Collective locomotion of two-dimensional lattices of flapping plates. Part 1. Numerical method, single-plate case and lattice input power

Abstract

We propose a model and numerical method for the propulsion of rectangular and rhombic lattices of flapping plates at (10-100) Reynolds numbers in incompressible flow. The numerical method uses an adaptive mesh to mitigate singularities at the plates' edges. We establish convergence rates and find good numerical accuracy in a test problem: Laplace's equation in the region exterior to a plate. We then use the method to establish benchmark results for a single flapping plate, including vortex wake characteristics and Froude efficiency over ranges of flapping amplitude, frequency and Strouhal number. As a prelude to a study of propulsive efficiency in Part 2 (J. Fluid Mech., vol. 915, 2021, A21), we study a key ingredient: the time-averaged input power in lattices of plates. Scaling laws for the mean input power are estimated in the limits of small and large streamwise spacings, using steady flow models with small-gap and Poiseuille-like flows between the plates respectively in the two limits. For both lattice types, the mean input power saturates as the lateral spacing becomes large (and thrust occurs). At small lateral spacings, the rhombic lattices' input power becomes much larger when the plates overlap. The time-averaged input power in flapping lattices agrees qualitatively with the steady models.