CFD-driven symbolic regression of RANS transition models

Louenas Zemmour; Xavier Gloerfelt; Paola Cinnella

physics.flu-dynNov 2, 2026

CFD-driven symbolic regression of RANS transition models

Authors: Louenas Zemmour, Xavier Gloerfelt, Paola Cinnella

Organizations: Sorbonne Université, CNRS, Institut Jean Le Rond D’Alembert, 4 Pl. Jussieu, Paris, 75005, France. · DynFluid Laboratory, Arts et Metiers Institute of Technology, 151 Bd. de l’Hôpital, Paris, 75013, France.

Abstract

Laminar-turbulent transition governs skin friction, heat transfer and losses in turbomachinery, yet Reynolds-averaged Navier-Stokes (RANS) transition models rely on empirical correlations and hand-calibrated activation functions whose structural deficiencies cannot be resolved by recalibration. We propose a CFD-driven sparse symbolic regression framework to learn interpretable, closed-form corrections to the production and destruction activation functions in the intermittency equation of the local, Galilean-invariant $k$ - $ω$ - $γ$ model of Ge et al. Each correction combines a polynomial of a local pressure-gradient parameter, controlling amplitude and sign, with a bounded exponential of local flow features, controlling spatial support. This construction bounds the corrections without ad hoc limiters. Coefficients are identified by gradient-free optimization with the RANS solver in the loop, accelerated by POD-Kriging field surrogates, using mean-velocity data from wall-resolved large-eddy simulations (LES) of the ERCOFTAC T3 flat-plate cases. Elastic-net hyperparameters are selected automatically through bi-level optimization with leave-one-case-out cross-validation. The resulting model retains 11 of 21 candidate terms, compared with 20 under conventional grid-search tuning, while achieving comparable or slightly better accuracy on training and unseen cases. Both approaches recover the same leading-order correction. Applied without retraining to the transonic LS89 turbine cascade, far beyond the training envelope, the augmented model remains numerically robust and improves heat-transfer predictions at all four operating points, reducing errors by up to 70% where the baseline predicts premature transition under high freestream turbulence. Residual errors in transition onset and length highlight the limits of training on flat-plate data alone.

Figures & tables

	T3A	T3B	T3C3	T3C5	T3C2	ZPG-val
Pressure gradient	zero	zero	variable	variable	variable	zero
$Tu^{x=0}_{\mathrm{exp}}$ (%)	3	6	3	3	3	–
$Tu^{x=0}_{x}$ (%)	3.2	6.2	2.7	3.1	2.7	4.0
$Re_{L_{f}}$	2261	7914	713	2319	1249	14350
$M_{\mathrm{in}}^{\mathrm{exp}}$	0.0157	0.0274	0.0108	0.0245	0.0146	–
$M_{\mathrm{in}}$	0.5	0.5	0.2	0.2	0.2	0.1

Table 1: Flow cases of the LES database. For each case: streamwise pressure gradient; turbulence intensity at the leading edge of the plate in the experiments (

Tu^{x=0}_{\mathrm{exp}}

) and streamwise turbulence intensity at the same location in the simulations (

Tu^{x=0}_{x}

); Reynolds number based on the integral length scale of the inflow turbulence,

Re_{L_{f}}

; inlet Mach number in the experiments (

M_{\mathrm{in}}^{\mathrm{exp}}

) and in the simulations (

M_{\mathrm{in}}

); number of grid points and near-wall resolution in wall units. Experimental values from Roach and Brierley (1992) ; Savill (1993) , numerical values from Zemmour et al. (2026) ; Bienner et al. (2024b) .

Figure 1: Structure of the models identified with the bi-level (

M_{B.L}

, top row) and grid-search (

M_{G.S}

, bottom row) strategies. Each column corresponds to one of the 21 candidate terms of Eqs. ( 16 )–( 17 ):

\lambda_{P}^{n}

and

\lambda_{E}^{n}

denote the

n

-th order terms of the production and destruction prefactors, and the remaining labels the monomials of the production (

\eta_{1}

\eta_{2}

) and destruction (

\eta_{1}

\eta_{3}

) exponents. White cells correspond to terms pruned by the sparse regression; colored cells to active terms, the color indicating the value of the coefficient. The color scale goes from -1 to 1 with a step of 0.1.

Cases	Model	$m_{\overline{u}}$ (%)	$m_{\partial\overline{u}/\partial y}$ (%)	$m_{C_{f}}$ (%)	$m_{\mathrm{model}}$ (%)
Training	$M_{G.S}$	23.83	30.46	21.26	25.18
	$M_{B.L}$	25.15	34.42	25.22	28.26
Test	$M_{G.S}$	16.46	40.70	13.01	23.39
	$M_{B.L}$	18.40	41.03	14.90	24.78

Table 2: Improvement scores of the grid-search (

M_{G.S}

) and bi-level (

M_{B.L}

) models with respect to the baseline

k

–

\omega

–

\gamma

model (Eqs. ( 22 )–( 23 )), averaged over the training cases (T3A, T3B, T3C3, T3C5) and test cases (T3C2, ZPG-val), for the mean streamwise velocity, its wall-normal derivative and the skin-friction coefficient;

m_{\mathrm{model}}

is the mean of the three scores. Positive values denote an improvement over the baseline, and

100\%

a perfect agreement with the LES. For

M_{B.L}

, T3A is the validation case of the leave-one-case-out procedure.

Panel of Figure 2 — Figure 2: Zero-pressure-gradient training cases T3A (top, validation case of $M_{B.L}$ ) and T3B (bottom): skin-friction coefficient along the plate (left) and mean-velocity profiles $u/u_{e}$ versus $y/\delta_{99}$ at three streamwise stations across the transitional region, each profile being shifted by its abscissa $x/L$ (right). LES: black line with squares; baseline $k$ – $\omega$ – $\gamma$ model: gray dashed line with circles; $M_{B.L}$ : orange line; $M_{G.S}$ : green line.

Panel of Figure 3 — Figure 3: Pressure-gradient training cases T3C3 (top, transition in the adverse-pressure-gradient region) and T3C5 (bottom, transition in the favorable-pressure-gradient region): skin-friction coefficient (left) and mean-velocity profiles (right). Lines as in Fig. 2 .

Panel of Figure 4 — Figure 4: Test cases, not used for training: T3C2 (top, different pressure-gradient history) and ZPG-val (bottom, zero pressure gradient with a large integral length scale of the freestream turbulence). Skin-friction coefficient (left) and mean-velocity profiles (right); lines as in Fig. 2 .

	MUR129	MUR116	MUR235	MUR241
$Tu$ (%)	0.8	0.8	6	6
$M_{\mathrm{in}}$	0.15	0.157	0.15	0.15
$M_{2}$	0.84	1.09	0.927	1.089
$T_{0}$ (K)	409.2	418.9	413.3	416.4
$T_{w}$ (K)	297.75	297.55	301.15	299.75
Inlet $\mu_{t}/\mu$	1.56	3	15.47	20

Table 3: LS89 operating points: inlet turbulence intensity, inlet and isentropic outlet Mach numbers, inlet total temperature, wall temperature and eddy-viscosity ratio imposed at the inlet of the RANS simulations. Last row: improvement score of

M_{B.L}

on the heat transfer coefficient

H

, relative to the baseline model of Ge et al. Ge et al. (2014) .

Panel of Figure 5 — Figure 5: Heat transfer coefficient $H$ (Eq. 29 ) along the LS89 blade surface, as a function of the curvilinear abscissa $s$ measured from the leading edge ( $s<0$ : pressure side; $s>0$ : suction side), for the four operating points. Symbols: experiments of Arts et al. Arts et al. (1990) ; gray line with squares: baseline model of Ge et al. ; orange line: augmented model $M_{B.L}$ . Transition appears as a sharp rise of $H$ on the suction side.

Panel of Figure 6 — Figure 6: Extrapolation range of the LS89 simulations in feature space: projection of the input features of the corrections onto the first two principal components of the training data (gray: T3A, T3B, T3C3 and T3C5; colors: LS89 cases). The ellipse encloses $95\%$ of the training points, and the fraction of LS89 points lying outside it is indicated in each legend. The dashed rectangle delimits the domain of the training data. Symmetric logarithmic axes.

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Case	QoI	Kriging model	$R^{2}$ (%)	$Q^{2}$ (%)
T3A	$\overline{u}$	Kriging	99.9953	99.9856
	$\partial\overline{u}/\partial y$	Kriging	99.9914	99.9632
T3B	$\overline{u}$	KPLSK	99.9978	99.9887
	$\partial\overline{u}/\partial y$	KPLSK	99.9981	99.986
T3C3	$\overline{u}$	Kriging	99.9685	99.3043
	$\partial\overline{u}/\partial y$	Kriging	99.966	99.7159

Appendix

Table 4: Accuracy of the POD–Kriging surrogates for each training case and QoI (mean streamwise velocity

\overline{u}

and its wall-normal derivative

\partial\overline{u}/\partial y

): selected Kriging model, coefficient of determination on the training (

R^{2}

) and on the test samples (

Q^{2}

Figure 7: Coefficients of the 72 candidate models obtained by the elastic-net grid search on the ZPG training cases T3A and T3B (one row per hyperparameter pair; one column per candidate term, with the notation of Fig. 1 ). Red and blue shades correspond to positive and negative coefficients, respectively; white cells correspond to inactive terms.

Figure 8: Projection of the standardized descriptors of the 72 candidate models of Fig. 7 onto their first two principal components. Dots: candidate models, colored by

k

-means cluster; diamonds: the two cluster centroids,

M_{1}^{\mathrm{ZPG}}

(1) and

M_{2}^{\mathrm{ZPG}}

(2), propagated through the RANS solver; shaded ellipses: extent of each cluster.

Figure 9: Coefficients of the two cluster centroids propagated through the RANS solver, $M_{1}^{\mathrm{ZPG}}$ (model 1) and $M_{2}^{\mathrm{ZPG}}$ (model 2); color code as in Fig. 7 .

Panel of Figure 10 — Figure 10: Cross-validation of the two cluster centroids of the ZPG grid search: skin-friction coefficient predicted on T3A and T3B by $M_{1}^{\mathrm{ZPG}}$ (red solid line) and $M_{2}^{\mathrm{ZPG}}$ (green dashed line), compared with the LES (black line with squares) and the baseline model (gray dashed line with circles).

Subset	$S^{(1)}$	$S^{(2)}$	$S^{(3)}$	$S^{(4)}$
Training cases	T3B, T3C3, T3C5	T3A, T3C3, T3C5	T3A, T3B, T3C5	T3A, T3B, T3C3
Validation case	T3A	T3B	T3C3	T3C5
$m_{\overline{u}}$ (%)	27.15	4.09	1.3	20.4
$m_{\partial\overline{u}/\partial y}$ (%)	37.81	5.78	2.4	33.97
$m_{C_{f}}$ (%)	24.82	2.59	1.14	23.69
$m_{\mathrm{model}}$ (%)	29.92	4.15	1.61	26.02

Appendix

Table 5: Leave-one-case-out splits of the four training cases used in the bi-level optimization, and improvement scores of the resulting models

M_{S^{(k)}}

, each evaluated with the RANS solver on its own validation case. The model obtained with subset

S^{(1)}

is retained as

M_{B.L}

Computational Fluid Dynamics

Computational fluid dynamics numerically models fluid motion, turbulence, heat and mass transport, and fluid–structure interactions.

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Physics-Informed ML Computational Fluid Dynamics

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Turbulence is ubiquitous in engineering and science, yet direct simulation is prohibitively expensive. The Reynolds-averaged Navier-Stokes (RANS) equations provide savings exceeding ten orders of magnitude but introduce unclosed terms (the closure problem). Offline-trained machine-learning (ML) closures suffer distribution shift in predictive simulations, while ML methods that bypass the governing equations struggle to generalise from scarce high-fidelity data. We develop a physics-derived deep learning closure model for RANS, the Deep Algebraic Reynolds Stress Model (DARSM), which can be trained on small datasets and accurately generalise across Reynolds numbers, to unseen geometries, and to different flow regimes. A neural network maps flow invariants to empirical parameters in an implicit algebraic Reynolds stress equation, derived from the Reynolds stress transport equations under the weak-equilibrium assumption, imposing physics-based structure on the ML closure. End-to-end optimisation through the governing PDEs and the coupled implicit closure eliminates distribution shift, but both unrolled and implicit automatic differentiation fail on the stiff coupled solver. We derive adjoint equations that exploit the solver's implicit-explicit structure for efficient optimisation. On canonical square-duct and periodic-hill benchmarks, DARSM reduces average test velocity error over baseline RANS by

2

4\times

across Reynolds number, geometries, and flow regimes, with peak case-level reductions of

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Abstract

Computational Fluid Dynamics

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