Figures and data
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.
Emergent equilibria in pairwise formations.
A. Time evolution of scaled streamwise separation 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, 26, 46, 51] and numerical [47–50] 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. Snapshots 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). Thrust parameter T is shown at ℓ = 0 and ℓ = 0.5L. A negative slope ∂T/∂d indicates stability of the predicted equilibria.
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 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. 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. D-F. Predictions of equilibrium locations, hydrodynamic benefits, and cohesion based on the wake of a solitary swimmer following the approach in Figs. 4 and 5. In all of the simulations, we set A = 15◦, f = 1 and τdiss = 2.45.
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.
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. 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 forth 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. 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.
Pairs of swimmers in CFD simulations
Vorticity field (left column) and flow agreement parameter V (right column) in the wake of a pair of inline and inphase swimmers Reynolds number ReA = 206, 308, 411, 1645, respectively. The pitching amplitude of leader and follower is set to A = 15◦, except in A, where the follower is pitching at A = 13.5◦.
Pairs of swimmers in VS simulations
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 τdiss = 2.45T, 3.45T, 4.45T, and 9.45T, respectively. The pitching amplitude is set to A = 15◦, and the pitching frequency to f = 1.
Influence of fluid property.
A. Separation distance versus time in a pair of swimmers in the CFD model for five values of Reynolds numbers ReA = 206, 308, 411, 822, 1645 (Fig. S.1). B. Separation distance versus time in a pair of swimmers in the VS model for five values of dissipation time τdiss = 2.45T, 3.45T, 4.45T, 9.45T, (Fig. S.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◦, except in ReA = 206, where the follower is pitching at A = 13.5◦ to avoid collision.
Hydrodynamic torque for pair of swimmers at different spatial configurations.
A. inline (ℓ = 0, ϕ = 0), B. diagonal (ℓ = L/4, 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. For each simulation, we show the active torque Ma exerted by the swimmers after a time tss, 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 confirgurations, 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.
Time-delay particle model and stability of swimmer.
A. Schematics of time-delayed particle model [26, 46] and its extension to laterally-offset swimmers. Each swimmer generates hydrodynamic thrust via oscillating vertically, which also leaves a wake behind it. The follower swimmer interacts with the wake of the leader. B. For an inphase pair, starting at initial distance d/UT = 1.15, we incrementally increase the lateral offset from ℓ = 0 to ℓ = L. (inset) Lateral decay of flow speed in the wake of a solitary swimmer in the vortex sheet model (blue line) and the fitted exponential curve (red line). The lateral exponential decay in the time-delay particle model takes the form exp ℓ/1.6 2.73. C. The hydrodynamic force F2 acting on the follower as a function of the corresponding distance from the equilibrium. Due to the decay of the leader’s wake in the lateral direction (see inset in A), the hydrodynamic force F2 experienced by the follower decreases in magnitude at increasing lateral offset ℓ. Orange lines show the linear change in force at the corresponding equilibria. The slope δF/δd is a measure of linear stability. Negative slopes imply stable formations for all ℓ L, but these slopes become more shallow as we increase ℓ.
Inline formations.
Snapshots of inline formations composed of 2, 3, 4, 5 and 6 swimmers in VS simulations at steady state; time-evolution of pairwise distances is shown on the right. A and B. Formations composed of 2 or three swimmers are stable with at consecutive spacing d/UT = 1. C. For a trail of 4 swimmers, the group splits into a leading subgroup of 3 swimmers while the fourth swimmer separates from the rest. D and E. For formation of 5 or 6 swimmers, the group splits into a leading subgroup of 3 swimmers and another subgroup containing the remaining 2 or 3 swimmers. F. reports recent savings in COT for each swimmer and the average of the whole group.
Side-by-side formations.
Vortex sheet simulation of side-by-side formations with 2, 3, 4, 5 and 6 swimmers, from top to bottom, respectively. On right hand side, we report pairwise spacing between them. In all of the groups the formations are stable and the distances between every pair are close to zero. F. reports recent savings in COT for each swimmer and the average of the whole group.
