Span Efficiencies of Wings at Low Reynolds Numbers

Abstract

Elegant and inviscid analytical theory can predict the induced drag on lifting wings of finite span. The theoretical prediction is then often modified by multiplication with a dimensionless coefficient for which the departure from a value of 1 is used as a way to incorporate realistic and necessary departures from the idealized model. Unfortunately, there are conflicting definitions of these dimensionless coefficients, often known as span efficiencies, so that even if numerical values are assigned in a clear and transparent fashion, their application and validity remain unclear. Here, the differences between two commonly used definitions of span efficiency are identified and it is shown that for the case of airfoil sections and finite wings at chordwise Reynolds numbers less than 10 5 , neither one has values close to those commonly assumed in the aeronautics literature. The cause of these significant viscous modifications to inviscid theory is traced to the movement of separation points from the trailing edge of real airfoils. A modified nomenclature is suggested to reduce the likelihood of confusion, and appropriate formulations for the drag of streamlined bodies in viscous flows at moderate Reynolds number are considered, with application to small-scale flying devices, both natural and engineered.