Experimental observations and modelling of sub-Hinze bubble production by turbulent bubble break-up

We present experiments on large air cavities spanning a wide range of sizes relative to the Hinze scale $d_{H}$, the scale at which turbulent stresses are balanced by surface tension, disintegrating in turbulence. For cavities with initial sizes $d_0$ much larger than $d_{H}$ (probing up to $d_0/d_{H} = 8.3$), the size distribution of bubbles smaller than $d_{H}$ follows $N(d) \propto d^{-3/2}$, with $d$ the bubble diameter. The capillary instability of ligaments involved in the deformation of the large bubbles is shown visually to be responsible for the creation of the small bubbles. Turning to dynamical, three-dimensional measurements of individual break-up events, we describe the break-up child size distribution and the number of child bubbles formed as a function of $d_0/d_{H}$. Then, to model the evolution of a population of bubbles produced by turbulent bubble break-up, we propose a population balance framework in which break-up involves two physical processes: an inertial deformation to the parent bubble that sets the size of large child bubbles, and a capillary instability that sets the size of small child bubbles. A Monte Carlo approach is used to construct the child size distribution, with simulated stochastic break-ups constrained by our experimental measurements and the understanding of the role of capillarity in small bubble production. This approach reproduces the experimental time evolution of the bubble size distribution during the disintegration of large air cavities in turbulence.