Computation of turbulent natural convection with buoyancy corrected second moment closure models
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1 Computation of turbulent natural convection with buoyancy corrected second moment closure models S. Whang a, H. S. Park a,*, M. H. Kim a, K. Moriyama a a Division of Advanced Nuclear Engineering, POSTECH, Pohang, Gyeongbuk 37673, Republic of Korea Abstract In IVR-ERVC circumstances, a highly turbulent natural convection with volumetric heating is expected in the molten pool and its convection is one of the most important aspect of the IVR-ERVC evaluation. The maor phenomena of turbulent natural convection should be simulated to understand thermal behavior of the oxide pool. The buoyancy corrected second moment closure models are implemented into OpenFOAM and validated with experiments. From the differentially heated cavity case, the primary behavior of turbulent natural convection such as a transition along the wall and an anisotropy behavior near wall is simulated with the suggested model. In case of the natural convection with internal heat source, limited models are implemented and validated so far. Preliminary numerical test with oxide melts was conducted from the perspective of the fluid Prandtl number effect. Keywords: Buoyancy, Turbulence modeling, Natural convection 1. Introduction In-vessel retention by external reactor vessel cooling (IVR-ERVC) is one of the promising accident strategies to mitigate or terminate the severe accident. The safety evaluation of IVR-ERVC has been conducted conservatively, and there is less safety margin for the high power reactor such as APR1400 (>1,000MWe). At the same time, understanding of failure mode of reactor vessel is important as it becomes an initial condition of ex-vessel accident scenarios. The crucial phenomena contain the molten pool convection, the crust formation, the conduction through the vessel wall and the external two-phase natural circulation flow. Among those phenomena, the molten pool convection is important because the heat is generating from the oxide layer and the distribution of the generated heat is determined by its behavior. It is reported that the oxide pool is highly turbulent natural convection with decay heat. Figure 1 shows the expected maor phenomena of the natural convectionin the melt pool: stratification, upper unstable zone and stratified zone; cold plume along the cold top wall; laminar to turbulent transition along the side cold wall [1]. Since buoyancy plays an important role in the flow, buoyancy production term for second moments should be considered, it is called buoyancy corrected for the rest of this paper. The anisotropic behavior in the vicinity of the wall should be reproduced properly for the turbulent flow. Eddy viscosity model () and eddy diffusivity model (EDM) which is an isotropic turbulence model is not appropriate for this case. Second moment closure (SMC) model is applicable because it can resolve the anisotropic behavior of turbulent fluxes. Nourgaliev et al. [2,3] investigated the characteristic of volumetrically heated liquid pools by numerical method. Various aspects were considered which influence to the thermal behavior of the pools, such as fluid Prandtl number, temperature-dependent viscosity and mushy region effects. Especially, the Prandtl number effect, so-called α-phenomenon and υ-phenomenon, was analyzed in various geometries and Rayleigh numbers up to They concluded that the influence of Prandtl number on the thermal behavior of the liquid pool is dependent on the Rayleigh number and the cavity geometry, and similar analysis is required for high-ra-number conditions. Dihn and Nourgaliev [4] emphasized the importance of modeling of the anisotropic turbulence in the vicinity of the wall to simulate the natural convection phenomena. They proposed phenomenological corrections for eddy viscosity and turbulent Prandtl number to the existing standard low Reynolds number turbulence model. Fukasawa et al. [5] studied on the natural convection with internal source with the LES and several Low- Reynolds number RANS models. They tried to modify the original RANS and LES turbulence model to take account the anisotropy effect by the buoyancy and chose the BALI experiment to validate their model. The hypothetical reactor simulation was performed with modified model and the heat flux distribution was given as * Corresponding author address: hesunny@postech.ac.kr (H. S. Park)
2 one of the results. The simulations so far were conducted based on isotropic models (/EDM) or buoyancy production term was not properly considered in the second moments equations. Therefore, the understanding of the thermal behavior is required with an anisotropy and buoyancy corrected model. In this study, the buoyancy corrected SMC models are implemented into OpenFOAM which is open source CFD code. The selected models are validated with differentially heated cavity case, King s cavity [6] and a natural convection with internal heat source, BALI [1]. For the former case, the implemented models simulate successfully maor phenomena of the natural convection and anisotropic behavior in the vicinity of the wall. The latter case is required to confirm its applicability of implemented models to analze the molten pool of the IVR- ERVC. Preliminary test with oxide melts was analyzed from the fluid Prandtl number point of view. Figure 1 Maor phenomena of the oxide pool [1] 2. Governing equations and numerical methodology 2.1. Governing equation The turbulent natural convection is described by the Reynolds equations which is assumed the incompressible fluid and the Boussinesq approximation. These Reynolds equations are derived by timeaveraging the fluctuations from the Navier-Stokes equations. Continuity: U x i i 0 ; (1) Momentum equations: U U P U U U u u ( T T ) g t x x x x x i i i i 0 i i i ; (2) Temperature equation: T T T q U u t x x x C v p ; (3) In the equation (2) and (3), the unknown variables; second moments, u are contained which should be modelled to close the equations set. In this study, we consider second-moment closure models (SMC) which require separate equations for each component and the equations are expressed in symbolic form, i.e.: Turbulent momentum flux transport equation, uu i :
3 u u u u U D D P G ; (4) i i T * k t xk where, D, T D, *, P, G and represent viscous diffusion, turbulent diffusion, velocity-pressuregradient correlation (redistribution), production by the mean velocity gradient, production by buoyancy force and dissipation, respectively. Turbulent heat flux transport equation, u : ui u U P P G D D t i T U * t k i i i i i i i xk ; (5) * i, i D T U where, P i, P i, G i, and i denote thermal production, mechanical production, gravitational production, pressure-temperature-gradient correlation (scrambling), dissipation and diffusion transport (superscripts stand for: molecular and t by turbulent velocity) Turbulence models Reynolds stress model [7-9] EBRSM (Elliptic blending Reynolds stress model) Manceau et al. [7] proposed EBRSM to predict the turbulence anisotropy in the vicinity of the wall. They analyzed the near wall behavior and figured out that and should be reproduced to simulate an anisotropic near wall behavior. In this model, the near-wall and the far-from-the-wall model of those terms are blended with a single elliptic operator. The elliptic operator which is relying on the geometry configuration and the length scale, is zero at the wall and going to 1 far from the wall. The models for near-wall and far-from-the-wall are followed by their approaches [7-9]. * 2 2 L x x 1 ; (6) 3/2 3/4 k L CL max(, C ) 1/4 ; (7) (1 a ) ; (8) * 2 wall 2 bulk (1 a ) ; (9) 2 wall 2 bulk CL 0.15, 50 C ; (10) Turbulent heat flux model [9-11] (Elliptic blending algebraic heat flux model) and (Elliptic blending differential heat flux model) Dehoux et al. [10] proposed and by applying the elliptic blending approach to the turbulent heat flux modeling. Later, Choi et al. [9] modified the turbulence model constants to simulate natural convection more accurately. In the present study, Choi s model was adapted, i.e.: 2 2 L x x 1 ; (11) h R R (1 )Pr, L 3RL ; (12)
4 (1 a ) ; (13) * 2 wall 2 bulk i i i (1 a ) ; (14) 2 wall 2 bulk i i i k Ui 2 ui C uiuk u gi u nin x x k ; (15) h bulk 2 1 R 0.5, i 0, 5, 5, (1 )(1 C ), C 0.5(1 Pr ) ; (16) 2.3. Algorithm and Numerical discretization The OpenFOAM code, an open source code for CFD is used to perform the numerical simulation and the models described above were implemented in the OpenFOAM. The second order central difference scheme was applied for gradient, Laplacian, Reynolds stress and turbulent heat flux term. The first order upwind scheme which assure the high numerical stability was used for the rest terms. The second order Crank-Nicolson and the first order Euler schemes were blended and used to discretize the equations in time. A steady calculation was done for the case of King s cavity, and a transient calculation was conducted for the case of BALI. Two algorithms are applied to couple the momentum and pressure equations; the SIMPLE method for steady state and PIMPLE method which is a combination of PISO and SIMPLE algorithms for transient state 2.4. Validation cases and grid resolution In this study, two experiments are chosen to evaluate the implemented turbulence models. First experiment is the King s cavity [6] shown in figure 2 (a), the natural convection test case with differentially heated wall ( T is 43.8 in this case). And it is a rectangular cavity with aspect ratio five and the working fluid is air. The Rayleigh number is around in this case. The comprehensive turbulent data including the anisotropy turbulent heat flux in the vicinity of the wall can be obtained, and the laminar-to-turbulence transition can be induced from the result. Figure 2 (b) shows the BALI experiment [1], the natural convection case with internal heat generating. The radius of the facility is 2 meter which is a prototypical size of the reactor pressure vessel. It is 2D geometry with 15cm thick. The simulant fluid is water and the decay heat is simulated by direct current heating. The internal Rayleigh number is around which is comparable value with the real circumstance. The stratification phenomenon and local or averaged heat transfer correlation are presented as a main result. For the numerical simulation, the mesh was generated as shown in figure 2. The grid was clustered toward the wall in both cases, and the polyhedral mesh was adopted in the BALI case. Number of grid is about 30k, 46k and y + is below 0.8, 1.8 for the King and BALI, respectively.
5 (a) King (b) BALI Figure 2 Numerical grid and boundary conditions in the present study 3. Results The main goal of this study is to simulate the phenomena of the turbulence natural convection by the buoyancy corrected SMC models such as EBRSM/ and EBRSM/ (the notation of A/B means A model for Reynolds stress and B model for turbulent heat flux, respectively). The results are compared with experiment and other models: (/EDM), (SMC/EDM), (SMC/EDM) and (SMC/). All models were implemented and converged in the case of King s simulation. However, in the BALI simulation, the buoyancy corrected models were not converged at present. The calculation with these advanced models are in progress King s cavity: Natural convection in a rectangular cavity All validation result with the King s cavity is shown in Fig. 3. Firstly, the vertical mean velocity at y/h=0.5 near the hot wall is compared with the prediction by models in Fig. 3 (a). EBRSM/ shows the best agreement with the maximum value of vertical velocity and the behavior far from the wall, and seems to overestimate the peak velocity. In Fig. 3 (b), the calculated local Nusselt number was compared with the experiment and other researcher s work [12]. Barhaghi et al. [12] investigated the behavior of the turbulent natural convection boundary layer and the transition from laminar to turbulent along the wall by large eddy simulation (LES). The transition region is about ~0.5m as shown in the experiment result. Nusselt number is well predicted by EBRSM/ and EBRSM/, and the transition at the similar region is also well predicted with those models. In this respect, this shows the better result than LES works. However, there is no transition occurred by using the model without buoyancy corrected or. Figure 3 (c) and (d) show the comparison of the experiment and simulation data for the Reynolds shear stress and the vertical velocity fluctuation. In Fig. 3 (c), the experimental data shows asymmetry result, due to the insufficient insulation at the top. On the other hand, the numerical results present symmetric and underestimate near hot wall and overestimate near cold wall compared with the experiment. Vertical fluctuating velocity is well predicted with the buoyancy corrected model in Fig. 3 (d). Models without buoyancy effects, and, tend to overestimate the vertical velocity fluctuation far from the wall. Figure 3 (e) and (f)
6 Horizontal turbulent heat flux (m K/s) Vertical turbulent heat flux (m K/s) Reynolds shear stress (m 2 /s 2 ) Vertical velocity fluctuation (m/s) Vertical velocity (m/s) Local Nusselt number The 8 th European Review Meeting on Severe Accident Research -ERMSAR-2017 show that the turbulent heat fluxes near the hot wall are reproduced appropriately with the buoyancy corrected models, and the anisotropy behavior in the vicinity of the wall is captured properly by those models. On the other hand, eddy diffusivity model ( and SMC(TMF) in this study) fails to calculate the vertical turbulent heat flux as shown in Fig. 3 (f). 0.3 SMC(TMF) SMC(TMF/THF) LES (SMG) [12] LES (DYN) [12] LES (WALE) [12] SMC(TMF) SMC(TMF/THF) (a) Y (b) SMC(TMF) SMC(TMF/THF) SMC(TMF) SMC(TMF/THF) (c) (d) SMC (TMF) SMC (TMF/THF) SMC (TMF) SMC (TMF/THF) (e) (f) Figure 3 Natural convection in a rectangular cavity of King [6] at Ra ~ (a) Mean vertical velocity profiles at y/h=0.5 near the hot wall, (b) Local Nusselt number distribution along the hot wall, (c) Reynolds shear stress profiles at y/h=0.5, (d) Vertical velocity fluctuation at y/h=0.5, (e) Horizontal turbulent heat flux at y/h=0.5, (f) Vertical turbulent heat flux at y/h= BALI: Natural convection with internal heat source The instantaneous temperature contour is shown in Fig. 4 (a). The cold plumes are captured intermittently along the top, and the temperature of the upper region stays uniform. Figure 4 (b) shows the mean temperature profile along the center depth direction at x = 0.1m. Experiment result shows that the dimensionless depth of mixing region is about from the top, and it is well matched with the numerical results. However, the deviations are observed in the stratified zone depending on the models.
7 Heat flux / Maximum heat flux (-) Temperature (-) The 8 th European Review Meeting on Severe Accident Research -ERMSAR-2017 Heat flux distribution along the curved wall compared with the experiment is showed in Fig. 5. The heat flux decreases continuously from its peak value near top surface in case of model, while it shows almost uniform in the unstable zone which is convection dominant region in case of SMC ( and SSG) model. A similar tendency is observed in the stable zone with little convection Exp. [1] SMC (TMF) Depth/ Pool height (-) (a) (b) Figure 4 Natural convection with internal heat of BALI [1] at Ra ~ (a) Contour of the instantaneous temperature (), (b) Mean temperature profiles along the center depth direction at x=0.1m Exp. [1] SMC (TMF) SSG(A) Depth/ pool height (-) Figure 5 Heat flux profiles along the side wall Reactor case: low Prandtl number effect (Ra ~ ) In this study, properties are modified to simulate the oxide layer which has lower Prandtl number than water. Material properties in actual scenarios were used [13]. Figure 6 shows the contours of the instantaneous temperature for water (Pr = 5.4) and corium (Pr = 0.5) cases. For the low Pr case, it is observed that the intense cold plumes from the top surface and descending flow along the side wall penetrate into the stable zone, as a result the stable zone becomes more unstable than high Pr number case (it is called υ-phenomenon). Figure 7 shows the temperature distribution along the center line at x = 0.1m (a) and heat flux distribution along the side wall (b). Both figures show that low temperature and high heat flux distribution in the stably-stratified zone for the low Prandtl number case, and it means that the heat transfer in the lower cavity is enhanced due to relatively
8 Temperature (-) Heat flux / Maximum heat flux (-) The 8 th European Review Meeting on Severe Accident Research -ERMSAR-2017 high thermal diffusivity coefficient, so-called α-phenomenon. On the other hand, a similar trend is observed in the well-mixed convection dominant region. (a) (b) Figure 6 Contour of the instantaneous temperature (turbulence model: ). (a) Water (Pr = 5.4), (b) Corium (Pr = 0.5) Turbulence model: Exp. [1] Water (Pr = 5.4) Corium (Pr = 0.5) Turbulence model: Exp. [1] Water (Pr = 5.4) Corium (Pr = 0.5) Depth/ Pool height (-) Depth/ pool height (-) (a) (b) Figure 7 Comparison results between water and corium with same turbulence model (). (a) Mean temperature distribution along the center line, (b) Heat flux profiles along the side wall 4. Conclusion To understand thermal behavior of the oxide pool, dominant phenomena of turbulence natural convection should be resolved including stratification, laminar to turbulence transition, cold plumes and anisotropic behavior near wall. The buoyancy corrected SMC model was implemented in OpenFOAM which is open source CFD code. From the differentially heated natural convection cavity case, the predicted transition location along the vertical wall with EBRSM/ and EBRSM/ shows good agreement with experiment data. The anisotropic behavior near wall is also reproduced properly. In case of natural convection with internal source, the cold plume and the stratification phenomena is observed with elementary model which is isotropy or buoyancy not-corrected. Some discrepancy of temperature distribution is analyzed in the stratified zone and it is necessary to improve it with buoyancy corrected models. Finally, the properties of the prototypical oxide melts were used and the fluid Prandtl number effect which was proposed by Nourgaliev et al. was observed in Ra is around Further simulations with the buoyancy corrected turbulence models and several geometries, i.e. 2D, 3D, slice, hemisphere etc. are required to understand the thermal behavior of the real reactor case. Acknowledgements
9 This work was supported by the Nuclear Safety Research Program through the Korea Foundation Of Nuclear Safety(KOFONS), granted financial resource from the Nuclear Safety and Security Commission(NSSC), Republic of Korea (No SB120) References [1] J. M. Bonnet and J. M. Seiler, Thermal Hydraulic Phenomena in Corium Pools : the Bali Experiment., 7th Int. Conf. Nucl. Eng., no. 9, pp. 1 10, [2] R. R. Nourgaliev, Modeling and analysis of heat and mass transfer processes during in-vessel melt progression stage of light water reactor (LWR) severe accidents, KTH, [3] R. R. Nourgaliev, T. N. Dinh, and B. R. Sehgal, Effect of fluid Prandtl number on heat transfer characteristics in internally heated liquid pools with Rayleigh numbers up to 10(12), Nucl. Eng. Des., vol. 169, no. 1 3, pp , [4] T. N. Dinh and R. R. Nourgaliev, Turbulence modelling for large volumetrically heated liquid pools, Nucl. Eng. Des., vol. 169, no. 1 3, pp , [5] M. Fukasawa, S. Hayakawa, and M. Saito, Thermal-Hydraulic Analysis for Inversely Stratified Molten Corium in Lower Vessel, J. Nucl. Sci. Technol., vol. 45, no. 9, pp , [6] K. J. King, Turbulent natural convection in rectangular cavities, [7] R. Manceau and K. Hanalić, Elliptic blending model: A new near-wall Reynolds-stress turbulence closure, Phys. Fluids, vol. 14, no. 2, pp , [8] R. Manceau, Recent progress in the development of the Elliptic Blending Reynolds-stress model To cite this version : Recent progress in the development of the Elliptic Blending Reynolds-stress model, no. September, [9] S.-K. Choi, J.-W. Han, S.-O. Kim, and T.-H. Lee, Computation of turbulent natural convection with the elliptic-blending differential and algebraic flux models, Numer. Heat Transf. Part B Fundam., vol. 71, no. 1, pp , [10] F. Dehoux, Y. Lecocq, S. Benhamadouche, R. Manceau, and L. E. Brizzi, Algebraic modeling of the turbulent heat fluxes using the elliptic blending approach-application to forced and mixed convection regimes, Flow, Turbul. Combust., vol. 88, no. 1 2, pp , [11] F. Dehoux, S. Benhamadouche, and R. Manceau, An elliptic blending differential flux model for natural, mixed and forced convection, Int. J. Heat Fluid Flow, vol. 0, no. September, pp. 1 15, Oct [12] D. G. Barhaghi and L. Davidson, Natural convection boundary layer in a 5:1 cavity, Phys. Fluids, vol. 19, no. 12, pp. 1 15, [13] H. Esmaili and M. Khatib-Rahbar, Analysis of In-Vessel Retention and Ex-Vessel Fuel Coolant Interaction for AP1000, NUREG/CR-6849, 2004.
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