Comparison of the Average Lift Coefficient CL and Normalized Lift ηL for Evaluating Hovering and Forward Flapping Flight

The capability of flapping wings to generate lift is currently evaluated by using the lift coefficient CL, a dimensionless number that is derived from the basal equation that calculates the steady-state lift coefficient CL for fixed wings. In contrast to its simple and direct application to fixed wings, the equation for CL requires prior knowledge of the flow field along the wing span, which results in two integrations: along the wing span and over time. This paper proposes an alternate average normalized lift ηL that is easy to apply to hovering and forward flapping flight, does not require prior knowledge of the flow field, does not resort to calculus for its solution, and its lineage is close to the basal equation for steady state CL. Furthermore, the average normalized lift ηL converges to the legacy CL as the flapping frequency is reduced to zero (gliding flight). Its ease of use is illustrated by applying the average normalized lift ηL to the hovering and translating flapping flight of bumblebees. This application of the normalized lift is compared to the same application using two widely-accepted legacy average lift coefficients: the first CL as defined by Dudley and Ellington, and the second lift coefficient by Weis-Fogh. Furthermore, it is shown that the average normalized lift ηL has a physical meaning: that of the ratio of work exerted by the flapping wings onto the surrounding flow field and the kinetic energy available at the aerodynamic surfaces during the generation of lift. The working equation for the average normalized lift ηL is derived and is presented as a function of Strouhal number, St.