Abstract
Tandem configurations of two self-propelled flexible flappers of finite span are explored by means of numerical simulations. The same sinusoidal vertical motion is imposed on the leading edge of both flappers, but with a phase shift ( $\phi$ ). In addition, a vertical offset, $H$ , is prescribed between the flappers. The configurations that emerge are characterized in terms of their hydrodynamic performance and topology. The flappers reach a stable configuration with a constant mean propulsive speed and a mean equilibrium horizontal distance. Depending on $H$ and $\phi$ , two different tandem configurations are observed, namely compact and regular configurations. The performance of the upstream flapper (i.e. the leader) is virtually equal to the performance of an isolated flapper, except in the compact configuration, where the close interaction with the downstream flapper (i.e. the follower) results in higher power requirements and propulsive speed than an isolated flapper. Conversely, the follower's performance is significantly affected by the wake of the leader in both regular and compact configurations. The analysis of the flow shows that the follower's performance is influenced by the interaction with the vertical jet induced by the vortex rings shed by the leader. This interaction can be beneficial or detrimental for the follower's performance, depending on the alignment of the jet velocity with the follower's vertical motion. Finally, a qualitative prediction of the performance of a hypothetical follower is presented. The model is semi-empirical, and it uses the flow field of an isolated flapper.
Highlights
Nature provides a host of examples of interacting bodies through a fluid with surprising behaviours
The tandem configurations of two self-propelled flexible plates of finite span are explored by means of numerical simulations
We explore the stable tandem configurations that emerge in the parametric space of phase shift between the motion of each leading edge (φ ∈ [0◦ − 360◦]) and vertical offset between the mean vertical position of the flappers (H/C ∈ [0 − 0.6])
Summary
Nature provides a host of examples of interacting bodies through a fluid with surprising behaviours These range from a single, passive body like an auto-rotating maple seed (Lentink et al 2009) to a large number of synchronized, active bodies which interact with the surrounding fluid, like fish schooling or bird flocks (Weihs 1973; Mora et al 2016). The main reason why animals form schools or flocks may not be entirely clear yet, it is well known that animals benefit from collective motion in terms of flow interaction (Weimerskirch et al 2001; Becker et al 2015) This beneficial interaction is not restricted to a large number of bodies, but it is observed at its minimal expression for two-body configurations. Boschitsch, Dewey & Smits (2014), Muscutt, Weymouth & Ganapathisubramani (2017) and Kurt & Moored (2018) found that, for an inline tandem configuration of two oscillating foils, the distance and phase shift between the motion of the foils can always be adjusted such that the follower foil interacts with the oncoming vortices extracting energy from the flow, confirming the Kármán gait hypothesis proposed in Liao et al (2003) and Streitlien, Triantafyllou & Triantafyllou (1996)
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