Transcript Taylor Series Expansion- and Least Square

Taylor Series Expansion- and Least
Square- Based Lattice Boltzmann
Method
C. Shu
Department of Mechanical
Engineering
Faculty of Engineering
National University of Singapore

Standard Lattice Boltzmann
Method (LBM)

Current LBM Methods for Complex
Problems

Taylor Series Expansion- and Least
Square-Based LBM (TLLBM)

Some Numerical Examples

Conclusions
1. Standard Lattice Boltzmann
Method (LBM)

Particle-based Method (streaming & collision)
D2Q9
Model
Streaming process
Collision process
f ( x  ext , y  eyt , t  t )  f ( x , y, t ) 
[ feq ( x , y, t )  f ( x , y, t )] / 
eq
f
 1 1  e  U  e  U  2 U 2 
     2  2  4  2   2 
c 
 c 
 2 6  c
N
   f
 0
P  c / 2
2
N
U   f e
 0
(2  1) 2

c t
8

Features of Standard LBM
o Particle-based method
o Only one dependent variable
Density distribution function f(x,y,t)
o Explicit updating; Algebraic operation;
Easy implementation
No solution of differential equations and
resultant algebraic equations is involved
o Natural for parallel computing
 Limitation---Difficult for
complex geometry
and non-uniform
mesh
Mesh points
Positions from streaming
2. Current LBM Methods for
Complex Problems

Interpolation-Supplemented LBM
(ISLBM)
He et al. (1996), JCP
Features of ISLBM
 Large computational effort
 May not satisfy conservation
Laws at mesh points
 Upwind interpolation is needed
for stability
Mesh points
Positions from streaming

Differential LBM
Taylor series expansion to 1st order derivatives
f 
f 
f 
f  ( x, y, t )  f  ( x, y, t )
 ex
 ey

t
x
y
  t
eq
Features:
 Wave-like equation
 Solved by FD, FE and FV methods
 Artificial viscosity is too large at high Re
 Lose primary advantage of standard LBM
(solve PDE and resultant algebraic equations)
3. Development of TLLBM
•
Taylor series expansion
f ( P , t  t )
f ( P , t  t )
f ( A, t  t )  f ( P , t  t )  x A
 y A

x
y
2
2
1
1
2  f  ( P , t  t )
2  f  ( P , t  t )
( x A )
 ( y A )

2
2
2
x
2
y
 2 f  ( P , t  t )
x A y A
 O[(x A ) 3 , ( y A ) 3 ]
xy
P-----Green (objective point)
A----Red (neighboring point)
Drawback: Evaluation
of Derivatives

Idea of Runge-Kutta Method (RKM)
du
 f (u, t ),
dt
u  u0 , when t  0
Taylor series method:
n
n+1
2
3
du
1
d
u
1
d
u
n 1
n
2
3
u  u  h
 h
 h
   , h  t
2
3
dt 2 dt
6 dt
Need to evaluate high order derivatives
Runge-Kutta method:
n
Apply Taylor series expansion at
Points to form an equation system
n+1
Taylor series expansion is applied at 6
neighbouring points to form an algebraic
equation system
A matrix formulation obtained:

[S ]{V }  {g}
(*)
{V}  { f  , f  / x, f  / y,  f  / x ,  f  /  y,  f  / xy}
2
{g}  {gi }
Tg
i
 f  ( xi , yi , t ) 
2

2
2
2
T
/ 
eq
f  ( xi , yi , t )  f  ( xi , yi , t )
[S] is a 6x6 matrix and only depends on the geometric
coordinates (calculated in advance in programming)

Least Square Optimization
Equation system (*) may be ill-conditioned or
singular (e.g. Points coincide)
Square sum of errors
E 
M
2
 erri
i 0


   gi   si , jV j 
i  0
j 1

M
6
2
i  0,1,2,...,M (M  5 for 2D)
M is the number of neighbouring points used
Minimize error:
E / Vk  0, k  1,2,...,6
Least Square Method (continue)
The final matrix form:


1
{V }  [ S ] [ S ] [ S ] { g}  [ A]{ g}
T
T
[A] is a 6(M+1) matrix
The final explicit algebraic form:
M 1
f ( x0 , y0 , t  t )   a1,k gk 1
k 1
a1,k
are the elements of the first row of
the matrix [A] (pre-computed in program)

Features of TLLBM
o Keep all advantages of standard LBM
o Mesh-free
o Applicable to any complex geometry
o Easy application to different lattice
models
Flow Chart of Computation
Input
 , U , Re, e
M 1
f ( x0 , y0 , t  t )   a1,k gk 1
k 1
Calculating Geometric Parameter
and physical parameters
( a1, k , N=0 )
M
   f
 0
N=N+1
M
U   f e
 0
No
Convergence ?
Calculating
eq
f
YES
OUTPUT
Boundary Treatment
Bounce back from the wall

Fluid Field

Stream from fluid field
f  f 
Non-slip condition is exactly satisfied
4. Some Numerical Examples
Square Driven Cavity (Re=10,000, Nonuniform mesh 145x145)
0.6
1
Ghia's result
0.4
0.8
Present Result
0.2
0.6
Ghia's
result
Present
result
Y
0.4
0.2
0
-0.5 -0.25
V 0
-0.2
-0.4
-0.6
0
U
0.25 0.5
0.75
1
0
0.2
0.4 X 0.6
0.8
Fig.1 velocity profiles along vertical and horizontal central lines
1
Square Driven Cavity (Re=10,000,
Non-uniform mesh 145x145)
Fig.2 Streamlines (right) and Vorticity contour (left)
Lid-Driven Polar Cavity Flow

Fig. 3 Sketch of polar cavity and mesh
Lid-Driven Polar Cavity Flow
1
ur
uθ
0.75
—
Present 4949
– – – Present 6565
— – — Present 8181
uθ
0.5
■
▲
0.25
Num. (Fuchs
exp. Tillmark)
0
ur
-0.25
-0.5
0
0.2
0.4
0.6
0.8
r-r0
Fig.4 Radial and azimuthal velocity profile
along the line of =00 with Re=350
1
Lid-Driven Polar Cavity Flow
Fig. 5 Streamlines in Polar Cavity for Re=350
Flow around A Circular Cylinder
Fig.6 mesh distribution
Flow around A Circular Cylinder
SymbolsExperimental
5
L
4
Re=40
3
(Coutanceau et
al. 1982)
Lines-Present
2
1
Re=20
0
0
4
8
16
20
24
t 12
Fig.7 Flow at Re = 20
(Time evolution of the wake length )
Flow around A Circular Cylinder
Fig.8 Flow at Re = 20 (streamlines)
Flow around A Circular Cylinder
T=5
Re=3000
Fig. 9 Flow at Early stage at Re = 3000
(streamline)
Flow around A Circular Cylinder
T=5
Re=3000
Fig.10 Flow at Early stage at Re = 3000
(Vorticity)
Flow around A Circular Cylinder
U
SymbolsExperimental
(Bouard
&Coutanceau)
1
T=1
0.5
T=2
0
-0.5
Lines-Present
T=3
Re=3000
-1
-1.5
1
1.5
X 2
2.5
3
Fig.11 Flow at Early stage at Re = 3000 (Radial Velocity
Distribution along Cut Line)
Flow around A Circular Cylinder
t = 3T/8
t = 7T/8
Fig. 12 Vortex Shedding (Re=100)
Natural Convection in An Annulus
Fig. 13 Mseh in Annulus
Natural Convection in An Annulus
Fig. 14 Temperature Pattern
5. Conclusions

Features of TLLBM
 Explicit form
 Mesh free
 Second Order of accuracy
 Removal of the limitation of the
standard LBM
Great potential in practical
application
 Require large memory for 3D problem

Parallel computation