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Flow associated with Lighthill’s elongated-body theory

Published online by Cambridge University Press:  24 April 2025

Christophe Eloy Open the ORCID record for Christophe Eloy [Opens in a new window]  and
Sébastien Michelin Open the ORCID record for Sébastien Michelin [Opens in a new window]
Christophe Eloy*
Affiliation:
Aix Marseille Univ, CNRS, Centrale Med, IRPHE, Marseille, France
Sébastien Michelin
Affiliation:
LadHyX, CNRS - Ecole Polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France
*
Corresponding author: Christophe Eloy, [email protected]
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Abstract

The hydrodynamic forces acting on an undulating swimming fish consist of two components: a drag-based resistive force, and a reactive force originating from the necessary acceleration of an added mass of water. Lighthill’s elongated-body theory, based on potential flow, provides a framework for calculating this reactive force. By leveraging the high aspect ratio of most fish, the theory simplifies the problem into a series of independent two-dimensional slices of fluids along the fish’s body, which exchange momentum with the body and neighbouring slices. Using momentum conservation arguments, Lighthill’s theory predicts the total thrust generated by an undulating fish, based solely on the dimensions and kinematics of its caudal fin. However, the assumption of independent slices has led to the common misconception that the flow produced lacks a longitudinal component. In this paper, we revisit Lighthill’s theory, offering a modern reinterpretation using essential singularities of potential flows. We then extend it to predict the full three-dimensional flow field induced by the fish’s body motion. Our results compare favourably with numerical simulations of realistic fish geometries.

JFM classification

Aerodynamics: Flow-structure interactions Biological Fluid Dynamics: Swimming/flying
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1009 , 25 April 2025 , A72
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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References

Alexander, R.M. 1977 Swimming. In Mechanics and Energetics of Animal Locomotion (ed. Alexander, R.M. & Goldspink, G.), pp. 222–248. Chapman & Hall.Google Scholar
Anderson, E.J., McGillis, W.R. & Grosenbaugh, M.A. 2001 The boundary layer of swimming fish. J. Exp. Biol. 204 (1), 81–102.CrossRefGoogle ScholarPubMed
Borazjani, I. & Sotiropoulos, F. 2008 Numerical investigation of the hydrodynamics of carangiform swimming in the transitional and inertial flow regimes. J. Exp. Biol. 211 (10), 1541–1558.CrossRefGoogle ScholarPubMed
Borazjani, I. & Sotiropoulos, F. 2009 Numerical investigation of the hydrodynamics of anguilliform swimming in the transitional and inertial flow regimes. J. Exp. Biol. 212 (4), 576–592.CrossRefGoogle ScholarPubMed
Boyer, F., Porez, M. & Leroyer, A. 2010 Poincaré–Cosserat equations for the Lighthill three-dimensional large amplitude elongated body theory: application to robotics. J. Nonlinear Sci. 20, 47–79.CrossRefGoogle Scholar
Buchak, P., Eloy, C. & Reis, P.M. 2010 The clapping book: wind-driven oscillations in a stack of elastic sheets. Phys. Rev. Lett. 105 (19), 194301.CrossRefGoogle Scholar
Candelier, F., Boyer, F. & Leroyer, A. 2011 Three-dimensional extension of Lighthill’s large-amplitude elongated-body theory of fish locomotion. J. Fluid Mech. 674, 196–226.CrossRefGoogle Scholar
Candelier, F., Porez, M. & Boyer, F. 2013 Note on the swimming of an elongated-body in a non-uniform flow. J. Fluid Mech. 716, 616–637.CrossRefGoogle Scholar
Cheng, J.-Y., Pedley, T.J. & Altringham, J.D. 1998 A continuous dynamic beam model for swimming fish. Phil. Trans. R. Soc. Lond. B: Biol. Sci. 353 (1371), 981–997.CrossRefGoogle Scholar
Cheng, J.-Y., Zhuang, L.-X. & Tong, B.-G. 1991 Analysis of swimming three-dimensional waving plates. J. Fluid Mech. 232, 341–355.CrossRefGoogle Scholar
Crighton, D.G. 1985 The Kutta condition in unsteady flow. Ann. Rev. Fluid Mech. 17 (1), 411–445.CrossRefGoogle Scholar
Ehrenstein, U. & Eloy, C. 2013 Skin friction on a moving wall and its implications for swimming animals. J. Fluid Mech. 718, 321–346.CrossRefGoogle Scholar
Eloy, C. 2012 Optimal Strouhal number for swimming animals. J. Fluids Struct. 30, 205–218.CrossRefGoogle Scholar
Eloy, C. 2013 On the best design for undulatory swimming. J. Fluid Mech. 717, 48–89.CrossRefGoogle Scholar
Filella, A., Nadal, F., Sire, C., Kanso, E. & Eloy, C. 2018 Hydrodynamic interactions influence fish collective behavior. Phys. Rev. Lett. 120, 198101.CrossRefGoogle Scholar
Gray, J. & Hancock, G.J. 1955 The propulsion of sea-urchin spermatozoa. J. Exp. Biol. 32, 802–814.CrossRefGoogle Scholar
Hess, F. & Videler, J.J. 1984 Fast continuous swimming of saithe (Pollachius virens): a dynamic analysis of bending moments and muscle power. J. Exp. Biol. 109, 229–251.CrossRefGoogle Scholar
Lauder, G.V. & Tytell, E.D. 2005 Hydrodynamics of undulatory propulsion. Fish Biomech. 23, 425–462.CrossRefGoogle Scholar
Li, G., Kolomenskiy, D., Liu, H., Thiria, B. & Godoy-Diana, R. 2019 On the energetics and stability of a minimal fish school. PloS One 14 (8), 1–20.Google ScholarPubMed
Lighthill, M.J. 1960 Note on the swimming of slender fish. J. Fluid Mech. 9, 305–317.CrossRefGoogle Scholar
Lighthill, M.J. 1971 Large-amplitude elongated-body theory of fish locomotion. Proc. R. Soc. Lond. B 179 (1055), 125–138.Google Scholar
Michelin, S. & Doaré, O. 2013 Energy harvesting efficiency of piezoelectric flags in axial flows. J. Fluid Mech. 714, 489–504.CrossRefGoogle Scholar
Michelin, S., Llewellyn Smith, S.G. & Glover, B.J. 2008 Vortex shedding model of a flapping flag. J. Fluid Mech. 617, 1–10.CrossRefGoogle Scholar
Mougel, J., Doaré, O. & Michelin, S. 2016 Synchronized flutter of two slender flags. J. Fluid Mech. 801, 652–669.CrossRefGoogle Scholar
Munk, M.M. 1924 The aerodynamic forces on airship hulls. Tech. Rep. NACA-TR-184. National Advisory Committee for Aeronautics.Google Scholar
Pedley, T.J. & Hill, S.J. 1999 Large-amplitude undulatory fish swimming: fluid mechanics coupled to internal mechanics. J. Exp. Biol. 202 (23), 3431–3438.CrossRefGoogle ScholarPubMed
Porez, M., Boyer, F. & Ijspeert, A.J. 2014 Improved Lighthill fish swimming model for bio-inspired robots: modeling, computational aspects and experimental comparisons. Intl J. Robot. Res. 33 (10), 1322–1341.CrossRefGoogle Scholar
Sarpkaya, T. 1986 Force on a circular cylinder in viscous oscillatory flow at low Keulegan–Carpenter numbers. J. Fluid Mech. 165, 61–71.CrossRefGoogle Scholar
Shirgaonkar, A.A., MacIver, M.A. & Patankar, N.A. 2009 A new mathematical formulation and fast algorithm for fully resolved simulation of self-propulsion. J. Comput. Phys. 228 (7), 2366–2390.CrossRefGoogle Scholar
Singh, K., Michelin, S. & de Langre, E. 2012 Energy harvesting from axial fluid-elastic instabilities of a cylinder. J. Fluid. Struct. 30, 159–172.CrossRefGoogle Scholar
Taylor, G.I. 1952 Analysis of the swimming of long and narrow animals. Proc. R. Soc. Lond. A 214 (1117), 158–183.Google Scholar
Tytell, E.D. & Lauder, G.V. 2004 The hydrodynamics of eel swimming. I. Wake structure. J. Exp. Biol. 207 (11), 1825–1841.CrossRefGoogle ScholarPubMed
Videler, J.J. 1981 Swimming movements, body structure and propulsion in cod, Gadus morhua . Symp. Zool. Soc. Lond. 48, 1–27.Google Scholar
Videler, J.J. & Hess, F. 1984 Fast continuous swimming of two pelagic predators, saithe (Pollachius virens) and mackerel (Scomber scombrus): a kinematic analysis. J. Exp. Biol. 109 (1), 209–228.CrossRefGoogle Scholar
Webb, P.W. 1975 Hydrodynamics and energetics of fish propulsion. J. Fish. Res. Bd Can. 190, 1–158.Google Scholar
Weihs, D. 1975 Some hydrodynamical aspects of fish schooling. In Swimming and Flying in Nature (ed. T.YT. Wu, C.J. Brokaw & C. Brennen). Springer.Google Scholar
Wu, T.Y.-T. 1961 Swimming of a waving plate. J. Fluid Mech. 10 (3), 321–344.CrossRefGoogle 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.CrossRefGoogle Scholar
Yu, Z. & Eloy, C. 2018 Extension of Lighthill’s slender-body theory to moderate aspect ratios. J. Fluids Struct. 76, 84–94.CrossRefGoogle Scholar