RESULTS

Unified foundation model

We developed a unified turbulence modeling framework that is trained once and generalizes well across a broad range of flows, including attached boundary layers, separated, secondary, and free-shear flows. Based on this framework, we construct a unified foundation model that integrates multiple diverse flow mechanisms into a single formulation. Unlike most existing data-driven models, which are trained for narrow flow categories and tend to degrade outside their training regimes [13], our model performs well on the training flows and improves predictions for unseen flows exhibiting similar flow mechanisms. This directly addresses long-standing challenges emphasized in recent reviews and community discussions [11–13], particularly the need for general turbulence models that deliver consistent, broadly applicable improvements while remaining robust and free from manual zoning. Such requirements were central to the NASA 2022 Collaborative Testing Challenge [11]. Our unified foundation model makes substantial progress toward this goal, delivering robust and accurate predictions across multiple flow regimes. The NASA challenge cases, included in our evaluation, also demonstrate clear performance improvements (Section S5).

Model training and evaluation setup

The unified foundation turbulence model is trained on nine flows representing distinct flow mechanisms, identified using the distribution-based training set selection method. These include a curved step and a periodic hill for internal separated flows; a bump, a hump, and an S809 airfoil at high angle of attack for external separated flows; two square ducts at different Reynolds numbers and a rectangular duct for secondary flows; and a round jet for free-shear flows. Each case provides sparse, indirect, and heterogeneous observations tailored to its nature: sparse velocity measurements for the curved step, periodic hill, bump, hump, square and rectangular ducts, and round jet; and aerodynamic forces for the S809 airfoil. In total, 10 training objectives are defined, as the S809 airfoil case involves two competing objectives: lift and drag. Once trained, the model’s generalization is tested on 27 unseen cases without manual zoning or parameter tuning, as shown in Fig. 2a and Table 1 (Section S4).

Evaluation metric

The unified foundation model performs well across all training cases and generalizes effectively to unseen cases, including both those within the training flow categories and complex three-dimensional configurations. Its performance is evaluated on diverse test cases spanning attached boundary layers, separated flows, secondary flows, free-shear flows, and complex three-dimensional flows. The model’s performance across training flow categories is shown in Fig. 2b. It is evaluated using the normalized misfit \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\hat{e} = \frac{e_{\mathrm{unified}} - e_{\mathrm{single}}}{e_{\mathrm{base}} - e_{\mathrm{single}}}$\end{document}, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $e_{\mathrm{unified}}$\end{document} is the misfit of the unified foundation model relative to the ground truth, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $e_{\mathrm{base}}$\end{document} is the misfit of the baseline model, that is, Wilcox (1988) k–ω model [51], and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $e_{\mathrm{single}}$\end{document} is the misfit of a single-case trained model, which defines the upper performance limit. A value of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\hat{e} < 1$\end{document} indicates improved performance over baseline, while \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\hat{e} > 1$\end{document} indicates degradation.

Performance evaluation of the unified foundation model

Overall, the unified foundation model generalizes well, achieving lower misfits than the baseline in most cases and comparable performance otherwise. For attached boundary layers, such as the zero pressure gradient flat plate, the model matches the baseline model in regimes where linear eddy-viscosity models are already reliable, recovering the law of the wall and maintaining accuracy across geometries and Reynolds numbers. In free-shear flows, represented here by a round jet, the model provides more accurate predictions of centerline velocity decay, a key metric that reflects the jet spreading rate. Since this behavior is strongly influenced by the turbulent transport equations, the improved prediction highlights the importance of maintaining physical consistency between the constitutive relation and the transport equations. For secondary flows in ducts with different Reynolds numbers and aspect ratios, the model better captures in-plane flows, which require a nonlinear eddy-viscosity model to represent the anisotropic stress driving the secondary motion, and maintains accuracy under both geometric and Reynolds number extrapolation. For separated flows, the model substantially improves predictions of massive separation, such as periodic hills with varying slope steepness, bumps with different heights, the hump, the curved step, and S809 airfoil at high angles of attack. These improvements mainly arise from the spatial variation of the leading tensor-basis coefficient \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $g^{(1)}$\end{document} shown in Fig. 1a, which represents the eddy-viscosity contribution and directly governs separation and reattachment. For complex three-dimensional flows, including the generic car, three-dimensional diffuser, and generic aircraft, the model generally improves predictions across different flow mechanisms, while remaining comparable to the baseline otherwise. The generic car shows a modest improvement in separation prediction over the baseline; the three-dimensional diffuser recovers the ground-truth skin-friction coefficients along the bottom-wall midsection, where secondary flow interacts with separation; the generic aircraft exhibits improved skin-friction and drag predictions under large angles of attack, where multiple interacting flow mechanisms are present with separation being the dominating effect. The overall improvements for complex three-dimensional flows remain limited, with the diffuser separation location showing some improvement but still exhibiting noticeable discrepancies. Detailed performance of the unified foundation model on seven representative test cases is shown in Fig. 3. The unified foundation model demonstrates improved generalization across diverse flow regimes, achieving better, or at least comparable, accuracy to the baseline model in all test cases.

Figure 3.: Performance evaluation of the unified foundation model on representative canonical flows and complex three-dimensional flows, comparing the ground truth, baseline model, and unified foundation model. The flat plate represents the attached boundary layer, with the inner-scaled velocity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $u^+$\end{document} vs. the wall distance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $y^+$\end{document} and the logarithmic law shown for reference. The round jet represents the free-shear flow, with the normalized streamwise velocity along the jet centerline. The periodic hill represents the separated flow, where the shaded regions from the ground-truth streamwise velocity field indicates regions with u< 0, darker shading corresponds to stronger backflow, and contour lines mark u=0. The square duct represents the secondary flow, with cross-sectional in-plane velocity streamlines illustrating the secondary motion. The complex three-dimensional cases include the generic car, where the separated wake region is visualized using the same ground-truth u< 0 shading and u=0 contours; the three-dimensional diffuser, with the skin-friction coefficient \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $c_f$\end{document} along the bottom wall midsection; and the generic aircraft, with surface distributions of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $c_f$\end{document} and the drag coefficient.

Specialist model

The specialist model further improves prediction accuracy for targeted flows through additive fine-tuning of the unified foundation model on three representative cases with related flow mechanisms. Building on the unified foundation model, it preserves much of the foundation performance while reallocating model capacity toward the target mechanisms. Although some generalization capability may be reduced, this trade-off yields substantially improved accuracy for the targeted flows. Such specialization reflects common industrial practice, where high fidelity for specific applications is prioritized over uniform performance.

Model training and evaluation setup

The specialist model targets flows dominated by separation and secondary flows. Fine-tuning is performed using three canonical cases: the curved step representing separated flows, and rectangular ducts with aspect ratios of 3 and 10 representing secondary flows. These cases are selected as the three closest matches to the target flows using the distribution-based training case selection method. This combination enables the model to improve its performance for both secondary and separated flow mechanisms. The specialist model is evaluated on all benchmark canonical flows and further tested on an asymmetric three-dimensional diffuser flow, which involves interactions between the fine-tuned flow mechanisms. The three-dimensional diffuser is a challenging validation case featuring incompressible, asymmetric internal flow with strong adverse pressure gradients and significant Reynolds-stress anisotropy. The resulting three-dimensional separation closely mirrors practical diffuser behavior, making it an ideal benchmark for evaluating separation and secondary flow prediction capabilities.

Performance evaluation of the specialist model

The specialist model maintains good generalization across all benchmark canonical flows, while delivering clear performance enhancements for the fine-tuning flow categories and, in particular, for the complex three-dimensional diffuser case. With additive fine-tuning, the model reduces the over-prediction of separation in the curved step and accurately recovers in-plane secondary flows in rectangular ducts. While the unified foundation model improves predictions for both flows, fine-tuning delivers substantially higher accuracy. Moreover, the specialist model captures diffuser separation more accurately, correcting the baseline k–ω model’s sidewall separation and aligning it with the ground truth, where separation occurs along the upper wall in the expansion region. The evaluation results are summarized as follows:

(i) Generalization in benchmark flows. The specialist model remains robust across all benchmark canonical flow cases. Within the fine-tuning flow categories, including secondary and separated flows, most cases show clear performance improvements relative to the unified foundation model. Outside these categories, predictive accuracy is mildly degraded and remains comparable to, or slightly below, the baseline model. Nevertheless, the specialist model continues to satisfy essential physical constraints, including consistency with the logarithmic law of the wall (Section S5).

(ii) Generalization to complex 3D flow. In the three-dimensional diffuser, where secondary flows interact with separation, the specialist model suppresses spurious sidewall separation and correctly predicts the primary upper-wall separation observed in the ground truth, as shown in Fig. 4. Cross-sectional analyses show that the specialist model accurately captures the growth and reattachment of the separated region, while the baseline model misplaces both the location and extent of separation (Section S5). The specialist model’s success on this complex configuration demonstrates strong generalization beyond the training manifold and highlights its ability to capture interacting flow mechanisms in industrial applications.

Figure 4.: Significant performance improvement of the specialist turbulence model for a complex three-dimensional diffuser flow. (a) The diffuser geometry and sampling planes are shown for evaluation, with x and z denoting the streamwise and spanwise directions, respectively. Plane 1 is a spanwise cross-section at z/B=0.75, where B is the diffuser inlet width. Planes 2–4 are streamwise cross-sections extracted at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $x/h{\mathrm{d}}=5$\end{document}, 8, and 15, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $h_{\mathrm{d}}$\end{document} is the diffuser inlet height. The ground truth on Plane 1 is obtained from experiments [49], whereas the ground truth on Planes 2–4 is obtained from the DNS mean flow field [52]. (b) The separation predictions from the ground truth, baseline model, and specialist model are compared on these planes. Flow separation is identified by negative streamwise velocity u< 0: shading denotes flow separation in the ground truth, with darker shading indicating stronger reverse flow, while the u=0 contours enable clear comparison of the separation predictions._

Interpretation for model unification

The unified turbulence model combines two complementary strategies for flow mechanism unification: automatic balancing of overlapping input features and internal branching for case-specific features across multiple flows. In overlapping regions, where identical features occur in different flows, the model resolves conflicts through multi-objective learning. In case-specific regions, it learns distinct mappings for each flow, acting as a piecewise function with internal branching. Together, these mechanisms enable the model to generalize across flows governed by distinct yet interacting flow mechanisms. We next use the unified foundation model to interpret how these strategies enable generalization.

The unified foundation model’s input feature analysis reveals that different flows exhibit overlapping features while also maintaining distinct case-specific structures. We project the four frame-invariant input features into two dimensions using t-SNE for visualization [53]. The feature distributions of two representative flows, periodic hill and square duct, are shown in Fig. 5a. The distributions contain both overlapping and case-specific regions, indicating that the unified foundation model aims to capture shared structures across flows while preserving case-specific characteristics.

Figure 5.: Feature-space overlap and conflict resolution enabled by the unified foundation model. (a) The t-SNE projection reduces the four-dimensional input feature space to two dimensions for visualization and shows the feature distributions of the periodic hill and square duct, including case-specific regions as well as overlapping regions shared by both flows. (b) For overlapping features, the predicted coefficient functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $g^{(1)}$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $g^{(2)}$\end{document} are compared in the shared feature regions for the single-case trained models and the unified foundation model, showing how the unified foundation model balances conflicting model responses. (c) For non-overlapping features, neuron activation patterns in the sub-network associated with the quadratic coefficient \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $g^{(2)}$\end{document} are shown separately for periodic hill and square duct features, indicating case-dependent branching within the unified foundation model.

Balancing in overlapping regions

Overlapping regions in the feature space serve as critical zones where the model must resolve conflicts among cases to enable joint learning of interacting mechanisms. Because identical features in these regions can correspond to different outputs across cases, deterministic models alone cannot eliminate the resulting discrepancies. Multi-objective learning addresses this by adjusting the contribution of each case during training, promoting balanced optimization and compelling the model to reconcile conflicting objectives. As shown in Fig. 5b, the single-case trained models often map overlapping features to significantly different outputs, creating distinct regions in the output space and limiting generalization. The unified foundation model resolves these conflicts by maintaining a balanced representation, achieving an intermediate and more generalizable state. This balancing not only enhances generalization but also preserves the ability to capture genuine interactions where multiple mechanisms coexist.

Origins of feature space overlapping

Conflicts can originate not only from different flow mechanisms across different flows, but also from conflicting quantities of interest within the same flow. Quantities of interest from the same flow are not necessarily aligned: while some probe the same dominant mechanism and respond consistently to model corrections, others reflect different physical sensitivities. When training relies on only a subset of these quantities, the resulting model can neglect unresolved mechanisms, leading to degraded predictions for the others and the appearance of conflicting objectives. For example, in the curved-step flow, the velocity field and skin-friction coefficient are aligned, both governed by separation and reattachment dynamics. In contrast, for the S809 airfoil, lift and drag impose conflicting corrections on the dominant closure coefficient. This conflict is reflected not only at the level of integrated quantities but also in the spatial structure of the learning signals: the gradients with respect to the turbulent eddy viscosity induced by lift and drag exhibit opposing patterns across the flow domain (see Fig. S10), indicating that these two objectives drive the model toward fundamentally different flow adjustments. These competing directions correspond to distinct physical pathways, with lift favoring enhanced separation and drag favoring increased wake mixing (Section S3.2). Together, these observations provide a concrete physical basis for the conflicts observed in overlapping feature regions and motivate the need for multi-objective learning to reconcile them within a unified framework.

Branching in case-specific regions

In non-overlapping regions of the feature space, the network learns case-specific mappings, and effectively behaves like a piecewise function. To study this behavior, we extract case-specific features and pass them through the unified foundation model. For each neuron, we compute its activation rate, defined as the fraction of inputs producing a positive output. For rectified linear units (ReLU), this is equivalent to the proportion of non-zero outputs. The analysis is performed separately for the periodic hill and square duct. The activation patterns of the sub-network for the quadratic coefficient \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $g^{(2)}$\end{document} are shown in Fig. 5c. Dark neurons and connections represent pathways activated in both the periodic hill and square duct cases, highlighting their similarities. In the periodic hill, most first-layer neurons are active, with one additional neuron uniquely activated (blue), but the second layer channels information almost entirely through a single pathway. In contrast, in the square duct, second-layer activations are more widely distributed, with several unique pathways remaining active (green). These distinct connection patterns demonstrate how the model reconfigures its internal branching to adapt to the statistical and structural characteristics of each flow.