半古典波茲曼模型方程式在廣義座標下之相空間直接解法

Abstract

本文利用通量分離法來求解半古典波茲曼BGK模型方程式和半古典波茲曼橢球BGK模型方程式，利用橢球BGK模型中參數來修正BGK模型中的普郎特數，以及調整BGK模型中的鬆弛時間來改變流場的稀薄度，並以不同馬赫數的流場、文獻來驗證本研究模擬的正確性。另外，將卡式座標系統轉換至廣義座標系統，來解決結構網格在曲面型邊界的問題，而本文使用圓柱流場做驗證，並比較在量子氣體，玻色-愛因斯坦統計、費米-狄拉克統計與古典極限的馬克斯威爾-波茲曼統計有何不同。 空間離散所使用之數值方法為高解析算則中的全變量消逝法，其能夠結合高階與低階準確算則的好處，以解決各別在連續解與不連續解的問題；在初始值問題中使用加權型基本不震盪算則來提高在不連續解的準確性。在時間離散中，分別加入顯式算則與隱式算則，來求解在廣義座標系統下的半古典波茲曼BGK模型方程式，模擬震波暫態與穩態的物理問題，而本研究主要以圓柱震波繞射作為驗證。

I solved the Semiclassical Boltzmann BGK model equations and Semiclassical Boltzmann Ellipsoidal BGK model equations by flux vector splitting method, and we can adjust Prandtl number is correct by Ellipsoidal BGK model. And then, we can adjust the level of rarefied flow by relaxation time in BGK model. The result of simulation could be validated in different Mach numbers and literature. In addition, we transformed Cartesian coordinate system to generalized coordinate system in order to solve the curved boundary on structure mesh, and compared the difference in Bose–Einstein statistics, Fermi–Dirac statistics, and Maxwell-Boltzmann statistics. The present numerical methods combined total variation diminishing in discrete space and implicit methods in discrete time, and solved the Semiclassical Boltzmann BGK model equations in generalized coordinate system. Weighted Essentially Non-Oscillatory (WENO) are applied to initial value problem.