Hydrodynamics and wave energy extraction in passive propulsion of a flexible foil under water surface waves

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 O(1000). 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 (St) values of 0.2 and 0.4. At low St values, the scaling relationship of the mean thrust coefficient (C_T ) and propulsion efficiency (η_e ) with respect to 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 C_Tscales with St². At high St values, C_T scales with St³. When 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 C_T ∼ St³ scaling and the enhanced thrust force generated at high St.