Propulsive characteristics of heaving membrane wings with passive pitching*

Membrane wings possess several aerodynamic advantages over rigid wings due to their passive adaptation to the surrounding flow. The dynamic and lift responses of heaving membrane wings depend strongly on the reduced frequency of the prescribed motion, and are strongly enhanced near resonance. However, the unsteady fluid load on the airfoil also produces an aerodynamic pitch moment that incites passive pitch oscillations. In pursuit of a model to evaluate the benefits of passively pitching, heaving membrane wings, a theoretical framework is developed for the propulsive characteristics of two-dimensional flapping membrane wings in an inviscid flow. An extensible membrane of small camber is assumed with a constant tension along its length. The membrane is attached to a torsional spring at the leading edge, where the heaving motion is prescribed. The aerodynamic load on the airfoil is obtained using unsteady thin airfoil theory, which is coupled with the membrane dynamic motion and the passive pitch oscillations of the airfoil. Closed-form expressions are derived for the thrust of passively pitching heaving membrane wings and are compared to the results available for rigid thin airfoils to assess the effect of membrane flexibility on the propulsion potential of flapping wings.