Research Article: On the energetics and stability of a minimal fish school

Date Published: August 28, 2019

Publisher: Public Library of Science

Author(s): Gen Li, Dmitry Kolomenskiy, Hao Liu, Benjamin Thiria, Ramiro Godoy-Diana, Roi Gurka.


The physical basis for fish schooling is examined using three-dimensional numerical simulations of a pair of swimming fish, with kinematics and geometry obtained from experimental data. Energy expenditure and efficiency are evaluated using a cost of transport function, while the effect of schooling on the stability of each swimmer is examined by probing the lateral force and the lateral and longitudinal force fluctuations. We construct full maps of the aforementioned quantities as functions of the spatial pattern of the swimming fish pair and show that both energy expenditure and stability can be invoked as possible reasons for the swimming patterns and tail-beat synchronization observed in real fish. Our results suggest that high cost of transport zones should be avoided by the fish. Wake capture may be energetically unfavorable in the absence of kinematic adjustment. We hereby hypothesize that fish may restrain from wake capturing and, instead, adopt side-to-side configuration as a conservative strategy, when the conditions of wake energy harvesting are not satisfied. To maintain a stable school configuration, compromise between propulsive efficiency and stability, as well as between school members, ought to be considered.

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The behaviors of living beings provide amazing examples of aggregated dynamics that result from complex social reasons [1–4]. Depending on the species, animals aggregate and modulate group cohesion to improve foraging and reproductive success, avoid predators or facilitate predation. Global cohesive decision and action for the whole group result from different types of interaction at the local scale. Fish schools, for instance, are an archetypal example of how local interactions lead to complex global decisions and motions [5]. Fish interact through vision but also by sensing the surrounding flow using their lateral line system [6]. From the fluid dynamics perspective, hydrodynamic interactions between neighbors have often been associated with swimming efficiency strategies, considering how each individual in the school is affected by the vortical flows produced by its neighbors. Breder [7] already recognized the importance of this issue, and more recent works have described how fish make use of vortices when swimming through an unsteady flow, whether produced by neighboring fish or by other features in the environment (see e.g. the review by Liao [8]). Concerning collaborative interactions between swimming fish, the first clear picture was proposed in the early 70’s by Weihs’ pioneering work [9]. He focused on interactions within a two-dimensional layer of a three-dimensional school, and proposed an idealized two-dimensional model in which each individual in the fish school places itself to benefit from the wakes generated by its two nearest neighbors, giving rise to a precise diamond-like pattern.

We conducted a simulation of a solitary fish in self-propelled swimming mode and obtained its terminal speed of 9.25 cm s−1 with a tail beat frequency of 8 Hz, which agrees well with the experiments [29]. We then applied an oncoming uniform flow at that velocity U = 9.25 cm s−1 (which gives a Reynolds number Re = 3700) and the same tail beat frequency of f = 8 Hz for all the rest of simulations in tethered mode. Note that the speed and the kinematics are not chosen arbitrarily, but representatively: a range of speeds of approximately 3 to 15 cm s−1 has been observed in the experiments (see Fig 2 in [29]), and 9.25 cm s−1 is almost in the middle. The experiments also suggested that fish had preferred combinations of frequency and amplitude depending on the speed. One of those is used in the simulations.

Our results show that the spatial organization and the kinematic synchronization of a pair of swimming fish—the minimal school—have a clear effect on two crucial aspects of schooling: energy expenditure and fluctuation minimization. We have examined the effect of the hydrodynamic interaction between the two fish on several performance parameters by probing forces and consumed hydrodynamic power on a fish that we have called the protagonist fish, while placing it in different positions and with a kinematic phase shift with respect to its neighbor (the companion fish).

We developed an in-house three-dimensional overset grid numerical approach based on finite-volume method and programmed in FORTRAN 90 to simulate cyclic swimming of fish [30–32, 42]. The approach comprises surface models of the changing fish shape (dimension: 121 × 97), and local fine-scale body-fitted grids (dimension: 121 × 97 × 20) plus a large stationary global grid (dimension: various) to calculate the flow patterns around the fish with sufficient resolution (supportive information on grid resolution and size tests can be found in supplementary materials). As shown in Fig 10, to simulate a fish pair, two body-fitted grids were deployed, which deformed as the fish model deformed. The global grid surrounded the body-fitted grids and covered a sufficiently large domain to enclose the swimming fish and their wake. The ensemble of body-fitted grids and global grid was set up as a multi-blocked, overset-grid system based on a chimera grid scheme [43]. During the simulation, the body-fitted and global grids share values on their interfaces through inter-grid communication algorithm. The body was modelled on the silhouette of a Red nose tetra fish (Hemigrammus bleheri), with a body length of 4 cm, an average length measured in previous experimental study [29]. All cross-sections of the fish were modeled as ellipses. To reduce the complexity in modeling and computation, we assume that the hydrodynamic influence of all fins other than the tail fin is relatively minor, and neglect them in the model. Also, for the same reason, the gap of the fork-shaped tail fin is neglected, and the fish model has a triangle-shaped fin instead. The instantaneous body shape is driven by sinusoidal variation of the midline, cf. [32],
where l is the dimensionless distance from the snout along the longitudinal axis of the fish based on the length of the fish model L; H(l, t) is the dimensionless lateral excursion at time t;
is the dimensionless amplitude envelope function at l; λ is the length of the body wave and it is set as 1.2L; f is the tail beat frequency defined as f = 8 Hz. We use a = 0.11 in all simulations, unless stated. These values of the model parameters are based on data from experiments [29]. Eq 7 may cause total body length along the midline to vary during the tail beat; this variation is corrected by a procedure that preserves the lateral excursion H(l, t) while ensuring that the body length remains constant. The correction algorithm is explained in S1 File. Procedure flow of simulations is shown in Fig 1b. We conducted simulations in two modes. In free-swimming (self-propelled) mode simulation, we simulated single fish swims in the horizontal plane with its center-of-mass (CoM) movements and body orientation determined by the hydrodynamic forces on the body, while oncoming flow was set as zero. By using free-swimming simulation, we obtained the terminal speed in single fish swimming and apply to the rest simulations. All the rest simulations were conducted in fixed CoM mode: we simulated a single fish or fish pair swimming with CoM and relative position fixed, while the rotational degree of freedom was still available to model the rotational recoil effect during swimming. This means that the fish can rotate if the torque exerted on it is not zero. Such semi-tethered condition is necessary for producing the performance maps. Otherwise, in free swimming with no feedback control, the fish may not necessarily hold formation.




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