the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 12 Feb 2025
Effect of blockage on wind turbine power and wake development
Abstract. Recent work by Lanzilao & Meyers (J. Fluid Mech, 2024) has shown that wind-farm blockage introduces an unfavourable pressure gradient in front of the farm and a favourable pressure gradient in the farm, which are strongly correlated with the nonlocal efficiency and wake efficiency respectively. In particular, the favourable pressure gradient in the farm increases the farm wake efficiency, defined as the average farm power normalized by the average front-row power. Here, we investigate the impact of blockage on wake development and power of wind turbines using an idealized large-eddy simulation setup in which blockage conditions are artificially introduced using a rigid-lid, further also using neutral stratification and no wind veer. We simulate both infinite and finite single turbine rows, as well as a setup with two staggered rows. Blockage strength is adjusted by varying the boundary layer height (H) and turbine spacing (S). We find that blockage strongly affects near wake behaviour, altering Froude momentum theory, by introducing a favourable pressure difference (∆pNW) across the turbine row. The same setup also leads to an unfavourable pressure difference (∆pFW) in the far wake, which simply follows from the rigid-lid conditions and the change of momentum flux due to wake recovery. A strong positive correlation was observed of -∆pNW with both power coefficient (CP) and thrust coefficient (CT). Specifically, as S and H decrease, −∆pNW, CP and CT increase. At the same time a lower induction is observed at the rotor disk, and a lower wake deficit in the near wake. The reduction of near wake velocity deficit as a result of blockage also translates into lower deficits and wake widths in the far wake. When scaling the far wake development with initial far wake deficit and width, we do not see a direct effect of the adverse pressure gradient on the wake recovery. However, we do see a profound effect of H on the wake recovery, with higher boundary layers leading to faster recovery. This relates to the fact that, the wake can more freely expand vertically in high-boundary layer cases, into a larger region of high-speed flow than for shallow boundary layers. Finally, we introduce a simplified Froude-momentum balance to parametrize the relation between blockage, pressure gradient and near wake properties, and compare it to the LES results.
Competing interests: At least one of the (co-)authors is a member of the editorial board of Wind Energy Science. The authors have no other competing interests to declare.
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 preprint. The responsibility to include appropriate place names lies with the authors.- Preprint
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RC1: 'Comment on wes-2025-6', Anonymous Referee #1, 11 Mar 2025
Review report on the manuscript entitled “Effect of blockage on wind turbine power and wake development” by Olivier Ndindayino, Augustin Puel, and Johan Meyers submitted to the journal Wind Energy Science
In this work, the authors conducted 17 idealized large-eddy simulations (LES) with different blockage conditions by varying turbine spacing and boundary layer height to investigate the impact of blockage on wake development and power output of wind turbines. Additionally, the authors developed an analytical model based on the classic Froude momentum theory to predict the blockage effect on near-wake characteristics and compared the results obtained from the analytical model with the LES results. The work is well-conducted and of interest to the community. Some minor comments are as follows:
- The causes for the discrepancies shown in Fig. 11 need to be better explained, especially considering that the LES results are already employed for model closure.
- The order of Eqs. 4 and 5 does not match their order in the corresponding text.
- Lines 180–181: “is,er” appears to be a typographical error.
- The abscissa in Fig. 6 has dimensions, while it is nondimensionalized in Fig. 8. Ensure consistency throughout the paper.
- Lines 278–279: “Returning to Fig. 7, we further observe that for constant H, the C(x) and δy(x) curves align closely across varying Svalues, and thus varying adverse pressure strengths.” This is not the case in Fig. 7(f) at far-wake locations.
- Lines 283–284: "Finally for cases with S = 2.5D, (and also S = 5D for the two staggered rows) neighbouring wakes start touching at x ≈ 16D (see Fig. 6a)."Please verify if the authors are referring to Fig. 6a.
Citation: https://doi.org/10.5194/wes-2025-6-RC1 -
RC2: 'Comment on wes-2025-6', Anonymous Referee #2, 13 Mar 2025
In the manuscript, the authors conduct a series of LES calculations to understand blockage in wind farms. The simulation approach uses a slip wall top boundary condition to mimic the capping inversion and varies the boundary layer height and spanwise turbine spacing to change the overall blockage. Both spanwise infinite, spanwise finite, and staggered configurations are considered. In addition, the authors develop an analytical model to quantify blockage affects on turbine/farm performance.
Overall, the manuscript is technically sound, but there are several issues that must be addressed prior to publication:
1. The governing equations are missing the viscous term. Is this neglected in the approach?
2. The approach to provide an inflow profile from a precursor within a code with streamwise periodic boundary conditions seems needless complex. Why not simply use the finite difference method also in the streamwise direction with inflow/outflow boundary conditions? This almost certainly does not affect the simulation results, but it is cumbersome to explain.
3. In the Smagorinsky model, how is the scalar grid spacing computed?
4. Figure 3 is quite misleading in how it is drawn and took me several attempts to interpret it. I interpreted the figure as the x-z plane, which leads to confusion with the text with regards to the area for the different configurations (e.g., A1=A2 for the spanwise infinite farm). The authors should add additional information to this figure to help with interpretability including both the x-z and x-y planes as well as figures specific to spanwise infinite and finite cases.
5. Analysis of the spanwise finite case is rather interesting but underdeveloped. All of the turbines have induction factors considerably larger than the corresponding infinite case. How can this be? This suggests a collective blockage effect of the entire farm. Will a spanwise finite farm ever look infinite due to the collective blockage effect? If a finite farm can look infinite, how large does it need to be? (The implications of this question are rather significant given the number of studies using spanwise periodic farms!)
6. The most significant technical deficiency is the lack of explanation for the overprediction of power and thrust by the analytical model. No commentary is provided to explain the overprediction, which is somewhat puzzling since some of the inputs came directly from the LES. This discrepancy must be explained.
7. Finally, there are a few typographical errors, and some of the results are presented in mixed dimensional and non-dimensional form.
Citation: https://doi.org/10.5194/wes-2025-6-RC2 -
RC3: 'Comment on wes-2025-6', Anonymous Referee #3, 19 Mar 2025
The submitted paper investigates blockage in channel flow large eddy simulations (LES) with infinite (horizontally periodic) and finite wind farms. An extension of momentum theory is developed accounting for blockage, yielding a model that is similar to studies that model blockage ‘corrections.’ The model predictions are compared to the LES data for the infinite farm case where the model is a closed system of equations. The model yields very good agreement compared to the infinite wind farm LES data. But the model is not closed for the finite farm, and measurements from the LES are used to provide the closure, and the output is then compared back to the same LES. The model exhibits much worse performance compared to LES for the finite farm, even though it is provided the pressure from the same LES. Also, blockage appears to affect the initial velocity deficit and the horizontal wake width / wake shape, but not the wake recovery.
Overall, this is a useful and interesting study. I have comments and questions on the setup and assumptions made in the model development that should be further clarified. Further, given the poor agreement of the model in the finite farm case (Figure 11), even when provided the LES pressure data, it seems clear that the model is not adequate in this setting. It would be more helpful if the authors return to the fundamentals of the model derivation to reveal why the model is not accurate in this setting. This can guide improved modeling in this study and in future studies. In other words, the authors next goal is to couple the model from this paper with a separate pressure model for finite wind farms, but it appears that even with the right pressure, the model is not very accurate. More research should be performed in this paper to reveal the cause of the breakdown of the model.
I hope the authors can consider the following comments in a revision.
Point comments:
- Line 36: The sentence starting “Defining [...]” is very long and complex, consider re-phrasing for clarity.
- Line 51: “Looking at a control volume [...]”
I appreciate that the introduction utilizes fundamental arguments to justify the setup.This discussion can be improved, because it is not “directly clear” from the schematics that there should be an unfavorable pressure gradient. Rather, this is clear from an inspection of the conservation equations along with the geometry. - Line 54: “so that outflowing momentum is larger than inflowing momentum”
Why is this guaranteed to be the case for all parameter values? - Eq. 1: The analysis, boundary conditions, and assumptions to arrive at Equation 1 are not clear
- Line 61: How does Equation 1 tell you that |p_nw| >> |p_fw|?
- Line 80: How are the authors able to conclude from Figure 1 that the far wake pressure gradient will be much smaller than the near wake pressure gradient for all parameter values?
- Line 109: The Shapiro correction factor was derived based on Froude momentum theory without blockage, but the present simulations include blockage. This sensitivity should be expanded upon. Does the correction factor affect the results?
- Section 2.2: The authors have made the choice to fix the friction velocity in simulations with different domain heights, rather than fixing the pressure gradient. This choice could be justified in more detail.
- Figure 3: It is unclear to me why the p_side approximation is made in this ‘general setup’ figure. What is the justification for this approximation?
- Line 165: The geometry used in this modeling is non-standard and should be introduced more clearly. In classic axial momentum theory, the streamtube and control volumes are typically cylindrical. Here, the authors set A_1 = S*H (rectangular). The implications of this choice are not described.
- Figure 3 / Line 165: More generally, the choice of the larger control volume following streamlines is non-standard, and requires the authors to make arguments about the side pressure. I’m also having trouble understanding what it means for A_1 to be the “the ‘available’ inflow for the turbine,” equal to spacing S times domain height H, but then for the control volume to expand into A_2 (A_2 > A_1). Thus, for the finite farm, the streamtube for one wind turbine will overlap with the streamtube from another turbine. But streamlines cannot cross each other.
- Line 180: typographical error
- Line 183: The approximation of p_side requires much more justification based on theory and the LES data
- Line 223: Since U_d is rotor averaged, it seems logical that U_in should be the rotor average of the freestream wind from the precursor
- Line 230: In what spatial position is the maximum velocity upstream of the turbine occurring?
- Figure 5: As noted in the discussion, the finite and infinite farms differ in their induction factors, which is not explained by the A/A_d parameter, which only describes blockage on the turbine level. It seems natural to define a farm geometric blockage parameter that is the total cross section of the domain (Ly*H) divided by the total surface area of all turbines (N*pi*D^2/4 with N turbines).
- Section 4.1.2: “Here we observe that a classical Gaussian shape function provides good fits along the downstream direction.”
The fit should be quantitatively evaluated and those quantities should be reported, as in [1, 2]. - Line 271: “As H increases, C(x) remains approximately the same, while \delta_y(x) spreads faster.”
This is an interesting/unexpected result, as we typically think of the wake recovery rate and the wake spreading rate as connected quantities (more comments on this later in regards to interpretation of the results/conclusions).
Perhaps this is because the authors only evaluate horizontal wake width, rather than the three-dimensional wake shape? - Line 274: “In particular for cases H350 is observed that vertical spreading of the wake is limited by the presence of the rigid-lid.”
This especially motivates the need to evaluate the reliability of the Gaussian fitting, as in comment 17. - Line 315: The pressure is calculated at x=4D (what the authors say is the end of the near wake). But does the near wake length depend on the blockage?
- Figure 10 and associated discussion: this result suggests that the radial normal Reynolds stress is relevant to the analysis. But these Reynolds stresses were neglected in the model. Discussion of this would be useful.
- Figure 11, where the model is closed with LES pressure, implies that the model is not adequate for the finite farm (i.e. an assumption/approximation made in the model is not reliable for the finite farm)
- Section 5: Conclusions
“This study set out to analyse the effect of blockage on the wake development behind turbines and the turbine power. Recent research by Lanzilao and Meyers (2024) discovered a strong positive correlation between the favourable pressure gradient in the farm and the wake efficiency ηw. This implies that the favourable pressure gradient, induced by blockage, enhances the wake recovery mechanism.”
This framing of the conclusion is a little inconsistent with the paper to me. Lanzilao and Meyers (2024) may have found that correlation, and that may imply a pressure gradient that enhances wake recovery. Yet this study largely focuses on the effect of the pressure gradient on Cp and Ct of turbines in ‘freestream’ / not in wakes. Therefore, the pressure gradient effect on the wake recovery does not matter for most of the results. The primary result on wakes appears to be Figure 7, where it seems blockage / favorable pressure gradient has a minor impact on wake recovery (C(x)). - Line 367: “[...] and also \Delta p_nw increases towards the centre of the row” -> “[...] and also the magnitude of \Delta p_nw increases towards the centre of the row”
- Line 373: “However, we do see a profound effect of H on the wake recovery, with higher boundary layers leading to faster recovery.”
Where is this result shown? From Figure 7, it appears that C(x) (i.e. wake recovery) does not depend strongly on H. \delta_y may depend on H, but that is only the horizontal wake width, not the wake recovery, which is usually understood in terms of momentum deficit
References
[1] Brugger, Peter, Fernando Carbajo Fuertes, Mohsen Vahidzadeh, Corey D. Markfort, and Fernando Porté-Agel. "Characterization of wind turbine wakes with Nacelle-Mounted Doppler LiDARs and model validation in the presence of wind veer." Remote Sensing 11, no. 19 (2019): 2247.
[2] Dar, Arslan Salim, and Fernando Porté-Agel. "Influence of wind direction on flow over a cliff and its interaction with a wind turbine wake." Physical Review Fluids 9, no. 6 (2024): 064604.
Citation: https://doi.org/10.5194/wes-2025-6-RC3
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