Simulation of Micro-channel Flows by Lattice Boltzmann Method

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. Flow recoveries Equilibrium functions 1. Lattice Boltzmann Method 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

10. 2.1. Boundary Conditions • Equilibrium functions at openings • Specular bounce back at solid walls.

11. 2.2. Extrapolation Scheme • Another boundary treatment scheme • Approximating unknown f’s by their feq’s. • feq 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 / Pout, P*’ = P* - P*linear , u* = u / umax

14. 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.

15. 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.

16. 3.1. General Profiles • Pr=2.0, Kn=0.05. • Several magnitude smaller. • Anti-phase peaks, growing along the channel.

17. 3.2. Pressure Distributions • Non-linearity of pressure, P’. • Rarefaction negates compressibility on micro flow. • Less compressibility predicted by both models.

18. 3.2. Pressure Distributions • Slip flow: Pr=1.88, Kn=0.056. • Over-prediction by analytical solution • Due to insufficient rarefaction taken into account.

19. 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.

20. 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:

21. 3.3. Slip Velocities • where and • Slip is generally dependent on the Pr, Kn, and the pressure gradients dP*/dX.

22. 3.3. Slip Velocities (Spec) • Kn = 0.05 • Generally agree with analytical predictions. • Convergence of slip at outlet for different Pr’s.

23. 3.3. Slip Velocities (Spec) • Kn = 0.1 • Slip is enhanced by Rarefaction considerably. • Convergence of slip at outlet for different Pr’s.

24. 3.3. Slip Velocities (U Ext.) • Kn = 0.05 • Generally predicts less slip than Spec. • Convergence of outlet slip is seen.

25. 3.3. 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. 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. 4. Closure