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研究生:連邵瑜
論文名稱:以晶格波玆曼法結合散射反彈邊界配合壁面修正函數模擬微流道流體
論文名稱(外文):Numerical Simulations of Microflow by Lattice Boltzmann Method with Diffusive-Bounceback Boundary Condition and Wall Function
指導教授:林昭安
學位類別:碩士
校院名稱:國立清華大學
系所名稱:動力機械工程學系
學門:工程學門
學類:機械工程學類
論文種類:學術論文
論文出版年:2010
畢業學年度:98
語文別:英文
論文頁數:55
中文關鍵詞:晶格波茲曼法蒙地卡羅法
外文關鍵詞:Lattice Boltzmann methodDirect Simulation Monte Carlo
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In this thesis, we use Lattice Boltzmann method to simulate microflows. According to the previous work, the cubic form of the equilibrium distribution function, feq, is deemed to be able to improve the velocity prediction in microflow. Hence, we choose quadratic D2Q9 model and three cubic models, D2Q13, D2Q17 and D2Q21 as our bases to analyze effects of higher-order term in feq on velocity prediction. Moreover, modification is also applied to these models, such as Stops’ wall function (SWF). SWF can not only lower the slip velocity but also predict a nonlinear behavior in near-wall region. Here, wall function is applied to the modification of relaxation time. In order to predict the slip velocity, we use several different boundary conditions to simulate microflow, including bounceback boundary condition, diffuse-scattering boundary condition, and β-weighted diffusive-bounceback boundary condition. First of all, we use a constant force in the streamwise direction with periodic boundary condition at the inlet and outlet. The results show that when we use β-weighted diffusive-bounceback boundary condition with SWF, the slip velocity at the wall can be captured correctly. These results are compared with linearized Boltzmann solution data and DSMC results. Finally, Knudsen minimum effect is exhibited for these models in flow rate simulation. Second, we utilize extrapolated formulas for pressure boundary condition at the inlet and outlet. We test two different Knudsen number using LBM with those three different boundary conditions and SWF. These results are compared with Direct Simulation Monte Carlo (DSMC) data. However, both of the pressure distribution and the exit velocity profile simulated by the LBM model deviate from the DSMC data.
1 Introduction . . . . . . . . . . . . . . . . . . . . . .1
1.1 Introduction to Lattice Boltzmann method and microflow
Lattice Boltzmann method: . . . . . . . . . . . . . . 1
1.2 Literature survey . . . . . . . . . . . . . . . . . . 3
1.2.1 Lattice boltzmann microflow model: . . . . . . . . .3
1.3 Objective and motivation . . . . . . . . . . . . . . .6

2 Theory and governing equations . . . . . . . . . . . . .7
2.1 The Boltzmann equation . . . . . . . . . . . . . . . .7
2.2 The BGK and the low-Mach-number approximation . . . . 8
2.2.1 The BGK approximation . . . . . . . . . . . . . . . 8
2.2.2 The low-Mach-number approximation . . . . . . . . . 11
2.3 Higher-order expansion of equilibrium distribution for gas flow in the micro-channel . . . . . . . . . . . . . . 11
2.4 Discretization of the Boltzmann equation . . . . . . .12
2.4.1 Discretization of time . . . . . . . . . . . . . . .12
2.4.2 Discretization of phase space . . . . . . . . . . . 14
2.5 The lattice Boltzmann equation with an external force term . . . . . . . . . . . . . . . . . . . . . . . . . . .17
2.6 Modifications for lattice Boltzmann equation for simulating isothermal gas flow in microflow . . . . . . . 18
2.6.1 Relaxation time . . . . . . . . . . . . . . . . . . 18
2.6.2 Wall functions for bounded system . . . . . . . . . 19

3 Numerical algorithm . . . . . . . . . . . . . . . . . . 20
3.1 Simulation procedure . . . . . . . . . . . . . . . . .20
3.2 Periodic boundary condition . . . . . . . . . . . . . 21
3.3 Pressure boundary condition . . . . . . . . . . . . . 22
3.4 Boundary conditions for the computational domain . . .23
3.4.1 Bounceback boundary condition . . . . . . . . . . . 23
3.4.2 Diffuse-scattering boundary condition . . . . . . . 23
3.4.3 Diffusive-bounceback boundary condition . . . . . . 28

4 Numerical results . . . . . . . . . . . . . . . . . . . 29
4.1 Poiseuille microflow with periodic boundary condition . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 Poiseuille microflow with pressure boundary condition . . . . . . . . . . . . . . . . . . . . . . . . 32

5 Conclusions . . . . . . . . . . . . . . . . . . . . . . 34

6 Figures . . . . . . . . . . . . . . . . . . . . . . . . 36

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