Fate of Secondary Droplets Produced by High-speed Raindrops Interacting with a Liquid Pool

This study employs direct numerical simulations to reveal that secondary droplets generated by high-speed raindrop impacts on a liquid pool follow a universal size distribution scaling law of $N_d(r_s)\propto r_s^{-5/2}$ and are significantly influenced by the formation and breakup of a central liquid film, which modulates droplet capture and re-merging dynamics.

The Gist

Imagine you are standing by a calm pond on a rainy day. You watch a single raindrop hit the water. It makes a little splash, a tiny crown of water shoots up, and then it collapses back down. That's a simple story.

But what happens when two raindrops hit the water at the exact same time, right next to each other? Or what if a whole storm of them hits together?

This paper is like a high-speed, super-detailed movie camera that zooms in on that moment to see the invisible chaos happening. The researchers used powerful computer simulations to watch how raindrops interact with the ocean (or a pool) and, more importantly, how they create a "family" of tiny new droplets (secondary droplets) that fly into the air.

Here is the story of their findings, broken down with some everyday analogies:

1. The "Crowded Dance Floor" vs. The "Solo Dancer"

When a single raindrop hits the water, it's like a solo dancer spinning on a stage. It creates a hole (a cavity) and a ring of water (a crown) that shoots up. Eventually, that ring collapses inward, pinching off a bubble and shooting a jet of water back up.

But when two drops hit close together, it's like two dancers bumping into each other mid-spin.

• The Collision: As the two "crowns" rise, they crash into each other in the middle.
• The Wall: Instead of collapsing inward like a solo dancer, they form a vertical wall of water (called a "central liquid film") between them.
• The Result: This wall changes the whole dance. It stops the air from rushing in the way it usually does. Because the air can't rush in as fast, the water doesn't collapse inward as violently. The "dance" is slower, the hole in the water isn't as deep, and the splash behaves differently.

2. The "Popcorn" Rule (The Size of the Splashes)

One of the coolest things the researchers discovered is a mathematical rule for the size of the tiny droplets that fly off.

Think of the splash like a bucket of popcorn popping. You get a few huge kernels, a bunch of medium ones, and a massive amount of tiny, fluffy bits.

• The researchers found that the number of droplets follows a very specific pattern: The smaller the droplet, the more of them there are.
• Specifically, they found a "magic formula" (a scaling law) that predicts exactly how many tiny droplets you get based on the size of the droplet. It's like a recipe: if you know the size of the droplet, you can predict exactly how many of that size will be created.
• They also found that if the water is "stickier" (higher surface tension), you get fewer droplets. If the raindrop is bigger, you get a lot more.

3. The "Trap" and the "Escape"

The most fascinating part of the study is what happens to these tiny droplets after they are born. Do they fly away into the sky, or do they fall back into the pool?

• The Solo Drop (One Raindrop): When a single drop hits, it creates a strong "vacuum" effect. The air rushes in fast, creating a whirlwind that sucks the tiny droplets down into the hole, making them fall back into the water very quickly.
• The Double Drop (Two Raindrops): When two drops hit, that central wall of water acts like a traffic jam for the air. The air can't rush in as fast.
  • The Small Stuff: Because the air isn't rushing in as hard, the tiniest droplets (the "fluffy popcorn bits") are more likely to escape the trap and fly upward into the air.
  • The Medium Stuff: The slightly larger droplets still fall back in, but they take longer to sink because the "whirlwind" pulling them down is weaker.

Why Does This Matter?

You might think, "So what? It's just rain." But this is actually huge for our planet:

1. Ocean Health: When rain hits the ocean, it sprays salt and water into the air. This changes the temperature and saltiness of the ocean surface, which affects weather patterns and climate.
2. Measuring Rain: Sometimes, rain gauges get confused. They might count the tiny "secondary droplets" flying around as if they were new rain, making it look like it's raining harder than it actually is.
3. Oil Spills: If there's an oil spill, rain can break the oil into tiny droplets and mix them into the water. Understanding how rain creates these droplets helps us clean up spills better.

The Bottom Line

This paper is like a detective story solving the mystery of a raindrop's splash. They found that when raindrops hit the ocean together, they don't just make two separate splashes; they create a complex interaction where a "wall" of water forms between them. This wall changes the airflow, which decides whether the tiny spray droplets escape into the sky or get sucked back down into the ocean.

It turns out that the fate of a tiny drop of water depends entirely on whether it has a neighbor hitting the water at the same time!

Technical Summary

1. Problem Statement

The interaction between falling raindrops and a liquid pool is a fundamental process in engineering (e.g., spray cooling) and geophysics (e.g., ocean surface salinity, rainfall measurement bias, and acoustic monitoring). While single-drop impacts have been extensively studied, the dynamics of multi-drop interactions in the high-energy Bubble Canopy (BC) regime remain poorly understood.

• The Gap: Existing literature largely focuses on low-energy regimes or single-drop impacts. In high-speed rainfall (BC regime), neighboring raindrops interact, forming complex morphologies like central liquid films. The mechanisms governing the generation, size distribution, and ultimate fate (capture vs. re-merging) of secondary droplets in these multi-drop scenarios are not quantitatively characterized.
• Objective: To investigate the dynamics of secondary droplets generated by high-speed raindrop impacts on a deep liquid pool, specifically analyzing how inter-raindrop spacing, surface tension, and drop size influence droplet size distribution and spatial-temporal evolution.

2. Methodology

The study employs Direct Numerical Simulation (DNS) using the open-source solver Basilisk.

• Numerical Approach:
  • Governing Equations: Incompressible Navier-Stokes equations with surface tension modeled via the Continuum Surface Force (CSF) model.
  • Interface Tracking: Volume of Fluid (VOF) method with a split advection scheme.
  • Grid: Octree-structured adaptive mesh refinement (AMR) to resolve fine interfacial features and secondary droplets. The finest mesh size is $\Delta = 15.6 \, \mu\text{m}$.
• Simulation Setup:
  • Domain: 3D computational domain representing a deep pool ($H = 8d_0$).
  • Parameters: Raindrops are modeled as ellipsoids ($d_h=4.3$ mm, $d_v=3.8$ mm) impacting at $U_0 = 7.2$ m/s.
  • Dimensionless Numbers: $Re \approx 30,000$, $We \approx 2,964$, $Fr \approx 1,290$ (placing the impact firmly in the BC regime).
  • Cases:
    • Single-drop (SR): Reference case with varying surface tension ($\sigma$) and diameters ($d$).
    • Two-drop (D2, D3, D4): Two raindrops with center-to-center distances of $2d_h$, $3d_h$, and $4d_h$ to study interaction effects.
• Validation: Simulation results for cavity radius and depth in the single-drop case were validated against experimental data (Murphy et al., 2015) and previous numerical studies (Wang et al., 2023), showing excellent agreement.

3. Key Contributions

A. New Scaling Law for Secondary Droplet Size Distribution

The authors derived and validated a new scaling law for the number density of secondary droplets, $N_d(r_s)$, as a function of droplet radius $r_s$.

• Derivation: Based on the Kolmogorov–Hinze framework, assuming $N_d$ depends on the ejection rate $Q$, surface tension $\sigma$, water density $\rho_w$, and radius $r_s$.
• Result: The distribution follows a power law:
  $$N_d(r_s) \propto Q \left(\frac{\sigma}{\rho_w}\right)^{-1/2} r_s^{-5/2}$$
  Where $Q$ (volume of water ejected per unit air volume per second) scales with the square of the drop diameter ($Q \propto d^2$).
• Validation: When normalized by this scaling law, data from simulations with varying surface tensions and drop diameters collapse onto a single curve, confirming the universality of the mechanism in the high-$Re$, high-$We$ regime.

B. Identification of the Central Liquid Film Mechanism

The study elucidates the role of the central liquid film formed when the crowns of adjacent raindrops merge.

• Morphology: Unlike single drops where the crown converges radially to form a bubble canopy and a strong downward jet, multi-drop impacts form a vertical liquid sheet between drops.
• Dynamics: The rupture of this central film creates a recoil effect. In closely spaced drops (D2), this leads to a deep downward bulge and a secondary cavity. As spacing increases (D3, D4), the film rupture is delayed and less violent, resulting in weaker bulges and shallower cavities.

C. Fate of Secondary Droplets: Size-Dependent Behavior

The study reveals that the "fate" of droplets (whether they are captured by the cavity or escape) is highly size-dependent and modulated by the central film:

• Small Droplets ($r_s < 0.05$ mm): Generated early (prompt splash). In multi-drop cases, they are more likely to be captured by the secondary flows of the merging cavities compared to single-drop cases, though this capture efficiency decreases as drop spacing increases.
• Medium Droplets ($0.05 < r_s < 0.1$ mm): Generated later via crown breakup. In single-drop cases, strong downward airflow (vortex-induced low pressure) pulls these droplets rapidly back into the pool. In multi-drop cases, the central film blocks airflow, reducing the downward velocity and causing these droplets to remain suspended longer or escape.
• Large Droplets ($r_s > 0.1$ mm): In the D2 case, a significant population of large droplets is generated specifically by the violent disintegration of the central film, a mechanism absent in single-drop impacts.

4. Results and Analysis

• Impact Morphology:

  • D2 (Closest): Crowns merge quickly, forming a central film that ruptures violently, creating a deep secondary cavity and a "downward bulge."
  • D3/D4 (Wider): Merging is delayed. The central film rupture is less energetic, leading to weaker recoil and shallower penetration.
  • Analytical Comparison: The cavity depth evolution in single-drop cases matches potential flow theory (Bisighini et al., 2010) initially but diverges as the downward jet forms. Multi-drop cases deviate earlier due to film interactions.

• Spatial and Temporal Distribution:

  • Spatial: In D2, the spatial distribution of droplets evolves from two peaks (corresponding to the two drops) to a single central peak as the central film disintegrates and produces a burst of droplets at $x=0$.
  • Temporal: The number of small droplets remains stable in single-drop cases but vanishes faster in multi-drop cases due to cavity capture. Conversely, medium-sized droplets persist longer in multi-drop cases due to reduced downward airflow.

• Flow Dynamics:

  • Single Drop: Strong inward airflow creates a vortex that drives a high-velocity downward jet, rapidly re-merging droplets with the pool.
  • Two Drops: The central film acts as a barrier, limiting air inflow. This reduces the pressure differential and the strength of the downward jet, altering the trajectory and residence time of secondary droplets.

5. Significance

• Theoretical Advancement: Provides the first quantitative scaling law for secondary droplet size distributions in the high-energy Bubble Canopy regime, bridging the gap between experimental observations and theoretical models.
• Geophysical Implications: Improves the understanding of air-sea interaction mechanisms. The findings suggest that rainfall intensity and drop spacing significantly alter the production and retention of secondary droplets, which directly impacts:
  • Rainfall Measurement: Correcting biases in instruments that overestimate intensity due to secondary droplet detection.
  • Ocean Mixing: Understanding how rain-induced turbulence and bubble entrainment vary with drop clustering.
  • Acoustic Monitoring: Refining models for precipitation monitoring via underwater acoustics.
• Future Directions: The study establishes a baseline for simulating complex rainfall fields, suggesting that future models must account for inter-drop spacing and central film dynamics to accurately predict ocean surface processes.

In summary, this paper uses high-fidelity 3D DNS to demonstrate that raindrop interactions fundamentally alter the hydrodynamics of impact, creating unique droplet generation mechanisms (central film rupture) and modifying the aerodynamic forces that dictate the fate of secondary droplets.