Eulerian multi-fluid modeling for the numerical simulation of coalescence in polydisperse dense liquid sprays

Abstract

In this paper, we present a new Eulerian multi-fluid modeling for dense sprays of evaporating liquid droplets which is able to describe droplet coalescence and size polydispersion as well as the associated size-conditioned dynamics. It is an uncommon feature of Eulerian spray models which are required in a number of non-stationary simulations because of the optimization capability of a solver coupling a Eulerian description for both phases. The chosen framework is the one of laminar flows or the one of direct numerical simulations since no turbulence models are included in the present study. The model is based on a rigorous derivation from the kinetic level of description (p.d.f. equation) and can be considered as a major extension of the original sectional method introduced by Tambour et al. We obtain a set of conservation equations for each “fluid”: a statistical average of all the droplets in given size intervals associated to a discretization of the size phase space. The coalescence phenomenon appears as quadratic source terms, the coefficients of which, the collisional integrals, can be pre-calculated from a given droplet size discretization and do not depend on space nor time. We validate this Eulerian model by performing several comparisons, for both stationary and non-stationary cases, to a classical Lagrangian model which involves a stochastic algorithm in order to treat the coalescence phenomenon. The chosen configuration is a self-similar 2D axisymmetrical decelerating nozzle with evaporating sprays having various size distributions, ranging from smooth ones up to Dirac delta functions through discontinuous ones. We show that the Eulerian model, if the discretization in the size phase space is fine enough (the problem is then 3D unstationary, 2D in space and 1D in size), is able to reproduce very accurately the non-stationary coupling of evaporation, dynamics and coalescence. Moreover, it can still reproduce the global features of the behavior of the spray with a coarse size discretization, which is a nice feature compared to Lagrangian approaches. The computational efficiency of both approaches are then compared and the Eulerian model is proved to be a good candidate for more complex and realistic configurations.