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

The study investigates how collective movements of organisms emerge from interactions among individuals, emphasizing that bird flocks and fish schools involve not only social/behavioral interactions but also physical interactions mediated by aero- or hydrodynamics at intermediate to high Reynolds numbers. The complexity of unsteady, high-Re flows complicates understanding, especially with multiple bodies interacting through their flow fields. The authors advocate using physical analogues that abstract biological details to reveal generic phenomena. Robophysical experiments with flapping foils have replicated many aspects of biological propulsion and provide a controlled platform for multi-propulsor interactions. The work focuses on linear (in-line) formations—an idealized, one-dimensional configuration inspired by columnar formations of birds—where previous pairwise studies showed followers lock into discrete positions behind leaders. The central research question is how visco-inertial flow interactions in such chains determine group structure (ordering) and dynamics (stability, wave-like excitations), and under what conditions order is disrupted.

The paper situates itself at the intersection of collective animal behavior and active matter. Prior work has shown active swarms (e.g., bacteria) can be modeled and reproduced in abiotic systems, supporting the view of such collectives as active matter with exchange of ideas from statistical mechanics and condensed matter physics. Robophysical experiments on single and multiple flapping foils have established that mechanized propulsors replicate key features of animal locomotion, including reverse von Kármán wakes, forces, energetics, and dynamics. Studies on tandem foils found discrete stable follower positions, supporting the Lighthill conjecture that formations can be induced by flow interactions. Simulations and models (including vortex-based approaches) have examined arrays of flapping plates, hydrodynamic schooling, and efficiency. However, prior multi-foil experiments often used rigidly linked foils and could not reveal independent collective dynamics, and simulations at lower Reynolds numbers did not report the amplifying wave modes observed here, suggesting a role for higher Re and ambient disturbances. This work extends that literature by probing many-body effects, nonreciprocal interactions, and memory in high-Re flow-mediated collectives.

Robophysical experiments: A rotational “flight mill” apparatus houses up to five identical foils that fan out radially from a vertically oscillated shaft. The shaft, driven sinusoidally by a stepper motor, imparts identical vertical heaving motions with amplitude A=1.5 cm and frequency f=2.5 Hz to each foil while allowing free rotation about the shaft via low-friction bearings, enabling self-propelled orbital flight in a water tank. The tank (cylindrical inner wall diameter 60 cm, height 30 cm) is covered with a clear lid for top-view imaging. Each 3D-printed PLA wing has NACA0017 cross-section, chord c=4 cm, span s=8 cm (area sc=32 cm²); midspan radius R=19 cm. High-speed video captures support arm angles, converted to angular positions θ(t) and arc-length positions Xn(t)=Rθ(t). Derived quantities: flight speed Un=dXn/dt, trajectory wavelength λ=Un/f, inter-member gap gn=Xn−1−Xn−c, and dimensionless spacing Sn=gn/λn−1. Conditions yield Re_f≈1500 (based on Af) and Re≈1200 (based on mean U), with A/c=0.375 and Strouhal St=Af/U≈0.125. Groups were initialized with careful spacing and flapping frequency ramp-up to realize long-lived formations; dynamics recorded for tens of minutes (thousands of cycles). Perturbations: DC (steady) forces applied to a targeted member during flight by locking its bearing housing to the shaft and applying a controlled torque via a mass–string–pulley to the shaft. Effective midspan force F=Wr/R (weight W, spool radius r) is used to map force–displacement curves by measuring steady positions under varying loads in both directions. AC (oscillatory) perturbations applied to the leader by locking it to the shaft fitted with a motorized oscillator that sinusoidally drives a small mass, inducing back-and-forth leader motion; downstream spacing fluctuations measured versus forcing frequency. Modeling and simulations: A minimal 1D point-particle wake-interaction model captures nonreciprocal, nearest-neighbor interactions with memory. Each flyer n has prescribed vertical flapping speed Vn(t)=2πfA cos(2πft+φn), writes a wake Wn equal to its flapping speed at emission that decays exponentially with timescale τ, and experiences thrust proportional to the square of relative flapping speed (Vn−Wn−1 at the flyer’s location), with quadratic drag ∝U². Memory enters via a state-dependent delay tn(t) defined implicitly by Xn(t)=Xn−1(tn), representing when the upstream neighbor traversed the current location. Governing delay differential equations (for Xn, Un, tn) are integrated numerically (MATLAB ddesd, adaptive tolerance ≤1e−3). Parameters informed by experiments: effective mass M=I/R²≈20 g, sc=32 cm², ρ=1.0 g/cm³, thrust coefficient CT=0.96, drag coefficient CD=0.074, wake decay τ=0.5 s, A=1.5 cm, f=2.5 Hz. Closed (cyclic) or open domains modeled; cyclic case includes leader interaction with last member. External DC forcing adds constant F/M to Ûn; AC forcing on leader adds (FAC/M) sin(2πfp t) with frequency-dependent amplitude FAC=K fp (K≈442 g·cm) derived from the experimental oscillator analysis. Simulations initialize at equilibrium spacing/speeds; groups up to N=30 explored. Phase relationships φn varied to introduce defects (vacancies) or diversity (alternating or random phases).

• Spontaneous crystalline ordering: In experiments with up to N=5 foils, members self-organize into a lattice-like formation with approximately uniform spacing. The dimensionless gap Sn=gn/λn−1 has a time-averaged value ≈1 across members, identifying the undulatory trajectory wavelength λ as the lattice parameter.
• Emergence of flonons: Measurements reveal temporally correlated spacing fluctuations that propagate downstream with delays proportional to rank and increase in amplitude, forming longitudinal displacement waves—termed “flonons”—that are unidirectional and self-amplifying, leading to eventual collisions.
• Spring-like, nearest-neighbor, nonreciprocal interactions: DC perturbation experiments mapping force–displacement curves show restorative, spring-like responses stabilizing S≈1, with similar profiles for downstream members regardless of rank, indicating nearest-neighbor-dominated, one-way (leader-to-follower) hydrodynamic interactions. Excessive forcing causes collisions or detachment from the upstream neighbor.
• Resonant amplification and fragility: AC perturbations of the leader produce resonance in followers; for member 2, a resonance peak near ≈0.6 Hz is observed. Later members (n=3,4) show amplified oscillations, with the last member in N=4 oscillating wildly near resonance and colliding—demonstrating self-amplification as the root of fragility.
• Modeling reproduces phenomena: The wake-interaction model captures crystalline equilibria (including multiple stable positions near integer S), DC force–displacement behavior (primary S≈1 and secondary S≈2 with weaker spring), and downstream resonance amplification under AC forcing. Simulations show slight downstream shifts to lower resonant frequencies and, for large N (e.g., N=30), eventual collisions due to amplified waves.
• Mechanistic insight—resonance cascade: A simplified analysis of a one-way, nearest-neighbor mass–spring chain shows that each follower acts as a driven, damped oscillator forced by its leader, yielding an amplitude gain factor per pair. Iteration produces a resonance cascade explaining flonon amplification.
• Stabilization via disorder and diversity: Introducing a vacancy defect (placing a member at S=2 downstream) reduces oscillation amplitudes at the group’s tail in both experiments and simulations, effectively filtering waves due to mismatched resonances. Simulated phase diversity (alternating or random φn leading to varied spacings/spring constants) further suppresses fluctuations compared to identical in-phase chains, enabling longer, disordered yet more stable formations. Quantitative highlights: Re≈10³; primary lattice spacing S≈1; resonance near ≈0.6 Hz for member 2; downstream amplification increases with rank; simulations confirm multiple stable positions at integer S and tail amplification leading to collisions for large N.

The findings demonstrate that high-Re flow-mediated interactions alone can organize flapping flyers into crystalline lattices while simultaneously rendering long chains fragile due to self-amplifying, unidirectionally transmitted displacement waves (flonons). Pairwise follower–wake phasing creates effective spring-like restoring forces that stabilize discrete equilibrium spacings, but one-way nonreciprocity and flow memory convert small perturbations at the leader into resonance-amplified oscillations downstream, explaining metastability and eventual collisions in larger groups. These dynamics parallel phonon-like excitations in crystals but differ fundamentally in amplification due to non-potential, nonreciprocal hydrodynamic forces. The study connects to active matter by highlighting the interplay between cohesive ordering and nonreciprocal amplification of fluctuations. It identifies key generic ingredients—nonreciprocal interactions, memory (long-lived wakes), and phase locking—that likely extend to other high-Re collective locomotion systems (e.g., schooling fish, bird flocks). The observed mitigation of instabilities through structural defects and individual variability suggests that natural diversity in kinematics and morphology, combined with sensing and behavior, may stabilize very long formations in nature. The work prompts biological tests at higher Re and with more complex kinematics to assess the prevalence and consequences of flonons in animal groups.

The paper establishes that flow interactions in high-Re flapping ensembles produce self-organized, lattice-like formations with spacing set by the trajectory wavelength, but are disrupted by novel, self-amplifying longitudinal waves (flonons) arising from nonreciprocal, memory-bearing interactions. Robophysical experiments and a minimal wake-interaction model jointly reveal spring-like nearest-neighbor forces, resonance-driven amplification forming a cascade down the chain, and the resultant fragility of long formations. Introducing structural defects or diversity in flapping phase stabilizes groups by detuning resonances and suppressing flonons, enabling longer-lived, disordered formations. Future work should probe higher Reynolds number regimes relevant to birds and fish, incorporate richer flapping kinematics (e.g., pitching), perform direct numerical simulations of coupled flow–structure dynamics, and develop behavioral control models to manage or exploit flonons in biological and robotic collectives.

• Experimental geometry is rotational (flight mill), not translational; although validated qualitatively, self-interactions in the closed domain can perturb the leader and influence dynamics.
• Group sizes in experiments are limited to N≤5; larger-N behavior is inferred from simulations.
• Reynolds numbers (Re≈10³) are at the lower end for many birds and fish; quantitative extrapolation to higher Re is uncertain.
• The minimal model is idealized: point particles, nearest-neighbor “erase-and-replace” wake interaction, exponential wake decay, quadratic thrust/drag with fixed coefficients, and prescribed flapping kinematics. It omits 3D effects, complex body–wake interactions, and nonlocal hydrodynamics.
• Initial conditions and ambient disturbances affect metastability and collision onset; precise thresholds for sustained order in natural conditions remain undetermined.
• Direct quantitative correspondence to specific species or behaviors is not attempted; biological variability and active control are not included in the physical model.