Towards a full-scale version of Yakhot's model of strong turbulence

arxiv: 2602.10327 · v2 · submitted 2026-02-10 · ⚛️ physics.flu-dyn

Towards a full-scale version of Yakhot's model of strong turbulence

Christoph Renner This is my paper

Pith reviewed 2026-05-16 05:01 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn

keywords turbulence modelingstructure functionsYakhot modeldissipation rangeinertial rangeReynolds numberKolmogorov law

0 comments p. Extension

The pith

Yakhot's turbulence model is extended to full scales with closed-form expressions for second- and third-order structure functions that match experiments without free parameters.

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

The paper extends Yakhot's model of strong turbulence to small scales by using an empirically observed relation for even-order structure functions that holds from the inertial range into the dissipation range. Combined with Kolmogorov's four-fifth law, this relation produces models for the second- and third-order structure functions that recover the expected limits at dissipative scales and correctly capture the transition to inertial scaling. An extra length scale marking the crossover is expressed solely as a function of Reynolds number. When joined with an earlier large-scale extension, the result is a complete, closed-form description of these structure functions that agrees with data from the smallest dissipative scales up to the system scale.

Core claim

By applying an observed scaling relation for even-order structure functions that persists into the dissipation range and combining it with Kolmogorov's four-fifth law, closed-form expressions are obtained for the second- and third-order structure functions. These expressions correctly reproduce the dissipative and inertial scaling behaviors and include a Reynolds-number-dependent length scale that marks the crossover between the two regimes. Together with an earlier large-scale extension of the model, this provides complete, parameter-free formulas for the structure functions that are consistent with experimental data from the smallest dissipative scales up to the system size.

What carries the argument

The empirically observed relation for even-order structure functions that extends from the inertial into the dissipation range, used together with Kolmogorov's four-fifth law to derive the crossover scale as a function of Reynolds number.

If this is right

The models recover the expected small-scale limits and the correct transition between dissipative and inertial regimes.
The crossover length scale is fixed by Reynolds number alone, removing adjustable parameters.
Combined with the prior large-scale extension, the expressions cover the entire range from dissipative scales to system size.
The resulting closed-form formulas agree with experimental measurements across all scales.

Where Pith is reading between the lines

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

If the same empirical relation holds for higher-order structure functions, the method could be applied beyond orders two and three.
The explicit Reynolds-number dependence of the crossover scale supplies a prediction that can be checked in flows at widely different Reynolds numbers.

Load-bearing premise

The empirically observed relation for even-order structure functions continues to hold in the dissipation range and the crossover length scale depends only on the Reynolds number without extra fitting.

What would settle it

Direct measurements of even-order structure functions in the dissipation range that deviate from the assumed scaling relation, or data showing that the crossover length scale fails to follow the predicted dependence on Reynolds number.

Figures

Figures reproduced from arXiv: 2602.10327 by Christoph Renner.

**Figure 1.** Figure 1: Functions dn(r) as defined in eq. (20) implied from experimental data of structure functions of order two (dashed lines) and four (dotted lines) in linear (left) and log–linear (right) scale. Under hypothesis H2, the necessary additional term in (19) can approximately be implied from experimental data. We denote this term by Rn(r) and define it as the residual of the difference of the left and right hand s… view at source ↗

**Figure 2.** Figure 2: The residuals Rn(r) as defined in eq. (21) for orders n = 2 (dashed lines) and 4 (dotted lines) compensated by the structure functions Sn+1 (left) and additionally scaled by order n (right). The straight line in the right graph shows a fit according to eq. (22). A complete full–scale model of turbulence hence requires a further relation for odd order. Unfortunately, odd order residuals Rn(r) are found to n… view at source ↗

**Figure 3.** Figure 3: Compensated and scaled residuals Rn(r) for orders n = 3 (dashed lines) and n = 5 (dotted lines) with power law fits (solid lines). Left graph: Residuals scaled with order n in comparison to the power law fit of even–order residuals according to eq. (22). Right graph: Results for alternative scaling with 1/(n − 1) in comparison to a power–function fit of second order (the fitted pre–factor does not coincide… view at source ↗

**Figure 4.** Figure 4: Second order structure function determined from expe [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

**Figure 5.** Figure 5: The second order structure function (solid line) in compar [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗

**Figure 6.** Figure 6: Third order structure function determined from experim [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗

**Figure 7.** Figure 7: Even order compensated and scaled residuals [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗

**Figure 8.** Figure 8: Second order structure function (straight lines) for th [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗

**Figure 9.** Figure 9: Second order structure function (straight lines) for th [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗

**Figure 10.** Figure 10: Second–order structure function in the simplified (dott [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗

read the original abstract

We present first elements of an extension of Yakhot's model of strong turbulence towards small scales. The analysis is based on an empirically observed relation for even order structure functions which extends from the inertial into the dissipation range. With this relation and Kolmogorov's four-fifth law, models for structure functions of orders two and three can be derived that replicate expected small scale limits and describe the transition from dissipative to inertial range scaling regimes correctly. An additional length scale parameter is introduced by the extension. It marks the crossover point from the inertial to the dissipation range and can be expressed as a function of the Reynolds number. In combination with a recently proposed large-scale extension of Yakhot's model, we ultimately obtain full-scale models for structure functions of second and third order. These expressions are closed--form, do not contain free parameters and are in good agreement with experimental data from the smallest dissipative scales up to the system scale.

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.

Desk Editor's Note private letter to a colleague

Renner extends Yakhot to full scales with closed-form S2 and S3 via an empirical even-order relation and Re-dependent crossover, but the parameter-free claim needs checking on how l*(Re) is actually obtained.

read the letter

Renner takes Yakhot's model and adds a small-scale piece by assuming an empirical relation for even-order structure functions that keeps working into the dissipation range. Combined with the 4/5 law this produces closed expressions for the second- and third-order structure functions that recover the right small-scale limits and the transition to inertial scaling. Stacking that on the earlier large-scale extension gives full-range formulas claimed to have no free parameters and to line up with experimental data from dissipative scales out to the system size. That is the concrete advance: the full-scale closed forms are new. The construction is straightforward and the limits check out on paper. The data agreement is presented as supportive, at least at the level of the abstract. The soft spot is the crossover length l* that marks the inertial-to-dissipation transition. It is written as a function of Reynolds number only, but the abstract does not show the explicit steps that produce its functional form from the even-order relation alone. If that form is shaped by fitting to the same data used for validation, then the no-free-parameters statement and the independent agreement both weaken. The circularity risk is real and central; without the derivation or quantitative error metrics it is hard to judge how much weight the data comparison carries. This is for people who already work with Yakhot-type closures or structure-function modeling in turbulence. A reader who wants analytical expressions that run from dissipation to large scales will see a clean way to stitch the pieces together, even if they will want to verify the Re dependence themselves. I would send it to peer review. The core construction is simple enough to evaluate quickly and the payoff, if the derivation holds, is worth referee time.

Referee Report

1 major / 2 minor

Summary. The paper extends Yakhot's model of strong turbulence to small scales by invoking an empirically observed relation for even-order structure functions that is assumed to hold from the inertial into the dissipation range. Combining this relation with Kolmogorov's 4/5 law yields closed-form expressions for the second- and third-order structure functions S2(r) and S3(r). An auxiliary crossover length scale l* is introduced to mark the inertial-to-dissipation transition and is stated to depend only on the Reynolds number. When combined with a prior large-scale extension of Yakhot's model, the resulting full-scale expressions for S2 and S3 are claimed to be parameter-free and to agree with experimental data across all scales.

Significance. If the no-free-parameter claim can be substantiated, the work would supply analytically closed expressions for low-order structure functions from the dissipative range through the system scale, offering a compact description of the entire range of turbulent scales without scale-by-scale empirical tuning. This would be a useful contribution to turbulence modeling, particularly if the functional form of l*(Re) follows rigorously from the assumed even-order relation and the 4/5 law rather than from data calibration.

major comments (1)

[Abstract and the section deriving l*(Re)] Abstract: the central claim that the derived expressions 'do not contain free parameters' rests on the crossover scale l* being expressible as a function of Re alone. The manuscript must specify the explicit functional form of l*(Re) and demonstrate that it is obtained directly from the empirical even-order relation and the 4/5 law without additional fitting constants or data-driven selection of its shape. If the form of l*(Re) is calibrated to experimental profiles, the parameter-free assertion does not hold and the reported data agreement becomes partly tautological.

minor comments (2)

The abstract refers to 'a recently proposed large-scale extension'; the corresponding reference should be cited explicitly in the text.
All final expressions for S2(r) and S3(r) should be written out with explicit notation for the crossover scale l* and the Reynolds-number dependence to improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We address the single major comment below and will revise the manuscript to strengthen the presentation of the parameter-free claim.

read point-by-point responses

Referee: Abstract: the central claim that the derived expressions 'do not contain free parameters' rests on the crossover scale l* being expressible as a function of Re alone. The manuscript must specify the explicit functional form of l*(Re) and demonstrate that it is obtained directly from the empirical even-order relation and the 4/5 law without additional fitting constants or data-driven selection of its shape. If the form of l*(Re) is calibrated to experimental profiles, the parameter-free assertion does not hold and the reported data agreement becomes partly tautological.

Authors: We agree that an explicit functional form for l*(Re) must be stated to fully substantiate the no-free-parameter claim. In the revised manuscript we will add the derivation showing that l* follows directly by enforcing continuity of the even-order structure-function relation with Kolmogorov's 4/5 law at the inertial-to-dissipation crossover; the resulting expression contains only Re and no adjustable constants. The comparison with experimental data is performed after the functional form is fixed and therefore constitutes validation rather than calibration. We will update both the abstract and the dedicated section on l*(Re) accordingly. revision: yes

Circularity Check

1 steps flagged

Crossover length scale l*(Re) functional form may be chosen to fit data rather than derived from the empirical even-order relation alone

specific steps

fitted input called prediction [Abstract]
"An additional length scale parameter is introduced by the extension. It marks the crossover point from the inertial to the dissipation range and can be expressed as a function of the Reynolds number. [...] These expressions are closed--form, do not contain free parameters and are in good agreement with experimental data from the smallest dissipative scales up to the system scale."

The crossover length l* is introduced precisely to connect the inertial and dissipative regimes and is stated to depend only on Re. For the final S2 and S3 expressions to be parameter-free by construction, this l*(Re) dependence must be fixed by the even-order empirical relation alone. If instead the functional shape is selected or calibrated to reproduce the measured profiles, the 'no free parameters' claim and the data agreement become partly tautological.

full rationale

The paper's small-scale extension rests on an empirically observed even-order structure function relation extended into the dissipation range, combined with the 4/5 law to produce closed expressions for S2(r) and S3(r). An auxiliary crossover scale l* is introduced to mark the inertial-to-dissipation transition and is asserted to be expressible as a function of Re alone, enabling the claim of fully closed, parameter-free models. Because the explicit functional form of l*(Re) is not shown to follow uniquely from the even-order relation without reference to data calibration, the reported agreement with experiment reduces in part to a fitted-input construction rather than an independent derivation. No self-citation load-bearing or self-definitional steps appear in the abstract-level chain, so the circularity is partial rather than total.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

Central claim depends on standard turbulence law, one empirical domain assumption, and one new length scale whose functional dependence on Reynolds number is asserted but not derived in the abstract.

axioms (2)

standard math Kolmogorov's four-fifth law
Invoked to derive the structure function models.
domain assumption Empirically observed relation for even order structure functions extending into the dissipation range
Foundation for extending the model to small scales.

invented entities (1)

Additional length scale parameter for inertial-to-dissipation crossover no independent evidence
purpose: Marks the transition point between scaling regimes
Introduced in the extension and tied to Reynolds number.

pith-pipeline@v0.9.0 · 8970 in / 1324 out tokens · 211697 ms · 2026-05-16T05:01:12.811079+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

These expressions are closed-form, do not contain free parameters and are in good agreement with experimental data from the smallest dissipative scales up to the system scale.
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean alpha_pin_under_high_calibration refines

?

refines
Relation between the paper passage and the cited Recognition theorem.

An additional length scale parameter is introduced by the extension. It marks the crossover point from the inertial to the dissipation range and can be expressed as a function of the Reynolds number.

What do these tags mean?

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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

Probability density and scaling exponents o f the moments of longi- tudinal velocity diﬀerence in strong turbulence,

V. Yakhot, “Probability density and scaling exponents o f the moments of longi- tudinal velocity diﬀerence in strong turbulence,” Phys. Rev. E , vol. 57, pp. 1737– 1751, Feb 1998. 16

work page 1998
[2]

What is the turbule nce problem, and when may we regard it as solved?,

K. R. Sreenivasan and J. Schumacher, “What is the turbule nce problem, and when may we regard it as solved?,” Annual Review of Condensed Matter Physics , vol. 16, no. Volume 16, 2025, pp. 121–143, 2025

work page 2025
[3]

The local structure of turbulence in i ncompressible viscous ﬂuid for very large reynolds numbers,

A. N. Kolmogorov, “The local structure of turbulence in i ncompressible viscous ﬂuid for very large reynolds numbers,” Dokl. Akad. Nauk. SSSR , vol. 30, pp. 301– 305, 1941

work page 1941
[4]

Dissipation of energy in the locally i sotropic turbulence,

A. N. Kolmogorov, “Dissipation of energy in the locally i sotropic turbulence,” Proceedings: Mathematical and Physical Sciences , vol. 434, no. 1890, pp. 15–17, 1991

work page 1991
[5]

A reﬁnement of previous hypotheses co ncerning the local structure of turbulence in a viscous incompressible ﬂuid at high reynolds number,

A. N. Kolmogorov, “A reﬁnement of previous hypotheses co ncerning the local structure of turbulence in a viscous incompressible ﬂuid at high reynolds number,” Journal of Fluid Mechanics , vol. 13, pp. 82–85, May 1962

work page 1962
[6]

Analytic calculation of th e anomalous exponents in turbulence: Using the fusion rules to ﬂush out a small para meter,

V. S. L’vov and I. Procaccia, “Analytic calculation of th e anomalous exponents in turbulence: Using the fusion rules to ﬂush out a small para meter,” Phys. Rev. E, vol. 62, pp. 8037–8057, Dec 2000

work page 2000
[7]

The temperature of turbulent ﬂows,

B. Castaing, “The temperature of turbulent ﬂows,” Journal de Physique II, EDP Sciences, vol. 6, pp. 105–114, Jan 1996

work page 1996
[8]

A generalization of scaling mod els of turbulence,

C. Renner and J. Peinke, “A generalization of scaling mod els of turbulence,” Journal of Statistical Physics , vol. 146, pp. 25–32, Jan 2012

work page 2012
[9]

Empirical large-scale extension of yakhot’ s model of strong turbu- lence,

C. Renner, “Empirical large-scale extension of yakhot’ s model of strong turbu- lence,” Journal of Turbulence , vol. 0, no. 0, pp. 1–13, 2025

work page 2025
[10]

Inte rmittency in a tur- bulent low temperature gaseous helium jet,

O. Chanal, B. Chabaud, B. Castaing, and B. Hebral, “Inte rmittency in a tur- bulent low temperature gaseous helium jet,” The European Physical Journal B - Condensed Matter and Complex Systems , vol. 17, pp. 309–317, Sep 2000

work page 2000
[11]

Universality of small scale turbulence,

C. Renner, J. Peinke, R. Friedrich, O. Chanal, and B. Cha baud, “Universality of small scale turbulence,” Phys. Rev. Lett. , vol. 89, p. 124502, Sep 2002

work page 2002
[12]

Dynamics of three-dim ensional turbulence from navier-stokes equations,

K. R. Sreenivasan and V. Yakhot, “Dynamics of three-dim ensional turbulence from navier-stokes equations,” Phys. Rev. Fluids , vol. 6, p. 104604, Oct 2021

work page 2021
[13]

Analysis of struc- ture function equations up to the seventh order,

J. Boschung, F. Hennig, D. Denker, H. Pitsch, and R. J. Hi ll, “Analysis of struc- ture function equations up to the seventh order,” Journal of Turbulence , vol. 18, no. 11, pp. 1001–1032, 2017. 17

work page 2017

[1] [1]

Probability density and scaling exponents o f the moments of longi- tudinal velocity diﬀerence in strong turbulence,

V. Yakhot, “Probability density and scaling exponents o f the moments of longi- tudinal velocity diﬀerence in strong turbulence,” Phys. Rev. E , vol. 57, pp. 1737– 1751, Feb 1998. 16

work page 1998

[2] [2]

What is the turbule nce problem, and when may we regard it as solved?,

K. R. Sreenivasan and J. Schumacher, “What is the turbule nce problem, and when may we regard it as solved?,” Annual Review of Condensed Matter Physics , vol. 16, no. Volume 16, 2025, pp. 121–143, 2025

work page 2025

[3] [3]

The local structure of turbulence in i ncompressible viscous ﬂuid for very large reynolds numbers,

A. N. Kolmogorov, “The local structure of turbulence in i ncompressible viscous ﬂuid for very large reynolds numbers,” Dokl. Akad. Nauk. SSSR , vol. 30, pp. 301– 305, 1941

work page 1941

[4] [4]

Dissipation of energy in the locally i sotropic turbulence,

A. N. Kolmogorov, “Dissipation of energy in the locally i sotropic turbulence,” Proceedings: Mathematical and Physical Sciences , vol. 434, no. 1890, pp. 15–17, 1991

work page 1991

[5] [5]

A reﬁnement of previous hypotheses co ncerning the local structure of turbulence in a viscous incompressible ﬂuid at high reynolds number,

A. N. Kolmogorov, “A reﬁnement of previous hypotheses co ncerning the local structure of turbulence in a viscous incompressible ﬂuid at high reynolds number,” Journal of Fluid Mechanics , vol. 13, pp. 82–85, May 1962

work page 1962

[6] [6]

Analytic calculation of th e anomalous exponents in turbulence: Using the fusion rules to ﬂush out a small para meter,

V. S. L’vov and I. Procaccia, “Analytic calculation of th e anomalous exponents in turbulence: Using the fusion rules to ﬂush out a small para meter,” Phys. Rev. E, vol. 62, pp. 8037–8057, Dec 2000

work page 2000

[7] [7]

The temperature of turbulent ﬂows,

B. Castaing, “The temperature of turbulent ﬂows,” Journal de Physique II, EDP Sciences, vol. 6, pp. 105–114, Jan 1996

work page 1996

[8] [8]

A generalization of scaling mod els of turbulence,

C. Renner and J. Peinke, “A generalization of scaling mod els of turbulence,” Journal of Statistical Physics , vol. 146, pp. 25–32, Jan 2012

work page 2012

[9] [9]

Empirical large-scale extension of yakhot’ s model of strong turbu- lence,

C. Renner, “Empirical large-scale extension of yakhot’ s model of strong turbu- lence,” Journal of Turbulence , vol. 0, no. 0, pp. 1–13, 2025

work page 2025

[10] [10]

Inte rmittency in a tur- bulent low temperature gaseous helium jet,

O. Chanal, B. Chabaud, B. Castaing, and B. Hebral, “Inte rmittency in a tur- bulent low temperature gaseous helium jet,” The European Physical Journal B - Condensed Matter and Complex Systems , vol. 17, pp. 309–317, Sep 2000

work page 2000

[11] [11]

Universality of small scale turbulence,

C. Renner, J. Peinke, R. Friedrich, O. Chanal, and B. Cha baud, “Universality of small scale turbulence,” Phys. Rev. Lett. , vol. 89, p. 124502, Sep 2002

work page 2002

[12] [12]

Dynamics of three-dim ensional turbulence from navier-stokes equations,

K. R. Sreenivasan and V. Yakhot, “Dynamics of three-dim ensional turbulence from navier-stokes equations,” Phys. Rev. Fluids , vol. 6, p. 104604, Oct 2021

work page 2021

[13] [13]

Analysis of struc- ture function equations up to the seventh order,

J. Boschung, F. Hennig, D. Denker, H. Pitsch, and R. J. Hi ll, “Analysis of struc- ture function equations up to the seventh order,” Journal of Turbulence , vol. 18, no. 11, pp. 1001–1032, 2017. 17

work page 2017