Optimal Energy Harvesting from Vortex-Induced Vibrations of Cables

Transcription

1 Optimal Energy Harvesting from Vortex-Indced Vibrations of Cables Gillame O. Antoine, Emmanel de Langre, and Sébastien Michelin LadHyX Département de Mécaniqe, Ecole Polytechniqe CNRS, 928 Palaisea, France. (Dated: October 8, 26) Vortex-indced vibrations (VIV) of flexible cables are an example of flow-indced vibrations that can act as energy harvesting systems by converting energy associated with the spontaneos cable motion into electricity. This work investigates the optimal positioning of the harvesting devices along the cable, sing nmerical simlations with a wake oscillator model to describe the nsteady flow forcing. Using classical gradient-based optimization, the optimal harvesting strategy is determined for the generic configration of a flexible cable fixed at both ends, inclding the effect of flow forces and gravity on the cable s geometry. The optimal strategy is fond to consist systematically in a concentration of the harvesting devices at one of the cable s ends, relying on deformation waves along the cable to carry the energy toward this harvesting site. Frthermore, we show that the performance of systems based on VIV of flexible cables is significantly more robst to flow velocity variations, in comparison with a rigid cylinder device. This reslts from two passive control mechanisms inherent to the cable geometry : (i) the adaptability to the flow velocity of the fndamental freqencies of cables throgh the flow-indced tension and (ii) the selection of sccessive vibration modes by the flow velocity for cables with gravity-indced tension. sebastien.michelin@ladhyx.polytechniqe.fr

2 2 I. INTRODUCTION The field of renewable energies is gaining interest de to the limited availability and the environmental impact of fossil fels. Flow-indced vibrations, i.e. the motion of a solid strctre reslting from the destabilizing effect of forces applied by the srronding flow, have recently received an increased attention as a potential alternative to classical wind- and water-trbines to convert a fraction of the flid s kinetic energy into electricity []. From an energy point of view, flow-indced vibrations transfer a fraction of the kinetic energy of the incoming flow to the solid strctre that is set into motion. Vibrations can then be sed to power an electric generator and effectively convert some of the solid s mechanical energy into electrical form. Recently, many classical examples of flow-indced vibrations have been revisited as potential energy harvesting mechanisms, inclding galloping [2 6], copled-mode fltter of airfoil profiles [7], fltter of flexible cylinders or membranes in axial flow [8 ], flapping in nsteady wakes [2 5] and vortex-indced vibrations (VIV) of rigid and flexible strctres [, 6 22]. The present work focses on the possibility to harvest energy from VIV of long flexible cables. VIV reslt from the copling of a blff body to its nsteady vortex wake. The steady flow arond a fixed blff body at high Reynolds nmber (Re or inertial flows) is characterized by an nsteady shedding of vortex strctres. The wake strctre is then complex and differs from a classical Von-Karman vortex street observed at lower Re, bt its freqency spectrm is still dominated by a fndamental freqency known as the Strohal freqency f, proportional to the flow velocity U as f = S T U/D with S T the Strohal freqency and D the diameter (e.g. [23, 24]). This reslts in an nsteady lift force on the blff body. For a flexible or flexibly-monted strctre (see Fig. ), this nsteady force will force the solid body into so-called vortex-indced vibrations, and this nsteady motion of the body will also introdce a feedback copling on the vortex shedding and nsteady lift. VIV of a rigid cylinder monted on an elastic fondation have been widely stdied, both experimentally and nmerically (see [23 26]. A distinctive featre of VIV is the lock-in mechanism: when the fndamental freqency of the solid s vibrations is close to the Strohal freqency, the copling between the vibrations and the vortex shedding synchronizes both dynamics over an extended range of flow velocity. This reslts in self-sstained, self-limited large amplitde oscillations of the solid, typically of the order of one cross-flow diameter. This lock-in phenomenon is particlarly interesting for energy harvesting prposes since large amplitdes imply that the oscillating strctre has a large amont of kinetic energy that can potentially be harnessed to prodce electricity. The VIVACE system [] relies on that phenomenon. However, VIV of elastically-monted rigid cylinders also have intrinsic limitations. Lock-in occrs only when the Strohal freqency and the eigenfreqency of the strctre are sfficiently close, e.g. [24]. If this condition is not met ( lock-ot ), the rigid cylinder is not properly excited by the flow and its oscillations have negligible amplitde, which reslts in inefficient energy extraction. Geophysical flows (e.g. oceanic, tidal or river crrents) are characterized by an important variability in the flow velocity magnitde, and systems relying on VIV of rigid cylinders can only prodce energy when the flow velocity remains close to the velocity for which they are designed. FIG.. Vortex-indced vibrations of a rigid cylinder (left) and a flexible cable (right) in a steady and niform cross flow. A method to circmvent this isse is to se flexible strctres (cables, or beams) rather than rigid cylinders, as sketched in Fig. : cables have mltiple deformation modes which have their own eigenfreqencies and different modes can be locked-in for different vales of the flow velocity (and Strohal freqency). The selective excitation of a specific deformation mode of a flexible strctre when the eigenfreqency of the latter is close to the Strohal freqency was demonstrated experimentally [27, 28] and nmerically [29]. Rather than lock-ot, variations of the flow velocity indce a transition towards lock-in of higher or lower deformation modes. This advantage of flexible strctres was recently demonstrated for a system consisting of a hanging cable in a cross flow, attached to a local energy harvester at its pper end [2]. The experimental and nmerical reslts obtained for that system demonstrated its increased robstness to variations in the flow velocity in comparison with an elastically spported rigid system. A challenge and open qestion associated with harvesting energy from VIV of flexible strctres is to determine the optimal harvesting strategy, i.e. where and how the mechanical energy of the solid strctre shold be converted into

3 3 e Z O e Y e X e e r e FIG. 2. VIV of a flexible cable. The inset shows the definition of the local orthonormal basis (e r, e θ, e ϕ). electricity. With a rigid cylinder, the motion is a pre translation with a single degree of freedom, and determining the optimal strategy is relatively simple. For flexible strctres, the periodic motion is not niform throghot the strctre, which reslts in a mch larger configration space for the harvesting distribtion. Optimization of sch a system is a particlar challenge. Moreover, the geometry of the strctre and how it is held in the flow are additional design variables. The present work addresses these qestions and investigates the optimal rote to efficient energy harvesting sing VIV of flexible cables. The paper is organized as follows. Section II presents the mathematical and nmerical models sed here to analyse the VIV dynamics and reslting energy harvesting. In Section III, the distribtion of harvesting devices on a flexible cable is optimized, and the performance of sch system is analysed and compared to the reference rigid system in Section IV. Section V finally smmarizes the main conclsions of or work, stressing ot both its fndamental and engineering implications. II. MODEL FOR THE VIV OF HANGING CABLES A. Problem geometry In this work, we consider a long inextensible cable of circlar cross-section with density ρ S, length L and diameter D with a large aspect ratio Λ = L/D, see Fig. 2; its extremities O and O are fixed and aligned along e X at a distance L from each other. By cable (or eqivalently string ) we mean that the strctre has no bending stiffness, ths the only strctral force is the tension force which acts as a Lagrangian mltiplier that enforces the inextensibility condition. The reader is referred to Ref. [3] for a thorogh description of the string model. The cable is immersed in a niform horizontal cross flow = e Y. The dynamics of the cable are the reslt of a balance between the cable s inertia, the internal tension, gravity and boyancy, and flow forces. We focs on vortex-indced vibrations (VIV) for which the cable ndergoes small oscillations abot a steady mean position as a reslt of the nsteady vortex shedding on the strctre. Flow forces on the moving strctre can be modelled as the sperposition of for different components [3, 32]: (i) a drag force 2 ρc DD r r, (ii) a friction force π 2 ρc F D r r, (iii) an added mass and (iv) an nsteady lift force reslting from periodic vortex shedding in the strctre s wake. Here, ρ is the mass density of the flid and r = ẋ is the local relative velocity of the incoming flow to the cable whose position is noted x(s, t), and ( r,r ) are its components along and normal to the cable, respectively. In the following, the problem is written in non-dimensional form by choosing L, /(2πf) and 2 ρc DU 2 DL as characteristic length, time and force, respectively. B. Mean Position of the Strctre We first focs on the mean (i.e. time-averaged) steady position of the cable, obtained by balancing the effects of the steady flid forces, the cable s weight, boyancy and the internal tension (inertia, added-mass and wake effects

4 only contribte to the nsteady dynamics). A local orthonormal basis (e r, e θ, e ϕ ) is sed with e r = x/ s the local direction of the cable where x is the mean position of the cable, and e θ = (Fig. 2). The eqilibrim writes 4 (χ e r ) + β sin ϕ e r + cos 2 ϕ e ϕ Γ e Z = () where s is the dimensionless crvilinear coordinate along the cable and ( ) denotes the derivative with respect to s. The angles θ and ϕ determine the local orientation of the local basis with respect to fixed axes (see Fig. 2 and Appendix A), and χ(s) = T (s)/( 2 ρc DU 2 D) is the non-dimensional tension. Eqation (), together with the inextensibility condition e r = x/ s, is an implicit eqation for the mean position x(s) and tension χ(s). Bondary conditions frther impose that e r ds = δ e X. (2) The mean problem is ths completely determined by three non-dimensional parameters Γ = πdg ( ) ρs 2C D U 2 ρ, β = πc F, C D δ = L L. (3) Here Γ is a relative measre of the net gravity compared to drag forces, β is the ratio of the friction and drag coefficients, and δ is the relative span. Note that δ = corresponds to a straight cable, and decreasing δ corresponds to a larger sag of the cable in its mean position. Eqations () (2) are solved nmerically sing an iterative Broyden s method [33] for the mean shape and mean tension of the strctre. Depending on the vale of Γ, the physical origin of the tension varies from flow- to gravityindced regime. When Γ, gravity and boyancy effectively balance; as a reslt of the drag force, the cable is horizontal ( in-flow catenary ) and its tension is de to the sole effect of the flid ( flow-indced tension ). When Γ, gravity effects are dominant and the cable is vertical ( cross-flow catenary ) while its tension is de to the sole effect of gravity ( gravity-indced tension ). Conversely, when Γ = O(), gravity and flid effects are comparable in magnitde and they both significantly contribte to the mean position and tension of the cable. Whether flow-indced or gravity-indced, the tension within the cable provides it with the ability to carry waves. To be generic, the typical wave speed shold be defined (dimensionally) as C = max(t )/m T, with m T = πd 2 (ρ+ρ S )/4 the total lineic mass of the cable (inclding the flid added mass). This is effectively eqivalent to rescaling the nondimensional tension as χ(s) = χ(s)/max(χ) = T (s)/max(t ), which is done in the rest of the manscript. We define the characteristic freqency of the waves as f = C which corresponds to the fndamental freqency of a straight 2L cable with niform tension. C. Governing Eqations for VIV VIV correspond to O(D) displacements of the cable with respect to its mean position in response to the nsteady flow separation in the strctre s wake and the reslting nsteady flow forces. These displacements are typically small compared to the strctre s length (D L) which jstifies in the following the linearisation of the eqations of motion abot the mean position determined above. The cable s motion modifies both its position and orientation, with O(D/L) leading order corrections; the position of the cable is ths in non-dimensional form x(s) + (D/L)d(s, t) with d = O() the non-dimensional displacement (scaled by D). The relative flow velocity and internal tension are also modified both in orientation and in intensity. The non-dimensional tension magnitde in the cable can be decomposed into its steady and nsteady parts as χ(s) + (D/L) ξ(s, t) with ξ = O() the non-dimensional flctating tension. Linearising the eqations of motion abot the time-averaged static state determined above leads to the non-dimensional eqation for the cable s displacement d(s, t): d = f D + f F + f T + f L, (4)

5 where ( ) denotes the time derivative, and the flctating drag, friction and tension forces are obtained as f D = γ ( µ cos ϕ d θ e θ + 2 d ) ϕ e ϕ (5) f F = β γ ( ) d r + sin ϕ e Y µ ḋ e r, (6) f T = ( π 2 2 χ d + χ d + ξ e r + ξ ) e r (7) with (d r, d θ, d ϕ ) the components of d in the local basis, and, γ and µ, respectively defined as the redced velocity (or freqency ratio), damping and mass ratios [2, 29, 34] = f L U = 2S T f D C, γ = C D, µ = π ( + ρ ) S. (8) 4πS T 4 ρ The inextensibility of the cable frther imposes that d e r =. The effect of the nsteady wake on the strctre is modelled here as a flctating lift force f L which is orthogonal to both the direction of the flow and the axis of the cable. As shown by Franzini et al. [35], vortex shedding is mostly governed by the flow orthogonal to the cable s axis and, conseqently, the lift force is qadratic in the normal relative flow velocity (i.e. r 2 ) with a local flctating lift coefficient C L q(s, t)/2 (with C L the lift coefficient of a still cylinder [34]), so that in Eq. (4), f L is given by f L = M q(s, t) cos 2 ϕ e θ, with M = C L 6π 2 µst 2 (9) So-called wake oscillator models describe the dynamics of this flctating lift as a nonlinear van der Pol oscillator forced by the motion of the strctre in order to accont for the feedback copling of solid motion on vortex shedding, which is an essential ingredient to lock-in. Previos stdies have shown that a local inertial copling leads to a good agreement with experimental and nmerical stdies on rigid and flexible strctres in VIV [29, 34, 36, 37]. The dynamics of the wake variable q(s, t) are then governed by q + ɛ cos ϕ ( q 2 ) q + cos 2 ϕ q = A d θ. () The vales of the non-dimensional parameters ɛ and A of the wake oscillator model need to be determined sing experimental (or nmerical) data. More details abot this model for the wake effects and validation against experimental data for the VIV of rigid cylinders and straight strings are provided in [29, 34]. In this work, nless stated otherwise, the following nmerical vales are sed for the parameters of the model: S T =.7, C L =.6, C D = 2., C F =.83, ɛ =.3 and A = 2 (e.g. [34]). Two different vales of µ are sed in the paper: µ = 2.79, which is consistent with previos works on the topic (e.g. [2, 29, 34]), and µ = π/2.57, which corresponds to a netrally boyant cable. The extremities of the cable are fixed, which yields the bondary conditions d s= = d s= =. Eqations (4) (7), (9) and () are nmerically integrated in time starting from an initial state where the cable is at rest in its mean position (d(t = ) = ) while the wake is given a small pertrbation (at t =, q = q (s) and q = ) that triggers the oscillations of the system which eventally reaches a steady oscillatory regime that does not depend on the choice of initial conditions. 5 D. Modeling Energy Harvesting This work focses on the possibility of harvesting energy from a flow sing the cable s VIV, and on the optimal strategy to maximize the energy otpt. Energy harvesting amonts to converting some of the energy associated with the vibrations into a sable form typically electricity. As it effectively removes some mechanical energy from the strctre, energy harvesting mst be modelled explicitly in order to properly accont for its effect on the vibration itself. The simplest model is that of a pre linear damping force. This damping can be either distribted (i.e. present all along the strctre) or localized at some position along the cable. The former will be or initial focs, and corresponds to an additional non niform damping force f H in the eqations of motion of the cable, Eq. (4): f H = σ(s) ḋ. ()

6 6 O (s) FIG. 3. Energy harvesting of a flexible cable in VIV sing a continos distribtion of harvesting devices (i.e. dampers) of local intensity σ(s). For the sake of simplicity, we draw a single line of dampers in the figre. The damping intensity σ(s) (scaled by 2πfm T ) may depend on the position s (see Fig. 3), with the reqirement that σ(s) (passive energy harvester). The latter (i.e. localized harvesting) will be explored sbseqently. In both cases, the power P el dissipated in the dampers acts as a proxy for the power harvested from the flid flow by the device. Its efficiency is therefore defined as the ratio of P el to the reference kinetic energy flx throgh the reference srface A = LD η = P el 2 ρu 3 A = 6 µ π3 ST 3 σ(s) ḋ 2 ds, (2) with the time average, compted once the system reaches a permanent satrated regime. Note that η can also be nderstood as the non-dimensional power otpt. III. OPTIMAL ENERGY HARVESTING STRATEGY Finding the optimal harvesting strategy, i.e. determining where the vibration energy of the cable mst be harvested in order to maximize the otpt power, is eqivalent to determining the fnction σ(s) that maximizes the harvesting efficiency η defined in Eq. (2). The location of maximal vibration depends on the vibration mode selected by the flow velocity (see the experiments of Ref. [27]). The optimal harvesting strategy σ opt (s) and the corresponding efficiency η opt are therefore also expected to depend on the redced flow velocity. A. Optimal Harvesting from VIV of Straight Cables We first focs on the simplest geometry, namely that of a straight cable with niform tension (δ = ). In this special case, the only non-zero component of the displacement is d θ = z and the coordinate s matches x. In that limit, the eqations governing the VIV dynamics take a mch simpler form, which is amenable to a fll gradient-based optimization sing adjoint methods to compte the gradient of the harvesting efficiency with respect to σ(s). Adjoint methods represent a poplar and powerfl approach to optimization problems involving partial differential eqations, e.g. see [38]. Here, we follow an approach similar to [39]: the gradient σ [η](s) of the efficiency with respect to the damping distribtion σ(s) is a fnction of s and is compted as σ [η](s) = 6 µ π 3 S 3 T ż (ż z ) (3) where z is the adjoint displacement. The derivation of the adjoint eqations for (z, q ) from the direct eqations, Eqs. (4) and (), is presented in Appendix B. Starting from an initial gess for z(s, t) and q(s, t) withot any damping (σ = ), the direct and adjoint eqations are solved at each iteration to obtain σ [η] sing Eq. (3); a steepest-ascent algorithm is then sed ntil convergence to the optimal damping distribtion σ opt (s).

7 7 (a) y z x y z x (b) x x x (c) Optimal damping distrib. σ opt.25.5 Optimal damping distrib. σ opt x x Optimal damping distrib. σ opt x (d) x x x le in mode, FIG. from Figre 4. top Optimization to : bottom: Straight Figre ofinitial cable the : Straight harvesting y-contors, modes cable efficiency (left) optimal in mode and forσ2, VIV 2 distribtion, from (right), in modes top from toone final bottom: top (left) y-contors to bottom: and initial two y-contors, (right) initial y-contors, of aoptimal straight optimal cable σ distribtion, (δ = σ distribtions, From finaltop y-contors bottom: (a) sketch of the cable, (b) contors of the cable deflection z withot energy harvesting (σ = ), ) with final y-conto µ = (c) optimal damping distribtion σ opt, and (d) contors of z with σ = σ opt. In VIV of flexible strctres, the flow velocity selects and excites the eigenmode of the strctre whose freqency is the closest to the Strohal freqency (see experiments in [27]). For the wake oscillator model sed here, linear stability analysis indicates that the n-th mode of the straight cable is specifically excited when n [29]. For instance, when =, the cable deforms in mode one and σ opt (s) is expected to be large at the center of the cable, where the kinetic energy is the largest. When = 2 however, the cable deforms in mode two and there is little to no available energy at the center of the cable; ths the optimal harvesting strategy in mode one is nlikely to carry over to mode two, and the optimal harvesting strategy mst be determined for each individal mode by setting to the appropriate vale. Figre 4 presents the optimization reslts for a straight cable in mode and 2. For = (mode- lock-in), the distribtion of deflection z is qalitatively similar with no damping (initial gess)

8 and with an optimal damping (final reslt). The amplitde of the oscillations with the optimal harvesting strategy is nearly half that of the oscillations withot energy harvesting, indicating a significant energy extraction from the oscillations. The optimal damping distribtion σ opt respects the mode one symmetry. For = 2 (mode-2 lock-in), the distribtion of deflection z withot damping (initial gess) corresponds to a modetwo deformation which validates or criterion n =. This is not the case for the z-contors in the optimal-damping sitation: the optimal strategy corresponds to a non-symmetric damping distribtion σ opt (s) (Fig. 4), for which most of the energy extraction occrs on one half of the cable. Progressive waves carry energy from the ndamped part of the cable to the harvesting location. The optimal efficiency is similar for = and = 2, with η opt = 8.% and η opt = 8.6%, respectively. These reslts are in fact generic: for any n 2, the optimization algorithm leads to an optimal strategy with harvesting concentrated on a redced fraction of the cable near one of its fixed ends, while the rest of the cable is ndamped, and energy is carried to the harvesting region by travelling waves along the cable. Those reslts are particlarly relevant from a practical and engineering point of view: regardless of the selected mode except mode (and therefore for any sfficiently large velocity), the optimal strategy consists in restricting the harvesting system to a limited fraction of the cable near one of its attachment points, rather than distribting it along the entire cable. 8 B. Optimal Harvesting from VIV of Catenary Cables We can extend the previos approach and reslts to the general configration of Fig. 3. The optimal harvesting strategy σ opt (s) now depends on three parameters: δ and Γ, that set the mean position of the cable, and, which sets the vibration mode of the cable excited by the flow. Assming a piecewise constant damping σ(s), the damping distribtion σ opt (s) is compted for different vales of (δ, Γ, ) sing a steepest-ascent algorithm, now compting the gradient nmerically. Figre 5 shows the optimal distribtion of damping and reslting dynamics obtained for a cross-flow catenary cable (η opt = 7.6%). As for the straight cable, the optimal damping distribtion is not symmetric and waves are observed to carry energy toward the harvesting site. This behavior is in fact observed for all geometries provided the excited mode nmber is greater than 2. This generality therefore sggests to go one step frther and investigate the performance of harvesting energy at a single point located at the extremity of the cable rather than in its vicinity. C. Local Point-Wise Damper as Optimal Harvesting Strategy To this end, the fixed bondary condition at s = is now relaxed to allow for transverse displacements of the cable and energy extraction. More precisely, at s =, the cable can slide in the local (θ,ϕ) plane and energy harvesting is modelled as a linear viscos force resisting its velocity with a damping intensity σ (Fig. 6). Note that σ is a scalar vale here. The cable and wake dynamics are still governed by Eqs. (4) and () (withot an additional damping force) and the bondary condition at s = now balances local tension and viscos forces (the fixed bondary condition at s = remains nchanged): d r =, d e θ + π 2 2 σ d θ =, d e ϕ + π 2 2 σ d ϕ = (4) The harvesting efficiency of this system is still defined as the ratio of the average power dissipated in the damper to the kinetic energy flx throgh the area occpied by the cable: η = 6 µ π 3 ST 3 2 σ ḋ (5) s= The optimization now consists in maximizing η with respect to the scalar parameter σ. For a given geometry (i.e. given δ and Γ), the optimal damping strategy and efficiency are determined by directly compting η(, σ) and finding its absolte maximm. Figre 7 shows the evoltion of η(, σ) for the configrations of Figs. 4 (straight cable) and 5 (catenary). The maximm efficiency for the straight cable and the cross-flow catenary is slightly more than 9%, which exceeds the maximm performance obtained with distribted harvesters: the local harvesting strategy is therefore optimal for both configrations. This finding is again generic, and valid for most geometries: for δ ranging between.25 and and Γ, Γ and Γ we find that the optimal efficiency of a local damper is eqivalent or slightly higher than the optimal efficiency of distribted damping.

9 9 z x (a) y (b) s s Optimal damping distrib. σ opt s (c) Optimal damping distrib. σ opt s (d) s s.6 Figre FIG. 5. : Optimization Heavy cable of with the harvesting δ=.5 in efficiency mode 2, from for a mode top to2 bottom: VIV ( = initial 2) of adcross-flow Figre : Heavy cable with span.5 2, from top bottom: initial d θ -contors, θ -contors, catenary optimal (Γ σ distribtions, ) with δ =.5final and µ = (a) sketch of the cable in mode 2 lock-in, (b) contors of d optimal σ distribtion, d final d θ -contors θ -contors θ withot energy harvesting (σ = ), (c) optimal damping distribtion σ opt, (d) contors of d θ with σ = σ opt. We note that the contors of Fig.7 are qalitatively similar to the efficiency map obtained by Grothier et al. [2] with a hanging cable attached to a local harvester at its pper end. The agreement is also qantitative if we accont for the different vales for the lift coefficient C L sed in the works (.63 in the present work,.8 in [2]). Those similarities sggest that the inflence of the detailed geometry on the performance of cables with gravity-indced tension is weak. This is discssed frther in Section IVIV C. Besides, we note that the periodic cable stdied in

10 O e r e e FIG. 6. Energy harvesting of a flexible cable in VIV sing a local harvester of intensity σ at s =. (a) 4.8 (b) σ... σ FIG. 7. Efficiency Figre η as: a fnction η vs. of and theσ. damping Left: straight σ and the cable; redced Right: velocity heavy cable for (a) w/ theδ straight =.5. cable For both: and (b) µ = the2.79. cross-flow Left: catenary with span opt /π δ = 3.2,.5. σred opt = crosses 9.7% are =.6, placed η opt at= the 9.7% maximm ; Right: efficiency. opt /π = 3., (a) ησ opt = 9.7% 9.2%, = opt.8, = 3.2, η opt σ opt = 9.6% =.6, and (b) η opt = 9.%, opt = 3., σ opt =.8. For both contors, µ = [2] exhibits the same maximm efficiency as the hanging cable and catenary cables althogh its efficiency map is considerably different. IV. PEAK PERFORMANCE AND ROBUSTNESS Energy harvesting sing VIV was originally proposed sing a rigid cylinder system (VIVACE system []). The goal of this section is to compare the overall performance of the flexible cable system in Fig. 6 to this reference configration, with a particlar focs on its robstness to variability in the flow velocity. A. Harvesting performance and limitation of rigid cylinders in VIV Ref. [2] carried ot a detailed stdy of the rigid cylinder system, and we present only a brief smmary here, together with an pdate of the nmerical reslts (Fig. 8) for the particlar parameter vales considered here (in particlar, the Strohal nmber and lift coefficient are modified in comparison with Ref. [2]). This system extracts energy from VIV of a rigid cylinder and it is the origin of the VIVACE concept. The system is a rigid cylinder monted horizontally on an elastic fondation (stiffness k per nit length) and attached to an energy harvester of intensity R (per nit length). Its resonance freqency in still flow is therefore f = k/m T /2π. It is immersed in a horizontal cross-flow U and its motion is restricted to vertical vibrations along e Z. The harvesting efficiency η (defined as above) depends solely on the redced velocity (or freqency ratio) = f/f and damping intensity σ = R/(2πfm T ). Optimizing the design of sch a system, which is meant to be deployed in a crrent of typical mean velocity

11 σ Figre : Extracting energy from VIVs of rigid cylinder: sketch and contors. (a).2 (c)..8 (b) Efficiency.6.4 U U σ U /U FIG. 8. Energy harvesting Figre 2: by Extracting a rigid cylinder energy with from µ VIVs = of rigid (a) cylinder: is sketch sketch, of the contors, system, and (b) is isodamping. the harvesting efficiency η. The red cross is the optimal configration (η opt =.5% with opt =. and σ opt =.8), and the red line corresponds to variations of the dimensional flow velocity for an optimal system (all other qantities being fixed). (c) shows the evoltion of the efficiency with flow velocity arond the optimal configration (red line in (b)). U, amonts to choosing its dimensional characteristics sch that = opt and σ = σ opt nder normal conditions (U = U ) so that η = η opt =.5%. The velocity of actal geophysical flows typically exhibits a significant variability in magnitde (in particlar for tidal crrents). All other qantities being fixed (e.g. geometry, stiffness,...), a change in the dimensional flow velocity U modifies (, σ) and therefore indces a departre from the optimal operating conditions and a redction of the system s efficiency (Fig. 8, c). This redced efficiency reslts from a departre from the optimal lock-in conditions: when U U, the shedding freqency f differs from the fndamental freqency of the cylinder. The efficiency peak arond U is narrow: its width at half height, i.e. the range of U in which the efficiency is at least half its peak vale, is almost eqal to U. Ths, the system is efficient only when.67 (U/U ).7. In a practical sitation, this cold reslt in a negligible efficiency of the system for a significant fraction of its operating time. Below, we show that extracting energy from VIV of flexible cables rather than rigid cylinders provides a simple and efficient method to remedy this isse, while still maintaining comparable peak performance. B. Harvesting performance and robstness of flexible cables in VIV We now trn back to the case of a flexible cable with a local harvester at s = (Fig. 6). The performance of this system as well as its sensitivity to variations in the flow velocity depend on the shape of the cable determined by the span δ and redced gravity Γ. The performance of the flexible cable with respect to and σ is analysed for.25 δ and Γ (in-flow catenary) and Γ (cross-flow catenary) by compting an efficiency map similar to that obtained for the rigid cylinder in Fig. 8-b or those obtained for cables in Fig. 7. The optimal harvesting efficiency η opt of each particlar geometry is then determined (together with the corresponding optimal parameters opt and σ opt ), as well as the inflence on η of a change in the dimensional velocity U away from its design vale U. The performance and robstness of the system to flow flctations is characterized by two important qantities: the peak efficiency η opt and the peak width U/U measring the relative velocity range over which the efficiency is at least η opt /2. Both qantities are critical: the former characterizes the efficiency of the system and the latter its robstness to flow velocity variability. They are respectively reported in Fig. 9 and in Table I. Efficient energy harvesting can be achieved sing cross-flow catenaries (Γ, gravity-indced tension) of any span or in-flow catenaries (Γ, flow-indced tension) that are almost straight (δ ) as shown in Fig. 9. This efficiency is comparable to the rigid cylinder performance, albeit slightly lower. In-flow catenaries with larger sag only show

12 2 % (a) % (b) Maximm efficiency 8% 6% 4% 2% η > 5% η < 5% Maximm efficiency 8% 6% 4% 2% η > 5% η < 5% % Span % Span FIG. 9. Maximm harvesting efficiency as a fnction of the span δ for (a) cross-flow catenaries and (b) in-flow catenaries. Key to the plots: for µ = 2.79, for µ = π/2. In-flow catenary Γ µ = π/2 µ = 2.79 δ = δ =.75 δ =.5 δ = δ =.75 δ = Cross-flow catenary Γ Rigid cylinder.6. TABLE I. Width of efficiency peaks at half height for systems extracting energy from VIV of cables and rigid cylinders. poor peak performance. All configrations presented are nevertheless significantly more robst to variations of the flow velocity than their rigid conterpart (Table I). In particlar, the efficiency peak width of the straight in-flow catenary (δ, Γ ) is seven times as large as that of the rigid cylinder, which makes it an attractive candidate for an energy harvesting system based on VIV. The comparisons between these different systems is smmarized in Fig., by plotting the variation of the efficiency with the flow velocity U arond the optimal configration. C. Robstness of Energy Harvesting from VIV of Cross-Flow Catenaries The mean tension of the cross-flow catenaries is determined solely by gravity, and is ths independent of the flow velocity. Hence, the eigenmodes of the cable have constant freqencies. When the velocity of the flow increases, so does the Strohal freqency and as as a reslt higher modes are sccessively locked-in. The transitions between the different modes of the cable correspond to the kinks in the crves in Fig. -a. The robstness of cross-flow catenaries to flow velocity variations therefore arises from their ability to adapt their deformation mode to the srronding flow and remain at lock-in, which the rigid cylinder is nable to do since it only has a single resonance freqency. Noticeably, the vale of the relative span δ of cross-flow catenaries has only a minor impact on their efficiency in terms of optimal performance and robstness to flow velocity variations.

13 Rigid Rigid cylinder Cable Cable with with span span Cable Cable with with span span Cable Cable with with span span.5.5 Rigid Rigid cylinder and and cross-flow catenary, µ = µ 2.79 = 2.79 Rigid Rigid cylinder and and in-flow catenary, µ = µ = π/2 π/ (a) (b) 3 Efficiency Efficiency Efficiency Efficiency U /U U /U U /U U /U FIG.. EvoltionFigre of the: harvesting : Efficiency efficiency vs. vs. U Uratio ratio η as for for athe fnction the rigid rigid cylinder, of the cross-flow velocityand U/U and in-flow for (a) catenary rigid cables. cylinders and cross-flow catenaries with µ = 2.79 and (b) rigid cylinders and in-flow catenaries with µ = π/2. U denotes the flow velocity corresponding to the optimal design. Key to the plots: rigid cylinder (thick solid), catenary with span (thin solid), catenary with span.75 (dashed), catenary with span.5 (dash-dotted). Interestingly, reslts obtained by [2] show that the robstness to flow variations of cross-flow catenaries is comparable to that of hanging cables (width of efficiency peak is 3.6 for the former and 3.7 for the latter). That observation agrees with the minor effect of the span δ on the performance of cross-flow catenaries in sggesting that the robstness of energy harvesting from a cable with gravity-indced tension is only determined by the mechanism of transitioning between mode lock-in and is independent of the cable shape. D. Robstness of Energy Harvesting from VIV of In-Flow Catenaries For in-flow catenaries, boyancy and gravity balance and the tension is indced by the flow drag and friction forces. It is therefore set by the flow velocity magnitde and larger flow velocity reslts in a larger tension in the catenary, that scales as U 2. An increase in the flow velocity U now has two conseqences: (i) a linear increase in the Strohal freqency and (ii) a qadratic increase of the tension in the cable, reslting in a linear increase with U of its eigenfreqencies. Both flid and solid freqencies increase in the same proportion, and the system passively adapts to remain at lock-in despite the flow variations. The efficiency is therefore only weakly dependent on the dimensional vale of the flow velocity U (Fig. -b). The robstness of in-flow catenaries to flow velocity variations therefore arises from the passive adaptation of their internal tension to the srronding flow. Sch passive control is not possible with a rigid cylinder. V. CONCLUSION This work presented a detailed and systematic analysis of the energy harvesting performance of flexible cables in VIV. A fndamental symmetry-breaking in the optimal harvesting strategy was identified that leads to recommending a concentration of the energy harvesting devices near one of the fixed ends of the cable. We frther showed that, ot of all distribtions possible along the entire cable, the very simplest one performs best, namely a single energy harvester located at one of the extremity of the cable; beyond its fndamental importance, this reslt also has significant practical engineering implications. With similar peak performance, flexible cables in VIV are also significantly more robst than their rigid-cylinder conterpart with respect to variations in the flow velocity (their range of efficient operation is three to eight times larger). In particlar, a fndamental physical insight on this increased robstness was obtained and two different mechanisms were identified: (i) the passive control by the flow velocity of the deformation mode (for systems with gravity-indced tension) and (ii) the passive control of the internal tension of the cable (for systems with flow-indced tension); both maintain the system at lock-in with high energy harvesting efficiency. The best performing design is identified as an almost straight cable (i.e. with minimal sag) with flow-indced tension, which can be practically achieved with a netrally-boyant cable. From an engineering point of view, this system is not more difficlt to bild than a device based on a rigid cylinder, and Fig. smmarizes the comparison of its performance to the rigid-cylinder system. Frther, for a netrally-boyant cable the direction of gravity is irrelevant, and the system can be fixed vertically, providing an interesting design soltion to passively adapt to the variability in the flow direction.

14 Efficiency E/ciency U = / U FIG.. Evoltion of the harvesting efficiency with the flow velocity for harvesting systems based on VIV of rigid cylinders (solid) and straight cables with flow-indced tension (dashed). µ = π/2 for the rigid cylinder and the cable. Appendix A: Local frame definition The local orthonormal basis in Fig. 2 is given by e r e θ e ϕ = cos θ cos ϕ sin ϕ sin θ cos ϕ sin θ cos θ cos θ sin ϕ cos ϕ sin θ sin ϕ e X e Y e Z (A.) Appendix B: Straight Cable: Direct and Adjoint Eqations For a straight cable with distribted harvester, the only non-zero component of the displacement is z, and z and q are governed by ( z + σ + γ ) ż µ π 2 2 z = M q (B.) q + ɛ ( q 2 ) q + q = A z and the bondary and initial conditions are z t= = ż t= = q t= =, q t= = q (x) and z x= = z x= =. We choose a final time t f that is mch larger than the satration time and long enogh to ensre that the timeaverage operator involved in the definition of efficiency is converged: [ tf ] η = 6 µ π 3 ST 3 σ(x) ż 2 dt dx (B.3) The definition of η in Eq. (B.3) is sed here for its convenience to derive adjoint eqations. Using Eq. (B.3), the gradient of η with respect to the fnction σ is the fnction of x: t f t= (B.2) σ [η](x) = 6 µ π 3 S 3 T t f tf t= ż ( ż z ) dt (B.4) where the variables z and q satisfy the adjoint eqations ( z σ + γ ) ż ( z ) = A q µ π σ z q ɛ ( q 2 ) q + q = M z (B.5) (B.6)

15 with the final and bondary conditions z t=tf = q t=tf = q t=tf =, ż t=tf = 2 σ ż t=tf, z x= = z x= =. Note that the fnctions z and q appearing in Eqs. (B.5) and (B.6) and in the final condition for z are the soltions to the direct problem Eqs. (B.) and (B.2). 5 Appendix C: Rigid Cylinder For a rigid cylinder, the governing eqations for the cross-flow (vertical) displacement z and wake variable q are ( z + σ + γ ) ż + µ 2 z = M q (C.) q + ɛ ( q 2 ) q + q = A z (C.2) where γ, µ, M have the same definition as for the flexible strctre (Eqs. (8) and (9)) and and σ are defined consistently with the flexible case as = f/f and σ = R/(2πfm T ). [] M. Bernitsas, K. Raghavan, Y. Ben-Simon, and E. Garcia, VIVACE (Vortex Indced Vibration Aqatic Clean Energy): A new concept in generation of clean and renewable energy from flid flow, Jornal of Offshore Mechanics and Arctic Engineering, vol. 3, no. 4, p. 4, 28. [2] A. Barrero-Gil, G. Alonso, and A. Sanz-Andres, Energy harvesting from transverse galloping, Jornal of Sond and Vibration, vol. 329, no. 4, pp , 2. [3] H.-J. Jng and S.-W. Lee, The experimental validation of a new energy harvesting system based on the wake galloping phenomenon, Smart Materials and Strctres, vol. 2, no. 5, p. 5522, 2. [4] A. Abdelkefi, Z. Yan, and M. R. Hajj, Modeling and nonlinear analysis of piezoelectric energy harvesting from transverse galloping, Smart materials and Strctres, vol. 22, no. 2, p. 256, 23. [5] D. Vicente-Ldlam, A. Barrero-Gil, and A. Velazqez, Optimal electromagnetic energy extraction from transverse galloping, Jornal of Flids and Strctres, vol. 5, pp , 24. [6] H. Dai, A. Abdelkefi, U. Javed, and L. Wang, Modeling and performance of electromagnetic energy harvesting from galloping oscillations, Smart Materials and Strctres, vol. 24, no. 4, p. 452, 25. [7] Q. Xiao and Q. Zh, A review on flow energy harvesters based on flapping foils, Jornal of Flids and Strctres, vol. 46, pp. 74 9, 24. [8] K. Singh, S. Michelin, and E. de Langre, Energy harvesting from axial flid-elastic instabilities of a cylinder, Jornal of Flids and Strctres, vol. 3, pp , 22. [9] K. Singh, S. Michelin, and E. de Langre, The effect of non-niform damping on fltter in axial flow and energy-harvesting strategies, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, vol. 468, no. 247, pp , 22. [] O. Doaré and S. Michelin, Piezoelectric copling in energy-harvesting flttering flexible plates: linear stability analysis and conversion efficiency, Jornal of Flids and Strctres, vol. 27, no. 8, pp , 2. [] S. Michelin and O. Doaré, Energy harvesting efficiency of piezoelectric flags in axial flows, Jornal of Flid Mechanics, vol. 74, pp , 23. [2] J. Allen and A. Smits, Energy harvesting eel, Jornal of Flids and Strctres, vol. 5, no. 3, pp , 2. [3] G. Taylor, J. Brns, S. Kammann, W. Powers, and T. Welsh, The energy harvesting eel: a small sbsrface ocean/river power generator, IEEE Jornal of Oceanic Engineering, vol. 26, no. 4, pp , 2. [4] H. Akaydın, N. Elvin, and Y. Andreopolos, Wake of a cylinder: a paradigm for energy harvesting with piezoelectric materials, Experiments in Flids, vol. 49, no., pp , 2. [5] D.-A. Wang, C.-Y. Chi, and H.-T. Pham, Electromagnetic energy harvesting from vibrations indced by kármán vortex street, Mechatronics, vol. 22, no. 6, pp , 22. [6] P. Meliga, J.-M. Chomaz, and F. Gallaire, Extracting energy from a flow: an asymptotic approach sing vortex-indced vibrations and feedback control, Jornal of Flids and Strctres, vol. 27, no. 5, pp , 2. [7] W. Hobbs and D. H, Tree-inspired piezoelectric energy harvesting, Jornal of Flids and Strctres, vol. 28, pp. 3 4, 22. [8] A. Barrero-Gil, S. Pindado, and S. Avila, Extracting energy from vortex-indced vibrations: a parametric stdy, Applied Mathematical Modelling, vol. 36, no. 7, pp , 22. [9] H. D. Akaydin, N. Elvin, and Y. Andreopolos, The performance of a self-excited flidic energy harvester, Smart Materials and Strctres, vol. 2, no. 2, p. 257, 22. [2] A. Mehmood, A. Abdelkefi, M. Hajj, A. Nayfeh, I. Akhtar, and A. Nhait, Piezoelectric energy harvesting from vortexindced vibrations of circlar cylinder, Jornal of Sond and Vibration, vol. 332, no. 9, pp , 23. [2] C. Grothier, S. Michelin, R. Borget, Y. Modarres-Sadeghi, and E. de Langre, On the efficiency of energy harvesting sing vortex-indced vibrations of cables, Jornal of Flids and Strctres, vol. 49, pp , 24.

16 [22] H. Dai, A. Abdelkefi, and L. Wang, Theoretical modeling and nonlinear analysis of piezoelectric energy harvesting from vortex-indced vibrations, Jornal of Intelligent Material Systems and Strctres, vol. 25, no. 4, pp , 24. [23] T. Sarpkaya, A critical review of the intrinsic natre of vortex-indced vibrations, Jornal of Flids and Strctres, vol. 9, no. 4, pp , 24. [24] C. Williamson and R. Govardhan, Vortex-indced vibrations, Annal Review of Flid Mechanics, vol. 36, pp , 24. [25] C. Williamson and R. Govardhan, A brief review of recent reslts in vortex-indced vibrations, Jornal of Wind Engineering and Indstrial Aerodynamics, vol. 96, no. 6, pp , 28. [26] M. P. Païdossis, S. J. Price, and E. de Langre, Flid-Strctre Interactions: Cross-Flow-Indced Instabilities. New York: Cambridge Univ Press, 2. [27] R. King, An investigation of vortex indced vibrations of sb-sea commnications cables, in Proceedings of the 6th International conference on Flow-Indced Vibration, London, UK: PW Bearman (ed), pp , 995. [28] J. Chaplin, P. Bearman, F. Heara Harte, and R. Pattenden, Laboratory measrements of vortex-indced vibrations of a vertical tension riser in a stepped crrent, Jornal of Flids and Strctres, vol. 2, no., pp. 3 24, 25. [29] R. Violette, E. de Langre, and J. Szydlowski, A linear stability approach to vortex-indced vibrations and waves, Jornal of Flids and Strctres, vol. 26, no. 3, pp , 2. [3] B. Adoly and Y. Pomea, Elasticity and geometry: from hair crls to the non-linear response of shells. Oxford Univ Press, 2. [3] R. Blevins, Applied flid dynamics handbook. New York: Van Nostrand Reinhold Co., 984. [32] R. Blevins, Flow-indced vibrations. New York: Van Nostrand Reinhold Co., 99. [33] C. G. Broyden, A class of methods for solving nonlinear simltaneos eqations, Mathematics of comptation, vol. 9, no. 92, pp , 965. [34] M. Facchinetti, E. de Langre, and F. Biolley, Copling of strctre and wake oscillators in vortex-indced vibrations, Jornal of Flids and Strctres, vol. 9, no. 2, pp. 23 4, 24. [35] G. Franzini, A. L. C. Fjarra, J. R. Meneghini, I. Korkischko, and R. Franciss, Experimental investigation of vortexindced vibration on rigid, smooth and inclined cylinders, Jornal of Flids and Strctres, vol. 25, no. 4, pp , 29. [36] M. Facchinetti, E. de Langre, and F. Biolley, Vortex-indced travelling waves along a cable, Eropean Jornal of Mechanics-B/Flids, vol. 23, no., pp , 24. [37] R. Violette, E. de Langre, and J. Szydlowski, Comptation of vortex-indced vibrations of long strctres sing a wake oscillator model: comparison with DNS and experiments, Compters & Strctres, vol. 85, no., pp. 34 4, 27. [38] Y. Cao, S. Li, L. Petzold, and R. Serban, Adjoint sensitivity analysis for differential-algebraic eqations: The adjoint dae system and its nmerical soltion, SIAM Jornal on Scientific Compting, vol. 24, no. 3, pp , 23. [39] P. Meliga, E. Bojo, G. Pjals, and F. Gallaire, Sensitivity of aerodynamic forces in laminar and trblent flow past a sqare cylinder, Physics of Flids, vol. 26, no., p. 4, 24. 6