Effects of flexibility on the aerodynamic performance of flapping wings

Abstract Effects of chordwise, spanwise, and isotropic flexibility on the force generation and propulsive efficiency of flapping wings are elucidated. For a moving body immersed in viscous fluid, different types of forces, as a function of the Reynolds number, reduced frequency (k), and Strouhal number (St), acting on the moving body are identified based on a scaling argument. In particular, at the Reynolds number regime of $O(1{0}^{3} \ensuremath{-} 1{0}^{4} )$ and the reduced frequency of $O(1)$, the added mass force, related to the acceleration of the wing, is important. Based on the order of magnitude and energy balance arguments, a relationship between the propulsive force and the maximum relative wing-tip deformation parameter ($\gamma $) is established. The parameter depends on the density ratio, St, k, natural and flapping frequency ratio, and flapping amplitude. The lift generation, and the propulsive efficiency can be deduced by the same scaling procedures. It seems that the maximum propulsive force is obtained when flapping near the resonance, whereas the optimal propulsive efficiency is reached when flapping at about half of the natural frequency; both are supported by the reported studies. The established scaling relationships can offer direct guidance for micro air vehicle design and performance analysis.