  For present study, setting Strouhal Number (St) as control parameter,
numerical simulations for flow past oscillating NACA-0012 airfoil at 1,000
Reynolds Numbers (Re) are performed. Temporal profiles of unsteady forces; lift
and thrust, and their spectral analysis clearly indicate the solution to be a
period-1 attractor for low Strouhal numbers. This study reveals that
aerodynamic forces produced by plunging airfoil are independent of initial
kinematic conditions of airfoil that proves the existence of limit cycle.
Frequencies present in the oscillating lift force are composed of fundamental
(fs), even and odd harmonics (3fs) at higher Strouhal numbers. Using numerical
simulations, shedding frequencies (f_s) were observed to be nearly equal to the
excitation frequencies in all the cases. Unsteady lift force generated due to
the plunging airfoil is modeled by modified van der Pol oscillator. Using
method of multiple scales and spectral analysis of steady-state CFD solutions,
frequencies and damping terms in the van der Pol oscillator model are
estimated. We prove the applicability of this model to all planar motions of
airfoil; plunging, pitching and flapping. An important aspect of
currently-proposed model is capturing the time-averaged value of aerodynamic
lift coefficient.
