Study of a low Mach nuclear core model for two-phase flows with phase transition I: stiffened gas law
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 48 (2014) no. 6, pp. 1639-1679.

In this paper, we are interested in modelling the flow of the coolant (water) in a nuclear reactor core. To this end, we use a monodimensional low Mach number model supplemented with the stiffened gas law. We take into account potential phase transitions by a single equation of state which describes both pure and mixture phases. In some particular cases, we give analytical steady and/or unsteady solutions which provide qualitative information about the flow. In the second part of the paper, we introduce two variants of a numerical scheme based on the method of characteristics to simulate this model. We study and verify numerically the properties of these schemes. We finally present numerical simulations of a loss of flow accident (LOFA) induced by a coolant pump trip event.

DOI : https://doi.org/10.1051/m2an/2014015
Classification : 35Q35,  35Q79,  65M25,  76T10
Mots clés : low Mach number flows, modelling of phase transition, analytical solutions, method of characteristics, positivity-preserving schemes
@article{M2AN_2014__48_6_1639_0,
author = {Bernard, Manuel and Dellacherie, St\'ephane and Faccanoni, Gloria and Grec, B\'er\'enice and Penel, Yohan},
title = {Study of a low Mach nuclear core model for two-phase flows with phase transition I: stiffened gas law},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
pages = {1639--1679},
publisher = {EDP-Sciences},
volume = {48},
number = {6},
year = {2014},
doi = {10.1051/m2an/2014015},
mrnumber = {3264368},
language = {en},
url = {archive.numdam.org/item/M2AN_2014__48_6_1639_0/}
}
Bernard, Manuel; Dellacherie, Stéphane; Faccanoni, Gloria; Grec, Bérénice; Penel, Yohan. Study of a low Mach nuclear core model for two-phase flows with phase transition I: stiffened gas law. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 48 (2014) no. 6, pp. 1639-1679. doi : 10.1051/m2an/2014015. http://archive.numdam.org/item/M2AN_2014__48_6_1639_0/

[1] TRACE V5.0 Theory Manual, Field Equations, Solution Methods and Physical Models. Technical report, U.S. Nuclear Regulatory Commission (2008).

[2] A. Acrivos, Method of characteristics technique. Application to heat and mass transfer problems. Ind. Eng. Chem. 48 (1956) 703-710.

[3] G. Allaire, G. Faccanoni and S. Kokh, A strictly hyperbolic equilibrium phase transition model. C. R. Acad. Sci. Paris Ser. I 344 (2007) 135-140. | MR 2288604 | Zbl 1109.35066

[4] A.S. Almgren, J.B. Bell, C.A. Rendleman and M. Zingale, Low Mach number modeling of type Ia supernovae. I. hydrodynamics. Astrophys. J. 637 (2006) 922.

[5] A.S. Almgren, J.B. Bell, C.A. Rendleman and M. Zingale, Low Mach number modeling of type Ia supernovae. II. energy evolution. Astrophys. J. 649 (2006) 927.

[6] M. Bernard, S. Dellacherie, G. Faccanoni, B. Grec, O. Lafitte, T.-T. Nguyen and Y. Penel. Study of low Mach nuclear core model for single-phase flow. ESAIM Proc. 38 (2012) 118-134. | MR 3006539

[7] D. Bestion. The physical closure laws in the CATHARE code. Nucl. Eng. Des. 124 (1990) 229-245.

[8] H. B. Callen, Thermodynamics and an Introduction to Thermostatistics. 2nd edition. John Wiley and sons (1985). | Zbl 0095.23301

[9] V. Casulli and D. Greenspan, Pressure method for the numerical solution of transient, compressible fluid flows. Int. J. Numer. Methods Fluids 4 (1984) 1001-1012. | Zbl 0549.76050

[10] S. Clerc, Numerical Simulation of the Homogeneous Equilibrium Model for Two-Phase Flows. J. Comput. Phys. 181 (2002) 577-616. | MR 1762085 | Zbl 1169.76407

[11] P. Colella and K. Pao, A projection method for low speed flows. J. Comput. Phys. 149 (1999) 245-269. | MR 1672739 | Zbl 0935.76056

[12] J.M. Delhaye, Thermohydraulique des réacteurs. EDP sciences (2008).

[13] S. Dellacherie, On a diphasic low Mach number system. ESAIM: M2AN 39 (2005) 487-514. | Numdam | MR 2157147 | Zbl 1075.35038

[14] S. Dellacherie, Numerical resolution of a potential diphasic low Mach number system. J. Comput. Phys. 223 (2007) 151-187. | MR 2314387 | Zbl 1163.76035

[15] S. Dellacherie, Analysis of Godunov type schemes applied to the compressible Euler system at low Mach number. J. Comput. Phys. 229 (2010) 978-1016. | MR 2576236 | Zbl pre05668155

[16] S. Dellacherie, On a low Mach nuclear core model. ESAIM Proc. 35 (2012) 79-106. | MR 3040775 | Zbl pre06023187

[17] S. Dellacherie, G. Faccanoni, B. Grec, F. Lagoutière, E. Nayir and Y. Penel, 2D numerical simulation of a low Mach nuclear core model with stiffened gas using Freefem++. ESAIM. Proc. (accepted).

[18] S. Dellacherie, G. Faccanoni, B. Grec and Y. Penel, Study of low Mach nuclear core model for two-phase flows with phase transition II: tabulated EOS. In preparation.

[19] M. Drouin, O. Grégoire and O. Simonin, A consistent methodology for the derivation and calibration of a macroscopic turbulence model for flows in porous media. Int. J. Heat Mass Transfer 63 (2013) 401-413.

[20] D.R. Durran, Numerical methods for fluid dynamics, With applications to Geophysics, vol. 32 of Texts in Applied Mathematics. Springer, 2nd edition. New York (2010). | MR 2723959 | Zbl 1214.76001

[21] P. Embid, Well-posedness of the nonlinear equations for zero Mach number combustion. Comm. Partial Differ. Equ. 12 (1987) 1227-1283. | MR 888460 | Zbl 0632.76075

[22] G. Faccanoni, Étude d'un modèle fin de changement de phase liquide-vapeur. Contribution à l'étude de la crise d'ébullition. Ph.D. thesis, École Polytechnique, France (2008).

[23] G. Faccanoni, S. Kokh and G. Allaire, Modelling and simulation of liquid-vapor phase transition in compressible flows based on thermodynamical equilibrium. ESAIM: M2AN 46 1029-1054 2012. | Numdam | MR 2916371 | Zbl 1267.76110

[24] P. Fillion, A. Chanoine, S. Dellacherie and A. Kumbaro, FLICA-OVAP: A new platform for core thermal-hydraulic studies. Nucl. Eng. Des. 241 (2011) 4348-4358.

[25] E. Goncalvès and R.F. Patella, Numerical study of cavitating flows with thermodynamic effect. Comput. Fluids 39 (2010) 99-113. | MR 2600812 | Zbl 1242.76331

[26] J.M. Gonzalez-Santalo and R.T. Jr Lahey, An exact solution for flow transients in two-phase systems by the method of characteristics. J. Heat Transfer 95 (1973) 470-476.

[27] W. Greiner, L. Neise and H. Stöcker, Thermodynamics and statistical mechanics. Springer (1997). | Zbl 0823.73001

[28] H. Guillard and C. Viozat, On the behaviour of upwind schemes in the low Mach number limit. Comput. Fluids 28 (1999) 63-86. | MR 1651839 | Zbl 0963.76062

[29] S. Jaouen, Étude mathématique et numérique de stabilité pour des modeles hydrodynamiques avec transition de phase. Ph.D. thesis, Université Paris 6, France (2001).

[30] M.F. Lai, J.B. Bell and P. Colella. A projection method for combustion in the zero Mach number limit, in Proc. of 11th AIAA Comput. Fluid Dyn. Conf. (1993) 776-783.

[31] O. Le Métayer, J. Massoni and R. Saurel, Elaborating equations of state of a liquid and its vapor for two-phase flow models. Int. J. Therm. Sci. 43 (2004) 265-276,.

[32] O. Le Métayer, J. Massoni and R. Saurel, Modelling evaporation fronts with reactive Riemann solvers. J. Comput. Phys. 205 (2005) 567-610. | MR 2134994 | Zbl 1088.76051

[33] E.W. Lemmon, M.O. Mclinden and D.G. Friend, Thermophysical Properties of Fluid Systems. National Institute of Standards and Technology, Gaithersburg MD, 20899.

[34] A. Majda and K.G. Lamb, Simplified equations for low Mach number combustion with strong heat release, Dynamical issues in combustion theory, vol. 35 of IMA Vol. Math. Appl. Springer-Verlag (1991). | MR 1119793 | Zbl 0751.76068

[35] A. Majda and J. Sethian, The derivation and numerical solution of the equations for zero Mach number combustion. Combust. Sci. Technol. 42 (1985) 185-205.

[36] R. Menikoff and B.J. Plohr, The Riemann problem for fluid flow of real materials. Rev. Modern Phys. 61 (1989) 75-130. | MR 977944 | Zbl 1129.35439

[37] S. Müller and A. Voss, The Riemann problem for the Euler equations with nonconvex and nonsmooth equation of state: construction of wave curves. SIAM J. Sci. Comput. 28 (2006) 651-681. | MR 2231725 | Zbl 1114.35127

[38] Y. Penel, An explicit stable numerical scheme for the 1D transport equation. Discrete Contin. Dyn. Syst. Ser. S 5 (2012) 641-656. | MR 2861831 | Zbl 1244.65131

[39] Y. Penel, Existence of global solutions to the 1D abstract bubble vibration model. Differ. Integral Equ. 26 (2013) 59-80. | MR 3058697 | Zbl 1289.35262

[40] R. Saurel, F. Petitpas and R. Abgrall, Modelling phase transition in metastable liquids: application to cavitating and flashing flows. J. Fluid Mech. 607 (2008) 313-350. | MR 2436919 | Zbl 1147.76060

[41] G.I. Sivashinsky, Hydrodynamic theory of flame propagation in an enclosed volume. Acta Astronaut. 6 (1979) 631-645. | Zbl 0397.76062

[42] G. Volpe, Performance of compressible flow codes at low Mach numbers. AIAA J. 31 (1993) 49-56. | Zbl 0775.76140

[43] A. Voss, Exact Riemann solution for the Euler equations with nonconvex and nonsmooth equation of state. Ph.D. thesis, RWTH Aachen (2005). | Zbl 1114.35127

[44] N. Zuber, Flow excursions and oscillations in boiling, two-phase flow systems with heat addition, in Symposium on Two-phase Flow Dynamics, Eindhoven EUR4288e (1967) 1071-1089.