Variational space–time (dis)continuous Galerkin method for nonlinear free surface water waves

Gagarina, Elena; Ambati, V.R.; Vegt, J.J.W. van der; Bokhove, Onno

doi:10.1016/j.jcp.2014.06.035

Cited by 30 publications

(27 citation statements)

References 41 publications

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“…Expressed in new variables φ s = φ s exp 3⌫t/(2l 2 ) ,φ = φ exp 3⌫t/(2l 2 ) and η =(h−H 0 )exp 3νt/(2l 2 ) , the variational principle (11) and energy no longer have any explicit time dependence anymore in this long-time limit. A detailed and analogous analysis for the damped nonlinear oscillator that motivated us is found in Gagarina et al [12]. A straightforward comparison is now made between the potential flow model arising from (10) and an experiment.…”

Section: Validating Wave Sloshing Experimentsmentioning

confidence: 85%

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Hele-Shaw beach creation by breaking waves: a mathematics-inspired experiment

et al. 2014

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Fundamentals of nonlinear wave-particle interactions are studied experimentally in a Hele-Shaw configuration with wave breaking and a dynamic bed. To design this configuration, we determine, mathematically, the gap width which allows inertial flows to survive the viscous damping due to the side walls. Damped wave sloshing experiments compared with simulations confirm that width-averaged potential-flow models with linear momen- 2A n t h o n y T h o r n t o n e t a l .tum damping are adequately capturing the large scale nonlinear wave motion. Subsequently, we show that the four types of wave breaking observed at realworld beaches also emerge on Hele-Shaw laboratory beaches, albeit in idealized forms. Finally, an experimental parameter study is undertaken to quantify the formation of quasi-steady beach morphologies due to nonlinear, breaking waves: berm or dune, beach and bar formation are all classified. Our research reveals that the Hele-Shaw beach configuration allows a wealth of experimental and modelling extensions, including benchmarking of forecast models used in the coastal engineering practice, especially for shingle beaches.

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Section: Validating Wave Sloshing Experimentsmentioning

confidence: 85%

“…Variation of (11) with respect to φ in the interior and the conjugate variables {h, φ s e 3⌫t/l 2 } at z = h yields (10). Principle (11) implies that a modified potential energy emerges as…”

Section: Validating Wave Sloshing Experimentsmentioning

confidence: 99%

Hele-Shaw beach creation by breaking waves: a mathematics-inspired experiment

et al. 2014

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“…Another class of models that use pertubative calculations is based on the High-Order Spectral (HOS) method (Dommermuth and Yue, 1987;Craig and Sulem, 1993;Guyenne and Nicholls, 2007;Gouin et al, 2016). Direct methods of solution of fully nonlinear FSPF equations have also been proposed (Bingham and Zhang, 2007;Gagarina et al, 2014;Grilli et al, 2001). An alternative approach that treats the fully nonlinear problem is the Hamiltonian Coupled-Mode Theory or HCMT.…”

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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“…The displacementX can be treated as a small perturbation and linear terms in in (18) translate to cubic terms inX,Ũ and ∂ iX j in (17). Therefore, leaving only quadratic terms and omitting for brevity the (·) E superscript, (17) becomes…”

Section: ∂X ∂Amentioning

confidence: 99%

“…However, once the temporal scheme is obtained, we would like to avoid the costly computation of the inverse of C. Therefore, guided by (49), see also [17], we re-introduce φ i in the interior as…”

Section: Appendix B: Derivation Of Temporal Discretizationmentioning

confidence: 99%

Variational modelling of wave–structure interactions with an offshore wind-turbine mast

2017

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We consider the development of a mathematical model of water waves interacting with the mast of an offshore wind turbine. A variational approach is used for which the starting point is an action functional describing a dual system comprising a potential-flow fluid, a solid structure modelled with nonlinear elasticity, and the coupling between them. We develop a linearized model of the fluid-structure or wave-mast coupling, which is a linearization of the variational principle for the fully coupled nonlinear model. Our numerical results for the linear case indicate that our variational approach yields a stable numerical discretization of a fully coupled model of water waves and an elastic beam. The energy exchange between the subsystems is seen to be in balance, yielding a total energy that shows only small and bounded oscillations amplitude of which tends to zero with the second-order convergence as the timestep approaches zero. Similar second-order convergence is observed for spatial mesh refinement. The linearized model so far developed can be extended to a nonlinear regime.

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Variational space–time (dis)continuous Galerkin method for nonlinear free surface water waves

Cited by 30 publications

References 41 publications

Hele-Shaw beach creation by breaking waves: a mathematics-inspired experiment

Hele-Shaw beach creation by breaking waves: a mathematics-inspired experiment

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

Variational modelling of wave–structure interactions with an offshore wind-turbine mast

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