Steady flow in rapidly rotating variable-area rectangular ducts

Abstract

Steady liquid flow in a variable-area rectangular duct rotating rapidly about an axis perpendicular to its centre-line is treated. This problem is significant because the idea of driving a liquid through a rotating system has been largely overlooked by rotating-fluid dynamicists and because it closely resembles the flow inside the impellers of centrifugal pumps and hydraulic turbines. For a prototype formed by joining a semi-infinite constant-area duct and a semi-infinite duct with straight diverging walls, the flow in the diverging duct is carried entirely by large, O(E^1/3) velocities in a boundary layer of thickness O(E^1/3) adjacent to one of the (side) walls parallel to the axis of rotation, where E is the (small) Ekman number. With a vertical axis of rotation this high-velocity boundary layer is adjacent to the side wall on the right when facing in the flow direction. For a diverging or converging duct placed between two semi-infinite constant-area ducts, large, O(E^1/3) velocities occur in side-wall boundary layers on both sides of the variable-area duct and on the left and right sides of the upstream and downstream constant-area ducts respectively. The existence of high-velocity side layers in rapidly rotating rectangular ducts should be relatively easy to prove experimentally and actual measurements of their velocity profiles would provide a good test of the present theory.