Numerical Modelling of Wave Energy Converters: State-of-the-Art Techniques for Single Devices and Arrays

Numerical Modelling of Wave Energy Converters: State-of-the Art Techniques for Single WEC and Converter Arrays presents all the information and techniques required for the numerical modelling of a wave energy converter together with a comparative review of the different available techniques. The authors provide clear details on the subject and guidance on its use for WEC design, covering topics such as boundary element methods, frequency domain models, spectral domain models, time domain models, non linear potential flow models, CFD models, semi analytical models, phase resolving wave propagation models, phase averaging wave propagation models, parametric design and control optimization, mean annual energy yield, hydrodynamic loads assessment, and environmental impact assessment.

Each chapter starts by defining the fundamental principles underlying the numerical modelling technique and finishes with a discussion of the technique’s limitations and a summary of the main points in the chapter. The contents of the chapters are not limited to a description of the mathematics, but also include details and discussion of the current available tools, examples available in the literature, and verification, validation, and computational requirements. In this way, the key points of each modelling technique can be identified without having to get deeply involved in the mathematical representation that is at the core of each chapter.

The book is separated into four parts. The first two parts deal with modelling single wave energy converters; the third part considers the modelling of arrays; and the final part looks at the application of the different modelling techniques to the four most common uses of numerical models. It is ideal for graduate engineers and scientists interested in numerical modelling of wave energy converters, and decision-makers who must review different modelling techniques and assess their suitability and output.

• Consolidates in one volume information and techniques for the numerical modelling of wave energy converters and converter arrays, which has, up until now, been spread around multiple academic journals and conference proceedings making it difficult to access
• Presents a comparative review of the different numerical modelling techniques applied to wave energy converters, discussing their limitations, current available tools, examples, and verification, validation, and computational requirements
• Includes practical examples and simulations available for download at the book’s companion website
• Identifies key points of each modelling technique without getting deeply involved in the mathematical representation

• Language: English
• Publisher: Academic Press
• Release date: Jun 14, 2016
• ISBN: 9780128032114

Book preview

Kingdom

Chapter 1

Introduction

M. Folley    School of Planning, Architecture and Civil Engineering, Queen's University Belfast, Belfast, Northern Ireland

Abstract

The challenge of developing numerical models for wave energy converters (WECs) is significant because of the wide range of WEC technologies that currently exists. This challenge is increased because the literature pertaining to the numerical modelling of WECs is spread over a range of publications that are often difficult or expensive to obtain. This book is intended to aggregate the information on the numerical modelling of WECs into a single resource. The history of the numerical modelling of WECs is relatively short, starting in 1974. Over the subsequent 40 years the range of techniques for the modelling of WECs and WEC arrays has slowly increased to provide the range of options that are currently available. The book consists of four parts. The first part contains details of numerical models whose hydrodynamics are based on linear potential flow, the most common type of numerical modelling techniques for WECs. The second part contains details of other modelling techniques used for individual WECs that are not based on linear potential flow theory. The third part deals with the modelling of arrays of WECs, whilst the fourth part considers how these models may be used in the design of WECs and wave farms. The contents of this book are the result of a large amount of collaborative effort and support, which is acknowledged.

Keywords

Wave energy; History; Future; Challenge; Motivation; Acknowledgements

1.1 The Challenge of Wave Energy

The potential for extracting and using the energy in ocean waves has been recognised for at least 200 years, with the first patent for a wave energy converter (WEC) being submitted by Monsieur Girard and his son in 1799. In this patent, energy was extracted from the waves by resisting the heaving motion of a ship using a lever mounted on the dockside. Undoubtedly, this idea would have worked if it had ever been constructed, but Monsieur Girard and his son would have lacked the numerical tools to estimate the power generation with any accuracy. Up to the 1970s, designs for WECs continued to be proposed, and some prototypes were even constructed at the beginning of the 20th century. However, in general these designs could be considered as fruits of intuition and empirical research, unsupported by any numerical analysis.

This chapter first provides a short history of the numerical modelling of WECs before looking at the current challenges and future developments in the field. The chapter then discusses why the book has been written and how it should be used. Finally, the chapter finishes with acknowledgement of the collaborative effort that has made this book possible.

1.2 A Short History of the Numerical Modelling of WECs

It was not until after the first oil crisis in 1974 that serious scientific attempts were made to numerically model the response of WECs and estimate their potential power capture. Although the first article on the potential for wave energy is generally attributed to Salter (1974), the fundamental theory for WECs was first produced independently by Evans (1976), Mei (1976) and Budal (1977). This theory was then effectively used over the next five years to develop numerical models of WECs and WEC arrays in the frequency domain (Chapters 2 and 8) and time domain (Chapters 3 and 8), as well as semianalytical methods for modelling arrays of WECs (Chapter 9).

During the next 15–20 years, up to 1997, numerical models of WECs and WEC arrays continued to be developed, but without any significant development in the types of modelling techniques used. Towards the end of this period sufficient computing power became available that the hydrodynamic coefficients for arbitrary shapes could be developed, the first example of this being the model of an oscillating water column (OWC) by Lee et al. (1996). However, this only increased the scope and accuracy of the possible numerical models rather than representing a fundamental development. Up to this time all of the models of WECs and WEC arrays had been based on linear potential flow theory; then in 1997 a model of a WEC based on fully nonlinear potential flow theory (Chapter 5) was published by Clément (1997), which coincidentally was also a model of an OWC.

The next significant advancement in the modelling of WECs came in 2004 with the use of a computational fluid model (CFD) of a WEC (Chapter 6) that solved the incompressible Euler equations for the flow around an OWC (Mingham et al., 2004). However, although other CFD models were also developed around that time (Alves and Sarmento, 2005), it was not really until 2016, with the increase in computing power, that the production of CFD models of WECs became more common-place. It would seem that this has been enabled significantly by the availability of the open source software OpenFOAM (www.openfoam.com), which allows developers to share their code and advancements, an advantage that was not previously available, as each developer worked on their own particular software tool.

2007 saw, to the author’s knowledge, the first implementations of WECs in wave propagation models that would allow the far-field effect of WECs and WEC arrays to be determined. Millar et al. (2007) produced the first example of a phase-averaging wave propagation model to include WECs (Chapter 11), whilst Venugopal and Smith (2007) produced the first example of a phase-resolving model to include WECs (Chapter 10). The representation of WECs in a phase-resolving model was subsequently improved by Beels and Troch (2009) to enable the modelling of array interactions in 2009, whilst Silverthorne and Folley (2011) did the same for phase-averaging models in 2011.

Most recently two additional modelling techniques for WECs have been developed. The first of these techniques, spectral-domain modelling (Chapter 4), was first implemented by Folley and Whittaker (2010), whilst the second of these techniques, model identification (Chapter 7), was first implemented by Davidson (2013). Both of these modelling techniques are focused on achieving a computationally more efficient WEC model, rather than increasing the model fidelity.

1.3 Current Challenges and Future Developments

The current key challenge in the numerical modelling of WECs (one that it is hoped this book will go some way in meeting) is to identify which, from the wide range of modelling techniques available, is most appropriate for a particular WEC concept and modelling objective. One reason for this wide range of potential modelling techniques is that there is a wide range of WEC concepts with very different sizes and operating principles, with each concept making potentially different demands on the modelling technique. A second reason is that the modelling of WECs is a relatively new field, compared to many other fields such as naval architecture, and so there is no canon of modelling techniques that have been publically acknowledged as acceptable by the wave energy community. Consequently, anyone new to wave energy is likely to find it difficult to determine how best to numerically model their particular WEC.

Intimately linked to the challenge of identifying an appropriate modelling technique is model validation. Fundamentally, without validation of a model it is difficult to fully assess its accuracy and true suitability. Unfortunately, at this point only a few WECs have been deployed at full scale and no, or very limited, data is publically available from these deployments that could be used to provide validation of a numerical model. An alternative potential source of validation data is wave-tank testing, with the clear understanding that scaling issues mean that some differences may exist between this data and what would be expected at full scale. However, even wave-tank data suitable for the validation of numerical models is relatively rare, with only a small number of cases being published. Moreover, the author is unaware of any comparative analysis of the validation of numerical models to determine their relative fidelity for a particular WEC and wave condition. It is clear that a lack of rigorous and critical validation of the numerical models limits our current ability to fully assess the potential and relative merits of the different numerical modelling techniques available. This makes it a clear and present challenge for the numerical modelling of WECs.

Considering future developments in the numerical modelling of WECs, it is obvious that the seemingly unrelenting increase in computing power will have an impact. The particular areas where this is likely to be significant is in the development of CFD models and the assessment of the mean annual energy production (MAEP), both of which are generally limited to some extent by the computational resources available to the modeller. However, whilst greater computing power may be expected to provide an incremental advancement in the numerical modelling of WECs, it is not considered likely to result in any kind of step change. It would seem that whenever computing power increases, the expectations of the numerical models also increase, whilst many of the underlying issues with the individual numerical modelling techniques will remain the same.

An exciting future development in the numerical modelling of WECs is likely to be the production of hybrid models that use the best elements from separate models and combine them to produce a higher fidelity or more efficacious model. This trend has already started with the inclusion of WECs in wave propagation models (Babarit and Folley, 2013); however, more hybrid models are to be expected. The key challenge with these hybrid models is to configure the inputs and outputs from each of the models so that there is a seamless transfer of information in both directions. For example, bidirectional coupling could be used to model a WEC array, with a CFD model that defines the local flow around each WEC, whilst a (non) linear potential flow model is used to propagate the waves between the WECs. This hybrid model may be expected to accurately model any flow separation around each WEC, whilst minimising the issues with numerical diffusion that can occur with the propagation of waves in a CFD model. Moreover, it would be expected that the model would be computationally less demanding. Of course, the concept of hybrid models is not novel; however, the author is unaware of their use in the modelling of WECs (although by the time you are reading this book, hopefully this will no longer be the case).

1.4 Why This Book

Currently, other than attending a specialist course on the modelling of WECs, the only option available to someone new to wave energy is to work through the current literature in the field. However, this wave energy literature is extremely dispersed, being contained in the proceedings of specialist conferences such as the highly recommended European Wave and Tidal Energy Conference (EWTEC) series and in academic journals such as Applied Ocean Research and Ocean Engineering. This complicates the work of studying the numerical modelling of WECs as these papers can be difficult and costly to obtain. A second issue with the current literature is that it can be opaque and difficult to interpret. This is because in many cases the literature describes ongoing research that may be incomplete or not fully validated. In the context of the conference in which the work was presented this may be acceptable; however, it can significantly complicate interpretation for a reader who did not attend the conference, or may be reading the paper a number of years after its publication. In extremis, the current wave energy literature can appear contradictory. Whether the literature is actually contradictory, or simply appears to be due to differing contexts, is to some extent irrelevant because for someone new to the field it is impossible to tell the difference.

This book has been written and compiled in response to all of the points discussed here. Thus, it is intended that this book will provide a single compendium of the techniques currently used for modelling WECs. Necessarily, if only to limit its size, it does not go into the intimate details of each method, but confines itself to a clear exposition of the different modelling techniques available. Thus, the focus is on the fundamental characteristics of each technique, together with its inherent limitations. From this it is anticipated that the reader will be able to assess whether the technique is suitable for the particular WEC and modelling objective they are interested in. However, each chapter contains extensive references that can be used by the reader for further investigation once the modelling technique of interest has been identified. In addition, the book does not go into detail in the modelling of components of WECs that are adequately covered in other texts, even though they may have a large influence on performance, but limits itself to how models of these components may be incorporated in a WEC model. Thus, the highly complex field of modelling moorings is not included in this book, nor does it include the modelling of pneumatic, hydraulic or electrical machines that may be used to extract power from a WEC; the reader is directed to the many books that already exist on these subjects where more in-depth information can be found. Consequently, the book can focus specifically on the modelling of WECs, which allows the text to be more succinct and hopefully more readable.

It may reasonably be questioned why WECs need to be treated differently to other marine structures, for which a number of books on the subject already exist. From a fundamental perspective it is true that a WEC can be considered as simply another marine structure; however, the modelling objective for WECs is generally very different. Specifically, it is typically the power extracted from the waves that is important in modelling a WEC, whilst for a general marine structure the structural responses or forces are typically required. Moreover, it is common in the modelling of marine structures to either assume that they are large with small motions so that a potential flow model can be used, or small with larger relative motions so that the Morison equation can be used. Neither of these assumptions is typically reasonable for WECs that are normally relatively large with relatively large motions. Notwithstanding this difference, all of the modelling techniques may also be suitable for modelling marine structures. However, the important distinction in the text is that in this book the focus is on their suitability for modelling WECs, rather than their general application.

1.5 How to Use This Book

This book is separated into four parts.

• Part A deals with the modelling of WECs where the hydrodynamic forces are based on linear potential flow, which is the most common method used for modelling the WEC hydrodynamics. Specifically, this part contains chapters on the frequency-domain modelling of WECs, the time-domain modelling of WECs and the spectral-domain modelling of WECs. Currently, WEC models whose hydrodynamics are based on linear potential flow probably represent over 90% of models produced. However, their ubiquity should not disguise the fact that they have a number of limitations that need to be recognised.

• Part B deals with the modelling of WECs using techniques other than linear potential flow. Thus, this part contains chapters on nonlinear potential flow models, computational fluid dynamics models and model identification. Naturally, there is no strong link between these chapters except that they describe models that are not based on linear potential flow.

• Part C deals with the modelling of arrays of WECs. This part contains chapters on the semi-analytical modelling of WEC arrays, the modelling of arrays as an extension of techniques used for single WECs, WEC array modelling based in phase-resolving wave propagation models and WEC array modelling based on phase-averaging wave propagation models. The modelling of WEC arrays is particularly important for large scale development of wave energy as it is required not only for the prediction of a wave farm’s energy yield, but also for assessing what the environmental impact may be of large-scale extraction of wave energy.

• Part D deals with the use of the WEC modelling techniques described in Parts A to C to achieve particular modelling objectives. This part contains chapters on the control of WECs, the calculation of the mean annual energy production (MAEP), the estimation of the structural loading and the modelling of the environmental impact of WECs. A particular focus of the chapters in this part is the identification of the appropriate WEC modelling technique for the particular modelling objective.

As much as is reasonably possible, all the modelling technique chapters in the book have the same layout. Thus, they typically start with an introduction to the particular modelling technique that contains details of its fundamental principles followed by the application of the technique to modelling WECs. At the end of each chapter there is typically a section on the limitations of the modelling technique and a bullet-point summary of the chapter.

This book has been written so that each chapter can be read largely independently of all the other chapters. However, there are clearly a number of links between the different chapters and these links are referenced in the text where appropriate. Although the chapters in the book can be read in any order it is anticipated that the order in which the chapters are read will depend on the perspective of the reader. Three distinct perspectives have been identified—the model specifier, the model developer and the model interpreter—although it is likely that any reader will actually have a combination of one or more of these perspectives.

A model specifier will typically know their modelling objectives and so is most likely to be a developer that is looking to identify the modelling technique that is most appropriate for the current challenges in their current device or project development. It is anticipated that this type of reader will start in Part D of the book, where the applications of the modelling techniques are detailed, and then refer to the appropriate models in Parts A to C as required. Conversely, it is expected that a model developer, most likely to be an engineer or computer programmer, will start with the chapter on the particular modelling technique that they intend to use in Parts A to C of the book and then subsequently look at Part D when considering how this model may be applied to achieve a particular modelling objective. Finally, a model interpreter, who is likely to be an assessor or potential investor, is likely to read both the chapter on the modelling technique used, in Parts A to C, and also the relevant chapter in Part D on the application to the particular modelling objective.

1.6 Acknowledgements

The production of this book has been a collaboration between many of the currently leading experts in wave energy. The knowledge that these experts have developed in this field over the years is a result of significant and sustained support from a range of sources that deserve acknowledgement for their contribution to wave energy in general and this book in particular.

Specifically, Chapter 3 has been largely based on research material previously developed by the author when working at Instituto Superior Técnico supported by the EC WAVETRAIN Research Training Network Towards Competitive Ocean Energy, contract No. MRTN-CT-2004-50166 and subsequently at Tecnalia Research and Innovation partially funded by the Department of Industry, Innovation, Commerce and Tourism of the Basque Government (ETORTEK Program). Although not directly involved in the writing of the Chapter 3, the author recognizes the contribution arising from fruitful collaboration with António Falcão, Jean-Baptiste Saulnier, Joseba Lopez Mendia and Imanol Touzon among others. Then Chapters 4 (Spectral Domain Models) and 8 (Conventional Multiple Degree-of-Freedom Array Models) have been produced by authors who were supported by the UK Engineering and Physical Science Research Council under the SuperGen Centre for Marine Energy Research project (grant EP/I027912/1).

The writing of Chapter 7 (Identifying Models Using Recorded Data) was supported by a project funded by Enterprise Ireland (Irish Government and the European Union under Ireland's EU Structural Funds Programme 2007–13) under Grant EI/CF/2011/1320, and Science Foundation Ireland under Grant No. 13/IA/1886.

The work presented in Chapter 10 (Phase-resolving wave propagation array models) has been supported by the PhD funding grant of Dr. Vasiliki Stratigaki by the Research Foundation Flanders (FWO), Belgium. Furthermore, part of the work presented in this chapter has been supported by the FWO research project 3G029114. The experimental data used for validating numerical methods presented in this chapter have been obtained during the ‘WECwakes’ project, supported by the European Community's Seventh Framework Programme through the grant to the budget of the Integrating Activity HYDRALAB IV within the Transnational Access Activities, Contract no. 261520, and by the Research Foundation Flanders (FWO)-Contract Number FWO-KAN-15 23 712N.

The writing for Chapter 12 (Control optimisation and parametric design) was supported by a project funded by Enterprise Ireland (Irish Government and the European Union under Ireland's EU Structural Funds Programme 2007–13) under Grant EI/TD/2009/0331, the Irish Research Council, and Science Foundation Ireland under Grant No. 12/RC/2302 for the Marine Renewable Ireland (MaREI) centre.

Finally, Chapter 15 (Environmental impact assessment) was written by authors supported by the UK Engineering and Physical Research Council grant EP/J010065/1, together with welcome contributions from Graham Savidge, formerly of Queen’s University Belfast.

References

Alves M., Sarmento A. Non-linear and viscous diffraction response of OWC wave power plants. In: 6th European Wave and Tidal Energy Conference, Glasgow; 2005.

Babarit A., Folley M. On the modelling of WECs in wave models using far field coefficients. In: 10th European Wave and Tidal Energy Conference, Aalborg, Denmark; 2013.

Beels C., Troch P. Numerical simulation of wake effects in the lee of a farm of wave dragon wave energy converters. In: 8th European Wave and Tidal Energy Conference, Uppsala, Sweden; 2009.

Budal K. Theory for absorption of wave power by a system of interacting bodies. J. Ship Res. 1977;21(4):248–253.

Clément A.H. Dynamic nonlinear response of OWC wave energy devices. Int. J. Offshore Polar Eng. 1997;7(2):264–271.

Davidson J. Linear parametric hydrodynamic models based on numerical wave tank experiments. In: 10th European Wave and Tidal Energy Conference, Aalborg, Denmark; 2013.

Evans D.V. A theory for wave-power absorption by oscillating bodies. J. Fluid Mech. 1976;77(1):1–25.

Folley M., Whittaker T. Spectral modelling of wave energy converters. Coast. Eng. 2010;57(10):892–897.

Lee C.H., Newman J.N., et al. Wave Interactions With an Oscillating Water Column, Los Angeles, CA, USA, vol. 1. Golden, CO: Int Soc of Offshore and Polar Engineers (ISOPE); 1996.82–90.

Mei C.C. Power extracted from water waves. J. Ship Res. 1976;20:63–66.

Millar D.L., Smith H.C.M., et al. Modelling analysis of the sensitivity of shoreline change to a wave farm. Ocean Eng. 2007;34(5-6):884–901.

Mingham C.G., Qian L., et al. Non-Linear Simulation of Wave Energy Devices. Cupertino, CA: ISOPE; 2004.

Salter S. Wave power. Nature. 1974;249:720–724.

Silverthorne K., Folley M. A new numerical representation of wave energy converters in a spectral wave model. In: 9th European Wave and Tidal Energy Conference, Southampton, UK; 2011.

Venugopal V., Smith G.H. Wave climate investigation for an array of wave power devices. In: 7th European Wave and Tidal Energy Conference, Porto, Portugal; 2007.

I

Wave Energy Converter Modelling Techniques Based on Linear Hydrodynamic Theory

Chapter 2

Frequency-Domain Models

M. Alves    WavEC - Offshore Renewables, Lisbon, Portugal

Abstract

This chapter discusses the frequency-domain modelling approach to model wave energy converters (WECs), and presents its advantages and limitations. It is shown that frequency-domain modelling requires the linearization of the forces acting on the WEC. This simplification is acceptable for waves and small-amplitude device oscillatory motions and whenever the mooring system can be modelled by a linear spring and the power take-off (PTO) using either a linear damper or a linear spring-damper system. Under these circumstances, the first step to model WECs is typically based on a frequency-domain approach, where the excitation is assumed to be of a simple harmonic form. Accordingly, all the physical quantities vary sinusoidally in time with the same frequency of the incident wave. Therefore, in the frequency domain the equations of motion become a system of algebraic linear equations that may be solved straightforwardly. However, this requires the calculation of the hydrodynamic coefficients, which can be produced relatively easily using a linear potential flow solver and boundary element methods (BEMs). Hence, frequency-domain models are relatively fast and so widely used to get a first insight on response and power capture of WECs. Nevertheless, despite the advantages of frequency-domain models, it is important to take into account that these models may be relatively inaccurate for large waves, at frequencies close to resonance or where viscous forces are significant.

Keywords

Frequency-domain; Linear theory; WEC; Diffraction; Hydrostatic; Excitation; Radiation; Power take-off; Mooring; Damping; Added mass; Power capture; BEM

2.1 Introduction and Fundamental Principles

The hydrodynamic interaction between wave energy converters (WECs) and ocean waves is a complex high-order nonlinear process that, under particular conditions, can be simplified. This is true for waves and small-amplitude device oscillatory motions. In this case the hydrodynamic problem is well characterized by a linear approach, which in general is acceptable throughout the device’s operational regime (Falnes, 2002). In addition, whenever a linear representation of reactive forces, such as moorings (using a linear spring) and power take-off (using either a linear damper or a linear spring-damper system) is used, the original nonlinear WEC dynamics are completed described by linear equations. This means that the superposition principle applies¹ (Denis, 1973), and linear combinations of simple solutions can be used to form more complex solutions.

Under these circumstances, the first step in modelling the WEC dynamics is traditionally carried out in the frequency domain, where the excitation is of a simple harmonic form. Accordingly, all the physical quantities vary sinusoidally in time with the same frequency of the incident wave. Therefore, the inhomogeneous equations of motion become a system of algebraic linear equations that may be solved straightforwardly. The main challenge in a frequency-domain analysis is the determination of the radiation and excitation loads on the captor (the body or bodies that interact directly with the waves). This typically relies on the application of boundary element methods (BEMs) (also referred to as boundary-integral equation methods (BIEMs) or panel methods) to estimate the hydrodynamic coefficients of added mass and damping and the excitation force per unit incident wave amplitude.

The BEM is used widely in computational solutions of a number of physical problems such as acoustics, stress analysis and potential flow. In wave-structure hydrodynamic interactions the fundamental basis of this method is a form of Green’s theorem, where the velocity potential at any point on the body wetted surface is represented by distributions of singularities (sources or dipoles) over the body discretised surface (Newman, 1977; Linton and McIver, 2001). This leads to an integral equation that must be solved for the unknown source strength or dipole moment. Global quantities, including the hydrodynamic added mass and wave radiation damping coefficients and exciting force components, can then be obtained from the velocity potentials (Hess and Smith, 1994) for specified modes, frequencies and wave headings.

Accurate numerical approximations of the free-surface Green function, which are valid for all ranges of frequency and water depth, were developed by Newman (1985, 1992). Based on this development, and with the use of an iterative method of solution of the linear system developed by Lee (1988) and Korsmeyer et al. (1988), it is possible to determine the hydrodynamics of complex offshore structures. The numerical methodologies of Newman and Lee led to the development of the frequency-domain, free-surface, radiation/diffraction code WAMIT (Lee and Newman, 2013), which has been widely used for offshore and naval problems, wave-structure interaction and wave energy conversion. Subsequently, other BEM codes, such as ANSYS-AQWA (http://www.ansys.com/Products/Other+Products/ANSYS+AQWA), Moses (http://www.ultramarine.com) and the open source code NEMOH (http://www.lheea.ec-nantes.fr/cgi-bin/hgweb.cgi/nemoh) dedicated to the computation of first-order wave loads on offshore structures, have been developed. These codes have been essential, and typically the first step, in the evaluation of WEC technologies, due to their satisfactory accuracy and relatively low computation effort.

Since the 1980s a wide range of WECs with different working principles have been modelled using linear potential theory and BEM codes. In 1980, Standing (1980) predicted the power absorption efficiency and the reaction forces of a submerged pitching ‘Duck’. In 1992, Pizer numerically modelled a pitching device called the Salter's Duck (Pizer, 1993). Later, Yemm et al. (1998) and Pizer et al. (2000) modelled the Pelamis wave energy converter, a hinged attenuator concept. In 1998, Brito-Melo et al. (1998) presented an adaptation of the BEM code AQUADYN (Delhommeau et al., 1992) to study the dynamic behaviour of oscillating water columns (OWCs). Babarit et al. (2005), Josset et al. (2007), and Ruellan et al. (2010), modelled the SEAREV, a floating oscillating body completely enclosed with an internal moving mass. Moreover, Folley et al. (2007a,b), and Renzi and Dias (2012) modelled a concept similar to Oyster, a bottom-hinged flap device. Farley et al. (2011) modelled the flexible Anaconda device, a submerged flooded rubber tube aligned with the predominant wave direction. Furthermore, Babarit et al. (2012) developed a numerical benchmark of a wide range of WECs.

The aforementioned list of works of modelling studies is nonexhaustive, since nowadays the first step in modeling WECs is almost universally a frequency domain analysis and the application of BEM codes. A more comprehensive review on the use of BEM codes to model wave energy devices is given by Payne et al. (2008).

2.2 Phenomenological Discussion

Essentially, the numerical modelling of WECs is based on Newton’s second law, which states that the inertial force is balanced by the whole forces acting on the WEC, schematically represented in Fig. 2.1. These forces are usually split into external (hydrodynamic/hydrostatic) loads and reaction forces. The external (hydrodynamic/hydrostatic) loads include:

Fig. 2.1 Schematic representation of a generic wave energy converter (WEC).

• Hydrostatic force caused by variation of the hydrostatic pressure distribution due to the oscillatory motion of the captor,

• Excitation loads due to the action of the incident waves on a motionless captor,

• Radiation force corresponding to the force experienced by the captor due to the pressure field alteration as a result of the fluid displaced by its own oscillatory movement, in the absence of an incident wave field.

Furthermore, depending on the type of WEC, the reaction forces may be caused by the

• Power take-off (PTO) equipment, which converts mechanical energy (captor motions) into electricity or other useful energy vector,

• Mooring/foundation system, responsible for the WEC station-keeping,

• End-stop mechanism, used to decelerate the captor at the end of its stroke in order to dissipate the kinetic energy gently, and so avoid mechanical damage in the device.

2.3 Potential Flow Theory

This section provides an overview of the most important considerations and the fundamental equations of potential flow theory. This theory is considered in some depth here because of its fundamental relevance to a large number of other models. The section starts by developing the Laplace equation, which is fundamental to solving the potential flow. This is followed by defining the boundary conditions for the water surface, the body surface and the seabed. The solution of the Laplace equation is then defined for sinusoidal waves and finally the decomposition of the solution into incident, diffracted and radiated waves is described.

Potential flow theory is based on the assumption of ideal flow, ie, inviscid (frictionless) and irrotational. An inviscid flow is a flow in which there are no viscous shear stresses to deform fluid elements or cause fluid particle rotation; only normal stresses are observed. Furthermore, an irrotational flow is a flow where the fluid elements do not rotate relative to their own centre of gravity (although, they can describe circular trajectories). Therefore, in essence the potential theory states that if an inviscid flow is initially irrotational then it remains irrotational at all subsequent