A flexible body can interact with an incident water wave passively and extract wave energy to generate a thrust force. To understand the hydrodynamic performance and the wave energy extraction mechanism in this passive propulsion, we use direct numerical simulation to investigate the two-dimensional unsteady motion of a passive flexible foil with a fixed leading edge under an incident wave superposed with a current flow at a Reynolds number of $\mathit{O}\left(1000\right)$. The results of kinematics studies show that the passive flapping motion is up–down asymmetric, with the trailing-edge displacement having a larger magnitude above the fixed leading edge. The peak-to-peak trailing-edge amplitude is a strong function of the reduced frequency, and a maximum trailing-edge amplitude is observed, indicating a resonance phenomenon. Passive propulsion is achieved under the wave with a maximum cruising speed of $4.5$ times of the chord length per unit time at a Reynolds number of 1000. The foil reaches a peak propulsive efficiency between the Strouhal number $\left(\text{St}\right)$ values of 0.2 and 0.4. At low $\text{St}$ values, the scaling relationship of the mean thrust coefficient $\left(\overline{C_T}\right)$ and propulsion efficiency $\left({\eta }_e\right)$ with respect to $\text{St}$ are similar between the passive propulsion of flexible foils under waves and the active heaving and pitching of rigid foils in free streams, and $\overline{C_T}$ scales with ${\text{St}}^2$. At high $\text{St}$ values, $\overline{C_T}$ scales with ${\text{St}}^3$. When $\text{St}$ is greater than a threshold value between 0.2 and 0.27, a passing-over leading-edge vortex is generated, which is responsible for the $\overline{C_T}\sim {\text{St}}^3$ scaling and the enhanced thrust force generated at high $\text{St}$.

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• Swimming
• Vortex flows
• Wave-structure interactions
• Waves and free surface flows