# Numerical Investigation of the Relationship between
## 3 Results
### Optimization of Anastomosis Angle
#### 3.1.2 Axial Velocity and Secondary Velocity  
The axial velocity profiles across the longitudinal section of the model across different anastomosis angles are shown in Figure 6. Linking the discussion to the pressure distributions observed, the velocity profile supported the idea of greater inflow resistance in larger anastomosis angles. At lower anastomosis angles, the flow achieved a faster velocity (orange) much earlier in the entry graft length compared to larger anastomosis angles, indicating less resistance to flow. The jump from orange to red values signified the graft inlet where the area was reduced due to the presence of the ridge, so the velocity increased based on the continuity equation. Following the pipe bend analogy, similar observations can be made regarding the velocity profile in the anastomosis region as those relating to the pressure profile. The flow separation observed in the pressure field was confirmed by the velocity profile, showing higher velocities at the host artery bed and a dead zone in the anastomosis toe. Likewise, \begin{table}
\begin{tabular}{c c} \hline \hline
**Angle (Degrees)** & **Total Pressure Drop (Pa)** \\ \hline
30 & 474.8 \\
45 & 498.39 \\
60 & 526.22 \\
75 & 554.61 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Total pressure drops experienced by blood, based on anastomosis angle. Figure 6: Axial velocity contours across the longitudinal section of the models. the same relationship between the anastomosis angle and velocity gradient was observed where a larger anastomosis angle resulted in higher velocity gradients. Furthermore, the axial velocity contours prove that spiral flow has indeed been developed as a result of the internal ridge within the graft. At the anastomosis, the blood flow shifted toward the artery bed, as indicated by the high-velocity region in the lower portion of the artery immediately after the anastomosis. Due to inertia, the flow was expected to stay within this region if the flow had been straight [34]. Looking further downstream, near the model outlet, the high axial velocity regions had instead shifted toward the upper region of the artery, which is only possible if the flow is spiral. Secondary velocity refers to the swirling characteristics of flow, consisting of the radial and tangential velocity components. Its magnitude is calculated using Equation (1). \[v_{secondary}=\sqrt{\left(v_{radial}\right)^{2}+\left(v_{tangential}\right)^{ 2}} \tag{1}\]  
A higher secondary velocity indicates the presence of greater spirality in flow, so it is expected to impart benefits to blood flow through higher, atheoprotective WSS and reduction in stagnation, recirculation, and turbulence [14; 15; 16; 17]. Figure 7 shows the secondary velocity contours and crossflow streamlines 50 mm from the anastomosis toe. The contours indicate the magnitude of the secondary velocity. Because of the spirality of flow, the overall bulk fluid motion was shifted from the center toward the upper region of the artery, causing asymmetry. This is consistent with the higher axial velocity regions in the upper region of the artery observed near the model outlet in Figure 6. This flow asymmetry brought the bulk motion nearer to the walls, allowing higher shear stress to be developed [25]. However, it is worth noting that compared to the axial velocity, the magnitude of the secondary velocity was much smaller. Meanwhile, the black lines in Figure 7 represent the surface streamline. This represents the paths through which zero particle masses would pass, and it is calculated using the Runge-Kutta method of vector variable integration [38]. Consistent with the secondary velocity profile, the cores of the surface streamlines were shifted toward the upper artery region. Single dominant spiral flows were observed in the 45\({}^{\circ}\), 60\({}^{\circ}\), and 75\({}^{\circ}\) grafts. Meanwhile, two concentric spiral flows, associated with significantly lower secondary velocity magnitudes, were observed for the 30\({}^{\circ}\) graft.