Stagnation point flow in a vortex core

Abstract

In an axisymmetric vortex which is in contact with a plane boundary perpendicular to the axis of symmetry, viscous effects will be significant in the vortex core and close to the boundary. Certain types of vortex core have been studied by Rott (1958, 1959) and Bellamy-Knights (1970, 1971), neglecting viscous effects near the boundary. The boundary layer region away from the axis of symmetry has been studied by Bellamy-Knights (1974). On the boundary and near the axis of symmetry, these two viscous regions will interact.

This paper sets out to unify the flows treated by Bellamy-Knights (1971, 1974), by obtaining similarity equations valid in this interaction region, which tend asymptotically to the boundary layer equations with increasing radius and also tend to the core equations with increasing axial distance from the boundary.

An exact solution of the unsteady, viscous, axisymmetric Navier–Stokes equations for an incompressible fluid, relevant to the core and interaction regions, is found as a special case of the solution of the general equations using a certain separation of variables. This reduces the Navier–Stokes equations to two coupled ordinary differential equations which are solved numerically.

The resulting solutions can be used to model the conditions existing at the central core of a tornado vortex. The direction of axial flow in such vortices has long been a matter of some controversy, there being observational evidence for both directions. The solutions described here have the interesting property that either is possible depending on the value of Ω, a parameter proportional to the angular velocity external to the boundary layer. In addition, in some circumstances, the existence of an axial stagnation point aloft is suggested.