Hydrodynamic schooling of flapping swimmers

Abstract

Fish schools and bird flocks are fascinating examples of collective behaviours in which many individuals generate and interact with complex flows. Motivated by animal groups on the move, here we explore how the locomotion of many bodies emerges from their flow-mediated interactions. Through experiments and simulations of arrays of flapping wings that propel within a collective wake, we discover distinct modes characterized by the group swimming speed and the spatial phase shift between trajectories of neighbouring wings. For identical flapping motions, slow and fast modes coexist and correspond to constructive and destructive wing-wake interactions. Simulations show that swimming in a group can enhance speed and save power, and we capture the key phenomena in a mathematical model based on memory or the storage and recollection of information in the flow field. These results also show that fluid dynamic interactions alone are sufficient to generate coherent collective locomotion, and thus might suggest new ways to characterize the role of flows in animal groups.

Figure 1. Flapping wings swimming in rotational orbits mimic an infinite array of locomotors.

(a) A motor heaves a wing or wing pair up and down at prescribed frequency f and peak-to-peak amplitude A, resulting in swimming of rotational frequency F around a cylindrical water tank. (b) The rotational geometry allows for interactions with the flows generated in previous orbits.

Figure 2. Dynamics of interacting wings.

(a) Swimming speed, as measured by the rotational frequency F, versus flapping frequency f. For each peak-to-peak amplitude A, an upward sweep of f is followed by a downward sweep (as indicated by arrows), and the data form a hysteresis loop. (b) Schooling number S, which represents the number of wavelengths separating successive wings. Each hysteresis loop is bounded by in-phase (integer value of S) and out-of-phase states (half-integer S). (c) A polar histogram of S mod 1 shows peaks corresponding to preferred spatial phase relationships between successive wings.

Figure 3. Flow visualization.

(a,b) Flow field around a non-interacting wing of chord length 3 cm extracted using particle image velocimetry and rendered in a schematic. The colour map indicates the vertical component of the velocity vector field, with red indicating upward and blue downward flows. The upstroke produces an upward flow and the downstroke a downward flow. (c,d) Slow mode of interacting wings: the downstroke of a wing (red path) occurs within the downward flow of its predecessor (blue path). (e,f) Fast mode: the downstroke occurs within the upward flow of its predecessor.

Figure 4. Simulations of interacting wings.

(a–d) An infinite array of synchronized or temporally in-phase wings is simulated by a single airfoil driven to flap up and down, and allowed to swim freely left to right across a periodic domain. (a) Schooling number S for increasing (blue) and decreasing (red) flapping Reynolds number, Re_f. A non-interacting wing (dashed curve) swims at a speed intermediate between the two schooling modes. (b) Input power normalized by that of an isolated wing. (c) Computed vorticity field for the slow mode (blue circle in a): The wing slaloms between vortices. (d) Fast mode (red circle in a): the wing intercepts each vortex core. (e,f) Schooling dynamics and power consumption for an array in which nearest neighbours flap temporally out-of-phase with one another.

Figure 5. Mathematical model.

(a) An infinite linear array of synchronized swimmers is represented by a single particle undergoing repeated passes across a domain specified by periodic boundary conditions. (b) Schooling number for a model with parameter values s=1.5, P=1.5, ɛ=1, τ=1 (see text for details).