# Two-Phase Flow Simulations Using 1D Centerline-Based C- and U-Shaped Pipe Meshes
## 1 Introduction  
Flows in pipes are most commonly observed in an industrial system (e.g., energy plant and refrigeration); hence, many studies on straight and curve pipelines for single- and two-phase flows have been presented. In straight pipes [1], the axial velocity gradient, called the primary flow, is the dominant property of the velocity field. In curve pipes [2, 3], the velocity field in the cross-section, called the secondary flow, is also an important characteristic in addition to the primary flow. Aside from the velocity field properties, curve pipes can also generate pressure gradients in the radial direction by flow inertia. A two-phase flow in a pipe [4, 5] appears in various industrial sites, such as steam pipes, boilers, refrigerant pipes, and refineries in power plants. In the case of a straight pipe with two-phase flows, the buoyancy caused by the density difference is the primary force determining the volume fraction of two-phase fluids at a low Reynolds number. In the case of a curve pipe with two-phase flows, the volume fraction of two-phase fluids could be determined by the balance between the centrifugal and buoyancy forces. Thus, investigating the effects of buoyancy and centrifugal forces on the two-phase flow is critical for understanding the flow physics in curve regions and the redistribution of volume fraction. Many studies on the curve pipe have been performed for the 90\({}^{\circ}\)- and 180\({}^{\circ}\)-curved cases. [PERSON] et al. [6] studied the two-phase flow for the right angle-bent pipe and found that the annular flow pattern was maintained in both before and after the curve pipe. [PERSON] et al. [7] studied the water-oil two-phase flow in the curve pipe, showing the optimal condition for reproducing the annular flow with less friction. Helical structures are also widely used at the industrial site, and two-phase flows frequently occur in thispipe type. In addition, [PERSON] et al. [8] performed experiments on the inner flow in a helical pipe with both single- and two-phase flows. They found that the fluid friction can be determined by the gas volume fraction at the two-phase flow, and drags are less than those observed for a single-phase flow. [PERSON] and [PERSON] [9] obtained both experimental and numerical results of two-phase flows in the helical pipe and utilized centrifugal force to separate the two phases. Meanwhile, [PERSON] [10] numerically investigated the bubbly flow in a U-shaped curve pipe. The secondary flow at the curve pipe was found to be less dominant in the two-phase flow compared to the single-phase flow. The secondary flow depends on the curvature of the pipe, which is a characteristic length of the pipe section [11]. Thus, the secondary flow that occurs in C- and U-shaped pipes causes a much more complex interphase interaction compared with that in L-shaped pipes. This phenomenon allows for unstable flow, which leads to danger at industrial sites. Several industrial applications of C- and U-shaped pipes have been investigated. For instance, [PERSON] et al. [12] studied the effect of U-shaped pipes on relative humidity and reported that a U-shaped pipe can reduce relative humidity by up to about 21%. Thus, the use of U-shaped pipe decreases the required work of heating, ventilation, and air conditioning (HVAC) systems by up to 647 W. Likewise, [PERSON] et al. [13] studied the performance of a C-shaped pipe in a typical HVAC system and concluded that the aspect ratio of the pipe could affect its performance in the heat exchanger. [PERSON] et al. [14] performed an experiment to investigate the flow behavior and pressure drop in air-water two-phase flows by imposing two opposite gravitational directions on a U-shaped pipe, i.e., a U-shaped pipe and an inverted-U-shaped pipe. They found that the inverted U-shaped pipe has a wider safety region, which is better than that of the U-shaped pipe at preventing drying out. Therefore, inverted-U-shaped pipes are preferable for industrial design.