the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 01 Sep 2025
Comment on "Glauert's optimum rotor disk revisited – a calculus of variations solution and exact integrals for thrust and bending moment coefficients" by Tyagi and Schmitz (2025)
Status: final response (author comments only)
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RC1: 'Comment on wes-2025-105', Anonymous Referee #1, 05 Sep 2025
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AC1: 'Reply on RC1', J. Gordon Leishman, 10 Sep 2025
I appreciate the reviewer’s effort in providing a reply to my Comment. However, it is immediately clear that this “reviewer” is in fact an editor, because only an editor would know which deficiencies in the original peer review “should have been addressed.” The reply, therefore, reads less like an impartial scientific review and more like an editorial defense of the decision to publish.
The reply dismisses my critique as “opinion,” while itself expressing many opinions about physical realizability, engineering usefulness, the scope of momentum theory, and the novelty of the calculus of variations approach. This is a double standard. I am told that I cannot raise questions about the practical or scientific value of the paper because such remarks are deemed “opinion,” yet the reply itself offers opinions on precisely those same matters in defense of the article.
The issues I raised remain factual and verifiable. In the high tip-speed ratio limit, the analysis forces the swirl factor to vanish, which also forces the torque to vanish. Since power is the product of torque and angular velocity, this implies zero power in the very regime where power is claimed. Whether the reply describes this as “zero” or “approaching zero,” the contradiction remains, and the result is physically unrealizable. In the low tip-speed ratio limit, the paper presents exact integrals, yet turbines in this regime are dominated by stall, separation, and unsteady three-dimensional flow. The assumptions of momentum theory, namely steady and axisymmetric flow with a uniform pressure jump, do not apply. Presenting closed-form solutions in conditions where the governing theory breaks down is misleading. These are not opinions but inconsistencies between assumptions and applications. The published article also contains problems of transparency and reproducibility.
It is also legitimate for a Comment in an engineering journal to note that extreme limiting cases add nothing to turbine design. Modern turbines are not operated at very high or very low tip-speed ratios. At one end, the geometry becomes impractical, and at the other, turbines idle or feather. To point out that the published results provide no guidance for design or performance optimization is not “opinion” but recognition of engineering practice.
My Comment identified contradictions, misapplied theory, unexplained constants, mathematical mistakes, and unacknowledged duplication. These are matters of fact. The reply does not rebut them but reframes them as subjective judgments. That is not a scientific response but an editorial attempt to shield a publication decision from scrutiny. The original reviews were superficial at best, and the record now contains results that are inconsistent, unrealizable, and duplicated from prior work. Post-publication commentary exists to correct such oversights. To deny these corrections by dismissing them as “opinion” undermines both the journal and the integrity of the scientific record.
Citation: https://doi.org/10.5194/wes-2025-105-AC1 -
CEC1: 'Reply on AC1', Carlo L. Bottasso, 11 Sep 2025
For clarity:
- The reviewers are independent experts who are not journal editors, were not involved in the review of the original paper, and have no conflicts of interest with any of the parties concerned. (Reviewer RC1 has self-identified within the review comments so the interested reader can see that this is not an editor.)
- Likewise, the peer review of Dr. Leishman’s comment is being managed by different Associate and Chief Editors than those who oversaw the review of the Tyagi and Schmitz paper.
- The Board of WES has recently published an Editorial Note, expressing its position on certain aspects of the Tyagi and Schmitz paper (please see https://wes.copernicus.org/articles/10/451/2025).
Carlo L. Bottasso
Editor in Chief
Wind Energy Science
Citation: https://doi.org/10.5194/wes-2025-105-CEC1 -
AC4: 'Reply on CEC1', J. Gordon Leishman, 11 Sep 2025
It seems to me that the Editorial Board is applying a double standard. The journal operates under a labyrinth of rules so opaque that even the editors themselves appear not to understand them. While the official response insists that the reviewers of my Comment are “independent experts” with no conflicts of interest, their reviews read less like neutral assessments and more like defenses of the journal’s original decision to publish a flawed article. The claim that independence is assured because different Associate and Chief Editors handled my Comment than the Tyagi and Schmitz paper does not address the deeper concern: that the journal has now invested more effort in policing my Comment than it did in evaluating the original article. The recently published Editorial Note only reinforces this impression, as it seeks to shore up the credibility of the article rather than acknowledge the errors and misrepresentations that my Comment has exposed.
Citation: https://doi.org/10.5194/wes-2025-105-AC4
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CEC1: 'Reply on AC1', Carlo L. Bottasso, 11 Sep 2025
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AC1: 'Reply on RC1', J. Gordon Leishman, 10 Sep 2025
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RC2: 'Comment on wes-2025-105', Anonymous Referee #2, 09 Sep 2025
This comment pertains to a recent paper published in Wind Energy Science (“Glauert’s optimum rotor disk revisited – a calculus of variations solution and exact integrals for thrust and bending moment coefficients” by Tyagi and Schmitz (2025)) and offers additional commentary after the review process has been completed and the paper has been published. The original paper in question was reviewed by two reviewers and members of the Editorial Board of Wind Energy Science before it was published.
The comment provides statements that describe the author’s opinion on the “practicality” or “engineering usefulness” of the published paper. Technical comments are provided regarding both the high and low tip speed ratio limit as well as the applicability of idealized momentum theory in general from a “practical standpoint” for “modern turbines.”
The author also questions the novelty of the adopted mathematical approach in the published paper, namely the use of calculus of variations and the application of L’Hopital rule to obtain limiting values of both the thrust and bending moment coefficients.Line comments:
- Line 9: “The authors introduce no new physical interpretation, favoring abstract derivations over engineering context. Although mathematically elaborate, their derivation of “exact” integrals for thrust and bending moment coefficients has limited practical relevance and does not materially advance rotor theory or wind turbine engineering.”
These statements are subjective, based on the opinion of the author of the comment, and are difficult to quantitatively evaluate. The author may consider framing critiques using measures that can be objectively evaluated in scientific peer review. - Line 18: “This issue introduces a fundamental contradiction in their theory, i.e., a turbine cannot extract power without torque. While the integrals they have derived may be mathematically consistent, they do not apply to a physically realizable situation. The authors neither resolve this mathematical inconsistency nor acknowledge this physical reality, which critically undermines the credibility of their high-λ results.”
This is an interesting observation on the divergence between mathematical consistency and physical realizability. In the original paper, a’ is approaching zero, rather than being equal to zero. It may be more appropriate to restate “implying that the turbine imparts no swirl and, consequently, no torque” as “implying the turbine swirl and torque are approaching zero.” Likewise with this statement: “This issue introduces a fundamental contradiction in their theory, i.e., a turbine cannot extract power without torque.” The torque appears to be approaching zero, and is not equal to zero. It is worth noting here that this apparent mathematical contradiction is not something that the authors of the published paper introduced, but rather an inherent characteristic of simple actuator disks. A finite power P, where P=Q*Omega such that Q is the rotor’s torque and Omega is the rotational speed, implies a drop in the torque Q when Omega increases in order to hold the power constant. In the limiting case when Omega approaches infinity, Q would approach 0 and so does a’. As the author of the comment mentioned, this is a mathematical approximation and real-world rotors can behave differently. - Paragraph starting on line 26: This paragraph outlines practical limitations of Glauert’s theory in the low tip-speed ratio limit. It appears that the practical limitations of the idealized aerodynamic theory are described by the references included in the original published work, along with comments in Section 7. The discussion in this paragraph is therefore largely a critique of this class of idealized aerodynamic models (i.e. momentum theory) and does not seem to be a critique specific to the paper of relevance. For example: “Under these conditions, the flow field is highly three-dimensional, non-uniform, and often dominated by large blade section angles of attack, rendering the momentum theory assumptions of steady, axisymmetric, and uniform flow with a constant pressure jump entirely inapplicable.”
- Line 36: “The analytical effort invested by the authors in characterizing this physically irrelevant limit offers no guidance for turbine design, control strategy, or performance optimization.”
This statement is a subjective opinion that is challenging to objectively evaluate or quantify. - Line 40: “Using the calculus of variations to rederive Glauert’s third-order polynomial equation offers a modest pedagogical novelty.”
It appears that the use of “modest pedagogical novelty” in this context is subjective, and may be accurate in the opinion of the author of the comment, but a different reader may reach a difficult conclusion as the statement is challenging to objectively evaluate. Aside from engineering application, using Lagrange multipliers subject to an equality constraint may entail redundancy. For example, the derivative with respect to the multiplier returns the constraint as is. This can be seen for equations 5 and 14 in (Tyagi and Schmitz, 2025) where the latter is a re-arrangement of the former. As such, the optimization process reduces to a simple zero-gradient problem, which is identical to Glauert’s original solution, except for the introduction of a redundant variable (the multiplier) and the subsequent rejection of its entailed solution (a=1). - Lines 42-46: Similar to our comment above, this is a general critique of the limitations of Glauert’s formulation. Additionally, we couldn’t find a reference to a “math problem” in the published paper.
- Paragraph starting on Line 48: The critiques on clarity and reproducibility would ideally be addressed during the original review process of an article. The editorial board of Wind Energy Science may consider a policy that outlines the use of numerical constants, including approximation, significant figures, etc. It should be noted these constants are a result of arithmetic work. While we agree this should have been summarized or omitted altogether in the published version of the paper, such a discussion should have taken place during the original review process.
- Line 62: “While the work may be of limited academic interest to those studying the historical development of rotor theory, it certainly falls short of the novelty, applicability, and physical relevance expected of contributions to Wind Energy Science.”
The paper in question was reviewed by two academic peers as well as members of the editorial board of Wind Energy Science before it was accepted and published. It therefore appears that this statement is not accurate as written. - Line 66: “Their work most certainly does not “unlock new possibilities in wind turbine design that Hermann Glauert did not consider” (Sliman, 2025).”
This sentence quotes a statement written by a university press office. This statement appears not to be written by the authors of this paper and this statement has also not been peer reviewed. It therefore seems appropriate to remove this sentence from this comment, as this would be a comment on that press article, rather than a comment on the Wind Energy Science academic paper. At minimum, this statement could be reworked to make it clear that it was not written by the authors of the paper in question. - Paragraph on Line 70: This aspect has been addressed by the editorial board of Wind Energy Science and could be removed from this comment, now that it has been addressed.
- On a separate note, the original paper by Tyagi and Schmitz (2025) contains two mathematical mistakes in equations 33 and 47. When the integrals in both equations are transformed from ‘lambda_r’ to ‘a’, the limits of integration should be changed accordingly.
It appears that aspects of this comment seek to clarify the “practical” and “engineering” usefulness of the results described by the original academic paper. In part, this comment appears to be responding to the press/media coverage of the original paper rather than the original paper’s academic qualities or the statements by the authors in the original work. The nature of media reports that are not peer reviewed cannot be solely placed in the responsibility of the original paper’s authors. In contrast, statements and quotes attributed to the authors may be a different matter, although such statements are often outside of the standard academic peer review process. The editorial board of Wind Energy Science may consider guidance and/or a policy for authors who choose to publish in this journal regarding their engagements with media and, specifically, their characterization of their peer reviewed work in external forums.
Citation: https://doi.org/10.5194/wes-2025-105-RC2 -
AC2: 'Reply on RC2', J. Gordon Leishman, 10 Sep 2025
I appreciate the effort in providing a reply to my Comment. However, it seems to me that the response did not come from an independent referee. Only an editor would invoke what “should have been handled during review” or state that issues had “already been addressed by the board.” This is not a neutral evaluation but an editorial defense of the journal’s own decision. That is a conflict of interest. The responsibility lies with the journal, both for allowing superficial review to miss obvious flaws and for permitting the paper to be glorified in the popular press without correction or context. Remarkably, more time and energy have now been spent attempting to dismiss my Comment than appears to have been spent reviewing the original paper. The result is that apparent contradictions, errors, and duplication were overlooked during superficial review, while my effort to correct the record is met with exhaustive resistance. This imbalance confirms the problem: the journal is investing effort in defending a weak publication with obvious errors and inapplicability rather than in safeguarding the scientific record.
Line 9
I wrote that the authors introduced no new physical interpretation and that their integrals have limited relevance. This is not an opinion but a statement of fact. The paper provides no new physical insight beyond Glauert’s original work and focuses on extreme tip-speed ratio limits that are irrelevant to turbine design. Dismissing this as “opinion” while offering judgments of your own on novelty and usefulness applies a double standard.Line 18
I pointed out that the analysis implies power extraction without torque. The reply insists torque merely “approaches zero.” This rhetorical shift does not resolve the contradiction. In the published formulation, torque vanishes while power is claimed. Without torque, there is no power. This is not an inherent feature of actuator disk theory; it is a flaw in how the limit was handled and published.Paragraph starting at line 26
The reply claims my critique of the low tip-speed ratio regime is a general criticism of momentum theory. That is incorrect. My Comment identifies a specific misuse: publishing closed-form integrals in a regime where the assumptions of momentum theory collapse. That is a direct critique of the paper. Superficial review allowed this to stand.Line 36
I wrote that the analysis offers no guidance for turbine design. This is not subjective. Modern turbines do not operate in these regimes. To deny this is to deny engineering practice.Line 40
The reply concedes that the calculus of variations treatment reduces to a trivial restatement of Glauert’s result with a redundant multiplier. That concession proves my point that the claimed novelty is minimal.Lines 42–46
The reply says there is no “math problem” in the paper. Yet the authors themselves, in associated publicity, pasted all over the internet, claimed to have solved a 100-year-old problem and to have unlocked new possibilities for wind turbine design. This is entirely false. The so-called “problem” never existed in the first place. Glauert did not see his formulation as a problem but as a solution, and it has long been accepted as such. The methods presented in the paper, therefore, add no new physical insight. The closed-form integrals in extreme limits have no practical relevance and cannot improve turbine design. The fact that such exaggerated claims were repeated in press coverage without correction is damaging. The journal bears responsibility as well: by permitting superficial review and then allowing the article to be publicized as a breakthrough, Wind Energy Science helped lend credibility to a false narrative.Paragraph starting at line 48
The reply admits the constants were unexplained but insists this “should have been handled during review.” Precisely. This is proof that the original reviews were superficial. The journal cannot now dismiss the problem by acknowledging that its own review process failed.Line 62
I wrote that the work falls short of novelty and relevance. The reply answers only that two reviewers and the board approved it. That is circular reasoning. Approval by superficial review is not evidence of novelty. The fact that errors, duplication, and contradictions slipped through demonstrates the weakness of that review.Line 66
The reply objects to my reference to a press release. Yet that release, based on author interviews, declared that the work “unlocked new possibilities.” The journal allowed those claims to circulate without correction. That is an editorial failure, not a problem with my Comment.Line 70
The reply asserts that overlap with the authors’ earlier AIAA paper “has been addressed.” It has not. The Wind Energy Science article still does not cite that earlier paper, which contained the same derivations and conclusions. That earlier paper has since been deleted from public access, which is troubling in itself. The absence of citation conceals duplication, and the deletion removes the evidence from view. Readers are left with no way to know that much of the analysis had already been published elsewhere.Mathematical errors
The reply acknowledges errors in Equations 33 and 47. This is further proof of the superficiality of the original reviews. Obvious mistakes passed unchecked into print.In summary, the published article contains physical contradictions, misapplied theory, unexplained constants, mathematical mistakes, and unacknowledged duplication. These are factual issues, not opinions. The reply has not rebutted them but has reframed them as subjective to shield the journal. The journal bears responsibility for permitting superficial review to miss these problems and for allowing exaggerated publicity that misrepresented the work. Post-publication commentary exists precisely to correct such failures. To suppress it undermines the journal and the integrity of the scientific record.
Citation: https://doi.org/10.5194/wes-2025-105-AC2 -
CEC2: 'Reply on AC2', Carlo L. Bottasso, 11 Sep 2025
For clarity:
- The reviewers are independent experts who are not journal editors, were not involved in the review of the original paper, and have no conflicts of interest with any of the parties concerned. (Reviewer RC1 has self-identified within the review comments so the interested reader can see that this is not an editor.)
- Likewise, the peer review of Dr. Leishman’s comment is being managed by different Associate and Chief Editors than those who oversaw the review of the Tyagi and Schmitz paper.
- The Board of WES has recently published an Editorial Note, expressing its position on certain aspects of the Tyagi and Schmitz paper (please see https://wes.copernicus.org/articles/10/451/2025).
Carlo L. Bottasso
Editor in Chief
Wind Energy Science
Citation: https://doi.org/10.5194/wes-2025-105-CEC2 -
AC5: 'Reply on CEC2', J. Gordon Leishman, 11 Sep 2025
It seems to me that the Editorial Board is applying a double standard. The journal operates under a labyrinth of rules so opaque that even the editors themselves appear not to understand them. While the official response insists that the reviewers of my Comment are “independent experts” with no conflicts of interest, their reviews read less like neutral assessments and more like defenses of the journal’s original decision to publish a flawed article. The claim that independence is assured because different Associate and Chief Editors handled my Comment than the Tyagi and Schmitz paper does not address the deeper concern: that the journal has now invested more effort in policing my Comment than it did in evaluating the original article. The recently published Editorial Note only reinforces this impression, as it seeks to shore up the credibility of the article rather than acknowledge the errors and misrepresentations that my Comment has exposed.
Citation: https://doi.org/10.5194/wes-2025-105-AC5
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CEC2: 'Reply on AC2', Carlo L. Bottasso, 11 Sep 2025
- Line 9: “The authors introduce no new physical interpretation, favoring abstract derivations over engineering context. Although mathematically elaborate, their derivation of “exact” integrals for thrust and bending moment coefficients has limited practical relevance and does not materially advance rotor theory or wind turbine engineering.”
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RC3: 'Comments by reviewer wes-2025-105', Anonymous Referee #3, 11 Sep 2025
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AC3: 'Reply on RC3', J. Gordon Leishman, 11 Sep 2025
Please see the attached PDF file.
It is refreshing to see a review from someone who has clearly studied both the article and my Comment in detail. Yet again, there is an admission by a reviewer that the original article is mathematically incremental, modest in its contribution, and of limited practical value. These reviews all reinforce my central point: the results have no design relevance because both limits treated are non-operational for wind turbines. Publishing exact integrals in limits where either the physics is invalid or the operating point is non-existent does not advance rotor engineering.
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RC4: 'Reply on AC3 by reviewer 3', Anonymous Referee #3, 22 Sep 2025
The comment was uploaded in the form of a supplement: https://wes.copernicus.org/preprints/wes-2025-105/wes-2025-105-RC4-supplement.pdf
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AC6: 'Reply on RC4', J. Gordon Leishman, 23 Sep 2025
Rebuttal to Reviewer #3 (Second Round)
Rather than address the deficiencies of the originally published article, this reviewer chooses to digress and introduce Glauert’s generalized momentum theory and its later extensions (e.g., Burton, Sørensen), citing this as evidence that my critique is “flawed.” This is not relevant to the matter at hand. The Tyagi & Schmitz article is not a study of generalized momentum theory. It is rooted in the simple actuator-disk model, the same Glauert limit that treats the rotor as an infinite-bladed, inviscid, tip-loss-free disk with uniform loading. That is the framework within which the “exact integrals” were derived, and that is the framework my Comment addresses.
My Comment critiques the physical irrelevance of applying this simple model at the asymptotic limits of high and low tip-speed ratios. Whether the generalized momentum theory exists in the literature is beside the point. Tyagi & Schmitz did not use it. The reviewer is shifting the discussion away from the actual scope of the article and defending a theory that the authors themselves did not employ. This is not a correction of my Comment. It is a misguided redirection that clouds the issues.
Similarly, the reviewer appeals to Glauert’s Section IV and later textbooks (Burton, Sørensen) to argue for “families of optimal rotors.” This, too, is wildly misplaced. Within the simple actuator-disk model used by Tyagi & Schmitz, there cannot be multiple optimal rotors. There is only a single idealized construct, and even this construct is nonphysical at the limits in question. Invoking generalized theory at this stage conflates frameworks and attempts to smuggle in legitimacy that the article itself never earned. My original point stands. Within the scope of the article under comment, the reviewer’s notion of a “family” of rotors is not afforded by the theory the authors applied.
The reviewer also rejects my observation that extraordinary effort has been spent searching for flaws in my Comment, insisting instead on impartiality. The record shows otherwise. My Comment has now been reviewed four times, line by line, while the original article passed with only two cursory reviews that overlooked contradictions, mathematical mistakes, unexplained constants, and nonphysical interpretations. That imbalance is obvious: my Comment has been dissected far more harshly than the paper it critiques, and the journal is now expending more energy defending a weak publication than it ever devoted to ensuring its quality. That is not impartiality. It is bias.
The reviewer suggests that my Comment will “benefit the community” only if reshaped along the lines proposed in this fourth review. I disagree entirely. With only minor corrections, my Comment already exposes the fundamental deficiencies of the Tyagi & Schmitz article. I already conceded ground under pressure from the editor-in-chief as a condition for my Comment to be accepted for review in the first place. Diluting the message further by importing defenses of theories the authors never used, or by conceding ground where none should be conceded, would defeat the purpose. The role of a Comment is to correct the record, not to rehabilitate a flawed article or to pacify its defenders.
Once again, the double standard is unmistakable. The original article was waved through review despite its obvious shortcomings, while my Comment has been subjected to extraordinary scrutiny. Most troubling is that the journal has already issued an editorial decision before the commentary period has even closed, signaling a rush to protect the article’s credibility rather than to acknowledge its deficiencies.
Citation: https://doi.org/10.5194/wes-2025-105-AC6 -
CEC3: 'Reply on AC6', Carlo L. Bottasso, 23 Sep 2025
The comment was uploaded in the form of a supplement: https://wes.copernicus.org/preprints/wes-2025-105/wes-2025-105-CEC3-supplement.pdf
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AC7: 'Reply on CEC3', J. Gordon Leishman, 26 Sep 2025
The editors of Wind Energy Science required me to make two rounds of revisions to my original Comment as a condition for review. Yet they have now published that same original version (second submission) alongside their email correspondence. This makes clear that the revisions were never about improving the Comment itself, but about protecting their earlier decision to publish the Tyagi & Schmitz article.
Instead of engaging and addressing the technical flaws in the Tyagi & Schmitz article, including contradictions, mathematical mistakes, unexplained constants, and nonphysical interpretations that were overlooked in the two superficial reviews, the editors have shifted the discussion toward defending their procedures. Publishing private email correspondence without my consent and framing it in this context is a serious breach of scholarly publishing standards. By COPE guidance, editor-author correspondence is confidential and intended only to facilitate review, not to be used after the fact to defend an editorial decision.
More troubling still, this Board of Editors has already issued an Editorial Note and announced an editorial decision before the commentary period has closed. This act demonstrates that the outcome of this commentary was already predetermined. Such actions are inconsistent with impartial peer review and appear to be an apparent effort to shield the journal’s reputation rather than confront the serious technical flaws in the Tyagi & Schmitz article.
What is now required is not further editorial maneuvering but a formal retraction or correction of the Tyagi & Schmitz article so that the scholarly record accurately reflects the contradictions, errors, and lack of technical relevance in that article.
Citation: https://doi.org/10.5194/wes-2025-105-AC7
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AC7: 'Reply on CEC3', J. Gordon Leishman, 26 Sep 2025
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CEC3: 'Reply on AC6', Carlo L. Bottasso, 23 Sep 2025
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AC6: 'Reply on RC4', J. Gordon Leishman, 23 Sep 2025
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RC4: 'Reply on AC3 by reviewer 3', Anonymous Referee #3, 22 Sep 2025
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AC3: 'Reply on RC3', J. Gordon Leishman, 11 Sep 2025
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see attached file