## Abstract

A mathematical model is described to analyse the hydrodynamic behaviour of a wave energy farm consisting of oscillating wave surge converters in oblique waves. The method is a highly efficient semi-analytical approach based on the linear potential flow theory. Wave farms with a large number of such devices are studied for various configurations. For an inline configuration with normally incident waves, the occurrence of a near-resonant behaviour, already known for small arrays, is confirmed. A strong wave focusing effect is observed in special configurations comprising a large number of devices. The effects of the arrangement and of the distance of separation between the flaps are also studied extensively. In general, the flaps lying on the front of the wave farm are found to exhibit an enhanced performance behaviour in average, owing to the mutual interactions arising within the array. A random sea analysis shows that a slightly staggered arrangement can be an ideal layout for a wave farm of this device. The hydrodynamics of two flaps that oscillate back to back is also discussed.

## 1. Introduction

Wave farms comprising a large number of wave energy converters (WECs) are planned at sites which have already been identified for the purpose of energy extraction (e.g. Lewis wave project, see [1]). The arrangement of the devices in such a farm can follow several possible configurations. This study analyses the interaction of waves with an array of oscillating wave surge converters (OWSCs) and the performance of such systems. The OWSC considered here is a bottom hinged flap-type WEC which extracts energy by virtue of its pitching motion and resembles the Oyster developed by Aquamarine Power.

Wave power absorption in an array has already been studied in the literature, starting with the seminal work of Budal [2]. However, the majority of the investigations deal with the hydrodynamics of point absorbers [3–5], which is based on the assumption that the body dimensions are much smaller than the wavelength of the incident wave field. Recent studies have shown that for OWSCs such as Oyster, the point absorber limit is no longer applicable and hence better and more accurate modelling of the device needs to be undertaken [6]. Some recent investigations also dealt with a detailed analysis of multiple WECs, but most of them did not go beyond three or four of such devices [7–10]. Indeed, in the literature, there have been very few attempts to understand the dynamics of large finite arrays. The analytical modelling of large and complex systems becomes difficult, whereas numerical approaches, on the other hand, are computationally expensive and performing such an analysis experimentally is quite challenging. Recently, Borgarino *et al.* [11] used a fast multi-pole accelerated linearized boundary element method to study large arrays of sparsely distributed generic WECs in deep water. However, despite the recent effort of Renzi *et al.* [12], who devised a new method to investigate the hydrodynamics of a small inline array of OWSCs, to date, there is still a need for an unifying theory of large arrays of OWSCs in any configuration and in oblique waves. The analysis in this paper extends the semi-analytical work of Renzi *et al.* [12] to investigate a large farm of OWSCs in any configuration under oblique incident waves.

A mathematical model is developed here within the framework of linear potential theory. The theory allows the analysis of arbitrary configurations of an array of OWSCs, the only constraint being that all the converters have parallel pitching axes (figure 1). The problem is formulated as a boundary value problem for the radiation and scattering potentials. The use of Green's integral theorem yields hypersingular integrals (HIs) in terms of the jump in potential across the sides of each flap, which are solved using a numerical approach in terms of the Chebyshev polynomial of the second kind. The derivation of the mathematical model is quite general: one can solve for the unknown jump in potential across each flap for arbitrary configurations of the array. A wave farm consisting of various layouts of a finite array of OWSCs is then studied considering complete hydrodynamic interaction among all the devices.

The first theoretical model based on HIs was developed for an OWSC in a channel [13] and was then extended to study the hydrodynamics of an infinite array of WECs [14], a single device in the open ocean [15] and a finite array of inline converters [12]. Recently, the same method was also used to analyse the hydrodynamics of a flap-type device near a straight coast [16]. Following the same approach, in this study, we develop a mathematical model to investigate the hydrodynamic behaviour for the most generalized case consisting of a large number of OWSCs in any configuration with oblique wave incidence.

The generalized mathematical model is derived in the first part of the paper (§§2 and 3). In §4, the effect of the separation distance is studied in detail using three flaps. This is followed by an analysis of both a wave farm comprising 13 flaps in various possible arrangements and a wave farm of 40 inline flaps. Finally, the semi-analytical model is used to study the hydrodynamics of two devices located back to back—a configuration which has intrigued many (see [17]).

## 2. Mathematical model

### (a) Governing equations

A wave farm of M OWSCs is considered to be located in an ocean of constant water depth *h*′. Waves are incident from the right making an angle *ψ* with the *x*′-axis as shown in figure 2. The origin is located on the mean free surface with *y*′ the pitching axis of the flaps and *z*′-directed upwards. Primes in this mathematical model are used to denote the physical variables. With the assumption of irrotational flow and inviscid, incompressible fluid, the velocity potential *Φ*′ satisfies the Laplace equation
*f*′=(*f*′_{,x′},*f*′_{,y′},*f*′_{,z′}) is the nabla operator; subscripts with commas denote differentiation with respect to relevant variables. The linearized kinematic–dynamic boundary condition on the free surface gives
*g* is the acceleration owing to gravity, whereas the no-flux boundary condition at the sea bed yields
*c*′ above the seabed (see again figure 2). The WECs are modelled using a thin-rigid plate approximation (see [18]), and the kinematic boundary condition on their surface is then expressed as
*x*_{m}′ is the *x*′ coordinate of the centre of the *m*th flap, *y*′ coordinates corresponding to the two edges of the device and *H* is the Heaviside step function.

Like in previous work [6,12–16], a non-dimensional system of variables is chosen as
*w*′ is the length scale of the system (e.g. the width of the largest flap), and *A*_{I}′ is the amplitude of the incident wave. Assuming the oscillation of the flaps to be simple harmonic in nature, the time dependence of the variables can be separated out as
*Θ*_{m} are respectively, the angular frequency and amplitude of oscillation of the *m*th flap, whereas *ϕ*(*x*,*y*,*z*) is the complex spatial velocity potential. The spatial potential can, in turn, be resolved into
*ϕ*^{D} is the diffracted wave potential, *ϕ*^{(β)} is the unit radiation potential induced by the motion of the *β*th flap, whereas the other flaps are held fixed and *V* _{β}=*iωΘ*_{β} is the complex angular velocity of the moving flap. Also note, in (2.8), *k* is the solution to the dispersion relation *ϕ*^{(β,D)} denotes either potential, the linearized free-surface boundary condition
*β*=1,2,…,*M*, *δ*_{nm} being the Kronecker delta. Finally, both *ϕ*^{(β)} and *ϕ*^{D} are required to be outgoing disturbances of the wave field [19]. The vertical dependence can now be isolated out of the three-dimensional governing system (2.9)–(2.13) by using the separation (see [12,15,19]):
*κ*_{0}=*k* and *κ*_{n}=*ik*_{n} are the solutions of the dispersion relation

Using the decomposition (2.14) and the orthogonality relation, (2.16) yields a two-dimensional governing system for *β*th mode radiation potential is obtained as
*m*th flap, *U*_{p} is the Chebyshev polynomial of the second kind and order *p*, *y*-coordinate of the centre of flap *m*, whereas *b*_{p0m} are the complex solutions of a system of equations, again solved numerically. Note that in *ϕ*^{D} (2.23) only the zeroth-order vertical mode is present, the flaps being walled structures in the scattering problem (i.e *n*>0). Using the above expressions (2.22) and (2.23), the velocity potential is known in the whole fluid domain. It gives access to the flaps' hydrodynamic coefficients, which enables solving the equation of motion for the flaps.

### (b) Hydrodynamic parameters

The solution for the velocity potential is then used to solve the equation of motion of each individual flap in the frequency domain. Suppose for the *α*th flap, *et al.* [12]
*et al.* [12] for details). Note that in (2.25)–(2.26), the *α*th flap and, following [7], is set equal to the optimal PTO damping for the same OWSC isolated in the open ocean
*ν*^{open}_{α} are respectively the added moment of inertia and radiation damping of the *α*th OWSC isolated in the open ocean. According to the theory of damped oscillating systems (see [20]), the average extracted power by the wave farm over a wave period is
*q*, defined as the ratio of total power captured by an array of *M* flaps to the power captured by an isolated WEC of the same type multiplied by *M*
*q*>1 implies that there is a gain in the net power output from an array because of constructive interaction among the flaps. On the other hand, *q*<1 indicates that mutual interactions have a cumulative destructive influence on the array efficiency. However, the interaction factor *q* (2.30) does not quantify the performance of individual array elements. In order to understand the performance dynamics of each WEC in an array, Babarit [7] defined a term *P*_{m} is the power captured by the *m*th flap while *P*_{single} in the considered range of incident wave periods. The parameter *q* and

## 3. Algorithm implementation and computational cost

An algorithm based on the mathematical model described here has been implemented through a code written in Mathematica 8. The algorithm and the code have been made as general as possible and can handle a large number of flaps in any staggered configuration. The code requires no modification if the number of flaps or their configuration/positions are changed. Only the coordinates of the flap centres need to be changed. The other required inputs to the code are the flap width, distance from the sea bottom to the hinge, water depth, incident wave amplitude, range and number of incident wave periods, angle of oblique wave incidence, moment of inertia and buoyancy torque of the flap and total number of vertical eigenmodes, order of Chebyshev polynomials and terms in the remainder of the Hankel function (see (A 8)). A relative error of *O*(10^{−3}) is obtained with the first three vertical eigenmodes and sixth-order Chebyshev polynomials (*P*=6). From a computational point of view, the semi-analytical approach described here is extremely efficient compared with a full numerical approach. The latter has been used in Renzi *et al.* [12] to model a three-flap inline and a two-flap staggered configurations. The computational expense associated with the full numerical approach was on an average 1 h for a single wave period evaluation performed on a computer equipped with an i7 2.67 GHz CPU and 12 GB RAM. Computations with the semi-analytical model presented here were performed with an i7 3.40 GHz CPU and 16 GB RAM-equipped computer. For the assessment of a system of 13 flaps, only 6 min are required in average for each wave period.

## 4. Results

The computations are performed for several configurations of OWSCs, each one closely resembling the Oyster800 WEC developed by Aquamarine Power. The parameters of the system are reported in table 1.

In the following, we validate the computational model with available theoretical and numerical results. Then, we discuss the interactions arising in a simple three-flap cluster and further show the potential of the model in handling more complex and populated arrays.

### (a) Validation

The solution obtained for an inline array (*x*_{m}=0, *m*=1,2,…,*M*) of flaps and normal incidence (*ψ*=0°) is exactly the same as shown in Renzi *et al.* [12], and consequently the same results are obtained for the two-flap inline and three-flap inline cases as presented in Renzi *et al.* [12]. For staggered configurations, results for only two flaps are available in the literature and have been obtained with a numerical tool [12]. Figure 3 shows the variation of the excitation torque versus time period of the incident wave for the two-flap staggered case of Renzi *et al.* [12]. A fairly consistent agreement is observed in the results obtained by the current model and the numerical approach of Renzi *et al.* [12]. Small discrepancies can be observed at around 7 s which are likely due to the thickness-induced effect. The latter is modelled in the numerical model, but not in the semi-analytical solution presented here (see [12] for details).

### (b) Three-flap cluster

In order to understand the effects of the mutual interactions arising in a wave farm, we first consider a basic array cluster comprising only three flaps, and we focus our attention on the performance of the flap positioned centrally among them. This central flap in a way represents an OWSC located well within an array, where the hydrodynamic influences of only its two neighbouring devices are dominant. We consider both symmetrical and non-symmetrical configurations of the three flaps with essentially uniform spacing between them in normally incident waves.

Let us first consider the case of the symmetrical configuration shown in figure 4*a*. Here, the distance *d*′, measured from the central flap, is positive in the positive *x*′-direction. Therefore, *d*′>0 *m* represents the case when the central flap is located behind the two lateral flaps, whereas *d*′<0 *m* indicates otherwise. Figure 5 plots the *q*^{mod}_{2} of the central OWSC for various distances of separation. Each of the subplots shows the behaviour for a particular value of the lateral distance *b*′ while varying *d*′. It can be observed that *d*′>0 *m* is associated with a strong destructive influence on the central flap's performance across the entire operating range of periods. On the other hand, for *d*′<0 *m*, positive interaction effects dominate and significantly enhance the performance of the central flap, suggesting that an OWSC will have better power absorption characteristics when located at the front of the cluster. The most important thing to note is that for the situations considered here, the qualitative behaviour of the *q*^{mod} variation is determined by *d*′, whereas *b*′ primarily dictates the extent of the peaks (see again figure 5). In general, as the distance *b*′ is increased, there is a shift in the *q*^{mod} variations towards higher periods, accompanied by a reduction in the magnitude of the peaks, which means a decrease in the interaction among the flaps. It can be inferred that as *b*′ is further increased, there would be a larger number of local maxima and minima of reduced magnitudes and so on average, there would be no distinctive positive or negative interaction effect on the device performance for any value of *d*′.

Now, we consider the case where the layout of the flaps with respect to the centreline of the middle OWSC is non-symmetrical, as shown in figure 4*b*. The notable difference with the previous arrangement is that the pitching axes of the extreme flaps are now separated from that of the central flap in opposite directions. The *q*^{mod}_{2} variation of the central flap for the various cases is plotted in figure 6. Almost ubiquitously for the range of distances considered, such a configuration has a negative influence on the WECs performance. This is likely due to the opposite interaction effects on the central OWSC by the two lateral flaps.

### (c) Wave farm of 13 oscillating wave surge converters

A wave farm consisting of 13 flaps in various configurations is shown in figure 7. Typically, even larger arrays could be studied using the same computational infrastructure mentioned previously within a reasonable time. The spacing between the flaps is chosen similar to the one planned for the proposed wave farm at the Isle of Lewis in Scotland [1]. For the purpose of identifying each individual converter, the flaps are numbered in an increasing order from right to left of the array with the OWSC located on the extreme right considered as flap 1. The distance between the edges of the neighbouring flaps is 20 m in the *x*′-direction for all the configurations shown in figure 7, whereas the pitching axes of the neighbouring flaps in the staggered configurations are separated by a distance of 15 m. The wave farms considered in the analysis are symmetrical about the central flap (flap 7), so for normal wave incidence the hydrodynamic behaviour is symmetric with respect to the *x*′-axis passing through the centre of the central flap.

*Inline*: the inline case corresponds to the configuration in which the pitching axes of all the flaps are orientated along the same *x*′-coordinate. As first described by Renzi *et al.* [12], a near-resonant behaviour is observed in this case which is similar to the resonance of an infinite array of inline OWSCs [14] or a single OWSC in an open channel [13] (figure 8*a*). At the near-resonant period, the performance of every individual OWSC is higher than in the isolated case, and *q*^{mod} has a peak for all the flaps. However, such a behaviour is also accompanied by destructive influences at higher periods. Among all the flaps, the outermost OWSC has a slightly distinguishable behaviour from the others. This is due to the fact that while all the other OWSCs have neighbouring flaps on both sides which generate the maximum influence, the outermost flap only experiences the hydrodynamic influence of the converters located on one side. Let us now consider a case of oblique wave incidence on inline OWSCs. As expected, the behaviour of the wave farm is no longer symmetrical about its innermost flap. Figure 9*a*,*b* shows the *q*^{mod} of all the 13 flaps when *ψ*=30°. A similar near-resonant behaviour is observed in this case as well. However, the strongest near-resonant behaviour occurs for flap 1, and the magnitude of the peaks reduces as one moves towards the other end of the array, with flap 13 showing a distinctively different behaviour.

*S1*: in such a configuration, the OWSCs are placed in a zigzag manner with the array comprising two rows of devices. The flaps located in the same row have similar hydrodynamic behaviour, as seen in figure 8*b*. Flaps 3, 5 and 7, which are positioned in the front, have almost the same *q*^{mod} variation and similar are the behaviours of flaps 2, 4 and 6. However, the performance characteristics of the flaps in the two rows are in striking contrast, with the maxima in *q*^{mod} of the OWSCs in the front row corresponding to the minima of the OWSCs in the back row and vice versa. This happens, because a flap in the front row experiences the maximum constructive interaction, as already anticipated in the cluster analysis of §4*b*. Figure 10 plots the response amplitude operator (RAO) of the free surface elevation (|*ζ*′/*A*′_{I}|) for an incident wave period of 5 s in the region surrounding the wave farm. There is hardly any notable change in the wave field in front of the array, but, immediately, behind the first row, there is a reduction in the wave elevation, meaning a less energetic wave field available for extraction by the second row. At the back of the second row, the energy reduction is stronger, but limited in extent to the first 15 m. At further distance, the reduction is as significant as in the front of the first row.

*S2*: here the devices are again placed in a zigzag distribution, but now there are three rows in this configuration. Flaps 3 and 7 are located in front of the array and experience a beneficial influence owing to constructive interactions leading to relatively high values of *q*^{mod} (figure 8c). One can again note the similarity in the behaviour of the OWSCs in the second row (flaps 2, 4 and 6). Finally, flap 5, the only non-external flap to be located on the last row, has a predominantly negative *q*^{mod} factor. The behaviour is indeed similar to that obtained from the corresponding configurations of the three-flap cluster of §4*b*.

*S3*: the layout of this array resembles an inverted ‘V’ shape, pointing away from the coast. Figure 8*d* shows the *q*^{mod} variation of the flaps in such a configuration. The most striking behaviour is of the foremost WEC (flap 7). Indeed, one could have expected it to have a positive *q*^{mod} factor based on the behaviour observed in the cluster model of §4*b*. However, the magnification of the *q*^{mod} factor, in this case, is further enhanced by what we believe to be a strong focusing effect. In the S3 configuration (see again figure 7), all the flaps behind the central one reflect back some amount of incident wave energy. As a consequence, more energy is available for extraction by the foremost device (flap 7), resulting in the peak of the relevant *q*^{mod} in figure 8*d*. A further insight into such dynamics is offered by figure 11, which shows the excitation torque on the flaps in the S3 configuration. The variation of the excitation torque is similar to that of the *q*^{mod} factor of figure 8*d,* and one can note a sharp increase in |*F*′| for flap 7 at the same peak period (*T*′∼7.2 *s*). Such a behaviour again corroborates the well-known fact that the dynamics of the OWSCs such as Oyster is primarily torque-driven (see [6,13,15]).

*S4*: here again, the outermost flaps, which are located in the front, record the highest peak in the *q*^{mod} factor (figure 8*e*). However, although the configuration mirrors to the previous one, there is no such equivalent constructive focusing effect on the central flap (flap 7).

An overview of the general behaviours of all the systems is provided in figure 12. Here, the variation of the global performance parameter *q* (2.30) is plotted against the period of the incident wave. Overall, the strongest constructive interaction is achieved in the inline system, whereas the staggered systems S1 and S2 show the least constructive interference between the flaps, mainly owing to the poor performance of the back row because of the sheltering effect of the front row (figure 10). This confirms the earlier findings of Renzi *et al.* [12] for a smaller system. Finally, the configurations S3 and S4, for which the net power output is the same, show a smaller peak than the inline configuration, but an overall better performance according to the *q* indicator. It is worth mentioning that our analysis is based purely on the hydrodynamic performance of the system. Other aspects (environmental impact, site bathymetry, etc.) could of course orient the designer towards a less effective configuration from the hydrodynamic viewpoint. Nevertheless, such a hydrodynamic analysis is a first step towards the effective design of such a costly system.

### (d) Forty flaps inline

The proposed 40 MW wave farm off the northwest coast of Lewis, Scotland, is expected to have a deployment of around 40–50 Oyster devices on an approximate 3.2 km stretch of coast. In order to check the reproducibility of the results obtained from the small wave farm cases in such large configurations, a simulation of 40 OWSCs in a simple inline configuration is performed. The general geometry is considered to be the same as in the 13 flap configuration. In figure 12, the variation of the *q* factor for the 40 flap configuration is plotted. The behaviour is indeed similar to that of the 13 flap inline case, and the near-resonant behaviour is again confirmed with a slightly larger spike tending towards that of an infinite number of OWSCs (see again [12,14]). It can be reasonably inferred that the general behaviour in other configurations would be similar to that in the smaller wave farm case with sharper spikes and troughs.

### (e) Random seas

A random sea analysis is performed in this section for the most probable sea-state at the Isle of Lewis with the significant wave period *T*_{1/3}=8.24 *s* and the significant wave height *H*_{1/3}=1 *m* (obtained through personal communication with Aquamarine Power 2013) with normal wave incidence. The standard Bretschneider spectrum, described in Goda [21], is used to model the wave climate at the location. Computations are performed for five possible layouts of a 13 OWSC wave farm as shown in figure 7. Table 2 shows the *q*-factor for the various array configurations. The *q*-factor for all the layouts are found to be less than 1, which indicates that the effect of the interactions on the net power output from the array in random seas are destructive in nature for the spectrum considered. The inline configuration which records the highest peak in the *q*-factor in monochromatic seas (figure 12) has the lowest values in irregular waves, whereas S1, the least staggered configuration of all, achieves the best performance in random seas. Note, S1 has the smallest spikes in *q*, but, at the same time, the destructive influences on its cumulative performance are the least as well (see again figure 12). The other staggered layouts S2, S3 and S4 have lower values of *q* than that of S1. As already noted, the net power output from the S3 and S4 configurations are the same, which consequently results in identical values of the *q*-factor in random seas as well. It is worth recognizing that the performance in random seas is strongly dependent on the description of the incident wave spectrum. For the sea-state considered in this analysis, its peak period *T*_{p}=8.65 *s* (note: *T*_{p}=1.05*T*_{1/3}, see [21]), did not coincide with any peak of the interaction factor *q*. Although it is not a rule of thumb, such a co-occurrence can help at attaining *q*-factors greater than 1. For example, a *q*-factor of 1.027 is observed for the inline layout in a sea-state with *T*_{p}=5.7 *s* and *H*_{1/3}=1 *m*, which also concurred with the peak in the interaction factor. In general, the peak period of the spectrum can vary significantly throughout the year because of the seasonal variations and therefore for practical purposes, *S1* would be the ideal layout.

### (f) Two flaps back to back

Two flaps with their centres along the same *y*′-coordinate are studied here (figure 13). It is expected that such a configuration would result in strong hydrodynamic interaction between the two devices. Srokosz & Evans [17] were the first to analyse the behaviour of two top hinged independently oscillating rolling plates in deep water as a WEC. The novel concept motivated a few other studies [22], where one of the major drawbacks of such a system was identified to be its strong directional sensitivity to wave incidence and the concept was thereafter shelved. Surprisingly, the idea was not pursued in shallow waters where the waves are predominantly directional. In this study, we are going to explore whether it is wise to place two OWSCs back to back.

Figure 14 plots the behaviour of the excitation torque (|*F*′|), radiation damping (*ν*′), added inertia (*μ*′) and the performance indicator *q*^{mod} versus the non-dimensional parameter *kd* for *d*′=50 *m*. The qualitative variation of the hydrodynamic parameters of the front flap resembles that observed in the case of an OWSC in front of a straight coast [16]. In the latter, periodic occurrences of extremes are observed in the variation of the excitation torque, with the minima occurring at integer values of *kd*/*π*. In addition, sharp spikes are observed in the variation of the radiation parameters at values a little less than *m*=1,2,…. In the case of the two flaps analysed here, the hydrodynamic behaviour of the front OWSC is similar to that of a flap in front of a straight coast, with, however, reduced peaks and a shift where the extremes occur.

As far as the performance of the devices is concerned, the average value of *q*^{mod} of flap 1 is higher than that of flap 2 (figure 14*b*). The constructive interference effects on flap 1 are very strong at *kd*≈5, where *q*^{mod} almost reaches a value of 0.5. Flap 2 (back OWSC) always captures less power than a single isolated OWSC, which means that the interaction effects are always destructive on its performance. The primary reason for such a behaviour is that the back flap lies in the hind side of the front flap where the wave energy is reduced. Figure 15 shows the variation of *q* for various values of the distance *d*′. For *d*′=25 *m*, the destructive interaction effects are quite significant and *q*≈0.5 at an incident wave period of about 6 s. This means that the total power captured by the two devices combined at that frequency is equivalent to the energy extracted by an isolated single device. As the distance *d*′ is increased, the occurrence of the humps in the variation of *q* increases, but the magnitude of such deviations also reduces. The most important thing to note is that the constructive interference effects are much weaker compared with destructive influences, and on an average, the two OWSCs in such a configuration capture less power than two isolated WECs.

### (g) Two wave farms

In reality, an ideal wave energy site may encourage the deployment of two consecutive wave farms for energy harvesting. It is important to understand the dynamics of the system in such cases especially with one of the wave farm lying in the energy shadow of the other. A simplified case of two inline wave farm configurations, each comprising 13 flaps is considered in normal wave incidence (figure 16). The analysis is performed in constant water depth to understand the dominant interaction effects between the systems, although, in reality, variations in depth are expected to modify the behaviour slightly. The term *q*_{farm} is used to understand the effect of the interaction on each of the wave farms and is defined as
*P*_{farm} is the total power captured by a particular wave farm, whereas *P*_{farm isolated} is that by the same farm in an isolated environment. *q*_{farm}>1 would mean that the presence of the other farm has a net beneficial influence on power absorption characteristics of the particular wave farm considered, whereas *q*_{farm}<1 indicates otherwise. Figure 17 plots the variation of *q*_{farm} versus the incident wave period of the two wave farms for various distances of separation. The oscillatory behaviour of the *q*_{farm} factor is similar to that of the *q* factor observed in the two back-to-back OWSCs case (figure 15), with a higher number of such oscillations occurring for larger distances of separation. For the range of distances considered, the *q*_{farm} factor of wave farm 1 is always less than 1, which indicates that such configurations will tend to have a detrimental influence on the farm located nearer to the shore. However, a steady upward shift in the *q*_{farm} factor of wave farm 1 is observed as the distance is increased, which can be explained owing to the energy recovery in the rear side of wave farm 2 (see [7]). The rate of energy recovery is in fact quite slow and even for a distance of 2000 m, the *q*_{farm} factor is still below 1. On the other hand, wave farm 2 has both detrimental and favourable interference effects. However, the magnitude of the oscillations in its *q*_{farm} factor is much higher than that in wave farm 1. It is interesting that the bandwidths of the oscillations are almost the same for the distances considered.

## 5. Conclusion

A mathematical model based on the linear potential flow theory has been used to analyse the hydrodynamic interaction between multiple flap-type WECs in a wave farm. The semi-analytical model can efficiently solve a reasonably sized OWSC wave farm which otherwise is difficult to evaluate with a complete numerical approach. It is shown that the dynamics of each individual OWSC in the wave farms considered in the analysis strongly depends on its location in the farm, the wave frequency and the angle of oblique wave incidence. As the distance between the flaps increases, the mutual hydrodynamic interaction between them reduces and the behaviour of the converters tends towards that of an isolated device. However, from an economic perspective, one would want to maximize the number of devices at a particular wave farm location to extract more power. This is important for near shore devices such as OWSCs as the space would be strictly limited unlike for offshore converters.

Wave absorption by an array of 13 OWSCs is studied for some of its possible layouts. For an inline configuration with normal incidence, a near resonant phenomenon is observed which becomes stronger as the number of flaps is increased. However, for oblique wave incidence, there is a shift in the frequency of occurrence of this phenomenon with a slight increase in the resonant bandwidth associated with it. In a particular configuration of the large array (S3), a large enhancement in the performance of the front-most flap is observed. Such a behaviour is attributed to a sharp increase in the excitation torque owing to the focusing of waves by the other devices in the array. In general, the converters which are located in front of the array experience a notable positive interaction effect leading to a gain in their power capture. Such a favourable behaviour in the performance of the foremost devices of the array is also reported in the recent study of Borgarino *et al.* [11]. An irregular wave analysis for the most probable sea at the Isle of Lewis reveals that *S*1—the least staggered configuration—is a suitable layout for an OWSC wave farm.

In the case of two back-to-back OWSCs located close to each other, the effect on the performance of the back flap is found to be detrimental across its entire operating range, whereas the front OWSC experiences regions of both positive and negative influences. And when two such flaps are considered as one system, the destructive interference effects are found to be more important than the constructive influences. Therefore, such a system of two OWSCs is not recommended in reality. In addition, it is shown that a system of two consecutive wave farms has, in general, a negative interaction effect on the net performance of the wave farm located downstream.

In a practical wave farm design however, the layout of an array configuration could be constrained by bathymetry variations which would affect the optimization process. Although no particular layout could be suggested which would lead to a gain in net wave farm energy output across the entire operating range of the device, because the constructive interference effects are usually accompanied by destructive also influences, the study can help understand what sort of variability in the performance of individual OWSCs one can expect.

## Acknowledgements

The authors thank Science Foundation Ireland for the financial support to the research project (no. 10/IN.1/I2996) ‘High-end computational modelling for wave energy systems’. Dr G. Bellotti and Mr A. Abdolali are kindly acknowledged for the provision of the numerical data.

## Appendix A. Semi-analytical solution

The procedure to obtain the solution to the two-dimensional spatial diffraction and radiation potential is described here. Consider the two-dimensional Green's function
*φ*_{n} and *G*_{n} for the whole fluid domain yields
*φ*_{nm}=*φ*_{n}(*x*_{m}−*ε*,*y*)−*φ*_{n}(*x*_{m}+*ε*,*y*) denotes the modal potential difference across the two sides of flap *m*. Applying the two-dimensional spatial potential on the kinematic boundary conditions on the flaps, gives (see [12])
*y* coordinate of the centre of flap *m*, *mϵ*[1,*M*]. Making the following change of variables

Now
*J*_{1} is the Bessel function of first kind and first order, and *χ*=0.577215… is the Euler constant [13]. Expanding the unknown jumps in potential across the two sides of the flap as
*b*_{pnm} are unknown complex coefficients to be determined and *U*_{p}(*u*) is the Chebyshev polynomial of second kind, finally gives

In the case of an inline array configuration, *x*_{α}=*x*_{γ} and the term *D*_{pnαγ} indeed reduces to
*et al.* [12]. Further, generalizing it for normal incidence reduces the system of equations (A 10) exactly to (A 10) of Renzi *et al.* [12].

- Received February 11, 2014.
- Accepted March 31, 2014.

- © 2014 The Author(s) Published by the Royal Society. All rights reserved.