Generation and breakup of Worthington jets after cavity collapse. Part 1. Jet formation

Abstract

At the beginning of the last century Worthington and Cole discovered that the high-speed jets ejected after the impact of an axisymmetric solid on a liquid surface are intimately related to the formation and collapse of an air cavity created in the wake of the impactor. In this paper, we combine detailed boundary-integral simulations with analytical modelling to describe the formation of such Worthington jets after the impact of a circular disk on water. We extend our earlier model in Gekle et al. (Phys. Rev. Lett., vol. 102, 2009a, 034502), valid for describing only the jet base dynamics, to describe the whole jet. We find that the flow structure inside the jet may be divided into three different regions: the axial acceleration region, where the radial momentum of the incoming liquid is converted to axial momentum; the ballistic region, where fluid particles experience no further acceleration and move constantly with the velocity obtained at the end of the acceleration region; and the jet tip region, where the jet eventually breaks into droplets. From our modelling of the ballistic region we conclude that, contrary to the case of other physical situations where high-speed jets are also ejected, the types of Worthington jets studied here cannot be described using the theory of hyperbolic jets of Longuet-Higgins (J. Fluid Mech., vol. 127, 1983, p. 103). Most importantly, we find that the velocity and the shape of the ejected jets can be well predicted at any instant in time with the only knowledge of quantities obtained before pinch-off occurs. This fact allows us to provide closed expressions for the jet velocity and the sizes of the ejected droplets as a function of the velocity and the size of the impactor. We show that our results are also applicable to Worthington jets emerging after the collapse of a bubble growing from an underwater nozzle, although this system creates thicker jets than the disk impact.

Simulation of the droplet-to-bubble transition in a two-fluid system

“…10 for D = 100 at t = 1.897 14. This behavior could indicate the initiation of a Worthington jet as reported in [17].…”

Bubble generation in quiescent and co-flowing liquids

“…8(h)). The reason is that during the bubble collapse stage, the negative radial pressure difference at the neck accelerates the liquid flow in radial direction around the neck and produces a high speed liquid jet penetrating into the newly formed bubble once the bubble detaches from the orifice [42]. The formation of a Worthington jet is physically realistic as described in Ref.…”

Volume entrained in the wake of a disk intruding into an oil-water interface

“…For this, we use the boundary integral method as described in [8,11,[17][18][19]. We performed boundary integral simulations of an impulsively started disk in an infinite bath of homogeneous fluid, moving at a constant speed V D .…”

“…We approach this problem experimentally by impulsively starting a disk from the interface of two immiscible liquids, oil and water, similarly and with the same setup as we did before when studying the dynamics of an air-water interface when a disk would impact it [7][8][9][10][11]. Besides being a practical solution to studying drift volumes, this method has relevance for, e.g., the mechanism of oil We start with the bottom of the disk at rest at the interface between a deep layer of oil (45 mm) on top of a deep layer of water (25 cm), after which we pull down the disk at constant speed V D .…”