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

This paper investigates the dynamics of Worthington jets produced by sphere impacts without cavity formation, identifying three distinct pinch-off modes driven by Rayleigh–Plateau instability and deriving a new universal scaling law for maximum jet height based on momentum and energy conservation.

The Gist

Imagine you are standing by a swimming pool. If you drop a heavy stone into the water, you usually see two things: a big splash that creates a hollow "bubble" or cavity under the surface, and then a thin, high jet of water that shoots straight up into the air. That upward jet is called a Worthington jet.

For over a century, scientists have focused on that "bubble" part. They thought the jet was like a spring being released—the bubble collapses, hits the bottom, and "kicks" the water upward.

But this new paper reveals a different story.

The researchers at Shanghai Jiao Tong University studied a specific scenario: what happens when you drop a sphere into water so carefully (or use a specific type of object) that no bubble forms at all. They wanted to know: if there is no collapsing bubble to "kick" the water up, where does the energy come from to make the jet shoot up?

Here is the breakdown of their discovery using everyday analogies:

1. The "Traffic Jam" Mechanism (How the jet is born)

Instead of a "spring" (the bubble), the researchers found that the jet is actually caused by a "collision of flows."

Imagine a busy intersection where cars are driving from all directions toward the center. In this experiment, as the sphere sinks, it pushes water out of its way. This water rushes around the sides of the sphere and meets at the back.

Think of it like a converging crowd of people rushing into a narrow hallway. When everyone meets in the middle, they can't go sideways anymore, so they are forced to surge forward in one direction. In the water, this "collision" of moving liquid at the back of the sphere creates a high-pressure zone that shoots a column of water upward. It’s not a "kick" from below; it’s a "surge" from the collision.

2. The Three "Break-up" Modes (The jet's personality)

The researchers noticed that depending on how high you drop the sphere, the jet behaves in three different ways, almost like different styles of dancing:

• The Smooth Slider (No pinch-off): If you drop the sphere from a low height, the jet rises and falls like a single, graceful ribbon of water without ever breaking.
• The Falling Split (Downward pinch-off): If you drop it a bit higher, the jet shoots up, but as it starts to fall back down, it "snaps" in the middle, leaving a little droplet floating at the top.
• The Early Breakup (Upward pinch-off): If you drop it from a great height, the jet is so energetic that it actually breaks apart while it is still moving upward, like a firework bursting mid-flight.

3. The "Universal Recipe" (The Math)

The most impressive part of the paper is that the scientists found a "Universal Scaling Law."

In science, a "universal law" is like a master recipe. Usually, if you change the weight of the ball, the thickness of the liquid, or the speed of the drop, the math gets messy and unpredictable. But these researchers found a specific mathematical formula that works for everything they tested.

Whether they used steel, glass, or plastic spheres, or whether they used pure water or thick, syrupy glycerol, their formula predicted exactly how high the jet would go. It’s like finding a single recipe that can predict exactly how much a cake will rise, whether you are using flour, almond meal, or chocolate powder.

Why does this matter?

Understanding these jets isn't just about playing with water. It helps us understand:

• Nature: How raindrops hit soil or how pollutants spread in the ocean.
• Industry: How to control sprays in painting or how to cool sensitive electronics with liquid droplets.
• Science: It provides a fundamental rulebook for how energy moves from a solid object into a liquid.

In short: The jet isn't a reaction to a collapsing void; it is a surge created by a liquid traffic jam.

Technical Summary

Technical Summary: Dynamics and Universal Scaling of Worthington Jets in the Cavity-Free Regime

1. Problem Statement

Worthington jets—upward liquid columns ejected following the impact of an object on a liquid surface—are traditionally studied in the context of cavity collapse. In those cases, an air cavity is entrained and subsequently implodes due to hydrostatic pressure, forcing a jet upward.

However, this paper addresses a distinct and less understood regime: the cavity-free regime. In this scenario, no air cavity is formed (often due to high wettability or specific impact velocities), and the jet is generated through a fundamentally different mechanism. The authors aim to clarify the generation mechanism, identify the governing physical parameters, derive a universal scaling law for the maximum jet height, and model the temporal evolution of the jet's shape and height.

2. Methodology

The study employs a dual approach combining high-speed experimental observation with theoretical derivation:

• Experimental Setup: Solid spheres of varying materials (steel, aluminum, glass, POM), diameters ($D_s$), and surface wettabilities (hydrophilic vs. hydrophobic) were dropped from various heights ($H_s$) into pure water and glycerol–water mixtures (to vary viscosity $\mu$ and surface tension $\sigma$). High-speed imaging (400 fps) was used to capture the jet's evolution.
• Theoretical Framework:
  • Regime Identification: The Rayleigh–Plateau instability was applied to categorize pinch-off behaviors.
  • Scaling Law Derivation: The authors used dimensional analysis combined with momentum and energy conservation principles. They incorporated the concept of "added mass" and assumed a momentum balance between the upward jet and the downward wake.
  • Jet Evolution Modeling: To predict the time-dependent height and shape, the authors utilized self-similar solutions derived from one-dimensional mass and momentum conservation equations, incorporating gravity and surface tension. A kinematic condition at the jet tip was used to bridge the gap between the self-similar profile and the temporal evolution of the jet height.

3. Key Contributions

• Identification of a New Generation Mechanism: The paper demonstrates that cavity-free jets are sustained by the collision of converging flows behind the sphere, rather than the implosion of a void.
• Classification of Pinch-off Modes: The authors identified three distinct modes: no pinch-off, downward pinch-off (detachment during the falling stage), and upward pinch-off (detachment before reaching maximum height).
• Derivation of a Universal Scaling Law: A new, a priori scaling law for the dimensionless maximum jet height ($H_j/D_s$) was derived, accounting for the density ratio ($\tilde{\rho}$), Froude number ($Fr$), Weber number ($We$), and Reynolds number ($Re$).
• Self-Similar Evolution Model: A mathematical framework was developed to predict the jet's vertical profile and its height over time, accounting for the restorative effects of surface tension.

4. Results

• Regime Boundaries: The three pinch-off modes were found to be independent of sphere density and wettability. The transitions between these modes are governed by the Rayleigh–Plateau instability, with theoretical boundaries ($H^{cr}_{j1}$ and $H^{cr}_{j2}$) matching experimental observations remarkably well.
• Universal Scaling Validation: The proposed scaling law:
  $$\frac{H_j}{D_s} = K \left[ \frac{\tilde{\rho}}{\tilde{\rho} + C_m} \right] Fr^{1.125} We^{-0.375} Re^{0.5}$$
  successfully collapsed all experimental data (across different materials, liquids, and impact conditions) onto a single master curve, proving its universality.
• Dynamics of Height and Shape:
  • The jet's motion is primarily gravity-dominated.
  • Surface tension acts as a downward restoring force that suppresses the rising motion and accelerates the descent. This effect is most significant in the "no pinch-off" regime; once a child droplet detaches, the motion becomes almost entirely governed by gravity.
  • The self-similar model accurately predicts the jet's slender shape and vertical trajectory.

5. Significance

This work provides a fundamental shift in the understanding of Worthington jets by decoupling the jet dynamics from cavity collapse. By establishing a universal scaling law and a predictive model for the cavity-free regime, the research offers valuable insights for:

• Industrial Applications: Improving spray cooling, painting, and droplet manipulation processes.
• Environmental Science: Understanding the spread of pollutants or pathogens during liquid impacts.
• Fluid Mechanics: Advancing the theoretical modeling of slender jet evolution and the complex interplay between inertia, gravity, and capillary forces.