|Abstract: ||摘要:本計劃將發展單一化分子力學算則(unified kinetic scheme)，適用於自由分子流至連續流 域的全Knudsen 流(all Knudsen flows)模擬。重要特點是在有限體積法的架構下，將BGK 模型 方程式的積分解，用於計算網格界面通量，亦即同時考量分子傳輸(transport)與碰撞 (collision)，重建數值解函數。可以視為一種動態複合算則(dynamic hybrid method)，由單一分 子力學方程式積分解的極限計算各種流域的流場參數，不同於一般複合算則將不同的控制方 程式分別應用到各流域的方法。因此稱為Unified BGK 算則或簡稱U-BGK 算則。U-BGK 算 則擴展BGK-NS 算則的適用流域，將分子速度空間離散，以分立座標法處理，適用於連續流 域黏性流與稀薄流域計算。U-BGK 算則可以採用比分子碰撞時間大的計算時步，獲得準確的 黏性流(Navier-Stokes)的解。也可以準確的獲得非平衡稀薄流或是自由分子流的解。由於 U-BGK 單一化算則本身具有多尺度應用的特性，在同一計算時步內耦合計算巨觀守恆量與微 觀分佈函數，且透過巨觀流場參數的計算，可以利用正確的Prandtl 數修正熱通量。整體而言， 可以證明U-BGK 算則是一種分子力學BGK 方程式的AP(asymptotic preserving)算則。 本計畫將發展一系列多維度分子力學算則U-BGK，全計畫分三年，逐年完成各算則發 展：(1)建立多維度U-BGK 算則，(2)高階Shakhov 與Ellipsoidal 模型算則U-Shk 與U-ES，(3) 多成分氣體化學反應分子力學算則，(4) 混合氣體分子力學算則，(5)內旋轉自由度之雙原子 氣體分子力學算則等，並應用於自由分子流至連續體流全Knudsen 數流場模擬。驗證測試算 例包括再入飛行器(re-entry vehicles)氣動熱力學分析、微結構流(微流道、微噴嘴)模擬等。 預期本計畫完成後將充分建立自由分子流至連續流模擬，與低次音速至極超音速氣動力 學分析的技術能量。本項新開發技術也可以推廣應用於衛星姿態控制噴嘴、真空泵浦、化學 氣相沉積、以及微機電系統等科學及工業上。在太空計畫、國防科技與產業發展等均能有所 助益。|
Abstract:The purpose of the project is to develop a class of unified kinetic schemes which will be applied to all Knudsen flows, from free molecular to continuum flow regimes. The validity of the approach is based on its full representation of particle movement, i.e., transport and collision. Different from many other approaches, the critical step is that the integral solution of kinetic model (BGK equation) is used in the flux evaluation across the cell interface. The present scheme can be considered as a dynamic hybrid method, where the different flow behavior is obtained through the different limits of the integral solution of a single kinetic equation, instead of solving different governing equations in different flow regimes. The present scheme is named unified BGK scheme, or U-BGK scheme. U-BGK scheme is an extension of the gas-kinetic BGK-NS scheme from the continuum flow to the rarefied regime with the discretization of particle velocity space by discrete ordinate method. In comparison with many existing kinetic schemes for the Boltzmann equation, the current method has no difficulty to get accurate Navier–Stokes (NS) solutions in the continuum flow regime with a time step being much larger than the particle collision time. At the same time, the rarefied flow solution, even in the free molecule limit, can be captured accurately. U-BGK scheme couples closely the update of macroscopic conservative variables with the update of microscopic gas distribution function within a time step. Due to its multiscale nature of the unified scheme, through the update of macroscopic flow variables the heat flux of the scheme can be modified according to correct Prandtl number. The period for this project is three years. The working iterms included are (1) the multi-dimensional U-BGK scheme, (2) the U-Shk and U-ES scheme based on Shakhov and Ellipsoidal model equation, respectively, (3) the unified kinetic scheme for muti-materials chemical reaction flows, (4) the unified kinetic scheme for mixing gas, (5) the unified kinetic scheme for diatomic gas with internal degree of freedom of rotation. Many numerical tests will be used to validate the unified method. The aerodynamics of re-entry vehicles and flow characteristics of micro-structures are studied. It is expectable that the new methods developed in this project can be used in many technological applications, like as altitude control nozzle of satellite, sounding rocket, tactical ballistic missile, vaccum pump, chemical vapor deposition in semiconductor manufacturing, and micro electro mechanical system. The results of this project will contribute to native space project, defence technology and industry.