## 2 Numerical Approach
\begin{table}
\begin{tabular}{c c c c c c} \hline  & **Drag** & **RMS Lift** & **Vortex** & **Rear Surface** & **Flow Separation** \\  & **Coefficient** & \(C_{D}\) & **Coefficient** & \(C_{L^{\prime}}\) & **Sheeding Sr** & **Coefficient** & \(C_{P_{h}}\) \\ \hline Other published & \multirow{2}{*}{1.74} & \multirow{2}{*}{1.42} & \multirow{2}{*}{0.26} & \multirow{2}{*}{\(-\)2.16} & \multirow{2}{*}{72\({}^{\circ}\) and 288\({}^{\circ}\)[25]} \\ results [25; 35] & & & & & \(75^{\circ}\) and 285\({}^{\circ}\)[35] \\ Present study & & & & & \\ Deviation & (\(-\)4\%) & (\(-\)6\%) & (\(+\)10\%) & (\(+\)4\%) & (\(+\)3\%) \\ \hline \end{tabular}
\end{table} 
Table 2: Comparison of the results from the present numerical model using the optimal mesh for a cylinder in cross-flow at \(Re=3900\) with results from [PERSON] et al. [25] and [PERSON] [35]. Figure 3: Computational grid for (**a**) the full domain and (**b**) a close-up view of the vicinity of the cylinder and synthetic jet actuator cavity and orifice. the performance of the present numerical model, the computational grid with the inclusion of the cavity and orifice was compared with the work of [PERSON] et al. [37], which focused on the formation criterion for synthetic jets in quiescent flow. Quiescent flow simulations were conducted with the present model at a jet Reynolds number \(Re_{\overline{U_{o}}}\) of 190, 255 and 479 and a fixed jet Stokes number of \(St_{o}\) = 15.8. Figure 4a shows how the vortex sheets produced at the orifice roll up during the blowing phase. When the vortical structures are strong enough to avoid re-entrainment into the cavity during the suction phase, the pair of counter-rotating vortices propagates away from the orifice and moves with a near constant velocity in a direction perpendicular to the plane of the orifice (Figure 4b). During the suction phase, a new vortical structure forms within the internal cavity and the process repeats itself. Subsequent periodic ejections of vortices from the orifice ultimately lead to a train of counter-rotating vortices and, after some periods, establish the fully formed synthetic jet flow field, including the internal flow dynamics within the cavity and orifice. Once the numerical model was shown to capture the formation of a synthetic jet, the operation of the actuation device in cross-flow was considered. A 'pseudo diaphragm' was used in the form of a velocity inlet boundary condition, located at the base of the cavity with a user-defined function (UDF) to produce the appropriate oscillating velocity (Equation (2)):  
\[U_{dia}=\frac{U_{p}}{10}\sin(2\pi f_{a}t) \tag{2}\]  
where \(U_{p}\) is the peak velocity at the orifice, \(t\) represents time and the factor 10 is the ratio of the diaphragm cross-sectional area to the orifice cross-sectional area. To verify the numerical model, the jet operating conditions (summarized in Table 3) are based on the jet Strouhal number \(Sr_{o}\), Stokes number \(St_{o}\) and Reynolds number \(Re_{\overline{U_{o}}}\), as used in the work of [PERSON] et al. [25]. All three dimensionless numbers are set to produce comparable synthetic jets, in spite of differing cavity and orifice geometries for actuation devices [38] and are defined by Equations (3)-(6). The synthetic jet orifice used by [PERSON] et al. [25] has a width of 0.135 mm, driven at frequencies between 50 Hz and 360 Hz. For the present study, a 1 mm wide synthetic jet orifice is used. To achieve the same values for \(Sr_{o}\), \(St_{o}\) and \(Re_{\overline{U_{o}}}\) as [PERSON] et al. [25], the synthetic jet is operated at lower actuation frequencies, between 0.9 Hz and 6.6 Hz. It should be noted that the work of [PERSON] et al. [25] used a velocity inlet boundary condition at the orifice, without internal cavity flow; hence, no realistic roll up of vortical structures would occur. The addition of the fluid-filled  
Figure 4: Vorticity plots for formation criteria set out by [PERSON] et al.