Prediction of the free-surface elevation for rotational water waves using the recovery of pressure at the bed

Henry, David; Thomas, Gareth

doi:10.1098/rsta.2017.0102

Cited by 22 publications

(17 citation statements)

References 36 publications

(65 reference statements)

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“…As shown in [39], the equations (88) admit a local conservation of energy, the energy density being here the sum of the energy density associated to the irrotational SGN equations and of a rotation (or turbulent) energy e rot ; a similar correction must also be made for the energy flux, so that (27) becomes (89) ∂ t e NSW + e rot + ∇ · F NSW + F rot = 0, where e rot = 1 2 TrE and F rot = 1 2 TrEV + EV .…”

Section: 32mentioning

confidence: 99%

“…quite obviously, enstrophy (or, equivalently, turbulent energy), is created in the vicinity of wave breaking, where the gradient of the velocity becomes important; the mechanical energy of the wave is consequently decreased. This mechanism restores the local conservation of the total energy (89). However, in a second step, the small scale dissipation of the total energy must be taken into account; there should therefore be a dissipation mechanism D such that ∂ t e NSW + e rot + ∇ · F NSW + F rot = −D.…”

Section: 4mentioning

confidence: 99%

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Modeling shallow water waves

Lannes

2020

Nonlinearity

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We review here the derivation of many of the most important models that appear in the literature (mainly in coastal oceanography) for the description of waves in shallow water. We show that these models can be obtained using various asymptotic expansions of the "turbulent" and non-hydrostatic terms that appear in the equations that result from the vertical integration of the free surface Euler equations. Among these models are the well-known nonlinear shallow water (NSW), Boussinesq and Serre-Green-Naghdi (SGN) equations for which we review several pending open problems. More recent models such as the multi-layer NSW or SGN systems, as well as the Isobe-Kakinuma equations are also reviewed under a unified formalism that should simplify comparisons. We also comment on the scalar versions of the various shallow water systems which can be used to describe unidirectional waves in horizontal dimension d = 1; among them are the KdV, BBM, Camassa-Holm and Whitham equations. Finally, we show how to take vorticity effects into account in shallow water modeling, with specific focus on the behavior of the turbulent terms. As examples of challenges that go beyond the present scope of mathematical justification, we review recent works using shallow water models with vorticity to describe wave breaking, and also derive models for the propagation of shallow water waves over strong currents.

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Section: 32mentioning

confidence: 99%

Section: 4mentioning

confidence: 99%

Modeling shallow water waves

Lannes

2020

Nonlinearity

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“…There are an infinite number of real positive roots ν j of (3.3) such that ν j ∼ 3j − 1/2 − √ 3/2πj as j → ∞. There are also two complex roots, approximately −1/2 ± i × 1.0714, that do not appear in the extreme wave (as they would yield an unbounded free surface); they do play a role in near extreme waves, however (Longuet-Higgins & Fox 1977). In order to derive expression (3.2), we use -cf.…”

Section: Stokes Extreme Wavementioning

confidence: 99%

“…Until recent years, most studies were based on a linear transfer function (Escher & Schlurmann 2008), primarily due to the intractability of the nonlinear governing equations, and even then for irrotational waves (cf. Henry & Thomas (2018) for linear, and weakly nonlinear, recovery formulae for waves with vorticity). However, the inadequacy of linear formulae for waves of even moderate amplitude is well known (Bishop & Donelan 1987;Tsai et al 2005), hence the importance of taking into account nonlinear effects.…”

Section: Introductionmentioning

confidence: 99%

Extreme water-wave profile recovery from pressure measurements at the seabed

Clamond

Henry

2020

J. Fluid Mech.

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“…Consequently, the investigation of the pressure field beneath a surface water wave is currently of great interest. This topic is covered by three papers in this issue: theoretical aspects are addressed in [36,37], while [38] presents experimental studies.…”

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confidence: 99%

Nonlinear water waves: introduction and overview

Constantin

2017

Phil. Trans. R. Soc. A.

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For more than two centuries progress in the study of water waves proved to be interdependent with innovative and deep developments in theoretical and experimental directions of investigation. In recent years, considerable progress has been achieved towards the understanding of waves of large amplitude. Within this setting one cannot rely on linear theory as nonlinearity becomes an essential feature. Various analytic methods have been developed and adapted to come to terms with the challenges encountered in settings where approximations (such as those provided by linear or weakly nonlinear theory) are ineffective. Without relying on simpler models, progress becomes contingent upon the discovery of structural properties, the exploitation of which requires a combination of creative ideas and state-of-the-art technical tools. The successful quest for structure often reveals unexpected patterns and confers aesthetic value on some of these studies. The topics covered in this issue are both multi-disciplinary and interdisciplinary: there is a strong interplay between mathematical analysis, numerical computation and experimental/field data, interacting with each other via mutual stimulation and feedback. This theme issue reflects some of the new important developments that were discussed during the programme ‘Nonlinear water waves’ that took place at the Isaac Newton Institute for Mathematical Sciences (Cambridge, UK) from 31st July to 25th August 2017. A cross-section of the experts in the study of water waves who participated in the programme authored the collected papers. These papers illustrate the diversity, intensity and interconnectivity of the current research activity in this area. They offer new insight, present emerging theoretical methodologies and computational approaches, and describe sophisticated experimental results. This article is part of the theme issue ‘Nonlinear water waves’.

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Prediction of the free-surface elevation for rotational water waves using the recovery of pressure at the bed

Cited by 22 publications

References 36 publications

Modeling shallow water waves

Modeling shallow water waves

Extreme water-wave profile recovery from pressure measurements at the seabed

Nonlinear water waves: introduction and overview

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