## Abstract

Understanding of flow patterns and their transitions is significant to uncover the flow mechanics of two-phase flow. The local phase distribution and its fluctuations contain rich information regarding the flow structures. A wire-mesh sensor (WMS) was used to study the local phase fluctuations of horizontal gas–liquid two-phase flow, which was verified through comparing the reconstructed three-dimensional flow structure with photographs taken during the experiments. Each crossing point of the WMS is treated as a node, so the measurement on each node is the phase fraction in this local area. An undirected and unweighted flow pattern network was established based on connections that are formed by cross-correlating the time series of each node under different flow patterns. The structure of the flow pattern network reveals the relationship of the phase fluctuations at each node during flow pattern transition, which is then quantified by introducing the topological index of the complex network. The proposed analysis method using the WMS not only provides three-dimensional visualizations of the gas–liquid two-phase flow, but is also a thorough analysis for the structure of flow patterns and the characteristics of flow pattern transition.

This article is part of the themed issue ‘Supersensing through industrial process tomography’.

## 1. Introduction

Multiphase flow is widely encountered in industrial processes such as power plants, chemical engineering and petroleum transportation; understanding of its flow mechanics will facilitate the safety design and optimization of the process engineering and facilities [1]. Gas–liquid two-phase flows, especially transient ones, are in general difficult to analyse, model and predict due to the complex fluid mechanical interactions of the phases. They consist of a variety of local phase fluctuations, phase slippage and interactions, and combined chaotic and dissipative flow behaviour. In a horizontal pipe, gravity acts on the fluids, making the mixture asymmetrically flowing with complex interfaces [2]. Analysing the flow patterns and their transitions with the overall phase fraction fluctuation only reflects the overall flow characteristics, but the local characteristics, especially the simultaneous fluctuations of local phase fraction of a two-phase flow, are difficult to analyse due to the lack of an effective sensing system and data analysis tools.

A wire-mesh sensor (WMS) measures the local capacitance/resistance of two-phase flow in the local area of two crossing electrodes, and thus provides a reliable estimate of the local phase fractions in a visualized manner [3]. Based on the sensitive dielectric properties of the fluid, the WMS can be divided into two modalities: conductance WMS [3] and capacitance WMS [4]. These two modalities can be combined to form a dual-modality WMS for three-phase flow monitoring [5]. The measurement principle of WMS has also been applied to temperature field reconstruction by replacing the electrodes with temperature mesh sensors [6]; the temperatures were measured by thermocouples placed in the crossing points of transmitters and receivers. WMS has been used in analysing the flow conditions of gas–liquid two-phase flow by using the void fraction profile [7]. A capacitance WMS was compared with optical images obtained with a high-speed camera. The results showed that the WMS can extract the gas void fraction distribution, and quantitative flow structure information, such as bubble size distributions, which cannot be obtained from optical images [8]. An experimental characterization of vertical gas–liquid pipe flow for annular and liquid loading conditions using dual WMS was conducted; the different characteristic frequencies have been identified for different liquid viscosities, and the periodicity of liquid loading flow structure increases as liquid viscosity increases [9]. A method was presented that determined the radial profiles of velocity and angular displacement of the gaseous phase using two conductivity WMS; the data analysis was based on two-dimensional cross-correlation within concentric cylindrical planes [10]. Another method of flow velocity measurement was to use a three-plane sensor; the bubble velocity was evaluated by cross-correlating the instantaneous gas fraction profiles [11].

WMS were also used specifically in gas–liquid two-phase flow investigations. For instance, gas–liquid bubbly flows have been studied in [12,13]; they found that the bubble break-up rate increased with bubble velocity, but the bubble deceleration did not depend on the bubble velocity. For flow pattern studies, the feature-based flow pattern analysis was conducted by using either average phase fraction or local phase fraction [12,14].

The local phase fractions have not been investigated by considering the relationships between each local fraction of typical flow patterns. This relation could discover the correlation of the phase fraction fluctuations at different locations during the transition of one flow pattern to another, which will facilitate the understanding of the flow mechanics and then modelling of two-phase flow. To characterize the correlations between local phase fractions, one effective tool is multivariate analysis by treating each local phase fraction fluctuation as an independent time series. Many methods have been introduced for such applications; for instance, the multivariate multiscale entropy was used in electrical resistance tomography (ERT) to analyse the gas–water two-phase flow transition by treating each electrode as a single sensor [15], but not many works have been reported to correlate the actual local phase fractions to study the flow patterns.

To account for the above issues, we presented a network-based method of synthesizing the multiple measurement of a WMS to study the inter-relation of the local phase fractions of horizontal gas–liquid two-phase flow. The relationships were established by the cross-correlation between each local phase fraction to characterize the flow patterns, and were validated through comparison with the reconstructed three-dimensional flow structures.

The gas–liquid two-phase flow is complex, and the WMS provides a set of high-dimension time series, so the detailed local phase fraction shows a very complex response with the change of flow velocity. The network-based analysis provides a comprehensive understanding of the flow conditions with multiple measurements. ‘Network’ describes the complex systems that contain a large number of components (or subsystems) correlated with each another. It can be abstracted as many vertices or nodes representing these components of a system, along with connections between them, which are called edges, representing their interactions. Networks can be roughly divided into four types according to the relationships between each node: binary directed networks, binary undirected networks, weighted directed networks and weighted undirected networks.

Complex network analysis is a subset of graph theory that focuses on topologically complex networks, performing on graphs. In 1735, the solution of the Konigsberg bridge problem by Euler was considered the first true proof in network theory [16]. In the late 1950s and 1960s, network theory was further developed after the emergence of the ER random graphs theory proposed by Erdös & Rényi [17]. Later, the ER random graph was used as the basic model for studying complex networks. A lot of important properties of ER random graphs would emerge with the increase in network scale. However, a random graph is inadequate to describe some important properties of real-world networks. In the 1990s, networks experienced a breakthrough due to the appearance of two network models, the small-world networks [18] and the scale-free networks [19]. In addition, a number of researches of real networks have proved that real-world networks are neither regular networks nor random networks, but have the properties of both small-world networks and scale-free networks. Since then, complex network theory has undergone a fast development and has been successfully applied to many complex systems [20–22].

In this work, we introduced the network analysis method to analyse the local phase fraction obtained by a 16×16 WMS to study the characteristics of the gas–liquid two-phase flow patterns and their transitions. By establishing the relations between the local phase fraction fluctuations, the inter-connections between each sensing node are presented by using the cross-correlation method. The structure of the network was then quantified with the network's features to reveal the distinct flow behaviours of gas–liquid two-phase flow.

## 2. Sensor and experiments

### (a) Wire-mesh sensor

A PXI-based conductivity WMS for laboratory research was developed for local phase fraction analysis [23]. It consists of 16×16 electrodes and 208 effective measurements of local phase fraction at a data acquisition rate of 200 frames of imaging data per second. The sensor was fabricated in a 50 mm inner diameter Plexiglas pipe, the electrodes are 0.18 mm in diameter and equally distributed at 3.1 mm separation in each plane, and the axial distance between two electrode planes is 2 mm. The signal generation and switching logic is controlled by an field-programmable gate array (FPGA), and the data are collected through the PXI bus to the computer and stored in the hard drive by dedicated software. The cross-talk between electrodes is eliminated due to the measurement principle (one electrode is excited by voltage while the others remain grounded), which has been verified by numerical simulations. The structure of the WMS sensor and system is shown in figure 1. The efficiency and accuracy of estimating the phase fraction of gas–liquid two-phase flow has been verified through static and dynamic experiments.

When using a WMS to analyse the flow patterns, one concern is the influence of the intrusive electrodes of the WMS on the bubble size and distribution. The distortion by the WMS of the bubble size in a 50×50 mm pipe has been studied by Nuryadin *et al*. [24], who found that the WMS had little effect (within ±10% deviation) on the bubble size through comparison with visual observations. In addition, the disturbance caused by the WMS to the gas–liquid two-phase flow was studied by using a WMS of 8×32 crossing points with spatial resolution of 2.22×3.03 mm and wire diameter of 0.125 mm [25]. It is found that the two-phase mixture is slightly decelerated by the sensor planes, and the deformation of the flow structure can only be considered when a quantified phase fraction is required.

### (b) Experimental facility

The flow experiments were conducted in a horizontal test pipe of the oil–gas–water multiphase flow facility of Tianjin University, as illustrated in figure 2. The testing pipeline is manufactured of steel tubing with an internal diameter of 50 mm and a total length of approximately 16.6 m. A 32 m height water tower provides a stable water head (about 0.32 MPa) to the testing pipeline. Tap water and air were used as the experimental fluids. Water was pumped into the entrance through the water tower, and air was pumped into the entrance through a refrigerated dryer after an air compressor. The two fluids were mixed by a nozzle at the beginning of the horizontal pipe. Standard single-phase flow meters with a precision of ±0.5% were installed on the air and water inlet pipes. The air after the experiments was released to the atmosphere, and water was led back into the water tank for re-use [15].

The WMS was implemented 15 m downstream from the inlet, as shown in figure 2, allowing a 300-diameter distance for the gas and water to settle and the flow patterns to fully develop before reaching the testing section. A transparent pipe section was inserted into the pipeline for the camera recording the flow process.

### (c) Experiments and observations

The flow patterns observed in the experiments include stratified/wavy stratified flow, bubbly flow, plug flow (which is also known as elongated bubble flow), slug flow and churn flow, as demonstrated by the photos in figure 3. The ideal form of annular flow was not observed during the experiments, because the highest air velocity in the experiments was restricted by the facility. The experiments were conducted by increasing the air flow rate while fixing the water flow rate in each test group. Then the water flow rate was increased by 4 m^{3} *h*^{−1} for the next group of experiments.

The flow structures reconstructed by the WMS are shown in figure 4. The interface between gas and liquid is seen clearly in the reconstructed three-dimensional images, especially for the stratified flow, the plug flow and the slug flow. For the bubbly flow, since the bubbles are small in size and accumulate at the top of the pipe, only the upper part of the pipe shows a small amount of bubbles. In the churn flow, the liquid was pushed to the pipe wall, forming a liquid film around the pipe perimeter, and the pipe appears to be filled with gas. The liquid film is thick at the bottom due to gravity, so the bottom shows accumulated liquid in the churn flow. Comparing with figure 3, the reconstructed flow structure is close to the actual flow structures, which validates the measured data that will be used for further flow pattern analysis.

## 3. Flow pattern analysis

The flow patterns are usually analysed with the images reconstructed by using the WMS, or the time–frequency representation of the local phase fraction. But the inter-relations between each local phase fraction have rarely been analysed. This section focuses on the methods of building and analysing the connections between local phase fractions, to explore the mechanisms regarding the flow patterns and their transitions.

### (a) Inter-connection analysis by using wire-mesh sensor

For the inter-connection analysis, a node is defined as the location of a sensing point of the WMS in figure 5*b*. It provides the local phase fraction measured in the said crossing area, and thus represents the location of each local phase fraction in figure 5*a*. Therefore, an inter-connection between two nodes reveals the similarity of the phase fluctuations in these two local areas. In this work, the 208 nodes correspond to the 208 measured phase fractions in each frame of the sliced tomography image in figure 4.

The connection between two nodes is named an ‘edge’ in network analysis, which is quantified by the cross-correlation coefficient of two time series of the phase fraction of two nodes. For each pair of local void fraction time series, *α*_{i}(*t*) and *α*_{j}(*t*), the Pearson correlation coefficient *C*(*i*,*j*) can be expressed as
3.1where *n* is the length of void fraction time series, *k* is the time step in each measurement time series, and and are the average value of time series *α*_{i} and *α*_{j}, respectively, i.e. and . The coefficient *C*(*i*,*j*) is between −1 and 1, describing the strength of correlation between nodes *i* and *j*, where *C*(*i*,*j*)=0 implies no correlation, and *C*(*i*,*j*)=±1 implies complete positive or negative correlation. *C* is a symmetric correlation matrix and its dimension is 208×208 according to the number of nodes. The correlation matrix is named the ‘weighted adjacency matrix’, which is depicted in figure 6*a* (warm and cool colours represent positive and negative correlation, respectively; warm (red) colours being towards bottom left and cool (blue) colours being at the top and the right).

The weighted adjacency matrix *C* can be converted into a binary adjacency matrix *A* by choosing a proper threshold *r*. If |*C*(*i*,*j*)|≥*r*, an edge is established connecting nodes *i* and *j*, correspondingly assigning *A*(*i*,*j*) to 1; otherwise, assigning *A*(*i*,*j*) to 0 and no edge exists between nodes *i* and *j*.

The elements of adjacency matrix *A* describe the similarity between two nodes, by justifying whether an edge exists. Many statistical characteristics can be extracted from this adjacency matrix, and a change of threshold will lead to different network topological structures and features. Consequently, it is important to choose an optimal threshold *r*_{o} for constructing a network. It is well known in practice that 0≤|*C*(*i*,*j*)|<0.1 can approximately be considered as no correlation, and 0.1≤|*C*(*i*,*j*)|<0.3, 0.3≤|*C*(*i*,*j*)|<0.5 and 0.5≤|*C*(*i*,*j*)|≤1 correspond to weak correlation, moderate correlation and strong correlation, respectively. Besides, if the threshold is too low, many weak connections may emerge between node pairs, resulting in the increase of noise, even though the network could retain more information. On the contrary, if the threshold is too high, useful information might be discarded with noise. Hence, the threshold is preliminarily restricted to the domain 0.5≤*r*≤0.8. The method to determine the optimal threshold *r*_{o} can be seen in §??sec4*c* for details. With this threshold, all the nodes and edges could construct a gas–liquid two-phase flow pattern network, and the topological structure and statistical characteristics of this network are stored in its adjacency matrix *A*, which is depicted in figure 6*b*. This connection (black dots) indicates that the fluctuations of two nodes are caused by the same fluid structure, therefore the two nodes present similar fluctuations.

### (b) Flow pattern networks

By establishing the connections and considering the spatial distribution of each node, the two-phase flow pattern networks for several typical flow patterns have been constructed in figure 7. The lines represent the connections between each node, so the structure of the network reveals the similarity of phase fraction fluctuations between each local area, and also the flow structure distributions.

In stratified flow, the upper part of the pipe is occupied by the gas, and the gas and the water form two continuous layers vertically in the pipe. The fluctuations only exist at the interface between gas and water. Therefore, edges exist only at the gas–water interface in figure 7*a*. The height of the accumulated area of the edges reflects the average height and position of the fluctuating waves in the stratified flow. It indicates that the fluctuations exist only around the interface of gas and water, and the shape of the fluctuations can be identified through the shape of the connections.

In the bubbly flow, the gas phase flows almost along the top centre of the pipe due to gravity. The size of the bubbles is usually small as the gas fraction is very low in this flow pattern. Therefore, as shown in figure 7*b*, the sparse connections are distributed on the locations with fluctuating phase fraction, which reveals the overall weak correlation (suggesting nearly random fluctuations of each bubble). Some connections only exist between two nodes, which are caused by the small gas bubbles that were not fluctuating with the big bubbles, but at their own frequencies; thus the fluctuations of these isolated nodes are not correlated with other nodes.

With increasing gas velocity, small bubbles start coalescing into big bubbles, and the flow pattern transitioned to plug flow. Many big bubbles or gas plugs flow along the upper part of the pipe at a certain frequency, and some entrained gas phase mixed in the liquid plug between two gas plugs is observed in figures 3 and 4. Therefore, as shown in figure 7*c*, the fluctuation frequencies of local phase fractions are uniform in the area where the gas plug exists, which can be reflected by a large number of connections between nodes in those areas, demonstrating the strong correlation of local phase fraction fluctuations in plug flow. At the bottom of the flow pattern network, some isolated connections exist, indicating that some small bubbles existed and fluctuated independently from the gas plug.

Further increasing the gas flow, the size of the gas bubbles increases accordingly. These large gas bubbles occupying the majority of the cross section of the pipe, and the liquid phase carrying many small bubbles and big bubbles appears alternately. However, liquid phase in the bottom is scooted up by these large gas slugs when they flow through the pipe, leading to the phase fluctuations at the bottom correlating with those above them, creating new connections of the nodes at the bottom with those at the top, as shown in figure 7*d*. Some isolated connections of dispersed bubbles also exist at the bottom.

With an increase in gas flow rate, slug–annular transitional flow (churn flow) occurred, where the high-speed gas flow created an impacting effect pushing the liquid to the pipe wall and forming a thin liquid film on the pipe wall. In a horizontal pipe, the liquid film at the bottom of the pipe is thicker than on other parts of the pipe, and the wave at the interface between the liquid film and the gas core is similar around the pipe wall (figure 4*e*). At the centre of the pipe, before forming an ideal annular flow at a much higher gas velocity, many liquid droplets are entrained in the gas core (figure 3*e*), causing complex fluctuations in the central area, which could also be synchronized with the fluctuations at the film surfaces. As shown in figure 3*e*, the churn flow pattern network has two major parts; the upper part consists of the connections caused by the gas core, and the lower part contains the connections caused by the waves on the liquid film. The correlations between the gas core and the liquid film at the bottom is weak, suggesting that slippage is significant and the similarities between gas and liquid are reduced. Compared with the strength of network connections of slug flow as depicted in figure 7*d*, the strength of related degree of churn flow is weaker (figure 7*e*), owing to the disordered fluctuations of entrained liquids in the gas core.

Figure 7 demonstrates that the flow pattern network provides a visual characterization to facilitate the understanding of the spatial correlation of air–water two-phase, reflected by the strength of the connections between each node. The more edges that appear in the flow pattern network, the stronger is the overall correlation of the phase fluctuations the flow pattern presents. Hence, network structures can effectively characterize the gas–liquid flow structure.

## 4. Flow pattern network analysis

### (a) Features for characterizing the networks

In order to further characterize the flow pattern networks, some features should be extracted to quantify the networks. Consider an undirected and unweighted network *G*=(*N*,*E*), which is abstracted as node set *N* and edge set *E*, where *N* and *E* are the number of nodes and edges in the network. The most convenient way of describing a network is the symmetric adjacency matrix *A*(*i*,*j*), *i*=1,2,…,*N*. The following quantification parameters can be calculated with these coefficients.

(1) *Degree*. In an undirected network, the degree of a node is the number of edges connected to it:
4.1Degree represents the number of connections between a single node and the other nodes. It is the most important statistical characteristic of inter-connected nodes in a network. The greater the degree of a node, the more important the node is in the network. Network average degree is the average of all nodes in the network:
4.2

(2) *Clustering coefficient*. The node clustering coefficient depicts the characteristic of the cluster degree of the nodes in the network. It characterizes the density of connections to a node. For node *i* of degree *k*_{i}, the local clustering coefficient of the node without degree correlation is defined as [26]
4.3where *R*_{i} is the numbered edges between neighbours of node *i*, and *E*_{i} is the maximum number of edges possible between the neighbours of node *i*. For nodes of degree *k*_{i}=0 or 1, the clustering coefficient is defined as *C*_{i}=0. A high clustering coefficient quantifies high intensity of connections between the neighbouring nodes of one specific node [27]. The network clustering coefficient is the average of local clustering coefficients for all nodes in the network.

(3) *Average path length* (*characteristic path length*). The distance between node *i* and node *j* is defined as the shortest path length *d*_{ij} between them, which is one of the paths connecting the two nodes with the minimum length. Consequently, the average path length *L* is the average value of the shortest path lengths *d*_{ij} of all pairs of nodes:
4.4

(4) *Global efficiency*. In many practical applications, there are also unconnected nodes in networks and their shortest path length is zero by definition. As a result, the average path length is invalid. Latora & Marchiori [28] put forward a measure called global efficiency:
4.5The global efficiency represents the efficiency of the network for sending information between nodes, which is proportional to the reciprocal of the nodes’ distance. So the average path length can be defined by the harmonic mean of the shortest path length, i.e. the reciprocal of the global efficiency:
4.6

(5) *Network density*. Network density is the ratio between the number of edges in the network and the maximum possible number of edges. It represents the sparseness of the network:
4.7

(6) *Network entropy*. Entropy characterizes the homogeneity of the system energy distribution or the stability of the object state. The more ordered a system is, the lower the entropy the system possesses. Introducing the concept of Shannon entropy [29], network entropy *E* is given by
4.8where *I*_{i} is the importance of node *i* with degree *k*_{i}. If the importance of nodes are the same in a network (unordered), the entropy value is high and vice versa. Network entropy *E* can be used to measure the ordering degree for the degree distribution in the network or the complexity of the corresponding system.

### (b) Evolution of flow pattern network features with threshold

Considering that the flow pattern network was constructed from the correlation of local phase fluctuations, it would present different statistical characteristics when the threshold changes. As a result, the influence of the threshold on the flow pattern network structure should be closely scrutinized. Since the threshold is to define the existence of an edge between two nodes, a high threshold will result in a simple structure of the flow pattern network since it eliminates a majority of the links, while a low threshold will result in a much more complex network by introducing more connections that may be caused by the noise. To achieve this, we constructed networks with thresholds ranging from 0.2 to 0.8 (step size defined as 0.01) to understand the evolution of the flow pattern networks, with special focus on the network statistical features at the flow pattern transition. Here, network measures such as network density, average path length, global efficiency, clustering coefficient and network entropy were used to characterize the flow pattern network.

There are disconnected parts or isolated nodes in flow pattern networks, which lead to infinite average path length. So it is necessary to calculate the average path length via the equations proposed by Latora & Marchiori [28] instead of the traditional method. In addition, a normalized *E* for the network entropy *E*_{global} is used in the study. The measures of network changes with the thresholds are shown in figures 8 and 9 for different flow patterns when changing the water and gas flow rates.

Figure 8 demonstrates the flow pattern changes at a fixed water flow rate of 0.5 m^{3} *h*^{−1}. At this flow rate, when the gas flow rate increases, the flow pattern transitioned from stratified flow to wavy flow (with more waves at the gas–water interface of the stratified flow), then to churn flow. During this process, the flow pattern becomes more complex. Figure 8*a* shows that the flow pattern network density decreases when increasing the threshold. As the threshold *r* increases, the number of connections among the nodes drops, and the network density obviously decreases. However, in the transition from stratified flow to wavy flow and churn flow, the network density becomes larger, and the fluctuation of the two-phase flow also affects more sensing points of the WMS, inducing more nodes to correlate with each other and forming more connections. In addition, the correlation strength of churn flow becomes higher than that of wavy flow and stratified flow, suggesting that the network density is sensitive to the flow pattern transition, and can characterize the spatial flow structure of gas–liquid two-phase flow.

In figure 8*b*, the average path length of each flow pattern increases with increasing threshold. This increasing trend becomes exponential when the threshold exceeds 0.7, which is caused by the increased length of the shortest path between two nodes. However, the average path length decreases when the gas flow rate increases, suggesting that the flow pattern network structure tends to be more compact and the global connectivity of the network is higher. The global efficiency of the flow pattern network, which is the reciprocal of the average path length, decreases quickly with the threshold, as shown in figure 8*c*, because shorter average path leads to a higher efficiency of connections.

The clustering coefficient and network entropy show similar trends with increasing threshold in figure 8*d*,*e*. The increase in the clustering coefficient during the transition of the flow pattern reflects that the nodes connected to the same node mutually correlate with each other at a high probability, i.e. the average connection density among the neighbouring nodes of each single node increases. For the network entropy, it decreases with increasing threshold, because increasing the threshold will enlarge the difference in the significance of a node, leading the degree distribution of the flow pattern network into order. But the network entropy increases as the flow pattern evolves from stratified flow to churn flow, which reveals the increased complexity of the corresponding two-phase flow when increasing the gas flow rate. Hence, the spatial flow structure and complexity of gas–liquid two-phase flow can be characterized by clustering coefficient and network entropy, which are sensitive to the flow pattern transition.

At a higher water flow rate, more flow patterns appeared. Figure 9 shows the flow pattern networks’ development with increasing gas flow rate while keeping a constant water flow rate of 8 m^{3} h^{−1}. Bubbly flow, plug flow, slug flow and slug–churn transitional flow were observed in this set of experiments. Bubbly flow has a distinctive trend with the threshold, owing to the sparse connections of its flow pattern network which are caused by the random distribution and flow process of bubbles in a high flow rate of water. During the transition from bubbly flow to slug flow, the distributions of network measures are similar to those in figure 8. However, when the gas velocity is high, the network measures of slug–churn flow distribute between that of bubbly flow and slug flow rather than continue to increase or decrease as the order from bubble flow to slug flow. Because edges are determined by the correlation strength among nodes through local phase fraction fluctuation, the overall strength of the related degree of churn flow pattern network falls between plug flow and slug flow (figure 7), which then leads to the network measures of transitional flow distributing between that of bubble flow and slug flow.

### (c) Choosing the optimal threshold

The above research suggested that an optimal threshold *r*_{o} should be determined in order to quantify the structure of flow pattern networks on the same basis. The threshold is preliminarily determined to be in the domain 0.5≤*r*≤0.8 with the aim to obtaining the binary adjacency matrix that can describe the characteristics of the flow pattern network in the previous section. However, there has not been a general criterion for selecting an optimal threshold *r*_{o}. We selected the optimal threshold by the criterion of relative stability of network modularity [30], as demonstrated in figure 10. A neighbourhood of *r*_{o} should exist in which the modularity or the structure of the flow pattern network remains stable. For the calculation of modularity [16], we applied the Louvain community structure algorithm to divide the vertices into groups; then the modularity was calculated by
4.9where elements *e*_{ii} are equal to the fraction of edges that fall within group *i*, and , where *e*_{ij} is one-half of the fraction of edges in the network that connect vertices in group *i* to those in group *j*. In this case, *r*_{o} can be found through simulating a set of flow pattern networks, by increasing the threshold while keeping the modularity almost unchanged. In this study, the optimal threshold *r*_{o}=0.62 is selected.

### (d) Node degree analysis of flow pattern network

Because the flow pattern networks are constructed by using the local phase fraction time series and their correlations, the spatial distributions of flow structure are consequently embedded into the flow pattern networks, which is reflected by the node degree of the network. We construct the networks of the typical flow patterns by using the optimal threshold *r*_{o}=0.62, and the node degree was selected as an index in the analysis, because the node degree directly reflects the number of connections of one node to other nodes. The statistics of the node degree of typical flow pattern networks are shown in figure 11. The stratified flow and the bubbly flow have only a small fraction of nodes that contribute to the network construction, and thus present a small number of non-zero-degree nodes. The plug flow, slug flow and churn flow all show intensive connections and the majority of the nodes are connected with some other nodes. The difference is that the slug flow has the highest node degree, suggesting more nodes were connected, and the churn flow possesses more non-zero-degree nodes in the networks, suggesting its large and impacting structure could induce large fluctuations that affect most of the sensing points of WMS.

In order to further reveal the underlying physics of each network, and investigate the characteristics of flow pattern networks during flow pattern transition, the texture of the binary adjacency matrix is employed in figure 12. There, a black dot means that the corresponding two nodes are related with each other.

Owing to the existence of weak oscillation in the air–water phase interface in stratified flow, only a few concentrated local phase fraction fluctuations relate with each other, so stability is the main property of a spatial flowing structure, which is reflected by the texture of the adjacency matrix, i.e. a small black block developed along the principal diagonal at the lower left corner, as shown in figure 12*a*. In bubbly flow (figure 12*b*), because many small bubble as well as some big bubbles flow along the top of the pipe, fewer scattered local phase fraction fluctuations relate with each other. So randomness is the main feature of bubbly flow's spatial structure, as reflected by the texture of its adjacency matrix, which presents a comparatively sparse point–lines structure. With the increase in gas flow velocity, as the smooth movement of big bubbles or gas plugs, and the entrained gas phase in the liquid phase between two gas plugs, the local phase fraction fluctuations of plug flow show periodic features, which is reflected in figure 12*c*, where a texture of black irregular leaf shape exists due to point–surfaces developing along the principal diagonal from the lower left corner. Hence, from figure 12*b* to *c*, it can be found that there are more high-degree nodes and fewer low-degree nodes in the plug flow pattern network than in the bubbly flow, indicating that the spatial correlation strength of plug flow is much higher than bubble flow.

Further increase in gas velocity induces the slug flow. Figure 12*d* shows the texture of slug flow's adjacency matrix transformed into a regular big black block from the irregular leaf shape of plug flow. More high-degree nodes and low-degree nodes are shown in the slug flow pattern network than in plug flow, which demonstrates the stronger correlation of inter-phase fluctuating void fraction at a larger flowing space of slug flow. When the gas velocity continues to increase, the flow pattern evolves from slug flow to churn flow, and the corresponding texture of the adjacency matrix evolves from figure 12*d* to *e*. The adjacency matrix texture in churn flow shows a significant difference from other flow patterns, exhibiting a banded dentate shape developing along the principal diagonal. The degree of each node is similar and relatively small, indicating that the local phase fraction fluctuation at one node is just strongly related to the nodes that are in the vicinity of that node rather than the nodes far away from it.

### (e) Discussions

The length of phase fraction time series *n* for each node is important to obtain a meaningful cross-correlation coefficient for an edge. If *n* is small, the dataset would be too short to reflect and distinguish the characteristics of flow regimes. On the contrary, a large *n* would cause a redundant data problem. So, we judge the best *n* value based on the perspective that the WMS data size is adequate to fully describe the flow structure of each flow regime through reconstructed images, while avoiding redundancy. During experiments, the data were collected for 30 s at an acquisition rate of 200 frames of imaging data per second. Three-dimensional flow structures reconstructed by the WMS data in 10 s show that the interface between gas and liquid could be clearly identified. Meanwhile, a small value of *n* could produce an inaccurate network structure due to a lack of completeness in the data description of the flow regime, especially for plug flow and slug flow. So we extended the length of *n* to 5000 for network construction, while keeping the computational cost safe.

The creation of the network is objectively based on the strict selection of the parameters introduced in this paper, and the node has a clear physical meaning as the physical location and phase fraction. So the network is not arbitrary. This flow pattern network reveals the similarity of phase fluctuations between each pair of nodes using multi-point measurements, and its statistical features can reflect various features of the flow patterns and their transitions. The node degree of the networks based on the texture of the adjacency matrix can also graphically describe the spatial structures of flow patterns. Then, the flow regime recognition can be achieved with these parameters and an identification algorithm such as an artificial neural network (ANN) or fuzzy clustering [31–33]. The identification algorithms adopt a supervision-based learning and testing with features extracted from the measured signals which are usually differential pressures [31,32] or the overall phase fraction [33]. Comparing with these features, the proposed method provides a new set of features (the quantification parameters of the flow pattern network) from the perspective of the similarity of local phase fraction fluctuations, so that the flow pattern can be clearly distinguished, and, more importantly, the underlying physics of local fluctuations can be revealed, which is the advantage of this method. However, as a detailed analysis tool, it requires a large amount of computational resources to calculate the cross-correlation between each pair of nodes. Therefore, it is not ready for online flow pattern recognition.

## 5. Conclusion

Understanding of the flow pattern structures and their transition mechanisms are of practical importance and academic value in two-phase flow studies. The flow patterns of gas–liquid two-phase flow are analysed with a 16×16 wire-mesh sensor. Instead of analysing the flow patterns through visual inspections on reconstructed images, the local phase fraction fluctuations provided by each crossing point of the WMS were analysed through multivariate time-series analysis. The sensing nodes were connected by cross-correlating with each other, and a flow pattern network was obtained for each flow condition to demonstrate the inter-connections between each local phase fraction. The change of the network's topology with the threshold was studied, and an optimal threshold was obtained to finalize the construction of the flow pattern network.

The statistics of the node degree of flow pattern networks and further physical implication analysis based on the texture of the adjacency matrix in different flow patterns are efficient to graphically describe the spatial structures of two-phase flow patterns and their transitions. The present method only presents an overall characterization of flow pattern networks by using the statistical features of networks. More detailed and local characteristics of the flow patterns and their transitions could be carried out based on the present work.

## Authors' contributions

W.L.L. carried out the experiments and data processing. C.T. performed the data analysis and drafted the manuscript. F.D. conceived of and designed the study. All authors read and approved the manuscript.

## Competing interests

The authors declare that they have no competing interests.

## Funding

This work is supported by the National Natural Science Foundation of China (nos. 61473206 and 61227006).

## Footnotes

One contribution of 10 to a theme issue ‘Super-sensing through industrial process tomography’.

- Accepted February 16, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.