Abstract
In this paper, a fully nonlinear Boussinesq model is presented and applied to the description of breaking waves and shoreline motions. It is based on Serre Green-Naghdi equations, solved using a time-splitting approach separating hyperbolic and dispersive parts of the equations. The hyperbolic part of the equations is solved using Finite-Volume schemes, whereas dispersive terms are solved using a Finite-Difference method. The idea is to switch locally in space and time to NSWE by skipping the dispersive step when the wave is ready to break, so as the energy dissipation due to wave breaking is predicted by the shock theory. This approach allows wave breaking to be handled naturally, without any ad-hoc parameterization for the energy dissipation. Extensive validations of the method are presented using laboratory data.References
Audusse, E., F. Bouchut, M.-0. Bristeau, R. Klein and B. Perthame. 2004. A fast and stable wellbalanced scheme with hydrostatic reconstruction for shallow water flows. SIAM Journal on Scientific Computing, 25(6), 2050-2065. http://dx.doi.org/10.1137/S1064827503431090
Berthon, C. and F. Marche, A positive preserving high order VFRoe scheme for shallow water equations: A class of relaxation schemes. 2008. SIAM Journal of Scientific Computing, 30(5), 2587-2612. http://dx.doi.org/10.1137/070686147
Bonneton, P. 2007. Modelling of periodic wave transformation in the inner surf zone, Ocean Engineering, 34, 1459-1471. http://dx.doi.org/10.1016/j.oceaneng.2006.09.002
Bonneton, P., F. Chazel, D. Lannes, F. Marche and M. Tissier. 2010a. A splitting approach for the fully nonlinear and weakly dispersive Green-Naghdi model, in correction to Journal of Computational Physics.
Bonneton, P., E. Barthélemy, J.D. Carter, F. Chazel, R. Cienfuegos, D. Lannes, F. Marche, M. Tissier. 2010b. Fully nonlinear weakly dispersive modelling of wave propagation, breaking and run-up, in correction to European Journal of Mechanics, B/Fluids.
Bonneton P., N. Bruneau, F. Marche, B. Castelle. 2010c. Large-scale vorticity generation due to dissipating waves in the surf zone, DCDS-S, 13(4), 729-738. http://dx.doi.org/10.3934/dcdsb.2010.13.729
Brocchini, M. and N. Dodd. 2008. Nonlinear Shallow Water Equation Modeling for Coastal Engineering, J. Wtrwy., Port, Coast., and Oc. Engrg., 134(2), 104-120.
Bruno, D., F. De Serio and M. Mossa. 2009. The FUNWAVE model application and its validation using laboratory data, Coastal Engineering, 56(7), 773-787. http://dx.doi.org/10.1016/j.coastaleng.2009.02.001
Castelle, B, H. Michallet, V. Marieu, F. Leckler, B. Dubardier, A. Lambert, C. Berni, P. Bonneton, E. Barthélemy and F. Bouchette, 2010. Laboratory experiment on rip current circulations over a moveable bed: drifter measurements, in revision to Journal of Geophysical Research.
Chazel, F., D. Lannes, F. Marche. 2010. Numerical simulation of strongly nonlinear and dispersive waves using a Green-Naghdi model, Journal of Scientific Computing, DOI: 10.1007/s10915-010-9395-9. http://dx.doi.org/10.1007/s10915-010-9395-9
Cienfuegos, R., E. Barthélemy, and P. Bonneton. 2006. A fourth-order compact finite volume scheme for fully nonlinear and weakly dispersive Boussinesq-type equations. Part I: Model development and analysis. Int. J. Numer. Meth. Fluids, 56, 1217-1253. http://dx.doi.org/10.1002/fld.1141
Cienfuegos, R., E. Barthélemy and P. Bonneton. 2010. A wave-breaking model for Boussinesq-type equations including roller effects in the mass conservation equation. J. Wtrwy., Port, Coast., and Oc. Engrg., 136(1), 10-26.
Cox, D. 1995. Experimental and numerical modelling of surf zone hydrodynamics. PhD thesis, University of Delaware, Newark, Del.
Kennedy, A.B., Q. Chen, J.T. Kirby and R.A. Dalrymple. 2000. Boussinesq modelling of wave transformation, breaking and runup. I:1D. J. Wtrwy.,Port,Coast., and Oc. Engrg.,119, 618-638.
Kobayashi, N., G. De Silva and K. Watson. 1989. Wave transformation and swash oscillation on gentle and steep slopes, Journal of Geophysical Research, 94, 951-966. http://dx.doi.org/10.1029/JC094iC01p00951
Lannes, D., and P. Bonneton. 2009. Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation, Physics of Fluids, 21(1). http://dx.doi.org/10.1063/1.3053183
Madsen, P.A., 0.R. Sørensen and H.A. Schäffer. 1997. Surf zone dynamics simulated by a Boussinesq type model. Part I. Model description and cross-shore motion of regular waves, Coastal Engineering, 32, 255-287. http://dx.doi.org/10.1016/S0378-3839(97)00028-8
Marche, M., P. Bonneton, P. Fabrie and N. Seguin. 2007. Evaluation of well-balanced bore-capturing schemes for 2D wetting and drying processes, Int. J. Num. Meth. Fluids, 53(5), 867-894. http://dx.doi.org/10.1002/fld.1311
Schäffer, H.A., P. A. Madsen and R. Deigaard. 1993. A Boussinesq model for waves breaking in shallow water, Coastal Engineering, 20, 185-202. http://dx.doi.org/10.1016/0378-3839(93)90001-O
Synolakis, C. E. 1987. The run-up of solitary waves. Journal of Fluid Mechanics, 185, 523-555. http://dx.doi.org/10.1017/S002211208700329X
Ting, F. and J. Kirby. 1994. Observation of undertow and turbulence in a laboratory surf zone. Coastal Engineering, 24, 51-80. http://dx.doi.org/10.1016/0378-3839(94)90026-4