Transition boundaries and an order-to-chaos map for the flow field past a flapping foil

Abstract

The present study focuses on identifying dynamical transition boundaries and presents an order-to-chaos map for the unsteady flow field of a flapping foil in the low Reynolds number regime. The effect of an extensive parametric space, covering a large number of kinematic conditions, has been investigated. It is shown that the conventional non-dimensional parameters cannot effectively capture the changes in the flow field due to the variations in the relevant kinematic parameters and are unable to demarcate the dynamical transition boundaries. Two new non-dimensional measures – maximum effective angle of attack and a leading-edge amplitude-based Strouhal number – are proposed here, which can capture the physical effect of the parametric variations on the wake dynamics. The study proposes generalised transition boundaries and an order-to-chaos map through a transitional regime in terms of these two newly proposed parameters. Published data from the existing literature have also been tested to verify the proposed transition model. It is seen that despite the wide variety of the parametric combinations, the dynamical states from both the new and the published data corroborate well the proposed boundaries, giving credibility to the order-to-chaos map.

Passive dynamics regulates aperiodic transitions in flapping wing systems

“…Majumdar et al ( 9 ), for a plunging configuration, reported quasi-periodicity around and chaos at . For actuated pitching–plunging, Bose et al ( 4 ) also reported a similar range: a quasi-periodic flow-field at , appearance of intermittency at , and sustained chaos at ; note that the reporting of Bose et al ( 4 ) was for zero phase difference between pitch and plunge motions, a non-zero phase-gap would have changed these onsets ( 10 ). On the other hand, as seen in the present results, aperiodic transition is completely absent (or delayed) due to passive pitch dynamics even at as high as ; see Fig.…”

“…2 C and F)]. The primary LEV for the AP–AP foil was also stronger (the circulation values should be noted in the figure), which approached the regime of aperiodicity ( 10 ). Furthermore, the shedding of a secondary vortex along with the main TEV contributed towards the loss of periodicity and influenced the aerodynamic loads; see supplementary Video SV2 .…”

“…Quasi-periodic route to chaos in the flow-field and loads was reported in Refs. ( 1 , 5 , 9 , 10 , 12 , 13 ). Chaotic transition through an intermittency (sporadic intermittent burst of chaotic windows interspersed by periodic/quasi-periodic states) route was also reported in Refs.…”

“…Within this context, the flow-field mechanism and the underlying physics that are played at the near-field to prevent the transition to aperiodicity need clarity. Recently, Majumdar et al ( 10 ) investigated an extensive set of wing kinematic and profile parameters, and observed that two key factors—the speed of the leading-edge movement and the relative orientation with which it interacts with the incoming flow—fundamentally govern the evolution of the primary LEV, and in turn, dictate the dynamical transitions between the flow regimes. The two-way fluid–structure interaction (FSI) effects associated with passive oscillations can potentially alter the LEV separation behavior and modify the dynamical transitions significantly—this hypothesis will be examined here by systematically examining the effect of passive oscillations on wing kinematics, and in turn, on the two above-mentioned quantities.…”

Chordwise flexible aft-tail suppresses jet-switching by reinstating wake periodicity in a flapping foil

“…The wake transition behind rigid flapping foils has been studied comprehensively in the recent literature for different kinematics, such as pure heaving (Lai & Platzer 1999; Lewin & Haj-Hariri 2003; Ashraf, Young & Lai 2012; Badrinath, Bose & Sarkar 2017; Majumdar, Bose & Sarkar 2020 a , b ) and pure pitching (Koochesfahani 1989; Godoy-Diana, Aider & Wesfreid 2008; Schnipper, Andersen & Bohr 2009; Shinde & Arakeri 2013), as well as simultaneous heaving–pitching (Lentink et al. 2010; Bose & Sarkar 2018; Bose, Gupta & Sarkar 2021; Majumdar, Bose & Sarkar 2022). The dynamic plunge velocity () or Strouhal number () was used as the control parameter to study the transition; here, , where and are the plunge/heave amplitude and frequency, respectively; is the non-dimensional heave amplitude; and is the reduced frequency, with being the chord length of the foil.…”