  We present numerical simulations of vortices that appear via primary
bifurcations out of the unstructured circular Couette flow in the
Taylor-Couette system with counter-rotating as well as with co-rotating
cylinders. The full, time dependent Navier-Stokes equations are solved with a
combination of a finite difference and a Galerkin method for a fixed axial
periodicity length of the vortex patterns and for a finite system of aspect
ratio 12 with rigid nonrotating ends in a setup with radius ratio eta=0.5.
Differences in structure, dynamics, symmetry properties, bifurcation and
stability behavior between spiral vortices with azimuthal wave numbers M=+-1
and M=0 Taylor vortices are elucidated and compared in quantitative detail.
Simulations in axially periodic systems and in finite systems with stationary
rigid ends are compared with experimental spiral data. In a second part of the
paper we determine how the above listed properties of the M=-1,0,1 vortex
structures are changed by an externally imposed axial through-flow with
Reynolds numbers in the range -40 <= Re <= 40. Among others we investigate when
left handed or right handed spirals or toroidally closed vortices are
preferred.
