## 3 Results and Discussion
### Comparison with Experiments and Other Data  
The time average aerodynamic lift and drag coefficients obtained from wind tunnel experiments (WT) and numerical simulations (VPM and XFOIL) are compared in Figures 15 and 16, respectively. The aerodynamic coefficients utilized in the FAST software for the NREL 5 MW baseline wind turbine [26] are given in the figures, as well. Since FAST is used for full-scale wind turbine simulation, the aerodynamic coefficients are associated with \(Re_{C}=750,\)000. In these figures, the blue-, red-, and black-colored symbols are used to represent the results of \(Re_{C}=4000\), 60,000, and 750,000, respectively. As can be seen in Figure 15, the highest mean lift coefficients can be observed at the highest Reynolds number for \(0^{\circ}\leq\alpha\leq 30^{\circ}\). The lift coefficients generally decrease as the Reynolds number decreases for \(0^{\circ}\leq\alpha\leq 30^{\circ}\). By comparing the results of \(Re_{C}=4000\) and 750,000, the maximum lift reduction is about \(\Delta C_{L}=1\) at \(\alpha=10^{\circ}\) for NACA64 and \(\Delta C_{L}=1.8\) at \(\alpha=20^{\circ}\) for DU40. For the mean drag coefficients shown in Figure 15, the drag coefficients generally decrease as the Reynolds number increases for low angles of  
Figure 16: Time-averaged drag coefficients: (a) NACA64 and (b) DU40. Figure 15: Time-averaged lift coefficients: (a) NACA64 and (b) DU40. attack \(-10^{\circ}\leq\alpha\leq 10^{\circ}\). Therefore, the effects of Reynolds number variations on the airfoil aerodynamic performance are significant. At \(Re_{C}=750,000\), agreements of \(C_{L}\) (Figure 15) and \(C_{D}\) (Figure 16) in XFOIL simulations and FAST can be found in \(-10^{\circ}\leq\alpha\leq 10^{\circ}\). At \(Re_{C}=60,000\) as well as \(\alpha=0^{\circ}\) and \(10^{\circ}\), VPM generally overestimates \(C_{L}\) (Figure 15) and underestimates \(C_{D}\) (Figure 16), which is obtained by wind tunnel experiments. On the other hand, the XFOIL predictions of \(C_{L}\) and \(C_{D}\) are generally closer to the measurements at \(Re_{C}=60,000\) for \(-10^{\circ}\leq\alpha\leq 10^{\circ}\). Deviations of XFOIL \(C_{D}\) from the measured values can be observed for \(\alpha>10^{\circ}\) or \(\alpha<-10^{\circ}\). At \(Re_{C}=4000\), better agreements can be found in VPM and XFOIL simulations for \(-10^{\circ}\leq\alpha\leq 10^{\circ}\). For larger angles of attack, \(\alpha>10^{\circ}\), VPM gives higher lifts and drags. These excessive lifts and drags may be caused by the missing 3D mechanisms in the 2D simulations for higher angles of attack. Figures 15 and 16 show that the force coefficients of wind tunnel measurements obtained at \(Re_{C}=60,000\) are closer to the coefficients of simulations (XFOIL and VPM) at the low Reynolds number, i.e., \(Re_{C}=4000\). This implies that the aerodynamics observed at \(Re_{C}=4000\) and 60,000 may be similar. On the other hand, significant differences in aerodynamics may exist in \(Re_{C}=60,000\) and \(Re_{C}=750,000\). To examine the progressive changes in the aerodynamics, XFOIL is applied to compare the steady-state integral forces coefficients (Figure 17a,b) and surface pressures coefficients (Figure 17c-f) at \(Re_{C}=4000\), 60,000, 120,000, 240,000, 480,000, and 750,000. The lift (Figure 17a) and drag (Figure 17b) coefficients of the NACA64 airfoil show that significant changes occur in \(60,000\leq Re_{C}\leq 120,000\). As the Reynolds number further increases, i.e., \(Re_{C}\geq 120,000\), the force coefficients remain constant, indicating that the effects of the Reynolds number variations saturate. On the other hand, for DU40 airfoil, larger variations of the force coefficients over the wider Reynolds number (\(12,000\leq Re_{C}\leq 480,000\)) are observed. As the Reynolds number increases, a significant increase in the auctions on the upstream portion of the upper surface (Figure 17c,e), as well as the increase in positive pressures on the lower surface (Figure 17d,f) contribute to the increase in lift force coefficients (Figure 17a).