Optimal propulsion performance of an elastically mounted flexible pitching foil

Abstract

The fluid-structure interaction of an incompressible and inviscid current with a flexible foil elastically mounted at the leading edge and actuated by an oscillating torque is analyzed. The passive pitch and flexural deformation of the foil are obtained analytically through the successive moments of the Euler- Bernoulli beam equation using a linearized foil deformation model that allow to capture analytically the first two bending resonant frequencies and that associated to the torsional spring. Thrust and propulsive efficiency are then obtained analytically. It is found that the optimal aquatic propulsion is associated with the natural frequency of the torsional spring, but for a reduced frequency higher than the resonant one, of about 1.4, for a non-dimensional torsional spring constant of approximately 2.5, and with the propulsive efficiency slowly increasing with stiffness of the foil. For aerial propulsion, the optimal performance is found at the first natural bending frequency for a non-dimensional torsional spring constant of order of tens, and with an increase in the propulsive efficiency as the bending stiffness decreases, up to the validity limit of the present linear theory.