## 3 Results and Discussion
### VPM and XFOIL Simulations--NACA64 Airfoil
For the low-Reynolds number flow shown in Figure 7a-d, the lower surface experiences higher suc  
Figure 6: Instantaneous fields of NACA64 airfoil at \(\alpha=0^{\circ}\). Vorticity: (**a**) \(R_{\text{C}}=4000\) and (**b**) \(R_{\text{C}}=60,000\). Velocity: (**c**) \(R_{\text{C}}=4000\) and (**d**) \(R_{\text{C}}=60,000\). Pressure coefficients, \(C\): (**e**) \(R_{\text{C}}=4000\) and (**f**) \(R_{\text{C}}=60,000\). Steady-state simulation of boundary layer streamlines by XFOIL: (**g**) \(R_{\text{C}}=4000\) and (**g**) \(R_{\text{C}}=60,000\). Note: (**a**,**c**,**e**) correspond to the first red star in Figure 5a (\(Ut/C=55.06\)) while (**b**,**d**,**f**) correspond to the first red star in Figure 5b (\(Ut/C=11.68\)). For (**e**–**h**), the associated mean force coefficients are attached on the right. For high Reynolds number flow, the VPM results in Figure 7e-h show higher suctions along the upper surface before separation, which leads to the higher mean lift coefficient, i.e., \(\overline{C_{L}}=0.34\). However, XFOIL shows reversed results for this Reynolds number, i.e., the upper surface has lower suctions than the lower surface. This leads to a lower mean lift coefficient, \(\overline{C_{L}}=-0.44\), as compared to its low Reynolds number result, \(\overline{C_{L}}=-0.13\) (Figure 5g,h). Significant spatial pressure fluctuations can be observed for \(x/C>0.65\) in VPM simulations, which are due to the shedding of small-scale vortices (see Figure 6b,f). Sudden drops of suctions at \(x/C>0.8\) can be observed. The steady-state pressure obtained from XFOIL has a similar pressure recovery at \(x/C>0.9\) on the lower surface. For \(\alpha=10^{\circ}\), a similar set of results is shown in Figures 8 and 9. Figure 8 shows the instantaneous flow fields associated with the peak lifts, i.e., at times \(Ut/D=51.26\) and \(4.49\), for \(Re_{C}=4000\) (Figure 5b) and \(60,000\) (Figure 5d), respectively. Under this angle  
Figure 7: Instantaneous distribution of surface pressure coefficient of NACA64 airfoil. For \(\alpha=0^{\circ}\), \(Re_{C}=4000\), \(Ut/C=\mathbf{(a)}\) 55.06, \(\mathbf{(b)}\) 55.23, \(\mathbf{(c)}\) 55.42, and \(\mathbf{(d)}\) 55.57. For \(Re_{C}=60,000\), \(Ut/C=\mathbf{(e)}\) 11.684, \(\mathbf{(f)}\) 11.89, \(\mathbf{(g)}\) 12, and \(\mathbf{(h)}\) 12.08. Note that the blue-colored symbols represent the upper surface while the red represents the lower surface. of attack, Figure 8 indicates that the flow separations only occur on the upper surface. As compared to the case of \(\alpha=0^{\circ}\), the higher angle of attack moves the separation points upstream. For \(Re_{C}=4000\), Figure 8a,c indicates that the separation occurs at 40% of the chord. No reattachment is observed in this Reynolds number. Under the separated shear layer, quiescent flow is observed. Since the wake vortices are significantly larger than those in the high Reynolds number flow, the frequency of the lift fluctuation shown in Figure 5b is lower than observed in Figure 5d. The instantaneous separation streamlines (Figure 8c) agree with the XFOIL simulation (Figure 8g) at the low Reynolds number. For the higher Reynolds number flow fields shown in Figure 8b,d, the higher inertia of the ambient flow still has an effect on moving the separation point downstream to about 45% of the chord. The separated flow is forced to reattach due to the high ambient flow inertia, resulting in small separation bubbles and roll-up vortices. Due to this flow pattern, the higher wind speed observed above the upper surface leads to a higher mean lift coefficient, \(\overline{C_{L}}=1.31\), as compared to the low Reynolds number result, \(\overline{C_{L}}=0.73\). As compared to the steady-state separating streamline of XFOIL in Figure 8h, the instantaneous streamlines in Figure 8d are more attached to the airfoil upper surface. Figure 8: Instantaneous fields of NACA64 airfoil at \(\alpha=10^{\circ}\). Vorticity: (**a**) \(Re_{C}=4000\) and (**b**) \(Re_{C}=60\),000. Velocity: (**c**) \(Re_{C}=4000\) and (**d**) \(Re_{C}=60\),000.