An experimental comparison of velocities underneath focussed breaking waves
Introduction
Rogue waves threaten safety and survivability of marine structures. The mechanisms leading to the formation of such extreme waves have been investigated and probabilistic descriptions derived to provide improved design criteria (e.g. Perlin et al., 2013; Bitner-Gregersen and Toffoli, 2014; Toffoli et al., 2012; Alberello et al., 2016a). Breaking of large waves is the most hazardous condition in terms of wave forces on marine structures (Faltinsen, 1993; Grue, 2002; Kim, 2008; Alberello et al., 2017). However, it remains elusive how the mechanism leading the formation of a rogue wave affects the wave shape at the breaking and the associated kinematic field.
Measurements under deep water breaking waves have shown that wave velocities, and associated forces, exceed those predicted by the potential flow theory in the crest region. Using Laser Doppler Anemometry (lda) under plungers, Easson & Greated (Easson and Greated, 1984) report velocities two times larger than those predicted by linear theory and forces fives times larger than those of an equivalent 5th order Stokes wave. Analogous results are reported in Kim et al. (1992) for a spiller in random sea. Measured particle velocities in the crest region exceed those predicted using equivalent Stokes wave and linear superposition of the spectral components. Kim et al. (1992) argue that the asymmetric shape (crest higher than the troughs with forward leaning wave front) associated to large transient waves as a result of energy focussing might affect the accuracy of the estimation of the velocity field.
Breaking waves have also been experimentally investigated by means of Particle Image Velocimetry (piv). This technique, compared to lda, offers the advantage of obtaining fluid velocities over a plane (unlike pointwise lda measurements). Under plungers, Skyner (1996) recorded particle velocities higher than the phase speed of the waves. Observations of velocities exceeding the phase speed were also made by Perlin et al. (1996), even though the fluid flow presents a different topology compared to Skyner (1996). Difference in the flow structure are most certainly related to a different underlying wave spectrum. piv was systematically employed by Grue et al. (Grue et al., 2003; Grue and Jensen, 2006, 2012) to investigate breaking waves in deep water conditions. Monochromatic wave trains, unidirectional focussed wave groups and unidirectional random seas were all considered. However, the role played by different focussing mechanism on the velocity profile has not been assessed.
Grue et al. (2003) observed that all velocity profiles could be described by an universal profile if opportune dimensionless parameters were chosen. The velocity profile beneath a wave can be approximated by a third order monochromatic Stokes wave with the same period and amplitude using the so-called Grue method (Grue et al., 2003). The wavenumber k and the steepness ε (product of the wavenumber and the linear wave amplitude a) are obtained numerically solving the system of equations:
The angular frequency is computed linearly from the trough-to-trough wave period (i.e. being the distance between the troughs around the crest) and is the maximum wave elevation. Once the solution is obtained the velocity profile has the exponential profile:
The Grue velocity profile matches previous breaking measurements reported in e.g. (Kim et al., 1992; Skyner, 1996; Baldock et al., 1996). Furthermore, the Grue method compares well with second order potential flow predictions (Stansberg et al., 2006; Johannessen, 2010). The good performance of the Grue method and its relative simplicity established it as one of the methods commonly accepted by industry standards to define the velocity profile under large waves (Stansberg et al., 2006).
Another method to estimate the velocity profile underneath a random wave field has been proposed by Donelan et al. (1992). The method is based on a superposition of wave components. Unlike traditional linear superposition, that has been found to overestimate crest velocities, in the Donelan method spectral wave components (surface and velocity corrections) are iteratively added to the perturbed solution. To compute the velocity profile the required steps are as follows. First a Fourier Transform algorithm is used to compute amplitudes, , and phases, , of the surface elevation. A vertical grid is defined, i.e. z. The successive velocity and amplitude increments ( and respectively) are computed iteratively as
Finally, the velocities for grid points outside the water domain have to be set to zero. From the iterative procedure it can be deduced that for the component the mean water level is the pre-existing wavy surface and the velocities are computed over a varying z. The Donelan method has been found to compare well with field data (Donelan et al., 1992), but it has not been applied to focussed breaking waves yet.
In this paper the predictive performances of the Grue and Donelan methods are tested against laboratory measurements of the velocity profile underneath breaking rogue waves. The formation of the breaking waves in the wave flume is controlled by wave focussing techniques, e.g. (Longuet-Higgins, 1974; Tromans et al., 1991). Two techniques commonly used in model tests are compared: the dispersive focussing (Longuet-Higgins, 1974; Tromans et al., 1991), using different underlying JONSWAP spectra, and the nonlinear Schrödinger equation (NLS) framework (Zakharov, 1968). Whereas the velocity field under breaking waves generated by dispersive focussing has been examined in the past, it is yet uncertain how it compares to the kinematic field of breaking events generated using breathers solutions of the NLS that more realistically replicate wave evolution at sea. Other physical mechanisms can trigger the formation of breaking rogue waves in the ocean (e.g. directional focussing, wave-current interaction and bathymetry, see Onorato et al., 2013a), but only unidirectional deep water waves are considered in the present study.
The paper is structured as follows. In the next Section we describe the experimental set-up. The wave generation mechanisms are presented in Section 3. The evolution in space of the wave group and its spectral properties are shown in the following Section. Description of the wave shape, velocity profiles and comparison with engineering methods are discussed in Section 5. Final remarks are reported in the Conclusions.
Section snippets
Experimental set-up
The purpose of the experiments is to monitor the spatial evolution of a steep wave group and measure water particle velocity at breaking. Experiments have been conducted in the Extreme Air-Sea Interaction facility (EASI) in the Michell Hydrodynamics Laboratory at The University of Melbourne (Australia). The wave flume is 60 2 m (length width). The water depth was imposed to be 0.9 m. At one end of the tank a computer-controlled cylindrical wave-maker produces user-defined wave forms. At the
Wave generation
The location of the breaking event in the wave tank is controlled deterministically by means of wave focussing techniques. In deep water conditions, the dispersive focussing has been commonly used in the past. This method relies on the differential celerity of wave components of the wave spectrum, i.e. longer waves propagate faster than shorter waves. By defining an initial phase shift at the wave-maker for each spectral component, it is possible to synchronise wave components at a specific
Dispersive focussing
The time-series of the surface elevation are presented in Fig. 3. The groups become more compact in time as they approach the breaking point (probe 5). After the breaking, the wave groups broaden again, i.e. the envelope is elongated.
The dimensionless spectra corresponding to the various stages of evolution are reported in Fig. 4. A spectral transformation is observed as the group propagates along the tank. An energy downshift occurs during the wave focussing (i.e. up to the breaking point).
Dispersive focussing
Although the spectral evolution provides fundamental information about the nonlinear wave interactions, wave breaking is a highly localised mechanism that strongly relates to the time series rather than the spectral characteristics, e.g. Chalikov and Babanin (2012). The asymmetry parameters can be used as an indication of proximity to breaking (Chalikov and Babanin, 2012). Following Babanin et al. (2010), these are defined as:where is the elevation of the crest, the
Conclusions
Two different focussing techniques, dispersive focussing and nonlinear Schrödinger mechanisms have been used to generate a breaking rogue wave event in an unidirectional wave flume and to compare the associated wave field. These focussing techniques are the two main mechanisms leading to the wave growth and, eventually, breaking in the ocean. The evolution has been recorded and the associated velocity field measured at the breaking by means of optical measurements, i.e. Particle Image
Acknowledgements
A.A. and F.N. acknowledge support from the Swinburne University of Technology Postgraduate Research Award (SUPRA).
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