# Conservative differencing procedures for rotationally symmetric flow with swirl

## Abstract

When swirling motion is accompanied by a rapid radial component of flow toward the axis, numerical procedures for handling the coupling term in the azimuthal velocity equation become unstable. The difficulty is averted by replacing this equation with one which governs the angular momentum per unit mass. Moreover, since this is a physically conserved quantity, differencing schemes can be constructed directly from a balance law. Conservative schemes thus emerge automatically. In plane motion of an incompressible fluid, the vorticity is a pseudo-conserved quantity, i.e., it is governed by a differential equation which resembles one derived from a physical balance law. This is not the case for the vorticity of the meridional motion of a rotationally symmetric flow. However, a little known theorem due to the 19th century hydrodynamicist A.F. Svanberg demonstrates that the vorticity divided by the distance from the axis of symmetry is pseudo-conserved. This is useful for deriving differencing schemes governing the meridional motion. In the present paper these concepts are used to obtain prognostic procedures which are then applied to a problem of flow in a crystal-growing crucible. Crystal and crucible rotation provide the swirl and thermocapillary flow provides the inward radial motion which renders earlier procedures unsatisfactory. © 1981.