A numerical dye method for the visualization of unsteady three-dimensional flow calculations is introduced by coupling the unsteady convection-diffusion equation to the Navier-Stokes equation for mass and momentum. This system of equations is descretized using a finite volume projection-like algorithm with generalized coordinates and overset grids. A powerful pressure prediction method is used to accelerate the convergence of the Pressure Poisson equation. To demonstrate the visualization technique, blood flow through the aortic arch region and the three main arterial branches is computed using various Womersley numbers. In this technique, parcels of fluid are followed in time as a function of the cardiac cycle without having to track individual particles, which in turn aids us to better understand some important aspects of the three-dimensionality of the developing unsteady flow. Using this numerical dye method we analyze the strength of the cross flow during the cardiac cycle, the relationship between the penetration of blood into the aortic branches from its relative position in the ascending aortic region and the effects of the Womersley parameter. This technique can be very useful in the design and development of stents where the topology of the device would require understanding where the blood emanating from the heart ends up at the end of the cardiac cycle. Moreover, this method could be useful in investigating the influence of flow and geometry on the local introduction of medication.