Wave Radiation Problem Research Articles

Hydrodynamic performance of vertical cylindrical wave energy absorbers in front of a vertical wall

This paper studies the hydrodynamic performance of vertical cylindrical absorbers in front of a vertical wall. All the absorbers are independent of each other and restricted to only the heave motion. Based on a linear potential flow theory, an analytical solution is developed for the problems of wave diffraction and radiation by absorbers. In the solving procedure, the hydrodynamic problem is first transformed into an equivalent problem in an open water domain using the image principle. The number of absorbers in the equivalent problem is twice that in the real problem, and the plane layout is symmetric about the original vertical wall. The velocity potential of the fluid domain is obtained using the method of variable separation, and the unknown expansion coefficients in the velocity potential are determined by the matched boundary conditions. The heave excitation force, added mass, radiation damping, motion response, and energy capture width of the absorbers are calculated. Case studies are presented to show the effects of the wall reflection and hydrodynamic interaction on the energy extraction performance of the wave energy converter (WEC) system. Subsequently, the WEC performance under the action of irregular waves is analyzed by considering an incident wave spectrum, and the mean annual absorbed power of the device is estimated by considering the wave data statistics at the actual sites. The results indicate that when the wave motion resonates with the absorber motion, the energy extraction performance of the absorbers is significantly improved. The performance of the absorbers can be effectively improved when the structures are close to the antinodes of a standing wave field. By designing a reasonable plane layout, the hydrodynamic interaction can play a constructive role in the performance of the WEC system.

Wave diffraction and radiation problems on a heteroaxial floating structure composed of two cylinders

To expand the application of the analytical method in dealing with the diffraction and radiation problems of the heteroaxial cylinder structure under waves, this paper takes the heteroaxial cylinder structure composed of two cylinders as an example to establish an analytical model. The wave diffraction and radiation problems of the structure were solved, and the wave force, added mass and radiation damping coefficients of the structure were obtained. The respective cylindrical coordinate systems were established in the axis of the upper and lower cylinders, and the water area was divided into three regions. Analytical solution of velocity potentials in different regions of wave diffraction and radiation problems are derived by using matched eigenfunction expansion method (MEEM) and Fourier series expansion method. After comparing with the boundary element method (BEM) software, the effect of the relative distance between the upper- and lower-cylindrical axes on the wave force and hydrodynamic coefficients of the structure were analyzed, and the effect of the height ratio of the upper- and lower-cylinder was discussed. The results show that different from coaxial structure, the wave exciting force of heteroaxial floating structure composed of two cylinders may have two peaks with the variation of wave number. The peak values of hydrodynamic coefficients A11, A22, B11 and B22 increase with the height ratio.

Linear diffraction and radiation theory of water waves by a truncated vertical circular cylinder with heave plate in deep and shallow drafts

A first-order analytical theory on a truncated and surface-piercing vertical circular cylinder with a circular plate mounted at the bottom of the cylinder was generalized to solve the linear wave diffraction and radiation problems based on potential flow and the hypothesis of small wave and motion amplitudes. The domain was decomposed, and the linearized velocity potentials were derived in each subdomain and matched on each subdomain interface employing pressure and normal velocity continuity. The linear hydrodynamic loads obtained with the analytical method were compared with the results of linear boundary element method (BEM) solvers. Dedicated experimental campaigns were performed on fixed models with regular waves (diffraction) and then on models oscillating in surge, heave, and pitch motions without incident waves (radiation). The analytical method describes well the wave excitation loads from the experiments for small wave steepness (H/λ=0.02). The predictions of added mass and damping are shown to be applicable to the surge motion, only for small Keulegan-Carpenter numbers (KC<1). On the other hand, the analytically predicted radiated waves demonstrate satisfactory agreement with experiments for all motions.

Semi-analytical and CFD formulations of a spherical floater

Today, humanity is facing the great pressure of fossil fuels exhaustion and environmental pollution. This obliges governments and industries to make accelerated efforts on producing green energy. The focus is spotted on marine environment which is a vast source of renewable energy. Among several classes of designs proposed for wave energy conversion, spherical Wave EnergyConverters (WECs) have received considerable attention. The problems of water wave diffraction and radiation by a sphere has been examined by a substantial amount of literature, i.e., [1]–[4], whereas in [5]–[8] linear hydrodynamic effects on a spherical WEC have been examined. All these research works are based on potential flow methodologies. Nevertheless, overthe last decade there has been a significant interest on Computational Fluid Dynamics CFD modelling due to its detailed results, focusing also to spherical WECs [9]–[10].In the present work a semi-analytical model is applied to solve the wave radiation problem around a spherical WEC (Figure 1), in the context of linear potential theory. The outcomes of the theoretical analysis are supplemented and compared with high fidelity CFD simulations (Figure 2 for a semi-submerged sphere). Furthermore, the two methodologies are compared with a theoretical approach for the hydrodynamic analysis of floating bodies with vertical axis as being presented in [11]. The method is based on the discretization of the flow field around the body using coaxial ring elements, which are generated from the approximation of the sphere’s meridian line by a stepped curve.Numerical results are given from the comparison of the three formulations, and some interesting phenomena are discussed concerning the viscous effects on the floater.&#x0D; [1] Havelock, T. H. 1955. Wave due to a floating sphere making periodic heaving oscillations. R. Soc. London,A231, 1-7.[2] Hulme, A. 1982. The wave forces acting on a floating hemisphere undergoing force periodic oscillation. J. FluidMech., 121, 443-463.[3] Wang, S. 1986. Motions of a spherical submarine in waves. Ocean Engng., 13, 249-271.[4] Wu, G.X. 1995. The interaction of water waves with a group of submerged spheres. Appl. Ocean. Res., 17, 165-184.[5] Srokosz, M.A. 1979. The submerged sphere as an absorber of wave power. J. Fluid Mech., 95, 717-741.[6] Thomas, G.P., Evans, D.V. 1981. Arrays of three-dimensional wave energy absorbers. J. Fluid Mech., 108, 67-88.[7] Linton, C.M. 1991. Radiation and diffraction of water waves by a submerged sphere in finite depth. Ocean Engng.,18, 61-74.[8] Meng, F., et al. Modal analysis of a submerged spherical point absorber with asymmetric mass distribution.Renew. Energy 130, 223-237.[9] Shami, E.A., et al. 2021. Non-linear dynamic simulations of two-body wave energy converters via identificationof viscous drag coefficients of different shapes of the submerged body based on numerical wave tank CFD simulation.Renew. Energy, 179, 983-997.[10] Katsidoniotaki, E., et al. 2023. Validation of a CFD model for wave energy system dynamics in extreme waves.Ocean Engng., 268, 113320.[11] Kokkinowrachos, K., et al. 1986. Behaviour of vertical bodies of revolution in waves. Ocean Engng., 13, 505-538

Wave radiation by a horizontal circular cylinder in a three-layer fluid

To construct cylindrical structures like circular pipe bridges or tunnels submerged in the almost still density-stratified ocean or seawater, the study of waves radiated by the cylinder is essential. This research solves the wave radiation problem by calculating non-dimensionalized added mass and damping coefficients to the mass of the fluid displaced by the submerged horizontal cylinder in either layer of a three-layer fluid, which is still otherwise. Under the linear theory of water waves, we investigate the circular cylinder's hydrodynamic forces by its swaying and heaving motion. The time-harmonic wave propagates with three distinct wavenumbers for a given frequency. The method of multipoles has been employed due to its rapid converging solutions by increasing the truncation limit. Potential functions are expressed into systems of linear algebraic equations, which are solved numerically for two sets of unknowns in each case by truncation. Then, the added mass and damping coefficients are obtained from the non-dimensionalized hydrodynamic forces when the horizontal circular cylinder is submerged in the lower, middle and upper layers, respectively. The obtained results are depicted graphically against wavenumber in numerous figures and analysed.

Hydrodynamic performance and energy absorption of multiple spherical absorbers along a straight coast

The development and utilization of wave energy have great potentiality to alleviate the urgent problem of global energy shortage. Spherical bodies can be used as point absorbers to extract wave energy, and much attention has been paid to the performance of spherical absorbers in an open water domain. This study focuses on the hydrodynamic performance and energy absorption of multiple spherical absorbers in front of a straight coast. The coast is assumed to be a fully reflecting vertical wall, and all the absorbers are restricted to only heave motion. An analytical solution based on linear potential flow theory is developed for the problem of wave diffraction and radiation by multiple absorbers. In the solution procedure, the hydrodynamic problem is transformed into an equivalent problem in an open water domain by applying the image principle. The velocity potential of the fluid motion is solved using the method of multipole expansions combined with the shift of local spherical coordinate systems. Then, the wave excitation force, added mass coefficient, radiation damping coefficient, and energy extraction performance of the absorbers are calculated. Case studies are presented to analyze the effects of the coastal reflection and hydrodynamic interaction among absorbers on the energy extraction performance of the wave energy converter (WEC) system. The effects of wave frequency, incident angle, spacing between the absorber and coast, submergence depth, absorber number, and plane layout are also clarified. The results suggest that the energy extraction performance of an isolated absorber is significantly improved when the motions of the waves and absorber are in resonance, and the coastal reflection can enhance the overall energy extraction performance for a WEC system with multiple absorbers. In addition, when the number of absorbers increases, the effects of the coastal reflection and hydrodynamic interaction become more complicated.

Borehole radiation and reception responses for azimuthal shear-wave reflection imaging with an off-centred dipole acoustic tool

Abstract Radiation and reception responses of a dipole acoustic logging tool placed eccentrically in borehole fluid are an interesting and important topic in acoustic reflection imaging. Herein, we present a thorough research study on these responses. We treat the wave incidence from the reflector as the radiation from a virtual source and use the cylindrical-wave expansion method to solve both the wave radiation and reception problems for the off-centred tool, which, by using the steepest-descent method, yields asymptotic solutions for modelling the radiation and reception wavefield characteristics. The modelling results from the analytical solution and the 3D finite-difference method were in good agreement. Specifically, we analysed the radiation directivity of an eccentric dipole source in a fluid-filled borehole. The results revealed that the radiation pattern was asymmetric with respect to the borehole, and the asymmetry was determined by the eccentric distance, source frequency and formation properties. In particular, for the typical 3 kHz dipole logging frequency, the radiation was stronger in the off-centred direction than in the opposite direction. The asymmetry of the eccentric radiation resulted in a significant amplitude difference relative to its centred counterpart, which provided a potential method for addressing the 180° azimuth ambiguity of the dipole source. We used a theoretical waveform modelling example to demonstrate this advantage. Therefore, the results of this study provide a theoretical foundation for the development and application of dipole shear-wave imaging technology.

Hydrodynamic performance of a horizontal cylinder wave energy converter in front of a partially reflecting vertical wall

Based on linear potential flow theory, this study investigates the hydrodynamic performance of a horizontal cylinder wave energy converter (WEC) in front of a wall. The wall is assumed to be a partially reflecting vertical one, and the circular cylinder is fully submerged and restricted to only heave motion. The structural hydrodynamic quantities, including wave excitation force, added mass coefficient and radiation damping coefficient, are estimated using the well-known wide-spacing approximation, and the wave energy capture performance of the device is investigated. The estimation only requires solving the problems of water wave diffraction and radiation by a submerged horizontal cylinder in an open fluid domain, which can be analytically solved using the multipole expansion method. The estimation is validated by comparison with numerical results based on the boundary element method (BEM), and analytical results for the limiting case of a fully reflecting wall. Case studies are presented to clarify the effects of wave frequency, spacing between cylinder and wall, wall reflection and cylinder size on the hydrodynamic quantities and energy capture performance of the WEC. The hydrodynamic problem for regular waves is also extended to irregular waves. The results indicate that the WEC's performance can be significantly improved owing to the presence of a wall, which provides several feasible approaches for capturing more wave energy, such as setting vertical plates behind WECs and installing WECs near caisson breakwaters. This study also provides a reliable method for evaluating the performance of other WECs.

Radiation of water waves by a heaving submerged disc in a three-layer fluid

A thin rigid disc is submerged in the infinitely deep lower layer of a three-layer fluid. Using linear water wave theory, the three-dimensional problem of wave radiation due to heaving motion of the disc is considered. The problem is first reduced to solving a hypersingular boundary integral equation. This integral equation is then reduced to a system of one-dimensional Fredholm integral equations of second kind. The solution of each integral equation is associated with a radial component of the difference of potential function across the plate. The solution of these integral equations has been used to compute added mass and damping coefficient numerically. Numerical results for a heaving disc submerged in a single layer, a two-layer, and a three-layer fluid have been depicted. Also, the effects of different parameters like depth of upper layer, middle layer, distance of the disc from the interface, and the density ratios of the fluid layers on these physical quantities have been demonstrated.

Three dimensional numerical modelling for wave radiation problem under arbitrary seabed condition

Three dimensional Rankine source model is utilized to investigate the hydrodynamic responses for wave radiation problems under arbitrary seabed condition in time domain. The free surface, seabed surface and body surface are presented by layer of continuous panels rather than a discretization by isolated points. These panels are positioned exactly on the fluid boundary surfaces and no desingularization technique required. To accommodate the sloping seabed profile, a straightforward source panel distribution on seabed surface is newly developed with two parameters determined by numerical tests. The accuracy and efficiency of the numerical solutions are validated by comparison with published data. The influences of flat seabed, uneven seabed and sloping seabeds are examined. Numerical simulations demonstrate that various seabed profiles have significant and complicated effects on the body hydrodynamic response characteristics.

Interactions of waves with a body floating in an open water channel confined by two semi-infinite ice sheets

Abstract

Numerical Study on Wave Radiation by a Barge with Large Amplitudes and Frequencies

A two-dimensional boundary element method is used to study the hydrodynamics of a single barge with prescribed motions of large amplitudes and high frequencies. The wave radiation problem is solved in the time domain based on the fully nonlinear potential flow theory. For numerical simulations, special treatments like plunging wave cutting and remeshing approaches are presented in detail. The numerical schemes are verified through comparing with analytical results. Both the generated outgoing wave amplitudes and hydrodynamic coefficients can be calculated with sufficient accuracy. Then, we focus on large heave, sway and roll motions to investigate the nonlinear effects on hydrodynamic forces, respectively. In particular, the heave motion with two frequencies is also simulated to study the interactions between results at different frequencies. It is interesting to see the sum and difference frequency components and the envelopes in time histories as a result. For forces caused by forced sway or roll motions, there are only even-order harmonics for vertical forces and only odd-order harmonics for horizontal forces. Finally, a single body with combined sway, heave and roll motion is studied to examine the interactions between motion modes.

A study on full interaction of water waves with moored rectangular floating breakwater by applying 2DV scaled boundary finite element method

The problem of interaction between water wave and moored rectangular floating breakwater is studied using the semi-analytical Scaled Boundary Finite Element Method (SBFEM) in a two-dimensional vertical plane. The current study applies a generalized solution process by locating the scaling center within the bounded sub-domains instead of the body of structure, resulting in a homogeneous SBFEM equation. In spite of the merits of such choice for the scaling center that makes possible the analysis of non-rectangular objects, the method is changed to numerical-analytical one with its consequences. Such numerical solution in the problem of water wave radiation and diffraction is validated by comparing the results with previous reported studies. The generality and robustness of the current SBFEM-based model is also examined for non-rectangular cross-sections of the floating breakwater. Focusing on the proposed SBFEM-based model, the full wave-rigid body interaction is analyzed for a moored rectangular floating structure. Moreover, in order to make the linear numerical modeling accurate, taking into account the nonlinear aspects of the interaction problem, the artificial damping coefficients are quantified, and the relevant formula is derived.

Moving Object: Spectra of Signals of Passive, Active Location and Transient Radiation

The paper presents an analysis of basic problems of signal formation by a moving object as problems of wave radiation and reflection with allowance for moving boundaries. The inverse problem is considered: reconstruction of the law of displacement of an object based on the known shape of the probing and reflected waveforms. The spectrum of the harmonics generated by reflection from an incrementally moving and oscillating obstacle is found. The correlation function of the noise emitted by a moving object is calculated. General considerations about the transient radiation of a flying body in restrained urban conditions are presented.

Hydrodynamic coefficients for a floating semi-porous compound cylinder in finite ocean depth

Employing the methods of separation of variables and matched eigenfunction expansions for velocity potential, analytical solutions are proposed for a water wave radiation problem of a floating semi-porous compound cylinder in finite ocean depth. The configuration of the semi-porous compound cylinder is such that it consists of an impermeable inner cylinder rising above the free surface and a coaxial truncated porous cylinder around the lower part of the inner cylinder with the top of the porous cylinder being impermeable. The condition on the porous boundary is defined by applying Darcy’s law as in Williams et al. (Ocean Eng 27:1–28, 2000) . The translational motions in the x- and z-directions, i.e., surge and heave motions, are investigated. A mathematical model is developed which can be considered as an extension of a number of the earlier works, e.g., Kokkinowrachos et al. (Ocean Eng 13:505–538, 1986) and Calisal and Subancu (Ocean Eng 11(5):529–542, 1984), in which significance of porosity of the structure was neglected. Numerical investigation is taken up here in order to examine the influence of submerged depth, radii, porous coefficient, and water depth on added mass and radiation damping, two most important entities in radiation problems, with respect to surge and heave motions. It is found that the variation of porous coefficient, radii, and depth has a significant influence on the added mass and damping coefficients for the semi-porous compound cylinder. The added mass is found not sufficiently affected by lower values of porous coefficient G, but exhibits significant variation corresponding to higher values of G. Another important observation is that the damping coefficients oscillate alternately between negative and positive values which can be attributed to coupled behavior between different motions. The results establish that an appropriate optimal ratio of various parameters may be considered in designing ocean structures with minimum adverse hydrodynamic effect. The effectiveness of the present model is validated by comparing it with an available result which shows an excellent agreement.

Wave power extraction from a hybrid oscillating water column-oscillating buoy wave energy converter

Oscillating water column (OWC) devices and oscillating buoys (OBs) are two of the main types of wave energy converters (WECs). In this paper a hybrid oscillating water column-oscillating buoy wave energy converter is proposed, which we have named OWCOB. The oscillating buoy is hinged at the outer wall of the oscillating water column. As waves propagate through the OWCOB, the water column within the OWC chamber moves up and down, producing air flow to propel a turbine. Meanwhile, the oscillation of the OB drives a separate hydraulic system. To solve the wave diffraction and radiation problems of the OWCOB and investigate its energy capture performance, an analytical model is developed based on linear potential flow theory and the eigenfunction matching method. Assuming that the PTOs of the OWC and OB are both linear, the wave power extraction of the OWCOB is evaluated in the frequency domain. Of the two configurations considered, the OWCOB with the OWC opening waveward and the OB hinged leeward is found to have a broader primary frequency band of wave power capture compared to the OWCOB with the OWC opening and the OB on the same side. Further, a thorough sensitivity analysis of power capture is carried out considering the main design parameters (size and submergence of the OWC opening, distance between the OWC and the OB, OB hinge elevation, OB radius), which can form the basis of an optimization study for a particular wave climate. Importantly, we find that the OWCOB performs generally better than stand-alone OWCs and OBs, not least in terms of frequency bandwidth.

Hydrodynamic performance of a multi-Oscillating Water Column (OWC) platform

A rectangular barge consisting of multiple oscillating water columns (OWCs) is considered in this paper, hereinafter referred to as a multi–OWC platform. Each OWC chamber is enclosed by two partially submerged vertical walls and the deck of the platform. An incident wave produces oscillation of the water column in each OWC chamber and hence air is pumped by the internal water surface to flow through a Wells turbine installed at the chamber top. The effect of the turbine is characterised as a linear power take–off (PTO) system. A semi–analytical model based on linear potential flow theory and the eigen–function expansion method is developed to solve the wave radiation and diffraction problems of the multi–OWC platform. The hydrodynamic coefficients evaluated with direct and indirect methods of the model are shown to be in excellent agreement, and the energy conservation relationship of the multi–OWC platform is satisfied. The validated model is then applied to predict wave motion, dynamic air pressure, wave power extraction, and wave reflection and transmission coefficients of the multi–OWC platform. The effects of the PTO strategies, the number of chambers, the overall platform dimensions and the relative dimensions of adjacent chambers on wave power extraction and wave attenuation are investigated. A smaller–draft front wall and a larger–draft back wall are found to be beneficial for broadening the range of high–efficiency performance of the platform. The same wave transmission coefficient can be obtained by two multi–OWC platforms with inverse geometric constructions.

A frequency-dependent absorbing boundary condition for numerically solving u-U elastic wave equations in layered and fluid-saturated porous media

An accurate absorbing boundary condition (ABC) is proposed to incorporate in the frequency-domain finite element simulation for numerically solving u-U elastic wave equations in horizontally multilayer fluid-saturated porous media. The u-U elastic wave equations are first discretized only along the depth direction in which the material properties are heterogeneous, analytical method using in the remaining coordinate direction. A general eigenvalue problem on the horizontal wavenumber is then solved with the frequency as known parameter. Dynamic stiffness on the artificial boundary of finite domain is finally obtained as ABC based on the solution to the above eigenvalue problem. The proposed ABC is mathematically derived without introducing any approximate assumptions, and the only approximation comes from the finite-element discretization of the fluid-saturated porous medium in the depth direction. The proposed ABC can also be coupled seamlessly with the finite element method (FEM). Numerical examples of wave radiation problems are given to demonstrate the effectiveness of the proposed ABC and its coupling with FEM. Finally, a practical application of proposed ABC to assess the topography effects of the fluid-saturated porous basin-type site is presented to show that the proposed ABC can also be used to solve wave scattering problems in geotechnical earthquake engineering.

A novel solution to the second-order wave radiation force on an oscillating truncated cylinder based on the application of control surfaces

A truncated cylinder in harmonic oscillation with a small amplitude at a finite water depth is considered. A perturbation procedure is applied to analyse the wave radiation problem up to the second order in frequency domain. The technique of control surfaces is used to overcome the difficulty associated with the singularity of the fluid velocity at the corner, and a novel solution to the double-frequency wave radiation force on a surging/heaving truncated cylinder is developed. The proposed solution achieves an obvious advantage in that the calculation of the singular terms owing to the singularity of the fluid velocity at a sharp corner is avoided. After examining the convergence and validity of the proposed model, numerical studies are conducted. Numerical results reveal that the force component associated with the body-surface forcing term predominates among the various force components of the total double-frequency wave radiation force. In addition, the force component from the quadratic products of the first-order quantities is in general out-of-phase with the one associated with the body-surface forcing term.

A hybrid method for linearized wave radiation and diffraction problem by a three dimensional floating structure in a polynya

A hybrid method is developed to solve the interaction problem of wave with a three dimensional floating structure in a polynya. The linearized velocity potential theory is used for fluid flow, and the thin elastic plate model is adopted for the infinitely extended ice sheet. Because of sudden change of the upper boundary of the computational domain, namely from the ice sheet to the free surface, the domain is divided into two sub-domains, one below free surface and the other below the ice sheet. The solution method is divided into three components. The first component is the integral equation over the structure surface and the interface of the two sub-domains. In the second component, the velocity potential is expanded into a series of eigenfunctions in the vertical directions, which avoids the numerical difficulty in calculation of the fifth derivatives. This is coupled with a series of integral equations along the edge of the ice sheet. In the third component of the method, two orthogonal inner products are used to impose the continuity conditions of the velocity and pressure on the interface, as well as the boundary conditions on the ice edge. The developed method is verified through comparison with the analytical solution for a circular cylinder. Case study is then made for a FPSO in a polynya with different shapes and floating positions. The hydrodynamic coefficients, wave exciting force and wave elevation in polynya are provided and analyzed.