In-line motion causes high thrust and efficiency in flapping foils that use power downstroke

We show experimentally that flapping foil kinematics consisting of a power downstroke and a feathering upstroke together with a properly timed in-line motion, similar to those employed in forelimb propulsion of sea turtles, can produce high thrust and be hydrodynamically as efficient as symmetrically flapping foils. The crucial parameter for such asymmetrically flapping foils is a properly sized and timed in-line motion, whose effect is quantified by a new parameter, the advance angle, defined as the angle of the foil trajectory with respect to the horizontal, evaluated at the middle of the power downstroke. We show, in particular, that optimal efficiency in high aspect ratio rigid foils, accompanied by significant thrust production, is obtained for Strouhal numbers in the range 0.2-0.6 for Reynolds number equal to 13,000, and for values of the advance angle around 0.55 pi (100 deg.). The optimized kinematics consist of the foil moving back axially during the downstroke, in the direction of the oncoming flow, and rotating with a large pitch angle. This causes the force vector to rotate and become nearly parallel to the steady flow, thus providing a large thrust and a smaller transverse force. During the upstroke, the foil is feathering while it moves axially forward, i.e. away from the vorticity shed during the power stroke; as a result, the transverse force remains relatively small and no large drag force is produced. Observations from turtles confirm qualitatively the findings from the foil experiments.