On the basis of the signal data acquired by the conductance sensor measurement system, we select 20,000 measured points to carry out the multi-scale morphological analysis of the first-order difference sequence, and the maximum scale is set to 80. For the flow where the gas phase distribution is relatively uniform, the scatter points in the first-order difference scatter plot are located more closely to the coordinate origin. The second-order moment is small and it does not change obviously with the increase of scale. However, for the flow where the gas phase distribution is non-uniform, the scatter points in the plot distribute far away from the coordinate origin, which contributes to a relatively large second-order moment. Meanwhile, as the scale increases, the details of flow characteristics are ignored. Gas phase distribution is considered to be more uniform than that at lower scale. Hence, the scatter points gather towards the coordinate origin, and therefore, the second-order moment decreases obviously. When the scale increases to a certain value, the distribution and morphology of scatter points remain basically unchanged, that is, the second-order moment turns almost constant. Thus it is feasible to extract indexes for indicating the inhomogeneous distribution characteristics of the dispersed phase, that is, the gas phase in gas–liquid two-phase flow.

The evolutions of a multi-scale first-order difference plot constructed with three typical measurement signals from A-channel are shown in Figures 7–9, representing bubbly flow, slug–bubble transition flow, and slug flow, respectively. The scales are set to 1, 20, 50, and 80. The bisector of the second-fourth quadrant is regarded as the reference line to observe the evolution of scatter points along with the increase of scale on the first-third quadrant. It can be seen from the figures that there is the largest distribution range of scatter points for slug flow, less for slug–bubble transition flow, and the least for bubbly flow. At the same time, with increasing scale, the degree of scatter points approaching the coordinate origin is the biggest for slug flow, less for slug–bubble transition flow, and the least for bubbly flow. Specific analysis is given as follows.

Figure 7: Multi-scale first-order difference scatter plot of bubbly flow (*U*_{sg}=0.147 m/s, *U*_{sw}=0.884 m/s).

Figure 8: Multi-scale first-order difference scatter plot of slug–bubble transition flow (*U*_{sg}=0.147 m/s, *U*_{sw}=0.442 m/s).

Figure 9: Multi-scale first-order difference scatter plot of slug flow (*U*_{sg}=0.147 m/s, *U*_{sw}=0.037 m/s).

The evolution of a multi-scale first-order difference plot of bubbly flow is presented in Figure 7. Combined with the analysis in Section 3.3, we know that when the water superficial velocity is high enough, the gas phase exists in the form of bubbles, most of which are almost with the same size except several a little larger ones appearing occasionally. As a result, the first-order difference signal presents a continuous fluctuation with the amplitude maintaining a low value, and the corresponding scatter points concentrate near the coordinate origin. Therefore, the distribution area of scatter points in bubbly flow shrinks to the smallest one with the largest degree of the scatter points dispersing towards the second and fourth quadrants. In addition, the scatter points corresponding to the relatively large bubbles appearing occasionally are located far away from the coordinate origin. When the scale *s* is <50, the distribution area reduces slightly as the scale increases. However, when the scale exceeds 50, with the increase of the scale, the scatter points exist in almost the same distribution range, which demonstrates the homogeneous distribution in the pipe under bubbly flow.

The result of the evolution of a multi-scale first-order difference scatter plot of slug–bubble transition flow is illustrated in Figure 8. The explicit behaviour of this flow is relatively short gas slugs flowing past with bubbles dispersed in long water slugs. As a result, the fluctuation of difference sequence exhibits a smaller frequency but larger amplitude. In terms of the corresponding first-order difference plot, for low scales, compared with the bubbly flow, the distribution of scatter points is almost identical with that of bubbly flow, except for its larger distribution area. In other words, the distribution area of scatter points shrinks in the second and fourth quadrants, but extends along with the bisector of the second-fourth quadrant, resulting in a stronger distribution trend than bubbly flow. Similarly, the corresponding scatter points of bubbles concentrate near the coordinate origin, but the corresponding scatter points of short gas slugs distribute away from the coordinate origin. Furthermore, it is noteworthy that the size discrepancies among short gas slugs and bubbles here are much larger than bubbly flow, that is, the degree of inhomogeneous distribution enhances. Thus, the distribution area of scatter points is larger. Meanwhile, with the increase of scale, the distribution area diminishes obviously.

The evolution of a multi-scale first-order difference plot of slug flow is shown in Figure 9. In slug flow, when there are no gas slugs flowing through the measuring area, the amplitude of the first-order difference sequence fluctuates slightly near zero; thus the corresponding scatter points concentrate near the coordinate origin. When gas slugs flow through the measuring area, the fluctuating range of difference sequence becomes severer; thus the distribution area extends along with the bisector of the first-third quadrant. Besides, with the increase of scale, the distribution area shrinks extremely obviously. That is because the gas distribution under slug flow is extremely non-uniform, where gas slugs and water slugs flow past alternately, resulting in the largest fluctuation range of difference sequence and the largest distribution area with the strongest distribution trend in the plot.

According to the characteristics of the foregoing distributions in the multi-scale first-order difference plot under three typical flow patterns, we extract two indexes, that is, the second-order moment *M*_{1,2}(*s*) based on the bisector of the second-fourth quadrant as the reference line and the slope of *M*_{1,2}(*s*) in all scales, to indicate the differences under different flow conditions and to investigate the inhomogeneous distribution characteristics of the gas phase in gas–liquid two-phase flow.

The curves of *M*_{1,2}(*s*) with increasing scale and its fitted slope in all scales under five typical flow conditions obtained via fixing the gas superficial velocity and increasing the water superficial velocity are shown in Figures 10–12.

Figure 10: Multi-scale *M*_{1,2}(*s*) and its fitted slope in the condition of increasing the water superficial velocity when fixing the gas superficial velocity at 0.055 m/s.

Figure 11: Multi-scale *M*_{1,2}(*s*) and its fitted slope in the condition of increasing the water superficial velocity when fixing the gas superficial velocity at 0.147 and 0.221 m/s, respectively.

Figure 12: Multi-scale *M*_{1,2}(*s*) and its fitted slope in the condition of increasing the water superficial velocity when fixing the gas superficial velocity at 0.368 and 0.589 m/s, respectively.

It can be concluded from Figure 10 that when fixing the gas superficial velocity at 0.055 m/s and then increasing the water superficial velocity gradually, long gas slugs are broken up into short ones, corresponding to the conversion process of almost all gas phase with little water flowing in the pipe to gas slugs and water slugs appearing alternately, resulting in more and more inhomogeneous distribution of the gas phase in the pipe and the larger slope value of the corresponding multi-scale second-order moment. In particular, the slope of the second-order moment reaches maximum at *U*_{sw}=0.147 m/s, implying the most inhomogeneous distribution of the gas phase under this flow condition. Continuing to increase the water superficial velocity, as the turbulent energy becomes high enough to break several gas slugs up into bubbles, the degree of inhomogeneous distribution of the gas phase becomes lower and the slope decreases accordingly. With the water superficial velocity further increasing to 0.442 m/s, all gas slugs are broken up into bubbles of almost the same size. As a result, the degree of inhomogeneous distribution of the gas phase becomes the lowest and the slope reaches minimum.

Figure 11 exhibits the curves of *M*_{1,2}(*s*) with increasing scale and its fitted slope when the gas superficial velocity is set to 0.147 m/s and 0.221 m/s, respectively. The variation trend of the curves is similar to that in Figure 10. However, the gas superficial velocity is higher, which means the gas phase is more difficult to be broken up; thus the slope of *M*_{1,2}(*s*) will reach the maximum at *U*_{sw}=0.295 m/s instead of at *U*_{sw}=0.147 m/s. As the water velocity continues to increase to 0.442 m/s, the gas slugs are shortened with more bubbles existing at the same time, and until the water superficial velocity is up to 0.589 m/s, the gas phase suspends in the water phase in the form of bubbles, which are almost with the uniform size. Therefore, the degree of inhomogeneous distribution of the gas phase presents the lowest and the slope of *M*_{1,2}(*s*) becomes the smallest.

The curves of same criterions with the gas superficial velocity set to 0.368 and 0.589 m/s are shown in Figure 12. Similarly, the slope of *M*_{1,2}(*s*) increases first and then decreases with the increase of water superficial velocity. But it is impossible for the gas phase to disintegrate into bubbles when the water superficial velocity is up to 0.147 m/s, and thus the flow pattern now is slug–churn transition flow instead of slug–bubble transition flow. Besides, it is difficult for the appearance of a very low slope, that is, the gas cannot be shattered into bubbles in the uniform size. As a result, the degree of inhomogeneous distribution of the gas phase is relatively high at all times.

In order to avoid overlapping of points with the same water superficial velocity and different gas superficial velocity, the relationship between the slope of *M*_{1,2}(*s*) and gas superficial velocity as well as mixture velocity is provided in Figure 13, in which the horizontal axis represents the mixture velocity and the vertical axis represents the slope of *M*_{1,2}(*s*). Additionally, the conditions with the same gas superficial velocity are presented with the same type of line, and different points with diverse shapes and colours symbolise different flow patterns. For the nine points in each line, the water superficial velocity is 0.037, 0.074, 0.147, 0.295, 0.442, 0.589, 0.737, 0.884, and 1.032 (m/s), respectively.

Figure 13: The slope of multi-scale *M*_{1,2}(*s*) versus water cut and mixture velocity.

It can be seen from Figure 13 that when the gas superficial velocity is between 0.055 and 0.221 m/s, before the water superficial velocity increases to 0.147 m/s, corresponding to the process of long gas slugs shortened to be short ones, the flow pattern is the slug pattern and the degree of inhomogeneous distribution of the gas phase as well as the corresponding slope of *M*_{1,2}(*s*) increases. When the water superficial velocity is up to 0.295 m/s, several gas slugs are broken up into bubbles, due to which the degree of inhomogeneous distribution of the gas phase continues to increase and the corresponding flow pattern is slug–bubble transition flow. Continuing to increase the water superficial velocity to 0.442 m/s, though the flow pattern is still slug–bubble transition flow, the gas slugs are so short that the degree of inhomogeneous distribution and the corresponding slope is lower. Until the water superficial velocity is up to 0.589 m/s, turbulent energy becomes high enough to break the gas slugs up into dispersed gas bubbles. Thereby, the degree of inhomogeneous distribution of the gas phase shows the lowest and the slope of *M*_{1,2}(*s*) becomes the smallest. Notably, when the gas superficial velocity is relatively low, particularly at 0.055 and 0.074 m/s, slug–bubble transition flow appears only at *U*_{sw}=0.295 m/s, and the bubbly flow occurs at a lower velocity of water.

When the gas superficial velocity is between 0.295 and 0.589 m/s, the flow pattern of the first three points is slug flow as well and the slope of *M*_{1,2}(*s*) increases. However, the turbulent energy breaks up the gas slugs into lots of churns with different sizes instead of bubbles when the water superficial velocity increases to 0.295 m/s. Thus the flow pattern is slug–churn transition flow, and the degree of inhomogeneous distribution of the gas phase increases as well and the slope becomes larger accordingly. With further increment of water superficial velocity, churn flow emerges and the degree of inhomogeneous distribution increases first, and then decreases as the gas churns tend to be with more uniform size, and the slope of *M*_{1,2}(*s*) becomes smaller accordingly, but still larger than that under bubbly flow.

To shed light on the variation tendency of the slope of multi-scale second-order moment under different flow conditions, we summarise the analysis as follows: (1) The degree of inhomogeneous distribution of the gas phase increases first and then decreases with the increase of water superficial velocity. At lower gas superficial velocity, the gas phase exists in the form of bubbles when water velocity is high enough, leading to the most homogeneous distribution of the dispersed phase in the pipe. As a result, the slope of multi-scale second-order moment is much lower and remains almost constant. However, at higher gas superficial velocity, the gas phase exists in the form of churns when water velocity is high enough. Thus the degree of inhomogeneous distribution is higher and the slope is larger accordingly. (2) The slope of multi-scale second-order moment is different with different gas superficial velocities. At lower gas superficial velocity, due to the uniform structure of the gas phase, the slope is relatively low. On the contrary, when gas superficial velocity is higher, the slope is higher as well. It indicates that gas holdup has a strong influence on the distribution of the gas phase in gas–liquid two-phase flow. (3) The variation tendency of the slope at different gas superficial velocities is similar, which illuminates that the flow structure of the gas phase possesses a certain principle with the increase of water superficial velocity.

To better investigate the inhomogeneous distribution characteristics of the gas phase in gas–liquid two-phase flow, we further analyse the conduction fluctuating signals from the energy point of view based on the adaptive optimal kernel time–frequency representation (AOK TFR) [43]. The AOK TFRs of five typical flow patterns are shown in Figure 14.

Figure 14: Time–frequency joint distribution of five typical flow patterns: (a) slug flow; (b) bubbly flow; (c) churn flow; (d) slug–bubble transitional flow; (e) slug–churn transitional flow.

For slug flow, the frequency mainly focuses on between 0 and 20 Hz with one or more peaks. The energy exhibits an intermittent feature in the time domain, which corresponds to the intermittent presence of long gas slugs. The frequency band of bubbly flow is quite wide. The energy distribution presents a dispersed characteristic in the time domain, which can be explained by the extremely dispersed bubble in the water phase. Meanwhile, the average energy of bubbly flow is relatively low. In churn flow, there exists obvious frequency peaks associated with the periodic oscillation. The energy distribution is continuous and uniform in the time domain, but have a high value, which corresponds to the flow behaviour that gas slugs pass through the sensor. In slug–bubble transitional flow, because of the movement of minute bubbles, there are no obvious peaks and a more wide frequency band than slug flow but the frequency band is not as wide as bubbly flow. In the AOK TFR plane, high energy and low energy appear alternatively, which can be explained by the behaviour that gas slugs and gas bubbles pass through the sensor alternatively. The frequency of slug–churn transitional flow mainly gathered around 0–20 Hz; however, there is a peak between 0 and 5 Hz. Besides, the frequency distribution is rare before the peak; normally there is a mutation when the signal comes to peak. The energy distribution presents a continuous feature but with occasional low energy. This explains the fact that the flow pattern is not stable in both churn flow and slug flow.

On the basis of the AOK TFR plane, we calculate the total energy *E*, whose value is low when the flow is stable, and vice versa. Correspondingly in Figure 15, the outcomes reveal that the increasing *U*_{sw} enhances the total energy in the AOK TFR plane first. The slug flow structure with higher energy leads to more violent signal fluctuations and more inhomogeneous distribution of the gas phase in water phase flow. Slug–bubble transitional flow occurs as the water superficial velocity increases when fixing the gas superficial velocity between 0.055 and 0.221 m/s. The energy increases first because the tremendous turbulent kinetic energy breaks long gas slugs up to short ones and blisters and the system is more unstable. This explains that the degree of inhomogeneous distribution of the gas phase increases at this moment. When the water superficial velocity continues to increase, the energy decreases, which illustrates that the flow tends to be more stable. There are few gas slugs and more gas bubbles. Though the flow pattern is still slug–bubble transitional flow, the degree of inhomogeneous distribution of the gas phase becomes lower relatively. For the gas superficial velocity between 0.295 and 0.589 m/s, the slug flow evolves to slug–churn transitional flow with the increase of water superficial velocity. The total energy is higher than before, which can be explained by the fact that at this moment the gas phase exists in the form of gas slugs and blocks, which corresponds to a more unstable flow structure, leading to higher energy and a more inhomogeneous distribution of the gas phase. For churn flow, the increasing water superficial velocity leads to breakup of gas blocks, resulting in uniform gas block sizes. Thus the flow becomes more and more stable, the energy and the degree of inhomogeneous distribution of the gas phase decrease gradually as well. In bubbly flow, numbers of gas bubbles with the uniform size are distributed randomly in the pipe. In this case, the system is so stable that the energy is the lowest and the degree of inhomogeneous distribution of the gas phase reaches the lowest value.

Figure 15: The total energy *E* versus water cut and mixture velocity.

From Figures 13 and 15, we find that the trends of the slope of *M*_{1,2}(*s*) and the total energy in the AOK TFR plane are accordant during the evolvement of gas–liquid two-phase flow structure. Besides, the energy is a dynamic factor indicating the system stability and the process of breakup and coalescence of the gas phase. Thus it can be seen that the multi-scale morphological analysis method based on the first-order difference scatter plot can effectively depict the system stability of two-phase flow and can indicate the inhomogeneous distribution of the gas phase in gas–liquid two-phase flow.

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