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Numerical simulation of nuclear pebble bed conﬁgurations A. Shams a,∗ , F. Roelofs a , E.M.J. Komen a , E. Baglietto b a b

Nuclear Research and Consultancy Group (NRG), Petten, The Netherlands Massachusetts Institute of Technology (MIT), United States

h i g h l i g h t s • • • • •

Numerical simulations of a single face cubic centred pebble bed are performed. Wide range of turbulence modelling techniques are used to perform these calculations. The methods include 1-DNS, 1-LES, 3-Hybrid (RANS/LES) and 3-RANS models, respectively. The obtained results are extensively compared to provide guidelines for such ﬂow regimes. These guidelines are used to perform reference LES for a limited sized random pebble bed.

a r t i c l e

i n f o

Article history: Received 5 October 2014 Accepted 5 November 2014

a b s t r a c t High Temperature Reactors (HTRs) are being considered all over the world. An HTR uses helium gas as a coolant, while the moderator function is taken up by graphite. The fuel is embedded in the graphite moderator. A particular inherent safety advantage of HTR designs is that the graphite can withstand very high temperatures, that the fuel inside will stay inside the graphite pebble and cannot escape to the surroundings even in the event of loss of cooling. Generally, the core can be designed using a graphite pebble bed. Some experimental and demonstration reactors have been operated using a pebble bed design. The test reactors have shown safe and efﬁcient operation, however questions have been raised about possible occurrence of local hot spots in the pebble bed which may affect the pebble integrity. Analysis of the fuel integrity requires detailed evaluation of local heat transport phenomena in a pebble bed, and since such phenomena cannot easily be modelled experimentally, numerical simulations are a useful tool. As a part of a European project, named Thermal Hydraulics of Innovative Nuclear Systems (THINS), a benchmarking quasi-direct numerical simulation (q-DNS) of a well-deﬁned pebble bed conﬁguration has been performed. This q-DNS will serve as a reference database in order to evaluate the prediction capabilities of different turbulence modelling approaches. A wide range of numerical simulations based on different available turbulence modelling approaches are performed and compared with the reference q-DNS. This paper reports a detailed comparison of LES, Hybrid (RANS/LES) and RANS models with the reference q-DNS. These simulations are performed for a well-deﬁned single face cubic centred pebble conﬁguration. The obtained ﬂow and thermal ﬁelds are extensively analyzed to understand the ﬂow physics in such complex ﬂow regime. Furthermore, lessons learned from these simulations are summarized in the form of guidelines for such complex ﬂow conﬁgurations. In addition, following these guidelines, a strategy has been developed to perform large eddy simulations of a realistic limited sized random pebble bed. © 2014 Elsevier B.V. All rights reserved.

1. Introduction

∗ Corresponding author. Tel.: +31 224568152. E-mail addresses: [email protected] (A. Shams), [email protected] (F. Roelofs), [email protected] (E.M.J. Komen), [email protected] (E. Baglietto).

High Temperature Reactors (HTR) are being considered for deployment around the world. An HTR uses helium gas as a coolant, while the moderator function is accomplished by graphite. The fuel is embedded in the graphite moderator and can sustain extremely high temperatures (Janse van Rensburg et al., 2006; Janse van Rensburg and Kleingeld, 2011), basically preventing the fuel from

http://dx.doi.org/10.1016/j.nucengdes.2014.11.002 0029-5493/© 2014 Elsevier B.V. All rights reserved.

Please cite this article in press as: Shams, A., et al., Numerical simulation of nuclear pebble bed conﬁgurations. Nucl. Eng. Des. (2014), http://dx.doi.org/10.1016/j.nucengdes.2014.11.002

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Fig. 1. CFD triangle (left) and different pebble bed distributions (right) (Shams et al., 2013d).

melting. An important feature associated with the HTRs, is the high coolant temperature that can be achieved and therefore high electric efﬁciency. The high coolant outlet temperature also leads to optimal suitability for coupling the reactor to an industrial process requiring high temperatures. The core can be designed using a graphite pebble bed. Some experimental reactors have been operated over the world using this design (IAEA, 2001) and have shown safe and efﬁcient operation, however questions have raised about the occurrence of potential local hot spots in the pebble bed, possibly affecting the pebble integrity (Moormann, 2008). Understanding this phenomenon is not an easy task and poses difﬁcult challenges. Heat transfer around a curved surface shows a complex behaviour, which can be affected by both laminar and turbulent characteristics and by the effect of ﬂow curvature. The narrow ﬂow passages through the gaps between the pebbles can have concave and convex conﬁgurations which will cause suppression or stimulation of the turbulence level (Hassan, 2008). In addition, pressure gradients strongly affect the boundary layers. Transition from laminar to turbulent, wake, ﬂow separation and its respective reattachment make this ﬂow conﬁguration very complex. Hence a detailed evaluation of the pebble bed ﬂow physics needs to be performed. Thermal-hydraulic aspects of a pebble bed conﬁguration can be predicted by a porous medium or realistic approach. In a porous medium approach, an averaged concept of porosity is applied to simulate the closely packed geometry (Janse van Rensburg and Kleingeld, 2011). Whereas, in a realistic approach, every pebble is realistically and separately modelled in a simulation using direct numerical simulation (DNS), large eddy simulation (LES), Reynolds averaged Navier–Stokes (RANS) calculations or hybrid RANS/LES. For a reactor scale simulation, probably only a porous medium approach is feasible the coming decades. Although such porous medium approaches have been developed, there is still room for improvement and furthermore, validation is still pending. Experimental studies or validated CFD approaches could be helpful in validating such a porous medium approach. On the experimental side, the phenomena cannot be easily studied, in the recent past, few attempts have been made but quite a small amount of information is still available (Dominguez et al., 2008; Lee and Lee, 2008, 2010). On the other hand, to obtain accurate CFD results based on DNS or LES for a realistic pebble bed ﬂow is computationally

too expensive and is not foreseeable in the near future. A number of attempts (In et al., 2005, 2006, 2008; Lee et al., 2005, 2007; Taylor et al., 2002; Matzner, 2004; PBMR, 2005; Wu et al., 2010; Yesilyurt, 2003) have been made to model the complex ﬂow around curved pebble surfaces by using Reynolds Averaged Navier Stokes (RANS) CFD approaches, but have not been particularly successful. The troubles evidenced by the RANS methods are not surprising, and related to the limitations imposed by the various closures, as explained in (Patel and Sotiropoulos, 1997; Wu et al., 2010). Hassan (2008) investigated the ﬂow distribution in an aligned pebble geometry consisting of 27 pebble spheres by using an LES modelling approach. In the recent past, Lee et al. (2007) also used LES to investigate idealized pebble bed conﬁgurations. In order to check the feasibility and prediction capabilities of LES for such complex ﬂow physics, unfortunately, no detailed validation with a reference database was provided. To gain trust in the numerical models and simulations, their validation is always an important step. Recently, Shams et al. (2012, 2013c) selected an arranged pebble bed in order to generate a benchmark numerical database to validate the numerical strategies for such complex ﬂow regimes. As a result, a quasi-direct numerical simulation (qDNS) of a single face cubic pebble bed has been performed and an extensive database of velocity and thermal ﬁeld is now available to assess the prediction capabilities of available different modelling approaches (Shams et al., 2013c,d). Moreover, Shams et al. (2013a) have proposed a stepwise validation procedure involving different types of pebble bed distributions including an idealized pebble bed, a limited size random pebble bed, a limited size random pebble bed with wall effects and a real pebble bed, respectively, as shown in Fig. 1. It is worth mentioning that the transfer to more realistic applications as well as the introduction of other relevant phenomena, such as transport and deposition of graphite dust, cannot be included in the given validation effort as these range beyond its scope. In the present study, a wide range of numerical simulations are performed for two different pebble bed conﬁgurations, i.e. (i) single cubic pebble bed and (ii) limited sized random pebble bed. Numerical methods and results related to the single cubic pebble bed are given in Section 2. Whereas, details regarding the limited size random pebble bed are reported in Section 3. This section is followed by the conclusions.

Please cite this article in press as: Shams, A., et al., Numerical simulation of nuclear pebble bed conﬁgurations. Nucl. Eng. Des. (2014), http://dx.doi.org/10.1016/j.nucengdes.2014.11.002

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Fig. 2. (Left) Pebbles distribution for a single face cubic centred domain. (Right) Isometric view of the computational domain.

2. Idealized pebble bed 2.1. Computational domain A well-deﬁned single Face Cubic Centred (FCC) arrangement of spherical pebbles is considered as a ﬂow conﬁguration. By following the speciﬁcation of PMBR-250MWth (Lee and Lee, 2008, 2010), the diameter of the pebbles is kept 60 mm. This single cubic FCC domain consists of half spherical pebbles at each face and 1/8th at each corner (see Fig. 2). The length of the domain is 91.92 mm. Furthermore, this conﬁguration has an inter-pebble gap of 5 mm which gives a porosity level of 0.42, close to various experiments performed worldwide as explained in Shams et al. (2012, 2013c,d). Nonetheless, in reality pebbles are randomly arranged, hence, the pebble bed conﬁguration selected in this part of the paper is called an idealized pebble bed. 2.2. Flow parameters and boundary conditions Following the PBMR-250MWth conﬁguration (Lee and Lee, 2008, 2010), helium is adopted as a working ﬂuid. In addition, all the ﬂow parameters which are considered for the present case are given in Table 1. It is worth mentioning that instead of using the original parameters, the mass ﬂow rate and heat ﬂux are scaled by the factor of seven (with respect to the original conﬁguration) in order to reach a feasible cell count of q-DNS for the available computing resources (for details see (Shams et al., 2013c,d)). The scaling is performed in such a way that the overall ﬂow topology is kept same, see (Shams et al., 2012). In addition, periodic boundary conditions (in all three directions) are imposed for mass ﬂow rate and temperature. 2.3. Numerical methods and turbulence modelling In order to generate guidelines for the use of turbulence models, a wide range of turbulence modelling strategies are considered and are discussed below. Table 1 Flow parameters. Properties

Value

Units

Density Viscosity Thermal conductivity Speciﬁc heat Mass ﬂow rate Inlet temperature Heat ﬂux Pressure

5.36 3.69 × 10−5 0.3047 5441.6 0.01607 773 8317 8.5

kg/m3 N s/m2 W/m K J/kg K kg/s K W/m2 MPa

3

As a ﬁrst step, quasi-direct numerical simulations (q-DNS) of FCC pebble bed are performed. Generally higher order numerical schemes are adapted to reduce the numerical errors to generate a high quality DNS. However for the present study, a second order central scheme with boundedness has been used for spatial discretization along with a second order implicit scheme for temporal discretization. As a result, it gives a “reasonable” but not fully veriﬁable conﬁdence of its accuracy like the DNS standards would require. Therefore, this method is called a quasi-direct numerical simulation (q-DNS). For details readers are referred to Shams et al. (2012, 2013c,d). As an outcome of this q-DNS a large database (qualitative and quantitative) has been generated. This q-DNS database is considered as a reference to validate available turbulence models. Three different turbulence modelling strategies are considered to perform the simulations of this FCC domain. These modelling approaches include LES, hybrid RANS/LES methods and URANS. A wall-adapting local eddy-viscosity (WALE) modelling approach is selected for LES calculations. The wall-adapting local eddy-viscosity (WALE) model developed by Nicoud and Ducros (1999) is selected. The WALE model is speciﬁcally designed to return the correct wall asymptotic y3 behaviour of the SGS viscosity (Nicoud and Ducros, 1999). The model is based on the square of the velocity gradient tensor and accounts for the effect of both the strain and the rotation rate to obtain the local eddy viscosity, which can be very important especially for wall-bounded ﬂows. Since no dynamic procedure or explicit ﬁltering is used near the wall, this model works with both structured and unstructured meshes. The equations solved for LES are obtained by applying a ﬁlter to the Navier–Stokes equations. The ﬁltering operation results in a turbulent stress tensor ( ij ) which represents the sub-grid scale stresses and is deﬁned as: ij −

1 ı = −2t S¯ ij 3 kk ij

where t is the sub-grid-scale (SGS) turbulent viscosity and S¯ ij is the rate of strain tensor. A SGS model is required to model the SGS turbulent viscosity t . In the present work, the eddy viscosity is modelled as:

t = LS2

Sijd =

1 2

g¯ ij =

∂U¯ i ∂xj

S¯ ij S¯ ij

Sijd Sijd

5⁄2

g¯ ij2 − g¯ ji2 −

3⁄2

+ S¯ ijd S¯ ijd

1 2 ıijg¯ kk 2

5⁄4

Ls is given by Cω V1/3 . Cω is a constant with value 0.544 and V represents the cell volume. From hybrid RANS/LES turbulence modelling view point, detached eddy simulation (DES) type method is considered for the present study. A DES formulation uses a blend of unsteady (U) RANS and LES, as it attempts to treat near-wall regions in a (U) RANSlike manner, and treat the rest of the ﬂow in an LES-like manner. A good overview of the development of the hybrid methods and variations to the theme is given by Spalart (2009) and Fröhlich and Terzi (2008). The RANS eddy viscosity transport equation contains a turbulence destruction term that is a function of the wall distance (d). In the subsequent DES approach developed by Spalart et al. (1997), the wall distance was replaced by a length scale dependent on the grid size () which modiﬁes the RANS model into a LESSGS model. In this DES model the length scale switches between wall distance (RANS) and grid size (LES). For the current ﬂow conﬁguration the latest formulations of DES are selected, which are

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delayed detached eddy simulation (DDES) and its improved variant, i.e. Improved DDES (IDDES). Moreover, for the IDDES formulation two combinations based on SST k–ω and Spalart Allmaras models are considered, and are denoted as IDDES-SST and IDDES-SA, respectively. Whereas, only a SST k–ω model is considered for DDES formulation (i.e. DDES-SST), for details see (Shams et al., 2013b). For URANS calculations, the selection of a turbulence model is very challenging. In the past (In et al., 2005, 2006, 2008; Lee et al., 2005, 2007; Taylor et al., 2002; Yesilyurt, 2003; Matzner, 2004; Wu et al., 2010), several turbulence models have been used for a similar type of ﬂow conﬁgurations and the results have concluded that among them SST k–ω performs generally well. However, no extensive evaluation and/or validation has been performed for pebble bed type applications. From ﬂow physics view point, the q-DNS study of Shams et al. (2013c,d) has shown that for the present ﬂow conﬁguration, although the pebbles are arranged in a structured manner, the ﬂow presents a complex, and difﬁcult to predict, behaviour, with peaked velocity and stagnation/separation regions in combination with ﬂow transition. Furthermore, heat transfer around a curved surface shows a complex behaviour, which can be affected by transitional ﬂow characteristics and by the effect of ﬂow curvature. This all together makes this ﬂow conﬁguration extremely challenging for RANS type turbulence models. Since the turbulence is strongly affected by the strong curvature of the pebbles and as a result non-linear RANS models are expected to respond better to the streamline curvature than eddy viscosity models. In addition, linear models do not tend to model normal stresses properly. This eventually results in production of turbulence which is non-physical. Hence, for the present study two different types of non-linear models are selected, i.e. a Reynolds stress model (RSM) and a cubic k–ε model, in combination with a linear SST k–ω model to provide an extensive evaluation of RANS models. For further details regarding these selected turbulence models, readers are referred to the STAR-CCM+ manual (STARCCM+, 2011). All the numerical calculations presented in this paper are performed by using the commercially available code STAR-CCM+ (STAR-CCM+, 2011). For q-DNS and LES cases, a second order central scheme with 5% boundedness has been used for spatial discretization. However, for hybrid (RANS/LES) and URANS calculations second order hybrid and upwind schemes are used, respectively. In addition, a second order implicit scheme is used for temporal discretization. The present discretization is performed by using collocated and a Rhie-and-Chow type pressure–velocity coupling combined with a SIMPLE-type algorithm. For more details regarding these selected turbulence models and the numerical schemes, reader are referred to see (STAR-CCM+, 2011). 2.4. Mesh generation Mesh generation for such a complex geometry is not an easy task and poses severe restrictions for structured type meshes. The commercial software STAR-CCM+, allowed generating a fully polyhedral mesh inside the ﬂow domain. This provides optimal cell quality for bulk solution, with the addition of an offset layer in the near wall region, which is fundamental to best capture the high gradient ﬂow. Fig. 3 displays the polyhedral mesh generated for URANS calculations with an off-set layer near the pebble walls. A similar meshing strategy has been adopted for the other mentioned simulations. Details of the meshes used to perform these calculations are given in Table 2. The use of such a meshing strategy (polyhedral) to meet the DNS requirement has been tested for a separate pipe ﬂow DNS case in Shams et al. (2012). The results suggest the feasibility of using a hybrid mesh approach with some precautionary measures, especially in the transition region: from the boundary layer mesh to

Fig. 3. Front view of the mesh (of URANS calculations) for a single cubic FCC domain (polyhedral mesh with an off-set layer).

the bulk. Finally, following the recent DNS mesh requirements as given in Georgiadis et al. (2009), a mesh has been generated and is of the order of ∼15 million grid points. Similarly, for LES, the generated mesh is relatively ﬁner than the recent mesh requirements for LES given by Georgiadis et al. (2009). Whereas, for hybrid URANS/LES methods, meshing strategy is kept same as of LES. However, the stretching ratio in the prism boundary layer region (where the hybrid method switches into URANS mode) is kept 1.2 instead of 1.07 for LES. This results in the overall mesh of 3.8 million grid points. In the case of URANS, the mesh consists of around ∼1.3 million grid points with a dimensionless mesh size of smaller than 1 in wall normal direction. An extensive mesh sensitivity study performed by Shams et al. (2012) has shown that even ∼0.3 M grid points are good enough to reproduce ﬂow ﬁeld correctly. However, in that study an inter-pebble gap of 1 mm was considered and hence reduced the total number of grid points. Whereas, in the present case this gap is 5 mm. This result in more grid points specially in the near wall region to predict correctly the high ﬂow gradients, as shown in Fig. 3. 2.5. Simulation parameters All the simulations are performed for a long integration time in order reach a sufﬁcient statistical convergence for mean and RMS ﬁelds. The integration time is deﬁned in terms of ﬂow through times as FTT = LD /Uinlet(ave) (i.e. 0.35 m/s), where LD is the length of the computational domain and Uinlet(ave) is average inlet velocity. For q-DNS, simulations are performed for approximately 90–95 FTT, with a time step of 5 × 10−5 s, and 8 sub-iterations per time step. The Kolmogorov time scale was calculated for this ﬂow conﬁguration and gives a minimum of =7.5 × 10−5 s. Hence, it is sufﬁce to say that the selected time step is sufﬁcient to resolve the time scale of the turbulence for the present case. A single ﬂow through time Table 2 Details of the meshes used for different calculations.

Q-DNS LES Hybrid (RANS/LES) URANS

+ Wall normal

+ Azimuthal

+ Xsectional

Mesh (million)

<1 <1 <1 <1

∼5 ∼10 ∼10 ∼30

∼5–7 ∼10–20 ∼10–20 ∼40–80

∼15 ∼6 ∼3.8 ∼1.3

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Table 3 Summary of the results in comparison with the reference q-DNS: Quant. % diff. = ( q-DNS − sim. )max. / q-DNS , where represents the property and max. correspond to the maximum value. q-DNS

U Mean

v/s

Qual.

Quant. % diff

T Mean Qual.

Quant. % diff

Qual.

Quant. % diff

Qual.

Quant. % diff

Quant. % diff

LES: WALE IDDES: SST k–ω Cubic k–ε

5 4 5

∼2 ∼4 ∼7

5 3–4 4

∼2 ∼3 ∼4

5 4 3

∼6 ∼8 ∼18

5 4 3

∼6 ∼5 ∼20

∼4 ∼5 ∼40

U RMS

requires ∼14,704 (i.e. 1838 × 8) iterations, which gives a total of 1.47 million iterations needed to perform the complete simulation. The simulation is performed on NRG’s computers cluster, by using 60 processors, each having a speed of 2.6 GHz resulting in a total computational time of 6072 h (∼253 days or ∼8 months). Similar strategy has been applied for other selected simulations, in order to get good statistically converged results. However, the total simulation time for the other approaches is obviously less than the q-DNS and is given in Table 3 (see Section 2.7). 2.6. Results and discussion The ﬂow along the arranged single cubic FCC pebble bed exhibits a complex behaviour, which varies drastically throughout the computational domain and poses various challenges. Hence, in order to

T RMS

Turb. heat ﬂux

TDNS /Tsim .

6 14 173

evaluate the prediction capabilities of used turbulence modelling strategies an extensive comparison on qualitative and quantitative basis has been made and is given in the following sections. As mentioned earlier three different models were selected for hybrid and URANS methods, respectively. An inter-comparison of these methods for mean velocity and temperature proﬁles along the centerline (see Fig. 10) is given in Figs. 4 and 5. By looking at the inter-comparison for the hybrid methods with the q-DNS, it can be clearly noticed that the IDDES-SST shows much better agreement than the rest of models used. Similarly, comparison of URANS is made with the reference database and the results indicate that k–ε model based on cubic formulation shows superiority to the SST k–ω and RSM models for this complex ﬂow conﬁguration. Based on these comparisons IDDES-SST and cubic k–ε models are selected for an extensive comparison with q-DNS and LES.

Fig. 4. Evolution of mean (left) velocity and (right) temperature, along the centre of the domain for hybrid (RANS/LES) methods.

Fig. 5. Evolution of mean (left) velocity and (right) temperature, along the centre of the domain for URANS models.

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models is in fairly good agreement with the q-DNS. Whereas, cubic k–ε shows high over-prediction in the centre region. Interestingly, two hot spots can be observed Fig. 8 soon after the narrow gap, which is a consequence of the ﬂow separation and is highlighted by two shear layers appearing in Fig. 7. The overall mean temperature distribution on the pebbles surfaces is also compared and given in Fig. 9. It is clearly noticeable that in comparison to the q-DNS the overall temperature distribution is very well reproduced by these models. However, cubic k–ε slightly over-predicts the mean temperature.

Fig. 6. Iso-metric view of computational domain with a mid-cross-sectional plane.

2.6.1. Qualitative comparison A mid cross section through the single cubic pebble bed geometry is considered for qualitative comparison, see Fig. 6. The mean and RMS of velocity magnitude and temperature ﬁeld are selected to observe the prediction (qualitative) capabilities of aforementioned turbulence modelling approaches. 2.6.1.1. Velocity ﬁeld: mean and RMS. Fig. 7 displays the isocontours of mean and RMS velocity magnitude for q-DNS, LES, IDDES-SST and cubic k–ε model. The velocity ﬁeld is nondimensionalized by using the magnitude of maximum velocity scale (i.e. Umax = 1.54 m/s) appearing in the computational domain. By looking at Fig. 7, it can be noticed that despite of the symmetric geometry conﬁguration, the predicted ﬂow ﬁeld is asymmetric. This ﬂow asymmetry is caused by the appearance of low and high scale vortices on either side of the pebbles. Such type of asymmetric ﬂow behaviour is believed to represent a Coanda like effect, which has also been previously reported for a rod bundle case (see Shams et al. (2013a,b,c) for details). Moreover, the iso-contours of mean velocity magnitude indicate the presence of stagnant regions appearing on the front and rear part of the pebbles. In addition, ﬂow of relatively high velocity from the narrow inter-pebble gap interacts with the wake region (low velocity region) and subsequently form the shear layers. It can depicted that the overall ﬂow topology is successfully reproduced by all models. LES shows an excellent comparison with the q-DNS. Moreover, the predicted ﬂow asymmetry is less pronounced for IDDES-SST method, whereas, cubic k–ε shows much better agreement with the q-DNS. The comparison of RMS of velocity magnitude is given in Fig. 7 and clearly illustrates that once again LES produces an excellent comparison with the q-DNS. On the other hand, IDDES-SST shows relatively better agreement than cubic k–ε model, which under-predicts the RMS values. 2.6.1.2. Temperature ﬁeld: mean and RMS. Fig. 8 presents the isocontours of time averaged temperature ﬁeld and its respective RMS for all used turbulence modelling approaches. The temperature ﬁeld is non-dimensionalized by using average temperature difference over a pebble (i.e. Tp = 32), and the minimum ﬂuid temperature (i.e. Tf min = 771). Comparison of the obtained results with the q-DNS indicates that the mean temperature ﬁeld has been successfully reproduced by LES, IDDES-SST and cubic k–ε. However, a slight over-prediction in the temperature scales is produced by IDDES-SST. By looking at the RMS ﬁeld of temperature (in Fig. 8), it can be depicted that the overall ﬁeld predicted by these

2.6.2. Quantitative comparison In the previous section, it is noticed that the complex ﬂow conﬁguration in a single cubic pebble bed is qualitatively very well reproduced by the used modelling approaches. Furthermore, it was observed that the centre region of the geometry exhibits a lot of ﬂow complexities in the form of ﬂow separation, reattachment and the successive interactions of low and high velocity regimes. Hence, to quantify the predictive capabilities of aforementioned methods three different proﬁles are selected in the centre region, as shown in Fig. 10. Top and bottom lines are 8 mm apart from the centre of the computational domain. 2.6.2.1. Velocity ﬁeld (mean and RMS). Fig. 11 displays the evolution of time averaged velocity magnitude and its respective RMS along the top, centre and bottom lines in the centre region. These results are extensively compared with q-DNS. By looking at mean velocity proﬁles, it is noticeable that LES shows good agreement with the q-DNS. In addition, IDDES-SST displays a better comparison than cubic k–ε, which slight over-predicts the mean velocity in the centre region. For the RMS of velocity magnitude, LES and IDDES-SST show good prediction. More precisely, the RMS values produced by IDDES-SST display a slight over-prediction in the near wall region. Whereas, cubic k–ε model shows a big deviation in comparison with the q-DNS, but reproduces the overall ﬂow topology fairly well. 2.6.2.2. Temperature ﬁeld: mean and RMS. The evolution of time averaged temperature and its respective RMS along the top, centre and bottom lines in the centre region are presented in Fig. 12. The evolution of the mean temperature proﬁle shows that all the modelling approaches are in close agreement with the q-DNS. For the case of temperature RMS, IDDES-SST demonstrates good comparison with the q-DNS. In the near wall region, the RMS value is slight over-predicted by the IDDES-SST. However, in the centre region the obtained results are quite close to q-DNS. Moreover, the results reproduced by LES are slightly under-predicting the temperature RMS in the centre region. Whereas, URANS cubic k–ε indicates a big deviation with respect to the reference q-DNS and shows high over-prediction. 2.6.2.3. Turbulent heat ﬂux. Finally turbulent heat ﬂux for the principal component (i.e. v) is compared with the reference q-DNS, and is given in Fig. 13. Interestingly, there appear two peaks in the turbulence heat ﬂux proﬁles. These correspond to the appearance of two shears layers in the centre region of Fig. 7. Soon after the narrow the ﬂow goes through curvature effects. Hence, resulting in the ﬂow separation and the formation of the shear layers. In this region, the principle velocity component is quite dominant and results in the peaks of its respective heat ﬂux proﬁles. It can be depicted that once again LES and IDDES-SST demonstrate much better prediction than k–ε cubic. More precisely, the considered URANS model fails to reproduce the turbulent heat ﬂux. It is evident that the URANS turbulence model are unable to predict the overall behaviour of turbulence heat ﬂux, as expected, given that the adopted models

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¯ U ¯ max ) and (right) RMS (Urms /U ¯ max ) of velocity magnitude for q-DNS, LES, IDDES-SST, cubic k–ε, top to bottom, respectively. Fig. 7. Iso contours of (left) Mean (U/

do not contain any explicit thermal time scale in their modelling. It is worthwhile to mention here that similar conclusions can also be drawn for other components of turbulent heat ﬂux.

2.7. Lessons learned The extensive inter-comparison of the results produced by the used turbulence modelling approaches is summarized in Table 3. This table gives an explicit overview of their performance on qualitative and quantitative basis. The quality of the results produced by these turbulence modelling approaches has been assigned grades to do the qualitative comparison, i.e. 1 – very bad, 2 – bad, 3 – fair, 4 – good and 5 – excellent. Whereas, quantitatively these results are assessed by computing the percentage difference in comparison with the reference q-DNS. This table also gives an overview of total

simulation time of these simulations with respect to the reference q-DNS. The results clearly indicate that LES performs much better than any other modelling approach used for this work. Apart from LES, hybrid (RANS/LES) methods also display good accuracy and consistency, however, it is observed to be less accurate than LES. In addition, the computational cost for the used hybrid methods is less than LES. While looking at URANS performance, it can noticed that the mean ﬁelds are in good agreement with the q-DNS. However, as may be expected for URANS, the ﬂuctuating parts are only reasonably well captured by URANS models. Although, URANS calculations are found to be much faster than the used LES and hybrid methods. This indicates that the cubic k–ε model seems to be a possible way to proceed for such complex ﬂow conﬁgurations, keeping in mind the discrepancies observed for the RMS ﬁelds by this very model.

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Fig. 8. Iso contours of (left) Mean and (right) RMS of temperature ﬁeld for q-DNS, LES, IDDES-SST, cubic k–ε, top to bottom, respectively.

Nonetheless, it should be kept in mind that these conclusions are based on an idealized pebble bed conﬁguration. The prediction capabilities of this model for a realistic pebble bed are yet to be tested and/or evaluated. Hence, as a next step to this validation procedure proposed by Shams et al. (2013a), it is planned to validate this cubic k–ε model for a realistic pebble bed conﬁguration. Unfortunately, for such a conﬁguration no reference database is available. Since a realistic pebble bed exhibits a lot of geometric complexities, it has not been feasible to perform DNS and/or q-DNS. On the other hand, the results obtained for an idealized pebble bed have shown that LES could be used as a reference for such type of ﬂow conﬁgurations. Considering the amount of computational cost and the provided resource, LES calculations can be used as a reference case to validate the IDDES-SST k–ω and cubic k–ε models for a realistic pebble bed conﬁguration.

3. Limited size random pebble bed This section will put emphasis on the use of the developed guidelines in the process of generating a reference LES database. Based on the lessons learned from the idealized pebble bed, the feasibility for a reference LES of a realistic random pebble bed has been performed. It is worthwhile to mention that only the calibration of a realistic limited sized random pebble bed is reported in this paper, since, the obtained results and the corresponding discussion has been extensively documented in Shams et al. (2014). 3.1. Computational domain The selected ﬂow domain is a randomly stacked spherical shaped pebble bed in a rectangular domain of x = 0.177 m, y = 0.354

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Fig. 9. Iso contours of Mean of temperature ﬁeld for on the pebbles q-DNS, LES, IDDES-SST, cubic k–ε, top: left to right and bottom: left to right, respectively.

and z = 0.177 m, which is shown in Fig. 14. This random pebble geometry is obtained from (Pavlidis and Lathouwers, 2013), and consists of approximately 30 pebbles, and exhibits an average porosity (ε = volume of voids/total volume) level of about 0.4. Following the speciﬁcation of PMBR-250MWth the diameter of the pebbles is 60 mm. In the obtained geometry, pebbles were clustered in a cubic domain of 0.177 m per side. However, for the current purpose inlet and outlet boundary conditions needed to be imposed. Hence, the domain was further extended in the ydirection, i.e. 0.0885 m for both inlet and outlet sections. As a result, the appearing pebbles on these sides are not truncated, as can be seen on x- and z-directions in Fig. 14. Moreover, in the original geometry the pebbles are in point contacts with each other. However, to model such point contacts is problematic from a meshing view point and consequently may induce numerical errors to the solution. Hence, in order to avoid such large numerical errors, these point contacts are converted into small area contacts by scaling the pebbles by a factor of 1.034. This gives a negligible maximum radial overlap of ∼1 mm between two pebbles. The corresponding small contact area is 0.0019 cm2 .

scenario of the core of a pebble bed reactor. Similar to the single cubic pebble bed, a simpliﬁed assumption of constant heat ﬂux of 8317 W/m2 is applied on the pebble surfaces. As a consequence of this simpliﬁcation, the temperature corresponding to the hot-spots will probably be over-predicted. 3.3. Mesh generation Fig. 15 displays the generated polyhedral mesh for mid crosssections with off-set layers near the pebble walls. This mesh consists of around 18 million grid cells with a dimensionless mesh size smaller than 1 in wall normal, around 10 in azimuthal and 12 in the cross-sectional directions. The generated mesh is ﬁner than the one used for single cubic pebble bed, see Section 2. It is worth mentioning that these mesh dimensions correspond to the pebble bed region. The mesh is reﬁned in the region of area contacts,

3.2. Flow/simulation parameters To mimic the core of an HTR realistically, the ﬂow is imposed via an inlet and outlet, from top to bottom in negative y-direction, as shown in Fig. 14. Following the single cubic pebble, similar ﬂow parameters are used for the random pebble bed, see Table 1. Based on the pebble diameter and the predicted maximum velocity, the estimated Reynolds number is 9753. The synthetic eddy method (SEM) is used to generate and sustain the turbulence level at the inlet section. The top view of these randomly stacked pebbles is given in Fig. 14. This suggests no passage for the straightforward ﬂow in the principle direction and the possible appearance of strong cross-ﬂow. Hence, in order to capture the three-dimensional effects, periodic boundary conditions are imposed for mass ﬂow rate and temperature on all four sides (i.e. in x- and z-directions) of the computational domain. This eventually replicates the real case

Fig. 10. Position of (top, centre and bottom) proﬁles extraction through the midcross-section plane.

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Fig. 11. Evolution of (left) Mean and (right) RMS of velocity magnitude, along the top, centre and bottom lines, respectively (top to bottom: q-DNS, LES, IDDES-SST and cubic k–ε).

as can be seen in Fig. 15. Whereas, in the outlet section the mesh is coarsened to avoid the reverse ﬂow effects. The extensive preceding comparative study of LES and q-DNS of an idealized pebble bed has shown that even a non-dimensional mesh size of 10 and 23 in azimuthal and cross-sectional directions, respectively, is good enough to obtain a well-resolved LES solution. The results obtained corresponding to the mentioned mesh are found to be in good agreement with the reference q-DNS solution, with a maximum deviation of ∼6% (see Table 3).

3.4. Numerical schemes and turbulence modelling A second order central scheme with 5% boundedness (i.e. of second order upwind scheme) has been used for spatial discretization. While the details of the scheme are available in the code manual (STAR-CCM+, 2011), it is worthwhile mentioning here that in STAR-CCM+, this boundedness is introduced only when the local Normalized-Variable Diagram (NVD) value is outside the range 0 < 1, which means in the cells where central differencing (CD)

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Fig. 12. Evolution of (left) Mean and (right) RMS of temperature ﬁeld, along the top, centre and bottom lines, respectively (top to bottom: q-DNS, LES, IDDES-SST and cubic k–ε).

would anyhow introduce an error. This suggests that for a ﬁne mesh, such as the present LES, the introduced boundedness is negligible, as it is only rarely introduced on very few cells, i.e. less than 1%. In addition a second order, implicit three time level scheme is used for temporal discretization. The present discretization is performed by using collocated and a Rhie-and-Chow type pressure-velocity coupling combined with a SIMPLE-type algorithm. Moreover, following the lesson learned from single cubic pebble bed (given in Section 2), LES based on a WALE model is selected to perform simulations for this random pebble bed conﬁguration. Thanks to the guidelines generated from an idealized pebble bed, the selection of the numerical methods has become quite straight-forward. Hence, these guidelines indicate that LES is found

to be in excellent agreement with the q-DNS with a maximum difference (for the ﬁrst and second order statistics) of less than 6 percent. Hence, it is sufﬁce to say that for such complex geometries where performing q-DNS calculations are not possible, LES can be used as a reference to validate the low order turbulence models. Lessons learned from step 1 have been carefully applied by Shams et al. (2014) for step 2 to generate a reference database for a limited sized random pebble bed. The selected ﬂow domain is a complex geometry and poses challenges in understanding the ﬂow, discussed in Shams et al. (2014). Fig. 16 displays an isometric view of the obtained numerical solution, highlighted by the instantaneous temperature distribution on the pebbles. The ﬂow from top to bottom is depicted by the streamlines. At a ﬁrst glimpse, the complexities appearing in the ﬂow regime are evident.

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Fig. 13. Evolution of turbulent heat ﬂux for the principal component (v) along the top: left, top: right and bottom lines, respectively.

Fig. 14. Schematic of the computational domain (left) iso-metric view (right) top view.

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4. Conclusions

Fig. 15. Polyhedral mesh across the computational domain in (top-left) z-, (topright) y-planes and (bottom) on a single pebble.

A good prediction of the ﬂow and heat transport in a pebble bed core is a challenge for available turbulence models. As a part of European project, named Thermal hydraulics of Innovative Nuclear Systems (THINS), a wide range of numerical simulations are performed for two different pebble bed conﬁgurations, i.e. a single cubic pebble bed and a limited sized random pebble bed. As a ﬁrst step, a benchmarking quasi-direct numerical simulation (q-DNS) of a well-deﬁned single cubic pebble bed conﬁguration has been performed. This q-DNS is used as a reference database in order to evaluate the prediction capabilities of LES, Hybrid (RANS/LES) and URANS modelling approaches. A detailed comparison of LES: WALE, Hybrid RANS/LES: DDES-SST k–ω and URANS: cubic k–ε models has been made with the reference q-DNS. The obtained results indicate that LES performs much better than any other modelling approach used for this work. Apart from LES, hybrid (RANS/LES) methods also display good accuracy and consistency, however, it was observed to be less accurate than LES. In addition, the computational cost for the used hybrid methods is less than LES. Whereas, the mean ﬁelds obtained by URANS models are in good agreement with the q-DNS. However, as may be expected for URANS, the ﬂuctuating parts are only reasonably well captured by URANS models. Although, URANS calculations are found to be much faster than the used LES and hybrid methods. This indicates that the cubic k–ε model seems to be a possible way to proceed for such complex ﬂow conﬁgurations, keeping in mind the discrepancies observed for the RMS ﬁelds by this model. In addition, the results obtained for an idealized pebble bed have shown that LES could be used as a reference for such type of ﬂow conﬁgurations. Considering the amount of computational cost and the provided resource, one may use LES calculations to validate the IDDES-SST k–ω and cubic k–ε model for a realistic pebble bed conﬁguration. In the second part of the paper, the lessons learned from an idealized pebble bed are applied for the preparation of the reference LES calculations of a limited sized random pebbled. These guidelines are carefully used for the LES case to meet the requirement of a reference database. Hence, this reference LES will be used to validate the lower order modelling approaches for such a ﬂow conﬁguration, which is of prime importance in the nuclear ﬁeld. Acknowledgments The work described in this paper is funded by the Dutch Ministry of Economic Affairs and the FP7 EC Collaborative Project THINS No. 249337. The limited size random pebble bed used in paper was provided by Dimitrios Pavlidis and Danny Lathouwers of the Technical University of Delft and is acknowledged here. References

Fig. 16. Isometric view of the predicted ﬂow features in the computational domain: streamlines coloured with velocity, pebbles are coloured with temperature.

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