TurtLES: A Large-Scale Benchmark for Turbulent 3D Neural PDE Surrogates

Jarno Platenburg; Armand Kassaï Koupaï; Paola Cinnella; Patrick Gallinari

physics.flu-dynNov 6, 2026

TurtLES: A Large-Scale Benchmark for Turbulent 3D Neural PDE Surrogates

Authors: Jarno Platenburg, Armand Kassaï Koupaï, Paola Cinnella, Patrick Gallinari

Organizations: Sorbonne Université, CNRS, ISIR, Paris, France · Delft University of Technology, Faculty of Aerospace Engineering, Delft, The Netherlands · Sorbonne Université, CNRS, Institut Jean Le Rond d’Alembert, Paris, France · Criteo AI Lab, Paris, France

Abstract

Machine-learning surrogates are increasingly used to accelerate computational fluid dynamics, yet progress is limited by the lack of benchmarks capturing realistic, time-dependent turbulent flows. We introduce TurtLES, a 14 TB dataset of 3D turbulent cylinder wakes in the shear-layer transition regime, computed via high-fidelity implicit large-eddy simulations. The dataset contains 400 trajectories of 400 time steps each, with 3-11 million points per frame, spanning variations in geometry, Reynolds number, and angle of attack. Unlike existing datasets, TurtLES combines (i) physically realistic turbulence with an active energy cascade driven by vortex stretching, (ii) dense irregular meshes, and (iii) long-horizon temporal dynamics. We benchmark state-of-the-art neural operators on autoregressive rollouts and identify key failure modes, including temporal instability, loss of small-scale energy, and challenges in learning on large irregular domains. TurtLES provides a new benchmark for evaluating surrogate models on 3D turbulent wakes at industrially relevant mesh resolution.

Figures & tables

Figure 1: TurtLES visualized through Q-criterion isosurfaces and its dataset parameters.

Figure 2: Overview of the parameter sampling space. (a) Definition of the super-elliptical geometry. (b) The

N_{s}=400

samples visualized in three dimensions, including projections on the

x,y,z

planes in gray for reference. (c-f) Histograms of the sampled parameters. Note that the colors used for different values of

n

in (f) correspond with those in (a) and (b).

Panel of Figure 3 — Figure 3: Schematic overview of the domain. (a) Three-dimensional view illustrating the physical boundaries, including the inlet ( A ), outlets ( B – D ), object surface ( E ), and periodic side-walls ( F , G ). (b) $xy$ -plane of the computational mesh, which is extended into three dimensions via a structured extrusion in the span-wise ( $z$ ) direction.

Figure 4: Overview of the flow case data structure. For each case, a single set of coordinates is stored with the solutions at 400 time-steps (e.g. $t_{0},t_{1},...,t_{f}$ , where $t_{f}=399$ ). A solution comprises the three-dimensional pressure ( $p$ ), density ( $\rho$ ), and velocity ( $u$ , $v$ , $w$ ) fields.

	Field error		Spectral error
Model	Rel- $L^{2}$ $\downarrow$	VRMSE $\downarrow$	Low $\downarrow$	Mid $\downarrow$	High $\downarrow$
UPT	7.01e-2	8.94e-2	4.69e-2	2.33e-2	4.42e-2
GINO	1.32e-1	1.21	1.32e-1	7.31e-2	4.48e-2
Transolver++	1.08e-1	1.06	8.02e-2	2.89e-2	4.41e-1
CALM-PDE	6.80e-2	7.26e-2	4.66e-2	2.38e-2	4.44e-2

Table 1: Next frame results on the test set. Lower is better for all metrics. Spectral errors are RMS log-power discrepancies computed over low-, mid-, and high-frequency spanwise wavenumber bands.

Model	$1$ – $50$	$51$ – $150$	$151$ – $300$	$301$ – $399$	Cost [GPU-s] $\downarrow$
	Rollout VRMSE $\downarrow$
Solver (PyFR)	–	–	–	–	13160
UPT	6.51e-1	1.56	3.31	5.12	835
GINO	3.18	10.32	24.31	33.12	4097
Transolver++	2.33	8.32	19.33	31.24	1148
CALM-PDE	4.61e-1	5.46e-1	6.52e-1	7.89e-1	1073

Table 2: Rollout stability and computational cost. Rollout columns: VRMSE averaged over windows of rollout steps. Cost: GPU time to generate (solver) or predict (surrogates) one full trajectory, i.e. wall-clock time

\times

GPUs used (four for the solver, one for each surrogate) on NVIDIA H100 80 GB.

Figure 5: Spectral fidelity of autoregressive rollouts. Panel (a) shows reference and predicted spanwise energy spectra averaged over test trajectories. Panel (b) shows the log-power discrepancy across wavenumbers.

Appendix figures & tables19 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 6 — Figure 6: Schematic overview of the computational domain. (a) Three-dimensional view illustrating the physical boundaries, including the inlet ( A ), outlets ( B – D ), object surface ( E ), and periodic side-walls ( F , G ). (b) 2D side view showing characteristic dimensions relative to the object thickness $D$ . Note that black lines have no physical implications but indicate regions used for mesh refinement.

Category	Symbol (in Figure 6 a )	Boundary condition
Inlet	A	Riemann invariant
Outlet	B – D	Riemann invariant
Wall	E	Adiabatic no-slip wall
Left side	F	Periodic
Right side	G	Periodic

Appendix

Table 3: Boundary conditions of the simulation. Surface symbols refer to boundaries indicated in Figure 6 a

Panel of Figure 7 — Figure 7: Two-dimensional ( $xy$ -)plane of the computational mesh. The full three-dimensional domain is a 30-layer extrusion in the $z$ direction. (a) Full domain. (b) Zoomed-in view of the geometric object.

Figure 8: Comparison between present simulations (solid black markers) and validation data from the literature (open and hatched markers) for (a) Strouhal number, (b) drag coefficient, and (c) recirculation length. Validation data are taken at discrete Reynolds numbers from the following sources: Re=1000 ( Jiang and Cheng, 2017 ) ; Re=1783 ( Bai and Alam, 2018 ) ; Re=2000 (hatched) ( Sharma and Dutta, 2022 ) and (open) ( Brun et al., 2008 ) ; Re=3000 ( Cimarelli et al., 2018 ) ; Re=3900 ( Lehmkuhl et al., 2013 ) .

Figure 9: Results for the coarse, medium and fine computational grids: (a) pressure coefficient distribution over the cylinder surface as a function of the angular position

\theta

, defined from the front stagnation point (

\theta=0^{\circ}

) and increasing along the cylinder surface in the clockwise direction; and (b) mean streamwise velocity along the centerline (

y/D=0

) downstream of the cylinder. Validation data are taken from Lehmkuhl et al. (2013) .

Figure 10: Results for the coarse, medium, and fine computational grids: mean wake velocity profiles of (a) streamwise component

u

and (b) transverse component

v

at downstream locations

x/D=1.06

1.54

, and

2.02

. For clarity, profiles at

x/D=1.54

and

2.02

are shifted downward. Validation data are taken from Lehmkuhl et al. (2013) .

Figure 11: Results for the coarse, medium, and fine computational grids: mean fluctuation profiles of (a) streamwise component

u^{\prime}u^{\prime}

and (b) transverse component

v^{\prime}v^{\prime}

at downstream locations

x/D=1.06

1.54

, and

2.02

. For clarity, profiles at

x/D=1.54

and

2.02

are shifted downward. Validation data are taken from Lehmkuhl et al. (2013) .

Figure 12: Comparison of velocity power spectra for (a) stream-wise and (b) cross-stream components at position

(x/D,y/D)=(0.71,0.66)

. Spectra are shown for both the full temporal resolution and the sub-sampled resolution

\Delta t_{s}

. The characteristic frequencies

{St}_{vs}

{St}_{sh}

, and

{St}_{KH}

are annotated for reference alongside the theoretical

-5/3

turbulence decay slope.

Figure 13: Comparison of velocity power spectra for (a) stream-wise and (b) cross-stream components at position

(x/D,y/D)=(3.0,0.0)

. Spectra are shown for both the full temporal resolution and the sub-sampled resolution

\Delta t_{s}

. The theoretical

-5/3

turbulence decay slope is indicated for reference.

Figure 14: Comparison of (a) mean pressure coefficient distribution, (b) root-mean-square pressure coefficient fluctuations, and (c) skin-friction coefficient over the rectangle surface. Validation data are taken from Cimarelli et al. (2018) .

Panel of Figure 15 — Figure 15: Comparison between the present study (top row) and Cimarelli et al. (2018) (bottom row) of the mean flow field above the rectangular surface. Streamlines of the mean velocity field are shown in (a), and the mean pressure field in (b). In (a), green lines indicate the primary vortex , red lines the secondary vortex , and black lines the wake vortex .

Figure 16: Mean velocity profiles in (a) from near to far downstream of the object, and Reynolds stress components: (b)

u^{\prime}v^{\prime}

close to the cylinder and (c)

u^{\prime}u^{\prime}

in the far wake region. Validation data are taken from Sharma and Dutta (2022) .

Figure 17: Numerical convergence criterion

\chi

for (a) streamwise and (b) vertical velocity at probes

P_{4}

P_{6.5}

, and

P_{9.5}

for the

[Re,\alpha,a,n]=[1796,11,0.424,1]

case. For

v

\chi

is computed using standard deviation only. Stabilization indicates the evacuation of numerical transients and the onset of statistically steady flow.

	Size		Shape family (% of split)
Split	Trajectories	Fraction (%)	$n=1$	$n=2$	$n=5$	$n=\infty$
Train	320	80.0	25.0	25.0	25.0	25.0
Validation	40	10.0	25.0	25.0	25.0	25.0
Test	40	10.0	25.0	25.0	25.0	25.0
All	400	100.0	25.0	25.0	25.0	25.0

Appendix

Table 4: Size and geometric composition of the TurtLES train/validation/test split. Each subset reproduces the composition of the full dataset.

	$Re$ [-]		$\alpha$ [deg]		$a$ [-]
Split	mean $\pm$ std	range	mean $\pm$ std	range	mean $\pm$ std	range
Train	2456 $\pm$ 839	[1001, 3889]	22.6 $\pm$ 13.2	[0, 45]	0.603 $\pm$ 0.365	[0.167, 1.499]
Validation	2434 $\pm$ 847	[1077, 3868]	22.5 $\pm$ 13.3	[2, 45]	0.614 $\pm$ 0.393	[0.172, 1.476]
Test	2431 $\pm$ 859	[1089, 3832]	22.4 $\pm$ 13.3	[0, 45]	0.604 $\pm$ 0.368	[0.178, 1.399]
All	2451 $\pm$ 839	[1001, 3889]	22.5 $\pm$ 13.2	[0, 45]	0.604 $\pm$ 0.367	[0.167, 1.499]

Appendix

Table 5: Parameter coverage of each split. Entries report the mean

\pm

standard deviation and the

[\min,\max]

range of the Reynolds number

Re

, the angle of attack

\alpha

, and the super-ellipse semi-axis parameter

a

Figure 18: Coverage of the parameter space by the train/validation/test split. The first three panels show the sampled cases projected onto the

(Re,\alpha)

(Re,a)

and

(\alpha,a)

planes, with training cases in gray, validation cases as blue squares and test cases as red triangles. The right-most panel shows the fraction of each subset belonging to each shape family

n

Geometry tier	Families $n$	T100	T200	T300	T400
G4	$\{1,2,5,\infty\}$	3.5 TB	7.0 TB	10.5 TB	14 TB
G3	$\{1,2,\infty\}$	2.6 TB	5.1 TB	7.7 TB	10.2 TB
G2	$\{2,\infty\}$	1.8 TB	3.6 TB	5.3 TB	7.1 TB
G1	$\{2\}$	0.8 TB	1.6 TB	2.4 TB	3.3 TB

Appendix

Table 6: Approximate download size of each mini-TurtLES subset. Geometry tiers reduce the number of shape families; temporal tiers truncate each trajectory.

Figure 19: Visualization of eight TurtLES cases. For each case, 3D, side and top views show isosurfaces of the Q-criterion ( Hunt et al., 1988 ) ( $Q^{*}=QD^{2}/U_{\infty}^{2}=1$ ) at an instantaneous snapshot, colored by the non-dimensional velocity magnitude $\|\mathbf{U}\|/U_{\infty}$ .

Hyperparameter	UPT	GINO	Transolver++	CALM-PDE
Trainable parameters	$61.96$ M	$3.10$ M	$21.79$ M	$45.14$ M
Hidden dim	$384$	$64$ (FNO)	$512$	$384$
Heads	$6$	–	$8$	$6$
MLP ratio	$4$	–	$4$	$4$
Depth	2/4/2 (enc/lat/dec)	$4$ (FNO)	$8$	$8$
Tokens / latent grid	$K{=}16{,}384$ , $L{=}4{,}096$	$12^{3}$	$128$ slices	$16^{3}{=}4{,}096$

Appendix

Table 7: Summary of baseline hyperparameters. Counts are per model.

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