Fig 1.
Numerical simulation methodology.
(a) An explanation diagram of performance maps in Figs 3–7. Simulations were implemented by varying the relative longitudinal and lateral positions between two fish. To test the influence of phase difference, for each position (circles) we implemented four simulations (δϕ = 0, T/4, T/2 and 3T/4, respectively). Based on simulation results and interpolation, maps for swimming performance parameters were drawn. This performance map provide the performance value of the protagonist fish with its companion fish located at the origin. (b) (LHS) We conducted simulations in two modes: free-swimming (self-propelled) mode and tethered (fixed CoM) mode; (RHS) Procedure flow of simulations. Firstly, we simulated free-swimming single fish, obtained the terminal speed and apply to the rest simulations; We then simulated single fish swim and fish pair swim with CoM fixed. The performance of protagonist fish in the pair relative to the performance of a single fish is used to draw a performance map to demonstrate the influence of relative position and phase shift comprehensively. Here U is the terminal speed in single fish free-swimming, Π represents swimming performance parameter, such as net force, power, cost of transport, etc. Note that the same diagrams provide the performance of each of the two fish in the pair: the reader should select which of the two is considered as a protagonist, calculate its relative position and its phase shift with respect to its companion, and look for the corresponding point value on the performance map.
Fig 2.
Three-dimensional flow visualization using an iso-surface of the Q-criterion [34].
(a) δx = 0.2L, δy = 1.25L, δϕ = 0; (b) δx = 0.5L, δy = 0, δϕ = T/2. For more examples of the wake topology, see [35].
Fig 3.
Performance maps of the protagonist fish in terms of normalized net longitudinal force.
Normalized net longitudinal force is calculated as ΔF∥/mg = (F∥ − F∥solo)/mg.
Fig 4.
Performance maps of the protagonist fish in terms of relative power consumption.
Relative power consumption is calculated as P/Psolo × 100%.
Fig 5.
Performance maps of the protagonist fish in terms of cost of transport.
Relative cost of transport is calculated as .
Fig 6.
Performance maps of the average cost of transport of the group.
Relative average cost of transport of the group is calculated as .
Fig 7.
Performance maps of the protagonist fish in terms of (a) normalized lateral force, ΔF⊥/mg, (b) relative standard deviation of lateral force, s.d.F⊥/s.d.F⊥solo × 100% and (c) relative standard deviation of longitudinal force, s.d.F∥/s.d.F∥solo × 100%.
Fig 8.
Time-average flow field near one fish.
(a) pressure coefficient, , (b) longitudinal velocity,
, (c) lateral velocity,
and (d) fluctuation kinetic energy in the horizontal plane,
. Vectors show the velocity in the horizontal plane.
Fig 9.
Interaction between the follower’s boundary layer vorticity with the leader’s wake.
(a) Single fish; (b) Tandem formation with δx = 0, δy = ±1.25L, δϕ = 0; (c) Tandem formation with δx = 0, δy = ±1.25L, δϕ = T/2. Color plots show the vertical component of the dimensionless vorticity sampled on a horizontal plane. Instantaneous snapshots are shown such that the leader’s midline deformation is the same in all three cases. The arrows show the region of vortex capture. Note that the case (b) corresponds to ΔF∥ > 0 and (c) corresponds to ΔF∥ < 0.
Fig 10.
Computational fluid dynamics model.
(a) Red nose tetra fish (Hemigrammus bleheri); (b) Surface model of Red nose tetra fish (dimension: 121 × 97); (c) A function (Eq 7) drives the instantaneous body shape. Variation of body length caused by this driving function was corrected to keep lateral excursion and body length constant at 1L. (d) Multi-blocked computational grid system consists of local fine-scale body-fitted grids (dimension: 121 × 97 × 20) plus a large stationary global grid (dimension: variant). Reprinted from [35] under a CC BY license, with permission from the Society of Aero Aqua Bio-mechanisms, original copyright 2019.