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

Stagnation points beneath solitary gravity-capillary waves in the weakly nonlinear weakly dispersive regime in a sheared channel with finite depth and constant vorticity are investigated. A Korteweg-de Vries equation that incorporates the surface tension and the vorticity effects is obtained asymptotically from the full Euler equations. The velocity field in the bulk fluid is approximated which allow us to compute stagnation points in the solitary wave moving frame. We show that stagnation points bellow the crest of elevation solitary waves exist for large values of the vorticity and Bond numbers less than a critical value that depends on the vorticity. Remarkably, the positions of these stagnation points do not depend on the surface tension. Besides, we show that when there are two stagnation points located at the bottom of the channel, they are pulled towards the horizontal coordinate of the solitary wave crest as the Bond number increases until its critical value.

Complex flow structures beneath rotational depression solitary waves in gravity-capillary flows

Flow structure beneath periodic waves with constant vorticity under strong horizontal electric fields

The motion of an interface separating two fluids under the effect of electric fields is a subject that has picked the attention of researchers from different areas. While there is an abundance of studies investigating the free surface wave properties, very few works have examined the associated velocity field within the bulk of the fluid. Therefore, in this paper, we investigate numerically the flow structure beneath solitary waves with constant vorticity on an inviscid conducting fluid bounded above by a dielectric gas under normal electric fields in the framework of a weakly nonlinear theory. Elevation and depression solitary waves with constant vorticity are computed by a pseudo-spectral method, and a parameter sweep on the intensity of the electric field is carried out to study its role in the appearance of stagnation points. We find that for elevation solitary waves, the location of stagnation points does not change significantly with a variation of the electric field. For depression solitary waves, on the other hand, the electric field acts as a catalyzer that makes possible the appearance of stagnation points. In the sense that in its absence, there are no stagnation points.

This paper considers permanent capillary–gravity waves on the free surface of a two–dimensional incompressible inviscid fluid of finite depth. It is shown that there are no solitary-wave solutions of the exact governing equations of the flow that decay to zero exponentially at infinity if the surface tension coefficient is less than its critical value and lies in some intervals. The proof is based upon an estimate of a constant that is related to the approximation of the solution, if it exists, near its singularity. The approximation satisfies a fourth–order nonlinear ordinary differential equation when the solution is extended to the complex plane. Then the non–existence of truly solitary waves is obtained by using a contradiction on this constant.

While some studies have investigated the particle trajectories and stagnation points beneath solitary waves with constant vorticity, little is known about the pressure beneath such waves. To address this gap, we investigate numerically the pressure beneath solitary waves in flows with constant vorticity. Through a conformal mapping that flats the physical domain, we develop a numerical approach that allows us to compute the pressure and the velocity field in the fluid domain. Our experiments indicate that there exists a threshold vorticity such that pressure anomalies and stagnation points occur when the intensity of the vorticity is greater than this threshold. Above this threshold, the pressure on the bottom boundary has two points of local maxima and there are three stagnation points in the flow, and below it the pressure has one local maximum and there is no stagnation point.

Particle paths beneath small amplitude periodic forced waves in a shallow water channel are investigated. The problem is formulated in the forced Korteweg-de Vries equation framework which allows to approximate the velocity field in the bulk fluid. We show that the flow can have zero, one or three stagnation points. Besides, differently from the unforced problem, stagnation points can arise for small values of the vorticity as long as the moving disturbance travels sufficiently fast.

In this paper we study the water wave problem with capillary effects and constant vorticity when stagnation points are not excluded. When the constant vorticity is close to certain critical values we show that there exist Wilton ripples solutions of the water wave problem with two crests and two troughs per minimal period. They form smooth secondary bifurcation curves that emerge from primary bifurcation branches that contain a laminar flow solution and consist of symmetric waves of half of the period of the Wilton ripples, at some nonlaminar solution. We also prove that any Wilton ripple contains an internal critical layer provided its minimal period is sufficiently small.

Hydrodynamic interaction between bodies in a perfect incompressible fluid and their motion in nonuniform streams: PMM vol. 37, n≗4, 1973, pp. 680–689

Waves with constant vorticity and electrohydrodynamics flows are two topics in fluid dynamics that have attracted much attention from scientists for both the mathematical challenge and their industrial applications. Coupling of electric fields and vorticity is of significant research interest. In this paper, we study the flow structure of steady periodic traveling waves with constant vorticity on a dielectric fluid under the effect of normal electric fields. Through the conformal mapping technique combined with pseudo-spectral numerical methods, we develop an approach that allows us to conclude that the flow can have zero, two, or three stagnation points according to variations in the voltage potential. We describe in detail the recirculation zones that emerge together with the stagnation points. In addition, we show that the number of local maxima of the pressure on the bottom boundary is intrinsically connected to the saddle points.

Solitary waves with constant vorticity in water of finite depth are calculated numerically by a boundary integral equation method. Previous calculations are confirmed and extended. It is shown that there are branches of solutions which bifurcate from a uniform shear current. Some of these branches are characterized by a limiting configuration with a 120° angle at the crest of the wave. Other branches extend for arbitrary large values of the amplitude of the wave. The corresponding solutions ultimately approach closed regions of constant vorticity in contact with the bottom of the channel. A numerical scheme is presented to calculate directly these closed regions of constant vorticity. In addition, it is shown that there are branches of solutions which do not bifurcate from a uniform shear flow.

On the symmetry of equatorial travelling water waves with constant vorticity and stagnation points

We present a numerical study of essentially nonlinear dynamics of surface gravity waves on deep water with constant vorticity using governing equations in conformal coordinates. The dispersion relation of surface gravity waves on shear flow is known to have two branches; one of which is weakly dispersive for long waves. Weakly nonlinear evolution of the waves of this branch can be described by the Benjamin–Ono equation, which is integrable and has soliton and multi-soliton solutions. Currently, the extent to which the properties of such solitary waves obtained within the weakly nonlinear model are preserved in the exact Euler equations is unknown. We investigate the behavior of this class of solitary waves without the restrictive assumption of weak nonlinearity by using the exact Euler equations. The evolution of localized initial perturbations leading to the formation of single or multiple solitary waves is modeled, and the properties of finite-amplitude solitary waves are discussed. We show that within the framework of the exact equations, two-soliton collisions are almost elastic, but in contrast to solutions of the Benjamin–Ono equation, the waves receive a phase shift as a result of the interaction.

Abstract

The purpose of this work is to explore in detail the structure of the interior flow generated by periodic surface waves on a fluid with constant vorticity. The problem is mapped conformally to a strip and solved numerically using spectral methods. Once the solution is known, the streamlines, pressure and particle paths can be found and mapped back to the physical domain. We find that the flow beneath the waves contains zero, one, two or three stagnation points in a frame moving with the wave speed, and describe the bifurcations between these flows. When the vorticity is sufficiently strong, the pressure in the flow and on the bottom boundary also has very different features from the usual irrotational wave case.