Bubble clustering and trapping in large vortices. Part 1: Triggered bubbly jets investigated by phase-averaging

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1 Intenational Jounal of Multiphase Flow 33 (2007) Bubble clusteing and tapping in lage votices. Pat 1: Tiggeed bubbly jets investigated by phase-aveaging Rade Ž. Milenković a, *, Beat Sigg b, Geoge Yadigaoglu b,1 a Laboatoium fü Themohydaulik, Paul Schee Institut, CH-5232 Villigen PSI, Switzeland b Fome Laboatoium fü Kentechnik, ETHZ, CH-8092 Züich, Switzeland Received 31 August 2006; eceived in evised fom 17 Mach 2007 Abstact The chaacteistics of lage coheent stuctues in tubulent, peiodically-excited, as well as natually-developing bubbly jets ae investigated, and inteactions between bubbles and lage-scale votices ae studied quantitatively. As infomation on size and development of the lage-eddy stuctues is vey difficult to obtain in natually-developing flows, peiodical excitation of the jet with fequencies in the neighbouhood of natual instabilities is applied to systematically ceate lage coheent stuctues. When bubbles ente such votices, they can be tapped if cetain conditions ae fulfilled; a tapping citeion is developed. To study tapping phenomena, simultaneous two-phase PIV (paticle image velocimety), DOS (double optical sensos), LIF (lase induced fluoescence) and photogaphic techniques wee applied fo tacking the lage votices and bubble stuctues. In ode to quantify the inteaction between bubbles and the lage votices that ae fomed in the shea laye, chaacteistic phase-aveaged quantities wee detemined by PIV and DOS. Ó 2007 Elsevie Ltd. All ights eseved. Keywods: Bubbly jets; Excited jets; Coheent stuctues; Bubble tapping; PIV; Optical pobes; Phase aveaging; Ensemble aveaging 1. Intoduction A numbe of expeimental, theoetical and computational investigations of usually quasi-fully-developed multiphase flows have aleady been conducted to impove the undestanding of complex multiphase flow phenomena. Even simplified topological sketches of the diffeent two-phase flow pattens immediately give ise to many questions that cannot be easily answeed. The mathematical teatment of such multidimensional gas liquid flows is vey complicated and equies adequate models fo the exchange of mass, momentum and heat * Coesponding autho. Pesent addess: Univesity of Applied Sciences Nothwesten Switzeland, Institute fo Themal and Fluid Engineeing, Klostezelgstasse 2, CH-5210 Windisch, Switzeland. Tel.: ; fax: addesses: vmil@bluewin.ch, ade.milenkovic@fhnw.ch (R.Ž. Milenković). 1 Pesent addess: ETH WEN B-13, Weinbegstasse 94, CH-8006 Züich, Switzeland /$ - see font matte Ó 2007 Elsevie Ltd. All ights eseved. doi: /j.ijmultiphaseflow

2 R.Ž. Milenković et al. / Intenational Jounal of Multiphase Flow 33 (2007) between the phases. In paticula, the modelling of tubulence (shea-induced and bubble-induced tubulence), of tubulence chaacteistics such as fluctuating velocities, of autocoelation functions and integal scales of tubulence, of foces acting on gas flow stuctues, of phase inteactions, of the time-dependent shape and size of gas liquid flow stuctues, of any effects of sufactants and of the inteaction between gas and liquid flow stuctues ae subjects of special inteest. Since one of the objectives of this pape is to analyze coheent stuctues of liquid and bubbles in bubbly jets, especially by making use of tiggeing techniques fo poducing peiodically developing votex stuctues in the jet, a shot suvey of petinent wok on the subject is given. Most eseach has, of couse, been conducted with single-phase jets; we eview these publications fist. Thee ae two types of jet instabilities, the shea-laye mode and the jet-column mode (Zaman and Hussain, 1980; Hussain, 1983). These two distinct modes chaacteize the conditions fo votex development in the nea field of cicula jets. The fist mode is attibuted to the instability of the shea laye foming at the jet exit. This mode may be convective o absolute (Huee and Monkewitz, 1985), whee absolute instabilities can be impotant fo jets with void factions above 30%. Fo the pesent expeiments, only the convective mode is of inteest. The second mode, called jet-column mode, is chaacteized by a Stouhal numbe scaled by the jet diamete D. The pefeed mode leads to the fomation of lage votices with Stouhal numbes St ¼ f D V jet 0:3 (f is the excitation fequency and V jet is the mean supeficial liquid velocity of the jet at the nozzle exit) in the downsteam egion of y/d between 2 and 6; y is the axial coodinate (Hussain and Zaman, 1981; Cow and Champagne, 1971). This mode is called pefeed because the fist maximum of the centeline veticalvelocity fluctuation, which occus at fou jet diametes fom the jet exit, coesponds to this mode. Peiodical tiggeing of this o any othe mode allows one to poduce lage coheent stuctues in the jet. By phase-locking the measuing equipment to the tiggeing mechanism, it is possible to analyze coheent stuctues at vaious phases of development (Hussain and Zaman, 1980; Zaman and Hussain, 1980). Expeiments with buoyant, tubulent, single-phase jets wee pefomed by Papanicolaou and List (1988), and the effect of peiodic focing on mixing in neutal and buoyant jets was analyzed by Mastoakos et al. (1996). It was found that highe levels of focing (tiggeing intensity) wee equied fo tubulent jets than fo lamina ones and that a maximum enhancement of jet mixing with ambient fluid esulted fo St =, which might have been the consequence of paiing induced by tiggeing a hamonic togethe with the basic mode. The impotance of the lage-scale coheent stuctues poducing inteactions between the dispesed and the continuous phase was ecognized long ago and was mainly investigated expeimentally in plana fee shea layes. Pevious analytical wok (Ruetsch and Meibug, 1993) on bubble motion in both isolated votices and in tempoally-evolving shea layes showed that thee is a stable equilibium point at the votex cente when the bubble ise velocity is equal to zeo. In analytical wok (Sene et al., 1994) the effects of the tapping paamete of the jet shea laye and of the Jet Foude numbe on the bubble-tapping condition wee examined. Moe ecently published expeimental (Rightley and Lasheas, 2000) and numeical wok (Duzhinin and Elghobashi, 2001) on bubble dispesion in plana fee shea layes have shown that the pesence of lage-scale stuctues dominates the dispesion of the bubbles and the enegy edistibution within the continuous phase. An expeimental study (Rightley and Lasheas, 2000) of the dispesion of mico bubbles (50 lm) inside a plana shea laye with vey weak focing (and theefoe, coesponding to the natually-developing shea laye) povided infomation on the distibution of the expeimentally detemined inte-phase momentum tansfe integal which includes the cumulative influence of all tems in the equation of bubble motion on to the continuous phase. It was shown that the pesence of bubbles enhances the kinetic enegy of the continuous phase and that its edistibution acoss the mixing laye occus inhomogeneously. The expeiment has not povided significant entapping of the bubbles inside lage coheent stuctues. The Stouhal numbe of the most amplified fequency was 32, which is below the focing fequencies used duing the expeimental investigations pesented hee. DNS simulation of a 3D bubbly mixing laye (Duzhinin and Elghobashi, 2001) showed an incement of bubble concentation close to the votex centes as well as inside the olles geneated by votex paiing in the coe of the mixing laye. In addition, analysis of the instantaneous two-way coupling souce tem in

3 1090 R.Ž. Milenković et al. / Intenational Jounal of Multiphase Flow 33 (2007) the equation of the spanwise voticity showed that bubbles modified the voticity field by ceating neighbouing egions of enhanced o educed voticity. To contibute to the undestanding of complex two-phase phenomena and to povide expeimental infomation, especially egading coheent stuctues, a completely new expeimental installation was built and used to collect extensive expeimental data sets. The main goal of ou investigations was to eveal the chaacteistics of lage coheent stuctues in tubulent, peiodically-excited as well as in natually-developed, bubbly jet flows. In the tiggeed-jet expeiments, the shea laye at the jet exit is excited by a small extenal suging flow with contollable fequency and amplitude. At the same time, bubbles of unifom size ae injected in a contollable jet flow and tacked with a novel method developed fo this pupose. Simultaneous two-phase PIV and photogaphic techniques open anothe pomising chapte in exploing two-phase flow phenomena. Thus the inteaction between coheent stuctues and bubbles, as well as the feedback of bubble agglomeation on the development of these stuctues can be studied. 2. The expeiment The basic idea of this expeiment (Milenkovic, 2005) is to ceate lage, odely stuctues in a bubbly jet with contollable fequency and phase, in ode to enable thei study by statistical means, in paticula by pefoming phase o conditional ensemble aveages, and to povide well-contolled conditions fo investigating the inteactions between the dispesed (bubbles) and the continuous phase (liquid). Peiodical excitation of the jet at fequencies in the neighbouhood of natual instabilities is applied to systematically ceate lage coheent stuctues and to demonstate thei impotance. When bubbles ente such votices, they can be tapped if cetain conditions ae fulfilled. The advantage of this appoach is that the lage, odely votices that appea with contollable fequency in the shea laye and the unifomly-sized bubbles that can be geneated with contolled diamete ceate bubble movements that can be systematically obseved. Such movements ae difficult to analyze in natually-developing two-phase jet flows. A photo that illustates a votex ing in a single-phase jet is pesented in Fig. 1. It was obtained by a LIF (Lase-Induced Fluoescence) technique (Milenkovic, 2005). The pixel gay level in the photogaph coesponds to dye concentation. Based on this photo the following conclusions can be dawn: The measued distance between two consecutive votex ings coesponds to the excitation peiod. Theefoe, it is possible to oughly estimate the votex vetical velocity and the size of the votices at diffeent positions. Fig. 1. Photos of the votex ings obtained by LIF. Pixel values ae inveted in ode to achieve bette contast. Two votex ings can be obseved. The small black dots ae the seeding paticles used fo the PIV measuements.

4 R.Ž. Milenković et al. / Intenational Jounal of Multiphase Flow 33 (2007) Thee is a peak of the pixel gey-level value close to the cente of the votices, which can be clealy obseved fo a second votex ing. The distibution of the gey-level values acoss these votices has an appoximately Gaussian shape (Milenkovic, 2005). The positions of the section of the lage votices futhe downsteam ae no longe symmetic, because of votex ing defomations caused by instabilities in the shea laye. In the case of bubbly flows, an inteesting question aising is whethe bubbles can be tapped in lage votices of the liquid flow field and if bubble clustes would be fomed by the votex stuctues. Thus, an attempt was made to show whethe bubble clusteing o tapping can be demonstated in case of natuallydeveloping jets by DOS and PIV. If these techniques ae applied in a standad way, only the spatial distibution of local statistical quantities can be measued. Since bubble clusteing and tapping in case of natually-developing jets ae stochastic pocesses, it is vey difficult to deduce elevant infomation egading size of the stuctues, azimuthal voticity distibution and bubble/liquid velocities acoss these stuctues fom distibutions of statistical popeties. Theefoe, it was decided to extenally and peiodically excite the bubbly jet. The expeiments have been caied out with a vetical wate jet (called hee single-phase jet) as well as a wate jet containing bubbles of vaious well-contolled sizes and volume factions (called hee bubbly jet); the jet is injected into a wate volume contained in a lage Plexiglas tank to minimize wall effects. Unpetubed flows, with constant inlet flow ate of liquid and gas ae hee efeed to as natually-developing jets. If the jet flow is peiodically excited with contolled fequency and amplitude, it is called tiggeed jet. The excitation is achieved by peiodically modulating the jet shea laye by means of a coaxial wate laye (EF) injected close to the jet exit though a sepaate nozzle (Fig. 2). The gas/liquid injecto and a schematic of the expeiment (Milenkovic, 2005) ae pesented in Fig. 2. To poduce the bubbly jet with unifom, but vaiable, bubble sizes, a special injecto has been developed afte a seies of expeiments caied out with diffeent configuations (Milenkovic and Fehlmann, 2005). The optimal configuation of tubes and needles fo foming bubbles with unifom size in the ange between 1 and 6 mm was pesented in moe detail by Milenkovic and Fehlmann (2005). Bubbles ae fomed by continuously injecting ai (AF) though the needles into the co-cuently flowing intenal liquid flow (ILF). The jet flow is fomed aftewads by adding the second, extenal liquid flow (ELF). The main dimensions of the expeimental setup ae: Diamete of the jet nozzle: D = 90 mm. Diametes of the injecto tubes: d o /d i = 4.0/3.4 mm. Diametes of the excitation ing-nozzle: D o /D i = 98/96 mm. Numbe of injecto tubes and needles: N it = 39. Hoizontal dimensions of the tank: L = mm. Height of the tank: H = 2240 mm. Wate level in the tank: H L = 1200 mm. 3. Expeimental techniques 3.1. PIV (paticle image velocimety) and photogaphic techniques In ode to obtain time-aveaged fields in case of natually-developing jets, as well as phase-aveaged velocity and voticity fields at vaious phases of the tiggeing peiod (i.e., positions of the votices in the flow field) in case of peiodically tiggeed jets, a standad, commecial PIV system was used. It consists of a dual-cavity lase and two CCD cameas that wok in a two-fame, double-exposue mode. A 2 mm lase light sheet illuminated a mid plane of the axisymmetic jet, while the two cameas with the beam splitte wee aimed pependiculaly to the lase light sheet. The fist camea with a geen filte captued eflections fom the bubbles, while the second camea with a ed filte acquied the images with light emitted fom the fluoescent seeding paticles used in the expeiment. These wee 40 lm fluoescent (Rhodamine-B) paticles. Theefoe, the fist

5 1092 R.Ž. Milenković et al. / Intenational Jounal of Multiphase Flow 33 (2007) Fig. 2. Schematic of bubbly jet poduction. The injecto shown at the bottom at a much lage scale has 39 injection needles (only one shown). camea was used fo the measuement of bubble velocities, wheeas the second one enabled measuement of liquid velocities. The dimensions of the fields of view used wee mm and mm in diffeent seies of tests. Fo the smalle field of view, the optical esolution of 1 pixel coesponded to about 140 lm, and the size of the chosen inteogation aea (IA) was pixels o about 9 9 mm. The same setting was used fo obtaining velocity fields of the liquid and of the bubbles. The second field of view was used fo estimating the velocity of the votex and of the bubble ing discussed late. The time between two lase pulses was adjusted to obtain a maximum paticle displacement of less than 16 pixel (i.e., 25% of the IA). The eo in velocity measuements coesponded to about 1% of the maximum value, if the esulting displacement uncetainty is 0.1 pixel (Dantec, 2000). Neithe CFD (Computational Fluid Dynamics) methods no expeimental techniques like PIV poduce esults with infinitesimal esolution in space o time; both imply some inheent fillteing. Fo a thoough discussion of expeimental esults obtained by PIV and thei compaison with CFD esults, it is theefoe necessay to popely define and distinguish diffeent filteed quantities. Filteing means that a time- and space-dependent quantity is multiplied with a weight function and integated ove an inteval in time o space. The PIV signals can be decomposed into aveage and fluctuating components. An inheent chaacteistic of PIV esults is that they ae both filteed in space as well as in time. Space filteing in PIV is applied ove the inteogation volume (aea lase light sheet thickness). In PIV the size of the inteogation volume is oughly constant, which means that a constant size of the inteogation mesh is chosen fo pocessing the

6 images. The successive image coelation technique applied fo acquiing the aveage velocity in the IA povides an inheent space and time filteing, with the chaacteistic lengths: size of IA and time delay between two successive lase flashes. The time steps in CFD and PIV have the same ode of magnitude, although the citeia fo thei selection ae, of couse, diffeent. In the case of two-phase flow a new fundamental poblem aises. The selection of the PIV inteogation aea depends on the bubble diametes and tubulent length scales to be esolved. The PIV method applied hee was used fo esolving the lage eddy stuctues. The IA was lage enough to contain one o moe bubbles, as the PIV mesh size should not be smalle than the chaacteistic length scale of the dispesed phase, i.e., the bubble diamete. Anothe question is whethe the measued bubble velocity is coect if the bubble size is compaable to the IA size. The size of the inteogation aea should be lage enough to contain the stongest eflections of the bubbles in ode to achieve sufficient statistical accuacy fo estimating the coss-coelation function. Fo the same total intensity of lase light eflected fom the bubbles in an IA, the eflections can consist of many small spots fom seveal small bubbles o, at the othe exteme, by one lage patch fom a lage unique bubble. The coss-coelation function poduced by the small bubbles is moe stongly peaked (povided that bubble eflections pattens ae vey simila in the two consecutive exposues) than in the case of one lage bubble. The statistical eo of the velocity calculations might, theefoe, incease with bubble size. Nevetheless, the eflections emitted by a elatively lage bubble that extents ove a consideable pat of the IA exhibit a chaacteistic intensity stuctue which still leads to meaningful esults of the peak of the coss-coelation function and thus the velocity calculation. On the othe hand, the size of the lage stuctues that ae visualized expeimentally by PIV must be consideably lage than the size of the IA as well as that of the bubbles in ode that inteaction effects between lage votices and bubbles can be investigated. In paticula, if the bubble tapping phenomenon is investigated, the size of the lage votices should obviously be consideably lage than the size of the bubbles to be tacked inside them. Consideing now the liquid velocity measuements fo a successful application of the coss-coelation algoithm, the numbe of paticles pe IA should be at least 5 (Dantec, 2000). If an adequate seeding density is ensued, i.e., 5 10 paticles pe IA, the void aeas caused by the pesence of bubbles do not contibute significantly to the measuement eos; the liquid velocities emain coss-coelated aveages ove the IA occupied by the liquid and coectly do not include missing data fom the aeas occupied by the bubbles. Such an adequate seeding density fo PIV measuements was achieved with ou dosing system (Milenkovic, 2005). A statistical analysis of PIV esults (Raffel et al., 1998) equies an ensemble consisting of a lage numbe of vecto maps. Depending on the flow conditions, the basic PIV settings and, especially, the quality of feeding the flow with seeding paticles, the numbe of ejected velocity vectos pe vecto map may vay. Fo the expeiments with natually-developing jets, pefomed at the constant acquisition fequency of 15 Hz, 1024 vecto maps wee acquied. Assuming that 50% of the vectos wee accepted fo a given point (an IA) in the shea laye (i.e., N v = 512), the theoetical statistical eo of the mean velocity is less than 2% fo a standad deviation of the velocity measuements of 30% and about equal to the esolution eo of the PIV measuements. The theoetical statistical eo of the vaiance is less than 10%. A simila eo analysis was pesented by Ullum et al. (1998). In addition to the PIV cameas, a thid digital camea was used to captue pictues of bubbles and bubble stuctues, especially fo the tiggeed expeiments Votex tacking R.Ž. Milenković et al. / Intenational Jounal of Multiphase Flow 33 (2007) In ode to obtain velocity and voticity fields at vaious phases within the tiggeing peiod, showing diffeent positions of the votices in the flow field, a data acquisition scheme has been devised that is synchonized with the excitation of the jet. PIV acquisition is stated by the peiodic extenal signal fom an encode, which coves two excitation peiods. The PIV ecodings wee acquied with highe fequency (in ou cases 12 Hz o 15 Hz, depending on the excitation fequency). Fo instance, at the excitation fequency of 2 Hz, six shots pe excitation peiod can be captued with a PIV acquisition ate of 12 Hz, allowing phase-aveaging at six diffeent times within the excitation peiod to be pefomed. The data acquied this

7 1094 R.Ž. Milenković et al. / Intenational Jounal of Multiphase Flow 33 (2007) way yield infomation on defomations, size modifications and velocity of the votex ing. This acquisition method, called hee votex tacking method, is a vey useful tool fo tacking lage votices in a flow field (Milenkovic, 2005; Milenkovic et al., 2005) Double optical senso (DOS) measuements A standad instument, often used in bubbly flows, is the double optical senso. Void faction (e), numbe of bubbles (N), bubble chod-length distibution and vetical bubble velocity ( ) can be measued by DOS. The double optical senso used in ou tests fo void faction and bubble velocity measuements was hoizontally oiented and had a vetical distance between tips of Dy tips = 5 mm; the tips wee vetically aligned. Ealie tests done with this configuation wee pesented by Kubasch (2001). It was shown that senso oientation has vey little influence on the void faction measuements and no effect on bubble fequency measuement fo the ange of bubble sizes investigated hee. Compaison of bubble velocity measuements by PIV and DOS fo tubulent flow conditions with high-bubble concentation (Milenkovic, 2005), showed that the velocities measued by DOS ae lage than those obtained with PIV. As epoted by Le Coe and Ishii (2002), an oveestimation of the bubble velocity by up to 30% is expected fo egions of the flow field whee the bubble velocity fluctuations cannot be neglected. In geneal, the DOS-measued bubble velocities may be influenced by many factos such as senso esponse, senso geomety, bubble shape and bubble velocity fluctuations (Le Coe and Ishii, 2002). The mean void faction at a local position in the flow field is defined as the atio between the time when ai is pesent at the upsteam senso tip and the total measuing time; the mean void faction is estimated by aveaging the void signal ove a long-time peiod. The vetical component of the bubble velocity,, is defined as the atio between the tip sepaation distance and the measued time of flight. The bubble velocity is obtained by aveaging ove a suitably long-time that must be popely chosen to obtain statistically secued esults. The data acquisition time vaied between 300 and 600 s, depending on bubble concentation, making the vetical bubble velocity measuements longtime, local aveages. In ode to pefom phase aveaging of the expeimental esults fo void faction and bubble velocity in case of tiggeed jets, the data acquisition was tiggeed again by the extenal encode signal. An intege numbe of excitation peiods (usually 6000 peiods containing data) wee ecoded. Each excitation peiod was subdivided into an intege numbe of equal intevals. Fo instance, the excitation peiod of s was divided into 25 intevals of 2 s. Bubble numbe, void faction and bubble velocity data wee aveaged ove the total numbe of peiods fo each of these intevals (ensemble aveage). The selected time esolution should be adequate fo distinguishing tapped fom non-tapped bubbles in phase-aveaged void faction and bubble velocity distibutions. If, fo instance, the velocity of the votex ing displacement is about 7 m/s, the time inteval of 2 s coesponds to a vetical displacement of a tapped bubble of about 5.5 mm. The void faction geneated by tapped bubbles would then be aveaged ove Dy = 5.5 mm, which is compaable to the bubble size. In addition, fo egions with low-bubble concentation, i.e., the jet shea laye, the total numbe of peiods fo phase-aveaging must be sufficient fo statistically meaningful esults. 4. Bubble movement and tapping in lage shea-laye stuctues In ode to study inteactions between bubbles and lage eddies fomed in the shea laye and especially bubble tapping phenomena, it is necessay to conside the equation of motion of a single bubble and to detemine the velocity field of the liquid. Despite the fact that flow conditions fo bubble tapping ae in eality moe complex, the appoach pesented hee can be used to check whethe the conditions fo bubbles to each an equilibium position in lage votices ae fulfilled Simplified condition fo bubble tapping Fo the deivation of a tapping citeion, a simplified cylindical votex with axis nomal to the buoyancy diection moving unifomly in the x y plane is consideed and the foces acting on a single bubble ae fomu-

8 R.Ž. Milenković et al. / Intenational Jounal of Multiphase Flow 33 (2007) lated. Tapping of the bubble is then defined as the existence of an equilibium position of the bubble inside the votex. A moe geneal fom of tapping is that of a bubble moving elative to the votex axis, but emaining inside the votex (Sene et al., 1994; Magnaudet and Eames, 2000). Moe infomation on these citeia can be found in Section 5.1. Futhemoe, in votices tilted with espect to the diection of gavity, a tapped bubble would move along the votex axis. The lage, coheent, tooidal votices suounding the jet tavel vetically with about half the mean jet velocity (V jet )(Cow and Champagne, 1971). Fo simplicity, these tooidal objects ae eplaced by cylindical votices with axis nomal to the x y plane in Fig. 3. It is assumed that thei voticity x z has an appoximately Gaussian shape, i.e., As x z ¼ x o e 2 R 2 v x z ¼ 1 o o ð U uþ ð1þ ð2þ one gets U u ¼ 1 Z 0 x z 0 d 0 ¼ x o R 2 v 2 1 e 2 R 2 v whee = D V /2 is the effective adius of the votex shown in Fig. 3. This Gaussian votex satisfies the timedependent Navie Stokes equations if x o R 2 v ¼ const. The simplified equation of bubble motion used (Sene et al., 1994; Auton et al., 1988) includes only fou vecto foces, buoyancy (B), dag (D), Inetia (I) and lift (L), which detemine bubble movement in the flow field (see Fig. 3). It eads ð3þ C m d dt ¼ g 3 4 C d d jwjwþð1þc mþ D LV L C l ½Wð$ V L ÞŠ Dt ð4þ y B Uϕ L D I z ϕ x D v Fig. 3. Foces acting on a single bubble in a votex.

9 1096 R.Ž. Milenković et al. / Intenational Jounal of Multiphase Flow 33 (2007) with D L Dt ¼ oðþ ot þ V L ðþ ð5þ whee the vectos V L, and W = V L ae the liquid, bubble and elative velocities, espectively. C m, C d and C l ae the added mass, dag and lift coefficients, espectively. The adial components of L, I, B acting on a bubble at an equilibium position ae L ¼ C l U u x z ¼ C l x o R 2 v 2 1 e 2 R 2 v I ¼ ð1þc m Þ U 2 u ¼ ð1þc mþ x2 o R4 v B ¼ g cos u p ¼ g sin u 2 x o e 2 R 2 v 2 1 e 2 R v ð7þ Neglecting the time dependence of the votex shape, the following condition must be fulfilled at an equilibium position of the bubble inside the votex L þ I þ B ¼ 0 ð9þ which esults in sin u ¼ð1þC m Þ x2 o R4 v R 1 e v g 2 C l x2 o R2 v 2 g 2 R 1 e 2 2 v e 2 R 2 v Similaly, the azimuthal components of B and D ae B u ¼ g cos u ð11þ D u ¼ 3 4 C d d U 2 u ¼ 3 4 C d d x2 o 2 R4 v 1 e 2 R v ð12þ Fo equilibium conditions, the dag coefficient C d, which, in geneal, depends on the elative velocity between bubble and liquid, may be eplaced by the dag coefficient C dt that coesponds to the teminal velocity of the bubble V T. Thus with 3 4 C d d ¼ g ð13þ V 2 T whee d is the bubble diamete, one obtains D u ¼ g x2 o 2 R4 v 1 e 2 R 2 V 2 4 T 2 v ð14þ Fo equilibium B u þ D u ¼ 0 ð15þ and theefoe cos u ¼ x2 o 2 R4 v R 1 e V 2 v ð16þ T 2 Reaangement of Eqs. (10) and (16) gives sin u ¼ x2 o R v 4 g f 1 ð17þ cos u ¼ x2 o R2 v f V 2 2 ð18þ R T v ð6þ ð8þ ð10þ

10 with f 1 ¼ R v R 1 e 2 2 v ð1 þ C m Þ R2 v R 1 e v 2 C l e 2 R 2 v ð19þ 2 f 2 ¼ R2 v R 1 e 2 2 v ð20þ 4 2 In ode to be stable, the equilibium points must lie at < 1 whee the function f 1 has its maximum. This can be shown by checking the diection of the net foce acting on a bubble slightly displaced fom the equilibium position. The geneal tapping citeion can be obtained by noting that i.e., sin 2 u þ cos 2 u ¼ 1 F 2 x f 2 1 þ C 4 x f 2 2 ¼ 1 have been into- Hee, the Votex Foude numbe F x ¼ x2 o Rv 4g duced. They ae elated by and the votex tapping paamete C x ¼ x o V T ð21þ ð22þ F x ¼ x2 o R2 v 4 g ¼ C2 x F b b ð23þ whee F b ¼ V 2 T d is the Bubble Foude numbe and b ¼ 2gd 2 the length scale atio. F b can be eplaced by 3 2C dt. Fom Eqs. (10) (23): 0 11=4 C x 1 A ð24þ F 2 b b2 f 2 1 R.Ž. Milenković et al. / Intenational Jounal of Multiphase Flow 33 (2007) þ f 2 2 F b b F x ¼ 1=2 ð25þ F 2 b b2 f1 2 þ f2 2 Eqs. (24) and (25) elate C x and F x to the equilibium adius. The minimum values of C x and F x equied fo tapping ae obtained if the maximum allowable value fo is used. Fo the usual values of C l = C m =, the functions f 1 and f 2 have thei maxima at 1 ¼ 1:027 and Votex Tapping paamete Votex Foude numbe Γ ω min, F ω min Fig. 4. Minimum values of the votex tapping paamete and of the Votex Foude numbe equied fo bubble tapping as functions of F b Æ b fo C l = C m =. F b β

11 1098 R.Ž. Milenković et al. / Intenational Jounal of Multiphase Flow 33 (2007) ¼ 1:121, espectively. Fo simplicity, f 1 ¼ 1 and f 2 ¼ 1 ae used now fo calculating the minima of C x and F x that ae equied fo tapping. The esults ae plotted in Fig. 4 as functions of F b Æ b. These esults ae used fo quantifying the bubble tapping condition inside lage votices at diffeent locations in the flow field. The tapping condition C ¼ DV V T > 2 given by Sene et al. (1994), whee DV is the velocity diffeence between two fluid steams foming a shea laye, is a simplified fom of the moe exact bubble-tapping equiement deived hee, which depends on bubble and votex size. The condition can be expessed in tems of eithe numbe, C x o F x, and the two quantities ae equivalent in this case. In the discussion by Sene et al. (1994) and Magnaudet and Eames (2000), the effect of votices on bubble motion ae govened by a tapping paamete and by a Foude numbe which ae defined somewhat diffeently. Moe explanations can be found in Section 5.1. With the definition of C x Eq. (24), the minimum value of x o,min equied fo tapping a bubble inside a Gaussian votex is x o;min ¼ C x;min V T ð26þ Fig. 5 shows x o,min as a function of the bubble diamete and the votex adius fo wate at 20 C and bubbles in the wobbly egime. Since the voticity of the liquid phase inside the votex stuctues can be diectly estimated fom PIV measuements, the data pesented in Fig. 5 can be used fo given bubble and votex size to estimate quickly whethe the conditions fo bubble tapping ae fulfilled o not. Magnaudet and Eames (2000) noted that C d and C l depend on shea and bubble defomation, so that the tapping condition (Eqs. (24) and (25)) could be affected. In ode to discuss the effect of changing C l, the appoximate ange within which C l may vay has to be estimated. Accoding to Magnaudet and Eames (2000), C l lies between about and 5 inside a votex. Expeimental data obtained in simple sheaed ai wate systems (Tomiyama et al., 2002) show that the lift coefficient of small bubbles (0 < d < 5 mm) takes values anging fom 0 to, wheeas fo big bubbles (d > 6 mm) the lift coefficient anges between 0.15 and. In ode to illustate the effect of diffeent values of C l on the tapping paamete and the Votex Foude numbe, a ange of 5 < C l < is chosen and the esults plotted in Fig. 6. To obtain the esults pesented in Fig. 6, the fact that the adius 1 whee the function f 1 eaches its maximum deceases with diminishing C l has been taken into account. The diagam shows that fo negative (C l = 5) o vanishing (C l = 0) lift foce, bubbles can be moe easily tapped than in the case of C l = +. The effect of shea on C d can be detemined based on esults by Legende and Magnaudet (1998), whee it is stated that the dag coefficient inceases with shea, i.e., C d ðre b ; SÞ ¼C d ðre b ; 0Þð1 þ 0:55 S 2 Þ ð27þ ω o,min [1/s] d [mm] = 5 mm =10 mm =15 mm =20 mm Fig. 5. The minimum value of x o equied fo tapping a single bubble with diamete d inside a Gaussian votex.

12 R.Ž. Milenković et al. / Intenational Jounal of Multiphase Flow 33 (2007) Fig. 6. Effect of the lift coefficient on the votex tapping paamete and on the Votex Foude numbe. whee Re b is the Bubble Reynolds numbe jw jd Re b ¼ ð28þ m W is the elative velocity and S, the non-dimensional shea ate, in case of plane shea, is S ¼ ov L ox d ð29þ jw j whee V L is the vetical velocity of the liquid. At an equilibium position inside a Gaussian votex, and fo cylindical geomety (Spuk, 1987) S ¼ o U u d ð30þ o U u and using Eq. (3) 2 3 S ¼ d 4 2 e 2 R 2 v R 2 5 ð31þ v 1 e 2 R 2 v The absolute value of S inceases almost linealy with and eaches the value of Æ b at ¼ 1. Thus, the maximum value of the coection facto fo C d is 1 þ 1:54 b 2 Fo a bubble with d = 2 mm inside a votex with = 10 mm (i.e., b = 0.1) one obtains ð1 þ 0:55 S 2 Þ < 1:0154 and fo a bubble with d = 6 mm (i.e., b = ) ð1 þ 0:55 S 2 Þ < 1:138 We conclude that the coection facto fo C d does not significantly change the tapping condition, but it facilitates tapping in all cases. The discussion of flow-induced changes of C l and C d shows that bubble defomation and shea tend to pomote tapping, although the votex egion whee equilibium of foces can exist deceases if C l is educed.

13 1100 R.Ž. Milenković et al. / Intenational Jounal of Multiphase Flow 33 (2007) Expeimental esults fo natually developing jets Expeimental esults fo natually developing jets ae pesented to make the advantages of tiggeing the jet evident and to compae data fo natually-developing and tiggeed jets. These esults also show what happens if the tapping conditions ae not fulfilled and theefoe epesent a baseline fo futhe study of tiggeed jets. The data wee obtained fo a tubulent bubbly jet with mean supeficial liquid velocity at the nozzle exit, V jet = 5 m/s, homogeneous void faction of 3.4% and bubble diamete d of about 2 mm. This test condition was initially chosen to ceate an inetia-dominated flow with the highest possible values of the non-dimensional paametes that could be achieved with the existing wate pump and the installed jet nozzle (D = 90 mm). The tapping conditions defined by Sene et al. (1994) wee not fulfilled at a downsteam distance fom the nozzle of 4D whee the lage votices each thei maximum stength (Cow and Champagne, 1971) because the Foude numbe F v ¼ V 2 jet 2gd (whee d is the chaacteistic votex diamete, which equals D/2 at a distance 4D fom the nozzle exit (Cow and Champagne, 1971)), was less than 1 and the tapping paamete C ¼ V jet V T (whee V T is the teminal bubble ise velocity), was only about 1.7. Despite this, the flow egime was investigated to examine whethe bubble tapping could be found close to the nozzle exit whee the votex diamete d is smalle and the local F x and C x would possibly be lage enough. Pofiles of the mean vetical velocity of the bubbles and of the liquid obtained by PIV ae pesented in Fig. 7a. Bubbles have highe velocity than the liquid at all downsteam positions in the shea laye, which appeas to indicate that they ae not tapped inside lage stuctues fomed in the mixing laye. One has to add, howeve, that only a faction of the bubbles that would be detected at adial positions of the jet whee lage votices occu can be tapped inside such votices. The azimuthal voticity pesented in Fig. 7b is poduced by the mean shea. Its maximum in the shea laye at y = 100 mm (y/d = 1.12) fom the nozzle exit is about 25 s 1. The thickness of the shea laye is about 25 mm. If one assumes that the adius of the lage stuctues in the shea laye coesponds to half its thickness, the non-dimensional paametes pesented above can be calculated in ode to check the tapping conditions. The teminal ise velocity fo 2 mm bubbles is about m/s. Based on these values, the votex tapping paamete is 3, while the Votex Foude numbe is The minimum values equied fo tapping bubbles inside a Gaussian votex with a adius of 12.5 mm calculated with the citeia pesented above, ae 2.9 fo the votex tapping paamete and 1.5 fo the Votex Foude numbe. Basically the tapping conditions ae not fulfilled. Howeve, as the andom fomation of lage votices is possible, the peak voticity inside these stuctues can be expected to be highe than the mean value. The esults thus pove that significant and evident bubble tapping could not be found in the chosen case. In ode to fulfil the tapping condition, jet flow conditions with highe values of Foude numbe and tapping paamete must be eached Statistical analysis of DOS esults In ode to test whethe bubble clusteing could be detected by the DOS, a statistical analysis was conducted to show whethe the numbe of bubbles hitting the pobe pe time inteval deviates fom that of a pobabilistically unifom bubble density distibution. A second goal was to find whethe the bubble velocity depended on the bubble density o the coesponding void faction obtained by aveaging ove small time intevals. Since the void faction is given by the time faction duing which bubbles contact the senso, it is obvious that, if the measuement-time intevals ae small, this void faction fluctuates stochastically between 0 and 1 and no longe has the oiginal physical meaning of volumetic void faction. Howeve, the numbe of bubbles hitting the senso pe time inteval can be pecisely counted and statistically analyzed. Coelations of detected void factions with appetaining bubble velocities ae pesented fo a location in the shea laye at x = +50 mm and y = 350 mm (y/d = 3.9), Fig. 8. The statistical analyses of the expeimental data have been pefomed by dividing the total measuing peiod into intevals of 2 s o 0.1 s. If the mean bubble velocity is about m/s, the DOS data, that ae aveaged ove a time inteval of 2 s ae thus effectively aveaged ove a vetical distance of 12 mm wheeas the

14 R.Ž. Milenković et al. / Intenational Jounal of Multiphase Flow 33 (2007) Bubbles Liquid y/d=3.9 y=350 mm y/d=1.12 y=100 mm y/d=3.43 y=310 mm ω z [1/s] y/d=1.44 y=130 mm V L -DOS -PIV V L, y/d=1.12 y=100 mm 5 4 US Channel DS Channel y/d=5 y=50 mm x [mm] x [mm] Fig. 7. (a) Downsteam development of the hoizontal pofiles of the vetical bubble and liquid velocities obtained by PIV. (b) Veticalvelocity pofiles of the bubbles, obtained by DOS and PIV, and of the liquid, obtained by PIV (middle); void faction pofiles obtained by downsteam optical pobe tip (DS) and upsteam optical pobe tip (US) (bottom); and azimuthal voticity pofile (top) at y = 100 mm fom the nozzle exit. time inteval of 0.1 s coesponds to a vetical distance of 60 mm. Thus, assuming that the diamete of the lage votices is about 25 mm, the time inteval of 0.1 s is too lage fo esolving bubble stuctues poduced by these votices, but as mentioned befoe, a smalle time inteval means that thee ae fewe events available fo aveaging and theefoe, void factions obtained by integating the bubble signals ove small time intevals vay within a vey wide ange. Since the total numbe of bubbles in the shea laye is consideably smalle than at the jet cente, the longe time intevals wee chosen thee to incease the pobability of detecting moe than one bubble pe time inteval. The coelation between bubble velocity and void faction is pesented in Fig. 8a fo a point in the shea laye. Although the void faction and bubble velocity data ae coelated in this case, this esult is not

15 1102 R.Ž. Milenković et al. / Intenational Jounal of Multiphase Flow 33 (2007) a b Pobability of bubble numbe pe time intevals Expeimental esults Poisson distibution (λ =) Numbe of bubbles, N Fig. 8. (a) Coelation between velocity and void faction at x = +50 mm, y = 350 mm (inteval of 0.1 s) and (b) pobability of detecting N bubbles pe time inteval of 0.1 s fo the point at x = +50 mm and y = 350 mm. poduced by the mean shea because the measued velocity deceases with e, opposite to the effect that the mean shea would have, because fluid lumps coming fom inside the jet contain moe bubbles and have highe axial velocity than those coming fom outside. The eal eason fo the obseved coelation lies in the fact that the bubble esidence times ae not shot compaed to the integation time inteval of 0.1 s. Slow bubbles with long esidence times give ise to lage e values since the pobability that moe than one bubble is detected in a time inteval is elatively small, independent of the velocity. The pobability of detecting N bubbles pe time inteval and a compaison with a Poisson distibution ae pesented in Fig. 8b. The figue does not show a significant deviation fom the Poisson distibution, although this may be caused by the too-small pobability of detecting moe than one bubble pe time inteval. In ode to find a deviation fom the Poisson distibution, a lage bubble density would have been equied fo the given time inteval. These statistical analyses of DOS measuements thus do not show any clusteing of bubbles in the shea laye of natually-developing jets and the appaent coelation between e and is only a fotuitous esult. It is obviously vey difficult to obtain infomation on the size and the development of the lage eddy stuctues in natually-developing flows, as well as to quantify the inteaction between bubbles and the lage votices that ae fomed in the shea laye. Fo coelating flow chaacteistics such as bubble and liquid velocities, as well as void faction, it would have been necessay to develop new o to impove existing expeimental techniques fo the simultaneous measuement of these quantities. The simultaneous PIV measuements that wee pefomed duing this study epesent a vey fist, basic step as they esolve only the velocity fields in two-phase flow. Pocessing of the images to also obtain infomation on bubble concentation inside the inteogation aea could, howeve, be a tempting task fo futue investigations. If, howeve, coheent stuctues ae peiodically poduced, bubble movements inside these stuctues can be systematically analysed. In ode to investigate the possibility of using the DOS technique to obtain phaseaveaged void faction and bubble velocity data, the tiggeed expeiment pesented in the next section was conducted. 6. Expeimental esults fo tiggeed jets The esults pesented now (St =, f = 2 Hz (Milenkovic, 2005)) wee obtained fo a tubulent bubbly jet with mean supeficial liquid velocity at the nozzle exit of V jet ¼2 m/s, homogeneous void faction of 3.4% and bubble diamete of about 4 mm. The analytical appoach (Section 4.1) shows that lage bubbles can be moe easily tapped inside Gaussian votices than small ones because they equie smalle voticity. This is mainly due to the educed stength of the lift foce (see discussion on the vaiation of the lift coefficient C l

16 R.Ž. Milenković et al. / Intenational Jounal of Multiphase Flow 33 (2007) Excitation peiod V L,t Fig. 9. Modulated excitation velocity at the exit of the tiggeing nozzle fo case f 2 (2 Hz). with bubble diamete in Section 4.1). Theefoe, bubbles with diamete of 4 mm instead of 2 mm, as in the expeiment with a natually-developing jet, wee used now. The tiggeing fequency f, which coesponds to a Jet Stouhal numbe St ¼ Df V jet ¼ 0:6, was 2 Hz. Ealie tests pefomed with tiggeed single-phase jets (Milenkovic, 2005) indicated that focing at highe Stouhal numbe aises the peak voticity and educes the distance between two consecutive votex ings. The Jet Reynolds numbe was Re jet ¼ V jetd m ¼ The vaiation of the tiggeing-velocity fo this test is pesented in Fig. 9. It is the velocity of the excitation flow at the exit of the tiggeing nozzle, obtained fom the measued modulated flow ate Bubble ing Bubble tapping is the condition of bubbles eaching an equilibium position within the votices (as defined in Section 4.1) and tavelling with them at the same velocity. If a sufficient numbe of bubbles each such positions in the tooidal votices, a bubble ing is fomed. The photogaph of Fig. 10 illustates this phenomenon. The geneation of a bubble ing (Fig. 11) that tavels with the same velocity as the votex is a clea indication of bubble tapping inside the votex. The velocities of both votex and bubble ing can be estimated fom the data acquied by the PIV votex tacking method with o without illumination of the flow field with back-light (fo moe details see Milenkovic et al., 2005; Milenkovic, 2005). Visualization tests wee conducted by injecting fluoescent dye in the votices (white clouds in Fig. 11). These pictues show stong defomations of the bubble ing (Fig. 11). Fig. 10. Pefectly aligned bubbles in a bubble ing.

17 1104 R.Ž. Milenković et al. / Intenational Jounal of Multiphase Flow 33 (2007) Fig. 11. Sequence of two photos of the votex ing with bubble tapped close to the votex cente. The pictues, acquied with time inteval of 66.7 ms, show almost the same position of the tapped bubble close to the votex cente, which means that it tavels downsteam at the same vetical velocity as the votex Phase-aveaged velocity fields obtained by PIV A typical esult fo the voticity field obtained by PIV when the cente of the votex ing lies at y mm is shown in Fig. 12. The position of the bubble ing fo this phase is shown in Fig. 11. Phase-aveaged pofiles of the vetical velocity of the bubbles ( ) and of the liquid (V L ), as well as of the azimuthal voticity below and acoss the votex ing (at y = 84, 91, 98 and 105 mm), ae pesented in Fig. 13. The votex cente on the ight side is at about y = 105 mm. These data ae pesented in ode to compae bubble and liquid velocities in and below the votex ing and to check whethe the simplified citeion fo bubble tapping (Section 4.1) is fulfilled o not. The velocity of the bubbles inside the votex ing at x 60 mm is in this case highe than the liquid velocity at the same x-coodinate and about the same as the vetical velocity of the votex cente (about 6 m/s), obtained by the votex tacking method mentioned above. This indicates that the bubbles ae tapped at this location and they tavel with the same velocity as the votex ing. In ode to check whethe the analytical Fig. 12. Phase-aveaged azimuthal voticity of the liquid. The jet centeline is at X = 108 mm.

18 R.Ž. Milenković et al. / Intenational Jounal of Multiphase Flow 33 (2007) a 0.9 y =84 mm y =91 mm y =98 mm y=105 mm b 0.9 y=84 mm y=91 mm y=98 mm y=105 mm Votex velocity 6 m/s V L Votex velocity 6 m/s 0.1 Position of the tapped bubbles 0.1 Votex cente c ω z [1/s] x [mm] y=84 mm y=91 mm y=98 mm y=105 mm x [mm] d ω z [1/s] x [mm] x [mm] 2a ψ 0 y =105 mm Fig. 13. Phase-aveaged velocity pofiles of the bubbles (a), of the liquid (b), and of the azimuthal voticity of the liquid (c). Gaussian fit of the azimuthal voticity of the liquid acoss the votex cente (d). tapping conditions ae fulfilled, the size of the votex and the maximum voticity at the votex cente have to be estimated. Fig. 13d shows a Gaussian fit of the azimuthal voticity of the liquid though the votex cente at y = 105 mm. Measued maximum voticity at the votex cente is about 48 s 1. The votex adius estimated fom this fit is about 15 mm. The teminal bubble ise velocity fo 4 mm bubbles is 4 m/s. Based on these values, the votex tapping paamete is 3 and the Votex Foude numbe 8. The minimum values equied fo tapping of bubbles inside a Gaussian votex with adius of 15 mm based on the citeia pesented above (Fig. 4) ae 3.1 fo the votex tapping paamete and 0.92 fo the Votex Foude numbe. Thus, the tapping conditions ae fulfilled Phase-aveaged void faction and vetical bubble velocity measued by DOS DOS data have been acquied fo the same flow conditions at the elevations y = 100 mm and y = 200 mm fo points in the jet cente, at the jet boundaies, and in the shea laye (Milenkovic, 2005). Only the esults fom the shea laye ae discussed hee. As aleady mentioned above, in ode to obtain a statistically sufficient numbe of events, especially fo points located in the low-bubble concentation egions (in the shea laye), the total numbe of excitation peiods used fo phase-aveaging must be high. This means vey long measuement times, faily high demands on DOS signal stability and high-pecision hadwae. Theefoe, data acquisition was caied out fo 6000 excitation peiods fo points with low-bubble concentation and fo 3000 peiods fo points with highe bubble concentation. Each excitation peiod of s was subdivided into 25 intevals of 2 s. The bubble numbe, void faction and bubble vetical-velocity data wee phase-aveaged fo each of the intevals. Expeimental esults at two diffeent locations, y = 100 mm and y = 200 mm at the bounday of the shea laye (x = 45 mm) ae pesented in Fig. 14. At this adius, shea-laye votices entain bubbles togethe with