1 Introduction and Kinetics of Particles ....................... 1
1.1 Introduction ............................................ 1
1.2 Kinetic Theory .......................................... 3
1.2.1 Particle Dynamics ................................ 4
1.2.2 Pressure and Temperature ......................... 5
1.3 Distribution Function ................................... 7
1.3.1 Boltzmann Distribution .......................... 12
2 The Boltzmann Equation ...................................... 15
2.1 Boltzmann Transport Equation ........................... 15
2.1.1 Example 2.1 ..................................... 17
2.2 The BGKW Approximation ................................. 18
2.3 Lattice Arrangements ................................... 19
2.3.1 One-Dimensional ................................. 19
2.3.2 Two-Dimensional ................................. 20
2.3.3 Three-Dimensional ............................... 21
2.4 Equilibrium Distribution Function ...................... 23
3 The Diffusion Equation ...................................... 25
3.1 Diffusion Equation ..................................... 25
3.1.1 Example 3.1 ..................................... 26
3.1.2 Example 3.2 ..................................... 26
3.2 Finite Differences Approximation ....................... 27
3.3 The Lattice Boltzmann Method ........................... 28
3.4 Equilibrium Distribution Function ...................... 30
3.5 Chapman-Enskog Expansion ............................... 31
3.5.1 Normalizing and Scaling ......................... 34
3.5.2 Heat Diffusion in an Infinite Slab Subjected
to a Constant Temperature ....................... 36
3.5.3 Boundary Conditions ............................. 37
3.5.4 Constant Heat Flux Example ...................... 37
3.6 Source or Sink Term .................................... 39
3.7 Axi-Symmetric Diffusion ................................ 40
3.8 Two-Dimensional Diffusion Equation ..................... 41
3.8.1 D2Q4 ............................................ 41
3.8.2 D2Q5 ............................................ 42
3.9 Boundary Conditions .................................... 43
3.9.1 The Value of the Function is Given at the
Boundary ........................................ 43
3.9.2 Adiabatic Boundary Conditions, for Instance ..... 43
3.9.3 Constant Flux Boundary Condition ................ 44
3.10 Two-Dimensional Heat Diffusion in a Plate .............. 44
3.10.1 D2Q9 ............................................ 44
3.10.2 Boundary Conditions ............................. 47
3.10.3 Constant Flux Boundary Conditions ............... 47
3.11 Problems ............................................... 48
4 Advection-Diffusion Problems ................................ 51
4.1 Advection .............................................. 51
4.2 Advection-Diffusion Equation ........................... 52
4.2.1 Finite Difference Method ........................ 52
4.2.2 The Lattice Boltzmann ........................... 54
4.3 Equilibrium Distribution Function ...................... 55
4.4 Chapman-Enskog Expansion ............................... 56
4.4.1 Two-Dimensional Advection-Diffusion Problems .... 60
4.5 Two-Dimensional Lattice Boltzmann Method ............... 61
4.5.1 D2Q4 ............................................ 61
4.5.2 D2Q9 ............................................ 62
4.6 Problems ............................................... 64
4.6.1 Combustion in Porous Layer ...................... 64
4.6.2 Cooling a Heated Plate .......................... 65
4.6.3 Coupled Equations with Source Term .............. 66
5 Isothermal Incompressible Fluid Flow ........................ 67
5.1 Navier-Stokes Equation ................................. 67
5.2 Lattice Boltzmann ...................................... 68
5.2.1 The BGK Approximation ........................... 68
5.3 Boundary Conditions .................................... 72
5.3.1 Bounce Back ..................................... 73
5.3.2 Boundary Condition with Known Velocity .......... 75
5.3.3 Equilibrium and Non-Equilibrium Distribution
Function ........................................ 78
5.3.4 Open Boundary Condition ......................... 79
5.3.5 Periodic Boundary Condition ..................... 80
5.3.6 Symmetry Condition .............................. 80
5.4 Computer Coding ........................................ 81
5.5 Examples ............................................... 81
5.5.1 Lid Driven Cavity ............................... 81
5.5 Developing Flow in a Two-Dimensional Channel ........... 82
5.5.3 Flow over Obstacles ............................. 84
5.6 Vorticity and Stream Function Approach ................. 89
5.7 Hexagonal Grid ......................................... 89
5.8 Problems ............................................... 90
6 Non-Isothermal Incompressible Fluid Flow .................... 91
6.1 Naiver-Stokes and Energy Equations ..................... 91
6.2 Forced Convection, D2Q9-D2Q9 ........................... 92
6.3 Heated Lid-Driven Cavity ............................... 93
6.4 Forced Convection Through a Heated Channel ............. 94
6.5 Conjugate Heat Transfer ................................ 95
6.6 Natural Convection ..................................... 96
6.6.1 Example: Natural Convection in
a Differentially Heated Cavity .................. 97
6.7 Flow and Heat Transfer in Porous Media ................. 99
7 Multi-Relaxation Schemes ................................... 101
7.1 Multi-Relaxation Method ............................... 101
7.2 Problem ............................................... 104
7.3 Two-Relaxation-Time ................................... 104
8 Complex Flows .............................................. 107
Appendix A: Computer Codes .................................... 109
Bibliography .................................................. 173
Index ......................................................... 177
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