Numerical study of combined overflow and wave overtopping over a smooth impermeable seawall

A numerical wave flume is used to investigate the discharge characteristics of combined overflow and wave overtopping of impermeable seawalls. The numerical procedure computes solutions to the Reynolds-averaged Navier–Stokes equations and includes the generation of an irregular train of waves, the simulation of wave breaking and interaction with a sloping, impermeable wall. The numerical model is first tested against published experimental observations, approximate analytical solutions and empirical design formulae for the cases of pure overflow and pure overtopping. A sequence of numerical experiments simulating combined overflow and overtopping are described. The results are used to determine empirical discharge formulae of the form used in current practice.     

A numerical model of wave overtopping on seadikes

This paper describes the development of a numerical model for wave overtopping on seadikes. The model is based on the flux-conservative form of the nonlinear shallow water equations (NLSW) solved with a high order total variation diminishing (TVD), Roe-type scheme. The goal is to reliably predict the hydrodynamics of wave overtopping on the dike crest and along the inner slope, necessary for the breach modelling of seadikes. Besides the mean overtopping rate, the capability of simulating individual overtopping events is also required. It is shown theoretically that the effect of wave breaking through the drastic motion of surface rollers in the surfzone is not sufficiently described by the conventional nonlinear shallow water equations, neglecting wave setup from the mean water level and thus markedly reducing the model predictive capacity for wave overtopping. This is significantly improved by including an additional source term associated with the roller energy dissipation in the depth-averaged momentum equation. The developed model has been validated against four existing laboratory datasets of wave overtopping on dikes. The first two sets are to validate the roller term performance in improving the model prediction of wave overtopping of breaking waves. The last two sets are to test the model performance under more complex but realistic hydraulic and slope geometric conditions. The results confirm the merit of the supplemented roller term and also demonstrate that the model is robust and reliable for the prediction of wave overtopping on seadikes.     

Numerical simulation of wave overtopping of coastal structures using the non-linear shallow water equations

A one-dimensional high-resolution finite volume model capable of simulating storm waves propagating in the coastal surf zone and overtopping a sea wall is presented. The model (AMAZON) is based on solving the non-linear shallow water (NLSW) equations. A modern upwind scheme of the Godunov-type using an HLL approximate Riemann solver is described which captures bore waves in both transcritical and supercritical flows. By employing a finite volume formulation, the method can be implemented on an irregular, structured, boundary-fitted computational mesh. The use of the NLSW equations to model wave overtopping is computationally efficient and practically flexible, though the detailed structure of wave breaking is of course ignored. It is shown that wave overtopping at a vertical wall may also be approximately modelled by representing the wall as a steep bed slope. The AMAZON model solutions have been compared with analytical solutions and laboratory data for wave overtopping at sloping and vertical seawalls and good agreement has been found. The model requires more verification tests for irregular waves before its application as a generic design tool.     

The role of offshore boundary conditions in the uncertainty of numerical prediction of wave overtopping using non-linear shallow water equations

The paper examines the variability of wave overtopping parameters predicted by numerical models based on non-linear shallow water equations, due to the boundary conditions obtained from wave energy density spectra. Free surface elevation time series at the boundary are generated using the principle of linear superposition of the spectral components. The components' phases are assumed to be random, making it possible to generate an infinite number of offshore boundary conditions from only one spectrum.A reference case was provided by carrying out overtopping tests on a simple concrete structure in a wave flume. Numerical tests using the measured free surface elevation at the toe of the structure were carried out. Three parameters were analysed throughout the paper: the overtopping discharge, the probability of overtopping and the maximum overtopping volume. These showed very good agreement between the numerical solver prediction and the overtopping measurements. Subsequently, the measured spectra at the toe were used to generate a population of reconstructed offshore boundary time series for each test, following a Monte Carlo approach. A sensitivity analysis determined that 500 tests were suitable to perform a statistical analysis on the predicted overtopping parameters. Results of these tests show that the variability in the predicted parameters is higher for the smaller number of overtopping waves in the modelled range and decreases significantly as overtopping becomes more frequent. The characteristics of the distributions of the predictions have been studied. The average value of the three parameters has been compared with the measurements. Although the accuracy is lower than that achieved by the model when the measured time series are used at the boundary, the prediction is still fairly accurate above all for the highest overtopping discharges. The distribution of the modelled probability of overtopping was found to follow a normal distribution, while the maximum value follows a GEV one. The overtopping discharge shows a more complex behaviour, values in the middle of the tested range follow a Weibull distribution, while a normal distribution describes the top end of the range better.Results indicate that when the probability of overtopping is smaller than 5%, a sensitivity analysis on the seeding of the offshore boundary conditions is recommended.     

Tsunami-like solitary waves impinging and overtopping an impermeable seawall: Experiment and RANS modeling

This study investigates tsunami-like solitary waves impinging and overtopping an impermeable trapezoidal seawall on a 1:20 sloping beach. New laboratory experiments are performed for describing three typical cases: a turbulent bore rushes inland and subsequently impacts and overtops the seawall (Type 1); a wave directly collapses on the seawall and then generates overtopping flow (Type 2); and, a wave straightforwardly overtops the seawall crown and collapses behind the seawall (Type 3). A two-dimensional volume of fluid (VOF) type model called the COBRAS (COrnell BReaking And Structure) model, which is based on the Reynolds-Averaged Navier–Stokes (RANS) equations and the k–ε turbulence closure solver, is validated by experimental data and then applied to investigate wave dynamics for which laboratory data are unavailable. Additionally, a set of numerical experiments is conducted to examine the dynamic wave acting force due to waves impacting the seawall. Effects of wave nonlinearity and freeboard are elucidated. Special attention is given to a distinct vortex evolutionary behavior behind the seawall, in which the dynamic properties of entrapped air-bubbles are briefly addressed experimentally and numerically.     

Field measurements of wave overtopping at the rubble mound breakwater of Rome–Ostia yacht harbour

This paper describes a new station for full-scale measurement of wave overtopping at the Rome yacht harbour rubble mound breakwater in Ostia (Italy) and the results of the successful first measurement campaign carried out during the winter season 2003–2004. The equipment and the research activities were supported by the EU project CLASH, focusing on scale effects for wave overtopping at coastal structures. The site is characterized by a very small tidal range, a long shallow foreshore and depth-limited breaking waves which interact with a shallow sloping porous rock structure. Overtopping water is collected by a steel tank installed on the crown slab behind the parapet wall. The measurement of water level variation inside the tank by means of two pressure transducers allows the calculation of individual overtopping volumes. Incident waves, sea levels and wind are also measured. During seven independent storms, more than 400 individual overtopping events were recorded and about 86 h of valid data are available. This extensive dataset is presented, discussed and then used for comparison with two commonly used overtopping prediction formulae based on small-scale model tests showing their tendency to underestimate the prototype results. A strong correlation between the hourly mean overtopping discharge and corresponding maximum volume is also presented. The paper generally confirms the validity of the approach used in Troch et al. (2004) [Troch, P., Geeraets, J., Van de Walle, B., De Rouck, J., Van Damme, L., Allsop, W., Franco, L., 2004. Full-scale wave overtopping measurements on the Zeebrugge rubble mound breakwater. Coastal Engineering 51, 609–628] for field measurement of wave overtopping.     

Unsteady ship waves in shallow water of varying depth based on Green–Naghdi equation

Based on Green–Naghdi equation this work studies unsteady ship waves in shallow water of varying depth. A moving ship is regarded as a moving pressure disturbance on free surface. The moving pressure is incorporated into the Green–Naghdi equation to formulate forcing of ship waves in shallow water. The frequency dispersion term of the Green–Naghdi equation accounts for the effects of finite water depth on ship waves. A wave equation model and the finite element method (WE/FEM) are adopted to solve the Green–Naghdi equation. The numerical examples of a Series 60 (C_B=0.6) ship moving in shallow water are presented. Three-dimensional ship wave profiles and wave resistance are given when the ship moves in shallow water with a bed bump (or a trench). The numerical results indicate that the wave resistance increases first, then decreases, and finally returns to normal value as the ship passes a bed bump. A comparison between the numerical results predicted by the Green–Naghdi equation and the shallow water equations is made. It is found that the wave resistance predicted by the Green–Naghdi equation is larger than that predicted by the shallow water equations in subcritical flow , and the Green–Naghdi equation and the shallow water equations predict almost the same wave resistance when , the frequency dispersion can be neglected in supercritical flows.     

Numerical simulation of irregular wave overtopping against a smooth sea dike

Based on the filtered Navier-Stokes equations and Smagorinsky turbulence model,a numerical wave flume is developed to investigate the overtopping process of irregular waves over smooth sea dikes.Simulations of fully nonlinear standing wave and regular wave’s run-up on a sea dike are carried out to validate the implementation of the numerical wave flume with wave generation and absorbing modules.To model stationary ergodic stochastic processes,several cases with different random seeds are computed for each specified irregular wave spectrum.It turns out that the statistical mean overtopping discharge shows good agreement with empirical formulas,other numerical results and experimental data.     

Numerical simulation of wave overtopping at coastal dikes and low-crested structures by means of a shock-capturing Boussinesq model

In this paper, a shock-capturing numerical model, based on the combined solution of Boussinesq and nonlinear shallow water equations is applied to the simulation of wave runup, overtopping and wave train propagation over impermeable, emerged and low-crested, structures. In order to improve the performances of the scheme at wet–dry interfaces, a numerical treatment is introduced to provide well-balancing of advective fluxes and source terms. One- and two-dimensional test cases are presented to validate the performances of the model under both regular and irregular wave conditions.     

非线性波传播的新型数值模拟模型及其实验验证 引入变换速度变量

以一种新型的含变换速度变量的Boussinesq型方程为控制方程组,采用五阶Runge-Kutta-England格式离散时间积分,采用七点差分格式离散空间导数,并采用恰当的出流边界条件,从而建立了非线性波传播的新型数值模拟模型.对均匀水深水域内波浪传播的数值模拟,说明在引入变换速度后进一步增大了模型的水深适用范围.对潜堤地形上波浪传播的数值模拟说明,在引入变换速度后进一步提高了模型的数值模拟精度.     

Random wave runup and overtopping a steep sea wall: Shallow-water and Boussinesq modelling with generalised breaking and wall impact algorithms validated against laboratory and field measurements

A semi-implicit shallow-water and Boussinesq model has been developed to account for random wave breaking, impact and overtopping of steep sea walls including recurves. At a given time breaking is said to occur if the wave height to water depth ratio for each individual wave exceeds a critical value of 0.6 and the Boussinesq terms are simply switched off. The example is presented of waves breaking over an offshore reef and then ceasing to break as they propagate inshore into deeper water and finally break as they run up a slope. This is not possible with the conventional criterion of a single onset of breaking based on rate of change of surface elevation which was also found to be less effective generally. The runup distribution on the slope inshore of the reef was well predicted. The model is tested against field data for overtopping available for Anchorsholme, Blackpool and corresponding 1:15 scale wave flume tests. Reflection of breaking waves impacting a steep sea wall is represented as a partial reversal of momentum flux with an empirically defined coefficient. Offshore to nearshore significant wave height variation was reasonably predicted although nearshore model spectra showed distinct differences from the experiments. The breaking wave shape described by a shape parameter was also not well represented as might be expected for such a simple model. Overtopping agreement between model, field and flume was generally good although repeatability of two nominally identical flume experiments was only within 25%. Different distributions of random phase between spectral components can cause overall overtopping rates to differ by up to a factor of two. Predictions of mean discharge by EurOtop methods were within a factor of two of experimental measurements.     

Numerical analysis of wave overtopping of rubble mound breakwaters

The paper describes the results of a two-dimensional (2-D) numerical modelling investigation of the functionality of rubble mound breakwaters with special attention focused on wave overtopping processes. The model, COBRAS-UC, is a new version of the COBRAS (Cornell Breaking Waves and Structures) based on the Volume Averaged Reynolds Average Navier–Stokes (VARANS) equations and uses a Volume of Fluid Technique (VOF) method to capture the free surface. The nature of the model equations and solving technique provides a means to simulate wave reflection, run-up, wave breaking on the slope, transmission through rubble mounds, overtopping and agitation at the protected side due to the combined effect of wave transmission and overtopping. Also, two-dimensional experimental studies are carried out to investigate the performance of the model. The computations of the free surface and pressure time series and spectra under regular and irregular waves, are compared with the experimental data reaching a very good agreement. The model is also used to reproduce instantaneous and average wave overtopping discharge. Comparisons with existing semi-empirical formulae and experimental data show a very good performance. The present model is expected to become in the near future an excellent tool for practical applications.     

一种新型二阶全非线性Boussinesq方程及其数值验证

通过改进二阶全非线性 Boussinesq 波浪方程中的色散项,得到了一组没有改变原方程的数学形式但适用于更大变化水深的新方程,其色散性能和变浅性能都比原方程有了很大改进,所适用的水深范围更大,能更好地描述从深水到近岸浅水处的波浪传播;并基于新方程建立了波浪数值模型,通过模拟波浪从浅水到深水的传播变形来验证新方程的有效性.     

Validation of the three-wave quasi-kinetic approximation for the spectral evolution in shallow water

This paper aims at validating the three-wave quasi-kinetic approximation for the spectral evolution of weakly nonlinear gravity waves in shallow water. The problem is investigated using a one-dimensional numerical wave propagation model, formulated in the spectral representation. This model includes both a nonlinear triad interactions term and a wave breaking dissipation term. Some numerical tests were carried out in order to show the importance of using the triad nonlinear term in wave propagation spectral models, particularly to describe both behavior of the spectral integral parameters and of the spectral shape evolution in shallow water depth. Furthermore; a comparison against different set of experimental observations was carried out. Comparing the numerical results with the experimental observations made it possible to show the modeling efficiency of the three-wave quasi-kinetic approximation.     

Spatial distribution of wave overtopping water behind coastal structures

Spatial distribution of random wave overtopping water behind coastal structures was investigated using a numerical model based on Reynolds-Averaged Navier-Stokes solver (RANS) and Volume of Fluid (VOF) surface capturing scheme (RANS-VOF). The computed spatial distributions of wave overtopping water behind the structure agree well with the measurements by Pullen et al (2008) for a vertical wall and Lykke Andersen and Burcharth (2006) for a 1:2 sea dike. A semi-analytical model was derived to relate spatial distribution of wave overtopping water behind coastal structures to landward ground level, velocity and layer thickness on the crest. This semi-analytical model agrees reasonably well with both numerical model results and measurements close to coastal structures. Our numerical model results suggest that the proportion of wave overtopping water passing a landward location increases with a seaward slope when it is less than 1:3 and decreases with a seaward slope when it gets steeper. The proportion of wave overtopping water passing a landward location increases with landward ground level and overtopping discharge. It also increases with the product of incident wave height and wavelength, but decreases with increasing relative structure freeboard and crest width. We also found that the extent of hazard area due to wave overtopping is significantly reduced by using a permeable structure crown. Findings in this study will enable engineers to establish the extent of hazard zones due to wave overtopping behind coastal structures.     

2-D numerical analysis of near-field flow at low-crested permeable breakwaters

This paper describes the capability of a numerical model named COrnell BReaking waves And Structures (COBRAS) [Lin, P., Liu, P.L.-F., 1998. A numerical study of breaking waves in the surf zone. Journal of Fluid Mechanics 359, 239–264; Liu, P.L.-F., Lin, P., Chang, K.A., Sakakiyama, T., 1999. Numerical modeling of wave interaction with porous structures. Journal of Waterway, Port, Coastal and Ocean Engineering 125, 322–330, Liu, P.L.-F., Lin, P., Hsu, T., Chang, K., Losada, I.J., Vidal, C., Sakakiyama, T., 2000. A Reynolds averaged Navier–Stokes equation model for nonlinear water wave and structure interactions. Proc. Coastal Structures '99, 169–174] based on the Reynolds Averaged Navier–Stokes (RANS) equations to simulate the most relevant hydrodynamic near-field processes that take place in the interaction between waves and low-crested breakwaters. The model considers wave reflection, transmission, overtopping and breaking due to transient nonlinear waves including turbulence in the fluid domain and in the permeable regions for any kind of geometry and number of layers. Small-scale laboratory tests were conducted in order to validate the model, with different wave conditions and breakwater configurations. In the present study, regular waves of different heights and periods impinging on a wide-crested structure are considered. Three different water depths were tested in order to examine the influence of the structure freeboard. The experimental set-up includes a flow recirculation system aimed at preventing water piling-up at the lee of the breakwater due to overtopping. The applicability and validity of the model are examined by comparing the results of the numerical computations with experimental data. The model is proved to simulate with a high degree of agreement all the studied magnitudes, free surface displacement, pressure inside the porous structure and velocity field. The results obtained show that this model represents a substantial improvement in the numerical modelling of low-crested structures (LCS) since it includes many processes neglected previously by existing models. The information provided by the model can be useful to analyse structure functionality, structure stability, scour and many other hydrodynamic processes of interest.     

斜向和多向不规则波对直立堤平均越浪量研究

通过三维波浪模型试验研究了斜向和多向不规则波对直立堤的越浪量。分别按平均越浪量和单波最大越浪量进行研究,探讨了平均越浪量随相对堤高、波浪方向、波浪方向分布宽度、波陡和相对水深等影响因素的变化规律,导得了斜向和多向不规则波作用于直立堤上的平均越浪量的计算公式。     

A new form of generalized Boussinesq equations for varying water depth

A new set of equations of motion for wave propagation in water with varying depth is derived in this study. The equations expressed by the velocity potentials and the wave surface elevations include first-order non-linearity of waves and have the same dispersion characteristic to the extended Boussinesq equations. Compared to the extended Boussinesq equations, the equations have only two unknown scalars and do not contain spatial derivatives with an order higher than 2. The wave equations are solved by a finite element method. Fourth-order predictor–corrector method is applied in the time integration and a damping layer is applied at the open boundary for absorbing the outgoing waves. The model is applied to several examples of wave propagation in variable water depth. The computational results are compared with experimental data and other numerical results available in literature. The comparison demonstrates that the new form of the equations is capable of calculating wave transformation from relative deep water to shallow water.     

Numerical simulation for the two-dimensional nonlinear shallow water waves

This study deals with the general numerical model to simulate the two-dimensional tidal flow, flooding wave (long wave) and shallow water waves (short wave). The foundational model is based on nonlinear Boussinesq equations. Numerical method for modelling the short waves is investigated in detail. The forces, such as Coriolis forces, wind stress, atmosphere and bottom friction, are considered. A two-dimensional implicit difference scheme of Boussinesq equations is proposed. The low-reflection outflow open boundary is suggested. By means of this model,both velocity fields of circulation current in a channel with step expansion and the wave diffraction behind a semi-infinite breakwater are computed, and the results are satisfactory.     

Hindcasting nearshore wind waves using a FEM code for SWAN

An improved SWAN model using the Finite Element Method (FEM) was developed for wind waves simulations in both large-scale oceanic deep water regions and small-scale shallow water regions. The model employs a Taylor–Galerkin finite element technique for the discretization of the modeled area, which makes it flexible to represent bottom topography and irregular boundaries. The fractional step numerical scheme was adopted to split the wave action balance equation into three one-dimensional space equations, which can be solved efficiently by one-dimensional algorithms. The Flux-Corrected Transport method was also applied to circumvent the steep-gradients of the action density in the frequency space. The FEM code with unstructured grids improves the numerical schemes in the original SWAN to maintain computational efficiency at the operational stage. A simulation of wind wave activities for the monsoon and the 2000 Typhoon Bilis were performed using the FEM and SWAN models. The simulated results were compared with field observations in order to verify the suitability of the method.