## 3 Results and Discussions
### Active Flow Control
#### 3.2.3 Influence of Control Frequency  
The final study was performed to evaluate the effect of the control frequency on the drag coefficient reduction in the actuated cases with synthetic and unsteady jets. The influence of the control frequency of f\({}_{\rm c}\) = 68 Hz and 272 Hz on the flow structure was considered, while the flow characteristics with a control frequency of f\({}_{\rm c}\) = 34 Hz have been presented in Section 3.2.1. The evolution of the drag coefficient on the model concerning the control frequencies is shown in Table 3. It is clear that, in both flow control techniques, the drag coefficient reduction increases when high control frequency is applied (\(\Delta\)C\({}_{\rm Dmax}\) = 5.62% with a synthetic jet at f\({}_{\rm c}\) = 272 Hz and (\(\Delta\)C\({}_{\rm Dmax}\) = 17.27% with an unsteady jet at f\({}_{\rm c}\) = 272 Hz). A significant drag reduction of 8.73% was also found with the synthetic jet at the high control frequency of 400 Hz and jet velocity 75 m/s on the small-scale square-back Ahmed model [30]. This result can be explained through the analysis of the time-averaged streamlines (Figures 21 and 22) and the normalized streamwise velocity profiles at position x/H = 0.5 on the symmetry plane (z = 0) (Figure 23). When a high control frequency is applied, the control jet deviated the streamlines, reducing the size of the near-wake region. In addition, the high introduced momentum decreased the size of the recirculation zones, while pushing them away from the rear surface of the model (Figures 21 and 22). This leads to a limitation of negative streamwise velocity, with u\({}_{\rm min}\) = \(-\)0.2U\({}_{0}\) for the synthetic jet and u\({}_{\rm min}\) = \(-\)0.19U\({}_{0}\) for the unsteady jet at control frequency of f\({}_{\rm c}\) = 272 Hz, as shown in Figure 23. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Synthetic Jet** & **f\({}_{\rm c}\) = 34 Hz** & **f\({}_{\rm c}\) = 68 Hz** & **f\({}_{\rm c}\) = 272 Hz** \\ \hline C\({}_{\rm D}\) & 0.246 & 0.242 & 0.235 \\ \(\Delta\)C\({}_{\rm D}\) (\%) & 1.2 & 2.81 & 5.62 \\ \hline
**Unsteady Jet** & **f\({}_{\rm c}\) = 34 Hz** & **f\({}_{\rm c}\) = 68 Hz** & **f\({}_{\rm c}\) = 272 Hz** \\ \hline C\({}_{\rm D}\) & 0.229 & 0.226 & 0.206 \\ \(\Delta\)C\({}_{\rm D}\) (\%) & 8.03 & 9.24 & 17.27 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Drag coefficient reduction with different actuator types at various control frequencies. Figure 20: Vertical normalized streamwise velocity profiles measured at x/H = 0.5 in the symmetry plane with different actuator types at various jet angles. (**a**) A steady blowing jet; (**b**) a synthetic jet; (**c**) an unsteady jet. A spectral analysis was performed to better investigate the control frequency effect when using the synthetic and unsteady jet actuators. Figures 24a and 25a show the evo  
Figure 23: Vertical normalized streamwise velocity profiles measured at \(\mathrm{x}/\mathrm{H}=0.5\) in the symmetry plane with different actuator types at various control frequencies. (**a**) A synthetic jet; (**b**) an unsteady jet. Figure 21: Time-averaged streamlines in the behind of the model, on the symmetry plane with a synthetic jet at different control frequencies. (**a**) \(\mathrm{f_{c}}=68\) Hz; (**b**) \(\mathrm{f_{c}}=272\) Hz. Figure 22: Time-averaged streamlines in the behind of the model, on the symmetry plane with an unsteady jet at different control frequencies. (**a**) \(\mathrm{f_{c}}=34\) Hz; (**b**) \(\mathrm{f_{c}}=68\) Hz; \(\mathrm{f_{c}}=272\) Hz. lution of the drag coefficient as function of flow unit t* (t* = t.U\({}_{\rm o}\)/L) with different control frequencies for synthetic and unsteady jets, respectively. The power density spectra (PSD) of the drag coefficient signal in these cases are calculated and plotted in Figures 24b and 25b.