Optimal Efficiency and Heaving Velocity in Flapping Foil Propulsion

A novel model is proposed to estimate the optimal propulsive efficiency and the optimal dimensionless heaving velocity of a flapping foil based on a lift–drag ratio under the assumption of negligible pitching power. The optimal efficiency increases with the lift–drag ratio. The optimal Strouhal number of sinusoidal heaving motion approaches 0.318, within the range of 0.25–0.35 for optimal efficiency. In addition, the thrust generation of a foil requires that the product of the lift–drag ratio and the dimensionless heaving velocity must be greater than unity. To demonstrate how to use the new model to estimate the optimal propulsive efficiency and the optimal dimensionless heaving velocity, the lift–drag ratio is estimated using static tests of an impulsively started uniform flow over an infinitely thin flat plate for a range of angle of attack. To illustrate how to design kinematics to achieve the estimated optimal efficiency, two examples of flapping motions of a flat plate at a Reynolds number of 2000 are considered: sinusoidal and nonsinusoidal. Results show that the efficiency of nonsinusoidal flapping motion approaches the optimal efficiency using the lift–drag ratio estimated from static tests over the effective angle of attack of 6–30°.