Three complementary approaches for the

development of a flexible membrane tidal energy

converter : analytical, experimental and numerical

Astrid Deporte∗†‡, Martin Trasch†, Gregory Germain∗, Alan Artaux∗†, Jean-Baptiste Drevet†

and Peter Davies∗∗IFREMER, Marine structures laboratory, 150 Quai Gambetta, 62200 Boulogne sur mer, France

E-mail: astrid.deporte@ifremer.fr / gregory.germain@ifremer.fr / peter.davies@ifremer.fr/†EEL Energy, 42 rue Monge, 75005 Paris, France

E-mail: mtrasch@eel-energy.fr / jbf.drevet@gmail.com‡ADEME, 20 avenue du Gresille, 49004 Angers, France

Abstract—The development and the comparison of threedevelopment tools: analytical, experimental and numerical arediscussed. The analytical model provides the operating forces andpredicts the general tends in order to evaluate the consistencyof optimization choices. The experimental prototype shows thereal behaviour and the test results give an important databaseto fit the models. The numerical model, validated with theexperimental results, gives access to many more parametersand configurations but it needs much time to process. Energyconversion is discussed.

Index Terms—Hydrodynamic, marine energy, experimentaltrials, flume tank, flexible membrane

I. INTRODUCTION AND PRESENTATION OF THE EEL

ENERGY CONCEPT

The use of renewable resources such as sunlight, wind or

water (wave, geothermal, tidal energy) is increasing. Among

them, tidal current energy has comparative advantages: the

resource is predictable, located and could provide enough

energy to be attractive [1], [2].

Currently, the main technology is based on the turbine (for

instance Kobold [3], Darrieus [4], Cormat [5], Open Hydro

[6], Tidal Generation Ltd [7], Seagen turbines [8], [9]). But

such machines are characterized by high costs of production,

installation and maintenance due to the complex geometry of

the blades, the high shear forces and the harsh exploitation

conditions in open sea, especially where tidal resource is

interesting [2], [10], [11].

In this severe environment, EEL ENERGY proposes a

different system, based on a flexible membrane set in motion

by currents [12], [13]. The system, described on the figure 1,

is a sort of flag constructed in an semi-rigid material: flexible

enough to undulate and rigid enough to transmit energy to

converters. A compression is applied between the frame and

the downstream extremity using a stretched cable. In this

configuration, the force of the current is transmitted to the

membrane and creates an undulating motion. A travelling wave

is observed in the axial direction. The front and rear flaps help

to bend the membrane at the beginning and extend its motion

at the end.

Fig. 1. EEL ENERGY tidal energy converter - CAD view

Low speed linear electromagnetic converters have been

specifically developed to convert these low frequency un-

dulations into electricity. Converters consist of independent

modules disposed along the length of the membrane, half

below and half above. The independence of the adjustable

modules allows an optimal capture of the fluid forces for each

flow velocity.

Much research has been done to study interactions between

slender bodies and axial flow [14], [15], [16]. Comparison

between experimental and numerical models have also been

made by Watanabe [17], [18], Paıdoussis [19] and Eloy [20].

But very few of these studies focus on a buckled membrane

undulating under the action of an axial flow.

To study this disruptive technology, we have been develop-

ing three tools: an analytical model, an experimental prototype

and a finite element model. In this article, we describe the

analytical model first. It allows the main parameters and the

operating forces to be examined in order to identify ways of

optimization. Then the 1/20th prototype is described with the

instrumentation used during trials. It is used to validate the

models. Its size is adapted to the French Marine Institute

(IFREMER) flume tank and limits the edge effects. But

once the prototype is built, the number of configurations is

limited. So we have also been developing a 2D finite element

model with the commercial sofware ADINA. Results in terms

of frequency, modes and amplitude are compared and the

discussion concludes on the advantages and limits of each

approaches.

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Proceedings of the 11th European Wave and Tidal Energy Conference 6-11th Sept 2015, Nantes, France

ISSN 2309-1983 Copyright © European Wave and Tidal Energy Conference 2015

II. DESCRIPTION OF MODELS

A. Analytical model

The aim is to reproduce EEL Energy converter motion

with a simple analytical model. To take into account the

fluid-structure interaction, we make several assumptions. 3D

effects are neglected and an inextensible membrane with small

deflections submitted to lateral loads is considered. This leads

to Euler Beam theory to describe the solid’s physics. The

material is modelled as a Kelvin-Voigt viscoelastic material.

The small movement at the leading edge caused by the

arms is represented by an additional length of the membrane

calculated to have equivalent stiffness. Effects of flaps are

represented as the lift force of an inclined plate, concentrated

at the beginning and the end of the membrane. Force depends

on local membrane angle with horizontal and it is described

by the Equation 1.

Fflapθ(s, t) =π

2ρfSflapU

2δ(s, Lflap)∂y

∂s(1)

with Sflap the flap surface and δ(i, j) the Kronecker delta.

We also choose to model the cables by a sum of a buckling

strength and drag force induced tension spread all over the

length of the membrane. The induced tension is describes by

the Equation 2 and represented on Figure 2.

T = Tbuckling + Tdrag =π2EI

(0.7L)2+

1

2CdρfSU

2 (2)

with Cd the drag coefficient.

Fig. 2. Sketch of the analytical model

The flow is considered incompressible and non-viscous.

With the assumption that the length of the beam is much

greater than its width, the slender-body theory can be used

[21]. Forces and moments of added mass derive from acceler-

ation that fluid particules undergo when they interact with the

membrane. We use the notion of fluid derivative: the operator

d/dt is the derivative in the moving frame. Local geometry

is defined by the deflection of the membrane, it has a vertical

position y(s, t) and an angle θ(s, t) (Figure 2).

The fluid velocity orthogonal to the solid body is noted wn.

wn(s, t) =∂y

∂tcos θ − U sin θ (3)

The first component is time derivative of solid frame, the

second is due to particle deflection by the inclined body.

Under the hypothesis of small deflection, we assume that the

curvilinear coordinate s is almost equal to linear coordinate x.

For small angles, one can write Equation 4, which is the fluid

derivative operator ddty.

wn(s, t) =∂y

∂t− U

∂y

∂s(4)

A classical result of slender-body theory [21] is Equation 5

where Φ is the fluid potential and mf = ρfλπ , approximation

of added mass for wide flags [22].

∆P (s, t) = mf [∂Φ

∂t+ U

∂Φ

∂s]y (5)

This leads us to:

∆P = −mf (∂

∂t+ U

∂

∂s)2y (6)

The previous hypotheses permit to represent the problem

as:

EI∂4y

∂s4+ms

∂2y

∂t2+D

∂5y

∂s4∂t+T

∂2y

∂s2+Fflap

∂y

∂s= ∆P (7)

The non-dimensional parameters used here are the same as

in Padoussis’ publications [23]:

η = yL : non-dimensional beam deflection

ξ = sL : non-dimensional curvilinear coordinates

β =mf

mf+ms: mass ratio

u = UL√

mf

EI : non-dimensional speed

µ = D

L2

√EI(mf+ms)

: non-dimensional material damping

Γ = L2TEI : non-dimensional cable compression

fflap =FflapL

3

EI : non-dimensional flap force coefficient

τ = t 1L2

√

EIms+mf

This leads to the following non-dimensional equation, where

the following notation is used: ()′ = ∂∂ξ and () = ∂

∂τ :

η′′′′ + (u2 +Γ)η′′ +2β1/2uη′ + µη′′′′ + η+ fflapη′ = 0 (8)

This system is solved by a particular solution, considering a

periodic motion (Equation 11) and a constant wave length λ.

The Galerkin solution of variable separation is then applied:

η(ξ, τ) = ΣNj=1ηj(ξ, τ) = ΣN

j=1φj(ξ)qj(τ) (9)

with:

φj(ξ) = α1 cosh(λjξ)+α2 sinh(λjξ)+α3 cos(λjξ)+α4 sin(λjξ)(10)

and

qj(τ) = aeiωjτ (11)

Coefficients αi and λj are found through the boundary

conditions, here we consider a cantilever beam (Equation 10).

In order to obtain an ordinary differential equation of order

2 in time, we then multiply by φi, and integrate on interval

[0, 1] [19]. The system to solve is then:

[M ]qj + [C]qj + [K]qj = 0 (12)

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Where mass matrix [M], damping matrix [C] and stiffness

matrix [K] are:

• Mij = dij• Cij = µλ4

jdij + 2β1/2ucij• Kij = λ4

jδij + (u2 + Γ)bij + fflapφ′j(ξ

eq.bras)φi(ξ

eq.bras)

And the coefficients are:

• bij =∫ 1

ξeq.bras

φ′′j φidξ

• cij =∫ 1

ξeq.bras

φ′jφidξ

• dij =∫ 1

ξeq.bras

φjφidξ

Solution is z = aeγt = aeiωt, which is equivalent to the

following eigenvalue problem: (γ[I]− [R])[A] = 0. We define

then the matrix:

R =

[

−M−1C −M−1K

I 0

]

Non-dimensional natural frequencies are R matrix eigenval-

ues divided by imaginary number i. Undulation mode is then

to be selected, knowing that unstable modes have a negative

imaginary part of their natural frequencies [23].

To calculate the undulation amplitude, we use the in-

extensibility of the solid:

(∂x

∂s)2 + (

∂y

∂s)2 = 1 (13)

And the condition brought by the cable:

x2L + y2L = d2 (14)

We do then a Taylor expansion of first order. Neglecting the

4th-order term, and dividing by L2, we get:

1−∫ 1

0

η′2ds+ η21 = ι2 (15)

We substitute η from Equation 9 in Equation 15. Terms

are separated between those constant in time and those time

dependent, the following equality is obtained:

1− ι2 = 2‖a‖2[∫ 1

0

‖φ′‖2ds− ‖φ1‖2] (16)

The non-dimensional amplitude formulation is then:

‖a‖ =

√

1− ι2

2(∫ 1

0‖φ′‖2ds− ‖φ1‖2)

(17)

And A = 2 × ‖a‖ × L is the dimensional amplitude at

downstream edge.

B. Experimental 1/20th prototype

The prototype presented in previous articles [12], [13] was a

1/6th prototype (Figure 3). Made with a carbon skeleton recov-

ered by rubber membrane, its stiffness was not homogeneous

so its Young Modulus has to be approximated to be compared

with the numerical models. This prototype was dedicated to

sea trials and it was not well-adapted to the flume tank of

Boulogne sur mer. Its size generates important side effects

making it impossible to study the flow around the structure.

Fig. 3. 1/6th prototype in the flume tank of Boulogne sur mer

To work on the development of the system, a 1/20th

prototype (Figure 5) has been developed to be used in the

IFREMER flume tank. The dimensions of the flume tank are

8 meters length, 4 meters width and 2 meters depth (Figure 4).

The fluid is unsalted water and the velocity can reach 2.2m/s.

With or without a honeycomb, the turbulence rate can be

changed from 3 % to 15%. Waves can be generated and an

hexapode can be used to impose specific motions.

Fig. 4. Sketch of the IFREMER flume tank in Boulogne sur Mer

The characteristic length of the prototype L was determined

to allow the wake observation in the flume tank. To do so, the

numerical model, validated by former tests on the 1/6th pro-

totype, was used. The fluid similarity is based on the Froude

number (Fr = U√gL

) to take into account the inertia effects

with a compatible flow speed range in the flume tank. The

Reynolds numbers similarity is then neglected even if it may

exist a factor 100 between the experimental scale (Re ≈ 106)

and the full scale. Studies on turbines have shown an influence

of Reynolds number on power coefficient, we will check if

it is also the case on the membrane system. The general

geometry is the same as the 1/6th prototype but it was made

with only one material: Polyacetal (POM-C). Constructed from

one block, some geometric discontinuities generate a more

important transverse stiffness and avoid twisting motions. The

prototype has then a two dimensional behaviour. The material

was chosen to be equivalent to the 1/6th prototype stiffness

using a scale factor. Bending tests were performed on the

prototype to evaluate material properties. Some samples of this

material have been tested separately. The prototype measures

about 1 meter square, so we can neglect the edge effects.

The prototype is linked to a frame by rigid arms. The

fixation between the arms and membrane is made by two thin

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Fig. 5. Picture of the 1/20 th prototype during trials

plates to ensure clamping between arms and membrane. The

framework is fixed to a beam, located above the free surface,

through a 6 components load-cell. The stretched cable used

to pre-stress the membrane is attached between the bottom of

the frame and the end of the membrane part of the prototype.

The instrumentation is composed of a 6 component load-

cell recording drag and lift forces and associated moments. A

tracking motion system following 6 targets motions (LED) is

used to characterize the motion of the membrane. The images

are captured by CCD video camera (Photon Focus MVI-

D1312-100-GB, 1082x1312 pixel2) and the image-capturing

rate is 20 frames/s. The images are then analysed with the

open-source software Blender.

The small size of the prototype allows us to test other

hanging conditions, arm rigidities and the length of cables.

On the flume tank, we use a speed range from 0.8m/s, the

beginning of the oscillation to 1m/s with a step of 0.05m/s.

At this time, there is no ”converter” installed on the device.

The installation of micro-dampers along the central line to

simulate the converters effects is planed for May 2015.

C. ADINA finite element model

Automatic Dynamic Incremental Nonlinear Analysis (AD-

INA) is a software developed by Dr. J.K. Bathe [24] and

associates since 1986. It is a commercial software for linear

and non linear analysis and multiphysic coupling. To deal with

strong coupling between fluid and structure, we use a direct

solver. Fluid and solid parts are interacting at the same time,

so we need to solve the equations together but the two parts

are defined separately [25].

Equations 18 and 19 correspond to nonconservative Navier-

Stokes equations with f b the body force vector of the fluid

medium and τ the stress tensor.

∂ρf∂t

+ v · ∇ρf + ρf∇ · v = 0 (18)

ρf∂v

∂t+ ρfv · ∇v −∇ · τ = fB (19)

The fluid part is 6L height and 14.5L length as shown on

Figure 6. The device is in the middle of a tunnel and the

velocity on the wall is specified with a no-slip condition.

The fluid is defined as laminar and sightly compressible

(Equation 20): the density depends on the bulk modulus κ.

There are 136054 fluid elements.A convergence mesh study

has been done on a similar model and this configuration was

efficient.

ρf = ρfo · (1 +p

κ) (20)

Fig. 6. Mesh of fluid part of the numerical model

The solid part includes all prototype elements: the arm, the

flaps, the cables and the membrane. All materials are supposed

to be elastic and isotropic: the equivalent Young modulus is

calculated by an equivalent stiffness. To simplify the model,

we avoid geometric non-linearities.

The cable is modelled by a rigid beam which is dis-

placed by its upstream extremity, the other being fixed on

the membrane’s end. The membrane is thereby pre-stressed.

Kinematic assumptions are large displacement and small strain

formulations. There are 1083 solid elements.

Fig. 7. Mesh of solid part of the 2D numerical model

The fluid-structure interactions are made only on the mem-

brane and flaps parts, in pink and red on the Figure 7, because

arms and cables don’t cover all the width of the prototype and

so don’t interact with the fluid.

The solver is a Sparse solver. The time integration method

uses a composite method. The time step is 0.01s. The model

is in two dimensions to reduce computational time cost. This

is realistic compared to the prototype behaviour.

III. RESULTS AND DISCUSSION

A. Variables for comparison and studied cases

To characterize the motion, we look for the oscillation

frequency and for the maximum of the amplitude envelope.

We study six targets separated by 0.2L from the beginning to

the end of the membrane part. Another way to characterize the

dynamic of the movement is to quantify the wave propagation

between each target.

We limit the comparison to experimental prototype config-

urations. The prototype begins to oscillate around 0.8m/s and

it has been tested up to 1m/s.

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B. Analysis in frequency and amplitude

On Figure 8, the comparison of oscillation frequency from

all models is quite good. We note that the main frequency is

the same for all targets. The analytical model starts to oscillate

for the same flow velocity but at a higher frequency. The

difference between analytical and experimental results tends to

disappear when the velocity increases. The numerical model

remains around the experimental curve, with a more significant

difference just for 0.9m/s.

Fig. 8. Oscillation frequencies for different flow velocity for the three models

On Figure 9, amplitudes are overall well represented and

the amplitude increases along the membrane length. The

final amplitude diminishes when the fluid velocity rises. The

numerical model shows less differences between the different

velocities while the analytical model accentuates spread. The

analytical model doesn’t show the slope at the beginning

because there is no discontinuity between the arm and the

membrane and the flap effect is simplified by a tangential

force. It extends the length but doesn’t help to initiate the

motion.

Fig. 9. Amplitude envelopes for velocities from 0.8 to 1m/s for the threemodels at 6 targets

The trajectories take into account the way the membrane

curves. Figure 10 shows the trajectory of each target for the

three models. Experimental and numerical results are very

close except at the beginning where the link between the

arms and the membrane isn’t exactly the same. Numerical

results have fluctuation close to the centreline, in particular for

targets P4 and P5, jolts are not damped by the material and

the link. In the analytical model, the abscissa is calculated a

posteriori with Equation 13, after solving the y-problem and

cable strength isn’t exactly in the attached point direction.

Fig. 10. Targets trajectories during oscillations for the three models(u=0.85m/s)

Finally, we compare the amplitude time signals for the

targets P3 and P6 on Figure 11. The analytical model proposes

a sinusoidal signal for both targets while experimental and

numerical signal differ. They look like sinusoidal signals for

P3 but look more like a triangular signal for P6. Perturbations

(parasite frequencies) are more present on the numerical result.

There is no material damping in the numerical model so the

slightest jerk spreads.

Fig. 11. Time Amplitude variation of targets P3 and P6 (u=0.85m/s)

C. Analysis of modes and natural frequencies

Figures 12, 13 and 14 show the Fourier spectrum of ampli-

tude oscillation of each target from the different models. The

analytical and numerical models don’t take into account the

same harmonics but the second mode frequency is largely the

most important in amplitude and quite similar. The analytical

model uses just the main frequency when the numerical and

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experimental results show four harmonics but not the same

ones. The second mode amplitude is increasing from the

beginning to the trailing edge of the membrane but the four

frequencies are found on all targets.

Fig. 12. Fourier spectrum of targets amplitude from analytical model(u=0.85m/s)

Fig. 13. Fourier spectrum of targets amplitude from experimental model(u=0.85m/s)

If we consider the main frequency as the second mode

frequency, during trials, excited frequencies correspond to the

harmonics 2, 3, 4 and 5. In the numerical simulation, the

excited harmonics are the 2nd 4th, 6th and 8th, the even

frequencies. The exited harmonics depends on the boundary

conditions and the cable length but the second mode is

always the most important. A possible explanation is that the

numerical model is not hung exactly as the prototype. The

clamping conditions of the prototype are not perfect as in the

numerical model.

D. Analysis in terms of wave propagation velocity

The wave propagation velocity corresponds to the distance

between targets divided by the delay time. The phase is

obtained with a Fourier series decomposition. Table I deals

Fig. 14. Fourier spectrum of targets amplitude from numerical model(u=0.85m/s)

with the propagation wave velocity between two consecutive

targets. For the experimental results and numerical model, the

wave accelerates until P4 and decelerates between P5 and P6.

On the analytical results, it accelerates from the beginning to

the end.

UP1−P2 UP2−P3 UP3−P4 UP4−P5 UP5−P6

Analy 0.6 0.64 0.69 0.84 1.18

Exp 0.38 0.77 0.96 0.73 0.50

Num 0.44 0.94 1.16 0.85 0.58

TABLE IPropagation wave velocity between 2 consecutive targets (m/s)

The small distance (0.2L) and short delay time could explain

differences in absolute results between the experimental results

and numerical model. For the analytical model, the direction

for cable forces could be a possible source of error.

The converters, placed above and below the membrane, will

slow down the membrane motion and specially the wave prop-

agation. This parameter could be useful in the optimisation

of the power take off with the adjustment of independent

converter modules.

E. Forces

Lift and drag force are represented on Figure 15 and 16 on a

representative period. We see double oscillation frequencies on

the drag force. The main trend is the same on all models but

experiments and numerical models show, as we have noted

previously, more harmonics. Both numerical and analytical

models are conservative compared to the experimental results.

We have good variation and the right order of magnitude.

Models provide higher forces so they can be used to dimension

the support.

IV. DISCUSSION

These models are compared in their capacity to represent a

specific phenomenon. The advantages and disadvantages refer

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Fig. 15. Representative non-dimensional drag force on a period (u=0.85m/s)

Fig. 16. Representative non-dimensional lift force on a period (u=0.85m/s)

to the development of EEL Energy tidal energy converters. The

constraints of cost and time are compared to effectiveness, an

extreme precision is not always needed.

The analytical model is very fast, a few seconds on Matlab

software and it gives critical velocity, frequency and amplitude

with a good accuracy. To do so, some parameters such as the

cable strength or the amplitude have to be adjusted. Once

adjusted, it could serve to optimize the behaviour quickly.

Concerning the cable, the direction of force application is not

the attachment point but spread all over the length. It applies a

compression but less than in the real case. The different parts,

other than the membrane, are defined by their impacts on the

membrane and not directly in the fluid-structure interactions.

Moreover, only the main frequency (corresponding to the

second mode) is taken into account, so we do not reproduce

the jerky parasite motion. The fluid is also simplified, only the

pressure on the beam is modelled, potential vortex detachment

are not taken into account.

Tests in flume tank generate a reference database which

is used to validate our models. Small size and one-piece

conception make it easy to modify and to place into water.

After conception, only small modifications could be made:

the arms attached and the cable length. But the material

(polyacetate) is quite cheap and other prototypes could be

made on the same model. It was conceived so that its wake

can be characterized from LDV or PIV measurement.

The numerical model developed with ADINA is a 2D

model. It shows good accuracy in terms of amplitude and

frequency. Not only the main frequency but the four first

harmonics are excited even if they are not exactly the same

as on the prototype. The model is less damped than in reality.

The fluid domain is limited by the tank walls, far enough to

be neglected but different of the free surface effects in the

flume tank. All parts are distinct from each other as in reality,

and the interactions are limited to the membrane and flap parts.

The main inconvenience is the computing time, a few days for

one configuration, on a HP Z820 Workstation (32 Go RAM,

processor: Intel(R) Xeon(R) CPU E5-2687W 0 @ 3.10 GHz

3,10 GHz), the licence limited the use to 8 processors.

Each model has its advantages in the industrial develop-

ments. The analytical approach, very fast, could be used to

identify the main parameters and find the best combinations.

It offers an overview of the behaviour over a large range of

configurations. The experimental prototype provides concrete

information on specific cases. The dynamics of the motion

and the forces on the devices are recorded and correlated with

model results to fit them. The size of the prototype is well-

adapted to the flume tank and will allow a characterization

of the flow around the structure. The numerical model then

serves to check with good accuracy the configuration selected

with the analytical model and, if necessary, to improve the

prototype. Both analytical and numerical models could be used

to extrapolate results to a higher scale.

Compared to real functioning condition, one of the main

differences here is the lack of a turbulence model. In the

flume tank, the turbulent intensity rate is about 3% while

previous studies show that at sea, it could reach 20% [26].

The imposed turbulent intensity level could influence not only

loads on the machine and power take-off but also the wake and

the farm arrangement [27], [28]. The flow is also limited in

two dimensions and delimited by walls, in the numerical and

experimental tests, which has an influence on hydrodynamic

and structure behaviour [29]. The free surface effect needs

to be investigated in details, as well as Reynolds effects

like in [30]. Contrary to rotating turbines for which the flow

behaviour around the blade profile influences the performances

of the turbine, the flow close to the structure plays a lower part

on the propagation speed of the membrane ondulation.

We also show that despite all the attention, the boundary

conditions are not exactly the same. They are efficient enough

to reproduce the motion on the second natural mode with

a good accuracy but the following natural frequencies are

not identical. Causes could be find in the lack of material

damping or boundary conditions.

With these three models, the system can be characterized

without power take-off system. We are actually working on

the simulation of the power take-off using each approach.

At full scale, linear electromagnetic converters are being

developed. The conversion is proportional to the relative

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velocity of the magnet compared to the coil (Equation 21).

Fconv = C · vmagnet/coil (21)

But, it is not possible to miniaturize this system. So, on

the 1/20th experimental model, we cannot use electromagnetic

devices. We decide to dissipate the energy into heat with

hydraulic linear micro-dampers. The commercial adjustable

available micro-dampers need to be characterized before inte-

gration on the membrane as shown on the Figure 1.

With the analytical model, the first evaluation of power take

off will be made by a dissipation due to material damping,

proportional to time curvature deformation. This term 22 can

be added in the Equation 7:

Fdissipation = Dpower∂5y

∂s4∂t(22)

To sum up, we accentuate the material dissipation like piezo-

electric device [31]. It gives a first idea of how much energy

we can produce and how to adjust the real converters.

For the numerical model, ADINA proposes some particular

elements called spring, they connect the attachment points

which are shifted away from the membrane centreline shown

on Figure 17.

Fig. 17. Numerical solid model with 12 converters

First results presented in [13] give good confidence for the

behaviour of the system and in the prospective production.

V. CONCLUSION

The aim of this comparison is to delimit the area of action

of each model to save time and money. The analytical model

runs very quickly and allows the behaviour to be approximated

for on a large range of parameters. Thereby, we can select

the main parameters and the best configurations. The 1/20th

prototype gives a database to fit models. Made in one material,

it is easier to compare with the analytical and numerical

model than the former 1/6th prototype. Attachment boundary

conditions are also better controlled even if we find differences

between higher natural frequencies. The numerical model is

very time consuming and, without a much more powerful

computer, it could not be used to optimize the system. Its

accuracy could be used to check some specific configurations

as was the case to work on the scale effect. Another advantage

is the access to forces and the fluid domain. Forces help in

the design of support and the fluid domain describes the wake

of the machine which is necessary for the farm design. We

can also access the fluid flow characterisation, with the PIV

method for instance, to validate the fluid part of the numerical

model and to study the wake of one machine.

We are now working on the power take-off modelling.

With the analytical model, we will accentuate the material

damping along the length so we will get the same distribution

of converters but the forces will be more spread. On the

prototype, adjustable linear micro dampers will be installed on

the centreline. The location will be the same as on the 1/6th

prototype. Concerning the numerical model, the first tests have

already been done, so we are testing more configurations and

waiting for experimental validation of the dynamics of the flow

and the power take-off.

The next step will be the characterization of the interaction

between several machines, as was done for horizontal axis

turbine in [32].

ACKNOWLEDGMENT

The authors gratefully acknowledge the financial support

of French Environment and Energy Management Agency

(ADEME) and EEL ENERGY. We would like to thank B.

Gaurier, J-V. Facq and T. Bacchetti for their assistance and

advice during trials. Finally we thank C. Eloy for his help

with the analytical part of this work.

NOMENCLATURE

A : Amplitude

a = A/L : Non-dimentional amplitude

d : Length of compression cable

D : Material damping

E : Young modulus

f : Frequency

I : Moment of inertia

L : Characteristic length

mf : Approximation of the added mass

ms : Linear mass

∆P : Differential fluid pressure

q(τ) : Non-dimentional time function

s : Curvilinear coordinates

T : Cable force

U : Flow velocity

u = U · L√

mf

E·I : Non-dimentional fluid velocity

y : Beam deflection

β =mf

ms+mf: Mass ratio

Γ = L2·TE·I : Non-dimentional cable compression force

δ(i, j) : Kronecker delta

η = y/L : Non-dimentional beam deflection

θ(s, t) : local membrane angle with horizontal

ι = dL : Non-dimentional cable length

κ : Fluid bulk modulus of elasticity

λ : Non-dimentional wave length

µ = D

L2·√

E·I·(mf+ms): Non-dimentional material damping

ξ = s/L : Non-dimentional curvilinear coordinates

ρs ρf : Solid and fluid density

τ = t∗ = t · 1L2 ·

√

E·Ims

: Non-dimentional time

φ(ξ) : Non-dimentional shape function

807B3-4-

ω : Angular frequency

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