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Ansys fluent 2020R1 theory guide.pdf

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    ANSYS Fluent Theory Guide
    Table of Contents
    Using This Manual
    1. The Contents of This Manual
    2. Typographical Conventions
    3. Mathematical Conventions
    Chapter 1: Basic Fluid Flow
    1.1. Overview of Physical Models in ANSYS Fluent
    1.2. Continuity and Momentum Equations
    1.2.1. The Mass Conservation Equation
    1.2.2. Momentum Conservation Equations
    1.3. User-Defined Scalar (UDS) Transport Equations
    1.3.1. Single Phase Flow
    1.3.2. Multiphase Flow
    1.4. Periodic Flows
    1.4.1. Overview
    1.4.2. Limitations
    1.4.3. Physics of Periodic Flows
    1.4.3.1. Definition of the Periodic Velocity
    1.4.3.2. Definition of the Streamwise-Periodic Pressure
    1.5. Swirling and Rotating Flows
    1.5.1. Overview of Swirling and Rotating Flows
    1.5.1.1. Axisymmetric Flows with Swirl or Rotation
    1.5.1.1.1. Momentum Conservation Equation for Swirl Velocity
    1.5.1.2. Three-Dimensional Swirling Flows
    1.5.1.3. Flows Requiring a Moving Reference Frame
    1.5.2. Physics of Swirling and Rotating Flows
    1.6. Compressible Flows
    1.6.1. When to Use the Compressible Flow Model
    1.6.2. Physics of Compressible Flows
    1.6.2.1. Basic Equations for Compressible Flows
    1.6.2.2. The Compressible Form of the Gas Law
    1.7. Inviscid Flows
    1.7.1. Euler Equations
    1.7.1.1. The Mass Conservation Equation
    1.7.1.2. Momentum Conservation Equations
    1.7.1.3. Energy Conservation Equation
    Chapter 2: Flows with Moving Reference Frames
    2.1. Introduction
    2.2. Flow in a Moving Reference Frame
    2.2.1. Equations for a Moving Reference Frame
    2.2.1.1. Relative Velocity Formulation
    2.2.1.2. Absolute Velocity Formulation
    2.2.1.3. Relative Specification of the Reference Frame Motion
    2.3. Flow in Multiple Reference Frames
    2.3.1. The Multiple Reference Frame Model
    2.3.1.1. Overview
    2.3.1.2. Examples
    2.3.1.3. The MRF Interface Formulation
    2.3.1.3.1. Interface Treatment: Relative Velocity Formulation
    2.3.1.3.2. Interface Treatment: Absolute Velocity Formulation
    2.3.2. The Mixing Plane Model
    2.3.2.1. Overview
    2.3.2.2. Rotor and Stator Domains
    2.3.2.3. The Mixing Plane Concept
    2.3.2.4. Choosing an Averaging Method
    2.3.2.4.1. Area Averaging
    2.3.2.4.2. Mass Averaging
    2.3.2.4.3. Mixed-Out Averaging
    2.3.2.5. Mixing Plane Algorithm of ANSYS Fluent
    2.3.2.6. Mass Conservation
    2.3.2.7. Swirl Conservation
    2.3.2.8. Total Enthalpy Conservation
    Chapter 3: Flows Using Sliding and Dynamic Meshes
    3.1. Introduction
    3.2. Dynamic Mesh Theory
    3.2.1. Conservation Equations
    3.2.2. Six DOF Solver Theory
    3.3. Sliding Mesh Theory
    Chapter 4: Turbulence
    4.1. Underlying Principles of Turbulence Modeling
    4.1.1. Reynolds (Ensemble) Averaging
    4.1.2. Filtered Navier-Stokes Equations
    4.1.3. Hybrid RANS-LES Formulations
    4.1.4. Boussinesq Approach vs. Reynolds Stress Transport Models
    4.2. Spalart-Allmaras Model
    4.2.1. Overview
    4.2.2. Transport Equation for the Spalart-Allmaras Model
    4.2.3. Modeling the Turbulent Viscosity
    4.2.4. Modeling the Turbulent Production
    4.2.5. Modeling the Turbulent Destruction
    4.2.6. Model Constants
    4.2.7. Wall Boundary Conditions
    4.2.7.1. Treatment of the Spalart-Allmaras Model for Icing Simulations
    4.2.8. Convective Heat and Mass Transfer Modeling
    4.3. Standard, RNG, and Realizable k-ε Models
    4.3.1. Standard k-ε Model
    4.3.1.1. Overview
    4.3.1.2. Transport Equations for the Standard k-ε Model
    4.3.1.3. Modeling the Turbulent Viscosity
    4.3.1.4. Model Constants
    4.3.2. RNG k-ε Model
    4.3.2.1. Overview
    4.3.2.2. Transport Equations for the RNG k-ε Model
    4.3.2.3. Modeling the Effective Viscosity
    4.3.2.4. RNG Swirl Modification
    4.3.2.5. Calculating the Inverse Effective Prandtl Numbers
    4.3.2.6. The R-ε Term in the ε Equation
    4.3.2.7. Model Constants
    4.3.3. Realizable k-ε Model
    4.3.3.1. Overview
    4.3.3.2. Transport Equations for the Realizable k-ε Model
    4.3.3.3. Modeling the Turbulent Viscosity
    4.3.3.4. Model Constants
    4.3.4. Modeling Turbulent Production in the k-ε Models
    4.3.5. Effects of Buoyancy on Turbulence in the k-ε Models
    4.3.6. Effects of Compressibility on Turbulence in the k-ε Models
    4.3.7. Convective Heat and Mass Transfer Modeling in the k-ε Models
    4.4. Standard, BSL, and SST k-ω Models
    4.4.1. Standard k-ω Model
    4.4.1.1. Overview
    4.4.1.2. Transport Equations for the Standard k-ω Model
    4.4.1.3. Modeling the Effective Diffusivity
    4.4.1.3.1. Low-Reynolds Number Correction
    4.4.1.4. Modeling the Turbulence Production
    4.4.1.4.1. Production of k
    4.4.1.4.2. Production of ω
    4.4.1.5. Modeling the Turbulence Dissipation
    4.4.1.5.1. Dissipation of k
    4.4.1.5.2. Dissipation of ω
    4.4.1.5.3. Compressibility Effects
    4.4.1.6. Model Constants
    4.4.2. Baseline (BSL) k-ω Model
    4.4.2.1. Overview
    4.4.2.2. Transport Equations for the BSL k-ω Model
    4.4.2.3. Modeling the Effective Diffusivity
    4.4.2.4. Modeling the Turbulence Production
    4.4.2.4.1. Production of k
    4.4.2.4.2. Production of ω
    4.4.2.5. Modeling the Turbulence Dissipation
    4.4.2.5.1. Dissipation of k
    4.4.2.5.2. Dissipation of ω
    4.4.2.6. Cross-Diffusion Modification
    4.4.2.7. Model Constants
    4.4.3. Shear-Stress Transport (SST) k-ω Model
    4.4.3.1. Overview
    4.4.3.2. Modeling the Turbulent Viscosity
    4.4.3.3. Model Constants
    4.4.3.4. Treatment of the SST Model for Icing Simulations
    4.4.4. Effects of Buoyancy on Turbulence in the k-ω Models
    4.4.5. Turbulence Damping
    4.4.6. Wall Boundary Conditions
    4.5. Generalized k-ω  (GEKO) Model
    4.5.1. Model Formulation
    4.5.2. Limitations
    4.6. k-kl-ω Transition Model
    4.6.1. Overview
    4.6.2. Transport Equations for the k-kl-ω Model
    4.6.2.1. Model Constants
    4.7. Transition SST Model
    4.7.1. Overview
    4.7.2. Transport Equations for the Transition SST Model
    4.7.2.1. Separation-Induced Transition Correction
    4.7.2.2. Coupling the Transition Model and SST Transport Equations
    4.7.2.3. Transition SST and Rough Walls
    4.7.3. Mesh Requirements
    4.7.4. Specifying Inlet Turbulence Levels
    4.8. Intermittency Transition Model
    4.8.1. Overview
    4.8.2. Transport Equations for the Intermittency Transition Model
    4.8.3. Coupling with the Other Models
    4.8.4. Intermittency Transition Model and Rough Walls
    4.9. The V2F Model
    4.10. Reynolds Stress Model (RSM)
    4.10.1. Overview
    4.10.2. Reynolds Stress Transport Equations
    4.10.3. Modeling Turbulent Diffusive Transport
    4.10.4. Modeling the Pressure-Strain Term
    4.10.4.1. Linear Pressure-Strain Model
    4.10.4.2. Low-Re Modifications to the Linear Pressure-Strain Model
    4.10.4.3. Quadratic Pressure-Strain Model
    4.10.4.4. Stress-Omega Model
    4.10.4.5. Stress-BSL Model
    4.10.5. Effects of Buoyancy on Turbulence
    4.10.6. Modeling the Turbulence Kinetic Energy
    4.10.7. Modeling the Dissipation Rate
    4.10.8. Modeling the Turbulent Viscosity
    4.10.9. Wall Boundary Conditions
    4.10.10. Convective Heat and Mass Transfer Modeling
    4.11. Scale-Adaptive Simulation (SAS) Model
    4.11.1. Overview
    4.11.2. Transport Equations for the SST-SAS Model
    4.11.3. SAS with Other ω-Based Turbulence Models
    4.12. Detached Eddy Simulation (DES)
    4.12.1. Overview
    4.12.2. DES with the Spalart-Allmaras Model
    4.12.3. DES with the Realizable k-ε Model
    4.12.4. DES with the BSL or SST k-ω Model
    4.12.5. DES with the Transition SST Model
    4.12.6. Improved Delayed Detached Eddy Simulation (IDDES)
    4.12.6.1. Overview of IDDES
    4.12.6.2. IDDES Model Formulation
    4.13. Shielded Detached Eddy Simulation (SDES)
    4.13.1. Shielding Function
    4.13.2. LES Mode of SDES
    4.14. Stress-Blended Eddy Simulation (SBES)
    4.14.1. Stress Blending
    4.14.2. SDES and SBES Example
    4.15. Large Eddy Simulation (LES) Model
    4.15.1. Overview
    4.15.2. Subgrid-Scale Models
    4.15.2.1. Smagorinsky-Lilly Model
    4.15.2.2. Dynamic Smagorinsky-Lilly Model
    4.15.2.3. Wall-Adapting Local Eddy-Viscosity (WALE) Model
    4.15.2.4. Algebraic Wall-Modeled LES Model (WMLES)
    4.15.2.4.1. Algebraic WMLES Model Formulation
    4.15.2.4.1.1. Reynolds Number Scaling
    4.15.2.4.2. Algebraic WMLES S-Omega Model Formulation
    4.15.2.5. Dynamic Kinetic Energy Subgrid-Scale Model
    4.15.3. Inlet Boundary Conditions for Scale Resolving Simulations
    4.15.3.1. Vortex Method
    4.15.3.2. Spectral Synthesizer
    4.15.3.3. Synthetic Turbulence Generator
    4.15.3.3.1. Limitations
    4.16. Embedded Large Eddy Simulation (ELES)
    4.16.1. Overview
    4.16.2. Selecting a Model
    4.16.3. Interfaces Treatment
    4.16.3.1. RANS-LES Interface
    4.16.3.2. LES-RANS Interface
    4.16.3.3. Internal Interface Without LES Zone
    4.16.3.4. Grid Generation Guidelines
    4.17. Near-Wall Treatments for Wall-Bounded Turbulent Flows
    4.17.1. Overview
    4.17.1.1. Wall Functions vs. Near-Wall Model
    4.17.1.2. Wall Functions
    4.17.2. Standard Wall Functions
    4.17.2.1. Momentum
    4.17.2.2. Energy
    4.17.2.3. Species
    4.17.2.4. Turbulence
    4.17.3. Scalable Wall Functions
    4.17.4. Non-Equilibrium Wall Functions
    4.17.4.1. Standard Wall Functions vs. Non-Equilibrium Wall Functions
    4.17.4.2. Limitations of the Wall Function Approach
    4.17.5. Enhanced Wall Treatment ε-Equation (EWT-ε)
    4.17.5.1. Two-Layer Model for Enhanced Wall Treatment
    4.17.5.2. Enhanced Wall Treatment for Momentum and Energy Equations
    4.17.6. Menter-Lechner ε-Equation (ML-ε)
    4.17.6.1. Momentum Equations
    4.17.6.2. k-ε Turbulence Models
    4.17.6.3. Iteration Improvements
    4.17.7. y+-Insensitive Wall Treatment ω-Equation
    4.17.8. User-Defined Wall Functions
    4.17.9. LES Near-Wall Treatment
    4.18. Curvature Correction for the Spalart-Allmaras and Two-Equation Models
    4.19. Production Limiters for Two-Equation Models
    4.20. Definition of Turbulence Scales
    4.20.1. RANS and Hybrid (SAS, DES, and SDES) Turbulence Models
    4.20.2. Large Eddy Simulation (LES) Models
    4.20.3. Stress-Blended Eddy Simulation (SBES) Model
    Chapter 5: Heat Transfer
    5.1. Introduction
    5.2. Modeling Conductive and Convective Heat Transfer
    5.2.1. Heat Transfer Theory
    5.2.1.1. The Energy Equation
    5.2.1.2. The Energy Equation in Moving Reference Frames
    5.2.1.3. The Energy Equation for the Non-Premixed Combustion Model
    5.2.1.4. Inclusion of Pressure Work and Kinetic Energy Terms
    5.2.1.5. Inclusion of the Viscous Dissipation Terms
    5.2.1.6. Inclusion of the Species Diffusion Term
    5.2.1.7. Energy Sources Due to Reaction
    5.2.1.8. Energy Sources Due To Radiation
    5.2.1.9. Energy Source Due To Joule Heating
    5.2.1.10. Interphase Energy Sources
    5.2.1.11. Energy Equation in Solid Regions
    5.2.1.12. Anisotropic Conductivity in Solids
    5.2.1.13. Diffusion at Inlets
    5.2.2. Natural Convection and Buoyancy-Driven Flows Theory
    5.3. Modeling Radiation
    5.3.1. Overview and Limitations
    5.3.1.1. Advantages and Limitations of the DTRM
    5.3.1.2. Advantages and Limitations of the P-1 Model
    5.3.1.3. Advantages and Limitations of the Rosseland Model
    5.3.1.4. Advantages and Limitations of the DO Model
    5.3.1.5. Advantages and Limitations of the S2S Model
    5.3.1.6. Advantages and Limitations of the MC Model
    5.3.2. Radiative Transfer Equation
    5.3.3. P-1 Radiation Model Theory
    5.3.3.1. The P-1 Model Equations
    5.3.3.2. Anisotropic Scattering
    5.3.3.3. Particulate Effects in the P-1 Model
    5.3.3.4. Boundary Condition Treatment for the P-1 Model at Walls
    5.3.3.5. Boundary Condition Treatment for the P-1 Model at Flow Inlets and Exits
    5.3.4. Rosseland Radiation Model Theory
    5.3.4.1. The Rosseland Model Equations
    5.3.4.2. Anisotropic Scattering
    5.3.4.3. Boundary Condition Treatment at Walls
    5.3.4.4. Boundary Condition Treatment at Flow Inlets and Exits
    5.3.5. Discrete Transfer Radiation Model (DTRM) Theory
    5.3.5.1. The DTRM Equations
    5.3.5.2. Ray Tracing
    5.3.5.3. Clustering
    5.3.5.4. Boundary Condition Treatment for the DTRM at Walls
    5.3.5.5. Boundary Condition Treatment for the DTRM at Flow Inlets and Exits
    5.3.6. Discrete Ordinates (DO) Radiation Model Theory
    5.3.6.1. The DO Model Equations
    5.3.6.2. Energy Coupling and the DO Model
    5.3.6.2.1. Limitations of DO/Energy Coupling
    5.3.6.3. Angular Discretization and Pixelation
    5.3.6.4. Anisotropic Scattering
    5.3.6.5. Particulate Effects in the DO Model
    5.3.6.6. Boundary and Cell Zone Condition Treatment at Opaque Walls
    5.3.6.6.1. Gray Diffuse Walls
    5.3.6.6.2. Non-Gray Diffuse Walls
    5.3.6.7. Cell Zone and Boundary Condition Treatment at Semi-Transparent Walls
    5.3.6.7.1. Semi-Transparent Interior Walls
    5.3.6.7.2. Specular Semi-Transparent Walls
    5.3.6.7.3. Diffuse Semi-Transparent Walls
    5.3.6.7.4. Partially Diffuse Semi-Transparent Walls
    5.3.6.7.5. Semi-Transparent Exterior Walls
    5.3.6.7.6. Limitations
    5.3.6.7.7. Solid Semi-Transparent Media
    5.3.6.8. Boundary Condition Treatment at Specular Walls and Symmetry Boundaries
    5.3.6.9. Boundary Condition Treatment at Periodic Boundaries
    5.3.6.10. Boundary Condition Treatment at Flow Inlets and Exits
    5.3.7. Surface-to-Surface (S2S) Radiation Model Theory
    5.3.7.1. Gray-Diffuse Radiation
    5.3.7.2. The S2S Model Equations
    5.3.7.3. Clustering
    5.3.7.3.1. Clustering and View Factors
    5.3.7.3.2. Clustering and Radiosity
    5.3.8. Monte Carlo (MC) Radiation Model Theory
    5.3.8.1. The MC Model Equations
    5.3.8.1.1. Monte Carlo Solution Accuracy
    5.3.8.2. Boundary Condition Treatment for the MC Model
    5.3.9. Radiation in Combusting Flows
    5.3.9.1. The Weighted-Sum-of-Gray-Gases Model
    5.3.9.1.1. When the Total (Static) Gas Pressure is Not Equal to 1 atm
    5.3.9.2. The Effect of Soot on the Absorption Coefficient
    5.3.9.3. The Effect of Particles on the Absorption Coefficient
    5.3.10. Choosing a Radiation Model
    5.3.10.1. External Radiation
    Chapter 6: Heat Exchangers
    6.1. The Macro Heat Exchanger Models
    6.1.1. Overview of the Macro Heat Exchanger Models
    6.1.2. Restrictions of the Macro Heat Exchanger Models
    6.1.3. Macro Heat Exchanger Model Theory
    6.1.3.1. Streamwise Pressure Drop
    6.1.3.2. Heat Transfer Effectiveness
    6.1.3.3. Heat Rejection
    6.1.3.4. Macro Heat Exchanger Group Connectivity
    6.2. The Dual Cell Model
    6.2.1. Overview of the Dual Cell Model
    6.2.2. Restrictions of the Dual Cell Model
    6.2.3. Dual Cell Model Theory
    6.2.3.1. NTU Relations
    6.2.3.2. Heat Rejection
    Chapter 7: Species Transport and Finite-Rate Chemistry
    7.1. Volumetric Reactions
    7.1.1. Species Transport Equations
    7.1.1.1. Mass Diffusion in Laminar Flows
    7.1.1.2. Mass Diffusion in Turbulent Flows
    7.1.1.3. Treatment of Species Transport in the Energy Equation
    7.1.1.4. Diffusion at Inlets
    7.1.2. The Generalized Finite-Rate Formulation for Reaction Modeling
    7.1.2.1. Direct Use of Finite-Rate Kinetics (no TCI)
    7.1.2.2. Pressure-Dependent Reactions
    7.1.2.3. The Eddy-Dissipation Model
    7.1.2.4. The Eddy-Dissipation Model for LES
    7.1.2.5. The Eddy-Dissipation-Concept (EDC) Model
    7.1.2.6. The Thickened Flame Model
    7.1.2.7. The Relaxation to Chemical Equilibrium Model
    7.2. Wall Surface Reactions and Chemical Vapor Deposition
    7.2.1. Surface Coverage Reaction Rate Modification
    7.2.2. Reaction-Diffusion Balance for Surface Chemistry
    7.2.3. Slip Boundary Formulation for Low-Pressure Gas Systems
    7.3. Particle Surface Reactions
    7.3.1. General Description
    7.3.2. ANSYS Fluent Model Formulation
    7.3.3. Extension for Stoichiometries with Multiple Gas Phase Reactants
    7.3.4. Solid-Solid Reactions
    7.3.5. Solid Decomposition Reactions
    7.3.6. Solid Deposition Reactions
    7.3.7. Gaseous Solid Catalyzed Reactions on the Particle Surface
    7.4. Electrochemical Reactions
    7.4.1. Overview
    7.4.2. Electrochemical Reaction Model Theory
    7.5. Reacting Channel Model
    7.5.1. Overview
    7.5.2. Reacting Channel Model Theory
    7.5.2.1. Flow Inside the Reacting Channel
    7.5.2.2. Surface Reactions in the Reacting Channel
    7.5.2.3. Porous Medium Inside Reacting Channel
    7.5.2.4. Outer Flow in the Shell
    7.6. Reactor Network Model
    7.6.1. Reactor Network Model Theory
    7.6.1.1. Reactor network temperature solution
    Chapter 8: Non-Premixed Combustion
    8.1. Introduction
    8.2. Non-Premixed Combustion and Mixture Fraction Theory
    8.2.1. Mixture Fraction Theory
    8.2.1.1. Definition of the Mixture Fraction
    8.2.1.2. Transport Equations for the Mixture Fraction
    8.2.1.3. The Non-Premixed Model for LES
    8.2.1.4. The Non-Premixed Model with the SBES Turbulence Model
    8.2.1.5. Mixture Fraction vs. Equivalence Ratio
    8.2.1.6. Relationship of Mixture Fraction to Species Mass Fraction, Density, and Temperature
    8.2.2. Modeling of Turbulence-Chemistry Interaction
    8.2.2.1. Description of the Probability Density Function
    8.2.2.2. Derivation of Mean Scalar Values from the Instantaneous Mixture Fraction
    8.2.2.3. The Assumed-Shape PDF
    8.2.2.3.1. The Double Delta Function PDF
    8.2.2.3.2. The β-Function PDF
    8.2.3. Non-Adiabatic Extensions of the Non-Premixed Model
    8.2.4. Chemistry Tabulation
    8.2.4.1. Look-Up Tables for Adiabatic Systems
    8.2.4.2. 3D Look-Up Tables for Non-Adiabatic Systems
    8.2.4.3. Generating Lookup Tables Through Automated Grid Refinement
    8.3. Restrictions and Special Cases for Using the Non-Premixed Model
    8.3.1. Restrictions on the Mixture Fraction Approach
    8.3.2. Using the Non-Premixed Model for Liquid Fuel or Coal Combustion
    8.3.3. Using the Non-Premixed Model with Flue Gas Recycle
    8.3.4. Using the Non-Premixed Model with the Inert Model
    8.3.4.1. Mixture Composition
    8.3.4.1.1. Property Evaluation
    8.4. The Diffusion Flamelet Models Theory
    8.4.1. Restrictions and Assumptions
    8.4.2. The Flamelet Concept
    8.4.2.1. Overview
    8.4.2.2. Strain Rate and Scalar Dissipation
    8.4.2.3. Embedding Diffusion Flamelets in Turbulent Flames
    8.4.3. Flamelet Generation
    8.4.4. Flamelet Import
    8.5. The Steady Diffusion Flamelet Model Theory
    8.5.1. Overview
    8.5.2. Multiple Steady Flamelet Libraries
    8.5.3. Steady Diffusion Flamelet Automated Grid Refinement
    8.5.4. Non-Adiabatic Steady Diffusion Flamelets
    8.6. The Unsteady Diffusion Flamelet Model Theory
    8.6.1. The Eulerian Unsteady Laminar Flamelet Model
    8.6.1.1. Liquid Reactions
    8.6.2. The Diesel Unsteady Laminar Flamelet Model
    8.6.3. Multiple Diesel Unsteady Flamelets
    8.6.4. Multiple Diesel Unsteady Flamelets with Flamelet Reset
    8.6.4.1. Resetting the Flamelets
    Chapter 9: Premixed Combustion
    9.1. Overview
    9.2. C-Equation Model Theory
    9.2.1. Propagation of the Flame Front
    9.3. G-Equation Model Theory
    9.3.1. Numerical Solution of the G-equation
    9.4. Turbulent Flame Speed Models
    9.4.1. Zimont Turbulent Flame Speed Closure Model
    9.4.1.1. Zimont Turbulent Flame Speed Closure for LES
    9.4.1.2. Flame Stretch Effect
    9.4.1.3. Wall Damping
    9.4.2. Peters Flame Speed Model
    9.4.2.1. Peters Flame Speed Model for LES
    9.5. Extended Coherent Flamelet Model Theory
    9.5.1. Closure for ECFM Source Terms
    9.5.2. Turbulent Flame Speed in ECFM
    9.5.3. LES and ECFM
    9.6. Calculation of Properties
    9.6.1. Calculation of Temperature
    9.6.1.1. Adiabatic Temperature Calculation
    9.6.1.2. Non-Adiabatic Temperature Calculation
    9.6.2. Calculation of Density
    9.6.3. Laminar Flame Speed
    9.6.4. Unburnt Density and Thermal Diffusivity
    Chapter 10: Partially Premixed Combustion
    10.1. Overview
    10.2. Partially Premixed Combustion Theory
    10.2.1. Chemical Equilibrium and Steady Diffusion Flamelet Models
    10.2.2. Flamelet Generated Manifold (FGM) Model
    10.2.2.1. Premixed FGMs in Reaction Progress Variable Space
    10.2.2.2. Premixed FGMs in Physical Space
    10.2.2.3. Diffusion FGMs
    10.2.2.4. Nonadiabatic Flamelet Generated Manifold (FGM)
    10.2.3. FGM Turbulent Closure
    10.2.3.1. Scalar Transport with FGM Closure
    10.2.4. Calculation of Mixture Properties
    10.2.5. Calculation of Unburnt Properties
    10.2.6. Laminar Flame Speed
    10.2.7. Generating PDF Lookup Tables Through Automated Grid Refinement
    Chapter 11: Composition PDF Transport
    11.1. Overview
    11.2. Composition PDF Transport Theory
    11.3. The Lagrangian Solution Method
    11.3.1. Particle Convection
    11.3.2. Particle Mixing
    11.3.2.1. The Modified Curl Model
    11.3.2.2. The IEM Model
    11.3.2.3. The EMST Model
    11.3.2.4. Liquid Reactions
    11.3.3. Particle Reaction
    11.4. The Eulerian Solution Method
    11.4.1. Reaction
    11.4.2. Mixing
    11.4.3. Correction
    11.4.4. Calculation of Composition Mean and Variance
    Chapter 12: Chemistry Acceleration
    12.1. Overview and Limitations
    12.2. In-Situ Adaptive Tabulation (ISAT)
    12.3. Dynamic Mechanism Reduction
    12.3.1. Directed Relation Graph (DRG) Method for Mechanism Reduction
    12.4. Chemistry Agglomeration
    12.4.1. Binning Algorithm
    12.5. Chemical Mechanism Dimension Reduction
    12.5.1. Selecting the Represented Species
    12.6. Dynamic Cell Clustering with ANSYS Fluent CHEMKIN-CFD Solver
    12.7. Dynamic Adaptive Chemistry with ANSYS Fluent CHEMKIN-CFD Solver
    Chapter 13: Engine Ignition
    13.1. Spark Model
    13.1.1. Overview and Limitations
    13.1.2. Spark Model Theory
    13.1.3. ECFM Spark Model Variants
    13.2. Autoignition Models
    13.2.1. Model Overview
    13.2.2. Model Limitations
    13.2.3. Ignition Model Theory
    13.2.3.1. Transport of Ignition Species
    13.2.3.2. Knock Modeling
    13.2.3.2.1. Modeling of the Source Term
    13.2.3.2.2. Correlations
    13.2.3.2.3. Energy Release
    13.2.3.3. Ignition Delay Modeling
    13.2.3.3.1. Modeling of the Source Term
    13.2.3.3.2. Correlations
    13.2.3.3.3. Energy Release
    13.3. Crevice Model
    13.3.1. Overview
    13.3.1.1. Model Parameters
    13.3.2. Limitations
    13.3.3. Crevice Model Theory
    Chapter 14: Pollutant Formation
    14.1. NOx Formation
    14.1.1. Overview
    14.1.1.1. NOx Modeling in ANSYS Fluent
    14.1.1.2. NOx Formation and Reduction in Flames
    14.1.2. Governing Equations for NOx Transport
    14.1.3. Thermal NOx Formation
    14.1.3.1. Thermal NOx Reaction Rates
    14.1.3.2. The Quasi-Steady Assumption for [N]
    14.1.3.3. Thermal NOx Temperature Sensitivity
    14.1.3.4. Decoupled Thermal NOx Calculations
    14.1.3.5. Approaches for Determining O Radical Concentration
    14.1.3.5.1. Method 1: Equilibrium Approach
    14.1.3.5.2. Method 2: Partial Equilibrium Approach
    14.1.3.5.3. Method 3: Predicted O Approach
    14.1.3.6. Approaches for Determining OH Radical Concentration
    14.1.3.6.1. Method 1: Exclusion of OH Approach
    14.1.3.6.2. Method 2: Partial Equilibrium Approach
    14.1.3.6.3. Method 3: Predicted OH Approach
    14.1.3.7. Summary
    14.1.4. Prompt NOx Formation
    14.1.4.1. Prompt NOx Combustion Environments
    14.1.4.2. Prompt NOx Mechanism
    14.1.4.3. Prompt NOx Formation Factors
    14.1.4.4. Primary Reaction
    14.1.4.5. Modeling Strategy
    14.1.4.6. Rate for Most Hydrocarbon Fuels
    14.1.4.7. Oxygen Reaction Order
    14.1.5. Fuel NOx Formation
    14.1.5.1. Fuel-Bound Nitrogen
    14.1.5.2. Reaction Pathways
    14.1.5.3. Fuel NOx from Gaseous and Liquid Fuels
    14.1.5.3.1. Fuel NOx from Intermediate Hydrogen Cyanide (HCN)
    14.1.5.3.1.1. HCN Production in a Gaseous Fuel
    14.1.5.3.1.2. HCN Production in a Liquid Fuel
    14.1.5.3.1.3. HCN Consumption
    14.1.5.3.1.4. HCN Sources in the Transport Equation
    14.1.5.3.1.5. NOx Sources in the Transport Equation
    14.1.5.3.2. Fuel NOx from Intermediate Ammonia (NH3)
    14.1.5.3.2.1. NH3 Production in a Gaseous Fuel
    14.1.5.3.2.2. NH3 Production in a Liquid Fuel
    14.1.5.3.2.3. NH3 Consumption
    14.1.5.3.2.4. NH3 Sources in the Transport Equation
    14.1.5.3.2.5. NOx Sources in the Transport Equation
    14.1.5.3.3. Fuel NOx from Coal
    14.1.5.3.3.1. Nitrogen in Char and in Volatiles
    14.1.5.3.3.2. Coal Fuel NOx Scheme A
    14.1.5.3.3.3. Coal Fuel NOx Scheme B
    14.1.5.3.3.4. HCN Scheme Selection
    14.1.5.3.3.5. NOx Reduction on Char Surface
    14.1.5.3.3.5.1. BET Surface Area
    14.1.5.3.3.5.2. HCN from Volatiles
    14.1.5.3.3.6. Coal Fuel NOx Scheme C
    14.1.5.3.3.7. Coal Fuel NOx Scheme D
    14.1.5.3.3.8. NH3 Scheme Selection
    14.1.5.3.3.8.1. NH3 from Volatiles
    14.1.5.3.4. Fuel Nitrogen Partitioning for HCN and NH3 Intermediates
    14.1.6. NOx Formation from Intermediate N2O
    14.1.6.1. N2O - Intermediate NOx Mechanism
    14.1.7. NOx Reduction by Reburning
    14.1.7.1. Instantaneous Approach
    14.1.7.2. Partial Equilibrium Approach
    14.1.7.2.1. NOx Reduction Mechanism
    14.1.8. NOx Reduction by SNCR
    14.1.8.1. Ammonia Injection
    14.1.8.2. Urea Injection
    14.1.8.3. Transport Equations for Urea, HNCO, and NCO
    14.1.8.4. Urea Production due to Reagent Injection
    14.1.8.5. NH3 Production due to Reagent Injection
    14.1.8.6. HNCO Production due to Reagent Injection
    14.1.9. NOx Formation in Turbulent Flows
    14.1.9.1. The Turbulence-Chemistry Interaction Model
    14.1.9.2. The PDF Approach
    14.1.9.3. The General Expression for the Mean Reaction Rate
    14.1.9.4. The Mean Reaction Rate Used in ANSYS Fluent
    14.1.9.5. Statistical Independence
    14.1.9.6. The Beta PDF Option
    14.1.9.7. The Gaussian PDF Option
    14.1.9.8. The Calculation Method for the Variance
    14.2. SOx Formation
    14.2.1. Overview
    14.2.1.1. The Formation of SOx
    14.2.2. Governing Equations for SOx Transport
    14.2.3. Reaction Mechanisms for Sulfur Oxidation
    14.2.4. SO2 and H2S Production in a Gaseous Fuel
    14.2.5. SO2 and H2S Production in a Liquid Fuel
    14.2.6. SO2 and H2S Production from Coal
    14.2.6.1. SO2 and H2S from Char
    14.2.6.2. SO2 and H2S from Volatiles
    14.2.7. SOx Formation in Turbulent Flows
    14.2.7.1. The Turbulence-Chemistry Interaction Model
    14.2.7.2. The PDF Approach
    14.2.7.3. The Mean Reaction Rate
    14.2.7.4. The PDF Options
    14.3. Soot Formation
    14.3.1. Overview and Limitations
    14.3.1.1. Predicting Soot Formation
    14.3.1.2. Restrictions on Soot Modeling
    14.3.2. Soot Model Theory
    14.3.2.1. The One-Step Soot Formation Model
    14.3.2.2. The Two-Step Soot Formation Model
    14.3.2.2.1. Soot Generation Rate
    14.3.2.2.2. Nuclei Generation Rate
    14.3.2.3. The Moss-Brookes Model
    14.3.2.3.1. The Moss-Brookes-Hall Model
    14.3.2.3.2. Soot Formation in Turbulent Flows
    14.3.2.3.2.1. The Turbulence-Chemistry Interaction Model
    14.3.2.3.2.2. The PDF Approach
    14.3.2.3.2.3. The Mean Reaction Rate
    14.3.2.3.2.4. The PDF Options
    14.3.2.3.3. The Effect of Soot on the Radiation Absorption Coefficient
    14.3.2.4. The Method of Moments Model
    14.3.2.4.1. Soot Particle Population Balance
    14.3.2.4.2. Moment Transport Equations
    14.3.2.4.3. Source Terms for the Moments Equations
    14.3.2.4.3.1. Detailed Soot Mechanism
    14.3.2.4.3.2. Built-In HACA Mechanism
    14.3.2.4.3.2.1. Nucleation
    14.3.2.4.3.2.2. Surface Growth and Oxidation
    14.3.2.4.4. Coagulation
    14.3.2.4.5. Soot Aggregation
    14.4. Decoupled Detailed Chemistry Model
    14.4.1. Overview
    14.4.1.1. Limitations
    14.4.2. Decoupled Detailed Chemistry Model Theory
    Chapter 15: Aerodynamically Generated Noise
    15.1. Overview
    15.1.1. Direct Method
    15.1.2. Integral Method by Ffowcs Williams and Hawkings
    15.1.3. Method Based on Wave Equation
    15.1.4. Broadband Noise Source Models
    15.2. Acoustics Model Theory
    15.2.1. The Ffowcs Williams and Hawkings Model
    15.2.2. Wave Equation Model
    15.2.2.1. Limitations
    15.2.2.2. Governing Equations and Boundary Conditions
    15.2.2.3. Method of Numerical Solution
    15.2.2.4. Preventing Non-Physical Reflections of Sound Waves
    15.2.2.4.1. Mesh Quality
    15.2.2.4.2. Filtering of the Sound Source Term
    15.2.2.4.3. Ramping in Time and Limiting in Space (Masking) of the Sound Source Term
    15.2.2.4.4. Damping of Solution in a Sponge Region Using Artificial Viscosity
    15.2.2.5. Kirchhoff Integral
    15.2.2.5.1. Compatibility and Limitations
    15.2.2.5.2. Mathematical Formulation
    15.2.3. Broadband Noise Source Models
    15.2.3.1. Proudman’s Formula
    15.2.3.2. The Jet Noise Source Model
    15.2.3.3. The Boundary Layer Noise Source Model
    15.2.3.4. Source Terms in the Linearized Euler Equations
    15.2.3.5. Source Terms in Lilley’s Equation
    Chapter 16: Discrete Phase
    16.1. Introduction
    16.1.1. The Euler-Lagrange Approach
    16.2. Particle Motion Theory
    16.2.1. Equations of Motion for Particles
    16.2.1.1. Particle Force Balance
    16.2.1.2. Particle Torque Balance
    16.2.1.3. Inclusion of the Gravity Term
    16.2.1.4. Other Forces
    16.2.1.5. Forces in Moving Reference Frames
    16.2.1.6. Thermophoretic Force
    16.2.1.7. Brownian Force
    16.2.1.8. Saffman’s Lift Force
    16.2.1.9. Magnus Lift Force
    16.2.2. Turbulent Dispersion of Particles
    16.2.2.1. Stochastic Tracking
    16.2.2.1.1. The Integral Time
    16.2.2.1.2. The Discrete Random Walk Model
    16.2.2.1.3. Using the DRW Model
    16.2.2.2. Particle Cloud Tracking
    16.2.2.2.1. Using the Cloud Model
    16.2.3. Integration of Particle Equation of Motion
    16.3. Laws for Drag Coefficients
    16.3.1. Spherical Drag Law
    16.3.2. Non-spherical Drag Law
    16.3.3. Stokes-Cunningham Drag Law
    16.3.4. High-Mach-Number Drag Law
    16.3.5. Dynamic Drag Model Theory
    16.3.6. Dense Discrete Phase Model Drag Laws
    16.3.7. Bubbly Flow Drag Laws
    16.3.7.1. Ishii-Zuber Drag Model
    16.3.7.2. Grace Drag Model
    16.3.8. Rotational Drag Law
    16.4. Laws for Heat and Mass Exchange
    16.4.1. Inert Heating or Cooling (Law 1/Law 6)
    16.4.2. Droplet Vaporization (Law 2)
    16.4.2.1. Mass Transfer During Law 2—Diffusion Controlled Model
    16.4.2.2. Mass Transfer During Law 2—Convection/Diffusion Controlled Model
    16.4.2.3. Mass Transfer During Law 2—Thermolysis
    16.4.2.4. Defining the Saturation Vapor Pressure and Diffusion Coefficient (or Binary Diffusivity)
    16.4.2.5. Defining the Boiling Point and Latent Heat
    16.4.2.6. Heat Transfer to the Droplet
    16.4.3. Droplet Boiling (Law 3)
    16.4.4. Devolatilization (Law 4)
    16.4.4.1. Choosing the Devolatilization Model
    16.4.4.2. The Constant Rate Devolatilization Model
    16.4.4.3. The Single Kinetic Rate Model
    16.4.4.4. The Two Competing Rates (Kobayashi) Model
    16.4.4.5. The CPD Model
    16.4.4.5.1. General Description
    16.4.4.5.2. Reaction Rates
    16.4.4.5.3. Mass Conservation
    16.4.4.5.4. Fractional Change in the Coal Mass
    16.4.4.5.5. CPD Inputs
    16.4.4.6. Particle Swelling During Devolatilization
    16.4.4.7. Heat Transfer to the Particle During Devolatilization
    16.4.5. Surface Combustion (Law 5)
    16.4.5.1. The Diffusion-Limited Surface Reaction Rate Model
    16.4.5.2. The Kinetic/Diffusion Surface Reaction Rate Model
    16.4.5.3. The Intrinsic Model
    16.4.5.4. The Multiple Surface Reactions Model
    16.4.5.4.1. Limitations
    16.4.5.5. Heat and Mass Transfer During Char Combustion
    16.4.6. Multicomponent Particle Definition (Law 7)
    16.4.6.1. Raoult’s Law
    16.4.6.2. Peng-Robinson Real Gas Model
    16.5. Vapor Liquid Equilibrium Theory
    16.6. Physical Property Averaging
    16.7. Wall-Particle Reflection Model Theory
    16.7.1. Rough Wall Model
    16.8. Wall-Jet Model Theory
    16.9. Wall-Film Model Theory
    16.9.1. Introduction
    16.9.2. Leidenfrost Temperature Considerations
    16.9.2.1. Default Wall Temperature Limiter
    16.9.2.2. Leidenfrost Temperature Reporting
    16.9.3. Interaction During Impact with a Boundary
    16.9.3.1. The Stanton-Rutland Model
    16.9.3.1.1. Regime Definition
    16.9.3.1.2. Rebound
    16.9.3.1.3. Splashing
    16.9.3.2. The Kuhnke Model
    16.9.3.2.1. Regime definition
    16.9.3.2.2. Rebound
    16.9.3.2.3. Splashing
    16.9.4. Separation and Stripping Submodels
    16.9.5. Conservation Equations for Wall-Film Particles
    16.9.5.1. Momentum
    16.9.5.2. Mass Transfer from the Film
    16.9.5.2.1. Film Vaporization and Boiling
    16.9.5.2.2. Film Condensation
    16.9.5.3. Energy Transfer from the Film
    16.10. Wall Erosion
    16.10.1. Finnie Erosion Model
    16.10.2. Oka Erosion Model
    16.10.3. McLaury Erosion Model
    16.10.4. Modeling Erosion Rates in Dense Flows
    16.10.4.1. Abrasive Erosion Caused by Solid Particles
    16.10.4.2. Wall Shielding Effect in Dense Flow Regimes
    16.10.5. Accretion
    16.11. Particle–Wall Impingement Heat Transfer Theory
    16.12. Atomizer Model Theory
    16.12.1. The Plain-Orifice Atomizer Model
    16.12.1.1. Internal Nozzle State
    16.12.1.2. Coefficient of Discharge
    16.12.1.3. Exit Velocity
    16.12.1.4. Spray Angle
    16.12.1.5. Droplet Diameter Distribution
    16.12.2. The Pressure-Swirl Atomizer Model
    16.12.2.1. Film Formation
    16.12.2.2. Sheet Breakup and Atomization
    16.12.3. The Air-Blast/Air-Assist Atomizer Model
    16.12.4. The Flat-Fan Atomizer Model
    16.12.5. The Effervescent Atomizer Model
    16.13. Secondary Breakup Model Theory
    16.13.1. Taylor Analogy Breakup (TAB) Model
    16.13.1.1. Introduction
    16.13.1.2. Use and Limitations
    16.13.1.3. Droplet Distortion
    16.13.1.4. Size of Child Droplets
    16.13.1.5. Velocity of Child Droplets
    16.13.1.6. Droplet Breakup
    16.13.2. Wave Breakup Model
    16.13.2.1. Introduction
    16.13.2.2. Use and Limitations
    16.13.2.3. Jet Stability Analysis
    16.13.2.4. Droplet Breakup
    16.13.3. KHRT Breakup Model
    16.13.3.1. Introduction
    16.13.3.2. Use and Limitations
    16.13.3.3. Liquid Core Length
    16.13.3.4. Rayleigh-Taylor Breakup
    16.13.3.5. Droplet Breakup Within the Liquid Core
    16.13.3.6. Droplet Breakup Outside the Liquid Core
    16.13.4. Stochastic Secondary Droplet (SSD) Model
    16.13.5. Madabhushi Breakup Model
    16.14. Collision and Droplet Coalescence Model Theory
    16.14.1. Introduction
    16.14.2. Use and Limitations
    16.14.3. Theory
    16.14.3.1. Probability of Collision
    16.14.3.2. Collision Outcomes
    16.15. Discrete Element Method Collision Model
    16.15.1. Theory
    16.15.1.1. The Spring Collision Law
    16.15.1.2. The Spring-Dashpot Collision Law
    16.15.1.3. The Hertzian Collision Law
    16.15.1.4. The Hertzian-Dashpot Collision Law
    16.15.1.5. The Friction Collision Law
    16.15.1.6. Rolling Friction Collision Law for DEM
    16.15.1.7. DEM Parcels
    16.15.1.8. Cartesian Collision Mesh
    16.16. One-Way and Two-Way Coupling
    16.16.1. Coupling Between the Discrete and Continuous Phases
    16.16.2. Momentum Exchange
    16.16.3. Heat Exchange
    16.16.4. Mass Exchange
    16.16.5. Under-Relaxation of the Interphase Exchange Terms
    16.16.6. Interphase Exchange During Stochastic Tracking
    16.16.7. Interphase Exchange During Cloud Tracking
    16.17. Node Based Averaging
    Chapter 17: Modeling Macroscopic Particles
    17.1. Momentum Transfer to Fluid Flow
    17.2. Fluid Forces and Torques on Particle
    17.3. Particle/Particle and Particle/Wall Collisions
    17.4. Field Forces
    17.5. Particle Deposition and Buildup
    Chapter 18: Multiphase Flows
    18.1. Introduction
    18.1.1. Multiphase Flow Regimes
    18.1.1.1. Gas-Liquid or Liquid-Liquid Flows
    18.1.1.2. Gas-Solid Flows
    18.1.1.3. Liquid-Solid Flows
    18.1.1.4. Three-Phase Flows
    18.1.2. Examples of Multiphase Systems
    18.2. Choosing a General Multiphase Model
    18.2.1. Approaches to Multiphase Modeling
    18.2.1.1. The Euler-Euler Approach
    18.2.1.1.1. The VOF Model
    18.2.1.1.2. The Mixture Model
    18.2.1.1.3. The Eulerian Model
    18.2.2. Model Comparisons
    18.2.2.1. Detailed Guidelines
    18.2.2.1.1. The Effect of Particulate Loading
    18.2.2.1.2. The Significance of the Stokes Number
    18.2.2.1.2.1. Examples
    18.2.2.1.3. Other Considerations
    18.2.3. Time Schemes in Multiphase Flow
    18.2.4. Stability and Convergence
    18.3. Volume of Fluid (VOF) Model Theory
    18.3.1. Overview of the VOF Model
    18.3.2. Limitations of the VOF Model
    18.3.3. Steady-State and Transient VOF Calculations
    18.3.4. Volume Fraction Equation
    18.3.4.1. The Implicit Formulation
    18.3.4.2. The Explicit Formulation
    18.3.4.3. Interpolation Near the Interface
    18.3.4.3.1. The Geometric Reconstruction Scheme
    18.3.4.3.2. The Donor-Acceptor Scheme
    18.3.4.3.3. The Compressive Interface Capturing Scheme for Arbitrary Meshes (CICSAM)
    18.3.4.3.4. The Compressive Scheme and Interface-Model-based Variants
    18.3.4.3.5. Bounded Gradient Maximization (BGM)
    18.3.5. Material Properties
    18.3.6. Momentum Equation
    18.3.7. Energy Equation
    18.3.8. Additional Scalar Equations
    18.3.9. Surface Tension and Adhesion
    18.3.9.1. Surface Tension
    18.3.9.1.1. The Continuum Surface Force Model
    18.3.9.1.2. The Continuum Surface Stress Model
    18.3.9.1.3. Comparing the CSS and CSF Methods
    18.3.9.1.4. When Surface Tension Effects Are Important
    18.3.9.2. Wall Adhesion
    18.3.9.3. Jump Adhesion
    18.3.10. Open Channel Flow
    18.3.10.1. Upstream Boundary Conditions
    18.3.10.1.1. Pressure Inlet
    18.3.10.1.2. Mass Flow Rate
    18.3.10.1.3. Volume Fraction Specification
    18.3.10.2. Downstream Boundary Conditions
    18.3.10.2.1. Pressure Outlet
    18.3.10.2.2. Outflow Boundary
    18.3.10.2.3. Backflow Volume Fraction Specification
    18.3.10.3. Numerical Beach Treatment
    18.3.11. Open Channel Wave Boundary Conditions
    18.3.11.1. Airy Wave Theory
    18.3.11.2. Stokes Wave Theories
    18.3.11.3. Cnoidal/Solitary Wave Theory
    18.3.11.4. Choosing a Wave Theory
    18.3.11.5. Superposition of Waves
    18.3.11.6. Modeling of Random Waves Using Wave Spectrum
    18.3.11.6.1. Definitions
    18.3.11.6.2. Wave Spectrum Implementation Theory
    18.3.11.6.2.1. Long-Crested Random Waves (Unidirectional)
    18.3.11.6.2.1.1. Pierson-Moskowitz Spectrum
    18.3.11.6.2.1.2. JONSWAP Spectrum
    18.3.11.6.2.1.3. TMA Spectrum
    18.3.11.6.2.2. Short-Crested Random Waves (Multi-Directional)
    18.3.11.6.2.2.1. Cosine-2s Power Function (Frequency Independent)
    18.3.11.6.2.2.2. Hyperbolic Function (Frequency Dependent)
    18.3.11.6.2.3. Superposition of Individual Wave Components Using the Wave Spectrum
    18.3.11.6.3. Choosing a Wave Spectrum and Inputs
    18.3.11.7. Nomenclature for Open Channel Waves
    18.3.12. Coupled Level-Set and VOF Model
    18.3.12.1. Theory
    18.3.12.1.1. Surface Tension Force
    18.3.12.1.2. Re-initialization of the Level-set Function via the Geometrical Method
    18.3.12.2. Limitations
    18.4. Mixture Model Theory
    18.4.1. Overview
    18.4.2. Limitations of the Mixture Model
    18.4.3. Continuity Equation
    18.4.4. Momentum Equation
    18.4.5. Energy Equation
    18.4.6. Relative (Slip) Velocity and the Drift Velocity
    18.4.7. Volume Fraction Equation for the Secondary Phases
    18.4.8. Granular Properties
    18.4.8.1. Collisional Viscosity
    18.4.8.2. Kinetic Viscosity
    18.4.9. Granular Temperature
    18.4.10. Solids Pressure
    18.4.11. Interfacial Area Concentration
    18.4.11.1. Transport Equation Based Models
    18.4.11.1.1. Hibiki-Ishii Model
    18.4.11.1.2. Ishii-Kim Model
    18.4.11.1.3. Yao-Morel Model
    18.4.11.2. Algebraic Models
    18.5. Eulerian Model Theory
    18.5.1. Overview of the Eulerian Model
    18.5.2. Limitations of the Eulerian Model
    18.5.3. Volume Fraction Equation
    18.5.4. Conservation Equations
    18.5.4.1. Equations in General Form
    18.5.4.1.1. Conservation of Mass
    18.5.4.1.2. Conservation of Momentum
    18.5.4.1.3. Conservation of Energy
    18.5.4.2. Equations Solved by ANSYS Fluent
    18.5.4.2.1. Continuity Equation
    18.5.4.2.2. Fluid-Fluid Momentum Equations
    18.5.4.2.3. Fluid-Solid Momentum Equations
    18.5.4.2.4. Conservation of Energy
    18.5.5. Interfacial Area Concentration
    18.5.6. Interphase Exchange Coefficients
    18.5.6.1. Fluid-Fluid Exchange Coefficient
    18.5.6.1.1. Schiller and Naumann Model
    18.5.6.1.2. Morsi and Alexander Model
    18.5.6.1.3. Symmetric Model
    18.5.6.1.4. Grace et al. Model
    18.5.6.1.5. Tomiyama et al. Model
    18.5.6.1.6. Ishii Model
    18.5.6.1.7. Ishii-Zuber Drag Model
    18.5.6.1.8. Universal Drag Laws for Bubble-Liquid and Droplet-Gas Flows
    18.5.6.1.8.1. Bubble-Liquid Flow
    18.5.6.1.8.2. Droplet-Gas Flow
    18.5.6.2. Fluid-Solid Exchange Coefficient
    18.5.6.3. Solid-Solid Exchange Coefficient
    18.5.6.4. Drag Modification
    18.5.6.4.1. Brucato et al. Correlation
    18.5.7. Lift Force
    18.5.7.1. Lift Coefficient Models
    18.5.7.1.1. Moraga Lift Force Model
    18.5.7.1.2. Saffman-Mei Lift Force Model
    18.5.7.1.3. Legendre-Magnaudet Lift Force Model
    18.5.7.1.4. Tomiyama Lift Force Model
    18.5.8. Wall Lubrication Force
    18.5.8.1. Wall Lubrication Models
    18.5.8.1.1. Antal et al. Model
    18.5.8.1.2. Tomiyama Model
    18.5.8.1.3. Frank Model
    18.5.8.1.4. Hosokawa Model
    18.5.9. Turbulent Dispersion Force
    18.5.9.1. Models for Turbulent Dispersion Force
    18.5.9.1.1. Lopez de Bertodano Model
    18.5.9.1.2. Simonin Model
    18.5.9.1.3. Burns et al. Model
    18.5.9.1.4. Diffusion in VOF Model
    18.5.9.2. Limiting Functions for the Turbulent Dispersion Force
    18.5.10. Virtual Mass Force
    18.5.11. Solids Pressure
    18.5.11.1. Radial Distribution Function
    18.5.12. Maximum Packing Limit in Binary Mixtures
    18.5.13. Solids Shear Stresses
    18.5.13.1. Collisional Viscosity
    18.5.13.2. Kinetic Viscosity
    18.5.13.3. Bulk Viscosity
    18.5.13.4. Frictional Viscosity
    18.5.14. Granular Temperature
    18.5.15. Description of Heat Transfer
    18.5.15.1. The Heat Exchange Coefficient
    18.5.15.1.1. Constant
    18.5.15.1.2. Nusselt Number
    18.5.15.1.3. Ranz-Marshall Model
    18.5.15.1.4. Tomiyama Model
    18.5.15.1.5. Hughmark Model
    18.5.15.1.6. Gunn Model
    18.5.15.1.7. Two-Resistance Model
    18.5.15.1.8. Fixed To Saturation Temperature
    18.5.15.1.9. User Defined
    18.5.16. Turbulence Models
    18.5.16.1. k- ε Turbulence Models
    18.5.16.1.1. k- ε Mixture Turbulence Model
    18.5.16.1.2. k- ε Dispersed Turbulence Model
    18.5.16.1.2.1. Assumptions
    18.5.16.1.2.2. Turbulence in the Continuous Phase
    18.5.16.1.2.3. Turbulence in the Dispersed Phase
    18.5.16.1.3. k- ε Turbulence Model for Each Phase
    18.5.16.1.3.1. Transport Equations
    18.5.16.2. RSM Turbulence Models
    18.5.16.2.1. RSM Dispersed Turbulence Model
    18.5.16.2.2. RSM Mixture Turbulence Model
    18.5.16.3. Turbulence Interaction Models
    18.5.16.3.1. Simonin et al.
    18.5.16.3.1.1. Formulation in Dispersed Turbulence Models
    18.5.16.3.1.1.1. Continuous Phase
    18.5.16.3.1.1.2. Dispersed Phases
    18.5.16.3.1.2. Formulation in Per Phase Turbulence Models
    18.5.16.3.2. Troshko-Hassan
    18.5.16.3.2.1. Troshko-Hassan Formulation in Mixture Turbulence Models
    18.5.16.3.2.2. Troshko-Hassan Formulation in Dispersed Turbulence Models
    18.5.16.3.2.2.1. Continuous Phase
    18.5.16.3.2.2.2. Dispersed Phases
    18.5.16.3.2.3. Troshko-Hassan Formulation in Per-Phase Turbulence Models
    18.5.16.3.2.3.1. Continuous Phase
    18.5.16.3.2.3.2. Dispersed Phases
    18.5.16.3.3. Sato
    18.5.16.3.4. None
    18.5.17. Solution Method in ANSYS Fluent
    18.5.17.1. The Pressure-Correction Equation
    18.5.17.2. Volume Fractions
    18.5.18. Algebraic Interfacial Area Density (AIAD) Model
    18.5.18.1. Modeling Interfacial Area
    18.5.18.2. Modeling Free-Surface Drag
    18.5.18.3. Modeling Sub-grid Wave Turbulence Contribution (SWT)
    18.5.18.4. Modeling Entrainment-Absorption
    18.5.19. Dense Discrete Phase Model
    18.5.19.1. Limitations
    18.5.19.2. Granular Temperature
    18.5.20. Multi-Fluid VOF Model
    18.5.21. Wall Boiling Models
    18.5.21.1. Overview
    18.5.21.2. RPI Model
    18.5.21.3. Non-equilibrium Subcooled Boiling
    18.5.21.4. Critical Heat Flux
    18.5.21.4.1. Wall Heat Flux Partition
    18.5.21.4.2. Flow Regime Transition
    18.5.21.5. Interfacial Momentum Transfer
    18.5.21.5.1. Interfacial Area
    18.5.21.5.2. Bubble and Droplet Diameters
    18.5.21.5.2.1. Bubble Diameter
    18.5.21.5.2.2. Droplet Diameter
    18.5.21.5.3. Interfacial Drag Force
    18.5.21.5.4. Interfacial Lift Force
    18.5.21.5.5. Turbulent Dispersion Force
    18.5.21.5.6. Wall Lubrication Force
    18.5.21.5.7. Virtual Mass Force
    18.5.21.6. Interfacial Heat Transfer
    18.5.21.6.1. Interface to Liquid Heat Transfer
    18.5.21.6.2. Interface to Vapor Heat Transfer
    18.5.21.7. Mass Transfer
    18.5.21.7.1. Mass Transfer From the Wall to Vapor
    18.5.21.7.2. Interfacial Mass Transfer
    18.5.21.8. Turbulence Interactions
    18.6. Wet Steam Model Theory
    18.6.1. Overview of the Wet Steam Model
    18.6.2. Limitations of the Wet Steam Model
    18.6.3. Wet Steam Flow Equations
    18.6.4. Phase Change Model
    18.6.5. Built-in Thermodynamic Wet Steam Properties
    18.6.5.1. Equation of State
    18.6.5.2. Saturated Vapor Line
    18.6.5.3. Saturated Liquid Line
    18.6.5.4. Mixture Properties
    18.7. Modeling Mass Transfer in Multiphase Flows
    18.7.1. Source Terms due to Mass Transfer
    18.7.1.1. Mass Equation
    18.7.1.2. Momentum Equation
    18.7.1.3. Energy Equation
    18.7.1.4. Species Equation
    18.7.1.5. Other Scalar Equations
    18.7.2. Unidirectional Constant Rate Mass Transfer
    18.7.3. UDF-Prescribed Mass Transfer
    18.7.4. Cavitation Models
    18.7.4.1. Limitations of the Cavitation Models
    18.7.4.1.1. Limitations of Cavitation with the VOF Model
    18.7.4.2. Vapor Transport Equation
    18.7.4.3. Bubble Dynamics Consideration
    18.7.4.4. Singhal et al. Model
    18.7.4.5. Zwart-Gerber-Belamri Model
    18.7.4.6. Schnerr and Sauer Model
    18.7.4.7. Turbulence Factor
    18.7.4.8. Additional Guidelines for the Cavitation Models
    18.7.4.9. Extended Cavitation Model Capabilities
    18.7.4.9.1. Multiphase Cavitation Models
    18.7.4.9.2. Multiphase Species Transport Cavitation Model
    18.7.5. Evaporation-Condensation Model
    18.7.5.1. Lee Model
    18.7.5.2. Thermal Phase Change Model
    18.7.6. Semi-Mechanistic Boiling Model
    18.7.7. Interphase Species Mass Transfer
    18.7.7.1. Modeling Approach
    18.7.7.2. Equilibrium Models
    18.7.7.2.1. Raoult’s Law
    18.7.7.2.2. Henry’s Law
    18.7.7.2.3. Equilibrium Ratio
    18.7.7.3. Mass Transfer Coefficient Models
    18.7.7.3.1. Constant
    18.7.7.3.2. Sherwood Number
    18.7.7.3.3. Ranz-Marshall Model
    18.7.7.3.4. Hughmark Model
    18.7.7.3.5. User-Defined
    18.8. Modeling Species Transport in Multiphase Flows
    18.8.1. Limitations
    18.8.2. Mass and Momentum Transfer with Multiphase Species Transport
    18.8.2.1. Source Terms Due to Heterogeneous Reactions
    18.8.2.1.1. Mass Transfer
    18.8.2.1.2. Momentum Transfer
    18.8.2.1.3. Species Transfer
    18.8.2.1.4. Heat Transfer
    18.8.3. The Stiff Chemistry Solver
    18.8.4. Heterogeneous Phase Interaction
    Chapter 19: Population Balance Model
    19.1. Introduction
    19.1.1. The Discrete Method
    19.1.2. The Inhomogeneous Discrete Method
    19.1.3. The Standard Method of Moments
    19.1.4. The Quadrature Method of Moments
    19.2. Population Balance Model Theory
    19.2.1. The Particle State Vector
    19.2.2. The Population Balance Equation (PBE)
    19.2.2.1. Particle Growth
    19.2.2.2. Particle Birth and Death Due to Breakage and Aggregation
    19.2.2.2.1. Breakage
    19.2.2.2.2. Luo and Lehr Breakage Kernels
    19.2.2.2.3. Ghadiri Breakage Kernels
    19.2.2.2.4. Laakkonen Breakage Kernels
    19.2.2.2.5. Parabolic PDF
    19.2.2.2.6. Generalized PDF
    19.2.2.2.7. Aggregation
    19.2.2.2.8. Luo Aggregation Kernel
    19.2.2.2.9. Free Molecular Aggregation Kernel
    19.2.2.2.10. Turbulent Aggregation Kernel
    19.2.2.2.11. Prince and Blanch Aggregation Kernel
    19.2.2.3. Particle Birth by Nucleation
    19.2.3. Solution Methods
    19.2.3.1. The Discrete Method and the Inhomogeneous Discrete Method
    19.2.3.1.1. Numerical Method
    19.2.3.1.2. Breakage Formulations for the Discrete Method
    19.2.3.2. The Standard Method of Moments (SMM)
    19.2.3.2.1. Numerical Method
    19.2.3.3. The Quadrature Method of Moments (QMOM)
    19.2.3.3.1. Numerical Method
    19.2.3.4. The Direct Quadrature Method of Moments (DQMOM)
    19.2.3.4.1. Numerical Method
    19.2.4. Population Balance Statistics
    19.2.4.1. Reconstructing the Particle Size Distribution from Moments
    19.2.4.2. The Log-Normal Distribution
    Chapter 20: Solidification and Melting
    20.1. Overview
    20.2. Limitations
    20.3. Introduction
    20.4. Energy Equation
    20.5. Momentum Equations
    20.6. Turbulence Equations
    20.7. Species Equations
    20.8. Back Diffusion
    20.9. Pull Velocity for Continuous Casting
    20.10. Contact Resistance at Walls
    20.11. Thermal and Solutal Buoyancy
    Chapter 21: The Structural Model for Intrinsic Fluid-Structure Interaction (FSI)
    21.1. Limitations
    21.2. The FSI Model
    21.3. Intrinsic FSI
    21.4. Linear Elasticity
    21.4.1. Equations
    21.4.1.1. Linear Isotropic and Isothermal Elasticity
    21.4.1.2. Evaluation of the von Mises Stress
    21.4.2. Finite Element Representation
    21.4.2.1. Construction of the Matrix of the System
    21.4.2.2. Dynamic Structural Systems
    21.4.2.2.1. The Newmark Method
    21.4.2.2.2. Backward Euler Method
    21.5. Nonlinear Elasticity
    21.5.1. Finite Element Geometric Nonlinearity
    21.5.2. Finite Element Nonlinear Discretization
    21.5.3. Constitutive Equations
    21.5.4. The Transient Scheme for the Nonlinear System
    Chapter 22: Eulerian Wall Films
    22.1. Introduction
    22.2. Mass, Momentum, and Energy Conservation Equations for Wall Film
    22.2.1. Film Sub-Models
    22.2.1.1. DPM Collection
    22.2.1.2. Particle-Wall Interaction
    22.2.1.3. Film Separation
    22.2.1.3.1. Separation Criteria
    22.2.1.3.1.1. Foucart Separation
    22.2.1.3.1.2. O’Rourke Separation
    22.2.1.3.1.3. Friedrich Separation
    22.2.1.4. Film Stripping
    22.2.1.5. Secondary Phase Accretion
    22.2.1.6. Coupling of Wall Film with Mixture Species Transport
    22.2.2. Partial Wetting Effect
    22.2.3. Boundary Conditions
    22.2.4. Obtaining Film Velocity Without Solving the Momentum Equations
    22.2.4.1. Shear-Driven Film Velocity
    22.2.4.2. Gravity-Driven Film Velocity
    22.3. Passive Scalar Equation for Wall Film
    22.4. Numerical Schemes and Solution Algorithm
    22.4.1. Temporal Differencing Schemes
    22.4.1.1. First-Order Explicit Method
    22.4.1.2. First-Order Implicit Method
    22.4.1.3. Second-Order Implicit Method
    22.4.2. Spatial Differencing Schemes
    22.4.3. Solution Algorithm
    22.4.3.1. Steady Flow
    22.4.3.2. Transient Flow
    22.4.4. Coupled Solution Approach
    Chapter 23: Electric Potential and Lithium-ion Battery Model
    23.1. Electric Potential
    23.1.1. Overview
    23.1.2. Electric Potential Equation
    23.1.3. Energy Equation Source Term
    23.2. Lithium-ion Battery Model
    23.2.1. Overview
    23.2.2. Lithium-ion Battery Model Theory
    Chapter 24: Modeling Batteries
    24.1. Single-Potential Empirical Battery Model Theory
    24.1.1. Introduction
    24.1.2. Computation of the Electric Potential and Current Density
    24.1.3. Thermal and Electrical Coupling
    24.2. Dual-Potential MSMD and Circuit Network Battery Models
    24.2.1. Battery Solution Methods
    24.2.1.1. MSMD Solution Method
    24.2.1.2. Circuit Network Solution Method
    24.2.2. NTGK Model
    24.2.3. ECM Model
    24.2.4. Newman’s P2D Model
    24.2.5. MSMD Solution Method: Coupling Between CFD and Submodels
    24.2.6. Simulating Battery Pack Using the MSMD Solution Method
    24.2.7. Reduced Order Solution Method (ROM)
    24.2.8. External and Internal Electric Short-Circuit Treatment
    24.2.9. Thermal Abuse Model
    Chapter 25: Modeling Fuel Cells
    25.1. PEMFC Model Theory
    25.1.1. Introduction
    25.1.2. Electrochemistry Modeling
    25.1.2.1. The Cathode Particle Model
    25.1.3. Current and Mass Conservation
    25.1.4. Water Transport and Mass Transfer in PEMFC
    25.1.4.1. The Dissolved Phase Model
    25.1.4.2. The Liquid Phase Model
    25.1.4.2.1. Liquid Water Transport Equation in the Porous Electrode and the Membrane
    25.1.4.2.2. Liquid Water Transport Equation in Gas Channels
    25.1.5. Heat Source
    25.1.6. Properties
    25.1.7. Transient Simulations
    25.1.8. Leakage Current (Cross-Over Current)
    25.1.9. Zones where User-Defined Scalars are Solved
    25.2. Fuel Cell and Electrolysis Model Theory
    25.2.1. Introduction
    25.2.1.1. Introduction to PEMFC
    25.2.1.2. Introduction to SOFC
    25.2.1.3. Introduction to Electrolysis
    25.2.2. Electrochemistry Modeling
    25.2.3. Current and Mass Conservation
    25.2.4. Heat Source
    25.2.5. Liquid Water Formation, Transport, and its Effects (PEMFC Only)
    25.2.6. Properties
    25.2.7. Transient Simulations
    25.2.8. Leakage Current (Cross-Over Current)
    25.3. SOFC Fuel Cell With Unresolved Electrolyte Model Theory
    25.3.1. Introduction
    25.3.2. The SOFC With Unresolved Electrolyte Modeling Strategy
    25.3.3. Modeling Fluid Flow, Heat Transfer, and Mass Transfer
    25.3.4. Modeling Current Transport and the Potential Field
    25.3.4.1. Cell Potential
    25.3.4.2. Activation Overpotential
    25.3.4.3. Treatment of the Energy Equation at the Electrolyte Interface
    25.3.4.4. Treatment of the Energy Equation in the Conducting Regions
    25.3.5. Modeling Reactions
    25.3.5.1. Modeling Electrochemical Reactions
    25.3.5.2. Modeling CO Electrochemistry
    Chapter 26: Modeling Magnetohydrodynamics
    26.1. Introduction
    26.2. Magnetic Induction Method
    26.2.1. Case 1: Externally Imposed Magnetic Field Generated in Non-conducting Media
    26.2.2. Case 2: Externally Imposed Magnetic Field Generated in Conducting Media
    26.3. Electric Potential Method
    Chapter 27: Modeling Continuous Fibers
    27.1. Introduction
    27.2. Governing Equations of Fiber Flow
    27.3. Discretization of the Fiber Equations
    27.3.1. Under-Relaxation
    27.4. Numerical Solution Algorithm of Fiber Equations
    27.5. Residuals of Fiber Equations
    27.6. Coupling Between Fibers and the Surrounding Fluid
    27.6.1. Momentum Exchange
    27.6.2. Mass Exchange
    27.6.3. Heat Exchange
    27.6.4. Radiation Exchange
    27.6.5. Under-Relaxation of the Fiber Exchange Terms
    27.7. Fiber Grid Generation
    27.8. Correlations for Momentum, Heat and Mass Transfer
    27.8.1. Drag Coefficient
    27.8.2. Heat Transfer Coefficient
    27.8.3. Mass Transfer Coefficient
    27.9. Fiber Properties
    27.9.1. Fiber Viscosity
    27.9.1.1. Melt Spinning
    27.9.1.2. Dry Spinning
    27.9.2. Vapor-Liquid Equilibrium
    27.9.3. Latent Heat of Vaporization
    27.9.4. Emissivity
    27.10. Solution Strategies
    Chapter 28: Solver Theory
    28.1. Overview of Flow Solvers
    28.1.1. Pressure-Based Solver
    28.1.1.1. The Pressure-Based Segregated Algorithm
    28.1.1.2. The Pressure-Based Coupled Algorithm
    28.1.2. Density-Based Solver
    28.2. General Scalar Transport Equation: Discretization and Solution
    28.2.1. Solving the Linear System
    28.3. Discretization
    28.3.1. Spatial Discretization
    28.3.1.1. First-Order Upwind Scheme
    28.3.1.2. Second-Order Upwind Scheme
    28.3.1.3. First- to Higher-Order Blending
    28.3.1.4. Central-Differencing Scheme
    28.3.1.5. Bounded Central Differencing Scheme
    28.3.1.6. QUICK Scheme
    28.3.1.7. Third-Order MUSCL Scheme
    28.3.1.8. Modified HRIC Scheme
    28.3.1.9. High Order Term Relaxation
    28.3.2. Temporal Discretization
    28.3.2.1. Implicit Time Integration
    28.3.2.2. Bounded Second-Order Implicit Time Integration
    28.3.2.2.1. Limitations
    28.3.2.3. Second-Order Time Integration Using a Variable Time Step Size
    28.3.2.4. Explicit Time Integration
    28.3.3. Evaluation of Gradients and Derivatives
    28.3.3.1. Green-Gauss Theorem
    28.3.3.2. Green-Gauss Cell-Based Gradient Evaluation
    28.3.3.3. Green-Gauss Node-Based Gradient Evaluation
    28.3.3.4. Least Squares Cell-Based Gradient Evaluation
    28.3.4. Gradient Limiters
    28.3.4.1. Standard Limiter
    28.3.4.2. Multidimensional Limiter
    28.3.4.3. Differentiable Limiter
    28.4. Pressure-Based Solver
    28.4.1. Discretization of the Momentum Equation
    28.4.1.1. Pressure Interpolation Schemes
    28.4.2. Discretization of the Continuity Equation
    28.4.2.1. Density Interpolation Schemes
    28.4.3. Pressure-Velocity Coupling
    28.4.3.1. Segregated Algorithms
    28.4.3.1.1. SIMPLE
    28.4.3.1.2. SIMPLEC
    28.4.3.1.2.1. Skewness Correction
    28.4.3.1.3. PISO
    28.4.3.1.3.1. Neighbor Correction
    28.4.3.1.3.2. Skewness Correction
    28.4.3.1.3.3. Skewness - Neighbor Coupling
    28.4.3.2. Fractional-Step Method (FSM)
    28.4.3.3. Coupled Algorithm
    28.4.3.3.1. Limitation
    28.4.4. Steady-State Iterative Algorithm
    28.4.4.1. Under-Relaxation of Variables
    28.4.4.2. Under-Relaxation of Equations
    28.4.5. Time-Advancement Algorithm
    28.4.5.1. Iterative Time-Advancement Scheme
    28.4.5.1.1. The Frozen Flux Formulation
    28.4.5.2. Non-Iterative Time-Advancement Scheme
    28.5. Density-Based Solver
    28.5.1. Governing Equations in Vector Form
    28.5.2. Preconditioning
    28.5.3. Convective Fluxes
    28.5.3.1. Roe Flux-Difference Splitting Scheme
    28.5.3.2. AUSM+ Scheme
    28.5.3.3. Low Diffusion Roe Flux Difference Splitting Scheme
    28.5.4. Steady-State Flow Solution Methods
    28.5.4.1. Explicit Formulation
    28.5.4.1.1. Implicit Residual Smoothing
    28.5.4.2. Implicit Formulation
    28.5.4.2.1. Convergence Acceleration for Stretched Meshes
    28.5.5. Unsteady Flows Solution Methods
    28.5.5.1. Explicit Time Stepping
    28.5.5.2. Implicit Time Stepping (Dual-Time Formulation)
    28.6. Pseudo Transient Under-Relaxation
    28.6.1. Automatic Pseudo Transient Time Step
    28.7. Multigrid Method
    28.7.1. Approach
    28.7.1.1. The Need for Multigrid
    28.7.1.2. The Basic Concept in Multigrid
    28.7.1.3. Restriction and Prolongation
    28.7.1.4. Unstructured Multigrid
    28.7.2. Multigrid Cycles
    28.7.2.1. The V and W Cycles
    28.7.3. Algebraic Multigrid (AMG)
    28.7.3.1. AMG Restriction and Prolongation Operators
    28.7.3.2. AMG Coarse Level Operator
    28.7.3.3. The F Cycle
    28.7.3.4. The Flexible Cycle
    28.7.3.4.1. The Residual Reduction Rate Criteria
    28.7.3.4.2. The Termination Criteria
    28.7.3.5. The Coupled and Scalar AMG Solvers
    28.7.3.5.1. Gauss-Seidel
    28.7.3.5.2. Incomplete Lower Upper (ILU)
    28.7.4. Full-Approximation Storage (FAS) Multigrid
    28.7.4.1. FAS Restriction and Prolongation Operators
    28.7.4.2. FAS Coarse Level Operator
    28.8. Hybrid Initialization
    28.9. Full Multigrid (FMG) Initialization
    28.9.1. Overview of FMG Initialization
    28.9.2. Limitations of FMG Initialization
    Chapter 29: Adapting the Mesh
    29.1. Adaption Process
    29.1.1. Hanging Node Adaption
    29.1.2. Polyhedral Unstructured Mesh Adaption
    29.2. Anisotropic Adaption
    29.3. Geometry-Based Adaption
    29.3.1. Geometry-Based Adaption Approach
    29.3.1.1. Node Projection
    29.3.1.2. Example of Geometry-Based Adaption
    Chapter 30: Reporting Alphanumeric Data
    30.1. Fluxes Through Boundaries
    30.2. Forces on Boundaries
    30.2.1. Computing Forces, Moments, and the Center of Pressure
    30.3. Surface Integration
    30.3.1. Computing Surface Integrals
    30.3.1.1. Area
    30.3.1.2. Integral
    30.3.1.3. Area-Weighted Average
    30.3.1.4. Custom Vector Based Flux
    30.3.1.5. Custom Vector Flux
    30.3.1.6. Custom Vector Weighted Average
    30.3.1.7. Flow Rate
    30.3.1.8. Mass Flow Rate
    30.3.1.9. Mass-Weighted Average
    30.3.1.10. Sum of Field Variable
    30.3.1.11. Facet Average
    30.3.1.12. Facet Minimum
    30.3.1.13. Facet Maximum
    30.3.1.14. Vertex Average
    30.3.1.15. Vertex Minimum
    30.3.1.16. Vertex Maximum
    30.3.1.17. Standard-Deviation
    30.3.1.18. Uniformity Index
    30.3.1.19. Volume Flow Rate
    30.4. Volume Integration
    30.4.1. Computing Volume Integrals
    30.4.1.1. Volume
    30.4.1.2. Sum
    30.4.1.3. Sum*2Pi
    30.4.1.4. Volume Integral
    30.4.1.5. Volume-Weighted Average
    30.4.1.6. Mass-Weighted Integral
    30.4.1.7. Mass
    30.4.1.8. Mass-Weighted Average
    Appendix A. Nomenclature
    Bibliography
    ANSYS Fluent Theory Guide ANSYS, Inc. Southpointe 2600 ANSYS Drive Canonsburg, PA 15317 ansysinfo@ansys.com http://www.ansys.com (T) 724-746-3304 (F) 724-514-9494 Release 2020 R1 January 2020 ANSYS, Inc. and ANSYS Europe, Ltd. are UL registered ISO 9001: 2015 companies.
    Copyright and Trademark Information © 2020 ANSYS, Inc. Unauthorized use, distribution or duplication is prohibited. ANSYS, ANSYS Workbench, AUTODYN, CFX, FLUENT and any and all ANSYS, Inc. brand, product, service and feature names, logos and slogans are registered trademarks or trademarks of ANSYS, Inc. or its subsidiaries located in the United States or other countries. ICEM CFD is a trademark used by ANSYS, Inc. under license. CFX is a trademark of Sony Corporation in Japan. All other brand, product, service and feature names or trademarks are the property of their respective owners. FLEXlm and FLEXnet are trademarks of Flexera Software LLC. Disclaimer Notice THIS ANSYS SOFTWARE PRODUCT AND PROGRAM DOCUMENTATION INCLUDE TRADE SECRETS AND ARE CONFID- ENTIAL AND PROPRIETARY PRODUCTS OF ANSYS, INC., ITS SUBSIDIARIES, OR LICENSORS. The software products and documentation are furnished by ANSYS, Inc., its subsidiaries, or affiliates under a software license agreement that contains provisions concerning non-disclosure, copying, length and nature of use, compliance with exporting laws, warranties, disclaimers, limitations of liability, and remedies, and other provisions. The software products and documentation may be used, disclosed, transferred, or copied only in accordance with the terms and conditions of that software license agreement. ANSYS, Inc. and ANSYS Europe, Ltd. are UL registered ISO 9001: 2015 companies. U.S. Government Rights For U.S. Government users, except as specifically granted by the ANSYS, Inc. software license agreement, the use, duplication, or disclosure by the United States Government is subject to restrictions stated in the ANSYS, Inc. software license agreement and FAR 12.212 (for non-DOD licenses). Third-Party Software See the legal information in the product help files for the complete Legal Notice for ANSYS proprietary software and third-party software. If you are unable to access the Legal Notice, contact ANSYS, Inc. Published in the U.S.A.
    Table of Contents Using This Manual ................................................................................................................................... xxxix 1. The Contents of This Manual .......................................................................................................... xxxix 2.Typographical Conventions ................................................................................................................. xl 3. Mathematical Conventions ............................................................................................................... xliii 1. Basic Fluid Flow ....................................................................................................................................... 1 1.1. Overview of Physical Models in ANSYS Fluent .................................................................................... 1 1.2. Continuity and Momentum Equations ............................................................................................... 2 1.2.1. The Mass Conservation Equation .............................................................................................. 2 1.2.2. Momentum Conservation Equations ........................................................................................ 3 1.3. User-Defined Scalar (UDS) Transport Equations .................................................................................. 4 1.3.1. Single Phase Flow .................................................................................................................... 4 1.3.2. Multiphase Flow ....................................................................................................................... 5 1.4. Periodic Flows .................................................................................................................................. 6 1.4.1. Overview ................................................................................................................................. 6 1.4.2. Limitations ............................................................................................................................... 7 1.4.3. Physics of Periodic Flows .......................................................................................................... 7 1.4.3.1. Definition of the Periodic Velocity .................................................................................... 7 1.4.3.2. Definition of the Streamwise-Periodic Pressure ................................................................ 8 1.5. Swirling and Rotating Flows .............................................................................................................. 8 1.5.1. Overview of Swirling and Rotating Flows .................................................................................. 9 1.5.1.1. Axisymmetric Flows with Swirl or Rotation ....................................................................... 9 1.5.1.1.1. Momentum Conservation Equation for Swirl Velocity ............................................. 10 1.5.1.2.Three-Dimensional Swirling Flows .................................................................................. 10 1.5.1.3. Flows Requiring a Moving Reference Frame ................................................................... 11 1.5.2. Physics of Swirling and Rotating Flows .................................................................................... 11 1.6. Compressible Flows ........................................................................................................................ 12 1.6.1. When to Use the Compressible Flow Model ............................................................................ 13 1.6.2. Physics of Compressible Flows ................................................................................................ 14 1.6.2.1. Basic Equations for Compressible Flows ......................................................................... 14 1.6.2.2. The Compressible Form of the Gas Law .......................................................................... 15 1.7. Inviscid Flows ................................................................................................................................. 15 1.7.1. Euler Equations ...................................................................................................................... 15 1.7.1.1.The Mass Conservation Equation .................................................................................... 16 1.7.1.2. Momentum Conservation Equations .............................................................................. 16 1.7.1.3. Energy Conservation Equation ....................................................................................... 16 2. Flows with Moving Reference Frames ................................................................................................... 17 2.1. Introduction ................................................................................................................................... 17 2.2. Flow in a Moving Reference Frame .................................................................................................. 19 2.2.1. Equations for a Moving Reference Frame ................................................................................ 19 2.2.1.1. Relative Velocity Formulation ......................................................................................... 20 2.2.1.2. Absolute Velocity Formulation ....................................................................................... 21 2.2.1.3. Relative Specification of the Reference Frame Motion ..................................................... 21 2.3. Flow in Multiple Reference Frames .................................................................................................. 22 2.3.1. The Multiple Reference Frame Model ...................................................................................... 22 2.3.1.1. Overview ....................................................................................................................... 22 2.3.1.2. Examples ....................................................................................................................... 23 2.3.1.3. The MRF Interface Formulation ...................................................................................... 24 2.3.1.3.1. Interface Treatment: Relative Velocity Formulation ................................................. 24 2.3.1.3.2. Interface Treatment: Absolute Velocity Formulation ............................................... 25 2.3.2. The Mixing Plane Model ......................................................................................................... 25 Release 2020 R1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. iii
    Theory Guide 2.3.2.1. Overview ....................................................................................................................... 26 2.3.2.2. Rotor and Stator Domains .............................................................................................. 26 2.3.2.3. The Mixing Plane Concept ............................................................................................. 27 2.3.2.4. Choosing an Averaging Method ..................................................................................... 28 2.3.2.4.1. Area Averaging ..................................................................................................... 28 2.3.2.4.2. Mass Averaging .................................................................................................... 29 2.3.2.4.3. Mixed-Out Averaging ............................................................................................ 29 2.3.2.5. Mixing Plane Algorithm of ANSYS Fluent ........................................................................ 29 2.3.2.6. Mass Conservation ........................................................................................................ 30 2.3.2.7. Swirl Conservation ......................................................................................................... 30 2.3.2.8. Total Enthalpy Conservation .......................................................................................... 31 3. Flows Using Sliding and Dynamic Meshes ............................................................................................ 33 3.1. Introduction ................................................................................................................................... 33 3.2. Dynamic Mesh Theory .................................................................................................................... 35 3.2.1. Conservation Equations ......................................................................................................... 36 3.2.2. Six DOF Solver Theory ............................................................................................................ 37 3.3. Sliding Mesh Theory ....................................................................................................................... 38 4.Turbulence ............................................................................................................................................. 39 4.1. Underlying Principles of Turbulence Modeling ................................................................................. 39 4.1.1. Reynolds (Ensemble) Averaging .............................................................................................. 40 4.1.2. Filtered Navier-Stokes Equations ............................................................................................. 40 4.1.3. Hybrid RANS-LES Formulations ............................................................................................... 41 4.1.4. Boussinesq Approach vs. Reynolds Stress Transport Models ..................................................... 42 4.2. Spalart-Allmaras Model ................................................................................................................... 42 4.2.1. Overview ............................................................................................................................... 43 4.2.2. Transport Equation for the Spalart-Allmaras Model ................................................................. 43 4.2.3. Modeling the Turbulent Viscosity ............................................................................................ 43 4.2.4. Modeling the Turbulent Production ........................................................................................ 44 4.2.5. Modeling the Turbulent Destruction ....................................................................................... 45 4.2.6. Model Constants .................................................................................................................... 45 4.2.7. Wall Boundary Conditions ...................................................................................................... 45 4.2.7.1. Treatment of the Spalart-Allmaras Model for Icing Simulations ....................................... 46 4.2.8. Convective Heat and Mass Transfer Modeling .......................................................................... 46 4.3. Standard, RNG, and Realizable k-ε Models ........................................................................................ 47 4.3.1. Standard k-ε Model ................................................................................................................ 47 4.3.1.1. Overview ....................................................................................................................... 47 4.3.1.2. Transport Equations for the Standard k-ε Model ............................................................. 48 4.3.1.3. Modeling the Turbulent Viscosity ................................................................................... 48 4.3.1.4. Model Constants ........................................................................................................... 48 4.3.2. RNG k-ε Model ....................................................................................................................... 49 4.3.2.1. Overview ....................................................................................................................... 49 4.3.2.2. Transport Equations for the RNG k-ε Model ..................................................................... 49 4.3.2.3. Modeling the Effective Viscosity ..................................................................................... 49 4.3.2.4. RNG Swirl Modification .................................................................................................. 50 4.3.2.5. Calculating the Inverse Effective Prandtl Numbers .......................................................... 50 4.3.2.6. The R-ε Term in the ε Equation ........................................................................................ 51 4.3.2.7. Model Constants ........................................................................................................... 51 4.3.3. Realizable k-ε Model ............................................................................................................... 51 4.3.3.1. Overview ....................................................................................................................... 51 4.3.3.2. Transport Equations for the Realizable k-ε Model ............................................................ 53 4.3.3.3. Modeling the Turbulent Viscosity ................................................................................... 54 4.3.3.4. Model Constants ........................................................................................................... 54 iv Release 2020 R1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates.
    Theory Guide 4.3.4. Modeling Turbulent Production in the k-ε Models ................................................................... 55 4.3.5. Effects of Buoyancy on Turbulence in the k-ε Models ............................................................... 55 4.3.6. Effects of Compressibility on Turbulence in the k-ε Models ...................................................... 56 4.3.7. Convective Heat and Mass Transfer Modeling in the k-ε Models ............................................... 56 4.4. Standard, BSL, and SST k-ω Models ................................................................................................... 58 4.4.1. Standard k-ω Model ............................................................................................................... 58 4.4.1.1. Overview ....................................................................................................................... 58 4.4.1.2. Transport Equations for the Standard k-ω Model ............................................................. 59 4.4.1.3. Modeling the Effective Diffusivity ................................................................................... 59 4.4.1.3.1. Low-Reynolds Number Correction ......................................................................... 59 4.4.1.4. Modeling the Turbulence Production ............................................................................. 60 4.4.1.4.1. Production of k ..................................................................................................... 60 4.4.1.4.2. Production of ω ..................................................................................................... 60 4.4.1.5. Modeling the Turbulence Dissipation ............................................................................. 60 4.4.1.5.1. Dissipation of k ..................................................................................................... 60 4.4.1.5.2. Dissipation of ω ..................................................................................................... 61 4.4.1.5.3. Compressibility Effects .......................................................................................... 61 4.4.1.6. Model Constants ........................................................................................................... 62 4.4.2. Baseline (BSL) k-ω Model ........................................................................................................ 62 4.4.2.1. Overview ....................................................................................................................... 62 4.4.2.2. Transport Equations for the BSL k-ω Model ..................................................................... 62 4.4.2.3. Modeling the Effective Diffusivity ................................................................................... 63 4.4.2.4. Modeling the Turbulence Production ............................................................................. 63 4.4.2.4.1. Production of k ..................................................................................................... 63 4.4.2.4.2. Production of ω ..................................................................................................... 63 4.4.2.5. Modeling the Turbulence Dissipation ............................................................................. 64 4.4.2.5.1. Dissipation of k ..................................................................................................... 64 4.4.2.5.2. Dissipation of ω ..................................................................................................... 64 4.4.2.6. Cross-Diffusion Modification .......................................................................................... 64 4.4.2.7. Model Constants ........................................................................................................... 64 4.4.3. Shear-Stress Transport (SST) k-ω Model ................................................................................... 65 4.4.3.1. Overview ....................................................................................................................... 65 4.4.3.2. Modeling the Turbulent Viscosity ................................................................................... 65 4.4.3.3. Model Constants ........................................................................................................... 65 4.4.3.4. Treatment of the SST Model for Icing Simulations ........................................................... 65 4.4.4. Effects of Buoyancy on Turbulence in the k-ω Models .............................................................. 66 4.4.5. Turbulence Damping .............................................................................................................. 67 4.4.6. Wall Boundary Conditions ...................................................................................................... 68 4.5. Generalized k-ω (GEKO) Model ........................................................................................................ 68 4.5.1. Model Formulation ................................................................................................................. 69 4.5.2. Limitations ............................................................................................................................. 71 4.6. k-kl-ω Transition Model ................................................................................................................... 71 4.6.1. Overview ............................................................................................................................... 71 4.6.2. Transport Equations for the k-kl-ω Model ................................................................................ 71 4.6.2.1. Model Constants ........................................................................................................... 74 4.7. Transition SST Model ....................................................................................................................... 74 4.7.1. Overview ............................................................................................................................... 75 4.7.2. Transport Equations for the Transition SST Model .................................................................... 75 4.7.2.1. Separation-Induced Transition Correction ...................................................................... 77 4.7.2.2. Coupling the Transition Model and SST Transport Equations ........................................... 78 4.7.2.3. Transition SST and Rough Walls ...................................................................................... 78 4.7.3. Mesh Requirements ............................................................................................................... 79 Release 2020 R1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. v
    Theory Guide 4.7.4. Specifying Inlet Turbulence Levels .......................................................................................... 82 4.8. Intermittency Transition Model ....................................................................................................... 83 4.8.1. Overview ............................................................................................................................... 83 4.8.2.Transport Equations for the Intermittency Transition Model ..................................................... 84 4.8.3. Coupling with the Other Models ............................................................................................. 88 4.8.4. Intermittency Transition Model and Rough Walls ..................................................................... 88 4.9. The V2F Model ................................................................................................................................ 88 4.10. Reynolds Stress Model (RSM) ......................................................................................................... 88 4.10.1. Overview ............................................................................................................................. 89 4.10.2. Reynolds Stress Transport Equations ..................................................................................... 90 4.10.3. Modeling Turbulent Diffusive Transport ................................................................................ 90 4.10.4. Modeling the Pressure-Strain Term ....................................................................................... 91 4.10.4.1. Linear Pressure-Strain Model ........................................................................................ 91 4.10.4.2. Low-Re Modifications to the Linear Pressure-Strain Model ............................................ 92 4.10.4.3. Quadratic Pressure-Strain Model .................................................................................. 92 4.10.4.4. Stress-Omega Model ................................................................................................... 93 4.10.4.5. Stress-BSL Model ......................................................................................................... 95 4.10.5. Effects of Buoyancy on Turbulence ........................................................................................ 95 4.10.6. Modeling the Turbulence Kinetic Energy ............................................................................... 95 4.10.7. Modeling the Dissipation Rate .............................................................................................. 95 4.10.8. Modeling the Turbulent Viscosity .......................................................................................... 96 4.10.9. Wall Boundary Conditions .................................................................................................... 96 4.10.10. Convective Heat and Mass Transfer Modeling ...................................................................... 97 4.11. Scale-Adaptive Simulation (SAS) Model ......................................................................................... 98 4.11.1. Overview ............................................................................................................................. 98 4.11.2.Transport Equations for the SST-SAS Model ........................................................................... 99 4.11.3. SAS with Other ω-Based Turbulence Models ........................................................................ 100 4.12. Detached Eddy Simulation (DES) ................................................................................................. 100 4.12.1. Overview ........................................................................................................................... 101 4.12.2. DES with the Spalart-Allmaras Model .................................................................................. 101 4.12.3. DES with the Realizable k-ε Model ....................................................................................... 102 4.12.4. DES with the BSL or SST k-ω Model ...................................................................................... 103 4.12.5. DES with the Transition SST Model ...................................................................................... 103 4.12.6. Improved Delayed Detached Eddy Simulation (IDDES) ........................................................ 104 4.12.6.1. Overview of IDDES ..................................................................................................... 104 4.12.6.2. IDDES Model Formulation .......................................................................................... 104 4.13. Shielded Detached Eddy Simulation (SDES) ................................................................................. 105 4.13.1. Shielding Function ............................................................................................................. 105 4.13.2. LES Mode of SDES .............................................................................................................. 107 4.14. Stress-Blended Eddy Simulation (SBES) ........................................................................................ 108 4.14.1. Stress Blending ................................................................................................................... 108 4.14.2. SDES and SBES Example ..................................................................................................... 109 4.15. Large Eddy Simulation (LES) Model .............................................................................................. 110 4.15.1. Overview ........................................................................................................................... 110 4.15.2. Subgrid-Scale Models ......................................................................................................... 111 4.15.2.1. Smagorinsky-Lilly Model ............................................................................................ 112 4.15.2.2. Dynamic Smagorinsky-Lilly Model .............................................................................. 113 4.15.2.3. Wall-Adapting Local Eddy-Viscosity (WALE) Model ...................................................... 113 4.15.2.4. Algebraic Wall-Modeled LES Model (WMLES) .............................................................. 114 4.15.2.4.1. Algebraic WMLES Model Formulation ................................................................ 115 4.15.2.4.1.1. Reynolds Number Scaling ......................................................................... 115 4.15.2.4.2. Algebraic WMLES S-Omega Model Formulation ................................................. 116 vi Release 2020 R1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates.
    Theory Guide 4.15.2.5. Dynamic Kinetic Energy Subgrid-Scale Model ............................................................. 116 4.15.3. Inlet Boundary Conditions for Scale Resolving Simulations .................................................. 117 4.15.3.1. Vortex Method ........................................................................................................... 118 4.15.3.2. Spectral Synthesizer ................................................................................................... 119 4.15.3.3. Synthetic Turbulence Generator ................................................................................. 120 4.15.3.3.1. Limitations ........................................................................................................ 122 4.16. Embedded Large Eddy Simulation (ELES) ..................................................................................... 123 4.16.1. Overview ........................................................................................................................... 123 4.16.2. Selecting a Model ............................................................................................................... 123 4.16.3. Interfaces Treatment ........................................................................................................... 123 4.16.3.1. RANS-LES Interface .................................................................................................... 124 4.16.3.2. LES-RANS Interface .................................................................................................... 124 4.16.3.3. Internal Interface Without LES Zone ........................................................................... 125 4.16.3.4. Grid Generation Guidelines ........................................................................................ 125 4.17. Near-Wall Treatments for Wall-Bounded Turbulent Flows .............................................................. 126 4.17.1. Overview ........................................................................................................................... 126 4.17.1.1. Wall Functions vs. Near-Wall Model ............................................................................. 127 4.17.1.2. Wall Functions ........................................................................................................... 129 4.17.2. Standard Wall Functions ..................................................................................................... 129 4.17.2.1. Momentum ............................................................................................................... 129 4.17.2.2. Energy ....................................................................................................................... 130 4.17.2.3. Species ...................................................................................................................... 132 4.17.2.4. Turbulence ................................................................................................................ 132 4.17.3. Scalable Wall Functions ....................................................................................................... 133 4.17.4. Non-Equilibrium Wall Functions .......................................................................................... 134 4.17.4.1. Standard Wall Functions vs. Non-Equilibrium Wall Functions ....................................... 135 4.17.4.2. Limitations of the Wall Function Approach ................................................................. 135 4.17.5. Enhanced Wall Treatment ε-Equation (EWT-ε) ...................................................................... 136 4.17.5.1.Two-Layer Model for Enhanced Wall Treatment ........................................................... 136 4.17.5.2. Enhanced Wall Treatment for Momentum and Energy Equations ................................. 138 4.17.6. Menter-Lechner ε-Equation (ML-ε) ...................................................................................... 140 4.17.6.1. Momentum Equations ............................................................................................... 141 4.17.6.2. k-ε Turbulence Models ............................................................................................... 141 4.17.6.3. Iteration Improvements ............................................................................................. 142 4.17.7. y+-Insensitive Wall Treatment ω-Equation ........................................................................... 142 4.17.8. User-Defined Wall Functions ............................................................................................... 142 4.17.9. LES Near-Wall Treatment ..................................................................................................... 142 4.18. Curvature Correction for the Spalart-Allmaras and Two-Equation Models ..................................... 143 4.19. Production Limiters for Two-Equation Models .............................................................................. 145 4.20. Definition of Turbulence Scales .................................................................................................... 147 4.20.1. RANS and Hybrid (SAS, DES, and SDES) Turbulence Models .................................................. 147 4.20.2. Large Eddy Simulation (LES) Models .................................................................................... 147 4.20.3. Stress-Blended Eddy Simulation (SBES) Model ..................................................................... 148 5. Heat Transfer ....................................................................................................................................... 149 5.1. Introduction ................................................................................................................................. 149 5.2. Modeling Conductive and Convective Heat Transfer ...................................................................... 149 5.2.1. Heat Transfer Theory ............................................................................................................. 150 5.2.1.1. The Energy Equation .................................................................................................... 150 5.2.1.2. The Energy Equation in Moving Reference Frames ........................................................ 150 5.2.1.3. The Energy Equation for the Non-Premixed Combustion Model .................................... 150 5.2.1.4. Inclusion of Pressure Work and Kinetic Energy Terms .................................................... 151 5.2.1.5. Inclusion of the Viscous Dissipation Terms .................................................................... 151 Release 2020 R1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. vii
    Theory Guide 5.2.1.6. Inclusion of the Species Diffusion Term ........................................................................ 152 5.2.1.7. Energy Sources Due to Reaction ................................................................................... 152 5.2.1.8. Energy Sources Due To Radiation ................................................................................. 152 5.2.1.9. Energy Source Due To Joule Heating ............................................................................ 152 5.2.1.10. Interphase Energy Sources ......................................................................................... 152 5.2.1.11. Energy Equation in Solid Regions ............................................................................... 153 5.2.1.12. Anisotropic Conductivity in Solids .............................................................................. 153 5.2.1.13. Diffusion at Inlets ....................................................................................................... 153 5.2.2. Natural Convection and Buoyancy-Driven Flows Theory ........................................................ 153 5.3. Modeling Radiation ...................................................................................................................... 154 5.3.1. Overview and Limitations ..................................................................................................... 154 5.3.1.1. Advantages and Limitations of the DTRM ..................................................................... 155 5.3.1.2. Advantages and Limitations of the P-1 Model ............................................................... 155 5.3.1.3. Advantages and Limitations of the Rosseland Model .................................................... 156 5.3.1.4. Advantages and Limitations of the DO Model ............................................................... 156 5.3.1.5. Advantages and Limitations of the S2S Model .............................................................. 157 5.3.1.6. Advantages and Limitations of the MC Model ............................................................... 157 5.3.2. Radiative Transfer Equation .................................................................................................. 158 5.3.3. P-1 Radiation Model Theory .................................................................................................. 160 5.3.3.1. The P-1 Model Equations ............................................................................................. 160 5.3.3.2. Anisotropic Scattering ................................................................................................. 161 5.3.3.3. Particulate Effects in the P-1 Model .............................................................................. 161 5.3.3.4. Boundary Condition Treatment for the P-1 Model at Walls ............................................. 162 5.3.3.5. Boundary Condition Treatment for the P-1 Model at Flow Inlets and Exits ...................... 163 5.3.4. Rosseland Radiation Model Theory ....................................................................................... 163 5.3.4.1. The Rosseland Model Equations ................................................................................... 163 5.3.4.2. Anisotropic Scattering ................................................................................................. 164 5.3.4.3. Boundary Condition Treatment at Walls ........................................................................ 164 5.3.4.4. Boundary Condition Treatment at Flow Inlets and Exits ................................................. 164 5.3.5. Discrete Transfer Radiation Model (DTRM) Theory ................................................................. 165 5.3.5.1. The DTRM Equations .................................................................................................... 165 5.3.5.2. Ray Tracing .................................................................................................................. 165 5.3.5.3. Clustering .................................................................................................................... 166 5.3.5.4. Boundary Condition Treatment for the DTRM at Walls ................................................... 167 5.3.5.5. Boundary Condition Treatment for the DTRM at Flow Inlets and Exits ............................ 167 5.3.6. Discrete Ordinates (DO) Radiation Model Theory ................................................................... 167 5.3.6.1. The DO Model Equations ............................................................................................. 168 5.3.6.2. Energy Coupling and the DO Model ............................................................................. 168 5.3.6.2.1. Limitations of DO/Energy Coupling ..................................................................... 170 5.3.6.3. Angular Discretization and Pixelation ........................................................................... 170 5.3.6.4. Anisotropic Scattering ................................................................................................. 173 5.3.6.5. Particulate Effects in the DO Model .............................................................................. 173 5.3.6.6. Boundary and Cell Zone Condition Treatment at Opaque Walls ..................................... 174 5.3.6.6.1. Gray Diffuse Walls ............................................................................................... 175 5.3.6.6.2. Non-Gray Diffuse Walls ........................................................................................ 175 5.3.6.7. Cell Zone and Boundary Condition Treatment at Semi-Transparent Walls ...................... 176 5.3.6.7.1. Semi-Transparent Interior Walls ........................................................................... 176 5.3.6.7.2. Specular Semi-Transparent Walls ......................................................................... 178 5.3.6.7.3. Diffuse Semi-Transparent Walls ............................................................................ 180 5.3.6.7.4. Partially Diffuse Semi-Transparent Walls ............................................................... 180 5.3.6.7.5. Semi-Transparent Exterior Walls ........................................................................... 181 5.3.6.7.6. Limitations .......................................................................................................... 183 viii Release 2020 R1 - © ANSYS, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates.
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