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.     

Interaction of waves with non-colinear currents

The present paper reports on a study of the interaction of a current-free monochromatic surface wave field with a wave-free uniform current field in a three-dimensional flow frame. The wave and the current fields are not necessarily collinear with each other. The formulation of the wave-current field is done under the assumption of irrotational and inviscid flow. We have developed the three dimensional expressions describing the characteristics of the combined flow in terms of mass, momentum and energy transport conservation. These equations are found efficient to describe the sought-for combined wave-current field. The parameters describing the wave-current field after the interaction are the surface disturbance amplitude and length, mean water depth, mean current-like parameter and direction of the combined flow, which would be calculated from a set of equations that satisfy conservation of mean mass, momentum and energy flux and a dispersion relation on the free surface before and after the interaction. The results are shown in terms of relative changes in wave heights and lengths, current-like parameters and final directions obtained for the combined wave-current field with respect to current-free wave and wave-free current pre-interaction parameters.     

An explicit finite difference model for simulating weakly nonlinear and weakly dispersive waves over slowly varying water depth

In this paper, a modified leap-frog finite difference (FD) scheme is developed to solve Non linear Shallow Water Equations (NSWE). By adjusting the FD mesh system and modifying the leap-frog algorithm, numerical dispersion is manipulated to mimic physical frequency dispersion for water wave propagation. The resulting numerical scheme is suitable for weakly nonlinear and weakly dispersive waves propagating over a slowly varying water depth. Numerical studies demonstrate that the results of the new numerical scheme agree well with those obtained by directly solving Boussinesq-type models for both long distance propagation, shoaling and re-fraction over a slowly varying bathymetry. Most importantly, the new algorithm is much more computationally efficient than existing Boussinesq-type models, making it an excellent alternative tool for simulating tsunami waves when the frequency dispersion needs to be considered.     

Influence of currents on weakly nonlinear topographic waves

In the Boussinesq approximation, we consider trapped topographic waves in an inhomogeneous current directed along isobaths. The influence of the current on the dispersion properties of trapped topographic waves in the Norwegian Sea is studied. We determine the mean currents and nonoscillatory (on the time scale of the waves) density corrections induced by the waves due to their nonlinearity. It is shown that the influence of currents is significant in the short-wave region. Its influence leads to a decrease in the wavelength for the constant period of waves, whereas the mean current caused by nonlinearity noticeably varies, especially in the bottom layer.     

Deterministic and stochastic evolution equations for fully dispersive and weakly nonlinear waves

This paper presents a new and more accurate set of deterministic evolution equations for the propagation of fully dispersive, weakly nonlinear, irregular, multidirectional waves. The equations are derived directly from the Laplace equation with leading order nonlinearity in the surface boundary conditions. It is demonstrated that previous fully dispersive formulations from the literature have used an inconsistent linear relation between the velocity potential and the surface elevation. As a consequence these formulations are accurate only in shallow water, while nonlinear transfer of energy is significantly underestimated for larger wave numbers. In the present work we correct this inconsistency. In addition to the improved deterministic formulation, we present improved stochastic evolution equations in terms of the energy spectrum and the bispectrum for multidirectional waves. The deterministic and stochastic formulations are solved numerically for the case of cross shore motion of unidirectional waves and the results are verified against laboratory data for wave propagation over submerged bars and over a plane slope. Outside the surf zone the two model predictions are generally in good agreement with the measurements, and it is found that the accuracy of e.g., the energy spectrum and of the third-order statistics is considerably improved by the new formulations, particularly outside the shallow-water range.     

Interaction mechanisms among waves,currents and a submerged plate

Wave-induced loads on a submerged plate, representative of submerged breakwater, coastal-bridge deck and a certain type of wave energy converter, in a uniform current are investigated in this study using fully nonlinear numerical wave tanks (NWTs) based on potential flow theory. The coupling effect of wave and current is explored, and the underlying interaction mechanisms of the hydrodynamic forces are described. The presence of a background current modifies the frequency dispersion. It produces changes of the water-surface elevation, and also has an effect on wave-induced loads. Depending on the nonlinearity, higher harmonic wave components are generated above the submerged plate. These contribute to the wave forces. It is found that the horizontal and the vertical force, hence the moment, are affected in the opposite way by the currents. The Doppler shifted effect dominates the vertical force and the moment on the plate. Whereas, the Doppler shifted effect and the generation of higher wave harmonics play opposite roles on the horizontal forces. The contribution of 2^nd order harmonics is found to be up to 30% of the linear component. The current-induced drag force, represented by the advection term ρU∂φ/∂x in the pressure equation, is found to lead to a decrease in the moment for the most range of wavelengths considered, and an increase in the moment for a small range of longer waves.     

Movable bed roughness and current profiles in the presence of irregular waves with an arbitrary angle to currents

The bed roughness k_s and current velocity profiles in the presence of waves with an arbitrary angle θ to currents are studied. It is found that the movable bed roughness is affected by both the wave and the current and only slightly by the angle θ between the wave propagation and the current, and that existing formulae derived in purely oscillatory flows generally fail to predict k_s. In the present study, a new formula which takes account the effect of the wave and the current on the bed roughness is suggested to calculate k_s in combined wave-current flows. With the present formula, the current profiles calculated by the model of You agree satisfactorily with the laboratory data of van Kampen and Nap and Havinga, and the field measurements of Grant and Williams and Drake et al.     

Interaction of water waves with vertical cylinders using null-field integral equations

The scattering of water waves by bottom-mounted vertical circular cylinders is solved by using the null-field integral equations in conjunction with degenerate kernels and Fourier series to avoid calculating the Cauchy and Hadamard principal values. In the implementation, the null-field point can be exactly located on the real boundary owing to the introduction of degenerate kernels for fundamental solutions. An adaptive observer system of polar coordinates is considered to fully employ the properties of degenerate kernels. For the hypersingular equation, vector decomposition for the radial and tangential gradients is carefully considered. This method can be seen as a semi-analytical approach since errors attribute from the truncation of Fourier series. Neither hypersingularity in the Burton and Miller approach nor the CHIEF concept was required to deal with the problem of irregular frequencies. Five advantages of free of calculating principal value, well-posed algebraic system, convergence rate of exponential order, meshfree and elimination of boundary-layer effect, are achieved by using the present approach. Numerical results are given for the forces and free-surface elevation around the circular boundaries. Also, the near-trapped behavior arisen from the physical resonance is detected. A general-purpose program for water wave impinging several circular cylinders with arbitrary number, radii, and positions was developed. Several examples of water wave structure interaction by vertical circular cylinders were demonstrated to see the validity of the present formulation.     

Interaction of flexural gravity waves with shear current in shallow water

In the present study, the effect of shear current on the propagation of flexural gravity waves is analyzed under the assumptions of linearized shallow-water theory. Explicit expressions for the reflection and transmission coefficients associated with flexural gravity wave scattering by a step discontinuity in both water depth and current speed are derived. Further, trapping and scattering of flexural gravity waves by a jet-like shear current with a top-hat profile are examined and certain limiting conditions for the waves to exist are derived. The effects of change in water depth, current speed, incident wavelength and the angle of incidence on the group and phase velocities as well as on the reflection and transmission characteristics are analyzed through different numerical results.     

Mild-slope equation for water waves propagating over non-uniform currents and uneven bottoms

I~IOXThe interaction between surface waves and ambient currents and nearshore topography lies atone of the heat of morphological medelling. Accurate predictions of how wave propagates overcurrents and topography, and of the consequent erosion and dePOSition of sand on a beach or tidalflat are vital when assessing how a coastline may be affected by changing conditions.The mild-slope equation was introduced by Berkhoff (1972) as a way of approximating therefraction-diffraction of linearized s…     

Numerical study on water waves and wave-induced longshore currents in Obaky coastal water

In this paper, the water waves and wave-induced longshore currents in Obaky coastal water which is located at the Mediterranean coast of Turkey were numerically studied. The numerical model is based on the parabolic mild-slope equation for coastal water waves and the nonlinear shallow water equation for the wave-induced currents. The wave transformation under the effects of shoaling, refraction, diffraction and breaking is considered, and the wave provides radiation stresses for driving currents in the model. The numerical results for the water wave-induced longshore currents were validated by the measured data to demonstrate the efficiency of the numerical model. Then the water waves and longshore currents induced by the waves from main directions were numerically simulated and analyzed based on the numerical results. The numerical results show that the movement of the longshore currents was different while the wave propagated to a coastal zone from different directions.     

VIV response of a long flexible riser fitted with strakes in uniform and linearly sheared currents

An experimental investigation was conducted on a flexible riser with and without various strake arrangements. The aim of the present work was to further improve the understanding of the response performance of the vortex-induced vibration (VIV) of a riser with helical strakes. Two current profiles, including uniform and linearly sheared flows, were simulated. The uniform current was simulated by towing the riser model in one direction using the towing carriage, and the linearly sheared current was simulated by fixing one end of the riser and using a driven cantilever to traverse a circular arc. Based on the modal superposition method, the displacement responses were obtained from the measured strain. Strakes with different heights and pitches were analysed, and response parameters such as the displacement response and fatigue damage were studied. The results of the bare model test show that the lock-in phenomenon displays multi-order characteristics, and the VIV displacement decreases with an increased order of the lock-in regime. The results of the straked model test indicate that the response characteristics of a bare riser can be quite distinct from those of a riser with helical strakes, and the response performance depends closely on the geometry of the strake configuration.     

Applicability of numerical models to nonlinear dispersive waves

The applicability of three different wave-propagation models in nonlinear dispersive wave fields has been investigated. The numerical models tested here are based on three different wave theories: a fully nonlinear potential theory, a Stokes second-order theory, and a Boussinesq-type theory with an improved dispersion relation. Physical experiments and computations were conducted for wave evolutions during passage over a submerged shelf under various wave conditions. As expected, the fully nonlinear solutions agree better with the measurements than do the other solutions. Although the second-order solution has sufficient accuracy for smaller-amplitude wave cases, the truncation after the third harmonics causes significant discrepancies in wave form for larger waves. In addition, the second-order model markedly overestimates the first- and second-harmonic amplitudes in transmitted waves. The Boussinesq model provides excellent predictions of wave profile over the shelf even in larger wave cases. However, this model also overestimates the magnitudes of several higher harmonics in transmitted waves. These facts may indicate that energy transfer from bound components into free waves in these higher harmonics cannot be accurately evaluated by the Boussinesq-type equations.     

Combined refraction and diffraction models for sea waves over complicated topography and with currents in shallow water

-Combined refraction and diffraction models in the form of linear parabolic approximation are derived through smallparameter method. More strictly theoretical basis and more accuracy in the models than Lozano's (1980) are obtained. Some theoretical defects in Liu's model (1985) with consideration of current are not only found but also eliminated. More strict and accurate models are, therefore, presented in this paper.The calculation results and analysis in applying the models to actual wave field with consideration of bottom friction will be given in the following paper.     

Solution of an unstable axisymmetric Cauchy–Poisson problem of dispersive water waves for a spectrum with compact support

 It has been known that the axisymmetric Cauchy–Poisson problem for dispersive water waves is well posed in the sense of stability. Thereby time evolution solutions of wave propagation depend continuously on initial conditions. However, in this paper, it is demonstrated that the axisymmetric Cauchy–Poisson problem is ill posed in the sense of stability for a certain class of initial conditions, so that the propagating solutions do not depend continuously on the initial conditions. In order to overcome the difficulty of the discontinuity, Landweber–Fridman's regularization, famous and well known in applied mathematics, are introduced and investigated to learn whether it is applicable to the present axisymmetric wave propagation problem. From the numerical experiments, it is shown that stable and accurate solutions are realized by the regularization, so that it can be applicable to the determination of the ill-posed Cauchy–Poisson problem.     

Dynamics of Rossby waves in an ocean with inhomogeneous currents

A model for a two-layer ocean is applied to consider, in terms of the geometrical optics approximation, the effect of mean flows propagating within the upper layer upon the dynamics of Rossby waves. The case is theoretically analysed, with the depth of the ocean's upper layer much smaller than that of the underlying layer. In this case, the flow's impact upon the baroclinic mode of Rossby waves is ubiquitous, with the exception of synchronicity. Depending on the parameters, four types of wave packets' behaviour in the vicinity of synchronicity points are singled out, namely, the elimination of the peculiarity, shadowing, and convective/absolute instability. For the mean flow profile simulating cyclonic and anticyclonic gyres, we have obtained wave packet trajectories and have studied the wave packet's interaction with the current. Specifically, it has been demonstrated that, given some modulus of the wave packet, vigorous energy exchange between the wave vector and the flow takes place. Translated by Vladimir A. Puchkin.     

Sediment waves on the tiber prodelta slope: Interaction of deltaic sedimentation and currents along the shelf

A regressive depositional sequence has been prograding on the northeastern Tyrrhenian Shelf since the establishment of the present high stand of sea level. Thickness and distribution of this prograding sequence are chiefly controlled by the Tiber Delta sediment source and the oceanographic conditions on the shelf. Wavy bedforms characterize the Tiber prodelta slope between 35 and 100 m water depth. On 3.5 kHz subbottom profiles, these bedforms show the same morphology and internal depositional geometry as most of the deep-water examples of sediment waves.     

Analysis and prediction of vortex-induced vibrations of variable-tension vertical risers in linearly sheared currents

Many studies have tackled the problem of vortex-induced vibrations (VIV) of a vertical riser with a constant tension and placed in uniform currents. In this study, attention is focused on the cross-flow VIV modelling, time-domain analysis and prediction of variable-tension vertical risers in linearly sheared currents. The partial-differential equation governing the riser transverse motion is based on a flexural tensioned-beam model with typical pinned-pinned supports. The hydrodynamic excitation model describing the modulation of lift force is based on a distributed van der Pol wake oscillator whose nonlinear equation is also partial-differential due to the implementation of a diffusion term. The variation of empirical wake coefficients with system parameters and the water depth-dependent Reynolds number is introduced. Based on the assumed Fourier mode shape functions obtained by accounting for the effect of non-uniform tension, the Galerkin technique is utilized to construct a low-dimensional multi-mode model governing the coupled fluid-riser interaction system due to VIV. Numerical simulations in the case of varying sheared flow profiles are carried out to systematically evaluate riser nonlinear dynamics and highlight the influence of fluid-structure parameters along with associated VIV aspects. In particular, the effects of shear and tensioned-beam (tension versus bending) parameters are underlined. Some comparisons with published experimental results and observations are qualitatively and quantitatively discussed. Overall parametric analysis and prediction results may be worthwhile for being a new benchmark against future experimental testing and/or numerical results predicted by an alternative model and methodology.     

Interaction of oblique waves with an infinite cylinder

A Green's function procedure is applied to compute the oblique wave interaction with a cylinder of arbitrary section on the free surface in water of infinite depth. Also, the hydrodynamic coefficients associated with the motion of the cylinder oscillating in its three degrees of freedom, periodic along its axis, are treated. A computer program based on the present procedure is found to be accurate and efficient. The results are applicable to the analysis of floating breakwaters, floating bridges, ship hulls and other elongated structures on a free surface.