Numerical studies of two-dimensional hydroelastic periodic and generalised solitary waves

Gao, Tao; Vanden‐Broeck, J.‐M.

doi:10.1063/1.4893677

Cited by 19 publications

(20 citation statements)

References 17 publications

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“…From then on, analytical and numerical investigations of this new model have been gradually carried out. Of note are the works of Toland (2008), who rigorously proved the existence of periodic hydroelastic waves, Guyenne & Pȃrȃu (2012), who discovered that both elevation and depression branches exist below the minimum of the phase speed at finite amplitude in deep water, Wang, Vanden-Broeck & Milewski (2013), who extended the branch of elevation solitary waves to the highly nonlinear regime with the wave profiles featuring multi-packet structure and computed periodic waves with an overhanging structure, Gao & Vanden-Broeck (2014), who investigated generalised solitary waves extensively, and Page & Pȃrȃu (2014), who considered nonlinear hydroelastic hydraulic falls past a submerged bottom obstruction.…”

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confidence: 99%

New hydroelastic solitary waves in deep water and their dynamics

2016

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A numerical study of fully nonlinear waves propagating through a two-dimensional deep fluid covered by a floating flexible plate is presented. The nonlinear model proposed by Toland (Arch. Rat. Mech. Anal., vol. 289, 2008, pp. 325-362) is used to formulate the pressure exerted by the thin elastic sheet. The symmetric solitary waves previously found by Guyenne & Pȃrȃu (J. 750-761) are briefly reviewed. A new class of hydroelastic solitary waves which are non-symmetric in the direction of wave propagation is then computed. These asymmetric solitary waves have a multi-packet structure and appear via spontaneous symmetry-breaking bifurcations. We study in detail the stability properties of both symmetric and asymmetric solitary waves subject to longitudinal perturbations. Some moderateamplitude symmetric solitary waves are found to be stable. A series of numerical experiments are performed to show the non-elastic behaviour of two interacting stable solitary waves. The large response generated by a localised steady pressure distribution moving at a speed slightly below the minimum of the phase speed (called the transcritical regime in the literature) is also examined. The direct numerical simulation of the fully nonlinear equations with a single load reveals that in this range the generated waves are of finite amplitude. This includes a perturbed depression solitary wave, which is qualitatively similar to the large response observed in experiments. The excitations of stable elevation solitary waves are achieved by applying multiple loads moving with a speed in the transcritical regime.

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confidence: 99%

New hydroelastic solitary waves in deep water and their dynamics

2016

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“…To carry out a more rigorous investigation on whether true elevation solitary waves exist in the presence of a normal electric field, we follow [13,30] to perform a numerical investigation by monitoring the curvature of the solution at the right end of the computational domain, denoted by κ 0 , for a fixed τ and various E b . As can be seen clearly from Fig.…”

Section: Travelling Wavesmentioning

confidence: 99%

Capillary–gravity waves on a dielectric fluid of finite depth under normal electric field

Gao

Doak

Vanden‐Broeck

et al. 2019

European Journal of Mechanics - B/Fluids

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a b s t r a c tIn this work we consider two-dimensional capillary-gravity waves propagating under the influence of a vertical electric field on a dielectric of finite depth bounded above by a perfectly conducting and hydrodynamically passive fluid. Both linear and weakly nonlinear theories are developed, and long-wave model equations are derived based on the analyticity of the Dirichlet-Neumann operator. Fully nonlinear computations are carried out by using a time-dependent conformal mapping method. Solitary waves are found, and their stability characteristics subject to longitudinal perturbations are studied numerically. The shedding of stable solitary waves is achieved by moving a Gaussian pressure on the free surface with the speed close to a phase speed minimum and removing the pressure after a period of time. The novel result shows that a depression bright solitary wave and an elevation generalized solitary wave co-exist in the solitary-wave excitation.

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“…14 Milewski et al 15 computed solitary waves in deep water using the same model for ice and performed direct time-dependent computations based on conformal mapping techniques. The fully nonlinear model for ice was considered in Guyenne and Părău 16,17 when computing solitary waves and by Gao and Vanden-Broeck 18 for both periodic and solitary waves. A more general discussion considering periodic interfacial waves with and without mass is seen in Akers et al 9,19 where a different parameterization of the problem was considered.…”

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confidence: 99%

Stability of periodic traveling flexural‐gravity waves in two dimensions

et al. 2018

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In this work, we solve the Euler's equations for periodic waves traveling under a sheet of ice. These waves are referred to as flexural‐gravity waves. We compare and contrast two models for the effect of the ice: a linear model and a nonlinear model. The benefit of this reformulation is that it facilitates the asymptotic analysis. We use it to derive the nonlinear Schrödinger equation that describes the modulational instability of periodic traveling waves. We compare this asymptotic result with the numerical computation of stability using the Fourier–Floquet–Hill method to show they agree qualitatively. We show that different models have different stability regimes for large values of the flexural rigidity parameter. Numerical computations are also used to analyze high‐frequency instabilities in addition to the modulational instability. In the regions examined, these are shown to be the same regardless of the model representing ice.

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Numerical studies of two-dimensional hydroelastic periodic and generalised solitary waves

Cited by 19 publications

References 17 publications

New hydroelastic solitary waves in deep water and their dynamics

New hydroelastic solitary waves in deep water and their dynamics

Capillary–gravity waves on a dielectric fluid of finite depth under normal electric field

Stability of periodic traveling flexural‐gravity waves in two dimensions

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