eulerian multi-fluid models for the numerical …
TRANSCRIPT
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
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
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
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
Application and physicsModels Hierarchie and numerical strategies
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
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
Application and physicsModels Hierarchie and numerical strategies
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
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
Application and physicsModels Hierarchie and numerical strategies
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
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
Application and physicsModels Hierarchie and numerical strategies
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
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
Application and physicsModels Hierarchie and numerical strategies
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
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
Application and physicsModels Hierarchie and numerical strategies
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
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
Application and physicsModels Hierarchie and numerical strategies
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
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
Application and physicsModels Hierarchie and numerical strategies
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
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
Application and physicsModels Hierarchie and numerical strategies
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
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
Application and physicsModels Hierarchie and numerical strategies
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
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
Application and physicsModels Hierarchie and numerical strategies
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
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
Application and physicsModels Hierarchie and numerical strategies
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
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
Application and physicsModels Hierarchie and numerical strategies
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
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
Application and physicsModels Hierarchie and numerical strategies
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
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
Application and physicsModels Hierarchie and numerical strategies
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
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
Application and physicsModels Hierarchie and numerical strategies
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
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
Application and physicsModels Hierarchie and numerical strategies
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
Simulations by A. Berlemont, S.Tanguy and T. Mnard
CORIA - Rouen France
Ghost fluid / VOF / Level set
Swirled Jet
Turbulent round jet
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
Application and physicsModels Hierarchie and numerical strategies
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
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
Application and physicsModels Hierarchie and numerical strategies
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
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
Application and physicsModels Hierarchie and numerical strategies
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)
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 3D
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
Application and physicsModels Hierarchie and numerical strategies
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.
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 3D
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
Application and physicsModels Hierarchie and numerical strategies
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.
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 3D
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
Application and physicsModels Hierarchie and numerical strategies
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.
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 3D
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
Application and physicsModels Hierarchie and numerical strategies
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.
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 3D
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
Application and physicsModels Hierarchie and numerical strategies
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.
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 3D
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
Application and physicsModels Hierarchie and numerical strategies
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
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
Application and physicsModels Hierarchie and numerical strategies
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
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
Application and physicsModels Hierarchie and numerical strategies
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
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
Application and physicsModels Hierarchie and numerical strategies
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
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
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
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
General 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
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
General 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
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
General 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
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
General 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
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
General 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
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
General pictureVarious Levels of Eulerian methodsSynthesis, advantages and drawback of the various methods
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
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
General pictureVarious Levels of Eulerian methodsSynthesis, advantages and drawback of the various methods
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
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
General pictureVarious Levels of Eulerian methodsSynthesis, advantages and drawback of the various methods
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
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
General pictureVarious Levels of Eulerian methodsSynthesis, advantages and drawback of the various methods
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
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
General pictureVarious Levels of Eulerian methodsSynthesis, advantages and drawback of the various methods
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
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
General pictureVarious Levels of Eulerian methodsSynthesis, advantages and drawback of the various methods
Exemple of the “Two-fluid” model for dilute flows
∂tnd + ∂x (nd vd )=0
∂t (αdρd ) + ∂x (αdρd vd )=−�
m
∂t (αdρd vd ) + ∂x
(αdρd vd
2)
=−βgdαdρd (vd − ug) +�
mvd
∂t (αdρd hd ) + ∂x (αdρd hd vd )=Φc −�
mhd
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
rd =
(3αd
4πnd
)1/3
,�
m = µ(Tg ,Td , rd , . . .) Φc = φ(Tg ,Td , rd , . . .)
No information about the size ditribution
No information about the size/velocity correlation
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
General pictureVarious Levels of Eulerian methodsSynthesis, advantages and drawback of the various methods
Exemple of the “Two-fluid” model for dilute flows
∂tnd + ∂x (nd vd )=0
∂t (αdρd ) + ∂x (αdρd vd )=−�
m
∂t (αdρd vd ) + ∂x
(αdρd vd
2)
=−βgdαdρd (vd − ug) +�
mvd
∂t (αdρd hd ) + ∂x (αdρd hd vd )=Φc −�
mhd
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
rd =
(3αd
4πnd
)1/3
,�
m = µ(Tg ,Td , rd , . . .) Φc = φ(Tg ,Td , rd , . . .)
No information about the size ditribution
No information about the size/velocity correlation
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
General pictureVarious Levels of Eulerian methodsSynthesis, advantages and drawback of the various methods
Exemple of the “Two-fluid” model for dilute flows
∂tnd + ∂x (nd vd )=0
∂t (αdρd ) + ∂x (αdρd vd )=−�
m
∂t (αdρd vd ) + ∂x
(αdρd vd
2)
=−βgdαdρd (vd − ug) +�
mvd
∂t (αdρd hd ) + ∂x (αdρd hd vd )=Φc −�
mhd
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
rd =
(3αd
4πnd
)1/3
,�
m = µ(Tg ,Td , rd , . . .) Φc = φ(Tg ,Td , rd , . . .)
No information about the size ditribution
No information about the size/velocity correlation
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
General pictureVarious Levels of Eulerian methodsSynthesis, advantages and drawback of the various methods
Exemple of the “Two-fluid” model for dilute flows
∂tnd + ∂x (nd vd )=0
∂t (αdρd ) + ∂x (αdρd vd )=−�
m
∂t (αdρd vd ) + ∂x
(αdρd vd
2)
=−βgdαdρd (vd − ug) +�
mvd
∂t (αdρd hd ) + ∂x (αdρd hd vd )=Φc −�
mhd
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
rd =
(3αd
4πnd
)1/3
,�
m = µ(Tg ,Td , rd , . . .) Φc = φ(Tg ,Td , rd , . . .)
No information about the size ditribution
No information about the size/velocity correlation
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
General pictureVarious Levels of Eulerian methodsSynthesis, advantages and drawback of the various methods
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.
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
General pictureVarious Levels of Eulerian methodsSynthesis, advantages and drawback of the various methods
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.
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
General pictureVarious Levels of Eulerian methodsSynthesis, advantages and drawback of the various methods
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.
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
General pictureVarious Levels of Eulerian methodsSynthesis, advantages and drawback of the various methods
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.
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
General pictureVarious Levels of Eulerian methodsSynthesis, advantages and drawback of the various methods
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.
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
General pictureVarious Levels of Eulerian methodsSynthesis, advantages and drawback of the various methods
General view
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
General pictureVarious Levels of Eulerian methodsSynthesis, advantages and drawback of the various methods
General view
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
General pictureVarious Levels of Eulerian methodsSynthesis, advantages and drawback of the various methods
General view
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
General pictureVarious Levels of Eulerian methodsSynthesis, advantages and drawback of the various methods
General view
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
General pictureVarious Levels of Eulerian methodsSynthesis, advantages and drawback of the various methods
General view
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
General pictureVarious Levels of Eulerian methodsSynthesis, advantages and drawback of the various methods
Key issues for Eulerian models
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
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
General pictureVarious Levels of Eulerian methodsSynthesis, advantages and drawback of the various methods
Key issues for Eulerian models
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
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
General pictureVarious Levels of Eulerian methodsSynthesis, advantages and drawback of the various methods
Key issues for Eulerian models
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
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
General pictureVarious Levels of Eulerian methodsSynthesis, advantages and drawback of the various methods
Key issues for Eulerian models
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
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays
General FrameworkNumerical strategies for solving Williams-Boltzmann NDF equation
General pictureVarious Levels of Eulerian methodsSynthesis, advantages and drawback of the various methods
General view
Marc Massot Eulerian Multi-fluid models for evaporating polydispersed srays