Pilot Study Of Discontinuous Galerkin Methods For Euler Equations

The second order accurate flow solvers are more and more helpless in resolving problems of increasing complexity. It also can't satisfy the needs of deeply recognizing the inbeing of flow. Results from AIAA CFD drag prediction works suggest that the order of present flow solvers can't satisfy the needs of many applications, especially in computing drag. It becomes important to develop high-order accurate methods. The discontinuous galerkin method (DGM) is one of those methods. It combines two advantages features commonly associated to finite element and finite volume methods. DGM achieves high-order accurate through increasing the number and degree of polynomial within an element. It can be used on unstructured grids and easily handle complex geometries and boundary.As the first step of DGM, this thesis researches the numerical method without considering viscosity. The second and third order DGM are developed on unstructured grids for the Euler governing equations. For the particularity of tetrahedron, the first and second level polynomial approximations are constructed using volume coordinate. It avoids solving the element-Jacobi matrix and improves the efficiency. The mass matrix of the first level polynomial is a diagonal matrix, so it is easy to get the degrees of freedom. Many non-diagonal elements of the second level polynomial are also zero. Because DGM is sensitive to the treatment and implementation of limiters, the discontinuity detector is developed to distinguish"troubled cell"and only the"troubled cells"are limited. The hermite weighted essentially non-oscillatory (HWENO) limiter is developed using weighted essentially non-oscillatory finite volume methodology. The explicit two-stage second-order Runge-Kutta scheme and the implicit LU-SGS scheme are used.The DGM is validated by M6 wing from flux scheme, grid, discontinuity detector, limiter and different order. The computed pressure coefficient distributions are compared with experimental data at six spanwise stations. The results compare closely with experimental data except at the root station, due to lack of viscous effects. The DGM results are compared with FV results under the same order and grids. The comparison results suggest the DGM is more perfect. Results under different orders are also compared, the high-order results on coarse grid is the same as the low order results on fine grids.The ability of computing complex geometries is qualitative validated by the wing-body configuration F4 and YF-16. The position of shock wave on the wing of F4 is behind the experimental data and reference data. It accords with the rule. The flowfiled of YF-16 are computed in subsonic, transonic, supersonic cases. The performance suggests the fine computation accuracy and the good robust.