Linear Water Wave Theory Research Articles

Surface gravity wave interaction with a horizontal flexible floating plate and submerged flexible porous plate

An analytical study associated with surface gravity wave interaction with a horizontal flexible floating and a submerged porous plate is presented under linearized water wave theory and small amplitude structural response. The expansion formulae of the referred problem based on Green's function technique is derived using the two-dimensional Laplace equation and fundamental solution of source potentials in finite and infinite water depths. The velocity potential for the free oscillations in a closed basin is derived and the hydroelastic wave characteristics in floating and submerged flexural modes are analyzed in specific cases. Further, the usefulness of the expansion formula is demonstrated by analyzing a real physical problem of wave interaction with a moored flexible floating plate and a moored submerged flexible porous plate of finite dimensions in finite water depth. The convergence of the present solution is checked and the effect of a moored finite submerged porous plate on the moored finite flexible floating plate with different design parameters are analyzed. It is observed that the present analysis may be helpful for better understanding in designing a floating and submerged flexible porous plate system for application as a breakwater.

Hydrodynamic Forces on a Submerged Circular Cylinder in Two-layer Fluid with an Ice-cover

We consider problems based on linear water wave theory concerning the interaction of wave with horizontal circular cylinder submerged in two-layer ocean consisting of a upper layer of finite depth bounded above by an ice-cover and below by an infinite layer of fluid of greater density, the ice-cover being modelled as an elastic plate of very small thickness. Using the method of multipoles, we formulate the problems of hydrodynamic forces on a submerged cylinder in either the upper or the lower layer. The vertical and horizontal forces on the circular cylinder are obtained and depicted graphically against the wave number for various values of flexural rigidity of ice-cover to show the effect of the presence of ice-cover on these quantities. Also when the flexural rigidity and surface density of the ice-cover are taken to be zero, the ice-cover tends to a free-surface. Then all the forces are the same as in the case of two-layer fluid with free surface.

Diffraction and Radiation of Water Waves by a Heaving Absorber in Front of a Bottom-Mounted, V-shaped Breakwater of Infinite Length

In the present study, the problems of diffraction and radiation of water waves by a cylindrical heaving wave energy converter (WEC) placed in front of a reflecting V-shaped vertical breakwater are formulated. The idea conceived is based on the possible exploitation of amplified scattered and reflected wave potentials originating from the presence of V-shaped breakwater, towards increasing the WEC’s wave power absorption due to the wave reflections. An analytical solution based on the method of images is developed in the context of linear water wave theory, taking into account the hydrodynamic interaction phenomena between the converter and the vertical wall. Numerical results are presented and discussed concerning the hydrodynamic forces on the absorber and its wave power efficiency for various examined parameters, namely, the breakwaters’ forming angle, the distance between the converter and the vertical walls and the wave heading angle. The results show that the amount of the harvested wave power by the WEC in front of the walls is amplified compared to the wave power absorbed by the same WEC in the open sea.

WATER WAVE SCATTERING BY A THIN VERTICAL SUBMERGED PERMEABLE PLATE

An alternative approach is proposed here to investigate the problem of scattering of surface water waves by a vertical permeable plate submerged in deep water within the framework of linear water wave theory. Using Havelock’s expansion of water wave potential, the associated boundary value problem is reduced to a second kind hypersingular integral equation of order 2. The unknown function of the hypersingular integral equation is expressed as a product of a suitable weight function and an unknown polynomial. The associated hypersingular integral of order 2 is evaluated by representing it as the derivative of a singular integral of the Cauchy type which is computed by employing an idea explained in Gakhov’s book [7]. The values of the reflection coefficient computed with the help of present method match exactly with the previous results available in the literature. The energy identity is derived using the Havelock’s theorems.

Scattering of Gravity Waves by a Rectangular Floating Flexible Porous Plate

Scattering of oblique surface gravity waves by a finite, floating porous-elastic plate is investigated, with assumptions of linear water wave theory and plate response. A boundary value problem is set up, wherein the thin plate equation together with a porosity parameter is used to formulate the condition on the floating plate. A matched eigenfunction approach is adopted for the solution of this problem, with roots of the dispersion relation being located with the aid of contour plots, and various hydrodynamic scattering quantities are computed. Energy dissipation due to plate porosity is seen to have a significant impact on both reflection and transmission of waves, while flexibility of plate only alters the extent of wave reflection by porous elastic plates. An oscillatory trend is shown by reflection coefficient for smaller values of relative plate width, and there is no variation in reflection or transmission coefficients when the plate width is increased beyond a certain cut-off value. Comparison of scattering properties of four different types of plates highlights the effects of porosity and flexibility and establishes the superiority of a flexible porous plate as a wave attenuating device, with moderate reflection, high energy dissipation and low transmission.

Scattering and trapping of water waves by a composite breakwater placed on an elevated bottom in a two-layer fluid flowing over a porous sea-bed

The present work is concerned with the scattering and trapping of oblique surface waves due to a vertical composite type porous breakwater placed on an elevated rigid block in a two-layer fluid. The ocean depth is taken to be finite and the sea-bed to be porous. The problem is studied by employing linearized water wave theory and eigenfunction expansion. The objective is to investigate the impact of various porous parameters in wave attenuation for surface and interface modes. Water waves propagate through the porous structure with distinct eigenvalues. Further, a perforated front wall on the structure is also considered to study its effect in wave trapping. In the scattering model, reflection and transmission coefficients are investigated for two propagating modes along with the waveloads on the breakwater in order to ascertain the appropriateness of the structures of various configurations. Similarly, appraisals of the reflection coefficients, their effects on the free surface and interface elevations and the waveload on the structure as well as on the wall are analyzed to review the impact of various structure characteristics in the trapping model. In choosing the present model, it is ensured that the model conforms to realistic physical scenario so that it is useful for applications. Also, in carrying out numerical experiments, the selection of values of parameters is done in such a way that they are physically meaningful. Before proceeding with the investigation, validation of the present model is carried out by comparing a present result with one available in the literature. It points to a very good agreement which establishes the efficiency of the present model and the associated study which paves the way for completing the present study. The investigation establishes that, for a suitable configuration of the porous structure, an optimum width can be worked out to design a breakwater of reasonable efficiency possessing the characteristics of both reflection and dissipation. The problems are solved analytically and the results are presented in graphical form. This kind of study is likely to have immense significance for designing of different types of coastal structures with respect to reflection and dissipation of wave energy at continental shelves which is influenced by a stratified fluid, which is modeled in this work as a two-layer fluid for convenience.

Wave power extraction analysis on a dual-chamber oscillating water column device composed by two separated units: An analytical study

The hydrodynamic characteristics of a dual-chamber oscillating water column device system with a gap between each unit is considered in this paper. A theoretical model is established to explore its wave power extraction performance under the framework of linear water wave theory and matched eigenfunction technique. A numerical strategy of successive approximation is utilized to seek for the optimal turbines combinations for the optimal energy absorption efficiency. Effects of the total chamber breadth, lip-wall drafts, breadth ratio for front and rear unit and gap length on the wave power extraction efficiency are explored and analysed. It is found that a larger total chamber breadth is more beneficial for the wave power extraction both for short and long wave regimes, and when the front and rear unit breadth ratio is 2/3, a more desired efficiency distribution could be obtained. Though the increase of lip-wall drafts (except for the most front one) could lead to a more satisfactory peak efficiency value and a much broader effective frequency bandwidth, the superiority of present system lies in the condition with relatively smaller lip-wall drafts in contrast to the single-chamber one. In addition, the gap length can effectively contribute to the formation of multiple peak efficiency values at the short wave regimes, especially under the condition that with a large gap length and significant improvement of efficiency could be achieved within a specified range of wave frequencies.

Hydrodynamic analysis of a seaside quarter-circular breakwater with an array of porous cages using DBEM

ABSTRACT The gravity wave interaction with an array of porous cages in the presence of a seaside submerged quarter-circular porous breakwater (QCPBW) is analyzed within the framework of linearized water wave theory. A quadratic pressure drop is adopted for the wave past the porous structures to capture the effect of wave height on the energy dissipation. The boundary value problem is solved using the iterative Dual Boundary Element Method (DBEM). The numerical code is validated with known results in the literature. One or upto four units of submerged and bottom fixed porous cages are considered. The effect of porosity on QCPBW and porous cages are investigated pragmatically to study the scattering and wave forces. It is found that introducing a single porous cage with 10% porosity at the rear side of a QCPBW with a porosity of 25% can reduce the wave transmission coefficient by about 50% for relative water depth . Increasing the porosity of the single cage from 10% to 40% has a significant reduction on the horizontal and vertical force coefficients on the cage, whereas the change is insignificant on QCPBW. The horizontal wave force is three to four times higher when compared to the vertical wave force for both QCPBW as well as on the porous cage. For a system comprising four units of cages, both the horizontal and vertical wave forces on cage 1, 2, 3, and 4 reduce progressively (cage 2 is at the lee side of cage 1 and so on). The results of the present study can be used for the optimal design of open sea porous systems.

Dynamic wave induced loads on a moored flexible cylindrical net cage with analytical and numerical model simulations

A mathematical model associated with wave interaction with a moored floating flexible cylindrical net cage is developed based on linearized water wave theory. The analytical solution is obtained using Fourier Bessel series solution and least squares approximation method along with matching the velocity and pressure at the fluid-structure interfaces. The analytical results of the horizontal displacement and acceleration for different design parameters are compared with numerical simulations and they have a good level of agreement. For validation, the numerical model results are compared with existing published experimental results. The effects of mooring stiffness, membrane tension, and cage height on horizontal wave loads are determined from the analytical solutions. The loads on the mooring lines are assessed with the numerical tool. The comparison results indicate that the analytical model can be used with confidence in the design of a flexible cylindrical net cage for application to offshore aquaculture.

Wave Trapping Efficiency of a Flexible Membrane Near a Partially Reflecting Seawall

Abstract The gravity wave interaction with a flexible membrane placed at a finite distance from the partially reflecting seawall is analyzed under the framework of linear water wave theory using the multi-domain boundary element method (BEM). The flow through a flexible membrane is assumed to follow Darcy’s law in addition to membrane displacements. As a viable alternative to the existing wave dampers, the flexible membrane is examined for the effective dampening of incident waves. The correctness of the numerical results is affirmed with the known results available in the literature. The effect of membrane tension, submergence depth, membrane width, porosity, angle of inclination, and confined chamber spacing on hydrodynamic coefficients is discussed as a function of dimensionless wavenumber. The partially reflecting harbor wall diminishes the wave reflection coefficient in the long-wave regime. The increase in the flexible membrane width does not necessarily ensure the ideal wave capturing performance. A shift in the peak of the maximum deflection is observed with the increase of membrane width while there is a shift in peak outward for the increase in the submergence depth. Moreover, the maximum deflection is found to decrease with the increase in porosity, and it is 62% reduction for membrane porosity b = 1 due to the significant wave damping. The wave run-up and the wall force coefficients are found to be minimum when the relative plate width is B/h = 1. The present study is expected to be useful for the design of cost-effective wave attenuating systems.

Surface wave scattering by multiple flexible fishing cage system

A study of the wave dynamics around a multiple cylindrical fishing cage system is carried out under the assumption of linear water wave theory and small-amplitude wave response. The Fourier–Bessel series expansion of the velocity potential is derived for the regions enclosed under the open-water and cage systems and the immediate vicinity. The scattering between the cages is accounted for by employing Graf's addition theorem. The porous flexible cage system is modeled using Darcy's law and the three-dimensional membrane equation. The edges of the cages are moored along their circumferences to balance its position. The unknown coefficients in the potentials are obtained by employing the matched eigenfunction method. In addition, the far-field scattering coefficients for the entire system are obtained by expanding the Bessel and Hankel functions in the plane wave representation form. Numerical results for the hydrodynamic forces, scattering coefficients, and power dissipation are investigated for various cage and wave parameters. The time simulation for the wave scattering from the cage system is investigated. The study reveals that wave loading on the cage system can be significantly reduced by the appropriate spatial arrangement, membrane tension, and porous-effect parameter. Moreover, the far-field results suggest that the cage system can also be used as a breakwater.

Hydrodynamic force and wave run-up due to diffraction of ocean water waves by a surface-piercing bottom-mounted compound partial-porous cylinder

Here we consider the case of a train of linear water waves incident on a bottom-mounted surface-piercing compound partial-porous cylinder consisting of two coaxial cylinders of which the upper cylinder is hollow with a thin porous side wall and the lower cylinder, with radius greater than that of the upper one, is rigid. Subsequently, we examine the associated hydrodynamic forces. Using linear water wave theory and eigenfunction expansion, the problem is developed in terms of suitable velocity potentials. The important boundary condition on the porous boundary is defined with the aid of Darcy’s law. The matching conditions across the linear interface between successive fluid domains arising due the continuity of pressure and velocity are suitably used. Thereafter, a system of linear equations arises in terms of the unknowns solving which the hydrodynamic force and wave run-up for the compound partial-porous cylinder are calculated. Various numerical experiments show the effect of different parameters, such as porosity of the upper cylinder, draft ratio, the ratio of radii of the upper and lower cylinders and the depth of water on hydrodynamic force and wave run-up. It is observed that the wave force takes higher values corresponding to lower values of radius ratio, draft and porous coefficients. Further, for fixed values of radii, porosity and depth, the wave run-up is observed to be more when the wavenumber takes increasing values. The obtained results establish that different parameters may be suitably chosen in order to design useful ocean structures which may be installed as wave-absorbers for various activities in ocean including capturing of wave energy. Further, such structures can also act as the base of windmills installed in oceans to extract wind energy. The efficiency of the model developed is validated by comparing it with an available established result from which an excellent agreement is observed.

Analytical study of wave diffraction by an irregular surface located on a flexible base in an ice-covered fluid

The reflection and transmission of surface waves propagating over an irregular surface located on a flexible base in an ice-covered fluid are analyzed within the context of linearized water wave theory. The ice-floe and flexible bed surface are assumed as narrow elastic sheets with different compositions. Under such circumstances, there are two types of proliferating waves that exist for any specific frequency. The proliferating waves having smaller wavenumber spread at just beneath the ice-floe (ice cover mode) and the other spreads over the flexible bottom of the fluid (flexural base mode). An elementary perturbation theory is used for reforming the governing boundary value problem (bvp) to a first-order bvp which is solved by utilizing the Green’s function technique. The first-order correction of the reflection and transmission coefficients are calculated in the form of integrals comprising of a function which represents the base deformation. A particular example of base deformation is taken to evaluate all these coefficients and the results are depicted graphically. The major strength of the recent study is that the results for the values of reflection and transmission coefficients for both the wavenumbers are established to meet the energy relation almost exactly.

Wave radiation by a sphere in three-layer fluid

Using linear water wave theory, we consider the wave radiation (both heave and sway) by a sphere submerged in a three-layer fluid consisting of a layer of finite depth bounded above by a fresh water of finite depth with free surface and below by an infinite layer of water of greater density. The sphere is submerged in either of the three-layers. Each problem is reduced to an infinite system of linear equations by employing the method of multipoles and the system of equations are solved numerically by standard technique. The hydrodynamic forces (added-mass coefficients and damping coefficients) for both the heaving and swaying problems are obtained and depicted graphically against the wave number. When the density ratio of upper and middle layer made to approximately one, curves for added-mass and damping coefficients almost coincide with the corresponding curves for the case of a two-layer fluid with free surface. This means that in the limit, the density ratio of upper and middle layer go to approximately one, the solution agrees with solution for the case of a two-layer fluid with free surface.

Interaction of flexural gravity wave in ice cover with a pair of bottom-mounted rectangular barriers

This paper considers the reflection and transmission of flexural gravity waves by a pair of bottom-mounted rectangular barriers of arbitrary height and width. Depth of the fluid considered here is finite but not necessarily uniform. The fluid region is bounded by an infinite long thin ice sheet of homogeneous properties. Using linearized water wave theory the velocity potential associated with the motion is formulated. Eigenfunction matching technique is used to convert the boundary value problem to a set of integral equations which are solved by using multi-term Galerkin method. The reflection and transmission coefficients are calculated and represented graphically. An expression for the energy balance relation is given and checked carefully. A model of the problem in which the fluid is bounded by a free surface is recovered. For a set of physical parameters the results are analyzed through a number of figures. The convergence and correctness of the method is confirmed by comparing the results with the one available in the literature. It is shown that a pair of thick bottom-mounted barriers have significant impact on reflected and transmitted flexural gravity waves. Resonance effects are observed and verified for different parameters of the current problem.

Integral equation technique for water wave interaction by an array of vertical flexible porous wave barriers

AbstractScattering of obliquely incident surface gravity waves by an array of partial flexible vertical porous wave barriers of varied configurations is studied in finite water depth and deep water cases. For the mathematical modeling purpose, linearized water wave theory is considered. Firstly, the problem associated with double vertical barriers is handled for a solution. To solve the problem, the associated BVP (boundary value problem) is transformed into a system of Fredholm integral equations of the second kind. By using suitable quadrature formulae, these integral equations are converted to a system of linear algebraic equations. These are solved to compute various important physical quantities associated with wave scattering. The energy identities for both finite water depth and deep water cases are derived. These derived energy identities are used to check the computational accuracy and to quantify the loss of wave energy by the barrier system. Subsequently, the study is extended to deal with wave scattering by multiple non‐identical unequally spaced barriers using wide‐spacing approximation and results for equally spaced identical barriers are obtained as a special case. The methodology is further generalized to deal with Bloch problem to incorporate a periodic array of infinite barriers. Occurrences of Bragg resonance in case of multiple barriers are analyzed. From the results associated with the linearized water wave theory, the second‐order mean wave forces are computed and compared with the first‐order wave forces. The phenomena of cloaking for wave scattering by multiple impervious barriers are discussed for a wide variety of wave and structural parameters. The methodology used in this present study can easily be generalized to solve similar problems that arise in mathematical physics.

Performance of a Shore Fixed Oscillating Water Column Device for Different Bottom Slopes and Front Wall Drafts: A Study Based on Computational Fluid Dynamics and BIEM

Abstract Oscillating water column (OWC) wave energy converters are one of the most widely researched devices for ocean wave energy harvesting. This study investigates the hydrodynamic performance of a shore-fixed OWC device for different bottom slopes using two numerical approaches, namely, computational fluid dynamics (CFD) and boundary integral equation method (BIEM). In the BIEM method, the boundary value problem is solved in two-dimensional Cartesian coordinates using the linear water wave theory. The CFD model uses a numerical wave tank (NWT) built using the volume of fluid (VOF) method. Numerical computations are carried out for different sloped bottom geometries and front wall drafts to analyze the hydrodynamic efficiency. There is a general agreement between CFD and BIEM results in terms of resonating behavior of the device. It is observed that the front wall draft has a more significant effect, a lower draft leading to a wider frequency band for optimum conversion at high efficiency. While the BIEM-based analysis resulted in improved performance curve for few of the steeper slopes, the CFD study predicted a lower peak efficiency for the same slopes due to the consideration of real fluid characteristics. Detailed performance comparisons are presented using the time histories of free surface elevation, chamber pressure, and streamlines at different time instants within the OWC chamber.

Mitigation of wave force on a circular flexible plate by a surface-piercing flexible porous barrier

Wave force acting on a circular flexible plate in the presence of a vertical surface-piercing flexible porous membrane is examined under the hypothesis of linear water wave theory. A semi-analytic solution is developed employing the Fourier-Bessel series expansion method along with the methods of separation of variables and least-squares approximation. To keep the outer flexible membrane at a desired position, clamped-moored condition is used. Appropriate orthogonal mode-coupling relationship for the eigenfunctions of the plate covered region is exploited in the expansion formulas together with the continuity of velocity and pressure for achieving the system of equations and to determine the unknowns. The effects of various physical parameters are analyzed by computing wave load on the structures, plate deflection and flow distribution. It is found that due to the presence of vertical flexible porous membrane, a significant amount of wave force on the plate is reduced. Thus, a cylindrical flexible porous membrane can be implemented to protect the inner circular flexible plate.

Wave attenuation by multiple thin vertical porous walls in water of uniform finite depth

The analysis for wave attenuation by multiple thin porous walls submerged in water of uniform finite depth is carried out under the acceptance of linear theory of water waves. Two different types of barrier configurations are considered, namely partially immersed walls and bottom-standing walls. A set of coupled Fredholm-type integral equations are obtained for the boundary value problem by employing Havelock's inversion formulae and continuity of water wave potential across the gap below or above the walls. The methodology adopted to solve these integral equations is multi-term Galerkin's approximation with Chebychev's polynomials as a set of basis functions multiplied by suitable weights. Also, the square-root singularity of fluid velocity at the edge of the porous walls is precisely handled. The numerical results are depicted for reflection and transmission coefficients, dissipation of wave energy and dynamic wave force against wavenumber for different parametric values involving in the barrier configurations. This theory demonstrates that the dynamic wave forces exerted on the porous walls are reduced significantly as the magnitude of the porous effect parameter is increased. The present study is ratified with the existing results in the literature of water waves for a single as well as double rigid barriers and single porous barrier.

Bragg scattering of long waves by an array of floating flexible plates in the presence of multiple submerged trenches

Bragg scattering of long gravity waves by an array of floating elastic plates in the presence of a series of rectangular trenches is studied under the assumption of linearized water wave theory and small amplitude structural response. Bragg reflection occurs in the case of an array of multiple floating flexible plates and trenches in isolation or combination, and the number of sub-harmonic peaks between two consecutive peaks is two less than the number of plates/trenches. In the case of long-wave scattering by an array of trenches in the absence of floating plates, wave reflection becomes zero minimum for a certain fixed wavenumber at the end of the first cycle (as referred to 0 < k1h1 < 0.32), irrespective of the even/odd number of trenches. Furthermore, the Bragg resonant reflection pattern of the second cycle is a mirror image of the reflection occurring in the first cycle. Conversely, the symmetric pattern of Bragg reflection exhibited in the case of an array of trenches does not occur for an array of plates in isolation or for plates and trenches in combination. Between consecutive harmonic peaks, the minima/maxima in the wave reflection occur in the case of the even/odd number of trenches/plates in isolation. In contrast, it occurs within the first cycle in the case of trenches and plates in combination. Between consecutive harmonic peaks, maxima/minima of wave reflection coincide for a certain wavenumber, irrespective of the even/odd number of trenches/plates in isolation. In addition, these maxima attained for a certain wavenumber for the even number of trenches/plates are 180° out of phase to that of minima for an odd number of trenches/plates. However, similar phenomena occur within the first cycle of Bragg resonance in the case of trenches and plates in combination. Moreover, the amplitude of oscillation in the wave reflection increases with an increase in plate rigidity. Time-dependent motion due to Bragg scattering by trenches and plates is demonstrated in different cases.