An approximate second-order wing theory.

Abstract

T note describes the development of a general but approximate second-order wing theory. In the absence of a numerically tractable, mathematically precise determination of supersonic flow over three-dimensional wings, several methods for improving on linear perturbation theory have been evolved. One method combines the first-order differential equation with an exact boundary condition; comparisons with experiment and theoretical results, where available, show that the method is of value. The present method takes this approach a stage further in that some account is taken of the second-order differential equation via an approximate particular integral, but only the second-order boundary condition is satisfied. The approximate particular integral may be interpreted as an approximation to the source distribution associated with the second-order effect. Comparison with an exact theory and a second-order theory available for the special case of a supersonic edge delta wing shows that, in these circumstances, the error arising from the approximate particular integral is not significant. However, it is possible that erroneous results could arise in other cases; the range of applicability has yet to be determined. In 1950, Sugo proposed a second-order theory for supersonic flow over three-dimensional wings of arbitrary planform. Lee has examined this paper and shown that Sugo's theory incorporated some errors and assumptions. Nevertheless, Sugo's method of developing an approximate particular integral for the nonhomogeneous second-order equation is of interest. Lee has used a similar approximate particular integral, and the results obtained so far from the modified theory are encouraging; although still semiempirical, they are certainly superior to those of the original work. Lee's theory, like Sugo's theory, probably approaches second-order accuracy only near the wing surface. This restriction, which was not altogether clear in Sugo's original work, seems to have been disregarded by Wallace and Clarke (Ref. 3, p. 184) in an application of Sugo's theory to cruciform wings of arbitrary profile; such an application requires a particular integral valid throughout the extended region of flow between the planar wing surfaces.