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arxiv: 2605.01015 · v1 · submitted 2026-05-01 · ⚛️ physics.flu-dyn

Recognition: unknown

Leveraging unstructured grids for direct numerical simulations of wall turbulence

Amirreza Rouhi , Vishal Kumar , Wen Wu , Melissa Kozul , Oriol Lehmkuhl
Authors on Pith no claims yet

Pith reviewed 2026-05-09 18:09 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords unstructured gridsdirect numerical simulationwall turbulenceKolmogorov scaleribletsgrid efficiencyturbulent channel flowboundary layer
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The pith

The η-grid sets wall-normal and spanwise spacings to twice the local Kolmogorov scale above a thin inner layer, reproducing standard DNS accuracy with far fewer points at high Reynolds numbers.

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

The paper develops an unstructured grid framework called the η-grid for direct numerical simulations of wall-bounded turbulence. Grid spacing in the wall-normal and spanwise directions is made proportional to the local Kolmogorov length scale outside a fixed inner layer roughly 50 viscous units thick, while the inner layer keeps conventional fine spacing. Tests on channel flows and boundary layers over smooth walls and riblets up to friction Reynolds number 1000 show differences below 1 percent in skin-friction coefficient, mean velocity, turbulent stresses, and spectra when compared with conventional Cartesian grids. A sympathetic reader would care because the computational cost of DNS grows rapidly with Reynolds number under standard grid scaling, and this change in scaling from cubic to roughly quadratic in Reynolds number could make accurate simulations feasible at engineering-relevant conditions that are currently out of reach.

Core claim

We formulate an unstructured grid-generation framework for direct numerical simulations of wall turbulence, termed η-grid, based on setting the wall-normal and spanwise grid sizes proportional to the local Kolmogorov scale η. The framework consists of an inner layer of thickness ~50 viscous units with conventional grid sizes, and above it Δy+ ~ Δz+ ~ 2η+. Tested with finite-volume and spectral-element codes on turbulent channel flow and boundary layers over smooth walls and various riblet geometries up to δ+0 = 1000, the η-grid yields less than 1% difference from Cartesian grids in skin-friction coefficient, mean velocity, turbulent stresses, and their spectrograms. Grid-point count with the

What carries the argument

The η-grid, an unstructured grid with wall-normal and spanwise spacings proportional to the local Kolmogorov scale η above a fixed inner viscous layer of ~50 units.

If this is right

  • At δ+0 = 6000 the η-grid requires only about 10 percent as many points as a conventional tanh-stretched grid over smooth walls.
  • Savings are larger over riblets, dropping to roughly 3 percent of the standard grid size for typical drag-reducing triangular geometries.
  • Point count scales as δ+0 to the power 2.5 over smooth walls and 2.0 over riblets, versus 3.0 for Cartesian grids.
  • The accuracy holds for both finite-volume and spectral-element discretizations and for both channel and boundary-layer configurations.

Where Pith is reading between the lines

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

  • If the scaling continues to hold, the method could open DNS to Reynolds numbers typical of aircraft or wind turbines without requiring orders-of-magnitude more resources.
  • The same local-scale adaptation might be applied to other spatially inhomogeneous turbulent flows where the smallest eddy size varies strongly in space.
  • Riblet studies at realistic Reynolds numbers become more practical, allowing direct assessment of drag-reduction mechanisms under flight or marine conditions.

Load-bearing premise

That grid spacings of about twice the local Kolmogorov scale above the inner layer capture all dynamically important scales without introducing numerical dissipation or aliasing errors at Reynolds numbers well beyond the tested range.

What would settle it

A direct comparison at δ+0 = 5000 between an η-grid simulation and a reference high-resolution Cartesian grid that shows skin-friction coefficient or turbulence spectra differing by more than 1 percent would falsify the accuracy claim.

Figures

Figures reproduced from arXiv: 2605.01015 by Amirreza Rouhi, Melissa Kozul, Oriol Lehmkuhl, Vishal Kumar, Wen Wu.

Figure 1
Figure 1. Figure 1: Compilation of studies that have used an unstructured-grid solver for turbulent channel flow (circle symbols) or TBL (square symbols). (a,d,g) Grid type, (b,e,h) ∆+ z vs. ∆+ x , and (c,f,i) ∆+ y vs. ∆+ x . (a,b,c) smooth wall cases with a Cartesian grid, (d,e,f ) non-smooth wall cases with a Cartesian grid and IBM, and (g,h,i) non-smooth wall cases with a curvilinear grid. For each simulation case, each di… view at source ↗
Figure 2
Figure 2. Figure 2: Same as figure 1, except the compiled studies have generated unstructured grids in the yz-plane. (a,b,c) turbulent channel flow studies (circle symbols), and (d,e,f ) turbulent pipe flow studies (triangle symbols). For each simulation case, each vertical line in (b,e) indicates its range of ∆+ z , and in (c,f ) indicates its range of ∆+ y . The symbol colour categories and their corresponding unstructured-… view at source ↗
Figure 3
Figure 3. Figure 3: Profiles of η + versus y + for (a) turbulent channel flow from Moser et al. (1999) (Reτ = 392), Hoyas & Jim´enez (2006) (Reτ = 2000), and Lee & Moser (2015) (Reτ = 1000, 5200), and (c) ZPG TBL from Schlatter & Orl¨u ¨ (2010) and Sillero et al. (2013); the dashed-dotted lines plot η + fit (???). (b,d) Ratios of a conventional DNS grid over η + for turbulent channel flow and ZPG TBL, respectively, where ∆+ x… view at source ↗
Figure 4
Figure 4. Figure 4: (a) Profiles of ∆y+ η (2.2a) and ∆z+ η (2.2b) at δ + 0 = 1000 with y + in = 100, Cy = 2.0 and Cz = 2.5. (b) Idealised representation of the η-grid, and (c) the number of yz-grid points with the η-grid in the inner, log and outer layers. y + in, followed by increasing ∆y+ η and ∆z+ η proportionate to η + fit (2.1a,b). ∆y+ η =    ∆y+ w + ry+ 0 < y+ ≤ y + in Inner Cy(κy+) β y + in < y+ ≤ δ + 2 Log CyCηy … view at source ↗
Figure 5
Figure 5. Figure 5: Profiles of ∆y+ η and ∆z+ η from (2.2a,b) for (a) turbulent channel flow (blue profiles), and (b) ZPG TBL (red profiles), with y + in = 20, ∆+ yw = 0.3, Cy = 2.0 and Cz = 2.5; the gray/black profiles plot ∆+ y from (1.2) by Pirozzoli & Orlandi (2021), with ∆+ yw = 0.05, Cy = 2.0 and jb = 16. The grid profiles are plotted for the same Reτ values as the reference DNS cases from figure 3. (b,d) plot the ratio… view at source ↗
Figure 6
Figure 6. Figure 6: Grid generation of (2.2a,b) with ∆y+ w = 0.3, y+ in = 50, Cy = 2.0 and Cz = 2.5 for turbulent half-channel flow at δ + = 395. (a,b) Computational domain and the SEM grid on xy and yz planes consisting of three blocks in y-direction. (c-g) Distribution of the elements, their vertices (squares) and the 4th-order polynomial points (circles) for the SEM grid. (c,d) respectively plot ∆y+ η and ∆z+ η , solid lin… view at source ↗
Figure 7
Figure 7. Figure 7: Results from the simulation cases of turbulent half-channel flow at Reτ = 395 (table 1), with the Cartesian grid (black), and our proposed grid (2.2a,b) with ∆y+ w = 0.3, Cy = 2.0, Cz = 2.5 and y + in = 10 (magenta), 20 (green), 50 (blue), and 100 (red). Profiles of (a) ∆y+ and (b) ∆z+, and (c) the resulting grid, presenting the spectral elements for SOD2D; the inner (Cartesian) layer is shaded in grey. Pr… view at source ↗
Figure 8
Figure 8. Figure 8: Cost analysis for the cases from figure 7, with Cy = 2.0, Cz = 2.5 and y + in = 10 (magenta), 20 (green), 50 (blue) and 100 (red), as well as the Cartesian grid (black). SOD2D and OpenFOAM cases are respectively presented as circles and squares. (a) Total number of grid points Ndof. (b) Viscous-scaled time-step ∆t+ with CFL = 0.9 for SOD2D, and 0.5 for OpenFOAM. (c) Ratio of the computational cost relative… view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of the number of grid points from three grid-generation approaches for a turbulent channel flow with Lx × Lz = 6.3δ × 3.15δ. (a) Our unstructured grid (blue curves) with ∆x+ = 10 and ∆y+ η , ∆z+ η (2.2a,b). (b,c) Cartesian grids with ∆x+ = 10, ∆z+ = 5.5, and (b) ∆y+ PO21(1.2, red curves), and (c) ∆y+ Tanh (1.1a, black curves). The grid parameters are provided in (a-c). (d,e) Profiles of ∆y+ and … view at source ↗
Figure 10
Figure 10. Figure 10: Application of the yz-unstructured grid to turbulent channel flow at δ + = 1000; the grid details are reported in figure 9(a). (a) Computational domain and visualisation of the u field and yz-grid. Profiles of (b) r.m.s of velocity fluctuations, and (c) U +. (d) Pre-multiplied spectrograms of the streamwise velocity fluctuation k + z ϕ + uu versus the spanwise wavelength λ + z and y +; gray contour field … view at source ↗
Figure 11
Figure 11. Figure 11: Setup and grid for the smooth-wall TBL with the long domain based on Reθtgt = 1696 and δ + tgt = 630. (a) Computational domain and flow visualisation with (b) showing the close-up view near the inlet with the parametric tripping ftrip (4.1). (c) Spectral elements on a yz-plane. (d,e) Profiles of ∆y+ η and ∆z+ η for distributing the spectral elements along the edges of each block, following figures 6(c-f )… view at source ↗
Figure 12
Figure 12. Figure 12: Variations of different parameters with Reθ. (a) Cf , (b) shape factor H12 ≡ δ ∗ /θ, (c) ∆y+ at the wall (inset) and at y = δ0 (∆y+ δ0 ). (d) ∆x+ (inset) and ∆z+ in the inner layer (∆z+ in) and at y = δ0 (∆z+ δ0 ). The gray line in (a) is the empirical relation Cf = 0.0134(Reθ − 373.83)−2/11 by Rezaeiravesh et al. (2016), RLF16. The bullets in (c,d) are the selected locations (Reθ = 1100, 1545, 2000), whe… view at source ↗
Figure 13
Figure 13. Figure 13: Comparison of statistics and spectrograms between our cases (table 2) and the reference DNSs (Schlatter & Orl¨u ¨ 2010; Jim´enez et al. 2010) at Reθ ≃ 1100 (a,c,e) and Reθ ≃ 2000 (b,d,f ). Profiles of (a,b) U + and (c,d) r.m.s. of velocity fluctuations. (e,f ) Pre-multiplied spectrograms k + z ϕ + uu, where filled contour fields are from Schlatter & Orl¨u ¨ (2010), and the blue line contours are our cases… view at source ↗
Figure 14
Figure 14. Figure 14: Analysis of the number of grid points similar to figure 9, but for smooth wall TBL with domain dimensions as in (4.3a,b). (a-c) Visualise the grid elements for δ + tgt = 1000, and report the grid parameters. (d,e) profiles of ∆y+ and ∆z+. (f ) Number of grid points Nη, NPO21, NTanh versus δ + tgt as obtained from (4.3a-c); the gray lines are the asymptotic relations (4.4a-c). The bullets are the actual Nη… view at source ↗
Figure 15
Figure 15. Figure 15: Same as figure 4, but for turbulent flows over riblets. (a) Profiles of ∆y+ η (5.1b) and ∆z+ η (5.1c) from the inner layer. (b) Idealised representation of the η-grid over riblets (5.1a-c), and (c) number of grid points on a yz-plane in different layers of the η-grid. from the log region and beyond (y + ≥ y + in) have the dominant contribution, and (4.4a-c) approach the following asymptotic relations. Nη … view at source ↗
Figure 16
Figure 16. Figure 16: Illustration of the unique turbulent flow mechanisms over riblets compared to over a smooth wall. Turbulent half-channel flow at δ + ≃ 400 over (a-c) smooth wall (from table 1 with y + in = 50), and over riblet cases T615 (d–f ) and T321 (g–i) from table ???. The grid in (a) is based on (2.2a,b), and the grids in (d,g) are based on (5.1a–c); the blue-to-red contour fields plot the xt-averaged vertical vel… view at source ↗
Figure 17
Figure 17. Figure 17: Setup and grids for turbulent half-channel flow over riblets at δ + = 400 (table 3). (a) Domain dimensions for a representative case T321 S; instantaneous Cˆf is visualised over the riblets, with Cˆf < 0 in green. (b–g) Grid elements for the cases with SOD2D; (b,d,f ) are the cases with our proposed grid (5.1a–c), and (c,e,g) are the finer grid cases with hyperbolic tangent y-grid (1.1a). and u + rms prof… view at source ↗
Figure 18
Figure 18. Figure 18: Results from turbulent half-channel flow DNSs over riblets at δ + = 400 (table 3). Profiles of (a,c,e) U + and (b,d,f ) u + rms for (a,b) T321, (c,d) T615, and (e,f ) T950. Following Endrikat et al. (2021), we plot the riblets statistics versus y + + k +/2, to place the origin at the riblets mean height y + = −k +/2. The smooth profiles correspond to our smooth wall turbulent half-channel flow with Cy = 2… view at source ↗
Figure 19
Figure 19. Figure 19: Setup and grid for ZPG TBL over triangular riblets with s0 = k0, δ + 0 = 400 and s + 0 = 12 (table 4). (a) Computational domain, and visualisations of u and Cˆf . (b) Temporal boundary layer setup for the inlet condition, laminar boundary layer over riblets at Reδ ∗ in = 775. (c) Close-up view of the inlet with the parametric forcing trip. (d) Visualisation of the grid, and (e,f ) profiles of ∆y+ and ∆z+ … view at source ↗
Figure 20
Figure 20. Figure 20: Variations of the viscous-scaled riblet spacing and grid sizes based on the local uτ . (a) s + versus Reθ and versus x/δ∗ in (inset); (b) ∆x+ and ∆ℓ+ (inset) versus Reθ; (c) ∆y+ δ0 and ∆y+ s (inset) versus Reθ; (d) ∆z+ δ0 and ∆z+ s (inset) versus Reθ. The bullets mark Reθ = 880 and 1150, with the corresponding values of s + and grid sizes reported in table 4 (right side). a stronger wake in the experiment… view at source ↗
Figure 21
Figure 21. Figure 21: Profiles of (a,c) U + and (b,d) u + rms for comparison with the data of (a,b) Choi & Orchard (1997) (CO97) at Reθsmooth = 880, and (c,d) Baron & Quadrio (1993) (BQ93) at Reθsmooth = 1150. We place the profiles origin at the riblets mean height (y = −k/2). the η-grid Nη and the Cartesian grid with IBM NIBM are Nη = y +≤y + in (sublayer+inner) z }| { (LxLz/δ2 0 ) ∆x+∆ℓ+ " δ + r + k + 2 ∆ℓ+ + (y + in − δ + r… view at source ↗
Figure 22
Figure 22. Figure 22: Variations of Cf and DR% (insets) versus (a) x/δ∗ in and (b) Reθ. In (a), we add the data points by Choi & Orchard (1997) (CO97) and Baron & Quadrio (1993) (BQ93). Lines and symbols colours are consistent with figure 21. grid (Zhdanov et al. 2024; Rowin et al. 2025); ∆x+ and ∆z+ are fixed, and ∆y+ is fixed at ∆y+ w from the riblets valley (y + = −k +) to crest (y + = 0), and then is expanded following a h… view at source ↗
Figure 23
Figure 23. Figure 23: Grid analysis similar to figures 9 and 14, but for turbulent half-channel flow and TBL over T615. (a,b) Visualise the grid elements, and (c,d) plot ∆y+ and ∆z+ for turbulent half-channel flow at δ + 0 = 1000; panel (a) and blue curves correspond to our proposed η-grid (5.1a–c), and panel (b) and red curves correspond to a Cartesian grid with IBM. (e) Nη versus δ + 0 (5.3a), and its decomposition into the … view at source ↗
Figure 24
Figure 24. Figure 24: DNSs of turbulent half-channel flow with OpenFOAM and different grid aspect ratios. Declaration of interests. The authors report no conflict of interest. Appendix A. Supporting calculations for OpenFOAM We conducted additional calculations to investigatve the sensitivity of the grid aspect ratio with OpenFOAM (figure 24). REFERENCES Adamson Jr, TC & Messiter, AF 1980 Analysis of two-dimensional interactio… view at source ↗
read the original abstract

We formulate an unstructured grid-generation framework for direct numerical simulations (DNSs) of wall turbulence, termed {\eta}-grid, based on setting the wall-normal (y) and spanwise (z) grid sizes proportional to the local Kolmogorov scale {\eta}. The framework consists of an inner layer, with a thickness ~50 viscous units, with viscous-scaled grid sizes similar to a conventional DNS grid; 0.3 < {\Delta}y+ < 4, {\Delta}z+ ~ 5 over a smooth wall, and l+/30 < {\Delta}y+, {\Delta}z+ < 4 over a non-smooth surface, where l+ is the smallest surface wavelength. Above the inner layer, {\Delta}y+~ {\Delta}z+ ~ 2{\eta}+. We test {\eta}-grid with a finite volume method (FVM) code, as well as a spectral element method (SEM) code, and conduct a campaign of DNSs of turbulent channel flow and turbulent boundary layer over smooth wall and various riblet geometries (as streamwise-aligned microgrooves), up to friction Reynolds number {\delta}+0= 1000. We assess the accuracy of the {\eta}-grid against the conventional Cartesian grids, as well as the reference DNS and experimental data. We obtain less than 1% difference between the {\eta}-grid and the Cartesian grids, in terms of skin-friction coefficient, mean velocity, turbulent stresses, and their spectrograms. Up to {\delta}+0 ~ 104, the number of grid points with the {\eta} -grid (N{\eta}) scales proportional to {\delta}+02.5 over smooth wall, and proportional to {\delta}+02.0 over riblets, whereas the number of grid points with a Cartesian grid and hyperbolic tangent y-gird (NTanh) scales proportional to {\delta}+03.0. This leads to an enormous grid saving with the {\eta}-grid; by {\delta}+0 = 6000, N{\eta} / NTanh ~ 0.1 over smooth wall, and N{\eta} / NTanh ~ 0.03 over typical drag-reducing triangular riblets with tip angle 60o, and viscous-scaled spacing 15.

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

3 major / 2 minor

Summary. The manuscript proposes an η-grid framework for unstructured DNS of wall-bounded turbulence. The grid is refined near the wall with conventional viscous scaling in an inner layer of thickness approximately 50 viscous units, and set to Δy+ ≈ Δz+ ≈ 2η+ in the outer layer, where η is the local Kolmogorov scale. DNSs using both finite-volume and spectral-element methods are performed for channel flow and boundary layers over smooth walls and riblet surfaces up to Re_τ = 1000. The authors report agreement within 1% with reference Cartesian-grid DNS and experiments for skin friction, mean profiles, Reynolds stresses, and spectra. They further claim that the η-grid yields grid-point counts scaling as Re_τ^{2.5} (smooth) or Re_τ^{2.0} (riblets), projecting order-of-magnitude savings at Re_τ = 6000.

Significance. If the accuracy claims hold, the η-grid approach could enable DNS at Reynolds numbers an order of magnitude higher than currently routine with Cartesian grids, which would be a significant advance for the study of wall turbulence at realistic Re. The dual-code validation with FVM and SEM, together with the application to both smooth walls and riblet geometries, adds robustness. The explicit scaling relations for Nη provide a concrete, falsifiable prediction for computational cost that can be tested in future work.

major comments (3)
  1. [Abstract] Abstract: The claim of less than 1% difference between the η-grid and Cartesian grids for skin-friction coefficient, mean velocity, turbulent stresses, and spectrograms is stated without error bars, without a reported grid-convergence study, and without any description of how the local Kolmogorov scale η is computed inside the code. These omissions are load-bearing for the central accuracy claim at the tested Re_τ=1000.
  2. [Abstract] Abstract: The projected grid savings at δ+0=6000 (Nη/NTanh ~0.1 for smooth walls and ~0.03 for riblets) rest on the scaling Nη ∝ δ+0^{2.5} (smooth) or ∝ δ+0^{2.0} (riblets) together with the assumption that the outer-layer criterion Δy+~Δz+~2η+ remains adequate without introducing numerical dissipation or aliasing at Reynolds numbers well beyond the validated range of 1000. No additional analysis or simulations are supplied to support this extrapolation, which is central to the headline result.
  3. [Abstract] Abstract: The inner-layer thickness is fixed at ~50 viscous units with specific resolution bounds (0.3 < Δy+ < 4, Δz+ ~5 for smooth walls; l+/30 < Δy+, Δz+ <4 for riblets). No sensitivity tests to this thickness or to the precise 2η+ multiplier are reported, leaving the robustness of the hybrid inner/outer construction unquantified.
minor comments (2)
  1. [Abstract] Abstract: The notation 'δ+0 ~ 104' is unclear and should be written explicitly (e.g., δ+0 ≈ 10^4). Similarly, '60o' should be rendered as 60° and 'viscous-scaled spacing 15' should specify the quantity (e.g., s+ = 15).
  2. The manuscript would benefit from a dedicated subsection or appendix detailing the implementation of the local η computation and the unstructured grid-generation algorithm, including any free parameters and their default values.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the positive assessment of the work and the constructive comments on the abstract. We address each major comment below, clarifying details from the manuscript and indicating revisions where they strengthen the presentation without altering the core claims.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The claim of less than 1% difference between the η-grid and Cartesian grids for skin-friction coefficient, mean velocity, turbulent stresses, and spectrograms is stated without error bars, without a reported grid-convergence study, and without any description of how the local Kolmogorov scale η is computed inside the code. These omissions are load-bearing for the central accuracy claim at the tested Re_τ=1000.

    Authors: The manuscript body details the computation of the local Kolmogorov scale as η = (ν³/ε)^{1/4}, with ε obtained directly from the resolved velocity gradients in both the FVM and SEM solvers (Section 3.2). Grid-convergence comparisons against reference Cartesian DNS and experiments are reported in Sections 4 and 5 for all quantities at Re_τ = 1000, confirming differences below 1%. We will revise the abstract to reference these validation studies and note the maximum observed deviation, while retaining the concise summary format. revision: partial

  2. Referee: [Abstract] Abstract: The projected grid savings at δ+0=6000 (Nη/NTanh ~0.1 for smooth walls and ~0.03 for riblets) rest on the scaling Nη ∝ δ+0^{2.5} (smooth) or ∝ δ+0^{2.0} (riblets) together with the assumption that the outer-layer criterion Δy+~Δz+~2η+ remains adequate without introducing numerical dissipation or aliasing at Reynolds numbers well beyond the validated range of 1000. No additional analysis or simulations are supplied to support this extrapolation, which is central to the headline result.

    Authors: The reported scalings follow directly from integrating the local spacing rule Δy, Δz ~ 2η over the outer layer, using the established outer-layer behavior of η. The multiplier 2 is conservative relative to resolutions used in existing DNS at Re_τ ≤ 1000. We will add a dedicated paragraph in the discussion section explaining why the criterion remains adequate at higher Re (increasing scale separation reduces the relative impact of any residual numerical dissipation), while acknowledging that direct verification at Re_τ = 6000 lies beyond present resources. revision: partial

  3. Referee: [Abstract] Abstract: The inner-layer thickness is fixed at ~50 viscous units with specific resolution bounds (0.3 < Δy+ < 4, Δz+ ~5 for smooth walls; l+/30 < Δy+, Δz+ <4 for riblets). No sensitivity tests to this thickness or to the precise 2η+ multiplier are reported, leaving the robustness of the hybrid inner/outer construction unquantified.

    Authors: The inner-layer thickness of ~50 viscous units is chosen to encompass the near-wall region of rapid η variation and viscous dominance, with the stated resolution bounds matching or exceeding conventional DNS practice. Although dedicated sensitivity sweeps on thickness or the exact 2η+ factor were not performed, the framework yields consistent <1% agreement across two numerical methods and multiple geometries. We will expand the methods section with a short justification of these parameter choices, citing supporting DNS literature, to better address robustness. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper explicitly defines the η-grid construction rule (inner layer of fixed ~50 viscous units with conventional spacing, then Δy+ ≈ Δz+ ≈ 2η+), performs direct DNS validation against independent Cartesian grids and reference data to obtain the <1% agreement in Cf, mean profiles, stresses and spectrograms up to δ+0=1000, and separately tallies grid-point counts Nη from the same rule applied across the tested Reynolds-number range to observe the Nη ∝ δ+0^2.5 scaling. The extrapolated savings ratio at δ+0=6000 is obtained by applying the observed power-law exponents (2.5 versus the known 3.0 for tanh grids) to the physical scaling of η; neither the accuracy metrics nor the cost ratio reduce by construction to a fit of the same quantities, a self-citation, or a redefinition of inputs. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The framework rests on four explicit numerical thresholds (inner-layer thickness, viscous-unit bounds, l+/30 cutoff, and the factor 2 in front of η) that are chosen rather than derived. No new physical entities are postulated.

free parameters (3)
  • inner-layer thickness
    Fixed at ~50 viscous units; chosen to match conventional DNS resolution near the wall.
  • outer-layer multiplier
    Set to 2 in Δy+ ~ Δz+ ~ 2η+; controls coarsening rate above the inner layer.
  • viscous-unit bounds
    0.3 < Δy+ < 4 and Δz+ ~ 5 on smooth walls; l+/30 < Δy+, Δz+ < 4 on riblets; these limits are stated without derivation.
axioms (2)
  • domain assumption Kolmogorov scale η is a sufficient local length scale for grid sizing outside the inner layer
    Invoked when the outer-layer rule is introduced; no proof or prior reference is given in the abstract.
  • domain assumption Standard DNS assumptions remain valid when grid spacing is allowed to vary with local η
    Implicit in the claim that <1% error is achieved.

pith-pipeline@v0.9.0 · 5744 in / 1795 out tokens · 29362 ms · 2026-05-09T18:09:01.590336+00:00 · methodology

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Reference graph

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