**Abstract**

We propose a data-driven correction reduced order model

(DDC-ROM) framework for the numerical simulation of fluid flows, which

can be formally written as DDC-ROM = Galerkin-ROM + Correction. The new

DDC-ROM is constructed by using ROM spatial filtering and data-driven

ROM closure modeling (for the Correction term). Furthermore, we propose

a physically-constrained DDC-ROM (CDDC-ROM), which aims at improving the

physical accuracy of the DDC-ROM. The new physical constraints require

that the CDDC-ROM operators satisfy the same type of physical laws

(i.e., the Correction term's linear component should be dissipative and

the Correction term's nonlinear component should conserve energy) as

those satisfied by the fluid flow equations. To implement these physical

constraints, in the data-driven modeling step, we replace the

unconstrained least squares problem with a constrained least squares

problem. We perform a numerical investigation of the new CDDC-ROM and

standard DDC-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 CDDC-ROM is significantly more accurate than the DDC-ROM in both

regimes.