Vortex-ring-induced internal mixing upon the coalescence of initially stationary droplets

This study numerically investigates the internal jetlike mixing upon the coalescence of two initially stationary droplets of unequal sizes by employing the volume of fluid method with an adaptive mesh refinement algorithm. The emergence of the internal jet is attributed to the formation of a main vortex ring, as the jetlike structure shows a strong correlation with the main vortex ring inside the merged droplet. By tracking the evolution of the main vortex ring together with its circulation, we identified two mechanisms that are essential to the internal-jet formation: the vortex-ring growth and the vortex-ring detachment. Recognizing that the initiation of the vortex-ring-induced jet energetically relies on the merged interface but the manifestation of the jet physically relies on the competition between the convection and viscous dissipation of the vortex ring, we further developed and substantiated a vortex-ring-based Reynolds number $\left(\mathrm{R}{\mathrm{e}}_J={\mathrm{\Gamma }}_{\mathrm{VD}}/4\pi \nu \right)$ criterion, where ${\mathrm{\Gamma }}_{\mathrm{VD}}$ is the circulation of the detached main vortex ring and $\nu$ is the liquid kinematic viscosity, to interpret the occurrence of the internal jet at various Ohnesorge numbers and size ratios. For the merged droplet with apparent jet formation, the average mixing rate after jet formation increases monotonically with $\mathrm{R}{\mathrm{e}}_J$, which therefore serves as an approximate measure of the jet strength. In this respect, stronger internal jet is responsible for enhanced mixing of the merged droplet.

Figure 1

Computational domain and adaptive mesh for the current droplet coalescence study. The symmetry boundary condition is specified on the axis of symmetry and free outflow boundary conditions are specified on the other three boundaries. The zoomed-in window shows the grid configuration of the interface at the initial time of contact.

Figure 2

Grid independence study for the simulation. (a) Comparison of droplet deformation and internal mixing pattern among different mesh levels, (4, 5, 6), (4, 6, 7), (4, 7, 8), and (4, 8, 9), where the three numbers in parenthesis are the mesh levels for the gas zone, the fluid zone, and the interface zone, respectively; (b) Comparison of the total kinetic energy, $E_K$, normalized by the initial surface energy, $E_{S0}$, for the different meshes presented in (a).

Figure 3

Comparison of the droplet mixing patterns between experiment [46] and simulation. (a) Jet forming for a mixture of silicone oil and bromobenzene (3.3 CP measured viscosity), $\mathrm{We}=0.0,\mathrm{Oh}=0.012,\mathrm{\Delta }=2.08,t_c=20.2\mathrm{ms}$, (b) no jet for a high viscous silicone oil (99.0 CP measured viscosity), $\mathrm{We}=0.0,\mathrm{Oh}=0.203,\mathrm{\Delta }=1.75,t_c=57.5\mathrm{ms}$.

Figure 4

Comparison of the droplet coalescence speed between experiment [29] and simulation. The controlling parameters are $\mathrm{Oh}=2.5\times {10}^{-3},\mathrm{\Delta }=1.06$. The red-dotted lines in the sub-images are droplet interfaces obtained from the present simulation. Note that although $\mathrm{\Delta }=1.06$ was reported in the original work, $\mathrm{\Delta }=1.02$ was measured from the experimental images and therefore adopted in the simulation.

Figure 5

Deformation and mixing upon the coalescence of initially stationary droplets of (a) $n$-decane with $t_c=0.50\mathrm{ms},$ (b) water with $t_c=0.33\mathrm{ms}$, and (c) $n$-tetradecane with $t_c=0.48\mathrm{ms}$, and for different $\mathrm{\Delta }$ $\left(\mathrm{\Delta }=1.8,2.2,2.8\right)$. The left contour shows the static pressure, with the magnitude ranging from 0.0 of blue to 2.0 of red. The right contour plots the tracer variable, with red and blue tracking the fluid initially from the small and large droplets, respectively.

Figure 6

Regime nomogram of initially stationary droplet coalescence in the $\mathrm{\Delta }-\mathrm{Oh}$ parametric space. The jetlike mixing is denoted by the square, the no-jet regime by the cross, and the transition by the circlet. The red-dashed fitting line is given by the fourth-order polynomial, $\mathrm{\Delta }=1.3691\times {10}^7\mathrm{O}{\mathrm{h}}^4-4.6129\times {10}^5\mathrm{O}{\mathrm{h}}^3+6.4108\times {10}^3\mathrm{O}{\mathrm{h}}^2+25.0344\mathrm{Oh}+1$ . The dashed area corresponds to the ranges of $\mathrm{\Delta }$ and $\mathrm{Oh}$ effecting droplet pinch-off, experimentally given by Zhang et al. [62].

Figure 7

Comparison between the tracer variable distribution (left) and vorticity contour (right) at $T=1.70$ for the cases of (a) $n$-decane, (b) water, and (c) $n$-tetradecane with $\mathrm{\Delta }=2.2$. The color bar applies to the vorticity contours throughout the paper.

Figure 8

The main plot shows the evolution of the nondimensional circulation of the main vortex for $n$-decane droplet coalescence with $\mathrm{\Delta }=2.2,$ and $\mathrm{Oh}=1.56\times {10}^{-2}$. The subfigures (a)–(e) are the tracer variable plot (left) and vorticity contour (right) at the typical times (red circlets) on the vorticity evolution curve. In each subfigure, the green contour on the left side of the droplet represents the boundary of the main vortex identified through the vorticity-contour approach. The green cross marks the center of the main vortex corresponding to the local maximum vorticity. For clarity of illustration, positive and negative vortices in the vorticity contour are encircled by solid and dashed contour lines, respectively.

Figure 9

(a) The tracer variable plot (left) and vorticity contour (right) at a representative instant of Stage I [Fig. 6] for $n$-decane droplet coalescence with $\mathrm{\Delta }=2.2,$ and $\mathrm{Oh}=1.56\times {10}^{-2}$. The green contour marks the boundary of the main vortex. (b) The vorticity distribution along the gas–liquid interface near the main vortex. The $s^{*}$ coordinate is along the surface of the droplet, as indicated by the black arrow in (a).

Figure 10

(a) The tracer variable plot (left) and vorticity contour (right) for a representative instant of Stage III [Fig. 6] for $n$-decane droplet coalescence with $\mathrm{\Delta }=2.2$ and $\mathrm{Oh}=1.56\times {10}^{-2}$. (b) The corresponding vorticity distribution along the gas–liquid interface near the main vortex.

Figure 11

(a) The tracer variable plot (left) and vorticity contour (right) for a representative instant of Stage IV [Fig. 6] for $n$-decane droplet coalescence with $\mathrm{\Delta }=2.2$ and $\mathrm{Oh}=1.56\times {10}^{-2}$. (b) The corresponding vorticity distribution along the gas–liquid interface near the main vortex.

Figure 12

Evolution of the nondimensional circulation around the main vortex for different cases of droplet coalescence. The subfigures (a)–(f) present the tracer variable plot (left) and vorticity contour (right) for the instants when the main vortex completely detaches from the surface of the droplet. The circulations corresponding to these instants are also marked on the evolution curves.

Figure 13

Validation of the vortex criterion for the jet formation for different cases of droplet coalescence. The slopes of the two dashed lines correspond to $\mathrm{R}{\mathrm{e}}_J=2.4$ and $\mathrm{R}{\mathrm{e}}_J=3.5$, marking the transition from the no-jet regime to the mushroom-like jet regime. Here, all data points correspond to the cases of different $\mathrm{\Delta }$ and $\mathrm{Oh}$ plotted in Fig. 6.

Figure 14

Time evolution of the mixing index for $n$-decane $\left(\mathrm{Oh}=1.56\times {10}^{-2}\right)$, water $\left(\mathrm{Oh}=8.29\times {10}^{-3}\right)$, and $n$-tetradecane $\left(\mathrm{Oh}=3.62\times {10}^{-2}\right)$ droplets of various size ratios. $\mathrm{R}{\mathrm{e}}_J$ for each case is presented in the parentheses to relate the performance of mixing to the formation of internal jet.