## 4 Results and Discussion  
### Validation of the Simulation  
#### 4.1.1 Comparison of Pressure Coefficients  
In order to validate the numerical model, the results were compared with experimental data. Due to the effect of AoA on the transition location, the validation was done for a range of AoAs for which experimental data were available. The computed pressure coefficient (C\({}_{\text{p}}\)) distributions along the blade chord at different AoAs of 0\({}^{\circ}\), 1.02\({}^{\circ}\), 5.13\({}^{\circ}\), 9.22\({}^{\circ}\), 14.24\({}^{\circ}\), and 20.15\({}^{\circ}\) for a Reynolds number of \(2.0\times 10^{6}\) are compared with the experimentally measured values in Figure 6[26, 29]. The blade chord presented is expressed as a normalized distance of X/C. The computed pressure coefficients calculated from the transition model show a very good agreement with the pressure measurements on the suction and pressure sides for all the angles. On the other hand, the simulation results of the standard \(k\)-\(\varepsilon\) turbulence model were overpredicted or underpredicted over the blade chord for the all the angles due to a delay in LS. It is noteworthy that small discrepancies between the pressure coefficients by the transition simulation exist near the suction side below X/C = 0.5 and over X/C = 0.8 for the high angle of 14.24\({}^{\circ}\). These discrepancies might be due to a stall by the vortex shedding and a strong vertical disturbance over the suction surface, occurring near a stall angle of 16\({}^{\circ}\)[32]. Similarly, discrepancies can be observed near the suction side for the high angle of 20.15\({}^{\circ}\). Nonetheless, these discrepancies are of such small magnitude that they are unlikely to affect the flow field caused by LSBs on the airfoil. The transition ramp, which is defined as the region over which the bubble moves with increasing AoA, was found on the suction side, as can be clearly seen in Figure 6a-c[26, 29]. It is interesting to  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{\begin{tabular}{c} **Grid** \\ **Name** \\ \end{tabular} } & \multirow{2}{*}{**Max. \(y^{\star}\)**} & \multirow{2}{*}{**\
\begin{tabular}{c} **Wall Normal** \\ **Expansion** \\ **Ratio** \\ \end{tabular} } & \multirow{2}{*}{**Number of Grid Elements**} & \multirow{2}{*}{**
\begin{tabular}{c} **Total** \\ **Cells** \\ \end{tabular} } \\ \cline{5-6}  & & & **Pressure** & **Suction** & & **Orthogonal** & **Wake** \\  & & & **Side** (Lower** & **Side** (Upper** & **Direction** & **Direction** & **Direction** \\ \hline Grid 1 & \textless{}1 & 1.1 & 95 & 95 & 150 & 35 & 33,750 \\ Grid 2 & \textless{}1 & 1.1 & 190 & 190 & 150 & 70 & 67,500 \\ Grid 3 & \textless{}1 & 1.1 & 380 & 380 & 150 & 140 & 135,500 \\ Grid 4 & \textless{}1 & 1.1 & 1000 & 1000 & 150 & 400 & 360,000 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Grid independence study.  
Figure 5: C\({}_{\text{d}}\) percent change based on C\({}_{\text{d}}\) = 0.006497 for the finest mesh (Grid 4) at each calculation as a function of mesh resolution for the NREL S809 airfoil with AoA = 0\({}^{\circ}\) and Re = \(2\times 10^{6}\).  
note that as the AoA is increased, the TO point moves forward as the suction side pressure gradient becomes more adverse. Since the flow field characteristics around an airfoil are closely related to the change in axial and angular momentum [33] that leads to the pressure distribution on the airfoil, accurate predictions of the pressure coefficients along the airfoil provide the necessary validation for the subsequent LSB analysis.