Pressure-Based Unstructured-Grid Algorithms Incorporating High-Resolution Schemes for All-speed Flow Calculations
|關鍵字:||壓力修正法;可壓縮流;通量限制函數;全速流;特徵變數;遲滯密度;遲滯壓力;正規化變數圖;無結構性網格;pressure correction scheme;compressible flow;flux limiter function;all speed flow;characteristic variable;retarded density;retarded pressure;normalized variable diagram;unstructured grid|
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.