After the flow simulation stops, the phase contours as well as the velocity vectors of the pipeline longitudinal section and representative cross-sections are extracted to analyze the entrainment behavior in the pipeline.
5.2.1. Entrainment Distribution
Figure 4 shows the phase distribution and velocity vector in the pipeline. It can be seen that as the upstream length increases, the gas in the pipe center moves much faster than that near the wall, the gas velocity fluctuation in the pipe becomes obvious, and the uniform liquid film starts waving and then breaks up into droplets (see
Figure 4b). Then, the droplets are gradually entrained into the pipe center by the high-speed gas. When the upstream length reaches 6
D, the droplet distribution seldom changes, and the entrainment process reaches dynamic equilibrium. After entering the elbow, most of the droplets hit the elbow extrados and coalesce into liquid film. Then, the film flows along the extrados side to the downstream pipe (see
Figure 4c).
Representative cross-sections of the upstream pipe are extracted to reveal more information (see
Figure 4d). At an upstream length of 2
D, there is a film wave at the right side of the cross-section (circled). The corresponding velocity vectors indicate that a turbulence burst here contributes to the wave. When the upstream length reaches 2.4
D, the film becomes discontinuous, and “liquid bridges” (circled) occur, which are also related to the turbulence bursts near the wall. As the flow develops, turbulence bursts become stronger and break the liquid bridges into droplets, and then, the droplets are entrained into the pipe center by the gas. At an upstream length beyond 6
D, the droplets are distributed uniformly in the pipe cross-section. Droplet coalescence (circled) is also found in the flow at an upstream length of 7
D.
Other researchers like Zahedi [
34] and Farokhipour [
21] also studied liquid distribution in annular flow pipelines. Zahedi [
34] only reported the liquid distribution in the middle of the elbow (elbow angle = 45°), and no liquid was found at the elbow extrados; however, this seems inaccurate, because droplets carried by the gas core will hit the extrados and form a liquid film. Farokhipour’s simulation [
21] showed the liquid film distribution in the elbow well, but no droplets can be observed in the figures. Compared with their studies, the simulation in this study provides more details of droplet distribution, like droplet coalescence, which can be seen as a remedy for entrainment simulation. Another difference is the grid. Other researchers use hexahedral cells in their simulations, but polygonal cells are adopted in this study to reduce skewness to obtain more accurate results. Additionally, droplet distribution and entrainment changes in the elbow are also investigated, which have not featured in other simulations.
5.2.2. Entrainment Characteristics
Further, the void fraction (area-weighted average) of the cross-sections (every 0.25
D) in the upstream section is extracted and summarized in
Figure 5.
It can be seen that for the upstream length of 2D, the void fraction is steady but fluctuates slightly around 0.95. As the upstream length increases to 6D, the void fraction rises to 0.978. Meanwhile, its fluctuation becomes stronger, and droplets start to take shape and enter the gas core. Then, after 6D, the void fraction becomes steady again, with its value fluctuating more vigorously between 0.978 and 0.981. It seems that for the annular liquid inlet, the liquid droplet entrainment and re-deposition process reach dynamic equilibrium when the upstream length is beyond 6D.
According to Ishii and Kataoka [
5,
35], for the case where the liquid is injected smoothly as a film at the inlet, the entrained fraction expression takes the form of exponential relaxation. For the case in this study, the trend of the curve in
Figure 5 is also exponential relaxation, which means the CFD results here are reasonable.
Then, the droplet information (droplet number, droplet diameter, droplet velocity, and droplet position (radial distance from the wall)) in the pipeline’s longitudinal section is collected to further analyze the entrainment. Droplet diameters and positions are measured by the scale of the phase distribution contour; droplet velocity is determined by the velocity vector in the center of the droplet.
Figure 6 shows a histogram of the relationship between droplet diameter and droplet number distribution in the longitudinal section in the developed annular flow. It can be seen that the droplet number roughly follows a negative skew distribution. Most droplet diameters are distributed in a 1 mm to 2.5 mm range, and the droplet number in the range of 2 mm to 2.5 mm is the largest, while in the range of 0 to 0.5 mm, it is the smallest.
In contrast to one study [
36], for the conditions of low-pressure (below 3 atm) and low-viscosity fluid (water), the droplet diameter distribution in this study is similar to the upper limit log-normal (ULLN) distribution given by Mugele and Evans [
37]. The maximum size of the droplet can be calculated based on the Kelvin–Helmholtz instability and critical Weber number.
where
is the maximum diameter of the droplet, and
is the critical Weber number, which equals 12. For this study, the maximum diameter of the droplet calculated using this equation is 2.7 mm, which is represented by the last bar in
Figure 6.
For this study, the calculated volume median diameter of the droplet is 1.1 mm, while the volume median droplet diameter predicted using CFD simulation is 1.7 mm (54.5% higher than the calculated value). Further improvement may be achieved by refining the mesh, which is limited by the current computational resources.
Figure 7 shows the relationship between the radial distance (from the wall) and the droplet diameter in a developed annular flow. As shown in this figure, the average value of all the droplet diameters is 1.81 mm. Droplet diameter fluctuates around the average value, and its amplitude increases with the radial distance.
Table 5 presents the variance distribution of droplet diameters. It can be seen that as the radial distance (from the wall) increases, the expectation of the droplet diameter varies very slightly around 1.81 mm, by less than ±0.05 mm. However, for the variance, it increases from 0.19 to 0.63. This means that the change in droplet diameter becomes increasingly unstable the closer the droplet is to the gas core. This may be because droplets closer to the pipe center are more likely to be broken up or coalesced by the gas core turbulence, which strengthens the fluctuation.
According to Kolev [
27], the droplet breakup mechanism can be expressed as a balance between the external stress force and the surface force. During the breakup, external stress force tries to disrupt the droplets, while surface tension force tries to avoid droplet deformation. From the critical Weber number, Equation (11), it can be seen that the droplet diameter varies with the inverse square of the gas velocity. As the radial distance from the wall increases, the gas velocity becomes higher, so the minimum value of droplet diameter decreases. On the other hand, according to the Kelvin–Helmholtz instability, as the gas velocity increases, the gas–liquid interface becomes more unstable. Additionally, droplet diffusion becomes stronger, so droplets are more likely to coalesce, which increases the maximum droplet diameter.
Figure 8 shows velocity distribution (both gas and droplet) according to radial distance (from the wall). It can be seen that both gas and droplet velocities rise and fluctuate as radial distance increases, while the velocity difference between gas and droplets also increases, which means that the slippage becomes more obvious. Moreover, the fluctuations in both gas and droplets become stronger.
In order to clearly express the droplet slippage, the droplet slippage ratio can be calculated with the following equation.
where
is the droplet slippage ratio, and
is the droplet velocity. It can be found that the slippage ratio near the wall (27.3%) is smaller than that in the pipeline center (33.8%).
5.2.3. Entrainment Mechanism
In this subsection, the mechanism of the entrainment process in annular flow is interpreted gradually, as shown in
Figure 9. In the initial state (see
Figure 9a), the liquid film and gas are uniform and there is no fluctuation. Since there is a velocity difference between gas and liquid film, vortices occur at the two-phase interface under the effect of shear and friction, which produce waves in the film and spread downward (see
Figure 9b). As the flow develops, the crest stretched by the vortices becomes higher, and its bottom is necking under the effect of surface tension (see
Figure 9c,d). Then, the necking breaks, the liquid entering the gas becomes droplets, and the rest of the liquid becomes film (see
Figure 9e). The high-speed gas moving in the pipe center lowers the dynamic pressure here, so the droplets can be entrained under the pressure difference (see
Figure 9f). This process reduces the film volume and makes the film discontinuous.
The Kelvin–Helmholtz instability is mainly associated with flows that have tangential variation in the velocity field [
38,
39,
40]. This instability is caused by the hydrodynamic amplification of perturbations that arise at the gas–liquid interface with a discontinuity in the velocity field. The entrained droplet size is approximately equal to the height of the most unstable wavelength.
where
is the height of the wavelength. For this study, the calculated droplet diameter is 1.8 mm, which is close to the average droplet diameter in
Figure 6. This indicates that the simulation results in this study fit the Kelvin–Helmholtz instability theory and are reasonable.
As the flow develops downstream, the entrained droplets travel to the wall and are deposited into the liquid film again. As the entrainment and deposition reach dynamic equilibrium, the flow is fully developed. The most important factor that affects droplet entrainment is the vortices near the gas–liquid interface.