Linear Water Wave Research Articles

Hydrodynamic responses and layout optimization of wave energy converter arrays consisting of five-degree-of-freedom truncated cylinders in front of a vertical wall

The hydrodynamic responses and layout optimization of a group of cylindrical wave energy conversion devices (WEC) in front of a fully reflecting vertical wall are investigated. Each truncated floating cylinder can oscillate with five degrees of freedom, i.e., surge, sway, heave, roll, and pitch. Based on the linear water wave theory, an analytical solution is developed for the hydrodynamic problem. The results of specific parameter studies suggest that the wall reflection effect significantly improves the energy extraction performance of the WEC array with the appropriate parameter conditions. A multi-level optimization method based on a genetic algorithm is developed. This paper investigates the optimal layout of the six WEC arrays, composed of 2–7 buoys, respectively. Additionally, the impact of other degrees of freedom (DOFs), besides the heave mode, on the hydrodynamic performance of the array is investigated. For β ≤ π/12, there is no need to consider the impact of other DOFs on the energy extraction in heave mode. The dimensionless amplitudes of other DOFs gradually decrease as the equivalent constraint stiffness increases. For k0a > 1.0, the heave amplitude and energy capture performance of the WEC array are significantly smaller. However, the amplitudes of other DOFs still have considerable magnitudes for k0a > 1.0. Therefore, for the sea area with high-frequency incident waves (k0a > 1.0), setting up a power takeoff system on other DOFs of each buoy to extract energy is a feasible solution to improve the performance of the WEC array.

The Moving Element Method for the Dynamic Responses of Floating Double-Plate Structures on the Shallow Water

In this paper, the dynamic behavior of floating double-plate structures under a moving load is investigated in great detail with different factors utilizing the moving element method (MEM). The structures in this study are very large floating structures (VLFSs). The Kirchhoff–Love theory is used to model two floating plates connected by a Winkler-type elastic core, whilst the water surrounding the structure is simulated by the shallow linear water wave theory. In order to improve the computational efficiency of moving load problems, the MEM is adopted. Since the MEM uses fewer discrete elements than those of the traditional finite element method (FEM), the computational cost is reduced. The proposed two-plate model helps to reduce the influence of the surrounding water on the upper plate. Several numerical results are presented to investigate the influence of the ratio of thickness between layers as well as core stiffness on the displacements with different water depths, and load speeds.

A semi-analytical model of an array of moored floating flexible offshore net cages under current loads

An analytical model of an array of moored floating flexible net cage systems is developed to study the dynamics and wave loads on the net cages under the linear water wave theory and small-amplitude wave response assumption. The Fourier–Bessel series expansion of the velocity potential is derived for the regions enclosed under the open-water and cage systems and their proximity. The theoretical solution is obtained by employing the least squares approximation method along with matching the velocity and pressure at the fluid-structure interfaces. The present result of horizontal displacement and horizontal wave load are compared with a published finite element analysis and with experimental data sets, respectively. Further, the results of cage displacements, cage accelerations, and wave loads on the different arrangements for mooring stiffness, cage radius, cage heights, and current speeds are compared and analysed. Moreover, the present analysis will be helpful to understand designing further a multiple floating fish cage system with flexible floaters application to offshore aquaculture.

Radiation study of coaxial porous cylindrical structures for a given oscillation

The paper uses the eigenfunction expansion method to investigate the radiation of coaxial porous cylindrical structures in the ocean for a given oscillation. The study of the coaxial porous cylindrical structures considers two cases: (1) a coaxial porous structure placed on the seabed, and (2) a floating coaxial porous structure. Based on the linear water wave theory and the eigenfunction expansion method, the equations for the velocity potential, the added mass, and the damping coefficient of the coaxial porous cylindrical structure are obtained. Darcy’s law is satisfied at the porous boundary. The continuity condition is satisfied between the fluid regions. The added mass and damping coefficient are calculated based on the velocity potential. The radiation results of the two cases are studied, and the influence of different parameters on the structures can be obtained. In addition, the porous parameters and radius ratio have a significant effect on the hydrodynamic coefficients (the added mass and the damping coefficient) of the structure, so the appropriate porous parameters and radius ratio can be selected to reduce the effect on the structure. The results in this paper provide clear information for the selection of appropriate parameters when engineering in offshore and coastal regions.

Wave power extraction from slanted oscillating water columns in the presence of an array of arbitrary trenches

This paper investigates the impact of arbitrary bottom topography on the efficiency of a slanted oscillating water column device located in near-shore areas. The device incorporates a turbine positioned at its opening, which harnesses the oscillation of the fluid at the free surface, created by recurrent incident waves, to propel an air volume in a uni-direction. To analyze the wave power absorption capability of the system, an eigenfunction expansion and a boundary element model based on linear water wave theory are employed. The notion is to use the anticipated amplification of scattered and reflected wave fields resulting from the existence of a sloping wall to improve the wave power absorption caused by the walls' wave reflections. The accuracy of the current study's numerical results is ratified by conducting a comparative study using the available data given by other researchers. The study demonstrates that multiple efficiencies occur in the case of the long-wave region as compared to the short-wave region. Additionally, both the shape and magnitude of the efficiency curves are influenced by the slanted design of the device's front wall and the number of trenches.

Hydrodynamic response of dual obliquely submerged non-uniform flexible thin plates

Examining inclined flexible plates with variable thickness becomes crucial when it comes to optimizing or achieving controlled reflection and transmission of waves, especially for the construction of breakwaters. Unlike horizontal breakwaters, inclined barriers have the ability to penetrate through numerous layers of the fluid, having different particle velocities and foster their interactions. This causes wave breaking which leads to loss of wave energy. Also, despite exhibiting a similar behavior, vertical structures do not attenuate waves as effectively as inclined ones do. Additionally, the resonant motion of the fluid trapped between two structures proves to enhance the attenuation, thus it is recommended to include dual structures in the model rather than just one. Therefore in the present study, we examine the water wave scattering phenomenon by a pair of symmetric flexible thin plates with non-uniform thickness obliquely submerged in deep water. We use the linear water wave theory and Kirchhoff’s thin plate theory to model the physical problem. The boundary value problem is converted into a system of coupled integral equations using repeated integration and Green’s integral theorem. Using appropriate approximations, this system is solved and its solutions are used to determine numerical values of different hydrodynamic quantities. Results of two horizontal as well as two vertical plates could be obtained from the present model, thus it is a very general model. Also results are illustrated to analyze the contribution of the thickness variation and the inclinations of the two flexible plates towards the wave scattering process and some related physical quantities.

Wave trapping by freely floating obstacles in a discretely stratified fluid

In the context of the linear water wave theory, we model the interaction of time‐harmonic water waves with an infinite array of three‐dimensional freely floating bodies in a multilayer fluid through a spectral boundary value problem, consisting of a differential equation coupled with an algebraic system. We present a variational and operator formulation for the problem and derive sufficient conditions for the existence of trapped waves. We give examples of floating obstacles supporting trapped waves in a discretely stratified fluid in different obstacle configurations with a variable number of layers. Some conclusions regarding the dependence of the sufficient condition on the number of fluid layers are drawn, considering whether the sufficient condition is of the potential or algebraic type.

Trapped modes along periodic structures submerged in a two‐layer fluid with background steady flow

The trapping of linear water waves by infinite arrays of three‐dimensional fixed periodic structures in a two‐layer fluid, where the layers are considered semi‐infinite in depth, have a common interface and move each with an independent uniform velocity with respect to the ground, is studied. The existence of real solutions to the dispersion relation demands a further stability condition on the layer velocities. From the variational formulation, after certain choices of background steady flow, results a nonlinear spectral problem, which upon a sensible linearization gives a geometric condition ensuring the existence of trapped modes (within the limits set by the stability condition). Symmetries reduce the global problem to the first quadrant of the velocity space. Examples are shown of configurations of obstacles that are both independent of the layer velocities and dependent only on their difference. Future developments are suggested.

A mathematical study of a two-layer fluid flow system in the presence of a floating breakwater in front of VLFS

The present study deals with the usefulness of a surface-piercing porous structure of definite width in a two-layer fluid placed at a finite distance from a floating elastic plate to mitigate the hydrodynamic response of the elastic plate. We use linear water wave theory with the eigenfunction expansion approach to develop analytical solutions to the model problem in finite water depth. Here, we identify and explore the complete parameter space associated with the problem by examining the hydrodynamic coefficients for waves in surface and internal modes, plate elevation, shear force, bending moment, wave loads on the elastic plate, flow amplitude in the plate-covered region and wave attenuation. We investigate the effectiveness of structures of various configurations and geometries on wave scattering. This study demonstrates an oscillatory pattern of wave reflection in the confined region between the plate and the breakwater. Moreover, optima in the oscillatory pattern can be observed due to the resonating interaction of waves and the breakwater-plate system. Furthermore, the research shows that an ideal width and distance between the breakwater and floating plate can be calculated while assuming an appropriate configuration of the porous structure to create a breakwater-plate system with a reasonable efficiency that possesses both reflection and dissipation properties. The design of various types of coastal structures employed in the marine environment for the reflection and dissipation of wave energy at continental shelves dominated by stratified fluid, described here as a two-layer fluid, will significantly benefit from the findings of the current study.

Performance of two vertically submerged piezoelectric plate wave energy converters in presence of a non-flat flexible barrier

Wave energy converters together with flexible breakwaters not only provide shelter to the energy devices against wave attacks but also offer improved performance, longer survivability and lower costs. Thus, in the present work we are interested to investigate the interaction of linear water waves with an array of three thin vertical plates, comprising of two plate wave energy converters and one flexible breakwater, submerged in water of finite depth in a two dimensional space. The plate in the middle of the setup is a flexible plate of variable thickness, and on either side of it are piezoelectric plate wave energy converters with constant thickness. The problem is solved by converting the associated boundary value problem into a system of coupled integral equations. A modified form of the energy balance equation is derived, which results due to the presence of the two piezoelectric plates. We investigate how the structural properties of the piezoelectric plates like their submergence depths, plate lengths, distance from the breakwater and different edge conditions can affect the wave attenuation and wave power absorption efficiency. Further, it is also examined how the presence of the non-uniform flexible plate with different flexibility and lengths can improve the performance of the system. It is analysed that shorter sized piezoelectric plate wave energy converters in the presence of non-flat flexible plate offer higher wave power absorption efficiency at lower frequencies.

Investigation of Bragg resonant reflection of linear and nonlinear water waves by multiple fixed floating structures based on viscous meshfree method

The Bragg resonant reflection of linear and nonlinear water waves over a classical Bragg structures is numerically studied. The Smoothed Particle Hydrodynamics (SPH) method is a Lagrangian meshfree method, and SPH is adopted based on the code DualSPHysics, which takes the effect of viscosity into consideration. The sensitivity of particle spacing and a benchmark test to the transmission coefficient is analyzed. In the propagation of both linear and nonlinear water waves over Bragg floating structures, the Bragg resonant reflection occurs when the wavelength (λ) is about double of the spacing (L). The phase upshift can be observed in the cases of linear and nonlinear water waves, which means the minimum transmission coefficient appears when 2L/λ>1.0, and the phase upshift occurs with the increase of the width of floating in the nonlinear water waves. Besides, in the resonant case with three or more floating structures, the transmitted wave height is reduced by more than 70.9% and 73.5% relative to the incident wave height under linear and nonlinear water waves, respectively. The results give a reference for the design of wave attenuation in floating periodic structures in both linear and nonlinear water waves based on the Bragg rules.

Approximate solutions to shallow water wave equations by the homotopy perturbation method coupled with Mohand transform

In this paper, the Mohand transform-based homotopy perturbation method is proposed to solve two-dimensional linear and non-linear shallow water wave equations. This approach has been proved suitable for a broad variety of non-linear differential equations in science and engineering. The variation trend of the water surface elevation at different time levels and depths are given by some graphs. Moreover, the obtained solutions are compared with the existing results, which show higher efficiency and fewer computations than other approaches studied in the literature.

Long-time asymptotics of a linear higher-order water wave model

Long-time asymptotics of a linear higher-order water wave model

Tsunami generation by a seabed deformation in the presence of a viscoelastic mud

In this work, under the assumption of linear water waves, we study tsunamis generated by a seabed deformation in the presence of viscoelastic mud. We divide the total control volume under study into a water layer, which is assumed to be an irrotational and inviscid flow, and a mud layer with viscoelastic properties that obeys a linear Maxwell rheological model. Considering that fluid layer thicknesses are of the same order of magnitude and that they are much smaller than the characteristic horizontal length of the seabed deformation, we obtain a semi-analytical solution that models the evolution of the free surface elevation. For the above limits, the fluid motion in the water layer is essentially horizontal. Passive and active tsunami generation cases are analyzed. The seabed deformation is modeled as a Heaviside step function. For an active generation case, when the mud layer thickness increases, the tsunami's maximum amplitude decreases. For the passive generation case, the tsunami's maximum amplitude remains constant in a finite time interval of the same order of magnitude as the characteristic time; this phenomenon does not occur for the active generation case.

Analytical solution to wave diffraction by a hemisphere in infinite water depth using the multipole expansion method

Linear water wave diffraction by a hemisphere in infinite water depth is investigated using the multipole expansion method in this paper. In the spherical coordinate system, the diffraction potential is expressed in the form of multipole wave potentials and the wave-free potential, as derived by Hulme (1982) for radiation potentials. The issues associated with the high-order multipole expansion are solved in this paper. The new method can calculate the multipole wave potentials at arbitrary orders and thus successfully calculate the diffraction potentials on the body surface and free surface around a hemisphere. The wave exciting forces acting on hemispheres are calculated using the near-field and far-field methods, which utilizes the radiation potential computed by Hulme's (1982) method to verify the accuracy of the analytical solution in this paper. Compared with the far-field method, the analytical solution in this paper converges more quickly as the truncation term N increases, especially when the wavenumber Ka is large. In this paper, the theory of applying the multipole expansion method to solve hydrodynamic problems with a hemisphere on the free surface is completed, which makes the multipole expansion method no longer limited to low-order expansion calculations and can be applied to more fields.

UBathy (v2.0): A Software to Obtain the Bathymetry from Video Imagery

UBathy is an open source software developed for bathymetry estimation from video images. The proposed scheme is based on extracting the wave modes from videos of the nearshore surface wave propagation. These videos can be formed either from raw camera images, which must have been previously calibrated, or from georeferenced planviews. For each wave mode extracted from the videos, the wave frequency and the spatially dependent wavenumbers are obtained. The frequencies and wavenumbers from different videos are used to estimate the bathymetry by adjusting the dispersion relationship for linear surface water waves. The bathymetry at different times can further be weighted and aggregated using the Kalman filter. The new software is suitable for Argus-type video monitoring stations and for moving cameras mounted on drones or satellites, and it is meant for users familiar with coastal image processing and suitable for non-experienced users. The software and an application example are available on the GitHub platform.

Asymptotic behaviour of a linearized water waves system in a rectangle1

We consider the asymptotic behaviour of small-amplitude gravity water waves in a rectangular domain where the water depth is much smaller than the horizontal scale. The control acts on one lateral boundary, by imposing the horizontal acceleration of the water along that boundary, as a scalar input function u. The state z of the system consists of two functions: the water level ζ along the top boundary, and its time derivative ∂ ζ ∂ t . We prove that the solution of the water waves system converges to the solution of the one dimensional wave equation with Neumann boundary control, when taking the shallowness limit. Our approach is based on a special change of variables and a scattering semigroup, which provide the possiblity to apply the Trotter–Kato approximation theorem. Moreover, we use a detailed analysis of Fourier series for the dimensionless version of the partial Dirichlet to Neumann and Neumann to Neumann operators.

Trapped waves within the blocking frequency under compressed sea ice and two-dimensional current

The existence of trapped modes due to a horizontal submerged rigid cylinder placed below a compressed ice-covered surface is shown to exist within the frequency band of wave blocking by employing linearized water wave theory and with the inclusion of the effects of a two-dimensional current and the obliqueness of the propagating wave. A new modified multipole expansion method catering to the multiple propagating modes is proposed. The impact of different choices of propagating modes at a fixed frequency within the above-mentioned band on the trapped modes is detailed. Multiple trapped modes whose number depends on the direction and magnitude of the current exist within the frequency band. Out of the three choices of current types considered, while two of them result in the generation of trapped modes for all three possible choices of the propagating wave, the remaining one shows that only one propagating mode can generate a trapped wave. A break in the continuous spectrum of the trapped mode frequency due to the wave blocking and at the point of inflexion is presented. There exists a region of angle of propagation within which an abundance of the trapped mode is shown. The surface profiles for different angles of propagation and choices of propagating modes are illustrated graphically. The trapped modes associated with a higher angle of propagation decay at a faster rate.

Numerical investigation of diffraction and radiation of surface gravity waves by a rectangular structure over a stepped bottom with and without a vertical wall

A numerical investigation based on the boundary element method is conducted to analyze the effect of step bottom on the radiation of surface gravity wave by the rectangular marine structure under the assumption of a two-dimensional linearized water wave theory. The investigation is supplemented by analyzing the presence of a back wall located near the floating structure. The accuracy of the method is validated using the known special case of published results. Hydrodynamic coefficients such as added mass and damping coefficients and wave excitation force coefficients are evaluated in both cases. The influence of depth ratio and the distance of step from the center of the structure is calculated to understand the role of the stepped bottom on the hydrodynamic coefficients. On many occasions, these results are compared with the flat bottom case. The study results are helpful to understand the influence of the step-type bottom profile on the design of floating rectangular structures in various marine structure applications and wave energy converter devices.

Wave-triggered breakup in the marginal ice zone generates lognormal floe size distributions: a simulation study

Abstract. Fragmentation of the sea ice cover by ocean waves is an important mechanism impacting ice evolution. Fractured ice is more sensitive to melt, leading to a local reduction in ice concentration, facilitating wave propagation. A positive feedback loop, accelerating sea ice retreat, is then introduced. Despite recent efforts to incorporate this process and the resulting floe size distribution (FSD) into the sea ice components of global climate models (GCMs), the physics governing ice breakup under wave action remains poorly understood and its parametrisation highly simplified. We propose a two-dimensional numerical model of wave-induced sea ice breakup to estimate the FSD resulting from repeated fracture events. This model, based on linear water wave theory and visco-elastic sea ice rheology, solves for the scattering of an incoming time-harmonic wave by the ice cover and derives the corresponding strain field. Fracture occurs when the strain exceeds an empirical threshold. The geometry is then updated for the next iteration of the breakup procedure. The resulting FSD is analysed for both monochromatic and polychromatic forcings. For the latter results, FSDs obtained for discrete frequencies are combined following a prescribed wave spectrum. We find that under realistic wave forcing, lognormal FSDs emerge consistently in a large variety of model configurations. Care is taken to evaluate the statistical significance of this finding. This result contrasts with the power law FSD behaviour often assumed by modellers. We discuss the properties of these modelled distributions with respect to the ice rheological properties and the forcing waves. The projected output can be used to improve empirical parametrisations used to couple sea ice and ocean wave GCM components.