Influence of the aspect ratio of the sheet for an electric generator utilizing the rotation of a flapping sheet
Mechanical Engineering Journal
The "Flutter-mill" is a power generation device that can be parallelized and downsized more easily than conventional wind-power generators with the added advantage of lower manufacturing costs. Flutter-mills comprise a flexible sheet with an electric power generator at its leading edge. Flutter-mills exhibit complex power generation performance characteristics that are highly dependent on the specifications of the flexible sheet and the inlet flow velocity. In particular, the span width of the
... span width of the sheet affects the stability and flapping behavior significantly. Using the numerical analysis model, these complex flutter-mill characteristics can be estimated without experiments, and thereby numerical model inform the choice of the sheet dimensions, which are critical for designing an effective flutter-mill. Here, we present a numerical model that can provide a preliminary survey of the power generation performance and attempt to clarify the relationship between the aspect ratio of the sheet and the harvested power. The equation of motion for a flexible sheet includes rotational damping at the leading edge of the sheet to emulate the coupling effect between the sheet and the energy harvesting circuit. We verified the validity of our numerical model by evaluating its performance against previously published experimental results, and thus established the relationship between the aspect ratio of the sheet and the harvested power. We found that the local minimum value of the harvested power of the flutter-mill may be caused by the vibration amplitude at the leading edge of the sheet decreasing in the transition domain of the flapping vibration mode if the aspect ratio is large. are associated with conventional wind turbines, and evaluated the power generation performance of the flutter-mill with respect to the inlet flow velocity through experiments (Bae et al., 2014) . These power generators produce electrical power by utilizing the bending deformation of a flexible sheet. Therefore, high sheet durability is required to ensure power generation longevity. However, to achieve power generation, flutter-mills require either a metal-coated sheet or a sheet covered with piezoelectric patches. Therefore, the choice of sheet materials for flutter-mills is restricted. By contrast, Kang et al. proposed a simple flutter-mill design that employs a power generator at the leading edge of the flexible sheet (Kang et al., 2015) . The flow of ambient fluid around the sheet induces flapping behavior, and the rotational vibration at the leading edge of the sheet is converted to electrical power by the power generator at the leading edge. This power generation mechanism does not demand a specific material for the flexible sheet; therefore, a wider selection of sheet materials are available, which can lead to improved sheet durability. The specifications for the flexible sheet and the inlet flow velocity can have complex effects on the power generation performance of the flutter-mill. To clarify these effects, Michelin et al. proposed a numerical model to survey the power generation performance for a small-span-width sheet in axial flow based on the slender body theory (Michelin and Doaré, 2013). Separate studies by Alben and Shelley, Chen et al., and Michelin et al. also proposed the numerical model to reproduce the developed amplitude of the flapping oscillation under the assumption of two-dimensional potential flow for a fluid and the beam approximation for a sheet, respectively (Alben and Shelley, 2008)(Chen et al., 2014)(Michelin, 2009b). However, the slender-body theory and two-dimensional flow approximation for fluids are valid for a sheet with a specific span width. Moreover, the beam approximation for a sheet cannot reproduce the deformation along the span direction. Computational fluid dynamics (CFD) analysis is more accurate than the above two approximations for fluidstructure interaction analysis (Connell and Yue, 2007)(Huang and Sung, 2010)(Engels et al., 2013). However, the choice of turbulence model, number of divisions of the fluid and solid meshes, and stability of the analysis must all be considered. Moreover, this method requires powerful computing resources. Tang et al. proposed combining the fluid-structure coupled model with the nonlinear structure model and the linearized unsteady vortex lattice method (UVLM), with the combined model assuming that vortex panels are on a stationary plane to reduce the computing cost (Tang and Dowell, 2018)(Tang et al., 2019) . We have also presented a numerical model to avoid these problems in previous studies (Yamano et al., 2018 (Yamano et al., , 2020 . In our approach, three-dimensional fluid flow was modeled using the UVLM without the above-mentioned assumption, which involves reduced computing costs compared with CFD analysis (Katz and Plotkin, 2001 )(Fritz and Long, 2004 )(Stanford and Beran, 2013 )(Abdelkefi et al., 2014 . Thereby, many parameter combinations including large deformation can be surveyed. A flexible sheet was modeled using the finite element method (FEM) with absolute nodal coordinate formulation (ANCF), thereby utilizing the shell element to reproduce the deformation of a plate while considering geometrical nonlinearity (Dufva and Shabana, 2005 )(Shabana, 2008 )(Hyldahl, 2013 . Therefore, our model can reproduce the spanwise deformation unlike the beam approximation in previous studies. We used this model to assess the performance of the generator under various boundary conditions at the leading edge (Yamano et al., 2020) as well as material dampings (Yamano et al., 2018), and verified it via comparisons with experimental results and equivalent simulations performed by other numerical models. Nevertheless, few of these flutter-mill studies report the relationship between the aspect ratio of the sheet and the power harvested from the fluid flow, despite it being known that the aspect ratio of a sheet affects the stability and flapping behavior significantly (Eloy et al., 2007) . Here, we verify the validity of our proposed numerical model by comparing simulations with the experimental results reported by Kang et al. (Kang et al., 2015) , which were obtained by the fluttermill targeted by our numerical model. Then, the relationship between the aspect ratio of a sheet and the harvested power is studied for a flutter-mill design based on our model.