optimal energy harvesting by vortex-induced vibrations … · optimal energy harvesting by...

OPTIMAL ENERGY HARVESTING BY VORTEX-INDUCED VIBRATIONS INCABLES

Clement Grouthier!Sebastien MichelinEmmanuel de Langre

LadHyXDepartment of MechanicsEcole Polytechnique

91128 Palaiseau, FranceEmail: [email protected]

ABSTRACTThe issue of global climate change and the grow-

ing energy demand induce a need for innovative energyharvesting devices. The possibility to harvest energy us-ing VIV of a long tensioned cable or of an elastically-mounted rigid cylinder is investigated throughout this pa-per. A simple wake-oscillator model is used to representthe major characteristics of the complex dynamics of suchstructures. The optimal efficiency of the two devices aresimilar and are reached when the solid and its fluctuatingwake are in lock-in condition. The sensitivity of the op-tima of such energy harvesters with flow velocity is alsodiscussed.

INTRODUCTIONGeophysical flows represent a widely available

source of clean energy, useful to tackle the global energydemand using for example wind turbines, marine turbinesor wave energy converters.

An original way to extract energy from these flowsis to take advantage of flow-induced vibrations, reviewedfor example in Ref. [1]. For instance, several devicesbased on fluid-elastic instabilities like transverse gallop-ing or flutter have already been introduced in Refs. [2–5].Another kind of flow-induced oscillations that can be use-ful to harvest energy from a flow is the vortex-induced vi-brations (VIV) of a bluff body [6]. The strong couplingbetween a solid and its fluctuating wake may lead to alock-in phenomenon between the solid dynamics and thevortex shedding, resulting in high amplitude oscillations,

!Address all correspondence to this author.

that can be used for energy harvesting [7, 8].Yet, the energy density in geophysical flows is small,

and large systems are required in order to harvest signifi-cant amount of energy. Besides, VIV of long cables likeoil rigs anchors or risers have been extensively investi-gated since they are of capital importance for offshore in-dustry [9]. The possibility to harvest energy from a flowusing VIV of long tensioned cables is consequently stud-ied in this paper [10].

Extensive experimental and numerical analysis haveshown that the dynamics of slender structures in VIV arevery rich and complex [11–13]. The most important fea-tures of these dynamics (frequencies, wavenumbers) arehowever well predicted by a simple wake oscillator ap-proach [14–16]. This approach is used in the present pa-per to investigate the energy harvesting using VIV of longtensioned cables.

In a first section, the model is presented and thegeneric case of energy extraction using VIV of anelastically-mounted short rigid cylinder is analyzed. En-ergy harvesting using VIV of an infinite tensioned cableis investigated in a second section.

THE ELASTICALLY-MOUNTED RIGID CYLINDERThe energy harvesting from an elastically-mounted

rigid cylinder VIV is first investigated, see Fig. 1. Thefluid density and velocity are respectively noted ! andU ,while D and ms stand for the rigid cylinder diameter andmass per unit length. Let also r and h be the damping andstiffness coefficients of the elastic support.

Flow-Induced Vibration, Meskell & Bennett (eds) ISBN 978-0-9548583-4-6

291

h

r

DU

X Y

Z

L

rU !

!

FIGURE 1. ENERGY HARVESTING FROM VIV OF(LEFT) AN ELASTICALLY-MOUNTED RIGID CYLINDERAND (RIGHT) AN INFINITE TENSIONED CABLE.

The fluid-solid model for energy harvestingThe cross-flow displacement Y of the cylinder is de-

scribed by

mt" 2Y"T 2

+(r+ ra)"Y"T

+hY = Fwake, (1)

where mt = ms+ma, ma = #$D2CM0/4 being the addedmass per unit length and CM0 the added mass coefficient.The fluid added damping is defined by ra = $DUCD/2,where CD is the cylinder drag coefficient and Fwake de-notes the wake forcing on the body [14]. Two frequen-cies appear : (i) the natural frequency of the solid instill fluid %s =

!

h/mt and (ii) the vortex shedding fre-quency behind a fixed cylinder, % f = 2#StU/D, St beingthe Strouhal number [1]. Using y = Y/D and t = % f T ,the dimensionless equation for the solid motion reads

y+"

& +'µ

#

y+( 2y= fwake, (2)

where ( = %s/% f is the frequency ratio and & = r/mt% fthe damping coefficient. The dimensionless fluid addeddamping coefficient '/µ is defined by the stall parameter' = CD/4#St [17] and the mass ratio µ = mt/$D2 [14].As far as the cylinder is concerned, harvesting energyfrom its motion comes to a loss of energy, which is mod-elled by the damping term r"Y/"T , or & y in dimension-less form. The efficiency of such an harvesting device is

defined as the ratio between the time-averaged extractedpower and the energy flux through the section of the cylin-der $DU3/2 [3, 8, 10, 18]. In dimensionless form, it re-duces to

) = 16µ#3St3$

& y2%

. (3)

Following [14], a wake-oscillator approach is used tomodel fwake. The fluctuating load due to the wake isassumed to be proportional to a fluctuating lift coeffi-cient q= 2CL/CL0, which satisfies a Van der Pol oscillatorequation,CL0 being a fluctuating lift coefficient. This sec-ond equation is then coupled with Eqn. (2) by an inertialforcing to define the fluid-solid model

y+"

& +'µ

#

y+( 2y=Mq, (4a)

q+ *&

q2!1'

q+q= Ay, (4b)

where ˙( ) denotes derivation with respect to dimen-sionless time t, coefficient M being defined as M =CL0/16µ#2St2 and A and * are parameters based on ex-periments. In all the paper, the values A = 12, * = 0.3,CD = 2, CL0 = 0.8, St = 0.17, µ = 2.79 and CM0 = 1 arefixed as in [10, 16], so that M = 0.06 and '/µ = 0.34.Equations (4) are integrated using finite differences andthe limit cycle is analyzed in terms of efficiency.

Optimal energy harvesting and lock-inThe map of the efficiency as a function of the fre-

quency ratio ( and the damping coefficient & is displayedon Fig. 2.

There is an optimal harvesting configuration leadingto maximum efficiency, corresponding to a frequency ra-tio of ( = 0.89 and a damping coefficient of & = 0.20.The corresponding optimal efficiency )opt2D = 0.23 is closeto the value found in [7] for a similar system.

The efficiency vanishes for both small and largedamping, as expected. For small damping, the amplitudesaturates according to the Skop-Griffin diagram [6] and) thus varies linearly with & . On the other hand, largedamping inhibits the solid motion leading to an amplitudevarying as 1/& : the efficiency ) + & y2 consequently alsovanishes as 1/& for large values of & .

The optimal frequency ratio ( = %s/% f = 0.89 isclose to 1, which corresponds to the synchronization ofthe wake and solid natural frequencies or lock-in [6],

10th International Conference on Flow-Induced Vibration (& Flow-Induced Noise)

292

10!2 10!1 1 10

1

2

3

0.1

0.2

&

(

FIGURE 2. (a) HARVESTING EFFICIENCYAS A FUNC-TION OF THE TWOHARVESTING PARAMETERS ( AND& (LEVEL STEP : 0.015).

known to lead to high amplitude oscillations. The opti-mal configuration for energy harvesting from VIV of anelastically-mounted short rigid cylinder corresponds to alock-in condition for the frequency ratio ( and a well-balanced value of the damping coefficient, or harvestingintensity, & .

THE INFINITE TENSIONED CABLEThe possibility to harvest energy from a flow using

VIV of an infinite tensioned cable is now investigated[10]. The same approach as in the previous section is fol-lowed, the vortex-induced vibrations of the structure aremodelled by a wake-oscillator. This model is identicalto the one used for the rigid cylinder case, except that thestiffness force hY is now replaced by the stiffness inducedby the uniform tension ! of the cable : !!" 2Y/"Z2.The spanwise coordinate Z is scaled using a characteristiclength based on the waves phase velocity z= Z% f

!

mt/!and the dimensionless form of the model reads

y+'µy! y"" =Mq, (5a)

q+ *&

q2!1'

q+q= Ay, (5b)

where ( )" denotes derivation with respect to dimension-less spanwise coordinate z. The extraction of energy froma flow is still modelled by damping. In this configuration,discrete harvesters/dashpots are periodically distributedover the span of the cable, see Fig. 1, with a distance Lbetween two harvesters of damping coefficient R. This

10!2 10!1 1 10

0.1

0.2

&

)

FIGURE 3. COMPARISON OF EFFICIENCIES OF (RED- DASHED) THE ELASTICALLY-MOUNTED RIGIDCYLINDER CASE )2D (& ) AND (BLUE - SOLID) THEINFINITE TENSIONED CABLE )3D (& ).

introduces a damping condition at each damper locationwhich reads in dimensionless form

y" (0, t)! y" (l, t) = l& y(l, t) , (6)

where & = R/Lmt% f is the dimensionless damping perunit length and l = L% f

!

mt/! the dimensionless dis-tance between two harvesters.The efficiency is again defined as the ratio between thetime-averaged power extracted by a dashpot and the en-ergy flux through a length L of the cable, which is stillwritten using the dimensionless variables as Eqn. (3). Thesystem of Eqns. (5) and (6) is integrated in space and timeusing finite differences on a spatially periodic domain,with the cable initially at rest y = 0 and a small randomperturbation of the fluctuating lift q. The harvesting isstudied as a function of the length between two dampersl and the damping per unit length & .

Optimal efficiencyIn order to compare the energy harvesting using ei-

ther an elastically-mounted rigid cylinder or an infinitetensioned cable, two optimal efficiency curves are de-rived, giving the maximum achievable efficiency for anyvalue of the damping density & . For the two-dimensionalcase of the rigid cylinder, )2D (& ) corresponds to the ef-ficiency obtained for an optimal value of the frequencyratio ( at a given damping density & , whereas for the infi-nite tensioned cable case, )3D (& ) is the optimal efficiencygiven by the choice of an optimal value of l at a given & .The two curves are shown on Fig. 3.


293

10!2 10!1 1 10

1

3

5

7

&

l#

0

0.1

0.2

A

B C

FIGURE 4. HARVESTING EFFICIENCY AS A FUNC-TION OF THE LENGTH BETWEEN TWO HARVESTERS lAND THEDAMPING PERUNIT LENGTH & (LEVEL STEP: 0.01).

The optimal performances of these two systems aresimilar, )opt2D = 0.23 while )opt3D = 0.19, justifying a deeperinvestigation of the new concept of energy harvesting us-ing a tensioned cable.

The differences in the shape of the two curves )2Dand )3D are yet striking. Contrary to the case of the rigidcylinder, the efficiency of energy harvesting via an infinitetensioned cable in VIV is far from a classical bell-shapecurve. The optimal damping per unit length is also verydifferent for the two configurations as, for the elastically-mounted rigid cylinder & opt2D = 0.20, whereas in the caseof an infinite tensioned cable, & opt3D = 3.65. A deeper anal-ysis of the efficiency dependence of the latter configura-tion with the two harvesting parameters l and & is thusnecessary. Figure 4 displays the map of the efficiency asa function of these two parameters.

This map is actually much more complex than that ofFig. 2. Three zones may be defined to explain its struc-ture.

The first one, zone A, corresponds to small distancesbetween two dashpots for any damping. Within this zone,the classical influence of damping on the efficiency is re-trieved. Actually, ) vanishes for small and large damp-ing, even if the peak is not at the same location for everyvalues of l. Moreover, the overall optimal harvesting con-figuration lies in this zone : & = 3.65 and l = 1.09# leadto ) = 0.19. As shown on Fig. 5, this zone explains mostof the efficiency curve )3D (& ).

The second zone, zone B, corresponds to the left partof the map, typically & < 0.4, and larger values of l. Theefficiency within this zone is rather small, it correspondsto the left low peak of the efficiency curve, Fig. 5. Within

10!2 10!1 1 10&

)

)

)

0

0

0

0.1

0.1

0.1

0.2

0.2

0.2

A

B

C

FIGURE 5. CONTRIBUTION OF EACH ZONE TO THEEFFICIENCY CURVE )3D (& ).

this zone, the motion of the cable is close to travellingwaves of wavelength , = 2# , 4# , 6# , etc... Yet, the ef-ficiency is low in zone B so it shall not be discussed anyfurther.

Within zone C, the efficiency depends very stronglyon the two parameters, especially on l. High efficiencytongues are surrounded with inefficient harvesting config-urations, resulting in the discontinuities that can be seenon the efficiency curve, Fig. 5.

Mode shapes and efficiencyTo gain some understanding of the efficient part of

this complex energy map (zone A and C), the dynamicsof the cable in these two zones are compared and shownon Fig. 6. Contrary to zone B, the motion of the cable isclose to stationary waves.

The optimal cable displacement for energy harvestingis shown on Fig. 6(a), and the corresponding values of theparameters are reported in Tab. 1. The motion resemblesa mode 1 vibration of a tensioned cable, with slight dis-placement of the dashpots.

If a fixed value of the damping & is considered, thecable motion evolves with an increasing length l, but it isalways close to one of the classical harmonics of a ten-sioned cable. The cable dynamics actually jumps fromeven modes in zones where the harvesting efficiency isvery low, Fig. 6(b), to odd modes in the high efficiencyregions of zone C, Fig. 6(c).

The motion of the dashpot is forced by the jump inthe cable slope between the two sides of the harvester,Eqn. (6). Odd modes are thus more likely than even onesto lead to high jumps in the cable slope, then to high effi-ciency, Fig. 7.

The mode number is defined as n = 2l/, , where ,denotes the wavelength of the cable motion. This mode


294

!1!1!1 000 111000

111

222

y(z, t)

zl

(a) (b) (c)

FIGURE 6. DYNAMICS OF THE CABLE IN ZONES AAND C, THE RED LINE STANDS FOR THE INSTANTA-NEOUS DISPLACEMENT OF THE CABLE, THE BLUEONE FOR THE POSITION OF ONE OF THE DASHPOTSAND THE BLACK LINE REPRESENTS THE ENVELOPEOF THE CABLE MOTION : (a) l = 3.43, & = 3.65, ) = 0.19,ZONE A (OPTIMAL CASE), (b) l = 6.80, & = 3.65, ) =

9.10!7, ZONE C, (c) l = 9.56, & = 3.65, ) = 0.07, ZONEC.

odd mode even mode

FIGURE 7. LOCAL MECHANISM OF FORCING OF ANHARVESTER BY STATIONARY MODES. LEFT : ODDMODES, HIGH EFFICIENCY, RIGHT : EVEN MODES,LOW EFFICIENCY.

number may not be an integer, in that case, the closest in-teger is used. Following Ref. [16], the linearized versionof the model is used to derive the characteristics of themost unstable linear mode which will dominate the non-linear response of the model. The linearized version ofthe model reads

y+'µy! y"" =Mq, (7a)

q! * q+q= Ay. (7b)

The dominance zones of every mode number n are re-ported on the efficiency map of the infinite tensioned ca-

10!2 10!1 1 10

1

3

5

7

&

l#

0

0.1

0.2

0# 10# 1

23

45

67

8

FIGURE 8. MODE NUMBERS ZONES REPORTED ONTHE EFFICIENCY MAP (LEVEL STEP : 0.01).

ble, see Fig. 8. There is a very good agreement betweenthe discontinuities of the efficiency map and the frontiersbetween the dominance zones of two different mode num-bers. In particular, the high efficiency tongues indeed cor-respond with regions where the dominant mode numberis odd. In zone B, only even modes exist, which are ex-pected to be less efficient.

Finally, the damping coefficient plays a double role inenergy harvesting from VIV of an infinite tensioned cableas it controls both the local dynamics of the harvesters andthe global mode shape of the solid displacement.

DISCUSSIONEnergy harvesting using VIV of (i) an elastically-

mounted rigid cylinder and (ii) an infinite tensioned cablewas investigated in this paper and exhibit similar optimalefficiencies, of the order of 0.2.

Optimal harvesting and lock-inFor the two devices, the optimal harvesting configura-

tion corresponds to a lock-in condition. For an elastically-mounted rigid cylinder, this lock-in condition takes theclassical form of synchronization between the vortexshedding frequency and the solid natural frequency, ( =%s/% f = 1.

In the case of an infinite tensioned cable, Refs. [15,16] showed that lock-in corresponds to the highest growthrates of the fluid/solid coupled mode instability. Themodes dominance zones shown on Fig. 8 can then alsobe regarded as the lock-in regions of each mode. Thedispersion relation between frequency % and wavenum-ber k derived from Eqn. 7 consequently results in a lock-in condition for an infinite cable, k = 1. This condi-


295

tion is yet modulated by the periodic boundary condition,Eqn. (6), leading to a lock-in condition for each mode.The solid adapts its own dynamics via its wavenumberin order to be always at lock-in, under the restrictions ofthe periodic harvesting boundary conditions. Consider-ing an ideal lock-in condition k = 1 and the definitionof the wavenumber, one may nevertheless derive optimallengths corresponding with the lock-in condition for ev-ery mode. For the odd modes, which were shown to bethe efficient ones, this results in the lock-in conditionsl = (2n+1)# , which agree well on the location of theseveral peaks on Fig.5. For each mode number 2n+1, italso defines a frequency ratio (2n+1 = (2n+1)#/l, play-ing a similar role as the frequency ratio ( = %s/% f ofthe rigid cylinder case. Reminding that the exact opti-mal configurations differ slightly from the exact lock-inconditions, see Tab. 1, the optimal frequency ratio indeedreads for the elastically-mounted rigid cylinder

( =

"

D2#StU

#

%s = 0.89, (8)

and for the infinite tensioned cable

(1 =#l=

"

D2#StU

#

#L

(

!mt

= 0.92. (9)

These two conditions are of the same form and actuallycorrespond to lock-in between the vortex shedding fre-quency and the natural frequency of the solid motion, thenatural frequency of the rigid cylinder %s being replacedby the natural frequency of the cable vibrations

%c =#L

(

!mt

. (10)

The major difference between the two optimal config-urations lies in the values of the optimal damping inten-sity & , which differ by more than an order of magnitudebetween the two analyzed devices. This is due to the factthat for the tensioned cable with periodically distributedharvesting devices, & does not only drive the local dy-namics of the dashpots, as it is the case for the short rigidcylinder, but it also controls the choice of the overall dy-namics of the cable. This double role of the harvesters isof capital importance since the harvesting efficiency de-pends a lot on the cable overall motion, especially on theselected mode number.

TABLE 1. OPTIMAL CONFIGURATIONS FOR THETWO CONSIDERED ENERGY HARVESTING DEVICESUSING VIV.

Rigid cylinder Tensioned cable

) 0.23 0.19

& 0.20 3.65

( 0.89 0.92

Optimal design of a real energy harvesterThe present study may then be used to design an en-

ergy harvester based on VIV, using Tab. 1. The definitionof the efficiency ) provides an estimation of the maxi-mum power extracted by a cable of length Ltot = 100 mand diameterD= 4 cm in a flow of mean velocityU = 1.5m.s!1, namely P= 1282.5 W. It is interesting to note thatthis is far from being negligible and that it can be achievedfor many different triplets (!, L, R).

For an elastically-mounted rigid cylinder of diameterD = 4 cm and mass ratio µ = 2.79 placed in a flow ofmean velocity U = 1.5 m.s!1, the dimensional optimaldamping and stiffness are r = 35.8 kg.s!1.m!1 and h =5673.2 kg.s!2.m!1. For the tensioned cable mentionedabove (Ltot = 100 m), under a tension ! = 10 kN, thedamping coefficient of each dashpot is R = 2645 kg.s!1and they are separated by a distance L= 4.05 m.

Influence of the velocity variations on the efficiencyin real operating conditions

The dimensional parameters are fixed once and forall, but the operating conditions may drift from the op-timal ones because of a varying flow velocity U . Theinfluence of a varying flow speed on the performancesof a harvesting device is now discussed. Three cases areconsidered : (i) an elastically-mounted rigid cylinder, (ii)an infinite cable with constant tension and (iii) an infi-nite cable with drag-induced tension. The values of theharvesting parameters are derived so that the optimal isreached for the mean flow speed U . The evolution of theefficiency with the flow speed for each case is then plot-ted on Fig.9. In order to quantify the influence of flowspeed fluctuations on the actual efficiency, the peak widthw is defined as the ratio between the length of the velocityinterval for which the efficiency is above 75% of the peakefficiency -U and the mean velocityU , see Tab. 2.

For the rigid cylinder, it comes from the dimension-less parameters definitions that ( + 1/U and & + 1/U .


296

0

0.1

0.2

0 2 4 6 8

U&

m.s!1'

)

FIGURE 9. EVOLUTION OF THE EFFICIENCY WITHFLOW VELOCITY. (RED - DASHED) RIGID CYLINDER,(BLUE - SOLID) CABLE WITH DRAG-INDUCED TEN-SION AND (BLACK - DASH-DOT) CABLE WITH FIXEDTENSION.

The curve of the evolution of ) withU hence correspondsto the values of ) along a curve ( + & passing throughthe optimal configuration. Even if the best efficiency isachieved as expected by this configuration, the width ofthe peak is small, see Tab. 2. The efficiency dramaticallydrops down as the current speed drifts away from its meanvalue.

For the long tensioned cable, if the tension is con-stant (induced for instance by a buoy on top of a cable an-chored at its bottom), & + 1/U and l varies linearly withU . The efficiency depends a lot on the flow speed, Fig.9.The peak width is even smaller than for the rigid cylinder,w= 0.54, and ) even falls down to zero for some veloci-ties. This solution should be avoided because of this highsensitivity to current velocity.

In the last case considered, the tension is due to a dragforce acting on a well-chosen area A, which may differfrom the cable area DLtot . The area A has here been cho-sen so that the optimal harvesting parameters are identicalto the constant tension case in order to compare this con-figuration with previous ones (!= 10 kN forU =U). Inthat case, ! +U2, & + 1/U and l is constant (l = 3.43),which sounds a valuable option as the efficiency dependsa lot more on l than on & , Fig. 4. It actually appears asthe best solution since the efficiency stays high for a widerange of velocity, Fig. 9. The peak value is lower thanfor the rigid cylinder, but the peak width is much larger,w = 2.47, see Tab. 2. The efficiency decrease as U devi-ates from U is slow, it even overtakes the rigid cylinderefficiency forU $ 1.1 m.s!1 andU % 2.2 m.s!1.

TABLE 2. WIDTH OF THE PEAK AROUND THE OPTI-MAL FLOW VELOCITY w= -U/U .

Configuration Peak width w

Rigid cylinder 0.73

Cable - constant tension 0.54

Cable - drag-induced tension 2.47

As a conclusion, the analysis performed throughoutthis paper has shown that energy harvesting using vortex-induced vibrations of a long tensioned cable seems, atleast, as promising as using those of an elastically-mounted short rigid cylinder as in [7] or [8]. This newconfiguration may even have some advantages like thepossibility to imagine very long cables able to harvestlarge amounts of energy, and to adapt their dynamics sothat they are always near lock-in conditions, if their ten-sion is induced by drag.

ACKNOWLEDGMENTS. Michelin would like to acknowledge the sup-

port of a Marie Curie International Reintegration Grantwithin the 7th European Community Framework Pro-gram. Clement Grouthier is also supported by a GaspardMonge scholarship of the Ecole Polytechnique.

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