On the energetics and stability of a minimal fish school

Abstract

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.

Burst-and-coast swimmers optimize gait by adapting unique intrinsic cycle

“…2 e. The cost of transport is defined as i.e., as the power normalized by the average swimming speed and the mass of the fish (see ref. 13 and “Methods” for details). The shaded area in each subplot of Fig.…”

On the Fluid Dynamical Effects of Synchronization in Side-by-Side Swimmers

“…Indeed, 2D models have been successful in describing other schooling configurations; see, e.g., [30]. However, the thrust production, energy expenditure, and stability of a fish school are very sensitive to the details of the pattern, as shown recently by Li et al [24] using 3D numerical simulations. Although 2D models and interpretations such as the one we have used to explain the advantage of the anti-phase synchronization can be useful to understand the basic physical mechanisms at play during schooling, the 3D reality needs to be always kept in mind, especially in a bio-inspired robotics perspective.…”

“…Conceptually, the setup with two swimmers that we study in the present paper is one of the simplest model realizations of the minimal school [24,25] and has been designed to examine the complex flow dynamics that arises due to the undulation kinematics of the two swimmers. Recent experiments on real fish in a swimming channel with imposed velocity have shown that fish favor a synchronized kinematics with their nearest neighbors as the swimming speed of the school increases [17,18].…”

Using a robotic platform to study the influence of relative tailbeat phase on the energetic costs of side-by-side swimming in fish

“…Although the bar may transmit force between the robotic fish while limiting lateral movements, most forces were predominantly directed forwards (see electronic supplementary material, movie S1), indicating that the bar had only a very limited influence on straight forward swimming of the robots. Therefore, our system is also similar to previous simulations [13,41] or physical models [18], where lateral movements of the models were also restrained.…”

“…To evaluate the hydrodynamic interactions of swimming in formation, we developed a method to place two robotic fish side-by-side at 0.33 BL. The formation and distance have been suggested as a configuration that could allow real fish, and experimental physical models of fish-like swimming, to save energy when in a group [12,14,16,41]. Because of the turbulence and backflow, it is difficult to dynamically control and keep a stable formation between the two robots.…”