Slugs and bubbles two-phase flow patterns dynamics in a minichannel are analysed. During the experiment, the volume flow rates of air and water were changed. We study transition of bubbles to slugs two-phase flow patterns using Fourier and multifractal approaches to optical transitivity signal. The sequences of light transmission time series are recorded by a laser-phototransistor sensor. Multifractal analysis helps to identify the two-phase structure and estimate the signal complexity. Especially, we discuss occurrence and identification of a self-aggregation phenomenon. These results are compared to corresponding Fourier spectra. The results indicate that the fractality is a an important factor influencing the distribution of the gas phase in water.
Gas-liquid two-phase flows play important role in production processes such as petroleum industry, chemical engineering, nuclear and thermal engineering , and proposed innovative techniques of thermal energy harvesting in two-phase oscillating structures of heat pipes . Such applications demand understanding and identification of flow dynamics and two-phase structures.
Usually, the two-phase flow patterns are identified by calculation of the average values of such parameters as mass or volume flux of phases, void fraction, phase momentum flux, dimensionless flow rate, superficial velocity, and others. In addition, the visual observations of gas and liquid arrangement inside the minichannel are used for flow pattern identifications. The alternative way of flow pattern identification is based on the analysis of dynamics of two-phase flow. Such analyses have been carried out recently by many scientists.
To increase the ability to identification and interpretation, Wang et al. , used selected nonlinear techniques including the Hurst and Lyapunov exponents, correlation dimension, and pseudo-phase-plane trajectory to analyse the pressure fluctuations of two-phase flow. The correlation dimension and Kolmogorov entropy were used by Jin et al.  to identify the oil-water flow patterns.
Novel measurement techniques and nonlinear results analysis led to the identification of flow patterns of the oil-gas-water mixture from mini-conductance probe array and vertical multielectrode array conductance sensor . The nonlinear analysis of the flow of oil–water also carried out  showing that such analysis is useful for identification of the flow patterns in inclined channel with larger diameters. The flow patterns in a vertical minichannel were studied by Faszczewski et al.  determining the borders between flow patterns. Stabilities of flows were also investigated by the largest Lyapunov exponent and correlation dimension . Recently, Gorski et al. ,  using the recurrence features studied two-phase flow bifurcation due to turbulence, self-aggregation phenomenon, and stable flow conditions in a two-phase flow through a mini-channel using the laser transitivity data. On the other hand, Lian et al.  suggested multiscale morphological analysis of conductance signals. In this article, we continue previous report  and apply the multifractal approach to study crossover from bubbles to slugs patterns.
2 Experimental Setup and Measurement Results
Flow patterns were recorded with using the Casio EX-F1 digital camera at 1200 fps (336×96 pixels). The content of the minichannel (bubbles or liquid) has been qualitatively assessed using the laserphototransistor sensor. The sensor consisted of a laser (which generated the laser beam with a diameter of 3 mm), the lens and silicon sensor (which were placed in focal point of lens). Bubbles inside the minichannel bend the light, which modifies the light intensity on the silicon sensor. Data from the sensors was acquired by the acquisition system (Data translation 9804, an accuracy of 1 mV for voltages in the range −10 to 10 V) at a sampling rate of 1 kHz. Finally, the pressure difference between the inlet and outlet of the minichannel was measured using the silicon pressure sensor MPX12DP.
Flows analysed in the work were configured with a vertical mini pipe of diameter 5 mm (see Fig. 1) Experiments were performed with a constant water flow rate 0.38 l/min, while in subsequent measurements, the change in air flow rates (l/min) was following: (a) 0.0025, (b) 0.0063, (c) 0.0105, (d) 0.0157, (e) 0.0216, (f) 0.0295, (g) 0.0381, (h) 0.0457, (i) 0.1, (j) 0.2, (k) 0.3, and (l) 0.4.
The corresponding camera shots of considered cases are presented in Figure 2. Note that increasing of air volume rate strongly influences on the size and distributions of bubbles. The evolution of the flow with increasing of air flow rate leads to an evolution of the flow pattern. In cases (a)–(c) (Fig. 2) in the minichannel, the flow of fairly small bubbles is observed; however, their variation in size is growing from case (a)–(c). In the case, medium size of bubbles dominates, whereas in cases (e)–(h), the number of bubbles is growing and they are changing their distribution. Finally, in the cases (i)–(j), the bubbles became very large, and in case (k), they coalesce to form the annular flow in case (l). Interestingly, in Figure 2h,i,j, we observe a region of self-organising of the bubbles phenomena covering the whole pipe cross-sectional area. In contrast, Figure 2l shows an almost continuous air slug with annular water flow. The aim of this article is to parametrise flow patterns by complexity measures.
Starting from the small level of air flow rate, we observe the highest level of light transmission and small fluctuations about this level. This is due to the high uniformity of the water volume. This uniformity is disturbed by number of small size bubbles which not always cross the limited vision area of laser-phototransistor detector. The main characteristics is expressed in the amplitude of fluctuations. This uniformity is destroyed with increasing the air flow rate. In the limit of larger air flow rate, this process is leading their merging to elongated bubble (slug).
To complete these observations, the average value of corresponding voltage at the phototransistor signal, <u>, together with the standard deviation are presented in Table 1. Note that the average value at first sightly increases with increasing of air flow rate. However, in fairly large air flow rate, it has the mostly opposite tendency. On the other hand, the standard deviation σu (Tab. 1) is initially increasing up to the case (i) with increasing air flow rate and then slowly decreasing.
|No||<u> [mV]||σu [mV]||h0||Δh|
Note also that the signal differs significantly from the original photographs (Fig. 2). Firstly, it is related to the limited penetration of the region of laser beam comparing to the whole cross-sectional visible in Figure 2. Secondly, the optical property of the moving borders between phases (air and water) is changing. This leads to additional laser light scattering due to variable dispersion properties. In the paper , an optical method based on an image processing was to measure the top and bottom film thicknesses for two-phase annular flow in a horizontal minichannel. It has been shown that fluctuations of gas film thickness have complex character. These fluctuations influence on the laser light beam refraction and reflection directions. Strictly speaking, the laser beam is changing the propagation direction. Such variations, during the flow, force the fluctuation in a refracted light direction and effectively scatter the laser beam producing fast oscillations in the photo-transistor sensor.
To show various timescales, we performed the Fourier transform of corresponding transitivity time series (Fig. 3). The current results are presented in Figure 4. Interestingly, the self-organisation phenomenon (Fig. 2h,i,j) is reflected by increase in Fourier components with given frequency. Obviously, the Fourier spectrum is smeared and not continues because of the fairly short time series and complex character of the flow. However, the height of of peaks is clearly maximal in Figure 4i (with the frequency about 40–50 Hz). In the annular flow case, we observe the flat distribution as expected for the Gaussian noise. This confirm that in that case variable refraction light phenomenon plays the main role.
3 Multifractal Analysis
In this section, we applied a multifractal analysis . It was used previously to characterise complex features  in many nonlinear dynamical systems. Especially, complex biological systems ,  were explored with multifractals. Recently, this concept has been applied in engineering systems, e.g. to examine seismic sequences ,  and study combustion engine dynamics , . Finally, this approach was used to characterise the structure of an inclined oil–water flow .
The main concept of multifractals is to analyse a spectrum of local exponential grows. Using a time series u(i) (Fig. 2), we study their evolution along the examined points i in time series u(i+δi). At vicinity each sampled voltage i, we look for the local characteristic exponent hi (usually noninteger), which estimates the corresponding difference:
Here, ah is a coefficient related to the exponent hi following the local internal separation.
Generally, the multifractal analysis of voltage u is based on a singularity spectrum D(h) of all hi exponents providing a precise quantitative description of the system behaviour , , . Formally, hi defines the Hölder exponent while the probability of its distribution D(h) coincides with the Hausdorff dimension of a dynamical system. The results of our calculations are shown in Figure 5. Note, the width of the spectrum D(h):
where hmin and hmax are defined by nodal values D(h)=0. Δh defines the complexity measure of the system response, whereas the h0, determined by the maximum of D(h), approximates the average exponent, indicating the randomness of pressure fluctuations. Note that the value h0=0.5 states the Brownian motion for which the consecutive steps are fully independent .
In other cases, for h0>0.5, the stochastic process is persistent (u(i) of neighbours i is correlated positively), whereas for h0<0.5, it is antipersistent (u(i) for neighbours i is correlated negatively). Summarising the interpretations of Δh and h0 one can say that the wider the range of possible fractal exponents, the richer the dynamical structure, while the larger h0 means more correlated fluctuations (less random).
The results of the mutifractal analysis are presented in Figure 5 and summarised in Table 1. Firstly, it should be noticed that h0 is continuously growing with increasing air flow rate. This monotonic increase is easy to clarify. Simply, in the limit of large air flow rate, the objects (air bubbles and slugs) are larger and, consequently, the laser beam test the same objects several times leading to persistent time series. On the other hand, the complexity measure Δh is changing in a nonmonotonous way. It decreases and increases again. The initial trend can be associated by the normalisation of the bubble sizes with increasing the air flow rate, while the final trend is based on the light beam scattering on variable borders between air and water media. The most spectacular is the increase in case (i) (see Fig. 5i). Its unexpected rise (Fig. 5i and Tab. 1 in ith row) could be associated with the self-aggregation phenomenon observed and discussed in our previous paper . Note that the greater the width of Δh becomes, the wider the probability distribution D(h) is and the more complex the self-aggregation phenomenon mode becomes. The Δh reaches the maximum in three cases (b), (f), (i). In these cases, the width of the multifractal spectrum becomes wider. From the physical point of view, it means that processes occurring in the different time scales influence on the flow pattern. In case (b) the flow of small bubbles of various diameters is observed, it seems that large number of bubbles and intensity of their coalescences are responsible for Δh increasing. In case (f), bubbles become larger in comparison with case (b). However, their size are too small to block the liquid flow inside the channel. Bubbles touch each other and coalesce. It seems that these processes are leading to increasing of Δh. On the other hand, in case (i), large bubbles are merging. This merging, signaled by a local increase of Δh, can be associated with self-organisation phenomenon possible for selected water flow rates. Furthermore, this result confirms the results in the Fourier spectrum where in Figure 4i we found the largest periodic components.
We presented arguments that the studied dynamical system of air–water flow can be described in terms of complexity parameters. Especially, the nonmonotonous behaviour of Δh signals the transition to self-aggregation. In this context, the spectacular increase in Figure 5i (see also case (i) in Tab. 1) is drawing attention. This transition is confirmed by observation in the Fourier spectrum (Fig. 4i), where we see the formation of a peak in the range [50–100] Hz and also by the maximum in terms of output voltage standard deviation σu (Tab. 1).
The drawback of the proposed method may be based on the laser-phototransistor measuring system. Firstly, the laser beam is penetrating the localised area. Secondly, the scattering of the laser beam on the air bubbles is very strongly influencing complexity, Δh, of the flow phenomenon. The later effect contributes positively to the index h0 which is the kind of a correlation measure. It worth to notice that laser beam fluctuations are themselves the consequence of the significant ratio between air and water and can be used to measure its effective value. Therefore, we conclude that both parameters Δh and h0, taken together, provide important information of the flow dynamics.
The presented research was funded by the National Science Centre, Poland – the number of decision: DEC-2013/09/B/ST8/02850.
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