Matched Eigenfunction Expansion Method Research Articles

Analysis of Wave-Induced Forces on a Floating Rectangular Box with Analytical and Numerical Approaches

A three-dimensional mathematical hydrodynamic model associated with surface wave radiation by a floating rectangular box-type structure due to heave, sway, and roll motions in finite water depth is investigated based on small amplitude water wave theory and linear structural response. The analytical expressions for the radiation potentials, wave forces, and hydrodynamic coefficients are presented based on matched eigenfunction expansion method (MEFEM). The correctness of the analytical results of wave forces is compared with the construction of a numerical model using the open-source boundary element method code NEMOH. In addition, the present result is compared with the existing published experimental results available in the literature. The effects of the different design parameters on the floating box-type rectangular structure are studied by analyzing the vertical wave force, horizontal wave force, torque, added mass, and damping coefficients due to the heave, sway, and roll motions, and the comparison analysis between the forces is also analyzed in detail. Further, the effect of reflection and transmission coefficients by varying the structural width and drafts are analyzed.

Second-order wave loads on a floating bridge of T-type cross section

An analytical model for the complete second-order wave loads on a floating bridge of T-type cross section is established within the framework of potential flow theory. The first-order problem is solved using the matched eigenfunction expansion method (MEEM). Meanwhile, the second-order wave loads are divided into two distinct parts which are computed separately. The first part is dependent on the quadratic products of the first-order velocity potential and its derivative. It has been evaluated through two different ways. One is based on a direct pressure integration, i.e., a near-field method. To ensure a high accuracy of the computation, a local expansion is applied to determine the spatial derivative of the first-order velocity potential at a sharp corner. The other is based on the momentum conservation over a control volume. The effect of the control surface location on the computation has been discussed. In addition, the two proposed methods have been applied to both the time-independent wave-load component as well as the second-harmonic component, respectively. The second part of the complete second-order wave loads is due to the second-order velocity potential. It is evaluated based on an indirect method by introducing an auxiliary potential. Detailed numerical investigation is then conducted. The influence law of the sectional geometric parameters on the second-order wave loads is investigated. The numerical results reveal that the cantilever plate imposes an apparent protection effect on the inner web beam. Such effect gets obviously enhanced as the section net width or section height decreases or as the plate width or draft increases.

Assessment of viscous damping of a solid heave disk for motion reduction of a floating circular cylinder

In the present study, we developed an analytical model incorporating viscous damping of a solid heave disk for the motion reduction of a floating circular cylinder in three degrees of freedom (3DOF) (surge, heave, pitch). The matched eigenfunction expansion method (MEEM) based on a potential theory is employed for solving the diffraction and radiation problem of a floating cylinder with a solid disk. To implement viscous damping due to vortex formation at the disk edge, a novel approach using the drag force and moment in the Morison equation is presented along with the conventional method using free decay tests. The developed analytical model is validated through the comparison with the regular wave experiments and computational fluid dynamics (CFD) turbulent model. Attaching the solid heave disk to the bottom of a cylinder reduces the maximum values of the heave and pitch RAOs (Response of Amplitude Operator) to 34 % and 53 %, compared to the cylinder model with no disk. Increasing the size of the heave disk decreases heave motion accordingly and significantly, however, its radius ratio must exceed the specified value of a/b=1.5 for affecting the reduction of a pitch motion.

Hydrodynamic Performance of an Array of Stratified Pile Rock Breakwaters Placed on Elevated Seabed

Abstract The present study outlines the hydrodynamic performance of stratified pile-rock breakwaters (SPRBs) in series using the analytical calculation under the framework of linearized potential flow theory. The rubble mounds are separated into two porous layers (surface and bottom layers) and tightly packed within the space available between the seaside and leeside vertical piles. The SPRB is installed on the elevated bed, and it is considered as a bottom rigid layer. The newly proposed breakwater is titled as stratified pile-rock breakwater and vertical piles are suggested to minimize the unwanted displacements of rubble mounds from frequent failures due to the incident wave stroke. The analytical model is developed based on the method of matched eigenfunction expansions (MMEEs) along with suitable boundary conditions to assess the hydrodynamic performance of the SPRB. The study results are compared with the literature based on experimental and analytical methods for specific conditions. The wave reflection, transmission, and energy damping by a series of SPRBs are reported for changes in incident wave properties and breakwater physical properties. The effect of layer porosity, angle of contact, free spacing, and number of breakwaters on the hydrodynamic coefficients is reported. The study suggested that a pair of SPRBs having 80% and 40% porosities for surface and bottom layers, with clear spacing, varied within 1 ≤ w/h1 ≤ 2, and the angle of contact varied within 30 deg ≤ θ ≤ 45 deg to achieve a 90% wave-damping when the relative wavenumber is k10h1 = 1.

Effect of vertical porous baffle on sloshing mitigation of two-layered liquid in a swaying tank

The sloshing behavior of two-layered liquid in a rectangular sloshing tank with a vertical porous baffle under sway excitation has been investigated using both analytical and numerical methods. The analytical model is developed using the matched eigenfunction expansion method in conjunction with linear potential flow theory. The existence of vertical porous baffle results in energy dissipation by applying the boundary condition that pressure drop across the baffle is proportional only to the quadratic drag term. For comparison, numerical simulations are conducted using computational fluid dynamics (CFD) with unsteady, implicit, and incompressible fluid equations. The study explores the effects of the liquid density ratio, the baffle's porosity, and its submergence depth, on various parameters including the free surface and interfacial elevations, hydrodynamic force on the tank surface, and horizontal force on the baffle. The trends observed in the analytical model align well with the CFD results and experiments. It is found that a fully submerged vertical porous baffle with a porosity close to P = 0.1 leads to stable and less dynamically active sloshing behavior in a two-layered liquid tank.

Hydrodynamic coefficients on a deeply submerged heaving solid/porous disk: in analytical and experimental approaches

ABSTRACT The hydrodynamic coefficients on a deeply submerged solid/porous disk subjected to forced heave oscillation have been investigated in analytical and experimental approaches. The analytical model is developed by means of a matched eigenfunction expansion method (MEEM) using a quadratic relationship between the pressure drop and traversing fluid velocity across the porous disk. For the improvement and validation of the analytical model, a series of experiments with the solid/porous disks is conducted for various porosities, Keulegan-Carpenter (KC) numbers, and oscillation periods. In addition to the small-scale hole vortices across the porous disk, the effect of edge vortices on the hydrodynamic coefficients is empirically implemented by curve-fitting with experimental data. The inclusion of edge vortices in the analytical model results in good agreement with experimental data of the damping coefficient for a deeply submerged disk. However, there are some discrepancies in the added mass, with noticeable differences occurring in the disk with low porosity.

Wave interactions with a cylinder surrounded by an arc-shaped breakwater

Inspired by the energy island used for energy storage, conversion, and transmission, this study explores the water wave interactions with a bottom-standing cylinder surrounded by an arc-shaped breakwater of negligible thickness. The arc-shaped breakwater can be either impermeable or porous. To analyze the problem, a semi-analytical matched eigenfunction expansion method is developed within the framework of linear potential flow theory. The fluid domain is divided into subdomains, with the velocity potentials in each subdomain expressed as eigenfunction expansions. Matching conditions, ensuring continuity in pressure and normal velocity, are imposed at the juncture boundary. The accuracy of the developed semi-analytical method is verified through comparison with results obtained using the numerically-based boundary element method. Extensive discussions are made on hydrodynamic responses, including wave exciting force and free surface elevations. When the breakwater is impermeable, fluid resonance occurs in the gap between the cylinder and the arc-shaped breakwater, leading to significantly large responses in both free surface elevation and hydrodynamic forces. A simple method is developed to estimate the resonant frequencies and mode shapes associated with these resonances. On the other hand, it is observed that the presence of the porous breakwater considerably reduces the free surface responses in the gap as well as the hydrodynamic loads on the cylinder. The porous breakwater effectively mitigates the occurrence of fluid resonance and offers an efficient means to dampen wave energy, providing valuable insights for wave energy dissipation and coastal engineering applications.

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.

Wave power extraction by an oscillating water column array embedded in comb-type breakwaters: Performance analysis and hydrodynamic mechanism

Cost-sharing, space-sharing, and multi-function can be achieved through integrating wave energy converters into coastal defense facilities. In this paper, we consider a periodical array of oscillating water columns (OWCs) embedded in the coast-based comb-type breakwater in the presence of the step bottom. Based on the linear potential flow theory and matched eigenfunction expansion method, a semi-analytical model for solving the diffraction and radiation problems of the periodic OWC array is developed. The mathematical model is verified using Haskind relations and energy conservation law. Parametrical studies are carried out to illustrate the hydrodynamic characteristics of the OWC array embedded in the comb-type breakwater. This study also reveals the constructive and destructive interference effects between the breakwater and OWCs. It is found that the wave amplification caused by the projecting caisson produces a constructive effect on the wave power extraction. However, the inherent strong wave reflection caused by the caisson array weakens the wave power extraction, particularly in the sensitive frequency range (i.e., 2 < kh1 < 5.5 in the present investigations).

New Strategy on Power Absorption of a Concentric Two-Body Wave Energy Converter

The wave energy converter (WEC) system, which extracts electricity through the relative heave motion of two concentric cylinders, comprises an inner cylinder and a torus-type outer cylinder sliding along the inner cylinder. To maximize the relative heave motion between two cylinders, the natural frequencies of the two cylinders must be precisely tuned to resonate and be situated on each side of the peak frequency of the wave spectrum. However, the demerit of this strategy is that it demands a deep draft of each cylinder for tuning, and a large-scale PTO damping device is necessary for mechanical power amplified by resonance. As an alternative to efficient and stable WECs, we adopt a new strategy in which the outer cylinder follows the incoming waves and the motion of the inner one is restricted to be minimal using a heave disk. The viscous damping due to formation of vortices at the disk edge is realized by the drag force in the Morison equation. The developed hydrodynamic model of two-body WEC based on a matched eigenfunction expansion method (MEEM) is applied to irregular waves characterized by significant wave height and peak period. It is found that the present two-body WEC with heave disk produces wave energy stably across a wide range of wave frequencies compared to the previous two-body WECs using resonance.

Oblique wave trapping by a permeable wall breakwater connected with thick surface porous layer

ABSTRACT The reflection of surface gravity waves by a vertical wall breakwater connected with a thick surface porous layer is analysed under the framework of potential flow theory. The thick surface porous layer is placed within the confined space available between the vertical permeable wall and the impermeable seawall. The analytical investigation is progressed by creating the hidden water chamber beneath the surface porous layer. The Matched Eigenfunction Expansion Method (MEEM) is used based on the continuity of velocity and pressure along with orthogonal mode-coupling relation. The hidden water chamber will be helpful to enhance the trapping of incident wave energy and also to reduce the quantity of construction material. The present study results are well agreed with the results available in the literature when a hidden water chamber is absent. The effect of permeable wall porosity, thick surface layer porosity, angle of wave contact, depth of the surface porous layer, thickness of the permeable wall, and hidden chamber spacing on wave reflection against incident wave properties and breakwater physical properties are studied. A comparative study is performed between three different types of porous breakwaters to investigate the integrity of the proposed breakwater. The minor value of wave reflection and gradual reduction of fluid force experienced by the rigid wall is achieved due to the presence of a surface layer, which maximises the incident wave damping, as most of the wave energy propagates near the free surface.

A mathematical study of water wave interaction with a thin perforated barrier in a two-layer fluid over a permeable bottom

This article studies the impact of a submerged interface-piercing perforated barrier in a two-layer fluid flowing over a permeable bottom. We investigate oblique wave scattering, trapping and radiation due to the structure focusing on the bottom permeability. The dead water phenomenon is analysed with the consideration of the bottom permeability, which results in a higher variation of the interfacial wave due to the bottom permeability. The matched eigenfunction expansion method and the least square technique are used to calculate various hydrodynamic coefficients. Wave energy identity relation is derived for the scattering scenario, and the associated energy loss due to the barrier is calculated. In order to attain the maximum wave dissipation, an ideal porous-effect parameter of the barrier is proposed for consideration, and it is observed that larger values of porous-effect parameter result in the lowest feasible pressure distribution. A good comparison with a prior result justifies the current semi-analytical procedure. Furthermore, the verification of the energy-identity terms aid in the validation of the computed results. Additionally, wave trapping in a confined region is examined by investigating reflection coefficients by considering a rigid wall. The thin perforated barrier model is further considered for examining the radiation aspect while considering its slow motion. For various porous-effect parameters of the barrier, the amplitude ratio of the radiated potential is investigated, and it is clearly observed that higher frequency significantly lowers the amplitude for both free surface and interfacial propagating modes. The impact of the perforated barrier is analysed by investigating the essential hydrodynamic coefficients, namely, added mass and damping coefficient.

Hydrodynamic behaviour of floating polygonal compound platforms with wide porous media

Presented herein are the formulation and semi-analytical solutions for the hydrodynamic characteristics and wave fields of floating polygonal compound platforms with wide porous media. The wetted porous medium is assumed to be homogeneous and fully saturated. Limiting cases of the semi-analytical solutions without the porous medium have been verified by the results obtained by using the commercial software ANSYS AQWA. The present semi-analytical solutions based on a matched eigenfunction expansion method are useful as benchmark results for analysts working on new numerical methods for the hydrodynamic analysis of floating platforms with porous media. Parametric studies were conducted to examine the effects of porous media on the wave exciting forces, RAOs, wave fields, porous damping and mooring loads. It is found that the wide porous medium with appropriate porosity and resistance coefficient contributes to reducing the wave exciting forces, RAOs, and overall level of wave fields in the long period region by attenuating the incident wave energy. Also, the porous medium helps to decrease the mooring loads, especially at resonance. Thus, a wide porous medium surrounding the compound platform is effective in both enhancing the motion performance of the floating platform and reducing the wave fields around it.

Design optimization of a multi-layer porous wave absorber using an artificial neural network model

The design optimization of a multi-layer porous wave absorber has been achieved through the supervised artificial neural network (ANN) and analytical model validated by CFD and experiments. The analytical model is established by means of a matched eigenfunction expansion method (MEEM) by applying the boundary condition with a quadratic relationship between the pressure drop and traversing fluid velocity at the porous plates. Along with the reflection coefficient as a target feature, input key features of the multi-layer porous wave absorber are selected through the parametric study. A dataset with 200 combinations of input design features is generated by the developed analytical model. Using the dataset, the ANN model is trained with a R2 score of 0.97. Predictions are generated for a large sample set with the trained ANN model. Top 1% combinations of input features that yield a minimum value of the reflection coefficient are used for the optimal design of the wave absorber. Based on the interquartile range (IQR) value of this 1% data, it is found that the most important design features are the porosity and submergence depth of the upmost plate and their optimal ranges for a double-layer wave absorber are 0.055≤d1/h≤0.067 and 0.117≤P1≤0.173. The optimal range of design features can be used as a guideline for the design of an effective wave absorber.

Scattering of Water Waves by Very Large Floating Structure in the Presence of a Porous Box

Abstract We investigate the effectiveness of a porous box in attenuating the structural response of a very large floating structure (VLFS). Assuming the water depth to be finite and small amplitude water wave theory, the physical problem is formulated by employing Darcy’s law for flow past a porous structure. The boundary value problem is reduced to a system of linear algebraic equations with the aid of matched eigenfunction expansion method. Further, these simultaneous equations are solved numerically to compute physical quantities. The mathematical model is validated through a comparison with the theoretical results available in the literature. The reflection, transmission, and dissipation coefficients, elastic plate deflection, forces acting on the box, and free-surface elevation are computed. The reflection, transmission, and dissipation coefficients exhibit an oscillatory pattern for large values of wavenumber irrespective of the structural parameters of the box. It is highlighted that a porous box with moderate values of the porous-effect parameter is effective in reducing strain and shear force on the VLFS. Further, the results on elevation and VLFS deflection depict that the suitable porous-effect parameter values for the box reduce the resultant wave amplitude of the elastic plate deflection as well as the amplitude in the lee side of the VLFS. The study reveals that the width and height of the porous box are critical toward the trapping of incident waves inside the box and dissipating the maximum amount of incident wave energy in reduction of wave transmission in the lee side of the structure and thereby attenuating the structural response of the VLFS.

Wave action of non-coaxial double-layer cylindrical structure

Based on the potential flow theory, the mathematical formula of the wave action of the non-coaxial double-layer cylinder structure is derived, and the effect caused by the non-coaxial of the upper and lower single cylinders is analyzed. In this paper, the water area is divided into two regions, and their respective coordinate systems are established in each region to describe the velocity potential function. The velocity potential of the two regions is linked by the matched eigenfunction expansion method (MEEM) at the interface of the two regions. The cross-sectional radius function of the interface is expressed by Fourier series. After verifying the accuracy of the mathematical formula, the effect of asymmetric structure is analyzed. The results show that when the relative angle of the two axes is 0°, the wave force on the structure can be reduced to a greater extent if the two axes are as far away as possible. At the same time, the effect of this reduction weakens with the increase of the relative angle, and even increases the wave force when α = 90°. When the lower-lawyer cylinder radius is 0.6 times the upper-lawyer cylinder radius, the effect of wave force reduction caused by the non-coaxial is the largest.

Influence of submerged annular rippled breakwater on hydrodynamic performance of column floating pier

Based on linear potential flow theory, a mathematical model of the interactions between a floating pier with a submerged annular rippled breakwater and linear monochromatic waves was established. The fluid domain was divided into four regions: an exterior region, a mild-slope region, a gap region, and an interior region. The modified mild slope equation (MMSE) was used to establish the velocity potential expression of the mild-slope region, where the rippled breakwater was located. The matched eigenfunction expansion method (MEEM) was then used to derive the semi-analytical solution of the diffraction and radiation problem of the floating pier under linear monochromatic waves. After the accuracy of the mathematical model was ensured by comparison with other results and convergence analysis, the effects of the amplitudes and lengths of the ripples, the distance between the floating pier and the breakwater, and the number of ripples of the submerged annular rippled breakwater was analyzed. The results showed that the increase in the ripple amplitude and number aggravated the fluctuations of the wave excitation forces and the hydrodynamic coefficients of the floating pier as the wavenumber and ripple length varied. The wave excitation force and hydrodynamic coefficient of the floating pier changed periodically with the increase in the distance between the breakwater and the floating pier. This period was only related to the wavenumber, which was slightly larger than half the wavelength. The valley value and the corresponding position of the valley that appeared with this periodic change were affected by the ripple amplitude and length of the submerged annular rippled breakwater.

On the scattering and radiation of water waves by a finite dock floating over a rectangular trench

In this study, the scattering and radiation of surface water waves by a finite dock floating over an asymmetric rectangular trench-type bottom topography are analyzed. The boundary value problems are handled for solutions analytically using a matched eigenfunction expansion method and numerically using the boundary element method. Various physical quantities associated with the scattering and radiation problems are studied for various values of wave and structural parameters. The results demonstrate that the scattering coefficients are significantly influenced due to the variation in dock length and depth ratios of the bottom trench. In the case of radiation problems, the dock length plays a crucial role in analyzing the radiation coefficients and wave excitation forces associated with each mode of motion. On the other hand, the effect of depth ratios of the bottom trench is negligible on the radiation coefficients. The present study reveals that the scattering of water waves by a floating dock over an abrupt change in bottom topography plays an important role in the construction of floating structures having more durability to resist harsh wave environments and reduce the wave impact near the seashore.

The influence of oblique waves on the hydrodynamic efficiency of an onshore OWC wave energy converter

In this work, the maximum theoretical hydrodynamic efficiency of an onshore Oscillating Water Column (OWC) device is investigated in relation to the incidence angle of the wave and the configuration of the chamber. Linearized water wave theory is used, and the full solution of the associated boundary value problem (BVP) is achieved by using the simple matched eigenfunction expansion method (EEM) and the boundary element method (BEM) with second-order elements. This study is novel in using these techniques to address the influence of oblique water waves on the efficiency of an OWC with a thick front barrier. The hydrodynamic efficiency is analyzed for different directions of wave incidence and OWC chamber geometrical configurations. The semi-analytical and numerical approaches were found to be in good agreement with cases published in the specialized literature. The results show that both a thick front wall and a large incidence angle of the wave can significantly narrow the hydrodynamic efficiency band and modify the resonant frequency.

A Theoretical Study of the Hydrodynamic Performance of an Asymmetric Fixed-Detached OWC Device

The chamber configuration of an asymmetric, fixed-detached Oscillating Water Column (OWC) device was investigated theoretically to analyze its effects on hydrodynamic performance. Two-dimensional linear wave theory was used, and the solutions for the associated radiation and scattering boundary value problems (BVPs) were derived through the matched eigenfunction expansion method (EEM) and the boundary element method (BEM). The results for the hydrodynamic efficiency and other important hydrodynamic properties were computed and analyzed for various cases. Parameters, such as the length of the chamber and the thickness and submergence of the rear and front walls, were varied. The effects on device performance of adding a step under the OWC chamber and reflecting wall in the downstream region were also investigated. A good agreement between the analytical and numerical results was found. Thinner walls and low submergence of the chamber were seen to increase the efficiency bandwidth. The inclusion of a step slightly reduced the frequency at which resonance occurs, and when a downstream reflecting wall is included, the hydrodynamic efficiency is noticeably reduced at low frequencies due to the near trapped waves in the gap between the OWC device and the rigid vertical wall.