the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 18 Jun 2026
Computationally efficient aerodynamic modeling of curved wind turbine blades and non-planar rotors using coupled near wake and vortex cylinder models
Abstract. Accurate aerodynamic modeling is essential for load calculation and optimization of modern wind turbine blades, which are increasingly flexible and may exhibit substantial sweep, prebend, coning, and deformation. Existing engineering models remain largely specialized to either sweep effects or prebend and non-planar effects, and therefore lack the ability to represent their combined influence within a single engineering formulation. This study develops a unified and computationally efficient framework for general curved and deformed blades by combining the coupled near and far wake model for swept blades with the vortex cylinder model for non-planar rotors. An idealized coupled formulation is first introduced as the highest-fidelity realization of the proposed framework before practical simplifications are introduced. Motivated by computational fluid dynamics (CFD) results showing that sweep and prebend effects can be modeled separately and then superimposed with good accuracy, simplified coupled models are then developed for practical applications. Comparisons with higher-fidelity free-wake lifting line (LL) and blade-resolved Reynolds-averaged Navier-Stokes (RANS) simulations show that the proposed models capture the aerodynamic load redistribution effects of curved blades with good accuracy across different blade configurations. In particular, for blades combining sweep and prebend, they close the gap between the previously separate sweep-only and prebend-only engineering approaches and provide clear improvements over existing specialized engineering models. The computational cost is higher than BEM but remains orders of magnitude lower than that of LL and CFD, making the simplified models attractive for time-domain aero-servo-elastic simulations and design optimization.
Competing interests: DTU Wind and Energy Systems develops and distributes the Navier-Stokes solver EllipSys3D on commercial and academic terms. DTU Wind and Energy Systems also develops, supports and distributes HAWC2 on commercial terms, and HAWC2 is available free of charge for educational and academic research purposes.
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Status: final response (author comments only)
- CC1: 'Comment on wes-2026-85', J. Gordon Leishman, 21 Jun 2026
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RC1: 'Comment on wes-2026-85', Anonymous Referee #1, 12 Aug 2026
The paper merges two engineering corrections for Blade Element Momentum Theory (BEMT) that have been studies separately up to now: the coupled near/far wake model for swept blades, and the vortex cylinder model for non-planar rotors. The result is one framework that handles sweep, prebend and coning together. Three models are presented; the first models the effects of prebend and sweep in a coupled manner, while the latter two model sweep and prebend separately. An important finding of the study is also the validation of the coupled and “uncoupled” models with respect to higher order codes. It implies that separate models for prebend and sweep correction can be reasonably “stacked upon each-other”. With respect to lifting line and blade-resolved RANS, the new models reproduce the spanwise load redistribution well, representing a significant step-up in capability with respect to BEM. The paper is clear and fills a real gap for BEM-level tools, which matters for the sweep and prebend of current large blades. My comments are mostly minor. Please find more details below.
- The abstract and conclusions say the cost stays "orders of magnitude lower than that of LL and CFD". The reader is referred to a reference for a computational cost comparison (p. 3 (L. 82–83) to Li et al. (2025a)), which I believe does not include the coupled model proposed herein. In addition, the models presented here are not the same as those in Li et al. (2025a), as explained in section 9.2. Moreover, computational efficiency with respect to lifting line is one of the core motivations of the study, making the proposed model more applicable to optimization studies than higher order theories such as Lifting-Line. Therefore, this deserves a bit of space and discussion. I would suggest to add a timing table for an exemplary case, normalized to BEM. It is cheap to produce and it supports the paper's main selling point. It would also show how much of the NW(ideal)-VC cost the simplifications actually save, which is currently only asserted.
- P21 (table 2): why is radial induction not considered in the reference BEM results?
- Figure 7: how do you explain the difference between the BEM models? For the straight blade shouldn’t they “collapse” to a single line? Perhaps this is worth stating in the manuscript.
- P 39 - "Can be seamlessly extended to account for blade deflections". If the authors intend “mean deflections” this is a reasonable claim. How the model behaves when accounting for dynamic deflections is probably a separate and wider topic. Please account for this in text. On a broader perspective, some discussion on the applicability of these methods to aeroelastic frameworks is mentioned in the introduction and conclusions. This aspect needs to be better introduced, since modelling a dynamically-deforming rotor is different than being able to model a geometry that is fixed in time. Despite these topics being deferred to future work, which is reasonable, I would suggest to introduce if the methodologies are expected to be able to intrinsically capture these time-varying phenomena and if they are, why. Moreover, the fact that the study focuses on steady-state conditions should be briefly stated in the abstract and conclusions too for quick reference.
- Sensitivity with respect to model inputs would be nice to see: 80 sections (Sect. 8.1) and the quarter-revolution near wake (p. 11) are used throughout with no sensitivity analysis. I imagine this is the best cost-performance tradeoff based on the authors experience, but a short statement, or perhaps a pointer to Li et al. (2025a), would help.
Minor comments:
- 21: “"a setup analog to the CFD setup" – analogous?
- Figure 7: please repeat in the caption why the NW(ideal)-VC and other models are omitted
- 33: "not in the the simplified coupled models" – “the” is repeated
Citation: https://doi.org/10.5194/wes-2026-85-RC1 -
RC2: 'Comment on wes-2026-85', Anonymous Referee #2, 06 Oct 2026
An interesting article which combines the previously published work from the authors to extend 'straight blade' BEM theory with modeling capabilities for in plane curvature (sweep) and out of plane blade geometries (prebend, cone, deformation). There is quite some difficult physics that needs to be considered here, and the authors succeed to thoroughly analyse the matter following up with a rigorous comparison exercise. As in the previous work of the authors, the article is on the long side for the message it brings, and therefore, combined with the level of detail, can be perceived difficult to consume. Therefore it is suggested to help the reader and restructure a bit e.g. by removing detail that is less relevant.
Although the authors mention that future work should investigate unsteady effects as well, it is a pity that unsteady effects and the performance of this model in IEC load cases like settings such as turbulent inflow are not evaluated in the current paper (the real proof of the pudding). One would wonder, considering the complexity of the engineering models, how straightforward it is to implement these into a stable time marching aero-elastic code? It could be argued that the real challenge in aero-elastic simulations is getting the unsteady aero right, and the small differences in average load levels are less of an issue and fall in the uncertainty band. Nonetheless it is believed the article provides an important building block to get there.
(As mentioned by other reviewers, the computational cost investigation needs to be detailed to highlight the importance of developing the engineering models in comparison to using higher fidelity vortex wake methods.)
Specific comments:
-p3 from line 61 onwards. This reads more like methodology than introduction, also considering the amount of detail given herein. Line 84 then contains conclusions on the unified modeling framework that is proposed in this paper. It seems quite early to conclude this from an introduction. Suggest to reword and restructure.-p4 line 98. Include ref to KJ theorem
-p4/5 section 2.1. Are these coordinate systems purposely chosen differently in comparison to the mainstream GL coordinate systems? Switching x/y does not facilitate interpretation in this respect. Also B is mentioned as the blade root coordinate system, but later is mentioned to originate in the rotational axis? That is quite confusing, and sounds more like a hub coordinate system?
-p11 section 4. Although the start of this section mentions that a repeat is not intended, about four pages are still covered to explain the coupled near and far wake method. Please consider in how far this is a repeat of previous published work and is justified for the sake of understanding the coupled model for both sweep and prebend.
-p15 section 5. Prebend is mentioned (also before in the text) and it is presumed that out of plane prebend is meant, but better to clarify the direction explicitly. Also, coning and deformations were mentioned earlier on, but now only prebend is mentioned?
-p23 section 8.4. Perhaps add an illustration to depict the resulting blade planforms (chord, twist) that are described in the first paragraph. Is this the same chord/twist modification that is referred to in section 8.5?
Also a table overview of the significance of the different blades mB/mW/mC1-5 would help in addition to Fig 5&6.p27 section 9.1 Fig 7. It is interesting to see the difference in fall-off towards the tip between CFD/LL and BEM to already commence from roughly midboard onwards. Any idea what is causing this difference to be larger than usual, assuming that BEM and LL are using the same airfoil coefficients? Not sure if this has been previously observed in the aerodynamic IEA Task where these type of comparisons are often made?
-p28/29 Fig. 8/9. The trend between fwd and bckwd is very similar. In that sense one could argue to explain that in the text and leave out fig. 9. The same could be argued for Fig 10 and 11 on p31.
Citation: https://doi.org/10.5194/wes-2026-85-RC2 -
EC1: 'Comment on wes-2026-85', Emmanuel Branlard, 07 Oct 2026
Dear authors, please consider the comments from this third reviewer in the review of your paper:
The paper is the latest in a series of studies by the authors on the engineering aerodynamic modelling of pre-bent and swept blades. In this final contribution, the authors effectively combine the vortex cylinder and dynamic near-wake methods to capture the aerodynamic effects associated with the combined sweep and pre-bend of wind turbine blades.
The engineering framework presented in the paper is relevant to the wind energy community, as computational cost remains a significant limitation/barrier when employing medium-fidelity lifting line models for the large number of turbulent wind simulations required by current certification standards. It is worth noting that, although the overall effect of pre-bend and sweep on the integrated loads appears to be relatively small, the resulting redistribution of aerodynamic loading along the blade can significantly affect the distribution of structural loads and elastic deformations. Accurate prediction of the spanwise load distribution is therefore essential. Furthermore, the paper is well written, and the comparison of the different engineering model variants against medium and high fidelity datasets is neat, comprehensive, and thorough.
Based on the above, I would recommend publication of the paper after minor revision, primarily concerning the structure and presentation of the methodology.
Presentation of the methods:
My main concern is that the description of the methods is unnecessarily long and, for a reader who is unfamiliar with the particular framework (although familiar with other aerodynamic modelling approaches, such as lifting line methods) it is rather difficult to follow the equations and understand the purpose and role of the different theoretical sections.
In my view, the authors would benefit from providing a concise, narrative description of the underlying methodology, supported by appropriate references, and focusing the detailed discussion primarily on the modifications and extensions introduced in the present work. The sections devoted to the methodological background are currently quite extensive, but they do not necessarily provide sufficient guidance to the reader regarding the motivation for, and role of, the individual equations.
Some more specific comments:
-In Section 8.5, further clarification would be helpful regarding the authors’ comment on the pitch angle and the reference to Li et al. (2025b). The relevance and implications of this point are not immediately clear to a reader who is not familiar with the referenced study.
-It would also be valuable to include some results demonstrating the improvement achieved by using the revised coupling factor. This would help quantify the benefit of the proposed modification and provide a clearer assessment of its impact on the model performance.
Citation: https://doi.org/10.5194/wes-2026-85-EC1
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I do not find the motivation for the proposed modeling strategy convincing. The paper devotes substantial effort to constructing a complex semi-analytical correction framework around vortex-cylinder and BEM-type assumptions. This includes near-wake corrections, far-wake approximations, empirical coupling factors, Prandtl-type tip-loss treatment, superposition of sweep and prebend effects, and separate treatment of axial, tangential, and radial induction. The resulting model may improve on standard BEM and on earlier sweep-only or prebend-only engineering models, but it remains a heavily corrected low-order framework. Many of the physical effects approximated here, including finite-blade trailed-vortex geometry, radial induction, wake skewing, sweep-induced spanwise coupling, and non-planar rotor geometry, are more naturally represented in modern free-wake lifting-line methods.
This raises a more fundamental question that the manuscript does not adequately answer: why is the wind energy field continuing to invest in increasingly elaborate vortex-cylinder/BEM-style corrections when free-wake lifting-line solvers are now computationally practical for many design, analysis, and research applications? The paper repeatedly appeals to computational efficiency, but the cost argument is not demonstrated with enough specificity for realistic aero-servo-elastic design workflows. A convincing case would require a clear cost-accuracy-robustness comparison against contemporary free-wake lifting-line methods across representative load cases, including curved blades, large deflections, off-design operation, and unsteady conditions. Without such a comparison, the proposed model appears as an incremental extension of an increasingly dated engineering-model architecture rather than a necessary advance.