Interfering or Nonplanar Lifting Surfaces in Subsonic Flow
A unified theory of interference for three-dimensional lifting surfaces in a subsonic main stream can be built up around the concept of pressure or acceleration-potential doublets. We begin by appealing to the Prandtl – Glauert-Gothert law, described in Section 7-1, which permits us to restrict ourselves to incompressible fluids. Granted the availability of high-speed computing equipment, it then proves possible to represent the loading distribution on an arbitrary collection of surfaces (biplane, multiplane, T-tail, V-tail, wing-stabilizer combination, etc.) by distributing appropriately oriented doublets over all of them and numerically satisfying the flow-tangency boundary condition at a large enough set of control points. The procedure is essentially an extension of the one for planar wings that is sketched in Section 7-6.
Two observations are in order about the method described below. First it overlooks two sometimes significant phenomena that occur when applied to a pair of lifting surfaces aligned stream wise (e. g., wing and tail). These are the rolling up of the wake vortex sheet and finite thickness or reduced dynamic pressure in the wake due to stalling. They are reviewed at some length in Sections C,2 and C,4 of Ferrari (1957).
The second remark concerns thickness. In what follows, we represent the lifting surfaces solely with doublets, which amounts to assuming negligible thickness ratio. When two surfaces do not lie in the same plane, however, the flow due to the thickness of one of them can induce interference loads on the other, as indicated in Fig. 11-1. The presence of this thickness and the disturbance velocities produced at remote points thereby may be represented by source sheets in extension of the ideas set forth in Section 7-2. Since the procedure turns out to be fairly straightforward, it is not described in detail here.
The necessary ideas for analyzing most subsonic interference loadings of the type listed under items 1, 2, and 3 can be developed by reference to the thin, slightly inclined, nonplanar lifting surface illustrated in Fig.
10- 2. We use a curvilinear system of coordinates x, s to describe the surface of S, and the normal direction n is positive in the sense indicated. The small camber and angle of attack, described by the vertical deflection Az(x, y) or corresponding small normal displacement Дn(x, s), are superimposed on the basic surface z0(y). The latter is cylindrical, with generators in the free-stream ж-direction. To describe the local surface slope in y, г-planes, we use