Viscous–inviscid interaction in transonic Prandtl–Meyer flow

Abstract

This paper presents a theoretical analysis of perfect gas flow over a convex corner of a rigid-body contour. It is assumed that the flow is subsonic before the corner. It accelerates around the corner to become supersonic, and then undergoes an additional acceleration in the expansion Prandtl–Meyer fan that forms in the supersonic part of the flow behind the corner. The entire process is described by a self-similar solution of the Kármán–Guderley equation. The latter shows that the boundary layer approaching the apex of the corner is exposed to a singular pressure gradient, ${\rm d} p / {\rm d} x \sim (-x)^{-3/5}$, where $x$ denotes the coordinate measured along the body surface from the corner apex. Under these conditions, the solution for the boundary layer also develops a singularity. In particular, the longitudinal velocity near the body surface behaves as $U \sim Y^{1/2}$. Here $Y$ is the normal coordinate scaled with the boundary-layer thickness $Re^{-1/2}$; $Re$ being the Reynolds number, assumed large in this theory.

As usual, the boundary layer splits up into two parts, a viscous near-wall sublayer and a locally inviscid main part of the boundary layer. The analysis of the displacement effect of the boundary layer shows that neither the viscous sublayer nor the main part determines the displacement thickness. Instead, the overlapping region situated between them proves to be responsible for the shape of the streamlines at the outer edge of the boundary layer. This leads to a significant simplification of the analysis of the flow behaviour in the viscous–inviscid interaction region that forms in a small vicinity of the corner. In order to describe the flow behaviour in this region, one has to solve the Kármán–Guderley equation for the inviscid part of the flow outside the boundary layer. The influence of the boundary layer is expressed through a boundary condition, that relates the streamline deflection angle $\vartheta $ at the outer edge of the boundary layer to the pressure gradient ${\rm d}p / {\rm d}x$ acting upon the boundary layer. The boundary-layer analysis leads to an analytical formula that relates $\vartheta $ and ${\rm d}p /{\rm d}x$ (unlike in previous studies of the viscous–inviscid interaction). The interaction problem was solved numerically to confirm that the solution develops a finite-distance singularity.