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Three-dimensional extension of Lighthill's large-amplitude elongated-body theory of fish locomotion

Published online by Cambridge University Press:  11 March 2011

FABIEN CANDELIER ,
FREDERIC BOYER  and
ALBAN LEROYER
FABIEN CANDELIER*
Affiliation:
Université Aix-Marseille 2, IUSTI, Technopôle de Château Gombert, 60 rue Joliot-Curie, 13453 Marseille CEDEX 13, France
FREDERIC BOYER
Affiliation:
Ecole des Mines de Nantes, La Chantrerie, 4 rue Alfred Kastler, BP 20722, 44307 Nantes, France
ALBAN LEROYER
Affiliation:
Laboratoire de Mécanique des Fluides, Ecole Centrale Nantes, 1 rue de la Noë, BP 92101, 44321 Nantes CEDEX 3, France
*
†Email address for correspondence: [email protected]
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Abstract

The goal of this paper is to derive expressions for the pressure forces and moments acting on an elongated body swimming in a quiescent fluid. The body is modelled as an inextensible and unshearable (Kirchhoff) beam, whose cross-sections are elliptic, undergoing prescribed deformations, consisting of yaw and pitch bending. The surrounding fluid is assumed to be inviscid, and irrotational everywhere, except in a thin vortical wake. The Laplace equation and the corresponding Neumann boundary conditions are first written in terms of the body coordinates of a beam treating the body as a fixed surface. They are then simplified according to the slenderness of the body and its kinematics. Because the equations are linear, the velocity potential is sought as a sum of two terms which are linked respectively to the axial movements of the beam and to its lateral movements. The lateral component of the velocity potential is decomposed further into two sub-components, in order to exhibit explicitly the role of the two-dimensional potential flow produced by the lateral motion of the cross-section, and the role played by the curvature effects of the beam on the cross-sectional flow. The pressure, which is given by Bernoulli's equation, is integrated along the body surface, and the expressions for the resultant and the moment are derived analytically. Thereafter, the validity of the force and moment obtained analytically is checked by comparisons with Navier–Stokes simulations (using Reynolds-averaged Navier–Stokes equations), and relatively good agreements are observed.

JFM classification

Biological Fluid Dynamics: Swimming/flying
Type
Papers
Information
Journal of Fluid Mechanics , Volume 674 , 10 May 2011 , pp. 196 - 226
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Anderson, J. 1996 Vorticity control for efficient propulsion. PhD thesis, Massachussets Institute of Technology and the Woods Hole Ocean Engineering.CrossRefGoogle Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Blake, R. W. 2004 Fish functional design and swimming performance. J. Fish Biol. 65 (5), 1193–1222.CrossRefGoogle 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.CrossRefGoogle Scholar
Boyer, F., Porez, M., Leroyer, A. & Visonneau, M. 2008 Macro-continuous computed torque algorithm for a three-dimensional eel-like robot. IEEE Trans. Robot. 22, 975–991.Google Scholar
Childress, S. 1981 Mechanics of Swimming and Flying. Cambridge University Press.CrossRefGoogle Scholar
Colgate, J. E. & Lynch, K. M. 2004 Mechanics and control of swimming: a review. IEEE J. Ocean. Engng 29, 660–673.CrossRefGoogle Scholar
Eldredge, J. D. 2008 Dynamically coupled fluid–body interactions in vorticity-based numerical simulations. J. Comput. Phys. 227, 9170–9194.CrossRefGoogle Scholar
Galper, A. & 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.CrossRefGoogle Scholar
Galper, A. R. & Miloh, T. 1999 Curved slender structures in non-uniform flows. J. Fluid Mech. 385, 21–40.CrossRefGoogle Scholar
Galper, A. R. & Miloh, T. 2000 Hydroelasticity of the Kirchhoff rod: buckling phenomena. J. Fluids Struct. 14, 1089–1100.CrossRefGoogle Scholar
Gray, J. 1933 Studies in animal locomotion. I. The movement of fish with special reference to eel. J. Expl. Biol. 10, 88–104.CrossRefGoogle Scholar
Kanso, E. 2009 Swimming due to transverse shape deformations. J. Fluid Mech. 631, 127–148.CrossRefGoogle Scholar
Kanso, E., Marsden, J. E., Rowley, C. W. & Melli-Huber, J. B. 2005 Locomotion of articulated bodies in a perfect fluid. J. Nonlinear Sci. 15 (4), 255–289.CrossRefGoogle Scholar
Kern, S. & Koumoutsakos, P. 2006 Simulation of optimized anguilliform swimming. J. Expl. Biol. 209, 4841–4857.CrossRefGoogle ScholarPubMed
Leroyer, A. & Visonneau, M. 2005 Numerical methods for RANSE simulations of a self-propelled fish-like body. J. Fluids Struct. 20, 763–775.CrossRefGoogle Scholar
Lighthill, M. J. 1960 Note on the swimming of slender fish. J. Fluid Mech. 9 (2), 305–317.CrossRefGoogle Scholar
Lighthill, M. J. 1970 Aquatic animal propulsion of high hydromechanical efficiency. J. Fluid Mech. 44, 265–301.CrossRefGoogle Scholar
Lighthill, M. J. 1975 Mathematical Biofluiddynamics. Regional Conference Series in Applied Mathematics, vol. 17. SIAM.CrossRefGoogle Scholar
Lopes, J.-L., Païdoussis, M. P. & Semler, C. 2002 Linear and nonlinear dynamics of cantilevered cylinders in axial flow. Part 2. The equations of motion. J. Fluids Struct. 16 (6), 715–737.CrossRefGoogle Scholar
Milne-Thomson, L. M. 1968 Theoretical Hydrodynamics. Dover.CrossRefGoogle Scholar
Muller, U. K., Stamhuis, E. J. & Videler, J. J. 2000 Hydrodynamics of unsteady fish swimming and the effects of body size: comparing the flow fields of fish larvae and adults. J. Expl. Biol. 203 (2), 193–206.CrossRefGoogle ScholarPubMed
Munnier, A. 2009 Locomotion of deformable bodies in an ideal fluid: Newtonian versus Lagrangian formalisms. J. Nonlinear Sci. 19 (6), 665–715.CrossRefGoogle Scholar
Newman, J. N. 1973 The force on a slender fish-like body. J. Fluid Mech. 58 (4), 689–702.CrossRefGoogle Scholar
Newman, J. N. & Wu, T. Y. 1973 A generalized slender-body theory for fish-like forms. J. Fluid Mech. 57 (4), 673–693.CrossRefGoogle Scholar
Reissner, E. 1973 On a one-dimensional large displacement finite-strain theory. Stud. Appl. Maths 52, 87–95.CrossRefGoogle Scholar
Sfakiotakis, M., Lane, D. M. & Davies, J. B. C. 1999 Review of fish swimming modes for aquatic locomotion. IEEE J. Ocean. Engng 24, 237–252.CrossRefGoogle 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
Singh, K. & Pedley, T. J. 2008 The hydrodynamics of flexible body manoeuvres in swimming fish. Physica D: Nonlinear Phenom. 237 (14–17), 2234–2239.CrossRefGoogle Scholar
Sparenberg, J. A. 2002 Survey of the mathematical theory of fish locomotion. J. Engng Maths 44, 395–448.CrossRefGoogle Scholar
Taylor, G. 1952 Analysis of the swimming of long and narrow animals. Proc. R. Soc. Lond. A 214, 158–183.Google Scholar
Triantafyllou, M. S., Triantafyllou, G. S. & Yue, D. K. P. 2000 Hydrodynamics of fishlike swimming. Annu. Rev. Fluid Mech. 32, 33–53.CrossRefGoogle Scholar
Tytell, E. D. 2004 The hydrodynamics of eel swimming. II. Effect of swimming speed. J. Expl. Biol. 207, 3265–3279.CrossRefGoogle ScholarPubMed
Weihs, D. 1972 A hydrodynamical analysis of fish turning manoeuvres. Proc. R. Soc. Lond. B 182, 59–72.Google Scholar
Weihs, D. 1973 The mechanism of rapid starting of slender fish. Biorheology 10, 343–350.CrossRefGoogle ScholarPubMed
Wolfgang, M. J. 1999 Hydrodynamics of flexible-body swimming motions. PhD thesis, Massachussets Institute of Technology.Google Scholar
Wu, T. Y. 1961 Swimming of the waving plate. J. Fluid Mech. 10, 321–344.CrossRefGoogle Scholar
Zhu, Q., Wolfgang, M. J., Yue, D. K. P. & Triantafyllou, M. S. 2002 Three-dimensional flow structures and vorticity control in fish-like swimming. J. Fluid Mech. 468, 1–28.CrossRefGoogle Scholar