Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-26T10:28:44.023Z Has data issue: false hasContentIssue false
  • English
  • Français

Clarifying the relationship between efficiency and resonance for flexible inertial swimmers

Published online by Cambridge University Press:  23 August 2018

Daniel Floryan Open the ORCID record for Daniel Floryan [Opens in a new window]  and
Clarence W. Rowley
Daniel Floryan*
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
Clarence W. Rowley
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
*
†Email address for correspondence: dfloryan@princeton.edu
Get access Rights & Permissions [Opens in a new window]

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).

JFM classification

Biological Fluid Dynamics: Propulsion Biological Fluid Dynamics: Swimming/flying
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 853 , 25 October 2018 , pp. 271 - 300
Copyright
© 2018 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alben, S. 2008a The flapping-flag instability as a nonlinear eigenvalue problem. Phys. Fluids 20 (10), 104106.Google Scholar
Alben, S. 2008b Optimal flexibility of a flapping appendage in an inviscid fluid. J. Fluid Mech. 614, 355–380.Google Scholar
Combes, S. A. & Daniel, T. L. 2003 Into thin air: contributions of aerodynamic and inertial-elastic forces to wing bending in the hawkmoth Manduca sexta . J. Exp. Biol. 206 (17), 2999–3006.Google Scholar
Dewey, P. A., Boschitsch, B. M., Moored, K. W., Stone, H. A. & Smits, A. J. 2013 Scaling laws for the thrust production of flexible pitching panels. J. Fluid Mech. 732, 29–46.Google Scholar
Eloy, C., Souilliez, C. & Schouveiler, L. 2007 Flutter of a rectangular plate. J. Fluids Struct. 23 (6), 904–919.Google Scholar
Ferreira de Sousa, P. J. S. A. & Allen, J. J. 2011 Thrust efficiency of harmonically oscillating flexible flat plates. J. Fluid Mech. 674, 43–66.Google Scholar
Floryan, D., Van Buren, T., Rowley, C. W. & Smits, A. J. 2017 Scaling the propulsive performance of heaving and pitching foils. J. Fluid Mech. 822, 386–397.Google Scholar
Garrick, I. E.1936 Propulsion of a flapping and oscillating airfoil. NACA Tech. Rep. 567.Google Scholar
Hua, R.-N., Zhu, L. & Lu, X.-Y. 2013 Locomotion of a flapping flexible plate. Phys. Fluids 25 (12), 121901.Google Scholar
Kang, C.-K., Aono, H., Cesnik, C. E. S. & Shyy, W. 2011 Effects of flexibility on the aerodynamic performance of flapping wings. J. Fluid Mech. 689, 32–74.Google Scholar
Katz, J. & Weihs, D. 1978 Hydrodynamic propulsion by large amplitude oscillation of an airfoil with chordwise flexibility. J. Fluid Mech. 88 (3), 485–497.Google Scholar
Katz, J. & Weihs, D. 1979 Large amplitude unsteady motion of a flexible slender propulsor. J. Fluid Mech. 90 (4), 713–723.Google Scholar
Michelin, S. & Llewellyn Smith, S. G. 2009 Resonance and propulsion performance of a heaving flexible wing. Phys. Fluids 21 (7), 071902.Google Scholar
Moore, M. N. J. 2014 Analytical results on the role of flexibility in flapping propulsion. J. Fluid Mech. 757, 599–612.Google Scholar
Moore, M. N. J. 2017 A fast Chebyshev method for simulating flexible-wing propulsion. J. Comput. Phys. 345, 792–817.Google Scholar
Moored, K. W., Dewey, P. A., Boschitsch, B. M., Smits, A. J. & Haj-Hariri, H. 2014 Linear instability mechanisms leading to optimally efficient locomotion with flexible propulsors. Phys. Fluids 26 (4), 041905.Google Scholar
Paraz, F., Schouveiler, L. & Eloy, C. 2016 Thrust generation by a heaving flexible foil: resonance, nonlinearities, and optimality. Phys. Fluids 28 (1), 011903.Google Scholar
Quinn, D. B., Lauder, G. V. & Smits, A. J. 2014 Scaling the propulsive performance of heaving flexible panels. J. Fluid Mech. 738, 250–267.Google Scholar
Quinn, D. B., Lauder, G. V. & Smits, A. J. 2015 Maximizing the efficiency of a flexible propulsor using experimental optimization. J. Fluid Mech. 767, 430–448.Google Scholar
Ramananarivo, S., Godoy-Diana, R. & Thiria, B. 2011 Rather than resonance, flapping wing flyers may play on aerodynamics to improve performance. Proc. Natl Acad. Sci. USA 108 (15), 5964–5969.Google Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Theodorsen, T.1935 General theory of aerodynamic instability and the mechanism of flutter. NACA Tech. Rep. 496; originally published as ARR-1935.Google Scholar
Vanella, M., Fitzgerald, T., Preidikman, S., Balaras, E. & Balachandran, B. 2009 Influence of flexibility on the aerodynamic performance of a hovering wing. J. Exp. Biol. 212 (1), 95–105.Google Scholar
Wu, T. Y.-T. 1961 Swimming of a waving plate. J. Fluid Mech. 10 (3), 321–344.Google Scholar
Zhu, X., He, G. & Zhang, X. 2014 How flexibility affects the wake symmetry properties of a self-propelled plunging foil. J. Fluid Mech. 751, 164–183.Google Scholar

Floryan et al. supplementary movie

Comparison between Euler-Bernoulli mode P2 and flutter mode S1 for $S = 0.1$. The modes have been normalized so that their second derivatives at the leading edge are real and equal to 1.

Download Floryan et al. supplementary movie(Video)
Video 1.1 MB