Numerical and experimental studies are conducted on flows developing between two concentric cylinders with vertical axes. The inner cylinder rotates and the outer and the lower end wall are fixed. The upper boundary is a free surface. The flow is at rest in an initial state, and the inner cylinder impulsively begins to rotate or its rotation speed linearly increases to a prescribed value. The acceleration rate of the inner cylinder changes the formation processes of flows and/or the final flow modes. Time-dependent flows appear at higher Reynolds numbers, and the numerical and experimental results of the power spectra show some agreements. It is suggested that critical Reynolds numbers appear, at which the fluctuations in the displacement of the free surface and the kinetic energy of a velocity component steeply increase.