Capillary driven fragmentation of large gas bubbles in turbulence

The bubble size distribution below a breaking wave is of paramount interest when quantifying mass exchanges between the atmosphere and oceans. Mass fluxes at the interface are driven by bubbles that are small compared with the Hinze scale $d_h$, the critical size below which bubbles are stable, even though individually these are negligible in volume. Combining experimental and numerical approaches, we report a power-law scaling $d^{-3/2}$ for the small bubble size distribution, for sufficiently large separation of scales between the injection size and the Hinze scale. From an analysis of individual bubble breakups, we show that small bubbles are generated by capillary effects, and that their breakup time scales as $d^{3/2}$, which physically explains the sub-Hinze scaling observed.

Figure 1

(a) Sketch of a breakup process, involving inertial deformations, followed by a capillary splitting event. (b) Schematic of a bubble of size $\mathrm{\Delta }$ splitting into a small bubble of size $\delta$ in a time $T(\mathrm{\Delta },\delta )$ and subsequent splittings.

Figure 2

DNS snapshots of a typical breakup sequence, with the initial bubble size $d_0/d_h=2.9$. The bubbles' interface is represented in white. The first images (a), (b) show large-scale deformation due to turbulence, happening over the eddy turn over time at the size of the initial bubble scale $d_0$, $t_c\left(d_0\right)={\epsilon }^{-1/3}{d}_0^{2/3}$, leading (c), (d) to the formation of thin filaments. (e)–(h) Successive splitting events of the filament are visible leading to multiple child bubbles. The filaments quickly break creating a wide range of bubble sizes, the smallest being orders of magnitude smaller than the initial one.

Figure 3

Lifetime of bubbles as a function of the size of the smallest bubble they split apart into. The size of the parent bubble is given by the color. Most splitting events occur on a timescale faster than the eddy turnover time $t_c\left(\delta \right)$, shown in dotted line, suggesting that the splitting events are not primarily instigated by turbulent deformations at the small child bubble size. The ensemble averaged time ${\langle T(\mathrm{\Delta },\delta )\rangle }_{\mathrm{\Delta }}$ is shown in black squares and follows the capillary timescale of the small child bubble $T_2\left(\delta \right)$ without any adjusting parameters (shown as the dashed line). Diamonds and circles denote two levels of resolution, associated with a grid size of $\ell =0.0075d_0$ and $\ell =0.015d_0$, respectively, with $d_0$ the initial bubble size.

Figure 4

(a) Sketch of the experimental setup. (b)–(e) Successive snapshots of the release of a large bubble into a turbulent background flow. (b) Before the crack opening, an air pocket is trapped within a extended thins rubber sheet. (c) Just after the membrane piercing, the membrane moves away, generating locally a high shear situation, but on a timescale much shorter than the typical turbulence time at the bubble scale (0.2 ms). (d) The small wavelength disturbances are then dissipated by viscosity while the interface deforms at larger scale under the action of the background turbulence. (e) Eventually the bubble interface experiences multibreaking events, generating a broad distribution of bubble size. ( f)–(h) Zoom in view of a typical breaking dynamics of a gas filament during the process. The time between images is 2 ms. The red rectangle highlights the region where ligament collapsed creating many sub-Hinze bubbles.

Figure 5

Bubble size distribution obtained both experimentally and numerically in two different geometries: a breaking wave ($⧫$, $•$) and a single bubble breaking ($\blacksquare$, $★$). The distribution exhibits two power laws: for $d>d_h$, $\mathcal{N}\left(d\right)\propto d^{-10/3}$ (dashed line), while $\mathcal{N}\left(d\right)\propto d^{-3/2}$ for $d<d_h$ (dotted line).