the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
A simple vortex model applied to an idealized rotor in sheared inflow
Mac Gaunaa
Niels Troldborg
Emmanuel Branlard
Abstract. A simple analytical vortex model is presented and used to study an idealized wind turbine rotor in uniform and sheared inflow, respectively. The work reveals that locally 1D momentum theory is valid for a nonuniformly loaded rotor in a sheared inflow. Hence the maximum local power coefficient (expressed with respect to the local, upstream velocity) of an ideal rotor is not affected by the presence of shear. We study the interaction between the wake vorticity generated by the rotor and the wind shear vorticity, and find that their mutual interaction results in no net generation of axial vorticity: the wake effects and the shear effects exactly cancel each other out. This means that there is no resulting crossshear induced velocities and therefore also no crossshear deflection of the wake in this model.
Mac Gaunaa et al.
Status: closed

RC1: 'Comment on wes202294', Joseph Saverin, 05 Dec 2022
General Comments:
This paper presents a simplified vortex model for the analysis of influence on local power coefficient and wake behaviour of a sheared inflow. The paper begins by describing some contradicting results in the literature regarding the influence of sheared inflow on global power generation and wake behaviour as predicted by medium and high order numerical methods. The paper presents a relatively detailed description of an analytical model based on the treatment of the wake as a set of disjoint wake regions separated by infinitely thin vortex sheets. The analysis shows that a number of the results predicted by classical 1D momentum theory can be reproduced within this theory and thus certainly has scientific significance.
It is concluded that the results from 1D momentum theory as used within a BEM approach be applied locally and that global corrections such as average disc loading and induction should be avoided. It is furthermore concluded that, according to the herein presented theory, there should be no crossshear deflection of the wake.
The reviewer finds that the conclusions made regarding model inconsistencies by applying rotoraveraged corrections are correct, and that under the assumptions of the model, no wake deflection should occur. In this sense this simpified theory represents a fundamental contribution to numerical treatments of wind turbine rotor and wake aerodynamics. Emphasis should be placed on the fact that this analysis is valid for an idealized rotor & conditions (for example the statement made in the abstract that locally 1D momentum theory is valid for nonuniformly loaded rotors may be somewhat ambitious). As with classical 1D momentum theory, underlying the simplified vortex model presented here are numerous physical assumptions. The vortex cylinder treament for example assumes a semiinfinite, inviscid vortex sheet. The analogies between the model types are certainly valuable, and as with classical 1D momentum theory, there is a suitable area of application for this theory.
Specific Comments:
 Perhaps more emphasis should be placed on the model assumptions, in order to ensure that the theory is correctly applied in future works.
 Is the wake sheet suggested in Section 3.3 consistent with regards to Helmholtz's theorems?
 Suggest to use a different term than cylindrical to decribe disjoint wake regions. The theory has been applied to cylindrical sections in the literature, however in the work here this is applied to general "extrusions" of areas in the rotor plane.
 There appears to be an averaging across a full revolution required for the derivation of equation (8). Does this contradict the approach that distinct wake regions have distinct convection velocities?
Technical Corrections:
There are a small number of typographical errors in the manuscript which are highlighted in the attached .pdf file.

RC2: 'Comment on wes202294', Eduardo Alvarez, 20 Dec 2022
The content of this paper has scientific relevance to theoretical and modeling advances in the field. I recommend this paper for publication, after minor revisions to help readibility.
Typos
• “recent validation studies[ _]Boorsma”
• “using various free vortex vortex tools”
Comments/suggestions to improve readibility
• The description of Figure 1 is clear in the text, but the diagram shown in the figure itself is confusing. Showing either the rotor disc or the position of the blade in the diagram would make it easier to understand.
• Eq. 2: What is d? Are the dr in numerator and denominator typos? They feel like it. Why are they there? Also, bold Gamma was defined, but what is the nonbold Gamma? The magnitude of bold Gamma? If so, please mention that in the text.
• “Letting â¦ go to infinity…” that would lead to an infinite Mach number along the blade, breaking the assumption of incompressible flow.
• How is Eq. 4 useful? It was introduced without further use at this point.
• “To do this we will be using the following result, which is proven in Appendix B and illustrated in Figure 2:” Is there something missing here or is that colon a typo?
• Swap Appendix A and Appendix B according to the order that they are mentioned in the text.
• “… any change in bound vorticity in the spanwise direction…” Fig 3a refers to it as radial direction. Change either the text or figure caption for consistency.
• Paragraph between Lines 120 and 125: What do you mean by “each side of the vortex sheet”?
• Fig. 7 is difficult to understand. Please add a 3D view (like in Fig 8) to make clear what is meant by the two different strengths.
• "...can readily be extended to a slightly more general case of two regions that both have borders with each other and with the outer flow...": I am confused. I thought that was what is shown in Fig. 7 already. Please make a figure that clarifies what is meant by this, and/or redo Fig. 7.
• "the following expression for the shear sheet condensation": What do you mean by condensation?
• "... determined by integrating the vorticity entering and exiting an infinitesimal cylinder enclosing the junction between the wake border and the shear sheet layer from 0 to z...": Could you please make a figure showing what is meant by this?
• Please add to Section 4 a discussion/summary of the assumption taken to develop this model and its limitations.
• The reader could benefit from adding a few sentences on future work and possible applications/extensions of this model.
Appendix
• “and [d]sθ is an infinitesimal…”
• “Since there for all segments on the cylinder surface is a corresponding segment…” grammatically wrong or at least confusing. Please rephrase.Evaluation
 Does the paper address relevant scientific questions within the scope of WES? Yes
 Does the paper present novel concepts, ideas, tools, or data? Yes
 Is the paper of broad international interest? Yes
 Are clear objectives and/or hypotheses put forward? This study puts forward the intent of developing a simple vortex model to analyse and model nonuniformlyloaded and sheared wakes.
 Are the scientific methods valid and clear outlined to be reproduced? The study could use better diction and figures to help the reader come along, but its mathematical derivations and logic are robust.
 Are analyses and assumptions valid? The analyses and assumptions are clearly stated and seem robust.
 Are the presented results sufficient to support the interpretations and associated discussion? Yes
 Is the discussion relevant and backed up? Yes
 Are accurate conclusions reached based on the presented results and discussion? Yes
 Do the authors give proper credit to related and relevant work and clearly indicate their own original contribution? Yes, to the best of my knowledge.
 Does the title clearly reflect the contents of the paper and is it informative? Yes
 Does the abstract provide a concise and complete summary, including quantitative results? Yes
 Is the overall presentation well structured? Yes
 Is the paper written concisely and to the point? Yes
 Is the language fluent, precise, and grammatically correct? Yes
 Are the figures and tables useful and all necessary? They are all necessary, but I'd recommend adding a few more to help the reader (see notes).
 Are mathematical formulae, symbols, abbreviations, and units correctly defined and used according to the author guidelines? Yes
 Should any parts of the paper (text, formulae, figures, tables) be clarified, reduced, combined, or eliminated? No
 Are the number and quality of references appropriate? It may be valuable to add a paragraph reviewing similar models in the literature.
 Is the amount and quality of supplementary material appropriate and of added value? N/A
Citation: https://doi.org/10.5194/wes202294RC2  AC1: 'Comment on wes202294', Niels Troldborg, 16 Feb 2023
Status: closed

RC1: 'Comment on wes202294', Joseph Saverin, 05 Dec 2022
General Comments:
This paper presents a simplified vortex model for the analysis of influence on local power coefficient and wake behaviour of a sheared inflow. The paper begins by describing some contradicting results in the literature regarding the influence of sheared inflow on global power generation and wake behaviour as predicted by medium and high order numerical methods. The paper presents a relatively detailed description of an analytical model based on the treatment of the wake as a set of disjoint wake regions separated by infinitely thin vortex sheets. The analysis shows that a number of the results predicted by classical 1D momentum theory can be reproduced within this theory and thus certainly has scientific significance.
It is concluded that the results from 1D momentum theory as used within a BEM approach be applied locally and that global corrections such as average disc loading and induction should be avoided. It is furthermore concluded that, according to the herein presented theory, there should be no crossshear deflection of the wake.
The reviewer finds that the conclusions made regarding model inconsistencies by applying rotoraveraged corrections are correct, and that under the assumptions of the model, no wake deflection should occur. In this sense this simpified theory represents a fundamental contribution to numerical treatments of wind turbine rotor and wake aerodynamics. Emphasis should be placed on the fact that this analysis is valid for an idealized rotor & conditions (for example the statement made in the abstract that locally 1D momentum theory is valid for nonuniformly loaded rotors may be somewhat ambitious). As with classical 1D momentum theory, underlying the simplified vortex model presented here are numerous physical assumptions. The vortex cylinder treament for example assumes a semiinfinite, inviscid vortex sheet. The analogies between the model types are certainly valuable, and as with classical 1D momentum theory, there is a suitable area of application for this theory.
Specific Comments:
 Perhaps more emphasis should be placed on the model assumptions, in order to ensure that the theory is correctly applied in future works.
 Is the wake sheet suggested in Section 3.3 consistent with regards to Helmholtz's theorems?
 Suggest to use a different term than cylindrical to decribe disjoint wake regions. The theory has been applied to cylindrical sections in the literature, however in the work here this is applied to general "extrusions" of areas in the rotor plane.
 There appears to be an averaging across a full revolution required for the derivation of equation (8). Does this contradict the approach that distinct wake regions have distinct convection velocities?
Technical Corrections:
There are a small number of typographical errors in the manuscript which are highlighted in the attached .pdf file.

RC2: 'Comment on wes202294', Eduardo Alvarez, 20 Dec 2022
The content of this paper has scientific relevance to theoretical and modeling advances in the field. I recommend this paper for publication, after minor revisions to help readibility.
Typos
• “recent validation studies[ _]Boorsma”
• “using various free vortex vortex tools”
Comments/suggestions to improve readibility
• The description of Figure 1 is clear in the text, but the diagram shown in the figure itself is confusing. Showing either the rotor disc or the position of the blade in the diagram would make it easier to understand.
• Eq. 2: What is d? Are the dr in numerator and denominator typos? They feel like it. Why are they there? Also, bold Gamma was defined, but what is the nonbold Gamma? The magnitude of bold Gamma? If so, please mention that in the text.
• “Letting â¦ go to infinity…” that would lead to an infinite Mach number along the blade, breaking the assumption of incompressible flow.
• How is Eq. 4 useful? It was introduced without further use at this point.
• “To do this we will be using the following result, which is proven in Appendix B and illustrated in Figure 2:” Is there something missing here or is that colon a typo?
• Swap Appendix A and Appendix B according to the order that they are mentioned in the text.
• “… any change in bound vorticity in the spanwise direction…” Fig 3a refers to it as radial direction. Change either the text or figure caption for consistency.
• Paragraph between Lines 120 and 125: What do you mean by “each side of the vortex sheet”?
• Fig. 7 is difficult to understand. Please add a 3D view (like in Fig 8) to make clear what is meant by the two different strengths.
• "...can readily be extended to a slightly more general case of two regions that both have borders with each other and with the outer flow...": I am confused. I thought that was what is shown in Fig. 7 already. Please make a figure that clarifies what is meant by this, and/or redo Fig. 7.
• "the following expression for the shear sheet condensation": What do you mean by condensation?
• "... determined by integrating the vorticity entering and exiting an infinitesimal cylinder enclosing the junction between the wake border and the shear sheet layer from 0 to z...": Could you please make a figure showing what is meant by this?
• Please add to Section 4 a discussion/summary of the assumption taken to develop this model and its limitations.
• The reader could benefit from adding a few sentences on future work and possible applications/extensions of this model.
Appendix
• “and [d]sθ is an infinitesimal…”
• “Since there for all segments on the cylinder surface is a corresponding segment…” grammatically wrong or at least confusing. Please rephrase.Evaluation
 Does the paper address relevant scientific questions within the scope of WES? Yes
 Does the paper present novel concepts, ideas, tools, or data? Yes
 Is the paper of broad international interest? Yes
 Are clear objectives and/or hypotheses put forward? This study puts forward the intent of developing a simple vortex model to analyse and model nonuniformlyloaded and sheared wakes.
 Are the scientific methods valid and clear outlined to be reproduced? The study could use better diction and figures to help the reader come along, but its mathematical derivations and logic are robust.
 Are analyses and assumptions valid? The analyses and assumptions are clearly stated and seem robust.
 Are the presented results sufficient to support the interpretations and associated discussion? Yes
 Is the discussion relevant and backed up? Yes
 Are accurate conclusions reached based on the presented results and discussion? Yes
 Do the authors give proper credit to related and relevant work and clearly indicate their own original contribution? Yes, to the best of my knowledge.
 Does the title clearly reflect the contents of the paper and is it informative? Yes
 Does the abstract provide a concise and complete summary, including quantitative results? Yes
 Is the overall presentation well structured? Yes
 Is the paper written concisely and to the point? Yes
 Is the language fluent, precise, and grammatically correct? Yes
 Are the figures and tables useful and all necessary? They are all necessary, but I'd recommend adding a few more to help the reader (see notes).
 Are mathematical formulae, symbols, abbreviations, and units correctly defined and used according to the author guidelines? Yes
 Should any parts of the paper (text, formulae, figures, tables) be clarified, reduced, combined, or eliminated? No
 Are the number and quality of references appropriate? It may be valuable to add a paragraph reviewing similar models in the literature.
 Is the amount and quality of supplementary material appropriate and of added value? N/A
Citation: https://doi.org/10.5194/wes202294RC2  AC1: 'Comment on wes202294', Niels Troldborg, 16 Feb 2023
Mac Gaunaa et al.
Mac Gaunaa et al.
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