Characterisation of aeroelastic harvester efficiency by measuring transient growth of oscillations

• A method for estimating harvesting performance from the free transient is developed. • Curved blade is shown to be an order of magnitude more efficient than square prism. • The optimal curvature range of the galloping curved blade harvester is established. • Flow attachment is shown to improve performance of galloping energy harvesters. • The cyclic formation and shedding of a leading edge vortex is documented.


Introduction
The objective of energy harvesting is to convert ambient energy to a useable form, typically electrical energy for use in small electrical devices. Interest in energy harvesting technologies has grown rapidly in recent years as demand for small autonomous electrical devices, such as those in wireless sensor networks, has increased. Energy harvesting systems can enable the avoidance of chemical batteries and their drawbacks [1], with applications varying from roadway engineering [2] to wearable devices [3].
Aeroelastic energy harvesting utilises fluid-structure interaction to transform kinetic energy in fluid flow into a useable form and can broadly be distinguished into three phenomenon, vortex induced vibrations [9], the flutter of aerofoils [10] and the galloping instability of prismatic sections [11]. Vortex induced vibrations occur due to the unsteady periodic shedding of vortices from a structure, which results in a periodic forcing that may lead to resonant oscillations. Aerofoil flutter occurs as the coupled two degree of freedom dynamics of the aerofoil results in the formation of self-sustained oscillations. The galloping instability only requires a single degree of freedom and can also lead to the formation of self-sustained oscillations under incident flow conditions [12]. These self-sustained oscillations can be exploited for the harvesting of flow energy by the implementation of an appropriate transduction mechanism, typically based on either the piezoelectric effect [13] or electromagnetic induction [14].
The tip geometry of the galloping energy harvester fundamentally determines the nature of the fluid-structure interaction and hence the potential energy harvesting performance. A variety of geometries have been considered, particularly the rectangular [15], isosceles triangular [16] and D-shaped [17] prismatic sections. Experimental comparison of these geometries has demonstrated the square prism to provide the greatest power output [18]. Modifications to the circular cylinder by the attachment of cylindrical rods has also been demonstrated to result in galloping oscillations [19], rather than oscillations purely induced by the periodic shedding of vortices. Other geometries, such as the forkshaped design developed by Liu, Zhang, Peng and Meng [20] and the curved plate geometry of Zhou, Zou, Wei and Liu [21], have more recently been demonstrated to be capable of high harvesting efficiencies.
Dissimilarly to the detached bluff body flows which are observed for the rectangular, isosceles triangular and D-shaped sections, in the case of the curved blade tip geometry suggested in Tucker Harvey, Khovanov and Denissenko [22], the flow becomes attached to the rear face of the geometry and results in improved harvesting efficiencies. The curvature of the blade is a key geometric parameter for defining its aerodynamics and hence whether the flow becomes attached. As previous investigations of the galloping curved blade considered only a single curvature, the role of curvature forms a significant knowledge gap in the development of these devices, which is addressed in the current study.
In this work, the influence of the curvature of the galloping curved blade on energy harvesting performance is evaluated experimentally. Dissimilarly to other experimental assessments of the performance of galloping energy harvesters, where a transduction mechanism was implemented and optimised to allow the measurement of harvested power [15], an approach is developed to the predict the performance from the free oscillation transient without a transduction mechanism. This avoids the influence of the particular implementation of a transduction mechanism as well as the requirement of optimising its parameters. This suggested alternative method could be applied to many other similar devices, while the results identify the optimal curvatures for the curved blade geometry.
The nature of the flow field around the blade is examined through flow visualisation. This methodology allows the different flow structures to be readily identified and related to the variation of performance and avoids the use of computationally expensive simulations. The visualisations demonstrate clear distinctions in the flow field between the lower and higher performing curvatures. The operational regime of the latter is related to that of wind turbines. To provide a reference to geometries which have been previously considered in the literature, a square prism geometry is also evaluated and compared to results obtained with the curved blade.
In this paper, the experimental methodology applied to capture the transient and steady state dynamics of the galloping oscillators is first described in Section 2. Following this the method for the prediction of the energy harvesting performance from the free oscillation transient is derived from the underlying mathematical model in Section 3, alongside the results of a numerical verification. The experimental results are presented in Section 4, with both the steady state dynamics and predicted energy harvesting performance. Finally, flow visualisations for the different geometries are illustrated in Section 5.

Experimental methodology
The galloping oscillator was constructed by fixing a 3D printed curved blade to the end of two cantilever beams. The curved blades of different curvatures were manufactured with an UP BOX 3D printer using acrylonitrile butadiene styrene (ABS) as the extruded material. The oscillator could then be mounted into a wind tunnel where flow conditions could be closely controlled, and aligned such that oscillations occur in a direction perpendicular to the freestream flow direction. Fig. 1 demonstrates the constructed galloping oscillator and its orientation with relation to the freestream flow velocity, which is denoted by U , while the detailed experimental setup within the wind tunnel test section is illustrated in Fig. 2.
The geometry of the curved blade was defined by a circular arc of base radius R and arc angle 2 . The curvature could hence be defined by   the half arc angle as demonstrated in Fig. 3. A range of angles from°9 0 to°25 were selected for the analysis. For a comparison to be made between blades of different curvature, the chord length, Ch, span, s, and thickness, t, of the blade were maintained constant between the geometries. The blade mass m tip was also kept constant to ensure that the oscillation frequency was the same throughout the experiments. This was achieved by varying the infill percentage in the 3D printing process. The parameters of the experimental system are presented in Table 1.
A simplified two dimensional illustration of the galloping oscillator is demonstrated in Fig. 4, which also shows the aerodynamic forces and velocity vectors. The galloping instability occurs due to the change in the aerodynamic forces acting on the blade with the effective angle of attack , which is defined as, where y denotes the velocity of the blade in the y direction and is the maximum slope angle of the beam. This relationship is illustrated in Fig. 4.
To provide a comparison to geometries which have been previously studied in the literature, a square prism geometry was constructed with width and length equal to the curved blade chord length. The mass and span of the square geometry were also fixed to that of the curved blades.
The dynamics of the oscillator were captured with the use of a single camera motion tracking system with four fluorescent markers under ultraviolet illumination. The positions of the camera and UV light within the wind tunnel test section are demonstrated in Fig. 2. The camera, which was positioned above the galloping oscillator, had a frame rate of 50 fps and a resolution of × 1920 1080 pixels. The first marker was positioned at the base of the cantilever beam, acting as a datum. The second marker was placed at the top of the cantilever beam, providing the position of the blade, while the two remaining markers provided the locations of the blade tips, which allowed the maximum slope angle of the cantilever beam to be determined. The maximum uncertainty of the motion tracking was estimated at 1.8 mm, which is equivalent to 0.026 when nondimensionalised with chord length Ch.
The flow velocity within the wind tunnel was varied from 0.9 ms 1 to 2.0 ms 1 corresponding to Reynolds numbers of × 4.0 10 3 and × 9.0 10 3 respectively. The Reynolds number is defined as, where and µ are the density and dynamic viscosity of air respectively. Flow velocity measurements were made within the test section of the wind tunnel with a pitot tube connected to a Furness Controls FCO560 digital manometer. Measurements were taken at a sampling frequency of 100 Hz with an estimated maximum uncertainty of 0.012 ms 1 . The dynamics of the galloping oscillator were acquired from rest under constant flow velocity conditions, providing both the transient and steady state dynamics.
Smoke wire flow visualisation was performed to allow the observation of the flow patterns around the different blade geometries. A Phantom high speed camera captured images at a frame rate of 600 fps and with a resolution of × 1152 896 pixels, while smoke was generated by a twisted pair of 0.255 mm diameter steel wires, which was coated with mineral oil and mounted horizontally across the wind tunnel test section. An electrical current was then put through the wires to vaporise the oil, hence seeding the flow. The location of the smoke wires relative to the galloping oscillator is illustrated in Fig. 2.

Predicting energy harvesting performance from the free oscillation transient
To demonstrate how the performance of a galloping energy harvester can be predicted from the free oscillation transient, the underlying mathematical model must first be considered. The dynamics of a galloping energy harvester with a piezoelectric harvesting circuit have been shown to be well described by the system of Ordinary Differential Equations (ODEs) [23], where C ( ) y is the force coefficient of the projection of the aerodynamic force in the y direction, A f is the projected frontal area of the tip geometry, C damp is the mechanical damping coefficient and denotes the oscillation frequency. In the second equation, which describes the harvesting circuit, represents the electromechanical coupling factor, V is the voltage generated across the piezoelectric, R l is the resistance of the load resistor and C p is the capacitance of the piezoelectric. An equation in terms of instantaneous power can be obtained by multiplying Eq. (3) by the linear velocity y and the tip mass m tip yielding,  Assuming the trajectory is periodic with a frequency of , Eq. (5) can be cycle averaged by integrating temporally between 0 and 2 yielding, The total energy in the galloping oscillator E tot can be defined as the sum of the kinetic and potential energies, Taking the time derivative and cycle averaging yields, Here P P , fs damp and P elec denote the cycle averaged power flows due to fluid-structure interaction, mechanical damping and electromechanical interaction respectively. Similar consideration of Eq. (4), which describes the harvesting circuit, provides a second cycle averaged power equation which can be written as, where P harv denotes the cycled averaged harvested power and Provided there is significant time scale separation between the conservative and non-conservative terms the system can be considered as weakly nonlinear and hence the trajectory can be approximated as sinusoidal, where A y denotes the amplitude of the y displacement, A V is the amplitude of the voltage in the harvesting circuit and is the phase difference between the displacement and the voltage. Under this assumption, it can be shown that all terms in Eq. (11) become functions of the displacement amplitude A y only and hence the power flow is defined by the oscillation amplitude and the parameters. When the system is operating on a limit cycle, hence with constant amplitude, and energy is harvested with > 0, there cannot be cycle averaged accumulation of energy within either the electrical circuit or the galloping oscillator resulting in, This results in the simplification of Eq. (11) to, Alternatively, considering a case without energy harvesting, hence = 0, and in which oscillations are growing in amplitude, = dE dt P P tot fs damp (15) Comparing Eqs. (14) and (15) and considering that the terms are functions of the oscillation amplitude A y only with the assumption of a sinusoidal trajectory, Under the condition that the steady state amplitude when > 0 is equal to the instantaneous amplitude when = 0, The cycle averaged accumulation of total energy within a galloping oscillator dE dt tot without a harvesting mechanism therefore provides an estimate of the power which could be harvested at the same instantaneous amplitude, A y , if a mechanism where to be implemented and harvested energy under steady state conditions, whilst maintaining constant amplitude.
The method for estimating power was verified with numerical simulations of the underlying ODEs, Eqs. (3) and (4), while utilising the polynomial approximation of C y for the square cylinder tip geometry developed by Parkinson [12]. Typically parameters for the galloping oscillator and harvesting circuit where taken from Yang, Zhao and Tang [18]. A transient with = 0 was first simulated and then used to predict the harvesting power P harv as a function of amplitude A y . These results were then compared to steady state simulations of the system with > 0. Both small perturbation and large displacement initial conditions were utilised for these simulations to ensure the discovery of all limit cycles. Fig. 5 demonstrates the verification results at three different reduced velocities. Reduced velocity is defined as, where Ch is the chord length of the blade. Similarly to wind turbines, a tip speed ratio can be defined as, ts ts (19) u ts represents the ratio based on instantaneous velocity y, whereas U ts is based on velocity amplitude Y and hence provides a maximum speed where A swept is the frontal area which the blade travels through. The prediction based on the free oscillation transient can be seen to accurately predict the harvested power when the model was simulated with > 0. where f s is the shedding frequency. In this region the frequency of vortex shedding f s becomes close to the oscillation frequency 2 , hence resulting in small resonant oscillations. After this region, the amplitude of the limit cycles can be seen to decrease with the curvature half arc angle , while the minimum flow velocity required to lead to the formation of galloping oscillations was greatest for the least curved geometries. As the range of evaluated reduced velocities is not significantly larger than St 1 , unsteady aerodynamic effects are likely to have a significant influence across the full range, while it is also possibly that vortex induced vibrations delayed the onset of galloping oscillations due to quenching. Fig. 7 demonstrates the maximum tip speed ratio with reduced velocity. For all of the considered geometries, it is noticeable that the maximum speed ratio U ts appears to tend to a constant value with reduced velocity. It can be shown that in the case of a galloping oscillator where the angle of attack is a function of u ts only with = u tan ( ) ts 1 and the cantilever beam orientated perpendicular to U , the amplitude U ts on the limit cycle will become constant with reduced velocity as aerodynamics dominates the influence of mechanical damping. This has also been demonstrated experimentally in previous investigations of the galloping curved blade [22]. As Fig. 7 shows a similar behaviour, it can be suggested that the most significant period of power flow, which defines the limit cycle amplitude, occurs when is close to zero. This would hence provide a similar dependence of U ts on U as when is a function of u ts only, where the deflection of the beam does not result in a rotation of the tip geometry which contributes to the angle of attack. Fig. 8 illustrates how the angle of attack varies with the tip speed ratio on the steady state limit cycle at a number of different reduced velocities. The cycles tend to a constant value at the u ts maximas, which is consistent with the tending of U ts to a constant value in Fig. 7. The maximum values of occur below the maximum value of u ts and hence correspond to the combined influence of both beam deflection and the velocity of the blade.

Experimental dynamics and estimated harvesting performance
The calculation of dE dt tot for the estimation of the energy harvesting performance is illustrated for an individual transient in Fig. 9. The top plot demonstrates the time series of the centre of mass displacement nondimensionalised by the chord length, = y y Ch / . Time is nondimensionalised by the oscillation time period, = t t 2 . The second plot presents the cycle averaged total energy E tot . This can be seen to grow from zero, where there are no oscillations, to a constant value, where the oscillations are on the limit cycle. The rate at which E    grows with time can be seen to be small at the start of the transient and small close to the limit cycle but with a maximum between. The third plot illustrates the cycle averaged time derivative dE dt tot , which provides the estimate of energy harvesting performance. The amplitude at which cycle averaged energy growth is fastest and hence the energy harvesting performance would be greatest is marked by the red line.
The dependence of the estimated coefficient of performance C p on the tip speed ratio for the different geometries considered is presented in Fig. 10. Results obtained at three different reduced velocities are shown, as illustrated by dotted, dashed and solid lines. The°°35 , 45 and°6 5 geometries can be seen to significantly outperform the other two. The curves for the three different reduced velocities also appear to be close to collapsing, especially for the°45 and°65 curvatures. This collapse would be expected in the case where the cantilever beam is perpendicular to U and hence suggests that the most significant point for power flow within the cycle is when is close to zero.
The optimal operating amplitude for energy harvesting is defined by the location of the C p maximum. The tip speed ratio U ts at the C p maximum was greatest for the°35 geometry with an average of 0.76, while the lowest values occurred with the°90 geometry with an average of 0.45.
The maxima of C p with reduced velocity is illustrated in Fig. 11 for the different curvature geometries. The°°35 , 45 and°65 geometries can be seen to substantially outperform the other two geometries for all considered reduced velocities. The leading of the two varied with reduced velocity, with the°35 geometry leading at lower velocities, while the°65 geometry performed best at the higher end of the reduced velocity range.
A comparison of performance of the curved blade and the square prism geometry is demonstrated in Fig. 12. All of the evaluated curved blade geometries can be seen to substantially outperform the square prism, which operates at a much lower tip speed ratio. Comparing the performance of the curved blade to that of rotating wind energy devices demonstrates it to be positioned at a C p value of close to half that of an American Multiblade turbine and only three to four times below Savonius turbines [24].

Flow visualisation
Flow visualisations for the five considered curvatures are illustrated in Fig. 13 at a reduced velocity of 8.4. The blade is viewed from above, with the freestream flow direction in the upward direction. The blade is   close to its zero displacement position and is traveling from left to right. The colourmap represents the pixel intensities obtained from the monochrome Phantom high speed camera and is used to improve the clarity of the visualisations. The twisted wires generated a series of filaments of smoke, rather than a continuous sheet. These filaments of smoke represent streaklines, defined as the loci of points of all the fluid particles which have passed through a particular point in the past, and are clearly illuminated in Fig. 13. When unperturbed by the presence of the blade, the streaklines appear straight, reflecting only the freestream uniform flow. This is evident in the right hand side of the visualisations. Closer to the blade the streaklines are deflected significantly. The deflection of the streaklines in the left direction on the rear surface of the blades is illustrative of the generation of a force in the y direction acting on the blade. Dark regions in the visualisations illustrate areas where the smoke is well mixed due to higher vorticity in these areas.
In subplot A, which relates to the least curved geometry with an arc angle of°25 , leading edge separation is evident as well as the formation of a large leading edge vortex (LEV). In subplots B, C and D, which relate to the higher performing°°35 , 45 and°65 half arc angle geometries, leading edge flow separation is not evident and the flow appears to be attached for the majority of the chord length. Subplot E demonstrates the geometry with the highest curvature, relating to a half arc angle of°90 . In this case the separation point is close to the leading edge resulting in the formation of a large wake.
In the square prism case, illustrated in subplot F of Fig. 13, the flow can be seen to separate on both of the front edges of the square without reattachment on the sides parallel to the freestream flow direction. The stagnation point can also be seen to be close to the centre of the front face. The flow in the case of the square prism is hence highly separated, differing significantly from the attached flows observed with the °°35 , 45 and°65 half arc angle curved blade geometries. It can hence be concluded that the attachment of the flow to the rear surface of the blade allows the°°35 , 45 and°65 arc angle geometries to outperform the others considered. The formation and shedding of a leading edge vortex during each half oscillation was evident for the°°°2 5 , 35 , 45 and°65 cases and could provide a lift enhancement in these cases by similar mechanisms to as has been shown to occur in flapping flight [25,26]. However, the unsteady forces relating to the leading edge vortex and how they relate to energy harvesting performance requires further investigation.

Conclusions
In this work, the influence of curvature on the energy harvesting performance of the galloping curved blade has been investigated experimentally by measuring the growth rate of total energy in the free oscillation transient. The developed method avoids the implementation of a transduction mechanism and hence the optimisation of the associated parameters, providing a convenient and fast means of evaluating the energy harvesting performance of galloping geometries. The method could hence be applied to similar geometries and aid in their experimental development.
The curved blades with curvatures in the medium range, with half arc angles of°°35 , 45 and°65 were found to considerably outperform curved blades with°25 and°90 half angles. As demonstrated by flow visualisations, the flow becomes attached in these three cases but not in the others. The higher performance in these cases can hence be attributed to the attachment of the flow to the rear surface of the blade when the velocity is close to its maxima. A maximum coefficient of performance of 0.08 was obtained with the°65 half arc angle geometry. The optimal operating amplitude of a wind energy harvester is defined by the amplitude of the coefficient of performance maximum. For the curved blades investigated, the tip speed ratio at the coefficient of performance maximum ranged from 0.32 to 0.74.
Comparison of the curved blade to the square prism geometry, which has been shown to relate to a highly separated flow, has demonstrated the curved blade to provide an order of magnitude higher performance within the considered range of reduced velocities.
Flow visualisations have also demonstrated the presence of a leading edge vortex, which is formed and shed during each half oscillation and occurs with four of the five investigated geometries. Although the leading edge vortex may provide a lift enhancement, further investigation is required to determine whether it is beneficial to energy harvesting performance.
The results hence demonstrate that when designing galloping energy harvesters based on the curved blade geometry, half arc angles of around°65 will provide the greatest harvesting efficiencies. The measured coefficient of performance of 0.08 places the curved blade harvester only three to fours times below Savonius turbines, making it a candidate for real world applications.