Hydrodynamics of an actively heaving flexible foil under an incident surface wave

We numerically investigate the hydrodynamics of an actively heaving flexible foil flapping under a wave surface. The coupled level set and volume-of-fluid method is used to capture the air–water interface, and the immersed-boundary method is used to capture the fluid–structure interaction. A sinusoidal heaving motion is imposed at the foil’s leading edge, and its posterior parts oscillate passively according to its flexible characteristics, allowing dynamic interactions with the wave-induced flow. The propulsive performance of the foil is examined for the influence of three main factors: the ratio of the heaving frequency ( $f_{\!f}$ ) to the wave frequency ( $f_w$ ), the phase difference between the heaving motion and the incident wave ( $\mathit \varPhi$ ) and the submergence depth of the foil ( $D$ ). At $\mathit \varPhi = 0$ , the results reveal that the propulsion of the flexible foil benefits from flapping near the wave surface when $f_{\!f}/f_w = 0.5$ , and the propulsive efficiency is optimised at $D/L = 1$ , where $L$ is the foil’s length. However, when $f_{\!f}/f_w$ = 1.0 and 2.0, the propulsion of the flexible foil is hindered near the wave surface. This hydrodynamic hindrance is closely related to vortex splitting and roll-up phenomena, which induce the formation of a drag wake. By adjusting the phase difference $\mathit \varPhi$ , the hindrance in the flexible foil propulsion can be mitigated to enhance propulsive performance. To further understand the relationship between the flapping kinematics and propulsive dynamics, we perform a scaling analysis based on lift force and added mass force, offering good quantification of propulsive performance.