Abstract: The viscous flow fields in the inflow axisymmetric passages of centrifugalcompressors are predicted with vorticity- stream function (ω-ψ) method. The expressions of q-ω turbulence model equations in orthogonal curvilinear coordinate system are derived here, and they are used to predict the turbulent flow fields. The q-τ turbulence model has an advantage in capability of treating the boundary conditions. The unsteady terms are introduced in the τ-equation and q, τ equations, and the time-marching method is applied to solution of these equations. In the case of turbulent flow prediction the wall function is used in the near-wall region. Several numerical examples of laminar and turbulent flows in a 2D channel and axisymmetric passages with complex boundaries show that their solutions obtained with the present method is in good agreeement with their exact solution or their experimental data.