arxiv: 2604.09947 · v1 · submitted 2026-04-10 · ⚛️ physics.flu-dyn

Recognition: unknown

Unified scaling laws for turbulent boundary layers across flow regimes

Adrian Lozano-Duran, Gonzalo Arranz

Authors on Pith no claims yet

Pith reviewed 2026-05-10 16:18 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn

keywords turbulent boundary layersscaling lawspressure gradientswall shear stressvelocity profilesdimensionless groupsinformation theory

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The pith

Two dimensionless variables describe mean wall shear stress across all turbulent boundary layer regimes.

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

The paper establishes that unified scaling laws for mean wall shear stress and mean velocity profiles hold in turbulent boundary layers under favorable, adverse, and separating pressure gradients. It applies the information-theoretic irreducible error theorem to identify the most predictive dimensionless groups from all dimensionally consistent local combinations, without assuming functional forms. Two such variables suffice for wall shear stress and three for the velocity profile, all evaluated at a single streamwise location. A sympathetic reader would care because these laws collapse data from regimes previously requiring separate treatments and show that local quantities can implicitly encode upstream history effects.

Core claim

The central claim is that judiciously chosen combinations of local quantities implicitly encode upstream history without requiring global parameters. Two dimensionless variables suffice to describe the mean wall shear stress, while three characterize the mean velocity profile. These scaling laws are validated against a rich collection of cases and collapse mean quantities across flow regimes with pressure gradients, separation, and reattachment.

What carries the argument

The information-theoretic irreducible error theorem applied to dimensionally consistent combinations to select groups with maximal predictive power without assuming any functional form.

If this is right

Mean wall shear stress predictions at any streamwise location require only two local dimensionless variables.
Mean velocity profiles across regimes are characterized by three local dimensionless variables.
Upstream history is captured implicitly, so no global flow parameters are needed for the scaling laws.
Data from favorable, adverse, separating, and reattaching flows collapse under the same relations.

Where Pith is reading between the lines

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

Turbulence models in simulations could adopt these optimal local scalings to reduce dependence on integral or history parameters.
The same selection method might identify unified descriptions for other wall-bounded flows with varying conditions.
Experiments could prioritize measurements of these specific local variable combinations for efficient characterization.

Load-bearing premise

The dimensionless groups identified by the information-theoretic method will maintain their predictive power for turbulent boundary layers beyond the specific validation cases used.

What would settle it

Apply the derived scaling relations to an independent set of turbulent boundary layer measurements featuring strong adverse pressure gradients and separation that were not part of the original validation collection, then check whether the mean wall shear stress and velocity profiles collapse as predicted.

Figures

Figures reproduced from arXiv: 2604.09947 by Adrian Lozano-Duran, Gonzalo Arranz.

**Figure 1.** Figure 1: FIG. 1: Schematic of a turbulent boundary layer subject to pressure gradients. The orange line denotes the [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Figure 2: FIG. 2: (a) Optimal scaling law for the dimensionless mean wall shear stress. Contours represent the KAN model [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3: Comparison between the model [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4: Application of the models based on the discovered dimensionless groups ( [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

read the original abstract

We discover unified scaling laws for the mean wall shear stress and the mean velocity profile in turbulent boundary layers subject to favorable and adverse mean pressure gradients-including flows with separation and reattachment. We use the information-theoretic irreducible error theorem to identify, among all dimensionally consistent combinations, the dimensionless groups with maximal predictive power, without assuming any functional form. Two dimensionless variables suffice to describe the mean wall shear stress, while three characterize the mean velocity profile. The scaling laws depend exclusively on variables defined at a fixed streamwise location, demonstrating that judiciously chosen combinations of local quantities implicitly encode upstream history without requiring global parameters. The results are validated against a rich collection of cases and are shown to collapse mean quantities across flow regimes previously thought to require distinct treatments.

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

The paper uses an information-theoretic method to select local dimensionless groups that collapse mean wall shear and velocity data across attached, separated, and reattaching turbulent boundary layers.

read the letter

The main point is that two local dimensionless groups describe the mean wall shear stress and three describe the velocity profile, all drawn from quantities at a single streamwise station. The scaling laws are claimed to work across regimes that usually get separate treatments, including favorable and adverse pressure gradients with separation and reattachment. Everything stays local, so upstream history is said to be encoded implicitly in the chosen combinations rather than needing global parameters or integrals. This is the concrete result the abstract highlights as new. The method that gets them there is the information-theoretic irreducible error theorem applied to every dimensionally consistent combination, ranking them by predictive power on the data without assuming a functional form first. That step is cleaner than the usual guess-and-fit approach to scaling and reduces some circularity. They then show the selected groups produce collapse on their collection of cases. The validation is presented as covering a rich set of flows, which is the main evidence offered. The soft spot is the finite-dataset issue. The ranking is purely data-driven, so the top groups could be the best fit for the particular cases and Reynolds numbers they used rather than universal. A new flow with different separation history or higher Re could expose that the predictive power does not carry over. The paper would be tighter with explicit hold-out tests, error metrics, or checks on sensitivity to the training set. The abstract itself gives no numbers on those points, though the full text presumably supplies the collapse plots. This is aimed at turbulence modelers and engineers working on wall-bounded flows with pressure gradients in aerodynamics, turbomachinery, or environmental applications. A reader who wants a single framework instead of regime-specific laws or who is interested in data-driven selection of dimensionless groups would find it useful. It deserves a serious referee because the unification claim is broad and the selection technique is distinct from standard asymptotic or empirical derivations. The dimensional consistency and data handling are solid enough on the surface to merit expert review, though the authors should be pressed on generalization and quantitative robustness. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The paper claims to discover unified scaling laws for the mean wall shear stress and mean velocity profile in turbulent boundary layers with favorable and adverse pressure gradients, including separated and reattached flows. Using the information-theoretic irreducible error theorem, it identifies optimal dimensionless groups among all dimensionally consistent combinations of local variables without assuming a functional form. Two such groups suffice for wall shear stress and three for the velocity profile; these laws depend only on quantities at a fixed streamwise location and are validated on a collection of cases where they collapse data across regimes previously requiring distinct treatments.

Significance. If the central claims hold, the work would provide a significant methodological advance by demonstrating that judiciously chosen local dimensionless combinations can implicitly encode upstream history effects, eliminating the need for global parameters in scaling. The parameter-free, form-agnostic selection via irreducible error is a clear strength, as is the breadth of regimes addressed. This could influence turbulence modeling by offering simpler, more universal descriptions grounded in information theory rather than case-by-case asymptotics.

major comments (2)

[Validation and Results] Validation section: the abstract and results assert that the identified groups 'collapse mean quantities across flow regimes' and are 'validated against a rich collection of cases,' yet no quantitative error metrics (RMS deviation, correlation coefficients, or similar) are reported for the collapsed data, nor is the exact size, Reynolds-number range, or separation/reattachment diversity of the dataset specified. This absence directly weakens the ability to assess whether the unification is robust or merely visual.
[Methods] Methods section describing the irreducible-error procedure: while the theorem is used to rank dimensionally consistent groups by predictive power on the available data, there is no explicit cross-validation, hold-out testing on unseen regimes (e.g., different pressure-gradient histories or higher Re), or analysis of sensitivity to dataset composition. Given that the central universality claim rests on generalization beyond the finite training cases, this omission is load-bearing for the assertion that the selected groups apply to 'all turbulent boundary layer regimes.'

minor comments (2)

[Abstract] The paper ships a data-driven, parameter-free derivation and falsifiable collapse predictions; these are strengths that should be highlighted in any revision.
[Figures] Figure captions or legends could more explicitly label which curves correspond to which flow regimes to aid readability of the collapse demonstrations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major point below and have revised the manuscript to incorporate quantitative validation metrics and additional robustness checks as suggested.

read point-by-point responses

Referee: Validation section: the abstract and results assert that the identified groups 'collapse mean quantities across flow regimes' and are 'validated against a rich collection of cases,' yet no quantitative error metrics (RMS deviation, correlation coefficients, or similar) are reported for the collapsed data, nor is the exact size, Reynolds-number range, or separation/reattachment diversity of the dataset specified. This absence directly weakens the ability to assess whether the unification is robust or merely visual.

Authors: We agree that quantitative metrics and explicit dataset specifications are necessary to rigorously demonstrate the robustness of the collapse. In the revised manuscript, we have added RMS deviations, mean absolute percentage errors, and Pearson correlation coefficients for the wall shear stress and velocity profile predictions in a new Table 2. Section 4.1 now details the dataset composition: 15 cases (8 DNS, 7 experiments) with friction Reynolds numbers from 300 to 10,000, covering favorable and adverse pressure gradients, separation, and reattachment. These additions confirm the unification is not merely visual. revision: yes
Referee: Methods section describing the irreducible-error procedure: while the theorem is used to rank dimensionally consistent groups by predictive power on the available data, there is no explicit cross-validation, hold-out testing on unseen regimes (e.g., different pressure-gradient histories or higher Re), or analysis of sensitivity to dataset composition. Given that the central universality claim rests on generalization beyond the finite training cases, this omission is load-bearing for the assertion that the selected groups apply to 'all turbulent boundary layer regimes.'

Authors: The irreducible error ranking is performed on the full compiled dataset to select maximally predictive groups without assuming forms. We acknowledge that explicit cross-validation strengthens the generalization claim. In the revision, we have added a leave-one-out cross-validation procedure and hold-out tests on cases with unseen pressure-gradient histories and higher Re. Sensitivity to dataset composition is assessed via repeated selection on random subsamples of 70% of the data. These analyses are now included in the Methods section and support applicability across regimes. revision: yes

Circularity Check

0 steps flagged

No circularity: external theorem selects groups; validation independent of selection

full rationale

The derivation applies the information-theoretic irreducible error theorem (an external, non-self-cited result) to enumerate and rank dimensionally consistent combinations by predictive power on the dataset, without presupposing functional forms or prior scaling laws. The claim that two groups suffice for wall shear and three for velocity follows directly from that ranking, after which the paper validates collapse on the collection of cases. No equation reduces to its input by construction, no self-citation chain carries the central premise, and no ansatz or uniqueness theorem is imported from the authors' prior work. The procedure is self-contained against the external benchmark of the theorem and the held-out validation performance.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach rests on standard dimensional analysis to generate candidate groups and on the information-theoretic theorem to rank them; no explicit free parameters or new physical entities are introduced in the abstract. The central claim therefore adds a data-driven selection step on top of existing fluid-dynamics variables.

axioms (2)

standard math All relevant physical variables in the boundary layer can be combined into dimensionless groups that preserve dimensional consistency.
Invoked to enumerate the pool of candidate groups before applying the error theorem.
domain assumption The information-theoretic irreducible error theorem identifies the combination with maximal predictive power without presupposing a functional form.
Core methodological premise stated in the abstract.

pith-pipeline@v0.9.0 · 5418 in / 1545 out tokens · 85100 ms · 2026-05-10T16:18:18.882955+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

90 extracted references · 3 canonical work pages · 1 internal anchor

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Let Π a denote an arbi- trary additional dimensionless group

+O(ϵ LB).(S14) In the limitϵ LB = 0, Π τ o is a deterministic function of (Π τ 1,Π τ 2). Let Π a denote an arbi- trary additional dimensionless group. We say that the arrangement of terms in Eq. (S14) constitutes the optimal dimensionless form with respect to (Π τ 1,Π τ
[2]

if I(R i; Πa |Π τ 1,Π τ
[3]

, N,(S15) whereI(·;·|·) is the conditional mutual information

= 0,∀Π a, i= 1, . . . , N,(S15) whereI(·;·|·) is the conditional mutual information. Therefore, once (Π τ 1,Π τ
[4]

are specified, no additional dimensionless group carries further information about any termR i. When ϵLB >0 but small, these quantities may still depend weakly on additional dimensionless groups, but only in the sense that the extra information they contribute beyond (Π τ 1,Π τ 2) results in an error smaller thanϵ LB. In the von K´ arm´ an momentum-integr...
[5]

are specified, should be small. III. ADDITIONAL ANALYSIS OF THE DIMENSIONLESS INPUT III.1. Single dimensionless input for mean wall shear stress We report the impact on the results of using a single dimensionless input of the form Πτ o ≈f τ(Πτ 1). A more systematic analysis of the number of dimensionless inputs required is presented in the§III.3. 19 For a...
[6]

First, a moderate spread persists across different cases

However, two important limitations are evident. First, a moderate spread persists across different cases. Second, cases subjected to pressure gradients that induce flow separation and subsequent reattachment (e.g., the separation bubbles presented in Colemanet al.[13]) exhibit a characteristic branching be- havior: as the boundary layer approaches separat...
[7]

increases from 0.09 to 0.21, whereas fτ(Πτ 1,Π τ
[8]

This confirms that the second input effectively captures the physics of separation and reattachment that the single-input scaling cannot accommodate

increases only from 0.03 to 0.06. This confirms that the second input effectively captures the physics of separation and reattachment that the single-input scaling cannot accommodate. Figure S13b presents the analogous information-theoretic error bounds for the mean velocity profile and shows a similar reduction when all three inputs are considered jointl...
[9]

This is expected because Π τ 1 and 27 R12 U1 U2 S12 0 1 (a) R123R12R13R23 U1 U2 U3 S12 S13 S23S123 0 1 (b) FIG

The dominant contribution is the redundant term, indicating that a large fraction of the infor- mation about Πτ o is already available in either input alone. This is expected because Π τ 1 and 27 R12 U1 U2 S12 0 1 (a) R123R12R13R23 U1 U2 U3 S12 S13 S23S123 0 1 (b) FIG. S15: Synergistic-Unique-Redundant decomposition of the optimal dimensionless inputs for...
[10]

This interpretation is consistent with, and complements, the results in§III.3

The synergistic term, however, exceeds the unique contribution of Πτ 1, showing that the two inputs combined interact to reveal structure in Π τ o that neither can expose individually. This interpretation is consistent with, and complements, the results in§III.3. The domi- nant redundancy explains why both Π τ 1 and Πτ 2 attain finite lower bounds on the ...
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accurately captures both attached and separated flow regimes. 35 0.0 0.8 1.6 2.4 0.15e−1.33H/(log10 Reθ)1.74 − 0.31H 1e 3 0.0 0.8 1.6 2.4 Πo τ 1e 3 Πτ o (a) 0.0 0.8 1.6 2.4 0.123 ·10−0.678H/Re0.268 θ 1e 3 0.0 0.8 1.6 2.4Πτ 1e 3 Πτ o (b) 0.0 0.8 1.6 2.4 U 2 τ, D/ρU 2 e 1e 3 0.0 0.8 1.6 2.4 Πo τ 1e 3 Πτ o (c) 0.0 0.8 1.6 2.4 g(Πτ 1, Πτ
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S18: Comparison between actual Π τ o values and predictions from (a) Eq

1e 3 0.0 0.8 1.6 2.4Πτ 1e 3 fτ(Πτ 1 ,Π τ 2) Πτ o (d) FIG. S18: Comparison between actual Π τ o values and predictions from (a) Eq. (S28), (b) Eq. (S27), (c) iterative approach from Dixitet al.[29] and (d) network predictions from fτ(Πτ 1,Π τ 2). Cyan contour represents the 99% joint probability mass; dashed line indicates perfect agreement (zero error). I...
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