Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-07T23:18:52.022Z Has data issue: false hasContentIssue false
  • English
  • Français

A second-order multi-fluid model for evaporating sprays

Published online by Cambridge University Press:  15 September 2005

Guillaume Dufour  and
Philippe Villedieu
Guillaume Dufour
Affiliation:
Laboratoire MIP, Université Paul Sabatier, 118 Route de Narbonne, 31062 Toulouse Cedex, France. [email protected] DMAE/Onera - 2, Avenue E. Belin, B.P 4025, 31055 Toulouse Cedex, France. [email protected]
Philippe Villedieu
Affiliation:
Laboratoire MIP, Université Paul Sabatier, 118 Route de Narbonne, 31062 Toulouse Cedex, France. [email protected] DMAE/Onera - 2, Avenue E. Belin, B.P 4025, 31055 Toulouse Cedex, France. [email protected]
Get access

Abstract

The aim of this paper is to present a method using both the ideas of sectionalapproach and moment methods in order to accurately simulate evaporationphenomena in gas-droplets flows. Using the underlying kinetic interpretation ofthe sectional method [Y. Tambour, Combust. Flame60 (1985)15–28] exposed in [F. Laurent and M. Massot, Combust. Theory Model.5 (2001) 537–572], we propose an extension of thisapproach based on a more accurate representation of the droplet size numberdensity in each section ensuring the exact conservation of two moments (asopposed to only one moment used in the classical approach). A correspondingsecond-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 maximumprinciple on the velocity and the mean droplet mass under a suitable CFL-likecondition. Numerical simulations have been performed and the results confirm theaccuracy of this new method even when a very coarse mesh for the droplet sizevariable (i.e.: a low number of sections) is used.

Keywords

Moment methodsectional methodspraysevaporationmulti-fluid.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 39 , Issue 5 , September 2005 , pp. 931 - 963
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

Archambault, M.R., Edwards, C.F. and MacCormack, R.W., Computation of spray dynamics by moment transport equations I: Theory and development. Atomization Spray 13 (2003) 63–87. CrossRef
Archambault, M.R., Edwards, C.F. and MacCormack, R.W., Computation of spray dynamics by moment transport equations II: Application to calculation of a quasi-one-dimensional spray. Atomization Spray 13 (2003) 89–115. CrossRef
Beck, J.C. and Watkins, A.P., On the development of spray submodels based on droplet size moments. J. Comput. Phys. 182 (2002) 1–36. CrossRef
Beck, J.C. and Watkins, A.P., 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. CrossRef
Bobylev, A.V. and Ohwada, T., The error of the splitting scheme for solving evolutionary equations. Appl. Math. Lett. 14 (2001) 45–48. CrossRef
F. Bouchut, On zero pressure gas dynamics. Advances in Kinetic Theory and Computing, Selected Papers, Ser. Adv. Math. Appl. Sci. 22 (1994) 171–190.
Bouchut, F., Jin, S. and Numerical, X. Li approximations of pressureless and isothermal gas dynamics. SIAM J. Numer. Anal. 41-1 (2003) 135–158. CrossRef
R. Clift, J.R. Grace and M.E. Weber, Bubbles, drops and particles. Academic Press (1978).
Crowe, C.T., Review - Numerical methods for dilute gas-particle flows. Trans. ASME J. Fluids Eng. 104 (1982) 297–303. CrossRef
Dafermos, C., Generalized characteristics in hyperbolic systems of conservation laws. Arch. Rational Mech. Anal. 107 (1989) 127–155. CrossRef
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).
Domelevo, K., The kinetic sectional approach for noncolliding evaporating sprays. Atomization Spray 11 (2001) 291–303. CrossRef
Drew, D.A., Mathematical modeling of two-phase flows. Annu. Rev. Fluid. Mech. 15 (1983) 261–291. CrossRef
G. Dufour, Un modèle multi-fluide Eulerien pour les écoulements diphasiques à inclusions dispersées. Ph.D. Thesis, Université Toulouse III, France (2005).
Dukowicz, J.K., A particle-fluid numerical model for liquid sprays. J. Comput. Phys. 35 (1980) 229–253. CrossRef
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).
Grad, H., On the kinetic theory of rarefied gases. Comm. Pure Appl. Math. 2 (1949) 331–407. CrossRef
Greenberg, J.B., Silverman, I. and Tambour, Y., On the origin of spray sectional conservation equations. Combustion Flame 93 (1993) 90–96. CrossRef
Harten, A., Engquist, B., Osher, S. and Chakravarthy, S., Uniformly high order accurate essentially non-oscillatory schemes III. J. Comput. Phys. 126 (1987) 231–303. CrossRef
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).
M. Ishii, Thermo-fluid dynamics of two-phase flows. Eyrolles, Paris (1975).
Laurent, F., 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. CrossRef
F. Laurent, Numerical analysis of Eulerian multi-fluid models in the context of kinetic formulations for dilute evaporating sprays. To be submitted.
Laurent, F. and Massot, M., 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. CrossRef
Laurent, F., Massot, M. and Villedieu, P., Eulerian multi-fluid modeling for the numerical simulation of coalescence in polydisperse dense liquid sprays. J. Comput. Phys. 194 (2004) 505–543. CrossRef
Laurent, F., Santoro, V., Noskov, M., Smooke, M.D., Gomez, A. and Massot, M., Accurate treatment of size distribution effects in polydisperse spray diffusion flames: multi-fluid modelling, computations and experiments. Combust. Theor. Model. 8 (2004) 385–412. CrossRef
Levermore, C.D., Moment closure hierarchies for kinetics theories. J. Statist. Phys. 83 (1996) 1021–1065. CrossRef
Levy, D., Puppo, G. and Russo, G., On the behavior of the total variation in CWENO methods for conservation laws. Appl. Numer. Math. 33 (2000) 407–414. CrossRef
Maxey, R. and Riley, J., Equation of motion of a small rigid sphere in a non-unifom flow. Phys. Fluids 26 (1983) 883–889. CrossRef
P.J. O'Rourke, Collective drop effects on vaporizing liquid sprays. Ph.D. Thesis, Los Alamos national Laboratory, New Mexico 87545 (1981).
Poupaud, F. and Rascle, M., Measure solutions to the linear multi-dimensional transport equation with non-smooth coefficients. Comm. P.D.E 22 (1997) 337–358. CrossRef
D. Ramkrishna and A.G. Fredrickson, Population balances: Theory and applications to particulate systems in engineering. Academic Press (2000).
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).
O. Simonin, Modélisation numérique des écoulements turbulents diphasiques à inclusions dispersées. École de Printemps de Mécanique des Fluides numériques, Aussois (1991).
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).
Strang, G., On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5 (1968) 507–517. CrossRef
Tambour, Y., Lagrangian, A sectional approach for simulating droplet size distribution of vaporizing fuel sprays in a turbulent jet. Combustion Flame 60 (1985) 15–28. CrossRef
Van Leer, B., A second-order sequel to Godunov's method. J. Comput. Phys. 32 (1979) 101–136. CrossRef
Williams, F.A., Spray combustion and atomization. Phys. Fluids 1 (1958) 541–555. CrossRef
F.A. Williams, Combustion Theory. Addison-Wesley Publishing (1985).
Wright, D.L., McGraw, R. and Rosner, D.E., 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. CrossRef