Self-directed propulsion of an unconstrained flapping swimmer at low Reynolds number: hydrodynamic behaviour and scaling laws

Lin, Xingjian; Wu, Jie; Zhang, Tongwei

doi:10.1017/jfm.2020.955

Cited by 25 publications

(21 citation statements)

References 44 publications

(103 reference statements)

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“…Nevertheless, our scaling exponent is in excellent agreement with the power-law fit of reported in the recent independent work on BF foils by Lin et al. (2021). Notably, using , the - power law can be recast into an equivalent form , where the scaling exponent of deduced from our runs is in excellent agreement with the exponent of reported in Lin et al.…”

Section: The - Relationship and The Drag–thrust Balancesupporting

confidence: 92%

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

2021

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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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Section: The - Relationship and The Drag–thrust Balancesupporting

confidence: 92%

“…2012; Zhu, He & Zhang 2014; Deng & Caulfield 2016; Verma et al. 2017; Das, Shukla & Govardhan 2019; Lin, Wu & Zhang 2021). This tendency emerges as the thrust produced from a sufficiently intense prescribed harmonic excitation surmounts the hydrodynamic resistance to the foil's motion.…”

Section: Introductionmentioning

confidence: 99%

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

2021

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“…As shown in figure 9(a,b), it can be seen that both Re ux and Re uy , respectively, can be couched as two simple scaling laws of Re ux ∼ Re 1.8 f and Re uy ∼ Re 3.2 f . Similar to the scaling law of a single swimmer (Gazzola et al 2014;Lin et al 2021), our results seem to indicate that the speed of multiple swimmers in the collective behaviour also can be couched as a simple scaling law. Moreover, it can be seen from figure 9(a) that, as compared with the performance of a single foil, the hydrodynamic benefit of the collective behaviour of two foils is enhanced as Re f increases.…”

Section: Collective Performance For Staggered Formationsupporting

confidence: 74%

“…where θ i is the instantaneous pitching motion of ith foil (i = 1, 2), θ m and f are, respectively, the pitching amplitude and frequency, ϕ is the phase difference between two foils, t is the time. In this paper, the pivot location of each foil is fixed at 0.05c, and its propulsion is controlled by Newton's second law, which can be described as follows (Lin, Wu & Zhang 2021):…”

Section: Problem Description and Methodologymentioning

confidence: 99%

“…To strike a balance between computational expense and accuracy that is related to the mesh, a mesh of x = 0.01c is chosen for the current simulations. In this study, the length is non-dimensionalized with c, the time is scaled by 10 −2 c 2 /(2ν), the velocity is non-dimensionalized with U = 10 2 ν/(0.5c), the force is scaled by 10 4 ρν 2 /(0.5c) and the power is scaled by 10 6 ρν 3 /(0.25c 2 ) (Lin et al 2021). In addition, the Reynolds number is set to be Re = Uc/ν = 200 in this work.…”

Section: Problem Description and Methodologymentioning

confidence: 99%

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