On coherent vortical structures in wave breaking

 

Figure 1: Underwater vortical structures generated during the breaking: vortex-tubes and vortex-sheets are drawn in yellow and gray, respectively.

The breaking of ocean waves is of significant interest due to its implications in various physical, chemical, and biological processes that occur at the ocean-atmosphere interface.  Wave breaking generates free-surface turbulence, dissipates wave energy, and enhances momentum, heat, and gas transfer between air and water. Over the years, a number of review articles and monographs have been published on the subject (Banner & Peregrine 1993; Melville 1996; Duncan 2001; Babanin 2011; Kiger & Duncan 2012; Perlin, Choi & Tian 2013; Lubin & Chanson 2017; Deike 2022), and these works call for more research into nearly every aspect of wave breaking. For these reason, the flow generated by the breaking of free-surface waves in a periodic domain is simulated numerically by means of a gas-liquid multiphase Navier Stokes solver. The solver relies on the Volume-of-Fluid (VOF) approach, and interface tracking is carried out by using a novel algebraic scheme based on a tailored TVD limiter (Pirozzoli et al., 2019). The solver is proved to be characterized by low numerical dissipation, thanks to the use of the MAC scheme, which guarantees discrete preservation of total kinetic energy in the case of a single phase. Both two- and three-dimensional simulations have been carried out, and the analysis is presented in terms of energy dissipation, air entrainment, bubble fragmentation, statistics and distribution. Particular attention is paid to the analysis of the mechanisms of viscous dissipation. For this purpose, coherent vortical structures (Horiuti and Takagi, 2005), are identified and the different behaviour of vortex sheets and vortex tubes are highlighted, at different Re. The correlation between vortical structures and energy dissipation demonstrates clearly their close link both in the mixing zone and in the pure water domain, where the coherent structures propagate as a consequence of the downward transport. Notably, it is found that the dissipation is primarily connected with vortex sheets, whereas vortex tubes are mainly related to flow intermittency.

In order to highlight the connections between air entrainment and viscous dissipation with vortical structures, in fig. 2, slices taken in the longitudinal symmetry plane are drawn. The results display a very close correlation between viscous dissipation and the vortex sheet indicator. Also, it is worth noticing that viscous dissipation is not confined about the free surface, but it is spread within the bubble cloud. Within the high-dissipation regions, marked by the vortex sheet indicator, vortex tubes also form in zones with high vorticity.

 

Figure 2: Longitudinal sections of the solutions computed at Re = 10000 (top) and Re = 40000 (bottom). The coloured contours denote the local values of the normal-to-plane vorticity components (left), and the local dissipation rate (right). The black solid lines in left and right figures denote the vortex-tube and vortex-sheet iso-lines, respectively. Note that the solutions in the water domain are shown only.