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

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

(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)

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 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 Method (LBM) Streaming process Collision process Particle-based Method (streaming & collision) D2Q9 Model

Isothermal Flows:

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

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

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

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

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)

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

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

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) (*)

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:

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)

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

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

Boundary Treatment 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 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. ?

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

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

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)

LBM Version Replace integral by summation

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

5. Numerical Examples High pressure Low pressure

Non-linear Pressure Distribution along the Channel

Non-linear Pressure Distribution along the Channel

Pressure-Driven Microchannel Flows Kn=0.053 Kn=0.165 Mass Flow(Kg/s) Mass Flow(Kg/s) Pressure ratio Pressure ratio

Comparison of slip magnitude at wall in the 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 Flow T U Y Velocity Profile High T Low T Y U T Velocity Profile Temperature Profile (Ec =10)

Slip length Temperature Jump Ls Kn Lj

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

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

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

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

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

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

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

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

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)