When an upstream steady uniform supersonic flow impinges onto a symmetric straight-sided wedge,governed by the Euler equations,there are two possible steady oblique shock configurations if the wedge angle is less than...When an upstream steady uniform supersonic flow impinges onto a symmetric straight-sided wedge,governed by the Euler equations,there are two possible steady oblique shock configurations if the wedge angle is less than the detachment angle—the steady weak shock with supersonic or subsonic downstream flow(determined by the wedge angle that is less than or greater than the sonic angle)and the steady strong shock with subsonic downstream flow,both of which satisfy the entropy condition.The fundamental issue—whether one or both of the steady weak and strong shocks are physically admissible solutions—has been vigorously debated over the past eight decades.In this paper,we survey some recent developments on the stability analysis of the steady shock solutions in both the steady and dynamic regimes.For the static stability,we first show how the stability problem can be formulated as an initial-boundary value type problem and then reformulate it into a free boundary problem when the perturbation of both the upstream steady supersonic flow and the wedge boundary are suitably regular and small,and we finally present some recent results on the static stability of the steady supersonic and transonic shocks.For the dynamic stability for potential flow,we first show how the stability problem can be formulated as an initial-boundary value problem and then use the self-similarity of the problem to reduce it into a boundary value problem and further reformulate it into a free boundary problem,and we finally survey some recent developments in solving this free boundary problem for the existence of the PrandtlMeyer configurations that tend to the steady weak supersonic or transonic oblique shock solutions as time goes to infinity.Some further developments and mathematical challenges in this direction are also discussed.展开更多
基金supported by the US National Science Foundation (Grant Nos. DMS0935967 and DMS-0807551)the UK Engineering and Physical Sciences Research Council (Grant Nos. EP/E035027/1 and EP/L015811/1)+1 种基金National Natural Science Foundation of China (Grant No. 10728101)the Royal Society-Wolfson Research Merit Award (UK)
文摘When an upstream steady uniform supersonic flow impinges onto a symmetric straight-sided wedge,governed by the Euler equations,there are two possible steady oblique shock configurations if the wedge angle is less than the detachment angle—the steady weak shock with supersonic or subsonic downstream flow(determined by the wedge angle that is less than or greater than the sonic angle)and the steady strong shock with subsonic downstream flow,both of which satisfy the entropy condition.The fundamental issue—whether one or both of the steady weak and strong shocks are physically admissible solutions—has been vigorously debated over the past eight decades.In this paper,we survey some recent developments on the stability analysis of the steady shock solutions in both the steady and dynamic regimes.For the static stability,we first show how the stability problem can be formulated as an initial-boundary value type problem and then reformulate it into a free boundary problem when the perturbation of both the upstream steady supersonic flow and the wedge boundary are suitably regular and small,and we finally present some recent results on the static stability of the steady supersonic and transonic shocks.For the dynamic stability for potential flow,we first show how the stability problem can be formulated as an initial-boundary value problem and then use the self-similarity of the problem to reduce it into a boundary value problem and further reformulate it into a free boundary problem,and we finally survey some recent developments in solving this free boundary problem for the existence of the PrandtlMeyer configurations that tend to the steady weak supersonic or transonic oblique shock solutions as time goes to infinity.Some further developments and mathematical challenges in this direction are also discussed.
