The interaction of a diffusing line vortex and an aligned shear flow

Abstract

An exact solution of the Navier─Stokes equations of incompressible flow, which represents the interaction of a diffusing line vortex and a linear shear flow aligned so that initially the streamlines in the shear flow are parallel to the line vortex, is presented. If Γ is the circulation of the line vortex and v the kinematic viscosity then, when Re ═ Γ/2π v is large, the vorticity of the shear flow is expelled from the circular cylinder 0 < r ≪ ( vt ) 1/2 Re 1/3 , where r is the distance from the axis of the diffusing line vortex and t the time since initiation of the flow. At larger radii a peak vorticity 0.903Ω Re 1/3 is found at a radial distance 1.26( vt )1/2 Re 1/3 , where Ω is the initial uniform vorticity in the shear flow. The vortex filament is embedded in a growing cylinder from which vorticity has been expelled, the cylinder itself being bounded by an annular region of thickness of order Re 1/3 ( vt ) 1/2 in which the vorticity is of order Ω Re 1/3 .