Simulation For The Two-dimensional Euler Equations On Adaptive Unstructured Grids

A method for the generation and remeshment of unstructured grids is presented. The grids are suitable for the simulation of two-dimensional Euler equations. In this method, point positions and connections are computed simultaneously. The approach with which the new points are positioned is based on the ideas that are commonly used in the advancing front techniques to ensure grid quality around the boundary. Utilizing Delaunay criterion, the corresponding connections are created to make automatic triangulation easy. A new method for constructing the background grid is also presented. In order to ensure the adequate resolution of the flow feature, the circumscribed radius for the final triangulation at certain point is prescribed by employing background grid. So every time a new point needs to be positioned, the background grid is searched. Thus the optimization of the searching operation is crucial for improving the efficiency of the overall grid generator. Here quadtree structure is introduced to organize background grid. Its introduction makes it possible to find the scale for any given point by checking several background cells. Information about the centroid position of every element is also stored on the node of the quadtree. It is helpful for finding the element that a given point lies in. Different data structures are designed for grid generating and gas dynamics computing. While the grid is being generated, grid elements should be added or deleted often. So dynamic structure is adopted for its flexibility. During the gas dynamic computation, the grid is unchanged. In order to improve the efficiency, static structure is adopted. Next, methods for the calculation of the numerical conservative fluxes passing through the zone edge are given. With the velocity tangent to the zone edge being treated as a passively advected quantity, the fluxes are obtained by solving Riemann problem for the projected equations along the normal to that zone edge. The methods yield accurate and stable results even with subsonic isentropic flow or strong shocks. A new monotonicity constraint condition are presented as a complementary to that introduced by Collella and Woodward[47]. This new condition subjects the average gradients in a cell to the gradients in the adjacent cells. Tests on the shocktube problem show that it performs well near the shock and does not change theaccuray in the smooth flow.