Mathematical modelling of a floating Clam-type wave energy converter

In this paper, wave power extraction from a floating Clam-type wave energy converter is investigated. The device is mainly composed of a Clam, which is formed from two pieces of floating flaps hinged at a submerged body. The Clam is closed by a flexible impermeable bag with the two hinged floating flaps kept apart by a Power Take-Off system. As waves propagate through the device, the Clam motion of the device is excited, which can be used to drive the Power Take-Off system to capture wave power. To evaluate the response and also the wave power absorption of the device, a mathematical model is developed based on the linear potential flow theory, in which a generalised mode method is adopted to model the Clam action. Theoretical expressions of the maximum wave power absorption and the corresponding optimised Power Take-Off system and mooring parameters are derived. Good agreement between the present numerical results of the device response and the physical observations is obtained. The validated model is then applied to do a series of case studies. It is revealed that the optimised Power Take-Off stiffness and mooring stiffness are independent of the Power Take-Off damping. The maximum wave power absorption can be achieved when the device is fixed in heave mode or free-floating without any constraints from the mooring system.


Introduction
To capture wave power from ocean waves, a large number of wave energy conversion concepts have been proposed since the 1790s [1]. In spite of the large variation in concepts and design, most of them are composed of rigid floats (e.g., see [2][3][4][5]). Apart from the traditional rigid floats based wave energy converters (WECs), a flexible deformable body can work as a WEC as well. Its mean shape can be changed so as to adjust its resonant periods, adapting itself to different environmental conditions, e.g., the sea sites where wave conditions change seasonally from summer to winter. A flexible deformable structured WEC is also expected to have a better survivability in extreme wave conditions [6].
The Lancaster Flexible Bag and Circular Clam developed in the early 1980s could be the first flexible WECs [7]. The devices are composed of a series of inflatable bags filled with air. The air bags would expand and contract following the oscillatory motion of the incoming waves, pumping air into a turbine connected to a generator. A similar technique is also adopted in the mWave WEC developed by the company Bombora Wave Power [8]. Anaconda and S3 are two attenuator-type flexible WECs consisting of a long submerged elastic tube filled with water [9,10]. Bulge waves propagating along the tube can be excited by the interaction between water waves and the flexible tube. The bulge waves then derive a power take-off system at the stern or distributed along the tube, e.g., electro-active polymers, to capture electricity. The Wave Carpet developed at UC Berkeley in 2012 is composed of a S. Zheng  Instantaneous displacement and timeindependent amplitude of displacement, respectively, of each flap in Mode (PTO) system, which is composed of hydraulic rams and tension springs holding the Clam sides apart. The Clam motion was treated as a generalised mode in their numerical model. For simplicity, the flap was assumed to be weightless in the analysis adopted in the frequency domain. The Clam was found to be most effective when the device was constrained in heave. More recently, Kurniawan et al. [19] turned their attention to an axisymmetric version of the floating device, in which the pair of rigid plates is replaced by a completely flexible bag. The static and dynamic behaviour of the device was investigated numerically and experimentally. The results showed that the bag shape and dynamic response could be changed by varying the amount of air in the bag. The dynamic response, but not the shape, also depended on the turbine damping. The resonance period of the device was found to be larger than that of a rigid device with the same geometry. An accompanied loss in bandwidth compared to the bandwidth of the rigid device, albeit not severe, was observed. Kurniawan et al. [20] proposed another variation of the device by arranging the flexible bag to be fully submerged and connected to a rigid ballasted float at the top and to a weighted cylinder at the bottom. Three different equilibrium device geometries were indicated in their study for the same bag pressure. Although the wave power extraction of different versions of the floating clam device have been investigated both numerically and experimentally, most of them have been focused on examining the performance of the device with an arbitrarily specified PTO damping, and the optimisation of their PTO for maximising power absorption has not been carried out. The aim of the present work is, therefore, to develop a mathematical model to seek the maximum wave power that can be captured by a floating clam device, and specify the corresponding optimised PTO/mooring system required theoretically. The model is then applied to study the performance of the device with different geometrical parameters.  The remainder of the paper is structured as follows. The mathematical model for the wave-structure interaction problem is described in Section 2. The expressions of the theoretical maximum wave power absorption, together with the corresponding optimised PTO parameters can be found in Section 3. The present model is validated by comparing the present results with measured physical data, and the validated model is applied to carry out a multi-parameter study on the performance of C-WEC, the results of which, together with discussions, are also presented in Section 4. Finally, conclusions are drawn in Section 5. Fig. 1 shows the schematic diagram of the problem. The C-WEC device consists of two rigid flaps, which are flexibly connected to two hinges at a submerged body and held apart by a PTO system. The flexible parts of the device are made from waterproof flexible deformable membranes, which allow the two flaps to freely rotate about the hinges. A potential benefit of the flexible parts is the possibility to integrate an alternative PTO system, e.g., a dielectric elastomer generator (DEG) [21], into it, helping reduce the amount of mechanical moving parts and ensure reliability and reduced/simplified maintenance costs of the device. Yet this is out of the scope of the present paper. The tension and deflection of the membranes are assumed to be small and neglected in the hydrodynamic problem. However, for the C-WEC of small width, the tension and deflection of the membranes may become much more significant and may have to be taken into account. A Cartesian coordinate system is adopted with the mean free-surface coinciding with the -plane, the -plane coinciding with symmetric plane of C-WEC, and measured vertically upwards. Therefore, the fluid bottom (i.e., the sea bed) is at = −ℎ. The dimension of the device in the -direction, i.e., the width, is denoted by . The length and the submerged depth of each flap are denoted by and , respectively. The C-WEC is subjected to regular waves of angular frequency and amplitude , propagating along the -axis. The motion of the C-WEC can be decomposed into 7 modes. The first 6 modes are the conventional rigid body modes. Modes 1 to 3 describe translational motion in the , , (surge, sway and heave) axes, respectively, and Modes 4, 5 and 6 represent rotation about the , , and axes (roll, pitch and yaw), respectively. Mode 7 is a 'generalised mode', i.e., the Clam mode, and it is defined by the movement of the PTO ram -the positive direction of movement being shown by the arrows as shown in Fig. 1c. This definition of the Clam mode is different from that adopted by Phillips [18], who defined the Clam mode as unit change of the distance between the two flaps along the line of action of PTO. Different definitions of the Clam mode would result in different hydrodynamic coefficients and different Clam mode related generalised mass and restoring stiffness, and PTO parameters as well.

Mathematical model
For regular incident waves propagating along the -axis, only surge, heave, pitch, and Clam modes are excited provided the = 0 is a plane of symmetry for both geometries and physical properties of the C-WEC. The present work is focused on the responses in heave and Clam modes, since the Clam mode is associated with wave power absorption of the C-WEC and coupled with the heave mode. Although the motions of the C-WEC in surge and pitch modes do exist, they are not considered hereinafter because they are decoupled with the heave and Clam modes.

Static shape of C-WEC
The static shape of the Clam in terms of (the opening angle between the two flaps is 2 ), together with the device position in terms of (see Fig. 1b), should be evaluated prior to the hydrodynamic analysis. Fig. 2 presents a schematic diagram of static force/moment acting on the device in still water.
denotes the moment acting on the flap provided by the compressed spring in the PTO system with = −2 ( − 0 ), in which is the stiffness of the PTO system and 0 represents the Clam position (i.e., the opening angle between the two flaps is 2 0 ) when no resistance is provided by the PTO stiffness; 0 is the mass of each flap; represents the total mass of the device with the two flaps excluded; denotes the net vertical buoyancy force due to the hydrostatic pressure acting on the wetted surface of the whole device except the two flaps; is the density of the water, and represents the gravitational acceleration. 0 and ,0 represent a constant moment acting on the flap, which could be provided by using a constant force spring, and the mooring force acting on the device in still water, respectively, and they can be used to adjust the static shape of C-WEC.
For a C-WEC placed at its equilibrium in still water, the following two relations should be satisfied: • The overall buoyancy of the device equals the weight of the device, and the mooring force • The buoyancy-moment acting on the two flaps about the hinges is equal to two times the moment induced by the PTO system, the weight of the flap and the constant force spring 2( + 0 ) + 0 sin − ( + 0 ∕2 sin ) 3 3 cos 2 ≈2( where 0 represents the thickness of the flap, which is assumed to be much smaller than .
It is noted that both and are unknown, nevertheless, they are dependent on each other, i.e., once one of them is known, the other one can be evaluated accordingly. can be evaluated iteratively with a bisection method by assuming an initial value between 0 and ∕2 and solving for the corresponding , until the computed value of is close to the target value within a given tolerance.

Dynamic model
All amplitudes are assumed to be small enough that linear theory applies, and the fluid is assumed to be inviscid, incompressible and irrotational. It is further assumed that all motion is time-harmonic with angular frequency , hence the fluid velocity potential and the displacement of the C-WEC about its static equilibrium position may be expressed by and where Re denotes the real part. The functions ( , , ) and represent the time-independent parts of the complex velocity potential and the complex displacement of the device in Mode , respectively. Under the assumptions above, the spatial velocity potential satisfies the Laplace equation in the fluid with the boundary conditions = i ( 3 3 + 7 7 ) on the mean wetted surface of the device (7) where ∕ = ⃗ ⋅ ▿, in which ⃗ represents the unit normal on the boundary curve directed away from the fluid domain. 3  Otherwise, if the flaps are allowed to rotate about two symmetrically deployed hinges separately rather than about the identical hinge, 0 will be split into 0 − and 0 + , which correspond to the horizontal position of the two flap hinges, with 0 − + 0 + = 0 and 0 + > 0. = 1 and −1 for the wetted surface of the right-hand side flap and the left-hand side one (see Fig. 1), respectively, and = 0 for the other wetted part of the device.

Hydrodynamic solutions
The velocity potential in the fluid domain can be further decomposed into incident wave, diffracted wave and radiated wave components as follows where is the velocity potential of undisturbed incident waves; represents the velocity potential of the diffracted wave due to the existence of the fixed device; 3 and 7 are the radiated velocity potentials induced by oscillation of the device in heave and Clam modes, respectively, with unit velocity amplitude. , 3 and 7 can be determined with the employment of a boundary element method (BEM)-based numerical code, e.g., WAMIT [22] and NEMOH [23].

Motion response
The motion response matrix equation of the C-WEC may be expressed as in which and are the matrices of the device mass and static stiffness, respectively; and represent the matrices of added-mass and wave radiation damping coefficients, respectively; and are the PTO system associated damping and stiffness matrices, respectively; represents the stiffness matrix of the mooring system; = [ 3 , 7 ] and = [ 3 , 7 ] represent the vector of the displacements of the device and the vector of the wave excitation forces acting on the device in heave mode and Clam mode, respectively.
, and can be obtained by using a BEM-based code. The specific expressions of the matrices at the left-hand side of Eq. (9) are given as follows: in which represents the total area of the cut water-plane; is adopted to consider the effect of the stiffness induced by a mooring system linking the submerged float to the seabed; = 2 0 + , = 0 2 ∕3 is the rotary inertia of each flap about the hinge; is the damping coefficient of the PTO system; ℎ denotes the external heave damping coefficient, which vanishes unless otherwise specified; , and , for , = 3, 7 represent the hydrodynamic coefficients of Mode due to the motion in Mode . It should be noted that the heave and Clam modes are not only coupled in terms of hydrodynamic coefficient matrices and , but also with regard to the matrices of the device mass and static stiffness, i.e., and , the detailed derivation of the expressions are given in Appendix. The total clam opening angle change (2 7 ) may be a natural parameter for a Clam with a single hinge, but the present device has two separate hinges, so the individual opening angle change ( 7 ) is used here in defining the generalised mode. Therefore the PTO damping and PTO stiffness in Eqs. (12) and (13) Box I.

Wave power absorption
Wave power is converted by the damping effect of the PTO system. The time-averaged power that C-WEC captures from regular waves can be written as The wave energy capture width ratio (or capture factor) can be calculated by where in represents the incident wave power per unit width of the wave front, and denotes the wave number.

Maximum wave power absorption
The subject of this subsection is to study, theoretically, the maximum wave power that the C-WEC can capture from regular water waves. Optimisation of three variables, i.e., PTO damping, PTO stiffness, and mooring stiffness, are considered.
In order to derive the expressions of the maximum wave power absorption, together with the corresponding optimised parameters, the variables should be separated and the motion equation may be rewritten as where the subscript ''0'' denotes the element is independent of both the PTO parameters and the stiffness of mooring system, 0 = 3,3 − 2 ( + 3,3 ) + i 3,3 , 0 = 3,7 − 2 ( 3,7 − 0 sin ) + i 3,7 , 0 = 7,3 − 2 ( 7,3 − 0 sin ) + i 7,3 , 0 = 7,7 − 2 (2 + 7,7 ) + i 7,7 . Correspondingly, the time-averaged wave power absorption can be expressed as For the device in a specified sea state, three parameters, i.e., PTO damping ( ), PTO stiffness ( ), and mooring stiffness ( ), could be optimised to improve the performance of the device in wave power absorption. Hereinafter, three different levels of optimisation principles are considered, i.e., levels I, II, and III. For level I optimisation, two of these three parameters are fixed with given values, and the remaining one is optimised. For the level II optimisation, one of them is fixed with a specified value, and the remaining two are optimised at the same time. For the level III optimisation, all these three parameters are optimised at the same time.
(1) Level I, optimisation of There is a corresponding maximum of absorbed wave power when = 0, which occurs if where the subscript ''opt'' denotes optimisation of PTO damping. The maximum absorbed wave power with can be further expressed as (see Eq. (22) which is given in Box I).
(2) Level I, optimisation of There is a corresponding maximum of absorbed wave power when = 0, which occurs if which indicates that the optimised PTO stiffness is affected by the stiffness of the mooring system, but independent of the PTO damping coefficient.
The maximum absorbed wave power with can be further expressed as A negative-stiffness mechanism is common in some control strategies of WECs, e.g., the reactive phase control strategy (e.g., see [24]). The negative-stiffness may be achieved by using some well-arranged classic positive-stiffness springs, e.g., two symmetrical oblique springs [25,26], or by adopting magnetic springs (e.g., see [27,28]). Nevertheless, the application of a negative-stiffness mechanism would increase the complexity of the device and may affect the stability as well. In the present work, the PTO stiffness is assumed to be non-negative. Hence, Eqs. (23) and (24) may be rewritten as: (3) Level I, optimisation of The expression of the wave power absorption may be written as The corresponding positive that satisfies = 0 is [29] (see Eq. (28) which is given in Box II) where However, it should be noticed that the that satisfies = 0 here results in the minimum of the wave power absorption, rather than the maximum.
The maximum wave power absorption may be achieved as the larger value between the results with = 0 or → ∞, i.e., This is in line with the research work reported by Phillips [18].
S. Zheng et al.
Box II.
(4) Level II, optimisation of and as given in Eq. (22) can be further enlarged with an optimal . If and can be chosen such that: for which we have .
The maximum absorbed wave power under the optimal combination of and is If < 0 is obtained with the utilisation of Eq. (32), then , , and , should be recalculated with and depending on which term in the brackets of Eq. (36) is larger. (6) Level II, optimisation of and The maximum wave power absorption and the corresponding optimised and may be expressed as and depending on which term in the brackets of Eq. (38) is larger. If < 0 is obtained with the utilisation of Eq. (39), then , and , should be recalculated with and depending on which term in the brackets of Eq. (40) is larger. (7) Level III, optimisation of , , and The maximum wave power absorption and the corresponding optimised , , and may be expressed as and depending on which term in the brackets of Eq. (42) is larger. It is noted that when the stiffness of the mooring system → ∞, this equation becomes = | 7 | 2 8 7,7 , which coincides with the maximum absorbed power that a Clam-motion only C-WEC, i.e., no motion in the heave mode, can achieve in resonant conditions as expected.
Again, if the value of evaluated with the employment of Eq. (43) is negative, , , and should be recalculated by and depending on which term in the brackets of Eq. (44) is larger.

Results and discussions
The geometry of the C-WEC considered hereinafter and the corresponding mesh arrangements of the wetted surface are presented in Fig. 3. The uprights connecting the keel to the wedge are thin and have little impact on hydrodynamics. Hence they are omitted in the hydrodynamic simulation. Following Phillips [18]'s work, the hinge point S. Zheng et al.
Figs. 4 and 5 present the frequency response of the wave excitation forces and the hydrodynamic coefficients evaluated by WAMIT. As expected, 7,3 = 3,7 and 7,3 = 3,7 are satisfied by the numerical results. To shorten the length of the paper, 7,3 and 7,3 are not plotted in Fig. 5.  measured physical data, which has been scaled up to the full prototype from the 1:50 experimental model. The 1:50 scale physical tests were carried out in the Ocean Basin at the Coastal, Ocean and Sediment Transport (COAST) laboratory, housed in the Marine Building at the University of Plymouth [18]. The numerical predictions are generally in good agreement with the experiment data, although there are some discrepancies between = 9 and 14 s. Fig. 7 presents the frequency response of and obtained by using the present analytical model. For comparison, the results predicted with the employment of the trial and error method (i.e., an exhaustive search method), which finds the optimisation solution by trying a finite number of attempts over a specified range of the value of a variable (e.g., the PTO damping coefficient) and then comparatively evaluating the solutions based upon some predefined set of criteria (e.g., the maximum wave power absorption), are also displayed. A good agreement of the present results with the trial and error numerical results is achieved, and it gives us confidence in the present model for solving wave diffraction/radiation problems and evaluating the maximum wave power absorption of the C-WEC.

Model validation
Hereinafter, the validated model is adopted to predict the potential maximum wave power absorption of the device and evaluate the corresponding optimised PTO system or/and mooring stiffness. Fig. 8 illustrates the frequency response of ( ) and̄( ) of the C-WEC with̄= 0, i.e., the device is free-floating. Different curves denote the device with different values of PTO stiffness (̄). For 1 rad/s < < 2 rad/s, in the examined five cases, the larger̄, the smaller ( ) and the larger the correspondinḡ( ) . At ≈ 0.87 rad/s, the change in̄has little impact on ( ) and̄( ) . This is because | 0 | is rather small at ≈ 0.87 rad/s, making ( ) and̄( ) insensitive S. Zheng et al.   to the change of̄(see Eqs. (21) and (22)). For̄≥ 0.4, a local peak of ( ) occurs at < 1.0 rad/s, where a local minimum of̄( ) is observed. This is because 0 − 0 0 ∕ 0 ≈ R − for < 1.0 rad/s. When is large enough, saȳ≥ 0.4, | 0 + 4 − 0 0 ∕ 0 | ≈ 0 can be achieved (see Eqs. (21) and (22)), leading to a local peak and valley of the frequency responses of ( ) and̄( ) , respectively. When the incident wavelength is large, say < 0.5 rad/s, the free-floating device moves up and down with the waves and the Clam-mode motion can be hardly excited to capture wave power.

Optimisation of the PTO damping coefficient
We select = 0.5, 1.0, 1.5, 2.0, and 2.5 rad/s as five typical wave conditions and plot the corresponding ( ) −̄and̄( ) −̄in Fig. 9. For intermediate wave frequencies in the examined five wave conditions, = 1.0 and 1.5 rad/s, the largest value of ( ) is achieved without a PTO stiffness. Whereas for the remaining low and high wave frequencies (i.e., = 0.5, 2.0, and 2.5 rad/s), there is an optimised non-vanishinḡto maximise ( ) . For any specified wave condition, the variation of̄( ) with̄shows an opposite trend to that of the ( ) −̄. For the lowest wave frequency, = 0.5 rad/s, a sharp peak  of ( ) −̄and a relatively linear̄( ) −̄are observed. This is because when → 0, wave radiation damping coefficients → 0, and 0 , 0 , 0 , and 0 all tend to be real numbers. Hence Eqs. (21) and (22) tend to be the amplitudes of a linear function and a reciprocal function of , respectively, leading to the results for small wave frequencies, e.g., = 0.5 rad/s, as plotted in Fig. 9. Fig. 10 displays the frequency response of ( ) and̄( ) of the C-WEC with a specified PTO damping,̄= 2.0. Different curves represent the device by applying the mooring system with different stiffness. When the device is free-floating, i.e.,̄= 0, very little wave power can be captured by the device for < 0.5 rad/s. As increases from 0.5 rad/s, ( ) shows a rapid increase and reaches the maximum value 0.4 around = 1 rad/s. As keeps increasing, ( ) starts decreasing. As̄increases from 0 to 3, the main peak of ( ) − curve shifts towards large wave frequencies and the bandwidth becomes smaller and smaller. Meanwhile, the curve for small wave frequencies, e.g., < 0.5 rad/s, raises up with the increase of̄. It is noted that for mooring stiffness̄= 1, 2, and 3, no wave power can be captured at = 1.1, 1.4, and 1.7 rad/s, respectively. This is because 0 − 0 3 ∕ 7 ≈ R − , and we have | + 0 − 0 3 ∕ 7 | ≈ 0 at specific wave conditions for these three cases, resulting in no wave power absorption regardless of the values of the PTO damping and stiffness (see Eq. (20)). When the wave frequencies become large enough, e.g., > 2.3 rad/s, ( ) and̄( ) are found to be insensitive to the change of̄. Fig. 11 exhibits the frequency response of ( ) and̄( ) , , and different curves represent the cases with different PTO damping. In long waves, e.g., < 0.7 rad/s, the maximum wave power absorption of the device is achieved when the mooring stiffness is infinite, i.e., the heave motion of the device is completely restricted (see Fig. 11b). Whereas for 1.0 rad/s < < 2.5 rad/s, the free-floating device without any constraint from the mooring system can capture more wave power than those with some mooring restraints. In the examined five cases, the smaller the PTO damping, the larger ( ) is obtained for most wave conditions, except around 1.0 rad/s. Fig. 12 shows the maximum wave power absorption of the C-WEC when the PTO damping and PTO stiffness are both optimised. Different curves denote the cases with different values of mooring stiffness. For long-wave conditions, e.g., < 0.5 rad/s, the longer the incident waves, the larger the maximum wave power capture factor, except the two cases with̄= 4 and 10 000, for which̄( ) = 0, and ∕ = 0 is not satisfied, meaning the resonance condition is not achieved. For > 0.5 rad/s, the ( ) − curves for the examined six cases almost overlap one another, except three steps observed at 1.1, 1.4, and 1.7 rad/s for = 1, 2, and 3, respectively, where we have | + 0 − 0 3 ∕ 7 | ≈ 0 as reported in Section 4.3. Fig. 12a exhibits that ( ) ≈ 0.5 for > 2.0 rad/s. This may be explained from the two-dimensional point of  view: when the incident wavelength is much smaller than the width of the device, the hydrodynamic problem becomes a two-dimensional case. In the two-dimensional case, because the device is symmetric so that waves are radiated from the device equally in opposite directions, meaning that not more than 50% of the incident wave energy can be captured.

Optimisation of PTO damping coefficient, PTO stiffness coefficient and mooring stiffness coefficient
Fig . 13 illustrates the frequency response of the maximum wave power capture factor of the device when the PTO damping, PTO stiffness, and the mooring stiffness are all optimised. As increases from 0.1 rad/s, ( ) exhibits a rapid decrease first and reaches the minimum around = 0.9 rad/s. As keeps increasing, ( ) increases slightly and remains at 0.5 when > 1.6 rad/s. ( ) > 1.0 is obtained for < 0.5 rad/s, and it should be pointed out that the correspondinḡ( ) is rather small, meaning that a large response in the Clam-mode is required. This, however, may not be achievable in practical applications since the C-WEC would have physical limitations placed upon excursions due to restraints such as pump stroke, and what is more, the present linear theory may not be valid when the device response is too severe.

Effect of the width
Wave energy could increase the power generated and reduce the LCoE (Levelised cost of energy) by increasing the device scale [30]. In the above subsections, it shows that the maximum wave power capture factor around = 1.0 rad/s can be achieved with̄= 2.0,̄= 0 and̄= 0. In this subsection, we keep adopting these settings and examine the performance of the device with different width,̄= 0.2, 0.3, 0.4, 0.5, and 0.6. The frequency response of the wave power capture factor is displayed in Fig. 14. The effect of device width on wave power absorption mainly happens at 0.8 rad/s < < 1.7 rad/s. For̄< 0.4, there are two peaks of observed in this range of wave conditions, and as̄keeps increasing towards 0.6, these two peaks merge into one single peak with the peak value increasing and approaching 0.5. Fig. 15 plots the maximum wave power absorption when the PTO damping can be optimised. It shows that the smaller width of the C-WEC, the larger the range of̄( ) is required around = 1.0 rad/s, making it more difficult to control in practice.
The maximum wave power absorption of a free-floating C-WEC when the PTO damping and PTO stiffness can be both optimised is displayed in Fig. 16. The smaller the width of the device, the larger ( ) is exhibited for < 0.8 rad/s. This is because when the device width gets small, the device acts as a point absorber. The point absorber effect (also known as the antenna effect), which is the ability to absorb power from a larger width than the physical dimension of the device, becomes stronger with the decrease of the device width. Whereas in short waves, e.g., > 2.0 rad/s, the performance of the C-WEC in terms of wave power capture factor remains 0.5 for the examined five cases with different width. As decreases from 1.0 rad/s to 0.1 rad/s, ( ) shows a more and more rapid increase. For < 0.5 rad/s, the narrower the device, the higher the ( ) . When is small enough, ( ) > 5 can be achieved for all the examined five cases. However, this may not be achievable in practical applications. To achieve such a high ( ) , a rather large device motion is required, and it could make the assumptions of linear theory invalid. Moreover, there are physical limitations to restricting the motion amplitude, e.g., pump stroke and mooring lines. It would be better to consider the effect of a constraint on the performance of the device. Similar studies were reported in the context of point absorbers (e.g., see [31]) and two-hinged-raft devices S. Zheng et al.  (e.g., see [32]). This evaluation is beyond the scope of this paper and, therefore, is left for future work. It should also be pointed out that the absorbed wave power is expressed in terms of wave energy capture width ratio (see Eq. (18)), meaning that plenty of wave power could be captured by the device with a large width even if the wave energy capture width ratio is low.

Effect of the opening angle
The opening angle of the device in still water in terms of 2 can be adjusted by changing the initial PTO moment ( 0 ) acting on the flap in still water. Note that the submergence of the clam ( ) at the equilibrium would also change accordingly with the change of should other parameters be fixed. For < 1.0 rad/s, the ( ) − curves for the examined seven cases overlap with one other, demonstrating the independence of ( ) on the opening angle of the device. Whereas for ≈ 2.0 rad/s, as increases from 20 • to 50 • , less and less wave power can be captured by the device. This may be because the larger the opening angle, the smaller the submerged depth, and the smaller both the hydrodynamic force acting on the flaps and the moment arm about the hinges, leading to smaller hydrodynamic moments about the hinges and resulting in a worse performance of the device in wave power absorption. A main peak of thē( ) * − curve is observed around = 1.0 rad/s regardless of the values of the device opening angle in still water. The peak value of̄( ) * is found to be strongly dependent on the value of , and the larger the opening angle, the smaller the peak value. This makes sense because when the optimum phase condition and optimum amplitude condition are satisfied simultaneously we havē( ) * ≈ 7,7 [33], in which 7,7 becomes smaller when the opening angle of the device is increased, i.e., the submerged depth is decreased, in still water.
To achieve the maximum wave power absorption as plotted in Fig. 18a, both PTO damping and PTO stiffness need to be adapted, and   a large variation range of wave frequency dependent̄( ) * and̄( ) * could be required over the examined wave conditions, especially for = 20 • (see Figs. 18b and c). This may increase the system complexity and costs of the device. For simplicity, choosing a proper PTO stiffness and optimising only the PTO damping to improve the power capture ability of the device may be welcomed. Fig. 19 presents the frequency response of ( ) and̄( ) * for seven different opening angles with̄ * = 0.3 and = 0. In the examined seven cases, a spike of the ( ) − curve and a valley of thē( ) * − curve are observed, respectively, between = 0.4 rad/s and 0.8 rad/s. This is because 0 , 0 , 0 , and 0 all tend to be real numbers as wave radiation damping → 0 when → 0, and meanwhile, 0 ( 0 + 4 − 0 0 ) ≈ 0 is satisfied for a specified wave frequency between = 0.4 rad/s and 0.8 rad/s for the examined seven cases. 0 ( 0 + 4 − 0 0 ) ≈ 0 is located at the denominator and numerator in the expressions of the maximum absorbed wave power and optimised PTO damping, respectively (see Eqs. (21) and (22)), resulting in a local peak of the ( ) − curve and a local valley of thē( ) * − curve between = 0.4 rad/s and 0.8 rad/s. The larger the , the larger the heaving stiffness ( 3,3 ) and the clam stiffness ( 7,7 ), making the local peak of the ( ) − curve and the local valley of thē ( ) * − curve move towards large wave frequencies.

Conclusions
In this paper, a Clam-type wave energy converter (C-WEC), which consists of a pair of flaps that are hinged at the bottom, is considered. The relative rotation of the flaps, i.e., the so-called Clam-motion, is utilised to capture energy from water waves. To investigate the performance of the C-WEC, a numerical model based on the linear potential flow theory is developed. What is more, the maximum wave power that can be captured by the device when the PTO/mooring system can be optimised is predicted. Some selected findings are listed below.
• When the incident wavelength is large, the Clam-mode motion of the free-floating device without any mooring restraints is hardly excited to capture wave power. • For some specific mooring stiffness, no wave power could be captured by the device for particular wave conditions regardless of the values of the PTO damping and stiffness. • When the PTO stiffness and mooring stiffness of the device are both variable, their optimised values are found to be independent of the PTO damping. • For the freely floating C-WEC with̄= 2.0 and̄= 0, there are two peaks of the frequency response of wave power capture factor when the device width is small, e.g.,̄< 0.4. Whereas when Fig. 18. Variation of ( ) ,̄( ) * and̄( ) * with for different values of (the opening angle between the two flaps is 2 ) in still water,̄= 0, and ℎ = 60 m. Fig. 19. Variation of ( ) and̄( ) * with for different values of (the opening angle between the two flaps is 2 ) in still water,̄= 0,̄ * = 0.3, and ℎ = 60 m.
the device width is large, there is only one single peak, and the peak value becomes larger with the increase of the device width. • A spike of the frequency response of the maximum wave power capture factor for a free-floating device with a constant PTO stiffness and an optimised PTO damping is observed. The spike moves towards large wave frequencies when the opening angle of the device in still water gets larger.
The present model is developed in the framework of potential flow theory, hence it may not be suitable for the extreme wave-structure interactions. A fully nonlinear hydrodynamic numerical model will be developed in the near future to predict the performance of C-WEC in extreme wave conditions and study the nonlinear characteristics of the mooring system and complex PTO system.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.  in which thė2 7 associated term is neglected in this paper provided the motion response is assumed to be small.