Thrust generation by a heaving flexible foil: Resonance, nonlinearities, and optimality

Abstract

International audienceFlexibility of marine animal fins has been thought to enhance swimming performance. However, despite numerous experimental and numerical studies on flapping flexible foils, there is still no clear understanding of the effect of flexibility and flapping amplitude on thrust generation and swimming efficiency. Here, to address this question, we combine experiments on a model system and a weakly nonlinear analysis. Experiments consist in immersing a flexible rectangular plate in a uniform flow and forcing this plate into a heaving motion at its leading edge. A complementary theoretical model is developed assuming a two-dimensional inviscid problem. In this model, nonlinear effects are taken into account by considering a transverse resistive drag. Under these hypotheses, a modal decomposition of the system motion allows us to predict the plate response amplitude and the generated thrust, as a function of the forcing amplitude and frequency. We show that this model can correctly predict the experimental data on plate kinematic response and thrust generation, as well as other data found in the literature. We also discuss the question of efficiency in the context of bio-inspired propulsion. Using the proposed model, we show that the optimal propeller for a given thrust and a given swimming speed is achieved when the actuating frequency is tuned to a resonance of the system, and when the optimal forcing amplitude scales as the square root of the required thrust

Clarifying the relationship between efficiency and resonance for flexible inertial swimmers

“…(Although exact definitions vary from one work to the other, they are all in the same spirit.) While thrust generally exhibits local maxima when actuating near natural frequencies (when the system is in resonance), efficiency has been observed to exhibit local maxima below natural frequencies, near natural frequencies, and above natural frequencies (Dewey et al 2013;Moored et al 2014;Quinn et al 2014Quinn et al , 2015Paraz et al 2016), as well as at frequencies relatively far from a natural frequency (Ramananarivo et al 2011;Kang et al 2011;Vanella et al 2009;Zhu et al 2014;Michelin & Llewellyn Smith 2009). This muddled relationship between efficiency and resonance can be partly explained by an ill-conceived notion of natural frequencies.…”

On the diverse roles of fluid dynamic drag in animal swimming and flying

“…As a model, we still consider a passive elastic body where the actuation is localized at one extremity as defined by equation (4.1). Practical examples of such systems have been investigated recently experimentally [51,88,90,92,[95][96][97] and numerically [66,86,89,98]. Equation (4.1) can be written in the weak amplitude approximation, except for the dissipation term, which holds due to the high transversal velocities involved along the body.…”

“…Our first example of fluid-structure interaction involving a passive structure concerns the thrust production of a slender rectangular flexible flapping plate in water, a basic model of a swimmer that has been the subject of several recent studies [51,64,[86][87][88][89][90][91][92][93]. The usual problem here is to determine how the local actuation imposed at one of the extremities of the plate gives rise to thrust production and consequently to locomotion.…”

A fast Chebyshev method for simulating flexible-wing propulsion

“…These studies have demonstrated that flexibility can drastically improve propulsive performance, especially when a wing or fin is driven near resonance [51,50,21,54,69]. Several studies even performed optimization to uncover wing properties and/or actuation strategies that deliver peak performance [94,2,57,73,69].…”

“…Several studies even performed optimization to uncover wing properties and/or actuation strategies that deliver peak performance [94,2,57,73,69]. All of the studies mentioned above, though, considered only the simplest material distributions, such as uniform flexibility [33,2,4,51,50,21,57,69] or flexibility that is localized through a torsional joint [84,54].…”