Eulerian and Lagrangian modelling of a solitary wave attack on a seawall

Abstract

The problem of a solitary wave attack on a vertical seawall is investigated by applying two approaches. The first is a semi-analytical method in which the Euler equations of motion are solved under the assumption of a potential flow by employing an approach based on the fast Fourier transform technique and a numerical time-stepping scheme. The second approach employed to analyse the problem considered is the smoothed particle hydrodynamics (SPH) method, in which the equations are solved in the Lagrangian coordinates. The two methods are used to simulate solitary wave propagation in water of uniform depth, followed by a wave impact on a vertical wall. The simulations focus on the determination of the maximum run-up of the wave on the wall, the calculation of pressures exerted by water on the structure, and the evaluation of water velocities in the vicinity of the structure. The predictions of the two approaches are compared to identify wave regimes for which both methods give satisfactory results. The results of numerical simulations have shown that both proposed methods predict practically the same free-surface profiles for waves of small and moderate amplitudes. For higher waves, some discrepancies between the results of the two methods occur. The two models results have been also compared with empirical data known from the literature, showing good agreement with experimental measurements in terms of the maximum wave run-up and the wave crest residence time at a wall.

Introduction

The problem of propagation of a solitary wave and its reflection at a vertical wall is of interest to ocean and coastal scientists and engineers. The dynamic forces acting on vertical walls during the solitary wave run-up may cause severe damages to this type of structures. Due to the importance of the problem of a solitary wave impact on a wall, and its intrinsic complexity associated with the occurrence of high pressure gradients and rapid changes in the water free surface, many theoretical and numerical models have already been developed, and substantial experimental work has been carried out.

Experimental studies have been conducted by Maxworthy (1976), who studied head-on collisions of two solitary waves. Park et al. (2012) used particle tracking techniques to measure velocity fields during a solitary wave reflection at a vertical wall. Chen et al. (2015) performed experiments in a wave tank to investigate the solitary wave reflection. They were particularly interested in determining characteristics of steep solitary waves reflecting at the wall, and were able to generate solitary waves of heights exceeding half of the still water depth.

More information on the physics underlying solitary wave collisions can be inferred from the theoretical analysis of the phenomenon. Byatt-Smith (1988) derived a perturbation expansion of the Euler equations to solve the problem of head-on collision of two equal-amplitude solitary waves. Their analytical solution showed that the amplitude of the solitary wave was reduced as a result of the collision. Su and Mirie (1980) also applied the perturbation approach to investigate the interaction of two solitary waves, and obtained a solution that is up to third-order accurate. Temperville (1979) derived a leading-order asymptotic formula for the phase shift between the incident and the reflected solitary waves. The analytical relation between the solitary wave run-up height at a vertical wall and the solitary wave amplitude was proposed by Pelinovsky et al. (1999), who used the shallow water wave theory to investigate the run-up of tsunami waves on a vertical wall in a bay. Craig et al. (2006) investigated the interactions of counter- and co-propagating solitary waves using a pseudo-spectral method applied to the Dirichlet-Neumann operator (Craig and Sulem, 1993).

The problem under investigation has also been treated numerically. It is likely that one of the first numerical models for simulating solitary waves was developed by Chan and Street (1970), who validated their model by comparing its predictions with experimental data. Fenton and Rienecker (1982) solved the problem of the solitary wave impact on a vertical wall by applying a Fourier method. Cooker et al. (1997) solved the similar problem by employing a Boundary Integral Equation method. The same Boundary Integral Equation approach was also used to solve the problem of a solitary wave run-up by Chambarel et al. (2009), and their results were later compared with the Green-Naghdi model by Touboul and Pelinovsky (2014). An alternative discrete approach was pursued by Shao (2005), who modelled solitary waves by an incompressible smoothed particle hydrodynamics method.

In this work, the problem of a solitary wave impact on a vertical seawall is solved by employing two methods. In the first method, the solution is derived by a semi-analytical approach developed by Sulisz and Paprota, 2004, Sulisz and Paprota, 2011. In this approach, water is treated as an incompressible fluid, and the governing equations are formulated in the Eulerian coordinates. In order to solve the wave propagation problem, the kinematic and dynamic free-surface boundary conditions are expanded in a Taylor series (Dean and Dalrymple, 1984). The solution is derived in the form of eigenfunction expansions which satisfy the fluid continuity equation and the bottom boundary condition. The calculation of unknown free-surface elevation and velocity field components is carried out by a time-stepping scheme.

In the second method followed in this work, the solitary wave propagation problem is solved in the Lagrangian coordinates by employing a discrete method based on the smoothed particle hydrodynamics (SPH) approach. In this approach (Monaghan, 1992, Liu and Liu, 2003, Violeau, 2012) water is treated as a weakly compressible liquid, for which the pressure is related to density by a constitutive equation. The latter assumption enables employing an efficient explicit time-stepping scheme for integration of the continuity and momentum equations describing the motion of the fluid. A so-called corrected version of the SPH method is employed, in which standard approximation functions (kernel smoothing functions in the SPH terminology) are modified to satisfy linear reproducing conditions (Belytschko et al., 1998). The initial conditions for the displacement, velocity and pressure fields in the fluid have been adopted in the form of a second-order analytical approximation to the solution of the Korteweg-de Vries equation describing waves in shallow water (Wehausen and Laitone, 1960). The SPH model based on the above assumptions was previously used by Staroszczyk (2011) to simulate the collision of two solitary waves.

The present investigation is a follow-up of the studies reported in a conference paper by Paprota et al. (2015). The two above methods have been used to simulate numerically the solitary wave propagation in water of uniform depth, followed by the wave impact on a vertical seawall. The simulations focus on the determination of the maximum wave run-up height on the wall, the calculation of time-histories of the pressures and total forces exerted by water on the structure, and the evaluation of the water velocities in the vicinity of the seawall. In particular, the effect of the wave height on the behaviour of solitary waves is investigated. The predictions of the two distinct approaches, one based on the Eulerian and the other on the Lagrangian description, are compared in order to identify solitary wave regimes for which both methods give satisfactory results. The results obtained are also compared with recent experimental measurements (Chen et al., 2015).