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Interaction of vortex rings generated by two unsynchronised drop impacts

This study experimentally demonstrates that the time difference between two unsynchronized drop impacts on a liquid pool dictates four distinct regimes of vortex ring interaction—from symmetric merging to independent evolution—by modulating the influence of capillary waves generated by the first impact.

Original authors: Anna A. L. Huttunen, Guilherme Moreira Bessa, Matilda Backholm

Published 2026-07-09

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Anna A. L. Huttunen, Guilherme Moreira Bessa, Matilda Backholm

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are standing by a calm pond. If you drop a single pebble in, it creates a ripple, but if you drop a specific type of water droplet with just the right speed and size, it doesn't just make a splash—it creates a hidden, spinning donut of water that dives deep underwater. Scientists call this a vortex ring. It's like a smoke ring, but made of water, and it's surprisingly stable and strong.

Now, imagine dropping two pebbles (or water droplets) into the pond at the same time, very close to each other. What happens to their underwater spinning donuts? Do they merge into one giant donut? Do they crash and bounce off? Or does one destroy the other?

This paper is all about figuring out exactly what happens when those two drops hit the water at slightly different times. The researchers found that the timing is everything. It's like a dance where the steps must be perfectly synchronized, or the whole performance falls apart.

Here is what they discovered, broken down into four simple scenarios based on how much time passes between the first drop and the second drop:

1. The Perfect Twin Dance (Time difference: Less than 0.5 milliseconds)

If the two drops hit the water almost instantly one after the other (so fast that a human eye couldn't tell the difference), the two underwater spinning donuts form side-by-side. Because they are so close and synchronized, they immediately lean toward each other, crash, and merge into a single, slightly squashed ring.

The Analogy: Think of two identical twins running toward each other and hugging perfectly in the middle. They become one unit.

2. The Clumsy Hug (Time difference: 0.5 to 7 milliseconds)

If the second drop hits just a tiny bit later (like a fraction of a blink), the dance gets messy. The first ring has already started moving, and the second ring is still forming. When they collide, they don't merge neatly. Instead, they crash asymmetrically. One ring survives the crash and keeps going, while the other ring gets torn apart or "stagnates" (stops moving effectively).

The Analogy: Imagine two people trying to high-five, but one is slightly late. Instead of a clean slap, they bump hands awkwardly, and one person stumbles while the other keeps walking.

3. The Spoiled Second Act (Time difference: 7 to 80 milliseconds)

If you wait a bit longer (about 7 to 80 milliseconds), the first drop has already made a big splash and sent out ripples (called capillary waves) across the surface. By the time the second drop hits, the water surface isn't flat anymore; it's wavy and moving.
Because the second drop hits a wavy, unstable surface, it can't form a strong, clean spinning ring. The first ring dives down perfectly fine, but the second ring is "impaired"—it's weak, wobbly, and can't dive deep.

The Analogy: Imagine trying to jump into a swimming pool to do a perfect dive, but someone just jumped in right before you, creating huge waves. You hit the water, but instead of a clean dive, you just get tossed around by the waves. Your dive is ruined.

4. The Solo Acts (Time difference: 80 milliseconds or more)

If you wait long enough (80+ milliseconds), the water from the first drop has completely settled down. The surface is calm again. When the second drop hits, it's as if the first drop never happened. Both rings form perfectly and dive down without bothering each other.

The Analogy: This is like two people jumping into a pool, but with a long pause between them. The first person dives, swims away, and the water calms down before the second person jumps. They are two separate, perfect dives.

The Secret Ingredient: The Ripples

The researchers realized that the key to understanding all these different outcomes is the ripples (capillary waves) created by the first drop.

If the second drop hits before the ripples arrive, the rings merge (or crash).
If the second drop hits while the ripples are there, the second ring gets ruined.
If the second drop hits after the ripples have passed, everything is fine.

They even did the math to predict exactly when these changes would happen based on how fast the ripples travel, and their predictions matched their experiments perfectly.

Why does this matter?

The paper concludes that if you want to mix liquids efficiently using drops (like in high-tech manufacturing or bioprinting), you have to be very careful about the timing. If you want the drops to merge and mix well, you need them to hit almost simultaneously. If you want them to dive deep and stay separate, you need to wait long enough for the water to calm down. If you get the timing wrong, you end up with a messy, weak mix instead of a controlled one.

Technical Summary: Interaction of Vortex Rings Generated by Two Unsynchronised Drop Impacts

Problem Statement
Vortex rings generated by single drop impacts are well-documented phenomena with applications in natural gas exchange and technological fluid mixing. While the dynamics of single rings and those formed by synchronised double drop impacts have been extensively studied, the interaction of rings generated by unsynchronised impacts remains unexplored. In practical scenarios, such as high-throughput mixing applications, neighbouring drops rarely impact a liquid surface at the exact same time. The lack of understanding regarding how the time difference ( $\Delta t$ ) between two adjacent impacts influences the formation, collision, and subsequent evolution of the resulting vortex rings limits the ability to achieve uniform and controlled mixing.

Methodology
The authors conducted experiments using two water drops impacting a deep water pool with a controlled time difference ( $\Delta t$ ).

Experimental Setup: Two blunt needles separated by $L = 6$ mm were used to generate drops of radius $R \approx 1.5$ mm from a height of $h = 2.9$ cm. The impact velocity was approximately $0.58$ m/s, with a Weber number ($We$) ranging from 16 to 17.
Visualization: Particle Image Velocimetry (PIV) was employed to quantify the flow field. The fluid was seeded with polyethylene microspheres (45–53 $\mu$ m). High-speed imaging (2000 fps) captured the drop formation, impact, and the evolution of the vortex rings up to 3 cm deep into the pool.
Analysis: Streamlines were plotted to locate vortex core centers. Key metrics analyzed included the trajectories of the cores, the tilt angle of the rings, the distance between cores, translational velocities, and local peak vorticities. The study varied $\Delta t$ from near-simultaneous ( $<0.5$ ms) to widely separated ( $>169$ ms).

Key Contributions and Results
The study identifies four distinct regimes of vortex ring evolution based on the time difference $\Delta t$ between the two impacts:

Symmetric Merging ( $\Delta t < 0.5$ ms):
When impacts are nearly simultaneous, the two vortex rings approach each other, collide, and merge symmetrically. The cores deform, vorticity annihilates at the contact point, and the rings reconnect to form a single, distorted ring. The trajectories and angle changes are symmetric, consistent with previous literature on synchronised impacts.
Asymmetric and Incomplete Merging ( $0.5 \text{ ms} < \Delta t < 7 \text{ ms}$ ):
As the time difference increases, the interaction becomes asymmetric. The first ring tilts more aggressively toward the second before collision. The merging is incomplete; the second ring's core often decays or stagnates, while the first ring survives the collision. A third core may reappear between the outer cores, but the resulting structure is unstable and does not form a clean single ring.
Impaired Second Ring ( $7 \text{ ms} \le \Delta t < 80 \text{ ms}$ ):
In this regime, the first impact significantly disrupts the formation of the second ring. By the time the second drop impacts, capillary waves generated by the first impact have reached the second impact site. This prevents the formation of a clean, deep-penetrating vortex ring. The second ring is severely impaired, often failing to detach properly or penetrate deeply, while the first ring remains relatively undisturbed, moving straight down.
Independent Evolution ( $\Delta t \ge 80 \text{ ms}$ ):
When the time difference is sufficiently large, the surface perturbations (capillary waves and crater dissipation) from the first impact have fully subsided before the second drop arrives. Consequently, the two rings form and evolve independently without influencing each other, behaving effectively as two single drop impacts.

Mechanism of Interaction
The authors attribute the transitions between these regimes to the propagation of capillary waves and the evolution of the surface crater.

The transition from asymmetric merging (Regime ii) to impaired formation (Regime iii) occurs when capillary waves from the first impact reach the second impact site before the second crater begins to form. Theoretical calculations based on capillary wave velocity ( $v_c \approx 0.44$ m/s) and crater formation time estimate this transition at $\Delta t \approx 6.3$ ms, which aligns well with the observed transition at $5.3 \pm 1.8$ ms.
The transition to independent evolution (Regime iv) occurs when the final waves of the dissipating crater have passed the second impact site. This is estimated to occur around $\Delta t \approx 57.5$ ms, consistent with the observed transition range of $56.5$ to $80$ ms.

Significance
The paper demonstrates that the time difference between drop impacts is a critical parameter governing the creation and interaction of adjacent vortex rings. The findings highlight that even small delays can drastically alter the symmetry and completeness of ring merging, while moderate delays can prevent the formation of a functional second ring entirely. These results are significant for applications requiring controlled fluid mixing, such as bioprinting or high-throughput processing, where precise spatial and temporal adjustment of drop impacts is necessary to ensure optimal vortex ring generation and deep penetration into the liquid pool. The study provides a framework for predicting interaction outcomes based on the timing of impacts relative to capillary wave propagation.