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, Volume 40 (2006) no. 3, p. 431-468
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.
Classification:  35L05,  65L20,  76T10
Keywords: spray, evaporation, multi-fluid method, kinetic schemes
     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},
     publisher = {EDP-Sciences},
     volume = {40},
     number = {3},
     year = {2006},
     pages = {431-468},
     doi = {10.1051/m2an:2006023},
     zbl = {1160.76380},
     zbl = {pre05122981},
     mrnumber = {2245317},
     language = {en},
     url = {}
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, Volume 40 (2006) no. 3, pp. 431-468. doi : 10.1051/m2an:2006023.

[1] A.A. Amsden, P.J. O'Rourke and T.D. Butler, 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] G. Chanteperdrix, P. Villedieu and J.P. Vila, A compressible model for separated two-phase flows computations, in ASME Fluids Engineering Division Summer Meeting, number 31141, Montreal (2002).

[3] K. Domelevo, The kinetic sectional approach for noncolliding evaporating sprays. Atomization Spray. 11 (2001) 291-303.

[4] K. Domelevo and L. Sainsaulieu, 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] D.A. Drew and S.L. Passman, Theory of multicomponent fluids. Applied Mathematical Sciences, Springer 135 (1999). | MR 1654261 | Zbl 0919.76003

[6] G. Dufour and P. Villedieu, A second-order multi-fluid model for evaporating sprays. ESAIM: M2AN 39 (2005) 931-963. | Numdam | Zbl 1075.35048

[7] J.K. Dukowicz, A particle-fluid numerical model for liquid sprays. J. Comput. Phys. 35 (1980) 229-253. | Zbl 0437.76051

[8] J.B. Greenberg, D. Albagli and Y. Tambour, An opposed jet quasi-monodisperse spray diffusion flame. Combust. Sci. Technol. 50 (1986) 255-270.

[9] J.B. Greenberg, I. Silverman and Y. Tambour, On the origin of spray sectional conservation equations. Combust. Flame 93 (1993) 90-96.

[10] H. Guillard and A. Murrone, A five equation reduced model for compressible two phase flow problems. Prepublication 4778, INRIA (2003). | Zbl 1061.76083

[11] A. Harten, J.M. Hyman and P.D. Lax, 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] J. Hylkema, Modélisation cinétique et simulation numérique d'un brouillard dense de gouttelettes. Application aux propulseurs à poudre. Ph.D. thesis, ENSAE (1999).

[13] F. Laurent, 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] F. Laurent, Modélisation mathématique et numérique de la combustion de brouillards de gouttes polydispersés. Ph.D. thesis, Université Claude Bernard, Lyon 1 (2002).

[15] F. Laurent and M. Massot, 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] F. Laurent, M. Massot and P. Villedieu, Eulerian multi-fluid modeling for the numerical simulation of polydisperse dense liquid spray. J. Comput. Phys. 194 (2004) 505-543. | Zbl 1100.76069

[17] F. Laurent, V. Santoro, M. Noskov, A. Gomez, M.D. Smooke and M. Massot, 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] R.J. Leveque, Numerical methods for conservation laws. Birkhäuser Verlag, Basel, second edition (1992). | MR 1153252 | Zbl 0847.65053

[19] D.L. Marchisio, R.D. Vigil and R.O. Fox, Quadrature method of moments for aggregation-breakage processes. J. Colloid Interf. Sci. 258 (2003) 322-334.

[20] M. Massot and P. Villedieu, 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] M. Massot, M. Kumar, A. Gomez and M.D. Smooke, 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] P.J. O'Rourke, Collective drop effects on vaporizing liquid sprays. Ph.D. thesis, University of Princeton (1981).

[23] D. Ramkrishna and A.G. Fredrickson, Population Balances: Theory and Applications to Particulate Systems in Engineering. Academic Press (2000).

[24] P.-A. Raviart and L. Sainsaulieu, 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] M. Rüger, S. Hohmann, M. Sommerfeld and G. Kohnen, Euler/Lagrange calculations of turbulent sprays : the effect of droplet collisions and coalescence. Atomization Spray. 10 (2000) 47-81.

[26] B. Van Leer, 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] P. Villedieu and J. Hylkema, 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] F.A. Williams, Spray combustion and atomization. Phys. Fluids 1 (1958) 541-545. | Zbl 0086.41102

[29] F.A. Williams, Combustion Theory (Combustion Science and Engineering Series). F.A. Williams Ed., Reading, MA: Addison-Wesley (1985).

[30] D.L. Wright, R. Mcgraw and D.E. Rosner, 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.