2ND-ORDER SMALL-DISTURBANCE SOLUTION FOR WING AT SUPERSONIC SPEEDS

An integral equation of the transonic small-disturbance differential e quation is extended to second order for supersonic flows around three- dimensional thin wings, allowing bow as well as embedded shock waves. An explicit solution of the second-order small-disturbance equations s atisfying the second-order boundary conditions of a wing is obtained. The particular solution is expressed as the differentiation of a volum e integral with specifically chosen boundaries, and the complementary solution satisfies the linearized velocity potential equation with app ropriate boundary conditions. Applied to two-dimensional airfoils, the method lends itself to a completely analytical calculation, and the r esults agree with the well-known solution. A numerical procedure based on this solution for three-dimensional wing calculations is proposed. The flows about a delta wing with a wedge cross section and a superso nic leading edge at zero attack angle are calculated by the numerical methods. In comparison with those obtained by the exact method and the Euler code, the present method yields a better approximation than the linear theory, except in the vicinity of the apex Mach cone and other regions near solution discontinuities and singularities, which are am enable to subsequent treatment via strained coordinate or other method s.