Numerical Simulation of Non-Linear Wave Propagation in Waters of Mildly Varying Topography with Complicated Boundary

In this paper, the characteristics of different forms of mild slope equations for non-linear wave are analyzed, and new non-linear theoretic models for wave propagation are presented, with non-linear terms added to the mild slope equations for non-stationary linear waves and dissipative effects considered. Numerical simulation models are developed of non-linear wave propagation for waters of mildly varying topography with complicated boundary, and the effects are studied of different non-linear corrections on calculation results of extended mild slope equations. Systematical numerical simulation tests show that the present models can effectively reflect non-linear effects.     

Propagation and interaction of non-linear surface and internal waves in a two-layer fluid

This paper focuses on the study of linear and non-linear surface and internal waves, in a complete setting, using a two-layer model of a stratified fluid. The respective Korteweg-de Vries evolutionary equations have been obtained, analysed, and compared with the rigid lid model data. Boussinesq-type equations have been derived for the interacting modes pertaining to one type and to different types. It is shown that in addition to the known mechanisms of interaction between internal and surface waves, interaction between long non-linear baroclinic modes and barotropic modes, propagating in the same direction, is likely in such a system.Translated by Vladimir A. Puchkin.     

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.     

Gerris在海洋大振幅内孤立波数值模拟中的应用

本文运用基于自适应网格的流体动力学开源软件Gerris,来建立基于Boussinesq近似下的二维不可压缩Euler方程组的数值模型,以模拟不同层化条件下稳定状态的完全非线性大振幅内孤立波。文中比较了完全非线性的用Gerris实现的Euler模型与弱非线性的KdV理论模型在刻画大振幅内孤立波结构及特征参数上的差异,说明在模拟大振幅内孤立波时,高阶非线性不应忽略。Euler模型模拟结果表明,完全非线性大振幅内孤立波的等密度面半宽度随深度变化,这使得基于KdV方程解析解、利用卫星SAR(Synthetic Aperture Radar)图像提取内孤立波极值间距来反演内波振幅的可行性存疑,需要重新评估。此外,本文用两组实测数据验证了用Gerris实现的Euler模型模拟大振幅内波的有效性。     

An investigation into parametric roll resonance in regular waves using a partly non-linear numerical model

A partly non-linear time-domain numerical model is used for the prediction of parametric roll resonance in regular waves. The ship is assumed to be a system with four degrees of freedom, namely, sway, heave, roll and pitch. The non-linear incident wave and hydrostatic restoring forces/moments are evaluated considering the instantaneous wetted surface whereas the hydrodynamic forces and moments, including diffraction, are expressed in terms of convolution integrals based on the mean wetted surface. The model also accounts for non-potential roll damping expressed in an equivalent linearised form. Finally, the coupled equations of motion are solved in the time-domain referenced to a body fixed axis system.This method is applied to a range of hull forms, a post-Panamax C11 class containership, a transom stern Trawler and the ITTC-A1 containership, all travelling in regular waves. Obtained results are validated by comparison with numerical/experimental data available in the literature. A thorough investigation into the influence of the inclusion of sway motion is conducted. In addition, for the ITTC-A1 containership, an investigation is carried out into the influence of tuning the numerical model by modifying the numerical roll added inertia to match that obtained from roll decay curves.     

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.     

Numerical experiments on estimating the effect of non-linearity on long wave propagation over a seamount

In the context of the nonlinear, theory of long waves, we solve, using numerical techniques, a non-stationary problem on the barotropic wave propagation over an isolated bottom elevation. The time interval during which the wave process over an obstacle sets in is determined. The relative contributions of the non-linear terms to the model equations have been analysed. A practical criterion of applicability of the linear approximation to the problem examined has been formulated.Translated by Vladimir A. Puchkin.     

The generation of non-linear internal waves by a barotropic tide in the area of a solitary bottom elevation

A mathematical model is suggested for calculating current, density, and pressure fields in the area of a solitary bottom rise (seamount). The model is based on a set of non-linear differential equations governing the motion of an inviscid continuously stratified fluid. The algorithm for solving the equations is based on the splitting technique. The model has been used to compute non-linear baroclinic waves generated by a barotropic tide in the seamount area. Translated by Vladimir A. Puchkin.     

两相流模型在南海内孤立波与沙波相互作用中的应用

为研究内孤立波与沙波的相互作用,本文对基于OpenFOAM的SedWaveFoam求解器进行改进,建立了内孤立波-泥沙运动欧拉两相流模型。在利用试验资料对模型进行验证的基础上,在南海北部典型代表性条件下,模拟分析了500 m水深位置沙波床面上内孤立波作用下的水动力变化和泥沙运动。结果表明,内孤立波逐渐离开沙波时,海底沙波背流面处出现与内孤立波背景流速反向的流速,在内孤立波导致的流场作用下,沙波床面上的泥沙悬起并运动到床面以上的水体中。振幅100 m的内孤立波可以导致床面以上14 m高的位置处出现约0.07 kg/m³的悬沙浓度。     

Observation and analysis of solitary internal waves in the coastal zone of the Sea of Okhotsk

Experimental data on the measurement of solitary internal waves in the coastal zone of the Sea of Okhotsk are reported. The non-linear and dispersion coefficients of the Korteweg-de Vries (KdV) equation are determined for weakly non-linear disturbances of the first mode. The shapes and parameters of solitary waves are compared with the corresponding parameters of solitons. It is shown that at least ten observed solitary waves are similar to the KdV solitons. Histograms of the distribution of the amplitudes, duration, and spatial dimensions are plotted for a large number of solitary waves.Translated by Mikhail M. Trufanov.     

基于FLUENT的内孤立波质量源造波方法

为研究内孤立波质量源数值造波方法,本文采用两个点源形式的质量源,分别放置于两层流体的上下层中作为内孤立波激发源。推导源项表达式,从不可压缩流体的Navier-Stokes方程出发,结合内孤立波Kd V、e Kd V理论,基于商业软件FLUENT发展了一种内孤立波质量源数值造波方法。通过数值模拟,分析了质量源造波过程中内孤立波的生成过程,并将数值模拟结果同理论及实验作对比。结果表明:基于此方法生成的内孤立波波形、波高及波致水平速度与理论及实验吻合度较好,该方法是可行的,并且耗时短、效率高。     

Energy conservation issues in sigma-coordinate free-surface ocean models

This paper focuses on the energy conservation properties of a hydrostatic, Boussinesq, coastal ocean model using a classic finite difference method. It is shown that the leapfrog time-stepping scheme, combined with the sigma-coordinate formalism and the motions of the free surface, prevents the momentum advection from exactly conserving energy. Because of the leapfrog scheme, the discrete form of the kinetic energy depends on the product of velocities at odd and even time steps and thus appears to be possibly negative when high-frequency modes develop. Besides, the study of the energy balance clarifies the numerical choices made for the computation of mixing processes. The time-splitting technique used to reduce the computation costs associated to the resolution of surface waves leads to the well-known external and internal mode equations. We show that these equations do not conserve energy if the coupling of these two modes is forward in time. Even if non-linear terms are negligible, this shortcoming can be significant regarding the pressure gradient term ‘frozen’ over a baroclinic time step. An alternative energy-conserving time-splitting technique is proposed in this paper. Discussion and conclusions are conducted in the light of a set of numerical experiments dedicated to surface and internal gravity waves.     

A coupled-mode model for water wave scattering by horizontal, non-homogeneous current in general bottom topography

A coupled-mode model is developed for treating the wave–current–seabed interaction problem, with application to wave scattering by non-homogeneous, steady current over general bottom topography. The vertical distribution of the scattered wave potential is represented by a series of local vertical modes containing the propagating mode and all evanescent modes, plus additional terms accounting for the satisfaction of the free-surface and bottom boundary conditions. Using the above representation, in conjunction with unconstrained variational principle, an improved coupled system of differential equations on the horizontal plane, with respect to the modal amplitudes, is derived. In the case of small-amplitude waves, a linearised version of the above coupled-mode system is obtained, generalizing previous results by Athanassoulis and Belibassakis [J Fluid Mech 1999;389:275–301] for the propagation of small-amplitude water waves over variable bathymetry regions. Keeping only the propagating mode in the vertical expansion of the wave potential, the present system reduces to an one-equation model, that is shown to be compatible with mild-slope model concerning wave–current interaction over slowly varying topography, and in the case of no current it exactly reduces to the modified mild-slope equation. The present coupled-mode system is discretized on the horizontal plane by using second-order finite differences and numerically solved by iterations. Results are presented for various representative test cases demonstrating the usefulness of the model, as well as the importance of the first evanescent modes and the additional sloping-bottom mode when the bottom slope is not negligible. The analytical structure of the present model facilitates its extension to fully non-linear waves, and to wave scattering by currents with more general structure.     

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.     

内孤立波作用下Spar平台动力响应特性

以三类内孤立波理论(Kd V、e Kd V和MCC)的适用性条件为依据,采用Morison和傅汝德-克雷洛夫公式分别计算Spar平台内孤立波水平力和垂向力,结合时域有限位移运动方程,建立了有限深两层流体中内孤立波与带分段式系泊索Spar平台相互作用的理论模型。以东沙群岛某海域实测内孤立波为对象,数值分析了在内孤立波作用下某经典式Spar平台的内孤立波动态载荷、运动响应及其系泊张力的变化特性。研究表明,内孤立波不仅会对Spar平台产生突发性冲击载荷,使其产生大幅度水平漂移运动,而且还会使其系泊张力显著增大。因此,在Spar平台等深海平台的设计应用中,内孤立波的影响不可忽视。     

内孤立波中半潜平台动力响应特性

基于内孤立波mKdV理论,采用时域有限位移运动方程,结合改进的Morison公式,研究了两层流体中内孤立波与带分段式悬链系泊约束半潜平台的相互作用问题。针对东沙群岛南部海域附近某实测内孤立波特征参数,计算分析了在该内孤立波作用下SEDCO-700型半潜平台的内孤立波载荷、运动响应及其系泊张力的变化特性。研究表明,内孤立波不仅会对半潜平台产生突发性冲击载荷作用,使其产生大幅度水平漂移运动,并导致其系泊张力显著增大,因此在半潜平台等深海平台的设计与应用中,内孤立波的影响是不可忽视的。     

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.     

Study of intensive internal waves in the shelf zone of Morocco

Conditions for generating a train of intensive internal waves and a solitary internal wave, as well as the evolution and dissipation of shelf waves have been studied using temperature measurements at six buoy moorings and an array of distributed temperature meters towed in the Morocco shelf zone. Waves of each type have been numerically simulated. The paper's focus is the generation of a baroclinic tide in a shelf zone and the occurrence of a packet of intensive internal waves on the back slope of a baroclinic wave. Besides, it studies the mechanism responsible for the generation of a solitary internal wave of soliton type. Thein situ data and the model data are matched up and shown to be consistent, in terms of quality and quantity. Translated by Vladimir A. Puchkin.     

Hydroelastic analysis of floating structures with liquid tanks and comparison with experimental tests

In this study, we develop a numerical method for a 3D linear hydroelastic analysis of floating structures with liquid tanks subjected to surface regular water waves and compare the numerical results with experimental tests. Considering direct couplings among structural motion, sloshing, and water waves, a mathematical formulation and a numerical method extended from a recent work [1] are developed. The finite element method is employed for the floating structure and internal fluid in tanks, and the boundary element method is used for the external fluid. The resulting formulation completely incorporates all the interaction terms including hydrostatic stiffness and the irregular frequency effect is removed by introducing the extended boundary integral equations. Through various numerical tests, we verify the proposed numerical method. We also performed 3D hydroelastic experimental tests of a floating production unit (FPU) model in an ocean basin. The measured dynamic motions are compared with the numerical results obtained using the proposed method.