使用漸近保守算則於半古典波茲曼方程式之數值模擬

Abstract

為探討稀薄氣體效應下的氣體流動特性，本研究設計數值方法有效求解半古典Boltzmann-BGK方程式，並利用一維非穩態Sod震波管與二維非穩態震波繞射方柱的問題加以測式，同時搭配適當的機制達到節省計算成本的目的。 為了解決不連續現象引發的數值模擬困難，乃將滿足半古典Boltzmann-BGK方程式的速度分佈函數，用平滑的狄拉克δ函數拆解成兩條速度分佈函數，利用原半古典Boltzmann-BGK方程式推導並求解此兩速度分佈函數必須滿足之統御方程式；而平滑轉換區內的速度分佈函數即為該兩條速度分佈函數相加，因此，無須考慮耦合介面的邊界條件，因而簡化了模擬上的難度。研究中共採用線性、餘弦及雙曲三種截斷函數測試平滑轉換區內的轉換效果。 在數值的離散化上，以離散座標法用於的速度空間中，並使用全變量消逝法與加權型基本不震盪法的高解析算則處理物理空間。最後，將漸近保守算則引入，使得半古典Boltzmann-BGK方程式的鬆弛時間能獨立於碰撞項，以節省計算時間成本。 本文以一維非穩態Sod震波管與二維非穩態震波繞射方柱為測試例，進行Maxwell-Boltzmann、Fermi-Dirac與Bose-Einstein三種不同統計量子氣體之流場模擬，藉以探討前述算則在不同截斷函數搭配下於平滑轉換區之表現與節省計算時間的成效。

This study is aimed at solving the semi-classical Boltzmann-BGK equation to figure out the characteristics of gas flow, especially for rarefied gases. The coupling transformation of both the unsteady one dimensional Sod shock tube and the unsteady two dimensional shock wave impinging upon a square cylinder were investigated numerically. In addition, in order to reduce the computational amount, an appropriated mechanism is applied in this study. To deal with the discontinuity existing in problem, the solution of the semi-classical Boltzmann-BGK equation, namely the velocity distribution function, was divided into two parts with the help of a smoothed dirac delta function. Modified semi-classical Boltzmann-BGK equations were derived and solved for them over the whole computational domain then; the sum of the two parts gives the velocity distribution function in the buffer region. Consequently no more interface conditions need considering and the simulation is largely simplified. Three types of the smoothing functions – linear, cosine, and hypertangent, were tested and the conversation effect in buffer zone were examined in this thesis. As far as numerical discretization is concerned, the discrete coordinate method is employed for the velocity space and a high resolution scheme, either Total Variation Diminishing (TVD) or Weighted Essentially Non Oscillatory (WENO), was utilized for the physical space. Finally the asymptotic preserving scheme is taken in this study as well, which makes the relaxation time independent of collision term of semi-classical Boltzmann-BGK equation, resulting in a significant reduction in the computational amount. Finally the flow fields of quantum gas described by Maxwell-Boltzmann, Fermi-Dirac and Bose-Einstein statistics were all simulated. From the test examples of one dimensional unsteady Sod shock wave tube and two dimensional unsteady shock wave impinging upon a square cylinder, the investigation shows a use of a smoothing function and a high resolution scheme combined with the asymptotic preserving scheme technique does help reducing the computational amount.