Pressure-Based Unstructured-Grid Algorithms Incorporating High-Resolution Schemes for All-speed Flow Calculations

• Main Authors: Wu, Tian-Cherng, 吳添成
• Other Authors: Tsui, Yeng-Yung
• Format: Others
• Language: zh-TW
• Published: 2009
• Online Access: http://ndltd.ncl.edu.tw/handle/54910872095858014200

博士 === 國立交通大學 === 機械工程系所 === 97 === 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.