Figures 2a and b show the time evolution of patterns of vorticity. In addition, Figure 2a exhibits corresponding patterns of instantaneous velocity and superposed streamlines, and Figure 2b gives close-ups of the patterns of velocity vectors near key vorticity concentrations. Contours of positive and negative vorticity are designated respectively by red and yellow lines. The time sequence of images is represented by excerpts from the sequence of frames N = 7 through 19, which extend over nearly one complete cycle of the wave motion. At the instants corresponding approximately to frames N = 7 and 15, the free-surface attains its maximum and minimum elevations respectively.
Frame N = 7 shows three major concentrations of vorticity. Negative vortex A is shed from the top of the cylinder, while positive vortices B and D develop respectively along the right and bottom sides of the cylinder. Positive vortex C was generated during the previous cycle of the wave motion and is located above the cylinder. The corresponding image of the velocity field at N = 7 shows that the instantaneous direction of the wave is essentially from left to right, i.e., the velocity vectors in the region of the undistorted wave away from the cylinder are horizontal and essentially parallel to the free-surface. The locus of vortex A is clearly identifiable in the instantaneous streamline pattern, which takes the form of a limit cycle. A saddle point (locus of intersecting streamlines) is located immediately to the right of this limit cycle. Moreover, an alleyway flow occurs in the region between: the right half of the cylinder; and the streamline that leads to the saddle point. A localized separation bubble is discernible along the right surface of the cylinder; it corresponds to the vorticity concentration B along that surface.
At a later instant, corresponding to frame N = 9, further development of vortices A through D is indicated. Vortex A has approximately attained its maximum displacement from the surface of the cylinder, vortices B and D have moved clockwise around the cylinder and vortex C has translated horizontally to the right. At this instant, the velocity field of the undisturbed portion of the wave is oriented in the downward, nearly vertical direction. The large-scale negative vortex A is again clearly defined by the pattern of streamlines. It exhibits an outward-spiral from its center, corresponding to an unstable focus. In contrast to the streamline topology at N = 7, the saddle point has now switched to the left side of vortex A. An alleyway flow exists between the top surface of the cylinder and the streamline connected to the saddle point. This alleyway flow is oriented in the leftward direction, which is compatible with the outward-spiraling motion of the velocity field and streamline pattern associated with vortex A.
At the instant corresponding to the frame N = 11, vortex A has moved back towards the cylinder and is about to collide with its surface and undergo severe distention. The vorticity concentration B indicated in frame N = 9 has further moved in the clockwise direction about the cylinder, separated from its surface and joined with the vorticity concentration D immediately adjacent to the left side of the cylinder. The consequence is formation of a new, two-part vorticity concentration, henceforth designated as B,D. Vortex C has moved still further to the right, and its peak vorticity level has decreased. The corresponding velocity-streamline pattern indicates that the direction of the undisturbed portion of the wave motion is downward to the left. The identities of vortex A and all remaining concentrations of vorticity are not indicated by the streamline topology.
Further development of the pattern of vorticity at frame N = 13 (not shown herein) involves interaction of vortex A with the cylinder and its severe distension about the cylinder surface. At a substantially later instant of time, represented by frame N = 15 (top of Figure 2b), a remnant of vortex A is evident, but the most predominant features are the large-scale vortex B,D that has separated from the surface of the cylinder and, simultaneously, onset of a vorticity concentration A' immediately adjacent to the left side of the cylinder. Vortex C has continued its orbital-like motion about the cylinder and translated downward relative to its position in frame N = 11. At this instant, N = 15, the free-surface attains its minimum elevation corresponding to the trough of the free-surface wave. The corresponding image of the velocity pattern shows an enlarged view of the jet-like flow between the counterrotating vortex system B,D and A'.
In frame N = 17, the large-scale positive vortex B,D has translated up and to the left of the cylinder. Vortex A¢ has continued to increase in scale while continuing to move in the clockwise direction about the surface of the cylinder and vortex C exhibits a further decrease in maximum vorticity level. The corresponding image of the velocity field again focuses on the jet-like flow associated with the system B,D and A'.
Finally, frame shows the continued upward movement of vorticity concentration B, along with the matured development of vortex A', which has migrated further in the clockwise direction about the surface of the cylinder. Furthermore, a new positive vorticity concentration B' develops along the lower right surface of the cylinder. The zoomed-in view of the velocity field reveals the large velocity gradient associated with the development of vortex A¢ in the near-wake of the cylinder. This region of high gradient is coincident with the extremum of vorticity concentration A'.
The trajectories of the major vorticity concentrations are shown on the xy plane in Figure 3. These diagrams were obtained from the entire succession of PIV images over approximately one cycle of the wave motion. The position of each vorticity concentration was determined by locating the coordinates of the maximum absolute value of vorticity. This extremum is not necessarily coincident with the centroid; moreover, the shape of the vorticity concentration distorts with time. As a consequence, the trajectories of Figure 3 must be viewed as approximate. Nevertheless, the generally orbital trajectories of positive vorticity concentrations are evident in the left diagram and negative concentrations in the right diagram. The direction of the vortex motion is indicated by the arrows.
Since the flow pattern was generally repeatable from cycle to cycle, the paths of positive vorticity concentrations B', B and E shown in the left diagram essentially represent the continuous trajectory of the vorticity concentration that is initially formed along the right side of the cylinder, moves along the cylinder surface, is shed from the bottom of the cylinder surface, and follows its orbit about the cylinder. Likewise, concentrations D and C represent the trajectory of the concentration that is originally formed along, then shed from the left side of the cylinder. Finally, the paths of the negative concentrations of vorticity A and A' given in the right diagram represent the trajectory of the concentration that is formed along the left side of the cylinder, moves along the cylinder surface, then eventually separates from the upper surface of the cylinder; its small orbital trajectory leads to eventual collision with the surface of the cylinder. Viewing all of the foregoing processes together, it is evident that, during a single cycle of the wave motion, vortex formation originates at three distinct sites, followed by eventual shedding from three locations along the surface of the cylinder, i.e. approximately on the upper, lower, and left surfaces.
The orbits of the positive vorticity concentrations shown in the left diagram of Figure 3 have ratios of major to minor axes of roughly 1.6:1 and 1.5:1 and are inclined at an angle of about 36° to the horizontal. These values compare with those of the wave orbit in the vicinity of the cylinder of 2.2:1 and 41°. No doubt, mutual induction and image effects influence the paths of the vortical structures. Yet the reasonable correspondence between the wave and vortex orbits suggests that the wave motion exerts a predominant influence.