Interaction Of Surface Gravity Waves Research Articles

Establishing patterns of change in the coefficients of reflection, transmission, and dissipation of wave energy depending on parameters of a permeable vertical wall

The object of research is permeable vertical walls (breakwaters) with different degrees of permeability. This paper reports the results of experimental research into the interaction of gravity waves with models of permeable vertical walls (breakwaters), which were formed from cylindrical piles of circular cross-section. With the help of visual and instrumental studies, the features of interaction of surface gravity waves with permeable vertical walls of different permeability have been identified. The degree of wave transformation by these walls was also determined in the form of reflection, transmission, and dissipation coefficients of wave energy. It was established that with a decrease in the permeability of the vertical wall and an increase in the steepness of the initial wave and a decrease in its period, the height of the reflected wave increased. The pattern of the transmitted wave height had the opposite trend. It was determined that the wave reflection coefficient increased with a decrease in the permeability of the vertical wall and the steepness of the initial wave. The wave transmission coefficient had the opposite trend, namely, it increased with increasing wall permeability and with decreasing steepness of the initial wave. The gravity wave energy dissipation coefficient decreased with increasing vertical wall permeability, but for waves with a significant steepness hi/λ>0.038, a decrease in the wave energy dissipation coefficient was observed for walls with low permeability and the appearance of extreme values of this coefficient. Thus, features in the interaction of surface gravity waves with permeable vertical walls (breakwaters) of different permeability have been researched and the degree of wave transformation by these walls has been determined, which could make it possible to effectively design and operate permeable vertical walls as coastal protection structures

Surface gravity wave interaction with a floating circular elastic plate in the presence of concentric porous arc walls

A Fourier Bessel series type expansion formula is used to investigate the interaction of surface gravity wave with two arc-shaped porous breakwaters that shield a circular floating flexible structure from wave action. The flexural gravity waves result from the interaction between the surface gravity waves and the circular floating structure. Additionally, when the floating structure encounters lateral compression force, flexural gravity wave blocking occurs. Moreover, the floating structure is triggered by the interaction of three unique wave modes for every wave frequency within the confines of the blocking frequencies. The two arc-shaped permeable breakwaters significantly reduce the wave force acting on the floating structure as compared to situations where there is just one arc wall or no breakwater at all. Resonant peaks in the wave forces arising within the impermeable breakwaters diminish with the introduction of structural permeability. The hydrodynamic load on the floating structure and the pair of porous arc breakwaters exhibit removable discontinuities near the blocking frequency. Surface plots illustrate the irregular distribution of plate deflection at certain angular frequencies. Contour plots are used to ascertain the spatial configuration of fluid flow around a structure. Furthermore, several wave and structural parameters, including the position of the breakwaters and their structural dimensions, breakwater porosity, annular radius, compressive force, incident angle, and wave forces experienced by the floating structure, contribute to mitigating wave-induced structural response and wave forces experienced by the floating structure. Time-dependent simulation of the surface displacement by the incident wave demonstrates the flow features.

Wavy approach for fluid–structure interaction with high Froude number and undamped structure

This paper addresses the fluid–structure interaction problem, with an interest on the interaction of a gravity wave with a flexible floating structure, anchored to a seabed of constant depth. To achieve this goal, we make use of the model equations, namely, the Navier–Stokes equations and the Navier–Lamé equation, as well as the associated the boundary conditions. Applying the multi-scale expansion method, these set of equations are reduced to a pair of nonlinearly coupled complex cubic Ginzburg–Landau equations (CCGLE). By applying the proposed modified expansion method, the group velocity dispersion and second-order dispersion relation are deduced. In the same vein, modulation instability (MI) is investigated as a mechanism of formation of pulse trains in fluid–structure system using a CCGLE. For the analytical analysis, we made use of the inverse scattering method to find analytical solutions to the coupled nonlinear equations. Through that method, the obtained solutions depict rogue-shaped waves. Our results suggest that uncontrolled MI within the interaction between a flexible body and gravity waves in viscous flow may be considered as the principal source of many structural ruptures, which are the first cause of critical damage due to the great energy and unpredictability of rogue waves. The present work aims to provide tools to model a wide range of physical problems regarding the interaction of surface gravity waves and an offshore-anchored structure, and it aims to be essential to our understanding of the nonlinear characteristics of offshore structures in real-sea states.

Bragg Resonance of Surface Gravity Waves by Surface-Piercing Vertical Multi-Plate Breakwaters

Abstract The interaction between surface gravity waves and surface-piercing vertical multi-plate breakwaters in finite water depth is investigated under the assumption of potential flow theory. The time-domain desingularized boundary integral equation method (DBIEM) is adopted to solve the boundary value problem. A decomposition method based on the superposition principle is used to obtain reflected waves. The effect of the vertical plates as a floating breakwater by making full use of Bragg resonance is analyzed. Initially, the waveforms calculated without breakwaters are compared with second-order Stokes regular wave to obtain the appropriate desingularization distance. Then, the interaction of surface gravity waves with the breakwaters with the single plate, double, and triple plates are analyzed. The transmission and reflection coefficients of the breakwater with the single plate are verified with the analytical solutions of previous literature. The transmission and reflection coefficients of the breakwaters with double and triple plates are verified with experimental results. Finally, the transmission and reflection coefficients of surface-piercing vertical multi-plate breakwaters with different plate numbers and drafts are also calculated. The occurring condition of Bragg resonance and its values affected by the plate number and the draft are analyzed. The present results are reference values for the optimal design of vertical plates as a floating breakwater for offshore and coastal engineering applications by fully using Bragg resonance.

Surface gravity wave interaction with a submerged tunnel in the presence of a submerged wavy porous plate

The present study deals with the surface gravity wave interaction of a submerged composite wavy porous plate in the presence of a tunnel placed at a finite distance under the assumption of small amplitude theory and a two-dimensional framework. To solve the boundary value problem, a numerical method using the multi-domain boundary element method (MDBEM) is implemented. Hydrodynamic characteristics of interest such as reflection coefficient and transmission coefficient are computed and studied for a wide range of dimensionless wavenumber and relative plate lengths. Various parametric studies are conducted to emphasize the effect of structural parameters like the number of relative ripple wavelengths, relative ripple amplitude, and relative submergence depth. Dimensionless wave force coefficients acting on the plate and the tunnel are calculated and investigated. The study reveals that the presence of composite wavy porous plate has a significant effect in the mitigation of wave-exerted force on the tunnel. Optimized results are shown for the efficient design of coupled breakwater and tunnel models, which can be used beneficially to protect against the severity of wave action in similar applications of ocean engineering in marine environments.

Performance of a two-body heaving cylindrical wave energy converter on stepped sea bottom

The present study deals with the surface gravity wave interaction with a heaving dual concentric cylindrical wave energy converter (WEC) buoy system on the stepped sea bottom. The energy captured by linear electric generator (LEG), is directly proportional to relative heave motion of the present WEC buoy system. In the present study, the analytical diffraction and radiation problems are solved using the Fourier–Bessel series expansion method and matched eigenfunction expansion method (MEFEM) under the assumption of small amplitude water wave theory and structural responses. Wave exciting force acting on both vertical cylinders is obtained. The role of the step type bottom on the relative heave response amplitude operator (RAO) of the buoy system, is analyzed for various structural configurations and wave characteristics. A series of experiments are carried out in a two-dimensional wave flume both with and without the presence of step-type bottom. The computed results are in good agreement with the experimental results. It is found that the inclusion of artificial step at the sea bottom leads to higher wave energy harnessing. The findings of the present study are likely to be of immense help in the design and development of various point absorber-type WEC buoy systems.

Surface gravity wave interaction with a floating dock in the presence of a submerged composite wavy porous plate

A numerical model based on the multi-domain boundary element method is adopted to study surface gravity wave interaction with a finite floating dock in the presence of a submerged composite wavy porous plate in the two-dimensional Cartesian coordinate system. Physical parameters of interest such as reflection coefficient, transmission coefficient, dissipation coefficient, horizontal and vertical wave force coefficients of the wavy plate, vertical force and moment coefficients about the centre of the dock are calculated. The influence of structural parameters such as the number of ripple wavelengths, relative ripple amplitude, structural porosity and relative submergence depth of wavy plate, relative space between the plate and dock are computed and studied. An insight into the comparison of various types of plates such as flat plate, wavy plate and composite wavy plate is also presented. The study reveals that the mitigation of wave force on the dock is significant due to a wavy porous plate. Besides, the study exhibits that placing the wavy porous plate at a finite distance from the floating dock has a significant role in dissipating incident wave energy compared to that placing the submerged plate below the dock. Bragg reflection occurs due to the presence of the submerged wavy plate, which diminishes with an increase in structural porosity and depth of submergence of the wavy plate.

Hydroelastic analysis of surface gravity wave interaction with multiple flexible porous structures

A solution to the problem of wave interaction with multiple flexible porous barriers is presented using the Galerkin approximation technique in deep water. The study has analyzed the reflection coefficient, dissipated energy, and wave forces on the barriers and presented results for different inclination angles, separation lengths, porosity, and flexural rigidity of the barriers. The comparison of the results with those available in the literature adds credibility to the study. The finding that the flexural rigidity of the porous barriers has a significant effect on wave reflection, dissipated energy, and wave forces is an important contribution to the field. The observation that an increase in the number of barriers reduces the wave load significantly is also noteworthy. Overall, this study provides valuable insights into the behavior of wave interaction with multiple flexible porous barriers, which could have practical implications for coastal protection and wave energy conversion systems.

Four-wave resonant interaction of surface gravity waves in finite water depth

We investigated the four-wave resonant and quasi-resonant interactions in a special degenerated case, wherein bichromatic mother waves are generated to give birth to a daughter wave. Through theoretical and numerical analyses, we first discover the threshold value of water depth when the four-wave resonance diminishes. Our work can shed light on the limiting shallow water depth where the energy exchange due to four wave resonance creases.

Numerical and Experimental Study on the Bragg Reflection of Water Waves by Multiple Vertical Thin Plates

The Bragg reflection of water waves by multiple vertical thin plates is investigated numerically and experimentally. The problem of surface gravity wave interaction with multiple vertical thin plates is formulated mathematically under the assumption of linear water wave theory. Numerical results of the reflection and transmission coefficients are obtained based on the dual boundary element method and compared with the results in the literature. The characteristics of Bragg reflection are represented by the occurring condition, the primary reflection coefficient and the effective bandwidth. The effects of the number, the immerged depth and the spacing of the plates on the characteristics of Bragg reflection are further analyzed systematically. Model experiments are conducted and the experimental results are used to compare with the numerical results of Bragg reflection. This study can provide guidance on the design of multiple vertical thin plates as effective breakwaters by taking advantage of Bragg reflection.

Gravity wave interaction with an articulated submerged plate resting on a Winkler foundation

Surface gravity wave interaction with an articulated flexible submerged plate in the presence of lateral compressive force is investigated within the framework of blocking dynamics, assuming that the submerged plate is resting on a uniform Winkler foundation. The problem is studied considering small-amplitude plate response and linearised water wave theory in finite water depth. The loci of the dispersion relation reveal the occurrence of wave blocking in the submerged plate mode for a range of wave frequency and compressive force when the submerged plate is resting on a Winker foundation. However, in the absence of the Winkler foundation, no wave blocking occurs in the plate mode irrespective of the values of compressive force, and plane wave propagation in plate mode does not happen for particular values of compressive force. Four propagating wavenumbers exist within the blocking frequencies, of which one is in the surface mode, and others are in the submerged plate mode. Thus, to account for the four propagating wave modes, the existing solution approach for a pair of propagating wave modes is modified in an appropriate manner within the limit of blocking frequencies. Consequently, the energy relation is obtained using Green’s identity, in which the reflection and transmission coefficients are a combination of the amplitudes of different reflected and transmitted wave modes and the associated energy transfer rates. The reflection coefficient in the submerged plate mode jumps from one to zero for very small values of frequency due to the introduction of the Winkler foundation in the presence of compressive force. In addition, zero wave reflection occurs for a higher frequency range with an increase in the values of the Winkler constant. Removable discontinuities are observed at the blocking and saddle points, whilst jump discontinuities occur in the reflection and transmission coefficients due to the change of incident wave mode within the blocking frequencies. Linear time-dependent displacements in surface and submerged plate modes are presented prior to the blocking limit.

Water wave interaction with dual unequal horizontal flexible porous barriers using integral equations

ABSTRACT Under the assumption of linear wave theory, this work investigates the interaction of surface gravity waves with two asymmetric horizontal flexible porous barriers. Three types of barriers are considered, namely, (i) step barriers, (ii) partially overlapped barriers and (iii) fully overlapped barriers. Green's integral theorem and the boundary condition on the barriers are utilised to produce four integral equations for functions related to potential difference and normal velocity of fluid across the barriers which are solved numerically by an expansion-collocation method. As a result, the role of asymmetric flexible porous barriers with an additional comparison between dual and single horizontal flexible porous barriers is studied by analysing numerical solutions of different hydrodynamic quantities. This study shows that the upper barrier plays a major role compared to the lower one in the propagation of surface waves. The current analysis is validated by reviving previous results and utilising energy balance relationship.

3D hydroelastic modelling of fluid–structure interactions of porous flexible structures

A 3D hydroelastic model subjected to linear wave interaction with horizontal flexible floating and submerged porous structures is developed. The dispersion relation of the aforesaid problem is derived based on Green’s functions approach in water of finite and infinite depths. Three-dimensional Fourier-type expansion formulae for the velocity potentials associated with orthogonal mode-coupling relations are derived in a channel of finite and infinite water depths of finite width. A real physical problem of surface gravity wave interaction with a moored floating and submerged porous plate in finite water depth is analysed by the expansion formula to validate the usefulness of the current method. The numerical accuracy of the present numerical results is checked via convergence test. Further, the effect of different design parameters on the hydroelastic response of floating and a submerged porous plate is studied by analysing the deflection amplitude and structural deformation. It is observed that the present analysis could be useful for a greater insight to design a multi-use flexible floating system.

Surface gravity wave interaction with a partial porous breakwater in a two-layer ocean having bottom undulations

Linear surface gravity wave scattering by a partial porous breakwater is studied in a two-layer ocean having an interface with bottom undulations. The partial breakwaters, namely surface-piercing breakwater (SPB) and bottom-standing breakwater (BSB), are considered. The continuity of fluid pressure and mass flux is applied at interface boundaries, while mass conserving conditions are employed to account for slope discontinuity in seabed geometry. Porous breakwaters' reflective and dissipative characteristics are studied under wave-wave interaction. The study reveals that when a mound bottom is present, more incident waves in interface mode are transmitted through BSB, while more waves in surface mode are dissipated. More incident waves in surface mode are transmitted through SPB, while more waves in interface mode are dissipated. In contrast, more transmission occurs in surface mode through both BSB and SPB for a slope-shelf bottom. However, the breakwater dissipates a substantial amount of incident-wave energy in interface mode. Further, a considerable reduction in wave transmission occurs due to higher reflection by the interface-piercing porous breakwaters, and the breakwaters experience high wave loads. Relatively, porous breakwater dissipates less wave energy in shallow depth. The study reveals that the dead water effect is weakened by the porous breakwater.

Surface Gravity Wave Interaction With a Submerged Composite Wavy Porous Plate Attached to a Vertical Wall

Abstract The hydrodynamic performance of composite wavy submerged porous plate attached to a wall is investigated using numerical method multi-domain boundary element method in the linearized water wave theory in which wave past the porous barriers are modeled using Darcy’s law. Effect of the presence of wall on the hydrodynamic parameters like reflection coefficient, dimensionless wave height, wave force and moment acting on the composite wavy porous structure, and horizontal force on the vertical rigid wall is analyzed for various physical parameters like the number of ripple wavelength, structural porosity, submergence depth, and relative amplitude of composite wavy porous plate. The study demonstrates that the efficiency of hydrodynamic characteristics of the composite wavy porous plate attached to the wall is better compared to a horizontal porous plate attached to the wall of the same applicability conditions (around 27% reduction in wave reflection). Moreover, optimum performance of this kind of breakwater system is increased by considering the lower submergence, higher relative ripple amplitude, appropriate relative amplitude, and suitable moderate porosity of the structure in the range of wavenumber 1 < k0h < 5. Dimensionless wave height, horizontal load on the impermeable wall are reduced to zero, while substantial minimization of vertical load on the structure with suitable consideration of the other influencing parameters of porosity Gp = 0.3, relative amplitude Ds/h = 0.1, relative ripple wavelength m > 3, and submergence depth h1/h = 0.3. The present structural arrangement will be useful for attenuating wave effects on the sea wall.

Bragg Scattering of Surface Gravity Waves Due to Multiple Bottom Undulations and a Semi-Infinite Floating Flexible Structure

Surface gravity wave interaction with a semi-infinite floating elastic plate in the presence of multiple undulations has been studied under the assumption of linearized water wave theory and small amplitude structural response. The elastic plate is modeled using the Euler-Bernoulli beam equation, whilst the multiple undulations are categorized as an array of submerged trenches or breakwaters. The numerical solution obtained in finite water depth using the boundary element method is validated with the semi-analytic solution obtained under shallow water approximation. Bragg resonance occurs due to the scattering of surface waves by an array of trenches or breakwaters irrespective of the presence of the floating semi-infinite plate. The zero-minima in wave reflection occur when the width of the trench and breakwater is an integer multiple of 0.6 and 0.35 times wavelength, respectively, as the number of trenches or breakwaters increases. In contrast to trenches and breakwaters in isolation, non-zero minima in wave reflection occur in the presence of a semi-infinite plate. Moreover, the number of complete cycles in trenches is less than the number of complete cycles in breakwaters, irrespective of the presence of the floating structure. The frequency of occurrence of zero minimum in wave reflection is reduced in the presence of the semi-infinite plate, and wave reflection increases with an increase in rigidity of the floating plate. Time-dependent simulation of free surface displacement and plate deflection due to multiple undulations of seabed in the presence of the semi-infinite floating plate is demonstrated in different cases.

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.

Attenuating surface gravity waves with mechanical metamaterials

Metamaterials and photonic/phononic crystals have been successfully developed in recent years to achieve advanced wave manipulation and control, both in electromagnetism and mechanics. However, the underlying concepts are yet to be fully applied to the field of fluid dynamics and water waves. Here, we present an example of the interaction of surface gravity waves with a mechanical metamaterial, i.e., periodic underwater oscillating resonators. In particular, we study a device composed of an array of periodic submerged harmonic oscillators whose objective is to absorb wave energy and dissipate it inside the fluid in the form of heat. The study is performed using a state-of-the-art direct numerical simulation of the Navier–Stokes equation in its two-dimensional form with free boundary and moving bodies. We use a volume of fluid interface technique for tracking the surface and an immersed boundary method for the fluid–structure interaction. We first study the interaction of a monochromatic wave with a single oscillator and then add up to four resonators coupled only fluid-mechanically. We study the efficiency of the device in terms of the total energy dissipation and find that by adding resonators, the dissipation increases in a nontrivial way. As expected, a large energy attenuation is achieved when the wave and resonators are characterized by similar frequencies. As the number of resonators is increased, the range of attenuated frequencies also increases. The concept and results presented herein are of relevance for coastal protection applications.

Surface Gravity Wave Interaction with a Partial Porous Breakwater in the Presence of Bottom Undulation

AbstractUsing potential flow theory, oblique wave scattering and trapping by a partial porous breakwater having finite width are studied under the assumption of small-amplitude water wave theory in...

Interaction of ocean waves with floating and submerged horizontal flexible structures in three-dimensions

A three-dimensional general mathematical hydroelastic model dealing with the problem of wave interaction with a floating and a submerged flexible structure is developed based on small amplitude wave theory and linear structural response. The horizontal floating and submerged flexible structures are modelled with a thin plate theory. The linearized long wave equations based on shallow water approximations are derived and results are compared. Three-dimensional Green’s functions are derived using fundamental source potentials in water of finite and infinite depths. The expansion formulae associated with orthogonal mode-coupling relations are derived based on the application of Fourier transform in finite and infinite depths in case of finite width in three-dimensions. The usefulness of the expansion formula is demonstrated by analysing a physical problem of surface gravity wave interaction with a moored finite floating elastic plate in the presence of a finite submerged flexible membrane in three-dimensions. The numerical accuracy of the method is demonstrated by computing the complex values of reflected wave amplitudes for different modes of oscillation and mooring stiffness. Further, the effect of compressive force and modes of oscillations on a free oscillation hydroelastic waves in a closed channel of finite width and length for floating and submerged elastic plate system is analysed.