ANSYS Fluent Theory Guide16.0.pdf

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Table of Contents
Using This Manual .................................................................................................................................... xxix
1.The Contents of This Manual ............................................................................................................ xxix
2.The Contents of the Fluent Manuals .................................................................................................. xxx
3. Typographical Conventions ............................................................................................................. xxxi
4. Mathematical Conventions ............................................................................................................ xxxiv
5.Technical Support ........................................................................................................................... xxxv
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 ................................................................................................ 13
1.6.2.1. Basic Equations for Compressible Flows ......................................................................... 14
1.6.2.2.The Compressible Form of the Gas Law .......................................................................... 14
1.7. Inviscid Flows ................................................................................................................................. 15
1.7.1. Euler Equations ...................................................................................................................... 15
1.7.1.1.The Mass Conservation Equation .................................................................................... 15
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 .................................................................................................. 18
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
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2.3.1.3.2. Interface Treatment: Absolute Velocity Formulation ............................................... 25
2.3.2. The Mixing Plane Model ......................................................................................................... 25
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 .................................................................................................... 28
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 .................................................................................................................... 34
3.2.1. Conservation Equations ......................................................................................................... 35
3.2.2. Six DOF (6DOF) Solver Theory ................................................................................................. 36
3.3. Sliding Mesh Theory ....................................................................................................................... 37
4.Turbulence ............................................................................................................................................. 39
4.1. Underlying Principles of Turbulence Modeling ................................................................................. 39
4.1.1. Reynolds (Ensemble) Averaging .............................................................................................. 39
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 ..................................................... 41
4.2. Spalart-Allmaras Model ................................................................................................................... 42
4.2.1. Overview ............................................................................................................................... 42
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 ........................................................................................ 43
4.2.5. Modeling the Turbulent Destruction ....................................................................................... 44
4.2.6. Model Constants .................................................................................................................... 45
4.2.7. Wall Boundary Conditions ...................................................................................................... 45
4.2.8. Convective Heat and Mass Transfer Modeling .......................................................................... 45
4.3. Standard, RNG, and Realizable k-ε Models ........................................................................................ 45
4.3.1. Standard k-ε Model ................................................................................................................ 46
4.3.1.1. Overview ....................................................................................................................... 46
4.3.1.2.Transport Equations for the Standard k-ε Model ............................................................. 46
4.3.1.3. Modeling the Turbulent Viscosity ................................................................................... 47
4.3.1.4. Model Constants ........................................................................................................... 47
4.3.2. RNG k-ε Model ....................................................................................................................... 47
4.3.2.1. Overview ....................................................................................................................... 47
4.3.2.2.Transport Equations for the RNG k-ε Model ..................................................................... 48
4.3.2.3. Modeling the Effective Viscosity ..................................................................................... 48
4.3.2.4. RNG Swirl Modification .................................................................................................. 49
4.3.2.5. Calculating the Inverse Effective Prandtl Numbers .......................................................... 49
4.3.2.6.The R-ε Term in the ε Equation ........................................................................................ 49
4.3.2.7. Model Constants ........................................................................................................... 50
4.3.3. Realizable k-ε Model ............................................................................................................... 50
4.3.3.1. Overview ....................................................................................................................... 50
4.3.3.2.Transport Equations for the Realizable k-ε Model ............................................................ 51
4.3.3.3. Modeling the Turbulent Viscosity ................................................................................... 52
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4.3.3.4. Model Constants ........................................................................................................... 53
4.3.4. Modeling Turbulent Production in the k-ε Models ................................................................... 53
4.3.5. Effects of Buoyancy on Turbulence in the k-ε Models ............................................................... 53
4.3.6. Effects of Compressibility on Turbulence in the k-ε Models ...................................................... 54
4.3.7. Convective Heat and Mass Transfer Modeling in the k-ε Models ............................................... 55
4.4. Standard, BSL, and SST k-ω Models ................................................................................................... 56
4.4.1. Standard k-ω Model ............................................................................................................... 57
4.4.1.1. Overview ....................................................................................................................... 57
4.4.1.2.Transport Equations for the Standard k-ω Model ............................................................. 57
4.4.1.3. Modeling the Effective Diffusivity ................................................................................... 57
4.4.1.3.1. Low-Reynolds Number Correction ......................................................................... 57
4.4.1.4. Modeling the Turbulence Production ............................................................................. 58
4.4.1.4.1. Production of k ..................................................................................................... 58
4.4.1.4.2. Production of ω ..................................................................................................... 58
4.4.1.5. Modeling the Turbulence Dissipation ............................................................................. 58
4.4.1.5.1. Dissipation of k ..................................................................................................... 58
4.4.1.5.2. Dissipation of ω ..................................................................................................... 59
4.4.1.5.3. Compressibility Effects .......................................................................................... 59
4.4.1.6. Model Constants ........................................................................................................... 60
4.4.2. Baseline (BSL) k-ω Model ........................................................................................................ 60
4.4.2.1. Overview ....................................................................................................................... 60
4.4.2.2.Transport Equations for the BSL k-ω Model ..................................................................... 60
4.4.2.3. Modeling the Effective Diffusivity ................................................................................... 61
4.4.2.4. Modeling the Turbulence Production ............................................................................. 61
4.4.2.4.1. Production of k ..................................................................................................... 61
4.4.2.4.2. Production of ω ..................................................................................................... 61
4.4.2.5. Modeling the Turbulence Dissipation ............................................................................. 62
4.4.2.5.1. Dissipation of k ..................................................................................................... 62
4.4.2.5.2. Dissipation of ω ..................................................................................................... 62
4.4.2.6. Cross-Diffusion Modification .......................................................................................... 62
4.4.2.7. Model Constants ........................................................................................................... 62
4.4.3. Shear-Stress Transport (SST) k-ω Model ................................................................................... 63
4.4.3.1. Overview ....................................................................................................................... 63
4.4.3.2. Modeling the Turbulent Viscosity ................................................................................... 63
4.4.3.3. Model Constants ........................................................................................................... 63
4.4.4.Turbulence Damping .............................................................................................................. 64
4.4.5. Wall Boundary Conditions ...................................................................................................... 64
4.5. k-kl-ω Transition Model ................................................................................................................... 65
4.5.1. Overview ............................................................................................................................... 65
4.5.2.Transport Equations for the k-kl-ω Model ................................................................................ 65
4.5.2.1. Model Constants ........................................................................................................... 68
4.6.Transition SST Model ....................................................................................................................... 68
4.6.1. Overview ............................................................................................................................... 69
4.6.2.Transport Equations for the Transition SST Model .................................................................... 69
4.6.2.1. Separation-Induced Transition Correction ...................................................................... 71
4.6.2.2. Coupling the Transition Model and SST Transport Equations ........................................... 71
4.6.2.3.Transition SST and Rough Walls ...................................................................................... 72
4.6.3. Mesh Requirements ............................................................................................................... 72
4.6.4. Specifying Inlet Turbulence Levels .......................................................................................... 75
4.7. Intermittency Transition Model ....................................................................................................... 76
4.7.1. Overview ............................................................................................................................... 76
4.7.2.Transport Equations for the Intermittency Transition Model ..................................................... 77
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4.7.3. Coupling with the Other Models ............................................................................................. 79
4.7.4. Intermittency Transition Model and Rough Walls ..................................................................... 79
4.8.The V2F Model ................................................................................................................................ 79
4.9. Reynolds Stress Model (RSM) ........................................................................................................... 79
4.9.1. Overview ............................................................................................................................... 80
4.9.2. Reynolds Stress Transport Equations ....................................................................................... 80
4.9.3. Modeling Turbulent Diffusive Transport .................................................................................. 81
4.9.4. Modeling the Pressure-Strain Term ......................................................................................... 82
4.9.4.1. Linear Pressure-Strain Model .......................................................................................... 82
4.9.4.2. Low-Re Modifications to the Linear Pressure-Strain Model .............................................. 82
4.9.4.3. Quadratic Pressure-Strain Model .................................................................................... 83
4.9.4.4. Low-Re Stress-Omega Model ......................................................................................... 84
4.9.5. Effects of Buoyancy on Turbulence ......................................................................................... 85
4.9.6. Modeling the Turbulence Kinetic Energy ................................................................................. 85
4.9.7. Modeling the Dissipation Rate ................................................................................................ 86
4.9.8. Modeling the Turbulent Viscosity ............................................................................................ 86
4.9.9. Wall Boundary Conditions ...................................................................................................... 87
4.9.10. Convective Heat and Mass Transfer Modeling ........................................................................ 87
4.10. Scale-Adaptive Simulation (SAS) Model ......................................................................................... 88
4.10.1. Overview ............................................................................................................................. 88
4.10.2.Transport Equations for the SST-SAS Model ........................................................................... 89
4.10.3. SAS with Other ω-Based Turbulence Models .......................................................................... 90
4.11. Detached Eddy Simulation (DES) ................................................................................................... 90
4.11.1. Overview ............................................................................................................................. 91
4.11.2. DES with the Spalart-Allmaras Model .................................................................................... 91
4.11.3. DES with the Realizable k-ε Model ......................................................................................... 92
4.11.4. DES with the BSL or SST k-ω Model ....................................................................................... 93
4.11.5. DES with the Transition SST Model ........................................................................................ 93
4.11.6. Improved Delayed Detached Eddy Simulation (IDDES) .......................................................... 93
4.11.6.1. Overview of IDDES ....................................................................................................... 93
4.11.6.2. IDDES Model Formulation ............................................................................................ 94
4.12. Large Eddy Simulation (LES) Model ................................................................................................ 94
4.12.1. Overview ............................................................................................................................. 95
4.12.2. Subgrid-Scale Models ........................................................................................................... 95
4.12.2.1. Smagorinsky-Lilly Model .............................................................................................. 97
4.12.2.2. Dynamic Smagorinsky-Lilly Model ............................................................................... 97
4.12.2.3.Wall-Adapting Local Eddy-Viscosity (WALE) Model ........................................................ 98
4.12.2.4. Algebraic Wall-Modeled LES Model (WMLES) ................................................................ 98
4.12.2.4.1. Algebraic WMLES Model Formulation .................................................................. 99
4.12.2.4.1.1. Reynolds Number Scaling ......................................................................... 100
4.12.2.4.2. Algebraic WMLES S-Omega Model Formulation ................................................. 101
4.12.2.5. Dynamic Kinetic Energy Subgrid-Scale Model ............................................................. 101
4.12.3. Inlet Boundary Conditions for the LES Model ....................................................................... 102
4.12.3.1.Vortex Method ........................................................................................................... 102
4.12.3.2. Spectral Synthesizer ................................................................................................... 103
4.13. Embedded Large Eddy Simulation (ELES) ..................................................................................... 104
4.13.1. Overview ........................................................................................................................... 104
4.13.2. Selecting a Model ............................................................................................................... 105
4.13.3. Interfaces Treatment ........................................................................................................... 105
4.13.3.1. RANS-LES Interface .................................................................................................... 105
4.13.3.2. LES-RANS Interface .................................................................................................... 106
4.13.3.3. Internal Interface Without LES Zone ........................................................................... 106
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4.13.3.4. Grid Generation Guidelines ........................................................................................ 107
4.14. Near-Wall Treatments for Wall-Bounded Turbulent Flows .............................................................. 107
4.14.1. Overview ........................................................................................................................... 107
4.14.1.1.Wall Functions vs. Near-Wall Model ............................................................................. 108
4.14.1.2. Wall Functions ........................................................................................................... 110
4.14.2. Standard Wall Functions ..................................................................................................... 110
4.14.2.1. Momentum ............................................................................................................... 110
4.14.2.2. Energy ....................................................................................................................... 111
4.14.2.3. Species ...................................................................................................................... 113
4.14.2.4. Turbulence ................................................................................................................ 113
4.14.3. Scalable Wall Functions ....................................................................................................... 114
4.14.4. Non-Equilibrium Wall Functions .......................................................................................... 114
4.14.4.1. Standard Wall Functions vs. Non-Equilibrium Wall Functions ....................................... 116
4.14.4.2. Limitations of the Wall Function Approach ................................................................. 116
4.14.5. Enhanced Wall Treatment ε-Equation (EWT-ε) ...................................................................... 116
4.14.5.1.Two-Layer Model for Enhanced Wall Treatment ........................................................... 117
4.14.5.2. Enhanced Wall Treatment for Momentum and Energy Equations ................................. 118
4.14.6. Menter-Lechner ε-Equation (ML-ε) ...................................................................................... 120
4.14.6.1. Momentum Equations ............................................................................................... 122
4.14.6.2. k-ε Turbulence Models ............................................................................................... 122
4.14.6.3. Iteration Improvements ............................................................................................. 122
4.14.7. y -Insensitive Wall Treatment ω-Equation ........................................................................... 122
4.14.8. User-Defined Wall Functions ............................................................................................... 123
4.14.9. LES Near-Wall Treatment ..................................................................................................... 123
4.15. Curvature Correction for the Spalart-Allmaras and Two-Equation Models ..................................... 123
4.16. Production Limiters for Two-Equation Models .............................................................................. 125
4.17. Definition of Turbulence Scales .................................................................................................... 127
4.17.1. RANS and Hybrid (SAS and DES) Turbulence Models ............................................................ 127
4.17.2. Large Eddy Simulation (LES) Models .................................................................................... 127
5. Heat Transfer ....................................................................................................................................... 129
5.1. Introduction ................................................................................................................................. 129
5.2. Modeling Conductive and Convective Heat Transfer ...................................................................... 129
5.2.1. Heat Transfer Theory ............................................................................................................. 129
5.2.1.1.The Energy Equation .................................................................................................... 129
5.2.1.2.The Energy Equation in Moving Reference Frames ........................................................ 130
5.2.1.3.The Energy Equation for the Non-Premixed Combustion Model .................................... 130
5.2.1.4. Inclusion of Pressure Work and Kinetic Energy Terms .................................................... 131
5.2.1.5. Inclusion of the Viscous Dissipation Terms .................................................................... 131
5.2.1.6. Inclusion of the Species Diffusion Term ........................................................................ 131
5.2.1.7. Energy Sources Due to Reaction ................................................................................... 132
5.2.1.8. Energy Sources Due To Radiation ................................................................................. 132
5.2.1.9. Interphase Energy Sources ........................................................................................... 132
5.2.1.10. Energy Equation in Solid Regions ............................................................................... 132
5.2.1.11. Anisotropic Conductivity in Solids .............................................................................. 133
5.2.1.12. Diffusion at Inlets ....................................................................................................... 133
5.2.2. Natural Convection and Buoyancy-Driven Flows Theory ........................................................ 133
5.3. Modeling Radiation ...................................................................................................................... 134
5.3.1. Overview and Limitations ..................................................................................................... 134
5.3.1.1. Advantages and Limitations of the DTRM ..................................................................... 135
5.3.1.2. Advantages and Limitations of the P-1 Model ............................................................... 135
5.3.1.3. Advantages and Limitations of the Rosseland Model .................................................... 136
5.3.1.4. Advantages and Limitations of the DO Model ............................................................... 136
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5.3.1.5. Advantages and Limitations of the S2S Model .............................................................. 136
5.3.2. Radiative Transfer Equation .................................................................................................. 137
5.3.3. P-1 Radiation Model Theory .................................................................................................. 138
5.3.3.1. The P-1 Model Equations ............................................................................................. 138
5.3.3.2. Anisotropic Scattering ................................................................................................. 140
5.3.3.3. Particulate Effects in the P-1 Model .............................................................................. 140
5.3.3.4. Boundary Condition Treatment for the P-1 Model at Walls ............................................. 141
5.3.3.5. Boundary Condition Treatment for the P-1 Model at Flow Inlets and Exits ...................... 142
5.3.4. Rosseland Radiation Model Theory ....................................................................................... 142
5.3.4.1.The Rosseland Model Equations ................................................................................... 142
5.3.4.2. Anisotropic Scattering ................................................................................................. 143
5.3.4.3. Boundary Condition Treatment at Walls ........................................................................ 143
5.3.4.4. Boundary Condition Treatment at Flow Inlets and Exits ................................................. 143
5.3.5. Discrete Transfer Radiation Model (DTRM) Theory ................................................................. 143
5.3.5.1.The DTRM Equations .................................................................................................... 143
5.3.5.2. Ray Tracing .................................................................................................................. 144
5.3.5.3. Clustering .................................................................................................................... 144
5.3.5.4. Boundary Condition Treatment for the DTRM at Walls ................................................... 145
5.3.5.5. Boundary Condition Treatment for the DTRM at Flow Inlets and Exits ............................ 145
5.3.6. Discrete Ordinates (DO) Radiation Model Theory ................................................................... 146
5.3.6.1. The DO Model Equations ............................................................................................. 146
5.3.6.2. Energy Coupling and the DO Model ............................................................................. 147
5.3.6.2.1. Limitations of DO/Energy Coupling ..................................................................... 148
5.3.6.3. Angular Discretization and Pixelation ........................................................................... 148
5.3.6.4. Anisotropic Scattering ................................................................................................. 151
5.3.6.5. Particulate Effects in the DO Model .............................................................................. 152
5.3.6.6. Boundary and Cell Zone Condition Treatment at Opaque Walls ..................................... 152
5.3.6.6.1. Gray Diffuse Walls ............................................................................................... 154
5.3.6.6.2. Non-Gray Diffuse Walls ........................................................................................ 154
5.3.6.7. Cell Zone and Boundary Condition Treatment at Semi-Transparent Walls ...................... 154
5.3.6.7.1. Semi-Transparent Interior Walls ........................................................................... 155
5.3.6.7.2. Specular Semi-Transparent Walls ......................................................................... 156
5.3.6.7.3. Diffuse Semi-Transparent Walls ............................................................................ 158
5.3.6.7.4. Partially Diffuse Semi-Transparent Walls ............................................................... 159
5.3.6.7.5. Semi-Transparent Exterior Walls ........................................................................... 159
5.3.6.7.6. Limitations .......................................................................................................... 161
5.3.6.7.7. Solid Semi-Transparent Media ............................................................................. 162
5.3.6.8. Boundary Condition Treatment at Specular Walls and Symmetry Boundaries ................. 162
5.3.6.9. Boundary Condition Treatment at Periodic Boundaries ................................................. 162
5.3.6.10. Boundary Condition Treatment at Flow Inlets and Exits ............................................... 162
5.3.7. Surface-to-Surface (S2S) Radiation Model Theory .................................................................. 162
5.3.7.1. Gray-Diffuse Radiation ................................................................................................. 162
5.3.7.2.The S2S Model Equations ............................................................................................. 163
5.3.7.3. Clustering .................................................................................................................... 164
5.3.7.3.1. Clustering and View Factors ................................................................................ 164
5.3.7.3.2. Clustering and Radiosity ...................................................................................... 164
5.3.8. Radiation in Combusting Flows ............................................................................................ 165
5.3.8.1. The Weighted-Sum-of-Gray-Gases Model ..................................................................... 165
5.3.8.1.1.When the Total (Static) Gas Pressure is Not Equal to 1 atm .................................... 166
5.3.8.2.The Effect of Soot on the Absorption Coefficient ........................................................... 166
5.3.8.3.The Effect of Particles on the Absorption Coefficient ..................................................... 167
5.3.9. Choosing a Radiation Model ................................................................................................. 167
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5.3.9.1. External Radiation ....................................................................................................... 168
6. Heat Exchangers .................................................................................................................................. 169
6.1.The Macro Heat Exchanger Models ................................................................................................ 169
6.1.1. Overview of the Macro Heat Exchanger Models .................................................................... 169
6.1.2. Restrictions of the Macro Heat Exchanger Models ................................................................. 171
6.1.3. Macro Heat Exchanger Model Theory .................................................................................... 172
6.1.3.1. Streamwise Pressure Drop ........................................................................................... 173
6.1.3.2. Heat Transfer Effectiveness ........................................................................................... 174
6.1.3.3. Heat Rejection ............................................................................................................. 175
6.1.3.4. Macro Heat Exchanger Group Connectivity .................................................................. 176
6.2. The Dual Cell Model ...................................................................................................................... 177
6.2.1. Overview of the Dual Cell Model ........................................................................................... 177
6.2.2. Restrictions of the Dual Cell Model ........................................................................................ 178
6.2.3. Dual Cell Model Theory ......................................................................................................... 178
6.2.3.1. NTU Relations .............................................................................................................. 179
6.2.3.2. Heat Rejection ............................................................................................................. 179
7. Species Transport and Finite-Rate Chemistry ..................................................................................... 181
7.1. Volumetric Reactions .................................................................................................................... 181
7.1.1. Species Transport Equations ................................................................................................. 181
7.1.1.1. Mass Diffusion in Laminar Flows ................................................................................... 182
7.1.1.2. Mass Diffusion in Turbulent Flows ................................................................................ 182
7.1.1.3.Treatment of Species Transport in the Energy Equation ................................................. 182
7.1.1.4. Diffusion at Inlets ......................................................................................................... 182
7.1.2.The Generalized Finite-Rate Formulation for Reaction Modeling ............................................ 183
7.1.2.1.The Laminar Finite-Rate Model ..................................................................................... 183
7.1.2.2. Pressure-Dependent Reactions .................................................................................... 186
7.1.2.3.The Eddy-Dissipation Model ......................................................................................... 187
7.1.2.4. The Eddy-Dissipation Model for LES ............................................................................. 188
7.1.2.5. The Eddy-Dissipation-Concept (EDC) Model ................................................................. 188
7.1.2.6.The Thickened Flame Model ......................................................................................... 189
7.1.2.7.The Relaxation to Chemical Equilibrium Model ............................................................. 191
7.2.Wall Surface Reactions and Chemical Vapor Deposition .................................................................. 192
7.2.1. Surface Coverage Reaction Rate Modification ....................................................................... 194
7.2.2. Reaction-Diffusion Balance for Surface Chemistry ................................................................. 194
7.2.3. Slip Boundary Formulation for Low-Pressure Gas Systems ..................................................... 195
7.3. Particle Surface Reactions ............................................................................................................. 197
7.3.1. General Description .............................................................................................................. 197
7.3.2. ANSYS Fluent Model Formulation ......................................................................................... 198
7.3.3. Extension for Stoichiometries with Multiple Gas Phase Reactants .......................................... 199
7.3.4. Solid-Solid Reactions ............................................................................................................ 200
7.3.5. Solid Decomposition Reactions ............................................................................................ 200
7.3.6. Solid Deposition Reactions ................................................................................................... 200
7.3.7. Gaseous Solid Catalyzed Reactions on the Particle Surface .................................................... 200
7.4. Reacting Channel Model ............................................................................................................... 201
7.4.1. Overview and Limitations ..................................................................................................... 201
7.4.2. Reacting Channel Model Theory ........................................................................................... 201
7.4.2.1. Flow Inside the Reacting Channel ................................................................................. 202
7.4.2.2. Surface Reactions in the Reacting Channel ................................................................... 203
7.4.2.3. Porous Medium Inside Reacting Channel ...................................................................... 204
7.4.2.4. Outer Flow in the Shell ................................................................................................. 204
7.5. Reactor Network Model ................................................................................................................ 205
7.5.1. Reactor Network Model Theory ............................................................................................ 205
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7.5.1.1. Reactor network temperature solution ......................................................................... 206
8. Non-Premixed Combustion ................................................................................................................. 207
8.1. Introduction ................................................................................................................................. 207
8.2. Non-Premixed Combustion and Mixture Fraction Theory ............................................................... 207
8.2.1. Mixture Fraction Theory ....................................................................................................... 208
8.2.1.1. Definition of the Mixture Fraction ................................................................................ 208
8.2.1.2.Transport Equations for the Mixture Fraction ................................................................ 210
8.2.1.3. The Non-Premixed Model for LES ................................................................................. 211
8.2.1.4. Mixture Fraction vs. Equivalence Ratio .......................................................................... 211
8.2.1.5. Relationship of Mixture Fraction to Species Mass Fraction, Density, and Temperature ..... 212
8.2.2. Modeling of Turbulence-Chemistry Interaction ..................................................................... 213
8.2.2.1. Description of the Probability Density Function ............................................................ 213
8.2.2.2. Derivation of Mean Scalar Values from the Instantaneous Mixture Fraction ................... 213
8.2.2.3. The Assumed-Shape PDF ............................................................................................. 214
8.2.2.3.1.The Double Delta Function PDF ........................................................................... 214
8.2.2.3.2.The β-Function PDF ............................................................................................. 215
8.2.3. Non-Adiabatic Extensions of the Non-Premixed Model .......................................................... 216
8.2.4. Chemistry Tabulation ........................................................................................................... 218
8.2.4.1. Look-Up Tables for Adiabatic Systems ........................................................................... 218
8.2.4.2. 3D Look-Up Tables for Non-Adiabatic Systems .............................................................. 220
8.2.4.3. Generating Lookup Tables Through Automated Grid Refinement .................................. 222
8.3. Restrictions and Special Cases for Using the Non-Premixed Model ................................................. 224
8.3.1. Restrictions on the Mixture Fraction Approach ...................................................................... 224
8.3.2. Using the Non-Premixed Model for Liquid Fuel or Coal Combustion ...................................... 227
8.3.3. Using the Non-Premixed Model with Flue Gas Recycle .......................................................... 228
8.3.4. Using the Non-Premixed Model with the Inert Model ............................................................ 228
8.3.4.1. Mixture Composition ................................................................................................... 229
8.3.4.1.1. Property Evaluation ............................................................................................. 230
8.4.The Diffusion Flamelet Models Theory ........................................................................................... 230
8.4.1. Restrictions and Assumptions ............................................................................................... 230
8.4.2.The Flamelet Concept ........................................................................................................... 230
8.4.2.1. Overview ..................................................................................................................... 230
8.4.2.2. Strain Rate and Scalar Dissipation ................................................................................. 232
8.4.2.3. Embedding Diffusion Flamelets in Turbulent Flames ..................................................... 232
8.4.3. Flamelet Generation ............................................................................................................. 233
8.4.4. Flamelet Import ................................................................................................................... 234
8.5. The Steady Diffusion Flamelet Model Theory ................................................................................. 236
8.5.1. Overview ............................................................................................................................. 236
8.5.2. Multiple Steady Flamelet Libraries ........................................................................................ 237
8.5.3. Steady Diffusion Flamelet Automated Grid Refinement ......................................................... 237
8.5.4. Non-Adiabatic Steady Diffusion Flamelets ............................................................................. 237
8.6. The Unsteady Diffusion Flamelet Model Theory ............................................................................. 238
8.6.1. The Eulerian Unsteady Laminar Flamelet Model .................................................................... 239
8.6.1.1. Liquid Reactions .......................................................................................................... 241
8.6.2. The Diesel Unsteady Laminar Flamelet Model ....................................................................... 241
8.6.3. Multiple Diesel Unsteady Flamelets ....................................................................................... 242
8.6.4. Multiple Diesel Unsteady Flamelets with Flamelet Reset ........................................................ 243
8.6.4.1. Resetting the Flamelets ................................................................................................ 243
9. Premixed Combustion ......................................................................................................................... 245
9.1. Overview and Limitations ............................................................................................................. 245
9.1.1. Overview ............................................................................................................................. 245
9.1.2. Limitations ........................................................................................................................... 246
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9.2. C-Equation Model Theory .............................................................................................................. 246
9.2.1. Propagation of the Flame Front ............................................................................................ 246
9.3. G-Equation Model Theory ............................................................................................................. 247
9.3.1. Numerical Solution of the G-equation ................................................................................... 248
9.4. Turbulent Flame Speed Models ..................................................................................................... 249
9.4.1. Zimont Turbulent Flame Speed Closure Model ...................................................................... 249
9.4.1.1. Zimont Turbulent Flame Speed Closure for LES ............................................................. 250
9.4.1.2. Flame Stretch Effect ..................................................................................................... 251
9.4.1.3. Gradient Diffusion ....................................................................................................... 251
9.4.1.4.Wall Damping .............................................................................................................. 252
9.4.2. Peters Flame Speed Model .................................................................................................... 252
9.4.2.1. Peters Flame Speed Model for LES ................................................................................ 253
9.5. Extended Coherent Flamelet Model Theory ................................................................................... 254
9.5.1. Closure for ECFM Source Terms ............................................................................................. 255
9.5.2.Turbulent Flame Speed in ECFM ............................................................................................ 258
9.5.3. LES and ECFM ...................................................................................................................... 258
9.6. Calculation of Properties ............................................................................................................... 260
9.6.1. Calculation of Temperature ................................................................................................... 260
9.6.1.1. Adiabatic Temperature Calculation ............................................................................... 260
9.6.1.2. Non-Adiabatic Temperature Calculation ....................................................................... 260
9.6.2. Calculation of Density .......................................................................................................... 261
9.6.3. Laminar Flame Speed ........................................................................................................... 261
9.6.4. Unburnt Density and Thermal Diffusivity ............................................................................... 262
10. Partially Premixed Combustion ........................................................................................................ 263
10.1. Overview .................................................................................................................................... 263
10.2. Limitations .................................................................................................................................. 263
10.3. Partially Premixed Combustion Theory ........................................................................................ 264
10.3.1. Chemical Equilibrium and Steady Diffusion Flamelet Models ............................................... 264
10.3.2. Flamelet Generated Manifold (FGM) model ......................................................................... 265
10.3.2.1. Premixed FGMs .......................................................................................................... 265
10.3.2.2. Diffusion FGMs .......................................................................................................... 267
10.3.3. FGM Turbulent Closure ....................................................................................................... 267
10.3.4. Calculation of Unburnt Properties ....................................................................................... 269
10.3.5. Laminar Flame Speed ......................................................................................................... 269
11. Composition PDF Transport .............................................................................................................. 271
11.1. Overview and Limitations ............................................................................................................ 271
11.2. Composition PDF Transport Theory ............................................................................................. 271
11.3.The Lagrangian Solution Method ................................................................................................. 272
11.3.1. Particle Convection ............................................................................................................ 273
11.3.2. Particle Mixing ................................................................................................................... 274
11.3.2.1.The Modified Curl Model ............................................................................................ 274
11.3.2.2.The IEM Model ........................................................................................................... 274
11.3.2.3. The EMST Model ........................................................................................................ 275
11.3.2.4. Liquid Reactions ........................................................................................................ 275
11.3.3. Particle Reaction ................................................................................................................. 275
11.4. The Eulerian Solution Method ..................................................................................................... 276
11.4.1. Reaction ............................................................................................................................. 277
11.4.2. Mixing ................................................................................................................................ 277
11.4.3. Correction .......................................................................................................................... 278
11.4.4. Calculation of Composition Mean and Variance ................................................................... 278
12. Chemistry Acceleration ..................................................................................................................... 279
12.1. Overview and Limitations ............................................................................................................ 279
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12.2. In-Situ Adaptive Tabulation (ISAT) ................................................................................................ 279
12.3. Dynamic Mechanism Reduction .................................................................................................. 281
12.3.1. Directed Relation Graph (DRG) Method for Mechanism Reduction ....................................... 282
12.4. Chemistry Agglomeration ........................................................................................................... 283
12.4.1. Binning Algorithm .............................................................................................................. 284
12.5. Chemical Mechanism Dimension Reduction ................................................................................ 286
12.5.1. Selecting the Represented Species ...................................................................................... 286
13. Engine Ignition .................................................................................................................................. 289
13.1. Spark Model ................................................................................................................................ 289
13.1.1. Overview and Limitations ................................................................................................... 289
13.1.2. Spark Model Theory ............................................................................................................ 289
13.1.3. ECFM Spark Model Variants ................................................................................................. 292
13.2. Autoignition Models ................................................................................................................... 293
13.2.1. Model Overview ................................................................................................................. 293
13.2.2. Model Limitations .............................................................................................................. 293
13.2.3. Ignition Model Theory ........................................................................................................ 294
13.2.3.1.Transport of Ignition Species ...................................................................................... 294
13.2.3.2. Knock Modeling ........................................................................................................ 294
13.2.3.2.1. Modeling of the Source Term ............................................................................. 295
13.2.3.2.2. Correlations ...................................................................................................... 295
13.2.3.2.3. Energy Release .................................................................................................. 296
13.2.3.3. Ignition Delay Modeling ............................................................................................. 296
13.2.3.3.1. Modeling of the Source Term ............................................................................. 296
13.2.3.3.2. Correlations ...................................................................................................... 297
13.2.3.3.3. Energy Release .................................................................................................. 297
13.3. Crevice Model ............................................................................................................................. 297
13.3.1. Overview ........................................................................................................................... 297
13.3.1.1. Model Parameters ...................................................................................................... 298
13.3.2. Limitations ......................................................................................................................... 299
13.3.3. Crevice Model Theory ......................................................................................................... 300
14. Pollutant Formation .......................................................................................................................... 301
14.1. NOx Formation ........................................................................................................................... 301
14.1.1. Overview ........................................................................................................................... 301
14.1.1.1. NOx Modeling in ANSYS Fluent .................................................................................. 301
14.1.1.2. NOx Formation and Reduction in Flames .................................................................... 302
14.1.2. Governing Equations for NOx Transport .............................................................................. 302
14.1.3.Thermal NOx Formation ...................................................................................................... 303
14.1.3.1. Thermal NOx Reaction Rates ...................................................................................... 303
14.1.3.2. The Quasi-Steady Assumption for [N] ......................................................................... 303
14.1.3.3.Thermal NOx Temperature Sensitivity ......................................................................... 304
14.1.3.4. Decoupled Thermal NOx Calculations ......................................................................... 304
14.1.3.5. Approaches for Determining O Radical Concentration ................................................ 304
14.1.3.5.1. Method 1: Equilibrium Approach ....................................................................... 304
14.1.3.5.2. Method 2: Partial Equilibrium Approach ............................................................. 305
14.1.3.5.3. Method 3: Predicted O Approach ....................................................................... 305
14.1.3.6. Approaches for Determining OH Radical Concentration .............................................. 305
14.1.3.6.1. Method 1: Exclusion of OH Approach ................................................................. 305
14.1.3.6.2. Method 2: Partial Equilibrium Approach ............................................................. 305
14.1.3.6.3. Method 3: Predicted OH Approach ..................................................................... 306
14.1.3.7. Summary ................................................................................................................... 306
14.1.4. Prompt NOx Formation ....................................................................................................... 306
14.1.4.1. Prompt NOx Combustion Environments ..................................................................... 306
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14.1.4.2. Prompt NOx Mechanism ............................................................................................ 306
14.1.4.3. Prompt NOx Formation Factors .................................................................................. 307
14.1.4.4. Primary Reaction ....................................................................................................... 307
14.1.4.5. Modeling Strategy ..................................................................................................... 307
14.1.4.6. Rate for Most Hydrocarbon Fuels ................................................................................ 308
14.1.4.7. Oxygen Reaction Order .............................................................................................. 308
14.1.5. Fuel NOx Formation ............................................................................................................ 308
14.1.5.1. Fuel-Bound Nitrogen ................................................................................................. 308
14.1.5.2. Reaction Pathways ..................................................................................................... 309
14.1.5.3. Fuel NOx from Gaseous and Liquid Fuels .................................................................... 309
14.1.5.3.1. Fuel NOx from Intermediate Hydrogen Cyanide (HCN) ....................................... 309
14.1.5.3.1.1. HCN Production in a Gaseous Fuel ............................................................ 310
14.1.5.3.1.2. HCN Production in a Liquid Fuel ................................................................ 310
14.1.5.3.1.3. HCN Consumption .................................................................................... 311
14.1.5.3.1.4. HCN Sources in the Transport Equation ..................................................... 311
14.1.5.3.1.5. NOx Sources in the Transport Equation ..................................................... 311
14.1.5.3.2. Fuel NOx from Intermediate Ammonia (NH3) ..................................................... 312
14.1.5.3.2.1. NH3 Production in a Gaseous Fuel ............................................................. 312
14.1.5.3.2.2. NH3 Production in a Liquid Fuel ................................................................ 312
14.1.5.3.2.3. NH3 Consumption .................................................................................... 313
14.1.5.3.2.4. NH3 Sources in the Transport Equation ..................................................... 313
14.1.5.3.2.5. NOx Sources in the Transport Equation ..................................................... 313
14.1.5.3.3. Fuel NOx from Coal ........................................................................................... 314
14.1.5.3.3.1. Nitrogen in Char and in Volatiles ............................................................... 314
14.1.5.3.3.2. Coal Fuel NOx Scheme A ........................................................................... 314
14.1.5.3.3.3. Coal Fuel NOx Scheme B ........................................................................... 314
14.1.5.3.3.4. HCN Scheme Selection ............................................................................. 315
14.1.5.3.3.5. NOx Reduction on Char Surface ................................................................ 315
14.1.5.3.3.5.1. BET Surface Area .............................................................................. 316
14.1.5.3.3.5.2. HCN from Volatiles ........................................................................... 316
14.1.5.3.3.6. Coal Fuel NOx Scheme C ........................................................................... 316
14.1.5.3.3.7. Coal Fuel NOx Scheme D ........................................................................... 317
14.1.5.3.3.8. NH3 Scheme Selection ............................................................................. 318
14.1.5.3.3.8.1. NH3 from Volatiles ........................................................................... 318
14.1.5.3.4. Fuel Nitrogen Partitioning for HCN and NH3 Intermediates ................................ 318
14.1.6. NOx Formation from Intermediate N2O ............................................................................... 319
14.1.6.1. N2O - Intermediate NOx Mechanism .......................................................................... 319
14.1.7. NOx Reduction by Reburning ............................................................................................. 320
14.1.7.1. Instantaneous Approach ............................................................................................ 320
14.1.7.2. Partial Equilibrium Approach ..................................................................................... 321
14.1.7.2.1. NOx Reduction Mechanism ............................................................................... 321
14.1.8. NOx Reduction by SNCR ..................................................................................................... 323
14.1.8.1. Ammonia Injection .................................................................................................... 323
14.1.8.2. Urea Injection ............................................................................................................ 324
14.1.8.3. Transport Equations for Urea, HNCO, and NCO ............................................................ 325
14.1.8.4. Urea Production due to Reagent Injection .................................................................. 326
14.1.8.5. NH3 Production due to Reagent Injection ................................................................... 326
14.1.8.6. HNCO Production due to Reagent Injection ................................................................ 326
14.1.9. NOx Formation in Turbulent Flows ...................................................................................... 327
14.1.9.1. The Turbulence-Chemistry Interaction Model ............................................................. 327
14.1.9.2. The PDF Approach ..................................................................................................... 328
14.1.9.3.The General Expression for the Mean Reaction Rate .................................................... 328
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14.1.9.4.The Mean Reaction Rate Used in ANSYS Fluent ........................................................... 328
14.1.9.5. Statistical Independence ............................................................................................ 328
14.1.9.6.The Beta PDF Option .................................................................................................. 329
14.1.9.7.The Gaussian PDF Option ........................................................................................... 329
14.1.9.8. The Calculation Method for the Variance .................................................................... 329
14.2. SOx Formation ............................................................................................................................ 330
14.2.1. Overview ........................................................................................................................... 330
14.2.1.1.The Formation of SOx ................................................................................................. 330
14.2.2. Governing Equations for SOx Transport ............................................................................... 331
14.2.3. Reaction Mechanisms for Sulfur Oxidation .......................................................................... 332
14.2.4. SO2 and H2S Production in a Gaseous Fuel ......................................................................... 333
14.2.5. SO2 and H2S Production in a Liquid Fuel ............................................................................. 334
14.2.6. SO2 and H2S Production from Coal ..................................................................................... 334
14.2.6.1. SO2 and H2S from Char .............................................................................................. 334
14.2.6.2. SO2 and H2S from Volatiles ........................................................................................ 334
14.2.7. SOx Formation in Turbulent Flows ....................................................................................... 335
14.2.7.1. The Turbulence-Chemistry Interaction Model ............................................................. 335
14.2.7.2. The PDF Approach ..................................................................................................... 335
14.2.7.3.The Mean Reaction Rate ............................................................................................. 335
14.2.7.4.The PDF Options ........................................................................................................ 335
14.3. Soot Formation ........................................................................................................................... 335
14.3.1. Overview and Limitations ................................................................................................... 336
14.3.1.1. Predicting Soot Formation ......................................................................................... 336
14.3.1.2. Restrictions on Soot Modeling ................................................................................... 336
14.3.2. Soot Model Theory ............................................................................................................. 337
14.3.2.1.The One-Step Soot Formation Model .......................................................................... 337
14.3.2.2.The Two-Step Soot Formation Model .......................................................................... 338
14.3.2.2.1. Soot Generation Rate ........................................................................................ 338
14.3.2.2.2. Nuclei Generation Rate ...................................................................................... 339
14.3.2.3. The Moss-Brookes Model ........................................................................................... 339
14.3.2.3.1.The Moss-Brookes-Hall Model ............................................................................ 341
14.3.2.3.2. Soot Formation in Turbulent Flows .................................................................... 342
14.3.2.3.2.1.The Turbulence-Chemistry Interaction Model ............................................ 342
14.3.2.3.2.2.The PDF Approach .................................................................................... 343
14.3.2.3.2.3. The Mean Reaction Rate ........................................................................... 343
14.3.2.3.2.4.The PDF Options ....................................................................................... 343
14.3.2.3.3.The Effect of Soot on the Radiation Absorption Coefficient ................................. 343
14.3.2.4.The Method of Moments Model ................................................................................. 343
14.3.2.4.1. Soot Particle Population Balance ....................................................................... 343
14.3.2.4.2. Moment Transport Equations ............................................................................ 345
14.3.2.4.3. Nucleation ........................................................................................................ 345
14.3.2.4.4. Coagulation ...................................................................................................... 347
14.3.2.4.5. Surface Growth and Oxidation ........................................................................... 350
14.4. Decoupled Detailed Chemistry Model ......................................................................................... 352
14.4.1. Overview ........................................................................................................................... 352
14.4.1.1. Limitations ................................................................................................................ 352
14.4.2. Decoupled Detailed Chemistry Model Theory ..................................................................... 353
15. Aerodynamically Generated Noise ................................................................................................... 355
15.1. Overview .................................................................................................................................... 355
15.1.1. Direct Method .................................................................................................................... 355
15.1.2. Integral Method Based on Acoustic Analogy ....................................................................... 356
15.1.3. Broadband Noise Source Models ........................................................................................ 357
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15.2. Acoustics Model Theory .............................................................................................................. 357
15.2.1. The Ffowcs-Williams and Hawkings Model .......................................................................... 357
15.2.2. Broadband Noise Source Models ........................................................................................ 360
15.2.2.1. Proudman’s Formula .................................................................................................. 360
15.2.2.2.The Jet Noise Source Model ........................................................................................ 361
15.2.2.3.The Boundary Layer Noise Source Model .................................................................... 362
15.2.2.4. Source Terms in the Linearized Euler Equations ........................................................... 363
15.2.2.5. Source Terms in Lilley’s Equation ................................................................................ 363
16. Discrete Phase ................................................................................................................................... 365
16.1. Introduction ............................................................................................................................... 365
16.1.1.The Euler-Lagrange Approach ............................................................................................. 365
16.2. Particle Motion Theory ................................................................................................................ 366
16.2.1. Equations of Motion for Particles ........................................................................................ 366
16.2.1.1. Particle Force Balance ................................................................................................ 366
16.2.1.2. Inclusion of the Gravity Term ...................................................................................... 366
16.2.1.3. Other Forces .............................................................................................................. 366
16.2.1.4. Forces in Moving Reference Frames ............................................................................ 367
16.2.1.5.Thermophoretic Force ................................................................................................ 367
16.2.1.6. Brownian Force .......................................................................................................... 368
16.2.1.7. Saffman’s Lift Force .................................................................................................... 368
16.2.2.Turbulent Dispersion of Particles ......................................................................................... 368
16.2.2.1. Stochastic Tracking .................................................................................................... 369
16.2.2.1.1. The Integral Time .............................................................................................. 369
16.2.2.1.2.The Discrete Random Walk Model ...................................................................... 370
16.2.2.1.3. Using the DRW Model ....................................................................................... 371
16.2.2.2. Particle Cloud Tracking ............................................................................................... 371
16.2.2.2.1. Using the Cloud Model ...................................................................................... 374
16.2.3. Integration of Particle Equation of Motion ........................................................................... 374
16.3. Laws for Drag Coefficients ........................................................................................................... 376
16.3.1. Spherical Drag Law ............................................................................................................. 376
16.3.2. Non-spherical Drag Law ..................................................................................................... 376
16.3.3. Stokes-Cunningham Drag Law ............................................................................................ 377
16.3.4. High-Mach-Number Drag Law ............................................................................................ 377
16.3.5. Dynamic Drag Model Theory .............................................................................................. 377
16.3.6. Dense Discrete Phase Model Drag Laws .............................................................................. 378
16.4. Laws for Heat and Mass Exchange ............................................................................................... 378
16.4.1. Inert Heating or Cooling (Law 1/Law 6) ............................................................................... 379
16.4.2. Droplet Vaporization (Law 2) ............................................................................................... 381
16.4.2.1. Mass Transfer During Law 2—Diffusion Controlled Model ........................................... 381
16.4.2.2. Mass Transfer During Law 2—Convection/Diffusion Controlled Model ........................ 382
16.4.2.3. Defining the Saturation Vapor Pressure and Diffusion Coefficient (or Binary Diffusivity) ......................................................................................................................................... 383
16.4.2.4. Defining the Boiling Point and Latent Heat ................................................................. 384
16.4.2.5. Heat Transfer to the Droplet ....................................................................................... 384
16.4.3. Droplet Boiling (Law 3) ....................................................................................................... 385
16.4.4. Devolatilization (Law 4) ...................................................................................................... 386
16.4.4.1. Choosing the Devolatilization Model .......................................................................... 387
16.4.4.2.The Constant Rate Devolatilization Model ................................................................... 387
16.4.4.3. The Single Kinetic Rate Model .................................................................................... 387
16.4.4.4.The Two Competing Rates (Kobayashi) Model ............................................................. 388
16.4.4.5. The CPD Model .......................................................................................................... 389
16.4.4.5.1. General Description .......................................................................................... 389
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16.4.4.5.2. Reaction Rates .................................................................................................. 390
16.4.4.5.3. Mass Conservation ............................................................................................ 390
16.4.4.5.4. Fractional Change in the Coal Mass .................................................................... 391
16.4.4.5.5. CPD Inputs ........................................................................................................ 392
16.4.4.5.6. Particle Swelling During Devolatilization ............................................................ 393
16.4.4.5.7. Heat Transfer to the Particle During Devolatilization ........................................... 393
16.4.5. Surface Combustion (Law 5) ............................................................................................... 394
16.4.5.1.The Diffusion-Limited Surface Reaction Rate Model .................................................... 394
16.4.5.2.The Kinetic/Diffusion Surface Reaction Rate Model ..................................................... 395
16.4.5.3. The Intrinsic Model .................................................................................................... 395
16.4.5.4.The Multiple Surface Reactions Model ........................................................................ 397
16.4.5.4.1. Limitations ........................................................................................................ 397
16.4.5.5. Heat and Mass Transfer During Char Combustion ....................................................... 397
16.4.6. Multicomponent Particle Definition (Law 7) ........................................................................ 398
16.4.6.1. Raoult’s Law .............................................................................................................. 400
16.4.6.2. Peng-Robinson Real Gas Model .................................................................................. 400
16.5.Vapor Liquid Equilibrium Theory .................................................................................................. 400
16.6.Wall-Jet Model Theory ................................................................................................................. 402
16.7.Wall-Film Model Theory ............................................................................................................... 403
16.7.1. Introduction ....................................................................................................................... 403
16.7.2. Interaction During Impact with a Boundary ......................................................................... 404
16.7.3. Splashing ........................................................................................................................... 406
16.7.4. Separation Criteria .............................................................................................................. 408
16.7.5. Conservation Equations for Wall-Film Particles .................................................................... 408
16.7.5.1. Momentum ............................................................................................................... 408
16.7.5.2. Mass Transfer from the Film ........................................................................................ 409
16.7.5.3. Energy Transfer from the Film ..................................................................................... 412
16.8. Particle Erosion and Accretion Theory .......................................................................................... 413
16.9. Particle–Wall Impingement Heat Transfer Theory ......................................................................... 415
16.10. Atomizer Model Theory ............................................................................................................. 416
16.10.1.The Plain-Orifice Atomizer Model ...................................................................................... 417
16.10.1.1. Internal Nozzle State ................................................................................................ 418
16.10.1.2. Coefficient of Discharge ........................................................................................... 419
16.10.1.3. Exit Velocity ............................................................................................................. 421
16.10.1.4. Spray Angle ............................................................................................................. 421
16.10.1.5. Droplet Diameter Distribution .................................................................................. 421
16.10.2. The Pressure-Swirl Atomizer Model ................................................................................... 423
16.10.2.1. Film Formation ........................................................................................................ 423
16.10.2.2. Sheet Breakup and Atomization ............................................................................... 424
16.10.3.The Air-Blast/Air-Assist Atomizer Model ............................................................................. 426
16.10.4.The Flat-Fan Atomizer Model ............................................................................................. 427
16.10.5.The Effervescent Atomizer Model ...................................................................................... 428
16.11. Secondary Breakup Model Theory ............................................................................................. 429
16.11.1.Taylor Analogy Breakup (TAB) Model ................................................................................. 429
16.11.1.1. Introduction ............................................................................................................ 429
16.11.1.2. Use and Limitations ................................................................................................. 430
16.11.1.3. Droplet Distortion .................................................................................................... 430
16.11.1.4. Size of Child Droplets ............................................................................................... 431
16.11.1.5.Velocity of Child Droplets ......................................................................................... 432
16.11.1.6. Droplet Breakup ...................................................................................................... 432
16.11.2.Wave Breakup Model ........................................................................................................ 433
16.11.2.1. Introduction ............................................................................................................ 433
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16.11.2.2. Use and Limitations ................................................................................................. 434
16.11.2.3. Jet Stability Analysis ................................................................................................. 434
16.11.2.4. Droplet Breakup ...................................................................................................... 435
16.11.3. KHRT Breakup Model ........................................................................................................ 436
16.11.3.1. Introduction ............................................................................................................ 436
16.11.3.2. Use and Limitations ................................................................................................. 436
16.11.3.3. Liquid Core Length .................................................................................................. 436
16.11.3.4. Rayleigh-Taylor Breakup ........................................................................................... 437
16.11.3.5. Droplet Breakup Within the Liquid Core .................................................................... 438
16.11.3.6. Droplet Breakup Outside the Liquid Core .................................................................. 438
16.11.4. Stochastic Secondary Droplet (SSD) Model ........................................................................ 438
16.11.4.1. Theory ..................................................................................................................... 438
16.12. Collision and Droplet Coalescence Model Theory ....................................................................... 439
16.12.1. Introduction ..................................................................................................................... 439
16.12.2. Use and Limitations .......................................................................................................... 440
16.12.3.Theory .............................................................................................................................. 440
16.12.3.1. Probability of Collision ............................................................................................. 440
16.12.3.2. Collision Outcomes .................................................................................................. 441
16.13. Discrete Element Method Collision Model .................................................................................. 442
16.13.1.Theory .............................................................................................................................. 442
16.13.1.1. The Spring Collision Law .......................................................................................... 443
16.13.1.2. The Spring-Dashpot Collision Law ............................................................................ 444
16.13.1.3.The Friction Collision Law ......................................................................................... 444
16.13.1.4. DEM Parcels ............................................................................................................. 446
16.13.1.5. Cartesian Collision Mesh .......................................................................................... 446
16.14. One-Way and Two-Way Coupling ............................................................................................... 447
16.14.1. Coupling Between the Discrete and Continuous Phases .................................................... 447
16.14.2. Momentum Exchange ...................................................................................................... 448
16.14.3. Heat Exchange ................................................................................................................. 449
16.14.4. Mass Exchange ................................................................................................................. 450
16.14.5. Under-Relaxation of the Interphase Exchange Terms ......................................................... 450
16.14.6. Interphase Exchange During Stochastic Tracking ............................................................... 451
16.14.7. Interphase Exchange During Cloud Tracking ..................................................................... 451
16.15. Node Based Averaging .............................................................................................................. 451
17. Multiphase Flows .............................................................................................................................. 453
17.1. Introduction ............................................................................................................................... 453
17.1.1. Multiphase Flow Regimes ................................................................................................... 453
17.1.1.1. Gas-Liquid or Liquid-Liquid Flows .............................................................................. 453
17.1.1.2. Gas-Solid Flows .......................................................................................................... 454
17.1.1.3. Liquid-Solid Flows ...................................................................................................... 454
17.1.1.4. Three-Phase Flows ..................................................................................................... 454
17.1.2. Examples of Multiphase Systems ........................................................................................ 455
17.2. Choosing a General Multiphase Model ........................................................................................ 456
17.2.1. Approaches to Multiphase Modeling .................................................................................. 456
17.2.1.1.The Euler-Euler Approach ........................................................................................... 456
17.2.1.1.1.The VOF Model .................................................................................................. 456
17.2.1.1.2. The Mixture Model ............................................................................................ 457
17.2.1.1.3.The Eulerian Model ............................................................................................ 457
17.2.2. Model Comparisons ........................................................................................................... 457
17.2.2.1. Detailed Guidelines ................................................................................................... 458
17.2.2.1.1.The Effect of Particulate Loading ........................................................................ 458
17.2.2.1.2.The Significance of the Stokes Number .............................................................. 459
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17.2.2.1.2.1. Examples .................................................................................................. 459
17.2.2.1.3. Other Considerations ........................................................................................ 460
17.2.3.Time Schemes in Multiphase Flow ....................................................................................... 460
17.2.4. Stability and Convergence .................................................................................................. 461
17.3.Volume of Fluid (VOF) Model Theory ............................................................................................ 462
17.3.1. Overview of the VOF Model ................................................................................................ 462
17.3.2. Limitations of the VOF Model .............................................................................................. 462
17.3.3. Steady-State and Transient VOF Calculations ....................................................................... 462
17.3.4.Volume Fraction Equation ................................................................................................... 463
17.3.4.1. The Implicit Formulation ............................................................................................ 463
17.3.4.2.The Explicit Formulation ............................................................................................. 464
17.3.4.3. Interpolation Near the Interface ................................................................................. 465
17.3.4.3.1. The Geometric Reconstruction Scheme ............................................................. 466
17.3.4.3.2.The Donor-Acceptor Scheme ............................................................................. 466
17.3.4.3.3.The Compressive Interface Capturing Scheme for Arbitrary Meshes (CICSAM) ..... 467
17.3.4.3.4.The Compressive Scheme and Interface-Model-based Variants ........................... 467
17.3.4.3.5. Bounded Gradient Maximization (BGM) ............................................................. 468
17.3.5. Material Properties ............................................................................................................. 468
17.3.6. Momentum Equation ......................................................................................................... 468
17.3.7. Energy Equation ................................................................................................................. 469
17.3.8. Additional Scalar Equations ................................................................................................ 469
17.3.9. Surface Tension and Adhesion ............................................................................................ 469
17.3.9.1. Surface Tension ......................................................................................................... 470
17.3.9.1.1. The Continuum Surface Force Model ................................................................. 470
17.3.9.1.2.The Continuum Surface Stress Model ................................................................. 471
17.3.9.1.3. Comparing the CSS and CSF Methods ................................................................ 471
17.3.9.1.4.When Surface Tension Effects Are Important ...................................................... 472
17.3.9.2.Wall Adhesion ............................................................................................................ 472
17.3.9.3. Jump Adhesion .......................................................................................................... 472
17.3.10. Open Channel Flow .......................................................................................................... 473
17.3.10.1. Upstream Boundary Conditions ............................................................................... 474
17.3.10.1.1. Pressure Inlet .................................................................................................. 474
17.3.10.1.2. Mass Flow Rate ................................................................................................ 474
17.3.10.1.3.Volume Fraction Specification .......................................................................... 475
17.3.10.2. Downstream Boundary Conditions ........................................................................... 475
17.3.10.2.1. Pressure Outlet ................................................................................................ 475
17.3.10.2.2. Outflow Boundary ........................................................................................... 475
17.3.10.2.3. Backflow Volume Fraction Specification ........................................................... 476
17.3.10.3. Numerical Beach Treatment ..................................................................................... 476
17.3.11. Open Channel Wave Boundary Conditions ........................................................................ 477
17.3.11.1. Airy Wave Theory ..................................................................................................... 478
17.3.11.2. Stokes Wave Theories ............................................................................................... 479
17.3.11.3. Cnoidal/Solitary Wave Theory ................................................................................... 480
17.3.11.4. Choosing a Wave Theory .......................................................................................... 481
17.3.11.5. Superposition of Waves ............................................................................................ 483
17.3.11.6. Modeling of Random Waves Using Wave Spectrum ................................................... 484
17.3.11.6.1. Definitions ...................................................................................................... 484
17.3.11.6.2.Wave Spectrum Implementation Theory .......................................................... 484
17.3.11.6.2.1. Long-Crested Random Waves (Unidirectional) ......................................... 484
17.3.11.6.2.1.1. Pierson-Moskowitz Spectrum ......................................................... 485
17.3.11.6.2.1.2. JONSWAP Spectrum ....................................................................... 485
17.3.11.6.2.1.3. TMA Spectrum ............................................................................... 485
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17.3.11.6.2.2. Short-Crested Random Waves (Multi-Directional) .................................... 486
17.3.11.6.2.2.1. Cosine-2s Power Function (Frequency Independent) ....................... 486
17.3.11.6.2.2.2. Hyperbolic Function (Frequency Dependent) ................................. 487
17.3.11.6.2.3. Superposition of Individual Wave Components Using the Wave Spectrum ........................................................................................................................... 487
17.3.11.6.3. Choosing a Wave Spectrum and Inputs ............................................................ 488
17.3.11.7. Nomenclature for Open Channel Waves .................................................................... 490
17.3.12. Coupled Level-Set and VOF Model .................................................................................... 491
17.3.12.1. Theory ..................................................................................................................... 491
17.3.12.1.1. Surface Tension Force ...................................................................................... 492
17.3.12.1.2. Re-initialization of the Level-set Function via the Geometrical Method ............. 493
17.3.12.2. Limitations .............................................................................................................. 495
17.4. Mixture Model Theory ................................................................................................................. 495
17.4.1. Overview ........................................................................................................................... 495
17.4.2. Limitations ......................................................................................................................... 495
17.4.3. Continuity Equation ........................................................................................................... 496
17.4.4. Momentum Equation ......................................................................................................... 496
17.4.5. Energy Equation ................................................................................................................. 497
17.4.6. Relative (Slip) Velocity and the Drift Velocity ........................................................................ 497
17.4.7.Volume Fraction Equation for the Secondary Phases ............................................................ 499
17.4.8. Granular Properties ............................................................................................................ 499
17.4.8.1. Collisional Viscosity .................................................................................................... 499
17.4.8.2. Kinetic Viscosity ......................................................................................................... 500
17.4.9. Granular Temperature ......................................................................................................... 500
17.4.10. Solids Pressure ................................................................................................................. 501
17.4.11. Interfacial Area Concentration .......................................................................................... 501
17.4.11.1. Hibiki-Ishii Model ..................................................................................................... 502
17.4.11.2. Ishii-Kim Model ........................................................................................................ 502
17.4.11.3.Yao-Morel Model ...................................................................................................... 503
17.5. Eulerian Model Theory ................................................................................................................ 505
17.5.1. Overview of the Eulerian Model .......................................................................................... 505
17.5.2. Limitations of the Eulerian Model ........................................................................................ 506
17.5.3.Volume Fraction Equation ................................................................................................... 506
17.5.4. Conservation Equations ...................................................................................................... 507
17.5.4.1. Equations in General Form ......................................................................................... 507
17.5.4.1.1. Conservation of Mass ........................................................................................ 507
17.5.4.1.2. Conservation of Momentum .............................................................................. 507
17.5.4.1.3. Conservation of Energy ..................................................................................... 508
17.5.4.2. Equations Solved by ANSYS Fluent ............................................................................. 508
17.5.4.2.1. Continuity Equation .......................................................................................... 508
17.5.4.2.2. Fluid-Fluid Momentum Equations ...................................................................... 509
17.5.4.2.3. Fluid-Solid Momentum Equations ...................................................................... 509
17.5.4.2.4. Conservation of Energy ..................................................................................... 509
17.5.5. Interfacial Area Concentration ............................................................................................ 510
17.5.6. Interphase Exchange Coefficients ....................................................................................... 510
17.5.6.1. Fluid-Fluid Exchange Coefficient ................................................................................ 511
17.5.6.1.1. Schiller and Naumann Model ............................................................................. 511
17.5.6.1.2. Morsi and Alexander Model ............................................................................... 512
17.5.6.1.3. Symmetric Model .............................................................................................. 512
17.5.6.1.4. Grace et al. Model .............................................................................................. 513
17.5.6.1.5.Tomiyama et al. Model ....................................................................................... 514
17.5.6.1.6. Ishii Model ........................................................................................................ 514
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17.5.6.1.7. Universal Drag Laws for Bubble-Liquid and Droplet-Gas Flows ........................... 515
17.5.6.1.7.1. Bubble-Liquid Flow .................................................................................. 515
17.5.6.1.7.2. Droplet-Gas Flow ...................................................................................... 516
17.5.6.2. Fluid-Solid Exchange Coefficient ................................................................................ 517
17.5.6.3. Solid-Solid Exchange Coefficient ................................................................................ 519
17.5.6.4. Drag Modification ...................................................................................................... 520
17.5.6.4.1. Brucato Correlation ........................................................................................... 520
17.5.7. Lift Force ............................................................................................................................ 520
17.5.7.1. Lift Coefficient Models ............................................................................................... 521
17.5.7.1.1. Moraga Lift Force Model .................................................................................... 521
17.5.7.1.2. Saffman-Mei Lift Force Model ............................................................................ 522
17.5.7.1.3. Legendre-Magnaudet Lift Force Model .............................................................. 522
17.5.7.1.4.Tomiyama Lift Force Model ................................................................................ 523
17.5.8. Wall Lubrication Force ........................................................................................................ 523
17.5.8.1.Wall Lubrication Models ............................................................................................. 524
17.5.8.1.1. Antal et al. Model .............................................................................................. 524
17.5.8.1.2.Tomiyama Model ............................................................................................... 524
17.5.8.1.3. Frank Model ...................................................................................................... 525
17.5.8.1.4. Hosokawa Model .............................................................................................. 525
17.5.9. Turbulent Dispersion Force ................................................................................................. 526
17.5.9.1. Models for Turbulent Dispersion Force ....................................................................... 526
17.5.9.1.1. Lopez de Bertodano Model ............................................................................... 526
17.5.9.1.2. Simonin Model .................................................................................................. 527
17.5.9.1.3. Burns et al. Model .............................................................................................. 527
17.5.9.1.4. Diffusion in VOF Model ...................................................................................... 527
17.5.9.2. Limiting Functions for the Turbulent Dispersion Force ................................................ 528
17.5.10.Virtual Mass Force ............................................................................................................. 529
17.5.11. Solids Pressure ................................................................................................................. 529
17.5.11.1. Radial Distribution Function ..................................................................................... 530
17.5.12. Maximum Packing Limit in Binary Mixtures ....................................................................... 532
17.5.13. Solids Shear Stresses ......................................................................................................... 532
17.5.13.1. Collisional Viscosity .................................................................................................. 532
17.5.13.2. Kinetic Viscosity ....................................................................................................... 532
17.5.13.3. Bulk Viscosity ........................................................................................................... 533
17.5.13.4. Frictional Viscosity ................................................................................................... 533
17.5.14. Granular Temperature ....................................................................................................... 534
17.5.15. Description of Heat Transfer .............................................................................................. 536
17.5.15.1. The Heat Exchange Coefficient ................................................................................. 536
17.5.15.1.1. Constant ......................................................................................................... 537
17.5.15.1.2. Nusselt Number .............................................................................................. 537
17.5.15.1.3. Ranz-Marshall Model ....................................................................................... 537
17.5.15.1.4.Tomiyama Model ............................................................................................. 537
17.5.15.1.5. Hughmark Model ............................................................................................ 537
17.5.15.1.6. Gunn Model .................................................................................................... 537
17.5.15.1.7. Two-Resistance Model ..................................................................................... 538
17.5.15.1.8. Fixed To Saturation Temperature ...................................................................... 538
17.5.15.1.9. User Defined ................................................................................................... 539
17.5.16. Turbulence Models ........................................................................................................... 539
17.5.16.1. k- ε Turbulence Models ............................................................................................. 540
17.5.16.1.1. k- ε Mixture Turbulence Model ......................................................................... 540
17.5.16.1.2. k- ε Dispersed Turbulence Model ..................................................................... 541
17.5.16.1.2.1. Assumptions .......................................................................................... 541
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17.5.16.1.2.2. Turbulence in the Continuous Phase ....................................................... 542
17.5.16.1.2.3.Turbulence in the Dispersed Phase .......................................................... 542
17.5.16.1.3. k- ε Turbulence Model for Each Phase ............................................................... 543
17.5.16.1.3.1.Transport Equations ................................................................................ 543
17.5.16.2. RSM Turbulence Models ........................................................................................... 544
17.5.16.2.1. RSM Dispersed Turbulence Model .................................................................... 544
17.5.16.2.2. RSM Mixture Turbulence Model ....................................................................... 545
17.5.16.3. Turbulence Interaction Models ................................................................................. 546
17.5.16.3.1. Simonin et al. .................................................................................................. 546
17.5.16.3.1.1. Formulation in Dispersed Turbulence Models .......................................... 546
17.5.16.3.1.1.1. Continuous Phase .......................................................................... 546
17.5.16.3.1.1.2. Dispersed Phases ........................................................................... 547
17.5.16.3.1.2. Formulation in Per Phase Turbulence Models ........................................... 548
17.5.16.3.2. Troshko-Hassan ............................................................................................... 548
17.5.16.3.2.1.Troshko-Hassan Formulation in Mixture Turbulence Models ..................... 548
17.5.16.3.2.2. Troshko-Hassan Formulation in Dispersed Turbulence Models ................. 549
17.5.16.3.2.2.1. Continuous Phase .......................................................................... 549
17.5.16.3.2.2.2. Dispersed Phases ........................................................................... 549
17.5.16.3.2.3.Troshko-Hassan Formulation in Per-Phase Turbulence Models .................. 549
17.5.16.3.2.3.1. Continuous Phase .......................................................................... 549
17.5.16.3.2.3.2. Dispersed Phases ........................................................................... 549
17.5.16.3.3. Sato ................................................................................................................ 550
17.5.16.3.4. None ............................................................................................................... 550
17.5.17. Solution Method in ANSYS Fluent ..................................................................................... 550
17.5.17.1.The Pressure-Correction Equation ............................................................................. 550
17.5.17.2. Volume Fractions ..................................................................................................... 550
17.5.18. Dense Discrete Phase Model ............................................................................................. 551
17.5.18.1. Limitations .............................................................................................................. 552
17.5.18.2. Granular Temperature .............................................................................................. 552
17.5.19. Multi-Fluid VOF Model ...................................................................................................... 553
17.5.20. Wall Boiling Models .......................................................................................................... 554
17.5.20.1. Overview ................................................................................................................. 554
17.5.20.2. RPI Model ................................................................................................................ 555
17.5.20.3. Non-equilibrium Subcooled Boiling .......................................................................... 557
17.5.20.4. Critical Heat Flux ...................................................................................................... 558
17.5.20.4.1.Wall Heat Flux Partition .................................................................................... 558
17.5.20.4.2. Flow Regime Transition ................................................................................... 559
17.5.20.5. Interfacial Momentum Transfer ................................................................................. 560
17.5.20.5.1. Interfacial Area ................................................................................................ 560
17.5.20.5.2. Bubble and Droplet Diameters ........................................................................ 560
17.5.20.5.2.1. Bubble Diameter .................................................................................... 560
17.5.20.5.2.2. Droplet Diameter .................................................................................... 560
17.5.20.5.3. Interfacial Drag Force ...................................................................................... 561
17.5.20.5.4. Interfacial Lift Force ......................................................................................... 561
17.5.20.5.5.Turbulent Dispersion Force .............................................................................. 561
17.5.20.5.6. Wall Lubrication Force ..................................................................................... 561
17.5.20.5.7. Virtual Mass Force ........................................................................................... 561
17.5.20.6. Interfacial Heat Transfer ............................................................................................ 561
17.5.20.6.1. Interface to Liquid Heat Transfer ...................................................................... 561
17.5.20.6.2. Interface to Vapor Heat Transfer ....................................................................... 562
17.5.20.7. Mass Transfer ........................................................................................................... 562
17.5.20.7.1. Mass Transfer From the Wall to Vapor ............................................................... 562
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17.5.20.7.2. Interfacial Mass Transfer .................................................................................. 562
17.5.20.8.Turbulence Interactions ............................................................................................ 562
17.6. Wet Steam Model Theory ............................................................................................................ 562
17.6.1. Overview of the Wet Steam Model ...................................................................................... 562
17.6.2. Limitations of the Wet Steam Model .................................................................................... 563
17.6.3.Wet Steam Flow Equations .................................................................................................. 563
17.6.4. Phase Change Model .......................................................................................................... 564
17.6.5. Built-in Thermodynamic Wet Steam Properties .................................................................... 566
17.6.5.1. Equation of State ....................................................................................................... 566
17.6.5.2. Saturated Vapor Line .................................................................................................. 567
17.6.5.3. Saturated Liquid Line ................................................................................................. 567
17.6.5.4. Mixture Properties ..................................................................................................... 567
17.7. Modeling Mass Transfer in Multiphase Flows ................................................................................ 567
17.7.1. Source Terms due to Mass Transfer ...................................................................................... 568
17.7.1.1. Mass Equation ........................................................................................................... 568
17.7.1.2. Momentum Equation ................................................................................................. 568
17.7.1.3. Energy Equation ........................................................................................................ 568
17.7.1.4. Species Equation ....................................................................................................... 569
17.7.1.5. Other Scalar Equations ............................................................................................... 569
17.7.2. Unidirectional Constant Rate Mass Transfer ......................................................................... 569
17.7.3. UDF-Prescribed Mass Transfer ............................................................................................. 569
17.7.4. Cavitation Models .............................................................................................................. 569
17.7.4.1. Limitations of the Cavitation Models .......................................................................... 570
17.7.4.2.Vapor Transport Equation ........................................................................................... 571
17.7.4.3. Bubble Dynamics Consideration ................................................................................ 571
17.7.4.4. Singhal et al. Model .................................................................................................... 572
17.7.4.5. Zwart-Gerber-Belamri Model ..................................................................................... 574
17.7.4.6. Schnerr and Sauer Model ........................................................................................... 575
17.7.4.7. Turbulence Factor ...................................................................................................... 576
17.7.4.8. Additional Guidelines for the Cavitation Models ......................................................... 576
17.7.4.9. Extended Cavitation Model Capabilities ..................................................................... 578
17.7.4.9.1. Multiphase Cavitation Models ........................................................................... 578
17.7.4.9.2. Multiphase Species Transport Cavitation Model ................................................. 579
17.7.5. Evaporation-Condensation Model ....................................................................................... 579
17.7.5.1. Lee Model ................................................................................................................. 579
17.7.5.2.Thermal Phase Change Model .................................................................................... 582
17.7.6. Interphase Species Mass Transfer ........................................................................................ 583
17.7.6.1. Modeling Approach ................................................................................................... 584
17.7.6.1.1. Equilibrium Model ............................................................................................. 584
17.7.6.1.2.Two-Resistance Model ....................................................................................... 585
17.7.6.2. Species Mass Transfer Models ..................................................................................... 587
17.7.6.2.1. Raoult’s Law ...................................................................................................... 587
17.7.6.2.2. Henry’s Law ...................................................................................................... 587
17.7.6.2.3. Equilibrium Ratio .............................................................................................. 588
17.7.6.3. Mass Transfer Coefficient Models ................................................................................ 589
17.7.6.3.1. Constant ........................................................................................................... 589
17.7.6.3.2. Sherwood Number ............................................................................................ 589
17.7.6.3.3. Ranz-Marshall Model ......................................................................................... 589
17.7.6.3.4. Hughmark Model .............................................................................................. 590
17.7.6.3.5. User-Defined ..................................................................................................... 590
17.8. Modeling Species Transport in Multiphase Flows ......................................................................... 590
17.8.1. Limitations ......................................................................................................................... 591
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17.8.2. Mass and Momentum Transfer with Multiphase Species Transport ....................................... 591
17.8.2.1. Source Terms Due to Heterogeneous Reactions .......................................................... 592
17.8.2.1.1. Mass Transfer .................................................................................................... 592
17.8.2.1.2. Momentum Transfer .......................................................................................... 592
17.8.2.1.3. Species Transfer ................................................................................................ 593
17.8.2.1.4. Heat Transfer ..................................................................................................... 593
17.8.3. The Stiff Chemistry Solver ................................................................................................... 594
17.8.4. Heterogeneous Phase Interaction ....................................................................................... 594
18. Solidification and Melting ................................................................................................................. 595
18.1. Overview .................................................................................................................................... 595
18.2. Limitations .................................................................................................................................. 596
18.3. Introduction ............................................................................................................................... 596
18.4. Energy Equation ......................................................................................................................... 596
18.5. Momentum Equations ................................................................................................................ 597
18.6.Turbulence Equations .................................................................................................................. 598
18.7. Species Equations ....................................................................................................................... 598
18.8. Back Diffusion ............................................................................................................................. 600
18.9. Pull Velocity for Continuous Casting ............................................................................................ 600
18.10. Contact Resistance at Walls ........................................................................................................ 602
18.11.Thermal and Solutal Buoyancy ................................................................................................... 602
19. Eulerian Wall Films ............................................................................................................................ 605
19.1. Introduction ............................................................................................................................... 605
19.2. Mass, Momentum, and Energy Conservation Equations for Wall Film ............................................. 606
19.2.1. Film Sub-Models ................................................................................................................. 607
19.2.1.1. DPM Collection .......................................................................................................... 607
19.2.1.2. Splashing .................................................................................................................. 607
19.2.1.3. Film Separation .......................................................................................................... 607
19.2.1.3.1. Separation Criteria ............................................................................................ 607
19.2.1.3.1.1. Foucart Separation ................................................................................... 608
19.2.1.3.1.2. O’Rourke Separation ................................................................................. 608
19.2.1.3.1.3. Friedrich Separation ................................................................................. 608
19.2.1.4. Film Stripping ............................................................................................................ 609
19.2.1.5. Secondary Phase Accretion ........................................................................................ 610
19.2.1.6. Coupling of Wall Film with Mixture Species Transport ................................................. 611
19.2.2. Boundary Conditions .......................................................................................................... 611
19.2.3. Obtaining Film Velocity Without Solving the Momentum Equations .................................... 611
19.2.3.1. Shear-Driven Film Velocity ......................................................................................... 612
19.2.3.2. Gravity-Driven Film Velocity ....................................................................................... 612
19.3. Passive Scalar Equation for Wall Film ............................................................................................ 613
19.4. Numerical Schemes and Solution Algorithm ................................................................................ 614
19.4.1.Temporal Differencing Schemes .......................................................................................... 614
19.4.1.1. First-Order Explicit Method ........................................................................................ 614
19.4.1.2. First-Order Implicit Method ........................................................................................ 615
19.4.1.3. Second-Order Implicit Method ................................................................................... 615
19.4.2. Spatial Differencing Schemes .............................................................................................. 616
19.4.3. Solution Algorithm ............................................................................................................. 617
19.4.3.1. Steady Flow ............................................................................................................... 617
19.4.3.2. Transient Flow ........................................................................................................... 617
20. Solver Theory .................................................................................................................................... 619
20.1. Overview of Flow Solvers ............................................................................................................ 619
20.1.1. Pressure-Based Solver ......................................................................................................... 620
20.1.1.1. The Pressure-Based Segregated Algorithm ................................................................. 620
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20.1.1.2.The Pressure-Based Coupled Algorithm ...................................................................... 621
20.1.2. Density-Based Solver .......................................................................................................... 622
20.2. General Scalar Transport Equation: Discretization and Solution ..................................................... 624
20.2.1. Solving the Linear System ................................................................................................... 626
20.3. Discretization .............................................................................................................................. 626
20.3.1. Spatial Discretization .......................................................................................................... 626
20.3.1.1. First-Order Upwind Scheme ....................................................................................... 627
20.3.1.2. Power-Law Scheme .................................................................................................... 627
20.3.1.3. Second-Order Upwind Scheme .................................................................................. 628
20.3.1.4. First-to-Higher Order Blending ................................................................................... 629
20.3.1.5. Central-Differencing Scheme ..................................................................................... 629
20.3.1.6. Bounded Central Differencing Scheme ....................................................................... 630
20.3.1.7. QUICK Scheme .......................................................................................................... 630
20.3.1.8.Third-Order MUSCL Scheme ....................................................................................... 631
20.3.1.9. Modified HRIC Scheme .............................................................................................. 631
20.3.1.10. High Order Term Relaxation ..................................................................................... 633
20.3.2. Temporal Discretization ...................................................................................................... 633
20.3.2.1. Implicit Time Integration ............................................................................................ 634
20.3.2.2. Bounded Second Order Implicit Time Integration ....................................................... 634
20.3.2.2.1. Limitations ........................................................................................................ 634
20.3.2.3. Explicit Time Integration ............................................................................................ 635
20.3.3. Evaluation of Gradients and Derivatives .............................................................................. 635
20.3.3.1. Green-Gauss Theorem ............................................................................................... 635
20.3.3.2. Green-Gauss Cell-Based Gradient Evaluation .............................................................. 636
20.3.3.3. Green-Gauss Node-Based Gradient Evaluation ............................................................ 636
20.3.3.4. Least Squares Cell-Based Gradient Evaluation ............................................................. 636
20.3.4. Gradient Limiters ................................................................................................................ 638
20.3.4.1. Standard Limiter ........................................................................................................ 638
20.3.4.2. Multidimensional Limiter ........................................................................................... 639
20.3.4.3. Differentiable Limiter ................................................................................................. 639
20.4. Pressure-Based Solver ................................................................................................................. 639
20.4.1. Discretization of the Momentum Equation .......................................................................... 640
20.4.1.1. Pressure Interpolation Schemes ................................................................................. 640
20.4.2. Discretization of the Continuity Equation ............................................................................ 641
20.4.2.1. Density Interpolation Schemes ................................................................................... 642
20.4.3. Pressure-Velocity Coupling ................................................................................................. 642
20.4.3.1. Segregated Algorithms .............................................................................................. 643
20.4.3.1.1. SIMPLE .............................................................................................................. 643
20.4.3.1.2. SIMPLEC ........................................................................................................... 644
20.4.3.1.2.1. Skewness Correction ................................................................................ 644
20.4.3.1.3. PISO .................................................................................................................. 644
20.4.3.1.3.1. Neighbor Correction ................................................................................. 644
20.4.3.1.3.2. Skewness Correction ................................................................................ 645
20.4.3.1.3.3. Skewness - Neighbor Coupling ................................................................. 645
20.4.3.2. Fractional-Step Method (FSM) .................................................................................... 645
20.4.3.3. Coupled Algorithm .................................................................................................... 645
20.4.3.3.1. Limitations ........................................................................................................ 646
20.4.4. Steady-State Iterative Algorithm ......................................................................................... 647
20.4.4.1. Under-Relaxation of Variables .................................................................................... 647
20.4.4.2. Under-Relaxation of Equations ................................................................................... 647
20.4.5.Time-Advancement Algorithm ............................................................................................ 647
20.4.5.1. Iterative Time-Advancement Scheme ......................................................................... 648
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20.4.5.1.1.The Frozen Flux Formulation .............................................................................. 649
20.4.5.2. Non-Iterative Time-Advancement Scheme .................................................................. 650
20.5. Density-Based Solver ................................................................................................................... 652
20.5.1. Governing Equations in Vector Form ................................................................................... 652
20.5.2. Preconditioning ................................................................................................................. 653
20.5.3. Convective Fluxes ............................................................................................................... 655
20.5.3.1. Roe Flux-Difference Splitting Scheme ......................................................................... 655
20.5.3.2. AUSM  Scheme ......................................................................................................... 655
20.5.3.3. Low Diffusion Roe Flux Difference Splitting Scheme ................................................... 656
20.5.4. Steady-State Flow Solution Methods ................................................................................... 656
20.5.4.1. Explicit Formulation ................................................................................................... 657
20.5.4.1.1. Implicit Residual Smoothing .............................................................................. 657
20.5.4.2. Implicit Formulation .................................................................................................. 658
20.5.4.2.1. Convergence Acceleration for Stretched Meshes ................................................ 658
20.5.5. Unsteady Flows Solution Methods ...................................................................................... 659
20.5.5.1. Explicit Time Stepping ............................................................................................... 659
20.5.5.2. Implicit Time Stepping (Dual-Time Formulation) ......................................................... 659
20.6. Pseudo Transient Under-Relaxation ............................................................................................. 661
20.6.1. Automatic Pseudo Transient Time Step ............................................................................... 661
20.7. Multigrid Method ........................................................................................................................ 663
20.7.1. Approach ........................................................................................................................... 663
20.7.1.1.The Need for Multigrid ............................................................................................... 663
20.7.1.2.The Basic Concept in Multigrid ................................................................................... 664
20.7.1.3. Restriction and Prolongation ...................................................................................... 664
20.7.1.4. Unstructured Multigrid .............................................................................................. 665
20.7.2. Multigrid Cycles .................................................................................................................. 665
20.7.2.1. The V and W Cycles .................................................................................................... 665
20.7.3. Algebraic Multigrid (AMG) .................................................................................................. 669
20.7.3.1. AMG Restriction and Prolongation Operators ............................................................. 669
20.7.3.2. AMG Coarse Level Operator ....................................................................................... 670
20.7.3.3. The F Cycle ................................................................................................................ 670
20.7.3.4. The Flexible Cycle ...................................................................................................... 670
20.7.3.4.1.The Residual Reduction Rate Criteria .................................................................. 671
20.7.3.4.2. The Termination Criteria .................................................................................... 672
20.7.3.5.The Coupled and Scalar AMG Solvers .......................................................................... 672
20.7.3.5.1. Gauss-Seidel ..................................................................................................... 673
20.7.3.5.2. Incomplete Lower Upper (ILU) ........................................................................... 673
20.7.4. Full-Approximation Storage (FAS) Multigrid ......................................................................... 674
20.7.4.1. FAS Restriction and Prolongation Operators ............................................................... 675
20.7.4.2. FAS Coarse Level Operator ......................................................................................... 675
20.8. Hybrid Initialization ..................................................................................................................... 675
20.9. Full Multigrid (FMG) Initialization ................................................................................................. 677
20.9.1. Overview of FMG Initialization ............................................................................................ 677
20.9.2. Limitations of FMG Initialization .......................................................................................... 678
21. Adapting the Mesh ............................................................................................................................ 681
21.1. Static Adaption Process ............................................................................................................... 681
21.1.1. Hanging Node Adaption ..................................................................................................... 681
21.1.1.1. Hanging Node Refinement ......................................................................................... 682
21.1.1.2. Hanging Node Coarsening ......................................................................................... 683
21.2. Boundary Adaption ..................................................................................................................... 683
21.3. Gradient Adaption ...................................................................................................................... 685
21.3.1. Gradient Adaption Approach .............................................................................................. 685
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21.3.2. Example of Steady Gradient Adaption ................................................................................. 687
21.4. Isovalue Adaption ....................................................................................................................... 689
21.5. Region Adaption ......................................................................................................................... 691
21.5.1. Defining a Region ............................................................................................................... 691
21.5.2. Region Adaption Example .................................................................................................. 692
21.6. Volume Adaption ........................................................................................................................ 693
21.6.1.Volume Adaption Approach ................................................................................................ 693
21.6.2.Volume Adaption Example .................................................................................................. 694
21.7.Yplus/Ystar Adaption ................................................................................................................... 695
21.7.1.Yplus/Ystar Adaption Approach .......................................................................................... 695
21.8. Anisotropic Adaption .................................................................................................................. 697
21.9. Geometry-Based Adaption .......................................................................................................... 697
21.9.1. Geometry-Based Adaption Approach .................................................................................. 698
21.9.1.1. Node Projection ......................................................................................................... 698
21.9.1.2. Example of Geometry-Based Adaption ....................................................................... 700
21.10. Registers ................................................................................................................................... 703
21.10.1. Adaption Registers ........................................................................................................... 703
21.10.2. Mask Registers .................................................................................................................. 704
22. Reporting Alphanumeric Data .......................................................................................................... 707
22.1. Fluxes Through Boundaries ......................................................................................................... 707
22.2. Forces on Boundaries .................................................................................................................. 708
22.2.1. Computing Forces, Moments, and the Center of Pressure ..................................................... 708
22.3. Surface Integration ..................................................................................................................... 710
22.3.1. Computing Surface Integrals .............................................................................................. 711
22.3.1.1. Area .......................................................................................................................... 711
22.3.1.2. Integral ...................................................................................................................... 712
22.3.1.3. Area-Weighted Average ............................................................................................. 712
22.3.1.4. Custom Vector Based Flux .......................................................................................... 712
22.3.1.5. Custom Vector Flux .................................................................................................... 712
22.3.1.6. Custom Vector Weighted Average .............................................................................. 712
22.3.1.7. Flow Rate ................................................................................................................... 712
22.3.1.8. Mass Flow Rate .......................................................................................................... 713
22.3.1.9. Mass-Weighted Average ............................................................................................ 713
22.3.1.10. Sum of Field Variable ................................................................................................ 713
22.3.1.11. Facet Average .......................................................................................................... 714
22.3.1.12. Facet Minimum ........................................................................................................ 714
22.3.1.13. Facet Maximum ....................................................................................................... 714
22.3.1.14.Vertex Average ......................................................................................................... 714
22.3.1.15. Vertex Minimum ...................................................................................................... 714
22.3.1.16.Vertex Maximum ...................................................................................................... 714
22.3.1.17. Standard-Deviation .................................................................................................. 714
22.3.1.18. Uniformity Index ...................................................................................................... 715
22.3.1.19. Volume Flow Rate .................................................................................................... 716
22.4. Volume Integration ..................................................................................................................... 716
22.4.1. Computing Volume Integrals .............................................................................................. 716
22.4.1.1.Volume ...................................................................................................................... 716
22.4.1.2. Sum .......................................................................................................................... 717
22.4.1.3. Sum*2Pi .................................................................................................................... 717
22.4.1.4. Volume Integral ......................................................................................................... 717
22.4.1.5.Volume-Weighted Average ......................................................................................... 717
22.4.1.6. Mass-Weighted Integral ............................................................................................. 717
22.4.1.7. Mass .......................................................................................................................... 718
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22.4.1.8. Mass-Weighted Average ............................................................................................ 718
A. Nomenclature ....................................................................................................................................... 719
Bibliography ............................................................................................................................................. 723
Index ........................................................................................................................................................ 753