Transient Starting Flow in a Cylinder With Counter-Rotating Endwall Disks

Abstract

Spin-up from rest in a cylinder with top and bottom endwall disks rotating in opposite directions (ΩT and ΩB are the respective rotation rate, but S[≡ ΩT /ΩB ] < 0) is investigated. The sidewall is fixed to the faster-rotating disk. A finite-difference numerical model is adopted to integrate the unsteady Navier-Stokes equations. We consider a cylinder of aspect ratio 0(1) and minute Ekman numbers. Numerical solutions are presented to show the transient azimuthal flow structures, axial vorticity profiles, and meridional flow patterns. An azimuthal velocity front, which separates the rotating from the nonrotating fluid, propagates radially inward from the sidewall. The appearance of the front is similar to the front propagation in spin-up in a rigid cylinder. As S decreases from zero, the direction of rotation in the bulk of the interior fluid becomes the same as that of the faster-rotating disk. The azimuthal velocities are still vertically uniform in the bulk of the interior. The scaled time to reach the steady state decreases. The angular velocities of the interior fluid near the central axis become very small. Under counter-rotation, the meridional circulation forms a two-cell structure. A stagnation point appears on the slower-rotating disk. During spin-up, the stagnation point moves from the sidewall to its steady-state position. As counter-rotation increases, the radial distance traveled by the stagnation point decreases.