Dynamics of vortex rings and spray-induced vortex ring-like structures

Analytical formulae, predicted by recently developed vortex ring models, in the limit of small Reynolds numbers (R e), are compared with numerical solutions of the underlying equation for vorticity and experimental data. Particular attention is focused on the recently developed generalised vortex ring model in which the time evolution of the thickness of the vortex ring core L is approximated as a t^b, where a and b are constants (1 / 4 ≤ b ≤ 1 / 2). This model incorporates both the laminar model for b = 1 / 2 and the fully turbulent model for b = 1 / 4. A new solution for the normalised vorticity distribution is found in the form ω₀ + R e ω₁, where ω₀ is the value of normalised vorticity predicted by the classical Phillips solution. This solution shows the correct trends in the redistribution of vorticity due to the Reynolds number effect, and it predicts the increase in the volume of fluid carried inside the vortex ring. It is emphasised that although the structures of vortex rings predicted by analytical formulae, based on the linear approximation, and numerical calculations for arbitrary R e are visibly different for realistic Reynolds numbers, the values of integral characteristics, such as vortex ring translational velocity and energy, predicted by both approaches, turn out to be remarkably close. The values of velocities in the region of maximal vorticity, predicted by the generalised vortex ring model, are compared with the results of experimental studies of vortex ring-like structures in gasoline engine-like conditions with a high-pressure (100 bar) injector. The data analysis is focused on the direct measurements of droplet axial velocities in the regions of maximal vorticity. Most of the values of these velocities lie between the theoretically predicted values corresponding to the later stage of vortex ring development between b = 1 / 4 (fully developed turbulence) and 1 / 2 (laminar case).