1 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

3 1. Lattice Boltzmann Method Originated from LGCA: i=0,1,…,k Collision term linearized, LBGK model:

4 1. Lattice Boltzmann Method This form is similar to Boltzmann equation with BGK collision term: In discrete velocity space:

5 1. Lattice Boltzmann Method Applying upwind scheme together with Lattice velocity, we have This is exactly standard LBM form is we set.

6 1. Lattice Boltzmann Method To determine, we assume linear relationship between and : We obtain this relationship: In our simplified analysis, we set:

7 1. Lattice Boltzmann Method D2Q9 lattice model is employed. Lattice vectors can be represented by:

8 1. Lattice Boltzmann Method Flow recoveriesEquilibrium functions i = 1, 3, 5, 7 i = 2, 4, 6, 8

9 2. Simulation of Micro Flow is unknown. Channel height, From Kn and relationship of we obtain

Boundary Conditions Equilibrium functions at openings Specular bounce back at solid walls.

Extrapolation Scheme Another boundary treatment scheme Approximating unknown f’s by their f eq ’s. f eq is function of local density and velocities.

12 2. Simulation of Micro Flow Simulation process involves only 2 updating steps: Local collision: Streaming: i = 1,…, 8

13 3. Results and Discussions Qualitative analyses: General profiles of flow properties. Quantitative analyses – pressure and velocity distributions. Normalising, P* = P / P out, P*’ = P* - P* linear, u* = u / u max

General Profiles Pressure distribution Pr=2.0, Kn=0.05. Pressure changes only along the channel, in X direction. Pressure is independent of Y.

General Profiles Pr=2.0, Kn=0.05 Increasing centerline and slip velocities along the channel. Parabolic profile of u across the channel.

General Profiles Pr=2.0, Kn=0.05. Several magnitude smaller. Anti-phase peaks, growing along the channel.

Pressure Distributions Non-linearity of pressure, P’. Rarefaction negates compressibility on micro flow. Less compressibility predicted by both models.

Pressure Distributions Slip flow: Pr=1.88, Kn= Over-prediction by analytical solution Due to insufficient rarefaction taken into account.

Pressure Distributions Transition regime Pr =2.05 and Kn= Over-prediction of analytical solution is more obvious. Present methods are more general.

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:

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

Slip Velocities (Spec) Kn = 0.05 Generally agree with analytical predictions. Convergence of slip at outlet for different Pr’s.

Slip Velocities (Spec) Kn = 0.1 Slip is enhanced by Rarefaction considerably. Convergence of slip at outlet for different Pr’s.

Slip Velocities (U Ext.) Kn = 0.05 Generally predicts less slip than Spec. Convergence of outlet slip is seen.

Slip Velocities (U Ext.) Kn = 0.1 seems to have better agreement at higher Kn. Slip is enhanced as Kn increases.

26 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.

27 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.