使用平滑轉換區於半古典跨流域稀薄氣體模擬

Abstract

在本研究當中求解半古典波茲曼BGK模型方程式(Semiclassical Boltzmann Bhatnagar-Gross-Krook)，使用Maxwell-Boltzmann, Fermi-Dirac和 Bose-Einstein三種不同統計，模擬量子氣體流場。並且使用將微觀尺度方程式與巨觀尺度方程式耦合的模型，該方法是藉由定義截斷函數，建立平滑轉換區於兩不同尺度的方程式之間。在平滑轉換區內，將速度分布函數由截斷函數分解成一個耦合方程組，其中一方程式投影至尤拉極限。平滑緩衝區內的解答，即為該耦合方程組的兩條方程式所求解出的分布函數相加而得。在本文當中考慮線性、餘弦和雙曲函數三種不同的截斷函數，測試不同截斷函數轉換的效果。藉由定義截斷函數，可以簡易的決定於哪一流域求解巨觀方程式、微觀方程式或者將兩者耦合的平滑轉換區。與過往的多尺度耦合方法相比，無須考慮兩尺度方程式之間交界面的邊界條件。在數值方法的部分，本文使用離散座標法，將速度空間從速度分布函數之中獨立出來，使得原半古典波茲曼BGK模型方程式於相空間成為數組具有源項的雙曲線型守恆律方程式。在物理空間部分，使用全變量消逝法和加權型基本不振盪法兩種高解析算則計算。本文中模擬數種量子氣體測試平滑緩轉換區之效果，包含一維震波管問題、二維非穩態震波繞射方柱及穩態流場流過圓柱之流場問題。各算例測試中平滑緩衝區表現良好，可做為多尺度問題耦合法使用。

This study solves Semiclassical Boltzmann-BGK equation (also called Uehling – Uhlenbeck Boltzmann-Bhatnagar-Gross-Krook equation) with Maxwell-Boltzmann, Fermi-Dirac and Bose-Einstein statistics which allows us to simulate quantum gas flow problem. In this work, we also implement a buffering zone model which provides a smooth transition between a kinetic and a hydrodynamic domain. The idea is to use buffer zone splitting the velocity distribution function into two coupled equations by cut-off function. The solution will be recovered as the sum of the two coupled equations. We can easily determine to solve kinetic, hydrodynamic or coupled equations via cut function. Three types of cut-function, linear, cosine and hypertangent, are considered in this article. The idea of buffer zone can avoid issue of interface boundary condition between macroscopic and microscopic equation. For numerical parts, we use discrete ordinate method to remove the velocity space dependency of the velocity distribution function which renders Boltzmann equation in phase space to a set of hyperbolic conservation laws with source terms in physical space. High resolution schemes, Total Variation Diminishing (TVD) and Weighted Essentially Non-Oscillatory (WENO), are applied to physical space. Several semiclassical gas flow problems including 1-D shock tube problem, 2-D unsteady shock wave impinging upon a square cylinder and steady flow over a cylinder have been simulated to test the buffer zone treatment. Buffer zone performed well in each test problem. It can be implemented in Multi-scale coupling method successfully.