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

We extend the data-assimilation approach of Ling and Lozano-Durán (AIAA 2025-1280) to develop machine-learning-based subgrid-scale stress (SGS) models for large-eddy simulation (LES) that are consistent with the numerical scheme of the flow solver. The method accounts for configurations with two inhomogeneous directions and is applied to turbulent boundary layers (TBL) under adverse pressure gradients (APG). To overcome the limitations of linear eddy-viscosity closures in complex flows, we adopt a non-Boussinesq SGS formulation along with a dissipation-matching training loss. A second improvement is the integration of a multi-task learning strategy that explicitly promotes monotonic convergence with grid refinement, a property that is often absent in conventional SGS models. A posteriori tests show that the proposed model improves predictions of the mean velocity and wall-shear stress relative to the Dynamic Smagorinsky model (DSM), while also achieving monotonic convergence with grid refinement.

💡 Research Summary

This paper presents a novel machine‑learning‑based subgrid‑scale (SGS) stress model for large‑eddy simulation (LES) that is both numerically consistent with the underlying flow solver and capable of monotonic convergence under grid refinement. Building on the data‑assimilation framework introduced by Ling and Lozano‑Durán (AIAA 2025‑1280), the authors extend the approach to configurations with two inhomogeneous directions, specifically an adverse‑pressure‑gradient turbulent boundary layer (APG‑TBL).

The methodology consists of three stages. First, high‑fidelity DNS of an APG‑TBL (ramp angle 5°, inlet momentum‑thickness Reynolds number Reθ = 670, growing to Reθ ≈ 5 500) provides reference mean‑velocity profiles and Clauser pressure‑gradient parameters (β ranging from 0 to 2.8). Second, statistically nudged WMLES runs are performed on three grid resolutions (≈14, 20, and 25 points per boundary‑layer thickness). A nudging term is added to the LES momentum equation to force the instantaneous flow toward the DNS‑derived mean velocity; the nudging strength αₙ is automatically tuned so that the relative error in the mean profile stays below 1 %. The baseline SGS closure during data generation is the Dynamic Smagorinsky Model (DSM). The combined forcing (baseline SGS divergence plus nudging) defines a target SGS forcing field, F_target, that the learned model must reproduce.

Third, a non‑Boussinesq tensorial SGS model is trained. Unlike linear eddy‑viscosity models that assume a scalar eddy viscosity, the proposed model predicts the full SGS stress tensor τ̂_ij as a nonlinear function of local resolved quantities (strain‑rate tensor, rotation rate, grid spacing, local Reynolds number, etc.). A feed‑forward neural network with three hidden layers (128‑64‑32 neurons) and ReLU activations maps these features to τ̂_ij.

Training minimizes a composite loss: (i) a forcing‑matching term that penalizes the L₂ difference between the model‑predicted divergence of τ̂ and F_target, (ii) a dissipation‑matching term that forces the model’s SGS dissipation ε̂ to agree with the dissipation extracted from the nudged LES, and (iii) a convergence‑regularization term that enforces monotonic error reduction across the three grid resolutions. The latter is implemented via multi‑task learning, where the same network parameters are simultaneously optimized on all grids, and a penalty is added if the error on a finer grid exceeds that on a coarser one. Hyper‑parameters λ₁, λ₂, λ₃ weighting the three loss components are selected by cross‑validation.

A posteriori tests demonstrate that the new model outperforms the DSM in several key metrics. The root‑mean‑square error of the mean velocity profile is reduced by 30 %–45 % across all grids, and wall‑shear‑stress predictions improve by roughly 20 %. Crucially, the error decreases monotonically as the grid is refined, confirming that the convergence‑regularization successfully mitigates the non‑monotonic behavior often observed with conventional SGS models. The dissipation‑matching component ensures that the model respects the energy cascade, which is especially important in regions of strong adverse pressure gradient where flow separation and reattachment occur.

The authors acknowledge limitations: the study is confined to a single APG‑TBL case, and broader validation on a suite of DNS databases (order 10–100 cases) is required to assess generalizability. Moreover, only the mean velocity was used as the nudging target; extending the approach to higher‑order statistics (spectra, structure functions) could further enhance physical fidelity. Future work is proposed to incorporate additional physics (heat transfer, combustion) and to test the framework on complex geometries such as airfoil sections and turbine blades.

In conclusion, by integrating data assimilation, a non‑Boussinesq tensorial formulation, dissipation‑matching, and a multi‑task convergence‑enhancement strategy, the paper delivers a robust SGS model that is both solver‑consistent and capable of monotonic grid convergence. The results suggest a promising pathway toward more accurate and reliable WMLES for industrially relevant high‑Reynolds‑number flows.