EULERIAN MULTI-FLUID MODELS FOR THE NUMERICAL …

General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation

EULERIAN MULTI-FLUID MODELS FOR THENUMERICAL SIMULATIONS OF

EVAPORATING POLYDISPERSED SPRAYSLecture 3 - Conclusion

Marc Massot

Professeur - Ecole Centrale Paris - Laboratoire EM2C - UPR CNRS 288Federation de Mathematiques de l’Ecole Centrale Paris - FR CNRS 3487

Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays

Application and physicsModels Hierarchie and numerical strategies

Outline

1 General FrameworkApplication and physicsModels Hierarchie and numerical strategies

2 Numerical strategies for solving Williams-Boltzmann NDF equationGeneral pictureVarious Levels of Eulerian methodsSynthesis, advantages and drawback of the various methods

Applications

Spray injection in a combustion chamber (Dieselengines)

Swirled spray injection for propulsion chambers

Solid propulsion in rocket boosters : aluminaparticles ejected from the energetic materialcombustion

Cryotechnic propulsion : oxygen droplets in anhydrogen gaseous stream

Meteorologie : water droplet formation in clouds

Planet formation in stellar nebulae

Applications

Physical Phenomena

Dispersed phase / Large size spectrum

Example of the combustion chamber

Injection

Primary Fragmentation

Secondary break-up

Turbulente dispersion

Droplets interactions(coalescence/rebound)

Droplet Wall interactions

Evaporation

Combustion

Physical Phenomena

Injection

Secondary break-up

Evaporation

Combustion

Physical Phenomena

Injection

Secondary break-up

Evaporation

Combustion

Physical Phenomena

Injection

Secondary break-up

Evaporation

Combustion

Physical Phenomena

Injection

Secondary break-up

Evaporation

Combustion

Physical Phenomena

Injection

Secondary break-up

Evaporation

Combustion

Physical Phenomena

Injection

Secondary break-up

Evaporation

Combustion

Physical Phenomena

Exemple of droplet/droplet interaction

Weber 23

Weber 40

Weber 105

Necessity of

interface and general fluid equationsresolutions

Type of fluid description

Separated phases

Physical Phenomena

Weber 23

Weber 40

Weber 105

Necessity of

Separated phases

Physical Phenomena

Weber 23

Weber 40

Weber 105

Necessity of

Separated phases

Physical Phenomena

Weber 23

Weber 40

Weber 105

Necessity of

Separated phases

Simulations by A. Berlemont, S.Tanguy and T. Mnard

CORIA - Rouen France

Ghost fluid / VOF / Level set

Swirled Jet

Turbulent round jet

Physical Phenomena

Weber 23

Weber 40

Weber 105

Necessity of

Separated phases

Three levels of descriptionFull two-phase DNS : Using Navier Stokes for both phases and interfaceresolution. Provides insight about the modeling with less details. Exemple :collisions. The liquid phase does not have to be dispersed.

Discrete particle simulations : of a dispersed liquid phase with discretedroplets. DNS of a gaseous phase coupled to this discrete particles with two-wayinteractions law issued from theoretical studies or Full two-phase DNS.Kinetic approach : Williams-Boltzmann equation on the number densityfunction : Eulerian description of a dispersed liquid phase. f (t , x , v , s, θ)

probable number of droplets satisfies :

∂f∂t

+ v · ∇x f︸︷︷︸transport

+∇v .(F f )︸︷︷︸forces

∂s(Kf )︸︷︷︸

size variation

∂θ(Rf )︸︷︷︸

heat exchange

= G(f )︸︷︷︸break-up

+ Q(f )︸︷︷︸coalescence

Phase space dimension : at least 8 in 3DMarc Massot Eulerian Multi-fluid models for evaporating polydispersed srays

Three levels of descriptionFull two-phase DNS : Using Navier Stokes for both phases and interfaceresolution. Provides insight about the modeling with less details. Exemple :collisions. The liquid phase does not have to be dispersed.Discrete particle simulations : of a dispersed liquid phase with discretedroplets. DNS of a gaseous phase coupled to this discrete particles with two-wayinteractions law issued from theoretical studies or Full two-phase DNS.Typically a Lagrangian approach. Global properties (motion of the center of mass/ fixed geometry / angular momentum ...)

Frozen turbulent flow

Kinetic approach : Williams-Boltzmann equation on the number densityfunction : Eulerian description of a dispersed liquid phase. f (t , x , v , s, θ)

∂f∂t

∂s(Kf )︸︷︷︸

size variation

∂θ(Rf )︸︷︷︸

heat exchange

Phase space dimension : at least 8 in 3D

Three levels of description

Full two-phase DNS : Using Navier Stokes for both phases and interfaceresolution. Provides insight about the modeling with less details. Exemple :collisions. The liquid phase does not have to be dispersed.

Discrete particle simulations : of a dispersed liquid phase with discretedroplets. DNS of a gaseous phase coupled to this discrete particles with two-wayinteractions law issued from theoretical studies or Full two-phase DNS.

∂f∂t

∂s(Kf )︸︷︷︸

size variation

∂θ(Rf )︸︷︷︸

heat exchange

∂f∂t

∂s(Kf )︸︷︷︸

size variation

∂θ(Rf )︸︷︷︸

heat exchange

∂f∂t

∂s(Kf )︸︷︷︸

size variation

∂θ(Rf )︸︷︷︸

heat exchange

∂f∂t

∂s(Kf )︸︷︷︸

size variation

∂θ(Rf )︸︷︷︸

heat exchange

∂f∂t

∂s(Kf )︸︷︷︸

size variation

∂θ(Rf )︸︷︷︸

heat exchange

Link between the three approaches

The full two-phase DNS as well as experimental measurements provide thecorrelations in order to use a DPS or a PDF approach. only conceivable for alimited number of droplets

The DPS allows to conduct interesting statistical studies in order to get insightabout closure and validation of PDF approaches (ex. : two-point or two-timecorrelations in turbulent flows)

In the framework of “DNS” studies without droplet interactions, the DPS andWilliams-Boltzmann equations are equivalent

Justification and convergence of DPS in the framework of interacting particle isdifficult but provides some information when stochastic methods are beyondcomputation capabilities

General pictureVarious Levels of Eulerian methodsSynthesis, advantages and drawback of the various methods

Outline

1 General FrameworkApplication and physicsModels Hierarchie and numerical strategies

2 Numerical strategies for solving Williams-Boltzmann NDF equationGeneral pictureVarious Levels of Eulerian methodsSynthesis, advantages and drawback of the various methods

Lagrangian versus Eulerian

Stochastic Monte-Carlo method

Based upon the NDF equation

Allow to model elementary processesat kinetic level

Trajectography of numerical particles

high computational cost for densesprays

slow convergence

Eulerian methods

System of conservation laws formacroscopic quantities.

Closure required

Discretisation by finite volumesmethods

Optimization/coupling abilities

Good precision for relatively low cost

slow convergence

Eulerian methods

Closure required

slow convergence

Eulerian methods

Closure required

slow convergence

Eulerian methods

Closure required

slow convergence

Eulerian methods

Closure required

Key issues for Eulerian models

Droplets have their dynamics conditioned by size, momentum and heatexchange vary depending on their size

There is a strong need to resolve properly the size/velocity correlations in orderto properly predict the gaseous fuel mass fraction topology

Modeling of collisions/break-up/wall interactions are given at the NDF level(kinetic level) and influence/are influenced strongly by the size distribution

Various approaches based on kinetic theory can be used in order to derive the“macroscopic conservation equations” especially for the velocity moments(Maxwell Transfer Equation/Chapman Enskog theory/Grad theory).

Size distribution

Exemple of the “Two-fluid” model for dilute flows

∂tnd + ∂x (nd vd )=0

∂t (αdρd ) + ∂x (αdρd vd )=−�

∂t (αdρd vd ) + ∂x

(αdρd vd

=−βgdαdρd (vd − ug) +�

∂t (αdρd hd ) + ∂x (αdρd hd vd )=Φc −�

Closure assumptionnecessary for :

βgd ,�

m, Φc - as well asfor the mean velocity !

Polydisperse character of the spray

Some size “in the mean” has to be recovered from the macroscopic quantities

m = µ(Tg ,Td , rd , . . .) Φc = φ(Tg ,Td , rd , . . .)

No information about the size ditribution

No information about the size/velocity correlation

(αdρd vd

βgd ,�

(αdρd vd

βgd ,�

(αdρd vd

βgd ,�

Treatment of polydispersion in size

Presumed PDF methods (Mossa-Babinsky/Sojka)

Presumed PDF in size with ad hoc size/velocitycorrelations

Moment methods (Fox, Marchisio, McGraw,Beck-Watkins)

Conservation of moments in size/velocity

Size moments

mk (t , x) =∫ +∞

0 sk f (t , x , s)ds

Class methods (Simonin - ONERA)

The original distribution is sampled into various“classes” and some bi-fluid model is used for each class

Multi-fluid model (Tambour, Laurent-Massot 2001)

The size phase space is discretized into size intervalscalled sections with velocity distribution conditioned bysize.

Size moments

mk (t , x) =∫ +∞

Size moments

mk (t , x) =∫ +∞

Size moments

mk (t , x) =∫ +∞

Size moments

mk (t , x) =∫ +∞

General view

The presumed PDF methods suffer from severe problems coming from both theevaporation process as well as the coupling with the velocity conditioned by size

Classes are well suited for situations for which the particles physics does notmodify the size distribution (problems with droplets and coalescence) andpartially describe the size/velocity correlations

Moment methods for sprays must be multivariate moment methods, thuspreventing from a “classical” use of QMOM (Fox - Marchisio)

The two methods which can lead to both accurate size distribution as well assize/velocity correlations are DQMOM and Eulerian multi-fluid

General view

EULERIAN MULTI-FLUID MODELS FOR THE NUMERICAL …

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