For a given time series {*x*_{1}, *x*_{2}, *x*_{3}, …, *x*_{n}}, after reconstructing the phase space, we can define the phase space trajectory of the recurrence plot as follows:

$$\begin{array}{l}{\text{X}}_{i}=\text{[}x\text{(}t\text{),\hspace{0.17em}}x\text{(}t+\tau \text{),\hspace{0.17em}}x\text{(}t+\text{2}\tau \text{),\hspace{0.17em}}\dots \text{,\hspace{0.17em}}x\text{(}t+\text{(}m-\text{1)}\tau \text{)]}\\ \text{\hspace{1em}}\text{(}i=\text{1,\hspace{0.17em}2,\hspace{0.17em}}\dots \text{,\hspace{0.17em}}{N}_{1}\text{)}\end{array}$$(7)

where *m* is the embedding dimension and *τ* represents the time delay, *N*_{1}=*n–*(*m–*1)*τ*. Based on (7), for another time series {*y*_{1}, *y*_{2}, *y*_{3}, …, *y*_{n}}, we can introduce a second phase trajectory:

$$\begin{array}{l}{Y}_{j}=[y\mathrm{(}t\mathrm{)},\text{\hspace{0.17em}}y\mathrm{(}t+\tau \mathrm{)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}y\mathrm{(}t+2\tau \mathrm{)},\text{\hspace{0.17em}}\dots ,\text{\hspace{0.17em}}y\mathrm{(}t+\mathrm{(}m-1\mathrm{)}\tau \mathrm{)}]\\ \text{\hspace{1em}}\mathrm{(}j=1,\text{\hspace{0.17em}}2,\text{\hspace{0.17em}}\dots ,\text{\hspace{0.17em}}{N}_{2}\mathrm{)}\end{array}$$(8)

Comparing each point of X_{j} and Y_{j}, by calculating its vector distance we can obtain the cross recurrence matrix *CR*. When the vector distance is larger than a defined threshold value, the corresponding position in *CR* is 0 and we draw a white point at (*i*, *j*) in the cross recurrence plot, while the vector distance being smaller than the threshold value results in the corresponding position in *CR* being 1 and we draw a black point at (*i*, *j*) in the cross recurrence plot. The cross recurrence matrix *CR* can be calculated as follows:

$$C{R}_{i,j}=\Theta \text{(}\epsilon -\Vert {X}_{i}-{Y}_{i}\Vert \text{)}$$(9)

where *i*=1, 2, …, *N*_{1}, *j*=1, 2, …, *N*_{2}, *ε* refers to the threshold value of distance, ||·|| is Euler norm. Θ(·) is defined as the *Heaviside* function with the calculation formula given as follows:

$$\Theta \text{(}x\text{)}=\{\begin{array}{c}\text{1\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(}x>\text{0)}\\ \text{0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(}x<\text{0)}\end{array}$$(10)

A cross recurrence plot denotes the similarity of two trajectories. When two trajectories are close enough, the cross recurrence plot presents a recursive state. According to the textural structure, we can identify the differences between the two trajectories. Unlike the traditional recurrence plot, a cross recurrence plot is not always square and the length of the two trajectories can be different. We utilise the signals of an eight-electrode rotating electric field conductance sensor to calculate the cross recurrence plot of the four channels at the cross section, which represents the distribution characteristics of the mixed fluid in different directions. Due to the orthogonality on the geometry position of channel A and C, conducting cross nonlinear analysis using the signals sampled by channel A and C is more persuasive at uncovering the turbulence characteristics in oil–gas–water three-phase flow.

Define A(t) and C(t) as signals collected by channel A and C, respectively. We can reconstruct a phase space from the time series A(t) and C(t), then the phase space trajectories X_{a}(*i*) and Y_{c}(*j*) can be obtained. Afterwards, a cross recurrence plot algorithm is applied to investigate the recursive features of channel A and C from the eight-electrode rotating electric field conductance sensor. The same method regarding a traditional recurrence plot is used to choose the suitable threshold value for the cross recurrence plot. The formula for determining the threshold value is given as follows:

$${\epsilon}_{\text{AC}}=\alpha \times \sqrt{\text{std(A)}\times \text{std(C)}}$$(11)

where std(A) and std(C) denote the standard deviations of A(t) and C(t), *α* is an empirical coefficient based on the value range of the practical signal. The bigger empirical coefficient leads to a bigger value in the threshold, which causes an increase in the number of points satisfying the criteria that the distances of two vectors are smaller than the threshold value. This leads to an increase in the number of recursive points and more black points in the cross recurrence plot. However, an excessively large threshold value will result in the cross recurrence plot being unclear due to too many recursive points. In order to improve the efficiency of the calculation, we choose 10,000 as the length of the original time series corresponding to the data sampled in 5 s.

For the six typical flow patterns aforementioned, there are significant discrepancies in the textures in the cross recurrence plots. As shown in Figure 7, above the diagonals, *j*>*i* denotes the similarity of channel A and C with a certain time lag in channel C. For instance, when *j*–*i*=1, the data on the lines represents the similarity in the mixed fluid of channels A and C with a lag of one sampling time. Similarly, under the diagonals, *j*<*i* illustrates the similarity of channels C and A after a certain time lag in channel A. *i*–*j*=1 demonstrates the similarity of mixed fluid when channel C is one sampling time ahead of channel A.

Figure 7: Textural structure of cross recurrence plots under typical flow patterns (*f*_{o}=0.02).

(a) Slug flow: *U*_{sg}=0.0552 m/s, *U*_{sl}=0.0368 m/s (b) Slug-bubble transition flow: *U*_{sg}=0.0552 m/s, *U*_{sl}=0.736 m/s (c) Bubble flow: *U*_{sg}=0.0552 m/s, *U*_{sl}=1.1776 m/s (d) Slug-churn transition flow: *U*_{sg}=0.4416 m/s, *U*_{sl}=0.5888 m/s (e) Churn flow: *U*_{sg}=0.4416 m/s, *U*_{sl}=0.736 m/s (f) Churn-bubble transition flow: *U*_{sg}=0.4416 m/s, *U*_{sl}=1.0304 m/s.

In this study, the embedding dimension *m* and delay time *τ* are set as 3 and 2, while the empirical coefficient *α* is equal to 0.25. As shown in Figure 7a, the cross recurrence plot of slug flow shows the texture of many intermittent black blocks. In addition, there exist obvious line textures parallel to the diagonal direction in the cross recurrence plot, which means that the phase space trajectories of the two channels are close enough. In the main diagonal, the phase space trajectories of the two channels have no time delay, and the continuous line texture indicates that the fluid structure is symmetrical. However, the blank textures appearing occasionally in the line texture also indicate that the fluid structure is not always symmetrical. In addition, the cross recurrence plot exhibits an intermittent line structure in both the horizontal and vertical directions, which signifies that when one channel is fixed at a certain time, the state of its phase space trajectory at this time intermittently has similar characteristics to the state of the phase space trajectory of the other channel. This can be explained by the alternating movement of the gas slug and the liquid slug. Besides, the intermittent black blocks show that in slug flow the gas slug and the liquid slug appear quasi-periodically.

Figure 7b presents the cross recurrence plot texture structures corresponding to slug–bubble transition flow. It can be seen that due to the increasing turbulent energy, some of the gas slugs are broken up into gas bubbles with obvious random motion. Therefore, scattering points appear in the cross recurrence plot. Notably, in the cross recurrence plot some intermittent rectangular outlines can also be found, which coincides with the characteristics of slug flow.

The cross recurrence plot of bubble flow consists of lots of scattering points (Fig. 7c), indicating that the similarity of the dynamic system states in the two channels is stochastic and will not last for a certain duration of time. This corresponds to the random motion of the gas bubbles which leads to the frequent fluctuation in signals.

The cross recurrence plot of slug–churn transition flow is illustrated in Figure 7d. As seen, the small rectangle blocks appearing intermittently represent the existence of slug flow. However, the rectangle blocks shrink obviously in comparison with the texture of gas slugs shown in Figure 7a. The line textures degenerated from rectangle blocks indicate that small gas blocks oscillate in churn flow.

For churn flow, the cross recurrence plot given in Figure 7e is composed of line textures parallel to the direction of the diagonal but the lines are usually shorter than those in slug flow. This is because churn flow has a characteristic of oscillating dramatically, which leads to a shorter time for gas blocks passing through the sensor and this corresponds to short line structures in the cross recurrence plot. According to the distribution of texture, we discover that the oscillation in churn flow is relatively uniform. In the horizontal or vertical direction, intermittent line textures as shown in slug flow do not exist, which indicates that when one channel is fixed at a certain time, the state of its phase space trajectory is not similar to that of other channels at any time. The foregoing results show that the fluid motion in churn flow is very complex and the flow structure oscillation is violent and not repetitive.

Figure 7f shows the cross recurrence plot texture structures in regard to churn–bubble flow. When both the liquid and gas superficial velocities are high enough, the turbulent energy of the mixed fluid starts to present the capacity of decomposing gas blocks into small gas bubbles. Hence, the moving randomness of gas phase enhances. The line structures further degenerate into scattering points corresponding to the random motion of gas bubbles, whilst the existing line structures indicate the complex flow characteristics in churn flow.

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