臺灣博碩士論文加值系統

本研究發展以Uehling-Uhlenbeck Boltzmann-BGK方程(Uehling-Uhlenbeck Boltzmann Bhatnagar-Gross-Krook Equation)與橢圓統計BGK方程(Ellipsoidal Statistical BGK Equation)與多鬆弛時間格子波茲曼方法(Multiple Relaxation Time Lattice Boltzmann Method，MRT-LBM)為基礎的多鬆弛時間半古典橢圓統計格子波茲曼方法。此方法利用Hermite展開法得到半古典橢圓統計平衡態分佈函數的Hermite展開式，並透過Chapman-Enskog展開得到鬆弛時間與黏滯係數間的關係。本文透過此方法，以D2Q9格子速度模型和反彈邊界為基礎，模擬方腔流流場問題。由不同雷諾數下模擬Bose-Einstein統計與Fermi-Dirac統計和Maxwell-Boltzmann統計的粒子展示此種方法，並由模擬結果比較單鬆弛時間半古典橢圓統計格子波茲曼方法(ES-SRT)與多鬆弛時間半古典橢圓統計格子波茲曼方法(ES-MRT)之差異性。同時，在OpenMP架構下建立平行化運算過程，達到降低計算時間的目的。

A Semiclassical Multiple Relaxation Time Lattice Boltzmann Ellipsoidal Statistical Method based on the Uehling-Uhlenbeck Boltzmann-BGK equation, Ellipsoidal Statistical BGK equation (ES-BGK) and Multiple Relaxation Time Lattice Boltzmann Method (MRT-LBM) is presented. The method is derived by expanding the Semiclassical equilibrium distribution function for Ellipsoidal Statistical method in term of Hermite polynomials, and the relationship between relaxation time and viscosity can be obtained by using Chapman-Enskog expansion. Simulations of the lid driven cavity flows based on D2Q9 lattice model, and Bounce-Back boundary condition are illustrated under Bose-Einstein, Fermi-Dirac and Maxwell-Boltzmann statistics in different Reynolds numbers in the thesis. Based on the result of simulations, a comparison between ES-SRT and ES-MRT is proposed. Also, in order to reduce computing time, this work establishes parallel computations based on OpenMP.

誌謝 I
中文摘要 II
ABSTRACT III
目錄 IV
圖目錄 VII
表目錄 X
符號 XI
第一章 緒論 1
1-1 計算流體力學 1
1-2 格子波茲曼法(LATTICE BOLTZMANN METHOD)簡介 1
1-3 格子波茲曼法(LATTICE BOLTZMANN METHOD)文獻回顧 2
1-4 本文目的 3
1-5 本文架構 3
第二章 理論與統御方程式 5
2-1 氣體動力學 5
2-2 分佈函數 7
2-3 波茲曼方程式 7
2-4 波茲曼H定理與MAXWELL分布 11
2-5 MAXWELL分布 12
2-6 波茲曼BGK方程 14
2-7 格子波茲曼方程與速度模型 15
2-8 平衡態分布函數HERMITE展開 17
第三章 半古典格子波茲曼法 21
3-1 理想量子氣體動力學 21
3-2 半古典格子波茲曼方程 22
3-2-1 平衡態分布函數Hermite的展開 22
3-2-2 巨觀量求法 27
3-2-3 Chapman-Enskog分析 29
3-3 半古典橢圓統計格子波茲曼方程 32
3-3-1 平衡態分布函數Hermite的展開 33
3-3-2 巨觀量求法 34
3-3-3 單鬆弛時間Chapman-Enskog分析 36
第四章 多鬆弛時間半古典格子波茲曼法理論 40
4-1 多鬆弛時間LBE原理 40
4-2 多鬆弛時間統計半古典格子波茲曼法 43
4-3 多鬆弛時間橢圓統計半古典格子波茲曼法 45
4-4 多鬆弛時間橢圓統計CHAPMAN-ENSKOG分析 47
第五章 基本模型與邊界處理方式 55
5-1 多鬆弛時間橢圓統計格子波茲曼法 55
5-2 平行化方法與架構 55
5-2-1 OpenMP平行化介紹 56
5-2-2 格子波茲曼法平行化架構 57
5-3 邊界條件 57
5-4 收斂條件與計算流程 59
第六章 模擬結果與討論 61
6-1 方腔流 61
6-2 問題描述 62
6-3 模擬結果討論 64
第七章 結論與展望 111
7-1 結論 111
7-2 未來展望 112
參考文獻 113

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