## 3 Validation  
During the numerical computation, the average tangential force coefficient of a blade per revolution was calculated to confirm a convergence. In addition, the criterion was defined as the relative error between two consecutive revolutions lower than 0.25%. Simulation results were obtained and then analyzed.  
A series of sensitivity analyses of the spatial discretization (mesh element size and blade element length) and temporal discretization (azimuthal angle step size) were carried out, and the instantaneous tangential force coefficients were compared. In terms of the meshes, besides a basic MI mesh, as mentioned above, a refined M2 mesh was used as a comparison, with a smaller element size of 0.4 \(c\)\(\times\) 0.4 \(c\)\(\times\) 0.4 \(c\). The total number of elements of M2 was 3,338,616, almost triple that of MI. The results with the two meshes exhibit good conformability in Figure 4a, with a relative error of 0.4% for the average tangential force coefficients. Therefore, the M1 mesh was selected for the following simulations. The blade element length is related to the Gaussian distribution width (\(\epsilon\)) in the ALM, which equaled the blade chord length in the present study. Three lengths of \(\epsilon\), 2/3 \(\epsilon\) and 1/2 \(\epsilon\) were chosen, and it was found that the relative errors of the two former lengths, compared to the latter, were 2.6% and 0.6%, respectively (Figure 4b). Thus, the blade element length was set to \(\epsilon\), corresponding to 27 elements in total for a blade. As for the azimuthal increments, simulations with azimuthal angle step sizes of 2.0\({}^{\circ}\) and 1.0\({}^{\circ}\) were executed. As shown in Figure 4c, the differences between the two results were quite tiny, with less than a 0.1% relative error. As such, an azimuthal angle step of 2.0\({}^{\circ}\) was determined in the present study.  
Figure 4: Instantaneous tangential force coefficients of a blade with different (**a**) meshes, (**b**) blade element lengths and (**c**) azimuthal angle step sizes.  
Figure 3: (**a**) Computational domain and (**b**) three mesh zones of the domain (not to scale).  
The ALM employed in the present study had been validated in the aspect of power coefficients (\(C_{P}\)) of a VAWT for a wide range of tip-speed ratios [33]. The simulation results by the ALM showed satisfying agreement with the experimental data, especially for moderate-to-high tip-speed ratios. Other than that, the \(C_{Fn}\) and \(C_{FI}\) of a blade on the equator were calculated and compared with the measured ones by [PERSON] et al. [36]. The wind turbine model used in the experiments had a diameter of 2 m and a span length of 1.2 m with two straight blades. The airfoil was NACA0021, and the chord length was 0.265 m. The inlet velocity of the wind tunnel was 8 m/s, and two moderate tip speed ratios, 2.58 and 2.19, were chosen to study. From Figure 5, it is clear that the calculated results fit well with experimental data. To be specific, the peaks of the \(C_{Fn}\) in the fore half, as well as the \(C_{FI}\) in the aft half, were predicted well, but the peaks of the \(C_{FI}\) in the fore half and the \(C_{Fn}\) in the aft half were overpredicted with acceptable errors.  
Therefore, the present ALM was proven to properly predict the aerodynamics of a VAWT, and the following studies were then carried out.