Aerodynamics simulations of three-dimensional inviscid flow using curvilinear discontinuous Galerkin method on unstructured meshes

Abstract

Over the last decades, the discontinuous Galerkin (DG) method has demonstrated its excellence in accurate, higher-order numerical simulations for a wide range of applications in aerodynamics simulations. However, the development of practical, computationally accurate flow solvers for industrial applications is still in the focus of active research, and applicable boundary conditions and fluxes are also very important parts. Based on curvilinear DG method, we have developed a flow solver that can be used for solving the three-dimensional subsonic, transonic and hypersonic inviscid flows on unstructured meshes. The development covers the geometrical transformation from the real curved element to the rectilinear reference element with the hierarchical basis functions and their gradient operation in reference coordinates up to full third order. The implementation of solid wall boundary conditions is derived by the contravariant velocities, and an enhanced algorithms of Harten-Lax-van Leer with contact (HLLC) flux based on curved element is suggested. These new techniques do not require a complex geometric boundary information and are easy to implement. The simulation of subsonic, transonic and hypersonic flows shows that the linear treatment can limit the accuracy at high order and demonstrates how the boundary treatment involving curved element overcomes this restriction. In addition, such a flow solver is stable on a reasonably coarse meshes and finer ones, and has good robustness for three-dimensional flows with various geometries and velocities. For engineering practice, a reasonable accuracy can be obtained at reasonably coarse unstructured meshes.