Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-23T07:02:46.480Z Has data issue: false hasContentIssue false
  • English
  • Français

Interface model coupling via prescribed local flux balance

Published online by Cambridge University Press:  24 April 2014

Annalisa Ambroso ,
Christophe Chalons ,
Frédéric Coquel  and
Thomas Galié
Annalisa Ambroso
Affiliation:
UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France
Christophe Chalons
Affiliation:
Laboratoire de Mathématiques de Versailles, UMR 8100, Université de Versailles Saint-Quentin-en-Yvelines UFR des Sciences, bâtiment Fermat, 45 avenue des Etats-Unis, 78035 Versailles Cedex, France. [email protected]
Frédéric Coquel
Affiliation:
CNRS & Centre de Mathématiques Appliquées, U.M.R. 7641 Ecole Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France; [email protected]
Thomas Galié
Affiliation:
UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France
Get access

Abstract

This paper deals with the non-conservative coupling of two one-dimensional barotropic Euler systems at an interface at x = 0. The closure pressure laws differ in the domains x < 0 and x > 0, and a Dirac source term concentrated at x = 0 models singular pressure losses. We propose two numerical methods. The first one relies on ghost state reconstructions at the interface while the second is based on a suitable relaxation framework. Both methods satisfy a well-balanced property for stationary solutions. In addition, the second method preserves mass conservation and exactly restores the prescribed singular pressure drops for both unsteady and steady solutions.

Keywords

Gas dynamics equationsinterfacial couplingmeasure valued loadrelaxation methodcoupled Riemann problem
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 3 , May 2014 , pp. 895 - 918
Copyright
© EDP Sciences, SMAI, 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adimurthi, M.S. and Gowda, G.D.V., Optimal entropy solutions for conservation laws with discontinuous flux-functions. J. Hyperbolic Differ. Equ. 2 (2005) 783–837. Google Scholar
Amadori, D., Gosse, L., Graziano, G., Godunov-type approximation for a general resonant balance law with large data. J. Differ. Equ. 198 (2004) 233–274. Google Scholar
Ambroso, A., Chalons, C., Coquel, F., Godlewski, E., Lagoutiere, F., Raviart, P.-A. and Seguin, N., The coupling of homogeneous models for two-phase flows. Int. J. Finite Volumes 4 (2007) 1–39. Google Scholar
Ambroso, A., Chalons, C., Coquel, F., Godlewski, E., Lagoutiere, F., Raviart, P.-A. and Seguin, N., Coupling of general Lagrangian systems. Math. Comput. 77 (2008) 909–941. Google Scholar
Ambroso, A., Hérard, J.-M. and Hurisse, O., A method to couple HEM and HRM two-phase flow models. Comput. Fluids 38 (2009) 738–756. Google Scholar
Audusse, E. and Perthame, B., Uniqueness for scalar conservation laws with discontinuous flux via adapted entropies. Proc. Roy. Soc. Edinburgh Sect. A 135 (2005) 253–265. Google Scholar
Bachmann, F. and Vovelle, J., Existence and uniqueness of entropy solution of scalar conservation laws with a flux function involving discontinuous coefficients. Commun. Partial Differential Equations 31 (2006) 371–395. Google Scholar
Bestion, D., Boucker, M., Boudier, P., Fillion, P., Grandotto, M., Guelfi, A., Hérard, J.M., Hervieu, E., Péturaud, P., Neptune: a new software platform for advanced nuclear thermal hydraulics. Nuclear Science and Engineering 156 (2007) 281–324. Google Scholar
F. Bouchut, Nonlinear stability of Finite volume methods for hyperbolic conservation laws, and well-balanced schemes for sources. Frontiers in Mathematics, Birkhauser (2004).
Boutin, B., Chalons, C. and Raviart, P.A., Existence result for the coupling problem of two scalar conservation laws with Riemann initial data. Math. Models Methods Appl. Sci. 20 (2010) 1859–1898. Google Scholar
Boutin, B., Coquel, F. and LeFloch, P.G., Coupling techniques for nonlinear hyperbolic equations. I. Self-similar diffusion for thin interfaces. Proc. Roy. Soc. Edinburgh Sect. A 141 (2011) 921–956. Google Scholar
Bucci, M. and Fillion, P., Analysis of the NUPEC PSBT Tests with FLICA-OVAP. Science and Technology of Nuclear Installations. Article ID 2012 (2012) 436142. Google Scholar
Bürger, R. and Karlsen, K.H., Conservation laws with discontinuous flux: a short introduction. J. Engrg. Math. 60 (2008) 241–247. Google Scholar
Bürger, R., Karlsen, K.H. and Towers, J.D., An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections. SIAM J. Numer. Anal. 47 (2009) 1684–1712. Google Scholar
Chalons, C., Raviart, P.-A. and Seguin, N., The interface coupling of the gas dynamics equations. Quaterly of Applied Mathematics 66 (2008) 659–705. Google Scholar
Dafermos, C.M., Solution of the Riemann problem for a class of hyperbolic systems of conservation laws by the viscosity method. Arch. Rational Mech. Anal. 52 (1973) 1–9. Google Scholar
Diehl, S., On scalar conservation laws with point source and discontinuous flux function. SIAM J. Math. Anal. 26 (1995) 1425–1451. Google Scholar
Diehl, S., Scalar conservation laws with discontinuous flux function. I. The viscous profile condition, Commun. Math. Phys. 176 (1996) 23–44. Google Scholar
W.H. Hager, Wastewater Hydraulics, Theory and Practice. Springer (2010).
T. Galié, Couplage interfacial de modèles pour la thermoohydraulique des réacteurs, Ph.D. thesis, Université Pierre et Marie Curie Paris 6 (2008).
Gimse, T. and Risebro, N.H., Solution of the Cauchy problem for a conservation law with a discontinuous flux function. SIAM J. Math. Anal. 23 (1992) 635–648. Google Scholar
Glimm, J., Marchesin, D. and McBryan, O., Numerical method for two phase flow with unstable interface. J. Comput. Phys. 39 (1981) 179–200. Google Scholar
Goatin, P. and LeFloch, P.G., The Riemann problem for a class of resonant hyperbolic systems of balance laws. Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004) 881–902. Google Scholar
Godlewski, E., Le Thanh, K.-C. and Raviart, P.-A., The numerical interface coupling of nonlinear hyperbolic systems of conservation laws. II. The case of systems. M2AN Math. Model. Numer. Anal. 39 (2005) 649–692. Google Scholar
Godlewski, E. and Raviart, P.-A., The numerical interface coupling of nonlinear hyperbolic systems of conservation laws. I. The scalar case. Numer. Math. 97 (2004) 81–130. Google Scholar
Gosse, L., A well-balanced scheme using non-conservative products designed for hyperbolic systems of conservation laws with source terms. Math. Models Methods Appl. Sci. 11 (2001) 339–365. Google Scholar
Gosse, L., Localization effects and measure source terms in numerical schemes for balance laws. Math. Comp. 71 (2001) 553–582. Google Scholar
Greenberg, J.M. and Leroux, A.Y., A well-balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 33 (1996) 1–16. Google Scholar
Isaacson, E. and Temple, B.J., Nonlinear resonance in systems of conservation laws. SIAM J. Appl. Math. 52 (1992) 1260–1278. Google Scholar
Greenberg, J.M., Roux, A.Y.L., Baraille, R. and Noussair, A., Analysis and approximation of conservation laws with source terms. SIAM J. Numer. Anal. 34 (1997) 1980–2007. Google Scholar
J.-M. Hérard, Schemes to couple flows between free and porous medium. Proceedings of AIAA (2005) 2005–4861.
J.-M. Hérard and O. Hurisse, Coupling two and one-dimensional models through a thin interface. Proceedings of AIAA (2005) 2005–4718.
J.-M. Hérard and O. Hurisse, Boundary conditions for the coupling of two-phase flow models. 18th AIAA CFD conference.
I.E. Idel’cik, Memento des pertes de charges. Coefficients de pertes de charges singulières et de pertes de charges par frottement. Collection Direction des Etudes et Recherches d’EDF. Eyrolles [in French] (1986).
Jin, S. and Xin, Z., The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Comm. Pure Appl. Math. 48 (1995) 235–276. Google Scholar
V.G. Kourakos, P. Rambaud, S. Chabane and J.M. Buchlin, Modeling of pressure drop in two-phase flow in singular geometries. 6th International Symposium on Multiphase Flow, Heat Mass Transfert and Energy Conservation. Xi’an, China, 11-15 July 2009, Paper No MN-30, 2009.
D.S. Miller (Ed.), Discharge Characteristics: IAHR Hydraulic Structures Design Manuals 8. Balkema: Rotterdam (1994).
Kruzkov, S.N., First order quasilinear equations with several independent variables. Mat. Sb. (N.S.) 81 (1970) 228–255. Google Scholar