## 4 Validation with a High Fidelity Solver
The concurrent drooping and thickening of the airfoil nose over its forward 10% leads to a smoother acceleration of the boundary layer (lower pressure gradient), which is retained over a larger portion of the suction side forming a high velocity pocket (flow velocities 2.5 times higher than the free stream velocity) up to 10-20% of the chord. Furthermore, the overall flattening of the whole suction surface, driven by the thickening of the forward 7-20% of the LE region leads to retarded separation of the flow, which, as also predicted by Foil2w code, at this high AoA (19\({}^{\circ}\)) is limited to the aft 20% of the airfoil section (low velocity pocket in the vicinity of the morphed TE part). Flow velocities depicted in the contour plot of Figure 15 exhibit a smooth variation at the connection parts, confirming again that the geometric constraints imposed in the optimization loop prevent geometric irregularities. In Figure 16a, the pressure coefficient distributions predicted by Foil2w and MaPFlow code at the C\({}_{\text{L,max}}\) AoA of the PSO airfoil are compared (19\({}^{\circ}\) for MaPFlow and 17.5\({}^{\circ}\) for Foil2w). Comparing the two distributions, two differences are distinct: (a) MaPFlow predicts slightly higher suction peak resulting in a higher C\({}_{\text{L,max}}\) and (b) Foil2w predicts a slightly more extended flow separation at the TE region (pressure plateau at the rear part of the airfoil) even at the lower angle of attack of 17.5\({}^{\circ}\). In Figure 16b, the pressure distribution predicted by MaPFlow is presented for the two optimized airfoils at the C\({}_{\text{L,max}}\) AoA (angle of C\({}_{\text{L,max}}\) 19\({}^{\circ}\) for both the PSO and the ES airfoils). Results of Figure 16b (predicted by MaPFlow code) are qualitatively very similar to those of Figure 12 (predicted by Foil2w). The main difference is that Foil2w predicts much lower suction peak for the ES airfoil, which justifies the lower C\({}_{\text{L,max}}\) predicted by the same code for the particular airfoil. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Solver**} & \multicolumn{3}{c}{**C\({}_{\text{L,max}}\)**} & \multicolumn{3}{c}{**\% Difference with NACA 64A010**} \\ \cline{2-6}  & **NACA 64A010** & **PSO** & **ES** & **PSO** & **ES** \\ \hline Foil2w & 1.098 & 2.271 & 2.067 & 106.96 & 88.31 \\ MaPFlow & 1.046 & 2.502 & 2.360 & 139.28 & 125.72 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Relative increase in CL,max for the PSO and ES airfoils, using Foil2w and MaPFlow. Figure 14: C\({}_{\text{L}}\) polars produced by Foil2w (dotted lines) and MaPFlow (continuous lines) for the PSO and ES geometries, along with the C\({}_{\text{L}}\) polars of the original airfoil. The marked regions denote the appearance of C\({}_{\text{L,max}}\) and the operation points of interest. MaPFlow predictions confirm that disturbance of the pressure at the connection points is lower for the PSO airfoil. The whole design process has been based on the \(\mathrm{C_{L,max}}\) optimization, which is critical during the take-off and landing segments of the flight. However, another important factor, especially during the take-off procedure, is the aerodynamic performance (\(\mathrm{C_{L}}\)/\(\mathrm{C_{D}}\)), which is related directly to the fuel consumption of the aircraft. Most commercial transport aircraft are optimized to fly with a small incidence at an altitude close to 11,000 m. While climbing to or descending from this optimal altitude the aircraft will travel with a large incidence, therefore, will operate inefficiently and as a result will consume a lot of fuel. It is expected that the designed airfoils will produce increased drag because of the highly deformed shape at the droop nose LE. This is confirmed by the MaPFlow predictions (Table 7) which suggest larger \(\mathrm{C_{D}}\) values for the morphed airfoils at the \(\mathrm{C_{L,max}}\) AoA.