A numerical method for the determination of weakly non‐hydrostatic non‐linear free surface wave propagation

A numerical method is described that may be used to determine the propagation characteristics of weakly non‐hydrostatic non‐linear free surface waves over a general, bottom topography. In shallow water of constant undisturbed depth, such waves are equivalent to the familiar cnoidal waves characterized by sharp crests and relatively flat troughs. For a certain range of parameters, these propagate without change of form by virtue of the weakly non‐hydrostatic balance in the vertical momentum equation. Effectively, this counters the tendency for the non‐linearity in a purely hydrostatic theory to lead to a continuously deforming surface wave profile. The realistic representation furnished by cnoidal wave theory of free surface waves in the shallow near‐shore zone has led to its utilization in evaluating their propagation characteristics. Nonetheless, the classic analytical theory is inapplicable to the case of wave propagation over a sloping beach or off‐shore sand bar topography. Under these conditions, a local change in form of the surface wave profile is anticipated before the waves break and knowing this is required in order to evaluate fully the propagation process. The efficacy of the numerical method is first demonstrated by comparing the solution for water of constant depth with the evaluation of the analytical solution expressed in terms of the Jacobian elliptic function cn. The general method described in the paper is then illustrated by experiments to determine the change in profile of weakly non‐hydrostatic non‐linear surface waves propagating over bed forms representative of those found in shallow coastal seas. Copyright © 2001 John Wiley & Sons, Ltd.     

非均匀水流水域波浪的传播变形

将两个不同的、考虑波流相互作用和能量耗散项的、依赖时间变化的双曲型缓坡方程分别化 为一组等价的控制方程组，具体分析了底摩阻项对相对频率和波数矢的影响，从而选择了合 适的数学模型. 将所选择的缓坡方程化为依赖时间变化的抛物型方程，并用ADI法进 行数值求解，建立了缓变水深水域非均匀水流中波浪传播的数值模拟模型. 通过和波流共线 的解析解的比较，说明数值解和解析解相一致. 结合Arthur(1950)水流这一经典算例，定 量地讨论了考虑联合折射-绕射作用后的波数和仅考虑折射作用的波数的差别及其对波高分 布的影响. 在基本同样的条件下, 本文的数值解与他人的计算结果一致.     

随机波浪作用下近岸波流场的数值模拟

结合近岸波浪抛物型缓坡模型和近岸波流场模型,对近岸不规则波浪及其破碎后所产生的流场进行了数值模拟. 在不规则波浪场的模拟中,采用JONSWAP波浪谱对入射单向不规则波浪要素按等分频率法进行离散,应用考虑波浪不规则性和破碎效应的抛物型缓坡方程对波浪场进行数值模拟,并基于抛物型缓坡方程中的波浪势函数等参数计算波浪辐射应力,以波浪辐射应力为主要动力因素基于近岸流数学模型对近岸波浪破碎所产生的近岸流场进行数值模拟,并对数值模拟结果进行了验证. 模拟结果表明该模型可有效地用于研究波浪破碎产生的近岸波流场.      

Numerical study of breaking waves by a two‐phase flow model

A two‐phase flow model, which solves the flow in the air and water simultaneously, is presented for modelling breaking waves in deep and shallow water, including wave pre‐breaking, overturning and post‐breaking processes. The model is based on the Reynolds‐averaged Navier–Stokes equations with the k ?ε turbulence model. The governing equations are solved by the finite volume method in a Cartesian staggered grid and the partial cell treatment is implemented to deal with complex geometries. The SIMPLE algorithm is utilised for the pressure‐velocity coupling and the air‐water interface is modelled by the interface capturing method via a high resolution volume of fluid scheme. The numerical model is validated by simulating overturning waves on a sloping beach and over a reef, and deep‐water breaking waves in a periodic domain, in which good agreement between numerical results and available experimental measurements for the water surface profiles during wave overturning is obtained. The overturning jet, air entrainment and splash‐up during wave breaking have been captured by the two‐phase flow model, which demonstrates the capability of the model to simulate free surface flow and wave breaking problems.Copyright © 2011 John Wiley & Sons, Ltd.     

Improvement of moving particle semi‐implicit method for simulation of progressive water waves

Precise simulation of the propagation of surface water waves, especially when involving breaking wave, takes a significant place in computational fluid dynamics. Because of the strong nonlinear properties, the treatment of large surface deformation of free surface flow has always been a challenging work in the development of numerical models. In this paper, the moving particle semi‐implicit (MPS) method, an entirely Lagrangian method, is modified to simulate wave motion in a 2‐D numerical wave flume preferably. In terms of consecutive pressure distribution, a new and simple free surface detection criterion is proposed to enhance the free surface recognition in the MPS method. In addition, a revised gradient model is deduced to diminish the effect of nonuniform particle distribution and then to reduce the numerical wave attenuation occurring in the original MPS model. The applicability and stability of the improved MPS method are firstly demonstrated by the calculation of hydrostatic problem. It is revealed that these modifications are effective to suppress the pressure oscillation, weaken the local particle clustering, and boost the stability of numerical algorithm. It is then applied to investigate the propagation of progressive waves on a flat bed and the wave breaking on a mild slope. Comparisons with the analytical solutions and experimental results indicate that the improved MPS model can give better results about the profiles and heights of surface waves in contrast with the previous MPS models. Copyright © 2017 John Wiley & Sons, Ltd.     

A compact numerical algorithm for solving the time‐dependent mild slope equation

The mild slope equation has been widely used to describe combined wave refraction and diffraction. In this study, a new numerical algorithm is developed to solve the time‐dependent mild slope equation in a second‐order hyperbolic form. The numerical algorithm is based on a compact and explicit finite difference method that is second‐order accurate in both time and space. The algorithm has the similar structure to the leap‐frog method but is constructed on three time levels for the second‐order time derivative term. The numerical model has the capability of simulating transient wave motion by correctly predicting the speed of wave energy propagation, which is important for the real‐time forecast of the arrival time of storm waves generated in the far field. The model is validated against analytical solution for wave shoaling and experimental data for combined wave refraction and diffraction over a submerged elliptic shoal on a slope (Coastal Eng. 1982; 6 :255). Lastly, the realistic scale Homma's island (Geophys. Mag. 1950; 21 :199) is studied with the use of various wave periods of T = 720s, T = 120 s, and T = 24 s. These wave periods correspond to long, intermediate, and short waves for the given topography, respectively. Comparisons are made between numerical results and existing analytical solutions in terms of the wave amplification around the island, which serves as the indicator for the potential wave runup. Excellent agreements are obtained. The model runs on a PC (Pentium IV 1.8GHz) and the computer capacity allows the computation of a mesh system up to 3000 × 3000, which is equivalent to about 150 × 150 waves or a large area of 540km × 540km for a wave train with the period of T = 60 s. Copyright 2004 John Wiley & Sons, Ltd.     

波浪作用下直立结构物附近强湍动掺气流体运动的数值模拟

波浪破碎卷入气体易对建筑物受力产生压力振荡, 了解波浪作用下建筑物附近掺气水流的运动特性是精确计算建筑物受力的前提. 基于OpenFOAM开源程序包和修正速度入口造波方法建立三维数值波浪水槽, 模型采用S-A IDDES湍流模型进行湍流封闭, 并采用修正的VOF 方法捕捉自由液面, 数值模拟了规则波在1:10的光滑斜坡上与直立结构物的相互作用过程, 重点分析了结构物附近的水动力和掺气水流运动特性. 结果表明, 建立的数值模型能精确地捕捉波浪作用下直立结构物附近的自由液面的变化以及气泡输运过程, 较好地描述气体卷入所形成的气腔形态以及多气腔之间的融合、分裂等过程; 波浪与直立结构物相互作用产生强湍动掺气水流, 其运动过程十分复杂; 掺气流体输运过程中水气界面周围一直伴随着涡的存在, 其中, 气泡的分裂与周围正负涡量剪切作用密切相关, 且其输运轨迹主要受周围流场的影响; 研究揭示了结构物附近湍动能与掺气特性的关系, 发现波浪作用下直立结构物附近湍动能的分布与掺气水流特征参数(气泡数量、空隙率)整体呈现一定的线性关系.      

SPH computation of plunging waves using a 2‐D sub‐particle scale (SPS) turbulence model

The paper presents a 2‐D large eddy simulation (LES) modelling approach to investigate the properties of the plunging waves. The numerical model is based on the smoothed particle hydrodynamics (SPH) method. SPH is a mesh‐free Lagrangian particle approach which is capable of tracking the free surfaces of large deformation in an easy and accurate way. The Smagorinsky model is used as the turbulence model due to its simplicity and effectiveness. The proposed 2‐D SPH–LES model is applied to a cnoidal wave breaking and plunging over a mild slope. The computations are in good agreement with the documented data. Especially the computed turbulence quantities under the breaking waves agree better with the experiments as compared with the numerical results obtained by using the k–ε model. The sensitivity analyses of the SPH–LES computations indicate that both the turbulence model and the spatial resolution play an important role in the model predictions and the contributions from the sub‐particle scale (SPS) turbulence decrease with the particle size refinement. Copyright © 2006 John Wiley & Sons, Ltd.     

Evolvement of tsunami waves on the continental shelves with gentle slope in the China Seas

Potential tsunami generated in the Okinawa Trench or the Manila Trench may attack the southeast coast of China. The continental shelves with extremely gentle slope in the China Seas affect the evolvement of tsunami waves. In this paper, we carry out the simulation of tsunami propagation based on the fully nonlinear and highly dispersive Boussinesq model, which could describe the nonlinearity and dispersion of water waves quite well. So the undulation characters could be well presented. In terms of the real topographies of the East China Sea and the South China Sea, we take some typical profiles to simulate the hypothetical tsunamis generated in the Okinawa Trench and the Manila Trench. Different waveforms in the near shore regions are obtained. The N-shape tsunami waves will evolve into long wave trains, undular bores or solitons near the coastal area. The numerical results of the near shore waveform provide essential conditions for the further studies of tsunami runup and inundation.     

破碎带波浪的数值模拟

基于一组色散关系得到改进的完全非线性Boussinesq方程建立了一个波浪模型可以模拟近岸水域的波浪变浅、破碎以及在海滩上的爬高等多种变形。波浪破碎引起的能量衰减是在动量方程中引入一个在空间和时间上都只作用于波前的涡粘项来模拟。动海岸线边界用窄缝法处理。波浪爬高用非线性浅水方程推导的非破碎波浪在斜坡上爬高的解析解来验证。本模型还模拟了波浪在斜坡上不同类型的破碎变形过程,并将其波高和平均水位的沿程变化和物理模型实验的结果比较,两者符合良好。     

Sensitivity of plume dynamics to the parameterization of vertical mixing

A three‐dimensional primitive equation, baroclinic numerical model incorporating a range of turbulence closure schemes is used to investigate the effects of vertical diffusion of momentum and density upon the spread of a freshwater plume, with particular reference to the Ebro plume. Initial calculations show that there are some differences in the horizontal spread and vertical mixing of the plume when diffusion coefficients are computed from a two‐equation turbulence energy model compared with a one‐equation model. To understand results from the turbulence energy models, the sensitivity of the plume dynamics to variations in the coefficient of vertical eddy viscosity and diffusivity is also considered, with increases in these parameters having a significant effect upon the cross‐shore and along‐shore spread of the plume. Also, increasing these parameters changes the plume characteristics from supercritical to subcritical and reduces the occurrence of meandering and baroclinic instability along the plume's off‐shore edge. However, differences in the southerly spread (the direction of Kelvin wave propagation in the model) of the plume (although not its northerly spread) produced by changes in diffusion coefficients are small compared with the influence of changes in the bottom slope, upon the along‐shore southerly spread of the plume, which moves in the direction of Kelvin wave propagation in the near coastal region. Results from the series of calculations are used as a guide in experimental design, with reference to a planned experiment in the Ebro region involving a coastal HF Radar deployment, as well as off‐shore measurements. Calculations suggest that surface current measurements from a coastal HF Radar, together with a detailed survey of the density field associated with the plume, may be an appropriate, although indirect, means of determining suitable mixing coefficients to use in plume discharge problems. Detailed measurements of water depth variation will also be required. Copyright © 1999 John Wiley & Sons, Ltd.     

Solution of the hyperbolic mild‐slope equation using the finite volume method

A finite volume solver for the 2D depth‐integrated harmonic hyperbolic formulation of the mild‐slope equation for wave propagation is presented and discussed. The solver is implemented on unstructured triangular meshes and the solution methodology is based upon a Godunov‐type second‐order finite volume scheme, whereby the numerical fluxes are computed using Roe's flux function. The eigensystem of the mild‐slope equations is derived and used for the construction of Roe's matrix. A formulation that updates the unknown variables in time implicitly is presented, which produces a more accurate and reliable scheme than hitherto available. Boundary conditions for different types of boundaries are also derived. The agreement of the computed results with analytical results for a range of wave propagation/transformation problems is very good, and the model is found to be virtually paraxiality‐free. Copyright © 2003 John Wiley & Sons, Ltd.     

A fully non‐linear model for three‐dimensional overturning waves over an arbitrary bottom

An accurate three‐dimensional numerical model, applicable to strongly non‐linear waves, is proposed. The model solves fully non‐linear potential flow equations with a free surface using a higher‐order three‐dimensional boundary element method (BEM) and a mixed Eulerian–Lagrangian time updating, based on second‐order explicit Taylor series expansions with adaptive time steps. The model is applicable to non‐linear wave transformations from deep to shallow water over complex bottom topography up to overturning and breaking. Arbitrary waves can be generated in the model, and reflective or absorbing boundary conditions specified on lateral boundaries. In the BEM, boundary geometry and field variables are represented by 16‐node cubic ‘sliding’ quadrilateral elements, providing local inter‐element continuity of the first and second derivatives. Accurate and efficient numerical integrations are developed for these elements. Discretized boundary conditions at intersections (corner/edges) between the free surface or the bottom and lateral boundaries are well‐posed in all cases of mixed boundary conditions. Higher‐order tangential derivatives, required for the time updating, are calculated in a local curvilinear co‐ordinate system, using 25‐node ‘sliding’ fourth‐order quadrilateral elements. Very high accuracy is achieved in the model for mass and energy conservation. No smoothing of the solution is required, but regridding to a higher resolution can be specified at any time over selected areas of the free surface. Applications are presented for the propagation of numerically exact solitary waves. Model properties of accuracy and convergence with a refined spatio‐temporal discretization are assessed by propagating such a wave over constant depth. The shoaling of solitary waves up to overturning is then calculated over a 1:15 plane slope, and results show good agreement with a two‐dimensional solution proposed earlier. Finally, three‐dimensional overturning waves are generated over a 1:15 sloping bottom having a ridge in the middle, thus focusing wave energy. The node regridding method is used to refine the discretization around the overturning wave. Convergence of the solution with grid size is also verified for this case. Copyright © 2001 John Wiley & Sons, Ltd.     

Standing waves for a two-way model system for water waves

In this paper, we prove the existence of a large family of nontrivial bifurcating standing waves for a model system which describes two-way propagation of water waves in a channel of finite depth or in the near shore zone. In particular, it is shown that, contrary to the classical standing gravity wave problem on a fluid layer of finite depth, the Lyapunov–Schmidt method applies to find the bifurcation equation. The bifurcation set is formed with the discrete union of Whitney's umbrellas in the three-dimensional space formed with 3 parameters representing the time-period and the wave length, and the average of wave amplitude.     

Experimental study and numerical simulation of mean longshore current for mild slope

In this paper, experimental and numerical studies were performed to investigate the characteristics of longshore current under two mild slopes, the results of which may complement the existing studies, which have mainly focused on steep slopes. The experimental results revealed that the average velocity distribution of the longshore current was significantly different under the two different mild slopes. Under the slope of 1:100, the distribution of the averaged longshore current velocity had a downward concave trend at the nearshore side, while under the slope of 1:40 the trend became convex. In addition, the analysis of the numerical results showed that the distribution of the averaged longshore current velocity was affected by the distribution of the wave height within the surf zone and the bottom friction equations, with the influence of the latter being more significant. For the slope of 1:100, the cross-shore variability of alongshore variability can be calculated using the flow-type bottom friction equation, while for the slope of 1:40, the wave-type bottom friction equation can be used instead. Finally, the secondary breaking wave heights for mild slopes and the distribution of wave set-up are also shown in the study.     

Large-scale vorticity generation by breakers in shallow and deep water

Water wave breaking is of considerable importance in the transfer of momentum, and in other transfers, between the atmosphere and oceans. Typically breaking occurs on deep water as events that have finite duration and finite spatial extent. Near shore lines most of the water motions are dominated by breaking waves. Recent work on the generation of vorticity by breaking waves and bores in the surf zone on beaches is considered and typical vortical structures are briefly discussed. Consideration of deep water breaking leads to the proposal that the end result of a breaking event in deep water may be a coherent structure within the resulting current field. Such a structure is topologically equivalent to half a vortex ring.     

Depth‐integrated free‐surface flow with a two‐layer non‐hydrostatic formulation

This paper presents the derivation of a depth‐integrated wave propagation and runup model from a system of governing equations for two‐layer non‐hydrostatic flows. The governing equations are transformed into an equivalent, depth‐integrated system, which separately describes the flux‐dominated and dispersion‐dominated processes. The depth‐integrated system reproduces the linear dispersion relation within a 5 error for water depth parameter up to kd = 11, while allowing direct implementation of a momentum conservation scheme to model wave breaking and a moving‐waterline technique for runup calculation. A staggered finite‐difference scheme discretizes the governing equations in the horizontal dimension and the Keller box scheme reconstructs the non‐hydrostatic terms in the vertical direction. An semi‐implicit scheme integrates the depth‐integrated flow in time with the non‐hydrostatic pressure determined from a Poisson‐type equation. The model is verified with solitary wave propagation in a channel of uniform depth and validated with previous laboratory experiments for wave transformation over a submerged bar, a plane beach, and fringing reefs. Copyright © 2011 John Wiley & Sons, Ltd.     

A new numerical model for simulations of wave transformation,breaking and long‐shore currents in complex coastal regions

In this paper, we propose a model based on a new contravariant integral form of the fully nonlinear Boussinesq equations in order to simulate wave transformation phenomena, wave breaking, and nearshore currents in computational domains representing the complex morphology of real coastal regions. The aforementioned contravariant integral form, in which Christoffel symbols are absent, is characterized by the fact that the continuity equation does not include any dispersive term. A procedure developed in order to correct errors related to the difficulties of numerically satisfying the metric identities in the numerical integration of fully nonlinear Boussinesq equation on generalized boundary‐conforming grids is presented. The Boussinesq equation system is numerically solved by a hybrid finite volume–finite difference scheme. The proposed high‐order upwind weighted essentially non‐oscillatory finite volume scheme involves an exact Riemann solver and is based on a genuinely two‐dimensional reconstruction procedure, which uses a convex combination of biquadratic polynomials. The wave breaking is represented by discontinuities of the weak solution of the integral form of the nonlinear shallow water equations. The capacity of the proposed model to correctly represent wave propagation, wave breaking, and wave‐induced currents is verified against test cases present in the literature. The results obtained are compared with experimental measures, analytical solutions, or alternative numerical solutions. Copyright © 2015 John Wiley & Sons, Ltd.     

基于VPM-THINC/QQ模型的波浪高保真模拟

为实现波浪传播的高保真数值模拟,采用包含单元均值和点值(volume-average/point-value method,VPM)的有限体积法求解纳维-斯托克斯方程和具有二次曲面性质和高斯积分的双曲正切函数(THINC method with quadratic surface representation and Gaussian quadrature,THINC/QQ)方法来重构自由面,建立以开源求解库OpenFOAM底层函数库为基础的VPM-THINC/QQ模型. 在本模型中添加推板造波法实现波浪的产生功能,采用松弛法实现消波功能,构建高精度黏性流数值波浪水槽. 分别采用VPM-THINC/QQ模型和InterFoam求解器(OpenFOAM软件包中广泛使用的多相流求解器)开展规则波的数值模拟,重点探究网格大小和时间步长等因素对波浪传播过程的影响,定量地分析波高衰减程度;为验证本模型的适应性,对长短波进行模拟. 结果表明,在相同网格大小或时间步长条件下,VPM-THINC/QQ模型的预测结果与参考值吻合较好,波高衰减较少,且无相位差,在波浪传播过程的模拟中呈现出良好的保真性. 本文工作 为波浪传播的模拟研究提供了一种高精度的黏性数值波浪水槽模型.      

Hybrid finite‐volume finite‐difference scheme for the solution of Boussinesq equations

A hybrid scheme composed of finite‐volume and finite‐difference methods is introduced for the solution of the Boussinesq equations. While the finite‐volume method with a Riemann solver is applied to the conservative part of the equations, the higher‐order Boussinesq terms are discretized using the finite‐difference scheme. Fourth‐order accuracy in space for the finite‐volume solution is achieved using the MUSCL‐TVD scheme. Within this, four limiters have been tested, of which van‐Leer limiter is found to be the most suitable. The Adams–Basforth third‐order predictor and Adams–Moulton fourth‐order corrector methods are used to obtain fourth‐order accuracy in time. A recently introduced surface gradient technique is employed for the treatment of the bottom slope. A new model ‘HYWAVE’, based on this hybrid solution, has been applied to a number of wave propagation examples, most of which are taken from previous studies. Examples include sinusoidal waves and bi‐chromatic wave propagation in deep water, sinusoidal wave propagation in shallow water and sinusoidal wave propagation from deep to shallow water demonstrating the linear shoaling properties of the model. Finally, sinusoidal wave propagation over a bar is simulated. The results are in good agreement with the theoretical expectations and published experimental results. Copyright © 2005 John Wiley & Sons, Ltd.