the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 10 Feb 2026
Edgewise instabilities of a wind turbine blade section in attached flow conditions
Abstract. With the persistent trend towards larger, lighter turbine blades and the resulting increase in blade flexibility, the risk of edgewise instabilities is elevated. This emphasizes the need for a better understanding of the accuracy of simulation models and the mechanisms affecting the damping of edgewise modes. This study investigates edgewise instabilities in wind turbines under attached flow conditions using a two-dimensional, three-degree-of-freedom typical section model. The primary goal of this work is to validate the stability analysis predictions from common low-fidelity aerodynamic models by comparing them with a high-fidelity computational fluid dynamics (CFD) reference result. In addition, the study aims to deepen the understanding of the fundamental mechanisms behind edgewise instabilities. The comparison of the stability analysis results obtained with aerodynamic models of different fidelity demonstrates excellent agreement in both the trends and quantitative damping predictions for the edgewise mode. An additional sensitivity study highlights the importance of the steady load vector on the damping prediction. The flutter mechanism analysis shows that the edgewise instabilities are caused by a coupling of flapwise motion with the structurally coupled edgewise-torsional motion. Edgewise instabilities can occur when there is sufficient structural edge-twist coupling, both in edge-twist coupling to stall or edge-twist coupling to feather. The findings in the paper increase the confidence in common low-fidelity aerodynamic models and contribute to a better understanding of edgewise instabilities.
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RC1: 'Comment on wes-2026-9', Anonymous Referee #1, 01 Apr 2026
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Dear authors,first of all I would like to say that I think this is a great and extremely detailed manuscript.However, in my opinion it would be easier to digest if it was split in 2 parts:* Edgewise instabilities of a blade section part 1: Validation of attached flow unsteady airfoil aerodynamics modeling* Sections 2 and 3 of the article* Edgewise instabilities of a blade section part 2: Analysis of the instability mechanism* Sections 4 and 5 of the articleThis is of course just a suggestion.I am not sure how the process for such a splitting would be at WES: reducing the length of the current article and resubmitting the second part?The article is very clearly written, and the plots are very clear as well.Apart from the above suggestion of splitting the article, I have some detailed comments for minor revisions below.Best regards,GeorgDetailed comments:* page 2, line 36 'an edgewise flutter-like instability was observed during overspeed operation in experimental tests': I would probably write 'dedicated experimental tests' to highlight that the tests were targeted towards the instability* page 2, line 53-54 'They furthermore showed that the aerodynamic lift and moment caused by harmonic edgewise motion did not significantly change the flutter speed.': This description is not fully clear to me, can you extend it a bit?* page 3, line 74 'but additionally showing that this sensitivity depends strongly on the operating point.' Is that due to geometric edge-twist coupling due to the flapwise deflection?* page 3, line 81: 'If the torsional motion leads flapwise motion, the flapwise force induced by torsional motion will perform positive work': maybe for clarity this could be extended to 'the flapwise force induced by torsional motion will be in phase with the flapwise velocity and thus perform positive work'* page 3: line 85 'in edgewise flutter, the torsional motion is introduced by the structural edge-twist coupling': I would think there is also - at least for a full blade - an aerodynamic coupling where edgewise motion causes the steady state flapwise force to move, which causes a torsional moment.* page 4, line 96: 'potential theory' => potential flow theory* page 4, line 111: 'This presented work serves a couple goals.' => The present work serves a couple of goals.* page 5, Figure 1: I think the collocation point and aerodynamic center should be defined somewhere (name and location)* page 6, line 159: 'For realistic, full turbine systems, the edge-twist coupling is mainly driven by the out-of-plane deflection of the blade, either by prebend or by flapwise deformation under load.': I think this could also be mentioned in the introduction, to clarify where the edge-twist coupling could come from.* page 7, line 182: 'As the model is not perfectly linear': Maybe write 'airfoil properties are not perfectly linear' instead.* page 8, Section 2.2.1: What timestep was used?* page 10, line 259 'effects due to the airfoil camber through both the tuned CLα and the effective angle of attack': how is the lift slope tuned?* page 13, section 2.3: I propose to slightly change the title to '2.3 Summary and corrections of the methodology by Stäblein et al. (2017a)* page 15, line 35: I believe the components should be Fyy, Fy \theta, Mθ y, M \theta \theta* page 16, Figure 4: In modern wind turbines, I believe the reduced frequency range for outboard parts of the blade would be roughly in the range of 0.05 to 0.1 for a first edgewise and maybe 0.1 to 0.2 for a second edgewise mode. Maybe it would make sense to mention this in the description of the figure, and to provide a plot ranging only from 0 to 0.2 in the appendix, where this very relevant area is visible in more detail?* page 16, Figure 4, Fx \theta: Might the increasing difference in the imaginary part for high reduced frequencies be due to neglecting the pitch rate squared term in the tangential force? (Equation ⑬ in https://doi.org/10.5194/wes-7-1341-2022)* page 16, line 397 'An even better agreement was found for a comparison with a thin, uncambered NACA 0006 profile.': Could an equivalent of Figure 4 for the NACA 0006 be included in an appendix? Also, Bergami et al have investigated changes of the circulatory response due to changing airfoil thickness in https://doi.org/10.1002/we.1516* page 19, line 477: 'In the six selected cases': Maybe you could write six selected coupling combinations? I would argue that it is 12 cases, 6 coupling combinations and 2 steady angles of attack.* page 25, table 3: 'Theodorsen - without steady load': maybe you could briefly describe how this is achieved in the model equations?* page 26, 27, Figures 11 and 12: using different types of lines (solid, dashed, dash-dotted,...) might be helpful in figures 11 and 12 to identify cases where several lines are on top of each other* page 28, line 561: 'edgewise induced forces': I understand what it means, but somehow the term is not totally clear to me. Maybe calling it edgewise-motion-induced forces or forces due to edgewise motion would be more clear?* page 29, Equations (42) to (43): could this reduction from 3 to 2 DOF be shown in more detail? for example, how is B chosen for the different cases that will be investigated in the following?* page 31, line 635 'moving from −360◦ (out-of-phase with torsion) to −180◦ (trailing torsion)': I guess -360 would be in-phase and -180 would be out-of-phase?* page 32, line 660 'altogether, these diagrams illustrate a straightforward mechanism': I just wanted to say that I think this is a great paragraph!* page 33, Figure 14 h): is the flapwise force amplitude due to torsion exactly on top of each other for the 2DOF and 3DOF model? If this is the case, please mention it* pages 33 and 35, Figures 14 and 15: maybe the total aerodynamic work line thickness could be reduced to make sure that they don't hide other lines* page 36, line 713: 'The simplified 3DOF model is still a good approximation of the full CFD reference result.': I am not sure I can agree with this. The 'critical speed' in the CFD result is at about 90 m/s, while the 3DOF simplified model predicts maybe 50 m/s (difficult to see)* page 36, line 717 'Edge-twist coupling to feather (gamma_x= -0.3)': I believe this should be: 'Edge-twist coupling to stall (gamma_x= 0.3)'* page 48, line 907, reference for Li et al. (2025): Sometimes in the text, I believe you refer to this, when you intend to refer to Li et al. (2022). Please double check.ReplyCitation: https://doi.org/
10.5194/wes-2026-9-RC1 -
CC2: 'Reply on RC1', J. Gordon Leishman, 22 Jun 2026
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I do not think the main issue is only the paper's length or organization. In particular, the comments about Figure 1, the definition of the aerodynamic center and collocation point, the force components in the aerodynamic matrix, the reduced-frequency range, the discrepancy in the imaginary part of (F_{x\theta}), the possible phase-sign confusion near line 635, the questionable agreement between the simplified 3-DOF model and the CFD reference near line 713, and the apparent error in the edge-twist coupling statement near line 717 all relate directly to the aerodynamic and aeroelastic formulation. These are not simply matters of wording or presentation. They affect the sign conventions, force projections, modal work interpretation, and ultimately the damping prediction.
I am especially concerned about the unsteady aerodynamic formulation. The manuscript relies on a Theodorsen/Greenberg-based reduced-order model and uses it to build the generalized aerodynamic matrix in Eq. (6). However, the force projections in Eqs. (8)–(16), the circulatory and non-circulatory terms in Eqs. (21)–(24), and the unsteady quarter-chord angle expression in Eq. (25) requires a more explicit derivation and verification. The manuscript does not sufficiently distinguish between the geometric angle used to rotate the steady load vector and the lagged circulatory response represented by Theodorsen’s function. If the Theodorsen function is embedded in the angle used for force projection, then the formulation may be mixing steady-load-vector rotation with circulatory wake lag.
The reviewer also asks how the lift-curve slope is tuned, what timestep was used in the CFD, whether an additional term might explain the (F_{x\theta}) discrepancy, and whether an equivalent NACA 0006 comparison should be shown. These are important questions. They reinforce the need for a more complete verification of both the CFD reference and the reduced-order aerodynamic implementation. In particular, the authors should document how the Theodorsen function (C(k)) and the Greenberg terms are implemented numerically, including the reduced-frequency definition, complex-sign convention, coordinate transformation, and assembly of the complex transfer functions into (A(p,\alpha_0)). A canonical flat-plate plunge/pitch verification case should be provided before applying the formulation to a thick cambered wind-turbine airfoil section.
I also agree with the reviewer’s concern that the statement that the simplified 3-DOF model remains a good approximation of the CFD reference is difficult to accept if the critical speeds differ substantially. This is not a minor issue. If the simplified model predicts instability at a substantially different speed than the CFD reference, then the manuscript should not use that result to support broad confidence in the reduced-order model without a much more careful explanation.
For these reasons, I would not characterize the required revisions as minor. The manuscript needs a careful rederivation of the unsteady aerodynamic terms, clearer correction of the blade-section geometry and angle definitions, better documentation of the Theodorsen/Greenberg implementation, stronger CFD verification, and more limited claims about validation.
Disclaimer: this community comment is written by an individual and does not necessarily reflect the opinion of their employer.Citation: https://doi.org/10.5194/wes-2026-9-CC2
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CC2: 'Reply on RC1', J. Gordon Leishman, 22 Jun 2026
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CC1: 'Comment on wes-2026-9', J. Gordon Leishman, 22 Jun 2026
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The manuscript addresses a relevant aeroelastic problem, but it is not publication-ready. The main issue is that the paper makes claims broader than the evidence supports, and unresolved issues in the aerodynamic formulation need to be checked before the stability results can be relied upon.
The study is based on a two-dimensional, three-degree-of-freedom typical section, one representative blade section, one airfoil, two angles of attack, and attached-flow conditions only. This is too limited to support broad conclusions about the reliability of classic reduced-order aerodynamic models for edgewise instabilities in wind turbine blades. The reduced-order model may be useful for mechanism studies, but the manuscript does not show that the conclusions transfer to a full rotating blade or turbine. Real edgewise instabilities are affected by radial structural variations, rotation, centrifugal stiffening, gravity, controller effects, blade twist, prebend, sweep, three-dimensional aerodynamics, and operating-point dependence. These effects appear mostly outside the scope of the present model. Therefore, the authors should narrow the title, abstract, and conclusions accordingly.
A major concern with this paper is the derivation of the unsteady aerodynamic model. The issues affect the generalized aerodynamic matrix used in the flutter calculation. In particular, the force projections in Eqs. (8)–(16), the Theodorsen and Greenberg terms in Eqs. (21)–(24), and the quarter-chord angle expression in Eq. (25) requires a careful rederivation. The manuscript does not sufficiently distinguish among circulatory lift, non-circulatory lift, Greenberg edgewise-velocity terms, and the rotation of the steady load vector. The angle used to rotate the steady lift vector should be a geometric or kinematic quantity, whereas the Theodorsen function belongs in the circulatory part of the lift response. The present formulation appears to mix these concepts, especially through the use of the Theodorsen function in the expression for the unsteady quarter-chord angle of attack. This raises the possibility of double-counting or misplacing the unsteady aerodynamic lag.
The manuscript repeatedly refers to the Theodorsen/Greenberg-based sectional model as a “low-fidelity” aerodynamic model. That terminology is perhaps acceptable in comparison with CFD, but it does not reduce the burden of mathematical correctness. A low-fidelity model can still be rigorous, internally consistent, and useful; conversely, if the force projections, angle definitions, non-circulatory terms, Greenberg terms, or sign conventions are incorrect, the model is not merely "low-fidelity" but incorrectly formulated. The authors should avoid treating agreement with selected CFD damping results as sufficient validation unless the underlying aerodynamic equations are first rederived and checked, and compared with standard check cases.
The non-circulatory terms also require a more rigorous derivation. The manuscript switches between inflow-fixed and chord-fixed coordinates, while the apparent-mass forces depend directly on the imposed acceleration direction and the direction in which the pressure force acts. The transformation between the inflow-fixed coordinates and the chord-fixed edgewise/flapwise coordinates should be written explicitly and carried through the derivation. The added chordwise apparent-mass term, proportional to the square of the airfoil thickness, is also not a classic unsteady aerodynamic or Theodorsen result. It may be a plausible engineering correction for a thick airfoil, but it should not be introduced as though it follows directly from the classical formulation. The authors need to provide a derivation or citation, the theoretical basis for the term, its sign convention, and an assessment of the sensitivity of the stability results to its inclusion. If the term is empirical or heuristic, that should be stated clearly.
There also appears to be a possible sign inconsistency in the perturbation terms associated with the relative velocity magnitude. The camber-moment perturbation and the drag perturbation should follow from the same expansion of the relative velocity squared. If the normal-velocity perturbation enters the camber-moment term with one sign but the corresponding drag perturbation with the opposite sign, the manuscript needs to explain why. Otherwise, this suggests a possible sign error. Because the stability result depends on small aerodynamic work terms and phase relationships, such sign issues cannot be treated as minor algebraic or notation details.
There is also a possible inconsistency in the coordinates of the Greenberg (unsteady free-stream) terms. The derivation states that the Theodorsen and Greenberg expressions are formulated in inflow-fixed coordinates, using motion components parallel and perpendicular to the relative wind. However, at least one expression appears to use an edgewise displacement amplitude without clearly indicating whether it is the inflow-fixed or chord-fixed component. If this is a typographical issue, it should be corrected. If not, then the derivation may be mixing coordinate systems.
The manuscript itself indicates that not all aerodynamic transfer functions agree well between the low-fidelity and CFD models. In particular, some edgewise-motion-induced components have poor agreement or even opposite signs in parts of the reduced-frequency range. The authors argue that these discrepancies have a limited effect on the final damping result for the selected case, but that does not establish that the low-fidelity aerodynamic formulation is generally reliable. These discrepancies are especially important because the paper’s stated purpose is to build confidence in low-fidelity aerodynamic models. The authors should identify which transfer-function discrepancies matter, which do not, and why. They should also avoid broad validation language unless the unexplained discrepancies are resolved.
The CFD comparison is also not sufficient to justify the validation language. The manuscript compares classic unsteady aerodynamics with a two-dimensional CFD calculation, but it does not provide enough verification of the CFD reference. More evidence is needed on grid convergence, time-step sensitivity, domain-size effects, boundary-condition effects, numerical dissipation, and extraction of the aerodynamic transfer functions. Without this, the CFD result cannot be treated as a definitive reference.
The blade-section kinematics and force definitions need to be corrected and redrawn. In particular, the angles in the blade-element/typical-section diagram appear to be defined incorrectly. For this problem, the angle of attack, inflow angle, chord line, lift direction, drag direction, and pitch/torsion convention must be geometrically consistent. Because the governing equations depend directly on these sign conventions and angle definitions, an incorrect angle construction relative to the chord undermines confidence in the subsequent force projections and physical interpretation. The authors should provide a corrected diagram showing the chord line, relative velocity vector, inflow angle, angle of attack, lift and drag directions, elastic axis, aerodynamic center, center of gravity, and positive directions of edgewise, flapwise, and torsional motion.
The implementation of the Theodorsen and Greenberg functions also needs to be documented. Given the noted ambiguity in the analytical formulation, it is difficult to know whether the numerical implementation uses the same sign conventions, coordinate definitions, reduced-frequency definition, and complex-valued transfer-function convention as the derivation. The authors should state explicitly how the Theodorsen function (C(k)) is evaluated, which branch/sign convention is used for the Hankel (or equivalent approximation if one is used), how the Greenberg terms are implemented for edgewise motion, and how the complex aerodynamic transfer functions are assembled into the matrix (A(p,\alpha_0)). They should also provide verification against a simple canonical case, such as a harmonic plunge and the pitch of a flat plate, before applying the formulation to the cambered thick airfoil section. Without such implementation checks, apparent agreement in selected modal damping results does not demonstrate that the Theodorsen/Greenberg model has been implemented correctly.
The paper also needs a clearer explanation of the physical mechanisms. The role of the steady load vector is important, but the manuscript does not clearly separate the effects of mean lift, zero-lift angle, camber moment, force-vector rotation, non-circulatory force, and structural edge-twist coupling. The discussion shows that these terms matter, but it does not explain the causal pathway with sufficient clarity. The authors should show explicitly how the aerodynamic work contribution changes when each term is retained or removed. The correction of the earlier Stäblein et al. methodology is potentially important, but it should be presented more systematically. The manuscript should state clearly which previous results remain valid, which results have changed, and which conclusions require reinterpretation. As written, the corrigendum aspect is mixed into the main analysis and is difficult to assess.
Overall, the paper is far too narrowly based for the strength of its conclusions, and the unsteady aerodynamic formulation contains issues that need a full and careful rederivation. The authors should substantially revise the manuscript by limiting the claims, avoiding broad validation language, strengthening the CFD verification, correcting the blade-element/typical-section diagrams and angle definitions, rederiving the circulatory, non-circulatory, Greenberg, and steady-load-vector terms, and clarifying the physical mechanisms. The paper is not ready for publication in its present form.
Disclaimer: this community comment is written by an individual and does not necessarily reflect the opinion of their employer.Citation: https://doi.org/10.5194/wes-2026-9-CC1
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