High order moment model for polydisperse evaporating sprays towards interfacial geometry

Mohamed Essadki (1, 2, 3) , Stephane de Chaisemartin (3) , Frédérique Laurent (1, 2) , Marc Massot (1, 2, 4)

1 EM2C - Laboratoire d'Énergétique Moléculaire et Macroscopique, Combustion
2 FR3487 - Fédération de Mathématiques de l'Ecole Centrale Paris
3 IFPEN - IFP Energies nouvelles
4 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

In this paper we propose a new Eulerian modeling and related accurate and robust numerical methods, describing polydisperse evaporating sprays, based on high order moment methods in size. The main novelty of this model is its capacity to describe some geometrical variables of the droplet-gas interface, by analogy with the liquid-gas interface in interfacial flows. For this purpose, we use fractional size-moments, where the size variable is taken as the droplet surface. In order to evaluate the evaporation of the polydisperse spray, we use a smooth reconstruction which maximizes the Shannon entropy. However, the use of fractional moments introduces some theoretical and numerical difficulties, which need to be tackled. First, relying on a study of the moment space, we extend the Maximum Entropy (ME) reconstruction of the size distribution to the case of fractional moments. Then, we propose a new accurate and realizable algorithm to solve the moment evolution due to evaporation, which preserves the structure of the moment space. This algorithm is based on a mathematical analysis of the kinetic evolution due to evaporation, where it shown that the evolution of some negative order fractional moments have to be properly predicted, a peculiarity related to the use of fractional moments. The present model and numerical schemes yield an accurate and stable evaluation of the moment dynamics with minimal number of variables, as well as a minimal computational cost as with the EMSM model, but with the very interesting additional capacity of coupling with diffuse interface model and transport equation of averaged geometrical interface variables, which are essential in oder to describe atomization.

Mots clés: interface geometry AMS subject classifications 76T10; entropy maximization; evaporation; AMS subject classifications 76T10; 65M99; 65D99 26; 65M12; 65M08; 35Q35; High order moment method; moment space; realizable high order numerical scheme; polydisperse spray; 65D99

Domaines: Analyse numérique [math.NA]; Milieux fluides et réactifs; Mécanique des fluides [physics.class-ph]; Equations aux dérivées partielles [math.AP]

Mohamed Essadki, Stephane de Chaisemartin, Frédérique Laurent, Marc Massot. High order moment model for polydisperse evaporating sprays towards interfacial geometry. SIAM Journal on Applied Mathematics, 2018, 78 (4), pp.2003-2027. ⟨10.1137/16M1108364⟩. ⟨hal-01355608v2⟩