Linear Approximation for Supersonic Flow Past a Delta Wing

Abstract

When a supersonic projectile moves in the air, a shock front generally appears ahead of it. If the head of the projectile has a sharp edge, the shock front is usually attached. Such a phenomenon is observed in physical experiment for both a wing and a cone-shape body. In this paper we are mainly concerned with the flow past a wing. In the case of a thin wing with a constant section, the corresponding boundary-value problem for the Euler system for compressible fluids has been solved explicitly in [1, 2]. Moreover, when the wing has variable sections or when its leading edge is a smooth curve, the local existence of the solution is obtained in [3]. The present paper studies supersonic flow past a triangular plate, as motivated by the modern design of delta wings. Much computational work has been done for the study of this problem (see [4–6]), but its mathematical theory is still open. In this paper we study the supersonic flow past a triangular plate with vertex angle near π, and take the point of view that locally the flow past such a triangular wing is a perturbation of the same flow past a plate in the same plane with its leading edge perpendicular to the oncoming flow. Under such assumptions we linearize the nonlinear problem at the unperturbed state, and solve the linearized problem as an approximation to the original nonlinear problem. Our linearization preserves the basic character of the equation as one of mixed type. Since the solution in the hyperbolic region and the boundary of degeneracy are easily determined, our problem is then reduced to a boundary-value problem for a degenerate elliptic equation, because in the hyperbolic region the solution is easily determined. The main result is