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arxiv: 2511.18566 · v2 · submitted 2025-11-23 · ⚛️ physics.flu-dyn

Effect of subgrid-scale anisotropy on wall-modeled large-eddy simulation of turbulent flow with smooth-body separation

Di Zhou , H. Jane Bae This is my paper

Pith reviewed 2026-05-17 05:52 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords subgrid-scale anisotropywall-modeled LESflow separationGaussian bumpReynolds stressfavorable pressure gradientnormal stresses
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The pith

Anisotropic SGS stresses yield consistent separation bubble sizes in WMLES

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

This paper shows that anisotropic subgrid-scale stress models give consistent predictions of flow separation in wall-modeled large-eddy simulation of a Gaussian bump, unlike eddy-viscosity models which vary non-monotonically with grid refinement. The effect is traced to the favorable pressure gradient region on the windward side, where normal SGS stresses alter Reynolds stress transport and separation onset. The findings indicate that including normal stress contributions improves reliability as grids are refined.

Core claim

Eddy-viscosity SGS models produce non-monotonic separation bubble sizes under refinement in Gaussian bump WMLES, while anisotropic models are consistent. Anisotropy in the favorable pressure gradient region modifies SGS dissipation and diffusion, affecting Reynolds stresses. Improvement comes from normal stress terms, confirmed by a priori filtered DNS showing high anisotropy under FPG.

What carries the argument

Anisotropic SGS stress tensor that modifies SGS dissipation and diffusion in the Reynolds stress equation via normal components.

If this is right

  • Separation predictions become consistent under grid refinement.
  • Favorable pressure gradient region controls downstream separation.
  • Normal SGS stresses are key to the improvement over eddy-viscosity models.
  • Resolved Reynolds stresses dominate at finer grids while SGS fluctuations influence dissipation.

Where Pith is reading between the lines

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

  • Anisotropic SGS modeling may benefit WMLES of other pressure-driven separated flows.
  • Canonical flow a priori tests can inform SGS choices for complex geometries.

Load-bearing premise

The SGS anisotropy from filtered Couette-Poiseuille flow under FPG represents that in the Gaussian bump, and consistency comes from anisotropy rather than other model aspects.

What would settle it

Non-monotonic separation bubble variation in WMLES even with anisotropic SGS models would falsify the claim that anisotropy ensures consistency.

Figures

Figures reproduced from arXiv: 2511.18566 by Di Zhou, H. Jane Bae.

Figure 1
Figure 1. Figure 1: Simulation set-up for flow over a Guassian-shaped b [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Isocontours of mean velocity /∞ from the medium-mesh simulations with the SM (a) and MSM (b) and from the reference DNS (Uzun & Malik 2022) (c). initial transients, and then another three flow-through times (6/∞) to obtain converged statistics. 3. Sensitivity of mean flow separation prediction to SGS model 3.1. Separation prediction and grid convergence test The flow field around the Gaussian-shaped bump, … view at source ↗
Figure 3
Figure 3. Figure 3: Isocontours of mean eddy viscosity / from the medium-mesh simulations with the SM (a) and MSM (b). -0.5 0 0.5 1 -1.6 -1.2 -0.8 -0.4 0 0.4 SM MSM DNS / [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Mean pressure coefficient on the bottom surface from [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The profiles of mean streamwise velocity at [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Mean streamwise velocity at the first off-wall cell c [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Mean separation bubble length on the leeward side o [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Virtual-interface setup dividing the domain into [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Mean separation bubble length on the leeward side o [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Mean streamwise velocity 1 (a), Reynolds shear stress ′ 1 [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Mean streamwise velocity 1 (a), Reynolds shear stress ′ 1 [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Mean streamwise momentum budget terms at [PITH_FULL_IMAGE:figures/full_fig_p016_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Mean pressure budget terms at / = 0.05 from medium-mesh simulations with the SM (a) and MSM (b). All terms are nondimensionalized using ∞, and . The line notations correspond to equations (4.6) and (4.7). 4.2. Mean pressure equation The Poisson equation for the mean pressure of the flow is given by − 1 [PITH_FULL_IMAGE:figures/full_fig_p017_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Reynolds shear stress ′ 1 [PITH_FULL_IMAGE:figures/full_fig_p018_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: SGS dissipation (solid square) and diffusion [PITH_FULL_IMAGE:figures/full_fig_p021_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: SGS dissipation (solid square) and diffusion [PITH_FULL_IMAGE:figures/full_fig_p022_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Mean separation point from simulations using the [PITH_FULL_IMAGE:figures/full_fig_p023_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Mean streamwise momentum budget terms at 0 [PITH_FULL_IMAGE:figures/full_fig_p024_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Mean streamwise momentum budget terms at 0 [PITH_FULL_IMAGE:figures/full_fig_p025_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Mean SGS stress tensor components sgs at / = −0.1 for medium-mesh simulations with the SM (a) and MSM (b). normal gradient of the streamwise velocity. In the SM case, the remaining components are negligible relative to this dominant shear stress. In contrast, the MSM produces two additional stress components of appreciable magnitude, sgs 11 and sgs 22 , which modify the principal directions of the mean SG… view at source ↗
Figure 21
Figure 21. Figure 21: Mean SGS stress tensor components ( sgs 11 , sgs 12 , and sgs 22 ) at / = −0.1 for simulations with the SM using the coarsest mesh (a), coarse mesh (b), medium mesh (c), and fine mesh (d). In contrast, the mean anisotropic stress exhibits a distinctly different behavior, where the shear stress component are negligible while the normal stress components dominate. Comparing these results with the total mean… view at source ↗
Figure 22
Figure 22. Figure 22: Mean SGS stress tensor components ( sgs 11 , sgs 12 , and sgs 22 ) at / = −0.1 for simulations with the MSM using the coarsest mesh (a), coarse mesh (b), medium mesh (c), and fine mesh (d). (a) -0.005 0 0.005 0 0.01 [PITH_FULL_IMAGE:figures/full_fig_p028_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Mean isotropic SGS stress tensor components ( [PITH_FULL_IMAGE:figures/full_fig_p028_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: SGS stress tensor r.m.s. components sgs , rms at / = −0.1 for medium-mesh simulations with the SM (a) and MSM (b). that this qualitative behavior of SGS stress fluctuations remains largely consistent throughout the FPG region. Figures 25 and 26 show the r.m.s. values of the SGS stress fluctuations at / = −0.1 for simulations using different mesh resolutions with the SM and MSM, respectively. Here, only th… view at source ↗
Figure 25
Figure 25. Figure 25: SGS stress tensor r.m.s. components ( sgs 11, rms, sgs 12, rms, and sgs 22, rms) at / = −0.1 for simulations with the SM using the coarsest mesh (a), coarse mesh (b), medium mesh (c), and fine mesh (d). the half-channel height and is the wall motion speed. Additional details of the DNS and the filtering operation are provided in Sections C and D. The a priori analysis focuses on the lower half of the chan… view at source ↗
Figure 26
Figure 26. Figure 26: SGS stress tensor r.m.s. components ( sgs 11, rms, sgs 12, rms, and sgs 22, rms) at / = −0.1 for simulations with the MSM using the coarsest mesh (a), coarse mesh (b), medium mesh (c), and fine mesh (d). (a) 0 0.001 0.002 0.003 0 0.01 [PITH_FULL_IMAGE:figures/full_fig_p031_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: Isotropic SGS stress tensor r.m.s. components ( [PITH_FULL_IMAGE:figures/full_fig_p031_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: Mean SGS stress tensor components sgs (a) and SGS stress tensor r.m.s. components sgs , rms (b) obtained from the Gaussian filtering of velocity field from the DNS of turbulent Couette-Poiseuille flow with the standard deviations of the Gaussian kernel /Δ = /Δ =2. deviation leads to larger mean SGS stress magnitudes, consistent with the fact that wider filters remove more turbulent scales and therefore at… view at source ↗
Figure 29
Figure 29. Figure 29: Curvature radius (a) of the Gaussian bump surface [PITH_FULL_IMAGE:figures/full_fig_p034_29.png] view at source ↗
Figure 30
Figure 30. Figure 30: Mean separation bubble length on the leeward side [PITH_FULL_IMAGE:figures/full_fig_p035_30.png] view at source ↗
Figure 31
Figure 31. Figure 31: Isocontours of the instantaneous streamwise vel [PITH_FULL_IMAGE:figures/full_fig_p036_31.png] view at source ↗
Figure 32
Figure 32. Figure 32: Mean streamwise velocity relative to the bottom w [PITH_FULL_IMAGE:figures/full_fig_p036_32.png] view at source ↗
Figure 33
Figure 33. Figure 33: Isocontours of the filtered streamwise velocity [PITH_FULL_IMAGE:figures/full_fig_p038_33.png] view at source ↗
read the original abstract

We examine the role of anisotropic subgrid-scale (SGS) stress in wall-modeled large-eddy simulation (WMLES) of flow over a spanwise-uniform Gaussian-shaped bump, with emphasis on predicting flow separation. The simulations show that eddy-viscosity-based SGS models often yield non-monotonic predictions of the mean separation bubble size on the leeward side under grid refinement, whereas models incorporating anisotropic SGS stress produce more consistent results. To identify where SGS anisotropy is most critical, we introduce anisotropic SGS stress in selected regions of the domain. The results reveal that the windward side, where a strong favorable pressure gradient (FPG) occurs, is crucial in determining downstream separation. Analysis of the Reynolds stress transport equation shows that fluctuations of anisotropic SGS stress modify SGS dissipation and diffusion in this region, thereby altering the Reynolds stress and the onset of separation. Examination of the mean streamwise momentum equation indicates that at coarse resolutions, the mean SGS shear stress dominates, and the differences between the eddy-viscosity-based and anisotropic models remain minor. With grid refinement, resolved Reynolds stresses increasingly govern the near-wall momentum transport, and the influence of SGS stress fluctuations grows as they determine the SGS dissipation and diffusion of Reynolds stresses. Component-wise analysis of the SGS stress tensor further shows that the improvement arises mainly from including significant normal stress contributions. An a priori study using filtered direct numerical simulation of turbulent Couette-Poiseuille flow confirms that wall-bounded turbulence under FPG is highly anisotropic and that anisotropic SGS models provide a more realistic SGS stress representation than eddy-viscosity-based models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript examines the role of anisotropic subgrid-scale (SGS) stress in wall-modeled large-eddy simulation (WMLES) of turbulent flow over a spanwise-uniform Gaussian bump. It reports that eddy-viscosity SGS models produce non-monotonic separation-bubble sizes under grid refinement, while anisotropic models yield more consistent results. Regional insertion of anisotropic stress identifies the windward favorable-pressure-gradient (FPG) region as critical; Reynolds-stress transport analysis links anisotropic SGS fluctuations to changes in SGS dissipation/diffusion and downstream separation; component-wise decomposition attributes the improvement mainly to normal-stress contributions. An a priori filtered-DNS study on Couette-Poiseuille flow under FPG is used to confirm that wall-bounded turbulence in FPG is highly anisotropic and that anisotropic models better represent the SGS stress.

Significance. If the central claim holds, the work would be significant for WMLES of separated flows: it supplies concrete numerical evidence that SGS anisotropy, particularly normal-stress components in FPG regions, can remove non-monotonic grid dependence in separation predictions. The regional-activation experiments and Reynolds-stress transport analysis are useful diagnostic tools that could be adopted more broadly. The a priori filtered-DNS check, while limited in scope, provides a falsifiable test of SGS anisotropy under FPG.

major comments (2)
  1. [a priori filtered DNS study] The a priori filtered-DNS study is performed on Couette-Poiseuille flow under FPG; this configuration lacks the bump curvature, adverse-pressure-gradient separation, and WMLES wall treatment of the target Gaussian-bump case. Consequently, the reported SGS anisotropy may not be representative of the anisotropy actually present in the WMLES, weakening the link between the a priori evidence and the main simulation results (abstract and a priori study section).
  2. [regional SGS insertion experiments] The regional SGS insertion experiments and the comparison between eddy-viscosity and anisotropic models do not explicitly state that all other model constants, numerical dissipation settings, and wall-model/SGS coupling parameters are identical. Without such controls, the observed monotonicity in separation-bubble size cannot be attributed unambiguously to the anisotropic stress tensor itself rather than to confounding differences in model formulation or dissipation (regional insertion experiments and grid-refinement studies).
minor comments (2)
  1. [grid-refinement studies] Quantitative error bars or uncertainty estimates on the reported separation-bubble lengths are not provided, making it difficult to judge the statistical significance of the monotonicity claim under refinement.
  2. [numerical methods] The manuscript would benefit from a short table summarizing the SGS model constants, filter widths, and wall-model parameters used for each simulation set to facilitate reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment below and indicate where revisions will be made to improve clarity and strengthen the presentation.

read point-by-point responses

  1. Referee: [a priori filtered DNS study] The a priori filtered-DNS study is performed on Couette-Poiseuille flow under FPG; this configuration lacks the bump curvature, adverse-pressure-gradient separation, and WMLES wall treatment of the target Gaussian-bump case. Consequently, the reported SGS anisotropy may not be representative of the anisotropy actually present in the WMLES, weakening the link between the a priori evidence and the main simulation results (abstract and a priori study section).

    Authors: We appreciate this point. The Couette-Poiseuille configuration was deliberately selected as a simplified, canonical wall-bounded flow that isolates the influence of a strong favorable pressure gradient on SGS anisotropy without the additional complexities of surface curvature or separation. While we acknowledge that this setup does not replicate the full bump geometry, adverse-pressure-gradient region, or wall-model implementation, it provides a controlled demonstration that wall-bounded turbulence under FPG is markedly anisotropic and that eddy-viscosity models under-represent the normal stress components. We will revise the a priori study section and abstract to explicitly state the rationale for this choice, note the differences from the target configuration, and clarify that the a priori results serve as supporting evidence rather than a direct replica of the WMLES conditions. revision: partial

  2. Referee: [regional SGS insertion experiments] The regional SGS insertion experiments and the comparison between eddy-viscosity and anisotropic models do not explicitly state that all other model constants, numerical dissipation settings, and wall-model/SGS coupling parameters are identical. Without such controls, the observed monotonicity in separation-bubble size cannot be attributed unambiguously to the anisotropic stress tensor itself rather than to confounding differences in model formulation or dissipation (regional insertion experiments and grid-refinement studies).

    Authors: We confirm that all regional insertion experiments and grid-refinement comparisons were performed with identical model constants, numerical dissipation settings, and wall-model/SGS coupling parameters; the sole difference was the form of the SGS stress tensor (eddy-viscosity versus anisotropic) within the designated regions. We regret that this was not stated explicitly and will add a dedicated paragraph in the methods section of the revised manuscript detailing these controls to remove any ambiguity. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected; claims rest on independent numerical experiments and transport analysis

full rationale

The paper's derivation chain consists of WMLES runs with eddy-viscosity versus anisotropic SGS models, selective regional activation of anisotropy, direct analysis of the Reynolds-stress transport equation, mean-momentum balance, and a separate a priori filtered-DNS study on Couette-Poiseuille flow under FPG. None of these steps reduce by construction to fitted parameters, self-definitions, or self-citation chains; the reported consistency under refinement and the localization to normal-stress contributions are outcomes of the simulations themselves rather than tautological re-statements of the model inputs. The a priori confirmation is performed on an independent configuration and therefore supplies external evidence rather than circular support.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard LES modeling assumptions and the representativeness of the chosen bump geometry and Couette-Poiseuille a priori case; no new free parameters, axioms, or invented entities are introduced in the abstract.

axioms (2)
  • standard math Filter commutes with derivatives in the LES equations
    Implicit in all SGS modeling and Reynolds stress transport analysis
  • domain assumption The filtered DNS of Couette-Poiseuille flow under FPG is representative of SGS anisotropy in the bump flow
    Used to confirm that wall-bounded turbulence under FPG is highly anisotropic

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