Eulerian Multi-Fluid models for the description of polydisperse coalescing sprays : evaluation of various numerical strategies F. Doisneau, F. Laurent
Context Coalescing sprays Astrophysics (planets, nebulae) Meteorology (raindrops, particles) Injection (diesel engine) Solid propellant combustion Aeronautical chambers 2 5ème Biennale de Mathématiques, Guidel 2011 Chemical synthesis (TiO2, CNT precursor)
Context Acknowledgements PhD Thesis 2009-2012 (DGA grant) «Modélisation et simulation d écoulements diphasiques chargés de particules polydispersées nanométriques dans les moteurs à propergol solide à l aide d une approche Eulérienne dite Multi-Fluide» Marc Massot, Frédérique Laurent (EM2C, Maths) Joël Dupays (ONERA, DEFA) PEA Nano (ONERA), trainee (EM2C) Industries Combustion DEFA computes SPS Transfers Plasmas Maths DSNA (Murrone 2011) distributes SNPE 3
Sprays I Physics conditionned by size Phenomena : Gas-droplet interactions (drag, heating, evaporation) Droplet-droplet interactions (coalescence, rebound, break-up) Subgridscale models (turbulence, acoustics, nanophysics ) Key role of droplet size: Modeling? NANO Coupled? MULTI-FLUID Multi-Velocity Lagrangian Agitation diffusion P230 granulometry Relaxation ū=u gaz stiff τ~r 2 crossings Coalescence brownian ballistic 0.1 1 10 100 radius (µm) 4
Sprays II Kinetic approach Huge number of droplets Few properties each Kinetic Modelling statistic description through a number distribution function (NDF) satisfies a Boltzmann like equation (mesoscopic scale) : free transport evaporation drag droplet size heat exchanges sources (coalescence ) coalescence collision partner concentration 5 5ème Biennale de Mathématiques, Guidel 2011 collision parameters
Sprays III Eulerian «Multi-Fluid» method Multi-Fluid (Massot et Laurent 01 and 04) : Size-velocity coupling : (choice = surface ) Size discretization: (finite volumes) Unique velocity per section : Size distribution in each section : (2 moments, Dufour 05 ) Sections (2 moments) 6 5ème Biennale de Mathématiques, Guidel 2011 Sections (1 moment)
Coalescence I Equations n coalescence (evaporation) gas coupling Transfers in phase space s k-1 s k s section (fixed bounds, one velocity) Size moments conservation eq. (pressureless fluid) for each section k 1 size moment 2 size moments 7
Coalescence II Computation Domains Number, mass and momentum creation and disappearance Between two sections i and j to form k : where NDF i NDF j cross section collision/coalescence efficiencies velocity difference mass 8
Coalescence III Integral computation methods Integrand with exponential functions ~3.N 2 double integral computations per cell and timestep Newton-Cotes quadrature (equidistributed, 25 to 81points) : Adaptive abscissa quadrature (4 points are enough) : tabulated Computation times on an academic test case (no transport) : 9
Coalescence IV Conclusion on the model Two Size moment MF with adaptive quadrature : Polydispersion ok Coalescence (+efficiency models) ok Validation? Computational efficiency? DNS point of view (no subgrid scale effects) is a first step before: Droplet crossings (Fréret 2008, Chalons 2010) LES modeling (Wunsch 2009) Nanometric modelling (Charles 2009) Brownian aspects (Friedlander 2000, Simoes 2006) Further work for comprehensive modeling 10
D herbigny I Experimental setup Droplet growth in a fog : D Herbigny experiment analytical solution simulation with : one size moment method two size moment method m m r Initially for collision efficiency laws : r D Herbigny experiment (ONERA) 11
D herbigny II Analytical model framework Kinetic modelling with size/velocity corellation assumption : Conclusions : Steady formulation Linearized coalescence Decoupling of velocity 12
D herbigny III Projection on size modes PDE becomes a system of ODEs : where is a length we define a coalescing length : Rem : link with classical approach (Smoluchowski 17)
D herbigny IV Constant kernel solution Poisson s law : Refined Two size moment simulation (green) Poisson s Law (+) Gaussian approximation (blue) Gaussian when > 5! Constant kernel model validation with ~ 10 5 14
D herbigny V General solution 15
D herbigny VI Simulations «Transport» in size phase space (Two size moment Multi-Fluid) Simulation Comparison : One Size Moment MF (200 sect.) Two Size Moment MF (80 sect.) radius (µm) Pseudo numerical diffusion lower with two size moments 16
D herbigny VII Conclusions Linearized Bimodal case : derivation of an analytical formula useful for chemical synthesis (Jeong 2005) code validation Experimental results (D Herbigny 2001) code validation collision efficiency models 17
Conclusions Our DNS polydisperse coalescing model : validated implemented in an industrial code (JCP 2011) SRM simulation (EUCASS 2011) Eulerian Lagrangian Average diameter (µm) and droplet trajectories Perspectives : effect of coalescence on instabilities (EUCASS 2011) num. Strategy for 2-way coupling (AIAA 2011) secondary break-up gaussian velocity coalescence kernel nanometric modeling 18 5ème Biennale de Mathématiques, Guidel 2011
Questions? References : J. Dupays, Y. Fabignon, P. Villedieu, G. Lavergne, and J. L. Estivalezes. Some aspects of two-phase flows in solid propellant rocket motors. Progress in Astronautics and Aeronautics, vol 185, AIAA, 2000. S. Friedlander. Smoke, Dust and Haze, Fundamentals of Aerosol Dynamics. Oxford University Press, 2000. F. X. D Herbigny and P. Villedieu. Etude expérimentale et numérique pour la validation d un modèle de coalescence. RF1/05166 DMAE, ONERA, 2001. F. Laurent, M. Massot, and P. Villedieu. Eulerian Multi-Fluid modeling for the numerical simulation of coalescence in polydisperse dense liquid sprays. J. Comp. Phys., 194:505 543, 2004. G. Dufour and P. Villedieu. A second-order Multi-Fluid model for evaporating sprays. M2AN Math. Model. Numer. Anal., 39(5):931 963, 2005. J. I. Jeong and M. Choi. A bimodal particle dynamics model considering coagulation, coalescence and surface growth, and its application to the growth of titania aggregates. Journal of Colloid and Interface Science, 281(2):351 359, 2005. D. Wunsch. Theoretical and numerical study of collision and coalescence - Statistical modeling approaches in gas droplet turbulent flows. PhD thesis, Institut de Mécanique des Fluides de Toulouse (IMFT), 2009. M. Simoes. Modélisation eulérienne de la phase dispersée dans les moteurs à propergol solide, avec prise en compte de la pression particulaire. PhD thesis, INP Toulouse, 2006. J. Mathiaud. Etude de systèmes de type gaz-particules. PhD thesis, ENS Cachan, 2006. L. Freret, S. de Chaisemartin, F. Laurent, P. Vedula, R.O. Fox, O. Thomine, J. Reveillon and M. Massot. Eulerian moment models for polydisperse weakly collisional sprays : model and validation. Proceedings of the Summer Program, CTR. 2008. F. Charles. Modélisation mathématique et étude numérique d un aérosol dans un gaz raréfié. Application à la simulation du transport de particules de poussière en cas d accident de perte de vide dans ITER. PhD thesis, ENS Cachan, 2009. A. Murrone and P. Villedieu. Numerical modeling of dispersed two-phase flows. Aerospace Lab, 2:1 13, 2011. Communications : F. Doisneau, F. Laurent, A. Murrone, J. Dupays, and M. Massot. Evaluation of Eulerian Multi-Fluid models for the simulation of dynamics and coalescence of particles in solid propellant combustion. To be submitted to J. Comp. Phys. 2011. F. Doisneau, F. Laurent, J. Dupays, and M. Massot. Two-way coupled simulation of acoustic waves in polydispersed coalescing twophase flows : application to Solid Rocket Motor instabilities. To appear in 8th European Conference on Aerospace Science EUCASS, St Petersburg 2011. F. Doisneau, A. Sibra, F. Laurent, J. Dupays, and M. Massot. Numerical strategy for two-way coupling in polydisperse dense sprays : application to solid rocket motor instabilities. To appear in 47th AIAA Joint Propulsion Conference, San Diego 2011. 19