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Simulation of Micro-channel Flows by Lattice Boltzmann Method

LIM Chee Yen, and C. Shu

National University of Singapore

2

Introduction

• 1. Lattice Boltzmann Method

• 2. Micro flow Simulation

• 3. Results and Discussions

• 4. Conclusions

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1. Lattice Boltzmann Method

• Originated from LGCA:

i=0,1,…,k

• Collision term linearized, LBGK model:

txftxftttcxf iiiii ,,,

txftxftxftttcxf eqiiiii ,,

1,,

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1. Lattice Boltzmann Method

• This form is similar to Boltzmann equation with BGK collision term:

• In discrete velocity space:

txvftxvftxvfvt

txvf eq ,,,,1

,,,,

txftxftxfct

txf eqiiii

i ,,1

,,

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1. Lattice Boltzmann Method

• Applying upwind scheme together with Lattice velocity , we have

• This is exactly standard LBM form is we set .

txci /

txftxft

txftttcxf eqiiiii ,,,,

t /

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1. Lattice Boltzmann Method

• To determine , we assume linear relationship between and :

• We obtain this relationship:

• In our simplified analysis, we set:

tO

x

xt

c xtc

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1. Lattice Boltzmann Method

• D2Q9 lattice model is employed.

• Lattice vectors can be represented by:

cii

c

c

i

4

1sin,

4

1cos

0,00

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1. Lattice Boltzmann Method

Flow recoveries Equilibrium functions

,0

k

iif

k

iii fcU

1

2

3

1cP

2

2

0 2

31

9

4

c

Uf eq

2

2

4

2

2 2

3

2

931

9

1

c

U

c

Uc

c

Ucf iieq

i

2

2

4

2

2 2

3

2

931

36

1

c

U

c

Uc

c

Ucf iieq

i

i = 1, 3, 5, 7

i = 2, 4, 6, 8

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2. Simulation of Micro Flow

• is unknown.

• Channel height,

• From Kn and relationship of

we obtainxt

)1( yNKn

yNH y )1(

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2.1. Boundary Conditions

• Equilibrium functions at openings

• Specular bounce back at solid walls.

f2

f0

f3

f1

f8f7f6

f5

f4

Outlet

f2

f0

f3

f1

f8f7f6

f5

f4

Inlet

f2

f0

f3

f1

f8f7f6

f5

f4

Lower Wall

f2

f0

f3

f1

f8f7f6

f5

f4

Upper Wall

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2.2. Extrapolation Scheme

• Another boundary treatment scheme

• Approximating unknown f’s by their feq’s.

• feq is function of local density and velocities.

f 2

f 0

f 3

f 1

f 8 f 7 f 6

f 5

f 4

Lower Wall

f 2

f 0

f 3

f 1

f 8 f 7 f 6

f 5

f 4

Upper Wall

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2. Simulation of Micro Flow

• Simulation process involves only 2 updating steps:

• Local collision:

• Streaming:

txftxftxftxf eqiiii ,,

1,,*

txftttcxf iii ,, *

i = 1,…, 8

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3. Results and Discussions

• Qualitative analyses: General profiles of flow properties.

• Quantitative analyses – pressure and velocity distributions.

• Normalising, P* = P / Pout, P*’ = P* - P*linear

, u* = u / umax

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3.1. General Profiles

• Pressure distribution Pr=2.0, Kn=0.05.

• Pressure changes only along the channel, in X direction.

• Pressure is independent of Y.

1

1.2

1.4

1.6

1.8

2

00.25

0.50.75

1 0

0.5

1

P*

X

Y

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3.1. General Profiles

• Pr=2.0, Kn=0.05

• Increasing centerline and slip velocities along the channel.

• Parabolic profile of u across the channel.

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3.1. General Profiles

• Pr=2.0, Kn=0.05.

• Several magnitude smaller.

• Anti-phase peaks, growing along the channel.

-0.002

-0.001

0

0.001

0.002

0

0.2

0.4

0.6

0.8

1

X-0.4

-0.2

0

0.2

0.4

Y

v

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3.2. Pressure Distributions

• Non-linearity of pressure, P’.

• Rarefaction negates compressibility on micro flow.

• Less compressibility predicted by both models.X

P'

0 0.25 0.5 0.75 10

0.01

0.02

0.03

0.04

0.05

0.06

Arkilic Kn=0.05Arkilic Kn=0.10Spec Kn=0.05Spec Kn=0.10U Ext. Kn=0.05U ext. Kn=0.10

Pr=2.0

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3.2. Pressure Distributions

• Slip flow: Pr=1.88, Kn=0.056.

• Over-prediction by analytical solution

• Due to insufficient rarefaction taken into account.0 0.25 0.5 0.75 1

0

0.025

0.05

Kn=0.056, Pr=1.88

P'

X

ArkilicSpec.U Ext.UCLA

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3.2. Pressure Distributions

• Transition regime Pr =2.05 and Kn=0.155.

• Over-prediction of analytical solution is more obvious.

• Present methods are more general.0 0.25 0.5 0.75 1

0

0.025

0.05

Kn=0.155, Pr=2.05

P'

X

ArkilicSpec.U Ext.UCLA

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3.3. Slip Velocities

• According to Arkilic et al, slip at outlet is only dependent on Kn:

, is set to 1.

• Slip along the channel can be written in term of outlet slip:

oos Kn

u41

11*

,

*,

*,

*ososs uU

rPu

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3.3. Slip Velocities

• where and

• Slip is generally dependent on the Pr, Kn, and the pressure gradients dP*/dX.

o

x

XP

XP

*

*

XKnPXKnKn

KnKn

X

P

x

1Pr1212162

Pr12Pr12122

2*

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3.3. Slip Velocities (Spec)

• Kn = 0.05

• Generally agree with analytical predictions.

• Convergence of slip at outlet for different Pr’s.

0 0.25 0.5 0.75 10

0.05

0.1

0.15

Specular Slip, Kn=0.05

2.02.252.5

2.753.0

Arkilic 2.0Arkilic 3.0

X

Us*

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3.3. Slip Velocities (Spec)

• Kn = 0.1

• Slip is enhanced by Rarefaction considerably.

• Convergence of slip at outlet for different Pr’s.0 0.25 0.5 0.75 1

0

0.05

0.1

0.15

0.2

0.25

Specular Slip, Kn=0.1

2.02.252.52.753.0

Arkilic 2.0Arkilic 3.0

Us*

X

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3.3. Slip Velocities (U Ext.)

• Kn = 0.05

• Generally predicts less slip than Spec.

• Convergence of outlet slip is seen.

0 0.25 0.5 0.75 10

0.05

0.1

0.15

U Ext., Kn=0.05

2.02.252.52.753.0

Arkilic 2.0Arkilic 3.0

X

Us*

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3.3. Slip Velocities (U Ext.)

• Kn = 0.1

• seems to have better agreement at higher Kn.

• Slip is enhanced as Kn increases.

0 0.25 0.5 0.75 10

0.05

0.1

0.15

0.2

0.25

U Ext., Kn=0.10

2.02.252.52.753.0

Arkilic 2.0Arkilic 3.0

X

Us*

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4. Closure

• Discuss the origin of LBM and its derivation from Boltzmann equation.

• Present an efficient LBM scheme for simulation of micro flows.

• Verify our numerical results by comparisons to experimental and analytical work.

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4. Closure

Pressure distribution

• Negation of compressibility by rarefaction.

• Insufficient consideration of rarefaction in N-S analytical solution.

Slip velocities

• Slip is function of u*

s,o, Pr, and dP*/dX.

• Convergence of outlet slip for different Pr’s.

• Kn enhances slip.