Hostname: page-component-848d4c4894-r5zm4 Total loading time: 0 Render date: 2024-06-20T13:23:13.319Z Has data issue: false hasContentIssue false
  • English
  • Français

An improved Lagrangian model for the time evolution of nonlinear surface waves

Published online by Cambridge University Press:  01 August 2019

Charles-Antoine Guérin Open the ORCID record for Charles-Antoine Guérin [Opens in a new window] ,
Nicolas Desmars ,
Stéphan T. Grilli Open the ORCID record for Stéphan T. Grilli [Opens in a new window] ,
Guillaume Ducrozet Open the ORCID record for Guillaume Ducrozet [Opens in a new window] ,
Yves Perignon  and
Pierre Ferrant
Charles-Antoine Guérin*
Affiliation:
Université de Toulon, Aix-Marseille Université, IRD, CNRS-INSU, Mediterranean Institute of Oceanography (MIO UM 110), 83957 La Garde, France
Nicolas Desmars
Affiliation:
Ecole Centrale Nantes, LHEEA Res. Dept. (ECN and CNRS), 44321 Nantes, France
Stéphan T. Grilli
Affiliation:
Department of Ocean Engineering, University of Rhode Island, Narragansett, RI 02882, USA
Guillaume Ducrozet
Affiliation:
Ecole Centrale Nantes, LHEEA Res. Dept. (ECN and CNRS), 44321 Nantes, France
Yves Perignon
Affiliation:
Ecole Centrale Nantes, LHEEA Res. Dept. (ECN and CNRS), 44321 Nantes, France
Pierre Ferrant
Affiliation:
Ecole Centrale Nantes, LHEEA Res. Dept. (ECN and CNRS), 44321 Nantes, France
*
†Email address for correspondence: guerin@univ-tln.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

Accurate real-time simulations and forecasting of phase-revolved ocean surface waves require nonlinear effects, both geometrical and kinematic, to be accurately represented. For this purpose, wave models based on a Lagrangian steepness expansion have proved particularly efficient, as compared to those based on Eulerian expansions, as they feature higher-order nonlinearities at a reduced numerical cost. However, while they can accurately model the instantaneous nonlinear wave shape, Lagrangian models developed to date cannot accurately predict the time evolution of even simple periodic waves. Here, we propose a novel and simple method to perform a Lagrangian expansion of surface waves to second order in wave steepness, based on the dynamical system relating particle locations and the Eulerian velocity field. We show that a simple redefinition of reference particles allows us to correct the time evolution of surface waves, through a modified nonlinear dispersion relationship. The resulting expressions of free surface particle locations can then be made numerically efficient by only retaining the most significant contributions to second-order terms, i.e. Stokes drift and mean vertical level. This results in a hybrid model, referred to as the ‘improved choppy wave model’ (ICWM) (with respect to Nouguier et al.’s J. Geophys. Res., vol. 114, 2009, p. C09012), whose performance is numerically assessed for long-crested waves, both periodic and irregular. To do so, ICWM results are compared to those of models based on a high-order spectral method and classical second-order Lagrangian expansions. For irregular waves, two generic types of narrow- and broad-banded wave spectra are considered, for which ICWM is shown to significantly improve wave forecast accuracy as compared to other Lagrangian models; hence, ICWM is well suited to providing accurate and efficient short-term ocean wave forecast (e.g. over a few peak periods). This aspect will be the object of future work.

JFM classification

Waves/Free-surface Flows: Surface gravity waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 876 , 10 October 2019 , pp. 527 - 552
Copyright
© 2019 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

ASC 2018 Flash LIDAR™, http://www.advancedscientificconcepts.com/products/overview.html.Google Scholar
Babarit, A. & Clément, A. 2006 Optimal latching control of a wave energy device in regular and irregular waves. Appl. Ocean Res. 28 (2), 77–91.Google Scholar
Belmont, M. R., Horwood, J. M. K., Thurley, R. W. F. & Baker, J. 2007 Shallow angle wave profiling LIDAR. J. Atmos. Ocean. Technol. 24 (6), 1150–1156.Google Scholar
Blondel-Couprie, E., Bonnefoy, F. & Ferrant, P. 2013 Experimental validation of non-linear deterministic prediction schemes for long-crested waves. Ocean Engng 58, 284–292.Google Scholar
Bonnefoy, F., Ducrozet, G., Le Touzé, D. & Ferrant, P.(Eds) 2009 Time-domain simulation of nonlinear water waves using spectral methods. In Advances in Numerical Simulation of Nonlinear Water Waves, Advances in Coastal and Ocean Engineering, vol. 11, pp. 129–164. World Scientific.Google Scholar
Clamond, D. 2007 On the Lagrangian description of steady surface gravity waves. J. Fluid Mech. 589, 433–454.Google Scholar
Dankert, H. & Rosenthal, W. 2004 Ocean surface determination from X-band radar-image sequences. J. Geophys. Res. 109, C04016.Google Scholar
Dannenberg, J., Hessner, K., Naaijen, P., van den Boom, H. & Reichert, K. 2010 The on board wave and motion estimator OWME. In The 20th International Offshore and Polar Engineering Conference, p. ISOPE-I-10-088. International Society of Offshore and Polar Engineers.Google Scholar
Dean, R. G. & Dalrymple, R. A. 1991 Water Wave Mechanics for Engineers and Scientists, vol. 2. World Scientific Publishing.Google Scholar
Dommermuth, D. 2000 The initialization of nonlinear waves using an adjustment scheme. Wave Motion 32 (4), 307–317.Google 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.Google Scholar
Ducrozet, G., Bonnefoy, F., Le Touzé, D. & Ferrant, P. 2007 3-d HOS simulations of extreme waves in open seas. Nat. Hazards Earth Syst. Sci. 7, 109–122.Google Scholar
Ducrozet, G., Bonnefoy, F., Le Touzé, D. & Ferrant, P. 2016 HOS-ocean: open-source solver for nonlinear waves in open ocean based on high-order spectral method. Comput. Phys. Commun. 203, 245–254.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.Google Scholar
Fenton, J. D. 1985 A fifth-order Stokes theory for steady waves. ASCE J. Waterway Port Coastal Ocean Engng 111 (2), 216–234.Google Scholar
Gerstner, F. J. 1809 Theorie der Wellen. Ann. Phys. 32 (8), 412–445.Google Scholar
Grilli, S. T., Guérin, C.-A. & Goldstein, B. 2011 Ocean wave reconstruction algorithms based on spatio-temporal data acquired by a Flash LIDAR camera. In The 21st International Offshore and Polar Engineering Conference, pp. 275–282. International Society of Offshore and Polar Engineers.Google Scholar
Hilmer, T. & Thornhill, E. 2014 Deterministic wave predictions from the WaMoS II. In OCEANS 2014-TAIPEI, pp. 1–8. IEEE.Google Scholar
Lindgren, G. 2009 Exact asymmetric slope distributions in stochastic Gauss–Lagrange ocean waves. Appl. Ocean Res. 31 (1), 65–73.Google Scholar
Lindgren, G. 2010 Slope distribution in front–back asymmetric stochastic Lagrange time waves. Adv. Appl. Probab. 42 (2), 489–508.Google Scholar
Lindgren, G. 2015 Asymmetric waves in wave energy systems analysed by the stochastic Gauss–Lagrange wave model. Proc. Estonian Acad. Sci. 64 (3), 291–296.Google Scholar
Lindgren, G. & Åberg, S. 2009 First order stochastic Lagrange model for asymmetric ocean waves. J. Offshore Mech. Arctic Engng 131 (3), 031602.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 (3), 459–480.Google Scholar
Monismith, S. G., Cowen, E. A., Nepf, H. M., Magnaudet, J. & Thais, L. 2007 Laboratory observations of mean flows under surface gravity waves. J. Fluid Mech. 573, 131–147.Google Scholar
Naaijen, P., Van Oosten, K., Roozen, K. & van’t Veer, R. 2018 Validation of a deterministic wave and ship motion prediction system. In ASME 2018 37th International Conference on Ocean, Offshore and Arctic Engineering. American Society of Mechanical Engineers.Google Scholar
Nieto Borge, J. C., RodrÍguez, G. R., Hessner, K. & González, P. I. 2004 Inversion of marine radar images for surface wave analysis. J. Atmos. Ocean. Technol. 21 (8), 1291–1300.Google Scholar
Nouguier, F., Chapron, B. & Guérin, C.-A. 2015 Second-order Lagrangian description of tri-dimensional gravity wave interactions. J. Fluid Mech. 772, 165–196.Google Scholar
Nouguier, F., Grilli, S. T. & Guérin, C.-A. 2014 Nonlinear ocean wave reconstruction algorithms based on spatiotemporal data acquired by a Flash LIDAR camera. IEEE Trans. Geosci. Remote Sens. 52 (3), 1761–1771.Google Scholar
Nouguier, F., Guérin, C.-A. & Chapron, B. 2009 ‘Choppy wave’ model for nonlinear gravity waves. J. Geophys. Res. 114, C09012.Google Scholar
Perez, T. 2006 Ship Motion Control: Course Keeping and Roll Stabilisation using Rudder and Fins. Springer Science & Business Media.Google Scholar
Perignon, Y.2011 Modélisation déterministe des états de mer – application à la rétrodiffusion d’ondes radar. PhD thesis, Ecole Centrale de Nantes (ECN).Google Scholar
Perignon, Y. L., Bonnefoy, F., Ferrant, P. & Ducrozet, G. 2010 Non-linear initialization in three-dimensional high order spectra deterministic sea state modeling. In ASME 2010 29th International Conference on Ocean, Offshore and Arctic Engineering, pp. 525–532. American Society of Mechanical Engineers.Google Scholar
Pierson, W. J.1961 Models of random seas based on the Lagrangian equations of motion. Tech. Rep. New York University College of Engineering Research Division/ONR.Google Scholar
Pierson, W. J. 1962 Perturbation analysis of the Navier–Stokes equations in Lagrangian form with selected linear solutions. J. Geophys. Res. 67 (8), 3151–3160.Google Scholar
Qi, Y., Wu, G., Liu, Y., Kim, M.-H. & Yue, D. K. P. 2018a Nonlinear phase-resolved reconstruction of irregular water waves. J. Fluid Mech. 838, 544–572.Google Scholar
Qi, Y., Wu, G., Liu, Y. & Yue, D. K. P. 2018b Predictable zone for phase-resolved reconstruction and forecast of irregular waves. Wave Motion 77, 195–213.Google Scholar
Qi, Y., Xiao, W. & Yue, D. K. P. 2016 Phase-resolved wave field simulation calibration of sea surface reconstruction using noncoherent marine radar. J. Atmos. Ocean. Technol. 33 (6), 1135–1149.Google Scholar
West, B. J., Brueckner, K. A., Janda, R. S., Milder, D. M. & Milton, R. L. 1987 A new numerical method for surface hydrodynamics. J. Geophys. Res. 92 (C11), 11803–11824.Google Scholar