Supersonic flow of a Chaplygin gas past a conical wing with $\Lambda$-shaped cross sections

Abstract

In this paper, by considering the anhedral angle, we for the first time study the problem of supersonic flow of a Chaplygin gas over a conical wing with $\Lambda$-shaped cross sections, where the flow is governed by the three-dimensional steady isentropic irrotational compressible Euler equations. This work is motivated by the design of the Nonweiler wing, which is one of the simplest waveriders. Mathematically, the problem reduces to a boundary value problem for a nonlinear mixed-type equation in conical coordinates. By introducing a viscosity parameter to treat the degenerate boundary, we use the continuity method to establish the existence of a piecewise smooth self-similar solution to the problem, in the case that the shock is attached to the leading edge of the conical wing. Our results verify part of K\"uchemann's speculation on the conical flow field structures of this type, and also find a new conical flow field structure.