Analytical model for curved-shock Mach reflection

Mach reflection (MR) is an essential component in the development of the shock theory, as the incident shock curvature is found to have a significant effect on the MR patterns. Curved-shock Mach reflection (CMR) is not yet adequately understood due to the rotational complexity behind curved shocks. Here, CMR in steady, planar/axisymmetric flows is analyzed to supplement the well-studied phenomena caused by oblique-shock Mach reflection (OMR). The solution from the von Neumann's three-shock theory does not fully describe the CMR case. A CMR structure is presented and characterized by an incident shock, reflected shock, Mach stem, and expansion/compression waves over the slipline or occasionally an absence of waves due to pressure equilibrium. On the basis of this CMR structure, an analytical model for predicting the Mach stem in the CMR case is established. The model reduces to the OMR case if the shock curvature is not applicable. Predictions of the Mach stem geometry and shock structure based on the model exhibit better agreement with the numerical results than predictions using previous models. It is found that the circumferential shock curvature plays a key role in the axisymmetric doubly curved CMR case, which results in a different outcome from the planar case.