# Analysis of Polygonal Vortex Flows in a Cylinder with a Rotating Bottom
### Numerical Modeling  
Analytical modeling for the problem in question is obviously limited. Thus, numerical simulation should be very appealing. Remarkably, the literature that might be considered relevant in this sense is rather limited. There exist numerical works on the configuration in question but they were focused on another phenomenon. It was experimentally observed, for \(Re<4000\) and \(h/R\leq 4\), where \(h\) is the initial water height, that the vortex, created by a rotating bottom disk, can be broken into recirculating bubbles [33]. This vortex breakdown is laminar and occurs under steady flow conditions. A possible bifurcation of streamline structures as function of \(Re\) and \(h/R\), assuming axisymmetric flow and _flat top_ surface was numerically derived in [34]. In [35], the flow was investigated numerically by direct solution of the Navier-Stokes equations without any symmetry assumption but again with _flat_ free surface. It was shown that at \(Re=6000\), the flow is divided into two regions: in the inner region, the flow behaves as solid body rotation, while the flow in the outer region behaves as a jet along the cylinder wall and turns into the radial direction towards the inner region, causing unstable azimuthal modes. The same problem but allowing the free surface to deform _slightly_, corresponding to \(Fr=0.1\) and relative surface deformation of 0.012, was modeled in [36]. Modeling the problem with a flat free surface imposes a reflection symmetry on the flow, where the free surface serves as the reflection plane, as studied in [37; 38]. Thus, a cylinder with double height (relative to the experimental cylinder) and with two co-rotating end-walls (\(Fr=0\)) was modeled. Two aspect ratios were considered: a deep system with \(h/R=2\) and a shallow system with \(h/R=0.25\). It is shown that in the shallow system, solid body rotation exists in the inner region and meridional flow occurs in the outer region, whereas the deep system has meridional flow throughout the cylinder. The area near the axis consists of recirculation bubbles, as observed in [33]. The behavior was assumed to be of Hopf bifurcation type, studied earlier in, e.g., [34; 39; 40] for similar cases, and in [41] for stationary and rotating disks. The mechanism for the symmetry breaking in the deep system is due to azimuthal instability of the shear layer that is turning to the interior from the stationary cylinder, as was argued in [42]. The symmetry breaking has a reflection symmetry, therefore the flat free surface modeling, which imposes that reflection symmetry, can predict symmetry breaking of the experiments. For the shallow system, the instability mechanism is due to azimuthal instability in the interface between the solid body rotation zone and the meridional flow zone. In this case, the symmetry breaking has no reflection plane and cannot be predicted by the flat free surface model. It is worth noting that as shown experimentally, increasing the surface tension could stabilize the flow into axisymmetric state [37]. Obviously, numerical modeling at low Reynolds numbers cannot describe the experimentally observed free surface symmetry breaking and polygon shape formation where the typical Reynolds numbers are of the order of \(10^{5}\), as described in detail above. However, to the best of our knowledge, only very few studies have started to go in that direction. Moreover, significant simplifications continue to be used. Experiments with small \(h/R\) ratio (see Table 1) were conducted in [9]. A polygonal flow structure was reported to be created by spiraling vortices in the corners of the polygon. Experiments reported in [9] were modeled in [43] with direct numerical simulation (DNS) at \(Re\) up to 27,000 without free surface distortion. The model predicted well the first symmetry breaking from the experimental work of [9] for large \(h/r\), but large discrepancies were obtained for small \(h/r\).