Axisymmetric direct numerical simulations (DNS) are performed to study the formation criterion and evolution of zero-net-mass-flux synthetic jets. Jet formation is characterised by an oscillating streamwise jet centreline velocity, showing net momentum flux away from the orifice. This momentum flux away from the orifice takes the form of a series of vortical structures, often referred to as a vortex train. Simulation of the jet actuator consists of a modified oscillating velocity profile applied to a wall boundary. The jet issues into quiescent air, and the Reynolds numbers used vary from 85 < Re < 1000. Variations to the input simulation parameters are carried out in order to determine the overall effects on the flow field. From these results the conditions necessary for the formation of the synthetic jet along with the input parameters that provide an optimal jet output are deduced. Jet optimisation is defined by both the vortical strength and longevity of the vortex train as it travels downstream. This study examines the vortical structures, the jet centreline velocities along with other flow characteristics in order to deduce and visualise the effects of the input parameters on the jet performance. The results attained on altering the oscillation frequency of the jet actuator indicated that synthetic jets with zero mean velocity at the inflow behave significantly differently from jets with non-zero mean velocity at the inflow. An evolution study into the formation of the train of vortical structures associated with the formation of a synthetic jet is performed. This study is accompanied with a time history of the jet centreline velocity, showing the net momentum flux of the fluid away from the orifice of a fully developed synthetic jet. Further details on the jet centreline velocity for all cases are also presented; along with a study on the effect on the vortical structures of altering the Reynolds number of the flow.