| 27 Dec 2025
Improved modeling of flow curvature effects and actuator line method with aerodynamic moment, with application to vertical-axis turbines
Abstract. We revisit the modeling, using actuator line methods (ALM), of flow-curvature effects on airfoils and their application to vertical-axis turbines (VAT); these effects being important when the ratio between the airfoil chord and the arm length is not small. The modeling can only use as input the aerodynamic coefficients of the airfoil in uniform flow and as function of the angle of attack (obtained here using wall-resolved CFD simulations). It then consists of analytical modifications of those coefficients. The models for the normal force and moment coefficients are based on the analogy with potential flow with curved streamlines past an airfoil; they are not new, although many authors have neglected the contribution of the aerodynamic moment. Moreover, their expression depends on both the airfoil pitch angle and the location of its attachment to the arm, and we develop and validate models that cover all possibilities, up to angles near stall. The model for the tangential force is new (the analogy with that of potential flow being flawed): its inviscid part correspond to thrust (i.e; negative drag) and is required to compensate for the aerodynamic moment.
The correction models are first validated against CFD data, obtained using wall-resolved simulations, of a steady flow: a rotating NACA0015 airfoil with ratio of airfoil chord to arm length of 2/7 and at various pitch angles, up to stall; also for two attachment points of the airfoil to the arm: at mid-chord and at quarter-chord.
The ALM with the improved models is then also implemented in the CFD framework, and is used to simulate the unsteady flow corresponding to a VAT configuration: the rotating NACA0015 airfoil without pitch placed in a free stream and operating at optimal tip speed ratio, and simulated for both attachment points. For that, a novel method is also developed to explicitly enforce the moment in the ALM. The various components of the ALM results are compared with those of the reference CFD data, and are found to be in good agreement throughout the rotation cycle.
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