Foil locomotion through non-sinusoidal pitching motion

We investigate the propulsive characteristics of a neutrally buoyant foil that self propels as it pitches about its quarter chord with a prescribed time periodic waveform. For a fixed pitching frequency and amplitude, we find the foil motion to be strongly dependent on the prescribed waveform. The maximum mean self-propelling speed corresponds to a square shaped waveform for which the foil is held at the two extremes for a significant part of the pitching cycle. In contrast, the cost of transport is minimal, and therefore the energetic efficiency of pitching induced self propulsion a maximum, for a triangle shaped profile in which case the foil's angular speed is largely held constant over the pitching period. Pitching waveform induced variations in the mean self-propelling speed ((u) over bar (p)) are surprisingly well predicted with a simple power-law relationship St(rms)similar to Re-0.37 , where Reynolds number Re is based on (u) over bar (p) and the Strouhal number St(rms) is the ratio of the root mean square foil trailing edge speed and (u) over bar (p). This power-law relationship is shown to arise naturally from a balance between the cycle-averaged inertial thrust generated from prescribed pitching and the enhanced drag owing to the boundary layer thinning. The imposed pitching motion induces a Reynolds number independent out of phase heave motion features of which are remarkably well predicted with a simple inviscid model for the foil's transverse motion. For all the prescribed pitching waveforms, a maximum in the energetic efficiency is always attained over the range 0.3 <= St <= 0.6. The close correspondence between this Strouhal number range for maximum energetic efficiency and the optimal Strouhal number range for swimming and flying animals is indicative of the similarity between self-propelling flapping foil configurations and undulatory biolocomotion. Our results point to the significance of rigid body pitching profile as an important input parameter that can be varied independently to achieve a desired outcome such as maximization of the mean self-propelling speed or energetic efficiency. (C) 2019 Elsevier Ltd. All rights reserved.