Determination of the instantaneous forces on flapping wings from a localized fluid velocity field

Abstract

Expressions are derived to relate the instantaneous pressure force on a flapping wing to the velocity field on a plane at the trailing edge and on a highly localized region around and near the wing, valid when the vortex sheet is thin. In its more practical version, the formalism is applicable to wings with close to two-dimensional geometry and has the advantage of not using spatial derivatives, but only a time derivative of a surface integral of the velocity. In the purely two-dimensional case, the expression obtained is used to justify a much simpler one that only requires the evaluation of the time derivative of the wing circulation. A comparison with a numerical simulation in a two-dimensional case shows a good representation of the forces, even with the most simplified expression, when the condition of a thin wake is met. Other examples are shown in which the wake is not thin in order to explore the limitations of the formalism. It is found in these cases that the thrust is sometimes not so well reproduced, with a tendency to be overestimated, while the lift is generally better reproduced. Remarkably, the simpler expression reproduces rather acceptably the phase and amplitude of both thrust and lift in all cases. © 2011 American Institute of Physics.