Effects of inertia on the time-averaged propulsive performance of a pitching and heaving foil

Abstract

The inertia may play an important role in the unsteady behavior of a flapping foil. Of particular relevance in forward flapping locomotion is how the foil inertia may affect its time-averaged propulsive performance. This is the question addressed by the present study from a general, nonlinear formulation of the unsteady interaction with the surrounding fluid of a thin, flexible, two-dimensional and non-uniform foil undergoing prescribed pitching and heaving motion of any amplitude about an arbitrary pivot axis. For a rigid foil it is shown that, although the unsteady inertial forces and moments may be much larger than the unsteady aerodynamic forces and moments if the fluid density is much smaller than the foil density, the inertia does not affect the cycle-averaged propulsive performance for harmonic pitching and heaving motion, independently of their amplitude. Inertia may affect only the cycle-averaged moment if the mean angle of attack is not zero, but without affecting the cycle-averaged power input, and therefore the propulsive efficiency. When a small flexural deflection of the foil is considered, although the cycle-averaged inertial thrust and lift also vanish for any amplitude of the pitching and heaving harmonic motion, the cycle-averaged power input does not. Thus, the foil inertia contributes through the moment and power to the time-averaged propulsive performance for any flexural deflection of the foil, obtained her e analytically in terms of the trailing edge location for general pitching and heaving motion of any amplitude and phase and small flexural deflection amplitude. Simple analytical results are also provided for inertia dominated deflection, characterizing the conditions that minimize the power consumption. The results are valid for arbitrary chord-wise distribution of mass, thickness and (sufficiently large) stiffness of the foil.