For the present study, setting Strouhal Number as the control parameter, we perform numerical simulations for the flow over oscillating NACA-0012 airfoil at Reynolds Number 103. Temporal profiles of the unsteady lift, and their respective spectra clearly indicate the solution to be a period-1 attractor for low Strouhal numbers. This study reveals that aerodynamic forces produced by the oscillating airfoils are independent of the initial kinematic conditions that proves existence of the limit cycle. Frequencies present in the oscillating lift force are composed of the fundamental harmonics, and its odd harmonics. Using these numerical simulations, we observe the shedding frequencies nearly equal to the excitation frequencies in all the cases. Hence, considering it as a primary resonance case, we model the unsteady lift force through a modified van der Pol oscillator. Using the method of multiple scales and spectral analysis of the steady-state CFD solutions, we estimate the frequencies and the damping terms in the reduced-order model. We prove the applicability of this model to all the planar motions of airfoil; heaving, pitching and flapping. With the increasing Strouhal number, the nonlinear damping terms for all types of motion approach similar magnitudes. Another important aspect in one of the currently-proposed model is capturing the time-averaged value of the aero-dynamic lift force.

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