A higher-order σ-coordinate non-hydrostatic model for nonlinear surface waves

A higher-order non-hydrostatic model in a σ-coordinate system is developed. The model uses an implicit finite difference scheme on a staggered grid to simultaneously solve the unsteady Navier-Stokes equations (NSE) with the free-surface boundary conditions. An integral method is applied to resolve the top-layer non-hydrostatic pressure, allowing for accurately resolving free-surface wave propagation. In contrast to the previous work, a higher-order spatial discretization is utilized to approximate the large horizontal pressure gradient due to steep surface waves or rapidly varying topographies. An efficient direct solver is developed to solve the resulting block hepta-diagonal matrix system. Accuracy of the new model is validated by linear and nonlinear standing waves and progressive waves. The model is then used to examine freak (extreme) waves. Features of downshifting focusing location and wave asymmetry characteristics are predicted on the temporal and spatial domains of a freak wave.     

A higher-efficient three-dimensional numerical model for small amplitude free surface flows

A higher-efficient three-dimensional non-hydrostatic model is developed to simulate small amplitude free surface flows based on a staggered unstructured grid. In this model, a fractional step algorithm is adopted to solve the Navier-Stokes equations in two major steps. A top-layer pressure method is proposed to minimize the number of vertical layers and subsequently the computational cost. Three classical examples of small amplitude free surface flows are used to demonstrate the capability and efficiency of the model. The satisfactory results demonstrated the capability and efficiency of modelling a range of small amplitude free surface flows with only a small number of vertical layers.     

Application of desingularized approach to water wave propagation over three-dimensional topography

A numerical approach based on desingularized boundary element method and mixed Eulerian–Lagrangian formulation [Zhang et al., 2006. Wave propagation in a fully nonlinear numerical wave tank: a desingularized method. Ocean Engineering 33, 2310–2331] is extended to solve the water wave propagation over arbitrary topography in a three-dimensional wave tank. A robust damping layer applicable for regular and irregular incident waves is employed to minimize the outgoing wave reflection back into the wave tank. Numerical results on the propagation of regular and irregular incident waves over the flat bottom and linear incident waves over an elliptical shoal show good concurrence with the corresponding analytical solutions and experimental data.     

Simulation of nearshore wave processes by a depth-integrated non-hydrostatic finite element model

This paper presents CCHE2D-NHWAVE, a depth-integrated non-hydrostatic finite element model for simulating nearshore wave processes. The governing equations are a depth-integrated vertical momentum equation and the shallow water equations including extra non-hydrostatic pressure terms, which enable the model to simulate relatively short wave motions, where both frequency dispersion and nonlinear effects play important roles. A special type of finite element method, which was previously developed for a well-validated depth-integrated free surface flow model CCHE2D, is used to solve the governing equations on a partially staggered grid using a pressure projection method. To resolve discontinuous flows, involving breaking waves and hydraulic jumps, a momentum conservation advection scheme is developed based on the partially staggered grid. In addition, a simple and efficient wetting and drying algorithm is implemented to deal with the moving shoreline. The model is first verified by analytical solutions, and then validated by a series of laboratory experiments. The comparison shows that the developed wave model without the use of any empirical parameters is capable of accurately simulating a wide range of nearshore wave processes, including propagation, breaking, and run-up of nonlinear dispersive waves and transformation and inundation of tsunami waves.     

Wave propagation in a fully nonlinear numerical wave tank: A desingularized method

The problem of wave propagation in a fully nonlinear numerical wave tank is studied using desingularized boundary integral equation method coupled with mixed Eulerian–Lagrangian formulation. The present method is employed to solve the potential flow boundary value problem at each time step. The fourth-order predictor–corrector Adams–Bashforth–Moulton scheme is used for the time-stepping integration of the free surface boundary conditions. A damping layer near the end-wall of wave tank is added to absorb the outgoing waves with as little wave reflection back into the wave tank as possible. The saw-tooth instability is overcome via a five-point Chebyshev smoothing scheme. The model is applied to several wave propagations including solitary, irregular and random incident waves.     

Non-hydrostatic pressure in σ-coordinate ocean models

Numerical ocean modelling is computationally very demanding. Traditionally, the hydrostatic approximation has been applied to reduce the computational burden. This approximation is valid in large scale studies with coarse grid resolution. With faster computers and gradually smaller grid sizes, we may expect that more studies will be performed with non-hydrostatic ocean models. In recent papers several methods for including non-hydrostatic pressure in σ-coordinate models have been suggested. In this paper the sensitivity of the non-hydrostatic pressure field, the velocity fields, and the density fields to changes in the method for computing non-hydrostatic pressure in σ-coordinate ocean models is addressed.The first test case used involves the propagation and breaking of an internal wave at an incline in a tank. The other test case concerns tidally driven flow over a sill in a stratified fjord. The results from our numerical exercises suggest that the velocity and density fields are very robust to the model choices investigated here. The differences between the model results are of the same order as the uncertainty due to the internal pressure gradient error, and they are smaller than an estimate of the uncertainty due to subgrid scale closure.     

A higher-order non-hydrostatic σ model for simulating non-linear refraction–diffraction of water waves

A higher-order non-hydrostatic σ model is developed to simulate non-linear refraction–diffraction of water waves. To capture non-linear (or steep) waves, a 4th-order spatial discretization is utilized to approximate the large horizontal pressure gradient. A higher-order top-layer pressure treatment is further implemented to resolve wave propagation. The model's characteristics including linear wave dispersion and non-linearity are carefully examined. The accuracy of the present model using only two vertical layers is validated by laboratory data and the available results predicted by the non-linear Schrödinger equation, Boussinesq-type equations, the non-linear mild slope equation, and the Laplace equation. Features of harmonic generation as well as the influences of dispersion and non-linearity on wave energy transfer processes are discussed.     

A non-hydrostatic algorithm for free-surface ocean modelling

An original implementation of a non-hydrostatic, free-surface algorithm based on a pressure correction method is proposed for ocean modelling. The free surface is implemented through an explicit scheme combined with a mode-spitting method but the depth-averaged velocity and the position of the free surface are updated at each non-hydrostatic iteration. The vertical momentum equation is also integrated up to the surface enabling a natural and accurate treatment of the surface layer. The consistent specification of the numerical schemes provides balanced transfers of potential and kinetic energy. This algorithm is well-suited for implementation as a non-hydrostatic kernel on originally hydrostatic free-surface ocean models such as Symphonie (http://poc.obs-mip.fr/pages/research_topics/modelling/symphonie/symphonie.htm) for which it has originally been developed.Energy balances associated with the propagation of short surface waves and solitary waves are presented for two dedicated well-documented configurations over closed domains. The buoyancy flux, the work rate of the pressure force together with the power of the advective terms are evaluated and discussed for the generation and the propagation of these two types of waves. The dissipation rate is in particular shown to be several orders of magnitude smaller than the work rates of the hydrostatic and non-hydrostatic pressure forces confirming the necessity for the exchanges of energy to be numerically balanced. The algorithm is subsequently applied to the complex generation of non-linear solitary internal waves by surface tides over Georges Bank, in the Gulf of Maine. The generation and the propagation of the observed non-linear and non-hydrostatic features in this region are correctly reproduced.     

Automated trinocular stereo imaging system for three-dimensional surface wave measurements

A novel automated trinocular stereo imaging system (ATSIS) is developed for non-intrusively measuring the temporal evolution of three-dimensional wave characteristics. The system consists of three progressive digital cameras to provide three independent stereo-pairs, i.e. left–right, left–center, and center–right, for accurately estimating depth of a scene. A third camera assists to resolve correspondence problems due to specular reflection on the water surface and provides additional constraints on image matching, dramatically reducing the chance of a mismatch. An oblique configuration for the trinocular system effectively increases spatial coverage, allowing observations of wave phenomena over a broad range of spatial scales. The height resolution is increased with the optical axes of the cameras pointed at an oblique angle with respect to vertical surface wave displacements. A new exterior calibration procedure is developed in this paper to determine the orientation of cameras in the field. Field experiments demonstrate that ATSIS can robustly measure hundreds of matched image points in seconds, allowing fast extraction of the temporal evolution of a three-dimensional surface wave field.     

Non-hydrostatic Kelvin waves in the ocean

A theory of the coastal Kelvin wave is presented in which the pressure is assumed not to be hydrostatic. The results show that the non-hydrostatic Kelvin wave is dispersive and that the e^–1 decay distance of the wave amplitude from the coast decreases as the wave length becomes shorter. Similar conclusions can be drawn on the equatorial Kelvin wave.     

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.     

Simulation of wave propagation over a submerged bar using the VOF method with a two-equation k– turbulence modeling

A two-equation k– turbulence model is used in this paper to simulate the propagation of cnoidal waves over a submerged bar, where the free surface is handled by the volume-of-fluid (VOF) method. Using a VOF partial-cell variable and a donor–acceptor method, the model is capable of treating irregular boundaries, including arbitrary bottom topography and internal obstacles, where the no-slip condition is satisfied. The model also allows the viscous sublayer to be modeled by a wall function approximation implemented in the grid nodes that are immediately adjacent to a wall boundary. The numerical model applied to the propagation of cnoidal waves over a submerged bar can produce results that are in general agreement with some laboratory measurements. Some remarks arising from the comparison between the computational and experimental results are presented.     

Algorithm for non-hydrostatic dynamics in the Regional Oceanic Modeling System

A non-hydrostatic algorithm for the Regional Oceanic Modeling System (ROMS) is proposed. It is based on a decomposition technique for hydrostatic and non-hydrostatic pressure. The algorithm has a pressure-correction scheme with split-explicit time-stepping for baroclinic and barotropic vertical modes with a free surface. The algorithm implementation requires solving a Poisson equation for a non-hydrostatic pressure that has a non-symmetric matrix in discrete form. The efficiency of a different class of solvers and preconditioners were tested. The algorithm is successfully implemented with several examples where non-hydrostatic effects are important. These include standing external gravity waves; strongly nonlinear internal wave generation and transformation; stratified shear instability and its associated mixing; and nonlinear internal tidal generation over a ridge. The corresponding changes in the pre-processing and post-processing infrastructure in the existing hydrostatic ROMS code were performed to implement parallel elliptic solvers and a new set of dynamical equations.     

Efficient computation of surf zone waves using the nonlinear shallow water equations with non-hydrostatic pressure

A numerical method for non-hydrostatic, free-surface, irrotational flow governed by the nonlinear shallow water equations including the effects of vertical acceleration is presented at the aim of studying surf zone phenomena. A vertical boundary-fitted grid is used with the water depth divided into a number of layers. A compact finite difference scheme is employed for accurate computation of frequency dispersion requiring a limited vertical resolution and hence, capable of predicting the onset of wave breaking. A novel wet–dry algorithm is applied for a proper handling of moving shoreline. Mass and momentum are strictly conserved at discrete level while the method only dissipates energy in the case of wave breaking. The numerical results are verified with a number of tests and show that the proposed model using two layers without ad-hoc assumptions enables to resolve propagating nonlinear shoaling, breaking waves and wave run-up within the surf and swash zones in an efficient manner.     

Coastal-Trapped Waves with Several-Day Period Caused by Wind along the Southeast Coast of Honshu, Japan

Generation and propagation of several-day period fluctuations along the southeast coast of Honshu, Japan, were investigated by analyzing sea level data and by using a numerical model. The sea level data obtained at twelve stations from Choshi to Omaezaki in fall in 1991, showed energy peaks at the 3–6 day period at the eastern stations in this coast. Time lags of the 3–6 day period fluctuations between station and station indicate westward propagation along the coast. However, the energy level of the 3–6 day period fluctuations suddenly decreased south of the Izu Peninsula. Numerical experiments using a two-layer model were performed to clarify the generation and propagation mechanism of the several-day period fluctuations by periodical wind in fall. The amplitude distributions of observed sea level were qualitatively explained by a coastal-trapped wave (CTW) in the numerical experiment. From the discussions on propagation of a free wave, CTW with the characteristics of a shelf wave generated by the wind along the northeast of the Boso Peninsula was separated into two types of wave at the southeast of the peninsula. One is an internal Kelvin wave with large interface displacement and the other is the shelf wave propagating over the northern part of the Izu Ridge. The sudden decrease in the surface displacement with the 3–6 day period observed at the western stations is considered to be due to the local effect of the wind and phase relation between the internal Kelvin wave and shelf wave.     

Discussion on the “Numerical simulation of wave-induced laminar boundary layers”

The effect of the shear and normal stress free surface boundary conditions on the laminar flow induced by wave propagation is discussed. The approximate form of the boundary conditions, as used by (Lin, P., Zhang, W., 2008. Numerical simulation of wave-induced laminar boundary layers. Coast. Eng. 55, 400–408), is valid only when the free surface slope is mild.     

A hybrid spectral/finite-difference large-eddy simulator of turbulent processes in the upper ocean

A three-dimensional numerical model for large-eddy simulation (LES) of oceanic turbulent processes is described. The numerical formulation comprises a spectral discretization in the horizontal directions and a high-order compact finite-difference discretization in the vertical direction. Time-stepping is accomplished via a second-order accurate fractional-step scheme. LES subgrid-scale (SGS) closure is given by a traditional Smagorinsky eddy-viscosity parametrization for which the model coefficient is derived following similarity theory in the near-surface region. Alternatively, LES closure is given by the dynamic Smagorinsky parametrization for which the model coefficient is computed dynamically as a function of the flow. Validation studies are presented demonstrating the temporal and spatial accuracy of the formulation for laminar flows with analytical solutions. Further validation studies are described involving direct numerical simulation (DNS) and LES of turbulent channel flow and LES of decaying isotropic turbulence. Sample flow problems include surface Ekman layers and wind-driven shallow water flows both with and without Langmuir circulation (LC), generated by wave effects parameterized via the well-known Craik–Leibovich (C–L) vortex force. In the case of the surface Ekman layers, the inner layer (where viscous effects are important) is not resolved and instead is parameterized with the Smagorinsky models previously described. The validity of the dynamic Smagorinsky model (DSM) for parameterizing the surface inner layer is assessed and a modification to the surface stress boundary condition based on log-layer behavior is introduced improving the performance of the DSM. Furthermore, in Ekman layers with wave effects, the implicit LES grid filter leads to LC subgrid-scales requiring ad hoc modeling via an explicit spatial filtering of the C–L force in place of a suitable SGS parameterization.     

不规则波在孔隙介质礁坪上传播过程的波高和增水变化数值研究

采用σ坐标系统下以体积平均的雷诺时间平均方程作为控制方程的三维非静压模型,对随机波浪在带有孔隙介质的岛礁地形上的传播过程进行了模拟,重点分析了礁坪上方波高和增水的变化。通过与多个组次工况的物理模型实验数据进行对比,结果显示,本文模型能很好地模拟波浪在孔隙介质上传播演化的过程,与实验结果吻合程度很高。分析结果表明,相比于光滑底床,孔隙介质的存在造成破碎点附近波高平均下降12%,礁坪上方波高平均下降28%。对于平均水位,孔隙底床条件下的最大减水幅值减小了43%,同时礁坪上方增水幅值上升6%。另外,孔隙率在0.47～0.87范围内变化时,对礁坪上方平均水位的变化基本无影响。     

Non-hydrostatic finite element model for coastal wave processes

This paper presents the application of the depth-integrated non-hydrostatic finite element model, CCHE2D-NHWAVE (Wei and Jia, 2014), for simulating several types of coastal wave processes. Specifically, the model is applied to (1) predict the swash zone hydrodynamics involving wave bore propagation, (2) resolve wave propagation, breaking, and overtopping in fringing reef environments, (3) study the vegetation effect on wave height reduction through both submerged and emergent vegetation zones using the drag force term technique, and (4) simulate tsunami wave breaking in the nearshore zone and inundation in the coastal area. Satisfactory agreement between numerical results and benchmark data shows that the non-hydrostatic model is capable of modeling a wide range of coastal wave processes. Furthermore, thanks to its simple numerical formulation, the non-hydrostatic model also demonstrates a better computation efficiency when comparing with other numerical models.     

礁坪上波浪传播的数值模拟

本文基于具备间断捕捉能力的二阶全非线性Boussinesq数值模型,对规则波和随机波在礁坪地形上的传播变形进行了数值模拟。该模型采用高阶有限体积法和有限差分方法求解守恒格式的控制方程,将波浪破碎视为间断,同时采用静态重构技术处理了海岸动边界问题。重点针对礁坪上波浪传播过程中的波高空间分布和沿程衰减,礁坪上的平均水位变化,以及波浪能量频谱的移动和空间差异等典型水动力现象开展数值计算。将数值结果与实验结果对比,两者吻合情况良好,验证了模型具有良好的稳定性,具备模拟破碎波浪和海-岸动边界的能力,能较为准确地模拟波浪在礁坪地形上的传播过程中发生的各种水动力现象。