Cross-correlation analysis of interfacial wave and droplet entrainment in horizontal liquid-liquid two-phase flows

Accepted Manuscript Cross-correlation analysis of interfacial wave and droplet entrainment in horizontal liquid-liquid two-phase flows Cong Yan, Lu-Sheng Zhai, Hong-Xin Zhang, Hong-Mei Wang, Ning-De Jin PII: DOI: Reference:

S1385-8947(17)30389-3 http://dx.doi.org/10.1016/j.cej.2017.03.044 CEJ 16640

To appear in:

Chemical Engineering Journal

Received Date: Revised Date: Accepted Date:

24 December 2016 11 March 2017 13 March 2017

Please cite this article as: C. Yan, L-S. Zhai, H-X. Zhang, H-M. Wang, N-D. Jin, Cross-correlation analysis of interfacial wave and droplet entrainment in horizontal liquid-liquid two-phase flows, Chemical Engineering Journal (2017), doi: http://dx.doi.org/10.1016/j.cej.2017.03.044

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Cross-correlation analysis of interfacial wave and droplet entrainment in horizontal liquid-liquid two-phase flows1 Cong Yan, Lu-Sheng Zhai*, Hong-Xin Zhang, Hong-Mei Wang, Ning-De Jin School of electrical and information engineering, Tianjin University, Tianjin 300072, China

ABSTRACT: Horizontal oil-water two-phase flows present complex temporal and spatial

structures. The cross-correlation analysis of the flow structures is of significance for uncovering the nonlinear dynamics of the oil-water flows. In this study, we first employ a detrended cross-correlation analysis (DCCA) method to investigate the cross-correlation characteristics of two series generated by two-component ARFIMA processes with an adjustable coupling strength. On this basis, to avoid spuriously high cross-correlations caused by noises, we conduct an anti-noise study applying a detrended partial cross-correlation analysis (DPCCA) to ARFIMA processes mixed with periodic signal, stochastic signal and chaotic signal, respectively. It’s found that the DPCCA can effectively reveal the intrinsic cross-correlations of coupled series. Through carrying out an experiment of horizontal oil-water two-phase flows, the upstream and downstream flow information is collected by a conductance cross-correlation velocity probe. The DPCCA algorithm is used to calculate the multi-scale cross-correlation coefficient of the upstream and downstream flow structures. The results indicate that the DPCCA cross-correlation coefficient is very sensitive to the dynamics of the oil-water interfacial wave and the entrained droplets, and can serve as an effective indicator of horizontal oil-water two-phase flow structures.

Keywords: Horizontal oil-water two-phase flow; Flow structure; Multi-scale; Crosscorrelation

∗

Corresponding author. Tel./fax: +86-22-27407641.

E-mail address: [email protected] 1

1. Introduction Horizontal oil-water two-phase flows are widely encountered in petroleum industries. The measurement of oil-water flow parameters is of great importance for the optimization of the oil production process. However, the complex dynamics of interfacial wave and droplet entrainment in the horizontal pipe put forward challenges for the monitoring of the flow parameters, such as total flow rate and phase volume fraction. Uncovering the dynamics of the wavy oil-water interface and the entrained droplets is of significance for improving the measurement accuracy of flow parameters. Nonlinear analysis methods based on observed time series are beneficial to reveal the dynamics of complex systems [1-6]. Multi-phase flow is a typical multi-scale nonlinear dynamic system. Zhao et al. [7] presented the multi-fractal characteristic of bubbling fluidized bed in micro-scale, meso-scale and macro-scale by wavelet method. Zheng et al. [8] used orthogonal wavelet multi-resolution analysis and power spectrum to reveal the multi-scale characteristics of particle fluctuation velocity in a horizontal self-excited gas-solid two-phase flow. Based on differential pressure signals, Wu et al. [9] extracted the multi-fractal characteristics of oil-gas-water flow patterns and realized the intelligent identification of flow regimes by a neural network method. Wu et al. [10] used chaos analysis to study the nonlinear and multi-scale flow behaviors in gas-solids two-phase flow systems. Notably, the internal variables or subsystem variables of a complex system usually present coupling characteristics, so the nonlinear analysis based on the causality, information transferring and cross-correlations of system internal variable have became a popular topic [11-14]. Thomas [15] proposed transfer entropy to quantify the information flow between two systems or between constituent subsystems of a complex system. Then, the transfer entropy was widely used to reveal the dynamics of complex nonlinear subsystems and uncover the causal

2

relationship in the information transferring [16-21]. Zhai et al. [22] employed the transfer entropy theory to investigate the information transferring characteristics of local gas-liquid flow structures in an annular space. Podobnik and Stanley [23] proposed a detrended cross-correlation analysis (DCCA) method to investigate long-range cross-correlations between two non-stationary time series. Horvatic et al. [24] applied the DCCA method with varying order of the polynomial to meteorological data and demonstrate its ability in subtracting periodic trends. Based on the DCCA, Zebende et al. [25] introduced a DCCA cross-correlation coefficient quantifying the cross-correlation level of two non-stationary series on different scales. The ability of the DCCA coefficient to measure correlation level between non-stationary series was validated by Podobnik et al. [26] and Ladislav [27]. The DCCA cross-correlation coefficient has been widely used in finance, meteorology and economy fields [28-30]. Zhou et al. [31] introduced the MF-DFA [32] and the DCCA into a multi-fractal detrended cross-correlation analysis (MF-DCCA), which can be used to reveal the multi-fractal features of two cross-correlated non-stationary signals. Ruan et al. [33] investigated the cross-correlations between price and volume in Chinese gold spot and futures markets using the MF-DCCA method. Oswiecimka et al. [34] proposed a multifractal cross-correlation analysis (MFCCA), which is a consistent extension of the detrended cross-correlation analysis and free of the limitations of MFDCCA. Jiang and Zhou [35] proposed another class of MF-DCCA algorithms termed multi-fractal detrending moving average (MFXDMA), which based on detrending moving average analysis (DMA) [36-37] and multi-fractal detrending moving average analysis (MF-DMA) algorithms [38]. There are a variety of other methods have been developed to investigate the cross-correlation characteristics of two time

series,

such as multifractal height

cross-correlation analysis (MF-HXA) [39], joint multifractal analysis based on the partition

3

function approach (MF-X-PF) [40-42], multifractal cross wavelet analysis (MF-X-WT) [43] and joint multifractal analysis based on wavelet leaders (MF-X-WL) [44]. Recently, a method of detrended partial cross-correlation analysis (DPCCA) [45-46] was proposed to remove the influences of shared noises and quantify the intrinsic relations of two non-stationary signals. Shen et al. [47] employed the DPCCA method to investigate the intrinsic cross-correlation characteristics of air pollution index (API) records and synchronously meteorological elements data. Lin et al. [48] used DPCCA method to analyze the intrinsic correlations between PNA/EPW and drought in the west United States. Coherent structures abundantly occur in multiphase flows [49], and have attracted great attention because of their complex self-similar, organized and multi-scale dynamic behavior [50-52]. The organized motion of coherent structures arising from the interfacial wave and droplet entrainment is a predominant feature of horizontal oil-water two-phase flows, and commonly present remarkable multi-scale cross-correlations characteristics. Since the multi-scale coherent structures in two-phase flows can be indentified by cross-correlation probe technologies [53-54], in this study we conduct an experiment of horizontal oil-water two-phase flows to collect the upstream and downstream flow information using a conductance cross-correlation velocity probe, and utilize the DPCCA algorithm to calculate multi-scale cross-correlation coefficient of coupled cross-correlated signals. The results indicate that the DPCCA cross-correlation coefficient can effectively indicate the coherent motion characteristics of the oil-water interface and the entrained droplets.

2. Multi-scale detrended cross-correlation analysis (DCCA) algorithms 2.1 Multi-scale DCCA algorithm and numerical experiment Consider two time series

{ x ( i )}

and

{ y (i )} , i = 1, 2,… , N ,

DCCA method can be introduced as follows [23]: 4

N is length of series. The

k

k

i =1

i =1

(1) Firstly, two integrated signals can be calculated by Rx ( k ) = ∑ xi and Ry ( k ) = ∑ yi ,

k = 1, 2,… , N , and then they are divided into

N − s overlapping boxes. Thus, there are s + 1

values in each box Rx ,i (k ) and Ry ,i ( k ) , i ≤ k ≤ i + s .  x ,i ( k ) and R  y , i ( k ) . The trend functions (2) Next, we define the local trend of each box as R

could be polynomials in the DCCA method, and the polynomial order is changeable. The ‘detrended walk’ is defined as differences of local trend values and local values:  x,i ( k ) , ε = R ( k ) − R  y ,i ( k ) , i ≤ k ≤ i + s . ε x = Rx ( k ) − R y y

(3) The detrended covariance of each box is calculated using the following equation: 2 f DCCA ( s , i) ≡

1 i+s ∑ ε x ⋅ε y s +1 i

(1)

(4) Finally, we can obtain the mean detrended covariance of N − s boxes: 2 FDCCA (s) ≡

1 N −s 2 ∑ f DCCA (s, i ) N − s i =1

(2)

If the self-affinity and long-range cross-correlation appear between these two series, the 2 ( s ) ∼ s 2 λ . λ is the long range power-law cross-correlations can be represented as FDCCA

power-law cross-correlations exponent. When x ( i ) = y ( i ) , DCCA reduces to DFA and the 2 2 detrended covariance FDCCA ( s ) reduces to FDFA ( s ) . The exponent λ enables to quantify

long-range power-law correlations, but fails to quantify the cross-correlation level between coupled non-stationary time series. To address this issue, Zebende [25] proposed a cross-correlation coefficient ρ DCCA which can be expressed as:

ρDCCA ≡

2 FDCCA FDFA { x ( i )} FDFA { y ( i )}

where FDFA { x ( i )} and FDFA { y ( i )} are the detrended variance function of

(3)

{ x ( i )}

and { y ( i )} , respectively. The DFA method used can refer to Peng et al. [55]. The DCCA cross-correlation coefficient ρDCCA is a dimensionless function depending on time scale s ,

5

and can quantity the cross-correlations on different time scales. The ρ DCCA ranges between

-1 ≤ ρDCCA ≤ 1 . ρDCCA = 0 means there is no cross-correlation between the two time series. 0 < ρ DCCA ≤ 1 present positive cross-correlation levels, whilst

-1<ρ DCCA < 0

present

negative case. A two-component fractionally autoregressive integrated moving average (ARFIMA) stochastic process [56-57] is used to evaluate the performance of the DCCA method in uncovering the cross-correlation characteristics of complex systems. Two time series and

{ y (i )}

{ x ( i )}

are generated by ARFIMA processes:  x ( i ) = WX (d1 , i) + (1 − W )Y (d 2 , i ) + ε x ( i )   y ( i ) = WY (d 2 , i ) + (1 − W ) X (d1 , i) + ε y ( i )

(4)

here, ∞

X ( d1 , i ) = ∑ a n ( d1 ) x ( i − n )

(5)

n =1

∞

Y (d 2 , i ) = ∑ a n ( d 2 ) y ( i − n )

(6)

n =1

an (d ) = d Γ(n − d ) Γ(1 − d )Γ(n + 1) 

(7)

where d ∈ (0, 0.5) is a memory parameter and related to the Hurst exponent, an ( d ) is a memory weight, ε x ( i ) and ε x ( i ) are independent distributed Gaussian variables with zero mean and unit variance, and W ∈ [ 0.5,1] is a cross-correlation strength controlling parameter

which can control the coupling strength between two series. When W = 1 , the two-component ARFIMA processes reduces to x ( i ) = X ( d1 , i) + ε x ( i )

(8)

y ( i ) = Y (d2 , i ) + ε y ( i )

In this case, the two series are independent in iteration.

{ x ( i )}

and

{ y (i )} are

fully

decoupled with no cross-correlation. When W increases from 0.5 to 1, the coupling strength of

{ x ( i )}

and

{ y (i )}

gradually decreases. 6

Six pairs of ARFIMA processes are generated with d1 = 0.1 , d 2 = 0.4 , ε x ( i ) ≠ ε y ( i ) and W = 0.5, 0.6, 0.7, 0.8, 0.9 and 1, respectively. The calculated ρ DCCA coefficients of each pair of ARFIMA processes are shown in Fig. 1. As can be seen, the ρ DCCA coefficient shows an increase tendency as the time scale s increases. In addition, the ρ DCCA gradually decreases when the cross-correlation strength controlling parameter W increases from 0.5 to 1. Note that when W=1, the ρDCCA fluctuates around zero over all time scales. 1.0

W=0.5 W=0.6 W=0.7 W=0.8 W=0.9 W=1

d1=0.1, d2=0.4

0.9 0.8 0.7

ρDCCA

0.6 0.5 0.4 0.3 0.2 0.1 0.0 -0.1 -0.2 10

100

1000

Scale s

Fig. 1. (color online) Cross-correlation levels of ARFIMA process with different cross-correlation strength controlling parameters. The length of data used is 213 .

In addition, the multi-scale sensibility of ρ DCCA in characterizing the cross-correlations of coupled non-stationary time series is investigated. We firstly generate two completely decoupled non-stationary ARFIMA processes d1 = 0.1 , d 2 = 0.4 ,

parameter

{ x (i )}

W = 1 and

and

{ y (i )}

ε x (i ) ≠ ε y (i ) .

through setting model Then,

sine

series

z ( i ) = sin ( 2π i / T ) , i = 1, 2,3, , N with different periods are produced, and are donated as

{ z ( i )} , { z ( i )} , { z ( i )} , { z ( i )} 1

2

3

4

and

{ z ( i )} 5

with period T equal to 10, 50, 100, 500

and 1000, respectively. The sine series are added into ARFIMA processes to access new coupled series

{ x ( i )} j

and

{ y ( i )} j

which can be described as

 x j ( i ) = x ( i ) + z j ( i ) j = 1, 2,3, 4, 5; i = 1, 2,3, N  j j  y ( i ) = y ( i ) + z ( i )

7

(9)

As shown in Fig. 2, because the scales, the cross-correlations of

{ x (i )}

and

{x ( i )} j

{ y (i )}

and

are completely uncorrelated over all

{ y ( i )} j

entirely depends on periodic

background sine series. Thus, for the coupling series mixed with different period sine series, the cross-correlation levels should be different and associated with the periods of the sine noises. 10

x(i)

5 0 -5 -10 10

y(i)

5 0

Test signals

-5 -10 8

z4(i)

4 0 -4 -8 12

4

4

4

4

x (i)=x(i)+z (i)

6 0 -6 -12 12

y (i)=y(i)+z (i)

6 0 -6 -12

0

1000

2000

3000

4000

5000

6000

7000

8000

Point number

Fig. 2. (color online) Overlap series of typical ARFIMA process series and sine series (T=500).

Then cross-correlation coefficient ρ DCCA of

{ x ( i )} j

and

{ y ( i )} j

calculated by the

DCCA method is shown in Fig. 3. As can be seen, the cross-correlation levels over all scales increase due to the mixed coupled sine series. Note that with increasing the scale s, ρDCCA ( s ) first increases to a peak value and then presents a decreasing tendency. The peak value is around 0.8, indicating that

{ x ( i )} j

and

{ y ( i )} j

have high cross-correlation levels at

certain scales, and then gradually lose cross-correlations when scale s is away from these scales. The particular scale corresponding to the highest cross-correlation level can be defined as smax . It can be seen from Fig. 3 that the smax values are different for the sine series with different periods and the period T is close to smax . In other words, the cross-correlation coefficient ρ DCCA of two series mixed with same sine series is particularly high at the scales nearby the period of the sine series. The result suggests that ρ DCCA can effectively reveal the 8

ρDCCA

cross-correlation levels of two non-stationary series at different scales. 1.0

x-y, DCCA

0.9

x -y , DCCA

1

1

0.8

2

x -y2, DCCA

0.7

x -y , DCCA

3

3

4

4

5

5

0.6

x -y , DCCA

0.5

x -y , DCCA

0.4 0.3 0.2 0.1 0.0 -0.1 -0.2 10

100

1000

Scale s

Fig. 3. (color online) The DCCA cross-correlation coefficients of ARFIMA processes mixed with sine series.

2.2 Multi-scale detrended partial cross-correlation analysis (DPCCA)

For m time series

{x } , { x } , {x } , 1 i

2 i

3 i

…

{ x } , where m i

,

i = 1, 2,3,  , N , the algorithm

of DPCCA can be briefly introduced as follows: 2 (1) Firstly, the mean detrended covariance Fj1 , j2 ( s ) between any two series is calculated by

DCCA method, where j1 , j2 = 1, 2,3, , m and s is the length of each box. Accordingly, we can obtain a covariance matrix F 2 ( s ) for every s :  F1,12 ( s ) F1,22 ( s )  F1,2m ( s )   2 2 F2,1 ( s ) F2,2 ( s )  F2,2m ( s )  2  F (s) =       2  2 2  Fm ,1 ( s ) Fm ,2 ( s )  Fm ,m ( s ) 

(10)

(2) Next ,we can calculate the cross-correlation coefficient ρ j1 , j2 ( s ) of any two time series:

ρ j , j (s) ≡ 1

2

F j21 , j2 ( s )

F j1 , j1 ( s ) ⋅ F j2 , j2 ( s )

(11)

Thus, a m × m coefficients matrix ρ ( s ) can be obtained as  ρ1,1 ( s ) ρ1,2 ( s )  ρ1, m ( s )    ρ2,1 ( s ) ρ 2,2 ( s )  ρ2,m ( s )   ρ (s) =         ρ m ,1 ( s ) ρ m,2 ( s )  ρ m , m ( s )  9

(12)

(3) An inverse matrix C ( s ) of cross-correlation coefficients matrix ρ ( s ) is calculated using the following equation:  C1,1 ( s ) C1,2 ( s )  C1, m ( s )    C ( s ) C2,2 ( s )  C2, m ( s )  C ( s ) = ρ −1 ( s ) =  2,1        Cm,1 ( s ) Cm ,2 ( s )  Cm , m ( s ) 

(13)

(4) Finally, we can obtain the partial-cross-correlation between two time series

{x } j1

i

and { xi 2 } by j

ρDPCCA ( j1, j2 ; s ) =

−C j1 , j2 ( s )

(14)

C j1 , j1 ( s ) ⋅ C j2 , j2 ( s )

Because the DPCCA can remove all possible influence of other time series

{x } , j ≠ j , j ≠ j j

i

1

2

, ρDPCCA ( j1 , j2 ; s ) can represent the intrinsic correlations between two

time series on time scale of s . The partial-cross-correlation coefficients on different time scales can be obtained by changing s . It should be noted that the DPCCA reduces to the DCCA when m = 2 , i.e., ρDPCCA ( j1 , j2 ; s ) = ρDCCA ( j1, j2 ; s ) . Two ARFIMA processes

{ x ( i )}

and

{ y ( i )}

with zero cross-correlation levels are

mixed with three different classes of noise signals, i.e., periodic process, stochastic process and chaotic system. The model parameters of the ARFIMA process are selected as d1=0.1, d 2=0.4 and W=1. The sine signal

process. A white noise

{ z ( i )} w

{ z ( i )} 3

with period T = 100 is chosen as the periodic

with zero mean and unit variance is chosen as the stochastic

process. The chaotic systems are represented by Logistic map and Schuster map. Logistic map z ( i + 1) = rz ( i ) (1 − z ( i ) ) generates a chaotic series value z (1) = 0.1 . Schuster map

pairs of correlated signals with noises using Eq. 9, i.e., w

l

and

{ y ( i )} , { x ( i )} l

l

r = 3.8 and initial

z ( i + 1) = z ( i ) + z a ( i ) , Mod 1 with a = 2 and initial value

z (1) = 0.05 generates a chaotic burst intermittent signal

{ y ( i )} , { x ( i )}

{ z ( i )} , where

s

and

{ y ( i )} . s

10

{ z ( i )} . Thus, we can obtain four s

{ x ( i )} 3

and

{ y ( i )} , { x ( i )} 3

w

and

In order to evaluate the advantage of the DPCCA in suppressing the noise influence and quantifying the intrinsic cross-correlations of two non-stationary signals, we calculate the multi-scale cross-correlation coefficients of original ARFIMA series and the series mixed with noises by the DCCA and the DPCCA method respectively, and the results are shown in Fig. 4. It can be seen that the multi-scale cross-correlation coefficients obtained by DCCA method increase significantly after adding common noise into

{ x ( i )}

and

{ y (i )} . And, increasing

the polynomial order of the DCCA can weaken the influence of sine noise and Schuster maps noise to some degree [24], as shown in Fig. 4(b) and Fig. 4(h), but it will enhance the influence of Logistic map noise and white noise, as shown in Fig. 4(d) and Fig. 4(f). Comparatively, DPCCA method can efficiently remove the effect of four classes of noises series, and thus can be used to reveal the multi-scale intrinsic cross-correlations between two considered variables. (a)

10

x(i)

5

(b)

-5 -10 10 0

Test signals

-5 -10 10

z3(i)

Sine signal

5 0 -5 -10 10

ρDCCA, ρDPCCA

y(i)

5

x3(i)=x(i)+z3(i)

5

3

x3-y3 , DCCA, order=2

0.7

x -y , DCCA, order=3

0.6

x3-y3 , DCCA, order=4

0.5

x3-y3 , DPCCA

3

3

0.4 0.3

0.0

3

y (i)=y(i)+z (i)

5

-0.1

0 -5

-0.2 0

500

1000

1500

10

2000

Scale s

Point number

(c) 10

x(i)

5

0.9

-10 10 0 -5 -10 10

w

z (i)

White noise

5 0 -5 -10 10

xw(i)=x(i)+zw(i)

5 0

1000

x-y, DCCA White noise

xw-yw, DCCA

0.8

xw-yw, DCCA, order=2

0.7

xw-yw, DCCA, order=3

0.6

xw-yw, DCCA, order=4

0.5

xw-yw, DPCCA

0.4 0.3 0.2 0.1

-5

0.0

-10

10

0

ρDCCA, ρDPCCA

y(i)

5

100

(d) 1.0

0 -5

Test signals

x3-y3 , DCCA

0.8

0.1

-10 10

5 0 -5 -10

x-y, DCCA

Sine signal

0.2

0 -5

-10

1.0 0.9

0

w

w

y (i)=y(i)+z (i)

-0.1 -0.2

500

1000

1500

2000

10

100

Scale s

Point number

11

1000

(e) 105

x(i)

(f)

1.0 0.9

0

x-y, DCCA Logistic map

l

-10 10 0

Test signals

-5 -10 10

l

Logistic map

5

z (i)

0 -5 -10 10

ρDCCA, ρDPCCA

y(i)

5

xl(i)=x(i)+zl(i)

5

0.8

l

x -yl, DCCA, order=2

0.7

x -y , DCCA, order=3

l l

x -yl, DCCA, order=4

0.5

xl-yl, DPCCA

0.4 0.3

0.1

-10 10

l

0.0

l

y (i)=y(i)+z (i)

5

-0.1

0 -5

-0.2 0

500

1000

1500

2000

10

100

Scale s

Point number

(g) 10

x(i)

5 0 -5 -10

(h)

zs(i)

Schuster map

3

0 10 5 0 -5 -10

xs(i)=x(i)+zs(i)

10 5 0 -5 -10

ys(i)=y(i)+zs(i)

ρDCCA, ρDPCCA

y(i)

1000

1.0 0.9

10 5 0 -5 -10

Test signals

l

0.6

0.2

0 -5

-10

l

x -y , DCCA

-5

x-y, DCCA Schuster map

xs-ys, DCCA s s

0.8

x -y , DCCA, order=2

0.7

x -y , DCCA, order=3

0.6

xs-ys, DCCA, order=4

0.5

xs-ys, DPCCA

s s

0.4 0.3 0.2 0.1 0.0 -0.1 -0.2 10

0

500

1000

1500

100

2000

1000

Scale s

Point number

Fig. 4. (color online) DCCA and DPCCA multi-scale cross-correlations for original ARFIMA processes and the processes contaminated by noisy signals. The length of data used is 213. (a) Sine noise and contaminated ARFIMA processes; (b) Periodic noise immunity analysis of DPCCA; (c) White noise and contaminated ARFIMA processes; (d) Stochastic noise immunity analysis of DPCCA; (e) Logistic map noise and contaminated ARFIMA processes; (f) Chaotic noise immunity analysis of DPCCA; (g) Schuster map noise and contaminated ARFIMA processes; (h) Chaotic noise immunity analysis of DPCCA.

3. The experiment for horizontal oil-water two-phase flows

The experiment of horizontal oil-water two-phase flows was carried out on the multi-phase flow loop facility of Tianjin University. The experimental facility is shown in Fig. 5. Tap water and No. 15 industry white oil with a density of 845kg/m3 and a viscosity of 11.984mPa·s are used as the experimental media, and the oil-water interfacial tension is 0.03 N/m. The oil-water flows were forced into a test section pipe with 20mm ID and 3.9m length from a large diameter pipe (ID = 125mm) by a diverter. The length of 125mm ID pipe before

12

the diverter is 1200 mm. The large diameter pipe plays an important role of wellbore storage effect and thus a relative short distance can make sure the flow fully developed [58]. A conductance cross-correlation velocity probe (CCVP) and a mini-conductance probe array are amounted on the flow test section. The mini-conductance probe array is used to identify the flow patterns, and its measurement principles can refer to Zhai et al [59]. The CCVP is used to detect the coupled upstream and downstream information of the flow structures. To avoid the ‘cross-talk’ [60] between the CCVP electrodes, the cross-correlation distance of the CCVP is selected as 30 mm using the finite element method [22]. 1200mm

100mm 900mm

1350mm

125mm

1650mm

Diverter Flow direction

Contraction Conductance orifice cross-correlation velocity probe

Mini-conductance probe array 3

Control System

1 2

F

F

7

6

5

4

1 oil/water separate tank; 2 water tank; 3 oil tank; 4 pump; 5 hand ball valve; 6 flow meter; 7 pneumatic diaphragm

Fig. 5. (color online) Experimental set up for horizontal oil-water two-phase flows.

The measurement system of the CCVP is shown in Fig. 6. The measurement system consists of an exciting module generating a sinusoidal 20 kHz voltage signal, a signal conditioning module and a data acquisition module (NI DAQ). In the horizontal oil-water two-phase flow experiment, because of the remarkable conductivity difference between oil and water phases, the equivalent conductivity of two-phase flows in the pipe would be changed by the variation of the flow structures. Through an I/V transition circuit, the sinusoidal exciting voltage as a carrier signal is modulated by conductivity fluctuations of the

13

oil-water flows. Then the modulated voltages are successively processed by a phase sensitive demodulator (AD630), a low-passing filter (SR-4BL1) and an amplifier (OP27G). After that, the signal conditioning system outputs upstream and downstream DC voltage signals Vm,up and Vm,down , which contain the flow structure information of oil-water phases.

Fig. 6. (color online) Measurement system of the conductance cross-correlation velocity probe.

In the experiment, we first fix the oil flow rate and then gradually increase the water flow rate. After the oil-water flows are stable, the responses of the CCVP and the mini-conductance probe array are collected using the NI DAQ device. Once the above process finished, the oil phase flow rates is increased to a next flow condition and the processes above is repeated. The upstream and downstream response signals of the CCVP are collected by channel 0 and channel 1 of acquisition card PXI 4472, and other channels (channel 2 to channel 7) are used to collect noise signals of the system. The sampling frequency is set as 600Hz. The mini-conductance probes are collected by channel 0 to 7 of data acquisition card PXI 6221, the sampling frequency is set as 2 KHz. Fig. 7(a) shows the experimental flow pattern map which consists of six flow patterns encountered in the horizontal pipe, i.e., stratified flow (ST), stratified flow with mixing at the interface (ST&MI), dispersion of water in oil and oil in water flow (DW/O&DO/W), dispersed flow includes dispersion of oil in water and water flow (DO/W&W), dispersion of oil in water flow (DO/W) and dispersion of water in oil flow (DW/O). In the experiment, the responses of the CCVP to five conductive flow patterns are collected. ST flow has an obvious 14

oil-water interface and the interface is relatively smooth and stable, as shown in Fig. 7(b). ST&MI flow is characterized by an oil-water interface with fluctuations, and the droplet entrainment is associated with the wavy interface. According to the different distributions of the oil and water droplets around the wavy interface, ST&MI flow is divided into four types. Specifically, the quantity of droplets appears around the oil-water interface in ST&MI flow of type I and type II is small, and the droplets symmetrically distribute along the oil-water interface, as shown in Fig. 7(c) and 7(d). When ST&MI flow gradually evolves to DO/W&W and DW/O flows, oil-water interfacial wave and oil droplets distribution exhibit two different forms, as shown in Fig. 7(e) and 7(f) respectively, which are defined as ST&MI flow of type III and type IV. The DW/O&DO/W flow indicates obvious droplet entrainments, which is characterized by dispersed water droplets in an upper oil layer and dispersed oil droplets in a lower water layer, and the multi-scale feature of the entrained droplets is remarkable. Meanwhile, the interface fluctuation becomes aggravated and complicated, and thus leads to a unstable alternation layer of oil-in-water and water-in-oil regions [58]. According to the location of the oil-water interface, DW/O&DO/W flow can be divided into three types, as shown in Fig. 7(g), 7(h) and 7(i). For DO/W&W flow, the water phase is continuous and flows in the lower part of pipe, and the dispersed oil droplets distribute in the continuous water layer at the upper part of the pipe. The high density distribution of the oil droplets in the upper layer leads to a ‘quasi-stratified’ flow structure in DO/W&W flows. For DO/W flow, dispersed oil droplets distribute in the continuous water phase, and the flow shows a great stability and tends to be a homogeneous flow. The detailed description about the flow structure of each flow pattern can refer to Zhai et al. [58].

15

ST ST&MI D W/O&D O/W

(a)

D O/W&W D O/W D W/O

3

Qw (m /day) 60 50 40

Usw (m/s)

1

30

Ⅲ

15

Ⅲ

10 8

Ⅱ

Ⅱ

Ⅰ

6 5 4

0.1

Ⅰ

20

Ⅳ

3 2.88 3.6 4.4 5.28 6.4

8

10 12

15

20

25 30 34 40 45 50 60 70

3

Qo (m /day)

0.1

Uso (m/s)

(b)

ST flow

(g)

(c)

ST&MI flow (Type I)

(h)

(d)

ST&MI flow (Type II)

(i)

(e)

ST&MI flow (Type III)

(j)

(f)

ST&MI flow (Type IV)

(k)

1

DW/O&DO/W flow (Type I)

DW/O&DO/W flow (Type II)

DW/O&DO/W flow (Type III)

DO/W&/W flow

DO/W flow

Fig. 7. (color online) Experimental flow pattern map and flow structures of oil-water two-phase flows.

The responses of the CCVP under typical flow conditions are shown in Fig. 8. Overall, the fluctuations of the upstream signal are similar to these of the downstream signals, and the downstream signals always follow the upstream ones. ST flow has a stable oil-water interface, so the probe signal fluctuates in a small range (see Fig. 8(a)).With increasing the oil flow rate, the oil-water interface gradually loses its stability and an interfacial wave with droplet 16

entrainment is observed, indicating the ST&MI flow pattern occurs in pipe. Meanwhile, as shown in Fig.8(a) and 8(b) (ST&MI of type I and type II), the amplitude of the probe response obviously increased, and with further increasing the oil rate, the number of oil and water droplets gradually increase, so does the fluctuation frequency. When the droplets around the oil-water interface show a unilateral diffusion distribution, the collected sensor responses have small fluctuations, as shown in Fig. 8(a) (ST&MI of type IV). Because there are intense oil-water interface fluctuations in ST&MI flow of Type III, the amplitude of the probe response is large and fluctuates widely, as shown in Fig. 8(b) (ST&MI of type III). For DW/O&DO/W flows in Fig. 8(b) and 8(c), the output signals of the CCVP fluctuate strongly, but the fluctuation amplitude gradually decrease and the fluctuation frequency gradually increase as oil flow rate increases. This phenomenon mainly arises from the complicated droplet distribution in DW/O&DO/W flow. For DO/W&W flows shown in Fig. 8(c), the signals of the CCVP represent fluctuating characteristics that similar to ST&MI flows and DW/O&DO/W flows. The result is supported by the fact that a large concentration of dispersed oil droplets flow in the upper part of the pipeline leads to a ‘quasi-stratified’ flow structures in DO/W&W flow. DO/W flows is characterized by the signal fluctuations with low amplitude and high frequency, as shown in Fig. 8(d), which is consistent with the random motion of the dispersed oil droplets with small sizes in the continuous water phase.

Probe signals (V)

0.2 0.1 0.0 -0.1 -0.2

downstream Qw=4 m3/day,Qo=2.88 m3/day,ST

0.2 0.1 0.0 -0.1 -0.2

3

3

Qw=4 m /day,Qo=4.4 m /day,ST

0.2 0.1 0.0 -0.1 -0.2

Qw=4 m3/day,Qo=6.4 m3/day,ST&MI type I

0.2 0.1 0.0 -0.1 -0.2

Qw=4 m /day,Qo=12 m /day,ST&MI type IV

0.2 0.1 0.0 -0.1 -0.2

Qw=4 m3/day,Qo=15 m3/day,ST&MI type IV

3

0

1

2

3

4

5

3

6

7

8

9

upstream

(b)

0.50 0.25 0.00 -0.25 -0.50 0.50 0.25 0.00 -0.25 -0.50 0.50 0.25 0.00 -0.25 -0.50 0.50 0.25 0.00 -0.25 -0.50 0.050 0.025 0.000 -0.025 -0.050

10

Time (s)

downstream

Qw=15 m3/day,Qo=2.88 m3/day,ST&MI, type III Qw=15 m3/day,Qo=10 m3/day,ST&MI, type II

Probe signals (V)

upstream

(a)

Qw=15 m3 /day,Qo=15 m3/day,DW/O&DO/W, type I

3

3

Qw=15 m /day,Qo=30 m /day,DW/O&DO/W, type I

3

3

Qw=15 m /day,Qo=45 m /day,DW/O&DO/W, type II

0

1

2

3

4

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Time (s)

17

6

7

8

9

10

upstream 0.10 0.05 0.00 -0.05 -0.10 0.10 0.05 0.00 -0.05 -0.10 0.10 0.05 0.00 -0.05 -0.10 0.10 0.05 0.00 -0.05 -0.10 0.10 0.05 0.00 -0.05 -0.10

downstream

Qw=30 m3/day,Qo=2.88 m3/day,DO/W&W

3

(d)

3

Qw=30 m /day,Qo=6.4 m /day,DO/W&W

3

3

Qw=30 m /day,Qo=12 m /day,DO/W&W

Qw=30 m3/day,Qo=20 m3/day,DW/O&DO/W, type III

Qw=30 m3/day,Qo=60 m3/day,DW/O&DO/W, type I

0

1

2

3

4

5

6

7

8

9

10

Probe signals (V)

Probe signals (V)

(c)

upstream 0.10 0.05 0.00 -0.05 -0.10 0.10 0.05 0.00 -0.05 -0.10 0.10 0.05 0.00 -0.05 -0.10 0.10 0.05 0.00 -0.05 -0.10 0.10 0.05 0.00 -0.05 -0.10

downstream Qw=40 m3/day,Qo=2.88 m3/day,DO/W

Qw=40 m3/day,Qo=6.4 m3 /day,DO/W

Qw=40 m3/day,Qo=15 m3/day,DO/W

3

3

Qw=40 m /day,Qo=34 m /day,DW/O&DO/W, type III

Qw=40 m3 /day,Qo=60 m3/day,DW/O&DO/W, type I

0

1

2

3

4

Time (s)

5

6

7

8

9

10

Time (s)

Fig. 8. (color online) Collected responses of the conductance cross-correlation velocity probe under typical flow conditions.

4. Multi-scale cross-correlation analysis of horizontal oil-water two-phase flow 4.1 Multi-scale cross-correlation coefficients of oil-water flows

Horizontal oil-water two-phase flows present a remarkable multi-scale structure feature, so investigating multi-scale cross-correlation characteristics of upstream and downstream flow structure is of great importance for uncovering the nonlinear dynamics of oil-water flows. The common noise of two nonlinear time series can lead to spuriously high cross-correlations, and DPCCA method can efficiently remove the effect of the noises so that reveal multi-scale intrinsic cross-correlations between two considered non-stationary time series. Hence, the DPCCA method in this section is used to investigate the multi-scale cross-correlation characteristics of oil-water flows, and the system noise used by DPCCA is collected by channel 2 of PXI 4472. Fig. 9 shows the DPCCA cross-correlation coefficients of the conductance signals under different flow patterns in horizontal oil-water two-phase. As can be seen, for Qw=4 m3/day, as the oil flow rate increases, ST flow and ST&MI flow successively occurs in test section. The DPCCA cross-correlation coefficients are negative at small scales but positive at large scales,

18

as show in Fig. 9(a). For Qw=15 m3/day, the flow pattern gradually transforms to DW/O&DO/W flow from ST&MI flow. The DPCCA cross-correlation coefficient of DW/O&DO/W flow is similar to ST flow and ST&MI flow, as shown in Fig. 9(b). For Qw=30 m3/day, the flow pattern transforms to DW/O&DO/W flow from DO/W&W flow with increasing the oil flow rate, as shown in Fig. 9(c). Accordingly, the cross-correlation level at small scales gradually change to negative from positive, while the cross-correlations at large scales are always positive. For Qw=40 m3/day, as the oil flow rate increases, the flow pattern changes to DW/O&DO/W flow from DO/W flow, and the cross-correlation characteristic of the flows shown in Fig. 9(d) present a similar evolution to that shown in Fig. 9(c). Qo(m3/day) Flow Pattern 2.88 ST 3.6 ST 4.4 ST 5.28 ST 6.4 ST&MI 8 ST&MI 10 ST&MI 12 ST&MI 15 ST&MI 20 ST&MI

Qw=4 m3/day

1.0 0.8 0.6

ρDPCCA

0.4 0.2 0.0 -0.2

(b) 1.2 1.0

0.6 0.4 0.2 0.0 -0.2

-0.4

-0.4

-0.6

-0.6

-0.8 10

100

3

-0.8 10

1000

Qw=30 m3/day

0.8

ρDPCCA

0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 10

100

Qo(m3/day) Flow Pattern 2.88 DO/W&W 3.6 DO/W&W 4.4 DO/W&W 5.28 DO/W&W 6.4 DO/W&W 8 DO/W&W 10 DO/W&W 12 DO/W&W 15 DO/W&W 20 DO/W&W 25 DW/O&DO/W 30 DW/O&DO/W 34 DW/O&DO/W 40 DW/O&DO/W 45 DW/O&DO/W 50 DW/O&DO/W 60 DW/O&DO/W

(d) 1.2 1.0

1000

100

Qo(m3/day) Flow Pattern 2.88 DO/W 3.6 DO/W 4.4 DO/W 5.28 DO/W 6.4 DO/W 8 DO/W 10 DO/W 12 DO/W 15 DO/W 20 DO/W 25 DO/W 30 DW/O&DO/W 34 DW/O&DO/W 40 DW/O&DO/W 45 DW/O&DO/W 50 DW/O&DO/W 60 DW/O&DO/W 1000 70 DW/O&DO/W

Qw=40 m3/day

0.8 0.6

ρDPCCA

1.0

100

Scale s

Scale s

(c) 1.2

Qo(m /day) Flow Pattern 2.88 ST&MI 3.6 ST&MI 4.4 ST&MI 5.28 ST&MI 6.4 ST&MI 8 ST&MI 10 ST&MI 12 ST&MI 15 ST&MI 20 ST&MI 25 DW/O&DO/W 30 DW/O&DO/W 34 DW/O&DO/W 40 DW/O&DO/W 45 DW/O&DO/W 50 DW/O&DO/W

Qw=15 m3 /day

0.8

ρDPCCA

(a) 1.2

0.4 0.2 0.0

-0.2 -0.4 -0.6 10

1000

Scale s

Scale s

Fig. 9. (color online) DPCCA cross-correlation coefficient for different flow pattern of horizontal oil-water two-phase flow.

4.2 Characterization of the multi-scale flow structures

The DPCCA results of the horizontal oil-water two-phase flows indicate that the multi-scale cross-correlation characteristics of different flow structures are quite different. We

19

extract the mean cross-correlation coefficients at small scales and large scales respectively, as shown in Fig. 10, in which the small scale is set as s ∈ (10,110) and the large scale is s ∈ (110,1000) . The mean cross-correlation coefficients at small and large scales indicate the

evolution characteristics of the flow structures. (1) For ST flows, small-scale flow structures are related to interfacial stochastic noises. Accordingly, the cross-correlations are negative at small scales, as shown in Fig. 10(a). However, the cross-correlations at large scales turn to be positive, which indicates that the stable structure of the large-scale interfacial wave of ST flow shows significant cross-correlation characteristics. (2) For ST&MI flows, as shown in Fig. 10(b) and Fig. 10(c), the upstream and downstream signals are negatively correlated at small scale and positively correlated at large scale, which are caused by a few droplets motion around the oil-water interface and obvious interfacial wave respectively (see Fig. 7(c)~ Fig. 7(f)). Notably, compared to ST flow, ST&MI flow is characterized by an obvious oil-water interfacial wave, which leads to a significant increase of cross-correlations at large scale. Meanwhile, the spatial distribution of the droplets in ST&MI flow of type I and type II are obviously different, and thus their cross-correlations show a certain degree of differences both at small scales and large scales (see Fig. 10(a)). (3) For DW/O&DO/W flows, oil-water interfacial wave become more remarkable and the wave structures are extremely complicated, leading to positive correlations of the flows at large scale. Meanwhile, the cross-correlations of the large-scale structures in DW/O&DO/W flow are affected by the position of the oil-water interface. Specifically, the cross-correlation level of the flow structures decreases when the oil-water interface approaches the bottom of pipe, as shown in Fig. 10(b). Notably, the mean cross-correlations at small scales of DW/O&DO/W flow are obviously dependent on droplets diffusion characteristics. When the

20

total flow rate is low, the symmetrical distribution region of the droplets around the interface is small, and thus the cross-correlations at small scales are negative (see type III in Fig. 10(b) and Fig. 10(c)). As the total flow rate increases, the droplet concentration obviously increases and the droplets diffuse in a large region of pipe. The motion of the dense droplets leads to a remarkable similarity of the upstream and downstream signals at small-scale fluctuations, so that the flow structures are positively correlated at small time scales. (4) For DO/W&W flows, there are both dispersed droplets motion and ‘quasi-stratified’ flow structures caused by high concentration oil droplets. Thus, the DO/W&W flow presents similar multi-scale cross-correlation characteristics with ST flow and ST&MI flow, as shown in Fig. 10(c). (5) When the total flow rate is high enough, the flow pattern evolves to DO/W flow. For DO/W flow, the dispersed oil droplets with random motion lead to low and negative cross-correlations at small scales. In addition, DO/W flow is characterized by a relatively homogeneous phase distribution, so the slippage effect between oil and water phases is weak, leading to similar flow structures passing by the upstream and downstream conductance probes. Thus, DO/W flows exhibit remarkable long-range cross-correlations (see Fig. 10(d)), which is consistent with our previous analysis results [61]. ST&MI, Small scale ST&MI, Large scale 3

Qw=4 m /day

0.8

Average value of ρDPCCA

0.6

ST&MI (type I)

0.4

(b)

ST&MI, Small scale ST&MI, Large scale

1.0

0.2 0.0 -0.2

ST&MI (type II)

ST&MI (type III)

0.6

ST&MI (type IV)

DW/O&DO/W, Small scale DW/O&DO/W, Large scale

Qw=15 m3/day

0.8

Average value of ρDPCCA

(a)

ST, Small scale ST, Large scale

1.0

0.4

DW/O&DO/W (type II) DW/O&DO/W (type I)

0.2 0.0

-0.2

-0.4

-0.4

-0.6

-0.6

-0.8

-0.8

-1.0

2

4

6

8

10

12

14

16

18

20

-1.0

22

0

3

Qo(m /day)

21

10

20

30

Qo(m3/day)

40

50

DW/O&DO/W, Small scale DW/O&DO/W, Large scale

Qw=30 m3/day

0.8

0.8 0.6

0.4

Average value of ρDPCCA

0.6

Average value of ρDPCCA

DO/W, Small scale DO/W, Large scale

(d) 1.0

DW/O&DO/W (type III)

0.2 0.0 -0.2

DW/O&DO/W (type I)

-0.4 -0.6 -0.8

DW/O&DO/W, Small scale DW/O&DO/W, Large scale

DW/O&DO/W (type III)

(c)

DO/W&W, Small scale DO/W&W, Large scale

1.0

0.4 0.2

Qw=40 m3/day

DW/O&DO/W (type I)

0.0 -0.2 -0.4 -0.6 -0.8

-1.0

0

10

20

30 3

40

50

-1.0

60

0

10

20

30

40

50

60

70

Qo(m3/day)

Qo(m /day)

Fig. 10. (color online) Average values of DPCCA coefficient at small and large scales for horizontal oil-water two-phase flows.

5. Conclusion

Through adding sine series with different periods to ARFIMA processes, the multi-scale cross-correlation characteristics of non-stationary time series are investigated using DCCA method, and it is found that the DCCA method can effectively reveal the cross-correlation level of non-stationary series at different scales. In addition, the ARFIMA processes mixed with periodic series, stochastic series and chaotic series are used in anti-noise analysis of DPCCA method. The result indicates that the DPCCA method can efficiently remove the effect of three classes of noises series so that reveal multi-scale intrinsic cross-correlations between two coupled series. Through collecting coupled upstream and downstream conductance signals from horizontal oil-water two-phase flow system and analyzing the cross-correlation of five conductive flow patterns, it is found that the multi-scale DPCCA cross-correlation coefficient can effectively uncover the cross-correlation characteristics of the flow structures with different scales, such as large-scale oil-water interfacial wave and small-scale entrainment of droplets. The extracted mean DPCCA cross-correlation coefficients are sensitive to the droplet distribution and the relative position of the oil-water interface, and can serve as a beneficial 22

indicator of the complex flow structures encountered in the horizontal pipe. In consideration of the obvious spatial distribution of the multi-scale structures of horizontal oil-water two-phase flow, in future work we plan to develop a distributed sensor that can detect the local flow motions and investigate the cross-correlation characteristics of local flow structures.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant Nos. 41504104, 11572220, 51527805), Natural Science Foundation of Tianjin, China (Grant Nos. 14JCQNJC04200) and China Scholarship Council (Grant Nos.201506255066).

References

[1] N. Scafetta, P. Grigolini, Scaling detection in time series: diffusion entropy analysis, Physical Review E 66 (2002) 036130. [2] J. Zhang, M. Small, Complex network from pseudoperiodic time series: topology versus dynamics, Physical Review Letters 96 (2006) 238701. [3] L.X. Zhuang, N.D. Jin, A. Zhao, Z.K. Gao, L.S. Zhai, T. Yi, Nonlinear multi-scale dynamic stability of oil-gas-water three-phase flow in vertical upward pipe, Chemical Engineering Journal 302 (2016) 595-608. [4] J.W. Kantelhardt, E.K. Bunde , H.H.A. Rego, S. Havlin, A. Bunde, Detecting long-range correlations with detrended fluctuation analysis, Physica A 295 (2001) 441-454. [5] N. Ellis, L.A. Briens, J.R. Grace, H.T. Bi, C.J. Lim, Characterization of dynamic behaviour in gas-solid turbulent fluidized bed using chaos and wavelet analyses, Chemical Engineering Journal 96 (2003) 105-116.

23

[6] M.R. Niu, Q.F. Liang, G.S. Yu, F.C. Wang, Z.H. Yu, Multifractal analysis of pressure fluctuation signals in an impinging entrained-flow gasifier, Chemical Engineering and Processing 47 (2007) 642-648. [7] G.B. Zhao, Y.R. Yang, Characteristics of multi-scale and multi-fractal of pressure signals in a bubbling fluidized bed, Journal of Chemical Engineering of Chinese Universities 17 (2003) 648-654. [8] Y. Zheng, A. Rinoshika, Multi-scale particle dynamics of low air velocity in a horizontal self-excited gas-solid two-phase pipe flow, International Journal of Multiphase Flow 53 (2013) 114-123. [9] H.J. Wu, F.D. Zhou, Y.Y. Wu, Intelligent identification system of flow regime of oil-gas-water, International Journal of Multiphase Flow 27 (2001) 459-475. [10] B. Wu, L. Briens, J.X. Zhu, Multi-scale flow behavior in gas–solids two-phase flow systems, Chemical Engineering Journal 117 (2006) 187-195. [11] M. Dhamala, G. Rangarajan, M. Ding, Estimating Granger causality from fourier and wavelet transforms of time series data, Physical Review Letters 100 (2008) 145-150. [12] D. Marinazzo, M. Pellicoro, S. Stramaglia, Kernel method for nonlinear granger causality, Physical Review Letters 100 (2008) 144103. [13] G. Sugihara, R. May, H. Ye, C.H. Hsieh, E. Deyle, M. Fogarty, S. Munch, Detecting causality in complex ecosystems, Science 338 (2012) 496-500. [14] E. Ghysels, J.B. Hill, K. Motegi, Testing for Granger Causality with Mixed Frequency Data, Journal of Econometrics 192 (2016) 207-230. [15] T. Schreiber, Measuring information transfer, Physical Review Letters 85 (2000) 461-464.

24

[16] M. Staniek, K. Lehnertz, Symbolic transfer entropy, Physical Review Letters 100 (2008) 158101. [17] J. Wang, Z.F. Yu, Symbolic transfer entropy-based premature signal analysis, Chinese Physics B 21 (2012) 018702. [18] S. Ito, M.E. Hansen, R. Heiland, A. Lumsdaine, A.M. Litke, J.M. Beggs, Extending transfer entropy improves identification of effective connectivity in a spiking cortical network model, Plos One 6 (2011) e27431. [19] X.S. Liang, R. Kleeman, Information transfer between dynamical system components, Physical Review Letters 95 (2005) 244101. [20] M. Wibral, B. Rahm, M. Rieder, M. Lindner, R. Vicente, J. Kaiser, Transfer entropy in magnetoencephalographic data: Quantifying information flow in cortical and cerebellar networks, Progress in Biophysics and Molecular Biology 105 (2011) 80-97. [21] R.E. Spinney, J.T. Lizier, M. Prokopenko, Transfer entropy in physical systems and the arrow of time, Physical Review E 94 (2016) 022135. [22] L.S. Zhai, P. Bian, Y.F. Han, Z.K. Gao, N.D. Jin, The measurement of gas–liquid two-phase flows in a small diameter pipe using a dual-sensor multi-electrode conductance probe, Measurement Science and Technology 27 (2016) 045101. [23] B. Podobnik, H.E. Stanley, Detrended Cross-Correlation Analysis: A New Method for Analyzing Two Non-stationary Time Series, Physical Review Letters 100 (2007) 084102. [24] D. Horvatic, H.E. Stanley, B. Podobnik, Detrended cross-correlation analysis for non-stationary time series with periodic trends, 94 (2011) 18007. [25] G.F. Zebende, DCCA cross-correlation coefficient: Quantifying level of cross-correlation, Physica A 390 (2011) 614-618.

25

[26] B. Podobnik, Z.Q. Jiang, W.X. Zhou, H.E. Stanley, Statistical tests for power-law cross-correlated processes, Physical Review E 84 (2011) 066118. [27] L. Kristoufek, Measuring correlations between non-stationary series with DCCA coefficient, Physica A 402 (2014) 291-298. [28] R.T. Vassoler, G.F. Zebende, DCCA cross-correlation coefficient apply in time series of air temperature and air relative humidity, Physica A 391 (2012) 2438-2443. [29] G.F. Zebende, M.F.D. Silva, A.M. Filho, DCCA cross-correlation coefficient differentiation: Theoretical and practical approaches, Physica A 392 (2013) 1756-1761. [30] S.H. Sardouie, M.B. Shamsollahi, L. Albera, I. Merlet, Interictal EEG noise cancellation: GEVD and DSS based approaches versus ICA and DCCA based methods, IRBM 36 (2015) 20-32. [31] W.X. Zhou, Multifractal detrended cross-correlation analysis for two nonstationary signals, Physical Review E 77 (2008) 066211. [32] J.W. Kantelhardt, S.A. Zschiegner, E. Koscielny-Bunde, S. Havlin, A. Bunde, H.E. Stanley, Multifractal detrended fluctuation analysis of nonstationary time series, Physica A 316 (2002) 87-114. [33] Q.S. Ruan, W. Jiang, G.F. Ma, Cross-correlations between price and volume in Chinese gold markets, Physica A 451 (2016) 10-22. [34] P. Oswiecimka, S. Drozdz, M. Forczek, S. Jadach, J. Kwapien, Detrended cross-correlation analysis consistently extended to multifractality, Physical Review E 89 (2014) 023305. [35] Z.Q. Jiang, W.X. Zhou, Multifractal detrending moving-average cross-correlation analysis, Physical Review E 84 (2011) 016106.

26

[36] E. Alessio, A. Carbone, G. Castelli, V. Frappietro, Second-order moving average and scaling of stochastic time series, The European Physical Journal B 27 (2002) 197-200 [37] J. Alvarez-Ramirez, E. Rodriguez, J.C. Echeverría, Detrending fluctuation analysis based on moving average filtering, Physica A 354 (2005) 199-219. [38] G.F. Gu, W.X. Zhou, Detrending moving average algorithm for multifractals, Physical Review E 82 (2010) 011136. [39] L. Kristoufek, Multifractal height cross-correlation analysis: A new method for analyzing long-range cross-correlations, Europhysics Letters 95 (2011) 68001. [40] C. Meneveau, K.R. Sreenivasan, Interface dimension in intermittent turbulence, Physical Review A 41 (1990) 2246-2248. [41] J. Wang, P.J. Shang, W.J. Ge, Multifractal cross-correlation analysis based on statistical moments, Fractals, 20 (2012) 271-279. [42] W.J. Xie, Z.Q. Jiang, G.F. Gu, X. Xiong, W.X. Zhou, Joint multifractal analysis based on the partition function approach: analytical analysis, numerical simulation and empirical application, New Journal of Physics 17 (2015) 103020. [43] Z.Q. Jiang, W.X. Zhou, H.E. Stanley, Multifractal cross wavelet analysis, Physical Review E, resubmitted (2017), http://arxiv.org/abs/1610.09519. [44] Z.Q. Jiang, Y.H. Yang, G.J. Wang, W.X. Zhou, Joint multifractal analysis based on wavelet leaders, Frontiers of Physics, in press (2017), http://arxiv.org/abs/1611.00897. [45] N.M. Yuan, Z. Fu, H. Zhang, P. Lin, X. Elena, L. Juerg, Detrended partial-cross-correlation analysis: a new method for analyzing correlations in complex system, Scientific Reports 5 (2015) 08143.

27

[46] X.Y. Qian, Y.M. Liu, Z.Q. Jiang, B. Podobnik, W.X. Zhou, H.E. Stanley, Detrended partial cross-correlation analysis of two nonstationary time series influenced by common external forces, Physical Review E 91 (2015) 062816. [47] C.H. Shen, C.L. Li, An analysis of the intrinsic cross-correlations between API and meteorological elements using DPCCA, Physica A 446 (2015) 100-109. [48] P. Lin, Z. Fu, N.M. Yuan, “Intrinsic” correlations and their temporal evolutions between winter-time PNA/EPW and winter drought in the west United States, Scientific Reports 6 (2016) 19958. [49] H.E.A.Van Den Akker, Coherent structures in multiphase flows, Powder Technology 100 (1998) 123-136. [50] R.Benzi, S.Patarnello, P.Santangelo, Self-similar coherent structures in two-dimensional decaying turbulence, Journal of Physics A: Mathematical and General 21 (1988) 1221-1237. [51] B. J. Cantwell, Organized motion in turbulent flow, Annual Review of Fluid Mechanics 13 (1981) 457-515. [52] J. H. Liu, N. Jiang, Z. D. Wang, W. Shu, Multi-scale coherent structures in turbulent boundary layer detected by locally averaged velocity structure functions, Applied Mathematics and Mechanics 26 (2005) 495-504. [53] J.S.Groen, R.G.C.Oldeman, R.F.Mudde, H.E.A. Van Den Akker, Coherent structures and axial dispersion in bubble column reactors, Chemical Engineering Science 51 (1996) 2511-2520. [54] H. Li, Identification of coherent structure in turbulent shear flow with wavelet correlation analysis, Journal of Fluids Engineering 120 (1998) 778-785.

28

[55]C.K. Peng, S. Havlin, H.E. Stanley, A.L. Goldberger, Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series, Chaos 5 (1995) 82. [56] J.R.M. Hosking, Fractional Differencing, Biometrika 68 (1981) 165-176. [57] B. Podobnik B, D. Horvatic D, A.L. Ng, H.E. Stanley, P.C. Ivanov, Modeling long-range cross-correlations in two-component ARFIMA and FIARCH processes, Physica A 387 (2008) 3954-3959. [58] L.S. Zhai, N.D. Jin, Y.B. Zong, Q.Y. Hao, Z.K. Gao, Experimental flow pattern map, slippage and time-frequency representation of oil-water two-phase flow in horizontal small diameter pipes, International Journal of Multiphase Flow 76 (2015) 168-186. [59] L.S. Zhai, N.D. Jin, Y.B. Zong, Z.Y. Wang, M. Gu, The development of a conductance method for measuring liquid holdup in horizontal oil-water two-phase flows, Measurement Science and Technology 23 (2012) 025304. [60] G.P. Lucas, J.C. Cory, R.C. Waterfall, A six-electrode local probe for measuring solids velocity and volume fraction profiles in solids-water flows, Measurement Science and Technology 11 (2000) 1498-1509. [61] L.S. Zhai, Y.B. Zong, H.M. Wang, C. Yan, Z.K. Gao, N.D. Jin, Characterization of flow pattern transitions for horizontal liquid-liquid pipe flows by using multi-scale distribution entropy in coupled 3D phase space, Physica A 469 (2017) 136-147.

29

Notation Qw

flow rate of the water phase

Qo

flow rate of the oil phase

Usw

superficial velocity of the water phase

Uso

superficial velocity of the oil phase

T

period of sine series

Ui

sinusoidal voltage exciting signal

Uo,up

upstream response signal of CCVP

Uo,down

downstream response signal of CCVP

s

time scale

smax

time scale corresponding to the maximum cross-correlation coefficient

{ x ( i )} , { y (i )}

non-stationary time series for the validity of cross-correlation analysis

{ z ( i )}

noisy signals

Greek letters

ρ DCCA

cross-correlation coefficient calculated by DCCA method

ρDPCCA cross-correlation coefficient calculated by DPCCA method

30

Figure captions Fig. 1. (color online) Cross-correlation levels of ARFIMA process with different cross-correlation strength controlling parameters. The length of data used is 213. Fig. 2. (color online) Overlap series of typical ARFIMA process series and sine series (T=500). Fig. 3. (color online) The DCCA cross-correlation coefficients of ARFIMA processes mixed with sine series. Fig. 4. (color online) DCCA and DPCCA multi-scale cross-correlations for original ARFIMA processes and the processes contaminated by noisy signals. The length of data used is 213. (a) Sine noise and contaminated ARFIMA processes; (b) Periodic noise immunity analysis of DPCCA; (c) White noise and contaminated ARFIMA processes; (d) Stochastic noise immunity analysis of DPCCA; (e) Logistic map noise and contaminated ARFIMA processes; (f) Chaotic noise immunity analysis of DPCCA; (g) Schuster map noise and contaminated ARFIMA processes; (h) Chaotic noise immunity analysis of DPCCA. Fig. 5. (color online) Experimental set up for horizontal oil-water two-phase flows. Fig. 6. (color online) Measurement system of the conductance cross-correlation velocity probe. Fig. 7. (color online) Experimental flow pattern map and flow structures of oil-water two-phase flows. Fig. 8. (color online) Collected responses of the conductance cross-correlation velocity probe under typical flow conditions. Fig. 9. (color online) DPCCA cross-correlation coefficient for different flow pattern of horizontal oil-water two-phase flow Fig. 10. (color online) Average values of DPCCA coefficient at small and large scales for horizontal oil-water two-phase flows.

31

1.0

W=0.5 W=0.6 W=0.7 W=0.8 W=0.9 W=1

d1=0.1, d2=0.4

0.9 0.8 0.7

ρDCCA

0.6 0.5 0.4 0.3 0.2 0.1 0.0 -0.1 -0.2 10

100

1000

Scale s

Fig. 1. (color online) Cross-correlation levels of ARFIMA process with different cross-correlation strength controlling parameters. The length of data used is 213 . 10

x(i)

5 0 -5 -10 10

y(i)

5 0

Test signals

-5 -10 8

z4(i)

4 0 -4 -8 12

x4(i)=x(i)+z4(i)

6 0 -6 -12 12

y4(i)=y(i)+z4(i)

6 0 -6 -12

0

1000

2000

3000

4000

5000

6000

7000

8000

Point number

Fig. 2. (color online) Overlap series of typical ARFIMA process series and sine series (T=500). 1.0

x-y, DCCA

0.9

x -y , DCCA

1

0.8

x -y2, DCCA

0.7

x -y , DCCA

0.6

ρDCCA

1

2 3

3

4

4

5

5

x -y , DCCA

0.5

x -y , DCCA

0.4 0.3 0.2 0.1 0.0 -0.1 -0.2 10

100

1000

Scale s

Fig. 3. (color online) The DCCA cross-correlation coefficients of ARFIMA processes mixed with sine series.

32

(a)

10

x(i)

5

(b)

1.0 0.9

0 -5 -10 10 0

Test signals

-5 -10 10

3

z (i)

Sine signal

5 0 -5 -10 10

ρDCCA, ρDPCCA

y(i)

5

x3-y3, DCCA, order=2

0.7

x -y , DCCA, order=3

3 3

x -y3, DCCA, order=4

0.5

x3-y3, DPCCA

0.2

0

0.1

-10 10

3

0.0

3

y (i)=y(i)+z (i)

5

-0.1

0 -5

-0.2 0

500

1000

1500

10

2000

Scale s

Point number

(c) 10

x( i)

5

0.9

-5 -10 10 0

Test signals

-5 -10 10

w

z (i)

White noise

5 0 -5 -10 10

ρDCCA, ρDPCCA

y(i)

5

x-y, DCCA White noise

xw-yw, DCCA

0.8

xw-yw, DCCA, order=2

0.7

xw-yw, DCCA, order=3

0.6

xw-yw, DCCA, order=4

0.5

x -y , DPCCA

w

0.2

0

0.1 0.0

-10

10

yw(i)=y(i)+zw(i)

-0.1 -0.2

500

1000

1500

10

2000

100

(e) 105

x(i)

1000

Scale s

Point number

(f)

1.0 0.9

0

x-y, DCCA Logistic map

l

-10 10 0

Test signals

-10 10

zl(i)

Logistic map

5 0 -5 -10 10

ρDCCA, ρDPCCA

y(i)

5 -5

xl(i)=x(i)+zl(i)

5

0.8

xl-yl, DCCA, order=2

0.7

x -y , DCCA, order=3

0.6

xl-yl, DCCA, order=4

0.5

xl-yl, DPCCA

l

l

0.4 0.3 0.2

0

0.1

-5 -10 10

l

0.0

l

y (i)=y(i)+z (i)

5

-0.1

0 -5

-0.2 0

500

1000

1500

2000

10

100

Scale s

Point number

(g) 10

x(i)

5 0 -5 -10

(h)

s

z (i)

Schuster map

3

0

xs(i)=x(i)+zs(i)

10 5 0 -5 -10

ρDCCA, ρDPCCA

y(i)

1000

1.0 0.9

10 5 0 -5 -10

Test signals

l

x -y , DCCA

-5

-10

w

0.3

-5

0

1000

0.4

xw(i) =x(i)+zw(i)

5

100

(d) 1.0

0

5 0 -5 -10

3

0.6

0.3

-5

-10

x3-y3, DCCA

0.8

0.4

x3(i)=x(i)+z3(i)

5

x-y, DCCA

Sine signal

x-y, DCCA Schuster map

xs-ys, DCCA

0.8

xs-ys, DCCA, order=2

0.7

xs-ys, DCCA, order=3

0.6

x -y , DCCA, order=4

0.5

x -y , DPCCA

s s s s

0.4 0.3 0.2 0.1 0.0

s

10 5 0 -5 -10

s

y (i)=y(i)+z (i)

-0.1 -0.2 10

0

500

1000

1500

100

2000

1000

Scale s

Point number

Fig. 4. (color online) DCCA and DPCCA multi-scale cross-correlations for original ARFIMA processes and the processes contaminated by noisy signals. The length of data used is 213. (a) Sine noise and contaminated

33

ARFIMA processes; (b) Periodic noise immunity analysis of DPCCA; (c) White noise and contaminated ARFIMA processes; (d) Stochastic noise immunity analysis of DPCCA; (e) Logistic map noise and contaminated ARFIMA processes; (f) Chaotic noise immunity analysis of DPCCA; (g) Schuster map noise and contaminated ARFIMA processes; (h) Chaotic noise immunity analysis of DPCCA.

1200mm

100mm 900mm

1350mm

125mm

1650mm

Diverter Flow direction

Contraction Conductance orifice cross-correlation velocity probe

Mini-conductance probe array 3

Control System

1 2

F

F

7

6

5

4

1 oil/water separate tank; 2 water tank; 3 oil tank; 4 pump; 5 hand ball valve; 6 flow meter; 7 pneumatic diaphragm

Fig. 5. (color online) Experimental set up for horizontal oil-water two-phase flows.

Fig. 6. (color online) Measurement system of the conductance cross-correlation velocity probe.

34

ST ST&MI D W/O&D O/W

(a)

D O/W&W D O/W D W/O

3

Qw (m /day) 60 50 40

Usw (m/s)

1

30

Ⅲ

Ⅰ

20 15

Ⅲ

10 8

Ⅱ

Ⅰ

6 5 4

0.1

Ⅱ

Ⅳ

3 2.88 3.6 4.4 5.28 6.4

8

10 12

15

20

25 30 34 40 45 50 60 70

3

Qo (m /day)

0.1

Uso (m/s)

(b)

ST flow

(g)

(c)

ST&MI flow (Type I)

(h)

(d)

ST&MI flow (Type II)

(i)

(e)

ST&MI flow (Type III)

(j)

(f)

ST&MI flow (Type IV)

(k)

1

DW/O&DO/W flow (Type I)

DW/O&DO/W flow (Type II)

DW/O&DO/W flow (Type III)

DO/W&/W flow

DO/W flow

Fig. 7. (color online) Experimental flow pattern map and flow structures of oil-water two-phase flows.

35

upstream

downstream

0.2 0.1 0.0 -0.1 -0.2 0.2 0.1 0.0 -0.1 -0.2

0.50 0.25 0.00 -0.25 -0.50 0.50 0.25 0.00 -0.25 -0.50 0.50 0.25 0.00 -0.25 -0.50 0.50 0.25 0.00 -0.25 -0.50 0.050 0.025 0.000 -0.025 -0.050

Qw=4 m3/day,Qo=4.4 m3/day,ST

0.2 0.1 0.0 -0.1 -0.2

Qw=4 m3/day,Qo=6.4 m3/day,ST&MI type I

0.2 0.1 0.0 -0.1 -0.2

Qw=4 m /day,Qo=12 m /day,ST&MI type IV

0.2 0.1 0.0 -0.1 -0.2

Qw=4 m3/day,Qo=15 m3/day,ST&MI type IV

3

0

1

2

3

4

3

5

6

7

8

9

upstream

(b)

Qw=4 m3/day,Qo=2.88 m3/day,ST

10

Qw=15 m3/day,Qo=10 m3/day,ST&MI, type II

Qw=15 m3 /day,Qo=15 m3/day,DW/O&DO/W, type I

3

3

3

(d)

3

Qw=30 m /day,Qo=6.4 m /day,DO/W&W

3

3

Qw=30 m /day,Qo=12 m /day,DO/W&W

Qw=30 m3/day,Qo=20 m3/day,DW/O&DO/W, type III

Probe signals (V)

Probe signals (V)

downstream

Qw=30 m3/day,Qo=2.88 m3/day,DO/W&W

Qw=30 m3/day,Qo=60 m3/day,DW/O&DO/W, type I

0

1

2

3

4

3

Qw=15 m /day,Qo=45 m /day,DW/O&DO/W, type II

0

1

2

3

4

5

6

7

8

9

10

Time (s)

upstream 0.10 0.05 0.00 -0.05 -0.10 0.10 0.05 0.00 -0.05 -0.10 0.10 0.05 0.00 -0.05 -0.10 0.10 0.05 0.00 -0.05 -0.10 0.10 0.05 0.00 -0.05 -0.10

3

Qw=15 m /day,Qo=30 m /day,DW/O&DO/W, type I

Time (s)

(c)

downstream

Qw=15 m3/day,Qo=2.88 m3/day,ST&MI, type III

Probe signals (V)

Probe signals (V)

(a)

5

6

7

8

9

10

upstream 0.10 0.05 0.00 -0.05 -0.10 0.10 0.05 0.00 -0.05 -0.10 0.10 0.05 0.00 -0.05 -0.10 0.10 0.05 0.00 -0.05 -0.10 0.10 0.05 0.00 -0.05 -0.10

downstream Qw=40 m3/day,Qo=2.88 m3/day,DO/W

Qw=40 m3/day,Qo=6.4 m3 /day,DO/W

Qw=40 m3/day,Qo=15 m3/day,DO/W

3

3

Qw=40 m /day,Qo=34 m /day,DW/O&DO/W, type III

Qw=40 m3 /day,Qo=60 m3/day,DW/O&DO/W, type I

0

1

2

3

4

Time (s)

5

6

7

8

9

10

Time (s)

Fig. 8. (color online) Collected responses of the conductance cross-correlation velocity probe under typical flow conditions.

1.0

Qo(m3/day) Flow Pattern 2.88 ST 3.6 ST 4.4 ST 5.28 ST 6.4 ST&MI 8 ST&MI 10 ST&MI 12 ST&MI 15 ST&MI 20 ST&MI

Qw=4 m 3/day

0.8

ρDPCCA

0.6 0.4 0.2 0.0 -0.2

(b) 1.2 1.0

0.6 0.4 0.2 0.0 -0.2

-0.4

-0.4

-0.6

-0.6

-0.8 10

100

1000

3

Qo(m /day) Flow Pattern 2.88 ST&MI 3.6 ST&MI 4.4 ST&MI 5.28 ST&MI 6.4 ST&MI 8 ST&MI 10 ST&MI 12 ST&MI 15 ST&MI 20 ST&MI 25 DW/O&DO/W 30 DW/O&DO/W 34 DW/O&DO/W 40 DW/O&DO/W 45 DW/O&DO/W 50 DW/O&DO/W

Qw=15 m3 /day

0.8

ρDPCCA

(a) 1.2

-0.8 10

Scale s

36

100

Scale s

1000

3

Qo(m /day) Flow Pattern 2.88 DO/W&W 3.6 DO/W&W 4.4 DO/W&W 5.28 DO/W&W 6.4 DO/W&W 8 DO/W&W 10 DO/W&W 12 DO/W&W 15 DO/W&W 20 DO/W&W 25 DW/O&DO/W 30 DW/O&DO/W 34 DW/O&DO/W 40 DW/O&DO/W 45 DW/O&DO/W 50 DW/O&DO/W 60 DW/O&DO/W

Qw=30 m3/day

1.0 0.8

0.4 0.2 0.0

-0.2 -0.4 -0.6 -0.8 10

100

Qo(m3/day) Flow Pattern 2.88 DO/W 3.6 DO/W 4.4 DO/W 5.28 DO/W 6.4 DO/W 8 DO/W 10 DO/W 12 DO/W 15 DO/W 20 DO/W 25 DO/W 30 DW/O&DO/W 34 DW/O&DO/W 40 DW/O&DO/W 45 DW/O&DO/W 50 DW/O&DO/W 60 DW/O&DO/W 1000 70 DW/O&DO/W

Qw=40 m3/day

1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 10

1000

100

Scale s

Scale s

Fig. 9. (color online) DPCCA cross-correlation coefficient for different flow pattern of horizontal oil-water two-phase flow.

ST, Small scale ST, Large scale

(a) 1.0

ST&MI, Small scale ST&MI, Large scale 3

Qw=4 m /day

0.8

ST&MI (type I)

0.4

ST&MI, Small scale ST&MI, Large scale

1.0

0.6

ST&MI (type IV)

0.2 0.0 -0.2

ST&MI (type II)

ST&MI (type III)

0.4

-0.4

DW/O&DO/W (type I)

0.2 0.0

-0.4

-0.6

-0.6 -0.8

-1.0

2

4

6

8

10

12

14

16

18

20

-1.0

22

0

10

20

Qo(m /day) DW/O&DO/W, Small scale DW/O&DO/W, Large scale

Qw=30 m3/day

0.8

DO/W, Small scale DO/W, Large scale

(d) 1.0 0.8 0.6

0.6 0.4

DW/O&DO/W (type III)

0.2 0.0 -0.2

DW/O&DO/W (type I)

-0.4 -0.6 -0.8

40

0.4 0.2

50

DW/O&DO/W, Small scale DW/O&DO/W, Large scale

DW/O&DO/W (type III)

DO/W&W, Small scale DO/W&W, Large scale

1.0

30

Qo(m3/day)

3

Average value of ρDPCCA

DW/O&DO/W (type II)

-0.2

-0.8

(c)

DW/O&DO/W, Small scale DW/O&DO/W, Large scale

Qw=15 m3/day

0.8

Average value of ρDPCCA

Average value of ρDPCCA

0.6

(b)

Average value of ρDPCCA

ρDPCCA

0.6

(d) 1.2

ρDPCCA

(c) 1.2

Qw=40 m3/day

DW/O&DO/W (type I)

0.0 -0.2 -0.4 -0.6 -0.8

-1.0

0

10

20

30 3

40

50

-1.0

60

0

10

20

30

40

50

60

70

Qo(m3/day)

Qo(m /day)

Fig. 10. (color online) Average values of DPCCA coefficient at small and large scales for horizontal oil-water two-phase flows.

37

Highlights (1) DPCCA quantifying cross-correlations of non-stationary signals is evaluated. (2) Conductance signals are collected from horizontal oil-water two-phase flows. (3) Multi-scale cross-correlations of flow structures are investigated using DPCCA. (4) Interfacial wave and droplet entrainment in flows are uncovered by DPCCA.

38