Numerical comparison among POD-ROM (3D)

We investigated the POD closure models for the moderate turbulent flow: 3D flow past a cylinder at Re = 1,000. Collecting 1,000 snapshots on the velocity filed over the time interval [0, 75], we employed the POD method and achieved the leading 6 POD modes that capture 84% of the system's kinetic energy. All POD-ROMs tested here were generated based on this set of basis and integrated over the time interval [0, 150].

Simulations are shown below. For clarity, only five isosurfaces are drawn.

Direct Numerical Simulation - obtained from CFD and considered as banchmark solutions.
POD-Galerkin ROM (no turbulence modeling) - it seems to add unphysical structures and, clearly, the numerical result is inappropriated.
Mixing-Length ROM (linear turbulence modeling) - dominated structures are missed after a while. It dues to the constant artificial viscosity (both spatially and temporally independent) that is added to the ROM eliminates some of the DNS structures.
VMS POD-ROM (nonlinear trubulence modeling) - Variational Multiscales mthod is extened to the POD setting. Artifical viscosity (spatially dependent) has been introduced to the small scales retained in the ROM. Dominated sturctures are clearly perserved.
DS POD-ROM (nonlinear trubulence modeling) - Dynamic subgrid-scale method is extended to the POD setting. Artifical viscosity (both spatially and temporally dependent) has been introduced to the ROM. Dominated sturctures are clearly perserved.