Fully appreciating the impossibility of shock wave regular reflection from the axis of symmetry in axisymmetric internal flows

Abstract

This study investigates the reflection of shock waves in internal axisymmetric flows and conclusively shows that regular reflection cannot occur over the axis of symmetry. The analysis employs the curved shock theory to examine flow gradients behind the axisymmetric shock. The curved shock equations are presented in explicit form using influence coefficients, which establish a direct relation between flow gradients in polar coordinates and shock curvatures. The gradient information reveals that the flow behind the incident shock around the reflection point is governed by the Taylor–Maccoll equation, indicating that the flow pattern is locally conical. By analyzing the singularity of internal conical flows, the only possible structure of conical shock reflection with a smooth singularity is constructed. After a thorough theoretical analysis of this conical shock reflection structure, the study concludes with definitive proof that this specific flow pattern cannot occur in practical applications because it requires the incident shock angle to be less than the freestream Mach angle. This suggests that regular reflection is impossible to occur over the axis of symmetry in internal axisymmetric flows.