A Detached Direct Numerical Simulation of Two-Phase Turbulent Bubbly Channel Flow

Transcription

1 A Detached Drect Numercal Smulaton of Two-Phase Turbulent Bubbly Channel Flow Igor A. Bolotnov, Kenneth E. Jansen, Donald A. Drew, Assad A. Obera, Rchard T. Lahey, Jr., and Mchael Z. Podowsk Center for Multphase Research Rensselaer Polytechnc Insttute th St., Troy, NY 1180, USA bolot@rp.edu Keywords: Two-phase bubbly flows; drect numercal smulaton Abstract Detached Drect Numercal Smulatons (DDNS) of a two-phase turbulent bubbly channel flow at Re η = 180 (based on frcton velocty and channel half-wdth) were performed usng a stablzed fnte element method (FEM), a level set algorthm to track the nterface, and subgrd wall models. Fully developed turbulent sngle-phase DNS results obtaned prevously (Trofmova et al., 009) wth the same stablzed FEM code, were used as the ntal flow feld, and a level-set dstance feld was ntroduced to resolve and track the gas bubbles. Surface tenson and gravty forces were used n the smulaton to physcally represent the behavor of a bubbly two-phase ar/water flow. The DDNS results were averaged to obtan the mean lqud and gas velocty dstrbutons, the local gas volume fractons, and the local turbulent knetc energy and dsspaton rate of the lqud phase. The lqud phase parameters were compared wth the correspondng sngle-phase turbulent channel flow to apprase the bubbles nfluence on the lqud s turbulence feld and to quantfy the mportance of bubble-nduced turbulence. Introducton The understandng of two-phase bubbly flows s mportant for scence and engneerng due to ther wdespread occurrence n natural and man-made systems. Recent advances n the development of massvely parallel computer systems and engneerng software makes drect numercal smulaton (DNS) of two-phase bubbly turbulent flows a vable way to study such complex flows. Tryggvason et al. (006) revewed the current state of the DNS of dspersed bubbly gas/lqud flows. Earler DNS applcatons to multphase flows resulted n the modelng of rsng bubbles n shear-free domans, such as: homogeneous bubbly flows (Bunner & Tryggvason, 003), bubbly flow n a lamnar channel (Nagrath et al., 005), and the decay of sotropc turbulence nteractng wth bubbles (Toutant et al., 008). Other DNS of two-phase turbulent flows have assumed neglgbly small bubble sze compared to the scales of turbulence (e.g., Nerhaus et al., 007). The two-phase bubbly turbulent channel flow smulaton presented n ths paper was performed usng a fnte element based approach coupled wth a level set method. The smulaton ncludes gravty, a realstc water/ar densty rato of 858.3, a surface tenson of N/m, and lqud and gas vscostes of kg/m-s and kg/m-s, respectvely, and a lqud phase turbulence Reynolds number of 180 (based on frcton velocty). The level set method also allowed for the modelng of possble coalescence and break-up of bubbles. In ths study, the results of a Detached Drect Numercal Smulaton (DDNS) are presented, n whch approprate subgrd wall models were used n order to control the computatonal costs. As t wll be dscussed later, the use of these subgrd models elmnated the need for an extensve near-wall nodalzaton whch would otherwse be requred to drectly apply a no-slp boundary condton at the condut walls, and to resolve the lubrcaton-type of force on the bubbles very close to the walls. Ths type of computatonal model allows for a DNS of the flud everywhere except very near the condut walls, and s analogous to the well known sngle-phase Detached Eddy Smulaton (DES) model (Spalart, 1997), n whch a subgrd wall model s used nstead of applyng a no-slp boundary condton at the walls, and a Large Eddy Smulaton (LES) s performed elsewhere n the flow feld. Nomenclature C D drag coeffcent (-) d level-set dstance feld (m) d w dstance to the wall (m) f body forces (N) F w wall force g gravtatonal constant (ms - ) k turbulent knetc energy (m s - ) p pressure (Nm - ) u velocty component (ms -1 ) R bubble radus (m) S j stran-rate tensor (s -1 ) t smulaton tme (s) X k phase ndcator functon (-) Greek letters gas volume fracton (-) δ channel half-wdth (m) 1

2 ε ls half of an nterface thckness (m) ε turbulence dsspaton rate for phase- (m s -3 ) μ dynamc vscosty (kg/m-s) ρ densty (kg m -3 ) ζ surface tenson (kg/s ) stress tensor (Nm - ) φ j level set varable (m) Subscrpts 1 frst phase lqud second phase gas Numercal Method Governng Equatons The spatal and temporal dscretzaton of the Incompressble Naver-Stokes (INS) equatons n PHASTA has been prevously descrbed by Whtng (1999) and Nagrath (004). The so-called strong form of these equatons s gven by: Contnuty: u, j 0 ( 1 ) Momentum: u, t u ju, j p, j, j f ( ) where ρ s densty, u s th component of velocty, p s the pressure, j s the stress tensor, and f represents body forces n the th coordnate drecton. For the ncompressble flow of a Newtonan flud, the stress tensor s related to the stran rate tensor, S j, as: S ( u u ) ( 3 ) j j, j j, Usng the Contnuum Surface Tenson (CST) model of Brackbll et al. (199), the surface tenson force was computed as a local nterfacal force densty, and ncluded n f. Level Set Method The level set method of Sussman (e.g., Sussman et al. (1998), Sussman and Fatem (1999), Sussman et al. (1999)) and Sethan (1999) nvolves modelng the nterface as the zero level set of a smooth functon, φ, where φ represents the sgned dstance from the nterface. Hence, the nterface s defned by φ = 0. The scalar φ, the so-called frst scalar, s convected wth the movng flud accordng to: D u 0 ( 4 ) Dt t where u s the flud s velocty vector. Phase-1, typcally the lqud phase, s ndcated by a postve level set, φ > 0, and phase- by a negatve level set, φ < 0. Evaluatng the jump n physcal propertes usng a step change across the nterface leads to poor computatonal results. Therefore, the propertes near an nterface were defned usng a smoothed Heavsde kernel functon, H ε, gven by (Sussman et al., 1999): 0, ls 1 1 H ( ) 1 sn, ls ls ls, ls 1 ( 5 ) and the flud propertes were defned as: ( ) 1H( ) (1 H( )) ( 6 ) ( ) 1H( ) (1 H( )) ( 7 ) Although the soluton wll be reasonably good n the mmedate vcnty of the nterface, the dstance feld may not be correct throughout the doman snce the varyng flud veloctes throughout the flow feld dstorts the level set contours. Thus the level set was corrected wth a re-dstancng operaton by solvng the followng PDE (Sussman and Fatem, 1999): d S( ) 1 d ( 8 ) where d s a scalar that represents the corrected dstance feld and η s a pseudo tme over whch the PDE s solved to steady-state. Ths may also be expressed as the followng transport equaton: d w d S( ) ( 9 ) The so-called second scalar, d, s orgnally assgned the level set feld, φ, and s convected wth a pseudo-velocty w, where, d w S( ) ( 10 ) d and S(φ) s defned as: 1, ls 1 S( ) sn, ls ( 11 ) ls ls, ls 1 Note that the zeroth level set, or nterface, φ = 0, does not move snce ts convectng velocty w, s zero. Solvng the second scalar to steady-state restores the dstance feld to d 1but does not alter the locaton of the nterface. The frst scalar, φ, s then updated usng the steady soluton of d. Sussman et al. (1999) and Sussman and Fatem (1999) proposed an addtonal constrant to be appled durng the re-dstance step to help ensure the nterface (φ = 0) does not move and thus to conserve mass. They found mposng ths constrant mproves the convergence of the re-dstancng step whle mantanng the orgnal zero level set. The essence of ths constrant, whch was used n our smulatons, s to preserve the orgnal volume and mass of each phase durng the re-dstance step. The mplementaton of the level set method nto the PHASTA code has been done by Nagrath (004) and Rodrguez (009). The presented smulatons were based on the PHASTA verson enhanced by Rodrguez (009) wth some modfcatons dscussed n the text. Dscusson Computatonal doman and grd The goal of the two-phase bubbly flow smulaton was to represent a number of adequately resolved bubbles n a turbulent channel flow. In order to keep the sze of the bubbles relatvely small compared to the sze of the channel the followng doman parameters were used:

3 stream wse (x) length: Lx channel wdth (y): Ly.0 span wse (z) length: L Ths represents a quarter of the doman sze used n prevous (Moser et al., 1999; Trofmova et al., 009) sngle-phase low Reynolds number DNS ( Lx 4, 4 L.0, L ). y z 3 The mesh resoluton used n the smulaton was chosen to accurately represent the bubble nterface by havng at least 18 elements across the dameter of an equvalent sphercal bubble. In addton, for the small bubble smulatons, the bubble dameter was chosen to represent one eghth of the channel wdth. The mesh parameters are summarzed n Table 1. The mesh was sotropc wth 9.8M hexahedral elements. Ths mesh was used to perform the followng smulatons n the turbulent channel flow: a sngle small bubble smulaton a multple (3) small bubbles smulaton a large sngle bubble smulaton Table 1. Hexahedral mesh parameters used for two-phase turbulent channel flow DDNS. Mesh parameters Resoluton n wall unts, x Number of nodes Stream-wse drecton, x z 3 Normal to the wall drecton, y Span-wse drecton, z In all cases, the bubbles were ntroduced nto a fully developed turbulent channel flow soluton by supplyng the sngle-phase soluton wth an analytcal expresson for an ntal level set dstance feld representng the bubble(s). Table summarzes the parameters of the three two-phase turbulent channel flow smulatons presented heren. Table. Bubble sze and volume fractons n the two-phase flow smulatons. Case number 1 3 Bubble dameter, number of hexahedral elements Bubble dameter, wall unts Bubble dameter, length unts Number of bubbles Volume of the bubbles Channel volume Bubble volume fracton 0.031% 1.0%.0% Sngle-phase turbulent channel flow The stablzed FEM-based code PHASTA-IC has been prevously used to perform sngle-phase turbulent channel flow DNS (Trofmova et al., 009). However, due to restrctons mposed on the doman and the grd by the desred mult-bubble two-phase flow smulaton we had to re-compute the sngle-phase turbulence feld for our two-phase problem setup. The smulaton of turbulence n the sngle-phase channel flow was approached dfferently n the present work. Snce, unlke Trofmova et al. (009), we dd not start wth a coarse mesh, an attempt to acheve turbulence usng a Poseulle flow (.e., parabolc mean velocty profle) wth random perturbatons resulted n flow re-lamnarzaton. Ths can be explaned by the fact that the random ntal perturbatons on a fner mesh wll excte small scales n the flow whch are very strongly nfluenced by the molecular vscosty compared to the larger scales. Anyway, n order to ntalze the sngle-phase turbulent channel flow run we ntroduced as an ntal condton a combnaton of the parabolc profle wth cosne waves whch had scales comparable wth the channel wdth. Snce we desgned the mesh wth unform spacng n all drectons to properly represent the bubbles we dd not have the ablty to apply a no-slp boundary condton at the wall snce the frst pont off the wall was at y 1.5 (.e., n the lamnar sublayer). Thus, we have appled a boundary condton at the walls usng an approprate tracton force (Spaldng, 1961). Fgure 1 shows the nstantaneous velocty feld n a sngle-phase turbulent channel used for ntalzng the two-phase flow DDNS. It was verfed that ths subgrd model yelds the expected law of the wall results for sngle-phase flows. Fgure 1. Sngle-phase turbulent channel flow soluton whch was used to ntalze the two-phase flow smulatons. Two-phase bubbly turbulent channel flows For all flows analyzed the followng steps were taken to ntalze the turbulent channel bubbly flow smulatons: a level set dstance feld was ntroduced wth an analytcal expresson representng the nterface of sphercal bubble(s) at prescrbed locaton(s) and sze. the turbulent flow smulaton was started ntally wth dentcal flud propertes (densty, ρ and molecular vscosty, μ) for both fluds the gas phase densty and vscosty are ramped down to the desred values over one flow-through n the perodc computatonal doman Let us next consder these steps n detals. A general form of the ntal dstance feld, 0, for the bubbly flow smulatons s: 3

4 0 mn x x y y z z R 1, Nb ( 1 ) where N b s the number of bubbles n the smulaton, ( x, y, z ) are the coordnates of the centers of the bubbles and R s the bubbles radus. The ntal bubbles dstrbutons for the smulatons are shown n Fgure. All three smulatons deal wth turbulent vertcal flow between parallel plates. The gas/lqud relatve velocty was acheved by applyng the gravty body force downwards whle havng a pressure gradent actng upward to overcome the gravty and frcton forces. nteractons (such as the contact angle between the wall and nterface) we ntally experenced bubbles attachng to the channel walls. Ths type of behavor can be physcal for certan fluds and certan condtons, but t s not expected for turbulent ar/water bubbly flows. Thus, n order to perform realstc smulatons of the ar/water flows presented n ths paper, we ntroduced a subgrd model for the lubrcaton-type of repellant wall force whch acts on the bubble s nterface near the wall. Ths type of subgrd wall force model s well known, and wdely used n computatonal multphase flud dynamcs (CMFD) smulatons (Antal et al., 1991). The followng expresson was used for the near-wall repellant force: a1 a F w lul R n w ( 15 ) dw dw where U l s the mean lqud velocty, R s the bubble s radus, d w s the dstance to the wall at the pont of applcaton, n s the normal to the wall, and a 1 = w and a = 35.0 are model coeffcents. Ths force s only appled on the lqud/gas nterface ( ) and s only sgnfcant very close to the wall ( dw 4.0 ls ). The regon of mportance of ths force s hghlghted n blue on the rght of the doman n Fgure 3. The force s only appled throughout the nterface thckness (bounded by the par of yellow lnes parallel to the black zero level set contour). In ths fgure the wall force s drected to the left, normal to the wall. Whle t would have been possble to resolve ths lubrcaton-type wall force by usng an adaptve fne grd mesh very near the condut walls, ths would have resulted n prohbtve computatonal costs and no mprovement n accuracy. Thus a subgrd wall force model was used. ls (a) (b) (c) Fgure. Intal bubble dstrbuton for the three cases of nterest. In order to ensure a smooth transton between the sngle and two-phase turbulent channel flow smulatons we used a numercal ramp to slowly change the propertes of the gas phase: l, f t t0 ( t) l 1 g, f t0 t t1 ( 13 ) g, f t t1 l, f t t0 ( t) l 1 g, f t0 t t1 ( 14 ) g, f t t1 t t0 where, s the lnear ramp parameter, t 0 s the t1 t0 tme of the start of the ramp, t 1 s the end tme of the ramp and t s the current smulaton tme. A typcal ramp nterval, t1 t0, s about one flow-through tme for the perodc doman. Ths method of two-phase flow ntalzaton was found to be the most effcent and stable way to acheve fully developed bubbly flows. Snce the nterface trackng method we used does not account for the physcal behavor of gas/lqud/wall Fgure 3. Bubble wall nteracton and the applcaton regon of the repellant wall force. Dmensonal nterpretaton of the two-phase smulatons The smulatons presented n ths paper were performed usng a non-dmensonal set of parameters. Whle for sngle-phase smulatons the flow between parallel plates s well characterzed by the Reynolds number, the complexty of the two-phase flow requres a more detaled descrpton of the physcal nterpretaton of the modeled flow. Ths 4

5 secton wll dscuss the sze, surface tenson, densty and other parameters for the cases under consderaton. For each parameter of nterest, β, we wll use the followng notaton: s the non-dmensonal parameter used n the PHASTA smulatons s the characterstc value of the parameter s the physcally sgnfcant dmensonal value of the parameter The sngle-phase smulatons we have performed used the followng non-dmensonal parameters (densty, dynamc vscosty, channel half-wdth, mean lqud velocty): 1 1.0, , 1.0, Um 1.0 ( 16 ) Ths set of parameters results n the followng Reynolds number based on the half-channel wdth and mean lqud velocty: U Re, 734 ( 17 ) If we assume that the lqud s water at atmospherc pressure and room temperature (STP), the dmensonal densty and dynamc vscosty are: kg, kg ( 18 ) m ms In ths case the characterstc densty and vscosty are: kg 1,.379 kg 3 ( 19 ) 1 m 1 ms In order to determne the three basc scalng quanttes (length, mass and tme) we need one more parameter. We chose the followng characterstc length: 3 l m ( 0 ) By performng a dmensonal analyss of the specfed characterstc quanttes we can dmensonalze all the parameters of the problem. Table 3 summarzes the relatons between the varous physcal quanttes. Gven the chosen flud propertes and characterstc length the present channel flow smulatons can be nterpreted as the flow of water at STP between vertcal parallel plates havng a spacng of 7.5 mm. The perodc doman s length was.8 mm and the mean velocty of the flow was m/s. Smulaton results The numercal smulatons were performed for several flow-through tmes of the perodc doman by the lqud phase after the ntal densty and vscosty rampng was completed. Fgures 4 6 demonstrate the bubble(s) evoluton n the channel flow. The fve snapshots shown n each fgure are equally spaced n tme. In each snapshot of the flow the left and rght boundares are the walls, the top and bottom boundares had nflow/outflow perodcty condtons and the velocty magntude colors are shown for one plane normal to the vewer s drecton. The bubble nterfaces presented are n three dmensons and don t necessarly ntersect wth the lqud flow feld shown. Table 3. Characterstc quanttes of nterest. Notaton Name Relaton Value Unts specfed kg densty lqud/gas property m l m t u p g dynamc vscosty length mass tme velocty pressure gradent gravty acceleraton surface tenson specfed lqud/gas property specfed channel half-wdth.379 kg m s m 3 l kg l l l 3 l 3 l s m s kg ms m s kg 1.5 s Fgure 4 shows the results of a smulaton representng an ar bubble n a turbulent flow of water between parallel plates. The equvalent bubble dameter was 0.9 mm whch corresponds to 1/8 of the dstance between the plates. The bubble s flow through the computatonal doman took about s. There s a strong tendency for the bubble to approach the wall and we can clearly observe bubble/wall nteractons as the bubble bounces off the wall and approaches t agan later n the run. The tme of these bubble/wall nteractons s not unform due to the turbulent fluctuaton s nfluence on the bubble s trajectory. In any event, the approxmate frequency of the bubbly/wall nteractons was about 30 Hz. Fgure 5 shows a mult bubble turbulent channel flow (3 bubbles ntally). Each bubble had an equvalent dameter of 0.9 mm and all the flow condtons were the same as n the sngle bubble case descrbed prevously. Snce we have used a unform mesh for all our smulatons, to resolve the bubbles n any doman locaton, the computatonal cost for the 3 bubble smulaton was about the same as for a sngle bubble. As expected (Lahey, 005), sgnfcant wall-peakng of the lateral bubble dstrbuton was observed. The thrd smulaton presented n ths paper deals wth a sngle large bubble n the same doman. The equvalent bubble dameter was 3.65 mm, whch represents the channel s half-wdth. However, once the flow becomes developed, the bubble develops a cap-lke shape and stretches tself across about 75% of the dstance between the parallel plates. Fgure 6 shows typcal large bubble behavor durng one flow-through of the smulaton. 5

6 Fgure 4. Sngle bubble turbulent channel flow evoluton. Computatonal cost The two-phase DDNS of turbulent channel flows were carred out for about 7 flow-through cycles n the perodc doman for each case. In ths way we could obtan statstcally statonary averaged results n each case. One flow-through cycle took approxmately 3000 tme steps. PHASTA ran n parallel on 048 processors on the BlueGene/L super computer at Computatonal Center for Nanotechnology Innovaton (CCNI) at Rensselaer Polytechnc Insttute. Two 1 hour runs were requred to go through 3000 tme steps and complete one flow-through Fgure 5. Multple bubbles turbulent channel flow smulaton. cycle n each case. Thus, approxmately 50,000 CPU-hours on a BlueGene/L machne were requred for each flow-through cycle. Therefore, to produce the fnal results we used about 350,000 CPU-hours for each of the three presented cases. To perform data averagng (dscussed below) we performed 5 smulatons (wth slghtly dfferent ntal bubble dstrbutons) of the 3 bubble case to mprove the statstcal results. Thus, the overall computatonal cost of the results presented n ths paper was about.45 mllon CPU-hours on a massvely parallel Blue Gene/L super computer. 6

7 Two-phase turbulence statstcs For each case presented heren tme and space averagng was performed to compute basc two-phase turbulent flow parameters such as the mean velocty for each phase, the gas volume fracton dstrbuton, and the turbulent knetc energy and turbulence dsspaton rate dstrbutons n the lqud phase. We perform sldng-wndow tme averagng to make sure a statstcally statonary soluton was obtaned after the ntal transent was washed out. As noted prevously, to obtan better statstcs we also employed an ensemble-averagng technque by performng several DNS runs of the same problem wth slghtly dfferent ntal condtons, and then averagng the results. The analyss resulted n the followng tme-dependent functons of the descrbed quanttes: Ne N U w k 1 1 () t X k u m t t j k N ( 1 ) e m1 N w j1 Ne N 3 k w k ( t ) X k u m( t t j ) k N ( ) e m1 N w j1 1 k () t Ne N 3 3 w 1 1 m Xk k N e m 1 N w j 1 1 k 1 xk Fgure 6. Large bubble turbulent channel flow smulaton. u ( t t j) ( 3 ) Ne N w 1 1 k ( t ) X k ( t t j ) N e m1 N ( 4 ) w j1 where X k s the phase ndcator functon for feld-k, u ( t t ) u ( t t ) U ( t ) s the fluctuaton of velocty m j m j component- computed durng ensemble run-m at tme nstant t + t j ; N e s the number of ensemble runs, N w s the number of velocty samples n each wndow, t s the current tme, t j N t s the local wndow j w tme, and t s the tme step. An example of mean phase veloctes and ther fluctuatons, and the gas phase ndcator functon (X ) s shown n Fgure 7. We note that for the global vod fractons smulated that the velocty statstcs are nherently much better for the lqud phase. u, m/s u 1 X tme, s Fgure 7. Example of DNS data produced by PHASTA for two-phase flow at a fxed pont whch was used for obtanng averaged data (u 1 s the axal velocty and X s the gas phase ndcator functon; the averaged values, and U 1, are shown n blue and orange, respectvely). Mult-bubble case Let us next consder the averaged statstcs for the mult-bubble flow case. We wll not present the sngle-bubble averaged result heren due to the very small volume fracton n the flow and thus, the very small U 1 1 7

8 statstcal sample n the gas phase from the sngle bubble smulatons. Fgure 8 shows the lateral dstrbuton of the lqud and gas mean veloctes, gas volume fracton and relatve velocty of the phases. Note that the equvalent bubble dameter s the same as the dstance between vertcal grdlnes shown n the plot. In order to gather more statstcal data for ths case we have performed 5 smulatons wth dfferent ntal perturbatons to obtan statstcal ndependence of the smulatons. To obtan ths average we used 4,000 tme steps averaged across 5 ensemble runs whch resulted n statstcs for 0,000 tme steps. Ths number of tme steps corresponds to approxmately one flow through of the lqud phase n each smulaton of the ensemble. U, m/s y, m α Mean velocty lqud Mean velocty gas Relatve velocty Vod fracton Fgure 8. Mean flow parameters for mult-bubble smulaton. We can see that the calculated volume fracton dstrbuton s consstent wth the expermental and theoretcal observatons for small bubble upflows n vertcal conduts (Lahey, 005). In partcular, the mean velocty gradent nduces a lft force on the bubbles whch moves them close to the walls (.e., wall peakng). Ths trend can also be observed n the smulaton snapshots (Fgure 5). Whle the lqud mean velocty shows a very smooth profle the gas velocty profle s less smooth. We can observe somewhat sngular behavor near the center lne, where the local gas volume fracton (.e., the local vod fracton, α) falls to just 0.07 %. Ths low value results only n capturng the statstcs for a few bubbles. In contrast, the near-wall gas velocty profle s qute unform, smlar to the lqud profle, snce each bubble occupes a dstance of ts dameter near the wall and those bubbles slde along the wall surface wth a sngle velocty whch s captured n ths statstc. The computed relatve velocty value s observed to be n the range of m/s, excludng the near-wall regon and the unresolved centerlne regon. We should note that a balance between a typcal drag force and the buoyancy force for a sphere n a quescent lqud results n 0.6 m/s as an estmate of relatve velocty. The dfference may be attrbuted to the fact the gas bubbles are generally not sphercal n the smulaton and may be travelng n regons wth sgnfcant mean lqud velocty gradents. Legendre & Magnaudet (1998) have observed that the drag coeffcent for a bubble n a lnear shear flow s hgher than n a non-shear case, whch would result n smaller relatve velocty. Large bubble case Fgure 9 shows the averagng results for the large bubble channel flow smulaton. We observe that the large bubbles are not nfluenced by the mean lqud velocty gradent n the same way as the small ones are. For example, the peak of the gas volume fracton s at the center lne n ths case. Also we observe a hgher relatve velocty of 0.7 m/s (when averaged across the channel). Joseph (003) proposed a drag coeffcent for a sphercal-cap bubble n the followng form: 3.0 CD Re ( 5 ) eq dequr where Reeq s the Reynolds number of the bubble based on relatve velocty and volume equvalent dameter (correspondng to the dameter of a sphercal bubble wth the same volume as the cap bubble under consderaton). In our case Reeq 60.6 (based on the observed relatve velocty) whch results n CD.69. Note that the Reynolds number used n the correlaton has a neglgble nfluence (less than 1%) on the overall result n the observed range of relatve veloctes. The use of ths analytcal drag coeffcent to estmate the theoretcal relatve velocty by balancng drag and buoyancy forces results n a relatve velocty value of 0.06 m/s whch s close to the computed value of 0.7 m/s. U, m/s Mean velocty lqud Relatve velocty y, m Mean velocty gas Vod fracton α 0.04 Fgure 9. Mean flow parameters for the large bubble turbulent channel flow case. Lqud phase analyss In ths secton we wll look at the changes n the lqud phase whch occur due to the presence of bubbles. We compare the mean flow parameters (.e., the mean velocty, turbulent knetc energy and turbulence dsspaton rate) obtaned from the mult-bubble and large bubble smulatons wth ther sngle-phase flow counterparts. Fgure 10 shows the mean lqud velocty profles for the

9 three cases of nterests. We can observe that the presence of the postve buoyant bubbles accelerate the lqud through the channel. Indeed, the change of the lqud velocty averaged across the channel s lnear wth respect to the gas volume fracton n the flow for the consdered cases (Table 4). U, m/s Sngle-phase flow Large bubble case y, m Mult bubble case Fgure 10. Mean lqud velocty profles for sngle-phase and two-phase flows. The presence of bubbles n the turbulent lqud flow ntroduces addtonal velocty fluctuatons nto the lqud turbulence. To quantfy the amount of ths gas-nduced addtonal energy we have computed the turbulent knetc energy of the lqud n the sngle-phase and two-phase channel flow cases. Table 4. Average lqud mean velocty summary Case Gas volume fracton Average lqud velocty, m/s Sngle-phase 0% Mult-bubble flow Large bubble flow 1% % Fgure 11 shows the lateral dstrbuton of the lqud s turbulent knetc energy for the three cases studed.. It s nterestng to note that mult-bubble case ncreases the lqud s turbulent knetc energy everywhere n the channel except at the centerlne locaton, where the observed volume fracton s close to zero. The amount of energy ncrease s notceable, but not really sgnfcant for the mult-bubble case. In contrast, for the large bubble case the lqud s turbulent knetc energy changes sgnfcantly. Ths s attrbuted to the shape of the large bubble and to ts sze. In fact, the whole pattern of turbulent fluctuatons n the lqud flow changes due to the presence of a bubble of ths sze snce the wake behnd t occupes a sgnfcant porton of the doman. To compare the lqud fluctuaton patterns, refer to Fgure 1 showng the sngle-phase nstantaneous velocty dstrbuton, Fgure 5 showng mult-bubble velocty feld and Fgure 6 showng the large bubble velocty feld. Whle the velocty feld structure s somewhat smlar n Fgure and Fgure 5, one can notce a large dfference between the velocty felds n Fgure and Fgure 6. The large changes n velocty magntude along the channel shown n Fgure 6 result n a sgnfcant turbulent knetc energy dfference (Fgure 11). It s also worth mentonng that the lateral shape of the turbulent knetc energy profles s smlar n all cases under consderaton. k 1, m /s y, m Sngle-phase flow Mult bubble case Large bubble case Fgure 11. Lqud turbulent knetc energy profles for sngle-phase and two-phase flows. ε 1, m /s y, m Sngle-phase flow Mult bubble case Large bubble case Fgure 1. Lqud turbulent dsspaton rate profles for sngle-phase and two-phase flows. Fgure 1 shows the turbulence dsspaton rate dstrbutons. We can observe that values ncrease across the channel consstent wth the ncrease of gas volume (.e., vod) fracton. The large bubble case exhbts a sgnfcant value of lqud turbulence dsspaton rate (ε) at the center lne compared wth the sngle and mult-bubble flow cases. Ths can be explaned by the presence of hgh mean shear rates n the wake of the large bubble at the center lne whle for the small bubbles at ths same locaton, the mean shear rate s very small. Conclusons The DDNS results presented heren demonstrate that the complex nteractons between the gas bubbles and the 9

10 turbulent lqud flow can be resolved and analyzed. The detaled two-phase DDNS data provded by ths study can also be used to spectrally analyze the behavor of turbulent two-phase bubble flows, as was done prevously for turbulent sngle-phase flow (Bolotnov et al., 010). These results are essental to provde new spectral data for the development and assessment of next-generaton spectral multphase turbulence models (Bolotnov et al., 009) for use n computatonal multphase flud dynamc (CMFD) codes. Future smulatons n conduts should nclude hgher vod fracton bubbly flows as well as dfferent flow regmes (e.g., slug, churn-turbulent, or annular flows). Hgh vod fracton bubbly flows are of partcular nterests for turbulence model development applcatons snce the expermental technques avalable today do not allow takng relable multphase flow data for flow wth volume fractons hgher than ~5% due to bubble nterference. Acknowledgements The authors would lke to acknowledge the support gven ths study by the U.S. Department of Energy (DOE) and the computatonal resources provded by the Computatonal Center for Nanotechnology Innovaton (CCNI) at Rensselaer Polytechnc Insttute (RPI). References Antal, S., Lahey, R.T., Jr. and Flaherty, J.E., Analyss of phase dstrbuton n fully developed lamnar bubbly two-phase flow, Int. J. Multphase Flow, 17(5), Bolotnov, I.A., Lahey, R.T., Jr., Drew, D.A., Jansen, K.E. and Obera, A.A., 009. Spectral Cascade Modelng of Turbulent Flow n a Channel, Japanese Journal of Multphase Flow, 3, No., Bolotnov, I.A., Lahey, R.T., Jr., Drew, D.A., Jansen, K.E. and Obera, A.A., 010. Spectral Analyss of Turbulence Based on the DNS of a Channel Flow, Computers & Fluds, 39, Brackbll, J. U., D. B. Kothe and C. Zemach, 199. "A contnuum method for modelng surface tenson.", Journal of Computatonal Physcs, 100 (), Bunner, B. & Tryggvason, G., 003. Effect of bubble deformaton on the propertes of bubbly flows, J. Flud Mech., 495, Joseph, D.D., 003. Rse velocty of a sphercal cap bubble, J. Flud Mech., 488, Lahey, R.T., Jr., 005. The Smulaton of Multdmensonal Multphase Flows, Nuc. Eng. & Desgn, 35, Legendre, D. & Magnaudet, J., The lft force on a sphercal bubble n a vscous lnear shear flow, J. Flud Mech., 368, Moser, R., Km, J. and Mansour, N., Drect numercal smulaton of turbulent channel flow up to Re η = 590, Phys. Fluds, 11, Nagrath, S., 004, "Adaptve stablzed fnte element analyss of mult-phase flows usng level set approach", Ph.D. thess, MANE, Rensselaer Polytechnc Insttute. Nagrath, S., Jansen, K.E. and Lahey, Jr., R.T., 005. Computaton of ncompressble bubble dynamcs wth a stablzed fnte element level set method, Computer Methods n Appled Mechancs and Engneerng, 194, 4-44, Nerhaus, T., Abeele, D.V. and Deconnck, H., 007. Drect numercal smulaton of bubbly flow n the turbulent boundary layer of a horzontal parallel plate electrochemcal reactor, Int. J. of Heat and Flud Flow, 8, Rodrguez, J.M., 009. "Numercal smulaton of two-phase annular flow", Ph.D. thess, MANE, Rensselaer Polytechnc Insttute, Troy, NY. Sethan, J. A., "Level Set Methods and Fast Marchng Methods", Cambrdge Unversty Press. Spalart, P.R., Comments on the feasblty of LES for wng and on a hybrd RANS/LES approach, 1st ASOSR Conference on DNS/LES, Arlngton, TX. Spaldng, D.B., 1961, A sngle formula for the law of the wall, ASME J. Appl. Math. 8 (3), Sussman, M., E. Fatem, P. Smereka and S. Osher., "An mproved level set method for ncompressble two-phase flows", Journal of Computers & Fluds, 7(5-6), Sussman, M. and E. Fatem., "An effcent, nterface-preservng level set re-dstancng algorthm and ts applcaton to nterfacal ncompressble flud flow", Sam Journal on Scentfc Computng, 0 (4), Sussman, M., A. S. Almgren, J. B. Bell, P. Colella, L. H. Howell and M. L. Welcome., "An adaptve level set approach for ncompressble two-phase flows", Journal of Computatonal Physcs, 148 (1), Toutant, A., Labourasse, E., Lebague, O., Smonn, O., 008. DNS of the nteracton between a deformable buoyant bubble and a spatally decayng turbulence: A pror tests for LES two-phase flow modellng, Computers & Fluds, 37, Trofmova, A., Tejada-Martnez, A.E., Jansen, K.E. and Lahey, R.T., Jr., 009. "Drect Numercal Smulaton of Turbulent Channel Flows usng a Stablzed Fnte Element Method", Computers and Fluds, 38 (4), Tryggvason, G., Esmaeel, A., Lu, J., Bswas, S., 006. Drect numercal smulatons of gas/lqud multphase flows, Flud Dynamcs Research, 38, Whtng, C. H., 1999, "Stablzed fnte element methods for flud dynamcs usng a herarchcal bass", Ph.D. thess, MEAE., Troy, NY, Rensselaer Polytechnc Insttute. 10