We derive general analytical expressions for the aerodynamic force and moment
on a flapping flexible foil undergoing a prescribed undulatory motion in a two-
dimensional, incompressible and linearized potential flow from the vortical impulse
theory. We consider a fairly broad class of foil motion, characterized by nine
non-dimensional parameters in addition to the reduced frequency. Quite simple
analytical expressions are obtained in the particular case when just a chordwise
flexure mode is superimposed to a pitching or heaving motion of the foil, for
which the optimal conditions generating a maximum thrust force and a maximum
propulsion efficiency are mapped in terms of the reduced frequency and the relative
amplitude and phase shift of the deflection of the foil. These results are discussed
in relation to the optimal conditions for a pitching or heaving rigid foil. The present
theoretical results are compared with available numerical data for some particular
undulatory motions of the flexible foil, with good agreement for small amplitudes of
the oscillations and sufficiently high Reynolds number.
