the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 03 Sep 2026
Brief communication: On the local disagreement of two engineering models for the radial induced velocity of non-planar wind turbine rotors
Abstract. Blade element momentum (BEM) codes traditionally neglect the radial induced velocity ur, although for the strongly deflected or prebent blades of current multi-MW turbines it produces an additional in-plane driving force. Two engineering models proposed by Li et al. (2022) — the smooth correction attributed to Madsen, driven by the area-averaged thrust coefficient, and the superposition of discrete semiinfinite vortex cylinders formed from the converged annulus inductions — were implemented independently in an open-source blade design and analysis code. For the NREL 5 MW reference rotor with an operational tip deflection of 6 m both models change thrust and power by less than 1 %, and their integral effects on the power coefficient agree to within 0.05 %. This agreement conceals a pronounced local disagreement: at mid-span the vortex-cylinder ur changes sign, following the non-monotonic radial derivative of the axial induction, and at the outermost section it grows logarithmically with the number of blade sections, ur ≃ 0.78 ln Nsec + 3.05 (m s−1), in close agreement with the analytic slope |γt tip|/2π = 0.82, reaching 6.6 m s−1 at Nsec = 70 against 2.3 m s−1 for the Madsen correction. We show that the integral agreement is systematic rather than coincidental, and that local load predictions near the tip of non-planar rotors are model- and grid-dependent and require an explicit, resolution-independent regularisation.
- Preprint
(626 KB) - Metadata XML
- BibTeX
- EndNote
Status: open (until 01 Oct 2026)
-
CC1: 'Comment on wes-2026-138', J. Gordon Leishman, 05 Sep 2026
reply
-
AC1: 'Reply on CC1', Alois Peter Schaffarczyk, 09 Sep 2026
reply
Author comment AC1 in reply to the community comment CC1 by Prof. J. Gordon Leishman on wes-2026-xx, "Brief communication: On the local disagreement of two engineering models for the radial induced velocity of non-planar wind turbine rotors"
A. P. Schaffarczyk and B. Masare, Kiel University of Applied Sciences – 5 September 2026
Dear Prof. Leishman,
We are grateful for your detailed community comment. It goes to the substance of what a brief communication of this kind can and cannot claim, and we agree with essentially all of it. Rather than replying with arguments, we have carried out the calculations your comment calls for – all at BEM cost with the same code – and we summarise here what they show and how we intend to revise the manuscript. A point-by-point response with the numerical results and three new figures is attached as a supplement to this comment; the scripts and raw results will be added to the code repository.
Table 1 (thrust). You are right that text and table were inconsistent. The "planar" column is the undeflected rotor, whereas the two corrected columns use the deflected geometry, in which the cone angle enters the inflow angle and the relative velocity through cos κ. A fourth run on the deflected geometry without the radial-velocity term separates the effects: coning reduces thrust by 0.7 % and power by 1.4 % (−67.9 kW), identically in all deflected variants, and the u_r term – which adds an in-plane force only – leaves the thrust untouched and adds +75.8 kW (Madsen) and +78.3 kW (vortex cylinder), i.e. +1.6–1.7 % of power. The "+0.2 %" of Sect. 3.1 was the net of these two effects and is wrong; the u_r effect is eight times larger than we stated.Generality of the integral agreement. We computed 5 operating points along the NREL 5 MW steady-state schedule (c_T from 0.78 to 0.24), 3 tip deflections (3, 6, 9 m) and 3 deflection shapes – the linear shape of the submission (constant κ), a cantilever bending line and an outboard-concentrated prebend shape. The 3 % agreement of the two models holds only for the constant-slope shape at high loading. For the bending line the vortex-cylinder model predicts 12–13 % more power gain than the Madsen correction, for the prebend shape 27–29 %, and at 15 m/s 14 % for every shape. The agreement we reported is a cancellation between a larger vortex-cylinder contribution outboard and a smaller, partly negative one inboard; it requires constant κ to be exact. The statement "systematic rather than coincidental" is withdrawn, as is "must coincide".
Local loads. This was your central point, and the answer reverses part of our emphasis. With a(r) and Γ(r) fixed and the vortex-cylinder sweep refined from 35 to 2240 sections, the additional sectional force on the outer 10 % of the blade converges (its peak changes by less than 2 % for N_sec ≥ 140), because the bound circulation vanishes at the tip faster than u_r grows; this holds also for a curved-tip case with κ = 30° over the outer 5 % of the blade. The logarithmic singularity of u_r is therefore not a source of grid-dependent loads, as you anticipated. What the local loads do show is the size of the model dependence: on the outer 5 % of the blade the two models differ by a factor of two in the moment contribution and in the peak sectional force (2.7 and 2.2 for the curved tip), at grid-converged values.
Regularisations. Evaluating u_r at a fixed offset δ from the tip and desingularising the sheet with a core radius r_c both give N_sec-independent tip values, but the values depend on the free parameter (6.6–8.1 m/s for δ = 1.0–0.05 m; 5.2–7.5 m/s for r_c = 1.0–0.1 m), and the implementation guard gives a plateau that corresponds to neither. We agree that the three options are different in kind and that only the coupled formulation removes the issue at the level of the formulation; the revised text says so.
Mid-span sign change. Repeating the sweep with a smoothed a(r) leaves the region of inward radial velocity (r ≈ 8–33 m) unchanged; its origin is the axial induction increasing outward at the lightly loaded root. It follows from the construction of the models, as you say – the point we will make is that the Madsen expression, driven by the cumulative c_T,av > 0, cannot represent inward flow at all, whereas the vortex-cylinder model responds to the loading gradient and hence also to any numerical waviness of the BEM solution.
The logarithmic tip behaviour and the guard are presented as what they are, a verification of the implementation and an implementation finding, in a shortened section.
We propose to keep the brief-communication format but to change its message from "integral agreement, local grid dependence" to "integral agreement only for constant slope; integral differences of 12–29 % for realistic and prebent shapes; local loads model-dependent by a factor of two but grid-independent; u_r worth +1.6 to +3.9 % of power". Concretely: an extended Table 1 with the deflected-no-u_r column and the corrected percentages; a short paragraph in Sect. 2 on how κ enters the BEM kinematics and on the three deflection shapes; Sect. 3.2 renamed "Verification of the tip behaviour" with the regularisation comparison as a new figure; two new short subsections with one figure each on the parametric study and on the local loads; abstract and conclusions rewritten accordingly; and the title changed to "… on the local and integral disagreement …". The linear 6 m case is kept as the reference case for continuity, but it is now labelled as the constant-slope shape, and the bending line and prebend shape are added. The revised manuscript is in preparation and will be submitted with the response to the referees.We thank you again for a comment that has substantially improved our understanding of what the manuscript can claim.
Alois Peter Schaffarczyk and Bhima Masare
Citation: https://doi.org/10.5194/wes-2026-138-AC1
-
AC1: 'Reply on CC1', Alois Peter Schaffarczyk, 09 Sep 2026
reply
Viewed
| HTML | XML | Total | BibTeX | EndNote | |
|---|---|---|---|---|---|
| 188 | 54 | 15 | 257 | 11 | 11 |
- HTML: 188
- PDF: 54
- XML: 15
- Total: 257
- BibTeX: 11
- EndNote: 11
Viewed (geographical distribution)
| Country | # | Views | % |
|---|
| Total: | 0 |
| HTML: | 0 |
| PDF: | 0 |
| XML: | 0 |
- 1
This communication addresses the behavior of two engineering models for radial induced velocity on non-planar wind turbine rotors. The subject is potentially useful, particularly the observation concerning the numerical behavior of the vortex-cylinder formulation near the blade tip. However, I do not think the manuscript in its present form provides a sufficiently complete or convincing contribution for publication in Wind Energy Science.
My main concern is that the conclusions go well beyond what is demonstrated. The analysis is based on a single rotor, a single operating condition, a single prescribed deflection distribution, and a single implementation of each model. From this, the authors conclude that the agreement of the Madsen and vortex-cylinder models in integral quantities is "systematic rather than coincidental." I don't think the conclusion follows from the results presented. A systematic result would require calculations over a meaningful range of rotor loading, operating conditions, non-planar geometries, and perhaps different induction distributions. The statement in the conclusions that extensions to larger prebend angles and to the fully coupled formulation are still in preparation suggests that the present study is incomplete.
There is also a basic inconsistency between the method description and Table 1. The authors state that the radial induced velocity is added after BEM convergence and that the axial momentum balance is left unchanged, "so thrust is identical in all variants by construction." Yet Table 1 gives 699.8 kN for the planar case and 695.1 kN for both corrected cases. This is not a round-off difference. Either the description of the calculation is incomplete, the table is wrong, or the cases are not being compared on the basis stated in the text. This needs to be resolved because it affects the interpretation of the principal results.
The logarithmic increase in radial velocity near the outer edge of the vortex-cylinder model is clearly shown. However, it appears to arise directly from the known singular behavior of the elliptic-integral solution as the evaluation point approaches the ideal vortex sheet. The authors themselves derive this behavior analytically and then demonstrate it numerically. That is a useful verification of the implementation, but I am not convinced that it constitutes a new aerodynamic result. If the outermost collocation point approaches the cylinder edge as the grid is refined, the divergence is the expected consequence of the idealized mathematical model.
More importantly, the paper does not establish that this local velocity singularity produces a significant error in a physically relevant load. The authors emphasize that the vortex-cylinder model reaches about 6.6 m/s in the outermost section, compared with about 2.3 m/s in the Madsen model. However, they also show that the circulation tends to zero toward the blade tip, so the region in which the velocity becomes singular contributes very little to the integrated force. The additional driving moment changes only from about 62.3 to 61.6 kNm over the very large refinement range from 35 to 2240 blade sections. In other words, the paper itself demonstrates that the singular velocity has very little effect on the integral aerodynamic result.
The authors then suggest that the problem is important for tip extensions, aeroelastic tailoring, and curved winglets. That may well be possible, but it is not demonstrated in this paper. If the physical significance is supposed to lie in local loads, then the consequences of local loads need to be shown. For example, the paper could demonstrate an effect on sectional force, local bending moment, structural response, or another quantity relevant to blade design calculations. At present, the large local difference in induced velocity is being used as a proxy for physical importance, but the manuscript does not show that the two are equivalent.
I am also not persuaded by the explanation given for the apparent agreement in the integral quantities. The authors state that because the Madsen expression is calibrated against vortex-cylinder flow for a uniformly loaded actuator disk, the two models "must coincide" where the local loading resembles the area average. That statement is too strong. The NREL 5 MW rotor is not uniformly loaded; the calculation includes tip loss, and the radial induction distribution is explicitly nonuniform. Indeed, the manuscript reports differences of 10-35% over part of the outer blade. Similarity of the integrated power correction in this one case does not establish a general equivalence between the models.
There is a related issue with the interpretation of the mid-span disagreement. In the vortex-cylinder model, the radial velocity follows radial changes in the induction distribution through the differences in annular induction, whereas the Madsen expression is driven by an area-averaged thrust coefficient and is necessarily smoother. It is therefore not surprising that one model can change sign locally while the other cannot. This appears largely to follow from the mathematical construction of the two models. The manuscript should more clearly explain what new physical insight is gained from observing this expected difference.
The proposed remedies for the tip behavior also need to be more carefully distinguished. Evaluation on a staggered grid, introduction of a finite vortex core, and use of the fully coupled formulation of Li et al. are not equivalent forms of regularization. Moving a collocation point is a numerical treatment. A vortex-core model changes the representation of the vortex sheet. A coupled solution changes the aerodynamic formulation itself. These approaches can give different answers and require different physical justification. It is not sufficient simply to list them as alternative ways of obtaining a resolution-independent result.
The assumed blade geometry is another limitation. The calculation uses an operational tip deflection of 6 m and an approximately 5.7-degree slope along the blade. It is not clear how representative this prescribed geometry is of the actual deflected NREL 5 MW blade under the stated operating condition. More generally, the paper draws conclusions about non-planar rotors, prebent blades, curved tips, and related applications based on a single idealized deflection case. Because the additional driving force is directly dependent on the local blade slope, the assumed shape is not a minor detail.
Finally, the numerical guard imposed on the elliptic-integral modulus is certainly worth pointing out. A hidden clipping condition that changes a mathematical divergence into an apparent grid-independent plateau can be important in code verification. In my view, however, that is a numerical implementation issue rather than sufficient evidence on its own for the broader aerodynamic conclusions drawn in the paper.
Overall, the manuscript identifies an interesting numerical feature of the uncoupled vortex-cylinder implementation, but it does not yet establish the broader physical significance claimed for that feature. The generality of the integral agreement is not demonstrated; the local singularity is largely an expected property of the ideal vortex-sheet representation; the physical consequences for blade loading are not established; and there is an unresolved inconsistency in the reported thrust results. Addressing these points would require a substantially broader study rather than a normal revision of this brief communication.