The occurrence of Taylor vortices between two conical cylinders of the inner one rotating and the outer one at rest is numerically simulated by the integration of time-dependent Navier-Stokes equations of the primitive variables. The motion is initiated by a progressive acceleration following a linear path of the inner conical cylinder rotation from the state of rest until its final rotation speed. In the integrated equations, the use of a coordinate transformation permitted to point out the apex angle alpha in the circular coordinates system and then to study the effects of alpha on the transient development of the Taylor-vortex structures, the steady states and the flow field characteristics. The numerical calculations have been achieved by the use of the SMAC method applied to a finite difference scheme for the case of a finite length. As a result, the symmetry of the Taylor vortices formed with regard to the mid plane for alpha = 0 does not exist anymore when the apex angle alpha becomes greater. The study of the flow field details showed that the contribution of the centrifugal forces becomes more important for increasing a due to the increase of the azimuthal component.