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Implementation of a fully nonlinear Hamiltonian coupled-mode theory, and application to solitary wave problems over bathymetry. (English) Zbl 1408.76084

Summary: This paper deals with the implementation of a new, efficient, non-perturbative, Hamiltonian coupled-mode theory (HCMT) for the fully nonlinear, potential flow (NLPF) model of water waves over smooth, single-valued, but otherwise arbitrary bathymetry [the first and last authors, “A new efficient Hamiltonian approach to the nonlinear water-wave problem over arbitrary bathymetry”, Preprint. arXiv:1704.03276]. Applications considered herein concern the interaction of solitary waves with bottom topographies and vertical walls both in two- and three-dimensional environments. The essential novelty of HCMT is a new representation of the Dirichlet-to-Neumann operator, which is needed to close the Hamiltonian evolution equations. This new representation emerges from the treatment of the substrate kinematical problem by means of exact semi-separation of variables in the instantaneous, irregular, fluid domain, established recently by the last and the first author [Proc. R. Soc. A, Math. Phys. Eng. Sci. 473, No. 2201, Article ID 20170017, 18 p. (2017; Zbl 1404.78031)]. The HCMT ensures an efficient dimensional reduction of the exact NLFP, being able to treat a variable bathymetry as simply as the flat-bottom case, without domain transformation. A key point for the efficient implementation of the method is the fast and accurate evaluation of the space-time varying coefficients appearing in some of its equations. In this paper, all varying coefficients are calculated analytically, resulting in a refined version of the theory, characterized by improved accuracy at significantly reduced computational time. This improved version of HCMT is first validated against existing experimental results and other computations, and subsequently applied to new solitary wave-bottom interaction problems. The latter include: (i) the investigation of a new type of “Bragg scattering” effect, appearing when a solitary wave propagates over a seabed with a sinusoidal patch, and (ii) the disintegration, focusing and reflection of a solitary wave moving over a three-dimensional bathymetry consisting of parallel banks and troughs, and impinging on a vertical wall.

MSC:

76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction

Citations:

Zbl 1404.78031
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References:

[1] Madsen, P.; Schaffer, H., Higher-order Boussinesq-type equations for surface gravity waves: derivation and analysis, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 356, 3123-3181, (1998) · Zbl 0930.35137
[2] Madsen, P.; Bingham, H.; Liu, H., A new Boussinesq method for fully nonlinear waves from shallow to deep water, J. Fluid Mech., 462, 1-30, (2002) · Zbl 1061.76009
[3] Madsen, P.; Bingham, H.; Schaffer, H., Boussinesq-type formulations for fully nonlinear and extremely dispersive water waves: derivation and analysis, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 459, 1075-1104, (2003) · Zbl 1041.76011
[4] Madsen, P.; Fuhrman, D.; Wang, B., A Boussinesq-type method for fully nonlinear waves interacting with a rapidly varying bathymetry, Coastal Eng., 53, 487-504, (2006)
[5] Zou, Z.; Fang, K., Alternative forms of the higher-order Boussinesq equations: derivations and validations, Coastal Eng., 55, 506-521, (2008)
[6] Karambas, T. V.; Memos, C. D., Boussinesq model for weakly nonlinear fully dispersive water waves, J. Waterw. Port Coast. Eng., 187-199, (2009)
[7] Adytia, D.; van Groesen, E., Optimized variational 1D Boussinesq modelling of coastal waves propagating over a slope, Coastal Eng., 64, 139-150, (2012)
[8] Chondros, M.; Memos, C., A 2DH nonlinear Boussinesq-type wave model of improved dispersion, shoaling, and wave generation characteristics, Coast. Eng., 91, 99-122, (2014)
[9] Filippini, A.; Bellec, A.; Colin, M.; Ricchiuto, M., On the nonlinear behaviour of Boussinesq type models: amplitude-velocity vs amplitude-flux forms, Coastal Eng., 99, 109-123, (2015)
[10] Lynett, P.; Liu, P., A two-layer approach to wave modelling, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 460, 2637-2669, (2004) · Zbl 1070.76009
[11] Chazel, B.; Benoit, M.; Ern, A.; Piperno, S., A double-layer model for nonlinear and dispersive waves highly, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 465, 2319-2346, (2009) · Zbl 1186.35156
[12] Zhao, B.; Duan, W.; Ertekin, R., Application of higher-level GN theory to some wave transformation problems, Coastal Eng., 83, 177-189, (2014)
[13] Lannes, D.; Marche, F., A new class of fully nonlinear and weakly dispersive Green-naghdi models for efficient 2 D simulations, J. Comput. Phys., 282, 238-268, (2015) · Zbl 1351.76114
[14] Mitsotakis, D.; Synolakis, C.; McGuiness, M., A modified Galerkin / finite element method for the numerical solution of the Serre-Green-naghdi system, Internat. J. Numer. Methods Fluids, 83, 755-778, (2017)
[15] Ma, Q., Advances in numerical simulation of nonlinear water waves, (2010), World Scientific Singapore
[16] Brocchini, M., A reasoned overview on Boussinesq-type models: the interplay between physics, mathematics and numerics, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 469, 20130496, (2013) · Zbl 1371.35245
[17] Kirby, J., Boussinesq models and their application to coastal processes across a wide range of scales, J. Waterw. Port Coast. Eng., 142, 1-29, (2016)
[18] Bingham, H.; Zhang, H., On the accuracy of finite difference solutions for nonlinear water waves, J. Eng. Math., 58, 211-228, (2007) · Zbl 1178.76256
[19] Cai, X.; Langtangen, P.; Nielsen, F.; Tveito, A., A finite element method for fully nonlinear waterwaves, J. Comput. Phys., 143, 544-568, (1998) · Zbl 0935.76042
[20] Ma, Q.; Yan, S., Quasi ALE finite element method for nonlinear water waves, J. Comput. Phys., 212, 52-72, (2006) · Zbl 1216.76032
[21] Gagarina, E.; Ambati, V. R.; van der Vegt, J. J.W.; Bokhove, O., Variational space-time (dis)continuous Galerkin method for nonlinear free surface water waves, J. Comput. Phys., 275, 459-483, (2014) · Zbl 1349.76204
[22] Brink, F.; Izsák, F.; van der Vegt, J., Hamiltonian finite element discretization for nonlinear free surface water waves, J. Sci. Comput., (2017) · Zbl 1433.76070
[23] A. Engsig-Karup, C. Eskilsson, D. Bigoni, A stabilised nodal spectral element method for fully nonlinear water waves. Part 2: Wave-body interaction, 2017, pp. 1-41. ArXiv Prepr.http://arxiv.org/abs/1512.02548; A. Engsig-Karup, C. Eskilsson, D. Bigoni, A stabilised nodal spectral element method for fully nonlinear water waves. Part 2: Wave-body interaction, 2017, pp. 1-41. ArXiv Prepr.http://arxiv.org/abs/1512.02548 · Zbl 1349.76570
[24] Grilli, S.; Skourup, J.; Svedsen, A., An efficient boundary element method for nonlinear water waves, Eng. Anal. Bound. Elem., 6, 97-107, (1989)
[25] Grilli, S.; Guyenne, P.; Dias, F., A fully non-linear model for three-dimensional overturning waves over an arbitrary bottom, Internat. J. Numer. Methods Fluids, 35, 829-867, (2001) · Zbl 1039.76043
[26] Fructus, D.; Clamond, D.; Grue, J.; Kristiansen, Ø., An efficient model for three-dimensional surface wave simulations. part I: free space problems, J. Comput. Phys., 205, 665-685, (2005) · Zbl 1087.76016
[27] Fructus, D.; Clamond, D.; Grue, J.; Kristiansen, Ø., An efficient model for three-dimensional surface wave simulations. part II: generation and absorption, J. Comput. Phys., 205, 686-705, (2005) · Zbl 1087.76015
[28] Fochesato, C.; Dias, F., A fast method for nonlinear three-dimensional free-surface waves, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462, 2715-2735, (2006) · Zbl 1149.76610
[29] Petrov, A., Variational statement of the problem of liquid motion in a container of finite dimensions, PMM, 28, 754-758, (1964)
[30] Zakharov, V., Stability of periodic waves of finite amplitude on the surface of a deep fluid, Zh. Prikl. Mekh. Tekh. Fiz., 9, 86-94, (1968)
[31] Craig, W.; Sulem, C., Numerical simulation of gravity waves, J. Comput. Phys., 108, 73-83, (1993) · Zbl 0778.76072
[32] Lannes, D., Water waves problem: mathematical analysis and asymptotics, (2013), American Mathematical Society Providence, Rhode Island · Zbl 1410.35003
[33] Bateman, W.; Swan, C.; Taylor, P., On the efficient numerical simulation of directionally spread surface water waves, J. Comput. Phys., 174, 277-305, (2001) · Zbl 1106.86300
[34] Nicholls, D. P.; Reitich, F., A new approach to analyticity of Dirichlet-Neumann operators, Proc. Roy. Soc. Edinburgh Sect. A, 131, 1411, (2001) · Zbl 1016.35030
[35] Nicholls, D. P.; Reitich, F., Stability of high-order perturbative methods for the computation of Dirichlet-Neumann operators, J. Comput. Phys., 170, 276-298, (2001) · Zbl 0983.65115
[36] Craig, W.; Nicholls, D. P., Traveling gravity water waves in two and three dimensions, Eur. J. Mech. B Fluids, 21, 615-641, (2002) · Zbl 1084.76509
[37] Nicholls, D. P., Boundary perturbation methods for water waves, GAMM-Mitt., 30, 44-74, (2007) · Zbl 1125.76008
[38] Schäffer, H., Comparison of Dirichlet-Neumann operator expansions for nonlinear surface gravity waves, Coastal Eng., 55, 288-294, (2008)
[39] Smith, R. A., An operator expansion formalism for nonlinear surface waves over variable depth, J. Fluid Mech., 363, 333-347, (1998) · Zbl 0911.76011
[40] Liu, Y.; Yue, D., On generalized Bragg scattering of surface waves by bottom ripples, J. Fluid Mech., 356, 297-326, (1998) · Zbl 0908.76014
[41] Guyenne, P.; Nicholls, D., Numerical simulation of solitary waves on plane slopes, Math. Comput. Simulation, 69, 269-281, (2005) · Zbl 1115.76019
[42] Craig; Guyenne, P.; Nicholls, D.; Sulem, C., Hamiltonian long-wave expansions for water waves over a rough bottom, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461, 839-873, (2005) · Zbl 1145.76325
[43] Guyenne, P.; Nicholls, D., A high-order spectral method for nonlinear water waves over moving bottom topography, SIAM J. Sci. Comput., 30, 81-101, (2007) · Zbl 1157.76034
[44] Gouin, M.; Ducrozet, G.; Ferrant, P., Development and validation of a non-linear spectral model for water waves over variable depth, Eur. J. Mech. B Fluid, 57, 115-128, (2016) · Zbl 1408.76066
[45] Athanassoulis, G.; Belibassakis, K., A consistent coupled-mode theory for the propagation of small-amplitude water waves over variable bathymetry regions, J. Fluid Mech., 389, 275-301, (1999) · Zbl 0959.76009
[46] Athanassoulis, G.; Belibassakis, K., A complete modal expansion of the wave potential and its application to linear and nonlinear water-wave problems, (Rogue Waves, (2000)), 73-90
[47] Athanassoulis, G.; Belibassakis, K., New evolution equations for non-linear water waves un general bathymetry with application to steady travelling solutions in constant, but arbritary depth, Discrete Contin. Dyn. Syst., 75-84, (2007) · Zbl 1163.76326
[48] Belibassakis, K.; Athanassoulis, G., A coupled-mode system with application to nonlinear water waves propagating in finite water depth and in variable bathymetry regions, Coastal Eng., 58, 337-350, (2011)
[49] Athanassoulis, G.; Papoutsellis, C., Exact semi-separation of variables in waveguides with nonplanar boundaries, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 473, (2017)
[50] C. Papoutsellis, G. Athanassoulis, A new efficient Hamiltonian approach to the nonlinear water-wave problem over arbitrary bathymetry, 2017, submited for publication. http://arxiv.org/abs/1704.03276; C. Papoutsellis, G. Athanassoulis, A new efficient Hamiltonian approach to the nonlinear water-wave problem over arbitrary bathymetry, 2017, submited for publication. http://arxiv.org/abs/1704.03276
[51] Tian, Y.; Sato, S., A numerical model on the interaction between nearshore nonlinear waves and strong currents, Coast. Eng. J., 50, 369-395, (2008)
[52] Yates, M.; Benoit, M., Accuracy and efficiency of two numerical methods of solving the potential flow problem for highly nonlinear and dispersive water waves, Internat. J. Numer. Methods Fluids, 77, 616-640, (2015)
[53] Raoult, C.; Benoit, M.; Yates, M., Validation of a fully nonlinear and dispersive wave model with laboratory non-breaking experiments, Coastal Eng., 114, 194-207, (2016)
[54] Mei, C.; Stiassnie, M.; Yue, D., Theory and applications of Ocean surface waves, advanced S, (2005), World Scientific Singapore
[55] Luke, J., A variational principle for a fluid with a free surface, J. Fluid Mech., 27, 395-397, (1967) · Zbl 0146.23701
[56] Whitham, G., Linear and nonlinear waves, (1974), John Wiley & Sons · Zbl 0373.76001
[57] Massel, S., Hydrodynamics of coastal zones, (Oceanography Series, vol. 48, (1989), Elsevier Netherlands)
[58] Isobe, M.; Abohadima, S., Numerical model of fully-nonlinear wave refraction and diffraction, (Coast. Eng., (1998))
[59] Klopman, G.; Van Groesen, B.; Dingemans, M., A variational approach to Boussinesq modelling of fully nonlinear water waves, J. Fluid Mech., 657, 36-63, (2010) · Zbl 1197.76026
[60] Hazard, C.; Luneville, E., An improved multimodal approach for non-uniform acoustic waveguides, IMA J. Appl. Math., 73, 668-690, (2008) · Zbl 1156.76050
[61] Mercier, J.-F.; Maurel, A., Acoustic propagation in non-uniform waveguides: revisiting Webster equation using evanescent boundary modes, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 469, (2013), 20130186-20130186. http://dx.doi.org/10.1098/rspa.2013.0186 · Zbl 1371.76122
[62] Papoutsellis, C., Nonlinear water waves over varying bathymetry: theoretical and numerical study using variational methods, (2016), National Technical University of Athens
[63] T. Papathanasiou, C. Papoutsellis, G. Athanassoulis, Semi-explicit solutions to the water-wave dispersion relation and their role in the nonlinear Hamiltonian coupled-mode theory, 2018, submitted for publication. https://arxiv.org/abs/1802.07963; T. Papathanasiou, C. Papoutsellis, G. Athanassoulis, Semi-explicit solutions to the water-wave dispersion relation and their role in the nonlinear Hamiltonian coupled-mode theory, 2018, submitted for publication. https://arxiv.org/abs/1802.07963
[64] Chamberlain, P. G.; Porter, D., The modified mild-slope equation, J. Fluid Mech., 291, 393, (1995) · Zbl 0843.76006
[65] Clamond, D.; Dutykh, D., Fast accurate computation of the fully nonlinear solitary surface gravity waves, Comput. Fluids, 84, 35-38, (2013) · Zbl 1290.76018
[66] Mitsotakis, D.; Dutykh, D.; Carter, J., On the nonlinear dynamics of the travelling-wave solutions of the Serre equations, Wave Motion, 70, 166-182, (2017)
[67] Cooker, M.; Weidman, P.; Bale, D., Reflection of a high-amplitude solitary wave at a vertical wall, J. Fluid Mech., 342, 141-158, (1997) · Zbl 0911.76012
[68] Chambarel, J.; Kharif, C.; Touboul, J., Head-on collision of two solitary waves and residual falling jet formation, Nonlinear Process. Geophys., 16, 111-122, (2009)
[69] Touboul, J.; Pelinovsky, E., Bottom pressure distribution under a solitonic wave reflecting on a vertical wall, Eur. J. Mech. B Fluids, 48, 13-18, (2014) · Zbl 1408.76112
[70] Craig, W.; Guyenne, P.; Hammack, J.; Henderson, D.; Sulem, C., Solitary water wave interactions, Phys. Fluids, 18, 57106, (2006) · Zbl 1185.76463
[71] Engsig-karup, A. P.; Hesthaven, J. S.; Bingham, H. B.; Madsen, P. A., Nodal DG-FEM solution of high-order Boussinesq-type equations, J. Eng. Math., 351-370, (2006) · Zbl 1200.76121
[72] Mitsotakis, D.; Synolakis, C.; McGuiness, M., A modified Galerkin/finite element method for the numerical solution of the Serre-Green-naghdi system, Internat. J. Numer. Methods Fluids, 83, 755-778, (2017)
[73] Chen, Y.; Kharif, C.; Yang, J.; Hsu, H.; Touboul, J.; Chambarel, J., An experimental study of steep solitary wave reflection at a vertical wall, Eur. J. Mech. B Fluids, 49, 20-28, (2015) · Zbl 06932535
[74] Su, H.; Mirie, R., On head-on collisions between two solitary waves, J. Fluid Mech., 98, 509-525, (1980) · Zbl 0434.76021
[75] Grilli, S. T.; Subramanya, R.; a. Svendsen, I.; Veeramony, J., Shoaling of solitary waves on plane beaches, J. Waterw. Port Coast. Ocean Eng., 120, 609-628, (1994)
[76] Kalisch, H.; Khorsand, Z.; Mitsotakis, D., Mechanical balance laws for fully nonlinear and weakly dispersive water waves, Physica D, 333, 243-253, (2016)
[77] Guyenne, P.; Grilli, S. T., Numerical study of three-dimensional overturning waves in shallow water, J. Fluid Mech., 547, 361-388, (2006) · Zbl 1082.76019
[78] Dodd, N., Numerical model of wave run-up, overtopping and regeneration, J. Waterw. Port Coast. Ocean Eng., 72-81, (1998)
[79] Walkley, M.; Berzins, M., A finite element method for the one-dimensional extended Boussinesq equations, Internat. J. Numer. Methods Fluids, 29, 143-157, (1999) · Zbl 0941.76055
[80] Antunes Do Carmo, J. S., Boussinesq and Serre type models with improved linear dispersion characteristics: applications, J. Hydraul. Res., 51, 719-727, (2013)
[81] Filippas, E.; Belibassakis, K., Hydrodynamic analysis of flapping-foil thrusters operating beneath the free surface and in waves, Eng. Anal. Bound. Elem., 41, 47-59, (2014) · Zbl 1297.76022
[82] Ducrozet, G.; Bingham, H.; Engsig-karup, A.; Ferrant, P., A comparative study of two fast nonlinear free surface water wave models, Internat. J. Numer. Methods Fluids, 2-29, (2012)
[83] Athanassoulis, G.; Papoutsellis, C., New form of the Hamiltonian equations for the nonlinear water-wave problem, based on a new representation of the dtn operator, and some applications, (Proc. 34th Int. Conf. Ocean. Offshore Arct. Eng., (2015), ASME, St. John’s, Newfoundland Canada), p. V007T06A029. http://dx.doi.org/10.1115/OMAE2015-41452
[84] Gasinski, L.; Papageorgiou, N., Nonlinear analysis, (2005), Chapman & Hall/CRC, Taylor & Francis Group
[85] Vatankhah, R.; Aghashariatmadari, Z., Improved explicit approximation of linear dispersion relationship for gravity waves: a discussion, Coastal Eng., 78, 21-22, (2013)
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