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Acta Mech. Sin. (2017) 33(4):647–662 DOI 10.1007/s10409-017-0683-6 REVIEW PAPER Analytical and computational modelling for wave energy systems: the example of oscillating wave surge converters 1 2 3 4 Frédéric Dias · Emiliano Renzi · Sarah Gallagher · Dripta Sarkar · 5 6 7 8 Yanji Wei · Thomas Abadie · Cathal Cummins · Ashkan Rafiee Received: 23 February 2017 / Revised: 25 March 2017 / Accepted: 17 April 2017 / Published online: 7 June 2017 © The Author(s) 2017. This article is an open access publication Abstract The development of new wave energy converters Keywords Wave energy · Wave energy converter · has shed light on a number of unanswered questions in fluid Slamming · Wave resource mechanics, but has also identified a number of new issues of importance for their future deployment. The main concerns relevant to the practical use of wave energy converters are 1 Introduction sustainability, survivability, and maintainability. Of course, it is also necessary to maximize the capture per unit area of Great prospects are offered by wave power devices for the structure as well as to minimize the cost. In this review, we the marine renewable energy sector. However, no well- consider some of the questions related to the topics of sustain- established wave energy industry is built anywhere in the ability, survivability, and maintenance access, with respect world at present. Ireland has the potential to become a world- to sea conditions, for generic wave energy converters with leading developer and manufacturer of the technologies that an emphasis on the oscillating wave surge converter. New will enable the harnessing of ocean energy resources. Since analytical models that have been developed are a topic of par- 2013, Science Foundation Ireland (SFI) has funded the ticular discussion. It is also shown how existing numerical MaREI Centre, which is a cluster of university and indus- models have been pushed to their limits to provide answers trial partners dedicated to solving scientific, technical, and to open questions relating to the operation and characteristics socio-economic challenges across the marine and renewable of wave energy converters. energy sectors. Earlier, from 2011 to 2016, SFI supported a research project led by University College Dublin, which focused on sustainability, survivability, and maintainabil- Frédéric Dias ity for generic wave energy converters (WECs) with an frederic.dias@ucd.ie emphasis on the oscillating wave surge converter (OWSC). School of Mathematics and Statistics, University College The project was undertaken in partnership with Aquamarine Dublin, MaREI Centre, Belfield, Dublin 4, Ireland Power Ltd. (APL), the company that developed the Oyster Department of Mathematical Sciences, Loughborough device. Unfortunately APL ceased to trade on 20 Novem- University, Leicestershire LE11 3TU, UK ber 2015. However, the project shed light on a number of Research, Environment and Applications Division, Met unanswered questions in fluid mechanics. Éireann, Glasnevin, Dublin 9, Ireland The aim of this review is to go through some of these Department of Engineering Sciences, University of Oxford, unanswered questions and the solutions that our group based Oxford, UK in University College Dublin provided over the period 2012– Advanced Production Engineering, University of Groningen, 2016. These questions fall into three major themes, which Groningen, The Netherlands range from local considerations for a single WEC, through Dublin City University, Glasnevin, Dublin 9, Ireland considerations on an array of WECs to finally considerations School of Engineering, University of Edinburgh, Edinburgh, on the global wave climate. UK Issues addressed within these themes include wave impact Carnegie Clean Energy Limited, Northam, Australia and pressure loads on a single WEC, interaction between 123 648 F. Dias, et al. Fig. 1 Artist’s sketch of the Oyster WEC concept waves and a single WEC, viscous and nonlinear effects, buoyant flap, hinged at the sea bed, whose pitching oscilla- slamming, device interactions for an array of WECs, opti- tions activate a set of double-acting hydraulic rams located on mal device spacing, wave climate prediction with improved the seabed that pump high pressure fluid ashore via a sub-sea coupling between wind modelling and wave modelling, pipeline, as shown in Fig. 1. The fluid flow is converted into and preferred geographical locations for nearshore WEC electric energy using a Pelton turbine. These bottom-hinged sites. devices are intended for deployment in the nearshore envi- The main sections of the paper are devoted to the three ronment, in relatively shallow water (ranging from 10 to 15 major themes (hydrodynamics and loading of one WEC; m). The Oyster WEC has a surface piercing flap that spans arrays of WECs; wave climate). In Sect. 2, we address the the entire water depth. survivability of ocean wave energy devices that are required Surface-piercing flap-type devices are designed to har- to operate in a harsh and violent environment, such as the vest wave energy in the nearshore environment. Established west coast of Ireland. They must be engineered in terms of mathematical theories of wave energy conversion, such as 3- their stability and structural strength to capture energy while D point-absorber and 2-D terminator theories, have proved operating under these extreme weather conditions. In Sect. 3, inadequate to accurately describe the behaviour of devices we address the efficiency of arrays of ocean wave energy like Oyster, leading to distorted conclusions regarding the devices. For wave energy to become commercially viable potential of such a concept to harness the power of ocean it is clear that WECs will have to be deployed in arrays. waves [1]. Accurate reproduction of the dynamics of Oyster We developed novel tools to provide a better understanding required the introduction of a new reference mathemati- of the behaviour of arrays of WECs. These tools allow an cal model, the “flap-type absorber”. A flap-type absorber is analysis of device interactions and of optimal device spac- a large thin device that extracts energy by pitching about ing for power production. In Sect. 4, we investigate the topic a horizontal axis parallel to the ocean bottom. It is now of wave climate from the perspective of wave energy, vital accepted that the wave capture rate is best for a wide flap- to address issues related to site selection, device and control type absorber [2] and the size of the flap drives, among specification, and access to maintenance of WECs. A detailed other factors, the capital expenditure (CAPEX): more mate- knowledge of the wave climate at the proposed deployment rial increases the CAPEX. One of the difficulties that led to sites is necessary not only for the capture of energy but also the failure of the Oyster WEC was its power take-off (PTO) for the maintenance of devices. Wave climate estimates rely system. The Finnish company AW-Energy is now work- largely on computer hindcast wind-wave models. We have ing on the WaveRoller WEC [3], with a supposedly better used improved models to obtain accurate annual wave cli- PTO than that of the Oyster WEC (see Fig. 2). Interestingly, mate predictions in terms of significant wave height, direction the first versions of the WaveRoller WEC were completely and mean wave period, with a focus on the nearshore wave submerged, which was not optimal from the power capture energy resource. perspective. Our group has collaborated closely with APL on the devel- The governing equations for the hydrodynamics of WECs opment of the Oyster WEC. The Oyster WEC comprises a are the continuity and Navier–Stokes momentum equations 123 Analytical and computational modelling for wave energy systems... 649 is driven by the strong exciting torque resulting from the pres- sure difference between its sides. Such a pressure difference originates because of the OWSC’s ability to favorably reflect, bend and shade waves in different areas of the surrounding Flap/powertrain PTO module module sea. This diffractive dynamics is much stronger than that for Power cable point absorbers, and, therefore, requires a non-point-absorber docking station explanation. Localised pressure points could compromise the struc- tural integrity of the device if not identified and factored WEC into the design. Details of pressure loads are, therefore, of Foundation foundation piles base great interest to the designer for both everyday wave climate conditions and in extreme/storm wave conditions. The lat- ter has a much more significant effect on device failure [6]. Fig. 2 Schematic drawing of the WaveRoller WEC concept As stated in Ref. [7], numerous coastal and marine struc- tures are damaged by wave action each year. The damage is with a free-surface. Typically, however, simplifications must often caused by the violent impacts of waves that are either be made to make the problem tractable. Such simplifications breaking or very close to breaking. Design formulae for esti- may be to consider only small perturbations to the free surface mating the magnitude of the impulsive pressures generated (linear waves), or to neglect the effects of viscosity by treating by breaking waves are presented, for example, in Ref. [8]. the fluid as an inviscid fluid. For the study of wave impact on These relationships are largely derived from the results of structures, one also needs to incorporate equations for fluid- laboratory tests rather than from an in-depth analysis of the solid interactions. One of the challenges, then, is to know fundamental mechanics. A review of the more theoretical which effects are important for the physical phenomenon one aspects of wave impacts on walls is provided in Ref. [9]. wishes to describe. Standard computational fluid dynamics Experimental scale model testing can assist with some of (CFD) tools are not suitable for farms of WECs, since typical these issues, but the results are much more uncertain under CFD computations require several hours of CPU time for a extreme wave conditions due to scale effects. One of the single wave period. At present, they can only be useful at most common difficulties of conducting experiments with the local level, for example, to understand the loads and the WECs is the presence of scale effects: the hydrodynamics viscous effects on a single WEC. requires different model scales and the influence of the vari- In the concluding Section, we will learn from the lessons ous effects is difficult to infer from small-scale experiments. of the past and give suggestions for the way forward. This makes numerical modelling a particularly valuable tool in the development of WECs. For OWSCs, inspiration was found in a field apparently disconnected from the field of 2 Survivability of wave energy systems wave energy: the transport of liquefied natural gas (LNG) in LNG carriers, where extreme loading (slamming) can occur Early research and development studies of WECs focused and damage the LNG tanks in extreme sea states. For nearly mainly on floating devices like point- and line-absorbers 10 years now, the LNG community has been developing a mathematical and computational modelling framework as [4]. Point absorbers are devices with dimensions that are much smaller than the incident wavelength (e.g., a heaving well as an experimental framework for the analysis of wave impacts arising in LNG carriers [10,11]. At least six physical buoy), while line absorbers have one dominant horizontal dimension, with an order of magnitude that is at least one phenomena were shown to be of importance during impact, wavelength (e.g., an articulated raft) [5]. Line absorbers can ranging from hydroelasticity and liquid/gas compressibility work either as terminators or attenuators, depending on their to interfacial instabilities and phase transition, with change alignment being, respectively, orthogonal or parallel to the in fluid momentum being the most important one. Both in direction of propagation of the incident waves. However, the ocean and in laboratory experiments, it has been observed driven by the need for more powerful WECs to decrease that OWSCs can pitch seaward violently before the next wave energy production costs, the wave energy sector has evolved crest arrives (see Fig. 3). We will see in Sect. 4, on the mod- towards the design of new large-scale WECs, which do not elling of the wave climate, that highly energetic sites off the west coast of Ireland are prone to slamming of OWSCs. For belong to the point- and line-absorber categories, namely Oscillating Wave Surge Converters such as the Oyster or the OWSCs, wave impacts on the flap during extreme sea states could have serious structural implications, which need to be WaveRoller devices. A key driver of the structural design of such devices is the pressure distribution and loading induced quantified. In particular, the following questions need to be on the flap by the incident waves [1]. ThepitchofanOWSC addressed: What are the local forces and pressure distribu- 123 650 F. Dias, et al. a e Wave b f c g d h 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 (m/s) Fig. 3 Typical slamming event of an OWSC observed both in numerical simulations (left column) and in laboratory experiments (right column). The wave propagates from left to right. The numerical results are colored by the velocity magnitude. The time difference between the first frame a and the last one h is 0.72 s. From Ref. [12] tions across the flap during wave impact from a large wave? for mapping the pressure on the flap. This pressure map is What are the local forces and pressures induced by the inci- useful to gain insight into the slamming phenomenon. dent waves at key geometric points on the flap structure? The first 3-D experiments, conducted in Queen’s Univer- What is the effect of entrained air pockets between the flap sity Belfast in collaboration with APL using a small-scale structure and the colliding wave? model of Oyster (1/25), revealed high impulsive loads, We addressed the impact question through computational associated with slamming events [13]. This study allowed and experimental modelling at the local level of a single identification of the different steps in a wave cycle leading to WEC. such slamming events, namely: (1) the flap is pushed towards the beach as the wave crest approaches; (2) the flap oscillates back seawards in the trough of the wave with the free sur- face lowering on the front face of the flap until it reaches a 2.1 Experiments maximum dry-out when the flap is almost vertical; (3) the flap re-enters the water with a high angular velocity and the The slamming of OWSCs was first observed during a set water level starts moving up the flap; until (4) the water is of 3-D experiments [13]. However, it was very difficult to ejected at the top of the front face and the cycle starts again. understand the characteristics of the violent flow near the flap. These observations showed that the impact is dominated Therefore it was decided to switch to 2-D experiments. As by the re-entry of the flap in the wave trough and that it opposed to 3-D experiments which are difficult to visualize, is the flap that impacts the wave rather than a classical wave 2-D experiments provide a better view to capture images of impact. The slamming of OWSCs has, therefore, been related the slamming process. Moreover, 2-D experiments are easier 123 Analytical and computational modelling for wave energy systems... 651 PT4 1.5 0.5 Pressure sensor array Immersion 17 sensor gauges locations front (IG01) −0.5 (PS01-PS17) and back 0 5 10 15 20 (IG02) Time (s) Fig. 4 Experiments in Ecole Centrale Marseille: small-scale model (1/40) of the Oyster WEC, location of pressure sensors and immersion gauges. Pressure signal at one pressure transducer (PT4) showing the impulsive load every period jr Fig. 5 Sketch of the free surface and the evolution of a jet during slamming. Pressure footprint on the front face of the flap as a function of time, following, with a slight delay, the immersion gauge measurement of the tip of the jet (red line). The oscillations, which are visible between time t = 40.65 s and t = 40.7 s and between 5 cm and 10 cm from the hinge, are typical of compressible air pocket oscillations to water entry problems. The subsequent 2-D experiments, the impact is not always well represented by the dynamic where the free surface is more easily tracked, showed sim- pressure (1) and similarities with flip-through impacts [19] ilar trends thereby confirming the earliest observations (see have been observed. In addition, although only bubble clouds Fig. 3). In the first set of 2-D experiments, conducted in Ecole and no big air pockets are observed (see Fig. 3), oscillations Centrale Marseille [12,14], only one pressure sensor was at the frequency of an air pocket can be seen in the pressure available, so a new set of experiments was conducted with a signals (see Fig. 5). These oscillations raise certain questions new small-scale Oyster model (1/40) [15] (Fig. 4). regarding the effects of compressibility that cannot yet be The 2-D experiments emphasized the importance of the answered for this type of impact. development of a jet in the pressure distribution along the flap, similar to a Wagner-type impact [16–18]. The pres- 2.2 Numerical simulations sure at a given transducer reaches a maximum when the jet root (defined in Fig. 5) passes in front of the sensor and the In the process of developing the numerical tools, several dif- maximum pressure at that instant can be estimated from the ficulties were encountered; a major one was the handling of jet-root dynamic pressure to a high degree of accuracy [15]: the mesh during large motions of the flap. One of the goals of the numerical simulations was to provide a full description p = ρu , (1) max of all forces acting on the device in extreme waves: inertial jr forces, drag forces, hydrodynamic radiation and diffraction where ρ is the density of water and u is the jet-root velocity effects. Scaling issues become more uncertain in extreme jr calculated from post-processed images. During a slamming wave conditions. event, the impulse pressure propagates towards the top of the CFD methods can take into account nonlinear effects nat- flap along the jet root (see Fig. 5). urally, e.g., flow separation, turbulence and wave impact, Even though the propagation of the jet is very similar to which may be important for predicting the hydrodynamic what happens in water entry problems, the initial stage of forces. CFD also can provide comprehensive flow details, and Distance from hinge (mm) Pressure (kPa) 652 F. Dias, et al. t t t t t t t t allow simulations at various scales, for various device shapes A 0.35 and wave conditions. These features make CFD an attractive method for studying wave-OWSC interaction problems. To 0.3 handle the large motions of the flap, there are three com- P1 0.25 putational techniques commonly employed: moving mesh P2 methods, fixed mesh methods, and meshless methods. The P3 0.2 P4 guideline for selecting a particular approach is that its algo- P5 0.15 rithm should be accurate, robust, and computationally inex- P6 P7 pensive. In Refs. [12,20], dynamic mesh methods were used 0.1 to describe the flap motion and investigate the viscous effects 0.05 and slamming on an OWSC. In Ref. [21], a model based on the immersed boundary method was developed to investigate 35.2 35.3 35.4 35.5 35.6 35.7 35.8 35.9 36 wave interactions between waves and a modular OWSC. In t (s) Ref. [13], the smoothed particle hydrodynamics (SPH) 0 0.5 1.0 1.5 2.0 method was used to investigate 2-D and 3-D slamming on Pressure (kPa) an OWSC (see below). Another difficulty is the high com- Fig. 6 Time histories of pressure on the seaward face of the flap during putational cost of slamming simulations in 3-D. In order an impact event, obtained by numerical simulations to reduce the computational cost while avoiding the re- reflection at the outer boundary, a “wavemaker-less” model with “relaxation zones” was developed to investigate the 3-D effects of wave slamming on an OWSC [22]. In addition, a computational nodes to simulate the domain; hence, they can hybrid model combining a Boussinesq model (FUNWAVE) model problems with large deformations such as wave inter- and a finite-volume model was proposed to simulate, at actions with OWSCs. Our group used the SPH method, which affordable computational cost, some of the conditions expe- is a meshless, purely Lagrangian technique originally devel- rienced by the full-scale Oyster 800 device, incorporating oped in 1977 [24–26]. It has subsequently been successfully real bathymetry at the deployment site [23]. employed in a wide range of problems [27]. With the numerical models, we first checked whether vis- In SPH, moving nodes (carrying field variables such cous effects played a role or not in the flow around the as pressure and density) are defined as the “particles”and device (turbulence, vortex shedding) [20]. Intensive simu- advected with the local velocity. Since the fields are defined lations demonstrated that vortex shedding from the flap is a only at a set of discrete points, to ensure differentiability, short-lived, periodic phenomenon, and that viscous scaling a continuous field is defined by interpolation kernels. In the effects are not an important issue for OWSCs. The con- weakly compressible SPH formulation (WCSPH), the fluid tinuous time–space distribution of the pressure on the flap is assumed compressible with a large sound speed (such that surface [12] demonstrated the “slosh-type” character of the the Mach number M ≈ 0.1 and the density of the fluid impacts and indicated the location of the strongest impact typically varies by less than 1%). The SPH method uses pressure on the flap (Fig. 6). In addition, the CFD results smoothing kernels to express a function in terms of its values helped us to understand the re-reflection effects in the 2- at a set of disordered points. The smoothing kernel function D slamming experiments. The main discovery was that the (or weighting function) specifies the contribution of a typical slamming intensity could be enhanced or suppressed due to field variable, A(r ), at position r in space. the re-reflection, depending on the wavelength and the dis- In order to model wave interactions with an OWSC, the tance between the wavemaker and the flap. Simulations of SPH particles were initially placed on a grid of squares with 3-D slamming events [22] showed the difference between 2- initial spacing of l = 0.033 m resulting in a total number of D and 3-D slamming. In 3-D slamming, water re-entry begins 3, 264, 668 particles [13,28]. The SPH smoothing length was at the sides and focuses into the centre, thus enhancing the set to h = 1.5l and the boundary particles were placed with a impact pressure there. spacing of l /3. Like the finite-volume simulations described It was assumed that the flow was incompressible, not- above, the SPH simulations were performed on ICHEC’s ing that the experiments did not show much evidence of (Irish Centre for High-End Computing) Stokes supercom- compressible effects. The elasticity of the WECs was not puter, which is an SGI Altix ICE 8200EX cluster with 320 considered either. Wave impact is such a complex problem compute nodes. The SPH simulations presented here used 72 that it is not possible to consider all phenomena together. processors and took ∼70 h for 13 s of physical simulation We also tested meshless methods. In contrast to classical time. methods such as finite-volume methods, meshless methods Figure 7 illustrates the simulation output of the entire wave do not need any grid or connectivity constraint between the tank. Waves progress from left to right past the OWSC at the y (m) Analytical and computational modelling for wave energy systems... 653 the cost of CFD would be prohibitive for arrays. In the next Section, we review alternative methods to model arrays. 3 Arrays of wave energy converters The commercial feasibility of wave energy demands a mod- elling environment that extends to multiple WECs. A single WEC, with a capacity comparable to a classic power plant (400 MW, say) is technologically impossible. Therefore, arrays of WECs, placed in a geometric configuration or farm, are needed. In a farm, WECs interact and the overall power absorption is affected. Determination of the optimal pattern of WECs in order to maximise power absorption is of major importance in the design of a wave farm, and this pattern would be expected to depend on the specific wave climate experienced at the site of interest (see Sect. 4). The funda- mental modelling of arrays of WECs, which is based on linear wave theory, was presented almost 40 years ago [29,30]. A comparison of the multi-scattering method, the plane wave method and the point-absorber approximation was presented in Ref. [31]. Analytic expressions for wave absorption by a periodic linear array were derived in Ref. [32]. A numeri- cal code based on the boundary element method (BEM) was used in Ref. [33] to study the impact on the absorbed wave power of the separation distance between two WECs and the wave direction. BEMs are powerful and used extensively in the study of floating bodies. However, they are computation- Fig. 7 SPH simulation of wave interaction with an OWSC. Particles ally expensive. An acceleration of the BEM code by a fast are coloured by their pressure, blue being low and yellow being high multipole algorithm was presented in Ref. [34]. However, the pressure. For clarity the OWSC position is highlighted by a red circle results were mixed because of the slow convergence of the expansions for large wavenumbers. centre of the images. It is crucial that simulations accurately We could also have used a BEM code given that our group predict the motion of the flap before any comparison of pres- has developed an efficient 3-D BEM code over the years sures is made. The simulated time variation of the flap angle [35,36]. However, we felt that the mixed results obtained in shows good agreement with experimental data (Fig. 8). Refs. [34,37] were not the best route to follow for the time Figure 9 presents the time history of the pressure exerted being. We also thought of Boussinesq modelling. For exam- on two pressure transducers located on the OWSC. The ple, a Boussinesq code with rectangular bottom-mounted model predictions are in good agreement with the experimen- (surface-piercing) structures has been used successfully in Ref. [38]. But the inclusion of structures in Boussinesq codes tal data. Slight discrepancies between numerical estimations of the maximum pressure peaks and the experimental data remains challenging. Instead, we decided to rely on analyti- are due to the well-known stochastic nature of wave impacts cal methods. Since no linear model existed for OWSCs, we which leads to scatter in the experimental results. In both had to derive such a model from first principles. cases, SPH simulations were capable of predicting the sharp pressure peaks. However, these peaks are stochastic, and 3.1 Mathematical model of a single wave energy therefore, their exact location in time and their peak value converter are not repeatable in the experiments. Therefore, slight dis- crepancies between the numerical values and experimental A mathematical model has been developed to study the data are to be expected. behaviour of an OWSC in a channel, noting that, during The 2-D and 3-D CFD simulations and experiments have laboratory tests in a wave tank, peaks in the hydrodynamic greatly improved the understanding of slamming and viscous actions on the converter occurred at certain frequencies of effects in a situation where only one OWSC is present. Next, the incident waves. This resonant mechanism is known to we must consider arrays of OWSCs. However, it is clear that be generated by the transverse sloshing modes of the chan- 123 654 F. Dias, et al. modes to the geometry of the device. Our analytical results agree very well with available experimental observations, see Refs. [39,40]. Our model shows that the initial two- dimensional motion of the incoming waves in the channel shatters into a series of three-dimensional sloshing waves. Each sloshing mode resonates at a specific wavelength λ , namely -20 -40 λ = b/n, n = 1, 2,..., SPH Experiment -60 024 6 8 10 where b is the channel width. We showed that when a slosh- Time (s) ing mode resonates, its energy becomes trapped near the flap and the system efficiency increases, up to 80% for a sim- Fig. 8 Comparison of numerical and experimental time histories of rotation angle of the OWSC ple device similar to the Oyster WEC. The model was then modified to deal with an OWSC in the ocean (no channel) 0.020 [41]. We showed that the behaviour in the ocean is substan- tially different from the one in the channel, because of the 0.015 different patterns of the radiated waves in the open ocean configuration. The results showed that the influence of the 0.010 lateral walls in the channel resulted in a 10% increase of the 0.005 wave torque acting on a device resembling Oyster 1, with respect to the open ocean scenario. This shows that extra 0.000 care should be taken when one uses the results obtained in a SPH wave tank to predict the behaviour of the OWSC in the open Experiment -0.005 024 6 8 10 ocean. A detailed analysis of the channel effect revealed that Time (s) a blockage ratio greater than 20% could significantly affect the performance of the device in the channel with respect to 0.030 its behaviour in the ocean [41]. The capture factor of a single 0.025 Oyster WEC has been given for six different sea states at the 0.020 European Marine Energy Centre (EMEC) test site [42]. 0.015 3.2 Model for arrays of wave energy converters 0.010 0.005 The modelling of wave farms falls within a framework that 0.000 must include wave interactions, wave reflection and diffrac- SPH Experiment -0.005 tion phenomena. We analysed the interaction of waves with 024 6 8 10 an array of WECs and determined the performance of the Time (s) array with respect to identifiable design parameters. In the Fig. 9 Comparison of the time history of pressure variations simple case (analytically convenient) of an inline periodic between numerical and measured values at two sensors on the flap array, comprising an infinite number of OWSCs, the rele- (1 bar = 100 kPa) vant parameter of interest is the single spacing parameter [39,40]. The first mathematical model for a finite number nel. However, the extent to which resonance would affect of OWSCs was restricted to an inline layout of up to three the behaviour of the device was not known. We developed a OWSCs [43], and it was shown that the inline configura- semi-analytical model to better understand the effect of such tion (see Fig. 11a) exhibited near-resonant behaviour, similar resonant peaks on the power production of the device. The to the resonant characteristics of an OWSC in a channel geometry is shown in Fig. 10. Within the framework of a which can enhance the performance at specific frequencies. linear inviscid potential-flow theory, application of Green’s Later, a semi-analytical approach was developed for an array theorem yields a hypersingular integral equation for the with an arbitrary number of OWSCs and arbitrary layouts. velocity potential in the fluid domain. The solution is found in This model facilitated the analysis of practical layouts of terms of a fast-converging series of Chebyshev polynomials WECs, and it was shown that a staggered configuration can of the second kind. The physical behaviour of the system was lead to a better performance than an inline one in random then analysed, showing sensitivity of the resonant sloshing seas. Pressure (bar) Pressure (bar) Angle (deg) Analytical and computational modelling for wave energy systems... 655 Fig. 10 Geometry of the 3-D analytical model of an OWSC. The incident waves are travelling from right to left.FromRef.[1] Arrays of flap-type systems have also been investigated in Fig. 11a) is 1-D with uniform spacing and, therefore, has other contexts, e.g., Venice gates (see Refs. [44,45]), and the one design parameter, while a staggered array (see Fig. 11b) possibility of using the latter for harnessing energy has been is symmetric and uniformly spaced with two design param- recently explored [46]. In some special designs of WECs, eters. A realistic array (see Fig. 11c) could have an arbitrary an individual device can itself comprise an array of smaller arrangement with constraints imposed by operational, eco- oscillating components. For example, the novel modular nomical and natural (bathymetric) limitations. We introduced OWSC concept [47] consists of multiple flap-type compo- a new approach for array optimization based on machine nents. The motivation for such a design was to address a learning techniques [50], which, to our knowledge, enabled shortcoming in the original OWSC configuration in the form optimization of large arrays for the first time. The main idea of large wave loads acting on its bottom foundation. The idea is to develop a cheap surrogate model for the performance is to distribute the wave loading on several flaps to mitigate function of the array, which then is used for optimization. The the detrimental effects. However, the new design leads to method first uses a statistical emulator [51], based on Gaus- more complex hydrodynamic interactions amongst the indi- sian processes (see Ref. [52]), to predict the performance vidual flaps of the device. We identified multiple resonating in small clusters. The original array is then formulated in frequencies of the system which could lead to large oscilla- terms of small clusters and a meta-model is derived for the tions of the flaps. We observed that the performance of the whole array. The high dimensional optimization is then per- device is strongly dependent on the PTO characteristics of formed using a custom genetic algorithm. The simplification the individual flaps due to the nature of the hydrodynamic of interactions is facilitated by an important, yet practical, interactions. The new concept provides flexibility in terms assumption that any particular WEC is largely influenced by of tuning the individual components of the system. However, only its nearest WECs. We optimized layouts for 40 WECs they need to be optimized together in order to maximize the under different constraints, but the approach would work power captured by the device as a whole. For PTO char- equally well for even larger arrays. The performance of arrays acteristics (per unit width) similar to that of the original of Oyster WECs has been evaluated for the most probable design, the total powers captured by the two systems are sea-state at the Isle of Lewis in Scotland [43,50] —see Sect. 4 comparable. on wave climate. A key challenge in the planning of wave energy farms is the identification of optimized array layouts. Although 3.3 Semi-analytical model for a single wave energy several mathematical models are available for the evalua- converter including viscous effects tion of hydrodynamic interactions, their computational costs are prohibitive for optimization purposes. The existing work Semi-analytical models [39,41,43,50,53] neglect the effects on array optimization was limited to small arrays [48,49], of viscous dissipation. However, experimental wave tank and in most cases constrained by symmetrical layouts and/or tests and CFD simulations have shown that flow separation uniform spacings, which reduced the dimension of the occurs at the edges of OWSC flaps [20]. Due to the time- optimization problem. For example, the inline layout (see consuming nature of wave tank testing and CFD simulation, 123 656 F. Dias, et al. Fig. 12 Dissipative surfaces: A bottom-hinged OWSC in a water depth Fig. 11 Finite array of OWSCs. a Inline layout (2-D symmetry and uni- h , hinged at a depth d beneath the water’s free surface, with dissipative form spacing); b staggered layout (1-D symmetry and uniform spacing); surface D extending from the OWSC’s edges. a Side, b front, and c c a realistic wave energy farm layout plan view. Waves are incoming in the negative x direction [62] drag laws in place of the standard Morison (quadratic) it is inefficient to use these tools to quantify the effect of drag law [54]. In each of Refs. [56–58,63], it is assumed this viscous dissipation on the hydrodynamic performance that the pressure drop across the screen/breakwater due of OWSCs in varieties of sea conditions and device orienta- to viscous effects is a linear function of the local flow tions. velocity. An alternative to CFD is to use a BEM approach com- More recently, in Ref. [64], the effect that viscous dis- bined with a Morison-type drag law [54]. Such an alternative sipation has on an OWSC is examined by modifying the is suited to bodies whose characteristic dimension w is semi-analytical theory of Renzi and Dias [39] to include the small compared to the incident wavelength λ , i.e., for small effects of viscous dissipation near the edge of the flap. This diffraction parameters: Kl = 2πw /λ 1; in addition, the is achieved by applying an effective pressure discharge Keulegan–Carpenter number KC = 2π A /w 1, where A is the amplitude of the incident wave, should be large. Δ P = f (v ), Generally speaking, the use of Morison’s equation is permit- ted when KC > 6, and Kl < 1[55]. Typically, neither of these hold in the current mathematical models of flap-type in the vicinity D of the edges of the flap, where Δ P denotes WECs, with Kl = O(1) and KC 1 due to the assumption the difference in the pressure P from the left to the right of linearity. side of the flap/dissipative surface in the wave direction (see Another alternative is to modify the inviscid theory in Fig. 12). The equation of motion of the flap is then solved in regions of the fluid domain where the effects of viscous dis- the frequency domain, and the solution is used to conduct sipation are non-negligible [56]. In the case of an OWSC, a parametric analysis of an OWSC for a variety of envi- this is near the edges of the flap [20]. Such an approach was ronmental conditions and device dimensions. We conclude adopted in Ref. [56] in a study of the free surface in a moon- that the effects of dissipation are to reduce the peak values pool. In Ref. [56], a control surface was defined from the of the hydrodynamic quantities, and that the dependence of moonpool’s sharp edge down to the seabed, across which a the hydrodynamic quantities on the dissipation is generally pressure discharge is imposed. The pressure discharge law weak when considering the environmental conditions typ- assumes a functional relationship between the pressure drop ically experienced by existing OWSC designs. The effects and the local flow velocity, which characterises the effects of dissipation are strongest near peaks in the hydrodynamic of dissipation. It is shown that such an approach eliminates quantities and for long-period waves. The effect of dissipa- unphysical spikes in the resonant free-surface of the moon- tion is negligible for short-period waves. The conclusion in pool (predicted by inviscid theory). Ref. [64] that viscous drag is more important for narrow flaps, The pressure discharge law typically takes the form of a and that the effects are amplified for long-period waves is in linear [56–58] or quadratic [59] function of the local flow agreement with existing numerical and physical modelling velocity. In addition, an effective linear law may be used data [20]. in place of the nonlinear one using the Lorentz principle The 2-D and 3-D analytical and computational mod- of equivalent work [60]. In the past, numerical models of els have greatly improved our understanding of arrays of OWSCs have used effective linear [54,61] and quasi-linear OWSCs. However, arrays must be placed in wave energetic 123 Analytical and computational modelling for wave energy systems... 657 areas and at the same time be accessible for maintenance. In We validated recent versions of WAVEWATCH III [4.18] the final section, we review wave climate assessment with the along the west coast of Ireland where wave data are avail- aim of addressing a range of critical dependent issues related able from several buoys. Of interest also is the question to wave energy applications. of whether the use of full-spectral third-generation wind- wave models such as WAVEWATCH III is sufficient for nearshore wave prediction, or if it is necessary to couple such spectral wave models with shallow-water type models 4 Wave climate assessment for wave energy in very shallow water where wave transformations and wave breaking are important. There is no consensus at the present systems off the coast of Ireland time (see for example Ref. [79]). However, recent improve- It is essential to understand the wave resource for at least ments in numerical wave models such as the development four reasons: (1) one needs to know what the average wave of better numerical methods, the inclusion of currents, and power is in the area where one wants to deploy WECs; water levels, and the better parameterization of nearshore (2) one needs to know the waves in more detail if one wave processes have enabled the increasingly accurate mod- wants to use control to optimize the efficiency of the WECs elling of coastal regions (see for example Ref. [80], or the [65–70]; (3) one wants to know when access to the WECs WAVEWATCH III Development Group [75]). We developed will be possible in case maintenance is needed [71,72]; and fruitful collaborations between atmospheric modellers and wave modellers. Indeed, the choice of wind forcing, such as (4) one wants to ensure the WECs can survive the more extreme wave conditions expected in the area of deployment. from the European Centre for Medium-Range Weather Fore- casts (ECMWF), is as important as the choice of the wave Ocean waves are created by wind and then propagate freely in the ocean, forming swells. When they enter coastal waters, model. the limited water depth affects the amplitude and direction of Met Éireann (the Irish Meteorological Service) developed ocean waves. Surprisingly, wave properties in shallow water techniques to produce optimal wind forecasts for driving are not yet fully understood. Better descriptions and measures a regional version of the WAVEWATCH III model, and to of local fluid particle velocities in shallow water waves, as improve the prediction of wind and wave conditions in the well as increased knowledge on waves close to breaking, are nearshore. The combination of state-of-the-art atmospheric needed [61]. Over the past two decades, the development of and wave models provided enhanced wave climate predic- increasingly accurate and efficient numerical models of non- tions, with an emphasis on preferred geographical locations for OWSCs in Ireland. In Ref. [72], a 14-year hindcast was linear surface waves has been a continuous challenge to the ocean and coastal engineering communities. Many types of carried out to create a wind and wave atlas for Ireland. wave and storm surge models have been developed to rep- The winds were dynamically downscaled from ERA-Interim resent conditions under major storms, either for real-time reanalysis to a 2.5 km horizontal resolution and 65 vertical forecasting or later hindcasting. These models are theoret- levels using the HARMONIE-AROME configuration of the ically applicable—given suitable forecast wind fields—for shared ALADIN-HIRLAM system (HARMONIE-37h1.1) spatial scales spanning at least four orders of magnitude: from [81,82]. For the wave hindcast, we used WAVEWATCH III ocean basin scales (thousands of kilometres) down to coastal on a triangular unstructured grid with resolution ranging scales (hundreds of metres). In practice, however, the com- between 10 km offshore and 225 m in the nearshore, forced by the downscaled HARMONIE 10 m winds and ERA-Interim putational efficiency of existing models severely limits the range of spatial scales accessible [73]. Moreover, given mod- wave spectra. The wind and wave hindcasts were thoroughly vali- els with fixed parameters may perform well on a given storm and not so well in other cases. There are a variety of classical dated against available buoy data, including wave buoys in spectral wave models available to scientists and engineers, nearshore locations and coastal synoptic stations. Significant including SWAN (Simulating WAves Near-shore) [74] and wave heights (H ) and winds from hindcasts were compared WAVEWATCH III [75] in the spirit of the WAM model [76]. against altimeter data from the CERSAT database at Ifremer. In these models, one solves the random phase spectral action An improvement in the wind and wave validation compared density balance equation for wavenumber-direction spectra. to an ERA-Interim driven hindcast in Ref. [77] was found, The implicit assumption of this equation is that properties of particularly in coastal regions where the orographic affects the medium (water depth and current) as well as the wave of bays, islands and coastline features were more accurately resolved. field itself vary on time and space scales that are much larger than the variation scales of a single wave. The models are The study examined the complementarity between the wind and wave energy resource around the coast of Ire- being constantly improved, and so we used the state-of-the- art versions to assess the nearshore wave resource of Ireland land. Joint wind and WEC farms could remove some of [71,72,77,78]. the high frequency variability of these renewable energy 123 658 F. Dias, et al. Fig. 13 Complementarity of the wind and wave energy resource on the 30 m and 60 m bathymetric contours around Ireland, adapted from [72]. a Mean annual wind power at the 100 m height level (W/m ); b mean annual wave power flux (kW/m); c correlation coefficient between the Fig. 14 Seasonal weather window analysis for accessibility on the wind and wave energy 30 m and 60 m bathymetric contours around Ireland. Average of the maximum waiting time for a weather window of at least 12 h duration satisfying the following criteria: (1) wind speed is less than 16 m/s; sources, which can create problems integrating these energy (2) H is less than 2 m; and the peak wave period is less than 13 s. JJA— June, July, August; MAM—March, April, May; DJF—December, sources into the power grid. This is the case, for exam- January, February; SON—September, October, November ple, when there are energetic waves but little wind or vice versa. The focus was on the complementarity without focus- ing on any particular technology, to find suitable locations for joint wind-wave farms. This could improve the viabil- (see Sect. 2), and, consequently, damage or destroy WECs. ity of future WEC deployments in such nearshore regions. Ireland has a long history of extreme waves [83]. H val- Wave and wind hindcasts were interpolated to points on the ues of over 15 m are regularly recorded by the Irish Marine 30 m and 60 m bathymetric contours, as can be seen in Buoy Network, by buoys located off the west coast of Ire- Fig. 13. The lower the level of correlation, the greater the land. Several maximum individual wave heights (trough to complementarity between the two resources. Along the west- crest) greater than 20 m have also been measured by the buoy ern seaboard, the lower correlation values indicate a higher network. Recent studies have emphasized how extremes vary occurrence of swell waves not generated by the local wind spatially off the west coast [84,85]. Large scale atmospheric conditions (i.e., higher complementarity for a joint wind- oscillations or teleconnections, such as the North Atlantic wave farm), than on the eastern seaboard, where wind-seas Oscillation, can also influence the likelihood of extreme wave dominate. events occurring, and more generally, the seasonal wave Another important concern for marine operations is site climatology and energy extraction potential of the North accessibility for installation and maintenance. In order to Atlantic for WECs [77,86,87]. assess weather windows around the coast of Ireland, and When planning long-term WEC installations, one must gain insight into the kind of operational planning required also consider the potential impact of global climate change to maintain such WECs, we created sample criteria (for a on the marine resource. An ensemble of wave climate pro- generic vessel) to estimate accessibility on the 30 m and jections was carried out for Ireland to investigate how the 60 m bathymetric contour [72]. As can be seen in Fig. 14, waves, wind climate and storm tracks over Ireland and the long waiting times were found for accessibility to sites in North Atlantic might change towards the end of the cen- winter along the western seaboard (>40 days in some north– tury [78,88], driven by EC-Earth wind and ice-fields [89]. west regions)—unfortunately, a common experience for most Although a small overall decrease in the mean annual H marine operators. was found, evidence for changes in wave extremes were less A major limitation for the development of WEC farms is robust—indicating that access for operational maintenance the issue of survivability. A combination of large swell and/or and survivability will continue to be an issue for WEC instal- extreme locally-generated sea waves could lead to slamming lations into the future. 123 Analytical and computational modelling for wave energy systems... 659 Another issue of importance for future deployments of WECs is the inclusion of future wave information [66]. This is a key issue for the real-time control of WECs. Short- term wave forecasting is still a largely open question, even if progress has been made recently [67,68]. One of the pressing questions is: How are free-surface evolutions best synthe- sized from wave spectra for power production assessment [69,70]? Acknowledgements The project was funded by the Science Founda- tion Ireland (SFI) under the research project “High-end Computational Modelling for Wave Energy Systems” (Grant SFI/10/IN.1/12996) in collaboration with Marine Renewable Energy Ireland (MaREI), the Fig. 15 An array of three CETO-5 units in operation. The CETO WEC SFI Centre for Marine Renewable Energy Research (SFI/12/RC/2302). concept is developed by Carnegie Clean Energy D. Sarkar acknowledges support from EPSRC through Project Grant EP/M021394/1. We also acknowledge support from the Sustain- able Energy Authority of Ireland (SEAI) through the Renewable Energy Research Development & Demonstration Programme (Grant RE/OE/13/20132074). The ESB, Met Éireann, the Marine Institute and Shell provided the buoy data for validation. The INFOMAR bathymet- 5 Concluding remarks ric datasets were provided by the Geological Survey Ireland (GSI) and the Marine Institute. The VORF software for tidal datum conversions Wave energy is still at its infancy. However, remarkable was obtained from the GSI. The UKHO bathymetry was provided by progress has been made in the development of analytical OceanWise Ltd. The authors thank the ECMWF for providing the ERA- Interim Re-analysis data. The altimeter derived wave data was obtained and computational models for wave energy systems. Lessons from the Centre ERS d’Archivage et de Traitement (CERSAT), at Ifre- must be learned from the recent failures of WECs and WEC mer, Plouzané, France in the frame of the Globwave project, funded by companies. Fortunately, a few companies over the world are the European Space Agency (ESA). The authors thank Dr. C. Sweeney and Prof. P. Lynch (UCD School of Mathematics and Statistics) for making progress towards making wave energy a reality. In the helpful discussions, Dr. F. Ardhuin (Ifremer) for his advice regarding introduction, we mentioned AW Energy, which is develop- the WAVEWATCH code, Dr. K. Doherty and Dr. A. Henry (Aquama- ing the WaveRoller OWSC. Another company is Carnegie rine Power) for providing useful information about the Oyster WEC, Wave Clean Energy Ltd, which is developing the CETO Dr. O. Kimmoun (Ecole Centrale Marseille) for his help with the 2-D experiments. Finally, the numerical simulations were performed on the WEC. CETO is a fully submerged point absorber device that Stokes and Fionn clusters at the Irish Centre for High-end Comput- converts ocean swell into zero-emission renewable power ing (ICHEC) and at the Swiss National Computing Centre under the and desalinated freshwater. Extensive numerical studies have PRACE-2IP project (Grant FP7 RI-283493) Nearshore wave climate been carried out on the CETO device. A prototype scale test analysis of the west coast of Ireland. of three of CETO units was installed recently and operated Open Access This article is distributed under the terms of the Creative along the west coast of Australia as part of the Perth Wave Commons Attribution 4.0 International License (http://creativecomm Energy Project (PWEP) (see Fig. 15). ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, Future WEC deployments will be required to survive in and reproduction in any medium, provided you give appropriate credit harsh ocean environments. Recent developments in WAVE- to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. WATCH III concerning an improved modelling of extremes [90] will further enable an understanding of the most extreme operational wave loads. References Recent work has shown that WAVEWATCH III spectra can be successfully coupled to the full Navier–Stokes (or 1. 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