Finite-Element Model for Modified Boussinesq Equations. I: Model Development
This article has a reply.
VIEW THE REPLYPublication: Journal of Waterway, Port, Coastal, and Ocean Engineering
Volume 130, Issue 1
Abstract
This paper and its companion paper describe the development of a finite-element model based on modified Boussinesq equations and the applications of the model to harbor resonance problems. This first of the two papers reports the model development and validations. A Galerkin finite-element method with linear elements is employed in the model. Auxiliary variables are introduced to remove the spatial third derivative terms in the governing equations, and an implicit predictor-corrector iterative scheme is used in the time integration. The treatments of various boundary conditions, including the perfect reflecting boundary with an irregular geometry, the absorbing (sponge layer) boundary, and the incident wave boundary, are described. Numerical results are obtained for several examples, and their accuracy is checked by comparing numerical results with either experimental data or analytical solutions.
Get full access to this article
View all available purchase options and get full access to this article.
References
Ambrosi, D., and Quartapelle, L.(1998). “A Taylor-Galerkin method for simulating nonlinear dispersive water waves.” J. Comput. Phys., 146, 546–569.
Antunes do Carmo, J. S., Seabra Santos, F. J., and Barthelemy, E.(1993). “Surface waves propagation in shallow water: A finite element model.” Int. J. Numer. Methods Fluids, 16, 447–459.
Beji, S., and Nadaoka, K.(1996). “A formal derivation and numerical modeling of the improved Boussinesq equations for varying depth.” Ocean Eng., 23(8), 691–704.
Berkhoff, J. C. W., Booy, N., and Radder, A. C.(1982). “Verification of numerical wave propagation models for simple harmonic linear water waves.” Coastal Eng., 6, 255–279.
Chen, Y., and Liu, P. L.-F.(1995). “Modified Boussinesq equations and associated parabolic models for water wave propagation.” J. Fluid Mech., 288, 351–381.
Engelman, M. S., Sani, J. L., and Gresho, P. M.(1982). “The implementation of normal and/or tangential boundary condition in finite element codes for incompressible fluid flow.” Int. J. Numer. Methods Fluids, 2, 225–238.
Katopodes, N. D., and Wu, C.-T.(1987). “Computation of finite-amplitude dispersive waves.” J. Waterw., Port, Coastal, Ocean Eng., 113(4), 327–346.
Kawahara, M., and Cheng, J. Y.(1994). “Finite element method for Boussinesq wave analysis.” Int. J. Comput. Fluid Dyn., 2, 1–17.
Langtangen, H. P., and Pedersen, G. (1996). “Finite elements for the Boussinesq wave equations.” Waves and nonlinear processes in hydrodynamics, J. Grue, G. Gjeviki, and J. E. Weber, eds., Kluwer Academic, Boston, 1–10.
Li, Y. S., Liu, S.-X., Yu, Y.-X., and Lai, G.-Z.(1999). “Numerical modeling of Boussinesq equations by finite element method.” Coastal Eng., 37, 97–122.
Liu, P. L.-F., (1994). “Model equations for wave propagations from deep to shallow water.” Advances in coastal and ocean engineering, P. L.-F. Liu, ed., Vol. 1, 125–157.
Lynett, P., and Liu, P. L.-F. (2002). “A numerical study of submarine landslide generated waves and runup.” Royal Society of London A, in press.
Madsen, P. A., and Sorensen, O. R.(1992). “A new form of the Boussinesq equations with improved linear dispersion characteristics. II: A slowly-varying bathymetry.” Coastal Eng., 18, 183–204.
Nwogu, O.(1993). “Alternative form of Boussinesq equations for nearshore wave propagation.” J. Waterw., Port, Coastal, Ocean Eng., 119(6), 618–638.
Peregrine, D. H.(1967). “Long waves on a beach.” J. Fluid Mech., 27, 815–882.
Shi, F., Dalrymple, R. A., Kirby, J. T., Chen, Q., and Kennedy, A.(2001). “A fully nonlinear Boussinesq model in generalized curvilinear coordinates.” Coastal Eng., 42, 337–358.
Skotner, C., and Apelt, C. J.(1999). “Internal wave generation in an improved two-dimensional Boussinesq model.” Ocean Eng., 26(4), 287–324.
Sorenson, O. R., and Sorensen, L. S. (2000). “Boussinesq type modelling using unstructured finite element technique.” Proc., 27th Int. Conf. on Coastal Engineering, 190–202.
Walkley, M. (1999). “A numerical method for extended Boussinesq shallow-water wave equations.” PhD thesis, Univ., of Leeds, Leeds, U.K.
Walkley, M., and Berzins, M.(1999). “A finite element method for the one-dimensional extended Boussinesq equations.” Int. J. Numer. Methods Fluids, 29, 143–157.
Walkley, M., and Berzins, M.(2002). “A finite element method for the two-dimensional extended Boussinesq equations.” Int. J. Numer. Methods Fluids, 39, 865–885.
Wei, G., and Kirby, J. T.(1995). “Time-dependent numerical code for extended Boussinesq equations.” J. Waterw., Port, Coastal, Ocean Eng., 121(5), 251–261.
Wei, G., Kirby, J. T., Grilli, S. T., and Subramanya, R.(1995). “A fully nonlinear Boussinesq model for surface waves. 1: Highly nonlinear unsteady waves.” J. Fluid Mech., 294, 71–92.
Woo, S.-B., and Liu, P. L.-F.(2001). “A Petrov-Galerkin finite element model for one-dimensional fully nonlinear and weakly dispersive wave propagation.” Int. J. Numer. Methods Fluids, 37, 541–575.
Woo, S.-B., and Liu, P. L.-F.(2004). “Finite-element model for modified Boussinesq equations. II: Applications to nonlinear harbor oscillations.” J. Waterw., Port, Coastal, Ocean Eng., 130(1), 17–28.
Information & Authors
Information
Published In
Copyright
Copyright © 2004 American Society of Civil Engineers.
History
Received: Nov 27, 2002
Accepted: Jun 9, 2003
Published online: Dec 15, 2003
Published in print: Jan 2004
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.