Stable formations of self-propelled fish-like swimmers induced by hydrodynamic interactions

Abstract

Fish schools are fascinating examples of macro-scale systems with collective behaviours. According to conventional wisdom, the establishment and maintenance of fish schools probably need very elaborate active control mechanisms. Sir James Lighthill posited that the orderly formations in fish schools may be an emergent feature of the system as a result of passive hydrodynamic interactions. Here, numerical simulations are performed to test Lighthill's conjecture by studying the self-propelled locomotion of two, three and four fish-like swimmers. We report the emergent stable formations for a variety of configurations and examine the energy efficiency of each formation. The result of this work suggests that the presence of passive hydrodynamic interactions can significantly mitigate the control challenges in schooling. Moreover, our finding regarding energy efficiency also challenges the widespread idea in the fluid mechanics community that the diamond-shaped array is the most optimized pattern.

Keywords: biolocomotion; computational fluid dynamics; energy efficiency; fish schooling; fluid–structure interaction.

Figure 1.

An FSI-driven swimmer which emulates a fish performing undulatory locomotion: (a) schematic diagram for the computational model; (b) wake structure represented by the vorticity contours. (Online version in colour.)

Figure 2.

Time histories of the instantaneous streamwise distance l between the head of the reference fish and those of other fishes. (a) Two fish under the in-line pattern; (b) three fish under the staggered II pattern; (c) four fish under the diamond-shaped pattern. Please note from (a) that no stable in-line formations can be achieved if the initial separation distances are larger than 2.8L. (Online version in colour.)

Figure 3.

A diagram for the stable formations observed in the simulations: (a) different configurations considered in this work; (b) head positions of the fish in the stable formations. The head positions of the selected fish for each stable formation (denoted as a star in (a)) are placed at the origin in (b). The red, blue and cyan colours in (b) denote the fish head positions in the schools composed of two, three and four members, respectively. For visualizability, the three fish in the staggered and echelon formations, the four fish in the rectangular and diamond formations are connected by a variety types of lines. In the right inset of (b), ‘IP’ and ‘AP’ denote the in-phase mode and anti-phase mode, respectively.)

Figure 4.

Flow structures represented by the vorticity contours for the stable formations composed of two, three and four fish: (a) two fish side-by-side (anti-phase); (b) two fish side-by-side (in-phase); (c) two fish in-line (compact); (d) two fish in-line (loose); (e) two fish staggered (compact); (f) two fish staggered (loose); (g) three fish side-by-side (anti-phase); (h) three fish side-by-side (in-phase); (i) three fish echelon; (j) three fish staggered (type I); (k) three fish staggered (type II); (l) four fish rectangular (compact, anti-phase); (m) four fish rectangular (loose, anti-phase); (n) four fish diamond. For all the formations listed here, the in-phase pattern is assumed if not specified. (Online version in colour.)

Figure 5.

The relative differences in COT between those in formation swimming and that in solitary swimming: (a) two-fish formations; (b) three-fish formations; (c) four-fish formations. For the side-by-side and the echelon formations, the lowest COT among all possible lateral spacings is shown. COT_s denotes the COT in solitary swimming.