the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 14 Nov 2025
A framework for the superposition of wind turbines wake properties
Abstract. Wind turbines extract kinetic energy from the wind flow and convert it to electric power, while leaving downstream complex, non-stationary, and meandering plumes of reduced wind speed and increased turbulence, called wakes. Several analytical models have been proposed in the literature to describe the structure of wake properties, such as wind speed deficit (∆U), turbulence intensity (TI), or turbulence kinetic energy (TKE), all of which have been expressed as a function of the upstream, or undisturbed, value of said properties. While this dependency on undisturbed values is natural and practical for a single wind turbine, it becomes less so in the presence of multiple turbines, to the point of being meaningless in large wind farms, where the distance between front- and last-row turbines may exceed tens of kilometers and therefore the very concept of undisturbed flow is moot for the majority of turbines. As such, superposition methods, which aim at obtaining the final distribution of wake properties in overlapping wakes from multiple wind turbines, struggle to provide consistent results, especially for turbulence properties. In addition, the literature is lacking coherent definitions of the superposition methods, with each study introducing its own naming convention and often slightly different formulations. Here we propose a framework to standardize wake superposition for use with any analytical model of wake properties, including ∆U, TI, or TKE (but not temperature or pressure). The framework is based on the concept of “inflow”, as opposed to undisturbed, values, which are in general affected by upstream turbines and are truly undisturbed only for front-row turbines. Within this framework, we propose that the inflow property should not be a constant, but rather a function of z, taken at a fixed distance upstream of each turbine, for example one diameter. Furthermore, the superposition methods are grouped into two main categories (i.e., simple or normalized summation), with the value of an exponent m controlling the linearity or non-linearity of the summed terms. The performance of the superposition methods within the framework is assessed for three analytical wake models (one for each of the three wake properties ∆U, TI, and TKE) against LES results from independent studies under various wind farm layouts, turbine models, and atmospheric stabilities. General findings are that the value of m controls the magnitude of the resulting wake property, with higher values for smaller m, and that, for the same value of m, the simple summation methods tend to create stronger overlapping wakes than the normalized ones in terms of turbulence properties, but the opposite is found for wind speed deficit. We conclude that the selection of the superposition method depends in part on the bias of the analytical wake model itself.
Competing interests: Prof. Crisitina Archer 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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- RC1: 'Comment on wes-2025-220', Anonymous Referee #1, 02 Dec 2025
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RC2: 'Comment on wes-2025-220', Anonymous Referee #2, 23 Jan 2026
The paper investigates different wake and turbulence superposition approaches used within analytical wind farm yield simulations and compare them to mean velocity and turbulence fields from LES simulations. Whilst generally an interesting topic, the paper unfortunately does not expand on the current level of knowledge in this field. Different to the title, no “framework of wake properties models” is provided, but instead a mix between a (incomplete) review of wake and added turbulence models and simple application of empirical superposition approach is presented. The issue with analytical wake, TI models and superposition is that a clear definition of each is needed to clearly show how they all interact (which is actually mentioned by the authors). In this context it is good that the authors highlight that scaling (called normalisation in the paper) of the deficits/added TI also comes into play. However this has been discussed and laid out much more clearly by other authors before (see Bastankhah et al 2021 or Zong et al 2020 for instance, both JFM).
Without the presentation of a framework the paper essentially applies a wake and TI model (both previously developed by one of the authors) to different LES cases and tries to see whether they can be made to agree with them, solely by changing the superposition model. A simple tuning factor “m” is introduced to do so, however this is exactly the issue many times when discussing engineering wake modelling, as many times the inadequacy of single wake models is compensated by tweaking the superposition model. However this only works in certain conditions and does not really advance the current field of research. There have been other papers that have provided proper frameworks for tuning (see for instance van Binsbergen et al 2024, WES), this should be used as a starting point to push beyond the status quo. A simple manual superposition model tuning is not sufficient for a journal paper with regard to the existing body of research.
Some detailed comments are given in the attached document.
- AC1: 'Comment on wes-2025-220', Ali Khanjari, 27 Feb 2026
Status: closed
-
RC1: 'Comment on wes-2025-220', Anonymous Referee #1, 02 Dec 2025
The article titled “A framework for the superposition of wind turbines wake properties” discusses the development of engineering wake models capable of representing both streamwise velocity and turbulence in three dimensions. In the abstract, the authors state that they “propose a framework to standardize wake superposition for use with any analytical model of wake properties”. In the concluding section, they further note that, relative to normalized summation (NS), simple summation (SS) might be a more suitable option for additive quantities such as turbulence within a wind farm. They also conclude that SS generally exhibits better numerical stability than NS, particularly when the exponent m is smaller than 1.
The topic of the study fits well within the scope of Wind Energy Science, as the wind energy industry has long relied on analytical and engineering models to efficiently predict flow fields within wind farms. For example, developers depend heavily on such models when optimizing wind-farm layouts.
In general, I found the manuscript difficult to follow overall, requiring a significant amount of effort to keep track of the authors’ reasoning. Moreover, I am not convinced that the authors have clearly established the “framework to standardize wake superposition” that they claim to propose. Even if such a framework is presented, it is not clear what is the concrete contribution of this work to the wind energy community or to the broader knowledge base. Several issues resulting in these concerns, along with additional comments, are outlined below.
- I am not convinced that the title “A framework for the superposition of wind turbines wake properties” accurately reflects the scope or contribution of the work. In my opinion, a title that can directly reflect the central of this study, which is studying the exponent of NSS, would be more proper. As it stands, the current title feels over-ambitious and gives the impression of presenting a completed or definitive framework, which I cannot endorse based on the current content. Moreover, the work does not appear to pioneer this area, and the current title risks overstating the contribution in a way that could unintentionally overshadow or absorb the contributions of earlier studies. Therefore, in my view, the title should be adjusted to one that more realistically conveys the incremental nature of the contribution. Additionally, I find several crucial prior works that are highly relevant to the current work are not mentioned, for example: Amin Niayifar and Fernando Porté-Agel (2016, https://doi.org/10.3390/en9090741), Haohua Zong and Fernando Porté-Agel (2020, https://doi.org/10.1017/jfm.2020.77 ), and Majid Bastankhah et al. (2021, https://doi.org/10.1017/jfm.2020.1037 ). These works, in my opinion, fulfill at least two out of three propositions motioned in the manuscript (perhaps missing the third). Comparing these prior results with the currently proposed framework may help advance the development of wake modeling.
- I find the overall structure of the manuscript to be relatively disorganized. This is particularly evident in the abstract and the introduction. As a reader, I often felt overwhelmed by fragmented pieces of information presented without a coherent narrative. Key ideas are scattered across multiple paragraphs, forcing the reader to piece together the intended meaning. I strongly recommend that the authors substantially restructure the manuscript to improve clarity, logical flow, and overall readability. Although this may be somewhat unconventional, may be even controversial, to note, I encourage the authors to consider make use of modern writing-assistance tools to help polish the writing, improve clarity, and reduce the fragmentation. This could substantially enhance the readability and accessibility of the manuscript.
- The presentation suffers from fragmentation and a lack of clean, consistent notations. A large number of equations, symbols, and acronyms are introduced throughout the manuscript, but many of them are not properly defined. Several variables appear abruptly or are only loosely described, which makes the mathematical development difficult to follow. While I recognize that some parameters are used only once and may not require extensive explanation, many important quantities are treated with the same level of brevity as those that are incidental. This often left me uncertain about the authors’ intent and the precise meaning of the expressions. I strongly recommend that the authors highlight the key components/building-blocks of the proposed framework more clearly, potentially through summary tables, as the literature the authors have cited (tables 2 and 3 of QI2021). Also, in my opinion, providing explicit definitions for all parameters that play a central role could definitely help. Additionally, the authors should reconsider the sequence in which equations and symbols are introduced to ensure a logical and accessible progression.
- Related to my previous comment, many important equations and parameters are introduced in the Introduction section, while another substantial portion is introduced later in the Methods section. This split presentation significantly disrupts the narrative flow and makes the manuscript difficult to follow. Consolidating the definitions and presenting the mathematical elements in a more coherent and centralized manner would greatly improve readability and comprehension.
- The figures in the appendix are presented without sufficient explanation or context.
With regard to the scientific methods and findings presented in the manuscript, I have several additional concerns, which are outlined below:
- It is unclear to me how broadly the “second proposition” is applied within the framework. For example, when using Eq. (24), should the formulation U_{infty} or U_{in, i} be employed? I recommend that the authors explicitly present the final set of equations and parameters used in the framework to avoid ambiguity and ensure reproducibility. This can be done in a dedicated section or an appendix.
- Related to the previous point, I am not sure whether the prediction of turbulence is actually coupled with the estimation of velocity in the current manuscript, as it is unclear to me whether the “second proposition” is applied universally across the framework. If the two fields are indeed coupled, it is not stated how the authors compute TI when predicting the velocity deficit field. Is it obtained using SS or NS, and what value of m is adopted in the framework? Clarifying these choices is essential for understanding the internal consistency of the framework and for assessing how the turbulence predictions relate to the velocity formulations. Moreover, physically/mathematically explaining how these engineering-model-predicted properties would, in my view, be very valuable to the wind energy community.
- I find that the conclusions drawn are rather superficial and may be highly case-dependent. In particular, it is unclear why the authors do not advocate using SS with m = 1.0 if the “second proposition” is to be fully upheld. My concern is based on the basic identity P_1(x, y, z) = P_{infty}(z) + Delta P_1(x, y, z), which holds regardless of whether SS or NS is used, and regardless of the value of m. By extension, one could equally argue for P_i(x, y, z) = P_{in, i}(z) + Delta P_i(x, y ,z). This is especially relevant if one assumes P_{in, i}(z) = P_{infty, i}(z), an equivalence that the “second proposition” appears to imply in my perspective. I therefore find it unclear why the formulation does not naturally lead to the use of SS with m = 1.0, as I think it is physically more sound. Clarification will be appreciated.
- It is not clear to me why Equations (3), (6), and (24) are given particular emphasis in the manuscript, especially given that there are many well-established alternatives in the literature (e.g., those appeared in the three references mentioned above). Their selection and role within the overall framework are not sufficiently explained, and it is unclear how they bridge to the subsequent analysis. I recommend that the authors clarify the specific importance of these equations and physically explain why they are central to the development of the proposed methodology/framework.
- Related to point 3, I currently do not clearly understand why NS or SS would be considered more physically sound than LSS (linear sum of squares). The manuscript does not provide sufficient justification for favoring NS or SS from a physical/mathematical/theoretical standpoint, nor does it explain under what conditions LSS would be inappropriate. I encourage the authors to elaborate on the physical rationale that distinguishes these approaches. Additionally, in the work that established QI2021, they have used an approach to superposition sigma_u (i.e., TI) that was different from NS and SS, therefore, I don’t think the comparison in this work, for example figure 12, is fair.
- U_{infty} for the LES benchmark cases is not explicitly provided, yet several parts of the analysis rely on absolute values that depend on this quantity. The absence of a precise values of hampers the reader’s ability to interpret and reproduce the results. I recommend that the authors explicitly specify the upstream wind speed used for each LES case and perhaps only presenting non-dimensionalized quantities.
- Even if the proposed framework does not perform well in predicting turbulence properties within wind farms in an absolute sense (which appears to be the case for the current formulation), it is important to articulate what can still be learned from the study (the authors have briefly discussed about the x-positions where the peak values of TKE appear). I fully recognize that accurately predicting both velocity deficit and turbulence simultaneously is extremely challenging for engineering models. However, I find that the manuscript lacks a meaningful discussion of the broader insights, implications, or limitations that emerge from the analysis. I encourage the authors to reflect more explicitly on what the community can take away from the work, even if the turbulence predictions remain only of limited accuracy.
- Lastly, if the objective of the current work is to establish a framework based on engineering or analytical models for predicting the flow field within wind farms, it would be greatly appreciated if the authors could share the code used in their implementation. Doing so would greatly improve transparency and reproducibility, and would strengthen the practical relevance of the current work for the community.
There are additional points highlighted in the attached PDF files that I encountered while reading the manuscript. Several of these are also important from my perspective, and I strongly encourage the authors to address them as part of a comprehensive revision. These comments are not exhaustive, and I believe it is reasonable to expect the authors to identify and resolve further issues beyond those explicitly noted. Please also note that some highlighted passages in the PDF are simply anchors for my own reference and should not be interpreted as specific concerns.
Overall, I can only reconsider on recommending this manuscript for publication after a substantial and thorough revision.
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RC2: 'Comment on wes-2025-220', Anonymous Referee #2, 23 Jan 2026
The paper investigates different wake and turbulence superposition approaches used within analytical wind farm yield simulations and compare them to mean velocity and turbulence fields from LES simulations. Whilst generally an interesting topic, the paper unfortunately does not expand on the current level of knowledge in this field. Different to the title, no “framework of wake properties models” is provided, but instead a mix between a (incomplete) review of wake and added turbulence models and simple application of empirical superposition approach is presented. The issue with analytical wake, TI models and superposition is that a clear definition of each is needed to clearly show how they all interact (which is actually mentioned by the authors). In this context it is good that the authors highlight that scaling (called normalisation in the paper) of the deficits/added TI also comes into play. However this has been discussed and laid out much more clearly by other authors before (see Bastankhah et al 2021 or Zong et al 2020 for instance, both JFM).
Without the presentation of a framework the paper essentially applies a wake and TI model (both previously developed by one of the authors) to different LES cases and tries to see whether they can be made to agree with them, solely by changing the superposition model. A simple tuning factor “m” is introduced to do so, however this is exactly the issue many times when discussing engineering wake modelling, as many times the inadequacy of single wake models is compensated by tweaking the superposition model. However this only works in certain conditions and does not really advance the current field of research. There have been other papers that have provided proper frameworks for tuning (see for instance van Binsbergen et al 2024, WES), this should be used as a starting point to push beyond the status quo. A simple manual superposition model tuning is not sufficient for a journal paper with regard to the existing body of research.
Some detailed comments are given in the attached document.
- AC1: 'Comment on wes-2025-220', Ali Khanjari, 27 Feb 2026
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The article titled “A framework for the superposition of wind turbines wake properties” discusses the development of engineering wake models capable of representing both streamwise velocity and turbulence in three dimensions. In the abstract, the authors state that they “propose a framework to standardize wake superposition for use with any analytical model of wake properties”. In the concluding section, they further note that, relative to normalized summation (NS), simple summation (SS) might be a more suitable option for additive quantities such as turbulence within a wind farm. They also conclude that SS generally exhibits better numerical stability than NS, particularly when the exponent m is smaller than 1.
The topic of the study fits well within the scope of Wind Energy Science, as the wind energy industry has long relied on analytical and engineering models to efficiently predict flow fields within wind farms. For example, developers depend heavily on such models when optimizing wind-farm layouts.
In general, I found the manuscript difficult to follow overall, requiring a significant amount of effort to keep track of the authors’ reasoning. Moreover, I am not convinced that the authors have clearly established the “framework to standardize wake superposition” that they claim to propose. Even if such a framework is presented, it is not clear what is the concrete contribution of this work to the wind energy community or to the broader knowledge base. Several issues resulting in these concerns, along with additional comments, are outlined below.
With regard to the scientific methods and findings presented in the manuscript, I have several additional concerns, which are outlined below:
There are additional points highlighted in the attached PDF files that I encountered while reading the manuscript. Several of these are also important from my perspective, and I strongly encourage the authors to address them as part of a comprehensive revision. These comments are not exhaustive, and I believe it is reasonable to expect the authors to identify and resolve further issues beyond those explicitly noted. Please also note that some highlighted passages in the PDF are simply anchors for my own reference and should not be interpreted as specific concerns.
Overall, I can only reconsider on recommending this manuscript for publication after a substantial and thorough revision.