Comparative numerical applications of the reverse-flow theorem to oscillating wings and control surfaces

Abstract

The reverse-flow theorem gives alternative integrals for generalized forces on oscillating wings by linearized theory. Applications to analytical and numerical solutions are discussed, and the latter are considered in some detail. Reverse-flow relations for plunging and pitching derivatives are formulated in terms of those for the reversed wing, and the accuracy of numerical solutions is examined thereby for wings having streamwise symmetry or mote general planform. The further relations required in the application of the reverse-flow theorem to cases of low frequency are given, and these formulae for the reversed wing are adapted to Multhopp's lifting-surface theory. Finally, for general frequency, treatment of oscillating control surfaces by smooth equivalent upwash functions is considered by means of reverse flow, with particular reference to rectangular wings with full-span controls. The various applications of the reverse-flow theorem are illustrated by calculated examples for a range of Mach number, frequency, wing planform and control surface. No firm conclusions regarding absolute accuracy are possible, nor can a preference be stated between numerical results by direct flow and reverse flow. Simple indications of inaccuracy due to inadequate collocation are illustrated. Convincing comparisons between the alternative calculations are found in most examples, and these include lift and pitching moment due to slowly oscillating part-span control surfaces on delta and arrowhead wings.