多鬆弛時間半古典橢圓統計格子波茲曼法之流場模擬

Abstract

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.