Hydrodynamic Analysis and Power Conversion for Point Absorber WEC with Two Degrees of Freedom Using CFD

Point absorber wave energy device with multiple degrees of freedom (DOF) is assumed to have a better absorption ability of mechanical energy from ocean waves. In this paper, a coaxial symmetric articulated point absorber wave energy converter with two degrees of freedom is presented. The mechanical equations of the oscillation buoy with power take-off mechanism (PTO) in regular waves are established. The three-dimensional numerical wave tank is built in consideration of the buoy motion based upon the CFD method. The appropriate simulation elements are selected for the buoy and wave parameters. The feasibility of the CFD method is verified through the contrast between the numerical simulation results of typical wave conditions and test results. In such case, the buoy with single DOF of heave, pitch and their coupling motion considering free (no PTO damping) and damped oscillations in regular waves are simulated by using the verified CFD method respectively. The hydrodynamic and wave energy conversion characteristics with typical wave conditions are analyzed. The numerical results show that the heave and pitch can affect each other in the buoy coupling motion, hydrodynamic loads, wave energy absorption and flow field. The total capture width ratio with two coupled DOF motion is higher than that with a single DOF motion. The wave energy conversion of a certain DOF motion may be higher than that of the single certain DOF motion even though the wave is at the resonance period. When the wave periods are high enough, the interaction between the coupled DOF motions can be neglected.


Introduction
Ocean energy is rich in renewable energy resources, and wave energy is one of them, which has the advantages of large reserves and easy development and utilization (Bernera, 2003). The wave energy can be converted into electrical energy through the three stage transformations by using a wave energy generator (Michael et al., 1982). The floater serves as an important part to trap energy in wave energy devices and its design of the hydrodynamic characteristics has a great influence on the motion, loads and the energy capture.
For a single-point wave energy device, though the floater has a complex motion of six degrees of freedom in waves, it mainly depends on a single degree of freedom motion to capture wave energy. So far, so many researches about single point wave energy converters by using numerical evaluation or experiments have been conducted. For example, in 1980, a spherical floating wave energy device was made in Norway (Budal et al., 1982), which was made of the oscillating buoy, the column and air turbine. In the wave energy conversion system, the buoy was a sphere with a radius of one meter, moving along the column that was hinged to the sea floor. And then the upper turbine was driven by the compressed air to carry out energy output. The device underwent a sea trial later in the Trondheim fjord. Power Buoy wave energy device was developed by the OPT company in the United States (Mekhiche and Edwards, 2014). It relied on the relative suspension of the floating body to generate electricity, and finished the sixth sea trials in Cornwall in the UK in 2006, officially launched in Hawaii in 2007. Caska and Finnigan (2008) used the analytic method to study the hydrodynamic characteristics of a cylindrical floater with the bottom hinged at the bottom of the sea, and some dimensionless numbers were also given to evaluate the performance of the buoy. Falnes and Budal (1978) presented that the point absorber wave power devices can absorb energy from a wave of a certain width. However, the width of the wave was greater than that of the model itself, and the maximum capture width was L/π (L for wave length) for the pitching wave power device as for its double degrees of freedom energy acquisition. Based upon the former research findings, the point absorber wave energy converters with single degree of freedom (heave or pitch) are widely studied.
The 10 kW nod duck device was developed by Guangzhou Energy Institute considering the China sea conditions based on the research conclusions of Salter (1974), who developed the British Stephen nod duck style device. The improved device swayed under the effect of waves and wave energy was changed into hydraulic energy, and then output by the hydraulic motor (Zhang et al., 2012). A pitch plate wave energy device with the bottom hinged at the bottom of the pool was theoretically deduced and tested in Portugal Lisbon Polytechnic University (Henriques et al., 2011). Jiang and Yeung (2012) used the FSRVM method to calculate the hydrodynamic forces of an asymmetrical cam wave energy device (nod duck) by controlling the generation and dissipation of vortices and simulating the viscous effects of fluids and the cam device of two shapes was analyzed. The results showed that the viscous force had a great influence on the additional moment of inertia and the device damping coefficient. Based upon the Navier-Stokes equation, Anbarsooz et al. (2014) used the finite volume method to study the motion performance of submerged cylinders under nonlinear incident waves, and the experimental data were compared with the numerical simulation results. The correctness of the method was verified, from which it was found that when the wave shape was thinner than the tip, the frequency regulation of the device was no longer adaptive to the linear wave theory. According to this conclusion, a numerical tank was built in order to study the wave-body interaction and to achieve the purpose of regulating the device frequency. Very recently, Guo et al. (2014) and Zhang et al. (2015) have explored the wave power conversion and fluid characteristics for WECs with single buoy or double floaters considering single heave motion by using ANSYS-CFD. The effect of the bottom shape of the oscillating buoy on the wave power conversion, especially the effect free surface and shipping sea were considered in their studies.
The hydrodynamic problems of the buoy with single heave or pitching have been extensively studied by researchers around the world, but those of the floating body with double or even multi degrees of freedom have not been extensively and deeply studied. Yeung et al. studied the three degrees of freedom coupling motion of the double cylindrical buoys (Chau and Yeung, 2010;Cochet and Yeung, 2010) according to a point absorber with Power Take-Off (PTO) Bachynski et al., 2010;Peiffer, 2009), and the frequency domain method based on the potential flow theory was used to solve the pitching and pitching coupling motion of the buoy. Here, the heave motion is regarded as an independent freedom and the viscosity factor was introduced as an approximation of viscosity to obtain the hydrodynamic characteristics of the buoy and the energy absorption of the device. This method considered that the heave and pitch motion were assumed not to affect each other. Similar theory without considering the fluid viscosity is also employed by Ye and Chen (2017), Abdelkhalik et al. (2017), Gao and Yu (2018) and Liu et al. (2018). However, in fact, the pitch and heave motions of the buoy both occur at the same time, the heave freedom would also affect the flow field of the pitching movement, and vice versa, which can be obtained in the study of Zhang and Guo (2016). The two degrees of freedom motions of the floating body together determined the details of the fluid field, the change of fluid flow state originated from the action of the buoy on the fluid, and then reflected the motion, loads and energy conversion of the floating body. In order to simulate accurately the flow field near the wall at low Reynolds number and when the buoy resonates at high Reynolds number, and to reveal the influence law of energy output between two principal degrees of freedom, the CFD simulation method was adopted. The time domain numerical simulation for the buoy motion of a wave energy device with the heave and pitch was carried out. The simulation results are illustrated and related phenomenon was described and explained in terms of the buoy motion, loads, energy conversion, and flow field.

Basic theories
The wave energy device considered in this paper is shown GUO Wei et al. China Ocean Eng., 2018, Vol. 32, No. 6, P. 718-729 719 in Fig. 1a, in which the two buoys were arranged in opposition and connected with the shafts at both ends of the box structure. In such case, the buoys could rotate in the axial direction of the box in waves, and the heave and pitch motions of the buoy both occur at the same time shown in Fig. 1b. The PTO mechanism is arranged in the box and activated by the relative motion between the buoy and box structure. The wave energy can be absorbed and transformed in this way. Energy output can be carried out by both the pitching and heaving of the buoy, based on the CFD, the mechanical equations for these motions are shown as follows, the heave motion direction is along the Y axis, and the pitching moment of the buoy is along the Z axis. Based upon the small amplitude wave assumption and Newton's second law, the equation of the heave and pitch motion of the buoy in regular waves is and are the forces directly obtained through the CFD simulations, it is the summation of the wave exciting, radiation and restoring forces, and it can be regarded as the integral of the fluid pressure on the surface of the buoy. The mass and moment of inertia of the buoy are donated by and , respectively. The PTO damping coefficients about the heave and pitch motion are denoted by and , respectively. and represent the wave excitation force and moment caused by incoming waves. The radiation forces caused by the radiated waves induced by the forced oscillation of the buoy are donated by in heave and in pitch motion respectively. Here the coefficients before the acceleration are the generalized added mass and those before the velocity are the generalized damping. The restore forces in the heave and pitch are denoted by and , respectively. Here the water density , gravity acceleration , water plane area , volume of displacement and metacentric height are assumed to be constants. The capture width ratio is important to measure the energy absorption performance of the buoy which can be expressed as follows: where, denotes the average absorbed wave power by the buoy and presents the average incident wave power in the width of the buoy. The wave height and period, buoy's H T D C diameter and PTO damping coefficient are denoted by , , and , respectively.

CFD simulation
In this paper, for the convenience of the CFD simulation, only a buoy of the wave energy device is considered. Fig. 2 shows the CFD calculation model with the geometrical parameters and Table 1 gives the detailed dimensions of the buoy.
Before using the CFD method to simulate the hydrodynamic performance of the buoy, the most significant work is to discretize the simulated area, namely grid partitioning. For all the CFD numerical simulations, the storage of structured grids can be recorded by the computer multidimensional array, and the simulation precision and speed are significantly superior to the unstructured grids. Therefore, the structured grids are employed to divide the calculation domain of the buoy. Figs. 3b and 3c show the mesh grids of the buoy and computational domain. The entire fluid domain employed in this paper is rectangular with the length 12D and width 4D, and the other parameters are shown in Fig. 3a.
The boundary conditions of the simulated tank model are set according to the real wave tank as shown in Fig. 3a. The wave-making board, side and bottom of the pool are set as walls because of their non-penetrability. The free surface is always kept at atmospheric pressure and the end of the tank is set as opening so that the water depth in the tank can be maintained.

Convergence precision control
In the process of CFD simulation, except for the convergence of fundamental fluid control equations should be guaranteed, and the calculated movement and force of the  buoy should also be convergent. For a buoy with the heave and pitch motions simulated by the CFD method, the convergence included the convergence of the force, moment and motion because the convergence criterion could greatly affect the length and accuracy of computation. In such case, three rigid body convergence schemes were selected and compared, as shown in Table 2. Fig. 4 is the calculated time-varying curves of the vertical displacement and pitching angle of the buoy when the wave height is 0.04 m and the wave period is 2 s. As seen from Fig. 4, though the curves for the three convergence criteria were consistent, they differ greatly in the peaks and troughs. In such case, No. 3 criterion is the best to be selec-ted according to the calculated results in Table 2.

CFD simulation validation
To verify the correctness of using the CFD to simulate the pitch motion of the buoy, a few wave conditions are selected for the simulation. And then the calculation results are compared with those from the model tests conducted in the towing tank (108 m×7 m×3.5 m) of Harbin Engineering University (HEU) in August 2014. In this test as shown in Fig. 5, to explore the motion effect of the rear buoy on the front buoy, the PTO mechanism was not considered. The results show that the rear buoy almost has no impact on the front buoy when the shaft between the two buoys was perpendicular to the wave direction. In such case, in this paper, only the front buoy is considered for the motion and power capture simulation. The comparisons of the pitching angles considering the wave heights of 0.1 and 0.15 m and the periods of 1.6 to 3.0 s are conducted and shown in Fig. 6.
As shown in Fig. 6, the time-varying pitching angles ob-    Table 3.
The compared results about the mean values in Table 3 further illustrate the feasibility of the CFD method in simulating the pitching motion of the buoy. As for the simulation of the buoy's heave motion by using the CFD method, it can be referred to the author's former works (Guo et al., 2014;Zhang et al., 2015;Zhang and Guo, 2016).

Numerical results
For the former buoy as shown in Fig. 2 and Table 1, the CFD simulations of the motions and hydrodynamic forces for the single degree of freedom (DOF, heave or pitch) and coupled DOFs (heave and pitch) are carried out in this part. In the section of single DOF, the motion RAOs of the buoy are first given to obtain the resonance periods of the buoy in the heave and pitch. In such case, the wave energy absorption, motion and load characteristics of the buoy with resonance are obtained in the heave and pitch respectively, and then are compared with those simulated in the section of the coupled DOFs.

Single DOF
By considering the linear regular wave period ranging from 1.0 to 3.0 s, the motion RAOs of the buoy are calculated by the CFD and shown in Fig. 7a. With the increase of the wave period, both the heave and pitch RAOs increase first and then decrease. When the wave period is 1.4 s, the heave RAO reaches the maximum value of 1.85, and at this point, the heave amplitude of the buoy is much larger than the wave amplitude, and when the wave period increases, the curve tends to be smooth. Similarly, for the pitch RAO, with the increase of the wave period, the peak value of 24 occurs at the period of 1.2 s.
As having obtained the resonance periods of the buoy in the heave and pitch, it is important to explore the wave energy absorption of the buoy considering these two degrees of freedom respectively. In such case, the capture width ratios of the buoy varying with the PTO damping coefficients are calculated and shown in Fig. 7b. Both the capture width ratios in the heave and pitch show the similar variation trend: increase first and then decrease and a peak value occurs. The difference is that the peak of the capture width ratio in the heave occurs at the PTO damping 500 Ns/m with the value of 0.503, while that in the pitch occurs at 2 Nms with the value of 0.747. In such case, we have obtained the optimum PTO damping coefficients and capture width ratios in the heave and pitch respectively. In the following section, we will give the motion and hydrodynamic force characteristics of the buoy with single DOF at the resonance period. For the convenience of description, the hydro-  dynamic and damping loads are expressed dimensionless as follows: where, and are the dimensionless hydrodynamic force coefficients of the heave and pitch, respectively, the dimensionless PTO damping force coefficients are denoted by and . The water plane area s of the buoy is 0.418 m 2 and the draft b is 0.12 m.
As shown in Fig. 8, the motion displacement amplitudes of the buoy in the heave and pitch show the similar change trend: decrease with the increase of the PTO damping coefficients. The difference is that the amplitudes of the heave decrease gradually in the whole range of the PTO damping coefficients, while those of the pitch decrease quickly and then gradually with the increase of the PTO damping coefficients. It means that the motion of the pitch is more sensitive than that of the heave when the PTO damping is small.
The dimensionless hydrodynamic and PTO damping force coefficients of the buoy in the heave and pitch are respectively depicted in Fig. 9. As depicted in Fig. 9a, the dimensionless hydrodynamic forces show opposite trends with the increase of the PTO damping coefficients. With the increase of the PTO damping coefficients, the dimensionless hydrodynamic forces of the heave increase and then stay stable, while those of the pitch decrease quickly and then stay stable. This is because that the PTO damping can affect the motion amplitudes directly, and then the restoring forces are also affected. In addition, the hydrodynamic forces of the heave are much larger than those of the pitch. When the PTO damping forces increase, the proportion of the restoring forces increase. In such case, when the PTO damping forces are large enough, the overdamped conditions for both the heave and pitch occur and their oscillating motions are more difficult. Therefore, with the increase of the PTO damping coefficients, the final movements of the heave and pitch remain stable. Similarly, the changing trends of the dimensionless PTO damping forces of the heave and pitch shown in Fig. 9b can also be explained with the same theories.
In this section, the motions, forces and wave energy absorptions of the buoy for single DOF of the heave and pitch are explored and discussed. The resonance periods, as well as the optimum PTO damping coefficients at which of the   GUO Wei et al. China Ocean Eng., 2018, Vol. 32, No. 6, P. 718-729 723 buoy for single DOF of the heave and pitch are obtained as shown in Table 4. These calculation results and changing laws of the wave energy absorption and hydrodynamic characteristics are important for the further study on the coupled DOF of the heave and pitch.

Coupled DOF
Based upon the previous exploration, the motions, forces and wave energy absorption of the buoy for coupled DOF of the heave and pitch are explored and discussed in this section. Though the motion equation expressions of the buoy in coupled DOF are the same as those in single DOF of the heave and pitch, the hydrodynamic forces in the formulas are different. Because they are caused by the coupled DOF of the heave and pitch, not by the single DOF of the heave or pitch. When the buoy moves with the coupling motion of the heave and pitch, the velocity and vorticity fields around the body's bottom change with the motion attitude of the buoy, and the flow field becomes complex and the hydrodynamic forces working on the buoy are impossible to be separated. In such case, the wave excitation forces, the additional mass (inertia) and viscous damping coefficients in the updated motion equations are difficult to be separated and obtained. Therefore, when the wave energy device works with multiple degrees of freedom, it is important to explore the coupled characteristics and their effects on the wave energy absorption. In order to study the effects of the coupled DOF on the motion and hydrodynamic forces of the buoy, the comparisons between the simulation results with double DOF and those with single heave or pitch considering their own resonance periods are conducted without the PTO damping. Fig.10 gives the comparisons about the time-varying motion displacement of the buoy with the coupled DOF and single DOF, respectively. In Fig. 10a, the wave period is set as 1.4 s so that the resonance occurs for the buoy with the single DOF of the heave. Under these circumstances, the displacement amplitude of the heave in the double DOF is smaller than that in the single DOF, which can further show that the pitching motion has an effect on the heave motion. Similarly, when the wave period is set as 1.2 s, the single pitch motion of the buoy will resonate as shown in Fig. 10b. The pitch motion amplitude for the single DOF is also larger than that for the coupled DOF.
To have a further understanding of the coupling effects on the hydrodynamic forces of the buoy, the buoy with free vibration with the single DOF and coupled DOF are simulated by CFD respectively. The dimensionless hydrodynamic forces of the buoy of the heave motion with the single DOF and coupled DOF are depicted in Fig. 10a and those in the pitch motion are depicted in Fig. 10b. As shown in Fig. 11,   Fig. 10. Comparisons of the time-varying displacements of the buoy.  The previous study in this section shows the motion and hydrodynamic force characteristics of the buoy with free vibration with the single DOF and double DOF of the heave and pitch with their own resonance periods. However, the main purpose is to explore the wave energy conversion of the buoy with the single DOF and coupled DOF. In such case, the PTO mechanism is indispensable in the following explorations.
As obtained in the former section, the optimum PTO damping coefficient for the considered buoy for single heave motion is 500 Ns/m and for singe pitch motion is 2 Nms. In such case, in Fig. 12a, the time-varying heave motion displacements of the buoy with the single DOF of heave and coupled DOF when the heave PTO damping coefficient is 500 Ns/m at the resonance period of 1.4 s are obtained. In Fig. 12b, the PTO damping coefficient for the pitch is 2 Nms and the wave period when the single pitch motion resonance occurs is 1.2 s. As depicted in Fig. 13, the damped coupling motion has no effect on the vibration frequencies of the heave and pitch motions. In addition, different from the free heave motion in Fig. 10, the gap of the heave motion displacement is bigger at the same time, though the displacement amplitude just changes a little. By comparing the calculation results about the displacement of the buoy with the single pitch and coupled DOF motions, the PTO damping makes the difference smaller as shown in Fig. 12b. Similarly, the hydrodynamic forces of the buoy with the single DOF and coupled DOF are shown in Fig. 13. The changing trends are nearly the same as those of the displacements.
However, the wave energy conversion of the buoy is affected not only by the PTO mechanism, but also by the hydrodynamic characteristics. In such case, we can conclude that the coupling motion has a great effect on the wave energy absorption of the buoy. If this is not enough, we can further study the effects of the coupling motion on the PTO damping forces as shown in Fig. 14. The PTO damping forces of the buoy with the single DOF and coupled DOF depicted in Fig. 14 further show that the coupling motion also has a great effect on the PTO damping force. It can further explain that the wave energy conversion ability of the buoy is affected by the coupling characteristics.
As the coupling motion has a great effect on the wave energy conversion of the buoy with single heave or pitch motion, then the capture width ratios of the buoy with the coupled DOF and single DOF motion are deserved to be explored. The capture width ratio is an important indicator to measure the energy output of wave energy device. Though the heaving and pitching amplitudes of the coupling motion are smaller than those of the single degree of freedom movement, from the energy absorption point of view, the coupling motion may obtain more energy than the total amount of the energy obtained by the single degree of free-  GUO Wei et al. China Ocean Eng., 2018, Vol. 32, No. 6, P. 718-729 dom. Without loss of generality, on the basis of the above analysis for the considered wave energy absorber, this section will study the wave energy conversion ability of the buoy under different wave conditions considering the single and coupled DOF.
To have a better understanding of the wave energy conversion ability of the buoy with different DOF, several cases for the same buoy with different wave periods are considered and shown in Table 5. As known before, for the single DOF of the heave, the optimum PTO damping coefficient is 500 Ns/m with the resonance period of 1.4 s, while for the single DOF of the pitch, the optimum PTO damping coefficient is 2 Nms with the resonance period of 1.2 s. In such case, in this section of all the designed cases, the PTO damping coefficient is set as 500 Ns/m for the heave and 2 Nms for the pitch. In Table 5, , and denote the wave energy conversion efficiency by the heave, pitch and their summation in the coupling motion separately. Similarly, the capture width ratios by single heave, pitch and their summation are denoted by , and , respectively. Table 5, when the wave period is 1.2 s, the single pitching motion is resonant and the capture width ratio is much higher than that of the coupling motion. In addition, for the coupling motion, the capture width ratio from the pitching is higher than that from the heaving. However, the total capture width ratio of the coupling motion is much higher than that of the single pitching motion. It means that when the wave period is low, the heave motion can inhibit the wave energy conversion of the pitch motion. When the wave period is 1.4 s, the capture width ratio of the heave in the coupling motion is higher than that of the single heave motion. In addition, at this wave period, the single heave motion is resonant. It means that the wave energy conversion of a certain DOF movement in the coupling motion can be more than that of the single freedom resonant motion. This is because the other DOF motions may have a positive effect on the wave energy conversion of the certain DOF motion.

As shown in
With the increase of the wave periods, by comparing the wave energy conversion efficiencies of heave in the coupled DOF and single DOF motions, as shown in Fig. 15a, we find that the efficiencies of the coupling motion are first higher than those of the single DOF motion, and then are nearly the same. Similarly, by comparing the efficiencies of pitch from the two motion forms depicted in Fig. 15b, we find that the efficiencies of the coupling motion are first smaller than those of the single DOF motion and then are nearly the same. As known to all, with the increase of wave period, the wave length also increases, while the wave steepness increases. In such case, for the considered buoy with the designed constant dimensions, its characteristic length becomes really smaller according to the increasing wave length. The pitch motion amplitude becomes much smaller and the interaction between the heave and pitch of coupling motion becomes insignificant compared with the single heave or pitch motion. In such case, we can conclude that at a much higher wave period which is far away from the resonance periods of heave and pitch, the heave and pitch motions are assumed to be dependent and the whole wave energy conversion can be regard as the summation of the single heave and pitch motion.
For more intuitive and comprehensive exposition, the influence mechanism of different degrees of freedom on the motion of the buoy, the velocity field as well as the vorticity field of the buoy with several typical positions are analyzed in the following part. The velocity field can directly reflect the size and direction of the fluid velocity around the buoy. In such case, the typical positions of the buoy such as the highest point (maximum displacement, non-speed) and   As shown in Fig. 16, when the buoy with the single heave motion is at its highest point, there is a small angle between the fluid velocity direction and the underside. In addition, the velocity at the bottom is rather large, which means that most of the wave energy is not absorbed by the buoy with the single heave motion. And when it is at the balance position, the direction of fluid near the bottom was deservedly along the outer normal direction of the bottom. When the buoy oscillates with the coupled heave and pitch, as shown in Figs. 16c and 16d , the fluid velocity direction basically follows the tangent direction at the bottom when the buoy is at the highest point. The most important thing is that no matter when the buoy with the coupling motion at its highest or balance position, the flow field around the buoy is not highly affected. In addition, the velocity at the body's bottom is rather small which means that most of the wave energy is absorbed by the buoy. In such case, when the wave period is 1.4 s, the capture width ratio in Table 5 can be reasonably explained.
Similarly, when the wave period is the pitching resonance period of 1.2 s (Fig. 17), the velocity field around the buoy with the single pitch and coupled motion are simulated and depicted. The main difference between the single DOF pitching motion and coupled DOF motion is reflected in the immersion depth. Compared with the buoy with the coupled DOF motion, the immersion depth of the buoy with the single DOF pitching is larger from the balance to the highest position and the wetted surface is larger. By comparing the velocity field around the buoy with these two types of the motion, we find that the velocity directions are nearly the same for the buoy at the balance and the highest position respectively. However, the velocity of the single DOF motion at the highest and balance positions both are larger than those of the coupled DOF motion at the con-   GUO Wei et al. China Ocean Eng., 2018, Vol. 32, No. 6, P. 718-729 727 sidered positions. These results show that the buoy with the coupled motion has a better wave energy conversion ability than the single DOF motion at the wave periods near the resonance period. The vorticity distributions around the buoy with the minimum velocity in the single pitch and coupled motion at the resonant period of the pitch motion (wave period equals 1.2 s) are shown in Fig. 18. The maximum vorticity of the single pitch and coupled motion both occurred at the junction near the wall and the distributions are relatively concentrated. However, the vorticity distribution area with relatively high vorticity (such as 5.0 s -1 ) of the coupled motion is smaller than that of the single pitch motion. In addition, though the maximum vorticity in the coupled motion is larger than that in the single pitch motion, the distribution area is much concentrated and small. It means that the buoy with the coupled motion can be better adapted to the change of the wave surface, which indicates that the wave power can be better absorbed.
Similar comparisons of the vorticity distributions of the buoy with the single heave and coupled motion at the resonant period of the heave (wave period equals 1.4 s) are shown in Fig. 19. The distribution characteristics are almost the same. The maximum vorticity both occur at both ends of the wave head side. However, in the middle part, the distribution is relatively uniform. By observing the maximum vorticity around the floating buoy, it can be found that the maximum vorticity in the coupled is far smaller than that in the single heave motion, though their distribution areas with relatively high vorticity (such as 7 s -1 in single heave and 3.5 s -1 in coupled motion) are conversed. The vorticity ranges in 1-8 s -1 in the single heave motion, while ranges in 0-4 s -1 in the coupled motion. These mean that when the period is at the resonant period of the heave motion, the movement of the buoy with the coupled motion has a smaller disturbance to the surrounding fluid. In such case, less wave energy is dissipated and much more wave energy can be obtained by the buoy, which is consistent with the statistical results in Table 5.

Conclusions
In this paper, a special point absorber wave energy converter with two degrees of freedom is presented. The equations for the heave and roll motions of the buoy in the device are established and simulated by using CFD. The hydrodynamic and wave energy conversion characteristics of the buoy with the coupling motion of the heave and roll are investigated and discussed. Based upon the motion equa-  Fig. 19. Vorticity field of the buoy with single heave and coupled motion at the wave period of 1.4 s.

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GUO Wei et al. China Ocean Eng., 2018, Vol. 32, No. 6, P. 718-729 tions, the difference of the added mass (inertia) and viscous damping coefficients between the single and coupling DOF motions are explained theoretically. The theoretical explanations are verified by analyzing the flow field and the essential reasons affecting the buoy motion and energy dissipation are explained. By calculating the wave energy conversion efficiency of the buoy with the single DOF heaving or pitching, the capture width ratios both increase first and then decrease with the increase of the PTO damping coefficient at the resonated wave periods, and the optimum PTO damping coefficients exist.
In the coupling motion, there is a great influence between different degrees of freedom on the wave energy conversion ability when the period is near the resonance period of the single DOF motion. When the wave period is near the single DOF pitching resonance period, the heaving DOF of the coupling motion has a negative effect on the pitching DOF. On the contrary, when the wave period is near the single DOF heaving resonance period, the pitching DOF of the coupling motion has a positive effect on the heaving DOF.
At a much higher wave period which is far away from the resonance periods of the single DOF of the heave and pitch, the capture width ratio of a certain DOF in the coupling motion is nearly the same as that of the single certain DOF motion. The heaving and pitching DOF of the coupling motion can be assumed to be dependent and the whole wave energy conversion can be regarded as the summation of the single DOF of the heave and pitch motion.
When the wave periods are between the single DOF resonance periods, the coupling motion shows a smaller disturbance on the flow field and the vorticity field shows a lower value and even distribution. The buoys of the coupling motion at these periods show a better wave energy conversion ability.