Dynamics and universal scaling of Worthington jets in the cavity-free regime

TL;DR

This study combines experiments and theory to analyze Worthington jets produced by sphere impacts without cavity formation, revealing universal scaling laws and distinct pinch-off modes governed by fluid dynamics principles.

Contribution

It introduces a new universal scaling law for jet height, identifies three pinch-off modes, and elucidates the jet generation mechanism in cavity-free impacts.

Findings

Three distinct pinch-off modes identified

Universal scaling law validated across impact conditions

Gravity-dominated jet dynamics with surface tension effects

Abstract

Worthington jets ejected after the impact of a solid or liquid object on a liquid surface have extensive applications in natural, industrial, and scientific contexts. Here, we present a combined experimental and theoretical investigation of the jet generated by sphere impact with no cavity formed. Experiments identify three distinct pinch-off modes, whose regime boundaries are independent of sphere wettability and density, and are theoretically determined by the Rayleigh--Plateau instability. From momentum and energy conservation, a new scaling law is derived for the dimensionless maximum jet height and agrees remarkably well with experiments across various impact conditions, thus validating its universal character and clarifying its dependence on the Froude, Weber, and Reynolds numbers as well as the density ratio. Coupling self-similar solutions with a kinematic condition at the jet tip yields good predictions for the evolution of jet height and shape, revealing gravity-dominated jet dynamics, with a modification from surface tension that is most pronounced without pinch-off. These findings demonstrate that the upward jet is sustained by the collision of converging flow behind the sphere, a generation mechanism fundamentally distinct from the cavity collapsing forced case.