Positive Tail Loads for Minimum Induced Drag of Subsonic Aircraft

By applying Prandtl's relation for the induced drag of a biplane to typical wing-tail combinations, it can be shown that the minimum induced drag occurs with a positive tail upload. This fact has been overlooked because the reduction in the total induced drag by a tail download was overestimated by using the total downwash of the wing on the tail, while neglecting the downwash produced on the wing by the tail. It is proved that, regardless of the relative size of the tail, the downwash produced by a tail download increases the induced drag of the wing so as to cancel the additional 'tail thrust,' and keep the mutually induced drag of a wing-tail combination the same as that induced upon the tail alone when it is in the wing's Trefftz-plane. At any finite tail length the bound circulation vortex of the wing produces a downwash that increases the induced drag of a tail upload. However, the circulation vortex system of the tail upload produces an upwash on the wing that results in a 'wing thrust' component that cancels the increased drag on the tail so that the total induced drag is a minimum with a positive tail load. In order to facilitate the calculation of the mutually induced drag of typical wing-tail combinations, an explicit relation is derived for the limiting case of a small-span tail at any distance above or below a large-span wing.