Integral Formulations for Lifting Surfaces in the Extreme Ground Effect

As mentioned in the Introduction, asymptotic analysis can be applied directly to the integral equation of the lifting surface in the ground effect. Essen­tially, in the extreme ground effect, we study the limiting process, when the distance between two double (vortex) layers, representing the wing and its mirror image, becomes vanishingly small. In the limit, the two double layers merge into a quadruple layer so that the procedure can be characterised as quadruplication.[58] The main result of quadruplication is the confluence (for h —> 0) of the integral equation of the wing-in-ground effect into a differen­tial equation (ordinary for two-dimensional flow, and in partial derivatives for three-dimensional flow). The resulting differential equation can be shown to be identical to that, obtained in the course of solving the corresponding boundary problem by the method of matched asymptotic expansions. The quadruplication approach in the aerodynamics of wings in the ground effect was first introduced by Panchenkov [64]. In what follows, all derivations will be based on a different scheme of quadruplication proposed in [66].