Fast droplet impact onto slowly moving deep pools

TL;DR

This study investigates how a slowly moving deep pool alters fast droplet impact ejecta, revealing that upstream outcomes become asymmetric and depend on both the Capillary number $Ca$ and the pool-to-droplet velocity ratio through the parameter $\sqrt{u_t/u_n}$. Using a combination of high-speed experiments and 3D direct numerical simulations, the authors map the upstream transition between separate ejecta sheets (SES) and upstream lamella for $Ca<0.2$, and show that the transition boundary is length-scale invariant. The results indicate a physical mechanism where pool motion constrains the evolution of the ejecta-sheet angle, promoting lamella formation upstream without a corresponding downstream reversal, and they extend to equivalent oblique impacts on static pools. These findings have implications for natural systems like ocean–rain interactions and industrial processes such as inkjet printing, where moving pools and oblique impacts are common.

Abstract

When a fast droplet impacts a pool, the resulting ejecta sheet dynamics determine the final impact outcome. At low Capillary numbers, the ejecta sheet remains separate from a deep static pool, whilst at higher viscosities it develops into a lamella. Here, we show that the common natural scenario of a slowly moving deep pool can change the upstream impact outcome, creating highly three-dimensional dynamics no longer characterised by a single descriptor. By considering how pool movement constrains the evolution of the ejecta sheet angle, we reach a length-scale invariant parameterisation for the upstream transition that holds for a wide range of fluids and impact conditions. Direct numerical simulations show similar dynamics for an equivalent oblique impact, indicating that the pool boundary layer does not play a decisive role for low pool-droplet speed ratios. Our results also provide insight into the physical mechanism that underpins pool impact outcomes more generally.

Figures (6)

• Figure 1: (a) $\hbox{We}$ versus $\hbox{Re}$ for all experiments reported. The purple line delineates the known vortex shedding boundary Agbaglah2015. (b) A rendering of the experimental setup.
• Figure 2: (a) Computational box highlighting adaptive grid refinement. (b) Experimental view of the case described by $\hbox{Ca} = 0.105$ (32 vol% fluid) and $u_t = 0.15m\per s$ ($u_n = 2.45m\per s$, $\sqrt{u_t/u_n} = 0.25$), at $t_\mu^\ast = 2$. The upstream outcome is a lamella. The orange arrow indicates the direction of pool movement and the scale bar is 2mm. (c) The result of a simulation matching the conditions in panel (b), with tracer fields used to visualise liquid originating from the droplet and the pool separately. As in the experiment, a lamella is seen upstream. (d) An equivalent oblique impact ($u_o = \sqrt{u_t^2 + u_n^2} = 2.455m\per s$; $\beta = \tan^{-1}(u_t/u_n) = \ang{3.5}$) on a static pool ($u_t=0m\per s$). The droplet falls from left to right here, in the direction indicated by the green arrow (dashed is vertical). A lamella seen on the leading side, which corresponds to upstream on a moving pool.
• Figure 3: (a)--(c) $\hbox{Ca}=0.132$ ($\hbox{We}=345$; $u_n = 3.10ms$) impact of a 32 vol% droplet onto a 32 vol% deep pool. (a) The pool is static: separate ejecta sheet, which is expected since $\hbox{Ca}<0.2$ (figure \ref{['fig:setup']}a). (b) The pool moves with $u_t = 0.17m\per s$ ($\sqrt{u_t/u_n}=0.23$): separate ejecta sheet, but the ejecta sheet dynamics are not axisymmetric. (c) The pool moves with $u_t = 0.26m\per s$ ($\sqrt{u_t/u_n}=0.29$): lamella upstream and a separate ejecta sheet downstream. (d) Sketch of an ejecta sheet, where $\theta$ is the ejecta sheet angle, as defined in Thoraval2012karman. Orange arrows indicate the direction of pool movement and all scale bars are 2mm.
• Figure 4: Upstream impact outcomes for normal droplet impact on a moving deep pool. Red circular markers indicate a lamella, while green triangular markers indicate a separate ejecta sheet. (a) This regime map includes all experimental conditions described in section \ref{['sec:exp']}: $\hbox{We}\in[134,450]$ and $\hbox{Re}\in[940,7930]$. The same regime map with markers coloured by the fluid involved is provided as supplementary material. The blue dashed line delineates a linear least squares fit to the transition. (b) $\hbox{Ca} = 0.072\pm0.002$ (21 vol% and 1 cSt fluids) with $r_n\in[1.11,1.87]~mm$ to assess the influence of length scale. The blue patch indicates the approximate upstream transition identified. For comparison, the yellow dashed line indicates a $1/4$ exponent (an arbitrary example of a weak dependence for demonstrative purposes) dependence on $r_n$ , i.e. $r_n^{1/4}\sqrt{u_t/u_n}$ for fixed $\hbox{Ca}$.
• Figure 5: (a)--(c) High resolution ($327\px\per mm$) images of the early-time ejecta sheet dynamics of $\hbox{Ca}~=~0.115\pm0.002$ impact (32 vol% fluid). (a) Static pool. (b)--(c) $u_t=0.20\pm0.01~m\per s$ ($\sqrt{u_t/u_n}=0.27$): (b) separate ejecta sheet upstream; (c) lamella upstream. Orange arrows indicate the direction of pool movement and all scale bars are 1mm. (d) Data indicating the difference between the horizontal extent of the ejecta sheet on moving and static pools, from the experiments in panels (a)--(c). $e_{x,p}$ is the horizontal position of the ejecta sheet tip from the original impact point (IP); $p=m$ for a moving pool (either (b) green or (c) red) and $p=s$ for a static pool. In one case ('moving IP', dashed line), the IP is translated with the moving pool speed over time.