3.1. Numerical method

We simulate a gas–liquid system with respective densities ${\it\rho}_{g}$ and ${\it\rho}_{l}$ and viscosities ${\it\mu}_{g}$ and ${\it\mu}_{l}$ using the Navier–Stokes equations:

where $\boldsymbol{u}$ is the fluid velocity, ${\it\rho}\equiv {\it\rho}(\boldsymbol{x},t)$ is the density, ${\it\mu}\equiv {\it\mu}(\boldsymbol{x},t)$ is the viscosity and $\unicode[STIX]{x1D63F}$ is the deformation tensor. The Dirac distribution function ${\it\delta}_{s}$ expresses the fact that the surface tension term is concentrated at the interface, ${\it\kappa}$ and $\boldsymbol{n}$ being the curvature and the normal of the interface respectively. We use the Gerris code, a quad/octree-based multilevel solver described in detail in Popinet (Reference Popinet2003). The interface is tracked by a volume-of-fluid method described in Popinet (Reference Popinet2009) and Tryggvason et al. (Reference Tryggvason, Scardovelli and Zaleski2011). The accuracy of the code has been validated for a number of multiphase flow problems (Fuster et al. Reference Fuster, Agbaglah, Josserand, Popinet and Zaleski2009; Popinet Reference Popinet2009; Agbaglah et al. Reference Agbaglah, Delaux, Fuster, Hoepffner, Josserand, Popinet, Ray, Scardovelli and Zaleski2011), including splashing (see e.g. Thoraval et al. Reference Thoraval, Takehara, Etoh, Popinet, Ray, Josserand, Zaleski and Thoroddsen2012, Reference Thoraval, Takehara, Etoh and Thoroddsen2013; Agbaglah & Deegan Reference Agbaglah and Deegan2014).

3.2. Numerical simulations

We model a drop with diameter $D$ and speed $U$ falling vertically through an initially quiescent gas and impacting on a deep pool of the same liquid with depth $H=5D$ . The simulation is characterized by five dimensionless parameters: the viscosity and density ratios ( ${\it\mu}_{l}/{\it\mu}_{g}$ and ${\it\rho}_{l}/{\it\rho}_{g}$ ), the liquid Reynolds number $Re={\it\rho}_{l}UD/{\it\mu}_{l}$ , the Weber number $We={\it\rho}_{l}U^{2}D/{\it\sigma}$ and the Froude number $U^{2}/(gD)$ . In our initial simulations we could discern no effect due to gravity, and thus it was omitted from subsequent simulations. We use the axisymmetric formulation of the Navier–Stokes equations to exploit the symmetry of the experimental features studied here. The computational domain is $5D\times 10D$ and is discretized with an adaptive mesh up to a maximum number $2^{14}$ of grid points along each dimension, corresponding to a minimum mesh size ${\rm\Delta}x=D/3277$ . The mesh is adapted based on four criteria: distance to the interface, curvature of the interface, vorticity and velocity magnitude.

Figure 3 compares simulation results with the equivalent experimental profiles obtained by high-speed X-ray imaging. The experimental profiles were obtained in the same data runs as reported in Zhang et al. (Reference Zhang, Toole, Fezzaa and Deegan2012a ). Briefly, the fluid was silicone oil with $D=0.2~\text{cm}$ , ${\it\mu}_{l}=1.3~\text{cP}$ , ${\it\rho}_{l}=0.851~\text{g}~\text{cm}^{-3}$ , ${\it\sigma}=17.6~\text{dyne}~\text{cm}^{-1}$ and $U=156~\text{cm}~\text{s}^{-1}$ , incidence was normal to the pool, the surrounding gas was air, and the pool depth was 5 cm. The excellent agreement between the experiments and simulations demonstrates the accuracy with which our simulations capture the dynamics.

Figure 3.

Comparison of X-ray phase contrast images from experiments (a) with simulation (b) for $We=292$ and $Re=2042$ at $\tilde{t}=tU/D=0.161$ , 0.261, 0.46, 0.811 (left to right). The experimental configuration and conditions are given in Zhang et al. (Reference Zhang, Toole, Fezzaa and Deegan2012a ) for the SO3 fluid listed therein; the scale bar corresponds to 0.2 mm. The colours in the simulation indicate the fluid from the drop (red) and the pool (blue), and the ambient gas (green).

We ran simulations in the parameter range $Re\leqslant 5000$ and $We\leqslant 900$ with air as the surrounding gas ( ${\it\mu}_{g}=1.94\times 10^{-2}~\text{cP}$ and ${\it\rho}_{g}=1.2\times 10^{-3}~\text{g}~\text{cm}^{-3}$ ). We fixed ${\it\rho}_{l}/{\it\rho}_{g}=709$ , $U=156~\text{cm}~\text{s}^{-1}$ and $D=0.2~\text{cm}$ , and varied ${\it\mu}_{l}$ and ${\it\sigma}$ to select $Re$ and $We$ . We observed the four distinct behaviours shown in figure 4. At high $Re$ where two jets emerge, the simulation results in figure 4(c) show a vortex ring driving the growth of the roll jet. This vortex ring originates at the corner where the ejecta and the drop meet, it detaches from the free surface along the downward direction, travels horizontal below the pool level, and resurfaces to form the roll jet. Figure 5 illustrates the flow field around the vortex-shedding event and shows that after the vortex sheds all the fluid entering the ejecta comes from the pool.

Figure 4.

Qualitatively different flows observed in simulations: (a) vortex separation forming a roll jet without a preceding ejecta ( $We=100,Re=2500$ ); (b) ejecta without vortex shedding leading to a lamella ( $We=292,Re=1000$ ); (c) ejecta plus vortex separation leading to separate roll jet ( $We=292,Re=2042$ ); (d) collision of the ejecta with the drop plus vortex shedding leading to a separate roll jet ( $We=700,Re=3000$ ). The columns correspond to equal times $\tilde{t}=0.211$ , 0.311, 0.511 (from left to right) except for ( $d_{3}$ ) which corresponds to $\tilde{t}=0.372$ . The colour indicates vorticity, with blue/red representing counterclockwise/clockwise rotation. The thin yellow line within the drop indicates the boundary between the fluid from the drop and the pool.

Irrespective of $Re$ , at early times there is always a large patch of vorticity near the point where the upper side of the jet and the drop meet. The large outward mean flow hides the fact that this vorticity patch is a vortex attached to the free surface (see, for example, figure 5). Below some critical $Re$ (see figure 4 b), we observe that the vortex ring remains attached to the interface at all times, becoming progressively larger but weaker, and eventually merges with the base of the ejecta to form the lamella. Thus, our simulations show that the transition from one jet (the lamella) to two jets (an ejecta and a roll jet) observed in experiments (Zhang et al. Reference Zhang, Toole, Fezzaa and Deegan2012a ) is dictated by whether the vortex remains attached to the free surface.

At the highest $Re$ and $We$ in our range of study the ejecta reconnects with the drop surface, capturing a bubble. This process leads to air entrainment and vortex separation, as shown in figure 4(d), and is the same as the ‘bumping’ regime described in Thoraval et al. (Reference Thoraval, Takehara, Etoh, Popinet, Ray, Josserand, Zaleski and Thoroddsen2012). At high $Re$ and low $We$ we observe vortex shedding and the formation of a lamella but no ejecta.

Figure 5.

Flow field and vorticity at the base of the jet in the laboratory frame (a,b) and in the reference frame comoving with the point $\boldsymbol{T}$ (c,d). (a–c) Vortex-shedding splash at $We=292$ and $Re=3000$ for $\tilde{t}=0.141$ (a,c) and $\tilde{t}=0.236$ (b). Note that in (b) the liquid entering the jet originates almost exclusively from the pool. (d) Non-shedding splash at $We=292$ and $Re=1500$ for $\tilde{t}=0.141$ . The flow field of non-shedding splashes at other parameter values at all times looks similar to (a) in the laboratory frame or (c or d) in the comoving frame.

The parametric dependence of the four regimes depicted in figure 4 as found in simulations is given by the phase diagram in figure 6. Each simulation point in figure 6 took two weeks of computation on seven cores using the MPI library combined with dynamic load balancing as described in Agbaglah et al. (Reference Agbaglah, Delaux, Fuster, Hoepffner, Josserand, Popinet, Ray, Scardovelli and Zaleski2011). For reference, the equivalent experimental observations from Zhang et al. (Reference Zhang, Toole, Fezzaa and Deegan2012a ) are also shown. The two datasets show strong agreement. The boundary between shedding and no shedding is reasonably captured by the relation $Re=5We$ , or equivalently $Ca\equiv {\it\eta}U/{\it\sigma}=0.2$ .

Figure 6.

Phase diagram for vortex shedding. Simulations (shown with large symbols) that exhibit vortex shedding are represented with circles, no vortex shedding with squares and bumping with triangles. Filled symbols indicate the presence of ejecta while open symbols have no ejecta. The experimental results reported in Zhang et al. (Reference Zhang, Toole, Fezzaa and Deegan2012a ) are plotted with small symbols using the same convention as for the simulations; the bumping regime was not distinguished in these experiments. The solid dark line represents the approximate limit of the vorticity-shedding regime: $Ca=0.2$ .

3.3. Characterizing the vortex shedding

In order to investigate the mechanism for vortex shedding, we measured various characteristics of the flow. Since the shedding event occurs in the vicinity where the jet emerges from the drop, our measurements focused on this zone and particularly on $\boldsymbol{T}$ , the point within this zone of maximum curvature of the air–liquid interface (see figure 5 a). We found that there are no simple characteristics that herald a shedding event. For example, as shown in figure 7, the time dependence and magnitude of the curvature ${\it\kappa}_{T}$ at $\boldsymbol{T}$ right up until the shedding event begins are indistinguishable from those of the non-shedding cases. In other words, we find no way to predict which configurations will shed a vortex. The same is true of the vorticity (see figure 8), the velocity (see the remark in the caption to figure 5), the local Reynolds number $Re_{L}={\it\rho}_{l}u{\it\kappa}_{T}^{-1}/{\it\mu}_{l}$ (see figure 9 a), where $u$ is the maximum speed tangential to the interface around $\boldsymbol{T}$ , and the local Weber number ${\it\rho}_{l}u^{2}{\it\kappa}_{T}^{-1}/{\it\sigma}$ .

Nonetheless, the initiation of shedding is clearly signalled by the vorticity and the curvature. Figure 7(b) shows that a subset of the simulations exhibit a minima in the curvature. All shedding cases exhibit such a minimum, and the minimum occurs prior to a visually discernible separation of the vortex. Thus, we take this minimum as a signal that the vortex separation process has started. Rescaling of the time by the visco-capillary time $t_{{\it\mu}{\it\sigma}}={\it\mu}_{l}D/{\it\sigma}$ as in figure 7(b) shows that the initiation of shedding occurs at a remarkably uniform time $t\approx 1.05t_{{\it\mu}{\it\sigma}}$ . These results hint at the importance of the capillary number $Ca={\it\mu}_{l}U/{\it\sigma}$ , because $t_{{\it\mu}{\it\sigma}}$ in dimensionless units $D/U$ is $Ca$ . Indeed, the local capillary number $Ca_{L}=u{\it\mu}_{l}/{\it\sigma}$ for vortex-shedding splashes is lower at all times than for non-shedding splashes (see figure 9 b). Moreover, taking the time averaging threshold value of $Ca$ indicated in figure 9(b) yields a value of 0.2. Plotting this value on the phase diagram (figure 6) fairly accurately captures the transition. Therefore, our data indicate that an understanding of the interplay between viscous and capillary forces in the vicinity of $\boldsymbol{T}$ is the most important ingredient for understanding the transition between one and two jets.

Figure 7.

(a) Mean curvature at $\boldsymbol{T}$ versus time for $We=300$ and $Re=1000$ (magenta), 1500 (red), 2000 (blue), 2500 (dark green) and 3000 (black). The vertical dashed lines indicate the time of the vortex separation chosen by eye when the material line defined by the pool–drop interface exhibits strong swirling such as in figure 4( $c_{2}$ ). The peak in the data indicates vortex shedding; conversely, its absence indicates that the vortex remains attached to the interface. (b) Mean curvature at $\boldsymbol{T}$ with time in units of the visco-capillary time ( $t_{{\it\mu}{\it\sigma}}={\it\mu}_{l}D/{\it\sigma}$ ) for $We=300$ and $Re=2000$ (blue), 2500 (dark green), 3000 (black); $We=400$ and $Re=2500$ (orange), 3000 (dark grey). The curves have been vertically shifted in order to avoid overlapping and clearly show the positions of the minima; their starting and ending values are listed with the same colour as the data.