Roland Borghi is Professor Emeritus at Ecole Centrale Marseille in France and works as a consultant in the space, petrol and automobile sectors. His research activities cover fluid mechanics, combustion and flames, and multi-phase and granular flows. He was a member of the CNRS scientific committee and a laureate of the French Academy of Science.Fabien Anselmet is Professor at Ecole Centrale Marseille in France. His research activities focus on the turbulence of fluids and its varied applications in industry and in fields linked to the environment.

• Acknowledgments xiIntroduction xiiiPart 1. Approach and General Equations 1Chapter 1. Towards a Unified Description of Multiphase Flows 31.1. Continuous approach and kinetic approach 31.2. Eulerian–Lagrangian and Eulerian formulations 7Chapter 2. Instant Equations for a Piecewise Continuous Medium 92.1. Integral and differential forms of balance equations 102.2. Phase mass balance equations in a piecewise continuous medium 132.3. Momentum balances 172.4. Energy balances 212.5. Position and interface area balance equations 232.6. Extension for a fluid phase that is a mixture 252.7. Completing the description of the medium 27Chapter 3. Description of a “Mean Multiphase Medium” 293.1. The need for a mean description 293.2. How are mean values defined? 313.2.1. Temporal average 313.2.2. Volumetric average 323.2.3. Statistical average 343.2.4. Filtered average 353.3. Which average to choose, according to their advantages and disadvantages? 37Chapter 4. Equations for the Mean Continuous Medium 394.1. Global balance equations for the mean medium 394.1.1. Total mass 394.1.2. Total momentum 404.1.3. Total energy 414.2. Balance equations for the phases of a mean medium 424.2.1. Phase mass 434.2.2. Phase momentum 444.2.3. Energies of each phase 474.2.4. Phase volume 494.3. Complete representation of the mean medium 494.3.1. Global representation 504.3.2. Multifluid representation 514.4. Mean equations of state 554.5. Extensions 584.5.1. Extension when a fluid phase is a mixture 584.5.2. Extension for dispersed media 594.6. Boundary conditions 61Part 2. Modeling: A Single Approach Adaptable To Multiple Applications 67Chapter 5. The Modeling of Interphase Exchanges 695.1. General methodology 695.2. Interface between phases and its mean area per unit of volume 715.2.1. Case of a suspension of liquid or solid particles 715.2.2. Case of a medium containing parcels of variable shapes and sizes 725.2.3. Case of a suspension of particles of constant and known sizes 745.3. Forces of contact and friction between phases 755.3.1. Pressure forces on spherical particles in a non-viscous flow 765.3.2. Friction on solid particles in steady flow 805.3.3. Slightly curved liquid–gas interfaces 875.3.4. Drops or bubbles 935.4. Heat transfers at the surface of a particle, without mass exchange 965.5. Heat and mass transfers during boiling 995.5.1. Slightly curved liquid–gas interfaces 995.5.2. Bubbles 1055.6. Mass and heat exchanges by vaporization 1075.6.1. Mass transfer by evaporation at a flat interface 1075.6.2. Evaporation of a drop 1135.6.3. Combustion of a drop 117Chapter 6. Modeling Turbulent Dispersion Fluxes 1196.1. Global modeling 1196.1.1. General information 1196.1.2. Kinetic energy of the “global fluctuations” 1236.1.3. Modeling the kinetic energy of the fluctuations 1286.1.4. Length scales for fluctuations and time scale for the dissipation of kinetic energy of fluctuations 1326.1.5. Further studies on the dispersion flux of a phase 1376.2. “Multifluid” modeling 1476.2.1. The kinetic energy of the fluctuations in each phase 1496.2.2. Modeling the balance equations of the kinetic energies of turbulence 1526.2.3. The modeling of time or spatial scales 1586.2.4. Modeling of the Reynolds tensor for every phase 162Chapter 7. Modeling the Mean Gas–Liquid Interface Area per Unit Volume 1657.1. Introduction 1657.2. Initial equation for the mean interface area per unit volume 1667.3. Model of the mean interface area during the “atomization” of a liquid jet 1687.4. Effects of vaporization on the interface area 172Chapter 8. “Large Eddy Simulation” Style Models 1758.1. Introduction 1758.2. Filtered equations and the nature of the models to be provided 1778.3. Classic LES modeling for SGS additional fluxes 1818.3.1. Reminder of LES in single-phase, constant density turbulent flows 1818.3.2. Toward an extension for multiphase flows 1838.4. Subgrid modeling of the interface area per unit volume 1858.5. Partially Integrated Turbulence Modeling 188Chapter 9. Contribution of Thermodynamics of Irreversible Processes 1919.1. Global two-phase medium models 1929.1.1. Entropy of a mean two-phase medium using the Prandtl model 1949.1.2. Entropy for the k–ε model, in a medium with a variable density 2009.2. Contribution of thermodynamics to multifluid models 206Chapter 10. Experimental Methods 21310.1. Introduction 21310.2. Intrusive methods 21410.2.1. Pitot tubes 21510.2.2. Hot films 21610.2.3. Optical needle probes (single probes, bi-probes and quadri-probes) 21910.2.4. Wire networks 22310.3. Non-intrusive methods 22410.3.1. Particle image velocimetry (PIV) 22510.3.2. Droplet tracking velocimetry 23010.3.3. Laser Doppler anemometry (LDA) 23410.3.4. Phase Doppler anemometry (PDA) 23710.3.5. Ultrasonic Doppler Anemometry 24110.3.6. Densimetry by attenuation of gamma, X-ray or neutron radiation 24310.4. Advanced optical methods 24510.4.1. Laser induced fluorescence 24510.4.2. Interferometric methods (digital inline holography, Fourier interferometric imaging, ILIDS/IPI, rainbow) 252Chapter 11. Some Experimental Results Pertaining to Multiphase Flow Properties that Are Still Little Understood 26511.1. Atomization/fragmentation of liquid jets 26511.2. Isolated bubbles, bubbles in swarm and their effects on carrier fluid 27411.3. Boiling crisis 285Part 3. From Fluidized Beds To Granular Media 297Chapter 12. Fluidized Beds 29912.1. Introduction 29912.1.1. Classification of different fluidization regimes 29912.1.2. Minimum fluidization and bubbling velocities 30412.2. Complete models for the dynamics of fluidized beds 30612.2.1. Bubbling fluidization regime 30712.2.2. Turbulent fluidization regime 31512.3. Global models for chemical conversion in fluidized beds 32112.3.1. Bubbling regime fluidizations 32112.3.2. Fast fluidization regime 32412.3.3. Turbulent fluidization regime 32512.4. Global models for heat transfers in fluidized beds 32812.4.1. Bubbling fluidization regime 32812.4.2. Fast fluidization regimes – circulating beds 33112.5. Conclusion 334Chapter 13. Generalizations for Granular Media 33513.1. Introduction 33513.2. Balance equations for mean granular media 33613.3. Necessary closure approximations 34213.4. Some already proposed methods 345Chapter 14. Modeling of Cauchy Tensor of Sliding Contacts 34914.1. Hypotheses and basic equations 34914.2. Unclosed balance equation for Cauchy tensor of sliding contact 35114.3. Closure approximations for irreversible terms 358Chapter 15. Modeling the Kinetic Cauchy Stress Tensor 36315.1. Prandtl–Bagnold modeling 36415.2. K-lt or “turbulent granular gas” modeling 36615.3. Toward a general model for all regimes 37115.4. Boundary conditions at walls 373Part 4. Studying Fluctuations and Probability Densities 377Chapter 16. Fluctuations of the Gas Phase in Reactive Two-Phase Media 37916.1. Specificities of reactive two-phase media 37916.2. Probability density of composition fluctuations of the gas phase 38016.2.1. Instant basic equations of the gas medium 38216.2.2. PDF equation 38516.3. Modeling the terms due to exchanges between phases 39016.3.1. Total mass exchange 39016.3.2. Mass exchange for species 39216.3.3. Heat exchange 39316.4. Modeling micromixing and turbulent dispersion 39516.4.1. The “micromixing” term in PDF equations 39516.4.2. Turbulent diffusion terms in PDF equations 39616.5. Practical use of PDF equations 397Chapter 17. Temperature Fluctuations in Condensed Phases 39917.1. Problems 39917.2. Instantaneous equation for the temperature of the liquid phase 40117.3. Equation for the PDF of the temperature of the liquid 40317.4. Closure of the equation of the temperature PDF 405Chapter 18. Study of the PDF for Velocity Fluctuations and Sizes of Parcels 40918.1. Phase velocity PDF equation 41018.2. Modeling the exchanges between phases and the internal interactions 41518.2.1. Terms of exchanges between phases 41518.2.2. Internal dissipation and production of fluctuations 41818.3. Practical calculation of PDF 41918.4. The study of the sizes of the dispersed phase parcels 42018.5. Eulerian–Lagrangian simulation of dispersed media 42318.5.1. Lagrangian equations of the parcels 42318.5.2. Stochastic simulations 426Bibliography 431Index 443

Roland Borghi is Professor Emeritus at Ecole Centrale Marseille in France and works as a consultant in the space, petrol and automobile sectors. His research activities cover fluid mechanics, combustion and flames, and multi-phase and granular flows. He was a member of the CNRS scientific committee and a laureate of the French Academy of Science.Fabien Anselmet is Professor at Ecole Centrale Marseille in France. His research activities focus on the turbulence of fluids and its varied applications in industry and in fields linked to the environment.

• Acknowledgments xiIntroduction xiiiPart 1. Approach and General Equations 1Chapter 1. Towards a Unified Description of Multiphase Flows 31.1. Continuous approach and kinetic approach 31.2. Eulerian–Lagrangian and Eulerian formulations 7Chapter 2. Instant Equations for a Piecewise Continuous Medium 92.1. Integral and differential forms of balance equations 102.2. Phase mass balance equations in a piecewise continuous medium 132.3. Momentum balances 172.4. Energy balances 212.5. Position and interface area balance equations 232.6. Extension for a fluid phase that is a mixture 252.7. Completing the description of the medium 27Chapter 3. Description of a “Mean Multiphase Medium” 293.1. The need for a mean description 293.2. How are mean values defined? 313.2.1. Temporal average 313.2.2. Volumetric average 323.2.3. Statistical average 343.2.4. Filtered average 353.3. Which average to choose, according to their advantages and disadvantages? 37Chapter 4. Equations for the Mean Continuous Medium 394.1. Global balance equations for the mean medium 394.1.1. Total mass 394.1.2. Total momentum 404.1.3. Total energy 414.2. Balance equations for the phases of a mean medium 424.2.1. Phase mass 434.2.2. Phase momentum 444.2.3. Energies of each phase 474.2.4. Phase volume 494.3. Complete representation of the mean medium 494.3.1. Global representation 504.3.2. Multifluid representation 514.4. Mean equations of state 554.5. Extensions 584.5.1. Extension when a fluid phase is a mixture 584.5.2. Extension for dispersed media 594.6. Boundary conditions 61Part 2. Modeling: A Single Approach Adaptable To Multiple Applications 67Chapter 5. The Modeling of Interphase Exchanges 695.1. General methodology 695.2. Interface between phases and its mean area per unit of volume 715.2.1. Case of a suspension of liquid or solid particles 715.2.2. Case of a medium containing parcels of variable shapes and sizes 725.2.3. Case of a suspension of particles of constant and known sizes 745.3. Forces of contact and friction between phases 755.3.1. Pressure forces on spherical particles in a non-viscous flow 765.3.2. Friction on solid particles in steady flow 805.3.3. Slightly curved liquid–gas interfaces 875.3.4. Drops or bubbles 935.4. Heat transfers at the surface of a particle, without mass exchange 965.5. Heat and mass transfers during boiling 995.5.1. Slightly curved liquid–gas interfaces 995.5.2. Bubbles 1055.6. Mass and heat exchanges by vaporization 1075.6.1. Mass transfer by evaporation at a flat interface 1075.6.2. Evaporation of a drop 1135.6.3. Combustion of a drop 117Chapter 6. Modeling Turbulent Dispersion Fluxes 1196.1. Global modeling 1196.1.1. General information 1196.1.2. Kinetic energy of the “global fluctuations” 1236.1.3. Modeling the kinetic energy of the fluctuations 1286.1.4. Length scales for fluctuations and time scale for the dissipation of kinetic energy of fluctuations 1326.1.5. Further studies on the dispersion flux of a phase 1376.2. “Multifluid” modeling 1476.2.1. The kinetic energy of the fluctuations in each phase 1496.2.2. Modeling the balance equations of the kinetic energies of turbulence 1526.2.3. The modeling of time or spatial scales 1586.2.4. Modeling of the Reynolds tensor for every phase 162Chapter 7. Modeling the Mean Gas–Liquid Interface Area per Unit Volume 1657.1. Introduction 1657.2. Initial equation for the mean interface area per unit volume 1667.3. Model of the mean interface area during the “atomization” of a liquid jet 1687.4. Effects of vaporization on the interface area 172Chapter 8. “Large Eddy Simulation” Style Models 1758.1. Introduction 1758.2. Filtered equations and the nature of the models to be provided 1778.3. Classic LES modeling for SGS additional fluxes 1818.3.1. Reminder of LES in single-phase, constant density turbulent flows 1818.3.2. Toward an extension for multiphase flows 1838.4. Subgrid modeling of the interface area per unit volume 1858.5. Partially Integrated Turbulence Modeling 188Chapter 9. Contribution of Thermodynamics of Irreversible Processes 1919.1. Global two-phase medium models 1929.1.1. Entropy of a mean two-phase medium using the Prandtl model 1949.1.2. Entropy for the k–ε model, in a medium with a variable density 2009.2. Contribution of thermodynamics to multifluid models 206Chapter 10. Experimental Methods 21310.1. Introduction 21310.2. Intrusive methods 21410.2.1. Pitot tubes 21510.2.2. Hot films 21610.2.3. Optical needle probes (single probes, bi-probes and quadri-probes) 21910.2.4. Wire networks 22310.3. Non-intrusive methods 22410.3.1. Particle image velocimetry (PIV) 22510.3.2. Droplet tracking velocimetry 23010.3.3. Laser Doppler anemometry (LDA) 23410.3.4. Phase Doppler anemometry (PDA) 23710.3.5. Ultrasonic Doppler Anemometry 24110.3.6. Densimetry by attenuation of gamma, X-ray or neutron radiation 24310.4. Advanced optical methods 24510.4.1. Laser induced fluorescence 24510.4.2. Interferometric methods (digital inline holography, Fourier interferometric imaging, ILIDS/IPI, rainbow) 252Chapter 11. Some Experimental Results Pertaining to Multiphase Flow Properties that Are Still Little Understood 26511.1. Atomization/fragmentation of liquid jets 26511.2. Isolated bubbles, bubbles in swarm and their effects on carrier fluid 27411.3. Boiling crisis 285Part 3. From Fluidized Beds To Granular Media 297Chapter 12. Fluidized Beds 29912.1. Introduction 29912.1.1. Classification of different fluidization regimes 29912.1.2. Minimum fluidization and bubbling velocities 30412.2. Complete models for the dynamics of fluidized beds 30612.2.1. Bubbling fluidization regime 30712.2.2. Turbulent fluidization regime 31512.3. Global models for chemical conversion in fluidized beds 32112.3.1. Bubbling regime fluidizations 32112.3.2. Fast fluidization regime 32412.3.3. Turbulent fluidization regime 32512.4. Global models for heat transfers in fluidized beds 32812.4.1. Bubbling fluidization regime 32812.4.2. Fast fluidization regimes – circulating beds 33112.5. Conclusion 334Chapter 13. Generalizations for Granular Media 33513.1. Introduction 33513.2. Balance equations for mean granular media 33613.3. Necessary closure approximations 34213.4. Some already proposed methods 345Chapter 14. Modeling of Cauchy Tensor of Sliding Contacts 34914.1. Hypotheses and basic equations 34914.2. Unclosed balance equation for Cauchy tensor of sliding contact 35114.3. Closure approximations for irreversible terms 358Chapter 15. Modeling the Kinetic Cauchy Stress Tensor 36315.1. Prandtl–Bagnold modeling 36415.2. K-lt or “turbulent granular gas” modeling 36615.3. Toward a general model for all regimes 37115.4. Boundary conditions at walls 373Part 4. Studying Fluctuations and Probability Densities 377Chapter 16. Fluctuations of the Gas Phase in Reactive Two-Phase Media 37916.1. Specificities of reactive two-phase media 37916.2. Probability density of composition fluctuations of the gas phase 38016.2.1. Instant basic equations of the gas medium 38216.2.2. PDF equation 38516.3. Modeling the terms due to exchanges between phases 39016.3.1. Total mass exchange 39016.3.2. Mass exchange for species 39216.3.3. Heat exchange 39316.4. Modeling micromixing and turbulent dispersion 39516.4.1. The “micromixing” term in PDF equations 39516.4.2. Turbulent diffusion terms in PDF equations 39616.5. Practical use of PDF equations 397Chapter 17. Temperature Fluctuations in Condensed Phases 39917.1. Problems 39917.2. Instantaneous equation for the temperature of the liquid phase 40117.3. Equation for the PDF of the temperature of the liquid 40317.4. Closure of the equation of the temperature PDF 405Chapter 18. Study of the PDF for Velocity Fluctuations and Sizes of Parcels 40918.1. Phase velocity PDF equation 41018.2. Modeling the exchanges between phases and the internal interactions 41518.2.1. Terms of exchanges between phases 41518.2.2. Internal dissipation and production of fluctuations 41818.3. Practical calculation of PDF 41918.4. The study of the sizes of the dispersed phase parcels 42018.5. Eulerian–Lagrangian simulation of dispersed media 42318.5.1. Lagrangian equations of the parcels 42318.5.2. Stochastic simulations 426Bibliography 431Index 443

Roland Borghi is Professor Emeritus at Ecole Centrale Marseille in France and works as a consultant in the space, petrol and automobile sectors. His research activities cover fluid mechanics, combustion and flames, and multi-phase and granular flows. He was a member of the CNRS scientific committee and a laureate of the French Academy of Science.Fabien Anselmet is Professor at Ecole Centrale Marseille in France. His research activities focus on the turbulence of fluids and its varied applications in industry and in fields linked to the environment.

• Acknowledgments xiIntroduction xiiiPart 1. Approach and General Equations 1Chapter 1. Towards a Unified Description of Multiphase Flows 31.1. Continuous approach and kinetic approach 31.2. Eulerian–Lagrangian and Eulerian formulations 7Chapter 2. Instant Equations for a Piecewise Continuous Medium 92.1. Integral and differential forms of balance equations 102.2. Phase mass balance equations in a piecewise continuous medium 132.3. Momentum balances 172.4. Energy balances 212.5. Position and interface area balance equations 232.6. Extension for a fluid phase that is a mixture 252.7. Completing the description of the medium 27Chapter 3. Description of a “Mean Multiphase Medium” 293.1. The need for a mean description 293.2. How are mean values defined? 313.2.1. Temporal average 313.2.2. Volumetric average 323.2.3. Statistical average 343.2.4. Filtered average 353.3. Which average to choose, according to their advantages and disadvantages? 37Chapter 4. Equations for the Mean Continuous Medium 394.1. Global balance equations for the mean medium 394.1.1. Total mass 394.1.2. Total momentum 404.1.3. Total energy 414.2. Balance equations for the phases of a mean medium 424.2.1. Phase mass 434.2.2. Phase momentum 444.2.3. Energies of each phase 474.2.4. Phase volume 494.3. Complete representation of the mean medium 494.3.1. Global representation 504.3.2. Multifluid representation 514.4. Mean equations of state 554.5. Extensions 584.5.1. Extension when a fluid phase is a mixture 584.5.2. Extension for dispersed media 594.6. Boundary conditions 61Part 2. Modeling: A Single Approach Adaptable To Multiple Applications 67Chapter 5. The Modeling of Interphase Exchanges 695.1. General methodology 695.2. Interface between phases and its mean area per unit of volume 715.2.1. Case of a suspension of liquid or solid particles 715.2.2. Case of a medium containing parcels of variable shapes and sizes 725.2.3. Case of a suspension of particles of constant and known sizes 745.3. Forces of contact and friction between phases 755.3.1. Pressure forces on spherical particles in a non-viscous flow 765.3.2. Friction on solid particles in steady flow 805.3.3. Slightly curved liquid–gas interfaces 875.3.4. Drops or bubbles 935.4. Heat transfers at the surface of a particle, without mass exchange 965.5. Heat and mass transfers during boiling 995.5.1. Slightly curved liquid–gas interfaces 995.5.2. Bubbles 1055.6. Mass and heat exchanges by vaporization 1075.6.1. Mass transfer by evaporation at a flat interface 1075.6.2. Evaporation of a drop 1135.6.3. Combustion of a drop 117Chapter 6. Modeling Turbulent Dispersion Fluxes 1196.1. Global modeling 1196.1.1. General information 1196.1.2. Kinetic energy of the “global fluctuations” 1236.1.3. Modeling the kinetic energy of the fluctuations 1286.1.4. Length scales for fluctuations and time scale for the dissipation of kinetic energy of fluctuations 1326.1.5. Further studies on the dispersion flux of a phase 1376.2. “Multifluid” modeling 1476.2.1. The kinetic energy of the fluctuations in each phase 1496.2.2. Modeling the balance equations of the kinetic energies of turbulence 1526.2.3. The modeling of time or spatial scales 1586.2.4. Modeling of the Reynolds tensor for every phase 162Chapter 7. Modeling the Mean Gas–Liquid Interface Area per Unit Volume 1657.1. Introduction 1657.2. Initial equation for the mean interface area per unit volume 1667.3. Model of the mean interface area during the “atomization” of a liquid jet 1687.4. Effects of vaporization on the interface area 172Chapter 8. “Large Eddy Simulation” Style Models 1758.1. Introduction 1758.2. Filtered equations and the nature of the models to be provided 1778.3. Classic LES modeling for SGS additional fluxes 1818.3.1. Reminder of LES in single-phase, constant density turbulent flows 1818.3.2. Toward an extension for multiphase flows 1838.4. Subgrid modeling of the interface area per unit volume 1858.5. Partially Integrated Turbulence Modeling 188Chapter 9. Contribution of Thermodynamics of Irreversible Processes 1919.1. Global two-phase medium models 1929.1.1. Entropy of a mean two-phase medium using the Prandtl model 1949.1.2. Entropy for the k–ε model, in a medium with a variable density 2009.2. Contribution of thermodynamics to multifluid models 206Chapter 10. Experimental Methods 21310.1. Introduction 21310.2. Intrusive methods 21410.2.1. Pitot tubes 21510.2.2. Hot films 21610.2.3. Optical needle probes (single probes, bi-probes and quadri-probes) 21910.2.4. Wire networks 22310.3. Non-intrusive methods 22410.3.1. Particle image velocimetry (PIV) 22510.3.2. Droplet tracking velocimetry 23010.3.3. Laser Doppler anemometry (LDA) 23410.3.4. Phase Doppler anemometry (PDA) 23710.3.5. Ultrasonic Doppler Anemometry 24110.3.6. Densimetry by attenuation of gamma, X-ray or neutron radiation 24310.4. Advanced optical methods 24510.4.1. Laser induced fluorescence 24510.4.2. Interferometric methods (digital inline holography, Fourier interferometric imaging, ILIDS/IPI, rainbow) 252Chapter 11. Some Experimental Results Pertaining to Multiphase Flow Properties that Are Still Little Understood 26511.1. Atomization/fragmentation of liquid jets 26511.2. Isolated bubbles, bubbles in swarm and their effects on carrier fluid 27411.3. Boiling crisis 285Part 3. From Fluidized Beds To Granular Media 297Chapter 12. Fluidized Beds 29912.1. Introduction 29912.1.1. Classification of different fluidization regimes 29912.1.2. Minimum fluidization and bubbling velocities 30412.2. Complete models for the dynamics of fluidized beds 30612.2.1. Bubbling fluidization regime 30712.2.2. Turbulent fluidization regime 31512.3. Global models for chemical conversion in fluidized beds 32112.3.1. Bubbling regime fluidizations 32112.3.2. Fast fluidization regime 32412.3.3. Turbulent fluidization regime 32512.4. Global models for heat transfers in fluidized beds 32812.4.1. Bubbling fluidization regime 32812.4.2. Fast fluidization regimes – circulating beds 33112.5. Conclusion 334Chapter 13. Generalizations for Granular Media 33513.1. Introduction 33513.2. Balance equations for mean granular media 33613.3. Necessary closure approximations 34213.4. Some already proposed methods 345Chapter 14. Modeling of Cauchy Tensor of Sliding Contacts 34914.1. Hypotheses and basic equations 34914.2. Unclosed balance equation for Cauchy tensor of sliding contact 35114.3. Closure approximations for irreversible terms 358Chapter 15. Modeling the Kinetic Cauchy Stress Tensor 36315.1. Prandtl–Bagnold modeling 36415.2. K-lt or “turbulent granular gas” modeling 36615.3. Toward a general model for all regimes 37115.4. Boundary conditions at walls 373Part 4. Studying Fluctuations and Probability Densities 377Chapter 16. Fluctuations of the Gas Phase in Reactive Two-Phase Media 37916.1. Specificities of reactive two-phase media 37916.2. Probability density of composition fluctuations of the gas phase 38016.2.1. Instant basic equations of the gas medium 38216.2.2. PDF equation 38516.3. Modeling the terms due to exchanges between phases 39016.3.1. Total mass exchange 39016.3.2. Mass exchange for species 39216.3.3. Heat exchange 39316.4. Modeling micromixing and turbulent dispersion 39516.4.1. The “micromixing” term in PDF equations 39516.4.2. Turbulent diffusion terms in PDF equations 39616.5. Practical use of PDF equations 397Chapter 17. Temperature Fluctuations in Condensed Phases 39917.1. Problems 39917.2. Instantaneous equation for the temperature of the liquid phase 40117.3. Equation for the PDF of the temperature of the liquid 40317.4. Closure of the equation of the temperature PDF 405Chapter 18. Study of the PDF for Velocity Fluctuations and Sizes of Parcels 40918.1. Phase velocity PDF equation 41018.2. Modeling the exchanges between phases and the internal interactions 41518.2.1. Terms of exchanges between phases 41518.2.2. Internal dissipation and production of fluctuations 41818.3. Practical calculation of PDF 41918.4. The study of the sizes of the dispersed phase parcels 42018.5. Eulerian–Lagrangian simulation of dispersed media 42318.5.1. Lagrangian equations of the parcels 42318.5.2. Stochastic simulations 426Bibliography 431Index 443