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On the best design for undulatory swimming

  • Christophe Eloy (a1) (a2)
Abstract
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.

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Corresponding author
Email address for correspondence: Christophe.Eloy@irphe.univ-mrs.fr
References
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Abramowitz M. & Stegun I. A. 1965 Handbook of Mathematical Functions. Dover.
Alexander R. M. 1977 Swimming. In Mechanics and Energetics of Animal Locomotion (ed. Alexander R. M. & Goldspink G.), pp. 222248. Chapman & Hall.
Anderson E. J., McGillis W. R. & Grosenbaugh M. A. 2001 The boundary layer of swimming fish. J. Expl Biol. 204, 81102.
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.
Bainbridge R. 1963 Caudal fin and body movement in the propulsion of some fish. J. Expl Biol. 40, 2356.
Blake R. W. 1983 Fish Locomotion. Cambridge University Press.
Blake R. W. 2004 Fish functional design and swimming performance. J. Fish Biol. 65, 11931222.
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.
Branke J., Deb K., Miettinen K. & Słowiński R. (Eds) 2008 Multiobjective Optimization: Interactive and Evolutionary Approaches. Springer.
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.
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.
Chopra M. G. & Kambe T. 1977 Hydromechanics of lunate-tail swimming propulsion. Part 2. J. Fluid Mech. 79, 4969.
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.
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.
D’Août K. & Aerts P. 1999 A kinematic comparison of forward and backward swimming in the eel Anguilla anguilla. J. Expl Biol. 202, 15111521.
Dodds P. S., Rothman D. H. & Weitz J. S. 2001 Re-examination of the ‘ inline-graphic $3/ 4$ -law’ of metabolism. J. Theor. Biol. 209, 927.
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.
Ehrenstein U. & Eloy C. 2012 Skin friction on a moving wall and its implications for swimming animals. J. Fluid Mech. (in press).
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.
Eloy C. 2012 Optimal Strouhal number for swimming animals. J. Fluids Struct. 30, 205218.
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.
Eloy C. & Schouveiler L. 2011 Optimisation of two-dimensional undulatory swimming at high Reynolds number. Intl J. Non-Linear Mech. 46, 568576.
Fish F. E. & Hui C. A. 1991 Dolphin swimming – a review. Mammal Rev. 21, 181195.
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.
Gray J. 1933 Studies in animal locomotion. I. The movement of fish with special reference to the eel. J. Expl Biol. 10, 88104.
Gray J. 1968 Animal Locomotion. Weidenfeld & Nicolson.
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.
Hoerner S. F. 1965 Fluid-dynamic Drag. Published by the author.
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.
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.
Kern S. & Koumoutsakos P. 2006 Simulations of optimized anguilliform swimming. J. Expl Biol. 209, 48414857.
Lauder G. V. & Tytell E. D. 2005 Hydrodynamics of undulatory propulsion. Fish Physiol. 23, 425468.
Lighthill M. J. 1960 Note on the swimming of slender fish. J. Fluid Mech. 9, 305317.
Lighthill M. J. 1969 Hydromechanics of aquatic animal propulsion. Annu. Rev. Fluid Mech. 1, 413446.
Lighthill M. J. 1970 Aquatic animal propulsion of high hydromechanical efficiency. J. Fluid Mech. 44, 265301.
Lighthill M. J. 1971 Large-amplitude elongated-body theory of fish locomotion. Proc. R. Soc. Lond. B 179, 125138.
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.
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.
Ota T. & Nishiyama H. 1984 Heat transfer and flow around an elliptic cylinder. Intl J. Heat Mass Transfer 27 (10), 17711779.
Rosen M. W. & Cornford N. E. 1971 Fluid friction of fish slimes. Nature 234, 4951.
Schlichting H. 1979 Boundary-layer Theory. McGraw-Hill.
Schneck D. J. 1992 Mechanics of Muscle, 2nd edn. New York University Press.
Taylor G. I. 1952 Analysis of the swimming of long and narrow animals. Proc. R. Soc. Lond. A 214, 158183.
Tokić G & Yue D. K. P. 2012 Optimal shape and motion of undulatory swimming organisms. Proc. R. Soc. Lond. B 279 (1740), 30653074.
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.
Triantafyllou M. S., Triantafyllou G. S. & Yue D. K. P. 2000 Hydrodynamics of fishlike swimming. Annu. Rev. Fluid Mech. 32 (1), 3353.
Tucker V. A. 1970 Energetic cost of locomotion in animals. Comp. Biochem. Physiol. 34 (4), 841846.
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.
Tytell E. D. & Lauder G. V. 2004 The hydrodynamics of eel swimming. I. Wake structure. J. Expl Biol. 207, 18251841.
Videler J. J. 1993 Fish Swimming, Fish and Fisheries Series , vol. 10. Chapman & Hall.
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.
Weihs D. 1973 Optimal fish cruising speed. Nature (London) 245, 4850.
White C. R. & Seymour R. S. 2005 Allometric scaling of mammalian metabolism. J. Expl Biol. 208, 16111619.
Wu T. Y.-T. 1971a Hydromechanics of swimming propulsion. Part 2. Some optimum shape problems. J. Fluid Mech. 46 (3), 521544.
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.
Wu T. Y. 2011 Fish swimming and bird/insect flight. Annu. Rev. Fluid Mech. 43, 25.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
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