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Kelvin wake pattern at large Froude numbers

  • Alexandre Darmon (a1) (a2), Michael Benzaquen (a1) and Elie Raphaël (a1)
Abstract

Gravity waves generated by an object moving at constant speed at the water surface form a specific pattern commonly known as the Kelvin wake. It was proved by Lord Kelvin that such a wake is delimited by a constant angle ${\simeq }19. 4{7}^{\circ } $ . However a recent study by Rabaud and Moisy based on the observation of airborne images showed that the wake angle seems to decrease as the Froude number $Fr$ increases, scaling as $F{r}^{- 1} $ for large Froude numbers. To explain such observations they make the strong hypothesis that an object of size $b$ cannot generate wavelengths larger than $b$ . Without the need of such an assumption and modelling the moving object by an axisymmetric pressure field, we analytically show that the angle corresponding to the maximum amplitude of the waves scales as $F{r}^{- 1} $ for large Froude numbers, whereas the angle delimiting the wake region outside which the surface is essentially flat remains constant and equal to the Kelvin angle for all $Fr$ .

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Email address for correspondence: elie.raphael@espci.fr
References
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Appel, W. 2007 Mathematics for Physics and Physicists. Princeton University Press.
Benzaquen, M., Chevy, F. & Raphaël, E. 2011 Wave resistance for capillary gravity waves: Finite-size effects. Europhys. Lett. 96 (3), 34003.
Benzaquen, M. & Raphaël, E. 2012 Capillary-gravity waves on depth-dependent currents: Consequences for the wave resistance. Europhys. Lett. 97 (1), 14007.
Casling, E. M. 1978 Planing of a low-aspect-ratio flat ship at infinite Froude number. J. Engng Maths 12, 4357.
Chepelianskii, A. D., Chevy, F. & Raphaël, E. 2008 Capillary-gravity waves generated by a slow moving object. Phys. Rev. Lett. 100 (7), 074504.
Crawford, F. 1984 Elementary derivation of the wake pattern of a boat. Am. J. Phys. 52, 782785.
Cumbertach, E. 1958 Two-dimensional planing at high Froude number. J. Fluid Mech. 4, 466478.
Darrigol, O. 2005 Words of Flow: A History of Hydrodynamics from the Bernoullis to Prandtl. Oxford University Press.
Falkovich, G. 2011 Fluid Mechanics — A Short Course for Physicists. Cambridge University Press.
Havelock, T. H. 1908 The propagation of groups of waves in dispersive media, with application to waves on water produced by a travelling disturbance. Proc. R. Soc. A 95, 354.
Havelock, T. H. 1919 Periodic irrotational waves of finite height. Proc. R. Soc. Lond. A 95, 3851.
Johnson, R. S. 1997 A Modern Introduction to the Mathematical Theory of Water Waves. Cambridge University Press.
Kelvin, Lord 1887 On ship waves. Proc. Inst. Mech. Engrs 38, 409434.
Lai, C. & Troesch, A. W. 1995 Modelling issues related to the hydrodynamics of three-dimensional steady planing. J. Ship Res. 39, 124.
Lamb, H. 1993 Hydrodynamics, 6th edn. Cambridge University Press.
Le Merrer, M., Clanet, C., Quéré, D., Raphaël, E. & Chevy, F. 2011 Wave drag on floating bodies. Proc. Natl Acad. Sci. 108 (30), 1506415068.
Lighthill, J. 1978 Waves in Fluids. Cambridge University Press.
Nakos, D. E. & Sclavounos, P. D. 1990 On steady and unsteady ship wave pattern. J. Fluid Mech. 215, 263288.
Parnell, K. E. & Kofoed-Hansen, H. 2001 Wakes from large high-speed ferries in confined coastal waters: Management approaches with examples from New-Zealand and Denmark. Coastal Management 29, 217237.
Rabaud, M. & Moisy, F. 2013a Ship wakes: Kelvin or mach angle? Phys. Rev. Lett 110, 214503.
Rabaud, M. & Moisy, F. 2013b Narrow ship wakes and wave drag for planning hulls. Innov-Sail, Lorient, 26–28 June .
Raphaël, E. & de Gennes, P.-G. 1996 Capillary gravity waves caused by a moving disturbance: wave resistance. Phys. Rev. E 53 (4), 34483455.
Suzuki, K., Nakata, Y., Ikehata, M. & Kai, H. 1997 Numerical prediction on wave making resistance of high speed trimaran. In Fourth International Conference on Fast Sea Transportation, Sydney, 21–23 July.
Tuck, E. O., Scullen, D. C. & Lazauskas, L. 2002 Wave patterns and minimum wave resistance for high-speed vessels. In 24th Symposium on Naval Hydrodynamics, Fukuoka, Japan, 8–13 July.
Voise, J. & Casas, J. 2010 The management of fluid and wave resistances by whirligig beetles. J. R. Soc. Interface 7, 343.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
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