# Analysis of Polygonal Vortex Flows in a Cylinder with a Rotating Bottom
## 2 Numerical Modeling
### Choice of the Numerical Approach  
For the range of experimental conditions analyzed in the present study, the free surface of the rotating fluid was strongly deformed. Hence, the errors associated with employing non-deformable free shear boundary conditions might be expected to be significant. Moreover, as the free surface for the range of the experimental conditions modeled in the present study has assumed, among other cases, the shapes of ellipses and triangles (as in [10]), full asymmetric 3D calculations are performed. The flow in question is expected to be a turbulent free surface flow. As the Reynolds number based on the disk radius is of the order of \(10^{5}\), direct numerical simulation (DNS) is impractical. Employing Reynolds-averaged Navier-Stokes (RANS) equations, unsteady simulations have proven to be unrealistic as, in a preliminary study, they revealed no asymmetry in the shape of the free surface (these results are not shown for the sake of brevity). It was thus decided to use Large Eddy Simulation (LES). The essential specific details of the adopted approach are presented below. Subgrid-Scale Stresses (SGS) model has to complement any LES calculation. A number of models for SGS calculation are available [53]. A localized dynamic model including an energy equation [54] was employed in the present study. The model solves a transport equation for the subgrid kinetic energy, while model coefficients are obtained using a localized dynamic procedure. The subgrid-scale kinetic energy, as implemented in Fluent, is defined as  
\[k_{sgs}=\frac{1}{2}\left(\overline{u_{k}^{2}}-\overline{u}_{k}^{2}\right) \tag{1}\]  
which is obtained by contracting the subgrid-scale stress  
\[\tau_{ij}\equiv\rho\overline{u_{i}u_{j}}-\rho\overline{u_{i}u_{j}} \tag{2}\]  
The subgrid-scale eddy viscosity, \(\mu_{t}\), is computed using \(k_{sgs}\) as  
\[\mu_{t}=C_{k}\rho k_{sgs}^{1/2}\Delta/\overline{S_{ij}} \tag{3}\]where \(\Delta_{f}\) is the filter size computed from the cell volume, \(\Delta_{f}\equiv V^{1/3}\). The subgrid-scale stress can be then written as follows:  
\[\tau_{ij}-\frac{2}{3}\rho k_{ggs}\delta_{ij}=-2C_{k}\rho k_{ggs}^{1/2}\Delta_{f }\overline{S_{ij}} \tag{4}\]  
The subgrid-scale kinetic energy is obtained solving the following transport equation:  
\[\rho\frac{\partial\overline{k_{ggs}}}{\partial t}+\rho\frac{\partial\overline{ u_{jj}}\overline{k_{ggs}}}{\partial x_{j}}=-\tau_{ij}\frac{\partial\overline{ u_{ij}}}{\partial x_{j}}-C_{\xi}\rho\frac{k_{ggs}^{3/2}}{\Delta_{f}}+\frac{ \partial}{\partial x_{j}}\left(\frac{\mu_{t}}{\sigma_{k}}\frac{\partial k_{ggs }}{\partial x_{j}}\right) \tag{5}\]  
where \(C_{k}\) and \(C_{\xi}\) are determined dynamically and \(\sigma_{k}=1\). The localized dynamic model was found to be particularly important since it implies that in high-Reynolds-number flows, a relatively coarser grid (when compared with the algebraic LES model) can be used [54]. This model was successfully applied for modeling rotating shear flows [55]. Free surface fluid flows are usually computed employing interface capturing methods [56]. Two methods are widely employed: volume-of-fluid (VOF) approach [57] and level set method (LSM, [58]). The main strength of VOF methods is their ability to preserve mass [59], while the advantage of those based on level set is the accurate representation of the free surface normals and curvature [60]. The ability of combined LES and free surface capturing methods to predict free surface turbulent flows is demonstrated in a number of publications (e.g., [56]). The LSM was employed in [61] in conjunction with LES for turbulent open channel flow over fixed dunes. Two subgrid-scale models, namely, dynamic Smagorinsky and dynamic two-parameter one, were used. It was observed that the method was able to accurately and realistically calculate the unsteady free surface motion and also provided evidence of boils, upwelling and downdraft at the water surface. The results were relatively insensitive to the choice of the subgrid-scale model.