The Ekman Matching Condition in a Partially Filled, Rapidly Rotating Cylinder

The use of an analytical matching condition in lieu of grid refinement and direct application of the no-slip boundary condition in a finite-difference calculation is considered for the case of a partially filled, rapidly rotating cylinder. The right circular cylinder is rotating fast enough that the liquid-air interface is nearly vertical. A non-wheel-flow velocity is induced by the differential rotation of the top lid. For the flow conditions of interest, the Ekman boundary layers on the horizontal surfaces are quite thin and their resolution using a very fine mesh makes the overall calculation very time-consuming and costly. We discuss the appropriate form of the Ekman matching condition, which has been widely used in rotating flow theory, to the case of a cylinder which is only partially full, and the fully implicit implementation of that condition into a MAC-derived, time-marching finite-difference calculation. The resulting algorithm is stable and efficient and the results compare quite well with calculations made using grid refinement and direct application of the no-slip condition and with recently published LDV measurements.