An analytic solution for two- and three-dimensional wings in ground effect

Abstract

The method of matched asymptotic expansions is applied to the problem of a wing of finite span in very close proximity to the ground. The general lifting surface problem is shown to be a direct problem, represented by a source-sink distribution on the upper surface of the wing and wake, with concentrated sources around the leading and side edges plus a separate confined channel flow region under the wing and wake. The two-dimensional flat plate airfoil is examined in detail and results for upper and lower surface pressure distribution and lift coefficient are compared with a numerical solution. A simple analytic solution is obtained for a flat wing with a straight trailing edge which has minimum induced drag. To lowest order, this optimally loaded wing has an elliptical planform and a lift distribution which is linear along the chord, resulting in a parabolic spanwise lift distribution. An expression for the lift coefficient at small clearance and angle of attack, valid for moderate aspect ratio, is derived. The analytic results show reasonable agreement when compared with numerical results from lifting surface theory.