Interaction of a free-surface wave with a horizontal cylinder is examined at a sufficiently high value of Keulegan-Carpenter number such that well-defined concentrations of vorticity are generated during a typical wave cycle. The mutual interaction of these vorticity concentrations at varying degrees of submergence beneath the free-surface provides a rich array of space-time patterns of the instantaneous vorticity field. Irrespective of the particular pattern, however, central elements of the vortex formation process appear to persist. In essence, due to orbital motion of the wave, the evolution of a given pattern of vorticity follows a well-defined sequence: (i) growth of a pronounced region of vorticity having an identifiable extremum on the surface of the cylinder; (ii) migration of this vorticity concentration about the surface of the cylinder in a direction compatible with the orbital motion of the wave; and (iii) separation of the vorticity concentration from the surface of the cylinder. This process of growth and migration of a vorticity concentration can originate from three sites, i.e. approximately on the upper, right, and lower surfaces of the cylinder, during a single oscillation cycle.
The foregoing process appears to be largely controlled by the orbital motion of the wave. In fact, the form and orientation of the orbital trajectories of the eventually-formed concentrations of vorticity resemble those of the particle trajectories of the wave. Basically, two types of orbits of the vorticity concentrations are discernible. The first is a relatively small amplitude orbit; in the limiting case of such an orbit, abrupt reversal in direction of movement of the vorticity concentration occurs, followed by collision of the concentration with the surface of the cylinder for sufficiently deep submergence. This type of small amplitude orbital motion is shown to actually give rise to relatively large moments of vorticity and large contributions to the instantaneous forces on the cylinder, as deduced by application of a concept described by Lighthill (1986). The second type of orbital motion of the vorticity concentrations exhibits a large amplitude trajectory. At a sufficiently large value of nominal submergence, the vorticity concentration navigates one complete loop about the cylinder. This type of large amplitude orbit, however, generally gives rise to smaller moments of vorticity. This is due to the fact that the large amplitude orbit involves a relatively long time scale, and thereby the opportunity for substantial decay of circulation of the vortex.
The nominal submergence of the cylinder beneath the wave has a pronounced effect on the pattern of vorticity concentrations. It has been demonstrated that generic concentrations of vorticity are common to all patterns of vorticity at various depths of submergence. For relatively deep submergence, as previously noted, the separated vorticity concentrations exhibit large-scale orbital motion about the cylinder. For intermediate submergence, however, the presence of the free-surface exerts a significant influence, and this large-scale orbital motion is substantially hindered. Moreover, the distorted velocity field of the wave, due to proximity of the free-surface of the cylinder, promotes formation of elongated vorticity concentrations from the top of the cylinder and, in addition, significantly enhances the vorticity level and scale of the vorticity concentrations from the bottom surface of the cylinder. In the extreme case of shallow submergence, whereby the cylinder pierces the free-surface during the wave cycle, the presence of the free-surface induces a substantial lag of the process of vortex formation from the surface of the cylinder. In addition, vortex formation is induced from the free-surface during a portion of the wave cycle. These fascinating processes of vortex formation and interaction yield an ordered array of as many as six counterrotating vortices immediately beneath the free-surface.
Finally, it has been demonstrated that, at relatively large submergence, the space-time evolution of the vorticity field is associated with identifiable topological features of the instantaneous streamline patterns. Changes in the predominant orientation of the velocity field of the undisturbed portion of the free-surface wave, in conjunction with the defined patterns of positive and negative vorticity concentrations about the cylinder, can induce well-defined singularities in the form of foci and saddle points. Even though the pattern of vorticity concentrations undergoes a very mild change from one instant to the next during the wave motion, nearly discontinuous changes in the streamline topology can occur. Such changes include, for example, abrupt switching of the location of a saddle point from one region of the flow to another. These changes are apparently due to the fact that the predominant direction of the velocity field of the wave varies substantially during the wave cycle.