A high order moment method simulating evaporation and advection of a polydisperse liquid spray

Abstract

In this paper, we tackle the modeling and numerical simulation of sprays and aerosols, that is dilute gasdroplet flows for which polydispersity description is of paramount importance. Starting from a kinetic description for point particles experiencing transport either at the carrier phase velocity for aerosols or at their own velocity for sprays as well as evaporation, we focus on an Eulerian high order moment method in size and consider a system of partial differential equations (PDEs) on a vector of successive integer size moments of order 0 to N , N > 2, over a compact size interval. There exists a stumbling block for the usual approaches using high order moment methods resolved with high order finite volume methods: the transport algorithm does not preserve the moment space. Indeed, reconstruction of moments by polynomials inside computational cells coupled to the evolution algorithm can create N -dimensional vectors which fail to be moment vectors: it is impossible to find a size distribution for which there are the moments. We thus propose a new approach as well as an algorithm which is second order in space and time with very limited numerical diffusion and allows to accurately describe the advection process and naturally preserves the moment space. The algorithm also leads to a natural coupling with a recently designed algorithm for evaporation which also preserves the moment space; thus polydispersity is accounted for in the evaporation and advection process, very accurately and at a very reasonable computational cost. These modeling and algorithmic tools are referred to as the EMSM (Eulerian Multi Size Moment) model. We show that such an approach is very competitive compared to multi-fluid approaches, where the size phase space is discretized into several sections and low order moment methods are used in each section, as well as with other existing high order moment methods. An accuracy study assesses the order of the method as well as the low level of numerical diffusion on structured meshes. Whereas the extension to unstructured meshes is provided, we focus in this paper on cartesian meshes and two 2D test-cases are presented: Taylor-Green vortices and turbulent free jets, where the accuracy and efficiency of the approach are assessed.

Eulerian multi-fluid models for the simulation of dynamics and coalescence of particles in solid propellant combustion

“…The choice of refining the size distribution description with a high order moment method for a finer resolution of polydispersity therefore increases the computational cost. In the context of evaporation, the generalization to more moments can allow to reduce the number of sections to one [37,48,39]. Regarding coalescence, only the one size moment method has been validated [43] so far as explained in the introduction.…”

Development of an evaporation model for the dense spray region in Eulerian-Eulerian multiphase flow simulations

“…sectional approaches [3]) to other built on the calculation of some moments of the spray probability density function (such as Quadrature Method Of Moments [4] or Direct Quadrature Method Of Moments [5]. Other approaches, such as entropy maximization [6] or moments with interpolative closure [7], can be also found in the literature but their description goes beyond the scope of the present work. Recent contributions in this area have shown the suitability of these methods in describing the most important features of liquid sprays [8].…”

Bivariate extensions of the Extended Quadrature Method of Moments (EQMOM) to describe coupled droplet evaporation and heat-up

“…Eulerian methods can be further subdivided into sectional or high-order moment methods (Kah, Laurent, Massot, & Jay, 2012;Massot, Laurent, Kah, & de Chaisemartin, 2010). In sectional methods the NDF phase space is discretized into a set of size classes which are then tracked solving at least one transport equation for each section (Greenberg, Alibagli, & Tambour, 1986;Laurent & Massot, 2001;Tambour, 1985).…”

“…Only a small set of statistical moments of the distribution are tracked (Kah et al, 2012;. The basic concept of moment methods is given in terms of transforming the PBE into moment equations, which, however, yields unclosed terms in the moment transport equations.…”

“…developed an improved approach, which introduces the reconstruction of the droplet NDF from its moments by applying entropy maximization (EM), allowing the disappearance flux to be determined accurately. This model was integrated into a modeling framework for Eulerian spray simulations called the Eulerian Multi Size Moment (EMSM) model, which was successfully applied to 2D and 3D CFD spray simulations (Emre et al, 2014;Kah et al, 2010Kah et al, , 2012Kah et al, , 2015Gumprich & Sadiki, 2013). Evaporation was modeled using the D 2 evaporation law.…”