Physical modelling of wave energy converters

Wanan Sheng n, Raymond Alcorn, Tony LewisUniversity College Cork, Hydraulics & Maritime Research Centre, Cork, Ireland

a r t i c l e i n f o

Article history:Received 23 August 2013Accepted 15 March 2014

Keywords:Physical modellingSimilarity lawDimensional analysisWave energy converterPower take-offWave-structure interaction

a b s t r a c t

In guiding the progression and implementation of wave energy converters in a more effective and solidway, stepwise protocols have been recommended for assessing and validating their performance,feasibility, reliability and survivability during the devices' progression stages from the concepts to full-scale commercial devices. One important aspect is scale model testing in different development stages asa path to solve the most important problems and to build confidence in the device development.Particularly, in the early development stages of the wave energy converters, small scale models are oftentested in well-controlled laboratory conditions in a manner that some dynamic effects can be isolated,hence the analysis and understanding of the dynamic process could be much simplified and specified.However, there is no theory or guideline developed for this scaling practice in explaining whether or notthe scaling is correct and how the test data can be used. In this paper, a theoretical analysis to therequirements and an explanation to the feasibilities of physical modelling/scaling, and some importantscaling issues on physical modelling of wave energy converters, are presented with an emphasis on thephysical modelling and scaling of power take-off systems. This theoretical analysis can help tounderstand why and how a small scale model can be tested and how the test data can be used.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Wave energy is a type of well-concentrated renewable energywhen compared to other renewable energy resources, such assolar, wind etc., and its potentials are huge. IEA estimates the totalwave energy is up to 80,000 TWh a year IEA-OES (2004), which isabout 5 times of the worldwide electricity production 17,400 TWhin the year of 2004. It is now recognised that efficiently utilisingwave energy may make significant contributions to achieve thetarget of green energy. For example, the World Energy Council hasestimated that there may be 140–750 TWh/year of wave energyelectricity production by the current technologies and designs ofdevices when they fully mature, and this figure could be as high as2000 TWh/year if the potential improvements can be realised, seeJolly (2010).

A wave energy converter (WEC) is a device for extractingenergy from waves and converting the extracted energy intouseful energy. Most WECs may have two or more energy conver-sion stages. Essentially, the first conversion stage is the primarywave energy conversion in which the wave-excited components ofthe device or the water bodies in oscillating water columns/overtopping devices convert wave energy into mechanical orpotential energy. In the second conversion stage, a power take-

off system, such as hydraulic pump/motor, direct electrical gen-erator, air turbine or water turbine (depending on the principle ofthe wave energy converter), is often applied to convert themechanical/potential energy into useful energy, see Cruz (2008),and Salter et al. (2002). For an efficient wave energy conversion,a device is frequently designed to have large-amplitude motions inwaves so that more wave energy can be converted into mechanicalenergy and thus extracted by the power take-off system. In manycases, the large-amplitude motions of the device and the powertake-off (PTO) introduce significant nonlinearities in the dynamicsystems of wave energy converters and create the difficulties forunderstanding and analysing them. Traditionally, those difficultieshave led to extensive model tests in wave tanks and in seas duringthe development of a device, as suggested by the stepwisedevelopment procedures and protocols by Holmes and Nielsen(2010). In the stepwise procedure, different scale model tests arerecommended in different stages as a path to solve the mostimportant problems in the development stages and to buildconfidence in the device development. For instance, most of thewell-progressed wave energy devices, such as Pelamis, Oyster,Wave Dragon and OE Buoy etc., have undergone significant wavetank tests and sea trials, from small scale models in early stages forfeasibility testing, to large models for performance testing, and tolarger sea-trial models for assessing the economic feasibility, andthe reliability and survivability in real sea conditions.

In developing a wave energy converter (WEC), the importantissues are the assessments of the device performances and its

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/oceaneng

Ocean Engineering

http://dx.doi.org/10.1016/j.oceaneng.2014.03.0190029-8018/& 2014 Elsevier Ltd. All rights reserved.

n Corresponding author. Tel./fax: þ353 21 4250038, þ353 21 4321003.E-mail address: w.sheng@ucc.ie (W. Sheng).

Ocean Engineering 84 (2014) 29–36

wave power capture capacity from seas. Principally, this can becarried out either by a numerical analysis or a physical modelling.In this investigation, our focus is on physical modelling of waveenergy converters.

It is well accepted that the stepwise development procedure/protocol is recommended by Holmes and Nielsen (2010) suggeststhat different scale models be tested for solving the most impor-tant problems in each development stage. This stepwise procedureshows the difficulties in wave energy development, and theuncertainties involved in the scale model tests. Those difficultiesand uncertainties may be related to how well the physicalmodelling can be conducted and how the data be used.

Although the St. Denis-Pierson's superimposition method,St Denis and Pierson (1953) and the relevant principle and theoryfor scale model tests and the data utilisation have been devel-oped and accepted for many years, see Chakrabarti (1998),Hughes (1994) and Vassalos (1999), it is basically only suitablefor linear dynamic systems. This may be justified for the conven-tional ocean platforms because of its inherent small-amplitudemotions in waves. For a nonlinear dynamic system, the scalingmethods may be different. Recently, a review carried out by BMT(2000) has clearly indicated the scaling issues for floating plat-forms that if the dynamic system is nonlinear, its well-knownresponse amplitude operators (RAOs) may not be meaningful asthose frequently used in the linear dynamic system. In sucha system, irregular wave tests may be conducted by scaling therelevant parameters of the specific waves, and the measured datamust be scaled and used but in a limited manner. For waveenergy converters, the nonlinear effects may be more evident,either from the designated large-amplitude motions of the deviceor from the nonlinear power take-off system or some othernonlinear sources. As proposed in Holmes and Nielsen (2010)),when an “appropriately large model” is used, the power matrix issuggested to be carried out in the accordingly scaled sea states.It can be seen that the scaling of the power matrix bears a similarprinciple to that shown in the literature (BMT 2000) for a non-linear dynamic system.

The scaling in physical modelling is practically accepted inmany cases, but the theory behind this is not well developed. Forexample, the Froude similarity is very preferable because it iswidely applied and factually the relevant requirements can beeasily satisfied for the reduced model. Another example is thedimensional analysis, which is very helpful in reducing thevariables in test, and thus test numbers via the so-called non-dimensional numbers (Froude number and Reynolds number arethe famous among those non-dimensional numbers), rather thanthe individual parameters. However, it is not clear so far why thephysical modelling is correct or under what conditions thephysical modelling can be correct or acceptable. To answer thosequestions, this investigation provides a theoretical analysis for thephysical scaling of wave energy converters and an answer to thequestion why the physical modelling can be conducted and howthe data can be used, and with an emphasis on the scaling of thepower take-off systems for wave energy converters.

2. Similarities

It is generally known that for a physical modelling, relevantsimilarities must be satisfied to ensure the meaningful and usefulscaling and modelling. For a meaningful physical modelling,geometrical similarity must be satisfied. That is, the scale modelmust be geometrically similar to the target of interest (prototype).If the physical modelling is made to be useful, for example, how toapply the data from a scaled model to the prototype, the importantkinematical and dynamic similarities must be partially or fullysatisfied, largely depending on the specific problems.

2.1. Geometrical similarity

A prerequisite of a meaningful physical modelling is the geome-trical similarity. Geometrical similarity can be defined as all linearlengths of one object have a fixed scale factor to the correspondinglinear lengths of the second object, see Hughes (1994). If a scale

Nomenclature

A regular wave amplitude/areaB width of wave energy converterbpto, Bpto (nonlinear) damping coefficient

F!

force vectorFpto force from power take-off systemFP average captured power functionFr Froude numberH wave heightHP power capture responseg gravitational accelerationkpto stiffness from PTOL characteristic lengthn! normal vectorm massmpto additional mass from PTOP powerp0 atmospheric pressurep, p0 pressure and non-dimensional pressure, respectivelyqp, qw flowrate through power take-off and driven by the

interior water surfaceRe Reynolds numberS0 non-dimensional wetted surfaceS wetted surface

Sw wave spectrumT wave periodU characteristic velocityu,v,w velocity components in a Cartesian coordinateu0,v0,w0 non-dimensional velocity components in a Cartesian

coordinatev! velocity vectorV0 volume of air chamber in calm waterW capture widthx! motion vectorγ specific heat ratio of airρ density of waterε scale factor (the ratio of the full-scale length over the

scaled length)μ dynamic viscosityω angular frequency

Superscripts/Subscripts

p full scale (prototype)m modelL large modelS small model0 denotation of non-dimensional parameter

W. Sheng et al. / Ocean Engineering 84 (2014) 29–3630

model is geometrically similar to the full-scale model, then the scalefactor, ε, can be defined as the ratio of the full-scale geometricallength, Lp, over the small scale geometrical length, Lm:

ε¼ LpLm

ð1Þ

As a result of the geometrical similarity, the area and volume scalefactors must be calculated as

εA ¼Ap

Am¼ ε2; εV ¼ Vp

Vm¼ ε3 ð2Þ

2.2. Kinematical similarity

Kinematical similarity requires that the motions of a scaleobject must be proportional to those in the prototype, i.e., thevelocity vectors must be proportional for the scale model and theprototype everywhere. The kinematical similarity can be achievedwhen the ratio between the components of all vectorial motionsfor the prototype and the model is the same for all particles at alltime, Hughes (1994). For instance, for a physical modelling, if theflow around the full scale device is often turbulent because of thelarge Reynolds number, the kinematical similarity requires thatthe flow around the scale model must also be turbulent. However,if a very small model (thus with a small Reynolds number if theFroude similarity is applied) is used in physical modelling, the flowaround the scale model could be laminar. Obviously, in this case,the kinematical similarity is not satisfied.

2.3. Dynamic similarity

Dynamic similarity requires that for two geometrically similarobjects, the force/moment vectors acting at the correspondinglocations on the two similar objects have a same ratio everywhere.For a floating structure, the major forces acting on the device comefrom the flow pressure around the structure, i.e., the forces can becalculated by integrating the flow pressure acting on the similarstructures. In addition, for wave energy converters, the dynamicsystems must include the force/moments from the power take-off(PTO) systems.

In the following analysis, these two similar dynamic systems ofwave energy converters will be dealt with separately: the fluiddynamic problem (i.e., interaction of floating structure and waves)and the power take-off problem.

3. Dimensional analysis

Dimensional analysis in the research aims to establish thedimensionless fluid dynamic equations which can be further usedfor the scaling analysis to show how the physical modelling can beestablished and justified and how the effects of Reynolds numbercan be minimised.

Dimensional analysis is a technique that is widely used fordifferent purposes. The simplest one could be for checking“dimensional consistency” of any given computational formula-dimensions and units of each separate term in the formula mustbe the same. Another application of dimensional analysis is toidentify the dimensionless physical parameters needed to depicta given physical phenomenon, so that the experiments can bedesigned in such a way that the collected data from the scalemodel can be used in predicting the behaviours of a corresponding(geometrically similar) full-scale object, McDonough (2012). Thelatter technique is normally called the Buckingham Π theory.However, in this investigation, dimensional analysis is directlyapplied to the governing equations of the wave energy converters,

both the fluid dynamic equation and power take-off dynamicequation.

3.1. Governing equations

The dynamic system of a wave energy converter is a system ofthe interaction of wave, structure and power take-off system.Under the excitations of waves, one or more bodies of the waveenergy device may move in waves so that a portion of the wavepower can be converted into mechanical energy. A power take-offsystem then converts the mechanical energy into useful energy.For example, a direct drive (a linear generator, see Chapter 6, Cruz(2008)) can be used to convert the mechanical energy intoelectricity. In this regard, for a wave energy converter, the dynamicsystem can be regarded as a combination of two dynamic systems:the dynamics of the wave-structure interaction for primary energyconversion and the dynamic system of the power take-off forsecondary energy conversion. Practically, these two dynamicprocesses are actually coupled together. In the following analysisthe two dynamic systems are dealt with separately, but they arephysically coupled.

3.2. Fluid dynamic equations

To study the interaction between the structure and waves forwave energy devices, the governing equations of the flow field arethe conservation of mass (continuity equation) and the conserva-tion of momentum (the Navier–Stokes equation).

For a convenience in describing the fluid dynamics, an earth-fixed coordinate system is defined with its origin located at themean position of the centre of gravity of the floating body. Itsx- and y-axes are on a horizontal plane, and z-axis vertically up. Inthis coordinate, the conventional 6-DOF motions of the floatingstructure can be easily defined: the angular motions, namely roll,pitch and yaw, are defined around x-, y- and z-axes, respectively,and the translational motions (surge, sway and heave) of thefloating body are given by the motions of the centre of gravityalong the x-, y- and z-axes respectively.

The flow around the device can be regarded as incompressible,and the corresponding continuity equation is given in the fixedcoordinate as

∂u∂x

þ∂v∂y

þ∂w∂z

¼ 0 ð3Þ

where u, v, and w are the velocity components on x-, y- and z-axes.The Navier–Stokes equation for wave energy devices has

a following form

∂u∂tþu∂u

∂xþv∂u∂yþw∂u∂z ¼ �1

ρ∂p∂xþμ

ρ∂2u∂x2 þ ∂2u

∂y2þ∂2u∂z2

� �

∂v∂tþu∂v

∂xþv∂v∂yþw∂v∂z ¼ �1

ρ∂p∂yþμ

ρ∂2v∂x2þ ∂2v

∂y2þ ∂2v∂z2

� �

∂w∂t þu∂w

∂xþv∂w∂yþw∂w∂z ¼ �1

ρ∂p∂z�gþμ

ρ∂2w∂x2 þ∂2w

∂y2 þ ∂2w∂z2

� �

8>>>>><>>>>>:

ð4Þ

where p is the pressure, g the gravity acceleration, ρ the density offluid, and μ the dynamic viscosity of the fluid.

3.3. Dimensionless fluid dynamic equations

For the dimensional analysis, the dimensionless parameters arefirst defined from the following (Kundu and Cohen 2002):

Dimensionless coordinates

x0 ¼ xL; y0 ¼ y

L; z0 ¼ z

Lð5Þ

W. Sheng et al. / Ocean Engineering 84 (2014) 29–36 31

Dimensionless time

t0 ¼ UtL

ð6Þ

Dimensionless velocities

u0 ¼ uU; v0 ¼ v

U; w0 ¼w

Uð7Þ

Dimensionless pressure

p0 ¼ p

1=2ρU2 ð8Þ

where L and U are the characteristic length and speed,respectively.

Fig. 1 shows a schematic drawing from the dynamic system ofinterest to a dimensionless dynamic system. Corresponding to thecharacteristic length (L) of the original dynamic system (S), thecharacteristic length of the dimensionless dynamic system (S0) isa unit.

Substituting (5) and (7) into (3), the continuity equationbecomes

∂u0

∂x0þ∂v0

∂y0þ∂w0

∂z0¼ 0 ð9Þ

The non-dimensional continuity equation has a same form as theoriginal continuity equation.

Substituting (5)–(8) into (4), the dimensionless Navier-Stokesequation is obtained

∂u0∂t0 þu0∂u0

∂x0 þv0∂u0

∂y0 þw0∂u0∂z0 ¼ �1

2∂p0∂x0 þ 1

Re

∂2u0∂x02 þ ∂2u0

∂y02 þ ∂2u0∂z02

� �

∂v0∂t0 þu0∂v0

∂x0 þv0∂v0

∂y0 þw0∂v0∂z0 ¼ �1

2∂p0∂y0 þ 1

Re

∂2v0∂x02 þ ∂2v0

∂y02þ ∂2v0∂z02

� �

∂w0∂t0 þu0∂w0

∂x0 þv0∂w0

∂y0 þw0∂w0∂z0 ¼ �1

2∂p0∂z0 � 1

Fr2þ 1

Re

∂2w0∂x02 þ∂2w0

∂y02 þ∂2w0∂z02

� �

8>>>>><>>>>>:

ð10Þ

with the Froude number

Fr ¼UffiffiffiffiffigL

p ð11Þ

and the Reynolds number

Re ¼ρLUμ

ð12Þ

From Eq. (10), it can be seen that the non-dimensional fluiddynamic equations are very similar to the original form of theN–S equation, but with two additional non-dimensional numbers:Froude number Fr and Reynolds number Re. A factor ½ in thepressure gradient term will disappear if the non-dimensionalpressure is defined by

p0 ¼ p

ρU2 ð13Þ

Nonetheless, in this paper, the definition of non-dimensionalpressure (8) is used, because it bears a same form as the definitionof the conventional pressure coefficient. Besides, the additional

factor ½ in the pressure term in Eq. (10) will not affect the overallanalysis in the following sections.

Next we will show how two geometrically similar objects fulfiltheir kinematical and dynamic similarities.

Suppose there are two geometrically similar objects (S1 and S2in Fig. 2), and their characteristic lengths are L1 and L2, respec-tively. Essentially, their dynamic behaviours will be different andare decided by the corresponding governing equations. Thegeometrical similarity itself cannot guarantee the kinematicaland dynamic similarities. It will be shown that their kinematicand dynamic similarities can only be established if only someconditions are met.

Following the non-dimensional parameters (5)–(8), the non-dimensional dynamic equations for the two dynamic system are

∂u0i∂x0i

þ ∂v0i∂y0iþ ∂w0

i∂z0i

¼ 0

∂u0i∂t0iþu0

i∂u0

i∂x0i

þv0i∂u0

i∂y0i

þw0i∂u0

i∂z0i

¼ �12∂p0i∂x0i

þ 1Rei

∂2u0i

∂x0i2þ ∂2u0i

∂y0i2þ∂2u0

i∂z0i2

� �∂v0i∂t0iþu0

i∂v0i∂x0iþv0i

∂v0i∂y0i

þw0i∂v0i∂z0i

¼ �12∂p0i∂y0i

þ 1Rei

∂2v0i∂x0i2

þ ∂2v0i∂y0i2

þ ∂2v0i∂z0i2

� �∂w0

i∂t0i

þu0i∂w0

i∂x0i

þv0i∂w0

i∂y0i

þw0i∂w0

i∂z0i

¼ �12∂p0i∂z0i� 1

F2riþ 1

Rei

∂2w0i

∂x0i2þ ∂2w0

i∂y0i2

þ ∂2w0i

∂z0i2

� �

8>>>>>>>>><>>>>>>>>>:

ð14Þ

with i¼1, 2 indicating two different dynamic systems.And the corresponding Froude number and Reynolds number

Fri ¼UiffiffiffiffiffiffiffigLi

p ð15Þ

Rei ¼ρiLiUi

μið16Þ

Comparing the two non-dimensional equations expressed byEq. (14), it can be seen that the two geometrically similar objectshave very similar non-dimensional equations, with only differentFroude and Reynolds numbers given by Eqs. (15) and (16),respectively.

The identical dimensionless dynamic equations for the twogeometrically similar systems can only be attainable if theirFroude numbers and Reynolds numbers are same. Under sucha condition, the dynamic behaviours of the two similar objects canbe uniquely decided by and scaled from their identical non-dimensional dynamic equations. If the dynamic behaviours areknown for the non-dimensional dynamic system, the dynamicbehaviours of the two similar objects can be deduced from thescaling relations given by Eqs. (5)–(8), which in turn ensure theFig. 1. Dynamic system (S) and its dimensionless form (S0).

Fig. 2. Two dimensionless dynamic systems (S01 and S02) of two geometricallysimilar systems (S1 and S2).

W. Sheng et al. / Ocean Engineering 84 (2014) 29–3632

kinematical and dynamic similarities between these two similarobjects. In short, they are kinematically and dynamically similar.

If the Froude numbers are designed to be same for the twosimilar objects, then the two systems satisfy the Froude law ofsimilarity. The corresponding requirement is

U1ffiffiffiffiffiL1

p ¼ U2ffiffiffiffiffiL2

p ð17Þ

Similar, if the Reynolds numbers are designed to be same for thetwo objects, the two objects satisfy the Reynolds law of similarity,and the corresponding relation is

ρ1L1U1

μ1¼ ρ2L2U2

μ2ð18Þ

From Eq. (17), the Froude similarity requires a smaller velocity fora smaller object, whilst the Reynolds similarity Eq. (18) requiresa larger velocity for a smaller object if the fluid viscosity and densityare same. These contradictory requirements make it only possibleto satisfy Froude similarity and Reynolds similarity at the sametime for some very specific cases. For example, same Froude andReynolds numbers may be produced if fluid media of differentviscosities are used. The Soding's “sauna tank” is a proposal forsolving the difficult problem, see Bertram (2000). In the ‘saunatank’, the water is heated to a temperature of 90 1C. At thistemperature, the kinematical viscosity of water is only 0.32 timesof that of water at 20 1C. As a result, a large Reynolds number canbe reproduced using a model of only half the length in theheated water.

The more practical approach to partially but effectively solvethe dilemma is use a large model to ensure an enough largeReynolds number for the model. From the non-dimensional N–SEq. (10) it can be seen that if the Reynolds number is large enough,the viscous terms in the right-hand side of the Eq. (10) willbecome minor and thus can be neglected. This implies that inthe dynamic system, the viscous forces are not comparable to thehydrodynamic and gravity forces if a large Reynolds number canbe attained in the model. Under this assumption, the dimension-less N–S equation has a following form

∂u0∂t0 þu0∂u0

∂x0 þv0∂u0

∂y0 þw0∂u0∂z0 ¼ �1

2∂p0∂x0

∂v0∂t0 þu0∂v0

∂x0 þv0∂v0

∂y0 þw0∂v0∂z0 ¼ �1

2∂p0∂y0

∂w0∂t0 þu0∂w0

∂x0 þv0∂w0

∂y0 þw0∂w0∂z0 ¼ �1

2∂p0∂z0 � 1

Fr2

8>>><>>>:

ð19Þ

Thus the kinematical and dynamic similarities can be guaranteedby the Froude similarity alone. This is the most popular andpractical similarity in physical modelling. The reasons for thisconvenient selection are as following:

– A relative large Reynolds number in a scale model can be easilyachieved in most practical cases. As stated by Salter (2003),

when a very small scale model is used, the worries may bethose coming from the interaction of the cables and some otherissues, such as instrumentations, balance of the model etc.,rather than the selection of the relevant similarity laws. ITTC(International Towing Tank Conference) reported that an extre-mely small model (scale factor ε¼170) has been used fora semi-submersible model, and it is found there is no signifi-cant scaling effect, (ITTC 1999). For wave energy devices, it hasbeen suggested that in the first development stage, a smallmodel (scale factor up to 100) can be used by Holmes andNielsen (2010).

– When a small scale model is used, the kinematical similaritybased on Froude similarity requires a small velocity for a smallmodel. This requirement is very in favour of laboratory testconditions.

– Relatively, physical modelling using Froude similarity has beenwidely accepted and the relevant method for utilising themodel test data has been developed by St Denis and Pierson(1953) although only for linear dynamic systems.

3.4. Froude similarity

Under the condition of a large Reynolds number in modelscaling, Froude similarity alone can ensure the correctness ofmodel scaling and of dynamic similarity of the fluid dynamics.The relevant scale factors for the dynamic system are given inTable 1 for reference (more scale factors can be found in Hughes(1994)).

4. Reynolds number

It has been shown the importance of the Reynolds number inthe scaling of the dynamic systems in the previous sections.However, the problems of Reynolds number are practically over-looked in many cases, not because they are not important, butmore likely the requirements for the Reynolds number may beactually satisfied. In this section, the problem is further addressedhow Reynolds number can be defined in the virtually stationaryplatforms and what is the critical Reynolds number in a good scalemodel testing.

4.1. Definition of Reynolds number

Reynolds number can be a well-defined non-dimensionalparameter in many situations, especially for those objects witha prescribed travelling velocity, U, see Eq. (12). For instance, thevelocity of a travelling ship can be taken as that for Reynoldsnumber calculation. However, for the virtually stationary floatingstructures, such as ocean platforms and wave energy converters,the Reynolds number is not well-defined. Newman (1977) definesthe Reynolds number for stationary structures as

Re ¼ρωL2

μð20Þ

where the characteristic velocity U is replaced by ωL (ω is thecircular frequency of the body oscillation).

Tao and Dray (2008) define a Reynolds number for an oscillat-ing circular plate as

Re ¼ρωALμ

ð21Þ

where A is the amplitude of the forced body oscillation, thecharacteristic length, L, is taken as the diameter of thecircular plate.

Table 1Scale factors by Froude similarity.

Quantity Scale factor

Acceleration ε0

Area ε2

Force ε3

Length ε

Mass/Volume ε3

Power ε3:5

Pressure ε

Time ε0:5

Velocity ε0:5

Volume flow rate ε2:5

Work/Energy ε4

W. Sheng et al. / Ocean Engineering 84 (2014) 29–36 33

The suitability of the definitions in (20) and (21) may beproblematic for the stationary floating structures and wave energyconverters. In (20), the characteristic velocity is replaced by ωL,which does not bear any obvious physical meaning. In addition,the motion amplitude or wave condition has not been included inconsiderations. In (21), the characteristic velocity is replaced bythe maximal velocity of the forced body motion. However, for thestationary floating structures or wave energy converters, themotion of the body is the answer which we need to solve, whichis not defined beforehand.

More sensibly, Reynolds number can be defined by employingwater particle velocity in waves (see Jonkman (2010) andFitzgerald and Bergdahl (2009)) as

Re ¼ρωAwL

μð22Þ

where Aw is the amplitude of the incoming wave.In this definition, the characteristic velocity is actually the

maximal horizontal velocity of water particles in the wave motion,which is taken as the relative velocity between the fluid and thestationary floating-structure. Its physical meaning is obvious.In the following analysis, this definition will be applied.

4.2. Critical Reynolds number

In physical modelling of a wave energy converter or a floatingstructure in waves, a small scale model is normally preferred,especially when the scaling is based on the Froude similarity. As aresult of this similarity, the Reynolds number of the small modelcan be much smaller than that of the full scale structure. However,to justify the Froude similarity alone can be enough for thekinematical and dynamic similarities, Reynolds number of thescale model must be large enough, and the viscous effect in thedynamic system must be minimised and therefore ignored. Fromthe standpoint of the physical modelling, to guarantee a turbulentflow around the model as that around the full scale model tosatisfy the kinematical similarity, the Reynolds number of themodel must be large enough.

Practically, the Reynolds number is a critical parameter indetermining laminar or turbulent flow. To ensure a turbulent flow,the Reynolds number is generally required to be larger than theupper critical one, above which the flow can no longer bemaintained as a laminar flow. However, there is no well acceptedupper critical Reynolds number, which is largely depended on theproblem of interest. Examples given by Eckhardt (2009) show thevalue could be from 13,000 to 100,000. Heller (2012) discussedthat the scale effects on the physical hydraulic models andsummarised the critical Reynolds numbers used for differentpurposes, which have a range of the critical Reynolds numberbetween 105 and 3�105. Fuchs and Wager (2012) have shown thatfor the overland flow the scale effect can be negligible if theReynolds number can be small as 6300, whilst for wave run-up,the critical Reynolds number will be much higher as to 70,000 fora negligible scale effect. Fitzgerald and Bergdahl (2009) haveshown that when the Reynolds number is in excess of 105, thedrag and inertial coefficients can be independent of Reynoldsnumber.

5. Scaling of wave energy devices

5.1. Scalable dynamic equation of WECs

For two geometrically similar objects, if their Reynolds num-bers are both large enough, then the viscous terms in the non-dimensional N–S equation can be neglected. Under such a condition,

the Froude similarity itself can make sure these two systemsdynamically similar. Fig. 3 shows the scaling relations. Twogeometrically similar systems (S1 and S2) share a same dimension-less dynamic system (S0). As a result of this, these two geome-trically similar systems (S1 and S2) are dynamically similareach other.

Suppose the non-dimensional pressure, p0, which is a solutionof the non-dimensional dynamic equation, then the pressures fortwo similar systems can be estimated by applying the scalerelations as

p1 ¼12ρU2

1p0 ð23Þ

p2 ¼12ρU2

2p0 ð24Þ

The forces acting on the wave energy devices by the fluid aroundthe devices can be calculated as

F!

1 ¼ZS1p1 n

!dS¼ 12ρU2

1L21

ZS0p0 n!dS0 ð25Þ

F!

2 ¼ZS2p2 n

!dS¼ 12ρU2

2L22

ZS0p0 n!dS0 ð26Þ

where S1 and S2 are the surface in fluid for the two similar objects,and S0 is the non-dimensional surface.

Hence the force scale factor is

εF ¼F!

2

F!

1

¼ U22L

22

U21L

21

¼ ε3 ð27Þ

where the scale factors of the velocity and the length according toFroude similarity (Table 1) have been invoked.

The dynamic equation of the wave energy devices (no powertake-off) can be expressed as

m1€x!1 ¼ F

!1 ð28Þ

m2€x!2 ¼ F

!2 ð29Þ

Obviously, these two equations are scalable dynamically.If the power take-off is applied, the dynamic equations for the

two similar wave energy devices become

m1€x!1 ¼ F

!1þ F

!pto1 ð30Þ

m2€x!2 ¼ F

!2þ F

!pto2 ð31Þ

Fig. 3. Scaling relations via a non-dimensional dynamic system.

W. Sheng et al. / Ocean Engineering 84 (2014) 29–3634

where F!

pto1 and F!

pto2 are the forces from thecorresponding PTOs.

To guarantee these two dynamic equations are scalable, thepower take-off forces must be scalable as

F!

pto2

F!

pto1

¼ ε3 ð32Þ

in the scale model test, if this relation can be maintained, then thePTO is said scalable.

Under such a condition, the power extractions by the PTOs arealso scalable, as

F!

pto2 U v!2

F!

pto1 U v!1

¼ ε3:5 ð33Þ

5.2. Scaling of power take-off system

5.2.1. Linear power take-off systemIf the power take-off is linear, a general expression can be

(following Babarit et al. (2012))

F!

pto ¼ �mpto€x!�bpto

_x!�kpto x! ð34Þ

A scalable power take-off force must have corresponding scalablePTO coefficients as following:

mpto2

mpto1¼ ε3 ð35Þ

bpto2bpto1

¼ ε2:5 ð36Þ

kpto2kpto1

¼ ε2 ð37Þ

5.2.2. Nonlinear power take-offIf the power take-off is nonlinear, the PTO force is supposed to

have an expression as

F!

pto ¼ �bpto_x!�Bpto

_x!j _x!j ð38ÞThen the scalable PTO coefficients must be

bpto2bpto1

¼ ε2:5 ð39Þ

Bpto2

Bpto1¼ ε2 ð40Þ

If the damping coefficients of the power take-off can be scaledaccording to Eqs. (39) and (40), the nonlinear PTO can be scalable.A practical problem may be how to produce the different dampingcoefficients.

5.2.3. Scaling of power take-off of oscillating water column (OWCs)There are normally two different types of power take-off

systems for OWC wave energy converters, namely the linear Wellsturbine and the nonlinear impulse turbine. For small model tests,physically scaling the air turbines is practically difficult, and inmost cases, it is not necessary. The important issue is the scaling ofair flow characteristics of the power take-off.

Linear power take-off has a relation as following

p¼ κ1qp ð41ÞBased on the Froude similarity, the damping coefficient κ1 shouldbe scaled by a scale factor

εk1 ¼ ε�1:5 ð42Þ

If the power take-off is nonlinear, for example, an orifice PTO, anexpression can be as following

p¼ κ2q2p ð43Þ

According to the Froude similarity, the damping coefficient κ2should be scaled by a scale factor

εk2 ¼ ε�4 ð44Þ

5.2.4. Air compressibilityOne important aspect for the OWC devices scaling is the air

compressibility, which may become important for the full scaledevice when the air chamber volume and chamber pressure arelarge enough.

Formulated by Sarmento et al. (1990), the linearised relation ofthe flowrate through the PTO is

qp ¼ qw� V0

γp0

dpdt

ð45Þ

It can be seen that the flowrate, qp, through the PTO is generallydifferent from that driven by the interior water surface, qw, due tothe air compressibility (the second term in RHS). To scale thecompressibility by the Froude similarity, it requires the chambervolume must be scaled by

εV0 ¼ ε2 ð46Þwhich is a different scale factor for the volume of the air chamberrather than the conventional scale factor of volume ðε3Þ requiredby the Froude similarity. This is why Weber (2007) claims that fora scale model of the air chamber, its height must be kept same, insuch a way that the air chamber modelling has reduced onedimension, hence Eq. (46) can be attained. In fact, the scale of thechamber volume required by Eq. (46) can be implemented inmany different ways. Keeping the height of the chamber regard-less of the scale factor is only one of the many options.

6. Scalability of power extraction

In model testing of wave energy devices, one important aspectis to assess the power extraction by the device in a small scaleratio. Generally, the model test of the floating structure can bemade by the Froude similarity, so does the power take-off system.Under such a condition, it is possible to measure the powerextraction of the wave energy converter model in the scaled seastates.

There may be some cases that the motions of wave energyconverter and the power take-off system are both nonlinear.However, its power capture response is close to linear (propor-tional to wave height squared). In this case, the power captureresponse curve can be very useful for assessing the powergeneration of the wave energy device, without performing therequired model tests in the corresponding sea states. In thissection, the assessment method is outlined (also see Sheng andLewis (2012)).

The power captured response is defined as

HP ¼Pre

H2 ð47Þ

where Pre is the average power of the device captured in regularwaves with a wave height H.

And the wave energy capture width, W, can be defined for thedevice

W ¼ Preρg232πH

2Tð48Þ

W. Sheng et al. / Ocean Engineering 84 (2014) 29–36 35

The capture width ratio (sometimes called the efficiency) of thedevice for capturing wave energy is given by

η¼WB

ð49Þ

In many cases, B is simply taken as the width of the device.If the device is deployed in irregular waves, then an average

captured wave power function, FP(ω), can be defined based on thepower capture response and the wave spectrum. Corresponding tothe frequency band, Δω, the device captures wave energy as

ΔP ¼HPH2 ¼Hp½2SwðωÞΔω� � 22 ð50Þ

Defining the average captured power function

FPðωÞ ¼ ΔPΔω¼ 8HPSwðωÞ ð51Þ

then the average captured power by the device in the sea state,SwðωÞ, is simply given by

Pirr ¼Z 1

0FPðωÞdω ð52Þ

7. Conclusions

The investigation has shown that the physical modelling andscaling may be feasible and acceptable if the Reynolds number ofscale model is large enough (probably larger than 105) for waveenergy converters and other offshore structures. Under sucha condition, the viscous forces on the marine structures is mini-mised and can be negligible. As a result of this, the Froudesimilarity alone can ensure the kinematical and dynamic simila-rities for the dynamic systems of offshore structures.

For wave energy converters, geometrical similarity is onlyrequired for the interaction of the converter structure and waves,whilst for the power take-off system, it is not necessary tophysically scale the power take-off system, but that of the relationbetween the forces and motions. The following conclusions can bedrawn from the study:

– To be scalable based on the Froude similarity, the physicalmodel must be large enough to ensure a large Reynoldsnumber of the model, such that the viscous forces in thedynamic system can be small and negligible. The recom-mended critical Reynolds number would be 105.

– Large Reynolds number for a scale physical model ensures theturbulent flow around the scale model, similar to that aroundthe prototype (kinematical similarity).

– To be scalable of a wave energy device, especially for the powerextraction assessment, power take-off system must be scaledaccording to the appropriate scale factors.

– To scale air compressibility, the air chamber must be scaled by ascale factor given in Eq. (46), which is larger than the volumerequired from the Froude similarity.

Acknowledgements

This material is based upon works supported by the ScienceFoundation Ireland, Ireland (SFI) under the Charles Parsons Award

at Beaufort Research-Hydraulics and Maritime Research Centre(HMRC). Statistics and data were correct at the time of writing thearticle; however the authors wish to disclaim any responsibilityfor any inaccuracies that may arise.

References

Babarit, A., Hals, J., Muliawan, M.J., Kurniawan, A., Moan, T., Krogstad, H.E., 2012.Numerical benchmarking study of a selection of wave energy converters.Renew. Energy 41, 44–63.

Bertram, V., 2000. Practical Ship Hydrodynamics. Butterworth-Heinemann, Oxford,United Kingdom.

BMT, 2000. Review of model testing requirements for FPSO's. Cited on: 03.08.13,website: ⟨http://www.hse.gov.uk/research/otopdf/2000/oto00123.pdf⟩.

Chakrabarti, S., 1998. Physical model testing of floating offshore structures.Dynamic Positioning Conference, October 13–14, 1998, Houston, USA.

Cruz, J.E., 2008. Ocean Wave Energy: Current Status and Future Perspectives.Springer.

Eckhardt, B., 2009. Introduction. Turbulence transition in pipe flow: 125th anni-versary of the publication of Reynolds' paper. Philos. Trans. R. Soc. A: Math.,Phys. Eng. Sci. 367, 449–455.

Fitzgerald, J., Bergdahl, L., 2009. Rigid moorings in shallow water: a wave powerapplication. Part I: experimental verification of methods. Mar. Struct. 22,809–835.

Fuchs, H., Wager, W.H., 2012. Scale effects of impulse wave run-up and run-over.J. Water, Port, Coastal, Ocean Eng. 138, 303–311.

Heller, V., 2012. Discussion: scale effects in physical hydraulic engineering models.J Hydraul. Res. 50, 244–246.

Holmes, B. and Nielsen, K., 2010. Task 2.1 guidelines for the development andtesting of wave energy systems (IEA-OES, ANNEX II). Cited on: 20.04.13,website: ⟨http://www.ocean-energy-systems.org/oes_documents/annex_ii_reports/⟩.

Hughes, S.A., 1994. Physical Models and Laboratory Techniques in Coastal Engineer-ing. World Scientific, Singapore.

IEA-OES, 2004. Ocean energy: opportunity, present status and challenges. Cited on:website: ⟨http://www.iea-oceans.org/_fich/6/Poster_Ocean_Energy.pdf⟩.

ITTC, 1999. The Specialist Committee on deep water mooring. 22nd ITTC, 5–11September, Seoul, Korea and Shanghai, China.

Jolly, A., 2010. Clean tech, clean profits: using effective innovation and sustainablebusiness practices to win in the new low-carbon economy. Kogan Page, London,United Kingdom.

Jonkman, J., 2010. Definition of the floating system for phase IV of OC3. NREL/TP-500-47535, National Renewable Energy Laboratory, USA.

Kundu, P.K., Cohen, I.M., 2002. Fluid Mechanics, Second Edition Academic Press.McDonough, J.M., 2012. Lectures in Elementary Fluid Dynamics: Physics, Mathe-

matics and Applications. Cited on: 10.05.12, website: ⟨http://www.engr.uky.edu/�acfd/me330-lctrs.pdf⟩.

Newman, J.N., 1977. Marine Hydrodynamics. The MIT Press, Cambridge, Massachu-setts, USA.

Salter, S., 2003. Research requirements for 4th generation wave energy devices.Cited on: 22.08.13, website: ⟨http://www.see.ed.ac.uk/�shs/DECC%20Marine%20Call%202010/2%20Necessary%20research%20for%20wave.pdf⟩.

Salter, S., Taylor, J.R.M., Caldwell, N.J., 2002. Power conversion mechanisms forwave energy. Proc. Inst. Mech. Eng., Part M: J. Eng. Marit. Environ. 216, 1–27.

Sarmento, A.J.N.A., Gato, L.M.C., de O. Falcao, A.F., 1990. Turbine-controlled waveenergy absorption by oscillating water column devices. Ocean Eng. 17,481–497.

Sheng, W., Lewis, A., 2012. Assessment of wave energy extraction from seas:numerical validation. J. Energy Res. Technol. 134, http://dx.doi.org/10.1115/1.4007193.

St. Denis, M., Pierson, W.J., 1953. On the motions of ships in confused seas. SNAMETrans. 61, 280–357.

Tao, L., Dray, D., 2008. Hydrodynamic performance of solid and porous heave plates.Ocean Eng. 35, 1006–1014.

Vassalos, D., 1999. Physical modelling and similitude of marine structures. OceanEng. 26, 111–123.

Weber, J., 2007. Representation of non-linear aero-thermodynamics effects duringsmall scale physical modelling of OWC WECs. In: Proceedings of the 7thEuropean Wave and Tidal Energy Conference, 11–14 September 2007, Porto,Portugal.

W. Sheng et al. / Ocean Engineering 84 (2014) 29–3636