平行化三維DSMC-NS偶合數值方法之發展及驗證

Abstract

本論文提出一個兼顧效率及準確度的平行化三維非結構性網格之直接蒙地卡羅法與Navier-Stokes偶合數值方法。本文中針對幾個同時涉及連續體及稀薄流域之高速氣流場進行模擬，以驗證該偶合數值方法之正確性。藉由使用直接蒙地卡羅法進行詳盡的分子動力學研究重新檢視，在靠近固體邊界區連續流假設失效參數的使用，並且提出新的連續體假設失效與否的新標準。以下，將本論文的工作分成兩個階段分別來介紹。 第一階段，本論文先提出一個平行化、使用三維非結構性網格的直接蒙地卡羅法與Navier-Stokes偶合數值方法。本方法是反覆地根據兩個連續體失效參數來決定在空間上各個區域其數值方法的選擇。利用非結構性網格的優點，採用數值方法區域重疊 (Domain Overlapping)策略，流場性質在邊界上的交換則採用固定式的邊界交換形式(Dirichlet-Dirichle type)。在本論文中所採用的兩個連續體失效參數包括: 1)區域性最大紐森數(local maximum Knudsen number)，其定義為，氣體分子平均自由徑除以流體性質梯度所定義之特徵長度，以及 2)熱力平衡指標數，其定義為: 移動動能分布溫度與旋轉動能分布溫度的差值除以移動動能分布溫度。為了驗證本論文所提出來之偶合方法，首先我們針對超音速(M□=4)氣流流過一個近似二維的25度角之楔形體作為驗證的第一個算例。在這個驗證算例中，未經偶合的直接蒙地卡羅法視為是對照用的正確解。利用本文提出之偶合方法跟未經偶合的直接蒙地卡羅法的比較結果極為吻合。關於數值方法重疊區域 (Overlapping Region) 的大小以及連續體失效參數標準大小的選用，對於這個偶合方法收斂性的影響也在本論文有所討論。這個算例結果顯示，相較於實務上常被使用的一次交換邊界的偶合 (One-shot coupled) 方法，目前本論文提出的反覆進行數值方法疊代之偶合數值方法可以更正確地預測流場結果。更進一步，將使用本論文目前所提的偶合數值方法，針對兩個真實三維氮氣場算例進行模擬：第一個是擴散到幾乎真空的環境的平行雙孔噴流，第二個則是衛星姿態控制器(Reaction control system)的噴嘴形成的氣流場，而模擬結果將與現有的實驗結果進行比較。 為了重新檢驗過去研究工作所建議的連續體失效參數的判斷標準，本論文的第二階段將以直接蒙地卡羅法，針對超音速氣流流經一個有限長度的楔形體的算例，進行詳盡的分子動力學研究。選用該流場作為分子動力學分析的原因是本算例的流場同時存在銳緣區(leading edge)、邊界層區、斜震波區以及扇形擴散區(expanding fan)，這些流場區域涵蓋了對於目前直接蒙地卡羅法與Navier-Stokes偶合數值方法模擬所遭遇的大多數關鍵性之流場現象。文中在流場各個不同特定位置上，針對卡氏坐標系之三個方向速度進行採樣，並與該位置所對應的馬斯威爾-波茲曼速度分布進行分析比較。此外，亦計算在這些位置上的各個不同溫度自由度之間熱力不平衡的程度。為了有效定義各種不同溫度自由度(這包括卡氏座標三個不同方向上的移動動能分布溫度、旋轉動能溫度以及振動動能溫度)之間熱力不平衡的程度，本文進一步重新定義一個更具一般性的熱力平衡指標數，並提出以0.03為其對應之熱力平衡臨界值。據結果顯示可發現，使用過去相關研究所建議的區域性最大紐森數0.05作為判定是否連續體失效標準，對於在邊界層區域裡連續體失效程度的判斷可能會過於高估。本論文提出針對於靠近固體邊界區域，採用與其他區域不同的區域性最大紐森數判斷標準0.8，作為該區域是否發生連續體失效之判定。

In this thesis, an efficient and accurate parallel coupled DSMC-NS method using three-dimensional unstructured grid topology is proposed and verified for the simulation of high-speed gas flows involving continuum and rarefied regimes. In addition, breakdown parameters near the solid wall are reinvestigated by a detailed kinetic study and a new criterion of breakdown parameter is proposed. Research in this thesis is divided into two phases, which are described in the following in turn. In the first phase, a parallel coupled DSMC-NS method using three-dimensional unstructured grid topology is proposed and verified. A domain overlapping strategy, taking advantage of unstructured data format, with Dirichlet-Dirichlet type boundary conditions based on two breakdown parameters is used iteratively to determine the choice of solvers in the spatial domain. The selected breakdown parameters for this study include: 1) a local maximum Knudsen number defined as the ratio of the local mean free path and local characteristic length based on property gradient and 2) a thermal non-equilibrium indicator defined as the ratio of the difference between translational and rotational temperatures to the translational temperature. A supersonic flow (M□=4) over a quasi-2-D 25□ wedge is employed as the first step in verifying the present coupled method. The results of simulation using the coupled method are in excellent agreement with those of the pure DSMC method, which is taken as the benchmark solution. Effects of the size of overlapping regions and the choice of breakdown parameters on the convergence history are discussed. Results show that the proposed iteratively coupled method predicts the results more accurately as compared to the “one-shot” coupled method, which has been often used in practice. Further, two realistic 3-D nitrogen flows are simulated using the developed coupled DSMC-NS scheme. The first one is a flow, which consists of two near-continuum parallel orifice jets underexpanding into a near-vacuum environment. The second one is a flow issuing from a typical RCS (Reaction Control System) thruster. Results are compared with experimental data wherever is available. In the second phase, a detailed kinetic investigation using the DSMC method is used to re-examine previous proposed criteria of continuum breakdown parameters by studying the supersonic flow past a finite-size wedge flow. Choice of this particular flow for kinetic study lies in the fact that it includes a leading edge, a boundary layer, an oblique shock and an expanding fan, which covers most critical flow phenomena for the present hybrid DSMC0-NS method. Velocities of three Cartesian directions at various critical locations in the flow field are sampled and compared with the corresponding local Maxwell-Boltzmann distribution. In addition, degree of thermal non-equilibrium among various degrees of freedoms is computed at these locations. To efficiently indicate the degree of thermal non-equilibrium among various degrees of freedoms, a general indicator of thermal non-equilibrium, with its threshold value 0.03, is used with the consideration of temperature deviations among different temperature modes, including translational temperatures in the three Cartesian directions, rotational temperature and vibrational temperature. From the results, it is found that the degree of the continuum breakdown in the boundary-lay region is overestimated with the previously recommended threshold value of 0.05. Revised criterion near the isothermal solid wall is proposed as 0.8 in the present study.