Reformulated dissipation for the free-stream preserving of the conservative finite difference schemes on curvilinear grids

In this paper, we develop a new free-stream preserving (FP) method for high-order upwind conservative finite-difference (FD) schemes on the curvilinear grids. This FP method is constrcuted by subtracting a reference cell-face flow state from each cell-center value in the local stencil of the original upwind conservative FD schemes, which effectively leads to a reformulated dissipation. It is convenient to implement this method, as it does not require to modify the original forms of the upwind schemes. In addition, the proposed method removes the constraint in the traditional FP conservative FD schemes that require a consistent discretization of the mesh metrics and the fluxes. With this, the proposed method is more flexible in simulating the engineering problems which usually require a low-order scheme for their low-quality mesh, while the high-order schemes can be applied to approximate the flow states to improve the resolution. After demonstrating the strict FP property and the order of accuracy by two simple test cases, we consider various validation cases, including the supersonic flow around the cylinder, the subsonic flow past the three-element airfoil, and the transonic flow around the ONERA M6 wing, etc., to show that the method is suitable for a wide range of fluid dynamic problems containing complex geometries. Moreover, these test cases also indicate that the discretization order of the metrics have no significant influences on the numerical results if the mesh resolution is not sufficiently large.