New trends in turbulence

References......Page 1
MEASURES OF ANISOTROPY AND THE UNIVERSAL
PROPERTIES OF TURBULENCE
......Page 53
1 Introduction......Page 56
2.1 The method of $SO(3)$ decomposition......Page 58
2.2 Foliation of the structure function into j-sectors......Page 62
2.3 The velocity structure functions......Page 63
2.3.1 The second-order structure function......Page 64
2.4 Dimensional estimates for the lowest-order anisotropic scaling exponents......Page 66
2.5 Summary......Page 68
3.2 Relevance of the anisotropic contributions......Page 69
3.3 The measurements......Page 70
4.1 General remarks on the data......Page 74
4.2.1 The anisotropic tensor component derived under the assumption of axisymmetry......Page 76
4.2.2 The complete $j = 2$ anisotropic contribution......Page 81
5.1 Extracting the $j = 1$ component......Page 85
6.1 Introduction......Page 88
6.2.1 The second-order structure function......Page 89
6.2.2 Higher-order structure functions......Page 92
6.3 Summary......Page 93
7 Conclusions......Page 98
A Full form for the $j = 2$ contribution for the homogeneous case......Page 99
B.1 Antisymmetric contribution......Page 105
B.2 Symmetric contribution......Page 107
C Tests of the robustness of the interpolation formula......Page 109
References......Page 110
LARGE-EDDY SIMULATIONS OF TURBULENCE
......Page 112
1 Introduction......Page 116
1.1 LES and determinism: Unpredictability growth......Page 117
2 Vortex dynamics......Page 118
2.1.3 The Q-criterion......Page 119
2.2 Vortex identification......Page 120
2.2.1 Isotropic turbulence......Page 121
2.2.2 Backward-facing step......Page 122
3.1 LES equations for a flow of constant density......Page 124
3.3 Eddy-viscosity and diffusivity assumption......Page 127
3.4 Smagorinsky’s model......Page 129
4.1 Spectral eddy viscosity and diffusivity......Page 130
4.2 EDQNM plateau-peak model......Page 131
4.2.1 The spectral-dynamic model......Page 133
4.2.2 Existence of the plateau-peak......Page 134
4.3 Incompressible plane channel......Page 136
4.3.2 Streaks and hairpins......Page 137
4.3.3 Spectral DNS and LES......Page 138
5.1.1 Formalism......Page 142
5.1.3 Structure-function versus Smagorinsky models......Page 144
5.2 Selective structure-function model......Page 145
5.4 A test case for the models: The temporal mixing layer......Page 146
5.5 Spatially growing mixing layer......Page 148
5.6 Vortex control in a round jet......Page 150
5.7 LES of spatially developing boundary layers......Page 152
6.1 Dynamic models......Page 157
7.1 Generalized hyperviscosities......Page 160
7.3 Scale-similarity and mixed models......Page 161
8 LES of rotating flows......Page 162
8.1.1 Free-shear flows......Page 163
8.1.2 Wall flows......Page 164
9.1 Baroclinic eddies......Page 168
9.1.1 Synoptic-scale instability......Page 170
9.1.2 Secondary cyclogenesis......Page 171
10 LES of compressible turbulence......Page 172
10.1 Compressible LES equations......Page 173
10.2 Heated flows......Page 174
10.2.1 The heated duct......Page 175
10.2.2 Towards complex flow geometries......Page 176
11 Conclusion......Page 180
References......Page 181
STATISTICAL TURBULENCE MODELLING
......Page 186
1 Approaches to characterising turbulence......Page 188
2 Some basic statistical properties of turbulence and associated implications......Page 195
3.1 The eddy-viscosity concept......Page 202
3.2 Model categories......Page 203
3.3 Model applicability......Page 206
4 Second-moment equations and implied stress–strain interactions......Page 211
4.1 Near-wall shear......Page 214
4.2 Streamline curvature......Page 216
4.3 Separation and recirculating flow......Page 217
4.4 Rotation......Page 218
4.5 Irrotational strain......Page 219
4.6 Heat transfer and stratification......Page 220
5 Second moment closure......Page 221
6 Non-linear eddy-viscosity models......Page 227
7.1 Overview......Page 232
7.3 Aerospatiale aerofoil......Page 234
7.4 Cascade blade......Page 237
7.5 Axisymmetric impinging jet......Page 238
7.6 Prolate spheroid......Page 239
7.7 Round-to-rectangular transition duct......Page 241
7.8 Wing/flat-plate junction......Page 244
7.9 Fin-plate junction......Page 245
8 Concluding remarks......Page 250
References......Page 252
1.1.1 General solution of the wave equation......Page 258
1.4.6 Overall pressure levels......Page 265
1.4.8 Perceived Noise Level (PNL)......Page 268
2.3 Engine noise......Page 269
3 Methodology for jet noise......Page 271
4.2.2 Outflow boundary treatment......Page 277
5.1.1 Filtering......Page 294
References......Page 300
THE TOPOLOGY OF TURBULENCE
......Page 317
“BURGULENCE”
......Page 339
1 Introduction......Page 341
1.1 The Burgers equation in cosmology......Page 342
1.3 The Burgers equation as testing ground for Navier–Stokes......Page 345
2.1 The Hopf–Cole transformation and the maximum representation......Page 346
2.2 Shocks in one dimension......Page 348
2.3 Convex hull construction in more than one dimension......Page 352
2.4 Remarks on numerical methods......Page 353
3 The Fourier–Lagrange representation and artefacts......Page 354
4 The law of energy decay......Page 356
5 One-dimensional case with Brownian initial velocity......Page 361
6 Preshocks and the pdf of velocity gradients in one dimension......Page 365
7 The pdf of density......Page 368
8.1 Forced Burgers equation and variational formulation......Page 371
8.2 Periodic kicks......Page 374
8.3 Connections with Aubry–Mather theory......Page 378
TWO-DIMENSIONAL TURBULENCE
......Page 382
1 Introduction......Page 384
2.1 Euler vs. Navier–Stokes equations......Page 388
2.2 Vorticity representation......Page 389
2.3 Conservation laws......Page 390
2.4 Steady solutions of the Euler equations......Page 393
3 Vortex dynamics......Page 394
3.1 Systems of discrete vortices......Page 395
3.2 Vortex pairs......Page 396
3.3 Instability of shear flows and vortex lattices......Page 400
3.4 Statistical mechanics of point vortices......Page 401
4 Spectral properties, energy and enstrophy cascade......Page 408
4.1 Spectrally truncated equilibrium states......Page 409
4.2 The enstrophy and inverse energy cascades of forced turbulence......Page 412
4.3 The enstrophy cascade of freely evolving turbulence......Page 419
4.4 The emergence and evolution of isolated vortices......Page 420
5.1 Statistical mechanics of non-singular vorticity fields......Page 422
5.2 The Gibbs states......Page 425
5.3 Tests and discussion......Page 429
6.1 Thermodynamic approach......Page 432
6.2 Kinetic models......Page 436
7 Conclusions......Page 439
References......Page 440
ANALYSING AND COMPUTING
TURBULENT FLOWS
USING WAVELETS
......Page 445
1 Introduction......Page 449
2 History......Page 452
3.1.1 Analyzing wavelet......Page 453
3.1.2 Wavelet analysis......Page 454
3.2 Higher dimensions......Page 455
3.3 Algorithm......Page 456
4.1.1 1D Multi-Resolution Analysis......Page 457
4.2.1 Tensor product construction......Page 459
4.2.2 2D Multi-Resolution Analysis......Page 460
4.2.3 Periodic 2D Multi-Resolution Analysis......Page 461
4.3 Algorithm......Page 462
5.1.2 Numerical experiments......Page 464
5.2 Averaging procedure......Page 465
5.3.1 Probability Distribution Function (PDF)......Page 466
5.3.4 Statistical moments......Page 467
5.3.7 Fourier spectrum......Page 468
6.1 Local and global wavelet spectra......Page 469
6.2 Relation with Fourier spectrum......Page 470
6.3 Application to turbulence......Page 471
7.1 Local and global wavelet spectra......Page 472
7.2 Relation with Fourier spectrum......Page 473
8.1 CVS filtering......Page 474
8.1.2 Nonlinear thresholding......Page 475
8.2 Application to a 3D turbulent mixing layer......Page 476
8.3 Comparison between CVS and LES filtering......Page 477
9.1.2 Vorticity–velocity formulation......Page 480
9.2.1 Direct Numerical Simulation (DNS)......Page 481
9.2.2 Modelled Numerical Simulation (MNS)......Page 482
9.3.1 Principle of CVS......Page 484
9.3.3 CVS with turbulence model......Page 485
10.1 Adaptive wavelet scheme for nonlinear PDE’s......Page 486
10.1.2 Wavelet decomposition......Page 487
10.1.3 Evaluation of the nonlinear term......Page 489
10.1.4 Substraction strategy......Page 490
10.2 Adaptive wavelet scheme for the 2D Navier–Stokes equations......Page 491
10.3.2 Comparison between CVS and Fourier pseudo-spectral DNS......Page 494
Part IV Conclusion......Page 497
References......Page 498
LAGRANGIAN DESCRIPTION OF TURBULENCE
......Page 501
1 Particles in fluid turbulence......Page 503
1.1 Single-particle diffusion......Page 504
1.2.1 General consideration......Page 506
1.2.2 Solvable cases......Page 511
1.3 Two-particle dispersion in a nonsmooth incompressible flow......Page 514
1.4 Multiparticle configurations and breakdown of scale-invariance......Page 521
1.4.1 Absolute and relative evolution of particles......Page 522
1.4.2 Multiparticle motion in Kraichnan velocities......Page 523
1.4.3 Zero modes and slow modes......Page 525
1.4.4 Perturbative schemes......Page 528
2.1 Unforced evolution of passive fields......Page 532
2.2 Cascades of a passive scalar......Page 534
2.2.1 Passive scalar in a spatially smooth velocity......Page 535
2.3 Passive scalar in a spatially nonsmooth velocity......Page 539
3.1 Enstrophy cascade in two dimensions......Page 542
3.2 On the energy cascades in incompressible turbulence......Page 545
4 Conclusion......Page 547
References......Page 548