The authors proposed a general approach to account for flow curvature effects in Actuator Line (ALM) simulations of Vertical-Axis Wind Turbines (VAWTs). The new methodology was validated against wall-resolved simulations of a fictitious, single blade VAWT, proving its effectiveness in improving load predictions over one rotor revolution. The Reviewer believes that the topic is very interesting and worthy of investigation. The study is extensive and is supported by a solid theoretical framework. Nonetheless, several aspects need improvement before publication:

1.  Abstract: the abstract should be more concise and better highlight the impact and novelty of the work;

2. Introduction: flow curvature in VAWT is an old problem and there are many approaches available in the literature. The authors should better emphasize the novelty and advantage given by their method over the existing ones, such as the conformal mapping by Migliore (which is relatively similar to this one as an approach);

3. The theoretical framework is quite hard to follow. It is recommended to discuss the general theory (Sec. 3.5) first and then present the specific cases. This would also reduce the word count, which is quite substantial for this manuscript;

4. The numerical set-up of both ALM and wall-resolved simulations (timestep, numerical settings, near-wall treatment, etc...) is too limited for quality assessment and repeatability. Please provide more details;

5. Section 4: given the problem at hand and the use of a more detailed smearing function as the one proposed for the pitching moment, a kernel of c/4 seems quite coarse. Was a sensitivity analysis performed?

6. Was an unsteady aerodynamics model used in the ALM simulations? Given the high c/R radius and so average reduced frequency, it is expected to have a notable impact on the results;

7. Section 6: It would be more effective to add the case without correction, to quantify the impact of the proposed method;

8. Section 6: The authors discussed that their method is valid for relatively thin airfoils. It would be interesting to test it on one of the thicker ones actually used in VAWT design, such as the NACA0018 or 0021

This paper presents a thorough and detailed explanation of an improved analytical model for flow curvature corrections of lift, drag, and moment coefficients for reduced-order modeling of cross-flow turbines. Detailed equations, as well as the approximations made, are presented in the body of the paper and supported in several appendices. 

Technical comments

- Lines 27 and 28 mention that high-fidelity numerical simulations are a precise approach for modeling cross-flow turbines, and CFD simulations are used later on for validation. However, CFD can still produce significant errors when modeling the unsteady dynamics of cross-flow turbines. Whether or not the CFD used for validation has been compared to experimental results should be addressed.

- Line 87: It is not clear which 2 stall angles are being referred to

- Lines 114-115: It would be helpful to show the normal forces being referred to here on the figure to illustrate the force directions being described

- Line 118: Clarify why the circulation is set as negative

- Equation 1: Clarify or introduce the "z" variable

- Figures 4 and 6: Clarify or explain the significantly larger error in the Cm_c/4 plots

Typographical comments

- Line 9: "inviscid part correspond to" should be "inviscid part corresponds to"

- Line 84: Missing a space in "Section3"

- Figure 1: Font sizes on axes should be larger

- Figure 5: Caption refers to plot (c) when there is no such plot shown

- Line 416: "Here, We use" should be "Here, we use"

- Line 662: "various pitch angles and for angle up to stall" should be "various pitch angles and for angles up to stall"