Solitary waves on constant vorticity flows with an interior stagnation point

Kozlov, Vladimir; Kuznetsov, N.; Lokharu, Evgeniy

doi:10.1017/jfm.2020.647

Cited by 13 publications

(12 citation statements)

References 46 publications

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“…It is well-known that the flow beneath a rotational water wave with constant vorticity is characterized mainly by: i) the appearance of stagtnation points -fluid particles with zero velocity in the wave's moving frame; ii) critical layers -a horizontal layer with closed streamlines separating the fluid into two disjoint regions; iii) the arising of pressure anomalies such as the occurrence of maxima and minima of pressure in locations other than below the crest and trough respectively. These results have been showed numerically [5,27,29,30], asymptotically [3,20] and also proved rigorously [14,33,24]. The constant vorticity also affects the shape of the free surface, for instance its profile can become rounder and possibly form an exotic overhanging wave [31,32,9,10].…”

Section: Introductionmentioning

confidence: 53%

Solitary waves on flows with an exponentially sheared current and stagnation points

Flamarion¹,

Ribeiro-Jr²

2022

Preprint

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While several articles have been written on water waves on flows with constant vorticity, little is known about the extent to which a nonconstant vorticity affects the flow structure, such as the appearance of stagnation points. In order to shed light on this topic, we investigate in detail the flow beneath solitary waves propagating on an exponentially decaying sheared current. Our focus is to analyse numerically the emergence of stagnation points. For this purpose, we approximate the velocity field within the fluid bulk through the classical Korteweg-de Vries asymptotic expansion and use the Matlab language to evaluate the resulting streamfunction. Our findings suggest that the flow beneath the waves can have zero, one or two stagnation points in the fluid body. We also study the bifurcation between these flows. Our simulations indicate that the stagnation points emerge from a streamline with a sharp corner.

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Section: Introductionmentioning

confidence: 53%

Solitary waves on flows with an exponentially sheared current and stagnation points

Flamarion¹,

Ribeiro-Jr²

2022

Preprint

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“…Both authors used in their works the Dubreil-Jacontin transformation which presupposes the nonexistence of stagnation points -fluid particles with zero velocity in the wave's moving frame. Only recently, Kozlov et al [2020] proved the existence of solitary waves for the Euler equations in flows with constant vorticity allowing stagnation points within the fluid bulk. However, asymptotic works from the 1980s have indicated the existence of solitary waves in flows with stagnation points in the interior Johnson [1986].…”

Section: Introductionmentioning

confidence: 99%

Numerical stability of solitary waves in flows with constant vorticity for the Euler equations

M.¹,

Flamarion²,

Ribeiro-Jr³

2022

Preprint

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The study of the Euler equations in flows with constant vorticity has piqued the curiosity of a considerable number of researchers over the years. Much research has been conducted in this subject under the assumption of steady flow. In this work, we provide a numerical approach that allows to compute solitary waves in flows with constant vorticity and analyse their stability. Through a conformal mapping technique, we compute solutions of the steady Euler equations, then feed them as initial data for the time-dependent Euler equations. We focus on analysing to what extent the steady solitary waves are stable within the time-dependent framework. Our numerical simulations indicate that although it is possible to compute solitary waves for the steady Euler equations in flows with large values of vorticity, such waves are not numerically stable for vorticities with absolute value much greater than one. Besides, we notice that large waves are unstable even for small values of vorticity.

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“…Very recently, overturning periodic waves with constant vorticity have been constructed with weak gravity and large or infinite depth by perturbing a family of explicit solutions with zero gravity [HW21]. Unlike in the periodic case, small-amplitude solitary waves cannot have stagnation points, but see [KKL20].…”

Section: Introductionmentioning

confidence: 99%

Large-amplitude steady solitary water waves with constant vorticity

Haziot¹,

Wheeler²

2021

Preprint

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This paper considers two-dimensional steady solitary waves with constant vorticity propagating under the influence of gravity over an impermeable flat bed. Unlike in previous works on solitary waves, we allow for both internal stagnation points and overhanging wave profiles. Using analytic global bifurcation theory, we construct continuous curves of large-amplitude solutions. Along these curves, either the wave amplitude approaches the maximum possible value, the dimensionless wave speed becomes unbounded, or a singularity develops in a conformal map describing the fluid domain. We also show that an arbitrary solitary wave of elevation with constant vorticity must be supercritical. The existence proof relies on a novel reformulation of the problem as an elliptic system for two scalar functions in a fixed domain, one describing the conformal map of the fluid region and the other the flow beneath the wave.

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Solitary waves on constant vorticity flows with an interior stagnation point

Abstract: Abstract

Cited by 13 publications

References 46 publications

Solitary waves on flows with an exponentially sheared current and stagnation points

Solitary waves on flows with an exponentially sheared current and stagnation points

Numerical stability of solitary waves in flows with constant vorticity for the Euler equations

Large-amplitude steady solitary water waves with constant vorticity

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