Self-similar structures based genuinely two-dimensional Riemann solvers in curvilinear coordinates

Abstract

Extension of the self-similar structures based genuinely two-dimension Riemann solver called MuSIC (Multidimensional, Self-similar, strongly-Interacting, Consistent) to curvilinear coordinate systems are conducted in this study. Built upon Balsara's work, the MuSIC1 (MuSIC considering the 1st order moment) scheme solves the compressible Euler equations in curvilinear coordinates by considering the first moment in the similarity variables, and the MuSIC2 (MuSIC considering the 2nd order moment) scheme considers the second moment in the similarity variables. Systematic numerical test cases are conducted. One dimensional cases indicate that both MuSIC1 and MuSIC2 are capable of accurately capturing one-dimensional shocks and expansion waves. Also, MuSIC2 is with a higher resolution than MuSIC1 in capturing linear contact discontinuities because it considers the linear variation of similarity variables in strongly-interacting zones. Two dimensional isotropic case shows that the self-similar structures based genuinely two-dimensional Riemann solvers can help improve the traditional one-dimensional Riemann solvers' mesh imprinting phenomenon remarkably. The other two-dimensional cases show that MuSIC2 is with a high resolution in simulating multidimensional complex flows in both Carestein coordinates and curvilinear coordinates, while MuSIC1 performs worse due to its lack of accuracy in resolving linear variation of similarity variables in resolved states.