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REVIEW 3 major objections 6 minor 22 references

Reconsiderations about inner layer of wall-bounded flows

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Wall-bounded turbulent flows carry a universal pure-log inner layer with $1/\kappa_{in}=2.6$, beyond which the log coefficient $\kappa_o$ of the extended overlap becomes flow-dependent.

desk verdict A plausible empirical revision of the overlap model with a genuine new split between κ_in and κ_o, but the universal κ_in claim is not yet robustly supported and needs a transparent fitting procedure. read the letter →

arxiv 2505.18718 v1 pith:UUIDMTAZ submitted 2025-05-24 physics.flu-dyn

Hassan Nagib This is my paper · ORCID
classification physics.flu-dyn
keywords wall-boundedturbulencelogarithmicpluslinearlawvonKármánconstantinnerlayeroverlapregionstreamwisenormalstressdefect-powermodelindicatorfunction
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper revises the account of the mean velocity profile in wall-bounded turbulent flows. It argues that the previously unified extended logarithmic overlap region actually consists of a thin pure-logarithmic inner layer, whose slope is universal at $1/\kappa_{in}=2.6$, followed by a logarithmic-plus-linear overlap whose logarithmic coefficient $\kappa_o$ depends on the flow geometry and pressure gradient. The paper also re-evaluates the two leading models for the streamwise normal stress and finds that an inviscid wall-scaled-eddies (logarithmic) model and a viscous bounded-dissipation (defect-power) model fit the data equally well just outside the viscous-dominated wall layer, with fitted parameters that differ from the accepted overlap-region values. The conclusions sharpen the Reynolds-number requirements for testing asymptotic wall-turbulence models and separate questions about the near-wall inner layer from questions about the outermost overlap and wake.

What carries the argument

The load-bearing object for the mean-velocity analysis is the indicator function $\Xi = y^+ \mathrm{d}U^+/\mathrm{d}y^+ = Y\,\mathrm{d}U^+/\mathrm{d}Y$, whose plateau in a pure-log layer equals $1/\kappa$. The revised model writes $\Xi_{OL} = 1/\kappa_o + S_o(Y - Y_{in})$, inserting the inner-layer boundary $Y_{in}\approx 0.11$ to $0.15$ and assigning the universal value $1/\kappa_{in}=2.6$ to the pure-log segment. For the normal-stress comparison, the key device is the ratio of viscous to total stress, which fixes the fitting region just outside the inner peak ($y^+\gtrsim 20$, viscous stress below 20% of total) and supplies the four fit parameters $A_1$, $B_1$, $\alpha_1$, and $\beta_1$ from DNS and experimental profiles.

What would settle it

Compute the indicator function $\Xi = y^+ \mathrm{d}U^+/\mathrm{d}y^+$ from a well-resolved DNS of an adverse-pressure-gradient boundary layer at $Re_\tau > 10{,}000$. If within $y^+$ beyond the viscous region up to $Y \approx 0.12$ the indicator function fails to reach a plateau at $2.6$, or if that plateau value changes with pressure gradient, the claimed universality of the inner pure-log layer is refuted.

Watch

Extended reading notes

Core claim

The central claim is that observed mean velocity profiles in channels, pipes, and zero-pressure-gradient boundary layers are best represented by a revised logarithmic-plus-linear law in which the single von Kármán coefficient is replaced by two: a universal inner-layer coefficient $\kappa_{in}\approx 1/2.6$ for a pure-log segment lying just above the viscous wall region at sufficiently high $Re_\tau$ (at least 10,000), and a non-universal coefficient $\kappa_o$ for the logarithmic term in the extended overlap region, whose value is set by the pressure gradient imposed by the flow geometry. A second claim is that the constant-shear-stress layer does not coincide with a constant indicator function, so the classical derivation of a logarithmic law from a constant-stress region is refuted by the DNS data examined here. For the streamwise normal stress, the paper claims that the inviscid logarithmic model and the viscous defect-power model both agree with experiments and DNS in a region close to the wall where viscous stress is a minority of the total stress, and that the four fitted parameters $A_1$, $B_1$, $\alpha_1$, and $\beta_1$ vary consistently across $200 < Re_\tau < 100{,}000$.

Load-bearing premise

The load-bearing premise is that a short pure-logarithmic inner layer exists above the viscous wall region at friction Reynolds numbers of roughly 10,000 or more and has the same universal slope $1/\kappa_{in}=2.6$ in every wall-bounded flow; the paper states this as an assumption rather than a derived result.

Editorial extensions

If this is right

  • If the inner pure-log layer is universal at $1/\kappa_{in}=2.6$, then the non-universality of the von Kármán coefficient in the literature is confined to the extended overlap coefficient $\kappa_o$, not to the inner layer.
  • The absence of a constant indicator function over the constant-shear-stress layer rules out the constant-stress derivation of the log law as a valid basis for the overlap region.
  • Because both normal-stress models fit equally well in the near-wall outer edge of the inner layer, fits there cannot discriminate between the inviscid logarithmic and viscous defect-power descriptions, and their parameters must not be compared to overlap-region values.
  • Genuine tests of the assumed pure-log inner layer require friction Reynolds numbers of at least 10,000, so many existing channel, pipe, and boundary-layer datasets fall short of the asymptotic regime.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The fixed value $1/\kappa_{in}=2.6$ could be used as a calibration anchor when extracting overlap-region parameters from measurements, reducing the fitted degrees of freedom by one in boundary-layer studies.
  • The 20% viscous-stress threshold provides a Reynolds-number-independent recipe for choosing the lower fitting bound for normal-stress models, a rule that might transfer to other near-wall quantities such as passive scalar fluctuations.
  • If $\kappa_o$ varies with pressure gradient, wall models in large-eddy simulation that assume a single constant log-law slope may need the two-layer split, especially in separated or strongly accelerated flows.
  • Applying the revised equation to adverse-pressure-gradient boundary layers at $Re_\tau > 10{,}000$ would test whether $Y_{in}$ shifts systematically with pressure-gradient strength, a prediction the paper leaves open.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. This paper proposes a revision of the logarithmic-plus-linear (log+lin) extended-overlap model for the mean velocity profile in wall-bounded turbulent flows. The key modification is to separate the logarithmic coefficient into an inner pure-log coefficient κ_in, claimed to be universal (1/κ_in = 2.6, or κ_in ≈ 0.385), and an outer overlap coefficient κ_o, claimed to depend on the flow geometry and pressure gradient. The revision is motivated by indicator-function analysis of DNS and experimental profiles for channel, pipe, and zero-pressure-gradient boundary-layer flows. The paper also re-evaluates two competing models for the streamwise normal stress (a logarithmic model, Eq. (1), and a defect-power model, Eq. (2)) in a region closer to the wall than the traditional overlap region, reporting that both models fit the data equally well there. The conclusions emphasize the need for higher Reynolds numbers and, for the mean-velocity model, defer a matched-asymptotic derivation to future work.

Significance. The paper offers a clearly testable and falsifiable prediction: if a universal inner pure-log coefficient 1/κ_in = 2.6 exists, it can be checked against high-Reτ data and would change how the von Kármán 'constant' is interpreted in wall-bounded turbulence. A strength of the work is its use of multiple high-quality DNS and experimental datasets and its explicit acknowledgment of the limitations of current Reynolds numbers and of the need for matched asymptotic analysis. However, the central evidence is based on fits with several free parameters and on visual reading of indicator functions; the paper does not provide a derivation of Eq. (7), a sensitivity analysis in the region boundary Yin, or quantitative measures of fit quality for the normal-stress comparisons. These gaps currently prevent the claims from being fully load-bearing.

major comments (3)
  1. [Section III, Eq. (7)] The revised log+lin model is proposed without a derivation, and the paper explicitly states that 'Additional matched asymptotic work is now under consideration, but is not covered here.' Because the central claims—the universality of κ_in and the non-universality of κ_o—rest entirely on this proposed functional form, the manuscript should either supply the matched-asymptotic justification or clearly frame the model as a conjecture and quantify how robustly the fitted parameters support it. As written, the reader is asked to accept a model on the basis of fits to a small number of high-Reτ profiles.
  2. [Section III, Fig. 6 and discussion of Yin] The pure-log region boundary Yin is chosen post hoc: Fig. 6 uses Yin = 0.12, and the text states that Yin 'may be in the range from 0.11 up to 0.15, and flow type or pressure gradient dependent.' Since κ_in is extracted from the segment below Yin and κ_o from the segment above Yin, a variation of Yin from 0.11 to 0.15 can shift both fitted coefficients. The paper reports no sensitivity analysis over this admitted range, nor does it list the individual data sets, Reynolds numbers, and scatter behind the proposed value 1/κ_in = 2.6. Without such evidence, the separation into a universal κ_in and a non-universal κ_o is not convincingly separated from the fitting procedure.
  3. [Section IV, Figs. 8–13] The claim that the logarithmic and defect-power models 'agree equally well' with the near-wall normal-stress data is not quantitatively supported. Each model is fitted with two free parameters (A1, B1 for Eq. (1); α1, β1 for Eq. (2)), but no residuals, goodness-of-fit statistics, or parameter uncertainties are reported; Figs. 12 and 13 show best-fit values without error bars. Since both functional forms are flexible two-parameter fits to the same data, the apparent agreement is not a strong test and should be quantified before drawing conclusions about the 'perplexing' behavior. In addition, the fitting region is defined by the 20% viscous-stress threshold, which is itself chosen without a robustness check.
minor comments (6)
  1. [Section III, Eq. (7)] Equation (7) has a notation inconsistency: the second equality writes κ^{-1} without the subscript o, although the first equality uses κ_o^{-1}.
  2. [Section IV, text near Fig. 7] The text states 'Figure 7 shows the normal stress...' but the Figure 7 caption (and the surrounding discussion) describes the ratio of viscous to total stress; the normal-stress fits appear in Figs. 8 and 9. Please correct the figure references.
  3. [Abstract and Section III] The abstract and Section III use both κ_in and k_in for the same quantity; please standardize the notation.
  4. [Figure captions 9 and 10] Figure captions 9 and 10 have typos in the citations: 'Pirozzoli (2024 [12]' and 'Samie et al. (2018 [9]' are missing closing parentheses.
  5. [Section V, Conclusions] Section V says the fits were made 'outside of the region dominated by viscous effects; that is, the ratio of viscous over total stresses greater than 20%'; this is inconsistent with the abstract, which describes the fitting region as where viscous stresses are less than 20% of total. Please clarify which side of the threshold is meant.
  6. [Section III, last paragraph] The sentence 'This is consistent with the result 0.38 < κ_in < 0.39 found for ZPG boundary layers by Monkewitz and Nagib (2023), ...' attributes to those references a quantity (κ_in) that, to my knowledge, those papers did not define; please clarify whether those values were originally reported as the coefficient of the pure-log segment or as the overlap-region κ.

Circularity Check

2 steps flagged · score 6.0 of 10

The universal 1/κin=2.6 is a fitted value promoted to a universal constant; the central claim is not independently predicted.

  1. fitted input called prediction [Section III, 'Revised logarithmic plus linear overlap model of mean velocity', paragraph following Eq. (7)]
    "Although this data set reveals a value for κin = 0.384, or 1/κin = 2.65, it is proposed based on all measurements from various wall-bounded flows at high Reτ conditions examined, that the best value to use for the coefficient of the limited range of pure-log layer is a universal value for all wall-bounded flows and given by 1/κin = 2.6."

    The central number 1/κin = 2.6 is not a predicted or derived constant: it is the best-fit value obtained from the same velocity profiles that also determine the fitting boundaries Yin and Yout. The paper explicitly defers the matched-asymptotic justification ('Additional matched asymptotic work is now under consideration, but is not covered here.'), so the universality of κin is asserted from the fits rather than demonstrated independently. Since κo is extracted from the same fits, the claimed contrast between a universal κin and a non-universal κo is a property of the chosen fitting windows, not an independent test.

  2. fitted input called prediction [Section IV, 'Evaluation of streamwise normal stress models closer to the wall', around Figs. 7-11]
    "With the same colors, the best fits of equation 1 are shown using the solid lines to represent the logarithmic trend. Similarly, the best fits of the defect-power trend are shown in dotted lines. The fits are carried out in the region outside the location of the peak of the normal stress ... It is perplexing that both an inviscid and a viscous model agree equally well with experiments and DNS data in this region closer to the wall."

    The finding that both the logarithmic and defect-power models 'agree equally well' with the data is by construction: all four parameters (A1, B1, α1, β1) are least-squares fitted to the same data being displayed, with no holdout set or out-of-sample prediction. Two flexible two-parameter curves will typically both match a smooth segment, so the agreement cannot discriminate between the models; it only reflects that each model was optimized to those very data points.

full rationale

The paper's main structural contribution—separating an inner pure-log layer (κin) from the extended log+lin region (κo)—is not itself definitionally circular: Eq. (7) is a representation of the indicator function with an added fitting parameter Yin, and the DNS data for channel and pipe flow are independent of the model form. However, the paper's headline claim that 1/κin is universal at 2.6 is a fitted numerical value, not a derived prediction. The paper acknowledges the inner-layer universality is assumed ('In this model, this inner layer is assumed to be universal among all wall-bounded turbulent flows') and defers matched-asymptotic justification to future work. Likewise, the normal-stress evaluation compares two models after fitting both to the same data, so the reported equal agreement is a consequence of the fitting procedure rather than a discriminating test. These issues make the central quantitative claims partially circular: the universal constant is an output of the fitting exercise, then treated as an established universal. The paper is not entirely circular because the data comparisons, parameter trends, and the refutation of the constant-stress derivation rest on independent DNS and experimental profiles; but the core numerical universality claim is not independently predicted.

Assumptions & free parameters 8 free parameters · 3 assumptions · 0 invented entities

The paper's claims rest on several fitted parameters and domain assumptions. κ_in and κ_o are extracted from fits; Y_in and S_o are chosen per flow; the normal stress model coefficients are all fitted. The revised model lacks a first-principles derivation at this stage, so its value is empirical.

free parameters (8)
  • κ_in (or 1/κ_in) = 1/2.6 ≈ 0.384
    Proposed universal value for the pure-log inner layer, fitted to high-Reτ DNS and experimental profiles.
  • κ_o = Flow-dependent; e.g., 0.41 for channel at Reτ≈5200, 0.44 for pipe at Reτ≈12055
    Extracted from fits of Eq. 7 to the extended overlap region; pressure-gradient dependent.
  • S_o = e.g., 1.1 for channel, 2.7 for pipe
    Linear-term coefficient in the log-plus-linear model, fitted per flow.
  • Y_in = ~0.12 (range 0.11 to 0.15)
    Boundary between the inner pure-log layer and the extended overlap region; chosen per flow.
  • A1 (logarithmic model coefficient) = Varied with Reτ; shown in figure 12
    Parameter of the logarithmic normal stress model, fitted to data in the near-wall region.
  • B1 (logarithmic model constant) = Varied with Reτ; shown in figure 12
    Parameter of the logarithmic normal stress model, fitted to data in the near-wall region.
  • α1 (defect-power model coefficient) = Varied with Reτ; shown in figure 13
    Parameter of the defect-power normal stress model, fitted to data in the near-wall region.
  • β1 (defect-power model coefficient) = Varied with Reτ; shown in figure 13
    Parameter of the defect-power normal stress model, fitted to data in the near-wall region.
assumptions (3)
  • ad hoc to paper A universal pure-log inner layer exists at high Reτ (≥10,000).
    Section III assumes this without derivation; the paper states that additional matched asymptotic work is under consideration.
  • domain assumption The indicator function approach yields reliable estimates of κ and S_o.
    Indicator functions require numerical derivatives; the paper notes this is unsuitable for experimental data and relies on DNS for these quantities.
  • domain assumption The region where viscous stress is less than 20% of total stress is a valid fitting region for both normal stress models.
    The 20% threshold is introduced as a criterion for separating the viscous-dominated wall region, but no justification for the specific value is provided.

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Cite this review

Pith. "Pith review of Reconsiderations about inner layer of wall-bounded flows." pith.science (2026). https://pith.science/paper/UUIDMTAZ

@misc{pith2026250518718,
  author       = {Pith},
  title        = {Pith review of: Reconsiderations about inner layer of wall-bounded flows},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UUIDMTAZ}},
  note         = {Machine review of arXiv:2505.18718}
}
abstract

Following recent evidence that even ZPG boundary layers do not exhibit a purely logarithmic extended overlap region, reconsideration of recently advanced logarithmic plus linear extended overlap region in wall-bounded flows leads to a revision of the model for the extended overlap region. The significant difference between the two representations is a separation between the inner layer and the extended overlap layer in the coefficient of the logarithmic term into $\kappa_{in}$ and k_o, respectively. From a wide range of data examined in wall-bounded flows, the value of k_in is universal and equal to 1/2.6 or in the range 0.38<k_in<0.39. The value of k_o depends on the pressure gradient imposed by the flow geometry. In regard to the trends of the streamwise normal stress, recent publications concluded that the defect-power model developed from bounded dissipation is in more agreement with experimental data from ZPG boundary layers and pipe flows, as well as DNS data for channel and pipe flows, than the logarithmic model developed with inviscid analysis based on wall-scaled eddies. For some recent investigations and the entire previous literature on this popular topic, the assessment is made in the overlap region between inner and outer flows, which has limited viscous effects and is essentially inviscid; i.e., y+_in>400 and Y_out~0.45. This appears to be counterintuitive and deserves further attention. Here, both models are reevaluated using the same data sets from recent investigations in a region closer to the wall but outside the region with viscous stresses exceeding 20% of the total stress; i.e., dominated by viscous effects. It is perplexing that both an inviscid and a viscous model agree equally well with experiments and DNS data in this region closer to the wall.

Figures

Figures reproduced from arXiv: 2505.18718 by the authors.

Figure 1
Figure 1. FIG. 1: Indicator function of mean velocity, Ξ, and turbulent stress, [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Indicator function of mean velocity, Ξ, versus outer-scaled wall distance, [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Indicator function of mean velocity, Ξ, and turbulent stress, [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Indicator function of mean velocity, Ξ, versus outer-scaled wall distance, [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Indicator function of mean velocity, Ξ, and turbulent stress, [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Depiction in indicator function , Ξ, versus outer-scaled wall distance, [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Ratio of viscous to total stress versus inner-scaled wall distance, [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Normal stress, [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Normal stress, [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: Normal stress, [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11: Normal stress, [PITH_FULL_IMAGE:figures/full_fig_p010_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12: Variation with [PITH_FULL_IMAGE:figures/full_fig_p011_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13: Variation with [PITH_FULL_IMAGE:figures/full_fig_p011_13.png]

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Reference graph

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Reviewed August 7, 2026 · model on record in the stance chip above.