Homogeneous Turbulence Dynamics

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 9
Abbreviations Used in This Book......Page 18
1.1 Scope of the Book......Page 19
1.2 Structure and Contents of the Book......Page 21
Bibliography......Page 27
2.1.1 Mass Conservation......Page 28
2.1.2 The Navier–Stokes Momentum Equations......Page 30
2.1.3 Incompressible Turbulence......Page 31
2.1.4 First Insight into Compressibility Effects......Page 32
2.1.5 Reminder About Circulation and Vorticity......Page 33
2.1.6 Adding Body Forces or Mean Gradients......Page 34
2.2.2 Single-Point and Multipoint Moments......Page 37
2.2.4 Application of Reynolds Decomposition to Dynamical Equations......Page 38
2.3.1 RST Equations......Page 40
2.3.2 The Mean Flow Consistent With Homogeneity......Page 42
2.3.3 Homogeneous RST Equations. Briefs About Closure Methods......Page 44
2.4 Anisotropy in Physical Space. Single-Point and Two-Point Correlations......Page 45
2.5.1 Second-Order Statistics......Page 46
2.5.2 Poloidal–Toroidal Decomposition and Craya–Herring Frame of Reference......Page 49
2.5.3 Helical-Mode Decomposition......Page 50
2.5.4 Use of Projection Operators......Page 51
2.5.5 Nonlinear Dynamics......Page 53
2.5.6 Background Nonlinearity in Different Reference Frames......Page 54
2.6.1 Second-Order Velocity Statistics......Page 56
2.6.1.1 Directional and Polarization Anisotropy – Intrinsic Form......Page 58
2.6.1.3 Bridging with Dimensionality and Componentality......Page 59
2.7 A Synthetic Scheme of the Closure Problem: Nonlinearity and Nonlocality......Page 61
Bibliography......Page 65
3.1.1 How to Generate Isotropic Turbulence?......Page 67
3.1.2 Main Observed Statistical Features of Developed Isotropic Turbulence......Page 69
3.1.3 Energy Decay Regimes......Page 75
3.1.4 Coherent Structures in Isotropic Turbulence......Page 76
3.2.1 Symmetries of Navier–Stokes Equations and Existence of Self-Similar Solutions......Page 77
3.2.2 Algebraic Decay Exponents Deduced From Symmetry Analysis......Page 80
3.2.3 Time-Variation Exponent and Inviscid Global Invariants......Page 82
3.2.4 Refined Analysis Without PLE Hypothesis......Page 83
3.2.5 Self-Similarity Breakdown......Page 84
3.2.6 Self-Similar Decay in the Final Region......Page 85
3.3 Reynolds Stress Tensor and Analysis of Related Equations......Page 86
3.4.1 Double Correlations and Typical Scales......Page 88
3.4.2 (Very Brief) Reminder About Kolmogorov Legacy, Structure Functions, “Modern” Scaling Approach......Page 89
3.4.3 Turbulent Kinetic-Energy Cascade in Fourier Space......Page 91
3.5.1 The Background Triadic Interaction......Page 94
3.5.2 Nonlinear Energy Transfers and Triple Correlations......Page 97
3.5.3 Global and Detailed Conservation Properties......Page 98
3.5.4 Advanced Analysis of Triadic Transfers and Waleffe’s Instability Assumption......Page 99
3.5.5 Further Discussions About the Instability Assumption......Page 103
3.5.6 Principle of Quasi-Normal Closures......Page 104
3.5.7 EDQNM for Isotropic Turbulence. Final Equations and Results......Page 107
3.5.7.2 Transfer Term at Increasing Reynolds Number......Page 108
3.5.7.3 Toward an Infinite Reynolds Number......Page 111
3.5.7.5 On Instantaneous Energy Transfers......Page 112
3.5.7.6 Nonlinear Cascade Time Scale, Equilibrium, and Dissipation Asymptotics......Page 113
3.6 Topological Analysis, Coherent Events, and Related Dynamics......Page 115
3.6.1 Topological Analysis of Isotropic Turbulence......Page 116
3.6.2 Vortex Tube: Statistical Properties and Dynamics......Page 120
3.6.3 Bridging with Turbulence Dynamics and Intermittency......Page 125
3.7.1 On Vortices, Scales, Wavenumbers, and Wave Vectors – What are the Small Scales?......Page 127
3.7.2 Is There an Energy Cascade in the Physical Space?......Page 129
3.7.3 Self-Amplification of Velocity Gradients......Page 130
3.7.4 Non-Gaussianity and Depletion of Nonlinearity......Page 134
3.8.1 Influence of the Space Dimension: Introduction to d-Dimensional Turbulence......Page 135
3.8.2 Pure 2D Turbulence and Dual Cascade......Page 136
3.8.3 Role of Pressure: A View of Burgers’ Turbulence......Page 138
3.8.4 Sensitivity with Respect to Energy-Pumping Process: Turbulence with Hyperviscosity......Page 140
Bibliography......Page 141
4.1 Physical and Numerical Experiments......Page 145
4.1.1 Brief Review of Experiments, More or Less in the Configuration of Homogeneous Turbulence......Page 147
4.2.2 Important Nondimensional Numbers. Particular Regimes......Page 149
4.3 Advanced Analysis of Energy Transfer by DNS......Page 151
4.4 Balance of RST Equations. A Case Without “Production.” New Tensorial Modeling......Page 153
4.5.1 Analysis of Deterministic Solutions......Page 157
4.5.2.1 Single-Time Second-Order Statistics......Page 161
4.5.2.2 Single-Time Third-Order Statistics......Page 162
4.6.1 Full Exact Nonlinear Equations. Wave Turbulence......Page 163
4.6.2 Second-Order Statistics: Identification of Relevant Spectral-Transfer Terms......Page 166
4.6.3 Toward a Rational Closure with an EDQNM Model......Page 167
4.6.4 Recovering the Asymptotic Theory of Inertial Wave Turbulence......Page 168
4.7.1 Eventual Two-Dimensionalization or Not......Page 171
4.7.2 Meaning of the Slow Manifold......Page 173
4.7.3 Are Present DNS and LES Useful for Theoretical Prediction?......Page 174
4.7.4 Is the Pure Linear Theory Relevant?......Page 175
4.7.5 Provisional Conclusions About Scaling Laws and Quantified Values of Key Descriptors......Page 176
4.8 Coherent Structures, Description, and Dynamics......Page 177
Bibliography......Page 182
5.1 Main Observations......Page 185
5.2 Experiments for Turbulence in the Presence of Mean Strain.Kinematics of the Mean Flow......Page 187
5.2.1 Pure Irrotational Strain, Planar Distortion......Page 188
5.2.2 Axisymmetric (Irrotational) Strain......Page 190
5.2.4 More General Distortions. Kinematics of Rotational Mean Flows......Page 191
5.3.1.1 Planar Strain......Page 192
5.3.1.3 More General Rotational Strains......Page 194
5.3.2 General Assessment of RST Single-Point Closures......Page 195
5.3.3 Linear Response of Turbulence to Irrotational Mean Strain......Page 196
5.4 The Fundamentals of Homogeneous RDT......Page 198
5.4.2 Results at Very Short Times. Relevance at Large Elapsed Times......Page 201
5.5.1 General RDT Solution......Page 202
5.5.2 Linear Response of Turbulence to Axisymmetric Strain......Page 203
5.6 First Step Toward a Nonlinear Approach......Page 204
5.7 Nonhomogeneous Flow Cases. Coherent Structures in Strained Homogeneous Turbulence......Page 205
Bibliography......Page 207
6.1 Physical and Numerical Experiments: Kinetic Energy, RST, Length Scales, Anisotropy......Page 210
6.1.2 Main Observations......Page 211
6.2 Reynolds Stress Tensor and Analysis of Related Equations......Page 215
6.3 Rapid Distortion Theory: Equations, Solutions, Algebraic Growth......Page 217
6.3.1 Some Properties of RDT Solutions......Page 219
6.3.2 Relevance of Homogeneous RDT......Page 222
6.4 Evidence and Uncertainties for Nonlinear Evolution: Kinetic-Energy Exponential Growth Using Spectral Theory......Page 224
6.5 Vortical–Structure Dynamics in Homogeneous Shear Turbulence......Page 225
6.6 Self-Sustaining Turbulent Cycle in Homogeneous Sheared Turbulence......Page 227
6.7 Self-Sustaining Processes in Nonhomogeneous Sheared Turbulence: Exact Coherent States and Traveling-Wave Solutions......Page 228
6.8 Local Isotropy in Homogeneous Shear Flows......Page 232
Bibliography......Page 235
7.1 Observations, Propagating and Nonpropagating Motion. Collapse of Vertical Motion and Layering......Page 237
7.2 Simplified Equations, Using Navier–Stokes and Boussinesq Approximations, With Uniform Density Gradient......Page 241
7.2.1 Reynolds Stress Equations With Additional Scalar Variance and Flux......Page 242
7.2.2 First Look at Gravity Waves......Page 243
7.3 Eigenmode Decomposition. Physical Interpretation......Page 244
7.4 The Toroidal Cascade as a Strong Nonlinear Mechanism Explaining the Layering......Page 247
7.5 The Viewpoint of Modeling and Theory: RDT, Wave Turbulence, EDQNM......Page 249
7.6.1 Simplified Scaling Laws......Page 253
7.6.2 Pancake Structures, Zig-Zag, and Kelvin–Helmholtz Instabilities......Page 255
Bibliography......Page 259
8.1 Rotating Stratified Turbulence......Page 261
8.1.1 Basic Triadic Interaction for Quasi-Geostrophic Cascade......Page 264
8.1.2 About the Case With Small but Nonnegligible f/N Ratio......Page 265
8.1.3 The QG Model Revisited. Discussion......Page 266
8.2 Rotation or Stratification With Mean Shear......Page 268
8.2.1 The Rotating-Shear-Flow Case......Page 271
8.2.3 Analogies and Differences Between the Two Cases......Page 273
8.3.1 Physical Context, the Mean Flow......Page 274
8.3.2 RDT Equations......Page 276
8.4 Elliptical Flow Instability From “Homogeneous” RDT......Page 277
8.5 Axisymmetric Strain With Rotation......Page 283
8.6 Relevance of RDT and WKB RDT Variants for Analysis of Classical Instabilities......Page 284
Bibliography......Page 288
9.1.1 Statement of the Problem......Page 291
9.1.2 Kovasznay’s Linear Decomposition......Page 292
9.1.3 Weakly Nonlinear Corrected Kovasznay Decomposition......Page 296
9.1.4 Helmholtz Decomposition and Its Extension......Page 297
9.2.1 Arbitrary Flows......Page 299
9.2.2 Simplifications in the Isotropic Case......Page 303
9.2.3 Quasi-Isentropic Isotropic Turbulence: Physical and Spectral Descriptions......Page 306
9.3 Different Regimes in Compressible Turbulence......Page 309
9.3.1.1 Linear Theory......Page 310
9.3.1.2 The Relevant Incompressible Limit for Both Spectra of Solenoidal Energy and Pressure Variance......Page 314
9.3.1.3 Quasi-Inviscid Limit: Toward an Extended Wave-Turbulence Model......Page 315
9.3.1.4 Introducing Relevant Eddy-Damping. Main Results......Page 316
9.3.1.5 Additional Discussion About the Modified Decorrelation Function......Page 319
9.3.1.6 Analytical Fauchet–Bertoglio Model......Page 321
9.3.1.7 Numerical Experiments......Page 326
9.3.2 Weakly Compressible Thermal Regime......Page 328
9.3.2.1 Asymptotic Analysis and Possible Thermal Regimes......Page 329
9.3.2.3 Numerical Observations......Page 332
9.3.3.1 Conditions for Occurrence of Shocklets......Page 333
9.3.3.2 Energy Budget and Shocklet Influence......Page 334
9.3.3.3 Enstrophy Budget and Shocklet Influence......Page 335
9.3.4 Supersonic Regime......Page 336
9.4 Structures in the Physical Space......Page 337
9.4.1 Turbulent Structures in Compressible Turbulence......Page 338
9.4.2 A Probabilistic Model for Shocklets......Page 339
Bibliography......Page 342
10.1 Effects of Compressibility in Free-Shear Flows. Observations......Page 345
10.1.1 RST Equations and Single-Point Modeling......Page 346
10.1.2 Preliminary Linear Approach: Pressure-Released Limit and Irrotational Strain......Page 348
10.2 A General Quasi-Isentropic Approach to Homogeneous Compressible Shear Flows......Page 350
10.2.1 Governing Equations and Admissible Mean Flows......Page 351
10.2.2 Properties of Admissible Mean Flows......Page 353
10.2.3 Linear Response in Fourier Space. Governing Equations......Page 354
10.2.3.2 Recovering the Solenoidal Limit......Page 357
10.2.3.3 Irrotational Mean-Strain Case......Page 358
10.3 Incompressible Turbulence With Compressible Mean-Flow Effects: Compressed Turbulence......Page 360
10.4.1 Qualitative Results......Page 362
10.4.2 Discussion of Results......Page 363
10.4.3 Toward a Complete Linear Solution......Page 366
10.5 Perspectives and Open Issues......Page 367
10.5.2 Perspectives Toward Inhomogeneous Shear Flows......Page 368
10.6 Topological Analysis, Coherent Events and Related Dynamics......Page 369
10.6.1 Nonlinear Dynamics in the Subsonic Regime......Page 370
10.6.2 Topological Analysis of the Rate-of-Strain Tensor......Page 372
10.6.3 Vortices, Shocklets, and Dynamics......Page 373
Bibliography......Page 374
11.1.1 Destructive Interactions......Page 376
11.1.2 Nondestructive Interactions......Page 377
11.2.1 Shock Modeling and Jump Relations......Page 378
11.2.2 Introduction to the Linear Interaction Approximation Theory......Page 379
11.2.3 Vortical Turbulence–Shock Interaction......Page 381
11.2.4 Acoustic Turbulence–Shock Interaction......Page 388
11.2.5 Mixed Turbulence–Shock Interaction......Page 391
11.2.5.1 Influence of the Upstream Entropy Fluctuations......Page 392
11.2.6 On the Use of RDT for Linear Nondestructive Interaction Modeling......Page 396
11.3.1 Turbulent Jump Conditions for the Mean Field......Page 397
11.3.2 Jump Conditions for an Incident Isotropic Turbulence......Page 399
Bibliography......Page 400
12.1 Shock Description and Emitted Fluctuating Field......Page 402
12.2.2 Incident Entropy and Vorticity Waves......Page 404
12.2.2.2 Emitted Acoustic Waves—Propagative and Nonpropagative Regimes......Page 405
12.2.3.2 Emitted Entropy and Vorticity Waves......Page 407
12.2.3.3 Emitted Acoustic Waves......Page 408
12.3.1 General Decompositions of the Perturbation Field......Page 409
12.3.2 Calculation of Amplitudes of Emitted Waves......Page 411
12.4 Reconstruction of the Second-Order Moments......Page 414
12.4.1 Case of a Single Incident Wave......Page 416
12.4.2 Case of an Incident Turbulent Isotropic Field......Page 417
12.5 A posteriori Assessment of LIA......Page 421
Bibliography......Page 423
13.1.1 Solutions for ODEs in Orthonormal Fixed Frames of Reference......Page 424
13.1.2 Using Solenoidal Modes for a Green’s Function with a Minimal Number of Components......Page 426
13.1.3 Prediction of Statistical Quantities......Page 427
13.1.3.1 Initial-Value Problem or Forcing?......Page 429
13.2 Zonal RDT and Short-Wave Stability Analysis......Page 430
13.2.2 Zonal Stability Analysis With Disturbances Localized Around Base-Flow Trajectories......Page 431
13.2.3 Using Characteristic Rays Related to Waves Instead of Trajectories......Page 433
13.3.1 Transport Models Along Mean Trajectories......Page 435
13.3.2 Semiempirical Transport “Shell” Models......Page 436
13.4 Conclusions, Recent Perspectives Including Subgrid-Scale Dynamics Modeling......Page 437
Bibliography......Page 439
14.1 Canonical HIT, Dependence on the Eddy Damping for the Scaling of the Energy Spectrum in the Inertial Range......Page 441
14.2.1 Random and Averaged Nonlinear Green’s Functions......Page 443
14.2.2 Homogeneous Anisotropic Turbulence with a Mean Flow......Page 444
14.3 A General EDQN Closure. Different Levels of Markovianization......Page 446
14.3.1 EDQNM2 Version......Page 447
14.3.2 A Simplified Version: EDQNM1......Page 448
14.3.3 The Most Sophisticated Version: EDQNM3......Page 449
14.4 Application of Three Versions to the Rotating Turbulence......Page 451
14.5.1.2 Pure Shear......Page 455
14.5.2.1 Buoyant Flows in a Stably Stratified Fluid......Page 456
14.5.2.2 Weakly Compressible Isotropic Turbulence......Page 457
14.5.3 Role of the Nonlinear Decorrelation Time Scale......Page 458
14.6 Connection with Self-Consistent Theories: Single Time or Two Time?......Page 459
14.7.1 A Self-Consistent Representation of the Spectral Tensor for Weak Anisotropy......Page 461
14.7.2 Brief Discussion of Concepts, Results, and Open Issues......Page 463
14.8 Open Numerical Problems......Page 464
Bibliography......Page 465
15.1 Homogenization of Turbulence. Local or Global Homogeneity? Physical Space or Fourier Space?......Page 467
15.2 Linear Theory, “Homogeneous” RDT, WKB Variants, and LIA......Page 469
15.3 Multipoint Closures for Weak and Strong Turbulence......Page 471
15.3.1 The Wave-Turbulence Limit......Page 472
15.3.3 Revisiting Basic Assumptions in MPC......Page 473
15.4 Structure Formation, Structuring Effects, and Individual Coherent Structures......Page 474
15.5 Anisotropy Including Dimensionality, a Main Theme......Page 475
15.6 Deriving Practical Models......Page 476
Bibliography......Page 478
Index......Page 479