# Two-Phase Flow Simulations Using 1D Centerline-Based C- and U-Shaped Pipe Meshes
## 3 Results and Discussion
### Flow Physics in the C-Shaped Pipe (Case 1)  
#### 3.4.1 Flow and Bubble Redistribution in the C-Shaped Pipe  
We focused on Case 1 (C-shaped pipe with an upward injection) to understand the flow physics in the curve pipe. Hereafter, the \(\alpha_{air}~{}=~{}0.94\%\) was used for the simulation. Note that r/R is scaled to the range of \(-0.5\) and \(0.5\), and thus \(-0.5\) means the inner wall and \(0.5\) means the outer wall. Figure 16 demonstrates the axial water velocity profiles when the fluid flowed through the curve pipe for different curvature ratios of the (a) two- and (b) single-phase flows, respectively. Their maximum velocity magnitudes were also shown in Figure 16C. At the entrance of the curve inlet (\(0^{\circ}\)), the fluid was accelerated near the inner wall before \(15^{\circ}\) in both single- and two-phase flows by the pressure gradient for the curve region caused by a centripetal force. A greater velocity could be observed near the outer wall in both cases after \(15^{\circ}\) of the curve region (see Figure 16C), implying that the water tended to move in the outer wall instead of following the inner wall curve portion. The characteristics were much clearer in the curved part between \(45^{\circ}\) and \(60^{\circ}\). The shifted value of the maximum velocity magnitude from the curve portion of \(30^{\circ}\) was mainly caused by the influence of the inertia force. After the fluid passed the curve portion of \(60^{\circ}\), the maximum velocity magnitude tended to keep its radial position until it reached the curve  
Figure 15: Dynamic pressure of (**A**) air and (**B**) water phases for three different volume fractions in the case 1: C-shaped upward injection. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Cases**} & **Inlet-to-Curve** & **Curve Inlet-to-Curve** & **Curve Outlet-to-Outlet** & \multirow{2}{*}{**Total [Pa]**} \\  & **Inlet [N/m\({}^{3}\)]** & **Outlet [N/m\({}^{3}\)]** & & **[N/m\({}^{3}\)]** & \\ \hline S1 & 210 (375) & 233 (599) & 252 (450) & 696 (464) \\ S2 & 233 (417) & 233 (599) & 252 (450) & 720 (480) \\ S3 & 213 (381) & 233 (598) & 252 (451) & 700 (466) \\ S4 & 213 (381) & 233 (598) & 252 (450) & 699 (466) \\ T1 & 233 (416) & 243 (624) & 250 (447) & 727 (485) \\ T2 & 238 (426) & 258 (663) & 276 (493) & 773 (515) \\ T3 & 279 (499) & 289 (741) & 292 (521) & 860 (573) \\ T4 & 269 (481) & 287 (736) & 284 (507) & 840 (560) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Total pressure drops (pressure gradient, \(\frac{\Delta p}{L}\)) in the straight and curve pipes in all eight cases. outlet. The maximum axial velocity was considered to have fully developed around 60\({}^{\circ}\), regardless of the type of flow. Figure 17 illustrates the Dean flow [20] in a curve pipe at six different bend angles (i.e., 15\({}^{\circ}\), 30\({}^{\circ}\), 45\({}^{\circ}\), 60\({}^{\circ}\), 75\({}^{\circ}\), and 90\({}^{\circ}\)). The transverse Dean flows that arose as a result of the inertia force experienced by water moving along the curve portion offered an enhanced mixing in a 2D section by introducing a curvature to the flow path. The effect was generally characterized herein by a dimensionless number, called the Dean number, computed as follows:  
\[Dean\ =\ \frac{\sqrt{\frac{1}{2}(inertia\ force)(centripetal\ force)}}{viscous\ forces}=Re\sqrt{\frac{D}{2R_{c}}}, \tag{9}\]  
where \(R_{c}\) is the curvature-radius of the curve pipe. In our simulation cases, the Dean number was computed as 13,000, implying that inertia and centripetal forces were dominant over the viscous force. Note that the critical Dean number is 956 [21]. The inertial effects induced a secondary flow field characterized by the presence of two counter-rotating vortices (Figure 17) according to the curvature. In our case, the secondary flow was the strongest at the 30\({}^{\circ}\) bend angle and became weaker after the angle. The effect of the secondary flow became weaker when the water passed after the 60\({}^{\circ}\) bend angle (Figure 16).