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.

Figures (15)

• Figure 1: Two-dimensional collocated control-volume layout. Cell-centered velocity and pressure are associated with the control volumes $P$ and $N$. Face fluxes $\mathbf{u}_f$ are indicated on the cell faces, and $\Delta x$, $\Delta y$ denote the grid spacing. The vector $\mathbf{d}$ connects adjacent cell centers.
• Figure 2: Detailed workflow of the proposed hybrid methodology combining a physics-based ”discretize- then-project” ROM with a data-driven turbulence closure which is trained offline but integrated directly into the online time-stepping loop.
• Figure 3: Computational domain and grid generation. The geometry of the cubic cavity is shown on the left, and the corresponding computational mesh is shown on the right
• Figure 4: Plot of the error with modes variation, between the FOM solution and the ROM solution.
• Figure 5: Plots of the errors, with time variation, for the velocity field between the FOM solution and the ROM solution with the three different neural networks used: on the left LSTM, in the center MLP, on the right Transformer.