# zbMATH — the first resource for mathematics

An implicit integral formulation to model inviscid fluid flows in obstructed media. (English) Zbl 07074186
Summary: We focus here on a technique to compute compressible fluid flows in physical domains cluttered up with many small obstacles. This technique, referred to here as the integral formulation, consists in integrating the flow governing equations over the fluid part of control volumes including both fluid and solid zones; doing so, the integral of fluxes over solid boundaries may appear, for which expressions as a function of discrete variables must be provided. The integral formulation presents two essential advantages: first, we naturally recover the standard fluid approach when the mesh is refined; second, fluid/solid interactions may be, to some extent, modelled to recover the singular head losses at the interface between a free and a congested part of the computational domain. We apply here this approach to the Euler equations, using a collocated space discretization and a pressure correction algorithm, preserving the positivity of both the density and the internal energy. Verification test cases are performed, including a Riemann problem in a free domain and a shock wave reflection on a wall, using an equation of state which is suitable for weakly compressible fluid flows. Finally, we address a two-dimensional situation, where a shock wave impacts a set of obstacles; we observe a very encouraging agreement between the integral approach results and a CFD reference solution obtained with a pure fluid approach on a fine mesh.
##### MSC:
 76-XX Fluid mechanics
FLICA-4; THYC
Full Text:
##### References:
 [1] Nuclear Energy Agency (NEA), Loss-Of-Coolant Accident (LOCA). https://www.oecd-nea.org/nsd/reports/2009/nea6846_LOCA.pdf; 2009. [2] Institut de Radioprotection et de Sûreté Nucléaire (IRSN), Reactivity-Initiated Accident (RIA). https://www.irsn.fr/FR/connaissances/Installations_nucleaires/Les-centrales-nucleaires/criteres_surete_ria_aprp/Pages/1-accident-reactivite-RIA.aspx; 2018. [3] Barre, F.; Bernard, M., The CATHARE code strategy and assessment, Nucl Eng Des, 124(3):257-284, (1990) [4] EDF R&D, Code_Saturne 5.2.0 Theory Guide. https://code-saturne.org/cms/sites/default/files/docs/5.2/theory.pdf; 2018. [5] Aubry, S.; Caremoli, C.; Olive, J.; Rascle, P., The THYC three-dimensional thermal-hydraulic code for rod bundles: recent developments and validation tests, Nucl Technol, 112, 3, 331-345, (1995) [6] Toumi, I.; Bergeron, A.; Gallo, D.; Royer, E.; Caruge, D., FLICA-4: a three-dimensional two-phase flow computer code with advanced numerical methods for nuclear applications, Nucl Eng Des, 200, 139-155, (2000) [7] Grandotto, M.; Obry, P., Steam generators two phase flows numerical simulation with liquid and gas momentum equations, Nucl Sci Eng, 151, 313-318, (2005) [8] Grandotto, M.; Obry, P., Calculs des écoulements diphasiques dans les échangeurs par une méthode aux éléments finis, Revue Européenne des Eléments Finis, 5, 1, 53-74, (1996) · Zbl 0924.76067 [9] Belliard, M., Méthodes de décomposition de domaine et de frontiére immergée pour la simulation des composants nucléaires, (2014), Aix-Marseille Université, Habilitation á diriger des recherches [10] FLACS Software. GEXCON; https://www.gexcon.com/products-services/FLACS-Software/22/en; 2018. [11] Davit, Y.; Bell, C. G.; Byrne, H.; Chapman, L. A.C.; Kimpton, L. S.; Lang, K. H.L.; Leonard, G. E.; Leonard, G. E.; Oliver, J. M.; Pearson, N. C.; Shipley, R. J.; Waters, S. L.; Whiteley, J. P.; Wood, B. D., Homogenization via formal multiscale asymptotics and volume averaging: how do the two techniques compare?, Adv Water Resour, 62, 178-206, (2013) [12] Girault, L.; Hérard, J.-M., Multidimensional computations of a two-fluid hyperbolic model in a porous medium, Int J Finite Volumes, 7, 1, 1-33, (2010) [13] Eymard, R.; Gallouët, T.; Herbin, R., Finite volume methods, (Ciarlet, P. G.; Lions, J. L., Handbook for Numerical Analysis, 7, (2006), North Holland), 713-1020 · Zbl 0981.65095 [14] Chorin, A.-J., Numerical solution of the Navier-Stokes equations, Math Comput, 22(104):745-762, (1968) · Zbl 0198.50103 [15] Temam, R., Sur l’approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires (II), Arch Ration Mech Anal, 33, 5, 377-385, (1969) · Zbl 0207.16904 [16] Van Der Heul, D.-R.; Vuik, C.; Wesseling, P., A conservative pressure-correction method for flow at all speeds, Comput Fluids, 32, 1113-1132, (2003) · Zbl 1046.76033 [17] Gallouët, T.; Gastaldo, L.; Latché, J.-C.; Herbin, R., An unconditionnally stable pressure correction scheme for compressible barotropic Navier-Stokes equations, ESAIM: Math Mod Numer Anal, 44, 2, 251-287, (2010) [18] Kheriji, W.; Herbin, R.; Latché, J.-C., Pressure correction staggered schemes for barotropic one-phase and two-phase flows, Comput Fluids, 88, 524-542, (2013) · Zbl 1391.76778 [19] Zaza, C., Contribution to numerical methods for all mach flow regimes and to fluid-porous coupling for the simulation of homogeneous two-phase flows in nuclear reactors, (2015), Aix-Marseille Université [20] Archambeau, F.; Hérard, J.-M.; Laviéville, J., Comparative study of pressure-correction and Godunov-type schemes on unsteady compressible cases, Comput Fluids, 38, 1495-1509, (2009) · Zbl 1242.76152 [21] Hérard, J.-M.; Martin, X., An integral approach to compute compressible fluid flows in domains containing obstacles, Int J Finite Volumes, 12, 1, 1-39, (2015) [22] Menikoff, R.; Plohr, B., The Riemann problem for fluid flow of real materials, Rev Mod Phy, 61, 1, 75-130, (1989) · Zbl 1129.35439 [23] Martin, X., Modélisation d’écoulements fluides en milieu encombré d’obstacles, (2015), Aix-Marseille Université, Chapter 3, pages 207-238 [24] Eymard, R.; Gallouët, T.; Guichard, C.; Herbin, R.; Masson, R., TP or not TP, that is the question, Comput Geosci, 18, 285-296, (2014) · Zbl 1378.76118 [25] Buffard, T.; Gallouët, T.; Hérard, J.-M., A sequel to a rough Godunov scheme: application to real gases, Comput Fluids, 29, 13-847, (2000) · Zbl 0961.76048 [26] Dubois, F., Boundary conditions and the Osher scheme for the Euler equations of gas dynamics, Internal Report CMAP 170, Ecole Polytechnique, Palaiseau, France, (1987) [27] Blondel, F.; Audebert, B.; Pasutto, T.; Stanciu, M., Condensation models and boundary conditions for non-equilibrium wet steam flows, Int J Finite Volumes, 10, 1-53, (2013) [28] Smoller, J., Shock waves and reaction-diffusion equations, A Series of Comprehensive Studies in Mathematics, 258, (1994), Springer-Verlag: Springer-Verlag New York · Zbl 0807.35002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.