Physically-Constrained Data-Driven, Filtered Reduced Order Modeling of Fluid Flows
For practitioners of reduced order modeling of fluid flows, this method improves physical fidelity and predictive accuracy without requiring additional data.
This work introduces a physically-constrained data-driven filtered reduced order model (CDDF-ROM) that enforces energy conservation and dissipation constraints, achieving significantly higher accuracy than the unconstrained DDF-ROM in both reproductive and predictive regimes (using as little as 50% training data) for 2D channel flow past a cylinder at Re=100, 500, 1000.
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.