Contrasting thrust generation mechanics and energetics of flapping foil locomotory states characterized by a unified - scaling

Abstract

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.

Numerical validation of simple non-stationary models for self-propelled pitching foils

“…For higher Reynolds numbers the results depend on the pitch amplitude. This Strouhal number's dependence on the Reynolds number agrees with that found by Das et al (2022), but with a different constant shown with a dash-and-dot line in Fig. 5, with the constant given in the legend, probably due to the use of a NACA0012 airfoil in their numerical simulations instead of the present flat plate.…”

“…It must be noted that the Reynolds numbers here are significantly larger than in most previous studies dealing with the skin friction on a flapping foil (Ehrenstein and Eloy, 2013;Ehrenstein et al, 2014;Das et al, 2016Das et al, , 2022Gross et al, 2021). These computationally works for 𝑅𝑒 ≲ 10 3 confirmed the Bone-Lighthill boundary layer thinning hypothesis for uniform flow past oscillatory plates, with skin friction drag coefficients proportional to 𝑅𝑒 −1∕2 , like in Blasius flat plate, but multiplied by a factor proportional to the square root of the ratio of the transverse and longitudinal velocities.…”

“…5, with the constant given in the legend, probably due to the use of a NACA0012 airfoil in their numerical simulations instead of the present flat plate. Also, our range of Reynolds numbers is quite different, reaching much higher values of 𝑅𝑒: here 100 ≲ 𝑅𝑒 ≲ 3 × 10 5 vs. 10 ≲ 𝑅𝑒 ≲ 2 000 in Das et al (2022). That is why we use a transition turbulence model for the numerical simulations.…”

“…The Reynolds number based on 𝑈 in this case is 𝑅𝑒 ≃ 45 599 and the Strouhal number 𝑆𝑡 ≃ 0.087. Following Das et al (2022), it is interesting to characterize the mean swimming velocity 𝑈 through 𝑅𝑒 in a 𝑆𝑡 − 𝑅𝑒 chart. This is made in Fig.…”

“…1); i.e., heaving motion of the foil is not considered. This inhibition of the heaving motion will obviously reduce the propulsion capabilities of the hydrofoil (Sánchez-Rodríguez et al, 2020;Das et al, 2022), but it will not affect to the validation purposes of the present work.…”