On the vortex ring formation and mixing in thin films upon droplet impact

TL;DR

This study experimentally investigates how vortex rings form, propagate, and destabilize when a droplet impacts liquid films with varying thickness. Using two-view PIV and LIF, the authors map regime behavior across $Re$, $We$, and $\delta$, identifying three outcomes: Expansion, Multiple vortex rings, and Instability, with wall interactions precipitating transitions at lower $Re$ for thinner films and none observed for $\delta>0.60$. They quantify vortex-circulation evolution, introduce an energy-based maximum circulation $\Gamma_{\max}=\sqrt{\xi}\Gamma_{\text{th}}$ with $\xi=0.222$, and develop a sigmoid-based model for $\Gamma_t$, capturing generation and decay phases. An azimuthal-fingering mechanism is linked to crater receding, yielding a linear fingering relation $F= B_f Re$ with $B_f=0.02573-0.0334\delta^{0.8398}$, highlighting how thinner films compress rings and enhance mixing. Together, these results provide a predictive framework for vortex-ring dynamics and mixing efficiency in thin-film systems relevant to spray coating, cooling, and fuel-injection processes, where wall effects and constrained geometry play pivotal roles.

Abstract

When a droplet impacts a liquid film, a vortex ring form and govern momentum and species transport. We experimentally investigate vortex ring formation, propagation and instability during droplet impact onto liquid films, with particular emphasis on vortex ring-wall interactions. Particle image velocimetry and laser-induced fluorescence are used to study the effects of Reynolds number Re, Weber number We and dimensionless film thickness δover ranges Re \leq 3900, We \leq 61 and 0.09 \leq δ\leq 1.35. As film thickness decreases, a transition from a single axisymmetric vortex ring to azimuthally unstable, multi-vortex structures is observed. A regime map in Re-δspace is constructed, showing that vortex ring instabilities occur at lower Re for thinner films, while no instabilities are detected for thick films up to the highest Re studied. The azimuthal wave number increases with Re and decreases with δ. Thinner films exhibit faster decay of primary vortex ring circulation due to wall interactions, accompanied by the formation of secondary vortex ring at lower Re. An empirical model is proposed to predict the temporal evolution of total vortex ring circulation, accounting for both generation and decay.

Figures (16)

• Figure 1: a) Schematic illustration of bottom view setup: (1) Syringe pump, (2) z-Traverse, (3) Cannula, (4) Thin liquid film on FTO glass substrate, (5) High power LED, (6) x,y,z-Traverse, (7) HS-Camera, (8) Dichroic mirror with bandpass filters, (9) Microscope Objective. b) Schematic illustration of side view setup (HS - camera orientated along y- axis facing x-z plane. The blue and red boxes represent the field of view during LIF and PIV measurements, respectively. The yellow box represent the region of interest during PIV measurements
• Figure 2: (Left) Visualization of the vortex ring during droplet impact on liquid film using Laser-Induced Fluorescence. (Right) Velocity vector field overlaid with $\Gamma_2$ scalar field extracted from PIV measurements. The green point indicates the center of the vortex. The case corresponds to $\delta = 0.90$, $We = 54$, $Re = 3900$ at $t=27ms$.
• Figure 3: a) Least-squares fit of azimuthal velocities of two superposed Lamb-Oseen vortices (red line) and tangential velocity profile measured along a horizontal line passing through the centers of the vortex pair for $\delta=0.90$, $We=54$ and $Re=3900$ at $t=27ms$ (blue dots). White shaded area represents the boundary of the vortex used for the fit. b) Time evolution of vortex circulation for the case $\delta=0.90$, $We=54$ and $Re=3900$ calculated by Stokes' Theorem (blue) and SLV method (red).
• Figure 4: Time evolution of droplet impact on thin films at $Re=1400$ and $We=7$ for different film thicknesses. a) $\delta=0.09$, b) $\delta=0.29$, and c) $\delta=0.45$. SW marks the surface wave visible at early times, while red arrows indicated the area where floating droplet liquid causes reduced fluorescence intensity, corresponding to the features indicated with red arrows in Fig. \ref{['fig:5']}b. Colors represent the fluorescence signal intensity $I_f$.
• Figure 5: Side-view LIF visualizations of droplet impacts on thin films at $Re = 1400$ and $We = 7$ for different film thicknesses. a) $\delta=0.29$, b) $\delta=0.45$, and c) $\delta=0.90$. Images in (a) and (b) are shown at 5, 10, 30, and 60 ms after impact, while for (c) an additional frame at 90 ms is included. The extended time range for $\delta = 0.90$ is required because the increased film thickness delays the interaction of the vortex ring with the wall. MB marks micro-bubbles visible at early stages, BLL denotes boundary-layer lift-off, and PVR and SVR label the primary and secondary vortex rings, respectively. Red arrows in (b) indicate portions of droplet liquid floating on the film surface outside the vortex ring region, corresponding to the ones indicated by red arrows in Fig. \ref{['fig:4']}c. Scale bar is equivalent to 1 mm.