## 2 Model Description  
### Physical Model  
An airfoil is the cross-section of a shaped surface which can generate significant lift, such as an airplane wing. An example wing including commonly used terminology is sketched in Figure 1. The chord is a straight line from the leading edge of the airfoil to the rear edge. The angle of attack \"\(\alpha\)\" is the angle between the chord line and the incoming air. A wing moving through air can generate a force perpendicular to this movement: the lift force. Another force is generated on the opposite of the direction of the flow: the drag force. By generating significant lift, heavier-than-air flight is possible.  
The National Advisory Committee for Aeronautics (NACA) has created several formulas that describe the shape of wing-shaped airfoils. In 1933, NACA released a report detailing the performance of 78 airfoils tested at high Reynolds numbers [22]. In this report they created a formula describing airfoils using a 4 digit system. The first digit indicates the amount of camber as a percentage of the chord length, the second the position of this camber in tenths of the chord length. The last two digits describe the maximum thickness of the airfoil as a percentage if the chord length. For example, a NACA 0012 has a thickness that is 12% of the chord. As the first two digits are zero, this airfoil has no camber, it is symmetrical.  
Performance Metrics  
To evaluate the airfoil performance, the lift coefficient \(C_{L}\), drag coefficient \(C_{D}\), and their ratio \(C_{L}/C_{D}\) are used. The lift force is calculated by integrating the pressure perpendicular to the airflow direction over the airfoil surface. The lift coefficient \(C_{L}\) is given by,  
\[C_{L}=\frac{F_{L}}{\frac{1}{2}\rho u^{2}A} \tag{1}\]  
where \(F_{L}\) is the lift force, \(\rho\) is the density, \(u\) is the flow speed, and \(A\) is the projected area. The drag force is calculated by integrating the total force opposite to the airflow over the airfoil surface. The drag coefficient \(C_{D}\) is calculated by,  
\[C_{D}=\frac{F_{D}}{\frac{1}{2}\rho u^{2}A} \tag{2}\]  
where \(F_{D}\) is the drag force. In 3D models, \(A\) is the platform area. In 2D models, the projected area \(A\) is the chord length \(c\), this is also known as the section drag or lift coefficient. The cross-sections of the 3D wings do not vary across the span; as such, the 2D and 3D NACA  
Figure 1: Sketch of a wing with the main parts identified.  
012 results are directly comparable. In turn, \(C_{L}\) and \(C_{D}\) are used to refer to both the 2D and 3D model. The Reynolds number for airfoils is calculated by,  
\[Re=\frac{\rho\textit{uc}}{\mu} \tag{3}\]  
where \(\rho\) is the density, \(u\) is the air velocity, and \(c\) is the chord length. The Mach number is defined as the ratio of the local velocity to the local speed of sound:  
\[Ma=\frac{u}{c} \tag{4}\]  
where \(Ma\) is the mach number, \(u\) is the velocity, and \(c\) is the speed of sound. The speed of sound, assuming an ideal gas, is calculated by,  
\[c=\sqrt{kR_{p}T} \tag{5}\]  
where \(k\) is the specific heat ratio, which is 1.4 for air, \(R_{p}\) is the particular gas constant, and \(T\) is the temperature [21; 23]. A high lift coefficient is desirable as it allows for more weight to be lifted, and a low drag coefficient is desirable as it reduces fuel costs [24]. The performance of an airfoil can be increased by adding so-called high-lift devices, for example, a flap [12].