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

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.

💡 Research Summary

This paper presents a comprehensive experimental and theoretical study of Worthington jets generated when a solid sphere impacts a liquid surface without forming an air cavity—a regime that has received comparatively little attention. By systematically varying sphere diameter (15–40 mm), material (steel, aluminum, glass, POM), surface wettability (hydrophilic and hydrophobic coatings), release height (0.1–1 m), and liquid properties (pure water and glycerol‑water mixtures of 10–30 % glycerol), the authors capture a broad range of impact conditions. High‑speed imaging (400 fps, 0.126 mm pixel⁻¹) records the jet formation, ascent, and eventual breakup.

Three distinct pinch‑off modes are identified based on the maximum jet height H_j: (i) no pinch‑off, (ii) downward pinch‑off (a child droplet separates after the jet reaches its apex), and (iii) upward pinch‑off (a droplet detaches before the apex). Remarkably, the boundaries between these modes are independent of sphere density, material, or wettability, indicating that the underlying physics is governed by the competition between inertia and surface tension rather than specific material properties.

The authors invoke the Rayleigh–Plateau instability to explain the pinch‑off thresholds. Assuming the jet column is cylindrical and its volume equals that of the sphere, they relate the jet radius D_j to H_j via D_j ≈ √(2 D_s³/3 H_j). The most unstable wavelength λ_m ≈ 4.5 D_j leads to a first critical height H_cr1 = (3/2) D_s, below which no pinch‑off occurs. By comparing the capillary‑inertial pinch‑off time T_p ≈ √(ρ_l D_j³/σ) with the time to reach the apex T_0 ≈ √(2 H_j/g), a second critical height H_cr2 is derived: H_cr2 = 5 ρ_l g D_s⁹/(27 σ²). These analytical curves partition the (D_s, H_j) space into three regimes and match the experimental data with high fidelity.

A universal scaling law for the dimensionless maximum jet height H* = H_j/D_s is derived from momentum and energy conservation. The law incorporates four dimensionless groups: the Froude number Fr = U_s²/(g D_s), the Weber number We = ρ_l U_s² D_s/σ, the Reynolds number Re = ρ_l U_s D_s/μ, and the density ratio β = ρ_s/ρ_l. The resulting expression

 H* ≈ C · Fr^{1/2} · We^{−1/4} · Re^{−1/8} · β^{1/2},

with C ≈ 0.9, collapses data from all experimental configurations onto a single curve, confirming its universal character. This scaling reveals that the jet height grows with the square root of the impact Froude number, diminishes with increasing Weber and Reynolds numbers, and is enhanced by a higher sphere‑to‑liquid density ratio.

To describe the temporal evolution of the jet, the authors couple a self‑similar solution (height ∝ t^{2/3}) with a kinematic condition at the jet tip. In the gravity‑dominated regime, the jet height follows H(t) ≈ H_j − ½ g t², while surface tension introduces only a modest correction during the early ascent. The model predicts that upward pinch‑off occurs when the capillary‑inertial time scale is shorter than the ascent time (T_p < T_0), leading to a droplet detaching before the apex; downward pinch‑off arises when T_p > T_0, so the droplet separates after the apex. The theoretical height‑time curves and jet shape profiles agree well with the high‑speed measurements.

Overall, the study demonstrates that Worthington jets in the cavity‑free regime are driven by the collision of converging flow behind the sphere, a mechanism fundamentally distinct from the cavity‑collapse driven jets traditionally studied. The Rayleigh–Plateau instability governs the onset of pinch‑off, and the derived scaling law provides a compact, dimensionless description of jet height that depends only on Fr, We, Re, and β. These insights broaden the understanding of high‑speed impact phenomena and have practical implications for applications such as spray cooling, inkjet printing, pesticide delivery, and biological fluid dynamics where controlled jet formation without cavity formation is desirable.