A 2D numerical model of wave run-up and overtopping

Abstract

A two-dimensional (2D) numerical model of wave run-up and overtopping is presented. The model (called OTT-2D) is based on the 2D nonlinear shallow water (NLSW) equations on a sloping bed, including bed shear stress. These equations are solved using an upwind finite volume technique and a hierarchical Cartesian Adaptive Mesh Refinement (AMR) algorithm. The 2D nature of the model means that it can be used to simulate wave transformation, run-up, overtopping and regeneration by obliquely incident and multi-directional waves over alongshore-inhomogeneous sea walls and complex, submerged or surface-piercing features. The numerical technique used includes accurate shock modeling, and uses no special shoreline-tracking algorithm or shoreline coordinate transformation, which means that noncontiguous flows and multiple shorelines can easily be simulated. The adaptivity of the model ensures that only those parts of the flow that require higher resolution (such as the region of the moving shoreline) receive it, resulting in a model with a high level of efficiency. The model is shown to accurately reproduce analytical and benchmark numerical solutions. Existing wave flume and wave basin datasets are used to test the ability of the model to approximate 1D and 2D wave transformation, run-up and overtopping. Finally, we study a 2D dataset of overtopping of random waves at off-normal incidence to investigate overtopping of a sea wall by long-crested waves. The data set is interesting as it has not been studied in detail before and suggests that, in some instances, overtopping at an angle can lead to more flooding than at normal incidence.

Introduction

Overtopping of coastal structures and inundation of coastal regions by waves is a constant hazard, the effects of which can be disastrous. A number of circumstances can contribute to such an event, including a high tide, storm surge, large waves (due to swell or wind) or a tsunami, as well as the failure of some form of sea defence, often in conjunction with the aforementioned events.

In an effort to provide guidelines for designing coastal structures that can withstand such occurrences, numerous flume and basin tests of run-up and overtopping have been performed (e.g. De Waal and Van Der Meer, 1992, Owen, 1980, Saville, 1955) and formulas and design curves for estimating maximum run-up and average overtopping rates constructed. Many of these methods are summarised in the Shore Protection Manual (1984) and the Manual on the Use of Rock on Coastal and Shoreline Engineering (1991), and a review of more recent work is presented in Overtopping of Sea Walls: Design and Assessment Manual (Besley, 1999). These empirical tools have proved successful. However, they are overwhelmingly based on flume experiments with monochromatic or spectrally (and directionally) narrow-banded waves, and their application is limited to a small number of breakwater and berm types.

In recognition of these limitations, computational models have been developed (see e.g. Hibberd and Peregrine, 1979, Kobayashi and Watson, 1987, Kobayashi et al., 1987, Kobayashi et al., 1989, Titov and Synolakis, 1995, Dodd, 1998, Hu et al., 2000). These are overwhelmingly based on the nonlinear shallow water (NLSW) equations, can be run at a fraction of the cost of flume tests, for any input spectrum, and are not limited by simplifications concerning the structure section or beach profile, as long as the assumptions underpinning the NLSW equations are not violated, and even in circumstances like these it has been shown (Hu et al., 2000) that the equations can give realistic predictions. They have been shown to work well in simulating wave run-up and swash motions, overtopping volumes and rates, and even regeneration of waves in the lee of a structure (see Dodd, 1998). These models are, however, all one-dimensional (1D). That is, they simulate on–offshore motions only (or in the terminology used by Titov and Synolakis, 1998, they are 1+1 models: one propagation direction plus time). Implicitly, therefore, they assume that a structure section or beach is alongshore-homogeneous, and that waves are shore-normal.

Fully two-dimensional (2D or 2+1) models of these equations and similar systems have been developed over the last 15 years. However, most of the resulting codes have been used to solve the Euler equations of gas dynamics, or to model dam-break problems or shallow water flows of hydraulics, typically without the demanding additional requirements of coastal engineering models. It is only comparatively recently that codes for coastal engineering problems have been developed. There are a number of reasons for this.

First, the calculation of the position of the time-varying shoreline has always caused problems, because of the very small depths in its vicinity. In 1D an algorithm has been developed (Hibberd and Peregrine, 1979) and refined (Kobayashi et al., 1987), which tracks the shoreline position, which is robust, but also unwieldy, and a generalisation to 2D would entail much book-keeping. A more satisfying approach (1D) was taken by Titov and Synolakis (1995) (applied to a 2D model by Titov and Synolakis, 1998), who impose a final, shoreline boundary point on the beach surface by a horizontal extrapolation of the free surface from the neighboring point (and by assuming the velocity at the shoreline node is equal to that at the neighbouring node). A simpler and more general approach was used by Watson et al. (1992), Dodd (1998), Hu et al. (2000) and Brocchini et al. (2001) (the last of these being a 2D implementation), who used Godunov-type numerical methods at all model locations. In such a treatment, the numerical fluxes are calculated by approximating a series of dam-break problems, and the shoreline becomes a special case of this, in which on one side all flow variables are zero. These methods can therefore be adapted to modeling multiple shorelines and shoreline interactions, and wave overtopping.

Second, the inevitable increase in computational expense that results from extending 1D codes to 2D can be prohibitively large. It is now becoming common for numerical models to incorporate adaptive grid techniques, or at least to use irregular grids, in recognition of this problem. Titov and Synolakis (1995) actually used an irregularly spaced grid in 1D, and reported that their model required only about half as many points per wavelength as models based on the Lax-Wendroff solver. In their 2D model, Titov and Synolakis (1998) use a finer nested grid within their main grid nearer to the shore wherever the depth falls below a certain threshold.

Finally, it is difficult to verify 2D NLSW models. This was noted by Titov and Synolakis (1998), who cited the 1D solution of Carrier and Greenspan (1958) as the only standard analytical solution (and this for strictly nonbreaking waves). Ryrie (1983) has extended this solution by assuming that the angle of propagation to the shore is small, from which assumption a new, but reduced, set of quasi-2D equations results. However, as such it is only an approximate solution to the 2D equations. The analytical solution of Thacker (1981), who considers a body of water in a parabolic bowl, is a more satisfying 2D test in that it is an exact solution of the 2D NLSW equations (without bed friction) including a shoreline, the latter aspect being a crucial test for accurate coastal engineering codes. Furthermore, good validatory wave basin data are also difficult to find. So far, only the test of Briggs et al. (1995) looks like becoming a standard data set.

Nevertheless, in recent years 2D codes for coastal hazard problems have been developed. Titov and Synolakis (1998) give a brief description of progress to date on 2D codes, and they, Liu et al. (1995), Takahashi et al. (1995), Özkan-Haller and Kirby (1997) and Brocchini et al. (2001) have all presented and validated codes. Most of these models have primarily been developed with the modeling of tsunami run-up in mind.

Here a fully adaptive mesh approach is undertaken, in which high grid resolution is used only where necessary, thus reducing computational times, and where mesh refinement is done automatically depending on the value of a user-defined parameter, thus avoiding the necessity of choosing nested grid areas beforehand for each problem. There is no limitation to the level of refinement that can be used—other than practical ones of computational expense and memory. Furthermore, mesh refinement is achieved based not only on local depth, so that the shoreline region is well resolved, but also on local gradients in the flow, so that waves and bores are well resolved. The algorithm used is the Adaptive Mesh Refinement (AMR) algorithm of Quirk (1991) (see also Quirk and Karni, 1996). It only superimposes finer meshes over coarser ones where required, solving on all meshes but using data from the finest mesh available in each region.

In the next section, the physical and theoretical framework for the model is set out. In Section 3, the numerical model (with boundary conditions) is developed. In Section 4, a number of case studies are presented, and the model validated. In Section 5, we use OTT-2D to simulate overtopping experiments. These experiments (see Owen, 1982) are of overtopping of a sea-wall by random waves at a number of angles. They are of interest not only from the point of view of verification, but also because they show, in some cases, a peak in measured mean overtopping at oblique angles, which can seem counter-intuitive. They are therefore presented separately from the case studies of Section 4. Finally (Section 6), conclusions are arrived at.