In this paper we propose a new Eulerian model 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 relies on the use of fractional droplet surface moments and their ability to predict some geometrical variables of the droplet-gas interface, by analogy with the liquid-gas interface in interfacial flows. Evaporation is evaluated by using a Maximum Entropy (ME) reconstruction. The use of fractional moments introduces some theoretical and numerical difficulties. First, relying on a study of the moment space, we extend the ME reconstruction to the case of fractional moments. Then, we propose a new high order and robust algorithm to solve the moment evolution due to evaporation, which preserves the structure of the moment space. It involves some negative order fractional moments for which a novel treatment is introduced. The present model and numerical schemes yield an accurate and stable evaluation of the moment dynamics with minimal number of variables, as well as computational cost, but also provides an additional capacity of coupling with diffuse interface model and transport equation of averaged geometrical interface variables, which are essential in order to describe atomization.

Predictive simulation of liquid fuel injection in automotive engines has become a major challenge for science and applications. The key issue in order to properly predict various combustion regimes and pollutant formation is to accurately describe the interaction between the carrier gaseous phase and the polydisperse evaporating spray produced through atomization. For this purpose, we rely on the EMSM (Eulerian Multi-Size Moment) Eulerian polydisperse model. It is based on a high order moment method in size, with a maximization of entropy technique in order to provide a smooth reconstruction of the distribution, derived from a Williams-Boltzmann mesoscopic model under the monokinetic assumption [O. ]. The present contribution relies on a major extension of this model [M. Essadki, S. de Chaisemartin, F. Laurent, A. Larat, M. Massot (2016) Submitted to SIAM J. Appl. Math.], with the aim of building a unified approach and coupling with a separated phases model describing the dynamics and atomization of the interface near the injector. The novelty is to be found in terms of modeling, numerical schemes and implementation. A new high order moment approach is introduced using fractional moments in surface, which can be related to geometrical quantities of the gas-liquid interface. We also provide a novel algorithm for an accurate resolution of the evaporation. Adaptive mesh refinement properly scaling on massively parallel architectures yields a precise integration of transport in physical space limiting both numerical dissipation as well as the memory trace of the solver. A series of test-cases is presented and analyzed, thus assessing the proposed approach and its parallel computational efficiency while evaluating its potential for complex applications. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Résumé -Raffinement de maillage adaptatif et méthode de moments géométriques d'ordre élevé pour la simulation des sprays en évaporation -La simulation prédictive de l'injection diphasique dans les chambres de combustion automobiles représente un enjeu majeur scientifique et applicatif. La description détaillée de l'interaction entre le brouillard de gouttes polydispersé produit par atomisation et l'écoulement gazeux est fondamentale pour prédire les régimes de combustion et la formation de polluant. Pour décrire la phase liquide, le modèle Eulérien polydisperse EMSM (Eulerian Multi-Size Moment) est choisi. Cette approche de type moments d'ordre élevé en taille avec reconstruction continue par maximisation d'entropie est construite à partir d'un modèle mésoscopique de Williams-Boltzmann sous l'hypothèse monocinétique [O. Emre (2014) PhD Thesis, ]. La présente contribution propose une extension majeure de ce modèle [M. Essadki, S. de Chaisemartin, F. Laurent, A. Larat, M. Massot (2016) Submitted to SIAM...