Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-11T01:41:36.920Z Has data issue: false hasContentIssue false
  • English
  • Français

On the statistical properties of surface elevation, velocities and accelerations in multi-directional irregular water waves

Published online by Cambridge University Press:  11 January 2021

Mathias Klahn Open the ORCID record for Mathias Klahn [Opens in a new window] ,
Per A. Madsen Open the ORCID record for Per A. Madsen [Opens in a new window]  and
David R. Fuhrman Open the ORCID record for David R. Fuhrman [Opens in a new window]
Mathias Klahn*
Affiliation:
Department of Mechanical Engineering, Technical University Of Denmark, DK-2800 Kgs. Lyngby, Denmark
Per A. Madsen
Affiliation:
Department of Mechanical Engineering, Technical University Of Denmark, DK-2800 Kgs. Lyngby, Denmark
David R. Fuhrman
Affiliation:
Department of Mechanical Engineering, Technical University Of Denmark, DK-2800 Kgs. Lyngby, Denmark
*
†Email address for correspondence: matkla@mek.dtu.dk
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper presents a detailed investigation of the role played by the wave steepness in connection with the statistical properties of the surface elevation and fluid kinematics in irregular, directionally spread, deep-water wave fields initially given by a JONSWAP spectrum. Using ensembles of large wave fields obtained from fully nonlinear simulations, we first consider the statistical properties of the surface elevation. In that connection we determine the probability density functions (PDFs) of the surface and crest elevations for wave fields of relatively small to unprecedentedly large steepness, and compare them with theoretical results from the literature in order to establish the latter's accuracy. We then consider certain statistical aspects of the fluid kinematics found at the surface and of the fluid kinematics accompanying large crests, which to our knowledge marks the first investigation of these properties in the literature. We first determine the PDFs of the horizontal fluid velocities and accelerations as well as the vertical fluid acceleration at the surface. Next, we investigate the joint PDF of the surface elevation and each of the velocities and accelerations at the surface, and use it to determine the surface elevations for which the velocities and accelerations at the surface are large. We then present an analysis of the largest fluid velocities and accelerations found in the vicinity of large crests, and compute the PDFs of these quantities. Finally, we consider the PDFs of the location at which the largest velocities and accelerations occur relative to the crest.

JFM classification

Waves/Free-surface Flows: Surface gravity waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 910 , 10 March 2021 , A23
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

Alberello, A., Chabchoub, A., Gramstad, O., Babanin, A. V. & Toffoli, A. 2016 Non-Gaussian properties of second-order wave orbital velocity. Coast. Engng 110, 42–49.CrossRefGoogle Scholar
Alberello, A., Chabchoub, A., Monty, J. P., Nelli, F., Elsnab, J. & Toffoli, A. 2018 An experimental comparison of velocities underneath focused breaking waves. Ocean Engng 155, 201–210.CrossRefGoogle Scholar
Annenkov, S. Y. & Shrira, V. I. 2009 Evolution of kurtosis for wind waves. Geophys. Res. Lett. 36, L13603.CrossRefGoogle Scholar
Annenkov, S. Y. & Shrira, V. I. 2013 Large-time evolution of statistical moments of wind-wave fields. J.Fluid Mech. 726, 517–546.CrossRefGoogle Scholar
Annenkov, S. Y. & Shrira, V. I. 2018 Spectral evolution of weakly nonlinear random waves: kinetic description versus direct numerical simulations. J.Fluid Mech. 844, 766–795.CrossRefGoogle Scholar
Boccoti, P. 2000 Wave Mechanics for Ocean Engineering. Elsevier Science.Google Scholar
Cartwright, D. E. & Longuet-Higgins, M. S. 1956 The statistical distribution of the maxima of a random function. Proc. R. Soc. Lond. 237, 212–232.Google Scholar
Dommermuth, D. 2000 The initialization of nonlinear waves using an adjustment scheme. Wave Motion 32 (4), 307–317.CrossRefGoogle Scholar
Dommermuth, D. G. & Yue, D. K. P. 1987 A high-order spectral method for the study of nonlinear gravity waves. J.Fluid Mech. 184, 267–288.CrossRefGoogle Scholar
Dysthe, K. B., Trulsen, K., Krogstad, H. E. & Socquet-Juglard, H. 2003 Evolution of a narrow-band spectrum of random surface gravity waves. J.Fluid Mech. 478, 1–10.CrossRefGoogle Scholar
Fedele, F., Brennan, J., Ponce de León, S., Dudley, J. & Dias, F. 2016 Real world ocean rogue waves explained without the modulational instability. Sci. Rep. 6, 27715.CrossRefGoogle ScholarPubMed
Fedele, F. & Tayfun, M. A. 2009 On nonlinear wave groups and crest statistics. J.Fluid Mech. 620, 221–239.CrossRefGoogle Scholar
Fuhrman, D. R. & Bingham, H. B. 2004 Numerical solutions of fully non-linear and highly dispersive Boussinesq equations in two horizontal dimensions. Intl J. Numer. Meth. Fluids 44, 231–255.CrossRefGoogle Scholar
Grue, J. & Jensen, A. 2006 Experimental velocities and accelerations in very steep wave events in deep water. Eur. J. Mech. B/Fluids 25 (C3), 554–564.CrossRefGoogle Scholar
Klahn, M., Madsen, P. A. & Fuhrman, D. R. 2020 Simulation of three-dimensional nonlinear water waves using a pseudospectral volumetric method with an artificial boundary condition. J.Fluid Mech. (submitted). Available at: https://files.dtu.dk/u/1CDK80-gyvfGIMrx/manuscript.pdf?lGoogle Scholar
Kopriva, D. A. 2009 Implementing Spectral Methods for Partial Differential Equations. Springer.CrossRefGoogle Scholar
Longuet-Higgins, M. S. 1952 On the statistical distribution of the heights of sea waves. J.Mar. Res. 11, 245–266.Google Scholar
Longuet-Higgins, M. S. 1963 The effect of non-linearities on statistical distributions in the theory of sea waves. J.Fluid Mech. 17, 459–480.CrossRefGoogle Scholar
Longuet-Higgins, M. S. 1964 Modified Gaussian distributions for slightly nonlinear variables. Radio Sci. J. Res. NBS/USNC-URSI 68D, 1049–1062.Google Scholar
Longuet-Higgins, M. S. 1986 Eulerian and Lagrangian aspects of surface waves. J.Fluid Mech. 173, 683–707.CrossRefGoogle Scholar
Nicholls, D. P. 2011 Efficient enforcement of far-field boundary conditions in the transformed field expansions method. J.Comput. Phys. 230, 8290–8303.CrossRefGoogle Scholar
Onorato, M., Cavaleri, L., Fouques, S., Gramstad, O., Janssen, P. A. E. M., Monbaliu, J., Osborne, A. R., Pakozdi, C., Serio, M., Stansberg, C. T. et al. . 2009 Statistical properties of mechanically generated surface gravity waves: a lobaratory experiment in a three-dimensional wave basin. J.Fluid Mech. 627, 235–257.CrossRefGoogle Scholar
Onorato, M. & Suret, P. 2016 Twenty years of progress in oceanic rogue waves: the role played by weakly nonlinear models. Nat. Hazards 84, S541–S548.CrossRefGoogle Scholar
Phillips, O. M., Gu, D. & Donelan, M. 1993 Expected structure of extreme waves in a Gaussian sea. Part 1. Theory and SWADE buoy measurements. J.Phys. Oceanogr. 23, 992–1000.2.0.CO;2>CrossRefGoogle Scholar
Saad, Y. & Schultz, M. H. 1986 GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869.CrossRefGoogle Scholar
Sergeeva, A. & Slunyaev, A. 2013 Rogue waves, rogue events and extreme wave kinematics in spatio-temporal fields of simulated sea states. Nat. Hazards Earth Syst. Sci. 13, 1579–1771.CrossRefGoogle Scholar
Socquet-Juglard, H., Dysthe, K., Truelsen, K., Krogstad, H. E. & Liu, J. 2005 Probability distibutions of surface gravity waves during spectral changes. J.Fluid Mech. 542, 195–216.CrossRefGoogle Scholar
Song, J. B. & Wu, Y.-H. 2000 Statistical distribution of water-particle velocity below the surface layer for finite water depth. Coast. Engng 40, 1–19.CrossRefGoogle Scholar
Tanaka, M. 2001 Verification of Hasselmann's energy transfer among surface gravity waves by direct numerical simulations. J.Fluid Mech. 444, 199–221.CrossRefGoogle Scholar
Tayfun, M. A. 1980 Narrow-band nonlinear sea waves. J.Geophys. Res. 85 (C3), 1548–1552.CrossRefGoogle Scholar
Toffoli, A., Bitner-Gregersen, E., Suslov, S. & Onorato, M. 2012 Statistics of wave orbital velocity in deep water random directional wave fields. In ASME 2012 31st International Conference on Ocean, Offshore and Arctic Engineering (OMAE), 469–475. American Society of Mechanical Engineers.CrossRefGoogle Scholar
Toffoli, A., Gramstad, O., Truelsen, K., Monbaliu, J., Bitner-Gregersen, E. & Onorato, M. 2010 Evolution of weakly nonlinear random directional waves: laboratory experiments and numerical simulations. J.Fluid Mech. 664, 313–336.CrossRefGoogle Scholar
Xiao, W., Liu, Y., Wu, G. & Yue, D. K. P. 2013 Rogue wave occurrence and dynamics by direct simulations of nonlinear wave-field evolution. J.Fluid Mech. 720, 357–392.CrossRefGoogle Scholar
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J.Appl. Mech. Tech. Phys. 9, 190–194.CrossRefGoogle Scholar