Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-25T00:38:23.661Z Has data issue: false hasContentIssue false
  • English
  • Français

On the propulsion efficiency of swimming flexible hydrofoils of finite thickness

Published online by Cambridge University Press:  28 March 2006

J. P. Uldrick
J. P. Uldrick
Affiliation:
Engineering Department, U.S. Naval Academy, Annapolis, Maryland, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper presents some recent theoretical results on the energy exchange between a swimming flexible two-dimensional hydrofoil of finite profile thickness and the inviscid incompressible fluid in which the body swims. The rate at which kinetic energy is transferred to the fluid by the undulating hydrofoil, the power required to maintain the prescribed motion, and the resulting power available for propulsion are calculated in terms of the thickness to chord ratio and the displacement and rate of displacement of the hydrofoil. With a small unsteady perturbation theory, the analysis is decomposed to show separately the effects of the circulatory and non-circulatory flows, both depending on the first-order terms of the unsteady perturbation velocity components. In addition, an analysis is presented showing the effect of the non-linear unsteady pressure distribution on the surface of the hydrofoil. Contrary to what might be expected, this latter effect is of the same order of magnitude for a thick rounded-nose profile as for the flat plate where the effect is concentrated at the sharp leading edge and is related to the so-called suction force. However, except for small values of the reduced frequency, the non-linear contribution is negligible in comparison with the linear contribution.

New functions associated with the retarded flow in the wake are introduced and special techniques for their solution are presented, these being related to Theodorsen's function of unsteady airfoil theory for the special case of the undulating flat plate.

The numerical results reveal that the kinetic energy imparted to the fluid, the power required to maintain the motion, and the resulting propulsive power, follow closely those of an infinitesimal model for small values of the reduced frequency of oscillation, but diverge somewhat from the classical thin plate analysis for large reduced frequencies. Of particular interest is the fact that a very large percentage of the power required to maintain the motion is used in the generation of the wake, whereas a very small percentage of the power available for propulsion comes from the wake. This indicates that, if some mechanism could be devised to control the wake, very high swimming efficiencies could be attained. Fish, in all probability, have been succeeding in doing this for millions of years.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 32 , Issue 1 , 9 April 1968 , pp. 29 - 53
Copyright
© 1968 Cambridge University Press

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

Bonthron, R. J. & Fejer, A. A. 1962 A hydrodynamic study of fish locomotion. Proc. 4th U.S. Nat. Congr. Appl. Mech. pp. 1249–55. Berkeley, California.Google Scholar
Gray, J. 1936 Studies of animal locomotion. VI J. Exp. Biol. 13, 192–9.Google Scholar
Gray, J. 1948 Aspects of locomotion of whales Nature, Lond. 161, 191–200.Google Scholar
Gray, J. 1957 How fish swim Sci. Am. 197, 48–54.Google Scholar
Kelly, H. R. 1961 Fish propulsion hydrodynamics. Proc. 7th Midwestern Conf. on Fluid Mech., Michigan State University. In Developments in Mechanics, 1, 442–50. New York: Plenum Press.
Kelly, H. R. & Bowlus, G. H. 1963 Swimming hinged hydrofoils. NAVWEPS Rept. NOTS TP 3342, China Lake, California.Google Scholar
Küssner, H. G. & Gorup, G. V. 1960 Instationäre linearisierte Theorie der Flügelprofile endlicher Diche in inkompressibler Strömung. Mitt. Max-Planck-Institut Ström. Forsch. 26, Göttingen.Google Scholar
Küssner, H. G. & Schwarz, L. 1940 The oscillating wing with aerodynamically balanced elevator Luftfahrforschung, 17, 337–54. English translation 1941: N.A.C.A. T.M. 991.Google Scholar
Lighthill, M. J. 1960a Mathematics and aeronautics J. Roy. Aero. Soc. 64, 373–94.Google Scholar
Lighthill, M. J. 1960b Note on the swimming of slender fish J. Fluid Mech. 9, 305–17.Google Scholar
Munk, M. M. 1924 The aerodynamic forces on airship hulls. N.A.C.A. Tech. Rept. no. 184.Google Scholar
Pao, S. K. & Siekmann, J. 1964 Note on the Smith-Stone theory of fish propulsion. Proc. Roy. Soc A 280, 398–408.Google Scholar
Schwarz, L. 1940 Berechnung der Druckverteilung einer harmonich sich verformenden Tragfläche in ebener Strömung Luftfahrforschung, 17, 379–86.Google Scholar
Serrin, J. 1959 Mathematical principles of classical fluid mechanics. Handbuch der Physik VIII/1.Google Scholar
Siekmann, J. 1962 Theoretical studies of sea animal locomotion, Part I. Ing. Arch. Bd. 31, H. 3, 214–28.Google Scholar
Siekmann, J. 1963a Theoretical studies of sea animal locomotion. Part II. Ing. Arch. Bd. 32, H.1, 40–50.Google Scholar
Siekmann, J. 1963b On a pulsating jet from the end of a tube, with application to the propulsion of certain aquatic animals J. Fluid Mech. 15, 399–418.Google Scholar
Smith, E. H. & Stone, D. E. 1961 Perfect fluid forces in fish propulsion: the solution of the problem in an elliptic cylinder coordinate system. Proc. Roy. Soc A 261, 316–28.Google Scholar
Taylor, G. 1951 Analysis of the swimming of microscopic organisms. Proc. Roy. Soc. A 209, 445.Google Scholar
Taylor, G. 1952a The action of waving cylindrical tails in propelling microscopic organisms. Proc. Roy. Soc A 211, 225–39.Google Scholar
Taylor, G. 1952b Analysis of the swimming of long and narrow animals. Proc. Roy. Soc A 214, 158–83.Google Scholar
Theodorsen, T. 1934 General theory of aerodynamic instability and the mechanism of flutter. N.A.C.A. Rept. no. 496.Google Scholar
Uldrick, J. P. & Siekmann, J. 1964 On the swimming of a flexible plate of arbitrary finite thickness J. Fluid Mech. 20, 1–33.Google Scholar
Wu, T. Y. 1961 Swimming of a waving plate J. Fluid Mech. 10, 321–44.Google Scholar