Analytical model for curved-shock Mach reflection

Abstract

Mach reflection (MR) is an essential component in the development of 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 von Neumann's three-shock theory does not fully describe the CMR case. A CMR structure is presented, 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.

A study of streamline geometries in subsonic and supersonic regions of compressible flow fields

“…4.1.3. Second-order properties of streamlines Following the idea of the second-order CST (Shi et al 2020), equations about the second-order properties of streamlines, κ and h , could also be obtained by taking the directional derivatives of (4.15) and replacing the η-derivatives with (3.25). The mathematical derivations are a little tedious, see Appendix C. Only the major results are outlined here.…”

“…Based on the curvatures solved from the curved-shock equations, coordinates of streamlines can be approximated by a Taylor series function (Mölder 2017;Filippi & Skews 2018;Surujhlal & Skews 2018). The second-order CST was proposed by Shi et al (2020) and the accuracy of predictions was improved. To further improve the accuracy, Shi et al (2021) developed a method of curved-shock characteristics (MOCC) for analysis and inverse design in supersonic flow fields.…”

“…In their method, the downstream of the shock wave was precisely solved by a modified method of characteristics, where the boundary conditions were given by the gradients from CST. Many flow patterns have been studied with MOCC, showing significant improvements in the accuracy, efficiency and adaptability (Cheng et al 2022;Shi et al 2023;Zhang et al 2023).…”