A genuinely two-dimensional Riemann solver for compressible flows in curvilinear coordinates

A genuinely two-dimension Riemann solver for compressible flows in curvilinear coordinates is proposed. Following Balsara's idea, this two-dimension solver considers not only the waves orthogonal to the cell interfaces, but also those transverse to the cell interfaces. By adopting the Toro–Vasquez splitting procedure, this solver constructs the two-dimensional convective flux and the two-dimensional pressure flux separately. Systematic numerical test cases are conducted. One dimensional Sod shock tube case and moving contact discontinuity case indicate that such two-dimensional solver is capable of capturing one-dimensional shocks, contact discontinuities, and expansion waves accurately. Two-dimensional double Mach reflection of a strong shock case shows that this scheme is with a high resolution in Cartesian coordinates. Also, it is robust against the unphysical shock anomaly phenomenon. Hypersonic viscous flows over the blunt cone and the two-dimensional Double-ellipsoid cases show that the two-dimensional solver proposed in this manuscript is with a high resolution in curvilinear coordinates. It is promising to be widely used in engineering areas to simulate compressible flows.