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Unsteady lifting-line theory as a singular-perturbation problem. (English) Zbl 0589.76027
Summary: Unsteady lifting-line theory is developed for a wing of large aspect ratio oscillating at low frequency in inviscid incompressible flow. The wing is assumed to have a rigid chord but a flexible span. Use of the method of matched asymptotic expansions reduces the problem from a singular integral equation to quadrature. The pressure field and airloads, for a prescribed wing shape and motion, are obtained in closed form as expansions in inverse aspect ratio. A rigorous definition of unsteady induced downwash is also obtained. Numerical calculations are presented for an elliptic wing in pitch and heave; compared with numerical lifting-surface theory, computation time is reduced significantly. The present work also identifies and resolves errors in the unsteady lifting-line theory of E. C. James [ibid. 70, 753-771 (1975; Zbl 0363.76006)], and points out a limitation in that of T. Van Holten [e.g.: ibid. 77, 561-579 (1976; Zbl 0338.76009)].
MSC:
76B10Jets and cavities, cavitation, free-streamline theory, water-entry problems, etc.
76B25Solitary waves (inviscid fluids)
76M99Basic methods in fluid mechanics