In the present work a numerical study of transitional pulsatile flow through planar and cylindrical constrictions is presented. First, a simulation carefully coordinated with an experiment is carried out for validation purposes and results are in good agreement with the experiment. The parametric space that we adopted is similar to the one reported in a variety of past experiments relevant to the flow through stenosed arteries. In general, the flow just downstream of the constriction is dominated by the dynamics of the accelerating/decelerating jet that forms during each pulsatile cycle. It is found that the disturbance environment upstream of the stenosis has an effect on the spatial and temporal localization of the transition process in the post-stenotic area. The flow in the reattached area further downstream, is also affected by the jet dynamics. A 'synthetic', turbulent-like, wall-layer develops, and is constantly supported by streamwise vortices that originate from the spanwise instabilities of the large, coherent structures generated by the jet. The relation of these structures to the phase-averaged turbulent statistics and the turbulent kinetic energy budgets is discussed. The flow physics in the cylindrical configuration are qualitatively similar to those in the planar cases. The effect of blood rheology on the flow characteristics is also assessed by employing a biviscosity model in the simulations and it is found not to have a big effect on the turbulent intensities.