Modélisation des écoulements diphasiques du régime dense au régime dilué / From dense to dilute two-phase flows

Résumé Anglais : Well posed two-phase flow models exist for dense granular flows as well as dilute (low dispersed phase concentration) ones. “Well posed models” designates hyperbolic set of non-linear partial differential equations admitting mixture conservation laws of mass, momentum and energy, these formulations being in agreement with the second law of thermodynamics. In the limit of dense granular mixtures the Baer and Nunziato (1986) model and its variants seem appropriate for most situations of non-equilibrium mixture flows. In the opposite limit of dilute suspensions the pressureless equations for the particles dynamics and the Euler equations of compressible fluids with source terms for the gas dynamics seem appropriate (Marble, 1963, Zeldovich, 1970). Between these two bounds no hyperbolic model is known. This is a serious limitation of actual knowledge. In particular a discontinuity in waves and sound propagation appears from these two limit models. This has been the subject of recent efforts by Lhuillier et al. (2013), McGrath et al. (2015), Houim and Oran (2015) and Chinnayya and Saurel (2015). In this last reference a clear link is given as well as an appropriate hyperbolic model making the missing link. Research subject, work plan: The aim of this thesis is to derive numerical schemes for the Baer-Nunziato model having asymptotic preserving properties, in particular those of the limit model of the last reference, in the limit of low particles concentration. References Baer, M. R., & Nunziato, J. W. (1986). A two-phase mixture theory for the deflagration-to- detonation transition (DDT) in reactive granular materials. International journal of multiphase flow, 12(6), 861-889 Chinnayya, A. and Saurel, R. (2015) From dense to dilute two-phase flows. Journal of Fluid Mechanics, in preparation Houim, R.W and Oran, E.S. (2015) A multiphase model for compressible granular-gaseous flows: Formulation and initial tests. Journal of Fluid Mechanics, submitted Lhuillier, D., Chang, C. H., & Theofanous, T. G. (2013). On the quest for a hyperbolic effective-field model of disperse flows. Journal of Fluid Mechanics,731, 184-194 Marble, F.E (1963) Dynamics of a gas containing small solid particles. Combustion and Propulsion (5th AGARD Colloquium), Pergamon Press McGrath, T., St. Clair, J.G. and Balachandar, S. (2015) A compressible multiphase model for dispersed particle flows with application from dense to dilute regimes. Physics of Fluids, submitted Ya. B. Zeldovich (1970), “Gravitational instability: An approximate theory for large density perturbations”, Astron. Astrophys, 5, 84–89