Wave interaction with low-mound breakwaters using a RANS model

Abstract

Using COBRAS-UC, a numerical model based on the Volume-Averaged Reynolds Average Navier–Stokes (VARANS) equations, 2-D wave interaction with low-mound breakwaters is analyzed. The model uses a Volume of Fluid (VOF) technique method to capture the free surface which allows the modeling of complicated processes such as breakwater overtopping. Furthermore, thanks to the VARANS equations, the flow in the permeable layers underneath the caisson is quantitatively correct. In order to validate the model's performance, a new set of experimental studies are carried out in a wave flume at a 1:20 scale using regular and irregular waves. Comparisons between numerical and experimental free surface, pressure time histories, and overtopping layer thickness for regular and irregular waves show a good agreement. Further comparisons of numerically predicted overtopping magnitudes with existing semi-empirical formulae and experimental data indicate that the model can be used as a complementary tool for the functional design of this kind of structures.

Introduction

Coastal structures are built to protect coastal areas from erosion and flooding, as well as to shelter ports and marinas from wave action. The wave and structure interaction process depends on incident wave conditions and the structure typology.

In order to provide design guidelines for the hydraulic response of coastal structures, many semi-empirical formulations based on flume and basin experiments have been developed in the past. Such formulae try to consider and parameterize the most relevant variables ruling the process which occurs in the wave and structure interaction phenomenon, mainly, reflection, run-up, rundown, overtopping and transmission. An extensive number of very successful semi-empirical formulations are available in the literature and can be found, as part of the state of the art in coastal structure designs, in several books and manuals (e.g. Burcharth and Hughes, 2006).

As previously stated, semi-empirical formulations, based on flume or wave basin experiments, cover a limited number of typologies and setups. Furthermore, many of them are based on a reduced set of incident wave conditions, which may lead to uncertainties and errors if used out of range, a problem often faced in current design or pre-design conditions.

In recent years, computational modeling of wave interaction with coastal structures is becoming an important complementary approach, especially when pre-designing the hydraulic response of some typologies of coastal structures.

The numerical model ability to successfully analyze wave and structure interaction problems depends mainly on the equations and solving techniques used, Losada (2003). Moreover, the leap to its application on real case studies requires computational robustness and efficiency in terms of computation as well as a thorough validation process which is only available today for a limited set of models.

According to the existing literature, the most relevant numerical models currently available for wave and coastal structures interaction are different versions of models solving the nonlinear shallow equations; models based on Reynolds Average Navier–Stokes equations (RANS) including any kind of free surface-tracking or capturing technique and, more recently, particle methods such as the Smoothed Particle Hydrodynamics (SPH). In this last model the governing partial differential equations of continuum fluid dynamics, are transformed into particle form by integral equations through the use of an interpolation function that gives kernel estimation of the field variables at a point.

Models based on nonlinear shallow water equations (NSWE), are derived on the assumption of hydrostatic pressure and are obtained by vertically integrating the Navier–Stokes equations (NSE), i.e. Kobayashi and Wurjanto (1989), Mingham and Causon (1998), Hu et al. (2000), Hubbard and Dodd (2002) and Stansby and Feng (2004). The use of NSWE places severe restrictions on real applications inherent, essentially, to the intrinsic requirement to satisfy the shallow water limit. This restriction is especially harsh when considering high frequency components in the incident wave spectrum and also requires locating the offshore boundary condition of the numerical model close to the structure. Other restrictions are associated with the semi-empirical introduction of breaking, porous flow modeling or the difficulty in simulating complicated free surfaces. However, they are computationally very efficient in providing the possibility to simulate wave trains including about 1000 waves very rapidly, which may be of importance, for example, when analyzing extreme statistics of wave overtopping.

The inherent limitation of the shallow water limit may be overcome considering the application of modified Boussinesq equations for wave and structure interactions (e.g. Lynett et al., 2000). However, the computational effort required to solve these equations has discouraged researchers in using them for real applications.

During the last years particle methods such as the Moving Particle Semi-Implicit (MPS) method of Koshizuka et al. (1995); or the SPH method in its different versions (i.e. Dalrymple et al., 2001; Gotoh et al., 2004; Shao, 2006) have become very popular. Being based on a grid-less Lagrangian approach, they are able to accurately track large deformations of the free surface. However, whether or not the weakly incompressible or the incompressible SPH method is used, this technique has several disadvantages. The high number of particles required to obtain a reliable solution; the use of fixed particle spacing; the limited validation available and a very low computational efficiency are some of the issues that need to be solved before considering SPH for practical purposes.

Free of some of the mentioned limitations above, numerical models based on the NSE have received considerable attention in the last decade applied to the analysis of wave interaction with coastal structures under breaking conditions, as they allow the calculation of the velocity field in the whole computational domain for any type of flow, whether rotational or irrotational.

Among existing NSE models, RANS-based equation models are nowadays the more suitable tool to analyze the wave–structure interaction problem. The analysis of wave interaction with porous structures lies in the fact that they take into account the turbulence generation/dissipation mechanisms inside the porous media and also in the wave breaking process. The introduction of the turbulent effects constitutes a true stumbling block for the modeling of wave–structure interaction with permeable breakwaters. Among other RANS models available in literature (i.e. Troch and de Rouck, 1998; Kawasaki, 1999; Li et al., 2004), COBRAS is probably the most extensively validated, especially for wave–structure interaction.

Lin and Liu (1998), based on a previously existing model called RIPPLE (Kothe et al., 1991), presented COBRAS (Cornell Breaking Waves and Structures) for simulating breaking waves and wave interaction with coastal structures. In this work, a modified and improved version of COBRAS, named COBRAS-UC (Losada et al., 2008; Torres-Freyermuth et al., 2007) is used to investigate the interaction of regular and irregular waves with low-mound breakwaters focusing on the hydraulic response of the structure. According to the PROVERBS parameter map, Oumeraci et al. (2001), low-mound breakwaters are characterized by a relative height $0.3<h_{\mathrm{b}}^{*}=(h_{\mathrm{b}}/h)<0.6$, where h_b is the mound foundation height and h is the water depth in front of the structure. Depending on the relative wave height, this structure typology might be subjected to quasi-standing wave loads or impact wave loads. Please note, that the quasi-standing wave load response is not usually expected for composite breakwaters. In terms of hydraulic response low-mound breakwaters are closer to vertical breakwaters than rubble-mound breakwaters and therefore reflection dominates the structure response and potential overtopping. Wave breaking is not relevant in this case and the porous flow is limited. Therefore, the modeling presented in this paper is very different to the case considered in Losada et al. (2008) in which the hydraulic response, run-up and overtopping are dominated by the wave breaking process on the breakwater rubble-mound layers.

The paper is organized as follows. The main characteristics of COBRAS-UC are presented in the following section. The experimental setup carried out for the model validation is presented followed by an explanation of the numerical setup and model calibration. The next section is devoted to the numerical model validation through the comparison of the numerical and experimental results of free surface and pressure time series, wave spectra and several overtopping related magnitudes. A specific section on the application of the model to wave overtopping analysis can be found next. The paper is completed with some comparisons between results of low-mound and rubble-mound breakwaters, recommendations for numerical modeling based on the experience achieved and conclusions.