### 1. Introduction

### 2. Computational Setup

### 2.1. Model Geometry

### 2.2 Governing Equation

*u*,

_{i}*ρ*,

*t*,

*x*,

_{i}*p*,

*μ*, and

*g*are the fluid velocity, fluid density, time, coordinate, pressure, kinematic viscosity of the fluid, and gravitational force, respectively. The Realizable

_{i}*k–∊*model was used to close the RANS equation throughout the study.

*E*,

*P*,

*R*,

*T*,

*e*, and

*C*are the energy, pressure, gas constant (287 J/kgK in this study), temperature, internal energy, and specific heat at a constant volume (0.717 kJ/kgK in this study)

_{v}*α*is the volume fraction of gas and

_{G}*α*is the volume fraction of the liquid. When the volume fraction of gas is 0 in the cell, it means it is in the liquid phase, where the volume fraction of liquid is 1. As a result, the gas-liquid interface exists in the cell where the volume fraction of gas is between 0 and 1.

_{L}### 2.3 Convergence and Turbulent Model Test

*Re*) of 325 was carried out, and the simulation results were compared with the available experimental data (Nogueira et al., 2006). The coarse and fine meshes were derived by decreasing and increasing the cell numbers per pipe diameter using a refinement factor (

_{UTB}*r*) of

_{k}*G*

_{1},

*G*

_{2}, and

*G*

_{3}, with a uniform parameter ratio chosen for

*R*), order of accuracy (

_{k}*P*), and grid uncertainty (

_{k}*U*) based on the Grid Convergence Index (GCI, Stern et al., 2006; Seo et al., 2016) were obtained using the Eqs. (8)–(10). where

_{k}*∊*

_{k}_{,}

_{G}_{12}=

*S*

_{k}_{,}

_{G}_{1}–

*S*

_{k}_{,}

_{G}_{2},

*∊*

_{k}_{,}

_{G}_{23}=

*S*

_{k}_{,}

_{G}_{2}–

*S*

_{k}_{,}

_{G}_{3}are the differences between coarse-base and base-fine solutions,

*Fs*is a factor of safety (1.25 was used in this study based on the recommendation by Roache (1998)), and

*p*

_{k}_{,}

*is an estimate of the limiting order of accuracy to 2. When*

_{est}*R*is in the range between 0 and 1, it is called the monotonic convergence.

_{k}*R*

_{k}_{,}

*= 0.333 (Table 2). The grid uncertainty(*

_{G}*U*

_{k}_{,}

*) of 1.540% also verified the monotonic convergence, which is when*

_{G}*U*

_{k}_{,}

*< 5.00 %. Based on the results of the grid convergence test, the medium mesh structure (*

_{G}*G*

_{2}) was chosen for further simulations in the study.

*R*

_{k}_{,}

*= 0.571 and time uncertainty with*

_{T}*U*

_{k}_{,}

*= 1.21% (Fig. 3). Both the grid convergence test and time convergence test results were in the monotonic convergence range, and the uncertainty was below 5.00%. Therefore, the base mesh system was chosen, and the time step was set to 1.00 ×10*

_{T}^{−3}regarding the computational time and accuracy of the result.

*k*−

*∊*model showed better agreement than the others. Therefore, the Realizable

*k*−

*∊*was used in this study.

### 2.4 Validation

*Re*= 325, Froud number (

_{UTB}*Fr*)) = 0.68, and Eotvos number (

_{UTB}*E*) = 167. Fig. 5 compares the simulation and the experimental result of the Taylor bubble shape and liquid film thickness;both are the main hydrodynamic features of the Taylor bubble mentioned by Araújo et al. (2012).

_{o}*g*,

*D*,

*C*

_{1}, and

*U*are the gravitational force, diameter, dimensionless coefficient, and mean flow, respectively. The value for the terminal velocity followed the theoretical value and experimental value well. The validation of the present numerical code was conducted around the main hydrodynamic features, such as the bubble shape, bubble terminal velocity, liquid film thickness, and liquid film velocity. All simulations showed good agreement with the experimental data.

_{m}### 2.5 Case Study

^{−4}m

^{3}, respectively, which are the same as the experimental data to investigate only the effects of the air injection velocity. The bending pipe was inevitable in the pipeline design for optimal space use under the limited design conditions. Therefore, the vertical pipe and the vertical to horizontal 90° bent pipe were investigated to examine the effects of geometry in the pipe. From Cases 6 to 10, the bent pipe was used to study the effect of geometry under the same conditions as the first five cases. In addition, the simulation considered the laminar, transient, and turbulent regimes by changing the water velocity while keeping the air injection velocity constant to compare the effect of the air injection velocity and the effect of the water velocity. Table 7 provides details of the simulation case.

### 3. Simulation Results

### 3.1 Slug Flow in a Vertical Pipe

*P*is the amplitude of the peak pressure, and

_{n}*A*and

_{n}*D*are the average and difference of the two successive peak pressure, respectively.

_{n}### 3.2 Slug Flow in a Bent Pipe

### 3.3 Effect of Water Velocity

### 4. Conclusion

The peak pressures were observed when the air was injected into the pipeline, and pressure oscillations were followed by fluctuations of the bubble length. The pressure oscillations occurred due to the compressibility of air, and the magnitude increased for high air injection velocities. Furthermore, the pressure oscillations showed the same trend for all the measurement positions because the bubbles affect the entire region of the pipeline.

The peak pressure showed the maximum at the PG2, which was located 20D from the inlet, and it increases with increasing air injection velocity. An opposite trend was shown after normalizing the pressure to the pressure coefficient using the air injection velocity. This means that the peak pressure is strongly affected at lower air injection velocities, because the bubbles can be compressed more at those velocities.

The pressure damping in the vertical pipe was investigated using the method to calculate the roll damping coefficient for the lateral motion of a ship. Damping is stronger at higher air injection velocities, while the distance from the inlet did not affect pressure damping.

When the pipe had a bend, the peak pressure showed a 1.5 times larger magnitude than that in the vertical pipeline in the same air and water injection velocity. The water has a stagnant point at the corner of the bend and causes more air compression in the bent pipe.

The compressibility of the air was also affected by the water velocity, and its effect was higher when the flow velocity was relatively small, showing a similar trend to the air injection velocity. The change in peak pressure with various water velocities was linear when the flow regime was laminar, but it fluctuated when the water velocity increased.