Effect of flexibility on the self-propelled locomotion by an elastically supported stiff foil actuated by a torque.

Abstract

A theoretical model is presented for the locomotion of an aquatic vehicle propelled by a flexible foil elastically mounted to translational and torsional springs and dampers at an arbitrary pivot axis and actuated by a harmonic torque. The work extends a previous model by the authors for an elastically mounted rigid foil [1], allowing for a passive flexural motion of the foil in addition to the passive pitching and heaving motions of the rigid foil, all of them generated by the actuating torque and the fluid-foil interaction. The Euler-Bernoulli beam equation is used together with linearized results from the potential flow theory, valid for small pitch, heave and flexural deflection amplitudes. The problem is governed by four ordinary differential equations (ODEs) for the temporal evolutions of the swimming velocity, and the pitch, heave and flexural motions of the flexible foil. In addition to numerical results of these ODEs, we also present an analytical perturbation solution which provides a valuable quick insight about the propulsion performance, but which is additionally restricted to very small swimming velocities. The vehicle’s propulsion performance is discussed in terms of the foil stiffness ratio and the remaining non-dimensional parameters, particularly the translational and torsional spring constants, the pivot axis location and the Lighthill number. It is found that, except for very low Lighthill numbers, the maximum swimming velocity is reached for a rigid foil actuated at the leading edge with the resonant combination of the translational and torsional springs constants for the given frequency. However, higher propulsive efficiencies and lower costs of transport, but with slightly smaller swimming velocities, are obtained for flexible foils with the same resonant combination of the elastic supports at the leading edge. As a validation of the model, the Strouhal number for optimal propulsion efficiency is found in a narrow band around 0.32, in agreement with many experimental and numerical works on optimal propulsion by flapping foils. Additionally, the relation between Strouhal and Lighthill numbers for optimal propulsion is favorably compared with experimental data for fishes where the primary mechanism for producing thrust is an oscillatory prominent caudal fin.