Flow-coupled swimmers self-organize into stable pairwise formations

A. inline (ℓ = 0, ϕ = π/2), B. diagonal (ℓ = L/2, ϕ = 0), C. inphase side-by-side (ℓ = L/2, ϕ = 0) and D. antiphase side-by-side (ℓ = L/2, ϕ = π) in CFD (left) and VS (right) simulations. Power savings at steady state relative to respective solitary swimmers are reported in Fig. 3. Parameter values are A = 15◦, Re=2πρAf L/µ = 1645 in CFD, and fτdiss = 2.45 in VS simulations. Corresponding hydrodynamic moments are given in Fig. 1.S4. Simulations at different Reynolds numbers and dissipation times are given in Figs. 1.S1, 1.S2 and 1.S3.

Emergent equilibria in pairwise formations

A. Time evolution of scaled streamwise separa-tion distance d/UT for a pair of inline swimmers at ϕ = 0. Depending on initial conditions, the swimmers converge to one of two equilibria at distinct separation distance. B. At ℓ = L/2, d/UT changes slightly compared to inline swimming in panel A. Importantly, a new side-by-side inphase equilibrium is now possible where the swimmers flap together at a slight shift in the streamwise direction. C. Starting from the first equilibrium in panel A, d/UT increases linearly as we increase the phase lag ϕ between the swimmers.

Hydrodynamic benefits and linear phase-distance relationship in pairs of swimmers

A. change in cost of transport compared to solitary swimmers for the inline, diagonal, side-by-side inphase and side-by-side antiphase formations shown in Fig. 1. B. Emergent formations in pairs of swimmers in CFD and VS models satisfy a linear phase-distance relationship, consistent with experimental [9, 11, 27, 47, 52] and numerical [48–51] studies. With the exception of the antiphase side-by-side formation, swimmers in these formations have a reduced average cost of transport compared to solitary swimming.

Predictions of equilibrium formations from the wake of a solitary swimmer

A, B. Snap-shots of vorticity and fluid velocity fields created by a solitary swimmer in CFD and VS simulations and corresponding flow agreement parameter 𝕍 fields for a virtual follower at ϕ = 0. Locations of maximum 𝕍-values (i.e., peaks in the flow agreement parameter field) coincide with the emergent equilibria in inphase pairwise formations (indicated by black circles). Contour lines represent flow agreement pa-rameter at ±0.25, ±0.5. Thrust parameter T is shown at ℓ = 0 and ℓ = 0.5L. A negative slope ∂𝕋/∂d indicates stability of the predicted equilibria. See also Figs. 1.S1 and 1.S2.

Predictions of equilibrium locations, power savings, and cohesion, from the wake of a solitary leader

A. Location of maximum 𝕍 as a function of phase lag ϕ in the wake of solitary leaders in CFD and VS simulations. For comparison, equilibrium distances of pairwise simulations in CFD, VS and time-delay particle models (Fig. 5.S1) are superimposed. Agreement between 𝕍-based predictions and actual pairwise equilibria is remarkable. B. 𝕍 values also indicate the potential benefits of these equilibria, here shown as a function of lateral distance ℓ for a virtual inphase follower in the wake of a solitary leader in CFD and VS simulations. The power savings of an actual follower in pairwise formations in CFD and VS simulations are superimposed. C. A negative slope ∂𝕋/∂d of the thrust parameter 𝕋 indicates stability and |∂𝕋/∂d| expresses the degree of cohesion of the predicted formations, here, shown as a function of ℓ for an inphase virtual follower. |∂F/∂d| obtained from pairwise formations in VS and time-delay particle models are superimposed (Fig. 5.S1). Results in D. and E. are normalized by the corresponding maximum values to facilitate comparison.

Equilibria are dense over the parameter space

For any given phase lag ϕ and at any lateral offset ℓ inside the wake, the pair reach equilibrium formations that are stable and power saving relative to a solitary swimmer. A. Equilibrium separation distances, B. average power saving, and C. stability as a function of phase lag and lateral distance in a pair of swimmers. Predictions of D. equilibrium locations, E. hydrodynamic benefits, and F. cohesion based on the wake of a solitary swimmer following the approach in Figs. 4 and 5. For comparison, the contour lines from panels B and C based on pairwise interactions are superimposed onto panels E and F (white lines). Simulations in panels A-C are based pairwise interactions and Simulations in panels D-F are based on the wake of a single swimmer, all in the context of the vortex sheet model with A = 15◦, f = 1 and τdiss = 2.45T.

Larger inline and side-by-side formations

A. Inline formations lose cohesion and split into two subgroups as depicted here for a group of six swimmers. B. Side-by-side formations remain cohesive. C. Power saving of each swimmer in inline and side-by-side formations. Dissipation time τdiss = 2.45T. Simulations of inline formations and side-by-side formations ranging from 2 to 6 swimmers are shown in Fig. 7.S1 and Fig. 7.S2.

Loss of cohesion in larger groups of inline swimmers

Number of swimmers that stay in cohesive formation depends on parameter values. A-C. For dissipation time τdiss = 2.45T, 3.45T and 4.45T, the 4th, 5th and 6th swimmers separate from the group, respectively. D. Power savings per swimmer in panels A-C, respectively. On average, all schools save equally in cost of transport, but the distribution of these savings vary significantly between swimmers. In all case, swimmer 3 receives the most hydrodynamic benefits.

Prediction of equilibrium formations, cohesion, and power savings from the wake of upstream swimmers

A., B. Snapshots of vorticity fields created by two inline inphase swimmers, and three inline inphase swimmers. C., D. shows the corresponding flow agreement parameter 𝕍 fields. Contour lines represent flow agreement parameter at ±0.25, ±0.5. E., F. plots the corresponding period-averaged streamwise velocity. Separation distances d/UT predicted by the locations of maximal 𝕍 are marked by circles in the flow agreement field. In the left column, separation distances d/UT based on freely swimming triplets are marked by black circles and coincide with the locations of maximal 𝕍. In the right column, the orange marker shows the prediction of the location of a fourth swimmer based on the maximum flow agreement parameter. In two-way coupled simulation, swimmer 4 actually separates from the leading 3 swimmers as illustrated in Fig 8A. CFD simulation shows swimmer 4 will collide with swimmer 3 as in Fig. 9.S1. G., H. shows the transverse flow velocity in a period at the location predicted by the maximum flow agreement parameter and with a lateral offset ℓ = 0, 0.5L, L, in comparison to the follower’s tailbeat velocity.

Passive and active methods for stabilizing an emergent formation of four swimmers

A. In an inline school of four-swimmers, the leading three swimmers flap inphase, but swimmer 4 actively controls its phase in response to the flow it perceives locally to match its phase to that of the local flow as proposed in [9]. The phase controller stabilizes swimmer 4 in formation but at no hydrodynamic benefit. B. Sequentially increasing the phase lag by a fixed amount Δϕ = −30o in an inline school of four-swimmers stabilizes the trailing swimmer but at no hydrodynamic benefit. C. Gradually tuning the phase lag Δϕ in a school of four swimmers as done in panel B. At moderate phase lags, the school stays cohesive (top plot) but swimmer 4 barely gets any power savings (bottom plot). D. By laterally offsetting the swimmers, four swimmers, all flapping inphase, form cohesive schools with different patterns, e.g. with side-by-side pairing of two swimmers, staggered, and diamond patterns. The time evolution of separation distances is shown in Fig. 10.S1. Individual in each pattern receive a different amount of hydrodynamic benefit. Diamond formation provides the most power saving for the school as anticipated in [1] for a school in a regular infinite lattice. In panels A, B and D, % values indicate the additional saving or expenditure in cost of transport relative to solitary swimming.