# Analysis of Polygonal Vortex Flows in a Cylinder with a Rotating Bottom
### Experimental Studies
With the same experimental device and flow parameters as of [8], the free surface cross section transition by visualizing the flow pattern using anisotropic flakes was studied in [13]. The vertical cross section along the centerline of the cylinder was recorded. The orientation of the flakes, which were arranged along the principal strain in the flow, was used to observe the flow patterns. The measurements of the free surface level at the cylinder axis and the radial velocity profile, and the computation of the turbulent intensity as function of \(Re\), were reported in [14]. The same experimental set-up as in [4; 6], but with several rotating disk diameters, was employed in [15]. Beside the main vortex, secondary vortices were observed in the corners of the rotating polygons, as was previously reported in [7]. It was suggested that when the velocity of the disk is increased, the main and the secondary spiraling vortices intensify and cause the liquid to recede, thus forming a dry polygon. In other words, a polygon with \(N\) corners is equivalent to \(N\) secondary point vortices rotating around the main polygon, forming the vortex. In the experiments of [4; 6; 7; 15], the highest wave number polygon observed was with \(N=6\). It was argued that the reason for not showing polygons with the wave numbers higher than \(6\) is that the secondary vortices are not stable according to [16; 17]. The ratio of the polygon to disk angular velocities was found to decrease with an increase in the Froude number (with the initial height as the characteristic length) and to increase with \(N\)[15]. Different parameters affecting creation and transition of the polygons were studied in [18; 19; 20]. In particular, it was indicated that a smoother surface delays the creation of the polygons. Then, the radial distance from the center to the pattern was determined for each azimuthal angle as function of time, using image processing. The functions were transformed to the frequency domain for each angle and averaged. It was found that before the transition, the dominant frequency corresponded to the rotation of the \(N\) polygon. Other frequencies were few orders of magnitude lower than the \(N\) frequency. Increasing the rotating disk frequency caused the frequency of \(N+1\) to be more dominant than the frequency of \(N\), whereas these two frequencies are coupled. The study [21] focused their attention on the formation of dry and wet polygons with \(N=3\), i.e., a triangle. Their experimental device was similar to that of [7]. Similar to [4; 15], it was observed that the formation of dry polygons did not depend on the initial conditions: the same shape was obtained when starting from the rest or from a circular shape, or when destroying the stable polygon without changing the disk rotation speed. To investigate the evolution of the dry triangle polygon, the Fourier coefficients of the contour radial coordinate were computed with time, and it was shown that the polygon formation is caused by linear instability. In the circular state prior to the formation of the triangle, the azimuthal and radial velocities on the free surface were measured. It was shown that the azimuthal velocity has a profile of Rankine vortex, as predicted analytically assuming the flow is inviscid. In the triangle state the angular velocity was much higher than the one that results from three-point vortices at the corners of the triangle in the triangle rotating reference frame. Thus, it was argued that the point secondary vortices used in the [PERSON]'s model [17] without a central vortex cannot describe the flow. In an additional communication, it was shown that the maximum vorticity was obtained in the center of the spiraling (secondary) vortices, and a plateau of the vorticity exists between them [21]. Using the experimental set-up of [15], it was argued in [22] that the velocity of the \(N\) spiraling vortices causes the circular shape of the primary vortex to deform to the polygon shapes.