## 4 Conclusions  
Using unsteady RANS simulations with the transition SST turbulence model, a significant modification of the drag coefficient \(C_{D}\) and RMS lift coefficient \(C_{L}^{\prime}\) for a circular cylinder in cross-flow has been observed, for a freestream Reynolds number \(Re=3900\) and a range of synthetic jet Stokes numbers from \(St_{o}=2.4\) to 6.4. Overall, the trend in \(C_{D}\) modification agrees with the work of [PERSON] et al. [25], with a general decrease in \(C_{D}\) achieved at \(St_{o}=4.9\) for a fixed jet Reynolds number of \(Re_{\overline{U}_{o}}=12\). It was also observed that the modification of the RMS lift coefficient \(C_{L}^{\prime}\) was closely coupled with the modification of the drag coefficient \(C_{D}\).  
Previous research has used the dimensionless actuation frequency \(F^{+}\) for comparison of synthetic jet operating conditions and the effects on flow control for NACA aerofoil geometries [3; 6; 16; 17; 42]. Despite a similar-sized cylinder and comparable freestream velocity, the results of the present study and [PERSON] et al. [25] are characterised by different dimensionless actuation frequencies \(F^{+}\), due to the different sized synthetic jet orifices used in both studies. The present study opted for an orifice width of 1 mm, identical to the size used in water tunnel experiments in the research group, but larger than the 0.13 mm wide orifice used by [PERSON] et al. [25]. However, the three dimensionless jet operating parameters \(Re_{\overline{U}_{o}}\), \(Sr_{o}\) and \(St_{o}\) were maintained the same as [PERSON] et al. [25] for comparison of the vortex formation process. The optimum drag coefficient \(C_{D}\) and RMS lift coefficient \(C_{L}^{\prime}\) reduction occurs at \(F^{+}=0.4\) for the present study and at \(F^{+}=19.5\) for [PERSON] et al. [25], but otherwise similar trends have been observed.  
Decreases in the magnitude of the adverse surface pressure gradient \(dP/d\theta\) across the synthetic jet orifice indicate an increase in the momentum of the viscous sublayer. When the magnitude of the pressure gradient across the orifice decreases, there is an increase in turbulence intensity at the rear of the cylinder. This increase in the strength of the vortical structures at the rear of the cylinder appears to be responsible for the decrease in drag coefficient \(C_{D}\).  
Additionally, an increase in turbulence intensity in the lee of the cylinder also corresponds to a _decrease_ in the turbulence intensity further downstream in the cylinder wake, beyond \(8D\) downstream. This could be a useful insight for structures or devices implemented in arrays, such as tidal turbines and wave energy devices, or the risers used for offshore platforms and deep-water wind turbines. By operating the synthetic jet actuator to minimise the mean drag and RMS lift forces experienced by the structure, the turbulence intensity downstream of the structure will also be minimised. Thus, the ideal synthetic jet operating condition for drag and lift reduction would have a beneficial effect on the flow conditions of downstream devices.  
The authors gratefully acknowledge the support of [PERSON] for the useful discussions on flow control, [PERSON] for the assistance with the numerical implementation and the resources made available by the Irish Centre for High-End Computing (ICHEC). This work builds on research supported by the Higher Education Authority of Ireland through the Graduate Research Education Programme in Engineering (Programme for Research in Third-Level Institutions--cycle 5 (PRTLI5), co-funded by the European Regional Development Fund) and the Irish Research Council through its INSPIRE International Mobility Fellowship programme (co-funded by Marie Curie Actions; grant No. PD/2010/INSP/1313) and the Government of Ireland Postdoctoral Fellowship programme (grant No. GOIPD/2016/216). [PERSON] carried out the research work under the supervision of [PERSON]; [PERSON] is the primary author.  
The authors declare no conflict of interest.