應用半古典橢圓統計格子波茲曼法模擬並排雙圓柱尾流流場特性

Abstract

流體力學在連體模型中多以Navier-Stokes方程式或Euler方程式求解，而當氣體流場的稀薄程度增加時，這兩種統御方程式皆已不符合稀薄氣體動力學之條件，此時適用的統御方程式為波茲曼方程，且在做數值模擬計算時，會將波茲曼方程經空間、時間和速度離散推導出格子波茲曼方程。 以Uehling-Uhlenbeck Boltzmann-BGK方程(Uehling-Uhlenbeck Boltzmann Bhatnagar-Gross-Krook Equation)以及橢圓統計BGK方程(Ellipsoidal Statistical BGK Equation)為基礎，發展出半古典橢圓統計格子波茲曼法，亦使用四階Hermite多項式展開將半古典橢圓統計平衡態分布函數展開成離散型式，求得分布函數後計算出流場的巨觀物理量，並可藉由Chapman-Enskog分析使格子波茲曼方程推導回Navier-Stokes方程，及得到鬆弛時間與黏滯係數間的關係。 使用D2Q9格子速度模型以及沉浸邊界速度修正法處理物體邊界，對均勻流流經並排雙圓柱的流場問題，模擬Bose-Einstein統計、Fermi-Dirac統計和Maxwell-Boltzmann統計的粒子以及不同普朗特數的修正量，在低雷諾數範圍和圓柱之不同間距值的條件下，圓柱尾流形狀依其流場流線、渦度、升力係數和阻力係數特性，分為兩種穩態和五種非穩態類型，依序為異形、同形、單一扁平、偏向、正反、同步同相和同步反相流型。

The Navier-Stokes equation or Euler equation is traditionally used to solve solutions in the continuum regime of fluid dynamics. However, while the gas flow is rarefied, the government equation should become Boltzmann equation. The lattice Boltzmann method is derived by discretizing with Boltzmann equation in physical and velocity space for numerical simulation. Based on the Uehling-Uhlenbeck Boltzmann-BGK equation and Ellipsoidal Statistical BGK equation, a semiclassical lattice Boltzmann ellipsoidal statistical method is developed. The semiclassical equilibrium distribution function for ellipsoidal statistical method can be expanded by fourth-order Hermite polynomials, and the relationship between relaxation time and viscosity can be obtained by using Chapman-Enskog analysis. Under D2Q9 lattice model, uniform flow past a pair of cylinders in side-by-side arrangements has been simulated with immersed boundary velocity correction method which is used to describe the boundary of cylinders. At low Reynolds numbers and by varying cylinders spacing ratios, seven different wake patterns were systematically categorized. Simulations under Bose-Einstein, Fermi-Dirac and Maxwell-Boltzmann statistics including the corrections of Prandtl numbers are also presented.