Skip to main content Accessibility help
×
Home

A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves

  • Ge Wei (a1), James T. Kirby (a1), Stephan T. Grilli (a2) and Ravishankar Subramanya (a2)

Abstract

Fully nonlinear extensions of Boussinesq equations are derived to simulate surface wave propagation in coastal regions. By using the velocity at a certain depth as a dependent variable (Nwogu 1993), the resulting equations have significantly improved linear dispersion properties in intermediate water depths when compared to standard Boussinesq approximations. Since no assumption of small nonlinearity is made, the equations can be applied to simulate strong wave interactions prior to wave breaking. A high-order numerical model based on the equations is developed and applied to the study of two canonical problems: solitary wave shoaling on slopes and undular bore propagation over a horizontal bed. Results of the Boussinesq model with and without strong nonlinearity are compared in detail to those of a boundary element solution of the fully nonlinear potential flow problem developed by Grilli et al. (1989). The fully nonlinear variant of the Boussinesq model is found to predict wave heights, phase speeds and particle kinematics more accurately than the standard approximation.

Copyright

References

Hide All
Abbott, M. B., Mccowan, A. D. & Warren, I. R. 1984 Accuracy of short-wave numerical model. J. Hydr. Engng 110, 12871301.
Chen, Y. & Liu, P. L.-F. 1995 Modified Boussinesq equations and associated parabolic models for wave propagation. J. Fluid Mech., in Press.
Demirbilek, Z. & Webster, W. C. 1992 Application of the Green-Naghdi theory of fluid sheets to shallow-water wave problems: Report I. Model development. Tech. Rep. CERC-92-11. US Army Waterways Experiment Station, Vicksburg, MS.
Elgar, S. & Guza, R. T. 1985 Shoaling gravity waves: comparisons between field observations, linear thoery and a nonlinear model. J. Fluid Mech. 158, 4770.
Goring, D. G. 1978 Tsunamis - the propagation of long waves onto a shelf. PhD dissertation, California Institute of Technology, Pasadena, California.
Green, A. E. & Naghdi, P. M. 1976 A derivation of equations for wave propagation in water of variable depth. J. Fluid Mech. 78, 237246.
Grilli, S. T. 1993 Modeling of nonlinear wave motion in shallow water. In Computational Methods for Free and Moving Boundary Problems in Heat and Fluid Flow (ed. L.C. Wrobel and C.A. Brebbia), pp. 3765. Elsevier Applied Sciences, London, UK.
Grilli S. T. Skourup J. & Svendsen, I. A. 1989 An efficient boundary element method for nonlinear water waves. Engng Anal. with Boundary Elements 6, 97107.
Grilli, S. T. & Subramanya, R. 1994 Numerical modeling of wave breaking induced by fixed or moving boundaries. J. Comput. Phys. (submitted).
Grilli, S. T., Subramanya, R., Svendsen I. A. & Veeramony, J. 1994a Shoaling of solitary waves on plane beaches.. J. Waterway Port Coastal & Ocean Engng. 120, 609628.
Grilli, S. T., Svendsen I. A. & Subramanya, R. 1994b Breaking criterion and characteristics for solitary waves on plane beaches. J. Waterway Port Coastal & Ocean Engng (submitted).
Kirby, J. T. & Wei, G. 1995 A fully nonlinear Boussinesq model for surface waves. II. Bound waves in intermediate water depths. in preparation.
Liu, P. L.-F., Yoon, S. B. & Kirby, J. T. 1985 Nonlinear refraction-diffraction of waves in shallow water. J. Fluid Mech. 153, 184201.
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. A350, 126.
Madsen, P. A., Murray, R. & SöThrensen, O. R. 1991 A new form of Boussinesq equations with improved linear dispersion characteristics. Coastal Engng 15, 371388.
Mei, C. C. 1989 The Applied Dynamics of Ocean Surface Waves. World Scientific.
Miles, J. & Salmon, R. 1985 Weakly dispersive nonlinear gravity waves. J. Fluid Mech. 157, 519531.
Nwogu, O. 1993 An alternative form of the Boussinesq equations for nearshore wave propagation. J. Waterway, Port, Coast. Ocean Engng. 119, 618638.
Peregrine, D. H. 1966 Calculations of the development of an undular bores. J. Fluid Mech. 25, 321330.
Peregrine, D. H. 1967 Long waves on a beach. J. Fluid Mech. 27, 815827.
Rygg, O. B. 1988 Nonlinear refraction-diffraction of surface waves in intermediate and shallow water. Coastal Engng 12, 191211.
Su, C. H. & Gardner, C. S. 1969 Korteweg-de Vries equation and generalizations. III. Derivation of the Korteweg-de Vries equation and Burgers equation. J. Math. Phys. 10, 536539.
Tanaka, M. 1986 The stability of solitary waves. Phys. Fluids 29, 650655.
Wei G. And Kirby J. T. 1995 A time-dependent numerical code for extended Boussinesq equations. J. Waterway Port Coastal Ocean Engng, in Press.
Witting, J. M. 1984 A unified model for the evolution of nonlinear water waves. J. Comput. Phys. 56, 203236.
Wu, T. Y. T. 1981 Long waves in ocean and coastal waters. J. Engng Mech. 107, 501522.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves

  • Ge Wei (a1), James T. Kirby (a1), Stephan T. Grilli (a2) and Ravishankar Subramanya (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed