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A generalised Spalart-Allmaras model with blended near-wall and free-shear coefficients: multi-objective calibration in an open-source framework

This paper presents a generalized Spalart-Allmaras turbulence model that employs a shielding function to blend near-wall and free-shear coefficients, achieving improved accuracy in separated and free-shear flows while maintaining robustness in attached flows through multi-objective calibration within an open-source framework.

Original authors: Tony Di Fabbio, Riccardo Vallino, Eric Segalerba, Viola Rossano, Joel Guerrero, Andrea Ferrero

Published 2026-10-23

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CC BY 4.0

Original authors: Tony Di Fabbio, Riccardo Vallino, Eric Segalerba, Viola Rossano, Joel Guerrero, Andrea Ferrero

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Fluids surround us, from the air rushing over an airplane wing to the water swirling in a pipe. Predicting how these fluids move is essential for designing efficient vehicles, safer buildings, and cleaner energy systems. To do this, engineers rely on computer simulations that solve complex equations describing the motion of every particle. However, these equations are too difficult to solve exactly for real-world problems, so scientists use simplified models. One of the most trusted tools for decades has been a model named after its creators, Spalart and Allmaras. It works remarkably well for smooth, attached flows, like air gliding along a wing, because it uses a single, fixed set of rules. But when the flow gets messy—separating from a surface, swirling in a corner, or mixing freely in a jet—those same fixed rules often fail. The challenge has been to make the model smarter without breaking the simplicity that makes it so useful.

A team of researchers from Italy has taken a significant step toward solving this dilemma. They have created a generalized version of the classic model that can adapt its behavior depending on where the fluid is and what it is doing. Instead of forcing the entire simulation to follow one rigid set of instructions, their new model allows the rules to change locally. In regions where the fluid clings to a wall, it uses one set of parameters; in regions where the fluid is detached or mixing freely, it switches to a different set. These two sets of rules are blended together smoothly by a digital sensor that detects the local flow conditions. The researchers did not guess these new rules; they used a powerful computer optimization process to find the best possible values for them, testing the model against a wide variety of benchmark flows, from flat plates to complex jets and separated regions.

The results show that this approach successfully improves the model's accuracy in difficult areas without ruining its performance in the easy ones. For instance, in tests involving a free jet of air, the new model reduced the prediction error by a factor of twenty compared to the original. In tests involving a hump on a wall that causes the flow to separate, the error was cut by one-fifth. Crucially, the model retained its high accuracy for the smooth, attached flows where the original version was already excellent. This suggests that the limitations of the old model were not due to a fundamental flaw in its structure, but rather to the fact that it tried to use the same constant settings for every type of flow. By allowing the settings to vary, the researchers unlocked a much higher level of precision.

However, the study also revealed the limits of what can be fixed just by changing numbers. In one specific test involving a square duct, the flow created a secondary motion in the corners that the standard model could not reproduce, no matter how the coefficients were adjusted. This is because the physics driving that motion requires a more complex relationship between the fluid's stress and its shape, which a simple linear model cannot provide. To solve this, the researchers added a specific nonlinear correction to the model. When they did this, the model successfully generated the swirling corner vortices seen in high-fidelity data, reducing the error in that specific case by 85 percent. This finding separates the problems that can be solved by tuning coefficients from those that require a deeper change in the model's structure.

The team also tested how well their new models would perform on flows they had never seen before, such as an airplane wing stalling or a shock wave interacting with a surface. The model calibrated on external flows performed very well on these unseen cases, maintaining the accuracy of the original model while offering improvements in specific areas. This indicates that the new approach is not just a method limited to the test cases used during training, but a robust improvement that can be applied to real-world engineering challenges. The researchers have released their code and data as open-source software, allowing other scientists to build upon this work. Their success demonstrates that turbulence modeling can evolve by becoming more flexible and adaptive, bridging the gap between simple, robust tools and the complex reality of fluid motion.

Technical Summary: A Generalised Spalart-Allmaras Model with Blended Near-Wall and Free-Shear Coefficients

Problem Statement
The Spalart-Allmaras (SA) model is an industrial standard for Reynolds-averaged Navier-Stokes (RANS) simulations due to its robustness and low computational cost. However, its accuracy relies on a single set of globally constant coefficients calibrated on equilibrium wall-bounded flows. This uniformity limits its performance in separated flows, free-shear regions, and complex geometries where turbulence physics differ significantly from the calibration baseline. While previous efforts have added targeted corrections (e.g., rotation/curvature corrections) or widened the calibration space for specific mechanisms, they often retain fixed global coefficients or require zonal switching. The central question addressed is how far the SA model can be generalised to adapt to local flow regimes without losing its inherent robustness or requiring complex user-defined switching criteria.

Methodology
The authors propose a Generalised Spalart-Allmaras (GSA) model where selected calibration coefficients ( $\sigma_{\tilde{\nu}}$ , $C_{b1}$ , $C_{b2}$ ) possess two distinct states: an Inner set for attached near-wall turbulence and an Outer set for detached and free-shear regions. These states are blended continuously in space using a shielding function ( $F_{SA}$ ) adapted from Delayed Detached-Eddy Simulation (DDES) and Improved DDES (IDDES) concepts. The blending ensures the model adapts to the local flow regime without zonal decomposition.

Key methodological components include:

Coefficient Blending: Each coefficient $\psi$ is computed as $\psi(x) = F_{SA}(\psi_I - \psi_O) + \psi_O$ . The wall-destruction coefficient $C_{w1}$ is reconstructed locally to preserve the logarithmic-layer consistency relation in both limiting states.
Shielding Function: A hybrid function combining a wall-distance branch (protecting attached boundary layers) and a vorticity-Reynolds number branch (detecting separation and free shear). A topology-sensitive extension was introduced to raise the "floor" of the vorticity sensor in rotation-dominated regions, addressing conflicts between separation prediction and secondary flow generation.
Nonlinear Extension: To address the inability of linear eddy-viscosity closures to reproduce anisotropy-driven secondary flows (e.g., in square ducts), the model incorporates the Quadratic Constitutive Relation (QCR2024) with calibrated coefficients.
Multi-Objective Calibration: The coefficients were calibrated using an open-source framework coupling OpenFOAM-13 with evolutionary optimisers (PSO for single-case seeding, NSGA-II for multi-case suites). The optimisation minimised a cost vector where every error term was normalised by the baseline SA error, ensuring that improvements in one flow class did not hide degradations in another.
Benchmark Suites: Calibration was performed on two distinct suites: the NASA Turbulence Modeling Resource (TMR) suite (flat plate, channel, jet, hump) and the ERCOFTAC suite (steep/gentle hills, square duct, curved backward-facing step).

Key Contributions and Results
The study systematically separates limitations arising from coefficient selection from those inherent to the closure structure.

Generalisation without Compromise: The GSA model preserves the accuracy of the baseline SA model for attached wall-bounded flows (flat plate, channel) while significantly improving performance in challenging regimes.
- On the NASA suite, the GSA-N variant reduced the jet objective by a factor of twenty and the wall-mounted hump objective by one-fifth compared to baseline SA.
- On the ERCOFTAC suite, the GSA-E variant improved predictions for gentle hills and curved steps.
Coefficient Conflict and Transferability: The study highlights that different flow classes require conflicting Outer states. For instance, a free jet requires reduced production ( $C_{b1O}$ ), while pressure-induced separation on a hump requires increased production. Consequently, a model calibrated on the NASA suite (GSA-N) transfers well to ERCOFTAC cases but fails on the NASA jet if calibrated on ERCOFTAC (GSA-E), where the Outer production grows too high, collapsing the jet potential core.
Structural Limitations and QCR: The paper demonstrates that no set of coefficients can generate the secondary flow in a square duct, as this is a Reynolds-stress anisotropy effect beyond linear closures.
- A topology-sensitive shielding floor partially relieved the trade-off between predicting separation and secondary flow.
- The calibrated QCR2024 extension (GSA-E+QCR2024) successfully generated the corner vortices in the square duct, reducing the duct objective by 85% compared to the blending-only model, while also improving hill predictions.
Validation on Unseen Cases: The models were tested on three unseen configurations: a NACA 0012 airfoil polar, an axisymmetric transonic bump (ATB), and a wing-body junction (WBJ).
- GSA-N preserved the stall characteristics of the baseline SA on the airfoil, whereas GSA-E delayed stall due to high Outer production (a deficit corrected by QCR2024).
- On the ATB, the blending models did not degrade shock-boundary-layer interaction physics.
- On the WBJ, GSA-N reduced velocity errors by 15–18% compared to baseline, though the vortex topology remained largely unchanged, suggesting the remaining error is due to the steady RANS assumption rather than coefficient calibration.
Realisability: An anisotropy-invariant analysis confirmed that the modelled Reynolds stress remains physically realisable (positive semi-definite) in the square duct and in shear/rotation-dominated regions of the junction, even with the nonlinear QCR terms.

Significance and Claims
The paper claims that the proposed generalisation extends the applicability of the SA model without sacrificing its robustness or portability. The primary significance lies in the systematic disentanglement of model deficiencies:

Coefficient vs. Structure: The work identifies which errors can be fixed by calibrating coefficients (e.g., jet spreading, separation length) and which are structural (e.g., secondary flows in ducts), necessitating nonlinear constitutive relations.
Regime-Adaptive Calibration: It demonstrates that a single model formulation can adapt to different flow regimes via local coefficient blending, avoiding the need for zonal switching.
Data-Driven Interpretability: By using multi-objective optimisation on benchmark suites, the study locates specific coefficient states driving gains and trade-offs, offering an interpretable alternative to black-box data-driven corrections.

The authors conclude that while the GSA model improves accuracy across a broad range of flows, the "best" model depends on the target application. GSA-N is preferred for external aerodynamics where stall prediction is critical, while GSA-E+QCR2024 is superior for internal flows with strong secondary motion. The work suggests that future turbulence modelling should focus on physically grounded, interpretable models where coefficients adapt to flow topology, rather than purely data-driven corrections. All software, data, and calibration harnesses are released as open-source to facilitate further development.

Technical Summary: A Generalised Spalart-Allmaras Model with Blended Near-Wall and Free-Shear Coefficients

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