Propulsive performance of a flapping plate near a free surface

The propulsive performance, i.e., the time-averaged thrust coefficient or the propulsive efficiency, of a flapping flat plate advancing near an otherwise quiescent free surface (liquid–gas interface) with Re of 1000, Fr of 0.2 and 0.8, and various submergence depths is numerically investigated by employing an adaptive Cartesian cut-cell/level-set method. The flapping kinematics parameters excluding the pitch-leading-heave phase angle were fixed as those commonly seen in literature. Results show that for submergence depth larger than the heave amplitude, the propulsive performance peaks at a smaller pitch-leading-heave phase angle with a shallower submergence for Fr of 0.2 but at the same phase angle for Fr of 0.8. Proximity to the free surface enhances the peak propulsive performance for Fr of 0.2 but the influence is minor for Fr of 0.8. The propulsive performance with Fr of 0.2 increases with decreasing chord-normalized submergence depth for the pitch-leading-heave phase angle smaller than 100°. The trend is reversed for the pitch-leading-heave phase angle larger than 100°. However, the propulsive performance with Fr of 0.8 hardly depends on the chord-normalized submergence depth. For submergence depth equal to the heave amplitude, the temporal variation in the thrust coefficient exhibits characteristics inherently different from those with other submergence depths for Fr of 0.2. Also, the time-averaged thrust coefficient exhibits a unique variation with the pitch-leading-heave phase angle. However, the various characteristics of the propulsive performance are similar to those with other submergence depths for Fr of 0.8. For submergence depth smaller than the heave amplitude and Fr of 0.2, the propulsive performance gains much from exposure of the upper surface of the plate to the gas phase. The efficiency enhancement is linked to the weakening of the leading edge vortices. A second harmonic with significant amplitude is found in the upstream wave for Fr of 0.2 with a typical pitch-leading-heave phase angle.