The stability of viscous axial flow in an annulus with a rotating inner cylinder

Abstract

Predictions by two methods are presented of the onset of instability in developed tangential flow in a concentric annulus due to inner cylinder rotation. The first formulation is as an initial-value problem in which the time evolution of initially-distributed small random vorticity perturbations of given axial wavelength is monitored by numerically integrating the unsteady perturbation equations by explicit finite-difference procedure. The second method is the Galerkin approach where an eigenvalue problem is formulated in which the linearized disturbance equations are solved to predict the neutral stability condition. Comparisons for a radius ratio N of 0.9 and Re up to 350 show that an averaged axial velocity distribution and the exact axial distribution yield similar predictions of Ta c and the corresponding critical wavelength; these however, differ markedly from previous narrow-gap predictions based on a parabolic approximation to the axial distribution. The current use of the exact developed tangential velocity distribution permits investigation by the Galerkin method for 0.9≽ N ≽ 0.1 and Re up to 2000. Computations of Ta c are in satisfactory agreement with earlier measurements for N of 0.95, 0.82 and 0.81 and accord well with current measurements over the range 50 ≼ Re ≼ 400 in an annulus of radius ratio 0.9.