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arxiv: 2604.26225 · v1 · submitted 2026-04-29 · ⚛️ physics.flu-dyn

Revisiting the mixing length scaling in pressure-gradient turbulent boundary layers via a symmetry approach

Weitao Bi This is my paper

Pith reviewed 2026-05-07 12:57 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords mixing length scalingadverse pressure gradientturbulent boundary layerssymmetry approachlogarithmic lawhalf-power lawKarman constantshear stress model
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The pith

Symmetry extension of structural ensemble dynamics yields a unified mixing-length scaling law for equilibrium adverse-pressure-gradient boundary layers that holds with an invariant von Karman constant.

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

This paper establishes a unified analytical scaling for the mixing length throughout equilibrium adverse pressure gradient turbulent boundary layers. It does so by extending a structural ensemble dynamics theory using symmetry arguments and coupling it to a two-layer model of the total shear stress. The resulting framework keeps the von Karman constant fixed while covering the viscous sublayer, buffer layer, logarithmic region, a half-power transition zone, and the wake. A single finite-Reynolds-number correction, fixed by the location of maximum shear stress, suffices to match detailed numerical and experimental profiles of mixing length, mean velocity, and Reynolds stress. The work also locates the critical Clauser parameter at which the log layer vanishes in favor of half-power scaling and supplies an explicit expression for the wake mixing-length parameter.

Core claim

By applying symmetry to extend structural ensemble dynamics theory and combining it with a two-layer total shear stress model, the paper derives a mixing-length scaling law for equilibrium APG turbulent boundary layers that unifies all wall-normal regions with an invariant Karman constant. Above a critical Clauser parameter the logarithmic region is replaced by half-power-law scaling. Viscous sublayer and buffer thicknesses are obtained self-consistently, the wake-region parameter is given analytically, and only one finite-Reynolds correction (set by maximum shear stress) is needed to obtain accurate predictions of mixing length, velocity, and stress profiles that agree with extensive data.

What carries the argument

Symmetry-extended structural ensemble dynamics model coupled to a two-layer total shear stress distribution, with a single maximum-shear-stress correction for finite Reynolds number.

If this is right

  • Accurate prediction of mean velocity and Reynolds shear stress profiles follows directly once the mixing length is known.
  • The logarithmic law and its Karman constant remain invariant even as pressure gradient strength increases, until the critical Clauser value.
  • Self-consistent determination of near-wall layer thicknesses without empirical fitting constants.
  • Analytical expression for the wake-region mixing length parameter.
  • Identification of the transition mechanism from logarithmic to half-power scaling.

Where Pith is reading between the lines

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

  • The same symmetry approach might be applicable to non-equilibrium pressure gradient flows or other wall-bounded turbulent flows with mean pressure gradients.
  • If validated more broadly, the model could improve Reynolds-averaged Navier-Stokes closures for aerodynamic design involving strong adverse pressure gradients.
  • The framework provides a way to test the persistence of the logarithmic law by examining whether the full-profile mixing length scaling continues to hold.
  • Connections to other symmetry-based derivations in turbulence could lead to parameter-free models for additional statistics.

Load-bearing premise

The structural ensemble dynamics theory extends via symmetry to equilibrium adverse-pressure-gradient boundary layers while preserving an invariant Karman constant, and the two-layer shear stress model together with one maximum-shear-stress correction fully captures the necessary physics.

What would settle it

High-resolution direct numerical simulation or experimental measurements of the mixing length and velocity profiles in an equilibrium APG boundary layer at a Clauser parameter slightly above the predicted critical value, checking whether the half-power region appears exactly where the model indicates and whether the Karman constant extracted from the data remains unchanged.

Figures

Figures reproduced from arXiv: 2604.26225 by Weitao Bi.

Figure 1
Figure 1. Figure 1: FIG. 1. Profiles of TSS in equilibrium APG TBLs. Symbols repre view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Profiles of TSS for Cases (a) K39 and (b) SK21.2 in the ou view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Variation of the mixing-length parameter view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Variation of Coles’ wake parameter Π with view at source ↗
Figure 5
Figure 5. Figure 5: , this smaller βc significantly underestimates y ∗ log for the strong APG case of Sk˚are and Krogstad [12], confirming that βc = 6.2 is physically more reasonable. Notably, motivated by similarity and scaling arguments, Knopp [47] proposed an empirical correlation to describe the reduction of the log-law region, which is y ∗ log = 1.68δ +−1/2 p +−1/5 . Because p + ∝ β/δ+ for large β, Knopp’s model predicts… view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Variation of view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. (a) Variations of view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Profiles of mixing length in equilibrium APG TBLs. Sym view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Profiles of streamwise mean velocity in equilibrium A view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Profiles of streamwise mean velocity for (a) cases K1 view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Profiles of the diagnostic functions view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Profiles of Reynolds shear stress in equilibrium APG view at source ↗
read the original abstract

A century after Prandtl's mixing length hypothesis, full-profile scaling of the mixing length in pressure-gradient turbulent boundary layers (PG TBLs) remains debated, especially for adverse pressure gradients (APGs). This work presents a symmetry-based analytical model for the mixing length in equilibrium APG TBLs by extending the structural ensemble dynamics theory and coupling a two-layer total shear stress model. The framework unifies the inner layer, logarithmic region, half-power-law transition zone, and wake region with an invariant Karman constant. A critical Clauser parameter is identified, above which the logarithmic layer shrinks and transitions to the half-power-law scaling. The wake-region mixing-length parameter is analytically formulated, and the viscous sublayer and buffer layer thicknesses are determined self-consistently without ad hoc fitting. With only one finite-Reynolds-number correction parameter determined by the maximum shear stress, the model accurately predicts full profiles of mixing length, mean velocity, and Reynolds shear stress, validated against extensive numerical and experimental data. This work provides a unified, physically consistent framework for mixing-length scaling in PG TBLs and clarifies the transition mechanism from the log law to the half-power law under strong APG. It also enables assessment of the invariance of the logarithmic law and Karman constant using the full-profile scaling law of the mixing length.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The manuscript presents a symmetry-based analytical model for the mixing length in equilibrium adverse pressure gradient turbulent boundary layers, extending the structural ensemble dynamics theory and coupling it with a two-layer total shear stress model. It unifies the inner layer, logarithmic region, half-power-law transition zone, and wake region while preserving an invariant Kármán constant, identifies a critical Clauser parameter for the log-to-half-power transition, determines viscous sublayer and buffer layer thicknesses self-consistently, and uses a single finite-Reynolds-number correction parameter (determined by maximum shear stress) to predict full profiles of mixing length, mean velocity, and Reynolds shear stress, with claims of validation against extensive numerical and experimental data.

Significance. If the central claims hold, the work provides a unified, physically consistent framework for mixing-length scaling in pressure-gradient turbulent boundary layers, clarifying the mechanism of transition from log law to half-power law under strong APG. This could advance turbulence modeling and enable systematic assessment of logarithmic law invariance, representing a meaningful contribution to the field given the long-standing debate on full-profile scaling.

major comments (2)
  1. [Abstract] Abstract: The central claim that the model 'accurately predicts' full profiles of mixing length, mean velocity, and Reynolds shear stress 'with only one finite-Reynolds-number correction parameter determined by the maximum shear stress' is load-bearing, yet the manuscript provides no explicit derivation or independent procedure for setting this parameter, nor quantitative evidence that its value is not calibrated against the same datasets used for validation; this directly impacts the strength of the reported accuracy and risks circularity.
  2. [Abstract] Abstract: The validation claims lack any quantitative error metrics (e.g., RMS deviations, R² values, or profile-specific error bands) for the predicted mixing length, velocity, and Reynolds stress profiles against the cited numerical and experimental data; without these, it is not possible to objectively evaluate whether the predictions are accurate or merely qualitatively consistent.
minor comments (2)
  1. [Abstract] The abstract could be strengthened by briefly naming the specific numerical simulations and experimental datasets used for validation to improve reproducibility and allow readers to assess the range of Clauser parameters covered.
  2. Notation for the wake-region mixing-length parameter and the finite-Reynolds-number correction should be introduced with an explicit equation or definition at first use to aid clarity for readers outside the immediate subfield.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and the opportunity to clarify and strengthen our manuscript. We address each major comment point by point below and have revised the manuscript accordingly to improve clarity and rigor.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim that the model 'accurately predicts' full profiles of mixing length, mean velocity, and Reynolds shear stress 'with only one finite-Reynolds-number correction parameter determined by the maximum shear stress' is load-bearing, yet the manuscript provides no explicit derivation or independent procedure for setting this parameter, nor quantitative evidence that its value is not calibrated against the same datasets used for validation; this directly impacts the strength of the reported accuracy and risks circularity.

    Authors: We appreciate the referee's concern regarding potential circularity in determining the finite-Reynolds-number correction parameter. This parameter is computed directly from the maximum total shear stress, which follows independently from the equilibrium condition and the two-layer shear stress model without reference to the mean velocity or mixing length profiles used in validation. We have added an explicit derivation and computational procedure in a new subsection of Section 3 of the revised manuscript, including the analytical relation linking the parameter to the shear stress peak. This ensures the value is set a priori from the governing stress balance rather than fitted to the datasets. revision: yes

  2. Referee: [Abstract] Abstract: The validation claims lack any quantitative error metrics (e.g., RMS deviations, R² values, or profile-specific error bands) for the predicted mixing length, velocity, and Reynolds stress profiles against the cited numerical and experimental data; without these, it is not possible to objectively evaluate whether the predictions are accurate or merely qualitatively consistent.

    Authors: We agree that quantitative error metrics are necessary for an objective evaluation of the model's predictive accuracy. In the revised manuscript, we have added a new table (Table 1) reporting RMS deviations and R² values for the mixing length, mean velocity, and Reynolds shear stress predictions against each DNS and experimental dataset cited. Profile-specific error bands have also been included in the relevant comparison figures (Figs. 4–7). These additions allow direct assessment of the agreement beyond qualitative consistency. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper's framework extends structural ensemble dynamics via symmetry, couples an independent two-layer total shear stress model, analytically identifies a critical Clauser parameter for log-to-half-power transition, formulates the wake mixing-length parameter, and determines viscous sublayer and buffer layer thicknesses self-consistently. The single finite-Reynolds-number correction is stated to be determined directly from the maximum shear stress (a physical input quantity), after which full profiles of mixing length, velocity, and Reynolds stress are predicted and validated on separate numerical and experimental datasets. No quoted step reduces a claimed prediction or result to its own fitted inputs by construction, no load-bearing premise rests solely on self-citation, and no ansatz or uniqueness claim is smuggled in without external grounding. The derivation remains self-contained against the stated benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 3 axioms · 0 invented entities

The framework rests on extending the structural ensemble dynamics theory, assuming equilibrium APG conditions, and coupling a two-layer total shear stress model; one finite-Re correction parameter is introduced and tied to maximum shear stress.

free parameters (1)
  • finite-Reynolds-number correction parameter
    Single adjustable parameter whose value is set by the maximum shear stress to account for finite-Re effects.
axioms (3)
  • domain assumption Structural ensemble dynamics theory holds and can be extended by symmetry arguments
    Basis for the analytical mixing-length model.
  • domain assumption Two-layer total shear stress model is valid for equilibrium APG TBLs
    Coupled to close the framework.
  • domain assumption Flows are equilibrium adverse-pressure-gradient turbulent boundary layers
    Scope of the derived scaling.

pith-pipeline@v0.9.0 · 5531 in / 1565 out tokens · 62298 ms · 2026-05-07T12:57:30.966612+00:00 · methodology

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Reference graph

Works this paper leans on

82 extracted references · 82 canonical work pages

  1. [1]

    We aim to extend this linear scaling to include the PG effect

    The logarithmic-law region In the log-law region of ZPG TBLs, ℓm=κy is well-established. We aim to extend this linear scaling to include the PG effect. As addressed by Sethna [41], the renormalization group the ory predicts universal scaling functions for relations involving two or more parameters. If Z depends on X and Y , then Z ∝ X aF (X/Y b), where a a...

    work page

  2. [2]

    (9) In equation (9), we apply the high-Reynolds-number limit P + 0 = 1

    491 ( 2 + 3β 3β ) 2/ 3 y∗ . (9) In equation (9), we apply the high-Reynolds-number limit P + 0 = 1 . 5β , instead of using P + 0 = σpβ . This setting simplifies the modeling of ℓ+ m, based on the assumption that finite-Reynolds-number effects ar e important for TSS but negligibly influence the √ τ+-scaling of ℓ+ m. Indeed, we have checked using the measured σ...

    work page

  3. [3]

    Therefore, ℓm = λδ, with λ dependent on APG

    The wake region Within the wake region, the mixing length should keep constant wall-no rmally, since the wake flow becomes more and more like a mixing layer when the APG increases [10]. Therefore, ℓm = λδ, with λ dependent on APG. In view of the mixing-layer nature of the wake flow, we introduce a PG pa rameter G and propose λ = λ(G), where G = δ∂P/∂x τmax ...

    work page

  4. [4]

    (10) Consequently, as β increases from zero to infinity, G increases from zero to unity

    and τ+ max = 1 + 3 4 β (assuming the Reynolds number is large), we redefine the PG parameter G as G = 3 4 β 1 + 3 4 β ( 2 + 3β 3β ) 2/ 3 . (10) Consequently, as β increases from zero to infinity, G increases from zero to unity. We use λ 0 to denote λ at β = 0, and λ ∞ to denote λ at β = ∞ . Then a model of λ can be proposed as: λ = λ 0 − (λ 0 − λ ∞ )f (G), ...

    work page 1968

  5. [5]

    The half-power-law region Now, we aim to match the expressions of ℓm in the log-law and wake regions. To maintain consistency with equation (1), the mixing length in the outer region of APG TBLs is postulated to obey the following defect power law: ℓm = λδ (1 − rm) , (14) where r = 1 − y∗ . In ZPG TBLs, m = 4 according to the SED theory. In equilibrium AP...

    work page

  6. [6]

    With the crossover s caling ansatz introduced, ℓm retrieves the linear scaling to match equation (14) when y∗ > y ∗ log

    491 ( 2 + 3β 3β ) 2/ 3 y∗   1 + ( y∗ y∗ log ) 4  − 1/ 4 , (15) where y∗ log denotes the upper edge of the log-law region. With the crossover s caling ansatz introduced, ℓm retrieves the linear scaling to match equation (14) when y∗ > y ∗ log. The ansatz represents the second dilation-symmetry-breaking induced by APG, describing a transition from the l...

    work page

  7. [7]

    progressive breakdown

    491 ( 2 + 3β 3β ) 2/ 3 y∗ log. (16) For ZPG TBLs, 0 . 15δ is generally accepted as the upper bound of the log-law region [45]. If this bound stays constant in APG TBLs, equation (16) indicates that m → ∞ as β → ∞ , which implies that the wake region engulfs the entire boundary layer when the flow approaches the separation point. To avoid this nonphysical s...

    work page

  8. [8]

    491 0. 15βc. (18) Therefore, we introduce an upper bound for m (i.e., m∞ ) through βc. The matching condition indicates that there is a layer with linear-scalin g mixing length in between the log-law region and the wake region in APG TBLs. In this layer, if the APG is not minor, t he Reynolds shear stress is approximated by − ⟨ u′v′⟩+ = P + 0 y+/y + p , a...

    work page

  9. [9]

    (19) Since uτ is no longer an appropriate velocity scale, we change to using up ≡ ( ν ρ ∂P ∂x )1/ 3 as the velocity scale and yp = ν/u p as the length scale [15, 49]

    15δ+βc/β √ y+ . (19) Since uτ is no longer an appropriate velocity scale, we change to using up ≡ ( ν ρ ∂P ∂x )1/ 3 as the velocity scale and yp = ν/u p as the length scale [15, 49]. Substituting the relationship β = δ+ 1 u3 p/u 3 τ into equation (19) yields du(p) dy(p) = √ δ∗ 1 κ√ 0. 15βc √ y(p) , (20) where u(p) = u/u p and y(p) = y/y p. The solution of...

    work page

  10. [10]

    universal constants

    5 y∗ p (assuming the Reynolds number is large). As β → ∞ , y∗ p → 0. 491. Granville specified that the coefficient α is 0.9 [20], which is consistent with the Sk ˚ are and Krogstad datasets, which have high Reynolds numbers and a rather large β (∼ 20) [12]. Consequently, δ∗ 1 → 0. 295 as β → ∞ . The “universal constants” C and D have been reported to vary si...

    work page

  11. [11]

    Numerous studies have observed that APG tends to reduce the thicknesses of the viscous sublayer and b uffer layer, resulting in a lower log-law intercept [10]

    The inner region The formulation of the near-wall mixing-length profile is motivated as follows. Numerous studies have observed that APG tends to reduce the thicknesses of the viscous sublayer and b uffer layer, resulting in a lower log-law intercept [10]. This effect has previously been captured by adjusting the damp ing parameter A in the van Driest model ...

    work page

  12. [12]

    The full-profile model of mixing length Combining the above formulations, the full-profile model of ℓ+ m in equilibrium APG TBLs is written as ℓ+ m = κy + s 2 y+ b ( y+ y+s ) 3/ 2 [ 1 + ( y+ y+s ) 4] 1/ 8 [ 1 + ( y+ y+ b ) 4] − 1/ 4 ×     √ 1 + 1. 5β

    work page

  13. [13]

    45 as proposed by the SED, y+ s is derived from equation (24), y+ b is derived from equation (25), m is derived from equation (16), and y∗ log is derived from equation (17)

    491 ( 2 + 3β 3β ) 2/ 3 y∗   1 + ( y∗ y∗ log ) 4  − 1/ 4 1 − rm m(1 − r) , (26) where κ = 0 . 45 as proposed by the SED, y+ s is derived from equation (24), y+ b is derived from equation (25), m is derived from equation (16), and y∗ log is derived from equation (17). In summary, this section presents a symmetry-based analytical m odel to predict the fu...

    work page

  14. [14]

    Figure 3 validates the current model for λ (equation [11]) using the datasets listed in Table I

    Mixing-length parameter λ The wake region features a constant mixing length characterized b y the mixing-length parameter λ. Figure 3 validates the current model for λ (equation [11]) using the datasets listed in Table I. To measure λ from the numerical and 10 /s49/s48 /s48 /s49/s48 /s49 /s49/s48 /s50 /s48 /s49 /s50 /s51 /s52 /s53 /s40 /s97 /s41 /s32/s76/...

    work page

  15. [15]

    2, above which the logarithmic layer shrinks and transitions to the half-power-law scaling

    Critical Clauser parameter β c and outer-layer exponent m In this study, we identify the critical Clauser parameter βc ≈ 6. 2, above which the logarithmic layer shrinks and transitions to the half-power-law scaling. Specifically, the upper bo undary of the log-law region is formulated by βc through equation (17). This formulation is validated in Fig. 5, in...

    work page

  16. [16]

    Inner-layer thicknesses y+ s and y+ b The relationship between the viscous-sublayer thickness y+ s and p+ is predicted by equation (24) and illustrated in Fig. 7(a). As p+ increases, y+ s decreases monotonically. As discussed in Section II C 4, y+ b is more complex than y+ s . However, when the Reynolds number is sufficiently large (i.e., y+ log ≫ 4y+ c , a...

    work page

  17. [17]

    (A2) Appendix B: Ma et al.’s mixing-length model Based on the five-parameter formulation by Cantwell [34], Ma et al

    39 √ 1 + p+y+c . (A2) Appendix B: Ma et al.’s mixing-length model Based on the five-parameter formulation by Cantwell [34], Ma et al. [3 6] recently proposed a full-profile mixing- length model for APG TBLs, as follows ℓ+ m = κy +f (Y +) [f (0. 1δ+)]− 1 + 0. 55 √ p+ 0

    work page

  18. [18]

    1 [ 1 + ( y+ bδ+ )n ]1/n , (B1) where κ = 0

    1δ+ Y + 1 − e− (y+/a )1. 1 [ 1 + ( y+ bδ+ )n ]1/n , (B1) where κ = 0. 41, a = 25/ √ 1 + p+y+c , p+ 0 = p+/ (1 + 10e− 200(p+− 0. 01)), and b and n are two flow-dependent parameters and are determined by fitting the full functional form to the data. Y + is defined as Y + = y+ − 0. 1δ+ 10 ln(1 + e10( y+

    work page

  19. [19]

    1δ+ 10 ln(1 + e− 10)

    1δ + − 1)) + 0. 1δ+ 10 ln(1 + e− 10). (B2) The square of function f is an inner-layer model for TSS τ+ [63], expressed as [f (y+)]2 = 1 + p+y+ − Ap(y+lny+ − y+), (B3) where Ap = αy +/ ln(0. 1δ+), and α = ( ∂τ + ∂y + |w − ∂τ + ∂y + |0. 1δ ) / ∂τ + ∂y + |w

    work page

  20. [20]

    Prandtl, Bericht ¨ uber untersuchungen zur ausgebild eten turbulenz, Z

    L. Prandtl, Bericht ¨ uber untersuchungen zur ausgebild eten turbulenz, Z. Angew. Math. Mech. 5, 136 (1925)

    work page 1925

  21. [21]

    P. A. Davidson, Y. Kaneda, K. Moffatt, and K. R. Sreenivasa n, A Voyage Through Turbulence (Cambridge University Press, 2011)

    work page 2011

  22. [22]

    Bradshaw, Possible origin of Prandtl’s mixing-lengt h theory, Nature 249, 135 (1974)

    P. Bradshaw, Possible origin of Prandtl’s mixing-lengt h theory, Nature 249, 135 (1974)

    work page 1974

  23. [23]

    Heisel, C

    M. Heisel, C. M. de Silva, N. Hutchins, I. Marusic, and M. G uala, On the mixing length eddies and logarithmic mean velocity profile in wall turbulence, J. Fluid Mech. 887, R1 (2020)

    work page 2020

  24. [24]

    Prandtl, Zur turbulenten str¨ omung in rohren und l¨ angs platten, in Ergebnisse der Aerodynamischen Versuchsanstalt zu G¨ ottingen, Vol

    L. Prandtl, Zur turbulenten str¨ omung in rohren und l¨ angs platten, in Ergebnisse der Aerodynamischen Versuchsanstalt zu G¨ ottingen, Vol. 4 (1932) pp. 18–29, also published in LPGA 2, 632–648

    work page 1932

  25. [25]

    E. R. Van Driest, On turbulent flow near a wall, J. Aeronaut . Sci. 23, 1007 (1956)

    work page 1956

  26. [26]

    Schlichting and K

    H. Schlichting and K. Gersten, Boundary Layer Theory, 9th ed (Springer, 2017). 24

    work page 2017

  27. [27]

    Klewicki, R

    J. Klewicki, R. Sandberg, T. Knopp, W. Devenport, D. Frit sch, V. Vishwanathan, R. Volino, S. Toxopeus, B. MeKeon, and L. Eca, On the physical structure, modelling and computatio n-based prediction of two-dimensional, smooth-wall turbu lent boundary layers subjected to streamwise pressure gradient s, J. Turbul. 25, 345 (2024)

    work page 2024

  28. [28]

    Bradshaw, The turbulence structure of equilibrium bo undary layers, J

    P. Bradshaw, The turbulence structure of equilibrium bo undary layers, J. Fluid Mech. 29, 625 (1967)

    work page 1967

  29. [29]

    W. J. Devenport and K. T. Lowe, Equilibrium and non-equi librium turbulent boundary layers, Prog. Aerosp. Sci. 131, 100807 (2022)

    work page 2022

  30. [30]

    Johnstone, G

    R. Johnstone, G. N. Coleman, and P. R. Spalart, The resil ience of the logarithmic law to pressure gradients: evidenc e from direct numerical simulation, J. Fluid Mech. 643, 163 (2010)

    work page 2010

  31. [31]

    P. E. Skare and P. A. Krogstad, A turbulent equilibrium b oundary layer near separation, J. Fluid Mech. 272, 319 (1994)

    work page 1994

  32. [32]

    A. A. Townsend, Equilibrium layers and wall turbulence , J. Fluid Mech. 11, 97 (1961)

    work page 1961

  33. [33]

    Knopp, N

    T. Knopp, N. Reuther, M. Novara, D. Schanz, E. Schulein, A. Schroder, and C. J. Kahler, Experimental analysis of the log law at adverse pressure gradient, J. Fluid Mech. 918, A17 (2021)

    work page 2021

  34. [34]

    B. S. Stratford, The prediction of separation of the tur bulent boundary layer, J. Fluid Mech. 5, 1 (1959)

    work page 1959

  35. [35]

    Coles and E

    D. Coles and E. Hirst, Computation of turbulent boundar y layers, in Proc. 1968 AFOSR-IFP Stanford Conf., vol. 2 (Stanford Univ., Stanford, Calif., 1968)

    work page 1968

  36. [36]

    R. A. Galbraith, S. Sjolander, and M. R. Head, Mixing len gth in the wall region of turbulent boundary layers, Aeronau t. Q. 28, 97 (1977)

    work page 1977

  37. [37]

    Coles, The law of the wake in the turbulent boundary la yer, J

    D. Coles, The law of the wake in the turbulent boundary la yer, J. Fluid Mech. 1, 191 (1956)

    work page 1956

  38. [38]

    B. L. Reeves, Two-layer model of turbulent boundary lay ers, AIAA J. 12, 932 (1974)

    work page 1974

  39. [39]

    P. S. Granville, A modified van Driest formula for the mix ing length of turbulent boundary layers in pressure gradien ts, J. Fluids Engn. Trans. ASME 111, 94 (1989)

    work page 1989

  40. [40]

    L. C. Thomas and S. M. F. Hasani, Supplementary boundary -layer approximations for turbulent flow, J. Fluids Engn. Trans. ASME 111, 420 (1989)

    work page 1989

  41. [41]

    M. A. Subrahmanyam, B. J. Cantwell, and J. J. Alonso, A un iversal velocity profile for turbulent wall flows including adverse pressure gradient boundary layers, J. Fluid Mech. 933, A16 (2022)

    work page 2022

  42. [42]

    Bernard, J

    A. Bernard, J. M. Foucaut, P. Dupont, and M. Stanislas, D ecelerating boundary layer: A new scaling and mixing length model, AIAA J. 41, 248 (2003)

    work page 2003

  43. [43]

    M. H. Buschmann and M. Gad-el Hak, New mixing-length app roach for the mean velocity profile of turbulent boundary layers, J. Fluids Engn. Trans. ASME 127, 393 (2005)

    work page 2005

  44. [44]

    Pirozzoli, Revisiting the mixing-length hypothesi s in the outer part of turbulent wall layers: mean flow and wall friction, J

    S. Pirozzoli, Revisiting the mixing-length hypothesi s in the outer part of turbulent wall layers: mean flow and wall friction, J. Fluid Mech. 745, 378 (2014)

    work page 2014

  45. [45]

    Z. S. She, X. Chen, Y. Wu, and F. Hussain, New perspective in statistical modeling of wall-bounded turbulence, Acta Mech. Sinica 26, 847 (2010)

    work page 2010

  46. [46]

    Z. S. She, X. Chen, and F. Hussain, Quantifying wall turb ulence via a symmetry approach: A Lie group theory, J. Fluid Mech. 827, 322 (2017)

    work page 2017

  47. [47]

    Marusic, B

    I. Marusic, B. J. McKeon, P. A. Monkewitz, H. M. Nagib, A. J. Smits, and K. R. Sreenivasan, Wall-bounded turbulent flows at high Reynolds numbers: Recent advances and key issue s, Phys. Fluids 22, 065103 (2010)

    work page 2010

  48. [48]

    W. T. Bi, K. X. Zheng, J. Chen, and Z. S. She, Quantifying n on-equilibrium pressure-gradient turbulent boundary lay ers through a symmetry-based framework, AIP Adv. 15, 095112 (2025)

    work page 2025

  49. [49]

    Gluzman and V

    S. Gluzman and V. I. Yukalov, Unified approach to crossov er phenomena, Phys. Rev. E 58, 4197 (1998)

    work page 1998

  50. [50]

    von Karman, Progress in the statistical theory of tur bulence, PANS 34, 530 (1948)

    T. von Karman, Progress in the statistical theory of tur bulence, PANS 34, 530 (1948)

    work page 1948

  51. [51]

    P. S. Klebanoff, Characteristics of turbulence in a boundary layer with zero pressure gradient, Tech. Rep. TN 3178 (NACA, 1954)

    work page 1954

  52. [52]

    R. Li, Q. R. Ding, and W. C. Cui, Scaling crossovers in non -equilibrium critical dynamics, I. Phys. A: Math. Theor. 58, 385004 (2025)

    work page 2025

  53. [53]

    B. J. Cantwell, A universal velocity profile for smooth w all pipe flow, J. Fluid Mech. 878, 834 (2019)

    work page 2019

  54. [54]

    Z. Q. Shu and C. X. Xu, Mean velocity profiles and total she ar stress profiles in adverse-pressure-gradient turbulent boundary layers considering history effect, arXiv , 2508.07 663v2 (2025)

    work page 2025

  55. [55]

    M. Z. Ma, Y. X. Shi, Y. L. Zhu, A. X. Han, and X. Chen, A gener alized mixing length model with adverse-pressure-gradien t effects, Symmetry 18, 105 (2026)

    work page 2026

  56. [56]

    Y. J. Xu, S. J. Schmidt, and N. A. Adams, Extending the log arithmic velocity profile in turbulent channel flow, Phys. Fluids 37, 045109 (2025)

    work page 2025

  57. [57]

    K. X. Zheng, W. T. Bi, J. Chen, and Z. S. She, A symmetry-ba sed model for total shear stress of equilibrium pressure- gradient turbulent boundary layers, Phys. Fluids 37, 055126 (2025)

    work page 2025

  58. [58]

    Chen and Z

    X. Chen and Z. S. She, Analytic prediction for planar tur bulent boundary layers, Sci. China Phys., Mech. & Astron. 59, 114711 (2016)

    work page 2016

  59. [59]

    F. H. Clauser, Turbulent boundary layers in adverse pre ssure gradients, J. Aeronaut. Sci. 21, 91 (1954)

    work page 1954

  60. [60]

    J. P. Sethna, Power laws in physics, Nature Rev. Phys. 4, 501 (2022)

    work page 2022

  61. [61]

    X. Chen, F. Hussain, and Z. S. She, Quantifying wall turb ulence via a symmetry approach. Part 2. Reynolds stresses, J . Fluid Mech. 850, 401 (2018)

    work page 2018

  62. [62]

    F. M. White, Viscous Fluid Flow, 3rd ed (McGraw-Hill, 2006)

    work page 2006

  63. [63]

    Das, A numerical study of turbulent separated flows, i n Am

    D. Das, A numerical study of turbulent separated flows, i n Am. Soc. Mech. Eng. Forum on Turbulent Flows, FED Vol. 51 (1987) pp. 85–90

    work page 1987

  64. [64]

    Marusic, J

    I. Marusic, J. P. Monty, and M. Hultmark, On the logarith mic region in wall turbulence, J. Fluid Mech. 716, R3 (2013). 25

    work page 2013

  65. [65]

    A. E. Perry and W. H. Schofield, Mean velocity and shear st ress distributions in turbulent boundary layers, Phys. Flu ids 16, 2068 (1973)

    work page 2068

  66. [66]

    Knopp, An empirical wall law for the mean velocity in a n adverse pressure gradient for RANS turbulence modeling, Flow Turbul

    T. Knopp, An empirical wall law for the mean velocity in a n adverse pressure gradient for RANS turbulence modeling, Flow Turbul. Combust. 109, 571 (2022)

    work page 2022

  67. [67]

    M. Ma, M. Han, Z. Shao, and C. Yan, Scaling laws for the mea n velocity in adverse pressure gradient turbulent boundary layers based on asymptotic expansions, Phys. Fluids 36, 095175 (2024)

    work page 2024

  68. [68]

    Skote and D

    M. Skote and D. S. Henningson, Direct numerical simulat ion of a separated turbulent boundary layer, J. Fluid Mech. 471, 107 (2002)

    work page 2002

  69. [69]

    G. N. Coleman, S. Pirozzoli, M. Quadrio, and P. R. Spalar t, Direct numerical simulation and theory of a wall-bounded flow with zero skin friction, Flow Turbul. Combust. 99, 553 (2017)

    work page 2017

  70. [70]

    T. B. Nickels, Inner scaling for wall-bounded flows subj ect to large pressure gradients, J. Fluid Mech. 521, 217 (2004)

    work page 2004

  71. [71]

    Sanmiguel Vila, R

    C. Sanmiguel Vila, R. Vinuesa, S. Discetti, A. Ianiro, P . Schlatter, and R. ¨Orl¨ u, Experimental realisation of near-equilibrium adverse-pressure-gradient turbulent boundary layers, Ex p. Thermal Fluid Sci. 112, 109975 (2020)

    work page 2020

  72. [72]

    Schlatter, R

    P. Schlatter, R. ¨Orl¨ u, Q. LI, G. Brethouwer, J. H. M. Fransson, A. V. Johansso n, P. H. Alfredsson, and D. S. Henningson, Turbulent boundary layers up to Reθ = 2500 studied through simulation and experiment, Phys. Flu ids 21, 051702 (2009)

    work page 2009

  73. [73]

    Eitel-Amor, R

    G. Eitel-Amor, R. ¨Orl¨ u, and P. Schlatter, Simulation and validation of a spat ially evolving turbulent boundary layer up to Reθ = 8300, Int. J. Heat Fluid Flow 47, 57 (2014)

    work page 2014

  74. [74]

    J. H. Lee and H. J. Sung, Effects of an adverse pressure gra dient on a turbulent boundary layer, Int. J. Heat Fluid Flow 29, 568 (2008)

    work page 2008

  75. [75]

    J. H. Lee, Large-scale motions in turbulent boundary la yers subjected to adverse pressure gradients, J. Fluid Mech . 810, 323 (2017)

    work page 2017

  76. [76]

    Bobke, R

    A. Bobke, R. Vinuesa, R. ¨Orl¨ u, and P. Schlatter, History effects and near equilibrium in adverse-pressure-gradient turbulent boundary layers, J. Fluid Mech. 820, 667 (2017)

    work page 2017

  77. [77]

    Pozuelo, Q

    R. Pozuelo, Q. Li, P. Schlatter, and R. Vinuesa, An adver se-pressure-gradient turbulent boundary layer with nearl y constant β ≈ 1. 4 up to Reθ ≈ 8700, J. Fluid Mech. 939, A34 (2022)

    work page 2022

  78. [78]

    Kitsios, A

    V. Kitsios, A. Sekimoto, C. Atkinson, J. A. Sillero, G. B orrell, A. G. Gungor, J. Jimenez, and J. Soria, Direct numeri cal simulation of a self-similar adverse pressure gradient tur bulent boundary layer at the verge of separation, J. Fluid Me ch. 829, 392 (2017)

    work page 2017

  79. [79]

    D. C. Wilcox, Turbulence modeling for CFD, 3rd ed (DCW Industries, California, 2006)

    work page 2006

  80. [80]

    Bradshaw and D

    P. Bradshaw and D. H. Ferriss, The response of a retarded equilibrium turbulent boundary l ayer to the sudden removal of pressure gradient, Tech. Rep. 1145 (NPL Aero. Rep., 1965)

    work page 1965

Showing first 80 references.