The Swirling Vortex

J. Serrin

Abstract

We consider an infinite vortex line in a viscous fluid interacting with a plane boundary surface at right angles to the line. If the boundary surface were absent, the vortex would impart to the fluid a circular motion about the vortex line with speed inversely proportional to the distance to the line. The presence of the boundary surface, however, leads to a secondary flow due to the forced adherence of the fluid at the surface. The purpose of the paper is to describe a family of exact solutions of the Navier-Stokes equations which applies to the above situation. Under quite general hypotheses, it is shown that there can exist only three types of motion compatible with the assumed structure. In the first kind, the radial velocity component (using spherical polar coordinates about the point where the vortex meets the plane surface) is directed inward along the plane surface and upward along the axis of the vortex. In the second type of motion the radial velocity component is directed inward along the plane surface and downward on the axis, with a compensating outflow at an intermediate angle. In the third kind the radial velocity is directed outward near the plane and downward on the central axis. The results can also be used as a basis for numerical calculations of the solutions in question, and several typical flow patterns have been explicitly computed in order to illustrate the theory. The paper concludes with a discussion of the relation between the theoretical solutions and observed phenomena near the point of contact of tornadoes with the ground; this requires that the flows under discussion be considered as mean motions in a turbulent flow with constant eddy viscosity. The present work adds theoretical weight to the argument that central downdrafts can occur in tornadoes. Moreover, the model provides an explanation, other than centrifugal action, for the frequent appearance of a cascade effect at the foot of both tornadoes and water-spouts; finally it offers a unified point of view from which to consider the diversity of flow patterns observed when vortex fields interact with a boundary surface.

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