Grid-optimized upwind DRP finite difference scheme on curvilinear grids for computational aeroacoustics

Based on the curvilinear grids, grid-optimized upwind dispersion-relation-preserving (GOUPDRP) finite difference schemes are studied in the paper. GOUPDRP finite difference schemes can automatically resolve waves with much shorter wavelengths and need not consider the filter or the explicit dissipation terms, and have much smaller dispersive and dissipative errors. Recently, GOUPDRP schemes on the non-uniform Cartesian meshes have been developed and tested by authors of this paper, others optimized upwind finite schemes just only on the uniform Cartesian grids also have been investigated; nevertheless, practical problems existing in aeroacoustics are unusually restricted to the uniform or non-uniform Cartesian grids, the corresponding computational grids could be curvilinear ones owing to the different complex geometries. In contrast to other optimized upwind schemes on the uniform Cartesian grids, GOUPDRP schemes can be easily extended to the curvilinear meshes. In order to determine the optimization coefficients of GOUPDRP schemes on the curvilinear grids, a general curvilinear transformation is introduced from physical domain to computational domain. Mathematic formulations related to GOUPDRP schemes on the curvilinear meshes are also derived in detail in the paper. To demonstrate the effectiveness of GOUPDRP schemes, benchmark problems are solved using the curvilinear grids; calculated solutions are compared with their analytical counterparts. Numerical results show that GOUPDRP schemes on the curvilinear grids are highly efficient and relatively potential to solve practical problems in aeroacoustics.