7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010
A New Perspective for Investigating Two-Phase Flow: Complex Networks
Zhongke Gao and Ningde Jin
School of Electrical Engineering and Automation, Tianjin University
Tianjin 300072, People's Republic of China
zhongkegao~tju.edu.cn and ndjinatju.edu.cn
Keywords: Two-phase flow, complex networks, flow pattern identification, nonlinear dynamics
Abstract
Complex networks have established themselves in recent years as being particularly suitable and flexible for representing and
modelling many complex natural and artificial systems since the publication of the seminal works of Watts and Strogatz as
well as Barabisi and Albert. A lot of breakthroughs from different research areas have been achieved with the help of complex
networks. We introduce complex network to the study of flow pattern nonlinear dynamics of two-phase flow and have
achieved many interesting results. Based on our previous works, we in this paper briefly introduce the new progress of
complex network in the study of two-phase flow. That is, we develop three types of two-phase flow complex network, i.e.,
flow pattern complex network, fluid dynamic complex network and fluid structure complex network, and their corresponding
applications to vertical gas-water two-phase flow. Furthermore, we indicate that complex networks, which provide us with a
new viewpoint and an effective tool for understanding a complex system from the relations between the elements in a global
way, not only may be a powerful tool for revealing information embedded in nonlinear time series but also can be used for
studying nonlinear dynamic systems that can not be perfectly described by theoretical model (e.g., two-phase flow system).
Introduction
Two-phase flow very often exists in industrial applications
such as filtration, lubrication, spray processes, natural gas
networks and nuclear reactor cooling. In the study of
two-phase flow, flow patterns indicate how the phases are
distributed and mixed due to the physical nature of the
system. Two-phase flow patterns depend on the type of
fluid-fluid combination, the flow rates and direction, the
conduit shape, size and inclination. Further, heat and mass
transfer rates, momentum loss, rate of back mixing and
pipe vibration all vary greatly with the flow patterns. Hence,
it is quite important and necessary to discern the flow
patterns and study the nonlinear dynamics in different flow
patterns. But influenced by many complex factors, such as
fluid turbulence, phase interfacial interaction and local
relative movement between phases, two-phase flow
presents highly irregular, random and unsteady flow
structure. Due to its complex nature, theoretical analyses
have not been able to describe the two-phase system
perfectly. Therefore, a new method is strongly required
which, whether experimentally or theoretically, will detect
and describe the flow pattern in conditions near patten
transitions and the nonlinear dynamics in different flow
patterns.
The past few years have witnessed dramatic advances in
the field of complex networks since the publication of the
seminal works of Watts and Strogatz (1998) as well as
Barabisi and Albert (1999). Complex networks, which
have been observed to arise naturally in a vast range of
physical phenomena, can describe any complex system that
contains massive units (or subsystems) with nodes
representing the component units and edges standing for
the interactions between them. A lot of complex systems
have been examined from the viewpoint of complex
networks. Examples include World Wide Web (Albert
1999), metabolic networks (Jeong 2000), protein networks
in the cell (Jeong 2001), traffic networks (Li 2007),
earthquake networks (Sumiyoshi 2006) and human
electroencephalogram networks (Cai 2007). These
empirical studies have inspired researchers to develop a
variety of techniques and models to help us understand or
predict the behavior of complex systems (Crucitti 2004,
Wang 2006). Complex networks, which provide us with a
new viewpoint and an effective tool for understanding a
complex system from the relations between the elements in
a global way, not only may be a powerful tool for revealing
information embedded in time series (Zhang 2006, Xu
2008) but also can be used for studying nonlinear dynamic
systems that can not be perfectly described by theoretical
model.
Quite recently, complex network theory has been
incorporated into the study of two-phase flow (Gao and Jin
2009). Based on the signals measured from gas-water
two-phase flow experiment, we proposed three different
methods to construct Flow Pattemn Complex Networ
(FPCN), Fluid Dynamic Complex Network (FDCN) and
Fluid Structure Complex Network (FSCN). Using our
proposed community-detection algorithm (ie
community-detection algorithm based on K-means
clustering), we analyzed the community property of
FPCN, and achieved good identification of flow pattern for
7th Intentoa CnenconMlihse Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010
is mainly constitutes by three modules, which are
differential amplifier, sensitive demodulation and low pass
filter. The data acquisition equipment is selected from the
National Instrument Company's product PXI 4472 data
acquisition card, which is based on the PXI main bus
technology, equipped with eight channels and synchronized
acquiring function. The data processing part is realized
through graphical programming language LABVIEW 7.1
wrapped in the data acquisition card, which can realize real
time data waveform displaying, storing and analyzing.
The experimental plan was such that first we put a fixed
water flow rate into the pipe, then we gradually increased
the gas flow rate; every time finishing the proportion of gas
flow rate and water flow rate, we acquired one conductance
fluctuating signal from VMEA, and observed the flow
pattemnvariation by high speed VCR. In the experiment, we
set the resolution at 640 x 480; the frame rate at 200 frames
s According to the definition of flow pattern proposed by
Hewitt (1980), five different gas-water two-phase flow
patterns were observed in our experiment, i.e., Bubble flow,
Bubble-slug transitional flow, Slug flow, Slug-chumn
transitional flow and Chumn flow. Fig. 2 shows the five
flow patterns of gas-water two-phase flow in a vertical
upward pipe recorded by our high speed VCR system. The
water phase flow rate was between 1 m3/hour to 14 m3/hour,
of which the gas phase was between 0.2 m3/hour to 130
m3/hour, and there are 90 different proportions of gas flow
rate and water flow rate in this experiment. The sampling
frequency was 400 Hz, and the sampling data recording
time for one measuring point was 60 s. We have acquired
90 conductance fluctuating signals in the experiment all
together. The conductance fluctuating signals in five flow
patterns, measured from sensor C, are shown in Fig. 3, in
which U,, and U,, represent gas superficial velocity and
water superficial velocity, respectively.
Because of the significant difference in electrical sensibility
between gas phase and water phase, the random flow of gas
phase will cause voltage fluctuation on the measuring
electrode under a certain sinusoidal input, which implies
that the conductance fluctuating signals measured from the
VMEA conductance sensor are related to the flow
transition. Thus, we construct complex networks from the
conductance fluctuating signals and study the gas-water
two-phase flow through analyzing the resulting networks.
lol
dieveion elH,
s... -O-,-
(a) (b)
Figure 1: The VMEA (Vertical Multi-Electrode Array)
conductance sensor. (a) The VMEA measurement section;
(b) The geometry of VMEA.
gas-water two-phase flow by finding the community
structures which correspond to different flow patterns.
Throuth investigating the degree distribution of the FDCN,
we find its degree distributions can be well fitted with a
power law, which indicates the scale-free property of the
FDCN. Moreover, we found that the power-law exponent
and network information entropy, which are sensitive to the
flow pattern transition, can both characterize the nonlinear
dynamics of gas-water two-phase flow. Based on the our
recently proposed approach that can extract phase space
complex network from time series, we constructed PSCN
and indicated that assortative mixing property of PSCN not
only can well distinguish the dynamical regimes associated
with unstable periodic orbits (UPOs), but also can
effectively reflect the bubble coalescence and bubble
collapse in gas-water fluid structure. Applying complex
networks to analyze the characteristics of two-phase flow
measurement fluctuant signals could be useful exploration
for revealing and understanding the flow patten
transmission mechanism which cannot be described
accurately by mathematical model because of the
complexity and uncertainty it owns.
The paper is organized as follows. In Sec. II, we show the
two-phase flow experimental flow loop facility and data
acquisition. In Sec. III, we briefly introduce how to
construct FPCN from measured signals and its applications
to flow pattern identification. In Sec. IV, we indicate how
to construct FDCN and its applications to flow pattern
dynamics analysis. In Sec. V, we demonstrate how to
construct FSCN and how the statistic of FSCN can be used
to uncover the fluid structure of gas-water two-phase flow.
In Sec. VI, we present the conclusions and discussions.
The experimental flow loop facility and data
acquisition.
The gas-water two-phase flow experiment in a 125 mm
diameter vertical upward pipe was carried out in the
multiphase flow loop of Tianjin University. The whole
measurement system can be divided into several parts,
including the VMEA (Vertical Multi-Electrode Array)
conductance sensor that was designed and optimized by our
research team (Jin 2008), high speed VCR (Video Camera
Recorder), exciting signals generating circuit, signal
modulating module, data acquisition device and signal
analysis software. The VMEA, as shown in Fig. 1, consists
of eight alloy titanium ring electrodes axially separated and
flush mounted on the inside wall of the flowing pipe. El
and E2 are exciting electrodes. C1-C2 and C3- 4 are two
pairs of upstream and downstream correlation electrodes
denoted as sensor A and sensor B, respectively. Based on
the cross-correlation technique, we can extract the axial
velocity of two-phase flow from fluctuating signals of
sensors A and B. H1-H2 iS the volume fraction electrodes
denoted as sensor C. The fluctuating signals of sensor C are
mainly correlated with the phase volume fraction. The
measurement circuit is embedded inside the instrument, and
signals were transmitted to the data acquisition and
processing system through cable connected outside the
circuit tube. The measurement system uses the 20 kHz
constant voltage sine wave to excite, and the virtual value
of exciting voltage is 1.4 V. The signal modulating module
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010
1980) to process the conductance fluctuating signals. That
is, we use C-C method (Kim 1999) to calculate the delay
time r, from 90 conductance fluctuating signals,
respectively, and choose the proper r, that can make the
FPCN modularity (Nei\ nlan1 2004) largest. Then we extract
six time-domain features and four frequency-domain
features from each processed conductance fluctuating
signal to form the characteristic vector (In the time domain,
we choose the maximum value, minimum value, average
value, standard deviation, asymmetry coefficient and
kurtosis function as the features of signals; in the frequency
domain, we select the four coefficients of the linear
prediction model with four orders. That is, there are 90
characteristic vectors and each vector contains ten elements.
For each pair of characteristic vectors, Ti and Tj, the
correlation coefficient can be written as:
C = k=1 1
TIh (k -(i)-T (ki) T
where, M is the dimension of the characteristic vector and
(Ti) = Ti (k) I ,T = T (k)/, I. Theelement
C, are TCStricted to the domain-1 E C,, I1, where C, = 1,
0 and -1 correspond to perfect correlations, no correlations
and perfect anti-correlations, respectively. C is a symmetric
matrix and C, describes the state of connection between
node i and j. Finally, based on the network modulanrit
(Nein\lllnl 2004), choosing a critical threshold >;, the
correlation matrix C can be converted into adjacent matrix
A, the rules of which read:
4 10 ( C lr) (2)
That is, there will be an edge connecting node i and nodej
if C Icr .1 On he other hand, there will not be an edge
connecting node i and node j if C~ *
edges form the FPCN, and the topological structure of this*
network can be described with the adjacent matrix A. The
conditions 4,,= 1 and 4, = correspond to connection
and disconnection, respectively. More details about how to
properly select the threshold see Reference (Gao and Jin
2009 Phys. Rev. E).
In our previous work about conununity structure, we have
proposed two different network conununity-detection
algoritluns, i.e., conununity detection based on k-means
clustering and conununity detection based on data fields
(Gao and Jin 2009 Chinese Journal of Control and
Decision). After constructing the FPCN containing 90
nodes, we apply conununity-detection algoritlun based on
K-means clustering to investigate the conununity structure
of FPCN. We show in Fig. 4 the distribution of the
corresponding elements of the three first non-trivial
eigenvectors. As can be seen, three different communities
can be clearly identified when the components of the first
non-trivial eigenvector al are plotted versus those of a2 and
a3. The conununity structure of the FPCN, detected by
On? Lsw=()18mlstsg=()()4mls Bubble-slug transtlonal now
LUsw=() 18m s Lsg-) 12m s Slug now
no Lsw=()18m/stUsg-)35ms Slug-churn transltional now
no L~~~~~sw-() 18m s LUsg-( 61m s CunHw
0 2 4 6 810 12 14 16 18 20
Time
Figure 3: The conductance fluctuating signals in five flow
patterns
Flow pattern complex network
Flow Pattemn Complex Network (FPCN), extracted from
the conductance fluctuating signals, is an abstract network,
in which each flow condition is represented by a single
node and the edge is determined by the strength of
correlation between nodes. Flow condition here means that
the flow behavior under different proportion of gas flow
rate and water flow rate in the pipe. Since we configured 90
different proportions of gas flow rate and water flow rate to
get 90 conductance fluctuating signals in our gas-water
two-flow experiment, there are 90 different flow conditions
(i.e., the number of nodes contained in FPCN is 90), and
each node corresponds to one of these 90 conductance
fluctuating signals.
Note that the correlation between two nodes means that the
correlation between two corresponding conductance
fluctuating signals. We now demonstrate how the strength
of correlation between conductance fluctuating signals can
be used to determine the edge. Considering the nonlinear
characteristics of the gas-water two-phase flow, we first
apply the method of Time-Delay Embedding (Packard
(d) (e)
Figure 2: The five vertical upward gas-water two-phase
flow patterns recorded by high speed VCR system. (a)
Bubble flow: (b) Bubble-slug transitional flow; (c) Slug
flow: (d) Slug-chumn transitional flow: (e) Chumn flow.
LUsh-=( 18m 5 Usg () ()1m s
Bubble now
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010
Fluid dynamic complex network
To cast light into the nonlinear dynamics of gas-water
two-phase flow, we constructed Fluid Dynamic Complex
Network (FDCN) from one conductance fluctuating signal
with each segment of signal time series represented by a
single node and edge determined by the strength of
correlation between segments. Considering a conductance
fluctuating signal (iea time series), denoted
as {k,,k,,k,,------,kx} we can obtain all the possible
segments with length L which read:
(S = (k~,kZ,.----- k~ ))
tS2 = (k2,k3,. -- -- -, k ,))
conununity-detection algorithm based on K-means
clustering, is shown in Fig. 5. The conununity structure is
drawn by the software "Ucinet" and "Netdraw".
02 '02 .03 1
Figure 4: Components of the first non-trivial eigenvector
al are plotted versus those of a2 and a3
tS3 = (k3,k4,,------k +)}
(Sm = (k,,, k,,, ,km+,,, ---- k,,, ,_)l n= 1,-----,N
For each pair of segments, Si and Sj the
coefficient can be written as:
[Si (k) -S3 (k)
C =k=1
[s] L L
k=1 k=1;.
(3)
-L+1)
correlation
(4)
Carm~lt c (Chara Rlow) Consmly dpb (Slug flowr) Conrmmm~y n ubbLe Ror)
Figure 5: Conununity structure of the FPCN.
From the detected conununity structure, as shown in Fig. 5-
three communities of 21, 30 and 39 nodes are found.
denoted as conununity a, conununity b and conununity c.
respectively. Furthermore, combining with the
experimental observations using high speed VCR, we find
that community a mainly corresponds to bubble flow, such
as node 2 (Qg= 0.2 m /h, Q,, = 2.0 m /h) and node 16 (Qg=
0.94 m /h, Q,, = 12.0 m /h) both corresponding to bubble
flow; conununity b mainly corresponds to slug flow, such
as node 31l(Qg= 2.1 m /h, Q, = 2.0 m /h) and node 44 (Qg=
4.1 m /h, Q,, = 6.0 m /h) both corresponding to slug flow:
conununity c mainly corresponds to chum flow, such as
node 70 (Qg= 69.0 m /h, Q, = 4.0 m /h) and node 90 (Qg=
139.0 m /h, Q, = 2.0 m /h) both corresponding to chum
flow; the nodes of the FPCN that connect tightly between
conununity a and community b correspond to bubble-slug
transitional flow, such as node 19 (Qg= 1.0 m /h, Q,, = 2.0
m /h) and node 26 (Qg = 1.7 m /h, Q,, = 4.0 m /h) both
corresponding to the bubble-slug transitional flow: the
nodes of the FPCN that connect tightly between
conununity b and community c correspond to slug-chum
transitional flow, such as node 32 (Qg= 38.0 m /h, Q, = 8.0
m /h) and node 58 (Qg = 25.0 m /h, Q,, = 4.0 m /h) both
corresponding to the slug-chum transitional flow. Hence,
through detecting the community structure of the FPCN by
the conununity-detection algoritlun based on K-means
clustering, we have achieved good identification of
gas-water two-phase flow pattern by finding the three
conununities which correspond to the bubble flow, slug
flow and chumn flow respectively and the nodes that
connect tightly between two conununities corresponding to
the transitional flow.
Choosing a critical threshold r,, the correlation matrix C
can be converted into adjacent matrix A, the rules of which
read: 4,= 1 if C,, I, and 4,,= if IC,,
parameter, i.e., the threshold r, and the length of a segment
L, should be properly selected. Through the analysis of the
evolution of FDCN containing 2000 nodes, we found that
the resulting network can well keep the physically
meaningful correlations when r, and L is chosen as 0.95
and 50, respectively. (see Reference (Gao and Jin 2009
Phys. Rev. E) for details).
By considering segments as nodes and associating network
connectivity with the correlation among segments, time
domain dynamics are naturally encoded into a network
configuration. We selected five types of flow pattern
conductance fluctuating signals for constructing five
FDCNs with each containing 2000 nodes. After
investigating the degree distribution of the five networks,
we find their degree distributions can be well fitted with a
power law, which indicates the scale-free property of the
FDCN, as shown in Fig. 6.
aal-
p(k)
1E-3-
r =0 95 L=50
o y1 138
o on~
OOKDtoE
1 1 o10
m bubble flow
O bubble-slug transitional flow
e slug flow
1.2-( a, V slug-churn transitional flow AA A
A churn flow A
1.0-I m
0.8 o o
0.-Usw=0 023~0 23 (m s)
0.01 0.1 1
Usg (m/s)
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010
exponent. To further explore the variations of the degree
distribution power-law exponent in flow pattern transition,
we constructed 50 FDCNs under different flow conditions
and calculate their corresponding power-law exponents.
From Fig. 7, we could see that, the power-law exponents of
bubble flow and bubble-slug transitional flow are usually
large, and the power-law exponent decreases as the flow
pattern evolves from bubble flow to slug flow. But with
further increase of gas superficial velocity, the power-law
exponent increases as the flow pattern evolves from slug
flow to chumn flow. Usg and Usw, in Fig.7, represent gas
superficial velocity and water superficial velocity,
respectively.
a01
p(k)
1E-3
S1~ r =0 95 L=50
y=0 7869
) 01
E-3l o
S10 160
0 Oh~~~y= :;29383 35
0 1 -
0 01-
p(k)
1E-3-
Figure 7: The power-law exponent distribution in semi-log
scale of 50 FDCNs with the increase of gas superficial
velocity.
According to the definition of Shannon's entropy, we define
network information entropy for the FDCN and expect to
investigate the nonlinear dynamics of gas- water flow
through analyzing the network information entropy.
Definition 1: Let P(i) be the importance of node i:
p~i) k, k(5)
where N is the number of nodes contained in the network:
k, (k,) is the degree of node i (f).
Definition 2: Let E be the network information entropy:
E = -C k, p(i)In p(i) (6)
where N is the number of nodes contained in the network
and ks is the Boltzmann constant In order to simplify the
calculation, here we let ks=1, that is:
E = -C p(i)1In p(i) (7)
In order to make the network information entropy not
affected by the number of nodes contained in the network,
we normalize E as follows:
S10 100
p(k)
1E-3-
(e)
Figure 6: Degree distribution of different types of FDCNs
in log-log scale. (a) Bubble flow (Q = 0.2 m /h, Q, = 2.0
m /h): (b) Bubble-slug transitional flow (Qg= 1.0 m /h, Q,
= 2.0 m /h): (c) Slug flow (Qg= 4.1 m /h, Q, = 6.0 m /h):
(d) Slug-chum transitional flow (Q = 25.0 m3/h, Q, = 4.0
m /hl): (e) Chum flow (Qg= 139.0 m l/h; Q, = 2.0 m r/hl).
More high degree nodes and less low degree nodes implies
that small degree distribution power-law exponent. On the
contrary, more low degree nodes and less high degree
nodes implies that large degree distribution power-law
Sp(i)In p(i)-In 4(N -1)
r~l 2
_E Em
E-E 1 1
m"E -p(i)1n p(i)-(-j AIn--
We calculated the network information entropy from the 50
constructed FDCNs. Fig. 8 shows the variations of network
a bubbleflow
-~ blubble-slug transitional flow
V slug-churn transitional flow
churn flow
O 'r
~, t"
Usw-0 023 O 23 (m/s)
0.01 0.1 1
Usg(rn/s)
(a>
H bubble flow
O bubble-slug transitional flow
e slug flow
v slug-churn transitional flow
--s
.Usw-0 023~0 23 (rn/s)
Usg(rn/s)
m bubble flow
O bubble-slug transitional flow
slugflow
V slug-churn transitional flow A
A churn flow A
0.875-
0.850-
0.800-
O 77E
I
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010
infonnation entropy with the change in flow pattern. As can
be seen, the network infonnation entropy decreases as the
flow pattern evolves from bubble flow to slug flow, but
increases as the flow pattern evolves from slug flow to
chum flow.
S0.30
E
a
o A
7 y,
Usw-0 023~0 23 (rn/s) g
Usg (m/s)
0.48
E 0.40
S0.32
0.24
Figure 8: The network infonnation entropy distribution in
semi-log scale of 50 FDCNs with the increase of gas
superficial velocity.
Representing the conductance fluctuation signal through a
corresponding FDCN, we can then explore the nonlinear
dynamics of gas-water two-phase flow from network
organization, which is quantified via a number of
topological statistics. The previous study of our research
team has indicated that the Lempel and Ziv complexity
(Lempel 1976), and approximate entropy (Pincus 1991) are
sensitive to the flow pattern transition in gas-water
two-phase flow. Here we show, in Fig. 9, the variations of
these two complexity measures with the change in flow
pattern. From Fig.7 and Fig.8 as well as Fig.9, we could see
that there are good corresponding relations between
complexity measures and the FDCN topological statistics
(i.e., power-law exponent and network infonnation
entropy). When the gas superficial velocity is low, due to
the stochastic motion of large number of small bubbles, the
dynamics of bubble flow are very complex, corresponding
to the large power-law exponent and network infonnation
entropy. In the transition from bubble flow to slug flow, the
dynamics of this transitional flow become relative simple'
resulting in the decrease of the two network statistics.
Owing to the periodic alternating movements between gas
plug and water plug, the dynamics of slug flow are the very
simple and that is why the two network statistics decrease
as the flow pattern evolves from bubble flow to slug flow.
When the gas superficial velocity is high, chumn flow
which is composed of discrete gas phase and continuous
water phase of high turbulent kinetic energy, gradually
appears with the phenomenon of fluctuation. Because of
the influence of the turbulence effect, the dynamics of
chum flow become more complex than that of slug flow,
corresponding to the increase of the two network statistics
as the flow pattern evolves from slug flow to chumn flow.
Hence, the power-law exponent and network infonnation
entropy, which are sensitive to the flow pattern transition,
can both characterize the nonlinear dynamics of gas-water
two-phase flow.
(b)
Figure 9: The complexity measures distributions in
semi-log scale with the increase of gas superficial velocity.
(a) Lempel and Zir complexity (using four-symbol coarse
graining); (b) Approximate entropy.
Fluid structure complex network
Recently, based on phase space resconstructiuon, we have
put forward a unique method for constructing phase space
complex network from time series (Gao and Jin 2009
Chaos). We first briefly introduce its general idea as
follows: We start from the phase space reconstruction, that
is, we using C-C method (Kim 1999) and FNN (False
Nearest Neighbors) method (Kennel 1992) reconstruct
phase space from time series. Considering an arbitrary time
series z(it), i 1,2 .. ,At (t is sampling interval, At is the
sample size), if the embedding delay time is selected as r,
and the embedding dimension as m, the vector point in
phase space can be represented as follow:
Xk k k (2,.... xk(9)
= (z(kt), z(kt + r), - ,z(kt + (m 1)r)}
where k = 1,2,...,N, N = Af- (m-1) v/t denotes the total
vector points of the reconstructed phase space. The
embedding delay and embedding dimension is determined
by C-C method and FNN method, respectively. After
reconstructing phase space from a given time series,
denoted as {z(1),z(2),---,z(Af)}, we proceed to construct
complex network by considering each vector point in
reconstructed phase space as a basic node and using the
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010
phase space distance to determine network connection. The
phase space distance between vector point XI and X in
this study, is defined as:
d~=c, ~i, = Nln)- ln (10)
where X, (n) = z(i +(n -1)r) and XJ (n) = z(j +(n -1)r) is
the nth element of XI and X, m and r is the embedding
dimension and delay time, respectively. The constructed
network contains N = M (m-1) v /t nodes. Choosing a
critical threshold re the distance matrix D = (d,,) can then
be converted into adjacent matrix A = (ax,), the rules of
which read: A,,= 1 if dl~r <: ad A,,= 0 if
Id~ >4. Details about how to choose an appropriate
threshold see Reference (Gao and Jin 2009 Phys. Rev. E).
All the nodes and edges form the PSCN, and the
topological structure of this network can be described with
the adjacent matrix A. The conditions A, = 1 and
Ag,= 0 correspond to connection and disconnection,
respectively.
Then, we extracted different types of complex network from
chaotic, periodic and noisy time series, and drawn their
corresponding phase space network structures by using
Kamada-Kawai spring embedding algorithm (Kamada
1989), as shown in Fig. 10. Through analyzing the resulting
networks, we found that periodic time series and noisy time
series converted into regular networks and random networks,
respectively, and the networks built from chaotic time series
typically exhibited small-world and scale-free features. By
exploring an extensive range of network topology statistics,
we in Reference (Gao and Jin 2009 Chaos) not only
indicated that the degree correlations and Pearson
coefficient can fundamentally reflect the assortative mixing
property associated with the Unstable periodic orbits
(UPOs), but also demonstrated that the betweenness
distribution, combined with the clustering
coefficient-betweenness correlations, can well distinguish
different dynamical regimes in time series. In addition, we
have also tested our method by analyzing the chaotic time
series corrupted by measurement noise, and indicated that
the proposed method has good anti-noise ability.
(d)
Figure 10: Complex network of 200 nodes from (a) Lorenz
system; (b) Rossler system; (c) periodic time series and (d)
white noise time series.
To cast light into the application of phase space complex
network in the study of two-phase flow, we constructed
Fluid Structure Complex Network (FSCN) of 2000 nodes
using the method mentioned above and made a further
investigation on their assortative mixing property. The
structure of FSCNs with 200 nodes generated from three
typical flow patterns are shown in Fig. 11.
We in Reference (Gao and Jin 2009 Phys. Rev. E) have
demonstrated how the statistic of FSCN can be used to
reveal the fluid structure of gas-water two-phase flow as
follows: In the bubble flow, the gas phase is approximately
uniformly distributed in the form of discrete small bubbles
in a continuum of water phase, and no obvious bubble
coalescence can be observed. While in the slug flow, due to
the bubble coalescence, many large bubbles which have a
diameter almost equal to the pipe diameter appear. Bubble
coalescence, which has an important impact on the fluid
structure of gas-water two-phase flow, can be reflected by
1ouu
Slug flow
600-
400-
40200-
U 200 400 600 800 1000 1200
I UUU I
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010
many unstable periodic orbits (UPOs) in the reconstructed
phase space. For a FSCN from slug flow, there are multiple
clusters which are caused by the bubble coalescence, and
most nodes connected to each other within the same cluster
have a roughly similar number of connected neighbors or
degrees. The common degree shared by the nodes from one
cluster may differ from that of another because of the
different stability of the center loop of UPOs. This has led
to the fact that the FSCN from slug flow possesses the
property of strong assortative mixing (Details see Fig.
12(b)), while no assortative mixing can be found in the
FSCN from bubble flow (see Fig. 12(a)). Chumn flow can
be interpreted as an irregular, chaotic and disordered slug
flow. Since bubble coalescence and bubble collapse both
exist in the fluid structure of chumn flow, the FSCN from
chum flow also possesses the property of assortative
mixing. But compared with the FSCN from slug flow, the
FSCN from chumn flow shows weak assortative mixing,
which may be caused by the bubble collapse (Details see
Fig. 12(c)). Therefore, we have associated the fluid
structure of gas-water two-phase flow with the topological
indices of FSCN, and indicate that the assortative nuxmng
property of FSCN can effectively reveal the gas-water fluid
structure to some extent.
0 200 400 600
800 10
Churn flow
oo o
B o
o"
800-
32 600
Y 400-
200-
0200 400 600 800 1000
(c)
Figure 12: The regular plots of the nearest neighbor
average connectivity of nodes with respect to connectivity
of the networks containing 2000 nodes from (a) Bubble
flow (Qg= 0.2 m3/h, Q,, = 2.0 m3/h), (b) Slug flow (Qg= 4.1
m'/ih, Q,= 6.0 m'/h), (c) Chumrflow (Qg= 139.0 ,m/h Q,,
= 2.0 m3/h).
Conclusions and discussions
In summary, we have introduced complex network theory
to the study of gas-water two-phase flow. Using a new
method based on Time-Delay Embedding and modularity,
we constructed FPCN from conductance fluctuating signals.
Then we put forward a new flow pattern identification
method of gas-water two-phase flow by combining the
community structure of FPCN.
In order to study the nonlinear dynamics of gas-water
two-phase flow, 50 FDCNs under different flow conditions
were constructed, and the nonlinear dynamics could then
be studied by investigating the statistical characteristics of
those networks. Based on the investigation of the statistical
(c)
Figure 11: The structure of FSCN containing 200 nodes
from (a) Bubble flow (Qg= 0.2 m3/h, Q,,= 2.0 m3/h) with
r,=0.0 12, (b) Slug flow (Qg= 4.1 m3/h, Q, = 6.0 m3/h) with
r,=0.16, (c) Chum flow (Qg = 139.0 m3/h, Q,, = 2.0 m3/h)
with r,=0.11.
800- Bubble flow
co0
600-
68)
400-
200-
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010
in complex networks based on K-means clustering and
data field theory. Chinese Journal of Control and
Decision, Vol. 24, 377-382 (2009).
Hewitt, G. F. Measurement of two-phase flow parameters
(London: Academic Press) (1980).
Kamada, T., Kawai, S. An algorithm for drawing general
undirected graphs. Inform. Process. Lett., Vol, 31, 7-15
(1989).
Kennel, M. B., Brown, R. and Abarbanel, H. D..
Determining embedding dimension for phase-space
reconstruction using a geometrical construction. Phys. Rev.
A, Vol. 45, 3403-3411 (1992).
Kim, H. S., Eykholt, R., Salas, J. D. Nonlinear dynamics,
delay times, and embedding windows. Physica D, vol. 127,
48-60 (1999).
Lempel, A. and Ziv, J. On the complexity of finite
sequences. IEEE Trans. Inform. Theory, Vol. 22, 75-81
(1976).
Li, X. G, Gao, Z. Y., Li, K. P. and Zhao, X. M.
Relationship between microscopic dynamics in traffic flow
and complexity in networks. Phys. Rev. E, Vol. 76,
016110-1-7 (2007).
Newman, M. E. J. Fast algorithm for detecting community
structure in networks. Phys. Rev. E, Vol. 69, 066133-1-5
Jeong, H., Tombor, B., Albert, R., Oltvai, Z. N. and
Barabisi, A. L. The large-scale organization of metabolic
networks. Nature, Vol. 407, 651-654 (2000).
Jeong, H., Mason, S., Barabisi, A. L. and Oltvai, Z. N.
Lethality and centrality in protein networks. Nature, Vol.
411, 41-42 (2001).
Jin, N. D., Xin, Z., Wang, J., Wang, Z. Y., Jia, X. H.
and Chen, W. P. Design and geometry optimization of a
conductivity probe with a vertical multiple electrode array
for measuring volume fraction and axial velocity of
two-phase flow. Meas. Sci. Technol. 19, 045403-1-19
(2008).
Packard, N. H., Crutehfield, J. P. and Farmer, J. D.
Geometry from a time series. Phys. Rev. Lett., Vol. 45,
712-716 (1980).
Pincus, S. M. Approximate entropy as a measure of system
complexity. Proc. Natl. Acad. Sci. USA, Vol. 88,
2297-2301 (1991).
Sumiyoshi, A. and Norikazu, S. Complex earthquake
networks: Hierarchical organization and assortative mixing.
Phys. Rev. E, Vol.74, 026113-1-5 (2006).
Watts, D. J. and Strogatz, S. H. Collective dynamics of
'small-world' Networks. Nature, Vol. 393, 440-442 (1998).
characteristics of FDCNs, we have confirmed that the
power-law exponent and network information entropy,
which are sensitive to the flow pattern transition, can both
characterize the nonlinear dynamics of gas-water two-phase
flow.
Furthermore, we have proposed a general method that can
be used for analyzing time series from any kind of complex
system, to study the fluid structure of gas-water two-phase
flow. We constructed FSCN using this method and
demonstrated that the distinction in topological structure of
FSCN could really reflect the bubble coalescence and
bubble collapse in gas-water fluid structure.
The last decade has witnessed the birth of a new movement
of interest and research in the study of complex networks,
and the massive and comparative analysis of networks from
different fields have produced a series of unexpected and
dramatic results. With the study presented in this paper, a
natural bridge between complex networks and two-phase
flow has now been built. Further research on the
self-organizing and self-evolution in these networks will
help us establish a flow-evolution network model to
explore the complex mechanism in two-phase flow.
Acknowledgements
This work has been supported by National Natural Science
Foundation of China (Grant No. 50974095) and National
High Technology Research and Development Program of
China (Grant No. 2007AAO6Z231).
References
Albert, R.., Jeong, H., Barabisi, A. L. The diameter of
the World Wide Web. Nature, Vol. 401, 130-131 (1999).
Barabisi, A. L. and Albert, R. Emergence of scaling in
randomnetworks. Science, Vol. 286, 509-512 (1999).
Crucitti, R., Latora, V., Marchiori, M. A topological
analysis of the Italian electric power grid. Physica A, Vol.
338, 92-97 (21 III4).
Cai, S. M., Jiang, Z. H., Zhou, T., Zhou, P. L., Yang, H. J.
and Wang, B. H. Scale invariance of human
electroencephalogram signals in sleep. Phys. Rev. E, Vol.
76, 061903-1-5 (2007).
Gao, Z. K. and Jin, N. D. Flow Pattern Identification and
Nonlinear Dynamics of Gas-Liquid Two-Phase Flow in
Complex Networks. Phys. Rev. E, Vol. 79, 066303-1-14
(2009).
Gao, Z. K. and Jin, N. D. Complex network from time
series based on phase space reconstruction. Chaos, Vol.
19, 033137-1-12 (2009).
Gao, Z. K. and Jin, N. D. Complex network analysis in
inclined oil-water two-phase flow. Chinese Physics B,
Vol.18, 5249-5258 (2009).
Gao, Z. K. and Jin, N. D. Detecting community structure
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010
Wang, W. X., Wang, B. H., Yin, C. Y., Xie, Y. B. and
Zhou, T. Traffic dynamics based on local routing protocol
on a scale-free networkPhys. Phys. Rev. E, Vol. 73,
026111-1-7 (2006).
Xu, X., Zhang, J., and Small, M. Superfamily phenomena
and motifs of networks induced from time series. Proc. Natl.
Acad. Sci. USA 105, Vol. 19601-19605 (2008).
Zhu, C. P., Xiong, S.J., Tian, Y. J., Li, N. and Jiang, K. S.
Scaling of Directed Dynamical Small-World Networks with
Random Responses. Phys. Rev. Lett., Vol. 92, 218702-1-4
Zhang, J. and Small, M. Complex Network from
Pseudoperiodic Time Series: Topology versus Dynamics.
Phys. Rev. Lett., Vol. 96, 238701-1-4 (2006).
*
* |