## 3 Results and Discussion  
### VPM and XFOIL Simulations--NACA64 Airfoil  
Segments of \(C_{L}\) time histories simulated for NACA64 airfoil using VPM for a duration \(U\Delta T/C=2\) are shown in Figure 5 for different Reynolds numbers and angles of attack. At the \(\alpha=0^{\circ}\), Figure 5a shows a negative time-averaged lift force, i.e., \(\overline{C_{L}}=-0.17\), for the lower Reynolds number, i.e., \(Re_{C}=4000\). As the Reynolds number increases to \(Re_{C}=60\),000, Figure 5c shows a significant increase in the mean lift coefficient, i.e., \(\overline{C_{L}}=0.34\), at the same angle of attack. Similar increasing trends can be observed for \(\alpha=10^{\circ}\), as well. At \(Re_{C}=4000\), the mean lift coefficient observed in Figure 5b is \(\overline{C_{L}}=0.73\), and is increased to \(\overline{C_{L}}=1.31\) at \(Re_{C}=60\),000 in Figure 5d. In addition to the lift enhancement, a higher Reynolds number flow is found to increase the irregularity of the lift time histories; namely, more regularized periodic lift histories are observed in \(Re_{C}=4000\) (Figure 5a,b) while less periodic histories are observed in \(Re_{C}=60\),000 (Figure 5c,d). Figure 4: Wind tunnel test setup for the airfoil aerodynamic force measurements. To examine the effects in detail, instantaneous flow fields and surface pressures are examined for \(\alpha=0^{\circ}\) and \(10^{\circ}\). For \(\alpha=0^{\circ}\), Figure 6 shows the flow fields at instants \(Ut/D=55.06\) and \(11.68\) for \(Re_{C}=4000\) and \(60{,}000\), respectively, which correspond to the instants labeled by the first red stars in Figure 5c. For the low Reynolds number case, the vorticity (Figure 6a) develops from the windward portion of the body wall, indicating the growth of laminar boundary layers in this region. At approximately 60% of the chord, the flow and the vorticity separate, feeding into the wake. The two laminar shear layers of opposite signs interact downstream and a Karman-type of wake forms at around 20% of the chord length downstream of the trailing edge. For the high Reynolds number case (Figure 6b), the upstream boundary layers become thinner. The higher inertia of the ambient flow forces the separation point to move slightly downstream to about 65% of the chord. Instantaneous separation bubbles form and the shear layers roll up to vortices of about 70-80% of the chord. For the low Reynolds number case, Figure 6c shows no flow reattachment at \(\alpha=0^{\circ}\) and quiescent flow beneath the separation. For the high Reynolds number flow, Figure 6d shows thinner boundary layers before separation. Above this, higher wind speeds can be observed. Figure 6e,f shows the corresponding instantaneous pressure fields with the mean force coefficients attached on the right. For the VPM, the reported mean drag coefficients consider only the pressure drag. The mean drag coefficients reported by XFOIL include both viscous and pressure components. According to [PERSON]'s equation, higher wind speed results in higher suction atop the upper surface for the higher Reynolds number flow. Below the upstream portion of the lower surface, the wind speed is reduced at the higher Reynolds number, leading to increased positive pressure in this region. The associated mean force coefficients are attached on the right of Figure 6e-h. The steady-state boundary layer streamlines obtained from XFOIL are shown in Figure 6g,h for comparison, as well. The results agree with the trend that the higher Reynolds number flow makes the boundary layer thinner and moves the separation point downstream, although the degree of the effects is not as significant as the VPM results. Figure 7 compares the instantaneous airfoil surface pressures of \(\alpha=0^{\circ}\) at the representative instants during one cycle of lift fluctuations, which are labeled by the red stars in the force histories in Figure 5a,c. Steady-state surface pressure solutions obtained from XFOIL are plotted in Figure 7, as well.