# Two-Phase Flow Simulations Using 1D Centerline-Based C- and U-Shaped Pipe Meshes
## 4 Conclusions  
We employed herein an automated grid generation algorithm to observe the flow and pressure patterns at four different gravitational directions for the single- vs. two-phase flows in C- and U-shaped pipes. Both single- and two-phase flows followed the same trends with regards to static and hydrostatic pressures. In contrast, dynamic pressures have quite different distributions in the two-phase flows compared to the single-phase flows. In particular, the dynamic pressure is strongly affected by the direction of the buoyancy force. If the buoyancy force is aligned with the flow direction, the air phase velocity will increase and the water phase velocity will decrease for mass conservation. In addition, Dean flows are observed on the transverse plane of the curve regions in both C-shaped and  
Figure 19: Schematic of the turbulent kinetic energy (TKE) distribution of the (**A**) single- and (**B**) two-phase flows.  
Figure 20: Schematic of the (**A**) turbulent kinetic energy distribution in the 2D slice. (**B**) First fluctuating eigenmode (POD-derived coherent vortical structure) with an isosurface of \(\lambda_{2}=-0.01\). (**C**) Total pressure contour in the 2D slice.  
U-shaped pipes 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. Additionally, due to the influence of the inertia force, the water and air phases will switch positions when they pass the curve inlet. The inertia force also causes the secondary motion of the flow structure, becoming the strongest at around 30\({}^{\circ}\) of the curve bend. Furthermore, the strong TKE appeared near the inner walls of the similar region, where a strong secondary motion was captured. We further found that the TKE was greater in the two-phase flows than in the single-phase flows, implying that two-phase flows are more unstable. These following results could be often observed at various industrial sites. For instances, the complex interphase interaction caused by the secondary flow and volume fraction inside the C- and U-shaped pipes may lead to a dangerous situation at the nuclear energy field, such as the water hammer effect.  
In our study, the Reynolds number was fixed for all simulations, because we focused on two-phase flow physics rather than turbulent flow physics. Thus, the physics of turbulence in curved pipes with various Reynolds numbers should be examined in future research. In addition, we only validated water velocity because air velocity has not been reported in the benchmark case. Thus, a prospective study for both computation and experiment is required to measure air/water velocity and static pressure. More future studies may be recommended to consider the phase change phenomenon (e.g., water hammer) and to apply machine learning techniques [22; 23; 24].  
Conceptualization: T.Q.D.P., J.J., and S.C.; methodology, T.Q.D.P. and J.J.; software: T.Q.D.P., formal analysis: T.Q.D.P.; writing--original draft preparation: T.Q.D.P.; writing--review and editing: [PERSON], S.C.; supervision: S.C. All authors have read and agreed to the published version of the manuscript.  
This work was supported by Korea Electric Power Corporation (KEPCO) with the grant number of R18 XA06-67.  
Not applicable.  
The authors declare no conflict of interest.