以高階通量限制函數之壓力修正法應用無結構性網格求解全速流流場

Abstract

本文以壓力基底法發展可執行低速不可壓縮流到高速超音速可壓縮流之全速流流場計算方法，運用有限體積法、重置變數及任意邊形之無結構性網格來離散統御方程式，為了處理震波附近陡峭的梯度變化，採用全變量消去法(TVD)或正規化變數(NV)法導出之高階對流通量限制函數，通量限制子則由偵測兩個連續的梯度比來自動調整。在梯度的計算採用二階線性重置限制修正法，以強化計算過程的穩定性及解的準確度。 本文分別使用原始變數及守恆變數作為求解變數，以原始變數求解時，壓力修正方程式係由密度變量及速度變量各別與壓力變量之關係式代入連續方程式所導出，並藉由流場局部馬赫數來自動調整橢圓型式或雙曲線型式之壓力方程式。以守恆變數求解時，壓力修正方程式只有守恆速度( )變量與壓力變量之關係式，為了模擬超音速區域之雙曲線型式之流場特性及穩定計算過程，則使用上風型式之遲滯密度或遲滯壓力作修正。 採用一些策略來增加求解過程的穩定性，諸如(1)對流項使用一階隱式上風差分混合顯式高階差分之遲緩修正法；(2)將擴散項分解為只包含相鄰網格之隱式正交項，以及非正交部份之顯式垂直導數修正項並將其置於源項中；(3)採用不同的局部時步方式，所有控容體之時步由固定的Courant數來決定，即網格較小則時步較小；(4)以鬆弛法來進行線性方程的疊代。上述的方法會導致對角係數值的擴大，因此會使得係數矩陣具有對角佔優。 本文發展之計算解子允許網格為任意多邊形，計算所用之網格可以使用不同來源的網格產生方式，在本研究中亦發展一套整合式網格介面處理程式，可以將不同方式產生之不同邊數之區塊網格予以結合，並轉換產出滿足吾人發展之計算解子所需計算域網格資料。 經由數種流場測試來驗證本文發展的方法，黏性流有(1)流經圓柱之低速流、(2)低速空穴流、(3)流經NACA 0012翼型外流場、(4)雙喉部噴嘴內流場等。非黏性流則包括(1)漸縮-漸擴噴嘴內流場、(2)流經壁面圓孤之渠道流、(3)流經 NACA 0012翼型外流場、(4)流經圓柱之高速流場、(5)流經三角柱之高速流場等。由測試結果證明，不論是原始變數或守恆變數求解方式所建構之流場解子，均能執行低速不可壓縮流到高速可壓縮流之層流流場計算，均能獲得準確的收歛解且能準確地捕捉高速流場中震波的位置及強度。

Pressure-based algorithms applicable to all-speed flows, ranging from incompressible to supersonic flows, are developed in this thesis. The finite volume method is employed for discretization. The grids, which can be of arbitrary topology, are arranged in collocated manner. To tackle the abrupt change of gradient in the region of shock, either the total variation diminishing (TVD) scheme or the normalized variable (NV) scheme can be incoporated via the use of flux limiting function. These flux limiters are determined from the ratio of two consecutive gradients. To enhance solution accuracy, the gradients are calculated using a second-order linear reconstruction approach. In this study, the mathematical formulation is based on either the primitive variables or the conservative variables. In the model using the primitive variables, a pressure-correction equation is obtained from the continuity equation by using the relations between the variations of the velocities and density and that of the pressure. The resulted equation is of mixed type, either elliptic or hyperbolic, depending on the local Mach number. The second model consider the variation of the pressure with the conserved velocities ( ). To account for the hyperbolic character of the supersonic flows, either the density or the pressure is retarded in the upwind direction. Several strategies are adopted to enhance the stability of the solution iteration procedure as follows: (1) The convective flux is composed of a upwind part and an anti-diffusion part. The upwind part is treated implicitly and the other part explicitly; (2) The diffusive flux is divided into a part in the direction directed from the considering node to the neighboring node and a part normal to this direction. The former is tackled in an implicit manner while the latter is absorbed into the source term; (3) The time step for each control volume is based on the cell Courant number. With a fixed Courant number for all control volumes, the time steps are different for the control volumes. The smaller the cell volume, the smaller the time step; (4) The difference equations are under-relaxed during iteration. The above methods can enlarged the diagonal coefficients and ,thus, make the coeffient matrix more diagonal dominant. The algorithm developed allows the control volumes of the meshes to be a polygon of arbitrary geometry. Different sources of grid generator can be adopted to generate computational meshes. An interface is developed to combine the meshes generated in different blocks using different grid generation methods and transfer the grid data into the format required by our computational code. The methodology is validated via testing on a number of flows. For viscous flows there are (1) low-speed flows over a cylinder, (2) low-speed flows in a cavity, (3) flows over a NACA 0012 airfoil and (4) flows in a double throats. In inviscid flow, test cases include (1) flows in a convergent-divergent nozzle, (2) flows in a channel with a circular arc bump, (3) flows over a NACA 0012 airfoil, (4) high-speed flows over a cylinder, (5) high-speed flows over a triangle. Accurate results can be obtained effectively using the developed methods, regardless of the use of primitive or conservative variables, for the flows ranging from the incompressible to high-speed compressible flows. It is seen that the location and the strength of the shock waves in high-speed flow can be accurately predicted.