Scaling Laws for Caudal Fin Swimmers Incorporating Hydrodynamics, Kinematics, Morphology, and Scale Effects

TL;DR

This work develops a leading-edge vortex–based framework to derive scaling laws for thrust, power, efficiency, and cost of transport in carangiform caudal-fin swimmers across wide Reynolds and Strouhal ranges. Using high-fidelity DNS of a mackerel-inspired model, it links morphology and midline kinematics through new parameters such as $A'^*$ and $K_{morph}$ to predict performance and wake structure, and it validates the scaling against published data. The results reveal how LEV dynamics govern thrust, how wake topology evolves with $St_A$ and $Re$, and how scale interacts with kinematics and morphology to set optimal swimming speed and efficiency. The framework provides mechanistic insight for understanding biological locomotion and guiding bioinspired underwater vehicle design, with explicit guidance on how fin size, shape, and kinematic phase relationships should scale with body size to preserve performance.

Abstract

Many species of fish, as well as biorobotic underwater vehicles, employ body caudal fin propulsion, in which a wave-like body motion culminates in high-amplitude caudal fin oscillations to generate thrust. This study uses high fidelity simulations of a mackerel-inspired caudal fin swimmer across a wide range of Reynolds and Strouhal numbers to analyze the relationship between swimming kinematics and hydrodynamic forces. Central to this work is the derivation and use of a model for the leading edge vortex on the caudal fin. This vortex dominates the thrust production from the fin and the LEV model forms the basis for the derivation of scaling laws grounded in flow physics. Scaling laws are derived for thrust, power, efficiency, cost-of-transport, and swimming speed, and are parameterized using data from high fidelity simulations. These laws are validated against published simulation and experimental data, revealing several new kinematic and morphometric parameters that critically influence hydrodynamic performance. The results provide a mechanistic framework for understanding thrust generation, optimizing swimming performance, and assessing the effects of scale and morphology in aquatic locomotion of both fish and biorobotic underwater vehicles.

Figures (25)

• Figure 1: 3D fish model of a carangiform swimmer employed in the present study. The model is based on the common Mackerel (Scomber scombrus).
• Figure 2: Three-dimensional vortical structures around the swimming fish visualized by the iso-surface of the second invariant of velocity gradient, $Q=0.1f^2$, colored by the lateral velocity ($v$) at various Reynolds numbers. $\textrm{Re}_L=$ (a) 1000, (b) 2000, (c) 5000, (d) 10000, (e) 25000, (f) 50000.
• Figure 3: Evolution of the vortical structure in the wake of a swimming fish at $\textrm{Re}_L=10000$. The vortical structure is visualized by the iso-surface of $Q=10f^2$ colored by the normalized depthwise vorticity, $\omega_z/f$. $T=1/f$ is the tail-beat period.
• Figure 4: Characterization of the wake structure. $\lambda_w$: wake wavelength. $\theta_w$: wake spreading angle. The vortical structure is visualized by the iso-surface of $Q$ along with the lateral velocity contours. $\lambda_w/A_F=1/\textrm{St}_A$, $\theta_w=\tan^{-1}(\textrm{St}_A/2)$.
• Figure 5: Wake characteristics as a function of Strouhal number. (a) Wake wavelength, $\lambda_w$. (b) Wake spreading angle, $\theta_w$. Sold line: Present scaling law, Circle: Present DNS data, Square: Data measured from the results of Borazjani and Sotiropoulosborazjani2008numerical (Figs. 8B and 8C). Triangle: Measured from the result of Maertens et al.maertens2017optimal (Fig. 19(c)).