Global vorticity shedding for a shrinking cylinder

Abstract

We study numerically the viscous flow around a steadily moving two-dimensional cylinder undergoing a rapid reduction in its diameter as a model problem for force production through shape change which is encountered in the locomotion of certain animals. We consider first the case of a rapidly collapsing circular cylinder in steady translation, starting from an original diameter and reaching a final, smaller diameter under prescribed kinematics. We show that the difference in added mass energy is recovered by the body, and the boundary layer vorticity is reduced through annihilation with opposite-sign vorticity generated during the reduction phase. Next we consider a steadily moving circular cylinder which undergoes rapid but orderly melting, resulting in the same reduction of its diameter but which exhibits radically different flow patterns compared to the collapsing cylinder. The original vorticity in the boundary layer is shed instantaneously and globally in the fluid at the start of the melting phase, and then rapidly rolls up to form a pair of strong vortices, which contain the energy difference between the original and final cylinder states. The formation of the vortices in the melting cylinder takes less than a third of the time required by a rigid translating cylinder to form such vortices.