Application of higher-level GN theory to some wave transformation problems

Diffraction of cnoidal waves by vertical cylinders in shallow water

“…High-level GN equations, applicable to wave propagation in any water depth, are obtained by assuming higher-order polynomials for the distribution of the vertical velocity along the water column. Among others, see Shields and Webster [64], Webster and Kim [72], Demirbilek and Webster [9], and more recently Zhao et al [83][84][85], for discussion on high-level GN equations.…”

“…The Level I Green-Naghdi equations are used, for example, to determine the hydroelastic response of VLFS to solitary and cnoidal waves by Ertekin and Kim [14], Ertekin et al [20], Xia et al [78,79], Ertekin and Xia [16], and by Demirbilek and Webster [10], Ertekin and Sundararaghavan [15] to study refraction and diffraction of nonlinear waves in coastal waters and to study solitary and cnoidal wave loads on horizontal decks by Hayatdavoodi and Ertekin [30][31][32], Hayatdavoodi et al [33]. Further discussion on the recent developments on the Green-Naghdi equations, with comparisons between high-level equations and laboratory experiments, can be found in Kim et al [38], Ertekin et al [21], Zhao et al [83][84][85], among others. Theoretical models based on the Green-Naghdi and the Boussinesq equations were recently developed by Neill et al [50] to study the interaction of solitary wave with multiple in-line, vertical cylinders.…”

High-level Green–Naghdi wave models for nonlinear wave transformation in three dimensions

“…Zhang et al (2013) showed that properties of the so-called Boussinesq-Green-Naghdi equations may be substantially improved for a given order of approximation using asymptotic rearrangements. However, the numerical results of the models developed by Chazel et al (2011) and Zhang et al (2013) did not provide as good results as the converged high-level GN models (Zhao et al 2014a). Xu et al (1993) applied the deep-water GN-2 model to three-dimensional irregular wave propagation problems.…”

“…Demirbilek and Webster (1992) applied the GN-2 model to time-domain simulation of wave transformation problems for the first time. Due to the rapid increase in algebraic complexity at higher levels, the GN models up to Level III have been derived (Shields and Webster 1988;Webster 1992, 1999), but the applications of the GN-3 model, and even higher models, were made in two-dimensions in recent years by Zhao et al (2014a) who showed that high-level GN models are strongly nonlinear, strongly dispersive wave models. We mention that a simplified form of the GN model, introduced by Webster et al (2011), was used by Zhao et al (2014a) in two dimensions.…”

“…Due to the rapid increase in algebraic complexity at higher levels, the GN models up to Level III have been derived (Shields and Webster 1988;Webster 1992, 1999), but the applications of the GN-3 model, and even higher models, were made in two-dimensions in recent years by Zhao et al (2014a) who showed that high-level GN models are strongly nonlinear, strongly dispersive wave models. We mention that a simplified form of the GN model, introduced by Webster et al (2011), was used by Zhao et al (2014a) in two dimensions. The traditional form of the high-level GN models (Demirbilek and Webster 1992) is quite complicated to be applied to wave analysis.…”