The purpose of this article is the analysis and the development of Eulerian multi-fluid models to describe the evolution of the mass density of evaporating liquid sprays. First, the classical multi-fluid model developed in [Laurent and Massot, Combust. Theor. Model. 5 (2001) 537-572] is analyzed in the framework of an unsteady configuration without dynamical nor heating effects, where the evaporation process is isolated, since it is a key issue. The classical multi-fluid method consists then in a discretization of the droplet size variable into cells called sections. This analysis provides a justification of the “right” choice for this discretization to obtain a first order accurate and monotone scheme, with no restrictive CFL condition. This result leads to the development of a class of methods of arbitrary high order accuracy through the use of moments on the droplet surface in each section and a Godunov type method. Moreover, an extension of the two moments method is proposed which preserves the positivity and limits the total variation. Numerical results of the multi-fluid methods are compared to examine their capability to accurately describe the mass density in the spray with a small number of variables. This is shown to be a key point for the use of such methods in realistic flow configurations.
@article{M2AN_2006__40_3_431_0,
author = {Laurent, Fr\'ed\'erique},
title = {Numerical analysis of eulerian multi-fluid models in the context of kinetic formulations for dilute evaporating sprays},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {40},
year = {2006},
pages = {431-468},
doi = {10.1051/m2an:2006023},
mrnumber = {2245317},
zbl = {pre05122981},
zbl = {1160.76380},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2006__40_3_431_0}
}
Laurent, Frédérique. Numerical analysis of eulerian multi-fluid models in the context of kinetic formulations for dilute evaporating sprays. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 40 (2006) pp. 431-468. doi : 10.1051/m2an:2006023. http://gdmltest.u-ga.fr/item/M2AN_2006__40_3_431_0/
[1] , and, Kiva II, a computer program for chemically reactive flows with sprays. Technical Report LA-11560-MS. Los Alamos National Laboratory, Los Alamos, New Mexico (1989).
[2] , and, A compressible model for separated two-phase flows computations, in ASME Fluids Engineering Division Summer Meeting, number 31141, Montreal (2002).
[3] , The kinetic sectional approach for noncolliding evaporating sprays. Atomization Spray. 11 (2001) 291-303.
[4] and, A numerical method for the computation of the dispersion of a cloud of particles by a turbulent gas flow field. J. Comput. Phys. 133 (1997) 256-278. | Zbl 0883.76065
[5] and, Theory of multicomponent fluids. Applied Mathematical Sciences, Springer 135 (1999). | MR 1654261 | Zbl 0919.76003
[6] and, A second-order multi-fluid model for evaporating sprays. ESAIM: M2AN 39 (2005) 931-963. | Numdam | Zbl 1075.35048
[7] , A particle-fluid numerical model for liquid sprays. J. Comput. Phys. 35 (1980) 229-253. | Zbl 0437.76051
[8] , and, An opposed jet quasi-monodisperse spray diffusion flame. Combust. Sci. Technol. 50 (1986) 255-270.
[9] , and, On the origin of spray sectional conservation equations. Combust. Flame 93 (1993) 90-96.
[10] and, A five equation reduced model for compressible two phase flow problems. Prepublication 4778, INRIA (2003). | Zbl 1061.76083
[11] , and, On finite-difference approximations and entropy conditions for shocks. Comm. Pure Appl. Math. 29 (1976) 297-322. With an appendix by B. Keyfitz. | Zbl 0351.76070
[12] , Modélisation cinétique et simulation numérique d'un brouillard dense de gouttelettes. Application aux propulseurs à poudre. Ph.D. thesis, ENSAE (1999).
[13] , Analyse numérique d'une méthode multi-fluide Eulérienne pour la description de sprays qui s'évaporent. C. R. Math. Acad. Sci. Paris 334 (2002) 417-422. | Zbl 1090.76055
[14] , Modélisation mathématique et numérique de la combustion de brouillards de gouttes polydispersés. Ph.D. thesis, Université Claude Bernard, Lyon 1 (2002).
[15] and, Multi-fluid modeling of laminar poly-dispersed spray flames: origin, assumptions and comparison of the sectional and sampling methods. Combust. Theor. Model. 5 (2001) 537-572.
[16] , and, Eulerian multi-fluid modeling for the numerical simulation of polydisperse dense liquid spray. J. Comput. Phys. 194 (2004) 505-543. | Zbl 1100.76069
[17] ,,,, and, Accurate treatment of size distribution effects in polydispersed spray diffusion flames: multi-fluid modeling, computations and experiments. Combust. Theor. Model. 8 (2004) 385-412.
[18] , Numerical methods for conservation laws. Birkhäuser Verlag, Basel, second edition (1992). | MR 1153252 | Zbl 0847.65053
[19] , and, Quadrature method of moments for aggregation-breakage processes. J. Colloid Interf. Sci. 258 (2003) 322-334.
[20] and, Modélisation multi-fluide eulérienne pour la simulation de brouillards denses polydispersés. C. R. Acad. Sci. Paris Sér. I Math. 332 (2001) 869-874. | Zbl 1067.76088
[21] ,, and, Counterflow spray diffusion flames of heptane: computations and experiments, in Proceedings of the 27th Symp. (International) on Combustion, The Comb. Institute (1998) 1975-1983.
[22] , Collective drop effects on vaporizing liquid sprays. Ph.D. thesis, University of Princeton (1981).
[23] and, Population Balances: Theory and Applications to Particulate Systems in Engineering. Academic Press (2000).
[24] and, A nonconservative hyperbolic system modeling spray dynamics. I. Solution of the Riemann problem. Math. Mod. Meth. Appl. S. 5 (1995) 297-333. | Zbl 0837.76089
[25] ,, and, Euler/Lagrange calculations of turbulent sprays : the effect of droplet collisions and coalescence. Atomization Spray. 10 (2000) 47-81.
[26] , Towards the ultimate conservative difference scheme v. a second order sequel to godunov's method. J. Comput. Phys. 32 (1979) 101-136. | Zbl 0939.76063
[27] and, Une méthode particulaire aléatoire reposant sur une équation cinétique pour la simulation numérique des sprays denses de gouttelettes liquides. C. R. Acad. Sci. Paris Sér. I Math. 325 (1997) 323-328. | Zbl 0897.76077
[28] , Spray combustion and atomization. Phys. Fluids 1 (1958) 541-545. | Zbl 0086.41102
[29] , Combustion Theory (Combustion Science and Engineering Series). F.A. Williams Ed., Reading, MA: Addison-Wesley (1985).
[30] , and, Bivariate extension of the quadrature method of moments for modeling simultaneous coagulation and sintering of particle populations. J. Colloid Interf. Sci. 236 (2001) 242-251.