the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Implementation of a NearWake Region within the CurledWake Model
Paul Hulsman
Luis A. MartínezTossas
Nicholas Hamilton
Martin Kühn
Abstract. Modelling the nearwake region becomes more important as turbines are positioned with a relatively smaller spacing due to site restrictions, leading to significant power losses and increased fatigue loading. These effects can be mitigated by actively steering the wake away from the downstream turbine. This paper presents an approach to analytically estimate the wake deficit within the nearwake region by modifying the curled wake model. This is done by incorporating a new initial condition at the rotor using an azimuthdependent Gaussian profile, an adjusted turbulence model in the nearwake region and the farwake region and an iterative process to determine the velocity field, while considering the relation of the pressure gradient and accounting the conservation of mass. Comparison with highfidelity simulations for a single turbine case shows a good correlation of the wake profile for both a nonmisaligned and a misaligned case. Validation is performed using field lidar data, where the wake is captured within the nearwake region. The model shows a good correlation with the measurement data. The performance of the modified curled wake model is further analysed within a fiveturbine array, where the determined power output shows a significant improvement in comparison to other existing models. The implemented modification indicates a better representation of the nearwake region and will improve the calculation of the optimum misalignment angles for closely spaced turbines. This will aid the process of developing more accurate controloriented wake steering models.
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Paul Hulsman et al.
Status: closed

RC1: 'Comment on wes2023112', Koen Devesse, 17 Oct 2023
The manuscript aims to develop a nearwake model for yawed turbines. To this end, they extend the curledwake model from MartínezTossas et al. (2019). They do this by improving the initial velocity field at the rotor, improving the turbulence parametrization, and including the pressure gradients in the near wake. The new model is extensively validated using both highfidelity CFD results and field measurements, and is compared to the original curledwake model and the Gaussian wake model (Bastankhah and PortéAgel, 2014). The authors find good agreement between their model and the validation data, indicating that the model performs well. This would make it a valuable tool for the windenergy community. However, the current manuscript is flawed, mainly in regards to the thoroughness of the model’s derivation, and the presentation and analysis of the validation campaign. The main comments and questions are listed below.
 Introduction: There is no clear overview of the important flow phenomena in the near wake for yawed turbines. Lines 2835 mention some, but this is mixed with an overview of the methods developed in literature. Further down, lines 4655 discuss earlier measurement campaigns, but it’s not clear how this relates to this work. The paper would benefit substantially from separating the overviews of the flow phenomena and the existing models, and discussing each of them more thoroughly.
 Section 2: The original model of MartínezTossas et al. (2019) is not clearly explained. Equation 1 shows the nonlinear NavierStokes equation, which is not used by the model in this form, and lines 7376 mention some of the assumptions made in the further derivation. This is not sufficient, as the remainder of section 2 further develops this model, without a comprehensive description of it ever being given. At minimum, the authors should provide the main equations of the model.
 Section 2: Equations are not shown as they are introduced, and are often presented a paragraph later. This makes the derivations very hard to follow.
 Section 2.1.1: I have two major comments concerning the momentum balance:
 The description of this derivation is not thorough enough. The authors only cite a Danish textbook from 1968 that is not easily accessible online. Additionally, they mention the pressure term as being “second on the right” in equation 6, which it isn’t. This would be a minor comment, but it worsens an already confusing section. An explanatory sketch of the used control volume could improve this.
 It does not seem valid to assume that the pressure is equal right before and right after the rotor. Even in simplified Betz theory, the thrust force of a turbine is typically balanced by a pressure drop across the rotor disk (Burton et al., Wind Energy Handbook, 2001). Instead, the authors assume that the thrust force is entirely balanced by a deceleration of the flow, which seems to discontinuously go from an unperturbed inflow to the doubleGaussian profile right behind the disk. Presumably, this results in a similar momentum deficit as the pressure drop, as the wake model is momentumconserving. However, if this is the approach taken by the authors, it should be made clear and argued for within the text. As it stands now, the presented derivation seems incorrect. This is my most important comment, and the assumption of constant pressure should be removed for the manuscript to be accepted.
 Section 2.1.3: Multiple issues concerning this section are listed:
 Why is an iterative procedure is needed to include the pressure? Conservation of mass and momentum should result a closed system of equation for incompressible flow, and be equivalent to using the Bernoulli equation.
 The model seems to consist of the momentum equation for u (equation 11), the continuity equation (equation 12), the (simplified) Bernoulli equation, and a weighting function (equation 13). This results in 6 variables (u, v, w, u_conservation, u_momentum, and p) and 4 equations. How is this system closed?
 Line 168169: How do the imposed vortices and made assumptions prohibit the divergence reaching zero? Which assumptions specifically cause issues, and how?
 Line 169: Which analytical model is meant?
 Equation 13: u_momentum and u_conservation are never defined in another equation. Based on figure 1, it can be inferred that they are the results of solving equations 11 and 12, respectively. This should be made clear in the text.
 Figure 2 provides a good overview of the relevant physical phenomena that are included in this work. I suggest moving it to the beginning of the section, or even to the introduction, where this is more necessary.
 Section 2.2: How were the atmospheric conditions chosen?
 Section 2.2: The spacing of 2.7D is very tight. This is not necessarily an issue, as the authors want to investigate the near wake, but a comment on this in the paper would be appreciated.
 Line 229231: If the assumption that the lateral and vertical velocity components are negligible is not valid, how is this addressed in the calculation?
 Section 3: Throughout the section, the new model is compared against the Gaussian wake model (Bastankhah and PortéAgel, 2014). However, it is wellknown that this model is not valid in the near wake, and newer models have addressed this. The authors use one of these, the doubleGaussian model (Keane et al., 2016), in their improvement of the curledwake model. Therefore, this section should compare the new model against this doubleGaussian model, instead of against the standard Gaussian model.
 Section 3: Is it necessary to discuss this many cases without turbine yaw? While the developed model is valid for these cases, it seems that the increased computational cost compared to the doubleGaussian wake model might make it unsuitable for analyzing them. This is also relates to the lack of comparison to a conventional nearwake model, as mentioned in the previous comment.
 Section 3: Could you comment on how important each of the newly included effects are to the improved performance of the model?
 Section 3.1: Could you comment on the resulting computation cost?
 Figure 6: From x/D≅2.5 onward, the lines drop to zero, resulting in half of the figure being empty. Additionally, using a logscale for the vertical axis seems more appropriate for a convergence plot.
 Section 4: Which superposition method is used for the wake model?
 Lines 414422: Why does the turbine power increase in the second half of the array?
Technical issues and comments:
 Line 19: Barthelmie et al. (2007) should be in brackets. Similar errors are present throughout the manuscript.
 Line 29: The citation is a preprint, not the final paper.
 Line 142143: The sentence states that it is applicable to the nearwake region twice. Similar errors are present throughout the manuscript.
 Section 2.1.3: Δp/Δx and dp/dx are used interchangeably (eg. line 166 and figure 1).
 Line 446: The wake deficit … underestimates the wake deficit.
Citation: https://doi.org/10.5194/wes2023112RC1 
RC2: 'Comment on wes2023112', Anonymous Referee #2, 19 Oct 2023
A novel extension to the curledwake model is proposed that aims to improve predictions of wake deficits in the near wake of wind turbines operating under yaw misalignment. This model is calibrated using data from largeeddy simulation and validated using data from a field experiment. This work thus aims to make a contribution to wake steering control for closelyspaced wind farms by enabling better estimates of wake deficits in the near wake. However, the presentation of the study makes the derivations hard to follow and the validation with simulation data and field measurements appear insufficiently quantified to support the conclusions.
General comments
The introduction omits a complete explanation of wake steering and yaw misalignment, which do need to be introduced in some sense for the presented work to make sense.
The presentation of the theoretical models is quite confusing and hard to follow. The decoupling of equations from their introduction leaves a reader searching for the relevant information. For example, line 84 introduces a function C(psi,theta) which is only defined a page later by line 107. Integrating equations in the paragraphs would make the derivations easier to follow.
The description of the setup for the simulation study and field experiment are thorough and seemingly complete. The validation and comparison against other models appears to indicate the added value of the novel model that has been developed, but are limited in the measures and illustrations used to convey its correspondence.
The hub height velocity deficits show how the novel model corresponds better with field data than the original curledwake model. Wakes from turbines operating under yaw misalignment are, however, poorly represented in a hub height slice of the flow. Could the correspondence between model and measurements be quantified in a way that accounts for the threedimensional structure of the wake? The contour plots provide some threedimensional analysis, but without quantification in a relevant measure, these could be interpreted both as reasonable or as poor correspondence to the reference simulations.
The analysis appears limited by its qualitative nature. The authors claim the results `correlate very well' without quantification of this correspondence with the data. Looking at Figure 11, it is unclear which model performs better from this qualitative comparison. Unless the results are quantified in some sense, the Gaussian wake model may be said to perform as well as the others? The comparison of the models with the data remains qualitative throughout the manuscript. The manuscript could be improved if some measure of error were to be introduced to quantify the accuracy of the different models.
The validation against field data considers only yawaligned operation. How does the model fare for predicting wakes from yawmisaligned operation, its intended use case? Next to that, the measurements are also really close to the rotor at 0.5 D and 1 D downstream. No downstream turbine is going to be positioned at <1 D. This is a controloriented model  how does it predict further downstream where a (closelyspaced) rotor might actually be positioned?
Given that the model is oriented for control optimisation, it would be valuable to see what the effect of the model extension is on power estimates in the wake. Perhaps the authors could quantify what the improvement in accuracy is in terms of available power for a downstream turbine? This could complement the hub height velocity deficits with a more relevant measure for the intended purpose of control optimisation for wake steering.
Figure 12 does show power estimates for a row of wind turbines. However, it remains unclear to what extent the reference models have been tuned, which leaves open the question whether the comparison is fair?
Specific comments
 In terms of language, the manuscript could use refinement, as there are quite some errors and imprecise indications of model performance.
 The citation style is author (year) throughout the paper, whereas often `(author, year)' would be more appropriate.
 The derivations are hard to follow with the equations at the end of paragraphs, please integrate them in the text where they are introduced.
 Additionally, please label equation elements instead of referring to, for example, `third on the right'.
 Parameter introductions may be clarified by using tables, such as for the PALM parameters starting at line 199.
 The line plots work poorly in grayscale and need refinement to be suitable for colourblind readers.
Below is a (nonexhaustive) list of smaller refinements.
 line 7. `accounting the conservation' > accounting for the conservation'
 line 9.  the abstract mentions a good `correlation', whereas no correlation measures are provided. What is good?
 line 28. `countervortex pair', suggest changing to counterrotating vortex pair'
 line 74. `a marching problem' please specify space or time iteration
 line 81. the variables r, theta are not introduced
 line 96. `During a yaw misaligmnent'  please rephrase, a yaw misalignment is not an event
 line 101. `$TSR$'  this is not three variables multiplied, either format as \mathrm{} or assign a symbol
 line 127. The process terminates until' > terminates when' or `runs until'
 line 153. What measure is used to determine the threshold for `acceptable predictions'?
 line 175. What tolerance is used to get a divergence `closest to zero'?
 line 186. `farwale' > farwake'
 line 194. `are existing' > exist'
 Figure 4. The quivers do not have clear arrows indicating direction and the colorbar is missing label. (same goes for Figures 8 and 9)
 Figure 5. This might be a good place to use a divergin colormap centred on zero. The black contour levels distract from the result in Figure 5(a).
 Figure 6. Why does the horizontal axis go beyond x/D=3 if there is no line to show?
 line 375. `more prevalent' > do you mean more important? Please rephrase
 Figure 11. mode l(red)' > model (red)'
 line 415 `corresponds very well' > please make specific
 line 444. correlate very well
Citation: https://doi.org/10.5194/wes2023112RC2
Status: closed

RC1: 'Comment on wes2023112', Koen Devesse, 17 Oct 2023
The manuscript aims to develop a nearwake model for yawed turbines. To this end, they extend the curledwake model from MartínezTossas et al. (2019). They do this by improving the initial velocity field at the rotor, improving the turbulence parametrization, and including the pressure gradients in the near wake. The new model is extensively validated using both highfidelity CFD results and field measurements, and is compared to the original curledwake model and the Gaussian wake model (Bastankhah and PortéAgel, 2014). The authors find good agreement between their model and the validation data, indicating that the model performs well. This would make it a valuable tool for the windenergy community. However, the current manuscript is flawed, mainly in regards to the thoroughness of the model’s derivation, and the presentation and analysis of the validation campaign. The main comments and questions are listed below.
 Introduction: There is no clear overview of the important flow phenomena in the near wake for yawed turbines. Lines 2835 mention some, but this is mixed with an overview of the methods developed in literature. Further down, lines 4655 discuss earlier measurement campaigns, but it’s not clear how this relates to this work. The paper would benefit substantially from separating the overviews of the flow phenomena and the existing models, and discussing each of them more thoroughly.
 Section 2: The original model of MartínezTossas et al. (2019) is not clearly explained. Equation 1 shows the nonlinear NavierStokes equation, which is not used by the model in this form, and lines 7376 mention some of the assumptions made in the further derivation. This is not sufficient, as the remainder of section 2 further develops this model, without a comprehensive description of it ever being given. At minimum, the authors should provide the main equations of the model.
 Section 2: Equations are not shown as they are introduced, and are often presented a paragraph later. This makes the derivations very hard to follow.
 Section 2.1.1: I have two major comments concerning the momentum balance:
 The description of this derivation is not thorough enough. The authors only cite a Danish textbook from 1968 that is not easily accessible online. Additionally, they mention the pressure term as being “second on the right” in equation 6, which it isn’t. This would be a minor comment, but it worsens an already confusing section. An explanatory sketch of the used control volume could improve this.
 It does not seem valid to assume that the pressure is equal right before and right after the rotor. Even in simplified Betz theory, the thrust force of a turbine is typically balanced by a pressure drop across the rotor disk (Burton et al., Wind Energy Handbook, 2001). Instead, the authors assume that the thrust force is entirely balanced by a deceleration of the flow, which seems to discontinuously go from an unperturbed inflow to the doubleGaussian profile right behind the disk. Presumably, this results in a similar momentum deficit as the pressure drop, as the wake model is momentumconserving. However, if this is the approach taken by the authors, it should be made clear and argued for within the text. As it stands now, the presented derivation seems incorrect. This is my most important comment, and the assumption of constant pressure should be removed for the manuscript to be accepted.
 Section 2.1.3: Multiple issues concerning this section are listed:
 Why is an iterative procedure is needed to include the pressure? Conservation of mass and momentum should result a closed system of equation for incompressible flow, and be equivalent to using the Bernoulli equation.
 The model seems to consist of the momentum equation for u (equation 11), the continuity equation (equation 12), the (simplified) Bernoulli equation, and a weighting function (equation 13). This results in 6 variables (u, v, w, u_conservation, u_momentum, and p) and 4 equations. How is this system closed?
 Line 168169: How do the imposed vortices and made assumptions prohibit the divergence reaching zero? Which assumptions specifically cause issues, and how?
 Line 169: Which analytical model is meant?
 Equation 13: u_momentum and u_conservation are never defined in another equation. Based on figure 1, it can be inferred that they are the results of solving equations 11 and 12, respectively. This should be made clear in the text.
 Figure 2 provides a good overview of the relevant physical phenomena that are included in this work. I suggest moving it to the beginning of the section, or even to the introduction, where this is more necessary.
 Section 2.2: How were the atmospheric conditions chosen?
 Section 2.2: The spacing of 2.7D is very tight. This is not necessarily an issue, as the authors want to investigate the near wake, but a comment on this in the paper would be appreciated.
 Line 229231: If the assumption that the lateral and vertical velocity components are negligible is not valid, how is this addressed in the calculation?
 Section 3: Throughout the section, the new model is compared against the Gaussian wake model (Bastankhah and PortéAgel, 2014). However, it is wellknown that this model is not valid in the near wake, and newer models have addressed this. The authors use one of these, the doubleGaussian model (Keane et al., 2016), in their improvement of the curledwake model. Therefore, this section should compare the new model against this doubleGaussian model, instead of against the standard Gaussian model.
 Section 3: Is it necessary to discuss this many cases without turbine yaw? While the developed model is valid for these cases, it seems that the increased computational cost compared to the doubleGaussian wake model might make it unsuitable for analyzing them. This is also relates to the lack of comparison to a conventional nearwake model, as mentioned in the previous comment.
 Section 3: Could you comment on how important each of the newly included effects are to the improved performance of the model?
 Section 3.1: Could you comment on the resulting computation cost?
 Figure 6: From x/D≅2.5 onward, the lines drop to zero, resulting in half of the figure being empty. Additionally, using a logscale for the vertical axis seems more appropriate for a convergence plot.
 Section 4: Which superposition method is used for the wake model?
 Lines 414422: Why does the turbine power increase in the second half of the array?
Technical issues and comments:
 Line 19: Barthelmie et al. (2007) should be in brackets. Similar errors are present throughout the manuscript.
 Line 29: The citation is a preprint, not the final paper.
 Line 142143: The sentence states that it is applicable to the nearwake region twice. Similar errors are present throughout the manuscript.
 Section 2.1.3: Δp/Δx and dp/dx are used interchangeably (eg. line 166 and figure 1).
 Line 446: The wake deficit … underestimates the wake deficit.
Citation: https://doi.org/10.5194/wes2023112RC1 
RC2: 'Comment on wes2023112', Anonymous Referee #2, 19 Oct 2023
A novel extension to the curledwake model is proposed that aims to improve predictions of wake deficits in the near wake of wind turbines operating under yaw misalignment. This model is calibrated using data from largeeddy simulation and validated using data from a field experiment. This work thus aims to make a contribution to wake steering control for closelyspaced wind farms by enabling better estimates of wake deficits in the near wake. However, the presentation of the study makes the derivations hard to follow and the validation with simulation data and field measurements appear insufficiently quantified to support the conclusions.
General comments
The introduction omits a complete explanation of wake steering and yaw misalignment, which do need to be introduced in some sense for the presented work to make sense.
The presentation of the theoretical models is quite confusing and hard to follow. The decoupling of equations from their introduction leaves a reader searching for the relevant information. For example, line 84 introduces a function C(psi,theta) which is only defined a page later by line 107. Integrating equations in the paragraphs would make the derivations easier to follow.
The description of the setup for the simulation study and field experiment are thorough and seemingly complete. The validation and comparison against other models appears to indicate the added value of the novel model that has been developed, but are limited in the measures and illustrations used to convey its correspondence.
The hub height velocity deficits show how the novel model corresponds better with field data than the original curledwake model. Wakes from turbines operating under yaw misalignment are, however, poorly represented in a hub height slice of the flow. Could the correspondence between model and measurements be quantified in a way that accounts for the threedimensional structure of the wake? The contour plots provide some threedimensional analysis, but without quantification in a relevant measure, these could be interpreted both as reasonable or as poor correspondence to the reference simulations.
The analysis appears limited by its qualitative nature. The authors claim the results `correlate very well' without quantification of this correspondence with the data. Looking at Figure 11, it is unclear which model performs better from this qualitative comparison. Unless the results are quantified in some sense, the Gaussian wake model may be said to perform as well as the others? The comparison of the models with the data remains qualitative throughout the manuscript. The manuscript could be improved if some measure of error were to be introduced to quantify the accuracy of the different models.
The validation against field data considers only yawaligned operation. How does the model fare for predicting wakes from yawmisaligned operation, its intended use case? Next to that, the measurements are also really close to the rotor at 0.5 D and 1 D downstream. No downstream turbine is going to be positioned at <1 D. This is a controloriented model  how does it predict further downstream where a (closelyspaced) rotor might actually be positioned?
Given that the model is oriented for control optimisation, it would be valuable to see what the effect of the model extension is on power estimates in the wake. Perhaps the authors could quantify what the improvement in accuracy is in terms of available power for a downstream turbine? This could complement the hub height velocity deficits with a more relevant measure for the intended purpose of control optimisation for wake steering.
Figure 12 does show power estimates for a row of wind turbines. However, it remains unclear to what extent the reference models have been tuned, which leaves open the question whether the comparison is fair?
Specific comments
 In terms of language, the manuscript could use refinement, as there are quite some errors and imprecise indications of model performance.
 The citation style is author (year) throughout the paper, whereas often `(author, year)' would be more appropriate.
 The derivations are hard to follow with the equations at the end of paragraphs, please integrate them in the text where they are introduced.
 Additionally, please label equation elements instead of referring to, for example, `third on the right'.
 Parameter introductions may be clarified by using tables, such as for the PALM parameters starting at line 199.
 The line plots work poorly in grayscale and need refinement to be suitable for colourblind readers.
Below is a (nonexhaustive) list of smaller refinements.
 line 7. `accounting the conservation' > accounting for the conservation'
 line 9.  the abstract mentions a good `correlation', whereas no correlation measures are provided. What is good?
 line 28. `countervortex pair', suggest changing to counterrotating vortex pair'
 line 74. `a marching problem' please specify space or time iteration
 line 81. the variables r, theta are not introduced
 line 96. `During a yaw misaligmnent'  please rephrase, a yaw misalignment is not an event
 line 101. `$TSR$'  this is not three variables multiplied, either format as \mathrm{} or assign a symbol
 line 127. The process terminates until' > terminates when' or `runs until'
 line 153. What measure is used to determine the threshold for `acceptable predictions'?
 line 175. What tolerance is used to get a divergence `closest to zero'?
 line 186. `farwale' > farwake'
 line 194. `are existing' > exist'
 Figure 4. The quivers do not have clear arrows indicating direction and the colorbar is missing label. (same goes for Figures 8 and 9)
 Figure 5. This might be a good place to use a divergin colormap centred on zero. The black contour levels distract from the result in Figure 5(a).
 Figure 6. Why does the horizontal axis go beyond x/D=3 if there is no line to show?
 line 375. `more prevalent' > do you mean more important? Please rephrase
 Figure 11. mode l(red)' > model (red)'
 line 415 `corresponds very well' > please make specific
 line 444. correlate very well
Citation: https://doi.org/10.5194/wes2023112RC2
Paul Hulsman et al.
Paul Hulsman et al.
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