PHYSICAL REVIEW E 84, 066604 (2011) Flapping dynamics of a flexible filament H. Ait Abderrahmane and M. P. Paidoussis Department of Mechanical Engineering, McGill University, Montreal, Quebec H3A2K6, Canada M. Fayed and H. D. Ng Department of Mechanical and Industrial Engineering, Concordia University, Montreal, Quebec H3G 1M8, Canada (Received 11 August 2011; revised manuscript received 16 November 2011; published 14 December 2011) This paper investigates the dynamics of the flapping regime of a filament placed in a two-dimensional soap-film flow for different filament lengths and flow speeds. It was found that the onset of flapping is quasiperiodic, with the main flapping amplitude and frequency modulated by low-amplitude, low-frequency oscillation. At higher flow velocities, the oscillation becomes chaotic. The transition to chaos occurs via the quasiperiodic route to chaos. A new bistability phenomenon was discovered in which the system alternates between the stretched-straight and oscillatory states, which is here referred to as “switching oscillation.” Unlike some previously reported forms of bistability, in this case the system alternates between the two states continuously, without any external perturbation. DOI: 10.1103/PhysRevE.84.066604 PACS number(s): 46.40.-f I. INTRODUCTION Oscillation or “flapping” of platelike structures in axial flow has been the subject of a great deal of study by the science and engineering research communities, because it arises in many situations in nature and in engineering applications. Examples are the oscillation of web in paper making, fluttering of flags, oscillation of plant leafs, vibration of the soft plate in snoring, vibration in parallel-plate-type heat exchangers, flutter in “flutter mills” used for power generation, and many others. Perhaps the first study of this subject was made by Rayleigh [1], who pointed out the similiarities between the flapping of a flag and jet undulations. A multitude of studies then followed; see, for instance, the monograph by Dowell [2], the book by Paidoussis [3], and the review by Shelley and Zhang [4]. Some theoretical studies that should be mentioned are those by Kornecki et al. [5] and Huang [6] using Theodorsen’s theory [7], Yamaguchi et al. [8] using a linearly varying vortex theory, Watanabe et al. [9] and Balint and Lucey [10] using respectively compressible and incompressible two- dimensional (2D) Navier-Stokes solvers, Guo and Pa¨ ıdoussis [11] and Eloy et al. [12,13] using potential flow theory, Argentina and Mahadevan [14] using a simplified Theodorsen model, Tang and Pa¨ ıdoussis [15] using vortex panel method, Alben and Shelley [16] using a nonlinear vortex sheet, and Michelin et al. [17] using discrete-point vortices with unsteady strengths. Several other numerical approaches had been developed for the flapping flag in viscous fluid flow. For instance, Zhu and Peskin [18] used the immersed boundary method, and Connell and Yue [19] developed a fluid-structure direct-simulation capability that couples a direct numerical simulation of the Navier-Stokes equations to a solver for thin-membrane dynamics. There exists a wealth of experimental studies in the literature as well, starting with Taneda [20], who had in- vestigated many types of flag, some made of loosely woven material. Other notable studies are the wind tunnel experiments conducted by Huang [6], Yamaguchi et al. [21], Wanatabe et al. [22], Tang et al. [23], and Eloy et al. [12] and the water tunnel experiments by Shelley et al. [24], as well as the soap-film tunnel experiments by Zhang et al. [25], which are of particular interest here. Most of these numerical and experimental studies reported three different states for the structure: the stretched-straight state where the structure is aligned with the flow and does not flap, and the regular flapping and irregular flapping states. They reported also a bistability phenomenon and hysteresis of the stretched-straight and flapping states. However, most of the past and recent studies mainly focused on the instability threshold; few showed much interest in the dynamics of the flapping and its underlying mechanisms, for example, Connell and Yue [19], Alben and Shelley [16], and Michelin et al. [17]. These authors attempted a characterization of the flapping in terms of spectral analysis and dynamical systems theory to describe the transition from a regular into an irregular state and capture the inherent bistability of flapping and stationary states. These theoretical studies have shown that it is possible to characterize the dynamics of the flapping through the evolution of frequency power spectra and of system attractors in phase space. This paper is an experimental study on the nonlinear dynamics of a flapping flag by experiments using filaments in soap-film flow. The dynamics is analyzed using a nonlinear time-series method. This paper reports for the first time a robust phenomenon of “switching oscillations,” where the system switches from one type of oscillatory state to static equilibrium and back again, which may be viewed as a form of bistability associated with hysteresis. The switching- oscillations phenomenon manifests itself without any external action; that is, the filament dynamics continuously switches between two states: the stretched-straight and flapping states. This manifestation of bistability differs from the one pre- viously reported in Taneda [20], Connell and Yue [19], and Shelley and Zhang [4], where an external perturbation is needed to shift the system from the stretched-straight state to the flapping one and the system remains in that 066604-1 1539-3755/2011/84(6)/066604(8) ©2011 American Physical Society