Numerical predictions are presented of swirling turbulent flows with and without axial motion in a concentric annulus in which the core tube rotates about its axis. Conventional zero equation model and two equation model have been modified to account for the swirl effect on turbulent characteristics in this type of flows. This was achieved by introducing a flux Richardson number correction in appropriate terms in the governing turbulence equations. Computational results are in good agreement with experiments for a turbulent circular Couette flow, but when there is an axial motion of fluid through the annulus, the computational accuracy decreases with increasing swirl rate.