This work focuses on the direct numerical simulation of the pulsatile flow through a bileaflet mechanical heart valve under physiological conditions and in a realistic aortic root geometry. The motion of the valve leaflets has been computed from the forces exerted by the fluid on the structure both being considered as a single dynamical system. To this purpose the immersed boundary method, combined with a fluid–structure interaction algorithm, has shown to be an inexpensive and accurate technique for such complex flows. Several complete flow cycles have been simulated in order to collect enough phase-averaged statistics, and the results are in good agreement with experimental data obtained for a similar configuration. The flow analysis, strongly relying on the data accessibility provided by the numerical simulation, shows how some features of the leaflets motion depend on the flow dynamics and that the criteria for the red cell damages caused by the valve need to be formulated using very detailed analysis. In particular, it is shown that the standard Eulerian computation of the Reynolds stresses, usually employed to assess the risk of haemolysis, might not be adequate on several counts: (i) Reynolds stresses are only one part of the solicitation, the other part being the viscous stresses, (ii) the characteristic scales of the two solicitations are very different and the Reynolds stresses act on lengths much larger than the red cells diameter and (iii) the Eulerian zonal assessment of the stresses completely misses the information of time exposure to the solicitation which is a fundamental ingredient for the phenomenon of haemolysis. Accordingly, the trajectories of several fluid particles have been tracked in a Lagrangian way and the pointwise instantaneous viscous stress tensor has been computed along the paths. The tensor has been then reduced to an equivalent scalar using the von Mises criterion, and the blood damage index has been evaluated following Grigioni et al. (Biomech. Model Mechanobiol., vol. 4, 2005, p. 249).