To understand the mechanics of fish swimming, we need to know the forces exerted by the fluid and how these forces affect the motion of the fish. To this end, we developed a 3-D computational approach that integrates hydrodynamics and body dynamics. This study quantifies the flow around a swimming zebrafish (Danio rerio) larva. We used morphological and kinematics data from actual fish larvae aged 3 and 5 days post fertilization as input for a computational model that predicted free-swimming dynamics from prescribed changes in body shape. We simulated cyclic swimming and a spontaneous C-start. A rigorous comparison with 2-D particle image velocimetry and kinematics data revealed that the computational model accurately predicted the motion of the fish's centre of mass as well as the spatial and temporal characteristics of the flow. The distribution of pressure and shear forces along the body showed that thrust is mainly produced in the posterior half of the body. We also explored the effect of the body wave amplitude on swimming performance by considering wave amplitudes that were up to 40% larger or smaller than the experimentally observed value. Increasing the body wave amplitude increased forward swimming speed from 7 to 21 body lengths per second, which is consistent with experimental observations. The model also predicted a non-linear increase in propulsive efficiency from 0.22 to 0.32 while the required mechanical power quadrupled. The efficiency increase was only minor for wave amplitudes above the experimental reference value, whereas the cost of transport rose significantly.

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.

This paper addresses the physical mechanism of intermittent swimming by considering the burst-and-coast regime of fish swimming at different speeds. The burst-and-coast regime consists of a cycle with two successive phases, i.e., a phase of active undulation powered by the fish muscles followed by a passive gliding phase. Observations of real fish whose swimming gait is forced in a water flume from low to high speed regimes are performed, using a full description of the fish kinematics and mechanics. We first show that fish modulate a unique intrinsic cycle to sustain the demanded speed by modifying the bursting to coasting ratio while maintaining the duration of the cycle nearly constant. Secondly, we show using numerical simulations that the chosen kinematics by correspond to optimized gaits for swimming speeds larger than 1 body length per second.

Energetic expenditure is an important factor in animal locomotion. Here we test the hypothesis that fishes control tail-beat kinematics to optimize energetic expenditure during undulatory swimming. We focus on two energetic indices used in swimming hydrodynamics, cost of transport and Froude efficiency. To rule out one index in favour of another, we use computational-fluid dynamics models to compare experimentally observed fish kinematics with predicted performance landscapes and identify energy-optimized kinematics for a carangiform swimmer, an anguilliform swimmer and larval fishes. By locating the areas in the predicted performance landscapes that are occupied by actual fishes, we found that fishes use combinations of tail-beat frequency and amplitude that minimize cost of transport. This energy-optimizing strategy also explains why fishes increase frequency rather than amplitude to swim faster, and why fishes swim within a narrow range of Strouhal numbers. By quantifying how undulatory-wave kinematics affect thrust, drag, and power, we explain why amplitude and frequency are not equivalent in speed control, and why Froude efficiency is not a reliable energetic indicator. These insights may inspire future research in aquatic organisms and bioinspired robotics using undulatory propulsion.