Energy harvesting based on bio-inspired fluid-structure interaction (Mathematical Aspects and Applications of Nonlinear Wave Phenomena)

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(1)209. 数理解析研究所講究録第2034巻 2017年 209-215. Energy harvesting. based. on. bio‐inspired. fluid‐structure interaction Mikael A.. Langthjem. Faculty of Engineenng, Yamagata University, Jonan 4‐chome, Yonezawa, 992‐8510 Japan Abstract a fluid flow. If a waving (fluttering, i.e. flag‐like linearly unstable) flag is covered with piezoelectric elements it can be used to harvest energy from the flow. This problem has been considered by many researchers and it is known that the power generation efficiency is not high. The purpose of the present work is to consider a large number of interacting flags and determine the optimum pattern (positions) of these flags under the assumption that they can utilize the vortex wake from the other flags to increase the efficiency of the power generation, just as individual fish in a large school of similar fish can utilize the surrounding wakes with benefit. The present paper first gives a review of the classical, fundamental flapping flag problem. It then goes on to develop a simple model for the main problem, as just described.. This work is concerned with. structures in. Introduction. 1. The interest in renewable energy is experiencing energy. sources. heat; but the. at. a. boom in recent years.. have been considered in the past, such. present the. most active fields. are. as. Many types of renewable. tides and water waves, and. those concerned with. sunlight and. geothermal. wind. As to. latter, large farms of wind turbines have been, or are being, constructed in many parts of Denmark, for example, 40% of the total electric power was generated by the wind. the World. In in. 2015, and this percentage is At the. same. time there is. scale. The energy in with. a. on. the rise.. growing. interest in. harnessing. for. the energy of the wind. be harnessed if the. on a. smaller. is accommodated. example, fluttering flag, flag piezoelectric elements that generate electricity when being bent. This is the subject of the. present. paper, which. The. a. reports. on an. can. ongoing work.. problem has, of course, been considered already by several researchers; and. that the. efficiency. of. work is to consider. a. piezo‐element‐covered flag. high.. it is known. The main purpose of the present. a large number of interacting flags and determine the optimum pattern flags under the assumption that they can utilize the vortex wake from the. (positions). of these. other. to increase the. flags. is not. school of similar fish‐. or. efficiency. of the power generation, just. the individual birds in. a. large. formation‐. as. individual fish in. can. utilize the. a. large. surrounding. wakes with benefit. The fundamental one, and on. we. physical problem‐. will start out. to the main. problem,. that of the. flapping flag‐ is, however,. a. most. by giving. a. historical review of works related to it. We. to outline. a. simple modehng approach and show. computational results related hereto.. some. fascinating we. then go. preliminary.

(2) 210. of mathematical models of the. 2. History. Lord. Rayleigh was,. related to. in. [1], apparently the first. paper from 1879. a. flapping flag. flapping flags. give. to. and sails. The main topic of the paper is. a. mathematical. instability. of. analysis. cylindrical jets,. layer (between the moving and the stationary fluid) modeled by a cylindrical Flapping flags and sails were mentioned as applications of the special case of a. with the shear vortex sheet.. (two‐dimensional). zero‐curvature. and. Shelley of. edge. Zhang, Rayleighs analysis than. flapping flag. a. the initial. Following. more a. pointed. as. out in. a. Rayleighs study [1]. itself.. Rayleigh, ninety study flapping flags, of. of. passed by. years. the. a. also found. was. flags made of various types of fabrics, placed working section. He found that for sufficiently that the trailing edge of the flags begin to flutter after onset of. that,. More than. thirty. to this. devoted to. time,. with the birth of. developed (e.g. stiffness. A a. our. knowledge,. [6,. and to. potential flow model. vortex sheet. (similarly. to. function. A Kutta condition. of Taneda. vertical wind tunmel. a. low flow. speeds the flags. higher. at. flow. speeds.. increases with the flow. the. the flutter. higher. by Zhang The. Pope [8]. analyze. a. use. [5],. et al.. It. speed;. frequency.. who utihzed. of soap film also made. of. had been. a. satisfied at the. apply. flag. on. trailing edge,. flag. was. with. bending. was. modeled. non‐local. a. but vortex. in flow. these studies to. the flow. amounts to. other paper. On the other. plates. on. two‐dimensional. The effect of the. no. published.. the first to. Rayleighs approach [1]) which. was. 1879 paper,. of literature. probably. dynamics. employed.. was. problem. large body. were. the. Rayleighs. since. of the. aeronautics,. Fitt and. flapping flag problem. the. flag. in. possible.. to the best of. 7. the. reconsidered. were. further‐going theoretical study. a. hand, starting. frequency. lighter. \mathrm{t}_{\mathrm{W} $\theta$\ve }\mathrm{d}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}1 motions.. purely. very detailed flow visualizations. Up. the. later, experiments. years. soap films to realize. flowing. the flutter. flutter, flag length and. the smaller the. furthermore,. is elaborated and. purely experimental study. 35 \mathrm{c}\mathrm{m}\times 35 cm. do not flutter and. by. trailing. before the appearance of. Taneda considered. with. recent review paper. model of the vortex wake shed from the. flag. model of the. a. study. the next serious, scientific. by. is. Hydrodynamcs [3].. discussed also in Lambs. [4].. vortex sheet. But. potential. therefrom. shedding. ignored.. was. These two papers, the and. Pope [8],. to have initiated. seem. boom in related papers. Argentina made. use. of. experimental. appeared. and Mahadevan. a more. elaborate. [9]. a. paper. by Zhang. et al.. [5]. revival of the interest in the. in the. following. and theoretical. one. problem, because. a. by. Fitt. veritable. years.. reconsidered the. problem. aerodynamic model, based. on. as. defined. the. theory. by Pope and. [8], but [10]. This. Fitt. of Theodorsen. non‐circulatory velocity potential representing the structure (flag or plate) and a circulatory velocity potential representing the wake, due to vortex shedding. While the non‐circulatory flow is represented by a non‐local potential function in Theodorsens original theory, Argentina and Mahadevan simplified this part by theory makes. employing. a. use. local. of two. potential functions,. formulation, originally due. that is, the solution of the a. hadevan. [9]. generated by. the. [11].. The. the. stability analysis, was. carried out in. program AUTO.. aerodynamic. model. developed by Argentina and. Ma‐. a standard Galerkin discretization approach. They investigated boundary conditions (pinned or clamped) and also computed the flapping flag. In a subsequent paper [13] they investigated the effect of a. use. the effect of the upstream sound. path‐following. [12] employed. but made. to Milne‐Thomson. coupled fluid‐structure boundary value problem,. non‐discretized fashion, using the Manela and Howe. a. of.

(3) 211. flagpole, and and. in. particular. developed downstream. the vortex street. from it,. the. on. flag stability. dynamics. to the work of. Returning. regarding. mates. found that the tions. Argentina. Eloy. [14]. et al.. tial flow model for. a. went. finite sized. a. initiated at. dimensional vibrations. A major result. the. was. finding. Papers ones are. on. the. of. dynamics. however based. flapping flags. on \mathrm{C}\mathrm{F}\mathrm{D}. was. that. fluid. flags. They. that. by. a. two‐dimensional. full three‐dimensional poten‐. assumed to perform only two‐. flags. continue to appear. (computational. esti‐. and thus makes the vibra‐. mass. a. presented. finite‐sized. of finite span. are more. This is in agreement with the. of infinite span. flags. developed. which however still. (two‐dimensional flags). findings of Argentina and Mahadevan [9]. than. the added. higher flow speed. step further and. flag,. these authors also. ubiquitous by. are. three‐dimensionality of the flow reduces. stable, that is, the flapping is. more. flow model.. [9],. and Mahadevan. three‐dimensional flow effects which. regularly.. stable. just mentioned. Most of the recent. dynamics) approaches.. Such studies will. not be considered here.. A. 3. simple flag. model for. interacting flags. group of. a. As discussed in section 1, the main purpose of this work is to a. large not. are. number of. high.. and their wakes. This is. flags. To get. an. understanding. desirable to consider the simplest discrete vortex method. Such. an. problem. problem. possible model. This. represented by articulated plates. are. of the. a. the interaction between. study. where. hopes. of. and its solution it appears to be. analytical. is described in the. instant,. model where the. a. and where the vortex‐dominated flow is. approach. progress. at first. is,. flags. represented by. a. following.. Point vortex model. 3.1. In terms of. complex variables,. $\Gamma$_{n} located. at. z_{n}=x_{n}+\mathrm{i}y_{n}. is. the. velocity potential. at. z=x+\mathrm{i}y for. termed. a. panel),. [16]. from. vortex of. strength. (1). .. discuss how the vorticity distribution a. point. given by [15]. $\phi$_{n}=\displaystyle \frac{$\Gam a$_{n} {2 $\pi$ \mathrm{i} \log(z- _{n}) Katz and Plotkin. a. far field point of view,. be. can. on a. flat. represented by. a. plate (in. the. point vortex,. following. as. specified. quarter‐chord point (\mathrm{i}. \mathrm{e} at the point from the if the measured has upstream edge, panel length \ell. ) It is shown in [16] that if a \ell/4 colocation point, at which the normal flow velocity is zero, is placed at the three‐quarter‐chord by (1), placed. at the center of pressure, which is at the. point 3\ell/4 then the Kutta condition will be satisfied ,. The. flag. model is here discretized into. the simulation of the. unsteady flow. is, from the trailing edge of the shed. point. vortices. distribution of shed. rest;. or. point. number,. equivalently,. at the. most downstream. panel. theorem, which. stiff. point. flag (a plate). vortices after the. the flow has been. plate. impulsively. panel.. vortex. panels.. of the. In. flag (that. step. The strength of each. says that the total. strength. of all. [15].. discretized into four. has been. of the. lumped. trailing edge. at each time. and shed /fxee) must remain constant a. trailing edge. N say, of such. vortex is shed from the. vortex is determined from Kelvins. (bound Figure 1(a) shows. point. a. a. .. lumped. impulsively. turned on.. vortex. stasted. panels, and the. (to move). from.

(4) 212. Total circulation and lift results for this the lift.. depict. result based. analytical lower. The full. curves. curve. on. .. curve. is. The agreement between both lift and circulation. [16]).. result is taken from ref.. represent the circulation around the plate. Here the dashed. result, while the finely dotted. numerical. [10] (this. Theodorsens model. Fig. 1(\mathrm{b}) The upper two curves result, while dashed curve is the. shown in. case are. shows the present numerical. curve. again representing the Theodorsen solution.. curves. is. seen. to be very. good.. (a) Figure. (a). 1:. A. plate impulsively. The. is for the present. (b) (b) Comparison. started from rest.. between discrete vortex. results and Theodorsens solution.. A. 3.2. simple. We consider here into. each. panels are. and. very. simple structural model of the flag. Employing the. flag. where. are. a. concentrated. mass. assumed to act also in the. constants which. represent dissipation in these springs. is the. panels,. each of. length. p. .. placed. panel. discretization. in the center of. center. are. T=\displayst le\sum_{q=1}^{N}\frac{1}2m_{q}l^{2}\{( sum_{\mathrm{p}=1^{q}$\epsilon$_{\mathrm{p}q \sin$\thea$_{p}\dot{$\thea$}_{p})^{2}+(\sum_{p=1}^{q}$\epsilon$_{\mathrm{p}q\cos$\thea$_{\mathrm{p}\dot{$\thea$}_{p})^{2}\. angle. of. panel. no.. p with. points. The. p>N The potential. energy of the. Rayleigh dissipation function. respectively.. Let. given by. ,. (2). vertical, and. A dot denotes differentiation with respect to the time t .. c_{q} and c_{\mathrm{q}+1} ,. Then its kinetic energy is. $\epsilon$_{pq}=\displaystle\frac{1}2( -$\delta$_{p\mathrm{q})=\left\{ begin{ar y}{l 1,\mathrm{f}\mathrm{o}\mathrm{}p\neq \ \frac{1}2,\mathrm{f}\mathrm{o}\mathrm{}p=q \end{ar y}\right.. The. same. interconnected via springs. Let m_{q} be the point mass attached to panel no. q and let be the stiffness of the spring at upstream and downstream end, respectively. The. be discretized into N. $\theta$_{p}. and for. is. k_{q+1}. damping the. a. described in the previous section,. as. panel. Concentrated lift forces. panels. k_{q}. structural model. flag. is. .. It is remarked that. (3) $\theta$_{p}=0. for p< 1. given by. V=\displaystyle\sum_{q=1}^{N}\frac{1}{2}k_{p}($\theta$_{p}-$\theta$_{p-1})^{2}. (4). D=\displayst le\sum_{q=1}^{N}\frac{1}2c_{\mathrm{p}(\dot{$\theta$}_{p}-\dot{$\theta$}_{p-1})^{2}. (5). is. given by.

(5) 213. Let. L_{q}= $\rho$ u_{\mathrm{t}\mathrm{q} $\Gamma$_{\mathrm{q} be the lift force on panel no. velocity, and $\Gamma$_{q} is again the vortex strength.. q , where $\rho$ is the fluid. Then the. density, u_{tq} is the tangential generalized force on panel no. p is. Q_{p}=l\displaystyle\sum_{q=p}^{N}L_{q}$\epsilon$_{pq}\cos($\theta$_{p}-$\theta$_{q}). (6). .. Employing Lagranges equations [17]. \displayt e\frac{d} t (\displayt e\frac{prtialT}{\partil\dot{$\heta$}_{p ) +\displayst le\frac{\parti lD}{\parti l\dot{$\thea$}_{\mathrm{p} -\frac{\parti lT}{\parti l$\thea$_{p}+\frac{\parti lV}{\parti l$\thea$_{p}=Q_{p}, and. linearizing,. we. p=1 2, ,. .. .. .. ,. (7). N,. obtain the linear equations of motion. \displayst le\sum_{\mathrm{q}=\mathrm{p}^{N}P^{2}m_{\mathrm{q}$\epsilon$_{p\mathrm{q}\sum_{n=1}^{q}$\epsilon$_{nq}\d ot{$\thea$}_{n}-c_{p}\dot{$\thea$}_{p-1}+(c_{p}+\mathrm{c}_{p+1})\dot{$\thea$}_{p}-c_{\mathrm{p}+1}\dot{$\thea$}_{p+1} -k_{p}$\theta$_{p-1}+(k_{p}+k_{\mathrm{p}+1})$\theta$_{p}-k_{p+1}$\theta$_{p+1}=P\displaystyle\sum_{q=\mathrm{p} ^{N}L_{q}$\epsilon$_{pq}, for p=1 , 2,. \cdots. ,. N,. again with $\theta$_{p}=0 for p<1 and p>N As .. an. (8). example, for. N=2. we. have. the system. \el^{2}\left{\begin{ar y}{l \frac{1}4m_{1}+m_{2}&\frac{1}2m_{2}\ frac{1}2m_{2}&\frac{1}4m2 \end{ar y}\right\} left\{begin{ar y}{l .\cdot\ $\thea$_{1}\ .\cdot\ $\thea$_{2} \end{ar y}\right\}+ left\{begin{ar y}{l \mathrm{c}_\mathrm{l}+c_{2}&-c_{2}\ -c_{2}&c_{2} \end{ar y}\right\} left\{begin{ar y}{l \dot{$\heta$}_{1\ dot{$\heta$}_{2 \end{ar y}\right\} +\left\{ begin{ar y}{l k_{1}+k_{2}&-k_{2}\ -k_{2}&k_{2} \end{ar y}\right\} left\{ begin{ar y}{l $\thea$_{1}\ $\thea$_{2} \end{ar y}\right\}= el\eft\{ begin{ar y}{l \frac{1}2L_{1}+L_{2}\ \frac{1}2L_{2} \end{ar y}\right\} In. (a). Fig. 2, examples. is for. a von. a. low flow. of simulation results based. on. (9). are. speed, much below the flutter threshold.. Kármán‐like vortex street.. In part. flutter. Here the wake pattern is govem. correspond qualitatively well. to the. (b). the flow. mostly by. the. shown.. The. 2:. (In. clockwise. example. speed corresponds. in part. develops. into. to the threshold of. motion of the flag. These results experimental (soap film) results of Zhang et al. [5].. (a) Vibrating flag (plate) at color (pdf/online), red points rotating vortices.). Figure. .. Here the wake. flapping. (a) flutter.. (9). a. low flow. (b) speed.. (b). The. flag. at the threshold. of. represent counter‐clockwise rotating vortices, and blue.

(6) 214. 4. remarks. Concluding. The present paper. has, firstly, reviewed works related to the classical problem of the flapping flag;. and, secondly, outlined many. to include. simple but efficient method the model the fluid‐structure interaction of. a. The model should be. flags. a. model of. developed further, to represent a large collection of flags and piezoelectric elements; this work is ongoing. At the same time, a number. by fundamental flapping flag problem (effects of non‐local velocity potential, etc.) that are very interesting and attractive. of open. problems. remain. structural. damping,. Acknowledgment This work is. supported by JSPS Kakenhi \mathrm{J}\mathrm{P}16\mathrm{K}06172.. References. [1] Rayleigh,. Lord. Society 10,. (1879). [2] Shelley, M., Zhang, Annual Review. [3] Lamb,. H.. [4] Taneda,. On the. of. instability. jets. Proceedings of the London Mathematical. 4‐13.. J.. (2011) Flapping. and. bending bodies interacting. (1993) Hydrodynamics. Cambridge: Cambridge University. (1968) Waving. S.. with fluid flows.. of Fluid Mechanics 43, 449‐465.. motions of. flags.. Journal. of. the. Press. (orig. 1932).. Physical Society of Japan 24,. 392‐401.. [5] Zhang, J., Childress, S., Libchaber, A., Shelley, film. Nature. 40S,. [6] Kornecki, A., Dowell, dimensional. M.. (2000). Flexible filaments in. flowing. soap. H., OBrien, J. O. (1976) On the aeroelastic instability of incompressible flow. Journal of Sound and Vibration 47,. two‐. S35‐839.. panels. E.. in uniform. 163‐. 178.. [7] Huang,. L.. (1995). Flutter in cantilevered. plates in axial flow. Journal of Fluids. and. Structures,. 9, 127‐147.. [8] Fitt,. A.. D., Pope,. M. P.. (2001). The. motion of two‐dimensional. unsteady. flags. with. bending. stiffness. Journal of Engineering Mathematics 40, 227‐248.. [9] Argentina, M., Mahadevan, National. (2005). Academy of Sciences 102,. [10] Theodorsen, flutter. Naca. T.. (1935) L. M.. [12] Manela, A., Howe, and. General. Report, 49ó,. [11] Milne‐Thomson, (orig. 1958). of Sound. L.. Fluid‐flow‐induced flutter of. a. flag. Proceedings of the. aerodynamic instability. and the mechanism of. 1829‐1834.. theory. of. 413‐433.. .. M. S.. (1973). Theoretical. (2009 a). Vibration, 321,. On the. 994‐1006.. Aerodynamics.. New York: Dover Publications. stability and sound. of. an. unforced. flag. Journal.

(7) 215. [13] Manela, A., Howe,. M. S.. (2009b). The forced motion of. a. flag.. Journal. of Fluid Mechanics,. 635, 439‐454.. [14] Eloy, C., Souilliez, C., Schouveiler,. L.. (2007). Flutter of. a. rectangular plate.. Journal. of. Fluids and Structures, 23, 904‐919.. [15] Milne‐Thomson, (orig. 1968).. L. M.. [16] Katz, J., Plotkin,. A.. (1996). Theoretical. Hydrodynamics.. New York: Dover Publications. (2001) Low‐Speed Aerodynamics. Cambridge: Cambridge University. Press.. [17] Bishop,. R. E.. D., Johnson,. bridge University. Press.. D. C.. (1960). The Mechanics. of. Vibration.. Cambridge: Cam‐.

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