## 2 Numerical Approach
The computational grid has been refined until the numerical model achieved results sufficiently close to those reported in literature (see Table 2), using \(C_{D}\), \(C_{L}^{\prime}\), \(Sr_{0}\) and the coefficient of pressure at the rear of the cylinder \(C_{P_{b}}\) as reference values. Utilising the URANS approach with the transition SST turbulence model allowed for savings in both computational time and solution file size. A time step of 0.1 ms was used to ensure a Courant-Friedrichs-Lewy (CFL) condition of less than unity. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Mesh Size**} & **Drag Coefficient** & **RMS Lift** & **Vortex Shedding** & **Calculation Time** \\  & \(C_{D}\) & **Coefficient**\(C_{L}^{\prime}\) & \(Sr\) & **(Approx.)** \\ \hline
66,000 & 1.63 (error = \(-6\%\)) & 1.22 (error = \(-14\%\)) & 0.36 (error = \(+37\%\)) & 24 hours \\
113,000 & 1.66 (error = \(-5\%\)) & 1.28 (error = \(-10\%\)) & 0.32 (error = \(+22\%\)) & 48 hours \\
253,000 & 1.67 (error = \(-4\%\)) & 1.33 (error = \(-6\%\)) & 0.29 (error = \(+10\%\)) & 96 hours \\
512,000 & 1.69 (error = \(-3\%\)) & 1.38 (error = \(-3\%\)) & 0.27 (error = \(+3\%\)) & 192 hours \\ Literature [25; 35] & 1.74 & 1.42 & 0.26 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Effect of mesh density on characteristic dimensionless numbers averaged over the quasi-steady portion of flow time after 20 seconds of simulated flow time, for a cylinder in cross-flow at \(Re=3900\). Percentage values in brackets represent deviations for all results to values reported by [PERSON] et al. [25] and [PERSON] and Moin [35]. The final computational grid contains 253,000 elements. The inclusion of the synthetic jet cavity and orifice further increased the total number of elements to 333,000. The density of the mesh was highest at the synthetic jet orifice and cavity, as well as at the surface of the cylinder. The mesh density was increased here to capture the small-scale vortical structures accurately. The mean aspect ratio of the mesh was 6.6, with a mean quality of 0.997. The dimensionless size of the first cell layer near the wall (\(y^{+}\)) did not exceed 1.0. A structured mesh was used to minimise computational time. The grid was aligned parallel to the freestream flow away from the cylinder surface and aligned with the cylinder surface in an O-grid structure close to the cylinder surface (Figure 3). In this way, all elements in these regions are orientated in the general direction of the flow. By aligning the orientation of the grid with the flow in that region, the risk of false diffusion when using the upwind scheme can be eliminated and a more physically accurate solution is obtained [36]. Upon achieving a computational grid capable of producing accurate results for a cylinder in cross-flow at \(Re=3900\), the synthetic jet actuator geometry was incorporated into the design (Figure 3). Previous research by [PERSON] et al. [25] excluded the synthetic jet cavity and orifice. Instead, a velocity inlet boundary condition was applied to a 0.135 mm section of the cylinder surface to act as a synthetic jet orifice outlet. By using an alternating velocity inlet boundary condition on the surface of the cylinder, as opposed to modelling an actual cavity and orifice, a realistic synthetic jet would be unable to form. In the absence of an orifice and internal cavity, the flow separation near the sharp orifice edge would not lead to a realistic vortex formation [10]. The inclusion of the fluid-filled cavity and orifice (see Figures 2 and 3) allows for a more accurate modelling of synthetic jet formation. To verify