應用守恆元素/解答元素方法與離散座標法於理想氣體動力學

Abstract

中文摘要 氣體動力學理論歸類為二種，第一種方法，從巨觀空氣力學特性著手，將密度、質量與溫度視為獨立變數，並且考慮黏度、熱傳導係數等等。另一方面，第二種方法，則是由微觀的基本方程式層面下手，來探討氣體分子在巨觀表現下的一般特性，是現今最為被廣泛的接受，是為考慮單一粒子，在滿足波茲曼積分微方程下的分佈函數。本文著重的特色在運用新式的守恆算則，計算氣體動力流體，並建立在波茲曼方程式與局部熱力學平衡的假設下，配合分立坐標法的觀念，將一個原本在位置空間、時間及速度空間均連續的分佈函數的積分方程式，轉換為一在位置空間與時間連續。另一方面，經分立坐標法處理後之聯立微分方程組為一組念源項之雙曲線守恆律。進而由二階準確度，顯性算則稱著保守元素/解答方法，引入計算此方程式。積分捕捉SOD 震波管內流動結構的結果與Sjogreen 展開式問題，並且將其結果與Riemann’s Euler problem 比較程式的準確度為何。本文著重於將CE/SE算則引入分立坐標計算法，計算波茲曼方程式，這是先前研究都沒有使用過的。而在未來工作上，。本研究是著重於固態的基礎，將會把提高維度的計算，與引入碰撞項至波茲曼方程式中。

ABSTRACT The dynamic theory of gases may be studied from two points of view. One may take as starting point the macroscopic equations of aerodynamics with the density, mass velocity, and temperature as independent variables and involving various coefficients, e.g., viscosity, heat conduction, etc. On the other hand, one may use a more fundamental and general microscopic formalism. The most fruitful of such formalisms available at present is that in terms of one particle distribution functions satisfying integro-differential equation of the Boltzmann type. This thesis features the usage of a novel Conservation Element/Solution Element scheme for solving gas dynamical flows. The Boltzmann equation approach is adopted and the local thermodynamic equilibrium distribution is assumed. The discrete ordinate method is first applied to remove the velocity space dependency of the distribution function which renders the model Boltzmann equation in phase space to a set of hyperbolic conservation laws. Then a two level accurate, explicit scheme called Conservation Element/Solution Element method is employed to solve those equations. The integrated results will capture the flow structure of a SOD’s shock tube and Sjögreen’s expansion problem and are to be compared to the results of Riemann’s Euler problem for the same condition. Having known that the usage of CE/SE scheme simultaneously with Discrete Ordinate Method formulation to solve equilibrium Boltzmann equation have never been done before, this study emphasizes on making a solid foundation on a more complicated computations in the future; either that be in dimensional expansion or inclusion of collision terms of Boltzmann equation.