Reduced-order time splitting for Navier-Stokes with open boundaries

In this work, we propose a Proper Orthogonal Decomposition-Reduced Order Model (POD-ROM) applied to time-splitting schemes for solving the Navier-Stokes equations with open boundary conditions. In this method, we combine three strategies to reduce the computing time to solve NSE: time splitting, reduction of the computational domain through non-standard treatment of open boundary conditions and reduced order modelling. To make the work self-contained, we first present the formulation of the timesplitting scheme applied to the Navier-Stokes equations with open boundary conditions, employing a first-order Euler time discretization and deriving the non-standard boundary condition for pressure. Then, we construct a Galerkin projection-based ROM using POD with two different treatments of the pressure boundary condition on the outlet. We propose a comparative performance analysis between the standard projection-based PODROM (fully intrusive) and a hybrid POD-ROM that combines a projectionbased approach (intrusive) with a data-driven technique (non-intrusive) using Radial Basis Functions (RBF). We illustrate this comparison through two different numerical tests: the flow in a bifurcated tube and the benchmark numerical test of the flow past cylinder, numerically investigating the efficiency and accuracy of both ROMs.