Plane vortex flow in a cylindrical layer

Abstract

This paper presents brief analysis of publications dealing with the issues of experimental and theoretical studies of spiral fluid flows. The flows of this kind occur, in particular, in the vicinity of drain holes and are also observed in nature as storms and tornadoes. A mathematical modeling of a plane flow in a cylindrical layer has been carried out assuming that the viscous incompressible fluid is supplied along the normal to its outer surface and, accordingly, the vortex flow happens through its inner surface. The well-known general solution of the problem of vortex flow in unlimited space has been taken as a basis. A variant setting of the two boundary conditions for determining the velocity azimuthal component is proposed. The first boundary condition is the requirement for the azimuthal velocity component absence at the cylindrical layer entrance. The second, less obvious, boundary condition is adopted in the form when the strain rate tensor second invariant value is set at the cylindrical layer entrance. The rationale for such variant setting of the second boundary condition is given. Finally, it became possible to find an acсurate solution to the problem in a general setting. It is shown that at the layer exit, the velocity azimuthal component, which was initially absent at the layer entrance, can exceed the radial velocity component by an order of magnitude or more. A family of streamlines is constructed for the examined flow in the polar coordinate system. It is shown that, in the general case, the streamlines for the flow under consideration must have an inflection point. The numerical dependence of the inflection point position radial coordinate on the Reynolds number was obtained. It is also shown that in the cylindrical layer exit vicinity the fluid pressure takes on lower values than for the purely radial flow case. An alternative formulation of the second boundary condition for the pressure is presented for determining the integration constants in the azimuthal velocity component expression.