Hostname: page-component-745bb68f8f-s22k5 Total loading time: 0 Render date: 2025-01-28T02:56:50.503Z Has data issue: false hasContentIssue false
  • English
  • Français

Kelvin wake pattern at small Froude numbers

Published online by Cambridge University Press:  31 March 2021

Ravindra Pethiyagoda Open the ORCID record for Ravindra Pethiyagoda [Opens in a new window] ,
Timothy J. Moroney Open the ORCID record for Timothy J. Moroney [Opens in a new window] ,
Christopher J. Lustri Open the ORCID record for Christopher J. Lustri [Opens in a new window]  and
Scott W. McCue Open the ORCID record for Scott W. McCue [Opens in a new window]
Ravindra Pethiyagoda
Affiliation:
School of Mathematical Sciences, Queensland University of Technology, BrisbaneQLD4001, Australia
Timothy J. Moroney
Affiliation:
School of Mathematical Sciences, Queensland University of Technology, BrisbaneQLD4001, Australia
Christopher J. Lustri
Affiliation:
Department of Mathematics, Macquarie University, SydneyNSW2109, Australia
Scott W. McCue*
Affiliation:
School of Mathematical Sciences, Queensland University of Technology, BrisbaneQLD4001, Australia
*
†Email address for correspondence: scott.mccue@qut.edu.au
Get access Rights & Permissions [Opens in a new window]

Abstract

The surface gravity wave pattern that forms behind a steadily moving disturbance is well known to comprise divergent waves and transverse waves, contained within a distinctive $V$-shaped wake. In this paper, we are concerned with a theoretical study of the limit of a slow-moving disturbance (small Froude numbers) in the absence of surface tension, for which the wake is dominated by transverse waves. Three configurations are considered: flow past a submerged source singularity, a submerged doublet and a pressure distribution applied to the surface. We treat the linearised version of these problems and use the method of stationary phase and exponential asymptotics to demonstrate that the apparent wake angle is less than the classical Kelvin angle and to quantify the decrease in apparent wake angle as the Froude number decreases. These results complement a number of recent studies for sufficiently fast-moving disturbances (large Froude numbers) where the apparent wake angle has been also shown to be less than the classical Kelvin angle. As well as shedding light on the issue of apparent wake angle, we also study the fully nonlinear problems for our three configurations under various limits to demonstrate the unique and interesting features of Kelvin wake patterns at small Froude numbers.

JFM classification

Waves/Free-surface Flows: Surface gravity waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 915 , 25 May 2021 , A126
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Arzhannikov, A.V. & Kotelnikov, I.A. 2016 Excitation of ship waves by a submerged object: new solution to the classical problem. Phys. Rev. E 94, 023103.CrossRefGoogle ScholarPubMed
Batchelor, G.K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Benzaquen, M., Darmon, A. & Raphaël, E. 2014 Wake pattern and wave resistance for anisotropic moving disturbances. Phys. Fluids 26, 092106.CrossRefGoogle Scholar
Berry, M. 1991 Asymptotics, superasymptotics, hyperasymptotics. In Asymptotics Beyond all Orders (ed. H. Segur, S. Tanveer & H. Levine), pp. 1–14. Springer.CrossRefGoogle Scholar
Binder, B.J., Dias, F. & Vanden-Broeck, J.-M. 2006 Steady free-surface flow past an uneven channel bottom. Theor. Comput. Fluid Dyn. 20, 125–144.CrossRefGoogle Scholar
Boyd, J.P. 1999 The devil's invention: asymptotic, superasymptotic and hyperasymptotic series. Acta Appl. Maths 56, 1–98.CrossRefGoogle Scholar
Buttle, N.R., Pethiyagoda, R, Moroney, T.J. & McCue, S.W. 2018 Three-dimensional free-surface flow over arbitrary bottom topography. J. Fluid Mech. 846, 166–189.CrossRefGoogle Scholar
Chapman, S.J. & Vanden-Broeck, J.-M. 2006 Exponential asymptotics and gravity waves. J. Fluid Mech. 567, 299–326.CrossRefGoogle Scholar
Dagan, G. 1975 Waves and wave resistance of thin bodies moving at low speed: the free-surface nonlinear effect. J. Fluid Mech. 69, 405–416.CrossRefGoogle Scholar
Darmon, A., Benzaquen, M. & Raphaël, E. 2014 Kelvin wake pattern at large Froude numbers. J. Fluid Mech. 738, R3.CrossRefGoogle Scholar
Dias, F. 2014 Ship waves and Kelvin. J. Fluid Mech. 746, 1–4.CrossRefGoogle 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. 1990 Flow caused by a point sink in a fluid having a free surface. J. Austral. Math. Soc. 32, 231–249.CrossRefGoogle Scholar
Havelock, T.H. 1919 Wave resistance: some cases of three-dimensional fluid motion. Proc. R. Soc. Lond. A 95, 354–365.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. 2014 Comparison of three simple models of Kelvin's ship wake. Eur. J. Mech. B/Fluids 49, 12–19.CrossRefGoogle Scholar
Hermans, A.J. & Brandsma, F.J. 1989 Nonlinear ship waves at low Froude number. J. Ship Res. 33 (3), 176–193.CrossRefGoogle Scholar
Hocking, G.H., Forbes, L.K. & Stokes, T.E. 2016 A note on steady flow into a submerged point sink. ANZIAM J. 57, 417–428.Google Scholar
Keller, J.B. 1979 Ray theory of ship waves and the class of streamlined ships. J. Fluid Mech. 91, 465–488.CrossRefGoogle Scholar
Kim, W.D. 1969 Nonlinear free-surface effects on a submerged sphere. J. Hydronaut. 3, 29–37.CrossRefGoogle Scholar
Lamb, H. 1916 Hydrodynamics, 4th edn. Cambridge University Press.Google Scholar
Li, Y. 2018 Wave-interference effects on far-field ship waves in the presence of a shear current. J. Ship Res. 62, 37–47.CrossRefGoogle Scholar
Li, Y. & Ellingsen, S.Å. 2016 Ship waves on uniform shear current at finite depth: wave resistance and critical velocity. J. Fluid Mech. 791, 539–567.CrossRefGoogle Scholar
Liang, H. & Chen, X. 2019 Viscous effects on the fundamental solution to ship waves. J. Fluid Mech. 879, 744–774.CrossRefGoogle Scholar
Lustri, C.J. 2013 Exponential asymptotics in unsteady and three-dimensional flows. DPhil Thesis, University of Oxford.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. & Chapman, S.J. 2014 Unsteady flow over a submerged source with low Froude number. Eur. J. Appl. Maths 25, 655–680.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
Lustri, C.J., Pethiyagoda, R & Chapman, S.J. 2019 Three-dimensional capillary waves due to a submerged source with small surface tension. J. Fluid Mech. 863, 670–701.CrossRefGoogle Scholar
Ma, C., Zhu, Y., Wu, H., He, J., Zhang, C., Li, W. & Noblesse, F. 2016 Wavelengths of the highest waves created by fast monohull ships or catamarans. Ocean Engng 113, 208–214.CrossRefGoogle Scholar
McCue, S.W., Pethiyagoda, R. & Moroney, T.J. 2019 Simplified models for waves due to steadily moving ships and submerged bodies. In 34th International Worshop on Water Waves and Floating Bodies. Newcastle, Australia.Google Scholar
Miao, S. & Liu, Y. 2015 Wave pattern in the wake of an arbitrary moving surface pressure disturbance. Phys. Fluids 27, 122102.CrossRefGoogle Scholar
Moisy, F. & Rabaud, M. 2014 a Mach-like capillary-gravity wakes. Phys. Rev. E 90, 023009.CrossRefGoogle ScholarPubMed
Moisy, F. & Rabaud, M. 2014 b The scaling of far-field wake angle of non-axisymmetric pressure disturbance. Phys. Rev. E 89, 063004.CrossRefGoogle 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. 1981 Alternative integral representations for the Green function of the theory of ship wave resistance. J. Engng Maths 15, 241–265.CrossRefGoogle Scholar
Noblesse, F. 1986 a Analytical approximation for steady ship waves at low Froude numbers. Tech. Rep. DTNSRDC-86/058. David W Taylor Naval Ship Research and Development Center, Bethesda, MD.CrossRefGoogle Scholar
Noblesse, F. 1986 b Steady ship waves at low Froude numbers (Part One). Tech. Rep. DTNSRDC-86/024. David W Taylor Naval Ship Research and Development Center, Bethesda, MD.CrossRefGoogle Scholar
Noblesse, F. 1986 c Steady ship waves at low Froude numbers (Part Two). Tech. Rep. DTNSRDC-86/078. David W Taylor Naval Ship Research and Development Center, Bethesda, MD.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.CrossRefGoogle Scholar
Noblesse, F., Zhang, C., He, J., Zhu, Y., Yang, C. & Li, W. 2016 Observations and computations of narrow Kelvin ship wakes. J. Ocean Engng Sci. 1, 52–65.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. 2014 a What is the apparent angle of a Kelvin ship wave pattern? J. Fluid Mech. 758, 468–485.CrossRefGoogle Scholar
Pethiyagoda, R., McCue, S.W. & Moroney, T.J. 2015 Wake angle for surface gravity waves on a finite depth fluid. Phys. Fluids 27, 061701.CrossRefGoogle Scholar
Pethiyagoda, R., McCue, S.W. & Moroney, T.J. 2017 Spectrograms of ship wakes: identifying linear and nonlinear wave signals. J. Fluid Mech. 811, 189–209.CrossRefGoogle Scholar
Pethiyagoda, R., McCue, S.W., Moroney, T.J. & Back, J.M. 2014 b Jacobian-free Newton–Krylov methods with GPU acceleration for computing nonlinear ship wave patterns. J. Comput. Phys. 269, 297–313.CrossRefGoogle Scholar
Pethiyagoda, R., Moroney, T.J., Macfarlane, G.J., Binns, J.R. & McCue, S.W. 2018 Time-frequency analysis of ship wave patterns in shallow water: modelling and experiments. Ocean Engng 158, 123–131.CrossRefGoogle 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.CrossRefGoogle Scholar
Părău, E., Vanden-Broeck, J.-M. & Cooker, M.J. 2005 Nonlinear three-dimensional gravity-capillary solitary waves. J. Fluid Mech. 536, 99–105.CrossRefGoogle Scholar
Rabaud, M. & Moisy, F. 2013 Ship wakes: Kelvin or Mach angle? Phys. Rev. Lett. 110, 214503.CrossRefGoogle ScholarPubMed
Schwartz, L.W. 1974 Computer extension and analytic continuation of Stokes expansion for gravity waves. J. Fluid Mech. 62, 553–578.CrossRefGoogle Scholar
Scullen, D.C. 1998 Accurate computation of steady nonlinear free-surface flows. PhD thesis, University of Adelaide, Australia.Google Scholar
Smeltzer, B.K. & Ellingsen, S.Å. 2017 Surface waves on currents with arbitrary vertical shear. Phys. Fluids 29, 047102.CrossRefGoogle Scholar
Soomere, T. 2007 Nonlinear components of ship wake waves. Appl. Mech. Rev. 60, 120–138.CrossRefGoogle Scholar
Stokes, G.G. 1847 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441–455.Google Scholar
Vanden-Broeck, J.-M. & Keller, J.B. 1997 An axisymmetric free surface with a 120 degree angle along a circle. J. Fluid Mech. 342, 403–409.CrossRefGoogle Scholar
Verberck, B. 2013 Hydrodynamics: wake up. Nat. Phys. 9, 390.CrossRefGoogle Scholar
Wu, H., Wu, J., He, J., Zhu, R., Yang, C.-J. & Noblesse, F. 2019 Wave profile along a ship hull, short farfield waves, and broad inner Kelvin wake sans divergent waves. Phys. Fluids 31 (4), 47102.CrossRefGoogle Scholar
Zhang, C., He, J., Zhu, Y., Yang, C.-J., Li, W., Zhu, Y., Lin, M. & Noblesse, F. 2015 Interference effects on the Kelvin wake of a monohull ship represented via a continuous distribution of sources. Eur. J. Mech. B/Fluids 51, 27–36.CrossRefGoogle Scholar
Zhu, Y., He, J., Wu, H., Li, W., Noblesse, F. & Delhommeau, G. 2018 Elementary ship models and farfield waves. Eur. J. Mech. B/Fluids 67, 231–241.CrossRefGoogle Scholar
Zhu, Y., He, J., Zhang, C., Wu, H., Wan, D., Zhu, R. & Noblesse, F. 2015 Farfield waves created by a monohull ship in shallow water. Eur. J. Mech. B/Fluids 49, 226–234.CrossRefGoogle Scholar
Zhu, Y., Wu, H., Ma, C., He, J., Li, W., Wan, D. & Noblesse, F. 2017 Michell and Hogner models of far-field ship waves. Appl. Ocean Res. 68, 194–203.CrossRefGoogle Scholar