## 3 Results and Discussion
Figure 6: Modification of (-\(\cdot\)-\(\cdot\)-) RMS lift coefficient \(C_{L}^{\prime}\) and (-\(\cdot\)-) drag coefficient \(C_{D}\) as a function of the synthetic jet Stokes number \(St_{o}\). yielded no change in values. Near-field locations of interest were: (i) the stagnation point at the front of the cylinder (\(\theta=0\)); (ii) the top and bottom separation points \(\theta_{sep_{1}}\) and \(\theta_{sep_{2}}\); (iii) and the angle between these two separation points \(\theta_{\Delta sep}\) (Figure 7). The position of a stagnation point was determined by examination of the pressure on the surface of the cylinder. To determine the point at which flow separates from the cylinder, the wall shear stress \(\tau\) was used to calculate the dimensionless skin friction as \(\tau\sqrt{Re}/\big{(}\rho U_{\infty}^{2}\big{)}\). The two separation points are determined by the location where the skin friction equals zero. Both surface pressure and wall shear stress were recorded at 1016 locations on the surface of the cylinder, allowing for sufficiently accurate positioning. Looking first at the stagnation point on the surface of the cylinder, there was small change in position and only a slight variation in pressure, \(P_{stag}=22.6\) Pa with a standard deviation of \(\sigma_{P_{stag}}\)= 0.1 Pa. Thus, the effect of the synthetic jet is weak at this location (\(\theta\cong 0\)), far upstream of the synthetic jet orifice. For the case of an aerofoil, it has been shown that synthetic jet actuation can delay separation and hence reduce drag [10; 15; 16; 17; 18]. For the present case, the positions of the top and bottom separation points \(\theta_{sep_{1}}\) and \(\theta_{sep_{2}}\) respond differently as the actuation frequency is varied. The top separation point location is only weakly correlated to an increase in actuation frequency (Figure 8a). For all cases, separation occurs downstream of the synthetic jet orifice, which is located at a fixed position of \(\theta=\frac{\tau}{2}\cong 1.57\) rad. Thus, the actuator is always acting on decelerating flow that has not yet separated, although subjected to an adverse pressure gradient. However, for the bottom separation point--located on the opposite side to the synthetic jet orifice--a strong correlation is found between the actuation frequency and the separation point location, as evidenced by a high coefficient of determination R\({}^{2}\) = 0.908 in Figure 8b. The higher the actuation frequency, the more the bottom separation point moves downstream, towards the rear of the cylinder (Figure 8b). Although the top separation point location \(\theta_{sep_{1}}\) is only weakly correlated to the actuation frequency, a strong correlation is observed between the top separation point  
the separation points \(\theta_{\Delta sep}=\theta_{sep_{2}}-\theta_{sep_{1}}\) is also just weakly correlated with the actuation frequency (Figure 10a). A stronger relationship is observed between \(\theta_{\Delta sep}\) and the drag force (R\({}^{2}=0.931\)). As \(\theta_{\Delta sep}\) increases, the drag force decreases (Figure 10b). This contrasts the known relationship between the separation point and drag for aerofoils, where delaying separation results in a drag reduction [10; 15; 16; 17; 18]. Ultimately, the modification in drag coefficient \(C_{D}\) is the result of a changing surface pressure distribution. Since the drag force integration contains \(\cos\theta\) (Equation (7)), any negative pressure between \(\theta=\pi/2\) and \(3\pi/2\) gives a positive force in the \(x\) direction. The drag force per meter span is given by:  
\[F_{x}=\frac{1}{2}\int_{0}^{2\pi}P(\theta)D\cos\theta d\theta \tag{7}\]  
A strong correlation is observed between the drag and the minimum pressure over the rear half of the cylinder (Figure 11a). As this pressure decreases, the drag increases. Figure 8: Comparison of the modification of the separation points as a function of actuation frequency: (**a**) Top separation point location \(\theta_{sep_{1}}\); (**b**) Bottom separation point location \(\theta_{sep_{2}}\).