Aquatic locomotion due to a flexible foil flapping in a perfect fluid

Can a fish-like body swim in a perfect fluid - one that is purely inviscid and does not release vorticity? This question was raised by Saffman over fifty years ago, and he provided a positive answer by demonstrating a possible solution for an inhomogeneous body. In this paper, we seek to determine a suitable deformation for oscillatory fish swimming that enables slight locomotion in a perfect fluid, relying solely on tail flapping motion. This swimming style, typical of carangiform and thunniform species, allows for a separate analysis of the tail's interaction with the surrounding fluid. As a preliminary approach, the tail is approximated as a rigid plate with prescribed heave and pitch motions, while the presence of a virtual body placed in front is considered to evaluate the locomotion. Analytical solutions provide exact results while avoiding singular behaviour at sharp edges. A phase shift is shown to be strictly necessary for generating locomotion. A more refined approximation of a real fish is achieved by modelling the tail as a flexible foil, connected to the main body via a torsional spring with tuneable stiffness at the peduncle. While the heave motion remains prescribed, the pitch amplitude and phase are passively determined by flow interaction. A plausible solution reveals an optimal stride length as a function of dimensionless stiffness, driven by resonance phenomena. A small structural damping must be considered to induce a phase shift - essential for self-propulsion in the absence of vorticity release.