the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Modeling unsteady loads on windturbine blade sections from periodic structural oscillations and impinging gusts
Abstract. Many traditional methods for wind turbine design and analysis assume quasisteady aerodynamics, but atmospheric flows are inherently unsteady and modern turbine blades are susceptible to aeroelastic deformations. This study therefore evaluates the effectiveness of simple analytical models for capturing the effects of such unsteady conditions on windturbine blades. We consider a pitching and plunging airfoil in a periodic transverse gust as an idealization of unsteady loading scenarios on a blade section. A potentialflow model derived from a linear combination of canonical problems is proposed to predict the unsteady lift on a twodimensional airfoil in the smallperturbation limit. We then perform highfidelity twodimensional numerical simulations of a NACA0012 airfoil over a range of periodic pitch, plunge, and gust disturbances, and quantify the amplitude and phase of the unsteady lift response. Good agreement with the model predictions is found for low to moderate forcing amplitudes and frequencies, while deviations are observed when the angleofattack amplitudes approach the static flowseparation limit of the airfoil. Potential explanations are given for the cases in which the idealflow theory proves insufficient. This theoretical framework and numerical evaluation motivate the inclusion of unsteady flow models in design and simulation tools in order to increase the robustness and operational lifespans of wind turbine blades in real flow conditions.
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RC1: 'Comment on wes2023164', David Wood, 07 Feb 2024
The classical, unsteady, linearized airfoil theories used in this study are very important in wind turbine aerodynamics, but unsteadiness is commonly ignored or represented by a quasisteady steady process in models like blade element theory. On the other hand, the classical theories have their limitations which are carefully identified in this study. Significant airfoil thickness, large amplitude pitching and plunging, and high reduced frequencies will all challenge the limits to linearity. Nevertheless, the authors, like others before them, e.g. Chiereghin (2017, extra references below), have found the linear regime surprisingly wide, and, therefore, useful in practice. It is relevant that Prandtl, when introducing lifting line theory to English audiences after the first world war, stressed the perturbative nature of the theory, e.g. Prandtl (1923). This has long since been forgotten because the theory is robust. The novelty of the manuscript is the combined study of pitching and plunging by exploiting the linearity of the classical theories. The results are compared to available experiments and to their own simulations of an unsteady NACA 0012 airfoil. Generally good agreement was found and the route to extension of the modelling was discussed. There is a huge literature on unsteady behaviour of airfoils, from which the authors have drawn a comprehensive and appropriate reference list.
In summary, the manuscript provides a useful and important contribution to wind turbine aerodynamics and can be accepted once the following issues are addressed. I will list these in order.
The authors carefully state that the effects of plunging and pitching are superimposes and then on L83 describe this as a “linear model that results in a linear combination…”. I think ”results” is misplaced as the model is linear by construction.
Small point: “infinitespan airfoil” is a tautology as an airfoil must have infinite span.
Figure 2 shows the twodimensional (2D) computational domain but then we are told that the unsteady vorticity field was modeled by large eddy simulation (LES) which is inherently three dimensional. How the LES is embedded in the 2D simulation is not described.
Small point: “to perform this transformation” on L190 is vague. I think you mean “to return to the inertial frame”?
The discussion of the Reynolds number (Re) should be improved. My judgement is that Re > 200,000 is a good compromise as it avoids complexities like leading edge separation bubbles, that occur at lower Re while not requiring very fine grids. A vague reference to “the nonlinear effects of high Reynoldsnumber …” whatever they are, is not needed.
A brief description of the error bars in Figure 3 and the line thickness of the simulations in Figure 4 would help interpret the results. Presumably the latter represent averages over a number of cycles starting after a specified time. These details should be given. Similar remarks apply to the later figures.
The effects of finite Re are mentioned briefly on L370 where the theory is said to be “inviscid”. Since the classical theories contain a model for the wake and use the Kutta condition, a better adjective would be “infinite Reynolds number”.
Extra References
Chiereghin, N., Cleaver, D., & Gursul, I. (2017). Unsteady Measurements for a periodically plunging airfoil. In 55th AIAA Aerospace Sciences Meeting (p. 0996).
Prandtl, L. (1923). Applications of modern hydrodynamics to aeronautics, NACATR116.
Citation: https://doi.org/10.5194/wes2023164RC1  AC1: 'Reply on RC1', Omkar Shende, 08 Apr 2024

RC2: 'Comment on wes2023164', Anonymous Referee #2, 24 Mar 2024
The authors have done an academic study of unsteady aerodynamics of a symmetric airfoil in inflow with transverse gust under attached flow conditions in 2D. They are considering the plunge and pitch (blade torsion) of the airfoil, and the Theodorsen effect of shed vorticity on the circulatory lift and Sears function for the unsteadiness of the lift variation from the transverse gust variation.
I have recommended that the manuscript is not published in WES because I fail to see the novelty in the work. The authors claim that their model “could yield farreaching benefits to the operational longevity of wind turbines by better accounting for unsteady fatigue loads in the design process.” However, the authors seem to be unaware of the status of the art within aeroelastic modelling of wind turbines. The last 25 years we have had research and commercially available aeroelastic codes that include the Theodorsen effect in the unsteady aerodynamic lift models, and we have shown that these codes are able to predict the blade and turbine component fatigue loads within 5% relative error to measurements for each wind speed over the entire operational range. The biggest uncertainty in these predictions is not the lack of transverse gust modelling (which is mainly important when the gust “wavelengths” are of the order of the blade chord) in these codes, but the uncertainty in the inflow modelling. To capture the flapwise fatigue loads on the blades, it is very important include the deterministic components of the inflow (vertical and horizontal shear profiles, veer profiles, and yaw and upflow angles) as well as the structures of the turbulence (at least the intensity variation with height, but new methods also include turbulence reconstruction).
The authors exclude the edgewise (leadlag) motion of the airfoil (affecting the downwash of the shed vorticity, an effect included in some aeroelastic codes). Edgewise blade vibrations due to negative aerodynamic are often driving the blade design. They are highly affected by the coupling between the edgewise airfoil motion and its pitch (blade torsion) through the lift force. An unsteady aerodynamic model for wind turbines must therefore include the effect of edgewise airfoil motion on the unsteady lift.
Citation: https://doi.org/10.5194/wes2023164RC2 
AC2: 'Reply on RC2', Omkar Shende, 08 Apr 2024
The comment was uploaded in the form of a supplement: https://wes.copernicus.org/preprints/wes2023164/wes2023164AC2supplement.pdf

AC2: 'Reply on RC2', Omkar Shende, 08 Apr 2024

RC3: 'Comment on wes2023164', Anonymous Referee #3, 30 Apr 2024
“Modeling unsteady loads on windturbine blade sections from periodic structural oscillations and impinging gusts”
General Comments
The paper proposes an unsteady aerodynamics model for combined pitching and plunging airfoil motion. The authors compare the predictions of their model, which is based in classical aerodynamics, to CFD simulations of the NACA 0012 airfoil, with some initial validation cases around the NACA 0006 airfoil. In general, the paper is written well.
The reviewer appreciates the idea as a potential contribution to be used in wind turbine performance codes and actuatorline methods. The originality of the idea is laudable in the sense that using concepts of classical aerodynamics to solve new problems efficiently can have notable impact.
In its current form, though, the reviewer feels that the work is not quite ready yet for publication, see Specific/Technical comments below. I encourage the authors to take the constructive feedback into account as they progress their work in the future.
Specific/Technical Comments:
 The NACA 0012 is irrelevant for modern utilityscale wind turbines. As there are no experimental data available (not quite certain even) for combined pitching and plunging motion, the authors should have considered a thick cambered wind turbine airfoil for comparing their model to simulations.
 The U. Glasgow database of unsteady airfoil data (among others) could have been used as a further validation case (with more appropriate airfoils) instead of a fairly recent study on the NACA 0006, which again is irrelevant for modern wind turbines.
 In general, the authors somewhat neglect decades of work being done in unsteady aerodynamics and more suitable test cases and data that would be helpful in verifying and validating their model.
 In its present form, the reviewer cannot implement the unsteady pitching and plunging model into a BEMT code as not enough information is given. There is not even a nomenclature in the paper.
 It is unclear in section 3.2 which part of the rotor disk (radius, azimuth) is most affected by plunging and pitching motion.
 Similar in section 4.1. Where on the rotor disk of a modern wind turbine are these scenarios relevant?
 The Discussion eludes to the fact that the model would have challenges for more relevant thick airfoils. The reviewer feels that there is just more work to be done to have a compelling case for publication.
The reviewer hopes that the authors understand the feedback above constructively and as a way to conduct more work, thereby making a compelling case that their new model is indeed relevant to modern wind turbines by actually using representative airfoils and providing more information as to the implementation of their theory. These combined can make an impactful paper in the future.
Citation: https://doi.org/10.5194/wes2023164RC3  AC3: 'Reply on RC3', Omkar Shende, 07 May 2024

RC4: 'Comment on wes2023164', Anonymous Referee #4, 03 May 2024
The manuscript presents an analytical model of the forces acting on an airfoil as it undergoes simultaneous pitching and plunging. This model is a linear combination of the Theodorsen function and the Sears function. The model results are validated against numerical simulations of a NACA 0012 airfoil.Unfortunately, I do not believe that the manuscript warrants publication in Wind Energy Science, as it lacks novelty and scientific insight. The model is a linear combination of two decadesold analytical models, the Theodorsen function for a pitching airfoil and the Sears function for a plunging airfoil. In the introduction, the authors nicely list the work that has been done in this field over the past decades. It is rather trivial that such a linear combination yields a reasonable description of combined pitching and plunging in cases where nonlinear effects are small. As the authors point out, the model fails when nonlinearities become important, which is stated to be outside the scope of this work. However, this is exactly the regime that would have been interesting to model. In addition, the manuscript does not provide insight into the flow physics to explain the observations provided herein. Instead, the authors vaguely allude to viscous effects and flow separation, but many of the explanations are postulative and unconvincing. While it may be true that empirical models like LeishmanBeddoes provide less physical insight than analytical ones, the model presented herein breaks down for more complex flow behavior, whereas the parameter space well described by the model is also already well understood, so that little novel insight is provided.Furthermore, the authors state that simplifying assumptions used for the numerical simulation limit its applicability to the smallamplitude perturbation regime. This however limits the ability of the numerical simulations to serve as validation for the analytical model, since a validation should reveal when these assumptions break down. Currently, both the analytical and numerical approaches in this manuscript rely on major assumptions that do not hold true for real wind turbines, but no reliable validation is provided to evaluate these assumptions.General comments:
 Could you comment on the importance of the center of rotation, specifically pitching around the quarter chord vs around another point? For real wind turbines, what would be the best approximation of the rotation point?
 Could you comment to what extent it is possible or appropriate to correct your model for effects like airfoil thickness, camber, nonzero mean angle of attack and finite span? All of these are crucial in moving away from the idealized case to real application. In particular, could you comment on 3D effects and the extent to which this model holds for real wind turbines given that their blades have finite length and radially varying chord and inflow velocity vector?
 Could you elaborate on what you mean when you say the Reynolds number is “low enough so that the nonlinear effects of high Reynoldsnumber turbulence are limited”? What are these Reynolds number effects you expect to not be present, and to what extent are the simulations applicable to wind turbine blades, given that real blades operate at Re_c about an order of magnitude higher than your study?
 Why do you investigate reduced frequencies up to k = 4 when you state that the most extreme cases in the real world are k = 1? And why do you not investigate k < 0.2 if that is the range typically observed in the real world? It seems that your parameter space is not directly relevant for wind turbines.
 You assume sinusoidal oscillations. Can you comment on how realistic this is and how feasible it is to use this approach for more complex oscillation patterns?
 In section 4, you state that the gust is felt by different parts of the airfoil at different times. However, in an incompressible flow, the gust should be felt everywhere in the flow field simultaneously. Thus the explanation is not convincing.
 In section 5, you state that the dominant source of error of the analytical model is flow separation and stall. However, if I understand correctly, your simulations do not have separation and stall, so how can those effects explain the discrepancy between the model and the simulations? In particular, you do not exceed the static stall angle in any of your numerical simulations, so flow separation should not be the source of discrepancy.
 In the last paragraph of the manuscript, you discuss dynamic stall. This is an entirely different topic from what is covered in this manuscript. Certainly the model you describe here, if unable to model simple nonlinearities in the superposition of pitching and plunging effects, would not be able to describe the nonlinear dynamics involved in dynamic stall. Therefore, the connection to this topic here does not make sense to me.
Citation: https://doi.org/10.5194/wes2023164RC4  AC4: 'Reply on RC4', Omkar Shende, 07 May 2024
Status: closed

RC1: 'Comment on wes2023164', David Wood, 07 Feb 2024
The classical, unsteady, linearized airfoil theories used in this study are very important in wind turbine aerodynamics, but unsteadiness is commonly ignored or represented by a quasisteady steady process in models like blade element theory. On the other hand, the classical theories have their limitations which are carefully identified in this study. Significant airfoil thickness, large amplitude pitching and plunging, and high reduced frequencies will all challenge the limits to linearity. Nevertheless, the authors, like others before them, e.g. Chiereghin (2017, extra references below), have found the linear regime surprisingly wide, and, therefore, useful in practice. It is relevant that Prandtl, when introducing lifting line theory to English audiences after the first world war, stressed the perturbative nature of the theory, e.g. Prandtl (1923). This has long since been forgotten because the theory is robust. The novelty of the manuscript is the combined study of pitching and plunging by exploiting the linearity of the classical theories. The results are compared to available experiments and to their own simulations of an unsteady NACA 0012 airfoil. Generally good agreement was found and the route to extension of the modelling was discussed. There is a huge literature on unsteady behaviour of airfoils, from which the authors have drawn a comprehensive and appropriate reference list.
In summary, the manuscript provides a useful and important contribution to wind turbine aerodynamics and can be accepted once the following issues are addressed. I will list these in order.
The authors carefully state that the effects of plunging and pitching are superimposes and then on L83 describe this as a “linear model that results in a linear combination…”. I think ”results” is misplaced as the model is linear by construction.
Small point: “infinitespan airfoil” is a tautology as an airfoil must have infinite span.
Figure 2 shows the twodimensional (2D) computational domain but then we are told that the unsteady vorticity field was modeled by large eddy simulation (LES) which is inherently three dimensional. How the LES is embedded in the 2D simulation is not described.
Small point: “to perform this transformation” on L190 is vague. I think you mean “to return to the inertial frame”?
The discussion of the Reynolds number (Re) should be improved. My judgement is that Re > 200,000 is a good compromise as it avoids complexities like leading edge separation bubbles, that occur at lower Re while not requiring very fine grids. A vague reference to “the nonlinear effects of high Reynoldsnumber …” whatever they are, is not needed.
A brief description of the error bars in Figure 3 and the line thickness of the simulations in Figure 4 would help interpret the results. Presumably the latter represent averages over a number of cycles starting after a specified time. These details should be given. Similar remarks apply to the later figures.
The effects of finite Re are mentioned briefly on L370 where the theory is said to be “inviscid”. Since the classical theories contain a model for the wake and use the Kutta condition, a better adjective would be “infinite Reynolds number”.
Extra References
Chiereghin, N., Cleaver, D., & Gursul, I. (2017). Unsteady Measurements for a periodically plunging airfoil. In 55th AIAA Aerospace Sciences Meeting (p. 0996).
Prandtl, L. (1923). Applications of modern hydrodynamics to aeronautics, NACATR116.
Citation: https://doi.org/10.5194/wes2023164RC1  AC1: 'Reply on RC1', Omkar Shende, 08 Apr 2024

RC2: 'Comment on wes2023164', Anonymous Referee #2, 24 Mar 2024
The authors have done an academic study of unsteady aerodynamics of a symmetric airfoil in inflow with transverse gust under attached flow conditions in 2D. They are considering the plunge and pitch (blade torsion) of the airfoil, and the Theodorsen effect of shed vorticity on the circulatory lift and Sears function for the unsteadiness of the lift variation from the transverse gust variation.
I have recommended that the manuscript is not published in WES because I fail to see the novelty in the work. The authors claim that their model “could yield farreaching benefits to the operational longevity of wind turbines by better accounting for unsteady fatigue loads in the design process.” However, the authors seem to be unaware of the status of the art within aeroelastic modelling of wind turbines. The last 25 years we have had research and commercially available aeroelastic codes that include the Theodorsen effect in the unsteady aerodynamic lift models, and we have shown that these codes are able to predict the blade and turbine component fatigue loads within 5% relative error to measurements for each wind speed over the entire operational range. The biggest uncertainty in these predictions is not the lack of transverse gust modelling (which is mainly important when the gust “wavelengths” are of the order of the blade chord) in these codes, but the uncertainty in the inflow modelling. To capture the flapwise fatigue loads on the blades, it is very important include the deterministic components of the inflow (vertical and horizontal shear profiles, veer profiles, and yaw and upflow angles) as well as the structures of the turbulence (at least the intensity variation with height, but new methods also include turbulence reconstruction).
The authors exclude the edgewise (leadlag) motion of the airfoil (affecting the downwash of the shed vorticity, an effect included in some aeroelastic codes). Edgewise blade vibrations due to negative aerodynamic are often driving the blade design. They are highly affected by the coupling between the edgewise airfoil motion and its pitch (blade torsion) through the lift force. An unsteady aerodynamic model for wind turbines must therefore include the effect of edgewise airfoil motion on the unsteady lift.
Citation: https://doi.org/10.5194/wes2023164RC2 
AC2: 'Reply on RC2', Omkar Shende, 08 Apr 2024
The comment was uploaded in the form of a supplement: https://wes.copernicus.org/preprints/wes2023164/wes2023164AC2supplement.pdf

AC2: 'Reply on RC2', Omkar Shende, 08 Apr 2024

RC3: 'Comment on wes2023164', Anonymous Referee #3, 30 Apr 2024
“Modeling unsteady loads on windturbine blade sections from periodic structural oscillations and impinging gusts”
General Comments
The paper proposes an unsteady aerodynamics model for combined pitching and plunging airfoil motion. The authors compare the predictions of their model, which is based in classical aerodynamics, to CFD simulations of the NACA 0012 airfoil, with some initial validation cases around the NACA 0006 airfoil. In general, the paper is written well.
The reviewer appreciates the idea as a potential contribution to be used in wind turbine performance codes and actuatorline methods. The originality of the idea is laudable in the sense that using concepts of classical aerodynamics to solve new problems efficiently can have notable impact.
In its current form, though, the reviewer feels that the work is not quite ready yet for publication, see Specific/Technical comments below. I encourage the authors to take the constructive feedback into account as they progress their work in the future.
Specific/Technical Comments:
 The NACA 0012 is irrelevant for modern utilityscale wind turbines. As there are no experimental data available (not quite certain even) for combined pitching and plunging motion, the authors should have considered a thick cambered wind turbine airfoil for comparing their model to simulations.
 The U. Glasgow database of unsteady airfoil data (among others) could have been used as a further validation case (with more appropriate airfoils) instead of a fairly recent study on the NACA 0006, which again is irrelevant for modern wind turbines.
 In general, the authors somewhat neglect decades of work being done in unsteady aerodynamics and more suitable test cases and data that would be helpful in verifying and validating their model.
 In its present form, the reviewer cannot implement the unsteady pitching and plunging model into a BEMT code as not enough information is given. There is not even a nomenclature in the paper.
 It is unclear in section 3.2 which part of the rotor disk (radius, azimuth) is most affected by plunging and pitching motion.
 Similar in section 4.1. Where on the rotor disk of a modern wind turbine are these scenarios relevant?
 The Discussion eludes to the fact that the model would have challenges for more relevant thick airfoils. The reviewer feels that there is just more work to be done to have a compelling case for publication.
The reviewer hopes that the authors understand the feedback above constructively and as a way to conduct more work, thereby making a compelling case that their new model is indeed relevant to modern wind turbines by actually using representative airfoils and providing more information as to the implementation of their theory. These combined can make an impactful paper in the future.
Citation: https://doi.org/10.5194/wes2023164RC3  AC3: 'Reply on RC3', Omkar Shende, 07 May 2024

RC4: 'Comment on wes2023164', Anonymous Referee #4, 03 May 2024
The manuscript presents an analytical model of the forces acting on an airfoil as it undergoes simultaneous pitching and plunging. This model is a linear combination of the Theodorsen function and the Sears function. The model results are validated against numerical simulations of a NACA 0012 airfoil.Unfortunately, I do not believe that the manuscript warrants publication in Wind Energy Science, as it lacks novelty and scientific insight. The model is a linear combination of two decadesold analytical models, the Theodorsen function for a pitching airfoil and the Sears function for a plunging airfoil. In the introduction, the authors nicely list the work that has been done in this field over the past decades. It is rather trivial that such a linear combination yields a reasonable description of combined pitching and plunging in cases where nonlinear effects are small. As the authors point out, the model fails when nonlinearities become important, which is stated to be outside the scope of this work. However, this is exactly the regime that would have been interesting to model. In addition, the manuscript does not provide insight into the flow physics to explain the observations provided herein. Instead, the authors vaguely allude to viscous effects and flow separation, but many of the explanations are postulative and unconvincing. While it may be true that empirical models like LeishmanBeddoes provide less physical insight than analytical ones, the model presented herein breaks down for more complex flow behavior, whereas the parameter space well described by the model is also already well understood, so that little novel insight is provided.Furthermore, the authors state that simplifying assumptions used for the numerical simulation limit its applicability to the smallamplitude perturbation regime. This however limits the ability of the numerical simulations to serve as validation for the analytical model, since a validation should reveal when these assumptions break down. Currently, both the analytical and numerical approaches in this manuscript rely on major assumptions that do not hold true for real wind turbines, but no reliable validation is provided to evaluate these assumptions.General comments:
 Could you comment on the importance of the center of rotation, specifically pitching around the quarter chord vs around another point? For real wind turbines, what would be the best approximation of the rotation point?
 Could you comment to what extent it is possible or appropriate to correct your model for effects like airfoil thickness, camber, nonzero mean angle of attack and finite span? All of these are crucial in moving away from the idealized case to real application. In particular, could you comment on 3D effects and the extent to which this model holds for real wind turbines given that their blades have finite length and radially varying chord and inflow velocity vector?
 Could you elaborate on what you mean when you say the Reynolds number is “low enough so that the nonlinear effects of high Reynoldsnumber turbulence are limited”? What are these Reynolds number effects you expect to not be present, and to what extent are the simulations applicable to wind turbine blades, given that real blades operate at Re_c about an order of magnitude higher than your study?
 Why do you investigate reduced frequencies up to k = 4 when you state that the most extreme cases in the real world are k = 1? And why do you not investigate k < 0.2 if that is the range typically observed in the real world? It seems that your parameter space is not directly relevant for wind turbines.
 You assume sinusoidal oscillations. Can you comment on how realistic this is and how feasible it is to use this approach for more complex oscillation patterns?
 In section 4, you state that the gust is felt by different parts of the airfoil at different times. However, in an incompressible flow, the gust should be felt everywhere in the flow field simultaneously. Thus the explanation is not convincing.
 In section 5, you state that the dominant source of error of the analytical model is flow separation and stall. However, if I understand correctly, your simulations do not have separation and stall, so how can those effects explain the discrepancy between the model and the simulations? In particular, you do not exceed the static stall angle in any of your numerical simulations, so flow separation should not be the source of discrepancy.
 In the last paragraph of the manuscript, you discuss dynamic stall. This is an entirely different topic from what is covered in this manuscript. Certainly the model you describe here, if unable to model simple nonlinearities in the superposition of pitching and plunging effects, would not be able to describe the nonlinear dynamics involved in dynamic stall. Therefore, the connection to this topic here does not make sense to me.
Citation: https://doi.org/10.5194/wes2023164RC4  AC4: 'Reply on RC4', Omkar Shende, 07 May 2024
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