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Taylor Series Expansion- and Least Taylor Series Expansion- and Least Square- Based Lattice Boltzmann Square- Based Lattice Boltzmann Method 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

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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

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Page 1: 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

Page 2: Standard Lattice Boltzmann Method (LBM) Current LBM Methods for Complex Problems

Standard Lattice Boltzmann Method (LBM)

Current LBM Methods for Complex Problems

Taylor Series Expansion- and Least Square-Based LBM (TLLBM)

Some Numerical Examples

Conclusions

Page 3: Standard Lattice Boltzmann Method (LBM) Current LBM Methods for Complex Problems

1. Standard Lattice Boltzmann 1. Standard Lattice Boltzmann Method (LBM)Method (LBM)

Particle-based Method (streaming & collision)

Streaming process Collision process

D2Q9Model

Page 4: Standard Lattice Boltzmann Method (LBM) Current LBM Methods for Complex Problems

/)],,(),,([

),,(),,(

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(

Page 5: Standard Lattice Boltzmann Method (LBM) Current LBM Methods for Complex Problems

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

Page 6: Standard Lattice Boltzmann Method (LBM) Current LBM Methods for Complex Problems

Limitation----Difficult for complex geometry and non-uniform mesh

Mesh points

Positions from streaming

Page 7: Standard Lattice Boltzmann Method (LBM) Current LBM Methods for Complex Problems

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

Page 8: Standard Lattice Boltzmann Method (LBM) Current LBM Methods for Complex Problems

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

),,(),,(

Page 9: Standard Lattice Boltzmann Method (LBM) Current LBM Methods for Complex Problems

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

Page 10: Standard Lattice Boltzmann Method (LBM) Current LBM Methods for Complex Problems

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

Page 11: Standard Lattice Boltzmann Method (LBM) Current LBM Methods for Complex Problems

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 }{}{

(*)

Page 12: Standard Lattice Boltzmann Method (LBM) Current LBM Methods for Complex Problems

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

Page 13: Standard Lattice Boltzmann Method (LBM) Current LBM Methods for Complex Problems

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)

Page 14: Standard Lattice Boltzmann Method (LBM) Current LBM Methods for Complex Problems

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

Page 15: Standard Lattice Boltzmann Method (LBM) Current LBM Methods for Complex Problems

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,,,

Page 16: Standard Lattice Boltzmann Method (LBM) Current LBM Methods for Complex Problems

Boundary TreatmentBoundary Treatment

Fluid Field

Stream from fluid field

Bounce back from the wall

ff

Non-slip condition is exactly satisfied

Page 17: Standard Lattice Boltzmann Method (LBM) Current LBM Methods for Complex Problems

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

Page 18: Standard Lattice Boltzmann Method (LBM) Current LBM Methods for Complex Problems

Square Driven Cavity (Re=10,000, Non-uniform mesh 145x145)

Fig.2 Streamlines (right) and Vorticity contour (left)

Page 19: Standard Lattice Boltzmann Method (LBM) Current LBM Methods for Complex Problems

Lid-Driven Polar Cavity FlowLid-Driven Polar Cavity Flow

Fig. 3 Sketch of polar cavity and mesh

Page 20: Standard Lattice Boltzmann Method (LBM) Current LBM Methods for Complex Problems

-0.5

-0.25

0

0.25

0.5

0.75

1

0 0.2 0.4 0.6 0.8 1

ur

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

ur

Lid-Driven Polar Cavity FlowLid-Driven Polar Cavity Flow

Page 21: Standard Lattice Boltzmann Method (LBM) Current LBM Methods for Complex Problems

Lid-Driven Polar Cavity FlowLid-Driven Polar Cavity Flow

Fig. 5 Streamlines in Polar Cavity for Re=350

Page 22: Standard Lattice Boltzmann Method (LBM) Current LBM Methods for Complex Problems

Fig.6 mesh distribution

Flow around A Circular CylinderFlow around A Circular Cylinder

Page 23: Standard Lattice Boltzmann Method (LBM) Current LBM Methods for Complex Problems

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

Page 24: Standard Lattice Boltzmann Method (LBM) Current LBM Methods for Complex Problems

Fig.8 Flow at Re = 20 (streamlines)

Flow around A Circular CylinderFlow around A Circular Cylinder

Page 25: Standard Lattice Boltzmann Method (LBM) Current LBM Methods for Complex Problems

Fig. 9 Flow at Early stage at Re = 3000(streamline)

T=5

Re=3000

Flow around A Circular CylinderFlow around A Circular Cylinder

Page 26: Standard Lattice Boltzmann Method (LBM) Current LBM Methods for Complex Problems

Fig.10 Flow at Early stage at Re = 3000(Vorticity)

T=5

Re=3000

Flow around A Circular CylinderFlow around A Circular Cylinder

Page 27: Standard Lattice Boltzmann Method (LBM) Current LBM Methods for Complex Problems

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

Page 28: Standard Lattice Boltzmann Method (LBM) Current LBM Methods for Complex Problems

Fig. 12 Vortex Shedding (Re=100)

t = 3T/8

t = 7T/8

Flow around A Circular CylinderFlow around A Circular Cylinder

Page 29: Standard Lattice Boltzmann Method (LBM) Current LBM Methods for Complex Problems

Natural Convection in An Annulus

Fig. 13 Mseh in Annulus

Page 30: Standard Lattice Boltzmann Method (LBM) Current LBM Methods for Complex Problems

Natural Convection in An Annulus

Fig. 14 Temperature Pattern

Page 31: Standard Lattice Boltzmann Method (LBM) Current LBM Methods for Complex Problems

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