A comparison of weakly and fully non-linear models of the shoaling of a solitary internal wave

This study investigates the behaviour of internal solitary waves crossing a continental slope in the presence of a seasonal thermocline. Comparisons are made between a fully non-linear computational fluid dynamics (CFD) model, and weakly non-linear theory. Previous observations suggested that the amplitudes of solitary waves are capped as they pass across the continental slope, which may be due to laminar dynamics, or due to the effect of turbulence. Across the continental slope, CFD and second order variable depth KdV (vEKdV) predictions agree well with observations of a limited change in solitary wave amplitude. First order variable depth KdV theory overpredicts the final amplitude significantly. In terms of the wave shape, the CFD modeled wave changes from a KdV shape in deep water towards an EkdV solution in shallow water, as observations suggest. The phase speed of the CFD and vEKdV waves are similar to that observed in waters of 400–500 m deep, but are slightly lower than observed in 140 m depth. CFD predictions using a standard k, turbulence model showed that turbulence had little effect on the amplitude. These preliminary results indicate that in this situation wave capping is due to laminar, large amplitude solitary wave dynamics and is independent of turbulent mixing.     

Wave run-up on slender piles in design conditions — Model tests and design rules for offshore wind

Wave run-up on foundations is a very important factor in the design of entrance platforms for offshore wind turbines. When the Horns Reef 1 wind turbine park in Denmark was designed the vertical wave run-up phenomenon was not well known in the industry, hence not sufficiently considered in the design of Horns Reef 1. As a consequence damage was observed on the platforms. This has been the situation for several sites and design tools for platform loads are lacking. As a consequence a physical model test study was initiated at Aalborg University to clarify wave run-up on cylindrical piles for different values of diameter to water depth ratios (D/h) and different wave heights to water depth ratios (H/h) for both regular and irregular waves. A calculation model is calibrated based on stream function theory for crest kinematics and velocity head stagnation theory. Due to increased velocities close to the pile an empirical factor is included on the velocity head. The evaluation of the calculation model shows that an accurate design rule can be established even in breaking wave conditions. However, calibration of a load model showed that it was necessary to increase the run-up factor on the velocity head by 40% to take into account the underestimation of run-up for breaking or nearly breaking waves given that they produce thin run-up wedges and air entrainment, two factors not coped with by the measurement system.     

Study of irregular wave run-up over fringing reefs based on a shock-capturing Boussinesq model

This paper presents the results of a parametric study of irregular wave run-up over fringing reefs using the shock-capturing Boussinesq wave model Funwave-TVD to better understand the role of fringing reefs in the mitigation of wave-driven flooding. Laboratory experiments were newly performed with a typical fringing reef profile and typical hydrodynamic conditions to validate the model. Experimental data shows irregular wave run-ups are dominated by the low-frequency motions and confirms the run-up resonant phenomenon over the back-reef slope, which has been revealed in previous numerical studies. It is demonstrated that irregular wave evolution and run-up over fringing reefs are reasonably reproduced by the present model with a proper grid size. However, the infragravity run-up height and highest 2% run-up height over the back-reef slope are under-predicted due to the underestimation of the infragravity wave height over the reef flat. The validated model was then utilized to model irregular wave transformations and run-ups under different conditions. Through a series of numerical experiments, the effects of key hydrodynamic and reef geometry parameters, including the reef flat width, water depth over the reef flat, fore-reef slope angle and back-reef slope angle, on the irregular wave run-up were investigated. Variations of spectral components of irregular wave run-ups were examined to better understand the physical process underlying the effect of each parameter.     

RANS v2–f simulation of a swash event: Detailed flow structure

The flow structure of a swash event over a uniform slope is studied using a RANS-VOF numerical model coupled with a v²–f turbulence closure. The model is compared with experimental data of recent laboratory experiments. The ability of the turbulence modelling for simulating swash flow and the evolution of the computed bed shear stress during run-up and run-down are investigated. The agreement between numerical results and measured data, such as water depth, depth-averaged velocity and bed shear stress is very good during run-up. Main discrepancies are found during run-down. The paper also examines the aeration of the water layer in the swash flow, taking advantage of the PLIC method for computing the air–water interfaces. Air is continuously entrapped in the swash front and released at its rear during run-up. A detailed analysis indicates that the flow reversal is initiated near the bottom at the outer boundary of the swash zone and progresses landward. The study highlights the asymmetry between run-up and run-down. During run-up, the swash front propagation determines the turbulence properties and the bed shear stress profile on the beach, whereas the flow properties are more homogeneously distributed in the swash area during run-down.     

Numerical Study of the Impact of Climate Change on Irregular Wave Run-up Over Reef-Fringed Coasts

Wave hydrodynamics over fringing reefs is largely controlled by the reef surface roughness and hydrodynamic forcing. It is believed that climate change will result in a net increase in the water depth over the reef flat, a degrading of the surface roughness of coral reefs and changes in extreme incident wave heights. For an accurate assessment of how climate change affects the safety of reef-fringed coasts, a numerical study of the impact of climate change on irregular wave run-up over reef-fringed coasts was carried out based on a Boussinesq wave model,FUNWAVE-TVD. Validated with experimental data, the present model shows reasonable prediction of irregular wave evolution and run-up height over fringing reefs. Numerical experiments were then implemented based on the anticipated effects of climate change and carried out to investigate the effects of sea level rise, degrading of the reef surface roughness and increase of extreme incident wave height on the irregular wave run-up height over the backreef beach respectively. Variations of run-up components(i.e., spectral characteristics of run-up and mean water level) were examined specifically and discussed to better understand the influencing mechanism of each climate change-related effect on the run-up.     

Stratigraphic Model Predictions of Geoacoustic Properties

Geoacoustic properties of the seabed have a controlling role in the propagation and reverberation of sound in shallow-water environments. Several techniques are available to quantify the important properties but are usually unable to adequately sample the region of interest. In this paper, we explore the potential for obtaining geotechnical properties from a process-based stratigraphic model. Grain-size predictions from the stratigraphic model are combined with two acoustic models to estimate sound speed with distance across the New Jersey continental shelf and with depth below the seabed. Model predictions are compared to two independent sets of data: 1) Surficial sound speeds obtained through direct measurement using in situ compressional wave probes, and 2) sound speed as a function of depth obtained through inversion of seabed reflection measurements. In water depths less than 100 m, the model predictions produce a trend of decreasing grain-size and sound speed with increasing water depth as similarly observed in the measured surficial data. In water depths between 100 and 130 m, the model predictions exhibit an increase in sound speed that was not observed in the measured surficial data. A closer comparison indicates that the grain-sizes predicted for the surficial sediments are generally too small producing sound speeds that are too slow. The predicted sound speeds also tend to be too slow for sediments 0.5-20 m below the seabed in water depths greater than 100 m. However, in water depths less than 100 m, the sound speeds between 0.5-20-m subbottom depth are generally too fast. There are several reasons for the discrepancies including the stratigraphic model was limited to two dimensions, the model was unable to simulate biologic processes responsible for the high sound-speed shell material common in the model area, and incomplete geological records necessary to accurately predict grain-size     

Numerical study of vegetation damping effects on solitary wave run-up using the nonlinear shallow water equations

Vegetation damping effects on propagating water waves have been investigated by many researchers. This paper investigates the effects of damping due to vegetation on solitary water wave run-up via numerical simulation. The numerical model is based on an implementation of Morison's formulation for vegetation induced inertia and drag stresses in the nonlinear shallow water equations. The numerical model is solved via a finite volume method on a Cartesian cut cell mesh. The accuracy of the numerical scheme and the effects of the vegetation terms in the present model are validated by comparison with experiment results. The model is then applied to simulate a solitary wave propagating on a plane slope with vegetation. The sensitivity of solitary wave run-up to plant height, diameter and stem density is investigated by comparison of the numerical results for different patterns of vegetation. The numerical results show that vegetation can effectively reduce solitary wave propagation velocity and that solitary wave run-up is decreased with increase of plant height in water and also diameter and stem density.     

Numerical study on cnoidal wave run-up around a vertical circular cylinder

A finite element model of Boussinesq-type equations was set up, and a direct numerical method is proposed so that the full reflection boundary condition is exactly satisfied at a curved wall surface. The accuracy of the model was verified in tests. The present model was used to further examine cnoidal wave propagation and run-up around the cylinder. The results showed that the Ursell number is a nonlinear parameter that indicates the normalized profile of cnoidal waves and has a significant effect on the wave run-up. Cnoidal waves with the same Ursell number have the same normalized profile, but a difference in the relative wave height can still cause differences in the wave run-up between these waves. The maximum dimensionless run-up was predicted under various conditions. Cnoidal waves hold entirely distinct properties from Stokes waves under the influence of the water depth, and the nonlinearity of cnoidal waves enhances rather than weakens with increasing wavelength. Thus, the variations in the maximum run-up with the wavelength for cnoidal waves are completely different from those for Stokes waves, and there are even significant differences in the variation between different cnoidal waves.     

On dynamic coupling effects between a spar and its mooring lines

The responses of a spar constrained by slack mooring lines to steep ocean waves and tensions in the mooring lines are simulated using two different numerical schemes: a quasi-static approach (SMACOS) and a coupled dynamic approach (COUPLE). The two approaches are the same in computing wave loads on the structure. Their difference is in modeling dynamic forces of mooring lines; that is the dynamic forces are included in the computation of COUPLE but neglected in SMACOS. The numerical simulation is examined against the laboratory measurements of the JIP Spar in a water depth of 318 m. The dynamic coupling effects between the JIP Spar and its mooring lines in different water depths (318, 618 and 1018 m) are investigated by the comparison of numerical simulations obtained using the quasi-static and coupled dynamic approaches. It is found that the damping of mooring lines reduces the slow-drift surge and pitch of the Spar, especially in deep water. The reduction in the amplitude of slow-drift surge can reach about 10% in a water depth of 1018 m. The tension in mooring lines may greatly increase in the wave frequency range when dynamic forces in mooring lines are considered. The mooring-line tension in the wave frequency range predicted by the coupled dynamic approach can be eight times as great as the corresponding prediction by the quasi-static approach in a water depth of 1018 m. This finding may have important implications for the estimation of the fatigue strength and life span of the mooring lines deployed in deep water oceans.     

SWAN predictions of waves observed in shallow water onshore of complex bathymetry

SWAN model predictions, initialized with directional wave buoy observations in 550-m water depth offshore of a steep, submarine canyon, are compared with wave observations in 5.0-, 2.5-, and 1.0-m water depths. Although the model assumptions include small bottom slopes, the alongshore variations of the nearshore wave field caused by refraction over the steep canyon are predicted well over the 50 days of observations. For example, in 2.5-m water depth, the observed and predicted wave heights vary by up to a factor of 4 over about 1000 m alongshore, and wave directions vary by up to about 10°, sometimes changing from south to north of shore normal. Root-mean-square errors of the predicted wave heights, mean directions, periods, and radiation stresses (less than 0.13 m, 5°, 1 s, and 0.05 m³/s² respectively) are similar near and far from the canyon. Squared correlations between the observed and predicted wave heights usually are greater than 0.8 in all water depths. However, the correlations for mean directions and radiation stresses decrease with decreasing water depth as waves refract and become normally incident. Although mean wave properties observed in shallow water are predicted accurately, nonlinear energy transfers from near-resonant triads are not modeled well, and the observed and predicted wave energy spectra can differ significantly at frequencies greater than the spectral peak, especially for narrow-band swell.     

Third-order interactions,wave run-up and hydrodynamic loading on a vertical plate in an infinite wave field

The nonlinear wave interaction problem with a vertical plate of finite length is considered. Reference is made to previous experimental and numerical studies reported in Molin et al. [1–3], where it was shown that the observed run-up phenomena are due to third-order (or tertiary) interactions between the incoming and reflected wave systems. In this paper a new numerical model is proposed where the presence of lateral walls is relaxed. Run-up computations, with and without confinement effects, are compared. It is found that, in the model tests reported in Molin et al. [3], the effect of confinement was relatively small. The time-varying and steady wave loads which are exerted on the plate are also investigated. The dedicated numerical predictions show that as the wave steepness is increased the response amplitude operators of the time-varying loads first increase, reach a maximum and then decrease dramatically, due to phasing effects.     

Wave Run-up on A Vertical Seawall Protected by An Offshore Submerged Barrier

Submerged barriers are constructed in coastal zones for shoreline or harbor protection or to prevent the beach erosion. In the present study, the wave run-up on a vertical seawall protected by a submerged barrier is analyzed. The physical configurations include a rigid barrier and a long channel of finite depth. For linear water waves, by matching the velocity along the barrier and along the gap, the systems of linear equations about the velocity potentials are obtained. The wave run-up is further analyzed for various settings of barrier height and distance between the barrier and the wall, i.e. the chamber length. For nonlinear waves and random sea waves, a numerical model is extended to investigate the effect parameters of the barrier on the wave run-up against the seawall. Not only the numerical simulations, but also the analytical results illustrate that the wave run-up on the seawall depends very much on the distance between the barrier and the vertical seawall.     

Numerical analysis of run-up oscillations under dissipative conditions

This paper presents laboratory and numerical simulations of run-up induced by irregular waves breaking on a gentle-sloping planar beach. The experimental data are well reproduced by a numerical model based on the nonlinear shallow water equations. By extending the incoming wave conditions considered in the laboratory experiments, the model is applied to study the run-up variability under highly energetic incoming conditions. The numerical results support the idea that, for cases characterized by the same incident peak frequency, infragravity run-up increases almost linearly with the offshore significant wave height. Moreover, the most energetic conditions lead to an upper limit of the swash similarity parameter of about 1.8.     

陡坡上孤立波爬高的数值模拟

波浪爬高是海岸工程中重要的水动力学问题之一,其数值模拟方法通常是通过离散Navier-Stokes方程或Boussinesq方程实现的,其中基于光滑粒子流体动力学方法是近年发展起来的。本文应用该方法模拟相同水深下,不同波高的孤立波在45(°)陡坡上的爬高,模拟结果与理论计算结果及已有物理模型试验结果进行了对比,并模拟出孤立波激散破碎过程及粒子分布和速度场的变化过程。结果表明,对密度近似方程进行重新初始化保持了流场内的质量守恒,同时整个计算域内的压力分布更加规则,说明光滑粒子流体动力学法在波浪爬高计算中的有效性。     

孤立波与带窄缝双箱相互作用模拟研究

针对孤立波与带窄缝双箱的作用问题,应用时域高阶边界元方法建立了二维数值水槽。其中,自由水面满足完全非线性运动学和动力学边界条件,对瞬时自由表面流体质点采用混合欧拉-拉格朗日法追踪,采用四阶龙格库塔法对下一时刻的自由水面的速度势和波面升高进行更新。采用加速度势法求解物体湿表面的瞬时波浪力。采用推板方法生成孤立波。通过模拟孤立波在直墙上的爬高以及施加在直墙上的波浪力,并与已发表的实验和数值结果对比,验证本数值模型的准确性。通过数值模拟计算研究了窄缝宽度、方箱尺寸对波浪在箱体迎浪侧爬高,窄缝内波面升高,箱体背浪侧透射波高及箱体受波浪荷载的影响。同时研究了有一定时间间隔的双孤立波与带窄缝双箱系统作用问题。     

A three-dimensional multi-level turbulence model for tidal motion

A three-dimensional multi-level turbulence model is developed to simulate tide induced circulation in coastal waters. Based on the bathymetry data, the coastal waters are divided into a number of layers. In every layer, the velocities are integrated along the layer depth. The eddy viscosity and diffusivity are computed from the Prandtl mixing length turbulence model. This multi-level model solves for the water surface elevations and currents in different water depths. Comparison of numerical results with the measured data shows good conformity.     

Dynamic pressures and run-up on semicircular breakwaters due to random waves

The dynamic pressures due to random waves of predefined spectral characteristics exerted on a semicircular breakwater model at five different elevations along the depth are measured. In addition, the wave run-up on the model and its reflection characteristics are measured. The results on the variation of the frequency pressure spectra along the depth and the run-up spectra are reported in this paper. The average spectral characteristics as well as statistical properties of the above two parameters are presented. The average reflection coefficient is reported as a function of the wave steepness, described as the ratio of the significant wave height to the square of the peak period.     

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.     

片状分布植被对孤立波爬高影响的数值模拟研究

近岸植被对波浪爬坡具有一定的衰减作用。在自然界中,由于植被的死亡、再生或人为破坏等原因,近岸植被通常呈片状分布,且其内部分布也是不均匀的。本文以完全非线性Boussinesq方程为基础,引入植被作用项,建立了模拟近岸植被区波浪传播的数值模型,验证了模型可靠性,进而采用该模型模拟分析了片状分布植被对孤立波爬高的影响。数值模拟结果表明,片状分布植被能有效减小孤立波爬高;对于均匀分布的片状植被,高密度片状植被对孤立波爬高的消减效果优于低密度片状植被;对于相同密度、不同分布形式的片状植被,均匀分布的片状植被对孤立波的消减效果优于不均匀分布的片状植被;对于不均匀分布的片状植被,前密后疏的片状植被对孤立波的消减效果优于前疏后密的片状植被。     

Modified Rayleigh distribution of wave heights in transitional water depths

This paper concerns the calculation of wave height exceedance probabilities for nonlinear irregular waves in transitional water depths, and a Transformed Rayleigh method is first proposed for carrying out the calculation. In the proposed Transformed Rayleigh method, the transformation model is chosen to be a monotonic exponential function, calibrated such that the first three moments of the transformed model match the moments of the true process. The proposed new method has been applied for calculating the wave height exceedance probabilities of a sea state with the surface elevation data measured at the Poseidon platform. It is demonstrated in this case that the proposed new method can offer better predictions than those by using the conventional Rayleigh wave height distribution model. The proposed new method has been further applied for calculating the total horizontal loads on a generic jacket, and its accuracy has once again been substantiated. The research findings gained from this study demonstrate that the proposed Transformed Rayleigh model can be utilized as a promising alternative to the well-established nonlinear wave height distribution models.