Unsteady lifting-line theory with applications

Unsteady lifting-line theory is developed for a flexible unswept wing of large aspect ratio oscillating at low frequency in inviscid incompressible flow. The theory is formulated in terms of the acceleration potential and treated by the method of matched asymptotic expansions. The wing displacements are prescribed and the pressure field, airloads, and unsteady induced downwash are obtained in closed form. Sample numerical calculations are presented. The present work identifies and resolves errors in the unsteady lifting-line theory of James and points out a limitation in that of Van Holten. Comparison of the results of Reissner's approximate unsteady lifting-surface theory with those of the present work shows favorable agreement. The present work thus provides some formal justification for Reissner's ad hoc theory. For engineering purposes, the region of applicability of the theory in the reduced frequency-aspect ratio domain is identified approximately and found to cover most cases of practical interest.