1 Introduction ................................................. 1
1.1 Outline of the Book ..................................... 1
1.2 History ................................................. 1
1.2.1 Cellular Automata ................................ 2
1.2.2 Lattice Gas Automata ............................. 2
1.2.3 Lattice Boltzmann Method ......................... 3
1.3 Advantages .............................................. 4
1.4 Objectives .............................................. 4
2 Shallow Water Flows .......................................... 7
2.1 Introduction ............................................ 7
2.2 General Flow Equations .................................. 7
2.3 Subgrid-Scale Stress Model .............................. 9
2.4 Shallow Water Equations ................................ 10
2.5 Various Numerical Methods .............................. 18
2.6 Closure ................................................ 18
3 Lattice Boltzmann Method .................................... 19
3.1 Introduction ........................................... 19
3.2 Lattice Boltzmann Equation ............................. 19
3.3 Lattice Pattern ........................................ 21
3.4 Local Equilibrium Distribution Function ................ 23
3.5 Macroscopic Properties ................................. 26
3.6 Recovery of the Shallow Water Equations ................ 27
3.7 Stability Conditions ................................... 29
3.8 Relation to Continuum Boltzmann Equation ............... 30
3.9 Discussions ............................................ 32
3.9.1 Two Variants of the LBE ......................... 32
3.9.2 Solution Strategies ............................. 32
3.9.3 Choice of the LBE ............................... 33
3.10 Closure ................................................ 33
4 Force Terms ................................................. 35
4.1 Introduction ........................................... 35
4.2 Motivation ............................................. 35
4.3 Centred Scheme ......................................... 36
4.4 Feature of the Centred Scheme .......................... 39
4.5 Discussions ............................................ 42
4.5.1 Properties of Force Term ........................ 42
4.5.2 Recommendation .................................. 43
4.5.3 Discretization Errors ........................... 43
4.6 Closure ................................................ 45
5 Turbulence Modelling ........................................ 47
5.1 Introduction ........................................... 47
5.2 SGS for Shallow Water Equations ........................ 47
5.3 LABSWETM ............................................... 48
5.4 Recovery of Turbulent SWE .............................. 50
5.5 Closure ................................................ 52
6 Boundary and Initial Conditions ............................. 53
6.1 Introduction ........................................... 53
6.2 Bounce-Back Scheme ..................................... 53
6.3 Elastic-Collision Scheme ............................... 54
6.3.1 Representation of Boundary ...................... 55
6.3.2 Slip Boundary Condition ......................... 55
6.3.3 Semi-Slip Boundary Condition .................... 57
6.4 Inflow and Outflow ..................................... 58
6.5 Periodic Boundary Condition ............................ 60
6.6 Initial Condition ...................................... 60
6.7 Solution Procedure ..................................... 60
6.8 Discussions ............................................ 61
6.9 Closure ................................................ 61
7 Applications ................................................ 63
7.1 Introduction ........................................... 63
7.2 Basic Tests ............................................ 63
7.2.1 Steady Flow over a Bump ......................... 63
7.2.2 Tidal Flow over a Regular Bed ................... 64
7.2.3 Flow around a Cylinder .......................... 67
7.3 Flows with Force Term .................................. 68
7.3.1 Stationary Case ................................. 69
7.3.2 Steady Flow over an Irregular Bed ............... 72
7.3.3 Tidal Flow over an Irregular Bed ................ 73
7.3.4 2D Steady Flow over a Bump ...................... 76
7.4 Turbulence Modelling ................................... 78
7.4.1 Flow in a Straight Channel ...................... 78
7.4.2 Flow over a Submerged Island .................... 81
7.5 More Examples .......................................... 82
7.5.1 Wind-Driven Circulation in a Dish-Shaped
Basin ........................................... 83
7.5.2 Flow in a Strongly Curved Channel ............... 88
7.5.3 Flow around Multiple Bodies ..................... 90
7.6 Closure ................................................ 9l
A LABSWE on Hexagonal Lattice ................................. 95
В LABSWE Code ................................................. 97
B.l LABSWE Module .......................................... 97
B.2 An Example ............................................ 102
References .................................................... 105
Index ......................................................... 109
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