Interaction of cnoidal waves with an array of vertical concentric porous cylinders

An array of large concentric porous cylinder arrays is mounted in shallow water exposed to cnoidal waves. The interactions between waves and cylinders are studied theoretically using an eigenfunction expansion approach. Semi-analytical solutions of hydrodynamic loads and wave run-up on each cylinder are obtained using first approximation to cnoidal waves. The square array configuration of four-legged identical concentric porous cylinder is investigated in present study. Numerical results reveal the variation of dimensionless wave force and wave run-up on individual cylinder with angle of incidence, porosity parameter, spacing between outer and inner cylinders, spacing between concentric porous cylinders and wave parameter. Different mechanism of wave force is found under different range of scattering parameter.     

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.     

The Nonlinear Wave Force on a Circular Cylinder in Shallow Water

Based on the 1st order cnoidal wave theory, the nonlinear wave diffraction around a circular cylinder in shallow water is studied in this paper. The equation of the wave surface around the cylinder is formulated and by using this formula the wave surface elevation on the cylinder surface can be obtained. In this paper, the formula for calculating the cnoidal wave force on a circular cylinder is also derived. For the wave conditions which are often encountered in practical engineering designs, the ratios of the nonlinear wave forces to the linear wave forces are calculated, and the results are plotted in this paper for design purposes. In order to verify the theoretical results, model tests are conducted. After comparing the test results with the theoretical ones, it is concluded that, in shallow water, for the case of T g / d~(1/2) > 8-10 and H / d > 0.3, the cnoidal wave theory should be used to calculate the wave action on a cylindrical pier.     

Solitary wave interactions with an array of two vertical cylinders

This paper addresses a numerical investigation of nonlinear waves interactions with an array of two surface-piercing vertical cylinders and the corresponding nonlinear hydrodynamic loads on each individual cylinder. The primary interest of this study is concentrated on the problem of three-dimensional scattering of solitary waves by cylinder arrays and the nonlinear interactions between scattered waves. The theoretical model adopted for simulation is the generalized Boussinesq two-equation model. The boundary-fitted coordinate transformation and multiple-grid technique are utilized here to simplify the computation domain and to facilitate the applications of the boundary conditions on the cylinder surfaces. The velocity potential, free-surface elevation and subsequent evolution of the scattered wave field are numerically evaluated. The hydrodynamic forces on each cylinder during wave impact are also determined. A study of the sheltering effect by the neighboring structures on wave loads is conducted. It is found that the presence of the neighboring cylinder has shown significant influence on the wave loads and the scattering of the primary incident waves. For two transversely arranged cylinders, the transverse force coefficient increases as the separation distance decreases.     

Nonlinear effect on inertia component of wave forces on a cylinder

Ｎｏｎｌｉｎｅａｒｅｆｆｅｃｔｏｎｉｎｅｒｔｉａｃｏｍｐｏｎｅｎｔｏｆｗａｖｅｆｏｒｃｅｓｏｎａｃｙｌｉｎｄｅｒ￥ＬｉＹａｎｂａｏａｎｄＳｏｎｇＲｅｎｇ（ＤｅｐａｒｔｍｅｎｔｏｆＨｙｄｒａｕｌｉｃＥｎｇｉｎｅｅｒｉｎｇ，ＴｉａｎｊｉｎＵｎｉｖｅｒｓｉ...     

A Boussinesq model for simulating wave and current interaction

A new formulation of a pair of Boussinesq equations for three-dimensional nonlinear dispersive shallow-water waves is presented. This set of model equations permits spatial and temporal variations of the bottom topography and the presence of uniform currents. The newly derived equations are used to simulate the propagation of cnoidal waves and their interactions with a uniform current in a wave channel. The modified Euler's predictor-corrector algorithm for time advancing and a central difference representation for the space derivatives are applied to the computation of the basic equations. A set of open boundary conditions is developed to effectively transmit the cnoidal waves out of the computational domain. It is found that, as expected, the wave length decreases with an opposing current and increases with a following current. The wave height increases in magnitude with an opposing current and decreases with a following current. The Mach reflection due to oblique cnoidal waves propagating into an open channel with an opposing current is also investigated. Due to the opposing current, the wave patterns are compressed into smaller saddle-like regions in comparison with the Mach reflection without current effect.     

椭圆余弦内波及其对墩柱作用数值模拟

基于FLUENT软件建立数值波浪水槽,研究椭圆余弦内波及其对墩柱的作用。椭圆余弦内波的生成采用推板造波方法,自由面捕获采用VOF方法。首先,模拟了椭圆余弦内波的生成,比较了不同周期和波高下椭圆余弦内波的波浪形态。然后,在特定的周期和波高条件下模拟了三维椭圆余弦内波对单个墩柱及多个敦柱的作用;分析了各墩柱上受到的惯性力和粘性力的变化趋势,并对不同墩柱下的总波浪力进行了对比;得出了墩柱上受到的波浪力也呈现周期性变化并且在波浪前进方向上后墩柱的受力小于靠前墩柱的结论。     

Generalized boussinesq model for periodic non-linear shallow-water waves

This paper presents the development of a generalized Boussinesq (gB) model for the periodic non-linear shallow-water waves. An incident cnoidal wave solution for the gB model is derived and applied to the wave simulation. A set of radiation boundary conditions is also established to transmit effectively the cnoidal waves out of the computational domain. The classical solutions of the second-order cnoidal waves are discussed within the content of the KdV equation and the generalized Boussinesq equations. An Euler's predictor-corrector finite-difference algorithm is used for numerical computation. The propagation of normally incident cnoidal waves in a channel is studied. The simulated wave profiles agree well with the analytical results. The temporal and spatial evolution of an obliquely incident cnoidal wave is also modelled. The phenomenon of Mach reflection is discussed.     

Wave types of landslide generated impulse waves

Subaerial landslide generated impulse waves were investigated in a prismatic wave channel. Seven governing parameters, namely the still water depth, slide impact velocity, slide thickness, bulk slide volume, bulk slide density, slide impact angle, and grain diameter, were systematically varied. The generated impulse waves are nonlinear, intermediate- to shallow-water waves involving a small to considerable fluid mass transport. The Stokes wave, cnoidal wave, solitary wave, and bore theories were applied to describe the observed maximum waves. The theoretical and observed features of these four wave types are highlighted. A diagram allows to predict the wave type directly as a function of the slide parameters, the slide impact angle, and the still water depth.     

Internal generation of waves for extended Boussinesq equations

It is studied whether the mass transport or energy transport is the proper viewpoint for internally generating waves in the extended Boussinesq equations of Nwogu [J. Waterw., Port, Coastal Ocean Eng. 119 (1993) 618–638]. Numerical solutions of the Boussinesq equations with the internal generation of sinusoidal waves show that the energy transport approach yields the required wave amplitude properly while the mass transport approach yields wave amplitude different from the required one by the ratio of phase velocity to energy velocity. The waves which pass through the wave generation point do not cause any numerical distortion while the incident waves are generated. The technique of internal generation of waves shows its capability of generating nonlinear cnoidal waves as well as linear sinusoidal waves.     

An experimental investigation of the velocity field under cnoidal waves over the asymmetric rippled bed

1 IntroductionIn coastal areas a ubiquitous phenomenon is theformation of ripples in the seabed. It is now widelyaccepted that the flow and sediment transport overseabed are vital in relation to erosion, surface wavedissipation and pollution dispersion et…     

驱动非线性浅水波的行波特征研究

采用带有外界强迫效应的浅水动力学模式研究非线性波动、获得了依赖于外界输入形式的驱动水波的行波解。研究结果表明,驱动水波仍具有非线性波动的一般性质,而当外界强迫波速与水波固有速度一致时,水波出现共振效应,并且外界强迫孤立子将导致驱动水波孤立子产生。     

Stem waves along vertical wall due to random wave incidence

This study investigates stem waves, propagating along a vertical wall, due to obliquely incident random waves through laboratory experiments and numerical simulations. Attention is paid to the difference or similarity between the stem waves due to periodic waves and random waves, the nonlinear and linear characteristics, and the effect of wave breaking on the evolution of stem waves. The following were found from this study: as the incident angle of waves become large or the nonlinearity of the incident waves become small, the significant stem wave height, normalized by the incident significant wave height, becomes large. This tendency is the same as that generated by the Stokes waves or cnoidal waves. However, regardless of the nonlinearity of incident waves, the width of stem waves is almost the same. This is a different point between the stem waves due to periodic and random waves. The wave breaking suppresses the growth of the stem waves.     

三阶Stokes型随机波浪载荷谱研究

应用非线性谱分析理论，对三阶Stokes型随机波浪载荷谱进行了分析，将波面方程及海水质点的水平速度用一阶波面的非线性组表示，导出了随机波浪谱的表达式。为了便于求解随机波浪的载荷谱，将阻力项展开为幂级数式，并应用非线性谱分析理论，确定了幂级数的系数，进而将波浪载荷表示为一阶波面及其导数的非线性组合，最后得出波浪载荷谱密度的表达式，并应用数值分析方法，得出单位桩柱波浪力及总波浪力谱密度。     

A coupled-mode system with application to nonlinear water waves propagating in finite water depth and in variable bathymetry regions

A non-linear coupled-mode system of horizontal equations is presented, modelling the evolution of nonlinear water waves in finite depth over a general bottom topography. The vertical structure of the wave field is represented by means of a local-mode series expansion of the wave potential. This series contains the usual propagating and evanescent modes, plus two additional terms, the free-surface mode and the sloping-bottom mode, enabling to consistently treat the non-vertical end-conditions at the free-surface and the bottom boundaries. The present coupled-mode system fully accounts for the effects of non-linearity and dispersion, and the local-mode series exhibits fast convergence. Thus, a small number of modes (up to 5–6) are usually enough for precise numerical solution. In the present work, the coupled-mode system is applied to the numerical investigation of families of steady travelling wave solutions in constant depth, corresponding to a wide range of water depths, ranging from intermediate depth to shallow-water wave conditions, and its results are compared vs. Stokes and cnoidal wave theories, as well as with fully nonlinear Fourier methods. Furthermore, numerical results are presented for waves propagating over variable bathymetry regions and compared with nonlinear methods based on boundary integral formulation and experimental data, showing good agreement.     

波浪动力因素变化对沙质岸滩演变的影响

通过波浪水槽实验,对海平面变化造成的波浪动力因素改变引起的沙质岸滩形态响应开展机理性研究。实验采用1∶10单一沙质斜坡概化岸滩,利用3种不同实验水深模拟海平面变化,考虑椭圆余弦波、非规则波、规则波和孤立波4种类型波浪作用。实验对波浪在斜坡上的传播变形、破碎、上爬和回落过程的波高及波浪作用后的岸滩地形进行了测量。实验结果表明,椭圆余弦波、规则波和非规则波作用下,平衡岸滩呈现出滩肩形态,孤立波作用下则呈沙坝形态。海平面上升造成波浪动力增强,沙质岸滩平衡剖面形状基本保持不变向岸平移,槽谷、滩肩、沙坝位置以及岸线蚀退距离,均呈现出良好规律性。     

Numerical simulation of wave-induced laminar boundary layers

The three-dimensional numerical model with σ-coordinate transformation in the vertical direction is applied to the simulation of surface water waves and wave-induced laminar boundary layers. Unlike most of the previous investigations that solved the simplified one-dimensional boundary layer equation of motion and neglected the interaction between boundary layer and outside flow, the present model solves the full Navier–Stokes equations (NSE) in the entire domain from bottom to free surface. A non-uniform mesh system is used in the vertical direction to resolve the thin boundary layer. Linear wave, Stokes wave, cnoidal wave and solitary wave are considered. The numerical results are compared to analytical solutions and available experimental data. The numerical results agree favorably to all of the experimental data. It is found that the analytical solutions are accurate for both linear wave and Stokes wave but inadequate for cnoidal wave or solitary wave. The possible reason is that the existing analytical solutions for cnoidal and solitary waves adopt the first-order approximation for free stream velocity and thus overestimate the near bottom velocity. Besides velocity, the present model also provides accurate results for wave-induced bed shear stress.     

Nonlinear wave interaction with a vertical circular cylinder. Part II: Wave run-up

Theoretical results for second-order wave run-up around a large diameter vertical circular cylinder are compared to results of 22 laboratory experiments conducted in regular nonlinear waves. In general, the second-order theory explains a significant portion of the nonlinear wave run-up distribution measured at all angles around the cylinder. At the front of the cylinder, for example, measured maximum run-up exceeds linear theory by 44% on average but exceeds the nonlinear theory by only 11% on average. In some cases, both measured run-up and the second-order theory exceed the linear prediction by more than 50%. Similar results are found at the rear of the cylinder where the second-order theory predicts a large increase in wave amplitude for cases where the linear diffraction theory predicts little or no increase. Overall, the nonlinear diffraction theory is found to be valid for the same relative depth and wave steepness conditions applicable to Stokes second-order plane-wave theory. In the last section of the paper, design curves are presented for estimating the maximum second-order wave run-up for a wide range of conditions in terms of the relative depth, relative cylinder size, and wave steepness.     

海底边坡稳定性分析中波浪力的求解

针对浅水区波浪的非线性特性,提出了在海底边坡稳定性分析中应用椭圆余弦波理论来研究波浪力的问题,利用非线性弥散关系建立了新的适用于整个水深范围的椭圆余弦波的近似求解方法.结合工程实例,确立了海底边坡波浪力的计算步骤,并编制了计算程序.