Preface ......................................................... v
Introduction ................................................... xi
1 Constructive Modeling of Free Developed Turbulence -
Coherent Structures, Laminar Turbulent Transition, Chaos ..... 1
1.1 Introduction ............................................ 1
1.2 Rational averaging of large vortex structures .......... 12
1.3 Some experimental and theoretical investigations ....... 29
1.4 General problem formulation ............................ 38
1.5 Simulation of coherent structures in turbulent flows ... 41
1.6 Correctness of the problem formulation ................. 50
1.7 Calculated results for coherent structures in the
wake behind a body ..................................... 53
1.8 On the analysis of spectral characteristics ............ 60
1.9 Numerical simulation of the random component of
turbulence ............................................. 71
1.10 Laminar turbulent transition. Simulation of three-
dimensional flows in clean rooms ....................... 76
1.11 Transition to chaos (numerical experiments) ............ 84
1.11.1 General aspects ................................. 84
1.11.2 "Kolmogoroff's flow'' of the viscous fluid at
subcritical and supercritical regimes.
Transition to chaos ............................. 86
1.11.3 Study of the large-scale turbulence in ocean ... 111
1.11.4 Numerical simulation of the internal waves in
a stratified fluid ............................. 128
1.11.5 Rayleigh-Taylor instability: evolution to the
turbulent stage ................................ 137
1.11.6 Numerical simulation of the convective flow
over large-scale source of energy (big fire
in the atmosphere) ............................. 143
1.12 Axiomatic model of fully developed turbulence ......... 152
1.13 Conclusion ............................................ 154
Acknowledgement ............................................ 157
References ................................................. 157
2 Modeling of Richtmyer-Meshkov Instability .................. 164
Introduction ............................................... 164
2.1 Numerical method ...................................... 169
2.2 Model calculations .................................... 171
2.2.1 The Couchy problem for one-dimensional
isotopic flow of an ideal gas .................. 174
2.2.2 Boundary conditions ............................ 176
2.2.3 Comparison of results by different models ...... 178
2.3 The analytical approach ............................... 181
2.4 Computational experiment .............................. 186
2.5 Physical mechanisms of the RMI evolution .............. 190
2.6 A sequential transition to turbulence in RMI
instability ........................................... 198
2.7 Three-dimensional numerical simulation of the RMI ..... 200
2.8 Conclusion ............................................ 208
Appendix ................................................... 209
References ................................................. 210
3 Rayleigh-Taylor Instability: Analysis and Numerical
Simulation ................................................. 214
3.1 The theory of Rayleigh Taylor instability:
modulatory perturbations and mushroom-flow dynamics ... 214
3.1.1 Introduction ................................... 215
3.1.2 Periodicity and symmetry of modulatory
perturbations .................................. 216
3.1.3 Cutting off the singularities associated with
jets ........................................... 220
3.1.4 Classification of perturbations ................ 223
3.1.5 Results ........................................ 227
3.1.6 Classification of stability problems ........... 230
3.1.7 Initiation of a mushroom structure ............. 232
3.1.8 The mushroom flow structure .................... 238
3.1.9 Numerical simulation ........................... 243
3.2 Development of the Rayleigh- Taylor instability:
numerical simulations ................................. 246
3.2.1 Introduction ................................... 246
3.2.2 Numerical simulation of RTI development by
the method of large particles .................. 247
3.2.3 Intermode interaction in RTI ................... 252
3.2.4 RTI simulation by the method of pseudo-
compressibility ................................ 257
3.2.5 Numerical simulation of the RTI development
by means of high-resolution Euler hydrocode .... 262
References ................................................. 280
4 Direct Statistical Approach for Aerohydrodynamic
Problems ................................................... 285
4.1 Statistical modeling in rarefied gas-dynamics ......... 285
4.1.1 Introduction ................................... 285
4.1.2 Stochastic analogue of the Boltzmann
equation ....................................... 287
4.1.3 Probabilistic approach to the basic equation
of the collision stage ......................... 291
4.1.4 Algorithms for modeling the collision
relaxation ..................................... 294
4.2 Direct statistical modeling of the shock wave in
gaseous flow with velocity pulsations ................. 299
4.2.1 Introduction ................................... 299
4.2.2 Problem formulation ............................ 300
4.2.3 Results of the numerical modeling .............. 303
4.2.4 Conclusion ..................................... 307
4.3 Direct statistical simulation for some problems of
turbulence ............................................ 307
4.3.1 Introduction ................................... 307
4.3.2 An application of the statistical method of
particles in cell for simulation of the
momentumless wake .............................. 308
4.3.3 An application of the statistical method of
particles in cell to the problem of
a turbulent spot ............................... 311
4.3.4 The direct statistical modeling of the
turbulence within a wake behind the cylinder ... 320
4.3.5 Simulation results ............................. 330
4.3.6 Conclusion ..................................... 331
References ................................................. 333
Appendix A Computational Experiment: Direct Numerical
Simulation of Complex Gas-Dynamical Flows on the
Basis of Euler, Navier-Stokes, and Boltzmann
Models ............................................ 336
A.l Introduction .......................................... 336
A.1.1 The use of numerical methods ................... 336
A.1.2 Numerical methods applicable to gas-dynamical
problems ....................................... 339
A.1.2.1 Method of finite differences .......... 340
A.1.2.2 Method of integral relations .......... 340
A.1.2.3 Method of characteristics ............. 341
A.l.2.4 Particle-in-cell (PIC) method ......... 341
A.1.3 Development of numerical algorithms ............ 342
A.1.3.1 Steady-state schemes .................. 342
A.1.3.2 Unsteady-state schemes ................ 344
A.1.3.3 Large-particle method ................. 344
A.1.4 Computational experiments ...................... 345
A.2 "Large-particles" method for the study of complex
gas flows ............................................. 347
A.2.1 Calculations ................................... 347
A.2.2 Boundary conditions ............................ 349
A.2.3 Viscosity effects .............................. 350
A.2.4 Stability of the scheme ........................ 352
A.2.5 Advantages ..................................... 354
A.2.6 Results ........................................ 355
A.3 Computation of incompressible viscous flows ........... 363
A.3.1 The problem .................................... 363
A.3.2 The difference scheme .......................... 364
A.3.3 Results ........................................ 367
A.4 Computation of viscous compressible gas flow
(conservative flow method) ............................ 369
A.4.1 The method ..................................... 369
A.4.2 Analysis ....................................... 372
A.4.3 Results ........................................ 374
A.5 Statistical model for the investigation of rarefied
gas flows ............................................. 376
A.5.1 The model ...................................... 376
A.5.2 The method ..................................... 378
A.5.3 Results ........................................ 383
A.6 Conclusion ............................................ 386
References ................................................. 386
Appendix В Formation of Large-Scale Structures in the Gap
Between Rotating Cylinders: the Rayleigh-
Zeldovich Problem ................................. 389
B.l Introduction .......................................... 389
B.2 Background ............................................ 391
B.3 Direct numerical simulation methodology ............... 392
B.4 Statement of the problem and results .................. 393
B.4.1 The inner cylinder is at rest and the outer
cylinder is rotating ........................... 394
B.4.2 The inner cylinder is at rest and the outer
cylinder is brought to rest .................... 398
B.4.3 The inner cylinder is rotating and the outer
cylinder is at rest ............................ 400
B.5 Conclusions ........................................... 403
References ................................................. 404
Appendix С Universal Technology of Parallel Computations
for the Problems Described by Systems of
the Equations of Hyperbolic Type: A Step
to Supersolver .................................... 405
C.l Introduction .......................................... 405
C.2 Unified methodics ..................................... 406
C.3 A method for using non-conservative variables ......... 410
C.4 Parallel program implementation ....................... 413
C.5 Results of numerical simulation ....................... 415
C.6 Conclusion ............................................ 418
References ................................................. 420
Appendix D Supercomputers in Mathematical Modeling of
the High Complexity Problems ...................... 422
D.l Introduction .......................................... 422
D.2 Turbulence and hydrodynamic instabilities ............. 424
D.3 Supersolver ........................................... 428
D.4 Applications .......................................... 429
D.4.1 Gas-dynamics (CFD) ............................. 429
D.4.2 Hydrodynamic instabilities ..................... 434
D.4.3 Seismic data processing ........................ 436
D.4.4 Safety of housing and industrial
constructions under intensive dynamic
loadings ....................................... 437
D.4.5 Nonlinear contact shell dynamics ............... 439
D.4.6 Computer models in medicine .................... 440
D.5 Conclusion ............................................ 444
References ................................................. 445
Appendix E On Nuts and Bolts of Structural Turbulence and
Hydrodynamic Instabilities ........................ 448
E.l Rational Constructivism ............................... 448
E.2 Back in Mechanics ..................................... 449
E.3 Large Vortices ........................................ 450
E.4 Structural Instabilities .............................. 452
E.5 Vortex Cascades ....................................... 453
E.6 Principal Modes ....................................... 453
References ................................................. 456
Appendix F List of the Main Publications of
О.М. Belotserkovskii .............................. 459
Monographs .................................................... 459
Papers ........................................................ 460
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