High-resolution direct simulation of deep water breaking waves: transition to turbulence, bubbles and droplets production

Abstract

We present high-resolution three-dimensional (3-D) direct numerical simulations of breaking waves solving for the two-phase Navier–Stokes equations. We investigate the role of the Reynolds number (Re, wave inertia relative to viscous effects) and Bond number (Bo, wave scale over the capillary length) on the energy, bubble and droplet statistics of strong plunging breakers. We explore the asymptotic regimes at high Re and Bo, and compare with laboratory breaking waves. Energetically, the breaking wave transitions from laminar to 3-D turbulent flow on a time scale that depends on the turbulent Re up to a limiting value $Re_\lambda \sim 100$ , consistent with the mixing transition in other canonical turbulent flows. We characterize the role of capillary effects on the impacting jet and ingested main cavity shape and subsequent fragmentation process, and extend the buoyant-energetic scaling from Deike et al. (J. Fluid Mech., vol. 801, 2016, pp. 91–129) to account for the cavity shape and its scale separation from the Hinze scale, $r_H$ . We confirm two regimes in the bubble size distribution, $N(r/r_H)\propto (r/r_H)^{-10/3}$ for $r>r_H$ , and $\propto (r/r_H)^{-3/2}$ for $r<r_H$ . Bubbles are resolved up to one order of magnitude below $r_H$ , and we observe a good collapse of the numerical data compared to laboratory breaking waves (Deane & Stokes, Nature, vol. 418 (6900), 2002, pp. 839–844). We resolve droplet statistics at high Bo in good agreement with recent experiments (Erinin et al., Geophys. Res. Lett., vol. 46 (14), 2019, pp. 8244–8251), with a distribution shape close to $N_d(r_d)\propto r_d^{-2}$ . The evolution of the droplet statistics appears controlled by the details of the impact process and subsequent splash-up. We discuss velocity distributions for the droplets, finding ejection velocities up to four times the phase speed of the wave, which are produced during the most intense splashing events of the breaking process.

Capillary driven fragmentation of large gas bubbles in turbulence

“…where Q is the volume of air injected to the breaking cascade per volume of water per second, and can be evaluated from the breaker geometry and energetics [21,22]. The prefactor for the sub-Hinze distribution can then be evaluated using the continuity of N at d h , and…”

“…We now aim to verify that an initial large separation of scales, namely for which d 0 /d h 1, indeed leads to a universal N (d ) ∝ d −3/2 in the sub-Hinze-scale regime. We will analyze laboratory and numerical data of bubble size distribution under breaking waves from previous work [18,22]. On top of this, we compare the results to a more idealized configuration consisting of a single large bubble injected in a turbulent flow, both numerically [39] and experimentally.…”

“…Figure 5 shows the bubble size distributions N (d ) obtained under breaking waves [18,22] and from a single large bubble breaking in a turbulent flow, both experimentally and numerically. Within all four data sets, for d > d h , the distributions exhibit N (d ) ∝ d −10/3 scaling, in agreement with other previous experimental measures below a breaking wave [17,19,20] and in agreement with the classic breakup cascade argument from Garrett et al [24].…”

“…As a consequence, the fragmentation of bubbles has been extensively studied in model experiments [13][14][15][16] as well as under breaking waves both experimentally [17][18][19][20] and numerically [21][22][23]. For large bubbles, a consensus has been reached on the bubble size distribution, described as N (d ) ∝ d −10/3 with d being the bubble volume equivalent diameter.…”

Simulation-based study of low-Reynolds-number flow around a ventilated cavity

“…Giorgio, Pirozzoli & Iafrati (2022) corroborated the findings that the dissipated energy fraction does not depend much on the Reynolds number or dimensionality. Mostert, Popinet & Deike (2022) observed that the Reynolds number does not affect the shape of the bubble size distribution after Re reaches a threshold, as the mean turbulent dissipation rate is not sensitive to Re. The ventilated cavity flows discussed in the present study and the wave breaking flows are both air-water turbulent mixed flow systems.…”