Modelling of Mach reflections in internal axisymmetric steady supersonic flow

Abstract

A new model to predict internal axisymmetric shocks with Mach reflections at their centreline is developed, and results are compared with an Eulerian flow solver, termed CFD herein. The model combines the method of characteristics with the equations for quasi-one-dimensional flow. Calculations are conducted for a range of axisymmetric wedge geometries, and the effects of wedge length and shock angle at the wedge leading edge are studied. These various wedge geometries are found to generate flowfields with similar flow features, with Mach discs that vary greatly in size. This observation forms the basis of a method that uses the results from a CFD mesh convergence study, conducted for a single wedge geometry, to determine the mesh resolution requirements and uncertainty due to finite mesh resolution for all other wedge geometries. When uncertainties due to the effects of finite mesh resolution are considered, it is shown that either the differences between model results and CFD results for Mach disc radius are small, or the results are almost indistinguishable from one another. The model indicates that these geometries generate a flowfield that can be treated as two separate parts: one supported by the wedge surface and another supported by the sharp corner at the wedge trailing edge. The influence of these separate flowfields on the incident shock, flow properties along the slip-line, and the size of the Mach disc is assessed.

Appendix

The accuracy of the MOC code is assessed by comparison against a theoretical internal conical flow (ICF) [22, 23]. ICF can be characterised by a constant strength axisymmetric conical shock, with a downstream pointing apex, and can be predicted by solving the Taylor-Maccoll equations. A conical shock angle is selected, and the Taylor-Maccoll equations are integrated numerically starting from the conditions downstream of the conical incident shock. The solution yields a streamline, which may be viewed as a solid axisymmetric surface supporting the conical shock, and this surface is used to generate MOC predictions.

An ICF solution was generated for a freestream Mach number $M_{\mathrm{1}}=3.0$, a shock angle $\beta _{\mathrm{t}}=150^{\circ }$, and for $\gamma =1.4$. MOC calculations were then generated for $\Delta c=0.001$, 0.002, 0.004, and 0.008. The average shock angle, $\overline{\beta }$, was then compared to the theoretical value. The difference between $\overline{\beta }$ and $\beta _{\mathrm{t}}$ is plotted in Fig. 11. A generalised Richardson extrapolation method [21] was applied to results from the three finest meshes, indicating an error in shock angle of $-1.3\times 10^{-5}$deg as $\Delta c$ tends to zero.