波茲曼模型方程式之高解析數值方法

Abstract

波茲曼方程式為一多變數且非線性的積分微分方程式，在數學上極難求解。故其碰撞項常用一碰撞模式取代，可使得在數學處理上較容易。本文應用分立座標法將速度空間離散化，移除了速度空間與分佈函數的關係，故分佈函數只需用適當的速度分立點來表示即可。如此一來，原本在位置空間、速度空間及時間軸上皆為連續的分佈函數，其運動方程式為一積分微分方程式，經由分立座標法的處理後，將其變成在位置空間及時間軸上為連續，而在速度空間為點函數的微分方程組，此處理後，將大大簡化數值計算的困難。本文將利用加權型基本不振盪算則(WENO算則)配合分立座標法，求解波茲曼模型方程式；並發展波茲曼模型方程式之隱式WENO算則，以求解穩態稀薄氣體流場問題。 本文首先將利用一維震波管算例，將波茲曼模型方程式，經由分立座標法將速度空間分立化，再利用WENO算則計算以測試其準確性，並與其它高解析算則比較。 由於不同氣體分子的碰撞行為極難描述，故本文將首次研發出不同氣體分子的碰撞頻率，並將此碰撞頻率代入波茲曼模型方程式中，以求解二元混合氣體流場的問題。我們也將利用一維震波管算例，並在低紐森數的條件下，與尤拉方程式的解析解比較，以測試碰撞頻率的適用性。從結果發現，我們研發的碰撞頻率確實能描述氣體分子的碰撞特性。 在二維流場算例中，我們計算圓柱及NACA 0012翼形流場問題。在圓柱流場時，我們探討不同馬赫數且不同紐森數時的流場特性，並在低紐森數時，與那維爾-史托克方程式的計算結果比較，我們比較了弓形震波及尾流附近的特性，其結果是相當符合的，同時我們也比較不同高解析隱式算則的收斂歷程，從收斂歷程圖可得到，本文發展出來的隱式WENO算則有較佳的收斂效果；在NACA 0012翼形的流場中，我們將與實驗結果做比較，發現利用WENO算則計算的結果，對於具有攻角的流場，亦有相當高的準確度。

The Boltzmann equation is a nonlinear, integral, and differential equation with many variables. It is difficult to be solved mathematically, so the collision term is usually replaced with a collision model. This will make it easier to deal with. In this paper, the velocity space will be discreted by applying discrete ordinate method. The relation between velocity space and distribution function is eliminated, so the distribution function can be represented as proper discrete velocity points. Therefore, the motion equation of distribution function, which is continuous in physical space, velocity space, and time, is an integral and differential equation, and by discrete ordinate method it becomes differential equations, which are continuous in physical space and time only and point-wise in velocity space. After this kind of treatment, the difficulties of numerical calculating will be greatly reduced. In this paper, the WENO scheme in conjuction with discrete ordinate method was applied to solve the model Blotzmann equation, and the implicit WENO scheme for the model Blotzmann equation was developed to solve the steady solutions of rarefied gas flows. First, the accuracy of the present scheme was verified by calculating the case of 1-D shock tube problem, which applied discrete ordinate method to discretize the velocity space of Blotzmann model equation and WENO scheme. The result of this case was also compared with results of other high resolution schemes. Because it is difficult to describe the behaviors of collisions between different species of gas molecule, the collision frequency of different species of gas molecule was first developed and substituted into Blotzmann model equation to solve the binary gas mixture flow problem. The suitability was verified by comparing the result of 1-D shock tube case with the analytic solution of Euler’s equation in low Knudsen number condition. The collision frequency developed in this paper can surely describe the behaviors of gas molecules via the result. In cases of 2-D flow problems, the external flows of cylinder and NACA 0012 airfoil were studied. For gas flow past cylinder, the characters of flow field in different Mach number and Knudsen number condition were investigated, and especially for low Knudsen number cases, the results were compared with calculating results of Euler’s equation. It showed that they are correspondent by comparing the characters of bow shock and wake. The convergence rates of different high resolution and implicit schemes were also investigated. The convergence behavior of the implicit WENO scheme developed in this paper is better than others. For gas flow past NACA 0012 airfoil, the calculating results were compared with results of experiment. It showed that the results of WENO scheme are of higher accuracy for the case with angle of attack.