Derivatives on the downstream side of a moving, curved shock

Abstract

Novel relations are developed for the tangential and normal derivatives of the pressure, density, and velocity components just downstream of a regular point on a curved shock wave. The perfect gas flow is three dimensional with a nonuniform freestream and the flow may be unsteady. The analysis starts with data in a fixed, laboratory-frame Cartesian coordinate system. By means of an orthogonal transformation, a coordinate system is introduced via a solid-body translation and two solid-body rotations. This Cartesian system is shock based and limits the analysis to a selected shock point at a given instant of time when the flow is unsteady. The analysis is a local one in that Taylor series expansions are utilized for the configuration of the shock and the upstream gradients of the velocity, pressure, and density. The coefficients in the last three expansions must satisfy Euler equation constraints. These expansions are performed in both Cartesian systems, with occasional preference given to the laboratory-frame expansions where there are known data. The derivatives are in terms of a third coordinate system that is shock based and utilizes the flow plane at the selected shock point. Exact, explicit results are provided for the tangential and normal derivatives in as simple a form as possible. In the Taylor series expansion version for the tangential derivatives, the relative importance of various factors can be assessed. These are the upstream velocity, pressure, and density gradients, curvatures, and shock-shape terms not associated with curvatures. A different type of discussion is provided for the more involved normal derivatives.