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On the best design for undulatory swimming

Published online by Cambridge University Press:  01 February 2013

Christophe Eloy*
Affiliation:
Aix-Marseille University, IRPHE UMR 7342, CNRS, 13013 Marseille, France Department of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA 92093, USA
*
Email address for correspondence: Christophe.Eloy@irphe.univ-mrs.fr

Abstract

Most aquatic vertebrates swim by passing a bending wave down their bodies, a swimming mode known as undulatory propulsion. Except for very elongated swimmers like eels and lampreys, these animals have generally evolved to a similar shape: an anterior streamlined region of large volume separated from a caudal fin by a caudal peduncle of reduced cross-section. However, the link between this particular shape and the hydrodynamical constraints remains to be explored. Here, this question is addressed by seeking the optimal design for undulatory swimmers with an evolutionary algorithm. Animals of varying elliptic cross-section are considered whose motions are prescribed by arbitrary periodic curvature laws. In the elongated-body limit, reactive and resistive forces can be formulated at any cross-section, allowing the recoil motion and the mean swimming speed of a given animal to be calculated. A bi-objective optimization problem then consists of finding body shapes and corresponding motions associated with the lowest energetic costs, the highest stride lengths (which is a dimensionless measure of swimming speed) or any trade-offs between the two. For biologically relevant parameters, this optimization calculation yields two distinct ‘species’: one specialized in economical swimming and the other in large stride lengths. By comparing the attributes and performance of these numerically obtained swimmers with data on undulatory-swimming animals, it is argued that evolution is consistent with the selection of species with low energetic costs.

Type
Papers
Copyright
©2013 Cambridge University Press

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References

Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.Google Scholar
Alexander, R. M. 1977 Swimming. In Mechanics and Energetics of Animal Locomotion (ed. Alexander, R. M. & Goldspink, G.), pp. 222248. Chapman & Hall.Google Scholar
Anderson, E. J., McGillis, W. R. & Grosenbaugh, M. A. 2001 The boundary layer of swimming fish. J. Expl Biol. 204, 81102.CrossRefGoogle ScholarPubMed
Bainbridge, R. 1958 The speed of swimming of fish as related to size and to the frequency and amplitude of the tail beat. J. Expl Biol. 35, 109133.Google Scholar
Bainbridge, R. 1963 Caudal fin and body movement in the propulsion of some fish. J. Expl Biol. 40, 2356.Google Scholar
Blake, R. W. 1983 Fish Locomotion. Cambridge University Press.Google Scholar
Blake, R. W. 2004 Fish functional design and swimming performance. J. Fish Biol. 65, 11931222.Google Scholar
Borazjani, I. & Sotiropoulos, F. 2008 Numerical investigation of the hydrodynamics of carangiform swimming in the transitional and inertial flow regimes. J. Expl Biol. 211, 1541.CrossRefGoogle ScholarPubMed
Branke, J., Deb, K., Miettinen, K. & Słowiński, R. (Eds) 2008 Multiobjective Optimization: Interactive and Evolutionary Approaches. Springer.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, 196226.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, 981997.CrossRefGoogle Scholar
Chopra, M. G. & Kambe, T. 1977 Hydromechanics of lunate-tail swimming propulsion. Part 2. J. Fluid Mech. 79, 4969.CrossRefGoogle Scholar
Corne, D. W., Jerram, N. R., Knowles, J. D. & Oates, M. J. 2001 PESA-II: region-based selection in evolutionary multiobjective optimization. In Proceedings of the Genetic and Evolutionary Computation Conference (GECCO2001). Morgan Kaufmann.Google Scholar
Curtin, N. A. & Woledge, R. C. 1993 Efficiency of energy conversion during sinusoidal movement of red muscle fibres from the dogfish Scyliorhinus canicula. J. Expl Biol. 185, 195206.Google Scholar
D’Août, K. & Aerts, P. 1999 A kinematic comparison of forward and backward swimming in the eel Anguilla anguilla. J. Expl Biol. 202, 15111521.Google Scholar
Dodds, P. S., Rothman, D. H. & Weitz, J. S. 2001 Re-examination of the ‘ $3/ 4$ -law’ of metabolism. J. Theor. Biol. 209, 927.Google Scholar
Donley, J. M. & Dickson, K. A. 2000 Swimming kinematics of juvenile kawakawa tuna (Euthynnus affinis) and chub mackerel (Scomber japonicus). J. Expl Biol. 203, 31033116.Google Scholar
Ehrenstein, U. & Eloy, C. 2012 Skin friction on a moving wall and its implications for swimming animals. J. Fluid Mech. (in press).CrossRefGoogle Scholar
Ellerby, D. J., Spierts, I. L. Y. & Altringham, J. D. 2001 Slow muscle power output of yellow-and silver-phase european eels (Anguilla anguilla L.): changes in muscle performance prior to migration. J. Expl Biol. 204, 13691379.CrossRefGoogle ScholarPubMed
Eloy, C. 2012 Optimal Strouhal number for swimming animals. J. Fluids Struct. 30, 205218.Google Scholar
Eloy, C., Doaré, O., Duchemin, L. & Schouveiler, L. 2010 A unified introduction to fluid mechanics of flying and swimming at high Reynolds number. Exp. Mech. 50, 13611366.CrossRefGoogle Scholar
Eloy, C. & Schouveiler, L. 2011 Optimisation of two-dimensional undulatory swimming at high Reynolds number. Intl J. Non-Linear Mech. 46, 568576.CrossRefGoogle Scholar
Fish, F. E. & Hui, C. A. 1991 Dolphin swimming – a review. Mammal Rev. 21, 181195.CrossRefGoogle Scholar
Gillis, G. B. 1998 Environmental effects on undulatory locomotion in the American eel Anguilla rostrata: kinematics in water and on land. J. Expl Biol. 201, 949961.CrossRefGoogle Scholar
Gray, J. 1933 Studies in animal locomotion. I. The movement of fish with special reference to the eel. J. Expl Biol. 10, 88104.CrossRefGoogle Scholar
Gray, J. 1968 Animal Locomotion. Weidenfeld & Nicolson.Google Scholar
Hess, F. & Videler, J. J. 1984 Fast continuous swimming of saithe (Pollachius virens): a dynamic analysis of bending moments and muscle power. J. Expl Biol. 109, 229251.CrossRefGoogle Scholar
Hoerner, S. F. 1965 Fluid-dynamic Drag. Published by the author.Google Scholar
Hoyt, J. W. 1975 Hydrodynamic drag reduction due to fish slimes. In Swimming and Flying in Nature (ed. Wu, T. Y.-T., Brokaw, C. J. & Brennen, C.), vol. 2, pp. 653672. Plenum.Google Scholar
Jayne, B. C. & Lauder, G. V. 1995 Speed effects on midline kinematics during steady undulatory swimming of largemouth bass, Micropterus salmoides. J. Expl Biol. 198, 585602.CrossRefGoogle Scholar
Kern, S. & Koumoutsakos, P. 2006 Simulations of optimized anguilliform swimming. J. Expl Biol. 209, 48414857.CrossRefGoogle ScholarPubMed
Lauder, G. V. & Tytell, E. D. 2005 Hydrodynamics of undulatory propulsion. Fish Physiol. 23, 425468.CrossRefGoogle Scholar
Lighthill, M. J. 1960 Note on the swimming of slender fish. J. Fluid Mech. 9, 305317.Google Scholar
Lighthill, M. J. 1969 Hydromechanics of aquatic animal propulsion. Annu. Rev. Fluid Mech. 1, 413446.Google Scholar
Lighthill, M. J. 1970 Aquatic animal propulsion of high hydromechanical efficiency. J. Fluid Mech. 44, 265301.CrossRefGoogle Scholar
Lighthill, M. J. 1971 Large-amplitude elongated-body theory of fish locomotion. Proc. R. Soc. Lond. B 179, 125138.Google Scholar
Lindsey, C. C. 1978 Form, function and locomotory habits in fish. In Fish Physiology VII (ed. Hoar, W. S. & Randall, D. J.), pp. 1100. Academic.Google Scholar
Long, J. H., Koob-Emunds, M., Sinwell, B. & Koob, T. J. 2002 The notochord of hagfish Myxine glutinosa: visco-elastic properties and mechanical functions during steady swimming. J. Expl Biol. 205, 38193831.Google Scholar
Ota, T. & Nishiyama, H. 1984 Heat transfer and flow around an elliptic cylinder. Intl J. Heat Mass Transfer 27 (10), 17711779.Google Scholar
Rosen, M. W. & Cornford, N. E. 1971 Fluid friction of fish slimes. Nature 234, 4951.Google Scholar
Schlichting, H. 1979 Boundary-layer Theory. McGraw-Hill.Google Scholar
Schneck, D. J. 1992 Mechanics of Muscle, 2nd edn. New York University Press.Google Scholar
Taylor, G. I. 1952 Analysis of the swimming of long and narrow animals. Proc. R. Soc. Lond. A 214, 158183.Google Scholar
Tokić, G & Yue, D. K. P. 2012 Optimal shape and motion of undulatory swimming organisms. Proc. R. Soc. Lond. B 279 (1740), 30653074.Google Scholar
Triantafyllou, G. S., Triantafyllou, M. S. & Grosenbaugh, M. A. 1993 Optimal thrust development in oscillating foils with application to fish propulsion. J. Fluids Struct. 7, 205224.Google Scholar
Triantafyllou, M. S., Triantafyllou, G. S. & Yue, D. K. P. 2000 Hydrodynamics of fishlike swimming. Annu. Rev. Fluid Mech. 32 (1), 3353.CrossRefGoogle Scholar
Tucker, V. A. 1970 Energetic cost of locomotion in animals. Comp. Biochem. Physiol. 34 (4), 841846.CrossRefGoogle ScholarPubMed
Tytell, E. D., Hsu, C. Y., Williams, T. L., Cohen, A. H. & Fauci, L. J. 2010 Interactions between internal forces, body stiffness, and fluid environment in a neuromechanical model of lamprey swimming. Proc. Natl Acad. Sci. USA 107, 1983219837.Google Scholar
Tytell, E. D. & Lauder, G. V. 2004 The hydrodynamics of eel swimming. I. Wake structure. J. Expl Biol. 207, 18251841.CrossRefGoogle ScholarPubMed
Videler, J. J. 1993 Fish Swimming, Fish and Fisheries Series , vol. 10. Chapman & Hall.Google Scholar
Webb, P. W., Kostecki, P. T. & Don Stevens, E. 1984 The effect of size and swimming speed on locomotor kinematics of rainbow trout. J. Expl Biol. 109, 7795.Google Scholar
Weihs, D. 1973 Optimal fish cruising speed. Nature (London) 245, 4850.CrossRefGoogle Scholar
White, C. R. & Seymour, R. S. 2005 Allometric scaling of mammalian metabolism. J. Expl Biol. 208, 16111619.Google Scholar
Wu, T. Y.-T. 1971a Hydromechanics of swimming propulsion. Part 2. Some optimum shape problems. J. Fluid Mech. 46 (3), 521544.Google Scholar
Wu, T. Y.-T. 1971b Hydromechanics of swimming propulsion. Part 3. Swimming and optimum movements of slender fish with side fins. J. Fluid Mech. 46 (3), 545568.Google Scholar
Wu, T. Y. 2011 Fish swimming and bird/insect flight. Annu. Rev. Fluid Mech. 43, 25.Google Scholar