Thrust generation from pitching foils with flexible trailing edge flaps

In the present experimental study, we investigate thrust production from a pitching flexible foil in a uniform flow. The flexible foils studied comprise a rigid foil in the front (chord length $c_{R}$ ) that is pitched sinusoidally at a frequency $f$ , with a flexible flap of length $c_{F}$ and flexural rigidity $EI$ attached to its trailing edge. We investigate thrust generation for a range of flexural rigidities ( $EI$ ) and flap length to total chord ratio ( $c_{F}/c$ ), with the mean thrust ( $\overline{C_{T}}$ ) and the efficiency of thrust generation ( $\unicode[STIX]{x1D702}$ ) being directly measured in each case. The thrust in the rigid foil cases, as expected, is found to be primarily due to the normal force on the rigid foil ( $\overline{C_{TN}}$ ) with the chordwise or axial thrust contribution ( $\overline{C_{TA}}$ ) being small and negative. In contrast, in the flexible foil cases, the axial contribution to thrust becomes important. We find that using a non-dimensional flexural rigidity parameter ( $R^{\ast }$ ) defined as $R^{\ast }=EI/(0.5\unicode[STIX]{x1D70C}U^{2}c_{F}^{3})$ appears to combine the independent effects of variations in $EI$ and $c_{F}/c$ at a given value of the reduced frequency ( $k=\unicode[STIX]{x03C0}fc/U$ ) for the range of $c_{F}/c$ values studied here ( $U$ is free-stream velocity; $\unicode[STIX]{x1D70C}$ is fluid density). At $k\approx 6$ , the peak mean thrust coefficient is found to be about 100 % higher than the rigid foil thrust, and occurs at $R^{\ast }$ value of approximately 8, while the peak efficiency is found to be approximately 300 % higher than the rigid foil efficiency and occurs at a distinctly different $R^{\ast }$ value of close to 0.01. Corresponding to these two optimal flexural rigidity parameter values, we find two distinct flap deflection shapes; the peak thrust corresponding to a mode 1 type simple bending of the flap with no inflection points, while the peak efficiency corresponds to a distinctly different deflection profile having an inflection point along the flap. The peak thrust condition is found to be close to the ‘resonance’ condition for the first mode natural frequency of the flexible flap in still water. In both these optimal cases, we find that it is the axial contribution to thrust that dominates ( $\overline{C_{TA}}\gg \overline{C_{TN}}$ ), in contrast to the rigid foil case. Particle image velocimetry (PIV) measurements for the flexible cases show significant differences in the