A Hybrid Discretize-then-Project Reduced Order Model for Turbulent Flows on Collocated Grids with Data-Driven Closure

Abstract: This study presents a hybrid reduced-order modeling (ROM) framework for turbulent incompressible flows on collocated finite volume grids. The methodology employs the "discretize-then-project" consistent flux strategy, which ensures mass conservation and pressure-velocity coupling without requiring auxiliary stabilization like boundary control or pressure stabilization techniques. However, because standard Galerkin projection fails to yield physically consistent results for the turbulent viscosity field, a hybrid strategy is adopted: velocity and pressure are resolved via intrusive projection, while the turbulent viscosity is reconstructed using a non-intrusive data-driven closure. We evaluate three neural network architectures, Multilayer Perceptron (MLP), Transformers, and Long Short-Term Memory (LSTM), to model the temporal evolution of the viscosity coefficients. Validated against a 3D Large Eddy Simulation of a lid-driven cavity, the LSTM-based closure demonstrates superior performance in capturing transient dynamics, achieving relative errors of 0.7\% for velocity and 4\% for turbulent viscosity. The resulting framework effectively combines the mathematical rigor of the consistent flux formulation with the adaptability of deep learning for turbulence modeling.