  The purpose of this work is to present a reduced order modeling framework for
parametrized turbulent flows with moderately high Reynolds numbers within the
variational multiscale (VMS) method. The Reduced Order Models (ROMs) presented
in this manuscript are based on a POD-Galerkin approach with a VMS
stabilization technique. Two different reduced order models are presented,
which differ on the stabilization used during the Galerkin projection. In the
first case the VMS stabilization method is used at both the full order and the
reduced order level. In the second case, the VMS stabilization is used only at
the full order level, while the projection of the standard Navier-Stokes
equations is performed instead at the reduced order level. The former method is
denoted as consistent ROM, while the latter is named non-consistent ROM, in
order to underline the different choices made at the two levels. Particular
attention is also devoted to the role of inf-sup stabilization by means of
supremizers in ROMs based on a VMS formulation. Finally, the developed methods
are tested on a numerical benchmark.
