Extending an earlier idea by Walin and Rahm, an efficient method of producing a stably stratified fluid system in a container is explored. By pumping a small vertically-upward mass flux through the system, it has been shown that the time to reach the final state is greatly reduced below the diffusive time-scale. The effect of a small pulsating component ε sin (ωt) embedded in the through-flux is investigated. Time-dependent solutions to the governing Navier-Stokes equations formulated for a vertically-mounted circular cylinder have been obtained. The overall time-scale for the global motion is still characterized by the usual convective time-scale. However, the velocity fluctuations in the interior core in the final state demonstrate marked variations with ω. The velocity fluctuations are magnified when the imposed pulsating frequency ω reaches a value close to the characteristic buoyancy frequency of the system. This behaviour is insensitive to the variations of the amplitude of the pulsating part; this suggests a resonance phenomenon in the system. Detailed descriptions are provided for the flow patterns inside the cylinder when the amplitude of the pulsating component is appreciable.