Applied Mathematical Sciences Volume 58

Applied Mathematical Sciences

1. John: Partial Differential Equations, 4th ed. 2. Sirovich: Techniques of Asymptotic Analysis. 3. Hale: Theory of Functional Differential Equations, 2nd ed. 4. Percus: Combinatorial Methods. 5. von Mises/Friedrichs: Fluid Dynamics. 6. Freiberger/Grenander: A Short Course in Computational Probability and Statistics. 7. Pipkin: Lectures on Viscoelasticity Theory. 9. Friedrichs: Spectral Theory of Operators in Hilbert Space.

11. Wolovich: Linear Multivariable Systems. 12. Berkovitz: Optimal Control Theory. 13. Bluman/Cole: Similarity Methods for Differential Equations. 14. Yoshizawa: Stability Theory and the Existence of Periodic Solutions and Almost

Periodic Solutions. 15. Braun: Differential Equations and Their Applications, 3rd ed. 16. Lefschetz: Applications of Algebraic Topology. 17. Collatz/Wetterling: Optimization Problems. 18. Grenander: Pattern Synthesis: Lectures in Pattern Theory, Vol I. 20. Driver: Ordinary and Delay Differential Equations. 21. Courant/Friedrichs: Supersonic Flow and Shock Waves. 22. Ftouche/Habets/Laloy: Stability Theory by Liapunov's Direct Method. 23. Lamperti: Stochastic Processes: A Survey of the Mathematical Theory. 24. Grenander: Pattern Analysis: Lectures in Pattern Theory, Vol. II. 25. Davies: Integral Transforms and Their Applications, 2nd ed. 26. Kushner/Clark: Stochastic Approximation Methods for Constrained and

Unconstrained Systems. 27. de Boor: A Practical Guide to Splines. 28. Keilson: Markov Chain Models—Rarity and Exponentiality. 29. de Veubeke: A Course in Elasticity. 30. Sniatycki: Geometric Quantization and Quantum Mechanics. 31. Reid: Sturmian Theory for Ordinary Differential Equations. 32. Meis/Markowitz: Numerical Solution of Partial Differential Equations. 33. Grenander: Regular Structures: Lectures in Pattern Theory, Vol. III. 34. Kevorkian/Cole: Perturbation Methods in Applied Mathematics. 35. Carr: Applications of Centre Manifold Theory. 36. Bengtsson/Ghil/Käll6n: Dynamic Meterology: Data Assimilation Methods. 37. Saperstone: Semidynamical Systems in Infinite Dimensional Spaces. 38. Lichtenberg/Lieberman: Regular and Stochastic Motion.

(continued on inside back cover)

Theoretical Approaches to Turbulence

Edited by D.L. Dwoyer M.Y. Hussaini R.G. Voigt

With 90 Illustrations

Springer Science+Business Media, LLC

D . L . Dwoyer M . Y . Hussaini R.G. Voigt ICASE NASA Langley Research Center Hampton, Virginia 23665 U.S.A.

AMS Subject Classification: 76FXX

Library of Congress Cataloging in Publication Data Main entry under title: Theoretical approaches to turbulence.

(Applied mathematical sciences; v. 58) Bibliography: p. 1. Turbulence—Addresses, essays, lectures.

I. Dwoyer, Douglas L. II. Hussaini, M. Yousuff. III. Voigt, Robert G. IV. Series: Applied mathematical sciences (Springer-Verlag New York Inc.); v. 58. QA1.A647 vol. 58 510 s [532'.0527] 85-14765 [QA913]

This is to certify that the papers authored by D.M. Bushneil, G.A. Chapman, and P. Moin were prepared in their roles as U.S. Government employees and are thus in the public domain.

© 1985 by Springer Science+Business Media New York Originally published by Springer-Verlag New York Inc. in 1985 All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag, 175 Fifth Avenue, New York, New York 10010, U.S.A.

9 8 7 6 5 4 3 2 1

ISBN 978-0-387-96191-0 ISBN 978-1-4612-1092-4 (eBook) DOI 10.1007/978-1-4612-1092-4

Preface

Turbulence is the lIDst natural nDde of fluid lIDtion, and has been the subject of

scientific study for all!Dst a century. During this period, various ideas and

techniques have evolved to nDdel turbulence. Following Saffman, these theoretical

approaches can be broadly divided into four overlapping categories -- (1) analytical

lIDdelling, (2) physical lIDdelling, (3) phenomenologicalllDdelling, and (4) nurerical

lIDdelling. With the purpose of stmtnarizing our =ent understanding of these

theoretical approaches to turbulence, recognized leaders (fluid dynamicists,

mathematicians and physicists) in the field were invited to participate in a formal

workshop during October 10-12, 1984, sponsored by The Institute for CooIputer

Applications in Science and Engineering and NASA Langley Research Center. Kraiciman,

McCcxnb, Pouquet and Spiegel represented the category of analytical nDdelling, while

Landahl and Saffman represented physical lIDdelling. The contributions of Latmder and

Spalding were in the category of phenanenological lIDdelling, and those of Ferziger

and Reynolds in the area of nurericalllDdelling. Aref, Cholet, Lumley, Moin, Pope

and Temam served on the panel discussions. With the care and cooperation of the

participants, the workshop achieved its purpose, and we believe that its proceedings

published in this vol\.llre has lasting scientific value.

The tone of the workshop was set by two introductory talks by Bushnell and

ChaImm. Buslmell presented the engineering viewpoint while Chapman reviewed from

a historical perspective developments in the study of turbulence. The remaining

talks dealt with specific aspects of the theoretical approaches to fluid turbulence.

We now stmtnarize these talks as reported in this vol\.llre.

Buslmell focuses attention on the control aspect of the turbulence problan.

First, he examines the canonical structure of turbulence in wall-bounded flows

gleaned from detailed flow visualization and conditional sarrpling IlE8.surements

conducted over the past twenty years. Then he discusses the sensitivities of these

flows to various control parameters (such as additives and micro and macro geometric

variations, etc.). Finally, he explains how these sensitivities provide the test

cases for turbulence theories.

Chapman and Tobak present a rather novel overview of turbulence theories in the

context of interactions between observations, theoretical ideas and nDdelling of

turbulent flows. With sc.me precaution and reservation, they asS\.llre that turbulence

could be studied within the frBlIle'iNOrk of Navier-Stokes equations. After providing

a brief historical background for turbulence studies, they lII9ke the case for three

distinct stages of development in the scientific study of turbulence. They call the

v

vi

earliest period the "statistical nr:wement" Mlen. turbulence was looked upon frem the

non-detenninistic point of view. nte second stage, called the "structural nr:wement",

started in the thirties and is essentially observational. Its principal contribution

is the recognition of the presence and iIqxJrtance of structures in turbulence. nte

third and IIDst recent stage originated in the sixties, and is called the "determi­

nistic nr:wement". lhis encc:.cpasses bifurcation and strange attractor theories,

theory of fractals and renorma1ization group theory. nteir article contains

c:oq>rehensive and pertinent references for anyone who wuld like to get acquainted

with various approaches to turbulence.

Ferziger has been part of the research program in large eddy sinulation (LES)

of turbulent flows since its inception at Stanford in 1974. His article provides an

excellent introduction and overview of LES. After presenting the historical

backgrOLmd, he lays down the fOLmdations of the subgrid scale !lDdelling. He then

proceeds to a critical review of various IIDde1s in vogue. nte inl>act of supercom­

puters on LES is discussed. A IlU!Iber of deve10pllEllts required to advance the field

including better !lDde1s, better derivations of initial conditions, and better

treatment of bOLmdary conditions are also discussed.

After a brief introduction to statistical and dynamical methods to treat

turbulence, Herring starts with a statement of the lID!IEnt closures, and then presents

a simple calculation illustrating the possible shortccmings of the second order

~- and one-point closure. Then scme successes of the ~-point second order

!lDdelling are discussed particularly in the case of turbulent convection at low

Reynolds number. The article closes with relevant cc.mnents on closure providing the

rational framework for the subgrid scale IIDdelling procechIre.

Kraichnan's work is one of the IIDst important contributions of the workshop.

The first half of this paper presents in a unified manner material not necessarily

new, but in his opinion, insufficiently appreciated in the turbulence cO!llll.lllity.

nte second half focuses on the teclmica1 aspects of Mlat he calls the decimation

approach to turbulence. lhis new nonperturbative approach focuses only on a certain

IlU!Iber of !lDdes, the effect of neglected IIDdeS being IIDde1ed by random forces with

specifically imposed dynamical and statistical symmetries.

Landahl's work ccmes under the category of coherent structure !lDdelling, or

Mlat Saffman calls physica111Ddelling of turbulence. His inviscid flat-eddy IIDde1

for coherent structures in the near wall region of a turbulent bOLmdary layer yields

flow structure surprisingly similar to Mlat has been observed in exper:inEnts.

Launder's article on phencmeno1ogical turbulence IIDde1s is a superb discussion

of the capabilities and limitations of single-point closures. He confines his

attention to three types of !lDde1s in this class -- a ~-equation eddy viscosity

IIDde1 (EVM), an algebraic stress IIDde1 (ASM), and a differential stress !lDde1

(DSM) -- which are the subject of IIDSt current activity in this area. nte author

makes an honest attempt to give an accurate flavour of what has been achieved and

vii

where IIPre needs to be achieved in single-point closure. He also discusses briefly

the efforts to develop a split-spectrum IIPdel in which the turbulence energy

spectrum is divided into two parts with separate equations provided for the energy

dissipation rate and the rate at which energy passes from large scales to small

scales. He further notes that such efforts will bridge the gap between single-point

and sub-grid scherres.

Renormalization group Irethods ( which have proved to be extrerrely useful in the

study of critical phenomena) have been recently applied to the study of transition

to turbulence, hydrodynamic turbulence in the similarity spectrum range, and subgrid

scale IIPdelling in the numerical simulation of fully developed turbulence. McConil

concentrates on his own contribution to the last category. What he calls the

"Iterative Averaging" technique appears to be a promising way of applying the

renormalization group Irethod to homogeneous isotropic turbulence. It remains to

be seen how such techniques could be extended to include wall regions without

compromising whatever rigor they lay clalin to.

Pouquet I S article on "Statistical Methods in Turbulence" starts with th~

description of the properties of a non-dissipative flow, and goes on to show where

the statistical closures have been useful. An interesting part of this paper is

its discussion of statistical Irethods and chaos.

Reynolds and Lee give a flavour of what impact full turbulence sirrulation (TIS)

can have on phenorrenological IIPdelling of turbulence. Their recent sirrulation of

homogeneous turbulence subject to irrotational strains and 1.mder relaxation from

these strains appears to have revealed rather controversial new physics regarding

the behaviour of the anisotropy of the Reynolds stress, dissipation and vorticity

fields.

Saffman provides cogent reasons for vortex dyrumri.cs constituting one of the

fundamental theoretical approaches to the 1.mderstanding of turbulence. Vortex

dyrumri.cs falls into the category of physical IIPdelling of turbulence, and is

based on the surmise that turbulent flows can be thought of as assemblies of

vortical states which are exact solutions to the Navier-Stokes equations or the

Euler equations. He lists the vortical states (relevant to the physical IIPdelling

of turbulence) as (1) two-diIrensional array of finite area vortices, (2) three­

dimensional stretched vortices, (3) vortex rings, (4) finite amplitude

Tollmien-Schlichting waves, and (5) vortex sheets. He confines his attention to the

first category of vortical states with a brief discussion of other states.

Spalding notes a number of defects (in the present phenomenological IIPdelling

of turbulence) essentially due to the neglect of spottiness of turbulent flows.

He provides a theoretical fonnulation of a two-fluid IIPdel of turbulence with

preillninary results in the case of the plane wake, the axisynIIEtric jet, and

one-dimensional laminar flarre propagation. Although sCIre qualitative agreerrent with

experiIrent has been obtained with respect to features which other IIPdels cannot

viii

predict at all, it DUSt be noted that the subject is very nuch in its infancy and

there are a rn.miJer of open questions. Nevertheless, the two-fluid concept can form

a basis for further advancement.

Spiegel dispels any doubts one may have that chaotic solutions of the fluid

dynamic equations exist. His article is a clear and cogent exposition of the view

point that chaos may not be turbulence but that "the !lI)re we learn about chaos, the

better we will U1derstand turbulence". He describes the approach which asSll!lEs

amplitude expansions near to the onset of instability. After providing the

background on linear stability, he discusses the amplitude equation for triple

instability; i.e., three ~es going U1stable alnDst s:im.!ltaneously. He goes on to

show that the onset of instability in a continuous band of wave numbers leads to

chaotic coherent structures. His proposal to look at data from experiments or

numerical sinulations, and calculate LiapU10v exponents and dimensions of attractors

as functions of relevant parameters such as Reynolds number, is worth serious

consideration. This is one way of separating the chaotic aspect of turbulence from

its other aspects. lhis U'lion of two disparate approaches might shed sane light

on new physics of turbulence.

DID, MYH, RGV

Contents

PREFACE

CHAPl'ER I.

CHAPl'ER II.

Turbulence Sensitivity and Control in Wall Flows

Dermis M. Bushnell

Observations, Theoretical Ideas, and Modeling of Turbulent

Flows -- Past, Present, and Future

Gary T. Chapman and M.Jrray Tobak

CHAPl'ER III. Large Eddy S:i.nu1ation: Its Role in Turbulence Research

JoelH. Ferziger

CHAPl'ER IV.

CHAPl'ER V.

CHAPl'ER VI.

An Introduction and Overview of Various Theoretical Approaches

to Turbulence

Jackson R. Herring

Decimated Arrplitude Equations in Turbulence Dynamics

Robert H. Kraichnan

Flat-Eddy Model for Coherent Structures in Boundary Layer

Turbulence

Marten T. Landahl

CHAPl'ER VII. Progress and Prospects in P~logical Turbulence Models

Page

v

1

19

51

73

91

137

B.E. Launder 155

CHAPl'ER VIII. Renonnalisation Group Methods Applied to the Numerical

S:i.nu1ation of Fluid Turbulence

CHAPl'ER IX.

W.D. McComb

Statistical Methods in Turbulence

A. Pouquet

ix

187

209

CHAPl'ER X.

CHAPl'ER XI.

CHAPl'ER XII.

x

The Structure of HOlIDgeneous Turbulence

William C. Reynolds and Moon J. Lee

Vortex Dynamics

P.G. Saffman

Two-Fluid Models of Turbulence

D. Brian Spalding

CHAPl'ER XIII. Chaos and Coherent Structures in Fluid Flaws

E.A. Spiegel

CHAPl'ER XI V. Connection Between Two Classical Approaches to Turbulence:

The Conventional Theory and the Attractors

R. Temam

POSITION PAPERS BY PANEL MEMBERS

CHAPl'ER }N.

CHAPl'ER }NI.

Remarks on Prototypes of Turbulence, Structures in Turbulence

and the Role of Chaos

Hassan Aref

Subgrid Scale Modeling and Statistical Theories in

Three-Dimensional Turbulence

Jean-Pierre Chollet

CHAPl'ER }NIl. Strange Attractors, Coherent Structures and Statistical

Approaches

Page

231

263

279

303

337

347

353

John L. Lumley 359

CHAPl'ER }WIll. A Note on the Structure of Turbulent Shear Flaws

Parviz Moin

CHAPl'ER XIX. Lagrangian Modelling for Turbulent Flaws

S.B. Pope

365

369

Contributors

I Hassan Aref, Division of Engineering, Brown University, Providence, RI 02912, U.S.A.

Detmis M. Bushnell, NASA Langley Research Center, Hampton, VA 23665, U.S.A.

Gary T. Chapman, NASA AIles Research Center, Moffett Field, CA 94035, U. S .A.

Jean-Pierre Chollet, Institut de ~canique de Grenoble, 38402 Saint-Martin d'Heres

Cedex, France.

Joel H. Ferziger, Department of Mechanical Engineering, Stanford University,

Stanford, CA 94305, U.S.A.

Jackson R. Herring, National Center for At:nnspheric Research, Boulder, CO 80309,

U.S.A.

Robert H. Kraichnan, 303 Potrillo Drive, Los AlanDS, NM 87544, U.S.A.

Marten T. Landahl, Department of Aeronautics and Astronautics, Massachusetts

Institute of Technology. Cambridge. MA 02139. U.S.A.

B.E. Launder, Department of Mechanical Engineering, University of Manchester,

Manchester M60 lQD, United Kingdom.

Moon J. Lee, Department of Mechanical Engineering, Stanford University, Stanford,

CA 94305, U.S.A.

John 1. Lumley, Sibley School of Mechanical and Aerospace Engineering, Cornell

University, Ithaca, NY 14853, U.S.A.

W.D. McComb, Department of Physics, University of Edinburgh, Edinburgh ER9 3JL,

United Kingdom.

Parviz Moin, NASA AIles Research Center, Moffett Field, CA 94035, U.S.A.

xi

xii

S.B. Pope, Sibley School of Mechanical and Aerospace Engineering, Cornell

University, Ithaca, NY 14853, U.S.A.

A. Pouquet, Centre de la Recherche Scientifique, Observatoire de Nice, 06007 Nice

Cedex, France.

William C. Reynolds, Depart:na1t of Mechanical Engineering, Stanford University,

Stanford, CA 94305, U.S.A.

P.G. Saffman, Depart:na1t of Applied Mathanatics, california Institute of

Technology, Pasadena, CA 91125, U.S.A.

D. Brian Spalding, Computational Fluid Dynamics Unit, Imperial College of

Science and Technology, London SW7 2BX, United Kingdom.

E.A. Spiegel, Depart:na1t of Astronany, Columbia University, New York, NY 10027,

U.S.A.

R. Temam, Laboratoire d'Analyse Nt.D.rerique, Universite Paris-Sud, 91405 Orsay

Cedex, France.

~ay Tobak, NASA Amas Research Center, Moffett Field, CA 94035, U.S.A.