Effects of power-law entrainment on bubble fragmentation cascades

We consider the evolution of the bulk bubble-size distribution N(a, t) of large bubbles (Weber number We >> 1) under free-surface entrainment described generally by an entrainment size distribution I(a) with power-law slope gamma and large-radius cutoff a(max). Our main focus is the interaction between turbulence-driven fragmentation and free-surface entrainment, and, for simplicity, we ignore other mechanisms such as degassing, coalescence and dissolution. Of special interest are the equilibrium bulk distribution N-eq(a), with local power-law slope (beta) over tilde (eq)(a), and the time scale tau(c) to reach this equilibrium after initiation of entrainment. For bubble radii a << a(max), we find two regimes for the dependence of N-eq(a) on the entrainment distribution. There is a weak injection regime for gamma >= -4, where (beta) over tilde (eq)(a) = -10/3 independent of the entrainment distribution; and a strong injection regime for gamma < -4, where the power-law slope depends on gamma and is given by <(beta)over tilde>(eq) = gamma + 2/3. The weak regime provides a general explanation for the commonly observed -10/3 power law originally proposed by Garrett et al. (J. Phys. Oceanogr., vol. 30 (9), 2000, pp. 2163-2171), and suggests that different weak entrainment mechanisms can all lead to this result. For a similar to a(max), we find that N-eq(a) exhibits a steepening deviation from a power law due to fragmentation and entrainment, similar to what has been observed, but here absent other mechanisms such as degassing. The evolution of N(a, t) to N-eq(a) is characterised by the critical time tau(c) proportional to C-f epsilon(-1/3) a(max)(2/3), where epsilon is the turbulence dissipation rate and C-f is a new constant that quantifies the dependence on the daughter size distribution in a fragmentation event. For typical breaking waves, tc can be quite small, limiting the time t less than or similar to tau(c) when direct measurement of N(a, t) might provide information about the underlying entrainment size distribution.

Automated summary

This study focuses on the evolution of the bulk bubble-size distribution under the influence of turbulence-induced fragmentation and free-surface entrainment, finding two regimes based on the entrainment distribution dependence. The weak injection regime is independent of the entrainment distribution, while the strong injection regime's power-law slope is determined by gamma.