Abstract
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 Prandtl-Meyer 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.
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Acknowledgements
This work was supported by the US National Science Foundation (Grant Nos. DMS- 0935967 and DMS-0807551), the UK Engineering and Physical Sciences Research Council (Grant Nos. EP/E035027/1 and EP/L015811/1), National Natural Science Foundation of China (Grant No. 10728101), and the Royal Society-Wolfson Research Merit Award (UK). The materials presented in this paper contain direct and indirect contributions of my collaborators Myoungjean Bae, Jun Chen, Beixiang Fang, Mikhail Feldman
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Dedicated to Professor LI TaTsien on the Occasion of His 80th Birthday
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Chen, GQ. Supersonic flow onto solid wedges, multidimensional shock waves and free boundary problems. Sci. China Math. 60, 1353–1370 (2017). https://doi.org/10.1007/s11425-016-9045-1
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DOI: https://doi.org/10.1007/s11425-016-9045-1
Keywords
- shock wave
- free boundary
- wedge problem
- transonic
- mixed type
- Euler equations
- static stability
- dynamic stability