Scaling macroscopic aquatic locomotion

Abstract

Inertial aquatic swimmers that use undulatory gaits range in length L from a few millimetres to 30 metres, across a wide array of biological taxa. Using elementary hydrodynamic arguments, we uncover a unifying mechanistic principle characterizing their locomotion by deriving a scaling relation that links swimming speed U to body kinematics (tail beat amplitude A and frequency ω) and fluid properties (kinematic viscosity ν). This principle can be simply couched as the power law Re ∼ Sw α , where Re = UL/ν 1 and Sw = ωAL/ν, with α = 4/3 for laminar flows, and α = 1 for turbulent flows. Existing data from over 1,000 measurements on fish, amphibians, larvae, reptiles, mammals and birds, as well as direct numerical simulations are consistent with our scaling. We interpret our results as the consequence of the convergence of aquatic gaits to the performance limits imposed by hydrodynamics. Aquatic locomotion entails a complex interplay between the body of the swimmer and the induced flow in the environment 1,2. It is driven by motor activity and controlled by sensory feedback in organisms ranging from bacteria to blue whales 3. Locomotion at low Reynolds numbers (Re = UL/ν 1) is governed by linear hydrodynamics and is consequently analytically tractable, whereas locomotion at high Reynolds numbers (Re 1) involves nonlinear inertial flows and is less well understood 4. Although this is the regime that most macroscopic creatures larger than a few millimetres inhabit, as shown in Fig. 1a, the variety of sizes, morphologies and gaits makes it difficult to construct a unifying framework across taxa. Thus, most studies have tried to take a more limited view by quantifying the problem of swimming in specific situations from experimental, theoretical and computational standpoints 5-7. Beginning more than fifty years ago, experimental studies 8-12 started to quantify the basic kinematic properties associated with swimming in fish, while providing grist for later theoretical models. Perhaps the earliest and still most comprehensive of these studies was performed by Bainbridge 8 , who correlated size and frequency f = ω/(2π) of several fish via the empirical linear relation U /L = (3/4)f − 1. However, he did not provide a mechanistic rationale based on fundamental physical principles. The work of Bainbridge served as an impetus for a variety of theoretical models of swimming. The initial focus was on investigating thrust production associated with body motion at high Reynolds numbers, wherein inertial effects dominates viscous forces 13,14. Later models also accounted for the elastic properties of the body and muscle activity 15-18. The recent advent of numerical methods coupled with the availability of fast, cheap computational resources has triggered a new generation of direct numerical simulations to accurately resolve the full three-dimensional problem 19-24. Although these provide detailed descriptions of the forces and flows during swimming, the large computational data sets associated with specific problems obscure the sea...

Gait and speed selection in slender inertial swimmers

“…2E we show that U=L = α a Ω=2π, where α a approaches the experimental value of ∼ 0:75 when the scaled amplitude of the swimmer approaches a = 4:75 (Supporting Information). Analyzing our results in terms of the Strouhal number St = λω=ð2πuÞ, which mixes the input variables λ; ω and output variable u, we find that St associated with the velocity peaks is also in agreement with experimental observations (25,36) that show that 0:1 K St K 0:4 (Supporting Information).…”

“…We start by showing that our theory is consistent with a recent comparative analysis of macroscopic swimming (25). In the laminar regime, balancing the propulsive term with the viscous drag in Eq.…”

“…For simplicity, we focus on 2D swimming gaits, neglecting 3D effects, an approximation that is effective in characterizing many of the salient aspects of aquatic locomotion (15,25). We model the fish as a neutrally buoyant, slender elastic sheet of density ρ s ,…”

Design and characterization of a miniature free-swimming robotic fish based on multi-material 3D printing

“…This equation is adapted and simplified from (Kopman et al 2015) for the case of straight swimming, neglecting the angle of attack and the body dynamic motions of yaw and sway. This equation should be considered as a first approximation of the tail beat undulation of live animals (Gazzola et al 2014). Therein, it was shown that a simple, monochromatic function could be used to accurately explain the dependence of swimming speed on tail beat frequency and amplitude, across a wide range of aquatic species from fish to mammals.…”

Coupling Motion and Energy Harvesting of Two Side-by-Side Flexible Plates in a 3D Uniform Flow

“…For instance, through its passive shape reconfiguration in the flow, a tree leaf can achieve substantial and beneficial drag reduction [1,2]. Similarly, a fish can exploit energy from the surrounding vortex street and achieve cost savings by undulating its body [3][4][5]. The famous experiment conducted by Liao et al revealed that a trout can decrease muscle activity by synchronizing its kinematics to incoming vortices appropriately [6].…”