Upload
gilda
View
282
Download
0
Tags:
Embed Size (px)
DESCRIPTION
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 - PowerPoint PPT Presentation
Citation preview
Taylor Series Expansion- and Least Taylor Series Expansion- and Least Square- Based Lattice Boltzmann Square- Based Lattice Boltzmann
MethodMethod
C. Shu
Department of Mechanical EngineeringFaculty 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 1. Standard Lattice Boltzmann Method (LBM)Method (LBM)
Particle-based Method (streaming & collision)
Streaming process Collision process
D2Q9Model
/)],,(),,([
),,(),,(
tyxftyxf
tyxfttteytexf
eq
yx
2
2
2242
6
1
2
1
cccf eq
2UUeUe
Nf
0
eU
Nf
0
2/2cP tc 2
8
)12(
Features of Standard LBM
o Particle-based methodo Only one dependent variable
Density distribution function f(x,y,t)o Explicit updating; Algebraic
operation; Easy implementationNo 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 2. Current LBM Methods for Complex ProblemsComplex Problems
Interpolation-Supplemented LBM (ISLBM)
He et al. (1996), JCPFeatures 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 LBMTaylor series expansion to 1st order
derivatives
Features:Wave-like equationSolved by FD, FE and FV methodsArtificial viscosity is too large at high ReLose primary advantage of standard LBM
(solve PDE and resultant algebraic equations)
t
tyxftyxf
y
fe
x
fe
t
f eq
yx
),,(),,(
3. Development of TLLBM3. Development of TLLBM
• Taylor series expansion
P-----Green (objective point) Drawback: Evaluation A----Red (neighboring point) of Derivatives
])(,)[(),(
),()(
2
1),()(
2
1
),(),(),(),(
332
2
22
2
22
AAAA
AA
AA
yxOyx
ttPfyx
y
ttPfy
x
ttPfx
y
ttPfy
x
ttPfxttPfttAf
Idea of Runge-Kutta Method (RKM)
Taylor series method:
Runge-Kutta method:
0 , ),,( 0 twhenuutufdt
du
thdt
udh
dt
udh
dt
duhuu nn ,
6
1
2
13
33
2
221
n n+1
n+1n
Need to evaluate high order derivatives
Apply Taylor series expansion atPoints to form an equation system
Taylor series expansion is applied at 6 neighbouring points to form an algebraic equation system
A matrix formulation obtained:
[S] is a 6x6 matrix and only depends on the geometriccoordinates (calculated in advance in programming)
}{}]{[ gVS TyxfyfxfyfxffV }/,/,/,/,/,{}{ 22222
/),,(),,(),,( tyxftyxftyxfg iiiieq
iiiT
igg }{}{
(*)
Least Square OptimizationEquation system (*) may be ill-conditioned or singular (e.g. Points coincide)
Square sum of errors
M is the number of neighbouring points usedMinimize error:
)2 5(,...,2,1,0 DforMMi
M
i jjjii
M
ii VsgerrE
0
26
1,
0
2
6,...,2,1,0/ kVE k
Least Square Method (continue)The final matrix form:
[A] is a 6(M+1) matrix
The final explicit algebraic form:
}]{[}{][][][}{1
gAgSSSV TT
1
1
1,100 ),,(
k
M
kk gattyxf
ka ,1 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 ComputationFlow Chart of Computation
Input
Calculating Geometric Parameterand physical parameters ( N=0 )
Calculating eqf
,,1 ka
N=N+1
1
1
1,100 ),,(
k
M
kk gattyxf
Mf
0
eU
Mf
0
Convergence ?
No
OUTPUT
YES
eU Re,,,
Boundary TreatmentBoundary Treatment
Fluid Field
Stream from fluid field
Bounce back from the wall
ff
Non-slip condition is exactly satisfied
4. Some Numerical Examples 4. Some Numerical Examples Square Driven Cavity (Re=10,000, Non- uniform mesh 145x145)
0
0.2
0.4
0.6
0.8
1
-0.5 -0.25 0 0.25 0.5 0.75 1U
YGhia'sresultPresentresult
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0 0.2 0.4 0.6 0.8 1X
V
Ghia's result
Present Result
Fig.1 velocity profiles along vertical and horizontal central lines
Square Driven Cavity (Re=10,000, Non-uniform mesh 145x145)
Fig.2 Streamlines (right) and Vorticity contour (left)
Lid-Driven Polar Cavity FlowLid-Driven Polar Cavity Flow
Fig. 3 Sketch of polar cavity and mesh
-0.5
-0.25
0
0.25
0.5
0.75
1
0 0.2 0.4 0.6 0.8 1
ur
uθ
Fig.4 Radial and azimuthal velocity profile along the line of =00 with Re=350
r-r0
■ Num. (Fuchs ▲ exp. Tillmark)
— Present 4949 – – – Present 6565 — – — Present 8181
uθ
ur
Lid-Driven Polar Cavity FlowLid-Driven Polar Cavity Flow
Lid-Driven Polar Cavity FlowLid-Driven Polar Cavity Flow
Fig. 5 Streamlines in Polar Cavity for Re=350
Fig.6 mesh distribution
Flow around A Circular CylinderFlow around A Circular Cylinder
Fig.7 Flow at Re = 20(Time evolution of the wake length )
0
1
2
3
4
5
0 4 8 12 16 20 24t
L
Re=20
Re=40
Symbols-Experimental(Coutanceau et al. 1982) Lines-Present
Flow around A Circular CylinderFlow around A Circular Cylinder
Fig.8 Flow at Re = 20 (streamlines)
Flow around A Circular CylinderFlow around A Circular Cylinder
Fig. 9 Flow at Early stage at Re = 3000(streamline)
T=5
Re=3000
Flow around A Circular CylinderFlow around A Circular Cylinder
Fig.10 Flow at Early stage at Re = 3000(Vorticity)
T=5
Re=3000
Flow around A Circular CylinderFlow around A Circular Cylinder
Fig.11 Flow at Early stage at Re = 3000 (Radial Velocity Distribution along Cut Line)
-1.5
-1
-0.5
0
0.5
1
1 1.5 2 2.5 3X
USymbols-Experimental(Bouard &Coutanceau)
Lines-Present
T=1
T=2
T=3Re=3000
Flow around A Circular CylinderFlow around A Circular Cylinder
Fig. 12 Vortex Shedding (Re=100)
t = 3T/8
t = 7T/8
Flow around A Circular CylinderFlow around A Circular Cylinder
Natural Convection in An Annulus
Fig. 13 Mseh in Annulus
Natural Convection in An Annulus
Fig. 14 Temperature Pattern
5. Conclusions5. ConclusionsFeatures of TLLBM
Explicit formMesh freeSecond Order of accuracyRemoval of the limitation of the
standard LBMGreat potential in practical
applicationRequire large memory for 3D problem
Parallel computation