Propulsion of a foil undergoing a flapping undulatory motion from the impulse theory in the linear potential limit

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

“…Here we follow a different approach by using the three moment equations (2.4)–(2.6) to obtain the passive flexural motion of the foil at its lowest order for given heaving and pitching motions about . In a previous work (Alaminos-Quesada & Fernandez-Feria 2020) we used a quadratic approximation for such as to compute the coefficients , and in terms of the heaving, , pitching, , and flexural quadratic, , motions. In this foil's kinematics, , like and , are assumed small, which would be a valid approximation for sufficiently large stiffness of the foil.…”

“…As already mentioned, for the harmonic motion of the foil with quadratic deflection, i.e. when (2.7) with (2.15 a – c ) are used, the coefficients , and have been obtained analytically in terms of , and ((4.1), (4.6) and (4.8), respectively, in Alaminos-Quesada & Fernandez-Feria (2020), with and in the present notation, and now is assumed positive when counterclockwise). The addition of cubic and quartic terms in (2.8) obviously modifies these expressions, which are obtained here following the same procedure developed in Alaminos-Quesada & Fernandez-Feria (2020).…”

“…Technically, the approach may be considered as a low-order expansion, or minimal model of a flexible foil. For this kinematics with low-amplitude flexural motion, which structurally is valid when the stiffness of the foil is sufficiently large, we use analytical results derived very recently for the propulsive performance of a pitching and heaving foil with a small flexural deflection (Alaminos-Quesada & Fernandez-Feria 2020), computing the thrust force from a general vortical impulse formulation (Fernandez-Feria 2016). These kinematic-based results are coupled with the dynamics of the foil through the Euler–Bernoulli beam equation to compute the amplitude and the phase shift of the flexural deflection in terms of , and the kinematic parameters of the prescribed heaving and pitching motions about any given pivot point.…”

“…These kinematic-based results are coupled with the dynamics of the foil through the Euler–Bernoulli beam equation to compute the amplitude and the phase shift of the flexural deflection in terms of , and the kinematic parameters of the prescribed heaving and pitching motions about any given pivot point. The thrust, lift and moment exerted by the fluid on the foil are then obtained using the analytic general results for prescribed kinematics and deformation based on the impulse formulation (Alaminos-Quesada & Fernandez-Feria 2020). Thus, the present paper puts in a proper dynamical context the results in Alaminos-Quesada & Fernandez-Feria (2020), which considered just the kinematic problem, i.e.…”

“…The thrust, lift and moment exerted by the fluid on the foil are then obtained using the analytic general results for prescribed kinematics and deformation based on the impulse formulation (Alaminos-Quesada & Fernandez-Feria 2020). Thus, the present paper puts in a proper dynamical context the results in Alaminos-Quesada & Fernandez-Feria (2020), which considered just the kinematic problem, i.e. these former results provided the propulsion performance of a flapping foil when not only the heaving and pitching components of the motion were prescribed (forced), but also its flexural deformation as a quadratic deflection.…”