Random Fields and Stochastic Lagrangian Models: Analysis and Applications in Turbulence and Porous Media

✍ Scribed by Karl K. Sabelfeld; Nikolai A. Simonov

• Publisher: De Gruyter
• Year: 2012
• Tongue: English
• Leaves: 416
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Preface
1 Introduction
1.1 Why random fields?
1.2 Some examples
1.3 Fundamental concepts
1.3.1 Random functions in a broad sense
1.3.2 Gaussian random vectors
1.3.3 Gaussian random functions
1.3.4 Random fields
1.3.5 Stochastic measures and integrals
1.3.6 Integral representation of random functions
1.3.7 Random trajectories
1.3.8 Stochastic differential, Ito integrals
1.3.9 Brownian motion
1.3.10 Multidimensional diffusion and Fokker-Planck equation
1.3.11 Central limit theorem and convergence of a Poisson process to a Gaussian process
2 Stochastic simulation of vector Gaussian random fields
2.1 Introduction
2.2 Discrete expansions related to the spectral representations of Gaussian random fields
2.2.1 Spectral representations
2.2.2 Series expansions
2.2.3 Expansion with an even complex orthonormal system
2.2.4 Expansion with a real orthonormal system
2.2.5 Complex valued orthogonal expansions
2.3 Wavelet expansions
2.3.1 Fourier wavelet expansions
2.3.2 Wavelet expansion
2.3.3 Moving averages
2.4 Randomized spectral models
2.4.1 Randomized spectral models defined through stochastic integrals
2.4.2 Stratified RSM for homogeneous random fields
2.5 Fourier wavelet models
2.5.1 Meyer wavelet functions
2.5.2 Evaluation of the coefficients and ℱmՓ and ℱmΨ
2.5.3 Cut-off parameters
2.5.4 Choice of parameters
2.6 Fourier wavelet models of homogeneous random fields based on randomization of plane wave decomposition
2.6.1 Plane wave decomposition of homogeneous random fields
2.6.2 Decomposition with fixed nodes
2.6.3 Decomposition with randomly distributed nodes
2.6.4 Some examples
2.6.5 Flow in a porous media in the first order approximation
2.6.6 Fourier wavelet models of Gaussian random fields
2.7 Comparison of Fourier wavelet and randomized spectral models
2.7.1 Some technical details of RSM
2.7.2 Some technical details of FWM
2.7.3 Ensemble averaging
2.7.4 Space averaging
2.8 Conclusions
2.9 Appendices
2.9.1 Appendix A. Positive definiteness of the matrix B
2.9.2 Appendix B. Proof of Proposition 2.1
3 Stochastic Lagrangian models of turbulent flows: Relative dispersion of a pair of fluid particles
3.1 Introduction
3.2 Criticism of 2-particle models
3.3 The quasi-1-dimensional Lagrangian model of relative dispersion
3.3.1 Quasi-1-dimensional analog of formula (2.14a)
3.3.2 Models with a finite-order consistency
3.3.3 Explicit form of the model (3.26, 3.27)
3.3.4 Example
3.4 A 3-dimensional model of relative dispersion
3.5 Lagrangian models consistent with the Eulerian statistics
3.5.1 Diffusion approximation
3.5.2 Relation to the well-mixed condition
3.5.3 A choice of the coefficients ai and bij
3.6 Conclusions
4 A new Lagrangian model of 2-particle relative turbulent dispersion
4.1 Introduction
4.2 An examination of Durbin’s nonlinear model
4.3 Mathematical formulation of a new model
4.4 A qualitative analysis of the problem (4.14) for symmetric £(r)
4.4.1 Analysis of the problem (4.14) in the deterministic case
4.4.2 Analysis of the problem (4.14) for stochastic £(r)
4.5 Qualitative analysis of the problem (4.14) in the general case
5 The combined Eulerian-Lagrangian model
5.1 Introduction
5.2 2-particle models
5.2.1 Eulerian stochastic models of high-Reynolds-number pseudoturbulence
5.3 A new 2-particle Eulerian-Lagrangian stochastic model
5.3.1 Formulation of 2-particle Eulerian-Lagrangian model
5.3.2 Models for the p.d.f. of the Eulerian relative velocity
5.4 Appendix
6 Stochastic Lagrangian models for 2-particle relative dispersion in high-Reynolds-number turbulence
6.1 Introduction
6.2 Preliminaries
6.3 A closure of the quasi-1-dimensional model of relative dispersion
6.4 Choice of the model (6.1) for isotropic turbulence
6.5 The model of relative dispersion of two particles in a locally isotropic turbulence
6.5.1 Specification of the model
6.5.2 Numerical analysis of the Q1D-model (6.30)
6.6 Model of the relative dispersion in intermittent locally isotropic turbulence
6.7 Conclusions
7 Stochastic Lagrangian models for 2-particle motion in turbulent flows. Numerical results
7.1 Introduction
7.2 Classical pseudoturbulence model
7.2.1 Randomized model of classical pseudoturbulence
7.2.2 Mean square separation of two particles in classical pseudoturbulence
7.3 Calculations by the combined Eulerian-Lagrangian stochastic model
7.3.1 Mean square separation of two particles
7.3.2 Thomson’s “two-to-one” reduction principle
7.3.3 Concentration fluctuations
7.4 Technical remarks
7.5 Conclusion
8 The 1-particle stochastic Lagrangian model for turbulent dispersion in horizontally homogeneous turbulence
8.1 Introduction
8.2 Choice of the coefficients in the Ito equation
8.3 2D stochastic model with Gaussian p.d.f
8.4 Numerical experiments
9 Direct and adjoint Monte Carlo for the footprint problem
9.1 Introduction
9.2 Formulation of the problem
9.3 Stochastic Lagrangian algorithm
9.3.1 Direct Monte Carlo algorithm
9.3.2 Adjoint algorithm
9.4 Impenetrable boundary
9.5 Reacting species
9.6 Numerical simulations
9.7 Conclusion
9.8 Appendices
9.8.1 Appendix A. Flux representation
9.8.2 Appendix B. Probabilistic representation
9.8.3 Appendix C. Forward and backward trajectory estimators
10 Lagrangian stochastic models for turbulent dispersion in an atmospheric boundary layer
10.1 Introduction
10.2 Neutrally stratified boundary layer
10.2.1 General case of Eulerian p.d.f
10.2.2 Gaussian p.d.f
10.3 Comparison with other models and measurements
10.3.1 Comparison with measurements in an ideally-neutral surface layer (INSL)
10.3.2 Comparison with the wind tunnel experiment by Raupach and Legg (1983)
10.4 Convective case
10.5 Boundary conditions
10.6 Conclusion
10.7 Appendices
10.7.1 Appendix A. Derivation of the coefficients in the Gaussian case
10.7.2 Appendix B. Relation to other models
11 Analysis of the relative dispersion of two particles by Lagrangian stochastic models and DNS methods
11.1 Introduction
11.2 Basic assumptions
11.2.1 Markov assumption
11.2.2 Consistency with the second Kolmogorov similarity hypothesis
11.2.3 Thomson’s well-mixed condition
11.3 Well-mixed Lagrangian stochastic models
11.3.1 Quadratic-form models
11.3.2 Quasi-1-dimensional models
11.3.3 3-dimensional extension of Q1D models
11.4 Stochastic Lagrangian models based on the moments approximation method
11.4.1 Moments approximation conditions
11.4.2 Realizability of LS models based on the moments approximation method
11.5 Comparison of different models of relative dispersion for the inertial subrange of a fully developed turbulence
11.5.1 Q1D quadratic-form model of Borgas and Yeung
11.5.2 Comparison of different models in the inertial subrange
11.6 Comparison of different Q1D models of relative dispersion for modestly large Reynolds number turbulence (Re? ? 240)
11.6.1 Parametrization of Eulerian statistics
11.6.2 Bi-Gaussian p.d.f
11.6.3 Q1D quadratic-form model
12 Evaluation of mean concentration and fluxes in turbulent flows by Lagrangian stochastic models
12.1 Introduction
12.2 Formulation of the problem
12.3 Monte Carlo estimators for the mean concentration and fluxes
12.3.1 Forward estimator
12.3.2 Modified forward estimators in case of horizontally homogeneous turbulence
12.3.3 Backward estimator
12.4 Application to the footprint problem
12.5 Conclusion
12.6 Appendices
12.6.1 Appendix A. Representation of concentration in Lagrangian description
12.6.2 Appendix B. Relation between forward and backward transition density functions
12.6.3 Appendix C. Derivation of the relation between the forward and backward densities
13 Stochastic Lagrangian footprint calculations over a surface with an abrupt change of roughness height
13.1 Introduction
13.2 The governing equations
13.2.1 Evaluation of footprint functions
13.3 Results
13.3.1 Footprint functions of concentration and flux
13.4 Discussion and conclusions
13.5 Appendices
13.5.1 Appendix A. Dimensionless mean-flow equations
13.5.2 Appendix B. Lagrangian stochastic trajectory model
14 Stochastic flow simulation in 3D porous media
14.1 Introduction
14.2 Formulation of the problem
14.3 Direct numerical simulation method: DSM-SOR
14.4 Randomized spectral model (RSM)
14.5 Testing the simulation procedure
14.6 Evaluation of Eulerian and Lagrangian statistical characteristics by the DNS-SOR method
14.6.1 Eulerian statistical characteristics
14.6.2 Lagrangian statistical characteristics
14.7 Conclusions and discussion
15 A Lagrangian stochastic model for the transport in statistically homogeneous porous media
15.1 Introduction
15.2 Direct simulation method
15.2.1 Random flow model
15.2.2 Numerical simulation
15.2.3 Evaluation of Eulerian characteristics
15.2.4 Evaluation of Lagrangian characteristics
15.3 Construction of the Langevin-type model
15.3.1 Introduction
15.3.2 Langevin model for an isotropic porous medium
15.3.3 Expressions of the drift terms
15.4 Numerical results and comparison against the DSM
15.5 Conclusions
16 Coagulation of aerosol particles in intermittent turbulent flows
16.1 Introduction
16.2 Analysis of the fluctuations in the size spectrum
16.3 Models of the energy dissipation rate
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
16.4.1 The total number of clusters and the mean cluster size
16.4.2 The functions N3(t) and N10(t)
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
16.5 The case of a coagulation coefficient with no dependence on the cluster size
16.6 Simulation of coagulation processes in turbulent coagulation regime
16.7 Conclusion
16.8 Appendix. Derivation of the coagulation coefficient
17 Stokes flows under random boundary velocity excitations
17.1 Introduction
17.2 Exterior Stokes problem
17.2.1 Poisson formula in polar coordinates
17.3 K-L expansion of velocity
17.3.1 White noise excitations
17.3.2 General case of homogeneous excitations
17.4 Correlation function of the pressure
17.4.1 White noise excitations
17.4.2 Homogeneous random boundary excitations
17.4.3 Vorticity and stress tensor
17.5 Interior Stokes problem
17.6 Numerical results
Bibliography
Index