Figure 1. Flow-coupled swimmers self-organize into stable pairwise formations.

(A) Inline ($\ell =0,\varphi =\pi /2$), (B) diagonal ($\ell =L/2,\varphi =0$), (C) inphase side-by-side ($\ell =L/2,\varphi =0$), and (D) antiphase side-by-side ($\ell =L/2,\varphi =\pi$) in computational fluid dynamics (CFD) (left) and vortex sheet (VS) (right) simulations. Power savings at steady state relative to respective solitary swimmers are reported in Figure 3. Parameter values are $A={15}^{\circ }$, Re = $2\pi \rho AfL/\mu =1645$ in CFD, and $f{\tau }_{\text{diss}}=2.45$ in VS simulations. Corresponding hydrodynamic moments are given in Figure 1—figure supplement 4. Simulations at different Reynolds numbers and dissipation times are given in Figure 1—figure supplements 1–3.

Figure 1—figure supplement 1. Computational fluid dynamics (CFD) simulations of pairwise formations.

Vorticity field (left column) and flow agreement parameter $𝕍$ (right column) in the wake of a pair of inline and inphase swimmers Reynolds number ${\displaystyle \text{Re}=206}$ (A), 308 (B), 411 (C), 1645 (D), respectively. The pitching amplitude of leader and follower is set to $A={15}^{\circ }$, except in (A), where the follower is pitching at $A=13.5^{\circ }$.

Figure 1—figure supplement 2. Vortex sheet (VS) simulations of pairwise formations.

Snapshots of the swimmers (left column) and flow agreement parameter $𝕍$ (right column) in the wake of a pair of swimmers in the VS model for dissipation time $2.45T$ (A), $3.45T$ (B), $4.45T$ (C), and $9.45T$(D), respectively. The pitching amplitude is set to $A={15}^{\circ }$, and the pitching frequency to $f=1$.

Figure 1—figure supplement 3. Computational fluid dynamics (CFD) and vortex sheet (VS) simulations at various flow properties.

(A) Separation distance versus time in a pair of swimmers in the CFD model for five values of Reynolds numbers ${\text{Re}}_A=206,308,411,822,1645$ (Figure 1—figure supplement 1). (B) Separation distance versus time in a pair of swimmers in the VS model for five values of dissipation time ${\tau }_{\text{diss}}=2.45T,3.45T,4.45T,9.45T,\infty$ (Figure 1—figure supplement 2). Separation distance $d$ is normalized by swimming speed $U$ for each cases, respectively. In (A, B), the swimmers stabilize near $d/UT=1$. The pitching amplitude of leader and follower is set to $A={15}^{\circ }$, except in ${\text{Re}}_A=206$ , where the follower is pitching at $A=13.5^{\circ }$ to avoid collision.

Figure 1—figure supplement 4. Hydrodynamic moment acting on pairs of flapping swimmers.

(A) Inline ($\ell =0,\varphi =0$), (B) diagonal ($\ell =L/4,\varphi =0$), (C) inphase side-by-side ($\ell =L/2,\varphi =0$), and (D) antiphase side-by-side ($\ell =L/2,\varphi =\pi$) in computational fluid dynamics (CFD) (left) and vortex sheet (VS) (right) simulations. For each simulation, we show the active torque $M_a$ exerted by the swimmers after a time $t_{ss}$, ensuring that steady state has been reached. The non-reciprocity in the effects of leader on follower in inline and diagonal configuration is apparent. In the side-by-side configurations, a simple shift of the data in the inphase flapping case and a simple mirror symmetry in the antiphase flapping case would show that flow coupling is reciprocal.