A Simple Analytical Vortex Loop Model for the Unsteady Lift of Rotating Wings

Mathematical modeling of unsteady aerodynamics, including flapping wings, has gained popularity because of its application to micro air vehicle design and flight control. However, most of the existing models are not purely analytical, very expensive to implement, or two-dimensional. We develop a simple 3D theoretical model for the lift force on a rotating wing starting from rest using vortex loop dynamics. Experiments and simulations have shown that the rotating wing flow exhibits a large-scale loop structure around the wing, consisting of the leading-edge vortex, tip vortex, root vortex, and starting vortex. The fluid inertial force is derived using potential flow theory. The loop circulation is found from an impulse balance, and through algebraic manipulation, the circulatory lift force is established. This model is compared with six different experimental cases with various aspect ratios and starting motions, and produces a satisfactory match. Further, the formation time of the vortex loops is determined from the time-varying circulation, and compared with the universal formation time.