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.

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Self-propelled flapping foils with distinct locomotion-enabling kinematic restraints exhibit a remarkably similar Strouhal number ( $St$ )-Reynolds number ( $Re$ ) dependence. This similarity has been hypothesized to pervade diverse forms of oscillatory self-propulsion and undulatory biolocomotion; however, its genesis and implications on the energetic cost of locomotion remain elusive. Here, using high-resolution simulations of translationally free and restrained foils that self-propel as they are pitched, we demonstrate that a generality in the $St$ - $Re$ relationship can emerge despite significant disparities in thrust generation mechanics and locomotory performance. Specifically, owing to a recoil reaction induced passive heave, the fluid's inertial response to the prescribed rotational pitch, the principal source of thrust in unidirectionally free and towed configurations, ceases to produce thrust in a bidirectionally free configuration. Rather, the thrust generated from the leading edge suction mechanics self-propels a bidirectionally free pitching foil. Owing to the foregoing distinction in the thrust generation mechanics, the $St$ - $Re$ relationships for the bidirectionally and unidirectionally free/towed foils are dissimilar and pitching amplitude dependent, but specifically for large reduced frequencies, converge to a previously reported unified power law. Importantly, to propel at a given mean forward speed, the bidirectionally free foil must counteract the out-of-phase passive heave through a more intense rotational pitch, resulting in an appreciably higher power consumption over the range $10 \leq Re \leq 10^3$ . We highlight the critical role of thrust in introducing an offset in the $St$ - $Re$ relation, and through its amplification, being ultimately responsible for the considerable disparity in the locomotory performance of differentially constrained foils.

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High efficiency thrust generating foils are extensively being researched for potential use as thrusters in micro air vehicles and biomimetic autonomous underwater vehicles. Here, we propose a simple reduced order model for prediction of thrust generation attributes of foils that are pitched either continuously or intermittently in a periodic and possibly asymmetric fashion. Our model accounts for the distinct thrust contributions from added mass, leading edge suction, quasi steady and wake terms, all deduced from a rigorous generalization of linearized potential theory to foils undergoing small amplitude multimodal flapping motion. Additionally, the model relies on Bone-Lighthill boundary layer thinning hypothesis to account for the pitching motion induced increase in the drag force exerted on the foil. We derive generic forms of the thrust coefficient for prescribed multimodal pitching motions and specifically in the limit of large reduced frequencies, demonstrate a convergence to rather simplified scaling laws that are functions of just the Reynolds number and Strouhal number based on root mean square of the foil’s trailing edge velocity. Comparisons with previously reported experimental and simulation-based investigations demonstrate that the scaling laws capture the influence of imposed pitch on thrust generation characteristics over a range of pitching waveforms ranging from sinusoidal to square or triangular-shaped waveforms and also waveforms corresponding to intermittent pitching. The generalized relations derived in our work and the asymptotic scaling laws deduced from them are applicable to a wide spectrum of self-propulsion enabling and thrust producing waveforms including the ones that can potentially be employed in burst and coast swimming.