Propulsion performance analysis of an elastically mounted flexible pitching foil actuated by a given torque

Analytical models of flapping-foil propulsion are useful to understand key features of bio-inspired flapping-foil thrusters and to guide the tuning of the relevant parameters to improve propulsion performance. The present paper considers an analytical model of the fluid–structure interaction of a flexible foil elastically mounted, whose oscillation and deformation are passive when actuated by a given torque, valid for small amplitudes and for high Reynolds numbers, and accurate for a wide range of the stiffness parameter, down to values of order 0.1. Optimal aquatic propulsion of the flexible pitching foil is found related to the natural frequency of its torsional spring support, with a reduced frequency about 1.4 in all cases. The maximum efficiency, of about 40%, is reached for a rigid foil when the non-dimensional torsional spring constant is approximately 2.5, slightly decreasing as the flexibility increases. For aerial propulsion, a maximum efficiency slightly above 40% is found at the first natural bending frequency of the foil for a non-dimensional torsional spring constant of order of tens, with the efficiency increasing as the bending stiffness decreases down to the validity limit of the theory. Passive heave slightly improves thrust and efficiency but at the cost of increasing the complexity of the propulsion mechanism. Non-linear, flow separation and three-dimensional effects, not taken into account, will modify the optimal parameters, but the present analytical results can serve as a first approximation to guide the development of these efficient propellers.