High-order accurate kinetic-energy and entropy preserving (KEEP) schemes on curvilinear grids

High-order accurate kinetic energy and entropy preserving (KEEP) schemes in generalized curvilinear coordinates are proposed for stable and non-dissipative numerical simulations. The proposed schemes are developed on the basis of the physical relation that the fluxes in the Euler equations in generalized curvilinear coordinates can be derived by taking the inner product between the inviscid fluxes and the area vectors used for the coordinate transformation. To satisfy this physical relation discretely, this study proposes to interpret the area vector components as another individual variable and discretize the area vectors in the same way as other physical variables, such as the density and velocity. Consequently, the convective and pressure-related terms are discretized in a new split convective form, “quartic split form”, and quadratic split form, respectively. The high-order extension is straightforward, referring to the high-order formulations proposed for kinetic energy preserving schemes in a previous study. Numerical tests of vortex convection, inviscid Taylor-Green vortex, and turbulent boundary layer flow are conducted to assess the order of accuracy, the kinetic energy and entropy preservation property, and numerical robustness of the proposed KEEP schemes. The proposed high-order accurate KEEP schemes successfully perform long-time stable computations without numerical dissipation by preserving the total kinetic energy and total entropy well, even on a largely-distorted computational grid.