Aerodynamic Cost of Flapping

A new modeling approach is presented for mathematically describing the drag due to wing flapping. It is shown that there is an aerodynamic cost of flapping in terms of an increase in the drag when compared with non-flapping. The drag increase concerns the induced drag which results from lift generation. There are two effects that yield the induced drag increase caused by flapping. The first effect is due to changes in the direction of the lift vectors at the left and right wings during the flapping cycle by tilting them according to the flapping angle of the wings. Because of tilting the lift vectors, more lift has to be generated than is required for the vertical force balance in flapping flight. This lift enlargement causes an increase of the induced drag. The second effect that increases the induced drag is due to changes in the magnitude of the lift vector in the course of the flapping cycle. Changes in the magnitude of the lift vector are necessary for generating thrust which is required for the longitudinal force balance. As a result, both effects of lift vector changes cause a drag increase when compared with non-flapping. Solutions on an analytical basis and as well as results using a computational fluid dynamics method are presented.