# Analysis of Polygonal Vortex Flows in a Cylinder with a Rotating Bottom
## 3 Results and Discussion
### Fourier Analysis
The dash lines relate Figure 14 to the first and last shapes in Figure 5. Similar to Figure 13b, Figure 14b shows that the shape is getting larger with the height while \(c_{3}\), which is the dominant coefficient (except of course \(c_{0}\)), is getting smaller. The appearance of the two main components of the Fourier series, \(c_{0}\) and \(c_{3}\), obtained numerically in the present study for 4.6 Hz where the triangle polygon shape prevails, was previously reported in [21]--see Figure 4 in that work. The triangle polygon shapes reported in [21] for the initial water height of 40 mm and the rotating disk frequency of 2.4 Hz (Figure 3 in their paper) closely resemble the shapes presented in Figure 5 of the present study. This points at a strong similarity between the triangle shapes obtained experimentally and numerically. To compute polygon rotation rate, the values of \(r(\theta=0)\) at the disk level were stored for each time step of 0.01s, and the FFT was applied to the function. The results in the frequency domain are plotted in Figures 15 and 16, for the disk rotation frequencies of 3.4 Hz and 4.6 Hz, respectively, and at the level of the rotating disk. Only at this height is it possible to compare the present results with the experimental findings from the literature, where image processing was used to compute the rotation rate of the polygons: it was used only on the exposed rotating disk where the contrast between the liquid and the exposed solid surface was large enough, see [10; 18; 19]. It is worth noting that the numerical calculations are not restricted by this limitation on the image processing. Figure 15 shows the amplitude \(Y\) of each mode for the rotating disk frequency of 3.4 Hz. The largest peak, 2.07, represents the rotation rate of the polygon (ellipse). Since the dominant Fourier coefficient is \(c_{2}\) (see Figure 13), the rotation rate of the polygon is computed by dividing the dominant frequency, 2.07 Hz, by the two polygon \"corners\" (\(N=2\)), which results in a rotation rate of 1.035 Hz and a ratio of 0.304 between the rotation velocities of the polygon and the disk. The second dominant frequency, 4.11 Hz by the results of Figure 15, corresponds to the next polygon--a triangle. Then, additional peaks, corresponding to higher values of \(N\), are observed. The relative magnitude of the second peak may indicate that the disk rotating frequency is close to a transition between an ellipse and a triangle, as indeed was demonstrated by [10] and can be seen from Figure 3. It may be noted that the computed ratio of frequencies in our case is also close to the value observed by [19]. For the disk frequency of 4.6 Hz, the dominant Fourier coefficient (other than \(c_{0}\)) is \(c_{3}\), and its peak frequency from the computation, 4.13 Hz, see Figure 16, should be divided by In fact, the numerical tool allows us to use the same procedure on different heights and not only at the exposed rotating disk as done by the experimental studies. The results show that the polygon frequency as well as the ratio between the dominant frequencies do not change with the height.