# Analysis of Polygonal Vortex Flows in a Cylinder with a Rotating Bottom
## 3 Results and Discussion
### Velocity Field  
Figure 8 presents top views of radial, axial and normalized tangential average velocities at four heights: 5, 10, 30 and 60 mm, above the disk rotating at 3.4 Hz, where ellipses on the disk surface are observed. Each view is a result of 1085 fields, taken from 108.5 s of computation time. Each field was rotated to best correlate to the \(t=0\) contour chosen arbitrarily. An average velocity value is shown at a given point if 50% or more frames are occupied by water, otherwise a white color is assigned. This threshold is arbitrary, setting the size of the white \"dry spot,\" but the mean velocity field around it is captured correctly. In order to expose the velocity field features, the color scales are adjusted to the specific cases, and generally are not kept constant. It can be seen from the left column in Figure 8 that the radial velocity vanishes near the stationary cylinder walls at all the four heights. Closer to the ellipse shape, there are two regions of positive radial velocity and two regions of negative radial velocity. The non-zero velocity regions are positioned such that every  
Figure 7: Free surface contours at six levels for 4.6 Hz disk rotation frequency. Each contour is rotated to the contour of \(t=0\) s using maximum correlation between the shapes. For each height 360 contours are plotted, representing results for 36 s.  
apex is surrounded by positive and negative velocity regions. As expected, closer to the rotated disk, the non-zero radial velocities are higher in their absolute values and decrease with height. The maximum absolute radial velocities in the non-zero regions are 0.5, 0.4, 0.2 and 0.1 m/s at 5, 10, 30 and 60 mm above the disk, respectively. The middle column in Figure 8 presents the average axial velocity at the same four heights. It can be seen that near the stationary walls inside the Stewartson layer the liquid is moving upward with decreasing velocity at greater heights. The upward velocity results from the momentum given to the liquid by the rotating disk in radial direction and the restraint by the stationary cylinder walls. The regions surrounding the ellipse shape are characterized by positive and negative axial velocities. At greater heights the regions of positive velocities become larger at the expense of the negative-velocity regions. The absolute maximum values of axial velocities increase with the increase in height. For example, at 5 mm the maximum value is 0.02 m/s, whereas at 60 mm the velocity is 0.06 m/s. It can be seen from observation of the radial and axial velocities, that near the apex of the ellipse, the liquid is moving radially outward/inward and at the same time upward/downward, correspondingly. Figure 8: Top view of the radial, axial (both in m/s) and normalized tangential average velocities at four heights (from top to bottom: 5, 10, 30 and 60 mm) above the 3.4 Hz rotating disk. Each contour is rotated to the contour of \(t=0\) s using maximum correlation between the shapes. Velocity values are averaged if the specific location was occupied by water more than 50% of the frames, otherwise white color is assigned. The right column of Figure 8 presents the normalized tangential velocity to highlight the difference between the solid body rotation of the fluid on the rotating disk and the vanishing velocity on the side wall. It can be seen that near the ellipse shape the tangential velocity is approaching the maximum tangential velocity, i.e., the local disk velocity. The tangential velocity of water decreases with radius and vanishes as expected at the stationary cylinder walls. The tangential velocity does not change significantly with height. Figure 9 presents front cross-sectional views of radial, axial and normalized tangential average velocities. Each view represents averaging of 950 frames which are 95 s of computation. An average velocity value is shown at a given point if 50% or more frames are occupied by water, otherwise a white color is assigned.