DANIEL, T.L.; COMBES, S.A.; TRIMBLE, A.C.: Swimming and flying with flexible wings: bending by inertia or fluid forcing?

Flapping flight in many organisms is accompanied by significant wing deformations. Many of these deformations arise, in part, from the passive mechanics of oscillating a flexible air- or hydro-foil. The instantaneous shape of wings has a profound effect on the fluid dynamic forces they can generate, with non-monotonic relationships between the pattern of deformation waves passing along the wing and the thrust developed. At the same time, however, their instantaneous shape may well emerge from details of the fluid loading. This issue — the extent to which there is feedback between the wing shape and the fluid dynamic loading — is core to understanding flight control. We ask to what extent surface shape is controlled by structural mechanics versus fluid dynamic loading? To address this issue, we use a combination of computational and analytic methods to explore how bending stresses arising from elastic-inertial mechanisms scale to those stresses that arise from fluid pressure forces. Our analyses suggest that there are combinations of wing stiffness patterns, wing motions, and fluid densities for which the fluid pressure stresses play a minor role in determining wing shape. Nearly all of the relevant combinations correspond to wings moving in air. For high Reynolds number motions where linear potential flow equations provide reasonable estimates of lift and thrust, we can therefore ask how wing structure affects flight performance. Armed with the above scaling argument, we then show how modest levels of passive elasticity can increase thrust for a given level of energy input in the form of an inertial oscillation of a compliant foil.