# In-Vitro Simulation of the Blood Flow in an Axisymmetric Abdominal Aortic Aneurysm
## 3 Results and Discussion  
The velocity profiles for the whole set of experiments are shown in Figure 4. These profiles refer to both the healthy aorta and the aneurysmatic dilatation in correspondence with its maximum diameter (section AA' of Figure 2); the analytical Poiseuille profile is plotted as well; we recall that when a velocity profile follows the parabolic law, the average velocity is known to be twice the maximum velocity. Profiles have been normalized by the maximum velocity measured in the healthy aorta, \(W_{\text{m}}\). In the healthy aorta, the agreement between the measured and the parabolic profiles is quite good for \(Re\leq 1100\) (cases 1, 2, 3); for \(Re\geq 1600\), the difference among these two curves increases with \(Re\). This trend was expected since the entry length \(l_{e}\) required for the flow to develop is a function of \(Re\); as stated by the expression \(l_{e}=0.056\times Re\times D\) proposed for axial pipe laminar flows [31]. Considering the length of the tube before the bulge, the estimated maximum \(Re\) that allows for a developed flow is found to be \(\sim\)1000. Furthermore, it must be considered that for \(Re\) below the onset of turbulence, a transition flow regime is, nevertheless, expected. The analysis of the profiles in the aneurysm evidences a retrograde flow; the magnitude of the peak (negative) retrograde velocity is found to increase with \(Re\). Noticeably, the profiles in the healthy aorta and in the aneurysm almost overlap for the highest \(Re\) values. Figure 3: Example of reconstructed trajectories, run 6. Figure 5 shows the streamlines and the distribution of the longitudinal velocity \(W\) calculated for the average flow field at various \(Re\). Basically, the flow pattern consists in a main flow channel with associated zones of low-velocity recirculation corresponding to flow detachment at the model inlet and reattachment at its distal end; as known, flow detachment is associated with positive pressure gradients. Figure 4: Normalized velocity profiles in correspondence of the maximum vessel diameter for various selected \(Re\): (**a**) \(Re\) = 410, (**b**) \(Re\) = 600, (**c**) \(Re\) = 1100, (**d**) \(Re\) = 1600, (**e**) \(Re\) = 2000, and (**f**) \(Re\) = 2650. Data are from the first set of experiments (ID 1 to 6, Table 1). The profiles refer to the healthy aorta (triangles), the aneurysmatic dilatation (circles) and the analytical Poiseuille profile (black continuous line); the x axis is normalized by the radius of the healthy aorta, \(r\). Flow direction is from up to down. Figure 5: Streamlines and longitudinal velocity (m/s) distribution of the average flow field for the selected \(Re\): (**a**) \(Re\) = 410, (**b**) \(Re\) = 600, (**c**) \(Re\) = 1100, (**d**) \(Re\) = 1600, (**e**) \(Re\) = 2000, and (**f**) \(Re\) = 2650. Data are from the second set of experiments (ID 1\({}^{\prime}\) to 6\({}^{\prime}\), Table 1). As a matter of fact, this region is believed to be a probable site of aneurysm rupture; nevertheless, the role of low WSS has to be taken into account to fully characterize the rupture event [14]. A vortex pattern characterized by retrograde flow with respect to the main flow channel lies in the recirculation zones. The motion of the fluid particles can be better visualized in the movies uploaded as Supplementary Materials showing the time evolution of the reconstructed particles trajectories for the runs at \(Re=1100\), \(2650\). In the bulge, particles are shown to follow elliptical type trajectories; as shown by several authors, i.e., in [9; 19; 20]. In analogous conditions, the center of the recirculation regions moves downstream toward the distal end of the aneurysm, and this displacement is found to increase with \(Re\). We have recovered an analogous behavior. Also, it must be noted that for increasing \(Re\), the flow tends to become unstable: several eddies form in the bulge, and their edges appear to be irregular. This behavior is clearly visible in the uploaded movies (movie_Re1100.avi, movie_Re2650.avi).