The authors proposed a modification of the classical momentum theory to account for the effect of streamwise velocity gradients on the rotor induction and global performance. The reviewer believes the topic and activity are very interesting, innovative and worthy of investigation. The adopted methodology is adequate for the scope of the study and consistent throughout the work. Modelling assumptions and limitations are clearly stated and discussed. Nonetheless, the presentation of the work and the organization of the manuscript need improvement.

Some specific considerations:

1. Abstract: although clearly stated in the rest of the paper, it would be more effective to also add to the abstract what is the field of application of this model (e.g., wind turbine in complex terrain);
2.  Line 60: there is a typo, it should be: "U and u are the [...]  velocities;
3. The current organization of the manuscript can be improved for clarity: the description of the numerical model should be fully contained in Section 3, while a dedicated Section should be created for the validation part before the current Section 4. Figs. 3 and 4 should be moved to this new section, as they are currently too far from the part where they are commented;
4. Figure 2: it would be nice to add a picture of the computational grid;
5. Figures 2b, 3, 4, 5, and 6 all have formatting issues: missing decimal separators in the axis ticks, missing legend entries, missing axis labels and units. Please revise this figures carefully;
6. Line 183: a practical definition of the length scale l is stated very late in the paper. It is recommended to already discuss this aspect in Section 2, where the model is presented.

The publication of the manuscript is recommended once the minor modifications described so far has been performed. 

Dear authors,

thanks for presenting this work to the community. The review is stated below and additional comments are given directly in the PDF.

Best regards,

Christian Schulz

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The paper presents a model for the relation between axial induction and thrust force of an actuator disk, considering the presence of an accelerated surrounding flow field. The model is derived analytically based on a momentum balance and tested with idealized RANS simulations. It is shown that the model improves the prediction of thrust force compared to the conventional model in the investigated test cases. However, it is noted that the model violates some physical aspects and lacks a clear determination of an introduced parameter, the length scale.

The model's simplicity and the manner in which it is tested with simplified yet suitable test cases are precise and convincing. In the end, a ‘reverse sensitivity analysis’ of the only introduced ‘tuning’ parameter of the model is performed, which is straightforward and clearly aligns with the paper's progression. Therefore, the work contains relevant contributions to the wind energy research community. However, the introduction, explanation, and discussion of the model need improvement from my point of view.

General remarks:

• The paper is quite focused. This is beneficial in some respects; however, there seems to be room for additional content. More detailed or a variation of comparison cases or a comparison with results from the literature would be valuable improvements.

• Introduction/Literature: The consideration of existing literature seems superficial. This does not necessarily mean that the authors did not consider it, but as a reader, I felt the literature was listed rather perfunctorily. The discussion of this literature in the context of the proposed model is very brief.

Abstract:

- The abstract is short and precise, which is good from my point of view. However, adding one or two sentences on the usefulness of the new development (e.g., which realistic flow situations can now be modeled?) might increase potential readers' interest.

- The last sentence is meaningful but indirect. It could be beneficial to state what the consequence of this finding would be for wind turbine design, for example.

Lines 27-48: As I understand, the sentences before were related to corrections/improvements of the momentum theory to capture the physical impacts of different flow situations. From the text, it seems a bit off-topic to cite a publication that deals with the improvement of the wind turbine performance rather than the improvement of a modelling approach.

Line 43: The term ‘linearized flow solver’ sounds rather general to me. What kind of model is used here? This information is crucial to understand the ‘computational burden’ mentioned late.

Model derivation:

• To gain a better understanding, it would be helpful to describe the basic idea of the approach before delving into the mathematical derivation. I feel that there should be included that you assume a stream tube and how it is modified due to the accelerated flow field.

• From my point of view, the shape of the stream tube is modified by the accelerated flow field. As a consequence, the sketch should be modified. Or is there any reason why this should not be the case?

• You state that the force acting on the stream tube is non-zero. On the other hand, you use the assumption of as zero pressure difference between inner and outer part of the stream tube (which is evident). Therefore, the force needs to act on the in and outlet. Is a non-zero force on the in- and outlet in accordance with your idea? Wouldn’t the stream tube itself be accelerated then? Please explain.

• Lines 100 ff.: Intermediate steps would be helpful to understand exactly what has been done here. I feel there are quite some steps to go. E.g. there is a T in equation 13 which vanished in equation 14. However, no T is given in 9,10,11 and 12. Maybe T = dp*Ar is used. It might be that I am mistaken. Please explain.

• Line 106: It seems thatL*beta equals the relative velocity increase over a certain distance l. Naming it like this could make it easier for the reader to understand the meaning of equation 15.

• When having an updated description (and sketch), I would like to check if I can follow the idea better and see if there are any concerns regarding the basic idea on my side.

Line 165: Doesn’t this contradict the sentence above? (“If it is desired to better agree with the momentum theory results, it is necessary to increase the grid resolution...”)

Line 207: What is the ‘first-order behaviour of an actuator disc’?

Line 211: It would be helpful to clearly write down which of the assumptions are inconsistent/wrong. Strictly speaking, even the assumption of an inviscid fluid is ‘wrong’. However, I believe that this is not one of the assumptions which is intended to be doubted here.

4.3 On the length scale:

In the whole text, it is somehow suggested that the length scale is something like a physical parameter, that needs to be estimated. I am not quite sure if this is really the case. One could also see it as an arbitrary value that linearly influences the induction on the new model. In fact, omitting the physical derivation, one could see the whole model as a linear deviation of the induction with the flow acceleration and a second linear tuning parameter. The main assumption of the model would than be that the deviation of the induction is linearly dependent on the flow acceleration while the extent can be tuned by the length scale. Figure 6 shows that the required length scale to match the simulation results is strongly dependent on different parameters. (Right?) Therefore, the figure gives a hint on how strong the assumptions made in the model derivation differ from the (simulated) reality. The fact that there are two branches and a range of L between 0.5 and 2 for realistic operation conditions gives an hint on how strong reality differs from the proposed model. This is not a problem in my eyes. But if this is correct, it should be stated as it is.

Conclusion:

I find the conclusion rather short. Actually, there are no conclusions on the major question of the paper: How did the proposed model work? What are the major limitations for practical use?