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Note on the swimming of an elongated body in a non-uniform flow

Published online by Cambridge University Press:  28 January 2013

Fabien Candelier ,
Mathieu Porez  and
Frederic Boyer
Fabien Candelier*
Affiliation:
Aix-Marseille Université – IUSTI (UMR CNRS 7343), 60 rue Joliot-Curie, 13 453 Marseille CEDEX 13, France
Mathieu Porez
Affiliation:
Ecole des Mines de Nantes – IRCCyN (UMR CNRS 6597), La Chantrerie, 4 rue Alfred Kastler, BP 2072, 44 307 Nantes, France
Frederic Boyer
Affiliation:
Ecole des Mines de Nantes – IRCCyN (UMR CNRS 6597), La Chantrerie, 4 rue Alfred Kastler, BP 2072, 44 307 Nantes, France
*
†Email address for correspondence: [email protected]
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Abstract

This paper presents an extension of Lighthill’s large-amplitude elongated-body theory of fish locomotion which enables the effects of an external weakly non-uniform potential flow to be taken into account. To do so, the body is modelled as a Kirchhoff beam, made up of elliptical cross-sections whose size may vary along the body, undergoing prescribed deformations consisting of yaw and pitch bending. The fluid velocity potential is decomposed into two parts corresponding to the unperturbed potential flow, which is assumed to be known, and to the perturbation flow. The Laplace equation and the corresponding Neumann’s boundary conditions governing the perturbation velocity potential are expressed in terms of curvilinear coordinates which follow the body during its motion, thus allowing the boundary of the body to be considered as a fixed surface. Equations are simplified according to the slenderness of the body and the weakness of the non-uniformity of the unperturbed flow. These simplifications allow the pressure acting on the body to be determined analytically using the classical Bernoulli equation, which is then integrated over the body. The model is finally used to investigate the passive and the active swimming of a fish in a Kármán vortex street.

JFM classification

Biological Fluid Dynamics: Biological Fluid Dynamics Biological Fluid Dynamics: Swimming/flying
Type
Papers
Information
Journal of Fluid Mechanics , Volume 716 , 10 February 2013 , pp. 616 - 637
Copyright
©2013 Cambridge University Press

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References

Alben, S. 2009 On the swimming of a flexible body in a vortex street. J. Fluid Mech. 635, 27–45.Google Scholar
Alben, S. 2010 Passive and active bodies in vortex-street wakes. J. Fluid Mech. 642, 95–125.Google Scholar
Alexander, R. Mc. N. 1988 Elastic Mechanisms in Animal Movements. Cambridge University Press.Google 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.Google Scholar
Bergmann, M. & Iollo, A. 2011 Modelling and simulation of fish like swimming. J. Comput. Phys. 230 (2).Google Scholar
Borisov, A. V., Mamaev, I. S. & Ramodanov, S. M. 2007 Dynamic interaction of point vortices and two-dimensional cylinder. J. Math. Phys. 48, 1–9.CrossRefGoogle Scholar
Boyer, F., Porez, M. & Khalil, W. 2006 Macro-continuous computed torque algorithm for a three-dimensional eel-like robot. IEEE Trans. Robot. 22, 763–775.Google Scholar
Boyer, F., Porez, M. & Leroyer, A. 2009 Poincaré–Cosserat equations for the Lighthill three-dimensional large amplitude elongated body theory: application to robotics. J. Nonlinear Sci. 20 (1), 47–79.Google Scholar
Candelier, F., Boyer, F. & Leroyer, A. 2011 Three-dimensional extension of the Lighthill’s large amplitude elongated-body theory of fish locomation. J. Fluid Mech. 674, 196–226.Google 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 353, 981–997.CrossRefGoogle Scholar
Childress, S. 1981 Mechanics of Swimming and Flying. Cambridge University Press.Google Scholar
Eldredge, J. D. & Pisani, D. 2008 Passive locomotion of a simple articulated fish-like system in the wake of obstacle. J. Fluid Mech. 607, 279–288.Google Scholar
Eloy, C. 2012 Optimal Strouhal number for swimming animals. J. Fluids Struct. 30, 205–218.Google Scholar
Galper, A. R. & Miloh, T. 1995 Dynamic equations of motion for a rigid or deformable body in an arbitrary non-uniform potential flow field. J. Fluid Mech. 295, 91–120.Google Scholar
Galper, A. R. & Miloh, T. 1999 Curved slender structures in non-uniform flows. J. Fluid Mech. 385, 21–40.Google Scholar
Hoerner, S. F. 1965 Fluid Dynamic Drag: Practical Information on Aerodynamic Drag and Hydrodynamic Resistance, 2nd edn. Hoerner Fluid Dynamics.Google Scholar
Kanso, E. & Oskouei, B. G. 2008 Stability of a coupled body–vortex system. J. Fluid Mech. 600, 77–94.Google Scholar
Katz, J. & Plotkin, A. 2001 Low-Speed Aerodynamics, 2nd edn. Cambridge University Press.CrossRefGoogle Scholar
Khalil, W., Gallot, G. & Boyer, F. 2007 Dynamic modelling and simulation of 3-D serial eel-like robot. IEEE Trans. Syst. Man Cybern. C 37, 1259–1268.Google Scholar
Lauder, G. V. & Madden, P. G. A. 2006 Learning from fish: kinematics and experimental hydrodynamics for roboticists. Intl J. Automat. Comput. 4, 325–335.Google Scholar
Liao, J. C. 2004 Neuromuscular control of trout swimming in a vortex street: implications for energy economy during the Kármán gait. J. Expl Biol. 207, 3495–3506.Google Scholar
Liao, J. C. 2007 A review of fish swimming mechanics and behaviour in altered flows. Phil. Trans. R. Soc. B 362, 1973–1993.Google Scholar
Liao, J. C., Beal, D. N., Lauder, G. V. & Triantafyllou, M. S. 2003 The Kármán gait: novel body kinematics of rainbow trout swimming in a vortex street. J. Expl Biol. 206, 1059–1073.CrossRefGoogle Scholar
Lighthill, M. J. 1960 Note on the swimming of slender fish. J. Fluid Mech. 9 (2), 305–317.Google Scholar
Lighthill, M. J. 1971 Large-amplitude elongated-body theory of fish locomotion. Proc. R. Soc. Lond. B 179, 125–138.Google Scholar
Lighthill, M. J. 1975 Mathematical Biofluiddynamics. CBMS–NSF Regional Conference Series in Applied Mathematics, Vol. 17 , Society for Industrial and Applied Mathematics.Google Scholar
Long, J. H. & Nipper, K. S. 1996 The importance of body stiffness in undulatory propulsion. Am. Zool. 36, 678–694.Google Scholar
Reissner, E. 1973 On a one-dimensional large displacement finite-strain theory. Stud. Appl. Maths 52, 87–95.CrossRefGoogle Scholar
Saffman, P. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Simo, J. C. & Vu-Locq, L. 1988 On the dynamics in space of rods undergoing large motions. A geometrically exact approach. Intl J. Numer. Meth. Engng 66, 125–161.Google Scholar
Tytell, E. D. & Lauder, G. V. 2004 The hydrodynamics of eel swimming: I. Wake structure. J. Expl Biol. 207, 1825–1841.Google Scholar
Weihs, D. 1973 Hydrodynamics of fish schooling. Nature 241, 290–291.Google Scholar

Candelier et al. supplementary movie

Passive swimming of a body in a vortex street. V_10 is the head velocity and Vf_10 is the unperturbed fluid velocity at the middle of the rows, along the axial direction. Distance between two consecutive vortices on the same row: 0.208 m. Distance between the two rows: 0.07 m. Length of the body: 0.178 m.

Download Candelier et al. supplementary movie(Video)
Video 8.7 MB