Force Measurements On Rigid And Flexible Oscillating Foils

Abstract

In the present work, we experimentally study thrust generation from sinusoidally pitched rigid and flexible foils immersed in a uniform flow. The flexible foils are made by attaching a flexible flap of known flexural rigidity and flap length to the trailing edge of a rigid foil. For such thrust generating systems, a propulsive efficiency (η) may be defined as the ratio of the useful work done to the input energy requirement. In the present experiments, the propulsive efficiency (η) of the flapping foil can be determined from direct measurement of the unsteady forces and torque on the foil. The effects of systematic variation of the flexural rigidity of the foil, from highly flexible to rigid, on the thrust and efficiency characteristics of the foil are investigated. Studying such oscillating foils helps one to understand and mimic the efficient thrust generating mechanism in fishes and other creatures that use flapping to locomote themselves. A strain guage based loadcell is used to measure the forces normal to the foil (N) and forces along the chord of the foil (A). With a potentiometer, the instantaneous angular position (θ) is also measured, so that instantaneous lift (L) and thrust (T ) can be calculated. The measured moment (M) is used to calculate the instantaneous power input (P = Mθ˙). The foil is immersed in a uniform flow (u) set in a water tunnel, and the sinusoidal pitching (θ = θmaxsinωt) is provided by a servo motor. The Reynolds number (Re = uc/ν) in the present study is in the range of 103 to 104 . For the case of the rigid foil, the thrust and efficiency characteristics are presented for variation of the non-dimensional flapping frequency called the ‘reduced frequency’ (k = πfc/u), which is varied in the range of 1 to 10. At small reduced frequency (k < 3), the foil experiences a mean drag, while at k > 3, the foil experiences a mean thrust that grows rapidly as the reduced frequency (k) is increased. The thrust characteristics of the rigid foil are decided mainly by the normal force’s phase with respect to θ (φCN ) and its magnitude ([CN ]), as the chord-wise force is very small compared to the normal force (A << N). The measurements show that the non-dimensional mean thrust coefficient (CT ) scales as k2 and non-dimensional mean power (CP ) scales as k3 for k Ҳ 4. The maximum efficiency for rigid foils is found to be 8 % and it occurs at k 6. For the flexible foil case, the effect of making a portion of the total foil flexible by means of attaching a flexible flap of known flexural rigidity (EI) and flap length (cF ) to a rigid foil of length (cR) is studied. Unlike the rigid foils, the chordwise force (A) becomes an important factor in determining the thrust and efficiency characteristics of the flexible foils, due to the bending of the flap. We present results for a broad range of flexural rigidities from highly flexible flaps to stiff flaps, with the extent of flexibility fixed at cF /cR =0.8. We find that there is an optimal flexural rigidity for which the efficiency (η) reaches a maximum of 28 %. This represents a 250 % improvement compared to the rigid foil. The flexible foils with stiff flaps show a strange behavior with all the mean thrust coming from chordwise forces (A), unlike other flexible foils where the contribution to mean thrust come from both normal and chordwise forces. The effect of varying the extent of flexibility (cF/cR) with fixed flexural rigidity has also been studied. We define a non-dimensional flexibility parameter, R∗ = EI/(0.5ρu2sc3F ), which can combine the effect of variations in EI and cF /cR. Using this non-dimensional flexibility parameter (R∗), we find out that mean thrust and efficiency data for both the EI and cF/cR variation study collapse onto a single curve, indicating that R∗ can indeed be a single parameter characterizing flexibility. The present work shows that flexible foils can improve efficiency over rigid foils. Efficiency improvements can come in two ways depending on the R∗ of the flexible foil. Flexible foils with R∗ in the range of 10−2 to 100 show nearly 250% improvement in efficiency, accompanied by nearly 70 % loss in thrust compared to an entirely rigid foil of the same total chord. Flexible foils with R∗ in the range of 100 to 101 show nearly 50 % improvement in efficiency accompanied by nearly 100% increase in thrust.