Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-24T02:58:37.738Z Has data issue: false hasContentIssue false
  • English
  • Français

A new model of shoaling and breaking waves. Part 2. Run-up and two-dimensional waves

Published online by Cambridge University Press:  20 March 2019

G. L. Richard Open the ORCID record for G. L. Richard [Opens in a new window] ,
A. Duran  and
B. Fabrèges
G. L. Richard*
Affiliation:
LAMA, UMR5127, Université de Savoie Mont-Blanc, CNRS, 73376 Le Bourget-du-Lac, France
A. Duran
Affiliation:
Institut Camille Jordan, CNRS UMR 5208, Université Claude Bernard Lyon 1, 69100 Villeurbanne, France
B. Fabrèges
Affiliation:
Institut Camille Jordan, CNRS UMR 5208, Université Claude Bernard Lyon 1, 69100 Villeurbanne, France
*
†Email address for correspondence: gael.richard@univ-grenoble-alpes.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

We derive a two-dimensional depth-averaged model for coastal waves with both dispersive and dissipative effects. A tensor quantity called enstrophy models the subdepth large-scale turbulence, including its anisotropic character, and is a source of vorticity of the average flow. The small-scale turbulence is modelled through a turbulent-viscosity hypothesis. This fully nonlinear model has equivalent dispersive properties to the Green–Naghdi equations and is treated, both for the optimization of these properties and for the numerical resolution, with the same techniques which are used for the Green–Naghdi system. The model equations are solved with a discontinuous Galerkin discretization based on a decoupling between the hyperbolic and non-hydrostatic parts of the system. The predictions of the model are compared to experimental data in a wide range of physical conditions. Simulations were run in one-dimensional and two-dimensional cases, including run-up and run-down on beaches, non-trivial topographies, wave trains over a bar or propagation around an island or a reef. A very good agreement is reached in every cases, validating the predictive empirical laws for the parameters of the model. These comparisons confirm the efficiency of the present strategy, highlighting the enstrophy as a robust and reliable tool to describe wave breaking even in a two-dimensional context. Compared with existing depth-averaged models, this approach is numerically robust and adds more physical effects without significant increase in numerical complexity.

JFM classification

Geophysical and Geological Flows: Coastal engineering Geophysical and Geological Flows: Shallow water flows Waves/Free-surface Flows: Wave breaking
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 867 , 25 May 2019 , pp. 146 - 194
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

Antuono, M. & Brocchini, M. 2013 Beyond Boussinesq-type equations: semi-integrated models for coastal dynamics. Phys. Fluids 25, 016603.Google Scholar
Audusse, E., Bouchut, F., Bristeau, M.-O., Klein, R. & Perthame, B. 2004 A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25 (6), 2050–2065.Google Scholar
Azerad, P., Guermond, J. L. & Popov, B. 2017 Well-balanced second-order approximation of the shallow water equation with continuous finite elements. SIAM J. Numer. Anal. 55, 3203–3224.Google Scholar
Barré de Saint Venant, A. J. C. 1871 Théorie du mouvement non-permanent des eaux, avec application aux crues des rivières et à l’introduction des marées dans leur lit. C. R. Acad. Sci. Paris 73, 147–154.Google Scholar
Beji, S. & Battjes, J. A. 1993 Experimental investigations of wave propagation over a bar. Coast. Engng 19, 151–162.Google Scholar
Bonneton, P., Chazel, F., Lannes, D., Marche, F. & Tissier, M. 2011 A splitting approach for the fully nonlinear and weakly dispersive Green–Naghdi model. J. Comput. Phys. 230, 1479–1498.Google Scholar
Boussinesq, J. 1872 Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. J. Math Pures et Appl. Deuxième Série 17, 55–108.Google Scholar
Briggs, M. J., Synolakis, C. E., Harkins, G. S. & Green, D. R. 1995 Laboratory experiments of tsunami runup on a circular island. Pure Appl. Geophys. 144, 569–593.Google Scholar
Brocchini, M. 2013 A reasoned overview on Boussinesq-type models: the interplay between physics, mathematics and numerics. Proc. R. Soc. Lond. A 469, 20130496.Google Scholar
Brocchini, M. & Dodd, N. 2008 Nonlinear shallow water equations modeling for coastal engineering. J. Waterways Port Coast. 134, 104–120.Google Scholar
Brocchini, M. & Peregrine, D. H. 2001 The dynamics of strong turbulence at free surfaces. Part 2. Free-surface boundary conditions. J. Fluid Mech. 449, 255–290.Google Scholar
do Carmo, J. S. A., Ferreira, J. A., Pinto, L. & Romanazzi, G. 2018 An improved Serre model: efficient simulation and comparative evaluation. Appl. Math. Model. 56, 404–423.Google Scholar
Castro, A. & Lannes, D. 2014 Fully nonlinear long-waves models in presence of vorticity. J. Fluid Mech. 759, 642–675.Google Scholar
Chazel, F., Lannes, D. & Marche, F. 2011 Numerical simulation of strongly nonlinear and dispersive waves using a Green–Naghdi model. J. Sci. Comput. 48, 105–116.Google Scholar
Chen, Q., Dalrymple, R. A., Kirby, J. T., Kennedy, A. B. & Haller, M C. 1999 Boussinesq modeling of a rip current system. J. Geophys. Res. 104, 20617–20637.Google Scholar
Chen, Q., Kirby, J. T., Dalrymple, R. A., Kennedy, A. B. & Chawla, A. 2000 Boussinesq modeling of wave transformation, breaking and runup. II: 2D. J. Waterways Port Coast. 126, 48–56.Google Scholar
Cienfuegos, R., Barthelemy, E. & Bonneton, P. 2010 Wave-breaking model for Boussinesq-type equations including roller effects in the mass conservation equation. J. Waterways Port Coast. 136, 10–26.Google Scholar
Cockburn, B. & Shu, C. W. 1998 The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 141, 2440–2463.Google Scholar
Cockburn, B. & Shu, C. W. 2001 Runge–Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16, 173–261.Google Scholar
Cox, D., Kobayashi, N. & Okayasu, A.1995 Experimental and numerical modeling of surf zone hydrodynamics. Tech. Rep. No. CACR-95-07, Center for applied coastal research, University of Delaware, Newark.Google Scholar
Debieve, J. F., Gouin, H. & Gaviglio, J. 1982 Evolution of the Reynolds stress tensor in a shock wave–turbulence interaction. Indian J. Technol. 20, 90–97.Google Scholar
Derakhti, M., Kirby, J. T., Shi, F. & Ma, G. 2016 Wave breaking in the surf zone and deep-water in a non-hydrostatic RANS model. Part 2. Turbulence and mean circulation. Ocean Model. 107, 139–150.Google Scholar
Duran, A. & Marche, F. 2015 Discontinuous Galerkin discretization of a new class of Green–Naghdi equations. Commun. Comput. Phys. 17, 721–760.Google Scholar
Duran, A. & Marche, F. 2017 A discontinuous Galerkin method for a new class of Green–Naghdi equations on simplicial unstructured meshes. Appl. Math. Model. 45, 840–864.Google Scholar
Filippini, A. G., Kazolea, M. & Ricchiuto, M. 2016 A flexible genuinely nonlinear approach for nonlinear wave propagation, breaking and run-up. J. Comput. Phys. 310, 381–417.Google Scholar
Fuhrman, D. R. & Madsen, P. A. 2008 Simulation of nonlinear wave run-up with a high-order Boussinesq model. Coast. Engng 55, 139–154.Google Scholar
Gavrilyuk, S. & Gouin, H. 2012 Geometrical evolution of the Reynolds stress tensor. Intl J. Engng Sci. 59, 65–73.Google Scholar
Green, A. E. & Naghdi, P. M. 1976 A derivation of equations for wave propagation in water of variable depth. J. Fluid Mech. 78, 237–246.Google Scholar
Hasselmann, K. 1971 On the mass and momentum transfer between short gravity waves and larger-scale motions. J. Fluid Mech. 50, 189–205.Google Scholar
Hattori, M. & Aono, T. 1985 Experimental study on turbulence structures under breaking waves. Coast. Engng J. 28, 97–116.Google Scholar
Higgins, C., Parlange, M. B. & Meneveau, C. 2004 Energy dissipation in large-eddy simulation: dependence on flow structure and effects of eigenvector alignments. In Atmospheric Turbulence and Mesoscale Meteorology (ed. Fedorovich, E., Rotunno, R. & Stevens, B.), pp. 51–69. Cambridge University Press.Google Scholar
Hinterberger, C., Fröhlich, J. & Rodi, W. 2007 Three-dimensional and depth-averaged large-eddy simulation of some shallow water flows. J. Hydraul. Engng ASCE 133, 857–872.Google Scholar
Kamath, A., Bihs, H., Chella, M. A. & Arntsen, Ø. A. 2015 CFD simulations of wave propagation and shoaling over a submerged bar. Aquat. Procedia 4, 308–316.Google Scholar
Kânoǧlu, U. & Synolakis, C. E. 1998 Long wave runup on piecewise linear topographies. J. Fluid Mech. 374, 1–28.Google Scholar
Kazakova, M. & Richard, G. L. 2019 A new model of shoaling and breaking waves: one-dimensional solitary wave on a mild sloping beach. J. Fluid Mech. 862, 552–591.Google Scholar
Kazolea, M., Delis, A. I., Nikolos, I. K. & Synolakis, C. E. 2012 An unstructured finite volume numerical scheme for extended 2D Boussinesq-type equations. Coast. Engng 69, 42–66.Google Scholar
Kazolea, M., Delis, A. I. & Synolakis, C. E. 2014 Numerical treatment of wave breaking on unstructured finite volume approximations for extended Boussinesq-type equations. J. Comput. Phys. 271, 281–305.Google Scholar
Kazolea, M. & Ricchiuto, M. 2018 On wave breaking for Boussinesq-type models. Ocean Model. 123, 16–39.Google Scholar
Kennedy, A. B., Chen, Q., Kirby, J. T. & Dalrymple, R. A. 2000 Boussinesq modeling of wave transformation, breaking and runup. I. 1D. J. Waterways Port Coast. 126, 39–47.Google Scholar
Kirby, J. T. 2016 Boussinesq models and their application to coastal processes across a wide range of scales. J. Waterways Port Coast. Ocean Div. ASCE 142, 03116005.Google Scholar
Klonaris, G. T., Memos, C. D. & Karambas, T. V. 2013 A Boussinesq-type model including wave-breaking terms in both continuity and momentum equations. Ocean Engng 57, 128–140.Google Scholar
Kolmogorov, A. N. 1941 The local structure of turbulence in incompressible viscous fluids for very large Reynolds numbers. Dokl. Akad. Nauk SSSR 30, 299–303.Google Scholar
Kolmogorov, A. N. 1942 The equations of turbulent motion in an incompressible fluid. Izvestia Akad. Sci. USSR Phys. 6, 56–58.Google Scholar
Krivodonova, L., Xin, J., Remacle, J. F., Chevaugeon, N. & Flaherty, J. E. 2004 Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws. Appl. Numer. Math. 48, 323–338.Google Scholar
Lannes, D. & Marche, F. 2015 A new class of fully nonlinear and weakly dispersive Green–Naghdi models for efficient 2D simulations. J. Comput. Phys. 282, 238–268.Google Scholar
Le Métayer, O., Gavrilyuk, S. & Hank, S. 2010 A numerical scheme for the Green–Naghdi model. J. Comput. Phys. 229, 2034–2045.Google Scholar
Liang, Q. & Marche, F. 2009 Numerical resolution of well-balanced shallow water equations with complex source terms. Adv. Water Resour. 32, 873–884.Google Scholar
Lilly, D. K. 1967 The representation of small-scale turbulence in numerical simulation experiments. In Proc. IBM Scientific Computing Symp. on Environmental Sciences (ed. Goldstine, H. H.), pp. 195–210. IBM.Google Scholar
Liu, P., Cho, Y. S., Briggs, M. J., Kanoglu, U. & Synolakis, C. E. 1995 Runup of solitary waves on a circular island. J. Fluid Mech. 302, 259–285.Google Scholar
Lu, X., Dong, B., Mao, B. & Zhang, X. 2015 A two-dimensional depth-integrated non-hydrostatic numerical model for nearshore wave propagation. Ocean Model. 96, 187–202.Google Scholar
Lynett, P. J., Wu, T. S. & Liu, P. L. F. 2002 Modeling wave runup with depth-integrated equation. Coast. Engng 46, 89–107.Google Scholar
Madsen, P. A., Murray, R. & Sørensen, O. R. 1991 A new form of the Boussinesq equations with improved linear dispersion characteristics. Coast. Engng 15, 371–388.Google Scholar
Madsen, P. A., Sørensen, O. R. & Schäffer, H. A. 1997 Surf zone dynamics simulated by a Boussinesq-type model. Part I. Model description and cross-shore motion of regular waves. Coast. Engng 32, 255–287.Google Scholar
Nadaoka, K. & Yagi, H. 1998 Shallow-water turbulence modeling and horizontal large-eddy computation of river flow. J. Hydraul. Engng ASCE 124, 493–500.Google Scholar
Nwogu, O. 1993 Alternative form of Boussinesq equations for nearshore wave propagation. J. Waterways Port Coast. Ocean Div. ASCE 119, 618–638.Google Scholar
Nwogu, O. K. 1996 Numerical prediction of breaking waves and currents with a Boussinesq model. In Coastal Engineering 1996, Proceedings of the 25th International Conference (ed. Billy, L.), vol. 25, pp. 4807–4820. ASCE.Google Scholar
Okamoto, T. & Basco, D. R. 2006 The relative trough froude number for initiation of wave breaking: theory, experiments and numerical model confirmation. Coast. Engng 53, 675–690.Google Scholar
Peregrine, D. H. 1967 Long waves on a beach. J. Fluid Mech. 27, 815–827.Google Scholar
Pope, S. B. Turbulent Flows, p. 2000. Cambridge University Press.Google Scholar
Ricchiuto, M. & Bollermann, A. 2009 Stabilized residual distribution for shallow water simulations. J. Comput. Phys. 228, 1071–1115.Google Scholar
Richard, G.2013 Élaboration d’un modèle d’écoulements turbulents en faible profondeur. Application au ressaut hydraulique et aux trains de rouleaux. PhD thesis, Université d’Aix-Marseille.Google Scholar
Richard, G. L. & Gavrilyuk, S. L. 2012 A new model of roll waves: comparison with Brock’s experiments. J. Fluid Mech. 698, 374–405.Google Scholar
Richard, G. L. & Gavrilyuk, S. L. 2013 The classical hydraulic jump in a model of shear shallow-water flows. J. Fluid Mech. 725, 492–521.Google Scholar
Richard, G. L. & Gavrilyuk, S. L. 2015 Modelling turbulence generation in solitary waves on shear shallow water flows. J. Fluid Mech. 773, 49–74.Google Scholar
Richardson, L. F. 1922 Weather Prediction by Numerical Process. Cambridge University Press.Google Scholar
Roeber, V. & Cheung, K. F. 2012 Boussinesq-type model for energetic breaking waves in fringing reef environments. Coast. Engng 70, 1–20.Google Scholar
Roeber, V., Cheung, K. F. & Kobayashi, M. H. 2010 Shock-capturing Boussinesq-type model for nearshore wave processes. Coast. Engng 57, 407–423.Google Scholar
Rotta, J. C. 1951 Statistische Theorie nichthomogener Turbulenz. Z. Phys. 129, 547–572.Google Scholar
Sharifian, M. K., Kesserwani, G. & Hassanzadeh, Y. 2018 A discontinuous Galerkin approach for conservative modelling of fully nonlinear and weakly dispersive wave transformations. Ocean Model. 125, 61–79.Google Scholar
Shi, F., Kirby, J. T., Harris, J. C., Geiman, J. D. & Grilli, S. T. 2012 A high-order adaptive time-stepping TVD solver for Boussinesq modeling of breaking waves and coastal inundation. Ocean Model. 43, 36–51.Google Scholar
Smagorinsky, J. 1963 General circulation experiments with the primitive equations. I. The basic equations. Mon. Weath. Rev. 91, 99–164.Google Scholar
Stelling, G. & Zijlema, M. 2003 An accurate and efficient finite-difference algorithm for non-hydrostatic free-surface flow with application to wave propagation. Intl J. Numer. Meth. Fluids 43, 1–23.Google Scholar
Svendsen, I. A. 1987 Analysis of surf-zone turbulence. J. Geophys. Res. 92, 5115–5124.Google Scholar
Svendsen, I. A., Veeramony, J., Bakunin, J. & Kirby, J. T. 2000 The flow in weak turbulent hydraulic jumps. J. Fluid Mech. 418, 25–57.Google Scholar
Swigler, D. T.2009 Laboratory study investigating the three-dimensional turbulence and kinematic properties associated with a breaking solitary wave. PhD thesis, Texas A&M University.Google Scholar
Synolakis, C. E. 1987 The runup of solitary waves. J. Fluid Mech. 185, 523–545.Google Scholar
Teshukov, V. M. 2007 Gas-dynamics analogy for vortex free-boundary flows. J. Appl. Mech. Tech. Phys. 48 (3), 303–309.Google Scholar
Ting, F. C. K. & Kirby, J. T. 1994 Observation of undertow and turbulence in a laboratory surf zone. Coast. Engng 24, 51–80.Google Scholar
Tissier, M., Bonneton, P., Marche, F., Chazel, F. & Lannes, D. 2012 A new approach to handle wave breaking in fully nonlinear Boussinesq models. Coast. Engng 67, 54–66.Google Scholar
Titov, V. V. & Synolakis, C. E. 1995 Modeling of breaking and nonbreaking long-wave evolution and runup using VTCS-2. J. Waterways Port Coast. 121, 308–316.Google Scholar
Tonelli, M. & Petti, M. 2010 Finite volume scheme for the solution of 2D extended Boussinesq equations in the surf zone. Ocean Engng 37, 567–582.Google Scholar
Tonelli, M. & Petti, M. 2011 Simulation of wave breaking over complex bathymetries by a Boussinesq model. J. Hydraul. Res. 49, 473–486.Google Scholar
Veeramony, J. & Svendsen, I. A. 2000 The flow in surf-zone waves. Coast. Engng 39, 93–122.Google Scholar
Wei, G., Kirby, J. T., Grilli, S. T. & Subramanya, R. 1995 A fully nonlinear Boussinesq model for surface waves. Part I. Highly nonlinear unsteady waves. J. Fluid Mech. 294, 71–92.Google Scholar
Zelt, J. A. 1991 The run-up of nonbreaking and breaking solitary waves. Coast. Engng 15, 205–246.Google Scholar
Zhang, X., Xia, Y. & Shu, C. W. 2012 Maximum-principle-satisfying and positivity-preserving high order discontinuous Galerkin schemes for conservation laws on triangular meshes. J. Sci. Comput. 50, 29–62.Google Scholar
Zhang, Y., Kennedy, A. B., Donahue, A. S., Westerink, J. J., Panda, N. & Dawson, C. 2014 Rotational surf zone modeling for O (𝜇4) Boussinesq–Green–Naghdi systems. Ocean Model. 79, 43–53.Google Scholar
Zhang, Y., Kennedy, A. B., Panda, N., Dawson, C. & Westerink, J. J. 2013 Boussinesq–Green–Naghdi rotational water wave theory. Coast. Engng 73, 13–27.Google Scholar
Zhang, Y., Kennedy, A. B., Tomiczek, T., Donahue, A. S. & Westerink, J. J. 2016 Validation of Boussinesq–Green–Naghdi modeling for surf zone hydrodynamics. Ocean Engng 111, 299–309.Google Scholar

Richard et al. supplementary movie

Numerical simulation of the laboratory experiments of Swigler (2009) using the 2D model: propagation of a solitary wave on a realistic coastal configuration including a reef and the run-up and run-down phenomena on a beach.

Download Richard et al. supplementary movie(Video)
Video 1.3 MB