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What is the apparent angle of a Kelvin ship wave pattern?

Published online by Cambridge University Press:  09 October 2014

Ravindra Pethiyagoda ,
Scott W. McCue Open the ORCID record for Scott W. McCue [Opens in a new window]  and
Timothy J. Moroney
Ravindra Pethiyagoda
Affiliation:
Mathematical Sciences, Queensland University of Technology, QLD 4001, Australia
Scott W. McCue*
Affiliation:
Mathematical Sciences, Queensland University of Technology, QLD 4001, Australia
Timothy J. Moroney
Affiliation:
Mathematical Sciences, Queensland University of Technology, QLD 4001, Australia
*
†Email address for correspondence: scott.mccue@qut.edu.au
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Abstract

While the half-angle which encloses a Kelvin ship wave pattern is commonly accepted to be 19.47°, recent observations and calculations for sufficiently fast-moving ships suggest that the apparent wake angle decreases with ship speed. One explanation for this decrease in angle relies on the assumption that a ship cannot generate wavelengths much greater than its hull length. An alternative interpretation is that the wave pattern that is observed in practice is defined by the location of the highest peaks; for wakes created by sufficiently fast-moving objects, these highest peaks no longer lie on the outermost divergent waves, resulting in a smaller apparent angle. In this paper, we focus on the problems of free-surface flow past a single submerged point source and past a submerged source doublet. In the linear version of these problems, we measure the apparent wake angle formed by the highest peaks, and observe the following three regimes: a small Froude number pattern, in which the divergent waves are not visible; standard wave patterns for which the maximum peaks occur on the outermost divergent waves; and a third regime in which the highest peaks form a V-shape with an angle much less than the Kelvin angle. For nonlinear flows, we demonstrate that nonlinearity has the effect of increasing the apparent wake angle so that some highly nonlinear solutions have apparent wake angles that are greater than Kelvin’s angle. For large Froude numbers, the effect on apparent wake angle can be more dramatic, with the possibility of strong nonlinearity shifting the wave pattern from the third regime to the second. We expect that our nonlinear results will translate to other more complicated flow configurations, such as flow due to a steadily moving closed body such as a submarine.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Wakes/Jets: Wakes/Jets Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 758 , 10 November 2014 , pp. 468 - 485
Copyright
© 2014 Cambridge University Press 

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References

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Bensabat, J., Zhou, Q. & Bear, J. 2000 An adaptive pathline-based particle tracking algorithm for the Eulerian–Lagrangian method. Adv. Water Resour. 23, 383–397.Google Scholar
Benzaquen, M., Darmon, A. & Raphaël, E. 2014 Wake pattern and wave resistance for anisotropic moving disturbances. Phys. Fluids 26, 092106.CrossRefGoogle Scholar
Carusotto, I. & Rousseaux, G. 2013 The C̆erenkov effect revisited: from swimming ducks to zero modes in gravitational analogues. In Lecture Notes in Physics (ed. Faccio, D., Belgiorno, F., Cacciatori, S., Gorini, V., Liberati, S. & Moschella, U.), Analogue Gravity Phenomenology, vol. 870, pp. 109–144. Springer.Google Scholar
Chapman, S. J. & Vanden-Broeck, J.-M. 2006 Exponential asymptotics and gravity waves. J. Fluid Mech. 567, 299–326.CrossRefGoogle Scholar
Chung, Y. K. & Lim, J. S. 1991 A review of the Kelvin ship wave pattern. J. Ship Res. 35, 191–197.CrossRefGoogle Scholar
Cokelet, E. D. 1977 Steep gravity waves in water of arbitrary uniform depth. Phil. Trans. R. Soc. Lond. A 286, 183–230.Google Scholar
Dallaston, M. C. & McCue, S. W. 2010 Accurate series solutions for gravity-driven Stokes waves. Phys. Fluids 22, 082104.Google Scholar
Darmon, A., Benzaquen, M. & Raphaël, E. 2014 Kelvin wake pattern at large Froude numbers. J. Fluid Mech. 738, R3.Google Scholar
Darrigol, O. 2003 The spirited horse, the engineer, and the mathematician: water waves in nineteenth-century hydrodynamics. Arch. Hist. Exact Sci. 58, 21–95.CrossRefGoogle Scholar
Dias, F. 2014 Ship waves and Kelvin. J. Fluid Mech. 746, 1–4.CrossRefGoogle Scholar
Doyle, T. B. & McKenzie, J. F. 2013 Stationary wave patterns in deep water. Quaes. Math. 36, 487–500.Google Scholar
Ellingsen, S. Å. 2014 Ship waves in the presence of uniform vorticity. J. Fluid Mech. 742, R2.CrossRefGoogle Scholar
Forbes, L. K. 1989 An algorithm for 3-dimensional free-surface problems in hydrodynamics. J. Comput. Phys. 82, 330–347.CrossRefGoogle Scholar
Forbes, L. K. & Hocking, G. C. 2005 Flow due to a sink near a vertical wall, in infinitely deep fluid. Comput. Fluids 34, 684–704.Google Scholar
Havelock, T. H. 1928 Wave resistance. Proc. R. Soc. Lond. A 118, 24–33.Google Scholar
Havelock, T. H. 1932 Ship waves: the calculation of wave profiles. Proc. R. Soc. Lond. A 135, 1–13.Google Scholar
He, J., Zhang, C., Zhu, Y., Wu, H., Yang, C.-J., Noblesse, F., Gu, X. & Li, W. 2015 Comparison of three simple models of Kelvin’s ship wake. Eur. J. Mech. (B/Fluids) 49, 12–19.Google Scholar
Keller, J. B. 1979 The ray theory of ship waves and the class of streamlined ships. J. Fluid Mech. 91, 465–488.CrossRefGoogle Scholar
Lamb, H. 1916 Hydrodynamics, 4th edn. Cambridge University Press.Google Scholar
Lighthill, J. 1978 Waves in Fluids. Cambridge University Press.Google Scholar
Lukomsky, V., Gandzha, I. & Lukomsky, D. 2002 Steep sharp-crested gravity waves in deep water. Phys. Rev. Lett. 89, 164502.Google Scholar
Lustri, C. J. & Chapman, S. J. 2013 Steady gravity waves due to a submerged source. J. Fluid Mech. 732, 660–686.CrossRefGoogle Scholar
Lustri, C. J., McCue, S. W. & Binder, B. J. 2012 Free surface flow past topography: A beyond-all-orders approach. Eur. J. Appl. Maths 23, 441–467.CrossRefGoogle Scholar
Moisy, F. & Rabaud, M. 2014a The scaling of far-field wake angle of non-axisymmetric pressure disturbance. Phys. Rev. E 89, 063004.Google Scholar
Moisy, F. & Rabaud, M. 2014b Mach-like capillary–gravity wakes. Phys. Rev. E 90, 023009.Google Scholar
Noblesse, F. 1978 The steady wave potential of a unit source, at the centerplane. J. Ship Res. 22, 80–88.CrossRefGoogle Scholar
Noblesse, F., He, J., Zhu, Y., Hong, L., Zhang, C., Zhu, R. & Yang, C. 2014 Why can ship wakes appear narrower than Kelvin’s angle? Eur. J. Mech. (B/Fluids) 46, 164–171.Google Scholar
Părău, E. & Vanden-Broeck, J.-M. 2002 Nonlinear two- and three-dimensional free surface flows due to moving disturbances. Eur. J. Mech. (B/Fluids) 21, 643–656.Google Scholar
Părău, E., Vanden-Broeck, J.-M. & Cooker, M. J. 2007 Nonlinear three-dimensional interfacial flows with a free surface. J. Fluid Mech. 591, 481–494.CrossRefGoogle Scholar
Peters, A. S. 1949 A new treatment of the ship wave problem. Commun. Pure Appl. Maths 2, 123–148.CrossRefGoogle Scholar
Pethiyagoda, R., McCue, S. W., Moroney, T. J. & Back, J. M. 2014 Jacobian-free Newton–Krylov methods with GPU acceleration for computing nonlinear ship wave patterns. J. Comput. Phys. 269, 297–313.Google Scholar
Rabaud, M. & Moisy, F. 2013 Ship wakes: Kelvin or Mach angle? Phys. Rev. Lett. 110, 214503.CrossRefGoogle ScholarPubMed
Rabaud, M. & Moisy, F. 2014 Narrow ship wakes and wave drag for planing hulls. Ocean Engng, doi:10.1016/j.oceaneng.2014.06.039.Google Scholar
Schwartz, L. W. 1974 Computer extension and analytic continuation of Stokes expansion for gravity waves. J. Fluid Mech. 62, 553–578.Google Scholar
Soomere, T. 2007 Nonlinear components of ship wake waves. Appl. Mech. Rev. 60, 120–138.Google Scholar
Thomson, W. 1887 On ship waves. Proc. Inst. Mech. Engrs 38, 409–434.CrossRefGoogle Scholar
Trinh, P. H., Chapman, S. J. & Vanden-Broeck, J.-M. 2011 Do waveless ships exist? Results for single-cornered hulls. J. Fluid Mech. 685, 413–439.Google Scholar
Tuck, E. O. 1965 The effect of nonlinearity at the free surface on flow past a submerged cylinder. J. Fluid Mech. 22, 401–414.Google Scholar
Tuck, E. O. & Scullen, D. C. 2002 A comparison of linear and nonlinear computations of waves made by slender submerged bodies. J. Engng Maths 42, 255–264.Google Scholar
Ursell, F. 1960 On Kelvin’s ship-wave pattern. J. Fluid Mech. 8, 418–431.Google Scholar
Williams, J. M. 1981 Limiting gravity waves in water of finite depth. Phil. Trans. R. Soc. Lond. A 302, 139–188.Google Scholar