使用半古典格子波茲曼法之軸對稱不可壓縮流體模擬

Abstract

在近十年間，格子Boltzmann法(Lattice Boltzmann Method，LBM)已發展成為相當重要的一項研究流體流動的工具。本文的研究中，使用格子Boltzmann法來計算流體在三維軸對稱管流，在不同的Knudsen數，包含了滑移區跟過渡流區，並使用新發展的半古典格子Boltzmann法，來模擬量子氣體。 半古典格子Boltzmann法是利用Uehling-Uhlenbeck Boltzmann-BGK方程式，藉由Hermite多項式展開推導而得到的。根據邊界上的滑移運動，採用了一個調和係數(accommodation coefficient)來模擬氣體在邊界上的交互作用。 不同的Knudsen數，包含了滑移區跟過渡流區中，模擬了三種不同的粒子統計，計算而得到質量流率跟速度分佈曲線，最後順利發現Knudsen minimum現象的存在。由發現Knudsen minimum現象的展現可做為演算法驗證的方式，並和本研究使用量子統計得出結果做為比較。 當流體流過一圓球，球後方會產生漩渦，模擬三種統計下，漩渦的差異；改變雷諾數，比較在相同統計下，漩渦的變化；最後，比較三維軸對稱圓球流與二維圓柱流的結果並且說明三維的減少效應。

In the last decade, Lattice Boltzmann Method, an useful and powerful tool for general fluid flow simulation, has been developed. The three-dimensional axisymmetric pipe flow of gas of arbitrary statistics in the slip and transition regimes as characterized by the Knudsen number are studied using a newly developed semiclassical lattice Boltzmann method. The semiclassical lattice Boltzmann method is derived by directly projecting the Uehling-Uhlenbeck Boltzmann-BGK equations onto the tensor Hermite polynomials using moment expansion method. To take into account the slip motion at wall surface, the Maxwellian scattering kernel is adopted to model the gas surface interactions with an accommodation coefficient. The mass flow rates and the velocity profiles are calculated for the three particle statistics over the slip and transition regimes Knudsen numbers. The results indicate that the Knudsen minimum can be captured and distinct characteristics of the effect of quantum statistics can be delineated. When the fluid flows through a sphere, it will generate vortices behind the sphere. We compare the difference of vortices under the simulation of kinds of statistic models. We change the Reynolds number and compare the change of vortex under the same statistic model. Eventually,we compare the results of three-dimensional axisymmetric sphere flow and two-dimensional circular cylindrical flow and the three-dimensional relieving effect is illustrated.