Improvement to a general methodology for free-stream preservation on curvilinear grids

Exact free-stream preservation is an important property for finite-difference schemes designed on curvilinear grids. Geometrically induced errors from a non-preserved free-stream will cause severe computational instability and numerical inaccuracy. We improve a general numerical strategy to eliminate the geometrically induced errors of finite-difference schemes on both stationary and dynamic curvilinear grids. The main idea is that, instead of using the standard Euler equations in the generalized coordinate system as the governing equations, we solve full forms of the transformed equations and modify the flux functions to share the Jacobian and metrics on the grid point where the flux derivative is located. The metrics and Jacobian are grid-related geometric parameters that appear in the transformed equations as derivatives. Properly handling these is critical for free-stream preservation. The results of our numerical tests show the excellent free-stream and vortex preservation properties and robust shock-capturing properties of the new strategy compared with those of the standard method.