Flow interaction of three-dimensional self-propelled flexible plates in tandem

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.