Capillary effects on wave breaking

Abstract

We investigate the influence of capillary effects on wave breaking through direct numerical simulations of the Navier-Stokes equations for a two-phase air-water flow. A parametric study in terms of the Bond number, Bo, and the initial wave steepness, , is performed at a relatively high Reynolds number. The onset of wave breaking as a function of these two parameters is determined and a phase diagram in terms of ( , Bo) is presented that distinguishes between non-breaking gravity waves, parasitic capillaries on a gravity wave, spilling breakers and plunging breakers. At high Bond number, a critical steepness c defines the wave stability. At low Bond number, the influence of surface tension is quantified through two boundaries separating, firstly gravity-capillary waves and breakers, and secondly spilling and plunging breakers; both boundaries scaling as ∼ (1 + Bo) −1/3 . Finally the wave energy dissipation is estimated for each wave regime and the influence of steepness and surface tension effects on the total wave dissipation is discussed. The breaking parameter b is estimated and is found to be in good agreement with experimental results for breaking waves. Moreover, the enhanced dissipation by parasitic capillaries is consistent with the dissipation due to breaking waves.

Air entrainment and bubble statistics in breaking waves

“…In contrast, DNS is an appealing tool since no parametrizations are used to solve the multi-phase flow. The DNS has been limited to two-dimensional evolution of periodic unstable waves with relatively small wavelengths, providing numerical data on wave dissipation and the splashing processes (Chen et al 1999;Song & Sirviente 2004;Iafrati 2011;Deike et al 2015). Three dimensional simulations of breaking waves have recently become available, both DNS (Fuster et al 2009) and LES (Derakhti & Kirby 2014;Lubin & Glockner 2015).…”

“…The Dirac delta, δ s , expresses the fact that the surface tension term is concentrated on the interface, where γ is the surface tension coefficient, κ and n the curvature and normal to the interface. This solver has been successfully used in multiphase problems like atomization (Fuster et al 2009;Agbaglah et al 2011;Chen et al 2013), the growth of instabilities at the interface (Fuster et al 2013), wave breaking in two (Deike et al 2015) and three dimensions (Fuster et al 2009), capillary wave turbulence (Deike et al 2014) and splashing (Thoraval et al 2012). …”

“…We study a single breaking wave as was done in a previous 2D study (Deike et al 2015), but now extending it to three-dimensions.…”

“…A third-order Stokes wave solution for the interface η(x, y) and the velocity potential φ(x, y, z) in the water are used as initial conditions in a square box of size λ on a side (see Deike et al (2015)). The wave propagates in the x direction.…”

“…The wave slope S = ak, with a the initial wave amplitude and k = 2π/λ the wavenumber, varies from 0.35 to 0.65, i.e. from incipient wave breaking to strongly plunging waves (Deike et al 2015). Note that since we are using only the third-order Stokes wave solution, slopes higher than the limiting slope for the full Stokes wave solution can be defined.…”