Asymptotic Theory of Flapping Wing Propulsion in Extreme Ground Effect

Abstract

This study, dedicated to flapping wing propulsion in immediate proximity to a wall or between closely spaced flat walls, makes use of the method of matched asymptotic expansions. Its purpose is to create a simplified parametric model of such a propulsion system based on a single major assumption: immediate closeness of oscillating wing to a solid wall. In the case of a rectangular finite-aspect ratio wing, analytical expressions have been obtained for the coefficients of instantaneous and period-averaged thrust force as well as for the efficiency of the propulsor as a function of distance from the wall, Strouhal number and wing aspect ratio for selected cases of heaving, pitching and combined oscillations. It is shown that for some oscillation modes closeness to the ground results in increases in thrust and efficiency, and that optimally combined (considering ratio of the amplitudes and phase shift of contributing motions) heave–pitch oscillations allow to maximize thrust or efficiency of the flapping-wing propulsor. Increase in aspect ratio and decrease in Strouhal number (relative frequency of oscillations) in the case of heaving invariably brings about the increase in the ideal efficiency. An example is provided of a non-planar extreme ground effect application considering oscillations of a ring-wing embracing circular cylinder. A rule is derived for recalculation of the characteristics of the flapping-wing propulsor near a flat wall onto the characteristics of the same wing operating in a narrow canal between parallel walls. This rule can also be applied to evaluate propulsive properties of a ring-wing oscillating between co-axial cylindric surfaces.