oRANS: Online optimisation of RANS machine learning models with embedded high-fidelity data generation

Daniel Dehtyriov; Jonathan F. MacArt; Justin Sirignano

physics.flu-dynOct 3, 2025

oRANS: Online optimisation of RANS machine learning models with embedded high-fidelity data generation

Authors: Daniel Dehtyriov, Jonathan F. MacArt, Justin Sirignano

Organizations: Mathematical Institute, University of Oxford, Andrew Wiles Building, Woodstock Rd, Oxford, OX2 6GG, United Kingdom · Department of Aerospace and Mechanical Engineering, University of Notre Dame, 369 Fitzpatrick Hall of Engineering, Notre Dame, IN 46556, USA

Abstract

Deep learning (DL) has shown promise for flow simulations, but progress is constrained by the scarcity of high-fidelity training data, which is costly to generate and limited to few flow conditions. Closures trained offline on such data tend to overfit and fail to generalise to new regimes. We introduce an online optimisation framework for DL-based Reynolds-averaged Navier-Stokes (RANS) closures. Training data is generated by a simulation at direct numerical simulation (DNS) resolution embedded in a subdomain of the RANS domain. The RANS solution supplies boundary conditions to the embedded simulation, which provides the mean velocity and turbulence statistics used to update the closure, which thereby adapts to the target flow without precomputed datasets. The approach is developed on the stochastically forced Burgers equation and on turbulent channel flow at $Re_τ=180$ , 270, 395 and 590 with embedded domain lengths $1\leq L_0/L\leq 8$ , and is then applied to spanwise-rotating channel flow at $Ro_b=0.15$ and $0.5$ , a flow in which the Coriolis force vanishes from the equations of a linear eddy-viscosity closure. Online-optimised RANS models outperform offline-trained and literature-calibrated closures with modest embedded subdomains, and in the rotating channel recover the rotation-induced asymmetry from non-rotating inflow turbulence and a rotation-blind closure. The learned coefficients are interpretable, recovering near-wall damping and, under rotation, the collapse of the eddy viscosity on the stabilised side. Performance degrades when boundary contamination dominates or domains are too short to capture low-wavenumber modes. This framework provides a route to data-adaptive machine learning closures without large precomputed training datasets.

Figures & tables

Figure 1: Schematic of the coupled RANS and embedded simulation framework. The primary RANS-ML domain provides boundary conditions and forcing (mean velocity, turbulent kinetic energy, Reynolds stresses) to the embedded high-resolution region, which in turn supplies high-fidelity data to update the RANS closure and boundary conditions. This coupling enables consistent training of turbulence models on statistically representative flow fields.

Figure 2: Solutions to the stochastically forced Burgers equation with embedded domains of size (left)

L_{e}/L_{0}=1/4

and (right)

L_{e}/L_{0}=1/8

. The panels compare the true means

\bar{u}_{\textrm{ref}}

k_{\textrm{ref}}

with online-trained model predictions (in-sample and out-of-sample), and no-model baseline. The results show that the trained model reproduces the true statistics more accurately than the no-model baseline, including in out-of-sample settings.

Domain Fraction	$\overline{J_{u}}$	$\overline{J_{k}}$
$L_{e}/L_{0}=1/4$	0.771	0.738
$L_{e}/L_{0}=1/8$	0.871	0.927

Table 1: Relative error in mean velocity and turbulent kinetic energy for the oRANS model, normalised by the no-model baseline. Values below unity indicate improvement, confirming the gains observed in figure 2 for both embedded domain sizes.

Figure 3: Learned turbulence closure functions for stochastically forced Burgers turbulence. The oRANS-ML trained mixing length

\ell_{m}^{\mathrm{NN}}

(black) and turbulence model coefficients

C_{\mu}^{\mathrm{NN}}

(red) and

C_{D}^{\mathrm{NN}}

(blue) are shown, with standard constants for Navier–Stokes turbulence (Pope, 2000) indicated by dotted lines (

C_{\mu}=0.55

C_{D}=C_{\mu}^{3}

). The learned parameters deviate from the classical values and vary spatially, reflecting adaptation to the flow.

$Re_{\tau}$	Domain size $(L_{1},L_{2},L_{3})/\delta$	Grid resolution $(N_{1},N_{2},N_{3})$	Grid spacing $\Delta x^{+},\Delta y^{+}_{\min},\Delta z^{+}$
160	$4\pi\times 2\times 2\pi$	$192\times 128\times 160$	$10.5,\ 0.16,\ 6.3$
180	$4\pi\times 2\times 2\pi$	$192\times 128\times 160$	$11.8,\ 0.18,\ 7.1$
270	$4\pi\times 2\times 2\pi$	$384\times 256\times 320$	$8.8,\ 0.13,\ 5.3$
395	$4\pi\times 2\times 2\pi$	$384\times 256\times 320$	$12.9,\ 0.19,\ 7.76$
590	$4\pi\times 2\times 2\pi$	$768\times 256\times 640$	$9.7,\ 0.28,\ 5.8$

Table 2: Simulation parameters for reference DNS turbulent channel flow at the friction Reynolds numbers

Re_{\tau}

considered. Domain sizes are given in units of the channel half-height

\delta

, and the resolutions ensure grid spacing within DNS standards for near-wall turbulence.

Figure 4: Relative adjoint gradient error as a function of the finite-difference perturbation size

\epsilon

. For small

\epsilon

, floating-point round-off error dominates, while for large

\epsilon

, finite-difference truncation error dominates. This characteristic curve, with low minimum error, confirms the correctness of the adjoint implementation.

Panel of Figure 5 — Figure 5: Comparison of RANS closure models at $Re_{\tau}=180$ . Top row: mean velocity $u^{+}$ (left) and turbulent kinetic energy $k^{+}$ (right). Bottom row: specific dissipation rate $\omega^{+}$ (left) and objective function $J$ decay during training (right). Results are shown for DNS, the default $k$ – $\omega$ model, a parametric closure (equation 62 ), and both local (equation 64 ) and global (equation 63 ) feature neural-network closures. The neural closures closely reproduce DNS statistics and converge effectively, in contrast to the default and parametric baselines; note that $J$ is optimised only over $u$ and $k$ , and $\omega^{+}$ is included only for reference.

u^{\prime}_{\mathrm{inlet}}(t_{n})=\sqrt{\tfrac{k^{\mathrm{RANS}}}{k^{\mathrm{eDNS}}}}\,u^{\prime\mathrm{eDNS}}(t_{n})+\overline{u}^{\mathrm{RANS}},

Algorithm 1 Online Training Algorithm for the RANS closure with an embedded simulation

Figure 6: Schematic of the oRANS computational workflow for turbulent channel flow. The RANS-ML solver (blue,

\Omega_{\mathrm{RANS}}

) runs concurrently with the embedded simulation (red,

\Omega_{\mathrm{eDNS}}

), exchanging boundary conditions and statistical feedback through a replay buffer.

Panel of Figure 7 — Figure 7: Comparison of velocity and turbulent kinetic energy profiles from DNS with periodic boundary conditions at $Re_{\tau}=180$ , using various shortened streamwise domain lengths $L_{x}$ . While domains of length $L_{x}\geq 2\pi\delta$ reproduce the expected turbulent statistics, excessively short domains (e.g. $L_{x}=\tfrac{2}{3}\,\pi\delta$ ) lead to laminarisation.

Case ( $Re_{\tau}$ )	$k$ – $\omega$	Periodic DNS, $L_{x}=$
		$4\pi\delta$	$2\pi\delta$	$\tfrac{4}{3}\pi\delta$	$\tfrac{2}{3}\pi\delta$	$\tfrac{1}{2}\pi\delta$
II (180)	1.0	0.00	0.10	0.20	131	131
III (270)	1.0	0.00	0.06	0.07	0.11	0.38
IV (395)	1.0	0.00	0.10	0.03	0.21	0.19
V (590)	1.0	0.00	0.89	0.28	0.63	0.61

Table 3: Normalised cost functional

J^{*}

of velocity and turbulent kinetic energy for baseline RANS (

k-\omega

), and periodic boundary condition DNS with different streamwise domain fractions

L_{x}

. Values below

1

indicate improvement over the baseline RANS. Very large values correspond to laminarisation.

Case	$Re_{b}$ of inflow DNS	Target $Re_{b}$	Target $Re_{\tau}$
I	5600	5050	160
II	5050	5600	180
III	5600	9000	270
IV	5600	13752	395
V	5600	21944	590

Table 4: Setup of the oRANS channel flow numerical experiments. Each case is defined by the bulk Reynolds number

Re_{b}

of the periodic reference DNS from which the inflow fluctuations are rescaled (second column) (the same

Re_{\tau}=180

simulation for cases I and III–V, and the

Re_{\tau}=160

simulation for case II), the target bulk Reynolds number and the corresponding target friction Reynolds number

Re_{\tau}

. The experiments span a range of

Re_{\tau}

from 160 to 590 to test model performance across increasing Reynolds numbers.

			oRANS, $L_{x}=$
Case ( $Re_{\tau}$ )	$k$ – $\omega$	Offline DL	$4\pi\delta$	$2\pi\delta$	$\tfrac{4}{3}\pi\delta$	$\tfrac{2}{3}\pi\delta$	$\tfrac{1}{2}\pi\delta$
I (160)	1.0	0.12	0.13	0.10	0.09	0.09	0.09
II (180)	1.0	0.23	0.16	0.26	0.21	0.28	0.30
III (270)	1.0	0.24	0.20	0.21	0.19	0.48	–
IV (395)	1.0	0.49	0.16	0.13	0.55	0.16	–
V (590)	1.0	1.17	0.14	0.13	0.89	–	–

Table 5: Normalised cost functional

J^{*}

of velocity and turbulent kinetic energy for baseline RANS (

k

–

\omega

), offline optimised RANS (trained on full-domain DNS statistics at

Re_{\tau}=180

), and online optimised RANS (oRANS) with different streamwise domain fractions

L_{x}

. Values below

1

indicate improvement over the baseline RANS. Entries marked “–” correspond to cases where oRANS training diverged due to insufficient inflow length.

Figure 8: Turbulent kinetic energy (left), mean velocity (centre) and inner-scaled mean velocity (right) for cases II (

Re_{\tau}=180

, top) and IV (

Re_{\tau}=395

, bottom) at

L_{x}/L_{x,0}=1/6

: default

k

–

\omega

, offline-trained DL–RANS (trained on the

Re_{\tau}=180

DNS), oRANS, and the reference DNS of each case;

u_{b}

is defined in section 4.4 . In the right column each profile is scaled with its own friction velocity, so that an error in wall shear stress displaces the profile; grey lines are

u^{+}=y^{+}

and the logarithmic law.

Figure 9: Comparison of turbulent kinetic energy

k

(left) and mean velocity

\bar{u}

(right) profiles for cases II (

Re_{\tau}=180

) and V (

Re_{\tau}=590

) with varying streamwise subdomain lengths

L_{x}

used for oRANS training. At

Re_{\tau}=180

, oRANS predictions remain accurate across all subdomain lengths. At

Re_{\tau}=590

, agreement with DNS degrades as

L_{x}

is reduced, with excessively short domains (

L_{x}/\delta=\tfrac{2}{3}\pi

and

\tfrac{1}{2}\pi

) exhibiting algorithmic divergence due to boundary condition data contamination.

Figure 10: Convergence of the normalised cost functional

J^{*}

with

t^{*}

, the wall-clock time scaled by that of the transient-plus-averaging window

T_{\mathrm{DNS}}

of the reference periodic DNS, for cases II (

Re_{\tau}=180

, left) and IV (

Re_{\tau}=395

, right). Black: the error of the running statistics of the periodic DNS itself, started from an unscaled, perturbed field (left, and right dashed) or from the

Re_{\tau}=180

field with the fluctuations rescaled by equation 67 (right, solid). Colours: oRANS with

L_{x}/L_{x,0}=1

1/8

	periodic DNS	oRANS, $L_{x}/L_{x,0}=$
		$1$	$1/2$	$1/3$	$1/6$	$1/8$
$Re_{\tau}=180$ , DNS unscaled start ( $t^{*}_{\mathrm{DNS}}=0.72$ )	1	0.71	0.48	0.33	0.31	0.18
$Re_{\tau}=395$ , DNS rescaled start ( $t^{*}_{\mathrm{DNS}}=0.37$ )	1	3.57	2.62	3.88	1.24	–
$Re_{\tau}=395$ , DNS unscaled start ( $t^{*}_{\mathrm{DNS}}=2.80$ )	1	0.47	0.34	0.51	0.16	–

Table 6: Wall-clock time to reach

J^{*}=0.2

, a fivefold reduction of the baseline error, relative to that of the reference periodic DNS, for cases II and IV. At

Re_{\tau}=180

the DNS is started from an unscaled, perturbed field; two DNS references are given at

Re_{\tau}=395

: started from the

Re_{\tau}=180

field with the fluctuations rescaled by equation 67 , the transformation used for the oRANS inflow, and started from the same field unscaled, which is the ordinary practice. ‘–’: the run diverged (table 5 ).

Figure 11: Learned closure functions of the online-trained models with

L_{x}/L_{x,0}=1/6

(the runs of figure 8 ): wall-normal profiles of

\beta^{*}

\gamma

\beta_{0}

\sigma_{k}

\sigma_{\omega}

and

\alpha

(with

\nu_{t}=\alpha k/\omega

) for

Re_{\tau}=160

590

, dark to light. Bottom row:

\beta^{*}

and

\alpha

in the lower half of the channel against

y^{+}

. Dashed lines: the constants of the standard

k

–

\omega

model. The variation of the learned terms illustrates how the closure adapts to different flow regimes across the channel.

Panel of Figure 12 — Figure 12: Turbulence statistics for oRANS case II ( $Re_{\tau}=180$ , left) and case IV ( $Re_{\tau}=395$ , right) at $y/\delta=0.829$ . Top row: streamwise spectra $E_{uu}(k_{x})$ . Middle row: spanwise spectra $E_{uu}(k_{z})$ . Bottom row: longitudinal two-point correlations $R_{uu}(r_{x})$ . Blue curves show reference periodic DNS data, and black curves correspond to the embedded simulation with domain lengths $L_{x}=4\pi\delta$ (solid), $2\pi\delta$ (dashed), $\tfrac{4}{3}\pi\delta$ (dash-dot), $\tfrac{2}{3}\pi\delta$ (dotted), and $\tfrac{1}{2}\pi\delta$ (loosely dashed). As $L_{x}$ decreases, $k_{\min}=2\pi/L_{x}$ increases and wavelengths longer than $L_{x}$ are represented only in part, producing the observed loss of large-scale content and shorter correlation lengths. Divergence is seen in the shortest domains, where boundary-condition contamination dominates.

Panel of Figure 13 — Figure 13: Streamwise evolution of mean flow statistics for case II ( $Re_{\tau}=180$ ), and for a full length embedded domain $L_{x}=4\pi\delta$ . From top to bottom: (a) mean streamwise velocity $\bar{u}$ , (b) turbulent kinetic energy $k$ , and (c) Reynolds shear stress $\overline{u^{\prime}v^{\prime}}$ , each as a function of the streamwise coordinate $x$ . Results from oRANS (blue), DNS (black dashed), and Dreze et al. (2023) (green) are shown. The oRANS predictions recover the true DNS statistics earlier in the streamwise direction compared to Dreze et al. (2023) , highlighting the effectiveness of the rescaled turbulent inflow procedure.

Figure 14: Reference DNS of spanwise-rotating channel flow at

Re_{\tau}=194

for

Ro_{b}=0

0.15

and

0.5

: mean velocity (left) and turbulent kinetic energy (right) in wall units, with the default

k

–

\omega

solution, which at a given bulk velocity is independent of

Ro_{b}

. The pressure side is at

y/\delta=-1

and the suction side at

y/\delta=1

$Ro_{b}$		+ S–S	+ PR	$Ro_{b}{=}0$	$1$	$1/2$	$1/3$	$1/6$	$1/8$
	$k$ – $\omega$	$k$ – $\omega$	$k$ – $\omega$	Offline DL	oRANS, $L_{x}/L_{x,0}=$
0.15	1.0	0.67	0.83	0.98	0.44	0.34	0.32	0.50	0.66
0.5	1.0	0.85	0.55	1.20	0.19	0.16	0.13	0.50	0.72

Table 7: Normalised cost functional

J^{*}

(equation 72 ) for the rotating channel. The third column is the

k

–

\omega

model with the Spalart–Shur correction (in the two-equation form of Smirnov & Menter 2009, original constants) and the fourth with the eddy-viscosity correction of Pettersson Reif et al. (1999). The offline model in the fifth column is trained on the non-rotating DNS and deployed at the rotating conditions. All oRANS runs start from the non-rotating solution and inflow. S–S: the Spalart–Shur correction; PR: the correction of Pettersson Reif et al. (1999) .

Figure 15: Mean velocity (left) and turbulent kinetic energy (right) for

Ro_{b}=0.15

(top) and

0.5

(bottom): DNS, default

k

–

\omega

k

–

\omega

with the Spalart–Shur correction ( Spalart and Shur, 1997 ) in the form of Smirnov and Menter (2009) and with the Pettersson Reif correction ( Pettersson Reif et al., 1999 ) , the offline model trained on the non-rotating DNS and deployed here (Offline DL), and oRANS at the end of training with

L_{x}=L_{x,0}

L_{x,0}/2

and

L_{x,0}/3

, all started from the non-rotating solution and inflow. The

L_{x,0}/6

and

L_{x,0}/8

runs, whose statistics are sampled within the developing region (section 6.3.3 ), are omitted for clarity. Their errors are in table 7 .

Figure 16: Mean velocity in inner scaling with the local friction velocity of each wall, pressure side (left) and suction side (right), for

Ro_{b}=0.15

(top) and

0.5

(bottom): DNS, default

k

–

\omega

k

–

\omega

with the Spalart–Shur and Pettersson Reif corrections, the offline model trained at

Ro_{b}=0

, oRANS with

L_{x}=L_{x,0}

at its state at the end of training, and the data of Kristoffersen and Andersson (1993) . Thin grey lines are

\bar{u}^{+}=y^{+}

and

\bar{u}^{+}=\ln y^{+}/0.41+5.2

Figure 17: Development of the embedded simulation along the embedded region at the end of training. Top: plane- and time-averaged

k

x/L_{x}=0.05

0.3

0.6

and

0.9

(light to dark) for

L_{x}=L_{x,0}

, with the rotating DNS and the non-rotating DNS from which the inflow fluctuations are drawn. Bottom: error of the embedded statistics against the rotating DNS,

J_{u}/J_{u0}

(solid) and

J_{k}/J_{k0}

(dashed), at stations along the embedded region for

L_{x}=L_{x,0}

L_{x,0}/2

and

L_{x,0}/3

, plotted against distance from the inlet in units of

L_{x,0}

. Symbols mark where the training statistics are sampled in each run.

Ro_{b}=0.15

left,

0.5

right.

$Ro_{b}$		$1$	$1/2$	$1/3$	$1/6$	$1/8$
		$L_{x}/L_{x,0}=$
0.15	Error $J^{*}$ at end of training (table 7 )	0.44	0.34	0.32	0.50	0.66
	oRANS	0.25	0.12	0.051	0.046	0.042
	DNS from perturbed RANS solution	0.18	0.21	0.22	0.16	0.14
	DNS from developed non-rotating field	0.022	0.031	0.034	0.018	0.012
0.5	Error $J^{*}$ at end of training (table 7 )	0.19	0.16	0.13	0.50	0.72

Table 8: Wall-clock time

t^{*}

of each oRANS run to the end of training and of the periodic DNS to reach the same accuracy, in units of the full reference DNS run (

t^{*}=1

, transient and averaging), as in figure 10 . The first row of each block gives the accuracy of the oRANS run in that column, which is the target for the two DNS rows. The DNS is started either from the default RANS solution with random spanwise perturbations, as it would be for a new flow, or from the developed non-rotating field. The

L_{x}/L_{x,0}=1

column is oRANS with a full-length embedded region, not the periodic DNS.

Figure 18: Eddy viscosity for

Ro_{b}=0.15

(left) and

0.5

(right): DNS, evaluated under the Boussinesq assumption as

-\overline{u^{\prime}v^{\prime}}/\partial_{y}\bar{u}

; default

k

–

\omega

and oRANS with

L_{x}=L_{x,0}

at the end of training,

\alpha k/\omega

. Dashed lines mark the velocity maximum of each solution (DNS, black; default

k

–

\omega

, at the centreline; oRANS, blue), where the eddy viscosity evaluated from the DNS diverges.

		DNS		Default $k$ – $\omega$		oRANS, $L_{x}=L_{x,0}$
$Ro_{b}$	region	$P^{+}$	$a_{1}$	$P^{+}$	$a_{1}$	$P^{+}$	$a_{1}$
0.15	pressure side	38	0.38	44	0.30	51	0.38
	core	4.2	0.27	1.6	0.16	6.2	0.35
	suction side	14	0.09	44	0.30	4.7	0.21
0.5	pressure side	49	0.31	41	0.30	59	0.34
	core	12	0.15	1.5	0.16	12	0.32

Table 9: Production

P^{+}=P\,\nu/u_{\tau}^{4}

and structure parameter

a_{1}

(equation 76 ) of the DNS, the default

k

–

\omega

model and the oRANS closure at the end of training, averaged over the pressure side (

-0.9<y/\delta<-0.3

), the core (

|y/\delta|<0.3

) and the suction side (

0.3<y/\delta<0.9

P^{+}

is in units of

10^{-3}

Figure 19: Closure coefficients of the

Ro_{b}=0.5

L_{x}/L_{x,0}=1/2

closure at the end of training (

J^{*}=0.16

). Pressure wall at

y/\delta=-1

. Dotted lines mark the standard

k

–

\omega

values. Compare figure 11 for the non-rotating channel.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 20: Validation of DNS implementation against the canonical turbulent channel flow dataset of Kim et al. (1987) for a target friction Reynolds number

Re_{\tau}=180

. Profiles of mean velocity

\bar{u}/u_{\tau}

, turbulent kinetic energy

k/u_{\tau}^{2}

, and Reynolds stress components

\overline{u^{\prime}u^{\prime}}/u_{\tau}^{2}

\overline{v^{\prime}v^{\prime}}/u_{\tau}^{2}

\overline{w^{\prime}w^{\prime}}/u_{\tau}^{2}

, and

-\overline{u^{\prime}v^{\prime}}/u_{\tau}^{2}

are shown. Present DNS results at

Re_{\tau}=175

(blue, dotted) are in close agreement with Kim et al. (1987) (black), confirming the accuracy of the simulation setup.

Quantity	DNS	Kim et al. (1987)	Lee and Moser (2015)
$Re_{c}$	3,265	3,300	3,326
$Re_{b}$	5,600	5,600	5,714
$Re_{\tau}$	175	180	182
$u_{b}/u_{\tau}$	16.15	15.63	15.69
$C_{f}$	$7.66\times 10^{-3}$	$8.18\times 10^{-3}$	$8.12\times 10^{-3}$
$u_{c}/u_{b}$	1.165	1.16	1.164

Appendix

Table 10: Comparison of channel flow statistics between the present DNS (

Re_{\tau}=175

) and the reference data of Kim et al. (1987) (

Re_{\tau}=180

) and Lee and Moser (2015) (

Re_{\tau}=182

). Bulk Reynolds number

Re_{b}

, friction Reynolds number

Re_{\tau}

, mean velocity ratio

u_{b}/u_{\tau}

, friction coefficient

C_{f}

, and centreline velocity

u_{c}

ratios are reported. The close agreement across all quantities confirms that the present DNS accurately reproduces canonical turbulent channel flow at this Reynolds number.

Figure 21: Validation of the DNS at

Re_{\tau}\simeq 180

against Lee and Moser (2015) : mean velocity, turbulent kinetic energy and the Reynolds stresses in wall units against

y^{+}

. Symbols, present DNS (both halves of the channel averaged); lines, Lee and Moser (2015) at

Re_{\tau}=182

Figure 22: Validation of the rotating-channel DNS against Kristoffersen and Andersson (1993) at

Re_{\tau}=194

for

Ro_{b}=0

(top),

0.15

(centre) and

0.5

(bottom): mean velocity, streamwise and wall-normal intensities and Reynolds shear stress in wall units based on the wall-averaged friction velocity. The pressure side is at

y/\delta=-1

. The reference streamwise intensity at

Ro_{b}=0.5

could not be read reliably from the published figure and is omitted.

$Ro_{b}$	$u_{b}/u_{\tau}$		$u_{\tau,p}/u_{\tau}$		$u_{\tau,s}/u_{\tau}$		$\tau_{w,p}/\tau_{w,s}$
	present	K&A	present	K&A	present	K&A	present	K&A
0	15.55	15.06	0.951	1.002	0.954	1.004	0.99	1.00
0.15	15.75	15.41	1.144	1.213	0.690	0.754	2.75	2.59
0.5	15.30	15.25	1.164	1.207	0.594	0.679	3.84	3.16

Appendix

Table 11: Integral quantities of the rotating-channel DNS at

Re_{\tau}=194

and the corresponding values of Kristoffersen and Andersson (1993) (K&A, their table 3).

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Apr 26, 2026

Deep Learning of Solver-Aware Turbulence Closures from Nudged LES Dynamics

The differentiable physics paradigm may be leveraged as an a-posteriori approach for discovering turbulence closure models by embedding a neural network parameterization directly inside the solver and optimizing it given potentially sparse target data. This addresses a key limitation of a-priori learning where direct numerical simulation (DNS) data is used to approximate the subgrid stress with the assumption of a low-pass filter. Closures trained in this a-priori manner frequently lead to unstable deployments due to the mismatch between the assumed filter and the effect of numerical discretizations and coarse-graining. In comparison, while typically stable during deployment, a-posteriori learning incurs high computational costs due to the need to backpropagate through a large eddy simulation (LES) solver. Furthermore, a-posteriori methods are challenging to apply broadly since they require significant modification of existing solvers. Finally, both approaches are limited when generalization is desired across different numerical schemes with their implicit filtering characteristics. In this work, we present a deep-learning approach for turbulence closure modeling built on the continuous data assimilation framework. Our approach enables the a-priori training of closures using sparsely observed DNS data without modifying or differentiating through the LES solver, while preserving stability during deployment for the recovery of invariant statistics. We focus on the model's ability to adapt to different discretizations by explicitly conditioning it on the numerical scheme. We use two- and three-dimensional canonical cases to test our framework and show that the learned correction systematically tracks the discretization error of the coarse solver.

Ashwin Suriyanarayanan, Dibyajyoti Chakraborty, Romit Maulik

School of Mechanical Engineering Purdue University West Lafayette, IN 47907 · College of Information Sciences and Technology Pennsylvania State University University Park, PA 16802

Physics-Informed ML Computational Fluid Dynamics

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Nov 2, 2026

CFD-driven symbolic regression of RANS transition models

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.

Louenas Zemmour, Xavier Gloerfelt, Paola Cinnella

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.

Computational Fluid Dynamics Model Calibration Symbolic Regression

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Abstract

Computational Fluid Dynamics

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