A unified unsteady lifting-line theory

A lifting-line theory is developed for wings of large aspect ratio oscillating in an inviscid fluid. The theory is unified in the sense that the wing may be curved or inclined to the flow, and the asymptotic expansion is uniformly valid with respect to the frequency. The method is based on the integral equation formulation of the problem. The technique, pioneered by Kida & Miyai (1978). consists of asymptotically solving the Fredholm equation of the first kind which links the unknown pressure jump and the normal velocity imposed on the wing. Use of the finite-part integral theory introduced by Hadamard (1932) and of a technique developed in Guermond (1987, 1988, 1990) yields an asymptotic expansion of the surface integral in terms of the inverse of the aspect ratio. At each approximation order, the problem reduces to a classical two-dimensional integral equation, whose unknown is the pressure jump, and whose right-hand side depends only on the previous approximation orders of the solution. The first finite-span correction is explicitly calculated. An extensive numerical study is carried out, and comparisons with published results are made.