On the role of shoreline boundary conditions in wave overtopping modelling with non-linear shallow water equations

The note extends and completes the analysis carried out by Briganti and Dodd [Briganti, R., Dodd, N., 2009. Shoreline motion in nonlinear shallow water coastal models. Coastal Eng. 56(5–6) (doi:101016/j.coastaleng.2008.10.008), 495–505.] on the performance of a state of the art Non-Linear Shallow Water Equations solver in common coastal engineering applications. The case of bore-generated overtopping of a truncated plane beach is considered and the performance of the model is assessed by comparing with the Peregrine and Williams [Peregrine, D., Williams, S.M., 2001. Swash overtopping a truncated beach. J. Fluid Mech. 440, 391–399.] analytical solution. In particular the influence of shoreline boundary conditions is investigated by considering the two best performing approaches discussed in Briganti and Dodd [Briganti, R., Dodd, N., 2009. Shoreline motion in nonlinear shallow water coastal models. Coastal Eng. 56(5–6) (doi:101016/j.coastaleng.2008.10.008), 495–505.]. Different distances of the edge of the beach from the bore collapse point are tested. For larger distances, the accuracy of the overtopping modelling decreases, as a consequence of the error in modelling the tip of the swash lens and, consequently, the run-up. A sensitivity analysis using the numerical resolution is carried out. This reveals that the approach in which cells shallower than a prescribed threshold are drained and wave propagation speeds for wet/dry Riemann problem are used at the interface between a wet and a dry cell (referred as Option 2ea in [Briganti, R., Dodd, N., 2009. Shoreline motion in nonlinear shallow water coastal models. Coastal Eng. 56(5–6) (doi:101016/j.coastaleng.2008.10.008), 495–505.]) performs consistently better than the other.     

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 applicability of the shallow water equations for modelling violent wave overtopping

The shallow water equations (SWE) have been used to model a series of experiments examining violent wave overtopping of a near-vertical sloping structure with impacting wave conditions. A finite volume scheme was used to solve the shallow water equations. A monotonic reconstruction method was applied to eliminate spurious oscillations and ensure proper treatment of bed slope terms. Both the numerical results and physical observations of the water surface closely followed the relevant Rayleigh probability distributions. However, the numerical model overestimated the wave heights and suffered from the lack of dispersion within the shallow water equations. Comparisons made on dimensionless parameters for the overtopping discharge and percentage of waves overtopping between the numerical model and the experimental observations indicated that for the lesser impacting waves, the shallow water equations perform satisfactorily and provide a good alternative to computationally more expensive methods.     

波浪在浅水传播中的弱非线性效应

在波浪从深水向浅水传播过程中,考虑弱非线性效应具有重要的实用价值,因此得到广泛的讨论和研究。本文根据文献「６」导出的考虑能耗的定常缓坡方程,结合文献「５」给出的显式非线弥散关系,得出了含弱非线性效应的缓坡方程,用该方程对浅水中波浪的传播 计算,将计算结果和试验数据进行了比较,结果表明,含弱非线性效应的缓坡方程可以用于讨论浅水中波浪传播的弱非线性效应,所得计算计算结果与试验结果更为吻合。     

Field investigation on the results of non-linear shallow water equations in the Swash Zone

For the study of the cross-shore wave-induced hydrodynamics in the swash zone, a numerical model is developed based on the one-dimensional non-linear shallow water (NSW) equations for prediction of hydrodynamic parameters in the swash zone. In order to evaluate the accuracy of the outputs of the numerical model, the model's predictions in terms of water surface elevations and cross-shore velocities, are compared to field data from full-scale experiments conducted on three sites with different beach slope; mild and steep, several bed particle sizes and under various incident wave conditions. The quantitative and qualitative comparison of the results of the numerical model and the full-scale data reveals that the model can generally predict many aspects of the flow in the surf and swash zone on both types of beach. The accuracy is adequate for application in a sediment transport study. Considering the time-history and probability distribution of water surface elevation, the model is generally more accurate on steep beaches than on the mild beach. The model can adequately simulate the dominant frequency across the beach and saturation of higher frequencies on both mild and steep beaches for various incident wave energy characteristics. With regard to the horizontal (cross-shore) velocity, the sawtooth shape of time-history and negative acceleration of water are well predicted by the model for both mild and steep beaches. Due to the uncertainties in maximum and minimum values of velocity data, clear judgement about the accuracy of the numerical model in this matter was not possible. However, the comparison of the minimum velocities (offshore direction) revealed that the application of friction factors below the range which is suggested by literature best match the data.     

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.     

Refined seaward boundary conditions for modelling shallow water waves in semi-enclosed water bodies

Based on the theory of characteristics, this research elaborates on the numerical treatment of two types of seaward boundary conditions for modelling long-wave dynamics in truncated estuarine and coastal domains. These seaward boundary conditions are devised for the solution of the fully non-linear shallow water equations in the time domain. The first type is the clamped boundary, at which the water level variation is given and the velocity is computed along the characteristic line going out of the domain. The second type is the non-reflecting boundary, where the incident wave information is introduced and the reflected waves from inside the computational domain are allowed to escape at the same time. The essence of its numerical implementation is to distinguish the inward and outward characteristics and to disconnect the incoming characteristic relation from the actual flow inside the domain. Compared with previous techniques, the present method includes extra terms in the derivation to account for the effects of the uneven bed, bottom friction and shape of the characteristic lines. A shock-capturing finite difference method is used to solve the shallow water equations in the deviatoric format, but the seaward boundary algorithms constructed herein are generic and applicable to other solvers. The necessity of these refinements is highlighted by simulating the tidal oscillation in the Persian/Arabian Gulf, periodic wave runup on the coastline and the wave resonance in a narrow harbour. It is found that neglecting the bed slope at the boundary may result in biased mean water levels in the prediction.     

A numerical method for the two layer shallow water equations with dry states

A numerical method is proposed for solving the two layer shallow water equations with variable bathymetry in one dimension based on high-resolution f-wave-propagation finite volume methods. The method splits the jump in the fluxes and source terms into waves propagating away from each grid cell interface. It addresses the required determination of the system’s eigenstructure and a scheme for evaluating the flux and source terms. It also handles dry states in the system where the bottom layer depth becomes zero, utilizing existing methods for the single layer solution and handling single layer dry states that can exist independently. Sample results are shown illustrating the method and its handling of dry states including an idealized ocean setting.     

Filtering inertia-gravity waves from the initial conditions of the linear shallow water equations

A method for filtering inertia-gravity waves from elevation and depth-averaged velocity is described. This filtering scheme is derived from the linear shallow water equations for constant depth and constant Coriolis frequency. The filtered solution is obtained by retaining only the eigenvectors corresponding to the geostrophic equilibrium and by discarding explicitly the eigenvectors corresponding to the fast moving inertia-gravity waves. An alternative formulation is derived using a variational approach. Both filtering methods are tested numerically for a periodic domain with constant depth and the variational approach is implemented for a closed domain with large topographic variations. The filtering methods significantly reduce the amplitudes of the inertia-gravity waves while preserving the mean flow. The variational method is compared to the Incremental Analysis Update technique and the benefits of the variational filter are presented.     

浅水区波浪非线性效应对砂质海床动力响应的影响

以广义Biot动力固结理论为基础,运用一阶椭圆余弦波和二阶Stokes波等非线性波浪理论考虑浅水区波浪荷载的非线性效应,在时域上采用有限元方法对非线性波浪力作用下饱和砂质海床的动力响应进行了数值求解,并与线性波浪作用下海床动力响应特性进行了对比分析。结果表明,随着波长与水深之比L／d及无量纲参数T(g/d)^1/2的增大,非线性波浪对海床动力响应的影响增大。与线性波浪理论相比,孔隙水压力与有效应力幅值的增大效应非常显著。因此在近海海洋建筑物设计与工程场地评价中,波浪力的非线性特性必须引起注意。     

The wave transformation and breaking phenomena in shallow water

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The dynamics of wave induced ship motions in shallow water

The analytical method developed by Svendsen (1968) for a forced heave motion is extended to the general problem of wave induced heave, roll and sway motions of a long ship at a depth of water which is only slightly larger than the draught of the ship. This corresponds, for example, to the situation of a fully loaded ship in a harbour area.After linearization of the problem, the water motion is considered for each of the three individual motions and for the wave reflection-transmission problem for a fixed ship. The ensuing results for the forces on the ship are then synthesized to form the equations of motion, which are presented with all coefficients given, including mooring forces.Analytical and numerical results are given for the three components of motion, for the associated resonance frequencies, and for the hydrodynamic masses and moments of inertia. Finally, the assumptions used are analyzed and evaluated by comparison with measurements and with other results for a special case.     

Applications of sediment sudden deposition model based on the third-generation numerical model for shallow water wave

The existing numerical models for nearshore waves are briefly introduced, and the third-generation numerical model for shallow water wave, which makes use of the most advanced productions of wave research and has been adapted well to be used in the environment of seacoast, lake and estuary area, is particularly discussed. The applied model realizes the significant wave height distribution at different wind directions. To integrate the model into the coastal area sediment, sudden deposition mechanism, the distribution of average silt content and the change of sediment sudden deposition thickness over time in the nearshore area are simulated. The academic productions can give some theoretical guidance to the applications of sediment sudden deposition mechanism for stormy waves in the coastal area. And the advancing directions of sediment sudden deposition model are prospected.     

Source localization in shallow water in the presence of sensorlocation uncertainty

The problem of source localization in shallow water in the presence of sensor location uncertainty is considered. The Cramer-Rao Bound is used to carry out a feasibility study for the joint source and sensor location problem when the multipath propagation channel is modeled as a known, deterministic waveguide. Unlike the free-space propagation channel, the boundedness of the shallow-water waveguide along its vertical axis provides the key to joint determination of the source and sensor location parameters. It is seen that, when a set of intuitive identifiability conditions are satisfied, numerical examples indicate that, for the scenarios considered, the resulting loss in accuracy with which the source location can be estimated due to sensor location uncertainty may be tolerable     

Major challenges of offshore platforms design for shallow water oil and gas field in moderate ice conditions

Offshore engineers and scientists face fascinating economical and technical challenges in designing offshore platforms for shallow water oil and gas fields in moderate ice conditions. Petroleum production systems in these ice-infested areas such as the Bohai Bay of China, Cook Inlet, Barent Sea, and Caspian Sea must be designed to accommodate the harsh environmental conditions, among which the first-year sea ice is one of the major design consideration. Extreme ice loads and ice-induced vibrations still remain an area of uncertainty in offshore platforms. This paper demonstrates the main technical aspects on the use of jackets in the Bohai Bay, with particular focus on ice loads and the failure modes of slender ice-resistant structures, which are the two key issues in design considerations. A design proposal and some considerations for economical ice-resistant structures for safe development in the cold region are also conceptually discussed.     

Estimation of wave directional spreading in shallow water

This paper describes wave directional spreading in shallow water. Waves were measured for a period of 2 months using the Datawell directional waverider buoy at 15 m water depth on the east coast of India in the Bay of Bengal. The study also showed that in shallow water wave directional spreading was narrowest at peak frequency and widened towards lower and higher frequencies. The wind direction was found to deviate from the wave direction during most of the time. The unidirectional spectrum was found to be satisfactorily represented by Scott spectra.     

基于全和谐型浅水方程的有限体积海啸数值模型开发与应用

数值模拟作为海啸预报的主要研究方法在海啸预警中起着关键作用。本文采用Godunov格式的有限体积方法,使用MUSCL-Hancock格式,并利用HLLC Riemann近似求解器计算单元界面上的流体通量,建立了球坐标系下二阶精度的海啸数值模型。模型所基于的全和谐型浅水方程保证了数值的稳定性,而地形重构方法实现了干湿边界的精准模拟。本文模拟了2015年9月16日智利M_w8.3级地震海啸,通过与智利近岸14个测站和环太平洋20个DART浮标实测数据比较,验证了模型对实际越洋海啸模拟预报的能力。     

浅水波长计算问题

浅水波周期保持常值,而波长缩减时其计算需进行迭代。提出了两个精度较高的经验公式供计算应用。水深波长比差值ΔLD0的变化规律反映出浅水波长缩减为一复杂过程,并与波能密度加大相关。LD0≤0.14时,特别是LD0=0.056前后,其特征会更加显著,由此可探讨波长变化的内在机理。     

Wave loads on rubble mound breakwater crown walls in deep and shallow water wave conditions

The semi-empirical formulae by Pedersen (1996) for wave loads on vertical front faces of stiff crown walls are based on model tests with deep and intermediate water wave conditions. A new series of model tests performed at the same test facility as used by Pedersen has revealed that the formulae by Pedersen overpredict the loads in shallow water wave conditions. This paper presents a modification/expansion of the formulae to cover loads in both deep and shallow water wave conditions. The modification is based on a series of 162 physical model tests on typical rubble mound breakwaters with crown wall superstructures. The implementation of shallow water wave conditions in the formulae is done by modifying the term for wave run-up to be dependent on the incident wave height distribution. Moreover, the adjusted formulae provide more accurate estimates of the wave loads on free walls without front armour protection. Pressure transducers with very high eigen-frequencies were used in the present model tests as opposed to the transducers applied by Pedersen which in some cases seem to have been affected by dynamic amplifications.