Study on the Stability of a Jet Flow with a Transonic Shock


Student thesis: Doctoral Thesis

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Award date20 Dec 2022


In this thesis, We study the stability of a subsonic jet flow with a transonic shock. We focus on the case that the subsonic jet forms on the convex corner, caused by pressure difference.

In Chapter 1, we introduce the Euler equations with Rankine-Hugoniot conditions, which is a kind of system equations of classical conservation law. Then, we state the specific properties of contact discontinuity which is a type of piecewise smooth solution of Euler equations. Besides, the formulation and some basic properties of the subsonic jet with a transonic shock are presented.

In Chapter 2, we state the free boundary problem and main theorem as well as define the weighted Hölder norm for the subsonic jet flow. The problem contains two parts-Problem SS and Problem WS, which correspond to the strong shock and weak shock in the subsonic jet. Later, we reformulate the problem and main theorem in the Lagrangian coordinates.

In Chapter 3, we prove the main theorem of Problem SS, the global existence, asymptotic behaviors, uniqueness, and stability of the subsonic jet with a strong transonic shock for the 2-D steady full Euler equations. We choose the pressure p as the main role and use the Euler equation and Rankine-Hugoniot conditions in the Lagrangian coordinates to formulate the elliptic equation and boundary conditions for δ (δp =p-p0, p0 is the background pressure$)$. Then the classical elliptic estimate with proper weighted Hölder norms will be applied to deal with the wedge corner singularity and the asymptotic behaviors for the Euler equations in the Lagrangian coordinates. The iteration scheme of other variables like velocity, density and the slope of the shock will depend on the estimate of δp.

In Chapter 4, we discuss the result of the stability of the subsonic jet with a weak shock for the 2-D steady full Euler equations. The difference between the weak and the strong transonic shock in the subsonic jet is revealed in terms of the corner and decay estimates and the asymptotic behaviors of the solution.