Simulation of Micro Flows by Taylor Simulation of Micro Flows by Taylor Series Expansion- and Least Square-Series Expansion- and Least Square-

based Lattice Boltzmann Methodbased Lattice Boltzmann Method

   C. Shu, X. D. Niu, Y. T. Chew and Y. Peng

Department of Mechanical EngineeringFaculty of Engineering

National University of Singapore

OutlineOutlineIntroductionStandard Lattice Boltzmann

method (LBM)TLLBM for Non-uniform Mesh

and Complex geometriesTLLBM for Micro FlowsNumerical ExamplesConclusions

1. Introduction1. Introduction Micro Flows in MEMS Classification of Flow Regimes according

to Knudsen number

Re2

Ma

LKn

0.001 0.1 3 (10) 100

Continuum Flow

(Ordinary Density Levels)Slip-Flow Regime

(Slightly Rarefied)

Transition Regime

(Moderately Rarefied)Free-Molecule Flow

(Highly rarefied)

22

1

n

Studies of Micro Flows– Experimental– Numerical and Theoretical

Navier-Stokes Solver with Slip Condition– Only applicable to slip flow regime

Particle-based Methods– MD (Molecular Dynamics)– DSMC (Direct Simulation Monte Carlo)– Applicable to slip flow and transition regimes– Huge computational effort

Demanding More Efficient Solver for Micro Flow Simulation

• Does LBM have Potential to be an Does LBM have Potential to be an Efficient Method?Efficient Method?

LBM is also a particle-based method Features of LBM and MD, DSMC

Micromechanics of real particles by MD and DSMC; micromechanics of fictitious particles by LBM

Local conservation laws are satisfied by all

Tracking particles by MD and DSMC (Lagrangian solver); Dynamics of particles at a physical position and a given velocity direction by LBM (Eulerian solver)

LBM More efficient than MD and DSMC

2. Standard Lattice Boltzmann 2. 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(

Isothermal Flows:

Thermal Flowso Difficulty (stability and fixed Pr)o A number of thermal models

availableMulti-speed model; Hybrid model; two distributions model (passive scalar; IEDDF)

o We prefer to use IEDDF for the thermal flows (He et al. 1998)

f (mass) is used for velocity field

g (energy) is used for temperature field

Features of Standard LBM

o Particle-based methodo Density distribution function as

dependent variableo 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

3. TLLBM for Non-uniform 3. TLLBM for Non-uniform Mesh and Complex GeometryMesh and Complex Geometry

Current LBM Methods for Complex Problemso 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

o Differential LBMTaylor series expansion to 1st order

derivatives

Features:Wave-like equationSolved by FD 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

),,(),,(

• Development of TLLBMDevelopment of TLLBM

o Taylor series expansion

P-----Green (objective point) Drawback: valuation 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

o 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

o 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 the geometriccoordinates (calculated in advance in programming)

}{}]{[ gVS TyxfyfxfyfxffV }/,/,/,/,/,{}{ 22222

/),,(),,(),,( tyxftyxftyxfg iiiieq

iiiT

igg }{}{

(*)

o Least Square OptimizationEquation system (*) may be ill-conditioned or singular (e.g. Point 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)

(Shu et al. 2002 PRE, Vol 65)

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

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

Streamlines (right) and Vorticity contour (left)

4. TLLBM for Micro Flows4. TLLBM for Micro Flows LBM for macro flows

is related to viscosity () through Chapman-Enskog expansion

Bounce back rule (non-slip condition) Features of micro flows

Knudsen number (Kn) dominantVelocity slip at wallTemperature jump at wall

LBM for micro flows relationship not valid. kn ?Bounce back rule not valid. New B. C. ?

• Relationship of Relationship of -Kn-Kn

Re2

Ma

LKn

Pf

Pg Pr

)3/5

In Kinetic Theory:

;/c(Pr p );1/(c Rp

Relaxation time in velocity field:

Relaxation time in thermal field:

Knudsen number:

• Relationship of Relationship of -Kn-Kn

s

ref

c

UMa

HU refRe

)3/1( sc

KnHf

Pr

KnHg

In LBM:

Mach number and Reynolds number:

We derived:

• Boundary Conditions For Micro FlowsBoundary Conditions For Micro Flows Specular boundary condition

Diffuse-Scattering Boundary Condition (DSBC)

Kinetic Theory:

Maxwell Kernel:

Normalized condition:

'''

0)(

' )()()()()('

ξξξξnuξξnuξnuξ

dffw

ww

w

eqw

w

D

www

w

w

w f

RTRTRTRT uu

ξnuξuξnuξ

ξξ

)())((

)2(

1)

2

)(exp(

)(2

)()(

2

2

2

2'

Fluid

Wall

(incoming mass fluxEquals to outgoingMass flux)

)0n)u((1)( '

0n)u(

'

w

wd

n

LBM VersionReplace integral by summation

0)(

')()()(nue

''

'

eenuenuew

ff fww

w

eqw

w

Nf f

Auu' nueee

))(()(

)92(36 QDBA NN

0)(

')()()(nue

''

'

eenuenuew

gg gww

Ww

eqw

w

Ng g

B

,

))(()(uu' nueee

Theoretical analysis of DSBC--a 2D constant density flow along an infinite plate with a moving velocity and a temperature

2

2

0 2 n

uKn

n

uKnUuU wS

2

2

0 Pr2Pr n

TKn

n

TKnTTT wS

0

0.1

0.2

0.3

0

0.25

0.5

0.75

1

-0.5

-0.25

0

0.25

0.5

Pressure-Driven, Isothermal Micro Channel Flow

YX

Kn=0.05Pr=2.0

U*

5. Numerical Examples5. Numerical Examples

High pressure Low pressure

-0.002

-0.001

0

0.001

0.002

0

0.2

0.4

0.6

0.8

1-0.4

-0.2

0

0.2

0.4

Pressure-Driven, Isothermal Micro Channel Flow

Y

X

Kn=0.05Pr=2.0

V*

0 0.25 0.5 0.75 10

0.025

0.05

ArkilicSpec.U. Ext.UCLA

P'

X

Kn=0.056, Pr=1.88

Non-linear Pressure Distribution along the Channel

0 0.25 0.5 0.75 1

0

0.025

0.05

ArkilicSpec.U. Ext.UCLA

P'

X

Kn=0.155, Pr=2.05

Non-linear Pressure Distribution along the Channel

Pressure-Driven Microchannel FlowsPressure-Driven Microchannel Flows

Kn=0.053 Kn=0.165

0

0.2

0.4

0.6

0.8

1

1.2

1 1.5 2 2.5 3 3.5 4 4.5 5

Exe. data, Argon [23]

No-Slip

Present LBM

Analytical [23]

Mas

s F

low

(Kg/

s)

Pressure ratio

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

1 1.5 2 2.5 3

Exe. data, Helium [24]

No-Slip

Analytical [23]

Present LBM

Mas

s F

low

(Kg/

s)

Pressure ratio

Comparison of slip magnitude at wall in the Comparison of slip magnitude at wall in the slip flow regime (shear-driven flow)slip flow regime (shear-driven flow)

Kn Analytic LBM

0.001 0.000998 0.0008628

0.00167 0.001667 0.0015939

0.002 0.001990 0.0019277

0.004 0.003968 0.0039492

0.005 0.004950 0.0049355

0.01 0.009800 0.0097963

0.02 0.019231 0.0192269

0.025 0.023800 0.0238064

0.03 0.028302 0.0280455

0.04 0.037037 0.0370351

0.05 0.045400 0.0454530

0.1 0.083300 0.0833333

Thermal Couette FlowThermal Couette Flow

Velocity Profile Temperature Profile (Ec =10)

-1

-0.5

0

0.5

1

-0.5 -0.25 0 0.25 0.5

No-Slip

LBM(Kn=0.05)

LBM(Kn=0.1)

LBM(Kn=0.3)

Y

U

0

0.5

1

1.5

2

-0.5 -0.25 0 0.25 0.5

No-Slip

Kn=0.05

Kn=0.1Kn=0.3

Y

T

High T

Low T

Slip length Temperature Jump

0

0.4

0.8

1.2

0.01 0.1 1

Present LBMMaxwel [19]MD [20]DSMC [20]

Ls

Kn0

0.5

1

1.5

2

2.5

0.01 0.1 1

Present LBM

Maxwell [19]

Lj

Kn

Thermal Developing FlowThermal Developing Flow

(Constant Temperature on the walls Tw=10Tin)

0.5

1

1.5

0

5

10

15

20

-0.5

0

0.5

XY

Z

U

velocity

5

10

0

5

10

15

20

-0.5-0.250

0.250.5

YX

ZTemperature

Thermal Developing FlowThermal Developing Flow

Thermal developing Channel flow---Wall coefficients along channel

15

25

35

45

55

0.1 1 10

Kn=0.015

Kn=0.03

Kn=0.046

Cf*Re

x/H6

9

12

15

18

0.1 1 10

Kn=0.015

Kn=0.03

Kn=0.046

x/H

Nu

Thin Film Gas Slider Bearing Thin Film Gas Slider Bearing LubricationLubrication

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.2 0.4 0.6 0.8 1

2nd-order model

stress-density ratio model

LBMP

X/L

Knudsen Number, Kn

Inverse Knudsen Number, D

Loa

dCarryingCap

acity

,W

10-1100101

10-1 100

10-3

10-2

10-1

100

Continuumstress-density ratio modelsecond-order modelLBM

Boltzmann

=10

x

y

x=0 x=L

Kn=1.24, =123.2

200/6 HPLU w

Three-dimensional Thermal Three-dimensional Thermal Developing Flow in DuctsDeveloping Flow in Ducts

H

YX

Z

Flow

Constant Heat flux at the wall;Initial flow is assumed static with a constant temperature; Re=0.1, Pr=0.7

Three-dimensional Thermal Three-dimensional Thermal Developing Flow in DuctsDeveloping Flow in Ducts

Y

X

Z

Y

X

Z

1.04621.02401.01101.00551.00371.00100.99980.99890.9979

Velocity distribution along duct

Thermal distribution along duct

5.006.0Kn

.

Velocity profile at different cross Velocity profile at different cross sectionssections

0.5

1

1.5

u/U

0 -0.5-0.25

00.25

0.5

-0.25-0.125

00.125

0.25

X Y

Z

0.5

1

1.5

u/U

0 -0.5-0.25

00.25

0.5

y/w

-0.25-0.125

00.125

0.25

z/w

X Y

Z

0.5

1

1.5

u/U

0 -0.5-0.25

00.25

0.5

y/w

-0.25-0.125

00.125

0.25

z/w

X Y

Z

0.5

1

1.5

u/U

0 -0.5-0.25

00.25

0.5

y/w

-0.25-0.125

00.125

0.25

z/w

X Y

Zx=0.0 x=0.3442

x=0.7826 x=1.230

0.5

1

1.5

T/T

0 -0.5-0.25

00.25

0.5

y/w

-0.25-0.125

00.125

0.25

z/w

X Y

Z

Temperature profile at different Temperature profile at different cross sectionscross sections

x=0.0 x=0.3442

x=1.230

0.5

1

1.5

T/T

0 -0.5-0.25

00.25

0.5

y/w

-0.25-0.125

00.125

0.25

z/w

X Y

Z

0.5

1

1.5

T/T

0 -0.5-0.25

00.25

0.5

y/w

-0.25-0.125

00.125

0.25

z/w

X Y

Z

0.5

1

1.5

T/T

0 -0.5-0.25

00.25

0.5

y/w

-0.25-0.125

00.125

0.25

z/w

X Y

Z

x=0.7826

Three-dimensional Thermal Three-dimensional Thermal Developing Flow in Ducts: Developing Flow in Ducts:

ComparisonComparison

Slip model Present Slip model Present

1 0.21 0.25 2.85 3.16

0.75 0.25 0.27 2.81 3.06

0.5 0.32 0.30 2.71 2.92

1 0.28 0.31 2.69 2.87

0.75 0.33 0.33 2.62 2.79

0.5 0.41 0.36 2.48 2.66

1 0.34 0.37 2.53 2.60

0.75 0.39 0.38 2.44 2.54

0.5 0.48 0.41 2.26 2.44

NusU

04.0Kn

06.0Kn

08.0Kn

6. Conclusions6. Conclusions TLLBM is an efficient method for

simulation of different micro flows New relationship of -Kn is valid Specular and diffuse-scattering

boundary conditions can capture velocity slip and temperature jump

Not accurate for cases with high pressure ratio (an incompressible model)