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Dieter A. Wolf-Gladrow

Alfred Wegener Institute for Polar and Marine

Research,

Postfach 12 01 61

D-27515 Bremerhaven

Germany

e-mail: dwolf@awi-bremerhaven.de

Version 1.05

Lattice-Gas Cellular

Automata and

Lattice Boltzmann Models

- An Introduction

June 26, 2005

Springer

Berlin Heidelberg NewYork

Hong Kong London

Milan Paris Tokyo

VI Contents

3 Lattice-gas cellular automata .............................. 37

3.1 The HPP lattice-gas cellular automata . . . . . . . . . . . . . . . . . . . . . 37

3.1.1 Model description ................................. 37

3.1.2 Implementation of the HPP model: How to code

lattice-gas cellular automata? . . . . . . . . . . . . . . . . . . . . . . 42

3.1.3 Initialization...................................... 46

3.1.4 Coarse graining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2 The FHP lattice-gas cellular automata . . . . . . . . . . . . . . . . . . . . . 51

3.2.1 The lattice and the collision rules . . . . . . . . . . . . . . . . . . . . 51

3.2.2 MicrodynamicsoftheFHP model .................. 57

3.2.3 The Liouville equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.2.4 Massandmomentum density ....................... 63

3.2.5 Equilibrium mean occupation numbers . . . . . . . . . . . . . . 64

3.2.6 Derivation of the macroscopic equations: multi-scale

analysis .......................................... 67

3.2.7 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.2.8 Inclusionof bodyforces ........................... 78

3.2.9 NumericalexperimentswithFHP ................... 79

3.2.10 The 8-bitFHP model ............................. 85

3.3 Lattice tensors and isotropy in the macroscopic limit . . . . . . . . 88

3.3.1 Isotropictensors .................................. 88

3.3.2 Lattice tensors:single-speedmodels.................. 89

3.3.3 Generalized lattice tensors for multi-speed models . . . . . 93

3.3.4 Thermal LBMs: D2Q13-FHP (multi-speed FHP model) 101

3.3.5 Exercises.........................................103

3.4 Desperately seeking a lattice for simulations in three

dimensions .............................................104

3.4.1 Threedimensions..................................104

3.4.2 Fiveand higherdimensions.........................106

3.4.3 Fourdimensions...................................108

3.5 FCHC .................................................109

3.5.1 Isometric collision rules for FCHC by H´enon..........110

3.5.2 FCHC, computers and modiﬁed collision rules . . . . . . . . 111

3.5.3 IsometricrulesforHPP and FHP ...................112

Contents VII

3.5.4 Whatelse? .......................................113

3.6 The pair interaction (PI) lattice-gas cellular automata . . . . . . . 115

3.6.1 Lattice,cells, andinteractionin2D..................115

3.6.2 Macroscopicequations .............................118

3.6.3 Comparison of PI with FHP and FCHC . . . . . . . . . . . . . . 121

3.6.4 The collision operator and propagation in C and

FORTRAN.......................................121

3.7 Multi-speed and thermal lattice-gas cellular automata . . . . . . . . 125

3.7.1 The D3Q19model.................................125

3.7.2 The D2Q9model..................................128

3.7.3 The D2Q21model.................................131

3.7.4 Transsonic and supersonic ﬂows: D2Q25, D2Q57,

D2Q129..........................................131

3.8 Zanetti (‘staggered’) invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

3.8.1 FHP ............................................132

3.8.2 Signiﬁcance of the Zanetti invariants . . . . . . . . . . . . . . . . . 132

3.9 Lattice-gas cellular automata: What else? . . . . . . . . . . . . . . . . . . 134

4 Some statistical mechanics .................................137

4.1 TheBoltzmannequation .................................137

4.1.1 Five collision invariants and Maxwell’s distribution . . . . 138

4.1.2 Boltzmann’s H-theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

4.1.3 The BGKapproximation ..........................141

4.2 Chapman-Enskog: From Boltzmann to Navier-Stokes . . . . . . . . 143

4.2.1 The conservationlaws..............................144

4.2.2 The Eulerequation................................145

4.2.3 Chapman-Enskogexpansion ........................145

4.3 The maximum entropy principle . . . . . . . . . . . . . . . . . . . . . . . . . . 151

5 Lattice Boltzmann Models ................................157

5.1 From lattice-gas cellular automata to lattice Boltzmann

models ................................................157

5.1.1 Lattice Boltzmann equation and Boltzmann equation . . 158

5.1.2 Lattice Boltzmann models of the ﬁrst generation . . . . . . 161

5.2 BGKlattice Boltzmannmodelin2D ......................163

VIII Contents

5.2.1 Derivation of the W

..............................168

5.2.2 Entropy and equilibrium distributions . . . . . . . . . . . . . . . 169

5.2.3 Derivation of the Navier-Stokes equations by

multi-scaleanalysis ...............................172

5.2.4 Storage demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

5.2.5 Simulation of two-dimensional decaying turbulence . . . . 181

5.2.6 Boundary conditions for LBM . . . . . . . . . . . . . . . . . . . . . . . 187

5.3 Hydrodynamiclattice Boltzmann models in 3D .............193

5.3.1 3D-LBMwith19velocities .........................193

5.3.2 3D-LBM with 15 velocities and Koelman distribution . . 194

5.3.3 3D-LBM with 15 velocities proposed by Chen et al.

(D3Q15) .........................................195

5.4 Equilibrium distributions: the ansatz method . . . . . . . . . . . . . . . 196

5.4.1 Multi-scale analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

5.4.2 Negative distribution functions at high speed of sound . 201

5.5 HydrodynamicLBMwithenergyequation .................203

5.6 Stability of lattice Boltzmann models . . . . . . . . . . . . . . . . . . . . . . 206

5.6.1 Nonlinear stability analysis of uniform ﬂows . . . . . . . . . . 206

5.6.2 The method of linear stability analysis (von Neumann) 208

5.6.3 Linear stability analysis of BGK lattice Boltzmann

models...........................................209

5.6.4 Summary.........................................213

5.7 Simulatingoceancirculationwith LBM ....................217

5.7.1 Introduction......................................217

5.7.2 The model of Munk (1950) . . . . . . . . . . . . . . . . . . . . . . . . . 217

5.7.3 The latticeBoltzmannmodel .......................220

5.8 Alattice Boltzmannequationfordiﬀusion .................230

5.8.1 Finite diﬀerences approximation . . . . . . . . . . . . . . . . . . . . . 230

5.8.2 The lattice Boltzmann model for diﬀusion . . . . . . . . . . . . 231

5.8.3 Multi-scale expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

5.8.4 The special case ω =1 .............................234

5.8.5 The generalcase ..................................234

5.8.6 Numericalexperiments.............................234

5.8.7 Summaryandconclusion...........................235

剩余310页未读，继续阅读

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yangdonghai12

2013-11-28

对元胞自动机感兴趣，多谢分享
lotussnow0

2015-01-06

非常满意，很好的深入浅出，易懂
liupta

2014-11-15

很好的入门教材，搞LBM一定要看

john20092010

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Lattice gas cellular automata and lattice Boltzmann models

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