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

TL;DR

This work addresses cavity-free Worthington jets generated by solid-sphere impact on a liquid surface. It combines experiments and theory to reveal three pinch-off regimes governed by Rayleigh–Plateau instability and derives a universal jet-height scaling: $H_j/D_s = K \frac{\tilde{\rho}}{\tilde{\rho}+C_m} Fr^{1.125} We^{-0.375} Re^{0.5}$, where $\tilde{\rho}=\rho_s/\rho_l$, $Fr=\tfrac{2H_s}{D_s}$, $We=\tfrac{2gH_s \rho_l D_s}{\sigma}$, and $Re=\sqrt{2gH_s}\,D_s/\nu$. The jet evolution is modeled by self-similar solutions augmented with a kinematic condition at the jet tip, revealing gravity-dominated dynamics with surface-tension corrections; the approach collapses data from diverse spheres, liquids, and coatings onto a single master curve, illustrating the universal character of cavity-free Worthington jets. The study clarifies a generation mechanism distinct from cavity-collapse jets, with implications for applications where controlled jetting is critical and for understanding high-speed liquid-jet phenomena across fluids and geometries.

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.

Figures (8)

• Figure 1: Temporal evolution of Worthington jets or cavity dynamics following the impact of steel spheres on a water surface. Rows (a--c) show the jet evolution for spheres of diameter $D_s=30\ \mathrm{mm}$ at release heights $H_s=0.1\ \mathrm{m},\ 0.4\ \mathrm{m},\ 0.7\ \mathrm{m}$, respectively. Row (d) shows cavity dynamics for a hydrophobic-coated sphere of $D_s=35\ \mathrm{mm}$ at $H_s=0.6\ \mathrm{m}$. Snapshots with blue timestamps indicate the instant when the jet reaches its maximum height. Red boxes highlight key features: the formation of a fine splash at the initial stage of the jet in (a), and the pinch-off of the liquid column in (b) and (c). The experimental GIF animations corresponding to cases (a)--(c) are provided in the Supplemental Material supplemental2025.
• Figure 2: (a) Regime diagrams illustrating the pinch-off modes of Worthington jets corresponding to four types of spheres. Hollow black dots with thick edges: no pinch-off; solid red dots: pinch-off during the falling stage; hollow blue dots with thin edges: pinch-off during the rising stage. Each dot corresponds to an averaged experimental data with at least 3 repeated tests. The dashed and chain lines are transition boundaries $H_j^{cr1}$ and $H_j^{cr2}$, respectively, which are theoretically obtained from Eq. (\ref{['eq:critical Hj']}). (b) Impact velocity for cavity formation versus contact angle $\theta_s$. Squares, triangles, circles, diamonds, and stars denote experimental results for spheres made of glass, steel, aluminum, POM, and hydrophobic coated spheres (any of the above materials with hydrophobic coating), respectively. Diameter of the sphere is not shown explicitly. The solid line corresponds to the theoretical curve given by duez2007making. The dashed boundary is obtained by adjusting the parameter $g_0$ in their mathematical model from 7 to 5 to fit the present experiments. In both panels (a) and (b), water is used as the working liquid, with its corresponding fluid properties employed to obtain the theoretical curves.
• Figure 3: Experimental relationship between the maximum jet height $H_j$ and the release height $H_s$. The data include all experiments conducted with steel, aluminum, glass, POM, and coated spheres impacting both water and three different glycerol--water mixtures.
• Figure 4: Sketches of the flow near the free surface based on the numerical simulation by do2009splash, illustrating (a) the initial impact stage when the sphere is partially immersed, and (b) the early jet stage after the sphere has moved away from the free surface. Blue arrows indicate the flow direction.
• Figure 5: Experimental relationship between the dimensionless maximum jet height $H_j/D_s$ and the combined dimensionless parameters defined in Eq. (\ref{['eq:scaling law']}). The data include all experimental configurations as described in Fig. \ref{['Fig:Hj-Hs']}.