Investigation of Crossflow Shocks on Delta Wings in Supersonic Flow

Abstract

In the study described, a finite difference approach to solving the rotational Euler equations, explicitly fitting shocks as a boundary, is applied to a variety of geometrical shapes in the lower supersonic Mach number regime. It is shown how special techniques based on the physics of the flow can be used to circumvent a variety of numerical difficulties encountered with the conical flow problem that are primarily associated with the initial value characteristics of the hyperbolic scheme, causing embedded shock-induced entropy and crossflow layers to develop on the body surface. UMEROUS investigations over the past several decades have dealt with the problem of supersonic flow over thin wings. Many of the earlier methods used linearized potential flow theory and slender body assumptions to obtain solutions. The failure of linearized potential flow solutions for flows where significant supersonic crossflow regions exist is fairly well known (e.g., Refs. 1-3). More recently, nonlinear methods, such as the method of characteristi cs4 and the method of lines, 5 were developed which were adequate at small angles of attack and simple, but not very thin, cross- sectional shapes. Nevertheless, these nonlinear methods did not broach the topic of crossflow embedded shocks which occur on thin bodies, even at relatively small angles of attack. Under certain geometric and flow conditions, a large ex- pansion occurs around the leading edge, causing an embedded region of supersonic crossflow to occur on the leeward side of the body. Since the supersonic region has no advance warning of the symmetry plane crossflow stagnation condition, an embedded shock forms, causing the crossflow to become subsonic and capable of satisfying this boundary condition. A shock-capturing finite-difference approach was developed and applied to complex shapes with embedded shocks in the high supersonic to hypersonic Mach number regime.6'7 This shock-capturing approach uses the con- servation form of the Euler governing equations in order to resolve the embedded shocks with a minimal amount of numerical instability. In contrast to the aforementioned nonlinear methods in- tegrating the rotational form of Euler equations, a technique was developed to integrate the irrotational equations for supersonic conical flow. A relaxation method was applied to the conical full potential flow equation in Ref. 3 that relied heavily on numerical transonic techniques. Numerical solutions were computed successfully for circular/elliptic cones and thin winglike cross sections from low to high in- cidence at freestream Mach numbers between 1.2 and 3. This method captures the shocks over a few mesh points, ap- parently without numerical instabilities. Using the nonlinear