## 3 Results and Discussion  
The work of [PERSON] et al. [25] at \(Re=\) 3900 was used as a reference for the synthetic jet operating parameters of the present study. To the best of the authors' knowledge, there are no DNS results or other LES simulation results available in the scientific literature for the case of a cylinder in cross-flow with the inclusion of a synthetic jet actuator. The results of [PERSON] et al. [25] for a synthetic jet orifice located at \(\theta=\) 90\({}^{\circ}\) to the freestream flow were not presented specifically for that location, but rather in a three-dimensional surface plot. The results for the drag coefficient \(C_{D}\) against orifice location and actuation frequency were inferred from these results (Figure 5). To produce the same jet Strouhal number \(Sr_{o}\), Stokes number \(St_{o}\) and Reynolds number \(Re\overline{U_{o}}\), the pseudo diaphragm for the present study was driven at frequencies \(f_{a}\) between 0.9 and 6.7 Hz, at similar conditions to [PERSON] et al. [25]. At these frequencies, a similar overall \(C_{D}\) modification trend was observed, with \(C_{D}\) decreasing overall from a jet Stokes number \(St_{o}=\) 2.43 to 4.86 and then increasing again from \(St_{o}=\) 4.86 to 6.43 (see Figure 5). However, a noticeable deviation from the \(C_{D}\) results presented by [PERSON] et al. [25] was observed at \(St_{o}=\) 4.21. For this case, the pseudo diaphragm was driven at a frequency of 2.83 Hz, the same frequency as the natural shedding of vortices from the cylinder in cross-flow. This deviation from the overall trend could be the result of a weak feedback loop [39] between the synthetic jet frequency and the cylinder's vortex shedding frequency. Operating at frequencies from 50 to 360 Hz, \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(Re\overline{U_{o}}\) & \(f_{a}\) & \(St_{o}\) & \(Re\overline{U_{o}}/St_{o}^{2}\) & \(Sr_{o}\) \\ \hline
12 & 0.9 Hz & 2.43 & 2.03 & 0.49 \\
12 & 1.9 Hz & 3.44 & 1.01 & 0.99 \\
12 & 2.8 Hz & 4.21 & 0.68 & 1.48 \\
12 & 3.8 Hz & 4.86 & 0.51 & 1.97 \\
12 & 4.7 Hz & 5.43 & 0.41 & 2.46 \\
12 & 5.7 Hz & 5.95 & 0.34 & 2.96 \\
12 & 6.6 Hz & 6.43 & 0.29 & 3.45 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Operating parameters of the synthetic jet actuator in the present study, in terms of the Reynolds number \(Re\overline{U_{o}}\), actuation frequency \(f_{a}\), Stokes number \(St_{o}\) and Strouhal number \(Sr_{o}\). [PERSON] et al. [25] would have been well beyond the vortex shedding frequency and thus no feedback loop would be present. It should be recalled that the results of [PERSON] et al. [25] were produced with an alternating velocity inlet boundary condition on the surface of the cylinder. The absence of internal vortex formation within the cavity could be an additional reason for the deviation between the results of [PERSON] et al. [25] and those of the present study in certain cases (Figure 5). The root-mean-squared lift coefficient \(C_{L}^{\prime}\) experienced by the cylinder was also altered by the variation of synthetic jet actuation frequency. The trend in the drag coefficient due to increasing the actuation frequency closely matched the modification in RMS lift coefficient (Figure 6). However, the same level of agreement in the \(C_{D}\) and \(C_{L}^{\prime}\) trends was not observed in experimental measurements in comparable conditions [23; 24] and was not presented in other numerical studies [25]. Time averaging of the near field and far field of the cylinder was carried out to identify why the values of the drag coefficient \(C_{D}\) and RMS lift coefficient \(C_{L}\)/ were modified. The averaging was carried out over three vortex shedding cycles, capturing the shedding of six vortices from the cylinder. A sampling rate of 33 Hz was used, since it was found that further increasing this sampling rate  
Figure 5: Modification in drag coefficient \(C_{D}\) as a function of the synthetic jet Stokes number \(St_{o}\) for (--\(\cdot\)-) [PERSON] et al. [25] and (-\(\cdot\)-) the present study.