Pitch and heave dynamics of an elastically-mounted cyber-physical hydrofoil

The energy harvesting performance of an elastically-mounted hydrofoil (chord, $c$, span, $s$) subject to a prescribed pitching motion is studied using a cyber-physical force-feedback control system. We vary the mass, $m$, the frequency of the pitching motion, $\omega$, the parameters of the elastic support (stiffness, $k$ and damping, $b$) and the Reynolds number, $Re$. The extracted energy is obtained from measured heave force and velocity, $F \dot{y}$. The ratio between the pitching frequency and the natural frequency of the system, $\omega/\sqrt{k/m}$, and the damping coefficient, $b/(0.5\rho U s c)$, are found to play a major role. In particular, the maximum power output is achieved at a frequency ratio of 1, which corresponds to an optimal phase difference of $90^{\circ}$ between the driven pitch and passive heave motions. At the resonance condition, the damping coefficient defines the heaving amplitude, $H$, and thus the width of the wake and the Strouhal number, $St = fH/U$. The power coefficient, $C_p = $, reaches a maximum of 0.65 at a damping coefficient around 1.5, regardless of the Reynolds number (Re = 20,000 - 55,000). The contribution of the pitch component to power extraction is found to be small ($<$ 10\% of the heave component).