## 2 Model Description
### Numerical Model
#### 2.3.3 Boundary Conditions  
For all 2D models, the reference domain length is \(L\) = 180 m. The full domain is depicted in Figure 7; the air flows from left to right, with the inlet depicted in brown and the outlet in blue. This domain shape was selected as it is a commonly used shape in CFD simulations of NACA 0012 airfoils, where a large domain is used to minimize the effect of the boundary conditions [36; 37; 38; 39]. Figure 8 depicts the 3D model domain. Here, the wing spans the entire domain, as such there is no wingtip. The reference length of the domain is reduced to 10 m in order to reduce the computational time and increase the mesh density.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Flap Angle** & **Wind Angle Adjustment** & **Chord Length (m)** \\ \hline No flap & 0.00 & 1.80 \\
10 & \(-3.55\) & 2.11 \\
15 & \(-4.78\) & 2.10 \\
20 & \(-6.01\) & 2.09 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Air angle adjustments and chord lengths.  
The \(y\) axis has a length of \(2\cdot c=3.6\) m and is equal to the span of the wing. The blue planes are inlets and the red \(yz\) plane in the back is the outlet. The second 3D model has a domain which extends beyond the wing on one side. In turn, this model has a wingtip instead of the wing spanning the entire domain; this model is called \"with wingtip\". The domain extends \(2\cdot c=3.6\) m beyond the wingtip in the \(+y\) direction, and Figure A3 in Appendix A depicts this domain. The airfoil geometry is assigned the no-slip condition. The parameters used in the simulations are listed in Table 2. The free stream velocity is 50, 75, or 100 m/s for the 2D models and 50 m/s for the 3D model. These values were chosen because the type of airplanes that use the NACA 0012 and similar wings are typically single-engine propeller airplanes, and these are typical speeds used by these types of planes. At higher velocities, the flow will approach Mach numbers of 1 or higher and shockwaves should be visible; however, these speeds are not achieved in practice by airplanes using the NACA 0012. Moreover, a flow model is not suitable for this flow regime. The free-stream turbulent kinetic energy is calculated by  
\[k_{\infty}=0.1\frac{\mu_{\infty}U_{\infty}}{\rho_{\infty}L} \tag{16}\]  
and the free-stream specific dissipation rate by  
\[\omega_{\infty}=10\frac{U_{\infty}}{L} \tag{17}\]  
Figure 8: Boundaries of the 3D model without air beyond the wingtip. The yz plane in the back (red) is the outlet and the other (blue) boundaries are inlets.  
Figure 7: Boundary conditions of the domain, where the brown boundary is an inlet and the blue boundary is an outlet.  
The free-stream turbulent kinetic energy and specific dissipation rate are used as boundary conditions for the inlet and outlet. The inlet velocity in the x-direction is calculated by  
\[U_{\infty,x}=U_{\infty}\cos\left(\frac{\alpha\pi}{180}\right) \tag{18}\]  
and in the y-direction by  
\[U_{\infty,y}=U_{\infty}\sin\left(\frac{\alpha\pi}{180}\right) \tag{19}\]  
where \(U_{\infty}\) is the free stream velocity and \(\alpha\) is the angle of attack. \(\alpha\) is increased in an auxiliary sweep to model an increasing angle of attack, where the converged results are used for the next step. The outlet is set to have a normal stress of zero N/m\({}^{2}\). A simple potential flow model is solved for the first angle of attack and used for the initial values of the field for the first step in the auxiliary sweep.