# Analysis of Polygonal Vortex Flows in a Cylinder with a Rotating Bottom
### Experimental Studies
Using PIV, the rotation rate of the polygon was measured and compared with [PERSON]'s model of point vortices [17]. It was found that taking into account the angular velocity of the flow from the solid body rotation region, in addition to the angular velocity proposed in [17], yields good agreement with the angular velocity at the center of the spiraling vortices extracted from PIV measurements. The formation of polygons in a cylinder in which the bottom plate and the vertical walls could be rotated independently was investigated experimentally in [10]. The results for stationary walls were compared with those of [4; 6; 7]. \"Excitability\" of the polygons was explored. Similar to the works [4; 6; 7], multi-stability was observed in the experiments: for most of the parameters, a stable state, which did not depend on the initial conditions, was found; however, in some of the transitions, two or more stable states were observed, for the same frequency. The suggested explanation for this effect was the dependence on initial conditions at these certain frequencies. The excitability was evident by a number of unstable states when perturbing the flow. An experimental set-up with a cylinder radius in between the radii used by [7] was employed in [23]. They also observed a linear dependence between the initial water height and the transition disk rotation frequency. Like in [4; 15], it was found that the transition between an ellipse and a triangle was unstable, with changes between a distorted ellipse and an elongated triangle, which also rotated at different rates. The issue of the viscosity effect regained attention in [11]. The reason is that the main assumption in most analytical works (surveyed below) is that the fluid is inviscid. It was argued that in addition to Froude number, the Taylor number (which is the ratio of the centrifugal force to the viscosity force) is governing the flow. It was found that increasing the viscosity of the fluid during rotation results in decreasing the range of disk velocities for patterns with higher \(N\). The viscosity had no effect on the ratio between the frequencies of the polygons and the disk. In [24], it was shown experimentally that increasing the viscosity of the fluid does not influence the transition mechanism presented by [18; 20], thus arguing that the behavior of the flow can be described as rotating point vortices, as shown in [15]. The above surveyed experimental studies on free surface flows in cylinders with rotating bottom disk show mostly qualitative rather than quantitative agreement in the observed phenomena: same polygon shapes in the stable states, linear dependence between the initial water height and the disk frequencies for the transition between the neighboring polygon shapes, and independence of the stable states of the initial conditions for the most of experimental conditions. In some studies, the transition between the shapes was gradual (as in [6; 15]) and in others it occurred in an oscillatory manner (as in [7]) or periodically (as in [25]). In any case, the transition was characterized by combined \(N\) and \(N+1\) shapes.