Eulerian Multi-Fluid models for the description of ...

Eulerian Multi-Fluid models for the description of 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)


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

4

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


free transport evaporation drag heat exchanges sources (coalescence…)

5

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


Sprays III – Eulerian « Multi-Fluid » method

6

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) :


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

Coalescence I – Equations

7

Transfers in phase space

n

ssk-1 section (fixed bounds, one velocity)

s k

1 size moment

gas coupling

coalescence

2 size moments

(evaporation)

5ème Biennale de Mathématiques, Guidel 2011 8 8

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

NDF i NDF j cross section

collision/coalescenceefficiencies

velocitydifference

mass

where

Coalescence II – Computation Domains


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


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


Droplet growth in a fog : D’Herbigny experiment analytical solution simulation with :

one size moment method two size moment method

11

D’Herbigny experiment (ONERA)

r

r

m

m

D’herbigny I – Experimental setup

Initially for collision efficiency laws :


D’herbigny II – Analytical model framework

Kinetic modelling with size/velocity corellation assumption :

Conclusions : Steady formulation Linearized coalescence Decoupling of velocity


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 :


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 !


D’herbigny V – General solution


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)


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


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


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.

Eulerian Multi-Fluid models for the description of ...

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