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 dWe 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.ata. The numerical investigation clearly shows that the new CDDC-ROM is significantly more accurate than the DDC-ROM in both regimes.

## Physically-Constrained Data-Driven Correction for Reduced Order Modeling of Fluid Flows

Research Group:

Traian Iliescu

Institution:

Virginia Tech

Location:

A-133

Schedule:

Tuesday, July 3, 2018 - 11:30

Abstract: