Analytical modelling of wind-turbine wake turbulence in neutral atmospheric boundary layers

Erwan J\'ez\'equel; Fr\'ed\'eric Blondel; Helen Schottenhamml; Majid Bastankhah

arxiv: 2508.20012 · v4 · submitted 2025-08-27 · ⚛️ physics.flu-dyn

Analytical modelling of wind-turbine wake turbulence in neutral atmospheric boundary layers

Fr\'ed\'eric Blondel , Erwan J\'ez\'equel , Helen Schottenhamml , Majid Bastankhah This is my paper

Pith reviewed 2026-05-18 21:06 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn

keywords wind turbine wakewake-added turbulenceturbulence intensityanalytical modellingTKE budgetReynolds stressRANS closureneutral atmospheric boundary layer

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The pith

A new model for wake-added turbulence intensity is derived from TKE and streamwise Reynolds stress budgets using RANS assumptions and far-wake approximations.

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

The paper develops an analytical expression for the extra turbulence created by a wind turbine wake. It starts from the equations governing turbulent kinetic energy and the streamwise Reynolds stress, then applies standard RANS closure relations and far-wake simplifications to obtain a closed-form result. This matters for wind-farm design because turbulence levels control how quickly wakes recover, how much power downstream turbines can extract, and how loads are distributed across a farm. The model stays compact enough for fast engineering solvers yet reproduces the main trends seen in large-eddy simulations and wind-tunnel tests.

Core claim

By analysing the TKE and streamwise Reynolds stress budgets under classical RANS modelling assumptions and far-wake approximations, the authors obtain a simple, physically consistent expression for wake-added turbulence intensity that agrees with LES and wind-tunnel data in neutral atmospheric boundary layers.

What carries the argument

The TKE budget combined with the streamwise Reynolds stress budget, closed via classical RANS modelling assumptions and far-wake approximations.

If this is right

The model can be inserted directly into existing analytical wind-farm flow solvers to replace empirical turbulence closures.
Local turbulence intensity estimates become available without running full CFD, enabling faster layout optimisation.
Because the expression follows from budget equations rather than curve fits, it should extrapolate more reliably to new turbine sizes or spacings.
The same budget approach offers a route to include additional effects such as ground shear or modest stability once the basic neutral case is validated.

Where Pith is reading between the lines

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

The derivation suggests that similar budget reductions could be attempted for wakes in stable or convective boundary layers if the appropriate production and dissipation terms are retained.
Because the model separates the streamwise stress contribution, it may help diagnose why some existing turbulence closures over-predict lateral spreading in large wind farms.
Integration into control algorithms that adjust turbine yaw or pitch to manage downstream turbulence becomes feasible once the expression is coded into real-time farm simulators.

Load-bearing premise

Classical RANS modelling assumptions and far-wake approximations remain valid for the inherently three-dimensional turbulent wake fields in neutral atmospheric boundary layers.

What would settle it

A systematic mismatch between the model's predicted turbulence intensity profiles and either high-resolution LES or full-scale measurements taken in neutral conditions at distances where the far-wake assumption is expected to hold would falsify the central claim.

read the original abstract

So-called engineering or analytical wind farm flow solvers typically build upon two submodels: one for the velocity deficit and one for the wake-added turbulence intensity. While velocity deficit modelling has received considerable attention, wake-added turbulence models are less prevalent in comparison. Yet, accurate estimates of local turbulence intensity are essential for predicting flow interactions and energy yield, as turbine wakes are both sensitive to, and sources of turbulence. Existing wake-added turbulence models are typically empirical or assume axial symmetry despite the inherently three-dimensional nature of turbulent wake fields. In this work, we present a new model for wake-added turbulence intensity. Our approach is based on the analysis of the TKE and the streamwise Reynolds stress budget, incorporating classical RANS modelling assumptions and far-wake approximations. The resulting model maintains a simple and practical form, demonstrating strong agreement with LES and wind tunnel measurements. Our model provides a more physically consistent and predictive tool for wind farm flow modelling and performance estimation.

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

Budget-based model for wake-added turbulence gives a simple practical form but leans on far-wake cuts that look questionable in sheared neutral ABL.

read the letter

This paper derives a closed-form model for wake-added turbulence intensity starting from the turbulent kinetic energy and streamwise Reynolds stress budgets. The authors apply standard RANS closures together with far-wake simplifications that drop streamwise diffusion, pressure-strain terms, and some mean-shear production. The result is a practical expression that they compare against LES and wind-tunnel measurements, reporting good agreement. What is new is the budget analysis route rather than direct empirical fitting or axial-symmetry assumptions used in earlier wake turbulence models. This gives the model a clearer link to the governing equations and avoids some of the symmetry limitations that do not hold in real atmospheric wakes. The work does well at keeping the final formula simple enough for use in engineering wind-farm solvers. Accurate local turbulence estimates matter for predicting turbine loads and array performance, so a better sub-model in that space has direct value. The soft spots center on the validity of the far-wake approximations inside a neutral atmospheric boundary layer. The incoming logarithmic shear and wall-normal variations do not disappear uniformly even at five to eight diameters downstream. Without a quantitative check showing that the neglected production and redistribution terms remain small compared with the wake-induced contributions, it is hard to know whether the model is truly predictive or mainly calibrated to the validation data. This paper is for researchers and engineers who develop or apply fast analytical tools for wind-farm flow and performance calculations. A reader looking for an incremental but more physically motivated update to existing wake-added turbulence prescriptions would get the most out of it. It deserves a serious referee. The budget-based derivation is a reasonable step beyond pure empiricism, and the topic is relevant to renewable energy modeling, so external review can test the approximation assumptions and the strength of the data support. I recommend sending it for peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript presents a new analytical model for wake-added turbulence intensity in wind-turbine wakes embedded in neutral atmospheric boundary layers. The model is derived from the turbulent kinetic energy and streamwise Reynolds-stress budgets by invoking classical RANS closures together with far-wake simplifications that drop streamwise diffusion, pressure-strain redistribution, and certain mean-shear production contributions. The resulting closed-form expression is reported to agree well with LES and wind-tunnel data while remaining simple enough for engineering wind-farm solvers.

Significance. If the far-wake approximations can be shown to remain valid inside a sheared ABL, the work supplies a physically grounded, low-complexity alternative to existing empirical wake-turbulence models. Accurate local turbulence intensity is essential for predicting wake recovery, turbine loading, and array energy yield; therefore a transparent derivation that avoids ad-hoc fitting would be a useful addition to the analytical-modeling literature.

major comments (2)

[budget analysis / model derivation] The central derivation invokes far-wake approximations that neglect mean-shear production arising from the incoming logarithmic ABL profile as well as wall-normal inhomogeneity. These terms remain active inside the wake even at 5–8 D downstream. In the budget-analysis section, please supply an order-of-magnitude estimate or direct comparison against full LES budgets demonstrating that the neglected contributions are small compared with the retained wake-induced terms; without this quantification the closed-form expression risks reducing to an empirical fit rather than a predictive model.
[validation / results] The abstract states 'strong agreement' with LES and wind-tunnel measurements, yet no quantitative error metrics, data-selection criteria, or statement on whether any constants are independently derived versus calibrated on the same datasets appear in the validation section. These details are required to substantiate the claim of physical consistency.

minor comments (2)

Notation for the added turbulence intensity and the various budget terms should be defined once in a dedicated nomenclature table or at first use to improve readability.
[introduction] A brief comparison paragraph placing the new model against the most common existing analytical wake-turbulence closures (e.g., those based on Gaussian or top-hat assumptions) would help readers gauge the incremental advance.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the justification of our approximations and strengthen the validation. We address each major comment below and will revise the manuscript to incorporate the requested additions.

read point-by-point responses

Referee: The central derivation invokes far-wake approximations that neglect mean-shear production arising from the incoming logarithmic ABL profile as well as wall-normal inhomogeneity. These terms remain active inside the wake even at 5–8 D downstream. In the budget-analysis section, please supply an order-of-magnitude estimate or direct comparison against full LES budgets demonstrating that the neglected contributions are small compared with the retained wake-induced terms; without this quantification the closed-form expression risks reducing to an empirical fit rather than a predictive model.

Authors: We agree that a quantitative justification of the neglected terms is required to establish the validity of the far-wake simplifications. In the revised manuscript we will add an order-of-magnitude analysis in the budget section that directly compares the magnitudes of the mean-shear production from the incoming logarithmic profile and the wall-normal inhomogeneity terms against the retained wake-induced production and dissipation terms, using the same LES data already employed for validation. This comparison will be shown at multiple downstream stations (including 5–8 D) to demonstrate that the neglected contributions remain small relative to the wake-induced terms in the region where the model is intended to apply. revision: yes
Referee: The abstract states 'strong agreement' with LES and wind-tunnel measurements, yet no quantitative error metrics, data-selection criteria, or statement on whether any constants are independently derived versus calibrated on the same datasets appear in the validation section. These details are required to substantiate the claim of physical consistency.

Authors: We accept that the validation section requires more quantitative detail. The revised manuscript will report explicit error metrics (root-mean-square error and Pearson correlation coefficient) between the model predictions and both the LES and wind-tunnel data sets. We will also state the data-selection criteria (neutral stratification, specific turbine thrust coefficients, and downstream distances considered) and clarify that all model constants are obtained from the classical RANS closure assumptions without calibration against the validation data. revision: yes

Circularity Check

0 steps flagged

Derivation from TKE and Reynolds-stress budgets uses standard RANS closures and far-wake approximations without reducing to fitted inputs or self-citations

full rationale

The paper starts from the TKE and streamwise Reynolds-stress transport equations, applies classical RANS modelling assumptions (eddy-viscosity closure, etc.) and far-wake simplifications (neglect of streamwise diffusion, pressure-strain, and certain production terms), and arrives at a closed-form expression for wake-added turbulence intensity. These steps rely on well-established fluid-dynamics closures that predate the present work and are not constructed from the LES or wind-tunnel data used for validation. The resulting model is then compared to independent LES and experimental measurements, providing an external benchmark rather than a tautological fit. No load-bearing self-citation, parameter renaming, or ansatz smuggling is indicated in the abstract or described derivation chain. The approach therefore remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review is limited to the abstract; specific free parameters, axioms, or invented entities cannot be extracted. The model invokes classical RANS assumptions and far-wake approximations whose details are not visible.

pith-pipeline@v0.9.0 · 5711 in / 1102 out tokens · 39966 ms · 2026-05-18T21:06:10.678926+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 unclear

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

We present a new model for wake-added turbulence intensity based on the analysis of the TKE and the streamwise Reynolds stress budget, incorporating classical RANS modelling assumptions and far-wake approximations.
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean J_uniquely_calibrated_via_higher_derivative unclear

?

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

The resulting model maintains a simple and practical form, demonstrating strong agreement with LES and wind tunnel measurements.

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