Three-dimensional structure of wave-induced momentum flux in irrotational waves in combined shoaling-refraction conditions

In the paper, the three-dimensional structure of the wave-induced momentum flux in irrotational waves propagating over a two-dimensional, irregular bathymetry is analyzed. The expansion method developed by de Vriend and Kitou [de Vriend, H.J., Kitou, N., 1990a. Incorporation of wave effects in a 3D hydrostatic mean current model. Delft Hydraulics Report H-1295. de Vriend, H.J., Kitou, N., 1990b. Incorporation of wave effects in a 3D hydrostatic mean current model. Proc. 22nd Int. Coast. Eng. Conf. ASCE, 1005–1018.] for unidirectional waves has been extended to derive expressions for velocity components in three-dimensional waves over sloping bottom. The vertical wave-induced momentum flux resulting from this solution has been shown to be vertically-varying (contrary to the 2D-V case) and to act as a counterbalance for the vertical variability of the other wave forcing terms in the momentum equations. Thus, the total wave forcing remains depth-invariant, but—contrary to the ‘traditional’ solution based on the radiation stress concept—it does not depend explicitly on the direction of wave propagation and is a simple function of gradients of wave energy and water depth only. One of the most important consequences of this fact is the lack of the longshore-current-generating force in the case of non-dissipative waves approaching a shore with a bottom profile uniform in the along-shore direction. To illustrate the meaning of the new solution, the wave forcing due to waves approaching a barred beach has been analysed in detail. Also, the present solution has been shown to give the same results as the one obtained by extending of the approach by Rivero and Arcilla [Rivero, F.J., Arcilla, A.S., 1995. On the vertical distribution of 〈ũw˜〉. Coast. Eng. 25, 137–152.] to three dimensions.     

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.     

High order Hamiltonian water wave models with wave-breaking mechanism

Based on the Hamiltonian formulation of water waves, using Hamiltonian consistent modelling methods, we derive higher order Hamiltonian equations by Taylor expansions of the potential and the vertical velocity around the still water level. The polynomial expansion in wave height is mixed with pseudo-differential operators that preserve the exact dispersion relation. The consistent approximate equations have inherited the Hamiltonian structure and give exact conservation of the approximate energy. In order to deal with breaking waves, we extend the eddy-viscosity model of Kennedy et al. (2000) to be applicable for fully dispersive equations. As breaking trigger mechanism we use a kinematic criterion based on the quotient of horizontal fluid velocity at the crest and the crest speed. The performance is illustrated by comparing simulations with experimental data for an irregular breaking wave with a peak period of 12 s above deep water and for a bathymetry induced periodic wave plunging breaker over a trapezoidal bar. The comparisons show that the higher order models perform quite well; the extension with the breaking wave mechanism improves the simulations significantly.     

An integral contravariant formulation of the fully non-linear Boussinesq equations

In this paper we propose an integral form of the fully non-linear Boussinesq equations in contravariant formulation, in which Christoffel symbols are avoided, in order to simulate wave transformation phenomena, wave breaking and nearshore currents in computational domains representing the complex morphology of real coastal regions. Following the approach proposed by Chen (2006), the motion equations retain the term related to the approximation to the second order of the vertical vorticity. A new Upwind Weighted Essentially Non-Oscillatory scheme for the solution of the fully non-linear Boussinesq equations on generalised curvilinear coordinate systems is proposed. The equations are rearranged in order to solve them by a high resolution hybrid finite volume–finite difference scheme. The conservative part of the above-mentioned equations, consisting of the convective terms and the terms related to the free surface elevation, is discretised by a high-order shock-capturing finite volume scheme in which an exact Riemann solver is involved; dispersive terms and the term related to the approximation to the second order of the vertical vorticity are discretised by a cell-centred finite difference scheme. The shock-capturing method makes it possible to intrinsically model the wave breaking, therefore no additional terms are needed to take into account the breaking related energy dissipation in the surf zone. The model is verified against several benchmark tests, and the results are compared with experimental, theoretical and alternative numerical solutions.     

Note on conservation equations for nonlinear surface waves

Euler's equations of motion in conjunction with the dynamic boundary condition are manipulated to obtain exact (and approximate) alternative momentum equations for nonlinear irrotational surface waves. The Airy and Boussinesq equations are re-derived as demonstrative examples. A fully nonlinear version of the improved Boussinesq equations is presented as a new application of the proposed equations. Further use of the equations in developing depth-integrated wave models, which are not necessarily restricted to finite depths, is also pointed out.     

Numerical modeling of nonlinear water waves over heterogeneous porous beds

The transformation of nonlinear water waves over porous beds is studied by applying a numerical model based on Chen's [2006. Fully nonlinear Boussinesq-type equations for waves and currents over porous beds. Journal of Engineering Mechanics, 132:2, 220–230] Boussinesq-type equations for highly nonlinear waves on permeable beds. The numerical model uses a high-order time-marching solution and fourth-order finite-difference schemes for discretization of first-order spatial derivatives to obtain a computational accuracy consistent with the model equations. By forcing the wave celerity and spatial porous-damping rate of the linearized model to match the exact linear theory for horizontal porous bed over a prescribed range of relative depths, the values of the model parameters are optimally determined. Numerical simulations of the damped wave propagation over finite-thickness porous layer demonstrate the accuracy of both the numerical model and governing equations, which have been shown by prior theoretical analyses to be accurate for both nominal and thick porous layers. These simulations also elucidate on the significance of the higher-order porous-damping terms and the influence of the hydraulic parameters. Application of the model to the simulation of the wave field around a laboratory-scale submerged porous mound provides a measure of its capability, as well as useful insight into the scaling of the porous-resistance coefficients. For application to heterogeneous porous beds, the assumption of weak spatial variation of the porous resistance is examined using truncated forms of the governing equations. The results indicate that the complete set of Boussinesq-type equations is applicable to porous beds of nonhomogeneous makeup.     

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.     

Capillary-Gravitational Waves of Finite Amplitude on the Surface of a Homogeneous Liquid

The method of multiple scales is used to deduce equations for three nonlinear approximations of the capillary-gravitational disturbances of the free surface of a layer of a homogeneous liquid of constant depth. In these equations, the space-time variations of the wave profile in the expression for the velocity potential on the liquid surface are taken into account. On this basis, we construct asymptotic expansions up to the quantities of the third order of smallness for the velocity potential and elevations of the liquid surface induced by running periodic waves of finite amplitude. Furthermore, we analyze the dependences of the amplitude-phase characteristics of wave disturbances on the surface tension, depth of the liquid, and the length and steepness of waves of the first harmonic. __________ Translated from Morskoi Gidrofizicheskii Zhurnal, No. 5, pp. 25–34, September–October, 2005.     

A finite difference method for effective treatment of mild-slope wave equation subject to non-reflecting boundary conditions

A numerical model for coastal water wave motion that includes an effective method for treatment of non-reflecting boundaries is presented. The second-order one-way wave equation to approximate the non-reflecting boundary condition is found to be excellent and it ensures a very low level of reflection for waves approaching the boundary at a fairly wide range of the incidence angle. If the Newman approximation is adopted, the resulting boundary condition has a unique property to allow the free propagation of wave components along the boundary. The study is also based on a newly derived mild-slope wave equation system that can be easily made compatible to the one-way wave equation. The equation system is theoretically more accurate than the previous equations in terms of the mild-slope assumption. The finite difference method defined on a staggered grid is employed to solve the basic equations and to implement the non-reflecting boundary condition. For verification, the numerical model is then applied to three coastal water wave problems including the classical problem of plane wave diffraction by a vertical circular cylinder, the problem of combined wave diffraction and refraction over a submerged hump in the open sea, and the wave deformation around a detached breakwater. In all cases, the numerical results are demonstrated to agree very well with the relevant analytical solutions or with experimental data. It is thus concluded that the numerical model proposed in this study is effective and advantageous.     

The eigenvalue equations of equatorial waves on a sphere

Zonally propagating wave solutions of the linearized shallow water equations (LSWE) in a zonal channel on the rotating spherical earth are constructed from numerical solutions of eigenvalue equations that yield the meridional variation of the waves' amplitudes and the phase speeds of these waves. An approximate Schrödinger equation, whose potential depends on one parameter only, is derived, and this equation yields analytic expressions for the dispersion relations and for the meridional structure of the waves' amplitudes in two asymptotic cases. These analytic solutions validate the accuracy of the numerical solutions of the exact eigenvalue equation. Our results show the existence of Kelvin, Poincaré and Rossby waves that are harmonic for large radius of deformation. For small radius of deformation, the latter two waves vary as Hermite functions. In addition, our results show that the mixed mode of the planar theory (a meridional wavenumber zero mode that behaves as a Rossby wave for large zonal wavenumbers and as a Poincaré wave for small ones) does not exist on a sphere; instead, the first Rossby mode and the first westward propagating Poincaré mode are separated by the anti-Kelvin mode for all values of the zonal wavenumber.     

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.     

Nonlinear Surface Waves in a Basin with Floating Broken Ice

The method of multiple scales is used to deduce equations for three nonlinear approximations of a wave disturbance in a basin of constant depth covered with broken ice. In deducing these equations, we take into account the space and time variability of the wave profile in the expression for the velocity potential on the basin surface. These equations are used to construct uniformly suitable asymptotic expansions up to quantities of the third order of smallness for the liquid-velocity potential and elevations of the basin surface formed by a periodic running wave of finite amplitude. We analyze the dependence of the amplitude-phase characteristics of elevations of the basin surface on the thickness of ice, nonlinearity of its vertical acceleration, and the amplitude and wavelength of the fundamental harmonic.     

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.     

Finite-amplitude analysis of some Boussinesq-type equations

Recent progress in formulating Boussinesq-type equations includes improved features of linear dispersion and higher-order nonlinearity. Nonlinear characteristics of these equations have been previously analysed on the assumption of weak nonlinearity, being therefore limited to moderate wave height. The present work addresses this aspect for an important class of wave problems, namely, regular waves of permanent form on a constant depth. Using a numerical procedure which is valid up to the maximum wave height, permanent-form waves admitted by three sets of advanced Boussinesq-type equations are analysed. Further, the characteristics of each set of the Boussinesq-type equations are discussed in the light of those from the potential theory satisfying the exact free-surface conditions. Phase velocity, amplitude dispersion, harmonic amplitudes (namely, second and third) and skewness of surface profile are shown over a two-parameter space of frequency and wave height.     

初始倾斜跃层所致内波的数值研究

The pycnocline in a closed domain is tilted by external wind forcing and tends to restore to a level posi- tion when the wind falls. An internal seiche oscillation exhibits if the forcing is weak, otherwise internal surge and internal solitary waves emerge, which serve as a link to cascade energy to small-scale processes. A two-dimensional non-hydrostatic code with a turbulence closure model is constructed to extend previous laboratory studies. The model could reproduce all the key phenomena observed in the corresponding labo- ratory experiments. The model results further serve as a comprehensive and reliable data set for an in-depth understanding of the related dynamical process. The comparative analyses indicate that nonlinear term favors the generation of internal surge and subsequent internal solitary waves, and the linear model predicts the general trend reasonably well. The vertical boundary can approximately reflect all the incoming waves, while the slope boundary serves as an area for small-scale internal wave breaking and energy dissipation. The temporal evolutions of domain integrated kinetic and potential energy are also analyzed, and the results indicate that about 20% of the initial available potential energy is lost during the first internal wave breaking process. Some numerical tactics such as grid topology and model initialization are also briefly discussed.     

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.     

Hybrid finite volume – finite difference scheme for 2DH improved Boussinesq equations

In this paper, a hybrid scheme based on a set of 2DH extended Boussinesq equations for slowly varying bathymetries is introduced. The numerical code combines the finite volume technique, applied to solve the advective part of the equations, with the finite difference method, used to discretize dispersive and source terms. Time integration is performed using the fourth-order Adams–Bashforth–Moulton predictor–corrector method; the Riemann problem is solved employing an approximate HLL solver, a fourth-order MUSCL-TVD technique is applied. Five test cases, for non-breaking and breaking waves, are reproduced to verify the model comparing its results to laboratory data or analytical solutions.     

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.     

Numerical Simulation of Wave Field near Submerged Bars by PLIC-VOF Model

Investigating the wave field near structures in coastal and offshore engineering is of increasing significance. In the present study, simulation is done of the wave profile and flow field for waves propagating over submerged bars, using PLIC-VOF （Pieeewise Linear Interface Construction） to trace the free surface of wave and finite difference method to solve vertical 2D Navier-Stokes （N-S） equations. A comparison of the numerical results for two kinds of submerged bars with the experimental ones shows that the PLIC-VOF model used in this study is effective and can compute the wave field precisely.     

Experimental study on internal waves in a stratified shear flow

This paper describes experiments on interfacial phenomena in a stratified shear flow having a sharp velocity shear at a density interface. The interface was visualized in vertical cross-section using dye, and the flow pattern was traced using aluminum powder. Two kinds of internal waves with different phase velocities and wave profiles were observed. They are here named p(positive)-waves and n(negative)-waves, respectively. By means of a two-dimensional visualization technique, the following facts have been confirmed regarding these waves. (1) The two kinds of waves propagate in the opposite direction relative to a system moving with the mean velocity at the interface, and their dispersion relations approximately agree with the two solutions of interfacial waves in a two-layer system of a linear basic shear flow. (2) The p-wave has sharp crests and flat troughs, and the n-wave has the reverse of this. This difference in wave profile is due to the finite amplitude effect. (3) Phase velocity of each wave lies within the range of the mean velocity profile, so that a critical layer exists and each wave has a “cat's eye” flow pattern in the vicinity of the critical layer, when observed in a system moving with the phase velocity. Consequently, these two waves are symmetrical with respect to the interface. The mechanisms of generation of these waves, and the entrainment process are discussed. It is inferred that when the “cat's eye” flow pattern is distorted and a stagnation point approaches the interface, entrainment in the form of a stretched wisp from the lower to the upper layer occurs for the p-wave, and from the upper to the lower layer for the n-wave.