Foil locomotion through non-sinusoidal pitching motion

Abstract

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¯p) are surprisingly well predicted with a simple power-law relationship Strms∼Re−0.37, where Reynolds number Re is based on u¯p and the Strouhal number Strms is the ratio of the root mean square foil trailing edge speed and u¯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.