the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 13 Feb 2026
Fast blockage models for wind-farm power prediction
Abstract. Large offshore wind farms can trigger atmospheric gravity waves, with the associated hydrostatic blockage effect impacting their energy yield. Unfortunately, to date no tools exist that can model this wind-farm gravity-wave interaction and blockage at a computational cost that is not drastically higher than conventional engineering wake models. To address this, this paper applies insights from two-scale momentum (2SM) theory to an atmospheric perturbation model (APM), thereby significantly speeding up the latter. This leads to two different models, with one using pre-computed farm-level coefficients to compute the turbine forces and power output, while the other relies on repeated wake model evaluations. The core 2SM hypothesis and both developed models are validated using a prior LES dataset of a large wind farm. Both fast 2SM—APMs perform well, predicting blockage-corrected farm power at a computational cost that is only a factor 40 slower than a standalone wake model.
Competing interests: At least one of the (co-)authors is a member of the editorial board of Wind Energy Science.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors. Views expressed in the text are those of the authors and do not necessarily reflect the views of the publisher.- Preprint
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Status: final response (author comments only)
- RC1: 'Comment on wes-2026-14', Anonymous Referee #1, 24 Mar 2026
- RC2: 'Comment on wes-2026-14', Anonymous Referee #2, 10 Apr 2026
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RC3: 'Comment on wes-2026-14', Anonymous Referee #3, 11 Apr 2026
This article has to do with the reduced-order modelling of wind farm flows. Specifically, it focuses on an effort to improve the efficiency of an atmospheric perturbation model AOM) developed by the authors and described in earlier publications. To achieve efficiency gains, the authors leverage insights from two-scale momentum theory. The authors offer two different approaches to speed up the APM and compare the results with a dataset of large-eddy simulations over a range of different inflow conditions.
I found the article to be well written, and for the most part I found the model descriptions to be clear. The validation indicates that the two approaches used to improve calculation efficiency hold promise. Some potentially important terms were left out of the APM to simplify this study. The terms will likely be restored in a future study; it will be interesting to learn how this affects the results.
There is a need for better engineering/reduced-order models for wind farm flows. This paper is a helpful contribution towards that end, and I recommend its publication.
I have a number of minor comments. I hope at least some of them can lead to improvements in the paper.
Line 30 “one to five minutes on a standard laptop”: Is that one to five minutes for each atmospheric condition simulated? Or is it one to five minutes to cover the full range of conditions, including wind speed and wind direction, that would be assessed during a typical energy yield analysis? This is probably clarified in the reference cited at the end of the sentence, but it is probably worth clarifying here.
Lines 68-72, where C_p,k’ and C_T,k’ are defined as disk-based turbine power and turbine thrust coefficients. While reading this paper, I was wondering where these values come from. I thought they might be simply inputs to the farm-scale model, but then line 250 indicates that C_T,k’ is “calculated using a standalone wake model”. This makes me think that I may not fully understand what C_T,k’ is. Some clarification would help. I am not confident I understand how the values are determined for the different farm-scale models (wake model with no induction model, wake model with induction model, and truly neutral LES).
Figure 1: For the y-axis results, is P_0 the power of a standalone turbine simulated in truly neutral conditions? Might be worth clarifying in the paper.
Line 143 “For the sake of simplicity, the latter two are not included in this study”: This comes up a few times in the paper. Are the authors able to reasonably speculate on the impact of this simplification on the results? I cannot and am left wondering.
Line 207 “(no perturbation above the 2SM volume)”: I originally misread this to mean that the flow is not perturbed upwards, but I am pretty sure that is not what you mean. This line instead refers to a perturbation in wind speed. Perhaps I should have understood the meaning right away, but you may want to clarify to avoid others having this experience.
Line 232 “we will assume that the distribution of the force in space, normalized by the total force F/h1, is independent of the APM state”: Meyers has published papers indicating that this is not a good assumption (i.e. the papers show that the details of the stratification, which directly influence the APM state, can have a large impact on the distribution of the force across the wind farm). Any thoughts on the consequences of this simplification?
Line 268 “We leave out the H150 cases…”: On line 86 and 87, the authors write, “This height (H_f) is above the turbine tip height but below the ABL height for all the cases in the dataset of Lanzilao and Meyers (2024).” When I first read this line, I was confused because I know the Lanzilao and Meyers paper included cases with ABL heights below H_f. It was not until I reached line 268 that the matter was clarified for me. This does not necessarily mean that a change needs to be made, but I wanted to let the authors know that at least this reader was confused by the sentence starting on line 86.
Line 318 “… calculating these coefficients with a wake model places requirements on said wake model’s performance…”: I think I missed what those requirements are. Was it the need to include a turbine induction model? Or perhaps something else related to calculating U_f?
Line 337 “This condition gives in an implicit equation for U_b…”: Should the word “in” be there?
Line 346 “For sufficiently small perturbations, we can new assume”: I think “new” should be “now” or perhaps it should be simply dropped.
Equation 41: H should be H_1, right?
Lines 476-477 “Tuning the underlying wake models could address this, as both the wake spreading coefficients and the farm-averaged power and thrust coefficients can be optimized”: How does this align with the earlier statement that the bias mostly has to with an overprediction of blockage loss (line 454)? This question mostly relates to the tuning of wake spreading coefficients. Separately, I’m not familiar with the tuning of the thrust coefficients used in wake models. How would that be done and how would it be justified? Perhaps this relates back to my question about C_T,k’? I am also not clear on what would be done to tune the “farm-averaged power” of the wake model. Is this just a consequence of tuning the wake spreading coefficients? To be clear, I have no problem with the idea of tuning engineering wake models, which are highly simplified representations of wind farm flow – I just want to better understand what the authors are suggesting here and why they are suggesting it.
Citation: https://doi.org/10.5194/wes-2026-14-RC3
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The paper describes how the recently proposed two-scale momentum (2SM) theory can be coupled to a atmospheric perturbation model (APM) developed by the authors. The approach is novel and the paper well written, mathematically sound and mostly easy to follow, however the structure, validation and “fast” element of the paper need improvement.
The only part that seems to be “fast” is hidden in subsection 4.3.1, so either the title needs to be adapted, as it is currently misleading, or more work needs to be spend on timing the three different flavours of the coupling that are presented and show how the currently not so “fast” approaches can be sped up. Whilst going through section 4 and 5 the motivation of using one approach over the other remains somewhat obscure and currently just seems like a list of possible methods.
While the paper is mathematically rigorous, some decisions and sensitivities need to be explored further, especially how to choose the size of the control volume in the 2SM approach such that the error in determining the average wind farm wind speed, Uf, does not depend on the wind farm layout. The authors currently are able to avoid addressing this issue, as they use idealised square wind farms, the same idealised conditions also the 2SM theory is based on. At least one non-rectangular wind farm (or at least run one different wind direction) test case need to be added to showcase that the proposed approach has more general applicability and provides similar results as for an idealised setup. Additionally, one case above rated wind speed could be added, such that the farm has some CT, CP variation (potentially already the case but not explicitly discussed in the paper.
Detailed comments are provided in the attached document.