Existence of a sharp transition in the peak propulsive efficiency of a low- $Re$ pitching foil

We perform a comprehensive characterization of the propulsive performance of a thrust generating pitching foil over a wide range of Reynolds ( $10\leqslant Re\leqslant 2000$ ) and Strouhal ( $St$ ) numbers using a high-resolution viscous vortex particle method. For a given $Re$ , we show that the mean thrust coefficient $\overline{C_{T}}$ increases monotonically with $St$ , exhibiting a sharp rise as the location of the inception of the wake asymmetry shifts towards the trailing edge. As a result, the propulsive efficiency too rises steeply before attaining a maximum and eventually declining at an asymptotic rate that is consistent with the inertial scalings of $St^{2}$ for $\overline{C_{T}}$ and $St^{3}$ for the mean power coefficient, with the latter scaling holding, quite remarkably, over the entire range of $Re$ . We find the existence of a sharp increase in the peak propulsive efficiency ${\it\eta}_{max}$ (at a given $Re$ ) in the $Re$ range of 50 to approximately 1000, with ${\it\eta}_{max}$ increasing rapidly from about 1.7 % to the saturated asymptotic value of approximately $16\,\%$ . The $St$ at which ${\it\eta}_{max}$ is attained decreases progressively with $Re$ towards an asymptotic limit of $0.45$ and always exceeds the one for transition from a reverse von Kármán to a deflected wake. Moreover, the drag-to-thrust transition occurs at a Strouhal number $St_{tr}$ that exceeds the one for von Kármán to reverse von Kármán transition. The $St_{tr}$ and the corresponding power coefficient $\overline{C_{p,}}_{tr}$ are found to be remarkably consistent with the simple scaling relationships $St_{tr}\sim Re^{-0.37}$ and $\overline{C_{p,}}_{tr}\sim Re^{-1.12}$ that are derived from a balance of the thrust generated from the pitching motion and the drag force arising out of viscous resistance to the foil motion. The fact that the peak propulsive efficiency degrades appreciably only below $Re\approx 10^{3}$ establishes a sharp lower threshold for energetically efficient thrust generation from a pitching foil. Our findings should be generalizable to other thrust-producing flapping foil configurations and should aid in establishing the link between wake patterns and energetic cost of thrust production in similar systems.