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| 29 May 2026
Investigation of higher order statistics in wind turbine wakes using Large Eddy Simulations
Abstract. Large eddy simulation (LES) of the turbulent wake behind a wind turbine is employed to obtain a more complete statistical characterisation. Both one- and two-point statistics are therefore considered, including velocity increments. The downstream evolution of the wake size is determined by different statistical quantities, including the intermittency ring. Higher-order statistics show that the wake expands beyond the region commonly defined using the mean wind speed. Regarding the numerical simulations, a new method is introduced to reduce the computational cost of high-resolution wake simulations.
Competing interests: At least one of the (co-)authors is a member of the editorial board of Wind Energy Science.
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RC1: 'Comment on wes-2026-71', Anonymous Referee #1, 04 Jul 2026
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The comment was uploaded in the form of a supplement: https://wes.copernicus.org/preprints/wes-2026-71/wes-2026-71-RC1-supplement.pdfReplyCitation: https://doi.org/
10.5194/wes-2026-71-RC1 -
RC2: 'Comment on wes-2026-71', Anonymous Referee #2, 04 Jul 2026
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Review comments for Bock and Peinke entitled "Investigation of higher order statistics in wind turbine wakes using Large Eddy Simulations" submitted to Wind Energy Science, 2026Overall:
The article presents LES using both actuator-line (AL) and blade-resolved (BR) approaches to model a wind turbine wake. The investigation of higher-order statistics, including skewness, kurtosis, and intermittency, is interesting and provides valuable insight into wake dynamics, particularly through the proposed intermittency-ring characterization. My primary concern is the inclusion of the proposed BR-based methodology, whose suitability for the present analysis of higher-order statistics is not sufficiently demonstrated. In addition, the manuscript provides limited information regarding turbine control under the selected inflow conditions, and a dedicated discussion of implications and limitations is lacking. I therefore recommend major revisions. In particular, the authors should either provide stronger justification and validation of the BR methodology or consider restricting the analysis to the AL results.General comments
1. BR simulation and "new method":
The paper is also occasionally unfocused, primarily due to the presence of the BR simulations. Novelty alone does not necessarily justify the inclusion of a new methodology. In its current form, I do not see a clear benefit of the proposed method for the scientific objectives of this study, and its added value should be better justified. People typically use BR to validate AL. The authors compare AL and BR results, however the BR approach relies on a linear superposition of a laminar BR wake and synthetic turbulence. The manuscript itself acknowledges limitations of this assumption in the near wake (line 284). Consequently, it is unclear to what extent the BR results can be considered representative and valid in the region where a blade-resolved simulation would normally provide the greatest value.That AL and BR reproduce the correct flow field in the near wake (line 155) depends entirely on the load distributions of AL and BR, and how the force smearing in AL is performed. The load distributions should be shown. The assumption of superposing a Mann-generated turbulent field with a BR wake generated under uniform inflow requires validation beyond being stated as an assumption. I consider the underlying assumption to be fundamentally problematic for the objectives of the present study. The methodology effectively neglects nonlinear wake–turbulence interactions while subsequently analyzing higher-order statistics that arise from nonlinear interactions. As a result, I am not convinced that the BR results can be interpreted as physically representative without substantially stronger validation. Obviously, such assumptions are frequently made for simple engineering models, but the BR is supposed to be the highest fidelity. If the BR was performed with inflow turbulence, there would be substantial variance of much larger time scales than 30 s (line 164).
The authors could strengthen the manuscript by demonstrating that this approximation does not affect the conclusions regarding the higher-order statistics. In the absence of such validation, I believe the manuscript would be stronger if the analysis focused solely on the AL simulations. AL is sufficient for the far-wake, where the most interesting aspects are shown by the authors.
2. Turbulent statistics
As mentioned above, I find the statistical analysis interesting, but I believe some important aspects deserve further discussion.For instance, this study follows the common approach of most studies to only investigate the streamwise velocity component (line 70). I believe the limitation associated with considering only the streamwise velocity component should be discussed more explicitly, as the lateral and vertical velocity components may also contribute to the higher-order statistical characteristics of the wake. Furthermore, turbulence characterization is not limited to probability distributions and their moments. Spectral characteristics are also fundamental descriptors of turbulent flow and deserve at least brief discussion in the context of the characterization framework presented in lines 65–70. For this reason, I find the statement that turbulence is "completely" or "fully" characterized by the probability density function overly strong, particularly in the non-Gaussian case that forms the focus of the present study.
Line 17 emphasizes that turbulence requires higher order statistics, which I agree with. However, the inflow turbulence is generated using the Mann model, which is fundamentally a second-order description and relies on a number of assumptions. The manuscript would benefit from a more detailed discussion of why these assumptions are considered acceptable in a study focused on higher-order statistics. In particular, the relevance of homogeneous and isotropic inflow assumptions to realistic atmospheric wind-turbine wakes should be discussed more critically. Many current LES studies of wind turbine wakes employ precursor simulations to generate more realistic boundary-layer inflow conditions, including shear and turbulence structures, which are subsequently used as inflow.
I am not fully convinced that 24 minutes of data is sufficient to ensure convergence of all investigated statistics, particularly those related to rare extreme events and intermittency. I follow the arguments based on integral time scales, but these are determined from the Mann box and might not be representative of a 5 MW turbine operating in the real atmosphere. While the argument based on integral time scales is reasonable, additional discussion of statistical convergence would strengthen the manuscript, especially given that substantial uncertainty may remain even for considerably longer simulations. I also find the connection to IEC recommendations somewhat indirect. The IEC standard recommends six 600 s simulations for load analysis, and even under those conditions substantial uncertainty in statistical convergence remains. See, for example, Figure 6 in Liew and Larsen (2022), https://iopscience.iop.org/article/10.1088/1742-6596/2265/3/032049/meta, and Mozafari et al. 2023, https://doi-org.proxy.findit.cvt.dk/10.1177/0309524X231163825.
3. Inflow conditions and control
The selected inflow velocity of 11.4 m/s corresponds to the rated operating condition of the NREL 5 MW turbine. Given the presence of turbulence, the turbine is expected to experience both below-rated and above-rated conditions during the simulation. Since turbine control can significantly influence wake development in this operating regime, the manuscript should provide additional information regarding the implemented controller. The use of a constant rotor speed alone is difficult to interpret and justify without further context.The turbulence intensity should also be stated explicitly and consistently.
4. Lack of discussion
The manuscript would benefit from a dedicated discussion section addressing both the implications and limitations of the findings.For example, the observation in line 339 regarding the potential influence of the intermittency ring on downstream turbines is particularly interesting, but it is introduced only in the conclusion. Such implications deserve discussion earlier in the manuscript.
5. References:
- Line 19 emphasize the importance of unsteady flow features, but also that the importance is increasing, yet 2 of 3 references are more than 10 years old, so it's hard to follow that it is increasing. I think it would be beneficial with more recent references to support this otherwise good point. Examples could be Biswas and Buxton, JFM, 2026, https://doi.org/10.1017/jfm.2026.11665, and Hodgson et al., Physics of Fluids, 2023, https://doi.org/10.1063/5.0162311
- Sorensen and Kock, 1995, deals with actuator disc, so it is not an appropriate reference for actuator lines (line 61). Actuator line modelling was introduced in Sorensen and Shen, 2002, https://doi.org/10.1115/1.1471361
- Line 200 states "is often evaluated in wake studies". Such statements require references.
- Wake meandering is more than just a result of a turbulent environment as assumed in Larsen et al., 2007 (line 214). I suggest the authors investigate the review paper by Yang and Sotiropoulos, 2019, https://doi.org/10.3390/en12244725
- On self-similarity (Section 4.2), the paper by Xie and Archer, 2014, is also relevant https://doi.org/10.1002/we.1792Minor comments:
The article could improve with general proof-reading, so here are just some examples:
- The blue line in Figure 10 is hard to see
- In Figure 9 and similar figures the caption reads that the distributions have been shifted horizontally. I believe the distributions have been shifted vertically rather than horizontally.
- line 347: "ware" should be "wear"Citation: https://doi.org/10.5194/wes-2026-71-RC2
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