A numerical investigation of oscillatory instability is presented for axisymmetric swirling flow in a closed cylinder with rotating top and bottom. The critical Reynolds number and frequency of the oscillations are evaluated as function of the ratio of angular velocities of the bottom and the top (xi=Omega(bottom)/Omega(top)). Earlier linear stability analysis (LSA) using the Galerkin spectral method by Gelfgat [Phys. Fluids, 8, 2614 (1996)] revealed that the curve of the critical Reynolds number behaves like an "S" around xi=0.54 in the co-rotation branch and around xi=-0.63 in the counter-rotation branch. Additional finite volume computations, however, did not show a clear S behavior. In order to check the existence of the S shape, computations are performed using an axisymmetric finite volume Navier-Stokes code at aspect ratios (lambda=H/R) 1.5 and 2.0. Comparisons with LSA at lambda=1.5 show that the S shape does exist. The S shape of the stability diagram predicted by LSA is thus confirmed by a finite-volume based Navier-Stokes solver. The additional computations at aspect ratio lambda=2 show that the curve of critical Reynolds number has a wider S shape in the co-rotating branch for xi about 0.7 whereas a sharp "beak" appears in the counter-rotating branch for xi approximately -0.5.