Estimating lift from wake velocity data in flapping flight

The application of the Kutta–Joukowski (KJ) theorem to estimating the lift of a flying animal based on wake velocity fields often leads to significant underprediction of the lift, which is known as the wake momentum paradox. This work attempts to answer the puzzling question on whether the KJ theorem is legitimate in its use for complex viscous unsteady wakes generated by flapping wings. The limitations in applying the KJ theorem to flapping wings are quantitatively examined through numerical simulations of viscous incompressible flows over three flapping wing models. The three flapping wing models studied in this work are a flapping wing with a fixed wingspan, a flapping wing with a dynamically changing wingspan and a dihedral flapping wing. The KJ theorem fails to give a satisfactory prediction of the time-averaged lift unless an effective span length is correctly computed. We propose a wake-sectional Kutta–Joukowski (WS-KJ) model to predict the time-averaged lift, where the effective span length is computed based on the time-averaged distance between the streamwise vorticity centroids in the right and left half sides of the Trefftz plane. The WS-KJ model incorporates the spatial evolutionary effects of the complex vortex structures in the wake and significantly improves the prediction of the time-averaged lift. The physical foundation for such improvement is explored. In addition, the time-dependent amplitude and phase changes of the unsteady lift are discussed as the fluid acceleration effect.