Second-Order Small-Perturbation Theory for Finite Wings in Incompressible Flow

Abstract

The incompressible second-order theory for two-dimensional aerofoils is extended to finite swept wings. The flow field is represented by distributions of sources and 'lifting singularities' on the 'chord surface' which contains the chord at each spanwise station. The strength of the source distribution is obtained as the sum of the distribution from first-order theory and a correction which is derived from the second-order boundary condition. This involves the computation of the velocity which planar singularity distributions induce on arid off the plane; the computation can be done by computer programs developed at the R.A.E. It is suggested that the determination of the strength of the lifting singularities aims from the start at the solution for the wing of finite thickness. First a generally applicable solution is derived. By means of Taylor series expansions, this solution is simplified for the part of the wing away from centre and tip. The problem of designing a wing, of given thickness distribution, which has a prescribed pressure distribution on the upper surface is also treated. The Report describes only the calculation procedures, but does not give actual sample calculations. One of the procedures suggested for determining the pressure distribution has been applied successfully by C, C. L. Sells to untwisted uncambered wings and to wings with camber and twist.