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

This study proposes a hybrid reduced-order modeling framework for turbulent flows on collocated grids that combines a consistent flux "discretize-then-project" strategy for mass and momentum conservation with an LSTM-based data-driven closure to accurately reconstruct turbulent viscosity, achieving superior performance in capturing transient dynamics compared to other neural network architectures.

The Gist

Imagine you are trying to predict how a complex, swirling fluid (like wind in a room or water in a pipe) will move. To do this perfectly, you need a super-computer simulation that tracks every single tiny particle of that fluid. This is called a Full-Order Model (FOM). It's incredibly accurate, but it's also like trying to count every grain of sand on a beach to predict the tide: it takes forever and requires massive amounts of memory.

To solve this, scientists use Reduced-Order Models (ROMs). Think of a ROM as a "highlight reel" or a "summary" of the fluid's behavior. Instead of tracking billions of particles, it tracks only the most important patterns (like the big swirls) to give you a fast, good-enough answer.

However, there's a catch. When the fluid is turbulent (chaotic and swirling wildly), this "highlight reel" method often breaks down. It gets the big picture (speed and pressure) right, but it fails to predict the "friction" or "stickiness" of the turbulence (called turbulent viscosity) correctly. It's like having a weather forecast that predicts the wind speed perfectly but gets the humidity completely wrong.

The Paper's Solution: A Hybrid Team-Up

The authors of this paper created a new "hybrid" system that combines the best of two worlds to fix this problem. They used a 3D Lid-Driven Cavity (a box where the top lid slides back and forth, dragging the fluid inside) as their test case.

Here is how their system works, using simple analogies:

1. The "Physics" Team (The Strict Accountant)

For the speed of the fluid (velocity) and the pressure, the team uses a method called "Discretize-then-Project."

• The Analogy: Imagine you are building a house. You have a strict blueprint (the laws of physics) that ensures the walls are straight and the roof doesn't leak. This team follows the blueprint exactly. They take the complex math of the fluid, shrink it down to the "highlight reel" size, but they do it in a way that guarantees the fluid doesn't magically appear or disappear (mass conservation).
• The Result: They get the speed and pressure of the fluid very accurately without needing extra "patches" or fixes.

2. The "Data-Driven" Team (The Intuitive Artist)

For the turbulent viscosity (the chaotic friction), the "Strict Accountant" method fails. So, the authors brought in a Data-Driven team.

• The Analogy: Instead of trying to calculate the chaos with a rigid blueprint, they hired an artist who has watched thousands of hours of this specific type of fluid swirling. This artist uses Machine Learning (specifically Neural Networks) to "learn" the pattern of the chaos from the data.
• The Tool: They tested three different types of "artists" (Neural Network architectures):
  • MLP: A basic artist who looks at the current moment but forgets the past.
  • Transformer: An artist who can look at the whole timeline at once but might get distracted.
  • LSTM (Long Short-Term Memory): An artist who has a great memory. They remember not just what is happening now, but what happened a few seconds ago. This is crucial because turbulence is a chain reaction; what happens now depends heavily on what happened just before.

3. The Final Result: The Perfect Duo

The paper combines these two teams. The "Strict Accountant" handles the speed and pressure, while the "Intuitive Artist" (specifically the LSTM model) predicts the turbulent friction.

Why did the LSTM win?
Turbulence is like a row of dominoes falling. If you only look at the first domino (the current moment), you can't predict the rest. You need to see the chain of falling dominoes (the history). The LSTM model is the best at remembering this chain of events.

The Outcome

When they tested this hybrid system against the super-computer simulation:

• Speed and Pressure: The model was incredibly accurate (only 0.7% error).
• Turbulent Friction: The model predicted the chaos with a 4% error, which was much better than the other AI models they tried (which had errors up to 14%).

In Summary

The paper presents a clever way to simulate chaotic fluids quickly. They didn't try to force one method to do everything. Instead, they used rigid math for the parts that need to be exact (speed/pressure) and smart AI memory for the part that is chaotic and hard to calculate (turbulence).

The result is a fast, accurate simulation that captures the "swirls" of a 3D turbulent flow without needing a supercomputer, proving that sometimes the best way to solve a hard problem is to let the math and the machine learning do what they do best.

Technical Summary

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

Problem Statement
Computational Fluid Dynamics (CFD) simulations of turbulent flows, particularly those involving high-resolution models, are often constrained by prohibitive computational costs and memory requirements. Reduced-Order Modeling (ROM) offers a strategy to mitigate these costs by projecting governing equations onto lower-dimensional subspaces. However, applying standard projection-based ROMs (specifically POD-Galerkin) to turbulent flows on collocated finite volume grids presents significant challenges. While the "discretize-then-project" strategy ensures mass conservation and pressure-velocity coupling for velocity and pressure fields, it fails to yield physically consistent results when applied directly to the turbulent viscosity field. Standard intrusive Galerkin projection cannot accurately resolve the complex temporal evolution and non-linear interactions of eddy viscosity within a reduced basis, leading to physical inconsistencies. Furthermore, collocated grids often suffer from numerical instabilities (e.g., checkerboard pressure modes) if not properly stabilized, typically requiring auxiliary techniques like Rhie-Chow interpolation or pressure stabilization that complicate the reduced-order formulation.

Methodology
The authors propose a hybrid ROM framework that combines the mathematical rigor of intrusive projection with the adaptability of non-intrusive data-driven methods. The methodology is structured as follows:

1. Full-Order Model (FOM) and Discretization:

   • The incompressible Navier-Stokes equations are solved using a Large Eddy Simulation (LES) approach with the Smagorinsky subgrid-scale model.
   • The FOM utilizes a collocated finite volume method (FVM) on a structured grid, implemented within the OpenFOAM framework.
   • To address stability on collocated grids without auxiliary stabilization, the study employs the "discretize-then-project" consistent flux method (CFM). This strategy formulates the operators directly at the discrete level, projecting full-order matrices and boundary contributions onto reduced spaces to ensure consistency between velocity and pressure representations.

2. Hybrid Reduced-Order Strategy:

   • Intrusive Path (Velocity & Pressure): The velocity and pressure fields are resolved using the POD-Galerkin projection of the consistent flux formulation. This involves extracting Proper Orthogonal Decomposition (POD) modes from high-fidelity snapshots and projecting the discrete operators (convective, diffusive, gradient, and Laplacian) onto these modes. This path ensures mass conservation and maintains the physical coupling of the Navier-Stokes equations.
   • Non-Intrusive Path (Turbulent Viscosity): Recognizing that intrusive projection fails for the turbulence field, the turbulent viscosity ($\nu_T$) is reconstructed using a data-driven closure. Instead of projecting the viscosity equation, the temporal coefficients of the turbulent viscosity are predicted using neural networks trained on the temporal coefficients of velocity and pressure.

3. Neural Network Architectures:

   • Three architectures were evaluated to model the temporal evolution of the viscosity coefficients: Multilayer Perceptron (MLP), Transformers, and Long Short-Term Memory (LSTM) networks.
   • The LSTM architecture was selected for its ability to propagate information over extended time horizons, which is critical for capturing the history-dependent nature of unsteady turbulent flows.

Key Contributions

• Extension to 3D Turbulence: The work extends previous "discretize-then-project" strategies (previously limited to 2D laminar flows) to fully three-dimensional turbulent configurations, addressing the increased physical complexity and computational demands.
• Hybrid Closure for Collocated Grids: The paper introduces a novel hybrid approach where the turbulent viscosity is handled non-intrusively via machine learning, while velocity and pressure remain governed by the physics-based consistent flux formulation. This avoids the need for pressure stabilization techniques at the reduced level.
• Comparative Analysis of Architectures: The study provides a systematic comparison of MLP, Transformer, and LSTM networks for turbulence closure, identifying the specific strengths of recurrent architectures in this context.

Results
The framework was validated against a 3D Large Eddy Simulation of a lid-driven cavity (unit cube, $50 \times 50 \times 50$ mesh, $Re$ based on lid velocity).

• Model Configuration: 10 POD modes were retained for velocity, pressure, and turbulent viscosity. The first 1800 time steps were used for training, and the remaining 200 for validation.
• Performance Metrics:
  • The LSTM-based closure demonstrated superior performance in capturing transient dynamics compared to MLP and Transformer models.
  • Relative Errors (LSTM):
    • Velocity: 0.7%
    • Pressure: 6.0% (consistent across all architectures)
    • Turbulent Viscosity: 4.0% (significantly lower than MLP at 14.0% and Transformer at 9.0%)
  • Energy and Enstrophy: The LSTM model showed slightly lower relative errors in energy (0.7%) and enstrophy (1.7%) compared to the other architectures.
• Visual Validation: Comparisons of velocity, pressure, and turbulent viscosity fields between the Full-Order Model and the Hybrid ROM showed that the approximation is accurate, with the hybrid technique successfully reproducing the flow structures.

Significance and Claims
The paper claims that the proposed framework effectively bridges the gap between the mathematical consistency of projection-based ROMs and the predictive flexibility of deep learning. By separating the treatment of the flow variables (intrusive for velocity/pressure) and the turbulence closure (non-intrusive for viscosity), the method achieves a stable and accurate reduced-order model for 3D turbulent flows on collocated grids without requiring complex stabilization techniques.

The authors modestly conclude that while the hybrid model successfully captures the main characteristics of the flow evolution for the specific lid-driven cavity test case, additional test cases are required to fully evaluate the approach's performance across different flow conditions and geometries. The study highlights that explicitly accounting for temporal correlations (as done by the LSTM) is crucial for modeling unsteady turbulent dynamics where the closure term depends on recent flow history.