Numerical model for fluid spin-up from rest in a partially-filled cylinder

A theoretical investigation is presented for the axisymmetric spin-up of fluid in a partially-filled cylindrical cavity. It is an extension of the earlier analyses of Wedemeyer, and of Goller and Ranov, to those cases where the liquid free surface intersects one or both endwalls. The simplifying assumptions of a columnar flow and the quasi-steady treatment of the Ekman layer pumping of the secondary flow are retained. Earlier estimates of the laminar Ekman layer pumping are modified heuristically for situations where the layer(s) no longer covers the entire wall. Also, due to the very steep free surface contour in the latter stages of spin-up, it was found advantageous to develop the free surface equations in an axial, rather than radial, coordinate frame. The equations are solved using a straightforward finite-difference algorithm. Numerical results for a range of Reynolds number, Froude number, and fill ratio are presented and discussed.