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.
@article{M2AN_2005__39_3_487_0, author = {Dellacherie, St\'ephane}, title = {On a diphasic low Mach number system}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique}, volume = {39}, year = {2005}, pages = {487-514}, doi = {10.1051/m2an:2005020}, mrnumber = {2157147}, zbl = {1075.35038}, language = {en}, url = {http://dml.mathdoc.fr/item/M2AN_2005__39_3_487_0} }
Dellacherie, Stéphane. On a diphasic low Mach number system. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 39 (2005) pp. 487-514. doi : 10.1051/m2an:2005020. http://gdmltest.u-ga.fr/item/M2AN_2005__39_3_487_0/
[1] A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys. 150 425-467 (1999). | Zbl 0937.76053
, ,[2] 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. | Zbl 1010.76055
, and ,[3] A five-equation model for the simulation of interfaces between compressible fluids. J. Comput. Phys. 181 (2002) 577-616. | Zbl pre01845987
, and ,[4] The Application of Preconditioning in Viscous Flows. J. Comput. Phys. 105 (1993) 207-223. | Zbl 0768.76032
, ,[5] A Mathematical Introduction to Fluid Mechanics. Springer-Verlag (1979). | Zbl 0417.76002
and ,[6] On relaxation schemes for the multicomponent Euler system. ESAIM: M2AN 37 (2003) 909-936. | Numdam | Zbl 1070.76037
,[7] Dérivation du système diphasique bas Mach. Simulation numérique en géométrie monodimensionnelle. CEA report, ref. CEA-R-6046 (2004).
,[8] Zero Mach Number Diphasic Equations for the Simulation of Water-Vapor High Pressure Flows2003) 248-255.
and ,[9] Well-posedness of the nonlinear equations for zero Mach number combustion. Comm. Partial Differential Equations 12 (1987) 1227-1283. | Zbl 0632.76075
,[10] Volume-of-Fluid interface tracking with smoothed surface stress methods for three-dimensional flows. J. Comput. Phys. 152 (1999) 423-456. | Zbl 0954.76063
, , , and ,[11] On the behavior of upwind schemes in the low Mach number limit: II. Godunov type schemes. Comput. Fluids 33 (2004) 655-675. | Zbl 1049.76040
and ,[12] On the behaviour of upwind schemes in the low Mach number limit. Comput. Fluids 28 (1999) 63-86. | Zbl 0963.76062
and ,[13] Numerical calculation of time-dependent viscous incompressible flow of fluid with free interface. Phys. Fluids 8 (1965) 2182-2189.
and ,[14] The second gradient method for the direct numerical simulation of liquid-vapor flows with phase change. J. Comput. Phys. 169 (2001) 624-651. | Zbl 1047.76098
, , and ,[15] Computations of boiling flows. Int. J. Multiphase Flow 24 (1998) 387-410. | Zbl 1121.76455
and ,[16] 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).
,[17] Modelling merging and fragmentation in multiphase flows with SURFER. J. Comput. Phys. 113 (1994) 134-147. | Zbl 0809.76064
, , , and ,[18] Modélisation mathématique et résolution numérique de problèmes de fluides compressibles à plusieurs constituants. Ph.D. thesis of Paris VI University (2000).
,[19] Interface tracking towards the direct numerical simulation of heat and mass transfer in multiphase flow. Internat. J. Heat Fluid Flow 23 (2002) 242-257.
, and ,[20] 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).
, , and ,[21] Equations for low mach number combustion. Center of Pure and Applied Mathematics, University of California at Berkeley, report No. 112 (1982).
,[22] The derivation and numerical solution of the equations for zero Mach number combustion. Combust. Sci. Tech. 42 (1985) 185-205.
and ,[23] Computing interface motion in compressible gas dynamics. J. Comput. Phys. 100 (1992) 209-228. | Zbl 0758.76044
, and ,[24] A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114 (1994) 146-159. | Zbl 0808.76077
, and ,[25] On the filtering of sound from the Navier-Stokes equations. Sandia National Laboratories report SAND82-8257 (1982).
,[26] Level Set Methods. Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press (1996). | MR 1409367 | Zbl 0859.76004
,[27] A fluid-mixture type algorithm for compressible multicomponent flow with van der Waals equation of state. J. Comput. Phys. 156 (1999) 43-88. | Zbl 0957.76039
,[28] A front-tracking method for viscous, incompressible, multi-fluid flows. J. Comput. Phys. 100 (1992) 25-37. | Zbl 0758.76047
and ,[29] Review of preconditioning methods for fluid dynamics. Appl. Numer. Math. 12 (1993) 257-284. | Zbl 0770.76048
,[30] A volume of fluid based method for fluid flows with phase change. J. Comput. Phys. 160 (2000) 662-682. | Zbl 0962.76068
and ,