## 2 Model Description
### Numerical Model
#### 2.3.4 Meshes  
The first meshes were fully generated by COMSOL based on the type of physics included in the model; the only setting changed was the element size setting. The element size setting was varied from extremely coarse to extremely fine. Table 3 lists the specifications of all the COMSOL-generated meshes that were able to converge. These meshes showed poor results compared to experimental results; additionally, finer meshes were unable to converge. Therefore, a reference user-controlled mesh was made. A mapped mesh was used close to the airfoil, and further out, an unstructured triangular mesh was used. This mesh is illustrated in Figure 9, with a close-up around the airfoil in Figure 9. To make the reference mesh, the outline of the airfoil was enlarged, forming a domain around the airfoil. This domain was split into different sections, allowing more precise control of the element distribution in each section. In each of the split domains, a structured mesh was used with an exponential distribution normal to the airfoil surface, creating thin elements close to the airfoil surface. The distribution of each split domain along the surface was set to have a higher density at the leading and trailing edge of the airfoil, with a lower density in the middle part. This mesh shows good agreement with experimental lift coefficient data. A similar meshing strategy was employed for the modified airfoil, shown in Figure 10, and a close-up around the airfoil is depicted in Figure 10. The 3D model uses a mix of mapped and unstructured meshes on the airfoil surface. Figure 11 depicts a cross-section of the mesh, and the full mesh is shown in Figure 11. Table 4 lists the specifications of all the user generated meshes.  
\begin{table}
\begin{tabular}{c c c} \hline
**Symbol** & **Value** & **Description** \\ \hline \(U_{\infty}\) & Variable m/s & Free-stream velocity \\ \(\rho_{\infty}\) & 1.2043 kg/m\({}^{3}\) & Free-stream density \\ \(\mu_{\infty}\) & 1.814 \(\times 10^{-5}\) Pa-s & Free-stream dynamic viscosity \\ P & 101,325 Pa & Free-stream pressure \\ T & 293.13 K & Free-stream temperature \\ C & 1.8 m & Chord length \\ \(\alpha\) & Variable & Angle of attack \\ \hline \end{tabular}
\end{table}
Table 2: Model parameters.  
\begin{table}
\begin{tabular}{c c c c} \hline
**COMSOL Element** & **Number of** & **Average Element** & **Number of** \\
**Size Setting** & **Elements** & **Skewness Quality** & **Boundary Layers** \\ \hline Extra coarse & 1742 & 0.870 & 12 \\ Coarser & 2688 & 0.881 & 12 \\ \hline \end{tabular}
\end{table}
Table 3: Meshes generated by COMSOL for the stock NACA 0012.  
The number of mesh boundary layers around the modified airfoil was altered and evaluated with respect to the \(C_{L}/C_{D}\) ratio. This was executed with an airspeed of 50 m/s and a flap angle of 10\({}^{\circ}\). The results for four different angles of attack are displayed in Figure 12. An increase from 50 to 75 layers shows a significantly different \(C_{L}/C_{D}\) ratio. Additional layers show little change, especially at higher angles of attack. It was decided to use 100 layers to assure accuracy. This also indicates why the COMSOL-generated meshes performed poorly; they employ a significantly lower number of boundary layers and elements.  
\begin{table}
\begin{tabular}{l r r r r} \hline
**Airfoil Geometry** & **Flap** & **Number of** & **Average Element** & **Number of** \\  & **Angle** & **Elements** & **Skewness Quality** & **Boundary Layers** \\ \hline
2D NACA 0012 & - & 52,240 & 0.796 & 100 \\ Modified with flap & 10 & 105,800 & 0.738 & 100 \\ Modified with flap & 15 & 105,752 & 0.743 & 100 \\ Modified with flap & 20 & 105,755 & 0.745 & 100 \\
3D no wingip & - & 665,131 & 0.750 & 25 \\
3D with wingtip & - & 1,169,579 & 0.735 & 40 \\ \hline \end{tabular}
\end{table}
Table 4: User-controlled meshes.  
Figure 10: Mesh of the modified airfoil.  
Figure 9: User-controlled mesh of the 2D NACA 0012.