Vortex drift. II: The flow potential surrounding a drifting vortical region

Abstract

The potential flowfield surrounding a vortical flow domain is investigated. The vortical region is restricted to a three-dimensional finite domain in an unbounded incompressible viscous fluid that is at rest at infinity. The vector streamfunction of the flow is obtained by the integrals over the given vorticity field, expressing the Biot–Savart law. In the approach proposed here, the asymptotic vector streamfunction in the far field is expressed by integrals that represent the moments of the vorticity vector field. The asymptotic far-field potential is then determined by an integration that is analogous to the procedure well known in two dimensions, where the conjugate potential is found in the case that the streamfunction is known. In three dimensions, no general formula was found but an explicit expression for the asymptotic potential was constructed for the far-field dipole and quadrupole. The integrability conditions that permit the determination of the potential from the vector streamfunction are interpreted as the consequence of the requirement that the underlying vorticity field has to be source free. The asymptotic potentials are discussed first using only the inherent symmetries of the general dipole and the quadrupole fields. The results are then specialized to the case of axial symmetry. It will be shown that (i) the dipole strength is invariant and that (ii) the strength of the flow quadrupole remains constant while it moves together with the dipole with the drift speed already postulated in Part I. The role of the pressure in the flow generating process is also briefly noted.