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Collective locomotion of two-dimensional lattices of flapping plates

We study the propulsive properties of rectangular and rhombic lattices of flapping plates at O(10--100) Reynolds numbers in incompressible flow. We vary five parameters: flapping amplitude, frequency (or Reynolds number), horizontal and vertical spacings between plates, and oncoming fluid stream velocity. Lattices that are closely spaced in the streamwise direction produce intense vortex dipoles between adjacent plates. The lattices transition sharply from drag- to thrust-producing as these dipoles switch from upstream to downstream orientations at critical flow speeds. Near these transitions the flows assume a variety of periodic and nonperiodic states, with and without up-down symmetry, and multiple stable self-propelled speeds can occur. As the streamwise spacing increases, the plates may shed typical vortex wakes that impinge on downstream neighbors. With small lateral spacing, rectangular lattices yield net drag, while rhombic lattices may generate net thrust, sometimes with high efficiency. As lateral spacing increases, rectangular lattices begin to generate thrust, eventually with slightly higher efficiencies than rhombic lattices, as the two types of lattice flows converge. At Re = 70, the maximum Froude efficiencies of time-periodic lattice flows are about twice those of an isolated plate. At lower Re, the lattices' efficiency advantage increases until the isolated flapping plate no longer generates thrust. The mean input power needed to generate the lattice flows can be estimated in the limits of small and large streamwise spacings, with small-gap and Poiseuille-like flows between the plates respectively in the two cases. For both lattices, the mean input power saturates as the lateral spacing becomes large (and thrust occurs). At small lateral spacings, the rhombic lattices' input power may be much larger when the plates overlap, leading to a decrease in Froude efficiency.