the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 28 Oct 2024
An analytical formulation for turbulent kinetic energy added by wind turbines based on large-eddy simulation
Abstract. Wind turbine wakes are plume-like regions characterized by reduced wind speed and enhanced turbulent kinetic energy (TKE) that form downstream of wind turbines. Numerical mesoscale models, like the Weather Research and Forecast (WRF) model, are generally effective at reproducing the wind speed deficit, but lack skills at simulating the TKE added by wind turbines. Here we propose an analytical formulation for added TKE by a wind turbine that reproduces, via least-square error parameter fitting, the main features of the three-dimensional structure of added TKE as simulated in previous large-eddy simulation (LES) studies, including: a streamwise peak at x = 4–6D (where D is the turbine diameter), a vertical peak near the upper rotor region, and an annular Gaussian-like distribution along the rotor edge. Validation of the proposed formulation against independent LES results and wind tunnel observations from the literature indicates a promising performance in the case of a single wind turbine wake. The ultimate goal is to insert the proposed formulation, after further improvements, in the WRF model for use within existing or new wind farm parameterizations.
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RC1: 'Comment on wes-2024-128', Anonymous Referee #1, 15 Nov 2024
The paper presents a new analytical formulation for turbulent kinetic energy in the wake of a single wind turbine. The model proposes a detailed three-dimensional description of the tke field, intended to remain valid in both the near and far wake regions. In total, 15 parameters are introduced and calibrated against the results of large-eddy simulations through a two-step least-squares method. The paper is clear, easy to read, and has the potential to contribute to the improvement of the latest wake models.
Major comments
- Line 189, 203: For both Weibull-like laws, the shape parameters are set before starting the fitting process. The authors should motivate the choice of the values kA=2 and kW=4, and clarify why those parameters are not fitted using the two-step least-squares method.
- Line 274: The authors claim that kr is independent of CT. However, Figure 2b shows differences up to 20% between the value of kr at CT=0.4 and at CT=0.9. It would perhaps be interesting to consider an expression other than CT^b for the fitting.
- Figure 2: Because the form of Eq. 15 cannot capture the stability conditions, the differences between the “direct fit” values and that of the functional relationships are often large in the stable and unstable cases. For consistency, shouldn’t only neutral conditions be used for calibration?
- Line 305: The authors should clarify what they mean by “entire wake regions” and specify the limits of the regions along y and z as well, as this will influence the value of the RMSE.
Minor comments
- Line 6: The notation x = 4 – 6D appears a bit confusing. It might be worth considering an alternative notation.
- P2: The introduction is rich and well-documented. However, the relevance of the section between lines 28 and 46 is questionable in the scope of this work as it addresses velocity deficit models.
- Line 76: “We note that also Eq. 1 and 2 can be reduced to the same form”. This is not true for CT in the case of Eq. 2. The authors should maybe re-phrase this sentence for consistency.
- Line 133: Typo “u, v, andw”
- Line 160: Different definitions of $\Delta TI$ exist in the literature. It is worth clarifying which one is used and its connection to $\Delta TKE$.
- Line 189: The expression selected for A(x) is very similar to the one proposed by T Delvaux et al2024 Phys.: Conf. Ser. 2767 092089. The authors could consider providing additional reference for it.
- Line 194: Missing space after the bracket.
- Line 277, 279: Missing space before the bracket.
- Line 351: Typo “citeWuetal2023”
- 3.2: The proposed model appears to outperform the model of Ishara and Qian (2018) in most of the validation cases. The comparison could be further enriched with the 3D model of Tian et al. (2022).
Citation: https://doi.org/10.5194/wes-2024-128-RC1 -
RC2: 'Comment on wes-2024-128', Anonymous Referee #2, 06 Jan 2025
The manuscript under review introduces a new empirical formulation to better capture the variation of turbulence intensity (TI) in the wake of a wind turbine. The formulation builds on the work of Ishihara and Qian (2018) (IQ2018), which is noted to overpredict turbulent kinetic energy (TKE) at the top-tip level and introduce an artificial peak near the low-tip. The proposed model employs a two-step fitting process, using a Weibull distribution in both axial and vertical directions, and initially involves 15 empirical parameters, later reduced to 13. These coefficients are calibrated using LES datasets from the literature, and the model's performance is validated against four additional independent datasets. The results indicate that the new model generally provides accurate predictions.
The paper is well-written and easy to follow, though it could benefit from greater conciseness (specific suggestions follow). While the authors rightly highlight the need for improved TKE models—since most existing engineering models focus on velocity deficit prediction—the model’s reliance on 13 tuning parameters may hinder its broader applicability for diverse case studies. The novelty of this work could be enhanced either by incorporating more advanced ML data-driven models or perhaps adopting a more rigorous physics-based approach. While the authors' efforts in accurate predictions of TKE are appreciated, the current formulation appears more like a straightforward curve fitting to existing datasets rather than a highly novel contribution.
More detailed comments can be found below:
- Title: The use of “analytical” in the title may be misleading. While “analytical” is sometimes used in the literature to broadly describe engineering models, it typically refers to models derived directly from flow governing equations. Given the empirical nature of the developed model, I recommend replacing “analytical” with “engineering” or “empirical” in the title and throughout the text where applicable to avoid confusion.
- Introduction: The discussion on available engineering velocity-deficit models and WRF modelling, while detailed, is not central to this work. These sections could be shortened and presented more concisely to maintain focus.
- Please clarify how the added TKE and added TI are computed in this work. Especially, this is not trivial when sigma_u in the wake is less than the one in the incoming flow and thus the added TI becomes negative.
- Please improve the quality of all figures, especially figure 2 where Greek symbols are written in Latin letters. This should be avoided and the authors are expected to use proper typesetting to generate figures.
- Line 65: The phrase “… flow has less energy” could be misleading, as the far wake typically exhibits more kinetic energy than the near wake due to wake recovery.
- Line 100: The statement, “Notably, Wu et al. (2023) conducted LES that included the effect of atmospheric stability to show that the wind speed deficit behaves differently from the ∆TKE and that the two are not co-located in the wake region,” is somewhat vague. Could you clarify what is meant by “wind speed deficit behaves differently from the ∆TKE”? For instance, does this refer to differences in spatial distribution, magnitude, or temporal evolution? A more precise paraphrasing would enhance clarity.
- Line 125: The statement, “the WRF will add some TKE on its own due to the resolved vertical shear,” is unclear. If this detail is not crucial to the discussion or the importance of accurate TKE prediction, consider removing it. Alternatively, if it is essential, please clarify how WRF contributes to TKE through resolved vertical shear and its relevance to the context.
- Line 133: Space is missing in “u,v, andw” . Also “and” should not be written in math mode.
- Line 139: The phrase “… is the mean wind speed” should likely be “the incoming wind speed”.
- This recently-published paper (Modelling turbulence in axisymmetric wakes: an application to wind turbine wakes, 2024) could be relevant to the discussion provided in this paper, so the authors may want to include it in their literature review.
- Line 141: “Typically the largest one is σu, followed by σv (approximately 0.75σu in neutral conditions) and then by σw (approximately 0.52σu in neutral conditions) (Arya, 2001).” This needs to be moved to the next paragraph after discussing that x is aligned with streamwise direction in this study. Otherwise, this statement is incorrect based on west-east definition for coordinates mentioned initially.
- Line 271: “The implication is that the magnitude of added TKE in the wake of a wind turbine is essentially independent of atmospheric properties (such as turbulence intensity or stability)” . Is that consistent with previous works? Normally, it is expected to observe a negative correlation between the added TKE and the ambient TKE as suggested in Crespo’s work and also observed in some numerical and experimental studies.
- Figure 2: For some cases such as WRFLES-S or WRFLES-N, there are only one dataset shown in figures. Fitting a line to only one data point sounds tricky. Can you please clarify this?
- In several places including but not limited to line 277, there is no space between the text and the parentheses including the citation.
- Line 297: “Once again, it is physically correct that a more turbulent atmosphere causes a rising of the location of the added TKE peak.” Can you please explain why is that?
- Line 303: repeated citation!
- Figures 3-5: I'm not sure if contours shown in these figures are really necessary and add value to the paper as both vertical and lateral profiles are provided later. I suggest removing these figures for brevity.
- All figures especially Figs 6-10 seem to be inconsistent with the main text in terms of font size. They are also too big. Please consider making them smaller and grouping them to make the paper more concise.
- Line 393: “is associated with the reduction in vertical wind shear due to the wind speed deficit, is not reproduced with the proposed fit because it is not accounted for in its equations”. The reduced TKE level at lower heights have been reported in several studies, but the current model does not capture that. Please comment on how it can be included in the empirical formulation and how important it is to be modelled.
Citation: https://doi.org/10.5194/wes-2024-128-RC2 - RC3: 'Comment on wes-2024-128', Anonymous Referee #3, 11 Jan 2025
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