Analysis of Flows in a Cylindrical Container with Rotating Bottom and Top Undeformable Free Surface

Abstract

Numerical study was conducted for steady swirling flows of viscous incompressible fluid confined within cylindrical containers driven by the bottom wall rotating at constant angular velocity with top free surface. The flow axisymmetry is assumed and the top wall is treated as an undeformable flat free surface. The physical parameters to govern the flow field are the Reynolds number Re defined by the angular velocity of the bottom wall and the radius to height container aspect ratio h. In the previous experimental and numerical investigations, richness of meridional flow patterns was observed and these secondary flow patterns are classified into several flow types. Numerically obtained steady solutions are synthetically analyzed by the estimation of major transport mechanism of momentum whose information is disregarded in the topological classification of the flow types for representative physical parameters, and they point to qualitatively different characteristics for the flows with small and large aspect ratio respectively. When h is small i.e., h«1, a substantial portion of the fluid inside the cylinder exhibits a near rigid body rotation and notable meridional fluid motion is limited to a region in the vicinity of the side wall. Richness of meridional flow patterns formed by multiple surface bubbles or cellular zones are shown to be created in the region surrounding the axis of rotation where the Taylor-Proudman theorem prevails. When h is large i.e., h»1 on the other hand, the outcome of the analysis indicates that the reduced quasi-cylindrical equations hold as a fair approximation to describe the momentum balance in the vicinity of the axis of rotation where recirculating axis bubbles are formed.