Selection of an Appropriate Turbulence Modeling in a CFD code for an Ultra-long Life Core for the IRIS Reactor

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1 GENES4/ANP00 Sep Kyoo JAPAN Paper 5 Selecion of an Appropriae Turbulence Modeling in a FD code for an Ulra-long Life ore for he IRIS Reacor Emilio Baglieo * Hisashi Ninoaa Research Laboraory for Nuclear Reacors Toyo Insiue of Technology Toyo Japan A comparaive sudy of differen urbulence models is presened o selec which one could be adoped for evaluaion of hermo-hydraulic performances of innovaive ulra-long life core designs. Various models are briefly oulined and discussed. Linear and non-linear low Reynolds -ε urbulence closures are applied on an isohermal benchmar o evaluae he capabiliies o correcly predic wall shear sresses and velociy disribuion in he channel. apabiliies o analyze problems involving hea ransfer are also esimaed on a benchmar in a heaed channel involving liquid sodium coolan. Two alernaive approaches are considered for his case he - and he Shear Sress Transpor model. The relaive performances of he differen models are assessed. The resuls show in boh cases ha he capabiliy o accoun for anisoropic effecs is he ey requiremen for correc modeling of a rod bundle. In his respec non-linear models presen promising capabiliies bu require furher improvemen for accurae predicion. KEYWORDS: Turbulence anisoropic -ε secondary flow I. Inroducion The Inernaional Reacor Innovaive and Secure (IRIS) is an advanced ligh waer reacor being developed by an inernaional consorium of indusry laboraory universiy and uiliy esablishmens led by Wesinghouse ). Is design is based on proven LWR echnology so ha no new echnology developmen is needed and near erm deploymen is possible ). A he same ime evoluionary core soluions are being developed for fuure reloads ). The firs IRIS core will employ sandard <5% UO fuel and a sandard PWR fuel assembly design herefore no foreseeing licensing issues. A he same ime wor is ongoing o address he desirable arge of a high conversion-raio ulra-long life core. Such a core will be characerized by a very igh laice wih relaively small waer volume fracion. In order o successfully acle his challenge we mus inroduce innovaive fuel configuraions o obain a more uniform uilizaion of he coolan. A main undesirable aspec of classic riangular laice is in fac he non-homogeneous disribuion of waer around fuel pins which resuls in a non-uniform circumferenial clad emperaure and wall shear sress disribuion 4). The arge is herefore o minimize he maximum clad and fuel cenerline emperaures eeping a he same ime he pressure drops hrough he core as low as possible. A firs comparison beween various fuel geomeries have been performed by Romano and Todreas 5) who inroduced a performance index mehodology o assess hermo-hydraulic parameers such as core pressure drop fuel cenerline emperaure and clad surface emperaure disribuion. This preliminary wor shows he possibiliy of highly increasing fuel performances hrough he inroducion * orresponding auhor Phone/Fax d576@nr.iech.ac.p of novel designs. A large range of arrangemens is aen ino consideraions for he fuure core and needs o be assessed wih a deailed analysis capable of correcly modeling hree dimensional phenomena. lassical lumped parameers codes adoped in nuclear indusries such as subchannel analysis codes are no appropriae for our purpose being calibraed on specific geomeries and operaional condiions. As i is well nown he fines insrumens for modeling urbulence are DNS and LES. The compuing requiremens however are exremely high; in he DNS case for example hey scale as Re 9/4 maing hem a he momen unsuiable for our analysis 6). Modern FD codes of pracical use are based on he so-called Reynolds Averaged Navier Soes equaions (RANS) models where he erms represening urbulence ineracions are modeled in erms of he mean flow variables as for example in he widely used -ε approach. However he reliabiliy and accuracy of he resuls canno easily be assured 7). Turbulence modeling is usually considered he weaes poin in FD analysis. For his reason a profound analysis of available models has been carried ou and he models which showed promising characerisics have been esed and evaluaed agains isohermal and heaed channel benchmars for igh laice geomeries. II. Turbulence models I is well nown ha among he various urbulence models he -ε is currenly he mos popular and applicable o a large variey of flow cases wih reasonable predicion accuracy. However here sill are fundamenals problems ha need o be resolved. In paricular i is well nown ha he usual formulaions of he model assume an isoropic eddy diffusiviy in modeling he Reynolds sress ensor. One imporan deficiency of such a model is he fac ha anisoropic effecs are no accouned for. In simple erms since he viscosiy is a scalar each componen of he sress

2 affecs is srain o he same exen. For his reason flows where urbulen inensiy gradiens and hence he augmenaion effecs of secondary srain raes are imporan canno be accuraely prediced. The mos quoed example is he secondary flow ha originaes in a square duc due o he gradiens of he urbulen shear sresses 8). Due o his imporan limiaion Reynolds Sress Models (RSM) are ofen used insead because hey offer a greaer poenial for predicing he anisoropic phenomena correcly. Anyhow alhough he second momen ranspor equaions can be derived in exac forms he higher order correlaions need o be modeled wih some drasic assumpion of unnown validiy; even he principles and basic echniques for heir modeling have no been esablished ye a leas in he near wall region as saed by Myong and Kasagi 9). The simulaneous soluion of a se of highly non-linear coupled differenial equaions for each individual componen of he Reynolds sress ensor has also a very high compuaional cos and ofen leads o unsable compuaions. In his respec he use of algebraic raher hen differenial equaions for he Reynolds sresses which leads o he Algebraic Reynolds Sress Model (ARSM) offers a more solid soluion scheme. These models are anyhow up o dae far from being assessed and he soluions ofen rely heavily on he user assumpions maing hem no appealing for our research. Our aenion in his wor is herefore direced owards a differen approach o realisically simulae he effecs of anisoropy in he flow which consiss of a non-linear exension of he sress-srain relaion. This approach forms a Non-Linear Eddy Viscosiy Model (NLEVM) someimes also refereed o as an explici ARSM. Alhough he idea of a NLEVM emerged in he 70 s 0-) only recenly hey have seen pracical applicaions -). These models have shown promising capabiliies in successfully reproducing urbulence driven secondary flows. The capabiliies of hese models o correcly predic shear sress and velociy disribuions in a igh laice fuel bundle are evaluaed in comparison wih he classic EVM. Furher in our fuel design we will also have o challenge he hermal analysis of he fuel bundle. For his reason we also analyzed he model performances on a benchmar dealing wih a heaed rod bundle. In his case i is fundamenal no only o sudy he predicion capabiliies of he various models bu also o consider which offer a good balance beween accuracy robusness and calculaion cos requiremen. We herefore considered wo alernaive approaches he -w 4) and he Shear Sress Transpor (SST) model 5). The - model in place of he dissipaion rae ε adops an equaion for he urbulen frequency of he large scales. The -equaion is claimed o have significan advanages near he surface and o accuraely predic he urbulen lengh scale which should improve wall shear sress and hea ransfer predicions. Moreover he model has a very simple Low-Reynolds formulaion which does no require addiional non-linear wall damping erms. The main advanage of he model has herefore shown o be is robusness and he reduced resoluion demands for inegraion o he wall. The SST model insead combines he use of he - model near he wall wih he -ε model away from he wall where he - has shown some weaness.. Wall Treamen The commonly used high Reynolds number versions of he -ε model as well nown canno be applied near he viciniy of he wall since he model neglecs he effecs of viscosiy. The ypical approach o avoid modeling hese effecs is hrough he use of empirical wall funcions o bridge he gap beween he solid boundary and he urbulen core. However he universaliy of such funcions breas down for complex flows. Anoher common approach is he use of a wo-layer formulaion where he ε-equaion is only solved in he ouer par of he boundary layer whereas he inner porion of he logarihmic layer and he viscous sub-layer are reaed by a mixing lengh formulaion 6). Again for complex flows he coupling beween he mixing lengh and he ε-equaion becomes problemaic and he soluion depends srongly on he specificaion of he maching locaion; he uniqueness of he soluion can herefore no be guaraneed. Since our sudy deals wih analyzing novel fuel geomeries boh hese approaches canno be considered accepable for our inen. Hence near wall -ε models or low Reynolds number models which aemp o model he direc influence of viscosiy are he only suiable approach. This formulaion uses he sandard ranspor equaion for bu augmens he equaion for ε wih an exra erm which depends on he disance o he wall o render he model valid up o solid wall. I is anyhow compuaionally more inensive and in paricular requires he values of he y+ in he near wall cell o be of he order of. This condiion produces exremely high meshing requiremens maing he simulaion of large geomeries problemaic. For his reason in he wor paricular aenion is posed on he balance beween accuracy and mesh requiremens of he differen low Reynolds approaches as compared o he high Reynolds form. A paricularly aracing wall reamen is represened by he VF model developed by Durbin 7). The VF model avoids explici low-re erms in he e equaion by using an ellipic relaxaion equaion near he wall. In addiion an equaion for he flucuaing ν is inroduced. This model does no require he calculaion of wall disance and does no require he use of dumping funcions or wall funcions o adus he behavior of urbulence quaniies. I is claimed o have reduced compuaional overhead and o produce beer fricion and hea ransfer predicions hrough he correc modeling of he viscous sub-layer. This model is no evaluaed in his wor bu will be esed in he near fuure. III. Model Descripions. Linear and Non-Linear Turbulence losure The calculaions have been performed wih he commercial code Sar-D version.5. The formulaion of he -ε model has been published in many papers and will

3 no be repeaed here for conciseness bu can be found in reference 8). We insead briefly describe he urbulence closure formulaions for he linear and non linear models which are of main ineres. Linear model For he sandard linear -ε model he consiuive relaions for he Reynolds sresses are as follows: ρ u u i s u ρ δ i + i ( ) where due o he linear correlaion each componen of he shear sress influences only one componen of he velociy gradien herefore being unable o accoun for anisoropy. Non linear models ry o caer for his defec by adoping non-linear relaionships beween Reynolds sresses and he rae of srain. Quadraic model For he quadraic model he consiuive relaions for he Reynolds sresses are as follows: SiS ε where u i u u ρ + ρ δ i S δis l ΩiΩ ε S + l NL ( c NL6 + c NL7 S ) δ iω i + [ ΩiS + Ω Si ]+ ε l Ω l ( ) c ( ) c ( 4 ) NL ( c NL6 + c NL7 S ) c ( 5 ) NL ( cnl6 + cnl7 S ) and A0 ( 6 ) A + A S + AΩ where A0 A A A cnl cnl cnl cnl 6 cnl 7 are empirical coefficiens whose values are shown in able. Si and Ωi are elemens of he mean srain and voriciy ensors respecively and are given by: S u u i i + ( 7 ) i u u i Ω i ( 8 ) i The oher wo erms required o complee he definiion of are as follows: S S S i i ( 9 ) ε Ω Ωi Ωi ( 0 ) ε The inroducion of hese quadraic erms has shown o be able o successfully reproduce urbulence driven secondary flows in various cases. Anyhow he various quadraic models have shown limis in he sensiiviy o minor srains. The quadraic models in paricular do no have sensiiviy o sreamline curvaure. ubic model In order o capure he sreamline curvaure effecs i is necessary o inroduce furher cubic erms. A cubic erm is herefore added o correlaion ( ) as follows: u i u u ρ + ρ δ i S SiS ε δ is l S + l ΩiΩ δ iωlωl + 4 ε ε + 5 ε ( S l S l Ω l Ω l ) S i 4 NL4 i + [ ΩiS + Ω Si ]+ ε ( ) ( S Ω + S Ω ) S + c ( ) c ( ) 5 NL5 4 and 5 are again empirical coefficiens given in able. Table oefficien for urbulence closures A 0 A A A c NL c NL c NL c NL4 c NL5 c NL6 c NL models For he - model as implemened in Sar-cd he general forms of he and equaions are as follows: Turbulence Energy g ρ ( ) g ρ + ρu ~ + P ρ Specific dissipaion rae g i l σ ( ) g ρ + ρu ~ + σ li l ( 4 ) ( 5 )

4 Sandard ρα P ρ + ρ S For he sandard model he closure coefficiens and auxiliary relaions are: α 0 f χ f + 80χ where f + 680χ + 400χ χ 0 f σ χ σ i s ( ) o χ 0 χ > 0 S 0 s i S i and he eddy viscosiy is given by i ( 6 ) ρ ( 7 ) SST model For he SST model he coefficiens are expressed in he following general form: where φ ( F ) F + ( 8 ) φ φ 4 ( ) F anh arg ( 9 ) arg 500ν min max ; ; 0.09 y y 4ρ σ D y ( 0 ) σ σ α S κ σ ( F ) x ( ) ( 4 ) σ The eddy viscosiy for his model is ρ max where 0. a ( a Ω F ) a ( ) anh F arg arg 500ν max 0.09 y y Ω ( 5 ) Ω Ω i i For he low Reynolds number adoped a he near-wall cell is fixed algebraically according o: 60ν p ( 6 ) y where ν is he ineic viscosiy y he normal disance from he wall and IV. Benchmar Problems Descripion. Air flow in a igh laice rod bundle The firs benchmar adoped for he urbulence models refers o he evaluaion of shear sress and velociy field disribuion in an hexagonal igh laice rod bundle. The p/d raio of he laice is.7 9). The experimens were performed in a wind unnel using a 9-rods model of he fuel assembly. Geomerical daa of he model cross secion are shown in figure. Lengh of he model is 6 m. Only a shor descripion of he resuls is given in his wor bu a more deailed analysis can be found in reference 0). ρ 0 D max 0 ( ) σ The coefficien se is.76 σ σ α κ σ κ 0. 4 ( ) Fig. Rod Bundle Geomery and he coefficien se is Measuremens in he experimen have been performed in

5 he region of hydraulically fully developed urbulen flow and wihou he bac effec of oule cross-secion change. Daa refer o he middle-cell and we herefore assume ha he influence of he walls around he rod bundle is negligible. Shear sress and axial mean velociies have been obained wih Pio and Preson probes. The following able shows he physical parameers of he experimen. Table Physical parameers Fluid Densiy Kinemaic Viscosiy Reynolds Number Air. g/m.59e-5 m /s Temperaure disribuion in a rod bundle wih sodium flow The second benchmar regards a heaed bundle wih sodium flow. The experimen named TEGENA ) was performed a he Karlsruhe Insiue. Once again we presen only a shor descripion of he resuls while a complee repor will be published in fuure wor. The es geomery is presened in fig.. I consiss of 4 heaer rods of 5 mm in diameer and 6 m in lengh wih a p/d raio of.47 suspended verically in a recangular channel. The emperaure values of he sodium have been measured in an horizonal plane 9 mm ahead of he end of he recangular channel in he zone of hermally fully developed flow. The main physical parameers are lised in able. Table Tegena parameers Sodium Fluid Re Pe Flow Hea Flux a rod surface Sodium g/s 50 W/cm V. Numerical Simulaion. Numerical procedure The differenial equaions governing he conservaion of mass momenum energy ec. wihin he fluid are discreized in he code by he finie volume (FV) mehod. The discreized equaions are solved in a segregaed manner wih he SIMPLE (Semi-Implici Mehod for Pressure-Lined Equaions) algorihm ). For he spaial discreizaion a firs order upwind differencing (UD) scheme has shown sufficien bu a quadraic upsream inerpolaion of convecive inemaics (QUIK) has also been applied in all cases o assess he independency of resuls from discreizaion pracices. Descripion of hese mehods can be found in reference ).. ompuaional grid In our simulaions paricular aenion is always posed on he dependency of he resuls on he grid resoluion o demonsrae he grid-convergence of he soluion. For his reason afer preliminary sensiiviy sudies hree differen grid resoluions are adoped for every geomery. Figures 4 and 5 show he various grids respecively for he firs and second benchmar. The grids shown are he one adoped for he low-re number models where he y+ value for he firs cell is of he order of. For he firs case due o he symmery of he problem we i has been sufficien o describe he elemenary cell as showed in figure wih he appropriae symmery boundaries. Elemenary "infinie" flow cell Fuel Rod Fig. Tegena es geomery Fig. Benchmar elemenary cell

6 oarse mesh Fine mesh Refined mesh Fig.4 Benchmar grids In he second benchmar he minimum symmeric geomery modeled is represened by one fourh of he bundle secion. We show a paricular o appreciae he differen mesh finesse. Refined mesh Fig.5 Benchmar grids VI. Presenaion and Discussion of Resuls In boh benchmars grid convergence of he resuls and independence of he resuls from discreizaion mehods have been assessed wih he mehodology previously presened bu will no be presened for breviy.. Benchmar Regarding benchmar experimenal daa were available for he wall shear sress disribuion along he weed surface of he rod versus he azimuh φ as shown in fig. 6. φ Fig.6 Sress disribuion These daa are compared o hose calculaed from he simulaion for he differen urbulen models. Figure 7 and 8 presen respecively he predicions of he linear and nonlinear models. oarse mesh Fig.7 Linear low-re model Fine mesh

7 Fig.8 Non-linear low-re models We clearly see ha he sandard -ε model even hough correcly predicing he shear sress in he small gap region highly over predics he values in he large gap region. In his respec he non-linear model shows a much flaer disribuion obaining an accepable quaniaive agreemen wih experimenal daa. This is due o he abiliy of hese models o accoun in some exen for anisoropic effecs hus predicing a secondary flow in he channel as shown in fig. 9. Sill hese models can no qualiaively reproduce he disribuion flaening ypical of he experimenal daa owards he cener of he channel. This migh be relaed o he fac ha he velociy scale of he secondary moion in our predicions is only less han % of he axial velociy. Fig. Velociies a φ0 locaion Fig. Velociies a φ5 locaion Fig.9 Secondary velociy field Experimenal velociy profiles were also available a hree differen locaions a 0 5 and 0 degrees locaion as shown in fig. 0. omparisons again wih he linear and non-linear models are shown in figures and. Fig. Velociies a φ0 locaion φ Fig.0 Velociy sensors locaion The comparison shows how he non-lineariy in he models allows o beer describe he velociy disribuions in he flow. In paricular his is clear in he narrow gap region ( 0 degrees) while a he 5 degrees locaion all models give good predicions. A paricular discussion is relaed o he velociy disribuion a he 0 degrees locaion. As shown he non-linear models are able o correcly predic velociies in he near wall and bul region bu show a differen

8 behavior in beween hese wo poins. The experimenal disribuion is very fla in his zone in conras wih he calculaed values. This migh be a limiaion of he models bu some doubs exis abou he correcness of he experimenal daa.. Benchmar Regarding benchmar experimenal disribuion of he sodium emperaures is available along he line shown in fig.4. Fig. Temperaure measuremen locaion The sodium emperaure minus he flow averaged emperaure in he measuring plane is graphed agains he disance from he channel wall. Fig.4 Temperaure measuremen locaion The low-reynolds form of -ε - and he SST model produced he same predicions for he sodium emperaure disribuion and herefore only he -ε is presened. All linear models over predic in he same measure he emperaure values in he small gap region beween pins. Only he non linear models are capable of producing a closer agreemen wih he experimenal daa (resuls of he cubic low-reynolds model are no shown since hey presen no difference from he cubic one). The figure furher confirms how even a hi-reynolds form of he quadraic model wih he use of a law-of-he-wall can benefi from he anisoropic modeling. The various linear models show herefore a srong consisency bu a he same ime being unable o accoun for anisoropy canno produce a qualiaively correc predicion in his respec he non-linear modeling correcly capures he rend bu sill canno ge a close agreemen in he narrow gap region. VII. onclusion A large number of isohermal and hea ransfer simulaions have been carried ou o es he performances of numerous urbulence models. The resuls have clearly shown ha o correcly model flow in igh laice bundle i is necessary o adop an approach capable of describing anisoropic urbulence. omparisons show ha he secondary flow which arises from his anisoropy seems o be responsible of an homogenizaion of velociy and emperaure disribuion in he channel. In paricular experimenal daa show a srong flaening of he wall shear sress owards he channel cener which numerical predicions could no correcly reproduce. In all cases he non-linear models could herefore show improved predicions due o heir abiliy o reproduce in some exen he urbulence driven secondary moion bu improvemens o he models are needed for a beer agreemen. From he poin of view of compuaional requiremens all low-re forms of he models have shown o require very fine grids in he near wall region o obain a correcly converged soluion. This requiremen does no allow us a he momen o adop hese models for he simulaion of large geomeries involving hea ransfer as for insance a complee assembly. For he benchmar case for example he model of ¼ of he channel for a heigh of 50 hydraulic diameers required he use of almos million cells which showed o be an upper limi for our compuaions. High Reynolds number version of he non-linear model is sill he only available approach for large geomeries. In his regard we inend o es he performances of he VF approach in a near fuure being claimed o produce a correc modeling of he viscous sub-layer wih reduced compuaional requiremens. References ) W. D. Magwood DOE NERI Agenda: Moving Forward on Generaion IV Reacor Design Proc. PHYSOR 000 Pisburgh PA May 000 (000). ) M. D. arelli K. Miller. V. Lombardi N. Todreas E. Greenspan H. Ninoaa F. Lopez L. inoi J.ollado F.Oriolo IRIS: Proceeding Towards he Preliminary Design Proceedings of he 0h Inernaional onference on Nuclear Engineering Arlingon VA April (00). ) B. Perovic M. D. arelli E. Greenspam M. Milosevic J. Vuic E. Padovani F. Ganda Firs ore and Refueling Opions for IRIS Proceedings of he 0h Inernaional onference on Nuclear Engineering Arlingon VA April (00). 4) A. Romano V. Krivensev H. Ninoaa N.E. Todreas Novel Fuel Geomeries for he Generaion IV IRIS Reacor Proceedings of he 9h Inernaional onference on Nuclear Engineering Nice France April 8-00 (00). 5) A. Romano N. E. Todreas Analysis of Novel Fuel

9 Geomeries for he Genearion IV IRIS Reacor Deparmen of Nuclear Engineering Massachuses Insiue of Technology MIT-ANP-TR-07 (000). 6). J. heng J. Shenq-Yuh Fundamenals of Turbulence Modeling Taylor and Francis (998). 7). J. Freias Seleced Benchmars from ommercial FD odes Journal of Fluid Engineering 7 (995). 8) H. K. Myong Numerical Invesigaion of Fully Developed Turbulen Fluid Flow and Hea Transfer in a Square Duc In. J. Hea and Fluid Flow Vol. No. 4 (99). 9) H. K. Myong N. Kasagi Predicion of Anisoropy of he Near Wall Turbulence wih an Anisoropic Low-Reynolds-Number K-ε Turbulence Model Journal of Fluids Engineering Vol. pp 5-54 (990). 0) J. L. Lumley Toward a Turbulen onsiuive Relaion J. Fluid Mechanics 4- pp (970). ) S. Pope A More General Effecive-Viscosiy Hypohesis J. Fluid Mechanics 7 pp. -40 (975). ) T. J. raf B. E. Launder K. Suga Predicion of Turbulen Transiional Phenomena wih a Non-Linear Eddy-Viscosiy Model J. of Hea and Fluid Flow8 pp. 5-8 (997). ) K. Suga Recen Developmens in Eddy Viscosiy Modelling of Turbulence R&D Review of Toyoa RDL Vol. No. (998). 4) D.. Wilcox Turbulence Modelling for FD nd ediion DW Indusries Inc. (998). 5) F. R. Mener Zonal wo equaion - urbulence models for aerodynamic flows Proc. 4h Fluid Dynamics onf. Orlando Florida USA 6-9 July Paper No. AIAA (99). 6) V.. Pael W. Rodi G. Scheuerer Turbulence Models for Near-Wall and Low Reynolds Number Flows a review AIAA Journal 08 (985). 7) P. A. Durbin Separaed Flow ompuaions wih he -e-vf model AIAA Journal Vol. No. 4 (995). 8) B. E. Launder D.B. Spalding The numerical compuaion of urbulen flows omp. Meh. in Appl. Mech. and Eng. pp (974). 9) Benchmar Problem Tas 9h IAHR Woring Group on Advanced Nuclear Reacor Thermal Hydraulics Grenoble France (998). 0) E.Baglieo H. Ninoaa Selecion of an Appropriae Turbulence Model o Evaluae Performances of Novel Fuel Geomeries for The IRIS Reacor 4TH ASME_JSME Join Fluids Engineering onference Honolulu Hawaii USA FEDSM (00). ) R. Moller TEGENA: Deailed Experimenal Invesigaions of Temperaure and Velociy Disribuions in Rod Bundle Geomeries wih Turbulen Sodium Flow Kernforschungszenrum Karlsruhe (989). ) S.V. Paanar D.B. Spalding A alculaion Procedure For Hea Mass And Momenum In Three-Dimensional Parabolic Flows In. J. Hea Mass Transf 5 (97). ) STAR-D Version.0 Manual Mehodology ompuaional Dynamics Limied (999).