The effects of the Reynolds number (Re) and thickness on an undulatory self-propelled foil were numerically investigated using the immersed boundary method. Re varied from 50 to 2 × 10⁵, which encompasses the viscous, intermediate, and inertial regimes using a NACA 0012 airfoil. An investigation of the thickness was performed on NACA airfoils with maximum thicknesses of 0.04 ∼ 0.24 at two Re values (5 × 10⁴ and 500). The results indicated that the foil can achieve a higher forward velocity, perform less work, and exhibit a higher propulsive efficiency with increasing Re. However, the effect of Re is asymptotic beyond 5 × 10⁴. Four types of vortex structures exist, and the transition from one regime to another is closely related to hydrodynamic changes. In the thickness study, thinner foils outperformed thicker foils in terms of the forward velocity and input power at both Re values. However, the efficiency related to the conversion of input power into kinetic energy for NACA 0004 was the lowest. An optimum thickness exists that depends on Re. At higher Re, the vortical structure differs for each thickness with the deflection angle, whereas at low Re, the location of the separation point strongly influences the hydrodynamics.

The impact of Strouhal number St (= 0.1–1.0), Reynolds number Re (= 50–2000) and dimensionless wavelength λ (= 0.5–2.0) on the hydrodynamic performance of a travelling wavy foil of a constant length is extensively investigated. The relationship of time-mean thrust with St, Re and λ is presented, suggesting that the propulsive force increases with increasing St, Re and λ. As such, the drag–thrust boundary advances as these parameters increase. A shorter λ makes the thrust steadier while a longer λ enhances the maximum instantaneous thrust. The latter is beneficial for prey to escape from a predator. The fluid added mass caused by the foil oscillation increases with St and λ but declines with Re (<500). Seven types of wake structures produced by the foil are identified, discussed and connected to thrust generation, showing how St, Re and λ affect the fluid dynamics, wake transition, vortex strength, wake jet, velocity, added mass, added damping, power input, efficiency and pressure profiles. The outcome of this work renders a physical basis for understanding the swimming of aquatic animals.

A numerical study is carried out to investigate the energy harvesting from an inverted foil undergoing flow-induced pitching oscillation for reduced velocity Ur  =  1–45 and damping ratio ζ = 0–0.295. The benchmark results with undamped foil (ζ = 0) indicate that the foil does not oscillate for Ur  ≤ 27 but does oscillate with increasing amplitude for 27 < Ur < 34 and with constant amplitude for Ur  ≥ 34. Lissajous diagrams of moment coefficient against the foil displacement are linked to the energy harvesting, showing how Ur and ζ affect the oscillating amplitude, reduced frequency, wake structures, and power exchange between the foil and the flow. The energy harvesting efficiency η up to 15.06% is achieved at Ur  =  37 and ζ = 0.130 with a reduced frequency f* = 0.151 that is used by the cruising aquatic animals. The foil oscillation with negative power enhances the growth of vortices while that with positive power weakens the growth.

This paper reviews recent developments in the understanding of underwater bio-mimetic propulsion. Two impressive models of underwater propulsion are considered: cruise and fast-start. First, we introduce the progression of bio-mimetic propulsion, especially underwater propulsion, where some primary conceptions are touched upon. Second, the understanding of flapping foils, considered as one of the most efficient cruise styles of aquatic animals, is introduced, where the effect of kinematics and the shape and flexibility of foils on generating thrust are elucidated respectively. Fast-start propulsion is always exhibited when predator behaviour occurs, and we provide an explicit introduction of corresponding zoological experiments and numerical simulations. We also provide some predictions about underwater bio-mimetic propulsion.