Hostname: page-component-745bb68f8f-grxwn Total loading time: 0 Render date: 2025-02-05T07:08:21.413Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear interactions between deep-water waves and currents

Published online by Cambridge University Press:  06 December 2011

R. M. Moreira  and
D. H. Peregrine
R. M. Moreira*
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK
D. H. Peregrine
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK
*
†Present address: Computational Fluid Dynamics Laboratory, Fluminense Federal University, Rua Passo da Pátria 156, bl.D, sl.563A, Niterói, RJ 24210-240, Brazil. Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The effects of nonlinearity on a train of linear water waves in deep water interacting with underlying currents are investigated numerically via a boundary-integral method. The current is assumed to be two-dimensional and stationary, being induced by a distribution of singularities located beneath the free surface, which impose sharp and gentle surface velocity gradients. For ‘slowly’ varying currents, the fully nonlinear results confirm that opposing currents induce wave steepening and breaking within the region where a high convergence of rays occurs. For ‘rapidly’ varying currents, wave blocking and breaking are more prominent. In this case reflection was observed when sufficiently strong adverse currents are imposed, confirming that at least part of the wave energy that builds up within the caustic can be released in the form of partial reflection and wave breaking. For bichromatic waves, the fully nonlinear results show that partial wave blocking occurs at the individual wave components in the wave groups and that waves become almost monochromatic upstream of the blocking region.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Wave breaking Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 691 , 25 January 2012 , pp. 1 - 25
Copyright
Copyright © Cambridge University Press 2011

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.)

Footnotes

‡

Professor Peregrine passed away before this paper was completed. This manuscript was prepared for publication by the first author.

References

1. Barnes, T. C. D., Brocchini, M., Peregrine, D. H. & Stansby, P. K. 1996 Modelling post-wave breaking turbulence and vorticity. In Proceedings of the 25th International Conference on Coastal Engineering, pp. 186–199. ASCE.Google Scholar
2. Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
3. Battjes, J. A. 1982 A case study of wave height variations due to currents in a tidal entrance. Coast. Engng 6, 47–57.CrossRefGoogle Scholar
4. Bretherton, F. P. & Garrett, C. J. R. 1968 Wavetrains in inhomogenous moving media. Proc. R. Soc. Lond. A 302, 529–554.Google Scholar
5. Chawla, A. 1999 An experimental study on the dynamics of wave blocking and breaking on opposing currents. PhD thesis, University of Delaware, USA.Google Scholar
6. Chawla, A. & Kirby, J. T. 1998 Experimental study of wave breaking and blocking on opposing currents. In Proceedings of the 26th International Conference on Coastal Engineering, pp. 759–772. ASCE.Google Scholar
7. Chawla, A. & Kirby, J. T. 2002 Monochromatic and random wave breaking at blocking points. J. Geophys. Res. 107 (C7), doi:10.1029/2001JC001042.Google Scholar
8. Chen, Q., Madsen, P. A., Schäffer, H. A. & Basco, D. R. 1998 Wave-current interaction based on an enhanced Boussinesq approach. Coast. Engng 33, 11–39.CrossRefGoogle Scholar
9. Crapper, G. D. 1972 Nonlinear gravity waves on steady non-uniform currents. J. Fluid Mech. 52, 713–724.Google Scholar
10. Dold, J. W. 1992 An efficient surface-integral algorithm applied to unsteady gravity waves. J. Comput. Phys. 103, 90–115.CrossRefGoogle Scholar
11. Dold, J. W. & Peregrine, D. H. 1986 An efficient boundary-integral method for steep unsteady water waves. In Numerical Methods for Fluid Dynamics II (ed. Morton, K. W. & Baines, M. J. ), pp. 671–679. Oxford University Press.Google Scholar
12. Donato, A. N., Peregrine, D. H. & Stocker, J. R. 1999 The focusing of surface waves by internal waves. J. Fluid Mech. 384, 27–58.Google Scholar
13. Hocking, G. C. & Forbes, L. K. 1992 Subcritical free-surface flow caused by a line source in a fluid of finite depth. J. Engng Maths 26, 455–466.Google Scholar
14. Jonsson, I. G. 1990 Wave-current interactions. In The Sea: Ocean Engineering Science 9A (ed. Le Mehaute, B. & Hanes, D. M. ), pp. 65–120. Wiley Interscience.Google Scholar
15. Kharif, C. & Pelinovsky, E. 2006 Freak waves phenomenon: physical mechanisms and modelling. In Waves in Geophysical Fluids: CISM Courses and Lectures 489 (ed. Grue, J. & Trulsen, K. ), pp. 107–172. Springer.Google Scholar
16. Lai, R. J., Long, S. R. & Huang, N. E. 1989 Laboratory studies of wave-current interaction: kinematics of the strong interaction. J. Geophys. Res. 94, 16201–16214.Google Scholar
17. Lamb, H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
18. Longuet-Higgins, M. S. 1975 Integral properties of periodic gravity waves of finite amplitude. Proc. R. Soc. Lond. A 342, 157–174.Google Scholar
19. Longuet-Higgins, M. S. & Stewart, R. W. 1960 Changes in the form of short gravity waves on long waves and tidal currents. J. Fluid Mech. 8, 565–583.Google Scholar
20. Longuet-Higgins, M. S. & Stewart, R. W. 1961 The changes in amplitude of short gravity waves on non-uniform currents. J. Fluid Mech. 10, 529–549.Google Scholar
21. Mallory, J. K. 1974 Abnormal waves in the south-east coast of South Africa. Intl Hydrog. Rev. 51, 99–129.Google Scholar
22. Mekias, H. & Vanden-Broeck, J.-M. 1989 Supercritical free-surface flow with a stagnation point due to a submerged source. Phys. Fluids A 1, 1694–1697.Google Scholar
23. Mekias, H. & Vanden-Broeck, J.-M. 1991 Subcritical flow with a stagnation point due to a source beneath a free surface. Phys. Fluids A 3, 2652–2658.CrossRefGoogle Scholar
24. Miloh, T. & Tyvand, P. A. 1993 Nonlinear transient free-surface flow and dip formation due to a point sink. Phys. Fluids A 5, 1368–1375.CrossRefGoogle Scholar
25. Moreira, R. M. 2001 Nonlinear interactions between water waves, free-surface flows and singularities. PhD thesis, University of Bristol, UK.Google Scholar
26. Moreira, R. M. & Peregrine, D. H. 2010 Nonlinear interactions between a free-surface flow with surface tension and a submerged cylinder. J. Fluid Mech. 648, 485–507.Google Scholar
27. Novikov, YE. A. 1981 Generation of surface waves by discrete vortices. Izv. Atmos. Ocean. Phys. 17, 709–714.Google Scholar
28. Peregrine, D. H. 1976 Interaction of water waves and currents. Adv. Appl. Mech. 16, 9–117.Google Scholar
29. Peregrine, D. H. & Smith, R. 1979 Nonlinear effects upon waves near caustics. Phil. Trans. R. Soc. Lond. A 292, 341–370.Google Scholar
30. Peregrine, D. H. & Thomas, G. P. 1979 Finite-amplitude deep-water waves on currents. Phil. Trans. R. Soc. Lond. A 292, 371–390.Google Scholar
31. Ris, R. C. & Holthuijsen, L. H. 1996 Spectral modelling of current induced wave-blocking. In Proceedings 25th International Conference on Coastal Engineering, pp. 1246–1254. ASCE.Google Scholar
32. Sakai, S. & Saeki, H. 1984 Effects of opposing current on wave transformation. In Proceedings 19th International Conference on Coastal Engineering, pp. 1132–1148. ASCE.Google Scholar
33. Smith, R. 1975 The reflection of short gravity waves on a non-uniform current. Proc. Camb. Phil. Soc. 78, 517–525.CrossRefGoogle Scholar
34. Stiassnie, M. & Dagan, G. 1979 Partial reflexion of water waves by non-uniform adverse currents. J. Fluid Mech. 92, 119–129.CrossRefGoogle Scholar
35. Stocker, J. R. & Peregrine, D. H. 1999 The current-modified nonlinear Schrödinger equation. J. Fluid Mech. 399, 335–353.CrossRefGoogle Scholar
36. Stokes, T. E., Hocking, G. C. & Forbes, L. K. 2003 Unsteady free-surface flow induced by a line sink. J. Engng Maths 47, 137–160.Google Scholar
37. Suastika, I. K. & Battjes, J. 2009 A model for blocking of periodic waves. Coast. Engng J. 51, 81–99.CrossRefGoogle Scholar
38. Suastika, I. K., de Jong, M. P. C. & Battjes, J. A. 2000 Experimental study of wave blocking. In Proceedings 27th International Conference on Coastal Engineering, pp. 223–240. ASCE.Google Scholar
39. Tanaka, M. 1983 The stability of steep gravity waves. J. Phys. Soc. Japan 52, 3047–3055.Google Scholar
40. Thomas, G. P. & Klopman, G. 1997 Wave-current interactions in the nearshore region. In Gravity Waves in Water of Finite Depth (ed. Hunt, J.N. ), pp. 215–319. Computational Mechanics Publications.Google Scholar
41. Tuck, E. O. & Vanden-Broeck, J.-M. 1984 A cusplike free-surface flow due to a submerged source or sink. J. Austral. Math. Soc. B 25, 443–450.Google Scholar
42. Tyvand, P. A. 1991 Motion of a vortex near a free surface. J. Fluid Mech. 225, 673–686.Google Scholar
43. Tyvand, P. A. 1992 Unsteady free-surface flow due to a line source. Phys. Fluids A 4, 671–676.Google Scholar
44. Vanden-Broeck, J.-M. & Keller, J. B. 1987 Free surface flow due to a sink. J. Fluid Mech. 175, 109–117.Google Scholar
45. Vincent, C. E. 1979 The interaction of wind-generated sea waves with tidal currents. J. Phys. Oceanogr. 9, 748–755.2.0.CO;2>CrossRefGoogle Scholar
46. Xue, M. & Yue, D. K. P. 1998 Nonlinear free-surface flow due to an impulsively-started submerged point sink. J. Fluid Mech. 364, 325–347.Google Scholar