Time-Area-Averaged Momentum Stream Tube Model for Flapping Flight

Abstract

This paper formulates a time-area-averaged momentum stream tube model to provide the useful functional relations for the cruise velocity, power, and propulsive efficiency in forward flapping flight. This model is a direct generalization of the classical actuator disk theory by taking into account the effects of the unsteadiness and spatial nonuniformity of velocity in a momentum stream tube. It is found that the functional relation for the time-areaaveraged power required in flapping flight is almost the same as that given by the classical lifting-line theory for a fixed wing. The flapping span efficiency is introduced, and its physical meaning and significant role in the power relationareinterpreted.The flappingpropulsiveefficiencyisrelatedtothe flappingspanefficiency,normalizedtotal fluctuating kinetic energy, and wing aspect ratio. The use of this model to fit the collected data for birds allows estimating the flapping span efficiency, parasite, and induced drag coefficients and propulsive efficiency. Nomenclature A = effective area of actuator disk or cross-section area of momentum stream tube Ab = circular area of a diameter of wingspan AR = wing aspect ratio b = wingspan CDPara = parasite drag coefficient