基於氣體動力論之稀薄流研究

The study based on kinetic theory for rarefied gas flows

洪鉦杰(Jeng-Jye Hung)

指導教授 ： 楊照彥

國立臺灣大學/工學院/應用力學研究所/博士(2006年)

https://doi.org/10.6342/NTU.2006.10537

摘要

當氣體的稀薄度提高後，巨觀連體模型的描述方法會失效，此時只能以微觀的分子動力模型來描述。在此微觀模型下，波茲曼方程式為其統御方程式，用以描述分佈函數的變化。在本文中，將以模擬分子運動的一種方法來求解稀薄流場的問題，此方法為直接模擬蒙地卡羅法(DSMC)。
關於直接模擬蒙地卡羅法，其主要的關鍵是：在一個足夠小的時間步前提下，分子本來耦合在一起的運動與碰撞行為，得以分開處理；並用機率的精神來處理分子的碰撞。本文發展出一個泛用型DSMC計算程式，對於對應不同稀薄度(Kn=0.001、0.01、0.1、1)的二維圓柱流場，做了一個完整的模擬計算，藉由觀察震波等流場結構及阻力係數的計算，與連體模型下那維爾-史托克方程式的解及實驗結果相比較，得到很好的結果。並且針對不同馬赫數下的NACA0012二維翼形之稀薄流場問題作模擬，將模擬結果與BGK模型方程式的解及實驗結果相比較，亦得到很高的一致。透過前述問題的模擬結果，可證明本計算程式的正確性。
文中的泛用型DSMC計算程式其計算能力可處理三維流場，是而本文中將研究三維三角翼稀薄流場問題。三角翼是一個具有大後掠角的三角形翼面，可用來作為飛行體的主升力面。從不同截面的巨觀特性等高線圖、翼表面壓力係數的分佈以及升阻力係數的模擬結果，討論稀薄效應對其流場的影響，可得知在稀薄效應影響下，三角翼在連體模型之高雷諾數下，其背風面上使機翼產生非線性升力的渦旋並不會出現。

關鍵字

稀薄流 ； 直接模擬蒙地卡羅法 ； 三角翼

並列摘要

When the degree of rarefaction of gases increases, the description of continuum model in macroscopic level become invalid and the use of microscopic or molecular model to describe the gas flow is necessary. The mathematic model at the microscopic level is Boltzmann equation. It governs the behavior and evolution of the gas distribution function. In this study, a particle simulation method was chosen to solve the problems of rarefied gas flow, namely, the Direct Simulation Monte Carlo method (DSMC) pioneered by Bird. The key of DSMC method to solve the Boltzmann equation is that the coupled behavior between molecular translation and collision can be decoupled when the time step is small enough, and a process of probability is employed to deal with intermolecular collisions.

In this study, a common used DSMC simulation program has been adopted and developed for studying general two- and three-dimensional rarefied gas flows. First, the problem of flow past a two-dimension cylinder with various degree of rarefaction (Kn=0.001、0.01、0.1、1) has been simulated. By observing the structures of various flow fields and the value of drag coefficient, the DSMC results are found in good agreement with the results of Navier-Stokes calculation and available experiments. Second, the problems of flow past a two-dimensional NACA0012 airfoil covering several Mach numbers and Knudsen numbers have been simulated. Compare with both the results of BGK model equation and experiment, good agreement in every case is obtained. These two cases validate the present DSMC code.

Finally, the developed simulation program in this study is extended to simulate the three-dimension flow field and the problems of gas flow past a delta wing at various degrees of rarefaction were studied here. A delta wing is a triangular airfoil of high sweepback angle, and it can offer aircrafts higher lift. The effects of rarefaction to flow field of flow past a delta wing were investigated through the results of the various macroscopic variables contours, the distribution of pressure coefficient on wing surface at different sections, and the value of lift and drag coefficients. Both high resolution Navier-Stokes solutions and DSMC solution are given and compared. It is found that the vortex which appears at leeside of delta wing at high Reynolds number from using continuum model and can offer an additional nonlinear lift may not appear when the degree of rarefaction increases to certain extent.

並列關鍵字

Rarefied Gas Flow ； DSMC ； Delta Wing

參考文獻

[108] 湯國樑 (2005), “波茲曼模型方程式之高解析數值方法,” 台灣大學應用力學研究所 博士論文

[1] Agrawal, Shreekant, Barnett, R. M., and Robinson, B. A. (1992), “Numerical Investigation of Vortex Breakdown on a Delta Wing,” AIAA Journal, Vol. 30, No. 3, pp. 584-591.

[2] Alder, B. J. and Wainwright, T. E. (1957), “Studies in Molecular Dynamics,” J. Chem. Phys., Vol. 27, pp. 1208-1209.

[3] Alexander, F. J., Garcia, A. L., and Alder, B. J. (1994), “Direct Simulation Monte Carlo for Thin-Film Bearings,” Physics of Fluids, Vol. 6, Issue 12, pp. 3854-3860.

[4] Alexeenko, A. A., Gimelshein, S. F., Muntz, E. P., and Ketsdever, A. (2005), “Modeling of Thermal Transpiration Flows for Knudsen Compressor Optimization,” AIAA Paper, 2005-963.