A SECOND ORDER TURBULENCE MODEL BASED ON A REYNOLDS STRESS APPROACH FOR TWO-PHASE BOILING FLOW AND APPLICATION TO FUEL ASSEMBLY ANALYSIS

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1 A SEOND ORDER TURBULENE MODEL BASED ON A REYNOLDS STRESS APPROAH FOR TWO-PHASE BOILING FLOW AND APPLIATION TO FUEL ASSEMBLY ANALYSIS S. Mimouni 1, F. Archambeau 1, M. Boucer 1, J. Laieie 1,. More stephane.mimouni@edf.fr, (1) Eectricité de France R&D Diision, 6 Quai Watier F hatou, France, () ommissariat à Energie Atomique, 17 rue des Martyrs, F-8000 Grenobe, France Abstract High-therma performance PWR (pressurized water reactor) spacer grids require both ow pressure oss and high critica heat fux (HF) properties. Numerica inestigations on the effect of anges and position of mixing anes and to understand in more detais the main physica phenomena (wa boiing, entrainment of bubbes in the waes, recondensation) are required. In the fied of fue assemby anaysis or design by means of FD codes, the oerwheming majority of the studies are carried out using two-equation Eddy Viscosity Modes (EVM), especiay the standard K ε mode, whie the use of Reynods Stress Transport Modes (RSTM) remain exceptiona. But extensie testing and appication oer the past three decades hae reeaed a number of shortcomings and deficiencies in Eddy Viscosity Modes. In fact, the K ε mode is totay bind to rotation effects and the swiring fows can be regarded as a specia case of fuid rotation. This aspect is crucia for the simuation of a hot channe in a fue assemby. In fact, the mixing anes of the spacer grids generate a swir in the cooant water, to enhance the heat transfer from the rods to the cooant in the hot channes and to imit boiing. First, we started to eauate computationa fuid dynamics resuts against the AGATE-mixing experiment: singe-phase iquid water tests, with Laser-Dopper iquid eocity measurements upstream and downstream of mixing bades. The comparison of computed and experimenta azimutha (circuar component in a horizonta pane) iquid eocity downstream of a mixing ane for the AGATE-mixing test shows that the rotating fow is quaitatiey we reproduced by NEPTUNE_FD but azimutha iquid eocity is underestimated with the K ε mode. Before comparing performance of EVM and RSTM modes, we see to appy the Best Practice Guideines 1 (BPG) to quantify the numerica errors. Due to the geometry and mesh size of cases featuring spacer grids, ony few sensitiity tests can be performed. Therefore, we appied some recommendations of the BPG to two cases with ery simiar conditions but with a simper geometry, the DEBORA-tube case and the ASU-annuar channe case. Then, a geometry coser to actua fue assembies is considered. It consists of a rectanguar test section in which a x rod bunde equipped with a simpe spacer grid with mixing anes is inserted. The infuence of the turbuence mode on target ariabes ined to HF imitation wi be discussed. Moreoer, the sensitiity to the mesh refinement wi be particuary examined. The study of this case is a further step towards the modeing of the two-phase boiing fow in rea-ife grids and rod bundes. 1 NOMENLATURE A i interfacia area concentration 1 Best Practice Guideines for the use of FD in Nucear Reactor Safety Appications, NEA/SNI/R(007)5 1

2 d t g K M p Pr R drag coefficient numerica time step graity acceeration iquid turbuent inetic energy interfacia momentum transfer per unit oume and unit time pressure iquid Prandt number Reynods stress tensor. Re b bubbe Reynods number t time u i fuctuation of the iquid eocity V aeraged eocity of phase interfacia-aeraged eocity V i α, ε denotes the time fraction of phase dissipation rate µ g gas moecuar iscosity ν T ν ρ, σ τ w Σ iquid inematic iscosity iquid turbuent eddy iscosity aeraged density of phase surface tension wa shear stress moecuar stress tensor Subscripts/Superscripts iquid state apour bubbes phase = or INTRODUTION In a Pressurized Water nucear Reactor, an optimum heat remoa from the surface of the nucear fue eements (rod bunde with spacer grids) is ery important for therma margins and safety. The geometry of the fue assemby spacer grids has a strong infuence on the safety and performance issues, in particuar the grid anes incination ange. In (Shin, 005), the ritica Heat Fux (HF) experiment on the effect of the ange and of the position of mixing anes was performed in a x rod bunde. The authors show that the mixing anes increase the aue of the HF and the resut is correated to the magnitude of the swir generated by the mixing anes. If the ange of the mixing anes is reatiey sma, the magnitude of the swiring fow is smaer because the rotating force created by the mixing anes is wea. If the ange of the mixing anes is reatiey arge, the mixing anes pay the roe of fow obstace under the Departure from Nuceate Boiing (DNB) condition. Therefore, it is important that the turbuence modeing dea with rotation effects. There hae been seera studies on fow mixing and heat transfer enhancement caused by a mixing-ane spacer grid in a rod bunde geometry. Lee et a. (Lee, 007) simuate the fow fied and heat transfer in a

3 singe phase fow for a 17x17 rod bunde with eight spans of mixing anes. The FLUENT commercia code is empoyed and a Reynods Stress Transport Mode (RSTM) is used for turbuence. According to the authors, RSTM is hepfu. Ieda et a. (Ieda, 006) study an assemby consisting of a 5x5 heater rod bunde and eight specific mixing ane grids. For Ieda et a., it might be insufficient to appy a standard K ε mode to swir-mixing fow and narrow-channe fow conditions that incude non-isotropic effects. Moreoer, In et a. (In, 008) hae performed a series of FD singe phase fow simuations to anayse the heat transfer enhancement in a fuy heated rod bunde with mixing-ane spacers. For future wor, In et a. recommend that a refined omputationa Fuid Dynamic (FD) mode be deeoped to incude detais of the grid structure and a higher-order turbuence mode be empoyed to improe the accuracy of such simuations. Additiona two-phase effects ie accumuation of bubbes in the centre of sub-channe or pocets of bubbes on the rods shoud be taen into account to improe the simuation of fow cose to DNB. Indeed, singe-phase simuations remain insufficient and boiing fows simuations are required. In (Krepper, 007), the authors describe FD approaches to subcooed boiing and inestigate their capabiity to contribute to fue assemby design. A arge part of their wor is dedicated to the modeing of boiing fows and to forces acting on the bubbes. The authors note that the size of bubbes in the bu is correated to the oca subcooing which is an important parameter (see (Shin, 005)). onsidering fow in a PWR core in conditions cose to nomina, when boiing occurs, a high eocity steady fow taes pace with ery sma times scaes associated to the passage of bubbes (10-4 s 10 - s) and with quite sma bubbe diameters (10-5 to 10 - m) compared to the hydrauic diameter (about 10 - m). According to the synthesis of the wor performed in WP. of the NURESIM project (Bestion, 007), these are perfect conditions to use a time aerage or ensembe aerage of equations as usuay done in the RANS approach. A turbuent fuctuations and two-phase intermittency scaes can be fitered since they are significanty smaer than the scaes of the mean fow. The Large Eddy Simuation is aso a possibe approach. In the context of the NURESIM project, seera studies hae been carried out with a Large Eddy Simuation to study the axia deeopment of air-water bubby fows in a pipe. But in the synthesis of the wor performed in WP., it is noted in (Bestion, 007) that seera open modeing and numerica issues sti remain. So, we wi focus on the RANS approach in this paper. According to a these recommendations, a better understanding of the detaied structure of a fow mixing and heat transfer downstream of a mixing-ane spacer in a nucear fue rod bunde has to be inestigated with a RSTM. Moreoer, the oerwheming majority of industria FD appications today are sti conducted with twoequation eddy iscosity mode, especiay the standard K ε mode, whie RSTM remain exceptiona. As an exampe of RSTM deeopment, a RSTM mode adapted to bubby fows is studied in (hahed, 1999) and used to perform simuations of three basic bubby fows (grid, uniform shear and bubby wae). The authors decomposed the Reynods stress tensor of the iquid into two independent parts: a turbuent part produced by the mean eocity gradient that aso contains the turbuence of the bubbe waes and a pseudo-turbuent part induced by bubbe dispacements; each part is predicted from a transport equation. This mode is interesting but has not been seected here for the foowing reasons. Firsty, the computation effort is doubed (turbuent and pseudo-turbuent parts). Secondy, considering the fow cose to nomina PWR core conditions, when boiing occurs, a high eocity steady fow taes pace and the bubbe diameter is quite sma (10-5 to 10 - m): therefore, the bubbes foow the iquid streamines and so the modeing of the pseudo-turbuent part induced by bubbe dispacements can be omitted. Thirdy, the two-phase fow modeing proposed in (hahed, 1999) does not tend to a singephase fow formuation when the oid fraction tends to zero. These three arguments hae imposed the choice of the higher-order turbuence mode described in the paper.

4 A second-order moment turbuence mode for simuating a bubbe coumn is aso proposed in (Zhou, 00). The authors defined a Reynods tensor for each phase. Furthermore, foowing a simiar method used in deriing and cosing the Reynods stress equations, a modeed transport equation of two-phase fuctuating eocity correations is aso soed. For the same reasons as those mentioned aboe for the mode proposed in (hahed, 1999), we hae not adopted the mode proposed in (Zhou, 00). The turbuence modeing described in the present paper taes into account the Reynods tensor for the iquid ony, whie a more basic modeing is used for the apour phase. Howeer, to the authors nowedge, no industria FD approach for boiing fows with a RSTM approach is aaiabe in the context of the fue assemby design. This may be due to the fact that numerica probems may occur when using a RSTM approach without caution. Furthermore, the turbuence modeing of boiing fows is not straightforward. The use of RSTM aso requires finer meshes than eddy iscosity modes (EVM) and RSTM may therefore be more time and storage consuming. Deeoping an industria RSTM approach is a quite chaenging tas but it is worth woring at it: RSTM and EVM resuts wi probaby differ and it is important to determine what consequences it may hae on therma margin and safety of reactors. In the framewor of a R&D programme carried out in the NEPTUNE project (EDF, EA, AREVA-NP, IRSN), the foowing strategy has been adopted: 1. aidation of the NEPTUNE_FD code with a RSTM approach on singe phase fow with mixing anes and on more academic cases of air-water adiabatic bubby fows in a pipe;. aidation of the NEPTUNE_FD code with a RSTM approach on boiing fows in a pipe and sensitiity to the ange of the anes for fue assemby spacer grids performed in a x rod bunde in a boiing fow configuration;. aidation of the NEPTUNE_FD code with a RSTM approach for a 5x5 rod bunde with mixing anes currenty used for commercia nucear fue. The step 1 is described in (Mimouni, 008b). The second step is deeoped in the present paper. The step is not yet started. Our main objectie in this paper is to chec that the simuation with the RSTM gies satisfactory resuts in a simpe geometry as compared to a EVM: this point is crucia before cacuating rod bunde geometries where the EVM mode may fai. This paper is organized as foows. In section the genera mode we use for two-phase boiing fow simuations is presented in detai. In section 4, we underine the weanesses of the EVM modes. In section 6, the second-moment cosure mode for high Reynods number two-phase fows is presented. In sections 7 and 8 respectiey, the DEBORA case and the ASU case are briefy described. The comparison of the resuts of NEPTUNE_FD cacuations and the experimenta data are presented. The sensitiity of the numerica resuts to the turbuence mode for the fuid and to the most important modes is studied. The section 9 is dedicated to cacuations of geometry cose to actua fue assembies in PWR conditions. Finay, concusions are drawn about our current capabiities to simuate boiing fows with a RSTM mode and perspecties for future wor are gien. THE NEPTUNE_FD SOLVER AND PHYSIAL MODELLING.1 Introduction NEPTUNE_FD is a three dimensiona two-fuid code deeoped more especiay for nucear reactor appications. This oca three-dimensiona modue is based on the cassica two-fuid one pressure approach, incuding mass, momentum and energy baances for each phase. The NEPTUNE_FD soer, based on a pressure correction approach, is abe to simuate muticomponent mutiphase fows by soing a set of three baance equations for each fied (fuid component 4

5 and/or phase) (Guefi, 007), (Mimouni, 006, 008, 008b). These fieds can represent many inds of mutiphase fows: distinct physica components (e.g. gas, iquid and soid partices); thermodynamic phases of the same component (e.g.: iquid water and its apour); distinct physica components, some of which spit into different groups (e.g.: water and seera groups of different diameter bubbes); different forms of the same physica components (e.g.: a continuous iquid fied, a dispersed iquid fied, a continuous apour fied, a dispersed apour fied). The soer is based on a finite oume discretization, together with a coocated arrangement for a ariabes. The data structure is totay face-based, which aows the use of arbitrary shaped ces (tetraedra, hexahedra, prisms, pyramids...) incuding non conforming meshes (meshes with hanging nodes).. Goerning equations and physica modeing The FD modue of the NEPTUNE software patform is based on the two-fuid approach (Ishii, 1975), (Dehaye, 1981). In this approach, a set of oca baance equations for mass, momentum and energy is written for each phase. These baance equations are obtained by ensembe aeraging of the oca instantaneous baance equations written for the two phases. When the aeraging operation is performed, the major part of the information about the interfacia configuration and the microphysics goerning the different types of exchanges is ost. As a consequence, a number of cosure reations (aso caed constitutie reations) must be suppied for the tota number of equations (the baance equations and the cosure reations) to be equa to the number of unnown fieds. We can distinguish three different types of cosure reations: those which express the inter-phase exchanges (interfacia transfer terms), those which express the intra-phase exchanges (moecuar and turbuent transfer terms) and those which express the interactions between each phase and the was (wa transfer terms). The baance equations of the twofuid mode we use for two-phase boiing fows and their cosure reations are described in the foowing subsections...1 Main set of baance equations The two-fuid mode we use for our two-phase boiing fow cacuations consists of the foowing baance equations. Two mass baance equations: α ρ. ( α ρ V ) = Γ =, (1) t where t is the time, α, ρ, V denote the time fraction of phase, its aeraged density and eocity. The phase index taes the aues for the iquid phase and for apour bubbes. Two momentum baance equations: ( α ρ V V ) = α p M α ρ g. [ ( Σ R )], αρv. α =, () t where p is the pressure, g is the graity acceeration, M is the interfacia momentum transfer per unit oume and unit time, and Σ and R denote the moecuar and turbuent stress tensors, the atter being aso caed the Reynods stress tensor. The wa friction terms for the two phases do not appear in the momentum baance equations because soid was are ony present at the boundaries of the fow domain and the wa friction is expressed through the wa boundary conditions. Two tota enthapy baance equations: α ρ h t Π ' V A q. i w. α ρ h V T [ α ( q q )] =, V p = α α ρ g. V t Γ h i V where h is the phase-aeraged enthapy for phase and h i is the interfacia-aeraged enthapy. We hae assumed that the two phases are goerned by the same aeraged pressure fied p and we mae no () 5

6 distinction between the pressures in the two phases or between the bu pressure and the interface ' pressure for simpicity. The three terms Γ, M and Π denote the interfacia transfer terms of mass, momentum and heat, the quantity A i being the interfacia area concentration. The terms q w denote the T wa-to-fuid heat transfer per unit oume and unit time for each phase. The two terms q and q denote the moecuar and turbuent heat fuxes inside phase. The interfacia transfer of momentum M appearing in the RHS of Eq. () is assumed to be the sum of four forces: D AM L TD M = M M M M (4) The four terms are the aeraged drag, added mass, ift and turbuent dispersion forces per unit oume. Now we wi gie the expressions we use for these forces and for their coefficients. Drag force: D D 1 M = M = Ai ρ 8 D V V ( V V ) A i, (5) where D is the drag coefficient for bubbes and can be determined experimentay. Added-mass force: AM AM 1 α V V M = M = A αρ V. V V. V 1 α t t, (6) where A is the added mass coefficient which is equa to ½ for a spherica bubbe and the factor (1α)/(1-α) taes into account the effect of the bubbes concentration (Zuber, 1964), (Ishii, 1990). Lift force: L L M = M = α ρ V V V, (7) L ( ) ( ) where L is the ift coefficient. This coefficient is equa to ½ in the particuar case of a weay rotationa fow around a spherica bubbe in the imit of infinite Reynods number (Auton, 1987). Turbuent dispersion force: TD TD M = M = ρ K α, (8) TD where K is the iquid turbuent inetic energy and TD is a numerica constant of order 1. This expression was proposed by Lance (Lance, 1994). An aternatie approach is proposed by (Deutsch, 1991) to mode the turbuence induced by bubbes: an agebraic mode deeoped in the framewor of Tchen s theory where the turbuent inetic energy for the dispersed phase and the coariance are cacuated from the turbuent inetic energy of the continuous phase. For the dispersed phase, the Reynods stress tensor is cosed using a Boussinesq-ie hypothesis: R T T T = ρ ν ( V V ) I( ρk ρν. V ) (9) where I is the identity tensor, 6

7 with: F F T 1 t Aτ τ 1 F ν = q τ Kτ, the turbuent iscosity for the dispersed phase, ρ A ρ K q b = b η r = K, the gas turbuent inetic energy, 1 ηr b η r = K, the coariance of the dispersed phase, 1 ηr 1 ρ ρ t F A A τ η r =, the ratio between the time scae of the continuous phase turbuence seen by the τ dispersed phase (that taes into account crossing trajectories effect) and the characteristic time scae of the momentum transfer rate between the iquid and dispersed phases: t t τ τ = σ τ F α ( 1 ξ ) A ρ = F D β r 1 withσ is the turbuent Schmidt or Prandt turbuent for the α continuous phase, taen equa to 1.8, β is the crossing trajectories coefficient A is the added mass coefficient, Vr with ξ r = and 1 K t τ = µ. ε It can be shown that the turbuent dispersion force is a particuar case of the Tchen s theory where TD is a function of the physica quantities defined aboe and does not need any parameters. For the sae of simpicity, the turbuent dispersion force is used in the DEBORA tests and ASU tests, but the Tchen s theory is used in the industria geometry studied in the paper and in (Mimouni, 008, 008b)... Turbuent transfer terms The K ε mode describes energy processes in terms of production and dissipation, as we as transport through the mean fow or by turbuent diffusion. The Komogoro spectra equiibrium hypothesis aso enabes one to predict a arge eddy ength-scae. On the other hand, the anisotropy of the stresses is quite crudey modeed. First of a the EVM mode assumes the Reynods stress tensor is aigned with the strain rate tensor (Boussinesq approximation): T T T R = ρ ν ( V V ) I( ρ K ρν. V ), (10) T where I is the identity tensor, K is the iquid turbuent inetic energy and ν is the iquid turbuent eddy iscosity. The iquid turbuent eddy iscosity is expressed by the foowing reation: T ν µ K =, (11) ε 7

8 where µ = The turbuent inetic energy K and its dissipation rate ε are cacuated by using the two-equations K ε approach... Interfacia transfer terms If the mechanica terms are negected in comparison to the therma terms in the aeraged form of the energy jump condition, this condition reduces to: ( Γ q A ) 0 h. (1) i i i This important reation (together with the mass jump condition Γ = Γ ) aows to compute the mass transfer terms as functions of the interfacia heat transfer terms q A i i and the interfacia-aeraged enthapies h i : q i q i Γ = Γ = Ai. (1) h h i i We hae no information about the dependence of the interfacia- aeraged enthapies h i. Therefore, two basic assumptions can be made: (1) the interfacia-aeraged enthapies h i are identified to the phaseaeraged ones h or () the interfacia-aeraged enthapies h i are gien by the saturation enthapies. Here we hae made the assumption (1). Each interfacia heat transfer term q A i i is the product of the interfacia heat fux density: i i ( T p T ) sat q = ( ), (14) where i, T and T sat (p) denote a heat transfer coefficient, the aeraged temperature of phase and the saturation temperature. The interfacia area concentration is expressed as A i = 6α / d, where α is the oid fraction and d is the mean bubbe diameter. The foowing heat transfer coefficient is used: i Nu 1 V V d ν = Nu = 0.6 Re Pr 1 Re b = ˆ Pr = ˆ, (15) d ν a where Re b is the bubbe Reynods number and Pr is the iquid Prandt number, ν being the iquid inematic iscosity. The heat transfer coefficient between the apour and the interface for the case of bubbes is written as: i A αρ p i =, (16) tc where p is the gas heat capacity at constant pressure and t c is a characteristic time gien by the users. This reation simpy ensures that the apour temperature T remains ery cose to the saturation temperature T sat, which is the expected resut for bubby fows with sufficienty sma bubbes (fow in a PWR core in conditions cose to nomina)...4 Wa transfer mode for nuceate boiing In a first simpified approach, and foowing the anaysis of Kuru (Kuru, 1990), the boiing heat fux is spit into three terms: a singe phase fow conectie heat fux q c at the fraction of the wa area unaffected by the presence of bubbes, a quenching heat fux q q where bubbes departure bring cod water in contact with the wa periodicay, a aporisation heat fux q e needed to generate the apour phase. 8

9 Each of these three phenomena is expressed by a heat fux density (per unit surface of the heated wa) which is reated to the oumetric heat fux by the foowing reation: ( q q q ) Aw Aw q w = q w = c q e, (17) V V where A w is the heated wa surface in contact to the ce haing oume V, therefore q w is expressed in W/m and q w as we as q c, q q and q e are expressed in W/m. The quantities q c, q q and q e denote the heat fux densities due to iquid conectie heat transfer, quenching and eaporation respectiey. The iquid conectie heat transfer per unit surface of the heated wa is written as: q = A hog T T, (18) c c ( ) w where T w is the wa temperature and h og is a heat exchange coefficient which is gien by: * u hog = ρ p, (19) T where u * is the wa friction eocity and T is the non-dimensiona iquid temperature. The eocity u * is cacuated from the ogarithmic aw of the wa written for the iquid eocity in the wa boundary ayer. The non-dimensiona temperature foows a simiar ogarithmic profie. The heat fux density due to quenching is written as: λ ( Tw T ) qq = Abt q f, (0) πa t q where A b is the wa fraction occupied by bubbe nuceation, f is the bubbe detachment frequency, t q is the quenching time and a is the iquid therma diffusiity. The two fractions A c and A b are gien by: A b = min( 1,nπd d / 4), (1) Ac = 1 A b where n is the actie nuceation sites density (per unit surface of the heated wa) and d d is the bubbe detachment diameter. The actie nuceation sites density is modeed according to (Kuru, 1990): n = [ 10( T )] 1. 8 w Tsat, () as a function of the wa superheating. The bubbe detachment diameter is gien by the correation from Una (Una, 1977). The Una s correation is aid for subcooed iquid but has been extended to saturated iquid. The bubbe detachment diameter is gien by: a d d =.410 p, () bϕ where p is the pressure and a, b and ϕ are gien by the foowing reations: ( Tw Tsat ) λs a =, (4) ρ πa s where λ s and a s denote the wa conductiity and therma diffusiity, ρ denotes the apour density and is the atent heat of aporisation. In the modified correation, b is gien by: ( Tsat T ) St < ( 1 ρ / ρ ) b =, (5) 1 qc qq qe St > ( 1 ρ / ρ ) ρ p V where V is the norm of the iquid eocity and St is the Stanton number which is defined by: qc qq qe St = ˆ, (6) ρ V p ( T T ) sat 9

10 and the quantity ϕ appearing in Eq. () is gien by : V 0.47 max ϕ = 1, V 0.61 m / 0 = s V, (7) 0 The quenching time and the bubbe detachment frequency are modeed as: t q = f f = 4 g ρ ρ ρ d 1 /, (8) The third heat fux density q e used for eaporation is gien by: q e d πdd = f ρn. (9) 6..5 Wa function mode for boiing fow In order to tae into account the infuence of bubbes in the near wa area, a modified ogarithmic aw of the wa was suggested by Ramstorfer et a. (005) and Koncar et a. (008), which is usuay used for turbuent fows oer rough was and reads 1 u = n( y ) B u (0) κ where u = u t / u w, y = ρ u w y / µ and u w = τ w / ρ (τ w is the wa shear stress). Here t u is the nown eocity tangentia to the wa and y is the distance from the wa. oefficients κ and B are standard singe-phase constants with the aues of 0.41 and 5., respectiey. The ast term represents the offset of u due to the wa roughness u 0; = 1 n(1 κ r r ); r 11. r > 11. ( r =0.5 for sand-grain roughness) and r where r is the so-caed roughness Reynods number: ρ r u wu * =, () µ 1/ 4 1/ u * cµ = and = αd. We hae aso tested r r ρ ru = µ w but resuts are not changed (figure 6). 4 EVM WEAKNESSES: THEORETIAL APPROAH. Fows encountered in ertica pipe are of great interest to aidate the most important heat, mass and momentum cosure reations. Howeer, some negigibe effects in simpe geometry sometimes become preponderant in compex geometries. For exampe, the modeing of two-phase fow in water cooed nucear reactors needs to tae into account swirs and stagnation points. Appications in compex geometries aso need to tae into account the compex features of the secondary motions which are obsered experimentay. These requirements highight the need for meticuous turbuence modeing. A reason for the persistent widespread use of ow ee turbuence modeing in two-phase FD is perhaps the fact that the use of two-phase FD in compex industria geometries is ony starting. Moreoer, many studies merey require order of magnitude or good tendencies answers. Howeer, extensie testing and appication oer the past three decades hae reeaed a number of (1) 10

11 shortcomings and deficiencies in EVM modes, and among them the K ε mode, such as: Limitation to inear agebraic stress-strain reationship (poor performances whereer the stress transport is important, e.g.: non equiibrium, fast eoing, separating and buoyant fows); Insensitiity to the orientation of turbuence structure and stress anisotropy (poor performances where norma stresses pay an important roe, e.g.: stress-drien secondary fows in noncircuar ducts); Inabiity to account for extra strain (streamine curature, sewing, rotation); Poor prediction particuary of fows with strong aderse pressure gradients and in reattachment regions. In a pane strain situation, such as upstream of a stagnation point on a buff body, the exact (as obtained by a RTSM) production and that obtained from an EVM are respectiey (Hanjaic, 00): ' ' V, x V, x Pexact = ( ux u y ) and PEVM = 4ν, t ( ). x x The difference between the norma stresses actuay grows sowy on the short time scae needed for the fow to trae around the stagnation point, so the production remains moderate, and in any case is bounded, whereas P EVM usuay yieds a seere oer-prediction when the strain is high. The simuation of swiring fow generated by the mixing anes is our main goa since it pays an important roe for the prediction of the HF for the fue assembies. For this reason, the rotation effects are more specificay addressed hereafter. It can be easiy shown (hassaing, 000) that in the presence of an initiay anisotropic turbuence, rotation wi cause a redistribution of energy between norma components without affecting the aue of this quantity. In fact, the anguar eocity Θ does not appear expicity in the K-equation, obtained by adding the norma stresses: dk = ε dt Thus, the K ε mode is totay bind to rotation effects. The swiring fows can be regarded as a specia case of fuid rotation with the axis usuay aigned with the mean fow direction so that the oriois force is zero. This aspect is crucia for the simuation of hot channe of a fue assemby. In fact, the mixing anes of the spacer grids generate a swir in the cooant water, to enhance the heat transfer from the rods to the cooants in the hot channes and to imit boiing. In the foowing section, we present some exampes of arge-scae industria appications, performed using eddy-iscosity modes, and subsequenty discuss areas of weaness of the modes, highighting some improements that can be obtained through the use of more adanced stress transport cosures. 5 ILLUSTRATION ON THE SINGLE-PHASE TEST AGATE-MIXING EXPERIMENT Keeping in mind the ong-term objectie (two-phase FD cacuations aidated under typica Pressurized Water Reactor (PWR) geometries and therma-hydrauic conditions), we started ery recenty to eauate NEPTUNE_FD against spacer grid type experiments. An experimenta deice representing three mixing bades (figure ) was introduced in a heated tube (diameter = 19. mm) and used for two different programmes: AGATE-mixing experiment (Fa, 00): singe-phase iquid water tests, with Laser-Dopper iquid eocity measurements upstream and downstream of mixing bades (for each of the 15 horizonta panes, the iquid eocity is measured aong 1 different diameters and there are 1 points for each radius); the eocity at inet is m/s and the pressure is bar. 11

12 DEBORA-mixing experiment (Fa, 00): boiing R1 refrigerant tests, on the same geometry but the tota ength of the cacuation domain is.5 m; the tube is heated and the uniform wa heat fux is W/m which generates about % of apour at outet; the outet pressure is 6. bar; the inet iquid temperature is 6. ; the inet iquid mass fow is 0.87 g/s. The main physica phenomena to reproduce are: wa boiing, entrainment of bubbes in the waes, and recondensation (Mimouni, 008). So the prediction of the swirs is crucia. For the mixing bades part (60 mm) ces are needed. The figure 1 compares computed and experimenta azimutha iquid eocity downstream of the mixing ane (AGATE test). One can notice the rotating fow is quaitatiey we reproduced by NEPTUNE_FD athough the eocity is underestimated. This is mainy due to the turbuence mode (standard K ε here) which is not optimum for this type of geometry. figure 1: Azimutha iquid eocity downstream of the mixing ane (Agate-mixing experiment). figure : View of the mixing deice The K ε mode underestimates azimutha eocities downstream of the bades but the resuts remain quaitatiey satisfactory. The R ε mode gies satisfactory resuts (figure 1). In the foowing section, we propose a second-moment cosure mode to tae into account the iquid turbuence in order to aidate, in the ong-term, cacuations in typica Pressurized Water Reactor (PWR) geometries and therma-hydrauic conditions. In the present paper, we suppose that RSTM is we-nown in singe phase fow (Hanjaic, 00). Now, our objectie is to test and if possibe to improe our RSTM mode adapted to bubby fows as compared to experimenta data and K ε resuts. Indeed, we are interested by two-phase high Reynods numbers fows, but beforehand, the mechanica modes impemented in the NEPTUNE_FD code must be aidated against boiing fows in anaytica geometry. 6 THE SEOND-MOMENT LOSURE MODEL FOR HIGH REYNOLDS NUMBER FLOWS DEDIATED TO THE ONTINUOUS PHASE (LIQUID). In this section we omit the subscript for the iquid and α is the oid fraction for the sae of simpicity. 1

13 6.1 Equation on R DρR K ( 1 α ) = ρν ρs R ((1 α) R ) (1 α)( P G Φ ε ) () Dt x ε x In this mode, the Reynods stress tensor of the continuous phase is spit into two parts: a turbuent dissipatie part produced by the gradient of mean eocity and by the waes of the bubbes and a pseudoturbuent non-dissipatie part induced by the dispacements of the bubbes. The dispacements of the bubbes shoud be taen into account in experiments where air is injected at the bottom of a water poo creating a arge, axi-symmetric bubbe pume with a arge-scae recircuation fow around the pume. But swiring fows and high Reynods number characterize our industria appications. Hence, we negected, in our approach and as a first anaysis, the non-dissipatie component caed "pseudo-turbuent". We consider ony the "turbuent" dissipatie part. Within this framewor, the term of production by the bubbes interfaces is written as (hahed, 1999) p ( u n p u n ) ( u u ) n ' ' I ' ' I i j j i δ ν i j δ (4) ρ ρ x where n indicates the norma to the interface and δ Ι a Dirac function on the interface. It was omitted in (hahed, 1999). Indeed, according to (hahed, 1999) dissipation in the waes is baanced by the interfacia production: the equation of transport of the Reynods stress tensor has the same form as in the singe-phase case and is gien by (10). When the oid fraction is anishing, the two-phase fow modeing naturay degenerates to the singe-phase fow modeing. Some terms of the equation of transport of the Reynods stress tensor cannot be computed directy and must be modeed. A modeing resuting from (Hanjaic, 00) is proposed beow. A common way to mode the iscous destruction of stresses for high Reynods number fows is: ε = εδ (5) The turbuent diffusion is of diffusie nature and the most popuar mode is the generaized gradient t diffusion: K uiu j D = s uu (6) x ε x Pressure fuctuations tend to disrupt the turbuent structures and to redistribute the energy to mae ω ω turbuence more isotropic: Φ = Φ Φ Φ Φ Φ, with:, 1,,,1, Φ u iu j =, 1 1ε δ with 1 = 1. 8 (7) K Φ Φ,, = = P G Pδ Gδ with 1 P = P 1 G = G and = 0.6 = 0.55 (8) ε Φ u u n n δ u u n n u u, = n n f (9) K ω ω 1 1 m m i j j i ω 1

14 ω ω Φ, = Φ m, n nmδ Φ i, n n j Φ j, n ni fω ω ω with: 1 = 0. 5 = 0. norma to the wa. The terms x n (40) 0.4. K fω = where x n is the distance to the wa and n the base ector ε. Φ Φ are impemented in the code but are not used in the cacuations ω ω, 1,, because they don t improe the resuts. G is the production by body force. 6. Equation on ε In the RTSM cosures the same basic form of mode equation for ε is used as in the that now ( u u ) is aaiabe, which has the foowing impications: K ε mode, except The production of inetic energy (P and G) in the source term of ε are treated in exact form; The generaized gradient hypothesis is used to mode turbuent diffusion. Hence, the mode equation for ε has the form: Dε K (1 α) ε V (1 α) ε 1 α) = ε uu ε P ε G ε K ε ε 1 4 Dt x ε x x (41) K ( The coefficients of the R ε mode are: s 1 ω 1 ω ε ε 1 ε ε ε 4 Tabe 1: coefficients of the R ε mode 7 DEBORA ASE 7.1 Brief description of the DEBORA experiment The tests were performed on the DEBORA oop at EA-Grenobe with R1 refrigerant as a cooing fuid (Manon, 000). Goba fow data are gien in Tabe. The cacuation domain is a ertica cyindrica tube of 19. mm interna diameter and.485 m heated ength. At the measuring station (end of the heated ength), we compare numerica resuts against experimenta data for the axia apour eocity, the oid fraction and the iquid temperature. P s (outet pressure) (MPa) fowrate G (g/m²s) Heat Fux (W/m ) T e (iquid temperature at inet) ( ) X s (outet quaity) Tabe : goba fow data of the DEBORA test 9GP6W16Te68_1 14

15 The bubbe diameter is equa to 0. mm in accordance with experimenta obserations. 7. Simuations of the DEBORA experiment The fow is assumed to be axisymmetric therefore a two-dimensiona axisymmetric meshing is used. omputations hae been performed on three inds of meshing with the R ε turbuence mode and the wa function mode dedicated to boiing fows and proposed in the paper: a coarse grid (0 ces in the radia direction and 100 ces in the axia direction), a medium grid (0 ces in the radia direction and 00 ces in the axia direction) and a fine grid (0 ces in the radia direction and 00 ces in the axia direction). Resuts are simiar (figure ), hence the subsequent cacuations were performed on the first grid. A difference on the iquid temperature is obsered, but its magnitude is sma. The oid fraction profie iustrates that apour bubbes are nuceated onto the heated wa surface and condense in the subcooed iquid in the core of the fow. The iquid temperature profie is the combination of seera phenomena: the iquid near the wa is heated by the singe phase fow conectie heat fux q c pus the quenching heat fux q q ; bubbes condense in the subcooed iquid in the core of the fow and heat the iquid; the moecuar and turbuent heat fuxes inside the iquid phase modify the temperature profie. The iquid turbuent heat fux depends on the aues of K and ε and therefore depends on the turbuence modeing: on the figure 4, the R ε mode gies better resuts than the K ε mode which means that K ε mode oer-estimates the iquid turbuent heat fux. 15

16 figure : Sensitiity to the mesh refinement figure 4: Sensitiity to the ift force and to the wa function (WF) mode. On the figure 4, we study the effect of the ift force on the oid fraction, iquid temperature and axia apour eocity cacuated with the K ε turbuence mode for the iquid phase. acuations with ift force ( K-eps WF LIFT ) and without ift force ( K-eps WF ) gie simiar resuts. Great aues of the iquid mass fow rate and bubbes diameter of order of 0. mm mae the bubbes foow the iquid stream ines. The reatie eocity is wea and so, the ift force is sma. As a consequence, the ift force is negected, which is a desirabe simpification since it may cause numerica instabiities when used in conjunction with the R ε turbuence mode. We ony consider the drag, added-mass and turbuent dispersion force in the foowing computations. Foowing Koncar et a (008), wa function mode hae been impemented to tae into account the modification of the singe phase ogarithmic wa aw for boiing cases. When using the K ε turbuence mode, the oid fraction is oerestimated in the core with the wa function mode (WF). Without this mode, the resuts of the standard wa function (standard) mode are in quite good agreement with the experimenta data. omputations with the R ε mode used in conjunction with the WF mode gie gobay a quite good agreement with the experimenta data een if they sighty oerestimate the oid fraction near the wa. 8 ASU ASE 8.1 Brief description of the ASU experiment The Arizona State Uniersity (ASU) experiment is described in (Hasan, 1990) and (Roy, 199) for twophase boiing fow measurements. The test section of the ASU experiment consists of a ertica annuar channe with a heated inner wa and an insuated outer wa. The inner tube is made of 04 stainess stee (i.d. = 14.6 mm, o.d. = 15.9 mm) and the outer pipe of transparent pyrex gass (inet diameter = 8.1 mm, outet diameter = 47 mm) except for a m ong measurement section which is made of quartz (i.d. = 7.7 mm, o.d. = 41.7 mm). The inner tube is resistiey heated, the upper.75 m of the.66 m ong test section being the heated ength. The ower 0.91 m seres as the hydrodynamic entrance ength. The woring fuid is refrigerant 11 (R11). When the inner wa heating is sufficienty high, heterogeneous nuceation appears onto this wa surface. As the inet iquid is subcooed, the bubbes condense into the coder iquid when they are far from the inner wa ayer. According to Roy et a. (Roy, 199), a two ayer is obsered: a boiing bubby ayer adjacent to the heated wa and an outer a iquid region. A dua-sensor optica fibre probe and a 16

17 microthermocoupe were instaed diametricay opposite each other in the measurement section. The measurement pane was approximatey ocated at 1.94 m downstream of the beginning of the heated ength. The dua-sensor optica fibre probe was used to measure the radia profies of the different quantities characterizing the dispersed phase (oid fraction, bubbe axia eocity and bubbe chord ength). The bubbe chord ength distribution aows to compute the bubbe diameter distribution, hence the oca interfacia area concentration, by maing seera assumptions (see (Roy,199) for more expanations). Howeer, these authors concude that further wor is needed before the oca interfacia area concentration can be determined with confidence. The bubbe diameter is equa to 1. mm in the cacuations in accordance with experimenta obserations. We hae the radia profies of the oid fraction, the mean axia iquid eocity, the mean iquid temperature and the radia fuctuations of the iquid eocity for two different experimenta cases we name TP5 and TP6. The controing parameters for these cases are indicated in tabe beow. ase Pressure, bar Wa heat fux, W/m Inet mass fow rate, g/m s Inet temperature, tp tp Tabe : Experimenta conditions retained for our simuations. 8. Simuations of the ASU experiment For the experimenters, the adantage of using an annuar geometry instead of a pipe is that the fow fied can be studied by intrusie probes (e.g. optica fibre probes, microthermocoupes) or non intrusie probes (e.g. Laser Dopper Veocimeter) without haing to disrupt the heated wa (Roy, 199). For us, the adantage of this particuar geometry is to test the aidity of our modes for a boiing fow around a conex heated wa instead of inside a concae one as it is the case for a simpe pipe. This change of wa conexity can hae some important effects on the fow eoution. In a genera manner, it is adantageous to test the modes we hae retained for boiing fows on different experiments characterized by different woring fuids (R11 for ASU / R1 for DEBORA), different wa heatings, different inet iquid fow rates and subcooings The ength of the ASU cacuation domain is equa to.0 m and the ast.9 m ength are heated. The fow is assumed to be axisymmetric therefore a two-dimensiona axisymmetric cacuation grid has been used. omputations hae been performed on three inds of meshing with the RTSM turbuence mode and the wa function mode: a coarse grid (0 ces in the radia direction and 100 ces in the axia direction), a medium grid (0 ces in the radia direction and 100 ces in the axia direction) and a fine grid (0 ces in the radia direction and 00 ces in the axia direction). Resuts are simiar. Hence, the subsequent computations were performed on the first grid (figure 5). Foowing Koncar (Koncar, 008) and the concusions of the preious test case (namey the DEBORA test), the forces acting on the bubbes that were retained are the drag, added-mass and turbuent dispersion forces. 17

18 figure 5: Sensitiity to the mesh refinement (RSTMWF). The iquid eocity profie cacuated with the K ε mode is improed when using the WF mode. We thin that the discrepancy obsered on the iquid mean axia eocity may be due to a bad estimation of the iquid axia eocity near the heated wa. This woud not be surprising since we use the cassica singe-phase ogarithmic eocity profie to express the boundary condition on the iquid eocity (wa friction aw). Void fraction and iquid temperature profies are simiar with or without the WF mode. Moreoer, a the profies computed the RSTM mode are practicay unchanged with the WF mode (figure 6). So, the WF mode is adopted by defaut in the subsequent computations ( K ε and R ε ). 18

19 figure 6: Sensitiity to the wa function (WF) mode. We compare the RSTM and K ε modes for the oid fraction (figure 7), the iquid temperature (figure 8), the iquid axia eocity (figure 9), the iquid turbuent inetic energy (figure 10) and the radia fuctuations of the iquid eocity (figure 11). The iquid temperature profies are in quite good agreement with the experimenta data with both turbuence modes. The oca oid fraction at a gien point is the resut of the competition between seera physica phenomena ie bubbe nuceation and condensation, bubbe atera migration, bubbe reatie axia eocity Liquid temperature and axia iquid eocity profies are simiar with both turbuence modes but the oid fraction profie is particuary we predicted with the RSTM mode. In fact, radia fuctuations of the iquid eocity gie satisfactory resuts near the wa where most of the bubbes are concentrated. For the K ε mode, radia and axia eocity fuctuations are the same. Particuary near the wa, we obsere a quite good agreement between the cacuations using the RSTM mode and the experimenta data (see the oid fraction for the TP5 and TP6 tests on the figure 7), which is crucia for the prediction of the boiing crisis in PWR conditions. 19

20 figure 7: Void fraction with the WF mode. figure 8: Liquid temperature with the WF mode. The iquid turbuent inetic energy are sighty more correcty predicted near the wa with the RSTM mode but cacuations using the K ε mode gie sighty more satisfactory resuts in the core of the fow (figure 10). The radia fuctuations of the iquid eocity gie better resuts with the RSTM mode (figure 11). figure 9: Liquid eocity with the WF mode. figure 10: Liquid turbuent inetic energy (WF). 0

21 figure 11: Radia fuctuations of the iquid eocity (WF). Een in this anaytica geometry, where the turbuent intensities cacuated by both turbuence modes are reatiey wea, we obsere a difference on ariabes ie the oid fraction or the iquid eocity. 9 ALULATIONS OF A X ROD BUNDLE A geometry coser to actua fue assembies is considered (Shin, 00). It consists of a rectanguar test section in which a x rod bunde equipped with a simpe spacer grid with mixing anes (figure 1- figure 1-figure 15). The geometry of the mixing anes is different from (Shin, 00) and so we don t hae experimenta data but the cacuations performed in this section wi aow estimating our current capabiities to simuate boiing fows with a RSTM with a geometry coser to actua fue assembies. The therma-hydrauics conditions are representatie of PWR core configurations cose to nomina. At inet, the iquid temperature is 60K and the axia eocity is 5m/s. The outet pressure is 155 bar. An adiabatic singe-phase iquid eocity profie estabishes in the 0. m inet section. Aboe this imit, a uniform heat fux is imposed aong the rods up to 0.9 m upstream of the mixing anes. This heat fux is equa to 1.6MW so that the oid fraction reaches a maximum aue of about 70%. Foowing Koncar (Koncar, 008) and the concusions of DEBORA test, the forces acting on the bubbes that are retained are the drag and the added-mass forces. The turbuent dispersion is taen into account by the Tchen s theory. acuations hae been performed with three inds of meshing with the RTSM turbuence mode and the K ε turbuence mode: a coarse grid (figure 16), a medium grid (figure 17) and a fine grid (figure 18). The mean axia iquid eocity (profie 1 and ) upstream of mixing bades are presented figure 19 and figure 0. Resuts are simiar and do not depend on the grid refinement. But for the profie 1, the K ε mode on the finest mesh gie sighty higher aues because the mesh near the wa is much too fine and then the usua ogarithmic aw is repaced by a condition of anishing eocity at the wa. Therefore, in order to consere the mass fowrate, the iquid eocity taes higher aues in the core of the fow. The height Z=0 mm corresponds to the beginning of the spacer grid. The mean azimutha iquid eocity (profie 1) just aboe the mixing bades at the height Z=40mm is iustrated on figure 1. Resuts are simiar and do not depend on the grid refinement. Downstream of the mixing bades, at Z=00mm, the mean azimutha eocity (profie 1) predicted using the K ε mode is much ower than that predicted with the R ε, as expected from the theory. This fact is aso cear on figure 5, figure 6, figure 7 and figure 8. Therefore, the mixing of the fow predicted with the R ε mode. K ε mode is smaer than with the 1

22 The iquid temperature in the fow is the combination of seera phenomena as aready discussed in the T K DEBORA test. The turbuent heat fux is directy proportiona to ν = µ and the K ε usuay ε yieds a seere oer-prediction of K downstream of the mixing bades. The K ε mode under estimates the mixing but oer-predicts the turbuent coefficient diffusion of heat fux. The iquid temperature fieds for both turbuence modes are presented figure and figure 4: the iquid temperature cacuated with the K ε mode is more homogeneous in the cacuation domain, i.e. the gradient of temperature is ower. As a consequence, the oid fraction fieds downstream of the mixing bades hae sighty ower aues with the K ε mode than the R ε mode. A second order cosure for the turbuent heat fux shoud be used in future cacuations to tae into account the anisotropy of the turbuent heat fuxes, which is needed to detect hot points on the rod surface in the frame of the boiing crisis. The time step is about 10-5 s to ensure a FL equa to 1. As for the computationa costs, the resuts are synthesized in the tabe 4. The RSTM oer-cost is about 6% for singe-phase fow but reaches 56% for boiing fow. The oer-cost with the super-computer with 64 processors. R ε mode for boiing fows is important but remains reasonabe when using a Singe-phase fow Boiing fow K ε ; coarse grid,0 s, s K ε ; medium grid 6,5 s 9,9 s K ε ; fine grid 0,1 s 5,5 s R ε ; fine grid 7,5 s 40,0 s Tabe 4: PU time by time step. H = mm L downstream =0.5 m L upstream =1. m figure 1: spacer grid with mixing anes figure 1: x rod bundes.

23 figure 14: iquid eocities are cacuated aong profies 1 and. figure 15: x rod bundes (Shin, 00). figure 16: oarse meshing : ces (zoom at right). figure 17: Medium meshing : ces (zoom at right).

24 figure 18: Finest meshing : ces (zoom at right). figure 19: mean axia iquid eocity (profie 1) upstream of mixing bades. figure 0: mean axia iquid eocity (profie ) upstream of the mixing bades. 4

25 figure 1: mean azimutha iquid eocity (profie 1) just aboe the mixing bades. figure : mean azimutha iquid eocity (profie 1) downstream of the mixing bades. figure : iquid temperature ( K ε ). figure 4: iquid temperature ( R ε ) figure 5: oid fraction just aboe the mixing bades at Z=40 mm ( K ε ). figure 6: oid fraction downstream of mixing bades at Z=00 mm ( K ε ). figure 7: oid fraction just aboe the mixing bades at Z=40 mm ( R ε ) figure 8: oid fraction downstream of mixing bades at Z=00 mm ( R ε ). 5