

0.4 Baseline + T St/Re* = 6.7 T+St/Re* = 2.9 T+St/Re* = 1.1

0.2

0 20

40

60

v

0.2

Baseline + T St/Re* = 6.7 T+St/Re* = 2.9 T+St/Re* = 1.1

0.15

Production

80

+

0.1

0.05

0

0

10

20

30

40

50

v+

Fig. 9. (a) Changes in Reynolds stress; (b) Turbulence production for a constant amplitude, St and changing frequencies, T + . Re∗ = 167.

The effect of actuation on the Reynolds stress, u′ v ′ , and on the primary term in the turbulent energy production: P ≈ −u′ v ′ ·

d¯ u dy

(4.1)


345

is shown in Figs. 9(a) and 9(b) respectively for the same forcing conditions. Here we see that in the case where the mean velocity was retarded, the Reynolds stress and the turbulent production were enhanced, while at T + St/Re∗ = 2.9, where the maximum increase in the velocity profile is observed, the Reynolds stress is almost completely inhibited near the wall, only rising to its uncontrolled levels well into the buffer region of the flow, beyond y + = 50. Similarly, we see that the production is substantially reduced to almost nothing below y + = 10 and above y + = 20. We recall that the vertical penetration depth of the actuator was characterized by Breuer et al. (2004) to be approximately 3 mm or approximately 20 l∗ . This coincides with the region of maximal reduction in the Reynolds stress (note the location of the minimum in −u′ v ′ in Fig. 9(a)). 4.4. Integrated flow rate and production The overall effects of the frequency and amplitude of forcing are effectively summarized for the Re∗ = 167 condition in Fig. 10. Here, the mean velocity and turbulence production profiles have been integrated from the wall to y + = 75 in order to define a characteristic value that captures the effect of ˜ and turbulence production, P˜ : actuation on the overall flow rate, Q 75 ˜= u ¯(y + )dy + , (4.2) Q 0

75

d¯ u + dy . (4.3) dy 0 The upper frame shows the percent change from the baseline for the flow rate, plotted against the Stuart number (x-axis) and the forcing period (y-axis). The lower frame shows the percent change in the integrated production, plotted in the same manner. The inverse correlation between the flow rate and integrated production is quite striking and this relationship holds over the entire range of parameters tested. A maximum increase in flow rate, coupled with a maximum decrease in turbulence production can be seen at T + = 32 and St = 17. This correlates to a T + St/Re∗ = 2.9, which corresponds to the profiles shown in Figs. 8, 9(a) and 9(b). We also see that the maximum suppression of turbulence is attained at the lower range of forcing periods (higher frequencies) and at higher forcing amplitudes. The tight correspondence between turbulence production and mean flow rate is expected since, if the turbulence production is suppressed, while maintaining the driving pressure head, the flow can only respond by accelerating in that area. P˜ =

−u′ v ′

346

K. S. Breuer

Fig. 10. (a) Percent change in integrated flow rate; (b) Production over a wide range of forcing frequencies and amplitudes. Re∗ = 167.

At a higher Reynolds number that is tested, a similar inverse correlation is observed, and we can illustrate the relative success of the scaling discussed earlier (Eq. (2.7)) by plotting the integrated flow rates and the integrated turbulence production obtained at all four flow rates against the similarity scaling T + St/Re∗ . The control was found to be less effective at the higher

347

Percent Change in Production (scaled)


80 60 40 20 0 −20 −40 Re* = 144 Re* = 167 Re* = 305 Re* = 330

−60 −80 −100

0

2

4

+

6

8

6

8

Percent Change in Flow Rate (scaled)

StT /Re*

10

5

0 Re* = 144 Re* = 167 Re* = 305 Re* = 330

−5

−10

0

2

4

+

StT /Re*

Fig. 11. (a) Scaled changes in the integrated turbulence production; (b) Flow rate as functions of the scaling parameter, StT + /Re∗ . Data from four different Reynolds numbers are shown with a wide range of forcing amplitudes and frequencies. The amplitude of the changes in both production and flow rate are referenced to the Re∗ = 167 case p using the scaling factor: Re∗ /167.

Reynolds numbers and this is accounted for in Fig. 11 by plotting the percent changes normalized relative to the Re∗ = 167 case. The best normalization factor was found empirically to be the square-root of the Reynolds number, and although there is scatter in the data, the trends are quite

348

K. S. Breuer

faithfully preserved over all the test conditions and all Reynolds numbers tested, supporting the scaling arguments presented. In all cases, the optimal effect on the flow is found in the range of T + St/Re∗ ≈ 3 at which point the turbulence production is every effectively suppressed and the flow rate correspondingly enhanced. The square-root scaling of the vertical axis is, as mentioned, completely arbitrary, but however, does have some (tenuous) justification. We know that √ the location of the maximum in the turbulence production scales with Re∗ . Since the actuator penetration depth in our experiment is fixed at higher Reynolds numbers, its effective penetration depth moves relative to the location of maximum turbulence production. Although the dependence on T + St/Re∗ is very similar to that found by Breuer et al. (2004), there appears to be a discrepancy in the values of St at which optimal control was achieved. The difference is most likely a result of the measurement of the current density, J. In the present experiments, current was carefully measured for each forcing condition. However, in the previous experiments, the current density was estimated from the driving voltage and fluid conductivity and hence could be in error by a constant factor. Despite this, the trend of the data here is the same as was found by Breuer et al. (2004), showing an intermediate range of T + St/Re∗ at which the maximum suppression of turbulence by the Lorentz force actuator is achieved. 4.5. Two-point correlations Figures 12–14 show the two-point correlations, computed for both the baseline velocities as well as the forcing condition for which the maximum suppression of turbulence was observed (T + St/Re∗ = 2.9). The two-point correlation, Rab , is defined as: Rab (∆x, y; yo ) =

a′ (x, yo ) · b′ (x + ∆x, y) . a′2 (yo ) b′2 (y)

(4.4)

We show the correlations computed at three locations in the wall-normal direction: close to the wall at y + = 5, the point of maximum turbulence production at y + = 13, and finally in the log-layer at y + = 76. In all cases, the mean velocity has been subtracted prior to the computation of the correlation. Turning our attention first to the streamwise velocity correlation, Ruu (Fig. 12), we see in the unforced case, that the characteristic correlation has the familiar shape of an elongated “banana”, inclined to the wall at a shallow angle. This is consistent with the image of typical coherent structures in the near-wall region being comprised of the inclined shear


Fig. 12. Ruu (∆x, y; yo ) at three values of yo : y + = 5, 13 and 76 (left, center and right respectively). The top row shows the two-point correlations for the baseline flow, while the lower row of plots show the corresponding correlation at the optimal forcing condition (T + St/Re∗ = 2.9). Re∗ = 167. The contour lines represent values ranging from zero to one at intervals of 0.1. 349

350 K. S. Breuer

Fig. 13. Rvv (∆x, y; yo ) at three values of yo : y + = 5, 13 and 76 (left, center and right respectively). The top row shows the two-point correlations for the baseline flow, while the lower row of plots show the corresponding correlation at the optimal forcing condition (T + St/Re∗ = 2.9). Re∗ = 167. The contour lines are equally spaced at intervals of 0.1, ranging from zero to one.


Fig. 14. Ruv (∆x, y; yo ) at three values of yo : y + = 5, 13 and 76 (left, center and right respectively). The top row shows the two-point correlations for the baseline flow, while the lower row of plots show the corresponding correlation at the optimal forcing condition (T + St/Re∗ = 2.9). Re∗ = 167. The contour lines are equally spaced at intervals of 0.1, starting from −0.55 to +0.55. 351

352

K. S. Breuer

layers tilted over by the mean shear that dominates the near-wall flow. The structure is more rounded in the log-layer reflecting the reduced mean shear and the decreased influence of the wall. We see that at all y-locations, the effect of the Lorentz forcing is to reduce the streamwise extent of the velocity correlations, and in the near-wall and buffer region, to reduce the characteristic height of the correlation surface. This is most obvious in the buffer layer, where the typically correlation length, as characterized by the 0.8 contour level, stretches over the complete streamwise extent of the window, from ∆x = ±50 in the unforced baseline flow, but only extends about half that length in the case of the forced flow. We can interpret this as an indication that one effect of the forcing is to disrupt the coherent streak structures, shortening their characteristic length and reducing their characteristic height. The effect is slightly different in the log-layer where the forcing is observed to shorten the typical coherence length, but also to make it slightly taller in the wall-normal direction. The effect of forcing on the vertical velocity correlation, Rvv is also quite dramatic (Fig. 13). First, we note that the unforced correlation is quite compact (note the reduced scales in x and y), and that the effect of forcing is to reduce this correlation scale even further. The dynamic difference between the streamwise velocity and vertical velocity correlations emphasizes the completely different character that these two components have in the strong turbulent shear flow. We can understand this by recalling that for the case of incompressible flow, the Navier–Stokes equations can be written as a pair of equations, one for the vertical velocity and a second for the vertical vorticity. The former evolves with a character quite different from the latter, and in particular the vertical vorticity equation includes all of the vorticity tilting and stretching terms due to the interactions with mean shear. It is this feature that results in the elongated structures observed in the Ruu plots, while the more “wavelike” character of the vertical velocity equation leads to the less elongated and less inclined structures observed in the Rvv plots. The correlation between the streamwise and vertical perturbation velocities, Ruv (Fig. 14) shows the most dramatic change due to the Lorentz forcing. The negative correlation typical for the Ruv correlation (Liu et al., 2001) is reduced significantly both in its amplitude and in its characteristic extent. The correlations Ruu and Rvv are, by definition, equal to one at the zero-separation point, ∆x = 0, y = yo . This is not true for the crosscomponent correlation, Ruv and we see that the amplitude of the maximum (negative) correlation, which reaches approximately −0.55 in the unforced


353

flow, is reduced to about −0.3 in the near-wall region, and about −0.4 in the buffer layer. Interestingly, the correlation becomes stronger in the loglayer, rising slightly by about 0.1, and shifting backward. The implications of this are still to be determined. The noise in the two-point correlation very near the wall is due to the fact that the local rms velocity, which is used as the normalizing velocity scale, is somewhat noisy close to the wall for the case of the controlled flow (Fig. 9(a)) due to the relatively poor statistics (200 image pairs at this condition). This is exacerbated at large values of ∆x where the number of velocity pairs contributing the correlation becomes even lower due to the relatively small streamwise extent of the PIV window. 4.6. Production probability The so-called “bursting” events are known to be related to the center of inclined shear layer structures and that are known to be associated with peaks in turbulence production in the near-wall region (Johansson et al., 1991). Rathnasingham and Breuer (2003) reported that in an actively controlled boundary layer flow, the burst frequency was observed to fall dramatically, suggesting that the incidence of these critical coherent structures was reduced by the effects of near-wall actuation (in their case, wall-normal blowing in response to shear sensor inputs). In the present case, we have computed the threshold probabilities for the Reynolds shear stress, −u′ v ′ , to determine how the flow forcing affects the density of high-amplitude turbulence production events. The production probability is defined as the cumulative probability that the instantaneous value of the Reynolds shear stress, normalized by its average value is greater than some threshold value k:

′ ′ −u v > k . (4.5) PP (k) = P −u′ v ′ Thus, for k large and negative, the probability will be unity, but will decrease as k increases and a smaller fraction of the Reynolds stress field has the sufficient amplitude. Since the mean shear is constant, it is sufficient to compute the probability based on the Reynolds stress alone. The production probability for the low Reynolds number case, Re∗ = 167, is shown in Fig. 15 for the baseline flow (black symbols) and for three forcing conditions. The results confirm what we have seen in previous figures, that for both low and high values of the similarity parameter, T + St/Re∗ , the distribution of Reynolds stress shifts such that there are more high amplitude events, while

354

K. S. Breuer

Probability, P

100

10-1 Baseline T+St/Re* = 6.7 + T St/Re* = 2.9 + T St/Re* = 1.1

10-2

2

0

2 4 6 Threshold parameter, k

8

10

Fig. 15. Production probability, Puv (k), plotted as a function of the threshold parameter, k. Four forcing conditions are shown including the baseline flow and three controlled flows achieved by varying the forcing frequency while keeping the amplitude constant. Re∗ = 167.

at the optimal forcing condition of T + St/Re∗ = 2.9, we find that the high amplitude events become less frequent. 5. Conclusions Lorentz force actuators have been shown to be quite effective in their ability to affect a low Reynolds number saltwater channel flow. In many ways, they are ideal for such applications — they present no mechanical intrusion to the flow and have no moving parts. The penetration into the flow is set by the electrode-magnet spacing and is well-suited for near-wall control which is based within a few millimeters of the surface. A range of forcing frequencies and amplitudes were explored based on the optimum forcing predicted by the dimensionless parameter T + St/Re∗ . Several Reynolds numbers were also tested, although still at relatively low values. In the case of direct measurements of drag, we show a significant reduction in turbulent skin friction over a range of forcing parameters. In the case of PIV measurements of the velocity statistics, both the mean flow and the fluctuation quantities are observed to be strongly affected by the forcing, which is most effective in the near-wall region, centered around 1 mm from the surface (y + = 7), and we confirm this in a strong correlation between the integrated flow


355

rate and the integrated turbulence production which show a strong anticorrelation over the entire range of forcing parameters. Scaling arguments suggest that the similarity variable T + St/Re∗ should capture the essential features of the control performance, and this is observed to be true to a large degree. The amplitude of drag reduction appeared to be independent of the Reynolds number, but the changes in flow rate and production were found to scale with the square root of the Reynolds number. This is still an issue to be resolved, but may be due to the fact that the ideal amplitude for effective forcing could not be maintained at the higher Reynolds number (the required current density increases like the square of the Reynolds number). There is some discrepancy in the value of T + St/Re∗ at which optimal control is achieved. For the first series of experiments, T + St/Re∗ ≈ 0.6 was found to be most effective. However in these cases the current density was not actually measured, but estimated from the driving voltage and known conductivity of the fluid. In a second series of experiments, the most effective control was achieved at T + St/Re∗ ≈ 2–3, and at this forcing condition, the two-point correlations between the streamwise and vertical velocity components were observed to become much more compact, suggesting that the control significantly reduces the spatial extent of the near-wall coherent structures that maintain the turbulent flow. Lastly, the frequency of high-amplitude turbulence production events was also observed to be sharply reduced at the optimal forcing condition. These experiments are being continued and the next series of measurements will study the response of the flow in phase with the forcing of the Lorentz actuator and also the persistence of the controlled flow after the forcing has been removed. I am indebted to Jinil Park, Charlie Henoch and Maureen McCamley who have been my collaborators during the course of this research. The work was supported by DARPA and the US Office of Naval Research. Parts of this chapter are adapted from work presented by Breuer et al. (2004) and McCamley et al. (2005). This paper was presented at a workshop on Turbulence Control sponsored by the Institute for Mathematical Sciences, National University of Singapore in December 2004. The author’s visit was supported by the Institute.

References 1. T. Berger, C. Lee, J. Kim and J. Lim, Turbulent boundary layer control utilizing the Lorentz force, Phys. Fluids 12 (2000) 631. 2. T. Bewley, Flow control: New challenges for a new renaissance, Progress in Aerospace Sciences 37 (2001) 21–58.

356

K. S. Breuer

3. T. Bewley, P. Moin and R. Temam, DNS-based predictive control of turbulence: An optimal benchmark for feedback algorithms, J. Fluid Mech. 447 (2001) 179–225. 4. K. S. Breuer, J. Park and C. Henoch, Actuation and control of a turbulent channel flow using Lorentz forces, Phys. Fluids 16(4) (2004) 897–907. 5. H. Choi, P. Moin and J. Kim, Active turbulence control for drag reduction in wall-bounded flows, J. Fluid Mech. 262 (1994) 75–110. 6. K.-S. Choi, J.-R. DeBisschop and B. Clayton, Turbulent boundary layer control by means of spanwise wall oscillation, AIAA Journal 36(7) (1998) 1157. 7. G. M. Di Cicca, G. Iuso, P. G. Spazzini and M. Onorato, Particle image velocimetry investigation of a turbulent boundary layer manipulated by spanwise wall oscillations, J. Fluid Mech. 467 (2002) 41–56. 8. Y. Du, V. Symeonidis and G. E. Karniadakis, Drag reduction in wall-bounded turbulence via a transverse traveling wave, J. Fluid Mech. 457 (2002) 1–34. 9. Y. Du and G. E. Karniadakis, Suppressing wall turbulence by means of a transverse traveling wave, Science 288 (2000) 1230–1234. 10. C. Henoch and J. Stace, Experimental investigation of a saltwater turbulent boundary layer modified by an applied streamwise magnetohydrodynamic body force, Phys. Fluids 7 (1995) 1371. 11. A. V. Johansson, P. H. Alfredsson and J. Kim, Evolution and dynamics of shear layer structure in near-wall turbulence, J. Fluid Mech. 224 (1991) 579–599. 12. W. Jung, N. Mangiavacchi and R. Akhavan, Suppression of turbulence in wall-bounded flows by high frequency oscillations, Phys. Fluids 4 (1992) 1605–1607. 13. C. Lee and D. Goldstein, DNS of microjets for turbulent boundary layer control, AIAA Paper (2001) 2001–1013. 14. C. Lee, J. Kim, D. Babcock and R. Goodman, Application of neural networks to turbulent control for drag reduction, Phys. Fluids 9(6) (1997) 1740–1747. 15. Z. Liu, R. J. Adrian and T. J. Hanratty, Large-scale modes of turbulent channel flow: Transport and structure, J. Fluid Mech. 448 (2001) 53–80. 16. M. McCamley, C. Henoch and K. S. Breuer, The structure of turbulent channel flow subject to Lorentz force actuation, Proc. 2nd Int. Conf. Seawater Drag Reduction (Busan, Korea, 2005). 17. D. Nosenchuck and G. Brown, Discrete spatial control of wall shear stress in a turbulent boundary layer, Near-Wall Turbulent Flows, eds. R. M. C. So, C. G. Speziale and B. E. Launder (Elsevier Science, New York, 1993), 689–698. 18. J. Pang and K.-S. Choi, Turbulent drag reduction by Lorentz force oscillation, Phys. Fluids 16 (2004) L35–L38. 19. J. Park, C. Henoch, M. McCamley and K. S. Breuer, Lorentz force control of turbulent channel flow, AIAA Paper (Orlando, Florida, 2003), 2003–4157. 20. R. Rathnasingham and K. S. Breuer, System identification and control of turbulent flows, Phys. Fluids 9(7) (1997) 1867–1869. 21. R. Rathnasingham and K. S. Breuer, Active control of turbulent boundary layers, J. Fluid Mech. 495 (2003) 209–233.

COMPLIANT COATINGS: THE SIMPLER ALTERNATIVE

Mohamed Gad-el-Hak Department of Mechanical Engineering Virginia Commonwealth University Richmond, VA 23284-3015, USA E-mail: [email protected] Compliant coating research is one of those areas that experienced its fair share of triumphs and debacles. For close to half a century, the subject has fascinated, frustrated and occasionally gratified scientists and engineers searching for methods to delay laminar-to-turbulence transition, to reduce skin-friction drag in turbulent wall-bounded flows, to quell vibrations and to suppress flow-induced noise. In its simplest form, the technique is passive, relatively easy to apply to an existing vehicle or device and perhaps, not too expensive. Through the years, however, claims for substantial drag and noise reductions were made, only to be refuted later when the results were more critically examined. There are several important issues with regard to the reliability of available analytical, numerical and experimental results. In this chapter, some of these issues, particularly as they relate to the search for drag-reducing compliant coatings, will be addressed with the objective of elucidating the potential pitfalls to newcomers of the field. Problem formulation with proper boundary conditions, impossibility of obtaining first-principles analytical solutions when the wall-bounded flow is turbulent and limitations of existing numerical simulations will be elaborated. The effects of background turbulence in a wind or water tunnel, accurate drag measurements, compliant wall motion, and the geometry and properties of the coating used will be among the outstanding experimental issues discussed. Attempts will be made to explain some of the seemingly contradictory results available in the open literature. Contents 1

Introduction

358

2

Compliant Coatings Prior to 1985

361 357

358

3 4 5 6

M. Gad-el-Hak

Free-Surface Waves System Instabilities The Kramer’s Controversy Transitional Flows 6.1 Linear stability theory 6.2 Coating optimization 6.3 Practical examples 6.4 The dolphin’s secret 7 Turbulent Wall-Bounded Flows 8 What Works and What Does Not? 8.1 Analytical research 8.2 Numerical research 8.3 Experimental research 9 The Future 10 Parting Remarks References

364 367 369 372 372 377 382 383 386 391 391 392 393 394 397 398

1. Introduction A compliant wall, as opposed to a rigid one, offers the potential for favorable interference with a wall-bounded flow. Laminar-to-turbulence transition may be delayed or advanced, boundary-layer separation may be prevented or triggered, flow-induced noise may be modulated, and skinfriction drag in both laminar and turbulent flows may be altered. The challenge is, of course, to find a coating with the optimum physical properties to achieve a desired goal. Would hydrodynamists be ever lucky or tenacious enough to discover the right kind of compliant coating to reduce skin friction? Certain existing experimental and analytical studies indicate that we can, at least in the laboratory, but field trials have yet to be conducted and many technical challenges still remain. The present chapter commences with a résumé of past and recent attempts to search for dragreducing compliant coatings and concludes with the future prospects of the research field. For better or worse, hydrodynamically speaking, the epidermis of most nekton is pliable at their typical swimming speeds. For close to half a century, the science and technology of compliant coatings has fascinated, frustrated and occasionally gratified Homo sapiens searching for methods to delay laminar-to-turbulence transition, to reduce skin-friction

Compliant Coatings: The Simpler Alternative

359

drag in turbulent wall-bounded flows, to quell vibrations and to suppress flow-induced noise. Compliant coatings offer a rather simple method to delay laminar-to-turbulence transition, and therefore to benefit from the order-of-magnitude lesser skin friction in a laminar, in contrast to a turbulent, wall-bounded flow, as well as to interact favorably with a turbulent boundary layer or pipe flow. In its simplest form, the technique is passive, relatively easy to apply to an existing vehicle or device and perhaps, not too expensive. Unlike other drag-reducing techniques such as suction, injection, polymer or particle additives, passive compliant coatings do not require slots, ducts or internal equipment of any kind. Likewise, unlike the exceedingly complex reactive control strategies now in vogue (Gad-el-Hak, 2000), passive compliant coatings do not require sensors, actuators, feedforward/feedback loops or control algorithms. Aside from reducing drag, other reasons for the perennial interest in studying compliant coatings are their many other useful applications. For example, as sound-absorbent materials in noisy flow-carrying ducts in aero-engines, and as flexible surfaces to coat naval vessels for the purposes of shielding their sonar arrays from the sound generated by the boundary-layer pressure fluctuations and of reducing the efficiency of their vibrating metal hulls as sound radiators. The original interest in the field was spurred by the experiments of Kramer (1957) who demonstrated a compliant coating design based on the dolphin’s epidermis and claimed substantial transition delay and drag reduction in hydrodynamic flows. Those experiments were conducted in the seemingly less-than-ideal environment of Long Beach Harbor, California. Subsequent laboratory attempts to substantiate Kramer’s results failed and the initial interest in the idea fizzled. A similar bout of excitement and frustration followed suit that dealt mostly with the reduction of skinfriction drag in turbulent flows for aeronautical applications. Those results were summarized in the comprehensive review by Bushnell et al. (1977). During the early 1980s, interest in the subject was rejuvenated mostly due to modest investment in resources by the Office of Naval Research in the United States and the Procurement Executive of the Ministry of Defence in Great Britain.a Significant advances were made during this

a Through

all the ups and downs in the West, compliant coating research continued at a more or less steady pace in the former Soviet Union, but open-literature publications resulting from this work, either in English or in Russian, are rather scarce. For some valuable references, see, for example, the books by Aleyev (1977), and Choi (1991), and the articles by Babenko and Kozlov (1972) and, Carpenter and Garrad (1985).

360

M. Gad-el-Hak

period in numerical and analytical methods to solve the coupled fluid– structure problem. New experimental tools were developed to measure the minute yet important surface deformation caused by the unsteady fluid forces. Coherent structures in turbulent wall-bounded flows were routinely identified, and their modulation by the surface compliance could readily be quantified. Careful analyses by Carpenter and Garrad (1985) and Willis (1986) as well as the well-controlled experiments reported by Daniel et al. (1987) and Gaster (1988)b have, for the first time, provided direct confirmation of the transition-delaying potential of compliant coatings and have convincingly made a case for the validity of Kramer’s original claims as well as offered a plausible explanation for the failure of subsequent laboratory experiments. There is little doubt now that compliant coatings can be rationally designed to delay transition and to suppress noise on marine vehicles and other practical hydrodynamic devices. Transition Reynolds numbers that exceed by an order of magnitude on rigid-surface boundary layers can be readily achieved. Although the number of active researchers in the field continues to dwindle, new and promising results are being produced. Recent theoretical work by Davies and Carpenter (1997a), and Carpenter (1998) indicate that transition to turbulence can be delayed indefinitely, at least in principle, provided that optimized multiple-panel compliant walls are used and that the freestream is a low-disturbance environment. There is also recent evidence of favorable interactions of compliant coatings even for air flows (Lee et al., 1995) and for turbulent boundary layers (Lee et al., 1993a; Choi et al., 1997). The present chapter emphasizes the significant compliant coating research that took place during the last 20 years and suggests avenues for future research. The reader is referred to prior reviews for more classical work on the subject, for example those by Bushnell et al. (1977), Gad-el-Hak (1986a; 1987; 1996a), Riley et al. (1988), Carpenter (1990) and Metcalfe (1994). The book by Gad-el-Hak (2000) places compliant coatings within the broader area of flow control and the more recent article by Carpenter et al. (2001) focuses on the use of compliant walls for laminarflow control. Following these introductory remarks, a somewhat sketchy history of the subject, particularly prior to 1985, is recalled. This will help place more recent developments in proper perspective.

b Originally reported in the thesis by Willis (1986) and later contrasted to theory in the paper by Lucey and Carpenter (1995).


361

2. Compliant Coatings Prior to 1985 Before embarking to describe on the recent accomplishments in the field of compliant coatings, we first elaborate on its history prior to 1985. This seemingly arbitrary date is chosen because it demarks the time after which tools became more readily available to rationally design a compliant coating to delay transition. The victories and defeats of the subject matter will become clear through the following discussion. The idea of using compliant coatings for drag reduction motivated much of the earlier work in this area and was first introduced by Kramer (1957) based on his earlier observation, while crossing the Atlantic ocean in 1946, of dolphins swimming in water. He advanced the concept that the stability and transition characteristics of a boundary layer may be influenced by coupling it hydroelastically to a compliant coating. In his pioneering paper and several subsequent publications, Kramer (1960a; 1960b; 1960c; 1961; 1962; 1965; 1969) reported substantial drag reduction for towed underwater bodies covered with compliant coating modeled after the dolphin skin. He hypothesized that by tuning the elastic wall damping to a frequency near that of the most unstable Tollmien– Schlichting wave, it would be possible to dissipate partially the instability waves, thus delaying the transition to turbulence. Kramer’s tests were performed by towing a test model behind a motor boat in Long Beach Harbor. Unfortunately, many attempts by other investigators to repeat Kramer’s experiments under more controlled conditions failed to yield similar conclusions (e.g., Puryear’s experiment in a towing tank, 1962). This so-called Kramer controversy will be revisited in Sec. 5. Theoretical work by Benjamin (1960a; 1960b), Betchov (1960), Landahl (1962) and Kaplan (1964) indicated that drag reduction by delaying transition is possible. However, the theoretically predicted successful coatings had specific characteristics that would be extremely difficult to match in practice. It is important to stress that almost all this early work addressed the delay of transition, and ignored the potential for reducing turbulence skin friction with compliant coatings. During the mid-1960s, Benjamin (1966) explored the possibility that a compliant coating may affect the skin-friction drag in a fully-developed turbulent boundary layer without necessarily delaying transition. Dinkelacker (1966) conducted careful tests of a compliant surface in a water pipe flow. He systematically attempted to determine the repeatability of rigidtube data, the influence of small steps in the tube wall and the possible occurrence of organ pipe acoustic modes. Dinkelacker’s results seemed to indicate a modest reduction in drag by using a compliant wall.

362

M. Gad-el-Hak

Blick and his co-workers at the University of Oklahoma experimentally demonstrated significant reductions in turbulence skin friction for compliant surfaces in air (Fisher and Blick, 1966; Looney and Blick, 1966; Smith and Blick, 1966; Blick and Walters, 1968; Chu and Blick, 1969). Subsequent tests by Lissaman and Harris (1969) who attempted to substantiate Blick’s conclusions, yielded only extremely modest gains. In another study, McMichael et al. (1980) demonstrated that the apparent reduction in turbulence skin friction in the University of Oklahoma’s experiments could be a consequence of experimental deficiencies coupled with the improper interpretation of data. McMichael et al. concluded that drag reduction via compliant coating in gaseous flows would not be as successful as in liquids. During the 1970s, various compliant materials were tested in water at the Naval Ocean Systems Center, the Naval Research Laboratory, the Naval Undersea Systems Center and the Advanced Technology Center of the LTV Corporation, all in the United States. In no case was a statistically significant reduction in drag measured. Fischer and Ash (1974) presented a general review of concepts for reducing skin friction, including the use of compliant coatings. Bushnell et al. (1977), in summarizing the work conducted at the NASA Langley Research Center and the general status of compliant surface drag reduction, stated that, while it was possible to increase the transition Reynolds number by perhaps a factor of two, there was no definitive reduction of drag for turbulent flows in air. They also stated that drag reduction in turbulent flows in water is potentially feasible and can be accomplished using surfaces that can be practically built. It is of particular interest to note that much of the research on compliant coatings has been based on materials that attempt to replicate dolphin skin. Yet, in the Russian book Nekton (Aleyev, 1977) it was indicated that the “wrinkling” (Fig. 1) of the dolphin skin has no hydrodynamic-drag advantage. Other characteristics of the dolphin’s skin may, however, be beneficial. This subject will be revisited in Sec. 6.4. Bushnell et al. (1977) put forward the possibility of a feedback mechanism in turbulent wall-bounded flows through which the quasi-periodic, coherent structures termed “bursts” regenerate. Older bursts grow, migrate away from the wall, and interact to produce a pressure field which contains pulses of sufficient duration and amplitude to induce new bursts in the near-wall region. This model is supported by the measurements of Burton (1974), who reported a strong correlation between the occurrence of a burst


363

Fig. 1. Wavelike folds in compliant skin appearing during rapid swimming of a delphinidae (top photograph) and a Homo sapiens (bottom photograph). From Aleyev (1977).

and the imposition on the wall flow of a large, moving adverse pressure gradient followed by a favorable pressure-gradient. Bushnell et al. hypothesized that a successful compliant coating would modulate the pre-burst flow in the turbulent boundary layer by providing a pressure field that would tend to block the feedback mechanism and, thus, inhibit burst formation. This would result in a reduction of the number of bursts occurring per unit time and also in the skin-friction drag. Orszag (1979) assumed this conceptual model and performed numerical calculations of wall boundary-layer instability to explore the effects of compliant surfaces. His results, although preliminary, indicated that turbulence drag reduction may be possible for certain classes of materials. He concluded that compliant walls which support only short wavelengths may have an appreciable effect in inhibiting further bursts in a turbulent boundary layer. During the U.S. Navy-sponsored research program conducted over the period 1980–1985, the subject of boundary-layer interaction with compliant coatings has been re-examined to answer the question of whether compliant coatings can delay transition and/or significantly reduce turbulence skin friction on bodies at high Reynolds numbers. Several significant

364

M. Gad-el-Hak

developments have been achieved by the many investigators who have participated in this research program. Although irrefutable experimental evidence of compliant-coating drag reduction was still lacking by 1985, our understanding of boundary-layer flow over a compliant surface has increased dramatically over this period. That understanding proved crucial to the subsequent successes in the field, a subject which will be emphasized throughout this chapter. 3. Free-Surface Waves Before addressing the complex issue of stability of the coupled fluid– structure system, it is instructive to study the solid as a wave-bearing medium when no fluid (another wave-bearing medium) is present. Some understanding of how a compliant surface will respond to the flow above it can be obtained by examining the free-surface waves of the coating. As pointed out by Rayleigh (1887), the surface waves can be modeled as a linear combination of waves having displacements perpendicular and parallel to the propagation direction. These are called transverse and longitudinal displacement waves, respectively. Assuming that the coating is a singlelayer, elastic solid of thickness d attached to a rigid half-space at its lower boundary and bounded by vacuum on its upper surface, both wave systems satisfy the wave equation (see, for example, Landau and Liftshitz, 1987):

2 ∂ ∂2 ∂2 2 η − c + η = 0, (3.1) ∂t2 ∂x2 ∂y 2 where η is a component of the displacement vector, and x and y are coordinates parallel and normal to the undisturbed surface, respectively. The = G/ρ for the transverse waves, and propagation velocities are c = c t s c = cℓ = (Θ + 2G)/ρs for the longitudinal waves, where G and Θ are elastic constants, and ρs is the density of the solid. The free-surface wave dispersion relationship is obtained by assuming that the wave solutions have exponential dependence on the distance measured normal to the surface y, and have harmonic dependence on the distance measured along the surface x, and on time t: ξ = (A sinh αy + B cosh αy)ei(kℓ x−ωt) , η = (C sinh αy + D cosh αy)e

i(kℓ x−ωt)

,

(3.2) (3.3)

where ξ and η are respectively the displacement in the x- and y-direction, and kℓ is the longitudinal wavenumber. For real α, the waves decay exponentially with depth; whereas for imaginary α, they oscillate. Substituting these


365

solutions into the wave equation (3.1), and applying the boundary conditions: no normal or tangential stress at the surface y = 0 and no displacement at the bottom y = −d, respectively, ∂ξ 2 ∂η + c2ℓ = 0 at y = 0, (3.4) cℓ − 2c2t ∂x ∂y ∂ξ ∂η + = 0 at y = 0, (3.5) ∂y ∂x ξ=η=0

at y = −d.

(3.6)

Gad-el-Hak et al. (1984) obtained the following dispersion relationship αt αℓ M (ζ) ≡ 4 2 (2 − ζ 2 ) kℓ

αt αℓ − 2 4 + (2 − ζ 2 )2 cosh αt d cosh αℓ d kℓ 2 2 α α + 4 t 4 ℓ + (2 − ζ 2 )2 sinh αt d sinh αℓ d = 0, (3.7) kℓ where αt = kℓ 1 − ζ 2 , αℓ = kℓ 1 − ζ 2 R2 , R is the ratio of transverse to longitudinal wave speed, and ζ = cp /ct is the ratio of the surface wave speed to transverse wave speed. Plots of the dispersion curves resulting from Eq. (3.7) are given in Fig. 2 for the case R = 0, which are close to typical experimental conditions. Only one solution has a surface wave speed below ct (ζ < 1). This solution approaches its asymptote ζ = 0.956 for large kℓ d, which is the value of Rayleigh’s infinite half-layer solution. Also note that as ζ increases for a given kℓ d, the possibility exists for a richer range of interactions between the coating and a flow. Typical values for kℓ d in experiments, with the wavelength 2π/kℓ taken as the boundary-layer thickness, are in the range of 1–5. Figure 2 indicates that interaction between a fluid flow and a compliant surface may not be expected for flow speeds much below ct , and that the best opportunity for interactions will be for flow speeds well above ct . Unfortunately, hydroelastic instability waves appear for freestream speeds somewhat above ct , thus causing large surface deformations and limiting the opportunities for favorable interactions. The waveform can be obtained by determining the constants A, B, C and D in Eqs. (3.2) and (3.3) using the four boundary conditions (3.4)–(3.6). Duncan and Hsu (1984) calculated the dispersion relations for a twolayer coating by finding the zeroes of the determinant of the boundary condition coefficients. Their results for a thin, stiff coating placed on a much thicker, soft coating are shown in Fig. 3. The upper coating has

366

M. Gad-el-Hak

Fig. 2. Solutions of the dispersion equation for free-surface waves on a single-layer, homogeneous coating. From Gad-el-Hak et al. (1984).

Fig. 3. Solutions of the dispersion equation for free-surface waves on a two-layer coating. From Duncan and Hsu (1984).


367

about 100 times the shear modulus of the lower surface. In this figure, the wavenumber is normalized by the total coating thickness, while the phase speed of the free-surface waves is normalized by the transverse wave speed of the lower layer. This normalization allows an easy comparison with the single-layer case when the upper portion of the coating is replaced by a layer of stiffer material. The gross shapes of the curves in Fig. 3 are similar to those in Fig. 2, although a wrinkle is clearly seen in the two-layer case. The dashed line in Fig. 3 connects the maxima of the wrinkle in each modal curve and is a solution of the equation for long-bending waves in a free plate. Duncan and Hsu (1984) have hypothesized that the upper coating acts like a plate that is influenced by the lower layer. They have attributed the cutoff of the response of the bending waves in the upper layer at small wavenumbers to the lack of slow-moving waves in the lower layer.

4. System Instabilities From a fundamental viewpoint, a rich variety of fluid–structure interactions exists when a fluid flows over a surface that is able to interact with the flow. Not surprisingly, instability modes proliferate when two wave-bearing media are coupled. Some waves are flow-based, some are wall-based and some are a result of the coalescence of both kind of waves. What is most appealing about compliant coatings is their potential to inhibit, or to foster, the dynamic instabilities that characterize both transitional and turbulent boundary-layer flows, and in turn to modify the mass, heat and momentum fluxes as well as change the drag and acoustic properties. While it is relatively easy to suppress a particular instability mode, the challenge is of course to prevent other modes from growing if the aim is, say, to delay laminar-to-turbulence transition. From a practical point of view, it is obvious that an in-depth understanding of the coupled system instabilities is a prerequisite to rationally design a coating that meets a given objective. There are at least three classification schemes for the fluid–structure waves, each with its own merits. The original scheme is attributable to Landahl (1962) and Benjamin (1963). It divides the waves into three classes according to their response to irreversible energy transfer to and from the compliant wall. Both class A and class B disturbances are essentially oscillations involving conservative energy exchanges between the fluid and solid, but their stability is determined by the net effect of irreversible processes such as dissipation in the coating or energy transfer to the solid by non-conservative hydrodynamic forces. Class A oscillations are

368

M. Gad-el-Hak

Tollmien–Schlichting-type waves in the boundary layer modified by the wall compliance, in other words by the motion of the solid in response to the pressure and shear-stress fluctuations in the flow. The disturbance eignenfunction for class A waves has its maximum amplitude within the fluid region. Such waves are stabilized by the irreversible energy transfer from the fluid to the coating, but destabilized by dissipation in the wall. Class B waves are to be found in both the fluid and the wall. However, the disturbance eignenfunction has its maximum amplitude at the fluid– solid interface and thus, those waves are principally wall-based modes of instability. Such instability would not exist had the wall been rigid. The instability is due to the downstream-running free wave in the solid being modified by the fluid loading. The destabilization of class B waves is effected by the phase-difference between the pressure perturbation and the wall deformation, which allows a flow of energy from the fluid into the wall. The behavior of class B waves is the reverse of that for class A waves, stabilized by wall damping but destabilized by the non-conservative hydrodynamic forces. Essentially, class B waves are amplified when the flow supplies sufficient energy to counterbalance the coating’s internal dissipation. Finally, class C waves are akin to the inviscid Kelvin–Helmholtz instability and occur when conservative hydrodynamic forces cause a unidirectional transfer of energy to the solid. The pressure distribution in an inviscid flow over a wavy wall is in exact antiphase with the elevation. In that case, class C waves can grow on the solid surface only if the pressure amplitude is so large as to outweigh the coating stiffness. Class C waves are the result of a modal-coalescence instability where the flow speed is sufficiently high that the originally upstream-running wall free waves are turned to travel downstream and merge with the modified downstream-running wall waves. Irreversible processes in both the fluid and solid have negligible effect on class C instabilities. If one considers the total disturbance energy of the coupled fluid–solid system, a decrease in that energy leads to an increase in the amplitude of class A instabilities, while class B is associated with an energy increase, and virtually no change in total energy accompanying class C waves. In other words, any non-conservative flow of activation energy from/to the system must be accompanied by disturbance growth of class A/B waves, while the irreversible energy transfer for class C instability is nearly zero. The second classification scheme is due to Carpenter and Garrad (1985; 1986). It simply divides the waves into fluid-based and solidbased. Tollmien–Schlichting instability (TSI) is an example of fluid-based waves. The solid-based, flow-induced surface instabilities (FISI) are closely


369

analogous to the instabilities studied in hydro- and aeroelasticity, and include both the traveling-wave flutter that moves at speeds close to the solid, free wave speed (class B) and the essentially static — and more dangerous — divergence waves (class C).c The main drawback of this classification scheme is that under certain circumstances the fluid-based T–S waves and the solid-based flutter can coalesce to form a powerful new instability termed the transitional mode by Sen and Arora (1988). According to the energy criterion advanced by Landahl (1962), this latest instability is a second kind of class C wave. In a physical experiment, however, it is rather difficult to distinguish between static divergence and the transitional mode. The third scheme to classify the instability waves considers whether they are convective or absolute (Huerre and Monkewitz, 1990). An instability mode is considered to be absolute if there is a pinch point in the Fourier contour that prevents the temporal amplification rate from being reduced down to zero. In this case, the unstable mode propagates upstream as well as downstream and often has a very small (or even zero) group velocity in comparison with the velocity of the mean flow. On the other hand, the unstable development of a disturbance is said to be convective when none of its constituent modes possesses zero group velocity. Both class A and class B are convective, while class C divergence and the transitional modes are absolute. As Carpenter (1990) points out, the occurrence of absolute instabilities would lead to profound changes in the laminar-toturbulence transition process. “It is therefore pointless to consider reducing their growth rate or postponing their appearance to higher Reynolds number; nothing short of complete suppression would work.” Figure 4 combines and summarizes all three classification schemes. 5. The Kramer’s Controversy It may be worth recalling in more details the pioneering work of Max O. Kramer and the controversy surrounding it. The entire field of compliant coatings became the Rodney Dangerfield of fluid mechanics research, getting no respect from a skeptical community, largely because of the loss of credibility of Kramer’s original experiments. However, as will be seen in the following, the most recent evidence resurrects the good name of this c Static-divergence waves were erroneously interpreted in the past by, for example, Gad-el-Hak et al. (1984a), Duncan et al. (1985) and Yeo (1992) as being class A. The confusion occurs when divergence is treated as a convective instability, when in fact it is an absolute one (Carpenter, 1990).

370

M. Gad-el-Hak

Fluid–solid instabilities

Fluid-based TSI (Class A) (convective)

Coa

lesc

ence

Solid-based FISI

TWF (Class B) (convective)

Static divergence (Class C) (absolute)

Transitional (Class C) (absolute)

Fig. 4.

Summary of all three classification schemes.

ingenious German–American engineer, and with it renewed confidence in this waning and waxing field. As already mentioned, Kramer (1957; 1960a; 1960b; 1960c; 1961; 1962; 1965; 1969) conducted his original experiments by towing a model behind an outboard motor boat in Long Beach Harbor, California. His early tests showed a drag reduction of more than 50% when a dolphin-like skin was used. A typical successful coating used by Kramer consisted of a flexible inner skin, an outer diaphragm and stubs; all made of soft natural rubber. The cavity between the outer diaphragm and the inner skin was usually filled with a highly viscous damping fluid, such as silicone oil, which in Kramer’s view damped out the Tollmien–Schlichting waves. Subsequent experiments to confirm Kramer’s findings were conducted in a towing tank, a lake or a water tunnel (Puryear, 1962; Nisewanger, 1964; Ritter and Messum, 1964; Ritter and Porteous, 1964). No significant drag reduction was observed in any of these investigations. Since then, many researchers have assumed that Kramer’s results were in error and that his observed drag reduction could have come about as a result of favorable changes to the form drag or the accidental excretion of the silicone oil used as the damping fluid during the tests. Surface discontinuities could have favorably altered the pressure drag, and the released oil could have acted


371

as a drag-reducing polymer when released into the boundary layer and the ambient fluid. Carpenter and Garrad (1985) stated that, “It is probably no exaggeration to suggest that the credibility of Kramer’s coatings is now rather low.” Acceptance of his results was not granted by the scientific community because the rigorous standards of scientific investigation were not met and the gradual improvements by Kramer to meet these standards were not adequate (Johnson, 1980). It did not help his cause any that Kramer’s explanations of his own empirical results, though intuitively appealing, were proven physically incorrect. For example, we now know that damping in the solid destabilizes TSI. Almost 30 years after Kramer’s original investigation, Carpenter and Garrad (1985) presented a very careful analysis of his experiments (e.g., Kramer, 1957) and the subsequent tests (Puryear, 1962; Nisewanger, 1964; Ritter and Messum, 1964; Ritter and Porteous, 1964) that attempted to provide independent evidence of the drag-reducing capabilities of Kramer’s coatings. Based on their own rigorous analysis of the hydrodynamic stability of flows over Kramer-type compliant surfaces, Carpenter and Garrad argue that Kramer’s coatings were only marginally capable of delaying transition. Any unfavorable factor such as an adverse pressure gradient, a step where the compliant surface is joined to a rigid surface or an unusually high freestream turbulence level, could badly affect the performance of the coating. Also, a particular coating was designed for a restricted range of Reynolds number and was therefore unlikely to delay transition outside that range. Carpenter and Garrad (1985) contend that one or more of the above adverse factors may have existed in the experiments conducted by Puryear (1962), Nisewanger (1964), Ritter and Messum (1964), and Ritter and Porteous (1964). Puryear’s (1962) experiments were conducted using a prolate spheroid in a towing tank. He did not use Kramer’s coating with the best performance, and serious problems were encountered in making a smooth joint between the rigid and compliant surfaces. Nisewanger’s (1964) tests were conducted by releasing a lighter-than-water body of revolution from the bottom of a lake. His Kramer-like coating contained a fluid with a viscosity below that of the optimum fluid as determined from Kramer’s results. Ritter and Messum (1964), and Ritter and Porteous (1964) conducted their experiments in a water tunnel using either a flat plate or a cylindrical model with an elliptical nose. The conventional flume had a relatively high freestream turbulence level, which may render the facility unsuited for transition experiments.

372

M. Gad-el-Hak

In addition to these adverse effects, some evidence existed in the tests conducted to confirm Kramer’s results for the onset of a hydroelastic instability in the coating. Such large-amplitude waves would certainly lead to drag increase, and their presence may indicate that the boundary layer was already turbulent. Based on these experiments, Carpenter and Garrad (1985) concluded that the results presented in these tests should not be taken as conclusive evidence that the Kramer coatings are not capable of delaying transition and that “the case against Kramer’s coating may not be so strong as popularly supposed.” Carpenter’s (1988) optimization procedure described in Sec. 6.2 results in a compliant coating that is capable of delaying transition by a factor of 4–6 in Reynolds number. It is therefore quite conceivable to design a Kramer-type coating that may lead to a drag reduction of the order reported by the original inventor himself. The above analysis of Kramer’s tests illustrates the importance of carefully selecting the flow facility to conduct compliant coating experiments. The background turbulence in the facility should particularly be monitored if transition delay is sought. This is precisely what was done in the seminal experiments conducted by Gaster (1988) to confirm the theoretical prediction of Carpenter and Garrad (1985), both described in more details in the following section. 6. Transitional Flows 6.1. Linear stability theory Both the hydrodynamic and the hydroelastic stability theories have reached an impressive level of maturity during the last two decades. The linear theories can be handled, for the most part, analytically, while the nonlinear stability theories are more computer intensive. Perhaps no one has contributed more to the recent application of the stability theory to compliant coatings than Peter W. Carpenter, originally with the University of Exeter and presently with the University of Warwick. His list of relevant publications includes 80 papers and growing; obviously only a selected few will be cited in the present short chapter. Within the framework of the linear stability theory, two-dimensional small disturbances are assumed to be superimposed upon a steady, unidirectional mean flow. The nonlinear, partial Navier–Stokes equations are then reduced to the well-known Orr–Sommerfeld equation which is a fourth-order, linear, ordinary differential equation. The order of this equation increases when additional complexities are included in the problem. For rotating-disk flows, for example, Coriolis and streamline-curvature terms are incorporated leading


373

to a sixth-order stability equation. The major difficulty in integrating the Orr–Sommerfeld equation is that it is highly stiff and unstable, which makes it virtually impossible to apply conventional numerical schemes. Explicit codes with step size that is commensurate with the global behavior of the solution will lead to numerical instabilities and alternative routines have been developed to handle this stiff eigenvalue problem. An added difficulty when the walls are compliant is the interfacial conditions which require continuity of velocity and stress. Those boundary conditions can also be linearized, but special care should still be exercised in handling them. Appropriate equations must be used for the compliant walls to be able to fully couple the fluid and solid dynamics. Many types of compliant surfaces exist, so that there are numerous models for the solid. Those models can be either surface-based or volume-based (Fig. 5). The former model reduces the spatial dimensions by one, and is therefore less computationally demanding. An example is the thin plate-spring model used by Garrad and Carpenter (1982), McMurray et al. (1983), Carpenter and Garrad (1985; 1986), Domaradzki and Metcalfe (1987), Metcalfe et al. (1991), and Davies and Carpenter (1997a; 1997b), among others, to simulate Kramer-type coatings. This model is relatively simple yet contains characteristics representative of a broad range of surfaces. If a coordinate system is chosen with the x-axis lying along the undisturbed free surface and the y-axis normal to this surface, then the equation for the y-componentd of the momentum of the compliant coating reads ∂2η F ∂ 4η Tℓ ∂ 2 η ∂η k − = − D − η+F 2 2 4 ∂t m ∂x ∂t m ∂x m

(6.1)

where η(x, t) is the y-displacement of the surface from its equilibrium state at time t and position x, Tℓ is the longitudinal tension and F the flexural rigidity of the thin plate, m is the mass per unit area, D is the damping coefficient, k is the spring constant and F is an external forcing term. The volume-based models are based on the Navier equation and include single and multi-layer coatings (Duncan et al., 1985; Fraser and Carpenter, 1985; Buckingham et al., 1985; Yeo, 1988) as well as isotropic and anisotropic materials (Yeo and Dowling, 1987; Yeo, 1990; 1992; Duncan, 1988). The equations describing the stability of the coupled system form a numerical eigenvalue problem for the complex wavenumber of the disturbance. Duncan (1987) offers a useful comparison between the results obtained from a surface-based model and a corresponding volume-based one. d The

x-component of momentum is usually neglected in such a model.

374

M. Gad-el-Hak

Fig. 5. Volume-based and surface-based models of compliant coatings. From Carpenter (1990).

Compliant walls do suppress the Tollmien–Schlichting waves due to the irreversible energy transfer from the fluid to the solid, but solidbased instabilities proliferate if the coating becomes too soft. For the class A T–S waves, the wall compliance reduces the rate of production (via Reynolds stress) of the disturbance kinetic energy. Simultaneously, the viscous dissipation is increased and thus, the balance between the energy production and removal mechanisms is altered in favor of wave suppression. Experimental validation of the stability calculations is rather difficult and requires well-controlled tests in a quiet water or wind tunnel. Several


375

careful experiments to test the flow stability to two-dimensional as well as three-dimensional controlled disturbances have been reported in the past few years (Daniel et al., 1987; Gaster, 1988; Lee et al., 1995; 1997). Figures 6 and 7 show the remarkable agreement between theory and towing tank

Fig. 6. Growth curves for the rigid-wall case. (a) Theoretical prediction; (b) Experimental results. From Willis (1986).

376

M. Gad-el-Hak

16 HZ

18 HZ 20 HZ

22 HZ

24 HZ

24 HZ 16 HZ

18 HZ

20 HZ

22 HZ

Fig. 7. Growth curves for the compliant-wall case. A two-layer coating with an elastic modulus of the lower layer of E = 5000 Nm−2 . (a) Theoretical prediction; (b) Experimental results. From Willis (1986).

experiments for both rigid-wall and compliant-wall cases. The top part of each figure is the predicted amplification factors as function of flow speed for a range of modal frequencies, while the bottom part is the measured growth-decay cycle of artificially induced T–S waves. A simple compliant


377

model predicts a dramatic decrease in the instability of the flow and this prediction agrees well with the experimental observations when a thick, soft coating is covered with a thin, stiff layer. The papers by Lee et al. (1995; 1997) reported the results of wind tunnel experiments and actually demonstrated the stabilizing potential of compliant coatings in aerodynamic flows; a remarkable achievement that has been deemed impractical in the past (Bushnell et al., 1977; Carpenter, 1990). Excellent agreements are reported between the results of the stability theory and the hydrodynamic experiments (Willis, 1986; Daniel et al., 1987; Gaster, 1988; Riley et al., 1988; Carpenter, 1990; Lucey and Carpenter, 1995). The paper by Lucey and Carpenter, in particular, applies the linear stability theory to predict the experimentally observed evolution of both Tollmien– Schlichting waves and traveling-wave flutter in water flows. For the wind tunnel experiments, Carpenter (1998) has conducted the corresponding calculations, but his preliminary results thus far are negative: the density of an effective coating must be comparable to the fluid density, otherwise no transition delaying benefits are observed. This theoretical result leaves open the question of explaining the positive experimental findings of Lee et al. (1995). In here, we show one example of the suppression of T–S waves in an air boundary layer developing on top of a silicone elastomer–silicone oil compliant surface. Figure 8 depicts the wind tunnel results of Lee et al. (1995). The coating in Fig. 8(a) was made by mixing 91% by weight of 100 mm2 /s silicone oil with 9% of silicone elastomer. In Figs. 8(b)–8(d), the corresponding mix was 90% and 10%, yielding about 35% higher shear modulus. As compared to the rigid wall, the single-layer, isotropic, viscoelastic compliant coating significantly suppresses the rms-amplitude of the artificially-generated Tollmien–Schlichting waves across the entire boundary layer, for a range of displacement-thickness Reynolds numbers. Reductions in the maximum rmsamplitude of as much as 40% are observed for the softer coating (Fig. 8(a)), which may lead to delayed transition.

6.2. Coating optimization If a compliant coating is to be designed for use on an actual vehicle, a relevant question may be: what are the optimum wall properties that will give the greatest transition delay? The large number of available parameters makes it imperative that a rational (i.e. one derived from first principles) selection process be conducted. For obvious reasons, the trial-anderror empirical approach used in the past (if it is soft, let us try it!)

378

M. Gad-el-Hak

Fig. 8. Comparison of distribution of rms-amplitude of the TSI of rigid surface and compliant surface across boundary layer. Wind tunnel experiments of Lee et al. (1995). , rigid surface; , compliant surface. (a) Reδ∗ = 1274; (b) Reδ∗ = 1105; (c) Reδ∗ = 1225; (d) Reδ∗ = 1350. Inset shows locations of ribbon (•) and probe (◦) relative to neutral-stability curve.

should not even be contemplated. This should be particularly true now that rational optimization procedures are becoming readily available as described below. A wall that is too compliant (i.e. too soft) can substantially delay transition via TSI by shrinking its unstable region in the frequency–Reynolds number plane, but rapid breakdown can occur through the amplification of wall-based instabilities (Lucey and Carpenter, 1995). Both kinds of FISI are potentially harmful. The divergence instabilities are absolute, nearly static and yield to wholesale deformations of the surface


379

Fig. 9. Static-divergence waves on a single-layer viscoelastic coating subjected to the pressure fluctuations of a turbulent boundary layer. Flow is from left to right. Freestream velocity is 6.3 times the transverse wave speed in the solid. From Gad-el-Hak et al. (1984).

which are likely to trigger premature transition due to a roughness-like effect (Fig. 9). Flutter instabilities, though convective, are also dangerous. As shown in the stability diagram in Fig. 10, their narrow band of unstable frequencies extends indefinitely as Reynolds number increases downstream. Thus, once these instabilities are encountered at some downstream location, sustained growth follows. This is unlike the broad-band Tollmien– Schlichting instabilities that grow then decay as the different waves travel downstream and pass through the lower and upper branches of their neutral-stability curve (Fig. 10). A workable strategy for coating optimization suggested by Carpenter (1988) is to choose a restricted set of wall properties such that the coating is marginally stable with respect to FISI (both flutter and divergence). The remaining disposable wall parameters can then be varied to obtain the greatest possible transition (via TSI) delay. For the plate-spring, surfacebased model, for example, there are two disposable parameters: the wall damping and the critical wavenumber for divergence. The downstream location of the transition region is estimated from an en -criterion, where n is typically chosen in the range of 7–10. The lower exponent represents

380

M. Gad-el-Hak

Fig. 10. Typical neutral-stability curves for Tollmien–Schlichting waves and for traveling-wave flutter. The solid is a double-layer, Gaster-type compliant coating. From Lucey and Carpenter (1995).

the approximate limit of validity of the linear stability theory for a lowdisturbance environment, and provides a rather conservative calculation. Although wall dissipation destabilizes Tollmien–Schlichting waves, a viscoelastic coating with moderate level of damping leads to greater delay in transition as compared with purely elastic surfaces. Apparently, the stabilizing effects of wall damping on traveling-wave flutter allow a softer wall to be used, and thus more than offset the adverse effects of coating dissipation on TSI. Coating optimization with respect to TSI growth rate is performed at a rather narrow range of Reynolds numbers. On a growing boundary layer, the Reynolds number increases monotonically, and a compliant coating will not be optimum over the whole length of a vehicle. Carpenter (1993) suggests that a multiple-panel wall, placed in series, with each panel optimized for a particular range of Reynolds numbers, is likely to produce larger transition delays than a single-panel wall. His calculations for a two-panel, plate-spring type compliant wall indicate an additional performance improvement of over 30% over an optimized single-panel wall. It seems reasonable that a large number of panels, say 10, in series would lead to superior performance, but of course the calculations involved become prohibitive very quickly. An additional benefit from using multi-panels is that shorter panels are more resistant to both static-divergence waves and


381

Fig. 11. Static-divergence waves are clearly seen under the turbulent wedge generated by a single roughness element embedded into an otherwise laminar boundary layer. Flow is from left to right. From Gad-el-Hak et al. (1984).

traveling-wave flutter (Carpenter, 1993; Lucey and Carpenter, 1993; Dixon et al., 1994), thus allowing softer panels to be used which further suppress TSI and improve the coating performance. In the flat-plate and similar boundary layers in low-disturbance environments, the quasi-two-dimensional Tollmien–Schlichting waves dominate the laminar-to-turbulence transition. Various receptivity processes are responsible for generating 2D instabilities that are probably initially threedimensional and randomly distributed. Low-disturbance environments could be realized, for example, in free flight and marine vehicles. Recently, Davies and Carpenter (1997a), and Carpenter (1998) have shown that in such environments, complete suppression of the Tollmien–Schlichting waves is possible provided that optimized multiple-panel compliant walls are used with each panel tailored to suit its local surrounding. Assured by the experimental observations that static-divergence waves are only observed when the wall-bounded flow is turbulent (see Fig. 11, reproduced from Gad-el-Hak et al., 1984),e Carpenter’s (1998) new assumptions are e In other words, one less instability mode to worry about when the flow is laminar and the objective is to keep it so.

382

M. Gad-el-Hak

somewhat less conservative than those used in earlier theories (Carpenter, 1993; Dixon et al., 1994). The new finding raises the possibility of maintaining laminar flow, in situations where the T–S instabilities are the primary cause of transition,f to indefinitely high Reynolds numbers, which is a very profound prospect indeed. Work on nonlinear stability theory has recently been in the forefront and has confirmed that transition-delaying coatings, optimized by using the linear theory, maintained their beneficial effects into the latter stages of transition to turbulence (Metcalfe et al., 1991; Joslin and Morris, 1992; Thomas, 1992a, 1992b). Lee et al. (1997) studied experimentally the effects of a compliant surface on the growth rates of both the subharmonic and three-dimensional fluid-based instabilities of a laminar boundary layer in air. Their results suggest that a delay of the excitement of the secondary instability can be achieved by suppressing the growth of the primary waves using surface compliance. 6.3. Practical examples Most of the theoretical as well as experimental compliant coating research has been concerned with canonical boundary layers. Nevertheless, an attempt has been made in here to estimate the potential benefit of applying the technique for field applications where strong three-dimensional and pressure-gradient effects and, for aeronautical applications, compressibility effects may be present. The typical Reynolds numbers, based on vehicle speed and overall length, for a hydrofoil, a torpedo and a nuclear submarine are, respectively, of the order of 10 million, 50 million and 1 billion. Applying an en -type calculations (with the exponent chosen conservatively to be n = 7) to an optimum two-panel, plate-spring-type compliant wall, Carpenter (1993) computes a transition Reynolds number of 13.62 × 106 , as compared with 2.25 × 106 for a rigid wall.g This means that the laminar region that would normally extend over 23%, 5% and 0.2% of the respective f Excludes, for example, flows where the walls and/or the streamlines are concave (G¨ ortler instabilities), where the boundary layer is three-dimensional and the cross-flow velocity profile is inflectional (cross-flow instabilities), or where the wall is rough or the freestream turbulence levels are unusually high (by-pass transition). g Contrast this six-fold increase to the 30% higher transition Reynolds number reported in the experiments of Gaster (1988), who did not attempt to optimize his single-panel, two-layer, silicone rubber/latex rubber coating. Theoretical calculations by Dixon et al. (1994) indicate that an optimum Gaster-type coating would provide a 500% higher transition Reynolds number.


383

vehicle lengths would, with the use of an optimum coating, extend over a larger length of 100%, 27% and 1%. Computing the corresponding overall drag coefficients using standard methods for a mixed laminar–turbulent boundary layer over a flat plate, the potential reduction in skin-friction drag using the optimum two-panel compliant wall can be as much as 83%, 19% and 0% for the three respective vehicles. Obviously the large submarine does not benefit, as far as drag reduction is concerned, from the use of transition-delaying compliant coating but the smaller vehicles do. However, extending the laminar region on a submarine even by 1 m can be significant for sonar applications requiring longer quiet regions of the boundary layer. For aeronautical applications, a cruising commercial jet aircraft has a fuselage Reynolds number of the order of 0.5 billion and a wing Reynolds number of the order of 50 million. Again, increasing the transition Reynolds number by a factor of 5 or so is significant for the wing, but not for the fuselage. Skin-friction reduction of the order of 20% is achievable for the wings (whose skin-friction drag accounts for about 50% of the skin friction of the entire aircraft and 25% of the total drag).h Finding a compliant coating that would reduce the turbulent skin-friction drag would of course be very beneficial for both the typical fuselage and long submarine. The estimates above were made for a simple plate-spring model. Using more than two panels can provide further transition delay. More complex compliant surfaces, particularly anisotropic ones designed specifically to suppress the Reynolds stress fluctuations, can conceivably offer more spectacular savings. Such custom-designed coatings can also favorably interact with fully-turbulent flows. Even for laminar flows, the calculations involved when complex wall-based models are used, though straightforward in principle, are quite demanding in practice. 6.4. The dolphin’s secret The ability to swim or to fly with minimum skin-friction and pressure drag is of extreme importance to the Darwinian survival of certain nektonic and avian species. Homo sapiens interested in building the fastest submarine or the most fuel-efficient aircraft have much to learn about alternative dragreducing approaches from their humble earthlings. As Kramer has remarked h Note that the dominant mechanism leading to transition on a swept wing is cross-flow instability and not T–S waves. However, recent evidence (Cooper and Carpenter, 1995; 1997a; 1997b; 1997c) indicates beneficial effects of compliant surfaces even for the former kind of instability.

384

M. Gad-el-Hak

close to half a century ago, a school of porpoises, including the young and old, the weak and strong, showing off its seemingly effortless glide along a fast ocean-liner is a sight to behold. Cetaceans appear to possess unusually low overall drag coefficients. This is the basis for the so-called Gray’s (1936) paradox, in which a steadystate energy balance based on the anticipated muscle power of various nekton, including the dolphin, failed to explain their unusually fast swimming speeds. Gray clocked bottlenose dolphins, Tursiops trucatus, swimming at speeds exceeding 10 m/s for a period of 7 s. If one assumes that cetaceans power output is equal to that of other mammals (∼35 W/kg of body weight), then such speeds are reached under turbulent flow conditions only if dolphins can expend several times more power than their muscles can generate. Lang (1963) concluded that based on energy considerations, dolphins could not exceed a speed of 6 m/s for periods greater than 2 hours. Transition delay is of course an obvious, albeit arduous, technique for achieving about an order of lower magnitude for skin-friction drag. However, does the dolphin posses an exotic means by which such difficult flow control goal can be accomplished? Obviously the dolphin is not sharing its secrets with other fellow mammals. Kramer’s (1957, 1961) invention of a compliant coating tried to mimic the dolphin’s epidermis and he claimed a drag reduction of as much as 60%. His explanation for the dolphin’s secret is that their skin, like his successful compliant coating, is capable of substantially delaying laminar-to-turbulence transition. Kramer’s work was discredited for a while, but now it seems to be back in vogue as remarked in Sec. 5. The calculations presented in Sec. 6.3 indicate that it is quite conceivable to design a Kramer-type coating that delays transition by a factor of 4–6 in Reynolds number and that drag reduction of the order reported by Kramer is also quite possible. Does the dolphin or other similar fast swimmers posses such a coating? In a recent article, Bushnell and Moore (1991) quote the relevant energetic and controlled swimming studies, but conclude by supporting the explanation offered by Au and Weihs (1980) that dolphins, which must periodically breath air, achieve high-speed swimming by simply porpoising, i.e. momentarily leaping out of the water thereby reducing their drag force by a factor of 800 (density ratio of air and water). This more than pays for the additional interfacial or wave drag and accounts for the abnormally low apparent drag-coefficients inferred from the assumption of fully-submerged travel.


385

The present author, however, does not concur with the above final solution to the Gray’s paradox. Dolphins have been clocked at sustained and burst speeds of close to 10 m/s and 20 m/s, respectively. Delphinus delphis has a typical length of 2 m. This leads to sustained and burst Reynolds numbers based on the overall length of the order of 20 million and 40 million, respectively. Carpenter (1993) reports the results of optimizing a rather simple plate-spring coating. Using a single panel, as compared with a rigid surface, a 4.6-fold increase in transition Reynolds number is estimated, which leads to a drag reduction of 36% at the typical dolphin’s sustained speed and 20% at burst speed. Using a mere two-panel coating, the transition Reynolds number becomes 6.1 times the value for a rigid surface, and the potential drag reductions for the sustained and burst speeds are now 52% and 30%, respectively. These lower levels of skin friction are compatible with the available muscle power for a dolphin of the size used above. Admittedly, the above estimates were made for a flat-plate boundary layer and may not hold when pressure-gradient and other shape effects are taken into account. Additionally, the dolphin has also pressure drag on top of the (much larger) skin friction. On the other hand, cetaceans have had millions of years of evolutionary adaptations to hone their coatings for maximum speed and efficiency, and it is quite conceivable that their epidermis is considerably more complex, and hydrodynamically beneficial, than the simple ones computed in the examples above. Moreover, each portion of the skin could have been optimized for the appropriate range of local Reynolds numbers. Therefore, the dolphin’s apparent success is not incompatible with having optimum compliant coatings to substantially delay laminar-to-turbulence transition, and therefore to attain inordinately low coefficients of drag. A very recent theoretical investigation by Carpenter et al. (2000) agrees with our basic conclusion. Finally, the chapter by Babenko and Carpenter (2003) provides much more details on the subject of dolphin hydrodynamics. Other fascinating questions related to the amazing swimming abilities of the dolphin include the possibility that its excreted mucin is a dragreducing additive. Is there a hydrodynamic advantage to the warm-blooded cetaceans because their epidermis temperature is higher than the ambient one (in which case the near-wall water viscosity is lowered and the turbulent boundary layer may be relaminarized)? Does the dolphin’s particular body shape during coasting (with no attendant overall body deformation) or actual swimming (accompanied by appropriate body oscillations) offer

386

M. Gad-el-Hak

additional drag-reducing advantages? Also, what are the potential benefits to the porpoise when it uses ship-generated bow waves for body surfing? These subjects, though related to the above discussion, are outside the scope of the present brief and are therefore, left for another circumstance.

7. Turbulent Wall-Bounded Flows Unlike the laminar and transitional flows investigated in Sec. 6, compliant coating effects on turbulent boundary layers are rather difficult to study theoretically. In fact, any turbulent flow is largely unapproachable analytically. For a turbulent flow, the dependent variables are random functions of space and time, and no straightforward method exists for analytically obtaining stochastic solutions to the governing nonlinear, partial differential equations. The statistical approach to solving the Navier–Stokes equations always leads to more unknowns than equations (the closure problem), and solutions based on first principles are again not possible. Direct numerical simulations (DNS) of the canonical turbulent boundary layer have thus far been carried out up to a very modest momentum-thickness Reynolds number of 1410 (Spalart, 1988). How would one go about rationally choosing a coating to achieve a particular control goal for a turbulent boundary layer? Analytical optimization procedures such as those used to delay transition (Sec. 6.2) would not work for fully-turbulent flows. In order to analyze the full problem, direct numerical simulations of the turbulent boundary layer should be coupled to a finiteelement model of the compliant coating. This is a task that is extremely time consuming, expensive and taxes the fastest supercomputer around. Modeling the turbulence by an eddy-viscosity or even a more sophisticated closure scheme is less computationally demanding, but there is no guarantee that turbulence models developed primarily for rigid surfaces would work for a compliant surface. In fact, it is not difficult to argue that closure models based on mean quantities miss completely the all-important spectral contents of a fluid–solid interaction, and will therefore never work. A turbulent boundary layer is characterized by a hierarchy of coherent structures. Near the wall, the dynamics are dominated by the quasi-periodic bursting events (Robinson, 1991). A crude, albeit resourceful, attempt to model a turbulent boundary-layer interaction with a single-layer, isotropic, viscoelastic coating has been advanced by Duncan (1986). He approximates the turbulent flow over the coating by a potential flow with a superimposed pressure pulse, convecting downstream, that mimics the pressure


387

footprint of a single bursting event. In order to relate the problem to a real turbulent flow, the pressure pulse characteristics are taken from actual boundary-layer measurements and the potential flow is modified to incorporate the reduced magnitudes and phase shift found experimentally in boundary-layer flows over moving wavy walls. At low flow speeds (relative to the transverse wave speed in the solid), the coating response to the pressure pulse is stable and primarily localized under it. At intermediate speeds, the response is still stable but includes a discernible wave pattern tagging along behind the pressure pulse. At the highest speed studied, large-amplitude, unstable waves develop on the compliant surface, much the same as the FISI observed experimentally. Duncan and Sirkis (1992) have recently extended the above model to anisotropic compliant coatings. They report that certain anisotropic surfaces provide more effective control over the amplitude and angular extent of the generated stable response pattern. Larger amplitudes are generated as compared with isotropic surfaces, thus providing for greater potential for modifying the turbulence. Whenever the flow speed in a turbulent boundary layer becomes sufficiently large compared with the transverse free wave speed in the solid, flowinduced surface instabilities proliferate. The pressure fluctuations within the flow are an order of magnitude larger than the normal and tangential viscous stresses, and drive the coating response. In laminar wall-bounded flows, it is difficult to observe the hydroelastic waves in their unstable state. As soon as flutter or divergence waves grow, rapid breakdown to turbulence takes place in the boundary layer and the flow is no longer laminar. Most of the experimental studies concerning compliant coating effects on turbulent boundary layers focused on documenting the unstable flowinduced surface instabilities. When divergence waves or flutter are unstable, the effects, though adverse, are pronounced and are somewhat easier to document. Only recently few hardy souls have attempted to investigate the wall-bounded flows when these FISI are stable or neutrally stable. Obviously the latter kind of studies have to await the development of refined techniques to measure the minuscule surface deformation and the associated coherent structure modulation when the FISI are neutrally stable. Both Gad-el-Hak (1986a; 1986b) and Hess et al. (1993) introduce nonintrusive methods for the point measurement of the instantaneous vertical surface-displacement of a compliant coating, while Lee et al. (1993b) offer an optical holographic interferometer, in connection with an interactive fringe-processing system, to capture whole-field random topographic features. The latter technique is more expensive to set up but offers higher

388

M. Gad-el-Hak

spatial resolution, of the order of 1 micron, and yield simultaneous surface displacement information on a large section of the compliant coating. Both the local and global methods were initially employed to document the unstable surface response to the pressure fluctuations in turbulent boundary layers. The holographic interferometer was recently used to record the surface topography in the presence of stable flow-induced deformations (Lee et al., 1993a). The onset speed and wave characteristics of the solid-based class B and class C instabilities were systematically documented in a series of towingtank experiments (Gad-el-Hak, 1986b). Divergence waves were observed on a single-layer viscoelastic coating made from a PVC plastisol. The flutter appeared on an elastic coating made from common household gelatin but, in the absence of damping, its threshold speed was consistently lower than that for divergence (Fig. 12). The damping in the PVC coating stabilized the traveling-wave flutter and hence only divergence was observed

Fig. 12. Onset speed dependence on thickness. , traveling-wave flutter on an elastic coating; , static-divergence waves on a coating with damping. From Gad-el-Hak (1986b).


389

Fig. 13. Typical unstable displacements of an elastic coating (———) and viscoelastic coating (−−−−). From Gad-el-Hak (1986b).

there.i For the elastic coating, flutter appeared first and dominated the observed surface deformation. For both kind of waves, the threshold speed decreases with coating thickness. In other words, thin surfaces (relative to the displacement thickness of the boundary layer) are less susceptible to hydroelastic instabilities than thick ones. Typical profiles of unstable class B and class C waves were also recorded in the same hydrodynamic experiments using a laser displacement gauge (Fig. 13). The vertical displacement at a point associated with the slow i Parenthetically,

this and similar earlier observations led Gad-el-Hak (1986a) and others to the wrong conclusion, as stated in the footnote in Sec. 4, regarding the classification of the divergence waves. To reiterate, the class B flutter is stabilized by damping, while the class C divergence is largely unaffected.

390

M. Gad-el-Hak

moving, asymmetric, large-amplitude divergence waves contrasts the faster, more-or-less symmetric, smaller-amplitude flutter. Both types of waves cause roughness-like effect, but the static divergence is the more dangerous instability. The phase speed of the static-divergence waves is of the order of 1% of the freestream speed, and their wavelength is about 5–10 times the coating thickness. The corresponding quantities for the flutter are 40% and 1.5–3, respectively. Hess (1990) and Lee et al. (1993a) also investigated compliant coating effects on turbulent boundary layers. Both experiments were conducted in the same water tunnel, but the second paper focused on the stable interaction between the fluid and a single-layer, homogeneous, viscoelastic coating made of a mixture of silicone rubber and silicone oil. Lee et al.’s coating was chosen based on the criterion established by Duncan (1986). In the presence of a stable wave pattern on the compliant surface, the flow visualization experiments indicated low-speed streaks with increased spanwise spacing (by as much as 80%; see Fig. 14) and elongated spatial coherence compared with those obtained on a rigid surface. More significantly, for the particular compliant coating investigated, an intermittent relaminarization-like phenomenon was observed at low Reynolds numbers. Lee et al. (1993a) also 220

180

λ+ 140

100

60 0

1000

2000

3000

4000

5000

6000

Rθ Fig. 14. Variations of mean dimensionless spanwise spacing of the low-speed streaks in a turbulent boundary layer. •, rigid surface; , compliant surface, friction velocity uτ obtained from rigid-surface mean-velocity measurements; , compliant surface, uτ obtained from compliant-surface mean-velocity measurements. These data are from the water tunnel experiments of Lee et al. (1993a). Other symbols in the figure are from classical rigid-surface measurements.


391

reported a slight thickening of the buffer region and viscous sublayer, and an upward vertical shift in the compliant law-of-the-wall. The streamwise turbulence intensity, the local skin-friction coefficient and the Reynolds stress across the boundary layer were all reduced, indicating a possible interruption of the feedback loop that allows the turbulence to be self-sustaining. Thus, potentially favorable interaction between a compliant coating and a turbulent boundary layer has been demonstrated for the first time. The more recent hydrodynamic experiments by Choi et al. (1997) provided additional evidence for favorable interaction and indicated a total drag reduction for a long, slender body of revolution of the order of 7%. 8. What Works and What Does Not? For close to half a century, compliant coating research has fascinated, frustrated and occasionally gratified scientists and engineers searching for methods to delay laminar-to-turbulence transition or to reduce skin-friction drag in turbulent wall-bounded flows. Through the years, claims for substantial drag reduction were made, only to be later refuted when the results were more critically examined. There are several important issues with regard to the reliability of available analytical, numerical and experimental results. Problem formulation with proper boundary conditions particularly at the fluid–solid interface, analytical solutions when the fluid flow is turbulent and limitations of existing numerical simulations are all problematic. The effects of background turbulence in a wind or water tunnel, accurate drag measurements, compliant wall motion, and the geometry and properties of the coating used are among the outstanding experimental issues. In here, we provide a brief listing of what works and what does not in compliant coating research. The articles by Gad-el-Hak (1987) and Carpenter (1990) provide more details of the pitfalls commonly encountered in, respectively, experiments and stability calculations related to compliant coating research aiming specifically for drag reduction. Many of the items listed in this section can be equally said about any research topic, but are particularly acute for compliant coating research because of its complexity as well as its history of false starts.

8.1. Analytical research • The unsteady, three-dimensional, nonlinear problem coupling the Navier– Stokes equations in the fluid side and the Navier equation in the solid side is intractable.

392

M. Gad-el-Hak

• There is no known methodology by which stochastic solutions can be gleaned from the closed set of nonlinear, partial differential equations describing a turbulent flow. Therefore, there is no possibility of being able to obtain first-principles (i.e. without reverting to heuristic models to close the non-stochastic Reynolds-averaged equations of fluid motion) analytical solutions when the wall-bounded flow is turbulent (i.e. instantaneously random and three-dimensional). • Matching velocity and stress components at the unknown fluid–solid interface is a highly complex, nonlinear problem. • Linearizing the interface conditions has to be done cautiously. • Provided that the steady, laminar base flow is first determined analytically or numerically, stability calculations are doable if done with lots of TLC. • The stability problem is extremely stiff particularly at high Re and high G, thus requiring specialized algorithms. • Both linear and nonlinear stability calculations can, in principle, be carried out. However, the nonlinear problem is extremely computer intensive. 8.2. Numerical research • Without sufficient validation, any numerical result of the fluid–structure interaction problem is suspect at best and wrong at worst. • Numerical schemes that couple direct numerical simulations (DNS) in the turbulent flow side and finite-element (FE) algorithms in the compliant wall side are extremely expensive, and for the foreseeable future, perhaps even impractical at field Reynolds numbers and realistic geometries. DNS–FE is possible, albeit expensive, at laboratory-scale Reynolds numbers and simple geometries. Strategies to extrapolate properly the low-Reynolds-number DNS to high Re are yet to be developed. • Periodic boundary conditions, while convenient, cannot be used since the compliant coating is capable of propagating waves upstream as well as downstream. One can argue that this restriction can be relaxed if the propagation speeds of the solid waves are much higher than the convective speeds in the fluid side. However, by its very definition, a solid is compliant when its characteristic wave speed is of the same order of magnitude as the flow speed. • Similar comments apply when attempting to parabolize the momentum equation in the fluid side. • Large-eddy simulations (LES) is unable to resolve the all important fluid– solid interface region. Hybrid DNS (at the interface region) and LES


393

(elsewhere in the flow) may be possible, but such strategies are yet to be developed properly. The hybrid DNS–LES in the fluid side must, of course, still be coupled with a finite-element algorithm in the solid side. • Averaging the instantaneous equations of fluid motion and modeling the Reynolds stresses to close the resulting Reynolds-averaged Navier– Stokes equations (RANS) will not work in principle — though is about the only analytical/numerical tool available in practice to handle highReynolds-number turbulent flows over compliant coatings — because the averaging process eliminates the all important frequency contents of the instantaneous flow. 8.3. Experimental research • Trial-and-error approach to finding a drag-reducing compliant coating does not work. • Optimization strategies for rationally selecting a compliant coating capable of substantially delaying laminar-to-turbulence transition do work. In principle at least, similar strategies can be developed to design the optimum compliant coating to achieve skin-friction reduction in a turbulent flow. • Sloppy experiments do not work. • Meticulously executed experiments do work if guided by analysis and keen understanding of the physics. • Choose test rig with care. Natural bodies of water offer relatively quiet background turbulence, but are more difficult to control. Water tunnels are perhaps too noisy to enable well-controlled transition experiments. Towing tank may be a better alternative when studying transitional flows. • The test rig’s background turbulence is less of an issue when studying the interaction of a compliant coating with a turbulent boundary layer. • Flat-plate geometry provides a well-documented flow in the rigid case and does not add unnecessary three-dimensional effects. • Rotating disks can also be used as they combine the test rig and the flow geometry into one relatively inexpensive, well-documented configuration. The resulting flow is three-dimensional, however. • The coating’s physical properties such as shear modulus and loss tangent have to be carefully measured and can change as the coating ages. • Particularly for field applications, wearing, aging and fouling are important issues that must be considered, monitored and controlled. • Pressure gradient has to be controlled and monitored as it affects the perceived drag reduction. Minute changes in the pressure distribution over the compliant body can lead to false reading of drag reduction.

394

M. Gad-el-Hak

• Edges, gaps and steps as the compliant slap joins the rigid surface have to be avoided. Texture and surface roughness of the coating have to be documented and taken into account when comparing to the drag of a smooth rigid surface. • Surface-displacement measurements have to be done with non-invasive probes, such as the laser-photodiode strategy advanced by Gad-el-Hak et al. (1984). • Local skin-friction measurements cannot be conducted with surfacemounted probes since these inevitably disturb the surface compliance. • Integrating mean velocity profiles at different downstream locations to compute the drag does not offer the required accuracy. • Computing the skin friction by measuring the velocity profile extremely close to the wall is problematic for rigid walls and is not any less so for moving, compliant surfaces. • Drag balances can be used over an area encompassing the entire compliant slap, but have to be very precise and unaffected by surface deformation.

9. The Future The diminishing pool of researchers active in the field of compliant coatings includes teams from the University of Warwick, University of Nottingham, Johns Hopkins University, University of Houston, University of Maryland and the Institute of Thermophysics in Novosibirsk. A larger pool was involved during the early 1980s, but the realities of research funding combined with the checkered past of the field led to the present decline. Few suggestions for future research are given in here. The optimization procedures discussed in Sec. 6.2 have not been validated experimentally. Gaster-type experiments should be repeated using optimized coatings, including multi-panel ones. The recent claims by Davies and Carpenter (1997a), and Carpenter (1998) regarding the possibility of maintaining laminar flow to indefinitely high Reynolds numbers are very profound. Experiments, particularly field ones in low-disturbance environments, specifically designed to test those claims would be extremely useful. The results of the transitional boundary layer, wind tunnel experiments reported by Lee et al. (1995, 1997) are intriguing and fly-in-the-face of the conventional wisdom. They indicate that compliant coatings are capable of delaying transition even for air flows. Past calculations using a platespring model and considering the extremely large density of typical walls


395

compared with the density of air indicated that very flimsy coatings would be required to achieve transition delay and that the situation gets worse as the air speed increases. This led Carpenter (1990) among others to conclude that the use of wall compliance is impractical for aeronautical applications.j But the plate-spring results do not apply in any straightforward way to the homogeneous, single-layer walls studied by Lee et al. (1995). Validating the recent favorable results using both independent experiments and numerical simulations would open the door for aerodynamic applications, something that was seriously considered but later abandoned by NASA and the aerospace industry. The optimization procedures developed by Carpenter (1988) for transitional hydrodynamic flows should be extended to air flows. Experiments should be conducted using the resulting optimized coatings, if it is feasible to construct such material. More complex coatings could potentially yield superior performance as compared with the relatively simple walls studied thus far. Multi-panels, multi-layers, anisotropic coatings and combinations thereof should be investigated. In any such research program, experiment has to be guided by theoretical results. As already mentioned, trying to pick a compliant coating by trial and error is a very inefficient use of limited resources and will perhaps never work. Favorably modulating a fully-turbulent flow, in contrast to merely delaying transition, is also of great practical importance. The experimental results reported by Lee et al. (1993a) are very encouraging, but the coating used was chosen based on a rather simplistic model of the turbulence pressure fluctuations. In order to custom-design compliant coatings to achieve particular control goals for turbulent wall-bounded flows, direct numerical simulations of the coupled fluid–structure system have to be performed. Turbulence modeling via classical closure schemes, while sufficient for some simple flows over rigid surfaces, will perhaps not yield reliable results for compliant walls. DNS, on the other hand, requires extensive computer resources and is quite expensive to carry out. The bottom line is that relatively large investment in resources are required for this task, but the enormous potential payoffs could easily justify the expenditure. Most of the research thus far has considered incompressible, zeropressure-gradient, flat-plate boundary layers. Effects of compressibility, pressure gradient and three-dimensionality on the performance of compliant j It should be noted that Carpenter’s calculations were based on the Navier equation modified for the viscoelasticity of the coating.

396

M. Gad-el-Hak

coatings are largely unknown. Such studies will yield invaluable information for field application of the control technique for both air and water flows. Most practical aerodynamic flows are in the moderate-to-high Mach number regime, and compressibility effects must therefore be investigated before compliant coatings are used on actual aircraft. Related to the pressuregradient effects is the question of separated flows: does compliant coating affect separation favorably or adversely? Other stability modifiers, such as favorable pressure-gradient, suction or heating/cooling, do delay transition as well as prevent separation. It is not known whether compliant coatings also have this dual benefit and it may be beneficial to research the possibility. Finally, real flows are three-dimensional and involve complex geometries. A model problem for three-dimensional flows is the rotating disk. Few experiments have been conducted using a rotating disk with a compliant face (Hansen and Hunston, 1974). More recently, Cooper and Carpenter (1995; 1997a; 1997b; 1997c) analyzed the cross-flow (type I; inviscid) as well as the viscous (type II) fluid-based instabilities which develop in the same three-dimensional flow. The preliminary results are encouraging and indicate that compliant coatings can suppress the more dangerous type I instabilities. Very recently, Colley et al. (1999) conducted an experiment to investigate transition over a rotating, compliant disk. Their results qualitatively agree with existing theories and show that the particular elastic coating they used has an overall destabilizing effect on the flow and results in substantially lower transitional Reynolds number compared to the case of a rigid disk. The lesson here is important: compliant coatings do not always yield beneficial results. Active compliant coatings, or more properly internally driven flexible coatings, though bringing us back to the extreme complexity of reactive control systems, is an emerging area deserving of further research. Energy expenditure is required to drive the wall, but the potential for significant net drag reduction is higher than that for passive coatings. The feasibility of the concept for stabilizing laminar boundary layers has been shown through numerical experiments (Metcalfe et al., 1986). Internally driven flexible coatings could also be used to suppress the Reynolds stress and reduce the skin-friction drag in turbulent wall-bounded flows, but any realistic field application of the technique has to await further development of reasonablypriced and rugged microfabricated sensors and actuators (Gad-el-Hak, 1994, 1996b, 2000, 2001).


397

10. Parting Remarks Passive compliant coatings present a much simpler alternative to reactive flow control strategies aimed at favorably interfering with wall-bounded flows. The last 10–15 years witnessed renewed interest in compliant coatings as a means to achieve beneficial flow control goals. Significant advances were made in numerical and analytical techniques to solve the coupled fluid– structure problem. Novel experimental tools were developed to measure the stable as well as the unstable surface deformations caused by the pressure fluctuations in the boundary layer. In turbulent wall-bounded flows, coherent structures were routinely identified and their modulation by wall compliance could be quantified. Most significant results in the research field thus far were obtained when a strong cooperation existed between theory and experiment. Recent theoretical work indicates that complete suppression of the Tollmien–Schlichting waves may be possible, provided that optimized multiple-panel compliant walls are used. The new finding raises the possibility of maintaining laminar flow to indefinitely high Reynolds numbers, which is a very profound prospect indeed. Recent experiments indicate favorable compliant coating interactions for aerodynamic flows and even for turbulent boundary layers. More research is needed, however, to confirm these latest results. The coupled system instabilities are now well understood, and compliant coatings can therefore be rationally designed to achieve substantial, perhaps even indefinite, transition delay in hydrodynamic flows. That recent shift from random to rational search for the right kind of coating, particularly if it can be extended to tubulent flows, is not unlike the great paradigm change in synthetic chemistry that took place near the beginning of the twentieth century. Increased understanding of the molecular geometry of organic compounds changed the scene from a hapless alchemist muddling around hoping to chance the right combination of ingredients, heat, pressure and catalysts to produce something useful, to a professional chemist figuring out what is wanted and working backward from the shape of a desired molecule for, say, a synthetic hormone. In fact, the present analogy is apt: the fluid dynamist working with the Navier–Stokes equations can tell the chemist the exact properties of the compliant coating to be synthesized to achieve a given goal. If the needed molecular structure is too complicated, futuristic nanoscale machines can assemble the required molecules directly, atom by atom.

398

M. Gad-el-Hak

Portions of the present chapter were presented as the keynote address at the IUTAM Symposium on Flow in Collapsible Tubes and Past Other Highly Compliant Boundaries, Coventry, England, 26–30 March 2001, and appeared as a journal article in Progress in Aeropsace Sciences 38 (2002) 77–99. References 1. Yu. G. Aleyev, Nekton (Dr. W. Junk b.v. Publishers, the Hague, the Netherlands, 1977). 2. D. Au and D. Weihs, At high speeds dolphins save energy by leaping, Nature 284(5756) (1980) 548–550. 3. V. V. Babenko and P. W. Carpenter, Dolphin hydrodynamics, Flow Past Highly Compliant Boundaries and in Collapsible Tubes, eds. P. W. Carpenter and T. J. Pedley (Kluwer, Dordrecht, The Netherlands, 2003), pp. 293–323. 4. V. V. Babenko and L. F. Kozlov, Experimental investigation of hydrodynamic stability on rigid and elastic damping surfaces, J. Hydraulic Res. 10 (1972) 383–408. 5. T. B. Benjamin, Effects of a flexible boundary on hydrodynamic stability, J. Fluid Mech. 9 (1960a) 513–532. 6. T. B. Benjamin, Fluid flow with flexible boundaries, Proc. 11th Int. Congr. Appl. Mech., ed. H. G¨ ortler (Springer-Verlag, Berlin, 1960b), pp. 109–128. 7. T. B. Benjamin, The threefold classification of unstable disturbances in flexible surfaces bounding inviscid flows, J. Fluid Mech. 16 (1963) 436–450. 8. T. B. Benjamin, Fluid flow with flexible boundaries, Proc. 11th Int. Congr. Appl. Mech., ed. H. G¨ ortler (Springer-Verlag, Berlin, 1966), pp. 109–128. 9. K. Betchov, Simplified analysis of boundary-layer oscillations, J. Ship Res. 4 (1960) 37–54. 10. E. F. Blick and R. R. Walters, Turbulent boundary-layer characteristics of compliant surfaces, J. Aircraft 5 (1968) 11–16. 11. A. C. Buckingham, M. S. Hall and R. C. Chun, Numerical simulations of compliant material response to turbulent flow, AIAA J. 23 (1985) 1046–1052. 12. T. E. Burton, The Connection Between Intermittent Turbulent Activity Near the Wall of a Turbulent Boundary Layer with Pressure Fluctuations at the Wall, Massachusetts Institute of Technology Report No. 70208-10 (Cambridge, Massachusetts, 1974). 13. D. M. Bushnell and K. J. Moore, Drag reduction in nature, Annu. Rev. Fluid Mech. 23 (1991) 65–79. 14. D. M. Bushnell, J. N. Hefner and R. L. Ash, Effect of compliant wall motion on turbulent boundary layers, Phys. Fluids 20(10) (1977) S31–S48.


399

15. P. W. Carpenter, The optimization of compliant surfaces for transition delay, Turbulence Management and Relaminarisation, eds. H. W. Liepmann and R. Narasimha (Springer-Verlag, Berlin, 1988), pp. 305–313. 16. P. W. Carpenter, Status of transition delay using compliant walls, Viscous Drag Reduction in Boundary Layers, eds. D. M. Bushnell and J. N. Hefner (AIAA, Washington DC, 1990), pp. 79–113. 17. P. W. Carpenter, Optimization of multiple-panel compliant walls for delay of laminar–turbulent transition, AIAA J. 31 (1993) 1187–1188. 18. P. W. Carpenter, Current status of the use of wall compliance for laminarflow control, Exp. Thermal & Fluid Sci. 16 (1998) 133–140. 19. P. W. Carpenter and A. D. Garrad, The hydrodynamic stability of flow over kramer-type compliant surfaces. Part 1. Tollmien–Schlichting instabilities, J. Fluid Mech. 155 (1985) 465–510. 20. P. W. Carpenter and A. D. Garrad, The hydrodynamic stability of flow over Kramer-type compliant surfaces. Part 2. Flow-induced surface instabilities, J. Fluid Mech. 170 (1986) 199–232. 21. P. W. Carpenter, C. Davies and A. D. Lucey, Hydrodynamics and compliant walls: Does the dolphin have a secret?, Current Science 79(6) (2000) 758–765. 22. P. W. Carpenter, A. D. Lucey and C. Davies, Progress on the use of compliant walls for laminar-flow control, J. Aircraft 38 (2001) 504–512. 23. K.-S. Choi (ed.), Recent Developments in Turbulence Management (Kluwer, Dordrecht, the Netherlands, 1991). 24. K.-S. Choi, X. Yang, B. R. Clayton, E. J. Glover, M. Atlar, B. N. Semenov and V. M. Kulik, Turbulent drag reduction using compliant surfaces, Proc. R. Soc. Lond. A453 (1997) 2229–2240. 25. H. H. Chu and E. F. Blick, Compliant surface drag as a function of speed, J. Spacecraft & Rockets 6 (1969) 763–764. 26. A. J. Colley, P. J. Thomas, P. W. Carpenter and A. J. Cooper, An experimental study of boundary-layer transition over a rotating, compliant disc, Phys. Fluids 11 (1999) 3340–3352. 27. A. J. Cooper and P. W. Carpenter, The effects of wall compliance on instability in rotating disc flow, AIAA Paper No. 95-2257 (Washington DC, 1995). 28. A. J. Cooper and P. W. Carpenter, The stability of rotating-disc boundarylayer flow over a compliant wall. Part 1. Type I and II instabilities, J. Fluid Mech. 350 (1997a) 231–259. 29. A. J. Cooper and P. W. Carpenter, The stability of rotating-disc boundarylayer flow over a compliant wall. Part 2. Absolute instability, J. Fluid Mech. 350 (1997b) 261–270. 30. A. J. Cooper and P. W. Carpenter, The effect of wall compliance on inflexion point instability in boundary layers, Phys. Fluids 9 (1997c) 468–470. 31. A. P. Daniel, M. Gaster and G. J. K. Willis, Boundary Layer Stability on Compliant Surfaces, Final Report No. 35020 (British Maritime Technology Ltd., Teddington, Great Britain, 1987).

400

M. Gad-el-Hak

32. C. Davies and P. W. Carpenter, Numerical simulation of the evolution of Tollmien–Schlichting waves over finite compliant panels, J. Fluid Mech. 335 (1997a) 361–392. 33. C. Davies and P. W. Carpenter, Instabilities in a plane channel flow between compliant walls, J. Fluid Mech. 352 (1997b) 205–243. 34. A. Dinkelacker, Preliminary experiments on the influence of flexible walls on boundary layer turbulence, J. Sound Vibr. 4 (1966) 187–214. 35. A. E. Dixon, A. D. Lucey and P. W. Carpenter, The optimization of viscoelastic compliant walls for transition delay, AIAA J. 32 (1994) 256–267. 36. J. A. Domaradzki and R. W. Metcalfe, Stabilization of laminar boundary layers by compliant membranes, Phys. Fluids 30 (1987) 695–705. 37. J. H. Duncan, The response of an incompressible, viscoelastic coating to pressure fluctuations in a turbulent boundary layer, J. Fluid Mech. 171 (1986) 339–363. 38. J. H. Duncan, A comparison of wave propagation on the surfaces of simple membrane walls and elastic coatings bounded by a fluid flow, J. Sound Vibr. 119 (1987) 565–573. 39. J. H. Duncan, The dynamics of waves at the interface between a two-layer viscoelastic coating and a fluid flow, J. Fluids Structures 2 (1988) 35–51. 40. J. H. Duncan and J. S. Sirkis, The generation of wave patterns on isotropic coatings by pressure fluctuations in a turbulent boundary layer, J. Sound Vibr. 157 (1992) 243–264. 41. J. H. Duncan and C. C. Hsu, The response of a two-layer viscoelastic coating to pressure disturbances from a turbulent boundary layer, AIAA Paper No. 84-0535 (New York, 1984). 42. J. H. Duncan, A. M. Waxman and M. P. Tulin, The dynamics of waves at the interface between a viscoelastic coating and a fluid flow, J. Fluid Mech. 158 (1985) 177–197. 43. M. C. Fischer and R. L. Ash, A General Review of Concepts for Reducing Skin Friction, Including Recommendations for Future Studies (NASA Technical Memorandum No. X-2894, Washington DC, 1974). 44. D. H. Fisher and E. F. Blick, Turbulent damping by flabby skins, J. Aircraft 3 (1966) 163–164. 45. L. A. Fraser and P. W. Carpenter, A numerical investigation of hydroelastic and hydrodynamic instabilities in laminar flows over compliant surfaces comprising one or two layers of viscoelastic material, Numerical Methods in Laminar and Turbulent Flow, eds. C. Taylor et al. (Pineridge Press, Swansea, Great Britain, 1985), pp. 1171–1181. 46. M. Gad-el-Hak, Boundary layer interactions with compliant coatings: an overview, Appl. Mech. Rev. 39 (1986a) 511–524. 47. M. Gad-el-Hak, The response of elastic and viscoelastic surfaces to a turbulent boundary layer, J. Appl. Mech. 53 (1986b) 206–212. 48. M. Gad-el-Hak, Compliant coatings research: A guide to the experimentalist, J. Fluids Structures 1 (1987) 55–70. 49. M. Gad-el-Hak, Interactive control of turbulent boundary layers: A futuristic overview, AIAA J. 32 (1994) 1753–1765.


401

50. M. Gad-el-Hak, Compliant coatings: A decade of progress, Appl. Mech. Rev. 49(10, Part 2) (1996a) S1–S11. 51. M. Gad-el-Hak, Modern developments in flow control, Appl. Mech. Rev. 49 (1996b) 365–379. 52. M. Gad-el-Hak, Flow Control: Passive, Active, and Reactive Flow Management (Cambridge University Press, London, United Kingdom, 2000). 53. M. Gad-el-Hak (ed.), The MEMS Handbook (CRC Press, Boca Raton, Florida, 2001). 54. M. Gad-el-Hak, R. F. Blackwelder and J. J. Riley, On the interaction of compliant coatings with boundary layer flows, J. Fluid Mech. 140 (1984) 257–280. 55. A. D. Garrad and P. W. Carpenter, A theoretical investigation of flowinduced instabilities in compliant coatings, J. Sound Vibr. 85 (1982) 483–500. 56. M. Gaster, Is the dolphin a red herring?, Turbulence Management and Relaminarisation, eds. H. W. Liepmann and R. Narasimha (Springer-Verlag, Berlin, 1988), pp. 285–304. 57. J. Gray, Studies in animal locomotion. VI. The propulsive powers of the dolphin, J. Exp. Biol. 13 (1936) 192–199. 58. R. J. Hansen and D. L. Hunston, An experimental study of turbulent flows over compliant surfaces, J. Sound Vibr. 34 (1974) 297–308. 59. D. E. Hess, An Experimental Investigation of a Compliant Surface beneath a Turbulent Boundary Layer, Ph.D. dissertation (Johns Hopkins University, Baltimore, Maryland, 1990). 60. D. E. Hess, R. A. Peattie and W. H. Schwarz, A nonintrusive method for the measurement of flow-induced surface displacement of a compliant surface, Exp. Fluids 14 (1993) 78–84. 61. P. Huerre and P. A. Monkewitz, Local and global instabilities in spatially developing flows, Annu. Rev. Fluid Mech. 22 (1990) 473–537. 62. R. P. Johnson, Review of Compliant Coating Research of M. O. Kramer, RPJ Associates Technical Report No. RPJA-TR-0955-001 (Palos Verdes Peninsula, California, 1980). 63. R. D. Joslin and P. J. Morris, Effect of compliant walls on secondary instabilities in boundary-layer transition, AIAA J. 30 (1992) 332–339. 64. R. E. Kaplan, The Stability of Laminar Incompressible Boundary Layers in the Presence of Compliant Boundaries, Sc.D. thesis (Massachusetts Institute of Technology, Cambridge, Massachusetts, 1964). 65. M. O. Kramer, Boundary-layer stabilization by distributed damping, J. Aeronaut. Sci. 24 (1957) 459–460. 66. M. O. Kramer, Boundary-layer stabilization by distributed damping, J. Aero/Space Sci. 27 (1960a) 69. 67. M. O. Kramer, Boundary layer stabilization by distributing damping, J. Am. Soc. Naval Engrs. 72 (1960b) 25–33. 68. M. O. Kramer, The dolphin’s secret, New Scientist 7 (1960c) 1118–1120. 69. M. O. Kramer, The dolphin’s secret, J. Am. Soc. Nav. Engrs. 73 (1961) 103–107.

402

M. Gad-el-Hak

70. M. O. Kramer, Boundary layer stabilization by distributed damping, J. Am. Soc. Naval Engrs. 74 (1962) 341–348. 71. M. O. Kramer, Hydrodynamics of the dolphin, Advances in Hydroscience, ed. V. T. Chow, Vol. 2 (Academic Press, New York, 1965), pp. 111–130. 72. M. O. Kramer, Die widerstandsverminderung schneller unterwasserkörper mittels k¨ unstlicher delphinhaut, Jahrbuch 1969 der Deutschen Gesellschaft f¨ ur Luft- und Raumfahrt, eds. H. Blenk and W. Schulz (K¨ oln, Federal Republic of Germany, 1969), pp. 1–9. 73. M. T. Landahl, On the stability of a laminar incompressible boundary layer over a flexible surface, J. Fluid Mech. 13 (1962) 609–632. 74. L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd edn. (Pergamon Press, Oxford, Great Britain, 1987). 75. T. G. Lang, Porpoise, whales, and fish: Comparison of predicted and observed speeds, Naval Eng. J. 75 (1963) 437–441. 76. T. Lee, M. Fisher and W. H. Schwarz, Investigation of the stable interaction of a passive compliant surface with a turbulent boundary layer, J. Fluid Mech. 257 (1993a) 373–401. 77. T. Lee, M. Fisher and W. H. Schwarz, The measurement of flow-induced surface displacement on a compliant surface by optical holographic interferometry, Exp. Fluids 14 (1993b) 159–168. 78. T. Lee, M. Fisher and W. H. Schwarz, Investigation of the effects of a compliant surface on boundary-layer stability, J. Fluid Mech. 288 (1995) 37–58. 79. T. Lee, M. Fisher and W. H. Schwarz, An experimental study of the boundary-layer nonlinear instability over compliant walls, unpublished manuscript originally submitted to J. Fluid Mech. (1997). 80. P. B. S. Lissaman and G. L. Harris, Turbulent skin friction on compliant surfaces, AIAA Paper No. 69-164 (New York, 1969). 81. R. W. Looney and E. F. Blick, Skin friction coefficients of compliant surfaces in turbulent flow, J. Spacecraft & Rockets 3 (1966) 1562–1564. 82. A. D. Lucey and P. W. Carpenter, The hydroelastic stability of threedimensional disturbances of a finite compliant panel, J. Sound Vibr. 165 (1993) 527–552. 83. A. D. Lucey and P. W. Carpenter, Boundary layer instability over compliant walls: comparison between theory and experiment, Phys. Fluids 7 (1995) 2355–2363. 84. J. M. McMichael, P. S. Klebanoff and N. E. Meese, Experimental investigation of drag on a compliant surface, Viscous Flow Drag Reduction, ed. G. R. Hough (AIAA, New York, 1980), pp. 410–438. 85. J. T. McMurray, R. W. Metcalfe and J. J. Riley, Direct numerical simulations of active stabilization of boundary layer flows, Proc. 8th Biennial Symp. Turbulence, eds. J. L. Zakin and G. K. Patterson (36) (University of Missouri, Rolla, Missouri, 1983).


403

86. R. W. Metcalfe, Boundary layer control: A brief review, in Computational Fluid Dynamics ’94, eds. J. Periaux and E. Hirschel (Wiley, New York, 1994), pp. 52–60. 87. R. W. Metcalfe, F. Battistoni, J. Ekeroot and S. A. Orszag, Evolution of boundary layer flow over a compliant wall during transition to turbulence, Proc. Boundary Layer Transition and Control Conf. (Royal Aeronautical Society, Cambridge, Great Britain, 1991), pp. 36.1–36.14. 88. R. W. Metcalfe, C. J. Rutland, J. H. Duncan and J. J. Riley, Numerical simulations of active stabilization of laminar boundary layers, AIAA J. 24 (1986) 1494–1501. 89. C. R. Nisewanger, Flow Noise and Drag Measurements of Vehicle with Compliant Coating, U.S. Naval Ordnance Test Station Report No. 8518, NOTS No. TP-3510 (China Lake, California, 1964). 90. S. A. Orszag, Prediction of Compliant Wall Drag Reduction, NASA Contractor Report No. 3071 (Washington DC, 1979). 91. F. W. Puryear, Boundary Layer Control: Drag Reduction by Use of Compliant Coatings, David Taylor Model Basin Report No. 1668 (Bethesda, Maryland, 1962). 92. Lord Rayleigh, On waves propagated along the plane surface of an elastic solid, Proc. Lond. Math. Soc. XVII (1887) 4–11. 93. J. J. Riley, M. Gad-el-Hak and R. W. Metcalfe, Compliant coatings, Annu. Rev. Fluid Mech. 20 (1988) 393–420. 94. H. Ritter and L. T. Messum, Water Tunnel Measurements of Turbulent Skin Friction on Six Different Compliant Surfaces of 1 Ft Length, British Admiralty Research Laboratory Report No. ARL/N4/GHY/9/7, ARL/G/N9 (London, Great Britain, 1964). 95. H. Ritter and J. S. Porteous, Water Tunnel Measurements of Skin Friction on a Compliant Coating, British Admiralty Research Laboratory Report No. ARL/N3/G/HY/9/7 (London, Great Britain, 1964). 96. S. K. Robinson, Coherent motions in the turbulent boundary layer, Annu. Rev. Fluid Mech. 23 (1991) 601–639. 97. P. K. Sen and D. S. Arora, On the Stability of laminar boundary-layer flow over a flat plate with a compliant surface, J. Fluid Mech. 197 (1988) 201–240. 98. R. L. Smith and E. F. Blick, Skin friction of compliant surfaces with foamed material substrate, J. Hydronautics 3 (1966) 100–102. 99. P. R. Spalart, Direct simulation of a turbulent boundary layer up to Rθ = 1410, J. Fluid Mech. 187 (1988) 61–98. 100. M. D. Thomas, On the resonant triad interaction in flows over rigid and flexible boundaries, J. Fluid Mech. 234 (1992a) 417–442. 101. M. D. Thomas, On the nonlinear stability of flows over compliant walls, J. Fluid Mech. 239 (1992b) 657–670. 102. G. J. K. Willis, Hydrodynamic Stability of Boundary Layers over Compliant Surfaces, Ph.D. thesis (University of Exeter, Exeter, Great Britain, 1986).

404

M. Gad-el-Hak

103. K. S. Yeo, The stability of boundary-layer flow over single- and multi-layer viscoelastic walls, J. Fluid Mech. 196 (1988) 359–408. 104. K. S. Yeo, The hydrodynamic stability of boundary-layer flow over a class of anisotropic compliant walls, J. Fluid Mech. 220 (1990) 125–160. 105. K. S. Yeo, The three-dimensional stability of boundary-layer flow over compliant walls, J. Fluid Mech. 238 (1992) 537–577. 106. K. S. Yeo and A. P. Dowling, The stability of inviscid flows over passive compliant walls, J. Fluid Mech. 183 (1987) 265–292.

NOISE SUPPRESSION AND MIXING ENHANCEMENT OF COMPRESSIBLE TURBULENT JETS

Dimitri Papamoschou Department of Mechanical and Aerospace Engineering University of California, Irvine Irvine, CA 92697-2700, USA E-mail: [email protected] New passive control schemes for reducing jet noise and thermal signature from aeroengines are described. Common to both techniques is the use of an axial flow to achieve the desired effect. Jet noise suppression is accomplished by reshaping the bypass plume of a turbofan engine so that downward sound emission from turbulent large-scale eddies is curtailed. Cumulative reduction in effective perceived noise level of around 8 decibels has been achieved in realistic sub-scale models of engines with bypass ratio 5. Mixing enhancement, required for reducing thermal signature, exploits instabilities associated with supersonic nozzle flow separation. Reduction of 60% in the potential core length, and substantial increase in the growth rate, has been demonstrated in round and 2D jets. Contents 1

Noise Suppression 1.1 Introduction 1.2 Noise reduction method 1.3 Sample results 1.4 Summary 2 Mixing Enhancement 2.1 Introduction 2.2 Supersonic nozzle flow separation 2.3 Mixing enhancement method 2.4 Sample results 2.5 Concluding remarks Acknowledgments References

405

406 406 407 409 412 412 412 413 415 417 419 419 420

406

D. Papamoschou

1. Noise Suppression 1.1. Introduction Aircraft noise is an issue of enormous environmental, financial and technological impact. There are three main sources of noise in today’s commercial aircraft engines: airframe noise, fan/compressor noise and jet noise. Jet noise comprises turbulent mixing noise and, in the case of imperfectly expanded jets, shock noise (Tam and Chen, 1994). Turbulent mixing noise is very difficult to control so its suppression remains a challenge. It is generally agreed that turbulent shear flow mixing causes two types of noise: sound produced by the large-scale eddies and sound generated by the fine-scale turbulence (Tam, 1998). The former is very intense and directional, and propagates at an angle close to the jet axis. The latter is mostly uniform and, affects the lateral and upstream directions. Sound generation from large-scale structures in the shear layer of a turbulent jet is analogous to sound generation by an instability wave (McLaughlin et al., 1975). Consider the following expression for an instability wave traveling with convective speed Uc : η(x, t) = A(x)ei(x−Uc t) .

(1.1)

The amplitude modulation A(x) accounts for the growth-decay behavior of disturbances in a growing (non-parallel) mean flow. Tam and Burton (1984) showed that A(x) creates a continuous spectrum of phase speeds. For finite convective Mach number Mc = Uc /a∞ , part of this spectrum is supersonic. Thus, even a subsonically traveling wave can generate Mach wave emission (Crighton and Huerre, 1990; Avital et al., 1998). Direct numerical simulation of a Mach 0.9 jet by Freund (2001) revealed that the radiating component of the noise source, at single frequency, is a modulated wave of the type captured by Eq. (1.1) and illustrated in Fig. 1. The creation of a continuous spectrum of phase speeds by the amplitude modulation A(x) becomes evident when we write η(x, t) in Fourier space, ∞ U 1 ˆ − 1)eik(x− kc ) dk. A(k (1.2) η(x, t) = 2π −∞ Equation (1.2) shows that the wavepacket is a superposition of individˆ − 1)dk/(2π) and ual simple waves, each with wavenunber k, amplitude A(k phase speed Uc /k. The phase Mach number of each individual wave is mc =

Mc Uc /k = . a∞ k

(1.3)

407

Noise Suppression and Mixing Enhancement of Jets

A(x)

Mc

x 2

mc A(k−1)

1 Radiating wavenumbers

−3

−2

0

−1

0

Mc

1

2

3

k

−1

mc −2

Fig. 1.

Phase Mach number versus wavenumber for Mc = 0.5.

Instabilities with |k| < Mc are supersonic and radiate to the far field; those with |k| > Mc are subsonic and decay rapidly with distance away from the jet axis. Figure 1 plots mc versus k for Mc = 0.5. Also shown is a ˆ − 1). The energy contained in the radiating sound sketch of a generic A(k ˆ − 1) from −Mc to Mc . field is governed by the integral of A(k Clearly, reducing Mc will reduce the amount of energy radiated to the far field. Source location experiments indicate that most of the large-scale turbulent mixing noise comes from the region near the end of the potential core (Panda and Seasholtz, 2002; Narayanan et al., 2002). Any scheme to reduce noise via reduction of the convective Mach number must take this fact into account. In other words, it is not sufficient to reduce Mc near the nozzle exit. It should be reduced throughout the high-speed region of the jet. The length of this region is on the order of 10–20 jet diameters. 1.2. Noise reduction method Today, practically all engines powering commercial and military aircraft are of the turbofan type. The existence of a secondary flow — the bypass

408

D. Papamoschou

Fig. 2. Basic elements of the mean flow in a dual-stream jet. i1 , i2 and i3 indicate the location of inflection points of the velocity profile.

stream — provides an opportunity for reduction of the convective Mach number Mc in order to suppress noise from large-scale turbulent structures. The initial region of a dual-stream jet consists of two shear layers: the primary shear layer between the primary and secondary streams, and the secondary shear layer between the secondary and ambient streams (see Fig. 2). The primary shear layer encloses the primary potential core. The region between the primary and secondary shear layers defines the secondary core which contains an initial potential region followed by a non-potential region. The extent of the secondary core can be defined in terms of the inflection points of the velocity profile, as illustrated in Fig. 2 (Papamoschou, 2004). The secondary core provides a buffer layer that reduces the convective Mach number Mc of the eddies in the primary shear layer. In coaxial jets produced by turbofan engines, the secondary core ends upstream of the end of the primary core. To reduce Mc throughout the jet, it is necessary to have a long secondary core that covers the entire primary potential core. Eccentric nozzle arrangements have proven very effective in this regard. The experiments of Murakami and Papamoschou (2002) showed that eccentric nozzles shorten the length of the primary potential core and double the


Fig. 3.

409

Principle of deflection of the bypass stream.

length of the secondary potential core on the underside of the jet. Acoustic experiments by Papamoschou and Debiasi (2001) demonstrated that downward noise emission of the eccentric jet was much lower than the noise emission of the equivalent coaxial jet. A more practical approach of stretching the secondary core involves tilting the secondary stream downward relative to the axis of the primary stream, while maintaining a coaxial nozzle exit. This is possible by means of fixed or variable vanes installed near the exit of the bypass duct, as illustrated in Fig. 3. The scheme is called Fan Flow Deflection. A comprehensive study of the mean flow of dual-stream jets by Papamoschou (2003) shows that tilting of the secondary flow produces an effect similar to that obtained by offsetting the nozzles. The vanes could be placed inside or outside the bypass duct. Placement inside the duct has the advantage of a subsonic environment and thus avoidance of serious shock losses. There is a limit as to how deep inside the duct one should place the vanes: the aerodynamic force of the vanes should be transmitted to the momentum flux exiting the duct and not to the duct walls. Otherwise, the effect of vane lift will be lessened or canceled by transverse forces acting on the duct walls. 1.3. Sample results The Fan Flow Deflection technique has been investigated extensively in the UCI Aeroacoustic Facility. Recent tests used a scaled-down version of the baseline separate-flow nozzle that was used in the Nozzle Acoustic Test Rig (NATR) of the NASA Glenn Research Center (Janardan et al., 2000). The

410

D. Papamoschou

flow lines and flow conditions were typical of those in modern turbofan engines with bypass ratio of around 5. The nozzle was attached to a dualstream apparatus that supplies mixtures of helium and air to the primary (core) and secondary (bypass) nozzles. The helium mass fraction and the total pressure of each mixture were determined by the desired exit velocity and Mach number. Comparison of the UCI acoustic results with those from the NATR facility (which runs large-scale hot jets) shows excellent agreement in all measures of noise: spectal shapes, spectral levels and overall sound pressure levels (Papamoschou, 2005). The Reynolds number of the jet, based on the fan diameter, was 0.6 × 106 . Far field noise measurements were conducted inside an anechoic chamber using a one-eighth inch condenser microphone (Br¨ uel & Kjær 4138) with a frequency response of 140 kHz. The microphone was mounted on a pivot arm and traced a circular arc centered at the jet exit. The polar angle θ ranged from 20◦ to 120◦ relative to the jet axis. Rotation of the nozzle assembly allowed variation of the azimuth emission angle. The azimuth angle typically took the values φ = 0◦ , 30◦ , 60◦ and 90◦ relative to the downward vertical. The sound spectra were corrected for the microphone frequency response, free field response and atmospheric absorption. Integration of the corrected spectrum yielded the overall sound pressure level (OASPL). The effective perceived noise level (EPNL), a metric used for aircraft noise certification, was also estimated. Figure 4 shows two types of deflectors used in the UCI tests. The first arrangement, denoted 4V, has two pairs of vanes installed inside the bypass duct. The azimuth angles of the vane pairs are φ = 70◦ and φ = 110◦ relative to the vertical. The angle of attach for all the vanes is 10◦ . The second arrangement, denoted Wi , uses a wedge-type deflector installed inside and at the top of bypass duct; the wedge half-angle is 17◦ . The OASPL directivity for case 4V is shown in Fig. 5. We note substantial reductions in the

(a)

(b)

Fig. 4. Illustrations of nozzles and deflectors used in acoustic testing. (a) Configuration with two pairs of vanes (4V); (b) Configuration with internal wedge-type deflector (W i ).


411

125 Baseline 4V (φ = 0o ) 4V (φ =60o)

OASPL (dB)

120

115

110

105

100 20

40

60

80

100

120

θ (deg) Fig. 5. Overall sound pressure levels for case 4V at microphone azimuth angles 0◦ and 60◦ .

downward OASPL for θ < 60◦ and in the sideline OASPL for θ < 40◦ . Small increases in OASPL are noted for θ > 90◦ . It is our experience that this “cross-over” in OASPL depends on the angle of attack of the vanes. As the angle of attack increases, the levels at low polar angles fall dramatically, but the levels in the forward arc increase. The sound increase at large θ, although moderate, can offset the EPNL benefit of deep reductions at low θ. As a result, the EPNL benefit reaches a plateau beyond a certain vane angle of attack. This observation pertains to vane configurations having the overall geometry of case 4V. Other vane arrangements may not exhibit the same trend. The EPNL reductions are 4.3 dB for flyover and 1.9 dB for sideline. The cumulative reduction is a respectable 6.2 dB but, if sideline noise reduction is a priority, this arrangement may not be optimal. On the other hand, this is a very promising configuration for reducing the EPNL measured by the takeoff monitor. We now examine the acoustics of the internal wedge deflector, case Wi . Figure 6 shows some fundamental differences between the sound emission of Wi and those of vane arrangement 4V. First, the sound reductions in the downward and sideline directions are roughly the same. Second, reductions persist up to large polar angles θ ≈ 80◦ . Third, there is very little crossover of OASPL at large θ. Even though the sound reduction at very low polar angles is not as dramatic as that of case 4V, the fact that significant sound reduction occurs over a large range of polar angles, combined with the little

412

D. Papamoschou

125 Baseline Wi (φ= 0o) Wi (φ=60o)

OASPL (dB)

120

115

110

105

100 20

40

60

80

100

120

θ (deg) Fig. 6. Overall sound pressure levels for case Wi at microphone azimuth angles 0◦ and 60◦ .

OASPL crossover for large θ, make this configuration acoustically superior to case 4V. This is evidenced by 4.5 dB flyover and 3.7 dB sideline. The cumulative EPNL reduction of 8.2 dB is very substantial. 1.4. Summary The technique presented here reshapes the mean flow field of the jet to create a long secondary core that reduces the convective Mach number throughout the noise source region of the primary jet. Downward tilting of the secondary plume relative to the primary plume is an effective means for such reshaping. Using the appropriate type of deflectors, noise suppression can be very substantial. Although the technique is intrinsically passive, it can be used in an actuated mode that provides the best noise benefit while minimizing aerodynamic losses. A lot more work is needed to go into understanding and modeling the mean flow field, the turbulent velocity field and the acoustic emission. 2. Mixing Enhancement 2.1. Introduction In aircraft exhaust systems, mixing enhancement has been motivated by the need to suppress noise and thermal emissions. Mechanical schemes have included lobe mixers (Westley et al., 1952) and vortex generators


413

(Zaman et al., 1994). The drawbacks are thrust penalty, weight penalty and complexity of the engine exhaust. Zaman (1999) has shown that thrust losses escalate with increasing jet Mach number and can easily reach the order of 10% in supersonic jets. Complexity of the engine exhaust means increased manufacturing and maintenance costs, and possibly increased radar cross-section. The desire to find alternatives to mechanical mixers has sparked research on fluidic control of jets. The counter-current shear layer of Strykowksi et al. (1996) demonstrated a substantial increase in the mixing rate compared to the classical flow. Application of a counter-flow on an engine exhaust has obvious drawback of momentum loss. Significant effort has also been directed to pulsed-jet control, where unsteady transverse jets, installed at the jet exit, destabilize the plume (Raman, 1997). While impressive increases in the jet entrainment rate have been recorded, practical implementation presents several challenges, including impact on system performance and loss of axial momentum. The mixing enhancement method described in this paper is different from the above techniques in two essential aspects: first, it does not use mechanical devices to directly disturb the flow; second, it uses parallel injection to destabilize the flow, in contrast with the counter-flow and transverseflow fluidic schemes mentioned above.

2.2. Supersonic nozzle flow separation The essential ingredient of the mixing enhancement technique in question is supersonic nozzle flow separation. This phenomenon occurs when a convergent-divergent nozzle is operated at pressure ratios well below its design point. A normal shock forms inside the nozzle and flow downstream of the shock separates from the nozzle walls. Even though separation is typically viewed as an undesirable occurrence, it is actually used here as a type of fluidic actuator. Papamoschou and Zill (1994) studied supersonic nozzle flow separation in a specially-designed facility at U.C. Irvine. Spark schlieren photography captured the instantaneous features of the shock system and the ensuing turbulent flow separation. Figure 7 shows an example. The shock has “lambda feet” near the walls. Each lambda foot comprising the incident shock reflected shock and the triple point where the incident and reflected shocks merge into the Mach stem. For moderate-to-large expansion angles and nozzle pressure ratios, the lambda feet occur asymmetrically, resulting in a large separation region on one side and small separation region on the

414

D. Papamoschou

Fig. 7. Details of shock and separated flow for nozzle area ratio 1.5 and nozzle pressure ratio 1.5.

other side. Another very important feature of the flow is the succession of weak normal shocks past main shock. The presence of shocks downstream of the main part of the separation shock indicates that flow accelerates to supersonic speed, recompresses, reaccelerates, etc. The above observations are summarized in the sketch of Fig. 8. The alternating expansion and compression waves interact with the separation shear layer and apparently destabilize it. The shear layer of the large separation zone is particularly Subsonic region

Triple point Slipstream

Shear layer Separation region

Fig. 8.

Conjecture on shock and fluid phenomena.


415

Fig. 9. Enhanced spark schlieren images of the external flow at nozzle pressure ratio around 1.5 and for nozzle area ratio: (a) 1.0; (b) 1.4.

unstable. This instability grows outside the nozzle and results in formation of very large eddies in the vicinity of the nozzle exit, as shown in Fig. 9. The mixing enhancement technique exploits this instability. 2.3. Mixing enhancement method As discussed in the previous section, the plume of a shock-containing, convergent-divergent nozzle exhibits strong unsteadiness which causes mixing enhancement in the plume. Importantly, the unsteadiness can be used to enhance mixing of an arbirtary flow adjacent to the plume. This phenomenon was discovered by Papamoschou (2000) in coannular jet experiments at U.C. Irvine and was subsequently confirmed in large-scale tests

416

D. Papamoschou

at NASA Glenn Research Center (Zaman and Papamoschou, 2000). The method of Mixing Enhancement via Secondary Parallel Injection (MESPI) is characterized by the simplicity of the mixer nozzles and is relevant to combustion and propulsion applications. More recently, MESPI was studied in high-aspect ratio rectangular jets using centerline pitot surveys, laserinduced fluorescence and surveys of the entire velocity field (Murakami and Papamoschou, 2001; Papamoschou et al., 2004). These types of jets have increasing relevance to the exhaust of modern military fighters and unmanned aerial vehicles. Figure 10 shows the generic geometry of a two-dimensional MESPI nozzle and of the reference nozzle against which the performance of the MESPI nozzle is judged. The primary (inner) flow is the same in both cases; its conditions are irrelevant to the instability mechanism. The secondary (outer) flows of the MESPI and reference nozzles are subjected to the same pressure ratio and are typically compared at the same mass flow rate. For the MESPI nozzle, the duct of the secondary flow is convergent-divergent with exit-to-throat area ratio H2 /H2∗ . Mixing enhancement occurs when the secondary flow reaches sonic speed at the throat and experiences an adverse pressure gradient near the nozzle exit. The secondary duct of the reference nozzle is convergent. For MESPI to work, the secondary nozzle pressure ratio must be such as to produce a normal shock (i.e. supersonic nozzle flow separation) inside the secondary nozzle. Separation models by Romine

H2*

H2

H1

R (a)

(b)

Fig. 10. Generic geometry of (a) MESPI nozzle and (b) reference nozzle. The inner streams are identical; the outer streams are supplied at the same pressure ratio and same mass flow rate.


417

(1998) help in arriving at a range of the desired nozzle pressure ratios. For a nozzle with exit-to-throat area ratio of 1.5, the nozzle pressure ratio ranges from about 1.2 to about 3.0 (Papamoschou, 2000). 2.4. Sample results This section presents some representative results of mixing enhancement in large-aspect ratio rectangular jets. The jets were generated in a triplestream apparatus exhausting a primary stream sandwiched between two secondary streams. The primary stream had exit height H1 = 6.35 mm and width of 50.8 mm, giving an aspect ratio of 8.0. The primary nozzle was designed by the method of characteristics for Mach 1.5 exit flow. The secondary ducts had variable exit height H2 , area ratio H2 /H2∗ and recess R (Fig. 10). Diagnostics included spark schlieren photography, planar laserinduced fluorescence and velocity field measurements using a rake of Pitot probes. Some of the most unstable flows were created by flowing only one of the secondary streams. Figure 11 shows illustrative spark schlieren images. The

Fig. 11. Spark schlieren images of rectangular jets. (a) From reference nozzle; (b) From MESPI nozzle.

418

D. Papamoschou

6

z=0 4

y/H1

2 0

0.9

0.8

0.7

6 0.

0.5

-2

0.4 0.3

-4 -6

0.1

0

10

20

30

40

0.2

50

60

x/H1 (a) 6

z=0 4

y/H1

2 0

0.9 0.8

0.7

0.6

0.5

-2

0.4

-4 -6 0

0.1 0.2

10

20

0.3

30

40

50

60

x/H1 (b)

Fig. 12. Isocontours of velocity (normalized by primary exit velocity). (a) For reference jet; (b) For MESPI jet.

reference jet is very stable and grows slowly. The MESPI jet exhibits a very strong instability that enhances mixing. Figure 12 shows the mean velocity field of reference and MESPI jets similar to those depicted in Fig. 12. The faster growth, and reduced potential core length, of the MESPI jet is evident. The decay of centerline velocity is a relatively simple metric for assessing the mixing rate of a jet. Figure 13 compares the centerline velocity

419


1.1 Reference

1.0

MESPI

u (x ,0,0)/U1

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0

10

20

30

40

50

60

x /H1 Fig. 13.

Centerline velocity distributions of reference jet and MESPI jet.

distributions of the velocity fields of Fig. 12. The potential core length of the MESPI jet is 60% shorter than that of the reference jet. The velocity decay past the end of the potential core is faster in the MESPI case. 2.5. Concluding remarks The instabilities arising from supersonic nozzle flow separation were exploited to generate a mixing enhancement scheme that uses simple and clean nozzles. The aerodynamics efficiency can be very high (Murakami and Papamoschou, 2001). Past works indicate that the area ratio of the secondary flow is the dominant variable that governs the intensity of mixing. However, other variables such as the recession of the secondary nozzle and the actual shape of the convergent-divergent passage also play a role. It is clear that we need to better understand the fundamental mechanisms in order to further improve this technique. We are currently engaged in such efforts by examining in detail the flow emerging from the separation region of a supersonic nozzle. Acknowledgments The author gratefully acknowledges the support from NASA Glenn Research Center, the National Science Foundation and the National University of Singapore.

420

D. Papamoschou

References 1. E. J. Avital, N. D. Sandham and K. H. Luo, Mach wave radiation in mixing layers. Part I. Analysis of the sound field, Theoret. Comput. Fluid Dyn. 12 (1998) 73–90. 2. D. G. Crighton and P. Huerre, Shear-layer pressure fluctuations and superdirective acoustic sources, J. Fluid Mech. 220 (1990) 355–368. 3. J. B. Freund, Noise sources in a low-Reynolds-number turbulent jet at Mach 0.9, J. Fluid Mech. 438 (2001) 277–305. 4. B. A. Janardan, G. E. Hoff, J. W. Barter, S. Martens, P. R. Gliebe, V. Mengle and W. N. Dalton, AST critical propulsion and noise reduction technologies for future commercial subsonic engines separate-flow exhaust system noise reduction concept evaluation, NASA CR 2000-210039 (2000). 5. D. K. McLaughlin, G. D. Morrison and T. R. Troutt, Experiments on the instability waves in a supersonic jet and their acoustic radiation, J. Fluid Mech. 69(11) (1975) 73–95. 6. E. Murakami and D. Papamoschou, Experiments on mixing enhancement in dual-stream jets, AIAA-2001-0668 (2001). 7. E. Murakami and D. Papamoschou, Mean flow development of dual-stream compressible jets, AIAA J. 40(6) (2002) 1131–1138. 8. S. Narayanan, T. J. Barber and D. R. Polak, High subsonic jet experiments: Turbulence and noise generation studies, AIAA J. 40(3) (2002) 430–437. 9. J. Panda and R. G. Seasholtz, Experimental investigation of density fluctuations in high-speed jets and correlation with generated noise, J. Fluid Mech. 450 (2002) 97–130. 10. D. Papamoschou, Mixing enhancement using axial flow, AIAA-2000-0093 (2000). 11. D. Papamoschou and M. Debiasi, Directional suppression of noise from a high-speed jet, AIAA J. 39(3) (2001) 380–387. 12. D. Papamoschou, A new method for jet noise reduction in turbofan engines, AIAA-2003-1059 (2003). 13. D. Papamoschou, New method for jet noise suppression in turbofan engines, AIAA J. 42(11) (2004) 2245–2253. 14. D. Papamoschou and A. Zill, Fundamental investigation of supersonic nozzle flow separation, AIAA-2004-1111 (2004). 15. D. Papamoschou, T. D. Dixon and K. A. Nishi, Mean flow of multi-stream rectangular jets under normal and mixing enhancement conditions, AIAA2004-0919 (2004). 16. D. Papamoschou, Acoustic simulation of hot coaxial jets using cold helium-air mixture jets, AIAA-2005-0208 (2005). 17. G. Raman, Using controlled unsteady fluid mass addition to enhance jet mixing, AIAA J. 35(4) (1997) 647–656. 18. G. L. Romine, Nozzle flow separation, AIAA J. 36(9) (1998) 1618–1625. 19. P. J. Strykowksi, A. Krothapalli and S. Jendoubi, The effect of counterflow on the development of compressible shear layers, J. Fluid Mech. 308 (1996) 63–96.


421

20. C. K. W. Tam, Jet noise: Since 1952, Theoret. Comput. Fluid Dyn. 10 (1998) 393–405. 21. C. K. W. Tam and D. E. Burton, Sound generated by instability waves of supersonic flows. Part 2. Axisymmetric jets, J. Fluid Mech. 138 (1984) 249–271. 22. C. K. W. Tam and P. Chen, Turbulent mixing noise from supersonic jets, AIAA J. 32(9) (1984) 1774–1780. 23. R. Westley and G. M. Lilley, An investigation of the noise field from a small jet and methods for its reduction, College of Aeronautics, Report 53, Cranfield University, England, UK (1952). 24. K. B. M. Q. Zaman, Jet spreading increase by passive control and associated performance penalty, AIAA-99-3505 (1999). 25. K. B. M. Q. Zaman and D. Papamoschou, Study of mixing enhancement observed with a co-annular nozzle configuration, AIAA-2000-0094 (2000). 26. K. B. M. Q. Zaman, M. F. Reeder and M. Samimy, Control of an axisymmetic jet using vortex generators, Phys. Fluids A6(2) (1994) 778–796.


INDEX

practical examples, 382 research prior to 1985, 361 turbulent flow over, 386 compressible flow, 179 conditional analysis, 257 conjugate gradient method, 151 conservative nonlinearity, 13, 31 continuous spectrum, 93–95 contribution componential, 303 convective, 309 cumulative, 306 laminar, 304, 319 turbulent, 304, 306, 307, 309, 319 viscoelastic, 309 control active feedback, 301 algorithm, 299 efficiency, 330 law, 300, 310 objectives, 145 optimal, 299 suboptimal, 300, 310 system, 301 theory, 299 cost functional, 145 counterclockwise vortices, 268 coupling, 103, 104 critical Reynolds number, 6

actuator, 299, 301 efficiency, 337 adjoint operator, 129 variables, 147 adjoint-based method, 142 algebraic growth, 88, 94 attached eddy, 289–292 attachment-line, 151 background subtraction, 214 balance breaking and exchange, 286 blowing/suction, 299, 303, 305, 307, 310 boundary layer, 304, 309 breathing mode, 94 bubble generation, 334 Burgers’ equation, 10, 17 bursting, 353 bypass transition, 120 channel flow, 302 coherent structure, 209 combined-plane PIV, 213 compliant coatings analytical research, 391 benefits, 359 dispersion curves, 365 dolphin’s secret, 383 experimental research, 393 future of, 394 instabilities of, 367 linear stability theory, 372 model for, 373, 374 numerical research, 392

detached eddy, 289–292 direct numerical simulation (DNS), 99, 177, 299, 305, 311 discrete mode, 87 dolphin’s secret, 383 423

424

double correlation function, 274 drag reduction, 142, 299, 306, 311 rate, 301, 312, 314 dynamic range, 212 ejection, 298 energy growth rate, 116, 130 stability analysis, 109 feedback control, 6, 7 FISI, 368 Floquet’s theorem, 126 flow control, 142 flow-induced surface instabilities, 368 fluid dynamical stability, 167 fluid–structure instabilities, 367 classification scheme, 367, 369, 370 footprint, 233 free-surface waves, 364 freestream turbulence, 123 functional gain, 8, 20, 21 G¨ ortler–H¨ ammerlin assumption, 153 global stability analysis, 134 Gray’s paradox, 384 grid turbulence, 95 hairpin packet, 207, 263 model, 226 signature, 265 structures, 249 vortex, 260 heat transfer, 317 hydrodynamic stability, 108 inclined-plane cross-stream PIV, 211 inner/outer interaction, 288, 290, 292 internal shear layer motion, 257 interrogation window, 212 jet noise reduction, 408 Kramer, 359, 369

Index

Lagrangian functional, 144 LEBU, 239 lift-up, 88, 102 linear stability theory, 372 applied to fluid–structure system, 373 linear stochastic estimation (LSE), 229 log region, 214 Lorentz force actuators, 330 low-speed regions, 215 streak strength, 278 width, 253, 277 LQR problem, 14 Lyapunov exponent, 130 matrix exponential, 111 mean velocity, 342 mean-momentum balance, 285, 289–293 meandering, 207, 235 MEMS, 299 mesolayer, 286, 287 mixing enhancement, 412, 413, 415 modal analysis, 111 multi-scale analysis, 284 Navier–Stokes equations, 2 non-modal analysis, 111 non-orthogonal expansion, 112 numerical range, 114 Nusselt number, 318 opposition control, 305 optimal control, 142, 159 energy growth, 156 wall properties, 377 optimality condition, 148 Orr–Sommerfeld equation, 91 Oseen operator, 5, 8 outer-scaling, 232

425

Index

Particle Image Velocimetry, 339 passive wakes, 231 pattern selection, 121 penetrating modes, 93 perturbation jets, 89, 94 pipe flow, 304 PIV, 207 pixel locking, 212 POD, 231 Poiseuille flow, 109 Prandtl number, 318 primal-dual formulation, 163 primary instabilities, 108 pseudo-wavepacket solution, 138 quadrant event, 264 quasi-streamwise vortex, 249, 298 rapid distortion analysis (RDT), 94 rapid distortion theory (RDT), 88 Rayleigh–Benard convection, 109 receptivity, 108 relaminarization, 142 resolvent norm, 116 resonant forcing, 97 Reynolds (shear) stress, 306, 311, 316, 319, 320 distribution, 301 instantaneous, 298 near-wall, 310–313, 319, 320 weighted, 306, 307, 309, 313 Reynolds number, 298 bulk, 302, 309 dependency, 314 effect, 302, 313, 314, 321 friction, 301, 313, 315 Reynolds stress, 236, 342 Reynolds-averaged Navier–Stokes equation, 302 robust control, 149 Schr¨ odinger-, 41 secondary instabilities, 108 sensor, 299, 301

shear layer structures, 353 sheltering, 91, 92 skin-friction coefficient, 301, 303 drag, 298, 307, 310, 317, 319 spanwise oscillation, 240, 327 repetition, 238 stripiness, 215 wall oscillation, 271 spatial Fourier spectrum, 253 theory, 131 spectral analysis, 253 Squire equation, 91, 93 stability theory, 109 stationary points, 146 steepest descent method, 151 streaks, 122 stress gradient balance layer, 284, 288 stripiness, 214 Stuart number, 329 sub/supercritical behavior, 118 superbursts, 216, 231 surfactant, 305, 309 sweep, 298 swept Hiemenz flow, 151 swirl, 221 swirling motion, 249 strength, 249 time-periodic flow, 126 time-resolved PIV, 270 Tollmien–Schlichting instabilities, 368, 374, 381 transient growth, 113 transition, 108, 170 bypass, 95 continuous mode, 104 thresholds, 123 TSI, 368, 371, 378, 381

426

turbulence near-wall, 298, 321 production, 342 wall-, 298, 301, 313 turbulent charge density, 294 heat flux, 318 mixing, 406 production positions, 47 spot, 97 two-point correlations, 218, 348

Index

streak spacing, 251, 277 streaks, 249 VISA event, 256, 263 VITA event, 255 vortex identification, 221 signature, 250 vortical disturbance, 91

unconstrained optimization, 145 uniform momentum zone, 266

wake region, 214 wakes incident, 101, 104 passing, 103 wall-parallel plane PIV, 210

velocity increment, 287, 288

zone of uniform momentum, 266

Transition and Turbulence Control

Turbulence

Turbulence

Turbulence

Turbulence

Turbulence

Turbulence

Turbulence

Turbulence

Turbulence

Turbulence

Turbulence

Turbulence

Plasma and fluid turbulence

Instabilities, Chaos and Turbulence

Instabilities, Chaos And Turbulence

Turbulence and Shell Models

Vorticity and turbulence

Vorticity and turbulence

Turbulence and Shell Models

Random walks and turbulence

Patterns and turbulence PR

Burgers turbulence

Turbulence Seminar

Intermittent turbulence

Burgers turbulence

Veibel turbulence

Interstellar turbulence

Magnetohydrodynamic turbulence

Transition and Beyond (Studies in Economic Transition)

Magnetohydrodynamic turbulence

Transition and Turbulence Control

Turbulence

Turbulence

Turbulence

Turbulence

Turbulence

Turbulence

Turbulence

Turbulence

Turbulence

Turbulence

Turbulence

Turbulence

Plasma and fluid turbulence

Instabilities, Chaos and Turbulence

Instabilities, Chaos And Turbulence

Turbulence and Shell Models

Vorticity and turbulence

Vorticity and turbulence

Turbulence and Shell Models

Random walks and turbulence

Patterns and turbulence PR

Burgers turbulence

Turbulence Seminar

Intermittent turbulence

Burgers turbulence

Veibel turbulence

Interstellar turbulence

Magnetohydrodynamic turbulence

Transition and Beyond (Studies in Economic Transition)

Magnetohydrodynamic turbulence

Recommend Documents