# Analysis of Polygonal Vortex Flows in a Cylinder with a Rotating Bottom
## 3 Results and Discussion
### Velocity Field
The velocities at the left column are shown in the vertical plane crossing the water volume through the minor axis of the ellipse, as illustrated at the head of the column. Similarly, the right column represents the velocities in the vertical plane crossing the major axis of the ellipse. It can be seen from the radial velocity distribution and superimposed flow field illustration that the liquid moves radially outward close to the surface of the rotating disk inside the Ekman boundary layer, and then stops at the stationary cylinder. The liquid near the stationary walls is directed upward, as can be seen from the axial velocity, forming a Stewartson boundary layer. The flow is then turning downwards and outward near the surface, as can be seen from the radial and axial velocities. The tangential velocity is getting its local maximum values at the rotating disk and near the axis of rotation; it decreases gradually towards the stationary walls. Figure 9: Front view contours of the radial, axial (both in m/s) and normalized tangential velocities in the two vertical cross-sections corresponding to the main axes of the ellipse at the disk rotation frequency of 3.4 Hz. Each velocity field is averaged over 950 frames which represent 95 s of a transient. Similar to Figure 8, Figure 10 shows the radial, axial and normalized tangential velocities at four heights above the disk rotating at 4.6 Hz. 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. The computation time is 36.5 s forming 365 datasets. The contours were correlated and rotated as described in the discussion of Figure 8 above. The radial velocity presented in the left column vanishes near the stationary walls for the four heights, similar to Figure 8. However, three regions of non-zero velocities are located aside each triangle arm. Each arm is surrounded by positive and negative radial velocity which decreases with height. For the height of 5 mm, the maximum absolute velocity is 0.5 m/s and for 60 mm the maximum velocity is 0.1 m/s. These radial velocities are similar to the radial velocities obtained for the ellipse shape with 3.4 Hz rotating disk. From the axial velocities in the middle column of Figure 10, it can be seen that the liquid moves upward in a thin layer near the stationary walls and almost remains stationary far from the cylinder walls. However, close to free surface the axial velocity attains both positive and negative values. These regions are located at the same spots around free surface as the non-zero regions of the radial velocity. Thus, it can be observed that the liquid moves inward and downward near each triangle arm and also outward and upward at the other side of each arm. The radial velocity values are quite similar to the ones obtained for the ellipse shape (3.4 Hz) but the axial velocity is higher, reaching values of 0.1 m/s at all heights. Figure 10: Top view contours of the radial, axial (both in m/s) and normalized tangential velocities at four heights (from top to bottom: 5, 10, 30 and 60 mm) above the 4.6 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 normalized tangential velocity presented on the right side of Figure 10 shows high values on the valleys of each triangle arm at 5 and 10 mm heights. For all heights, the tangential velocity decreases with radius and does not change significantly with height. Figure 11 presents front view contours of radial, axial and normalized tangential velocities for three vertical cross-sections positioned 120 degrees from one another. As in Figure 10, each picture is representing averages of 36.5 s of computation time, meaning 365 data sets. From each set, three planes are used for averaging.