Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-21T10:40:02.580Z Has data issue: false hasContentIssue false
  • English
  • Français

Predicting the breaking strength of gravity water waves in deep and intermediate depth

Published online by Cambridge University Press:  06 June 2018

Morteza Derakhti Open the ORCID record for Morteza Derakhti [Opens in a new window] ,
Michael L. Banner  and
James T. Kirby
Morteza Derakhti*
Affiliation:
Applied Physics Laboratory, University of Washington, Seattle, WA 98105, USA
Michael L. Banner
Affiliation:
School of Mathematics and Statistics, The University of New South Wales Sydney, Sydney 2052, Australia
James T. Kirby
Affiliation:
Center for Applied Coastal Research, Department of Civil and Environmental Engineerings, University of Delaware, Newark, DE 19716, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We revisit the classical but as yet unresolved problem of predicting the strength of breaking 2-D and 3-D gravity water waves, as quantified by the amount of wave energy dissipated per breaking event. Following Duncan (J. Fluid Mech., vol. 126, 1983, pp. 507–520), the wave energy dissipation rate per unit length of breaking crest may be related to the fifth moment of the wave speed and the non-dimensional breaking strength parameter $b$. We use a finite-volume Navier–Stokes solver with large-eddy simulation resolution and volume-of-fluid surface reconstruction (Derakhti & Kirby, J. Fluid Mech., vol. 761, 2014a, pp. 464–506; J. Fluid Mech., vol. 790, 2016, pp. 553–581) to simulate nonlinear wave evolution, with a strong focus on breaking onset and postbreaking behaviour for representative cases of wave packets with breaking due to dispersive focusing and modulational instability. The present study uses these results to investigate the relationship between the breaking strength parameter $b$ and the breaking onset parameter $B$ proposed recently by Barthelemy et al. (J. Fluid Mech., vol. 841, 2018, pp. 463–488). The latter, formed from the local energy flux normalized by the local energy density and the local crest speed, simplifies, on the wave surface, to the ratio of fluid speed to crest speed. Following a wave crest, when $B$ exceeds a generic threshold value at the wave crest (Barthelemy et al. 2018), breaking is imminent. We find a robust relationship between the breaking strength parameter $b$ and the rate of change of breaking onset parameter $\text{d}B/\text{d}t$ at the wave crest, as it transitions through the generic breaking onset threshold ($B\sim 0.85$), scaled by the local period of the breaking wave. This result significantly refines previous efforts to express $b$ in terms of a wave packet steepness parameter, which is difficult to define robustly and which does not provide a generically accurate forecast of the energy dissipated by breaking.

JFM classification

Geophysical and Geological Flows: Air/sea interactions Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Wave breaking
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 848 , 10 August 2018 , R2
Check for updates
Copyright
© 2018 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

Allis, M. J.2013 The speed, breaking onset and energy dissipation of 3d deep-water waves. PhD thesis, U. New South Wales.Google Scholar
Banner, M. L. & Peirson, W. L. 2007 Wave breaking onset and strength for two-dimensional deep-water wave groups. J. Fluid Mech. 585, 93–115.Google Scholar
Barthelemy, X., Banner, M. L., Peirson, W. L., Fedele, F., Allis, M. & Dias, F. 2018 On a unified breaking onset threshold for gravity waves in deep and intermediate depth water. J. Fluid Mech. 841, 463–488.CrossRefGoogle Scholar
Derakhti, M. & Kirby, J. T. 2014a Bubble entrainment and liquid bubble interaction under unsteady breaking waves. J. Fluid Mech. 761, 464–506.CrossRefGoogle Scholar
Derakhti, M. & Kirby, J. T.2014b Bubble entrainment and liquid bubble interaction under unsteady breaking waves. Research Report CACR-14-06. Center for Applied Coastal Research, University of Delaware, available at http://www.udel.edu/kirby/papers/derakhti-kirby-cacr-14-06.pdf.Google Scholar
Derakhti, M. & Kirby, J. T. 2016 Breaking-onset, energy and momentum flux in unsteady focused wave packets. J. Fluid Mech. 790, 553–581.Google Scholar
Ducrozet, G., Bonnefoy, F. & Perignon, Y. 2017 Applicability and limitations of highly non-linear potential flow solvers in the context of water waves. Ocean Engng 142, 233–244.CrossRefGoogle Scholar
Duncan, J. H. 1983 The breaking and non-breaking wave resistance of a two-dimensional hydrofoil. J. Fluid Mech. 126, 507–520.Google Scholar
Kirby, J. T. & Derakhti, M.2018 Short-crested wave breaking. Eur. J. Mech. (B/Fluids) (in press).Google Scholar
Phillips, O. M. 1985 Spectral and statistical properties of the equilibrium range in wind-generated gravity waves. J. Fluid Mech. 156, 505–531.CrossRefGoogle Scholar
Pomeau, Y., Le Berre, M., Guyenne, P. & Grilli, S. 2008 Wave-breaking and generic singularities of nonlinear hyperbolic equations. Nonlinearity 21 (5), T61.Google Scholar
Rapp, R. J. & Melville, W. K. 1990 Laboratory measurements of deep-water breaking waves. Phil. Trans. R. Soc. Lond. A 331, 735–800.Google Scholar
Romero, L., Melville, W. K. & Kleiss, J. M. 2012 Spectral energy dissipation due to surface wave breaking. J. Phys. Oceanogr. 42, 1421–1444.CrossRefGoogle Scholar
Saket, A., Peirson, W. L., Banner, M. L. & Allis, M. J. 2018 On the influence of wave breaking on the height limits of two-dimensional wave groups propagating in uniform intermediate depth water. Coast. Engng 133, 159–165.Google Scholar
Saket, A., Peirson, W. L., Banner, M. L., Barthelemy, X. & Allis, M. J. 2017 On the threshold for wave breaking of two-dimensional deep water wave groups in the absence and presence of wind. J. Fluid Mech. 811, 642–658.CrossRefGoogle Scholar
Seiffert, B. R. & Ducrozet, G. 2018 Simulation of breaking waves using the high-order spectral method with laboratory experiments: Wave-breaking energy dissipation. Ocean Dyn. 68, 65–89.CrossRefGoogle Scholar
Seiffert, B. R., Ducrozet, G. & Bonnefoy, F. 2017 Simulation of breaking waves using the high-order spectral method with laboratory experiments: Wave-breaking onset. Ocean Model. 119, 94–104.Google Scholar
Tian, Z., Perlin, M. & Choi, W. 2008 Evaluation of a deep-water wave breaking criterion. Phys. Fluids 20, 066604.Google Scholar
Wu, C. H. & Nepf, H. M. 2002 Breaking criteria and energy losses for three-dimensional wave breaking. J. Geophys. Res. 107, C10, 3177.Google Scholar