5. Lattice Boltzmann Models

Abstract

5.1 From lattice-gas cellular automata to lattice Boltzmann models 5.1.1 Lattice Boltzmann equation and Boltzmann equation 5.1.2 Lattice Boltzmann models of the first generation 5.2 BGK lattice Boltzmann model in 2D 5.2.1 Derivation of the W i 5.2.2 Entropy and equilibrium distributions 5.2.3 Derivation of the Navier-Stokes equations by multi-scale analysis 5.2.4 Storage demand 5.2.5 Simulation of two-dimensional decaying turbulence 5.2.6 Boundary conditions for LBM 5.3 Hydrodynamic lattice Boltzmann models in 3D 5.3.1 3D-LBM with 19 velocities 5.3.2 3D-LBM with 15 velocities and Koelman distribution 5.3.3 3D-LBM with 15 velocities proposed by Chen et al. (D3Q15) 5.4 Equilibrium distributions: the ansatz method 5.4.1 Multi-scale analysis 5.4.2 Negative distribution functions at high speed of sound 5.5 Hydrodynamic LBM with energy equation 5.6 Stability of lattice Boltzmann models 5.6.1 Nonlinear stability analysis of uniform flows 5.6.2 The method of linear stability analysis (von Neumann) 5.6.3 Linear stability analysis of BGK lattice Boltzmann models 5.6.4 Summary 5.7 Simulating ocean circulation with LBM 5.7.1 Introduction 5.7.2 The model of Munk (1950) 5.7.3 The lattice Boltzmann model 5.8 A lattice Boltzmann equation for diffusion 5.8.1 Finite differences approximation 5.8.2 The lattice Boltzmann model for diffusion 5.8.3 Multi-scale expansion 5.8.4 The special case \(\omega = 1\) 5.8.5 The general case 5.8.6 Numerical experiments 5.8.7 Summary and conclusion 5.8.8 Diffusion equation with a diffusion coefficient depending on concentration 5.8.9 Further reading 5.9 Lattice Boltzmann model: What else? 5.10 Summary and outlook