Taylor-Couette flows in the annular region between rotating concentric cylinders are studied numerically to determine the combined effects of the co- and counter-rotation of the outer cylinder and the radius ratio on the system response. The computational procedure is based on a finite volume method using staggered grids. The axisymmetric conservative governing equations are solved using the SIMPLER algorithm. One considers the flow confined in a finite cavity with radius ratios η = 0.25, 0.5, 0.8 and 0.97. One has determined the critical points and properties for the bifurcation from the basic circular Couette flow (CCF) to the Taylor Vortex Flow (TVF) state. Indeed, the results are presented in terms of the critical Reynolds number Rei of the inner cylinder that depends on the rotational Reynolds number of the outer cylinder Reo and η. To show the capability of the present code, excellent quantitative agreement has been obtained between the calculations and previous experimental measurements for a wide range of radius ratios and rotation rates.