## 4 Conclusions  
To study the Reynolds number effects on airfoil aerodynamics, the two-dimensional (2D) vortex particle method (VPM) is used to numerically simulate the unsteady flow around a thin (NACA64) and a thick (DU40) airfoil. Simulations obtained at two chord Reynolds numbers, i.e., \(Re_{C}\) = 4000 and 60,000, and two angles of attack, i.e., \(\alpha=0^{\circ}\) and\(10^{\circ}\), are compared in detail, where \(C\) is the airfoil chord length. Steady-state simulations by XFOIL at \(Re_{C}\) = 4000 and 60,000 and wind tunnel aerodynamic force measurements at \(Re_{C}\) = 60,000 are compared, as well. The conclusions and recommendations are as follows:  
First, the observations reported by the previous numerical and experimental studies that are replicated by the 2D VPM model are:  
* As the Reynolds number increases, the laminar boundary layers become thinner before separation, and the separated flow may reattach, forming separation bubbles, vortex roll-ups, and irregular wake vortices of reduced length scales. * These kinematic observations are related to variations of the surface pressure distributions, increased lifts, and reduced drags. The main contributions of this work are the explanation of the connection between the aerodynamic loads, surface pressures, and nearby flow:
* Above the upstream portion of the upper airfoil surface, higher normalized wind speed occurs as \(Re_{C}\) increases, leading to significant increases in suction coefficients. Below the lower airfoil surface, the reduced normalized wind speed occurs as \(Re_{C}\) increases, leading to increased positive pressure coefficients. These two observations are the main causes that increase the lift coefficients. * As \(Re_{C}\) increases, stronger pressure recoveries induced by the flow reattachments are observed downstream of the separation bubble, leading to reduced drag coefficients. * At the lower Reynolds number, i.e., \(Re_{C}\) = 4000, the VPM-simulated unsteady wake vortices are larger and more organized, leading to larger wavelengths of surface pressure fluctuations and lower frequencies of aerodynamic force fluctuations. * At the higher Reynolds number, i.e., \(Re_{C}\) = 60,000, small-scale roll-up vortices shed downstream to form the wake, causing significant spatial pressure fluctuations on the downstream surfaces and more irregular force fluctuations at higher frequencies. * The wind tunnel measurements of the mean integral forces at \(Re_{C}\) = 60,000 are similar to VPM and XFOIL simulations at \(Re_{C}\) = 4000, implying little changes in aerodynamics in this range of Reynolds numbers, i.e., 4000 \(<Re_{C}\) < 60,000. Based on the analyses, recommendations for future works are as follows:
* It is not clear at this point why the pressure recoveries are stronger as the \(Re_{C}\) increases. The 2D VPM simulations at intermediate Reynolds numbers, i.e., 4000 \(<Re_{C}\) < 60,000, may assist in explaining this mechanism. * To increase the airfoil performance at low Reynolds numbers, adding roughness near the separation point may be a feasible way, since it may promote the roll-up of separating vorticity and thus the reattachments, higher lift, and reduced drag coefficients. However, this mechanism needs to be validated. * The 2D VPM simulations are shown to 'overestimate' the effects of the increasing Reynolds number at \(Re_{C}\) = 60,000. At higher angles of attack, i.e., \(\alpha>10^{\circ}\), reduced accuracy of VPM simulations is observed. These deficiencies are possibly due to the missing consideration of the 3D effects, which may be considered in future models. Although less accurate, the 2D VPM model is valuable in terms of its capability at a reduced cost to reflect the aerodynamics with Reynolds numbers at low angles of attack. Conceptualization, C.-H.W., Y.-L.L. and C.-L.F.; methodology, C.-H.W. and J.-Z.C. ; software, C.-H.W. and [PERSON]-Z.C. ; validation, all authors; investigation, C.-H.W. and [PERSON]; resources, [PERSON]-[PERSON] and [PERSON]; writing--original draft preparation, C.-H.W.