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

    Xingsheng Li and Jing Li*

    • Marine Numerical Experimental Center, State Key Laboratory of Ocean Engineering, School of Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, People's Republic of China

    • *Contact author: lijing_@sjtu.edu.cn

    Phys. Rev. Fluids 11, 033604 – Published 6 March, 2026

    DOI: https://doi.org/10.1103/2vdb-tqj6

    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.

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