Estimating lift from unsteady wakes of flapping wings

The Kutta-Joukowski (KJ) theorem usually leads to puzzling results when it is applied to estimating the lift from the unsteady wakes generated by flapping wings. We investigate this puzzling problem by using three different flapping wing models, where the unsteady wakes are obtained by numerically solving the Navier-Stokes equations at a low Reynolds number. It is found that neither the unsteady nor the time-averaged lift coefficient is correctly predicted when the parameters for the KJ theorem are selected according to the widely accepted models in the literature. We propose a wake-sectional KJ model to predict the time-averaged lift, where the spanwise distance between the streamwise vorticity centriods is computed as a effective span length. Furthermore, we quantitatively identified that the phase difference of unsteady lift is caused by the quasi-steady assumption. We show the phase difference can be corrected by using an added mass lift model. This work is helpful to clarify the errors in estimating the lift from the wakes in animal flight.