Transversally periodic solitary gravity-capillary waves (original) (raw)

Dynamics of gravity–capillary solitary waves in deep water

Journal of Fluid Mechanics, 2012

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 using surface variables and taking a particular cubic truncation possessing a Hamiltonian with desirable properties. This approximation agrees remarkably well with the full equations for the bifurcation curves, wave profiles and the dynamics of solitary waves for a two-dimensional fluid domain, and with higher-order truncations in three dimensions. Fully localized solitary waves are then computed in the three-dimensional problem and the stability and interaction of both line and localized solitary waves are investigated via numerical time integration of the equations. There are many solitary wave branches, indexed by their finite energy as their amplitude tends to zero. The dynamics of the solitary waves is complex, involving nonline...

Dynamics of Three-Dimensional Gravity-Capillary Solitary Waves in Deep Water

SIAM Journal on Applied Mathematics, 2010

A model equation for gravity-capillary waves in deep water is proposed. This model is a quadratic approximation of the deep water potential flow equations and has wavepacket-type solitary wave solutions. The model equation supports line solitary waves which are spatially localized in the direction of propagation and constant in the transverse direction, and lump solitary waves which are spatially localized in both directions. Branches of both line and lump solitary waves are computed via a numerical continuation method. The stability of each type of wave is examined. The transverse instability of line solitary waves is predicted by a similar instability of line solitary waves in the nonlinear Schrödinger equation. The spectral stability of lumps is predicted using the waves' speed energy relation. The role of wave collapse in the stability of these waves is also examined. Numerical time evolution is used to confirm stability predictions and observe dynamics, including instabilities and solitary wave collisions. SOLITARY GRAVITY-CAPILLARY WAVES 2391

Fully nonlinear gravity-capillary solitary waves in a two-fluid system of finite depth

Journal of engineering mathematics, 2002

Large-amplitude waves at the interface between two laminar immisible inviscid streams of different densities and velocites, bounded together in a straight infinite channel are studied, when surface tension and gravity are both present. A long-wave approximation is used to develop a theory for fully nonlinear interfacial waves allowing amplitudes as large as the channel thickness. The result is a set of evolution equations for the interfacial shape and the velocity jump across it. Traveling waves of permanent form are studied and it is shown that solitary waves are possible for a range of physical parameters. All solitary waves can be expressed implicitly in terms of incomplete elliptic integrals of the third kind. When the upper layer has zero density, two explicit solitary-wave solutions have been found whose amplitudes are equal to h/4 or h/9, where 2h is the channel thickness. In the absence of gravity solitary waves are not possible but periodic ones are. Numerically constructed solitary waves are given for representative physical parameters.

Three-dimensional Localized Solitary Gravity-Capillary Waves

Communications in Mathematical Sciences, 2005

In a weakly nonlinear model equation for capillary-gravity water waves on a twodimensional free surface, we show, numerically, that there exist localized solitary traveling waves for a range of parameters spanning from the long wave limit (with Bond number B > 1/3, in the regime of the Kadomtsev-Petviashvilli-I equation) to the wavepacket limit (B < 1/3, in the Davey-Stewartson regime). In fact, we show that these two regimes are connected with a single continuous solution branch of nonlinear localized solitary solutions crossing B = 1/3.

Nonlinear three-dimensional gravity–capillary solitary waves

Journal of Fluid Mechanics, 2005

Steady three-dimensional fully nonlinear gravity-capillary solitary waves are calculated numerically in infinite depth. These waves have decaying oscillations in the direction of propagation and monotone decay perpendicular to the direction of propagation. They travel at a velocity U smaller than the minimum velocity c min of linear gravity-capillary waves. It is shown that the structure of the solutions in three dimensions is similar to that found by Vanden-Broeck & Dias (J. Fluid Mech. vol. 240, 1992, pp. 549-557) for the corresponding two-dimensional problem.

A plethora of generalised solitary gravity–capillary water waves

Journal of Fluid Mechanics, 2015

The present study describes, first, an efficient algorithm for computing solutions in terms of capillary–gravity solitary waves of the irrotational Euler equations with a free surface and, second, provides numerical evidences of the existence of an infinite number of generalised solitary waves (solitary waves with undamped oscillatory wings). Using conformal mapping, the unknown fluid domain, which is to be determined, is mapped into a uniform strip of the complex plane. In the transformed domain, a Babenko-like equation is then derived and solved numerically.

On weakly nonlinear gravity–capillary solitary waves

Wave Motion, 2012

As a weakly nonlinear model equations system for gravity-capillary waves on the surface of a potential fluid flow, a cubic-order truncation model is presented, which is derived from the ordinary Taylor series expansion for the free boundary conditions of the Euler equations with respect to the velocity potential and the surface elevation. We assert that this model is the optimal reduced simplified model for weakly nonlinear gravity-capillary solitary waves mainly because the generation mechanism of weakly nonlinear gravity-capillary solitary waves from this model is consistent with that of the full Euler equations, both quantitatively and qualitatively, up to the third order in amplitude.

Capillary‐gravity solitary waves on water of finite depth interacting with a linear shear current

Studies in Applied Mathematics, 2021

The problem of two-dimensional capillary-gravity waves on an inviscid fluid of finite depth interacting with a linear shear current is considered. The shear current breaks the symmetry of the irrotational problem and supports simultaneously counter-propagating waves of different types: KdV-type long solitary waves and wave-packet solitary waves whose envelopes are associated with the nonlinear Schrödinger equation. A simple intuition for the broken symmetry is that the current modifies the Bond number differently for left-and right-propagating waves. Weakly nonlinear theories are developed in general and for two particular resonant cases: the case of second harmonic resonance and long-wave/short-wave interaction. Travelling-wave solutions and their dynamics in the full Euler equations are computed numerically using a time-dependent conformal mapping technique, and compared to some weakly nonlinear solutions. Additional attention is paid to branches of elevation generalized solitary waves of KdV type: although true embedded solitary waves are not detected on these branches, it is found that periodic wave-trains on their tails can be arbitrarily small as the vorticity increases. Excitation of waves by moving pressure distributions and modulational instabilities of the periodic waves in the resonant cases described above are also examined by the fully nonlinear computations.

A NOTE ON SOLITARY WAVES WITH VARIABLE SURFACE TENSION IN WATER OF INFINITE DEPTH

Two-dimensional gravity-capillary solitary waves propagating at the surface of a fluid of infinite depth are considered. The effects of gravity and of variable surface tension are included in the free-surface boundary condition. The numerical results extend the constant surface tension results of Vanden-Broeck and Dias to situations where the surface tension varies along the free surface. 2000 Mathematics subject classification: primary 74J35, 76B45, 76D45, 76B15.

Dynamics of fully nonlinear capillary–gravity solitary waves under normal electric fields

Journal of Engineering Mathematics, 2017

Two-dimensional capillary-gravity waves travelling under the effect of a vertical electric field are considered. The fluid is assumed to be a dielectric of infinite depth. It is bounded above by another fluid which is hydrodynamically passive and perfectly conducting. The problem is solved numerically by time-dependent conformal mapping methods. Fully nonlinear waves are presented, and their stability and dynamics are studied. Keywords Surface wave • Solitary wave • Wave interactions 1 Introduction Water waves propagating on the interface between two fluids have been studied intensively using either analytical or numerical methods. Many different mathematical methods have been introduced to study the steady or timedependent solutions both in shallow and deep waters (for review, see, e.g. [1,2] and the references therein). In the case of deep water, it is well acknowledged that there exist two families of capillary-gravity solitary waves bifurcating from the minimum of phase speed-denoted elevation and depression waves. In [3], the stability of these waves was studied using a numerical spectral analysis. It was found that depression waves with single-valued profiles were stable, whereas there was a stability exchange on the branch of elevation waves. These results were later verified numerically by Milewski et al. [1]. Recently, the problem of dynamics and stability was investigated systematically by Wang [4] where the depression waves with overhanging structure were proved to be also stable. On the experimental side, early experiments on three-dimensional capillary-gravity waves in a wind-wave tank were carried out by Zhang [5]. Fully localised lumps were observed. Later wavepacket solitary waves were generated in deep water by Diorio et al. [6]. In the presence of electric fields, this topic attracted much attention because it has many physical and industrial applications such as cooling systems and coating processes. In [7,8], capillary waves on a fluid sheet under the effects