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Passive bionic motion of a flexible film in the wake of a circular cylinder: chaos and periodicity, flow–structure interactions and energy evolution
Published online by Cambridge University Press: 10 May 2024
Abstract
The self-sustained interactions between a flexible film and periodic vortices epitomize the spirit of fish swimming and flag flapping in nature, involving intricate patterns of flow–structure coupling. Here, we comprehensively investigate the multiple coupling states of a film in the cylinder wake mainly with experiments, complemented by theoretical solutions and nonlinear dynamical analyses. Four regimes of film motion states are identified in the parameter space spanned by the reduced velocity and the length ratio. These regimes are (i) keeping stationary, (ii) deflection flutter, (iii) hybrid flutter and (iv) periodic large-amplitude flapping, each governed by a distinct coupling mechanism, involving regular and irregular Kármán vortices, local instability of the elongated shear layers and 2P mode vortex shedding. The film futtering in regimes (ii) and (iii) is substantiated to be chaotic and bears a resemblance to the ‘entraining state’ of fish behind an obstacle in the river. The periodic flapping in regime (iv) manifests itself in an amalgam of standing and travelling waves, and has intrinsic relations to the ‘Kármán gaiting’ of fish in periodic vortices. With the spatiotemporal reconstruction for the periodic flapping, we procure the energy distributions on the film, revealing the energy transfer processes between the film and the large-scale vortices. The findings unequivocally indicate that the flow–structure interaction during the energy-release stage of the film is more intense than that during the energy-extraction stage. Given the similarities, the mathematical and physical methods presented in this work are also applicable to the research on biological undulatory locomotion.
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Duan and Wang supplementary movie 1
(Left) Small-amplitude deflection flutter of the film with the length of L/D=1.0 in 2 seconds (about 180 periods of vortex shedding), corresponding to regime (ii) in figure 1(a) of the main text. (Right) Phase trajectory of the trailing edge, which evolves simultaneously with the film flutter shown in the left panel. The frame rate of the movie is 15 frames per second and the physical time interval between two successive frames is 1/800 s; thus, the flutter is slowed down by about 50x in this movie. The color of the instantaneous deformation of the film in the left panel AND the instantaneous phase point in the right panel is changed abiding by the color scale, which visually represents the physical time variation. For clarity, in the right panel, we just retain the color of the phase trajectory within about 12 periods of the vortex shedding, beyond which the phase trajectory is prescribed to be gray.
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10.1 MB
Duan and Wang supplementary movie 2
(Left) Small-amplitude hybrid flutter of the film with the length of L/D=3.0 in 2 seconds (about 150 periods of vortex shedding), corresponding to regime (iii) in figure 1(a) of the main text. (Right) Phase trajectory of the trailing edge, which evolves simultaneously with the film flutter shown in the left panel. Other details of this movie are the same as those described in the caption for movie 1.
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6.3 MB
Duan and Wang supplementary movie 3
(Left) Large-amplitude periodic flapping of the film with the length of L/D=5.0 in 2 seconds (about 60 periods of vortex shedding), corresponding to regime (iv) in figure 1(a) of the main text. (Right) Phase trajectories of the trailing edge (point of s/D=5, the red point) and the point of s/D=3 (the blue point), which evolve simultaneously with the film flapping shown in the left panel. Other details of this movie are the same as those described in the caption for movie 1.
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10.3 MB
Duan and Wang supplementary movie 4
The complete coupling interactions between the film with L/D=5.0 and the flow field are shown by the evolution of the Lagrangian coherent structures (LCSs) over 10 vortex shedding periods. The LCSs are extracted by finite time Lyapunov exponent. The frame rate of this movie is 10 frames per second and the physical time interval between two successive frames is 1/800 s; thus, the interactions between the film and the flow field are slowed down by 80x in this movie.
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7.7 MB