Hostname: page-component-76fb5796d-x4r87 Total loading time: 0 Render date: 2024-04-25T19:51:28.732Z Has data issue: false hasContentIssue false
  • English
  • Français

A new model of shoaling and breaking waves: one-dimensional solitary wave on a mild sloping beach

Published online by Cambridge University Press:  15 January 2019

M. Kazakova  and
G. L. Richard
M. Kazakova
Affiliation:
Institut de Mathématiques de Toulouse; UMR5219, Université de Toulouse; CNRS, UPS, F-31062 Toulouse CEDEX 9, France
G. L. Richard*
Affiliation:
LAMA, UMR5127, Université de Savoie Mont-Blanc, CNRS, 73376 Le Bourget-du-Lac, France
*
†Email address for correspondence: gael.loic.richard@orange.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

We present a new approach to model coastal waves in the shoaling and surf zones. The model can be described as a depth-averaged large-eddy simulation model with a cutoff in the inertial subrange. The large-scale turbulence is explicitly resolved through an extra variable called enstrophy while the small-scale turbulence is modelled with a turbulent-viscosity hypothesis. The equations are derived by averaging the mass, momentum and kinetic energy equations assuming a shallow-water flow, a negligible bottom shear stress and a weakly turbulent flow assumption which is not restrictive in practice. The model is fully nonlinear and has the same dispersive properties as the Green–Naghdi equations. It is validated by numerical tests and by comparison with experimental results of the literature on the propagation of a one-dimensional solitary wave over a mild sloping beach. The wave breaking is characterized by a sudden increase of the enstrophy which allows us to propose a breaking criterion based on the new concept of virtual enstrophy. The model features three empirical parameters. The first one governs the turbulent dissipation and was found to be a constant. The eddy viscosity is determined by a turbulent Reynolds number depending only on the bottom slope. The third parameter defines the breaking criterion and depends only on the wave initial nonlinearity. These dependences give a predictive character to the model which is suitable for further developments.

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 862 , 10 March 2019 , pp. 552 - 591
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

Antunes do Carmo, J. S. 2013 Boussinesq and Serre type models with improved linear dispersion characteristics: applications. J. Hydraul Res. 51 (6), 719–727.Google Scholar
Antunes do Carmo, J. S., Seabra Santos, F. J. & Almeida, A. B. 1993 Numerical solution of the generalized Serre equations with the MacCormack finite difference scheme. Intl J. Numer. Meth. Fluids 16 (8), 725–738.Google Scholar
Antuono, M., Bardazzi, A., Lugni, C. & Brocchini, M. 2014 A shallow-water sloshing model for wave breaking in rectangular tanks. J. Fluid Mech. 746, 437–465.Google Scholar
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
Bacigaluppi, P., Ricchiuto, M. & Bonneton, P. 2014 A 1D stabilized finite element model for nonhydrostatic wave breaking and run-up. In Finite Volumes for Complex Applications VII (ed. Ohlberger, M., Fuhrmann, J. & Rohde, C.), Springer Proceedings in Mathematics and Statistics, vol. 77, pp. 779–790. Springer.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
Bellotti, G. & Brocchini, M. 2002 On using Boussinesq-type equations near the shoreline: a note of caution. Ocean Engng 29, 1569–1575.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
Briganti, R., Musumeci, R. E., Bellotti, G., Brocchini, M. & Foti, E. 2004 Boussinesq modeling of breaking waves: description of turbulence. J. Geophys. Res. 109, C07015.Google Scholar
Brocchini, M. 2002 Free surface boundary conditions at a bubbly/weakly splashing air–water interface. Phys. Fluids 14, 1834–1840.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. Ocean Div. ASCE 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
Castro, A. & Lannes, D. 2014 Fully nonlinear long-waves models in presence of vorticity. J. Fluid Mech. 759, 642–675.Google Scholar
Chanson, H. 2011 Hydraulic jumps: turbulence and air bubble entrainment. La Houille Blanche 3, 5–16.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
Cienfuegos, R., Barthelemy, E. & Bonneton, P. 2006 A fourth-order compact finite volume scheme for fully nonlinear and weakly dispersive Boussinesq-type equations. Part I. Model development and analysis. Intl J. Numer. Meth. Fluids 51, 1217–1253.Google Scholar
Cienfuegos, R., Barthelemy, E. & Bonneton, P. 2007 A fourth-order compact finite volume scheme for fully nonlinear and weakly dispersive Boussinesq-type equations. Part II. Boundary conditions and validation. Intl J. Numer. Meth. Fluids 53, 1423–1455.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. Ocean Div. ASCE 136, 10–26.Google Scholar
Corrsin, S. & Kistler, A. L.1955 Free-stream boundaries of turbulent flows. Report 1244, NACA.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
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
Dimas, A. A. & Dimakopoulos, A. S. 2009 Surface roller model for the numerical simulation of spilling wave breaking over constant slope beach. J. Waterways Port Coast. Ocean Div. ASCE 135, 235–244.Google Scholar
Dong, H. & Li, M. 2016 A reconstructed central discontinuous Galerkin-finite element method for the fully nonlinear weakly dispersive Green–Naghdi model. Appl. Numer. Maths. 110, 110–127.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
Dutykh, D., Clamond, D., Milewski, P. & Mitsotakis, D. 2013 Finite volume and pseudo-spectral schemes for the fully nonlinear 1D Serre equations. Eur. J. Appl. Maths 24, 761–787.Google Scholar
Favrie, N. & Gavrilyuk, S. 2017 A rapid numerical method for solving Serre–Green–Naghdi equations describing long free surface gravity waves. Nonlinearity 30, 2718–2736.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
Freeman, J. C. & Le Méhauté, B. 1964 Wave breakers on a beach and surges on a dry bed. ASCE J. Hydraul. Engng 90, 187–216.Google Scholar
Fuchs, H. & Hager, W. H. 2015 Solitary impulse wave transformation to overland flow. J. Waterways Port Coast. Ocean Div. ASCE 141, 04015004.Google Scholar
Gavrilyuk, S. L., Liapidevskii, V. Y. & Chesnokov, A. A. 2016 Spilling breakers in shallow water: applications to Favre waves and to the shoaling and breaking of solitary waves. J. Fluid Mech. 808, 441–468.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
Grilli, S. T., Svendsen, I. A. & Subramanya, R. 1997 Breaking criterion and characteristics for solitary waves on slopes. J. Waterways Port Coast. Ocean Div. ASCE 123, 102–112.Google Scholar
Hafsteinsson, H. J., Evers, F. M. & Hager, W. H. 2017 Solitary wave run-up: wave breaking and bore propagation. J. Hydraul Res. 55, 1–12.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
Heitner, K. L. & Housner, G. W. 1970 Numerical model for tsunami run-up. J. Waterways Port Coast. Ocean Engng 96, 701–719.Google Scholar
Hibbert, S. & Peregrine, D. H. 1979 Surf and run-up on a beach: a uniform bore. J. Fluid Mech. 95, 323–345.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. ASCE J. Hydraul. Engng 133, 857–872.Google Scholar
Hsiao, S. C., Hsu, T. W., Lin, T. C. & Chang, Y. H. 2008 On the evolution and run-up of breaking solitary waves on a mild sloping beach. Coast. Engng 55, 975–988.Google Scholar
Iwasaki, T. & Togashi, H. 1970 On the shoreline and leading front conditions of tsunami waves in the light of the method of characteristics. Coast. Engng J. 13, 113–125.Google Scholar
Karambas, T. V. & Tozer, N. P. 2003 Breaking waves in the surf and swash zone. J. Coast. Res. 19, 514–528.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. Ocean Div. ASCE 126, 39–47.Google Scholar
Kim, D.-H., Lynett, P. J. & Socolofsky, S. A. 2009 A depth-integrated model for weakly dispersive, turbulent, and rotational fluid flows. Ocean Model. 27, 198–214.Google Scholar
Kimmoun, O. & Branger, H. 2007 A particle image velocimetry investigation on laboratory surf-zone breaking waves over a sloping beach. J. Fluid Mech. 588, 353–397.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
Kobayashi, N., Otta, A. K. & Roy, I. 1987 Wave reflection and run-up on rough slopes. J. Waterways Port Coast. Ocean Div. ASCE 113, 282–298.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
Lannes, D. 2013 The Water Waves Problem: Mathematical Analysis and Asymptotics, Mathematical Surveys and Monographs, vol. 188. American Mathematical Society.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
Lannes, D. & Marche, F. 2016 Nonlinear wave–current interactions in shallow water. Stud. Appl. Maths 136, 382–423.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
Lilly, D. K. 1967 The representation of small-scale turbulence in numerical simulation experiments. In Proceedings of the IBM Scientific Computing Symp. on Environmental Sciences (ed. Goldstine, H. H.), pp. 195–210. IBM.Google Scholar
Lin, P. & Liu, P. L.-F. 1998 A numerical study of breaking waves in the surf zone. J. Fluid Mech. 359, 239–264.Google Scholar
Liu, P. L.-F., Synolakis, C. E. & Yeh, H. H. 1991 Report on the international workshop on long-wave run-up. J. Fluid Mech. 229, 675–688.Google Scholar
Longuet-Higgins, M. S. 1992 Capillary rollers and bores. J. Fluid Mech. 240, 659–679.Google Scholar
Lubin, P., Vincent, S., Abadie, S. & Caltagirone, J. P. 2006 Three-dimensional large-eddy simulation of air entrainment under plunging breaking waves. Coast. Engng 53, 631–655.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., Rugbjerg, M. & Warren, I. R. 1988 Subgrid modelling in depth integrated flows. In Coastal Engineering 1996 (ed. Edge, B. L.), Proceedings of the 21st Internatonal Conference, vol. 21, pp. 505–511. ASCE.Google Scholar
Madsen, P. A. & Sørensen, O. R. 1992 A new form of the Boussinesq equations with improved linear dispersion characteristics. Part II. A slowly varying bathymetry. Coast. Engng 18, 183–204.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
McCowan, J. 1894 On the highest wave of permanent type. Phil. Mag. 38, 351–358.Google Scholar
Meyer, R. E. & Taylor, A. D. 1972 Run-up on beaches. Waves on Beaches and Resulting Sediment Transport, pp. 357–411. Academic Press.Google Scholar
Mitsotakis, D., Ilan, B. & Dutykh, D. 2014 On the Galerkin/finite-element method for the Serre equations. J. Sci. Comput. 61, 166–195.Google Scholar
Mitsotakis, D., Synolakis, C. & McGuinness, M. 2016 A modified Galerkin/finite element method for the numerical solution of the Serre-Green–Naghdi system. Intl J. Numer. Meth. Fluids 83, 755–778.Google Scholar
Musumeci, R. E., Svendsen, I. A. & Veeramony, J. 2005 The flow in the surf zone: a fully nonlinear Boussinesq-type of approach. Coast. Engng 52, 565–598.Google Scholar
Nadaoka, K. & Yagi, H. 1998 Shallow-water turbulence modeling and horizontal large-eddy computation of river flow. J. Hydraul. Engng 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 (ed. Edge, B. L.), Proceedings 25th International Conference, vol. 25, pp. 4807–4820. ASCE.Google Scholar
Packwood, A. R. & Peregrine, D. H.1981 Surf and run-up on beaches: models of viscous effects. Rep. AM-81 7, University of Bristol.Google Scholar
Panda, N., Dawson, C., Zhang, Y., Kennedy, A. B., Westerink, J. J. & Donahue, A. S. 2014 Discontinuous Galerkin methods for solving Boussinesq–Green–Naghdi equations in resolving non-linear and dispersive surface water waves. J. Comput. Phys. 273, 572–588.Google Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.Google Scholar
Popinet, S. 2015 A quadtree-adaptative multigrid solver for the Serre-Green–Naghdi equations. J. Comput. Phys. 302, 336–358.Google Scholar
Prandtl, L. 1945 Über ein neues Formelsystem für die ausgebildete Turbulenz. Nachr. Akad. Wiss. Göttingen Math-Phys. K1, 6–19.Google Scholar
Rayleigh 1876 On waves. Phil. Mag. 5, 257–279.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
Richard, G. L., Rambaud, A. & Vila, J. P. 2017 Consistent equations for open-channel flows in the smooth turbulent regime with shearing effects. J. Fluid Mech. 831, 289–329.Google Scholar
Richardson, L. F. 1922 Weather Prediction by Numerical Process. Cambridge University Press.Google Scholar
Russell, J. S. 1844 Report on waves. Brit. Assoc. Adv. Sci. 14, 311–390.Google Scholar
Schäffer, H. A., Madsen, P. A. & Deigaard, R. 1993 A Boussinesq model for waves breaking in shallow water. Coast. Engng 20, 185–202.Google Scholar
Serre, F. 1953 Contribution à l’étude des écoulements permanents et variables dans les canaux. La Houille Blanche 8, 374–388.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
Stansby, P. K. & Feng, T. 2005 Kinematics and depth-integrated terms in surf zone waves from laboratory measurement. J. Fluid Mech. 529, 279–310.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
Stocker, J. J. 1957 Water Waves – The Mathematical Theory with Applications. Interscience.Google Scholar
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, 536–539.Google Scholar
Svendsen, I. A. 1984 Wave heights and set-up in a surf zone. Cost. Engng 8, 303–329.Google Scholar
Svendsen, I. A. & Madsen, P. A. 1984 A turbulent bore on a beach. J. Fluid Mech. 148, 73–96.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
Svendsen, I. A., Yu, K. & Veeramony, J. 1996 A Boussinesq breaking wave model with vorticity. In Coastal Engineering 1996 (ed. Edge, B. L.), Proceedings of the 25th International Conference, vol. 25, pp. 1192–1204. ASCE.Google Scholar
Synolakis, C. E. 1989 Discussion of ‘Wave reflection and run-up on rough slopes’ by Nobuhisa Kobayashi, Ashwini K. Otta, and Indarajut Roy (May, 1987, Vol. 113, No. 4). J. Waterways Port Coast. Ocean Div. ASCE 115, 139–143.Google Scholar
Synolakis, C. E. & Skjelbreia, J. E. 1993 Evolution of maximum amplitude of solitary waves on plane beaches. J. Waterways Port Coast. Ocean Div. ASCE 119, 323–342.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. 2006 Large-scale turbulence under a solitary wave. Coast. Engng 53, 441–462.Google Scholar
Tissier, M., Bonneton, P., Marche, F., Chazel, F. & Lannes, D. 2012 A new approach to handle wave breaking in fully non-linear Boussinesq models. Coast. Engng 67, 54–66.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
Viviano, A., Musumeci, R. E. & Foti, E. 2015 A nonlinear rotational, quasi-2DH, numerical model for spilling wave propagation. Appl. Math. Model. 39, 1099–1118.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
Wei, G., Kirby, J. T. & Sinha, A. 1999 Generation of waves in Boussinesq models using a source function method. Coast. Engng 36, 271–299.Google Scholar
Yamazaki, Y., Kowalik, Z. & Cheung, K. F. 2009 Depth-integrated, non-hydrostatic model for wave breaking and run-up. Intl J. Numer. Meth. Fluids 61, 473–497.Google Scholar
Zelt, J. A. 1991 The run-up of nonbreaking and breaking solitary waves. Coast. Engng 15, 205–246.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