Time-domain Higher-order Boundary Element Method Research Articles

Numerical and Experimental Investigations on Non-Linear Wave Action on Offshore Wind Turbine Monopile Foundation

Monopiles are commonly utilized in offshore wind farms but are prone to non-linear wave loads and run-ups, significantly affecting their engineering design. Therefore, it is crucial to pursue a complete understanding of the non-linear wave action on monopile foundations. Both numerical and experimental investigations on the non-linear wave loads and run-ups on an offshore wind turbine monopile foundation are performed in this paper. The experiment is carried out at a scale of 1/30 in a wave flume at the State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, in which the wave loads and run-ups along the monopile are measured. Based on the second-order potential flow model and time-domain higher-order boundary element method (HOBEM), the related numerical tests are conducted to study the non-linear effects further. It is found that the present non-linear potential theory is sufficient for the simulation of wave force and run-ups on the monopile in the range of wave slope kA < 0.15 before wave breaking. “W” type distribution of wave run-up along the monopile is found, in which the peak value occurs at the frontward side (i.e., θ =180°) and is the maximum due to full reflection; the two symmetrical minimum amplitudes lie in the zones of (45° ≤ θ ≤ 90°) and (270° ≤ θ ≤ 315°), whose positions shift downward with the increase of wave non-linearity. Energy transfer among the fundamental wave component and higher-order components is also found, which is most apparent on the backward side. Besides, the transverse resonance occurs in the wave flume due to the wavelength being near the flume width, which induces the wave run-up at the backward position larger than that at the frontward position.

Hydrodynamic Investigation on a Land-Fixed OWC Wave Energy Device under Irregular Waves

The hydrodynamic response of a land-based oscillating water column (OWC) wave energy converter under various irregular wave conditions is investigated numerically. Based on the potential flow theory, a two-dimensional fully nonlinear numerical wave flume (NWF) is developed to model the hydrodynamic characteristics using the time-domain higher-order boundary element method (HOBEM). The inner-domain sources method and JONSWAP energy spectrum is used to generate the irregular incident waves, and a linear pneumatic model is used to determine the pneumatic pressure inside the chamber. The free surface elevations at the chamber centre and the oscillatory pneumatic pressures induced by the vertical motion of the water column are recorded. The influence of irregular waves on the hydrodynamic characteristics of the OWC device is carried out by comparison with the regular waves, and a number of significant wave heights and peak wave periods are considered. The hydrodynamic efficiency of the OWC device in irregular wave conditions is observed to be lower than that in regular waves for most wave frequencies, especially near the resonant frequency.

Nonlinear wave loads on an offshore oscillating-water-column wave energy converter array

This paper studies the nonlinear wave loads on a stationary platform that consists of four cylindrical Oscillating Water Column (OWC) Wave Energy Converters (WECs). The nonlinear wave action on the quad-OWC platform was simulated using a 2nd-order time-domain Higher-Order Boundary Element Method (HOBEM). A carefully instrumented experiment was carried out to validate HOBEM simulations. A circular orifice was introduced to model the nonlinear Power Take-Off (PTO) system on the ceiling of the OWC chamber. Subsequently, the sensitivities of the nonlinear wave loads to the incident wave direction and array spacing were investigated. The optimum angle of wave incidence was found to be π/4, i.e., when the quad-OWC platform is diagonally aligned with the incoming waves. A 2nd-order sloshing resonant mode was found inside each chamber of the OWC array, which leads to an increase of the wave loads. A good selection of the column and row spacing contributes to the reduced wave forces on both the front and rear OWC devices. The directions of the wave loads on the front device were analyzed. The rotational-angle peaks of the wave force and moment were always observed at high frequencies.

Water entry of a floating body into waves with air cavity effect

Based on a fully nonlinear time-domain higher-order boundary element method (HOBEM), the present paper investigates a semi-circular floating body entering obliquely into Stokes wave with air entrapment. In this model, a fifth-order analytical solution is used to simulate the Stokes wave as incident boundary condition. A long and thin jet layer is simulated and assumed to separate from wetted body surface along its tangential direction, which avoids fluid particular leaving or invading body surface. When inner free surfaces on both sides of body collide with each other, a closured air cavity is formed based on the assumption of adiabatic. Dual local stretched coordinate systems are applied to resolve the impact with air cavity effect and downward motion of inner jet due to the local impacts between two water surface which is mainly dynamic effects. Some auxiliary functions are employed to obtain the pressure distribution induced by wave impact. The developed model is verified against the published numerical results for water entry with air cavity effect in the absence of waves. Then, numerical simulations are undertaken to investigate the mechanism of wave entry with air cavity formation through designed parameters, i.e. wave amplitude, wave length, initial wave phase, initial air pressure, prescribed floating body velocity. Numerical results indicate that the presence of wave delays the formation of air pocket and leads to more asymmetric shape of air pocket with the increase of wave nonlinearity when the body impacts the wave peak. After the occurrence of air pocket, the fluid pressure sharply increases.

Nonlinear hydrodynamic analysis of an offshore oscillating water column wave energy converter

The hydrodynamic performance of a floating cylindrical oscillating water column (OWC) wave energy converter is investigated experimentally and numerically. The physical experiment is carried out in a wave flume at Dalian University of Technology. The floating cylindrical OWC device is constrained by springs and only moves vertically. A second-order time-domain Higher-Order Boundary Element Method, based on the perturbation expansion technique, is used to simulate the nonlinear wave interaction with the floating OWC device. The nonlinear terms concerning the pneumatic and viscous damping are introduced to the free surface boundary conditions inside the OWC chamber. The chamber surface elevation and air pressure, the hydrodynamic efficiency, and the vertical displacement of the OWC device are examined in detail. Good agreements are obtained between experimental data and numerical results. Then, the effects of opening ratio, wave steepness, mooring stiffness and chamber draft on the hydrodynamic performance are then investigated. It is found that the optimal opening ratio is between 0.02 and 0.03. The mooring stiffness plays an important role on the hydrodynamic response of the OWC device. The hydrodynamic efficiency and effective frequency bandwidth increase with the mooring stiffness.

Wave loads on a land-based dual-chamber Oscillating Water Column wave energy device

Wave energy is one of renewable energy resources with great potential. Due to the mechanical and structural simplicity, Oscillating Water Column (OWC) Wave Energy Converter (WEC) is considered to be one of the most promising marine renewable energy devices. However, OWC remains not commercialized mainly due to its complex hydrodynamic performance and uncertainty in wave loads. In the present study, based on potential flow theory and time-domain higher-order boundary element method (HOBEM), a fully nonlinear numerical model is developed and used to investigate the wave-induced force and bending moment on a land-fixed dual-chamber OWC device. The Bernoulli equation is used to calculate the wave force and bending moment. The equation is modified by accounting for the pneumatic pressure in the air chamber and the viscosity effect and then solved using an acceleration-potential method. The numerical model was compared with the experiment carried out in a wave-current flume at the State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, China, and good agreement between the simulation and experimental results was observed. The horizontal components of wave forces on the two curtain walls of the dual-chamber OWC WEC were found to be much larger than the corresponding vertical components. The seaside curtain wall suffered much larger wave loads in comparison with the inner curtain wall. Therefore, the wave force on the seaside curtain wall is the dominant force. The largest wave-induced bending moment occurs at the joint of device and seabed. The effects of the sub-chamber width ratio and curtain-wall draft on the wave-induced force and bending moment are investigated. The dominant wave force and moment increase with curtain wall draft. And the peak wave loads can be reduced by moving the internal curtain wall close to the seaside curtain wall.

Geometrical investigation of a U-shaped oscillating water column wave energy device

Hydrodynamic performance of a fixed U-shaped oscillating water column (U-OWC) wave energy converter is numerically investigated. Based on the time-domain higher-order boundary element method (HOBEM), a two-dimensional fully nonlinear numerical model is implemented to simulate the nonlinear wave interaction with a U-OWC device. In the model, the inner-domain-source method is adopted to generate incident waves and a linear pneumatic model is used to determine the air pressure which is imposed on the free surface inside the chamber. The numerical model is well validated against the published experimental data of the free surface elevation at the chamber centre, air pressure inside the chamber and hydrodynamic efficiency. Further, the present model is applied to study the effects of geometrical parameters (including the vertical duct height, vertical duct width and wall thickness) on the hydrodynamic performances of the U-OWC device. The results indicate that geometrical parameters of the vertical duct have significant influence on the air pressure inside the chamber and the hydrodynamic efficiency. The hydrodynamic efficiency and air pressure inside the chamber are found to increase with the increase of both vertical duct height and thickness of wall I. However, the increasing rate of the efficiency slows down gradually with the thickness of wall I increasing, which indicates that a comprehensive consideration of the construction cost and hydrodynamic performance is needed in the design and construction of a U-OWC device.

Numerical investigation of waves slamming on a rotating flap in free motion using a fully nonlinear HOBEM

A time-domain higher-order boundary element method (HOBEM) with fully non-linear boundary conditions is established to investigate the slamming problem on a rotating flap in free motion. The mixed Eulerian-Lagrangian (MEL) boundary conditions with numerical damping layers are adopted to track the time-dependent free surface nodes, through which the influences of coupled disturbed waves from the outer boundaries are eliminated. The potential of each element within the thin long jet is calculated in the rotating coordinate system to overcome numerical instability induced by the local highly oscillatory pressure. The domain decomposition method and the unequal distribution scheme of meshes are simultaneously adopted to capture the falling jet with strong non-linearity. The auxiliary function method is then employed to calculate the pressure distribution, through which the mutual dependence between the body motion and the fluid flow is decoupled. The present numerical model is verified through comparing the flap motion response, dynamic characteristics and free surface elevations against the published experimental data and the Computational Fluid Dynamics (CFD) results. The parameter studies are performed to investigate the influences of flap properties and wave parameters. Detailed results are provided through free surface profile, rotation angle, angular velocity, torque response and pressure distribution.

A numerical study of second-order springing of an elastic body using higher-order boundary element method (HOBEM)

Second-order springing on an elastic body with forward speed is analyzed by numerical simulations. The boundary-value problem for the velocity potential is solved by means of the direct time-domain higher-order boundary element method (HOBEM). The free-surface boundary condition in the boundary-value problem is approximated on the mean surface up to second order by use of perturbation and Taylor-series expansion methods. The body boundary condition for an elastic body is derived with various quantities which are redefined in the generalized mode. These variables such as mode shape, normal vector, etc. are obtained by using directional derivative and continuum mechanics, and the same mathematical expressions are used to obtain several second-order generalized forces. To validate the numerical results, the second-order hydrodynamic force on the bottom-mounted rigid/elastic cylinders without forward speed is compared with other semi-analytic results. The property of second-order forces on an elastic ship is studied by changing the flexural rigidity and forward speed with elastic response. It is confirmed that the second-order velocity potential is important for a body with forward speed and investigation should be made more on numerical methods for accurate computation of the second-order velocity-potential force with forward speed.

Solitary wave slamming induced by an asymmetric wedge through three degrees of freedom free motions

A time-domain higher-order boundary element method with fully nonlinear boundary conditions is developed to simulate the slamming of an asymmetric wedge entering freely and obliquely into a solitary wave in three degrees of freedom (3DOF). A third order analytical solution based on the Korteweg-de Vries equation is used to simulate the solitary wave incident boundary conditions. In the numerical model of slamming, a stretched coordinate system is applied to maintain numerical accuracy and stability at the initial stage. The thin long jet layer is generated along the wedge surface by assuming linear variation of the jet layer potential. A rotation scheme of the stretched coordinate system is adopted to avoid fluid particle leaving or entering the wedge surface. Some auxiliary functions are employed to decouple the intercoupling motions in 3DOF. The present model is verified by comparing with the published numerical results. Various parametric studies are carried out. Detailed results through the free surface, pressure distribution, accelerations, and velocities are provided to show the slamming effects, and their physical implications are discussed.

Fully nonlinear analysis incorporating viscous effects for hydrodynamics of an oscillating wave surge converter with nonlinear power take-off system

WaveRoller can be classified as an effective Oscillating wave surge converter (OWSC) installed in nearshore coastal areas. In this study, the hydrodynamic performance of a two-dimensional (2-D) WaveRoller is investigated based on a fully nonlinear time-domain higher-order boundary element method (HOBEM). The mechanism through which the wave power is extracted, is analyzed in a nonlinear power take-off (PTO) by coupling the time-varying motion of the flap and hydro-electric generator. A simplified Morison model is used to determine the viscosity term in the structural motion equation. The present model is validated against the published experimental and numerical results for an OWSC with its top edge piercing through the water surface. Numerical simulations are undertaken to investigate the mechanism of the nonlinear phenomenon including wave nonlinearity and PTO nonlinearity. The influence of the viscosity is also quantified by comparing the numerical results with and without the quadratic damping term. Numerical results indicate that when the incident wave amplitude is large, the capture efficiency is reduced by the strengthen higher order free wave components, and the PTO nonlinearity becomes more prominent. Furthermore, the maximum discrepancy induced by the viscosity occurs in the resonant zone.

Fully nonlinear simulation of wave–current interaction with an oscillating wave surge converter

On the basis of a time-domain HOBEM (higher-order boundary element method), a two-dimensional (2D) fully nonlinear numerical model is developed to investigate the wave energy capturing efficiency of an OWSC (oscillating wave surge converter) in the presence of uniform currents. In the model, the wave energy is extracted through a linear external damping coefficient, and the natural frequency of the OWSC is tuned to the wave characteristics by introducing an elastic and inertia coefficient. On the free surface, a five-point fourth order Lagrangian interpolation method is applied to determine the redistribution of meshes, and a rotation scheme of spatial coordinate system is employed to tackle the discontinuity of flux at the intersection points. The wave loads on the wetted body surface are then evaluated using two auxiliary functions through which the mutual dependence of body motion and fluid flow is decoupled. The present model is validated against the published experimental and numerical results for an OWSC in large motion excited by nonlinear waves. Simulations are performed to capture the hydrodynamic behavior of the OWSC with different currents. Numerical results are also provided to show the behavior dependence on the design parameters, such as current velocities, wave conditions, geometric characteristics and mechanical properties.

Numerical investigation of solitary wave slamming on an oscillating wave surge converter

A solitary wave slamming on an Oscillating Wave Surge Converter (OWSC) is numerically investigated using a time-domain higher-order boundary element method with fully non-linear boundary conditions. A stretched coordinate is implemented to improve the numerical efficiency as long as the slamming peak pressure occurs. The potential of the thin long jet is assumed to vary linearly, while the process of jet detaching is simulated through the domain decomposition method so that the local highly oscillatory pressure can be avoided. Two auxiliary functions are applied simultaneously to decouple the mutual dependence between the flap motion and the fluid flow. A unique mesh scheme is employed to simulate the free surface with strong deformation, through which the smallest meshes are distributed near the largest pressure gradient on the body and the mesh size increases gradually at a ratio. The validity of the present model to simulate the solitary wave and the slamming event is verified, respectively, based on which relatively comprehensive parameter studies are then performed. Through analyzing the flap’s motion states, the free surface profiles, and the pressure distributions, it is found that several unique phenomena and mechanisms exist in the solitary waves slamming on an OWSC.

Current effects on nonlinear wave slamming by an Oscillating Wave Surge Converter

A time-domain higher-order boundary element method (HOBEM) based on potential theory with fully nonlinear boundary is developed to simulate an Oscillating Wave Surge Converter (OWSC) rotating in waves in the presence of uniform currents. A domain decomposition method which neglects impact of jet flow to the main flow field is adopted. Auxiliary function is used to calculate the pressure distribution on body surface. The rotating coordinate system with origin fixed at the rotating center is established to avoid the water particle moving into or away from the body during the simulation. An iteration method is employed to assess the values of coupling wave characteristics affected by currents. Good agreement is found between present model and published fist-order boundary element method (FOBEM) model for the interaction of flap and current-free waves. Numerical simulations are then conducted in the presence of following current, opposing current and zero current, respectively. Several parametric studies, such as angle velocity and angular acceleration of the flap, initial phase, wave height and wave length, are performed to study the characteristics of the slamming problem with taking into account currents effects. The numerical results are presented through free surface profiles and pressure distributions.

Motion response analysis of large-scale structures with small-scale cylinders under wave action

The influence of small-scale cylinders is always ignored in classical models for hydrodynamic calculations of truss spar platforms and other large-scale structures in published research. This study is focused on this influence on the motion response under the wave action so that the response can be forecast more accurately. A numerical model is proposed to calculate the wave-body interactions under the second-order frame by utilizing the time-domain higher-order boundary element method. The wave loads on small-scale cylinders are calculated by the Morison formula, in which the wave field disturbed by a large-scale structure is taken into account. Three-dimensional elastic rod theory is chosen to analyze the cable force to a platform, and the motion equation for the mooring system and hull are solved simultaneously using the Newmark-β method. A moored truss spar platform is modelled, and the numerical results demonstrate that the existence of small-scale cylinders may lead to a decrease of the motion response of the platform, while the natural frequency will move to a lower frequency. This phenomenon explains why small-scale cylinders, which can be modelled as truss spar play an important role in damping and change the added mass of the platform. Therefore, in order to get more accurate results for motion response, the wave loads of small-scale components cannot be ignored.

Numerical investigation of water entry of a wedge into waves with current effects using a fully nonlinear HOBEM

This paper concerns the hydrodynamic performance of a two-dimensional asymmetric wedge entering waves obliquely with gravity effect in the presence of a uniform current. A time-domain higher-order boundary element method (HOBEM) based on potential theory with fully nonlinear boundary conditions is developed to solve this problem. A stretched coordinate system is adopted in the spatial domain to avoid a large number of elements discretized in fixed computational domain at the initial stage. During mesh regriding and interpolation, a rotation scheme of the stretched coordinate system is employed to meet the requirement of continuity of flux at the intersection points. An auxiliary function is introduced to calculate pressure distribution on body surface. The present model is validated against the published numerical results for vertical wave entry in the absence of currents. Numerical calculations are conducted to analyze the dependence of the free surface elevation and the pressure distribution on the physical parameters, such as current velocity, incident wave amplitude, entry location, heel angle and oblique motion. It is found that the variation of current velocity has obvious effect on both the horizontal velocity of fluid and the wave steepness, which play a major role on free surface and pressure.

Nonlinear and viscous effects on the hydrodynamic performance of a fixed OWC wave energy converter

The hydrodynamic performance of a fixed Oscillating Water Column (OWC) device is experimentally and numerically investigated. Based on the time-domain higher-order boundary element method (HOBEM), by introducing an artificial viscosity term in the dynamic free surface boundary condition, a fully nonlinear numerical wave model is used to simulate the hydrodynamic performance of an OWC device. A set of comprehensive experiments for regular waves is carried out to validate the numerical results as well as to investigate the nonlinear effects on the hydrodynamic performance of OWC. The mechanism of the nonlinear phenomenon is investigated based on the analysis of the experimental and numerical results. The influence of the wave nonlinearity and the viscosity on the hydrodynamic efficiency is quantified by comparing the linear and nonlinear numerical results. It was found that the hydrodynamic efficiency increases with the nonlinearity and viscosity when the incident wave amplitude is small. When the incident wave amplitude is large, the hydrodynamic efficiency is reduced by the weakened transmission of the second-order harmonic wave component due to the strong wave nonlinearity. However, when the wave amplitude is between these two regimes, the wave is weakly nonlinear, the efficiency decreases with the wave amplitude due to the combined effect of the nonlinearity and viscosity.

Numerical Simulation of a Dual-Chamber Oscillating Water Column Wave Energy Converter

The performance of a dual-chamber Oscillating Water Column (OWC) Wave Energy Converter (WEC) is considered in the present study. The device has two sub-chambers with a shared orifice. A two-dimensional (2D) fully nonlinear numerical wave flume based on the potential-flow theory and the time-domain higher-order boundary element method (HOBEM) is applied for the simulation. The incident waves are generated by using the immerged sources and the air-fluid coupling influence is considered with a simplified pneumatic model. In the present study, the variation of the surface elevation and the water column volume in the two sub-chambers are investigated. The effects of the chamber geometry (i.e., the draft and breadth of two chambers) on the surface elevation and the air pressure in the chamber are investigated, respectively. It is demonstrated that the surface elevations in the two sub-chambers are strongly dependent on the wave conditions. The larger the wavelength, the more synchronous motion of the two water columns in the two sub-chambers, thus, the lager the variation of the water column volume.

Extreme wave run-up and pressure on a vertical seawall

The performance of coastal vertical seawalls in extreme weather events is studied numerically, aiming to provide guidance in designing and reassessing coastal structures with vertical wall. The extreme wave run-up and the pressure on the vertical seawall are investigated extensively. A time-domain higher-order boundary element method (HOBEM) is coupled with a mixed Eulerian-Lagrangian technique as a time marching technique. Focused wave groups are generated by a piston wave-maker in the numerical wave tank using a wave focusing technique for accurately reproducing extreme sea states. An acceleration-potential scheme is used to calculate the transient wave loads. Comparisons with experimental data show that the extended numerical model is able to accurately predict extreme wave run-ups and pressures on a vertical seawall. The effects of the wave spectrum bandwidth, the wall position and the wave nonlinearity on the wave run-up and the maximum wave load on the vertical seawall are investigated by doing parametric studies.

Numerical and experimental investigation of nonlinear focused waves-current interaction with a submerged plate

This paper concerns the interaction of focused waves with uniform current and a submerged horizontal plate in the context of the two-dimensional nonlinear potential theory. A fully nonlinear NWT (numerical wave tank) based on a time domain HOBEM (higher-order boundary element method) is applied to the numerical solutions. In the model, the Goda's method is modified to discriminate incident and reflected waves, whereas the acceleration potential approach is introduced to determine the hydrodynamic forces on the structure. The corresponding experiments are also conducted in a two-dimensional glass-walled wave tank at Jiangsu University of Science and technology. The developed model is verified with published numerical results and validated with experimental test. Through a systematic parametric study, it is found that the maximum reflection or minimum transmission for different currents occur at the same plate width because of the compensating Doppler effects of current. The spectral analysis reveals that the energy dissipation occurs primarily in the fundamental waves when focused waves propagating over the plate. The current has a stronger influence on the wave forces than on wave surface elevation. Furthermore, the 'secondary loading cycle' phenomenon is also observed in the time series of the wave forces for a following current.