Sabelfeld / Simonov Random Fields and Stochastic Lagrangian Models

Analysis and Applications in Turbulence and Porous Media
1. Auflage 2012
ISBN: 978-3-11-029681-5
Verlag: De Gruyter
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 414 Seiten

The book presents advanced stochastic models and simulation methods for random flows and transport of particles by turbulent velocity fields and flows in porous media. Two main classes of models are constructed: (1) turbulent flows are modeled as synthetic random fields which have certain statistics and features mimicing those of turbulent fluid in the regime of interest, and (2) the models are constructed in the form of stochastic differential equations for stochastic Lagrangian trajectories of particles carried by turbulent flows.The book is written for mathematicians, physicists, and engineers studying processes associated with probabilistic interpretation, researchers in applied and computational mathematics, in environmental and engineering sciences dealing with turbulent transport and flows in porous media, as well as nucleation, coagulation, and chemical reaction analysis under fluctuation conditions. It can be of interest for students and post-graduates studying numerical methods for solving stochastic boundary value problems of mathematical physics and dispersion of particles by turbulent flows and flows in porous media.

Zielgruppe

Researchers, Advanced and Post-graduate Students in Mathematics, Specialists in Environment; Academic Libraries

Autoren/Hrsg.

Sabelfeld, Karl K.

Simonov, Nikolai A.

Fachgebiete

• Naturwissenschaften Physik Mechanik Kontinuumsmechanik, Strömungslehre
• Technische Wissenschaften Maschinenbau | Werkstoffkunde Technische Mechanik | Werkstoffkunde Strömungslehre
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle

Inhaltsverzeichnis

1;Preface;5
2;1 Introduction;17
2.1;1.1 Why random fields?;17
2.2;1.2 Some examples;19
2.3;1.3 Fundamental concepts;24
2.3.1;1.3.1 Random functions in a broad sense;25
2.3.2;1.3.2 Gaussian random vectors;29
2.3.3;1.3.3 Gaussian random functions;30
2.3.4;1.3.4 Random fields;32
2.3.5;1.3.5 Stochastic measures and integrals;33
2.3.6;1.3.6 Integral representation of random functions;35
2.3.7;1.3.7 Random trajectories;37
2.3.8;1.3.8 Stochastic differential, Ito integrals;38
2.3.9;1.3.9 Brownian motion;38
2.3.10;1.3.10 Multidimensional diffusion and Fokker-Planck equation;41
2.3.11;1.3.11 Central limit theorem and convergence of a Poisson process to a Gaussian process;42
3;2 Stochastic simulation of vector Gaussian random fields;45
3.1;2.1 Introduction;45
3.2;2.2 Discrete expansions related to the spectral representations of Gaussian random fields;46
3.2.1;2.2.1 Spectral representations;46
3.2.2;2.2.2 Series expansions;47
3.2.3;2.2.3 Expansion with an even complex orthonormal system;47
3.2.4;2.2.4 Expansion with a real orthonormal system;48
3.2.5;2.2.5 Complex valued orthogonal expansions;49
3.3;2.3 Wavelet expansions;49
3.3.1;2.3.1 Fourier wavelet expansions;50
3.3.2;2.3.2 Wavelet expansion;51
3.3.3;2.3.3 Moving averages;52
3.4;2.4 Randomized spectral models;53
3.4.1;2.4.1 Randomized spectral models defined through stochastic integrals;53
3.4.2;2.4.2 Stratified RSM for homogeneous random fields;55
3.5;2.5 Fourier wavelet models;55
3.5.1;2.5.1 Meyer wavelet functions;56
3.5.2;2.5.2 Evaluation of the coefficients and Fm. and Fm.;56
3.5.3;2.5.3 Cut-off parameters;58
3.5.4;2.5.4 Choice of parameters;59
3.6;2.6 Fourier wavelet models of homogeneous random fields based on randomization of plane wave decomposition;63
3.6.1;2.6.1 Plane wave decomposition of homogeneous random fields;63
3.6.2;2.6.2 Decomposition with fixed nodes;66
3.6.3;2.6.3 Decomposition with randomly distributed nodes;68
3.6.4;2.6.4 Some examples;70
3.6.5;2.6.5 Flow in a porous media in the first order approximation;72
3.6.6;2.6.6 Fourier wavelet models of Gaussian random fields;73
3.7;2.7 Comparison of Fourier wavelet and randomized spectral models;74
3.7.1;2.7.1 Some technical details of RSM;74
3.7.2;2.7.2 Some technical details of FWM;76
3.7.3;2.7.3 Ensemble averaging;78
3.7.4;2.7.4 Space averaging;78
3.8;2.8 Conclusions;79
3.9;2.9 Appendices;81
3.9.1;2.9.1 Appendix A. Positive definiteness of the matrix B;81
3.9.2;2.9.2 Appendix B. Proof of Proposition 2.1;81
4;3 Stochastic Lagrangian models of turbulent flows: Relative dispersion of a pair of fluid particles;86
4.1;3.1 Introduction;86
4.2;3.2 Criticism of 2-particle models;89
4.3;3.3 The quasi-1-dimensional Lagrangian model of relative dispersion;93
4.3.1;3.3.1 Quasi-1-dimensional analog of formula (2.14a);94
4.3.2;3.3.2 Models with a finite-order consistency;96
4.3.3;3.3.3 Explicit form of the model (3.26, 3.27);99
4.3.4;3.3.4 Example;104
4.4;3.4 A 3-dimensional model of relative dispersion;106
4.5;3.5 Lagrangian models consistent with the Eulerian statistics;108
4.5.1;3.5.1 Diffusion approximation;108
4.5.2;3.5.2 Relation to the well-mixed condition;110
4.5.3;3.5.3 A choice of the coefficients ai and bij;111
4.6;3.6 Conclusions;113
5;4 A new Lagrangian model of 2-particle relative turbulent dispersion;114
5.1;4.1 Introduction;114
5.2;4.2 An examination of Durbin’s nonlinear model;114
5.3;4.3 Mathematical formulation of a new model;116
5.4;4.4 A qualitative analysis of the problem (4.14) for symmetric £(r);118
5.4.1;4.4.1 Analysis of the problem (4.14) in the deterministic case;118
5.4.2;4.4.2 Analysis of the problem (4.14) for stochastic £(r);119
5.5;4.5 Qualitative analysis of the problem (4.14) in the general case;124
6;5 The combined Eulerian-Lagrangian model;129
6.1;5.1 Introduction;129
6.2;5.2 2-particle models;133
6.2.1;5.2.1 Eulerian stochastic models of high-Reynolds-number pseudoturbulence;133
6.3;5.3 A new 2-particle Eulerian-Lagrangian stochastic model;136
6.3.1;5.3.1 Formulation of 2-particle Eulerian-Lagrangian model;136
6.3.2;5.3.2 Models for the p.d.f. of the Eulerian relative velocity;139
6.4;5.4 Appendix;141
7;6 Stochastic Lagrangian models for 2-particle relative dispersion in high-Reynolds-number turbulence;145
7.1;6.1 Introduction;145
7.2;6.2 Preliminaries;146
7.3;6.3 A closure of the quasi-1-dimensional model of relative dispersion;147
7.4;6.4 Choice of the model (6.1) for isotropic turbulence;148
7.5;6.5 The model of relative dispersion of two particles in a locally isotropic turbulence;151
7.5.1;6.5.1 Specification of the model;151
7.5.2;6.5.2 Numerical analysis of the Q1D-model (6.30);153
7.6;6.6 Model of the relative dispersion in intermittent locally isotropic turbulence;155
7.7;6.7 Conclusions;157
8;7 Stochastic Lagrangian models for 2-particle motion in turbulent flows. Numerical results;158
8.1;7.1 Introduction;158
8.2;7.2 Classical pseudoturbulence model;159
8.2.1;7.2.1 Randomized model of classical pseudoturbulence;159
8.2.2;7.2.2 Mean square separation of two particles in classical pseudoturbulence;162
8.3;7.3 Calculations by the combined Eulerian-Lagrangian stochastic model;165
8.3.1;7.3.1 Mean square separation of two particles;165
8.3.2;7.3.2 Thomson’s “two-to-one” reduction principle;168
8.3.3;7.3.3 Concentration fluctuations;170
8.4;7.4 Technical remarks;172
8.5;7.5 Conclusion;174
9;8 The 1-particle stochastic Lagrangian model for turbulent dispersion in horizontally homogeneous turbulence;175
9.1;8.1 Introduction;175
9.2;8.2 Choice of the coefficients in the Ito equation;178
9.3;8.3 2D stochastic model with Gaussian p.d.f;180
9.4;8.4 Numerical experiments;183
10;9 Direct and adjoint Monte Carlo for the footprint problem;187
10.1;9.1 Introduction;187
10.2;9.2 Formulation of the problem;188
10.3;9.3 Stochastic Lagrangian algorithm;189
10.3.1;9.3.1 Direct Monte Carlo algorithm;190
10.3.2;9.3.2 Adjoint algorithm;192
10.4;9.4 Impenetrable boundary;194
10.5;9.5 Reacting species;196
10.6;9.6 Numerical simulations;199
10.7;9.7 Conclusion;203
10.8;9.8 Appendices;204
10.8.1;9.8.1 Appendix A. Flux representation;204
10.8.2;9.8.2 Appendix B. Probabilistic representation;204
10.8.3;9.8.3 Appendix C. Forward and backward trajectory estimators;205
11;10 Lagrangian stochastic models for turbulent dispersion in an atmospheric boundary layer;209
11.1;10.1 Introduction;209
11.2;10.2 Neutrally stratified boundary layer;213
11.2.1;10.2.1 General case of Eulerian p.d.f;213
11.2.2;10.2.2 Gaussian p.d.f;216
11.3;10.3 Comparison with other models and measurements;217
11.3.1;10.3.1 Comparison with measurements in an ideally-neutral surface layer (INSL);217
11.3.2;10.3.2 Comparison with the wind tunnel experiment by Raupach and Legg (1983);220
11.4;10.4 Convective case;223
11.5;10.5 Boundary conditions;227
11.6;10.6 Conclusion;228
11.7;10.7 Appendices;229
11.7.1;10.7.1 Appendix A. Derivation of the coefficients in the Gaussian case;229
11.7.2;10.7.2 Appendix B. Relation to other models;231
12;11 Analysis of the relative dispersion of two particles by Lagrangian stochastic models and DNS methods;234
12.1;11.1 Introduction;234
12.2;11.2 Basic assumptions;236
12.2.1;11.2.1 Markov assumption;237
12.2.2;11.2.2 Consistency with the second Kolmogorov similarity hypothesis;237
12.2.3;11.2.3 Thomson’s well-mixed condition;238
12.3;11.3 Well-mixed Lagrangian stochastic models;238
12.3.1;11.3.1 Quadratic-form models;239
12.3.2;11.3.2 Quasi-1-dimensional models;240
13;11.3.3 3-dimensional extension of Q1D models;241
13.1;11.4 Stochastic Lagrangian models based on the moments approximation method;242
13.1.1;11.4.1 Moments approximation conditions;242
13.1.2;11.4.2 Realizability of LS models based on the moments approximation method;243
13.2;11.5 Comparison of different models of relative dispersion for the inertial subrange of a fully developed turbulence;245
13.2.1;11.5.1 Q1D quadratic-form model of Borgas and Yeung;245
13.2.2;11.5.2 Comparison of different models in the inertial subrange;247
13.3;11.6 Comparison of different Q1D models of relative dispersion for modestly large Reynolds number turbulence (Re? ? 240);248
13.3.1;11.6.1 Parametrization of Eulerian statistics;248
13.3.2;11.6.2 Bi-Gaussian p.d.f;250
13.3.3;11.6.3 Q1D quadratic-form model;252
14;12 Evaluation of mean concentration and fluxes in turbulent flows by Lagrangian stochastic models;254
14.1;12.1 Introduction;254
14.2;12.2 Formulation of the problem;255
14.3;12.3 Monte Carlo estimators for the mean concentration and fluxes;259
14.3.1;12.3.1 Forward estimator;260
14.3.2;12.3.2 Modified forward estimators in case of horizontally homogeneous turbulence;261
14.3.3;12.3.3 Backward estimator;266
14.4;12.4 Application to the footprint problem;267
14.5;12.5 Conclusion;269
14.6;12.6 Appendices;269
14.6.1;12.6.1 Appendix A. Representation of concentration in Lagrangian description;269
14.6.2;12.6.2 Appendix B. Relation between forward and backward transition density functions;271
14.6.3;12.6.3 Appendix C. Derivation of the relation between the forward and backward densities;271
15;13 Stochastic Lagrangian footprint calculations over a surface with an abrupt change of roughness height;274
15.1;13.1 Introduction;274
15.2;13.2 The governing equations;275
15.2.1;13.2.1 Evaluation of footprint functions;276
15.3;13.3 Results;279
15.3.1;13.3.1 Footprint functions of concentration and flux;279
15.4;13.4 Discussion and conclusions;292
15.5;13.5 Appendices;293
15.5.1;13.5.1 Appendix A. Dimensionless mean-flow equations;293
15.5.2;13.5.2 Appendix B. Lagrangian stochastic trajectory model;294
16;14 Stochastic flow simulation in 3D porous media;296
16.1;14.1 Introduction;296
16.2;14.2 Formulation of the problem;299
16.3;14.3 Direct numerical simulation method: DSM-SOR;300
16.4;14.4 Randomized spectral model (RSM);302
16.5;14.5 Testing the simulation procedure;304
16.6;14.6 Evaluation of Eulerian and Lagrangian statistical characteristics by the DNS-SOR method;308
16.6.1;14.6.1 Eulerian statistical characteristics;308
16.6.2;14.6.2 Lagrangian statistical characteristics;310
16.7;14.7 Conclusions and discussion;314
17;15 A Lagrangian stochastic model for the transport in statistically homogeneous porous media;316
17.1;15.1 Introduction;316
17.2;15.2 Direct simulation method;317
17.2.1;15.2.1 Random flow model;317
17.2.2;15.2.2 Numerical simulation;319
17.2.3;15.2.3 Evaluation of Eulerian characteristics;322
17.2.4;15.2.4 Evaluation of Lagrangian characteristics;326
17.3;15.3 Construction of the Langevin-type model;330
17.3.1;15.3.1 Introduction;330
17.3.2;15.3.2 Langevin model for an isotropic porous medium;332
17.3.3;15.3.3 Expressions of the drift terms;335
17.4;15.4 Numerical results and comparison against the DSM;337
17.5;15.5 Conclusions;337
18;16 Coagulation of aerosol particles in intermittent turbulent flows;342
18.1;16.1 Introduction;342
18.2;16.2 Analysis of the fluctuations in the size spectrum;345
18.3;16.3 Models of the energy dissipation rate;348
18.3.1;16.3.1 The model by Pope and Chen (P&Ch);348
18.3.2;16.3.2 The model by Borgas and Sawford (B&S);350
18.4;16.4 Monte Carlo simulation for the Smoluchowski equation in a stochastic coagulation regime;351
18.4.1;16.4.1 The total number of clusters and the mean cluster size;353
18.4.2;16.4.2 The functions N3(t) and N10(t);355
18.4.3;16.4.3 The size spectrum N; for different time instances;356
18.4.4;16.4.4 Comparative analysis for two different models of the energy dissipation rate;357
18.5;16.5 The case of a coagulation coefficient with no dependence on the cluster size;358
18.6;16.6 Simulation of coagulation processes in turbulent coagulation regime;359
18.7;16.7 Conclusion;361
18.8;16.8 Appendix. Derivation of the coagulation coefficient;362
19;17 Stokes flows under random boundary velocity excitations;365
19.1;17.1 Introduction;365
19.2;17.2 Exterior Stokes problem;368
19.2.1;17.2.1 Poisson formula in polar coordinates;369
19.3;17.3 K-L expansion of velocity;372
19.3.1;17.3.1 White noise excitations;372
19.3.2;17.3.2 General case of homogeneous excitations;377
19.4;17.4 Correlation function of the pressure;382
19.4.1;17.4.1 White noise excitations;382
19.4.2;17.4.2 Homogeneous random boundary excitations;384
19.4.3;17.4.3 Vorticity and stress tensor;384
19.5;17.5 Interior Stokes problem;388
19.6;17.6 Numerical results;390
20;Bibliography;397
21;Index;413

Über Autor(innen)

Karl K. Sabelfeld, Institute of Computational Mathematics and Geophysics, Russian Acacemy of Sciences, Novosibirsk, Russia.