三維多鬆弛時間半古典格子波茲曼法之流場模擬

Abstract

發展基於Uehling-Uhlenbeck Boltzmann-BGK(Uehling-Uhlenbeck Boltzmann Bhatnagar-Gross-Krook Equation)方程的三維多鬆弛時間半古典格子Boltzmann法。此種方法藉由Hermite展開法得到離散半古典平衡態分布函數的Hermite展開形式，並且引入多鬆弛時間的概念，推導出三維半古典平衡態分布函數的矩空間形式，最後藉由此半古典矩空間平衡態分布函數加以計算便可得到各種巨觀物理量。 以D3Q19速度模型模擬方腔流，使用單一Re數，比較三維多鬆弛時間半古典格子Boltzmann法和三維單鬆弛時間半古典格子Boltzmann法的結果後，可以發現多鬆弛時間模擬的流場結構更為完整；並且為了更加了解多鬆弛時間半古典格子Boltzmann法的可行性，以二維D2Q9速度模型模擬長寬比例為1:2與2:1長方腔流流場問題，經由幾種Re數與三種遵循Bose-Einstein統計、Fermi-Dirac統計與Maxwell-Boltzmann統計的粒子模擬下，由結果分析可以明確展示下游次渦流在不同量子效應下產生不同的變化形態；最後經由各種邊界條件的計算後確定了此種方法的穩定性及準確性。

According to two dimensional multiple-relaxation-time semiclassical lattice Boltzmann method, a three dimensional multiple-relaxation-time semiclassical lattice Boltzmann based on the Uehling-Uhlenbeck Boltzmann-BGK equation has been successfully developed by using D3Q19 lattice model. The method is derived by expanding the equilibrium distribution function in term of Hermite polynomials. Then, we combine the concept of multiple-relaxation-time with lattice Boltzmann method to obtain the moment matrix and various physical quantities. Simulations with SRT and MRT for the lid-driven cavity flows based on D3Q19 lattice model have been carried out for three quantum particles including Bose-Einstein , Fermi-Dirac and Maxwell-Boltzmann statistics. Simulations show the streamline and pressure contour of 3D lid-driven cubic cavity flows. The result can be found in the multi-relaxation time simulation of the flow field structure is more complete. Moreover, comparing with different length and width of cavity flows such as 1:2 and 2:1, it is easy to notice that the shape of downstream eddy of rectangle flows are different for three quantum particles. Finally, after testing various boundary conditions, the stability and the accuracy of this method is illustrated.