Layout and Rotation Effect on Aerodynamic Performance of Multi-Rotor Ducted Propellers

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The paper novel approach is the employing of statistical method ART-ANOVA to analyze the correlation between propeller phase, rotational direction, and a combination of both.

The paper also presented the CFD results of side-by-side ducted propellers in various relative positioning, direction, and rotational phases. The CFD solutions were conducted using proven modeling and simulation methods from a previous publication.

Figure 1b need to specify which one is 2D-2P as the main text description was not very clear on the difference between 1D-2P and 2D-2P.

Eq. 3, forgot to define C_Q.

Eq. 5,6 should be taken out if they're not used anywhere else .

Figure 3 should show how far downstream the spanwise measurements were made.

Figure 9 is relatively poorly labeled. Randomness should probably be phrased as variability/uncertainty of the phase gap. This might show up better as line and error bar. Furthermore, while the figure showed the difference between the output, they do not give good context on how much influence those differences have on overall performance.

Please specify why the other factors with CFD solutions are not considered in the ART-ANOVA analysis. Also, specify what the ART-ANOVA tells us that not available by observing the results from the preceding sections.

Author Response

Cover Letter

Dear Editor Barry Yin and Reviewers,

We sincerely appreciate the editor's assistance in organizing the review process and the reviewers' thorough examination of our manuscript (Manuscript ID: drones-3697066). We have carefully considered each reviewer's comment and have made every effort to improve and refine the manuscript, with changes highlighted in the text. The following responses address the specific comments provided by the reviewers.

 

Comment 1

The paper novel approach is the employing of statistical method ART-ANOVA to analyze the correlation between propeller phase, rotational direction, and a combination of both.

The paper also presented the CFD results of side-by-side ducted propellers in various relative positioning, direction, and rotational phases. The CFD solutions were conducted using proven modeling and simulation methods from a previous publication.

Response 1

We appreciate the reviewer's recognition and have addressed all comments with corresponding revisions highlighted in yellow throughout the manuscript.

 

Comment 2

Figure 1b need to specify which one is 2D-2P as the main text description was not very clear on the difference between 1D-2P and 2D-2P.

Response 2

We acknowledge the reviewer's suggestion and have added labels for three cases in Fig. 1.

 

Comment 3

Eq. 3, forgot to define C_Q.

Response 3

We thank the reviewer for meticulous examination. The parameter in this equation was mislabeled and has been corrected to power coefficient (C_P) in the text.

 

Comment 4

Eq. 5,6 should be taken out if they're not used anywhere else.

Response 4

We appreciate the reviewer's suggestion. Although Equations 5 and 6 don't explicitly show calculation processes, they are implemented in parameter plots. Equation 5 is utilized for Mach number calculations in Fig. 4, Fig. 7, Fig. 10, line 199 (Section 3.1), and Table 3. Equation 6 applies to Vorticity Magnitude computations in Figs. 5, 8, and 11.

 

Comment 5

Figure 3 should show how far downstream the spanwise measurements were made.

Response 5

We thank the reviewer for the suggestion. Model setup descriptions in Fig. 3 were insufficiently detailed. The parametric study methodology is specified in lines 138-140: "When studying the influence of the spanwise parameter, ξ is set to zero. Likewise, γ adopts 1.10 when analyzing the influence of the streamwise parameter." We have supplemented this in Fig. 3's caption (line 260).

 

Comment 6

Figure 9 is relatively poorly labeled. Randomness should probably be phrased as variability/uncertainty of the phase gap. This might show up better as line and error bar. Furthermore, while the figure showed the difference between the output, they do not give good context on how much influence those differences have on overall performance.

Response 6

We acknowledge the reviewer's reminder. In studying phase gap effects, we treated phase gap as an independent variable over one rotation cycle. Dependent variables represent time-averaged values at specific phase gaps, reflecting overall performance. Thus, a time-averaged reference value was determined to precisely evaluate phase gap impacts on aerodynamic characteristics. This reference accommodates arbitrary phase differences (0°-120°), aligning with stochastic research objectives. Given its time-averaged nature, we represent it with a horizontal line rather than error bars, signifying the overall performance reference across rotational consistencies.

 

Comment 7

Please specify why the other factors with CFD solutions are not considered in the ART-ANOVA analysis. Also, specify what the ART-ANOVA tells us that not available by observing the results from the preceding sections.

Response 7

We appreciate the reviewer's suggestion. Our ART-ANOVA analysis focuses specifically on rotational phase gap and rotational consistency. Sections 3.1-3.2 investigate spanwise/streamwise parameters' aerodynamic effects via CFD, establishing variation patterns and empirical formulas. Section 3.3 examines rotational phase gap and consistency as independent variables. Crucially, spanwise and streamwise studies maintain parameter independence, while rotational characteristics exhibit coupling effects. Therefore, ART-ANOVA exclusively targets rotational properties to quantify main/interaction effects on aerodynamic performance metrics. Beyond trends identified in CFD (Section 3.3), ART-ANOVA reveals two key advances: 1) quantification of effect dominance showing rotational consistency dominates all performance metrics, while phase gap significantly affects only specific propeller parameters; 2) exposure of strong interaction effects between rotational characteristics undetectable by CFD, enhancing scientific rigor in rotational consistency research.

 

We believe that the revisions made to the manuscript, as detailed above, effectively address the reviewers' comments. Once again, we would like to express our sincere gratitude to the editor and reviewers for their valuable contributions to improving the quality of our paper.

 

Best regards,

The authors.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

 

Authors - I commend you for the thorough evaluation of the different propulsion system layouts in a hovering scenario combining a relatively high order numerical computational model and statistical modeling approaches to support your conclusions. Some comments about the work are below for your consideration: 

1) Was the ART-ANOVA analysis run separately for each of the dependent variables? If so, was some control/correction applied to reduce the risk of Type I errors? 
2) If my current understanding is correct, and the ANOVA was run for each dependent variable separately, how do you account for possible correlation or interdependence between them? I was not sure why a MANOVA analysis type was not performed. Are the raw statistics, such as reported F-value, meaningful if each variable was considered in a separate ANOVA?
3) The demonstration of the impact of rotational consistency in section 3.3 is a unique contribution, in my view. I recommend the authors add units the y-axis labels or legend labels for the thrust and power plots in Figure 9. 
4) For Section 2.4, I recommend the authors add the methodology for computing total thrust coefficient and efficiency metrics. I recognize the equations presented for a single propeller configuration - for completeness, document the reference dimensions used for total thrust coefficient, for example. 

Author Response

Cover Letter

Dear Editor Barry Yin and Reviewers,

We sincerely appreciate the editor's assistance in organizing the review process and the reviewers' thorough examination of our manuscript (Manuscript ID: drones-3697066). We have carefully considered each reviewer's comment and have made every effort to improve and refine the manuscript, with changes highlighted in the text. The following responses address the specific comments provided by the reviewers.

 

Comment 1

Authors - I commend you for the thorough evaluation of the different propulsion system layouts in a hovering scenario combining a relatively high order numerical computational model and statistical modeling approaches to support your conclusions.

Response 1

We appreciate the reviewer's recognition and have addressed all comments with corresponding revisions highlighted in green throughout the manuscript.

 

Comment 2

Was the ART-ANOVA analysis run separately for each of the dependent variables? If so, was some control/correction applied to reduce the risk of Type I errors?

Response 2

We thank the reviewer for profound statistical suggestions. ART-ANOVA was independently performed for each dependent variable, following multi-factor analysis standards to capture parameter-specific mechanisms while avoiding Type I error accumulation. Two safeguards were implemented: 1) stringent significance threshold (p<0.001 versus conventional p<0.05) to minimize false positives; 2) dual validation requiring both statistical significance (p<0.001) and practical significance (partial η²>0.13) for result interpretation, ensuring engineering relevance.

 

Comment 3

If my current understanding is correct, and the ANOVA was run for each dependent variable separately, how do you account for possible correlation or interdependence between them? I was not sure why a MANOVA analysis type was not performed. Are the raw statistics, such as reported F-value, meaningful if each variable was considered in a separate ANOVA?

Response 3

We appreciate the reviewer's suggestion. ART-ANOVA strictly adheres to physical mechanism independence principles. Our objective is quantifying rotational characteristics' effects on ducted propeller components, which require component-level analysis due to fundamentally different flow mechanisms. MANOVA's global effects would obscure component-specific mechanisms crucial for engineering optimization. Moreover, CFD-derived samples violate MANOVA's core assumptions: aerodynamic parameters exhibit significant skewness/kurtosis (Shapiro-Wilk test rejects multivariate normality), and variable correlations (Eqs. 1-4) exceed MANOVA's optimal range. Conversely, ART-ANOVA maintains robustness without distributional assumptions by preserving data order relationships. Despite inter-variable correlations, ART-ANOVA's F-statistic remains valid as it represents variance explained by independent variables relative to error variance. All statistical conclusions are corroborated by flow mechanisms, confirming they aren't correlation artifacts.

 

Comment 4

The demonstration of the impact of rotational consistency in section 3.3 is a unique contribution, in my view. I recommend the authors add units the y-axis labels or legend labels for the thrust and power plots in Figure 9.

Response 4

We thank the reviewer's suggestion. All y-axis parameters in Fig. 9 are dimensionless quantities and therefore unitless.

 

Comment 5

For Section 2.4, I recommend the authors add the methodology for computing total thrust coefficient and efficiency metrics. I recognize the equations presented for a single propeller configuration - for completeness, document the reference dimensions used for total thrust coefficient, for example.

Response 5

We acknowledge the reviewer's reminder. For consistent comparison across configurations and diameters, all aerodynamic parameters were non-dimensionalized and normalized per single ducted propeller. For Case-1D1P, parameters were computed normally. For multi-propeller cases (Case-1D2P/Case-2D2P), forces/moments were averaged (sum divided by 2) before non-dimensionalization. This ensures equivalent non-dimensional parameter ranges. Relevant descriptions were added in lines 192-198.

 

We believe that the revisions made to the manuscript, as detailed above, effectively address the reviewers' comments. Once again, we would like to express our sincere gratitude to the editor and reviewers for their valuable contributions to improving the quality of our paper.

 

Best regards,

The authors.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

General concept comments.
The article is well written and clear. Nevertheless, I think it lacks more explanation about the methods used in CFD. Drones is a generalist journal, and audience's backgrounds are diverse. I would explain better how simulations are done and why the parameters and geometry are chosen the way they are, for example.

On the other hand, I believe the data analysis is poor. The parametric modelling does not make much sense. Choosing functions ad-hoc to fit data without any physical meaning with parameters up to six figures add little knowledge to the problem studied. I think a deeper analysis must be performed.

Specific comments. Data analysis in subsections 3.1, 3.2 and 3.3 should be rethought. What is the point of choosing the functions that way? Why that level of accuracy? What is the purpose of modelling like that? Subsection 2.2 should be enhanced. The scope of the journal must be considered when explaining the models used, like turbulence, mesh, gas, etc. 

Author Response

Cover Letter

Dear Editor Barry Yin and Reviewers,

We sincerely appreciate the editor's assistance in organizing the review process and the reviewers' thorough examination of our manuscript (Manuscript ID: drones-3697066). We have carefully considered each reviewer's comment and have made every effort to improve and refine the manuscript, with changes highlighted in the text. The following responses address the specific comments provided by the reviewers.

 

Comment 1

The article is well written and clear. Nevertheless, I think it lacks more explanation about the methods used in CFD. Drones is a generalist journal, and audience's backgrounds are diverse. I would explain better how simulations are done and why the parameters and geometry are chosen the way they are, for example.

On the other hand, I believe the data analysis is poor. The parametric modelling does not make much sense. Choosing functions ad-hoc to fit data without any physical meaning with parameters up to six figures add little knowledge to the problem studied. I think a deeper analysis must be performed.

Response 1

We appreciate the reviewer's recognition of our manuscript and have implemented all suggestions with revisions highlighted in blue. The reviewer's comments are highly constructive.

 

Comment 2

Data analysis in subsections 3.1, 3.2 and 3.3 should be rethought. What is the point of choosing the functions that way? Why that level of accuracy? What is the purpose of modelling like that?

Response 2

We acknowledge insufficient physical justification for selected function forms in the original manuscript. Our core objective (Section 3) is revealing universal flow mechanisms (e.g., vortex evolution, pressure redistribution) influenced by layout parameters and rotational characteristics, while establishing lightweight empirical models for optimization frameworks. Function selection prioritized alignment with physical intuition underlying parameter-flow relationships. In the revised manuscript, we have comprehensively addressed this limitation.

Regarding Section 3.1's Boltzmann function (Eqs. 6-8): its S-shaped transition effectively characterizes the macro-trend from strong to weak interference in Fig. 4b, though we don't claim underlying statistical thermodynamics mechanisms. The reviewer's critique of fifth-order polynomial fitting in Section 3.2 is valid - we removed these non-extendable fits, retaining data figures with qualitative trend descriptions. For Section 3.3, sinusoidal fitting has strong physical basis: rotational phase gap (0°-120°) induces periodic unsteady aerodynamic loads and tip vortex interactions when blades traverse proximity zones, naturally producing periodic performance coefficient variations. High goodness-of-fit values reinforce this trigonometric characteristic.

Precision levels were determined through data analysis. Table 3 shows reference dimension D maintains 4 significant figures, while γ retains 3. Spanwise distance (D × γ) yields 3-7 significant figures depending on γ variation. Dependent variables preserve 4-6 significant figures to distinguish patterns. Origin Pro fitting employed maximum precision to ensure error tolerance. Formula precision in Sections 3.1/3.3 was adjusted based on fitting confidence intervals.

 

Comment 3

Subsection 2.2 should be enhanced. The scope of the journal must be considered when explaining the models used, like turbulence, mesh, gas, etc.

Response 3

We thank the reviewer for CFD methodology suggestions. Although references detail our methods, methodological transparency remains crucial. We supplemented specific settings and rationales in Section 2.2 (lines 154-171) to facilitate cross-disciplinary understanding.

 

We believe that the revisions made to the manuscript, as detailed above, effectively address the reviewers' comments. Once again, we would like to express our sincere gratitude to the editor and reviewers for their valuable contributions to improving the quality of our paper.

 

Best regards,

The authors.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

I believe the more common coefficients formulations are omitted in most publications. Please check with the editor on that.

In Figure 9, the vertical axis on both side of the plots showed the same scale for the same output variable (either Cp or Ct). It might be less busy without losing information if one of the vertical axis is removed.

Regarding the dash line for Cp/Ct with random phase gap is time averaged to get the phase-independant (?) coefficient, how was the randomness in phase gap generated, how many random combinations were tested? It might be easier to show it as the ensemble average of all Cp/Ct at all phase gaps tested?

The section on ART-ANOVA should include the advantage of using ART-ANOVA from the author's response. As currently presented, the authors simply stated the results without justifying the emphasis given to the statistical approach in the abstract. Otherwise, change it so ART-ANOVA is presented as a side-tool that shows the same trend as CFD results but offered a more quantifiable output. The authors might also want to discuss how the ART-ANOVA approach might alter future propulsion layout variation study design.

Author Response

Cover Letter

Dear Editor and Reviewers,

We sincerely appreciate the editor's assistance in organizing the review process and the reviewers' thorough examination of our manuscript (Manuscript ID: drones-3697066). We have carefully considered each reviewer's comment and have made every effort to improve and refine the manuscript, with changes highlighted in the text. The following responses address the specific comments provided by the reviewers.

 

Comment 1

I believe the more common coefficients formulations are omitted in most publications. Please check with the editor on that.

Response 1

We appreciate the reviewer's reminder. After verification, the Mach number—a commonly used coefficient—indeed lacked its defining equation in our manuscript. Since other equations typically appear in aerodynamic literature, we have removed the Mach number equation from this document.

Comment 2

In Figure 9, the vertical axis on both side of the plots showed the same scale for the same output variable (either Cp or Ct). It might be less busy without losing information if one of the vertical axis is removed.

Response 2

We thank the reviewer for the suggestion and have revised Figure 9 accordingly, enhancing its conciseness and clarity.

Comment 3

Regarding the dash line for Cp/Ct with random phase gap is time averaged to get the phase-independant (?) coefficient, how was the randomness in phase gap generated, how many random combinations were tested? It might be easier to show it as the ensemble average of all Cp/Ct at all phase gaps tested?

Response 3

We acknowledge the reviewer's rigor. Your suggestion aligns with our findings, particularly regarding periodic trigonometric results where averaged data effectively represents overall performance. Consequently, we recalculated the averages, with revisions highlighted in blue at lines 375–376.

Comment 4

The section on ART-ANOVA should include the advantage of using ART-ANOVA from the author's response. As currently presented, the authors simply stated the results without justifying the emphasis given to the statistical approach in the abstract. Otherwise, change it so ART-ANOVA is presented as a side-tool that shows the same trend as CFD results but offered a more quantifiable output. The authors might also want to discuss how the ART-ANOVA approach might alter future propulsion layout variation study design.

Response 4

We have incorporated the constructive feedback by adding content in the Abstract (lines 31–35) and Section 3.4 (lines 461–472), highlighted in blue.

 

We believe that the revisions made to the manuscript, as detailed above, effectively address the reviewers' comments. Once again, we would like to express our sincere gratitude to the editor and reviewers for their valuable contributions to improving the quality of our paper.

 

Best regards,

The authors.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Authors - Thank you for thoroughly addressing my comments.

Author Response

Cover Letter

Dear Editor and Reviewers,

We sincerely appreciate the editor's assistance in organizing the review process and the reviewers' thorough examination of our manuscript (Manuscript ID: drones-3697066). We have carefully considered each reviewer's comment and have made every effort to improve and refine the manuscript, with changes highlighted in the text. The following responses address the specific comments provided by the reviewers.

Comment

Authors - Thank you for thoroughly addressing my comments.

Response

We sincerely thank the reviewer for their recognition, suggestions, and assistance.

 

Best regards,

The authors.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

Thanks for revising the manuscript. Unfortunately, I still have the main concerns I pointed out in my first review. 

The parametric modelling does not make much sense. Choosing functions ad-hoc to fit data without any physical meaning add little knowledge to the problem studied. I think a deeper analysis must be performed. What is the point of choosing the functions that way? Why that level of accuracy? What is the purpose of modelling like that?

Author Response

Cover Letter

Dear Editor and Reviewers,

We sincerely appreciate the editor's assistance in organizing the review process and the reviewers' thorough examination of our manuscript (Manuscript ID: drones-3697066). We have carefully considered each reviewer's comment and have made every effort to improve and refine the manuscript, with changes highlighted in the text. The following responses address the specific comments provided by the reviewers.

Comment 1

Thanks for revising the manuscript. Unfortunately, I still have the main concerns I pointed out in my first review.

Response 1

We appreciate the reviewer's acknowledgment of our Round 1 revisions. New modifications are highlighted in yellow.

Comment 2

The parametric modelling does not make much sense. Choosing functions ad-hoc to fit data without any physical meaning add little knowledge to the problem studied. I think a deeper analysis must be performed. What is the point of choosing the functions that way? Why that level of accuracy?

Response 2

We recognize the critical need to clarify the rationale behind model selection. For the Boltzmann function, we explicitly define it as a phenomenological model in Section 3.1 (lines 241–258). We emphasize its purpose: to provide a computationally efficient and lightweight surrogate model for engineering applications, rather than to uncover fundamental physical laws. For the Sinusoidal function, we strengthened its physical basis in Section 3.3 (lines 381–401). We demonstrate that periodic blade motion inherently induces periodic unsteady aerodynamic interactions between rotors, thereby justifying sinusoidal modeling as physically grounded rather than ad hoc. Our focus lies in capturing physical trends over numerical precision, with parameters provided for reproducibility and consistency.

Comment 3

What is the purpose of modelling like that?

Response 3

To address "modeling purpose" queries and delineate our contribution scope, we added limitations and future work in the Conclusion (lines 561–570). We explicitly state that these models aim to furnish lightweight surrogates for complex multi-ducted fan systems in preliminary design and optimization.

 

We believe that the revisions made to the manuscript, as detailed above, effectively address the reviewers' comments. Once again, we would like to express our sincere gratitude to the editor and reviewers for their valuable contributions to improving the quality of our paper.

 

Best regards,

The authors.

Author Response File: Author Response.pdf

Round 3

Reviewer 1 Report

Comments and Suggestions for Authors

I have no further comments.

Reviewer 3 Report

Comments and Suggestions for Authors

Thanks for improving the manuscript. I think there is much room for a better modelling of the results, but this is a first step worth to be published.