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Using This Manual
1. The Contents of This Manual
2. Typographical Conventions
3. Mathematical Conventions
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
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
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
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. Remarks on the Linearization of Destruction Terms
4.3.6. Effects of Buoyancy on Turbulence in the k-ε Models
4.3.7. Turbulence Damping
4.3.8. Effects of Compressibility on Turbulence in the k-ε Models
4.3.9. 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
4.5. 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. Algebraic Transition Model
4.9.1. Overview
4.9.2. Model Formulation
4.9.3. Transition Correlation
4.9.4. Pressure Gradient Function
4.9.5. Bubble Separation
4.9.6. Coefficients for the Algebraic Transition Model
4.9.7. Algebraic Transition Model and Rough Walls
4.10. Explicit Algebraic Reynolds Stress Model (EARSM)
4.11. Reynolds Stress Model (RSM)
4.11.1. Overview
4.11.2. Reynolds Stress Transport Equations
4.11.3. Modeling Turbulent Diffusive Transport
4.11.4. Modeling the Pressure-Strain Term
4.11.4.1. Linear Pressure-Strain Model
4.11.4.2. Low-Re Modifications to the Linear Pressure-Strain Model
4.11.4.3. Quadratic Pressure-Strain Model
4.11.4.4. Stress-Omega Model
4.11.4.5. Stress-BSL Model
4.11.5. Effects of Buoyancy on Turbulence
4.11.6. Modeling the Turbulence Kinetic Energy
4.11.7. Modeling the Dissipation Rate
4.11.8. Modeling the Turbulent Viscosity
4.11.9. Wall Boundary Conditions
4.11.10. Convective Heat and Mass Transfer Modeling
4.12. Scale-Adaptive Simulation (SAS) Model
4.12.1. Overview
4.12.2. Transport Equations for the SST-SAS Model
4.12.3. SAS with Other ω-Based Turbulence Models
4.13. Detached Eddy Simulation (DES)
4.13.1. Overview
4.13.2. DES with the Spalart-Allmaras Model
4.13.3. DES with the Realizable k-ε Model
4.13.4. DES with the BSL or SST k-ω Model
4.13.5. DES with the Transition SST Model
4.13.6. Improved Delayed Detached Eddy Simulation (IDDES)
4.13.6.1. Overview of IDDES
4.13.6.2. IDDES Model Formulation
4.14. Shielded Detached Eddy Simulation (SDES)
4.14.1. Shielding Function
4.14.2. LES Mode of SDES
4.15. Stress-Blended Eddy Simulation (SBES)
4.15.1. Stress Blending
4.15.2. SDES and SBES Example
4.16. Large Eddy Simulation (LES) Model
4.16.1. Overview
4.16.2. Subgrid-Scale Models
4.16.2.1. Smagorinsky-Lilly Model
4.16.2.2. Dynamic Smagorinsky-Lilly Model
4.16.2.3. Wall-Adapting Local Eddy-Viscosity (WALE) Model
4.16.2.4. Algebraic Wall-Modeled LES Model (WMLES)
4.16.2.4.1. Algebraic WMLES Model Formulation
4.16.2.4.1.1. Reynolds Number Scaling
4.16.2.4.2. Algebraic WMLES S-Omega Model Formulation
4.16.2.5. Dynamic Kinetic Energy Subgrid-Scale Model
4.16.3. Inlet Boundary Conditions for Scale Resolving Simulations
4.16.3.1. Vortex Method
4.16.3.2. Spectral Synthesizer
4.16.3.3. Synthetic Turbulence Generator
4.16.3.3.1. Limitations
4.17. Embedded Large Eddy Simulation (ELES)
4.17.1. Overview
4.17.2. Selecting a Model
4.17.3. Interfaces Treatment
4.17.3.1. RANS-LES Interface
4.17.3.2. LES-RANS Interface
4.17.3.3. Internal Interface Without LES Zone
4.17.3.4. Grid Generation Guidelines
4.18. Near-Wall Treatments for Wall-Bounded Turbulent Flows
4.18.1. Overview
4.18.2. Wall Treatment for ε-based Models
4.18.2.1. Standard Wall Functions
4.18.2.1.1. Momentum
4.18.2.1.2. Energy
4.18.2.1.3. Species
4.18.2.1.4. Turbulence
4.18.2.2. Scalable Wall Functions
4.18.2.3. Non-Equilibrium Wall Functions
4.18.2.3.1. Standard Wall Functions vs. Non-Equilibrium Wall Functions
4.18.2.3.2. Limitations of the Wall Function Approach
4.18.2.4. Enhanced Wall Treatment ε-Equation (EWT-ε)
4.18.2.4.1. Two-Layer Model for Enhanced Wall Treatment
4.18.2.4.2. Enhanced Wall Treatment for Momentum and Energy Equations
4.18.2.5. Menter-Lechner ε-Equation (ML-ε)
4.18.2.5.1. Momentum Equations
4.18.2.5.2. k-ε Turbulence Models
4.18.2.5.3. Iteration Improvements
4.18.2.6. User-Defined Wall Functions
4.18.3. y+-Insensitive Near-Wall Treatment for ω-based Turbulence Models
4.18.3.1. Momentum Equations
4.18.3.2. Turbulence
4.18.3.3. Energy
4.18.3.4. Limitations
4.18.4. LES Near-Wall Treatment
4.18.4.1. Kader Blending Wall Functions
4.18.4.2. Harmonic Blending Wall Functions Based on r+
4.18.4.3. Werner and Wengle Wall Treatment
4.18.5. Wall Roughness Effects in Turbulent Wall-Bounded Flows
4.19. Curvature Correction for the Spalart-Allmaras and Two-Equation Models
4.20. Corner Flow Correction
4.21. Production Limiters for Two-Equation Models
4.22. Turbulence Damping
4.23. Definition of Turbulence Scales
4.23.1. RANS and Hybrid (SAS, DES, and SDES) Turbulence Models
4.23.2. Large Eddy Simulation (LES) Models
4.23.3. Stress-Blended Eddy Simulation (SBES) Model
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 outlets
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 Outlets
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
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
7. Modelling with Finite-Rate Chemistry
7.1. Species Transport and Finite-Rate Chemistry
7.1.1. Volumetric Reactions
7.1.1.1. Species Transport Equations
7.1.1.1.1. Mass Diffusion in Laminar Flows
7.1.1.1.2. Mass Diffusion in Turbulent Flows
7.1.1.1.3. Full Multicomponent Diffusion
7.1.1.1.3.1. General Theory
7.1.1.1.3.2. Maxwell-Stefan Equations
7.1.1.1.4. Kinetic Theory Parameters for Diffusion Coefficients
7.1.1.1.5. Unity Lewis Number
7.1.1.1.6. Anisotropic Species Diffusion
7.1.1.1.7. Thermal Diffusion Coefficients
7.1.1.1.8. Treatment of Species Transport in the Energy Equation
7.1.1.1.9. Diffusion at Inlets
7.1.1.2. The Generalized Finite-Rate Formulation for Reaction Modeling
7.1.1.2.1. Direct Use of Finite-Rate Kinetics (no TCI)
7.1.1.2.2. Pressure-Dependent Reactions
7.1.1.2.3. The Eddy-Dissipation Model
7.1.1.2.4. The Eddy-Dissipation Model for LES
7.1.1.2.5. The Eddy-Dissipation-Concept (EDC) Model
7.1.1.2.5.1. The Standard EDC Model
7.1.1.2.5.2. The Partially Stirred Reactor EDC Model
7.1.1.2.6. The Thickened Flame Model
7.1.1.2.7. The Relaxation to Chemical Equilibrium Model
7.1.1.3. Finite-Rate Chemistry with the Two-Temperature Model
7.1.2. Wall Surface Reactions and Chemical Vapor Deposition
7.1.2.1. Surface Coverage Reaction Rate Modification
7.1.2.2. Reaction-Diffusion Balance for Surface Chemistry
7.1.2.3. Slip Boundary Formulation for Low-Pressure Gas Systems
7.1.3. Particle Reactions
7.1.3.1. Combusting Particle Surface Reactions
7.1.3.1.1. General Description
7.1.3.1.2. Ansys Fluent Model Formulation
7.1.3.1.3. Extension for Stoichiometries with Multiple Gas Phase Reactants
7.1.3.1.4. Solid-Solid Reactions
7.1.3.1.5. Solid Decomposition Reactions
7.1.3.1.6. Solid Deposition Reactions
7.1.3.1.7. Gaseous Solid Catalyzed Reactions on the Particle Surface
7.1.3.2. Multicomponent Particles with Chemical Reactions
7.1.4. Electrochemical Reactions
7.1.4.1. Overview
7.1.4.2. Electrochemical Reaction Model Theory
7.1.5. Reacting Channel Model
7.1.5.1. Overview
7.1.5.2. Reacting Channel Model Theory
7.1.5.2.1. Flow Inside the Reacting Channel
7.1.5.2.2. Surface Reactions in the Reacting Channel
7.1.5.2.3. Porous Medium Inside Reacting Channel
7.1.5.2.4. Outer Flow in the Shell
7.1.6. Reactor Network Model
7.1.6.1. Reactor Network Model Theory
7.1.6.1.1. Reactor network temperature solution
7.2. Composition PDF Transport
7.2.1. Overview
7.2.2. Composition PDF Transport Theory
7.2.3. The Lagrangian Solution Method
7.2.3.1. Particle Convection
7.2.3.2. Particle Mixing
7.2.3.2.1. The Modified Curl Model
7.2.3.2.2. The IEM Model
7.2.3.2.3. The EMST Model
7.2.3.2.4. Liquid Reactions
7.2.3.3. Particle Reaction
7.2.4. The Eulerian Solution Method
7.2.4.1. Reaction
7.2.4.2. Mixing
7.2.4.3. Correction
7.2.4.4. Calculation of Composition Mean and Variance
7.3. Chemistry Acceleration
7.3.1. Overview and Limitations
7.3.2. In-Situ Adaptive Tabulation (ISAT)
7.3.3. Dynamic Mechanism Reduction
7.3.3.1. Directed Relation Graph (DRG) Method for Mechanism Reduction
7.3.4. Chemistry Agglomeration
7.3.4.1. Binning Algorithm
7.3.5. Chemical Mechanism Dimension Reduction
7.3.5.1. Selecting the Represented Species
7.3.6. Dynamic Cell Clustering with Ansys Fluent CHEMKIN-CFD Solver
7.3.7. Dynamic Adaptive Chemistry with Ansys Fluent CHEMKIN-CFD Solver
8. Modelling of Turbulent Combustion With Reduced Order
8.1. Non-Premixed Combustion
8.1.1. Introduction
8.1.2. Non-Premixed Combustion and Mixture Fraction Theory
8.1.2.1. Mixture Fraction Theory
8.1.2.1.1. Definition of the Mixture Fraction
8.1.2.1.2. Transport Equations for the Mixture Fraction
8.1.2.1.3. The Non-Premixed Model for LES
8.1.2.1.4. The Non-Premixed Model with the SBES Turbulence Model
8.1.2.1.5. Mixture Fraction vs. Equivalence Ratio
8.1.2.1.6. Relationship of Mixture Fraction to Species Mass Fraction, Density, and Temperature
8.1.2.2. Modeling of Turbulence-Chemistry Interaction
8.1.2.2.1. Description of the Probability Density Function
8.1.2.2.2. Derivation of Mean Scalar Values from the Instantaneous Mixture Fraction
8.1.2.2.3. The Assumed-Shape PDF
8.1.2.2.3.1. The Double Delta Function PDF
8.1.2.2.3.2. The β-Function PDF
8.1.2.3. Non-Adiabatic Extensions of the Non-Premixed Model
8.1.2.4. Chemistry Tabulation
8.1.2.4.1. Look-Up Tables for Adiabatic Systems
8.1.2.4.2. 3D Look-Up Tables for Non-Adiabatic Systems
8.1.2.4.3. Generating Lookup Tables Through Automated Grid Refinement
8.1.3. Restrictions and Special Cases for Using the Non-Premixed Model
8.1.3.1. Restrictions on the Mixture Fraction Approach
8.1.3.2. Using the Non-Premixed Model for Liquid Fuel or Coal Combustion
8.1.3.3. Using the Non-Premixed Model with Flue Gas Recycle
8.1.3.4. Using the Non-Premixed Model with the Inert Model
8.1.3.4.1. Mixture Composition
8.1.3.4.1.1. Property Evaluation
8.1.4. The Diffusion Flamelet Models Theory
8.1.4.1. Restrictions and Assumptions
8.1.4.2. The Flamelet Concept
8.1.4.2.1. Overview
8.1.4.2.2. Strain Rate and Scalar Dissipation
8.1.4.2.3. Embedding Diffusion Flamelets in Turbulent Flames
8.1.4.3. Flamelet Generation
8.1.4.4. Flamelet Import
8.1.5. The Steady Diffusion Flamelet Model Theory
8.1.5.1. Overview
8.1.5.2. Multiple Steady Flamelet Libraries
8.1.5.3. Steady Diffusion Flamelet Automated Grid Refinement
8.1.5.4. Non-Adiabatic Steady Diffusion Flamelets
8.1.6. The Unsteady Diffusion Flamelet Model Theory
8.1.6.1. The Eulerian Unsteady Laminar Flamelet Model
8.1.6.1.1. Liquid Reactions
8.1.6.2. The Diesel Unsteady Laminar Flamelet Model
8.1.6.3. Multiple Diesel Unsteady Flamelets
8.1.6.4. Multiple Diesel Unsteady Flamelets with Flamelet Reset
8.1.6.4.1. Resetting the Flamelets
8.2. Premixed Combustion
8.2.1. Overview
8.2.2. C-Equation Model Theory
8.2.2.1. Propagation of the Flame Front
8.2.3. G-Equation Model Theory
8.2.3.1. Numerical Solution of the G-equation
8.2.4. Turbulent Flame Speed Models
8.2.4.1. Zimont Turbulent Flame Speed Closure Model
8.2.4.1.1. Zimont Turbulent Flame Speed Closure for LES
8.2.4.1.2. Flame Stretch Effect
8.2.4.1.3. Wall Damping
8.2.4.2. Peters Flame Speed Model
8.2.4.2.1. Peters Flame Speed Model for LES
8.2.5. Calculation of Properties
8.2.5.1. Calculation of Temperature
8.2.5.1.1. Adiabatic Temperature Calculation
8.2.5.1.2. Non-Adiabatic Temperature Calculation
8.2.5.2. Calculation of Density
8.2.5.3. Laminar Flame Speed
8.2.5.4. Unburnt Density and Thermal Diffusivity
8.3. Partially Premixed Combustion
8.3.1. Overview
8.3.2. Partially Premixed Combustion Theory
8.3.2.1. Chemical Equilibrium and Steady Diffusion Flamelet Models
8.3.2.2. Flamelet Generated Manifold (FGM) Model
8.3.2.2.1. Premixed FGMs in Reaction Progress Variable Space
8.3.2.2.2. Premixed FGMs in Physical Space
8.3.2.2.3. Diffusion FGMs
8.3.2.2.4. Nonadiabatic Flamelet Generated Manifold (FGM)
8.3.2.3. FGM Turbulent Closure
8.3.2.3.1. Scalar Transport with FGM Closure
8.3.2.4. Calculation of Mixture Properties
8.3.2.5. Calculation of Unburnt Properties
8.3.2.6. Laminar Flame Speed
8.3.2.7. Strained Laminar Flame Speed
8.3.2.7.1. Strained Non-Adiabatic Flame Speed
8.3.2.8. Generating PDF Lookup Tables Through Automated Grid Refinement
9. Pollutant Formation
9.1. NOx Formation
9.1.1. Overview
9.1.1.1. NOx Modeling in Ansys Fluent
9.1.1.2. NOx Formation and Reduction in Flames
9.1.2. Governing Equations for NOx Transport
9.1.3. Thermal NOx Formation
9.1.3.1. Thermal NOx Reaction Rates
9.1.3.2. The Quasi-Steady Assumption for [N]
9.1.3.3. Thermal NOx Temperature Sensitivity
9.1.3.4. Decoupled Thermal NOx Calculations
9.1.3.5. Approaches for Determining O Radical Concentration
9.1.3.5.1. Method 1: Equilibrium Approach
9.1.3.5.2. Method 2: Partial Equilibrium Approach
9.1.3.5.3. Method 3: Predicted O Approach
9.1.3.6. Approaches for Determining OH Radical Concentration
9.1.3.6.1. Method 1: Exclusion of OH Approach
9.1.3.6.2. Method 2: Partial Equilibrium Approach
9.1.3.6.3. Method 3: Predicted OH Approach
9.1.3.7. Summary
9.1.4. Prompt NOx Formation
9.1.4.1. Prompt NOx Combustion Environments
9.1.4.2. Prompt NOx Mechanism
9.1.4.3. Prompt NOx Formation Factors
9.1.4.4. Primary Reaction
9.1.4.5. Modeling Strategy
9.1.4.6. Rate for Most Hydrocarbon Fuels
9.1.4.7. Oxygen Reaction Order
9.1.5. Fuel NOx Formation
9.1.5.1. Fuel-Bound Nitrogen
9.1.5.2. Reaction Pathways
9.1.5.3. Fuel NOx from Gaseous and Liquid Fuels
9.1.5.3.1. Fuel NOx from Intermediate Hydrogen Cyanide (HCN)
9.1.5.3.1.1. HCN Production in a Gaseous Fuel
9.1.5.3.1.2. HCN Production in a Liquid Fuel
9.1.5.3.1.3. HCN Consumption
9.1.5.3.1.4. HCN Sources in the Transport Equation
9.1.5.3.1.5. NOx Sources in the Transport Equation
9.1.5.3.2. Fuel NOx from Intermediate Ammonia (NH3)
9.1.5.3.2.1. NH3 Production in a Gaseous Fuel
9.1.5.3.2.2. NH3 Production in a Liquid Fuel
9.1.5.3.2.3. NH3 Consumption
9.1.5.3.2.4. NH3 Sources in the Transport Equation
9.1.5.3.2.5. NOx Sources in the Transport Equation
9.1.5.3.3. Fuel NOx from Coal
9.1.5.3.3.1. Nitrogen in Char and in Volatiles
9.1.5.3.3.2. Coal Fuel NOx Scheme A
9.1.5.3.3.3. Coal Fuel NOx Scheme B
9.1.5.3.3.4. HCN Scheme Selection
9.1.5.3.3.5. NOx Reduction on Char Surface
9.1.5.3.3.5.1. BET Surface Area
9.1.5.3.3.5.2. HCN from Volatiles
9.1.5.3.3.6. Coal Fuel NOx Scheme C
9.1.5.3.3.7. Coal Fuel NOx Scheme D
9.1.5.3.3.8. NH3 Scheme Selection
9.1.5.3.3.8.1. NH3 from Volatiles
9.1.5.3.4. Fuel Nitrogen Partitioning for HCN and NH3 Intermediates
9.1.6. NOx Formation from Intermediate N2O
9.1.6.1. N2O - Intermediate NOx Mechanism
9.1.7. NOx Reduction by Reburning
9.1.7.1. Instantaneous Approach
9.1.7.2. Partial Equilibrium Approach
9.1.7.2.1. NOx Reduction Mechanism
9.1.8. NOx Reduction by SNCR
9.1.8.1. Ammonia Injection
9.1.8.2. Urea Injection
9.1.8.3. Transport Equations for Urea, HNCO, and NCO
9.1.8.4. Urea Production due to Reagent Injection
9.1.8.5. NH3 Production due to Reagent Injection
9.1.8.6. HNCO Production due to Reagent Injection
9.1.9. NOx Formation in Turbulent Flows
9.1.9.1. The Turbulence-Chemistry Interaction Model
9.1.9.2. The PDF Approach
9.1.9.3. The General Expression for the Mean Reaction Rate
9.1.9.4. The Mean Reaction Rate Used in Ansys Fluent
9.1.9.5. Statistical Independence
9.1.9.6. The Beta PDF Option
9.1.9.7. The Gaussian PDF Option
9.1.9.8. The Calculation Method for the Variance
9.2. Soot Formation
9.2.1. Overview and Limitations
9.2.1.1. Predicting Soot Formation
9.2.1.2. Restrictions on Soot Modeling
9.2.2. Soot Model Theory
9.2.2.1. The One-Step Soot Formation Model
9.2.2.2. The Two-Step Soot Formation Model
9.2.2.2.1. Soot Generation Rate
9.2.2.2.2. Nuclei Generation Rate
9.2.2.3. The Moss-Brookes Model
9.2.2.3.1. The Moss-Brookes-Hall Model
9.2.2.3.2. Soot Formation in Turbulent Flows
9.2.2.3.2.1. The Turbulence-Chemistry Interaction Model
9.2.2.3.2.2. The PDF Approach
9.2.2.3.2.3. The Mean Reaction Rate
9.2.2.3.2.4. The PDF Options
9.2.2.3.3. The Effect of Soot on the Radiation Absorption Coefficient
9.2.2.4. The Method of Moments Model
9.2.2.4.1. Soot Particle Population Balance
9.2.2.4.2. Moment Transport Equations
9.2.2.4.3. Nucleation
9.2.2.4.4. Coagulation
9.2.2.4.5. Surface Growth and Oxidation
9.2.2.4.6. Soot Aggregation
9.3. Decoupled Detailed Chemistry Model
9.3.1. Overview
9.3.1.1. Limitations
9.3.2. Decoupled Detailed Chemistry Model Theory
10. Engine Ignition
10.1. Spark Model
10.1.1. Overview and Limitations
10.1.2. Spark Model Theory
10.2. Autoignition Models
10.2.1. Model Overview
10.2.2. Model Limitations
10.2.3. Ignition Model Theory
10.2.3.1. Transport of Ignition Species
10.2.3.2. Knock Modeling
10.2.3.2.1. Modeling of the Source Term
10.2.3.2.2. Correlations
10.2.3.2.3. Energy Release
10.2.3.3. Ignition Delay Modeling
10.2.3.3.1. Modeling of the Source Term
10.2.3.3.2. Correlations
10.2.3.3.3. Energy Release
10.3. Crevice Model
10.3.1. Overview
10.3.1.1. Model Parameters
10.3.2. Limitations
10.3.3. Crevice Model Theory
11. Aerodynamically Generated Noise
11.1. Overview
11.1.1. Direct Method
11.1.2. Integral Method by Ffowcs Williams and Hawkings
11.1.3. Method Based on Wave Equation
11.1.4. Broadband Noise Source Models
11.2. Acoustics Model Theory
11.2.1. The Ffowcs Williams and Hawkings Model
11.2.2. Wave Equation Model
11.2.2.1. Limitations
11.2.2.2. Governing Equations and Boundary Conditions
11.2.2.3. Method of Numerical Solution
11.2.2.4. Preventing Non-Physical Reflections of Sound Waves
11.2.2.4.1. Mesh Quality
11.2.2.4.2. Filtering of the Sound Source Term
11.2.2.4.3. Ramping in Time and Limiting in Space (Masking) of the Sound Source Term
11.2.2.4.4. Damping of Solution in a Sponge Region Using Artificial Viscosity
11.2.2.5. Kirchhoff Integral
11.2.2.5.1. Compatibility and Limitations
11.2.2.5.2. Mathematical Formulation
11.2.3. Broadband Noise Source Models
11.2.3.1. Proudman’s Formula
11.2.3.2. The Jet Noise Source Model
11.2.3.3. The Boundary Layer Noise Source Model
11.2.3.4. Source Terms in the Linearized Euler Equations
11.2.3.5. Source Terms in Lilley’s Equation
12. Discrete Phase
12.1. Introduction
12.1.1. The Euler-Lagrange Approach
12.2. Particle Motion Theory
12.2.1. Equations of Motion for Particles
12.2.1.1. Particle Force Balance
12.2.1.2. Particle Torque Balance
12.2.1.3. Inclusion of the Gravity Term
12.2.1.4. Other Forces
12.2.1.5. Forces in Moving Reference Frames
12.2.1.6. Thermophoretic Force
12.2.1.7. Brownian Force
12.2.1.8. Saffman’s Lift Force
12.2.1.9. Magnus Lift Force
12.2.2. Turbulent Dispersion of Particles
12.2.2.1. Stochastic Tracking
12.2.2.1.1. The Integral Time
12.2.2.1.2. The Discrete Random Walk Model
12.2.2.1.3. Using the DRW Model
12.2.3. Integration of Particle Equation of Motion
12.3. Laws for Drag Coefficients
12.3.1. Spherical Drag Law
12.3.2. Non-spherical Drag Law
12.3.3. Stokes-Cunningham Drag Law
12.3.4. High-Mach-Number Drag Law
12.3.5. Dynamic Drag Model Theory
12.3.6. Dense Discrete Phase Model Drag Laws
12.3.7. Bubbly Flow Drag Laws
12.3.7.1. Ishii-Zuber Drag Model
12.3.7.2. Grace Drag Model
12.3.8. Rotational Drag Law
12.4. Laws for Heat and Mass Exchange
12.4.1. Inert Heating or Cooling (Law 1/Law 6)
12.4.2. Droplet Vaporization (Law 2)
12.4.2.1. Mass Transfer During Law 2—Diffusion Controlled Model
12.4.2.2. Mass Transfer During Law 2—Convection/Diffusion Controlled Model
12.4.2.3. Mass Transfer During Law 2—Thermolysis
12.4.2.4. Defining the Saturation Vapor Pressure and Diffusion Coefficient (or Binary Diffusivity)
12.4.2.5. Defining the Boiling Point and Latent Heat
12.4.2.6. Heat Transfer to the Droplet
12.4.3. Droplet Boiling (Law 3)
12.4.4. Devolatilization (Law 4)
12.4.4.1. Choosing the Devolatilization Model
12.4.4.2. The Constant Rate Devolatilization Model
12.4.4.3. The Single Kinetic Rate Model
12.4.4.4. The Two Competing Rates (Kobayashi) Model
12.4.4.5. The CPD Model
12.4.4.5.1. General Description
12.4.4.5.2. Reaction Rates
12.4.4.5.3. Mass Conservation
12.4.4.5.4. Fractional Change in the Coal Mass
12.4.4.5.5. CPD Inputs
12.4.4.6. Particle Swelling During Devolatilization
12.4.4.7. Heat Transfer to the Particle During Devolatilization
12.4.5. Surface Combustion (Law 5)
12.4.5.1. The Diffusion-Limited Surface Reaction Rate Model
12.4.5.2. The Kinetic/Diffusion Surface Reaction Rate Model
12.4.5.3. The Intrinsic Model
12.4.5.4. The Multiple Surface Reactions Model
12.4.5.5. Heat and Mass Transfer During Char Combustion
12.4.6. Multicomponent Particle Definition (Law 7)
12.4.6.1. Raoult’s Law
12.4.6.2. Peng-Robinson Real Gas Model
12.5. Vapor Liquid Equilibrium Theory
12.6. Physical Property Averaging
12.7. Wall-Particle Reflection Model Theory
12.7.1. Rough Wall Model
12.8. Wall-Jet Model Theory
12.9. Lagrangian Wall-Film Model Theory
12.9.1. Introduction
12.9.2. Leidenfrost Temperature Considerations
12.9.2.1. Default Wall Temperature Limiter
12.9.2.2. Leidenfrost Temperature Reporting
12.9.3. Interaction During Impact with a Boundary
12.9.3.1. The Stanton-Rutland Model
12.9.3.1.1. Regime Definition
12.9.3.1.2. Rebound
12.9.3.1.3. Splashing
12.9.3.2. The Kuhnke Model
12.9.3.2.1. Regime definition
12.9.3.2.2. Rebound
12.9.3.2.3. Splashing
12.9.3.3. The Stochastic Kuhnke Model
12.9.4. Separation and Stripping Submodels
12.9.5. Conservation Equations for Wall-Film Particles
12.9.5.1. Momentum
12.9.5.2. Mass Transfer from the Film
12.9.5.2.1. Film Vaporization and Boiling
12.9.5.2.1.1. Diffusion Controlled Model
12.9.5.2.1.2. Convection/Diffusion Controlled Model
12.9.5.2.1.3. Thermolysis Model
12.9.5.2.1.4. Gas-Side Boundary Layer Model
12.9.5.2.1.5. Film Boiling Model with Conduction Heat Transfer Model
12.9.5.2.1.6. Film Boiling Model with Convective Heat Transfer Model
12.9.5.2.2. Film Condensation
12.9.5.2.3. Multicomponent Film Models
12.9.5.3. Energy Transfer from the Film
12.9.5.3.1. Conduction Heat Transfer Model
12.9.5.3.2. Convective Heat Transfer Model
12.10. Wall Erosion
12.10.1. Finnie Erosion Model
12.10.2. Oka Erosion Model
12.10.3. McLaury Erosion Model
12.10.4. DNV Erosion Model
12.10.5. Modeling Erosion Rates in Dense Flows
12.10.5.1. Abrasive Erosion Caused by Solid Particles
12.10.5.2. Wall Shielding Effect in Dense Flow Regimes
12.10.6. Accretion
12.11. Particle–Wall Impingement Heat Transfer Theory
12.12. Atomizer Model Theory
12.12.1. The Plain-Orifice Atomizer Model
12.12.1.1. Internal Nozzle State
12.12.1.2. Coefficient of Discharge
12.12.1.3. Exit Velocity
12.12.1.4. Spray Angle
12.12.1.5. Droplet Diameter Distribution
12.12.2. The Pressure-Swirl Atomizer Model
12.12.2.1. Film Formation
12.12.2.2. Sheet Breakup and Atomization
12.12.3. The Air-Blast/Air-Assist Atomizer Model
12.12.4. The Flat-Fan Atomizer Model
12.12.5. The Effervescent Atomizer Model
12.13. Secondary Breakup Model Theory
12.13.1. Taylor Analogy Breakup (TAB) Model
12.13.1.1. Introduction
12.13.1.2. Use and Limitations
12.13.1.3. Droplet Distortion
12.13.1.4. Size of Child Droplets
12.13.1.5. Velocity of Child Droplets
12.13.1.6. Droplet Breakup
12.13.2. Wave Breakup Model
12.13.2.1. Introduction
12.13.2.2. Use and Limitations
12.13.2.3. Jet Stability Analysis
12.13.2.4. Droplet Breakup
12.13.3. KHRT Breakup Model
12.13.3.1. Introduction
12.13.3.2. Use and Limitations
12.13.3.3. Liquid Core Length
12.13.3.4. Rayleigh-Taylor Breakup
12.13.3.5. Droplet Breakup Within the Liquid Core
12.13.3.6. Droplet Breakup Outside the Liquid Core
12.13.4. Stochastic Secondary Droplet (SSD) Model
12.13.5. Madabhushi Breakup Model
12.13.6. Schmehl Breakup Model
12.14. Collision and Droplet Coalescence Model Theory
12.14.1. Introduction
12.14.2. Use and Limitations
12.14.3. Theory
12.14.3.1. Probability of Collision
12.14.3.2. Collision Outcomes
12.14.3.2.1. Bouncing
12.14.3.2.2. Bouncing and Coalescence
12.15. Discrete Element Method Collision Model
12.15.1. Theory
12.15.1.1. The Spring Collision Law
12.15.1.2. The Spring-Dashpot Collision Law
12.15.1.3. The Hertzian Collision Law
12.15.1.4. The Hertzian-Dashpot Collision Law
12.15.1.5. The Friction Collision Law
12.15.1.6. Rolling Friction Collision Law for DEM
12.15.1.7. DEM Parcels
12.15.1.8. Cartesian Collision Mesh
12.16. One-Way and Two-Way Coupling
12.16.1. Coupling Between the Discrete and Continuous Phases
12.16.2. Momentum Exchange
12.16.3. Heat Exchange
12.16.4. Mass Exchange
12.16.5. Under-Relaxation of the Interphase Exchange Terms
12.16.6. Interphase Exchange During Stochastic Tracking
12.17. Node Based Averaging
12.18. Blockage Effect
13. Modeling Macroscopic Particles
13.1. Momentum Transfer to Fluid Flow
13.2. Fluid Forces and Torques on Particle
13.3. Particle/Particle and Particle/Wall Collisions
13.4. Field Forces
13.5. Particle Deposition and Buildup
14. Multiphase Flows
14.1. Introduction
14.1.1. Multiphase Flow Regimes
14.1.1.1. Gas-Liquid or Liquid-Liquid Flows
14.1.1.2. Gas-Solid Flows
14.1.1.3. Liquid-Solid Flows
14.1.1.4. Three-Phase Flows
14.1.2. Examples of Multiphase Systems
14.2. Choosing a General Multiphase Model
14.2.1. Approaches to Multiphase Modeling
14.2.1.1. The Euler-Euler Approach
14.2.1.1.1. The VOF Model
14.2.1.1.2. The Mixture Model
14.2.1.1.3. The Eulerian Model
14.2.2. Model Comparisons
14.2.2.1. Detailed Guidelines
14.2.2.1.1. The Effect of Particulate Loading
14.2.2.1.2. The Significance of the Stokes Number
14.2.2.1.2.1. Examples
14.2.2.1.3. Other Considerations
14.2.3. Time Schemes in Multiphase Flow
14.2.4. Stability and Convergence
14.3. Volume of Fluid (VOF) Model Theory
14.3.1. Overview of the VOF Model
14.3.2. Limitations of the VOF Model
14.3.3. Steady-State and Transient VOF Calculations
14.3.4. Volume Fraction Equation
14.3.4.1. The Implicit Formulation
14.3.4.2. The Explicit Formulation
14.3.4.3. Interpolation Near the Interface
14.3.4.3.1. The Geometric Reconstruction Scheme
14.3.4.3.2. The Donor-Acceptor Scheme
14.3.4.3.3. The Compressive Interface Capturing Scheme for Arbitrary Meshes (CICSAM)
14.3.4.3.4. The Compressive Scheme and Interface-Model-based Variants
14.3.4.3.5. Bounded Gradient Maximization (BGM)
14.3.5. Material Properties
14.3.6. Momentum Equation
14.3.7. Energy Equation
14.3.8. Additional Scalar Equations
14.3.9. Surface Tension and Adhesion
14.3.9.1. Surface Tension
14.3.9.1.1. The Continuum Surface Force Model
14.3.9.1.2. The Continuum Surface Stress Model
14.3.9.1.3. Comparing the CSS and CSF Methods
14.3.9.1.4. When Surface Tension Effects Are Important
14.3.9.2. Wall Adhesion
14.3.9.3. Jump Adhesion
14.3.10. Open Channel Flow
14.3.10.1. Upstream Boundary Conditions
14.3.10.1.1. Pressure Inlet
14.3.10.1.2. Mass Flow Rate
14.3.10.1.3. Volume Fraction Specification
14.3.10.2. Downstream Boundary Conditions
14.3.10.2.1. Pressure Outlet
14.3.10.2.2. Outflow Boundary
14.3.10.2.3. Backflow Volume Fraction Specification
14.3.10.3. Numerical Beach Treatment
14.3.11. Open Channel Wave Boundary Conditions
14.3.11.1. Airy Wave Theory
14.3.11.2. Stokes Wave Theories
14.3.11.3. Cnoidal/Solitary Wave Theory
14.3.11.4. Choosing a Wave Theory
14.3.11.5. Superposition of Waves
14.3.11.6. Modeling of Random Waves Using Wave Spectrum
14.3.11.6.1. Definitions
14.3.11.6.2. Wave Spectrum Implementation Theory
14.3.11.6.2.1. Long-Crested Random Waves (Unidirectional)
14.3.11.6.2.1.1. Pierson-Moskowitz Spectrum
14.3.11.6.2.1.2. JONSWAP Spectrum
14.3.11.6.2.1.3. TMA Spectrum
14.3.11.6.2.2. Short-Crested Random Waves (Multi-Directional)
14.3.11.6.2.2.1. Cosine-2s Power Function (Frequency Independent)
14.3.11.6.2.2.2. Hyperbolic Function (Frequency Dependent)
14.3.11.6.2.3. Superposition of Individual Wave Components Using the Wave Spectrum
14.3.11.6.3. Choosing a Wave Spectrum and Inputs
14.3.11.7. Nomenclature for Open Channel Waves
14.3.12. Coupled Level-Set and VOF Model
14.3.12.1. Theory
14.3.12.1.1. Surface Tension Force
14.3.12.1.2. Re-initialization of the Level-set Function via the Geometrical Method
14.3.12.2. Limitations
14.4. Mixture Model Theory
14.4.1. Overview
14.4.2. Limitations of the Mixture Model
14.4.3. Continuity Equation
14.4.4. Momentum Equation
14.4.5. Energy Equation
14.4.6. Relative (Slip) Velocity and the Drift Velocity
14.4.7. Volume Fraction Equation for the Secondary Phases
14.4.8. Granular Properties
14.4.8.1. Collisional Viscosity
14.4.8.2. Kinetic Viscosity
14.4.9. Granular Temperature
14.4.10. Solids Pressure
14.4.11. Interfacial Area Concentration
14.4.11.1. Transport Equation Based Models
14.4.11.1.1. Hibiki-Ishii Model
14.4.11.1.2. Ishii-Kim Model
14.4.11.1.3. Yao-Morel Model
14.4.11.2. Algebraic Models
14.4.12. Flow Regime Modeling
14.4.12.1. Phase Pair Interaction for Hybrid Morphology
14.4.12.2. Algebraic Interfacial Area Density Method for Flow Regime Blending
14.5. Eulerian Model Theory
14.5.1. Overview of the Eulerian Model
14.5.2. Limitations of the Eulerian Model
14.5.3. Volume Fraction Equation
14.5.4. Conservation Equations
14.5.4.1. Equations in General Form
14.5.4.1.1. Conservation of Mass
14.5.4.1.2. Conservation of Momentum
14.5.4.1.3. Conservation of Energy
14.5.4.2. Equations Solved by Ansys Fluent
14.5.4.2.1. Continuity Equation
14.5.4.2.2. Fluid-Fluid Momentum Equations
14.5.4.2.3. Fluid-Solid Momentum Equations
14.5.4.2.4. Conservation of Energy
14.5.5. Surface Tension and Adhesion for the Eulerian Multiphase Model
14.5.6. Interfacial Area Concentration
14.5.7. Interphase Exchange Coefficients
14.5.7.1. Fluid-Fluid Exchange Coefficient
14.5.7.1.1. Schiller and Naumann Model
14.5.7.1.2. Morsi and Alexander Model
14.5.7.1.3. Symmetric Model
14.5.7.1.4. Grace et al. Model
14.5.7.1.5. Tomiyama et al. Model
14.5.7.1.6. Ishii Model
14.5.7.1.7. Ishii-Zuber Drag Model
14.5.7.1.8. Universal Drag Laws for Bubble-Liquid and Droplet-Gas Flows
14.5.7.1.8.1. Bubble-Liquid Flow
14.5.7.1.8.2. Droplet-Gas Flow
14.5.7.2. Fluid-Solid Exchange Coefficient
14.5.7.3. Solid-Solid Exchange Coefficient
14.5.7.4. Drag Modification
14.5.7.4.1. Brucato et al. Correlation
14.5.7.4.2. Near-Wall Drag Enhancement
14.5.8. Lift Coefficient Correction
14.5.8.1. Shaver-Podowski Correction
14.5.9. Lift Force
14.5.9.1. Lift Coefficient Models
14.5.9.1.1. Moraga Lift Force Model
14.5.9.1.2. Saffman-Mei Lift Force Model
14.5.9.1.3. Legendre-Magnaudet Lift Force Model
14.5.9.1.4. Tomiyama Lift Force Model
14.5.9.1.5. Hessenkemper et al. Lift Force Model
14.5.10. Wall Lubrication Force
14.5.10.1. Wall Lubrication Models
14.5.10.1.1. Antal et al. Model
14.5.10.1.2. Tomiyama Model
14.5.10.1.3. Frank Model
14.5.10.1.4. Hosokawa Model
14.5.10.1.5. Lubchenko Model
14.5.11. Turbulent Dispersion Force
14.5.11.1. Models for Turbulent Dispersion Force
14.5.11.1.1. Lopez de Bertodano Model
14.5.11.1.2. Simonin Model
14.5.11.1.3. Burns et al. Model
14.5.11.1.4. Diffusion in VOF Model
14.5.11.2. Limiting Functions for the Turbulent Dispersion Force
14.5.12. Virtual Mass Force
14.5.13. Solids Pressure
14.5.13.1. Radial Distribution Function
14.5.14. Maximum Packing Limit in Binary Mixtures
14.5.15. Solids Shear Stresses
14.5.15.1. Collisional Viscosity
14.5.15.2. Kinetic Viscosity
14.5.15.3. Bulk Viscosity
14.5.15.4. Frictional Viscosity
14.5.16. Granular Temperature
14.5.17. Description of Heat Transfer
14.5.17.1. The Heat Exchange Coefficient
14.5.17.1.1. Constant
14.5.17.1.2. Nusselt Number
14.5.17.1.3. Ranz-Marshall Model
14.5.17.1.4. Tomiyama Model
14.5.17.1.5. Hughmark Model
14.5.17.1.6. Gunn Model
14.5.17.1.7. Two-Resistance Model
14.5.17.1.8. Fixed To Saturation Temperature
14.5.17.1.9. Constant Time Scale Method
14.5.17.1.10. User Defined
14.5.18. Turbulence Models
14.5.18.1. k- ε Turbulence Models
14.5.18.1.1. k- ε Mixture Turbulence Model
14.5.18.1.2. k- ε Dispersed Turbulence Model
14.5.18.1.2.1. Assumptions
14.5.18.1.2.2. Turbulence in the Continuous Phase
14.5.18.1.2.3. Turbulence in the Dispersed Phase
14.5.18.1.3. k- ε Turbulence Model for Each Phase
14.5.18.1.3.1. Transport Equations
14.5.18.2. RSM Turbulence Models
14.5.18.2.1. RSM Dispersed Turbulence Model
14.5.18.2.2. RSM Mixture Turbulence Model
14.5.18.3. Turbulence Interaction Models
14.5.18.3.1. Simonin et al.
14.5.18.3.1.1. Formulation in Dispersed Turbulence Models
14.5.18.3.1.1.1. Continuous Phase
14.5.18.3.1.1.2. Dispersed Phases
14.5.18.3.1.2. Formulation in Per Phase Turbulence Models
14.5.18.3.2. Troshko-Hassan
14.5.18.3.2.1. Troshko-Hassan Formulation in Mixture Turbulence Models
14.5.18.3.2.2. Troshko-Hassan Formulation in Dispersed Turbulence Models
14.5.18.3.2.2.1. Continuous Phase
14.5.18.3.2.2.2. Dispersed Phases
14.5.18.3.2.3. Troshko-Hassan Formulation in Per-Phase Turbulence Models
14.5.18.3.2.3.1. Continuous Phase
14.5.18.3.2.3.2. Dispersed Phases
14.5.18.3.3. Sato
14.5.18.3.4. None
14.5.19. Solution Method in Ansys Fluent
14.5.19.1. The Pressure-Correction Equation
14.5.19.2. Volume Fractions
14.5.20. Algebraic Interfacial Area Density (AIAD) Model
14.5.20.1. Modeling Interfacial Area
14.5.20.2. Modeling Free-Surface Drag
14.5.20.3. Modeling Sub-grid Wave Turbulence Contribution (SWT)
14.5.20.4. Modeling Entrainment-Absorption
14.5.21. Generalized Two Phase (GENTOP) Flow Model
14.5.21.1. Interface Detection of the GENTOP Phase
14.5.21.2. Clustering Force for the GENTOP Phase
14.5.21.3. Surface Tension for the GENTOP-Primary Phase Pair
14.5.21.4. Interface Momentum Transfer
14.5.21.5. Complete Coalescence Method
14.5.22. The Filtered Two-Fluid Model
14.5.23. Dense Discrete Phase Model
14.5.23.1. Limitations
14.5.23.2. Granular Temperature
14.5.24. Multi-Fluid VOF Model
14.5.25. Wall Boiling Models
14.5.25.1. Overview of Wall Boiling Models
14.5.25.2. RPI Model
14.5.25.3. Non-equilibrium Subcooled Boiling
14.5.25.4. Critical Heat Flux
14.5.25.4.1. Wall Heat Flux Partition
14.5.25.4.2. Coupling Between the RPI Boiling Model and the Homogeneous or Inhomogeneous Discrete PBM
14.5.25.4.3. Flow Regime Transition
14.5.25.5. Interfacial Momentum Transfer
14.5.25.5.1. Interfacial Area
14.5.25.5.2. Bubble and Droplet Diameters
14.5.25.5.2.1. Bubble Diameter
14.5.25.5.2.2. Droplet Diameter
14.5.25.5.3. Interfacial Drag Force
14.5.25.5.4. Interfacial Lift Force
14.5.25.5.5. Turbulent Dispersion Force
14.5.25.5.6. Wall Lubrication Force
14.5.25.5.7. Virtual Mass Force
14.5.25.6. Mass Transfer
14.5.25.6.1. Mass Transfer From the Wall to Vapor
14.5.25.6.2. Interfacial Mass Transfer
14.5.25.7. Turbulence Interactions
14.6. Wet Steam Model Theory
14.6.1. Overview of the Wet Steam Model
14.6.2. Limitations of the Wet Steam Model
14.6.3. Wet Steam Flow Equations
14.6.4. Phase Change Model
14.6.5. Built-in Thermodynamic Wet Steam Properties
14.6.5.1. Equation of State
14.6.5.1.1. Virial Equation Developed by Vukalovich
14.6.5.1.2. Virial Equation Developed by Young
14.6.5.2. Saturated Vapor Line
14.6.5.3. Saturated Liquid Line
14.6.5.4. Mixture Properties
14.6.6. Real Gas Property (RGP) Table Files for the Wet Steam Model
14.6.7. Computing Stagnation Conditions for the Wet Steam Model
14.7. Modeling Mass Transfer in Multiphase Flows
14.7.1. Source Terms due to Mass Transfer
14.7.1.1. Mass Equation
14.7.1.2. Momentum Equation
14.7.1.3. Energy Equation
14.7.1.4. Species Equation
14.7.1.5. Other Scalar Equations
14.7.2. Unidirectional Constant Rate Mass Transfer
14.7.3. UDF-Prescribed Mass Transfer
14.7.4. Cavitation Models
14.7.4.1. Limitations of the Cavitation Models
14.7.4.2. Vapor Transport Equation
14.7.4.3. Bubble Dynamics Consideration
14.7.4.4. Schnerr and Sauer Model
14.7.4.5. Zwart-Gerber-Belamri Model
14.7.4.6. Expert Cavitation Model
14.7.4.6.1. Limitations of the Singhal et al. Model
14.7.4.6.2. Singhal et al. Model Theory
14.7.4.7. Turbulence Factor
14.7.4.8. Additional Guidelines for the Cavitation Models
14.7.4.9. Extended Cavitation Model Capabilities
14.7.4.9.1. N-phase Cavitation Models
14.7.4.9.2. Multiphase Species Transport Cavitation Model
14.7.5. Evaporation-Condensation Model
14.7.5.1. Lee Model
14.7.5.2. Thermal Phase Change Model
14.7.6. Semi-Mechanistic Boiling Model
14.7.7. Interphase Species Mass Transfer
14.7.7.1. Modeling Approach
14.7.7.2. Equilibrium Models
14.7.7.2.1. Raoult’s Law
14.7.7.2.2. Henry’s Law
14.7.7.2.3. Equilibrium Ratio
14.7.7.3. Mass Transfer Coefficient Models
14.7.7.3.1. Constant
14.7.7.3.2. Sherwood Number
14.7.7.3.3. Ranz-Marshall Model
14.7.7.3.4. Hughmark Model
14.7.7.3.5. Higbie Model
14.7.7.3.6. User-Defined
14.8. Modeling Species Transport in Multiphase Flows
14.8.1. Limitations
14.8.2. Mass and Momentum Transfer with Multiphase Species Transport
14.8.2.1. Source Terms Due to Heterogeneous Reactions
14.8.2.1.1. Mass Transfer
14.8.2.1.2. Momentum Transfer
14.8.2.1.3. Species Transfer
14.8.2.1.4. Heat Transfer
14.8.3. The Stiff Chemistry Solver
14.8.4. Heterogeneous Phase Interaction
14.9. Population Balance Model
14.9.1. Introduction
14.9.1.1. The Discrete Method
14.9.1.2. The Inhomogeneous Discrete Method
14.9.1.3. The Standard Method of Moments
14.9.1.4. The Quadrature Method of Moments
14.9.2. Population Balance Model Theory
14.9.2.1. The Particle State Vector
14.9.2.2. The Population Balance Equation (PBE)
14.9.2.2.1. Particle Growth
14.9.2.2.2. Particle Birth and Death Due to Breakage and Aggregation
14.9.2.2.2.1. Breakage
14.9.2.2.2.2. Luo and Lehr Breakage Kernels
14.9.2.2.2.3. Ghadiri Breakage Kernels
14.9.2.2.2.4. Laakkonen Breakage Kernels
14.9.2.2.2.5. Liao Breakage Kernel
14.9.2.2.2.6. Parabolic PDF
14.9.2.2.2.7. Generalized PDF
14.9.2.2.2.8. Martinez-Bazan Breakage Kernel
14.9.2.2.2.9. Inverted U-PDF
14.9.2.2.2.10. Aggregation
14.9.2.2.2.11. Luo Aggregation Kernel
14.9.2.2.2.12. Free Molecular Aggregation Kernel
14.9.2.2.2.13. Turbulent Aggregation Kernel
14.9.2.2.2.14. Prince and Blanch Aggregation Kernel
14.9.2.2.2.15. Liao Aggregation Kernel
14.9.2.2.3. Particle Birth by Nucleation
14.9.2.3. Solution Methods
14.9.2.3.1. The Discrete Method and the Inhomogeneous Discrete Method
14.9.2.3.1.1. Numerical Method
14.9.2.3.1.2. Breakage Formulations for the Discrete Method
14.9.2.3.2. The Standard Method of Moments (SMM)
14.9.2.3.2.1. Numerical Method
14.9.2.3.3. The Quadrature Method of Moments (QMOM)
14.9.2.3.3.1. Numerical Method
14.9.2.3.4. The Direct Quadrature Method of Moments (DQMOM)
14.9.2.3.4.1. Numerical Method
14.9.2.4. Population Balance Statistics
14.9.2.4.1. Reconstructing the Particle Size Distribution from Moments
14.9.2.4.2. The Log-Normal Distribution
15. Solidification and Melting
15.1. Overview
15.2. Limitations
15.3. Introduction
15.4. Energy Equation
15.5. Momentum Equations
15.6. Turbulence Equations
15.7. Species Equations
15.8. Back Diffusion
15.9. Pull Velocity for Continuous Casting
15.10. Contact Resistance at Walls
15.11. Thermal and Solutal Buoyancy
16. The Structural Model for Intrinsic Fluid-Structure Interaction (FSI)
16.1. Limitations
16.2. The FSI Model
16.3. Intrinsic FSI
16.4. Linear Elasticity
16.4.1. Equations
16.4.1.1. Linear Isotropic and Isothermal Elasticity
16.4.1.2. Evaluation of the von Mises Stress
16.4.2. Finite Element Representation
16.4.2.1. Construction of the Matrix of the System
16.4.2.2. Dynamic Structural Systems
16.4.2.2.1. The Newmark Method
16.4.2.2.2. Backward Euler Method
16.4.2.2.3. Rayleigh Damping
16.5. Nonlinear Elasticity
16.5.1. Finite Element Geometric Nonlinearity
16.5.2. Finite Element Nonlinear Discretization
16.5.3. Constitutive Equations
16.5.4. The Transient Scheme for the Nonlinear System
16.6. Thermoelasticity Model
16.6.1. Constitutive Equations
16.6.2. Finite Element Discretization
16.7. Coupling of the Structural Model with the Battery Swelling Model
17. Eulerian Wall Films
17.1. Introduction
17.2. Mass, Momentum, and Energy Conservation Equations for Wall Film
17.2.1. Film Sub-Models
17.2.1.1. DPM Collection
17.2.1.2. Particle-Wall Interaction
17.2.1.3. Film Separation
17.2.1.3.1. Separation Criteria
17.2.1.3.1.1. Foucart Separation
17.2.1.3.1.2. O’Rourke Separation
17.2.1.3.1.3. Friedrich Separation
17.2.1.4. Film Stripping
17.2.1.5. Secondary Phase Accretion
17.2.1.6. Coupling of Wall Film with Mixture Species Transport
17.2.1.7. Coupling of Eulerian Wall Film with the VOF Multiphase Model
17.2.2. Partial Wetting Effect
17.2.3. Boundary Conditions
17.2.4. Obtaining Film Velocity Without Solving the Momentum Equations
17.2.4.1. Shear-Driven Film Velocity
17.2.4.2. Gravity-Driven Film Velocity
17.3. Passive Scalar Equation for Wall Film
17.4. Numerical Schemes and Solution Algorithm
17.4.1. Temporal Differencing Schemes
17.4.1.1. First-Order Explicit Method
17.4.1.2. First-Order Implicit Method
17.4.1.3. Second-Order Implicit Method
17.4.2. Spatial Differencing Schemes
17.4.3. Solution Algorithm
17.4.3.1. Steady Flow
17.4.3.2. Transient Flow
17.4.4. Coupled Solution Approach
18. Electric Potential and Electrochemistry Models
18.1. Electric Potential
18.1.1. Overview
18.1.2. Electric Potential Equation
18.1.3. Energy Equation Source Term
18.2. Lithium-ion Battery Model
18.2.1. Overview
18.2.2. Lithium-ion Battery Model Theory
18.3. Electrolysis and H2 Pump Model
18.3.1. Overview
18.3.2. Resolved Modeling Approach
18.3.2.1. Electrochemistry Modeling
18.3.2.2. Multiphase Modeling
18.3.2.3. Heat Source
18.3.3. Unresolved 0D Modeling Approach
19. Battery Model
19.1. Battery Solution Methods
19.1.1. CHT Coupling Method
19.1.2. FMU-CHT coupling method
19.1.3. Circuit Network Solution Method
19.1.4. MSMD Solution Method
19.2. Electro-Chemical Models
19.2.1. NTGK/DCIR Model
19.2.2. ECM Model
19.2.3. Newman’s P2D Model
19.3. MSMD Solution Method: Coupling Between CFD and Submodels
19.4. Simulating Battery Pack Using the MSMD Solution Method
19.5. Reduced Order Solution Method (ROM)
19.6. External and Internal Electric Short-Circuit Treatment
19.7. Thermal Abuse Model
19.8. Battery Venting Model
19.9. Battery Life Models
19.9.1. Empirical Battery Life Model
19.9.2. Battery Capacity Fade Model
19.9.3. Physics-Based Battery Life Model
19.10. Battery Swelling Model
19.10.1. Empirical-Based Swelling Model
19.10.2. Physics-Based Swelling Model
19.10.3. Coupling between the Swelling and FSI Structural Models
19.10.4. Modeling Swelling in the E-Chem Standalone Mode
20. Modeling Fuel Cells
20.1. PEMFC Model Theory
20.1.1. Introduction
20.1.2. Electrochemistry Modeling
20.1.2.1. The Cathode Particle Model
20.1.3. Current and Mass Conservation
20.1.4. Water Transport and Mass Transfer in PEMFC
20.1.4.1. The Dissolved Phase Model
20.1.4.2. The Liquid Phase Model
20.1.4.2.1. Liquid Water Transport Equation in the Porous Electrode and the Membrane
20.1.4.2.2. Liquid Water Transport Equation in Gas Channels
20.1.4.3. The Ice Phase Model
20.1.5. Heat Source
20.1.6. Properties
20.1.7. Transient Simulations
20.1.8. Leakage Current (Cross-Over Current)
20.1.9. Zones Where User-Defined Scalars Have Physically Meaningful Values
20.2. Fuel Cell and Electrolysis Model Theory
20.2.1. Introduction
20.2.1.1. Introduction to PEMFC
20.2.1.1.1. Low-Temperature PEMFC
20.2.1.1.2. High-Temperature PEMFC
20.2.1.2. Introduction to SOFC
20.2.1.3. Introduction to SOEC
20.2.2. Electrochemistry Modeling
20.2.3. Current and Mass Conservation
20.2.4. Heat Source
20.2.5. Liquid Water Formation, Transport, and its Effects (Low-Temperature PEMFC Only)
20.2.6. Properties
20.2.7. Transient Simulations
20.2.8. Leakage Current (Cross-Over Current)
20.3. SOFC Fuel Cell With Unresolved Electrolyte Model Theory
20.3.1. Introduction
20.3.2. The SOFC With Unresolved Electrolyte Modeling Strategy
20.3.3. Modeling Fluid Flow, Heat Transfer, and Mass Transfer
20.3.4. Modeling Current Transport and the Potential Field
20.3.4.1. Cell Potential
20.3.4.2. Activation Overpotential
20.3.4.3. Treatment of the Energy Equation at the Electrolyte Interface
20.3.4.4. Treatment of the Energy Equation in the Conducting Regions
20.3.5. Modeling Reactions
20.3.5.1. Modeling Electrochemical Reactions
20.3.5.2. Modeling CO Electrochemistry
20.3.5.3. Modeling Electrolysis
21. Modeling Magnetohydrodynamics
21.1. Introduction
21.2. Magnetic Induction Method
21.2.1. Case 1: Externally Imposed Magnetic Field Generated in Non-conducting Media
21.2.2. Case 2: Externally Imposed Magnetic Field Generated in Conducting Media
21.3. Electric Potential Method
22. Modeling Continuous Fibers
22.1. Introduction
22.2. Governing Equations of Fiber Flow
22.3. Discretization of the Fiber Equations
22.3.1. Under-Relaxation
22.4. Numerical Solution Algorithm of Fiber Equations
22.5. Residuals of Fiber Equations
22.6. Coupling Between Fibers and the Surrounding Fluid
22.6.1. Momentum Exchange
22.6.2. Mass Exchange
22.6.3. Heat Exchange
22.6.4. Radiation Exchange
22.6.5. Under-Relaxation of the Fiber Exchange Terms
22.7. Fiber Grid Generation
22.8. Correlations for Momentum, Heat and Mass Transfer
22.8.1. Drag Coefficient
22.8.2. Heat Transfer Coefficient
22.8.3. Mass Transfer Coefficient
22.9. Fiber Properties
22.9.1. Fiber Viscosity
22.9.1.1. Melt Spinning
22.9.1.2. Dry Spinning
22.9.2. Vapor-Liquid Equilibrium
22.9.3. Latent Heat of Vaporization
22.9.4. Emissivity
22.10. Solution Strategies
23. Solver Theory
23.1. Overview of Flow Solvers
23.1.1. Pressure-Based Solver
23.1.1.1. The Pressure-Based Segregated Algorithm
23.1.1.2. The Pressure-Based Coupled Algorithm
23.1.2. Density-Based Solver
23.2. General Scalar Transport Equation: Discretization and Solution
23.2.1. Solving the Linear System
23.3. Discretization
23.3.1. Spatial Discretization
23.3.1.1. First-Order Upwind Scheme
23.3.1.2. Second-Order Upwind Scheme
23.3.1.3. First- to Higher-Order Blending
23.3.1.4. Central-Differencing Scheme
23.3.1.5. Bounded Central Differencing Scheme
23.3.1.6. QUICK Scheme
23.3.1.7. Third-Order MUSCL Scheme
23.3.1.8. Modified HRIC Scheme
23.3.1.9. High Order Term Relaxation
23.3.2. Temporal Discretization
23.3.2.1. Implicit Time Integration
23.3.2.2. Bounded Second-Order Implicit Time Integration
23.3.2.2.1. Limitations
23.3.2.3. Second-Order Time Integration Using a Variable Time Step Size
23.3.2.4. Explicit Time Integration
23.3.3. Evaluation of Gradients and Derivatives
23.3.3.1. Green-Gauss Theorem
23.3.3.2. Green-Gauss Cell-Based Gradient Evaluation
23.3.3.3. Green-Gauss Node-Based Gradient Evaluation
23.3.3.4. Least Squares Cell-Based Gradient Evaluation
23.3.4. Gradient Limiters
23.3.4.1. Standard Limiter
23.3.4.2. Multidimensional Limiter
23.3.4.3. Differentiable Limiter
23.4. Pressure-Based Solver
23.4.1. Discretization of the Momentum Equation
23.4.1.1. Pressure Interpolation Schemes
23.4.2. Discretization of the Continuity Equation
23.4.2.1. Density Interpolation Schemes
23.4.3. Pressure-Velocity Coupling
23.4.3.1. Segregated Algorithms
23.4.3.1.1. SIMPLE
23.4.3.1.2. SIMPLEC
23.4.3.1.2.1. Skewness Correction
23.4.3.1.3. PISO
23.4.3.1.3.1. Neighbor Correction
23.4.3.1.3.2. Skewness Correction
23.4.3.1.3.3. Skewness - Neighbor Coupling
23.4.3.1.4. Fractional-Step Method (FSM)
23.4.3.2. Coupled Algorithm
23.4.3.2.1. Limitation
23.4.4. Steady-State Iterative Algorithm
23.4.4.1. Under-Relaxation of Variables
23.4.4.2. Under-Relaxation of Equations
23.4.5. Time-Advancement Algorithm
23.4.5.1. Iterative Time-Advancement Scheme
23.4.5.1.1. The Frozen Flux Formulation
23.4.5.2. Non-Iterative Time-Advancement Scheme
23.4.6. Correction Form Discretization of the Momentum Equations
23.5. Density-Based Solver
23.5.1. Governing Equations in Vector Form
23.5.2. Preconditioning
23.5.3. Convective Fluxes
23.5.3.1. Roe Flux-Difference Splitting Scheme
23.5.3.2. AUSM+ Scheme
23.5.3.3. Low Diffusion Roe Flux Difference Splitting Scheme
23.5.4. Steady-State Flow Solution Methods
23.5.4.1. Explicit Formulation
23.5.4.1.1. Implicit Residual Smoothing
23.5.4.2. Implicit Formulation
23.5.4.2.1. Convergence Acceleration for Stretched Meshes
23.5.5. Unsteady Flows Solution Methods
23.5.5.1. Explicit Time Stepping
23.5.5.2. Implicit Time Stepping (Dual-Time Formulation)
23.6. Pseudo Time Method Under-Relaxation
23.6.1. Local Time Step Method
23.6.2. Global Time Step Method
23.7. Multigrid Method
23.7.1. Approach
23.7.1.1. The Need for Multigrid
23.7.1.2. The Basic Concept in Multigrid
23.7.1.3. Restriction and Prolongation
23.7.1.4. Unstructured Multigrid
23.7.2. Multigrid Cycles
23.7.2.1. The V and W Cycles
23.7.3. Algebraic Multigrid (AMG)
23.7.3.1. AMG Restriction and Prolongation Operators
23.7.3.2. AMG Coarse Level Operator
23.7.3.3. The F Cycle
23.7.3.4. The Flexible Cycle
23.7.3.4.1. The Residual Reduction Rate Criteria
23.7.3.4.2. The Termination Criteria
23.7.3.5. The Coupled and Scalar AMG Solvers
23.7.3.5.1. Gauss-Seidel
23.7.3.5.2. Incomplete Lower Upper (ILU)
23.7.4. Full-Approximation Storage (FAS) Multigrid
23.7.4.1. FAS Restriction and Prolongation Operators
23.7.4.2. FAS Coarse Level Operator
23.8. Hybrid Initialization
23.9. Full Multigrid (FMG) Initialization
23.9.1. Overview of FMG Initialization
23.9.2. Limitations of FMG Initialization
24. Adapting the Mesh
24.1. Adaption Process
24.1.1. Hanging Node Adaption
24.1.2. Polyhedral Unstructured Mesh Adaption
24.2. Geometry-Based Adaption
24.2.1. Geometry-Based Adaption Approach
24.2.1.1. Node Projection
24.2.1.2. Example of Geometry-Based Adaption
25. Reporting Alphanumeric Data
25.1. Fluxes Through Boundaries
25.2. Forces on Boundaries
25.2.1. Computing Forces, Moments, and the Center of Pressure
25.3. Surface Integration
25.3.1. Computing Surface Integrals
25.3.1.1. Area
25.3.1.2. Integral
25.3.1.3. Area-Weighted Average
25.3.1.4. Custom Vector Based Flux
25.3.1.5. Custom Vector Flux
25.3.1.6. Custom Vector Weighted Average
25.3.1.7. Flow Rate
25.3.1.8. Mass Flow Rate
25.3.1.9. Mass-Weighted Average
25.3.1.10. Sum of Field Variable
25.3.1.11. Facet Average
25.3.1.12. Facet Minimum
25.3.1.13. Facet Maximum
25.3.1.14. Vertex Average
25.3.1.15. Vertex Minimum
25.3.1.16. Vertex Maximum
25.3.1.17. Standard-Deviation
25.3.1.18. Uniformity Index
25.3.1.19. Volume Flow Rate
25.4. Volume Integration
25.4.1. Computing Volume Integrals
25.4.1.1. Volume
25.4.1.2. Sum
25.4.1.3. Sum*2Pi
25.4.1.4. Volume Integral
25.4.1.5. Volume-Weighted Average
25.4.1.6. Mass-Weighted Integral
25.4.1.7. Mass
25.4.1.8. Mass-Weighted Average
25.5. Efficiency Calculation
25.5.1. Isentropic Efficiency
25.5.2. Polytropic Efficiency
A. Nomenclature
Bibliography