Turbulence analysis in breaking waves
Authors: Padova Diana De 
Published: May 2022

Two different types of breaking waves, a spilling and a plunging one, are examined, while propagating in the
surf zone along a sloped (1:20) impermeable and fixed bottom. They were reproduced in the laboratory wave
channel of the DICATECh Department, that is 45 m long and 1 m wide, with the water depth close to the
generator kept equal to 0.7 m. The main features of the waves are shown in Table 1, being H0 the wave height
at the wave paddles, T the wave period, L0 the wavelength according to Airy theory and ξ0 the Irribarren
number. The velocity field was measured by using a backscatter, two-component four-beam fibre-optic LDA
system, and the free surface profile was assessed using resistance probes. Figures 1a and 1b show a part of
the wave channel and the measuring system.

Specifically, we analyze the power spectra of the turbulent longitudinal and vertical velocities (u’ and w’
respectively) assessed in selected vertical sections of the surf zone, at three different depths, i.e., close to
the bottom, at mid-depth and near the surface (Figure 2a). In this way, we can detect some frequency regions
where: i) the decrease in energy almost follows a slope -1, which is indicative of a coexisting energy
production and cascade energy transfer (Nikora, 2005); ii) the spectrum shows peaks with a -3 slope,
corresponding to quasi 2D large coherent structures (Truong et al., 2018); iii) the slope of the spectrum
almost follows -5/3, thus is typical of the inertial subrange. Such information, examined along the longitudinal
and vertical directions in the breaking zone, seems useful to describe the turbulent behavior of the waves.
Furthermore, by considering the autocorrelation functions of the fluctuating velocities, the integral and
turbulent length scales are also computed.
As a second step, we use the Weakly-Compressible Smoothed Particle (WCSPH) model, coupled with a SubParticle Scale (SPS) approach to modeling turbulence, for reproducing numerically the waves. Preliminary
comparisons of the above-mentioned spectra show a good agreement (Figure 2b). The Okubo–Weiss parameter
W rated by W0=0.2σW, with σW being the standard deviation of W in the whole domain, allows to separate the
flow into different regions, i.e., a vorticity-dominated region, a strain-dominated region, and a background field.
Furthermore, the obtained numerical flow fields can therefore be used to investigate the characteristics of
breaking vorticity within these flows. In Figures 3 and 4 snapshots of W/ W0 and vorticity ω are shown in the 2-D
longitudinal section for the plunging case, during and after breaking. During breaking, due to the impact of the jet
on the surface, positive vorticity is generated and propagates with the uprush flow, while negative vorticity
spreads in the return flow at intermediate depths. After breaking (Figure 4b), positive vorticity largely persists at
the surface in the wave bores, while negative vorticity rapidly decreases. The maps of W/W0 highlight the
dominance of strain during breaking (Figure 3a); instead, after breaking (Figure 4a) some coherent structures
seem to propagate obliquely towards the bottom.