Wave Kinematics


Determination of the velocity associated with wave motion is paramount for the offshore and shipping industry which require accurate prediction of the shape of the largest waves and associated velocities to design and operate marine structures. In this case, of particular interest are the so-called rogue waves, these waves are at least twice the height of the surrounding waves and often prone to breaking. Equally important is the identification of the statistical properties of wave induced velocity under breaking and non-breaking waves needed in climatology and coastal engineering to quantify the momentum exchange between the ocean and the atmosphere and its propagation to the sea bottom.
The kinematic field underneath breaking and non-breaking waves is was analysed via laboratory experiments and numerical simulations.

Observation of turbulence and intermittency in wave-induced oscillatory flows

The dynamic and thermal regimes of the climate are regulated by an exchange of energy and momentum between the atmosphere and the ocean. Waves induce turbulence in the upper ocean by breaking and through Langmuir circulations enhancing the momentum exchange, however, there is evidence that the wave oscillatory flow itself can be turbulent and directly inject mixing into subsurface layers.
Measurements of the velocity field of oscillatory flows, which are induced by mechanically generated random wave fields in a large scale experimental facility are used to investigate the existence, extent and properties of this non-breaking wave-induced turbulent flow.

Velocity spectrum highlighting the turbulent tail.

We demonstrate that the spectral tail of the velocity field follows a power-law scaling close to classical Kolmogorov turbulence theory. Higher order analysis, based on rigorous statistical analysis of the structure functions, highlights the emergence of intermittency in oscillatory flows.

Structure functions using the Extended Self Similarity (left) and corresponding exponents (i.e. slope) highlighting the deviation from Kolmogorov predictions (right).

Our experiments suggest that turbulence induced by the wave oscillatory flow exists in the ocean, together with other sources of turbulence, further justifying the previous working hypothesis adopted in ocean modelling. At oceanic scales a wider inertial range would develop making the energy cascade even more evident. In limited water depth wave induced turbulence can extend to the sea bottom affecting sediment resuspension. In deep water conditions, however, the rapid exponential decay of the velocity profile limits the turbulent layer to a depth comparable to half of the wavelength.

Alberello, A., Onorato, A., Frascoli, F. and Toffoli, A., 2019. Observation of turbulence and intermittency in wave-induced oscillatory flows. Wave Motion, 184, pp.81-89.

The velocity field underneath a breaking rogue wave: laboratory experiments vs numerical simulations

In this work we perform a direct comparison of the velocity filed obtained from numerical simulations and experimental measurements is presented. Numerical simulations of a breaking rogue wave are performed with a Level-Set Navier–Stokes (LS-NS) solver developed by Alessandro Iafrati. To speed-up the simulations, Navier–Stokes (NS) are initialised from a pre-breaking solution obtained using the Higher Order Spectral Method (HOSM). The coupling of a “fast” method (HOSM) with a “slow” method (NS) allows one to reduce the computational time and, at the same time, accurately reproduce the breaking stage starting from a fully nonlinear wave condition that better reproduces ocean conditions. Corresponding experiments are conducted in the wave flume at The University of Melbourne where PIV measurements provide the velocity field.

Experimental and numerical breaking wave

Alberello, A. and Iafrati, A., 2019. The velocity field underneath a breaking rogue wave: laboratory experiments versus numerical simulations. Fluids, 4(2), 68.


An experimental comparison of velocities underneath focussed breaking waves

Nonlinear wave interactions affect the evolution of steep wave groups, their breaking and the associated kinematic field. Laboratory experiments are performed to investigate the effect of the underlying focussing mechanism on the shape of the breaking wave and its velocity field. In this regard, it is found that the shape of the wave spectrum plays a substantial role. Broader underlying wave spectra leads to energetic plungers at a relatively low amplitude. For narrower spectra waves break at higher amplitudes but with a less energetic spiller. Comparison with standard engineering methods commonly used to predict the velocity underneath extreme waves shows that, under certain conditions, the measured velocity profile strongly deviates from engineering predictions.

Breaking wave and reconstructed velocity field


Experimental PIV set-up

The formation of breaking rogue waves was controlled using different underlying physical mechanisms and the corresponding velocity field measured using Particle Image Velocimetry (PIV). The simulation methodology is based on the coupling of the Higher Order Spectral Method (HOSM) for the evolution stage and a Level-Set Navier-Stokes solver (LS-NS) for the breaking stage only. It was found that the simple linear potential wave theory is not a reliable tool to predict wave induced velocities, since they are likely to underestimate their values especially in breaking wave crests, and consequently hydrodynamic forces during storm conditions when rogue waves are more likely to occur.

Alberello, A., Chabchoub, A., Monty, J.P., Nelli, F., Lee, J.H., Elsnab, J. and Toffoli, A., 2018. An experimental comparison of velocities underneath focussed breaking waves. Ocean Engineering, 155, pp.201-210.


Non-Gaussian properties of second-order wave orbital velocity

A stochastic second-order wave model is applied to assess the statistical properties of wave orbital velocity in random sea states below the water surface. Directional spreading effects as well as the dependency of the water depth are investigated by means of a Monte-Carlo approach. Unlike for the surface elevation, sub-harmonics dominate the second-order contribution to orbital velocity. We show that a notable set-down occurs for the most energetic and steepest groups. This engenders a negative skewness in the temporal evolution of the orbital velocity. A substantial deviation of the upper and lower tails of the probability density function from the Gaussian distribution is noticed; velocities are faster below the wave trough and slower below the wave crest when compared with linear theory predictions.

Probability density function of wave orbital velocities for different wave directional spreading and water depth conditions.

Second-order nonlinearity effects strengthen with reducing the water depth, while weaken with the broadening of the wave spectrum. The results are confirmed by laboratory data. Corresponding experiments have been conducted in a large wave basin taking into account the directionality of the wave field. As shown, laboratory data are in very good agreement with the numerical prediction.

Alberello, A., Chabchoub, A., Gramstad, O., Babanin, A.V. and Toffoli, A., 2016. Non-Gaussian properties of second-order wave orbital velocity. Coastal Engineering, 110, pp.42-49.