Standing wave field observations at a vertical wall
Introduction
Coasts and seas are populated with protective and commercial structures designed to withstand the local sea and wave climate. When waves encounter a coastal or offshore structure, the properties of the obstruction determines the fraction of wave energy reflected, transmitted and absorbed, to create a (partial) standing wave. If fully reflected, the water surface and horizontal orbital velocity are, when linear wave theory is concerned, elevated by a factor of two at the anti-nodes and nodes, respectively (e.g. Holthuijsen, 2010).
Linear theory is, however, insufficient to capture the complex behaviour of standing waves at the wall. Non-breaking wave conditions require higher-order approximations for accurate representation of the pressure forces on the wall (e.g. Goda, 1967), whereas instantaneous impact forces related to breaking or focusing of the incident wavefront onto the wall can exceed the hydrostatic pressure by an one order of magnitude (Peregrine, 2003, Cuomo et al., 2010). When nonlinear interactions and instabilities are considered, the wave height at the wall can even be four times as high as the incoming wave if the nonlinear processes are promoted by the relative scales of the structure (Zhao et al., 2019). This not only impacts the hydrodynamic forces on the structure, but also the stability of the sediment bed through enhanced near bed velocity and liquefaction by pressure induced seepage (e.g. Gislason et al., 2009, Jeng et al., 2013). As these nonlinearities, and hence the resulting forces on the wall, are unambiguously associated with the statistical properties of the wave field, it is critical to be properly informed of the wave characteristics observed directly at the structure wall.
Kurtosis and skewness serve as important indicators of the relative importance of wave nonlinearities and therefore, the distribution of extreme waves (Janssen, 2003, Fedele et al., 2016). To the first order, standing waves do not exhibit different kurtosis or skewness than its undisturbed counterpart persé. Standing waves are, however, more susceptible to nonlinear processes as the wave steepness of a standing wave is doubled compared to the far field (Romolo and Arena, 2008). Kurtosis is affected by the properties of the wave field, where directional spreading of the wave field components can suppress instabilities leading to a decrease of kurtosis and extreme waves observed (Onorato et al., 2009, Luxmoore et al., 2019). Bi-modality, also know as crossing seas, can promote instabilities and therefore enhance kurtosis and the occurrence of extreme waves (e.g. Onorato et al., 2006, Toffoli et al., 2011).
Standing waves near a vertical wall are, effectively, bi-modal, where the wave direction relative to the wall defines the angle of crossing . Numerical and experimental observations of Toffoli et al. (2011) suggest that kurtosis of crossing seas increases with crossing angle and attains a maximum around (though varied from 0 to 90° only in this study). Gramstad and Bitner-Gregersen (2018) shows through numerical simulations that the maximum growth rate of instabilities attain a maximum at very large () and very small angles, i.e. counter crossing seas and a unimodal wave field, where the kurtosis of counter crossing waves is slightly larger than that of the incoming wave field. These results seem, however, inconsistent with the laboratory study of Støle-Hentschel et al. (2018), who observed that the kurtosis, skewness and normalized extreme wave heights of a partial standing wave field perpendicular to a vertical wall are smaller than those of the incoming unimodal wave field. This would imply that nonlinear effects are effectively suppressed by the presence of the vertical wall.
A possible explanation for the discrepancy between the numerical results of Gramstad and Bitner-Gregersen (2018) (see their Figure 7a, with peak enhancement factor ) and experimental observations of Støle-Hentschel et al. (2018) is, perhaps, the location of observations relative to the reflective wall. For instance, Romolo and Arena (2008) observe that standing wave nonlinear effects are largest at the wall itself. Moreover, Klopman and van der Meer (1999) show that a standing wave pattern for a JONSWAP incident wave spectrum extends to only about two spectral peak wavelengths from the wall; further away from the wall, the significant wave height is still elevated but nodes and antinodes cannot be identified anymore. As Støle-Hentschel et al. (2018) measured wave statistics at a minimum distance of from the wall, it is possible that the statistical properties and distributions observed there are not representative for the properties at the wall itself. Hence, not only is there still uncertainty in the exact behaviour of standing waves at a vertical wall, it appears that wave measurements even in close vicinity of the wall may not be representative for the wave statistics at the wall itself. While numerical and laboratory studies are of critical importance to gain a better understanding of the wave properties at a vertical wall, full-scale wave observations in the field would be a more direct approach in studying the wave properties at the vertical wall.
Despite the vast number of protective structures covering our coasts, detailed wave observations in the field at the structure wall are virtually absent (Laface et al., 2018a, Laface et al., 2018b). One of the main reasons of the lack of field observations is, perhaps, the difficulty of performing measurements close to the structure wall due to instrument and wall interference. The objective of this study is to address the knowledge gap of standing wave behaviour at a vertical structure wall by determining the wave characteristics and statistics at a vertical breakwater wall exposed to sea states of pure wind waves.
Section snippets
Methods
The data used in this paper have been recorded at Natural Ocean Engineering Laboratory (NOEL, noel.unirc.it) during the experiment on a small scale model of a vertical breakwater (Boccotti et al., 2012, Romolo and Arena, 2013). The NOEL laboratory is located in Reggio Calabria (Italy), in the Strait of Messina strait. The peculiarity of the site is that local wind generates sea states of pure wind waves which represent small scale models in a Froude Similarity of sea storms, thus allowing the
Results and discussion
Integral wave characteristics at the wall and of the incoming wave field are shown in Fig. 2. Note that the properties and statistics of the incoming wave field are identified by the subscript . The significant wave height and mean wave steepness at the wall are on average 1.8 larger than those observed in the undisturbed wave field (Fig. 2a and c). This suggests near-full reflection of the incoming wave field. The peak wave numbers are close to 1 and therefore does not seem to be
Conclusions
Standing waves are rarely measured at the wall itself. In this study we show that, while the integral properties of the wave field at the wall are consistent with linear wave theory, the wave statistics suggest that the observed standing waves are highly nonlinear. Waves at the wall have considerably higher kurtosis and skewness than the incoming wave field. The exceedance probability of crest height and wave trough of the undisturbed wave field are well approximated by the Tayfun and Rayleigh
CRediT authorship contribution statement
Joey J. Voermans: Conceptualization, Formal analysis, Writing - original draft. Valentina Laface: Methodology, Validation, Writing - review & editing. Alexander V. Babanin: Conceptualization, Supervision, Writing - review & editing. Alessandra Romolo: Validation, Writing - review & editing. Felice Arena: Validation, Writing - review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
J. J. V. and A. V. B. were supported by the DISI Australia–China Centre, Australia through Grant ACSRF48199. A. V. B. acknowledges support from the U.S. Office of Naval Research Grant N00014-17-1-3021.
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