Clarifying the relationship between efficiency and resonance for flexible inertial swimmers

Abstract

We study a linear inviscid model of a passively flexible swimmer, calculating its propulsive performance, eigenvalues, and eigenfunctions with an eye towards clarifying the relationship between efficiency and resonance. The frequencies of actuation and stiffness ratios we consider span a large range, while the mass ratio is mostly fixed to a low value representative of swimmers. We present results showing how the trailing edge deflection, thrust coefficient, power coefficient, and efficiency vary in the stiffness-frequency plane. The trailing edge deflection, thrust coefficient, and power coefficient show sharp ridges of resonant behaviour for mid-to-high frequencies and stiffnesses, whereas the efficiency does not show resonant behaviour anywhere. For low frequencies and stiffnesses, the resonant peaks smear together and the efficiency is high. In this region, flutter modes emerge, inducing travelling wave kinematics which make the swimmer more efficient. We also consider the effects of a finite Reynolds number in the form of streamwise drag. The drag adds an offset to the net thrust produced by the swimmer, causing resonant peaks to appear in the efficiency (as observed in experiments in the literature).

Distributed flexibility in inertial swimmers

“…The parameters we use in the following sections are summarized in Table 1. As shown in Floryan and Rowley [2018], the value of the mean mass ratio R qualitatively changes the propulsion of a flapping plate. At low values, however, the mass of the plate is dominated by the mass of the fluid.…”

“…Passive flexibility changes the thrust that a flapping plate produces, as well as the efficiency of thrust production. It has generally been found that, compared to rigid plates, uniformly flexible plates produce greater thrust when actuated near a fluidstructure natural frequency, and less thrust otherwise, but the efficiency of uniformly flexible plates is greater than that of rigid plates over a broad range of frequencies and stiffnesses [Alben, 2008b, Ferreira de Sousa and Allen, 2011, Dewey et al, 2013, Katz and Weihs, 1978, 1979, Quinn et al, 2014, Floryan and Rowley, 2018. While thrust generally exhibits local maxima when actuating near natural frequencies, 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, 2014, 2015, Paraz 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 and Llewellyn Smith, 2009.…”

“…It is very important to note, however, that in the cited works, the plates with uniform and distributed flexibilities had different mean stiffnesses, making it difficult to distinguish between the effects of changes in mean stiffness and changes in stiffness distribution. Being able to make the distinction is important because, as we will show later, changing the mean stiffness can significantly change natural frequencies, which have significant effects on thrust and efficiency, and changing the mean stiffness also changes the off-resonance behaviour in efficiency [Alben, 2008b, Floryan andRowley, 2018].…”

Connections between resonance and nonlinearity in swimming performance of a flexible heaving plate

“…The dual aims of understanding fish swimming and developing agile, efficient underwater vehicles have propelled research on the canonical problem of flow past a flapping flexible plate. Experimental (Ramananarivo et al 2011;Alben et al 2012;Dewey et al 2013;Quinn et al 2014Quinn et al , 2015, computational (Vanella et al 2009;Hua et al 2013;Zhu et al 2014;Zhang et al 2017), and theoretical work (Alben 2008;Ramananarivo et al 2011;Alben et al 2012;Floryan & Rowley 2018) on this problem has revealed propulsive benefits to flexibility over a range of Reynolds numbers (and for inviscid flows), plate parameters, and amplitudes and frequencies associated with the plate's kinematics. Many studies considered actuation frequencies and material parameters for which the plate exhibits 'first-mode' flapping; i.e., with the plate shape similar to that of the first mode of a clamped Euler-Bernoulli beam.…”

“…Many studies considered actuation frequencies and material parameters for which the plate exhibits 'first-mode' flapping; i.e., with the plate shape similar to that of the first mode of a clamped Euler-Bernoulli beam. However, propulsive benefits were shown to persist for parameters that led to 'higher mode' shapes (Alben 2008;Alben et al 2012;Quinn et al 2014;Floryan & Rowley 2018). Because this canonical problem involves the passive motion of deformable structures, a prevailing question is the extent to which performance is connected to structural resonance.…”