Adaptive Mesh Refinement and High Order Geometrical Moment Method for the Simulation of Polydisperse Evaporating Sprays

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. Emre (2014) PhD Thesis , Ecole Centrale Paris; O. Emre, R.O. Fox, M. Massot, S. Chaisemartin, S. Jay, F. Laurent (2014) Flow, Turbulence and Combustion 93 , 689-722; O. Emre, D. Kah, S. Jay, Q.-H. Tran, A. Velghe, S. de Chaisemartin, F. Laurent, M. Massot (2015) Atomization Sprays 25 , 189-254; D. Kah, F. Laurent, M. Massot, S. Jay (2012) J. Comput. Phys. 231 , 394-422; D. Kah, O. Emre, Q.-H. Tran, S. de Chaisemartin, S. Jay, F. Laurent, M. Massot (2015) Int. J. Multiphase Flows 71 , 38-65; A. Vie, F. Laurent, M. Massot (2013) J. Comp. Phys. 237 , 277-310]. 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.

[1]  A. Burluka,et al.  DEVELOPMENT OF A EULERIAN MODEL FOR THE “ATOMIZATION” OF A LIQUID JET , 2001 .

[2]  T. Ménard,et al.  Coupling level set/VOF/ghost fluid methods: Validation and application to 3D simulation of the primary break-up of a liquid jet , 2007 .

[3]  Richard Saurel,et al.  Simple and efficient relaxation methods for interfaces separating compressible fluids, cavitating flows and shocks in multiphase mixtures , 2009, J. Comput. Phys..

[4]  F. X. Demoulin,et al.  Numerical simulation of primary break-up and atomization: DNS and modelling study , 2009 .

[5]  D. Drew.  Evolution of geometric statistics , 1990 .

[6]  J. Réveillon,et al.  Effects of the preferential segregation of droplets on evaporation and turbulent mixing , 2007, Journal of Fluid Mechanics.

[7]  Marc Massot,et al.  A high order moment method simulating evaporation and advection of a polydisperse liquid spray , 2012, J. Comput. Phys..

[8]  F. Harlow,et al.  Numerical Calculation of Time‐Dependent Viscous Incompressible Flow of Fluid with Free Surface , 1965 .

[9]  Marc Massot,et al.  Combustion for aerospace propulsion Eulerian models for turbulent spray combustion with polydispersity and droplet crossing , 2009 .

[10]  L. Mead,et al.  Maximum entropy in the problem of moments , 1984 .

[11]  Y. Tambour,et al.  On the origins of spray sectional conservation equations , 1993 .

[12]  Marc Massot,et al.  On the Anisotropic Gaussian velocity closure for inertial- particle laden flows , 2015 .

[13]  B. M. Devassy,et al.  Atomization modelling of liquid jets using a two-surface density approach. , 2015 .

[14]  C. W. Hirt,et al.  An Arbitrary Lagrangian-Eulerian Computing Method for All Flow Speeds , 1997 .

[15]  D. Drew,et al.  Theory of Multicomponent Fluids , 1998 .

[16]  F. Laurent,et al.  Multi-fluid modelling of laminar polydisperse spray flames: origin, assumptions and comparison of sectional and sampling methods , 2001 .

[17]  Marc Massot,et al.  Solution of population balance equations in applications with fine particles: Mathematical modeling and numerical schemes , 2016, J. Comput. Phys..

[18]  M. Baer,et al.  A two-phase mixture theory for the deflagration-to-detonation transition (ddt) in reactive granular materials , 1986 .

[19]  R. Abgrall,et al.  A Multiphase Godunov Method for Compressible Multifluid and Multiphase Flows , 1999 .

[20]  S. Chaisemartin,et al.  EULERIAN MOMENT METHODS FOR AUTOMOTIVE SPRAYS , 2015 .

[21]  Oguz Emre,et al.  Modélisation de la polydispersion des brouillards de gouttes sous l'effet des interactions two-way turbulentes pour l'injection directe à haute pression dans les moteurs. (Modeling of spray polydispersion with two-way turbulent interactions for high pressure direct injection in engines) , 2014 .

[22]  D. Kah.  Taking into account polydispersity for the modeling of liquid fuel injection in internal combustion engines , 2010 .

[23]  Boniface Nkonga,et al.  Towards the direct numerical simulation of nucleate boiling flows , 2014 .

[24]  Marc Massot,et al.  Size-velocity correlations in hybrid high order moment/multi-fluid methods for polydisperse evaporating sprays: Modeling and numerical issues , 2013, J. Comput. Phys..

[25]  S. Jay,et al.  Combined surface density concepts for dense spray combustion , 2006 .

[26]  Towards Eulerian Modeling of a Polydisperse Evaporating Spray Under Realistic Internal-Combustion-Engine Conditions , 2014 .

[27]  Marc Massot,et al.  High order moment method for polydisperse evaporating sprays with mesh movement: Application to internal combustion engines , 2015 .

[28]  M. Massot,et al.  On the Development of High Order Realizable Schemes for the Eulerian Simulation of Disperse Phase Flows: A Convex-State Preserving Discontinuous Galerkin Method , 2014 .

[29]  Marc Massot,et al.  Operator splitting for nonlinear reaction-diffusion systems with an entropic structure : singular perturbation and order reduction , 2004, Numerische Mathematik.

[30]  Roy G. Gordon,et al.  Error Bounds in Equilibrium Statistical Mechanics , 1968 .

[31]  Marc Massot,et al.  A Robust Moment Method for Evaluation of the Disappearance Rate of Evaporating Sprays , 2010, SIAM J. Appl. Math..

[32]  Carsten Burstedde,et al.  p4est: Scalable Algorithms for Parallel Adaptive Mesh Refinement on Forests of Octrees , 2011, SIAM J. Sci. Comput..

[33]  Mark F. Adams,et al.  Chombo Software Package for AMR Applications Design Document , 2014 .

[34]  Heinz Pitsch,et al.  Combination of 3D unsplit forward and backward volume-of-fluid transport and coupling to the level set method , 2013, J. Comput. Phys..

[35]  Shi Jin,et al.  Numerical Approximations of Pressureless and Isothermal Gas Dynamics , 2003, SIAM J. Numer. Anal..

[36]  Alexandru Fikl,et al.  Experimenting with the p4est library for AMR simulations of two-phase flows , 2016, ArXiv.

[37]  F. Williams.  Spray Combustion and Atomization , 1958 .

[38]  G. Godsave.  Studies of the combustion of drops in a fuel spray—the burning of single drops of fuel , 1953 .

[39]  Max Duarte,et al.  Adaptive numerical methods in time and space for the simulation of multi-scale reaction fronts. (Méthodes numériques adaptatives pour la simulation de la dynamique de fronts de réaction multi-échelles en temps et en espace) , 2011 .

[40]  Frédérique Laurent,et al.  NUMERICAL ANALYSIS OF EULERIAN MULTI-FLUID MODELS IN THE CONTEXT OF KINETIC FORMULATIONS FOR DILUTE EVAPORATING SPRAYS , 2006 .

[41]  Richard Saurel,et al.  A multiphase model for compressible flows with interfaces, shocks, detonation waves and cavitation , 2001, Journal of Fluid Mechanics.

[42]  Siegfried Müller,et al.  Adaptive Multiscale Schemes for Conservation Laws , 2002, Lecture Notes in Computational Science and Engineering.

[43]  Macole Sabat.  Eulerian modeling and numerical methods for the description of turbulent polydisperse sprays. (Modèles euleriens et méthodes numériques pour la description des sprays polydisperses turbulents) , 2016 .

[44]  A. Harten.  Multiresolution algorithms for the numerical solution of hyperbolic conservation laws , 2010 .

[45]  W. Steckelmacher.  Molecular gas dynamics and the direct simulation of gas flows , 1996 .

[46]  Frédérique Laurent,et al.  Two-size moment multi-fluid model: a robust and high-fidelity description of polydisperse moderately dense evaporating sprays , 2016 .

[47]  Marc Massot,et al.  A hierarchy of simple hyperbolic two-fluid models for bubbly flows , 2016, 1607.08233.