A problem of minimal induced resistance in unsteady motion

Abstract Energy considerations are used to find the spanwise circulatory lift distribution yielding minimal induced resistance for a harmonically oscillating planar wing in a steady incompressible flow. Under the restriction that the time-averaged and oscillatory constituents of the wing's circulatory lift are known, the optimal distribution is such that its time-averaged constituent is elliptical, but the oscillatory one is generally not. The latter varies from elliptical to rectangular as the frequency of the oscillations increases. When the oscillatory part of the lift is not prescribed, present results infer that its reduction — as, for example, by an elastic twist — will typically reduce the drag, even if the twist yields a non-optimal (for that lift) lift distribution.