Random Fields and Stochastic Lagrangian Models (eBook)

lt;!doctype html public "-//w3c//dtd html 4.0 transitional//en"> Karl K. Sabelfeld, Institute of Computational Mathematics and Geophysics, Russian Acacemy of Sciences, Novosibirsk, Russia.

Preface 5
1 Introduction 17
1.1 Why random fields? 17
1.2 Some examples 19
1.3 Fundamental concepts 24
1.3.1 Random functions in a broad sense 25
1.3.2 Gaussian random vectors 29
1.3.3 Gaussian random functions 30
1.3.4 Random fields 32
1.3.5 Stochastic measures and integrals 33
1.3.6 Integral representation of random functions 35
1.3.7 Random trajectories 37
1.3.8 Stochastic differential, Ito integrals 38
1.3.9 Brownian motion 38
1.3.10 Multidimensional diffusion and Fokker-Planck equation 41
1.3.11 Central limit theorem and convergence of a Poisson process to a Gaussian process 42
2 Stochastic simulation of vector Gaussian random fields 45
2.1 Introduction 45
2.2 Discrete expansions related to the spectral representations of Gaussian random fields 46
2.2.1 Spectral representations 46
2.2.2 Series expansions 47
2.2.3 Expansion with an even complex orthonormal system 47
2.2.4 Expansion with a real orthonormal system 48
2.2.5 Complex valued orthogonal expansions 49
2.3 Wavelet expansions 49
2.3.1 Fourier wavelet expansions 50
2.3.2 Wavelet expansion 51
2.3.3 Moving averages 52
2.4 Randomized spectral models 53
2.4.1 Randomized spectral models defined through stochastic integrals 53
2.4.2 Stratified RSM for homogeneous random fields 55
2.5 Fourier wavelet models 55
2.5.1 Meyer wavelet functions 56
2.5.2 Evaluation of the coefficients and Fm. and Fm. 56
2.5.3 Cut-off parameters 58
2.5.4 Choice of parameters 59
2.6 Fourier wavelet models of homogeneous random fields based on randomization of plane wave decomposition 63
2.6.1 Plane wave decomposition of homogeneous random fields 63
2.6.2 Decomposition with fixed nodes 66
2.6.3 Decomposition with randomly distributed nodes 68
2.6.4 Some examples 70
2.6.5 Flow in a porous media in the first order approximation 72
2.6.6 Fourier wavelet models of Gaussian random fields 73
2.7 Comparison of Fourier wavelet and randomized spectral models 74
2.7.1 Some technical details of RSM 74
2.7.2 Some technical details of FWM 76
2.7.3 Ensemble averaging 78
2.7.4 Space averaging 78
2.8 Conclusions 79
2.9 Appendices 81
2.9.1 Appendix A. Positive definiteness of the matrix B 81
2.9.2 Appendix B. Proof of Proposition 2.1 81
3 Stochastic Lagrangian models of turbulent flows: Relative dispersion of a pair of fluid particles 86
3.1 Introduction 86
3.2 Criticism of 2-particle models 89
3.3 The quasi-1-dimensional Lagrangian model of relative dispersion 93
3.3.1 Quasi-1-dimensional analog of formula (2.14a) 94
3.3.2 Models with a finite-order consistency 96
3.3.3 Explicit form of the model (3.26, 3.27) 99
3.3.4 Example 104
3.4 A 3-dimensional model of relative dispersion 106
3.5 Lagrangian models consistent with the Eulerian statistics 108
3.5.1 Diffusion approximation 108
3.5.2 Relation to the well-mixed condition 110
3.5.3 A choice of the coefficients ai and bij 111
3.6 Conclusions 113
4 A new Lagrangian model of 2-particle relative turbulent dispersion 114
4.1 Introduction 114
4.2 An examination of Durbin’s nonlinear model 114
4.3 Mathematical formulation of a new model 116
4.4 A qualitative analysis of the problem (4.14) for symmetric £(r) 118
4.4.1 Analysis of the problem (4.14) in the deterministic case 118
4.4.2 Analysis of the problem (4.14) for stochastic £(r) 119
4.5 Qualitative analysis of the problem (4.14) in the general case 124
5 The combined Eulerian-Lagrangian model 129
5.1 Introduction 129
5.2 2-particle models 133
5.2.1 Eulerian stochastic models of high-Reynolds-number pseudoturbulence 133
5.3 A new 2-particle Eulerian-Lagrangian stochastic model 136
5.3.1 Formulation of 2-particle Eulerian-Lagrangian model 136
5.3.2 Models for the p.d.f. of the Eulerian relative velocity 139
5.4 Appendix 141
6 Stochastic Lagrangian models for 2-particle relative dispersion in high-Reynolds-number turbulence 145
6.1 Introduction 145
6.2 Preliminaries 146
6.3 A closure of the quasi-1-dimensional model of relative dispersion 147
6.4 Choice of the model (6.1) for isotropic turbulence 148
6.5 The model of relative dispersion of two particles in a locally isotropic turbulence 151
6.5.1 Specification of the model 151
6.5.2 Numerical analysis of the Q1D-model (6.30) 153
6.6 Model of the relative dispersion in intermittent locally isotropic turbulence 155
6.7 Conclusions 157
7 Stochastic Lagrangian models for 2-particle motion in turbulent flows. Numerical results 158
7.1 Introduction 158
7.2 Classical pseudoturbulence model 159
7.2.1 Randomized model of classical pseudoturbulence 159
7.2.2 Mean square separation of two particles in classical pseudoturbulence 162
7.3 Calculations by the combined Eulerian-Lagrangian stochastic model 165
7.3.1 Mean square separation of two particles 165
7.3.2 Thomson’s “two-to-one” reduction principle 168
7.3.3 Concentration fluctuations 170
7.4 Technical remarks 172
7.5 Conclusion 174
8 The 1-particle stochastic Lagrangian model for turbulent dispersion in horizontally homogeneous turbulence 175
8.1 Introduction 175
8.2 Choice of the coefficients in the Ito equation 178
8.3 2D stochastic model with Gaussian p.d.f 180
8.4 Numerical experiments 183
9 Direct and adjoint Monte Carlo for the footprint problem 187
9.1 Introduction 187
9.2 Formulation of the problem 188
9.3 Stochastic Lagrangian algorithm 189
9.3.1 Direct Monte Carlo algorithm 190
9.3.2 Adjoint algorithm 192
9.4 Impenetrable boundary 194
9.5 Reacting species 196
9.6 Numerical simulations 199
9.7 Conclusion 203
9.8 Appendices 204
9.8.1 Appendix A. Flux representation 204
9.8.2 Appendix B. Probabilistic representation 204
9.8.3 Appendix C. Forward and backward trajectory estimators 205
10 Lagrangian stochastic models for turbulent dispersion in an atmospheric boundary layer 209
10.1 Introduction 209
10.2 Neutrally stratified boundary layer 213
10.2.1 General case of Eulerian p.d.f 213
10.2.2 Gaussian p.d.f 216
10.3 Comparison with other models and measurements 217
10.3.1 Comparison with measurements in an ideally-neutral surface layer (INSL) 217
10.3.2 Comparison with the wind tunnel experiment by Raupach and Legg (1983) 220
10.4 Convective case 223
10.5 Boundary conditions 227
10.6 Conclusion 228
10.7 Appendices 229
10.7.1 Appendix A. Derivation of the coefficients in the Gaussian case 229
10.7.2 Appendix B. Relation to other models 231
11 Analysis of the relative dispersion of two particles by Lagrangian stochastic models and DNS methods 234
11.1 Introduction 234
11.2 Basic assumptions 236
11.2.1 Markov assumption 237
11.2.2 Consistency with the second Kolmogorov similarity hypothesis 237
11.2.3 Thomson’s well-mixed condition 238
11.3 Well-mixed Lagrangian stochastic models 238
11.3.1 Quadratic-form models 239
11.3.2 Quasi-1-dimensional models 240
11.3.3 3-dimensional extension of Q1D models 241
11.4 Stochastic Lagrangian models based on the moments approximation method 242
11.4.1 Moments approximation conditions 242
11.4.2 Realizability of LS models based on the moments approximation method 243
11.5 Comparison of different models of relative dispersion for the inertial subrange of a fully developed turbulence 245
11.5.1 Q1D quadratic-form model of Borgas and Yeung 245
11.5.2 Comparison of different models in the inertial subrange 247
11.6 Comparison of different Q1D models of relative dispersion for modestly large Reynolds number turbulence (Re? ? 240) 248
11.6.1 Parametrization of Eulerian statistics 248
11.6.2 Bi-Gaussian p.d.f 250
11.6.3 Q1D quadratic-form model 252
12 Evaluation of mean concentration and fluxes in turbulent flows by Lagrangian stochastic models 254
12.1 Introduction 254
12.2 Formulation of the problem 255
12.3 Monte Carlo estimators for the mean concentration and fluxes 259
12.3.1 Forward estimator 260
12.3.2 Modified forward estimators in case of horizontally homogeneous turbulence 261
12.3.3 Backward estimator 266
12.4 Application to the footprint problem 267
12.5 Conclusion 269
12.6 Appendices 269
12.6.1 Appendix A. Representation of concentration in Lagrangian description 269
12.6.2 Appendix B. Relation between forward and backward transition density functions 271
12.6.3 Appendix C. Derivation of the relation between the forward and backward densities 271
13 Stochastic Lagrangian footprint calculations over a surface with an abrupt change of roughness height 274
13.1 Introduction 274
13.2 The governing equations 275
13.2.1 Evaluation of footprint functions 276
13.3 Results 279
13.3.1 Footprint functions of concentration and flux 279
13.4 Discussion and conclusions 292
13.5 Appendices 293
13.5.1 Appendix A. Dimensionless mean-flow equations 293
13.5.2 Appendix B. Lagrangian stochastic trajectory model 294
14 Stochastic flow simulation in 3D porous media 296
14.1 Introduction 296
14.2 Formulation of the problem 299
14.3 Direct numerical simulation method: DSM-SOR 300
14.4 Randomized spectral model (RSM) 302
14.5 Testing the simulation procedure 304
14.6 Evaluation of Eulerian and Lagrangian statistical characteristics by the DNS-SOR method 308
14.6.1 Eulerian statistical characteristics 308
14.6.2 Lagrangian statistical characteristics 310
14.7 Conclusions and discussion 314
15 A Lagrangian stochastic model for the transport in statistically homogeneous porous media 316
15.1 Introduction 316
15.2 Direct simulation method 317
15.2.1 Random flow model 317
15.2.2 Numerical simulation 319
15.2.3 Evaluation of Eulerian characteristics 322
15.2.4 Evaluation of Lagrangian characteristics 326
15.3 Construction of the Langevin-type model 330
15.3.1 Introduction 330
15.3.2 Langevin model for an isotropic porous medium 332
15.3.3 Expressions of the drift terms 335
15.4 Numerical results and comparison against the DSM 337
15.5 Conclusions 337
16 Coagulation of aerosol particles in intermittent turbulent flows 342
16.1 Introduction 342
16.2 Analysis of the fluctuations in the size spectrum 345
16.3 Models of the energy dissipation rate 348
16.3.1 The model by Pope and Chen (P& Ch)
16.3.2 The model by Borgas and Sawford (B& S)
16.4 Monte Carlo simulation for the Smoluchowski equation in a stochastic coagulation regime 351
16.4.1 The total number of clusters and the mean cluster size 353
16.4.2 The functions N3(t) and N10(t) 355
16.4.3 The size spectrum N for different time instances
16.4.4 Comparative analysis for two different models of the energy dissipation rate 357
16.5 The case of a coagulation coefficient with no dependence on the cluster size 358
16.6 Simulation of coagulation processes in turbulent coagulation regime 359
16.7 Conclusion 361
16.8 Appendix. Derivation of the coagulation coefficient 362
17 Stokes flows under random boundary velocity excitations 365
17.1 Introduction 365
17.2 Exterior Stokes problem 368
17.2.1 Poisson formula in polar coordinates 369
17.3 K-L expansion of velocity 372
17.3.1 White noise excitations 372
17.3.2 General case of homogeneous excitations 377
17.4 Correlation function of the pressure 382
17.4.1 White noise excitations 382
17.4.2 Homogeneous random boundary excitations 384
17.4.3 Vorticity and stress tensor 384
17.5 Interior Stokes problem 388
17.6 Numerical results 390
Bibliography 397
Index 413