NACA-TN-1555

Abstract

INTRODUCTION <br>Although a number of treatments of the problem of the lift on a thin flat surface in supersonic flaw have recently been published (see bibliography of reference 1, and. references 2 and 3) the lift distribution over a wing with both leading and trailing edges swept behind their respective Mach lines has not, at the present writing, been determined. The only explicitly formulated solutions (all based on the linearized form of the flow equations) are essentially so-called "conical" type, introduced by Baseman (reference 4); that is, solutions in which such quantities as the velocities and pressure are constant along lines radiating from a single point. The limitation is therefore automatically imposed that the boundary conditions to be satisfied must also be constant along such lines. Under certain circumstances, conical fields may be superposed to give surfaces of no conical plan form. Figure 1(a) shows one such case, a finite-span trapezoidal wing swept only slightly compared to the mach lines. The basic solution here is that for a flat, symmetrical unyawed triangle at an angle of attack, and is conical with respect to the apex 0.The triangle may extend to infinity. The pressure distribution over the shaded portion is constant; the values vary between the mach lines from 0, but are constant along any ray drawn through 0. In order to obtain a finite wing, it is necessary to cancel the pressure beyond the tips AB and A'B' by superposing negatively loaded triangular surfaces with apexes at A and A' and one edge parallel to the stream. The conditions to be satisfied by the supplementary ;solution are that (1) the pressure be constant over the subtracted surfaces in order to cancel the constant pressure in the basic solution, and (2) the downwash induced inboard of the streamwise edge be zero in order that the surface may remain flat. Both of these conditions are conical with respect to A or A', so that a conical solution may be used. The area behind BCB' may, of course, be subtracted without affecting the wing itself in any way, since the wing lies entirely ahead of the mach lines originating from any point on the surface BCB. Consider, however, the case shown in figure l(b) . Here the Mach number is such that the Mach lines from A intersect the tips of the wing. starting with the same basic triangle, we find that the area t o be removed outboard of A includes, in this case, a region over which the pressure varies, pressure varies, and is conical with respect to 0. Since the boundaries of the region are, however, conical with respect to A, no one conical solution can satisfy the conditions. the preceding one can be treated by the point-source method of reference 2.