We report computational results of blood flow through a model of the human aortic arch and a vessel of actual diameter and length. A realistic pulsatile flow is used in all simulations. Calculations for bifurcation type vessels are also carried out and presented. Different mathematical methods for numerical solution of the fluid dynamics equations have been considered. The non-Newtonian behaviour of the human blood is investigated together with turbulence effects. A detailed time-dependent mathematical convergence test has been carried out. The results of computer simulations of the blood flow in vessels of three different geometries are presented: for pressure, strain rate and velocity component distributions we found significant disagreements between our results obtained with realistic non-Newtonian treatment of human blood and the widely used method in the literature: a simple Newtonian approximation. A significant increase of the strain rate and, as a result, the wall shear stress distribution, is found in the region of the aortic arch. Turbulent effects are found to be important, particularly in the case of bifurcation vessels.