Interaction of higher-order water waves with uniform currents

The interaction between current-free higher-order water waves with a wave-free uniform current normal to the wave crests is considered. The combined wave-current motion resulting from the interaction is assumed stable and irrotational. The velocity potential, dispersion relation, the particle kinematics and pressure distribution up to the third order in wave amplitude are developed. The conservation of mean mass, momentum and energy, together with the dispersion relation on the free surface are used to derive a set of four nonlinear equations, through which the relationship between wave-free current, current-free wave and the combined wave-current parameters is established. Numerical results for a range of current values are also presented.     

Effect of vertically logarithmic steady currents on shallow surface waves

The combined wave-current flow has been solved by researchers by assuming wave over either depthwise constant or linear current profile. Some complicated nonlinear current profiles have also been considered to simulate various shear currents. We consider a nonlinear current vertically logarithmic in nature and examine its interaction with a periodic surface wave. The Navier-Stokes equations for incompressible flow are solved for the current part and by using periodic boundary conditions. The effect of logarithmic current on wave components is assessed. The corresponding celerity and dispersion equation yields a close-form solution for the shallow-wave approximation. Several comparative trends between wave-only, wave with log current, and wave with constant current for the wave following/opposing these currents have been discussed. The flow properties of the first order are presented which can be applicable to the real inland and coastal flows, where progressive waves are ubiquitous over a depthwise logarithmic current. The work is further extended to the second-order semiempirical wave component by using past experimental data on the wave spectrum of combined flow. Published in Morskoi Gidrofizicheskii Zhurnal, No. 3, pp. 20–40, May–June, 2008.     

Current-depth refraction of water waves

Surface gravity waves propagating on large-scale currents over a gently sloping sea bed are considered. Assuming irrotational flow a general proof is given of the existence of a mean energy level, leading directly to the current wave set-down. The complete set of conservation equations describing combined current-depth refraction is presented and solved in two special ssituations. Also a case with a rotational current is considered. Dissipation is neglected.     

Explicit wave-averaged primitive equations using a generalized Lagrangian mean

The generalized Langrangian mean theory provides exact equations for general wave–turbulence–mean flow interactions in three dimensions. For practical applications, these equations must be closed by specifying the wave forcing terms. Here an approximate closure is obtained under the hypotheses of small surface slope, weak horizontal gradients of the water depth and mean current, and weak curvature of the mean current profile. These assumptions yield analytical expressions for the mean momentum and pressure forcing terms that can be expressed in terms of the wave spectrum. A vertical change of coordinate is then applied to obtain glm2z-RANS equations with non-divergent mass transport in cartesian coordinates. To lowest order, agreement is found with Eulerian mean theories, and the present approximation provides an explicit extension of known wave-averaged equations to short-scale variations of the wave field, and vertically varying currents only limited to weak or localized profile curvatures. Further, the underlying exact equations provide a natural framework for extensions to finite wave amplitudes and any realistic situation. The accuracy of the approximations is discussed using comparisons with exact numerical solutions for linear waves over arbitrary bottom slopes, for which the equations are still exact when properly accounting for partial standing waves. For finite amplitude waves it is found that the approximate solutions are probably accurate for ocean mixed layer modelling and shoaling waves, provided that an adequate turbulent closure is designed. However, for surf zone applications the approximations are expected to give only qualitative results due to the large influence of wave nonlinearity on the vertical profiles of wave forcing terms.     

波流共同作用下海底人工边坡动态响应分析

为了研究波流共同荷载作用下开挖基槽附近海床动态响应和液化破坏情况,提出一个二维耦合计算模型,采用雷诺时均纳维-斯托克斯（RANS）方程描述波浪运动情况,通过设定侧边界条件实现稳定流场。海床部分通过求解Biot固结方程,得到波流荷载下海床中的应力和位移情况。将模型计算结果与水槽试验数据和解析解进行比较,验证了波流模型和海床模型的有效性。在此模型基础上,分析得到了开挖之后海床新的应力和固结状态。同时,通过参数分析得到了波流耦合情况下波浪形态的变化,以及海流对海床液化情况和孔压情况的影响。最后,通过线性回归计算得到最大液化深度与流速的拟合关系曲线。计算结果可用于判断基槽开挖后不规则海床的液化情况,对相关研究和实际工程具有一定参考意义。     

基于FLOW-3D的三维数值波流水槽的构建及应用研究

为获得具有自由液面的三维波流水槽,基于FLOW-3D有限差分程序,选用RNG?-?紊流模型,基于线性波速度入口法造波,采用VOF方法对自由液面进行追踪。构建具有自由液面的三维波流联合作用数值水槽,对比压力出流边界与Sammerfeld辐射边界发现,在波流水槽内采用压力出流边界能够保持流体体积守恒,波形稳定;加入的孔隙结构,不仅能保证良好的消波效果,而且流体能够顺利通过,对消除反射波影响效果良好;与实验结果对比发现,垂向时均流速与实验数据拟合较好。波流场中桩柱前形成明显下潜水流及漩涡,是形成局部冲刷主要原因。本文所构建的波流数值模型波形稳定、具有良好的波流特性,能较好反应波浪与水流之间的相互作用,可进一步模拟波流场与结构物之间的相互作用或波流作用下泥沙冲刷等研究。     

Internal wave generation in an improved two-dimensional Boussinesq model

A set of Boussinesq-type equations with improved linear frequency dispersion in deeper water is solved numerically using a fourth order accurate predictor-corrector method. The model can be used to simulate the evolution of relatively long, weakly nonlinear waves in water of constant or variable depth provided the bed slope is of the same order of magnitude as the frequency dispersion parameter. By performing a linearized stability analysis, the phase and amplitude portraits of the numerical schemes are quantified, providing important information on practical grid resolutions in time and space. In contrast to previous models of the same kind, the incident wave field is generated inside the fluid domain by considering the scattered wave field in one part of the fluid domain and the total wave field in the other. Consequently, waves leaving the fluid domain are absorbed almost perfectly in the boundary regions by employment of damping terms in the mass and momentum equations. Additionally, the form of the incident regular wave field is computed by a Fourier approximation method which satisfies the governing equations accurately in water of constant depth. Since the Fourier approximation method requires an Eulerian mean current below wave trough level or a net mass transport velocity to be specified, the method can be used to study the interaction of waves and currents in closed as well as open basins. Several computational examples are given. These illustrate the potential of the wave generation method and the capability of the developed model.     

A swell model of the German Bight

A numerical model is presented, which investigates the contribution of wave-induced currents to the tidal residual circulation in the German Bight. The momentum transfer, by swell decaying to the mean circulation, is calculated including wave-current interaction without refraction. The model couples deep-water and shallow-water energy dissipation mechanisms such as wave attenuation and wave breaking. The model computes wave set-up, wave set-down and a longshore current as well. The pure wave-induced circulation is calculated, and also the residual circulation due to the interaction between waves and tide. The results suggest that the wave-induced currents can be neglected for the calculation of transport of near-surface pollutants.     

潮流波浪联合输沙及海床冲淤演变的理论体系与其数学模拟

根据近岸带及河口区潮流、波浪、湍流各自物理尺度的不同,从Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程和质量传输方程出发,利用Ｒｅｙｎｏｌｄｓ分解的方法,建立了模拟波浪 流联合输沙及海床冲淤演变的理论体系,给出了潮流作用下近岸波浪传播方程、波浪作用下潮流运动方程并通过利用波流合成底部切应力、底层湍流脉动随机特性,得出了波流联合作用下不平衡沙计算中泥沙起悬与沉降量的确定方法。本文模型应用于“广西合浦围垦工程潮流波浪     

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.     

浪、潮、风暴潮联合作用下的底应力效应

运用建立的二维非线性浪,潮和风暴耦全模式分析了波流相互作用下的底应力及其对耦合波浪场和流场的影响。由渤海的两次强寒潮过程的数值实验表明,在波流相互作用下,底应力明显增大,增大的底应力对波浪场影响甚微,但将明显改变水位和流速的大小,这种影响在近岸浅水区更加显著。     

A note on the horizontal momentum exchange in combined waves and current

The horizontal exchange of momentum due to the organized motion in combined waves and current has been analyzed. The combination of the vertical orbital wave motion and the mean current gives a periodic variation in the horizontal velocity in addition to the wave orbital motion. This periodic variation, combined with the wave orbital motion, gives a significant contribution to the momentum exchange. Two examples are considered, the interaction of a pure wave motion and a current normal to the direction of wave propagation, and a wave driven longshore current with an undertow velocity profile. It is demonstrated that the new contribution changes the resulting momentum exchange considerably.     

Numerical Modeling of Wave and Current Interaction in the Tidal Inlet Area

This paper investigates the phenomena of wave refraction and diffraction in the slowly varying topography, as well as the current deflection due to wave actions. A numerical model is developed based on depth integrated mean continuity equation and momentum equations, and a 3rd-order wave equation which governs the wave diffraction, refraction and interaction with current. Examples to examine the above model are given comparing with the laboratory data or the numerical results of other researchers. An example simulating the inlet area shows the interesting velocity field which may be used as a pioneer to further study on the nearshore hydrodynamics and sedimentation.     

内波对风浪中短波波面斜率分布的影响

在小风速条件下,分层水中,水下运动物体所产生的内波,通过改变表面流场,使得风浪中的短波波面斜率分布发生了改变。由此,可以通过观察表面风浪中短波波面斜率分布的变化,识别出水下运动物体的移动轨迹     

Transient effects in wave-current boundary layer flow

Previous studies of combined wave and current bottom boundary layer flow have concentrated on the final converged state of the flow following the addition of waves to a current. While this final state is of primary interest to modellers and engineers, it pre-supposes that such a state is actually attained in reality, and this may not always be the case. In addition, it overlooks the interesting and complicated transient effects which occur as a wave-current flow evolves from one state to another. The present study concentrates attention on the transient effects predicted by a “one-equation” turbulence closure model. Results of case studies are presented in which waves are superimposed co-linearly on a current (“forward problem”), and are then removed from the converged wave-current flow (“backward problem”). Two formulations of the “forward” and “backward” problems are discussed. In the first the steady component of the pressure gradient driving the mean flow is held constant throughout, and in the second the steady component of the mass flux is held constant. In each case the detailed evolution of the profiles of mean velocity, turbulent energy, mixing length, eddy viscosity and shear stress are discussed. More generally, the question of the convergence timescale of a combined wave-current flow is considered, and a convergence criterion is proposed.     

Interaction of dispersive water waves with weakly sheared currents of arbitrary profile

A set of depth-integrated equations describing combined wave–current flows is derived and validated. To account for the effect of turbulence induced by interactions between waves and currents with arbitrary horizontal vorticity, new additional stress terms are introduced. These stresses are functions of a parameter b that relates the relative importance of wave radiation stress and bottom friction stress to the wave–current interaction. To solve the equations, a fourth-order MUSCL-TVD scheme with an approximate Riemann solver is adopted. As a first-order check of the model, the Doppler shift effect and wave dispersion over linearly sheared currents are analytically shown to be retained appropriately in the equation set. The model results are then validated through comparisons with three experimental data sets. First, based on the experiments of Kemp and Simons (1982, 1983), a reasonable functional form of b is estimated. Second, simulations examining the propagation of a weakly dispersive wave over a depth-uniform or linearly sheared current are performed. Finally, the model is applied to a more complex configuration where bichromatic waves interact with spatially varying currents. Simulated results indicate that the model is capable of predicting nearshore interactions of waves with currents of arbitrary vertical structure. One of the unique properties of the developed model is its ability to assimilate an external current field from any source, be it from a circulation model or an observation, and predict the interaction of a nonlinear and dispersive wave field with that current.     

港珠澳大桥青州航道桥桥墩基础冲刷试验研究

桥墩基础冲刷是桥梁毁坏的重要因素,是桥梁基础设计的关键指标之一。目前国内外对于桥墩基础在复杂动力条件下冲刷深度的研究常采用物理模型试验方法,利用正态系列模型方法,在波流水槽中研究了水流、潮流和波流共同作用下青州航道桥索塔基础周围流态变化和局部冲刷特征。研究结果表明,桥墩最大冲刷深度和冲淤范围与水流流速、桥墩轴线与水流夹角和波浪等因素有关;在潮流最大流速和恒定流流速一致情况下,桥墩局部冲刷深度达到平衡后,将会达到与恒定流基本一致的最大冲深;波流共同作用下的最大冲刷深度比恒定流增加10%左右。设计桥墩形状在100年一遇水流和波浪共同作用下桥墩基础局部最大冲刷深度为13.7 m。     

Internal wave generation in an improved two-dimensional Boussinesq model

A set of Boussinesq-type equations with improved linear frequency dispersion in deeper water is solved numerically using a fourth order accurate predictor-corrector method. The model can be used to simulate the evolution of relatively long, weakly nonlinear waves in water of constant or variable depth provided the bed slope is of the same order of magnitude as the frequency dispersion parameter. By performing a linearized stability analysis, the phase and amplitude portraits of the numerical schemes are quantified, providing important information on practical grid resolutions in time and space. In contrast to previous models of the same kind, the incident wave field is generated inside the fluid domain by considering the scattered wave field in one part of the fluid domain and the total wave field in the other. Consequently, waves leaving the fluid domain are absorbed almost perfectly in the boundary regions by employment of damping terms in the mass and momentum equations. Additionally, the form of the incident regular wave field is computed by a Fourier approximation method which satisfies the governing equations accurately in water of constant depth. Since the Fourier approximation method requires an Eulerian mean current below wave trough level or a net mass transport velocity to be specified, the method can be used to study the interaction of waves and currents in closed as well as open basins. Several computational examples are given. These illustrate the potential of the wave generation method and the capability of the developed model.     

Numerical simulation of wave interaction with porous structures using an improved smoothed particle hydrodynamic method

A smoothed particle hydrodynamic (SPH) model is developed to simulate wave interaction with porous structures. The mean flow outside the porous structures is obtained by solving Reynolds Averaged Navier–Stokes (RANS) equations and the turbulence field is calculated by a large eddy simulation (LES) model. The porous flow is described by the spatially averaged Navier–Stokes type equations with the resistance effect of the porous media being represented by an empirical frictional source term. The interface boundaries between the porous flow and the outside flow are modeled by means of specifying a transition zone along the interface. The model is validated against other available numerical results and experimental data for wave damping over porous seabed with different levels of permeability. The validated model is then employed to investigate wave breaking over a submerged porous breakwater and good agreements between the SPH model results and the experimental data are obtained in terms of free surface displacement. In addition the predicted velocity, vorticity and pressure fields near the porous breakwater and in the breaking wave zone are also analyzed.     

Forces on a single pile caisson in breaking waves and current

The effect of breaking and steep non-breaking waves on a vertical pile such as found in minimal caisson structures in the Gulf of Mexico shallow water environment was studied in a wave tank testing. The waves generated were both regular and irregular. The irregular waves were of Bretschneider type. In some tests current was combined in the direction of waves. Specially steep waves in the recorded wave profile were chosen for the analysis of wave forces. It was observed that the forces on the caisson at the wave frequency due to breaking waves were no higher in general than those in nonbreaking steep waves. The drag coefficients in waves alone were much higher than those in a combined wave-current field. The wave-current drag coefficients approached those found in the steady current alone.