Span Efficiencies of Wings at Low Reynolds Numbers

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 idealizedmodel. 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 andvalidity 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, neither one has values close to those commonly assumed in the aeronautics literature. The cause of these significant viscousmodifications to inviscid theory is traced to themovement of separation points from the trailing edge of real airfoils. Amodified 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.