Regular Wave Groups Research Articles

Sand suspension and fluxes by wave groups and equivalent monochromatic waves

Sand transport dynamics under non-breaking regular waves and wave groups are studied in a large-scale wave flume. The water surface elevation, flow velocity, suspended sand concentration (SSC) distributions, and bedforms were measured over a horizontal test section of medium sand. The experiment involved five wave groups that had equal root-mean-square water surface elevation and mean period, but which differed in terms of modulation, wave group length, and short wave sequencing. In addition, three monochromatic wave conditions were generated which were equivalent to the wave groups in terms of root-mean-square water surface elevation or maximum wave height. Temporal and vertical distributions of SSC suggest a more effective sediment pumping for symmetric and backward leaning ‘waxing’ wave groups. Vertical distributions of the SSC phase lag indicate a more rapid vertical pumping for the short-duration wave group. Overall, the differences between wave group conditions are small and the time-averaged SSC and sediment diffusivity profiles are very similar. The total net suspended sand flux consists of current-related, long-wave-related and short-wave-related contributions that are all of similar magnitude. The net total suspended sand flux was always offshore-directed, but its magnitude depends on the vertical and temporal variation of SSC at short- and long-wave time scales. The results are used to discuss the applicability of the ‘equivalent wave concept’ for time-averaged SSC distributions and net suspended sand transport.

Ship dynamic stability assessment based on realistic wave group excitations

Ship propensity for stability failure in irregular beam seas is addressed. A new method, based on the combination of the Karhunen–Loève theorem with the theory of Markov processes, is employed in order to construct realistic wave groups of varying heights and periods. The focus is set on waveforms with high probability of occurrence, given a sea-state. These waveforms are then applied, as beam-sea excitation, to a modern container vessel. Safe basin integrity diagrams and transient capsize diagrams are produced through short numerical simulations. For comparison purposes, alternative forms of excitation, such as wave groups of varying heights but fixed periods; and regular wave groups of fixed heights and periods, are also considered. Differences in the system's transient response, specifically on the integrity of its safe basin and on the transient capsize diagram, are discussed. The proposed method advances the so called “critical wave groups method” and it can be used for a swift assessment of ship stability.

High‐order oscillatory contributions to shear stress under standing regular wave groups: Theory and experimental evidence

In this paper, the oscillatory contributions to the steady shear stress of regular wave groups propagating over plane beds in the laminar regime are investigated theoretically and experimentally. Assuming weak nonlinearity, it was found that weak‐unsteady motions induced by regular wave groups and second‐order harmonics induced by nonlinear bottom boundary layer (BBL) dynamics contribute to the same order. Wave reflection was found to be an additional mechanism to generate high‐order contributions to the steady shear stress. High‐order oscillatory contributions to the steady shear stress generate small‐amplitude perturbations to the steady streaming through the BBL. As a result, steady secondary circulation cells can be distinguished in the wave group–induced BBL. The magnitude and length scale of the measured small‐amplitude perturbations are consistent with the theoretical predictions. Moreover, it is shown that the oscillatory contributions are selected according to their characteristic timescale with distance to the bed. The results could have important implications for understanding BBL dynamics and transport and the transition to the perturbed laminar regime in multiple‐scale oscillatory flows.

Standing regular wave groups with oblique incidence: Phase averaging and bottom boundary layer in constant depth

This article analyzes the phase‐averaged hydrodynamics induced by regular wave groups and the related long waves under reflective conditions outside of the wave‐breaking zone. The solutions of the inviscid region and wave‐group bottom boundary layer (WGBL) are analytically investigated for a regular wave group at a constant depth, impinging obliquely on an alongshore uniform reflective wall. The problem is formulated and solved by using the classical multiple‐scale approach. Phase averaging is defined on the basis of the cascade of spatial and temporal scales of the problem leading to a weak unsteady (hydrodynamic) regime (WUR), a steady regime (SR), weak nonhomogeneous regime (WNHR), and homogeneous regime (HR) of instantaneous quantities. The analysis of the inviscid region hydrodynamics shows that depending on the time scale used for the averaging, modulation scales of O(100 m) and O(1 km) may appear in the time‐averaged pressure and the free stream velocity fields. These modulations are related to the nonlinear interference of the incoming and reflected wave‐group motions. Supermodulation scales of O(10 km) are also possible. Such scales result from the superposition of the bound long waves and the oscillatory free long wave mode released during the reflection of the wave train. The SR of the total pressure field and the WUR of the radiation stresses are also analyzed. The WGBL solution shows that multiple‐scale modulation of the inviscid region hydrodynamics can be fully transmitted to the Eulerian drift in the viscous region. We also found that under reflective conditions and nearly normal wave incidence, the WUR of the Eulerian drift may present standing modulated patterns with scales of O(100 m), O(1 km), and O(10 km). In the SR, there can be scales of O(100 m) and O(1 km). The mechanisms which determine the vertical structure and direction of the WUR and SR of the Eulerian drift beneath wave groups are also investigated as a function of the main parameters of the WGBL model. Physical implications of and qualifications to the theory presented on nearshore sediment transport are discussed.

Resonant response of harbours to short-wave groups

A non-linear short-wave propagation model is used for the simulation of low-frequency waves induced by short-wave groups. The model is based on the higher order Boussinesq-type equations for the propagation of breaking and non-breaking waves in nearshore areas. The 2DH (two horizontal dimensions) Boussinesq model is used to calculate the resonant response of harbours due to bound waves generated by short-wave groups. The results are compared with experimental data available in the literature. In these experiments, bound long waves are generated by carrier bichromatic short waves, which form regular wave groups. The results of the model were found to be in good agreement with the experimental data. The 1DH Boussinesq model is applied in the surf zone to simulate surf beat generation by a time-varying breakpoint and the reflected long wave induced by bichromatic wave groups. A new breaking criterion is introduced, based on the relative trough Froude number. Model generation of the surf beat is also shown to be in good agreement with experimental data.

Modelling Sand Resuspension by Waves over a Rippled Bed

With a two-dimensional k-ϵ model and equations to describe sediment entrainment, diffusion, convection and settling, wave-induced flow velocities and sediment concentration fields over rippled beds have been calculated. Results from the simulations are compared with time-averaged vertical suspended sediment concentration profiles, C-profiles, measured in a large wave flume using the autonomous multi-sensor instrument STABLE. The simulations make it possible to examine in detail the mechanisms of sediment entrainment, vortex formation and advection. The resulting animations of model output allow a quasi-3-D visualization of sediment resuspension processes and aid quantification of the resulting vertical suspended sediment concentration field due to regular waves, asymmetric waves and wave groups. The work contributes therefore, to knowledge about wave-sediment interactions and to understanding of sediment transport mechanics.

Simulation and prediction of swash oscillations on a steep beach

This paper presents numerical simulations and analytical predictions of key aspects of swash oscillations on a steep beach. Simulations of the shoreline displacement based on bore run-up theory are found to give excellent agreement with recent experimental data for regular waves, wave groups and random waves. The theory is used to derive parameters that predict the onset of swash saturation and the spectral characteristics of the saturated shoreline motion. These parameters are again in good agreement with the measured laboratory data and are also consistent with previous experimental data. Simulation of irregular wave run-up using a series of overlapping monochromatic swash events is found to reproduce typical features of swash oscillations and can accurately describe both the low and high frequency spectral characteristics of the swash zone. In particular, the low frequency components of the run-up can be modelled directly using a sequence of incident short wave bores, with no direct long wave input to the numerical simulations. This suggests that wave groupiness must be accounted for when modelling shoreline oscillations.

Influence of group-bounded long wave on sediment transport rate

The formula predicting the sediment transport rate induced by the interactions between short waves and group-bounded long wave is mathematically derived using Bailard's immersed weight sediment transport rate formula and the expansion of flow moments. As the first step for studying the influence of group-bounded long wave on the sediment transport rate, a regular wave groups which may be easily generated by the linear combination of monochromatic waves is used. From the numerical experiments, it is found that the group-bounded long wave plays an important role in the bottom sediment transport mechanism under wave groups. In particular, the group-bounded long wave tends to move the bottom sediment toward the direction of down slope or offshore. Also, the suspended load is more affected than the bed load by the group-bounded long wave. These are consistent with the field data which wave groups results in higher concentrations of suspended sand.

Low frequency swash motion induced by wave grouping

This paper concerns the low frequency motion of swash directly induced by wave grouping on a steep beach. A new experimental investigation is presented which considers the hydrodynamics of the inner surf zone and swash zone using vertical wave gauges and a run-up wire. Results for regular waves, wave groups and random waves are discussed, with particular reference to low frequency motions. The inner surf zone and swash zone are found to be unsaturated at incident short wave frequencies and, as a result, significant wave grouping is apparent at the shoreline. The low frequency motion in the surf zone is found to be in phase with the incident wave grouping and may therefore be regarded as a time varying set-up (Watson and Peregrine, 1992). The low frequency motion of the swash is shown to be an order of magnitude greater than that in the inner surf zone, inconsistent with cross-shore standing long waves, for which no evidence is found. We demonstrate that the low frequency motion of the shoreline provides an excellent approximation to the run-up of individual bores and therefore describes the run-up envelope. Spectral analysis shows that the low frequency motion of the swash is directly linked to the modulations in offshore wave height i.e. the low frequency energy in the incident wave envelope. In addition, the random wave run-up spectra show an f −4 high frequency roll-off, as found by Huntley et al. (1977). The accumulated data show that, unless the surf zone is totally saturated, a significant proportion of the low frequency swash motion may be directly due to incident wave grouping and not standing long waves.

An experiment on harbour resonance induced by incident regular waves and irregular short waves

It is generally accepted that bound long waves generated by wind waves excite harbour resonance. Nevertheless it is not clear if there are differences in the harbour response when resonance is induced by incident regular free long waves or by incident bound long waves. This experimental investigation aims at determining these differences for a narrow long bay. The finite water depth inside and outside the narrow bay is constant. For incident bound long waves, two cases have been studied: in the first case bound long waves are generated by carrier bichromatic short waves which form regular wave groups; in the second case bound long waves are generated by irregular short waves. The experimental setup allows the characteristics of the forcing waves to be defined with a good degree of accuracy. The bay response is obtained for the three cases of regular free long waves and bound long waves for regular and irregular groups.

Second-order wavemaker theory for irregular waves

Through the last decade the theory for second-order irregular wave generation was developed within the framework of Stokes wave theory. This pioneering work, however, is not fully consistent. Furthermore, due to the extensive algebra involved, the derived transfer functions appear in an unnecessarily complicated form. The present paper develops the full second-order wavemaker theory (including superharmonics as well as subharmonics) valid for rotational as well as translatory wave board motion. The primary goal is to obtain the second-order motion of the wave paddle required in order to get a spatially homogeneous wave field correct to second order, i.e. in order to suppress spurious free-wave generation. In addition to the transfer functions developed in the line of references on which the present work is based, some new terms evolve. These are related to the first-order evanescent modes and accordingly they are significant when the wave board motion makes a poor fit to the velocity profile of the desired progressive wave component. This is, for example, the case for the high-frequency part of a primary wave spectrum when using a piston-type wavemaker. The transfer functions are given in a relatively simple form by which the computational effort is reduced substantially. This enhances the practical computation of second-order wavemaker control signals for irregular waves, and no narrow band assumption is needed. The software is conveniently included in a PC-based wave generation system—the DHI Wave Synthesizer. The validity of the theory is demonstrated for a piston type wavemaker in a number of laboratory wave experiments for regular waves, wave groups and irregular waves.

A mathematical model for set down in harbours

This paper describes the first stage in the development of a numerical model of waves in harbours which includes the propagation of set down. A description is given of the theoretical background to the use of the Boussinesq equations in modelling set down beneath regular wave groups. The finite-difference scheme which is used to solve the governing Boussinesq equations is implemented on a parellel processing computer. This scheme has been found to provide an accurate solution method. The results from the model are compared with theoretical solutions and with data from physical model tests. In particular, the model is shown to be capable of representing accurately the diffraction of set down.

On the Increased Damping of a Moored Body During Low-Frequency Motions in Waves

Abstract An estimation method for the increased damping on a moored body in waves which allow simulation of the decay increase of low-frequency motion in waves is described. Emphasis is put on the analytical estimation method for the increased damping due to waves (wave drift damping) on a moored body during low-frequency motion. The estimated increased damping is compared with that of free oscillation tests in waves, and it is shown that the estimation method presented herein is acceptable qualitatively. The low-frequency resonant sway motions of a two-dimensional moored body in regular wave groups are also compared, and the numerical results show a tendency similar to the experiments when the wave drift damping is included in the equation of motion.

On the Increased Damping for a Moored Semi-Submersible During Low-frequency Motions in Waves

In order to investigate the increased damping of a moored semisubmersible during low-frequency motions in waves, free oscillation tests were conducted both in still water and regular waves. Test results show the influence of both wave and viscous forces on the low-frequency damping, and it seems that the interaction effects among columns play an important role in the increased damping due to wave effects in short waves. Calculations and experiments in regular wave groups were also compared to investigate the effect of increased damping due to waves on the low-frequency motions.

An experimental and analytical study of the effects of wave grouping on the slowly varying drift oscillation of a floating rectangular barge

An experimental study of the effects of wave grouping on the slowly varying drift oscillations of a floating moored rectangular barge has been presented. The work incorporates existing techniques to simulate wave groups with those that predict the slowly varying drift response of a moored model. The analytical model assumes that each pair of frequencies in a discrete wave spectrum will give rise to a regular wave group. Each of these regular wave groups will produce a low frequency force that is proportional to the product of the wave amplitudes in the group and varies at a frequency equal to the difference in the frequency pair. Furthermore, this low frequency force is related to the envelope of the regular wave group by a constant phase angle and the resultant force is determined by superposition of these regular slow-varying forces. Subsequently, the ensuing slowly oscillating motions are related to the envelope of the waves in a spectrum by a quadratic transfer function. A series of experimental tests were carried out on a model of a rectangular barge to evaluate the analytical model. The results demonstrate that the slowly varying drift response in linear waves can be related to the wave envelope by means of the transfer function.

Interaction of Waves With a Moored Semisubmersible

A semisubmersible moored in waves experiences a steady offset and two types of motion—a first-order motion at frequencies corresponding to the incident wave frequencies and a slowly oscillating motion near the natural frequency of the semisubmersible/mooring system. An extensive wave tank testing of a semisubmersible model was undertaken in which the motions of the semisubmersible and the loads in the mooring lines were measured. The semisubmersible was tested in the tank in a head sea as well as a beam sea heading in a series of regular waves, regular wave groups and irregular waves. The test results of the steady offset and first-order and slowly oscillating motions are presented for each heading and for each of these wave series as functions of the wave period. The experimental results are correlated with theoretical results based on a 3-D diffraction theory which takes into account the appropriate first and second-order terms. It is found that the nonlinear damping terms are quite important in explaining the behavior of the moored semisubmersible in waves and that the steady drift loads in wave groups can be determined from results based on regular waves.

LONG PERIOD DISTURBANCES DUE TO WAVE GROUPS

Measurements of long period disturbances associated with wave groups are compared with theoretical predictions. The experiments were carried out in a wave flume with an absorbing beach at one end. Tests were carried out using both regular wave groups (made up of two wave frequencies so that groups occur at a single beat frequency) and a random sea. The theoretical predictions are based on a Stokes expansion of the basic wave equations up to second order. The results are consistent with having three components in the long period disturbance at wave group periods. The first is set-down beneath wave groups, a disturbance that is tied to the wave groups travelling towards the beach. The second is a surf beat, a free long wave propagating back from the beach that is generated by reflection of the set-down from the beach. The third component is a secondary wave, a free long wave propagating towards the beach, that is generated by the wave-maker when it is only programmed to produce the primary wave system without its associated set-down beneath wave groups. The experimental results show that this third component can be minimised for both regular wave groups and a random sea by adding an appropriate long period movement of the wave-maker. Also, results are presented that indicate that long cycles of truly random waves, lasting many hours in model terms, are required to obtain reliable estimates of the effect of wave grouping on marine structures.

Mean and low frequency wave drifting forces on floating structures

A recently developed method, based on three-dimensional potential theory, to compute the mean wave drifting forces on a free floating structure in regular waves, is extended to include low frequency oscillatory components which arise when the structure is floating in regular wave groups consisting of two regular waves with small difference frequency. This completes the information necessary for the determination of the wave drifting forces under arbitrary irregular wave conditions. In regular wave groups the drifting forces not only depend on the first order velocity potential and the first body motions, but also on the wave exciting forces due to the low frequency part of the second order potential. For the general three-dimensional case the latter contribution can only be determined numerically and at the expense of long computation times. Since this contribution is generally not large compared to components which may be determined using linear potential theory it is included using a simple approximation. Results of the method of approximation are compared with some two-dimensional cases for which exact solutions are known. Results of computations of the total mean and low frequency surge forces on a rectangular barge and a column stabilized semi-submersible platform are presented. For both structures, the computed mean surge drifting forces in regular head waves are compared with results of model tests. The computed components of the total mean drifting forces are presented. It appears that for both the barge and the semi-submersible the same components are of importance. For the semi-submersible, the computed low frequency second order surge forces in head waves are compared with results obtained from a test in irregular head waves using cross-bispectral analysis methods.

The Slow Drift Oscillations of a Moored Object in Random Seas

Abstract The phenomenon of the slowly varying drifting force on a mowed object in a random sea is explained and illustrated from the results of several model tests with a rectangular barge. These tests, conducted at the Netherlands Ship Model Basin, were an extension of an object executed program. Using the results of measured or calculated drifting forces on an object moored in regular waves, a prediction can be Made of the drifting forces induced by wave trains consisting of regular wave groups. Also, for an irregular wave train the drifting force on the barge can be computed as a function of time, which makes it possible to calculate the surge motion of the barge. The results of tests and calculations show a reasonable agreement. Introduction In the last few years the problems concerning the mooring of objects in random seas have gained much attention as a result of the necessity to load and discharge big tankers in open sea, or because the sea bottom has to be explored and exploited by vessels operating from the water surface. Generally a floating object moored in waves will be subjected to forces causing horizontal and vertical motions and to moments causing angular motions about the horizontal and vertical axes. Here we will deal with the horizontal surge motion of a rectangular barge moored by means of linear springs in head waves. The surge motion can be split up into a mean excursion, a slowly varying motion, and a higher frequency oscillation around the slowly varying position. The period of the higher frequency oscillation is equal to that of the wave motion; and since a considerable amount of literature is available concerning this part of the motion, it will not be treated in this paper. From the results of model tests in regular waves the mean drifting force on the barge could be determined as a function of the wave frequency. Using these data, the long-periodical surge motion of the barge was calculated for different stiffnesses of the mooring system for the condition in which the barge was moored in a wave train consisting of regular wave groups. The results of these calculations are compared with model test results. From these and earlier executed tests it is clear that resonance phenomena may occur when the period with which the wave groups encounter the barge equals the natural period of the surge motion of the moored barge. period of the surge motion of the moored barge. It also appears to be possible to calculate the drifting force induced by regular wave groups when such a wave train is taken to consist of two regular waves with a small difference in frequency. The regular wave groups, used for a clear demonstration of certain long-periodical phenomena, have mainly educational value. Regular wave groups will seldom occur. Generally the wave height changes irregularly. To estimate the drifting forces exerted on an object in a particular irregular wave train as a function of time, a method exists which produces reasonable results. This method, based on the principle of known drifting force in regular waves, principle of known drifting force in regular waves, will be dealt with. Starting from the obtained drifting force, the surge motion of the object moored in this particular wave train can be calculated. This is illustrated by comparison of some calculated surge motion records with those of measured ones.