Hostname: page-component-745bb68f8f-l4dxg Total loading time: 0 Render date: 2025-02-05T07:13:42.687Z Has data issue: false hasContentIssue false
  • English
  • Français

A high-order spectral method for the study of nonlinear gravity waves

Published online by Cambridge University Press:  21 April 2006

Douglas G. Dommermuth  and
Dick K. P. Yue
Douglas G. Dommermuth
Affiliation:
Department of Ocean Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Dick K. P. Yue
Affiliation:
Department of Ocean Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

We develop a robust numerical method for modelling nonlinear gravity waves which is based on the Zakharov equation/mode-coupling idea but is generalized to include interactions up to an arbitrary order M in wave steepness. A large number (N = O(1000)) of free wave modes are typically used whose amplitude evolutions are determined through a pseudospectral treatment of the nonlinear free-surface conditions. The computational effort is directly proportional to N and M, and the convergence with N and M is exponentially fast for waves up to approximately 80% of Stokes limiting steepness (ka ∼ 0.35). The efficiency and accuracy of the method is demonstrated by comparisons to fully nonlinear semi-Lagrangian computations (Vinje & Brevig 1981); calculations of long-time evolution of wavetrains using the modified (fourth-order) Zakharov equations (Stiassnie & Shemer 1987); and experimental measurements of a travelling wave packet (Su 1982). As a final example of the usefulness of the method, we consider the nonlinear interactions between two colliding wave envelopes of different carrier frequencies.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 184 , November 1987 , pp. 267 - 288
Copyright
© 1987 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

Benney, D. J. 1962 Nonlinear gravity wave interactions. J. Fluid Mech. 14, 577–589.Google Scholar
Bryant, P. J. 1983 Cyclic gravity waves in deep water. J. Austral. Math. Soc. B 25, 2–15.Google Scholar
Cohen, B. I., Watson, K. M. & West, B. J. 1976 Some properties of deep water solitons. Phys. Fluids 19, 345–354.Google Scholar
Crawford, D. R., Lake, B. M., Saffman, P. G. & Yuen, H. C. 1981 Stability of weakly nonlinear deep-water waves in two and three dimensions. J. Fluid Mech. 105, 177–191.Google Scholar
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. K. W. Morton & M. J. Baines), pp. 671–679. Oxford University Press.
Dysthe, K. B. 1979 Note on a modification to the nonlinear Schrödinger equation for application to deep water waves. Proc. R. Soc. Lond. A 369, 105–114.Google Scholar
Fornberg, B. & Whitham, G. B. 1978 A numerical and theoretical study of certain nonlinear wave phenomena. Phil. Trans. R. Soc. Lond. A 289, 373–404.Google Scholar
Lo, E. & Mei, C. C. 1985 A numerical study of water-wave modulation based on a higher-order nonlinear Schrödinger equation. J. Fluid Mech. 150, 395–416.Google Scholar
Longuet-Higgins, M. S. 1974 Breaking waves in deep or shallow water. In Proc. 10th Symp. Naval Hydro., Cambridge, Mass. (ed. R. D. Cooper & S. W. Doroff), pp. 597–605. Government Printing Office, Washington.
Longuet-Higgins, M. S. & Cokelet, E. D. 1976 The deformation of steep surface waves on water. I. A numerical method of computation. Proc. R. Soc. Lond. A 350, 1–26.Google Scholar
Longuet-Higgins, M. S. & Phillips, O. M. 1962 Phase velocity effects in tertiary wave interactions. J. Fluid Mech. 12, 333–336.Google Scholar
Martin, D. U. & Yuen, H. C. 1980 Quasi-recurring energy leakage in the two-space-dimensional nonlinear Schrödinger equation. Phys. Fluids 23, 881–883.Google Scholar
Oikawa, M. & Yajima, N. 1974 A perturbation approach to nonlinear systems. II. Interaction of nonlinear modulated waves. J. Phys. Soc. Japan 37, 486–496.Google Scholar
Orszag, S. A. 1971 Numerical simulation of incompressible flows within simple boundaries: accuracy. J. Fluid Mech. 49, 75–112.Google Scholar
Phillips, O. M. 1960 On the dynamics of unsteady gravity waves of finite amplitude. Part 1. The elementary interactions. J. Fluid Mech. 9, 193–217.Google Scholar
Rienecker, M. M. & Fenton, J. D. 1981 A Fourier approximation method for steady water waves. J. Fluid Mech. 104, 119–137.Google Scholar
Schwartz, L. W. 1974 Computer extension and analytic continuation of Stokes' expansion for gravity waves. J. Fluid Mech. 62, 553–578.Google Scholar
Stiassnie, M. & Shemer, L. 1984 On modifications of the Zakharov equation for surface gravity waves. J. Fluid Mech. 143, 47–67.Google Scholar
Stiassnie, M. & Shemer, L. 1987 Energy computations for evolution of class I and class II instabilities of Stokes waves. J. Fluid Mech. 174, 299–312.Google Scholar
Su, M. Y. 1982 Evolution of groups of gravity waves with moderate to high steepness. Phys. Fluids 25, 2167–2174.Google Scholar
Vinje, T. & Brevig, P. 1981 Nonlinear ship motions. Proc. 3rd Intl Symp. Num. Ship Hydro., Paris.
West, B. J. 1982 Statistical properties of water waves. Part 1. Steady-state distribution of wind-driven gravity-capillary waves. J. Fluid Mech. 117, 187–210.Google Scholar
West, B. J., Watson, K. M. & Thomson, A. J. 1974 Mode coupling description of ocean wave dynamics. Phys. Fluids 17, 1059–1067.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.
Yuen, H. C. & Ferguson, W. E. 1978 Fermi-Pasta-Ulam recurrence in the two-space dimensional nonlinear Schrödinger equation. Phys. Fluids 21, 2116–2118.Google Scholar
Yuen, H. C. & Lake, B. M. 1982 Nonlinear dynamics of deep-water gravity waves. Adv. Appl. Mech. 22, 67–229.Google 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. (English Translation.)Google Scholar
Zakharov, V. E. & Shabat, A. B. 1972 Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys., J. Exp. Theor. Phys. 34, 62–69.Google Scholar