Nonlinearly Enhanced Vortex Induced Vibrations for Energy Harvesting

Transcription

1 2015 IEEE International Conference on Advanced Intelligent Mechatronics (AIM) July 7-11, Busan, Korea Nonlinearly Enhanced Vortex Induced Vibrations for Energy Harvesting BH. Huynh, T. Tjahjowidodo, ZW. Zhong, Y. Wang, N. Srikanth Abstract In order to enhance the performance of an energy converter based on a vortex induced vibration (VIV) system, a model of nonlinear springs, the so-called hardening springs, is applied to widen the resonance range. A nonlinear spring is introduced by utilizing cantilever beams with additional plates to harden the structure partially. An experimental investigation is carried out to analyze the potential of hardening stiffness spring in widening the resonance range. A fluid-structure interaction (FSI) simulation is performed to validate the experimental results. A comparison to the experimental results from a similar system that utilized linear springs shows the potential of hardening springs in widening the resonance range of VIV energy converters. I. INTRODUCTION Studies on harvesting energy from water flows based on VIV of an elastically mounted cylinder have intensively developed in recent years since the appearance of the VIVACE converter introduced by Bernitsas et al. [1] in 2008 that is due to the purity, plenty and sustainability of this source of energy. The working principle of this kind of energy converters is based on the VIV phenomenon. When a blunt structure is immersed perpendicularly to a fluid flow, the flow will generate alternate vortices shedding to two sides of the wake region behind the structure to form a von Kármán vortex street. These shedding vortices cause periodic alternating pressure distributions along the surface of the structure. Consequently, the structure will experience periodic drag forces and lift forces caused by shedding vortices. If the structure is elastically mounted by a spring and constrained to one degree-of-freedom in the cross-flow direction, it will vibrate under the effect of the lift forces. The structure is subsequently connected to a transmission mechanism to convert its kinetic energy into rotations of a generator shaft. In this way, the energy from the fluid flow is taken-off and converted into electrical energy. The power harvested by a VIV converter is related to oscillation amplitudes of the structure (please note that in our Huynh Bao Huy is with the Interdisciplinary Graduate School, Nanyang Technological University, 50 Nanyang Avenue, Singapore Tegoeh Tjahjowidodo is with the School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore ( ttegoeh@ntu.edu.sg). Zhong Zhaowei is with the School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore Wang Youyi is with the School of Electrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore Narasimalu Srikanth is with the Energy Research NTU, Nanyang Technological University, 50 Nangyang Avenue, Singapore system a cylindrical structure is chosen as the blunt structure). A VIV structure can only vibrate with large amplitudes when resonance occurs, i.e. the frequency of vortex shedding, which is determined by velocity of the fluid flow, is equal or close to the natural frequency of the structure. When the fluid flow induces vortex shedding at a frequency apart from the resonant frequency, the structure will vibrate with low amplitudes. This results in a poor performance of the VIV converter, where most of the cases, in natural conditions, the water flows vary significantly. Improving performance of VIV converters in terms of adapting the working condition to unstable natural water flows by enhancing oscillation amplitudes and widening the resonance range can be carried out by adjusting the governing parameters: (1) stiffness of mounting springs, k, (2) mass ratio, m*, damping ratio, ζ, and mass-damping parameter, m*ζ, and (3) Reynolds number, Re. Studies in this area have continuously focused on characterization of VIV structures under changes of the governing parameters. Some researchers tried to figure out the optimum values of the governing parameters for energy generation. These studies can be generally classified into two main classes, namely experimental model based studies and mathematical model based studies. These approaches complement and have become benchmarks for each other. There are many methods that have been proposed to improve the performance of VIV converters in experimental model based studies. The technique of utilizing surface roughness control or passive turbulence control (PTC) using waterproof sandpaper strips attached along the length of the cylinder was initiated in the study of Bernitsas and Raghavan [2] in 2008 and subsequently studied in [3], [4] and [5]. The effect of attached strips is suddenly turning the flow in the boundary layer around the cylinder from laminar regime to turbulence regime where the higher amplitude oscillations occur. Before the VIVACE converter appears, the maximum amplitude recorded of an elastically mounted cylinder is 1.19 times of the cylinder diameter [6], while PTC technique can increase the oscillation amplitude up to 2.9 times of the cylinder diameter [5]. Another method was proposed to improve the efficiency of a VIV converter in [7] and [8]. In this method, instead of sandpaper trips, wires are attached along the length of the cylinder to widen the wake region behind the cylinder. Wider wake region results in larger vortices that cause stronger lift forces and finally enlarges oscillation amplitudes. Experiments showed that tripping wires can maximize the efficiency up to 12.47% which is four times of the maximum efficiency in the case using a smooth cylinder. Recently, Bernitsas and Kumar [9] have conducted an experiment to confirm whether it is possible to integrate many cylinders together to maximize VIV for /15/$ IEEE 91

2 energy generation [9]. In this experiment, four elastically mounted cylinders are integrated together in a VIVACE converter with the distance between two cylinders from 0.5D 1.5D. This experiment showed a positive result that the oscillation amplitudes of all four cylinders are from 2.2D 2.8D. It means integration of many cylinders in a VIV converter is also a potential approach to improve its performance. Mathematical studies have focused on analyzing the effects of the governing parameters on the capability in harvesting energy of VIV converters. An asymptotic expansion for the VIV of a circular cylinder was implemented to investigate the effects of natural frequency, f n, water, mass ratio, m*, and mass-damping parameter, m*ζ, on energy generation [10]. This study showed that there is a configuration of these structural parameters that can maximize harnessed energy from fluid flows. In 2012, Barrero-Gil et al. [11] developed a mathematical model to comprehensively conduct a parametric study on the effects of the mass-damping parameter, m*ζ, on oscillation amplitudes, the resonance range and efficiency of a VIV converter. The results from this study are consistent with experimental results from previous studies about VIV of an elastically mounted cylinder. In particular, Barrero-Gil et al. [11] have successfully revealed the trend of influence of the massdamping parameter, m*ζ, on VIV response and efficiency of a VIV converter as well. It can be seen that the decrease in the mass-damping parameter, m*ζ, results in increasing of oscillation amplitudes as well as the resonance range and the optimum value of m*ζ (approximately 0.25) for maximum generated energy was also figured out. Recently, Dhanwani et al. [12] have developed a lumped parameter model of VIV to be applied in the VIVACE converter. With the objective of maximizing oscillation amplitudes, they have successfully figured out the optimal values for the stiffness of the mounting spring in the VIV structure. From the literature, it can be seen that broadening the resonance range to improve the performance of a VIV converter through maintaining its operation under severe changes of natural water flows has not been sufficiently researched. Moreover, all the reviewed studies have utilized linear springs to support the cylinder in the experiments and theoretical analysis. Meanwhile, Mackowski and Williamson [13] have experimentally proved that resonance range of an oscillating structure can be broadened if the structure is mounted by a nonlinear spring that can adjust its natural frequency as the oscillation amplitude varies to lock-in different values of frequency of exciting forces. However, the physical model of a VIV converter supported by nonlinear springs and the application of nonlinear spring structures to enhance the performance of the VIV energy converters in a sense of extending the resonance range have never been analyzed before. In this paper, a parametric experiment is conducted to investigate the effects of a prevalent kind of nonlinear springs, which is the so-called hardening spring on the ability in extending the resonance range of a VIV converter to improve its performance. Hardening stiffness is implemented by using cantilever beams hardened partially by additional plates. Details of this experimental model are presented in Section II. A numerical model based on the FSI simulation is built to validate the experimental results. Discussions on this numerical model are presented in Section III. As a comparison with the results from an experiment with linear springs, the results from this study consolidate the benefit of nonlinear springs in widening the resonance range. Details of this comparison are presented in IV. Finally, some conclusions and future directions of this study are presented in Section V. II. EXPERIMENTAL MODEL A. Background In order to respond to the main objective in extending the resonance range of the VIV converters for the purpose of maximizing harvested energy, a hardening spring element will be investigated. Hardening springs have been investigated in many literatures and some showed that the element demonstrates significant jump phenomenon in its dynamic property [14], [15]. Fig. 1 presents a mechanical structure supported by a nonlinear spring and its hardening stiffness characteristic. For a relatively low excitation, the structure with a hardening spring element will be more flexible than at a higher excitation and it is known that a mechanical structure with a hardening spring component exhibits a dependency of the (nonlinear) natural frequency to the amplitude of excitation, in such the resonance effect occurs at different frequencies depending on amplitudes. An experimental model is designed to investigate the effects of hardening springs on the capability in broadening the resonance range of a cylinder mounted by a cantilever beam structure and exposed to VIV from a water flow. A simple model of a hardening spring, which is the so-called piecewise stiffness spring, is configured by additional plates mounted appropriately to the structure to introduce nonlinearity on the spring stiffness (see Fig. 2). As can be seen in Fig. 2, a cylinder mounted to cantilever beams at its two ends is immersed perpendicularly to a water flow. Additional plates are mounted at the fixed support points to harden the beams partially. In this way, the beams manifest themselves as nonlinear hardening springs as their equivalent stiffness in transversal direction is increased proportionally to the sine of the angular position of the cylinder with respect to the fixed support points. The degree of hardening stiffness in the spring can be changed by adjusting lengths of the additional plates and materials of the cantilever beams. Introducing cantilever beams as the spring elements brings another advantage. This will minimize the frictional effect compared to a mechanism with a common longitudinal spring. The application of a longitudinal spring will require guide ways to support the cylinder motion. As a consequence, some power losses will be attributed to the dissipative energy from the frictional forces on the elements [16], [17], [18]. B. Stiffness characterization of hardening springs Different structures with different lengths of additional plates and different materials of cantilever beams including aluminum and aluminum alloy are utilized to capture the hardening spring effect with different stiffness characteristics. 92

3 k c m f(t) F k x Figure 1. Water flow A mechanical system with a hardening spring component Fixed support Fixed support Additional plates Additional plates Cantilever beam Cantilever beam y z x Cylinder Figure 2. Experimental model of a hardening spring VIV structure An investigation of the relationship between load and displacement is conducted to characterize stiffness properties of these structures. The illustration of the test can be seen in Fig. 4. Weights representing vertical forces are gradually imposed at the free tip of the cantilever beam and displacements at the free tip are measured. The results are presented in a load-displacement graph at Fig. 3. The structures of an aluminum beam, an aluminum beam with a short plate and an aluminum beam with a long plate exhibit the same slight nonlinear stiffness at the initial phase (k L ). The stiffness response trifurcates at the contacting points of the cantilever beams and additional plates. The structure of an aluminum beam maintains its slight nonlinear stiffness, while stiffness responses of the structures of an aluminum beam with a short plate and an aluminum beam with a long plate are corresponding with k 1 and k 2, respectively. The stiffness values can be obtained by calculating the slopes of theses characteristic lines. The structure of an aluminum beam with a long plate has a more severe change in stiffness corresponding to the steeper plot than the structure of an aluminum beam with a short plate. It is also observed that the stiffness of the structure of an aluminum alloy with a long plate exhibits a strong nonlinearity during the initial phase. After contacting with the additional plate, its stiffness is similar to the stiffness of the structure of an aluminum beam (k 3 ). Hereafter, based on the stiffness characteristics, the structure of an aluminum beam, the structure of an aluminum beam with a short plate, the structure of an aluminum beam with a long plate and the structure of an aluminum alloy with a long plate are represented by slight nonlinear spring, low hardening spring, high hardening spring and strong nonlinear spring, respectively. C. VIV of spring-mounted-cylinder investigation The effects of hardening stiffness on the spring configurations that have been characterized in the previous section on VIV responses including amplitudes, oscillation frequencies and the resonance range are investigated to correlate to their capabilities in harvesting energy. This experiment is conducted in a free surface circulating water channel at Aerodynamics Laboratory, Nanyang Technological University (see Fig. 5). The width, the depth and the length of the testing section are 0.3 m, 0.4 m and 1.0 Figure 3. Load displacement graph of different spring configurations m, respectively. The water channel is able to provide water flows with velocity in the range from m/s. A hollow acrylic cylinder with the diameter of 50 mm and the length of 250 mm is utilized in this experiment. The mass ratio of the oscillating structures is approximately equal to unity (m* 1). Oscillations of the cylinder are recorded by a camera and post-processed by the image processing software Tracker. A free decay test is conducted to measure free-vibration natural frequencies of the structures submerged in water. The results are presented in Table 1. Displacement δ Equilibrium of the cantilever beam Load Additional plate Figure 4. The illustration of the stiffness characterization test TABLE I. Figure 5. Experimental setup in a free surface recirculated water channel Fixed support FREE-VIBRATION NATURAL FREQUENCE IN WATER OF DIFERENT SPRING CONFIGURATIONS Spring configuration Free-vibration natural frequency in water (Hz) Slight nonlinear spring 0.86 Low hardening spring 0.87 High hardening spring 0.86 Strong nonlinear spring

4 III. FSI SIMULATION An FSI simulation model based on Computational Fluid Dynamics (CFD) and Finite Element Analysis (FEA) is built to validate the experimental results. The FSI simulation model relies on the transient coupling scheme of CFD and FEA modules in ANSYS Workbench environment. The unstructured mesh is applied since the geometry of the model is relatively complex (a cylinder mounted to cantilever beams). A mesh independence check is conducted to assure the independence of the accuracy in simulation results on the mesh size. The SST (Shear Stress Transport) turbulence model is selected since it is suitable to model the external flows involving strong adverse pressure gradients and separation and for most general-purpose commercial CFD codes [19]. These features meet the purpose of this simulation model, which is to simulate the flows in VIV phenomenon. The time step size needs to be determined properly for producing sufficiently accurate results but not resulting in time-consuming problem. In FSI simulation, the time step size, t, is determined based on its relation with the water flow velocity, u, and the mesh size, x, which is the socalled Courant number, C. The Courant number must satisfy the condition for stability in solution which is expressed in the following formula [20]: C = u t/ x 1 (1) Therefore, the maximum time step size is determined as: t max x min /u (2) The simulation reveals all the physical specifications of the VIV experiment with the structure of an aluminum beam that has the slight nonlinear characteristic. The modeling for other nonlinear structures is much more complicated and has not been completed at this point. IV. RESULTS AND DISCUSSIONS A. Experimental results Oscillations of the structures are recorded by a camera and the image processing software Tracker is used to quantify the displacement of the cylinder over time under different water flow velocities. Fig. 6 shows the time trace of cylinder displacement of the strong nonlinear spring structure at water flow velocity of 0.40 m/s. The values of the amplitudes and frequencies of these oscillations are averaged. Finally, amplitudes, A, oscillation frequencies, f osc, and water flow velocities, U, are normalized into amplitude ratio, A*, frequency ratio, f*, and normalized velocity, U*, for further considerations by the following formula: A* = A/D (3) f* = f osc /f n, water (4) U* = U/(f n, water D) (5) Where f n, water represents the free-vibration natural frequency of each spring configuration that was measured in the free decay test and D represents diameter of the cylinder. The final results are presented in Fig. 7 and Fig. 8. As can be seen in Fig. 7, the strong nonlinear structure oscillates at significantly larger amplitudes than other structures. For the strong nonlinear structure, the range of oscillation amplitudes is from 0.1D 0.75D, while the range of oscillation amplitudes of other structures is from 0.02D 0.52D. Therefore, the strong nonlinear structure takes the advantage for energy generation rather than other structures in a sense of increasing oscillation amplitudes. However, the resonance range of the strong nonlinear structure starts later than those of other structures. The strong nonlinear structure starts to oscillate with large amplitudes around the normalized water flow velocity of 7, while other structures start to oscillate with large amplitudes around the normalized water flow velocity of 6 (see Fig. 8). Figure 6. Time trace of cylinder displacement of the strong nonlinear structure at water flow velocity of 0.40 m/s Figure 7. Amplitude responses of four spring configurations at different values of normalized velocity of water flow Figure 8. Frequency ratios in the range of resonance of four spring configurations 94

5 A further observation is found from Fig. 8. There are no significant differences between frequency ratios in the resonance range of four different structures. Therefore, it can be deduced that imposing nonlinearity on the structures does not lower the oscillation frequencies of the structures. B. Simulation results and validation The simulation process is carried out at different values of water flow velocity corresponding to the values of water flow velocity in the experiment. In addition, the investigating range is broadened until the end of the resonance range is reached. The displacement values of the cylinder in y- direction (cross-flow direction) over time were recorded for different water flow velocities. Fig. 9 shows time trace of y- direction displacement of the cylinder at water flow velocity of 0.27 m/s. The values of amplitudes of these oscillations are averaged. Then, the results are also normalized to compare with the experimental results for validation and further considerations. The final results are presented in Fig. 10. As can be seen in Fig. 10, the simulation results show good agreement with the experimental results. For the experimental results, the range of oscillation amplitudes is from 0.02D 0.52D, while the range of oscillation amplitudes from the simulation results is from 0.01D 0.48D. In both experiment and simulation, the resonance range starts around the normalized water flow velocity of 6. However, at some values of water flow velocity, deviations between these two sets of results are observable. One thing has to be noted that is due to the limitation of the water tunnel specification, the experiment could not be performed at wider flow velocity range. However, based on the simulation results, the end point of the resonance range can still be observed. The resonance range of the slight nonlinear structure ends at the normalized water flow velocity of 22. C. Discussions For further considerations, a comparison is implemented between the results from this study, including experimental results and simulation results, and another comparable study utilizing linear springs. As mentioned before, the mass ratio of the cylinder in this study is approximately equal to unity (m* 1). Govardhan and Williamson [6] conducted an experiment to investigate the VIV of a cylinder, which had the mass ratio of 1.19 (m* = 1.19) and was mounted by linear springs. In addition, these two studies were conducted with the equivalent damping factor imposing on the structures since they were both carried out in water. Figure 10. Amplitude response of the slight nonlinear spring structure from experimental results and simulation results at different values of normalized velocity of water flow Therefore, results from these two studies are comparable to each other. Fig. 11 shows the amplitude and frequency responses in the resonance range of the following structures: the strong nonlinear structure from experimental results, the slight nonlinear structure from simulation results and the linear structure from Govardhan and Williamson [6] s experimental results. As can be seen in Fig. 11, at the normalized velocity of 8, the maximum amplitude of the linear structure reaches the normalized amplitude of D. This value is larger than the maximum amplitude of the strong nonlinear structure, which is 0.75D. The range of resonance of the linear structure is from the normalized velocity of 5 18, while for the strong nonlinear structure, the range of resonance starts later, at the normalized velocity of 7. Nonetheless, it is observable that at the normalized velocity around 16, the oscillation remains. In this case, we expect that the strong nonlinear structure may be promising in optimizing the resonance range. In addition, even though the oscillation amplitudes of the slight nonlinear structure are lower than that of the linear structure, the resonance range is extended up to the normalized water flow velocity of 22, while for the linear structure, the resonance range is from This observation consolidates the capability of the nonlinear structures in widening the resonance range. V. CONCLUSIONS In this experiment, additional stiffness was imposed on linear springs to introduce piecewise or non-smooth hardening stiffness. From the experimental results, the strong nonlinear structure has shown the advantage in harvesting energy rather than other structures that is due to its larger oscillation amplitudes. In the strong nonlinear structure, the material of the cantilever beam is aluminum alloy, while the material of cantilever beams in other structures is aluminum. Therefore, changing the materials is an effective approach to adjust nonlinearity in spring stiffness to improve capability in harvesting energy. Comparing to an experiment that Figure 9. Time trace of y-direction displacment of the cylinder at water flow velocity of 0.27 m/s 95

6 Figure 11. Comparison between VIV responses in resonance range of the linear and nonlinear structures utilized linear springs and based on the simulation results, the nonlinear structures have shown the potential in widening the resonance range. In terms of oscillating magnitude and occurrence of resonance, the simulation showed good agreement in the results to the experiment although there are noticeable discrepancies at a few points. These discrepancies are attributed to the selection of turbulence models and the meshing quality. In addition, the nonlinearity in spring stiffness has not been implemented in the simulation at this moment. In the future, other turbulence models will be applied to improve the modeling of the scenario in this study where the flow in subcritical regime is dominant. Although structured mesh is challenging to be applied for complex geometries, it can produce more stable and accurate solutions, particularly for the FSI simulation, where the mesh will be largely deformed in the solving process. Therefore, structured mesh will be considered to improve the simulation accuracy. A proper method will also be developed to implement nonlinearity in spring stiffness in the FEA module to simulate VIV responses of nonlinear systems. In this parametric experiment, there is a limitation in the investigating range of stiffness due to time-consuming construction of physical springs. In the future, a combination of a physical system and a computer-based force-feedback controller will be applied to facilitate parametric investigations due to appending the spring stiffness parameter in a wide investigating range by virtually defining its characteristics in the controller instead of building mechanical configurations. 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