## 3 Results and Discussion
### VPM and XFOIL Simulations--DU40 Airfoil  
Similar to NACA64, the set of the results of the thick airfoil, i.e., DU40, are presented in Figures 10-13. For the lower Reynolds number (\(Re_{C}=4000\)), the lift force history shown in Figure 10a,c resembles a regular sine wave, with relatively low fluctuation frequency and larger magnitude as compared to the high Reynolds number scenario in Figure 10b,d. Instantaneous flow fields when peak lifts are observed are shown in Figures 11 and 12 and compared to the XFOIL steady-state boundary layer streamlines. Figure 11: Instantaneous fields of DU40 airfoil at \(\alpha=0^{\circ}\). Vorticity: (**a**) \(Re_{C}\) = 4000 and (**b**) \(Re_{C}\) = 60,000. Velocity: (**c**) \(Re_{C}\) = 4000 and (**d**) \(Re_{C}\) = 60,000. Pressure coefficients, Cp: (**e**) \(Re_{C}\) = 4000 and (**f**) \(Re_{C}\) = 60,000. Steady-state simulation of boundary layer streamlines by XFOIL: (**g**) \(Re_{C}\) = 4000 and (**h**) \(Re_{C}\) = 60,000. Note: (**a**,**c**,**e**) correspond to the first red star in Figure 10a (\(Ut/C\) = 53.365) while (**b**,**d**,**f**) correspond to the first red star in Figure 10b (\(Ut/C\) = 3.396). For (**e**–**h**), the associated mean force coefficients are attached on the right. Figure 12: Instantaneous fields of DU40 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. Pressure coefficients, \(Cp\): (**e**) \(Re_{C}\) = 4000 and (**f**) \(Re_{C}\) = 60,000. Steady-state simulation of boundary layer streamlines by XFOIL: (**g**) \(Re_{C}\) = 4000 and (**h**) \(Re_{C}\) = 60,000. Note: (**a**,**c**,**e**) correspond to the first red star in Figure 9b (\(Ut/C\) = 59.315) while (**b**,**d**,**f**) correspond to the first red star in Figure 9d (\(Ut/C\) = 3.05). For (**e**–**h**), the associated mean force coefficients are attached on the right. The slower and larger lift fluctuation are linked to the larger wake vortices observed in the low-Reynolds number flow (see Figures 11a and 12a). No reattachment is found in the low-Reynolds number flow. On the other hand, the higher Reynold number flow makes the turbulent-like boundary layer more attached to the surface, as can be seen in Figures 11b and 12b. Similar Reynolds number effects can be observed in the XFOIL simulated steady-state boundary layer streamlines (Figures 11g,h and 12g,h). However, VPM simulations give thinner boundary layers in high Reynolds number flow (Figures 11d and 12d). By comparing the pressure fields in Figures 11e,f and 12e,f, it can be seen that the pressure recovers faster downstream in the high Reynolds number flow, leading to a smaller mean drag as compared to the low Reynolds number pressure fields. The surface pressures at representative instants of the lift history, i.e., the red stars labeled in Figure 10, are shown in Figures 13 and 14 and compared to the XFOIL steady-state solutions. At the low Reynolds number (Figures 13a-d and 14a-d), the upstream  
Figure 13: Instantaneous distribution of surface pressure coefficient of DU40 airfoil. For \(\alpha=0^{\circ}\), \(Re_{C}=4000\) and \(Ut/C=\mathbf{(a)}\) 56.365, \(\mathbf{(b)}\) 56.775, \(\mathbf{(c)}\) 57.115, and \(\mathbf{(d)}\) 57.66. For \(\alpha=0^{\circ}\), \(Re_{C}=60\),000 and \(Ut/C=\mathbf{(e)}\) 3.396, \(\mathbf{(f)}\) 3.472, \(\mathbf{(g)}\) 3.608, and \(\mathbf{(h)}\) 3.708. Note that the blue-colored symbols represent the upper surface while the red represents the lower surface. portion of the surface pressures remains steady for most of the time while more fluctuations can be observed on the leeward surface due to the unsteady wake. The mean value of surface pressures agrees with the XFOIL steady-state result in low-Reynolds number flow. For the high Reynolds number flow (Figures 13e-h and 14e-h), significant pressure fluctuations caused by flow separation and the roll-up vortices can be observed on the downstream portion of the surfaces.