Tailoring the bending pattern of non-uniformly flexible pitching hydrofoils enhances propulsive efficiency

We present new measurements of non-uniformly flexible pitching foils fabricated with a rigid leading section joined to a flexible trailing section. This construction enables us to vary the bending pattern and resonance condition of the foils independently. A novel effective flexibility, defined as the ratio of added mass forces to elastic forces, is proposed and shown to provide a scaling for the natural frequencies of the fluid-structural system. Foils with very flexible trailing sections of EI < 1.81 × 10 −5 N m 2 do not show a detectable resonance and are classified as ‘non-resonating’ as opposed to ‘resonating’ foils. Moreover, the non-resonating foils exhibit a novel bending pattern where the foil has a discontinuous hinge-like deflection instead of the smooth beam-like deflection of the resonating foils. Performance measurements reveal that both resonating and non-resonating foils can achieve high propulsive efficiencies of around 50% or more. It is discovered that non-uniformly flexible foils outperform their rigid and uniformly flexible counterparts, and that there is an optimal flexion ratio from 0.4 ⩽ λ ⩽ 0.7 that maximizes the efficiency. Furthermore, this optimal range coincides with the flexion ratios observed in nature. Performance is also compared under the same dimensionless flexural rigidity, R *, which highlights that at the same flexion ratio more flexible foils achieve higher peak efficiencies. Overall, to achieve high propulsive efficiency non-uniformly flexible hydrofoils should (1) oscillate above their first natural frequency, (2) have a flexion ratio in the range of 0.4 ⩽ λ ⩽ 0.7 and (3) have a small dimensionless rigidity at their optimal flexion ratio. Scaling laws for rigid pitching foils are found to be valid for non-uniformly flexible foils as long as the measured amplitude response is used and the deflection angle of the trailing section β is < 45°. This work provides guidance for the development of high-performance underwater vehicles using simple purely pitching bio-inspired propulsive drives.