# Analysis of Polygonal Vortex Flows in a Cylinder with a Rotating Bottom
### Experimental Studies  
The major experimental studies from the literature are summarized in Table 1. First experiments with water inside a stationary cylinder with a rotating bottom were reported in [4]. Under certain conditions, it was observed that the free surface reached the rotating disk and partially exposed it, producing shapes of polygons with varying numbers of corners, \(N\). These shapes were stable for up to \(N=6\), such that if perturbed or even fully destroyed, they would restore their initial configurations. In [4], the polygons were mapped by their number of corners, \"wave numbers\", as a function of the disk velocities and initial water heights. A range of disk velocities was found for each wave number. The experimental work [6] dealt with the impact of liquid viscosity on the phenomenon of free surface polygon formation, as two additional liquids, with the viscosities of 56.7 and 640 times higher than that of water, were used. The lower and upper boundaries of disk velocity for each range were found to increase linearly with the initial water height. Between two adjacent stable states, a mixed, time-dependent state was observed. This state was a superposition of the corresponding wave numbers, and was not stable, because each shape, with different wave number, was rotating at a different speed. Figure 1: The system addressed in the present study: (**a**) physical model and an example of a polygon created on the disk surface; (**b**) computational domain and grid (front and top views). Free surface polygon formation, similar to the one earlier reported in [4; 6], was independently confirmed in [7]. The experiments were conducted with water and ethylene glycol which has a \(15\) times higher viscosity and similar density, as compared with water. A phase diagram, showing the wave numbers as function of the initial height and the disk frequency, was reported. Linear dependence between the initial height and the limit frequencies for each range reported in [6] was observed in [7]. Close to the upper limit of the frequency range for each polygon, a transition state, in which the corners of the polygon changed periodically from \(N\) to \(N+1\) (in particular between \(N=2\) and \(N=3\)), was observed. For \(N>2\), the free surface exposed typically the rotating disk and thus created a \"dry\" polygon [7]. For \(N=2\) and in some sub-cases of \(N=3\), the rotating disk was completely covered with water, creating a \"wet\" polygon. However, in their phase diagram, dry and wet polygons were not distinguished. Surface tension effects by adding surfactant into the fluid, also reported in [7], did not change the phase diagram significantly. Spiraling vortices, observed in the corners of the polygons, were interpreted as Gortler vortices. It was argued that the secondary flow was highly dependent on the singular corners created by the proximity of the rotating disk and the cylinder fixed walls. This makes the details of the flow very complicated, as noted in [7]. In a similar but smaller experimental set-up, the switching between circular and elliptical cross sections, \(N=0\) and \(N=2\), was investigated in [8]. Measuring the free surface height above the center of the rotating disk, several free surface states depending on the Reynolds number (defined as \(Re=\omega R^{2}/\
u\), where \(\omega\) is the disk angular velocity, \(R\) is the cylinder radius and \(\
u\) is the kinematic viscosity) were identified [8]. The results from the above studies indicate that the creation of the polygons, their shapes, stability and rotating velocity depend on the geometry of the experiment and on the Reynolds number. In an additional communication, the use of the Froude number, defined as \(Fr=\omega r/(gr)^{0.5}\), where \(r\) is the bottom disk radius, was suggested in [12], which looks quite reasonable because free surface flows are considered.