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E-raamat: Homogeneous, Isotropic Turbulence: Phenomenology, Renormalization and Statistical Closures

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(Emeritus Professor of Physics, Edinburgh University)
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Homogeneous, Isotropic Turbulence: Phenomenology, Renormalization and Statistical ClosuresSuurem pilt
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Fluid turbulence is often referred to as `the unsolved problem of classical physics'. Yet, paradoxically, its mathematical description resembles quantum field theory. The present book addresses the idealised problem posed by homogeneous, isotropic turbulence, in order to concentrate on the fundamental aspects of the general problem. It is written from the perspective of a theoretical physicist, but is designed to be accessible to all researchers in turbulence, both theoretical and experimental, and from all disciplines. The book is in three parts, and begins with a very simple overview of the basic statistical closure problem, along with a summary of current theoretical approaches. This is followed by a precise formulation of the statistical problem, along with a complete set of mathematical tools (as needed in the rest of the book), and a summary of the generally accepted phenomenology of the subject. Part 2 deals with current issues in phenomenology, including the role of Galilean invariance, the physics of energy transfer, and the fundamental problems inherent in numerical simulation. Part 3 deals with renormalization methods, with an emphasis on the taxonomy of the subject, rather than on lengthy mathematical derivations. The book concludes with some discussion of current lines of research and is supplemented by three appendices containing detailed mathematical treatments of the effect of isotropy on correlations, the properties of Gaussian distributions, and the evaluation of coefficients in statistical theories.

Arvustused

At the end of reading this book I felt exuberant in several distinct ways: progress has been made, but there are still vast areas of turbulence that are hardly understood; there is beautiful mathematics underpinning the analyses in HIT in particular and turbulence in general; and one has a sense of optimism that theoretical progress is gaining momentum. * Thomas J. Bridges, Contemporary Physics *

Notation xviii
PART I THE FUNDAMENTAL PROBLEM, THE BASIC STATISTICAL FORMULATION, AND THE PHENOMENOLOGY OF ENERGY TRANSFER
1 Overview of the statistical problem
3(33)
1.1 What is turbulence?
4(3)
1.1.1 Definition and characteristic features
4(1)
1.1.2 The development of turbulence
5(1)
1.1.3 Homogeneous, isotropic turbulence (HIT)
6(1)
1.2 The turbulence problem
7(3)
1.2.1 The turbulence problem in real flows
7(2)
1.2.2 Formulation of the turbulence problem in HIT
9(1)
1.3 The characteristics of HIT
10(2)
1.4 Turbulence as a problem in quantum field theory
12(2)
1.5 Renormalized perturbation theory (RPT): the general idea
14(9)
1.5.1 Primitive perturbation series of the Navier--Stokes equations
15(2)
1.5.2 Application to the closure problem: the response function
17(2)
1.5.3 Renormalization
19(1)
1.5.4 Vertex renormalization
20(1)
1.5.5 Physical interpretation of renormalized perturbation theory
21(2)
1.6 Renormalization group (RG) and mode elimination
23(9)
1.6.1 RG as stirred hydrodynamics at low wavenumbers
28(1)
1.6.2 RG as iterative conditional averaging at high wavenumbers
28(3)
1.6.3 Discussion
31(1)
1.7 Background reading
32(4)
References
33(3)
2 Basic equations and definitions in x-space and k-space
36(18)
2.1 The Navier--Stokes equations in real space
37(1)
2.2 Correlations in x-space
38(4)
2.2.1 The two-point, two-time covariance of velocities
38(1)
2.2.2 Correlation functions and coefficients in isotropic turbulence
39(2)
2.2.3 Structure functions
41(1)
2.3 Basic equations in k-space: finite system
42(2)
2.3.1 The Navier--Stokes equations
42(1)
2.3.2 The symmetrized Navier--Stokes equation
43(1)
2.3.3 Moments: finite homogeneous system
44(1)
2.4 Basic equations in k-space: infinite system
44(4)
2.4.1 The Navier--Stokes equations
45(1)
2.4.2 Moments: infinite homogeneous system
46(1)
2.4.3 Isotropic system
47(1)
2.4.4 Stationary and time-dependent systems
47(1)
2.5 The viscous dissipation
48(1)
2.6 Stirring forces and negative damping
48(2)
2.7 Fourier transforms of isotropic correlations, structure functions, and spectra
50(4)
References
52(2)
3 Formulation of the statistical problem
54(22)
3.1 The covariance equations
54(2)
3.1.1 Off the time-diagonal: C(k; t, t!)
55(1)
3.1.2 On the time diagonal: C(k; t, t) = C(k, t)
55(1)
3.2 Conservation of energy in wavenumber space
56(3)
3.2.1 Equation for the energy spectrum: the Lin equation
56(2)
3.2.2 The effect of stirring forces
58(1)
3.3 Conservation properties of the transfer spectrum T(k, t)
59(2)
3.4 Symmetrized conservation identities
61(1)
3.5 Alternative formulations of the triangle condition
62(3)
3.5.1 The Edwards (k, j, μ) formulation
62(1)
3.5.2 The Kraichnan (k, j, l) formulation
63(1)
3.5.3 Conservation identities in the two formulations
63(2)
3.6 The L coefficients of turbulence theory in the (k, j, μ) formulation
65(1)
3.7 Dimensions of relevant spectral quantities
66(1)
3.7.1 Finite system
66(1)
3.7.2 Infinite system
66(1)
3.8 Some useful relationships involving the energy spectrum
67(1)
3.9 Conservation of energy in real space
68(2)
3.9.1 Viscous dissipation
68(2)
3.10 Derivation of the Karman--Howarth equation
70(6)
3.10.1 Various forms of the KHE
72(1)
3.10.2 The KHE for forced turbulence
73(1)
3.10.3 KHE specialized to the freely decaying and stationary cases
74(1)
References
75(1)
4 Turbulence energy: its inertial transfer and dissipation
76(37)
4.1 The test problems
76(3)
4.1.1 Test Problem 1: free decay of turbulence
77(1)
4.1.2 Test problem 2: stationary turbulence
78(1)
4.2 The Lin equation for the spectral energy balance
79(2)
4.2.1 The stationary case
80(1)
4.2.2 The global energy balances
80(1)
4.3 The local spectral energy balance
81(7)
4.3.1 The energy flux
83(1)
4.3.2 Local spectral energy balances: stationary case
84(2)
4.3.3 The limit of infinite Reynolds number
86(1)
4.3.4 The peak value of the energy flux
87(1)
4.4 Summary of expressions for rates of dissipation, decay, energy injection, and inertial transfer
88(2)
4.5 The Karman--Howarth equation as an energy balance in real space
90(5)
4.6 The Kolmogorov (1941) theory: K41
95(4)
4.6.1 The `2/3' law: K41A
95(2)
4.6.2 The `4/5' law
97(2)
4.6.3 The `2/3' law again: K41B
99(1)
4.7 The Kolmogorov (1962) theory: K62
99(2)
4.8 Some aspects of the experimental picture
101(4)
4.8.1 Spectra
101(2)
4.8.2 Structure functions
103(2)
4.9 Is Kolmogorov's theory K41 or K62?
105(8)
References
106(7)
PART II PHENOMENOLOGY: SOME CURRENT RESEARCH AND UNRESOLVED ISSUES
5 Galilean invariance
113(30)
5.1 Historical background
114(1)
5.2 Some relativistic preliminaries
115(2)
5.3 Galilean relativistic treatment of the Navier--Stokes equation
117(3)
5.3.1 Galilean transformations and invariance of the NSE
119(1)
5.4 The Reynolds decomposition
120(3)
5.4.1 Galilean transformation of the mean and fluctuating velocities
121(1)
5.4.2 Transformation of the mean-velocity equation to S
121(1)
5.4.3 Transformation of the equation for the fluctuating velocity to S
122(1)
5.5 Constant mean velocity
123(1)
5.6 Is vertex renormalization suppressed by GI?
124(2)
5.7 Extension to wavenumber space
126(4)
5.7.1 Invariance of the NSE in k-space
127(2)
5.7.2 The Reynolds decomposition
129(1)
5.8 Moments of the fluctuating velocity field
130(1)
5.9 The covariance equations
131(4)
5.9.1 Covariance equation for t ≠ t'
132(1)
5.9.2 The covariance equation for t = t'
133(2)
5.10 Two-time closures
135(1)
5.11 Filtered equations of motion: LES and RG
136(2)
5.12 Concluding remarks
138(5)
References
140(3)
6 Kolmogorov's (1941) theory revisited
143(45)
6.1 Standard criticisms of Kolmogorov's (1941) theory
143(7)
6.1.1 The effect of intermittency
144(1)
6.1.2 Local cascade or `nonlocal' vortex stretching?
145(2)
6.1.3 Problems with averages
147(2)
6.1.4 Anomalous exponents
149(1)
6.2 The scale-invariance paradox
150(7)
6.2.1 Scale invariance
151(1)
6.2.2 The paradox
152(2)
6.2.3 Resolution of the paradox
154(3)
6.3 Scale invariance and the `-5/3' inertial-range spectrum
157(4)
6.3.1 The scale-invariant inertial subrange
158(1)
6.3.2 The inertial-range energy spectrum
159(1)
6.3.3 Calculation of the Kolmogorov prefactor
159(1)
6.3.4 The limit of infinite Reynolds number
160(1)
6.4 Finite-Reynolds-number effects on K41: theoretical studies
161(17)
6.4.1 Batchelor's interpolation function for the second-order structure function
162(1)
6.4.2 Effinger and Grossmann (1987)
163(3)
6.4.3 Barenblatt and Chorin (1998)
166(2)
6.4.4 Qian (2000)
168(2)
6.4.5 Gamard and George (2000)
170(4)
6.4.6 Lundgren (2002)
174(4)
6.5 Finite-Reynolds-number effects on K41: experimental and numerical studies
178(4)
6.6 Discussion
182(6)
References
183(5)
7 Turbulence dissipation and decay
188(44)
7.1 The mean dissipation rate
189(2)
7.2 Dependence on the Taylor--Reynolds number
191(5)
7.3 The behaviour of the dissipation rate according to the Karman--Howarth equation
196(3)
7.3.1 The dependence of the dimensionless dissipation rate on Reynolds number
197(2)
7.4 A reinterpretation of the Taylor dissipation surrogate
199(5)
7.4.1 Reinterpretation of Taylor's expression based on results from DNS
200(4)
7.5 Freely decaying turbulence: the background
204(5)
7.5.1 Variation of the Taylor microscale during decay
205(1)
7.5.2 The energy spectrum at small wavenumbers
206(1)
7.5.3 The final period of the decay
207(1)
7.5.4 The Loitsiansky and Saffman integrals
207(2)
7.6 Free decay: the classical era
209(7)
7.6.1 Taylor (1935)
209(1)
7.6.2 Von Karman and Howarth (1938)
210(2)
7.6.3 Kolmogorov's prediction of the decay exponents
212(1)
7.6.4 Batchelor (1948)
213(2)
7.6.5 The non-invariance of the Loitsiansky integral
215(1)
7.7 Theories of the decay based on spectral models
216(4)
7.7.1 Two-range spectral models
216(4)
7.7.2 Three-range spectral models
220(1)
7.8 Free decay: towards universality?
220(12)
7.8.1 The effect of initial conditions
221(5)
7.8.2 Fractal-generated turbulence
226(2)
References
228(4)
8 Theoretical constraints on mode reduction and the turbulence response
232(41)
8.1 Spectral large-eddy simulation
234(4)
8.1.1 Statement of the problem
234(2)
8.1.2 Spectral filtering to reduce the number of degrees of freedom
236(2)
8.2 Intermode spectral energy fluxes
238(3)
8.2.1 Low-k partitioned energy fluxes
239(1)
8.2.2 High-k partitioned energy fluxes
239(1)
8.2.3 Energy conservation revisited
239(2)
8.3 Semi-analytical studies of subgrid modelling using statistical closures
241(7)
8.4 Studies of subgrid models using direct numerical simulation
248(2)
8.5 Stochastic backscatter
250(3)
8.6 Conditional averaging
253(2)
8.7 A statistical test of the eddy-viscosity hypothesis
255(5)
8.8 Constrained numerical simulations
260(6)
8.8.1 Operational LES
261(5)
8.9 Discussion
266(7)
References
267(6)
PART III STATISTICAL THEORY AND FUTURE DIRECTIONS
9 The Kraichnan--Wyld--Edwards covariance equations
273(26)
9.1 Preliminary remarks
274(3)
9.1.1 RPTs as statistical closures
274(1)
9.1.2 Perceptions of RPTs
274(3)
9.1.3 Some general characteristics of RPTs
277(1)
9.2 The problem restated: the exact covariance equations
277(4)
9.2.1 The general inhomogeneous covariance equation
277(2)
9.2.2 Centroid and difference coordinates
279(2)
9.2.3 The exact covariance equations for HIT
281(1)
9.3 A short history of closure approximations
281(2)
9.4 The KWE covariance equations: the problem reformulated
283(4)
9.4.1 Comparison of quasi-normality with perturbation theory
284(1)
9.4.2 The KWE covariance equations
285(2)
9.5 Renormalized response functions as closure approximations
287(5)
9.5.1 Failure of the EFP and DIA closures
288(2)
9.5.2 The Local Energy Transfer (LET) theory
290(2)
9.6 Numerical assessment of closure theories
292(2)
9.6.1 Some recent calculations of LET and EDQNM
292(2)
9.7 Conclusions
294(5)
References
296(3)
10 Two-point closures: some basic issues
299(18)
10.1 Perturbation theory and renormalization
299(4)
10.2 Quantum-style formalisms: Wyld--Lee and Martin--Siggia--Rose
303(7)
10.2.1 The improved Wyld--Lee formalism
305(2)
10.2.2 The Martin--Siggia--Rose formalism
307(3)
10.3 How general are the formalisms?
310(1)
10.4 Galilean invariance and the DIA
311(3)
10.5 Lagrangian-history theories
314(3)
References
315(2)
11 The renormalization group applied to turbulence
317(39)
11.1 Formulation of conditional mode elimination for turbulence
318(3)
11.2 Renormalization group
321(1)
11.3 Forster--Nelson--Stephen theory of stirred fluid motion
322(5)
11.3.1 Application of the RG to stirred fluid motion with asymptotic freedom as k → 0
322(3)
11.3.2 Differential RG equations
325(1)
11.3.3 FNS theory in terms of conditional averaging
326(1)
11.4 Turbulence RG theories based on filtered averages
327(5)
11.4.1 Iterative averaging: McComb (1982)
328(1)
11.4.2 Iterative averaging in wavenumber space
329(1)
11.4.3 Relationship of iterative averaging to Rose's (1977) method
330(1)
11.4.4 Improved iterative averaging
331(1)
11.5 Problems with filtered averages
332(2)
11.6 The two-field theory
334(7)
11.6.1 The hypothesis of local chaos
336(3)
11.6.2 The recursion relations of two-field theory
339(2)
11.7 Improved two-field theory
341(5)
11.7.1 Non-Gaussian perturbation theory
344(2)
11.8 Applications and developments of iterative averaging
346(2)
11.9 Is field-theoretic RG a theory of turbulence?
348(8)
11.9.1 Differential recursion relations
351(1)
References
352(4)
12 Work in progress and future directions
356(23)
12.1 Turbulence response
356(2)
12.1.1 Fluctuation-response relations (FRRs)
356(1)
12.1.2 Numerical assessment
357(1)
12.2 Renormalized perturbation theories
358(7)
12.2.1 Extension of Edwards' (1964) theory to the two-time covariance C(k; t, t')
359(3)
12.2.2 Recovering the LET theory
362(3)
12.3 Renormalization group
365(3)
12.3.1 Power-law forcing and the renormalization group
365(1)
12.3.2 Application of the Edwards (1964) pdf to RG mode elimination
366(2)
12.4 Towards shear flows
368(4)
12.4.1 Application of the two-field theory to LES of shear flows
368(4)
12.5 Postscript: The nature of the problem
372(7)
References
375(4)
PART IV APPENDICES
Appendix A Implications of isotropy and continuity for correlation tensors
379(4)
References
382(1)
Appendix B Properties of Gaussian distributions
383(16)
B.1 Discrete systems: real scalar variables
383(1)
B.1.1 Two-point correlations
384(5)
B.2 Discrete systems: complex scalar variables
389(1)
B.3 Scalar fields
390(2)
B.3.1 Extension to wavenumber and time
392(1)
B.3.2 The generating functional
393(2)
B.4 Vector fields
395(1)
B.5 Isotropic fields
396(2)
B.6 Inhomogeneous vector fields
398(1)
References
398(1)
Appendix C Evaluation of the L(k, j) coefficient
399(6)
C.1 Derivation of the closed covariance equation
399(2)
C.2 Evaluation of L(k, j)
401(2)
C.2.1 A note on numerical evaluation in closures
403(1)
References
403(2)
Index 405
The author has had wide experience in both engineering and physics departments. After early career experience in research and development in the nuclear power industry, he returned to university to study theoretical physics. Following the completion of a PhD in turbulence theory, he took up the post of Senior Scientific Officer in the Theoretical Physics Division at AERE, Harwell. Thereafter he held successively lectureships in engineering science and physics, a readership in physics, and a personal chair in statistical physics at Edinburgh University. On his retirement in 2006, he was appointed Professor Emeritus, and now also holds a Senior Professorial Fellowship. He has been guest professor at the University of Delft, and visiting fellow at Wolfson College and the Isaac Newton Institute, Cambridge. During the period 2007-09 he held a Leverhulme Emeritus Fellowship.
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