Numerical studies of fluid spin-up from rest in a partially-filled cylinder

This is the final report describing a 3 year theoretical investigation of the dynamics of fluid spin-up in a partially-filled cylindrical cavity. The effort was divided into two phases. The first phase consisted of developing 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 quasisteady treatment of the Ekman layer pumping of the secondary flow are retained. Earlier estimates of the 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 to advantageous to develop the free surface equations in an axial, rather than radial, coordinate frame. This model and the resulting computer code are the subject of an earlier interim report and a subsequent paper submitted for journal publication. Accordingly, only a summary of this phase and its results is presented here. No experimental or numerical data were available against which to compare the simplified model's predictions for cases where the free surface intersects one or both endwalls. Accordingly, in the second phase of the program a more refined numerical model was developed, based on the full nonlinear Navier-Stokes equations. This is a finite difference code in which the primitive variable equations are solved in conservative form. The development of the governing equations, and the semiimplicit predictor-corrector scheme used to solve them, are presented in some detail.