Vortex drift. I: Dynamic interpretation

Abstract

Vortical flow, restricted to a finite domain (in three dimensions) in an unbounded incompressible viscous fluid that is at rest at infinity, is investigated by the consideration of the dynamics in the potential flow region that surrounds the vortical domain. The evolution equations are considered for a flow that is given at an initial time t. The potential change in the far field is connected to the pressure, which in turn is expressed as the solution of a Poisson equation with sources distributed over the whole flow field. The leading term of the pressure at infinity is a quadrupole, which is caused by a drifting dipole field with a constant strength that is given by the impulse. This ‘‘dynamic’’ value of the drift is then identified with the classical ‘‘kinematic’’ definition as the speed of the impulse centroid. The main new result obtained by this method is the solution of the asymptotic drift problem in three dimensions, complementing the corresponding solution of Cantwell and Rott [Phys. Fluids 31, 3213 (1988)] for plane flow. The connection to the solution of the classical drift problem for a vortex ring is also established.