Passive and active bodies in vortex-street wakes

Abstract

We model the swimming of a finite body in a vortex street using vortex sheets distributed along the body and in a wake emanating from its trailing edge. We determine the magnitudes and distributions of vorticity and pressure loading on the body as functions of the strengths and spacings of the vortices. We then consider the motion of a flexible body clamped at its leading edge in the vortex street as a model for a flag in a vortex street and find alternating bands of thrust and drag for varying wavenumber. We consider a flexible body driven at its leading edge as a model for tail-fin swimming and determine optimal motions with respect to the phase between the body's trailing edge and the vortex street. For short bodies maximizing thrust or efficiency, we find maximum deflections shifted in phase by 90° from oncoming vortices. For long bodies, leading-edge driving should reach maximum amplitude when the vortices are phase-shifted from the trailing edge by 45° (to maximize thrust) and by 135° (to maximize efficiency). Optimal phases for intermediate lengths show smooth transitions between these values. The optimal motion of a body driven along its entire length is similar to that of the model tail fin driven only at its leading edge, but with an additional outward curvature near the leading edge. The similarity between optimal motions forced at the leading edge and all along the body supports the high performance attributed to fin-based motions.