Random Fields and Stochastic Lagrangian Models

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