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

Eulerian Multi-Fluid models for the description

• f polydisperse coalescing sprays :

evaluation of various numerical strategies

• F. Doisneau, F. Laurent

5ème Biennale de Mathématiques, Guidel 2011 2

Context – Coalescing sprays

Meteorology (raindrops, particles) Astrophysics (planets, nebulae) Injection (diesel engine) Aeronautical chambers Solid propellant combustion Chemical synthesis (TiO2, CNT precursor)

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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)

Maths Combustion Transfers Plasmas SPS SNPE …

Industries

DEFA DSNA

computes distributes (Murrone 2011)

5ème Biennale de Mathématiques, Guidel 2011

diffusion ū=ugaz brownian

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Phenomena :



Gas-droplet interactions (drag, heating, evaporation)



Droplet-droplet interactions (coalescence, rebound, break-up)



Subgridscale models (turbulence, acoustics, nanophysics…)

Key role of droplet size:

radius (µm) 1 10 100

τ~r2 stiff

Relaxation Agitation

MULTI-FLUID ?

Modeling Coalescence

ballistic

?

0.1

Multi-Velocity

P230 granulometry

Lagrangian

crossings

Coupled MULTI-FLUID

NANO

Sprays I – Physics conditionned by size

5ème Biennale de Mathématiques, Guidel 2011

free transport evaporation drag heat exchanges sources (coalescence…)

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Huge number of droplets Few properties each Kinetic Modelling



statistic description through a number distribution function (NDF)



satisfies a Boltzmann like equation (mesoscopic scale) :

coalescence

collision partner concentration collision parameters droplet size

Sprays II – Kinetic approach

5ème Biennale de Mathématiques, Guidel 2011

Sprays III – Eulerian « Multi-Fluid » method

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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)

Multi-Fluid (Massot et Laurent 01 and 04) :

5ème Biennale de Mathématiques, Guidel 2011

Size moments conservation eq. (pressureless fluid) for each section k

Coalescence I – Equations

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Transfers in phase space

n s sk-1

section (fixed bounds, one velocity)

s

k

1 size 
 moment

gas coupling coalescence

2 size 
 moments

(evaporation)

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Number, mass and momentum creation and disappearance Between two sections i and j to form k :

NDF i NDF j cross section collision/coalescence
 efficiencies velocity
 difference mass

where

Coalescence II – Computation Domains

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Coalescence III – Integral computation methods

~3.N2 double integral computations per cell and timestep



Newton-Cotes quadrature (equidistributed, 25 to 81points) :



Adaptive abscissa quadrature (4 points are enough) :

Integrand with exponential functions

Computation times on an academic test case (no transport) :

tabulated

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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

5ème Biennale de Mathématiques, Guidel 2011

Droplet growth in a fog :

 D’Herbigny experiment  analytical solution  simulation with :

 one size moment method  two size moment method

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D’Herbigny experiment (ONERA) r r m m

D’herbigny I – Experimental setup

Initially for collision efficiency laws :

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D’herbigny II – Analytical model framework

Kinetic modelling with size/velocity corellation assumption :

Conclusions : Steady formulation Linearized coalescence Decoupling of velocity

5ème Biennale de Mathématiques, Guidel 2011

D’herbigny III – Projection on size modes

PDE becomes a system of ODEs : where is a length

Rem : link with classical approach (Smoluchowski 17)

we define a coalescing length :

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D’herbigny IV – Constant kernel solution

Poisson’s law :

Refined Two size moment simulation (green) Poisson’s Law (+) Gaussian approximation (blue)

Constant kernel model validation with ~ 105 Gaussian when > 5 !

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D’herbigny V – General solution

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Simulation Comparison : One Size Moment MF (200 sect.) Two Size Moment MF (80 sect.)

D’herbigny VI – Simulations

Pseudo numerical diffusion lower with two size moments « Transport » in size phase space (Two size moment Multi-Fluid)

radius (µm)

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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

5ème Biennale de Mathématiques, Guidel 2011 18

Conclusions

Our DNS polydisperse coalescing model :



validated



implemented in an industrial code (JCP 2011)



SRM simulation (EUCASS 2011)

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

Average diameter (µm) and droplet trajectories Eulerian Lagrangian

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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 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.