Velocity distribution on thin tapered wings with fore-and-aft symmetry and spanwise constant thickness ratio at zero incidence

Abstract

This report is a continuation of three earlier ones by the present authors (1947-9) and contains a theoretical investigation of subsonic flow past thin tapered unswept wings (of full or cropped-rhombus plan form), at zero incidence. Only the case of spanwise constant thickness ratio is considered in this first attempt although alternative cases also merit attention. The first order method of linear perturbation based on continuous systems of sources and sinks is shown to be still applicable to tapered wings, although mathematical difficulties are greatly increased. These have been overcome, at least in the simple case of the biconvex parabolic profile, so as to give general solutions and computable formulae for the velocity distribution over the entire wing area. Complete detailed solutions for the mid-chord line have been worked out numerically and two examples of complete numerical solutions, with corresponding isobar patterns, for the entire wing area are presented. These results are sufficient to illustrate the effect of uniform taper on the velocity field of unswept wings, and lead to a number of general conclusions. The most important of these is that, although taper brings about noticeable decrease of supervelocities at the centre, higher values are encountered further outboard so that, for cropped plan forms, two symmetrically placed maximum suction areas arise inside the two half-wings. These are relevant for determining critical Mach numbers, and the effect of taper may be, according to choice of geometrical parameters, either beneficial or detrimental as to the values of Motet, but practically never very considerable. The method wilt still be applicable to the more general, and more important, case of tapered swept-back wings, especially for delta wings, and a general solution for the velocity distribution in the central sections of such wings is given in Appendix I and shown to be consistent with the earlier solution for untapered swept wings. However, for applying the method successfully (up to detailed numerical investigation) to the more general case, automatic high-speed integrating machinery seems indispensable--to replace classical methods of transforming integrals and manual computing, as used in the past and in the present report.