Origin and dynamics of vortex rings in drop splashing

A vortex is a flow phenomenon that is very commonly observed in nature. More than a century, a vortex ring that forms during drop splashing has caught the attention of many scientists due to its importance in understanding fluid mixing and mass transport processes. However, the origin of the vortices and their dynamics remain unclear, mostly due to the lack of appropriate visualization methods. Here, with ultrafast X-ray phase-contrast imaging, we show that the formation of vortex rings originates from the energy transfer by capillary waves generated at the moment of the drop impact. Interestingly, we find a row of vortex rings along the drop wall, as demonstrated by a phase diagram established here, with different power-law dependencies of the angular velocities on the Reynolds number. These results provide important insight that allows understanding and modelling any type of vortex rings in nature, beyond just vortex rings during drop splashing.

V ortical flow is a main component of turbulence, and it is very commonly observed in nature, such as in winds surrounding a hurricane or a tornado, whirlpools in the wake of boats and paddles and nebulae in space. Specifically, a vortex ring (that is, a torus-shaped vortex) is a fascinating flow phenomenon that is generated when a fluid is rapidly injected into another fluid 1,2 , and it is a notable phenomenon in many fields of science [3][4][5][6] . A vortex ring that forms during drop splashing has caught the particular attention of a number of scientists over the past century 7-20 ever since the pioneering work by Thomson and Newall was published in the 19th century 7 . This phenomenon is interesting as a result of its fundamental properties in addition to its importance in fluid mixing and mass transport processes. Figure 1a illustrates the conceptual process for vortex ring formation due to a drop impact. When a liquid drop impinges on a liquid pool, the pool fluid climbs up the drop wall due to surface tension. At the same time, the impact velocity causes the drop fluid not to spread on the pool surface and instead to penetrate the pool. A strong velocity gradient results across the liquidliquid boundary and then generates azimuthal vorticity that evolves into a vortex ring enveloping the drop [10][11][12][13][14][15] . Vortex rings are generally known to form when the speed of the impact is sufficiently low or zero 9,13 . However, the lack of appropriate visualization has hindered the exact criterion for the vorticity formation and the dynamics of the vortex rings to still remain to be determined even though many efforts have been recently made [18][19][20][21][22] .
In this study, we elucidate new details of the physical origin for vortex ring formation and its dynamics, and prove and disprove our hypotheses and earlier findings, respectively, based on irrefutable experimental evidence. We also find a row of vortex rings along the drop wall, as demonstrated by the phase diagram established here. Moreover, the vorticity behaviour and the spiral geometry of the vortex rings are characterized in detail.

Results
X-ray imaging for vortex ring formation. To obtain a clear visualization of the vortical flows during drop splashing, we utilized ultrafast X-ray imaging coupled with a drop-impact set-up 20,23 . Figure 1b provides in-line projection images of the system with high temporal and spatial resolutions. The refraction-enhanced phase contrast 24 and the addition of a contrast agent (see the Methods section) enable an unprecedented visualization of the vortical flows during drop impact, as shown in Fig. 1c and in the Supplementary Movie 1. The pool fluid is clearly shown to rapidly climb up the drop wall (166-184 ms) and sharply penetrate into the drop (184-203 ms), forming a spiral. It is remarkable that not only the coiling core, but also the horizontal lines at the crests and troughs that indicate an azimuthally symmetric vortex ring could be clearly resolved at the initial stage of the vortex formation. Such highly vivid imagery enables us to quantitatively analyse the evolution dynamics of the vortices, which has been unreachable through conventional methods.
Criterion for vortex ring formation. First, we solve the long-standing problem of determining when the vortex ring has formed. Over the past two decades, the formation criterion for a vortex ring has been characterized by its Weber number 9,13 We where r is the liquid density, D is the drop diameter, U is the drop impact speed and g is the surface tension of the liquid. Our results show that this criterion is not correct and that a vortex ring can actually form, even for a large We (464), as is clearly seen in Fig. 2a  Weber number is only a criterion for ring penetration, irrespective of the ring formation. We have found that the criterion to determine whether a vortex ring forms is the Ohnesorge number, which relates the viscous forces to the inertial and capillary forces as Oh ¼ m/(rDg) 0.5 (with m as the dynamic viscosity of liquid), rather than the Weber number. As shown in the phase diagram of Fig. 3h, an investigation of a wide range of liquids using water, glycerol and ethanol mixtures shows that the vortex rings formed only for Oh o0.011. Furthermore, we find that a row of vortex rings forms along the drop wall, which correspond to single, double and triple rings, respectively, as shown in Fig. 3a-c (see red arrows) (Supplementary Movies 5-7). When the Reynolds number increases (Re ¼ rDU/m ¼ We 1/2 /Oh), we specifically observe single rings at Re o2,000, double rings at Re 42,000, triple rings at Re 43,000 and quadruple rings at Re 45,000 (Fig. 3h).
Here the Oh dependency of the ring formation is explained to be a result of the capillary waves that are known to develop azimuthally at the impact moment and subsequently travel along the drop wall 25 . When the capillary waves propagate about one wavelength, the pool fluid penetrates sharply into the drop as a result of the momentum that is transported by the waves, followed by the vortices coiling, as clearly shown with yellow arrows in Fig. 3d. However, for a high Oh (40.011), the onset of the vortices is suppressed by a strong viscous dissipation of the momentum transfer, as shown in Fig. 3e.
The Re dependency of multiple vortex rings shown in Fig. 3a-c,h can be explained to be a result of the capillary waves as well. The wave velocities measured from the X-ray images are comparable to the group velocities of the capillary waves, u g ¼ 1.5(2pg/rl) 1/2 , where l is the wavelength measured over a wide range of Oh (Fig. 3f) and impact speeds (Fig. 3g). The dependencies of the wave velocity on the hydrodynamic factors are roughly estimated as u g pm -1 and u g pU, as fitted in Fig. 3f,g, respectively. These suggest that the long-range energy transfer by capillary waves is more favourable in a large Re with u g pU/m Re, which explains the increase in the number of the rings with Re in Fig. 3h. In fact, the formation of a row of multiple vortex rings is observed in the simulation 26 and with X-ray imaging experiments 20 . Here, we are the first to find the criteria in terms of dimensionless numbers to determine the number of vortex rings and to roughly explain the hydrodynamics of the phenomenon. Hydrodynamics of the vortex rings. For the first time, we have measured the circulation dynamics of the vortices, specifically, the time-dependent circulation angles of the vortex cores, as shown in Fig. 4a,b. Figure 4c is rescaled in time and shows that the circulation dynamics are not simply described by D/U, as previously reported in the literature 15,27 . As seen for the total circulation angle Y and average angular velocity O (obtained by dividing the total angle by the time), which are measured over a wide range of impact conditions (Fig. 5), Y and O consistently decrease as Oh increases. Figure 6 is plotted again as a function of Re, showing that the angular velocity of spiral increases as Re increases in a different manner with ring order. Specifically, O 1 pRe 2 for the first ring, O 2 pRe 1 for the second and O 3 pRe 0.5 for the third, which is largely different from the previous simple prediction OpRe 0.5 (refs 17,26). How does the angular velocity change over time? In general, the vorticity decreases with time as a result of the viscosity 17,26 .
Here we have measured the instantaneous angular velocity of the vortex ring (the angular velocity of the circulating vortex tip at the centre of the spiral) as O i ¼ 2kq, with k as the interface curvature and q as the tangential flow velocity 11 . Interestingly, we find an  Fig. 7a. In particular, the strong oscillation during the initial stage is conceivably a result of an elliptical deformation of the vortex (Fig. 3d), due to the high-speed expansion of the vortex ring on the pool surface from the spreading of the impacting drop. The eccentricity of the vortex e, shown in Fig. 7b (obtained from the same image data for Fig. 7a), decreases as the expansion speed (v) decreases, that is, with an increasing time (inset of Fig. 7b), which explains the weakening of the oscillation in a later stage.  We were able to measure the spiral geometry of the vortex streamlines, specifically the radius of each spiral layer for the first and the second rings (Fig. 7c). The vortex rings show logarithmic spirals similar to those in various vortical systems in nature, such as a nautilus hemishell, tornados and galaxies 28 , and these can be represented in a polar coordinate system (R, f) as RBR 0 e rf , where R 0 is the initial radius and r is the growth-rate coefficient. The best-fit values of r, which are estimated for the two subsequent rings (r 1 and r 2 ) in various impact conditions (Fig. 7d), are almost invariant in each ring with r 1 almost twice that of r 2 (r 1 B0.12 ± 0.02 and r 2 B0.06 ± 0.01) regardless of the liquid viscosity or the drop impact speed. The r values are notably much smaller than those of typical logarithmic spirals in nature. For example, rB0.18 for a nautilus shell,B0.21 for the Milky Way and B0.31 for typical golden spirals 28 .
Relation of the vortex ring and the liquid jet formation. We finally discuss a relation between the vortex ring formation and the jetting phenomena. Two kinds of jets were ejected after a drop impact on a liquid pool 17,[29][30][31] . The ejecta is an inertia-induced thin liquid sheet that emerges rapidly from the neck between the drop and the pool surface, and the lamella is a thicker capillarityinduced liquid sheet that propagates outwards from the impact site. As shown in the phase diagram in Fig. 8a 31 , three regimes exist for the jetting phenomena, including the (i) lamella emerging at a low We (Fig. 9a), (ii) the ejecta merging with the lamella into one jet at a high We and a low Re (Fig. 9b) and (iii) the ejecta and lamella developing separately at a high We and high Re (Fig. 9c). These three regimes are shown in different colours, and we plotted the number of the vortex rings in terms of the Re and the We in Fig. 8b. For regimes i and iii, the vortex rings form robustly as long as Oh o0.011. In regime iii, we note that the vortex ring forms within the lamella that has developed separately from the ejecta (Fig. 9c). In regime ii, however, the vortex ring never forms despite having a small Oh (o0.011), indicating that the ejecta, once merged with the lamella, suppresses the ring formation. This suggests that the lamella should not be merged with the ejecta as an additional condition for vortex ring formation.

Discussion
In this study, we have experimentally revealed the dynamics of vortex rings in a drop impact that is unresolvable with optical imaging, and we have therefore mostly relied on theoretical studies. In contrast with recent studies on the vortex rings 18,20,26 , we have made a great advance in addressing issues that had been unsolved for a long time. The criterion for vortex ring formation, the ring morphology dependence on the hydrodynamic conditions, the vorticity or energy of the vortex rings and the relation with an external splashing shape were investigated using a novel ultrafast X-ray imaging technique. This study therefore offers substantial insight for further analytical, numerical and experimental work on vortex ring dynamics in fluid mechanics.

Methods
X-ray imaging. Ultrafast X-ray imaging experiments were carried out using the XSD 32-ID beamline of the Advanced Photon Source, Argonne National Laboratory. An intense white (full energy spectrum) X-ray beam with a peak irradiance of B10 14 ph s À 1 mm À 2 per 0.1% bandwidth (bw) was used to conduct the high imaging speed. The detector system (Fig. 1b) is composed of a fast scintillator (LuAG:Ce, decay time B50 ns) and a right-angle mirror coupled to a high-speed camera (Photron Fastcam SA 1.1) via a long-working distance microscope objective. The detector system is synchronized to the X-ray pulses to enable the direct visualization of the ultrafast dynamics of the liquid-air and liquid-liquid interfaces at up to B270,000 frames per a second, with a 472-ns exposure time for each frame.
Experimental set-up. Liquid drops were dispensed from a 26-G syringe needle (outer diameter B0.46 mm and inner diameter B0.26 mm) at heights from 30 to 300 mm. A laser triggering system was installed to sense the falling drop and to trigger the camera and the fast shutter to take the images. The liquid pool was prepared in a cylinder made of Kapton with a diameter of 20 mm and a depth of 50 mm. ARTICLE Materials. The liquids were prepared by mixing water, ethanol and glycerol with various compositions. For all experiments, the liquids for the falling drop and the pool were prepared with identical compositions, except for the addition of zinc iodide (10 wt%) in the drop as a contrast agent. The addition of the small fraction of zinc iodide can increase the X-ray absorption and can therefore improve the image contrast between the drop and the pool fluids. This small concentration of zinc iodide (B0.006 M for water and B0.014 M for ethanol) can slightly increase the viscosity of the fluid (B1% for water and B3% for ethanol) 32 , but we assume that this change does not have a significant influence on the general trend of the hydrodynamics.