Skip to main content

Swimming performance, resonance and shape evolution in heaving flexible panels

  • Alexander P. Hoover (a1), Ricardo Cortez (a1), Eric D. Tytell (a2) and Lisa J. Fauci (a1)

Many animals that swim or fly use their body to accelerate the fluid around them, transferring momentum from their flexible bodies and appendages to the surrounding fluid. The kinematics that emerge from this transfer result from the coupling between the fluid and the active and passive material properties of the flexible body or appendages. To elucidate the fundamental features of the elastohydrodynamics of flexible appendages, recent physical experiments have quantified the propulsive performance of flexible panels that are actuated on their leading edge. Here we present a complementary computational study of a three-dimensional flexible panel that is heaved sinusoidally at its leading edge in an incompressible, viscous fluid. These high-fidelity numerical simulations enable us to examine how propulsive performance depends on mechanical resonance, fluid forces, and the emergent panel deformations. Moreover, the computational model does not require the tethering of the panel. We therefore compare the thrust production of tethered panels to the forward swimming speed of the same panels that can move forward freely. Varying both the passive material properties and the heaving frequency of the panel, we find that local peaks in trailing edge amplitude and forward swimming speed coincide and that they are determined by a non-dimensional quantity, the effective flexibility, that arises naturally in the Euler–Bernoulli beam equation. Modal decompositions of panel deflections reveal that the amplitude of each mode is related to the effective flexibility. Panels of different material properties that are actuated so that their effective flexibilities are closely matched have modal contributions that evolve similarly over the phase of the heaving cycle, leading to similar vortex structures in their wakes and comparable thrust forces and swimming speeds. Moreover, local peaks in the swimming speed and trailing edge amplitude correspond to peaks in the contributions of the different modes. This computational study of freely swimming flexible panels gives further insight into the role of resonance in swimming performance that is important in the engineering and design of robotic propulsors. Moreover, we view this reduced model and its comparison to laboratory experiments as a building block and validation for a more comprehensive three-dimensional computational model of an undulatory swimmer that will couple neural activation, muscle mechanics and body elasticity with the surrounding viscous, incompressible fluid.

Corresponding author
Email address for correspondence:
Hide All
Alben, S. 2008 Optimal flexibility of a flapping appendage in an inviscid fluid. J. Fluid Mech. 614, 355380.
Alben, S. & Shelley, M. 2005 Coherent locomotion as an attracting state for a free flapping body. Proc. Natl Acad. Sci. USA 102 (32), 1116311166.
Alben, S., Witt, C., Baker, T. V., Anderson, E. & Lauder, G. V 2012 Dynamics of freely swimming flexible foils. Phys. Fluids 24 (5), 051901.
Andersen, A., Bohr, T., Schnipper, T. & Walther, J. H. 2017 Wake structure and thrust generation of a flapping foil in two-dimensional flow. J. Fluid Mech. 812, R4.
Bainbridge, R. 1958 The speed of swimming of fish as related to size and to the frequency and amplitude of the tail beat. J. Expl Biol. 35 (1), 109133.
Balay, S., Abhyankar, S., Adams, M., Brown, J., Brune, P., Buschelman, K., Eijkhout, V., Gropp, W., Kaushik, D., Knepley, M. & others2009 PETSc: Web page
Balay, S., Gropp, W. D., McInnes, L. Curfman & Smith, B. F. 1997 Efficient management of parallelism in object-oriented numerical software libraries. In Modern Software Tools for Scientific Computing (ed. Arge, E. et al. ), pp. 163202. Springer.
Bhalla, A. P. S., Bale, R., Griffith, B. E. & Patankar, N. A. 2013 A unified mathematical framework and an adaptive numerical method for fluid–structure interaction with rigid, deforming, and elastic bodies. J. Comput. Phys. 250, 446476.
Bonet, J. & Wood, R. D. 1997 Nonlinear Continuum Mechanics for Finite Element Analysis. Cambridge University Press.
Borazjani, I. & Sotiropoulos, F. 2008 Numerical investigation of the hydrodynamics of carangiform swimming in the transitional and inertial flow regimes. J. Expl Biol. 211 (10), 15411558.
Borazjani, I. & Sotiropoulos, F. 2009 Numerical investigation of the hydrodynamics of anguilliform swimming in the transitional and inertial flow regimes. J. Expl Biol. 212 (4), 576592.
Buchholz, J. H. J. & Smits, A. J. 2006 On the evolution of the wake structure produced by a low-aspect-ratio pitching panel. J. Fluid Mech. 546, 433443.
Buchholz, J. H. J. & Smits, A. J. 2008 The wake structure and thrust performance of a rigid low-aspect-ratio pitching panel. J. Fluid Mech. 603, 331365.
Chu, W.-S., Lee, K.-T., Song, S.-H., Han, M.-W., Lee, J.-Y., Kim, H.-S., Kim, M.-S., Park, Y.-J., Cho, K.-J. & Ahn, S.-H. 2012 Review of biomimetic underwater robots using smart actuators. Intl J. Precis. Engng Manuf. 13 (7), 12811292.
Colin, S. P., Costello, J. H., Dabiri, J. O., Villanueva, A., Blottman, J. B., Gemmell, B. J. & Priya, S. 2012 Biomimetic and live medusae reveal the mechanistic advantages of a flexible bell margin. PloS One 7 (11), e48909.
Demont, M. E. & Gosline, J. M. 1988 Mechanics of jet propulsion in the hydromedusan jellyfish, Polyorchis penicillatus. Part III. A natural resonating bell: the presence and importance of a resonant phenomenon in the locomotor structure. J. Expl Biol. 134 (1), 347361.
Den Hartog, J. P. 1985 Mechanical Vibrations. Courier Corporation.
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, 2946.
Dong, H., Mittal, R. & Najjar, F. M. 2006 Wake topology and hydrodynamic performance of low-aspect-ratio flapping foils. J. Fluid Mech. 566, 309343.
Eldredge, J. D., Toomey, J. & Medina, A. 2010 On the roles of chord-wise flexibility in a flapping wing with hovering kinematics. J. Fluid Mech. 659, 94115.
Falgout, R. D. & Yang, U. M. 2002 Hypre: a library of high performance preconditioners. In Computational Science ICCS 2002 (ed. Sloot, P. M. A. et al. ), pp. 632641. Springer.
Fauci, L. J. & Peskin, C. S. 1988 A computational model of aquatic animal locomotion. J. Comput. Phys. 77 (1), 85108.
Fish, F. E. & Beneski, J. T. 2014 Evolution and bio-inspired design: natural limitations. In Biologically Inspired Design: Computational Methods and Tools (ed. Goel, A. K. et al. ), pp. 287312. Springer.
Green, M. A., Rowley, C. W. & Smits, A. J. 2011 The unsteady three-dimensional wake produced by a trapezoidal pitching panel. J. Fluid Mech. 685, 117145.
Green, M. A. & Smits, A. J. 2008 Effects of three-dimensionality on thrust production by a pitching panel. J. Fluid Mech. 615, 211220.
Griffith, B. E. & Luo, X. 2016 Hybrid finite difference/finite element version of the immersed boundary method. Int. J. Numer. Meth. Engng 33, 126.
Hamlet, C., Fauci, L. J. & Tytell, E. D. 2015 The effect of intrinsic muscular nonlinearities on the energetics of locomotion in a computational model of an anguilliform swimmer. J. Theor. Biol. 385, 119129.
Hamlet, C., Santhanakrishnan, A. & Miller, L. A. 2011 A numerical study of the effects of bell pulsation dynamics and oral arms on the exchange currents generated by the upside-down jellyfish Cassiopea xamachana . J. Expl Biol. 214 (11), 19111921.
Herschlag, G. & Miller, L. 2011 Reynolds number limits for jet propulsion: a numerical study of simplified jellyfish. J. Theor. Biol. 285 (1), 8495.
Hoover, A. & Miller, L. 2015 A numerical study of the benefits of driving jellyfish bells at their natural frequency. J. Theor. Biol. 374, 1325.
Hoover, A. P., Griffith, B. E. & Miller, L. A 2017 Quantifying performance in the medusan mechanospace with an actively swimming three-dimensional jellyfish model. J. Fluid Mech. 813, 11121155.
Hornung, R. D., Wissink, A. M. & Kohn, S. R. 2006 Managing complex data and geometry in parallel structured AMR applications. Engng Comput. 22 (3–4), 181195.
Hover, F. S., Haugsdal, Ø. & Triantafyllou, M. S. 2004 Effect of angle of attack profiles in flapping foil propulsion. J. Fluids Struct. 19 (1), 3747.
HYPRE2011 Hypre: high performance preconditioners.
IBAMR2014 IBAMR: An adaptive and distributed-memory parallel implementation of the immersed boundary method.
Jones, S. K., Laurenza, R., Hedrick, T. L., Griffith, B. E. & Miller, L. A. 2015 Lift vs. drag based mechanisms for vertical force production in the smallest flying insects. J. Theor. Biol. 384, 105120.
Kirk, B. S., Peterson, J. W., Stogner, R. H. & Carey, G. F. 2006 libMesh: a C++ library for parallel adaptive mesh refinement/coarsening simulations. Engng Comput. 22 (3–4), 237254.
Leftwich, M. C., Tytell, E. D., Cohen, A. H. & Smits, A. J. 2012 Wake structures behind a swimming robotic lamprey with a passively flexible tail. J. Expl Biol. 215 (3), 416425.
Lehn, A. M., Thornycroft, P. J., Lauder, G. V. & Leftwich, M. C. 2017 Effect of input perturbation on the performance and wake dynamics of aquatic propulsion in heaving flexible foils. Phys. Rev. Fluids 2 (2), 023101.
Licht, S. C., Wibawa, M. S., Hover, F. S. & Triantafyllou, M. S. 2010 In-line motion causes high thrust and efficiency in flapping foils that use power downstroke. J. Expl Biol. 213 (1), 6371.
Liu, P. & Bose, N. 1997 Propulsive performance from oscillating propulsors with spanwise flexibility. Proc. R. Soc. Lond. A 453, 17631770.
Lucas, K. N., Johnson, N., Beaulieu, W. T., Cathcart, E., Tirrell, G., Colin, S. P., Gemmell, B. J., Dabiri, J. O. & Costello, J. H. 2014 Bending rules for animal propulsion. Nat. Commun. 5, 3293.
Lucas, K. N., Thornycroft, P. J., 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.
Megill, W. M., Gosline, J. M. & Blake, R. W. 2005 The modulus of elasticity of fibrillin-containing elastic fibres in the mesoglea of the hydromedusa Polyorchis penicillatus . J. Expl Biol. 208 (20), 38193834.
Michelin, S. & Llewellyn Smith, S. G. 2009 Resonance and propulsion performance of a heaving flexible wing. Phys. Fluids 21 (7), 071902.
Miller, L. A. & Peskin, C. S. 2004 When vortices stick: an aerodynamic transition in tiny insect flight. J. Expl Biol. 207 (17), 30733088.
Miller, L. A. & Peskin, C. S. 2005 A computational fluid dynamics of ‘clap and fling’ in the smallest insects. J. Expl Biol. 208 (2), 195212.
Miller, L. A. & Peskin, C. S. 2009 Flexible clap and fling in tiny insect flight. J. Expl Biol. 212 (19), 30763090.
Mittal, R. & Iaccarino, G. 2005 Immersed boundary methods. Annu. Rev. Fluid Mech. 37, 239261.
Moore, M. N. J. 2014 Analytical results on the role of flexibility in flapping propulsion. J. Fluid Mech. 757, 599612.
Moore, M. N. J. 2015 Torsional spring is the optimal flexibility arrangement for thrust production of a flapping wing. Phys. Fluids 27 (9), 091701.
Moore, M. N. J. 2017 A fast Chebyshev method for simulating flexible-wing propulsion. J. Comput. Phys. 345, 792817.
Paraz, F., Eloy, C. & Schouveiler, L. 2014 Experimental study of the response of a flexible plate to a harmonic forcing in a flow. C. R. Méc. 342 (9), 532538.
Peskin, C. S. 1977 Numerical analysis of blood flow in the heart. J. Comput. Phys. 25 (3), 220252.
Peskin, C. S. 2002 The immersed boundary method. In Acta Numerica (ed. Iserles, A.), vol. 11, pp. 479517. Cambridge University Press.
Piñeirua, M., Thiria, B. & Godoy-Diana, R. 2017 Modelling of an actuated elastic swimmer. J. Fluid Mech. 829, 731750.
Quinn, D. B., Lauder, G. V. & Smits, A. J. 2014a Flexible propulsors in ground effect. Bioinspir. Biomim. 9 (3), 036008.
Quinn, D. B., Lauder, G. V. & Smits, A. J. 2014b Scaling the propulsive performance of heaving flexible panels. J. Fluid Mech. 738, 250267.
Quinn, D. B., Lauder, G. V. & Smits, A. J. 2015 Maximizing the efficiency of a flexible propulsor using experimental optimization. J. Fluid Mech. 767, 430448.
Raj, A. & Thakur, A. 2016 Fish-inspired robots: design, sensing, actuation, and autonomy. A review of research. Bioinspir. Biomim. 11 (3), 031001.
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. 108 (15), 59645969.
Richards, A. J. & Oshkai, P. 2015 Effect of the stiffness, inertia and oscillation kinematics on the thrust generation and efficiency of an oscillating-foil propulsion system. J. Fluids Struct. 57, 357374.
Root, R. G. & Courtland, H.-W. 1999 Swimming fish and fish-like models: the harmonic structure of undulatory waves suggests that fish actively tune their bodies. In International Symposium on Unmanned Untethered Submersible Technology, pp. 378388. University of New Hampshire, Marine Systems.
SAMRAI2007 SAMRAI: Structured Adaptive Mesh Refinement Application Infrastructure
Shoele, K. & Zhu, Q. 2012 Leading edge strengthening and the propulsion performance of flexible ray fins. J. Fluid Mech. 693, 402432.
de Sousa, P. J. F. & Allen, J. J. 2011 Thrust efficiency of harmonically oscillating flexible flat plates. J. Fluid Mech. 674, 4366.
Spagnolie, S. E., Moret, L., Shelley, M. J. & Zhang, J. 2010 Surprising behaviors in flapping locomotion with passive pitching. Phys. Fluids 22 (4), 041903.
Taylor, G. K., Nudds, R. L. & Thomas, A. L. 2003 Flying and swimming animals cruise at a Strouhal number tuned for high power efficiency. Nature 425 (6959), 707711.
Triantafyllou, G. S., Triantafyllou, M. S. & Grosenbaugh, M. A. 1993 Optimal thrust development in oscillating foils with application to fish propulsion. J. Fluids Struct. 7, 205224.
Tytell, E. D., Hsu, C.-Y. & Fauci, L. J. 2014 The role of mechanical resonance in the neural control of swimming in fishes. Zoology 117 (1), 4856.
Tytell, E. D., Hsu, C.-Y., Williams, T. L., Cohen, A. H. & Fauci, L. J. 2010 Interactions between internal forces, body stiffness, and fluid environment in a neuromechanical model of lamprey swimming. Proc. Natl Acad. Sci. 107 (46), 1983219837.
Tytell, E. D., Leftwich, M. C., Hsu, C.-Y., Griffith, B. E., Cohen, A. H., Smits, A. J., Hamlet, C. & Fauci, L. J. 2016 Role of body stiffness in undulatory swimming: insights from robotic and computational models. Phys. Rev. Fluids 1 (7), 073202.
Van Buren, T., Floryan, D., Brunner, D., Senturk, U. & Smits, A. J. 2017 Impact of trailing edge shape on the wake and propulsive performance of pitching panels. Phys. Rev. Fluids 2 (1), 014702.
Van Buren, T., Floryan, D., Wei, N. & Smits, A. J. 2018 Flow speed has little impact on propulsive characteristics of oscillating foils. Phys. Rev. Fluids 3, 013103.
Van Eysden, C. A. & Sader, J. E. 2006 Resonant frequencies of a rectangular cantilever beam immersed in a fluid. J. Appl. Phys. 100 (11), 114916.
Van Eysden, C. A. & Sader, J. E. 2007 Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope: arbitrary mode order. J. Appl. Phys. 101 (4), 044908.
Van Eysden, C. A. & Sader, J. E. 2009 Frequency response of cantilever beams immersed in compressible fluids with applications to the atomic force microscope. J. Appl. Phys. 106 (9), 094904.
Vandenberghe, N., Childress, S. & Zhang, J. 2006 On unidirectional flight of a free flapping wing. Phys. Fluids 18 (1), 014102.
Weaver, W., Timoshenko, S. P. & Young, D. H. 1990 Vibration Problems in Engineering. Wiley.
Williamson, C. H. K. & Roshko, A. 1988 Vortex formation in the wake of an oscillating cylinder. J. Fluids Struct. 2 (4), 355381.
Yeh, P. D. & Alexeev, A. 2014 Free swimming of an elastic plate plunging at low Reynolds number. Phys. Fluids 26 (5), 053604.
Zhang, C., Guy, R. D., Mulloney, B., Zhang, Q. & Lewis, T. J. 2014 Neural mechanism of optimal limb coordination in crustacean swimming. Proc. Natl Acad. Sci. USA 111 (38), 1384013845.
Zhu, X., He, G. & Zhang, X. 2014 Numerical study on hydrodynamic effect of flexibility in a self-propelled plunging foil. Comput. Fluids 97, 120.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

JFM classification

Type Description Title

Hoover et al. supplementary movie 5
Isocontour plots of the dimensionless y-vorticity for three panels that have a high dimensionless swimming speed, but differ in effective flexibility (Π≈17), at the same points of the phase. The resulting flow structures generated by all three panels vary from one another.

 Video (17.1 MB)
17.1 MB

Hoover et al. supplementary movie 1
Kinematics of untethered panels with fixed bending moduli (EI=1e-7 Nm2) but varying heaving frequency (Φ=0.5,1.0,2.0,3.0 s-1, from left to right)

 Video (8.0 MB)
8.0 MB

Hoover et al. supplementary movie 6
Isocontour plots of the dimensionless y-vorticity for three panels of similar effective flexibility (Π≈6) at the same points of the phase. Note that the bending modulus and heaving frequency for all three panels vary.

 Video (19.7 MB)
19.7 MB

Hoover et al. supplementary movie 3
Isocontours of dimensionless y-vorticity at the same points of the heaving cycle for three panels of fixed heaving frequency (Φ), but differing bending moduli (EI).

 Video (18.7 MB)
18.7 MB

Hoover et al. supplementary movie 4
Isocontours of dimensionless y-vorticity, flow velocity, and pressure for a panel (Φ=.625 s-1, EI=1e-7 Nm2).

 Video (8.5 MB)
8.5 MB

Hoover et al. supplementary movie 7
Isocontour plots of the dimensionless y-vorticity for three panels of similar effective flexibility (Π≈17) at the same points of the phase. Note that the bending modulus and heaving frequency for all three panels vary.

 Video (17.9 MB)
17.9 MB

Hoover et al. supplementary movie 2
Isocontours of dimensionless y-vorticity at the same points of the heaving cycle for three panels of differing heaving frequency (Φ), but fixed bending moduli (EI).

 Video (16.3 MB)
16.3 MB


Altmetric attention score

Full text views

Total number of HTML views: 3
Total number of PDF views: 258 *
Loading metrics...

Abstract views

Total abstract views: 562 *
Loading metrics...

* Views captured on Cambridge Core between 23rd May 2018 - 19th September 2018. This data will be updated every 24 hours.