# Numerical Investigation of Jet Angle Effect on Airfoil Stall Control
## 3 Results and Discussion  
### Static Airfoil Simulations  
#### 3.1.1 Baseline (No Flow Control)  
The OA209 airfoil is computed with the compressible Navier-Stokes solver SU2 in rotorcraft-relevant flow conditions: \(Ma=0.3,Re=1.15\) M and \(0^{\circ}\leq\alpha\leq 20^{\circ}\). The current simulation predicts the static stall angle \(\alpha\simeq 14^{\circ}\), the maximum lift coefficient \(c_{l,max}\simeq 1.3\), and the post-stall lift, as shown in Figure 3. It can be conjectured that the noticeable difference to the test in the linear lift region is related to 3D effects in the wind tunnel test, which cannot be modeled in the current 2D simulation (see Figure 3). The current simulation follows the theoretical lift slope \(\frac{dc_{l}}{da}=\frac{2\pi}{\sqrt{1-Ma^{2}}}\) in the linear region, whereas the test data slightly deviate from the theoretical line. Considering 2D environment, it should be note that the agreement to the test data ([PERSON] et al., 2017) is remarkably improved in the current simulation, compared to other simulations for the same \begin{table}
\begin{tabular}{c c c} \hline \hline
\begin{tabular}{c} **Case** \\ \end{tabular} & \(\Delta t\) & \(t_{c}/\Delta t\) \\ \hline \multirow{2}{*}{Static airfoil} & No actuation & \(10^{-4}\) s & 30 \\  & Jet actuation & \(10^{-5}\) s & 300 \\ \hline \multirow{2}{*}{Pitching airfoil} & No actuation & \(2\times 10^{-5}\) s & 150 \\  & Jet actuation & \(10^{-5}\) s & 300 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Time steps of the current simulation. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{\
\begin{tabular}{c} Freestream \\ (characteristic \\ condition) \\ \end{tabular} } & \(P_{\infty}\) & 56 kPa \\ \cline{2-3}  & \(T_{\infty}\) & 304 K \\ \cline{2-3}  & \(\rho_{\infty}\) & 0.64 kg/m\({}^{3}\) \\ \cline{2-3}  & \(\psi_{\infty}/\
u_{\infty}\) & 4 \\ \hline \multirow{4}{*}{
\begin{tabular}{c} Actuator \\ (characteristic \\ inflow) \\ \end{tabular} } & \(P_{t}\) & 290 kPa \\ \cline{2-3}  & \(T_{t}\) & 300 K \\ \cline{2-3}  & \(u_{i}/|u_{i}|\) & surface normal \\ \cline{2-3}  & \(\psi_{jet}/\
u_{\infty}\) & 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Boundary conditions in the current simulation. airfoil [19; 37] in Figure 3. The agreement on the aerodynamic moment \(c_{m}\) is acceptable. The drag coefficients are also well-predicted with values before the stall angle which was tested. Pressure profiles between the test and the current computations are compared, as shown in Figure 4. The current pressure profiles at \(\alpha=5,14\), and \(19^{\circ}\) are agreed well to the available test data. The current simulation slightly overestimate the suction peak, resulting in a slightly higher lift in the linear slop region in Figure 3. Based on 3D simulation of [PERSON] et al. [37], the wind tunnel experiments [36] involve small, but non-negligible 3D efforts primarily due to a tip flow near side walls, which slightly reduces the lift compared to the 2D case. Current stall phenomena are related to a supersonic pocket near the leading edge at high angles of attack (see Figure 5). The presence of the shock in the accelerated flow is expected because the suction peak at the stall angle \(\alpha=14^{\circ}\) is lower than the critical pressure coefficient \(c_{p}^{*}=-6.9\) in both the experiment and the simulation (see Figure 4). The shock near the leading edge thickens the boundary layer, if it does not immediately cause the flow separation, which is prone to separation under an adverse pressure gradient downstream. It should be emphasized that the current good agreement to test data do not imply that the current SA RANS approach should be used for prediction of general turbulent flow separation. As [PERSON] et al. [41] suggested, the limitation of typical RANS computations over various flow separation cases would require advanced, higher-fidelity approaches for general turbulent flow simulation.