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REVIEW 3 major objections 5 minor 259 references

Wall-modelled LES of boundary layers is accurate only when the subgrid-scale and wall closures are trained together inside the solver, and a model trained that way extrapolates well beyond its training Reynolds numbers.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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Joint end-to-end training of neural SGS and wall closures in a differentiable CFD solver outperforms conventional WMLES baselines and generalizes across Reynolds numbers, grids, and domains.

T0 review reviewed 2026-08-01 challenge →

load-bearing objection Solid demonstration of joint end-to-end training of SGS and wall closures in a differentiable WMLES solver; the extrapolation claim leans on the prescribed skin-friction law, but the coupling evidence holds. the 3 major comments →

arxiv 2607.17357 v1 pith:DRTRBBPE submitted 2026-07-19 physics.flu-dyn

Differentiable Hybrid Neural-CFD Modelling of Wall-Bounded Turbulence: Coupled Learning of Subgrid-Scale and Wall Closures

Xiantao Fan , Yi Liu , Meng Wang , Jian-Xun Wang This is my paper
classification physics.flu-dyn PACS 47.27.Eq
keywords wall-modelled large-eddy simulationsubgrid-scale closurewall-stress modeldifferentiable CFDneural closureend-to-end trainingturbulent boundary layerturbulence statistics
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that in coarse wall-modelled large-eddy simulation the subgrid-scale (SGS) closure, the wall-stress model, and the numerical discretization are not separable: through the resolved velocity field each one shapes what the others see, so calibrating any of them in isolation cannot fix the coupled errors. To make the coupling learnable, the authors build a differentiable finite-volume solver in which both closures are neural operators constrained to the physical structure of their conventional counterparts, a U-Net feeding a fixed eddy-viscosity form for the SGS stress and a U-Net adding fluctuations to an empirical mean wall-stress law. Both are optimized end-to-end by back-propagating a loss on low-order statistics (mean velocity, r.m.s. fluctuations, friction velocity) through a rollout of the coupled solver. Across nine a posteriori tests of a zero-pressure-gradient turbulent boundary layer, the jointly trained model beats two standard WMLES baselines, reproduces the logarithmic mean-velocity region and resolved energy spectra even though neither is in the training objective, and extrapolates to more than four times the highest training Reynolds number. Ablations in which only one closure is learned fail in complementary ways, which the authors take as direct evidence that joint optimization is necessary.

Core claim

The central claim is that the SGS closure, the wall closure, and the discretization are coupled through the resolved field, and that only their joint, in-the-loop optimization recovers the full set of turbulence statistics. The authors demonstrate this with Hybrid-Joint: a differentiable solver in which a structure-constrained neural SGS closure (a trained coefficient field inserted into the Vreman eddy-viscosity form) and a structure-constrained neural wall closure (a learned correction and fluctuation added to an empirically scaled mean wall stress) are trained together from statistics alone. The resulting model outperforms conventional WMLES, transfers to unseen grids and domains, and — w

What carries the argument

The carrying mechanism is the composed neural operator: for each closure, a trainable convolutional U-Net (3-D for SGS, 2-D for wall) supplies only the degrees of freedom that the conventional form leaves open, followed by a fixed differentiable physical layer that guarantees structure. For the SGS stress the fixed layer is the Vreman eddy-viscosity functional, which auto-switches off in laminar and pure-shear regions; for the wall stress it is an empirical skin-friction law for the mean plus the non-negative wall-eddy-viscosity imposition of the total stress. Because every operation in the solver is differentiable, the statistics-based loss is back-propagated through a 100-step rollout of t

Load-bearing premise

The load-bearing premise is that the mean wall shear stress is reliably anchored by an empirical skin-friction correlation evaluated from an online running-averaged velocity profile, with the training loss additionally pinning the friction velocity; if that empirical input drifts at extrapolated conditions, the claimed generalization rests entirely on the learned correction, which is not independently verified.

What would settle it

Run the trained closures on a zero-pressure-gradient boundary layer at Re_theta around 10,000, well beyond the tested range, and check whether the mean-velocity log region and the Cf–Re_theta trend still hold; alternatively, retrain with the empirical mean wall-stress law replaced by a constant or deliberately biased value and see whether the emergent log region collapses — if it does, the empirical anchor, not the joint learning, is carrying the extrapolation.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • If the coupling claim is right, a posteriori WMLES accuracy cannot be guaranteed by improving either the SGS model or the wall model in isolation; both must be co-optimized in the discretized solver.
  • Low-order statistics, not pointwise SGS-stress labels, are sufficient training signals for the coupled closures, making the method applicable where only statistical reference data exist.
  • The trained closures are frozen and reused across Reynolds numbers, domains, and grids with no retraining, so the one-time training cost is amortized over many deployments.
  • Recovering the logarithmic velocity region and resolved spectra as emergent, un-forced behavior implies the closure pair is not overfitting the single-point statistics it was trained on.
  • Because the neural SGS closure inherits the eddy-viscosity form, it cannot represent backscatter; representing that would require relaxing the structure constraint, a limitation the paper itself states.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The coupling argument generalizes: if SGS and wall closures are coupled through the resolved field, then the numerical discretization is a third partner in the same loop, and one could co-learn discretization parameters such as stencil weights, time-step weighting, or numerical viscosity alongside the closures.
  • A sharper falsifier of the claimed emergent log law would be to train with the empirical mean wall-stress law replaced by a deliberately wrong scaling; if the log region persists, the structure constraint is doing the work, whereas if it collapses, the learned correction is merely compensating around the empirical anchor.
  • The framework's reliance on a running-averaged momentum-thickness Reynolds number suggests that an online estimator of the same integral from wall quantities could be learned, removing the empirical correlation from the loop and testing how much of the extrapolation was due to the scaling law.
  • A natural stress test beyond ZPG TBL is to apply the same joint-training recipe to an adverse-pressure-gradient or separated boundary layer, where the equilibrium wall-stress assumption is known to fail, and ask whether joint optimization of the same two structured closures still recovers the statistics.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper presents a differentiable hybrid neural–CFD framework for wall-modelled LES of zero-pressure-gradient turbulent boundary layers. The SGS and wall closures are composed operators: a 3-D U-Net feeding a fixed Vreman-type eddy-viscosity layer, and a 2-D U-Net supplying a fluctuating/corrective wall stress on top of a Smits et al. skin-friction scaling law. Both closures are trained jointly and end-to-end through a differentiable finite-volume solver against a low-order statistics loss (mean velocity, r.m.s. velocity and pressure fluctuations, and friction velocity). Training is performed at Re_theta = 600 and 1200; deployment covers nine a posteriori cases up to Re_theta ≈ 6500, including changed domains, grids, and a spatially developing boundary layer. The authors report that Hybrid-Joint outperforms Smagorinsky and Vreman baselines, recovers the log region and resolved spectra, and that ablations learning only one closure fail in complementary ways, supporting the central claim that SGS closure, wall closure, and discretization must be trained jointly.

Significance. If the results hold, this is a substantive advance: it demonstrates stable end-to-end training of two coupled closures inside a differentiable solver from low-order statistics only, without pretraining, and with genuine deployment from a random Blasius inflow. The complementary failure pattern of the single-closure ablations is credible evidence for the coupling claim, and the paper is refreshingly explicit about limitations (ZPG-only scope, no backscatter, empirical wall-stress anchor). However, the headline extrapolation and log-region claims are weaker than the abstract suggests, because the mean wall stress is prescribed by Eq. (2.11) and u_tau is anchored in the loss, Eq. (2.16). Figure 5(b) is therefore not an independent test of learned Reynolds-number scaling. The central joint-training conclusion survives this concern, but the manuscript needs revision to separate constrained generalization from fully learned generalization, and to support single-rollout quantitative claims with uncertainty estimates.

major comments (3)
  1. [§2.2.3 and §2.3, Eqs. (2.11)–(2.12), (2.15)–(2.16)] The extrapolation/log-region claim is substantially predetermined by the empirical Smits et al. skin-friction law. Equation (2.11) sets the mean wall stress at every Reynolds number, Eq. (2.12) evaluates Re_theta online, and Eq. (2.16) explicitly anchors u_tau to the ground truth. Thus the Cf–Re_theta agreement in Fig. 5(b) and the inner scaling that produces the log region are to first order consequences of a correlation calibrated for ZPG TBLs, not of the learned closures. I recommend relabelling this as “constrained extrapolation around an empirical anchor” and quantifying the learned correction’s contribution, e.g. by comparing against the mean-only wall model at Re_theta = 5000 or by showing sensitivity to replacing Eq. (2.11) with a different correlation. This does not undermine the joint-training conclusion, but it changes what Fig. 5 demonstrates.
  2. [§3, Figs. 2, 5, 8, 10] No uncertainty estimates are reported. All statistics are based on single rollouts (final 20 flow-through times), and no ensemble of initial conditions or training runs is shown. Given the chaotic dynamics and the stochastic training procedure, statements such as “agrees closely” and “extrapolates without appreciable degradation” need run-to-run variability or at least a sensitivity study to the averaging window. Without error bars, it is difficult to judge whether Hybrid-Joint’s advantage over Baseline: Vreman in Figs. 2, 8, and 20 is significant relative to natural fluctuations.
  3. [§3, Case 8 and Appendix C, Fig. 20] The Re_theta = 5000 reference is described in Fig. 20 as “wall-resolved-quality fine-mesh WMLES,” not DNS. This is a model-dependent target, so the claim that Hybrid-Joint “agrees closely” and the quantitative extrapolation assessment are conditional on the fidelity of that reference. Please state explicitly how this reference is validated and, where possible, compare against DNS at overlapping lower Reynolds numbers. The Case 4 reference-domain mismatch is acknowledged in §3.3, but the same caution should be stated for Case 8.
minor comments (5)
  1. [§2.2.3, Eq. (2.12)] The averaging window T is defined as 10 delta_99 / U_infinity, but delta_99 is not defined before this equation, and the integral upper limit is written as infinity rather than delta_99. Please clarify notation.
  2. [§3.2, first paragraph] The sentence “This agreement is a genuine test of the model” should be softened or qualified in view of Eq. (2.11), which fixes the mean wall stress and hence the inner scaling. See major comment 1.
  3. [§3.2, Fig. 5(b)] Because Cf is prescribed by Eq. (2.11), the comparison with Fernholz & Finley data and Smits/Nagib correlations is not independent. Consider plotting the learned correction (Cf_model − Cf_Smits)/Cf_Smits to show what the network actually adds.
  4. [§4.4, Table 3] The CPU-to-GPU cost comparison is labeled indicative, which is appropriate. However, the claim of a “540× reduction” combines solver efficiency, mesh coarsening, and amortization; this should be broken down more clearly to avoid overstatement.
  5. [General] No code or data availability statement is included. Given the complexity of the framework, releasing the differentiable solver wrapper and trained weights would materially help reproducibility.

Circularity Check

3 steps flagged

Extrapolation and skin-friction evidence are largely predetermined by the empirical Smits law and the u_tau anchor; the joint-training conclusion itself is independently supported.

specific steps
  1. self definitional [§2.2.3, Eqs. (2.11)–(2.12); presented as extrapolation evidence in §3.2, Fig. 5(b)]
    "Cf = ⟨τ_w,1⟩/(1/2 ρ U∞^2) = 0.024 Re_theta^{-1/4}, ⟨τ_w,3⟩=0 ... The resulting Re_theta is substituted into equation (2.11) to give the reference mean stress."

    The mean wall shear stress used by the neural wall closure is prescribed at every time step by the Smits et al. empirical skin-friction law, with Re_theta evaluated online from the running-averaged profile. Therefore the simulated Cf is not an independent output of the learned closures: it is an input of the closure. The paper's claim that the predicted Cf follows the expected Re_theta scaling across 600–6500 is thus largely a restatement of Eq. (2.11), not a test of learned Reynolds-number generalization. The paper itself states this: the scaling law 'provides the correct Re-scaling, which the network need not relearn.'

  2. fitted input called prediction [§2.3, Eq. (2.16); invoked in §3.2 as evidence of accurate inner scaling and extrapolation]
    "L(θ) = w1‖⟨u+1⟩−⟨u+1,GT⟩‖^2 + ... + w4‖uτ − uτ,GT‖ ... the final term anchors the absolute magnitude of uτ, which inner scaling alone leaves undetermined."

    The friction velocity u_tau is directly included in the training loss, and u_tau is derived from the wall stress produced by the wall closure. Hence the accurate inner-scaled mean profile and the u_tau values at the training Reynolds numbers are fitted quantities, not predictions. At extrapolated Reynolds numbers, the same u_tau is dominated by the prescribed Smits-law mean (Eq. 2.11), so the Reynolds-number trend of the inner scaling is inherited from the empirical law rather than learned. Reporting this agreement as evidence of extrapolation is therefore circular with respect to the training objective.

  3. other [§4.1.2, Fig. 16]
    "The mean-only model gives a reasonable, slightly overpredicted mean velocity profile, since the imposed mean wall stress still provides the correct inner scaling."

    This ablation explicitly shows that the reasonable mean velocity profile — the basis for the log-region and inner-scaling claims — is obtained even when the learned wall-stress fluctuations are removed, because the mean wall stress is imposed by the scaling law. It confirms that the mean-profile agreement is not produced by the learned correction but by the empirical input. The learned fluctuations are needed for the Reynolds shear stress, which is genuine, but the mean-profile and Cf-based extrapolation evidence is tied to the prescribed input.

full rationale

The paper's central methodological claim — that the SGS closure, wall closure, and discretization are coupled and must be trained jointly — is independently supported by the ablation experiments (Hybrid-WallOnly and Hybrid-SGSOnly fail in complementary ways), and several reported results are genuinely emergent: the SGS coefficient near c≈0.07, the resolved spectra, the wall-stress fluctuation PDFs, and the need for fluctuations to recover the Reynolds shear stress. However, the Reynolds-number extrapolation claim is substantially prebuilt. The mean wall stress is set by the Smits et al. skin-friction law (Eq. 2.11) with online Re_theta (Eq. 2.12), and the loss contains an explicit u_tau anchor (Eq. 2.16). Consequently, the Cf–Re_theta agreement, the inner-scaled mean profile, and the log-region behavior at extrapolated Reynolds numbers are not independent tests of a learned Reynolds-number-scaling law; they largely reflect the empirical input plus a small trained correction. The paper openly acknowledges that the scaling law 'provides the correct Re-scaling, which the network need not relearn', and the mean-only ablation corroborates that the mean profile is reasonable even without the learned fluctuation. This is partial circularity in the generalization evidence, not in the joint-optimization conclusion, so a score of 6 is appropriate rather than higher.

Axiom & Free-Parameter Ledger

6 free parameters · 7 axioms · 0 invented entities

The central new content is the joint training procedure and the network corrections; the physical structure (Vreman form, Cf scaling law) is imported from prior literature and is part of why the 'predicted' friction scaling is not fully emergent. No new physical entities are postulated.

free parameters (6)
  • Neural SGS U-Net weights (θ_sgs) = trained (architecture in Table 4)
    3-D U-Net maps resolved fields to Vreman coefficient; all parameters optimized against the statistics loss; this is the learned part of the SGS closure.
  • Neural wall U-Net weights (θ_wall) = trained (architecture in Table 4)
    2-D U-Net produces wall-stress correction and fluctuations; joint optimization of this network with the SGS network is the central claim.
  • Smits et al. skin-friction law constants (0.024, -1/4) = 0.024 Re_theta^-1/4
    Imposed mean wall stress via Eq. (2.11); empirical constants adopted from prior literature make the reported Cf-Re_theta agreement partly by construction.
  • Training Reynolds number set = {600, 1200}
    Cases chosen for training; all extrapolation and interpolation claims are defined relative to this set.
  • Momentum-thickness averaging window T = 10 δ99/U∞
    Eq. (2.12) uses this ad hoc running-average window to estimate Re_theta online, which in turn sets the imposed mean wall stress.
  • Loss weights w_k = 1/|L_k| = recomputed per epoch, detached
    Inverse-magnitude weighting balances the four loss terms; procedural but affects the training trajectory and final closure.
axioms (7)
  • domain assumption Filtered incompressible Navier-Stokes equations (2.1) govern the resolved field with implicit filtering by the mesh
    Basis for both closures; no explicit filter is applied, so commutation errors are neglected.
  • domain assumption Eddy-viscosity form for SGS stress, τ_ij = -2ν_t S_ij
    Eq. (2.5); inherited from conventional closures and fixes the sign of SGS energy transfer to non-negative; paper notes it excludes backscatter.
  • domain assumption Vreman functional (2.7)-(2.8) as fixed output layer remains a suitable tensor structure for the learned coefficient
    Chosen for realizability and near-wall vanishing; an architectural assumption, not proved for all learned coefficients.
  • domain assumption Empirical skin-friction law of Smits et al. (2.11) gives accurate mean wall shear stress
    Mean wall closure relies on this correlation; the paper's Cf predictions inherit its accuracy.
  • domain assumption Local resolved fields at the third off-wall cell contain enough information for the wall network to predict stress fluctuations
    Eq. (2.13); a modeling assumption on the receptive field of the wall closure.
  • domain assumption Statistics accumulated over the final 20 flow-through times are converged without error bars
    All comparisons use single-rollout statistics; no confidence intervals or ensemble averaging are reported.
  • domain assumption Recycling-rescaling inflow (Lund et al. 1998) provides sufficient turbulence for autonomous transition and development
    Inference starts from Blasius plus random perturbations and relies on inlet recycling; no sensitivity analysis is given.

reviewed 2026-08-01 · how reviews work

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Cite this review

Pith. "Pith review of Differentiable Hybrid Neural-CFD Modelling of Wall-Bounded Turbulence: Coupled Learning of Subgrid-Scale and Wall Closures." pith.science (2026). https://pith.science/paper/DRTRBBPE

@misc{pith2026260717357,
  author       = {Pith},
  title        = {Pith review of: Differentiable Hybrid Neural-CFD Modelling of Wall-Bounded Turbulence: Coupled Learning of Subgrid-Scale and Wall Closures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DRTRBBPE}},
  note         = {Machine review of arXiv:2607.17357}
}
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read the original abstract

Wall-modelled large-eddy simulation (WMLES) treats the subgrid-scale (SGS) closure, wall closure and numerical discretization as independent components, although their effects are coupled through the same resolved field. We present a differentiable hybrid neural--CFD framework in which the SGS and wall closures are learned jointly, end-to-end, within a differentiable flow solver, using only low-order statistics as training targets. Each closure is a composed neural operator: a trainable neural network followed by a fixed differentiable layer that preserves the structure of its conventional counterpart, so that the network learns only the functions left undetermined by the conventional form. Because every operation is differentiable, gradients of the training loss are back-propagated through the coupled solver, allowing both neural closures to be optimized consistently against the flow field, rather than fitted offline or in isolation. We demonstrate the framework, denoted Hybrid-Joint, on a zero-pressure-gradient turbulent boundary layer across a posteriori tests spanning Re_\theta = 600--6500, computational domains and mesh resolutions. The model outperforms WMLES baselines, extrapolates to more than four times the highest training Reynolds number, and transfers to grids and domains absent from training. It recovers a logarithmic mean-velocity region, not imposed by the wall closure, and reproduces the resolved energy spectra accurately, although spectral information is excluded from the training objective. Ablation studies show that learning either closure alone is insufficient and that only joint optimization recovers the full set of statistics, confirming that SGS closure, wall closure and discretization are coupled and must be trained jointly. Once trained, the closures are reused without retraining across all cases, so that training cost is amortized over repeated deployment.

Figures

Figures reproduced from arXiv: 2607.17357 by Jian-Xun Wang, Meng Wang, Xiantao Fan, Yi Liu.

Figure 1
Figure 1. Figure 1 [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Turbulence statistics for Case 1 (𝑅𝑒 𝜃 = 600), compared with the two baselines and the ground truth (DNS). (a) Mean streamwise velocity profile. (b)–(d) Root-mean-square (RMS) velocity fluctuations 𝑢 + 𝑖,rms. (e) Reynolds shear stress −⟨𝑢 ′ 1 𝑢 ′ 2 ⟩ + . All statistics are normalized by the friction velocity 𝑢𝜏 . lead to an underprediction of the friction velocity 𝑢𝜏 by approximately 12% and hence to the o… view at source ↗
Figure 3
Figure 3. Figure 3: One-dimensional wavenumber energy spectra of the velocity components for Case 1 (𝑅𝑒 𝜃 = 600) at 𝑦 + = 138. (a) Streamwise velocity. (b) Wall-normal velocity. (c) Spanwise velocity. The spectra are normalized by 𝑢𝜏 . For consistency, the GT is filtered at the cut-off wavenumber associated with the same mesh. Baseline: Smagorinsky (c) Ground truth (a) (b) Hybrid-Joint (d) Baseline: Vreman [PITH_FULL_IMAGE:f… view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of the instantaneous velocity field predicted by different models for Case 1 (𝑅𝑒 𝜃 = 600). (a) Baseline: Smagorinsky (b) Baseline: Vreman (c) Hybrid-Joint (d) Ground truth, obtained by filtering the DNS solution and downsampling it onto the same coarse mesh. The isosurfaces are defined by the 𝑄 criterion at 𝑄 = 10−3 , and the colour denotes the velocity magnitude [PITH_FULL_IMAGE:figures/full_f… view at source ↗
Figure 5
Figure 5. Figure 5: (a) shows the mean velocity profiles over the range. A clear logarithmic region develops at every Reynolds number, in agreement with the law of the wall, with a slight departure only at the first off-wall point. This agreement is a genuine test of the model: unlike conventional equilibrium wall models, which invert an assumed law of the wall at a matching location, the present wall closure never matches th… view at source ↗
Figure 6
Figure 6. Figure 6: One-dimensional wavenumber energy spectra of the velocity components for Case 2 (𝑅𝑒 𝜃 = 800) at 𝑦 + = 170. (a) Streamwise velocity. (b) Wall-normal velocity. (c) Spanwise velocity. The spectra are normalized by 𝑢𝜏 . For consistency, the GT is filtered at the cut-off wavenumber associated with the same mesh. (a) (b) (c) [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: One-dimensional wavenumber energy spectra of the velocity components for Case 8 (𝑅𝑒 𝜃 = 5000) at 𝑦 + = 600. (a) Streamwise velocity. (b) Wall-normal velocity. (c) Spanwise velocity. The spectra are normalized by 𝑢𝜏 . For consistency, the GT is filtered at the cut-off wavenumber associated with the same mesh. The corresponding spanwise wavenumber spectra are shown in figures 6 and 7. At 𝑅𝑒 𝜃 = 800, Hybrid-J… view at source ↗
Figure 8
Figure 8. Figure 8: Turbulence statistics at 𝑅𝑒 𝜃 = 1200 for Case 3 (uniform grid, 100𝜃 domain) and Case 4 (wall￾normal stretched grid, 400𝜃 domain), compared with the two baselines and the ground truth (DNS). (a) Mean streamwise velocity profile. (b)–(d) Root-mean-square velocity fluctuations 𝑢 + 𝑖,rms. All statistics are normalized by the friction velocity 𝑢𝜏 . The reference and baselines are obtained in the same domain of … view at source ↗
Figure 9
Figure 9. Figure 9: Streamwise velocity for Case 9 (𝑅𝑒 𝜃,in = 5000). (a) Mean streamwise velocity contour, boundary￾layer thickness, and velocity profiles at selected streamwise locations. For clarity, the velocity profiles are scaled by a factor of 50. (b) Instantaneous streamwise velocity contour. (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Streamwise development of wall-friction quantities for Case 9 (𝑅𝑒 𝜃,in = 5000). (a) Skin-friction coefficient along the streamwise direction. (b) Predicted friction Reynolds number 𝑅𝑒𝜏 as a function of the Reynolds number based on momentum thickness, 𝑅𝑒 𝜃 . only the jointly trained closures keep the inner scaling, reflected in 𝑅𝑒𝜏, and the integral development of the layer, reflected in 𝑅𝑒 𝜃 , simultaneou… view at source ↗
Figure 11
Figure 11. Figure 11: Prediction of the neural SGS and wall models. (a) Mean subgrid viscosity 𝜈𝑡 over wall-normal distance, combining the interior SGS contribution and the wall contribution ⟨𝜈 wall 𝑡 ⟩ (first point). (b) Time history of the wall-parallel-averaged Vreman coefficient 𝑐 [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Instantaneous subgrid viscosity 𝜈𝑡 predicted by the neural SGS model. (a) Case 1, 𝑅𝑒 𝜃 = 600. (b) Case 2, 𝑅𝑒 𝜃 = 800. (c) Case 3, 𝑅𝑒 𝜃 = 1200. (d) Case 5, 𝑅𝑒 𝜃 = 2400. Bae 2025) and used as the fixed coefficient in Baseline: Vreman. This is an emergent outcome of training on turbulence statistics, not a value the network was steered toward: nothing in the loss or the architecture references this constant,… view at source ↗
Figure 13
Figure 13. Figure 13 [PITH_FULL_IMAGE:figures/full_fig_p024_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Instantaneous streamwise wall-shear stress 𝜏𝑤,1 predicted by different models for Case 1 (𝑅𝑒 𝜃 = 600). (a) Hybrid-Joint model. (b) Ground truth obtained by filtering the DNS solution and downsampling it to the same coarse mesh. (c) Baseline: Smagorinsky. (d) Baseline: Vreman. The values are normalized by the mean streamwise wall-shear stress of the ground truth. (e) PDF of streamwise wall-shear stress. (a… view at source ↗
Figure 15
Figure 15. Figure 15: PDF of the instantaneous streamwise wall-shear stress 𝜏𝑤,1 across all Reynolds numbers tested. (a) Hybrid-Joint. (b) Two WMLES baselines, Smagorinsky (solid) and Vreman (dashed), which are nearly indistinguishable at every 𝑅𝑒 𝜃 . To isolate the contribution of the fluctuating wall-stress component, we perform an ablation in which the network correction is removed and only the scaling-law mean wall shear s… view at source ↗
Figure 16
Figure 16. Figure 16: Effect of the wall-stress fluctuation, evaluated at 𝑅𝑒 𝜃 = 5000 (Case 8). Hybrid-Joint (Mean only) retains the scaling-law mean wall stress but removes the network-predicted stress fluctuation. (a) Mean velocity profile. (b) Reynolds shear stress 𝜏12. the mean wall stress alone is therefore not sufficient to recover the near-wall momentum exchange, and the wall-stress fluctuations provide unsteady forcing… view at source ↗
Figure 17
Figure 17. Figure 17: Turbulence statistics for case 8 (𝑅𝑒 𝜃 = 5000) under different ablation settings. (a) Mean streamwise velocity profile. (b)–(d) Root-mean-square velocity fluctuations 𝑢 + 𝑖,rms. (panel a), and the outer-layer turbulent kinetic energy is strongly underestimated (panels b– d), even though its neural SGS closure is identical in form to that of Hybrid-Joint. With the wall model fixed to the conventional equil… view at source ↗
Figure 18
Figure 18. Figure 18: shows the training histories for several choices of 𝑁1 and 𝑁2. Increasing the tracking window to 𝑁2 = 150 or 200 leads to unstable training in both cases: the gradient norm eventually grows by several orders of magnitude and the loss plateaus at an elevated level instead of continuing to decrease. This non-monotonic sensitivity to the window length is itself indicative of the difficulty of controlling gra… view at source ↗
Figure 19
Figure 19. Figure 19 [PITH_FULL_IMAGE:figures/full_fig_p034_19.png] view at source ↗
Figure 20
Figure 20. Figure 20 [PITH_FULL_IMAGE:figures/full_fig_p035_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Instantaneous streamwise velocity contour for Case 9 from two baselines (𝑅𝑒 𝜃,in = 5000): (a) Baseline: Smagorinsky. (b) Baseline: Vreman. (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p036_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Energy transfer between the resolved and subgrid scales for a priori test at 𝑅𝑒 𝜃 = 600. (a) Mean SGS energy-transfer rate ⟨𝛱⟩ in the wall-normal direction. (b) Probability density function (PDF) of the instantaneous energy-transfer rate 𝛱 at 𝑥2/𝛿99 = 0.2. 0 X0-36 [PITH_FULL_IMAGE:figures/full_fig_p036_22.png] view at source ↗

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This paper was first reviewed by deepseek-v4-flash on August 1, 2026.