Numerically Consistent Non-Boussinesq Subgrid-scale Stress Model with Enhanced Convergence

This paper presents a numerically consistent, non-Boussinesq subgrid-scale stress model for large-eddy simulation that leverages data assimilation and multi-task learning to achieve enhanced convergence and improved predictions of turbulent boundary layers under adverse pressure gradients compared to the Dynamic Smagorinsky model.

The Gist

Imagine you are trying to predict how a river flows around a rocky bend. If you try to calculate the movement of every single water molecule, it would take a supercomputer centuries to finish the job. This is what scientists call "Direct Numerical Simulation" (DNS). It's perfect but too slow for real-world engineering.

So, engineers use a shortcut called Large-Eddy Simulation (LES). Think of this like watching the river from a helicopter. You can clearly see the big whirlpools and the main current, but the tiny ripples and eddies are too small to see. To make the simulation work, you need a "Subgrid-Scale (SGS) Model." This model is a guess-keeper that says, "Okay, I can't see the tiny ripples, but I know they exist, so I'll add a little bit of 'friction' to my calculation to account for them."

For decades, these models have been like using a generic, one-size-fits-all friction setting. They work okay in simple rivers, but when the water gets turbulent, hits a ramp, or flows around a complex shape, these generic models often fail. They might predict the water slowing down when it should speed up, or they might get confused as you try to make the simulation more detailed.

The Problem: The "Zoom" Paradox
Usually, if you make a computer simulation more detailed (add more grid points, like zooming in with a camera), the answer should get better. But with these old models, sometimes making the grid finer actually makes the prediction worse. It's like trying to fix a blurry photo by adding more pixels, but the software just adds more noise. This is called "non-monotonic convergence," and it's a major headache for engineers.

The Solution: A Smart, Customized Coach
The authors of this paper, Ling and Lozano-Duran, created a new kind of SGS model using Machine Learning. Instead of guessing the friction, they taught a computer to learn the exact "friction" needed by watching a perfect simulation (the DNS) and then trying to mimic it.

Here is how they did it, using three simple analogies:

1. The "Nudging" Coach
Imagine you are trying to learn to ride a bike, but you only have a blurry map of the path. A "nudging" approach is like having a coach standing next to you. Every time you drift off the perfect path (the DNS data), the coach gives you a gentle push (a "nudge") to get you back on track.
In this paper, the computer runs a simulation and gets nudged toward the perfect data. The computer then records how hard it had to push to stay on track. This "pushing force" becomes the training data for the new model. The model learns: "When the water looks like this, I need to push that hard."

2. The "Non-Boussinesq" Toolbox
Old models were like a hammer: they only knew how to push things in one direction (like simple friction). But real water flow is complex; it twists, turns, and rotates.
The authors built a new model that is more like a Swiss Army Knife. Instead of just one tool, it uses a "tensorial" approach, meaning it has different tools for different directions. It can handle the twisting and turning of the water much better than the old "hammer" models. They call this a "non-Boussinesq" formulation, which is just a fancy way of saying, "We stopped assuming the water behaves simply and started treating it like the complex, twisting fluid it really is."

3. The "Multi-Task" Student
This is the paper's biggest breakthrough. Usually, when you train a machine learning model, you just tell it to be "accurate." But the authors realized that for this to work, the model needs to learn a specific lesson: "As you get more detailed, you must get more accurate."
They used a strategy called "Multi-Task Learning." Imagine a student taking three exams: an easy one (coarse grid), a medium one, and a hard one (fine grid).

• Old way: The teacher grades them all equally. The student might ace the easy test but fail the hard one.
• New way: The teacher tells the student, "The hard test matters 100 times more than the easy one."
  By weighting the training data this way, the model is forced to prioritize getting the fine details right. This ensures that as you zoom in (refine the grid), the answer gets better and better, never worse.

The Results
They tested this new model on a turbulent flow hitting a ramp (like air flowing over a wing that is tilted up).

• Accuracy: It predicted the speed of the air and the pressure on the wall much better than the standard "Dynamic Smagorinsky" model (the current industry standard).
• Convergence: Most importantly, when they made the grid finer, the error went down steadily. The "zoom paradox" was solved. The model behaved exactly as a good simulation should: more detail equals better results.

In Summary
The authors built a smarter, more flexible AI model for simulating turbulent fluids. By teaching it with a "nudging" technique, giving it a complex "Swiss Army Knife" toolkit instead of a simple hammer, and forcing it to prioritize fine details through special training weights, they created a model that is both more accurate and more reliable as simulations get more detailed.

Technical Summary

Technical Summary: Numerically Consistent Non-Boussinesq Subgrid-Scale Stress Model with Enhanced Convergence

Problem Statement
Wall-modeled large-eddy simulation (WMLES) offers a computationally efficient alternative to direct numerical simulation (DNS) for high-Reynolds-number industrial flows. However, conventional subgrid-scale (SGS) models often rely on physical approximations (e.g., linear eddy-viscosity closures) and empirical tuning that fail in complex flow scenarios, such as those with adverse pressure gradients (APG) or separation. A critical limitation in current WMLES practice is the non-monotonic convergence of solutions as grids are refined; errors arising from the interplay between SGS modeling, numerical discretization, and wall models can cause predictions to deteriorate rather than improve with mesh refinement. Furthermore, existing machine learning (ML)-based SGS models often struggle in a posteriori tests because they are typically trained on filtered DNS data (a priori training), which is inconsistent with the numerical errors inherent in LES solvers. While data-assimilation approaches like the Building-Block Flow (BFM) model have shown promise, previous iterations were restricted to configurations with a single inhomogeneous direction (e.g., channel flows) and struggled with separated flows.

Methodology
This study extends the data-assimilation framework of Ling and Lozano-Duran (2025) to develop a numerically consistent, ML-based SGS model for turbulent boundary layers (TBL) under APG, a configuration featuring two inhomogeneous directions. The methodology consists of three primary steps:

1. Reference Statistics and Nudging: The authors utilize a DNS of an APG TBL (with a ramp angle of $5^\circ$ and $Re_\theta = 670$) to define target statistics. Instead of assimilating full trajectories, which is ill-posed for coarse-grained systems, they employ a statistical nudging approach. A corrective forcing term ($F_{nudging}$) is added to the WMLES momentum equations to drive the mean velocity profile toward the DNS target. This generates a "numerically consistent" target forcing ($F_{target}$) that accounts for the specific discretization errors of the solver (charLES, a GPU-accelerated finite-volume code).
2. Non-Boussinesq Model Formulation: To address the limitations of linear eddy-viscosity models in complex flows, the authors adopt a non-Boussinesq tensorial SGS formulation. The model predicts the SGS stress tensor ($\tau^{SGS}$) using two terms: a standard strain-rate term and a rotation-tensor term ($SR - RS$). The coefficients for these terms are parameterized by a multi-layer perceptron (MLP) neural network. The network inputs are dimensionless invariants derived from the strain ($S$) and rotation ($R$) tensors, selected via an information-theoretic approach (Buckingham-$\pi$ theorem) to maximize predictive power.
3. Multi-Task Learning with Enhanced Convergence: The model is trained using a supervised learning strategy that minimizes a combined loss function. This loss includes a forcing-matching term (to reproduce $F_{target}$) and a dissipation-matching term (to ensure the model dissipates energy consistent with the baseline Dynamic Smagorinsky Model, DSM). Crucially, the authors implement a multi-task learning strategy where training data from three different grid resolutions (coarse, medium, fine) are weighted differently. By assigning higher weights to finer grids, the training process explicitly enforces monotonic convergence behavior, addressing the issue where coarser grids sometimes yield lower errors than finer ones in conventional models.

Key Contributions

• Extension to Two Inhomogeneous Directions: The framework is generalized from single-direction channel flows to APG TBLs, accommodating richer physics and secondary mean motions.
• Non-Boussinesq Formulation: The adoption of a tensorial SGS model with a rotation term moves beyond the Boussinesq hypothesis, better capturing complex stress anisotropies.
• Dissipation-Matching Loss: A specific loss component is introduced to constrain the model's energy dissipation, a statistical requirement shown to be essential for capturing smooth-body separation.
• Convergence-Enhanced Training: The integration of resolution-dependent weighting in the loss function is proposed as a mechanism to enforce monotonic grid convergence, a property often absent in standard data-driven SGS models.

Results
Posteriori tests were conducted on the APG TBL case using the trained model compared against the Dynamic Smagorinsky Model (DSM) and the optimal nudged simulation.

• Accuracy: The proposed model significantly outperforms the DSM in predicting mean velocity profiles and wall-shear stress (skin friction), achieving results comparable to the optimal nudged prediction. In some regions, the model appeared to outperform the nudged simulation, likely due to error cancellation between the SGS and wall models.
• Convergence: The study demonstrated that the "equal weights" training strategy resulted in non-monotonic convergence (where the coarsest grid had the lowest error). In contrast, the proposed "unequal weights" strategy (prioritizing finer grids) successfully recovered a consistent, monotonic convergence trend as the grid was refined, although the finest grid remained under-resolved.

Significance and Claims
The paper presents a proof-of-concept for a data-driven SGS modeling approach that is numerically consistent with the flow solver and capable of handling complex, multi-directional flow configurations. The authors claim that by integrating statistical nudging with a non-Boussinesq formulation and a multi-task learning strategy, they have addressed two persistent challenges in WMLES: the inaccuracy of linear closures in complex flows and the non-monotonic convergence of numerical solutions. The work suggests that explicitly promoting monotonic convergence through training loss weighting is a viable path toward more robust, production-ready ML-based SGS models. The study acknowledges limitations, noting that the current nudging scheme is scalar (streamwise only) and that future work is needed to extend this to vector-valued nudging for fully separated flows and to automate the weighting schedule for broader applicability.