During two additional experiments, the cylinder underwent a small-amplitude orbital motion obtained by programming sinusoidal and cosinusoidal motions in the horizontal and vertical directions respectively. The period of the orbital motion of the cylinder was 2 sec, and the diameter of the orbit was 0.127 m. The depth of submergence of the cylinder at the highest point of its trajectory was 20 mm for both experiments. The phase shift between the cylinder motion and the wave motion was approximately zero radians in the first case and approximately p radians in the second case. The parameters of the motion were chosen such as to match the Keulegan-Carpenter number of the cylinder motion, given by
Two cases were considered. During the first experiment, the phase shift between the cylinder motion and the wave motion was approximately p radians, meaning that a trough of the propagating wave was directly above the cylinder when the cylinder was at the highest point of its trajectory. The top graph of Figure D-1 shows the trace of measured horizontal force with respect to time. In this case (so-called maximum Fx case), the peak-to-peak amplitude of the signal is approximately 0.3 Volts, and the period is equal to 2 sec (same as the period of the wave motion and the period of the cylinder motion).
During the second experiment, the cylinder was moving in-phase with the propagating wave, meaning that a crest of the wave was directly above the cylinder when it was at its highest position. In this case (so-called minimum Fx case), the value of the measured horizontal force is almost zero throughout the whole wave cycle, as shown on the bottom graph of Figure D-1.
Figure D-2 shows images of the flow field in the maximum Fx case. The left column of images represents the instant t1 corresponding to the maximum negative value of the Fx. The right column represents the instant t2 corresponding to the maximum positive value of the Fx.
Top images show the instantaneous vorticity distributions in the flow field. Contours of constant positive and negative vorticity are represented by the solid and dashed lines respectively. There are well-defined concentrations of vorticity with high values of circulation at both instants of time. Positive and negative vortices develop on the surface of the cylinder and interact with the vortices shed during previous wave cycles.
Lower images show instantaneous velocity fields and streamline patterns. At both instants, the flow velocity experiences significant changes in direction and magnitude throughout the field of view. The singular points in streamline patterns indicate positions of the vortices.
The left image in Figure D-3 shows an instantaneous vorticity distribution for the minimum Fx case. Vortices that are developing on the surface of the cylinder have low values of circulation. There are no vortices shed during previous wave cycles.
The right image shows the corresponding velocity field and streamline pattern. The velocity field is very uniform, and the streamline pattern is not distorted and has no singular points.
Hence, moving the cylinder in-phase with the propagating wave significantly reduces the magnitude of the horizontal force acting on the cylinder and reduces the vortex shedding.