Hydrodynamics of a self-propelled undulating hydrofoil at fixed submergence near a free surface

Bioinspired propulsion demonstrates broad potential for engineering applications. In this study, the hydrodynamic behaviour of an undulating hydrofoil operating at a fixed submergence depth near the free surface is numerically investigated using the smoothed particle hydrodynamics method. The body-wave wavelength lamda λ $\lambda$ is introduced to characterise the spatial pattern of the prescribed undulatory motion of the hydrofoil, and the effects of submergence depth and wavelength on propulsion performance are systematically analysed together with flow field structures. The results show that cruising speed decreases monotonically with decreasing submergence depth, whereas propulsion efficiency exhibits a non-monotonic trend, indicating an optimal submergence depth for maximum efficiency. Propulsion efficiency decreases monotonically with increasing wavelength, whereas, at shallow submergence depths, an optimal wavelength exists that maximises the cruising speed. When operating close to the free surface, the hydrofoil experiences an attractive force towards the surface; therefore, a negative angle of attack is required to maintain a fixed depth. Free-surface wave generation intensifies with decreasing submergence depth and increasing wavelength; however, under mild wave-breaking conditions, the cruising efficiency of near-surface hydrofoils may still exceed that achieved in deep-water operation. In addition, a scaling relationship between cruising speed, submergence depth and wavelength is obtained through curve fitting and can be used to predict cruising speed. These results provide useful guidance for the optimisation and control of undulating hydrofoils operating near the free surface.