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arxiv: 2604.11859 · v1 · submitted 2026-04-13 · ⚛️ physics.flu-dyn · stat.ME

Generalised least squares approach for estimation of the log-law parameters of turbulent boundary layers

M. Aguiar Ferreira , B. Ganapathisubramani This is my paper

Pith reviewed 2026-05-10 16:22 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn stat.ME
keywords turbulent boundary layerlog lawgeneralised least squaresuncertainty quantificationvelocity profilehot-wire anemometrywall turbulenceparameter estimation
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The pith

Generalised least squares provides a standardised framework for estimating log-law parameters with quantified uncertainties in turbulent boundary layers.

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

The paper seeks to address uncertainty in log-law parameters by applying generalised least squares regression to the velocity profile. Unlike ordinary least squares, this method uses a full covariance matrix of residuals derived from uncertainties in the measured variables. The approach is tested with synthetic data mimicking hot-wire anemometer measurements on a linear traverse. It offers a way to identify dominant uncertainty sources beforehand and introduces a fitting method that does not require specifying the log region's extent. This enables more reliable comparisons across different experimental datasets.

Core claim

By incorporating the full covariance matrix of residuals propagated from primitive variable uncertainties consistent with the experimental methods, the generalised least squares regression model applied to the log-law velocity profile quantifies uncertainty in the parameters and supports a new fitting procedure that eliminates the need to prescribe the location and extent of the log region, as demonstrated through analysis of synthetic data emulating hot-wire measurements.

What carries the argument

The generalised least squares principle applied to the log-law velocity profile, using a covariance matrix of residuals propagated from experimental uncertainties.

If this is right

  • GLS yields more accurate uncertainty estimates for the log-law parameters compared to OLS and WLS by accounting for error correlations.
  • The systematic analysis with synthetic data serves as a predictive tool for experimental design to mitigate dominant uncertainty sources.
  • New insights are gained into the correlation between log-law parameters such as the von Karman constant and the additive constant.
  • The open-source Python implementation allows standardised uncertainty analysis across various datasets.

Where Pith is reading between the lines

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

  • Adopting this method could help resolve debates on the universality of the log-law constants by enabling direct comparisons.
  • Extending the GLS framework to other turbulence models or flow configurations might reveal similar benefits in uncertainty handling.
  • The synthetic data generation could be adapted to simulate different measurement techniques for broader applicability.

Load-bearing premise

Uncertainties in the primitive variables can be accurately propagated into a covariance matrix of residuals that is consistent with the experimental methods, and the synthetic data sufficiently captures the dominant real-world error sources.

What would settle it

Performing repeated independent experiments on the same turbulent boundary layer setup and checking if the observed scatter in fitted log-law parameters matches the uncertainty intervals predicted by the GLS method.

Figures

Figures reproduced from arXiv: 2604.11859 by B. Ganapathisubramani, M. Aguiar Ferreira.

Figure 1
Figure 1. Figure 1: (a) Synthetic hot-wire measurements of the viscous-normalised velocity profile and log-law fit, with parameters listed in table 1, friction Reynolds number Reτ = 104 , and boundary-layer thickness δ = 0.1 m. The dashed vertical lines indicate the location and extent of the log region 3Re1/2 τ < z+ < 0.15Reτ . fall within the log region 3Re1/2 τ < z+ < 0.15Reτ (Marusic et al. 2013), as indicated by the vert… view at source ↗
Figure 2
Figure 2. Figure 2: Reduced chi-squared merit function and joint uncertainty region of the log-law fit parameters at a confidence level of 95% for the velocity profile in figure 1. The dashed vertical and horizontal lines denote a ±5% deviation from the fitted values. Uncertainties in the primitive variables correspond to the baseline values listed in table 2. 2012). Based on these reported values, a relatively optimistic bas… view at source ↗
Figure 3
Figure 3. Figure 3: Joint uncertainty regions of the log-law fit parameters at a confidence level of 95%, for varying uncertainty in the primitive variables: (a) Uτ and U and (b) z0 and ∆z. Dashed horizontal and vertical lines denote a ±10 % deviation from the fitted values A = 4.094 and κ = 0.3815. The velocity profile corresponds to Reτ = 104 and δ = 0.1 m. Uncertainties in the friction velocity, mean velocity, and referenc… view at source ↗
Figure 4
Figure 4. Figure 4: Maps of the positive marginal uncertainties in the log-law fit parameters σ +∆ κ /κ and σ +∆ A /A, as defined in figure 2b. The maps are presented as functions of the friction Reynolds number Reτ and the boundary-layer thickness δ, along the horizontal and vertical axes, respectively. Panels (a) and (c) correspond to σ∆z = 10 µm, while panels (b) and (d) correspond to σ∆z = 100 µm. Test cases from studies … view at source ↗
Figure 5
Figure 5. Figure 5: Synthetic hot-wire measurements of the viscous-normalised velocity profile with parameters listed in table 1. Coloured markers denote data points within the log region 3Re1/2 τ < z+ < 0.15Reτ . (a) Profiles with varying friction Reynolds number Reτ , ranging from 103 to 106 . Each profile comprises 30 logarithmically spaced data points in the range 10 < z+ < 0.3Reτ , followed by 10 linearly spaced points e… view at source ↗
Figure 6
Figure 6. Figure 6: Maps of the positive marginal uncertainties in the log-law fit parameters σ +∆ κ /κ and σ +∆ A /A, as defined in figure 2b, overlaid with contours of the number of data points in the log regions N. The maps are presented as functions of the friction Reynolds number Reτ and the number of logarithmically-spaced data points along the velocity profile nlog, represented along the horizontal and vertical axes, r… view at source ↗
Figure 7
Figure 7. Figure 7: Estimates of the log-law parameters and the corresponding joint-uncertainty regions at a confidence level of 95 % for the bounds of the log region listed in table B.1. The dashed vertical and horizontal lines denote a ±5 % deviation from the nominal values. The velocity profile corresponds to Reτ = 104 and δ = 0.1 m. conservative bounds set a narrower log region, reducing the systematic error at the expens… view at source ↗
Figure 8
Figure 8. Figure 8: Estimates of the log-law parameters and the corresponding joint-uncertainty regions at a confidence level of 95 % for systematic variations of the lower and upper bounds of the log region, z + l (a) and z + u (b), respectively. The reference range of the log region is [z + l , z+ u ] = [3Re1/2 τ , 0.15Reτ ] (Marusic et al. 2013). The dashed vertical and horizontal lines denote a ±5 % deviation from the nom… view at source ↗
Figure 9
Figure 9. Figure 9: A–κA relationships from Nagib & Chauhan (2008) (equation 4.1, −·), Baxerres et al. (2024) (equation 4.2, − · ·), and Zanoun & Durst (2026) (equation 4.3, −−), overlaid with joint uncertainty regions of the log-law parameters at 95 % confidence level for the baseline uncertainty values in the primitive variables and σ A U /U ranging from 0.1 to 1.5 %. The velocity profile corresponds to Reτ = 104 and δ = 0.… view at source ↗
Figure 10
Figure 10. Figure 10: A–κA relationships from Nagib & Chauhan (2008) (equation 4.1, −·), Baxerres et al. (2024) (equation 4.2, − · ·), and Zanoun & Durst (2026) (equation 4.3, −−), overlaid with joint uncertainty regions of the log-law parameters at 95 % confidence level for the baseline uncertainty values in the primitive variables, σ A U /U = 1 %, and σUτ /Uτ ranging from 0.5 to 5 %. The velocity profile corresponds to Reτ =… view at source ↗
Figure 11
Figure 11. Figure 11: A–κA relationships from Nagib & Chauhan (2008) (equation 4.1, −·), Baxerres et al. (2024) (equation 4.2, − · ·), and Zanoun & Durst (2026) (equation 4.3, −−), overlaid with the major axis of the joint-uncertainty regions for varying Reτ , as expressed by equation 4.6. The velocity profile corresponds to δ = 0.1 m. 4.2. As certain as uncertainty gets How close can estimates of the log-law parameters get? T… view at source ↗
Figure 12
Figure 12. Figure 12: Solutions to the minimisation problem defined in equation 4.9 as a function of the statistical uncertainty in the mean velocity σ A U,i/Ui. (a) Velocity profile at friction Reynolds number Reτ = 104 and boundary-layer thickness δ = 0.1 m, with data points falling within the optimal fitting window coloured as listed in table 3. Vertical dashed lines mark the log-law region, ranging 3Re1/2 τ < z+ < 0.15Reτ … view at source ↗
Figure 13
Figure 13. Figure 13: GLS fit of the log law to the velocity profile of ZPG boundary layers from Samie et al. (2018), across the range of Reynolds numbers Reτ = 6 500 to 20 000. (a) and (b) optimal bounds of the log region are determined via the minimisation problem defined in section 4.2, whereas (c) and (d) fits over the empirically prescribed log region 3Re1/2 τ < z+ < 0.15Reτ (Marusic et al. 2013), indicated by the coloure… view at source ↗
read the original abstract

Uncertainty in estimating the log-law parameters is arguably the greatest obstacle to establishing definitive conclusions regarding their numerical values and universality. This challenge is exacerbated by the limited number of studies that provide thorough uncertainty analyses of experimental data and fitting procedures, and those that do often adopt different approaches, undermining direct comparisons. The present study applies the generalised least squares (GLS) principle to the log-law velocity profile to establish a standardised, comprehensive framework for quantifying uncertainty in the log-law parameters across datasets. GLS contrasts with ordinary least squares (OLS) and weighted least squares (WLS), which do not account for correlation in errors across measured quantities, as well as with alternative heuristic methods that independently sample primitive variables. Instead, it incorporates a full covariance matrix of the residuals, propagated from the uncertainties in the primitive variables and consistent with the experimental methods employed. The study presents a systematic analysis of the response of the log-law regression model using synthetic data, emulating measurements from a hot-wire anemometer mounted on a linear traverse. This analysis serves as a predictive tool for experimental design, identifying a priori the dominant sources of uncertainty in the log-law parameters and potential mitigation strategies. The study also provides new insights into the correlation between the log-law parameters and proposes a new fitting procedure that eliminates the need to prescribe the location and extent of the log region. The open-source Python implementation of the log-law regression model is available for download on GitHub at https://github.com/ma2ferreira/gls_loglaw.git.

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 / 3 minor

Summary. The paper applies the generalised least squares (GLS) principle to the log-law velocity profile to establish a standardised, comprehensive framework for quantifying uncertainty in the log-law parameters across datasets. It incorporates a full covariance matrix of residuals propagated from uncertainties in primitive variables and consistent with experimental methods, contrasts this with OLS/WLS and heuristic approaches, presents a systematic analysis using synthetic data emulating hot-wire anemometer measurements on a linear traverse as a predictive tool for experimental design, proposes a new fitting procedure that eliminates the need to prescribe the location and extent of the log region, and provides an open-source Python implementation.

Significance. If the GLS framework with accurate covariance propagation and the new fitting procedure hold, the work would provide a valuable standardized approach for uncertainty quantification in log-law parameter estimation, facilitating direct comparisons across studies and improving experimental design by identifying dominant uncertainty sources a priori. The open-source code and reproducible synthetic analysis are explicit strengths that support reproducibility and practical adoption.

major comments (2)
  1. [§4] §4 (Synthetic data generation and validation): The central claim that GLS delivers reliable uncertainty quantification rests on the residual covariance matrix being consistent with experimental methods. However, the matrix is constructed and validated exclusively via synthetic data emulating hot-wire anemometer measurements on a linear traverse. This emulation appears to omit key real-world error sources (e.g., time-varying calibration drift, spatial filtering by the sensor, traverse-induced vibrations, or non-stationary flow effects), which directly risks the generalizability of the reported parameter uncertainties and the new fitting procedure.
  2. [§5] §5 (New fitting procedure): The proposed procedure that avoids prescribing the log-region location and extent is demonstrated only on the synthetic datasets. Without explicit validation against multiple independent real experimental boundary-layer datasets (with their full error structures), it is unclear whether the procedure remains unbiased or merely exploits the idealized error model used in the synthetics.
minor comments (3)
  1. [Abstract] Abstract: The claim of 'new insights into the correlation between the log-law parameters' is stated but not summarized; a single sentence outlining the nature of the observed correlation would improve clarity.
  2. [Notation] Notation: Ensure consistent use of symbols for the covariance matrix of residuals and the primitive-variable uncertainties throughout; a table of symbols would aid readability.
  3. [Code availability] Code repository: The GitHub link is welcome, but the repository should include a README with installation instructions, example scripts reproducing the synthetic cases, and clear licensing.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed review. The comments correctly identify that our validation relies on synthetic data, and we address each point below with explanations of the methodological rationale and proposed revisions to improve clarity and transparency.

read point-by-point responses

  1. Referee: §4 (Synthetic data generation and validation): The central claim that GLS delivers reliable uncertainty quantification rests on the residual covariance matrix being consistent with experimental methods. However, the matrix is constructed and validated exclusively via synthetic data emulating hot-wire anemometer measurements on a linear traverse. This emulation appears to omit key real-world error sources (e.g., time-varying calibration drift, spatial filtering by the sensor, traverse-induced vibrations, or non-stationary flow effects), which directly risks the generalizability of the reported parameter uncertainties and the new fitting procedure.

    Authors: We agree that the synthetic emulation is idealized and excludes certain real-world effects such as calibration drift or vibrations. The synthetic framework was deliberately chosen to provide known ground-truth parameters, enabling direct assessment of estimator bias and uncertainty calibration under a controlled, fully specified error model—something impossible with real data alone. The covariance propagation method is modular by design and can incorporate additional error sources once their statistics are measured in a given experiment. We will add a new subsection in §4 explicitly discussing the assumptions of the synthetic model, its limitations, and practical guidance for extending the covariance matrix to include other experimental uncertainties. revision: partial

  2. Referee: §5 (New fitting procedure): The proposed procedure that avoids prescribing the log-region location and extent is demonstrated only on the synthetic datasets. Without explicit validation against multiple independent real experimental boundary-layer datasets (with their full error structures), it is unclear whether the procedure remains unbiased or merely exploits the idealized error model used in the synthetics.

    Authors: The new procedure follows directly from the GLS objective applied to the entire measured profile; it does not rely on the specific synthetic error model but on the supplied covariance matrix to weight all points appropriately. Synthetic data were used to verify unbiased recovery when the model is correctly specified. We acknowledge that real datasets may introduce model mismatch or unaccounted correlations. We will revise §5 to add an explicit discussion of this limitation, state the conditions required for unbiased performance, and provide concrete guidance (with code examples) on constructing suitable covariance matrices from real hot-wire or PIV data so that users can perform their own validation. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper applies the established GLS statistical framework to the log-law velocity profile by propagating a covariance matrix of residuals directly from uncertainties in primitive variables, using synthetic data that emulates hot-wire measurements on a linear traverse. This propagation and the subsequent systematic analysis of parameter response constitute an independent derivation that does not reduce to self-definition, fitted inputs renamed as predictions, or load-bearing self-citations. The proposed fitting procedure that avoids prescribing the log-region location emerges as an output of the GLS analysis rather than an input assumption. No uniqueness theorems or ansatzes are smuggled via self-citation, and the work remains self-contained against external benchmarks such as OLS/WLS comparisons.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard statistical assumptions of generalised least squares and domain knowledge of turbulent boundary layers; no new entities are postulated and the only free parameters are the log-law constants being estimated.

axioms (2)
  • domain assumption The mean velocity profile follows the logarithmic law in a portion of the turbulent boundary layer.
    Required to justify applying the regression model to the data.
  • domain assumption Uncertainties in primitive measured variables can be propagated to construct an accurate covariance matrix of residuals.
    Core premise of the GLS implementation described.

pith-pipeline@v0.9.0 · 5580 in / 1408 out tokens · 55339 ms · 2026-05-10T16:22:12.793327+00:00 · methodology

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Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

  1. [1]

    C.1935 On fitting polynomials to data with weighted and correlated errors

    Aitken, A. C.1935 On fitting polynomials to data with weighted and correlated errors. Proceedings of the Royal Society of Edinburgh54, 12–16. Alfredsson, P. H., Imayama, S., Lingwood, R. J.,¨Orl¨u, R. & Segalini, A.2013 Turbulent boundary layers over flat plates and rotating disks — The legacy of von k´ arm´ an: A stockholm perspective.European Journal of...

    work page 1935

  2. [2]

    & Smits, A.J.2014 Estimating the value of von k´ arm´ an’s constant in turbulent pipe flow.Journal of Fluid Mechanics749, 79–98

    Bailey, S.C.C., Vallikivi, M., Hultmark, M. & Smits, A.J.2014 Estimating the value of von k´ arm´ an’s constant in turbulent pipe flow.Journal of Fluid Mechanics749, 79–98. Baxerres, Victor, Vinuesa, Ricardo & Nagib, Hassan2024 Evidence of quasiequilibrium in pressure-gradient turbulent boundary layers.Journal of Fluid Mechanics987, R8. Bourassa, C. & Tho...

    work page 2014

  3. [3]

    inverse cascades

    Deissler, R. G.1950 Analytical and experimental investigation of adiabatic turbulent flow in smooth tubes. Technical Note NACA-TN-2138. National Advisory Committee for Aeronautics, Lewis Flight Propulsion Laboratory. Dixit, S. A. & Ramesh, O. N.2008 Pressure-gradient-dependent logarithmic laws in sink flow turbulent boundary layers.Journal of Fluid Mechan...

    work page 1950

  4. [4]

    canonical

    K¨ahler, C. J., McKenna, R. & Scholz, U.2005 Wall-shear-stress measurements at moderate re-numbers with single pixel resolution using long distanceµ-PIV–an accuracy assessment. In6th International Symposium on Particle Image Velocimetry Pasadena, California, USA. Kiefer, J. & Wolfowitz, J.1959 Optimum designs in regression problems.The annals of mathemati...

    work page 2005

  5. [5]

    Monkewitz, P

    – A generic feature of turbulent wall layers in ducts.Journal of Fluid Mechanics910, A45. Monkewitz, P. A., Chauhan, K. A. & Nagib, H. M.2007 Self-consistent high-reynolds- number asymptotics for zero-pressure-gradient turbulent boundary layers.Physics of Fluids19(11). Monkewitz, P. A. & Nagib, H. M.2023 The hunt for the K´ arm´ an ‘constant’ revisited.Jo...

    work page 2007

  6. [6]

    ¨Orl¨u, R., Fransson, J. H. M. & Alfredsson, P. H.2010 On near wall measurements of wall bounded flows — The necessity of an accurate determination of the wall position. Progress in Aerospace Sciences46(8), 353–387. ¨Osterlund, J. M.1999 Experimental studies of zero pressure-gradient turbulent boundary layer flow. PhD thesis, Royal Institute of Technology...

    work page 2010

  7. [7]

    Segalini, A., ¨Orl¨u, R

    Journal of Fluid Mechanics851, 391–415. Segalini, A., ¨Orl¨u, R. & Alfredsson, P. H.2013 Uncertainty analysis of the von k´ arm´ an constant.Experiments in Fluids54, 1–9. Sprent, P.1966 A generalized least-squares approach to linear functional relationships.Journal of the Royal Statistical Society: Series B28(2), 278–288. Thibault, R. & Poitras, G. J.2017...

    work page 2013