On the stability of an in-line formation of hydrodynamically interacting flapping plates

TL;DR

This study investigates the stability of in-line formations of hydrodynamically interacting flapping plates using numerical vortex sheet models, revealing conditions for stable schooling and methods to control and stabilize these formations.

Contribution

It introduces a numerical analysis of schooling modes in flapping plates, identifies destabilizing factors, and proposes a simple control mechanism to maintain stable formations.

Findings

Schooling modes can be stable under certain parameters.

Destabilization occurs with increased number of plates or decreased oscillation amplitude.

A control mechanism stabilizes formations and improves vortex wake regularity.

Abstract

The motion of several plates in an inviscid and incompressible fluid is studied numerically using a vortex sheet model. Two to four plates are initially placed in-line, separated by a specified distance, and actuated in the vertical direction with a prescribed oscillatory heaving motion. The vertical motion induces the plates' horizontal acceleration due to their self-induced thrust and fluid drag forces. In certain parameter regimes, the plates adopt equilibrium "schooling modes," wherein they translate at a steady horizontal velocity while maintaining a constant separation distance between them. The separation distances are found to be quantized on the flapping wavelength. As either the number of plates increases or the oscillation amplitude decreases, the schooling modes destabilize via oscillations that propagate downstream from the leader and cause collisions between the plates, an instability that is similar to that observed in recent experiments on flapping wings in a water tank (Newbolt et al., 2024). A simple control mechanism is implemented, wherein each plate accelerates or decelerates according to its velocity relative to the plate directly ahead by modulating its own flapping amplitude. This mechanism is shown to successfully stabilize the schooling modes, with remarkable impact on the regularity of the vortex pattern in the wake. Several phenomena observed in the simulations are obtained by a reduced model based on linear thin-airfoil theory.