The objective is to investigate numerically the motion of a rotating fluid (confined in a finite-length circular cylinder) by computing the axisymmetric, unsteady Navier-Stokes equations. The overall work is in two parts, the development of the numerical method and its application. In the first three chapters (2 to 4), we deal with computational techniques, giving the Navier-Stokes equations in the form of stream function, vorticity, and angular momentum. The finite-difference approximations enable the numerical scheme to be of second-order accuracy both spatially and temporally. The ADI method is applied to solve the discrete vorticity and angular momentum equations, and the SOR method is adopted to solve the discrete stream function equation. We then discuss fundamental aspects of the numerical method, including the structure of the boundary layers, the design of a computational grid, a plan for changing the grid and time-step, and the validation of the method. The application of numerical computations is mainly involved in the following three aspects in the remaining chapters. First, we study the flow induced by counter-rotating the endwall(s) and/or sidewall, with an aim at seeking a steady-state solution of the Navier-Stokes equations. We find that the separated flow and "vortex breakdown"-like bubbles can appear in the meridional stream function structure of the flow when a steady state is approached. Second, we study the confined rotating flow during spin-down process, particularly during spin-down to rest. Details of the generation of Taylor-Gortler (TG) vortices are investigated in the temporal evolution of the flow, including the onset, onset time and growth of TG vortices, and the effect of the TG vortices on the torque of wall- surface, the angular momentum of container and the spectral amplitude of flow. Third, we study the flow during both spin-up and spin-down processes, where the phenomenon associated with recirculation zones is examined. The numerics show that the onset of recirculation zones always sets out somewhere on the central axis of rotation, and their transient behavior reveals similarities between spin-up and spin-down at early stages of the temporal evolution but differs subsequently. Finally, we discuss the reliability of the assumption of a midplane symmetry used in the axisymmetric flow computation and show that the symmetry can normally hold well in our applications.