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Title: | 半古典波茲曼方程與尤拉方程耦合之稀薄流模擬 A Semi-Classical Kinetic and Hydrodynamic Coupling Method for Rarefied Flow Computation |
Authors: | Yun-Da Tsai 蔡昀達 |
Advisor: | 楊照彥(Jaw-Yen Yang) |
Keyword: | 尤拉方程,半古典波茲曼BGK,緩衝區,耦合法,離散座標法,全變量消逝法, Euler,Semi-Classical Boltzmann BGK,Buffer Zone,Couple Method,Discrete Ordinate Method,Total Variation Diminishing, |
Publication Year : | 2013 |
Degree: | 碩士 |
Abstract: | 針對稀薄流問題求解時,通常選擇波茲曼BGK模型來求解。但當BGK的鬆弛時間很小時,也就是當碰撞頻率變成很大時將會造成BGK方程的計算成本非常巨大。因此本文主要在研究當流場遇到此類型問題時,能透過適當的機制將巨觀Euler與微觀的BGK做轉換以達到節省計算成本的目的。
本文利用直接解數值方法來求解半古典波茲曼方程與Euler方程組成的耦合系統以模擬稀薄流。數值方法方面使用離散座標法(D.O.M.)與迎風法(Upwind),並加入高解析算則如全變量消逝法(Total Variation Diminishing, TVD)。 在傳統耦合模型求解稀薄流時。常使用介面邊界條件來轉換。但此方法必須添加額外的邊界條件,在維度拓展到二維時會非常複雜。因此耦合方法選擇加入緩衝區的設定,透過此緩衝區的設定能平滑的將巨觀與微觀尺度做連結。尺度轉換的關鍵在於緩衝區的設計,因此本文也針對三種不同類型的緩衝區做尺度轉換的比較。三種緩衝區分別是線性緩衝區、餘弦緩衝區、雙曲正切函數的緩衝區。 做耦合模型時需要求兩種分布函數。一種為BGK系統下的分布函數,另一種則為巨觀尺度下所求得的分布函數-平衡態分布函數。透過此兩種分布函數相加可得到整體系統的分布函數,進而得到系統的巨觀物理量。使用耦合尺度來求解震波管問題,可成功透過緩衝區捕捉震波、膨脹波。然而,計 顯示緩衝區大小若低於無因次長度0.25時模擬效果較容易出現不守恆量,而當緩衝設定位置遠離震波時三種型態的緩衝區就無明顯分別。 For targeting flows problem with different degrees or regimes of rarefaction, we can rely on Boltzmann-BGK model to find tractable method to solve this kind of problems [15] [17] [19]. But it is also well known that when collision source term becomes ‘large’ , i.e. the collisions become more frequent, however, its computational cost to use BGK model will increase. Hence, the primary goal of this work is to introduce an appropriate strategy to combine an Euler solver and a BGK solver and achieve cost reduction. In this work we follow and look for ways to improve upon ideas of previous approaches [1] [2]. We also extend the method to semiclassical Boltzmann-BGK equation. The numerical method is based on discrete ordinate method to approach the recovery of macroscopic quantities in velocity space, and upwind method, high-resolution methods such as TVD, to evolve information in physical space. Traditional coupled methods [3], rely on boundary conditions for the coupling two different hydrodynamic solvers, This strategy becomes more complicated when space domain expands to 2-dimensional problems. In this work, the idea is to use a transition region called buffer zone, in which, both semiclassical hydrodynamic and kinetic equations will be solved simultaneously. A buffer zone using three types of cut functions was designed and tested, namely a linear transition, trigonometric transition and hyper-tangent transition function were used. For our coupling method, two distribution functions are employed. Through the sum of both two distribution functions, we can recover total distribution function for the whole system and therefore recovered physical quantities. Applying the present couple method to solve sod’s problem, it is shown that shock wave and expansion wave can all be captured successfully. However, our results also show that there is a minimum dimensionless length of 0.25 for the buffer zone to work properly. If the buffer zone is far away from the shock region, results from the three transition functions show no differences. |
URI: | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/62218 |
Fulltext Rights: | 有償授權 |
Appears in Collections: | 應用力學研究所 |
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ntu-102-1.pdf Restricted Access | 8.07 MB | Adobe PDF |
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