In this work, we consider the blood fluid-dynamics in the ascending aorta in presence of a normally functioning bicuspid aortic valve (BAV). In particular, we perform an unsteady finite element study in real geometries with physiological velocity boundary conditions at the inlet to assess the effect of the inclusion of the leaflets on the fluid-dynamic abnormalities characterizing BAV cases. To this aim, we perform a comparison in two geometries (a dilated and a non-dilated ones) among three scenarios which are built up for each geometry: BAV without leaflets, BAV with leaflets, and tricuspid case with leaflets. For each case, we compute four indices quantifying flow asymmetry, reversal flows, helical patterns, and wall shear stresses. Our results show that the inclusion of the leaflets increases the fluid-dynamics abnormalities, especially for the non-dilated configuration, which presents a greater increment of the indices. In particular, we observe that the values of the time-averaged wall shear stress and of the systolic jet asymmetry increase by approximatively 100 and 40 %, respectively, when considering the leaflets.