Eulerian Multi-Fluid models for the description of polydisperse - PowerPoint PPT Presentation

Eulerian Multi-Fluid models for the description of polydisperse coalescing sprays : evaluation of various numerical strategies F. Doisneau, F. Laurent

Context – Coalescing sprays Astrophysics Meteorology (planets, nebulae) (raindrops, particles) Injection (diesel engine) Solid propellant combustion Chemical synthesis Aeronautical chambers (TiO 2 , CNT precursor) 5 ème Biennale de Mathématiques, Guidel 2011 2

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 computes SPS DEFA Combustion Maths SNPE Transfers distributes DSNA … Plasmas (Murrone 2011) 5 ème Biennale de Mathématiques, Guidel 2011 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: Coupled MULTI-FLUID MULTI-FLUID ? NANO ? Multi-Velocity Modeling Lagrangian P230 granulometry diffusion Agitation τ ~r 2 stiff ū =u gaz Relaxation crossings brownian ballistic Coalescence 0.1 1 10 100 radius (µm) 5 ème Biennale de Mathématiques, Guidel 2011 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) : droplet size heat exchanges free transport evaporation sources drag (coalescence…) coalescence collision partner concentration collision parameters 5 ème Biennale de Mathématiques, Guidel 2011 5

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) Sections (1 moment) 5 ème Biennale de Mathématiques, Guidel 2011 6

Coalescence I – Equations coalescence n (evaporation) gas coupling Transfers in phase space k s s k-1 s section (fixed bounds, one velocity) Size moments conservation eq. (pressureless fluid) for each section k 1 size 
 moment 2 size 
 moments 5 ème Biennale de Mathématiques, Guidel 2011 7

Coalescence II – Computation Domains Number, mass and momentum creation and disappearance Between two sections i and j to form k : where cross velocity 
 collision/coalescence 
 mass NDF i NDF j section difference efficiencies 5 ème Biennale de Mathématiques, Guidel 2011 8 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) :  tabulated Adaptive abscissa quadrature (4 points are enough) :  Computation times on an academic test case (no transport) : 5 ème Biennale de Mathématiques, Guidel 2011 9 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 5 ème Biennale de Mathématiques, Guidel 2011 10 10

D’herbigny I – Experimental setup Droplet growth in a fog :  D’Herbigny experiment m  analytical solution  simulation with :  one size moment method r  two size moment method m Initially for collision efficiency laws : r D’Herbigny experiment (ONERA) 5 ème Biennale de Mathématiques, Guidel 2011 11

D’herbigny II – Analytical model framework Kinetic modelling with size/velocity corellation assumption : Conclusions : Steady formulation Linearized coalescence Decoupling of velocity 5 ème Biennale de Mathématiques, Guidel 2011 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) 5 ème Biennale de Mathématiques, Guidel 2011

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 5 ème Biennale de Mathématiques, Guidel 2011 14

D’herbigny V – General solution 5 ème Biennale de Mathématiques, Guidel 2011 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 5 ème Biennale de Mathématiques, Guidel 2011 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  5 ème Biennale de Mathématiques, Guidel 2011 17

Conclusions Our DNS polydisperse coalescing model : validated  implemented in an industrial code (JCP 2011)  SRM simulation (EUCASS 2011)  Eulerian Lagrangian Average diameter ( µ m) 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  5 ème Biennale de Mathématiques, Guidel 2011 18

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 two- phase flows : application to Solid Rocket Motor instabilities. To appear in 8 th 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 47 th AIAA Joint Propulsion Conference , San Diego 2011. 5 ème Biennale de Mathématiques, Guidel 2011 19