Flow interactions lead to self-organized flight formations disrupted by self-amplifying waves

The collective movement of animals, such as bird flocks and fish schools, presents a fascinating challenge in understanding how complex group-level behaviors emerge from individual interactions. These interactions are not solely social or behavioral; they are significantly influenced by the physical interactions mediated by the aerodynamic or hydrodynamic forces generated during locomotion. The study of these flow interactions is complex due to the unsteady nature of locomotion at intermediate to high Reynolds numbers (Re), which is further complicated when multiple bodies interact through their flow fields. To address this complexity, the researchers employ a robophysical approach, using robotic "mock flocks" to abstract elements of biological systems while providing advantages in terms of control, characterization, and observation. This approach allows for a detailed investigation of the underlying physical mechanisms without the complexities of biological systems. Previous research on single locomotors has demonstrated that mechanized wings share many flow, force, and dynamic characteristics with biological propulsors. Furthermore, recent studies on multi-propulsor interactions highlight the potential of using this system to study collective locomotion at high Re. This study focuses on linear formations, a simplified yet informative configuration amenable to detailed analysis of flow-structure interactions, drawing inspiration from columnar formations observed in certain bird species. The research builds upon previous studies of tandem locomotion in pairs of foils, where a follower locks into specific positions behind a leader, to explore the group-wide consequences of collective flow interactions within longer chains.

The paper draws upon a substantial body of prior research on collective animal behavior, fluid dynamics of locomotion, and active matter. It references studies on the dynamical aspects of animal grouping, phase transitions in self-driven particle systems, and the theoretical frameworks for understanding flocking, schooling, and swarming behavior. The authors highlight previous experimental work and modeling efforts concerning the hydrodynamics of bacterial turbulence, active suspensions, and active matter. They also cite studies utilizing robotic systems and robophysics in understanding locomotion, including work on flapping foils and wings and the fluid dynamics of flapping propulsion. Research on tandem locomotion and the Lighthill conjecture (that locomotion formations are induced or assisted by flow interactions) are also critically reviewed, providing a foundation for the current investigation.

The research employed a robophysical approach using a novel experimental setup (Fig. 2c). Multiple flapping foils, arranged radially around a central shaft in a water tank, were used to simulate a linear flock. The shaft's sinusoidal up-and-down oscillations imparted flapping motions to the foils, allowing for independent horizontal movement. The foils' positions were tracked using high-speed video analysis. Experiments were conducted at Reynolds numbers (Re) on the lower end of the range observed in schooling fish and flocking birds (~10³). To probe flow interactions, two types of controlled perturbations were applied: (1) Steady (DC) forces were applied to individual foils using a mass-string-pulley system (Fig. 3a) to map the force-displacement relationship and characterize the interactions; (2) Oscillatory (AC) perturbations were applied to the leader foil using a motor-driven oscillator (Fig. 3b) to study the response of downstream members and the propagation of disturbances. A wake interaction model was developed to understand the observed phenomena. This model represents each flyer as a point particle emitting a wake flow, interacting with the wakes of others. The model incorporates nonreciprocal interactions with memory, capturing the essential features of flow interactions. The model equations were numerically solved to simulate flock dynamics and compare with experimental results.

The experiments revealed several key findings: (1) Foils spontaneously self-organize into a crystalline or lattice-like formation with uniform spacing (Fig. 2d, f); (2) This ordering is disrupted by self-amplifying positional waves, termed "flonons" (Fig. 2e, f), which propagate down the group, growing in amplitude and causing collisions; (3) Force measurements (Fig. 3c) indicated spring-like restorative forces due to nearest-neighbor, nonreciprocal hydrodynamic interactions, stabilizing the equilibrium positions; (4) Oscillatory perturbations applied to the leader caused resonant amplification in downstream members (Fig. 3d), confirming the self-amplifying nature of flonons; (5) The wake interaction model (Fig. 4) successfully reproduced the crystalline formation, flonon propagation, and resonant amplification observed in experiments; (6) Simulations further showed that the instability is amplified in larger groups; (7) Introducing variability among individuals, either through vacancy defects in the crystal structure or by varying the temporal phases of flapping oscillations, significantly suppressed flonon amplification and stabilized larger flocks (Fig. 5).

The findings demonstrate that collective flight formations in flapping flyers arise from a balance between self-ordering through pairwise flow interactions and destabilization through the amplification of longitudinal waves (flonons). The spring-like behavior observed in the force-displacement curves highlights the role of nearest-neighbor, nonreciprocal interactions. The flonons represent a novel type of collective excitation, distinct from conventional phonons due to their one-way amplification. The ability to stabilize larger flocks by introducing diversity underscores the importance of individual variations in maintaining the integrity of the group. The model's success in reproducing the key experimental observations suggests that the essential physics are captured by the simplified representation of wake interactions. The nonreciprocal nature of interactions, memory effects in the wake flows, and the phase-locking between flyer motion and wake patterns are identified as crucial ingredients in generating these phenomena.

This study reveals the intricate interplay between flow interactions and collective dynamics in flapping flyer formations. The discovery of self-amplifying flonons and the stabilizing effect of individual diversity provide novel insights into the mechanisms governing flocking behavior. Future research could focus on exploring the influence of higher Reynolds numbers, more complex flapping kinematics, and active sensing mechanisms on these phenomena, particularly in relation to the observed behavior in larger, more complex animal groups. Furthermore, the model could be extended to investigate other formations, such as circular milling.

The study used a simplified experimental system with idealized flapping kinematics and a limited number of foils. The model also incorporates several simplifying assumptions, such as the exponential decay of wake strength and the nearest-neighbor interaction assumption. While the Reynolds number range is relevant to biological systems, it is at the lower end of the range, and the results may not fully extrapolate to the higher Reynolds numbers found in natural flocks. The experimental setup is also constrained to a rotational geometry, although the validation against previous translational studies provides confidence in the qualitative relevance.