基於半古典橢圓統計波茲曼方程之格子波茲曼法

Abstract

基於Uehling-Uhlenbeck Boltzmann-BGK方程（Uehling-Uhlenbeck Boltzmann Bhatnagar-Gross-Krook Equation）和橢圓統計BGK方程（Ellipsoidal Statistical BGK equation，ES-BGK）為基礎發展出半古典橢圓統計波茲曼方程之格子波茲曼法。此方法從氣體動力學理論出發，成功地藉由Hermite多項式與D2Q9格子速度模型近似推導而得到。本文利用此方法模擬二維方腔流，雙平板流流場問題， 由不同的雷諾數和三種不同的量子氣體分別遵循Bose-Einstein統計與Fermi-Dirac統計和Maxwell-Boltzmann統計呈現了此方法，並從不同的量子統計以及雷諾數的模擬結果分析比較BGK碰撞模型和ES-BGK碰撞模型的差異。 從模擬方腔流場(Cavity Flows)結果可以清楚地看出方腔流場中流線函數在BGK和ES-BGK碰撞模型(b值為-0.5 ,0 , 0.5)的差異在於上游次渦漩的呈現，隨著b值增加，上游次渦漩會越趨於完整；而考慮三種量子統計下之流線函數圖則就無明顯地不同。方腔流場中等壓圖以及壓力張量圖的低壓區域隨著b值增加，形狀與位置則會有明顯地不同；而考慮三種量子統計下之等壓圖以及壓力張量圖的低壓區域的形狀與位置亦有明顯地不同。此外，模擬雙平板流場(Couette Flows)結果可以看出b值增加或是依序BE、MB、FD量子統計下，流場的最高溫均會增大。值得注意的是ES-BGK碰撞模型在b值為-0.5時，可以修正BGK碰撞模型的普朗特數(1→2/3)，因此，對於MB統計下可以比較雙平板流場在上下邊界上有速度與溫差下之解析解，可以發現在靠近邊界上時有些許偏移。

A semiclassical lattice Boltzmann–Ellipsoidal Statistical method based on the Uehling-Uhlenbeck Boltzmann-BGK equation and Ellipsoidal Statistical BGK equation is presented. According to gas kinetic theories, the kinetic governing equation for Ellipsoidal Statistical method is directly derived by the Hermite polynomials expansion and lattice velocity model. By using lattice Boltzmann method, this work successfully demonstrates the lid driven cavity flows and Couette flows with different collision operator, BGK and ES-BGK models. Simulations not only shows the similarity and the difference between BGK and ES-BGK collision models but also presents the result for different Reynolds numbers and three quantum particles that obeying Bose-Einstein and Fermi-Dirac and Maxwell-Boltzmann statistics. It is clear to notice that the shapes of the upper upstream secondary eddy of streamlines for three quantum particles from cavity simulation are different between BGK and ES-BGK collision models (the value of b equal -0.5, 0, 0.5), the shape of the upper upstream secondary eddy is more complete as the value of b increases, the shape and position of low pressure center and pressure tensor for three quantum particles from cavity simulations are also different as the value of b is varied. Moreover, simulation for Couette flows not only shows the velocity and temperature distribution but also presents the pressure and pressure tensor contour for three quantum particles. Because ES-BGK collision models (when the value of b equal -0.5) will recover the correct Prandtl number (1→2/3), we also compare our simulations for Maxwell-Boltzmann statistics with exact solution of Couette flows with velocity and temperature difference boundary condition and the result can be found slightly error near the boundary.