A second-order multi-fluid model for evaporating sprays

Dufour, Guillaume; Villedieu, Philippe

ESAIM: Modélisation mathématique et analyse numérique, Tome 39 (2005) no. 5, pp. 931-963.

The aim of this paper is to present a method using both the ideas of sectional approach and moment methods in order to accurately simulate evaporation phenomena in gas-droplets flows. Using the underlying kinetic interpretation of the sectional method [Y. Tambour, Combust. Flame 60 (1985) 15-28] exposed in [F. Laurent and M. Massot, Combust. Theory Model. 5 (2001) 537-572], we propose an extension of this approach based on a more accurate representation of the droplet size number density in each section ensuring the exact conservation of two moments (as opposed to only one moment used in the classical approach). A corresponding second-order numerical scheme, with respect to space and droplet size variables, is also introduced and can be proved to be positive and to satisfy a maximum principle on the velocity and the mean droplet mass under a suitable CFL-like condition. Numerical simulations have been performed and the results confirm the accuracy of this new method even when a very coarse mesh for the droplet size variable (i.e.: a low number of sections) is used.

[1] M.R. Archambault, C.F. Edwards and R.W. Maccormack, Computation of spray dynamics by moment transport equations I: Theory and development. Atomization Spray 13 (2003) 63-87.

[2] M.R. Archambault, C.F. Edwards and R.W. Maccormack, Computation of spray dynamics by moment transport equations II: Application to calculation of a quasi-one-dimensional spray. Atomization Spray 13 (2003) 89-115.

[3] J.C. Beck and A.P. Watkins, On the development of spray submodels based on droplet size moments. J. Comput. Phys. 182 (2002) 1-36. | Zbl

[4] J.C. Beck and A.P. Watkins, The droplet number moments approach to spray modeling: The development of heat and mass transfer sub-models. Int. J. Heat Fluid Flow 24 (2003) 242-259.

[5] A.V. Bobylev and T. Ohwada, The error of the splitting scheme for solving evolutionary equations. Appl. Math. Lett. 14 (2001) 45-48. | Zbl

[6] F. Bouchut, On zero pressure gas dynamics. Advances in Kinetic Theory and Computing, Selected Papers, Ser. Adv. Math. Appl. Sci. 22 (1994) 171-190. | Zbl

[7] F. Bouchut, S. Jin and X. Li, Numerical approximations of pressureless and isothermal gas dynamics. SIAM J. Numer. Anal. 41-1 (2003) 135-158. | Zbl

[8] R. Clift, J.R. Grace and M.E. Weber, Bubbles, drops and particles. Academic Press (1978).

[9] C.T. Crowe, Review - Numerical methods for dilute gas-particle flows. Trans. ASME J. Fluids Eng. 104 (1982) 297-303.

[10] C. Dafermos, Generalized characteristics in hyperbolic systems of conservation laws. Arch. Rational Mech. Anal. 107 (1989) 127-155. | Zbl

[11] K. Domelevo, Analyse mathématique et numérique d'une modélisation cinétique d'un brouillard de gouttelettes dans un écoulement gazeux turbulent. Ph.D. Thesis, École Polytechnique, France (1996).

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

[13] D.A. Drew, Mathematical modeling of two-phase flows. Annu. Rev. Fluid. Mech. 15 (1983) 261-291. | Zbl

[14] G. Dufour, Un modèle multi-fluide Eulerien pour les écoulements diphasiques à inclusions dispersées. Ph.D. Thesis, Université Toulouse III, France (2005).

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

[16] J. Dupays, Contribution à l'étude du rôle de la phase condensée dans la stabilité d'un propulseur à propergol solide pour lanceur spatial. Ph.D. Thesis, Institut National Polytechnique de Toulouse, France (1996).

[17] H. Grad, On the kinetic theory of rarefied gases. Comm. Pure Appl. Math. 2 (1949) 331-407. | Zbl

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

[19] A. Harten, B. Engquist, S. Osher and S. Chakravarthy, Uniformly high order accurate essentially non-oscillatory schemes III. J. Comput. Phys. 126 (1987) 231-303. | Zbl

[20] J.J. Hylkema, Modélisation cinétique et simulation numérique d'un brouillard dense de gouttelettes. Applications aux propulseurs à poudre. Ph.D. Thesis, ENSAE, France (1999).

[21] M. Ishii, Thermo-fluid dynamics of two-phase flows. Eyrolles, Paris (1975).

[22] F. Laurent, Analyse numérique d'une méthode multi-fluide Eulérienne pour la description de sprays qui s'évaporent. C. R. Acad. Sci. Paris Ser. I 334 (2002) 417-422. | Zbl

[23] F. Laurent, Numerical analysis of Eulerian multi-fluid models in the context of kinetic formulations for dilute evaporating sprays. To be submitted. | Numdam

[24] F. Laurent and M. Massot, Multi-fluid modeling of laminar poly-disperse spray flames: origin, assumptions and comparison of sectional and sampling methods. Combust. Theor. Model. 5 (2001) 537-572.

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

[26] F. Laurent, V. Santoro, M. Noskov, M.D. Smooke, A. Gomez and M. Massot, Accurate treatment of size distribution effects in polydisperse spray diffusion flames: multi-fluid modelling, computations and experiments. Combust. Theor. Model. 8 (2004) 385-412.

[27] C.D. Levermore, Moment closure hierarchies for kinetics theories. J. Statist. Phys. 83 (1996) 1021-1065. | Zbl

[28] D. Levy, G. Puppo and G. Russo, On the behavior of the total variation in CWENO methods for conservation laws. Appl. Numer. Math. 33 (2000) 407-414. | Zbl

[29] R. Maxey and J. Riley, Equation of motion of a small rigid sphere in a non-unifom flow. Phys. Fluids 26 (1983) 883-889. | Zbl

[30] P.J. O'Rourke, Collective drop effects on vaporizing liquid sprays. Ph.D. Thesis, Los Alamos national Laboratory, New Mexico 87545 (1981).

[31] F. Poupaud and M. Rascle, Measure solutions to the linear multi-dimensional transport equation with non-smooth coefficients. Comm. P.D.E 22 (1997) 337-358. | Zbl

[32] D. Ramkrishna and A.G. Fredrickson, Population balances: Theory and applications to particulate systems in engineering. Academic Press (2000).

[33] 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).

[34] O. Simonin, Modélisation numérique des écoulements turbulents diphasiques à inclusions dispersées. École de Printemps de Mécanique des Fluides numériques, Aussois (1991).

[35] O. Simonin, Continuum modeling of dispersed two-phase flows. Combustion and turbulence in two-phase flows. Lecture Series 1996-02, Von Karman Inst. for fluid dyn. (1996).

[36] G. Strang, On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5 (1968) 507-517. | Zbl

[37] Y. Tambour, A Lagrangian sectional approach for simulating droplet size distribution of vaporizing fuel sprays in a turbulent jet. Combustion Flame 60 (1985) 15-28.

[38] B. Van Leer, A second-order sequel to Godunov's method. J. Comput. Phys. 32 (1979) 101-136.

[39] F.A. Williams, Spray combustion and atomization. Phys. Fluids 1 (1958) 541-555. | Zbl

[40] F.A. Williams, Combustion Theory. Addison-Wesley Publishing (1985).

[41] 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.