# Analysis of Polygonal Vortex Flows in a Cylinder with a Rotating Bottom
## 2 Numerical Modeling
### Computational Details  
The experimental apparatus employed in [10] allowed simultaneous rotation of both the bottom disk and the cylindrical wall. In the present study, we focus on the cases where the cylindrical wall was kept static. The cylinder inner radius in the experiment was 145 mm. Experiments with only bottom plate rotation were performed with varying filling heights, from 30 to 80 mm. This study focuses on modeling the case with the initial water height of 60 mm. The experiments with bottom disk rotation frequencies of 3.0, 3.4, 4.0 and 4.6 Hz are simulated. The chosen frequency range encompasses the regions of ellipse and triangle shapes (two frequencies for each case) as observed in the experiment--see [10]. The explored values of governing dimensionless groups and other parameters are listed in Table 1, along with the corresponding data for the existing works. The computational domain and grid employed in the present study are presented in Figure 1b. The base grid consisted of 1.1 million hexahedral cells built employing the O-grid technique. Appropriate mesh morphology generation is vital for an accurate solution, faster convergence and reduction in numerical diffusion. Hexahedra structured O-grid employed in the present study is characterized by regular connectivity and high space efficiency. The chosen grid is aligned in the main rotational direction of the flow in question, where possible, leading to more accurate results and a better convergence. The minimal orthogonal quality of the computational domain was 0.75 with the maximal grid aspect ratio being 12 in the limited region of the stationary cylinder boundary layer, where the prevailing flow direction is azimuthal, this aligned with the longer edge of the computational cells. Such a grid resolution, being fine in the radial direction, ensures correct representation of the flow in the boundary layer at the stationary cylinder walls. The grid was refined close to the rotating bottom disk and vertical stationary cylinder wall to capture better the flow details in the Ekman and Stewartson layers, respectively. It is obvious that grid and time-step choice are of crucial importance for a reliable simulation of the expected complex flow fields. In this work, the following approach was adopted: the grid was refined using the LES index of quality (LESIQ\({}_{v}\)), introduced in [68], as the criterion. The LESIQ\({}_{v}\) is a dimensionless number between zero and unity, reflecting the ratio between resolved and total turbulent kinetic energy. According to [68; 69], based on [70], an index of quality greater than 0.8 is considered a good LES, while 0.95 and higher may be considered a DNS. Figure 2a,b show the contours of instantaneous values of LESIQ\({}_{v}\) for the fastest case of the present investigation, namely, of 4.6 Hz. Figure 2a shows the values of LESIQ\({}_{v}\) for the free surface of the liquid, while Figure 2b shows its values in the vertical cross-section of the flow. One can see from the figures that LESIQ\({}_{v}\) is above 0.8 practically everywhere, and in most locations, it is close to or above 0.9. A parameter, which may be used for ensuring temporal resolution of the simulation, is the Courant number. One can see from Figure 2c, for the same complex case of Figure 2b, that the Courant number is smaller than unity practically all over the simulated flow field. The only exceptions are the \"corners\", i.e., the locations where the rotating disk meets with the stationary wall, but this effect does not influence the outcome of the simulations. Calculation using a grid refined further by the factor of 1.5 in all the three directions (3.3 times global refinement) was done with no appreciable difference in the results. No-slip and no-penetration boundary conditions are applied on the rotating bottom disk and the vertical cylinder wall. The top boundary of the computational domain was defined as pressure outlet.