Asymptotic study of the propulsive motion of a flapping cylindrical wing with an elliptical cross-section

The propulsive motion of a cylindrical flapping wing with an elliptical cross-section in a viscous incompressible fluid is investigated to develop an analytical model for predicting the cruising speed of such a wing without resorting to computationally expensive numerical methods. The mathematical formulation of the problem is based on the unsteady Navier–Stokes equations. The wing motion is described as a planar translational-rotational oscillation with prescribed velocities. The problem is solved using an asymptotic approach, under the assumption of high-frequency and low-amplitude oscillations. A structural formula is derived that describes the variation of the cruising speed in relation to the angle of translational oscillations, the phase shift between translational and rotational oscillations, the amplitude of rotational oscillations, and the aspect ratio of the elliptical cross-section. The consistency of the results with known analytical solutions for a circular cylinder is demonstrated. The limits of the model’s applicability with respect to the oscillation frequency are considered.