Instability of Strong Regular Reflection and Counterexamples to the Detachment Criterion

Abstract

We consider a particular instance of reflection of shock waves in self-similar compressible flow. We prove that local self-similar regular reflection (RR) cannot always be extended into a global flow. Therefore the detachment criterion is not universally correct. More precisely, consider the following angle condition: the tangent of the strong-type reflected shock meets the opposite wall at a sharp or right downstream side angle. In cases where the condition is violated and the weak-type reflected shock is transonic, we show that global RR does not exist. Combined with earlier work we have shown that none of the classical criteria for RR$\rightarrow$MR transition is universally correct. A new criterion is proposed. Moreover, we have shown that strong-type RR is unstable, in the sense that global RR cannot persist under perturbations to one side. This yields a definite answer to the weak-strong problem because earlier work shows stability of weak RR in the same sense.