INVESTIGATION OF THE TURBULENT MASS TRANSPORT DURING THE MIXING OF A STABLE STRATIFICATION WITH A FREE JET USING CFD-METHODS

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1 INVESTIGATION OF THE TURBULENT MASS TRANSPORT DURING THE MIXING OF A STABLE STRATIFICATION WITH A FREE JET USING CFD-METHODS I1. Armin Zirkel, I2. Eckart Laurien Intitute of Nuclear Technology and Energy Sytem, Univerität Stuttgart Abtract During a evere accident, hydrogen can be produced by a chemical reaction between the Zircaloy cladding and water and ecape into the containment through a leak in the primary circuit. The prediction of the ma tranport of hydrogen i vital for an optimized poitioning of countermeaure like recombiner. It i poible that a table tratification of hydrogen and air occur, due to the different denitie of thoe fluid. Thi paper dicue the imulation of ma tranport procee during the mixing of uch a table tratification with a free jet. The mixing of a table tratification with a free jet i characterized by the time dependency of the flow, harp velocity and denity gradient a well a the non-iotropy of Reynold tree and turbulent ma fluxe. Thi paper dicue reult of the eddy diffuivity model for the turbulent ma flux a well a improvement with the ue of a newly implemented non-iotropic model. 1. INTRODUCTION Thi paper preent the current tate of an ongoing analyi and validation of turbulence model for computational fluid dynamic (CFD) for containment flow after a lo-of-coolant accident (LOCA) in an exiting light water reactor. The topic of thi work i to invetigate turbulence modeling for the mixing of a table tratification with an intruding turbulent free jet. Both, the free jet and the table tratification are cauing non-iotropic turbulence. Buoyancy turbulence production alo ha an influence, a turbulence i not only produced by hear tree. Fig. 1: Stable tratification inide a containment 1

2 Lumped parameter model have been developed, verified and ued to analyze and predict tranport procee within a containment (Fig. 1) (Allelein, 2005; Allelein, 2008). Thee model are baed on ma and energy budget between given control volume inide a containment building. They can provide valuable information about complex flow, uch a mixing, condenation and aerool tranport. However, flow model are often pecialized to a narrow range of application and the uer influence i rather large. Recently, method of computational fluid dynamic have alo been ued to imulate containment flow (Kljenak, 2006; Houkema, 2008). They are baed on temporally averaged ma, momentum and energy conervation equation, which appear a a et of coupled partial differential equation. Their turbulence model, ee Laurien (2007) for an overview, can be applied to a wide range of flow ituation and geometrie, repreented by a fine boundary fitted numerical grid. The Reynold tre turbulence model are capable of yielding good reult for flow with non-iotropic turbulent tree. Model to calculate non-iotropic turbulent fluxe are available (Rodi, 1993) but not implemented in commercial oftware. Currently, the mixing of a table tratification calculated by CFD take longer time than meaured mixing due to the iotropic calculation of the turbulent fluxe. The aim of thi work i to develop a formulation of a non-iotropic turbulent flux model that i able to predict the mixing of a table tratification. Thi new turbulent flux model can replace the eddy diffuivity concept in the Reynold tre turbulence model. In order to do thi, the influence of buoyancy turbulence production a well a the impact of the iotropic approach of eddy vicoity model i invetigated. To get detailed information about the turbulence, a theoretical teady tate tet cae i deigned. A large eddy imulation (LES) i performed to be the reference cae. 2. EXPERIMENT Experiment for containment flow are the THAI-experiment (Thermohydraulic, Hydrogen/Helium, Aerool and Iodine) performed by Becker Technologie. The experiment referred to in thi paper are the TH20 experiment (Kanzleiter, 2007). Fig. 2 how a model of the geometry and a chematic repreentation of the experiment. honeycomb cone Helium + Air Denity Layer Interaction Area fan Air inner cylinder Fan Fig. 2: left: TH20 geometry model; right: TH20 cheme 2

3 The height of the veel i 9.2m with a diameter of 3.16m. The configuration of the THAI-veel for TH20 contain a hort inner cylinder in the lower veel with a diameter of 1.4m. Inide the inner cylinder i the fan that generate the free jet. For afety reaon, helium i ued intead of hydrogen for the TH20 experiment. Before the actual experiment, the table tratification i created. Helium i carefully brought into the THAI-veel to prevent mixing due to the incoming ma flow of helium. Thi reult in a table tratification, becaue the le dene fluid accumulate in the upper veel and i on top of the dener fluid. The harp denity gradient between the light ga cloud on top and air below i denoted a denity layer. It i the reaon for the non-iotropic turbulence becaue the gradient only appear normal to the denity layer. After the creation of the tratification i complete, a free jet i generated by the fan inide the inner cylinder of the veel. The jet tend to penetrate the denity layer but i reflected becaue the layer i table. During the redirection proce, the jet take in helium and erode the denity layer. The duration of the experiment, from the tart of the fan until the helium ditribution inide the veel i homogeneou, i 13 minute. 3. GOVERNING EQUATIONS The governing equation for CFD with turbulence model are the Reynold averaged Navier-Stokeequation (RANS) with the concentration equation (1) for the ma tranport. t u i = u i ' ' (1) The concept of Reynold averaging i the averaging of the turbulent value over time, which are then determined by an appropriate turbulence model (Rodi, 1993). In commercial CFD code, the turbulent ma fluxe ui ' ' are calculated with the iotropic eddy diffuivity model (2). u i ' ' = t The eddy diffuivity i the eddy vicoity divided by a contant turbulent Schmidt number t = t t. Thi value i ued to determine the turbulent ma flux in every direction. The turbulent calar flux model (TSF) ha a tranport equation for each patial direction (3) and the variance (4). With thi four additional equation, the non-iotropy of turbulent ma fluxe can be calculated. t u ' i ' u ' x j ui ' = u ' ' i u j ' u u j x j ' i j x j p tat 1 C ' 2 3Y x j [ 2 3 C Y k 2 x j ref g i u ' i ' ] C 1Y ui k ' ' C 2Yu ' u j ' i C x 4Y u ' j ' j u j (2) (3) 3

4 t ' 2 u x j '2 = 2 ' u j ' l j x j x j [ 2 3 C YY k 2 x j ' ] 2 2C 1YY k ' 2 (4) The model contain everal coefficient, C Y, C 1Y, C 2Y, C 3Y, C 4Y, C YY and C 1YY, which are to be adjuted a part of thi work. 4. STEADY STATE TEST CASE An important factor for turbulence modeling i the comparability of different calculation. For the preent cae of eroding a table tratification with a free jet, patial comparion i neceary becaue the mixing happen locally. So a tranient cae i diadvantageou, becaue the time ha alway to be conidered when comparing patial ditribution of turbulent value. Therefore a theoretical teady tate tet cae i deigned (Fig. 3) He-inlet 500 Z [m] normal thin broad 0.2 Air-inlet Outlet Helium ma fraction Fig. 3: left: Scheme of the teady tate tet cae; right: initial helium ditribution The geometry i an open cylinder with height h = 1000 mm and radiu r = 500 mm. Air i flowing with 2 m/ into the domain through the air-inlet, generating a free jet. To create a teady tate, an equal amount of helium that left the domain through the outlet i reinerted through the helium-inlet. The outgoing maflow of helium i alo a value for the mixing quality of the ued turbulence model. A larger helium outflow mean a better mixing of the turbulence model. A table tratification i given through an initial helium ditribution. Fig. 3 how three different initial ditribution, where the ma of helium i g for the thin, g for the normal and g for the broad initial ditribution. A preliminary invetigation how no influence of the initial ditribution on the hape or thickne of the final tratification. The height of the layer varie with the amount of helium. 4

5 5. DIMENSION ANALYSIS 5.1 Dimenion Matrix To enure that the tet cae i a reaonable repreentation of the TH20 cae, a dimenion analyi i performed to determine independent dimenionle number to compare thoe cae (Szirte, 2007). A et of even parameter decribe the TH20 cae at a given time. Thoe parameter are the length L of the jet which i the ditance between the nozzle and the table tratification, the velocity u of the jet, the denity ρ, the denity difference Δρ between the jet and the light-ga cloud, the vicoity v, the gravity g and the diffuion contant D. The three bae unit for thoe parameter are m, and kg. Seven parameter with three bae unit mean that there are four independent dimenionle number. The expected relevant dimenionle number for thi dimenion analyi are the Reynold number Re, the Richardon number Ri, the Archimede number Ar and the Schmidt number Sc. The Reynold number i the ratio of inertial force to vicou force. The Richardon number i the ratio of potential to kinetic energy. The Archimede number i the ratio of buoyancy force to friction force. The Schmidt number i the ration of convective to diffuive ma tranport. Table 1: Paramter for the dimenion analyi L u ρ Δρ ν g D Re Ri Ar Sc m m kg m 3 kg m 2 m 3 m m 2 2 L u g L u 2 g L 3 2 D To prove that the dimenional number are independent, the determinant of the dimenion matrix mut not be zero. The dimenion matrix (Fig. 4) i the correlation of the decribing parameter and the bae unit and dimenionle number. L u g D m kg Re Ri Ar Sc Fig. 4: Dimenion matrix Every parameter ha it column and every bae unit and dimenionle number ha it row. The value of the matrix are the exponent with which the bae unit appear in the dimenion of the parameter or the exponent with which the parameter appear in a dimenionle number. For example, velocity ha the dimenion m/. So the value for meter i 1 and for econd -1. Velocity appear with the exponent 1 in the Reynold number and with -2 in the Richardon number. The determinant of the dimenion matrix i -56, o it i demontrated, that the dimenionle parameter are independent. 5

6 5.2 Parameter The length of the free jet, which i the ditance between the nozzle outlet and the table tratification, i the characteritic length. L tet =0.66 m; L TH20 =2.27 m The maximum velocity of the free jet i taken a the characteritic velocity. u tet =2.013 m ; u TH20=1.827 m Reference denity i the denity of air, Air =1.185 kg m. For the denity difference, the denity of the 3 light ga cloud i neceary. In the tet cae, the light ga cloud i pure helium with a denity of He = kg. In the TH20 cae, the cloud ha 36% helium and a denity of m He=0.84 kg 3 m 3 The characteritic denity difference i the difference between air and the light ga cloud. tet = Air He = kg m 3 ; TH20= Air Mix =0.345 kg m 3 The kinematic vicoity i tet = 1 2 Air He = 1 2 TH20 = 1 2 Air Mix = m2 m m2 5 m m2 = m2 = The gravity contant i g=9.81 m 2 cm 2 and the diffuion contant for helium and air D= Dimenionle number Table 2 how the dimenionle number for both cae. The Reynold number indicate a fully turbulent flow for both cae. The Richardon number how a light dominance of the potential energy in both cae.. The Archimede number in both cae i very large. Thi tand for a domination of the buoyancy force. The Schmidt number i almot the ame for both cae. It i poible to have an even better match of both cae, but thi would lead to a larger computational domain which increae the computational effort. Table 2: Dimenionle number Re Ri Ar Sc Tet cae TH IMPACT OF THE EDDY DIFFUSIVITY MODEL All two-dimenional imulation are carried out on a two-dimenional equiditant tructured grid which meet all criteria of the bet practice guideline for CFD code validation for reactor-afety application (Menter, 2002). A grid dependency tudy how no grid influence on the reult. To invetigate the impact of the turbulence model, two imulation are performed, one with a eddy vicoity model (kω) and on with a Reynold tre model (ωrsm). Both imulation ue the eddy 6

7 diffuivity model to calculate the turbulent ma fluxe. The imulation are carried out The helium outflow of the imulation uing the kω-model i He out =3.31 g. With the ωrsm He out =3.37 g. Fig. 5 give a comparion of the reult for the helium ma fraction and the vertical velocity. Both ditribution are almot identical for both turbulence model. Thi how that the reult i dominated by the model for the ma flux and the impact of the turbulence model i negligible. Fig. 5: Comparion ωrsm kω, left: helium ma fraction; right: vertical velocity 7. Large Eddy Simulation A Large Eddy Simulation directly imulate the large, energy bearing eddie. A ubgrid-cale model i ued for the eddie that are too mall to be reolved by the grid. Therefore a LES ha high demand on the quality of the grid. To get meaningful reult, the inertial range of Kolmogorov' energy cacade ha to be reolved (Fröhlich, 2007). The left hand ide of Fig. 6 how a generic turbulent pectrum (energy E over wave-number k). The three important region are the large eddie with lower wave-number, the inertial range and the diipation range with high wave-number. The typical lope of the inertial range i k 5 3. To check the quality of the LES, turbulent pectra mut be analyzed a part of the pot-proceing. The right hand ide of Fig. 6 how a repreentative turbulent pectrum of the performed LES for the teady tate tet cae. It how that the inertial range with it characteritic lope i reolved by the grid. Therefore it can be aumed, that the grid quality i ufficient. To verify thi aumption, a econd LES with a coniderably finer grid will be performed to invetigate a poible impact of a imulation of maller eddie on the reult. 7

8 : large eddie 2: inertial range 3: diipation range E 1E-02 1E-03 1E-04 1E-05 1E-06 1E-07 1E-08 1E-09 1E-10 1E-11 1E+00 1E+01 1E+02 1E+03 Fig. 6: Turbulent pectrum k E=k 5 3 The grid ued for thi LES i a three-dimenional block-tructured grid with node. The central difference advection cheme i ued. The time tep i with the 2 nd order backward Euler tranient cheme. Small eddie are modeled with the Smagorinki ubgrid-cale model. The imulation i carried out on 64 CPU. It take 24 hour to imulate 0.8 phyical time. After 10 econd phyical time, a tatitically teady tate i reached. The tranient averaged reult are averaged over additional 10 econd phyical time. The helium outflow i He out, LES =3.8 g, which i 0.49 g more than with the eddy diffuivity and kω-model. Fig. 7: Vertical velocity, denity layer and ma flux; left: tranient averaged; right: intantaneou 8

9 Fig. 7 how the helium layer, which i defined a an iourface of 50% helium. It i colored with the turbulent ma flux of helium u' '. The xz-plane below the layer how the vertical velocity. A comparion of the averaged (Fig. 7, left) and intantaneou (Fig. 7, right) helium layer how trong threedimenional, tranient effect, depite the ymmetric geometry and boundary condition. The interaction between the jet and the table tratification caue the jet to move unpredictable around the ymmetry axi of the geometry without a preferred direction. However, tranient averaging how that the reult i ymmetric. Fig. 8: Tranient averaged value; left: Vertical velocity; right: Turbulent ma flux The left hand ide of Fig. 8 how the tranient averaged vertical velocity of the LES. A oppoed to the free jet calculated with the eddy diffuivity model (Fig. 5), the down-flow region of the jet i le ditinct. It i broader and reache to the wall of the veel. The maximum velocity of the down-flow i 0.5 m where imulation with the eddy diffuivity model yield a fater down-flow with 1 m. Thi i an effect of the three-dimenional movement of the flow in the LES cae, which can not be captured in the twodimenional RANS imulation. Fig. 7 (left) and Fig. 8 (right) how, that the primary turbulent ma flux i in the redirection region of the free jet. The poible impact of the three-dimenional movement of jet and layer on the mixing i not yet determined. 8. CONCLUSION Simulation of the TH20 experiment ha hown that currently available commercial cfd-oftware ha no turbulence model to predict the mixing of a table tratification with a free jet. To improve thi ituation, the Turbulent Scalar Flux model will be teted and adjuted for thi cae. To accomplih thi, a theoretical teady tate tet cae wa deigned to improve the comparability of different imulation. Thi alo provide a more detailed view on pecific phyical apect like the turbulent ma fluxe. 9

10 A dimenion analyi ha hown, that the tet cae i a reaonable repreentation of the TH20 experiment. The defining dimenionle number are the Reynold, Richardon, Archimede and Schmidt number. Preliminary, two-dimenional imulation were performed to invetigate the impact of the turbulence model compared to the model for the turbulent ma fluxe. An eddy vicoity model (kω) and a Reynold tre model (ωrsm) were ued for thi invetigation. Both imulation were uing the eddy diffuivity model to calculate the turbulent ma fluxe. It turned out, that the eddy diffuivity model dominate the reult. A large eddy imulation wa performed a the reference cae. The ued grid ha node, which i ufficient to reolve the inertial range of the energy cacade. The geometry a well a the boundary condition are ymmetric, but the interaction of the free jet and the table tratification caue an unpredictable three-dimenional, tranient behavior of the flow. The tranient averaged reult i ymmetric. The calculated helium outflow i up to 0.49 g larger than the outflow of the preliminary imulation which indicate that the helium mixing i better in the LES cae. Another difference between the preliminary RANS imulation and the LES i the down-flow between helium layer and outlet boundary. A econd LES on a finer grid will be performed to invetigate a poible impact of the reolution of maller eddie on the reult. For a more detailed invetigation of the three-dimenional behavior, two- and three-dimenional RANSimulation will be carried out. The eddy diffuivity model and the new turbulent calar flux model will be ued. A part of thi invetigation, the model coefficient of the turbulent calar flux model will be adjuted. The adjuted model will then be ued to calculate the original TH20 experiment. ACKNOWLEDGMENTS Thi work ha been upported by the German Bundeminiterium für Wirtchaft und Technologie (BMWi), GRS REFERENCES H.-J. Allelein, S. Arndt, W. Klein-Heßling, S. Schwarz, C. Spengler, G. Weber: COCOSYS: Statu of development and validation of the German containment code ytem, Nuclear Engineering and Deign, 238, (2008) H.-J. Allelein, K. Neu, J.P. Van Dorelaere: European validation of the Integral Code ASTEC (EVITA) firt experience in validation and equence calculation, Nuclear Engineering and Deign, 235, (2005) I. Kljenak, M. Mabic and I. Bajic: Modeling of Containment Atmophere Mixing and Stratification Experiment uing a CFD Approach, Nuclear Engineering and Deign, 236, (2006) M. Houkema, N.B. Siccama, J. A. Lycklama à Nijeholt, E. M. J. Komen: Validation of the CFX4 CFD code for containment thermal-hydraulic, Nuclear Engineering and Deign, 238, (2008) E. Laurien: Turbulence Modeling for CFD in Reactor Safety, Proc. Ann. Meeting on Nuclear Techology, Karlruhe, May 22-24, 2007, Inforum GmbH, Bonn 10

11 W. Rodi: Turbulence Model and Their Application in Hydraulic, A. A. Balkema (1993) T. Kanzleiter, A. Kühnel. K. Ficher, M. Heitch, B. Schramm: THAI Blower Tet TH20, GRS Reactor Safety Reearch, Report No TH20 (2007) T. Szirte: Applied Dimenional Analyi and Modeling, Elevier (2007) F. Menter: CFD Bet Practice Guideline for CFD Code Validation for Reactor-Safety Application, EVOL-ECORA-D01 (2002) J. Fröhlich: Large Eddy Simulation turbulenter Strömungen, Teubner (2006) 11