Cruising speed of a cylindrical wing performing small translational-rotational oscillations

This work considers the propulsive motion of a flapping wing of a circular cross section. The problem of harmonic translational-rotational oscillations of the wing with an arbitrary phase shift in a viscous incompressible fluid, the motion of which is described by the non-stationary Navier–Stokes equation, is handled. An analytical solution of the problem is obtained in the first two terms by using the method of successive asymptotic expansions for the case of small oscillation amplitudes. It is shown that the nonlinear interaction of time harmonics of translational and rotational oscillations creates secondary flows that make the wing to move in the direction perpendicular to the axis of translational oscillations. For the cruising motion regime, when the average hydrodynamic force acting on the wing is equal to zero, the dependences of the average speed on the parameters of dimensionless oscillation are found.