Analytical results for the propulsion performance of a flexible foil with prescribed pitching and heaving motions and passive small deflection

The propulsive performance of a flexible foil with prescribed pitching and heaving motions about any pivot point location and passive chordwise flexural deflection is analysed within the framework of the linear potential flow theory and the Euler–Bernoulli beam equation using a quartic approximation for the deflection. The amplitude of the flexural component of the deflection and its phase, the thrust force, input power and propulsive efficiency are computed analytically in terms of the stiffness and mass ratio of the plate, frequency, pivot point location and remaining kinematic parameters. It is found that the maximum flexural deflection amplitude, thrust and input power are related to the first fluid–structure natural frequency of the system, corresponding to the deflection approximation considered. The same relation is observed for the propulsive efficiency when an offset drag is included in the analytical expressions. These results, which are valid for small amplitude and sufficiently large stiffness of the foil, are compared favourably with previous related results when the foil pivots about the leading edge. The configurations generating maximum thrust and efficiency enhancement by flexibility are analysed in relation to those of an otherwise identical rigid foil.Funding for open access charge: Universidad de Málaga. This research has been supported by grants from the Ministerio de Ciencia e Innovación of Spain (DPI2016-76151-C2-1-R and PID2019-104938RB-I00) and from the Junta de Andalucía, Spain (UMA18-FEDER-JA-047 and P18-FR-1532)