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Propulsion of a flexible foil in a wavy flow: resonance, antiresonance and destructive self-interference

Published online by Cambridge University Press:  14 November 2025

Abdur Rehman Open the ORCID record for Abdur Rehman [Opens in a new window]  and
Daniel Floryan Open the ORCID record for Daniel Floryan [Opens in a new window]
Abdur Rehman*
Affiliation:
Department of Mechanical and Aerospace Engineering, University of Houston , Houston, TX 77204, USA
Daniel Floryan*
Affiliation:
Department of Mechanical and Aerospace Engineering, University of Houston , Houston, TX 77204, USA
*
Corresponding authors: Abdur Rehman, abdurrehman.92@gmail.com; Daniel Floryan, dfloryan@uh.edu
Corresponding authors: Abdur Rehman, abdurrehman.92@gmail.com; Daniel Floryan, dfloryan@uh.edu
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Abstract

Swimming and flying animals demonstrate remarkable adaptations to diverse flow conditions in their environments. In this study, we aim to advance the fundamental understanding of the interaction between flexible bodies and heterogeneous flow conditions. We develop a linear inviscid model of an elastically mounted foil that passively pitches in response to a prescribed heaving motion and an incoming flow that consists of a travelling wave disturbance superposed on a uniform flow. In addition to the well-known resonant response, the wavy flow induces an antiresonant response for non-dimensional phase velocities near unity due to the emergence of non-circulatory forces that oppose circulatory forces. We also find that the wavy flow destructively interferes with itself, effectively rendering the foil a low-pass filter. The net result is that the waviness of the flow always improves thrust and efficiency when the wavy flow is of a different frequency than the prescribed heaving motion. Such a simple statement cannot be made when the wavy flow and heaving motion have the same frequency. Depending on the wavenumber and relative phase, the two may work in concert or in opposition, but they do open the possibility of simultaneous propulsion and net energy extraction from the flow, which, according to our model, is impossible in a uniform flow.

JFM classification

Aerodynamics: Flow-structure interactions Biological Fluid Dynamics: Propulsion Biological Fluid Dynamics: Swimming/flying

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Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1023 , 25 November 2025 , A11
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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References

Alben, S. 2008 Optimal flexibility of a flapping appendage in an inviscid fluid. J. Fluid Mech. 614, 355–380.10.1017/S0022112008003297CrossRefGoogle Scholar
Anderson, J.M., Streitlien, K., Barrett, D.S. & Triantafyllou, M.S. 1998 Oscillating foils of high propulsive efficiency. J. Fluid Mech. 360, 41–72.10.1017/S0022112097008392CrossRefGoogle Scholar
Ayancik, F., Zhong, Q., Quinn, D.B., Brandes, A., Bart-Smith, H. & Moored, K.W. 2019 Scaling laws for the propulsive performance of three-dimensional pitching propulsors. J. Fluid Mech. 871, 1117–1138.10.1017/jfm.2019.334CrossRefGoogle Scholar
Baddoo, P.J., Moore, N.J., Oza, A.U. & Crowdy, D.G. 2023 Generalization of waving-plate theory to multiple interacting swimmers. Commun. Pure Appl. Maths 76 (12), 3811–3851.10.1002/cpa.22113CrossRefGoogle Scholar
Beal, D.N., Hover, F.S., Triantafyllou, M.S., Liao, J.C. & Lauder, G.V. 2006 Passive propulsion in vortex wakes. J. Fluid Mech. 549, 385–402.10.1017/S0022112005007925CrossRefGoogle Scholar
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, 331–365.10.1017/S0022112008000906CrossRefGoogle ScholarPubMed
Catlett, M.R., Anderson, J.M., Badrya, C. & Baeder, J.D. 2020 Unsteady response of airfoils due to small-scale pitching motion with considerations for foil thickness and wake motion. J. Fluids Struct. 94, 102889.10.1016/j.jfluidstructs.2020.102889CrossRefGoogle Scholar
Chopra, M.G. 1974 Hydromechanics of lunate-tail swimming propulsion. J. Fluid Mech. 64 (2), 375–392.10.1017/S002211207400245XCrossRefGoogle 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, 29–46.10.1017/jfm.2013.384CrossRefGoogle Scholar
Edwards, J.W. 1977 Unsteady aerodynamic modeling and active aeroelastic control. Tech. Rep. NASA-CR-148019. NASA.Google Scholar
Fernandez-Feria, R. 2016 Linearized propulsion theory of flapping airfoils revisited. Phys. Rev. Fluids 1 (8), 084502.10.1103/PhysRevFluids.1.084502CrossRefGoogle Scholar
Fernandez-Feria, R. 2017 Note on optimum propulsion of heaving and pitching airfoils from linear potential theory. J. Fluid Mech. 826, 781–796.10.1017/jfm.2017.500CrossRefGoogle Scholar
Floryan, D. & Rowley, C.W. 2018 Clarifying the relationship between efficiency and resonance for flexible inertial swimmers. J. Fluid Mech. 853, 271–300.10.1017/jfm.2018.581CrossRefGoogle Scholar
Floryan, D. & Rowley, C.W. 2020 Distributed flexibility in inertial swimmers. J. Fluid Mech. 888, A24.10.1017/jfm.2020.49CrossRefGoogle 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, 386–397.10.1017/jfm.2017.302CrossRefGoogle 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. USA 115 (32), 8116–8118.10.1073/pnas.1805941115CrossRefGoogle ScholarPubMed
Floryan, D., Van Buren, T. & Smits, A.J. 2019 Large-amplitude oscillations of foils for efficient propulsion. Phys. Rev. Fluids 4 (9), 093102.10.1103/PhysRevFluids.4.093102CrossRefGoogle Scholar
Floryan, D., Van Buren, T. & Smits, A.J. 2020 Swimmers’ wake structures are not reliable indicators of swimming performance. Bioinspir. Biomim. 15 (2), 024001.10.1088/1748-3190/ab6fb9CrossRefGoogle Scholar
Garrick, I.E. 1936 Propulsion of a flapping and oscillating airfoil. Tech. Rep. 567. NACA.Google Scholar
Gazzola, M., Argentina, M. & Mahadevan, L. 2014 Scaling macroscopic aquatic locomotion. Nat. Phys. 10 (10), 758–761.10.1038/nphys3078CrossRefGoogle Scholar
Goldstein, M.E. & Atassi, H. 1976 A complete second-order theory for the unsteady flow about an airfoil due to a periodic gust. J. Fluid Mech. 74 (4), 741–765.10.1017/S0022112076002036CrossRefGoogle Scholar
Gordillo, J.M. 2025 A note on the thrust of airfoils. J. Fluid Mech. 1012, A6.10.1017/jfm.2025.10177CrossRefGoogle Scholar
Gordnier, R.E., Chimakurthi, S.K., Cesnik, C.E.S. & Attar, P.J. 2013 High-fidelity aeroelastic computations of a flapping wing with spanwise flexibility. J. Fluids Struct. 40, 86–104.10.1016/j.jfluidstructs.2013.03.009CrossRefGoogle Scholar
Goza, A., Floryan, D. & Rowley, C. 2020 Connections between resonance and nonlinearity in swimming performance of a flexible heaving plate. J. Fluid Mech. 888, A30.10.1017/jfm.2020.60CrossRefGoogle Scholar
Heathcote, S., Wang, Z. & Gursul, I. 2008 Effect of spanwise flexibility on flapping wing propulsion. J. Fluids Struct. 24 (2), 183–199.10.1016/j.jfluidstructs.2007.08.003CrossRefGoogle Scholar
von Kármán, T.H. & Sears, W.R. 1938 Airfoil theory for non-uniform motion. J. Aeronaut. Sci. 5 (10), 379–390.10.2514/8.674CrossRefGoogle Scholar
Liao, J.C. 2007 A review of fish swimming mechanics and behaviour in altered flows. Phil. Trans. R. Soc. B: Biol. Sci. 362 (1487), 1973–1993.10.1098/rstb.2007.2082CrossRefGoogle ScholarPubMed
Liu, P. & Bose, N. 1997 Propulsive performance from oscillating propulsors with spanwise flexibility. Proc. R. Soc. Lond. A: Math. Phys. Engng Sci. 453 (1963), 1763–1770.10.1098/rspa.1997.0095CrossRefGoogle Scholar
Lysak, P.D., Capone, D.E. & Jonson, M.L. 2013 Prediction of high frequency gust response with airfoil thickness effects. J. Fluids Struct. 39, 258–274.10.1016/j.jfluidstructs.2013.02.006CrossRefGoogle Scholar
Massaro, M. & Graham, J.M.R. 2015 The effect of three-dimensionality on the aerodynamic admittance of thin sections in free stream turbulence. J. Fluids Struct. 57, 81–90.10.1016/j.jfluidstructs.2015.05.012CrossRefGoogle Scholar
Michelin, S. & Llewellyn Smith, S.G. 2009 Resonance and propulsion performance of a heaving flexible wing. Phys. Fluids 21 (7), 071902.10.1063/1.3177356CrossRefGoogle Scholar
Moore, M.N.J. 2014 Analytical results on the role of flexibility in flapping propulsion. J. Fluid Mech. 757, 599–612.10.1017/jfm.2014.533CrossRefGoogle Scholar
Moore, M.N.J. 2015 Torsional spring is the optimal flexibility arrangement for thrust production of a flapping wing. Phys. Fluids 27 (9), 091701.10.1063/1.4930235CrossRefGoogle Scholar
Newton, I. 2024 Migratory flight. In The Migration Ecology of Birds, 2nd edn, chap. 3, pp. 29–50. Academic Press.10.1016/B978-0-12-823751-9.00021-XCrossRefGoogle Scholar
Paraz, F., Schouveiler, L. & Eloy, C. 2016 Thrust generation by a heaving flexible foil: resonance, nonlinearities, and optimality. Phys. Fluids 28 (1), 011903.10.1063/1.4939499CrossRefGoogle Scholar
Quinn, D.B., Lauder, G.V. & Smits, A.J. 2014 Scaling the propulsive performance of heaving flexible panels. J. Fluid Mech. 738, 250–267.10.1017/jfm.2013.597CrossRefGoogle Scholar
Senturk, U. & Smits, A.J. 2019 Reynolds number scaling of the propulsive performance of a pitching airfoil. AIAA J. 57 (7), 2663–2669.10.2514/1.J058371CrossRefGoogle Scholar
Smits, A.J. 2019 Undulatory and oscillatory swimming. J. Fluid Mech. 874, P1.10.1017/jfm.2019.284CrossRefGoogle Scholar
Spagnolie, S.E., Moret, L., Shelley, M.J. & Zhang, J. 2010 Surprising behaviors in flapping locomotion with passive pitching. Phys. Fluids 22 (4), 041903.10.1063/1.3383215CrossRefGoogle Scholar
Taha, H. & Rezaei, A.S. 2019 Viscous extension of potential-flow unsteady aerodynamics: the lift frequency response problem. J. Fluid Mech. 868, 141–175.10.1017/jfm.2019.159CrossRefGoogle Scholar
Taylor, G.K., Nudds, R.L. & Thomas, A.L.R. 2003 Flying and swimming animals cruise at a Strouhal number tuned for high power efficiency. Nature 425 (6959), 707–711.10.1038/nature02000CrossRefGoogle Scholar
Theodorsen, T. 1935 General theory of aerodynamic instability and the mechanism of flutter. Tech. Rep. 496. NACA.Google Scholar
Triantafyllou, M.S., Triantafyllou, G.S. & Gopalkrishnan, R. 1991 Wake mechanics for thrust generation in oscillating foils. Phys. Fluids A: Fluid Dyn. 3 (12), 2835–2837.10.1063/1.858173CrossRefGoogle Scholar
Van Buren, T., Floryan, D. & Smits, A.J. 2019 Scaling and performance of simultaneously heaving and pitching foils. AIAA J. 57 (9), 3666–3677.10.2514/1.J056635CrossRefGoogle Scholar
Verma, S., Novati, G. & Koumoutsakos, P. 2018 Efficient collective swimming by harnessing vortices through deep reinforcement learning. Proc. Natl Acad. Sci. 115 (23), 5849–5854.10.1073/pnas.1800923115CrossRefGoogle ScholarPubMed
Weihs, D. 1973 Hydromechanics of fish schooling. Nature 241 (5387), 290–291.10.1038/241290a0CrossRefGoogle Scholar
Whittlesey, R.W., Liska, S. & Dabiri, J.O. 2010 Fish schooling as a basis for vertical axis wind turbine farm design. Bioinspir. Biomim. 5 (3), 035005.10.1088/1748-3182/5/3/035005CrossRefGoogle ScholarPubMed
Wu, T.Y.-T. 1961 Swimming of a waving plate. J. Fluid Mech. 10 (3), 321–344.10.1017/S0022112061000949CrossRefGoogle Scholar
Wu, T.Y.-T. 1971 Hydromechanics of swimming propulsion. Part 1. Swimming of a two-dimensional flexible plate at variable forward speeds in an inviscid fluid. J. Fluid Mech. 46 (2), 337–355.10.1017/S0022112071000570CrossRefGoogle Scholar
Wu, T.Y.-T. 1972 Extraction of flow energy by a wing oscillating in waves. J. Ship Res. 16 (01), 66–78.10.5957/jsr.1972.16.1.66CrossRefGoogle Scholar
Wu, T.Y.-T. 2011 Fish swimming and bird/insect flight. Annu. Rev. Fluid Mech. 43, 25–58.10.1146/annurev-fluid-122109-160648CrossRefGoogle Scholar
Young, J. & Lai, J.C.S. 2007 Mechanisms influencing the efficiency of oscillating airfoil propulsion. AIAA J. 45 (7), 1695–1702.10.2514/1.27628CrossRefGoogle Scholar
Yudin, D., Floryan, D. & Van Buren, T. 2023 Propulsive performance of oscillating plates with time-periodic flexibility. J. Fluid Mech. 959, A31.10.1017/jfm.2023.166CrossRefGoogle Scholar
Zhang, J. 2017 Footprints of a flapping wing. J. Fluid Mech. 818, 1–4.10.1017/jfm.2017.173CrossRefGoogle Scholar
Zhu, Y., Tian, F.-B., Young, J., Liao, J.C. & Lai, J.C.S. 2021 A numerical study of fish adaption behaviors in complex environments with a deep reinforcement learning and immersed boundary–lattice Boltzmann method. Sci. Rep. UK 11 (1), 1691.10.1038/s41598-021-81124-8CrossRefGoogle ScholarPubMed