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Skin friction on a moving wall and its implications for swimming animals

Published online by Cambridge University Press:  08 February 2013

Uwe Ehrenstein  and
Christophe Eloy
Uwe Ehrenstein*
Affiliation:
Aix-Marseille Univ, IRPHE UMR 7342, CNRS, F-13384 Marseille, France
Christophe Eloy
Affiliation:
Aix-Marseille Univ, IRPHE UMR 7342, CNRS, F-13384 Marseille, France Department of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA 92093, USA
*
†Email address for correspondence: ehrenstein@irphe.univ-mrs.fr
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Abstract

Estimating the energetic costs of undulatory swimming is important for biologists and engineers alike. To calculate these costs it is crucial to evaluate the drag forces originating from skin friction. This topic has been controversial for decades, some claiming that animals use ingenious mechanisms to reduce the drag and others hypothesizing that undulatory swimming motions induce a drag increase because of the compression of the boundary layers. In this paper, we examine this latter hypothesis, known as the ‘Bone–Lighthill boundary-layer thinning hypothesis’, by analysing the skin friction in different model problems. First, we study analytically the longitudinal drag on a yawed cylinder in a uniform flow by using the approximation of the momentum equations in the laminar boundary layers. This allows us to demonstrate and generalize a result first observed semi-empirically by G.I. Taylor in the 1950s: the longitudinal drag scales as the square root of the normal velocity component. This scaling arises because the fluid particles accelerate as they move around the cylinder. Next we propose an analogue two-dimensional problem where the same scaling law is recovered by artificially accelerating the flow in a channel of finite height. This two-dimensional problem is then simulated numerically to assess the robustness of the analytical results when inhomogeneities and unsteadiness are present. It is shown that spatial or temporal changes in the normal velocity usually tend to increase the skin friction compared with the ideal steady case. Finally, these results are discussed in the context of swimming energetics. We find that the undulatory motions of swimming animals increase their skin friction drag by an amount that closely depends on the geometry and the motion. For the model problem considered in this paper the increase is of the order of 20 %.

JFM classification

Biological Fluid Dynamics: Biological Fluid Dynamics Boundary Layers: Boundary Layers Biological Fluid Dynamics: Swimming/flying
Type
Papers
Information
Journal of Fluid Mechanics , Volume 718 , 10 March 2013 , pp. 321 - 346
Copyright
©2013 Cambridge University Press

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References

Alexander, R. M. 1977 Swimming. In Mechanics and Energetics of Animal Locomotion (ed. Alexander, R. M. & Goldspink, G.). pp. 222–248. Chapman & Hall.Google Scholar
Aleyev, Y. G. 1977 Nekton. Junk Publishers.CrossRefGoogle Scholar
Anderson, E. J., McGillis, W. R. & Grosenbaugh, M. A. 2001 The boundary layer of swimming fish. J. Exp. Biol. 204 (1), 81–102.CrossRefGoogle ScholarPubMed
Bainbridge, R. 1963 Caudal fin and body movement in the propulsion of some fish. J. Exp. Biol. 40, 23–56.Google Scholar
Barrett, D. S., Triantafyllou, M. S., Yue, D. K. P., Grosenbaugh, M. A. & Wolfgang, M. J. 1999 Drag reduction in fish-like locomotion. J. Fluid Mech. 392, 183–212.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Bechert, D. W., Bruse, M., Hage, W., van Der Hoeven, J. G. T. & Hoppe, G. 1997 Experiments on drag-reducing surfaces and their optimization with an adjustable geometry. J. Fluid Mech. 338, 59–87.CrossRefGoogle Scholar
Bhushan, B. 2009 Biomimetics: lessons from nature – an overview. Phil. Trans. R. Soc. A 367, 1445–1486.CrossRefGoogle ScholarPubMed
Borazjani, I. & Sotiropoulos, F. 2008 Numerical investigation of the hydrodynamics of carangiform swimming in the transitional and inertial flow regimes. J. Exp. Biol. 211, 1541.Google Scholar
Carpenter, P. W., Davies, C. & Lucey, A. D. 2000 Hydrodynamics and compliant walls: does the dolphin have a secret. Curr. Sci. 79, 758–765.Google Scholar
Choi, H., Moin, P. & Kim, J. 1993 Direct numerical simulation of turbulent flow over riblets. J. Fluid Mech. 255, 503–539.CrossRefGoogle Scholar
Churchill, S. W. & Bernstein, M. 1977 A correlating equation for forced convection from gases and liquids to a circular cylinder in crossflow. J. Heat Transfer 99, 300.CrossRefGoogle Scholar
Ehrenstein, U. & Gallaire, F. 2008 Two-dimensional global low-frequency oscillations in a separating boundary-layer flow. J. Fluid Mech. 614, 315–327.Google Scholar
Eloy, C. 2012 Optimal Strouhal number for swimming animals. J. Fluids Struct. 30, 205–218.CrossRefGoogle Scholar
Fish, F. E. 2006 The myth and reality of grey’s paradox: implication of dolphin drag reduction for technology. Bioinspir. Biomim. 1, R17–R25.Google Scholar
Fish, F. E. & Hui, C. A. 1991 Dolphin swimming – a review. Mammal Rev. 21, 181–195.CrossRefGoogle Scholar
Fish, F. E. & Lauder, G. V. 2006 Passive and active flow control by swimming fishes and mammals. Annu. Rev. Fluid Mech. 38, 193–224.CrossRefGoogle Scholar
Gero, D. R. 1952 The hydrodynamic aspects of fish propulsion. Am. Mus. Novit. 1601, 1–32.Google Scholar
Gobert, M.-L., Ehrenstein, U., Astolfi, J. A. & Bot, P. 2010 Nonlinear disturbance evolution in a two-dimensional boundary-layer along an elastic plate and induced radiated sound. Eur. J. Mech. B/Fluids 29, 105–118.CrossRefGoogle Scholar
Gray, J. 1936 Studies in animal locomotion: VI. The propulsive powers of the dolphin. J. Exp. Biol. 13, 192–199.CrossRefGoogle Scholar
Hertel, H. 1966 Structure, Form, Movement. Reinhold.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. Exp. Biol. 109, 229–251.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. 653–672. Plenum.Google Scholar
Khan, W. A., Culham, J. R. & Yovanovich, M. M. 2005 Fluid flow around and heat transfer from elliptical cylinders: analytical approach. J. Thermophys. Heat Transfer 19 (2), 178–185.Google Scholar
Kramer, M. O. 1960 Boundary layer stabilization by distributed damping. J. Am. Soc. Nav. Engng 72, 25–34.Google Scholar
Lighthill, M. J. 1971 Large-amplitude elongated-body theory of fish locomotion. Proc. R. Soc. Lond. B 179, 125–138.Google Scholar
Marquillie, M. & Ehrenstein, U. 2003 On the onset of nonlinear oscillations in a separating boundary-layer flow. J. Fluid Mech. 490, 169–188.Google Scholar
Ota, T. & Nishiyama, H. 1984 Heat transfer and flow around an elliptic cylinder. Intl J. Heat Mass Transfer 27 (10), 1771–1779.Google Scholar
Peyret, R. 2002 Spectral Methods for Incompressible Flows. Springer.Google Scholar
Pohlhausen, K. 1921 Zur näherungsweisen integration der differentialgleichung der laminaren grenzschicht. Z. Angew. Math. Mech. 1, 252–290.CrossRefGoogle Scholar
Poll, D. I. A. 1985 Some observations of the transition process on the windward face of a long yawed cylinder. J. Fluid Mech. 150, 329–356.Google Scholar
Relf, E. F. & Powell, C. H. 1917 Tests on smooth and stranded wires inclined to the wind direction, and a comparison of results on stranded wires in air and water. Rep. Memor. Aero. Res. Comm. Lond. 307, 1–8.Google Scholar
Rosen, M. W. & Cornford, N. E. 1971 Fluid friction of fish slimes. Nature 234, 49–51.Google Scholar
Schlichting, H. 1979 Boundary-layer Theory. McGraw-Hill.Google Scholar
Schultz, W. W. & Webb, P. W. 2002 Power requirements of swimming: Do new methods resolve old questions?. Integr. Comp. Biol. 42, 1018–1025.Google Scholar
Sears, W. R. 1948 The boundary layer of yawed cylinders. J. Aeronaut. Sci. 15, 49–52.CrossRefGoogle Scholar
Shirgaonkar, A. A., MacIver, M. A. & Patankar, N. A. 2009 A new mathematical formulation and fast algorithm for fully resolved simulation of self-propulsion. J. Comput. Phys. 228, 2366–2390.CrossRefGoogle Scholar
Stoer, J. & Bulirsch, R. 1992 Introduction to Numerical Analysis. Springer.Google Scholar
Taylor, G. I. 1952 Analysis of the swimming of long and narrow animals. Proc. R. Soc. Lond. Ser. A 214, 158–183.Google Scholar
Videler, J. J. 1981 Swimming movements, body structure and propulsion in cod, Gadus morhua . Symp. Zool. Soc. Lond. 48, 1–27.Google Scholar
Žukauskas, A. & Žiugžda, J. 1985 Heat Transfer of a Cylinder in Crossflow. Hemisphere Publishing.Google Scholar
Walsh, M. J. 1990 Riblets. In Viscous Drag Reduction in Boundary Layers, Progress in Astronautics and Aeronautics, vol. 123 , pp. 203–261. AIAA.Google Scholar
Webb, P. W. 1975 Hydrodynamics and energetics of fish propulsion. J. Fish. Res. Bd Can. 190, 1–158.Google Scholar
Weihs, D. 1974 Energetic advantages of burst swimming of fish. J. Theor. Biol. 48 (1), 215–229.CrossRefGoogle ScholarPubMed
Weihs, D. 2004 The hydrodynamics of dolphin drafting. J. Biol. 3 (2), 8–16.CrossRefGoogle ScholarPubMed
Wild, J. M. 1949 The boundary layer of yawed infinite wings. J. Aeronaut. Sci. 16, 41–45.Google Scholar
Weis-Fogh, T. & Alexander, R. M. 1977 The sustained power output from striated muscle. In Scale Effects in Animal Locomotion (ed. Pedley, T. J.). pp. 511–525. Academic.Google Scholar
Williams, T. M., Friedl, W. A., Fong, M. L., Yamada, R. M., Sedivy, P. & Haun, J. E. 1992 Travel at low energetic cost by swimming and wave-riding bottlenose dolphins. Nature 355, 821–823.Google Scholar