臺灣博碩士論文加值系統

基於Uehling-Uhlenbeck Boltzmann-BGK方程（Uehling-Uhlenbeck Boltzmann Bhatnagar-Gross-Krook Equation）和多鬆弛時間格子波茲曼方法（Multi Relaxation Time Lattice Boltzmann Mathed，MRT-LBM）為基礎的多鬆弛時間半古典格子波茲曼方法。此方法成功地從動力學統御方程式，藉由Hermite多項式與各種氣體動力學近似推導而得到。本文藉由此方法，並以D2Q9格子模型為基礎模擬方腔流流場問題，由多個雷諾數和三種不同的粒子分別遵循Bose-Einstein統計與Fermi-Dirac統計和Maxwell-Boltzmann統計展示了這個方法。模擬結果指出在量子統計中特殊特性的結果。

A Multi Relaxation Time Semiclassical Lattice Boltzmann Method based on the Uehling-Uhlenbeck Boltzmann-BGK equation （Uehling-Uhlenbeck Boltzmann Bhatnagar-Gross-Krook Equation）and Multi Relaxation Time Lattice Boltzmann Method（MRT-LBM）is presented. The method is directly derived by projecting the kinetic governing equation onto the tensor Hermite polynomials and various hydrodynamic approximation orders can be achieved. Simulations of the lid　driven cavity flows based on D2Q9 lattice model for several Reynolds numbers and three different particles that obey Bose-Einstein and Fermi-Dirac and Maxwell-Boltzmann statistics are shown to illustrate the method. The results indicate distinct characteristics of the effects of quantum statistics.

中文摘要 I
Abstract II
誌謝…………………………………………………………………………………….III
目錄…………………………………………………………………………………….IV
圖目錄………..……………………………………………………………………….VII
表目錄……………………………….………………………………………………….X
符號…………………………………………...………………………………………..XI
第一章 緒論 1
1-1 格子Boltzmann 法(Lattice Boltzmann Method)簡介 1
1-2 格子Boltzmann 法文獻回顧 2
1-3 本文目的 3
1-4 本文架構 3
第二章 Boltzmann方程式 5
2-1 氣體運動理論(Gas Kinetic Theory) 5
2-2 分布函數(Distribution Function) 7
2-3 Liouville 方程 8
2-4 Boltzmann方程 9
2-5 Boltzmann H定理 11
2-6 Maxwell分布 14
2-7 Boltzmann BGK方程 16
2-8 格子Boltzmann方程與速度模型 17
2-9 平衡態分布函數的Hermite展開 19
第三章 半古典格子Boltzmann法的理論 24
3-1 理想量子氣體平衡態分布函數 24
3-2 半古典格子Boltzmann方程 25
3-3 宏觀物理量的求法 32
3-4 單鬆弛時間Chapman-Enskog分析 33
第四章 多鬆弛時間半古典格子Boltzmann法的理論 37
4-1 Multiple-Relaxation-Time LBE基本原理 37
4-2 使用 D2Q9 模型的多鬆弛時間半古典格子Boltzmann法 41
4-3 多鬆弛時間Chapman-Enskog分析 44
4-4 雙鬆弛時間半古典格子Boltzmann法 51
第五章 基本模型與邊界處理方法 52
5-1 多鬆弛時間格子Boltzmann法 52
5-2 邊界條件 52
5-3 收斂條件與計算流程 55
第六章 模擬結果與討論 56
6-1 方腔流 56
6-2 問題描述 57
6-3 模擬結果分析與討論 59
第七章 結論與展望 94
7-1 結論 94
7-2 展望 95
參考文獻 96

[1]Shan, X., Yuan, X.F., Chen, Y. H., “Kinetic Theory Representation of Hydrodynamics: A Waybeyond The Navier–Stokes Equation,” J. Fluid Mech., vol. 550, pp.413–441, (2006).

[2]He, X., Luo, L.S., “Theory of The Lattice Boltzmann Method: From The Boltzmann Equation to The Lattice Boltzmann Equation,” Phys. Rev. E, vol. 56, pp.6811–6817, (1997).

[3]Lutsko, J.F., “Chapman-Enskog Expansion about Nonequilibrium States With Application to The Sheared Granular Fluid,” Phys. Rev. E, vol. 73, p.021302, (2006).

[4]Yang, J.Y., Hung, L.H., Hu, S.H., “Semiclassical Lattice Hydrodynamics of Rarefied Channel Flows,” Applied Mathematics and Computation, vol. 217, Issue 11, pp.5151–5159, (2011).

[5]Du, R., Shi, B., Chen, X.W., “Multi-Relaxation-Time Lattice Boltzmann Model for Incompressible Flow,” Physics Letters A, vol. 359, Issue 6, pp.564–572, (2006).

[6]Luo, L.S., Liao, W., Chen, X.W., Peng, Y., Zhang, W., “Numerics of The Lattice Boltzmann Method: Effects of Collision Models on The Lattice Boltzmann Simulations,” Phys. Rev. E, vol. 83, p.056710, (2011).

[7]Lallemand, P., Luo, L.S., “Theory of The Lattice Boltzmann Method: Dispersion, Dissipation,Isotropy,Galilean Invariance, and Stability,” Phys. Rev. E, vol. 61, pp.6546–6562, (2000).

[8]Mezrhab, A., Moussaoui, M. A., Jami, M. et al., “Double MRT Thermal Lattice Boltzmann Method for Simulating Convective Flows,” Physics Letters A, vol. 374, pp.3499–3507, (2010).

[9]Yang, H.B., Liu, Y., Xu, Y.S., Kou, J.L., “Numerical Simulation of Two - Dimensional Flow over Three Cylinders by Lattice Boltzmann Method,” Commun. Theor. Phys., vol. 54, pp.886–892, (2010).

[10]Wu, S.J., Shao, Y.L., “Simulation of Lid-Driven Cavity Flows by Parallel Lattice Boltzmann Method Using Multi-Relaxation-Time Scheme,” Int. J. Numer. Meth. Fluids, vol. 46, pp.921–937, (2004).

[11]Ghia, U., Ghia, K.N., Shin, C.T., “High-Re Solutions for Incompressible Flow Using The Navier-Stokes Equations and A Multigrid Method,” J. Comput. Phys., vol. 48, pp.387–411, (1982).

[12]Lin, L.S., Chen, Y.C., Lin, C.A., “Multi Relaxation Time Lattice Boltzmann Simulations of Deep Lid Driven Cavity Flows at Different Aspect Ratios,” Computers & Fluids, vol. 45, pp.233–240, (2011).

[13]Ginzburg, I., Verhaeghe, F., d’Humieres, D., “Two-Relaxation-Time Lattice Boltzmann Scheme:About Parametrization,Velocity, Pressure and Mixed Boundary Conditions,” Commun. Comput. Phys., vol. 3, pp.427–478, (2008).

[14]Ginzburg, I., “Equilibrium-Type and Link-Type Lattice Boltzmann Models for Generic Advection and Anisotropic-Dispersion Equation,” Advances in Water Resources, vol. 28, pp.1171–1195, (2005).

[15]Qian, Y., d’Humieres, D., Lallemand, P., “Lattice BGK Models for Navier - Stokes Equation,” Europhys. Lett., vol.17, pp.479–484, (1992).

[16]Chen, H., Chen, S., Matthaeus, W.H., “Recovery of The Navier-Stokes Equation Using A Lattice Boltzmann Method,” Phys. Rev. E, vol. 45, pp.5339–5342, (1992).

[17]He, X., Chen, S., Doolen, G.D., “A Novel Thermal Model for The Lattice Boltzmann Method in Incompressible Limit,” J. Comput. Phys., vol. 146, pp.282–300, (1998).

[18]Ansumali, S., Karlin, I.V., “Single Relaxation Time Model for Entropic Lattice Boltzmann Methods,” Phys. Rev. E, vol. 65, p.056312, (2001).

[19]Patil, D.V., Lakshmisha, K.N., Rogg, B., “Lattice Boltzmann Simulation of Lid-Driven Flow in Deep Cavities,” Computers & Fluids, vol. 35, pp.1116–1125, (2006).

[20]McNamara, G., Zanetti, G., “Use of The Boltzmann Equation to Simulate Lattice-Gas Automata,” Phys. Rev. E, vol. 61, pp.2332–2335, (1988).

[21]Higuera, F., Jimenez, J., “Boltzmann Approach to Lattice Gas Simulation,” Europhys. Lett., vol.9, pp.663–668, (1989).

[22]Yang, J. Y., Hung, L.H., “Lattice Uehling-Uhlenbeck Boltzmann Bhatnagar- Gross-Krook Hydrodynamics of Quantum Gases.” Phys. Rev. E, vol. 79, p.056708, (2009).

[23]Chen, S.Y., Martinez, D., Mei, R.W., “On Boundary Conditions in Lattice Boltzmann Methods.” Physics of Fluids, vol. 8, pp.2257–2536, (1996).

[24]Zou, Q.S., He, X.Y., “On Pressure and Velocity Boundary Conditions for The Lattice Boltzmann BGK Model.” Physics of Fluids, vol. 9, pp.1591–1598, (1997).

[25]d’Humieres, D., Ginzburg, I., et al., “Multiple-Relaxation-Time Lattice Boltzmann Models in Three Dimensions.” Phil. Trans. R. Soc., vol. 360, pp.437–451, (2002).

[26]Bird, G.A., Molecular Gas Dynamics and The Direct Simulation of Gas Flows, Clarendon Press Oxford（1994）.

[27]Chen, G., Nanoscale Energy Transport and Conversion, Oxford University Press （2005）.

[28]沈清 稀薄氣體動力學（Rarefied Gas Dynamics），國防工業出版社（2003）。

[29]郭照立、鄭楚光 格子Boltzmann方法的原理及應用（Theory and Applications of Lattice Boltzmann Method），科學出版社（2009）。

[30]何雅玲、王勇、李慶 格子Boltzmann方法的原理及應用（Lattice Boltzmann Method: Theory and Applications），科學出版社（2009）。

[31]洪立昕 半古典晶格波滋曼方法，國立台灣大學工學院應用力學所博士論文，台北（2011）。