Evaluation of rotated upwind schemes for contact discontinuity and strong shock

Abstract

Rotated upwind schemes have been evaluated as carbuncle-free shock-capturing methods. In order to reveal the relation between the rotated upwind-differencing direction and the performance of rotated upwind schemes, theoretical and numerical investigations are made. Three commonly used upwind schemes, including Roe scheme, AUSM+ scheme and Van Leer scheme are investigated. According to the theoretical analysis, the dissipative terms of each upwind scheme are changed by the rotation procedure. Especially, the dissipative density term in the numerical flux function of rotated upwind schemes introduces extra dissipation. The numerical cases include simulations of laminar boundary layer, steady twin vortices and strong shock. The numerical results show that the carbuncle phenomenon can be eliminated by rotated upwind schemes, and the contact resolving capability of rotated upwind schemes is deteriorated by the rotation procedure. Monotonicity of the dissipation with rotation angle is also shown in the numerical results. Therefore, a novel rotation strategy that defines the rotation angle by a pressure weight function is introduced for tuning the dissipative effects adaptively. This strategy shows not only carbuncle-free result in hypersonic inviscid flow simulations, but also accurate results in viscous flow simulations.