# Application of Simultaneous Symmetric and Cambered Airfoils in Novel Vertical Axis Wind Turbines
## 3 Methodology
### Mesh and Timestep Independency  
Due to the type of study, 2D simulations are used (Figure 3a). Structural meshes are applied for the boundary layers while the rest of the domain is covered by triangular cells. Figure 3b,c show the four layers of the Cartesian mesh as a boundary layer around the S815 airfoil; the other airfoils have the same boundary layer meshes. The quadrilateral boundary layer cells have a height of 4 \(\times\) 10\({}^{-5}\) m for the first layer. This mesh has orthogonal quality greater than 0.2, and all of the boundary layer cells have orthogonality equal to one. The maximum wall y\({}^{+}\) is less than four in all cases.  
For an appropriate mesh, mesh independency analysis is carried out for the power coefficient (Figure 4a). The results are consistent for meshes denser than mesh_2 for all cases. Therefore, mesh_2 with 2.4 \(\times\) 10\({}^{5}\) cells is used, which results in low computational costs. For the denser meshes, C\({}_{\text{P}}\) deviation from mesh_2 is less than 1.5%.  
Figure 4: (**a**) Grid independency, (**b**) time step independency.  
Figure 3: VAWT mesh around S815 airfoil: (**a**) rotating domain; (**b**) close view of trailing edge; (**c**) close view of the leading edge.  
Two-time step values are considered for dynamic and static simulations. For the dynamic simulation, the value of the time step was based on 0.5\({}^{\circ}\) of the rotor rotation. It means that, by considering the mentioned rotational speed of VAWTs (\(\omega\) = 76 rad/s), the time step is equal to 1.1482 \(\times\) 10\({}^{-4}\) s. Therefore, it agrees with the Courant-Friedrichs-Lewy (CFL) criteria, as mentioned by [PERSON] et al. [54] for all of the studied tip speed ratios (TSRs) in this work. It should be noted that the value of the mentioned CFL number depends on freestream velocity. Therefore, for every TSR, the value of the CFL number is different from others and should be evaluated. However, according to the CFL of Equation (1), if the CFL number for the highest freestream velocity satisfies our criteria for all other cases, CFL criteria will be fulfilled. Therefore, the mentioned CFL criteria are checked for the highest inlet wind speed, presenting the highest CFL number in this study as follows:  
\[\text{CFL}(\Delta\alpha)=\frac{\text{R}_{\text{g}}\cdot\Delta\alpha}{\Delta \text{x}}\Bigg{(}1+\frac{1}{\text{R}_{\text{g}}\cdot\frac{\omega\alpha}{\text {V}_{\infty}}}\Bigg{)}=0.1. \tag{1}\]  
where \(\text{R}_{\text{g}}\) is 1.18 m, \(\text{V}_{\infty}\) is 29.9 m/s, \(\Delta\alpha\) is 0.5\({}^{\circ}\), and \(\Delta\text{x}\) is 0.12 m. The calculated CFL number for the worst case is about 0.1, lower than the theoretically allowable limit, which was 0.5. Therefore, this time step is suitable for all considered TSRs of this dynamic simulation.  
To select the appropriate time step value for the static simulation, static torque coefficient independency is checked for different time steps; then, the biggest time step, through which the results are independent, is employed. According to Figure 4, for the time step of 0.004 (s), the results are independent of the time step value. The static torque coefficients for the smaller time steps deviate less than 1%.