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arxiv: 2604.09454 · v1 · submitted 2026-04-10 · ⚛️ physics.flu-dyn · physics.comp-ph

Revisit eddy viscosity in pressure-driven wall turbulence at high Reynolds number

Ben-Rui Xu , Ao Xu This is my paper

Pith reviewed 2026-05-10 16:48 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn physics.comp-ph
keywords eddy viscositywall turbulenceopen-channel flowpipe flowclosed-channel flowturbulence modelingReynolds number effectsBoussinesq hypothesis
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The pith

An outer correction function added to the classical Cess eddy-viscosity model captures configuration-specific behavior in wall-bounded turbulent flows.

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

The paper examines how eddy viscosity varies in three types of pressure-driven wall turbulence: closed channels, open channels with a free surface, and pipes, using high-Reynolds-number DNS data. It finds that the outer region of the flow shows differences depending on the geometry, so a single expression does not fit all. The authors extend the log-law idea into the outer region with a stress-based velocity scale and a correction function, then insert a simple parametric version of that correction into the standard Cess framework that already includes van Driest near-wall damping. This yields a full-depth model that improves predictions notably for open-channel flow while staying as good as the original for the other two configurations. The work highlights how the outer boundary conditions affect the eddy viscosity and thus the mean flow.

Core claim

By inferring eddy viscosity from DNS statistics via the Boussinesq hypothesis across Re_tau from 2000 to 12000, the authors observe configuration dependence in the outer region. They introduce an outer correction function based on extending log-law scaling with a stress-based velocity scale, and embed a compact parametric form of it into the Cess-van Driest model. The resulting model shows improvement in eddy-viscosity profiles, log-law indicator, and skin friction for open-channel flow, while remaining comparable for closed-channel and pipe flows.

What carries the argument

The compact parametric outer correction function embedded in the Cess-type eddy-viscosity framework that includes van Driest near-wall damping.

Load-bearing premise

That a single compact parametric outer correction function works uniformly across different flow geometries without needing separate adjustments for each configuration.

What would settle it

Direct comparison of the model's predicted skin friction or mean velocity profiles against independent DNS or experimental data for open-channel flow at Reynolds numbers outside 2000-12000, or for a new geometry, would show whether the improvement holds or breaks down.

read the original abstract

We investigate eddy-viscosity distributions in pressure-driven wall turbulence for three canonical configurations: plane closed-channel flow, open-channel flow with a free-slip surface, and pipe flow. Using direct numerical simulation (DNS) databases spanning friction Reynolds numbers $Re_{\tau}=$ 2000--12000, we infer the eddy viscosity from one-point statistics through the Boussinesq relation. The DNS-inferred eddy viscosity displays configuration-dependent behavior in the outer region, indicating that a single full-depth expression is not uniformly accurate for all three configurations. Building on the interpretation of eddy viscosity as the product of a velocity scale and a length scale, we extend the log-law scaling into the outer region. Specifically, we adopt a stress-based velocity scale and introduce an outer correction function to capture the remaining dependence on the outer coordinate. We then embed a compact parametric form of this correction into a Cess-type framework with van Driest near-wall damping, yielding a full-depth eddy-viscosity model. We assess the model using eddy-viscosity profiles, the log-law indicator function, and skin friction. The results show that the proposed model yields noticeable improvement for open-channel flow while remaining comparable to the classical Cess model for closed-channel flow and pipe flow. These findings underscore the role of outer boundary conditions in shaping the outer-region eddy viscosity and, consequently, mean-flow predictions.

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 paper claims that DNS-inferred eddy viscosity (via Boussinesq) in closed-channel, open-channel, and pipe flows at Re_tau up to 12000 exhibits configuration-dependent outer-region behavior. It extends the Cess-van Driest framework by introducing a stress-based velocity scale plus a compact parametric outer correction function, embeds this once into a full-depth model, and reports that the resulting predictions improve noticeably for open-channel flow while remaining comparable to the classical Cess model for the other two geometries, as judged by eddy-viscosity profiles, indicator-function collapse, and skin-friction values.

Significance. If the central claim holds after clarification of the fitting procedure, the work would usefully demonstrate that outer boundary conditions shape eddy viscosity and that a single parametric correction can be embedded in an existing framework to improve mean-flow predictions selectively for open-channel flow without degrading performance elsewhere. The high-Re DNS range and use of standard, reproducible assessment metrics from public databases are strengths.

major comments (3)
  1. [Abstract and outer-correction derivation] Abstract and the section introducing the outer correction: the compact parametric form is stated to be chosen to match the same DNS-inferred eddy-viscosity profiles used later for assessment; this creates a circularity burden for the claimed improvement, and no details are given on the fitting procedure, coefficient values, residuals, or sensitivity to post-hoc choices.
  2. [Results and assessment section] Assessment of model performance: no error bars, uncertainty estimates, or quantitative error metrics (e.g., integrated L2 deviation or skin-friction prediction errors) are supplied for the DNS-inferred profiles or the model–DNS comparisons, so the statistical significance of the 'noticeable improvement' for open-channel flow cannot be judged.
  3. [Model embedding and multi-geometry comparison] The weakest assumption—that one fixed parametric outer correction suffices uniformly across geometries without per-configuration retuning—is load-bearing for the central claim yet is supported only by visual profile comparisons; an explicit cross-geometry validation or held-out Re_tau test would be required to substantiate it.
minor comments (2)
  1. [Model formulation] Notation for the outer coordinate and stress-based velocity scale should be defined once at first use and used consistently.
  2. [Figures] Figure captions should explicitly state the Re_tau values and flow configurations shown in each panel to aid readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed review of our manuscript. We address each major comment point by point below, acknowledging where clarifications and additions are needed. We propose specific revisions to strengthen the presentation and address the concerns regarding circularity, quantitative assessment, and validation of the uniform correction assumption.

read point-by-point responses

  1. Referee: [Abstract and outer-correction derivation] Abstract and the section introducing the outer correction: the compact parametric form is stated to be chosen to match the same DNS-inferred eddy-viscosity profiles used later for assessment; this creates a circularity burden for the claimed improvement, and no details are given on the fitting procedure, coefficient values, residuals, or sensitivity to post-hoc choices.

    Authors: We acknowledge the referee's concern about potential circularity in the model development. The compact parametric form of the outer correction was indeed motivated by the configuration-dependent features observed in the DNS-inferred eddy-viscosity profiles across the three geometries. However, the claimed improvement is evaluated relative to the classical Cess model (which lacks this correction) rather than as an absolute fit. To eliminate any ambiguity, we will add a new subsection in the revised manuscript that explicitly details the fitting procedure (including the objective function, fitting range in wall units, and optimization method), reports the resulting coefficient values, quantifies the fit residuals for each configuration, and includes a sensitivity analysis to variations in the fitting range and any post-hoc choices. This will allow independent assessment of robustness. revision: yes

  2. Referee: [Results and assessment section] Assessment of model performance: no error bars, uncertainty estimates, or quantitative error metrics (e.g., integrated L2 deviation or skin-friction prediction errors) are supplied for the DNS-inferred profiles or the model–DNS comparisons, so the statistical significance of the 'noticeable improvement' for open-channel flow cannot be judged.

    Authors: We agree that the absence of quantitative error metrics limits the ability to judge the significance of the reported improvements. In the revised manuscript, we will incorporate error bars on all DNS-inferred eddy-viscosity profiles (derived from statistical convergence of the underlying DNS databases) and provide explicit quantitative metrics. These will include integrated L2 deviations between model and DNS eddy-viscosity profiles over the full depth, as well as absolute and relative errors in predicted skin-friction coefficients for each geometry and Reynolds number. We will also tabulate these metrics to facilitate direct comparison between the proposed model and the classical Cess model. revision: yes

  3. Referee: [Model embedding and multi-geometry comparison] The weakest assumption—that one fixed parametric outer correction suffices uniformly across geometries without per-configuration retuning—is load-bearing for the central claim yet is supported only by visual profile comparisons; an explicit cross-geometry validation or held-out Re_tau test would be required to substantiate it.

    Authors: The central claim rests on the observation that a single, fixed parametric outer correction (derived from the collective DNS data) can be embedded uniformly without geometry-specific retuning. While the current evidence is based on visual profile comparisons and indicator-function collapse across the three configurations, we recognize that this is insufficient for full substantiation. In the revision, we will add an explicit cross-validation analysis: parameters will be fitted using data from two geometries and tested on the held-out third geometry, with performance quantified via the same L2 and skin-friction metrics mentioned above. We will also include tests on held-out Reynolds numbers within each geometry to assess extrapolation. These additions will provide a more rigorous test of the uniformity assumption. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation begins from DNS-inferred eddy viscosity via the Boussinesq relation, extends log-law scaling with a stress-based velocity scale and outer correction motivated by outer-region dependence, then embeds a compact parametric form of that correction into the pre-existing Cess-van Driest framework. The resulting single model is applied uniformly to all three geometries and assessed on standard metrics (profiles, indicator function, skin friction) drawn from the same DNS databases. No equation reduces to its input by construction, no parameter is fitted on a subset and then relabeled a prediction, and no self-citation or uniqueness theorem supplies the load-bearing step. The configuration dependence is addressed by one shared parametric expression rather than geometry-specific retuning, so the reported improvement for open-channel flow and parity with Cess for the other cases constitute ordinary model validation rather than a tautology.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the Boussinesq hypothesis to infer eddy viscosity, the extension of log-law scaling into the outer layer via a stress-based velocity scale, and the assumption that a single parametric correction suffices for three geometries. One free parameter set is introduced for the outer correction; no new physical entities are postulated.

free parameters (1)
  • coefficients of outer correction function
    Compact parametric form introduced to capture remaining outer-coordinate dependence after stress-based scaling.
axioms (2)
  • domain assumption Boussinesq relation holds for inferring eddy viscosity from one-point statistics
    Explicitly used to obtain eddy-viscosity profiles from DNS.
  • domain assumption Log-law scaling can be extended into the outer region with a stress-based velocity scale
    Basis for constructing the outer correction.

pith-pipeline@v0.9.0 · 5542 in / 1507 out tokens · 53670 ms · 2026-05-10T16:48:36.069340+00:00 · methodology

discussion (0)

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Works this paper leans on

52 extracted references · 52 canonical work pages

  1. [1]

    Xu, B.-Q

    C.-X. Xu, B.-Q. Deng, W.-X. Huang and G.-X. Cui, Coherent structures in wall turbulence and mechanism for drag reduction control, Sci. China-Phys. Mech. Astron. https://doi.org/10.1007/s11433-013-5087-4 56 , 1053 (2013)

    work page doi:10.1007/s11433-013-5087-4 2013

  2. [2]

    Procaccia, V.S

    I. Procaccia, V.S. L'vov, and R. Benzi, Colloquium: Theory of drag reduction by polymers in wall-bounded turbulence, Rev. Mod. Phys. https://doi.org/10.1103/RevModPhys.80.225 80 , 225 (2008)

    work page doi:10.1103/revmodphys.80.225 2008

  3. [3]

    Zhang, Structure and coupling characteristics of multiple fields in dust storms, Acta Mech

    H. Zhang, Structure and coupling characteristics of multiple fields in dust storms, Acta Mech. Sin. https://doi.org/10.1007/s10409-023-23339-x 40 , 123339 (2024)

    work page doi:10.1007/s10409-023-23339-x 2024

  4. [4]

    Tian, X.-R

    H.-P. Tian, X.-R. Yi, F. Xu, F. Li and N. Jiang, Lagrangian-based spatial-temporal topological study on the evolution and migration of coherent structures in wall turbulence, Acta Mech. Sin. https://doi.org/10.1007/s10409-021-09006-1 38 , 321465 (2022)

    work page doi:10.1007/s10409-021-09006-1 2022

  5. [5]

    Huang, Y.-C

    J.-D. Huang, Y.-C. Jie, Z.-W. Cui, C.-X. Xu, H.I. Andersson and L.-H. Zhao, How large-scale flow structures affect particle transport in wall turbulence, J. Fluid Mech. https://doi.org/10.1017/jfm.2025.10284 1015 , A23 (2025)

    work page doi:10.1017/jfm.2025.10284 2025

  6. [6]

    Chen, Y.M

    X. Chen, Y.M. Chung and M.-P. Wan, Backflow structures in turbulent pipe flows at low to moderate Reynolds numbers, J. Fluid Mech. https://doi.org/doi:10.1017/jfm.2023.461 966 , A38 (2023)

    work page doi:10.1017/jfm.2023.461 2023

  7. [7]

    Nezu, Open-channel flow turbulence and its research prospect in the 21st century, J

    I. Nezu, Open-channel flow turbulence and its research prospect in the 21st century, J. Hydraul. Eng.-ASCE https://doi.org/10.1061/(ASCE)0733-9429(2005)131:4(229) 131 , 229 (2005)

    work page doi:10.1061/(asce)0733-9429(2005)131:4(229 2005

  8. [8]

    Jim\'enez, Cascades in wall-bounded turbulence, Annu

    J. Jim\'enez, Cascades in wall-bounded turbulence, Annu. Rev. Fluid Mech. https://doi.org/10.1146/annurev-fluid-120710-101039 44 , 27 (2012)

    work page doi:10.1146/annurev-fluid-120710-101039 2012

  9. [9]

    Kim, Physics and control of wall turbulence for drag reduction, Philos

    J. Kim, Physics and control of wall turbulence for drag reduction, Philos. Trans. R. Soc. A-Math. Phys. Eng. Sci. https://doi.org/10.1098/rsta.2010.0360 369 , 1396 (2011)

    work page doi:10.1098/rsta.2010.0360 2010

  10. [10]

    Canton, R

    J. Canton, R. Örlü, C. Chin, N. Hutchins, J. Monty and P. Schlatter, On Large-Scale Friction Control in Turbulent Wall Flow in Low Reynolds Number Channels, Flow Turbul. Combust. https://doi.org/10.1007/s10494-016-9723-8 97 , 811 (2016)

    work page doi:10.1007/s10494-016-9723-8 2016

  11. [11]

    J. Yao, X. Chen and F. Hussain, Drag control in wall-bounded turbulent flows via spanwise opposed wall-jet forcing, J. Fluid Mech. https://doi.org/10.1017/jfm.2018.553 852 , 678 (2018)

    work page doi:10.1017/jfm.2018.553 2018

  12. [12]

    Lee and R

    M. Lee and R.D. Moser, Direct numerical simulation of turbulent channel flow up to Re_ 5200 , J. Fluid Mech. https://doi.org/10.1017/jfm.2015.268 774 , 395 (2015)

    work page doi:10.1017/jfm.2015.268 2015

  13. [13]

    Pirozzoli, J

    S. Pirozzoli, J. Romero, M. Fatica, R. Verzicco and P. Orlandi, One-Point Statistics for Turbulent Pipe Flow up to Re_ 6000 , J. Fluid Mech. https://doi.org/10.1017/jfm.2021.727 926 , A28 (2021)

    work page doi:10.1017/jfm.2021.727 2021

  14. [14]

    J. Yao, X. Chen and F. Hussain, Direct numerical simulation of turbulent open channel flows at moderately high Reynolds numbers, J. Fluid Mech. https://doi.org/10.1017/jfm.2022.942 953 , A19 (2022)

    work page doi:10.1017/jfm.2022.942 2022

  15. [15]

    Z.-H. Xia, P. Zhang and X.I.A. Yang, On skin friction in wall-bounded turbulence, Acta Mech. Sin. https://doi.org/10.1007/s10409-020-01024-4 37 , 589 (2021)

    work page doi:10.1007/s10409-020-01024-4 2021

  16. [16]

    Reichardt, Vollst\"andige Darstellung der turbulenten Geschwindigkeitsverteilung in glatten Leitungen, ZAMM-Z

    H. Reichardt, Vollst\"andige Darstellung der turbulenten Geschwindigkeitsverteilung in glatten Leitungen, ZAMM-Z. Angew. Math. Mech. https://doi.org/10.1002/zamm.19510310704 31 , 208 (1951)

    work page doi:10.1002/zamm.19510310704 1951

  17. [17]

    Z.-S. She, X. Chen and F. Hussain, Quantifying Wall Turbulence via a Symmetry Approach: A Lie Group Theory, J. Fluid Mech. https://doi.org/10.1017/jfm.2017.464 827 , 322 (2017)

    work page doi:10.1017/jfm.2017.464 2017

  18. [18]

    B. J. Cantwell, A universal velocity profile for smooth wall pipe flow, J. Fluid Mech. https://doi.org/10.1017/jfm.2019.669 878 , 834 (2019)

    work page doi:10.1017/jfm.2019.669 2019

  19. [19]

    Cheng, X.-L

    C. Cheng, X.-L. Chen, W.-K. Zhu, W. Shyy, and L. Fu, Progress in physical modeling of compressible wall-bounded turbulent flows, Acta Mech. Sin. https://doi.org/10.1007/s10409-024-23663-x 40 , 323663 (2024)

    work page doi:10.1007/s10409-024-23663-x 2024

  20. [20]

    Pope, Turbulent Flows (Cambridge University Press https://doi.org/10.1017/CBO9781316179475, Cambridge, 2000)

    S.B. Pope, Turbulent Flows (Cambridge University Press https://doi.org/10.1017/CBO9781316179475, Cambridge, 2000)

    work page doi:10.1017/cbo9781316179475 2000

  21. [21]

    Zeitschrift Angewandte Mathematik und Mechanik , author =

    L. Prandtl, Bericht \"uber Untersuchungen zur ausgebildeten Turbulenz, ZAMM-Z. Angew. Math. Mech. https://doi.org/10.1002/zamm.19250050212 5 , 136 (1925)

    work page doi:10.1002/zamm.19250050212 1925

  22. [22]

    von K\'arm\'an, Mechanical similitude and turbulence, NACA Tech

    T. von K\'arm\'an, Mechanical similitude and turbulence, NACA Tech. Mem. Report No. 611 (1931)

    work page 1931

  23. [23]

    Smits, B.J

    A.J. Smits, B.J. McKeon and I. Marusic, High-Reynolds Number Wall Turbulence, Annu. Rev. Fluid Mech. https://doi.org/10.1146/annurev-fluid-122109-160753 43 , 353 (2011)

    work page doi:10.1146/annurev-fluid-122109-160753 2011

  24. [24]

    Baldwin and H

    B. Baldwin and H. Lomax, Thin-layer approximation and algebraic model for separated turbulent flows, AIAA 16th Aerospace Sciences Meeting https://doi.org/10.2514/6.1978-257 78-257 (1978)

    work page doi:10.2514/6.1978-257 1978

  25. [25]

    Coles, The law of the wake in the turbulent boundary layer, J

    D. Coles, The law of the wake in the turbulent boundary layer, J. Fluid Mech. https://doi.org/10.1017/S0022112056000135 1 , 191 (1956)

    work page doi:10.1017/s0022112056000135 1956

  26. [26]

    A. A. R. Townsend, The structure of turbulent shear flow (Cambridge University Press, Cambridge, 1976)

    work page 1976

  27. [27]

    Shan, X.-X

    X.-L. Shan, X.-X. Sun, W.-B. Cao, W.-W. Zhang and Z.-H. Xia, Modeling Reynolds stress anisotropy invariants via machine learning, Acta Mech. Sin. https://doi.org/10.1007/s10409-024-23629-x 40 , 323629 (2024)

    work page doi:10.1007/s10409-024-23629-x 2024

  28. [28]

    Ciofalo, Thermofluid Dynamics of Turbulent Flows, (Springer https://doi.org/10.1007/978-3-030-81078-8, Cham, 2022)

    M. Ciofalo, Thermofluid Dynamics of Turbulent Flows, (Springer https://doi.org/10.1007/978-3-030-81078-8, Cham, 2022)

    work page doi:10.1007/978-3-030-81078-8 2022

  29. [29]

    L. Guo, D. Li, X. Zhang and G.-W. He, LES prediction of space-time correlations in turbulent shear flows, Acta Mech. Sin. https://doi.org/10.1007/s10409-012-0088-5 28 , 993 (2012)

    work page doi:10.1007/s10409-012-0088-5 2012

  30. [30]

    Liu, C.-X

    H.-C. Liu, C.-X. Xu and W.-X. Huang, A total-shear-stress-conserved wall model for large-eddy simulation of high-Reynolds number wall turbulence, J. Comput. Phys. https://doi.org/10.1016/j.jcp.2025.114029 534 , 114029 (2025)

    work page doi:10.1016/j.jcp.2025.114029 2025

  31. [31]

    van Driest, On turbulent flow near a wall, J

    E.R. van Driest, On turbulent flow near a wall, J. Aeronaut. Sci. https://doi.org/10.2514/8.3713 23 , 1007 (1956)

    work page doi:10.2514/8.3713 1956

  32. [32]

    R. D. Cess, A survey of the literature on heat transfer in turbulent tube flow. Research Report 8-0529-R24. Westinghouse (1958)

    work page 1958

  33. [33]

    Reynolds and W.G

    W.C. Reynolds and W.G. Tiederman, Stability of Turbulent Channel Flow, With Application to Malkus's Theory, J. Fluid Mech. https://doi.org/10.1017/S0022112067000308 27 , 253 (1967)

    work page doi:10.1017/s0022112067000308 1967

  34. [34]

    Cossu, Onset of large-scale convection in wall-bounded turbulent shear flows, J

    C. Cossu, Onset of large-scale convection in wall-bounded turbulent shear flows, J. Fluid Mech. https://doi.org/10.1017/jfm.2022.585 945 , A33 (2022)

    work page doi:10.1017/jfm.2022.585 2022

  35. [35]

    Symon, S.J

    S. Symon, S.J. Illingworth and I. Marusic, Energy Transfer in Turbulent Channel Flows and Implications for Resolvent Modelling, J. Fluid Mech. https://doi.org/10.1017/jfm.2020.929 911 , A3 (2021)

    work page doi:10.1017/jfm.2020.929 2020

  36. [36]

    Y.-T. Fan, M. Kozul, W.-P. Li and R.D. Sandberg, Eddy-viscosity-improved resolvent analysis of compressible turbulent boundary layers, J. Fluid Mech. https://doi.org/10.1017/jfm.2024.174 983 , A46 (2024)

    work page doi:10.1017/jfm.2024.174 2024

  37. [37]

    Ying, X.-L

    A.-J. Ying, X.-L. Chen, Z.-G. Li and L. Fu, Optimisation and modelling of eddy viscosity in the resolvent analysis of turbulent channel flows, J. Fluid Mech. https://doi.org/10.1017/jfm.2024.1099 1001 , A20 (2024)

    work page doi:10.1017/jfm.2024.1099 2024

  38. [38]

    Sun, Turbulent Poiseuille Flow Modeling by Modified Prandtl-van Driest Mixing Length, Acta Mech

    B.-H. Sun, Turbulent Poiseuille Flow Modeling by Modified Prandtl-van Driest Mixing Length, Acta Mech. Sin. https://doi.org/10.1007/s10409-022-22066-x 39 , 322066 (2023)

    work page doi:10.1007/s10409-022-22066-x 2023

  39. [39]

    Hoyas and J

    S. Hoyas and J. Jim\'enez, Scaling of the velocity fluctuations in turbulent channels up to Re_ =2003 , Phys. Fluids https://doi.org/10.1063/1.2162185 18 , 011702 (2006)

    work page doi:10.1063/1.2162185 2003

  40. [40]

    Hoyas and J

    S. Hoyas and J. Jim\'enez, Reynolds number effects on the Reynolds-stress budgets in turbulent channels, Phys. Fluids https://doi.org/10.1063/1.3005862 20 , 101511 (2008)

    work page doi:10.1063/1.3005862 2008

  41. [41]

    Bernardini, S

    M. Bernardini, S. Pirozzoli and P. Orlandi, Velocity statistics in turbulent channel flow up to Re_ =4000 , J. Fluid Mech. https://doi.org/10.1017/jfm.2013.674 742 , 171 (2014)

    work page doi:10.1017/jfm.2013.674 2013

  42. [42]

    Hoyas, M

    S. Hoyas, M. Oberlack, F. Alc\'antara-\'Avila, S.V. Kraheberger and J. Laux, Wall turbulence at high friction Reynolds numbers, Phys. Rev. Fluids https://doi.org/10.1103/PhysRevFluids.7.014602 7 , 014602 (2022)

    work page doi:10.1103/physrevfluids.7.014602 2022

  43. [43]

    Oberlack, S

    M. Oberlack, S. Hoyas, S.V. Kraheberger, F. Alcántara-Ávila, and J. Laux, Turbulence Statistics of Arbitrary Moments of Wall-Bounded Shear Flows: A Symmetry Approach, Phys. Rev. Lett. https://doi.org/10.1103/PhysRevLett.128.024502 128 , 024502 (2022)

    work page doi:10.1103/physrevlett.128.024502 2022

  44. [44]

    Pirozzoli, Searching for the Log Law in Open Channel Flow, J

    S. Pirozzoli, Searching for the Log Law in Open Channel Flow, J. Fluid Mech. https://doi.org/10.1017/jfm.2023.616 971 , A15 (2023)

    work page doi:10.1017/jfm.2023.616 2023

  45. [45]

    J. Yao, S. Rezaeiravesh, P. Schlatter and F. Hussain, Direct numerical simulations of turbulent pipe flow up to Re_ 5200 , J. Fluid Mech. https://doi.org/10.1017/jfm.2022.1013 956 , A18 (2023)

    work page doi:10.1017/jfm.2022.1013 2022

  46. [46]

    Pirozzoli, On the Streamwise Velocity Variance in the Near-Wall Region of Turbulent Flows, J

    S. Pirozzoli, On the Streamwise Velocity Variance in the Near-Wall Region of Turbulent Flows, J. Fluid Mech. https://doi.org/10.1017/jfm.2024.467 989 , A5 (2024)

    work page doi:10.1017/jfm.2024.467 2024

  47. [47]

    Launder and D.B

    B.E. Launder and D.B. Spalding, The numerical computation of turbulent flows, Comput. Methods Appl. Mech. Eng. https://doi.org/10.1016/0045-7825(74)90029-2, 3 , 269 (1974)

    work page doi:10.1016/0045-7825(74)90029-2 1974

  48. [48]

    Cui, X.-L

    Y.-K. Cui, X.-L. Tan, H. Zhang and X.-J. Zheng, Electrically driven relaminarisation of particle-laden turbulent channel flow, J. Fluid Mech. https://doi.org/10.1017/jfm.2025.10990 1025 , A53 (2025)

    work page doi:10.1017/jfm.2025.10990 2025

  49. [49]

    Chen and K.R

    X. Chen and K.R. Sreenivasan, Reynolds number scaling of the peak turbulence intensity in wall flows, J. Fluid Mech. https://doi.org/10.1017/jfm.2020.991 908 , R3 (2021)

    work page doi:10.1017/jfm.2020.991 2020

  50. [50]

    Chen and K.R

    X. Chen and K.R. Sreenivasan, Law of bounded dissipation and its consequences in turbulent wall flows, J. Fluid Mech. https://doi.org/10.1017/jfm.2021.1052 933 , A20 (2022)

    work page doi:10.1017/jfm.2021.1052 2021

  51. [51]

    Chen and K.R

    X. Chen and K.R. Sreenivasan, Reynolds number asymptotics of wall-turbulence fluctuations, J. Fluid Mech. https://doi.org/10.1017/jfm.2023.928 976 , A21 (2023)

    work page doi:10.1017/jfm.2023.928 2023

  52. [52]

    Chen and K.R

    X. Chen and K.R. Sreenivasan, Bounded dissipation law and profiles of turbulent velocity moments in wall flows, Proc. Natl. Acad. Sci. U.S.A. https://doi.org/10.1073/pnas.2502265122 122 , e2502265122 (2025)

    work page doi:10.1073/pnas.2502265122 2025