The present paper provides a physically sound numerical modeling of liquid flows experimentally observed inside a vertical circular cylinder with a stationary envelope, rotating bottom and open top. In these flows, the resulting vortex depth may be such that the rotating bottom disk becomes partially exposed, and rather peculiar polygon shapes appear. The parameters and features of this work are chosen based on a careful analysis of the literature. Accordingly, the cylinder inner radius is 145 mm and the initial water height is 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 as observed in the experimental studies reported in the literature. The free surface flow is expected to be turbulent, with the Reynolds number of O(105). The Large Eddy Simulation (LES) is adopted as the numerical approach, with a localized dynamic Subgrid-Scale Stresses (SGS) model including an energy equation. Since the flow obviously requires a surface tracking or capturing method, a volume-of-fluid (VOF) approach has been chosen based on the findings, where this method provided stable shapes in the ranges of parameters found in the corresponding experiments. Expected ellipse and triangle shapes are revealed and analyzed. A detailed character of the numerical results allows for an in-depth discussion and analysis of the mechanisms and features which accompany the characteristic shapes and their alterations. As a result, a unique insight into the polygon flow structures is provided.