Vortex ring formation from the interaction of a cavitation bubble with a confined air bubble: experiments and a timing criterion

This study investigates the formation of vortex rings resulting from the interaction between a collapsing cavitation bubble and a confined air bubble in a cylindrical blind hole, identifying specific geometric and temporal conditions through experiments and modeling that distinguish regimes where coherent rings are produced from those where they are not.

The Gist

Imagine a tiny, invisible spring trapped inside a narrow, deep well (a blind hole) drilled into a block of wood. Now, imagine a powerful, expanding balloon (a cavitation bubble) suddenly inflating just above the mouth of that well.

This paper explores what happens when these two things interact. The researchers found that instead of the usual messy splash or a jet of water shooting down, this specific setup can launch a perfect, donut-shaped ring of swirling water (a vortex ring) straight up into the air.

Here is the story of how they discovered this, broken down into simple steps:

The Setup: The Balloon and the Spring

Think of the cavitation bubble as a balloon that inflates and then violently pops (collapses). Think of the air bubble inside the hole as a compressed spring.

• When the balloon inflates above the hole, it pushes down on the water sitting on top of the spring.
• This squeezes the spring (the air bubble) tight.
• When the balloon starts to shrink (collapse), the squeezed spring suddenly snaps back, shooting the water above it upward like a cork popping out of a bottle.

The Three Outcomes: Timing is Everything

The researchers tested many different setups by changing two things:

1. How high the balloon was above the hole (Stand-off distance).
2. How much of the hole was filled with the spring (Air bubble size).

They found three possible results, like three different ways a race can end:

1. The Perfect Race (Vortex Ring Forms)

• The Scenario: The balloon is close enough to the hole to squeeze the spring hard, but not too close. The spring is also big enough to push a good chunk of water, but not so big that it runs out of water to push.
• The Result: The spring shoots the water upward as a solid, fast-moving "slug" (like a bullet made of water). Just as this water slug is flying up, the balloon above it is shrinking. The water slug hits the bottom of the shrinking balloon right at the perfect moment.
• The Magic: This collision creates a perfect, spinning donut of water (a vortex ring) that flies away. It's like a drummer hitting a drum skin at the exact right moment to create a perfect ripple.

2. The Late Arrival (No Ring)

• The Scenario: The balloon is too far away. It doesn't squeeze the spring hard enough.
• The Result: The spring pushes the water, but it's too weak and slow. By the time the water slug finally reaches the balloon, the balloon has already finished shrinking and is collapsing on its own. The water hits a mess of collapsing water instead of a clean surface.
• The Outcome: No ring forms. It's like trying to catch a ball after the game has already ended.

3. The Bypass (No Ring)

• The Scenario: The spring (air bubble) is huge, filling almost the whole hole, leaving very little water on top of it.
• The Result: When the spring snaps back, it expands so fast that it shoots through the tiny bit of water left on top. The air bubble itself hits the balloon directly.
• The Outcome: The water never gets a chance to form a solid slug. The air hits the balloon, but no ring is created. It's like a runner sprinting past a baton instead of carrying it.

The "Timing Rule"

The scientists created a simple math rule (a timing criterion) to predict if a ring will form.

• Imagine the water slug has to travel a certain distance to hit the balloon.
• The balloon has a specific amount of time to shrink before it disappears.
• The Rule: For a ring to form, the water slug must arrive at the balloon while the balloon is shrinking, but not too early (when it's still growing) and not too late (when it's already gone).
• If the timing is just right (between 1 and 1.5 times the "half-life" of the bubble), you get a ring.

What Happens to the Ring?

Once the ring is formed, it shoots upward at about 5 meters per second (roughly 11 mph). However, it doesn't last long. Because it's moving so fast and is relatively small, it becomes unstable. Within a few milliseconds, the ring starts to wobble and break apart, much like a smoke ring that eventually dissipates into the air.

Why Does This Matter?

The paper explains that this is a new way to make vortex rings. Usually, you need a special nozzle or a jet of water to make a ring. Here, nature does it all by itself using the interaction between an expanding bubble and a trapped pocket of air.

The researchers used high-speed cameras (taking thousands of pictures per second) to watch this happen and built computer models to prove that their "timing rule" works. They confirmed that if you get the distance and the air bubble size right, you can reliably create these swirling water rings.

Technical Summary

Technical Summary: Vortex Ring Formation from the Interaction of a Cavitation Bubble with a Confined Air Bubble

Problem Statement
The collapse of a cavitation bubble near a rigid boundary is a well-established phenomenon in fluid mechanics, typically characterized by the formation of a high-speed liquid jet directed toward the wall. This jet is responsible for cavitation erosion but also underpins applications like ultrasonic cleaning and targeted drug delivery. While the influence of planar boundaries, free surfaces, and elastic materials on this jet has been extensively studied, the dynamics of a cavitation bubble interacting with a confined gas pocket within a blind hole remain largely unexplored. The central question addressed by this study is how the confinement of an air bubble within a cylindrical blind hole modifies the collapse dynamics of a nearby cavitation bubble. Specifically, the authors investigate whether this specific geometric constraint can suppress the canonical liquid jet and instead generate a fundamentally different coherent structure: a vortex ring propagating away from the surface.

Methodology
The study employs high-speed shadowgraphy imaging to visualize the interaction between a spark-generated cavitation bubble and a confined air bubble located at the base of a blind hole (diameter 5 mm, depth 15 mm) in a solid acrylic block submerged in distilled water.

• Experimental Setup: A low-voltage spark discharge nucleates a cavitation bubble at a controlled stand-off distance ($h$) above the blind hole. An air bubble is introduced at the bottom of the hole with a controlled fill height.
• Parameters: The interaction is characterized by two dimensionless parameters:
  1. Stand-off distance ($H$): The ratio of the bubble generation height to the maximum bubble radius ($H = h/R_{max}$).
  2. Air bubble fill fraction ($B$): The fraction of the hole height occupied by the air bubble ($B = (d_{hole} - d_{top})/d_{hole}$).
• Data Analysis: High-speed video sequences (20,000–40,000 fps) are processed using ImageJ and custom MATLAB scripts to track the positions of the cavitation bubble boundary, the air bubble interface, and the liquid column.
• Theoretical Modeling: Two one-dimensional models are developed to explain the dynamics:
  1. Liquid Column Model: Based on the Rayleigh–Plesset equation for the cavitation bubble growth and a momentum balance equation for the liquid column driven by the pressure difference between the cavitation bubble and the compressed air bubble.
  2. Air Bubble Expansion Model: An isentropic expansion model for the air bubble that predicts the terminal speed of the liquid column ($U_{lc}$) and establishes a timing criterion.

Key Results and Regimes
Parametric experiments across the $H-B$ space reveal three distinct interaction regimes:

1. Liquid Column Impact (L-B Impact): Occurs when $H \lesssim 0.5$ and $B \lesssim 0.5$.

   • Mechanism: The growing cavitation bubble drives a downward flow, compressing the confined air bubble. The air bubble rebounds, expelling the overlying liquid column upward as a coherent slug. This slug impacts the far boundary of the collapsing cavitation bubble during its collapse phase.
   • Outcome: The impact generates an axisymmetric impulse that creates azimuthal vorticity, which rolls up and detaches as a coherent vortex ring propagating away from the surface.
   • Dynamics: The ring initiates at approximately 5 m/s and decelerates quadratically. It exhibits a Reynolds number of $Re \approx 4500$, leading to azimuthal instabilities and eventual breakup.

2. Late Impact: Occurs at large $H$ (e.g., $H = 0.72$).

   • Mechanism: The air bubble is less compressed and expands more slowly. The liquid column arrives at the far boundary only near the very end of the cavitation bubble's collapse.
   • Outcome: By the time the column arrives, the collapse is dominated by the inward jet, which deforms the column into a toroidal shape rather than pinching off a ring. No vortex ring forms.

3. Direct Air Bubble Impact (B-B Impact): Occurs at large $B$ (e.g., $B = 0.63$).

   • Mechanism: The liquid column above the air bubble is too short. The rapidly expanding air bubble overtakes and fragments the liquid column before it can act as a coherent slug.
   • Outcome: The air bubble itself directly impacts the far boundary of the cavitation bubble, causing only weak interfacial roll-up without ring formation.

Timing Criterion
The authors propose a dimensionless timing parameter, $\Pi$, to distinguish these regimes:
$$\Pi = \frac{h + R_{max}}{U_{lc} \cdot (t_{cav}/2)}$$
where $U_{lc}$ is the liquid column speed and $t_{cav}/2$ is the collapse half-period.

• Ring Formation: Observed when $1 \lesssim \Pi \lesssim 1.5$. This indicates the liquid column arrives during the collapse phase but not too late.
• No Ring: Occurs when $\Pi > 1.5$ (Late Impact) or when the physical coherence of the column is lost due to large $B$ (Bypass failure).

Significance and Claims
The paper claims to identify a new mechanism for vortex ring generation distinct from previous observations near through-holes (where rings were attributed to jet circulation). Here, the ring arises specifically from the impulsive impact of a directed liquid slug on a curved, collapsing interface, enabled by the unidirectional confinement of the air bubble.

The study provides a quantitative framework linking geometric parameters ($H, B$) to flow outcomes via the timing parameter $\Pi$. While the theoretical models are simplified (neglecting wall friction, liquid compressibility, and complex interfacial instabilities), they successfully predict the impact location and the existence of the timing window required for ring formation. The authors note that the aspect ratio of the hole was fixed at 3, and future work could explore how different geometries might suppress the observed instabilities. The findings offer a potential method for controlling flow structures in applications involving confined cavities, such as microfluidic transport or ultrasonic cleaning of porous surfaces.