New trends in turbulence Turbulence: nouveaux aspects: 31 July – 1 September 2000

564 Pages·2001·8.62 MB·English

by  A. Yaglom (auth.), M. Lesieur, A. Yaglom, F. David (eds.)

a NATO Advanced Study Institute
LES HOUCHES
SESSION LXXIV
3 1 July - 1 September 2000
New trends in turbulence
Turbulence : nouveaux aspects
Edited by
M. LESIEUR, A. YAGLOM and F. DAVID
Springer
Les Ulis, Paris, Cambridge Berlin. Heidelberg, New York,
Barcelona, Hong Kong, London
Milan, Paris, Tokyo
Published in cooperation with the NATO Scientific Affair Division
Preface
The phenomenon of turbulence in fluid mechanics has been known for many
centuries. Indeed, it was for instance discussed by the Latin poet Lucretius who
described in "de natura rerum" how a small perturbation ("clinamen") could be
at the origin of the development of a turbulent order in an initially laminar river
made of randomly agitated atoms. More recently, Leonardo da Vinci drew
vortices. Analogous vortices were sketched by the Japonese school of artists
called Utagawa in the 19th century, which certainly influenced van Gogh in "The
Starry Night". However, and notwithstanding decisive contributions made by
Benard, Reynolds, Prandtl, von Karman, Richardson and Kolmogorov, the
problem is still wide open: there is no exact derivation of the famous so-called
Kolmogorov k-5'3 cascade towards small scales, nor of the value of the
transitional Reynolds number for turbulence in a pipe. Besides these fundamental
aspects, turbulence is associated with essential practical questions in hydraulics,
aerodynamics (drag reduction for cars, trains and planes), combustion
(improvement of engine efficiency and pollution reduction), acoustics (the
reduction of turbulence-induced noise is an essential issue for plane reactors),
environmental and climate studies (remember the huge damage caused by severe
storms in Europe at the end of 1999), and astrophysics (Jupiter's Great Red Spot
and solar granulation are manifestations of turbulence). Therefore, there is an
urgent need to develop models that allow us to predict and control turbulence
effects.
During the last 10 years, spectacular advances have been made towards a
better physical understanding of turbulence, concerning in particular coherent-
vortex self-organization resulting from strong nonlinear interactions. This is due
both to huge progress in numerical simulations (with the development of large-
eddy simulations in particular) and the appearance of new experimental
techniques such as digitized particle-image velocimetry methods. New
theoretical tools for the study of fluid turbulence in three and two dimensions
have appeared, sometimes borrowed fiom statistical thermodynamics or
condensed-matter physics: multifractal analysis, Lagrangian dynamics and
mixing, maximum-entropy states, wavelet techniques, nonlinear amplitude
equations. .. These advances motivated the organization of the Les Houches 2000
School on "New Trends in Turbulence", and this book. Theoretical aspects have
been blended with more practical viewpoints, where the influence on turbulence
of boundaries, compressibility, curvature and rotation, helicity, and magnetic
fields is looked at. Various applications of turbulence modelling and control to
certain industrial or environmental issues are also considered.
The first introductory course by Akiva Yaglom deals with a "century of
turbulence theory". Here the problem of turbulence is reviewed from both the
linear and nonlinear points of view. Then the two main achievements of 20th
xxiv
century turbulence theory are considered, namely the logarithmic velocity
profiles (found by T. von Karman and L. Prandtl) for the intermediate layers of
flows in circular pipes, plane channels and boundary layers without pressure
gradients, and the Kolmogorov-Obukhov theory of the universal statistical
regime of small-scale turbulence in any flow with a large enough Reynolds
number (revised in 1962 by the authors to consider intermittency, which led to
predictions of the scaling for structure functions of various orders). Since
Yaglom was a student of Kolmogorov and participated in the developments of
the second of the above-mentioned subjects, his presentation is based on first-
hand knowledge of the history of these classic discoveries and of their present
status.
The basis of turbulence theory is complemented by Katepalli Sreenivasan and
S. Kurien (Course 2), who provide also a very complete account of fully
developed turbulence experimental data (concerning structure functions and
spectra of velocity and passive scalars) both in the laboratory and the atmosphere
at very high Reynolds number (up to Rh = 20 000). Sreenivasan insists on
departures from Kolmogorov scaling in the light of anisotropy effects, which are
studied with the aid of a SO(3) decomposition. This allows us to extract an
isotropic component from inhomogeneous turbulence.
Olivier Metais presents in detail "Large-eddy simulations of turbulence"
(LES) in Course 3. These new techniques are powerful1 tools allowing us to
simulate deterministically the coherent vortices formation and evolution. He
reviews also methods for vortex identification based upon vorticity, pressure and
the second invariant of the velocity-gradient tensor. He applies direct-numerical
simulations and LES to free-shear flows and boundary layers. Metais is also
involved in particular aspects related to thermal stratification and rotation, with
application to deep-water formation in the ocean and atmospheric severe storms.
He concludes with applications of LES to compressible turbulence in gases, with
reentry of hypersonic bodies into the atmosphere, and cooling of rocket engines.
In Course 4, Michael Leschziner describes methods used for predicting
statistical industrial flows, where the geometry is right now too complex to allow
the use of LES. He shows how these methods can be, in simple geometric cases,
assessed by comparison with DNS and LES methods. He presents very
encouraging results obtained with nonlinear eddy-viscosity models and second-
moment closures. Interesting applications to turbomachinery flows are provided.
Still on the industrial-application side, Reda Mankbadi (NASA) provides in
Course 5 a very informative review of computational aeroacoustics, with many
applications to aircraft noise (in particular jet noise in plane engines). He shows
linear, nonlinear and full LES methods.
The remaining chapters are more fundamental. Keith Moffatt (Course 6)
presents the basis of topological fluid dynamics and stresses the importance of
helicity in neutral and magnetohydrodynamic (MHD) flows. He shows in the
latter case how helicity at small scales can generate a large-scale magnetic field
(a-effect), a mechanism which might explain the magnetic-field generation in
planets and stars. Moffatt discusses also the possibility of a finite-time
singularity within Euler and even Navier-Stokes equations. During his oral
presentation, he made an analogy with Euler's disk. His practical demonstration
of the latter was one of the School's highlights, especially when people realized
that the experiment could be done as well with the restaurant plates, which was
less appreciated by the cook. Uriel Frisch and J. Bec (Course 7) speak also of
finite-time singularities, but mostly on the basis of Burgers equations in one or
several dimensions, with the formation of multiple shocks. They describe
methods that allow us to solve these equations analytically and numerically, and
the kinetic-energy decay problem. He shows how Burgers equations (in three
dimensions) can, in cosmology, apply approximately to the formation of large
scales in the universe.
Course 8 is a very complete account of two-dimensional turbulence provided
by Joel Sommeria (Grenoble). He first presents numerous examples of 2D
turbulence in the laboratory (rotating or MHD flows, plasmas), in the ocean and
in planetary atmospheres (Jupiter), insisting in particular on the absence of
kinetic-energy dissipation in these flows. He reviews the double cascade of
enstrophy (direct) and energy (inverse) proposed by Kraichnan and Batchelor.
He presents also very clearly the point-vortex statistical-thermodynamics
analysis of Onsager, and the generalization of this model to fmite blobs of
vorticity (maximum-entropy principle). Sommeria shows applications of this
model to mixing layers and to 2D oceanic flows with differential rotation.
Course 9 (Marie Farge and K. Schneider) is a useful presentation of wavelet
techniques, a further interesting application of which (not detailed in the book)
concerns data compression. In Course 10 (Gregory Falkovich, K. Gawedzki and
M. Vergassola), the Lagrangian mixing of passive scalars and the relative
dispersion of several particles is discussed. For two particles in particular,
Richardson's pioneering law (predicting a dispersion rate proportional to the
separation raised to the 413 power) is revisited in the light of Kraichnan's
renormalized analysis.
Let us finish this preface with some information regarding the Les Houches
2000 School "New Trends in Turbulence". A computing centre was set up
thanks to Patrick Begou during the duration of the programme. The machines
(Compaq ds2012proc, Sun ultra80/4proc, IBM 44p/2proc, HP j560012proc,
HP kayak linux.2proc) were kindly lent by the respective companies. With this
important computing power, we could organize for students five working groups
under the direction of a professor or senior researcher, and using databases
generated using several computational programs. These groups were: (1) Jets and
wakes (Olivier Mdtais); (2) Isotropic turbulence (Marcel Lesieur); (3) Industrial
applications (Franco Magagnato); (4) Transition (Laurette Tuckerman); and
(5) Environment (Elisabeth Wingate). Among the topics treated in these groups
were: analysis of helicity in a wake (group l), vortices in LES of 3D isotropic
turbulence (group 2), simulation of flows in complex geometries (around cars)
using Magagnato's SPARC code (group 3), transitional plane Couette flow
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(group 4), and dispersion of tracers in the stratosphere (group 5). The computing
centre was always crowded in the non-course periods. It also gave students a
very good formation in scientific computing and image processing.
The level of the students (who came fiom a wide range of countries) was
extremely good, and their sharp questions embarrassed sometimes even the best
professors. Relations between the students and the professors were excellent, and
discussions continued between the courses and during the meals, in the evenings,
and even on the mountain trails above Le Prarion. We had a wide participation of
students coming fiom countries in eastern Europe, and it gave the School a
distinct flavour. Contacts between the students were numerous, fliendly and
extremely tolerant.
Acknowledgments
We thank very much the lecturers for their efforts in preparing and delivering
their courses, and then writing the above lecture notes. We thank also Compaq,
Sun, IBM and Hewlett-Packard, who lent and maintained the (extremely
efficient) computers. Particular thanks go to Patrick Begou (LEGI-Grenoble) for
the huge energy he spent contacting the computer suppliers, setting up the
computing centre, installing the machines, making them work perfectly during
the programme, and (last but not least) de-installing everything at the end.
The practical organization provided by the Les Houches Physics School was
very good (especially thanks to the efforts and the patience of Ghyslaine
&Henry, Isabel Lelievre and Brigitte Rousset), with excellent housing
conditions. The food was great, and the restaurant offered a very fliendly
atmosphere, thanks to Claude Cauneau and his staff, who prepared unforgettable
fondues, raclettes and tartiflettes. The latter dish, made of Reblochon, an
excellent and not very well known cheese, is excellent.
The School was sponsored by Grenoble University, the European
Commission as a ,High-Level Scientific Conference, NATO as an Advanced
Study Institute, the CEA, CNRS (fixmation permanente), and the French
Ministry for Defense. We thank them very much for their support, without which
"New Trends in Turbulence" could never have been organized. We hope that the
reader will enjoy this book which provides an up-to-date account on what
Feynman called the last unsolved problem of classical physics.
Marcel Lesieur
Akiva Yaglom
Frangois David
CONTENTS
Lecturers xi
.. .
Participants Xll l
Prkface xvii
Preface xxiii
Contents xxvii
Course 1. The Century of Turbulence Theory:
The Main Achievements and Unsolved Problems
by A. Yaglom 1
1 Introduction 3
2 Flow instability and transition to turbulence 6
3 Development of the theory of turbulence in the 20th century:
Exemplary achievements 11
3.1 Similarity laws of near-wall turbulent flows . . . . . . . . . . . . . 11
3.2 Kolmogorov's theory of locally isotropic turbulence . . . . . . . . . 30
4 Concluding remarks; possible role of Navier-Stokes equations 39
Course 2. Measures of Anisotropy and the Universal Properties
of Turbulence
by S. Kurien and K.R . Sreenivasan 53
1 Introduction 56
2 Theoretical tools 58
2.1 The method of SO(3) decomposition . . . . . . . . . . . . . . . . . 58
2.2 Foliation of the structure function into j-sectors . . . . . . . . . . . 62
2.3 The velocity structure functions . . . . . . . . . . . . . . . . . . . . 63
2.3.1 The second-order structure function . . . . . . . . . . . . . 64
xxviii
2.4 Dimensional estimates for the lowest-order anisotropic scaling
exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3 Some experimental _considerations 69
3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.2 Relevance of the anisotropic contributions . . . . . . . . . . . . . . 69
3.3 The measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4 Anisotropic contribution in the case of homogeneity 74
4.1 General remarks on the data . . . . . . . . . . . . . . . . . . . . . 74
4.2 The tensor form for the second-order structure function . . . . . . 76
4.2.1 The anisotropic tensor component derived under the
assumption of axisymmetry . . . . . . . . . . . . . . . . . . 76
4.2.2 The complete j = 2 anisotropic contribution . . . . . . . . . 81
4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5 Anisotropic contribution in the case of inhomogeneity 85
5.1 Extracting the j = 1 component . . . . . . . . . . . . . . . . . . . 85
6 The higher-order structure functions 88
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.2 Method and results . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.2.1 The second-order structure function . . . . . . . . . . . . . 89
6.2.2 Higher-order structure functions . . . . . . . . . . . . . . . 92
6.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7 Conclusions 98
Appendix 99
A Full form for the j = 2 contribution for the homogeneous case 99
B The j = 1 component in the inhomogeneous case 105
B.l Antisymmetric contribution . . . . . . . . . . . . . . . . . . . . . . 105
B.2 Symmetric contribution . . . . . . . . . . . . . . . . . . . . . . . . 1 07
C Tests of the robustness of the interpolation formula 109
Course 3 . Large-Eddy Simulations of Turbulence
by 0. Me'tais 113
1 Introduction 117
1.1 LES and determinism: Unpredictability growth . . . . . . . . . . . 118
2 Vortex dynamics 119
2.1 Coherent vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
2.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 20
2.1.2 Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 20
2.1.3 The Q-criterion . . . . . . . . . . . . . . . . . . . . . . . . . 120
2.2 Vortex identification . . . . . . . . . . . . . . . . . . . . . . . . . . 121
2.2.1 Isotropic turbulence . . . . . . . . . . . . . . . . . . . . . . 122
2.2.2 Backward-facing step . . . . . . . . . . . . . . . . . . . . . 1 23
3 LES formalism in physical space 125
3.1 LES equations for a flow of constant density . . . . . . . . . . . . . 125
3.2 LES Boussinesq equations in a rotating frame . . . . . . . . . . . . 1 28
3.3 Eddy-viscosity and diffusivity assumption . . . . . . . . . . . . . . 128
3.4 Smagorinsky's model . . . . . . . . . . . . . . . . . . . . . . . . . .1 30
4 LES in Fourier space 131
4.1 Spectral eddy viscosity and diffusivity . . . . . . . . . . . . . . . . 131
4.2 EDQNM plateau-peak model . . . . . . . . . . . . . . . . . . . . . 132
4.2.1 The spectral-dynamic model . . . . . . . . . . . . . . . . . 134
4.2.2 Existence of the plateau-peak . . . . . . . . . . . . . . . . . 1 35
4.3 Incompressible plane channel . . . . . . . . . . . . . . . . . . . . . 1 37
4.3.1 Wall units . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
4.3.2 Streaks and hairpins . . . . . . . . . . . . . . . . . . . . . . 138
4.3.3 Spectral DNS and LES . . . . . . . . . . . . . . . . . . . . 139
5 Improved models for LES 143
5.1 Structure-function model . . . . . . . . . . . . . . . . . . . . . . . 143
5.1.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.1.2 Non-uniform grids . . . . . . . . . . . . . . . . . . . . . . . 1 45
5.1.3 Structure-function versus Smagorinsky models . . . . . . . 145
5.1.4 Isotropic turbulence . . . . . . . . . . . . . . . . . . . . . . 146
5.1.5 SF model, transition and wall flows . . . . . . . . . . . . . . 146
5.2 Selective structure-function model . . . . . . . . . . . . . . . . . . 1 46
. 5.3 Filtered structure-function model . . . . . . . . . . . . . . . . . . . 147
5.3.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.4 A test case for the models: The temporal mixing layer . . . . . . . 147
5.5 Spatially growing mixing layer . . . . . . . . . . . . . . . . . . . . 1 49
5.6 Vortex control in a round jet . . . . . . . . . . . . . . . . . . . . .1 51
5.7 LES of spatially developing boundary layers . . . . . . . . . . . . . 153
6 Dynamic approach in physical space 158
6.1 Dynamic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
7 Alternative models 161
7.1 Generalized hyperviscosities . . . . . . . . . . . . . . . . . . . . . 1 61
7.2 Hyperviscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
7.3 Scale-similarity and mixed models . . . . . . . . . . . . . . . . . . 162
7.4 Anisotropic subgrid-scale models . . . . . . . . . . . . . . . . . . . 163
8 LES of rotating flows 163
8.1 Rotating shear flows . . . . . . . . . . . . . . . . . . . . . . . . . . 164
8.1.1 Free-shear flows . . . . . . . . . . . . . . . . . . . . . . . . . 164
8.1.2 Wall flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
8.1.3 Homogeneous turbulence . . . . . . . . . . . . . . . . . . . 169
9 LES of flows of geophysical interest 169
9.1 Baroclinic eddies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
9.1.1 Synoptic-scale instability . . . . . . . . . . . . . . . . . . . 171
9.1.2 Secondary cyclogenesis . . . . . . . . . . . . . . . . . . . . . 172
10 LES of compressible turbulence 173
10.1 Compressible LES equations . . . . . . . . . . . . . . . . . . . . . . 174
10.2 Heated flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 75
10.2.1 The heated duct . . . . . . . . . . . . . . . . . . . . . . . . 1 76
10.2.2 Towards complex flow geometries . . . . . . . . . . . . . . . 177
11 Conclusion 181
Course 4 . Statistical Turbulence Modelling for the Computation
of Physically Complex Flows
by M.A. Leschziner
1 Approaches to characterising turbulence 189
2 Some basic statistical properties of turbulence and associated
implications 196
3 Review of "simple" modelling approaches 203
3.1 The eddy-viscosity concept . . . . . . . . . . . . . . . . . . . . . . 203
3.2 Model categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 04
3.3 Model applicability . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
4 Second-moment equations and implied stress-strain
interactions 212
4.1 Near-wall shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
4.2 Streamline curvature . . . . . . . . . . . . . . . . . . . . . . . . . . 217
4.3 Separation and recirculating flow . . . . . . . . . . . . . . . . . . . 218
4.4 Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
xxxi
4.5 Irrotational strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
4.6 Heat transfer and stratification . . . . . . . . . . . . . . . . . . . . 2 21
5 Second moment closure 222
6 Non-linear eddy-viscosity models 228
7 Application examples 233
7.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
7.2 Asymmetric diffuser . . . . . . . . . . . . . . . . . . . . . . . . . .2 35
7.3 Aerospatiale aerofoil . . . . . . . . . . . . . . . . . . . . . . . . . . 235
7.4 Cascade blade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 38
7.5 Axisymmetric impinging jet . . . . . . . . . . . . . . . . . . . . . . 2 39
7.6 Prolate spheroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
7.7 Round-terectangular transition duct . . . . . . . . . . . . . . . . . 2 42
7.8 Winglflat-plate junction . . . . . . . . . . . . . . . . . . . . . . . . 245
7.9 Fin-plate junction . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
7.10 Jet-afterbody combination . . . . . . . . . . . . . . . . . . . . . . 2 51
8 Concluding remarks 251
Course 5 . Computational Aeroacoustics
by R . Mankbadi
1 Fundamentals of sound transmission 261
1.1 One-dimensional wave analysis . . . . . . . . . . . . . . . . . . . . 2 62
1.1.1 General solution of the wave equation . . . . . . . . . . . . 263
1.1.2 The particle velocity . . . . . . . . . . . . . . . . . . . . . . 2 63
1.2 Three-dimensional sound waves . . . . . . . . . . . . . . . . . . . . 2 64
1.3 Sound spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 65
1.3.1 Spectral composition of a square pulse . . . . . . . . . . . . 2 66
1.3.2 Spectral composition of a harmonic signal . . . . . . . . . . 266
1.4 Logarithmic scales for rating noise . . . . . . . . . . . . . . . . . . 2 66
1.4.1 The Sound Power Level (PWL) . . . . . . . . . . . . . . . . 2 66
1.4.2 Sound Pressure Levels (SPL) . . . . . . . . . . . . . . . . . 267
1.4.3 Pressure Band Level (PBL) . . . . . . . . . . . . . . . . . . 267
1.4.4 Pressure Spectral Level (PSL) per unit frequency . . . . . . 267
1.4.5 Acoustic intensity . . . . . . . . . . . . . . . . . . . . . . . 2 69
1.4.6 Overall pressure levels . . . . . . . . . . . . . . . . . . . . . 2 69
1.4.7 Subjective noise measures . . . . . . . . . . . . . . . . . . . 269
1.4.8 Perceived Noise Level (PNL) . . . . . . . . . . . . . . . . . 270
1.4.9 Tone Corrected Perceived Noise Level (PNLT) . . . . . . . 270
1.4.10 Effective Perceived Noise Level (EPNL) . . . . . . . . . . . 2 70