2017/03/11 by Gui‐Qiang Chen, Chen, Gui-Qiang G.
Engineering · Mathematics · Physics and Astronomy · #35-02 #35B30 #35B35 #35B40 #35L60 #35L65 #35L67 #35M12 #35Q31 #35Q35 #35R35 #76H05 #76L05 #76N10 #76N15 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Black Holes and Theoretical Physics #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #Navier-Stokes equation solutions
paper · pdf · doi:10.48550/arxiv.1703.03997
openalex publication_date 2017/03/11 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
When an upstream steady uniform supersonic flow impinges onto a symmetric\nstraight-sided wedge, governed by the Euler equations, there are two possible\nsteady oblique shock configurations if the wedge angle is less than the\ndetachment angle -- the steady weak shock with supersonic or subsonic\ndownstream flow (determined by the wedge angle that is less or larger than the\nsonic angle) and the steady strong shock with subsonic downstream flow, both of\nwhich satisfy the entropy condition. The fundamental issue -- whether one or\nboth of the steady weak and strong shocks are physically admissible solutions\n-- has been vigorously debated over the past eight decades. In this paper, we\nsurvey some recent developments on the stability analysis of the steady shock\nsolutions in both the steady and dynamic regimes. For the static stability, we\nfirst show how the stability problem can be formulated as an initial-boundary\nvalue type problem and then reformulate it into a free boundary problem when\nthe perturbation of both the upstream steady supersonic flow and the wedge\nboundary are suitably regular and small, and we finally present some recent\nresults on the static stability of the steady supersonic and transonic shocks.\nFor the dynamic stability for potential flow, we first show how the stability\nproblem can be formulated as an initial-boundary value problem and then use the\nself-similarity of the problem to reduce it into a boundary value problem and\nfurther reformulate it into a free boundary problem, and we finally survey some\nrecent developments in solving this free boundary problem for the existence of\nthe Prandtl-Meyer configurations that tend to the steady weak supersonic or\ntransonic oblique shock solutions as time goes to infinity. Some further\ndevelopments and mathematical challenges in this direction are also discussed.\n