1. Simulation of Micro Flows by Taylor Series Expansion- and Least Square-based Lattice Boltzmann Method   C. Shu, X. D. Niu, Y. T. Chew and Y. Peng Department of Mechanical Engineering Faculty of Engineering National University of Singapore

2. Outline Introduction Standard Lattice Boltzmann method (LBM)
TLLBM for Non-uniform Mesh and Complex geometries
TLLBM for Micro Flows
Numerical Examples
Conclusions

3. (Ordinary Density Levels) (Moderately Rarefied)
1. Introduction
Micro Flows in MEMS
Classification of Flow Regimes according to Knudsen number
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)

4. 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

5. Does LBM have Potential to be an 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

6. 2. Standard Lattice Boltzmann Method (LBM)
Streaming process Collision process
Particle-based Method (streaming & collision)
D2Q9
Model

7. Isothermal Flows:

8. Thermal Flows Difficulty (stability and fixed Pr)
A number of thermal models available
Multi-speed model; Hybrid model; two distributions model (passive scalar; IEDDF)
We prefer to use IEDDF for the thermal flows (He et al. 1998)
fa (mass) is used for velocity field
ga (energy) is used for temperature field

9. Features of Standard LBM
Particle-based method
Density distribution function as dependent variable
Explicit updating; Algebraic operation; Easy implementation
No solution of differential equations and resultant algebraic equations is involved
Natural for parallel computing

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

11. 3. TLLBM for Non-uniform Mesh and Complex Geometry
Current LBM Methods for Complex Problems
Interpolation-Supplemented LBM (ISLBM)
He et al. (1996), JCP
Features of ISLBM
Large computational effort
May not satisfy conservation
Laws at mesh points
Upwind interpolation is needed
for stability

12. Differential LBM Features:
Taylor series expansion to 1st order derivatives
Features:
Wave-like equation
Solved by FD and FV methods
Artificial viscosity is too large at high Re
Lose primary advantage of standard LBM (solve PDE and resultant algebraic equations)

13. Taylor series expansion
Development of TLLBM
Taylor series expansion
P-----Green (objective point) Drawback: valuation
A----Red (neighboring point) of Derivatives

14. Idea of Runge-Kutta Method (RKM)
Taylor series method:
Runge-Kutta method:
n+1 n
Need to evaluate high order derivatives n
n+1
Apply Taylor series expansion at
Points to form an equation system

15. A matrix formulation obtained:
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 geometric
coordinates (calculated in advance in programming)
(*)

16. Least Square Optimization
Equation system (*) may be ill-conditioned or singular (e.g. Point coincide)
Square sum of errors
M is the number of neighbouring points used
Minimize error:

17. Least Square Method (continue)
The final matrix form:
[A] is a 6(M+1) matrix
The final explicit algebraic form:
(Shu et al. 2002
PRE, Vol 65)
are the elements of the first row of
the matrix [A] (pre-computed in program)

18. Features of TLLBM Keep all advantages of standard LBM Mesh-free
Applicable to any complex geometry
Easy application to different lattice models

19. Flow Chart of Computation
Input
Calculating Geometric Parameter
and physical parameters
( N=0 )
N=N+1 No
Convergence ?
Calculating
YES
OUTPUT

20. Boundary Treatment
Non-slip condition is exactly satisfied

21. Square Driven Cavity (Re=10,000, Non-uniform mesh 145x145)
Streamlines (right) and Vorticity contour (left)

22. 4. TLLBM for Micro Flows LBM for macro flows Features of micro flows
t is related to viscosity (m) through Chapman-Enskog expansion
Bounce back rule (non-slip condition)
Features of micro flows
Knudsen number (Kn) dominant
Velocity slip at wall
Temperature jump at wall
LBM for micro flows
t-m relationship not valid. t-kn ?
Bounce back rule not valid. New B. C. ?

23. Relationship of t-Kn In Kinetic Theory:
Relaxation time in velocity field:
Relaxation time in thermal field:
Knudsen number:

24. Relationship of t-Kn In LBM: Mach number and Reynolds number:
We derived:

25. Boundary Conditions For Micro Flows
Fluid
Wall n
Specular boundary condition
Diffuse-Scattering Boundary Condition (DSBC)
Kinetic Theory:
Maxwell Kernel:
Normalized condition:
(incoming mass flux
Equals to outgoing
Mass flux)

26. LBM Version
Replace integral by summation

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

28. 5. Numerical Examples
High pressure
Low pressure

30. Non-linear Pressure Distribution along the Channel

31. Non-linear Pressure Distribution along the Channel

32. Pressure-Driven Microchannel Flows
Kn= Kn=0.165
Mass Flow(Kg/s)
Mass Flow(Kg/s)
Pressure ratio
Pressure ratio

33. Comparison of slip magnitude at wall in the slip flow regime (shear-driven flow)Kn
Analytic
LBM
0.001
0.002
0.004
0.005
0.01
0.02
0.025
0.03
0.04
0.05
0.1

34. Thermal Couette Flow T U Y Velocity Profile
High T
Low T Y U T
Velocity Profile
Temperature Profile (Ec =10)

35. Slip length
Temperature Jump Ls Kn Lj

36. Thermal Developing Flow
(Constant Temperature on the walls Tw=10Tin) U
velocity
Temperature

37. Thermal Developing Flow
Thermal developing Channel flow---Wall coefficients along channel
Cf*Re
x/H Nu

38. Thin Film Gas Slider Bearing Lubricationx y
x=0
x=L P
X/L
Kn=1.24, =123.2

39. Three-dimensional Thermal Developing Flow in Ducts
Constant Heat flux at the wall;
Initial flow is assumed static with a constant temperature;
Re=0.1, Pr=0.7
Flow

40. Three-dimensional Thermal Developing Flow in Ducts.
Velocity distribution along duct
Thermal distribution along duct

41. Velocity profile at different cross sections
x=0.3442
x=0.0
x=0.7826
x=1.230

42. Temperature profile at different cross sections
x=0.0
x=0.3442
x=0.7826
x=1.230

43. Three-dimensional Thermal Developing Flow in Ducts: Comparison
Slip model
Present 1
0.21
0.25
2.85
3.16
0.75
0.27
2.81
3.06
0.5
0.32
0.30
2.71
2.92
0.28
0.31
2.69
2.87
0.33
2.62
2.79
0.41
0.36
2.48
2.66
0.34
0.37
2.53
2.60
0.39
0.38
2.44
2.54
0.48
2.26

44. 6. Conclusions
TLLBM is an efficient method for simulation of different micro flows
New relationship of t-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)