使用波茲曼模型方程式模擬自由分子流到近連續流的正方形空穴流場

Abstract

上板瞬間抽動的空穴流場在計算流體力學上是ㄧ個很基準的問題，因為它具有簡單的幾何結構並且在其角落上有差異性很大的奇異點。在流場模擬上,時常使用它來驗證不同的數值方法。然而,過去的學習都著重在連續流場上的研究。關於稀薄流場與近連續流流場的研究則是相當的稀少。因此本文的動機是模擬空穴流場在稀薄區域上的應用。 本文內容是利用建構在有限差分法(finite-difference scheme),速度空間上應用分立座標法(discrete ordinate method)的波茲曼模型方程式(MBE)來模擬二維的上板瞬間啟動之四方形穴流從自由分子流到近連續流的流場。本文中使用BGK model與Shakov model兩種模型來近似碰撞積分項。為了驗證波茲曼模型方程式(MBE)的正確性,我們比較直接蒙地卡羅模擬法(DSMC)與波茲曼模型方程式(MBE)在模擬穴流Kn=0.0033 與 Ma=2.0的結果。本文中空穴流場的模擬範圍包含了Kn=10~0.0033 與 Ma=0.5~2的流場。 在模擬的結果中顯示,相同的Mach number隨著氣體的稀薄化,速度滑動現象與溫度跳動現象會更加明顯。在各種的Mach number下 (M=0.5, 0.9, 1.1, 2), 隨著Kn的變小,渦流中心會朝著右上方移動,但是當M=0.5,Kn=0.0033與M=2,Kn=0.0033時渦流中心會改變成朝左下方移動。此外當Kn小於0.01時,在主要渦流的兩側角落都會產生出次渦流。當Kn大於0.01時,除了M=2,Kn=10這個情況下會在右下角落產生一個次渦流以外,其他的情況則不會產生次渦流。

The driven cavity flow is one of the benchmark problems often used in computational fluid dynamics due to its simple geometry but highly singular points at the corners. It is often used to verify different numerical methods for fluid-flow simulation. However, past studies in this regard focused on flows in thc continuum regime. Very few researches have been done systematically in the rarefied or near-contiuum regime. Several applications require consideration of rarefaction, which motivates the present thesis to focus on simulation of driven cavity flows in this region. This thesis reports the simulation of a two-dimensional top driven square cavity flows from free-molecular to near-continuum regime using a model Boltzmann equation (MBE) solver. The MBE was discretized using finite-difference scheme and discrete ordinate method for the configuration and velocity space, respectively. The collision integral was approximated by either the BGK or Shakov model. The MBE solver was first verified by comparing the results to those obtained using direct simulation Monte Carlo method for a driven cavity flow at Kn=0.0033 and Ma=2.0. Simulation conditions include Knudsen number and speed of the top driven plate in the range of Kn=10-0.0033 and Ma=0.5-2, respectively. Results show that the velocity slips and temperature jumps increase at the solid walls with increasing rarefaction at the same Mach number. The vortex center move toward left and down as Knudsen number (Kn=10, 1, 0.1, 0.01) decreasing for M=0.5, 0.9, 1.1, and 2, when Kn=0.0033 is opposite. But the vortex center move toward the opposite way for M=0.5, Kn=0.0033 and M=2, Kn=0.0033. For Kn=0.01, and 0.0033, under the main vortex secondary eddies have been created at the two bottom corners. Only in this special example for M=2, Kn=10, unnder the main vortex secondary eddie have been created at the right bottom corners.