Early-time jet formation in liquid–liquid impact problems: theory and simulations

Abstract

We perform a thorough qualitative and quantitative comparison of theoretical predictions and direct numerical simulations for the two-dimensional, vertical impact of two droplets of the same fluid. In particular, we show that the theoretical predictions for the location and velocity of the jet root are excellent in the early stages of the impact, while the predicted jet velocity and thickness profiles are also in good agreement with the computations before the jet begins to bend. By neglecting the role of the surrounding gas both before and after impact, we are able to use Wagner theory to describe the early-time structure of the impact. We derive the model for general droplet velocities and radii, which encompasses a wide range of impact scenarios from the symmetric impact of identical drops to liquid drops impacting a deep pool. The leading-order solution is sufficient to predict the curve along which the root of the high-speed jet travels. After moving into a frame fixed in this curve, we are able to derive the zero-gravity shallow-water equations governing the leading-order thickness and velocity of the jet. Our numerical simulations are performed in the open-source software Gerris, which allows for the level of local grid refinement necessary for a problem with such a wide variety of length scales. The numerical simulations incorporate more of the physics of the problem, in particular the surrounding gas, the fluid viscosities, gravity and surface tension. We compare the computed and predicted solutions for a range of droplet radii and velocities, finding excellent agreement in the early stage. In light of these successful comparisons, we discuss the tangible benefits of using Wagner theory to confidently track properties such as the jet-root location, jet thickness and jet velocity in future studies of splash jet/ejecta evolution.

On the splashing of high-speed drops impacting a dry surface

“…Thoroddsen et al (2012) demonstrated that the droplets decelerate by 30 % just 40 µs after the pinch-off due to the aerodynamic forces acting on the droplets. Since our experimental set-up does not allow us to capture the ejection velocity precisely at the beginning of splashing, the velocities measured in this study are lower than those calculated by the inviscid theories (Roisman 2009;Cimpeanu & Moore 2018) at the pinch-off moment (Thoroddsen et al 2012). The ejection angle of the droplets is small at the beginning of splashing but later on increases drastically depending on the splashing regime.…”

“…To estimate the ejection time of the lamella τ e , its thickness h l , and velocity u l , Riboux & Gordillo (2014) considered the liquid flow as both inviscid and incompressible, allowing the use of potential and Wagner's theory (Wagner 1932). It is important to mention that several other studies have been carried out using adaptations of Wagner's theory to analyse multiple scenarios, such as the drop-liquid interaction (Howison et al 2005;Cimpeanu & Moore 2018) FIGURE 12. The arithmetic mean diameter of the ejected droplets scaled by h µ = DRe −1/2 .…”

“…This is because the lamella spreading velocity is much higher than the relative Taylor velocity (Taylor 1959), and the lamella is thicker than the viscous boundary layer, as accurately described by the inviscid solutions (Roisman 2009;Riboux & Gordillo 2014. More recently, Cimpeanu & Moore (2018) showed that the root thickness and velocity of the ejected lamella can be well predicted within their proposed approximation using a variation of Wagner's theory, which neglects the surface tension and viscosity. This shows that the velocity of the FIGURE 14.…”

Universality in the viscous-to-inertial coalescence of liquid droplets

“…To this end, the volume of fluid (VOF) simulation (41,42) was implemented using the open source code Gerris (43,44). This numerical approach has been demonstrated to be suitable for multiphase flow involving the dynamics of droplets (12,(45)(46)(47)(48)(49)(50)(51).…”

Boundary layers in Helmholtz flows

“…In a recent analysis on droplet-droplet impact problems, Cimpeanu & Moore (2018) noted that the Helmholtz solution over-predicts the thickness of the splash jet close to its root and speculate that it is the neglect of other physical effects that leads to this discrepancy. This has lead to a desire to find the viscous perturbation to the free surface location predicted by the Helmholtz flow.…”