Hostname: page-component-77c89778f8-n9wrp Total loading time: 0 Render date: 2024-07-17T11:57:18.287Z Has data issue: false hasContentIssue false
  • English
  • Français

Distributed flexibility in inertial swimmers

Published online by Cambridge University Press:  11 February 2020

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@alumni.princeton.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We study a linear inviscid model of a passively flexible swimmer with distributed flexibility, calculating its propulsive performance and optimal distributions of flexibility. The frequencies of actuation and mean stiffness ratios we consider span a large range, while the mass ratio is fixed to a low value representative of swimmers. We present results showing how the trailing edge deflection, thrust coefficient, power coefficient and efficiency vary with frequency, mean stiffness and stiffness distribution. Swimmers with distributed flexibility have the same qualitative features as those with uniform flexibility. Significant gains in thrust can be made, however, by tuning the stiffness such that a resonant response is triggered, or by concentrating stiffness towards the leading edge if resonance cannot be triggered. To minimize power, the opposite is true. Meaningful gains in efficiency can be made at low frequencies by concentrating stiffness away from the leading edge, since doing so induces efficient travelling wave kinematics. We also speculate on 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 efficiency-maximizing distributions of flexibility to tend towards thrust-maximizing ones, representative of what is found in nature.

JFM classification

Biological Fluid Dynamics: Swimming/flying Biological Fluid Dynamics: Propulsion
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 888 , 10 April 2020 , A24
Copyright
© The Author(s), 2020. Published by 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.CrossRefGoogle Scholar
Alben, S. 2008b Optimal flexibility of a flapping appendage in an inviscid fluid. J. Fluid Mech. 614, 355–380.CrossRefGoogle Scholar
Alben, S. 2009 Simulating the dynamics of flexible bodies and vortex sheets. J. Comput. Phys. 228 (7), 2587–2603.CrossRefGoogle Scholar
Bergmann, C. 1839 Über die bewegungen von radius und ulna am vogelflügel. Arch. Anat. Physiol. wiss Med. 6, 296–300.Google Scholar
Combes, S. A. & Daniel, T. L. 2003 Flexural stiffness in insect wings II. Spatial distribution and dynamic wing bending. J. Expl Biol. 206 (17), 2989–2997.CrossRefGoogle ScholarPubMed
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.CrossRefGoogle Scholar
Eloy, C., Souilliez, C. & Schouveiler, L. 2007 Flutter of a rectangular plate. J. Fluids Struct. 23 (6), 904–919.CrossRefGoogle Scholar
Fish, F. E. & Lauder, G. V. 2006 Passive and active flow control by swimming fishes and mammals. Annu. Rev. Fluid Mech. 38, 193–224.CrossRefGoogle Scholar
Floryan, D. & Rowley, C. W. 2018 Clarifying the relationship between efficiency and resonance for flexible inertial swimmers. J. Fluid Mech. 853, 271–300.CrossRefGoogle 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.CrossRefGoogle Scholar
Floryan, D., Van Buren, T. & Smits, A. J. 2018 Efficient cruising for swimming and flying animals is dictated by fluid drag. Proc. Natl Acad. Sci. 115 (32), 8116–8118.CrossRefGoogle ScholarPubMed
Goza, A., Floryan, D. & Rowley, C. W. 2020 Connections between resonance and nonlinearity in swimming performance of a flexible heaving plate. J. Fluid Mech.; accepted.Google Scholar
Kancharala, A. K. & Philen, M. K. 2016 Optimal chordwise stiffness profiles of self-propelled flapping fins. Bioinspir. Biomim. 11 (5), 056016.CrossRefGoogle ScholarPubMed
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.CrossRefGoogle 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.CrossRefGoogle Scholar
Katz, J. & Weihs, D. 1979 Large amplitude unsteady motion of a flexible slender propulsor. J. Fluid Mech. 90 (4), 713–723.CrossRefGoogle Scholar
Lucas, K. N., Thornycroft, P. J. M., Gemmell, B. J., Colin, S. P., Costello, J. H. & Lauder, G. V. 2015 Effects of non-uniform stiffness on the swimming performance of a passively-flexing, fish-like foil model. Bioinspir. Biomim. 10 (5), 056019.CrossRefGoogle ScholarPubMed
2016 MATLAB and Optimization Toolbox Release 2016a. The Mathworks, Inc.Google Scholar
Michelin, S. & Llewellyn Smith, S. G. 2009 Resonance and propulsion performance of a heaving flexible wing. Phys. Fluids 21 (7), 071902.CrossRefGoogle Scholar
Moore, M. N. J. 2015 Torsional spring is the optimal flexibility arrangement for thrust production of a flapping wing. Phys. Fluids 27 (9), 091701.CrossRefGoogle Scholar
Moore, M. N. J. 2017 A fast Chebyshev method for simulating flexible-wing propulsion. J. Comput. Phys. 345, 792–817.CrossRefGoogle 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.CrossRefGoogle Scholar
Paraz, F., Schouveiler, L. & Eloy, C. 2016 Thrust generation by a heaving flexible foil: resonance, nonlinearities, and optimality. Phys. Fluids 28 (1), 011903.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle ScholarPubMed
Riggs, P., Bowyer, A. & Vincent, J. 2010 Advantages of a biomimetic stiffness profile in pitching flexible fin propulsion. J. Bionic Engng 7 (2), 113–119.CrossRefGoogle Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Shoele, K. & Zhu, Q. 2013 Performance of a wing with nonuniform flexibility in hovering flight. Phys. Fluids 25 (4), 041901.CrossRefGoogle 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.CrossRefGoogle Scholar
Triantafyllou, M. S., Triantafyllou, G. S. & Yue, D. K. P. 2000 Hydrodynamics of fishlike swimming. Annu. Rev. Fluid Mech. 32 (1), 33–53.CrossRefGoogle Scholar
Vanella, M., Fitzgerald, T., Preidikman, S., Balaras, E. & Balachandran, B. 2009 Influence of flexibility on the aerodynamic performance of a hovering wing. J. Expl Biol. 212 (1), 95–105.CrossRefGoogle ScholarPubMed
Wang, Z. J. 2005 Dissecting insect flight. Annu. Rev. Fluid Mech. 37, 183–210.CrossRefGoogle Scholar
Wu, T. Y.-T. 1961 Swimming of a waving plate. J. Fluid Mech. 10 (3), 321–344.CrossRefGoogle Scholar
Wu, T. Y.-T. 2011 Fish swimming and bird/insect flight. Annu. Rev. Fluid Mech. 43, 25–58.CrossRefGoogle Scholar
Yeh, P. D., Li, Y. & Alexeev, A. 2017 Efficient swimming using flexible fins with tapered thickness. Phys. Rev. Fluids 2 (10), 102101.CrossRefGoogle 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.CrossRefGoogle Scholar