Theoretical study of an inviscid transonic flow near a discontinuity in wall curvature (Part 1)

The work provides an extensive theoretical study of an inviscid transonic flow in the vicinity of a wall curvature discontinuity. Depending on the ratio of the curvatures upstream and downstream of the break, several physically different regimes can exist, including a special type of supersonic flows which decelerate to subsonic speeds without a shock wave, transonic Prandtl--Meyer flow and supersonic flows with a weak shock. Using a new numerical technique of solving the Karman--Guderley equation in the ODE form, we perform computations and then employ the \emph{hodograph method} along with the \emph{phase portrait technique} to obtain a complete theoretical description of the flow. It appears that if the flow can be extended beyond the \emph{limiting characteristic}, it subsequently develops a shock wave. As a consequence, a fundamental link between the local and the global flow patterns is observed in our problem (a detailed description of this is given in Part 2). The curvature discontinuity leads to singular pressure gradients $\partial p/\partial x \sim G_{\mp}\, (\mp x)^{-1/3}$ upstream and downstream of the break point, respectively. Analytical expressions for the amplitude coefficients $G_{\mp}$ are derived as functions of the ratio of the curvatures. These results are important for a subsequent study of the boundary layer separation.