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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.