  In our earlier work, we proposed a data-driven filtered reduced order model
(DDF-ROM) framework for the numerical simulation of fluid flows, which can be
formally written as \begin{equation*}
  \boxed{
  \text{
  DDF-ROM = Galerkin-ROM + Correction
  }
  } \end{equation*} The new DDF-ROM was constructed by using ROM spatial
filtering and data-driven ROM closure modeling (for the Correction term) and
was successfully tested in the numerical simulation of a 2D channel flow past a
circular cylinder at Reynolds numbers $Re=100, Re=500$ and $Re=1000$.
  In this paper, we propose a {\it physically-constrained} DDF-ROM (CDDF-ROM),
which aims at improving the physical accuracy of the DDF-ROM. The new physical
constraints require that the CDDF-ROM operators satisfy the same type of
physical laws (i.e., the nonlinear operator should conserve energy and the ROM
closure term should be dissipative) as those satisfied by the fluid flow
equations. To implement these physical constraints, in the data-driven modeling
step of the DDF-ROM, we replace the unconstrained least squares problem with a
constrained least squares problem. We perform a numerical investigation of the
new CDDF-ROM and standard DDF-ROM for a 2D channel flow past a circular
cylinder at Reynolds numbers $Re=100, Re=500$ and $Re=1000$. To this end, we
consider a reproductive regime as well as a predictive (i.e., cross-validation)
regime in which we use as little as $50\%$ of the original training data. The
numerical investigation clearly shows that the new CDDF-ROM is significantly
more accurate than the DDF-ROM in both regimes.
