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

Abstract

Abstract

Propulsion and energy harvesting performances of a flexible thin airfoil undergoing forced heaving motion with passive pitching and deformation of small amplitude

“…In addition, we shall consider the flexibility of the foil assuming a large stiffness, so that its deflection in relation to a rigid foil is also very small. Thus, the motion of the plate is governed by the Euler-Bernoulli beam equation, which in dimensionless form can be written as (e.g., Moore, 2017;Floryan and Rowley, 2018;Fernandez-Feria and Alaminos-Quesada, 2021) 2R…”

“…( 1), thus obtaining useful analytical solutions, but limited to small flexural deflections. This approach has been recently used for the propulsion problem of a foil with prescribed heaving and pitching motion and passive flexural deflection (Fernandez-Feria and Alaminos-Quesada, 2021), without considering the energy harvesting problem (i.e., with…”

“…The coefficients C L , C M and C F may be derived for a given harmonic kinematics of the foil using the vortex impulse theory within the linear potential flow limit (Fernandez-Feria, 2016;Alaminos-Quesada and Fernandez-Feria, 2020). Following (Fernandez-Feria and Alaminos-Quesada, 2021) we shall use a quartic polynomial approach for z s (x, t), which constitutes a minimal model of the flexible foil since for a lower polynomial approach the stiffness term in Eq. (1) vanishes, and, consequently, all the corresponding terms in the moment Eqs.…”

“…The following expressions for the coefficients corresponding to the quartic foil's deflection (11) are derived in Fernandez-Feria and Alaminos-Quesada (2021). Here are written in a more straightforward form in terms of h(t), α(t), d(t) and their temporal derivatives: where the superscript F refers to the contributions to these coefficients from the fluid-structure interaction (i.e., from C M and C F ).…”

“…Otherwise, the problem is formulated with the greatest generality: for any foil mass distribution and arbitrary pivot point location, so that any distance between the center of mass and the pivot point is allowed, any stiffness distribution, and of course, for any spring stiffness and operating frequency. To that end, general results recently obtained from the vortical impulse theory for the lift, thrust, moment and flexural moment exerted by the fluid on a foil undergoing arbitrary pitching and heaving motions coupled with arbitrary quartic flexural deformations are used (Alaminos-Quesada and Fernandez-Feria, 2020;Fernandez-Feria and Alaminos-Quesada, 2021). The limitation to (small amplitude) quartic deformations restricts the theory to include only the first resonant frequency of the system, but, as shown by Alben (2008) from a general linear potential flow theory, and by Moore (2015) including passive pitching like the present one, the maximum possible thrust coefficient is always achieved by the flexible foil operating at or near its first resonance.…”