Spin-up from rest in a differentially rotating cylinder

Spin-up from rest in a cylinder with top and bottom end-wall disks rotating at different rates, denoted by Omega(T) and Omega(B), respectively, is investigated by integrating numerically the unsteady Navier-Stokes equations. The approximate analytical model developed by Wedemeyer (1964) and Venezian (1970) is outlined. Numerical solutions are presented for three sets of Omega(T)/Omega(B) for a cylinder of a given aspect ratio and a minute Ekman number. The results are detailed flowfield data displaying the transient azimuthal velocity profiles, axial vorticity distributions, and the meridional flow patterns. It is pointed out that an azimuthal velocity shear front that separates the rotating from the nonrotating fluid propagates radially inward. As Omega(T)/Omega(B) approaches unity, the propagation speed of the front increases and the spin-up in the interior is more effective. The axial vorticity is zero ahead of the front and is more than twice as large as the final-state angular speed of the fluid behind the front. As Omega(T)/Omega(B) deviates from unity, the meridional circulation is no longer antisymmetric about the cylinder mid-depth. The final-state meridional circulation in the interior comprises purely vertical flows pulled toward the faster rotating end-wall disk.