A Model Equation for Wavepacket Solitary Waves Arising from Capillary‐Gravity Flows

Akers, Benjamin F.; Milewski, Paul A.

doi:10.1111/j.1467-9590.2009.00432.x

Cited by 50 publications

(50 citation statements)

References 19 publications

(42 reference statements)

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“…In the analysis of their experiments adjusted a simple one-way model proposed in Akers & Milewski (2009) by adding forcing and a viscous damping. The more accurate cDtNE equations can be modified to include small viscous damping effects by using the approximation presented in Dias et al (2008) whereby dissipation is modelled through the modification of both kinematic and dynamic boundary conditions: η t − G 0 ξ = 2Re −1 ∆η + (G 1 + G 2 )ξ (4.1) ξ t + (1 − ∆)η = 2Re −1 ∆ξ + P (x, y, t) + ∇ · ∇η 1 + |∇η| 2 − ∇η…”

Section: Extensionsmentioning

confidence: 99%

“…Line solitary waves are unstable to transverse perturbations of sufficiently long wavelength. Fully 2D dynamics have been considered by Akers & Milewski (2010) in a one-way simplified model and in a quadratic isotropic model in Akers & Milewski (2009). The main goal of this paper is to study 2D dynamics within a close approximation of the Euler equations.…”

Section: Introductionmentioning

confidence: 99%

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Dynamics of gravity–capillary solitary waves in deep water

Wang

Milewski

2012

J. Fluid Mech.

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The dynamics of solitary gravity-capillary water waves propagating on the surface of a three-dimensional fluid domain is studied numerically. In order to accurately compute complex time dependent solutions, we simplify the full potential flow problem by taking a cubic truncation of the scaled Dirichlet-to-Neumann operator for the normal velocity on the free surface. This approximation agrees remarkably well with the full equations for the bifurcation curves, wave profiles and the dynamics of solitary waves for a twodimensional fluid domain. Fully localised solitary waves are then computed in the threedimensional problem and the stability and interaction of both line and localized solitary waves are investigated via numerical time integration of the equations. The solitary wave branches are indexed by their finite energy at small amplitude, and the dynamics of the solitary waves is complex involving nonlinear focussing of wave packets, quasi-elastic collisions, and the generation of propagating, spatially localised, time-periodic structures (breathers).

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Section: Extensionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Dynamics of gravity–capillary solitary waves in deep water

Wang

Milewski

2012

J. Fluid Mech.

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“…Finally, combining Eq. (2) with a moving pressure forcing, Apð ; yÞ, ¼ x þ t, and assuming quadratic nonlinearity [19], the following equation for the free-surface elevation ð ; y; tÞ is obtained:…”

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confidence: 99%

“…On kinematic grounds [18], a forcing disturbance moving along x with speed near c min generates waves with wave vectors k ¼ ðk; lÞ close to k min ¼ ð1; 0Þ. For a leftgoing source as in the experiment, expanding the disper- sion relation (1) to second order around k min [19] and adding an imaginary part representing the wave decay rate due to viscous damping [20] yields…”

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Gravity-Capillary Lumps Generated by a Moving Pressure Source

et al. 2009

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The nonlinear wave pattern generated by a localized pressure source moving over a liquid free surface at speeds below the minimum phase speed (c min ) of linear gravity-capillary waves is investigated experimentally and theoretically. At these speeds, freely propagating fully localized solitary waves, or ''lumps,'' are known theoretically to be possible. For pressure-source speeds far below c min , the surface response is a local depression similar to the case with no forward speed. As the speed is increased, a critical value is reached c c % 0:9c min where there is an abrupt transition to a wavelike state that features a steady disturbance similar to a steep lump behind the pressure forcing. As the speed approaches c min , a second transition is found; the new state is unsteady and is characterized by continuous shedding of lumps from the tips of a V-shaped pattern. DOI: 10.1103/PhysRevLett.103.214502 PACS numbers: 47.35.Fg, 47.10.AÀ, 47.20.Ky Solitary waves are prominent nonlinear features of dispersive wave systems in various areas of physics [1][2][3]. Many key advances in understanding solitary waves were initiated by studies of waves on the free surface of a liquid. Most attention has been paid to solitary waves of the Korteweg-De Vries type [4,5], which bifurcate in the shallow-water limit. A different class of solitary waves, however, is possible on deep water [6][7][8][9] in the presence of gravity and surface tension; they bifurcate from linear sinusoidal waves at the minimum gravity-capillary phase speed, c min ¼ ffiffiffi 2 p ð g= Þ 1=4 , and propagate at speeds less than c min (g is the gravitational acceleration, the fluid density, and the surface tension). In dimensionless form, using L ¼ ð = gÞ 1=2 as the length scale and L=c min as the time scale, the dispersion relation for waves with frequency ! and wave number k isand the minimum of the phase speed !=k occurs at k ¼ k min ¼ 1. [14,15] and are stable at finite steepness [16]. On the experimental side, 2D steep-depression solitary waves have been generated by using an air jet from a narrow slit as forcing over a current with speed below c min [17]; however, persistent unsteadiness of the response due to cross-stream disturbances was also noted. In this Letter, we report on an experimental and theoretical investigation of gravity-capillary lumps generated by a localized pressure source moving at speeds below c min .The experiments were carried out in a water tank 7.3 m long and 76 cm wide with water depth of approximately 60 cm. The tank has glass walls and bottom for optical access. The surface tension was measured in situ with a Willhelmy plate and remained at ¼ 73 dyn=cm throughout the experiments. A pipe with inner diameter D ¼ 2:5 mm was mounted vertically to a carriage that rides on top of the tank. The open end of the pipe was positioned 1 cm above the water surface. The carriage was towed at various speeds by a servomotor with a precision of better than 0.3%. We define a speed parameter, ¼ c=c min , where c is the speed of the pipe-...

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“…For a three-dimensional fluid domain, fully localized two-dimensional steadily travelling waves, often called lumps, were first computed in the fluid equations by Pȃrȃu et al [9], with related work in [10][11][12] on reduced models. These localized states are predicted to be unstable at small amplitude based on the behaviour of the two-dimensional focusing nonlinear Schrödinger equation which approximates these solutions (see below).…”

Section: Introductionmentioning

confidence: 99%

Transversally periodic solitary gravity–capillary waves

Milewski

Wang

2014

Proc. R. Soc. A.

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When both gravity and surface tension effects are present, surface solitary water waves are known to exist in both two-and three-dimensional infinitely deep fluids. We describe here solutions bridging these two cases: travelling waves which are localized in the propagation direction and periodic in the transverse direction. These transversally periodic gravity-capillary solitary waves are found to be of either elevation or depression type, tend to plane waves below a critical transverse period and tend to solitary lumps as the transverse period tends to infinity. The waves are found numerically in a Hamiltonian system for water waves simplified by a cubic truncation of the Dirichlet-to-Neumann operator. This approximation has been proved to be very accurate for both two-and three-dimensional computations of fully localized gravity-capillary solitary waves. The stability properties of these waves are then investigated via the time evolution of perturbed wave profiles.

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A Model Equation for Wavepacket Solitary Waves Arising from Capillary‐Gravity Flows

Cited by 50 publications

References 19 publications

Dynamics of gravity–capillary solitary waves in deep water

Dynamics of gravity–capillary solitary waves in deep water

Gravity-Capillary Lumps Generated by a Moving Pressure Source

Transversally periodic solitary gravity–capillary waves

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