  In this paper, we investigate steady inviscid compressible flows with radial
symmetry in an annulus. The major concerns are transonic flows with or without
shocks. One of the main motivations is to elucidate the role played by the
angular velocity in the structure of steady inviscid compressible flows. We
give a complete classification of flow patterns in terms of boundary conditions
at the inner and outer circle. Due to the nonzero angular velocity, many new
flow patterns will appear. There exists accelerating or decelerating smooth
transonic flows in an annulus satisfying one side boundary conditions at the
inner or outer circle with all sonic points being nonexceptional and
noncharacteristically degenerate. More importantly, it is found that besides
the well-known supersonic-subsonic shock in a divergent nozzle as in the case
without angular velocity, there exists a supersonic-supersonic shock solution,
where the downstream state may change smoothly from supersonic to subsonic.
Furthermore, there exists a supersonic-sonic shock solution where the shock
circle and the sonic circle coincide, which is new and interesting.
