A slender–wing theory in potential flow

Edmund Chadwick

Abstract

Consider uniform, steady potential flow past a slender wing. By considering a horseshoe vortex in the limit as γ/Us → ∞, where γ is the circulation, U is the uniform stream velocity and s is the span, a model representing a vortex sheet is obtained from which the lift on the slender wing can be determined. (This is in contrast to the textbook approach of Batchelor and Katz & Plotkin, who discretize the vortex sheet with horseshoe vortices in the limit as γ/Us → ∞, but then relate the vortex strength to lift by using the two–dimensional limit γ/Us → 0. We shall argue that using these different limits in the same analysis is inconsistent and leads to an incorrect result.)

The resulting potential term is shown to be the same as the potential term of the lift Oseenlet in Oseen flow. In the limit of high–Reynolds–number flow, only half the contribution to the lift integral comes from the potential–velocity part of the lift Oseenlet. The other half comes from the vortex–wake–velocity part of the lift Oseenlet. We therefore assume potential flow everywhere except at the vortex sheet, along which we allow a singular vortex–wake–velocity term of the lift Oseenlet.

From this, a slender–wing theory is presented together with integral expressions for the lift and change in lift over the wing surface. Applications to slender bodies and large–aspect–ratio wings, in particular, the Lanchester–Prandtl lifting line, are then considered.