Direct Numerical Simulation of Turbulent Channel Flow with Deformed Bubbles

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1 Progress n NUCLEAR SCIENCE and TECHNOLOGY, Vol., () ARTICLE Drect Numercal Smulaton of Turulent Channel wth Deformed Bules Yoshnou YAMAMOTO * and Tomoak KUNUGI Kyoto Unversty, Yoshda, Sakyo, Kyoto, 66-85, Jaan In ths study, the drect numercal smulaton of a fully-develoed turulent channel flow wth deformed ules were conducted y means of the refned MARS method, turulent Reynolds numer 5, and Bule Reynolds numer. As the results, large-scale wake motons were oserved round the ules. At the ule located regon, mean velocty was degreased and turulent ntenstes and Reynolds shear stress were ncreased y the effects of the large-scale wake motons round ules. On the other hands, near wall regon, ules mght effect on the flow lamnarlze and drag reducton. Two tyes of drag coeffcent of ule were estmated from the accelerated velocty of ule and correlaton equaton as a functon of Partcle Reynolds numer. Emrcal correlaton equaton mght e overestmated the drag effects n ths Partcle Reynolds numer range. KEYWORDS: DNS, uly flow, turulent shear flow, drag frcton, gas entranment I. Introducton Turulent flows wth ules are very often found n engneerng devces such as a lght-water nuclear reactor and a chemcal reactor. Therefore, oth exermental and numercal nvestgatons have een conducted, extensvely. In the vew onts of understandng coherent structures and turulent statstcs ehavors, the DNS (Drect Numercal Smulaton) s exected to have advantages over the exermental aroaches. Recently, DNS of the turulent shear flows wth deformalty ules have een conducted y Kawamura and Kodama, ) Lu et al., ) and Lu and Tryggvason, 3) y means of the front-trackng method. These revous DNSs focused on nteracton etween turulent structures near wall and ules to nvestgate frcton drag reducton effects on the wall. In the vew onts of the comutatonal model 4) for the gas and ule ehavor n the rmary coolant system of a sodum-cooled Fast Reactor, ugradng of not only the wall drag coeffcent ut also the drag coeffcent acted on the deformale ules s ndsensale. In ths study, we aled the MARS method 5) as the drect numercal nvestgaton scheme for the drag coeffcent acted on the deformale ules n the turulent shear flows and the DNS dataase of a fully-develoed turulent channel flow wth ules were estalshed. II. Numercal Method. MARS Method Numercal rocedure was ased on the MARS method; 5) the governng equatons are conssted of Naver-Stokes equatons, contnuty equaton and transort equaton of a volume fracton functon (VOF) F. Regardng the dscretza- *Corresondng author, E-mal: yyama@nucleng.kyoto-u.ac.j c Atomc Energy Socety of Jaan, All Rghts Reserved. 543 ton of the velocty felds on the Cartesan coordnate system, the second-order scheme for the satal dfferencng terms s used on the staggered grd system. The hyscal rolem treated here s the moton of two Newtonan ncomressle fluds allowed the nterface deformaton etween them. To adat the MARS method for the DNS of turulent uly flows, two mrovements of the orgnal MARS method were conducted. One was the refnement of the calculaton of a surface tenson force and another was the ugradng the tme ntegraton schemes. In the orgnal MARS method, the surface tenson forces (:F v, the -th comonents of surface tenson force, =,,3) accordng to the CSF model, 6) are defned y the followng equatons: F F v x, n x, F, / m m g l, where ρ, σ, κ, and n denote densty, coeffcent of surface tenson, nterface curvature, and -th comonents of the nterface normal vector, resectvely. In ths study, to refne the calculaton of the surface tenson acted on the ules, a knd of the level-set functon (:) 7) estmated y usng the VOF functon, was used for the calculaton of the surface tenson forces, as followng, 8) () Lqud F, ( x, y, z) Interface () Gas Fv,. x x A second mrovement was adatng the hgh-accuracy tme ntegraton schemes for evaluatng the acceleraton of u- (3)

2 544 Yoshnou YAMAMOTO et al. 3 U u rms, vrms, wrms U, u rms, v rms, w rms Symol: Present, Lne: Kuroda et al. Loss Gan Producton Turulent dff. Vscous dff. Pressure dff. Dssaton Symol: Present, Lne: Kuroda et al. 5 5 Fg. Rsng ules smulaton, flow geometry, numercal condton, and tme seres of ule ehavors Fg. Verfcatons of the modfed MARS method comared wth results y Kuroda et al., mean velocty and turulent ntenstes, and udget of turulent knetc energy. les n the turulent shear flows. In the orgnal MARS method, the st-order mlct scheme was used ut low-storage 3rd-order Runge-Kutta scheme for convecton terms, Crank-Ncolson scheme for vscous terms and Euler Imlct scheme for Pressure terms were mlemented. As the results, we can ensure the over second-order accurate tme dscretzaton.. Sngle-Phase Turulent Channel Smulaton The accuracy of the modfed MARS method was examned for the sngle-hase turulent channel flow at a turulent Reynolds numer 5 ased on frcton velocty channel half heght and knetc vscosty. Comutaton was carred out usng 7x8x7 grds for the streamwse, vertcal, and sanwse drectons resectvely. Grd resoluton normalzed y the frcton velocty and the knetc vscosty was 6.7(x ),.5-.( ), and 8.3(z ) for streamwse, vertcal, and sanwse drecton, resectvely. Comutatonal tme ste was adjusted corresond to the maxmum CFL was.. Fgure shows the mean velocty and turulent ntensty rofles; the udget of the turulent knetc energy was also shown n Fg.. Deste the large-tme stes n the resent case, not only low order statstcs (: mean velocty and turulent ntenstes) ut also hgh order statstcs (: energy dssaton and turulent dffuson) agreed well wth the DNS results of Kuroda et al. 9) 3. Rsng Bules Smulaton To check the ugradng of the tme ntegraton and surface tenson forces, two rsng ule smulaton was also carred out. Comutatonal geometry, examle of the tme seres ehavors of ules and numercal condton, were Fg. 3 Comarson of the ule shaes at T =. (efore coalescng) n case of the st order tme ntegraton schemes and the hgh-order schemes wth varous tme stes summarzed n Fg.. In ths smulaton, ules were rsng drven y uoyancy and the undersde ule caught the uer ule and they coalesced nto the sngle ule as shown n Fg.. Fgure 3 shows the comarson of the ule shaes efore coalescng tme T =.. The uer fgures were calculated y usng the st order tme ntegraton schemes and the lower fgures were calculated y the hgh-order schemes, wth varous tme ste ntervals. Usng the resent new calculaton for surface tenson forces, unhyscal ehavors caused form the lacks of the surface tenson accuracy cannot e oserved n oth schemes. In cases of the st-order tme ntegraton scheme (t = -5 ) and the hgh-order scheme (t = -4, -4, and 4-4 ), rsng ule veloctes were corresondng to others wthn an uncertanty of aout 6% and ule shaes gves close agreement wth each other; These ndcate that the hgh-order scheme has more than tenfold advantage n the tme ste nterval comared wth the st-order scheme. III. Numercal Condton of a Turulent Channel wth Bules Fgure 4 shows flow geometry and coordnate systems. Numercal condton was taled n Tale. In ths study, PROGRESS IN NUCLEAR SCIENCE AND TECHNOLOGY

3 Drect Numercal Smulaton of Turulent Channel wth Deformed Bules 545 Fg. 4 B3 B B4 small comutatonal doman as same as the mnmum flow unts ) was emloyed. Thermal roertes at natural ar-water system were adated. As the ntal velocty felds, a fully develoed sngle-hase turulent channel flow data was used and 4 ules whch dameter was mm were entraned n channel center regon. Thus, vod fracton was only.5%. To otan the fully-develoed status, gravty effect was gnored and the streamwse constant ressure gradent was forced on the water hase. Non-sl at the walls and erodc condtons for the stream and sanwse drectons were mosed for the oundary condtons. From the ntal condton, tme ntegraton was conducted durng 4 non-dmensonal tmes (:T = T u / w T : tme, u frcton velocty at the wall, w : knetc vscosty of the water hase), to otaned the fully-develoed status. Then, t was confrmed that flow felds reached the fully-develoed status After that, T =, non-dmensonal tme ntegraton was conducted to otan the mean and statstcs data. B geometry and coordnate system Tale Numercal condton CASE CASE CASE Re 5(=h ) r 3(=.h) ρ w /ρ g 84. Doman, 4h, h, 4/3h L x, L y,l z Grd numer, 5, 34, 5 3, 8, N x,n y,n z Resoluton, 4.,.5-4., 4..,.5-.,. Δx,Δ,Δz T 4 4 Re τ = u τ h/ν w : Turulent Reynolds numer, u τ : Frcton velocty at wall, h: water deth, ν w : Knetc vscosty of water, r: Intal radus of ules, ρ w : Densty of water, ρ g :Densty of ar, L x, L y, L z : Comutatonal doman, N x, N y, N z : Grd numer, Δx,Δ,Δz : Grd resoluton for stream (x)-, vertcal (y)-, and san (z)- wse drectons, resectvely. Suer-scrt denotes the nondmensonal quanttes normalzed y frcton velocty and knetc vscosty used Reynolds numer defnton. T : Tme ntegraton length from ntal condton to fully-develoed status. IV. Numercal Results. Effect of Grd Resoluton Fgures 5 and show the nstantaneous turulent velocty vector lots round ules n CASE (Fg. 5) and CASE (Fg. 5). Here, turulent velocty denotes the velocty dfference from ts mean value. Large-scale wake motons round the ules were oserved. These wake motons round ules were no much dfference etween CASE and CASE. Fgure 5 shows the turulent ntensty rofles n CASE and. Note that these turulent ntenstes were averaged wthout the dstncton etween ar and water hases. Near channel center regon ( > 5), these dfferences of turulent ntenstes were ndfference. On the other hands, there were the slghtly dfferences near wall regon ( < 3). It was guessed that these dfferences near wall regon were not deended on the grd resoluton ut the ule detenton tme n ths regon. Therefore, to dscuss the grd resoluton effects of uly flows on near wall turulent structures, many numers of ules or very long tme ntegraton mght e requred to gnore the effects of the ule detenton tme. However, at least, we can confrm that the grd resoluton effect on turulent structures near channel center was small. Hereafter, to dscuss the drag coeffcent acted on the deformale ules n the channel center regon and the results of CASE were used as the DNS dataase.. Bule Behavor and Dstruton Fgure 6 shows nstantaneous ule ehavors wth streamwse turulent velocty contour lots n tme erods T =3-5. After flow have reached the fully-develoed status, large- wake moton round ules were constantly oserved and ule deformaton scale was aout 7% of the ntal ule dameter. In ths study, ules were laeled as B, B, B3, and B4 to dstngush from others as shown n Fg. 4. Fgure 7 shows the dstruton of the mean volume-rate functon of ule B and B3. Vertcal motons of ule B and B3 were confned n the channel center regon and fluctuatons of vertcal ule ostons were wthn n wall unts, durng the whole tme ntegraton length (T =,). On the other hands, after T = 6, the ule laeled as B moved to the near wall regon and nteracton etween ths ule and near wall streaky structures were oserved as shown n Fg. 8. In Fg. 8, near wall regon, ule mght e led to the low-ressure regons of the hgh-seed streaky structures. Wakes ehnd the ule were oserved oth n Fgs. 8 and, ut the hgh-seed regon n front of the ule was dmnshed as shown n Fg. 8. These mght ndcate that ules dstur the near wall coherent turulent moton. ) 3. Mean Velocty and Statstcs Water-hase mean velocty rofle was shown n Fg. 9. Mean velocty at the channel center regon were decreased comared wth the sngle-hase flow; water-hase dscharge was also decreased. Ths ndcates that ules works as the VOL., OCTOBER

4 546 Yoshnou YAMAMOTO et al. Sde vew To vew u rms,vrms,wrms 3 u rms (CASE) v rms (CASE) w rms (CASE) u rms (CASE) v rms (CASE) w rms (CASE) 5 5 (d) Fg. 5 Grd resoluton deendency, nstantaneous turulent velocty vector lots, sde vew, CASE, nstantaneous turulent velocty vector lots, sde vew and to vew, CASE, turulent ntensty rofles, CASE and. Fg. 6 Instantaneous ule ehavours wth streamwse turulent velocty contour lots, rd vew, CASE, -3.5(lue) < u < 3.5(red), T = 3., T = 38.3, T = 44., and (d) T = 5.. PROGRESS IN NUCLEAR SCIENCE AND TECHNOLOGY

5 Drect Numercal Smulaton of Turulent Channel wth Deformed Bules 547 F m Fg. 7 Mean volume-rate functon dstruton of ule B and B flow resstance n ths flow condton. Fgures 9 and show the water-hase turulent ntenstes and Reynolds shear stress rofles. Near channel center, turulent ntenstes and Reynolds shear stress were ncreased comared wth n case of the sngle-hase flow. Increase of the turulence s caused from the large-scale wake motons round ules as shown n Fg. 5. On the other hands, near wall regon, eak oston of the streamwse turulent ntensty was slghtly shfted to the channel center and the Reynolds stress near wall was decreased. Consequently, t seems that ules act on the flow lamnarlze and the drag reducton near the wall regon. 5 U 5 U U (sngle) 5 5 y Mean velocty rofles 3 u rms, v rms, w rms u rms v rms w rms u rms (sngle) v rms (sngle) w rms (sngle) u v -u v (sngle) -u v Fg. 8 Interacton etween ules and hgh-seed streaky structures, streamwse velocty contour, -3.5(lue) < u < 3.5(red) at = 8, T = 6.8, T = 63., and T = Fg. 9 Mean velocty and statstcs rofles, CASE, mean velocty, turulent ntenstes, and Reynolds stress. VOL., OCTOBER

6 548 Yoshnou YAMAMOTO et al. 4. Partcle Reynolds Numer and Drag Coeffcent In ths secton, we dscussed the drag coeffcent acted on the deformed ules laeled as B and B3 n CASE. These ules were confned n the channel center regon, durng the whole tme ntegraton length (T =,). The drag coeffcent acted on the ules was defned as: d X F / m, D dt CD FD a D u u u u, where X s the oston of ule center, F D s the forces actng on ules, m s the mass of ule, C D s the drag coeffcent of ule, u s the ule velocty, u s the water-hase velocty and D s the ntal ule dameter. From Eqs. (4) and (5), drag coeffcent of ule can e estmated y the followng equaton, C D 4 8Da, (6) 6 u u where a(=d X/dt ) s the ule acceleraton. If the artcle Reynolds numer (Re ) was gven and ule deformaton can e gnored, the drag coeffcent was also estmated y usng the followng emrcal correlaton equaton. C D Re / u u D, Re, Re, Partcle Reynolds numer was drectly otaned y DNS data as shown n Fg., and the two drag coeffcents of ule were estmated y usng Eqs. (6) and (7). Fgure shows the tme seres of the nstantaneous artcle Reynolds numer. Instantaneous artcle Reynolds numer changed wthn the range from to 35 and ts range was wthn the assumton of Eq. (7). Averaged artcle Reynolds numer was. Fgure shows the tme seres of the drag coeffcent of ule. In Fg., the local mean water-hase velocty at the ule center oston was used as the water-hase velocty (:u f = u ). On the other hands, the ulk water-hase velocty (:u ) was used n Fg.. In Fg., the drag coeffcents estmated y Eqs. (6) and (7) were conssted well. However, n Fg., the drag coeffcent (shown as the lne) estmated y the emrcal correlaton equaton (7) were overestmated one (shown as roken lne) evaluated y Eq. (6). Ths s caused from the ncrease of the relatve ule velocty. If the relatve ule velocty was estmated y the ulk velocty, nteractons etween ules and the large-scale wake motons seemed to e gnored and under estmated the drag effects. In ste of the defnton of water-hase velocty, drag coeffcent estmated y Eq. (6) was elow the results estmated y the correlaton equaton (7). Ths mles that emrcal correlaton equaton mght e overestmated the drag effects n ths Partcle Reynolds numer range. (4) (5) (7) Re Fg. Tme seres of nstantaneous artcle Reynolds numer C D C D 4 3 Re = u -u *D/ V. Concluson In ths study, the drect numercal smulatons of a fully-develoed turulent channel flow wth ules were conducted y means of the refned MARS method. DNS dataase such as the mean velocty, turulent statstcs and drag coeffcent of deformed ule were otaned. Man results are summarzed: ) In ths flow condton, average artcle Reynolds numer was aout, large-scale wake motons effected y ules were oserved. T C D =(.554.8/Re.5 ), Re = u -u f *D/ C D =8/6*D*a/ u -u f u :Bule velocty u f : Water-velocty at ule oston T C D =(.554.8/Re.5 ), Re = u -u a *D/ C D =8/6*D*a/ u -u a u :Bule velocty u :Bulk velocty Fg. Tme seres of drag coeffcent of ule, estmated y usng the local mean water-hase velocty, and estmated y usng the ulk water-hase velocty. T PROGRESS IN NUCLEAR SCIENCE AND TECHNOLOGY

7 Drect Numercal Smulaton of Turulent Channel wth Deformed Bules 549 ) At the ule located regon, mean velocty was degreased and turulent ntenstes and Reynolds shear stress were ncreased y the effects of the large-scale wake motons round ules. 3) The eak oston of the streamwse turulent ntensty was slghtly shfted to channel center and Reynolds stress near wall was decreased; near wall regon, ules mght effect on the flow lamnarlze and drag reducton. 4) Emrcal correlaton equaton mght e overestmated the drag effects n ths Partcle Reynolds numer range. Acknowledgment A art of the resent DNS were conducted y usng SX-9 at the Cyer Scence Center, Tohoku Unversty, and ths the study was suorted y the Gloal COE rogram Energy Scence n the Age of Gloal Warmng and a Grant-n-ad for Young Scentsts (B), KAKENHI (7656) MEXT, Jaan. References ) T. Kawamura, T. Kodama, Numercal smulaton method to resolve nteractons etween ules and turulence, Int. J. Heat Flud, 3, (). ) J. Lu, A. Fernandez, G. Tryggvason, The effect of ules on the wall shear n a turulent channel flow, Phys. Flud, 7, 95 (5). 3) J. Lu, G. Tryggvason, Effect of Bule Deformalty n Turulent Buly Uflow n a Vertcal Channel, Phys. Flud, 47 (8). 4) A. Yamaguch, A. Hashmoto, A Comutatonal Model for Dssolved Gas and Bule Behavor n the Prmary Coolant System of Sodum-Cooled Fast Reactor, The th Internatonal Tocal Meetng on Nuclear Reactor Thermal-Hydraulcs (NURETH-), Avgnon, France, Oct. -6 (5). 5) T. Kunug, MARS for multhase calculaton, Comut. Flud Dynam. J., 9, (). 6) J. U. Brackll, D. B. Kothe, C. Zemach, A contnuum method for modelng surface tenson, J. Comut. Phys.,, 335 (99). 7) M. Sussman, P. Smereka, S. Osher, A Level Set Aroch for Comutng Soluton to Incomressle Two-Phase, J. Comut. Phys., 4, (994). 8) T. Hmeno, T. Watanae, "Numercal analyss of sloshng and wave reakng n a vessel y CIP-LSM," Proc. of APCOM 7 n conjuncton wth EPMESC XI, Kyoto, 3-6 Decemer, 7. No. MS7-7- (7) [CD-ROM]. 9) A. Kuroda, N. Kasag, M. Hrata, Investgaton of Dynamcal Effects of the Mean Shear Rate on the Wall Turulence va Drect Numercal Smulaton, 7th Natonal Heat Transfer Symosum of Jaan, (99). ) J. Jmenez, P. Mon, The mnmal flow unt n near-wall turulence, J. Flud Mech., 5, 3-4 (99). VOL., OCTOBER