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Application of an algebraic turbulent heat flux model to a backward facing step flow at low Prandtl number
Annals of Nuclear Energy 117 (2018) 32–44
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Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene
Application of an algebraic turbulent heat flux model to a backward facing step flow at low Prandtl number Andrea De Santis, Afaque Shams ⇑ Nuclear Research and Consultancy Group NRG, Westerduinweg 3, 1755 LE Petten, The Netherlands
a r t i c l e
i n f o
Article history: Received 16 November 2017 Received in revised form 6 March 2018 Accepted 8 March 2018
Keywords: Turbulent heat transfer Low Prandtl fluids Algebraic heat flux model Forced convection Mixed convection
a b s t r a c t Turbulent heat transfer represents a considerably challenging phenomenon from the modelling point of view. In the RANS framework, the classical Reynolds analogy provides a simple and robust approach which is widely employed for the closure of the turbulent heat flux term in a broad range of applications. At the same time, there is an ever growing interest in the development and assessment of advanced models which would allow, at least to some extent, for the relaxation of the simplifying assumptions underlying the Reynolds analogy. In this respect, the use of algebraic closures for the turbulent heat flux has been proposed in the literature by different authors as a viable approach. One of these algebraic closures has been extended for its application to low Prandtl number fluids in various flow regimes, by means of calibration and assessment of the model against some basic test cases, in what is known as the AHFMNRG+ model. In the present work the AHFM-NRG+ is applied for the first time to a relatively complex configuration, i.e. a backward facing step in both forced and mixed convection regimes with a low Prandtl working fluid, and assessed against reference DNS data. The obtained results suggest that the AHFMNRG+ is able to provide more accurate predictions for the thermal field within the domain and for the heat transfer at the wall in comparison to the Reynolds analogy assumption. These encouraging results indicate that the AHFM-NRG+ can be considered as a promising model to improve the accuracy in the simulation of the turbulent heat transfer in industrial applications involving low Prandtl fluids. Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction Due to their high thermal conductivity, liquid metals represent an attractive option as a coolant fluid. In fact, such fluids are currently considered in a broad range of industrial applications including the production of steel (Felten et al., 2004) and semiconductors (Raufeisen et al., 2009), and in thermal solar plants (Frazer et al., 2014). With respect to nuclear fission applications, liquid metals are foreseen as the primary coolant fluid in two of the six concepts considered within the Generation IV International Forum (GIF), i.e. the Lead Fast Reactor (LFR) and the Sodium Fast Reactor (SFR) (GIF, 2010). In this context, thermal– hydraulics is regarded as one of the key issues in the development of liquid metal cooled reactors. The complexity of the thermal–hydraulics in such reactors stems from a number of factors, including (Roelofs et al., 2015):
⇑ Corresponding author. E-mail address: [email protected] (A. Shams). https://doi.org/10.1016/j.anucene.2018.03.016 0306-4549/Ó 2018 Elsevier Ltd. All rights reserved.
Complex flow field with significant turbulence anisotropy. Non-negligible buoyancy influence. Low Prandtl number (in the order of 0.01–0.001) of the working fluid. Computational Fluid Dynamics (CFD) is regarded as a tool of great importance in order to overcome the aforementioned issues (Grötzbach, 2013). In the framework of the RANS approach and in light of the points stressed above, it is worth to point out that a broad range of models have been developed in the past decades with respect to the turbulent momentum transfer (Shams, 2017a). This allows for the choice of suitable models to account for turbulence anisotropy (e.g. Reynolds Stress Models (Hanjalic´, 1999) or non-linear Eddy Viscosity Models (Bauer et al., 2000)). Therefore, at least in the forced convection flow regime, the uncertainties associated with the modelling of the turbulent flow field can be minimised by resorting to an appropriate turbulence model. Furthermore, most turbulence models available in CFD codes also feature dedicated source terms in the momentum and in the turbulence equations in order to account, at least partially, for buoyancy effects (Grötzbach, 2013). Nevertheless, when buoyancy
A. De Santis, A. Shams / Annals of Nuclear Energy 117 (2018) 32–44
has a strong influence on the flow, the temperature cannot be considered as a passive scalar since it can be expected to have a significant impact on the flow field. As a consequence, an accurate evaluation of the thermal field is of significant importance in such cases (Grötzbach, 2013). Unfortunately, differently from the turbulent momentum transport, only a limited number of models are available for the closure of the turbulent heat flux (THF) term (Grötzbach, 2007). Very often the only option available in commercial CFD codes relies on a Simple Gradient Diffusion Hypothesis (SGDH), in which the THF is assumed to be proportional to the mean temperature gradient through the turbulent thermal diffusivity at . Furthermore, in order to evaluate at , the so called Reynolds analogy is almost universally invoked. Under this hypothesis, similarity in the turbulent transport of momentum and heat is assumed, and the turbulent thermal diffusivity is evaluated from the momentum diffusivity by introducing the turbulent Prandtl number Pr t . The assumption of similarity between turbulent momentum and heat transport allows to simplify the problem significantly, but it can result in an oversimplification with respect to the actual physics of the flow in some configurations (Roelofs et al., 2015). In addition, in most cases Prt is assumed to be a constant (usually having a value around 0.9– 1.0), although broad experimental and numerical evidence has shown that this is a major simplification with respect to the actual physics (Grötzbach, 2013). Despite this substantial simplifying assumptions, the Reynolds analogy can represent a reasonable compromise for a broad range of applications and has been the workhorse for RANS turbulent heat transfer modelling for decades. Nevertheless, the nuclear community is well aware of its possible shortcomings when considering complex applications as those involving low Prandtl fluids and/or buoyant flows. As a consequence, in the European project THINS (Thermal–Hydraulics of Innovative Nuclear Systems), more advanced closures for the THF have been proposed in order to improve the accuracy of the CFD models dealing with the turbulent heat transfer at low Prandtl number (Shams, 2017a). The models considered within this project include a local turbulent Prandtl number model ( Kays, 1994; Duponcheel et al., 2014), and both an explicit (Manservisi and Menghini, 2014) and an implicit Algebraic Heat Flux Model (AHFM) (Kenjereš et al., 2005; Shams et al., 2014). The general conclusion within the THINS project was that the efforts towards the development of new THF models should be limited, and the focus should be put on further validation of the proposed models in more complex configurations (Shams, 2017a). With respect to the implicit AHFM approach, the model originally proposed in Kenjereš et al. (2005) for natural and mixed convection with unity Prandtl number fluids has been extended for the application to forced convection and low Prandtl fluids in the socalled AHFM-NRG model (Shams et al., 2014). The calibration and assessment of the model have been performed against a set of simple test cases (i.e. forced planar and wavy channel flows, Rayleigh-Bnard convection, vertical heated channel in mixed convection). In addition, this model has been further extended to natural convection at high Rayleigh number Ra in Shams (2017b), resulting in the so called AHFM-NRG+ model. One of the main hampering factors with respect to the further assessment of the proposed THF models in complex configurations is the lack of reference data, both experimental and numerical, for their validation. Therefore, the two European projects SESAME and MYRTE, which are currently ongoing, have the aim of generating such a reference database in order to be able to further assess the proposed THF models against comprehensive validation data (Shams, 2017a). In this context, the present work aims at contribute to the further understanding of the performance of different THF closures in a relatively complex case that is relevant for nuclear applications.
33
In particular the classical Reynolds analogy and the AHFM-NRG+ model have been employed to simulate the turbulent heat transfer in a backward facing step (BFS) flow in both forced and mixed convection with a low Prandtl working fluid. The reference database is a recently published DNS reported in Niemann and Fröhlich (2016). A description of the computational domain and of the numerical settings is reported in Section 2, followed by an overview of the turbulence models adopted in the present RANS calculations in Section 3. Successively, the results obtained for both the forced and the mixed convection cases are illustrated and compared against the reference DNS data in Section 4. Finally, conclusive remarks along with future perspectives are provided in Section 5.
2. Computational domain and simulation settings The computational domain, which has been employed for both the forced and the mixed convection calculations, reproduces the reference DNS domain Niemann and Fröhlich (2016) and is sketched in Fig. 1. The domain represents a vertically oriented BFS geometry with a step height equal to h and an expansion ratio of 1.5. In contrast with the reference DNS, in which a periodic condition is imposed in the spanwise z direction, the computational domain considered in this work is two-dimensional. In terms of boundary conditions, similarly to what has been done in the reference DNS, a fully developed inflow condition has been generated through a periodic simulation of an inlet channel. The velocity and turbulence profiles obtained from the periodic calculation have been imposed at the BFS inlet. For the temperature, a uniform inlet value of 423.15 K has been used. At the outlet section, a Dirichlet condition has been used for the pressure, whilst a homogeneous Neumann condition has been adopted for all the other variables. All the remaining boundaries have been treated as adiabatic no-slip walls, with the exception of the bottom wall downstream of the step, where a heat flux of 41 kW/m2 has been imposed for x < 20h. The working fluid is liquid sodium. The changes in the physical properties of the sodium with respect to the temperature have been neglected, and all the properties have been evaluated from Sobolev (2011) at the reference temperature T ref , which has been taken equal to the inlet temperature of 423.15 K. At this temperature the Prandtl number of the sodium is equal to 0.0088. In addition to the Prandtl number, the other non-dimensional groups defining the problem are the Reynolds number Re ¼ 2hU b =m and the Richardson number Ri ¼ gbDTh=U 2b . The Reynolds number is calculated with respect to the inlet channel width 2h and the bulk inlet velocity U b and is equal to 9610. The temperature difference DT is defined with respect to the imposed heat flux q00 as DT ¼ q00 h=k, where k is the thermal conductivity of the sodium, and is equal to 20 K. In the DNS reference data, two different values of the Richardson number have been considered. The first one is Ri ¼ 0, which is enforced by setting the thermal expansion coefficient b to zero, and corresponds to the forced convection regime. The second Ri value, which is equal to 0.338, corresponds to the mixed convection regime. Both cases have been modelled in the present calculations. All the simulations have been performed using the commercially available software STAR-CCM+ version 11.06 (SIEMENS PLM Software, 2016). It is worth to point out that the considered AHFM model has been implemented within this code in the framework of the THINS project. The governing equations have been solved in their steady-state formulation and the SIMPLE algorithm has been used for pressure–velocity coupling (Patankar and Spalding, 1972). A second order scheme has been employed for the spatial discretisation of all the equations.
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Fig. 1. Computational domain employed in the present RANS calculations.
3. Turbulence models In the present study, a linear low-Reynolds k e closure has been adopted for the modelling of the turbulent momentum transfer (Lien et al., 1996). This model has been employed since it represents a building block for the implementation of the AHFM-NRG+ model in STAR-CCM+. The transport equations for the turbulent kinetic energy and its dissipation rate are expressed as follows:
DðqkÞ @ ¼ Pk þ Pb qe þ Dt @xj
lt @k rk @xj
and two different closures have been employed for the unknown THF term hui appearing in Eq. (10). The first one is the classical SGDH assumption, which results in:
hui ¼ at
@T @xi
ð11Þ
where the unknown turbulent thermal diffusivity at has been evaluated by invoking the Reynolds analogy, i.e.:
mt
ð1Þ
at ¼
ð2Þ
k Pwall ¼ DF 2 P k þ 2l 2 exp ERe2d d
and the turbulent Prandtl number Pr t has been assumed to be equal to 0.9. The other closure employs an implicit algebraic expression, which is derived from the full second moment transport equation for the THF under the hypothesis of local equilibrium between production and dissipation (Dol et al., 1997). The resulting algebraic expression has the form (Kenjereš and Hanjalic´, 2000):
ð3Þ
hui ¼ C t0
lt ¼ mt q ¼ qf l C l ks
ð4Þ
where aij ¼ ik j 23 dij is the Reynolds stress anisotropy tensor. The three terms in brackets on the right hand side of Eq. (13) correspond to the three physical mechanisms involved in the THF generation, i.e. (Kenjereš and Hanjalic´, 2000):
lþ
DðqeÞ C e1 e 1 ¼ P þP þ C P qC e2 F 2 þ qSy Dt s k wall e3 b s s @ l @e þ lþ t @xj re @xj where
h pffiffiffiffiffiffiffiffi i f l ¼1 exp C d0 Red þ C d1 Red þ C d2 Re2d h i F 2 ¼ 1 C exp Re2t Sy ¼ C w
e2 k
ð5Þ ð6Þ
" max
2 # ‘ ‘ 1 ;1 ‘e ‘e
ð7Þ
@T @U i C t1 ui uj þ C t2 huj þ C t3 bg i h2 þ C t4 aij huj @xj e @xj
k
ð13Þ
uu
Production due to the mean temperature gradient @T=@xj . Production due to the mean rate of strain @U i =@xj . Impact of buoyancy on the turbulent fluctuations through the term bg i h2 .
3=2
‘¼
ð12Þ
Pr t
k
ð8Þ
e
‘e ¼ C ‘ d
ð9Þ
and P k ¼ qui uj @U i =@xj is the mechanical turbulent production, Pb ¼ qbg i hui is the turbulence production due to buoyancy, pffiffiffi pffiffi s ¼ max ke ; C t me ; d is the wall distance, Red ¼ kd=m; Ret ¼ k2 =me and Sy is a source term introduced in the transport equation for e following Yap (1987). The model coefficients have been taken from Shams et al., 2014 and SIEMENS PLM Software, 2016 and are summarised in Table 1. In addition, the energy transport equation has the form:
D qc p T @ @T ¼ qcp hui k Dt @xi @xi
In the formulation reported in Kenjereš et al. (2005), the model has been calibrated for its application in the natural and mixed convection regimes at unity Prandtl number. In the present work, the extended version of the model proposed in Shams et al. (2014) and Shams (2017b), and henceforth referred to as AHFMNRG+, is employed. In the AHFM-NRG+ the model coefficients C t1 and C t3 are expressed as a function of the product of the Prantl and Reynolds number and of the Prandtl and the Rayleigh number, respectively, as detailed in Eqs. (14) and (15).
C t1 ¼ 0:053lnðRePr Þ 0:27
with
C t3 ¼ 4:5 109 ðlog ðRaPrÞÞ þ 2:5 7
RePr > 180 with
ð14Þ
1 > RaPr > 1017 ð15Þ
ð10Þ
Table 1 Coefficients for the low-Reynolds k e model.
rk
re
Cl
Ct
C e1
C e2
C e3
C d0
C d1
C d2
C‘
Cw
C
E
D
1
1.3
0.09
1
1.44
1.92
0.33
0.091
0.0042
0.00011
2.55
0.83
0.3
0.00375
1
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Due to these modifications to the model coefficients, the applicability of the AHFM-NRG+ has been extended to the forced convection regime and to buoyant flows with high Rayleigh number. Furthermore, the dependency of the coefficients on Pr makes the model suitable for the application to fluids characterised by a broad range of Prandtl numbers, including liquid metals. The logarithmic dependency of C t1 and C t3 on RePr and RaPr, respectively, is depicted in Fig. 2. In addition, it should be noted that values of RePr lower than the limit of 180 reported in Eq. (14) can be encountered. This is often the case when comparing to DNS databases with low Prandtl fluids, where relatively low Reynods number are usually adopted in order to reduce the computational costs associated with the calculations. In the present case, from the settings reported in Section 2, it results RePr 84:6. Therefore, the value of C t1 for the present calculations has been evaluated by considering RePr ¼ 180 in Eq. (14). Further calibration of the model in order to extend the correlation for this coefficient to lower values of RePr would be desirable in the future. Furthermore, it should be noted that the closure reported in
where
2
sh ¼ 2heh and sm ¼ k=e. This assumption has been observed to
be reasonable in several cases, as reported in Shams (2017a), Hanjalic´ et al. (1996), Kozuka et al. (2009), and greatly simplifies the application of the AHFM to the simulation of complex flows. A summary of the coefficients for the k e h2 AHFM-NRG+ model employed in the present work is reported in Table 2. 4. Results This section discusses the results obtained for both the forced convection and the natural convection cases. As a first step, a mesh sensitivity study has been carried out for the forced convection case and is detailed in Section 4.1. Successively, the results obtained for the flow and the thermal fields in both regimes are reported and critically analysed by means of a detailed comparison with the reference DNS data. 4.1. Mesh sensitivity
2
Eq. (13) requires the evaluation of the temperature variance h . Therefore, a transport equation for this quantity is introduced, having the form
Dqh2 @ ¼ 2Pt 2qeh þ @xi Dt
"
k l þ t cp rh2
!
@h2 @xi
#
ð16Þ
@T where P t ¼ qhui @x is the temperature variance production term i
and eh is its dissipation rate. A suitable model for the latter variable is needed in order to close the model. In the approach followed in Kenjereš and Hanjalic´ (2000), an additional transport equation is solved for the temperature variance dissipation, resulting in a four-equation k e h2 eh model. However, such a transport equation presents a number of unclosed terms and its closure results in the introduction of significant additional uncertainty in the model (Kenjereš et al., 2005; Otic´ et al., 2005). Also, the fourequation model has shown convergence problems in some cases, as reported in Shams (2017a). In order to overcome these issues, a simplified approach has been employed in both the AHFM-NRG (Shams et al., 2014) and the AHFM-NRG+ (Shams, 2017b) formulations, and retained in the present work. In this approach, the temperature variance dissipation is evaluated through the assumption of a constant thermal to mechanical time scale ratio R (Hanjalic´ et al., 1996), resulting in a three-equation k e h2 model. The thermal to mechanical time scale ratio is defined as:
R¼
sh sm
ð17Þ
All the numerical grids used in the present study have been generated in STAR-CCM+ version 11.06 and consist of structured quadrilateral cells. The mesh sensitivity study has been carried out by employing four different grids (see Fig. 3) ranging from about 11 k–175 k elements, as reported in Table 3. All the considered meshes enforce the condition yþ < 1 at the walls everywhere in the domain. A typical feature of the BFS flow in the forced convection regime is the flow detachment and reattachment that takes place behind the step. The streamwise location of the reattachment point xr is used as a parameter to evaluate the sensitivity of the numerical results to the mesh resolution and the calculated xr values for the four different grids are reported in Table 3. In addition, the table reports the relative deviation of the calculated xr with respect to the most refined mesh, i.e. Mesh 4. In general, the relative deviation / is defined as:
/ ¼
/calc /ref /ref
ð18Þ
Table 2 Coefficients of the AHFM-NRG+ model. C t0
C t1
C t2
C t3
C t4
rh2
R
0.2
Eq. (14)
0.6
Eq. (15)
0.0
1
0.5
Fig. 2. C t1 as a function of RePr (left) and C t3 (right) as a function of RaPr in the AHFM-NRG+ model.
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Fig. 3. View of the four grids considered in the mesh sensitivity study.
Table 3 Mesh sensitivity: cell count and predicted reattachment point location.
Cell count xr =h
x
r
Mesh 1
Mesh 2
Mesh 3
Mesh 4
10875 6.62 2.5%
20771 6.70 1.4%
44370 6.76 0.4%
177480 6.79 –
where /calc is the calculated value for the generic quantity / and /ref is the reference value for the same quantity. From Table 3, it can be seen that the relative deviation of xr for Mesh 3 with respect to Mesh 4 is below 1%. In order to further assess the impact of the mesh resolution on the calculated flow and thermal fields, the profiles of the streamwise U and wall-normal V velocity components
evaluated at different streamwise locations are shown in Fig. 4. In addition, the same profiles for the turbulent kinetic energy k and the non-dimensional temperature T^ ¼ ðT T ref Þ=DT are reported in Fig. 5. It can be seen that the differences in the calculated values between Mesh 3 and Mesh 4 tend to be practically negligible, even for the most sensitive variable with respect to mesh resolution, which appear to be the turbulent kinetic energy. Therefore, Mesh 3 has been employed for all the calculations performed in the present work.
4.2. Flow field The results obtained for the turbulent flow field in forced and mixed convection are reported in Sections 4.2.1 and 4.2.2, respectively.
Fig. 4. Mesh sensitivity: calculated streamwise (top) and wall-normal (bottom) velocity components for different mesh resolutions.
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37
Fig. 5. Mesh sensitivity: calculated turbulent kinetic energy (top) and temperature (bottom) for different mesh resolutions.
Fig. 6. Forced convection case: streamwise velocity contours together with streamlines in the recirculation zone region.
Fig. 7. Forced convection case: skin friction coefficient along the bottom wall – solid lines current calculations, symbols: DNS.
4.2.1. Forced convection In the forced convection case, the typical flow features for a BFS configuration are observed, i.e. flow separation taking place behind the step, and reattachment of the flow and development of an
attached boundary layer further downstream. The calculated flow field is represented in Fig. 6 in terms of the streamwise velocity contours. The contours are shown on a section of the computational domain ranging from x=h ¼ 2:0 to x=h ¼ 20:0. It can be seen that the presence of a large clockwise recirculation zone behind the step is reproduced by the present RANS simulation. The calculated streamwise reattachment location is equal to 6.76h, compared with a value of 7.01h reported in the reference DNS. Furthermore, from the streamlines reported in Fig. 6, it can be inferred that a smaller counter-clockwise recirculation zone is present in the corner between the step and the bottom wall. In the reference DNS this secondary recirculation zone extends for the entire step height in the wall-normal direction, whilst its extent in the present RANS calculations appears reduced to roughly half of the step height. The calculated skin friction coefficient C f along the bottom wall is reported in Fig. 7. It can be observed that the results obtained with the linear low-Reynolds k e model in the present calculations are in a fair agreement with the DNS data for this quantity, with the largest discrepancies being found in the region very close to the step and in the value of the negative dip present within the recirculation zone. A more detailed picture of the predicted flow field at different streamwise locations within the domain is given in Figs. 8 and 9,
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Fig. 8. Forced convection case: streamwise (top) and wall normal (bottom) velocity components – solid lines: current calculations, symbols: DNS.
Fig. 9. Forced convection case: turbulent kinetic energy (top) and uv Reynolds stress component (bottom) – solid lines: current calculations, symbols: DNS.
which show the calculated streamwise and wall-normal velocity components, and the turbulent kinetic energy and the uv Reynolds stress, respectively. The streamwise positions included in these plots correspond to the locations where the reference DNS results are made available. A close agreement between the present calculations and the reference data is observed for the velocity field, whilst slightly larger discrepancies are present in the turbulent field. In particular, the most evident deviations with respect to the reference DNS are observed in the turbulent kinetic energy profiles within the recirculation zone at x=h ¼ 3 and x=h ¼ 6. Nevertheless, overall it can be concluded that the RANS approach
employed in the present calculations is able to provide a satisfactory representation of the turbulent flow field observed in the forced convection regime. 4.2.2. Mixed convection The calculated flow field for the mixed convection case is depicted in Fig. 10 in terms of the streamwise velocity contours. Since in this case the temperature cannot be regarded as a passive scalar, the figure includes the results obtained with both the SGDH and the AHFM-NRG+ closures for the THF. In contrast with the forced convection regime, the flow field in this case is charac-
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39
Fig. 10. Mixed convection case: streamwise velocity contour plot – SGDH(top) and AHFM-NRG+ (bottom).
to the turbulent fields, the overall prediction for k and uv is still satisfactory, although some non-negligible deviations from the reference data are observed in the turbulent variables, especially within the recirculation zone at x=h ¼ 3 and in the locations further downstream (x=h ¼ 12 and x=h ¼ 15). Moreover, more significant differences between the SGDH and the AHFM-NRG+ results are observed for these variables with respect to the predictions for the velocity field. Overall, it can be concluded that both turbulence models are able to provide a reasonable prediction of the flow field for the mixed convection case. 4.3. Thermal field
Fig. 11. Mixed convection case: skin friction coefficient along the bottom wall – solid line: SGDH, dashed line: AHFM-NRG+, symbols: DNS.
terised by the development of a jet at the bottom wall, due to the acceleration of the fluid caused by buoyancy. As a consequence, away from the bottom wall the streamwise velocity is reduced significantly, until the flows detaches from the upper wall forming a small recirculation zone. Furthermore, the recirculation zone found behind the step is significantly affected by the buoyancy forces, its extent in the streamwise direction being noticeably smaller with respect to the Ri ¼ 0 case. In addition, this recirculation zone is completely detached from the bottom wall, as it can be inferred from the positive values of the skin friction coefficient reported in Fig. 11. From the same figure, it can be observed that both turbulence models result in a fairly accurate prediction of C f , with the SGDH model being slightly closer to the DNS data in the dip found around x=h ¼ 2 and the AHFM-NRG+ model performing better further downstream. The predicted flow field within the domain at various streamwise locations is represented in Figs. 12 and 13, which depict the streamwise and wall-normal velocity components, and the turbulent kinetic energy and the uv Reynolds stress, respectively. Similarly to what has been observed in the forced convection regime, also in this case the predicted velocity field appears to be in a good agreement with the DNS data. Also, the SGDH and the AHFM-NRG+ models give quite similar results for the velocity field. With respect
4.3.1. Forced convection A contour plot of the calculated temperature field for the forced convection case is shown in Fig. 14. The SGDH and AHFM-NRG+ models are qualitatively in good agreement with each other, with the hot fluid located in the corner between the step and the heated wall. A strong temperature increase is observed moving from the top wall towards the heated bottom wall. A relatively low wall temperature is observed within the recirculation zone, whilst downstream of the reattachment point the temperature along the heated wall increases due the imposed heat flux. More insights with respect to the quantitative differences between the SGDH and the AHFM-NRG+ results can be gained from the temperature profiles reported in Fig. 15. From these profiles it can be observed that the SGDH approach consistently predicts a lower temperature with respect to the reference data in the near-wall region, whilst the algebraic model is in a closer agreement with the DNS results in all the considered axial locations with the exception of x=h ¼ 0. A quantification of the deviations of the two models with respect to the reference data is given in Table 4, reporting the calculated average deviation T^ at different streamwise locations. The average deviation along the vertical direction for a generic quantity / is evaluated as:
/ ¼
1 ba
Z
b a
/ ðyÞ dy
ð19Þ
where the local deviation / ðyÞ is defined according to Eq. (18). It is evident how the AHFM-NRG+ model gives consistently more accurate results with respect to the SGDH assumption at the different streamwise locations, with the exception of the x=h ¼ 0 position. Overall, the arithmetic average of the temperature deviation over
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Fig. 12. Mixed convection case: streamwise (top) and wall normal (bottom) velocity components – solid lines: current calculations, symbols: DNS.
Fig. 13. Mixed convection case: turbulent kinetic energy (top) and uv Reynolds stress component(bottom) – solid line: SGDH, dashed line: AHFM-NRG+, symbols: DNS.
the considered streamwise locations is reduced from 11.97% with the latter model to 8.88% with the AHFM-NRG+. It is worth to point out that the figures resulting from the arithmetic averaging in the streamwise direction can be influenced by the dimensions of the domain and the selection of the streamwise locations at which the comparison takes place. Nevertheless, these streamwiseaveraged values can be regarded as an approximate estimation of the difference in the overall accuracy between the two models for a given quantity. In addition, it should be noted that a reliable prediction of the turbulent flow field is a prerequisite for an
accurate evaluation of the turbulent heat transfer, regardless of the modelling approach employed for the THF. In fact, the overall accuracy in the prediction of the thermal field is affected by the uncertainties associated with both the turbulent momentum and the heat flux closures. Therefore, it is extremely difficult to single out the contribution of the sole THF model to the overall deviation from the reference data, and compensating errors might occur. This has to be kept in mind when performing a comparison between the results obtained with two different THF models and evaluating the resulting discrepancies with respect to the reference data.
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Fig. 14. Forced convection case: temperature contour plot – SGDH (top) and AHFM-NRG+ (bottom).
Fig. 15. Forced convection case: temperature profiles – solid line: SGDH, dashed line: AHFM-NRG+, symbols: DNS.
Table 4 Forced convection case: deviation of the predicted temperature with respect to the reference data at different streamwise locations. ^ SGDH ^ AHFM-NRG+ T T x=h ¼ 0 x=h ¼ 3 x=h ¼ 6 x=h ¼ 9 x=h ¼ 12 x=h ¼ 15
3.45% 22.37% 16.82% 12.09% 8.97% 8.14%
11.54% 14.59% 11.57% 8.02% 4.96% 2.61%
Average
11.97%
8.88%
A detailed analysis of the predicted heat transfer at the wall can be obtained from the wall temperature T^ w and the Nusselt number Nu profiles along the heated wall, which are shown in Fig. 16. The averaged deviation for these two quantities is reported in Table 5. From the figure, it can be observed how the wall temperature, and consequently the Nusselt number, predicted by the AHFM-NRG+ is in a significantly better agreement with the reference data with respect to the SGDH approach, resulting in an average deviation from the DNS results around 5%. 4.3.2. Mixed convection The iso-contours of the temperature for the mixed convection case are reported in Fig. 17. The most evident impact of the buoyancy effects on the the temperature is a general increase in the heat transfer, which results in lower temperature values with respect to the forced convection case. As a result of the changes in the flow field behind the step, the higher temperature region
Table 5 Forced convection case: average deviation of the predicted wall temperature and Nusselt number with respect to the reference data.
T^ w Nu
kk SGDH
kk AHFM-NRG+
13.81%
5.50%
15.90%
5.25%
is no longer located in the step corner, but it is shifted slightly downstream. Moreover, the thickness of the thermal boundary layer is reduced with respect to the Ri ¼ 0 case. Overall, the SGDH and the AHFM-NRG+ models are in a good qualitative agreement. A more detailed quantitative comparison is depicted in Fig. 18, showing the calculated temperature profiles at various streamwise locations. From this figure, it can be observed that both turbulence models are in a good agreement with the reference DNS data. Also, the differences between the results obtained with the two models are less evident with respect to the forced convection regime. In fact, due to the very low Pr value and the relatively low Reynolds number of the considered case, the mean diffusive and advective heat fluxes are significantly higher than the THF, and this is particularly evident in the mixed convection case (Niemann and Fröhlich, 2016). As a consequence, the impact of the considered THF model on the predicted temperature field is less evident in the mixed convection case with respect to the forced convection regime. Nevertheless, non-negligible discrepancies in the calculated temperature profiles are observed, as highlighted by the average temperature deviation at different streamwise locations reported in Table 6. In this case, for the reasons stressed above, the superiority of the AHFM-NRG+ over the Reynolds analogy
42
A. De Santis, A. Shams / Annals of Nuclear Energy 117 (2018) 32–44
Fig. 16. Forced convection case: wall temperature (left) and Nusselt number (right) at the heated wall – solid line: SGDH, dashed line: AHFM-NRG+, symbols: DNS.
Fig. 17. Mixed convection case: temperature contour plot – SGDH (top) and AHFM-NRG+ (bottom).
Fig. 18. Mixed convection case: temperature profiles – solid line: SGDH, dashed line: AHFM-NRG+, symbols: DNS.
assumption is less evident with respect to the Ri ¼ 0 case, although the results obtained with the former model are still in an overall better agreement with the reference DNS data. The heat transfer at the wall is analysed in Fig. 19 in terms of the calculated wall temperature and the Nusselt number profiles. Both the considered turbulence models are in a fairly good agreement with the reference DNS data. More in detail, it can be observed that the AHFM-NRG+ model is able to correctly predict the location and the value of the wall temperature peak located at x=h 2:5. Between x=h 3:0 and x=h 10:0 the SGDH is closer to the reference data, whilst the AHFM-NRG+ gives a more accurate wall temperature prediction further downstream. The same beha-
Table 6 Mixed convection case: deviation of the predicted temperature with respect to the reference data at different streamwise locations. ^ SGDH ^ AHFM-NRG+ T T x=h ¼ 0 x=h ¼ 3 x=h ¼ 6 x=h ¼ 9 x=h ¼ 12 x=h ¼ 15
16.09% 5.47% 1.63% 2.20% 7.46% 15.94%
14.90% 10.09% 5.58% 2.71% 1.47% 5.79%
Average
8.13%
6.76%
A. De Santis, A. Shams / Annals of Nuclear Energy 117 (2018) 32–44
43
Fig. 19. Mixed convection case: wall temperature (left) and Nusselt number (right) at the heated wall – solid line: SGDH, dashed line: AHFM-NRG+, symbols: DNS.
Table 7 Mixed convection case: average deviation of the predicted wall temperature and Nusselt number with respect to the reference data.
T^ w Nu
kk SGDH
kk AHFM-NRG+
4.75%
3.16%
4.50%
3.21%
availability of such reference data is regarded as of paramount importance in order to perform a thorough assessment of different turbulent heat flux closures.
Conflict of interest None declared.
viour is observed for the prediction of the Nusselt number. Also for these two quantities, the differences between the two THF closures are less evident with respect to the forced convection regime. Nevertheless, the AHFM-NRG+ still displays a lower average deviation from the reference data, as shown in Table 7.
5. Conclusions The paper focuses on the assessment of a recently proposed turbulent heat flux algebraic closure, the so called AHFM-NRG+ model, in its application to the modelling of a vertical backward facing step flow in both forced and mixed convection at low Prandtl number. The results obtained with the AHFM-NRG+ have been compared with a reference DNS database, along with the classical Reynolds analogy assumption. A low-Reynolds linear k e model has been employed for the turbulent momentum transfer, and a reasonable prediction of the turbulent flow field has been obtained in both the forced and the mixed convection regimes. In the former case, the AHFM-NRG+ showed a significant improvement in the prediction of the thermal field with respect to the Reynolds analogy. Moreover, the heat transfer at the heated wall has been predicted with a considerable accuracy with the proposed algebraic model. In the mixed convection regime, the superiority of the AHFM-NRG+ with respect to the Reynolds analogy assumption is less evident, and this is mainly due to the reduced contribution of the turbulent heat flux to the overall heat transfer in this case. Nevertheless, an improvement in the accuracy of the predicted thermal field and wall heat transfer has been observed in the mixed convection regime as well. The present work represents the first application of the AHFMNRG+ model to a complex flow configuration such as the vertical backward facing step at low Prandtl number, and confirms the promising potential of the AHFM-NRG+ as a tool to improve the prediction of the turbulent heat transfer in challenging flows of industrial interest involving low Prandtl fluids. The main bottleneck in the further assessment of the model is the lack of reliable reference databases for complex configurations, especially including low Prandtl fluids and/or buoyancy effects. Therefore, the
Acknowledgements The work described in this paper has been funded by the Dutch Ministry of Economic Affairs and from the Euratom research and training programme 2014–2018 under grant agreements Nos. 654935 (SESAME) and 662186 (MYRTE). The authors are also grateful to Prof. M. Niemann and Prof. J. Fröhlich from TU Dresden for kindly providing the reference DNS database. References Bauer, W., Haag, O., Hennecke, D.K., 2000. Accuracy and robustness of nonlinear eddy viscosity models. Int. J. Heat Fluid Flow 21 (3), 312–319. Dol, H.S., Hanjalic´, K., Kenjereš, S., 1997. A comparative assessment of the secondmoment differential and algebraic models in turbulent natural convection. Int. J. Heat Fluid Flow 18 (1), 4–14. Duponcheel, M., Bricteux, L., Manconi, M., Winckelmans, G., Bartosiewicz, Y., 2014. Assessment of RANS and improved near-wall modeling for forced convection at low Prandtl numbers based on LES up to Ret=2000. Int. J. Heat Mass Transf. 75, 470–482. Felten, F., Fautrelle, Y., Du Terrail, Y., Metais, O., 2004. Numerical modelling of electromagnetically-driven turbulent flows using LES methods. Appl. Math. Model. 28 (1), 15–27. Frazer, D., Stergar, E., Cionea, C., Hosemann, P., 2014. Liquid metal as a heat transport fluid for thermal solar power applications. Energy Procedia 49, 627– 636. GIF, Introduction to Generation IV nuclear energy system and the international forum.,http://www.gen-4.org (2010). Grötzbach, G., 2007. Anisotropy and buoyancy in nuclear turbulent heat transfer – critical assessment and needs for modelling. Forschungszentrum Karlsruhe FZKA 7363. Grötzbach, G., 2013. Challenges in low-Prandtl number heat transfer simulation and modelling. Nucl. Eng. Des. 264, 41–55. Hanjalic´, K., 1999. Second-moment turbulence closures for CFD: needs and prospects. Int. J. Comput. Fluid Dyn. 12 (1), 67–97. Hanjalic´, K., Kenjereš, S., Durst, F., 1996. Natural convection in partitioned twodimensional enclosures at higher Rayleigh numbers. Int. J. Heat Mass Transf. 39 (7), 1407–1427. Kays, W.M., 1994. Turbulent Prandtl number where are we? J. Heat Transfer 116 (2), 284–295. Kenjereš, S., Hanjalic´, K., 2000. Convective rolls and heat transfer in finite-length Rayleigh-Bénard convection: a two-dimensional numerical study. Phys. Rev. E 62 (6), 7987–7998. Kenjereš, S., Gunarjo, S., Hanjalic´, K., 2005. Contribution to elliptic relaxation modelling of turbulent natural and mixed convection. Int. J. Heat Fluid Flow 26 (4), 569–586.
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Kozuka, M., Seki, Y., Kawamura, H., 2009. DNS of turbulent heat transfer in a channel flow with a high spatial resolution. Int. J. Heat Fluid Flow 30 (3), 514– 524. Lien, F.S., Chen, W.L., Leschziner, M.A., 1996. Low-Reynolds number eddy-viscosity modelling based on non-linear stress-strain/vorticity relations. Eng. Turbul. Model. Exp., 91–100 Manservisi, S., Menghini, F., 2014. Triangular rod bundle simulations of a CFD k kh h heat transfer turbulence model for heavy liquid metals. Nucl. Eng. Des. 273, 251–270. Niemann, M., Fröhlich, J., 2016. Buoyancy-affected backward-facing step flow with heat transfer at low Prandtl number. Int. J. Heat Mass Transf., 1237–1250 Otic´, I., Grötzbach, G., Wörner, M., 2005. Analysis and modelling of the temperature variance equation in turbulent natural convection for low-Prandtl number fluids. J. Fluid Mech. 525, 237–261. Patankar, S.V., Spalding, D.B., 1972. A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. Int. J. Heat Mass Transf. 15 (10), 1787–1806. Raufeisen, A., Breuer, M., Botsch, T., Delgado, A., 2009. Transient 3D simulation of Czochralski crystal growth considering diameter variations. J. Cryst. Growth 311 (3), 695–697.
Roelofs, F., Shams, A., Otic, I., Böttcher, M., Duponcheel, M., Bartosiewicz, Y., Lakehal, D., Baglietto, E., Lardeau, S., Cheng, X., 2015. Status and perspective of turbulence heat transfer modelling for the industrial application of liquid metal flows. Nucl. Eng. Des. 290, 99–106. Shams, A., (U)RANS: turbulence modelling, SESAME Project: Lecture Series on Thermohydraulics and Chemistry of Liquid Metal Cooled Reactors, von Karman Institute for Fluid Dynamics, 10th-14th April 2017 (2017). Shams, A., Assessment and calibration of an algebraic turbulent heat flux model for high Rayleigh number flow regimes. In: 8th Conference on Severe Accident Research, May 16th–18th, Warsaw, Poland, 2017. Shams, A., Roelofs, F., Baglietto, E., Lardeau, S., Kenjeres, S., 2014. Assessment and calibration of an algebraic turbulent heat flux model for low-Prandtl fluids. Int. J. Heat Mass Transf. 79, 589–601. SIEMENS PLM Software, STAR-CCM+ User Guide – Version 11.06 (2016). Sobolev, V., 2011. Database of thermophysical properties of liquid metal coolants for GEN-IV, Technical Report 1069. Centre d’Etude de l’Energie Nucléaire, Belgium. Yap, C.J., 1987. Modelling convective heat transfer in recirculating and impinging flows Ph.D. thesis. University of Manchester, UK.
Contents lists available at ScienceDirect
Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene
Application of an algebraic turbulent heat flux model to a backward facing step flow at low Prandtl number Andrea De Santis, Afaque Shams ⇑ Nuclear Research and Consultancy Group NRG, Westerduinweg 3, 1755 LE Petten, The Netherlands
a r t i c l e
i n f o
Article history: Received 16 November 2017 Received in revised form 6 March 2018 Accepted 8 March 2018
Keywords: Turbulent heat transfer Low Prandtl fluids Algebraic heat flux model Forced convection Mixed convection
a b s t r a c t Turbulent heat transfer represents a considerably challenging phenomenon from the modelling point of view. In the RANS framework, the classical Reynolds analogy provides a simple and robust approach which is widely employed for the closure of the turbulent heat flux term in a broad range of applications. At the same time, there is an ever growing interest in the development and assessment of advanced models which would allow, at least to some extent, for the relaxation of the simplifying assumptions underlying the Reynolds analogy. In this respect, the use of algebraic closures for the turbulent heat flux has been proposed in the literature by different authors as a viable approach. One of these algebraic closures has been extended for its application to low Prandtl number fluids in various flow regimes, by means of calibration and assessment of the model against some basic test cases, in what is known as the AHFMNRG+ model. In the present work the AHFM-NRG+ is applied for the first time to a relatively complex configuration, i.e. a backward facing step in both forced and mixed convection regimes with a low Prandtl working fluid, and assessed against reference DNS data. The obtained results suggest that the AHFMNRG+ is able to provide more accurate predictions for the thermal field within the domain and for the heat transfer at the wall in comparison to the Reynolds analogy assumption. These encouraging results indicate that the AHFM-NRG+ can be considered as a promising model to improve the accuracy in the simulation of the turbulent heat transfer in industrial applications involving low Prandtl fluids. Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction Due to their high thermal conductivity, liquid metals represent an attractive option as a coolant fluid. In fact, such fluids are currently considered in a broad range of industrial applications including the production of steel (Felten et al., 2004) and semiconductors (Raufeisen et al., 2009), and in thermal solar plants (Frazer et al., 2014). With respect to nuclear fission applications, liquid metals are foreseen as the primary coolant fluid in two of the six concepts considered within the Generation IV International Forum (GIF), i.e. the Lead Fast Reactor (LFR) and the Sodium Fast Reactor (SFR) (GIF, 2010). In this context, thermal– hydraulics is regarded as one of the key issues in the development of liquid metal cooled reactors. The complexity of the thermal–hydraulics in such reactors stems from a number of factors, including (Roelofs et al., 2015):
⇑ Corresponding author. E-mail address: [email protected] (A. Shams). https://doi.org/10.1016/j.anucene.2018.03.016 0306-4549/Ó 2018 Elsevier Ltd. All rights reserved.
Complex flow field with significant turbulence anisotropy. Non-negligible buoyancy influence. Low Prandtl number (in the order of 0.01–0.001) of the working fluid. Computational Fluid Dynamics (CFD) is regarded as a tool of great importance in order to overcome the aforementioned issues (Grötzbach, 2013). In the framework of the RANS approach and in light of the points stressed above, it is worth to point out that a broad range of models have been developed in the past decades with respect to the turbulent momentum transfer (Shams, 2017a). This allows for the choice of suitable models to account for turbulence anisotropy (e.g. Reynolds Stress Models (Hanjalic´, 1999) or non-linear Eddy Viscosity Models (Bauer et al., 2000)). Therefore, at least in the forced convection flow regime, the uncertainties associated with the modelling of the turbulent flow field can be minimised by resorting to an appropriate turbulence model. Furthermore, most turbulence models available in CFD codes also feature dedicated source terms in the momentum and in the turbulence equations in order to account, at least partially, for buoyancy effects (Grötzbach, 2013). Nevertheless, when buoyancy
A. De Santis, A. Shams / Annals of Nuclear Energy 117 (2018) 32–44
has a strong influence on the flow, the temperature cannot be considered as a passive scalar since it can be expected to have a significant impact on the flow field. As a consequence, an accurate evaluation of the thermal field is of significant importance in such cases (Grötzbach, 2013). Unfortunately, differently from the turbulent momentum transport, only a limited number of models are available for the closure of the turbulent heat flux (THF) term (Grötzbach, 2007). Very often the only option available in commercial CFD codes relies on a Simple Gradient Diffusion Hypothesis (SGDH), in which the THF is assumed to be proportional to the mean temperature gradient through the turbulent thermal diffusivity at . Furthermore, in order to evaluate at , the so called Reynolds analogy is almost universally invoked. Under this hypothesis, similarity in the turbulent transport of momentum and heat is assumed, and the turbulent thermal diffusivity is evaluated from the momentum diffusivity by introducing the turbulent Prandtl number Pr t . The assumption of similarity between turbulent momentum and heat transport allows to simplify the problem significantly, but it can result in an oversimplification with respect to the actual physics of the flow in some configurations (Roelofs et al., 2015). In addition, in most cases Prt is assumed to be a constant (usually having a value around 0.9– 1.0), although broad experimental and numerical evidence has shown that this is a major simplification with respect to the actual physics (Grötzbach, 2013). Despite this substantial simplifying assumptions, the Reynolds analogy can represent a reasonable compromise for a broad range of applications and has been the workhorse for RANS turbulent heat transfer modelling for decades. Nevertheless, the nuclear community is well aware of its possible shortcomings when considering complex applications as those involving low Prandtl fluids and/or buoyant flows. As a consequence, in the European project THINS (Thermal–Hydraulics of Innovative Nuclear Systems), more advanced closures for the THF have been proposed in order to improve the accuracy of the CFD models dealing with the turbulent heat transfer at low Prandtl number (Shams, 2017a). The models considered within this project include a local turbulent Prandtl number model ( Kays, 1994; Duponcheel et al., 2014), and both an explicit (Manservisi and Menghini, 2014) and an implicit Algebraic Heat Flux Model (AHFM) (Kenjereš et al., 2005; Shams et al., 2014). The general conclusion within the THINS project was that the efforts towards the development of new THF models should be limited, and the focus should be put on further validation of the proposed models in more complex configurations (Shams, 2017a). With respect to the implicit AHFM approach, the model originally proposed in Kenjereš et al. (2005) for natural and mixed convection with unity Prandtl number fluids has been extended for the application to forced convection and low Prandtl fluids in the socalled AHFM-NRG model (Shams et al., 2014). The calibration and assessment of the model have been performed against a set of simple test cases (i.e. forced planar and wavy channel flows, Rayleigh-Bnard convection, vertical heated channel in mixed convection). In addition, this model has been further extended to natural convection at high Rayleigh number Ra in Shams (2017b), resulting in the so called AHFM-NRG+ model. One of the main hampering factors with respect to the further assessment of the proposed THF models in complex configurations is the lack of reference data, both experimental and numerical, for their validation. Therefore, the two European projects SESAME and MYRTE, which are currently ongoing, have the aim of generating such a reference database in order to be able to further assess the proposed THF models against comprehensive validation data (Shams, 2017a). In this context, the present work aims at contribute to the further understanding of the performance of different THF closures in a relatively complex case that is relevant for nuclear applications.
33
In particular the classical Reynolds analogy and the AHFM-NRG+ model have been employed to simulate the turbulent heat transfer in a backward facing step (BFS) flow in both forced and mixed convection with a low Prandtl working fluid. The reference database is a recently published DNS reported in Niemann and Fröhlich (2016). A description of the computational domain and of the numerical settings is reported in Section 2, followed by an overview of the turbulence models adopted in the present RANS calculations in Section 3. Successively, the results obtained for both the forced and the mixed convection cases are illustrated and compared against the reference DNS data in Section 4. Finally, conclusive remarks along with future perspectives are provided in Section 5.
2. Computational domain and simulation settings The computational domain, which has been employed for both the forced and the mixed convection calculations, reproduces the reference DNS domain Niemann and Fröhlich (2016) and is sketched in Fig. 1. The domain represents a vertically oriented BFS geometry with a step height equal to h and an expansion ratio of 1.5. In contrast with the reference DNS, in which a periodic condition is imposed in the spanwise z direction, the computational domain considered in this work is two-dimensional. In terms of boundary conditions, similarly to what has been done in the reference DNS, a fully developed inflow condition has been generated through a periodic simulation of an inlet channel. The velocity and turbulence profiles obtained from the periodic calculation have been imposed at the BFS inlet. For the temperature, a uniform inlet value of 423.15 K has been used. At the outlet section, a Dirichlet condition has been used for the pressure, whilst a homogeneous Neumann condition has been adopted for all the other variables. All the remaining boundaries have been treated as adiabatic no-slip walls, with the exception of the bottom wall downstream of the step, where a heat flux of 41 kW/m2 has been imposed for x < 20h. The working fluid is liquid sodium. The changes in the physical properties of the sodium with respect to the temperature have been neglected, and all the properties have been evaluated from Sobolev (2011) at the reference temperature T ref , which has been taken equal to the inlet temperature of 423.15 K. At this temperature the Prandtl number of the sodium is equal to 0.0088. In addition to the Prandtl number, the other non-dimensional groups defining the problem are the Reynolds number Re ¼ 2hU b =m and the Richardson number Ri ¼ gbDTh=U 2b . The Reynolds number is calculated with respect to the inlet channel width 2h and the bulk inlet velocity U b and is equal to 9610. The temperature difference DT is defined with respect to the imposed heat flux q00 as DT ¼ q00 h=k, where k is the thermal conductivity of the sodium, and is equal to 20 K. In the DNS reference data, two different values of the Richardson number have been considered. The first one is Ri ¼ 0, which is enforced by setting the thermal expansion coefficient b to zero, and corresponds to the forced convection regime. The second Ri value, which is equal to 0.338, corresponds to the mixed convection regime. Both cases have been modelled in the present calculations. All the simulations have been performed using the commercially available software STAR-CCM+ version 11.06 (SIEMENS PLM Software, 2016). It is worth to point out that the considered AHFM model has been implemented within this code in the framework of the THINS project. The governing equations have been solved in their steady-state formulation and the SIMPLE algorithm has been used for pressure–velocity coupling (Patankar and Spalding, 1972). A second order scheme has been employed for the spatial discretisation of all the equations.
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A. De Santis, A. Shams / Annals of Nuclear Energy 117 (2018) 32–44
Fig. 1. Computational domain employed in the present RANS calculations.
3. Turbulence models In the present study, a linear low-Reynolds k e closure has been adopted for the modelling of the turbulent momentum transfer (Lien et al., 1996). This model has been employed since it represents a building block for the implementation of the AHFM-NRG+ model in STAR-CCM+. The transport equations for the turbulent kinetic energy and its dissipation rate are expressed as follows:
DðqkÞ @ ¼ Pk þ Pb qe þ Dt @xj
lt @k rk @xj
and two different closures have been employed for the unknown THF term hui appearing in Eq. (10). The first one is the classical SGDH assumption, which results in:
hui ¼ at
@T @xi
ð11Þ
where the unknown turbulent thermal diffusivity at has been evaluated by invoking the Reynolds analogy, i.e.:
mt
ð1Þ
at ¼
ð2Þ
k Pwall ¼ DF 2 P k þ 2l 2 exp ERe2d d
and the turbulent Prandtl number Pr t has been assumed to be equal to 0.9. The other closure employs an implicit algebraic expression, which is derived from the full second moment transport equation for the THF under the hypothesis of local equilibrium between production and dissipation (Dol et al., 1997). The resulting algebraic expression has the form (Kenjereš and Hanjalic´, 2000):
ð3Þ
hui ¼ C t0
lt ¼ mt q ¼ qf l C l ks
ð4Þ
where aij ¼ ik j 23 dij is the Reynolds stress anisotropy tensor. The three terms in brackets on the right hand side of Eq. (13) correspond to the three physical mechanisms involved in the THF generation, i.e. (Kenjereš and Hanjalic´, 2000):
lþ
DðqeÞ C e1 e 1 ¼ P þP þ C P qC e2 F 2 þ qSy Dt s k wall e3 b s s @ l @e þ lþ t @xj re @xj where
h pffiffiffiffiffiffiffiffi i f l ¼1 exp C d0 Red þ C d1 Red þ C d2 Re2d h i F 2 ¼ 1 C exp Re2t Sy ¼ C w
e2 k
ð5Þ ð6Þ
" max
2 # ‘ ‘ 1 ;1 ‘e ‘e
ð7Þ
@T @U i C t1 ui uj þ C t2 huj þ C t3 bg i h2 þ C t4 aij huj @xj e @xj
k
ð13Þ
uu
Production due to the mean temperature gradient @T=@xj . Production due to the mean rate of strain @U i =@xj . Impact of buoyancy on the turbulent fluctuations through the term bg i h2 .
3=2
‘¼
ð12Þ
Pr t
k
ð8Þ
e
‘e ¼ C ‘ d
ð9Þ
and P k ¼ qui uj @U i =@xj is the mechanical turbulent production, Pb ¼ qbg i hui is the turbulence production due to buoyancy, pffiffiffi pffiffi s ¼ max ke ; C t me ; d is the wall distance, Red ¼ kd=m; Ret ¼ k2 =me and Sy is a source term introduced in the transport equation for e following Yap (1987). The model coefficients have been taken from Shams et al., 2014 and SIEMENS PLM Software, 2016 and are summarised in Table 1. In addition, the energy transport equation has the form:
D qc p T @ @T ¼ qcp hui k Dt @xi @xi
In the formulation reported in Kenjereš et al. (2005), the model has been calibrated for its application in the natural and mixed convection regimes at unity Prandtl number. In the present work, the extended version of the model proposed in Shams et al. (2014) and Shams (2017b), and henceforth referred to as AHFMNRG+, is employed. In the AHFM-NRG+ the model coefficients C t1 and C t3 are expressed as a function of the product of the Prantl and Reynolds number and of the Prandtl and the Rayleigh number, respectively, as detailed in Eqs. (14) and (15).
C t1 ¼ 0:053lnðRePr Þ 0:27
with
C t3 ¼ 4:5 109 ðlog ðRaPrÞÞ þ 2:5 7
RePr > 180 with
ð14Þ
1 > RaPr > 1017 ð15Þ
ð10Þ
Table 1 Coefficients for the low-Reynolds k e model.
rk
re
Cl
Ct
C e1
C e2
C e3
C d0
C d1
C d2
C‘
Cw
C
E
D
1
1.3
0.09
1
1.44
1.92
0.33
0.091
0.0042
0.00011
2.55
0.83
0.3
0.00375
1
35
A. De Santis, A. Shams / Annals of Nuclear Energy 117 (2018) 32–44
Due to these modifications to the model coefficients, the applicability of the AHFM-NRG+ has been extended to the forced convection regime and to buoyant flows with high Rayleigh number. Furthermore, the dependency of the coefficients on Pr makes the model suitable for the application to fluids characterised by a broad range of Prandtl numbers, including liquid metals. The logarithmic dependency of C t1 and C t3 on RePr and RaPr, respectively, is depicted in Fig. 2. In addition, it should be noted that values of RePr lower than the limit of 180 reported in Eq. (14) can be encountered. This is often the case when comparing to DNS databases with low Prandtl fluids, where relatively low Reynods number are usually adopted in order to reduce the computational costs associated with the calculations. In the present case, from the settings reported in Section 2, it results RePr 84:6. Therefore, the value of C t1 for the present calculations has been evaluated by considering RePr ¼ 180 in Eq. (14). Further calibration of the model in order to extend the correlation for this coefficient to lower values of RePr would be desirable in the future. Furthermore, it should be noted that the closure reported in
where
2
sh ¼ 2heh and sm ¼ k=e. This assumption has been observed to
be reasonable in several cases, as reported in Shams (2017a), Hanjalic´ et al. (1996), Kozuka et al. (2009), and greatly simplifies the application of the AHFM to the simulation of complex flows. A summary of the coefficients for the k e h2 AHFM-NRG+ model employed in the present work is reported in Table 2. 4. Results This section discusses the results obtained for both the forced convection and the natural convection cases. As a first step, a mesh sensitivity study has been carried out for the forced convection case and is detailed in Section 4.1. Successively, the results obtained for the flow and the thermal fields in both regimes are reported and critically analysed by means of a detailed comparison with the reference DNS data. 4.1. Mesh sensitivity
2
Eq. (13) requires the evaluation of the temperature variance h . Therefore, a transport equation for this quantity is introduced, having the form
Dqh2 @ ¼ 2Pt 2qeh þ @xi Dt
"
k l þ t cp rh2
!
@h2 @xi
#
ð16Þ
@T where P t ¼ qhui @x is the temperature variance production term i
and eh is its dissipation rate. A suitable model for the latter variable is needed in order to close the model. In the approach followed in Kenjereš and Hanjalic´ (2000), an additional transport equation is solved for the temperature variance dissipation, resulting in a four-equation k e h2 eh model. However, such a transport equation presents a number of unclosed terms and its closure results in the introduction of significant additional uncertainty in the model (Kenjereš et al., 2005; Otic´ et al., 2005). Also, the fourequation model has shown convergence problems in some cases, as reported in Shams (2017a). In order to overcome these issues, a simplified approach has been employed in both the AHFM-NRG (Shams et al., 2014) and the AHFM-NRG+ (Shams, 2017b) formulations, and retained in the present work. In this approach, the temperature variance dissipation is evaluated through the assumption of a constant thermal to mechanical time scale ratio R (Hanjalic´ et al., 1996), resulting in a three-equation k e h2 model. The thermal to mechanical time scale ratio is defined as:
R¼
sh sm
ð17Þ
All the numerical grids used in the present study have been generated in STAR-CCM+ version 11.06 and consist of structured quadrilateral cells. The mesh sensitivity study has been carried out by employing four different grids (see Fig. 3) ranging from about 11 k–175 k elements, as reported in Table 3. All the considered meshes enforce the condition yþ < 1 at the walls everywhere in the domain. A typical feature of the BFS flow in the forced convection regime is the flow detachment and reattachment that takes place behind the step. The streamwise location of the reattachment point xr is used as a parameter to evaluate the sensitivity of the numerical results to the mesh resolution and the calculated xr values for the four different grids are reported in Table 3. In addition, the table reports the relative deviation of the calculated xr with respect to the most refined mesh, i.e. Mesh 4. In general, the relative deviation / is defined as:
/ ¼
/calc /ref /ref
ð18Þ
Table 2 Coefficients of the AHFM-NRG+ model. C t0
C t1
C t2
C t3
C t4
rh2
R
0.2
Eq. (14)
0.6
Eq. (15)
0.0
1
0.5
Fig. 2. C t1 as a function of RePr (left) and C t3 (right) as a function of RaPr in the AHFM-NRG+ model.
36
A. De Santis, A. Shams / Annals of Nuclear Energy 117 (2018) 32–44
Fig. 3. View of the four grids considered in the mesh sensitivity study.
Table 3 Mesh sensitivity: cell count and predicted reattachment point location.
Cell count xr =h
x
r
Mesh 1
Mesh 2
Mesh 3
Mesh 4
10875 6.62 2.5%
20771 6.70 1.4%
44370 6.76 0.4%
177480 6.79 –
where /calc is the calculated value for the generic quantity / and /ref is the reference value for the same quantity. From Table 3, it can be seen that the relative deviation of xr for Mesh 3 with respect to Mesh 4 is below 1%. In order to further assess the impact of the mesh resolution on the calculated flow and thermal fields, the profiles of the streamwise U and wall-normal V velocity components
evaluated at different streamwise locations are shown in Fig. 4. In addition, the same profiles for the turbulent kinetic energy k and the non-dimensional temperature T^ ¼ ðT T ref Þ=DT are reported in Fig. 5. It can be seen that the differences in the calculated values between Mesh 3 and Mesh 4 tend to be practically negligible, even for the most sensitive variable with respect to mesh resolution, which appear to be the turbulent kinetic energy. Therefore, Mesh 3 has been employed for all the calculations performed in the present work.
4.2. Flow field The results obtained for the turbulent flow field in forced and mixed convection are reported in Sections 4.2.1 and 4.2.2, respectively.
Fig. 4. Mesh sensitivity: calculated streamwise (top) and wall-normal (bottom) velocity components for different mesh resolutions.
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Fig. 5. Mesh sensitivity: calculated turbulent kinetic energy (top) and temperature (bottom) for different mesh resolutions.
Fig. 6. Forced convection case: streamwise velocity contours together with streamlines in the recirculation zone region.
Fig. 7. Forced convection case: skin friction coefficient along the bottom wall – solid lines current calculations, symbols: DNS.
4.2.1. Forced convection In the forced convection case, the typical flow features for a BFS configuration are observed, i.e. flow separation taking place behind the step, and reattachment of the flow and development of an
attached boundary layer further downstream. The calculated flow field is represented in Fig. 6 in terms of the streamwise velocity contours. The contours are shown on a section of the computational domain ranging from x=h ¼ 2:0 to x=h ¼ 20:0. It can be seen that the presence of a large clockwise recirculation zone behind the step is reproduced by the present RANS simulation. The calculated streamwise reattachment location is equal to 6.76h, compared with a value of 7.01h reported in the reference DNS. Furthermore, from the streamlines reported in Fig. 6, it can be inferred that a smaller counter-clockwise recirculation zone is present in the corner between the step and the bottom wall. In the reference DNS this secondary recirculation zone extends for the entire step height in the wall-normal direction, whilst its extent in the present RANS calculations appears reduced to roughly half of the step height. The calculated skin friction coefficient C f along the bottom wall is reported in Fig. 7. It can be observed that the results obtained with the linear low-Reynolds k e model in the present calculations are in a fair agreement with the DNS data for this quantity, with the largest discrepancies being found in the region very close to the step and in the value of the negative dip present within the recirculation zone. A more detailed picture of the predicted flow field at different streamwise locations within the domain is given in Figs. 8 and 9,
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Fig. 8. Forced convection case: streamwise (top) and wall normal (bottom) velocity components – solid lines: current calculations, symbols: DNS.
Fig. 9. Forced convection case: turbulent kinetic energy (top) and uv Reynolds stress component (bottom) – solid lines: current calculations, symbols: DNS.
which show the calculated streamwise and wall-normal velocity components, and the turbulent kinetic energy and the uv Reynolds stress, respectively. The streamwise positions included in these plots correspond to the locations where the reference DNS results are made available. A close agreement between the present calculations and the reference data is observed for the velocity field, whilst slightly larger discrepancies are present in the turbulent field. In particular, the most evident deviations with respect to the reference DNS are observed in the turbulent kinetic energy profiles within the recirculation zone at x=h ¼ 3 and x=h ¼ 6. Nevertheless, overall it can be concluded that the RANS approach
employed in the present calculations is able to provide a satisfactory representation of the turbulent flow field observed in the forced convection regime. 4.2.2. Mixed convection The calculated flow field for the mixed convection case is depicted in Fig. 10 in terms of the streamwise velocity contours. Since in this case the temperature cannot be regarded as a passive scalar, the figure includes the results obtained with both the SGDH and the AHFM-NRG+ closures for the THF. In contrast with the forced convection regime, the flow field in this case is charac-
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Fig. 10. Mixed convection case: streamwise velocity contour plot – SGDH(top) and AHFM-NRG+ (bottom).
to the turbulent fields, the overall prediction for k and uv is still satisfactory, although some non-negligible deviations from the reference data are observed in the turbulent variables, especially within the recirculation zone at x=h ¼ 3 and in the locations further downstream (x=h ¼ 12 and x=h ¼ 15). Moreover, more significant differences between the SGDH and the AHFM-NRG+ results are observed for these variables with respect to the predictions for the velocity field. Overall, it can be concluded that both turbulence models are able to provide a reasonable prediction of the flow field for the mixed convection case. 4.3. Thermal field
Fig. 11. Mixed convection case: skin friction coefficient along the bottom wall – solid line: SGDH, dashed line: AHFM-NRG+, symbols: DNS.
terised by the development of a jet at the bottom wall, due to the acceleration of the fluid caused by buoyancy. As a consequence, away from the bottom wall the streamwise velocity is reduced significantly, until the flows detaches from the upper wall forming a small recirculation zone. Furthermore, the recirculation zone found behind the step is significantly affected by the buoyancy forces, its extent in the streamwise direction being noticeably smaller with respect to the Ri ¼ 0 case. In addition, this recirculation zone is completely detached from the bottom wall, as it can be inferred from the positive values of the skin friction coefficient reported in Fig. 11. From the same figure, it can be observed that both turbulence models result in a fairly accurate prediction of C f , with the SGDH model being slightly closer to the DNS data in the dip found around x=h ¼ 2 and the AHFM-NRG+ model performing better further downstream. The predicted flow field within the domain at various streamwise locations is represented in Figs. 12 and 13, which depict the streamwise and wall-normal velocity components, and the turbulent kinetic energy and the uv Reynolds stress, respectively. Similarly to what has been observed in the forced convection regime, also in this case the predicted velocity field appears to be in a good agreement with the DNS data. Also, the SGDH and the AHFM-NRG+ models give quite similar results for the velocity field. With respect
4.3.1. Forced convection A contour plot of the calculated temperature field for the forced convection case is shown in Fig. 14. The SGDH and AHFM-NRG+ models are qualitatively in good agreement with each other, with the hot fluid located in the corner between the step and the heated wall. A strong temperature increase is observed moving from the top wall towards the heated bottom wall. A relatively low wall temperature is observed within the recirculation zone, whilst downstream of the reattachment point the temperature along the heated wall increases due the imposed heat flux. More insights with respect to the quantitative differences between the SGDH and the AHFM-NRG+ results can be gained from the temperature profiles reported in Fig. 15. From these profiles it can be observed that the SGDH approach consistently predicts a lower temperature with respect to the reference data in the near-wall region, whilst the algebraic model is in a closer agreement with the DNS results in all the considered axial locations with the exception of x=h ¼ 0. A quantification of the deviations of the two models with respect to the reference data is given in Table 4, reporting the calculated average deviation T^ at different streamwise locations. The average deviation along the vertical direction for a generic quantity / is evaluated as:
/ ¼
1 ba
Z
b a
/ ðyÞ dy
ð19Þ
where the local deviation / ðyÞ is defined according to Eq. (18). It is evident how the AHFM-NRG+ model gives consistently more accurate results with respect to the SGDH assumption at the different streamwise locations, with the exception of the x=h ¼ 0 position. Overall, the arithmetic average of the temperature deviation over
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Fig. 12. Mixed convection case: streamwise (top) and wall normal (bottom) velocity components – solid lines: current calculations, symbols: DNS.
Fig. 13. Mixed convection case: turbulent kinetic energy (top) and uv Reynolds stress component(bottom) – solid line: SGDH, dashed line: AHFM-NRG+, symbols: DNS.
the considered streamwise locations is reduced from 11.97% with the latter model to 8.88% with the AHFM-NRG+. It is worth to point out that the figures resulting from the arithmetic averaging in the streamwise direction can be influenced by the dimensions of the domain and the selection of the streamwise locations at which the comparison takes place. Nevertheless, these streamwiseaveraged values can be regarded as an approximate estimation of the difference in the overall accuracy between the two models for a given quantity. In addition, it should be noted that a reliable prediction of the turbulent flow field is a prerequisite for an
accurate evaluation of the turbulent heat transfer, regardless of the modelling approach employed for the THF. In fact, the overall accuracy in the prediction of the thermal field is affected by the uncertainties associated with both the turbulent momentum and the heat flux closures. Therefore, it is extremely difficult to single out the contribution of the sole THF model to the overall deviation from the reference data, and compensating errors might occur. This has to be kept in mind when performing a comparison between the results obtained with two different THF models and evaluating the resulting discrepancies with respect to the reference data.
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Fig. 14. Forced convection case: temperature contour plot – SGDH (top) and AHFM-NRG+ (bottom).
Fig. 15. Forced convection case: temperature profiles – solid line: SGDH, dashed line: AHFM-NRG+, symbols: DNS.
Table 4 Forced convection case: deviation of the predicted temperature with respect to the reference data at different streamwise locations. ^ SGDH ^ AHFM-NRG+ T T x=h ¼ 0 x=h ¼ 3 x=h ¼ 6 x=h ¼ 9 x=h ¼ 12 x=h ¼ 15
3.45% 22.37% 16.82% 12.09% 8.97% 8.14%
11.54% 14.59% 11.57% 8.02% 4.96% 2.61%
Average
11.97%
8.88%
A detailed analysis of the predicted heat transfer at the wall can be obtained from the wall temperature T^ w and the Nusselt number Nu profiles along the heated wall, which are shown in Fig. 16. The averaged deviation for these two quantities is reported in Table 5. From the figure, it can be observed how the wall temperature, and consequently the Nusselt number, predicted by the AHFM-NRG+ is in a significantly better agreement with the reference data with respect to the SGDH approach, resulting in an average deviation from the DNS results around 5%. 4.3.2. Mixed convection The iso-contours of the temperature for the mixed convection case are reported in Fig. 17. The most evident impact of the buoyancy effects on the the temperature is a general increase in the heat transfer, which results in lower temperature values with respect to the forced convection case. As a result of the changes in the flow field behind the step, the higher temperature region
Table 5 Forced convection case: average deviation of the predicted wall temperature and Nusselt number with respect to the reference data.
T^ w Nu
kk SGDH
kk AHFM-NRG+
13.81%
5.50%
15.90%
5.25%
is no longer located in the step corner, but it is shifted slightly downstream. Moreover, the thickness of the thermal boundary layer is reduced with respect to the Ri ¼ 0 case. Overall, the SGDH and the AHFM-NRG+ models are in a good qualitative agreement. A more detailed quantitative comparison is depicted in Fig. 18, showing the calculated temperature profiles at various streamwise locations. From this figure, it can be observed that both turbulence models are in a good agreement with the reference DNS data. Also, the differences between the results obtained with the two models are less evident with respect to the forced convection regime. In fact, due to the very low Pr value and the relatively low Reynolds number of the considered case, the mean diffusive and advective heat fluxes are significantly higher than the THF, and this is particularly evident in the mixed convection case (Niemann and Fröhlich, 2016). As a consequence, the impact of the considered THF model on the predicted temperature field is less evident in the mixed convection case with respect to the forced convection regime. Nevertheless, non-negligible discrepancies in the calculated temperature profiles are observed, as highlighted by the average temperature deviation at different streamwise locations reported in Table 6. In this case, for the reasons stressed above, the superiority of the AHFM-NRG+ over the Reynolds analogy
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Fig. 16. Forced convection case: wall temperature (left) and Nusselt number (right) at the heated wall – solid line: SGDH, dashed line: AHFM-NRG+, symbols: DNS.
Fig. 17. Mixed convection case: temperature contour plot – SGDH (top) and AHFM-NRG+ (bottom).
Fig. 18. Mixed convection case: temperature profiles – solid line: SGDH, dashed line: AHFM-NRG+, symbols: DNS.
assumption is less evident with respect to the Ri ¼ 0 case, although the results obtained with the former model are still in an overall better agreement with the reference DNS data. The heat transfer at the wall is analysed in Fig. 19 in terms of the calculated wall temperature and the Nusselt number profiles. Both the considered turbulence models are in a fairly good agreement with the reference DNS data. More in detail, it can be observed that the AHFM-NRG+ model is able to correctly predict the location and the value of the wall temperature peak located at x=h 2:5. Between x=h 3:0 and x=h 10:0 the SGDH is closer to the reference data, whilst the AHFM-NRG+ gives a more accurate wall temperature prediction further downstream. The same beha-
Table 6 Mixed convection case: deviation of the predicted temperature with respect to the reference data at different streamwise locations. ^ SGDH ^ AHFM-NRG+ T T x=h ¼ 0 x=h ¼ 3 x=h ¼ 6 x=h ¼ 9 x=h ¼ 12 x=h ¼ 15
16.09% 5.47% 1.63% 2.20% 7.46% 15.94%
14.90% 10.09% 5.58% 2.71% 1.47% 5.79%
Average
8.13%
6.76%
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Fig. 19. Mixed convection case: wall temperature (left) and Nusselt number (right) at the heated wall – solid line: SGDH, dashed line: AHFM-NRG+, symbols: DNS.
Table 7 Mixed convection case: average deviation of the predicted wall temperature and Nusselt number with respect to the reference data.
T^ w Nu
kk SGDH
kk AHFM-NRG+
4.75%
3.16%
4.50%
3.21%
availability of such reference data is regarded as of paramount importance in order to perform a thorough assessment of different turbulent heat flux closures.
Conflict of interest None declared.
viour is observed for the prediction of the Nusselt number. Also for these two quantities, the differences between the two THF closures are less evident with respect to the forced convection regime. Nevertheless, the AHFM-NRG+ still displays a lower average deviation from the reference data, as shown in Table 7.
5. Conclusions The paper focuses on the assessment of a recently proposed turbulent heat flux algebraic closure, the so called AHFM-NRG+ model, in its application to the modelling of a vertical backward facing step flow in both forced and mixed convection at low Prandtl number. The results obtained with the AHFM-NRG+ have been compared with a reference DNS database, along with the classical Reynolds analogy assumption. A low-Reynolds linear k e model has been employed for the turbulent momentum transfer, and a reasonable prediction of the turbulent flow field has been obtained in both the forced and the mixed convection regimes. In the former case, the AHFM-NRG+ showed a significant improvement in the prediction of the thermal field with respect to the Reynolds analogy. Moreover, the heat transfer at the heated wall has been predicted with a considerable accuracy with the proposed algebraic model. In the mixed convection regime, the superiority of the AHFM-NRG+ with respect to the Reynolds analogy assumption is less evident, and this is mainly due to the reduced contribution of the turbulent heat flux to the overall heat transfer in this case. Nevertheless, an improvement in the accuracy of the predicted thermal field and wall heat transfer has been observed in the mixed convection regime as well. The present work represents the first application of the AHFMNRG+ model to a complex flow configuration such as the vertical backward facing step at low Prandtl number, and confirms the promising potential of the AHFM-NRG+ as a tool to improve the prediction of the turbulent heat transfer in challenging flows of industrial interest involving low Prandtl fluids. The main bottleneck in the further assessment of the model is the lack of reliable reference databases for complex configurations, especially including low Prandtl fluids and/or buoyancy effects. Therefore, the
Acknowledgements The work described in this paper has been funded by the Dutch Ministry of Economic Affairs and from the Euratom research and training programme 2014–2018 under grant agreements Nos. 654935 (SESAME) and 662186 (MYRTE). The authors are also grateful to Prof. M. Niemann and Prof. J. Fröhlich from TU Dresden for kindly providing the reference DNS database. References Bauer, W., Haag, O., Hennecke, D.K., 2000. Accuracy and robustness of nonlinear eddy viscosity models. Int. J. Heat Fluid Flow 21 (3), 312–319. Dol, H.S., Hanjalic´, K., Kenjereš, S., 1997. A comparative assessment of the secondmoment differential and algebraic models in turbulent natural convection. Int. J. Heat Fluid Flow 18 (1), 4–14. Duponcheel, M., Bricteux, L., Manconi, M., Winckelmans, G., Bartosiewicz, Y., 2014. Assessment of RANS and improved near-wall modeling for forced convection at low Prandtl numbers based on LES up to Ret=2000. Int. J. Heat Mass Transf. 75, 470–482. Felten, F., Fautrelle, Y., Du Terrail, Y., Metais, O., 2004. Numerical modelling of electromagnetically-driven turbulent flows using LES methods. Appl. Math. Model. 28 (1), 15–27. Frazer, D., Stergar, E., Cionea, C., Hosemann, P., 2014. Liquid metal as a heat transport fluid for thermal solar power applications. Energy Procedia 49, 627– 636. GIF, Introduction to Generation IV nuclear energy system and the international forum.,http://www.gen-4.org (2010). Grötzbach, G., 2007. Anisotropy and buoyancy in nuclear turbulent heat transfer – critical assessment and needs for modelling. Forschungszentrum Karlsruhe FZKA 7363. Grötzbach, G., 2013. Challenges in low-Prandtl number heat transfer simulation and modelling. Nucl. Eng. Des. 264, 41–55. Hanjalic´, K., 1999. Second-moment turbulence closures for CFD: needs and prospects. Int. J. Comput. Fluid Dyn. 12 (1), 67–97. Hanjalic´, K., Kenjereš, S., Durst, F., 1996. Natural convection in partitioned twodimensional enclosures at higher Rayleigh numbers. Int. J. Heat Mass Transf. 39 (7), 1407–1427. Kays, W.M., 1994. Turbulent Prandtl number where are we? J. Heat Transfer 116 (2), 284–295. Kenjereš, S., Hanjalic´, K., 2000. Convective rolls and heat transfer in finite-length Rayleigh-Bénard convection: a two-dimensional numerical study. Phys. Rev. E 62 (6), 7987–7998. Kenjereš, S., Gunarjo, S., Hanjalic´, K., 2005. Contribution to elliptic relaxation modelling of turbulent natural and mixed convection. Int. J. Heat Fluid Flow 26 (4), 569–586.
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