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arxiv: 2606.06876 · v1 · pith:5AIUI3LZnew · submitted 2026-06-05 · ⚛️ physics.flu-dyn

A Wall Function for Turbulent Boundary Layers under Rotation via Symbolic Regression

Yao Ma , Zhi Tao , Ruquan You , Haiwang Li This is my paper

Pith reviewed 2026-06-27 21:16 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords turbulent boundary layerwall functionsymbolic regressionsystem rotationCoriolis forceleading sidetrailing siderelaminarization
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The pith

Symbolic regression derives compact wall-function expressions that capture how Coriolis forces deflect turbulent boundary layers differently on leading and trailing sides under system rotation.

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

This paper applies symbolic regression to measured deflection patterns in rotating turbulent boundary layers to obtain analytical wall functions. The classical law of the wall assumes no rotation and fails when Coriolis forces are present. The resulting expressions remain valid across a wide range of rotation numbers and reproduce the observed contraction of the leading-side boundary layer, expansion on the trailing side, and the approach to relaminarization on the leading side. These closed-form relations supply a white-box replacement or complement for existing wall treatments in turbulence models.

Core claim

The central claim is that symbolic regression, trained on the deflection behavior of turbulent boundary layers, yields physically interpretable wall-function expressions valid over a wide range of rotation numbers; these expressions show that increasing rotation contracts the leading-side boundary layer, expands the trailing-side layer, and drives the leading side toward relaminarization, matching high-fidelity numerical results.

What carries the argument

Symbolic regression applied to the deflection behavior of turbulent boundary layers to generate rotation-dependent analytical wall functions.

If this is right

  • The wall functions are compact and remain interpretable across rotation numbers.
  • They reproduce the contraction of the leading-side boundary layer and expansion on the trailing side.
  • The leading side shows a tendency toward relaminarization as rotation intensifies.
  • The expressions complement conventional wall functions and supply a new route for turbulence-model closure under system rotation.

Where Pith is reading between the lines

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

  • The same regression procedure could be applied to other body-force effects such as buoyancy or curvature to produce analogous closed-form corrections.
  • Integration of these explicit functions into RANS codes would allow rotation effects to be included without additional sub-grid modeling.
  • The approach supplies a route to test whether other non-equilibrium wall laws can be recovered in closed form from limited simulation data.

Load-bearing premise

The symbolic regression produces expressions that remain valid and physically interpretable over a wide range of rotation numbers.

What would settle it

Comparison of the derived wall functions against independent DNS or LES data at rotation numbers outside the range used to generate the expressions would confirm or refute their accuracy.

Figures

Figures reproduced from arXiv: 2606.06876 by Haiwang Li, Ruquan You, Yao Ma, Zhi Tao.

Figure 1
Figure 1. Figure 1: Comparison of the present DNS results with those of Kim [22] and the theoretical solution. The parameters adopted in the present work are listed in table I. 𝑅𝑒𝑏 is the Reynolds number based on the bulk velocity, and 𝑅𝑒𝜏,L and 𝑅𝑒𝜏,T are the friction Reynolds numbers at the leading and trailing sides, respectively. TABLE I. CALCULATION PARAMETERS. Case 𝑹𝒐𝝉 𝑹𝒆𝒃 𝑹𝒆𝝉,𝐋 𝑹𝒆𝝉,𝐓 ST 0 2850 180 180 A1 0.1 2840 177 18… view at source ↗
read the original abstract

This study employs symbolic regression to derive physically interpretable, white-box wall-function expressions for turbulent boundary layers under system rotation. Flows in a rotating frame are subject to Coriolis forces, which deflect the boundary layer profile from static case. The classical law of the wall, formulated under non-rotating conditions, is ill-suited to describing the effects of rotation. To obtain the wall function under rotation, we examine the deflection behavior of the turbulent boundary layers on the leading and trailing sides, and construct wall functions that are valid over a wide range of rotation numbers. The analytical expressions show that, as the rotation effect intensifies, the boundary layer on the leading side contracts whereas that on the trailing side expands, and the leading side exhibits a tendency towards relaminarization, consistent with high-fidelity numerical results. The resulting symbolic expressions are compact and interpretable. The wall functions obtained in this study complement conventional wall functions, and provide a new avenue for turbulence model closure subject to system rotation.

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. This paper employs symbolic regression to derive physically interpretable wall-function expressions for turbulent boundary layers under system rotation. By examining deflection behavior on leading and trailing sides from simulation data, it constructs expressions claimed to be valid over a wide range of rotation numbers. These show boundary-layer contraction on the leading side, expansion on the trailing side, and a relaminarization tendency on the leading side, stated to be consistent with high-fidelity numerical results. The expressions are presented as compact, white-box complements to conventional wall functions for turbulence closure under rotation.

Significance. If the expressions prove generalizable, the work supplies an interpretable, data-derived alternative for incorporating Coriolis effects into near-wall modeling, which could aid RANS closures in rotating machinery flows. The emphasis on physical interpretability distinguishes it from purely black-box approaches. However, because the derivation is entirely data-driven, significance depends on demonstrated generalization beyond training data, which is not established in the manuscript.

major comments (2)
  1. [Method (symbolic regression procedure)] The manuscript provides no details on the symbolic regression algorithm employed, the criteria for selecting training data (e.g., specific rotation numbers or deflection metrics), or quantitative validation metrics such as hold-out error or cross-validation scores. This information is load-bearing for the central claim that the expressions remain valid over a wide range of rotation numbers and reproduce high-fidelity behavior.
  2. [Results and validation] No independent hold-out tests, cross-validation across rotation numbers, or extrapolation checks for rotation numbers outside the training range are reported. Without these, the consistency statements with high-fidelity results and the relaminarization tendency risk being circular, as the fitted expressions may simply reproduce the input data rather than predict new behavior.
minor comments (2)
  1. [Abstract] The abstract refers to 'high-fidelity numerical results' without citing the specific simulations, parameters, or references used for comparison.
  2. [Results] Notation for the derived wall functions (e.g., any new symbols for rotation-modified velocity or length scales) should be introduced with explicit definitions and units in the first results section.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments highlighting the need for greater methodological transparency and validation. We agree these elements are essential to substantiate the claims of generalizability and will incorporate the requested details and tests in the revised manuscript.

read point-by-point responses

  1. Referee: [Method (symbolic regression procedure)] The manuscript provides no details on the symbolic regression algorithm employed, the criteria for selecting training data (e.g., specific rotation numbers or deflection metrics), or quantitative validation metrics such as hold-out error or cross-validation scores. This information is load-bearing for the central claim that the expressions remain valid over a wide range of rotation numbers and reproduce high-fidelity behavior.

    Authors: We agree that the original manuscript omitted key details on the symbolic regression procedure. In the revision we will add a dedicated methods subsection specifying the algorithm (including any software implementation), the exact criteria used to select training data (rotation numbers, deflection metrics, and data sources), and quantitative metrics such as hold-out error and cross-validation scores. These additions will directly support the claim of validity across a wide range of rotation numbers. revision: yes

  2. Referee: [Results and validation] No independent hold-out tests, cross-validation across rotation numbers, or extrapolation checks for rotation numbers outside the training range are reported. Without these, the consistency statements with high-fidelity results and the relaminarization tendency risk being circular, as the fitted expressions may simply reproduce the input data rather than predict new behavior.

    Authors: We acknowledge the validity of this concern. The revised manuscript will include independent hold-out tests, cross-validation performed across distinct rotation numbers, and extrapolation checks for rotation numbers beyond the training range. These results will be presented to demonstrate that the derived expressions capture the observed physical trends (boundary-layer contraction/expansion and relaminarization) rather than merely reproducing the training data. revision: yes

Circularity Check

1 steps flagged

Symbolic regression wall functions fitted to high-fidelity data reproduce its behaviors by construction

specific steps
  1. fitted input called prediction [Abstract]
    "The analytical expressions show that, as the rotation effect intensifies, the boundary layer on the leading side contracts whereas that on the trailing side expands, and the leading side exhibits a tendency towards relaminarization, consistent with high-fidelity numerical results."

    Expressions are obtained by symbolic regression trained on the examined deflection behavior of the same turbulent boundary layer data. Their reproduction of contraction/expansion and relaminarization is therefore a direct encoding of the training patterns, not an independent prediction or derivation.

full rationale

The paper derives wall-function expressions exclusively via symbolic regression on deflection data from high-fidelity simulations of rotating boundary layers. Its central claim—that the resulting expressions demonstrate leading-side contraction, trailing-side expansion, and relaminarization tendency, and remain valid over wide rotation numbers—is presented as an analytical finding. Because the regression directly encodes the input data patterns, this consistency is tautological rather than independently derived. No hold-out validation, extrapolation tests, or external first-principles justification is described that would break the reduction to the training inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; no explicit free parameters, axioms, or invented entities listed. Symbolic regression implies data-fitted coefficients whose values and selection criteria are unknown.

pith-pipeline@v0.9.1-grok · 5705 in / 1006 out tokens · 14487 ms · 2026-06-27T21:16:04.423612+00:00 · methodology

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Reference graph

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