## 4 Validation with a High Fidelity Solver
However, the substantial  
Figure 16: (a) Comparison between \(\mathrm{C_{P}}\) distributions along PSO-airfoil, simulated by Foi2w and MaPFlow, at the maximum \(\mathrm{C_{L}}\) angles of attack;(b) \(\mathrm{C_{P}}\) distribution along the PSO and ES airfoils, simulated by MaPFlow, at the maximum \(\mathrm{C_{L}}\) angle of attack. Figure 15: (a) Velocity flow field of the PSO-airfoil at AoA = 19\({}^{\circ}\) which corresponds to the \(\mathrm{C_{L,max}}\). Velocities are normalized with the free stream velocity of 75 m/s. (b) Focus on the LE region. lift increase mentioned in the above analysis compensates for the drag increase, resulting in improved aerodynamic performance as reported in Table 7. This confirms the overall aerodynamic effectiveness of the designed airfoils. In all aforementioned simulations, transitional conditions are considered. Potential increase of the surface roughness would force transition of the flow from laminar to turbulent to occur earlier, which generally leads to reduced maximum lift. In order to assess performance deterioration of the morphed sections due to earlier (forced) transition, fully turbulent flow simulations are performed with MaPFlow code. As seen in the marked regions of Figure 17 changing flow conditions from transitional to fully turbulent does not significantly affect the C\({}_{\text{L,max}}\) value of the baseline NACA 64A010 airfoil, but causes an appreciable reduction in the C\({}_{\text{L,max}}\) of both PSO and ES optimized morphed airfoils. Moreover, the C\({}_{\text{L,max}}\) AoA decreases by 1\({}^{\circ}\) for both airfoils. Overall, as opposed to the transitional case, in turbulent flow conditions the ES airfoil outperforms the PSO airfoil over the whole range of the simulated AoAs. As noted above, this effect could be important to wind energy applications, in which the blade airfoil encounters a broad range of AoAs. Table 8 summarizes the C\({}_{\text{L,max}}\) predictions for transitional and turbulent conditions and the relative increase with respect to the baseline airfoil. Changing from transitional to fully turbulent conditions mitigates the relative increase of 139% and 126%, for the PSO and ES airfoils, respectively, to the same level of 118% for both airfoils. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Airfoil** & **C\({}_{\text{L,max}}\)** & **AoA** & **C\({}_{\text{D}}\)** & **C\({}_{\text{L}}\)/C\({}_{\text{D}}\)** \\ \hline NACA 64A010 & 1.05 & \(11^{\circ}\) & 0.030 & 34.33 \\ \hline ES & 2.36 & \(19^{\circ}\) & 0.055 & 42.81 \\ PSO & 2.50 & \(19^{\circ}\) & 0.043 & 58.72 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Comparison of aerodynamic characteristics at the C\({}_{\text{L,max}}\) angle of attack. Lift, drag, angle of attack, and aerodynamic performance are presented for the initial NACA 64A010 and the morphed PSO and ES airfoils. Figure 17: Comparison between fully turbulent and transitional flow C\({}_{\text{L}}\) polars, for the PSO and ES airfoils, as well as for the NACA 64A010. The marked regions denote the appearance of C\({}_{\text{L,max}}\) and the operation points of interest. Flow simulations performed using MaPFlow.