Collective locomotion of two-dimensional lattices of flapping plates. Part 2. Lattice flows and propulsive efficiency

摘要

We study propulsion of rectangular and rhombic lattices of flapping plates at O(10-100) Reynolds numbers in incompressible flow. The fluid dynamics often converges to time periodic in 5-30 flapping periods, facilitating accurate computations of time-averaged thrust force and input power. We classify the propulsive performances of the lattices and the periodicities of the flows with respect to flapping amplitude and frequency, horizontal and vertical spacings between plates, and oncoming flow velocity. Non-periodic states are most common at small streamwise spacing, large lateral spacing and large Reynolds number. Lattices that are closely spaced in the streamwise direction produce intense vortex dipoles between adjacent plates. The flows 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 pass through a variety of periodic and non-periodic states, with and without up-down symmetry, and multiple stable self-propelled speeds can occur. As the streamwise spacing increases (and with large lateral spacing), the plates may shed vortex streets that impinge on downstream neighbours. The most efficient streamwise spacing increases with flapping amplitude. With small lateral spacing, the rectangular lattices have Poiseuille-type flows that yield net drag, while the rhombic lattices may shed vortices and generate net thrust, sometimes with relatively high efficiency. As lateral spacing increases to one plate length and beyond, the rectangular lattices begin to shed vortices and generate thrust, eventually with efficiencies similar to the rhombic lattices', as the two types of flows converge. At Re = 70, the lattices' maximum Froude efficiencies are approximately twice those of an isolated plate (only considering nearly periodic lattice flows). As Re decreases, the lattices' efficiency advantage increases further.