On the far wake and induced drag of aircraft

Abstract

A set of matched asymptotic expansions is proposed for the flow far behind an aircraft, with the primary purpose of identifying lift, thrust and drag, particularly induced drag, in a unified manner in integral statements of the momentum equation. The fluid in the far wake is inviscid and incompressible, and variations of total pressure are allowed, as are vortex sheets. A notable feature is that the Trefftz-plane approximation is not invoked; instead the wake is taken as fully rolled-up, and the analysis proceeds without the assumption of light loading. Attention is paid to the absolute convergence of integrals over infinite domains and handling of discontinuities. The expansion includes a sink term, which appears new, so that the mass flux through a transverse plane is non-zero, as is the flux of mechanical energy. The lift can be formally attributed to the velocity induced by the bound vortex of the wing, which is at odds with some treatments, although consistent with Prandtl's analysis over a ground plane. The drag contains the integral of ρ(v2 + w2 − u2)/2, as in many treatments of the subject, u being the perturbation velocity along the wake. The negative sign for u2 appears paradoxical on two counts, one of which is resolved here. First, its very presence instead of the + sign, which would lead to the perturbation kinetic energy and therefore a compelling explanation of induced drag, is explained by the longitudinal energy flux. This energy, the integral of ρu2, is continuously provided by the unsteady starting-vortex system and was deposited earlier by the aircraft. Second, it appears that negative drag could be predicted by this equation. This is shown to be impossible, because of inequalities between the integrals of (v2 + w2) and of u2, but the proof is valid only if the vorticity is of only one sign on each side. A general proof of positivity has not been derived, because of nonlinearities, but neither has a counter-example.