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E-raamat: Random Fields and Stochastic Lagrangian Models: Analysis and Applications in Turbulence and Porous Media [De Gruyter e-raamatud]

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  • Formaat: 414 pages, 87 Illustrations; 9 Tables, black and white
  • Ilmumisaeg: 28-Nov-2012
  • Kirjastus: De Gruyter
  • ISBN-13: 9783110296815
Teised raamatud teemal:
  • De Gruyter e-raamatud
  • Hind: 155,94 €*
  • * hind, mis tagab piiramatu üheaegsete kasutajate arvuga ligipääsu piiramatuks ajaks
Random Fields and Stochastic Lagrangian Models: Analysis and Applications in Turbulence and Porous MediaSuurem pilt
  • Formaat: 414 pages, 87 Illustrations; 9 Tables, black and white
  • Ilmumisaeg: 28-Nov-2012
  • Kirjastus: De Gruyter
  • ISBN-13: 9783110296815
Teised raamatud teemal:
Wiley OnlineRohkem infot De Gruyter e-raamatute kohta
Raamatu kodulehekülg: http://dx.doi.org/10.1515/9783110296815
Probabilistic approach and stochastic simulation become more and more popular in all branches of science and technology, especially in problems where the data are randomly fluctuating, or they are highly irregular in deterministic sense. As a rule, in such problems it is very difficult and expensive to carry out measurements to extract the desired data. As important examples the book mentions the turbulent flow simulation in atmosphere, and construction of flows through porous media. The temporal and spatial scales of the input parameters in this class of problems are varying enormously, and the behaviour is very complicated, so that there is no chance to describe it deterministically.

Karl K. Sabelfeld, Institute of Computational Mathematics and Geophysics, Russian Acacemy of Sciences, Novosibirsk, Russia.
Preface v
1 Introduction
1(28)
1.1 Why random fields?
1(2)
1.2 Some examples
3(5)
1.3 Fundamental concepts
8(21)
1.3.1 Random functions in a broad sense
9(4)
1.3.2 Gaussian random vectors
13(1)
1.3.3 Gaussian random functions
14(2)
1.3.4 Random fields
16(1)
1.3.5 Stochastic measures and integrals
17(2)
1.3.6 Integral representation of random functions
19(2)
1.3.7 Random trajectories
21(1)
1.3.8 Stochastic differential, Ito integrals
22(1)
1.3.9 Brownian motion
22(3)
1.3.10 Multidimensional diffusion and Fokker-Planck equation
25(1)
1.3.11 Central limit theorem and convergence of a Poisson process to a Gaussian process
26(3)
2 Stochastic simulation of vector Gaussian random fields
29(41)
2.1 Introduction
29(1)
2.2 Discrete expansions related to the spectral representations of Gaussian random fields
30(3)
2.2.1 Spectral representations
30(1)
2.2.2 Series expansions
31(1)
2.2.3 Expansion with an even complex orthonormal system
31(1)
2.2.4 Expansion with a real orthonormal system
32(1)
2.2.5 Complex valued orthogonal expansions
33(1)
2.3 Wavelet expansions
33(4)
2.3.1 Fourier wavelet expansions
34(1)
2.3.2 Wavelet expansion
35(1)
2.3.3 Moving averages
36(1)
2.4 Randomized spectral models
37(2)
2.4.1 Randomized spectral models defined through stochastic integrals
37(2)
2.4.2 Stratified RSM for homogeneous random fields
39(1)
2.5 Fourier wavelet models
39(8)
2.5.1 Meyer wavelet functions
40(1)
2.5.2 Evaluation of the coefficients Fm(φ) and Fm
40(2)
2.5.3 Cut-off parameters
42(1)
2.5.4 Choice of parameters
43(4)
2.6 Fourier wavelet models of homogeneous random fields based on randomization of plane wave decomposition
47(11)
2.6.1 Plane wave decomposition of homogeneous random fields
47(3)
2.6.2 Decomposition with fixed nodes
50(2)
2.6.3 Decomposition with randomly distributed nodes
52(2)
2.6.4 Some examples
54(2)
2.6.5 Flow in a porous media in the first order approximation
56(1)
2.6.6 Fourier wavelet models of Gaussian random fields
57(1)
2.7 Comparison of Fourier wavelet and randomized spectral models
58(5)
2.7.1 Some technical details of RSM
58(2)
2.7.2 Some technical details of FWM
60(2)
2.7.3 Ensemble averaging
62(1)
2.7.4 Space averaging
62(1)
2.8 Conclusions
63(2)
2.9 Appendices
65(5)
2.9.1 Appendix A: Positive definiteness of the matrix B
65(1)
2.9.2 Appendix B: Proof of Proposition 2.1
65(5)
3 Stochastic Lagrangian models of turbulent flows: Relative dispersion of a pair of fluid particles
70(28)
3.1 Introduction
70(3)
3.2 Criticism of 2-particle models
73(4)
3.3 The quasi-1-dimensional Lagrangian model of relative dispersion
77(13)
3.3.1 Quasi-1-dimensional analog of formula (2.14a)
78(2)
3.3.2 Models with a finite-order consistency
80(3)
3.3.3 Explicit form of the model (3.26, 3.27)
83(5)
3.3.4 Example
88(2)
3.4 A 3-dimensional model of relative dispersion
90(2)
3.5 Lagrangian models consistent with the Eulerian statistics
92(5)
3.5.1 Diffusion approximation
92(2)
3.5.2 Relation to the well-mixed condition
94(1)
3.5.3 A choice of the coefficients ai and bij
95(2)
3.6 Conclusions
97(1)
4 A new Lagrangian model of 2-particle relative turbulent dispersion
98(15)
4.1 Introduction
98(1)
4.2 An examination of Durbin's nonlinear model
98(2)
4.3 Mathematical formulation of a new model
100(2)
4.4 A qualitative analysis of the problem (4.14) for symmetric ξ(τ)
102(6)
4.4.1 Analysis of the problem (4.14) in the deterministic case
102(1)
4.4.2 Analysis of the problem (4.14) for stochastic ξ (τ)
103(5)
4.5 Qualitative analysis of the problem (4.14) in the general case
108(5)
5 The combined Eulerian-Lagrangian model
113(16)
5.1 Introduction
113(4)
5.2 2-particle models
117(3)
5.2.1 Eulerian stochastic models of high-Reynolds-number pseudoturbulence
117(3)
5.3 A new 2-particle Eulerian-Lagrangian stochastic model
120(5)
5.3.1 Formulation of 2-particle Eulerian-Lagrangian model
120(3)
5.3.2 Models for the p. d. f. of the Eulerian relative velocity
123(2)
5.4 Appendix
125(4)
6 Stochastic Lagrangian models for 2-particle relative dispersion in high-Reynolds-number turbulence
129(13)
6.1 Introduction
129(1)
6.2 Preliminaries
130(1)
6.3 A closure of the quasi-1-dimensional model of relative dispersion
131(1)
6.4 Choice of the model (6.1) for isotropic turbulence
132(3)
6.5 The model of relative dispersion of two particles in a locally isotropic turbulence
135(4)
6.5.1 Specification of the model
135(2)
6.5.2 Numerical analysis of the Q1D-model (6.30)
137(2)
6.6 Model of the relative dispersion in intermittent locally isotropic turbulence
139(2)
6.7 Conclusions
141(1)
7 Stochastic Lagrangian models for 2-particle motion in turbulent flows. Numerical results
142(17)
7.1 Introduction
142(1)
7.2 Classical pseudoturbulence model
143(6)
7.2.1 Randomized model of classical pseudoturbulence
143(3)
7.2.2 Mean square separation of two particles in classical pseudoturbulence
146(3)
7.3 Calculations by the combined Eulerian-Lagrangian stochastic model
149(7)
7.3.1 Mean square separation of two particles
149(3)
7.3.2 Thomson's "two-to-one" reduction principle
152(2)
7.3.3 Concentration fluctuations
154(2)
7.4 Technical remarks
156(2)
7.5 Conclusion
158(1)
8 The 1-particle stochastic Lagrangian model for turbulent dispersion in horizontally homogeneous turbulence
159(12)
8.1 Introduction
159(3)
8.2 Choice of the coefficients in the Ito equation
162(2)
8.3 2D stochastic model with Gaussian p. d. f.
164(3)
8.4 Numerical experiments
167(4)
9 Direct and adjoint Monte Carlo for the footprint problem
171(22)
9.1 Introduction
171(1)
9.2 Formulation of the problem
172(1)
9.3 Stochastic Lagrangian algorithm
173(5)
9.3.1 Direct Monte Carlo algorithm
174(2)
9.3.2 Adjoint algorithm
176(2)
9.4 Impenetrable boundary
178(2)
9.5 Reacting species
180(3)
9.6 Numerical simulations
183(4)
9.7 Conclusion
187(1)
9.8 Appendices
188(5)
9.8.1 Appendix A: Flux representation
188(1)
9.8.2 Appendix B: Probabilistic representation
188(1)
9.8.3 Appendix C: Forward and backward trajectory estimators
189(4)
10 Lagrangian stochastic models for turbulent dispersion in an atmospheric boundary layer
193(25)
10.1 Introduction
193(4)
10.2 Neutrally stratified boundary layer
197(4)
10.2.1 General case of Eulerian p. d. f.
197(3)
10.2.2 Gaussian p. d. f.
200(1)
10.3 Comparison with other models and measurements
201(6)
10.3.1 Comparison with measurements in an ideally-neutral surface layer (INSL)
201(3)
10.3.2 Comparison with the wind tunnel experiment by Raupach and Legg (1983)
204(3)
10.4 Convective case
207(4)
10.5 Boundary conditions
211(1)
10.6 Conclusion
212(1)
10.7 Appendices
213(5)
10.7.1 Appendix A: Derivation of the coefficients in the Gaussian case
213(2)
10.7.2 Appendix B: Relation to other models
215(3)
11 Analysis of the relative dispersion of two particles by Lagrangian stochastic models and DNS methods
218(20)
11.1 Introduction
218(2)
11.2 Basic assumptions
220(2)
11.2.1 Markov assumption
221(1)
11.2.2 Consistency with the second Kolmogorov similarity hypothesis
221(1)
11.2.3 Thomson's well-mixed condition
222(1)
11.3 Well-mixed Lagrangian stochastic models
222(4)
11.3.1 Quadratic-form models
223(1)
11.3.2 Quasi-1-dimensional models
224(1)
11.3.3 3-dimensional extension of Q1D models
225(1)
11.4 Stochastic Lagrangian models based on the moments approximation method
226(3)
11.4.1 Moments approximation conditions
226(1)
11.4.2 Realiazability of LS models based on the moments approximation method
227(2)
11.5 Comparison of different models of relative dispersion for the inertial subrange of a fully developed turbulence
229(3)
11.5.1 Q1D quadratic-form model of Borgas and Yeung
229(2)
11.5.2 Comparison of different models in the inertial subrange
231(1)
11.6 Comparison of different Q1D models of relative dispersion for modestly large Reynolds number turbulence (Reλ 240)
232(6)
11.6.1 Parametrization of Eulerian statistics
232(2)
11.6.2 Bi-Gaussian p.d.f.
234(2)
11.6.3 Q1D quadratic-form model
236(2)
12 Evaluation of mean concentration and fluxes in turbulent flows by Lagrangian stochastic models
238(20)
12.1 Introduction
238(1)
12.2 Formulation of the problem
239(4)
12.3 Monte Carlo estimators for the mean concentration and fluxes
243(8)
12.3.1 Forward estimator
244(1)
12.3.2 Modified forward estimators in case of horizontally homogeneous turbulence
245(5)
12.3.3 Backward estimator
250(1)
12.4 Application to the footprint problem
251(2)
12.5 Conclusion
253(1)
12.6 Appendices
253(5)
12.6.1 Appendix A: Representation of concentration in Lagrangian description
253(2)
12.6.2 Appendix B: Relation between forward and backward transition density functions
255(1)
12.6.3 Appendix C: Derivation of the relation between the forward and backward densities
255(3)
13 Stochastic Lagrangian footprint calculations over a surface with an abrupt change of roughness height
258(22)
13.1 Introduction
258(1)
13.2 The governing equations
259(4)
13.2.1 Evaluation of footprint functions
260(3)
13.3 Results
263(13)
13.3.1 Footprint functions of concentration and flux
263(13)
13.4 Discussion and conclusions
276(1)
13.5 Appendices
277(3)
13.5.1 Appendix A: Dimensionless mean-flow equations
277(1)
13.5.2 Appendix B: Lagrangian stochastic trajectory model
278(2)
14 Stochastic flow simulation in 3D porous media
280(20)
14.1 Introduction
280(3)
14.2 Formulation of the problem
283(1)
14.3 Direct numerical simulation method: DSM-SOR
284(2)
14.4 Randomized spectral model (RSM)
286(2)
14.5 Testing the simulation procedure
288(4)
14.6 Evaluation of Eulerian and Lagrangian statistical characteristics by the DNS-SOR method
292(6)
14.6.1 Eulerian statistical characteristics
292(2)
14.6.2 Lagrangian statistical characteristics
294(4)
14.7 Conclusions and discussion
298(2)
15 A Lagrangian stochastic model for the transport in statistically homogeneous porous media
300(26)
15.1 Introduction
300(1)
15.2 Direct simulation method
301(13)
15.2.1 Random flow model
301(2)
15.2.2 Numerical simulation
303(3)
15.2.3 Evaluation of Eulerian characteristics
306(4)
15.2.4 Evaluation of Lagrangian characteristics
310(4)
15.3 Construction of the Langevin-type model
314(7)
15.3.1 Introduction
314(2)
15.3.2 Langevin model for an isotropic porous medium
316(3)
15.3.3 Expressions of the drift terms
319(2)
15.4 Numerical results and comparison against the DSM
321(1)
15.5 Conclusions
321(5)
16 Coagulation of aerosol particles in intermittent turbulent flows
326(23)
16.1 Introduction
326(3)
16.2 Analysis of the fluctuations in the size spectrum
329(3)
16.3 Models of the energy dissipation rate
332(3)
16.3.1 The model by Pope and Chen (P&Ch)
332(2)
16.3.2 The model by Borgas and Sawford (B&S)
334(1)
16.4 Monte Carlo simulation for the Smoluchowski equation in a stochastic coagulation regime
335(7)
16.4.1 The total number of clusters and the mean cluster size
337(2)
16.4.2 The functions N3(t) and N10(t)
339(1)
16.4.3 The size spectrum Nl for different time instances
340(1)
16.4.4 Comparative analysis for two different models of the energy dissipation rate
341(1)
16.5 The case of a coagulation coefficient with no dependence on the cluster size
342(1)
16.6 Simulation of coagulation processes in turbulent coagulation regime
343(2)
16.7 Conclusion
345(1)
16.8 Appendix. Derivation of the coagulation coefficient
346(3)
17 Stokes flows under random boundary velocity excitations
349(32)
17.1 Introduction
349(3)
17.2 Exterior Stokes problem
352(4)
17.2.1 Poisson formula in polar coordinates
353(3)
17.3 K-L expansion of velocity
356(10)
17.3.1 White noise excitations
356(5)
17.3.2 General case of homogeneous excitations
361(5)
17.4 Correlation function of the pressure
366(6)
17.4.1 White noise excitations
366(2)
17.4.2 Homogeneous random boundary excitations
368(1)
17.4.3 Vorticity and stress tensor
368(4)
17.5 Interior Stokes problem
372(2)
17.6 Numerical results
374(7)
Bibliography 381(16)
Index 397
Karl K. Sabelfeld, Institute of Computational Mathematics and Geophysics, Russian Acacemy of Sciences, Novosibirsk, Russia.
Püsilink: https://www.kriso.ee/db/9783110296815_pe.html