Nonlinear Supersonic Potential Flow over Delta Wings

Abstract

A numerical procedure has been developed for the computation of the steady, inviscid supersonic flow over aircraft configurations. The technique accounts for major nonlinear effects (shock waves, blunt leading edges) at low to moderate supersonic speeds. A fully implicit marching technique for the full potential equation is utilized in a stereographically projected, conformally mapped, spherical coordinate frame. Cross-flow planes are efficiently solved by type-dependen t relaxation techniques. Results are presented for several delta wing configurations and bodies of revolution, and are compared with existing experimental data, Euler's equations solutions, and results from linearized theories (panel methods). I. Introduction T HE goal of the present investigation is the development of a highly efficient and general numerical procedure for the computation of the steady, inviscid supersonic flow over aircraft configurations. In particular, this paper will concentrate on the development and verification of the method through application to isolated delta wings and bodies of revolution. Emphasis will be given to the prediction of nonlinear effects at imbedded shock waves and blunt leading edges in the low to moderate supersonic Mach number range. Existing computational techniques in this regime may be broadly characterized as panel methods and finite-difference methods. Panel methods, which have been demonstrated to be widely versatile and quite efficient, have reached a very advanced stage of development and utility for subsonic flows. Efforts are continuing to improve their capabilities at supersonic speeds (i.e., Ehlers et al. *). These methods address the solution to the first-order linear small-disturbance equation with various approximations for the boundary conditions. Finite-difference techniques for wing planforms in the supersonic regime concentrate on solutions to Euler's equations with differing treatments of shock waves, shock capturing2 or shock fitting,3'5 and a variety of methods for grid generation (i.e., Moretti5) and geometry input. The major drawbacks of these methods are associated with their large computational times which are due primarily to the fine grids necessary to resolve thin wings and the accompanying strict stability requirements. Other problem areas for the Euler solutions in the Mach number range considered here are due to subsonic Mach numbers in the axial marching direction, initial conditions at the apex of the wing or body, and vortical singularities of the leeward surface. In a recent paper by Mason and daForno6 the opportunities for performance gains through nonlinear aerodynamic effects are discussed including the impact on nonlinear computational analysis and design. A convincing case is made for the necessity of the inclusion of nonlinear effects particularly for wing leading edges and imbedded shock waves for the aerodynamic design and analysis of wings.