On the nonlinear dynamics of self-sustained limit-cycle oscillations in a flapping-foil energy harvester

The nonlinear dynamics of an airfoil at Reynolds number Re = 10, 000
constrained by two springs and subject to a uniform oncoming flow is studied
numerically. The studies are carried out using open source computational
fluid dynamics toolbox OpenFOAM. Under certain conditions related to aerodynamic flutter, this two-degree-of-freedom system undergoes self-sustained limit-cycle oscillations (LCOs) with potential application as an energy harvester. When the system is given a small initial perturbation, it is seen that
the response of the system decays to zero at flow velocities below the flutter
velocity, or oscillates in a limit cycle at velocities greater than the flutter
velocity. The flutter velocity at Re = 10, 000 is shown to deviate significantly
from the theoretical prediction (which is derived with an assumption
of infinite Reynolds number) owing to the effect of viscosity. The LCOs
at freestream velocities higher than the flutter velocity result in unsteady
flows that are heavily influenced by leading-edge vortex shedding as well as
trailing-edge flow separation. The influence of different system parameters on the onset of flutter and on the limit-cycle response characteristics is investigated
in this research. This is done by defining a baseline case and
examining the effects of varying aerodynamic parameters such as freestream
velocity, and structural parameters such as the pitch-to-plunge frequency
ratio and the type of spring stiffnesses. The conditions corresponding to
the lowest flutter velocities (and consequently the lowest “cut-in” speeds
for power extraction) and the parameter space that provide single-period,
single-amplitude and harmonic LCOs (ideal for power extraction) are identified.
Calculation of instantaneous and time-averaged power is presented by
modeling the extraction of energy through a viscous damper. The highest
power coefficients and efficiencies are obtained at velocities just higher than
the flutter velocity. Introduction of positive cubic stiffening in the system
springs is seen to make the system more stable, LCOs more harmonic and
single-period, and to potentially increase power extraction efficiency of the
system.