Rising bubbles draw surface patterns: a numerical study

TL;DR

This study addresses pattern formation on a liquid surface induced by chains of rising bubbles. It employs direct numerical simulations with the Aphros code and a volume-of-fluid formulation to resolve bubble interactions and surface dynamics, revealing disordered, two-armed, five-armed, and spiral three-armed surface configurations as the bubble release period $T$ varies. A simple heuristic model connects the divergence angle $\varphi$ to $T$ via $\varphi = \frac{\pi}{2} + \arcsin(a T - b)$ (with fitted $a,b$), and it reproduces both numerical and experimental results, including arm-number probabilities via $n\varphi = 2\pi m$. The work bridges near-surface hydrodynamics and macroscopic self-organization, offering predictive insights for interfacial transport, pattern control, and monitoring of gas-liquid systems.

Abstract

Small bubbles rising in a chain can self-organize into regular patterns upon reaching a liquid's free surface. This phenomenon is investigated through direct numerical simulations. By varying the bubble release period, distinct branching patterns characterized by different numbers of arms are observed. These macroscopic regular configurations arise from localized non-contact repulsion and pair collisions between bubbles as they arrive at the free-surface emergence site. A theoretical model is proposed to quantitatively relate the number of branches to the bubble release period. The model also predicts probabilities of observing specific arm counts in reality. This study provides insights into broader nonlinear pattern formation and self-organization phenomena.

Figures (16)

• Figure 1: Top view of the surface pattern formed by periodically injected air bubbles into a vessel partially filled with silicone oil. The image on the right is the side view of the bubble chain at the same instant.
• Figure 2: Bubble's terminal rise velocity versus equivalent diameter. The black hollow symbols represent the experimental data sets S3 (diamonds), S5 (triangles), S6 (squares), and S8 (circles) from Ref. Raymond2000. The Morton numbers for these four cases are $0.11$, $9\times10^{-4}$, $1\times10^{-4}$, and $9\times10^{-7}$, respectively. The red solid symbols with the same shapes are the corresponding simulation results using Aphros.
• Figure 3: (a) Illustration of the computational domain (bubble size not to scale) and the coordinate system. (b) Simulated bubbles rising in a chain, with the free surface (black) and streamlines (gray) shown in a cross section through the $z$-axis. Only a small region close to the bubble chain is displayed.
• Figure 4: (a) to (d): Four typical surface patterns observed in the numerical simulations (top view). The free surface is colored in gray. The dashed lines serve as visual guides to distinguish between different branches. The numbers indicate the sequence in which successive bubbles join these branches. From left to right, the release periods are 100, 50, 42, and 36 ms, respectively. (e) to (h): Divergence angle as a function of bubble release sequence number for the corresponding cases in the top row. The red dashed lines show the averaged values.
• Figure 5: (a) Free surface (gray) and trajectory (cyan) of a single bubble in a five-armed case. (b) Trajectories of three consecutive bubbles projected onto the horizontal plane. The green dashed lines indicate the emission directions of bubbles $-2$ and $-1$. The inset is an amplified view of the trajectories in the vicinity of the bubble emergence site.