Global solutions of shock reflection by large-angle wedges for potential flow

Abstract

When a plane shock hits a wedge head on, it experiences a reflection-diffraction process and then a self-similar reflected shock moves outward as the original shock moves forward in time. Experimental, computational, and asymptotic analysis has shown that various patterns of shock reflection may occur, including regular and Mach reflection. However, most of the fundamental issues for shock reflection have not been understood, including the global structure, stability, and transition of the different patterns of shock reflection. Therefore, it is essential to establish the global existence and structural stability of solutions of shock reflection in order to understand fully the phenomena of shock reflection. On the other hand, there has been no rigorous mathematical result on the global existence and structural stability of shock reflection, including the case of potential flow which is widely used in aerodynamics. Such problems involve several challenging difficulties in the analysis of nonlinear partial differential equations such as mixed equations of elliptic-hyperbolic type, free boundary problems, and corner singularity where an elliptic degenerate curve meets a free boundary. In this paper we develop a rigorous mathematical approach to overcome these difficulties involved and establish a global theory of existence and stability for shock reflection by large-angle wedges for potential flow. The techniques and ideas developed here will be useful for other nonlinear problems involving similar difficulties.

Accuracy of the Adaptive GRP Scheme and the Simulation of 2-D Riemann Problems for Compressible Euler Equations

“…The conservation interpolation used in [25] is 16) where n c = c x µ+c y ν, (c x ,c y ) = (x−x,y−ỹ), and (n c U ) ℓ k denotes the value of the n c U through the boundary ℓ k . In practice, we always use the following upwind approximation to define (U n c ) ℓ k : 17) where U m,k , m = L or R, is defined similar to (2.5).…”

“…Comprehensive descriptions can be found in [8]. Recently, some progresses have been made for the regular reflection by using the potential flow equation [16] and the stability of the Mach reflection structure [17] by using the full Euler equations. …”

Transonic shock and supersonic shock in the regular reflection of a planar shock

“…Experiments, computations and mathematical analysis ( [2], [12], [7], [11], [4]) have shown us a plenty of results on the patterns of shock reflection. In 1878 ( [6]), Mach studied the oblique-shock-reflection problem in his experiments.…”

Uniqueness and instability of subsonic–sonic potential flow in a convergent approximate nozzle

“…The key point is to prove a version of the Hopf boundary point lemma applicable to points on characteristic degenerate boundaries for degenerate elliptic equations, which is Theorem 2.1 stated in Section 2.1. Although there is much impressive progress in the study of degenerate elliptic equations and mixed type equations in gas dynamics and other fields in these years (see [1,2,6,7,8,9,12,13,19] and the references therein), this generalized Hopf lemma seems to be new. It captures the remarkable property that for subsonic-sonic flow or transonic flow, the potential flow equation behaves like the heat equation −∂ 1 ϕ + ∂ 22 ϕ = 0 near the sonic line (the line where the equation is degenerate), and hence the lower order term −∂ 1 ϕ is essential in studying these degenerate elliptic or mixed type equations (cf.…”

“…It captures the remarkable property that for subsonic-sonic flow or transonic flow, the potential flow equation behaves like the heat equation −∂ 1 ϕ + ∂ 22 ϕ = 0 near the sonic line (the line where the equation is degenerate), and hence the lower order term −∂ 1 ϕ is essential in studying these degenerate elliptic or mixed type equations (cf. [2,18]). This observation, obtained by studying special subsonic-sonic flows and transonic flows in an approximate nozzle, would be important for studying the flows in a physical nozzle.…”