Hostname: page-component-84b7d79bbc-4hvwz Total loading time: 0 Render date: 2024-07-26T22:25:22.687Z Has data issue: false hasContentIssue false
  • English
  • Français

On a diphasic low mach number system

Published online by Cambridge University Press:  15 June 2005

Stéphane Dellacherie
Stéphane Dellacherie*
Affiliation:
Commissariat à l'Énergie Atomique, 91191 Gif sur Yvette, France. Centre de Recherches Mathématiques, Université de Montréal, C.P. 6128-succursale Centre-ville, Montréal, H3C 3J7, Canada. stephane.dellacherie@cea.fr; dellache@crm.umontreal.ca
Get access

Abstract

We propose a Diphasic Low Mach Number (DLMN) system for the modelling of diphasic flows without phase change at low Mach number, system which is an extension of the system proposed by Majda in [Center of Pure and Applied Mathematics, Berkeley, report No. 112] and [Combust. Sci. Tech.42 (1985) 185–205] for low Mach number combustion problems. This system is written for a priori any equations of state. Under minimal thermodynamic hypothesis which are satisfied by a large class of generalized van der Waals equations of state, we recover some natural properties related to the dilation and to the compression of bubbles. We also propose an entropic numerical scheme in Lagrangian coordinates when the geometry is monodimensional and when the two fluids are perfect gases. At last, we numerically show that the DLMN system may become ill-posed when the entropy of one of the two fluids is not a convex function.

Keywords

Diphasic flowlow mach number systemthermodynamic equilibriumentropyvan der Waals equations of state.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 39 , Issue 3: Special issue on Low Mach Number Flows Conference , May 2005 , pp. 487 - 514
Copyright
© EDP Sciences, SMAI, 2005

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

R. Abgrall, R. Saurel, A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys. 150 425–467 (1999).
Allaire, G., Clerc, S. and Kokh, S., A five-equation model for the numerical simulation of interfaces in two-phase flows. C. R. Acad. Sci. Paris Ser. I 331 (2000) 1017–1022.
Allaire, G., Clerc, S. and Kokh, S., A five-equation model for the simulation of interfaces between compressible fluids. J. Comput. Phys. 181 (2002) 577–616. CrossRef
Choi, Y.-H., Merkle, C.L., The Application of Preconditioning in Viscous Flows. J. Comput. Phys. 105 (1993) 207–223. CrossRef
A.J. Chorin and J.E. Mardsen, A Mathematical Introduction to Fluid Mechanics. Springer-Verlag (1979).
Dellacherie, S., On relaxation schemes for the multicomponent Euler system. ESAIM: M2AN 37 (2003) 909–936. CrossRef
S. Dellacherie, Dérivation du système diphasique bas Mach. Simulation numérique en géométrie monodimensionnelle. CEA report, ref. CEA-R-6046 (2004).
S. Dellacherie and A. Vincent, Zero Mach Number Diphasic Equations for the Simulation of Water-Vapor High Pressure Flows, in Proc. of the 11th conference of the CFD Society of Canada, Vancouver (2003) 248–255.
Embid, P., Well-posedness of the nonlinear equations for zero Mach number combustion. Comm. Partial Differential Equations 12 (1987) 1227–1283. CrossRef
Gueyffier, D., Li, J., Nadim, A., Scardovelli, R. and Zaleski, S., Volume-of-Fluid interface tracking with smoothed surface stress methods for three-dimensional flows. J. Comput. Phys. 152 (1999) 423–456. CrossRef
Guillard, H. and Murrone, A., On the behavior of upwind schemes in the low Mach number limit: II. Godunov type schemes. Comput. Fluids 33 (2004) 655–675. CrossRef
Guillard, H. and Viozat, C., On the behaviour of upwind schemes in the low Mach number limit. Comput. Fluids 28 (1999) 63–86. CrossRef
Harlow, F.H. and Welch, J.E., Numerical calculation of time-dependent viscous incompressible flow of fluid with free interface. Phys. Fluids 8 (1965) 2182–2189. CrossRef
Jamet, D., Lebaigue, O., Coutris, N. and Delhaye, J.M., The second gradient method for the direct numerical simulation of liquid-vapor flows with phase change. J. Comput. Phys. 169 (2001) 624–651. CrossRef
Juric, D. and Tryggvason, G., Computations of boiling flows. Int. J. Multiphase Flow 24 (1998) 387–410. CrossRef
S. Kokh, Aspects numériques et théoriques de la modélisation des écoulements diphasiques compressibles par des méthodes de capture d'interface. Ph.D. thesis of Paris VI University (2001).
Lafaurie, B., Nardone, C., Scardovelli, R., Zaleski, S. and Zanetti, G., Modelling merging and fragmentation in multiphase flows with SURFER. J. Comput. Phys. 113 (1994) 134–147. CrossRef
F. Lagoutière, Modélisation mathématique et résolution numérique de problèmes de fluides compressibles à plusieurs constituants. Ph.D. thesis of Paris VI University (2000).
Lakehal, D., Meier, M. and Fulgosi, M., Interface tracking towards the direct numerical simulation of heat and mass transfer in multiphase flow. Internat. J. Heat Fluid Flow 23 (2002) 242–257. CrossRef
J.M. Le Corre, E. Hervieu, M. Ishii and J.M. Delhaye, Benchmarking and improvements of measurement techniques for local time-averaged two-phase flow parameters. Fourth International Conference on Multiphase Flows (ICMF 2001), New-Orleans, USA (2001).
A. Majda, Equations for low mach number combustion. Center of Pure and Applied Mathematics, University of California at Berkeley, report No. 112 (1982).
Majda, A. and Sethian, J.A., The derivation and numerical solution of the equations for zero Mach number combustion. Combust. Sci. Tech. 42 (1985) 185–205. CrossRef
Mulder, W., Osher, S. and Sethian, J.A., Computing interface motion in compressible gas dynamics. J. Comput. Phys. 100 (1992) 209–228. CrossRef
Osher, S., Sussman, M. and Smereka, P., A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114 (1994) 146–159.
S. Paolucci, On the filtering of sound from the Navier-Stokes equations. Sandia National Laboratories report SAND82-8257 (1982).
J.A. Sethian, Level Set Methods. Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press (1996).
Shyue, K.M., A fluid-mixture type algorithm for compressible multicomponent flow with van der Waals equation of state. J. Comput. Phys. 156 (1999) 43–88. CrossRef
Tryggvasson, G. and Unverdi, S.O., A front-tracking method for viscous, incompressible, multi-fluid flows. J. Comput. Phys. 100 (1992) 25–37.
Turkel, E., Review of preconditioning methods for fluid dynamics. Appl. Numer. Math. 12 (1993) 257–284. CrossRef
Welch, S.W.J. and Wilson, J., A volume of fluid based method for fluid flows with phase change. J. Comput. Phys. 160 (2000) 662–682. CrossRef