ENERGY HARVESTING FROM AXIAL FLOW INDUCED INSTABILITIES IN SLENDER STRUCTURES

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1 Flow-Induced Vibration, Meskell & Bennett (eds) ISBN ENERGY HARVESTING FROM AXIAL FLOW INDUCED INSTABILITIES IN SLENDER STRUCTURES Kiran Singh Sébastien Michelin Emmanuel de Langre Department of Mechanics, LadHyX École Polytechnique, Palaiseau, France ABSTRACT In this work we examine the prospects for harvesting energy from flutter instabilities of a slender, flexible cylinder in an axial flow stream. We couple the fluid-solid model and include energy harvesting as curvature based damping. In the first instance we model the reduced order system comprising of an articulated rigid cylinder pair with discrete springs and dampers and demonstrate the scope for harvesting energy from flutter instabilities. Extending the study to a continuously varying system confirms the scope for energy harvesting. The numerical models are employed to find the optimal damping distributions and we see that the two configurations give very different results. We examine the subtle differences between the two apparently canonical configurations that lead to these differing optimals. INTRODUCTION The projected growth in global energy demands strongly motivates the interest in energy harvesting concepts, where the idea is to harness unconventional and previously untapped sources of energy. Concepts range from energy harvesting from tidal currents [1] and ocean waves [] to energy scavenging from ambient vibrations in structures such as buildings and bridges and oscillatory motion of wheels in automobiles or turbines in engines to power sensors and mobile electronic devices [3]. Energy harvesting from fluid-structure interactions (FSI) include concepts such as vortex induced vibrations (VIV) of bluff bodies in a cross-flow [4], resonant vibrations induced in aerofoils mounted on elastic supports [5] and flutter of flexible plates [6, 7]. In this work we focus on harvesting Present address: OCCAM, Maths Institute, 4-9 St Giles, Oxford OX13LB, UK author for correspondence energy from flutter instabilities of slender structures in an axial flow. The classical description of flutter instabilities is selfsustained oscillations that arise due to the unstable coupling of fluid dynamic pressure and structural bending modes, where for undamped structures the critical speed at flutter onset depends on fluid as well as structural properties [8]. Flapping flags [9,10] and fluttering panels [11] in a steady flow fall under the broad category of external flow based instabilities. Recent work by de Langre et al. [1] examined the onset and evolution of fluid-induced instabilities in slender structures (also see [13]). In this work our objective is to evaluate the scope for harvesting energy from fluttering slender elastic structures. From the perspective of the fluid-solid system, energy harvesters act essentially as an energy sink, and in line with the theoretical point of view of this work, are modelled as curvature based damping. Noting that for the undamped case the reduced order system is a canonical model for a continuously varying system [14], in [15] we examined the scope to harvest energy in fluttering structures using a reduced-order model of a pair of slender cylinders connected with a curvature-based springdamper pair. In [16] we analysed the continuous energy harvester configuration consisting of an elastic beam with a non-uniform distribution of damping. These studies led to the surprising result that the optimal for the biarticulated configuration (peak damping at the fixed end) is very different from the continuous optimal (a damping distribution increasing with distance from the fixed end). In this paper we interpret and reconcile the results from these two harvester configurations. This paper is organised as follows: we first examine the reduced order system and briefly develop the governing equations for a nonlinear articulated spring-dampercylinder pair. We develop the fluid dynamics model for a 735

2 slender flexible structure, which includes the inviscid and viscous drag components. The power harvested is computed numerically and results from the numerical optimisation study are examined. Next we consider the continuous configuration: we develop the governing equations and model for nonlinear damping distributions and examine the optimal linear damping distribution. Finally we examine the origin of the subtle differences in configuration between the discrete and continuous systems, which lead to the different optimals. BI-ARTICULATED SYSTEM We consider the planar motion of a pair of rigid cylinders: the first cylinder is allowed to rotate about the fixed joint O and is connected to the second cylinder through the articulation P (see Fig. 1). These articulated joints give us a two degree of freedom configuration where θ 1, θ are the angles through which the first and second cylinders rotate about an axis perpendicular to plane (e x,e y ) at joints O and P respectively. The energy harvesters are modelled as viscous dashpots with damping coefficients (c 1,c ), and we seek the combination that corresponds to the peak power. Cylinders of cross-sectional area A, are modelled with equal lengths, L, torsional spring stiffness, K, and mass per unit length, ρ s A. The slender structure is immersed in a stream of fluid of density ρ moving at mean speed U, the motion of the structure is confined to the (e x,e y ) plane. In this section we develop the system of equations required to model the fluid-structure system combined with the dampers/energy harvesters. Since fully developed flutter is a nonlinear phenomenon, the equations of motion are developed for large rotation angles. We nondimensionalise the system of equations in terms of the characteristic length, L, mass, ρ s AL, and time, t = (ρ s AL 3 /K) 1/, the nondimensional flow speed is given as u = U (ρ s AL/K) 1/. (1) Governing equations We apply the conservation of momentum principle to derive the equations of motion for the pair of rotating rigid cylinders in contact [17]. This requires that the rates of change of linear momentum (dp/dt = m r) and angular momentum (dl/dt = (r mṙ)/ t) respectively balance the external forces and torques acting on a rigid body of mass m, instantaneously at position vector r (defined with respect to the fixed origin). Momentum balance leads to FIGURE 1: Model of the bi-articulated cylindrical energy harvester (see text for description). a pair of equations that in vector form give: dl dt = T k (X)+T c (Ẋ)+T f (X,Ẋ,Ẍ). () L is the angular momentum vector for the bi-articulated cylindrical system that balances the external forces and torques. T k and T c are the system stiffness and damping torques, respectively, and T f corresponds to the torque generated due to the locally acting fluid-dynamic force f. These expressions are defined in (1)-(15) (see [15] for a complete derivation). Solutions are sought in terms of variable X =[θ 1,θ ] T, where the dotted terms are differentiated in time. Fluid dynamic model For slender structures of diameter D L, the fluid force, f, can be decomposed into the reactive and resistive components, respectively f i and f v. The reactive force f i is obtained from Lighthill s Large Amplitude Elongated Body Theory [18] as: f i = m a (un n) t (u nu τ n) s + 1 (u nτ), (3) s and f v is obtained from Taylor s resistive force theory [19]: f v = M Cf u τ u C d u n u n n, (4) where u = u τ τ + u n n is the solid s local velocity relative to the incoming flow. We define the nondimensional mass 736

3 ratio, M = ρdl/ρ s A, and nondimensional added mass, m a = M a /ρ s A, with M a as the dimensional added mass per unit length. For details on the fluid model refer to [15, 16] and references cited therein. In the remainder of the paper, we assume a neutrally buoyant circular cylinder (ρ = ρ s, A = πd /4, m a = 1) with D/L = 0.1 (M 1.7) for all subsequent calculations. Model of energy harvesting As noted energy harvesters are modelled in this system as viscous dashpots, with nondimensionalised coefficient c i. To quantify the energy harvesting action we parametrise damping coefficients, c i, and compute the nondimensional power harvested: (a) P = P ρdlu 3 = c 1 θ 1 + c ( θ θ 1 ), (5) where P is the mean dimensional harvested power and operator indicates time-averaging over a limit cycle oscillation of period T. In the next section we seek optimal values of the damping coefficients that yield maximum harvested power, P. Results: Eqn. () corresponds to a pair of second-order nonlinear ordinary differential equations in time with variables θ 1, θ. The equations are solved numerically using a Runge-Kutta ode45 solver in MATLAB with a relative error tolerance of We fix the flow speed as u =.45, calculations are initiated with a perturbation from the rest state, the integration is stopped if either condition θ < π/ or θ i < π/ is violated. (b) FIGURE : (a) Flutter amplitude dependence on the bifurcation parameter, u (θ : solid curves; θ 1 : dotted curves). (b) Flutter response indicated by superimposed snapshots at discrete time instants during a limit cycle period (at u =.). Flutter instabilities in the undamped system Prior to the analysis of energy harvesting we examine the flutter response of the undamped system (c i = 0) in the absence of fluid dissipation (C f = C d = 0) and in Fig. (a) we show the system response to increasing flow speed, u. We calculate a critical flow speed of u cr.13, below which the fluid acts to damp out disturbances imparted to the system. At u cr the system undergoes a Hopf bifurcation and at higher velocities we see steady flutter oscillations. A physical picture of these oscillations are indicated in Fig. (b) where we superimpose snapshots of the bi-articulated cylinder at different instants during the oscillation cycle. Power harvesting We now introduce structural damping in the configuration and calculate the power P FIGURE 3: Power dependence on damping coefficients c 1 and c (at u =.45, C f = 0.01, C d = 1.0). for combinations of (c 1,c ) that yield limit cycle oscillations. We fix the flow speed at u =.45, and select representative values for the drag coefficients (C f = 0.01, C d = 1.0). In Fig. 3 we plot the power map in the (c 1,c )- parameter space; also indicated is the stability boundary 737

4 force, F, is: 1 r M t = ντ s 1 M U θ s + ξ (s) θ n + F, s s t (6) FIGURE 4: Model of the continuously varying energy harvesting system (see text for description). that shows combinations of damping coefficients above which flutter oscillations are damped out. A key point to note is that peak power lies on the c 1 axis, indicating that a dashpot at the moving articulation is far less effective in harvesting power than one at the fixed end. The expression (5) indicates that structural damping increases the proportion of power harvested, however the very process of energy harvesting necessarily mitigates the oscillatory response. The challenge is in identifying an optimal balance between these competing effects. The results show the scope for harvesting energy from slender fluttering structures. For this system the optimal configuration that maximises power is one with harvesters concentrated at the fixed end of the configuration, and this result is invariant to variations in flow speed, mass ratio (m a ) and viscous drag coefficients. In the next section we use these insights to examine a fluttering beam with a continuous distribution of damping. CONTINUOUS SLENDER STRUCTURE We consider a cantilevered (clamped-free, fixed at O) slender structure (see Fig. 4) of length L with crosswise dimension D, density ρ s, stiffness B, and nonuniformly varying structural damping B (s). The equations for the continuous system, are nondimensionalised by the system scales: ρ, L, U. Governing equations The flexible structure is modelled as an inextensible Euler-Bernoulli beam, where r(s) is the position vector in the fixed coordinate system (e x,e y ), and s is the curvilinear coordinate. At each point along the beam, the orientation θ(s,t) is defined as the angle of the tangent vector τ(s) with the horizontal; n is the local normal. The nonlinear equation of motion for the beam subjected to a fluid where the inextensibility condition r/ s = τ is satisfied with the internal tension, ν(s, t). The nondimensional damping is given by ξ (s) =U B (s)/(bl). The clamped-free boundary conditions must also be satisfied, namely at the fixed end (s = 0): and at the free end (s = 1): θ = 0, r = 0, (7) θ s +ξ θ s t = 0, θ s + ξ θ = 0, ν = 0. (8) s s t For consistency with the bi-articulated system we define an equivalent nondimensional flow speed, U = U L ρ s A/B 1/ ; as before A is the cross-sectional area of the structure, and M the nondimensional mass ratio. We employ the same fluid model as for the discrete system (3)-(4), the nondimensionalised fluid forces for the two configurations are related by F =(f i +f v )/M. Moreover in [15] we found that friction drag has a rather small effect on the system response; therefore here we only include the parasitic drag (C f = 0, C d = 1). Energy harvester model The energy harvesting is modelled as strain-based damping ξ (s), and we obtain a similar expression for nondimensional power harvested at the damper as for the discrete case: P = P ρdlu 3 = 1 1 M U ξ (s) κ ds, (9) 0 where as before P is the mean dimensional harvested power, and κ is the time derivative of the local curvature κ. We characterise the intensity and distribution of damping with: 1 ξ 0 = ξ (s)ds, and ξ (s)=ξ(s)/ξ0. (10) 0 Eqn. (10) allows us to independently evaluate the impact on the system response of the amount of damping (ξ 0 ) and its distribution in the structure. 738

5 (a) (a) (b) (b) FIGURE 5: (a) Maximum deflection y max (solid) and orientation θ max (dashed) of the free end as a function of flow speed U. (b) Snapshots of the beam response for U = 10, 13 and 19 (from top to bottom). Results Numerical model Expanding θ in terms of orthogonal Chebyshev polynomials in s, Eq. (6) is solved numerically together with Eqs. (7) (8), using an iterative second-order implicit method in time [0]. In the numerical implementation the beam and the fluid are initially at rest; the flow speed is ramped up to its steady state value and a small perturbation is applied to the vertical flow, and at each time step we ensure conservation of energy is satisfied within an acceptable tolerance (see [16] for details). Flutter response for an undamped beam As for the discrete case we first examine the undamped flutter response and plot the system response for increasing nondimensional flow speed in Fig. 5. The critical flow speed at which flutter ensues is verified from linear stability analysis. For the energy harvesting computations we choose a flow speed of U = 13. FIGURE 6: Power computations for a linear damping distribution (11): (a) Power contours P for varying ξ 0 and ξ 1. (b) Rescaled harvested power P/Ph max as a function of the damping intensity ξ 0 for linearly decreasing (ξ 1 =, dashed), constant (ξ 1 = 0, dotted) and linearly increasing (ξ 1 =, solid) damping distributions. Model of energy harvesting Based on the results from the bi-articulated configuration we seek distributions that are allowed to vary along the length of the beam. Here we examine a simple nonhomogeneous distribution of damping of the form: ξ (s)=ξ ξ1 (s 1/) (11) characterised by two coefficients, the total damping, 10 3 < ξ 0 < 10, and slope, ξ 1. Operating on the (ξ 0,ξ 1 )-parameter space, Fig. 6(a) shows the variation of harvested power P. In Fig. 6(b) we plot the rescaled power P/Ph max, where Ph max is the peak harvested power for a constant distribution of damping (ξ 1 = 0). The figure shows that for ξ 1 = the peak harvested power is enhanced by almost 50% compared to the constant distribution case. Thus the optimal linear damping distribution is one that increases with distance from the free end. 739

6 We note that the function family (11) corresponds physically to a dispersed distribution, and the damping is significant over the entire length of the structure. Inspired by the discrete optimal that has damping concentrated at the fixed end, in [16] we optimised on a family of Gaussian distributions, where the mean position on the beam and the Gaussian spread were the optimisation parameters. Despite these different families of functions we converged on a dispersed distribution that confirms the results in Fig. 6. Therefore in contrast to the bi-articulated system, for a continuously varying system we find that a dispersed distribution of damping generates higher peak power than a focused distribution. COMPARISONS AND CONTRASTS: Discrete vs. continuous systems In this work, we considered the possibility of harvesting energy from a slender body fluttering in an axial flow, in particular the impact of harvester-distribution on the performance of the system and resulting optimisation strategies. To this end, a simplified fluid-solid model was proposed with energy harvesting represented as curvature based damping. We first investigated the reduced-order system comprising of a spring-damper-cylinder pair in an axial flow stream. The model demonstrates the scope for energy harvesting, furthermore we find that a single harvester positioned at the fixed end can optimally extract power from the system. Conversely, an additional dashpot at the moving articulation suppresses the flutter instability and considerably diminishes the total harvested power. An investigation of the continuously varying system confirms the scope to harvest energy. However we find that to maximally harvest power, damping that is dispersed along the length of the beam is superior to a focused distribution. Furthermore, an increasing distribution of damping enhances the peak harvested power by approximately 50%, over that for a constant distribution. To reconcile the differing optimal solutions between these two apparently canonical systems, we note that curvature in the system drives the instability for both configurations. Whilst the bi-articulated system has only one source of instability (the second articulation), deformations can occur all along the length of the beam in the continuous system. As damping acts to suppress the instability locally, for the continuous system the flutter instability can be maintained as long as the fluttering beam can adapt to locally rigidify the structure at the position of high damping. Conversely for the discrete case where there is one sole source of curvature that sustains the flutter instability in the entire system, oscillations are quickly suppressed if damping is added at the same curvature. Thus although the system response to damping is identical in both systems, a careful examination of the dynamics of the different configurations is necessary to predict the optimal harvester distribution. ACKNOWLEDGMENT The authors gratefully acknowledge the support of Electricité de France (EDF) for their support through the Chaire Energies Durables at Ecole Polytechnique. S. M. was also supported by a Marie Curie International Reintegration Grant within the 7 th European Community Framework Program. REFERENCES [1] Westwood, A., 004. Ocean power: Wave and tidal energy review. Refocus, 5, pp [] Falcao, A., 010. Wave energy utilization: A review of the technologies. Renew. Sust. Energ. Rev., 10, pp [3] Anton, S., and Sodano, H., 007. A review of power harvesting using piezoelectric materials ( ). Smart Mater. Struct., 16, pp [4] Grouthier, C., Michelin, S., and de Langre, E., 01. Optimal energy harvesting by vortex-induced vibrations in cables. pre-print, arxiv: v1. [5] Peng, Z., and Zhu, Q., 009. Energy harvesting through flow-induced oscillations of a foil. Phys. Fluids, 1, p [6] Tang, L., Païdoussis, M., and Jiang, J., 009. Cantilevered flexible plates in axial flow: Energy transfer and the concept of flutter-mill. Journal of Sound and Vibration, 36, pp [7] Doaré, O., and Michelin, S., 011. Piezoelectric enegy harvesting from flutter instability: Local/global linear stability and efficiency. J. Fluids Struct., 7, pp [8] Païdoussis, M. P., 004. Fluid-Structure Interactions: Slender structures and axial Flow, Volume. London: Elsevier Academic Press. [9] Alben, S., and Shelley, M., 008. Flapping states of a flag in an inviscid fluid: bistability and the transition to chaos. Phys. Rev. Lett., 100, p [10] Michelin, S., Llewellyn Smith, S. G., and Glover, B., 008. Vortex shedding model of a flapping flag. J. Fluid Mech., 617, pp [11] Crighton, D. G., and Oswell, J. E., Fluid loading with mean flow. (I). Response of an elastic 740

7 plate to localized excitation. Philos. Trans. R. Soc. London, A, 335, pp [1] de Langre, E., Doaré, O., Païdoussis, M. P., and Modarres-Sadeghi, Y., 007. Flutter of long flexible cylinders in axial flow. J. Fluid Mech., 571, pp [13] Païdoussis, M. P., Grinevich, E., Adamovic, D., and Semler, C., 00. Linear and nonlinear dynamics of cantilevered cylinders in axial flow. Part 1. Physical dynamics. J. Fluids Struct., 16, pp [14] Païdoussis, M. P., Fluid-Structure Interactions: Slender structures and axial flow, Volume 1. London: Academic Press. [15] Singh, K., Michelin, S., and de Langre, E., 01. Energy harvesting from fluid-elastic instabilities of a cylinder. J. Fluids Struct., 30, pp [16] Singh, K., Michelin, S., and de Langre, E., 01. Effect of damping on flutter in axial flow and optimal energy harvesting. (under review). [17] Landau, L. D., and Lifshitz, E. M., Mechanics, third ed., Vol. Volume 1 of Course of Theoretical Physics. Elsevier Ltd. [18] Lighthill, M. J., Large-amplitude elongatedbody theory of fish locomotion. Proc. R. Soc. B, 179, pp [19] Taylor, G. I., 195. Analysis of the swimming of long and narrow animals. Proc. R. Soc. A, 14, pp [0] Alben, S., 009. Simulating the dynamics of flexible bodies and vortex sheets. J. Comp. Phys., 8, pp Appendix A: Expressions for the discrete system The terms in Eqs. () are defined in this section. We define the rate of change of angular momentum about point P and O respectively as: θ L p = 3 + θ 1 cos(θ θ 1 )+ θ 1 sin(θ θ 1 ), 4 L o = L p θ θ cos(θ θ 1 ) θ sin(θ θ 1 ), (1) and dl/dt =[ L o L p, L p ] T. Note that this recombination of the two equations leads to significantly simpler expressions. The contribution from the restoring torsional springs is linear thus the angular moment transferred to the joints is given as, T p k = (θ θ 1 ), Tk o = θ 1 = T p k (θ 1 θ ). (13) and T k =[T o k T p k,t p k ]T. From the Kelvin-Voigt model the angular moment transferred to the joints for dampers with coefficients, c 1 and c at P and O respectively are: T p c = c ( θ θ 1 ), T o c = c 1 θ 1, = T p c c 1 θ 1 c ( θ θ 1 ), (14) and T c =[T o c T p c,t p c ] T. Fluid torque vector T f = T f,v + T f,i, where the first term is the contribution from the viscous drag or resistive forces and the second term is the inviscid or reactive force contribution. For brevity we do not include the viscous components. The contribution of the reactive flow forces to the fluid torques at the two articulations is given by, T p f,i = m θ 1 a θ 1 θ θ sin(θ θ 1 )+ 1 cos(θ θ 1 ) T o f,i = T p f,i m a θ ucosθ θ, 1 θ cos (θ θ 1 ) + θ 1 θ 1 θ sin(θ θ 1 ) + θ cos(θ θ 1 ) + u θ 1 cosθ 1 + θ cosθ cos(θ θ 1 ) + T jump, (15) where m a is the nondimensionalised added mass. T jump is the contribution of the inviscid fluid dynamics over the corner, P, is: T jump = m a ao + a 1 u + a u (16) 741

8 where a o = θ 1 cos (θ θ 1 )sin(θ θ 1 ), a 1 = θ 1 cos (θ θ 1 )cosθ cosθ 1 a = 1 sinθ cosθ sinθ cos(θ θ 1 ) sinθ 1 (17) and the inviscid fluid moment vector is, T f,i =[Tf o,i T p f,i,t p f,i ]T. 74