  Oscillating foils in self-propelled mode are the simplest model for
investigating oscillatory locomotion in cruising fishes. In this investigation,
we explore the self-propulsion characterisitics of a NACA0015 section airfoil,
with chord length $C$, subjected to sinusoidal pitching using a rotary
apparatus. A power-spring-based crank-rocker mechanism actuates the airfoil. We
examine the effect of pitching frequency ($f$), amplitude ($A$), and the
pitching point location ($p$) on the self-propulsion speed, $U_s$. We present
the results in terms of self-propulsion Reynolds number ($Re_s = U_sC/\nu$),
reduced frequency ($k_s$), Strouhal number ($St$), and two non-dimensional
speeds, $U^*_{BL}$ (body length per oscillation) and $U^*_{AL}$ (forward speed
in terms of trailing edge excursion per oscillation). $U_s$ increases with
frequency and amplitude, but a diminishing effect of amplitude was noted at
larger amplitudes. Highest speeds were achieved when pitched closest to the
leading edge, with $St$ in the range of $0.2 - 0.4$. Three distinct scaling
regimes are identified, each characterized by a specific relationship between
$Re_s$ and trailing-edge Reynolds number, $Re_{TE} = fAC/\nu$. When pitching at
low-amplitude, close to leading-edge, $Re_s \sim (1-2p) Re_{TE}^{3/2}$ (power
scaling). For higher amplitude pitching, $Re_s \sim Re_{TE} (1-2p)^{1/2}
(A/C)^{-1/2}$ (separable scaling). When pitched close to the midpoint, the
airfoil propels beyond a threshold $Re_{TE,0}$, and $Re_s$ increases linearly
with $Re_{TE}$ (linear scaling). Notably, in separable and linear regimes,
self-propulsion is independent of viscosity. These relations collectively offer
a comprehensive framework for understanding the self-propulsion of rigid
pitching airfoils across a wide range of parameters.
