Single Mode Simulation Calculation of Oscillating Buoy Wave Energy Converter with A Slider

This paper presents an oscillating slider wave energy device which is based on a seabed anchoring and uses eagle beak as the absorber. The self-compiled program uses the boundary element theory based on the simple Green’s function to solve the wave forces and hydrodynamic parameters. And the equation of motion, the oscillation of the float and the capture width ratio are obtained by the modal method. The influences of the shape of the eagle beak, the angle of the slider and the wave heading on the capture ability of the device are investigated. According to the calculation results and the wave resources in the sea area, the optimal shape of the eagle beak and external damping can be selected to maximize the wave energy capture capability.


Introduction
With the continuous development of world economy and the improvement of people's living standards, the demand for energy in modern society is growing (You et al., 2003). The traditional non-renewable resources, such as oil and natural gas, will be gradually exhausted. And the excessive use of fossil fuel leads to excessive carbon emissions, aggravation of the greenhouse effect and destruction of the ecological environment. On the other hand, China has vast sea area with many islands which face varying degrees of power shortages because most islands are remote from the mainland, small in size and having limited maximum load. As the successful development of these islands is beneficial to marine military and economic power but the conventional power generation method cannot be effectively applied, wave energy provides an alternative.
Wave energy, an important part of marine renewable energy, is widely distributed and sustainable. It also has little impact on the environment and does not occupy the precious land resources of the island. The ever-growing wave energy industry is also likely to promote the development of a large number of industries and technologies such as marine equipment, special materials, transportation, aquaculture, marine anti-corrosion, and marine engineering, etc. On the other hand, the wave resources in the world are totally different, resulting in the different point of view of wave energy utilization, which makes the wave energy technology divergence, and the researchers' views on wave energy technology are not uniform. Waves near China are mostly caused by monsoons. Resources are relatively lower than that in Europe and America, so the power generation units are required to have high conversion efficiency. Moreover, most of China's sea areas are typhoon-prone areas, which mean that the equipment must have the ability to resist typhoons. Therefore, these situations force researchers to consider the particularity of sea conditions and develop wave energy converters suitable for China's specific sea areas.
Sharp Eagle wave energy converter developed independently by the Guangzhou Institute of Energy Conversion is an organic combination of wave energy device and semi-submersible barge. The semi-submersible platform ensures that it is easy to tow, place and maintain. Its large amount of space makes it possible to subsequently install marine instruments and photovoltaic panels. In 2012, the Ocean Energy Research Group of the Guangzhou Institute of Energy Conversion developed the first generation wave energy converter concept prototype 10 kW "Sharp Eagle I", which successfully resisted the typhoon 201330 (Haiyan) and obtained a series of sea trial results (Sheng et al., 2015). It is preliminarily verified that the Sharp Eagle wave energy converter technology has good environmental adaptability, high efficiency and good stability. In 2015, the 100 kW wave energy converter Sharp Eagle "Wanshan" with twoway four eagle beaks arrangement was successfully developed (Sheng et al., 2017). This device has a conversion efficiency of more than 20% during the period of 4−6.5 s and a wave height of 0.6−2.5 m, and has experienced several tropical storms. Furthermore, a semi-submersible wave energy culture cage "Penghu" was completed in June 2019 and will be put into use. The Sharp Eagle wave energy converter has stood the test of time, showing its powerful power generation, environmental adaptability and reliability. Its performance makes China take the lead in wave energy applications. The Sharp Eagle wave energy converter can be compartmentalized into three parts by function, the eagle beak, which is used to capture the wave energy, an energy conversion system and a semi-submersible hull. The eagle beak is the core component, and the cross-sectional profile is mainly composed of three arcs. The surface curve is a structural line conforming to the design concept obtained by the analytical solution. The eagle beak has its special design requirements: the front side absorbs the waves, the back side does not wave, the mass is light, the inertia is small, the reaction is quick, and the movement is synchronized.
In essence, the idea of wave energy technology optimization is to minimize the average power generation cost of a wave energy device over its entire life cycle. Therefore, the optimization should increase the ability to capture wave energy as much as possible and generate more power in a given situation. This requires continuous optimization of the absorber to improve the response agility and capture width ratio of the device. This paper uses eagle beak as the absorber, as shown in Fig. 1. The sliding bar passes through the eagle beak. When the inclination angle is fixed, the floating body has only one motion mode that reciprocates along the sliding rod under the action of the wave, and drives the hydraulic cylinder to perform energy conversion. In this system, the following four optimization directions were tried: (1) shape; (2) slider tilt angle; (3) wave heading; and (4) external damping. Based on the theory of potential flow superposition and modal analysis, the hydrodynamic response and the capture width ratios at different wave frequencies are calculated by the self-compiled program for optimization analysis.
Here, we start with brief presentation of the governing equations (Section 2), followed by details of our shape optimization methodology and definition of the objective function (Section 3); we then present and discuss results of optimization under four different shapes of eagle beaks (Section 4).

Fundamentals
This paper aims to study the hydrodynamics of multi-rigid coupling in deep seas or in isometric seas. Water is the ideal fluid, which is incompressible and non-rotating. In this paper, the fluid calculation domain is divided into two parts: the inner domain and the outer domain. The incident wave is assumed to be a known small sine regular wave. In order to prevent the relationship between multiple rigid bodies from being destroyed, energy and force are used to describe the relationship between motion and interaction. Lagrange dynamics is established by d'Alembert principle and virtual displacement theorem equation. To simplify the calculation process, only one mode of the float moving along the fixed slider is considered here.

Φ Φ
The velocity potential function is used to describe the fluid state. And time factor is separated from , so that calculation is performed in the frequency domain. Re where, represents the real part; is the potential function of the fluid space velocity dependent of time t; e is the natural exponent; i is the imaginary unit, ; is T the incident wave angular frequency; and is the wave period.
A wave generated by an incident wave acting on a stationary object is defined as a diffracted wave, and a wave caused by a dynamic pressure of a rigid body by a small sinusoidal motion is defined as a radiated wave. These two are collectively referred to as scattered waves.
According to the superposition of potential flow theorem and modal analysis method, basic control equations and boundary conditions are established (Yu and Falnes, 1995).
where, , and are, respectively, the incident wave velocity potential, the wave diffraction velocity potential, and radiation velocity potential. And the diffraction velocity potential and the radiation velocity potential are expressed by . Incident wave velocity potential: where, A is the incident wave amplitude(half of the wave height); k is the wave number; g is the acceleration of gravity; h is the water depth. And wave diffraction velocity potential and radiation velocity potential both satisfy the following boundary conditions.
In the calculation domain, (4) Free surface boundary condition, Seabed surface boundary condition, ∂ϕ l ∂n In order to ensure the closedness of the computational domain, the infinity surface boundary conditions are also given. Assuming a local disturbance in the infinite flow field, the most intuitive reflection is that the disturbance amplitude gradually decreases as the distance increases, which can be recorded as: Decompose the motion mode of the device according to the constraint conditions, and establish the device surface boundary condition that the constraint is not destroyed.
where, n represents the device generalized surface normal vector and the normal direction point to the inside of the device. At the finite distance, the near-field expression of the velocity potential in the cylindrical coordinates is used to establish the boundary conditions of the radiating surface, and the computational domain is constrained. Thus, the velocity potential technique expansion on the radiant cylindrical surface at a finite distance is used as a diffusion condition of the velocity potential, instead of the Sommerfeld diffusion condition at infinity as the radiation boundary condition of the internal computational domain. Through the boundary element method based on the simple Green's function, the boundary integral equation of the velocity potential is established and the constant unit is discretized, and then the coefficient matrix and the system of equations are obtained.
where, P and Q are, respectively, the field points and source points on the boundary surfaces; S is the boundary surfaces of the entire computational domain and can be written as ; is simple Green function, where .
The above equation can be written as follows (He and Dai, 1992): Then the forces received by the float in the system can be expressed as: XIAO Lei et al. China Ocean Eng., 2020, Vol. 34, No. 4, P. 547-557 549 where, and are the excitation force and radiation force, respectively; and are the acceleration and velocity of the motion, respectively. The radiative force acting on the floating body consists of two parts, one part is proportional to the acceleration, and the proportional coefficient has a mass dimension called the additional mass . The other part is proportional to the speed, and the proportionality factor is called the damping coefficient . It can be known that the additional mass and damping coefficient depend on the shape and motion pattern of the object (Wen, 2000). Added mass and damping coefficient can be written as: nds.
(17) The calculation of the wave force is transformed into the solution of the velocity potential in the hydrodynamics, that is, the solution to the flow is the solution of the Laplace equation. ξ j Finally the equation of motion is obtained. The displacement is a complex number that cotains the amptitude and the phase of the motion.: (18) The power absorbed by the device from the wave energy is .
(19) The input power of the wave within the unit width is λ η where, is the incident wavelength. First-order converts the wave energy into the mechanical energy, the capture width ratio is used to evaluate the first-order hydrodynamic performance of the device generally. The capture width ratio is as follows: where, B is the width of the floater. Owing to the "antenna effect", the capture width ratio of the float could be even larger than 100%. It is the fact that the floater can absorb a larger fraction of the power than what is available over its diameter. In a regular wave, the maximum absorption width of heaving point floater is theoretically (with linear theory) equal to the wave length divided by 2π (De Backer, 2009). For example, a regular wave with the height of 1 m and the period of 4 s, the maximum absorption width of the floater is 1.91. The program is compiled according to the above theory. Zhang et al. (2018) conducted trial calculations on two typical examples of cuboid and floating hemisphere (Vugts, 1968;Andersen and He, 1985). The results are very consistent and the program accuracy is verified.

Optimization of design principle
From the vibration mechanics, we can see that the equation of the single degree of freedom system motion excited by harmonic force is: where, x is the displacement; m is the object mass; c is the external damping; k is the system elasticity coefficient; is the excitation force amplitude; and is excitation frequency. Therefore, the motion amplitude expression of the system is: The expression for the phase difference is: .
(24) It can be seen from the analysis that the phase difference between the motion of the object and the exciting force can be reduced by increasing the modulus of elasticity and reducing the mass. According to the principle, the wave energy conversion system is optimized to improve the wave energy capture efficiency

Methodology
The motion of the wave energy device is a typical damped harmonic forced vibration under the action of wave excitation. It can be seen from the vibration characteristics that the steady-state forced vibration response of the system to the simple harmonic excitation is a simple harmonic vibration whose frequency is equal to the excitation frequency and whose phase lags behind the excitation force. Also, the amplitude and phase difference of the steady-state response depend on the physical properties (mass, stiffness, and damping) of the system itself, regardless of the initial conditions of the system (the way it enters).
In an oscillating buoy wave energy conversion with a slider system, the dissipation of energy captured by the absorber is related to damping, and the additional mass is the sum of the masses of the fluid pushed by the object as it moves in the fluid. Under the same incident conditions, the smaller the additional mass is, the more agile the buoyant motion response will be. As a mechanism for capturing wave energy, the absorber is required to have a small additional mass and radiation damping, which can increase its agility to wave response, reduce energy dissipation and improve capturing efficiency. According to the theoretical derivation of the optimal design principle, the added mass and damping coefficient of the wave energy device are related to the shape of the absorber. The design of the shape of eagle beak determines the performance of the wave energy device.
In this paper, four different shapes of eagle beaks as shown in Fig. 2 were selected for hydrodynamic calculation. In the simulation, the incident wave was set as regular wave with the height of 1 m. And the depth of water is 10 m. The effect of the shape on the hydrodynamic performance and energy conversion efficiency of the device is verified by calculation to optimize the shape of the device. According to different working conditions, design requirements and hydrodynamic characteristics of calculation results, the eagle beak was designed to improve the capture width ratio and optimize the performance of the device. By changing the structural shape of Model 1, 2 and 3 back wave surfaces of the model, the influence of the shape of the back structure on the performance of the device during the working process is verified. In addition, in order to verify the influence of the draft depth on the performance of the device, Model 4 increases the draught depth of the device along the direction of the slider, reducing the projection of the wave. In order to study the effects of the absorber shape, the mass, the length and the front arc of the absorber are all kept constant. Slider tilt angle is 30°, the wave heading is 0, and the external damping is the optimal damping for every model.
In order to make it easier to compare the data with previous results, we hereby transform the data to be dimensionless.
Dimensionless added mass: Dimensionless damping coefficient: Dimensionless force: Dimensionless displacement: where, V is the volume of displacement of absorber; L is the characeteristic length of absorber, 6 m; i and j depend on the degree of freedom.
(29) The following three aspects are also optimized and analyzed: In this case, Model 2 is used to study the effect of changing slider tilt angle, wave heading and external damping.
(1) Slider tilt angle n θ In the interval of 30°−90°, the calculation is carried out in steps of 15°. In the normal direction of the floating surface unit, the relationship between the component and the horizontal angle of the unit speed of the absorber along the tilting motion mode of the slider is: where, and are the partial derivatives of the normal direction of the object plane (pointing to the inside of absorber) in the x-axis and z-axis direction, respectively; In this case, slider tilt angle is the only change in the calculation.
(2) Wave heading Wave heading is the angle between the incident direction of the wave and the normal direction of the front surface of the absorber.
The relationship between the device-related parameters and the angle of attack is analyzed in the range of 0°−180°w ith a step of 45°. In this case, wave heading is the only change in the calculation.
(3) External damping The matching of external damping is the key to the energy conversion process of the wave energy device. The calculation of the floats moving along different tilting angles varies with external damping under different working conditions. In order to verify the adaptability of the device under different damping to different wave conditions, external damping are set to 80000, 160000, 240000, 320000, 400000 and 500000 kg/s, respectively. In this case, external damping is the only change in the calculation.

Float shape optimization analysis
(1) Influence of the float shape on the hydrodynamic parameters As can be seen from Fig. 3, the added masses of the four XIAO Lei et al. China Ocean Eng., 2020, Vol. 34, No. 4, P. 547-557 551 models are similar in general. The additional mass of Model 4 has an evident change with the change of wave frequency. Compared with the other three models, the damping coefficient of Model 4, which increases the draft depth, is significantly smaller.
(2) Influence of the float shape on the wave force As shown in Fig. 4 and Fig. 5, the trend of wave force changes in four models is basically the same, and the stability of Model 4 under different working conditions is poor. It can be seen that the wave frequency has a significant effect on the wave force acting on the absorber, with two orders of magnitude difference between the maximum and minimum values. Relative to the radiation force, the influence of the wave frequency on the excitation force is more significant, and the change of the excitation force is more complicated.
(3) Influence of the float shape on capture width ratio ω 1.6 rad/s As shown in Fig. 6, the amplitude and phase of the oscillation displacement of the four models are basically the same as the frequency, and the highest value is about 0.6 m, which appears when is nearby . It can be seen that the optimal capture width ratio of the four floats is different from the corresponding frequency, which theoretically proved the necessity of adjusting the shape of the float according to the different sea conditions.
The influence of the shape on the optimal damping and the capture width ratio of the system is shown in Fig. 7. The variation trend of the optimal damping corresponding to the four absorbers with different shapes is the same. The variation trend of the optimal capture width ratio realized by the four models is basically the same, but the frequency corresponding to the optimal value will be adjusted according to the shape, which theoretically verifies that the targeted op-timal design of the absorber for different wave conditions in different regions can be realized by adjusting the shape. Due to the influence of "antenna effect" in the water wave dynamics, the capture width ratio of absorber appears to be more than 100% in a certain range.

Slider tilt angle optimization analysis
(1) Influence of the slider tilt angle on the hydrodynamic parameters As shown in Fig. 8, the variation trend of the additional mass and damping coefficient of the device under different

552
XIAO Lei et al. China Ocean Eng., 2020, Vol. 34, No. 4, P. 547-557 θ=45 • θ=60 • slider tilt angles is consistent and the size is similar. When or the value of the added mass is the largest, and the damping coefficient continues to increase as the tilt angle increases until it reaches the maximum.
(2) Influence of the slider tilt angle on the wave force ω < 2.75 As shown in Fig. 9, the variation trend of the amplitude and phase of the incident wave force under the selected five inclination angles are basically the same, but the difference in numerical values is very obvious. When , the wave excitation force of the eagle beak is the largest when . However, as the wave frequency is larger than 2.75 rad/s, the wave excitation force increases as the slider tilt angle increases. In general, the phase of the excitation force under different inclination angles of the slider maintains a similar trend, and does not show obvious numerical difference, indicating that the slider tilt angle has little influence on the phase of the wave excitation force.
As shown in Fig. 10, unlike the wave excitation force, the influence of the inclination of the slider on the radiation force is only reflected in the magnitude of the amplitude. The smaller the angle of inclination, the larger the amplitude of the wave radiation wave force acting on float, which is consistent with the phenomenon that the effect of the sway in the water is more pronounced than the heave. However, this effect gradually decreases with the increase of frequency. When the frequency is larger than 1.75 rad/s,    XIAO Lei et al. China Ocean Eng., 2020, Vol. 34, No. 4, P. 547-557 553 the radiation force at different dip angles gradually converges, and the amplitude decreases rapidly, but the phase changes largely, and a process of decreasing first and then increasing.
(3) Influence of the slider tilt angle on capture width ratio As shown in Fig. 11, similar to the radiation force, the amplitude of the oscillation displacement at different inclination angles is consistent with the variational frequency, and the influence of the inclination of the slider on the oscillation displacement of the float is mainly reflected in the amplitude. The data show that when the frequency is between 1.25 rad/s to 2.25 rad/s, the displacement amplitude is negatively correlated with the tilt angle. The larger the tilt angle, the smaller the displacement amplitude. When the frequency is larger than 2.25 rad/s, the influence of the tilt angle on the displacement is significantly reduced. When the frequency is 1.57 rad/s, the displacement amplitude reaches the maximum of 0.58 m.
As shown in Fig. 12, the optimal external damping at different tilt angles maintains the consistency, indicating that the influence of the slider tilt angle on the optimal external damping of the system is negligible. However, the capture width ratio under different tilt angles shows a large difference. The capture width ratio decreases with the increase of the inclination angle. When the tilt angles are 30°a nd 45°, the capture width ratio is higher than others, and the highest value exceeds 100%.

Wave heading
(1) Influence of the wave heading on the hydrodynamic parameters As shown in Fig. 13, at different angles of wave heading, the higher the wave heading in the high-frequency wave when the frequency is smaller than 2.75 rad/s, the lar-ger the amplitude of the wave excitation force. when the frequency is larger than 2.75 rad/s, the excitation force amplitude begins to decrease as the wave heading increases. As shown in Fig. 14, it can be seen that the amplitude of the radiated force decreases as the wave heading increases. When the wave heading is 0° or 45°, the amplitude of the radiation force change is small. When the wave heading is larger than 45°, the amplitude of the radiation wave force decreases rapidly with the increase of the wave heading, indicating that the device is sensitive in this range. With the symmetry, the oscillating slider bar wave energy device has a good wave heading between −45° and 45°.
(2) Influence of the wave heading on capture width ratio As shown in Fig. 15, the variation trend of the displacement amplitude at different wave headings is basically the  same, and decreases as the wave heading increases. The overall trend of the displacement phase increases as the wave frequency increases. As shown in Fig. 16, the capture width ratio of the device decreases as the wave heading increases, and the high energy capture performance can be maintained when the wave heading is between 0° and 45°. With a further increase in the wave heading, the capture width ratio drops rapidly. With the symmetry of the device, the calculation shows that the optimal wave heading of the eagle beak is between −45° and 45°, and the device can maintain high energy capture performance within this range.

External damping
It can be seen from Figs. 17 and 18 that the wave radiation force amplitude and the displacement amplitude are consistent with the external damping. The difference occurs when the external damping is smaller than the optimal external damping. When the external damping is larger than the optimal damping, the amplitude and displacement amplitude of the radiation force converge rapidly, and the consistency increases with the increase of external damping. But the amplitude is gradually decreasing. At the same time, the figure shows that the phase of the radiation force and displacement gradually decreases with increasing damping when the period is smaller than 3.5 s, which is opposite to the trend when the period is larger than 3.5 s. However, the phase change is mainly concentrated near the optimal external damping point, and eventually stabilizes as the damping increases.
As shown in Fig. 19, the variation trend of the capture width ratio under various working conditions is substantially the same as the external damping: starts to decrease after reaching the maximum value at the optimal external damping point. The data show that the device has the best energy capture performance and the widest range of adaptive wave conditions at a period of 4 s and an inclination angle of 60°. As shown in Fig. 20, the device has the highest    XIAO Lei et al. China Ocean Eng., 2020, Vol. 34, No. 4, P. 547-557 555 capture width ratio when the external damping is 80000 kg/s, but the passband is narrower. And as the external damping increases, the passband becomes wider, but the highest capture width ratio decreases. Regardless of the size of the external damping, the device is capable of achieving a good response in high frequency waves of 3−6 s.

Conclusions
In this paper, the oscillating slider bar wave energy conversion system is a single-mode motion with a fixed angle of the sliding bar. For different shapes of eagle beaks, this article uses the self-compiled program to calculate the hydrodynamic coefficients such as additional mass and damping coefficient. Different slider tilt angles were used to calculate the effects of shape and wave heading on incident wave force, diffraction wave force, radiation wave force, oscillation displacement, optimal external damping, and capture width ratio. According to the above research, the following conclusions can be drawn.
(1) Oscillating buoy wave energy converter with a slider has a good capacity to capture wave power and it can meet the demand for electricity.
(2) With the calculation method of this paper, according to the calculation results and the wave resources in the sea area, it is convenient to select the best shape of the eagle beak and the working state after the device is put into operation, so as to maximize the wave energy capture.
(3) The matching of the external damping is the key to the energy conversion process of the wave energy device. With the external damping as the variable, the variation of the float with different slider tilt angles under different working conditions is calculated. The calculation results show that the wave energy capture system can absorb wave energy efficiently near the optimal external damping of the design, which proves the importance of the matching of external damping in the design of wave energy devices.
The research on the sharp eagle device in this paper enriches the theoretical calculation of this type of device. It has certain guiding significance for the design and optimization of the energy conversion system of the device, and

556
XIAO Lei et al. China Ocean Eng., 2020, Vol. 34, No. 4, P. 547-557 provides a theoretical reference for the design of the sharp eagle device. It also lays a foundation for the future calculation of wave energy devices with more complex constraints such as multi-floating and multi-modality. It should be pointed out that hydrodynamic calculations in this paper are only carried out for the case of single-modal motion with a fixed angle of the slider bar. However, the existing wave energy devices mostly adopt multi-rigid body coupled motion, and thus further calculation and optimization of the multi-structure multi-modal wave energy device are needed.