Homogeneous, isotropic turbulence: phenomenology, renormalization, and statistical closures
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Oxford Univ. Press
2014
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| Edition: | 1. ed. |
| Series: | International series of monographs on physics
162 |
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| 245 | 1 | 0 | |a Homogeneous, isotropic turbulence |b phenomenology, renormalization, and statistical closures |c W. David McComb |
| 250 | |a 1. ed. | ||
| 264 | 1 | |a Oxford |b Oxford Univ. Press |c 2014 | |
| 300 | |a XX, 408 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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| adam_text | Contents
Notation xviii
PART
I
THE FUNDAMENTAL PROBLEM, THE BASIC
STATISTICAL
FORMULATION, AND THE PHENOMENOLOGY
OF ENERGY TRANSFER
1
Overview of the statistical problem
3
1.1
What is turbulence?
4
1.1.1
Definition and characteristic features
4
1.1.2
The development of turbulence
5
1.1.3
Homogeneous,
isotropie
turbulence (HIT)
б
1.2
The turbulence problem
7
1.2.1
The turbulence problem in real flows
7
1.2.2
Formulation of the turbulence problem in HIT
9
1.3
The characteristics of HIT
10
1.4
Turbulence as a problem in quantum field theory
12
1.5
Renormalized perturbation theory (RPT): the general idea
14
1.5.1
Primitive perturbation series of the Navier-Stokes equations
15
1.5.2
Application to the closure problem: the response function
17
1.5.3
Renormalization
19
1.5.4
Vertex renormalization
20
1.5.5
Physical interpretation of renormalized perturbation theory
21
1.6
Renormalization group (RG) and mode elimination
23
1.6.1
RG as stirred hydrodynamics at low wavenumbers
28
1.6.2
RG as iterative conditional averaging at high wavenumbers
28
1.6.3
Discussion
31
1.7
Background reading
32
References
33
2
Basic equations and definitions in
х
-space and k-space
36
2
Л
The Navier-Stokes equations in real space
37
2.2
Correlations in x-space
38
2.2.1
The two-point, two-time covariance of velocities
38
2.2.2
Correlation functions and coefficients in
isotropie
turbulence
39
2.2.3
Structure functions
41
2.3
Basic equations in k-space: finite system
42
2.3.1
The Navier-Stokes equations
42
2.3.2
The symmetrized Navier-Stokes equation
43
2.3.3
Moments: finite homogeneous system
44
xii Contents
2.4 Basic
equations
in k-space: infinite
system
44
2.4.1 The Navier-Stokes
equations
45
2.4.2 Moments: infinite
homogeneous system
46
2.4.3 Isotropie
system
47
2.4.4
Stationary and time-dependent systems
47
2.5
The viscous dissipation
. 48
2.6
Stirring forces and negative damping
48
2.7
Fourier transforms of
isotropie
correlations, structure
functions, and spectra
50
References
52
3
Formulation of the statistical problem
54
3.1
The covariance equations
54
3.1.1
Off the time-diagonal: C(fc;i,
* )
55
3.1.2
On the time diagonal: C{k;t,t)
ξ
CCA;,
í)
55
3.2
Conservation of energy in wavenumber space
56
3.2.1
Equation for the energy spectrum: the Lin equation
56
3.2.2
The effect of stirring forces
58
3.3
Conservation properties of the transfer spectrum T(k, i)
59
3.4
Symmetrized conservation identities
61
3.5
Alternative formulations of the triangle condition
62
3.5.1
The Edwards (k, j,
μ)
formulation
62
3.5.2
The Kraichnan {k,j,V) formulation
63
3.5.3
Conservation identities in the two formulations
63
3.6
The
L
coefficients of turbulence theory in the (k,j,
μ)
formulation
65
3.7
Dimensions of relevant spectral quantities
66
3.7.1
Finite system
66
3.7.2
Infinite system
66
3.8
Some useful relationships involving the energy spectrum
67
3.9
Conservation of energy in real space
68
3.9.1
Viscous dissipation
68
3.10
Derivation of the
Kármán-Howarth
equation
70
3.10.1
Various forms of the KHE
72
3.10.2
The KHE for forced turbulence
73
3.10.3
KHE specialized to the freely decaying and stationary cases
74
References
75
4
Turbulence energy: its
inerţial
transfer and dissipation
76
4.1
The test problems
76
4.1.1
Test Problem
1:
free decay of turbulence
77
4.1.2
Test problem
2:
stationary turbulence
78
4.2
The Lin equation for the spectral energy balance
79
4.2.1
The stationary case
80
4.2.2
The global energy balances
80
4.3
The local spectral energy balance
81
4.3.1
The energy flux
83
Contents xiii
4.3.2
Local spectral energy balances: stationary case
84
4.3.3
The limit of infinite Reynolds number
86
4.3.4
The peak value of the energy flux
87
4.4
Summary of expressions for rates of dissipation, decay, energy
injection, and
inerţial
transfer
88
4.5
The
Kármán-Howarth
equation as an energy balance in real space
90
4.6
The Kolmogorov
(1941)
theory: K41
95
4.6.1
The
2/3
law: K41A
95
4.6.2
The
4/5
law
97
4.6.3
The
2/3
law again: K41B
99
4.7
The Kolmogorov
(1962)
theory: K62
99
4.8
Some aspects of the experimental picture
101
4.8.1
Spectra
101
4.8.2
Structure functions
103
4.9
Is Kolmogorov s theory K41 or K62?
105
References
106
PART II PHENOMENOLOGY: SOME CURRENT RESEARCH AND
UNRESOLVED ISSUES
5
Galilean
invariance
113
5.1
Historical background
114
5.2
Some relativistic preliminaries
115
5.3
Galilean relativistic treatment of the Navier-Stokes equation
117
5.3.1
Galilean transformations and
invariance
of the NSE
119
5.4
The Reynolds decomposition
120
5.4.1
Galilean transformation of the mean and fluctuating velocities
121
5.4.2
Transformation of the mean-velocity equation to
S
121
5.4.3
Transformation of the equation for the fluctuating
velocity to
S
122
5.5
Constant mean velocity
123
5.6
Is vertex renormalization suppressed by GI?
124
5.7
Extension to wavenumber space
126
5.7.1
Invariance
of the NSE in
fc-space
127
5.7.2
The Reynolds decomposition
129
5.8
Moments of the fluctuating velocity field
130
5.9
The covariance equations
131
5.9.1
Covariance equation for
t
φ
ť
132
5.9.2
The covariance equation for
t— ť
133
5.10
Two-time closures
135
5.11
Filtered equations of motion:
LES
and RG
136
5.12
Concluding remarks
138
References
140
6
Kolmogorov s
(1941)
theory revisited
143
6.1
Standard criticisms of Kolmogorov s
(1941)
theory
143
xiv Contents
6.1.1
The effect of intermittency
144
6.1.2
Local cascade or nonlocal vortex stretching?
145
6.1.3
Problems with averages
147
6.1.4
Anomalous exponents
149
6.2
The scale-invariance paradox
150
6.2.1
Scale
invariance
151
6.2.2
The paradox
152
6.2.3
Resolution of the paradox
154
6.3
Scale
invariance
and the
—5/3
inerţial-
range spectrum
157
6.3.1
The scale-invariant
inerţial
subrange
158
6.3.2
The inertial-range energy spectrum
159
6.3.3
Calculation of the Kolmogorov prefactor
159
6.3.4
The limit of infinite Reynolds number
160
6.4
Finite-Reynolds-number effects on K41: theoretical studies
161
6.4.1
Batchelor s interpolation function for the second-order
structure function
162
6.4.2
Effinger and
Grossmann (1987) 163
6.4.3
Barenblatt and Chorin
(1998) 166
6.4.4
Qian
(2000) 168
6.4.5
Gamard and George
(2000) 170
6.4.6
Lundgren
(2002) 174
6.5
Finite-Reynolds-number effects on K41: experimental
and numerical studies
178
6.6
Discussion
182
References
183
7
Turbulence dissipation and decay
188
7.1
The mean dissipation rate
189
7.2
Dependence on the Taylor—Reynolds number
191
7.3
The behaviour of the dissipation rate according
to the
Karman—
Howarth equation
196
7.3.1
The dependence of the dimensionless dissipation rate
on Reynolds number
197
7.4
A reinterpretation of the Taylor dissipation surrogate
199
7.4.1
Reinterpretation of Taylor s expression based on
results from DNS
200
7.5
Freely decaying turbulence: the background
204
7.5.1
Variation of the Taylor
microscale
during decay
205
7.5.2
The energy spectrum at small wavenumbers
206
7.5.3
The final period of the decay
207
7.5.4
The Loitsiansky and Saffman integrals
207
7.6
Free decay: the classical era
209
7.6.1
Taylor
(1935) 209
7.6.2 Von
Karman
and Howarth
(1938) 210
7.6.3
Kolmogorov s prediction of the decay exponents
212
Contents xv
7.6.4 Batchelor (1948) 213
7.6.5 The non-invariance
of the Loitsiansky
integral 215
7.7
Theories of the decay based on spectral models
216
7.7.1
Two-range spectral models
216
7.7.2
Three-range spectral models
220
7.8
Free decay: towards universality?
220
7.8.1
The effect of initial conditions
221
7.8.2
Fractal-generated turbulence
226
References
228
8
Theoretical constraints on mode reduction and the
turbulence response
232
8.1
Spectral large-eddy simulation
234
8.1.1
Statement of the problem
234
8.1.2
Spectral filtering to reduce the number of degrees of freedom
236
8.2
Intermode
spectral energy fluxes
238
8.2.1
Low-¿
partitioned energy fluxes
239
8.2.2
High-fc partitioned energy fluxes
239
8.2.3
Energy conservation revisited
239
8.3
Semi-analytical studies of subgrid modelling using statistical closures
241
8.4
Studies of subgrid models using direct numerical simulation
248
8.5
Stochastic backscatter
250
8.6
Conditional averaging
253
8.7
A statistical test of the eddy-viscosity hypothesis
255
8.8
Constrained numerical simulations
260
8.8.1
Operational
LES
261
8.9
Discussion
266
References
267
PART III STATISTICAL THEORY AND FUTURE DIRECTIONS
9
The Kraichnan-Wyld—Edwards covariance equations
273
9.1
Preliminary remarks
274
9.1.1
RPTs as statistical closures
274
9.1.2
Perceptions of RPTs
274
9.1.3
Some general characteristics of RPTs
277
9.2
The problem restated: the exact covariance equations
277
9.2.1
The general inhomogeneous covariance equation
277
9.2.2
Centroid and difference coordinates
279
9.2.3
The exact covariance equations for HIT
281
9.3
A short history of closure approximations
281
9.4
The K VE covariance equations: the problem reformulated
283
9.4.1
Comparison of quasi-normality with perturbation theory
284
9.4.2
The KWE covariance equations
285
9.5
Renormalized response functions as closure approximations
287
9.5.1
Failure of the EFP and
DIA
closures
288
xvi Contents
9.5.2
The Local Energy Transfer (LET) theory
290
9.6
Numerical assessment of closure theories
292
9.6.1
Some recent calculations of LET and EDQNM
292
9.7
Conclusions
294
References
296
10
Two-point closures: some basic issues
299
10.1
Perturbation theory and renormalization
299
10.2
Quantum-style formalisms: WyId-Lee and Martin-Siggia-Rose
303
10.2.1
The improved Wyld-Lee formalism
305
10.2.2
The Martin-Siggia-Rose formalism
307
10.3
How general are the formalisms?
310
10.4
Galilean
invariance
and the
DIA
311
10.5
Lagrangian-history theories
314
References
315
11
The renormalization group applied to turbulence
317
11.1
Formulation of conditional mode elimination for turbulence
318
11.2
Renormalization group
321
11.3
Forster-Nelson-Stephen theory of stirred fluid motion
322
11.3.1
Application of the RG to stirred fluid motion with
asymptotic freedom as
к
—* 0 322
11.3.2
Differential RG equations
325
11.3.3
FNS theory in terms of conditional averaging
326
11.4
Turbulence RG theories based on filtered averages
327
11.4.1
Iterative averaging: McComb
(1982) 328
11.4.2
Iterative averaging in wavenumber space
329
11.4.3
Relationship of iterative averaging to Rose s
(1977)
method
330
11.4.4
Improved iterative averaging
331
11.5
Problems with filtered averages
332
11.6
The two-field theory
334
11.6.1
The hypothesis of local chaos
336
11.6.2
The recursion relations of two-field theory
339
11.7
Improved two-field theory
341
11.7.1
Non-Gaussian perturbation theory
344
11.8
Applications and developments of iterative averaging
346
11.9
Is field-theoretic RG a theory of turbulence?
348
11.9.1
Differential recursion relations
351
References
352
12
Work in progress and future directions
356
12.1
Turbulence response
356
12.1.1
Fluctuation-response relations (FRRs)
356
12.1.2
Numerical assessment
357
12.2
Renormalized perturbation theories
358
Contents xvii
12.2.1
Extension
of Edwards
(1964)
theory to the two-time
covariance C(k;
t, ť)
359
12.2.2
Recovering the LET theory
362
12.3
Renormalization group
365
12.3.1
Power-law forcing and the renormalization group
365
12.3.2
Application of the Edwards
(1964) pdf
to RG mode elimination
366
12.4
Towards shear flows
368
12.4.1
Application of the two-field theory to
LES
of shear flows
368
12.5
Postscript: The nature of the problem
372
References
375
PART IV APPENDICES
Appendix A Implications of isotropy and continuity for
correlation tensors
379
References
382
Appendix
В
Properties of Gaussian distributions
383
B.I Discrete systems: real scalar variables
383
B.I.I Two-point correlations
384
B.2 Discrete systems: complex scalar variables
389
B.3 Scalar fields
390
B.3.1 Extension to wavenumber and time
392
B.3.
2
The generating functional
393
B.4 Vector fields
395
B.5
Isotropie
fields
396
B.6 Inhomogeneous vector fields
398
References
398
Appendix
С
Evaluation of the
L(k,j)
coefficient
399
C.I Derivation of the closed covariance equation
399
C.2 Evaluation of ¿(k, j)
401
C.2.1 A note on numerical evaluation in closures
403
References
403
Index
405
|
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| id | DE-604.BV041768132 |
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| indexdate | 2024-07-10T01:04:57Z |
| institution | BVB |
| isbn | 9780199689385 |
| language | English |
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| owner_facet | DE-19 DE-BY-UBM DE-703 DE-11 DE-20 |
| physical | XX, 408 S. graph. Darst. |
| publishDate | 2014 |
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| publisher | Oxford Univ. Press |
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| series2 | International series of monographs on physics |
| spelling | McComb, William D. 1940- Verfasser (DE-588)1158082231 aut Homogeneous, isotropic turbulence phenomenology, renormalization, and statistical closures W. David McComb 1. ed. Oxford Oxford Univ. Press 2014 XX, 408 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier International series of monographs on physics 162 Turbulenz (DE-588)4061232-6 gnd rswk-swf Turbulenz (DE-588)4061232-6 s DE-604 International series of monographs on physics 162 (DE-604)BV000106406 162 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027214170&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | McComb, William D. 1940- Homogeneous, isotropic turbulence phenomenology, renormalization, and statistical closures International series of monographs on physics Turbulenz (DE-588)4061232-6 gnd |
| subject_GND | (DE-588)4061232-6 |
| title | Homogeneous, isotropic turbulence phenomenology, renormalization, and statistical closures |
| title_auth | Homogeneous, isotropic turbulence phenomenology, renormalization, and statistical closures |
| title_exact_search | Homogeneous, isotropic turbulence phenomenology, renormalization, and statistical closures |
| title_full | Homogeneous, isotropic turbulence phenomenology, renormalization, and statistical closures W. David McComb |
| title_fullStr | Homogeneous, isotropic turbulence phenomenology, renormalization, and statistical closures W. David McComb |
| title_full_unstemmed | Homogeneous, isotropic turbulence phenomenology, renormalization, and statistical closures W. David McComb |
| title_short | Homogeneous, isotropic turbulence |
| title_sort | homogeneous isotropic turbulence phenomenology renormalization and statistical closures |
| title_sub | phenomenology, renormalization, and statistical closures |
| topic | Turbulenz (DE-588)4061232-6 gnd |
| topic_facet | Turbulenz |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027214170&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000106406 |
| work_keys_str_mv | AT mccombwilliamd homogeneousisotropicturbulencephenomenologyrenormalizationandstatisticalclosures |