Undulatory underwater swimming: linking vortex dynamics, thrust and wake structure with a biorobotic fish

Abstract

Flapping-based propulsive systems rely on fluid–structure interactions to produce thrust. At intermediate and high Reynolds numbers, vortex formation and organisation in the wake of such systems are crucial for the generation of a propulsive force. In this work, we experimentally investigate the wake produced by a tethered robotic fish immersed in a water tunnel. By systematically varying the amplitude and frequency of the fish tail as well as the free stream speed, we are able to observe and characterise different vortex streets as a function of the Strouhal number. The produced wakes are three-dimensional and exhibit a classical V-shape, mainly with two oblique trains of vortex rings convecting outward. Using two-dimensional particle image velocimetry in the mid-span plane behind the fish and through extensive data processing of the velocity and vorticity fields, we demonstrate the strong couplings at place between vortex dynamics, thrust production and wake structure. The main results are twofold. First, by accounting for the obliqueness of the vortex trains, we quantify in experiments the evolution of vortex velocity components in both streamwise and transverse directions. We also measure key geometrical and dynamical properties such as wake angle, vortex ring orientation, diameter and vorticity. Remarkably, all of these quantities collapse onto master curves that also encompass data from previous studies. Second, we develop a quasi-two-dimensional model that incorporates both components of the momentum balance equation and introduces an effective spanwise thickness of the wake structure. This additional dimension, which scales with the physical thickness of the fish, captures the fine features of the three-dimensional wake. The model successfully explains the experimental master curves and highlights the links between vortex dynamics, thrust and wake geometry. Together, this framework offers a comprehensive understanding of the influence of the Strouhal number, providing universal insights relevant for both biological locomotion and bio-inspired propulsion systems.