On the accurate and efficient calibration of a 3D wave basin

This paper describes the calibration procedure adopted for the new 3D wave basin located in the Hydrodynamics Laboratory at Imperial College London. Unlike traditional calibrations, based on observations of regular wave trains, the method described herein uses a focused wave approach. Such waves, produced by the constructive interference of freely propagating wave components, have led to a number of recent advances in theoretical wave modelling in which it was essential to know the underlying linear components. In the context of a laboratory study, similar advantages can be realised provided the linear wave components generated by the wave paddles are well defined. This, in turn, can only be achieved if the calibration is sufficiently accurate. The current study provides a calibration based upon a realistic JONSWAP spectrum, describes the details of the methodology employed, and highlights how the application of focused wave techniques eliminates spurious calibration effects due to unwanted reflections from the boundaries of the basin. The final calibration is verified through the generation of test cases, involving linear and nonlinear, unidirectional and directionally spread waves. These confirm both the accuracy of the calibration and the suitability of the methods employed.     

Deterministic combination of numerical and physical coastal wave models

A deterministic combination of numerical and physical models for coastal waves is developed. In the combined model, a Boussinesq model MIKE 21 BW is applied for the numerical wave computations. A piston-type 2D or 3D wavemaker and the associated control system with active wave absorption provides the interface between the numerical and physical models. The link between numerical and physical models is given by an ad hoc unified wave generation theory which is devised in the study. This wave generation theory accounts for linear dispersion and shallow water non-linearity. Local wave phenomena (evanescent modes) near the wavemaker are taken into account. With this approach, the data transfer between the two models is thus on a deterministic level with detailed wave information transmitted along the wavemaker.     

Wave crest and trough distributions in a broad-banded directional wave field

It is well established that the modulational instability enhances the probability of occurrence for extreme events in long crested wave fields. Recent studies, however, have shown that the coexistence of directional wave components can reduce the effects related to the modulational instability. Here, numerical simulations of the Euler equations are used to investigate whether the modulational instability may produce significant deviations from second-order statistical properties of surface gravity waves when short crestness (i.e., directionality) is accounted for. The case of a broad-banded directional wave field (i.e. wind sea) is investigated. The analysis is concentrated on the wave crest and trough distribution. For completeness a comparison with a unidirectional wave field is presented also. Results will show that the distributions based on second-order theory provide a good estimate for the simulated crest and trough height also at low probability levels.     

Diffraction of multidirectional random waves by multiple rectangular submarine pits

A numerical model is presented to predict the interaction of multidirectional random surface waves with one or more rectangular submarine pits. The water depth is assumed uniform and the method involves the superposition of diffraction solutions based on linearized shallow water wave theory obtained by a two-dimensional boundary integral approach. The incident wave conditions are specified using a discrete form of the Mitsuyasu directional spectrum. The present numerical model has been validated through comparisons with previous theoretical results for regular waves. Good agreement was obtained in all cases. Based on these comparisons it is concluded that the present numerical model is an accurate and efficient tool to predict the wave field around multiple submarine pits and navigation channels in many practical situations.     

Diffraction of a directionally spread wave group by a cylinder

The problem of diffraction of a directionally spread focused wave group by a bottom-seated circular cylinder is considered from the viewpoint of second-order perturbation theory. After applying the time Fourier transform and separation of vertical variable the resulting two-dimensional non-homogeneous Helmholtz equations are solved numerically using finite differences. The detailed formulation of the second-order radiation condition is presented. Numerical solutions of the problem are obtained for JONSWAP amplitude spectra for the incoming wave group with various types of directional spreading. The results are compared with the corresponding results for a unidirectional wave group of the same amplitude spectrum. Finally we discuss the applicability of the averaged spreading angle concept for practical applications.     

Wave propagation in a fully nonlinear numerical wave tank: A desingularized method

The problem of wave propagation in a fully nonlinear numerical wave tank is studied using desingularized boundary integral equation method coupled with mixed Eulerian–Lagrangian formulation. The present method is employed to solve the potential flow boundary value problem at each time step. The fourth-order predictor–corrector Adams–Bashforth–Moulton scheme is used for the time-stepping integration of the free surface boundary conditions. A damping layer near the end-wall of wave tank is added to absorb the outgoing waves with as little wave reflection back into the wave tank as possible. The saw-tooth instability is overcome via a five-point Chebyshev smoothing scheme. The model is applied to several wave propagations including solitary, irregular and random incident waves.     

Laboratory investigations on impulsive waves caused by underwater landslide

Laboratory investigations have been performed on the submarine landslide generated waves by performing 120 laboratory tests. Both rigid and deforming-slide masses are considered. The effects of bed slope angle, initial submergence, slide geometry, shape and deformation on impulse wave characteristics have been inspected. Impulse wave amplitude, period, energy and nonlinearity are studied in this work. The effects of bed slope angle on energy conversion from slide into wave are also investigated. Laboratory-based prediction equations are presented for impulse wave amplitude and period in near and far-field and are successfully verified using the available data in previous laboratory and numerical works.     

Scattering of obliquely incident water waves by partially reflecting non-transmitting breakwaters

In the present paper, analytic solutions are derived for scattering of water waves obliquely incident to a partially reflecting semi-infinite breakwater or breakwater gap. In order to examine the correctness of the derived solutions, they are compared with the solutions derived by McIver (1999) and Bowen and McIver (2002) for a semi-infinite breakwater and a breakwater gap, respectively, in the case of perfect reflection. The derived analytic solutions are used to investigate the effect of reflection coefficient of the breakwater and wave incident angle upon the tranquility at harbor entrance. The tranquility is deteriorated by the reflected waves as the reflection coefficient increases and as waves are incident more obliquely.     

Exact distributions for apparent waves in irregular seas

We discuss the long-run distributions of several characteristics for the apparent waves in a Gaussian sea. Three types of one-dimensional wave records are considered: 1) the seaway in time at a fixed position; 2) the instantaneous profile along a horizontal line; 3) the encountered seaway. Exact integral forms of the joint long run distributions are derived for the apparent periods, lengths, and heights. Results of numerical approximations of these distributions are presented in examples. For the computations we considered, as the input spectra, empirical estimates of the frequency spectra as well as JONSWAP type spectra. Effective algorithms are discussed and utilized in the form of a comprehensive computer package of numerical routines.     

Estimation of wave directional spreading in shallow water

This paper describes wave directional spreading in shallow water. Waves were measured for a period of 2 months using the Datawell directional waverider buoy at 15 m water depth on the east coast of India in the Bay of Bengal. The study also showed that in shallow water wave directional spreading was narrowest at peak frequency and widened towards lower and higher frequencies. The wind direction was found to deviate from the wave direction during most of the time. The unidirectional spectrum was found to be satisfactorily represented by Scott spectra.     

Nearshore directional wave measurements by surface-following buoy and acoustic Doppler current profiler

Directional energy spectra of nearshore surface waves were measured for a 3-year period (2004–2007) at a site with mean depth 14 m and mean tidal range 2.1 m. Triaxys surface-following wave buoys reported hourly directional wave energy spectra and wave parameters near the offshore end of the Savannah River Entrance Channel, Georgia, USA. An acoustic Doppler current profiler (ADCP) was located beside the wave buoy for 3 months. Directional and non-directional surface wave energy spectra and the corresponding bulk wave parameters (height, period, and direction) are compared for the two systems. Most parameters derived from the spectra agree closely; the most significant differences were found at the upper and lower frequency measurement limits, where signal-to-noise ratios were lower. The wave buoy consistently reports a small amount of energy below 0.05 Hz that does not appear in the ADCP-derived spectra and does not appear to be related to the mooring system. This leads to larger mean and peak periods reported by the buoy. All directional spectra were computed using the Maximum Entropy Method for both instruments, but the buoy, with spectra derived from six independent time series, provides lower directional resolving power than the ADCP, which utilizes twelve time series. Both systems gave similar results defining mean and peak wave directions, with the primary difference being that the ADCP indicates energy to be more tightly concentrated around the peak direction.     

Efficient computation of cross-spectral densities in the stochastic modelling of waves and wave loads

A new method is presented for efficient calculation of auto- and cross-spectral densities in the stochastic modelling of ocean waves and wave loads. As part of the short-term response analyses, the method may contribute to more efficient long-term response prediction. Specifically the cross-spectral densities of the first order wave excitation forces are considered, but the method is straightforwardly generalized to other cross-spectral densities, e.g. for wave elevation, wave kinematics or second order loads. The method can be used with any choice of directional spreading function, but special attention is given to the commonly used cos-2s type directional distribution. In addition to the development of the new method, the traditional method using the trapezoidal rule for numerical quadrature is improved by developing an adaptive way of choosing the number of integration points. The accuracy of the adaptive method and the new method is investigated, revealing rapid convergence for both methods. However, the new method appears more robust as it avoids so-called spurious hat errors. When applied to two different pontoon type floating bridges the adaptive method and the new method both achieve a great improvement in computational effort compared to the traditional trapezoidal rule method. When the dimensions of the floating bridge increase, i.e. the number of pontoons and their relative distances increase, the new method is superior with respect to computation time.     

On a near-optimal control approach for a wave energy converter in irregular waves

Real-time smooth reactive control and optimal damping of wave energy converters in irregular waves is difficult in part because the radiation impulse response function is real and causal, which constrains the frequency-dependent added mass and radiation damping according to the Kramers–Kronig relations. Optimal control for maximum energy conversion requires independent synthesis of the impulse response functions corresponding to these two quantities. Since both are non-causal (one being odd and other even), full cancellation of reactive forces and matching of radiation damping requires knowledge or estimation of device velocity into the future. To address this difficulty and the non-causality of the exciting force impulse response function, this paper investigates the use of propagating-wave surface elevation up-wave of the device to synthesize the necessary forces. Long-crested waves are assumed, and the approach is based on the formulations of Naito and Nakamura [2] and Falnes [22]. A predominantly heaving submerged device comprised of three vertically stacked discs driving a linear power take-off is studied. The overall formulation leads to smooth control that is near-optimal, given the approximations involved in the time-shifting of the non-causal impulse response functions and the consequent up-wave distances at which wave surface elevation is required. Absorbed power performance with the near-optimal approach is compared with two other cases, (i) when single-frequency tuning is used based on non-real time adjustment of the reactive and resistive loads to maximize conversion at the spectral peak frequency, and (ii) when no control is applied with damping set to a constant value. Simulation results for wave spectra over a range of energy periods and significant wave heights are compared for the three situations studied. While practical implementation presents engineering challenges, in terms of time-averaged absorbed power, unconstrained near-optimal control is found to perform significantly better than single-frequency tuning in the spectra with longer energy periods (>10 s for the present device), and somewhat better in the spectra with shorter energy periods (here ≤10 s).     

Kinematics of extreme waves in deep water

The velocity profiles under crest of a total of 62 different steep wave events in deep water are measured in laboratory using particle image velocimetry. The waves take place in the leading unsteady part of a wave train, focusing wave fields and random wave series. Complementary fully nonlinear theoretical/numerical wave computations are performed. The experimental velocities have been put on a nondimensional form in the following way: from the wave record (at a fixed point) the (local) trough-to-trough period, T_TT and the maximal elevation above mean water level, η_m of an individual steep wave event are identified. The local wavenumber, k and an estimate of the wave slope, ε are evaluated from ω²/(gk)=1+ε², where ω=2π/T_TT and g denotes the acceleration of gravity. A reference fluid velocity, is then defined. Deep water waves with a fluid velocity up to 75% of the estimated wave speed are measured. The corresponding kη_m is 0.62. A strong collapse of the nondimensional experimental velocity profiles is found. This is also true with the fully nonlinear computations of transient waves. There is excellent agreement between the present measurements and previously published Laser Doppler Anemometry data. A surprising result, obtained by comparison, is that the nondimensional experimental velocities fit with the exponential profile, i.e. e^ky, y the vertical coordinate, with y=0 in the mean water level.     

Profiles of fully developed (Airy) waves in different water depths

Airy waves have a sinusoidal profile in deep water that can be modeled by a time series at any point x and time t, given by η(x,t) = (H_o/2) cos[2πx/L_o − 2πt/T_w], where H_o is the deepwater height, L_o is the deepwater wavelength, and T_w is the wave period. However, as these waves approach the shore they change in form and dimension so that this equation becomes invalid. A method is presented to reconstruct the wave profile showing the correct wavelength, wave height, wave shape, and displacement of the water surface with respect to the still water level for any water depth.     

Separation of incident and reflected waves in wave–current flumes

A technique is developed to separate the incident and reflected waves propagating on a known current in a laboratory wave–current flume by analyzing wave records measured at two or more locations using a least squares method. It can be applied to both regular and irregular waves. To examine its performance, numerical tests are made for waves propagating on quiescent or flowing water. In some cases, to represent the signal noise and measurement error, white noise is superimposed on the numerically generated wave signal. For all the cases, good agreement is observed between target and estimation.     

Approximate Stream Function wavemaker theory for highly non-linear waves in wave flumes

An approximate Stream Function wavemaker theory for highly non-linear regular waves in flumes is presented. This theory is based on an ad hoc unified wave-generation method that combines linear fully dispersive wavemaker theory and wave generation for non-linear shallow water waves. This is done by applying a dispersion correction to the paddle position obtained for non-linear long waves. The method is validated by a number of wave flume experiments while comparing with results of linear wavemaker theory, second-order wavemaker theory and Cnoidal wavemaker theory within its range of application.     

Simulation of small-amplitude frequency-dispersive transient waves by means of the mild-slope equation

A numerical model based on the mild-slope equation is applied to reproduce the propagation of small-amplitude transient waves. The model makes use of the Fourier Transform to convert the time-dependent hyperbolic equation into a set of elliptic equations in the frequency domain. The results of two available experimental studies on tsunamis generated by landslides are used to validate the model, which appears to be able of carefully reproducing the effects of the frequency dispersion. An example application of tsunamis propagating around the Stromboli island is also presented to show the applicability of the present approach to real life scenarios. It is finally discussed how this model could be applied as support to a tsunami early warning system.     

Linear irregular wave generation in a numerical wave tank

In the design of any floating or fixed marine structure, it is vital to test models in order to understand the fluid/structure interaction involved. A relatively inexpensive method, compared to physical model testing, of achieving this is to numerically model the structure and the wave conditions in a numerical wave tank. In this paper, a methodology for accurately replicating measured ocean waves in a numerical model at full scale is detailed. A Fourier analysis of the measured record allows the wave to be defined as a summation of linear waves and, therefore, Airy's linear wave theory may be used to input the wave elevation and associated water particle velocities. Furthermore, a structure is introduced into the model to display the ability of the model to accurately predict wave–structure interaction. A case study of three individual measured waves, which are recorded at the Atlantic marine energy test site, off the west coast of Ireland, is also presented. The accuracy of the model to replicate the measured waves and perform wave–structure interaction is found to be very high. Additionally, the absolute water particle velocity profile below the wave from the numerical model is compared to a filtered analytical approximation of the measured wave at a number of time-steps and is in very good agreement.     

Unsteady ship waves in shallow water of varying depth based on Green–Naghdi equation

Based on Green–Naghdi equation this work studies unsteady ship waves in shallow water of varying depth. A moving ship is regarded as a moving pressure disturbance on free surface. The moving pressure is incorporated into the Green–Naghdi equation to formulate forcing of ship waves in shallow water. The frequency dispersion term of the Green–Naghdi equation accounts for the effects of finite water depth on ship waves. A wave equation model and the finite element method (WE/FEM) are adopted to solve the Green–Naghdi equation. The numerical examples of a Series 60 (C_B=0.6) ship moving in shallow water are presented. Three-dimensional ship wave profiles and wave resistance are given when the ship moves in shallow water with a bed bump (or a trench). The numerical results indicate that the wave resistance increases first, then decreases, and finally returns to normal value as the ship passes a bed bump. A comparison between the numerical results predicted by the Green–Naghdi equation and the shallow water equations is made. It is found that the wave resistance predicted by the Green–Naghdi equation is larger than that predicted by the shallow water equations in subcritical flow , and the Green–Naghdi equation and the shallow water equations predict almost the same wave resistance when , the frequency dispersion can be neglected in supercritical flows.