Assessment and calibration of an algebraic turbulent heat flux model for high Rayleigh number flow regimes

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1 Assessment and calibration of an algebraic turbulent heat flux model for high Rayleigh number flow regimes A. Shams Nuclear Research and Consultancy Group (NRG), Petten, The Netherlands Abstract The knowledge of heat transfer in corium pools is one of the important issues for corium retention. It defines the safety margin for vessel integrity and provides the necessary inputs for the core catcher design. In this regard, in the 90 s the BALI experimental program was performed at the CEA, France. The principle idea was to create a database regarding the heat transfer distribution at corium pool boundaries for in-vessel and exvessel configurations at high internal Rayleigh number (10 15 to ). One of the tasks within the ongoing IVMR project, the HORIZON 2020 program, is to assess the up-to-date turbulence models of CFD over a wide range of Rayleigh number for the homogenous pool tests of BALI experiments. In the present study, the assessment of three different turbulence models is performed for two BALI test cases. These turbulence models include a linear k- model, a non-linear Reynolds stress model and an advanced turbulent heat flux model developed by NRG, known as AHFM-NRG. After an extensive and careful assessment of these models it has been found that none of these models is able to correctly predict the complex heat transfer phenomena appearing in natural convection flow regimes at such high Ra numbers. Hence, a new model is proposed to deal with a wide range of high Ra number flow regimes. The obtained results are found to be in a very good agreement with the available experimental data. Keywords: Heat transfer, high Rayleigh number, turbulence models. 1. Introduction Numerical prediction of turbulent natural convection flows requires an especial attention with respect to the selection of the turbulence model. The limitations of most commonly used turbulence models with Eddy Diffusivity approach have become more evident, particularly for natural and mixed convection flows [1]. There exists a number of situations in the nuclear industry where turbulent natural convection heat transfer occurs. One such situation correspond to a hypothetical severe accident scenario in a light water reactor, in which the melt core may relocate and accumulate in the lower plenum of the reactor pressure vessel (RPV). The decay heating in the core melt may cause the formation of a melt core (corium) pool and the subsequent natural convection. This turbulent natural convection in the corium pool could have a profound impact on the thermal loading in the vessel. A proper knowledge of the heat transfer in corium pools is one of the important issues for corium retention, as it defines the safety margin for the vessel integrity. In addition, it provides the necessary inputs for the design of the core catcher. In this regard, in 1993 the BALI experiment has been designed to create a database about the heat transfer distribution at boundaries of corium pools [2]. In the BALI experimental series different pool configurations were studied, such as homogenous, stratified and porous pools. The BALI homogeneous pool test has been designed to simulate the natural convection flow and heat transfer phenomena in corium pools in the lower head of the RPV [2]. The test section is a full-scale 2D slice of a typical hemispherical pressurized water reactor (PWR) lower head, see Fig. 1 (top). A water-salt solution is used as a simulant fluid. The BALI tests reached prototypical values of the internal Rayleigh number, from to over The experiments were equipped with Particle Image Velocimetry (PIV), yielding information on the flow structure as shown in Fig. 1 (bottom). In principle three major zones can be distinguished. Starting from the bottom part of the pool, a thermally stratified zone appears with low ascending velocities. In the upper part, an iso-thermal appears, which is unstable in nature, with sizeable eddies caused by cold plumes descending from the upper cooled surface. In addition, a downward flow region along the cooled curved wall appears, where velocity reaches its maximum values. * Corresponding author address: shams@nrg.eu (A. Shams)

2 The 8 th European Review Meeting on Severe Accident Research - ERMSAR-2017 Figure 1 (Top) BALI experiment test section (Bottom) temperature profile and flow structure observed in the BALI tests [2]. A number of numerical studies have been performed in the past in order to reproduce similar experiments and also to provide an insight of the phenomena associated with the molten core. However, it is certain that numerical simulations for such flow configurations are challenging. To account for the effects of turbulence, a great amount of CFD research has been focused on the methods, which make use of the turbulence models for molten core and/or similar natural convection flow configurations. Dol and Hanjalic [3] performed 2D and full 3D computations with low Reynolds k-ε models to simulate the natural convection in a side-heated near-cubic enclosure containing dry air (Pr = 0.71) under the Rayleigh number equal to Ra = Aounallah et al. [4] performed the validation study of the k-ε and the k-ω SST models for turbulent a natural convection flow in a differentially heated cavity containing a fluid with Pr = 0.71 and Rayleigh numbers ranging from to The authors concluded that the k-ω SST model provided superior results compared to the considered k-ε

3 The 8 th European Review Meeting on Severe Accident Research -ERMSAR-2017 model. Nevertheless, it was not able to correctly reproduce the mean flow. Similarly, Nourgaliev et al. [5] have shown that the standard low Reynolds k-ε models are not sufficient for such high Rayleigh number flows in a reactor. However, a modification of these models with buoyancy anisotropic effects improves the analysis. Fukasawa et al. [6] have performed the assessment of several turbulence models for one of the homogenous BALI test cases. They have found the similar conclusions as Nourgaliev et al. [5]. In the present study, an assessment of four different turbulence models is performed for two BALI test cases. These turbulence models include i) a linear k- model ii) a non-linear Reynolds stress model iii) an advanced turbulent heat flux model developed by NRG, known as AHFM-NRG [7] and iv) a newly proposed model, which is a modification of AHFM-NRG, and is called as AHFM-NRG+. Details regarding this modification along with the flow configuration and the adopted numerical strategies are reported in Section 2. In Section 3, the obtained results are extensively discussed and compared with the experimental results. This section is followed by the conclusions. 2. Flow Configuration and Numerical Strategies 2.1. Selection of the Test Cases The BALI experimental program was comprised of four test series. The first three test series were dedicated to the homogenous corium pools, to study the effect of the Rayleigh number, the Prandtl number and to observe heat transfer in a porous medium. Whereas, the fourth series was performed on a different test section. Nevertheless, the main focus of the present study is to assess different RANS turbulence models for the homogenous pools. Test Pool depth (m) P tot (kw) Table 1 List of selected test cases Boundary conditions Internal Rayleigh number I 1 9 Uniform 1.00 x II Uniform 2.27 x In this regard, two of the available BALI tests are considered and details are given in Table 1. The test case BALI-I has a pool depth of 1m and consists of uniform temperature boundary conditions. Whereas, the case BALI-II has a pool depth of 2m and a high internal Rayleigh number compare to BALI-I Flow Configuration Figure 2 Geometric configuration of the (left) initial test section and (right) modified computational domain.

4 The 8 th European Review Meeting on Severe Accident Research - ERMSAR-2017 Following the BALI experiments, the selected flow configuration is a quarter slice of a hemisphere and representing a three dimensional geometric configuration, as shown in Fig. 2 (left). In the present study, the focus is on single phase computations, hence the ice formation, which was observed in the experiments, is not taken into account. Accordingly, based on the approximate thickness of the ice formed at the bottom of the pool, the resulting computational domain has been modified (see Fig. 2: right). It is worthwhile to mention that the thickness of the ice formation has been approximated based on the experimental results Flow Parameters The boundary conditions used for the considered BALI test cases are given in Table 2 and the corresponding boundaries are also highlighted in Fig. 3. Table 2 List of the boundary conditions Boundary Conditions Top Surface T = K Curved Surface T = K Figure 3 Graphical representation of the boundary conditions. Front Face Back Face Left Vertical Surface Adiabatic surface Adiabatic surface Adiabatic surface In addition it is worthwhile to remind that no-slip boundary conditions are adopted for all the walls. For all the simulations presented in this article, constant physical properties of water at 20 C are used and are given in Table 3. Tabel 3 Thermal properties of water at 20 C. Property Value Density kg/m 3 Dynamic viscosity Specific heat Thermal conductivity Pa-s 4182 J/kg-K 0.6 W/m-K Thermal expansion coefficient /K 2.4. Mesh Generation The commercially available software STAR-CCM+ [8], is used to generate the 3D mesh of the BALI geometry. The mesh consists of a structured prism layer close to the wall region, in order to capture the strong gradients, and an unstructured polyhedral mesh in the bulk region (see Fig. 4). An advanced layer meshing technique is used to generate the near wall structure mesh involving 21 prism layers. Following the work of Shams et al. [9], the y+ value of the first cell is kept 0.2 to obtain a better resolution for thin thermal boundary layer. Two different meshes are generated for both considered cases. The overall meshes for the BALI-I and BALI-II consist of 1.3 M and 2.94 M grid points, respectively.

5 The 8 th European Review Meeting on Severe Accident Research -ERMSAR-2017 Figure 4 A cross-section of the mesh for the BALI-II test case. It is worthwhile to mention that a mesh sensitivity study for the case BALI-I is performed for three different meshes, i.e. 0.5 M, 1.3 M and 2 M. The obtained results, not presented in this article, have shown that a mesh of 1.3 M grid points is sufficient enough to reproduce the overall flow and thermal field. Furthermore, similar meshing parameters are translated for the BALI-II case and resulted in a mesh of 2.94 grid points Turbulence Modelling As mentioned earlier, a water salt solution is used as a simulant for the BALI experiments. This simulant, at the operating conditions, has a non-unity Pr number. Hence, a special attention is required in selecting a turbulence model for such non-unity Prandlt fluid. It is worth mentioning that most of the commercially available codes use the Reynolds analogy for computing the turbulent heat transfer. This assumption has been successfully adopted for a wide range of CFD applications, which are based on Eddy Diffusivity models and has provided reasonable predictions of global parameters such as Nusselt numbers and mean temperature distributions [7]. Nevertheless, to study this phenomenon, four different turbulence models are selected to perform these computations and are given below: 1) Low Reynolds k-ε model with the Yap correction 2) Reynolds-stress model (RSM) with elliptical blending (RSM-EB) 3) Algebraic heat flux model (AHFM) version of NRG, i.e. AHFM-NRG 4) AHFM-NRG+, which is a calibrated and new version of AHFM-NRG STAR-CCM+ is considered as a CFD tool, in which all these models are available. Details of the first two models and the AHFM, which is the base for AHFM-NRG, are explicitly given in [8]. More details about the AHFM-NRG can be found in [7]. It is worth mentioning that the low Reynolds k- ε model is the building block of the AHFM-NRG. In addition, an algebraic formulation is used to model the turbulent heat flux and is given as: k θu i = C (C t0 ε t1u i u j T + C x t2 θu U j j + C j x t3 βg i θ ) 2 + C t4 a ij θu j (1) j where θu i is the turbulent heat flux, g i is the gravity vector, a ij is the Reynolds stress anisotropy tensor, β is the thermal expansion co-efficient, k is the turbulence kinetic energy and ε is its dissipation. The co-efficients of AHFM and AHFM-NRG are given in Table 4. Tabel 4 Model co-efficients of different algebraic heat flux models Model C t0 C t1 C t2 C t3 C t4 AHFM AHFM-NRG ln (Re.Pr) AHFM-NRG See Eq. 2 0

6 The 8 th European Review Meeting on Severe Accident Research - ERMSAR-2017 Figure 5 Graphical representation of a new correlation for natural convection flows This AHFM-NRG model has been extensively tested and calibrated by NRG for low Prandtl fluids in the framework of previous European project THINS [10]. Although, the model was calibrated for natural, mixed and forced convected flow regimes. However, the Ra number of the selected natural convection cases was relatively low, i.e. in the range of In the framework of the IVMR project, an attempt has been made to further calibrate this model for high Ra number cases, as in the BALI experiments. An extensive assessment of the AHFM-NRG has shown that the co-efficient C t3 has a significant influence on the prediction of natural convection flows for a wide range of Ra number. Hence, instead of a constant value, a new correlation is proposed. This correlation forms a logarithmic function for the value of Ra and Pr numbers and is given as: C t3 = a 1. log n (Ra. Pr) + a 2 with 10 0 < Ra.Pr < (2) Where a 1 = , a 2 = 2.5, n = 7, Ra and Pr are the Rayleigh and Prandtl numbers, respectively. A graphical representation of this correlation is shown in Fig. 5 and highlights the logarithmic decrease of C t3 with the increasing Ra.Pr. The introduction of the Pr in the correlation makes this model useful for different working fluids. Furthermore, it is worth reminding that in Shams at al. [7], it was observed that for natural convection flow regimes the co-efficient C t1 is found to be less sensitive and was fixed to 0.25, as given in Table 4. Nevertheless, hereafter this new formulation of AHFM-NRG model is called as AHFM-NRG Solver Settings The internal heating and cooling of the walls in BALI experiments generates a highly turbulent flow and is intrinsically unsteady. Fukasawa et al. [6] have performed an extensive steady state RANS and URANS computations for one such BALI test case by using two different variants of the low-reynolds k-ε model. The authors concluded that no significant difference was observed between RANS and URANS results for the prediction of the pool temperature and the surface heat flux. Nevertheless, in the present work, a detailed comparison of RANS and URANS computations (not shown in this article because of the page limitations) is performed for the BALI-I case by using all aforementioned four models. Once again, the obtained results have shown no significant difference for the prediction of the pool temperature and the surface heat flux. Therefore, hereafter the results obtained from steady state RANS computations are shown for the considered test cases. In addition, a segregated flow model with a constant density model is used. The constant density model has been chosen, as it required for using gravitational and buoyancy effects that are important for heat convection in the flow. Furthermore, the segregated fluid temperature model is used that solves the total energy equation with temperature as the solved variable. Enthalpy is then computed from temperature according to the equation of state. Moreover, a second order upwind scheme is used for spatial discretization [8]. The discretization is performed by using collocated and the Rhie-Chow type pressure-velocity coupling combined with a SIMPLEtype algorithm. Further details of the selected numerical schemes can be found in the code manual [8].

7 3. Results and Discussions The 8 th European Review Meeting on Severe Accident Research -ERMSAR-2017 Figures 6 and 7 display the iso-contours of flow and temperature fields for the BALI-I test case. The flow field is highlighted by a line integration convolution (LIC) technique coloured with velocity magnitude. This LIC technique is a texture-based technique for visualizing vector fields. In the resulting image (as shown in Figure 6), these streamlines cover the entire domain of the vector field. In addition, Fig. 8 displays the evolution of temperature profile along the center depth direction and the heat flux distribution along the curved wall. By looking at the results, it is interesting to notice that from qualitative view point, all four models are able to predict three regions of convection, i.e. the upper well mixed region, the downward accelerated flow along the wall and the stratified zone at the bottom of the pool. Nevertheless, these models exhibit different results. Figure 6 BALI-I: Integrated line convolution contours for (top-left) low Re k-ε, (top-right) RSM-EB, (bottom-left) AHFM-NRG and (bottom-right) AHFM-NRG+. Figure 7 BALI-I: Temperature contours for (top-left) low Re k-ε, (top-right) RSM-EB, (bottom-left) AHFM-NRG and (bottom-right) AHFM-NRG+. The low Reynolds k-ε model under-predicts the stratified zone and is highly unstable due to the excessive production of turbulence (see Figs. 6 & 7). It predicts the flow separation of the downward flow along the curved

8 The 8 th European Review Meeting on Severe Accident Research - ERMSAR-2017 wall similar to AHFM-NRG+ model. Moreover, it over-predicts the pool temperature in comparison with the experiments. The predicted lateral heat flux is in fair agreement with the experiments specifically in the center region. However, it shows a strange peak in the lower part of the vessel. The RSM-EB model also predicts a quasi-unstable stratified zone, which is small compared to the first model. The velocity magnitude of the downward flow predicted by RSM-EB model is high with respect to the four models and estimates the flow separation similar to AHFM-NRG model. In comparison to the experimental data it under-predicts the pool temperature and the lateral heat flux. The AHFM-NRG display a very good mixing zone at the top with a number of eddies. It shows almost no stratified region, nevertheless, the zone seems to be stable compare to the first two models. It is worthwhile to remind that the boundary condition is strongly dependent on whether or not there is an ice layer on the vessel wall. In the case of ice formation, the boundary temperature is uniform. Whereas, without the ice formation, which is considered in the present study, the boundary temperature is higher than the melting temperature of the ice. In addtion, the boundary temparature is not uniform because of different heat flux along the vessel wall. Nevertheless, it must be kept in mind that the boundary temperature definition can play a role on the extent of temperature stratification. Furthermore, the AHFM-NRG predicts the flow separation of the downward flow almost similar to the RSM-EB model, however, with lower velocity magnitude. In comparison with the experiments, the AHFM-NRG highly under-predicts the pool temperature and the lateral heat flux. This highlights the importance of the model calibration for high Ra number flow regimes. Accordingly, the model is further calibrated, as described in Section 2. The AHFM-NRG+, predicts a stable stratified zone. However, the thickness of the zone is under-predicted like the other models. Although, the model shows a good mixing zone at the top of the pool, it displays fewer eddies compare to the other models. The flow separation for this model is observed to be in between the first two models. This results in a very good prediction of the pool temperature in comparison with the experimental data. In addition, the model improvement has resulted in a very good prediction of the lateral heat flux. Figure 8 BALI-I: Evolution of axial temperature profile and lateral heat flux.

9 The 8 th European Review Meeting on Severe Accident Research -ERMSAR-2017 Furthermore, a thorough quantitative comparison of these models against the experimental data is performed and is given in Table 5. All four models are explicitly compared for Rayleigh and Nusselt numbers obtained from the experiments. In Table 6, a percentage error of these models for the aforementioned quantities is given. The formulae for internal and external Rayleigh numbers are given as: Ra internal = gβq vh 5 kυα ; Ra external = gβδth4 υα (3) where g is the gravity, β is the thermal expansion co-efficient, Q v is the volumetric flux dissipation, H is the characteristic length (here it is the height of the pool), k is the thermal conductivity, υ is the kinematic viscosity, α is the thermal diffusivity and ΔT is the temperature between bulk and the wall. The bulk temperature is calculated by taking the surface average of the top 60% height of the volume. In addition, the formula for the Nusselt number is given as: Nu = φh kδt (4) where φ is average surface heat flux and H is the characteristic length (here it is the length of the wall, as per the experiments). The Nu_up is computed for the top wall and Nu_down is computed for the curved wall. By looking at the predicted non-dimensional numbers of all four models it is clearly noticeable that the AHFM- NRG+ shows an excellent agreement with the experiments. The Rayleigh numbers are predicted within 1%, which is quite remakable. However, for the Nusselt numbers the error goes up to 7%, which still highlights a very good agreement. Whereas, other models show relatively high errors. Nevertheless, among these models, the RSM-EB model gives relatively beter results. The maximum error goes up to ~ 22 %, which can be considered as a fair agreement for a RANS type turbulence modelling approach. Table 5 BALI-I: Quantitative analysis of the non-dimensional numbers Experiements Low Re k- RSM-EB AHFM-NRG AHFM-NRG+ Ra_internal Ra_external Nu_up Nu_down Table 6 BALI-I: Percentage error of the computed non-dimensional numbers Experiements Low Re k- RSM-EB AHFM-NRG AHFM-NRG+ Ra_internal Ra_external Nu_up Nu_down Furthermore, Figs. 9 and 10 display the iso-contours of flow and temperature fields for the BALI-II test case. Once again, the low Reynolds k-ε and the AHFM-NRG models show similar results compare to the BALI-I case. Whereas, the RSM-EB model has shown some improvement in the prediction of the stratified zone, which is thicker compare to previous case. However, the zone is still under-estimated and quasi-unstable. On the other hand, AHFM-NRG+ has shown an improvement in predicting more enhanced mixing zone, as highlighted by the presence of more eddies at the top of the pool.

10 The 8 th European Review Meeting on Severe Accident Research - ERMSAR-2017 Figure 9 BALI-II: Integrated line convolution contours for (top-left) low Re k-ε, (top-right) RSM-EB, (bottom-left) AHFM-NRG and (bottom-right) AHFM-NRG+. Figure 10 BALI-II: Temperature contours for (top-left) low Re k-ε, (top-right) RSM-EB, (bottom-left) AHFM- NRG and (bottom-right) AHFM-NRG+.

11 The 8 th European Review Meeting on Severe Accident Research -ERMSAR-2017 Moreover, Fig. 11 displays the evolution of temperature profile along the center depth direction and the heat flux distribution along the curved wall for the BALI-II test case. It is clearly noticeable that, as observed in the BALI- I case, the low Reynolds k-ε over-predicts the pool temperature, whereas, the lateral heat flux is in fair agreement with the experiments. Figure 11 BALI-II: Evolution of axial temperature profile and lateral heat flux. Table 7 BALI-II: Quantitative analysis of the non-dimensional numbers Experiements Low Re k- RSM-EB AHFM-NRG AHFM-NRG+ Ra_internal Ra_external Nu_up Nu_down Table 8 BALI-II: Percentage error of the computed non-dimensional numbers Experiements Low Re k- RSM-EB AHFM-NRG AHFM-NRG+ Ra_internal Ra_external Nu_up Nu_down The RSM-EB model shows surprisingly good agreement for the pool temperature. However, it highly underpredicts the heat flux, as seen before for the BALI-I test case. In addition, it show a strange peak in the upper

12 The 8 th European Review Meeting on Severe Accident Research - ERMSAR-2017 part of the vessel. The AHFM-NRG model, like the previous case, under-predicts the pool temperature and the lateral heat flux. Finally, the AHFM-NRG+ model, once again has shown good agreement for the pool temperature and the lateral heat flux. The results obtained for this model are consistent for the both test cases and highlights the reliability and need of such advanced turbulent heat flux models. Furthermore, a quantitative comparison of the computed non-dimensional numbers and their respective percentage errors for all four models is given in Tables 7 & 8. Once again, it can been seen that the AHFM-NRG+ is in excellent agreement with the experiments with a maximum error of 5.4% for Nu_up. On the other hand, like the previous case, rest of the models show relatively high errors. Suprisingly, the RSM-EB model has shown some improvement ansd displays a relatively low error for Ra_interanl, Ra_external and Nu_up. Since, it under-predicts the lateral heat flux, hence, the corresponding Nu_down is highly over-predicted. 4. Conclusions Three-dimensional RANS computations are performed to assess different turbulence models for a high Rayleigh number flow regime as observed in homogenous pool tests of BALI experiments. In this regard, two test cases of the BALI experiments are considered with a Ra number of the order of and These assessed turbulence models include a linear k- model, a non-linear Reynolds stress model and an advanced turbulence model developed by NRG, known as AHFM-NRG. After an extensive and careful assessment of these models it has been found that none of these models, in their considered formulation, is able to correctly predict the complex heat transfer phenomena appearing at such high Ra number regimes. Accordingly, a thorough calibration of AHFM-NRG model is performed to deal with wide range of Ra number flow regimes irrespective of any working fluid. Hence, a new version of AHFM-NRG, i.e. AHFM-NRG+ is proposed. The obtained results have shown that the AHFM-NRG+ is capable of correctly predicting the complex heat transfer phenomena and is in excellent agreement with the experimental data. As an outlook, to gain more confidence further assessment of AHFM-NRG+ needs to be performed. Acknowledgement The work described in this paper is funded by the Dutch Ministry of Economic Affairs and the HORIZON 2020 IVMR Project No References [1] OECD/NEA, Handbook on Lead bismuth Eutectic Alloy and Lead Properties, Materials Compatibility, Thermal-hydraulics and Technologies. OECD NEA No. 6195, ISBN , [2] J. M. Bonnet, Thermal hydraulic phenomena in corium pools: the BALI experiment, Proc. Workshop on Severe Accident Research (SARJ-98), Tokyo, Japan, Nov. 4 6, 1998, 79 86, [3] H. Dol and K., Hanjalic, Computational study of turbulent natural convection in a side-heated near cubic enclosure at a high Rayleigh number, International Journal of Heat and Mass Transfer, Vol. 44, pp , [4] M. Aounallah, Y. Addad, S. Benhamadouche, O. Imine, L. Adjlout and D. Laurence, Numerical investigation of turbulent natural convection in an inclined square cavity with a hot wavy wall, International Journal of Heat an Mass Transfer, Vol. 50, pp , [5] R. Nourgaliev, T. Dinh and B. Sehgal, Effect of fluid Prandlt number on heat transfer characteristics in internally heated liquid pools with rayleigh numbers up to Nuclear Engineering and Design, Vol. 169, pp , [6] M. Fukasawa, S. Hayakawa and M. Saito, Thermal-hydraulic analysis for inversely stratified molten corium in lower vessel, Journal of Nuclear Science and Technology, Vol. 45, No. 9, pp , [7] A. Shams, F. Roelofs, E. Baglietto, S. Lardeau and S. Kenjeres, Assessment and calibration of an algebraic turbulent heat flux model for low-prandtl fluids, International Journal of Heat and Mass Transfer 79, , [8] CD-adapco, STAR-CCM+ Documentation, Version 10.02, [9] A. Shams, D. Visser and F. Roelofs, Influence of Numerical Tools on the Flow and Heat Transfer of Supercritical Water, NURETH14, Toronto, Canada, [10] F. Roelofs, A. Shams, I. Otic, M. Böttcher, M. Duponcheel,Y. Bartosiewicz, D. Lakehal, E. Baglietto, S. Lardeau and X. Cheng, Status and perspective of turbulence heat transfer modelling for the industrial application of liquid metal flows, Nuclear Engineering and Design, Vol. 290, pp , 2015.