Standard Lattice Boltzmann Method (LBM) Current LBM Methods for Complex Problems

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