On the scaling of propagation of periodically generated vortex rings

Abstract

The propagation of periodically generated vortex rings (period$T$) is numerically investigated by imposing pulsed jets of velocity$U_{jet}$and duration$T_{s}$(no flow between pulses) at the inlet of a cylinder of diameter$D$exiting into a tank. Because of the step-like nature of pulsed jet waveforms, the average jet velocity during a cycle is$U_{ave}=U_{jet}T_{s}/T$. By using$U_{ave}$in the definition of the Reynolds number ($Re=U_{ave}D/\unicode[STIX]{x1D708}$,$\unicode[STIX]{x1D708}$: kinematic viscosity of fluid) and non-dimensional period ($T^{\ast }=TU_{ave}/D=T_{s}U_{jet}/D$, i.e. equivalent to formation time), then based on the results, the vortex ring velocity$U_{v}/U_{jet}$becomes approximately independent of the stroke ratio$T_{s}/T$. The results also show that$U_{v}/U_{jet}$increases by reducing$Re$or increasing$T^{\ast }$(more sensitive to$T^{\ast }$) according to a power law of the form$U_{v}/U_{jet}=0.27T^{\ast 1.31Re^{-0.2}}$. An empirical relation, therefore, for the location of vortex ring core centres ($S$) over time ($t$) is proposed ($S/D=0.27T^{\ast 1+1.31Re^{-0.2}}t/T_{s}$), which collapses (scales) not only our results but also the results of experiments for non-periodic rings. This might be due to the fact that the quasi-steady vortex ring velocity was found to have a maximum of 15 % difference with the initial (isolated) one. Visualizing the rings during the periodic state shows that at low$T^{\ast }\leqslant 2$and high$Re\geqslant 1400$here, the stopping vortices become unstable and form hairpin vortices around the leading ones. However, by increasing$T^{\ast }$or decreasing$Re$the stopping vortices remain circular. Furthermore, rings with short$T^{\ast }=1$show vortex pairing after approximately one period in the downstream, but higher$T^{\ast }\geqslant 2$generates a train of vortices in the quasi-steady state.