Fishes regulate tail-beat kinematics to minimize speed-specific cost of transport.

Gen Li,Cees J Voesenek,Hao Liu,Ulrike K Müller,Johan L Van Leeuwen

doi:10.1098/rspb.2021.1601

Abstract

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.

Highlights

Undulatory swimming is the most common swimming style in fishes across a wide range of body sizes and shapes [1,2,3,4,5]
In answer to the question of why the optimal trajectory of the Ωmin strategy requires lower tail-beat amplitudes than the ηmax strategy, we found that the Ωmin strategy constrains amplitude whereas the ηmax strategy does not: drag on an undulatory swimmer depends not solely on swimming speed, and the kinematics of the undulating body; the swimmer needs to optimize its body kinematics to prevent excessive energy consumption by limiting drag according to the Ωmin strategy
Our results show that fishes minimize speed-specific fluiddynamic cost of transport (Ωmin strategy) rather than maximize speed-specific Froude efficiency

Summary

Introduction

Undulatory swimming is the most common swimming style in fishes across a wide range of body sizes and shapes [1,2,3,4,5]. We used computational fluid dynamics (CFD) to simulate experimental outcomes and computed counterfactual cases to describe the performance surface of fishes as a function of body-wave frequency and amplitude for two mathematically modelled body-wave types (carangiform, anguilliform) and one experimentally observed body wave (based on a larval zebrafish) (figures 2a, 3a and 4a). Combining these highresolution performance maps with our extensive experimental dataset on larval fishes allowed us to go beyond previous numerical studies [17,18] and test hypotheses about optimization strategies used by actual fishes. We examined two optimization strategies to find optimal combinations of A and f as a function of swimming speed U: one for minimizing cost of transport Ω and another for maximizing Froude efficiency η (see the electronic supplementary material, figure S9 for a detailed explanation on how we found optimal trajectories)

Results

Findings

Discussion