# Analysis of Polygonal Vortex Flows in a Cylinder with a Rotating Bottom
## 2 Numerical Modeling
### Choice of the Numerical Approach
Calculations on the flow past a cylinder protruding from the water surface at the Reynolds number \(Re=2.7\times 10^{4}\) and the Froude numbers \(Fr=0.2\) and \(0.8\) were reported in [62]. A Lagrangian dynamic subgrid-scale model, complemented with LSM for capturing the free surface, was employed. The mean interface level and the root mean square (r.m.s.) of interface fluctuations from the simulation were reported to be in excellent agreement with the experimental results. Along with VOF and LSM methods, a number of hybrid techniques were developed for interface capturing. The coupled level set/volume-of-fluid (CLSVOF) method, as developed in [63], is applied when the free surface is expected to be highly complex. Results of a very recent work which deals with mechanisms of air-core vortex evolution in an intake flow are reported in [64]. Based on the literature analysis, the localized dynamic SGS model for LES turbulence model was used in the present study for simulating flows in an open static cylinder with a rotating bottom, partially filled with water. As for resolving the interface, this study started with employing the CLSVOF approach. However, an extensive investigation for the frequencies where stable ellipse and triangle polygons were experimentally observed in [10] revealed that the hybrid CLSVOF, while successfully capturing these shapes, could not ensure their stability. For this reason, the VOF method was used instead. This method was recently successfully applied in [65] to modeling of rotating free surface turbulent flow in a whirlpool including the phenomenon of phase separation. The tracking of the air-water interface is accomplished by the solution of an explicitly discretized continuity equation for the volume fraction. Geometric reconstruction scheme is used for the cells near the water-air interface. The scheme represents the interface between fluids using a piecewise-linear approach. The VOF technique is employed to air and water immiscible fluids by solving a single set of momentum equations and tracking the volume fraction of each of the fluids throughout the domain. The tracking of the interface between the phases is accomplished by the solution of a continuity equation for the volume fraction of one of the two phases. For one of the phases, assuming incompressible fluids without mass transfer between the phases and without sources, this equation has the following form:  
\[\frac{\partial}{\partial t}\alpha_{a}+\overline{v}_{a}\cdot\
abla\alpha_{a}=0 \tag{6}\]  
As only air and water phases are present, the volume fractions of the two phases sum up to unity in each control volume:  
\[\alpha_{w}+\alpha_{a}=1 \tag{7}\]  
Implicit volume fraction formulation is chosen for time discretization in the present study. The following equation is solved for the secondary phase (water in this case) at each time step to obtain the face fluxes for all cells, including the interface:  
\[\frac{\alpha_{a}^{n+1}-\alpha_{a}^{n}}{\Delta t}V+\sum_{f}\Bigl{(}U_{f}^{n+1} \alpha_{a,f}^{n+1}\Bigr{)}=0 \tag{8}\]  
A single momentum equation is solved for the two phases with the properties, e.g., density, being calculated as void fraction weighted average of the two components. In the present work, VOF method is combined with a constant contact angle treatment (found appropriate in a number of applications--e.g., [66]) for the moving water boundaries on vertical walls and rotating bottom. Alternative treatment of the contact angle parameter can also be considered--e.g., recently published work [67] on application of dynamic contact angle formulation along with VOF method. As will be demonstrated below, the chosen approach makes it possible to obtain stable shapes while all other simulated features correspond well with the experimental observations.