Examination of empirical formulas for wave shoaling and breaking on steep slopes

Estimation of the wave height transformation of shoaling and breaking is essential for the nearshore hydrodynamics and the design of coastal structures. Many empirical formulas have been well recognized to the wave height transformation, but most of them were only applicable for gentle slopes. This paper reports the experimental results of wave shoaling and breaking over the steep slopes to examine the applicability of the previous empirical formulas. Two steep bottom slopes of 1/3 and 1/5, and one gentle slope of 1/10 were conducted in the present experiments. Experimental results show that the shoaling distance of steep slopes become short and the surface waves may be partially reflected from the steep bottom, thus the estimation of wave shoaling using the well-known previous formula did not conform completely to the experimental results. The previous empirical formulas for the wave breaking criteria were also examined, and the modified equations to the steep beaches were proposed in this work. A numerical model was finally adopted to calculate the wave height transformation in the surf zone by introducing the modified breaking index.     

基于Boussinesq方程近岸波浪演变的数值研究

首先对目前描述近岸波浪传播变形的数学模型进行了回顾与总结;对不同数学模型的特点、适用范围和发展情况进行了阐述与对比。应用基于Boussinesq方程的Coulwave模式针对几个经典实验地形进行了数值实验,数值结果和实验实测数据吻合较好。此外,分别采用不同的近岸波浪模型模拟了某渔港附近波浪的传播变形,结果表明:当考虑波浪的折射、绕射、反射联合作用时,Coulwave模式计算结果明显较缓坡方程及SWAN模型计算结果更加合理。     

Effects of wave breaking on wave statistics for deep-water random wave train

The experimental studies of the breaking effects on wave statistics for deep-water random waves are presented. It is especially focused on the behavior of kurtosis of surface elevations due to wave breaking. Wave breaking suppresses the maximum limit of kurtosis of the surface elevation, although skewness depends on characteristic wave steepness. The mean instantaneous wave steepness of breaking waves defined using the zero-down-crossing method was much lower than expected from the Stokes waves.     

The wave transformation and breaking phenomena in shallow water

Ｔｈｅｗａｖｅｔｒａｎｓｆｏｒｍａｔｉｏｎａｎｄｂｒｅａｋｉｎｇｐｈｅｎｏｍｅｎａｉｎｓｈａｌｌｏｗｗａｔｅｒ￥ＬｉＹｕｃｈｅｎｇ（１．ＤａｌｉａｎＵｎｉｖｅｒｓｉｔｙｏｆＴｅｃｈｎｏｌｏｇｙ，Ｄａｌｉａｎ１１６０２３，Ｃｈｉｎａ）Ａｂｓｔｒａｃｔ：...     

Shock-capturing Boussinesq-type model for nearshore wave processes

This paper describes the formulation and validation of a nearshore wave model for tropical coastal environment. The governing Boussinesq-type equations include the conservative form of the nonlinear shallow-water equations for shock capturing. A Riemann solver supplies the inter-cell flux and bathymetry source term, while a Godunov-type scheme integrates the evolution variables in time. The model handles wave breaking through momentum conservation with energy dissipation based on an eddy viscosity concept. The computed results show very good agreement with laboratory data for wave propagation over a submerged bar, wave breaking and runup on plane beaches as well as wave transformation over fringing reefs. The model accurately describes transition between supercritical and subcritical flows as well as development of dispersive waves in the processes.     

On using Boussinesq-type equations near the shoreline: a note of caution

We briefly analyze some characteristics of the behavior in very shallow waters i.e. near the shoreline of high-order (dispersive-nonlinear) Boussinesq-type equations. By using the Carrier and Greenspan (1958) solution as test flow conditions we illustrate the behavior of both purely dispersive and dispersive-nonlinear contributions near the shoreline. It is also shown that Boussinesq-type equations can be more usefully handled in the swash zone if written in terms of the total water depth.     

Simulation of wave breaking effects in two-dimensional elliptic harbor wave models

A technique is developed for including the effects of dissipation due to wave breaking in two-dimensional elliptic models based on the mild-slope wave equation. This involves exploration of convergence properties pertaining to iteration due to presence of the nonlinear wave breaking parameter in the governing equations as well as new boundary conditions that include wave-breaking effects. Five wave-breaking formulations are examined in conjunction with the resulting model, which is applied to tests involving a sloping beach, a bar-trough bottom configuration, shore-connected and shore-parallel breakwaters on a sloping beach, and two real-world cases. Model results show that three of the formulations, when used within the context of the modeling scheme presented here, provide excellent results compared to data.     

Numerical models for evolution of extreme wave groups

The paper considers the application of two numerical models to simulate the evolution of steep breaking waves. The first one is a Lagrangian wave model based on equations of motion of an inviscid fluid in Lagrangian coordinates. A method for treating spilling breaking is introduced and includes dissipative suppression of the breaker and correction of crest shape to improve the post breaking behaviour. The model is used to create a Lagrangian numerical wave tank, to reproduce experimental results of wave group evolution. The same set of experiments is modelled using a novel VoF numerical wave tank created using OpenFOAM. Lagrangian numerical results are validated against experiments and VoF computations and good agreement is demonstrated. Differences are observed only for a small region around the breaking crest.     

A Boussinesq-type wave driver for a morphodynamical model to predict short-term morphology

The prediction of near-shore morphology on the time scale of a storm event and the length scale of a few surf zone widths is an active area of research. Intense wave breaking drives offshore-directed currents (undertow) carrying sediment seawards, resulting in offshore bar migration. In contrast, higher order nonlinear properties, such as wave asymmetry (velocity skewness) and velocity asymmetry, are drivers for shoreward transport. These wave processes are included in phase-resolving models such as Boussinesq-type wave models (e.g., TRITON). Short-wave averaging in the wave model yields wave-induced forces (e.g., radiation stress gradients) and a wave asymmetry term. The wave-induced forces are used in a hydrostatic model (e.g., Delft3D flow module) to drive the current and undertow, resulting in a 3D velocity profile. The wave model and hydrostatic model are coupled online with a morphodynamic model (e.g., Delft3D morphology module). The latter computes, based on the 3D flow profile and the wave asymmetry term, the sediment transport and performs the bathymetry updates. The updates are transferred directly back to the hydrodynamic models. The coupling of the wave model TRITON and the Delft3D modules is validated by comparing against extensive laboratory data sets (LIP and Boers) and a field case (Duck94), and show a good performance for the hydrodynamics and a reasonable/fair performance for the bar movements.     

Improved representation of breaking wave energy dissipation in parametric wave transformation models

An improved formulation to describe breaking wave energy dissipation is presented and incorporated into a previous parametric cross-shore wave transformation model [Baldock, T.E., Holmes, P., Bunker, S., Van Weert, P., 1998. Cross-shore hydrodynamics within an unsaturated surf zone. Coastal Engineering 34, 173–196]. The new formulation accounts for a term in the bore dissipation equation neglected in some previous modelling, but which is shown to be important in the inner surf zone. The only free model parameter remains the choice of γ, the ratio of wave height to water depth at initial breaking, and a well-established standard parameter is used for all model runs. The proposed model is compared to three sets of experimental data and a previous version of the model which was extensively calibrated against field and laboratory data. The model is also compared to the widely used model presented by Thornton and Guza (1983) [Thornton, E.B., Guza, R.T., 1983. Transformation of wave height distribution. Journal of Geophysical Research 88 (No.C10), 5925–5938].     

Testing and calibrating parametric wave transformation models on natural beaches

To provide coastal engineers and scientists with a detailed inter-comparison of widely used parametric wave transformation models, several models are tested and calibrated with extensive observations from six field experiments on barred and unbarred beaches. Using previously calibrated (“default”) values of a free parameter γ, all models predict the observations reasonably well (median root-mean-square wave height errors are between 10% and 20%) at all field sites. Model errors can be reduced by roughly 50% by tuning γ for each data record. No tuned or default model provides the best predictions for all data records or at all experiments. Tuned γ differ for the different models and experiments, but in all cases γ increases as the hyperbolic tangent of the deep-water wave height, H_o. Data from two experiments are used to estimate empirical, universal curves for γ based on H_o. Using the new parameterization, all models have similar accuracy, and usually show increased skill relative to using default γ.     

Implementation and evaluation of alternative wave breaking formulas in a coastal spectral wave model

This paper describes methods and results of research for incorporating four different parameterized wave breaking and dissipation formulas in a coastal wave prediction model. Two formulations assume the breaking energy dissipation to be limited by the Rayleigh distribution, whereas the other two represent the breaking wave energy by a bore model. These four formulations have been implemented in WABED, a directional spectral wave model based on the wave action balance equation with diffraction, reflection, and wave–current interaction capabilities. Four parameterized wave breaking formulations are evaluated in the present study using two high-quality laboratory data sets. The first data set is from a wave transformation experiment at an idealized inlet entrance, representing four incident irregular waves in a slack tide and two steady-state ebb current conditions. The second data set is from a laboratory study of wave propagation over a complex bathymetry with strong wave-induced currents. Numerical simulation results show that with a proper breaking formulation the wave model can reproduce laboratory data for waves propagating over idealized or complicated bathymetries with ambient currents. The extended Goda wave breaking formulation with a truncated Rayleigh distribution, and the Battjes and Janssen formulation with a bore model produced the best agreement between model and data.     

Velocity potential formulations of highly accurate Boussinesq-type models

The highly accurate Boussinesq-type equations of Madsen et al. (Madsen, P.A., Bingham, H.B., Schäffer, H.A., 2003. Boussinesq-type formulations for fully nonlinear and extremely dispersive water waves: Derivation and analysis. Proc. R. Soc. Lond. A 459, 1075–1104; Madsen, P.A., Fuhrman, D.R., Wang, B., 2006. A Boussinesq-type method for fully nonlinear waves interacting with a rapidly varying bathymetry. Coast. Eng. 53, 487–504); Jamois et al. (Jamois, E., Fuhrman, D.R., Bingham, H.B., Molin, B., 2006. Wave-structure interactions and nonlinear wave processes on the weather side of reflective structures. Coast. Eng. 53, 929–945) are re-derived in a more general framework which establishes the correct relationship between the model in a velocity formulation and a velocity potential formulation. Although most work with this model has used the velocity formulation, the potential formulation is of interest because it reduces the computational effort by approximately a factor of two and facilitates a coupling to other potential flow solvers. A new shoaling enhancement operator is introduced to derive new models (in both formulations) with a velocity profile which is always consistent with the kinematic bottom boundary condition. The true behaviour of the velocity potential formulation with respect to linear shoaling is given for the first time, correcting errors made by Jamois et al. (Jamois, E., Fuhrman, D.R., Bingham, H.B., Molin, B., 2006. Wave-structure interactions and nonlinear wave processes on the weather side of reflective structures. Coast. Eng. 53, 929–945). An exact infinite series solution for the potential is obtained via a Taylor expansion about an arbitrary vertical position z = zˆ. For practical implementation however, the solution is expanded based on a slow variation of zˆ and terms are retained to first-order. With shoaling enhancement, the new models obtain a comparable accuracy in linear shoaling to the original velocity formulation. General consistency relations are also derived which are convenient for verifying that the differential operators satisfy a potential flow and/or conserve mass up to the order of truncation of the model. The performance of the new formulation is validated using computations of linear and nonlinear shoaling problems. The behaviour on a rapidly varying bathymetry is also checked using linear wave reflection from a shelf and Bragg scattering from an undulating bottom. Although the new models perform equally well for Bragg scattering they fail earlier than the existing model for reflection/transmission problems in very deep water.     

The effect of relative crest submergence on wave breaking over submerged slopes

Experiments were performed in a wave flume to measure the intensity, transmission and reflection of waves breaking over a submerged reef with an offshore gradient of 1:10. The results demonstrate that the relative water depth over the reef crest (h_c/H_o) is a dominant factor affecting the breaking characteristics. In particular it is found that as the relative crest submergence is reduced, there is a considerable increase in the intensity of wave breaking over the reef that can be quantified through measurements of the air cavity enclosed beneath the plunging jet. It is also shown that there is a corresponding decrease in wave transmission and reflection as the submergence is reduced.     

Calibration and modification of energy dissipation models for irregular wave breaking

This study is undertaken to recalibrate eight existing energy dissipation models and find out the suitable models, which can be used to compute H_rms for a wide range of experimental conditions. The examination shows that the coefficients in the existing models are not the optimal values for a wide range of experimental conditions. Using the new calibrated coefficients, all existing models can be used for computing H_rms and the model of Battjes, J.A., Stive, M.J.F. [1985. Calibration and verification of a dissipation model for random breaking waves. Journal of Geophysical Research 90 (C5), 9159–9167] gives the best predictions. The existing models are also modified by changing the breaker height formulas in the dissipation models. The accuracy of most existing models is improved significantly by using the suitable breaker height formula.     

Intermittency of wind wave group breaking

Knowledge on intermittency of wave breaking is so far limited to a few summary statistics, while the probability distribution of time interval between breaking events can provide a full view of intermittency. Based on a series of experiments on wind wave breaking, such probability distributions are investigated. Breaking waves within a wave group were taken as a single breaking event according to recent studies. Interval between successive wave groups with breaker is the focus of this paper. For intervals in our experiments with different fetch and wind conditions, their distributions are all skewed and weighted on small intervals. Results of Kolmogorov-Smirnov tests on time series of these intervals indicate that they all follow gamma distribution, and some are even exponential type. Average breaking-group-interval decreases with friction velocity and significant steepness until the wind is strong enough;most of them are more than 10 times the dominant wave period. Group breaking probability proposed by Babanin recently and the average number of breaking waves in wave groups are also discussed, and they are seemingly more reasonable and sensitive than traditional breaking probability defined in terms of single wave.     

Characteristics of breaking irregular wave forces on a monopile

The substructures of offshore wind turbines are subjected to extreme breaking irregular wave forces. The present study is focused on investigating breaking irregular wave forces on a monopile using a computational fluid dynamics (CFD) based numerical model. The breaking irregular wave forces on a monopile mounted on a slope are investigated with a numerical wave tank. The experimental and numerical irregular free surface elevations are compared in the frequency-domain for the different locations in the vicinity of the cylinder. A numerical analysis is performed for different wave steepness cases to understand the influence of wave steepness on the breaking irregular wave loads. The wave height transformation and energy level evolution during the wave shoaling and wave breaking processes is investigated. The higher-frequency components generated during the wave breaking process are observed to play a significant role in initiating the secondary force peaks. The free surface elevation skewness and spectral bandwidth during the wave transformation process are analysed and an investigation is performed to establish a correlation of these parameters with the breaking irregular wave forces. The role of the horizontal wave-induced water particle velocity at the free surface and free surface pressure in determining the breaking wave loads is highlighted. The higher-frequency components in the velocity and pressure spectrum are observed to be significant in influencing the secondary peaks in the breaking wave force spectrum.     

Dynamic analysis of a vertical plate exposed to breaking wave impact

From the experimental studies in recent years, it has become known that when a wave breaks directly on a vertical faced coastal structure, high magnitude impact pressures are produced. The theoretical and experimental studies show that the dynamic response of such structures under wave impact loading is closely dependent on the magnitude and duration of the load history. The dynamic analysis and design of a coastal structure can be succeeded provided the design load history for the wave impact is available. Since these types of data are very scarce, it is much more convenient to follow a method which is based on static analysis for the dynamic design procedure. Therefore, to facilitate the dynamic design of a vertical plate that is exposed to breaking wave impact, a multiplication factor called “dynamic magnification factor” is herein presented which is defined as the ratio of the maximum value of the dynamic response to that found by static analysis. The computational results of the present study show that the dynamic magnification factor is a useful ratio to transfer the results of static analysis to the dynamic design of a coastal plate for the maximum impact pressure conditions of p_max/γH₀≤18.     

Determining the onset and strength of unforced wave breaking in a numerical wave tank

A numerical wave tank is used to investigate the onset and strength of unforced wave breaking, and the waves have three types of initial spectra： constant amplitude spectrum, constant steepness spectrum and Pierson-Moscowitz spectrum. Numerical tests are performed to validate the model results. Then, the onset of wave breaking is discussed with geometric, kinematic, and dynamic breaking criteria. The strength of wave breaking, which is always characterized by the fractional energy loss and breaking strength coefficient, is studied for different spectra. The results show how the energy growth rate is better than the initial wave steepness on estimating the fractional energy losses as well as breaking strength coefficient.     

基于水槽实验的近岸波浪破碎计算研究

将前人波浪破碎计算及判定公式进行汇总、整理并分成3类;同时,为了对选取公式进行比较,文还将上述公式分为两类破碎波高计算模型,一类是直接计算模型;另一类是联立拍岸浪波高计算模型与破碎判据,建立的间接计算模型。在此基础上,结合实验数据,对破碎波高和破碎指标进行误差分析得出,间接计算模型中,Goda提出的破碎判据效果最好,为下一步拍岸浪精细化预报以及其区域计算模型的建立提供参考。