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TitleLattice Boltzmann Method: Fundamentals and Engineering Applications with Computer Codes
Author
LanguageEnglish
File Size1.9 MB
Total Pages195
Table of Contents
                            Cover
Lattice Boltzmann Method
ISBN 9780857294548
Preface
Acknowledgments
Contents
1 Introduction and Kinetics of Particles
	1.1…Introduction
	1.2…Kinetic Theory
		1.2.1 Particle Dynamics
		1.2.2 Pressure and Temperature
	1.3…Distribution Function
		1.3.1 Boltzmann Distribution
2 The Boltzmann Equation
	2.1…Boltzmann Transport Equation
		2.1.1 Example 2.1
	2.2…The BGKW Approximation
	2.3…Lattice Arrangements
		2.3.1 One-Dimensional
			2.3.1.1 D1Q3 and D1Q2
			2.3.1.2 D1Q5
		2.3.2 Two-Dimensional
			2.3.2.1 D2Q5 and D2Q4
			2.3.2.2 D2Q9
		2.3.3 Three-Dimensional
			2.3.3.1 D3Q15
			2.3.3.2 D3Q19
	2.4…Equilibrium Distribution Function
3 The Diffusion Equation
	3.1…Diffusion Equation
		3.1.1 Example 3.1
		3.1.2 Example 3.2
	3.2…Finite Differences Approximation
	3.3…The Lattice Boltzmann Method
	3.4…Equilibrium Distribution Function
	3.5…Chapman--Enskog Expansion
		3.5.1 Normalizing and Scaling
		3.5.2 Heat Diffusion in an Infinite Slab Subjected to a Constant Temperature
		3.5.3 Boundary Conditions
			3.5.3.1 Constant Temperature Boundary Condition, Dirichlet Boundary Condition
			3.5.3.2 Adiabatic Boundary Condition, Zero Flux Condition
		3.5.4 Constant Heat Flux Example
	3.6…Source or Sink Term
	3.7…Axi-Symmetric Diffusion
	3.8…Two-Dimensional Diffusion Equation
		3.8.1 D2Q4
		3.8.2 D2Q5
	3.9…Boundary Conditions
		3.9.1 The Value of the Function is Given at the Boundary
		3.9.2 Adiabatic Boundary Conditions, for Instance
		3.9.3 Constant Flux Boundary Condition
	3.10…Two-Dimensional Heat Diffusion in a Plate
		3.10.1 D2Q9
		3.10.2 Boundary Conditions
		3.10.3 Constant Flux Boundary Conditions
	3.11…Problems
4 Advection--Diffusion Problems
	4.1…Advection
	4.2…Advection--Diffusion Equation
		4.2.1 Finite Difference Method
		4.2.2 The Lattice Boltzmann
			4.2.2.1 The Lattice Boltzmann Method for Advection--Diffusion Problems
	4.3…Equilibrium Distribution Function
	4.4…Chapman--Enskog Expansion
		4.4.1 Two-Dimensional Advection--Diffusion Problems
	4.5…Two-Dimensional Lattice Boltzmann Method
		4.5.1 D2Q4
		4.5.2 D2Q9
	4.6…Problems
		4.6.1 Combustion in Porous Layer
		4.6.2 Cooling a Heated Plate
		4.6.3 Coupled Equations with Source Term
5 Isothermal Incompressible Fluid Flow
	5.1…Navier--Stokes Equation
	5.2…Lattice Boltzmann
		5.2.1 The BGK Approximation
			5.2.1.1 D2Q9, BKG--LBM
			5.2.1.2 Mach and Reynolds Numbers
			5.2.1.3 Two Dimensional with Nine Velocities, D2Q9
			5.2.1.4 Mass and Momentum Conservations
	5.3…Boundary Conditions
		5.3.1 Bounce Back
		5.3.2 Boundary Condition with Known Velocity
			5.3.2.1 West Boundary
			5.3.2.2 East Boundary
			5.3.2.3 North Boundary
			5.3.2.4 South Boundary
		5.3.3 Equilibrium and Non-Equilibrium Distribution Function
		5.3.4 Open Boundary Condition
		5.3.5 Periodic Boundary Condition
		5.3.6 Symmetry Condition
	5.4…Computer Coding
	5.5…Examples
		5.5.1 Lid Driven Cavity
		5.5.2 Developing Flow in a Two-Dimensional Channel
			5.5.2.1 The Computer Code for Flow in a Channel
		5.5.3 Flow over Obstacles
	5.6…Vorticity and Stream Function Approach
	5.7…Hexagonal Grid
	5.8…Problems
6 Non-Isothermal Incompressible Fluid Flow
	6.1…Naiver--Stokes and Energy Equations
	6.2…Forced Convection, D2Q9--D2Q9
	6.3…Heated Lid-Driven Cavity
	6.4…Forced Convection Through a Heated Channel
	6.5…Conjugate Heat Transfer
	6.6…Natural Convection
		6.6.1 Example: Natural Convection in a Differentially Heated Cavity
	6.7…Flow and Heat Transfer in Porous Media
7 Multi-Relaxation Schemes
	7.1…Multi-Relaxation Method (MRT)
	7.2…Problem
	7.3…Two-Relaxation-Time (TRT)
8 Complex Flows
Appendix A
	A.1.2 The LBM Code (2DQ4)
	A.1.3 The Finite Difference Code (2-D)
	A.1.4 Chapter 4, Advection-Diffusion
	A.1.5. Chapter Six
	A.1.6 Computer Code:
	A.1.7 Computer Code
Bibliography
Index
                        
Document Text Contents
Page 2

Lattice Boltzmann Method

Page 97

5.3.5 Periodic Boundary Condition

Periodic boundary condition become necessary to apply to isolate a repeating flow
conditions. For instance flow over bank of tubes as shown in Fig. 5.8. The flow
conditions above the line a and below the line b are the same. Hence, it is sufficient
to model the flow between these two lines and use periodic boundary conditions
alone these boundaries. The distribution functions that are leaving line a are the
same as the distribution functions entering from line b and visa versa.

The distributions functions, f4; f7 and f8 are unknown on the line a and f2; f5 and
f6 are unknown on the line b. The periodic boundary is as follows:

along line a:
f4;a ¼ f4;b; f7;a ¼ f7;a and f8;a ¼ f8;b
along line b:
f2;b ¼ f2;a; f5;b ¼ f5;a and f6;b ¼ f6;a:

5.3.6 Symmetry Condition

Many practical problems show a symmetry along a line or a plane. Then, it is
beneficial to find a solution for only one part of the domain, which saves computer
resources. For example, flow in a channel, Fig. 5.9, the flow above the symmetry
line is mirror image of the flow below the symmetry line. Therefore the integration
should be carried only for one part of the domain, and symmetry condition need to
be applied along the symmetry line.

The distribution functions, f5; f2 and f6 are unknowns. The way to construct
these functions is to set them equal to their mirror images, i.e., f5 ¼ f8; f2 ¼ f4 and
f6 ¼ f8:

Fig. 5.8 Illustrates the
periodic boundary conditions

80 5 Isothermal Incompressible Fluid Flow

Page 98

5.4 Computer Coding

Indeed the coding is quite simple and similar to the codes developed in Chap. 4 for
D2Q9, except that extra terms should be added to f eq as represented in Eq. 5.5.

The algorithm is as follows

5.5 Examples

5.5.1 Lid Driven Cavity

Lid driven cavity is used as test benchmark problem to test CFD codes. A square
cavity of 0.20 m side is filled with engine oil at 15�C (kinematic viscosity is
1:2� 10�3 m2=s). The lid is set to motion with a speed of 6 m/s.

In isothermal fluid flow it is important to match Reynolds number and
geometrical aspect ratio. For the above problem, Re ¼ Ulid � H=m ¼
6 � 0:2=0:00012 ¼ 1; 000:

In LBM, we are free to use any values for Ulid and viscosity provided that
Re ¼ 1; 000: In order to reduce compressibility effect of LBM, set Ulattice to 0.1
and viscosity to 0.01, then we need 100 lattices in H direction because Re ¼
1; 000 ¼ UlatticeN=mlattice ¼ 0:1 � N=0:01; then N = 100. Since the cavity is square,
i.e., aspect ratio of unity, then 100 nodes need to be adopted in y-direction too.

The computer code is given in the Appendix.

Fig. 5.9 Illustrates the
symmetry boundary
conditions

5.4 Computer Coding 81

Page 194

Index

A
Adiabatic, 37
Advection, 51–52
Advection–Diffusion, 52
Advection–diffusion, 23, 64
Avogadro’s number, 6, 10
Axi-Symmetric, 40

B
BGK, 18, 28, 62, 69
BGKW, 18
Boltzmann, 12–14, 17
Boltzmann constant, 6, 10
Bosensque, 92
Bounce back, 73
Boundary condition, 37
Boussinesq, 96
Buoyancy, 96

C
Chapman Enskog, 30
Chapman-Enskog, 30, 58
Collision, 5
Collisions, 28
Combustion, 65
Conjugate, 95

D
D2Q4, 20, 61
D2Q5, 20
D2Q9, 20, 44
D3Q15, 21
D3Q19, 21
Darcy, 100

Differentially heat cavity, 97
Diffusion, 23, 25–26, 29
Diffusion coefficient, 34, 60
Dirichlet, 37, 47
Distribution

function, 2, 7–10, 15, 28–29, 72, 78
DnQm, 19

E
Equilibrium distribution, 30
Equilibrium distribution

function, 19, 23–24, 29
Equilibrium

distribution functions, 55

F
False diffusion, 53
Finite difference, 27, 35
Force term, 96
Forchheimer, 100

G
Grashof, 92

H
Hexagonal lattice, 89

K
Kinetic energy, 5–6, 12
Kinetic theory, 3, 17
Knudson number, 31

177

Page 195

L
Laplace, 68
Lattice speed, 20
Lid driven, 93

M
Mach number, 68
Macro-scale, 35
Macroscopic, 2, 7, 15, 17
Maxwell, 7, 13
Maxwell’s, 23
Maxwell–Boltzmann, 18
Maxwell–Boltzmann distribution, 9
Maxwellian distribution

function, 11
Meso-scale, 2, 34
Micro-scale, 1
Microscopic, 15
Multi-scale, 31
Multi-scale expansion, 30

N
Natural convection, 91, 96
Navier Stokes, 67
Newton’s second law, 1, 4–5, 18
Numerical diffusion, 68
Nusselt number, 98

O
Obstacles, 84

P
Peclet, 53
Periodic boundary condition, 80
Porous, 64, 99
Prandtl, 92
Pressure, 5–6, 72
Probability distribution, 9

Q
Q, 20

R
Rayleigh, 92
Relaxation parameter, 36
Relaxation time, 19, 29
Reynolds, 92
Reynolds number, 69–70, 72
Root-mean-average speed, 11

S
Soret, 66
Source, 39
Stability condition, 28
Statistical mechanics, 2
Symmetry, 80

T
Taylor series, 23
Temperature, 5, 10, 28, 32

U
Up-wind scheme, 53, 61, 68

V
Velocity space, 8
Viscosity, 27
Vorticity, 89
Vorticity-stream, 68, 89

W
Weight factors, 32
Weighting factor, 29
Weighting factors, 20–21
Welander, 18

178 Index

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