AirfRANS: High Fidelity Computational Fluid Dynamics Dataset for Approximating Reynolds-Averaged Navier-Stokes Solutions

1 Introduction

• Navier–Stokes simulations for fluid analysis can require thousands of CPU hours, motivating surrogate models for optimization.Reference datasets and evaluation protocols are needed to compare and validate those models.Source: §1 · pp. 1–2
• The dataset extends earlier AirfRANS work with finer meshes intended to reduce numerical diffusion and improve flow fields, airfoil trails, and force coefficients.The authors describe the resulting dataset as a high-fidelity version.Source: §1 · p. 2
• The target task is to predict steady-state two-dimensional subsonic airfoil fields and forces, with lift-to-drag ratio as the design objective.The passage identifies turbulence and mesh engineering for accurate force coefficients as challenges.Source: §1 · p. 2
• The paper introduces a high-fidelity dataset of RANS airfoil solutions and presents its generation, force coefficients, four machine-learning tasks, evaluation protocol, and geometric deep-learning baselines.Evaluation includes force-coefficient ranks and boundary-layer accuracy.Source: §1 · p. 2

2 Related Work

• Few physics datasets and a lack of shared evaluation protocols impede rigorous comparison among physics-informed, neural-operator, and physically guided surrogate methods.The passage notes datasets across several physical domains, including fluid flow and structural mechanics.Source: §2 · p. 2
• The paper contributes a steady-state aerodynamics dataset for realistic flight scenarios and validates models on meaningful dynamics quantities, not only mean-square field loss.The authors situate this contribution among efforts to establish physics benchmarks with evaluation protocols.Source: §2 · p. 2
• Grid-based CNNs and Fourier neural operators require regular grids, while geometric learning can operate on unstructured meshes and arbitrary shapes.The passage says grid-based methods are not designed for direct operation on CFD meshes and can give inaccurate surface predictions.Source: §2 · p. 3

Dataset scope

• The dataset targets two-dimensional turbulent airflows at Reynolds numbers of 2–6 million, treated as incompressible below Mach 0.3, using RANS to control resolution complexity.The specified setting is steady subsonic flow at sea level and 298.15 K.Source: §3 · p. 3

Airfoil design space

• Airfoils come from NACA 4- and 5-digit families, whose parameterized shapes support automated generation; Table 1 gives the sampling strategy.For 4-digit profiles, values of P sampled below 1.5 are set to 0 to avoid geometries with maximum camber too near the leading edge.Source: §3 · p. 4, Table 1 · p. 4

Mesh generation

• Meshes use OpenFOAM blockMesh to create C-grid multi-block meshes, with boundaries 200 chords away and 250,000–300,000 cells per mesh.The first surface cells are 2 µm high, yielding y+ around 1 in the worst case of the design space.Source: §3 · p. 4

Dataset generation

• The dataset comprises 1,000 simulations defined by an airfoil, Reynolds number, and angle of attack, run with simpleFOAM and the SIMPLEC algorithm using k−ω SST until lift and drag converge.The Reynolds-number range is 2–6 million, and the angle-of-attack range is −5° to 15° to avoid stall and unsteady airfoil trails.Source: §3 · pp. 4–5

Force coefficients

• Drag and lift are the force components parallel and perpendicular to freestream, respectively, and their dimensionless coefficients are normalized by dynamic pressure and characteristic area.The work also computes wall shear stress from the surface velocity gradient using a ParaView gradient filter.Source: §3 · pp. 5–6

Benchmarking Setup

• Models regress mean velocity, reduced pressure, and turbulent viscosity, then derive wall shear stress and forces as post-processing.The four setups test full data, scarce data, Reynolds-number extrapolation, and angle-of-attack extrapolation.Source: §4 · p. 6
• The four setups are full data, scarce data, Reynolds-number extrapolation, and angle-of-attack extrapolation.The full-data regime uses 800 training and 200 test samples; scarce data uses 200 training samples and the same test set.Source: §4 · p. 6
• Training samples 32,000 nodes per epoch from cropped meshes and reconstructs a radius graph with 5 cm radius and at most 643 neighbors.Inference uses multiple subsamples to cover the initial mesh and averages predictions for nodes sampled more than once.Source: §4 · p. 6
• Mesh-density bias may hinder generalization to other mesh types, although all dataset meshes use the same generation procedure.The method also cannot sample surface and volume nodes independently to emphasize surface accuracy, although force depends strongly on surface pressure.Source: §4 · p. 7
• Main-paper results use classical MSE rather than the proposed loss separating surface and volume errors; the authors affirm that qualitative insights remain valid.The proposed loss is not necessarily a good proxy for accurate wall shear stress or RANS-equation satisfaction.Source: §4 · p. 7
• Evaluation pairs field MSE and drag/lift relative errors with Spearman rank correlation, the key force metric for ranking airfoils.Qualitative checks visualize boundary-layer velocity and turbulent viscosity, surface pressure and skin friction, and predicted versus true force coefficients.Source: §4 · p. 7

• The benchmark compares MLP and graph-learning baselines using field MSE, force-coefficient errors and rank correlations, and runtime.Models were trained five times; Tables 2–4 present the full-data regime results.Source: §5 · pp. 7–8, Table 2 · p. 8, Table 3 · p. 8
• Graph U-Net leads on surface pressure and lift-related metrics, while GraphSAGE achieves nearly equivalent performance with almost 20 times faster inference and half the parameters.These comparisons hold in the reported setting.Source: §5 · p. 8
• Near-wall velocity overestimation makes wall-shear-stress prediction difficult, affecting drag accuracy more than lift, whose rank is better predicted through surface pressure.The predicted-versus-true drag and lift coefficients are plotted in Figure 3.Source: §5 · p. 8, Figure 3 · p. 9
• Full-data training improves field MSE over scarce-data training, but force-coefficient scores are slightly higher in the scarce regime.Surface integration can accumulate or compensate for local errors, so field MSE is not necessarily a proxy for force-coefficient accuracy.Source: §5 · p. 10
• Extrapolation has significantly higher field MSE and lower lift Spearman correlation than interpolation, though the regimes use different test sets.Table 5 reports GraphSAGE scores for the four task-specific test sets.Source: Table 5 · p. 9, §5 · p. 10
• Model training cost is amortized after at most a dozen simulations, even in the reported worst case.Table 4 compares simulation, training, and inference times.Source: Table 4 · p. 9, §5 · p. 10

• Drag prediction is more challenging than lift prediction because wall shear stress is derived from velocity, while pressure is directly regressed.The authors also introduce baselines spanning MLPs, GNNs, point-cloud networks, and multi-scale GNNs.Source: §6 · p. 10
• AirfRANS presents high-fidelity 2D RANS solutions around NACA airfoils, four machine-learning tasks, and metrics for fields and force coefficients.The tasks span scarce-data regimes and extrapolation challenges.Source: §6 · p. 10
• For design, Spearman correlation is the primary force-coefficient metric because it measures preservation of rankings for finding high lift-over-drag airfoils.Relative errors and field MSE assess accuracy; minimizing force-coefficient relative errors is secondary, and 5% is a commonly used accuracy bound.Source: §6 · p. 10
• The dataset covers only NACA 4- and 5-digit airfoils and 2D flows; it has no out-of-distribution airfoil test, and 3D transfer is uncertain.The authors expect difficulty generalizing to more exotic shapes.Source: §6 · p. 10
• GNNs can be heavy on large graphs and require downsampling; they also rely on input CFD meshes for numerical derivatives during inference.MLPs permit more flexible sampling and do not require the input CFD mesh for derivatives, but their generalization across examples requires conditioning or hypernetworks.Source: §6 · pp. 10–11
• Unstructured-data neural operators and equivariant networks are identified as promising directions for surrogate models in these tasks.The authors describe AirfRANS as a first step toward generic treatment of real-world physical phenomena in ML and hope to encourage models that predict meaningful derived quantities as well as fields.Source: §6 · p. 11

7 Paper Checklist

• The checklist reports publicly available dataset access and code for reproducing experiments, ML tasks, and quantitative and qualitative results.It also points to training details and compute-resource information in the paper and appendix.Source: §7 · p. 18

A Resources Availability and Licensing

• The dataset-generation and ML assets are released under Open Data Commons Open Database License v1.0, which permits commercial use, distribution, modification, and private use under stated conditions.The listed conditions include source disclosure and licensing and copyright notices.Source: A Resources Availability and Licensing · p. 20

B Broader impact

• Proposed uses include testing GDL models, evaluating physical predictions, and extending graph-mesh and point-cloud benchmarks to physics problems.The authors also list surrogate solvers for design cycles and studies of generalization to large or unseen domains.Source: B Broader impact · p. 20
• The paper identifies reduced prototyping costs and avoidance of dangerous real-world studies as potential benefits of testing multiple plane configurations.Source: B Broader impact · p. 20

C Reproducibility statement

• The repository provides code to reproduce ML experiments and generate figures, and links to preprocessed and raw OpenFOAM data.The ML experiments used one NVIDIA RTX 3090 24Go.Source: C Reproducibility statement · p. 20
• Simulation-generation resources include extensible mesh-generation code and code to run simulations or build the dataset.The simulations used 16 CPU cores of an AMD Ryzen Threadripper 3960X.Source: C Reproducibility statement · p. 20
• The AirfRANS Python library is provided to facilitate manipulation of simulations from the dataset.Source: C Reproducibility statement · p. 20

D Description Of Software

• OpenFOAM v2112 was used for simulations; the paper describes it as C++ software for custom numerical solvers in continuum mechanics and CFD.The authors set absolute pressure to 0 N m−2 for the incompressible case because only differential pressure matters.Source: D Description Of Software · pp. 20–21
• ParaView visualized point clouds, meshes, and predicted and ground-truth fields, while PyVista was used to build deep-learning inputs from mesh analysis.Source: D Description Of Software · p. 21
• PyTorch was used for the training protocol, and PyTorch Geometric for designing message-passing schemes.Source: D Description Of Software · p. 21

E Constant and Dimensionless Quantities

• For air at 298.15 K and sea level, the incompressible case uses Reynolds number, while the compressible case also uses Mach and Prandtl numbers.Re compares convective and diffusive terms; Mach quantifies compressibility, and Prandtl compares energy variation via momentum with variation via heat transfer.Source: E Constant and Dimensionless Quantities · pp. 21–22

F Reynolds-Averaged Navier–Stokes Equations

• RANS averages turbulent flow quantities, introduces Reynolds stress, and requires a turbulence model to close the equations; the study uses Spalart–Allmaras and k−ω SST.The k−ω SST model blends k−ε and k−ω models, switching between them according to where each is more relevant.Source: F Reynolds-Averaged Navier–Stokes… · pp. 23–24

G Force Coefficients

• Drag and lift are the components of the surface stress force parallel and perpendicular to the main flow direction, respectively.For RANS, the effective mean-field pressure is used; turbulent viscosity is null on the airfoil, so wall shear stress reduces to the molecular-viscosity form.Source: G Force Coefficients · p. 24

H Airfoil Generation and Statistics

• NACA 4-digit profiles encode camber and thickness in MPXX, while NACA 5-digit profiles use LPQXX, with the first digit setting an optimal-lift-based camber and Q distinguishing standard from reflex camber.Both families use a camber line and an envelope; the 5-digit reflex camber is described as a double-cambered line that sets pitch coefficient to zero.Source: H Airfoil Generation and Statistics · pp. 24–25
• The NACA 4-digit sampling is slightly biased toward symmetrical profiles because sampled P values below 1.5 are set to zero.The authors note that this bias could support a smaller dataset of symmetrical profiles for testing models’ plane-symmetry invariance or equivariance.Source: H Airfoil Generation and Statistics · p. 26

I Meshing Procedure

• Steady-state RAS is used to recover mean steady-state fields around airfoils while modeling turbulence length scales through ensemble averaging.The section contrasts this approach with DNS and LES, which can require prohibitively many mesh cells.Source: I Meshing Procedure · p. 26
• The mesh resolves the boundary layer with a low-Reynolds strategy, targeting y+ around 1 in the worst case and setting the first-cell height to 2 µm.The authors choose this strategy to avoid modeling close to the wall when computing accurate airfoil force coefficients.Source: I Meshing Procedure · pp. 26, 28
• A blockMesh C-grid inspired by the NASA mesh places boundaries 200 chords from the airfoil, which the authors found large enough to be insensitive to boundary conditions.The template defines near-wall and airfoil-surface cell grading; Figure 5 shows the block layout, and Figure 4 gives parameter histograms across the dataset regimes.Source: Figure 4 · p. 27, I Meshing Procedure · p. 28

J Boundary Conditions

• Boundary-condition settings cover the fields needed for incompressible or compressible simulations and depend on the chosen turbulence model.Tables 7–9 give the settings and define the quantities and constants used for air at sea level and 298.15 K.Source: J Boundary Conditions · p. 28, Table 7 · p. 29, Table 8 · p. 30
• The study uses SIMPLEC for the incompressible setup and classical SIMPLE for compressible simulations because SIMPLEC was not stable in that case.The discretization schemes and linear-solver settings are listed in the fvSchemes and fvSolution dictionaries rather than presented in this section.Source: J Boundary Conditions · p. 28
• The boundary-condition tables distinguish compressible from incompressible simulations and document the mesh patches to which conditions are applied.Figure 5 identifies the airfoil, freestream, and internal patches in the mesh template.Source: Table 7 · p. 29, Figure 5 · p. 29

K Simulation Validation

• Surface pressure coefficients show no significant differences between turbulence models or between compressible and incompressible cases, and all simulations agree well with experiments.The comparisons cover NACA 0012 at 0° and 10° angles of attack.Source: K Simulation Validation · p. 32, Figure 7 · p. 32, Figure 8 · p. 32
• Both turbulence models agree qualitatively with experiments, but k−ω SST is more stable, faster to converge, and more accurate than Spalart–Allmaras.The study retains the incompressible k−ω SST setup to generate the dataset.Source: K Simulation Validation · pp. 32–33, Figure 9 · p. 33
• For NACA 4412, both models’ pressure coefficients agree with experiments, but the authors treat this case as qualitative validation because reference values and normalization differ.The TMR values differ slightly from the original papers, and the pressure-coefficient normalization is noted as a concern.Source: K Simulation Validation · p. 33
• Both turbulence models have difficulty matching experimental boundary-layer data, though k−ω SST appears more realistic than Spalart–Allmaras.The authors report that this behavior agrees with the TMR study of the NACA 4412.Source: K Simulation Validation · p. 33

L Models architecture

• Each model uses a shared 7-64-64-8 encoder and 8-64-64-4 decoder, with encoding dimension 8.Both are MLPs with ReLU activations and no batch normalization, and they are trained together with the chosen model.Source: L Models architecture · p. 33
• Baselines span an MLP, radius-graph GraphSAGE, PointNet, and a five-scale Graph U-Net.GraphSAGE uses a 5 cm radius graph; the Graph U-Net uses radius graphs at five scales and random downsampling.Source: L Models architecture · pp. 33–34
• Training uses a one-cycle cosine learning-rate schedule and matches gradient-update counts across tasks.The maximum learning rate is 0.001; simulations are fed one by one.Source: L Models architecture · p. 34
• The task-specific training budgets are 400 epochs for full data, 1600 for scarce data, 635 for Reynolds extrapolation, and 398 for angle-of-attack extrapolation.Models are trained on 90% of each predefined training set, with the remaining 10% used for validation.Source: L Models architecture · pp. 34, 36
• In scarce-data evaluation, field MSE is directly regressed, while force-coefficient relative errors and Spearman correlations are computed after postprocessing.Only reduced pressure is reported at the surface because other quantities are null under the boundary conditions.Source: Table 10 · p. 36, Table 11 · p. 36

M Additional Results

• In the full-data regime, wall shear stress is largely overestimated because the first inferred point after the surface is poorly approximated, leading to high drag errors.The passage links the near-wall velocity-profile error to the wall-shear and drag errors.Source: M Additional Results · p. 36
• Scarce-data evaluation uses the full-data test set but trains on 200 simulations rather than 800.The results include predicted force coefficients, boundary-layer profiles, and surface coefficients.Source: M Additional Results · pp. 36, 38
• The Reynolds- and angle-of-attack-extrapolation evaluations test on out-of-distribution values of those respective variables.The reported diagnostics include force-coefficient predictions and boundary-layer and surface-coefficient profiles.Source: M Additional Results · p. 38, N Results for loss 3 · p. 49
• MSE scores in the scarce-data, Reynolds-extrapolation, and angle-of-attack-extrapolation regimes are an order of magnitude above those in the full-data regime.The summary also reports higher force-coefficient relative errors and lower Spearman scores in the harder regimes, while noting the relationship is not direct in at least one case.Source: M Additional Results · p. 38
• Boundary-layer profiles are inaccurate close to geometries, while pressure coefficients suffer model-dependent out-of-distribution effects.These are reported as limitations across the additional regimes.Source: M Additional Results · p. 42

N Results for loss 3

• With the weighted loss, surface pressure is better regressed, but wall shear stress remains largely overestimated and high drag errors persist.The authors attribute the shear error to poor approximation of the first inferred boundary-layer point after the surface.Source: N Results for loss 3 · p. 49
• The scarce-data regime uses the full-data test set but trains on 200 simulations instead of 800.The passage identifies MSE and force-coefficient metrics alongside force, boundary-layer, and surface-profile plots.Source: N Results for loss 3 · p. 49
• The Reynolds-extrapolation regime tests on out-of-distribution Reynolds numbers.Its reported evaluations include field MSE, force-coefficient metrics, and predicted force and profile plots.Source: N Results for loss 3 · p. 49
• The angle-of-attack extrapolation regime tests on out-of-distribution angles of attack.The section reports field MSE, force-coefficient metrics, and predicted force and profile plots for this regime.Source: N Results for loss 3 · p. 53