1 Introduction

Extracting energy from ocean waves has beenan idea to put wave energy converters (WECs) into practical use for more than two centuries. Extensive work has been done on WECs, especially on the heaving type and oscillating water columns (OWCs) ^{[1, 2, 3]} ,while few literature is available concerning the oscillating wave surge converters. Whittaker et al. ^[4] presented the evolution of Oyster. A bottom hinged flap device and some project results were given ^{[4, 5, 6]} . Henriques et al.^[7] performed designing and testing for a bottom hinged plate wave energy converter with eddy current brakes simulatingas a non-linear power take-off system,gave a novel power taking-off type converter,and presented an experimental study. Under the action of incident waves,the flap was forced to move pitchingly about a bottom hinge. A generator was connected to the bottom shaft by matching the gearing-up or hydraulic booster,and then load resistances were used to the generator for electricity output.

In reality,the flap activates as a point absorber with a small dimension relative to the incoming wavelength,and the converter can capture wave energy from a width greater than its own. This is known as the point-absorber theory ^{[8, 9]} . However,for simplicity,it is common to consider the two-dimensional case where the flap is severalwavelengths wide. Renzi and Dias^{[10, 11, 12]} and Renzi et al. ^[13] developed several new modes to investigate the hydrodynamics of the flap type wave energy converters. In the open ocean case,the flap width effect was investigated on the hydrodynamics of a single oscillating wave surge converter (OWSC). It was found that increasing the flap width had a beneficial effect on enlarging the bandwidth of the capture factor curve.

The effect of the flap width on the hydrodynamic performance of a flap type OWSC (a pendulum wave energy converter specifically named in our model) is presented. The parameters such as pitching amplitude,added inertial moment,radiation damping,and conversion efficiency are fully investigated within the framework of a linear inviscid potential flow theory. From the above literatures,it can be found that,in general,only the radiation damping coefficient is included in the calculation of the hydrodynamic performances of the wave energy converter acting with the water particles. However,for the pendulum wave energy converter,the viscous damping effect plays an important role,especially in the vicinity of the natural frequency. The results without the viscous damping may be almost unreal in some cases. Unfortunately,no theoretical method is appropriate to obtain the solution of the viscous damping coefficient. The viscous damping coefficient has been estimated with the data from a wave tank test of a flap in a set of polychromatic seas (see Ref. ^[6]),but no expression has been given.

In the present paper,the nonlinear viscous damping coefficient is considered by referring to the ship rolling,and an expression is given. A theoretical model based on the eigenfunction expansion method,which is widely used in the study of the hydrodynamics of breakwaters, is used to obtain the hydrodynamic performance ^{[14, 15]} . It is shown that some hydrodynamic parameters show marked sensitivity to the converter width and that the three-dimensional results with width tend to be infinity agree well with those of two-dimensional cases. The same conclusions have been obtained by Folley et al. ^[4] ,who noted that the wave surge force was proportional to the width square in three-dimensional cases while linear to the width in two-dimensional cases ^[6] . Finally,it reveals that the point absorber effect for a pendulum wave energy converter is less prominent with the increase in the converter width,and the maximum primary energy conversion efficiency is less than 50% in the two-dimensional cases. An expected 100% energy conversion efficiency cannot be achieved even for a narrow pendulum wave energy converter because of the negative influence of the nonlinear viscous damping coefficient,which is considered as a dominant factor in the analysis of the following sections. 2 Modeltest

The model of a pendulum wave energy converter is shown in Fig. 1,whose height is 710 mm, width is 630 mm,thickness is 70 mm at the bottom and 145 mm at the top. The borders with the width of 35 mm and the thickness of 1 mm are attached on both sides of the pendulum flap. Two strengthen skeletons are settled at a distance of 210 mm from the outer side symmetrically. The bottom shaft is fixed on the top of an underwater rail,which spans the whole width of the flume and with a height of 590 mm from the channel bottom. The flap moves forward and backward to absorb the wave energy excited by the waves with an inertial moment of 2.38 kg·m² against the bottom shaft.

3 Motionequationinregularwaves

It is common that the dynamics of a fixed bottom-hinged pendulum wave energy converter can be modeled by a single degree of freedom system. Assume that the displacement is small, and the response amplitude of the model can be given by

where I₄₄ is the inertial moment, C₄₄ is the restoring moment coefficient,B_p is an external applied damping in regular waves,andωis the wave frequency,A₄₄ is the added inertia moment, B₄₄ is the radiation damping coefficient,and F₄ is the excited wave moment which can be obtained by two- or three-dimensional potential flow theory ^[16] .

The two-dimensional potential flow theory will be discussed in detail in the next section. B_v is the nonlinear viscous damping coefficient,which can be evaluated as a portion of the critical damping coefficient C_cr as follows:

where κ=0.5. The natural frequency ω pof the model is given by

The critical damping coefficientCcr canbeexpressedas ^[17]

Therefore,the critical damping coefficient in the pitch motion of the converter is

The external damping coefficient B_p can also be rewritten as where δ is the damping factor.

R_ao is defined as the response amplitude ratio by

where H is the wave height.

In addition,(8) gives the average input power in regular waves within the model scale

where L is the converter width,dis the water depth,ρ is the water density,and k is the wave number which can be determined by

The average power captured by the converter is estimated by

Thus,the primary energy conversion efficiency is given by 4 Two-dimensional eigenfunction expansion method

To calculate the hydrodynamic characteristics based on the two-dimensional eigenfunction expansion method when the converter width tends to be infinity,it is assumed that the converter section is rectangular (see Fig. 2). In Fig. 2,a 2Bwide rectangular pendulum wave energy converter is deployed in a water tank with the depth of d. The fluid is assumed inviscid, incompressible,and irrotational. Then,a velocity potential can be found in the flow region. The total velocity potential Φ is

where Φ_s is a scattering mode exited by the incident waves with a fixed converter ,and Φ_r is a radiation potential due to the motion of the converter against the bottom shaft.

For regular waves with the frequency of ω,Φ_s and Φ_r canbeexpressedas

where

The velocity potential must satisfy the Laplace equation,the linearized free surface conditions,the sea bottom condition,the Sommerfeld condition at far field,and the following solid boundary conditions:

The complete fluid domain is decomposed into three sub-regions: Sub-region 1,the left region (y≤−B); Sub-region 2,the fluid domain below the converter (−B≤y≤B); and Sub-region 3,the right region (y≥B). The velocity potentialsΦ_sj andΦ_rj in different regions are given by

where δ_j1 is the Kronecker delta. Let μ₀=−ik. Then,the incident wave velocity potential φ_i can be written as The general solutions for the velocity potential φ^g_qj (q=s,r)ineachregion Ω_j are as follows: whereA^q_jm (j =1,2,3,4;m=0,1,···,∞) are unknown complex expansion coefficients. The particular solution for the radiation velocity potential φ^p_rj may be written as and μ_m are the positive real roots of the following dispersion relations: The eigenvalues λ_m are given by

On the common boundaries between different sub-regions,the velocity potentials must satisfy the appropriate transmission conditions as follows:

where 0≤z≤S.

Substitute the velocity potentials into (27),multiply both sides of the obtained equations by the eigenfunction in Region Ω₂,and integrate with respect to zfrom zero to S. Substitute the velocity potentials into (28) and the solid boundary condition into (17),multiply both sides of the equations by the eigenfunction in Region Ω₁,and integrate with respect to z in the whole water depth. Then,four sets of equations can be obtained. All the unknown expansion coefficients in the velocity potentials can be determined by solving the system of linear algebraic equations.

The dynamic pressure caused by the scattering mode can be obtained by

Integrate the dynamic pressure along the wetted surface of the converter. Then,the magnitude of the excited wave moment F₄ against the rotating shaft can be obtained by

whereps =ρiωCΦ_s(y,z),n₄=yn_z−(z−S₀)n_y,S_B is the wetted surface of the converter,and n_y and n_z are the projections of the surface normal vector on they-andz-axes,respectively.

The magnitude of the radiation moment F_r4 due to unit pitching amplitude of the converter against the bottom shaft is

The radiation moment includes an added mass component and a radiation damping term,i.e., Then,the added inertial moment A₄₄ and the radiation damping moment coefficient B₄₄ can be obtained as follows: 5 Results analysis 5.1 Motion response for undamped converter

Model tests are conducted in a wave flume for the undamped condition (B_p=0). Figure 3 shows the incident wave height separated by the analytical method ^[18] . The incident wave height (see Fig. 3) is used for validating the numerical analysis,and a comparison of the motion response of the model is made between the numerical analysis and the experimental results (see Fig. 4). The motion response amplitude decreases with the increase in the incident wave height because of the nonlinear damping effect. Corresponding to the wave periods 3.6 s,3.8 s, 4.0 s,and 4.2 s,the measured incident wave heights in the experiments are 13.5 cm,14.8 cm, 14.1 cm,and 12.5 cm,respectively. Therefore,there is a small dip at periods of 3.8 s and 4.0 s because of the relatively larger wave heights (see Fig. 4). It is shown clearly that the motion response is much larger than the experimental results if the viscous damping is not taken into account in the numerical analysis,especially near the resonance region,while the numerical results including the viscous damping match the experimental results fairly well. Therefore,it is concluded that the nonlinear viscous damping must be considered in the numerical analysis for the pendulum wave energy converters.

5.2 Width effect on performance of converter

In order to study the width effect on the hydrodynamic performance of the converter,the added inertia moment A₄₄,the radiation damping B₄₄,and the exciting moment F₄ are nondimensionlized by

where V and T are the displacement and the draftof the converter,respectively.

The width of the converter model in the experiment isl =0.63 m. In the numerical analysis, it is chosen as L=l,2l,4l,8l,or∞.InthecaseofL=∞,the two-dimensional eigenfunction expansion method mentioned above is used to obtain the results,while the three-dimensional potential wave theory is used to analyze the other cases. The dimensionless added inertia moments,radiation damping coefficient,and wave excited force with the wave height of 15 cm are shown in Figs. 5,6,and 7,respectively. The non-dimensional inertial moment of the model is 0.154. It is shown in Fig. 5 that the added inertia moment is much larger than the inertia moment of the model itself for the majority of wave periods. It indicates that the natural period is slightly long for the pendulum wave energy converter. Meanwhile,the peak amplitude of the added inertia moment tends to long wave periods with the increase in the converter width,and approaches the two-dimensional results whereL/l=∞. It is shown in Fig. 6 that the radiation damping coefficient is insensitive to the converter width in short wave periods,increases with the increase in the converter width and wave periods,and finally maintains relatively small values in long periods. It can be concluded that the power output will not be high even in the vicinity of the resonant period because of the small radiation damping at the long wave period. The non-dimensional wave excited force shown in Fig. 7 has similar conclusions to those in Fig. 6.

The calculated motion response amplitudes of the converter for the undamped condition (Bp= 0) in regular waves with the wave height ofH= 15 cm are shown in Fig. 8. The natural frequencyωpfor the model with the width L=l,2l,4l,or8l are 1.7 rad/s,1.4 rad/s,1.2 rad/s, or 1.1 rad/s. The natural frequencyωpfor the two-dimensional case,namely,when the width tends infinity,is 0.954 rad/s. It is shown from Fig. 8 that the natural periodTp(2π/ωp)ofthe model increases with the increase in the converter width.

The pitching amplitudes of the converter under different external damping coefficients in regular waves with the wave height H= 15 cm are shown in Figs. 9-11. They are relatively small under large external damping coefficients. The results approach those of the two dimensional case with the increase in the converter width.

The primary energy conversion efficiency is shown in Figs. 12-14. It is noted from the figures that the maximum primary energy conversion efficiency descends with the increase in the converter width. In the case ofL=8l,the efficiency calculated by the three-dimensional method is in good agreement with that obtained by the two-dimensional method forL=∞, which gives the highest efficiency of less than 50%.

6 Conclusions

Analytical solutions are sought for a pendulum wave energy converter in the two- and threedimensional regions,respectively. The nonlinear viscous damping coefficient is included in the calculation of the converter response. An expression is given based on the theory of ship rolling. Some important parameters are obtained. The performance is known clearly. In particular, the width effect on the performance of the wave energy converter is given more close attention. Some conclusions can be made by means of the analysis on a bottom-hinged pendulum wave energy converter in regular waves mentioned above as follows:

(i) To obtain reasonable results,the nonlinear viscous damping must be taken into account in the analysis of the pendulum wave energy converter.

(ii) The hydrodynamic parameters obtained from the three-dimensional method approach the results of the two-dimensional method with the increase in the converter width,which implies that both the two methods mentioned above are reasonable.

(iii) The primary wave energy conversion efficiency goes down with the increase in the converter width in short wave periods,while it is converse in long wave periods.

(iv) The point absorber effect is less prominent with the increase in the converter width. The maximum primary energy conversion efficiency is less than 50% for the pendulum wave energy converter with relatively larger width in the two-dimensional analysis,while 100% energy conversion efficiency cannot be achieved by a narrow converter because of the negative influence of the nonlinear viscous damping coefficient.