- 學位論文
半古典波茲曼 BGK 方程式之任意統計稀薄流模擬
Modeling Rarefied Gas Flows of Arbitrary Statistics with the Semiclassical Boltzmann-BGK equation
摘要
半古典波茲曼方程式可適用較波茲曼方程式廣泛的傳輸現象, 即適用於遵從量子統計的粒子傳輸現象。半古典波茲曼方程式最初由Nordheim 於 1928 年推導出, 在 1933 年 Uehling 和 Uhlenbeck 擴充。它是能在同一框架下描述古典和量子氣體運動的合適的數學模型。這統一框架雖然形式上簡單, 但實質上卻不是簡單的,並具有非線性量子關係產生很多數值上的挑戰。雖然已有文獻報導許多方法來模擬量子系統的流體問題,量子熱力學關係的非線性性質阻礙了許多方法的運用機會。尤其在量子效應起主要作用,即近簡併狀態。在本文提出一種新的數值方法求解波茲曼 BGK 方程的古典和量子 (Bose-Einstein 和Fermi-Dirac 統計) 氣體問題。此外,利用不連續多項式所發展的數值直解法已完成平行化的工作,應用於模擬量子氣體粒子系統中將大幅增加其計算效率。本研究所提出的數值方法已由氣體動力學基準測試問題得到驗證。本研究是首篇能夠成功呈現包含內部自由度的理想量子氣體運動現象的模式。
並列摘要
The semiclassical Boltzmann equation is a generalization of the classical Boltzmann Equation intended to describe the dynamics of quantum particle system in phase space. Originally formulated by Nordheim in 1928 and extended by Uehling and Uhlenbeck in 1933, it is a suitable mathematical model capable of describing classical and quantum gases under a single framework. This general framework although simple it is not simplistic and possess many numerical challenges due to the non-linear quantum relations that arise in it. Although many approaches to model the hydrodynamic of quantum systems have been reported throughout the literature, the non-linear nature of the quantum thermodynamic relations hinders many methodological opportunities. This is especially true when considering conditions where the quantum effects play a major role, the near-degenerate regimes. In this dissertation, a new numerical methodology is presented to solve the Boltzmann-BGK equation of gas dynamics for the classical and quantum gases described by the Bose-Einstein and Fermi-Dirac statistics. Moreover, an efficient direct solver based on discontinuous polynomial representation is investigated to explore the parallel computing opportunities in modeling quantum gas systems. In this work, the proposed numerical methodology is validated by benchmark problems of gas dynamics. Descriptions of ideal quantum gases including internal degrees of freedom are successfully achieved and reported for the first time.