## 3 Results and Discussion
Figure 9: Correlation between the separation point locations and the drag force, for the range of actuation frequencies: (**a**) Top of the cylinder \(\theta_{sep_{1}}\); (**b**) Bottom of the cylinder \(\theta_{sep_{2}}\). The near-wall turbulent kinetic energy \(k\) directly at the rear of the cylinder (\(\theta=\pi\)) has been monitored. As before, averaging was carried out over three vortex shedding cycles at a sampling rate of 33 Hz. This was done to determine how the jet operating conditions altered the turbulence intensity (Equations (8) and (9)) and if indeed any correlation exists between the turbulence intensity \(I\) and the drag coefficient \(C_{D}\). The turbulent quantities are defined as follows:  
\[u^{\prime}=\sqrt{\frac{1}{3}\Big{(}{u^{\prime}_{x}}^{2}+\ {u^{\prime}_{y}}^{2}+{u^ {\prime}_{z}}^{2}\Big{)}}=\sqrt{\frac{2}{3}k} \tag{8}\]  
\[I=\frac{u^{\prime}}{U} \tag{9}\]  
Figure 11: Change in drag force \(F_{x}\) and minimum pressure over the rear of the cylinder as a function of \(St_{o}\): (**a**) Change in pressure and drag with \(St_{o}\); (**b**) Change in drag with pressure. Figure 10: Comparison of the magnitude of the separation angle \(\theta_{\Delta{sep}}\) (i.e., angle between top and bottom separation points) and the drag force, for the range of actuation frequencies: (**a**) Change in \(\theta_{\Delta{sep}}\) with \(St_{o}\); (**b**) Change in drag force \(F_{x}\) with \(\theta_{\Delta{sep}}\). where \(u^{\prime}\) represents the standard deviation of the flow velocity and \(U\) is the mean flow velocity, with subscripts \(x\), \(y\) and \(z\) denoting the three spatial directions. \(k\) is the turbulent kinetic energy obtained from the two-dimensional CFD calculation. This relationship between turbulence intensity at the rear of the cylinder and the drag coefficient is found to be particularly strong (Figure 12b). For all actuated and unactuated cases, whenever the turbulence intensity at \(\theta=\pi\) increases, there is a corresponding decrease in the drag coefficient (Figure 12a). This suggests that high momentum fluid is being injected into the near-wall sections of the boundary layer, which is in agreement with other studies [9; 40]. It is worth noting that the freestream turbulence intensity applied at the velocity inlet boundary condition is fixed at 5% in the present study, in contrast to [PERSON] and [PERSON] [41] who varied the inlet turbulence intensity to investigate its effect on the lift and drag force on a circular cylinder in cross-flow. During the suction phase, retarding flow would be accelerated towards the orifice and entrained into the cavity with higher momentum fluid in the freestream outside the viscous sublayer being drawn into the boundary layer. During the ejection phase, further momentum is added to the retarding viscous sublayer. As previously discussed, the synthetic jet orifice for all cases modelled was upstream of the separated flow region. This means that the synthetic jet would be operating in a region with an adverse pressure gradient and so an increase in momentum would be indicated by a decrease in the magnitude of this adverse pressure gradient across the jet orifice. A positive correlation is found between the reduction in magnitude of the surface pressure gradient \(\frac{dP}{d\theta}\) across the orifice (at \(\theta=\frac{\pi}{2}\)) and the turbulence intensity \(I\) (at \(\theta=\pi\)). The perceived link between the two variables is most pronounced at the higher end of the frequency range under investigation, from a jet Stokes number \(St_{o}\) between 4.2 and 6.4. Where the pressure gradient magnitude across the orifice decreases, a higher turbulent intensity is observed at the rear of the cylinder, suggesting agreement with other published results [9; 40] on the advantageous increase in momentum of the fluid downstream of the orifice, thus modifying \(C_{D}\) (Figure 13a).