Existence of three distinct scaling regimes in self-propelled rigid pitching airfoil

Abstract

We investigate the effect of imposed kinematics on the self-propulsion of the NACA0015 symmetric airfoil section subject to sinusoidal pitching. We employ a rotary apparatus capable of achieving self-propulsion. A power-spring-based crank-rocker mechanism actuates the airfoil. Three distinct scaling relations emerge, which relate the self-propulsion Reynolds number $Re_s$ to the frequency Reynolds number $Re_f$, the amplitude of pitching $\theta_0$, and the location of the pitching point, $p$. When pitched near the center, a \textit{linear} scaling emerges with $Re_s \sim Re_f \theta_0$. When pitched near the leading edge, a \textit{power} scaling emerges with $Re_s \sim (1-2p)(Re_f \theta_0)^{3/2}$ for low amplitude pitching and a \textit{separable} scaling emerges with $Re_s \sim (1-2p)^{1/2}Re_f\theta_0^{1/2}$ for moderate to high amplitude pitching. These relations are consistent with the scaling relations derived from balancing inviscid thrust with viscous drag, pressure drag, and enhanced pressure drag for the \textit{power}, \textit{separable}, and \textit{linear} regimes, respectively. We find that different vortical patterns in the wake are directly correlated to the airfoil's self-propulsion speed which essentially determines the spatial separation between the shed vortices. Our findings provide a comprehensive framework for understanding the self-propulsion of rigid pitching airfoils across a wide range of parameters validated experimentally.