# Aerodynamic Efficiency Improvement on a NACA-8412 Airfoil via Active Flow Control Implementation
## 2 Numerical Method
### Mesh Assessment  
Due to the fact that no previous experimental or numerical studies are available in the literature, it is essential to validate the numerical model. In order to study the baseline case and the later AFC configurations, a mesh independence test was performed to guarantee independence between the mesh resolution and the simulated results. A hybrid mesh configuration, composed by a rectangular structured and a triangular unstructuredmesh region, was implemented in order to capture the boundary layer while saving computational power required on the outer regions of the domain. The surrounding region of the airfoil consists on two body-fitted structured sub-meshes with a progressive growth of 1% for the one closest to the airfoil and 3% for the one surrounding the first sub-mesh. The remaining portion of the domain was filled with an unstructured mesh, the resolution was particularly high on the wake region. An overall view of the computational domain is presented in Figure 3, a close up view of the regions surrounding the airfoil for the baseline case and for the corresponding AFC mesh implementation is shown in Figure 4 and Figure 5, respectively. Four mesh candidates were designed according to four decreasing \(y^{+}\) values, and consequently an increasing mesh resolution, as it can be observed in Table 2 from the number of mesh cells \(N_{cell}\) employed in the CFD models. Due to the lack of experimental data for the NACA-8412 airfoil at \(Re=68.5\times 10^{3}\), the evaluation of the proper mesh could not be based on the relative error between experimental and computational data. Alternatively, the strategy followed was to study the temporal averaged \(C_{l}\) and \(C_{d}\), as well as \(C_{P}\) and \(C_{P}\) distributions along the chord for each mesh candidate, and to compare them to observe for which \(y^{+}\) the results did not significantly change. This approach was also applied to evaluate whether the relative error between the results compensated in terms of computational cost. A baseline case simulation was performed for each mesh candidate for a computational time of 30 s, which was determined based on the evaluation of the velocity and pressure residuals. The time step was of \(2\times 10^{-4}\) s. In order to avoid the transient phase, the resulting time averaged for the lift \(C_{l}\) and drag \(C_{d}\) coefficients were computed considering the last 10 s of each simulation. The resulting averaged aerodynamic coefficients together with the relative error \(C_{l}Error\), \(C_{d}Error\) given in percentage are presented in Table 2. Calculated \(y^{+}\) values were assessed on the upper side of the airfoil due to its importance in the later AFC implementation, and its maximum value obtained for each case is also included in Table 2. The resulting pressure and friction coefficients are introduced in Figures 6 and 7, respectively. Table 2 shows that the time averaged aerodynamic coefficients stabilize as \(y^{+}\) decreases. For instance, the relative error between \(y^{+}\) = 0.3 and \(y^{+}\) = 1 is 14.93% for \(C_{l}\) and 9.14% for \(C_{d}\). On the other hand, the relative error between \(y^{+}\) = 0.3 and \(y^{+}\) = 0.5 is 3.35% for \(C_{l}\) and 1.97% for \(C_{d}\). As the difference among the aerodynamic coefficients for \(y^{+}\) = 0.3 and \(y^{+}\) = 0.5 can be considered relatively small (\(\Delta C_{l}=-0.0425\) and \(\Delta C_{d}=4.8\times 10^{-3}\), with reference values corresponding to the \(y^{+}\) = 0.3 results), and the computational time for the \(y^{+}\) = 0.3 resulted in nearly the double than the one required for \(y^{+}\) = 0.5, the results corresponding to the mesh with an estimated \(y^{+}\) = 0.5 were considered acceptable for the present simulations. It is relevant to highlight that the calculated maximum \(y^{+}\) values on the airfoil upper surface were slightly lower than the estimated ones used to determine  
Figure 5: Mesh modification for the AFC jet implementation. (**a**) Close up view of the structured sub-mesh. (**b**) Close up view of the jet area.