The mechanism of vortex connection at a free surface

Abstract

Vortex connections at the surface are fundamental and prominent features in free-surface vortical flows. To understand the detailed mechanism of such connection, we consider, as a canonical problem, the laminar vortex connections at a free surface when an oblique vortex ring impinges upon that surface. We perform numerical simulations of the Navier–Stokes equations with viscous free-surface boundary conditions. It is found that the key to understanding the mechanism of vortex connection at a free surface is the surface layers: a viscous layer resulting from the dynamic zero-stress boundary conditions at the free surface, and a thicker blockage layer which is due to the kinematic boundary condition at the surface. In the blockage layer, the vertical vorticity component increases due to vortex stretching and vortex turning (from the transverse vorticity component). The vertical vorticity is then transported to the free surface through viscous diffusion and vortex stretching in the viscous layer leading to increased surface-normal vorticity. These mechanisms take place at the aft-shoulder regions of the vortex ring. Connection at the free surface is different from that at a free-slip wall owing to the generation of surface secondary vorticity. We study the components of this surface vorticity in detail and find that the presence of a free surface accelerates the connection process. We investigate the connection time scale and its dependence on initial incidence angle, Froude and Reynolds numbers. It is found that a criterion based on the streamline topology provides a precise definition for connection time, and may be preferred over existing definitions, e.g. those based on free-surface elevation or net circulation.

Experiments on free-surface turbulence

“…The velocity profiles in spanwise and streamwise direction, in figure (a), show that the location where the angular velocity equals zero coincides with the point of lowest elevation, but that the vortex is slightly asymmetric. Profiles of the vorticity in (b) show that the vorticity distribution in the vortex is not completely cylindrically symmetric, and that, consequently, the point of maximum vorticity does not coincide with the minimum in elevation, as predicted by Zhang et al (1999). Figure (c) shows profiles (in spanwise direction) of the slope h y = ∂h/∂y derived from PIV by means of the Euler equations, equation (2.32), as well as the directly measured slope, showing that both are broadly the same.…”

“…By setting a threshold for the vorticity, Dabiri excludes smaller and less intense sub-surface vortices. As was already mentioned in chapter 2, Zhang et al (1999) have studied the interaction between a vortex-ring and a free surface by means of numerical simulations. They conclude that the correlation between the surface-normal component of vorticity and the surface elevation is relatively poor, unless the vortex associated with the depression in the surface has a cylindrically symmetric distribution of surface-normal vorticity.…”

“…Dabiri notes a strong (≈ 0.8) correlation between the magnitude of the vertical vorticity and the surface elevation. However, Zhang et al (1999) note that the locations of the maxima of vorticity at the surface and the mimimal surface elevation do not coincide, even for these very localised structures, unless the distribution of vertical vorticity at the surface associated with the structure, is cylindrically symmetric. They show this by calculating the free surface shape above a columnar vortex, the axis of which is not normal to the surface.…”

“…commonly used in order to study vortex (dis)connection, both in experiments, for instance by Bernal & Kwon (1989), Song et al (1992), Gharib & Weigand (1996) and Weigand (1996), and in numerical simulations by Zhang et al (1999) among others. As a vortex ring approaches a free surface, it too tends to break up into smaller vortex tubes that end at the surface.…”

Statistics of surface renewal of passive scalars in free-surface turbulence

“…Figure 5(c) shows an upwelling near a strong surface-connected vortex (for the dynamics of the vortex, see e.g. Handler et al 1993;Pan & Banerjee 1995;Zhang, Shen & Yue 1999). In this case, the advection induced by the surface-connected vortex pushes some fluid particles over the upwelling as well.…”