An axisymmetric boundary layer solution for an unsteady vortex above a plane

Abstract

A given axisymmetric potential swirling flow is bounded by a plane perpendicular to the axis of symmetry. By the no slip condition, viscous effects will be important over the plane and since the circumferential velocity must be zero on the axis of symmetry, viscous effects will also be important in the core of the vortex. These two viscous regions will overlap near the intersection of the axis of symmetry with the plane. Thus the flow field can be divided into four regimes, the viscous core, the boundary layer on the plane, a 'stagnation point' regime and the given potential flow which provides the outer boundary conditions for each of the first two regimes. Such a model could be useful for studying meteorological flow systems such as tornadoes. The viscous core regime has already been studied (Rott, 1958(Rott, , 1959Bellamy-Knights, 1970, 1971).

Stagnation point flow in a vortex core

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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 (1958Rott ( , 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).

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“…Certain types of vortex core have been studied by Rott (1958Rott ( , 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.…”

“…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.…”

Boundary layer dynamics of a decaying vortex core

“…The simultaneous existence of positive and negative vertical velocities results in stagnation at a point along the axis. This region is called the stagnation point region (Bellamy-Knights, 1974;Hatton, 1975). The viscous core region overlies the boundary layer.…”

“…The decaying viscous core was studied earlier by Oseen (19 1 l), Rott (1958Rott ( , 1959 and Bellamy-Knights (1970, 197 1). This core region is surrounded by potential flow which serves as an upper bound for the boundary layer flow at very large radii (Bellamy-Knights, 1974).…”