large-eddy simulation of the triga mark ii reactor core

Large-Eddy Simulation of the TRIGA Mark II Reactor Core

Carolina IntroiniPolitecnico di Milano, Department of Energy

via La Masa 34I-20156, Milan, Italy

[email protected]

Antonio CammiPolitecnico di Milano, Department of Energy

via La Masa 34I-20156, Milan, Italy

[email protected]

ABSTRACT

Complex industrial applications still handle turbulence modelling using Reynolds Aver-aged Navier Stokes (RANS) models. Based on local-averaged quantities, this approach reducesthe required computational effort with respect to more accurate modelling strategies for turbu-lence, such as Large-Eddy Simulation (LES). Despite its limitations, RANS is still the mostpopular approach in use for CFD analysis in the nuclear field. However, in recent years thesafety requirements for nuclear installations have become even more restrictive, and accuratefluid-dynamics simulation for safety-related purposes must take into account the impact thatturbulence can have on the diffusion of momentum and energy and any local phenomena thatmay contribute to instabilities. As RANS modelling cannot provide the required level of accu-racy and rigour in the treatment of turbulence, reactor modelling using LES approaches is anongoing field of research. This work further explores this field by proposing a full LES analysisof the reactor core of the TRIGA Mark II reactor at University of Pavia, also providing somebest practices for LES simulations. Preliminary results of the proposed model are presented andcompared with RANS ones previously obtained.

1 INTRODUCTION

Complex industrial applications still handle turbulence modelling using Reynolds Aver-aged Navier-Stokes models. This approach reduces the required computational effort by elim-inating all turbulent fluctuation through the time-averaging process. Therefore, fluid equationsare in terms of local-averaged flow field quantities. With RANS, all turbulence is modelled;the main issue is that there is no universal RANS model for turbulent behaviour, nor there is afundamental physical law underlying all RANS models. A certain model may be adequate for aspecific application but unsuitable for another. Despite these limitations, RANS is still the mostpopular approach in use CFD analysis in nuclear reactors [1].

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Large-Eddy Simulation approaches lie between Direct Numerical Simulation (DNS) andRANS ones. Whereas DNS directly reproduces all the scales of turbulence without any mod-elling assumption, the LES approach resolves only the larger scales of turbulence, while stillmodelling the smaller ones. This method uses a low pass filtering based on grid size on theequations, eliminating the smaller length and time scales. Compared to DNS, this approach re-duces the required computational effort. As DNS modelling is feasible only for small test casesand simple geometries, for Computational Fluid Dynamics (CFD) analysis LES modelling rep-resents the reference approach.

Accurate simulations in CFD must take into account the impact that turbulence can haveon the diffusion of momentum and energy. For wall-bounded flows such as the cooling channelsof nuclear reactors, additional turbulence terms may cause a significant increase in heat transferand wall stresses. Turbulent phenomena may also be responsible for instabilities. An accuratesafety analysis of nuclear systems requires a very precise and rigorous treatment of turbulence.Even though RANS models still represent the state-of-the-art in this field, the feasibility tomore accurate approaches, in light of the fast improvements in terms of computational speed,represents an ongoing field of research. This is especially true for more complex reactor designswhere turbulence may play a significant role.

Reactor modelling using LES is still a field of research (as in [2], for example), and itis useful to test it on smaller-scale nuclear systems such as research reactors, in order to iden-tify the modelling best practices and bottlenecks. The use of research reactors as benchmarkfor LES modelling allows to evaluate their performance and required computational resourceswith respect to a full system, for example a whole reactor core compared to a single fuel as-sembly. The TRIGA (Training Research and Isotope production General Atomics) studied inthis work is a pool-type research reactor with maximum power 250 kW with an asymmetriccore configuration (Figure 1). The fluid-dynamics of the TRIGA reactor at University of Paviawas previously studied by the authors adopting a RANS approach for turbulence [3], and thesestudies represent the basis for this work. In particular, the CFD model of the reactor core [4]was taken as starting point from which the present LES model is built.

This paper is organised as follows. Section 2 introduces the model equations and brieflydescribes the two different approaches for the modelling of turbulence (RANS and LES), focus-ing on their main differences. Section 3 describes the case setup and the adopted strategies forthe discretization, mesh generation and initialisation of the LES simulation. Section 4 reportsthe preliminary results of the model, focusing on its differences with respect to the RANS one.Section 5 summarises the findings of the paper.

2 THERMAL-HYDRAULICS AND TURBULENCE MODELLING

The water flow in the TRIGA reactor is assumed to be incompressible, Newtonian andwith constant physical properties (taken at 20 ◦C). Buoyancy is modelled using the Boussinesqapproximation, meaning that density variations are taken into account only when they appearas a source term in the momentum equation:

∇ · u = 0,

∂u

∂t+∇ · (uu)−∇ · (ν∇u) = −∇p− gβ(T − TR),

∂T

∂t+∇ · (uT ) = α∆T,

(1)

Proceedings of the International Conference Nuclear Energy for New Europe, Portoroz, Slovenia, September 7–10, 2020

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Figure 1: Core configuration. Different colours correspond to different type of elements: white- irradiation channel (2), light grey - steel-clad fuel elements (27), gold - instrumented steel-clad fuel elements (3), red - control rods (3), blue - aluminium-clad fuel elements (50), grey -graphite elements (5), yellow - neutron source (1). In black the XY mid-plane.

where u represents the velocity vector, ν is the kinematic viscosity of the fluid, p is the nor-malised pressure with respect to density, g is the gravity acceleration and α is the effectivethermal diffusivity of the fluid.

2.1 Turbulence Modelling

The RANS approach is based on the assumption that the velocity field u can be writtenas a sum of its average behaviour U and a oscillating component u′ which describe the chaoticbehaviour of the flow. By substituting this decomposition to Equation 1 the Reynolds-AveragedNavier-Stokes equations are obtained, and the substitution gives rise to six additional termsin the form Rij = ρu′iu

′j . These terms are associated with the momentum exchange due to

convective transport by the turbulent eddies. They represent six additional unknowns, and hencerequires six additional equations. Standard RANS approaches computes the value of these termsstarting from an aptly-defined turbulent viscosity νt and transported variables [5]. For example,considering the κ− ε RANS model:

Rij = νt

(∂Ui∂xj

+∂Uj∂xi

)− 2

3ρκδij

νt = Cµρκ2

ε

(2)

(3)

where Cµ is an empirical constant, κ is the turbulent kinetic energy and ε is the turbulent dis-sipation rates. These two quantities are determined by solving additional transport equations,which, for sake of brevity, are not reported here. The main limitation of the RANS approach isthat, by performing an averaging operation, it neglects the different behaviour of different-scaleturbulent eddies.

LES approaches introduce a distinction between the different turbulence scales. Whereasthe large anisotropic eddies, which contains most of the turbulent energy, are directly computed,


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the small isotropic and less energetic ones are modelled, as is easier to accurately capture thebehaviour of the latter with models. Instead of averaging, an operation of space-filtering isperformed to obtain Navier-Stokes equations that governs the dynamics of the large eddiesonly. The filtered velocity can be written as:

u = G~ u (4)

where G is the convolution kernel, which changes according to the filter type used, and ~denotes the convolution operation. The filter kernel uses cut-off length and time scales ∆ andτ ; scales smaller than these are eliminated from u and collected in the sub-filtered portion u′.By applying the filtering procedure to the Navier-Stokes equations six additional terms in theform τij = ˜uiuj − uiuj arise, which represent the sub-grid scales of velocity [6]. Using anappropriate sub-grid model, such as the Smagorinsky one, this term can be modelled as:

τij = −2(Cs∆2)|S|Sij +

1

3δijτkk

|S| =√

2SijSij

Sij =1

2

(∂ui∂xj

+∂uj∂xi

)(5)

(6)

(7)

where Cs is an empirical constant. Van-Driest damping is used in order to have zero turbulentviscosity at solid boundaries.

3 CASE SETUP AND DISCRETISATION

All calculations are performed using the Finite Volume open-source code OpenFOAM.Mesh generation has been carried out using the commercial software ANSYS Workbench, usingas basis geometry a high-fidelity CAD (Computer-Aided Design) model of the reactor core. Toreduce the computational times involved in a LES computation, the PIMPLE (merged PISO-SIMPLE) algorithm for pressure-velocity with the Boussinesq approximation for temperature isused. The main advantage of this method over the standard PISO is that it allows for larger timesteps, overcoming the limitation of the Courant number, and permits to achieve better stabilityby iterating in SIMPLE mode over a single time step until convergence is achieved. For thepresent case, the convergence criteria were (a) a decrease of the numerical residuals by fiveorders of magnitude and (b) continuity errors in the mass flux below 10−8.

The mesh is made mainly of orthogonal hexahedra (except for the narrower regions be-tween fuel elements) to remove the need of non-orthogonal correctors and gradient reconstruc-tion sweeps and also improving stability. As previously done in [4], the two supporting grids andthe upper and lower spacers, which from a geometrical standpoint are the most complex andasymmetric elements in the core, were modelled using a porous medium approach. Second-order schemes were chosen for the discretization to limit numerical diffusion. The mesh has791’566 elements (with average non-orthogonality less than 5% and max skewness equal 0.88).

For the LES simulation, small grid sizes are required mainly for two reasons: (1) forbetter accuracy, it is desirable that the sub-grid scale modelling is reduced, by keeping ∆ aslow as reasonable considering the computational times; (2) the finer the grid, the higher is theaccuracy in capturing the large-scale flow structures. Clearly, the finer the grid the higher thecomputational times, hence in the present work it has been chosen to focus more on meshingin the wall-normal direction whilst using a coarser discretization in the expected flow direction.


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Figure 2: Adopted mesh for the reactor core (full and detailed view).

Cells in the latter are kept equidistant, while in the former they vary stretching up to one-thirdof the channel half-width. Figure 2 shows the generated mesh.

Finally, to improve convergence and run-time, the LES simulation was not started fromstagnant conditions. Rather, at first the mean flow was computed using a steady-state RANSon a coarse grid. Convergence of this preliminary simulation is not relevant, but it should bequick enough to do trial-and-error on it in order to identify the best settings for the test case.Once a satisfactory RANS solution is obtained, the mean velocity and temperature fields aremapped on the finer grid, and synthetic, randomly-generated turbulence is then superimposedon the mean flow. This represents the initial condition for the LES simulation. This way, caseoptimisation is made on the quick RANS, and only minor tweaks will be needed for the LEScase. The Smagorinsky SGS model was adopted for sub-grid modelling. The adopted boundaryconditions are summarised in Table 1.

Table 1: Boundary conditions for the core model. The power produced by the fuel elements istaken as input data according to Monte Carlo calculations [4].

Inlet Outlet Fuel Adiabatic surfacesPressure ∂p

∂n= 0 1.5 bar ∂p

∂n= 0 ∂p

∂n= 0

Velocity calculated ∂u∂n

= 0 u = 0 u = 0

Temperature T = TR∂T∂n

= 0 q′′ = Asin(Bz + C) +D ∂T∂n

= 0

Turbulence Freestream ∂∂n

= 0 ∂∂n

= 0 ∂∂n

= 0

4 RESULTS AND DISCUSSION

The results obtained with the LES simulation are compared with those previously ob-tained in [4]. Figure 3 shows the instantaneous temperature field in the core XY mid-plane forthe LES and the RANS case. The first difference is that the LES model is able to predict theupward diffusion of heat even in the upper porous region. The hot water exiting the core fromthe top is responsible for the heating of the reactor pool, thus creating the conditions for naturalconvection. The stand-alone RANS model fails at predicting this upward diffusion. However,


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the LES model seems to predict lower temperature values in the peripheral region outside theexternal fuel ring, only slightly above the ambient one. Interestingly enough, both models donot predict any downward diffusion of heat.

Figure 3: Temperature field in the XY mid-plane of the core (left - LES ; right - RANS).

Figure 4 shows the velocity profile for the LES and RANS simulations using the LIC(Line Integral Convolution) and vector glyphs representation. Only the LES model is ableto identify the recirculation region in the upper porous region, which are caused by naturalcirculation. Indeed, the RANS model detects almost neither recirculating nor cross-flow, butinstead water flows in a single direction from bottom to top. This is not compatible with thenatural circulation regime expected in the TRIGA core, for which downward flow from thepool and cross-flow from the peripheral to the central regions are significant. Water flowingdownward in the peripheral zone also explains why the LES model predicts lower temperaturesin this region. Interestingly enough, even without simulating the reactor pool the LES model isable to partly reconstruct its effect on the core flow regime.

Figure 5, which shows the velocity profile in the active zone outlet section for the LES andRANS cases, confirms the cross-flow of colder water from the peripheral to the central regions,as well as some recirculation zones outside the fuel rings. These small vortexes, not observedwith the RANS simulation, are likely due to the downward flow of water coming from the top.In terms of computational times, it is worth noting the influence of the initialisation throughRANS simulation: without it, acceptable convergence for the LES case was obtained after 351CPU-hours; with it, this time was reduced to 242 CPU-hours plus 35 CPU-hours for the RANS.

5 CONCLUSIONS

This paper proposes a LES model of the reactor core of the TRIGA Mark II reactor at Uni-versity of Pavia, aiming to contribute to the literature on LES modelling for nuclear reactors byproviding a test case of an entire reactor core. In particular, some strategies for LES modellingwere presented, such as the initialisation of the test case through a coarse RANS simulation,the use of the PIMPLE algorithm and second-order schemes, and the use of porous media forthe most complex geometries in the model. The preliminary results presented in this work aimat highlighting the difference between LES and RANS simulation of the same case and on the


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Figure 4: Velocity field in the XY mid-plane with the LIC (Line Integral Convolution) andvector representation (left - LES ; right - RANS).


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same numerical grid. In particular, the LES model is able to predict upward diffusion of heatand recirculation vortexes in the core, not seen by the RANS model. These results prove theneed for accurate turbulence modelling even for a relative simple case of a small-scale reactor.

Figure 5: Velocity field in the active zone outlet section (top - LES ; bottom - RANS).

REFERENCES

[1] Nuclear Energy Agency, Best Practice Guidelines for the Use of CFD in Nuclear ReactorSafety Application - Revision, Technical Report NEA/CSNI/R(2014)11, 2015.

[2] A. S. Silva, L. Y. R. Mazaira, D. S. Dominguez, ‘’Recent Advances on ThermohydraulicSimulation of HTR-10 Nuclear Reactor Core Using Realistic CFD Approach”, Proc. Int.Nuclear Atlantic Conf. (INAC), Sao Paulo, Brazil, October 4–9, ABEN, 2015.

[3] C. Introini, S. Lorenzi, A. Cammi, ‘’Complete Thermal-Hydraulic Modelling of the PaviaTRIGA Mark II Research Reactor for the Study of the Natural Circulation Regime”,Proc. Int. Topical Meeting on Nuclear Reactor Thermal-Hydraulics, Operation and Safety(NUTHOS), Qingdao, China, Octover 14-18, 2018, pp. 409–417.

[4] C. Introini, S. Lorenzi, A. Cammi, ‘’A 3D CFD Model for the Study of Natural Circulationin the Pavia TRIGA Mark II Research Reactor”, Proc. Int. Conf. Nuclear Energy for NewEurope (NENE), Bled, Slovenia, 11–13 September, 2017, pp. 451–459.

[5] H. K. Versteeg, W. Malalasekera, An Introduction to Computational Fluid Dynamics,Prentice Hall, 1996.

[6] P. Sagaut, Large Eddy Simulation for Incompressible Flows, Scientific Computation,Springer, 2006.


large-eddy simulation of the triga mark ii reactor core

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