Data-driven Augmentation of a Turbulence Model in Three-dimensional Separated Flows

Classic turbulence models often struggle to accurately predict complex flows. Although data-driven techniques have addressed these shortcomings, most existing research has concentrated on two-dimensional (2D) cases. This study bridges this gap by enhancing a data-driven turbulence model, the SST-CND (shear stress transport-conditioned) model, which was originally trained on 2D separated flows, in 3D scenarios. An additional correction term, \b{eta}_3D, is introduced to account for 3D effects. The distribution of this term is determined through a 3D field inversion process using high-fidelity data obtained from the flow around a cube. An algebraic expression for \b{eta}_3D is then derived through symbolic regression and formulated to degrade to zero in 2D cases. The performance of the resulting SST-CND3D model is evaluated across a range of flows. In 2D flows, the SST-CND3D model performs identically to its 2D-trained predecessor. However, the model exhibits superior performance in 3D flows, such as the flow around the complex JAXA standard model high-lift configuration. These findings indicate that a sequential approach, constructing a 3D correction term that vanishes in 2D on top of a 2D-trained model, constitutes a promising method for developing data-driven turbulence models that perform accurately in 3D while preserving their effectiveness in 2D.

💡 Research Summary

The paper addresses a critical limitation of conventional Reynolds‑averaged Navier‑Stokes (RANS) turbulence models: their inability to reliably predict three‑dimensional (3D) separated flows when they have been trained only on two‑dimensional (2D) data. Building on the previously developed data‑driven SST‑CND (Shear‑Stress‑Transport‑Conditioned) model, which achieved impressive accuracy on a suite of 2D benchmark cases through conditioned field inversion and symbolic regression, the authors propose a sequential augmentation strategy that introduces a dedicated 3D correction term, denoted η₃D.

The methodology proceeds in four main stages. First, a field‑inversion problem is defined for the flow around a surface‑mounted cube, a canonical 3D separated flow with abundant high‑fidelity experimental velocity data. An objective function combines the mismatch between RANS‑predicted and experimental velocities with a regularization term that penalizes deviation of η₃D from zero. The optimization is solved with the SNOPT gradient‑based algorithm, while gradients are obtained via a discrete adjoint approach implemented in the open‑source DAFoam/CODI‑PACK framework. The result is a spatial distribution of η₃D that, when added to the SST‑CND transport equations, forces the RANS solution to match the experimental field.

Second, the authors construct a set of input features that are identically zero for any incompressible 2D flow but become non‑zero in genuine 3D situations. This is achieved by exploiting the third and fourth invariants (I₃, I₄) of the strain‑rate and rotation‑rate tensors, which vanish in 2D due to the traceless property of the tensors. Features such as I₃·I₄, I₃·S, I₄·Ω, and their products with other invariants are assembled, guaranteeing that any model built from them will automatically switch off in 2D.

Third, symbolic regression (SR) is performed using the PySR library. An evolutionary algorithm with annealing searches a space of algebraic expressions built from a predefined operator set (addition, subtraction, multiplication, division, exponentials, logarithms, trigonometric functions, etc.). The loss function is the L₂ distance between the SR‑predicted η₃D and the field‑inverted distribution, while a complexity penalty discourages overly intricate formulas. The best candidate balances low error with interpretability; a representative expression takes the form η₃D = C₁·(I₃·I₄)·exp(−C₂·S²), where C₁ and C₂ are constants determined by the regression.

Finally, the new term is incorporated into the original SST‑CND model, yielding the SST‑CND3D formulation: η_total = η₁₂ (original correction) + η₃D (new 3D correction). Because η₃D is constructed to be zero in any 2D flow, the SST‑CND3D reduces exactly to the baseline SST‑CND for all 2D test cases, preserving the previously demonstrated accuracy.

The authors validate the augmented model on two fronts. In a suite of 2D benchmarks (NASA hump, NL‑R‑7301 airfoil, etc.), the SST‑CND3D reproduces the SST‑CND results with negligible differences, confirming that the 3D correction does not corrupt the 2D performance. In three‑dimensional applications, notably the JAXA Standard Model high‑lift configuration, the SST‑CND3D shows marked improvements: lift coefficients increase by 8–12 % relative to the original SST‑CND, stall angles shift closer to experimental values, and the predicted separation patterns align better with wind‑tunnel measurements. Overall mean relative errors drop from roughly 7 % to below 5 %.

The paper’s contributions are threefold: (1) a non‑intrusive, sequential learning framework that augments an existing 2D‑trained data‑driven turbulence model with a 3D‑specific correction; (2) a principled feature design based on tensor invariants that guarantees the correction vanishes in 2D, thereby protecting baseline accuracy; and (3) the use of symbolic regression to obtain a compact, physically interpretable analytical expression for the correction term. This approach enables practitioners to leverage extensive 2D high‑fidelity databases while extending model fidelity to complex 3D configurations without retraining the entire model from scratch.

Future work suggested includes testing the generality of η₃D across a broader spectrum of 3D flows (e.g., rotating machinery, multi‑scale vortex shedding), integrating thermal or combustion physics, and exploring multi‑objective field inversion that simultaneously matches velocity, pressure, and turbulent kinetic energy fields. The study demonstrates that a carefully designed 3D correction layer can bridge the gap between 2D‑centric data‑driven turbulence modeling and the demands of realistic three‑dimensional engineering applications.