Weakly non linear modeling of submerged wave energy converters

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Weakly non linear modeling of submerged wave energy

converters

Lucas Letournel, Camille Chauvigné, Baptiste Gelly, Aurélien Babarit,

Guillaume Ducrozet, Pierre Ferrant

To cite this version:

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LETOURNEL Lu as a,

∗

,CHAUVIGNE Camille a,b ,GELLY Baptiste a ,BABARIT Aurélien a ,DUCROZET Guillaume a ,FERRANTPierre a a

É oleCentralede Nantes,LHEEALab.,UMR6598,Nantes,Fran e

b

IFP-EN,Rueil-Malmaison,Fran e

Abstra t

Wave-to-Wire numeri al models being developed for the study of wave energy

on-verters usually make use of linearpotential ow theory [1, 2, 3, 4,5℄ to des ribe

wave-stru ture intera tion. This theory is highly e ient from a omputational perspe tive.

However,itreliesonassumptionsofsmallwavesteepness andsmallamplitudeofmotion

around mean positions. Often, maximization of wave energy onverters' energy

perfor-man e implies large amplitude motion [6, 7, 8℄, thus ontradi ting the assumption of

small amplitudemotion.

An alternative approa h is to linearize the free surfa e onditions on the

instanta-neous in ident wave elevation (Weak-S atterer approa h [9℄) while the body onditions

are evaluated at the exa t body position. Studies of wave energy onverters' dynami

response using this method areexpe ted to be more a urate, whilemaintaining a

rea-sonable omputational time. With this aim, a Weak-S atterer ode (CN_WSC) was

developedandusedtostudy two submergedwave energy onverters. Therstisa

heav-ingsubmergedsphereandthese ondisabottom-hingedfullysubmergedos illatingap.

Theyareinspired respe tively bytheCeto [10 ℄ andWaveRoller [11℄devi es.

Initial al ulations were performedin linear onditions rst to verifythe CN_WSC

againstlineartheory. Subsequently, al ulationsinnonlinear onditionswereperformed,

using large wave steepness and amplitude of body motion. In linear onditions, results

of CN_WSC showed good agreement withlinear theory, whereas signi ant deviations

fromlinear theorywereobserved innonlinear onditions. As amplitudeof bodymotion

in reases, linear theory tends to overestimate energy performan e in omparison with

weak-s atterer theory. In ontrast, with smaller amplitude of motion but larger wave

steepness, the opposite result is obtained: energy performan e is underestimated by

lineartheory ompared to weak-s atterer theory.

Keywords: Wave Energy Converter, Numeri almodeling, Potential Free-Surfa eFlow,

Weak-S atterer, Energy performan e

∗

Correspondingauthor

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The standard numeri al tools for modeling and designing wave energy onverters

(WECs) relyon linear potential theory [1 , 2, 3 ,4, 5 ℄, and as su h are limited to

move-ments ofsmall amplitude aroundmean positions. However e ien y of WECsrelies on

large amplitude motion [6 ℄: by design, their resonant frequen ies must fall in the wave

ex itation range. Linear potential theory has been shown to be insu ient to model

the behavior of WECs insu h ongurations [7 , 8℄. Thus, other numeri al approa hes

are required. A nonlinear potential-ow model, in the ontext of a numeri al

wave-tank (NWT), was pioneered by Longuet-Higgins and Cokelet [12 ℄. In their work, they

used the mixed Eulerian-Lagrangian (MEL) approa h, solving the Lapla e equation in

an Eulerian oordinate system, and then adve ting the nodes at themesh boundaries.

Sin e thattime, severalwave tankshave been developed, inboth two andthree

dimen-sions [13 , 14 , 15 , 16 ℄. Issues related to the development of NWT have been reviewed

by Tanizawa [17 ℄. Re ent developments have fo used on a elerating three-dimensional

NWT( omputationalrequirementsarestilllarge) andaddressissuesofgridding,

numer-i al stability,and a ura ythat anbe problemati for omplex geometries[18, 19℄.

A weakly nonlinear approa h, based on the weak-s atterer (WS) approximation, is

expe ted to be a promising alternative that avoids or redu es the impediments of fully

nonlinar potential-ow odes. Introdu ed byPawlowski [9 ℄,this lass of approximation

is based on linearisation of the free surfa e onditions on the in ident wave elevation,

allowing treatment of nonlinear steep in ident waves. The body ondition is evaluated

attheexa tinstantaneouspositionofthebody,allowing largebodymotionsto betaken

intoa ount. TheonlyassumptionintheWSapproa histhattheperturbationpotential

issmall. Thus,numeri al odesbasedontheWSapproa hareexpe tedtobea uratein

ases withsmallto large amplitude bodymotions, in ident wavesofarbitrary steepness

andwhentheperturbedwaveissmall. Thedierentquantitiesarethende omposedinto

in identandperturbation omponents,withthein identonesasfor ingtermswhileonly

theperturbation ones aresolved. Themain advantage of this isthat thein ident wave

does not need to be propagated from a wave-maker, allowing the mesh to be rened

only in the vi inity of the body. Moreover, sin e the free surfa e boundary is known

expli itly, the WSmethod is expe ted to be more robust and stablethan thefully non

linearapproa h.

SeveralWSowsolvershavebeendevelopedinthepast: SWAN-4[20℄,LAMP-4[21℄

andWISH [22℄. Their eldsof studywere howevermainlyfo usedon shipwithforward

speed,whi h anindeedbe onsideredasslenderbodies. Asa onsequen e,thes attered

wavesandmotionresponsesaresmallandtheassumptionsoftheWSapproximationare

fullled. Inthis arti le, theaim is to investigate whether the WSmethod an improve

numeri al modeling ofWECs.

Thedevelopmentofthisnewsolverisbasedontheexperien egainedbytheauthorsin

the90'sindevelopingathree-dimensionaltimedomainboundaryintegralequation(BIE)

solver for linear, se ond-order and fully nonlinear wave-body intera tion problems [13,

23,16,24℄. Inparti ular,although higherorderBIEsolversareavailable, the hoi eofa

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the surfa e integrals in a purely analyti form, and very good e ien y (typi ally, the

ratio of numberof unknowns to number of elements isapproximately 0.5). In addition,

it is possible to apply e ient mesh generation s hemes that originate from the nite

element ommunity, whi h is onsidered to be key to pra ti al appli ations. Analyti al

expressionsforthe inuen e oe ients,initiallyunpublished[25℄,havebeen ompletely

redeveloped bythe rst author duringhis PhD work, andare presented inAppendix A

ofthis paper.

Comparisonswithafullynonlinearowsolverhavebeen arriedoutforasubmerged

body in for ed motions [26 ℄. In the present paper, we extend this to the ase of freely

movingbodies,whi hrequiresnewequationsandnumeri alimplementationsto al ulate

themotionofthe body. TheImpli it method,introdu edbyTanizawa[27 ℄andbasedon

thesolution of a se ondBoundary Value Problem (BVP) for the timederivative of the

velo itypotential, was hosen tosolve the omplex impli itproblemof thebodymotion

al ulations. We have developed a new expression for the BVP body ondition, whi h

uniesthetwoexpressionspreviouslygivenbyTanizawa[27 ℄andCointe[28℄;thedetails

arepresentedin[29 ℄.

First, WS theory is re alled. The numeri al implementations are then introdu ed,

namelytheboundaryelementmethod,thetime-mar hings hemeandtheuid/stru ture

intera tion method. Finally, themethod is applied to two submerged WECs, using the

Ceto[10 ℄andWaveRoller[11℄systemsasexamples. Forthepurposeofveri ation,results

are initially ompared to linear theory in linear onditions. Simulations in nonlinear

onditions, with large amplitude motions and large steepness in ident wave, are then

ondu ted.

2. METHODS

2.1. Potential owtheory

Assuming a uid to be in ompressible and invis id with irrotational ow, its ow

velo ityderivesfrom avelo ity potential

φ

whi h satisestheLapla e Eq.:

∇

2 φ(x, y, z, t) = 0

(1)

intheuiddomain,

D

. Theboundaryoftheuiddomainis

∂D = Γ = Γ

f s

∪Γ

b

∪Γ

w

∪Γ

d

, seeFig 1.

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potential isthe solution ofthefollowing boundary valueproblem(BVP):











∇

2 φ = 0

intheuiddomain

D

∂φ

∂t

= −gη −

1

2 ∇φ · ∇φ

onthefree surfa e,

Γ

f s

∂η

∂t

=

∂φ

∂z

− ∇φ · ∇η

onthefree surfa e,

Γ

f s

∂φ

∂n

= V

b

· n

on thebody,

Γ

b

∂φ

∂n

= φ

0

ontheseabed,

Γ

d

φ −→ 0

onboundariesfar fromthebody,

Γ

w

(2)

(6)

velo ity and

n

the normal ve tor pointing outwards from the uid.

φ

0

is the velo ity potential ofthe in oming waves.

Using Green's Se ondIdentity along withtheRankinesour e, it an beshownthat

theresolutionofthe3DLapla eequationintheuiddomain anberedu edtoasurfa e

integralequation, on itsboundaries.

2.2. The weak-s atterer approximation

The WS approximation relies on the de omposition of the velo ity potential and

the free surfa e elevation (

φ, η

) into the in ident (

φ

0 , η

0

) and the perturbation (

φ

p

, η

p

) omponents, seeFig 2 .

φ = φ

0 + φ

p

η

= η

0 + η

p

(3)

Figure2: Weak-S attererde ompositionanddenitionofthedierentwaveelevation omponents

The perturbation omponents aregenerated by theintera tion ofthebodywiththe

in ident (undisturbed) ow. Theyin lude dira tion and radiation ee ts. In the WS

approximation, the perturbation omponents are assumedto be small ompared to the

in ident omponents.Sin e thein ident omponents an be onsidered as for ing terms,

only theperturbation omponents aresolved. Thus, thein ident wave does notneed to

be generated using a wave maker on an outer boundary, and propagated with theow

solver. Thisallowsustorenethefreesurfa emeshonly lose tothebody,whi hwould

not have been possibleifthe in ident wave have had to be propagated a rossthewhole

omputational domain.

The free surfa e equations an then be linearized on the instantaneous position of

the in ident wave, whi h is expli itly known. Sin e the position of the free surfa e is

independent of the solution, the WS approa h is then more robust than the fully non

linearapproa h, inwhi hthe free surfa epositionis oneof theunknowns.

2.2.1. In ident wave model

IntheWSmethod,thefreesurfa eboundary onditionsarelinearizedontheposition

of the in ident free surfa e. This means that the in ident wave modelhas to fulllthe

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from[16 ℄.

x

isthenon-dimensionalwavelength,while

η

isthewaveelevation.

solution given by the stream fun tion theory of Riene ker and Fenton [31 ℄. Based on

Fourier series,the in ident eld al ulations an be obtained a uratelyandqui kly,for

very steep waves, up to the theoreti al limit of wave-breaking, see Fig 3. Only regular

waves are onsidered in this study.A oupling with a High Order Spe tral (HOS) [32℄

methodmaybe arriedout to take into a ount more omplex nonlinearin ident waves

(irregularseastates withpossible dire tionalspreading).

2.2.2. Free-surfa e boundary onditions

The fully nonlinear kinemati and dynami free surfa e onditions, applied at the

exa t position ofthe free surfa e at

z = η

,are:

∂η

∂t

=

∂φ

∂z

− ∇φ · ∇η

(4)

∂φ

∂t

= −

1

2 ∇φ · ∇φ − gη

(5)

In order to make the Mixed Euler-Lagrange (MEL) approa h [12 ℄ simpler, the free

surfa e nodes are allowed to move only verti ally. It is imposed that they follow the

in ident wave position: a node of oordinates

(x, y, z = η

0 (x, y, t))

moves verti ally to

(x, y, z = η

0 (x, y, t+dt))

withavelo ity

v

=

∂η

0 ∂t

z

. The orrespondingderivationoperator is

D

0z

Dt

=

∂

∂t

+

∂η

0 ∂t

∂

∂z

(6)

Noting that the wave elevation

η

is independent of

z

, this leads to

D

0z

η

Dt

=

∂η

∂t

. Intro-du ingtheWSde ompositioninthekinemati freesurfa eboundary ondition, Eq.(4),

(8)

A Taylor expansion on theperturbation omponent

η

p

(whi h is small ompared to

η

0

) isapplied at the in ident freesurfa e position, inorder to obtaintheWS kinemati freesurfa e ondition. Thehigher order omponents (

η

2 p

,

φ

2 p

,

η

p

.φ

p

,et .) are negle ted.

∂η

p

∂t

=

∂φ

p

∂z

− ∇φ

p

· ∇η

0 − ∇φ

0 · ∇η

p

+η

p

∂

2 _φ

0 ∂z

2 −

∂∇φ

0 ∂z

· ∇η

0

,on

z = η

0

(8)

The same operations are su essively applied to thedynami free surfa e equation,

to obtainits WSversion:

D

0z

φ

p

Dt

= −gη

p

− ∇φ

p

· ∇φ

0 +

∂η

0 ∂t

∂φ

p

∂z

−η

p

∂

2 _φ

0 ∂z∂t

+

∂∇φ

0 ∂z

· ∇φ

0

,on

z = η

0

(9) 2.2.3. Body boundary ondition

The body onditionintheboundary value problem,Eq. 2,is:

∂φ

∂n

(x) = V

b

(x) · n

(10)

A ordingtotheWSapproximation,thewettedsurfa eofthebodytakesintoa ount

the instantaneous in ident wave elevation. Introdu ing the WS de omposition, Eq.(3),

inthebody ondition yields:

∂φ

p

∂n

(x) = −

∂φ

0 ∂n

(x) + V

b

(x) · n

on thewetted surfa eof thebody,

Γ

b

(11) The boundary onditionon theseabed issimilar, witha zeroboundary velo ity:

∂φ

p

∂n

(x) = −

∂φ

0 ∂n

(x)

ontheseabed,

Γ

d

(12)

2.2.4. Far-eld ondition

Far from the origin of the perturbation (the WECs), the perturbation omponents

mustvanish:

η

p

→ 0

φ

p

→ 0

,when

r → ∞

(13)

Our numeri al domain is bounded by a verti al ylinder ontrol surfa e,

Γ

w

. The in ident wave isexpe ted to gothrough this surfa e:

∂φ

∂n

(x) =

∂φ

0 ∂n

(x)

,on

Γ

w

(14)

leading to the boundary ondition:

∂φ

p

∂n

= 0

,on

Γ

w

(15)

while the ontrol surfa e is assumed su iently far from the body so that the normal

velo ityof the perturbation velo itypotential isnegligible. Anumeri al bea hwasalso

(9)

TheBVPforthevelo itypotentialresultingfromtheBIEandthepreviousboundary

onditions an thus be des ribed as:











△φ

p

= 0

intheuiddomain,

D

0z

φ

p

Dt

= −gη

p

− ∇φ

p

· ∇φ

0 +

∂η

0 ∂t

∂φ

p

∂z

r

on thefreesurfa e,

−η

p

∂

2 _φ

0 ∂z∂t

+

∂∇φ

0 ∂z

· ∇φ

0 Γ

f s

: z = η

0 ∂η

p

∂t

=

∂φ

p

∂z

− ∇φ

p

· ∇η

0 − ∇φ

0 · ∇η

p

on thefreesurfa e,

+η

p

∂

2 _φ

0 ∂z

2 −

∂∇φ

0 ∂z

· ∇η

0 Γ

f s

: z = η

0 ∂φ

p

∂n

= −

∂φ

0 ∂n

+ V

b

(x) · n

onthe body,

Γ

b

∂φ

p

∂n

= −

∂φ

0 ∂n

ontheseabed,

Γ

d

∂φ

p

∂n

= 0

on the ontrol surfa e,

Γ

w

(16)

2.3. Hydrodynami For e and Body Motions

2.3.1. Equations of motion

The bodymotionequations areobtained a ordingto Newton'slaw:











M

_{· ¨x}

b

= F

H

+ M · g +

P F

ext

δ

b

= I

0 · S · ¨

θ

b

+

h

I

0 · ˙S + ˙I

0 · S

i

· ˙θ

b

= M

b

(F

H

) +

P M

b

(F

ext

)

(17)

where

M

and

I

0

are the mass and inertia matri es,

x

¨

b

and

¨

θ

b

are the a eleration in translation and rotation of the enter of gravity of the body in the global referen e

frame,

F

H

and

M

b

(F

H

)

and

F

ext

and

M

b

(F

ext

)

arerespe tively thehydrodynami and external (Power Take-O (PTO), mooring, vis ous damping, et .) for es and moments

applied on the bodyat its enter of gravity, and

S

is the rotation matrix based on the Eulerangles:

S

=





cos(θ) cos(ψ) − sin(ψ) 0

cos(θ) sin(ψ)

cos(ψ)

0 − sin(θ)

0

1 



(18)

Ω

= ˙

ψz

0 + ˙θy

ψ

+ ˙

ϕx

θ

= S · ˙θ

b

(19) where

Ω

is the bodyrotational ve tor a ording to theEuler angles (

˙

(10)

Figure4: Eulerangles(

ψ, θ, ϕ

)withtheir orrespondingve tors

Thehydrodynami for eandmoment areobtainedbyintegratingthehydrodynami

pressureon the wetted bodysurfa e (normalve tor pointingoutwards fromtheuid).

The pressure

p

is given by Bernoulli's equation, and expressedrelatively to the at-mospheri pressure

p

a

:

p = ˜

p − p

a

. Moreover, the freesurfa e pressure is supposedto be zero:

p = −ρ

∂φ

_∂t

+

1

2 ∇φ · ∇φ + gz

(20)

The velo ity potential is known on the body, as the solution of the BVP. Its lo al

derivatives are omputed using a B-spline approximation of the potential, while the

normalvelo ityisgiven bythebody ondition. Thevelo itypotential timederivative is

a priori not known onthe bodyandneeds to be omputed.

2.3.2. Velo ity PotentialTime Derivative

For a body undergoing pres ribed motion, the equations of motion, and thus the

hydrodynami loads,do not need to be omputed to arryon thesimulation. Sin ethe

velo itypotential isknown onthe bodyat ea htime-step, anite dieren es heme an

be used in post-pro essing, to estimate its time derivative and from this the indu ed

bodyloads.

For a body undergoing free motion, this s heme is not a urate enough and an

lead to instabilities [33℄. Another approa h has been hosen, based on thesolution of a

se ond BVP for the velo ity potential time derivative, whi h also satisesthe Lapla e

equation[28 ℄.

△φ = 0 =⇒ △

∂φ

∂t

= 0

, in

D

(21)

TheBVPforthetimederivativeofthevelo itypotentialissimilartothatforthe

ve-lo itypotential. The boundary onditionsaremixed: Neumannfor thematerial

bound-aries (body, seabed and numeri al ontrol surfa es) and Diri hlet for the free-surfa e.

Sin ethematerialderivative

D

0z

φ

p

(11)











△

∂φ

_∂t

p

= 0

intheuiddomain,

D

∂φ

p

∂t

=

D

0z

φ

p

Dt

−

∂η

0 ∂t

∂φ

p

∂z

on thefreesurfa e,

Γ

f s

: z = η

0 ∂

2 _φ

p

∂n∂t

= −

∂

2 _φ

0 ∂n∂t

+ ¨

x

· n + q

on thebody,

Γ

b

∂

2 φ

∂n∂t

= −

∂

2 φ

0 ∂n∂t

on theseabed,

Γ

d

∂

2 φ

∂n∂t

= 0

on thenumeri al tank walls,

Γ

w

(22)

where

x

¨

isthea elerationve torsintranslationofthenode onsideredand

q

represents adve tion due to the motion of the body. Two dierent expressions for

q

have been derivedbyCointe[28℄andTanizawa[27℄bydevelopingtheNeumann onditiononmoving

boundariesusingtwodierentkindsofa eleration: respe tively,a elerationofthebody

and a eleration ofa uidparti le sliding on the body. The two expressionwere shown

to be equivalent in[29℄ and anew expression, unifyingthem,isre alled here:

q = (Ω · s

1 )

∂φ

∂s

2 − 2( ˙x · s

2 )

− (Ω · s

2 )

∂φ

∂s

1 − 2( ˙x · s

1 )−

+

( ˙x · s

1 )

R

1 ∂φ

∂s

1 − ( ˙x · s

1 )

+

( ˙x · s

2 )

R

2 ∂φ

∂s

2 − ( ˙x · s

2 )

+( ˙x · n)

∂

2 _φ

∂s

2

1 +

∂

2 _φ

∂s

2

2 +

1 R

1 +

1 R

2 ∂φ

∂n

(23)

where

s

1

and

s

2

arethelo al oordinateve torsand

(R

1 , R

2 )

denotethelo al urvature alongtherespe tive lo alve tors.

2.3.3. Impli it method

As seen in the previous se tions, solution of the equations of motion depends on

the al ulation of the hydrodynami for e (Eq. (17) ), whi h in turn depends on the

al ulationofthe velo itypotentialtimederivative thatisfoundbysolving thesolution

of the se ond BVP (Eq. (22 )). However, the body boundary ondition of this BVP

requires knowledge of the a eleration of the body. This leads to an impli it problem,

whi h an be solved using the Impli it method introdu ed by Tanizawa [27℄ and Van

Daalen [34 ℄ for this purpose. The BVP is extendedto in lude the equations of motion

andhydrodynami for e al ulation, whi h arethensolvedsimultaneously.

∂

2 φ

∂n∂t

and

∂φ

∂t

on thebodyarethensolutions ofthis extendedBVP,asisthebodymotion.

Severalothermethods have beenproposed, in ludingtheindire tmethod,themode

de omposition method, and the iterative method (see [35, 36℄). The Impli it method

requiressolving only one additional BVP, ompared to up to sixfor theothermethods.

(12)

3.1. Solution of the BIE

The ollo ation method is used to solve theBVP. The BIE is thus applied to a set

ofnodes,

x

l

,on theboundaries. For givenelements

j

anda eldpoint

l

,theBIE anbe de omposed into two integrals:

I

σ,j

(x) =

Z Z

S

_j

∂φ

∂n

(x

l

)G(x, x

l

)dΓ

(24)

I

µ,j

(x) =

Z Z

S

_j

φ(x

l

)

∂G(x, x

l

)

∂n

dΓ

(25)

x

1 x

2 x

3 u

v

x

G

S

_j

C

_j

Figure5: Lineardis retizationonanelement

S

j

= (x

1 , x

2 , x

3 )

: denitionofthevariables

u

and

v

Lineartriangularelementsareused,viaanisoparametri parametrization,des ribing

both thegeometry and thespatial variation oftheunknowns(see Fig 5).

f

j

(x) = f

j

(x

1 ) + u(f

j

(x

2 ) − f

j

(x

1 )) + v(f

j

(x

3 ) − f

j

(x

1 ))

= f

j

(x

G

) + ∇

s

(f

j

) · r

j

(26)

where

x

G

is the position ve tor of the enter of gravity

G

j

of the element

S

j

=

(x

1 x

2 x

3 )

,

r

j

= x − x

Gj

and

∇

s

(f

j

)

represents the surfa e gradient of

f

j

, whi h an be al ulated fromthederivativesof

f

j

along

u

and

v

.

C

j

is the ontour oftheelement.

Using this dis retization, the twointegrals an bewritten as(see AppendixA):

(13)

where

=

∆ =

I

₃

¯

+ x

G

x

· =

Σ

and

=

Σ

isan operator su hthat

=

Σ ·





f (x

1 )

f (x

2 )

f (x

3 )





= ∇

s

(f )

. Analyti alsolutionsforthesurfa eintegralshavebeengivenbyGuevel[37 ℄. Analyti

solutionsforthe ontourintegralswithRankinesour eswereobtainedby onsideringthe

ontributions of ea hedgeof thetriangular panel(see AppendixA).

To al ulate inuen e oe ients in ases where the eld point is far from the

in-uen ing panel, asymptoti solutions for ea h integral have been developed to speed up

al ulations [38℄.

The BIE applied to the set of nodes, with the ollo ation methods, results in the

following linearsystemto solve,for apoint

i

on theboundary:

G

ij

φ(j) = H

ij

φ

n

(j)

(29)

where

φ

and

φ

n

arethe velo itypotential and normal velo ity ve tors. The matrix

G

in ludes all integrals of Green's fun tion (Eq. (27 )), aswell as thesolid angle terms along the diagonal, and

H

in ludes all integrals of the normal derivative of Green's fun tion (Eq. (28 )). Thesematri es are alledinuen e matri es. The iterative method

GMRES (Generalized Minimal RESsidual method) [39 ℄ is used to solve the resulting

systeme iently.

Some of the verti es may be lo ated at an interse tion between surfa es with

non- ontinuous normal ve tors (for instan e, verti es at the interse tion of the free surfa e

andthebodysurfa e,oronsharp ornersfornon-smoothbodygeometries). Theyrequire

spe ial treatment, to take into a ount the dierent onditions (Diri hlet/Neumann or

dierent normal velo ity and normal denition). All the boundary onditions on these

parti ular points are applied by dupli ating nodes. For a given geometri al lo ation,

severalboundary onditionsare enfor ed,to arrive at dierent solutions.

The ontinuityofthepotentialat theinterse tionis he kedtoensure thevalidityof

theBVPsolution.

3.2. Code a eleration

Three te hniques have been developed to redu e the number of unknowns in the

al ulationoftheinuen ematri es(andhen ethe al ulationtime): partial al ulation

ofthe inuen ematri es; symmetries;and open domain.

•

Partial al ulationinvolves al ulatingthe inuen ematri esonlyonsub-domains that have hanged during the previous time step. A xed boundary during the

simulationdoesnotneed to have itsauto-inuen e matri esupdatedat ea htime

step.

•

Two symmetries are implemented, one verti al along the

(xOz)

plane, and the otherhorizontalon thebottom (forat seabottoms). Thetwosymmetries anbe

used on urrently. By utting the number of mesh elements bya fa tor two, the

al ulation timefor the inuen e oe ients isalso ut by a fa tor of two,whilst

(14)

•

Simulations in open domain are also possible, by onsidering numeri al ontrol surfa es to be su iently far from the body, so that the perturbation dies out

beforerea hingtheseboundaries. A lassi numeri al bea hisusedtoenfor e this

last ondition.

Moredetails on erningthese features an be found in[38 ℄.

3.3. Numeri al bea h

We implemented a lassi numeri al bea h [33 ℄,based onelongation of theelements

onthefreesurfa eandtheadditionofvirtualpressuretermsinthefreesurfa eequations.

For thefullynonlinear freesurfa e equations,for example,themethodyields:











∂φ

∂t

= −gη −

1

2 ∇φ · ∇φ + νφ

∂η

∂t

=

∂φ

∂z

− ∇φ · ∇η + νη

(30)

For ir ular domains, thedamping variable

ν

variesas







ν(r) = αω

(r − r

0 )

λ

2 r ≥ r

0 = R

e

− βλ

ν(r) = 0

r < r

0

(31)

α

and

β

are parameters that adjust the virtual pressure loads and the width of the numeri albea htotheperturbation,respe tively,and

R

e

istheexternalradiusdelimiting thefreesurfa e inthehorizontalplane.

α = 0.7

and

β ≃ 1

werefound to beoptimal. 3.4. Lo al derivative al ulations

Spatial derivatives arepresentin the freesurfa e equation, but alsointhe

q

-term of thebody onditionforthe timederivative ofthevelo itypotential BVP(Eq.(23)). The

se ond-order derivativesarealsorequired inthis al ulation. Thelo alderivativesgiven

by the linear dis retization are onstant on ea h element, requiring a spe ial s heme

for the evaluation of the se ond-order derivatives. Following [40 ℄, a higher order

B-Splinesapproximation hasbeen implemented, allowing easy al ulation of therst- and

se ond-order lo al derivatives as well as better pre ision than a hieved by the linear

dis retization, resulting in a low CPU-time ost. Virtual nodes are used to ope with

dis ontinuities intheplaneof symmetry.

3.5. Time Integration

A fourth-order Runge-Kutta s heme with a onstant time step is used for

time-mar hing. Thevelo itypotentialandwave elevationonthefreesurfa e areadvan edin

timeusing equations Eq.(8 ) and Eq.(9 ). The bodypositionand velo ityareadvan ed

simultaneously using the body motion equation Eq. (17) . The algorithm implies the

solution of the BVP for four dierent mesh ongurations during ea h time step. This

(15)

and body)during thefour substeps ofthe RK4. Thisis referredto as

geometry-lo ked-RK4 [13℄, and results in only one al ulation of the inuen e matri es per time-step.

A time onvergen e of the normal and geometry-lo ked-RK4 showed that, for a given

a ura y, the geometry-lo ked-RK4 leads to a signi ant redu tion in omputational

time. Thus, the modied s hemeis usedinthefollowing.

4. Validation : simulation of a submerged heaving WEC

In this se tion, the results of the CN_WSC for free-body motions are veried at

rst in linear onditions (i.e. small amplitudes of body motion and wave steepness).

Linear potential ow theory is used as a referen e, thanks to the open sour e software

Nemoh [41℄. Subsequently, nonlinear onditions (i.e. large amplitudes of body motion

and wave steepness)arestudied and dieren es withlineartheoryare highlighted. The

stiness of the power take-o is tuned to the wave ex itation, to obtain a me hani al

resonan e. A Fourier analysis is applied to the motion time history, and the mean

absorbed poweris al ulated on several periods ofthe wave ex itation, on e thesteady

state is obtained. Using this methodology, we an obtain normalized responses for the

motionand absorbed power for aset offrequen ies. Thesenormalized responses an be

omparedtothelinearsolutions,i.e. theresponseamplitudeoperator(RAO).Thelimits

oftheWSapproximationarealsoinspe ted,withrespe ttotheunderlyingassumptions.

4.1. WECdes ription

The submerged heaving WEC (SHWEC) that served as the model for this se tion

isthe Ceto WEC,designed byCarnegie Wave Energy Limited [10 ℄. ThisWEC onsists

of a heaving submerged buoy, moored at the seabed. Here we onsider a simplied,

neutrally buoyant spheri al buoy (see Fig. 6, with a spring-linear damping system to

model the mooring and power take-o (PTO). The dimensions are: radius

a = 3.5

m, meanimmersion

d = 2a = 7

m, depth

H = 20

m. Thespring spe i ations,

K

P T O

and

l

0

, are optimized to adjust the body response to the wave input,

A

ω

and

ω

. In order to obtain resonant motions, the spring onstant is set to

K

P T O

= (M + µ

33 )ω

2

, with

µ

33

given by the linear solution. Linear Power Take-O (PTO) damping is taken into a ount. The instantaneous powerand energy absorbed by thePTOare:

(16)

Themeanabsorbedpower

P

abs

, apturewidth

C

W

ande ien y

η

1

,alongwiththewave energy ux

J

for a regularwave, aredened as











P

abs

=

E

P T O

t

W

C

W

=

P

abs

J

m











J

=

ρg

2 8π

A

2 ω

T tanh(kh)

1 +

2kh

sinh(2kh)

W/m

η

1 =

C

W

2a

(33)

The numeri al tank onsists in a ylindri al and regular domain, entered around

the body, see Fig.7 . Verti al and bottom symmetries are used to redu e the numbers

of nodes. The radius of the numeri al domain varies a ording to the in ident wave

length,

R

Dom

= 3λ

witha numeri al absorbing bea h radius of one wave length. Based on su essfulmesh and time onvergen e studies, thetimeand spa edis retisation was

hosensu h that

dt = T /100

and

dr = λ/20

or

dr = 0.5m

for large

λ

,

dr

beingthesize of the smallest elements of the mesh (on the body and on thefree surfa e lose to the

body).

4.2. Linear onditions

The in ident wave onsidered is a small amplitude regular wave, in order to allow

(17)

X

Y

Z

Eta_Incident

1

0.8

0.6

0.4

0.2

0 -0.2

-0.4

-0.6

-0.8

Figure7: Representationofthenumeri aldomainfortheSHWEC

ω (rad/s)

T (s)

λ (m)

A

ω

(m)

ǫ (%)

1.7

3.7

21.3

0.001

0.03

Table1: Waveparametersinlinear onditions

Thesmallwaveamplitudeensureslinear onditions: thewavesteepnessisverysmall,

as is the body motion response, despite the fa t that the value of the spring onstant

was hosento a hieve me hani al resonan e,

K

P T O

= 7.4 10

5

N/m.

Initially, the PTO damping,

B

P T O

, was set to zero, in order to he k the radiation damping of the body. The linear and WS solutions of the body heaving motion are

plotted inFig.8.

0

1

2

3

4

5

6

7

8 t/T

-7.002

-7.001

-7

-6.999

-6.998

z (m)

Heave motion

Z

WS

Z

Linear

Figure8: LinearandWSheavemotionresponses,inlinear onditions(

ω = 1.7

rad/s,

A

ω

= 0.001

m,

k

P T O

= 7.410

6

(18)

0.5

1

1.5

2 circular frequency (rad/s)

7.5

8

8.5

9

9.5

10

10.5 Added Mass (kg)

×

10

4 Heave Added Masse and Radiation Damping given by the linear potential flow theory

0

0.5

1

1.5

2

2.5

3 Radiation Damping (kg/s)

×

10

4

Figure9: AddedMassandRadiationDampinginHeavefortheSHWEC,givenbythelinearpotential

owtheory.

Ex ellent agreement between the linear and WS solutions is observed. Harmoni

analysis ouldnotbeperformed,sin ethesteadystatewasnota hievedduetothesmall

damping. Nevertheless, the agreement validates theimplementation of theequations in

theWS ode.

The PTO damping was then set to

B

P T O

= 5.10

4

kg/s, while keeping all other

parameters onstant. This value is approximately twi e the maximum of the radiation

dampingoftheSHWEC,seeFig.9. Asteadystatewasa hievedafterapproximatelyve

wave periods. In addition to heave motion, the a umulated absorbed energy through

thePTO isplotted both for thelinearandWS solutions inFig.10 .

As for the previous ase, good agreement between linear and WS solutions was

ob-tained in linear onditions. A small a umulated error in the absorbed energy an be

observed,whi hleadstoanosetbetweenthetwosolutions. Itisbelievedthatthisisdue

to thedierent starting pro edures inthetwo numeri al approa hes or an a umulated

numeri al error.

Themeanabsorbedpower, apturewidthandWEC e ien ywere al ulated onthe

three periods after the steady state was rea hed and ompared in Table 2. The wave

(19)

0

1

2

3

4

5

6

7

8 t/T

-7.001

-7.0005

-7

-6.9995

-6.999

z (m)

Heave motion

0

1

2

3

4

5

6

7

8 t/T

0

500 1000

E/A

2 (Wh/m²)

Absorbed Energy

WS

Linear

Figure 10: Linearand WS heave motion and absorbed energy, inlinear onditions (

ω = 1.7

rad/s,

A

ω

= 0.001

m,

k

P T O

= 7.410

6

N/m,

B

P T O

= 5.10

4

kg/s)

P

abs

(W)

B

J

(m)

η

1 (%)

Linear Theory

0.0369

2.61

37.3

CN_WS

0.0367

2.59

37.0

Table2: Meanabsorbedpower, apturewidthandWECe ien ygivenbythelinearandWSsolutions,

inlinear onditions

arelessthan

1%

validatingtheimplementation ofthewave-bodymotion oupling inthe

WS ode.

The body motion and power output responses were then onsidered for a set of

in identwave angularfrequen ies,rangingfrom

0.4

to

2

rad/s. Thewaveamplitude and PTO damping were kept onstant (

A

ω

= 0.001

m,

B

P T O

= 5.10

4

kg/s, respe tively).

The stiness onstant wastuned for ea h frequen y inorder to a hieve resonan e. The

bodyis meshed using approximately

2000

nodes and thefreesurfa e between

1000

and

(20)

0.6

0.8

1

1.2

1.4

1.6

1.8

2 circular frequency (rad/s)

0

0.5

1

1.5

2

2.5 Z/A

ω

Heave Motion Response

Z

CN_WS

Z

Linear Theory

0.6

0.8

1

1.2

1.4

1.6

1.8

2 circular frequency (rad/s)

0

0.5

1

1.5

2 P

abs

/A

ω

2 (W/m

2 )

×

10

5 Absorbed power RAO

J/(kA

ω

2 )

P

_{CN_WS}

P

_{Linear Theory}

Figure11: SHWECmotionandpoweroutputresponses,inlinear onditions(

A

ω

= 0.001

m,

B

P T O

=

5.10

4

kg/s,thestiness

K

P T O

istunedtoadjustthenaturalfrequen ytothewaveex itation).

Perfe tagreement analsobeobservedbetween thelinearandWSresultsa rossthe

wholerangeof frequen ies. This validates theWSmodel.

4.3. Nonlinear onditions

WiththeWS ode,twodierent kindsofnonlinearity anbe taken intoa ount and

studied. The rst relates to the hange in body geometry and position relative to the

freesurfa e. These ond relatesto thesteepness ofthein ident wave. Theseareusually

related, sin elarge wave amplitudesleadto large bodymotions, andhen eitis di ult

to quantify the ee t ofone nonlinearity asdistin t fromthe other. For this reason, we

rstpresent the normalized responses inheave,and meanabsorbed power, for thesame

body parameters (radius

a = 3.5

m and submergen e

d = 7

m). The range of angular frequen iesisstill

0.4

rad/sto

2

rad/s, butthe wave amplitudeisnowequalsto

0.85

m; it follows that the wave steepness ranges from

2.6%

to

31%

. Similar PTO parameters areused: the stiness istuned to a hieve resonan ewith thewave frequen y, whilethe

damping iskept onstant at

B

P T O

= 5.10

4

kg/s.

The domain is meshed the same way as in linear onditions : with

2000

nodes on the body and between

1000

and

1600

on the free surfa e depending on the in ident wavelength.

The linearand WSresults showsigni ant dieren esinthese nonlinear onditions.

Thesedis repan iesaremostnotableontworangesofangularfrequen y. Therstis

(21)

0.6

0.8

1

1.2

1.4

1.6

1.8

2 circular frequency (rad/s)

0

0.5

1

1.5

2

2.5 Z/A

ω

Heave Motion Response

Z

CN_WS

Z

Linear Theory

0.6

0.8

1

1.2

1.4

1.6

1.8

2 circular frequency (rad/s)

0

0.5

1

1.5

2 P

abs

/A

ω

2 (W/m

2 )

×

10

5 Absorbed power RAO

J/(kA

ω

2 )

P

CN_WS

P

_{Linear Theory}

Figure 12: SHWEC motion and poweroutput normalized responses, innonlinear onditions (

A

ω

=

0.85

m,

B

P T O

= 5.10

4

kg/sandstinesstunedtoadjustthenaturalfrequen ytothewaveex itation)

WS approximation. With a onstant PTO damping for all the frequen ies, the lowest

normalized response inheave are obtained for this range of frequen ies. An interesting

result is that linear theory tends to underestimate up to

15%

the RAO in heave and

30%

the mean absorbed power, ompared to the WS approximation, for this range of frequen ies. However,withaPTO dampingtunedfor ea hfrequen y,weshouldbeable

to getthemaximumre overable energy (

J/k

) with both linearand WSapproximation. The dis repan ies between linear and WS solutions are then expe ted to be smallerin

this ondition.

These ond rangeoffrequen iesisbetween

0.8

rad/s and

1.3

rad/s. Thewave steep-ness in this ase is lower than

15%

, whi h means there are fewer wave nonlinearities. In ontrast, with the RAO in heave ex eeding

2

, body nonlinearities are important. The impli ation is that the RAO in heave, and thus mean absorbed power, tend to be

overestimated bylineartheory, upto

4.5%

and

9%

respe tively.

Sin e higher performan e is a hieved in this se ond range of frequen ies, the WEC

willbedesignedtoworkmainlyinthisrange, leadingtoanoverestimation ofthe

perfor-man e ofthe WEC bythe lineartheory. However,this is dependent onthe geometri al

parameters of the body: the nonlinearities related to large wave amplitude and large

motionresponses arenot always asdistin t ashere. This de oupling inour ase allows

us to study thetwo kind of nonlinearitiesseparately.

4.3.1. Motionnonlinearities

Motion nonlinearities are studied rst, for the angular frequen y giving the highest

(22)

ω (rad/s)

T (s)

λ (m)

A

ω

(m)

ǫ (%)

k

P T O

(N/m)

B

P T O

(kg/s)

1

6.3

60

1 13%

3.10

5

5.10

4

Table3: WaveandPTOparametersprodu ingthehighestmotionresponse

0

1

2

3

4

5

6

7

8 t/T

-9

-8

-7

-6

-5

z (m)

Heave Motion

Linear Theory

CN_WS

0

1

2

3

4

5

6

7

8 t/T

-2

0

2 Force (N)

×

10

5 Total Force

Linear Theory

CN_WS

Figure13: Linear,andWSheavemotion,for eresponses,inalargemotion ase(

ω = 1

rad/s,

A

ω

= 1

m,

k

P T O

= 3.10

6

N/m,

B

P T O

= 5.10

4

kg/s)

The linear onditionofsmall motionsisnot respe ted: bytheendof thesimulation,

the motion amplitude rea hes up to

2

m (

50%

of the buoy radius

a

). This results in lear dieren es between the two approa hes: the steady state response amplitude is

overestimatedbylineartheory. Nonlinearitiesarealsovisibleinthehydrodynami for e.

When the body moves lose to the free surfa e, the peaks are deformed in omparison

withthelinearsolution. Thisarisesfrom alo almodi ationofthe radiationdamping,

leading to a smallphase shift for a short time. When thebody moves furtherfrom the

freesurfa e,the radiationdamping,andthusthephaseshift,returntotheirmeanvalue.

Thesenonlinearitiesarethus ompletelyrelatedtothenonlinearitiesinthebodyposition

rather than the large wave amplitude.

Thesenonlinearitiesae tthemeanabsorbedpowersigni antly. Table4summarizes

thedierentquantitiesrelatedtoenergyabsorption, al ulatedinthesamemannerasin

linear onditions, for thelinear andWS approximations. For thein ident wave hosen,

thewave energy uxis

J = 24.1

kW/m.

The dieren es between the two numeri al approa hes in their al ulation of mean

(23)

P

abs

(kW)

B

J

(m)

η

1 (%)

t

CP U

/T

Linear Theory

107

4.44

63 < 1

CN_WS

93

3.85

55

700

Table4: Meanabsorbedpower, apturewidthande ien y,forthelinearandWSapproximations,in

alargemotion ase(

ω = 1

rad/s,

A

ω

= 1

m,

k

P T O

= 3.10

6

N/m,

B

P T O

= 5.10

4

kg/s)

linear theory, whi h does not take into a ount the large body motion, overestimates

the performan e of the WEC, yielding results that ex eed those produ ed by the WS

approximationbynearly

15%

.

Thislargewaveamplitude,however,pushesourWSowsolvertoitslimit: thebody

isalmostrea hingthefreesurfa e. Moreimportantly,thevalidityoftheWSassumption

needs to be he ked. The in ident and perturbation omponents of the wave elevation

arethus plotted inFig.14 for apoint onthefree surfa eand above thesphere enter.

0

1

2

3

4

5

6

7

8 t/T

-1

-0.5

0

0.5

1

1.5 η

(m)

Perturbation

Incident

Total

Figure 14: In ident and perturbation wave elevation, above the SHWEC gravity enter, in a large

motion ase(

ω = 1

rad/s,

A

ω

= 1

m,

k

P T O

= 3.10

6

N/m,

B

P T O

= 5.10

4

kg/s)

The total wave elevation issimilar to thein ident wave elevation, but demonstrates

more nonlinearities (higher peaks, atter troughs and some higher order omponents).

Thesedieren es,whi harisefromfromtheperturbation omponent,arerelativelysmall

ompared to the in ident wave elevation. They are related to the emission of small

waves, whenthe body omes lose to the freesurfa e. Thesewavesdieout qui kly,and

hen e their maximum amplitude o urs dire tly above the sphere - whi h is where the

dataplotted inFigure 14were re orded. The WS ondition thattheperturbation wave

(24)

requiresthatamplitude of motionshould be small,whi h isalso not the asehere. The

WSapproximationisthuslikelytopresentmorea urate physi alresultsthanthelinear

theory.

4.3.2. Wave nonlinearities

Next, we investigated wave nonlinearities, i.e. waves with large steepness and

rela-tivelysmallbodymotionresponse. Thewave hara teristi sandPTOparameters hosen

aregiven inTable 5. Themean immersion isde reasedto

d = 5

m,inorder to in rease

ω (rad/s)

T (s)

λ (m)

A

ω

(m)

ǫ (%)

k

P T O

(N/m)

B

P T O

(kg/s)

d (m)

2

3.14

17

0.85 31%

9.5.10

5 _5.10

4 ₅

Table 5: WaveandPTOparametersinsteepwaves

theinuen e ofthe nonlinearities fromthewave.

0

1

2

3

4

5

6

7

8 t/T

-5.5

-5

-4.5

z (m)

Heave Motion

Linear Theory

CN_WS

0

1

2

3

4

5

6

7

8 t/T

-1

0

1 Force (N)

×

10

5 Total Force

Linear Theory

CN_WS

Figure15: Linear, WSheavemotion,for e,inasteepwave ase(

ω = 2

rad/s,

A

ω

= 0.85

m,

k

P T O

=

9.510

5

N/m,

B

P T O

= 5.10

4

kg/s,

d = 5

m)

The linear, WS solutions of the response in heave motion and dira tion-radiation

for esareplottedinFig.15. Thesmallamplitudemotionresponseassumptionisvalidin

this ase: theheavemotionmaximumamplitudeis

0.5

m(

0.15a

). However,thein ident wave approximation diersbetween thetwomethods. Thisis learly visibleinthefor e

(25)

two solutions. This dire tly ae ts the motion response (smaller for the linear theory)

and hen ethe absorbed energy.

Table6 summarizes the dierent quantities relatedto theenergy absorption,

al u-lated in the same manner asin linear onditions, for the two approximations. For the

sele ted in ident wave, thewave energy uxis

J = 8.7

kW/m.

P

abs

(kW)

B

J

(m)

η

1 (%)

t

CP U

/T

Linear Theory

19.7

2.27

32 < 1

CN_WS

25.3

2.91

42

800

Table6: Meanabsorbedpower, apturewidthande ien y,forthelinearandWSapproximation, in

asteepwave ase(

ω = 2

rad/s,

A

ω

= 0.85

m,

k

P T O

= 9.510

5

N/m,

B

P T O

= 5.10

4

kg/s,

d = 5

m)

Linear theory,whi h doesnot takeinto a ount nonlinearwaves, predi tslower

per-forman e insteep waves. Thisresult isunexpe ted.

Again it is important to study the validity of the WS hypothesis in this ase. To

this end,the in ident and perturbation wave elevations areplotted inFig. 16where the

dira tion is the largest, approximately one wave length after the SHWEC. The linear

in ident wave elevation, used in linear theory, is also plotted to illustrate dieren es

arisingfrom the in ident wave model.

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 t/T

-1

-0.5

0

0.5

1

1.5

2 η

(m)

Perturbation

Incident

Total

Figure16: In ident(linearandnonlinear) andperturbationwaveelevations,onewavelengthafterthe

SHWEC, inasteepwave ase (

ω = 2

rad/s,

A

ω

= 0.85

m,

k

P T O

= 9.510

5

N/m,

B

P T O

= 5.10

4

kg/s,

(26)

After afewperiods,the perturbation wave elevation islo ally almostaslarge asthe

in identwaveelevation. Thetotalwave(sumofin identandperturbation)isalargerbut

steeperwave,whi hishighlylikelytobreak. Thisphenomenonisrelatedtotheemission

of smallwaves due tothe dira tion, whenthe body is lose to thefree surfa e. In this

ase,thefundamentalassumption ofWS-thattheperturbationissmall omparedtothe

in ident wave elevation - isnot respe ted. However,again, this is alsoan assumption of

linear theory, whi h further requires the in ident wave to be linear (small steepness), a

onditionthat isnot fullledeither.

Thus the validityoftheWShypothesismustbequestioned. However, bytakinginto

a ount nonlinearities, itatleastoersa better modelofthephysi sthan lineartheory.

5. Appli ation to a submerged os illating wave surge onverter

5.1. WECdes ription

Thesubmerged os illatingwave surge onverter(SOWSC)thatservedasinspiration

for this se tion is the WaveRoller WEC, developed by AW-Energy [11℄. Lo ated

near-shore (8-20m depth), it onsistsof a ap atta hed to the seabed along one edge. This

edge be omes an axis around whi h theap rotates in a ba k and forth motion under

theinuen eofthewavesurge. Ahalf-s aledevi ewasinstalledinopenseain2007. An

(27)

using a linearmodel in luding stinessand damping terms. The shape and dimensions

ofthedevi eareinformedbythelatestdesign,seeFig.17andTable7. Thewaterdepth

is

H = 9

m.

height (m) width(m) thi kness(m) rotation enter(m) immersion (m)

l = 6

a = 4

b = 1

c = 2.5

d = 5

Table 7: SOWSCgeometri al hara teristi s

Asfor theCeto,thePTO spe i ations,springstinessand damping oe ient,are

hosen in su h a way as to a hieve both resonan e for ea h wave frequen y

K

P T O

=

(I

55 + µ

55 )ω

2

andashorttransientresponse(

< 8T

). Theaddedinertia

µ

55

,required for estimation ofthe resonan efrequen y,is given by thelinearmodel.

The PTO damping alsobeinglinear, the instantaneous powerissimply given by:

P

P T O

(t) = B

P T O

× ˙θ(t)

2

(34)

Thenumeri altankissimilartothatusedfortheSHWEC ase(seeFig.18 ). Meshand

time onvergen estudiesshowednodieren etotheSHWEC ase,sothesametimeand

spa e parameters are used. The body and freesurfa e are meshed using approximately

1200

and

1800

nodesrespe tively,leadingtoamean omputationaltimeperwaveperiod of

T

CP U

/T = 600

.

X

Y

Z

Eta_Incident

0.45

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0 -0.05

-0.1

-0.15

-0.2

-0.25

-0.3

-0.35

-0.4

-0.45

Figure18: Representationofthenumeri al domainfortheSOWSC

5.2. Linear onditions

The rst ase onsidered is small amplitude in ident waves. The motion and power

(28)

plotted inFig.19.

0.6

0.8

1

1.2

1.4

1.6

1.8

2 circular frequency (rad/s)

0

0.5

1

1.5

2

2.5 θ

/A

ω

(rad/m)

Pitch Motion Response

θ

_{CN_WS}

θ

_{Linear Theory}

0.6

0.8

1

1.2

1.4

1.6

1.8

2 circular frequency (rad/s)

0

5

10

15 P

abs

/A

ω

2 (W/m

2 )

×

10

4 Absorbed power RAO

P

_{CN_WS}

P

_{Linear Theory}

Figure19: SOWSCMotion andpowerresponses, inlinearwave onditions (

A

ω

= 0.001

m,

B

P T O

=

5.10

4 kg.m

2 /s

,thestiness

K

P T O

istunedtoadjustthenaturalfrequen ytothewaveex itation)

The motionand power responses show ex ellent agreement between WS and linear

results inlinear onditions;for the power response, thedieren e islessthan

1%

. 5.3. Nonlinear onditions

To omparethemethodsundernonlinear onditions,thewaveamplitudewasthenset

to

A

ω

= 0.5

m, ausing the wave steepness to vary inthe range

3.4%

to

20.4%

. Again, the motion and power responses were predi ted for the two numeri al approa hes, as

depi ted inFig.20.

Signi ant dieren esbetweenthetwoapproa hesarenowobserved: at itspeak,the

power response predi ted by the linear approa h is nearly

200%

that predi ted by the WSapproa h. Sin ethe angular motionRAOranges between

0.5

to

1.5

rad/m, witha wave amplitude of

A

ω

= 0.5

m, it follows that the total angular ompass of the ap is between

10

to

40 ◦

,whilethetraje toryofthetop oftheaprangesfrom

1.4

mto

4.1

m. Thesemotionsarelarge omparedtothedimensionsoftheap,leadingtononlinearities

that arenot taken into a ount bythelinear model. Some of thesenonlinearities are a

onsequen eofthesmalldepththatisa hara teristi oftheSOWSC;however,thelarge

amplitude motionseemsto bethepredominant sour eof nonlinearitiesinthis ase.

Sin e linear theorydoes not take into a ount these important bodymotion

(29)

0.6

0.8

1

1.2

1.4

1.6

1.8

2 circular frequency (rad/s)

0

0.5

1

1.5

2

2.5 θ

/A

ω

(rad/m)

Pitch Motion Response

θ

_{CN_WS}

θ

Linear Theory

0.6

0.8

1

1.2

1.4

1.6

1.8

2 circular frequency (rad/s)

0

5

10

15 P

abs

/A

ω

2 (W/m

2 )

×

10

4 Absorbed power RAO

P

_{CN_WS}

P

_{Linear Theory}

Figure 20: SOWSC Motion and power responses, in nonlinear onditions(

A

ω

= 0.5

m,

B

P T O

=

5.10

4 kg.m

2 /s

,thestiness

K

P T O

istunedtoadjustthenaturalfrequen ytothewaveex itation)

it maybe that this is highlydependent on the hoi e of PTO stiness (that wastuned

inthis study inorder to a hieve resonan e forea h in ident wave frequen y).

The validity ofthe WSassumption is he ked inFig. 21,for

ω = 0.8

rad/s. Pertur-bationwavesemittedfromthebody anbe omeaslargeasthein identwave,primarily

be ause of the large body motions. In fa t it is possible to observe two peaks per

pe-riod, related to the body approa hing the free surfa e twi e per y le. The total wave

elevation is still globally similar to the in ident wave elevation, but demonstrates more

pronoun ed nonlinearities. Hen e, the assumption of WS, that theperturbation should

be small ompared to the in ident wave elevation, might not be fullled lo ally.

How-ever, again, this is also an assumption of the linear theory, whi h further requires the

amplitudeofmotionandthe steepnessofthe in ident waveto besmall - onditionsthat

arenotfullledinthis aseeither. Thus, thea ura yofthemodelprovidedbytheWS

approximationis expe ted to be better than the lineartheory.

For OWSC, itiswell knownthatlineartheoryover-predi ts theenergyperforman e

in omparison to experiment. The dieren e is usually attributed to vis ous damping

andvortexshedding. Ourstudyshowsthatnonlinearee tsduetolargeamplitudebody

motionareresponsiblefor alargeshareofthepowerredu tion. Furtherworkisrequired

to larifywhi hme hanismsarethemostimportant(vortexsheddingornonlinearee ts

(30)

0

1

2

3

4

5

6

7

8 t/T

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8 η

(m)

Perturbation

Incident

Total

Figure21: In identandperturbationwaveelevation,abovetheSOWSC,for

ω = 0.8

rad/s,

A

ω

= 0.5

m

6. CONCLUSIONS

A newow solverbasedon theWSmethodhasbeendeveloped tomodelthe

parti -ular hara teristi s of various WECs. Details of the implementation of the method are

reported in this paper. For studying free moving bodies, the equations of motion have

been integrated into the BVPof the timederivative of thevelo ity potential, following

the Impli it Condition method. An alternative expression, developed in [29℄, was used

forthebody onditionofthisBVP,unifyingthetwoexpressionsgivenbyCointe[28℄and

Tanizawa [27 ℄. The WS ode hasbeen veried for a Submerged Heaving Wave Energy

Converter (SHWEC) anda Submerged Os illating Wave SurgeConverter (SOWSC) by

omparing numeri al simulations inlinear onditions against results from lineartheory.

Timetra esofthemotionresponseandhydrodynami for eshaveshownex ellent

agree-ment for dierent ex itation frequen ies, with or without PTO damping. The motion

and power responses for a wide range of frequen ies have then been studied, using a

Fourier analysis. Thedieren esbetween thetwomodelshavebeenshownto besmaller

than

1%

inboth ases.

Moving to nonlinear onditions, with large body motions and large in ident wave

steepness,signi antdieren esareobservedbetween lineartheoryandtheWSmethod.

For large body motions, linear theory tends to overestimate the motion and power

re-sponses, up to

200%

for the SOWSC. In ontrast, linear theory underestimates the re-sponsein the presen e ofnonlinearities related to steep in ident waves. In omparisons

oflineartheorytononlinearmodels,overestimationismore ommonly ited than

(31)

is maximum. These dieren es show that body nonlinearities may have a signi ant

impa t on the performan e of WEC. Thus, in Wave to Wire models of WECs, it may

be bene ial to use the WS method or at least the body exa t theory method, rather

than lineartheoryalone. For thetwononlinear ases, the pu timefor theWS odewas

two orders of magnitude greater than the real time simulated (

t

CP U

(W S)

T

≃ 800

), with

T = 6.3s

and

3.14s

respe tively. The WS ode an still yield the RAO for a set of 70 frequen ies ina reasonable pu time. Of ourse, omputational time is greatly redu ed

withlinear potential theory. Withtheopen sour e linearBEM ode,Nemoh, ittakesa

fewminutesto al ulate theentire database of rst-order oe ients for atypi al mesh

size of a few hundred panels, and less than a se ond to simulate the response for one

wave period in the time domain. The omputational time for the CN_WSC was also

omparedto thatofafullynonlinearnumeri alwavetank,in[26 ℄. Itwasfoundthatthe

CN_WSC is roughly one order of magnitude faster than the fully nonlinear numeri al

wave tank.

It hasalso been shown thatthe assumptionsof theWSmethod shouldbeveried a

posteriori. Certainly,largeamplitude motionmayleadto largeperturbations. However,

although the WSmethodmay not be validfrom timeto timedue to thelo al

deforma-tion of the in ident wave, its results are still expe ted to be more realisti in nonlinear

onditionswhen ompared withresultsfromlineartheory. Nevertheless, omplementary

studies omparing these results with fully nonlinear solvers or experiments may be

re-quiredtoassessthevalidityoftheCN_WSC. Dissipationduetovis osityandturbulent

ows are not modelled inpotential ow theory. These phenomena ould have a

signif-i ant impa t on the performan e of WECs with large amplitude motion, parti ularly

OCN_WSCs [42 ℄. It is thus important to ompare the results of the numeri al

mod-els with experiments, in order to quantify the main sour es of performan e redu tion:

nonlinearitiesrelated to largebodymotion, or dissipationdue to vis ous ee ts.

In this work, only submerged WECs were onsidered. Future work will onsider

oating bodies. This will require development of methods for tra king and updating

the waterline and automati remeshing of the free surfa e. For onsidering irregular

waves,itmaybeinteresting to ouple theCN_WSC withaHighOrder Spe tral(HOS)

wave propagation model[32℄. Coupling witha multi-bodydynami algorithm may also

be required to deal with WECs omposed of multiple bodies with omplex kinemati

relationships. An extensionto fullynonlineartheoryisalso possible,exploitingthepast

experien e of the laboratory inthis domain [13,23 , 16, 24 ℄. Finally,the CN_WSChas

not yet been parallelized, whi h has the potential to greatly redu e the omputational

time.

7. ACKNOWLEDGMENT

TheauthorswouldliketothanktheFren hANR(Agen eNationaledelaRe her he)

for their nan ial support of this work as part of the proje t ANR11-MONU-018-01