# Progress in the Development and Application of Μηχανική

22 Φεβ 2014 (πριν από 7 χρόνια και 6 μήνες)

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Progress in the Development and Application of

Lattice Boltzmann Method

S. Y. Lin, C. T. Lin, and Y. C. Chen

Department of Aeronautics and Astronautics

National Cheng Kung University

Tainan, Taiwan, ROC

2009 International Workshop on Differential Equations
and Their Applications

Dec 2009

2

Outline

1. Objective

2. LBM: Mathematics and Physics

3. LBM: Computation

3.1 Pressure
-
Based LBM

3.2 Immersed Boundary (IB) Method

3.3 Compressibility Effect

3.4 Force Evaluation

4. Numerical Tests

5. Fluid
-
Particles Interaction Problems

6. Conclusions

3

1. Objective

1.
Develop an Incompressible Flow Solver to Study Debris
Flows. (Goal)

==================================================

1.
Combine LBM and IB Method to Simulate Fluid
-
Particle
Interaction Problems. (First Step)

2.
Introduce a Direct
-
Forcing Method for IB Method.

3.
Sedimentation of a Large Number Spherical Particles in an
Enclosure.

==================================================

1.
Investigate Solid
-
fluid Mixture Flows. (Second Step, Two
-
Phase Flow)

4

1. Objective

Liquefaction

Sedimentation

Solid
-
fluid Mixture
Flows

5

2. LBM: Mathematics and Physics

Incompressible Navier
-
Stokes equations
: PDE

0

i
i
u
x

0
2

i
i
j
i
j
i
u
x
p
u
u
x
t
u

LBM: D3Q15 LBM: Algebra Equation for density distribution function

t
x
f
t
x
f
t
x
f
t
t
e
t
x
f
eq
i
i
i
i
i
,
,
1
,
,

u
u
u
e
u
e
w
f
i
i
i
eq
i

2
3
2
9
3
1
2

i
eq
i
i
i
f
f
)
(

i
eq
i
i
i
i
i
f
e
f
e
u
)
(

t

6
1
2

6

2.1 LBM: Mathematics and Physics

Applications:

1.
Incompressible Navier
-
Stokes Solver.

2.
Two
-
Phase flow

3.
Micro Flow

Kn: Knudsen #

7

2.2 LBM: Mathematics and Physics

Boltzmann Equation + BGK Collision:

Maxwell
-
Boltzmann Equilibrium Distribution

)
,
,
(
),
(
1
t
x
f
f
f
f
f
t
f
eq

+ First
-
order finite difference scheme in time

+ First
-
order upwind scheme for convective term

))
,
(
)
,
(
(
1
,
,
t
x
f
t
x
f
t
x
f
t
t
te
x
f
eq

)
2
)
(
exp(
)
2
(
2
2
/
RT
u
RT
f
n
eq

8

2.3 LBM: Mathematics and Physics

LBM to Navier
-
Stokes Equation: Chapman Enskog Expansion

...
...
...;
)
2
(
2
)
1
(
)
0
(
1
2
1
2
2
1

f
f
f
f
x
x
t
t
t
x
x
t
t
t
2
2
)
(
)
(
u
A
u
e
C
u
e
B
A
f
eq

9

2.4 LBM: Mathematics and Physics

Deriving of Equilibrium Particle Distribution Function:

10

2.5 LBM: Mathematics and Physics

t
x
f
t
x
f
t
x
f
t
t
e
t
x
f
eq
i
i
i
i
i
,
,
1
,
,

u
u
u
e
u
e
w
f
i
i
i
eq
i

2
3
2
9
3
1
2

i
eq
i
i
i
f
f
)
(

Pressure
-
Based LBM:

t
x
p
t
x
p
t
x
p
t
t
e
t
x
p
eq
i
i
i
i
i
,
,
1
,
,

]
2
3
2
9
3
[
2
0
u
u
u
e
u
e
p
p
w
p
i
i
i
eq
i

0
2
0
2
,

s
s
c
p
c
p

i
i
i
i
i
p
e
u
p
p
p

0
,
Density
-
Based LBM:

i
eq
i
i
i
i
i
f
e
f
e
u
)
(

t

6
1
2

pressure
p
p
:
;
1
0

11

2. Boltzmann Distribution Function:

(2)

1
)
(

:
condition

)
exp(
)
(

2)

)
(

1)

:
Solutions
'
'
any
for

))
'
(
log(
))
'
(
log(
))
(
log(
))
(
log(

satisfies

:

function

a

Find
-
2
2
2
2
2

dx
x
f
x
A
x
f
C
x
f
y
x
y
x
y
f
x
f
y
f
x
f
R
R
f

Question: Uniqueness?

In Engineering: No question

In Physics: Yes, I need to think about it.

In Mathematics: O! It’s very important.

In general (No Physical sense), How can I solve?

12

3. LBM Computation

1.
Pressure
-
Based LBM

2.
Immersed Boundary Method

Direct
-
Forcing Method

3.

Compressibility Effect

13

3.1

Pressure
-
Based LBM

:

1
)

Incompressible

Flow

2
)

Inlet

Boundary

Condition
:

Velocity

Given

3
)

Outlet

Boundary

Condition
:

Pressure

Given

n
u

u
14

3.2 Immersed Boundary Method (IB)

f
G
U

)
(
v
v
v
z
F
y
E
x
G
z
F
y
E
x
t
The total force, f, on a particle:

1. external forces: gravity force

2. internal forces: collision forces between
particles and walls, drag/lift forces in the
fluid
-
particles interaction motion.

The drag/lift forces are obtained by the
immersed boundary method.

15

3.2.1 Immersed Boundary Method (IB)

Direct
-
Forcing

Method

B

= 1, in the particle cells

= 0, in the fluid cells

1

n
P
U
Particle velocity

]
2
3
2
9
3
[
)
,
(
2
0
*
,*
B
B
B
i
B
i
i
eq
i
U
U
U
e
U
e
p
t
x
p
w
p

,*)
(
*
1
)
(
)
,
(
))
(
1
(
)
,
(
eq
i
B
i
B
n
i
p
x
t
x
p
x
t
x
p

Step1: Define Volume fraction function to identify the body cells :

Step2: LBM to obtain

Step3: Direct
-
forcing to update

*
,

and

*
,
t
x
p
t
x
p
i

,
1

n
i
t
x
p

16

3.3 Compressibility Effect

The sound speed in LBM is

Non
-
dimensional Velocity <

In non
-
dimensional unit:

c
L
c
L
c
L
T
t
t
U
U
U
X
D
D

,
,
larger

c
T
3
1
0.1

t
z
y
x

17

3.4 Force Evaluation

S
S
d
σ
t
S
n
F
)
(
)
Re
1
(
2
u
P
V
cell
B



Ease for moving particle and complicated geometry

18

4. Test Problems

1.
Flow over a fixed sphere

2.
Sedimentation of one spherical particle in a
viscous fluid

3.
Sedimentation of two spherical particles in a
viscous fluid

19

Y
L

Z
L

X
L

U

Re=UD/

C
D
=

D=8~12 grids

Grids system:
(
10D,20D,10D)

19

A
ρU
f
IB
2
2
1
4.1
Flow over a fixed sphere(1)

20

4.1
Flow over a fixed sphere(2)

20

present work

Johnson & Patel

Marella et al.

Re

C
D

Lw/D

C
D

Lw/D

C
D

Lw/D

50

1.51/ 3.5%

0.386

1.57

0.4

1.56

0.39

100

1.02 /4.6%

0.832

1.08

0.86

1.06

0.88

150

0.858 /2%

1.129

0.9

1.2

0.85

1.19

200

0.777 /1%

1.324

0.771

1.439

0.768

1.436

Table 1.

21

4.1
Flow over a fixed sphere(3)

Re=50

Re=100

Re=150

Re=200

22

4.2
Sedimentation of one spherical
particle in a viscous fluid(1)

1.
Dimensions of the enclosure is 10
-
cm long, 10
-
cm wide and
16
-
cm high. The particle commences its motion at a height
H = 12 cm from the bottom.

2. Particle has density 1120 kg/m**3 and radius 1.5 cm.

3. Computational Grid: 80
×
80
×
128 grid points and the particle
is outlined by 12 grid points.

Table 2:

The fluid properties used:

Case number

f

[kg/m
3
]

m
f

[10
-
3

Ns/m
2
]

Re

t

[10
-
4
s]

Case 1

970

353

1.5

4.06

Case 2

965

212

4.1

7.11

Case 3

962

113

11.6

13.33

Case 4

960

58

32.2

25.86

23

4.2
Sedimentation of one spherical
particle in a viscous fluid(2)

Fig 1: Comparisons between trajectories
by simulations and measurements.

Fig 2: Comparisons of the measured
and simulated settling velocities.

24

4.3
Sedimentation of two circular
particles in a viscous fluid(1)

1.
The enclosure is 10 cm long, 4 cm wide, and 40 cm high.

2.
The fluid density is 960 and the dynamic viscosity of the
fluid is 0.113.

3.
Each particle density is 1120 and diameter 2 cm.

4.
A uniform grid of 50
×
20
×
200 is used and the diameter of
spherical particle is 10 grid points.

40 cm

33 cm

3 cm

2 cm

10 cm

25

4.3
Sedimentation of two circular
particles in a viscous fluid(2)

Fig. 3:

Flow velocity fields in the X
-
Z plane along Y=2 cm
at t =
0
,
0.51
,
1.02
,
1.53
,
2.04
,
2.55
,
3.06
,
3.57

[sec].

drafting
-
kissing
-
tumbling process

P2

P1

P2

P2

26

4.3

Sedimentation of two spherical particles
in a viscous fluid(3)

Fig 4: The histories of two particles: (a) x
-
position and (b) z
-
position.

27

4.3
Sedimentation of two spherical particles in
a viscous fluid(4)

Fig 5: The velocities of two particles: (a) u
-
velocity and (b) w
-
velocity.

28

5.
Sedimentation of a large number
spherical particles in an enclosure

Fig 6: 1260 particles in an enclosure

29

5.
Sedimentation of a large number spherical particles

(a)
t=0 sec

(b) t=2.13sec

(c) t =4.26 sec

(d) t =6.39 sec

(e) t =8.52 sec

(f) t =10.65sec

Rayleigh
-
Taylor

30

5.
Sedimentation of a large number spherical particles

(g) t =12.78 sec

(h) t =14.91sec

(i) t =17.04 sec

(j) t =19.17 sec

(k) t =21.03 sec

(l) t =23.43sec

31

5.
Sedimentation of a large number circular particles

(m) t =25.56 sec

(n) t =27.69 sec

(o) t =29.82 sec

32

6. Conclusions:

1. LBM: Navier
-
Stokes equation, compressibility effect, bound conditions.

2. The direct
-
forcing method is very efficient and stable forcing method for

the immersed boundary method.

3. Force evaluation method is efficient and accurate.

4.
Sedimentation of a large number circular particles in an enclosure is
simulated. This shows that the IB
-
LBM can simulate complicated
fluid
-
particles interaction problems
.

33

THANKS