Lateral flow interactions enhance speed and stabilize formations of flapping swimmers

While classic hydrodynamic models predict ordered formations for fish schools, observations show that schools are seemingly disordered. Our experiments on robotic swimmers may help to reconcile this discrepancy by showing that many different formations all emerge spontaneously and are stabilized due to flow interactions. Surprisingly, these locked states extend almost twice as far downstream for laterally displaced swimmers as for those in line. We also observe significant boosts in swimming speed—up to 60% faster than an isolated swimmer—for side-by-side formations. These findings demonstrate that benefits such as group cohesion and speed enhancement arise naturally via flow interactions and for the diverse relative arrangements seen in schools.

Figure 1

(a) Sketch of two fish swimming in diagonal formation while interacting through vortex wakes. (b) Experiment consisting of two hydrofoils that are flapped up and down in a water tank (not shown), with each foil free to revolve about an axle. Flapping leads to forward propulsion, which takes the form of large orbits around the tank. (c) Definitions of the chord length $c$, lateral spacing $L$, streamwise displacement $d$, and trajectory wavelength $\lambda$. (d) For prescribed but variable lateral spacing, possible emergent formations include side-by-side (red), tandem (blue), and diagonal (purple) states.

Figure 2

(a) Multiple stable formations are observed in experiments of two swimmers with fixed lateral spacing and flapping kinematics. (b) Time series of streamwise separation distance normalized by wavelength $d/\lambda$. Side-by-side (red) and multiple, diagonal (purple) formations are attained by varying initial separation $d_0$, and four trials are shown for each state. (c) A histogram of the distance between foils shows that more separated formations exhibit greater positional fluctuation. (d) An external impulse is applied to the follower in a diagonal formation at $t=0$, forcing a transition from diagonal ($d>0$) to side-by-side ($d=0$) formation. The side-by-side pair experiences an increase in swimming speed $U$. (e) Swimming speeds for formations with different lateral spacing $L/c$ compared to the speed $U_0$ of an isolated foil.

Figure 3

Locking region for a follower relative to a leader. Spatial maps of stable formations across streamwise and lateral spacings $(d,L)$ are shown for fixed flapping phase $\varphi =\pi$ and two amplitudes $A/c$. The shaded regions show states accessible by changing the relative flapping phase $\varphi$.