## 4 Results and Discussion
### Laminar Separation-Induced Transition  
#### 4.2.1 Laminar Separation-Induced Transition  
The skin friction coefficient (\(C_{f,x}\)) and TI on the airfoil's lower surface (pressure side) and upper surface (suction side) as a function of X/C are shown in Figure 7. The predictions from the \(\gamma\)-\(Re_{\theta}\) transition model are compared at two representative angles of 0\({}^{\circ}\) and 5.13\({}^{\circ}\). The \(C_{f,x}\) and TI curves for the pressure airfoil surface are shown as the blue lines, whereas those for the suction airfoil surface as the red line. The distributions on each surface clearly depicts the points of LS, TO, TR, and SBL. Figure 7a,b indicate that at an AoA of 0\({}^{\circ}\), the laminar boundary layers on the pressure and suction airfoil surfaces separate at X/C = 0.455 and 0.50, respectively. In addition, the shear layers reattach at X/C = 0.499 and 0.557, respectively. This results in SBLs of 4.44% and 5.74% the chord length, respectively. As the AoA is increased to 5.13\({}^{\circ}\), the LS on the pressure and suction airfoil surfaces is generated at X/C = 0.471 and 0.496 and is reattached at X/C = 0.523 and 0.518, resulting in SBLs of 5.17% and 2.24% the chord length, respectively. Therefore, as the AoA is increased, the bubble on the suction surface moves closer to the leading edge due to an earlier LS caused by the increased adverse pressure gradients, whereas the bubble on the pressure surface moves downstream due to a later LS. In addition, for Re = 2.0 \(\times\) 10\({}^{6}\), the bubble on the pressure side is longer by 0.73% (5.17% - 4.44%) of the chord length (compared to an AoA of 0\({}^{\circ}\)), whereas on the suction side the bubble is shorter by 3.5% (5.74% - 2.24%) of the chord length (than for an AoA of 0\({}^{\circ}\)). Similar observations were made in the experiments by [26]. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**AoA** & \multicolumn{8}{c}{**Drag Coefficient (C\({}_{d}\))**} \\ \cline{2-7}  & \multicolumn{2}{c}{**Simulation Results**} & \multicolumn{2}{c}{**Exp.**} & \multicolumn{2}{c}{**\% Error**} \\ \cline{2-7}  & \multicolumn{2}{c}{**Standard \(k\)-\(\epsilon\)**} & \multicolumn{1}{c}{**Transition Model**} & \multicolumn{1}{c}{**Maxed Laminar/ Turbulent**} & \multicolumn{1}{c}{**Standard \(k\)-\(\epsilon\)**} & \multicolumn{1}{c}{**Transition Model**} & \multicolumn{1}{c}{**Maxed Laminar/ Turbulent**} \\ \hline
0\({}^{\circ}\) & 0.0303 & 0.00646 & 0.0062 & 0.007 & 333 & \(-\)8 & \(-\)11 \\
1.02\({}^{\circ}\) & 0.0125 & 0.00653 & 0.0062 & 0.0072 & 74 & \(-\)9 & \(-\)14 \\
5.13\({}^{\circ}\) & 0.0342 & 0.0073 & 0.0069 & 0.007 & 389 & 5 & \(-\)1 \\
9.22\({}^{\circ}\) & 0.0259 & 0.0186 & 0.0416 & 0.0214 & 21 & \(-\)13 & 94 \\
14.24\({}^{\circ}\) & 0.0589 & 0.0605 & 0.0675 & 0.09 & \(-\)35 & \(-\)33 & \(-\)25 \\
20.15\({}^{\circ}\) & 0.119 & 0.114 & 0.1784 & 0.1851 & \(-\)36 & \(-\)38 & \(-\)4 \\ \hline \end{tabular}
\end{table} 
Table 2: Comparison between the simulation and experimental results for the drag coefficient at selected AoAs from different studies for the S809 airfoil (Re = 2 \(\times\) 106). \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**AoA** & \multicolumn{8}{c}{**Lift Coefficient (C\({}_{j}\))**} \\ \cline{2-7}  & \multicolumn{2}{c}{**Simulation Results**} & \multicolumn{2}{c}{**Exp.