the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 07 Feb 2025
Spectral proper orthogonal decomposition of active wake mixing dynamics in a stable atmospheric boundary layer
Abstract. Recent advancements in the use of active wake mixing (AWM) to reduce wake effects on downstream turbines open new avenues for increasing power generation in wind farms. However, a better understanding of the fluid dynamics underlying AWM is still needed to make wake mixing a reliable strategy for wind farm flow control. In this work, a spectral proper orthogonal decomposition (SPOD) is used to analyze the dynamics of coherent flow structures that are induced in the wake through blade pitch actuation. The data are generated using the Exawind software suite to perform a large eddy simulation of an AWAKEN 2.8 MW turbine operating in a stable atmospheric boundary layer. SPOD tracks the modal behavior of flow structures from their generation in the turbine induction field, through their growth in the near wake region, and to their subsequent evolution and energy transfers in the far wake. SPOD is shown to be a useful tool in the context of AWM because it translates the wavenumber and frequency inputs to the turbine controller to structures in the wake. A decomposition of the radial shear stress flux in the wake is also developed using SPOD to measure the contribution of coherent flow structures to mean flow turbulent entrainment and wake recovery. The effectiveness of AWM is connected to its ability to excite inherent structures in the wake of the turbine that arise using baseline controls. The effects of AWM on blade loading are also analyzed by connecting the axial force along the blade to the SPOD analysis of the turbine induction field. Lastly, the performance of different AWM strategies is demonstrated in a two-turbine array.
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RC1: 'Comment on wes-2025-14', Anonymous Referee #1, 27 Feb 2025
The manuscript "Spectral Proper Orthogonal Decomposition of Active Wake Mixing Dynamics in a Stable Atmospheric Boundary Layer" by Yalla et al. analyzes four different AWM strategies using SPOD in a stable ABL, relying on numerical simulation results. Apart from the modal analysis, it also investigates the effect of each strategy on the DEL, making an important contribution to further understand and optimize AWM. However, several issues, including one major concern, require attention before publication:Major Issue:1. The right panel of Figure 1 exhibits periodic patterns in the free flow around the wake and streamwise streaks that appear unphysical or at least uncommon. Since the entire study relies on these flow fields, their reliability should be validated. Therefore, the origin of these features needs to be thoroughly investigated, and their potential impact on the results must be assessed. One possible cause could be the grid refinements, whose exact locations should still be clarified in the manuscript. Around the assumed positions of these refinements, small-scale waves can be observed in the flow fields, resembling typical dispersion errors in numerical schemes. To verify the correctness of the flow fields, the authors could compare the current refined cases with a case using a uniform grid throughout the entire domain.However, whether the streamwise streaks are also related to grid refinements remains uncertain and might require further explanation. Flow fields showing a larger area around the turbine and its wake could help clarify this issue.Further Issues:2. Amplitude Comparison (Lines 142–143 and Figure 5)The "up-and-down" and "side-to-side" cases use double the pitch amplitude compared to the other cases, raising concerns about the fairness of the comparison. This is also critical for the practical implementation of AWM, as the pitch amplitude directly affects the feasibility of applying AWM strategies to real turbines. A stronger initial perturbation likely results in more pronounced wake dynamics while also influencing turbine loads. This discrepancy appears to favor the "up-and-down" and "side-to-side" cases over the helix and pulse strategies. For instance, Figure 12 suggests that the wake differences are more pronounced in these cases. Surprisingly, the "up-and-down" and "side-to-side" cases do not exhibit significantly higher turbine loads, despite the stronger excitation. To ensure a consistent comparison, all cases it would be preferred to maintain the same pitch amplitude, particularly when evaluating turbine performance and structural loads. The authors should further justify their choice of pitch amplitude and, if necessary, provide additional cases with uniform amplitudes for comparison.3. Several quantities are computed using a rotor disk centered around the wake center instead of the turbine center. Since the wake center also displaces in the vertical direction —something a downstream turbine cannot follow— a turbine-centered approach may provide more applicable insights. This would also help relate the results from Section 3.2 to those in Section 3.3. Would the conclusions change if the rotor disk were centered around the turbine?4. The chosen SPOD parameters resolve Strouhal numbers up to St = 0.15. Could the observed noise in the higher eigenvalues in Figure 20 be related to unresolved frequencies beyond this range? A sensitivity analysis on the SPOD parameters, such as the length of the time series and their overlap, would help determine whether adjustments can mitigate noise and furthermore strengthen the paper’s overall contribution.5. Figure 23: Power Improvements at A = 0.5 DegreesThe reported power improvements at the actuated turbine at A = 0.5 degrees require further verification. The authors suggest that this might be a regime where the wake does not yet respond to the actuation, yet the downstream turbine loses power, which indicates that the wake reacts, but in an unexpected and unwanted way. Additional analysis, such as energy entrainment or rotor-averaged velocity through the wake, might help clarify this behavior. If verification is not possible, the authors should consider removing the A = 0.5-degree case from the manuscript.6. The DEL study does not examine the effects on pitch bearings, which have been identified as a limiting factor in AWM application. Including this additional load channel would enhance the paper’s contribution.7. Section 3.3 presents the effect of AWM on the DEL. However, the manuscript does not describe how the DEL is computed. Please add this information to the Methodology or the Appendix, or at least include a reference.8. Consider restructuring the order of the Figures to align more logically with the text’s progression.9. Minor Issues and Clarity Improvements:- Line 34: The term "turbine layout" likely refers to "wind farm layout."- Line 35: The statement about AWM having a "Additionally, the design space for AWM is considerably larger than that of other WFFC strategies. Common implementations rely on at least four relevant design parameters to control..." is unclear. Further elaboration is needed.- Line 207: Planes are sampled at a 1.25m resolution, while the grid resolution is at least in parts coarser (2.5m or more). Is it correct that the planes are sampled at a higher resolution than the grid, and if yes, why?- Line 212: "Each" should be lowercase ("each").- Lines 470 ff: The term "DLC 1.2-like simulations" is not explained. Further clarification is necessary.- Figure 6: What causes the fluctuations in modal blade loads around x = 0.6 r/R, which are absent in the baseline but appear in all AWM cases?- Figure 10: The Greek "kappa" symbol is not properly compiled in the caption.- Figures: Throughout the manuscript, the font size and resolution vary from figure to figure. Ensure that the font size is consistent with the text and that the resolution is sufficient.- Figure 24: The upper limit of the y-axis is significantly higher than the highest bar, creating excessive white space and making it difficult to distinguish between the bars for the upstream and downstream turbines. Adjust the upper limit for clearer presentation of the results.- Reevaluate whether all figures contribute meaningfully to the study (e.g., Figure 2, left panel).- Consider citing the work by Muscari et al. "Physics-Informed DMD for Periodic Dynamic Induction Control of Wind Farms" (DOI: 10.1088/1742-6596/2265/2/022057) in the Introduction.Citation: https://doi.org/
10.5194/wes-2025-14-RC1 -
RC2: 'Comment on wes-2025-14', Anonymous Referee #2, 07 Mar 2025
The manuscript by Yalla et al analyzes different AWM strategies by using SPOD to isolate different modes in the flow at different frequencies related to the AWM excitation frequency. The work has significant scientific value as it furthers our understanding of the flow dynamics associated with AWM. On the other hand, the current manuscript leaves some room for improvement in terms of structure, and some of the results presented require additional analysis or explanation. In short, I feel like this manuscript is a very rough diamond. It should in my opinion be published after the authors perform some serious polishing. I voted to accept subject to minor revisions, because I believe in the value of this manuscript, but please note that some of the changes I feel are necessary are leaning towards “major revisions”.
Major comments:
- You mention in the second line of the introduction that power losses are particularly problematic in stable ABL, most commonly found in offshore wind farms. Yet, in section 2.1, you derive the simulations from measurements from an onshore farm. Please explain or justify why you did not choose to run simulations to match offshore conditions, where AWM would be expected to be most effective.
- Similarly, the choice of wind speeds, wind directions, TI’s, and veer that is studied needs more justification. I can see how these conditions would correspond to a stable ABL, but I don’t see how these specific conditions are necessarily prerequisites for a stable ABL. Is there literature available describing ABL’s in the AWAKEN experiment that supports this choice? Or are there other reasons you have chosen these conditions specifically? Also, please define the definition of wind direction and why these directions are more likely to result in stable ABL’s than other WD’s. Furthermore, please clarify whether the 230 minute dataset used is continuous or a combination of different subsets. Finally, in line 102, you mention that the resulting veer is 9 degrees, which is substantially lower than the threshold you mention earlier. Please explain why this is the case and how this still accurately represents the dataset, as we have seen in recent publications by Brown et al (2025) and Frederik et al (2025) that lower veer can have a large impact on how well different AWM strategies work.
- I agree with Reviewer 1 that not all figures add value to the paper. Consider removing Figure 2, 4 (unless you add the Cp/Ct curves of the turbines in the actual WF for comparison), 7, 21, and 22. Furthermore, I would be interested to also see a comparison between data and simulation in the time and/or frequency domain in Figure 3, not just a comparison of averages.
- I have concerns about the azimuth angle approximation stated on line 151-153. I have seen simulations in similar wind conditions, and especially if you are implementing AWM, I would expect non-negligible changes in rotor speed. Even tiny changes could affect the outcome of your Fourier approximation as you are in practice taking the transformation of a mode/azimuthal wave frequency that is variable over time. I would like to see what effect these variations have on the Fourier transformation, i.e., how well of an approximation Eq.(3) is w.r.t. Eq.(2). In fact, my guess would be that most of the non-periodicity that you describe in line 156 is caused by this fluctuation in rotor speed, not by the non-uniform inflow. Have you checked the periodicity when you don’t use the rotor speed approximation? How wide is your window now? I believe the rotor speed approximation might be reasonable to make in combination with a windowed transform, but would like to see this better studied or more accurately described in this section.
- In Section 3.1, I would be very interested to see an analysis of the eigenvalues of the baseline case for different values of St. The main hypothesis of why St=0.3 is optimal, is that it excites natural modes in the flow. Therefore, I would expect that taking the Fourier transform at different frequencies would result in lower values. Adding an analysis of the magnitude of different wavenumbers as a function of Strouhal numbers (say St=0.1 to 0.6) could reinforce or disprove this hypothesis and thus add significant value to the scientific contribution of this paper.
- I suggest restructuring Section 3.1 and its figures, as it is very hard to follow now. I keep having to scroll through pages to get to the figures that are covered in the text. For example, perhaps you should redo figures 9 and 10 so one of them shows the global EVs for both ranges, and the other the baseline-normalized EVs, as this is also how it’s discussed in the text. Similarly, perhaps put fig 11a and 12 together for better comparison. All the other panels of fig 11 don’t seem to be discussed, but could probably do with some more explanation as these results are probably interesting but not self-explanatory. In general, I would restructure this section so the analysis is grouped in the same order as the figures. That way, the reader no longer needs to scroll back and forth between pages every couple of lines. I would also consider splitting this subsection into multiple (sub)subsections to better separate different points that you are trying to make.
- I question the fidelity of the results presented in Figure 23 and surrounding text. The different pitch amplitude cases raise more questions to me than they answer. First, the upstream power gain at low amplitudes is very peculiar and should be studied. Same goes for the fact that helix and up-down lose downstream power when the amplitude increases. These results make me question the fidelity of the standalone OpenFAST model approach used, as the results do not align with similar studies performed using higher fidelity tools. I think you need to choose to either remove this analysis from the paper, or dive in deeper to explain why these results are different from literature. I recommend doing the former, as this analysis does not align with the main findings of the paper to begin with. Same can be said about Figure 24.
Minor comments:
Throughout: you use “(see Figure X)” almost consistently throughout the paper. This is just a personal preference, but I would like the text to revolve more around the figures, instead of being an afterthought. For example, I would suggest using phrases like “Figure X shows how …”, “As can be seen in Figure Y, …”. Again, just a personal preference, so do with it as you like.
All figures: Not sure how much attention you want to pay on this, but I printed the paper in BW and close to all the figures are very hard to impossible to interpret that way. Consider plotting lines in different styles or using more distinct colors.
Line 54: “phenomenon”, not “phenomena”
Line 56-57: I agree with the first reviewer that Muscari et al, 2022 and/or 2025, and/or Gutknecht et al, 2023 should be cited and discussed here, or in line 61-62. It seems to me that these studies use a slightly different method to achieve a similar goal. It would therefore be worthwhile to discuss the differences between and/or advantages and disadvantages of both methods.
Line 71: Add comma after “2”
Line 92-93: One citation seems incorrect (“b, a”), and all citations here are missing publication year.
Figure 1: Please include labels on the axes. Consider normalizing by the rotor diameter.
Line 106: change gradient to 7.5 \cdot 10^{-4}
Line 111: consider normalizing, at least in x- and y-direction, the domain by the turbine hub (or tower) location.
Line 117: Please define all parameters.
Line 123: This equation is not self-explanatory. Although I can see the value of this definition in light of the SPOD modes later on, I feel like this definition of the blade pitching is far less intuitive than the one used in publications by other groups, that uses the MBC/Coleman transformation. Consider relating this equation to that definition, or at least adding blade number subscripts and explaining what the definition of the clocking angle is.
Line 126: Double use of the word “structure”
Line 128: Consider changing to U_{\text{\infinity}}.
Line 140, Table 1: I do not understand how the clocking angle creates a non-uniform thrust force across the rotor disk, as it is constant. If I understand correctly, the clocking angle is only relevant for the side-to-side and up-and-down strategies, in which case it is used to have the two modes negate each other in horizontal and vertical direction, respectively. For the other strategies, it only changes the phase of the excitation, which is why I do not understand why it is defined at 90 degrees. For the helix and pulse, the pitch angle can equivalently be written as (when \phi_{clock} = 90 degrees):
$\Theta(t) = \Theta_0(t) - A \sin( \omega_e t - \kappa_{\theta} \psi(t) )$
Please clean up this definition.
Line 162: It might be worth mentioning that as per the MBC transformation, the collective component (pulse) would be expected to be 1.5x larger than the vertical and horizontal component (helix), which roughly corresponds to your findings here.
Line 170 onwards: Consider moving the first paragraph of this section to the introduction.
Line 189 (and various other places): WES uses Eq. and Fig., not Equation and Figure as used by the authors throughout the paper.
Line 199: Shouldn’t it be \hat{u}_j(r, \kappa_{\theta}, \omega) instead of \theta?
Figure 7: As mentioned before, this figure might not be necessary: A list/table of cross section locations would probably be clearer to me. Otherwise, the 3D representation does not seem to add much value to the figure and might even diminish clarity. Also, this paper uses normalized distance and centers around the turbine, whereas earlier figures used absolute distance. Please choose one or the other and apply throughout the paper (I would suggest using the one used here).
Line 212: period missing after “5”.
Figures 9 and 10: the panels in these figures are identical, but the subcaptions have very different sizes. Consider making these uniform.
Figure 10, caption: typo, “w” instead of “\” before “kappa”
Line 262: Referencing to future figures does not help the readability of this section. Consider limiting these references to where you do the analyses of these figures.
Line 269: I can see how the swirl due to blade rotation can induce some contribution to the -1 mode, but I would expect it mostly influences this mode at the 1P frequency, not the St=0.3 frequency. Have you investigated this?
Line 270: Similarly, I can see how the veer contributes to the 1 mode, but would love to see a similar analysis in low-veer conditions to confirm this.
Figure 12: consider changing the bins plotted here to match the instance plotted in Figure 11 (for example, up-down is reversed w.r.t. Fig 11).
Line 307: change “as shown below” to “as shown in Fig. 14), as the figure is actually on the following page.
Figure 13: I’m not sure how much value this figure adds. Panel b looks very similar to 11a, as one would expect. Panel a shows that your SPOD is working, but does not really match the overall point you are trying to make in this section. Furthermore, it makes me question: do we not have the same leading blade-rotation defined SPOD modes at 0.1D downstream? I would expect so, but you do not show that.
Line 338: not “as”, but “as opposed to”
Line 345: I’m not sure I agree that the runoff is steep enough that you only need the first 1-2 SPOD modes to accurately represent the flow. If you want to make this claim, you should plot the flow according to the first 2 SPOD modes and compare it to the actual flow.
Line 352: “other wind farm control strategies”
Figure 17: What does the tau in the subcaption of fig a mean?
Figures 16 and 18: consider making the limits of these figures a little smaller, as these figures show a lot of grey area that provide no information now.
Line 405: I would indeed be very interested to see how the CW helix performs here, as you suggest. I would highly recommend adding this case to the paper. If the main contribution of this paper was to find the best AWM strategy for power extraction, it makes complete sense to ignore this case. However, as this paper is more about understanding the aerodynamics of AWM, I think adding this case to the paper would strengthen the findings.
Figure 20: Wouldn’t it make more sense to plot |tau| here to make the figure smoother and more easily interpretable? Even if tau is negative, it still represents an eigenvalue, right?
Figures 21 and 22: if you want to keep these figures in the paper, you should include additional analysis of what you are showing here. It seems to me though that this analysis would not be in line with the main contribution of this paper, and is similar to previous LES studies performed on AWM. Furthermore, if you keep the figures, I would suggest using the same colormap throughout the paper (also applies to Fig 8).
Line 435: Please show some verification (or refer to a paper that shows) that this approach yields reliable results. I understand that it is not feasible to run simulations at each downstream distance and for each case, but you could at least use say two cases to show how well this method estimates the power capture of an LES with 2 turbines. You might also want to add a description of this approach in Section 2.1.
Figure 23: consider adding in some way the results from Figure 17 in Figure 23 to make it easier for the reader to verify how well the rotor averaged velocity predicts downstream power. For this purpose, you might need to plot just the downstream power separately. Also, there seems to be a very thin grey line around this figure, is that done on purpose?
Figure 24: If you do keep this figure, it makes no sense to me to unify the y-axes like you did here. The top 4 panels now provide close to no information as the differences are indistinguishably small. But similar to my main comment about Figure 23, I do not think this analysis adds much value to the paper and perhaps should be cut altogether.
Citation: https://doi.org/10.5194/wes-2025-14-RC2
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