Flapping strategies for flying formations

Long arrays of identical, self-propelling flapping flyers are inherently unstable and thus unlikely to exist without active control mechanisms. One approach to enable long in-line formations is to enforce a constant separation between the group members. The objective then becomes to determine the flapping strategies the flyers should adopt to achieve a certain separation. Using an aerodynamic model of vortex wake production and inter-flyer effects, we explore different flapping strategies for followers given the motion of the leader. The choice of tactic is dependent upon the aerodynamic, kinematic, and physical parameters of the system, and reflects an interplay between efficiency and stability. We find that whether a flyer flaps in or out of phase with its upstream neighbour, together with the target separation, strongly affect the flapping amplitude and, therefore, the energetic cost of group flight. In certain regimes, group flight is energetically favourable compared to isolated flight, while in others, flying in formation becomes less efficient. We also identify "goldilocks zones", ranges of separation in which one of the in- or out-of-phase motions can be simultaneously energetically efficient and dynamically stable. Outside these regions, energetically favourable flight is unstable and therefore unlikely to occur.