Eigenfunction Expansion Matching Method Research Articles

Hydrodynamic performance of multi-chamber oscillating water columns in a caisson array

A theoretical model based on the linear potential flow theory and eigenfunction expansion-matching method is developed to analyze the interaction between water waves and multi-chamber oscillating water columns (OWCs) embedded in a caisson array. The semi-analytical solution consists of the diffraction and radiation components of periodic OWCs. The velocity singularity at the tip of the OWC chamber walls is resolved by implementing a Chebyshev polynomial expansion. The unknown expansion coefficients are determined by the continuity equations of velocity and pressure at the interface of subdomains. The semi-analytical solution is verified using the Haskind relation and the conservation law of wave energy flux. It is found that multi-chamber OWCs have higher hydrodynamic efficiency over a broader frequency bandwidth than traditional single- and dual-chamber OWCs. The maximum hydrodynamic efficiency η increases with increasing of chamber number (J), from 0.5 (J = 1) to nearly 1.0 (J = 8 and 12). It is also found that a larger incident wave angle leads to a narrower frequency bandwidth for energy capturing. Increasing the incident wave angle from π/3 to 5π/12 results in a decrease of the first cutoff frequency kc from 2.41 to 2.02, and therefore results in the energy capture bandwidth narrowing accordingly. Both constructive and destructive hydrodynamic interactions between caissons array and OWCs are observed over the tested frequency range. Interestingly, when configuration of the OWC chambers is reversed, the transmission coefficient remains unchanged.

Analytical investigation on a wave energy converter-dual-arc breakwater integration system

A wave energy converter (WEC)-breakwater integration system consisting of a heaving WEC and a bottom-mounted dual-arc breakwater is proposed. The energy extraction performance and wave attenuation performance of the integration system were investigated using a theoretical model based on linear potential flow theory. The permeability of the dual-arc breakwater is considered in the theoretical model. The singularity at the edges of the perforated arc-shaped wall was solved using the multiterm Galerkin method. The unknowns in the expressions of velocity potential were solved using the eigenfunction expansion matching method. After successful validation, a comparative study between the models with dual-arc breakwater of different permeability was carried out first. Then, a parametric study on the configuration with a solid inner and a perforated outer arc-shaped wall is carried out. The performance of the proposed integration system is further investigated in multi-directional irregular waves, and it was found that the fluctuations of the average power capture width ratio are much lower compared with that under unidirectional regular wave conditions. The overall performance of the integration system is better with a larger directional spreading parameter.

Nonlinear wave energy dissipator with wave attenuation and energy harvesting at low frequencies

It is important to determine how to reduce the wave load and utilize the ocean energy through the hybrid wave attenuation and energy harvesting structure (WAAEHS) as it is the basic scientific problem in the field of ocean engineering. The existing linear hybrid devices have poor performance due to the mismatch between the natural frequency and the excitation frequency in the low frequency region. The method of increasing the mass is unreliable and low economic, and the introduction of nonlinear mechanism to reduce the stiffness has complex fluid-structure coupling problem between waves and structures. In this paper, a new type of negative stiffness mechanism with a simple structure and good stability has been proposed and applied to a hybrid WAAEHS, and a nonlinear frequency domain method combining eigenfunction expansion matching method(EEMM)and multi-harmonic balance method(MHBM) has been proposed to solve the nonlinear wave–structure interaction dynamic model. The influence of the key parameters of the mechanism on the wave attenuation and energy harvesting performance is examined, and the “phase control” mechanism of the negative stiffness mechanism is revealed to improve the low-frequency wave attenuation and energy harvesting performance.

Hydrodynamic analysis of resonant waves within the gap between a vessel and a vertical quay wall

Concerning the side-by-side offloading operations at gravity-based terminals, the wave interaction with a vessel alongside a long vertical wall in both normal and oblique seas was investigated. The imaging principle was applied to convert the original to an equivalent wave-diffraction problem by two symmetrical vessels in open seas exposed to bi-directional incident waves. The boundary integral equation method, in conjunction with a higher-order boundary element method (HOBEM), was used to solve the equivalent problem. Besides the linear free-surface elevation and wave loads, the second-order mean wave drift loads on the vessel was calculated based on either the direct pressure integration or using a semi-cylindrical control surface. Moreover, the solution was also developed for the oblique wave interaction with a two-dimensional barge-wall system using the eigenfunction expansion matching method. Systemic numerical studies were then conducted for the cases of a rectangular barge and a Wigley hull in close proximity to a long wall, respectively. Under oblique incidence, both the odd and even modes of fluid motion within the gap can be induced. In the case of a rectangular barge, the linear wave force can be obviously amplified at the location of the (1, 0) resonant mode, while the linear wave moment at the (2, 0) mode. In addition, the negative mean force along the transverse direction, which tends to push the barge away from the wall, can be apparently enhanced at both the odd and even resonant modes. Besides, in oblique seas, the (1, 0) resonant mode can give rise to the significant amplification of the positive mean yaw moment around the centre of the cross-sectional area of the barge, which tends to pull the stern of the barge (that faces incident waves) closer to the wall, while pushing the bow away from the wall. Such amplification gets less noticeable for higher modes. Numerical results also indicate that the hull geometry imposes a significant impact on the resonant waves in the gap. In the case of a Wigley hull, the amplification of the free-surface response is much less apparent when compared with a rectangular barge.

Low-frequency energy capture and water wave attenuation of a hybrid WEC-breakwater with nonlinear stiffness

Wave energy is attractive for its generous, sustainable and clean, the combination of floating breakwater (FB) and wave energy converter (WEC) is an economical approach to capture wave energy and attenuate waves. The conventional hybrid WEC-FB system is ineffective for low-frequency waves. To overcome this shortcoming, this paper proposed a novel WEC-FB with nonlinear stiffness mechanism to improve its wave attenuation and power capture performance. A hybrid semi-analytical nonlinear frequency-domain approach including the eigenfunction expansion matching method (EEMM) and multi-harmonic balance method (MHBM) is proposed to solve the nonlinear waves-structures interaction problem. The performance of the nonlinear WEC-FB is demonstrated by comparing with conventional linear counterpart, and the underlying reason for the effect of nonlinear mechanism is explored. The phase control mechanism for improving low frequency performance of wave elimination and energy capture is revealed by using the analytical method. This study shows that the introduction of the nonlinear mechanism can both improve the wave attenuation and energy absorption performance, especially in the low-frequency wave region.

Investigation on motion characteristics of an Anti-pitching Generating WEC (AGWEC) considering the viscous effect

Anti-pitching Generating Wave Energy Converter (AGWEC) uses the relative pitch motion between floating bodies to push the hydraulic energy storage PTO movement, which reduces the relative pitch between floating bodies while generating power. In this paper, based on the CFD method and the potential flow theory with viscosity modification, the motion characteristics of AGWEC in waves are studied. The viscous dissipation term is introduced into the boundary condition of potential flow, and the viscous modified potential flow theory method is established. The Eigenfunction Expansion Matching (EEM) method and Impulse Response Function (IRF) method are used to analyze the motion response of the device in frequency domain and time domain respectively. At the same time, compared with the results of CFD method based on Star-CCM+, the universality of the two methods is proved, and the motion characteristics of the AGWEC based on (non)linear PTO system and the variation law with the relevant parameters are obtained under different floating body axial spacing and wave frequency.

Analytical investigation on wave attenuation performance of a floating breakwater with nonlinear stiffness

Floating breakwaters play an important role in protecting harborage and marine structures from wave impact. Traditional floating breakwaters (FBs) are not applicative for their ineffectiveness in low frequency waves attenuation. Based on the idea of reducing equivalent stiffness, this paper proposes a FB with nonlinear stiffness to improve the wave attenuation performance. The eigenfunction expansion matching method (EEMM) and multi-harmonic balance method (MHBM) are corporately applied to solve the new model involving nonlinear coupling problem of water waves and floating structures. A pile restrained FB with quasi-zero-stiffness (QZS) is taken as an example to verify the feasibility of the hybrid analytical model and to reveal the wave attenuation mechanism of the nonlinear breakwater. The comparison between nonlinear FB and linear FB shows that nonlinear FB can effectively widen the attenuation frequency bandwidth. The conception of nonlinear floating breakwater proposed in this study provides a new technical idea for attenuating low frequency waves and the hybrid solution method adopted in this paper may be extensible to other wave-structure nonlinear coupling problems.

Hydroelastic Response of VLFS with An Attached Submerged Horizontal Solid/Porous Plate Under Wave Action

The hydroelastic responses of a submerged horizontal solid/porous plate attached at the front of a very large rectangular floating structure (VLFS) under wave action has been investigated in the context of linear water wave theory. Darcy’s law is adopted to represent energy dissipation in pores. It is assumed that the porous plates are made of material with very fine pores so that the normal velocity across the perforated porous is linearly associated with the pressure drop. In the analytic method, the eigenfunction expansion-matching method (EEMM) for multiple domains is applied to solve the hydrodynamic problem and the elastic equation of motion is solved by the modal expansion method. The performance of the proposed submerged horizontal solid/porous plate can be significantly enhanced by selecting optimal design parameters, such as plate length, horizontal position, submerged depth and porosity. It is concluded that good damping effect can be achieved through installation of solid and porous plate. Porous plate has better damping effect at low frequencies, while solid plate has better damping effect at high frequencies. The optimal ratio of plate length to water depth is 0.25–0.375, and the optimal ratio of submerged depth to water depth is 0.09–0.181.

Hydrodynamic performance of a novel WEC-breakwater integrated system consisting of triple dual-freedom pontoons

A breakwater consisting of three pontoon-type wave energy converters equipped with Power Take-Off (PTO) systems to extract wave energy from pitch and heave motions is proposed. The proposed integrated system has the potential to reduce the cost of the wave energy conversion system by sharing the essential infrastructure with the floating breakwater. In the model, the eigenfunction expansion matching method and technique of variables separation are used. The effects of the geometrical parameters (including pontoon width, draft and spacing) on the hydrodynamic performance characterized by the wave energy conversion efficiency, transmission and reflection coefficients are investigated, respectively. It is found that the effective bandwidth (transmission coefficient KT < 0.5 and hydrodynamic efficiency Cw > 0.3) and the peak efficiency are enhanced when the pitch and heave motions are considered simultaneously, compared with the single DoF motion system. The effective bandwidth increases with the decrease of width and draft of the front pontoon. And the variation of pontoon spacing affects significantly the distribution of the efficiency due to the Bragg-type reflection. Additionally, it is found that the variation of the geometrical parameters of the front pontoon on wave transmission is limited by changing the geometrical parameters of the front pontoon only.

Water wave interaction with a pair of floating and submerged coaxial cylinders in uniform water depth

Surge radiation problem due to two coaxial cylinders—one a riding hollow cylinder and the other a submerged solid cylinder of greater radius at some distance above an impermeable horizontal bottom—is solved analytically. Linear water wave theory and the method of separation of variables are used to obtain analytical expressions of the radiated velocity potential. In order to determine the unknown coefficients in the expressions, the eigenfunction expansion matching method is employed. Sets of hydrodynamic coefficients, namely, added mass and damping coefficient, for surge motion are obtained for different radii of the cylinders and for different gap between the cylinders. It is observed that change in radius, draft, and gap between the cylinders has significant effect on the hydrodynamic coefficients. The effect of the interior region of the hollow cylinder on the added mass and damping coefficient are presented. The hydrodynamic coefficients show very high oscillating behavior at a particular wavenumber which can be attributed to the occurrence of resonance in the interior region of the hollow cylinder. All the observations are presented through suitable graphs and compared with available results. Good agreement is observed with some established available results.

Hydrodynamic forces on blocks and vertical wall on a step bottom

A study, using potential water wave theory, is conducted on the oblique water wave motion over two fixed submerged rectangular blocks (breakwaters) placed over a finite step bottom. We have considered infinite and semi-infinite fluid domains. In both domains, the Fourier expansion method is employed to obtain the velocity potentials explicitly in terms of the infinite Fourier series. The unknown coefficients appearing in the velocity potentials are determined by the eigenfunction expansion matching method at the interfaces. The derived velocity potentials are used to compute the hydrodynamic horizontal and vertical forces acting on the submerged blocks for different values of block thickness, gap spacing between the two blocks, and submergence depth of the upper block from the mean free surface. In addition, the wave load on the vertical wall is computed in the case of the semi-infinite fluid domain for different values of blocks width and the incident wave angle. It is observed that the amplitudes of hydrodynamic forces are negligible for larger values of the wavenumber. Furthermore, the upper block experiences a higher hydrodynamic force than the lower block, regardless of the gap spacing, submergence depth, and block thickness.

Prediction on hydroelastic responses of very large floating structure with the eigenfunction expansion-matching method

This paper investigates the hydroelastic responses of a mat-like, rectangular very large floating structure (VLFS) with analytical method. The VLFS is considered as a two-dimensional thick plate and the wave linearity theory is employed. The flow field is divided into three regions and the eigenfunction expansion-matching method is applied in determining the velocity potential of each region, and then the Rayleigh-Ritz method is used to solve the elastic equation of motion. The analytical method is verified by comparing with the published numerical and experimental results. The elastic deformation and wave exciting force of the VLFS are computed and the effects of water depth, wavelength, draft and stiffness on the elastic deformation of the VLFS are further discussed. The deformation increases with the increase of wavelength in general, decreases with the increase of stiffness, increases slightly with the increase of draught, and does not change significantly with the depth of water.

Prediction on hydroelastic responses of very large floating structure with the eigenfunction expansion-matching method

This paper investigates the hydroelastic responses of a mat-like, rectangular very large floating structure (VLFS) with analytical method. The VLFS is considered as a two-dimensional thick plate an...

Wave diffraction from multiple truncated cylinders of arbitrary cross sections

Many marine structures are supported by piles or caissons which, from a mathematical point of view, can be assimilated to an array of truncated cylinders of arbitrary cross sections. The focus of this paper is such an array subjected to harmonic waves of small steepness. We develop an analytic method based on linear potential flow theory to solve the diffraction problem and evaluate the excitation forces and moments acting on each cylinder. The water domain is divided into the interior regions below each cylinder and an exterior region extending to infinity in the horizontal plane. A series of eigen-functions are applied to express the velocity potential in each region. The Fourier series method combined with the eigen-function expansion matching method is used to satisfy the wetted surface body conditions and continuity conditions between adjacent regions. The analytic model is validated by comparing its results with numerical modelling results and published data. It is then applied to two truncated cylinders with caisson cross sections, and results are given for the excitation forces and moments on each cylinder for different values of incident wave direction and spacing between the cylinders, and for different configurations.

Theoretical modeling of hydrodynamic characteristics of a compound column-plate structure based on a novel derivation of mean drift force formulation

To improve the seakeeping capability, some devices, such as submerged plates, are often installed on floating structures. The attached plate can not only suppress the motion response but also provide an additional immersed body surface that receives fluid action, aggravating the wave loads. In this study, a theoretical model is developed within the context of linear potential theory to study the hydrodynamic characteristics of a floating column with a submerged plate attached at the bottom. The eigenfunction expansion matching method is applied to obtain the velocity potential, based on which the linear wave force and wave runup can be found immediately. A novel derivation of the mean drift force formulation is developed via the application of Green’s second identity to the velocity potential and its derivative in finite fluid volume surrounding the body. Mean drift force formulation that involves control surfaces is then obtained. With the availability of the velocity potential, semi-analytical solution of the mean drift force on the compound column-plate structure is developed based on, respectively, the derived and the classic far-field formulations. After conducting convergence tests and validating the theoretical model, detailed numerical analysis is performed thereafter based on the theoretical model. The influence of the plate size, such as the radius and height, on the wave force and the associated wave runup is assessed.

Analytical study on wave power extraction from a hybrid wave energy converter

In this paper, a hybrid wave energy converter (WEC) is proposed, consisting of a fixed inverted flume with long length and a bottom hole, and a long floating cube hinged with the flume. The inverted flume and the long floating cube works as an oscillating water column (OWC) and a rotational float, respectively, to capture power from incident waves. To study the performance of this hybrid WEC, analytical solution of the wave diffraction/radiation problems, considering the hydrodynamic interaction between the OWC and the float, is derived based on linear potential flow theory and eigen-function expansion matching method in the two-dimensional Cartesian coordinate systems. The corresponding hydrodynamic coefficients, such as wave excitation forces, added mass and wave radiation damping, are also obtained, which can be further used in evaluation of the maximum theoretical power absorption of the hybrid WEC. Results are compared with a parallel study of an isolated OWC and an isolated float. Additionally, analytical study on power capture capability of the device for various geometrical parameters is then carried out.

Wave load on a navigation lock sliding gate

The wave load on a navigation lock sliding gate equipped with a ballast tank is investigated assuming the linear wave theory is valid and using the eigenfunction expansion matching method to solve the governing equations. The correctness and the limits of the solution have been checked by comparing the results with those of a numerical code which solve the full non linear Navier-Stokes equations. The results relating to the effect of the geometrical parameters and wave characteristics on the load acting on the gate are presented and discussed.The ballast tank is found to increase the complexity of the phenomenon in relation to the wave interaction with the gate. The results indicate that the peak value of the vertical force occurs for wave numbers mostly dependent on the tank length and on the tank position along the depth, while the thickness has a smaller influence. The ballast tank has also a significant effect on the horizontal dynamic force acting on the gate, which vanishes when the wave number takes particular values. Finally, the moment applied to the gate shows a dependency on the geometrical and hydrodynamic parameters similar to that of the forces.

Study on the performance analysis and optimization of funnel concept in wave-energy conversion

Wave-energy converters (WECs) usually comprise a wave-energy absorber and a reaction supporter on which the wave forces can work. The reaction supporter can promote the accumulation and conversion of wave energy, making WECs more feasible. In this article, we propose a novel WEC structure consisting of a submerged disk and a coaxially moon-pool paddock. In this structure, the disk functioned as the energy absorber and the paddock functioned as the corbelled and wave-energy accumulation device. The power take-off system comprises a linear damper and a spring, which can be activated by the heaving motion of the bodies. The heaving motion equation of the corresponding system in frequency domain is given according to the literature. The optimal reactive spring stiffness and resistive damping coefficients are determined by their vibration characteristics. Furthermore, the corresponding oscillation amplitudes and the wave-energy absorption capture width ratios are also determined approximately. Based upon eigenfunction expansion matching method in the linear potential theory, the analytical expressions for the diffraction and radiation are obtained. In addition, this research proposed the theoretical expressions for added mass, damping coefficient, and exciting force. Finally, the effects of the geometrical parameters and submerged depth of the moon-pool paddock on the wave-energy conversion are illustrated and discussed. It is found that the moon-pool paddock can assist accumulate wave energy and increase the energy conversion. The results also demonstrate the importance of tuning the geometries of the paddock to maximize total energy capture.

Hydrodynamic analysis and shape optimization for vertical axisymmetric wave energy converters

The absorber is known to be vertical axisymmetric for a single-point wave energy converter (WEC). The shape of the wetted surface usually has a great influence on the absorber’s hydrodynamic characteristics which are closely linked with the wave power conversion ability. For complex wetted surface, the hydrodynamic coefficients have been predicted traditionally by hydrodynamic software based on the BEM. However, for a systematic study of various parameters and geometries, they are too multifarious to generate so many models and data grids. This paper examines a semi-analytical method of decomposing the complex axisymmetric boundary into several ring-shaped and stepped surfaces based on the boundary discretization method (BDM) which overcomes the previous difficulties. In such case, by using the linear wave theory based on eigenfunction expansion matching method, the expressions of velocity potential in each domain, the added mass, radiation damping and wave excitation forces of the oscillating absorbers are obtained. The good astringency of the hydrodynamic coefficients and wave forces are obtained for various geometries when the discrete number reaches a certain value. The captured wave power for a same given draught and displacement for various geometries are calculated and compared. Numerical results show that the geometrical shape has great effect on the wave conversion performance of the absorber. For absorbers with the same outer radius and draught or displacement, the cylindrical type shows fantastic wave energy conversion ability at some given frequencies, while in the random sea wave, the parabolic and conical ones have better stabilization and applicability in wave power conversion.

Semi-Analytical Solution of Optimization on Moon-Pool Shaped WEC

Abstract In order to effectively extract and maximize the energy from ocean waves, a new kind of oscillating-body WEC (wave energy converter) with moon pool has been put forward. The main emphasis in this paper is placed on inserting the damping into the equation of heaving motion applied for a complex wave energy converter and expressions for velocity potential added mass, damping coefficients associated with exciting forces were derived by using eigenfunction expansion matching method. By using surface-wave hydrodynamics, the exact theoretical conditions were solved to allow the maximum energy to be absorbed from regular waves. To optimize the ability of the wave energy conversion, oscillating system models under different radius-ratios are calculated and comparatively analyzed. Numerical calculations indicated that the capture width reaches the maximum in the vicinity of the natural frequency and the new kind of oscillating-body WEC has a positive ability of wave energy conversion.