## 3 Results and Discussion
### Discussion  
At the lower Reynolds number, i.e., \(Re_{C}\) = 4000, the flow separation in both thin (NACA64) and thick (DU40) airfoils does not reattach, as observed in the VPM simulations. For the higher Reynolds number, i.e., \(Re_{C}\) = 60,000, the separation bubbles and roll-up vortices are observed in the VPM simulations. This trend is consistent with the smoke flow visualization results of NACA0025 by [15]. However, the authors of [15] mentioned that the minimum Reynolds number required to make the separated flow reattach is about \(Re_{C}\) = 135,000, 175,000, and 200,000 for \(\alpha=0^{\circ},5^{\circ}\), and \(10^{\circ}\), respectively. These Reynolds numbers are about 2-3 times the current observations from VPM simulations. From the PIV measurements of GA (W)-1 airfoil at \(Re_{C}\) = 70,000 and \(\alpha=10^{\circ}\), the authors of [14] observed flow separation bubbles and Kelvin-Helmholtz vortices near the separation point. In the numerical simulations of the flow around the Eppler-387 airfoil by [18], instantaneous roll-up vortices are observed at \(\alpha=4^{\circ}\) for \(Re_{C}\) = 60,000, 100,000, and 200,000. Similar to the current VPM simulations, their results show that the sizes of the roll-up vortices become smaller and the vortex trajectories become closer to the upper surface as the Reynolds number increases. However, as compared to the numerical simulations by [18] at the \(Re_{C}\) = 60,000, the current VPM (Figures 6b, 8b, 11b and 12b) shows that the roll-up vortices are relatively smaller and more 'attached' to the surface. As the Reynolds number increases, both VPM and XFOIL show increases in auctions on the upper surface upstream to the separation, and sudden pressure recoveries occur downstream of the separation (Figures 9e,f, 13e,f, and 14e,f). The higher upstream suction is due to the increased wind speed and thinner boundary layer Figures 6d, 8d, 11d and 12d) while the sudden pressure recovery is related to the flow reattachment. Surface pressures observed in the numerical [18] and experimental [15] tests show a similar variation trend with the Reynolds number. As a result, as the Reynolds number increases, the increases in \(\overline{C_{L}}\) (Figure 15) are due to the increased suction on the upstream upper surface while decreases in \(\overline{C_{D}}\) (Figure 16) are due to the pressure recovery caused by flow reattachment. As the reattached separated flows are generally associated with higher lift and lower drag, adding roughness near the separation point in the low-Reynolds number flow (e.g., \(Re_{C}\) = 4000) may promote early roll-ups of the shear layer and thus increase the performance of the airfoil. However, this mechanism has not been proven and merits future investigations. As the Reynolds number increases, the vortices near the airfoil become smaller and more irregular (Figures 6b, 8b, 11b and 12b), leading to more disorganized lift force fluctuations at higher frequencies (Figures 5 and 10). The wake velocity measurements by [15] show narrow peaks in the wake velocity spectra at lower Reynolds numbers, indicating organized structures of the wake flow. As the Reynolds number increases, the authors of [15] showed a spectral peak occurring at higher frequencies with the wider bandwidths, indicating a more irregular wake flow. Smoke wake flow visualizations by [15] showed narrower wakes in the higher Reynolds number flow, indicating that the length scales of the wake vortices are reduced. Although the VPM simulations can reflect the trends of Reynolds number variations, comparisons to the experimental data in Figures 15 and 16 show 'overestimations' of the Reynolds number effects. As a result, the wind tunnel measurement of the aerodynamic forces at \(Re_{C}\) = 60,000 is closer to the VPM and XFOIL simulations at \(Re_{C}\) = 4000 and small angles of attack, \(-10^{\circ}\leq\alpha\leq 10^{\circ}\). At higher larger angles of attack, both VPM and XFOIL are unable to predict the measurement results.