Nonlinear Fluid-Elastic Behavior of a Flapping Wing With Low-Order Chord-Wise Flexibility

Chord-wise flexibility of wings of natural flyers can play a pivotal role in achieving high propulsive efficiency. However, high-fidelity simulations of the fluid-elastic behavior of chord-wise flexible wings involve prohibitive computational cost. Hence, the present study attempts to capture the essential dynamics of the fluid-structure interaction (FSI) system using a limited mode chord-wise flexible structural model coupled with a high-fidelity Navier-Stokes (N-S) solver. The wing is modeled as two elliptic rigid foils connected by a non-linear torsional spring which incorporates the chord-wise bending stiffness of the flapping foil. While the front link is subjected to an active pitching-plunging motion, the rear link undergoes flow-induced passive oscillations. The present work aims to understand the role of passive flexion on the dynamic transition of a flapping foil. The structural governing equation for the rear link takes the form of a Duffing oscillator equation subjected to base excitation and external aerodynamic forcing. The aerodynamic loads on the foil are computed using a discrete forcing Immersed Boundary Method (IBM) based in-house N-S solver which is coupled in a staggered manner with the structural solver to simulate the FSI response. A bifurcation study has been performed considering the free-stream velocity as the control parameter in the presence of both structural and aerodynamic nonlinearities. The simulations are carried out in a low flapping Reynolds number (Re_f = V_f c/n, where V_f is maximum plunge velocity of the leading foil, c is the chord length of each foil and n is the kinematic viscosity of the fluid) regime. As the ratio (R) of the free stream velocity (U) to V_f, R = U/V_f is varied, a periodic reverse- Kármán wake at higher R transition into an entirely aperiodic trailing wake at lower R. The same periodic to aperiodic transition is also observed in the passive oscillation of the rear foil when analyzed with tools from the dynamic systems theory. In the lower R regime, the aerodynamic forces are less and the motion of the rear body is mainly dominated by the inertia. In this regime, the rear foil oscillates aperiodically with a higher amplitude leading to significant flow separations and aperiodic shedding of the vortices from the trailing-edge of the rear foil. On the other hand, at high R value the aerodynamic forces start to dominate and the oscillation amplitude of the rear body comes down.  As a result, the flow around the body remains mostly attached and a periodic vortex shedding takes place.   The flow-field around the body corresponding to the different dynamical states of the system and the associated vortex interactions are examined in detail and will be discussed in the full length paper.