The focus of the present study is the simulations of micro-channel flow at different Knudsen numbers using lattice Boltzmann method. There are two issues to be addressed, which are the influences of the boundary implementations and wall models. In order to capture the Knudsen layer near the wall, three different wall models are
adopted here, i.e. Stop wall function(SWF), Lockerby wall
function(LWF), and Guo wall function(GWF). As for the boundary condition implementation, this is achieved by the combinations of the bounce-back rule and diffusive-scattering boundary condition. The weighting of the boundary conditions combined with different wall models is obtained by optimizations with the fully developed
channel flow results from linearized Boltzmann equations at Knudsen number ranging from 0.1 to 10. The adopted lattice models are D2Q9, D2Q13, D2Q17 and D2Q21. Among the wall models, Stop's wall model
outperforms the other two. Using the Stop wall model, all the lattice models can be optimized to produce correct linearized Boltzmann solutions. The model results also compares favorably with the DSMC results for Knudsen number from 0.1 to 10. The capability of the models to predict developing channel flow are further examined. Due to the difficulty in implementing inlet and outlet
boundary conditions for D2Q13, D2Q17 and D2Q21, only D2Q9 is
adopted. Results show that simulations with Stop wall model can produce correctly the pressure variations in response the change of the Knudsen number and reproduce the results from DSMC simulations.

Contents
1 Introduction 1
1.1 Introduction to Lattice Boltzmann method and microflow Lattice
Boltzmann method: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Lattice Boltzmann method: . . . . . . . . . . . . . . . . . . . 1
1.1.2 Micro-flows: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Literature survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Wall models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 Boundary condition . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Objective and motivation . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Theory and governing equations 8
2.1 The Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 The BGK approximation . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 The low-Mach-number approximation . . . . . . . . . . . . . . . . . . 12
2.4 Higher-order expansion of equilibrium distribution for gas flow in the
micro-channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5 Discretization of the Lattice Boltzmann equation for phase space . . . 14
2.6 Discretization of the Lattice Boltzmann equation for time . . . . . . . 17
2.7 Modifications for simulating isothermal gas flow in microflow by using
lattice Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . 19
2.7.1 Relaxation time . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.7.2 Wall functions for bounded system . . . . . . . . . . . . . . . 20
3 Numerical algorithm 22
3.1 Simulation procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Pressure boundary condition . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 Boundary conditions in LBE simulations for microflow . . . . . . . . 25
3.3.1 Bounceback boundary condition . . . . . . . . . . . . . . . . . 25
3.3.2 Diffuse-scattering boundary condition . . . . . . . . . . . . . . 26
3.3.3 Diffusive-bounceback boundary condition . . . . . . . . . . . . 29
3.4 Periodic boundary condition . . . . . . . . . . . . . . . . . . . . . . . 32
4 Numerical results 34
4.1 Poiseuille microflow with periodic boundary condition . . . . . . . . . 34
4.2 Poiseuille microflow with pressure boundary condition . . . . . . . . . 38
5 Conclusions 41
6 Figures 43

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