Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-18T03:17:56.234Z Has data issue: false hasContentIssue false
  • English
  • Français

On the swimming of a flexible body in a vortex street

Published online by Cambridge University Press:  10 September 2009

SILAS ALBEN
SILAS ALBEN*
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We formulate a new theoretical model for the swimming of a flexible body in a vortex street. We consider the class of periodic travelling-wave body motions, in the limit of small amplitude. We calculate the output power provided to the body by thrust forces, and the input power done against pressure forces, as functions of the aspect ratio and strength of the vortex street. We then formulate two optimization problems. In the first, we determine the body wave which provides maximum output power for fixed amplitude. We find a closed-form solution with a transition from power law to exponential decay of output power as the vortex street widens. In the second problem, we incorporate internal viscoelasticity to the swimming body and compute its contribution to the input power. We find the body wave which maximizes efficiency for a given output power. The body shape and resulting efficiency are found in closed form and simple approximate formulas are given. We find that efficiency scales as the inverse of the damping parameter. Finally, we compare our results with previous experiments and simulations. We find agreement in some aspects and disagreement in others. We give physical interpretations for agreements and disagreements in terms of the phase between the body wave and vortex street.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 635 , 25 September 2009 , pp. 27 - 45
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Abrahams, M. V. & Colgan, P. W. 1987 Fish schools and their hydrodynamic function: a reanalysis. Environ. Biol. Fishes 20 (1), 79–80.CrossRefGoogle Scholar
Acton, E. & Dhanak, M. R. 1993 The motion and stability of a vortex array above a pulsed surface. J. Fluid Mech. 247, 231–245.CrossRefGoogle Scholar
Ahlfors, L. V. 1979 Complex Analysis. McGraw-Hill.Google Scholar
Alben, S. & Shelley, M. J. 2008 Flapping states of a flag in an inviscid fluid: bistability and the transition to chaos. Phys. Rev. Lett. 100, 074301.CrossRefGoogle Scholar
Banks, H. T. & Inman, D. J. 1991 On damping mechanisms in beams. J. Appl. Mech. 58, 716.CrossRefGoogle Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. 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.CrossRefGoogle Scholar
Bearman, P. W. & Zdravkovich, M. M. 1978 Flow around a circular cylinder near a plane boundary. J. Fluid Mech. 89, 33–47.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 353, 981–997.CrossRefGoogle Scholar
Drucker, E. G. & Lauder, G. V. 2001 Locomotor function of the dorsal fin in teleost fishes: experimental analysis of wake forces in sunfish. J. Exp. Biol. 204 (17), 2943–2958.CrossRefGoogle ScholarPubMed
Eldredge, J. D. & Pisani, D. 2008 Passive propulsion of a simple articulated system in the wake of an obstacle. J. Fluid Mech. 607, 279–288.CrossRefGoogle Scholar
Kanso, E. & Oskouei, B. G. 2008 Stability of a coupled body-vortex system. J. Fluid Mech. 600, 77–94.CrossRefGoogle Scholar
Liao, J. C., Beal, D. N., Lauder, G. V. & Triantafyllou, M. S. 2003 Fish exploiting vortices decrease muscle activity. Science 302 (5650), 1566–1569.CrossRefGoogle ScholarPubMed
Lighthill, M. J. 1960 Note on the swimming of slender fish. J. Fluid Mech. 9 (02), 305–317.CrossRefGoogle Scholar
Lissaman, P. B. S. & Shollenberger, C. A. 1970 Formation flight of birds. Science 168 (3934), 1003–1005.CrossRefGoogle ScholarPubMed
Long, J., Hale, M., Mchenry, M. & Westneat, M. 1996 Functions of fish skin: flexural stiffness and steady swimming of longnose gar, Lepisosteus osseus. J. Exp. Biol. 199 (10), 2139–2151.CrossRefGoogle ScholarPubMed
Saffman, P. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Segel, L. 1977 Mathematics Applied to Continuum Mechanics. Macmillan.Google Scholar
Shelley, M., Vandenberghe, N. & Zhang, J. 2005 Heavy flags undergo spontaneous oscillations in flowing water. Phys. Rev. Lett 94, 094302.CrossRefGoogle ScholarPubMed
Shukla, R. K. & Eldredge, J. D. 2007 An inviscid model for vortex shedding from a deforming body. Theor. Comput. Fluid Dyn. 21 (5), 343–368.CrossRefGoogle Scholar
Streitlien, K., Triantafyllou, G. S. & Triantafyllou, M. S. 1996 Efficient foil propulsion through vortex control. AIAA J. 34 (11), 2315–2319.CrossRefGoogle Scholar
Videler, J. J. 1993 Fish Swimming. Springer.CrossRefGoogle Scholar
Wainwright, S. A. 2000 The animal axis. Am. Zool. 40 (1), 19–27.Google Scholar
Weihs, D. 1973 Hydromechanics of fish schooling. Nature 241 (5387), 290–291.CrossRefGoogle Scholar
Weimerskirch, H., Martin, J., Clerquin, Y., Alexandre, P. & Jiraskova, S. 2001 Energy saving in flight formation. Nature 413 (6857), 697–698.CrossRefGoogle ScholarPubMed
Wu, T. Y. 1961 Swimming of a waving plate. J. Fluid Mech. 10 (3), 321–344.CrossRefGoogle Scholar
Wu, T. Y. & Chwang, A. T. 1975 Extraction of flow energy by fish and birds in a wavy stream. In Swimming and Flying in Nature (ed. Wu, T. Y. T., Brokaw, C. J. & Brennen, C.), vol. 2, pp. 687–702. Plenum Press.CrossRefGoogle Scholar
Zdravkovich, M. M. 1983 Observation of vortex shedding behind a towed circular cylinder near a wall. In Flow Visualization III: Proceedings of the Third International Symposium. Ann Arbor, MI.Google Scholar