A Theory for Rectangular Wings of Small Aspect Ratio

Abstract

T theory for flow about wings of large aspect ratio has been worked out rather completely along the lines suggested by Prandtl. This flow is of interest for the main lifting surfaces of present day airplanes. However, for some unconventional airplane wings and especially for the control surfaces of dirigibles, the lifting surfaces have generally a very small aspect ratio and the existing theories have so far been inadequate. Moreover, the dirigible as a whole exhibits very much the same behavior, at least as far as the dynamic lift is concerned, as a wing of small aspect ratio. In view of this, Th. von Karman suggested to the author making a theoretical analysis of wings of small aspect ratio. The first step was the limiting case of a rectangular wing of zero aspect ratio, that is, finite span but infinite chord. Such a wing is hydrodynamically equivalent to a constant distribution of bound vortices whose axes lie along the span (Fig. 1). At the ends of the span these vortices separate, the flow downstream following the resultant velocity vector. The inclination of the vortices to the wing was found to be about half the angle of attack. As an approximation, in order to simplify the calculation of the induced velocities, it was assumed that the trailing vortices formed straight lines at some effective inclination to the wing. This effective inclination was taken equal to the initial angle at which the vortices separate, on the assumption that the part of the vortex nearest the wing is the most influential in inducing downwash at the wing. With this approximation, three important results were obtained for the wing of zero aspect ratio: