##### 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

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