A generalized lifting-line theory for curved and swept wings

Abstract

A generalized lifting-line theory is developed in inviscid, incompressible, steady flow for curved, swept wings of large aspect ratio. It is shown in this paper that by using the integral formulation of the problem instead of the partial differential equation formulation, it is possible to circumvent the algebraic complications encountered by the previous approaches using the method of the matched asymptotic expansions. At each approximation order the problem is reduced to inverting a classical Carleman type integral equation. The asymptotic solution in terms of circulation is found up to A−1 and A−1 In (A−1). It is very convenient for illustrating the major three-dimensional effects induced on the flow by curvature and yaw angle. The concept of the finite part integrals, introduced by Hadamard (1932), is shown to be very useful for handling elegantly singularities like 1/x|x| or 1/|x| which occur in the course of our developments. Comparisons of the new, simple approach with lifting-surface theories reveal excellent agreements in terms of circulation. Furthermore, a consistent calculation of the three components of the total force acting on the wing is done in the lifting-line context without re-introducing the inner geometry of the wing.