CHAOTIC HYDRODYNAMICS IN HORIZONTAL GAS-LIQUID BUBBLY FLOW

Journal of hydrodynamics, Ser. B, 2(2003),77--82China Ocean Press, Beijing- Printed in ChinaCHAOTIC HYDRODYNAMICS IN HORIZONTAL GAS-LIQUID BUBBLYFLOWSu Yu-liangDepartment of Energy and power Engineering, Xi'an Jiaotong University, Xi'an 710049, ChinaDepartment of petroleum Engineering, Petroleum University, Dongying 257061. ChinaZhang Ming-yuan, Yang Jian, Zhang Chao-jieDepartment of Energy and power Engineering, Xi'an Jiaotong University, Xi'an 710049, ChinaLi Dong-xiaDepartment of petroleum Engineering, Petroleum University, Dongying 257061, China(Received Sept. 25, 2001)ABSTRACT: To investigate the non-linear hydrodynamics s-standard deviation horizontal gas-liquid bubbly flow, time-series data of localH- Hurst exponent, dimensionlessvoid fraction were obtained using a double sensor probe. BothL— pipe length,mdeterministic chaos and stochastic analyses were used to dD— pipe diameter,mnose the dynamic behaviors in the system. The power spectraof the local void fraction exhibited no single distinct peakliquid superficial velocitydicating the aperiodic nature of the variation of the local voidJGfraction. The kolmogorov entropy and correlation dimep- power spectral densityof the attractor reconstructed from the time-series data by thexi--reconstructed state space vector, di-embedding method were to be positive and very large, the mensionlessHurst exponent computed from the rescaled range analyyas found to be larger than 0. 5, indicating the non-linear Greek letterschaotic feature of the system. The Kolmogorov entropy andlocal void fractiothe correlation dimension decrease with increasing liquid su-perficial velocity, and the correlation dimension changes lessdelay time, sif the gas superficial velocity is high. On the other hand, the9--the Heaviside function, dimensionlessfluctuation of local void fraction in horizontal gas-liquid bubtime lag,sbly flow could be considered to take place due to the zigzagmotion of bubbles with different shape, size and velocity.1. INTRODUCTIONKEY WORDS: chaos Kolmogorov entropy, correlation diIn gas-liquidmension, Hurst exponent, local void fraction, gas-liquidic and transport characteristics are closely related tothe bubble flow behavior. And bubbles exhibitNOMENCLATUREnon-linear hydrodynamic behavior. Therefore it isd--embedding dimension, dimensionless important to characterize the flow structure of theC(r)-correlation sum. dimensionlesscomplicated non-linear system using different kindsber of pairs of points on the recon- of methodsstructed attractor, dimensionlessYHahe deterministic chaos analysesr-scaling distance, dimensionlesshavel中国煤化工 many fields of sciDCN MH GsSential tool to deteKolmogorov entropy, bits/smine complex phenomena and structures. ConvinR-range of the cumulative time seriesing evidence for deterministic chaos has come fromP原存数据。 orted by the National Natural science foundation of chin.(tN:59960a time series of recent experiments in multiphaseflow systems. Among those, Fan et al. (1990)an- :=Lx(ti ).(t;+n),",(t;+(d-1)y)(1)alyzed the times series of pressure fluctuation within three-phase fluidized beds in terms of Hurst's where d is an integral dimension of embedding spacerescaled range analysis, and characterized the oper- and y is a delay time that represents the time incre-ating regimes using Hurst exponent and the local ment between the successively sampled points usedfractal dimension. Franca et al. (1991) identified for the attractor reconstruction Hilborn 1994)the flow patterns within horizontal gas-liquid flow The Correlation sum Cd(r)is definedby the same method. Drahos et al.(1992) also apof pressure fluctuation to distinguish flow regimes C(r)=lim 1 Ser,x, I)plied Hurst (1951)exponent to the fractal analysis(2)in bubble columns. Luewisutthichat et al. (1995)employed the fractal analysis of particle trajectoriesin three-phase systems, and analyzed the particle where the Heaviside step function O(x)=ifx<0trajectories obtained from frame-by-frame analysishile e(x)=l if x>0, ris the radius of a hyper-of cine-films using the box-counting method. Lu- SPhere centred on either the point i j,andTewisutthichat et al. (1997)applied kolmogorov en- I is the scalar distance between two points on thetropy and correlation dimension obtained from the Solution attractor.time-series data of bubble-shape indices and velociFor the deterministic chaos system, C(r)ty components to determine chaotic hydrodynamics Deys the power law asof continuous single-bubble flow system. Previousnly focused on the charactC(r)rde(3)istic of pressure fluctuation and particle (or bubble)trajectories using fractal analysis. As an importantThe correlation dimension, D, can be detersource of information reflecting the hydrodynamic mined by fitting a straight line to a log-log plot offeature in gas-liquid two-phase flow the local void(r) against r within the scaling regionfraction fluctuation has not been studied extensive-known that Kolmogorov entropy reply. The objective of present study is, therefore, to sents the rate of information loss of the system dyapply deterministic chaos analysis of the time series namics. For a chaotic system Kolmogorov entropyof local void fraction to determine the feature of possesses a finite and positive value. On the othernon-linear hydrodynamics in horizontal gas-liquid hand, for dissipative ordered and random systemsbubbly flow systemthe value of Kolmogorov entropy appears to be zeroand infiniteting that the system is com-2. MEHTODpletely predictable and unpredictable, respectively.Two-phase flow is a non-linear dissipative dy- The Kolmogorov entropy, K, is also esti-namic system, which may exhibit a class of motion mated based on the extension of the correlation sumthat is chaotic. The most common manifestation of( Grassberger and Procaccia 1983)chaotic systems is the so-called strange attractorse space.If only one variable related to the flow is availx+m-x+m1|2)12(4)able, the state-space trajectories of the motiestrange attractor, can be reconstructed using apseudo-phase-space technique. In a system having d K(TH中国煤化工CNMHGtate variables, the attractor normally studied is asubset of a d-dimensional state spacelimK“(r)~Ka d dimensional pseudo-phase-space vector x, donis reconstucted from the time-series data of the instantanedi方教据 void fraction x(t;) as followsThe rescaled range(R/S) analysis was originally proposed by Hurst (1951) and it gives us a through an A/D converter to digitize and werepossibility to estimate the value of Hurst exponent, stored for later analysis. A high sampling rate ofH, for a given time series. For a fractal process, 10K Hz was used to ensure a sufficient resolution inH, has been found to be >0.5, while for inde- analyzing the highly speed small bubble two-phasependent random processes with finite variance, it is condition. 40000 signal points were obtained at awell correlated by H= 0measurement location and local void fraction of eachccordingR/S is w100 points was calculated in sequence, thereforescribed by the following empirical relation.r 400 points of loid fraction time series were1988)taken at given measurement location. The measureR2=(5(7)ment point was selected at the vertical distance of0. 0025m to the top of the inner pipe-surface, sowherethat enough information of local void fraction wasx(t;)(8) achieved, Detailed construction of the double sen-orIgnal processing are given elsewhere (Sun Kexia 2000)X(t,)=∑[x(t)-(9)4. RESULTS AND DISCUSSIONThe behavior of the local void fraction fluctuaR(r)= maxX (t r)- minX(tt(10) tion in the bubbly flillustrated in Fig. 1. Itobserved that the void fraction change erraticallythx(t,)-](11)The Power Spectral Density ( PSD) of eachIn the equation above r is the number of points of a state is computed, and showed in Fig. 2. No singleet of the time seriesdistinct peak in the power spectrum is observed, in-dicating the aperiodic feature of local void fraction3. EXPERIMENTALat meAn air/water test loop was built in the stateTo further characterize such the chaotic sys-tem, the time-delay embedding method is appliedKey laboratory of Multiphase Flow in Power Engl for reconstructing the pseudo-phase space attractorzontally positioned plexiglass tube with a ID of from time-series data of the local void fraction. The35mm and length of 10000mm was used as develop- correlation dimension D, and the Kolmogorov entroment section, so that a fully development flow was py K of the reconstructed attractor are calculated asachieved in the section. The experiments were con- shown in Table 1ducted at ambient temperature. At the start of theAs seen in Table 1. the value of the Kolmogexperiments, air and water were introduced into olov entropy for all the cases studied is be found tothe horizontal tube at adiusted rates to create bub- be positive, indicating that the temporal fluctuationbly flows. The experimental parameter as follows: of the local void fraction is chaotic. It is wellLiquid superficial velocity, ji=3.5 m/s, j.= known that the magnitude of local void fraction in4.5 m/s respectively;bubbly flows is closely related to the frequencyAir superficial velocity, jG=0.116 m/s, jGshape, size, velocity, and orientation of bubbles4385 m/s respectivelyand therefore can be considered to be one of the inTwo electrical resistivity probes were installed formation sources reflecting the characteristics oforizontally on section at LA circuit the motion of bubbles. The chaotic feature of localwas used to measure the potential difference be- void中国煤化工 tic motion of bubblthe probe tip and the ground. When the ItCNMHGlue of K decrease withprobe tip was in contact with the liquid, the circuit the increasing of liquid superfical velocity, aswas closed. A bubble opened the circuit to generate shown in Table 1a nearly binary signal. The probe analog signalsThis indicates that the increase of liquid velociwerefedriosthy into an IBM-PC-AT computer ty suppresses the irregularly motion of bubbles and0.80.40.200.20.30.40.1(a)j1=3.5m/s;j=0.132m/s(b)j=3.5m/s:ja=0.438m/86420864200.0.0.30.4(d)j=4.5m/sFig. 1 Local void fraction fluctuation in horizontal gas-liquid flow0000054320.20200600(a)j2=3.5m/s:j=0.132a/s(b)j=3.5m/s;j=0.438m/s0.60.05ILLML lulL.dJJym‰z(c)J=4.5m/s;j=0.116m/sd)J1=4.5m/s:J=0.385m/sFig 2 Power spectra of the local void fraction in horizontal gas-liquid flowTable 1 Correlation dimension, Kolmogorov entropy and Hurst exponent for various conditionsK(bits/s)0.1325.45/10.100.6840.4385.93/9.500.6550.10.385中国煤化工4.56/8.91HCNMHGprevents the bubbles from agglomeration and frag- velocity.mentation,consequently leads to a remarkable reThe correlation dimensions for embeddingduction in the degree of chaos at the higher liquid space dimension, d ranging from 2 to 20, for alld=2d=4d=8d=2750.9ogr(a)j=3.5m/s;Ji=0.132m/(b)j=3.5m/s;j=0.438m/sd=2d=4d=12d=16d=16d=20d=200.60.4L(c)j=4.5m/s;J=0.116m/s(d)j4=4.5m/s;j=0.385m/sFig 3 Correlation dimension(D plots for various velocities1.20.80.80.40000.5115(a)j4=3.5m/s;c=0.132m/s(b)j=3.5m/s:j=0.438m/s1.6200.511.522.500.5122.5Logo(r/2)(c)jt=4.5m/s;jc=0.116m/s(d)j4=4.5m/s;jc=0.385m/sFig 4 R/ S as a function of the time lag r for various velocity conditionsthe cases studied are shown in Fig 3, and, a dis- correlation dimension corresponds to low frequencytinct double-sloped curve was found under various fluctuation, reflecting large-scale phenomena. Thisflow conditions, indicating the existence of two reve中国煤化工 ture of the local voidnon-integer correlation dimensions in the system, fractCNMH onStrates the chaoticshown in Table 1. Multiple slopes in correlation feature in horizontal gas-llquld bubbly flow systemdimension plots have also been found by franca etThere is a similar trend betal. (1991), they found two-sloped curve for wavy tion dimension, D and Kolmogolov entropyand annular flows. One correlation dimension cor- der various flow conditions. However. the increasresponds#A-scale fluctuations, and, another ing of liquid velocity has less effect on the correla-tion dimension, D when the gas superficial velocity bedding method. The rescaled range analysis wasis high, as seen in Table 1. Since the irregular mo- also be used to study the nature of the flow. Finitetion of bubbles is concerned with momentum they positive values of the Kolmogorov entropy and two-possessed and obtained, when the gas superficial integer correlation dimensions were found in thevelocity is low, which means each bubble has rela- system. The Hurst exponent computed from thetively less momentum, even small momentum rescaled range analysis was found to be larger thanchange can cause more significant effects on the 0.5, indicating that the fluctuation of the loacalbubble's motion; on the other hand, if the gas su- void fraction occurs in a chaotic manner. The cherficial velocity is high, it requires much more mo- otic feature of local void fraction implies the erraticmentum to reach the same level. Whereas, the lig- motion of bubbles. The Kolmogorov entropy anduid superficial velocity is one of the significant con- the correlation dimension decrease with increasingtributions to the bubble s momentum change. liquid velocity, and, the correlation dimensionThus, when the gas superficial velocity is higher, changes less if the gas velocity is highthe bubbles 'motion is less affected if the liquid superficial velocity changes to the smae scale. It isobvious that correlation dimension. D of local void REFERENCESfraction indicates the minimum number of the variables that describe the systems's state. According- [1] FAN L. T, NEOGI D, YASHIMA M. and NAS-chastic analysis of a three-phasethe increasing of the liquid superficial velocity if thebed: fractal approach[J].A.I.Ch.E.J,1990,36gas superficial velocity is high.The results obtained from rescaled range [2] FRANCA F, ACIKGOZ M, LAHEY R. TJrandCLAUSSE A. The use of fractal techniques for flow(R/S) analysis reinforced the deterministic natureregime identification [J]. Int. 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