On the Weis-Fogh mechanism of lift generation

TLDR

Weis‑Fogh’s 1973 lift‑generation mechanism, which can operate in inviscid two‑dimensional flows starting from rest, predicts that a body breaking into two pieces can acquire equal and opposite circulations that generate lift. The study investigates how this mechanism extends to three dimensions, particularly in hovering insects such as *Encarsia formosa* that move wing tips in circular paths. The authors model the far‑field flow of small hovering insects as the exact Navier–Stokes solution for a concentrated force (the insect’s weight) acting on a viscous fluid, valid when \(mg/(p v^2)\) is sufficiently small. They find that this jet‑type induced motion is stable under those conditions.

Abstract

Weis-Fogh (1973) proposed a new mechanism of lift generation of fundamental interest. Surprisingly, it could work even in inviscid two-dimensional motions starting from rest, when Kelvin's theorem states that the total circulation round a body must vanish, but does not exclude the possibility that if the body breaks into two pieces then there may be equal and opposite circulations round them, each suitable for generating the lift required in the pieces’ subsequent motions! The ‘fling’ of two insect wings of chord c (figure 1) turning with angular velocity Ω generates irrotational motions associated with the sucking of air into the opening gap which are calculated in § 2 as involving circulations −0·69Ω c 2 and + 0.69Ω c 2 around the wings when their trailing edges, which are stagnation points of those irrotational motions, break apart (position ( f )). Viscous modifications to this irrotational flow pattern by shedding of vorticity at the boundary generate (§ 3) a leading-edge separation bubble, and tend to increase slightly the total bound vorticity. Its role in a three-dimensional picture of the Weis-Fogh mechanism of lift generation, involving formation of trailing vortices at the wing tips, and including the case of a hovering insect like Encarsia formosa moving those tips in circular paths, is investigated in § 4. The paper ends with the comment that the far flow field of such very small hovering insects should take the form of the exact solution (Landau 1944; Squire 1951) of the Navier-Stokes equations for the effect of a concentrated force (the weight mg of the animal) acting on a fluid of kinematic viscosity v and density p , whenever the ratio mg/pv 2 is small enough for that jet-type induced motion to be stable.