New Form of the Hamiltonian Equations for the Nonlinear Water-Wave Problem, Based on a New Representation of the DTN Operator, and Some Applications

Athanassoulis, G. A.; Papoutsellis, Christos E.

doi:10.1115/omae2015-41452

Cited by 11 publications

(17 citation statements)

References 33 publications

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“…This is achieved through a rapidly convergent vertical series expansion of the velocity potential that is valid in the entire non-uniform fluid domain up to the boundaries (Belibassakis and Athanassoulis, 2011;. The resulting equations retain the dimensionally-reduced Hamiltonian structure of the water wave system (Zakharov, 1968;Craig and Sulem, 1993) and give accurate predictions of fully nonlinear and strongly dispersive waves over variable bathymetry up to breaking (Athanassoulis and Papoutsellis, 2015;Papoutsellis and Athanassoulis, 2017;Papoutsellis et al, 2018). Wave models with similar mathematical structure and capabilities have also been developed using Chebyshev series representations of the potential in the vertical direction (Tian and Sato, 2008;Yates and Benoit, 2015;Raoult et al, 2016Raoult et al, , 2019.…”

Section: Introductionmentioning

confidence: 99%

Modelling of depth-induced wave breaking in a fully nonlinear free-surface potential flow model

Papoutsellis

Yates

Simon

et al. 2019

Coastal Engineering

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Two methods to treat wave breaking in the framework of the Hamiltonian formulation of free-surface potential flow are presented, tested, and validated. The first is an extension of Kennedy et al. (2000)'s eddyviscosity approach originally developed for Boussinesq-type wave models. In this approach, an extra term, constructed to conserve the horizontal momentum for waves propagating over a flat bottom, is added in the dynamic free-surface condition. In the second method, a pressure distribution is introduced at the free surface that dissipates wave energy by analogy to a hydraulic jump (Guignard and Grilli, 2001). The modified Hamiltonian systems are implemented using the Hamiltonian Coupled-Mode Theory, in which the velocity potential is represented by a rapidly convergent vertical series expansion. Wave energy dissipation and conservation of horizontal momentum are verified numerically. Comparisons with experimental measurements are presented for the propagation of a breaking dispersive shock wave following a dam break, and then incident regular waves breaking on a mildly sloping beach and over a submerged bar.Keywords: wave breaking, Hamiltonian formulation of water waves, eddyviscosity, fully nonlinear water waves Highlights• Two wave breaking techniques are tested in a fully nonlinear model. • Breaking is implemented by extending the Hamiltonian Coupled-Mode

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Section: Introductionmentioning

confidence: 99%

Modelling of depth-induced wave breaking in a fully nonlinear free-surface potential flow model

Papoutsellis

Yates

Simon

et al. 2019

Coastal Engineering

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“…The novel Hamiltonian Coupled-Mode Theory, proposed by [1], [16], is a nonperturbative, coupled-mode approach able to solve fully nonlinear water-wave problems in one or two horizontal dimensions, over varying bathymetry. In this approach, the evolution of the nonlocal Hamiltonian system requires the consecutive solution of a linear coupled-mode system with (x, t)-varying coefficients.…”

Section: Discussionmentioning

confidence: 99%

“…Recently, [1], see also [15], proposed a new formulation of the exact waterwave problem over arbitrary (smooth) bathymetry, providing both dimensional reduction and high accuracy, comparable with that ensured by DNM. This formulation is based on the exact semi-separation of variables in the instantaneous (non-canonical) fluid domain, established in [16], referred subsequently as AP17.…”

Section: Introductionmentioning

confidence: 99%

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Semi-explicit solutions to the water-wave dispersion relation and their role in the non-linear Hamiltonian coupled-mode theory

Papathanasiou¹,

Papoutsellis²,

Athanassoulis³

2019

J Eng Math

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The Hamiltonian Coupled-Mode Theory (HCMT), recently derived by Athanassoulis and Papoutsellis [1], provides an efficient new approach for solving fully nonlinear water-wave problems over arbitrary bathymetry. This theory exactly transforms the free-boundary problem to a fixedboundary one, with space and time varying coefficients. In calculating these coefficients, heavy use is made of the roots of a local, water-wave dispersion relation with varying parameter, which have to be calculated at every horizontal position and every time instant. Thus, fast and accurate calculation of these roots, valid for all possible values of the varying parameter, are of fundamen-tal importance for the efficient implementation of HCMT. In this paper, new, semi-explicit and highly accurate root-finding formulae are derived, especially for the roots corresponding to evanescent modes. The derivation is based on the successive application of a Picard-type iteration and the Householders root finding method. Explicit approximate formulae of very good accuracy are obtained, and machineaccurate determination of the required roots is easily achieved by no more than three iterations, using the explicit forms as initial values. Exploiting this procedure in the HCMT, results in an efficient, dimensionallyreduced, numerical solver able to treat fully non-linear water waves over arbitrary bathymetry. Applications to four demanding nonlinear problems demonstrate the efficiency and the robustness of the present approach. Specifically, we consider the classical tests of strongly nonlinear steady wave propagation and the transformation of regular waves due to trapezoidal and sinusoidal bathymetry. Novel results are also given for the disintegration of a solitary wave due to an abrupt deepening. The derived root-finding formulae can be used with any other multimodal methods as well.

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“…The reason of non-hydrostaticity of mSV equations lies in the pressure term and not in the frequency dispersion. Of course, the proposed model has to be further tested and validated by making direct comparisons with state-of-the-art numerical wave models [5,32].…”

Section: Discussionmentioning

confidence: 99%

Modified shallow water equations for significantly varying seabeds

Dutykh

Clamond²

2016

Applied Mathematical Modelling

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Abstract. In the present study, we propose a modified version of the Nonlinear Shallow Water Equations (Saint-Venant or NSWE) for irrotational surface waves in the case when the bottom undergoes some significant variations in space and time. The model is derived from a variational principle by choosing an appropriate shallow water ansatz and imposing some constraints. Our derivation procedure does not explicitly involve any small parameter and is straightforward. The novel system is a non-dispersive non-hydrostatic extension of the classical Saint-Venant equations. A key feature of the new model is that, like the classical NSWE, it is hyperbolic and thus similar numerical methods can be used. We also propose a finite volume discretisation of the obtained hyperbolic system. Several test-cases are presented to highlight the added value of the new model. Some implications to tsunami wave modelling are also discussed.

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New Form of the Hamiltonian Equations for the Nonlinear Water-Wave Problem, Based on a New Representation of the DTN Operator, and Some Applications

Cited by 11 publications

References 33 publications

Modelling of depth-induced wave breaking in a fully nonlinear free-surface potential flow model

Modelling of depth-induced wave breaking in a fully nonlinear free-surface potential flow model

Semi-explicit solutions to the water-wave dispersion relation and their role in the non-linear Hamiltonian coupled-mode theory

Modified shallow water equations for significantly varying seabeds

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