Free boundary problems for the unsteady transonic small disturbance equation: Transonic regular reflection

Abstract

We formulate and solve a transonic regular reflection problem for the unsteady transonic small disturbance equation, using a free boundary problem approach. Our method applies to self-similar shock reflection when the incident shock angle is large enough to permit a regular reflection configuration with a subsonic state behind the reflected shock. For the small-disturbance approximation in weak shock reflection, this corresponds to relatively large wedge angles. One contribution of this paper is the development of an asymptotic formula for the reflected shock, far from the reflection point, and for the subsonic state far downstream. These asymptotic series are valid for the small-disturbance approximation, for any incident shock angles. The main result in the paper is an existence theorem for the nonuniform subsonic flow behind the reflected shock. The flow velocity satisfies a quasilinear elliptic equation which is coupled to the Rankine-Hugoniot equations for the reflected shock, forming a free boundary problem on part of the boundary. Because the equation is not uniformly elliptic in the entire domain, we introduce a cut off to give a bounded domain, and also cut offs to the coefficients. Our result is incomplete in the following sense: we have been unable to remove the cut offs entirely. However, we prove that the flow we have constructed solves the original problem in a domain of finite size around the reflection point.