Unsteady laminar flow between a pair of disks corotating in a fixed cylindrical enclosure

Abstract

The unsteady streamlined motion of a constant property fluid in the unobstructed space between a pair of disks corotating at angular velocity Ω in a fixed cylindrical enclosure is investigated numerically. Two-dimensional (axisymmetric) and three-dimensional calculations are performed using a second-order accurate time-explicit algorithm. The flow configuration corresponds to that investigated experimentally by Schuler et al. [Phys. Fluids A 2, 1760 (1990)]. The steady flow solutions are characterized by a symmetrical pair of counter-rotating toroidal vortices in the cross-stream (r-z) plane. This secondary motion is driven by the radial imbalance between the outward-directed centrifugal force and the inward-directed pressure gradient force. Axisymmetric calculations predict a flow that is steady for Re<22 200, where Re is the Reynolds number based on the disk radius, the tip speed of the disks, and the kinematic viscosity of the fluid. Above this value the motion is unsteady periodic and, while the features of the cross-stream flow pattern are broadly preserved, the symmetry of the motion about the midplane is broken by alternating periodic crossings of the toroidal vortices. This instability is maintained through an interaction that arises between outward-directed fluid in the disk Ekman layers and inward-directed fluid in the return core flow. Three-dimensional calculations at Re=22 200 and 44 400 show that the toroidal vortices acquire a time-varying sinuous shape in the circumferential direction. These calculations reveal circumferentially periodic reversals of the axial velocity component in the cross-stream plane, including the detached shear layer separating the region of motion in solid-body rotation near the hub from the potential core, in agreement with the flow visualization observations of Humphrey and Gor [Phys. Fluids A 5, 2438 (1993)]. The wavelength of this oscillation is shown to be twice that of the circumferential velocity component which is responsible for the nodal distribution of axial vorticity. When plotted on the interdisk midplane, the axial component of vorticity manifests itself as an even integer number, 2n (n=1,2,...), of circumferentially periodic foci. Experiments show that the number of foci decreases in a stepwise manner with increasing Reynolds number. For the conditions of this study, the calculated dimensionless angular velocity of the foci, ΩF/Ω, ranges from 0.55 at Re=22 200 to 0.44 at Re=44 400. These values are close to the present experimental estimate ΩF/Ω=0.5.