Conservative high-order finite-difference cut-cell approach for shock-dominated flows

Cut-cell methods provide an efficient framework to handle practical geometries using structured grids. They are widely used for fluid flow calculations; however, their application to high-fidelity direct numerical or large-eddy simulations is limited by the stability, conservation, and accuracy issues at the cut-cell boundary. In this study, we apply energy stability concepts to derive stable cut-cell boundary closures for supersonic/hypersonic flows that involve shock-turbulence interactions. High-resolution computations of these flows require upwind (or biased) schemes around flow discontinuities and non-dissipative centered schemes in the smooth flow regions. Cut-cell boundary closures are, therefore, derived for an adaptive central-upwind sixth-order weighted essentially non-oscillatory (WENO) scheme, where the solutions are stored and advanced at the grid points (or the cell centers), while the fluxes are computed at cell interfaces (or the flux points) to ensure conservation by construction. The small-cell problem when the cut cell adjoining the embedded boundary is significantly smaller than a regular cell is addressed by choosing the flux point spacings that do not vanish even when the grid points at the embedded boundary coincide. This procedure avoids additional steps, e.g. cell mixing/merging, flux redistribution, etc., that are difficult to automate. The derived scheme is dimensionally-split and, hence, does not require geometry/solution reconstructions, making it easy to incorporate into an existing solver. Various linear and non-linear, inviscid and viscous numerical tests are performed to demonstrate the stability and accuracy of the proposed method.