# Analysis of Polygonal Vortex Flows in a Cylinder with a Rotating Bottom
## 3 Results and Discussion
### Fourier Analysis  
In addition to the visual representation of Figures 4 and 5, results of Fourier analysis, presented in Figures 13 and 14, enable the identification of the polygon patterns for the frame sequences deduced from the numerical computation. In order to quantify the polygon patterns, a Fourier decomposition is employed, following the procedure of [21]. The contours of the free surface at the level of the rotating disk (zero height) are analyzed by computing their Fourier coefficients as a function of time, where each coefficient corresponds to one polygon pattern expressed by the number of corners, \(N\). The Fourier analysis enables the identification of the dominant pattern at a given instant, and analysis of the stability of the patterns over time. The results are then compared with the frames in Figures 4 and 5 and with the literature. The polygon rotation frequency is computed for every rotating disk frequency. This is done by analyzing the radius of the free surface contour at the rotating disk level, at one static azimuthal angle as function of time. A Fast Fourier Transform (FFT) is then applied for this function to identify the frequency of the polygon and other features, like the transition between polygon shapes. It is worth noting that in the literature, the polygon frequency measurements were limited to the exposed rotating disk. In the present work,  
Figure 12: Instantaneous tangential velocity contours: (**a**) elliptical polygon for 3.4 Hz, (**b**) triangular polygon for 4.6 Hz. we also computed the frequencies for different heights above the disk, which can be used to quantify the illustrative results of Figures 6 and 7. The Fourier coefficients \(|c_{n}|=\left|\frac{1}{2\pi}\int_{0}^{2\pi}r(\theta)e^{in\theta}d\theta\right|\) are computed for the rotating disk frequencies of 3.4 and 4.6 Hz, where \(r(\theta)\) is the radius of the free surface contour at the angle \(\theta\) for selected height. Each coefficient corresponds to a specific polygon pattern. Thus, \(c_{0}\) is the average radius of the polygon; \(c_{1}\) corresponds to the transition of the center of the shape; \(c_{2}\) represents an ellipse and \(c_{3}\) a triangle. In Figures 13 and 14, the dominant and adjacent coefficients are presented for each rotating disk frequency. In order to juxtapose the values of the coefficients and the visual representation of the free surface shapes, Figures 13a and 14a are accompanied with top views at certain times. The dashed lines show the time interval chosen for the free surface level contours presented in Figures 4 and 5. Figure 13: The Fourier coefficients, \(c_{0}\) to \(c_{3}\) for 3.4 Hz rotating disk frequency: (**a**) the Fourier coefficients as function of time at the surface of the rotating disk. (**b**) The average and standard deviation of the Fourier coefficients for the time interval presented in (**a**) at different heights above the rotating disk. The left vertical axis corresponds to \(c_{0}\) and the right vertical axis corresponds to \(c_{2}\) and \(c_{3}\). larger and the ellipse coefficient, \(c_{2}\), becomes smaller. This corresponds to the results in Figure 6 where the ellipse transforms into a larger circle when the height increases. In addition, the standard deviations decrease with the height, indicating that the shapes are more stable as the height increases. Accordingly, the widths of the contours (\"braids\") in Figure 6 decrease with the height. Regarding Figure 14a that shows Fourier series coefficients for the rotating disk frequency of 4.6 Hz, it can be seen that the values of \(c_{3}\) become higher than \(c_{2}\). On the other hand, coefficient \(c_{4}\) that corresponds to the next polygon (square) is much smaller than \(c_{3}\), too. This shows quantitatively that the dominant pattern is triangle, as it was shown in Figure 5. In addition, all coefficients remain around their certain values for all computation time. This shows again that the patterns are stable.