# CFD modeling of liquid film reversal of two-phase flow in vertical pipes

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## Abstract

The phenomenon of two-phase flows is characterized by its wide range of presence in nature and industrial applications. In gas and oil production, the simultaneous two-phase flow of gases and liquids often occurs in gas wells. The produced gases flow rate decreases over the years until reaching a critical flow rate when they lose their ability to lift the associated liquids upward and the liquid film reverses direction, which triggers liquid loading. Liquid loading causes the produced liquids to accumulate in the bottom of the wellbore, which causes a high back pressure that reduces the well production rate till production is ceased eventually. The critical gas velocity exists in the churn flow regime which is mainly characterized by the oscillatory behavior of the liquids flow field. This study employs CFD techniques to model the churn flow in a 3-inch-diameter vertical pipe near the critical gas flow rate for different liquid flow rates. This work utilizes the two-fluid Eulerian model along with the RNG (Re-Normalization Group) k-ε turbulence model to investigate the behavior of the flow field in a two-dimensional axisymmetric computational domain. Simulations were carried out using the commercial software ANSYS Fluent 18.2 with air and water as the two working fluids. The model results showed a good agreement with the experimental data and proved the mesh and time independence of the model. Oscillatory behaviors of the liquid film flow rate, shear stress, and pressure were observed along with the formation of interfacial waves. Detailed information about the velocity, shear stress, and pressure behaviors is presented. Accordingly, recommendations are suggested for future considerations.

## Keywords

CFD Two-phase flow Churn flow Vertical pipes Film reversal## Introduction

Research in the area of two-phase flows continues to be one of the hot research areas due to its wide presence in nature and a broad range of industrial applications. Liquid–gas two-phase flow is widely present in gas wells where gases and liquids flow simultaneously. Most of the time, the present flow regime in a given gas well is the annular flow where gases flow at high velocity in the core region while liquids form a film flowing on the pipe wall. In the upward vertical two-phase flow in a typical gas well, the pressure difference along the pipe is reduced gradually over time which, in turn, reduces the gas velocity. The gas velocity is kept in continuous reduction till it reaches a velocity where the gases cannot drive the liquids to flow up. Afterward, the liquids start falling and accumulating in the wellbore which progressively reduces the liquids production and eventually results in blocking the flow channel. This imposes a back pressure as the accumulated liquids impeding the flowing of gases. Accordingly, the flow in the wellbore is no more an annular flow and high fluctuations and drop in the production rate are observed. The production rate fluctuation and sudden drop are the first common symptom of what is called “liquid loading” that can be observed and measured. However, the onset of this phenomenon occurs beforehand and is not usually predicted.

The initiation of this entire process that eventually leads to loading the gas well is the reversing of the liquid flow direction from upward to starting falling down, the onset of film reversal. At the onset of film reversal, part of the liquid film flows upward, while the remaining liquids keep flowing downward. Eventually, and after the gas velocity reaches the critical velocity, the entire liquid film starts flowing downward. The present flow regime is the annular flow and its transition to churn flow. Two major methods have been approached to determine the gas critical velocity: entrained droplet movement and liquid film stability.

*U*

_{SG,cr}is the critical gas velocity and \(\rho\) donates the density. An initial look at Eq. (1) indicates that the critical velocity depends on the surface tension, gas density, and liquid density. It has been suggested to apply an upward adjustment of 20% because the authors found that the model underestimates the minimum gas flow rate. By applying such an adjustment, additional 11 wells best fitted to the entrained drop transport model. Nevertheless, Turner et al.’s model had the disadvantage of assuming that the droplet shape is perfectly spherical and does not change during the flow. Moreover, the majority of the data fitted to the model were for wells having high wellhead pressure, mainly above 1000 psi.

Coleman et al. (1991) tested the entrained drop transport model of Turner et al. on the field data of gas wells with low wellhead pressure, typically below 500 psi. They concluded that the model accurately predicts the critical gas velocity without the need of the 20% adjustment for low wellhead pressure wells. Moreover, Coleman et al. reported, according to their work, that the results are significantly impacted by the pressure and wellbore diameter, while parameters such as temperature, gravity, and interfacial surface tension have negligible effects.

Nosseir et al. (1997) argued that the reason why Turner et al.’s model had some discrepancies is that they did not take the flow regime into consideration. The authors considered two flow regimes: transition flow regime and highly turbulent flow regime. Each of the two regimes has a different expression of the critical gas velocity. For the highly turbulent flow, the critical velocity reported by Nosseir et al. is Turner et al.’s equation but adjusted by 21%. Nosseir et al. also reported that adjusting Eq. (1) by 21% reduced the absolute cumulative error from 23.5 to 8.3%.

Unlike Turner et al. (1969), Li et al. (2001) did not assume a spherical shape of a liquid drop. Instead, they suggested that an entrained droplet has the shape of an ellipsoid on which their proposed model is based. The reason for such an assumption is that the pressure difference, under high gas velocity, between the aft and fore of the droplet results in a deformation which converts the spherical droplet into an ellipsoid. Consequently, the drag coefficient, *C*_{D}, would be unity unlike the value proposed by Turner et al., 0.44. Hence, Li et al.’s model resulted in a lower critical gas velocity formula, which was validated by comparison with 16 gas wells data in China. However, this model did not consider possible coalescence or separation of droplets.

In (2005), Guo et al. claimed that the model of Turner et al. (1969) did not predict the critical gas flow rate accurately even with the 20% adjustment because it neglected the bottom hole conditions which are more dominant than the wellhead conditions. They developed a new method based on the minimum kinetic energy criterion which requires the gas to reach a minimum value of kinetic energy to carry the droplets up. As a result, they came up with a set of equations which can be solved in an iterative manner to obtain the critical gas flow rate.

Experimentally, Westende et al. (2007) measured the size of droplets at the onset of film reversal using a Phase Doppler Anemometry (PDA). They used a 5-cm-diameter pipe and investigated annular and churn two-phase flow with air and water as the two phases. The authors did not observe any droplets falling, which makes the physical background on which Turner et al. model is based doubtful. Hence, this led the authors to conclude that film instability is what characterizes liquid loading. As previous works did not consider the liquid holdup (the amount of liquid in a gas stream) in their analysis, Zhou and Yuan (2010) proposed a new empirical model that is dependent on the liquid holdup as a major factor in liquid loading. The authors used the gas wells’ data provided by Turner et al. (1969) and also used data from Coleman et al. (1991) for validation. The model of Zhou and Yuan consists of a set of two equations with each equation being valid for a specific range of liquid holdup values.

Fadairo et al. (2015) proposed a four-phase gas–oil–water–solid mist flow model and compared it to the field data. The study showed that the model fitted more data points than those fitted by (Taitel et al. 1980; Anderson and Jackson 1967). Later, Bolujo et al. (2017) proposed another improved four-phase model, which fitted more data points than Fadairo et al.’s model.

Other researchers characterized liquid loading by the liquid film stability. Zabaras et al. (1986) were experimentally able to study the two-phase annular flow and monitoring the film flow. They observed that the liquid film reversed its direction at low gas velocity while the flow was completely co-current at high gas velocity. The wall shear stress, liquid film thickness, and pressure gradient were constantly monitored. Furthermore, they noticed that the shear stress direction is downward when the film was flowing upward, while it was periodically changing its direction after the film reversal.

Belt (2007) studied the phenomenon of the film reversal in a 0.05-m-diameter pipe using water and air as the two phases. He concentrated on studying (1) the interfacial friction as it is the dominant driving force for the liquid film to move upward and (2) explaining the secondary flow in the gas core. Belt (2007) concluded that the roll waves play a significant role in determining the film thickness.

Guner (2012) experimentally investigated the initiation of liquid loading in 3-inch 0°, 15°, 30° and 45° deviated pipes. The author measured the liquid holdup and pressure gradient and adopted the concept of film reversal in characterizing liquid loading. Moreover, she conducted a Computational Fluid Dynamics (CFD) simulation for the vertical pipe case using the commercial software ANSYS^{®} Fluent. Main recommendations included improving the inlet geometry of the CFD simulation, modifying the numerical scheme in the Volume of Fluid (VOF) method to Geo-reconstruction and considering the surface tension in the VOF model.

Similar to the work of Guner (2012), Alsaadi et al. (2015) conducted experiments to investigate the initiation of liquid loading but for a highly deviated 3-inch pipe (60°–88°). They measured the pressure gradient and the liquid holdup for different liquid and gas velocities and determined the critical gas velocity for each case. The author concluded that the critical gas velocity was not always at the point where the pressure gradient is minimum.

Most of the researchers tend to use empirical and mechanistic models to model the key parameters of the flow, such as the pressure drop and the average velocities, due to the simplicity of such approaches compared to others. With the current advances in CFD, numerical simulations provide numerous flow details which are not captured by mechanistic models. Jayanti and Hewitt (1997) employed CFD techniques to estimate the flow field in a liquid wavy film of an annular flow. The study considered a liquid film with a fixed interface profile (including a disturbance sinusoidal wave) on the one side, while bounded on the other side by a moving wall at the speed of a disturbance wave in the direction opposite to the flow direction. The boundary condition at the interface was a fixed interfacial shear stress profile obtained from correlations, whereas the inlet and outlet are constrained by periodic boundary conditions. The standard k-ε and the low Reynolds number k-ε turbulence models were implemented. Results showed the low Reynolds number k-ε model gave a more accurate prediction of the substrate film Reynolds number than the standard k-ε model. According to the results, the authors concluded that laminar flow existed in the substrate region, while turbulent flow dominated the disturbance wave region.

Han and Gabriel (2007) and Han (2005) conducted a CFD numerical simulation of annular flow in order to study the phenomena of liquid entrainment and the effect of disturbance waves on its progression. The one-fluid model with the VOF method implemented in the commercial software Fluent 6.18 was utilized with the RNG turbulence k-ε model. The authors identified two entrainment mechanisms that are partially similar in that the growing liquid wave is being sheared off at the crest. The major outcome is confirming the role that the liquid disturbance waves play in the liquid entrainment process.

Kishore and Jayanti (2004) established a new CFD model to simulate gas–liquid annular flow in ducts. The governing equations were developed only for the gas phase with the interfacial effect of the liquid film on the gas taken into account by considering a rough-walled duct. The deposition and entrainment rates were considered by adopting empirical correlations of Govan (1990). The results showed good predictions of the pressure gradient and the film thickness of the steady annular flow, while the variation of the gas density affected the results of the developing annular flow.

Liu et al. (2011) developed a two-fluid model based on the VOF method that included the effect of liquid roll waves. The gas phase was modeled as a homogeneous mixture of pure gas and liquid droplets with no slip between the droplets and the gas. The mass transfer between the liquid and the homogeneous mixture was taken into account by including the appropriate source terms in the governing equations. The study showed good predictions of the pressure gradient, film thickness, and shear stress. The authors noted the significant contribution of wave crests to the entrainment process, which was also observed by Han and Gabriel (2007), Han (2005).

Instead of modeling the two-phase flow using a two-fluid model, the flow can be modeled by utilizing a three-fluid model as implemented in the works of Stevanovic and Studovic (1995), Alipchenkov et al. (2004). In this case, the three distinct phases are the continuous liquid film, the gas stream occupying the core region and the dispersed liquid droplets entrained by the gas. Introducing the third phase to the model results in an additional computational effort, especially with the existence of turbulence, compared to the two-fluid modeling and may lead to further conversion issues.

Vieiro et al. (2013) conducted a CFD study using ANSYS CFX 13.0 to investigate the two-phase behavior of the annular flow near the transition to churn flow. They adopted the homogeneous model in a two-dimensional (2D) axisymmetric computational domain. Surprisingly, the study over-predicted the pressure drop by as high as 100%.

Modeling the behavior of the liquid film is the key to understand the phenomena of film reversal and liquid loading. Experimentally, it is difficult to observe the precise behavior of the liquid film due to the fact that it covers the channel walls. As formerly discussed, most of the analytical studies concentrated on the force balance for a liquid droplet with the use of field data, with little work spent in the film stability criteria. Recently, there have been no studies tending to improve the analytical approaches for the film stability and liquid drop transport criteria which were initiated by Turner et al. (1969). With the current advances in computational modeling, the concentration is expected to shift to CFD modeling. As a result, very few studies have been recently emerged in modeling the annular two-phase flow using CFD techniques such as the 3D work done by Karami et al. (2014) to simulate liquid loading in a horizontal pipe and the study performed by Han (2005) to investigate the liquid entrainment of annular two-phase flow of water and air in a vertical tube. The ultimate objectives of this study are to perform a CFD simulation to investigate the flow behavior in the vicinity of the liquid film. The study focuses on the characteristics of film reversal in order to examine the flow behavior over the range in which the critical gas velocity exists (the velocity corresponding to complete film reversal).

## Problem statement

Computationally model the churn two-phase flow with negligible mass transfer between the two phases.

Analyze and solve for the velocity field and the shear stress of the flow field.

Study the conditions of the complete film reversal.

## Solution methodology

The objective of this section is to present the model utilized including the governing equations and the closure relations. The two-fluid Eulerian model has been implemented as the governing model of the fluid motion, while the RNG k-ε turbulence model for each phase is used to model turbulence.

### Assumptions

The flow is 2D axisymmetric and incompressible.

The mass exchange between the two phases is neglected.

The flow is transient due to its time dependence.

The flow regime is turbulent.

The liquid droplets in the gas phase have an extremely small diameter.

### Governing equations

The conservation equations are derived by time averaging the local instantaneous balance for each phase (Anderson and Jackson 1967) or by implementing the mixture theory approach (Bowen 1976). Each phase has its phase-weighted conservation equations describing its distribution in the domain, velocity field, and pressure variation.

*L*and

*G*donate the liquid phase and the gas phase, respectively.

*is the velocity vector and \(\alpha\) refers to the volume fraction.*

**V***p*is the pressure,

*is the gravitational acceleration vector,*

**g**

**F**_{D}is the interfacial drag force, and

**F**_{s}is the surface tension force. \(\overline{\overline{\varvec{\tau}}}_{{v,{\text{L}}}}\) and \(\overline{\overline{\varvec{\tau}}}_{{t,{\text{L}}}}\) are the viscous shear stress tensor and the turbulent shear stress tensor of the liquid, respectively. Similarly, \(\overline{\overline{\varvec{\tau}}}_{{v,{\text{G}}}}\) and \(\overline{\overline{\varvec{\tau}}}_{{t,{\text{G}}}}\) are the viscous shear stress tensor and the turbulent shear stress tensor of the gas, respectively. The viscous shear stress tensors of both phases are defined as

### Interfacial drag

^{−5}m. Notice that \(F_{{D,{\text{GL}}}} = - F_{{D,{\text{LG}}}}\). The symmetric drag model is recommended when the secondary phase in a region of the computational domain becomes the continuous phase in another.

### Surface tension

**F**_{s}, is given by

In the work of Bartosiewicz et al. (2008), the authors showed that splitting the surface tension force by volume did not result in significant differences with the analytical solution. Hence, it is implemented in the current study.

### Interface tracking

### Turbulence closure

Turbulence disturbance is generated in both phases due to high velocity gradient at the interface, and thus the flow regime is turbulent (Chen et al. 2015). Han (2005) concluded that the RNG k-ε turbulence model is the convenient model for such applications since it is adjusted to account for the low Reynolds number effects and because the liquid film region is similar to the flow in the near-wall region. The RNG k-ε model for each phase is adopted in this work as it is the most general approach (transport equations are solved for each phase) and as it captures the turbulent interactions between the two phases while being adjusted to capture the low Reynolds number effects. Detailed transport equations can are discussed in ANSYS Fluent theory guide (Ansys Inc 2009).

### Meshing and computational domain

Generated meshes

Mesh | No. of elements |
---|---|

1 | 74,295 |

2 | 110,109 |

3 | 176,403 |

### Test matrix

Table of the input test matrix

U | ||||||
---|---|---|---|---|---|---|

| 23.99 | 21.17 | 18.03 | 16.01 | 13.46 | 10.72 |

| 23.77 | 21.77 | 19.11 | 16.72 | 14.44 | 10.99 |

| 24.23 | 22.33 | 19.96 | 17.41 | 15.00 | 12.35 |

### Initial and boundary conditions

The computational domain is initialized from the gas inlet with a velocity equal to the superficial gas velocity. In other words, the domain is initially filled with air flowing at *U*_{SG}.

Boundary conditions

Boundary | Condition |
---|---|

Wall | No slips |

Liquid inlet | \(\varvec{V}_{\text{L}} = U_{{{\text{L}},{\text{in}}}} \,\widehat{{\mathbf{j}}}\) |

Gas inlet | \(\varvec{V}_{\text{G}} = U_{{{\text{G}},{\text{in}}}}\, \widehat{{\mathbf{i}}}\) |

Pipe axis | Axis |

Top outlet | Outflow |

Bottom outlet | Outflow |

This study implemented an overall improved geometry and locations of boundary conditions in order to model the actual case of churn flow with film reversal existing at the considered conditions. Hence, the liquid inlet length is extended such that the liquid velocity at the inlet approaches zero in order to approach the conditions of the actual case and at the same time allowing for complete film reversal. Extending the liquid inlet length along the whole pipe length would make it more justified to logically approximate the liquid inlet region as wall since the velocity would be much smaller. However, it has been found that further extension of the liquid inlet length did not result in a change in the flow field in this region. This improvement in the locations and the geometry of the boundary conditions is implemented in order to obtain a solution that simulates the real actual case of the flow considering there is still so much room for improvement.

### Convergence criteria

Problems of multiphase flows do not usually converge to extremely low residuals, such as 10^{−6}. The convergence criteria are set to be 10^{−4} for the default residual parameters and equations: continuity, axial velocities of both phases, radial velocities of both phases, turbulent kinetic energies of both phases, turbulent dissipation rate of both phases and the volume fraction of the secondary phase. Furthermore, the parameters of the mass imbalance, the volume fraction at the outlet section and the average wall shear stress above the level at which the liquid is injected are monitored during the simulation.

## Results and discussion

### Updated test matrix

Table of the obtained test matrix

U | ||||||
---|---|---|---|---|---|---|

| 13.26 | 11.80 | 10.77 | 9.29 | 8.71 | 6.93 |

| 12.81 | 12.15 | 10.55 | 9.69 | 8.37 | 6.63 |

| 12.78 | 11.81 | 10.80 | 9.53 | 8.39 | 7.06 |

### Mesh independence

*U*

_{SG}= 13.26 m/s and

*U*

_{SL}= 0.1 m/s and reported in Table 5. Results of mesh 2 and mesh 3 ensure a grid-independent solution. Hence, mesh 2 is selected due to the computational time limitation.

Pressure gradient results for different grid sizes (*U*_{SG} = 13.26 m/s and *U*_{SL} = 0.1 m/s)

Mesh | No. of elements | − dp/dL (Pa/m) |
---|---|---|

1 | 74,295 | 1005.1 |

2 | 110,109 | 981.3 |

3 | 176,403 | 990.6 |

### Time step independence

Liquid holdup results for time steps of 0.0005, 0.001 and 0.002 s (*U*_{SG} = 13.26 m/s and *U*_{SL} = 0.10 m/s)

Time step |
| |Relative error| (%) |
---|---|---|

0.0005 | 0.08938 | 6.1 |

0.001 | 0.09742 | 2.4 |

0.002 | 0.10261 | 7.8 |

The time step independence test depicted in Table 6 implies the model independence of the time step with relative errors less than 8% for time steps 0.0005, 0.001 and 0.002 s. Hence, a time step of 0.002 s is chosen for the rest of the study due to the computational time limitation.

### Model validation

The pressure gradient resulted from the current study shows a good match with the nominal case with a maximum error of 13.4%.

### Phase distribution

*t*= 10 s for the cases when

*U*

_{SL}= 0.1 m/s,

*U*

_{SL}= 0.05 m/s and

*U*

_{SL}= 0.01 m/s, respectively. Unless explicitly mentioned, the axial dimension in all the contour plots is scaled down by a factor of 0.15 in order to show the entire length of the pipe and all plots are taken at

*t*= 10 s. The phase distribution results show a typical behavior of the churn flow for all the considered cases. Gas occupied most of the pipe core with little liquid entrained captured by the model. Results show that continuous liquid entrainment and deposition take place simultaneously in the vicinity of the liquid–gas interface.

Oscillatory behavior of the formed disturbance waves is observed at the liquid inlet, which is one of the churn flow characteristics. The number of the interfacial waves continuously changes as superposition and separation of these waves occurs continuously. The results indicate that the amplitude of the disturbance waves increases when the gas flow rate is reduced. Along the pipe wall, a wavy liquid film traveling upward covers the pipe wall downstream of the entrance zone. However, at the low liquid flow rate (*U*_{SL} = 0.01 m/s), the amplitude of the interfacial waves was found to be extremely small and becomes more pronounced as the gas velocity increases. In the case of *U*_{SG} = 6.93 m/s and at the lowest considered gas velocity (*U*_{SL} = 0.1 m/s), most of the liquid is observed to flow downward and leave the computational domain through the bottom outlet, while some liquid is still flowing upward and entrained by the gas, unlike the other cases where the liquid is all discharged from the top outlet.

### Streamlines

#### Gas streamlines

*U*

_{SL}= 0.1 m/s contoured with the axial gas velocity for the given range of superficial gas velocities. Several key characteristics of the flow field can be noticed. After the gas enters the flow domain, some of it leaves through the bottom outlet section in a steady manner, while the remaining part flows upward to interact with the laterally injected liquid. As a result of the gas–liquid interactions, the axial velocity distribution of the gas phase is considerably influenced by the shape of the gas–liquid interface which sometimes forms a nozzle-like shape causing the gas velocity to increase at the nozzle throat followed by the formation of small vortices near the gas–liquid interface. The high shear stress caused by the high gas velocity in the vicinity of the thick liquid layer (throat of the nozzle-like liquid layer) generates interfacial waves due to the abrupt contraction of the turbulent gas flow. Downstream of that region, the gas flow stabilizes with little disturbance resulted from the thin liquid film that keeps the gas streamlines wavy because of its continuous thickness change. However, due to the periodic large entrainment in the liquid inlet region, the streamlines downstream are periodically disturbed. At lower gas superficial velocities, the increased amplitude of the disturbance waves causes a larger rise in the gas velocity near the liquid inlet region than that of higher gas superficial velocities. For all cases, except for

*U*

_{SL}= 0.1 m/s and

*U*

_{SG}= 6.93 m/s, the flow below the liquid injection region is steady, but unsteady elsewhere.

### Liquid streamlines

*U*

_{SL}=0.1 m/s. The figures indicate the presence of liquid vortices near the liquid entrance region. This is manifested by the negative (downward) velocity of the liquid layer in the vicinity of the pipe wall and the positive (upward) velocity near the gas–liquid interface. It is important to mention that the downward velocity near the pipe wall is caused by gravity while the upward velocity near the gas–liquid interface is caused by the shear stress created by the high gas velocity. In the gas phase, circulating flow regions are also formed due to the curvatures of the gas–liquid interface. The formation of the roll waves phenomenon at the gas–liquid interface may result in the formation of regions of circulation with liquid at the outskirt and gas in the core region. Also, the solution indicated that the resulting gas vortices contain some liquid. The existence of such a phenomenon in a two-phase flow does undoubtedly confirm that churn flow is taking place. Notice that some liquid droplets that have been sheared off by the gas from the crests of interfacial waves rejoin the liquid film again.

#### Liquid inlet region

Figure 16b depicts the velocity vectors near the liquid film. It shows two regions of the liquid film where in the first region (near the pipe wall) the liquid film is moving downward due to both effects of gravity as well as the flow field created by the enclosed gas vortex. In the lower part of the same region, a stagnation point is clearly visible and separates the upper vortical motion from the lower stream moving upward which is mainly driven by the shearing force at the gas–liquid interface. The above-mentioned interaction between the liquid and gas phases results in the disturbance wave shown in Fig. 16b. The crest of the wave separates from the main film body, thus forming a roll wave that is clearly visible in Fig. 16c. The roll wave phenomenon is normally associated with the formation of a separate liquid droplet that travels with the gas stream and rejoins the liquid layer again at some distance downstream. Some of the separated droplets continue to travel with the gas stream until reaching the discharge section of the pipe.

Figure 16c shows a representative sketch of the flow field in the film region. Axial velocity near the pipe centerline is almost constant due to the turbulent mixing, but it start to sharply decrease near the interface. The wet gas reverses its direction between the roll wave and the base liquid film, which forms gas vortices.

### Pressure variation

*U*

_{SL}= 0.01 m/s and

*U*

_{SG}= 11.81 m/s. The figure shows a sharp rise in the pressure starting from the gas inlet. This is because of the abrupt expansion of the flow passage and the decrease in gas velocity. The pressure rise continues till the gas flow develops just before interacting with the liquid starting from

*x*= 10

*D*. Within the portion of the domain surrounded by the liquid inlet, the interfacial waves cause pressure disturbances, and thus a wavy pressure variation takes place. Downstream of the liquid inlet port, the pressure starts to drop in a steady manner due to the very thin liquid film. Notice that a little pressure rise exists just after the flow passes the liquid inlet region due to the drop in the film thickness which gives more cross-sectional area for the gas to flow through.

### Shear stress variation

#### Shear stress variation at the inlet region

*U*

_{SL}= 0.1 m/s and

*U*

_{SG}= 13.26 m/s. The graph shows a continuous oscillation of the shear stress. However, it oscillates around a positive average value (0.1 Pa) that is close to 0 which implies that the direction of the shear stress exerted on the liquid is upward most of the time. This, in turn, indicates that the liquid on the wall is flowing downward for most of the time. For all the considered cases, the average value is positive.

*t*= 10 s for the cases of

*U*

_{SL}= 0.1 m/s,

*U*

_{SL}= 0.05 m/s and

*U*

_{SL}= 0.01 m/s, and different gas velocities in each plot. In these figures, the gas viscous shear stress is defined as

*x*donates the component along the axial direction. The radial variation of the shear stress shows negative values for most of the radial distance. Starting from the centerline (

*y*= 0 m), the gas shear stress drops from 0 and bottoms out, and then it keeps undulating as approaching the liquid inlet port. Furthermore, the magnitude of the fluctuation decreases as the liquid film is approached. The fluctuation stretches closer toward the centerline as the superficial liquid velocity increases. This fluctuation behavior was also observed by Adaze (2018) in the vicinity of the liquid film in his study of CFD annular flow. In some cases, the shear stress takes positive values near the disturbance waves due to the gas vortices.

#### Shear stress variation near the outlet region

*D*before the top outlet versus time for the case of

*U*

_{SL}= 0.1 m/s and

*U*

_{SG}= 13.26 m/s. The figure shows vanishing wall shear stress at small times following the start of computations due to the absence of liquid near at the wall region. At later time steps, the liquid fills the annular region and the local wall shear stress starts oscillating around a negative value indicating that the direction of the shear stress exerted on the liquid layer is always downward. This, in turn, implies that the liquid on the wall is always flowing upward, which ensures the presence of the annular flow regime.

*x*= 35

*D*when

*t*= 10 s for the cases of

*U*

_{SL}= 0.1 m/s,

*U*

_{SL}= 0.05 m/s, and

*U*

_{SL}= 0.01 m/s, and different gas velocities in each plot. The radial variation of the shear stress shows negative values for most of the radial distance similar to that at the liquid inlet region. However, their magnitudes of the fluctuation are lower than those observed in the liquid inlet region. A severe rise in the magnitude of the gas shear stress is observed near the pipe wall. This may be attributed to the presence of the viscous sublayer. Furthermore, it is observed that the liquid volume fraction at the pipe wall never becomes unity since most of the liquid is concentrated within the churn flow domain in the vicinity of the liquid inlet port.

### Gas velocity profiles

#### Gas velocity profiles at the inlet region

*x*= 12.5

*D*for cases of

*U*

_{SL}= 0.1 m/s,

*U*

_{SL}= 0.05 m/s and

*U*

_{SL}= 0.01 m/s, respectively. In the core region, the velocity distribution is almost uniform due to the high mixing of the turbulent gas flow and the exchange of momentum between the gas layers. Near the gas–liquid interface, the gas velocity declines as a response to the interfacial friction caused by the liquid film and the interfacial waves. The velocity profile is continuously changing with time because of the presence of the roll waves, which is by nature an unsteady phenomenon. For a given gas superficial velocity, the gas centerline velocity varies in space and time depending on the geometry of the gas–liquid interface and its axial variation. One can easily see that the maximum gas velocity (centerline velocity) depends not only on the gas superficial velocity but also on the radial velocity distribution that in terms depends on the shape of the local gas–liquid interface. As the film thickness decreases by reducing the liquid superficial velocity, the uniform velocity distribution portion dominates.

#### Gas velocity profiles near the outlet region

*D*before the top outlet section for all cases are presented in Figs. 31, 32 and 33 for the cases of

*U*

_{SL}= 0.1 m/s,

*U*

_{SL}= 0.05 m/s and

*U*

_{SL}= 0.01 m/s, respectively. It should be emphasized that the velocity profiles are continuously changing with time due to the time variation of the annular liquid layer. Accordingly, the plotted profiles in the above figures represent only samples of the axial gas velocity distributions at time

*t*= 10 s. The figures indicate that the higher the gas flow rate, the higher centerline gas velocity. Because large amounts of liquid are periodically sheared off by the gas in the liquid inlet region and flowing upward, thus causing a wavy behavior of the gas velocity toward the pipe exit. Figure 33 has the largest number of smooth curves due to the extremely low liquid flow rate.

### Liquid velocity profiles

#### Liquid velocity profiles at the inlet region

*x*= 12.5

*D*for the cases of

*U*

_{SL}= 0.1 m/s,

*U*

_{SL}= 0.05 m/s,

*U*

_{SL}= 0.01 m/s, respectively. The liquid axial velocity curves do not extend for the whole radial distance since the liquid does not exist along the whole cross section and accordingly \(\alpha_{L} = 0\) in the core region. Some velocity distributions show negative values near the wall indicating the downward flow of the liquid phase. The magnitudes of liquid velocities are similar to those of the gas velocities but with small slip especially when approaching the core region. Liquid entrainment is most pronounced downstream that region.

#### Liquid velocity profiles near the outlet region

*x*= 35

*D*for the cases of

*U*

_{SL}= 0.1 m/s,

*U*

_{SL}= 0.05 m/s and

*U*

_{SL}= 0.01 m/s, respectively. It is clear from the figures that the liquid velocity extends more toward the pipe centerline indicating higher entrainment downstream the liquid inlet port. No negative velocities are observed near the wall and thus no film reversal. By comparing these figures with Figs. 31, 32 and 33 of the gas velocity for the same section, the same patterns are observed as the gas velocity ones.

### Critical gas velocity

Guner (2012) experimentally determined the gas velocity at the onset of film reversal and the critical gas velocity at the complete film reversal. On the other hand, Adaze (2018) performed a CFD study to investigate the flow field and reported values of the gas velocity at the onset of liquid loading which sufficiently match Guner’s results.

According to the current study of churn flow and the shear stress results, the superficial gas velocities corresponding to a complete film reversal are higher than the considered range, i.e., the critical gas velocities for the cases of *U*_{SL} = 0.1 m/s, *U*_{SL} = 0.05 m/s and *U*_{SL} = 0.01 m/s are greater than 13.26 m/s, 12.81 m/s and 12.78 m/s, respectively, which agrees with the results of Guner (2012), Adaze (2018). Fluctuation of the liquid wall shear stress around a positive value implies a complete liquid film reversal.

*U*

_{SG}= 13.92 m/s and

*U*

_{SL}= 0.1 m/s,

*U*

_{SG}= 14.58 m/s and

*U*

_{SL}= 0.05 m/s, and

*U*

_{SG}= 15.79 m/s and

*U*

_{SL}= 0.01 m/s) and the time variation of the liquid wall shear stress for these cases at section

*x*= 12.5

*D*is shown in Figs. 40, 41 and 42. It is observed that the liquid wall shear stress in these cases oscillates around a negative value, which indicates that the average shear stress is downward. Hence, the critical gas velocities are identified as

*U*

_{SG}= 13.92 m/s,

*U*

_{SG}= 14.58 m/s and

*U*

_{SG}= 15.79 m/s for the cases of

*U*

_{SL}= 0.1 m/s,

*U*

_{SL}= 0.05 m/s and

*U*

_{SL}= 0.01 m/s, respectively.

## Conclusion

In summary, the current work presents a CFD modeling of churn flow in a 3-inch vertical pipe near the critical gas flow rates. The study implements the two-fluid Eulerian model along with the RNG k-ε model for each phase using the commercial software ANSYS Fluent 18.2 (2009). Surface tension is included and the symmetric drag model is adopted for the interfacial momentum exchange.

The axisymmetric geometry was generated by ANSYS Design Modeler 18.2, and the mesh was constructed by ANSYS meshing tool. The geometry and the boundary condition are improved in order to model the liquid film reversal of the two-phase flow involved. The conservation equations were solved using the finite volume method embedded in the software with a convergence criterion set to 10^{−4} for all involved PDEs.

The model sufficiently captured the flow field and the nature of churn flow including the interfacial waves, the oscillatory behavior of the liquid, and the pressure and shear stress fluctuations.

The computational domain surrounded by the liquid inlet port shows the actual characteristics of the churn flow, while the domain above this area is mainly annular flow with a thin liquid film. Consequently, the liquid holdup is accurately predicted by the model within the domain near the liquid inlet port.

The interfacial waves play a significant role in disturbing the velocity, shear stress, and pressure fields. Their presence retains the flow unsteadiness. Moreover, the simultaneous processes of liquid entrainment and deposition are highly activated by the disturbance waves.

Radial fluctuations of the axial gas velocity and shear stress become more pronounced and affected by the disturbance waves as the film thickness increases.

Gas and liquid vortices are observed in the vicinity of the liquid film. Nevertheless, the liquid film is flowing downward most of the time in the middle section of the liquid inlet region.

According to the shear stress transient behavior, the critical gas velocities are in agreement with what has been reported by Guner (2012), Adaze (2018), i.e., all the considered cases of gas velocities lay below the critical gas velocities except the cases presented in "Critical gas velocity" section.

A three-dimensional domain can be considered to more accurately capture the flow details and observe the non-uniformity of the liquid film thickness in the angular dimension.

The turbulence model is extremely critical and can significantly affect the results. Therefore, the selection of the model should be carefully evaluated before performing the study.

A further improvement in the boundary conditions is possible. The problem configuration that allows film reversal and results in the entire computational domain to resemble the actual case is highly recommended. For example, instead of injecting the liquid within a height of 5D, it is recommended to inject the liquid uniformly over the entire pipe wall.

The interface can be captured more accurately by using the Geo-reconstruction scheme for constructing the interface. The implementation of this scheme is only possible when using an explicit discretization with a smaller time step that meets the stability criteria and leads to convergence.

Sufficient refinement of the computational mesh in the core region and near the interface might result in a more accurate modeling of the liquid droplets entrained in the gas core. Here, due to the computational time limitation, the elements are coarse within the core region.

The current study can also be expanded to include inclined pipes with different deviations in order to study the effect of the deviation angle on the flow field.

## Notes

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