ANSYS Fluent Theory Guide18.1.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 ............................................................................................................ xxxii
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 .................................................................................. 3
1.3.1. Single Phase Flow .................................................................................................................... 4
1.3.2. Multiphase Flow ....................................................................................................................... 5
1.4. Periodic Flows .................................................................................................................................. 5
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 ................................................................ 7
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 ................................................................... 10
1.5.2. Physics of Swirling and Rotating Flows .................................................................................... 10
1.6. Compressible Flows ........................................................................................................................ 11
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 ......................................................................... 13
1.6.2.2.The Compressible Form of the Gas Law .......................................................................... 14
1.7. Inviscid Flows ................................................................................................................................. 14
1.7.1. Euler Equations ...................................................................................................................... 14
1.7.1.1.The Mass Conservation Equation .................................................................................... 15
1.7.1.2. Momentum Conservation Equations .............................................................................. 15
1.7.1.3. Energy Conservation Equation ....................................................................................... 15
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 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
4.2.7.1.Treatment of the Spalart-Allmaras Model for Icing Simulations ....................................... 45
4.2.8. Convective Heat and Mass Transfer Modeling .......................................................................... 46
4.3. Standard, RNG, and Realizable k-ε Models ........................................................................................ 46
4.3.1. Standard k-ε Model ................................................................................................................ 47
4.3.1.1. Overview ....................................................................................................................... 47
4.3.1.2.Transport Equations for the Standard k-ε Model ............................................................. 47
4.3.1.3. Modeling the Turbulent Viscosity ................................................................................... 47
4.3.1.4. Model Constants ........................................................................................................... 48
4.3.2. RNG k-ε Model ....................................................................................................................... 48
4.3.2.1. Overview ....................................................................................................................... 48
4.3.2.2.Transport Equations for the RNG k-ε Model ..................................................................... 48
4.3.2.3. Modeling the Effective Viscosity ..................................................................................... 49
4.3.2.4. RNG Swirl Modification .................................................................................................. 49
4.3.2.5. Calculating the Inverse Effective Prandtl Numbers .......................................................... 50
4.3.2.6.The R-ε Term in the ε Equation ........................................................................................ 50
4.3.2.7. Model Constants ........................................................................................................... 51
4.3.3. Realizable k-ε Model ............................................................................................................... 51
4.3.3.1. Overview ....................................................................................................................... 51
4.3.3.2.Transport Equations for the Realizable k-ε Model ............................................................ 52
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4.3.3.3. Modeling the Turbulent Viscosity ................................................................................... 53
4.3.3.4. Model Constants ........................................................................................................... 54
4.3.4. Modeling Turbulent Production in the k-ε Models ................................................................... 54
4.3.5. Effects of Buoyancy on Turbulence in the k-ε Models ............................................................... 54
4.3.6. Effects of Compressibility on Turbulence in the k-ε Models ...................................................... 55
4.3.7. Convective Heat and Mass Transfer Modeling in the k-ε Models ............................................... 56
4.4. Standard, BSL, and SST k-ω Models ................................................................................................... 57
4.4.1. Standard k-ω Model ............................................................................................................... 58
4.4.1.1. Overview ....................................................................................................................... 58
4.4.1.2.Transport Equations for the Standard k-ω Model ............................................................. 58
4.4.1.3. Modeling the Effective Diffusivity ................................................................................... 58
4.4.1.3.1. Low-Reynolds Number Correction ......................................................................... 58
4.4.1.4. Modeling the Turbulence Production ............................................................................. 59
4.4.1.4.1. Production of k ..................................................................................................... 59
4.4.1.4.2. Production of ω ..................................................................................................... 59
4.4.1.5. Modeling the Turbulence Dissipation ............................................................................. 59
4.4.1.5.1. Dissipation of k ..................................................................................................... 59
4.4.1.5.2. Dissipation of ω ..................................................................................................... 60
4.4.1.5.3. Compressibility Effects .......................................................................................... 60
4.4.1.6. Model Constants ........................................................................................................... 61
4.4.2. Baseline (BSL) k-ω Model ........................................................................................................ 61
4.4.2.1. Overview ....................................................................................................................... 61
4.4.2.2.Transport Equations for the BSL k-ω Model ..................................................................... 61
4.4.2.3. Modeling the Effective Diffusivity ................................................................................... 62
4.4.2.4. Modeling the Turbulence Production ............................................................................. 62
4.4.2.4.1. Production of k ..................................................................................................... 62
4.4.2.4.2. Production of ω ..................................................................................................... 62
4.4.2.5. Modeling the Turbulence Dissipation ............................................................................. 63
4.4.2.5.1. Dissipation of k ..................................................................................................... 63
4.4.2.5.2. Dissipation of ω ..................................................................................................... 63
4.4.2.6. Cross-Diffusion Modification .......................................................................................... 63
4.4.2.7. Model Constants ........................................................................................................... 63
4.4.3. Shear-Stress Transport (SST) k-ω Model ................................................................................... 64
4.4.3.1. Overview ....................................................................................................................... 64
4.4.3.2. Modeling the Turbulent Viscosity ................................................................................... 64
4.4.3.3. Model Constants ........................................................................................................... 64
4.4.3.4.Treatment of the SST Model for Icing Simulations ........................................................... 64
4.4.4.Turbulence Damping .............................................................................................................. 65
4.4.5. Wall Boundary Conditions ...................................................................................................... 66
4.5. k-kl-ω Transition Model ................................................................................................................... 66
4.5.1. Overview ............................................................................................................................... 67
4.5.2.Transport Equations for the k-kl-ω Model ................................................................................ 67
4.5.2.1. Model Constants ........................................................................................................... 69
4.6.Transition SST Model ....................................................................................................................... 70
4.6.1. Overview ............................................................................................................................... 70
4.6.2.Transport Equations for the Transition SST Model .................................................................... 71
4.6.2.1. Separation-Induced Transition Correction ...................................................................... 73
4.6.2.2. Coupling the Transition Model and SST Transport Equations ........................................... 73
4.6.2.3.Transition SST and Rough Walls ...................................................................................... 73
4.6.3. Mesh Requirements ............................................................................................................... 74
4.6.4. Specifying Inlet Turbulence Levels .......................................................................................... 77
4.7. Intermittency Transition Model ....................................................................................................... 78
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4.7.1. Overview ............................................................................................................................... 78
4.7.2.Transport Equations for the Intermittency Transition Model ..................................................... 79
4.7.3. Coupling with the Other Models ............................................................................................. 81
4.7.4. Intermittency Transition Model and Rough Walls ..................................................................... 81
4.8.The V2F Model ................................................................................................................................ 81
4.9. Reynolds Stress Model (RSM) ........................................................................................................... 82
4.9.1. Overview ............................................................................................................................... 82
4.9.2. Reynolds Stress Transport Equations ....................................................................................... 83
4.9.3. Modeling Turbulent Diffusive Transport .................................................................................. 84
4.9.4. Modeling the Pressure-Strain Term ......................................................................................... 84
4.9.4.1. Linear Pressure-Strain Model .......................................................................................... 84
4.9.4.2. Low-Re Modifications to the Linear Pressure-Strain Model .............................................. 85
4.9.4.3. Quadratic Pressure-Strain Model .................................................................................... 86
4.9.4.4. Stress-Omega Model ..................................................................................................... 86
4.9.4.5. Stress-BSL Model ........................................................................................................... 88
4.9.5. Effects of Buoyancy on Turbulence ......................................................................................... 88
4.9.6. Modeling the Turbulence Kinetic Energy ................................................................................. 88
4.9.7. Modeling the Dissipation Rate ................................................................................................ 89
4.9.8. Modeling the Turbulent Viscosity ............................................................................................ 89
4.9.9. Wall Boundary Conditions ...................................................................................................... 90
4.9.10. Convective Heat and Mass Transfer Modeling ........................................................................ 90
4.10. Scale-Adaptive Simulation (SAS) Model ......................................................................................... 91
4.10.1. Overview ............................................................................................................................. 91
4.10.2.Transport Equations for the SST-SAS Model ........................................................................... 92
4.10.3. SAS with Other ω-Based Turbulence Models .......................................................................... 94
4.11. Detached Eddy Simulation (DES) ................................................................................................... 94
4.11.1. Overview ............................................................................................................................. 94
4.11.2. DES with the Spalart-Allmaras Model .................................................................................... 95
4.11.3. DES with the Realizable k-ε Model ......................................................................................... 95
4.11.4. DES with the BSL or SST k-ω Model ....................................................................................... 96
4.11.5. DES with the Transition SST Model ........................................................................................ 97
4.11.6. Improved Delayed Detached Eddy Simulation (IDDES) .......................................................... 97
4.11.6.1. Overview of IDDES ....................................................................................................... 97
4.11.6.2. IDDES Model Formulation ............................................................................................ 98
4.12. Shielded Detached Eddy Simulation (SDES) ................................................................................... 98
4.12.1. Shielding Function ............................................................................................................... 99
4.12.2. LES Mode of SDES .............................................................................................................. 100
4.13. Stress-Blended Eddy Simulation (SBES) ........................................................................................ 101
4.13.1. Stress Blending ................................................................................................................... 102
4.13.2. SDES and SBES Example ..................................................................................................... 102
4.14. Large Eddy Simulation (LES) Model .............................................................................................. 103
4.14.1. Overview ........................................................................................................................... 103
4.14.2. Subgrid-Scale Models ......................................................................................................... 104
4.14.2.1. Smagorinsky-Lilly Model ............................................................................................ 105
4.14.2.2. Dynamic Smagorinsky-Lilly Model .............................................................................. 106
4.14.2.3.Wall-Adapting Local Eddy-Viscosity (WALE) Model ...................................................... 107
4.14.2.4. Algebraic Wall-Modeled LES Model (WMLES) .............................................................. 107
4.14.2.4.1. Algebraic WMLES Model Formulation ................................................................ 108
4.14.2.4.1.1. Reynolds Number Scaling ......................................................................... 108
4.14.2.4.2. Algebraic WMLES S-Omega Model Formulation ................................................. 109
4.14.2.5. Dynamic Kinetic Energy Subgrid-Scale Model ............................................................. 110
4.14.3. Inlet Boundary Conditions for the LES Model ....................................................................... 110
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4.14.3.1.Vortex Method ........................................................................................................... 111
4.14.3.2. Spectral Synthesizer ................................................................................................... 112
4.15. Embedded Large Eddy Simulation (ELES) ..................................................................................... 113
4.15.1. Overview ........................................................................................................................... 113
4.15.2. Selecting a Model ............................................................................................................... 113
4.15.3. Interfaces Treatment ........................................................................................................... 114
4.15.3.1. RANS-LES Interface .................................................................................................... 114
4.15.3.2. LES-RANS Interface .................................................................................................... 114
4.15.3.3. Internal Interface Without LES Zone ........................................................................... 115
4.15.3.4. Grid Generation Guidelines ........................................................................................ 115
4.16. Near-Wall Treatments for Wall-Bounded Turbulent Flows .............................................................. 116
4.16.1. Overview ........................................................................................................................... 116
4.16.1.1.Wall Functions vs. Near-Wall Model ............................................................................. 117
4.16.1.2. Wall Functions ........................................................................................................... 119
4.16.2. Standard Wall Functions ..................................................................................................... 119
4.16.2.1. Momentum ............................................................................................................... 119
4.16.2.2. Energy ....................................................................................................................... 120
4.16.2.3. Species ...................................................................................................................... 122
4.16.2.4. Turbulence ................................................................................................................ 122
4.16.3. Scalable Wall Functions ....................................................................................................... 123
4.16.4. Non-Equilibrium Wall Functions .......................................................................................... 123
4.16.4.1. Standard Wall Functions vs. Non-Equilibrium Wall Functions ....................................... 125
4.16.4.2. Limitations of the Wall Function Approach ................................................................. 125
4.16.5. Enhanced Wall Treatment ε-Equation (EWT-ε) ...................................................................... 125
4.16.5.1.Two-Layer Model for Enhanced Wall Treatment ........................................................... 126
4.16.5.2. Enhanced Wall Treatment for Momentum and Energy Equations ................................. 127
4.16.6. Menter-Lechner ε-Equation (ML-ε) ...................................................................................... 129
4.16.6.1. Momentum Equations ............................................................................................... 131
4.16.6.2. k-ε Turbulence Models ............................................................................................... 131
4.16.6.3. Iteration Improvements ............................................................................................. 131
4.16.7. y -Insensitive Wall Treatment ω-Equation ........................................................................... 131
4.16.8. User-Defined Wall Functions ............................................................................................... 132
4.16.9. LES Near-Wall Treatment ..................................................................................................... 132
4.17. Curvature Correction for the Spalart-Allmaras and Two-Equation Models ..................................... 132
4.18. Production Limiters for Two-Equation Models .............................................................................. 134
4.19. Definition of Turbulence Scales .................................................................................................... 136
4.19.1. RANS and Hybrid (SAS, DES, and SDES) Turbulence Models .................................................. 136
4.19.2. Large Eddy Simulation (LES) Models .................................................................................... 136
4.19.3. Stress-Blended Eddy Simulation (SBES) Model ..................................................................... 137
5. Heat Transfer ....................................................................................................................................... 139
5.1. Introduction ................................................................................................................................. 139
5.2. Modeling Conductive and Convective Heat Transfer ...................................................................... 139
5.2.1. Heat Transfer Theory ............................................................................................................. 139
5.2.1.1.The Energy Equation .................................................................................................... 139
5.2.1.2.The Energy Equation in Moving Reference Frames ........................................................ 140
5.2.1.3.The Energy Equation for the Non-Premixed Combustion Model .................................... 140
5.2.1.4. Inclusion of Pressure Work and Kinetic Energy Terms .................................................... 141
5.2.1.5. Inclusion of the Viscous Dissipation Terms .................................................................... 141
5.2.1.6. Inclusion of the Species Diffusion Term ........................................................................ 141
5.2.1.7. Energy Sources Due to Reaction ................................................................................... 142
5.2.1.8. Energy Sources Due To Radiation ................................................................................. 142
5.2.1.9. Energy Source Due To Joule Heating ............................................................................ 142
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5.2.1.10. Interphase Energy Sources ......................................................................................... 142
5.2.1.11. Energy Equation in Solid Regions ............................................................................... 142
5.2.1.12. Anisotropic Conductivity in Solids .............................................................................. 143
5.2.1.13. Diffusion at Inlets ....................................................................................................... 143
5.2.2. Natural Convection and Buoyancy-Driven Flows Theory ........................................................ 143
5.3. Modeling Radiation ...................................................................................................................... 144
5.3.1. Overview and Limitations ..................................................................................................... 144
5.3.1.1. Advantages and Limitations of the DTRM ..................................................................... 145
5.3.1.2. Advantages and Limitations of the P-1 Model ............................................................... 145
5.3.1.3. Advantages and Limitations of the Rosseland Model .................................................... 146
5.3.1.4. Advantages and Limitations of the DO Model ............................................................... 146
5.3.1.5. Advantages and Limitations of the S2S Model .............................................................. 146
5.3.1.6. Advantages and Limitations of the MC Model ............................................................... 147
5.3.2. Radiative Transfer Equation .................................................................................................. 148
5.3.3. P-1 Radiation Model Theory .................................................................................................. 149
5.3.3.1. The P-1 Model Equations ............................................................................................. 150
5.3.3.2. Anisotropic Scattering ................................................................................................. 151
5.3.3.3. Particulate Effects in the P-1 Model .............................................................................. 151
5.3.3.4. Boundary Condition Treatment for the P-1 Model at Walls ............................................. 152
5.3.3.5. Boundary Condition Treatment for the P-1 Model at Flow Inlets and Exits ...................... 153
5.3.4. Rosseland Radiation Model Theory ....................................................................................... 153
5.3.4.1.The Rosseland Model Equations ................................................................................... 153
5.3.4.2. Anisotropic Scattering ................................................................................................. 154
5.3.4.3. Boundary Condition Treatment at Walls ........................................................................ 154
5.3.4.4. Boundary Condition Treatment at Flow Inlets and Exits ................................................. 154
5.3.5. Discrete Transfer Radiation Model (DTRM) Theory ................................................................. 154
5.3.5.1.The DTRM Equations .................................................................................................... 154
5.3.5.2. Ray Tracing .................................................................................................................. 155
5.3.5.3. Clustering .................................................................................................................... 155
5.3.5.4. Boundary Condition Treatment for the DTRM at Walls ................................................... 156
5.3.5.5. Boundary Condition Treatment for the DTRM at Flow Inlets and Exits ............................ 156
5.3.6. Discrete Ordinates (DO) Radiation Model Theory ................................................................... 157
5.3.6.1. The DO Model Equations ............................................................................................. 157
5.3.6.2. Energy Coupling and the DO Model ............................................................................. 158
5.3.6.2.1. Limitations of DO/Energy Coupling ..................................................................... 159
5.3.6.3. Angular Discretization and Pixelation ........................................................................... 159
5.3.6.4. Anisotropic Scattering ................................................................................................. 162
5.3.6.5. Particulate Effects in the DO Model .............................................................................. 163
5.3.6.6. Boundary and Cell Zone Condition Treatment at Opaque Walls ..................................... 163
5.3.6.6.1. Gray Diffuse Walls ............................................................................................... 165
5.3.6.6.2. Non-Gray Diffuse Walls ........................................................................................ 165
5.3.6.7. Cell Zone and Boundary Condition Treatment at Semi-Transparent Walls ...................... 165
5.3.6.7.1. Semi-Transparent Interior Walls ........................................................................... 166
5.3.6.7.2. Specular Semi-Transparent Walls ......................................................................... 167
5.3.6.7.3. Diffuse Semi-Transparent Walls ............................................................................ 169
5.3.6.7.4. Partially Diffuse Semi-Transparent Walls ............................................................... 170
5.3.6.7.5. Semi-Transparent Exterior Walls ........................................................................... 170
5.3.6.7.6. Limitations .......................................................................................................... 172
5.3.6.7.7. Solid Semi-Transparent Media ............................................................................. 173
5.3.6.8. Boundary Condition Treatment at Specular Walls and Symmetry Boundaries ................. 173
5.3.6.9. Boundary Condition Treatment at Periodic Boundaries ................................................. 173
5.3.6.10. Boundary Condition Treatment at Flow Inlets and Exits ............................................... 173
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5.3.7. Surface-to-Surface (S2S) Radiation Model Theory .................................................................. 173
5.3.7.1. Gray-Diffuse Radiation ................................................................................................. 173
5.3.7.2.The S2S Model Equations ............................................................................................. 174
5.3.7.3. Clustering .................................................................................................................... 175
5.3.7.3.1. Clustering and View Factors ................................................................................ 175
5.3.7.3.2. Clustering and Radiosity ...................................................................................... 175
5.3.8. Monte Carlo (MC) Radiation Model Theory ............................................................................ 176
5.3.8.1. The MC Model Equations ............................................................................................. 176
5.3.8.1.1. Monte Carlo Solution Accuracy ............................................................................ 176
5.3.8.2. Boundary Condition Treatment for the MC Model ......................................................... 177
5.3.9. Radiation in Combusting Flows ............................................................................................ 177
5.3.9.1. The Weighted-Sum-of-Gray-Gases Model ..................................................................... 177
5.3.9.1.1.When the Total (Static) Gas Pressure is Not Equal to 1 atm .................................... 178
5.3.9.2.The Effect of Soot on the Absorption Coefficient ........................................................... 179
5.3.9.3.The Effect of Particles on the Absorption Coefficient ..................................................... 179
5.3.10. Choosing a Radiation Model ............................................................................................... 179
5.3.10.1. External Radiation ...................................................................................................... 180
6. Heat Exchangers .................................................................................................................................. 181
6.1.The Macro Heat Exchanger Models ................................................................................................ 181
6.1.1. Overview of the Macro Heat Exchanger Models .................................................................... 181
6.1.2. Restrictions of the Macro Heat Exchanger Models ................................................................. 183
6.1.3. Macro Heat Exchanger Model Theory .................................................................................... 184
6.1.3.1. Streamwise Pressure Drop ........................................................................................... 185
6.1.3.2. Heat Transfer Effectiveness ........................................................................................... 186
6.1.3.3. Heat Rejection ............................................................................................................. 187
6.1.3.4. Macro Heat Exchanger Group Connectivity .................................................................. 188
6.2. The Dual Cell Model ...................................................................................................................... 189
6.2.1. Overview of the Dual Cell Model ........................................................................................... 189
6.2.2. Restrictions of the Dual Cell Model ........................................................................................ 190
6.2.3. Dual Cell Model Theory ......................................................................................................... 190
6.2.3.1. NTU Relations .............................................................................................................. 191
6.2.3.2. Heat Rejection ............................................................................................................. 191
7. Species Transport and Finite-Rate Chemistry ..................................................................................... 193
7.1. Volumetric Reactions .................................................................................................................... 193
7.1.1. Species Transport Equations ................................................................................................. 193
7.1.1.1. Mass Diffusion in Laminar Flows ................................................................................... 194
7.1.1.2. Mass Diffusion in Turbulent Flows ................................................................................ 194
7.1.1.3.Treatment of Species Transport in the Energy Equation ................................................. 194
7.1.1.4. Diffusion at Inlets ......................................................................................................... 194
7.1.2.The Generalized Finite-Rate Formulation for Reaction Modeling ............................................ 195
7.1.2.1. Direct Use of Finite-Rate Kinetics (no TCI) ...................................................................... 195
7.1.2.2. Pressure-Dependent Reactions .................................................................................... 197
7.1.2.3.The Eddy-Dissipation Model ......................................................................................... 199
7.1.2.4. The Eddy-Dissipation Model for LES ............................................................................. 200
7.1.2.5. The Eddy-Dissipation-Concept (EDC) Model ................................................................. 200
7.1.2.6.The Thickened Flame Model ......................................................................................... 202
7.1.2.7.The Relaxation to Chemical Equilibrium Model ............................................................. 203
7.2.Wall Surface Reactions and Chemical Vapor Deposition .................................................................. 205
7.2.1. Surface Coverage Reaction Rate Modification ....................................................................... 206
7.2.2. Reaction-Diffusion Balance for Surface Chemistry ................................................................. 207
7.2.3. Slip Boundary Formulation for Low-Pressure Gas Systems ..................................................... 207
7.3. Particle Surface Reactions ............................................................................................................. 209
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7.3.1. General Description .............................................................................................................. 209
7.3.2. ANSYS Fluent Model Formulation ......................................................................................... 210
7.3.3. Extension for Stoichiometries with Multiple Gas Phase Reactants .......................................... 211
7.3.4. Solid-Solid Reactions ............................................................................................................ 212
7.3.5. Solid Decomposition Reactions ............................................................................................ 212
7.3.6. Solid Deposition Reactions ................................................................................................... 212
7.3.7. Gaseous Solid Catalyzed Reactions on the Particle Surface .................................................... 212
7.4. Electrochemical Reactions ............................................................................................................. 213
7.4.1. Overview and Limitations ..................................................................................................... 213
7.4.2. Electrochemical Reaction Model Theory ................................................................................ 213
7.5. Reacting Channel Model ............................................................................................................... 216
7.5.1. Overview and Limitations ..................................................................................................... 216
7.5.2. Reacting Channel Model Theory ........................................................................................... 217
7.5.2.1. Flow Inside the Reacting Channel ................................................................................. 217
7.5.2.2. Surface Reactions in the Reacting Channel ................................................................... 218
7.5.2.3. Porous Medium Inside Reacting Channel ...................................................................... 219
7.5.2.4. Outer Flow in the Shell ................................................................................................. 219
7.6. Reactor Network Model ................................................................................................................ 220
7.6.1. Reactor Network Model Theory ............................................................................................ 220
7.6.1.1. Reactor network temperature solution ......................................................................... 221
8. Non-Premixed Combustion ................................................................................................................. 223
8.1. Introduction ................................................................................................................................. 223
8.2. Non-Premixed Combustion and Mixture Fraction Theory ............................................................... 223
8.2.1. Mixture Fraction Theory ....................................................................................................... 224
8.2.1.1. Definition of the Mixture Fraction ................................................................................ 224
8.2.1.2.Transport Equations for the Mixture Fraction ................................................................ 226
8.2.1.3. The Non-Premixed Model for LES ................................................................................. 227
8.2.1.4. Mixture Fraction vs. Equivalence Ratio .......................................................................... 227
8.2.1.5. Relationship of Mixture Fraction to Species Mass Fraction, Density, and Temperature ..... 228
8.2.2. Modeling of Turbulence-Chemistry Interaction ..................................................................... 229
8.2.2.1. Description of the Probability Density Function ............................................................ 229
8.2.2.2. Derivation of Mean Scalar Values from the Instantaneous Mixture Fraction ................... 229
8.2.2.3. The Assumed-Shape PDF ............................................................................................. 230
8.2.2.3.1.The Double Delta Function PDF ........................................................................... 230
8.2.2.3.2.The β-Function PDF ............................................................................................. 231
8.2.3. Non-Adiabatic Extensions of the Non-Premixed Model .......................................................... 232
8.2.4. Chemistry Tabulation ........................................................................................................... 234
8.2.4.1. Look-Up Tables for Adiabatic Systems ........................................................................... 234
8.2.4.2. 3D Look-Up Tables for Non-Adiabatic Systems .............................................................. 236
8.2.4.3. Generating Lookup Tables Through Automated Grid Refinement .................................. 238
8.3. Restrictions and Special Cases for Using the Non-Premixed Model ................................................. 240
8.3.1. Restrictions on the Mixture Fraction Approach ...................................................................... 240
8.3.2. Using the Non-Premixed Model for Liquid Fuel or Coal Combustion ...................................... 243
8.3.3. Using the Non-Premixed Model with Flue Gas Recycle .......................................................... 244
8.3.4. Using the Non-Premixed Model with the Inert Model ............................................................ 244
8.3.4.1. Mixture Composition ................................................................................................... 245
8.3.4.1.1. Property Evaluation ............................................................................................. 246
8.4.The Diffusion Flamelet Models Theory ........................................................................................... 246
8.4.1. Restrictions and Assumptions ............................................................................................... 246
8.4.2.The Flamelet Concept ........................................................................................................... 246
8.4.2.1. Overview ..................................................................................................................... 246
8.4.2.2. Strain Rate and Scalar Dissipation ................................................................................. 248
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8.4.2.3. Embedding Diffusion Flamelets in Turbulent Flames ..................................................... 248
8.4.3. Flamelet Generation ............................................................................................................. 249
8.4.4. Flamelet Import ................................................................................................................... 250
8.5. The Steady Diffusion Flamelet Model Theory ................................................................................. 251
8.5.1. Overview ............................................................................................................................. 252
8.5.2. Multiple Steady Flamelet Libraries ........................................................................................ 252
8.5.3. Steady Diffusion Flamelet Automated Grid Refinement ......................................................... 253
8.5.4. Non-Adiabatic Steady Diffusion Flamelets ............................................................................. 253
8.6. The Unsteady Diffusion Flamelet Model Theory ............................................................................. 254
8.6.1. The Eulerian Unsteady Laminar Flamelet Model .................................................................... 254
8.6.1.1. Liquid Reactions .......................................................................................................... 256
8.6.2. The Diesel Unsteady Laminar Flamelet Model ....................................................................... 257
8.6.3. Multiple Diesel Unsteady Flamelets ....................................................................................... 257
8.6.4. Multiple Diesel Unsteady Flamelets with Flamelet Reset ........................................................ 258
8.6.4.1. Resetting the Flamelets ................................................................................................ 258
9. Premixed Combustion ......................................................................................................................... 261
9.1. Overview and Limitations ............................................................................................................. 261
9.1.1. Overview ............................................................................................................................. 261
9.1.2. Limitations ........................................................................................................................... 262
9.2. C-Equation Model Theory .............................................................................................................. 262
9.2.1. Propagation of the Flame Front ............................................................................................ 262
9.3. G-Equation Model Theory ............................................................................................................. 264
9.3.1. Numerical Solution of the G-equation ................................................................................... 265
9.4. Turbulent Flame Speed Models ..................................................................................................... 265
9.4.1. Zimont Turbulent Flame Speed Closure Model ...................................................................... 265
9.4.1.1. Zimont Turbulent Flame Speed Closure for LES ............................................................. 266
9.4.1.2. Flame Stretch Effect ..................................................................................................... 267
9.4.1.3. Gradient Diffusion ....................................................................................................... 267
9.4.1.4.Wall Damping .............................................................................................................. 268
9.4.2. Peters Flame Speed Model .................................................................................................... 268
9.4.2.1. Peters Flame Speed Model for LES ................................................................................ 269
9.5. Extended Coherent Flamelet Model Theory ................................................................................... 270
9.5.1. Closure for ECFM Source Terms ............................................................................................. 272
9.5.2.Turbulent Flame Speed in ECFM ............................................................................................ 274
9.5.3. LES and ECFM ...................................................................................................................... 274
9.6. Calculation of Properties ............................................................................................................... 276
9.6.1. Calculation of Temperature ................................................................................................... 277
9.6.1.1. Adiabatic Temperature Calculation ............................................................................... 277
9.6.1.2. Non-Adiabatic Temperature Calculation ....................................................................... 277
9.6.2. Calculation of Density .......................................................................................................... 277
9.6.3. Laminar Flame Speed ........................................................................................................... 278
9.6.4. Unburnt Density and Thermal Diffusivity ............................................................................... 278
10. Partially Premixed Combustion ........................................................................................................ 279
10.1. Overview .................................................................................................................................... 279
10.2. Limitations .................................................................................................................................. 279
10.3. Partially Premixed Combustion Theory ........................................................................................ 280
10.3.1. Chemical Equilibrium and Steady Diffusion Flamelet Models ............................................... 280
10.3.2. Flamelet Generated Manifold (FGM) model ......................................................................... 281
10.3.2.1. Premixed FGMs .......................................................................................................... 281
10.3.2.2. Diffusion FGMs .......................................................................................................... 283
10.3.3. FGM Turbulent Closure ....................................................................................................... 283
10.3.4. Calculation of Mixture Properties ........................................................................................ 285
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10.3.5. Calculation of Unburnt Properties ....................................................................................... 286
10.3.6. Laminar Flame Speed ......................................................................................................... 286
11. Composition PDF Transport .............................................................................................................. 289
11.1. Overview and Limitations ............................................................................................................ 289
11.2. Composition PDF Transport Theory ............................................................................................. 289
11.3.The Lagrangian Solution Method ................................................................................................. 290
11.3.1. Particle Convection ............................................................................................................ 291
11.3.2. Particle Mixing ................................................................................................................... 292
11.3.2.1.The Modified Curl Model ............................................................................................ 292
11.3.2.2.The IEM Model ........................................................................................................... 292
11.3.2.3. The EMST Model ........................................................................................................ 293
11.3.2.4. Liquid Reactions ........................................................................................................ 293
11.3.3. Particle Reaction ................................................................................................................. 293
11.4. The Eulerian Solution Method ..................................................................................................... 294
11.4.1. Reaction ............................................................................................................................. 295
11.4.2. Mixing ................................................................................................................................ 295
11.4.3. Correction .......................................................................................................................... 295
11.4.4. Calculation of Composition Mean and Variance ................................................................... 296
12. Chemistry Acceleration ..................................................................................................................... 297
12.1. Overview and Limitations ............................................................................................................ 297
12.2. In-Situ Adaptive Tabulation (ISAT) ................................................................................................ 297
12.3. Dynamic Mechanism Reduction .................................................................................................. 299
12.3.1. Directed Relation Graph (DRG) Method for Mechanism Reduction ....................................... 300
12.4. Chemistry Agglomeration ........................................................................................................... 301
12.4.1. Binning Algorithm .............................................................................................................. 302
12.5. Chemical Mechanism Dimension Reduction ................................................................................ 304
12.5.1. Selecting the Represented Species ...................................................................................... 304
12.6. Dynamic Cell Clustering with ANSYS CHEMKIN-CFD Solver ........................................................... 305
13. Engine Ignition .................................................................................................................................. 307
13.1. Spark Model ................................................................................................................................ 307
13.1.1. Overview and Limitations ................................................................................................... 307
13.1.2. Spark Model Theory ............................................................................................................ 307
13.1.3. ECFM Spark Model Variants ................................................................................................. 310
13.2. Autoignition Models ................................................................................................................... 311
13.2.1. Model Overview ................................................................................................................. 311
13.2.2. Model Limitations .............................................................................................................. 311
13.2.3. Ignition Model Theory ........................................................................................................ 312
13.2.3.1.Transport of Ignition Species ...................................................................................... 312
13.2.3.2. Knock Modeling ........................................................................................................ 312
13.2.3.2.1. Modeling of the Source Term ............................................................................. 313
13.2.3.2.2. Correlations ...................................................................................................... 313
13.2.3.2.3. Energy Release .................................................................................................. 314
13.2.3.3. Ignition Delay Modeling ............................................................................................. 314
13.2.3.3.1. Modeling of the Source Term ............................................................................. 314
13.2.3.3.2. Correlations ...................................................................................................... 315
13.2.3.3.3. Energy Release .................................................................................................. 315
13.3. Crevice Model ............................................................................................................................. 315
13.3.1. Overview ........................................................................................................................... 315
13.3.1.1. Model Parameters ...................................................................................................... 316
13.3.2. Limitations ......................................................................................................................... 317
13.3.3. Crevice Model Theory ......................................................................................................... 318
14. Pollutant Formation .......................................................................................................................... 319
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14.1. NOx Formation ........................................................................................................................... 319
14.1.1. Overview ........................................................................................................................... 319
14.1.1.1. NOx Modeling in ANSYS Fluent .................................................................................. 319
14.1.1.2. NOx Formation and Reduction in Flames .................................................................... 320
14.1.2. Governing Equations for NOx Transport .............................................................................. 320
14.1.3.Thermal NOx Formation ...................................................................................................... 321
14.1.3.1. Thermal NOx Reaction Rates ...................................................................................... 321
14.1.3.2. The Quasi-Steady Assumption for [N] ......................................................................... 321
14.1.3.3.Thermal NOx Temperature Sensitivity ......................................................................... 322
14.1.3.4. Decoupled Thermal NOx Calculations ......................................................................... 322
14.1.3.5. Approaches for Determining O Radical Concentration ................................................ 322
14.1.3.5.1. Method 1: Equilibrium Approach ....................................................................... 322
14.1.3.5.2. Method 2: Partial Equilibrium Approach ............................................................. 323
14.1.3.5.3. Method 3: Predicted O Approach ....................................................................... 323
14.1.3.6. Approaches for Determining OH Radical Concentration .............................................. 323
14.1.3.6.1. Method 1: Exclusion of OH Approach ................................................................. 323
14.1.3.6.2. Method 2: Partial Equilibrium Approach ............................................................. 323
14.1.3.6.3. Method 3: Predicted OH Approach ..................................................................... 324
14.1.3.7. Summary ................................................................................................................... 324
14.1.4. Prompt NOx Formation ....................................................................................................... 324
14.1.4.1. Prompt NOx Combustion Environments ..................................................................... 324
14.1.4.2. Prompt NOx Mechanism ............................................................................................ 324
14.1.4.3. Prompt NOx Formation Factors .................................................................................. 325
14.1.4.4. Primary Reaction ....................................................................................................... 325
14.1.4.5. Modeling Strategy ..................................................................................................... 325
14.1.4.6. Rate for Most Hydrocarbon Fuels ................................................................................ 326
14.1.4.7. Oxygen Reaction Order .............................................................................................. 326
14.1.5. Fuel NOx Formation ............................................................................................................ 327
14.1.5.1. Fuel-Bound Nitrogen ................................................................................................. 327
14.1.5.2. Reaction Pathways ..................................................................................................... 327
14.1.5.3. Fuel NOx from Gaseous and Liquid Fuels .................................................................... 327
14.1.5.3.1. Fuel NOx from Intermediate Hydrogen Cyanide (HCN) ....................................... 328
14.1.5.3.1.1. HCN Production in a Gaseous Fuel ............................................................ 328
14.1.5.3.1.2. HCN Production in a Liquid Fuel ................................................................ 328
14.1.5.3.1.3. HCN Consumption .................................................................................... 329
14.1.5.3.1.4. HCN Sources in the Transport Equation ..................................................... 329
14.1.5.3.1.5. NOx Sources in the Transport Equation ..................................................... 329
14.1.5.3.2. Fuel NOx from Intermediate Ammonia (NH3) ..................................................... 330
14.1.5.3.2.1. NH3 Production in a Gaseous Fuel ............................................................. 330
14.1.5.3.2.2. NH3 Production in a Liquid Fuel ................................................................ 330
14.1.5.3.2.3. NH3 Consumption .................................................................................... 331
14.1.5.3.2.4. NH3 Sources in the Transport Equation ..................................................... 331
14.1.5.3.2.5. NOx Sources in the Transport Equation ..................................................... 331
14.1.5.3.3. Fuel NOx from Coal ........................................................................................... 332
14.1.5.3.3.1. Nitrogen in Char and in Volatiles ............................................................... 332
14.1.5.3.3.2. Coal Fuel NOx Scheme A ........................................................................... 332
14.1.5.3.3.3. Coal Fuel NOx Scheme B ........................................................................... 332
14.1.5.3.3.4. HCN Scheme Selection ............................................................................. 333
14.1.5.3.3.5. NOx Reduction on Char Surface ................................................................ 333
14.1.5.3.3.5.1. BET Surface Area .............................................................................. 334
14.1.5.3.3.5.2. HCN from Volatiles ........................................................................... 334
14.1.5.3.3.6. Coal Fuel NOx Scheme C ........................................................................... 334
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14.1.5.3.3.7. Coal Fuel NOx Scheme D ........................................................................... 335
14.1.5.3.3.8. NH3 Scheme Selection ............................................................................. 336
14.1.5.3.3.8.1. NH3 from Volatiles ........................................................................... 336
14.1.5.3.4. Fuel Nitrogen Partitioning for HCN and NH3 Intermediates ................................ 336
14.1.6. NOx Formation from Intermediate N2O ............................................................................... 337
14.1.6.1. N2O - Intermediate NOx Mechanism .......................................................................... 337
14.1.7. NOx Reduction by Reburning ............................................................................................. 338
14.1.7.1. Instantaneous Approach ............................................................................................ 338
14.1.7.2. Partial Equilibrium Approach ..................................................................................... 339
14.1.7.2.1. NOx Reduction Mechanism ............................................................................... 339
14.1.8. NOx Reduction by SNCR ..................................................................................................... 341
14.1.8.1. Ammonia Injection .................................................................................................... 341
14.1.8.2. Urea Injection ............................................................................................................ 342
14.1.8.3. Transport Equations for Urea, HNCO, and NCO ............................................................ 343
14.1.8.4. Urea Production due to Reagent Injection .................................................................. 344
14.1.8.5. NH3 Production due to Reagent Injection ................................................................... 344
14.1.8.6. HNCO Production due to Reagent Injection ................................................................ 344
14.1.9. NOx Formation in Turbulent Flows ...................................................................................... 345
14.1.9.1. The Turbulence-Chemistry Interaction Model ............................................................. 345
14.1.9.2. The PDF Approach ..................................................................................................... 346
14.1.9.3.The General Expression for the Mean Reaction Rate .................................................... 346
14.1.9.4.The Mean Reaction Rate Used in ANSYS Fluent ........................................................... 346
14.1.9.5. Statistical Independence ............................................................................................ 346
14.1.9.6.The Beta PDF Option .................................................................................................. 347
14.1.9.7.The Gaussian PDF Option ........................................................................................... 347
14.1.9.8. The Calculation Method for the Variance .................................................................... 347
14.2. SOx Formation ............................................................................................................................ 348
14.2.1. Overview ........................................................................................................................... 348
14.2.1.1.The Formation of SOx ................................................................................................. 348
14.2.2. Governing Equations for SOx Transport ............................................................................... 349
14.2.3. Reaction Mechanisms for Sulfur Oxidation .......................................................................... 350
14.2.4. SO2 and H2S Production in a Gaseous Fuel ......................................................................... 351
14.2.5. SO2 and H2S Production in a Liquid Fuel ............................................................................. 352
14.2.6. SO2 and H2S Production from Coal ..................................................................................... 352
14.2.6.1. SO2 and H2S from Char .............................................................................................. 352
14.2.6.2. SO2 and H2S from Volatiles ........................................................................................ 352
14.2.7. SOx Formation in Turbulent Flows ....................................................................................... 353
14.2.7.1. The Turbulence-Chemistry Interaction Model ............................................................. 353
14.2.7.2. The PDF Approach ..................................................................................................... 353
14.2.7.3.The Mean Reaction Rate ............................................................................................. 353
14.2.7.4.The PDF Options ........................................................................................................ 353
14.3. Soot Formation ........................................................................................................................... 353
14.3.1. Overview and Limitations ................................................................................................... 354
14.3.1.1. Predicting Soot Formation ......................................................................................... 354
14.3.1.2. Restrictions on Soot Modeling ................................................................................... 354
14.3.2. Soot Model Theory ............................................................................................................. 355
14.3.2.1.The One-Step Soot Formation Model .......................................................................... 355
14.3.2.2.The Two-Step Soot Formation Model .......................................................................... 356
14.3.2.2.1. Soot Generation Rate ........................................................................................ 356
14.3.2.2.2. Nuclei Generation Rate ...................................................................................... 357
14.3.2.3. The Moss-Brookes Model ........................................................................................... 357
14.3.2.3.1.The Moss-Brookes-Hall Model ............................................................................ 359
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14.3.2.3.2. Soot Formation in Turbulent Flows .................................................................... 360
14.3.2.3.2.1.The Turbulence-Chemistry Interaction Model ............................................ 360
14.3.2.3.2.2.The PDF Approach .................................................................................... 361
14.3.2.3.2.3. The Mean Reaction Rate ........................................................................... 361
14.3.2.3.2.4.The PDF Options ....................................................................................... 361
14.3.2.3.3.The Effect of Soot on the Radiation Absorption Coefficient ................................. 361
14.3.2.4.The Method of Moments Model ................................................................................. 361
14.3.2.4.1. Soot Particle Population Balance ....................................................................... 361
14.3.2.4.2. Moment Transport Equations ............................................................................ 363
14.3.2.4.3. Nucleation ........................................................................................................ 363
14.3.2.4.4. Coagulation ...................................................................................................... 365
14.3.2.4.5. Surface Growth and Oxidation ........................................................................... 368
14.3.2.4.6. Soot Aggregation .............................................................................................. 370
14.4. Decoupled Detailed Chemistry Model ......................................................................................... 374
14.4.1. Overview ........................................................................................................................... 374
14.4.1.1. Limitations ................................................................................................................ 375
14.4.2. Decoupled Detailed Chemistry Model Theory ..................................................................... 375
15. Aerodynamically Generated Noise ................................................................................................... 377
15.1. Overview .................................................................................................................................... 377
15.1.1. Direct Method .................................................................................................................... 377
15.1.2. Integral Method Based on Acoustic Analogy ....................................................................... 378
15.1.3. Broadband Noise Source Models ........................................................................................ 379
15.2. Acoustics Model Theory .............................................................................................................. 379
15.2.1. The Ffowcs-Williams and Hawkings Model .......................................................................... 379
15.2.2. Broadband Noise Source Models ........................................................................................ 382
15.2.2.1. Proudman’s Formula .................................................................................................. 382
15.2.2.2.The Jet Noise Source Model ........................................................................................ 383
15.2.2.3.The Boundary Layer Noise Source Model .................................................................... 384
15.2.2.4. Source Terms in the Linearized Euler Equations ........................................................... 385
15.2.2.5. Source Terms in Lilley’s Equation ................................................................................ 385
16. Discrete Phase ................................................................................................................................... 387
16.1. Introduction ............................................................................................................................... 387
16.1.1.The Euler-Lagrange Approach ............................................................................................. 387
16.2. Particle Motion Theory ................................................................................................................ 388
16.2.1. Equations of Motion for Particles ........................................................................................ 388
16.2.1.1. Particle Force Balance ................................................................................................ 388
16.2.1.2. Particle Torque Balance .............................................................................................. 388
16.2.1.3. Inclusion of the Gravity Term ...................................................................................... 389
16.2.1.4. Other Forces .............................................................................................................. 389
16.2.1.5. Forces in Moving Reference Frames ............................................................................ 389
16.2.1.6.Thermophoretic Force ................................................................................................ 390
16.2.1.7. Brownian Force .......................................................................................................... 390
16.2.1.8. Saffman’s Lift Force .................................................................................................... 391
16.2.1.9. Magnus Lift Force ...................................................................................................... 391
16.2.2.Turbulent Dispersion of Particles ......................................................................................... 392
16.2.2.1. Stochastic Tracking .................................................................................................... 392
16.2.2.1.1. The Integral Time .............................................................................................. 393
16.2.2.1.2.The Discrete Random Walk Model ...................................................................... 393
16.2.2.1.3. Using the DRW Model ....................................................................................... 394
16.2.2.2. Particle Cloud Tracking ............................................................................................... 395
16.2.2.2.1. Using the Cloud Model ...................................................................................... 397
16.2.3. Integration of Particle Equation of Motion ........................................................................... 397
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16.3. Laws for Drag Coefficients ........................................................................................................... 399
16.3.1. Spherical Drag Law ............................................................................................................. 400
16.3.2. Non-spherical Drag Law ..................................................................................................... 400
16.3.3. Stokes-Cunningham Drag Law ............................................................................................ 400
16.3.4. High-Mach-Number Drag Law ............................................................................................ 401
16.3.5. Dynamic Drag Model Theory .............................................................................................. 401
16.3.6. Dense Discrete Phase Model Drag Laws .............................................................................. 401
16.3.7. Bubbly Flow Drag Laws ...................................................................................................... 402
16.3.7.1. Ishii-Zuber Drag Model .............................................................................................. 402
16.3.7.2. Grace Drag Model ...................................................................................................... 403
16.3.8. Rotational Drag Law ........................................................................................................... 403
16.4. Laws for Heat and Mass Exchange ............................................................................................... 404
16.4.1. Inert Heating or Cooling (Law 1/Law 6) ............................................................................... 404
16.4.2. Droplet Vaporization (Law 2) ............................................................................................... 406
16.4.2.1. Mass Transfer During Law 2—Diffusion Controlled Model ........................................... 407
16.4.2.2. Mass Transfer During Law 2—Convection/Diffusion Controlled Model ........................ 408
16.4.2.3. Defining the Saturation Vapor Pressure and Diffusion Coefficient (or Binary Diffusivity) ......................................................................................................................................... 408
16.4.2.4. Defining the Boiling Point and Latent Heat ................................................................. 409
16.4.2.5. Heat Transfer to the Droplet ....................................................................................... 410
16.4.3. Droplet Boiling (Law 3) ....................................................................................................... 412
16.4.4. Devolatilization (Law 4) ...................................................................................................... 413
16.4.4.1. Choosing the Devolatilization Model .......................................................................... 413
16.4.4.2.The Constant Rate Devolatilization Model ................................................................... 413
16.4.4.3. The Single Kinetic Rate Model .................................................................................... 414
16.4.4.4.The Two Competing Rates (Kobayashi) Model ............................................................. 415
16.4.4.5. The CPD Model .......................................................................................................... 415
16.4.4.5.1. General Description .......................................................................................... 415
16.4.4.5.2. Reaction Rates .................................................................................................. 416
16.4.4.5.3. Mass Conservation ............................................................................................ 417
16.4.4.5.4. Fractional Change in the Coal Mass .................................................................... 417
16.4.4.5.5. CPD Inputs ........................................................................................................ 418
16.4.4.5.6. Particle Swelling During Devolatilization ............................................................ 419
16.4.4.5.7. Heat Transfer to the Particle During Devolatilization ........................................... 420
16.4.5. Surface Combustion (Law 5) ............................................................................................... 420
16.4.5.1.The Diffusion-Limited Surface Reaction Rate Model .................................................... 421
16.4.5.2.The Kinetic/Diffusion Surface Reaction Rate Model ..................................................... 421
16.4.5.3. The Intrinsic Model .................................................................................................... 422
16.4.5.4.The Multiple Surface Reactions Model ........................................................................ 423
16.4.5.4.1. Limitations ........................................................................................................ 424
16.4.5.5. Heat and Mass Transfer During Char Combustion ....................................................... 424
16.4.6. Multicomponent Particle Definition (Law 7) ........................................................................ 424
16.4.6.1. Raoult’s Law .............................................................................................................. 426
16.4.6.2. Peng-Robinson Real Gas Model .................................................................................. 426
16.5.Vapor Liquid Equilibrium Theory .................................................................................................. 426
16.6. Physical Property Averaging ........................................................................................................ 428
16.7.Wall-Particle Reflection Model Theory .......................................................................................... 430
16.7.1. Rough Wall Model .............................................................................................................. 432
16.8.Wall-Jet Model Theory ................................................................................................................. 434
16.9.Wall-Film Model Theory ............................................................................................................... 435
16.9.1. Introduction ....................................................................................................................... 435
16.9.2. Interaction During Impact with a Boundary ......................................................................... 436
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16.9.2.1. The Stanton-Rutland Model ....................................................................................... 437
16.9.2.1.1. Regime Definition ............................................................................................. 437
16.9.2.1.2. Rebound ........................................................................................................... 438
16.9.2.1.3. Splashing .......................................................................................................... 438
16.9.2.2.The Kuhnke Model ..................................................................................................... 443
16.9.2.2.1. Regime definition ............................................................................................. 443
16.9.2.2.2. Rebound ........................................................................................................... 446
16.9.2.2.3. Splashing .......................................................................................................... 446
16.9.3. Separation Criteria .............................................................................................................. 449
16.9.4. Conservation Equations for Wall-Film Particles .................................................................... 449
16.9.4.1. Momentum ............................................................................................................... 449
16.9.4.2. Mass Transfer from the Film ........................................................................................ 450
16.9.4.2.1. Film Vaporization and Boiling ............................................................................ 450
16.9.4.2.2. Film Condensation ............................................................................................ 453
16.9.4.3. Energy Transfer from the Film ..................................................................................... 454
16.10. Wall Erosion .............................................................................................................................. 456
16.10.1. Finnie Erosion Model ........................................................................................................ 457
16.10.2. Oka Erosion Model ........................................................................................................... 457
16.10.3. McLaury Erosion Model .................................................................................................... 458
16.10.4. Accretion ......................................................................................................................... 459
16.11. Particle–Wall Impingement Heat Transfer Theory ....................................................................... 460
16.12. Atomizer Model Theory ............................................................................................................. 462
16.12.1.The Plain-Orifice Atomizer Model ...................................................................................... 462
16.12.1.1. Internal Nozzle State ................................................................................................ 464
16.12.1.2. Coefficient of Discharge ........................................................................................... 465
16.12.1.3. Exit Velocity ............................................................................................................. 466
16.12.1.4. Spray Angle ............................................................................................................. 467
16.12.1.5. Droplet Diameter Distribution .................................................................................. 467
16.12.2. The Pressure-Swirl Atomizer Model ................................................................................... 468
16.12.2.1. Film Formation ........................................................................................................ 469
16.12.2.2. Sheet Breakup and Atomization ............................................................................... 470
16.12.3.The Air-Blast/Air-Assist Atomizer Model ............................................................................. 472
16.12.4.The Flat-Fan Atomizer Model ............................................................................................. 473
16.12.5.The Effervescent Atomizer Model ...................................................................................... 474
16.13. Secondary Breakup Model Theory ............................................................................................. 475
16.13.1.Taylor Analogy Breakup (TAB) Model ................................................................................. 475
16.13.1.1. Introduction ............................................................................................................ 475
16.13.1.2. Use and Limitations ................................................................................................. 476
16.13.1.3. Droplet Distortion .................................................................................................... 476
16.13.1.4. Size of Child Droplets ............................................................................................... 477
16.13.1.5.Velocity of Child Droplets ......................................................................................... 478
16.13.1.6. Droplet Breakup ...................................................................................................... 478
16.13.2.Wave Breakup Model ........................................................................................................ 479
16.13.2.1. Introduction ............................................................................................................ 479
16.13.2.2. Use and Limitations ................................................................................................. 480
16.13.2.3. Jet Stability Analysis ................................................................................................. 480
16.13.2.4. Droplet Breakup ...................................................................................................... 481
16.13.3. KHRT Breakup Model ........................................................................................................ 482
16.13.3.1. Introduction ............................................................................................................ 482
16.13.3.2. Use and Limitations ................................................................................................. 482
16.13.3.3. Liquid Core Length .................................................................................................. 482
16.13.3.4. Rayleigh-Taylor Breakup ........................................................................................... 483
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16.13.3.5. Droplet Breakup Within the Liquid Core .................................................................... 484
16.13.3.6. Droplet Breakup Outside the Liquid Core .................................................................. 484
16.13.4. Stochastic Secondary Droplet (SSD) Model ........................................................................ 484
16.13.4.1. Theory ..................................................................................................................... 484
16.14. Collision and Droplet Coalescence Model Theory ....................................................................... 485
16.14.1. Introduction ..................................................................................................................... 485
16.14.2. Use and Limitations .......................................................................................................... 486
16.14.3.Theory .............................................................................................................................. 486
16.14.3.1. Probability of Collision ............................................................................................. 486
16.14.3.2. Collision Outcomes .................................................................................................. 487
16.15. Discrete Element Method Collision Model .................................................................................. 488
16.15.1.Theory .............................................................................................................................. 488
16.15.1.1. The Spring Collision Law .......................................................................................... 489
16.15.1.2. The Spring-Dashpot Collision Law ............................................................................ 490
16.15.1.3. The Hertzian Collision Law ....................................................................................... 490
16.15.1.4. The Hertzian-Dashpot Collision Law ......................................................................... 491
16.15.1.5.The Friction Collision Law ......................................................................................... 491
16.15.1.6. Rolling Friction Collision Law for DEM ....................................................................... 492
16.15.1.7. DEM Parcels ............................................................................................................. 493
16.15.1.8. Cartesian Collision Mesh .......................................................................................... 493
16.16. One-Way and Two-Way Coupling ............................................................................................... 494
16.16.1. Coupling Between the Discrete and Continuous Phases .................................................... 494
16.16.2. Momentum Exchange ...................................................................................................... 495
16.16.3. Heat Exchange ................................................................................................................. 495
16.16.4. Mass Exchange ................................................................................................................. 496
16.16.5. Under-Relaxation of the Interphase Exchange Terms ......................................................... 496
16.16.6. Interphase Exchange During Stochastic Tracking ............................................................... 498
16.16.7. Interphase Exchange During Cloud Tracking ..................................................................... 498
16.17. Node Based Averaging .............................................................................................................. 498
17. Multiphase Flows .............................................................................................................................. 501
17.1. Introduction ............................................................................................................................... 501
17.1.1. Multiphase Flow Regimes ................................................................................................... 501
17.1.1.1. Gas-Liquid or Liquid-Liquid Flows .............................................................................. 501
17.1.1.2. Gas-Solid Flows .......................................................................................................... 502
17.1.1.3. Liquid-Solid Flows ...................................................................................................... 502
17.1.1.4. Three-Phase Flows ..................................................................................................... 502
17.1.2. Examples of Multiphase Systems ........................................................................................ 503
17.2. Choosing a General Multiphase Model ........................................................................................ 504
17.2.1. Approaches to Multiphase Modeling .................................................................................. 504
17.2.1.1.The Euler-Euler Approach ........................................................................................... 504
17.2.1.1.1.The VOF Model .................................................................................................. 504
17.2.1.1.2. The Mixture Model ............................................................................................ 505
17.2.1.1.3.The Eulerian Model ............................................................................................ 505
17.2.2. Model Comparisons ........................................................................................................... 505
17.2.2.1. Detailed Guidelines ................................................................................................... 506
17.2.2.1.1.The Effect of Particulate Loading ........................................................................ 506
17.2.2.1.2.The Significance of the Stokes Number .............................................................. 507
17.2.2.1.2.1. Examples .................................................................................................. 507
17.2.2.1.3. Other Considerations ........................................................................................ 508
17.2.3.Time Schemes in Multiphase Flow ....................................................................................... 508
17.2.4. Stability and Convergence .................................................................................................. 509
17.3.Volume of Fluid (VOF) Model Theory ............................................................................................ 510
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17.3.1. Overview of the VOF Model ................................................................................................ 510
17.3.2. Limitations of the VOF Model .............................................................................................. 510
17.3.3. Steady-State and Transient VOF Calculations ....................................................................... 510
17.3.4.Volume Fraction Equation ................................................................................................... 511
17.3.4.1. The Implicit Formulation ............................................................................................ 511
17.3.4.2.The Explicit Formulation ............................................................................................. 512
17.3.4.3. Interpolation Near the Interface ................................................................................. 513
17.3.4.3.1. The Geometric Reconstruction Scheme ............................................................. 514
17.3.4.3.2.The Donor-Acceptor Scheme ............................................................................. 515
17.3.4.3.3.The Compressive Interface Capturing Scheme for Arbitrary Meshes (CICSAM) ..... 515
17.3.4.3.4.The Compressive Scheme and Interface-Model-based Variants ........................... 516
17.3.4.3.5. Bounded Gradient Maximization (BGM) ............................................................. 516
17.3.5. Material Properties ............................................................................................................. 517
17.3.6. Momentum Equation ......................................................................................................... 517
17.3.7. Energy Equation ................................................................................................................. 517
17.3.8. Additional Scalar Equations ................................................................................................ 518
17.3.9. Surface Tension and Adhesion ............................................................................................ 518
17.3.9.1. Surface Tension ......................................................................................................... 518
17.3.9.1.1. The Continuum Surface Force Model ................................................................. 518
17.3.9.1.2.The Continuum Surface Stress Model ................................................................. 519
17.3.9.1.3. Comparing the CSS and CSF Methods ................................................................ 520
17.3.9.1.4.When Surface Tension Effects Are Important ...................................................... 520
17.3.9.2.Wall Adhesion ............................................................................................................ 520
17.3.9.3. Jump Adhesion .......................................................................................................... 521
17.3.10. Open Channel Flow .......................................................................................................... 522
17.3.10.1. Upstream Boundary Conditions ............................................................................... 522
17.3.10.1.1. Pressure Inlet .................................................................................................. 522
17.3.10.1.2. Mass Flow Rate ................................................................................................ 523
17.3.10.1.3.Volume Fraction Specification .......................................................................... 523
17.3.10.2. Downstream Boundary Conditions ........................................................................... 523
17.3.10.2.1. Pressure Outlet ................................................................................................ 523
17.3.10.2.2. Outflow Boundary ........................................................................................... 523
17.3.10.2.3. Backflow Volume Fraction Specification ........................................................... 524
17.3.10.3. Numerical Beach Treatment ..................................................................................... 524
17.3.11. Open Channel Wave Boundary Conditions ........................................................................ 525
17.3.11.1. Airy Wave Theory ..................................................................................................... 527
17.3.11.2. Stokes Wave Theories ............................................................................................... 527
17.3.11.3. Cnoidal/Solitary Wave Theory ................................................................................... 528
17.3.11.4. Choosing a Wave Theory .......................................................................................... 530
17.3.11.5. Superposition of Waves ............................................................................................ 532
17.3.11.6. Modeling of Random Waves Using Wave Spectrum ................................................... 533
17.3.11.6.1. Definitions ...................................................................................................... 533
17.3.11.6.2.Wave Spectrum Implementation Theory .......................................................... 533
17.3.11.6.2.1. Long-Crested Random Waves (Unidirectional) ......................................... 533
17.3.11.6.2.1.1. Pierson-Moskowitz Spectrum ......................................................... 533
17.3.11.6.2.1.2. JONSWAP Spectrum ....................................................................... 534
17.3.11.6.2.1.3. TMA Spectrum ............................................................................... 534
17.3.11.6.2.2. Short-Crested Random Waves (Multi-Directional) .................................... 534
17.3.11.6.2.2.1. Cosine-2s Power Function (Frequency Independent) ....................... 535
17.3.11.6.2.2.2. Hyperbolic Function (Frequency Dependent) ................................. 535
17.3.11.6.2.3. Superposition of Individual Wave Components Using the Wave Spectrum ........................................................................................................................... 536
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17.3.11.6.3. Choosing a Wave Spectrum and Inputs ............................................................ 537
17.3.11.7. Nomenclature for Open Channel Waves .................................................................... 539
17.3.12. Coupled Level-Set and VOF Model .................................................................................... 540
17.3.12.1. Theory ..................................................................................................................... 540
17.3.12.1.1. Surface Tension Force ...................................................................................... 541
17.3.12.1.2. Re-initialization of the Level-set Function via the Geometrical Method ............. 542
17.3.12.2. Limitations .............................................................................................................. 543
17.4. Mixture Model Theory ................................................................................................................. 543
17.4.1. Overview ........................................................................................................................... 544
17.4.2. Limitations of the Mixture Model ........................................................................................ 544
17.4.3. Continuity Equation ........................................................................................................... 545
17.4.4. Momentum Equation ......................................................................................................... 545
17.4.5. Energy Equation ................................................................................................................. 546
17.4.6. Relative (Slip) Velocity and the Drift Velocity ........................................................................ 546
17.4.7.Volume Fraction Equation for the Secondary Phases ............................................................ 548
17.4.8. Granular Properties ............................................................................................................ 548
17.4.8.1. Collisional Viscosity .................................................................................................... 548
17.4.8.2. Kinetic Viscosity ......................................................................................................... 548
17.4.9. Granular Temperature ......................................................................................................... 549
17.4.10. Solids Pressure ................................................................................................................. 549
17.4.11. Interfacial Area Concentration .......................................................................................... 550
17.4.11.1. Hibiki-Ishii Model ..................................................................................................... 550
17.4.11.2. Ishii-Kim Model ........................................................................................................ 551
17.4.11.3.Yao-Morel Model ...................................................................................................... 552
17.5. Eulerian Model Theory ................................................................................................................ 553
17.5.1. Overview of the Eulerian Model .......................................................................................... 554
17.5.2. Limitations of the Eulerian Model ........................................................................................ 554
17.5.3.Volume Fraction Equation ................................................................................................... 555
17.5.4. Conservation Equations ...................................................................................................... 556
17.5.4.1. Equations in General Form ......................................................................................... 556
17.5.4.1.1. Conservation of Mass ........................................................................................ 556
17.5.4.1.2. Conservation of Momentum .............................................................................. 556
17.5.4.1.3. Conservation of Energy ..................................................................................... 557
17.5.4.2. Equations Solved by ANSYS Fluent ............................................................................. 557
17.5.4.2.1. Continuity Equation .......................................................................................... 557
17.5.4.2.2. Fluid-Fluid Momentum Equations ...................................................................... 557
17.5.4.2.3. Fluid-Solid Momentum Equations ...................................................................... 558
17.5.4.2.4. Conservation of Energy ..................................................................................... 558
17.5.5. Interfacial Area Concentration ............................................................................................ 558
17.5.6. Interphase Exchange Coefficients ....................................................................................... 559
17.5.6.1. Fluid-Fluid Exchange Coefficient ................................................................................ 560
17.5.6.1.1. Schiller and Naumann Model ............................................................................. 560
17.5.6.1.2. Morsi and Alexander Model ............................................................................... 561
17.5.6.1.3. Symmetric Model .............................................................................................. 561
17.5.6.1.4. Grace et al. Model .............................................................................................. 562
17.5.6.1.5.Tomiyama et al. Model ....................................................................................... 563
17.5.6.1.6. Ishii Model ........................................................................................................ 564
17.5.6.1.7. Universal Drag Laws for Bubble-Liquid and Droplet-Gas Flows ........................... 564
17.5.6.1.7.1. Bubble-Liquid Flow .................................................................................. 565
17.5.6.1.7.2. Droplet-Gas Flow ...................................................................................... 565
17.5.6.2. Fluid-Solid Exchange Coefficient ................................................................................ 566
17.5.6.3. Solid-Solid Exchange Coefficient ................................................................................ 569
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17.5.6.4. Drag Modification ...................................................................................................... 569
17.5.6.4.1. Brucato et al. Correlation ................................................................................... 569
17.5.7. Lift Force ............................................................................................................................ 570
17.5.7.1. Lift Coefficient Models ............................................................................................... 570
17.5.7.1.1. Moraga Lift Force Model .................................................................................... 571
17.5.7.1.2. Saffman-Mei Lift Force Model ............................................................................ 571
17.5.7.1.3. Legendre-Magnaudet Lift Force Model .............................................................. 572
17.5.7.1.4.Tomiyama Lift Force Model ................................................................................ 572
17.5.8. Wall Lubrication Force ........................................................................................................ 573
17.5.8.1.Wall Lubrication Models ............................................................................................. 573
17.5.8.1.1. Antal et al. Model .............................................................................................. 573
17.5.8.1.2.Tomiyama Model ............................................................................................... 574
17.5.8.1.3. Frank Model ...................................................................................................... 574
17.5.8.1.4. Hosokawa Model .............................................................................................. 574
17.5.9. Turbulent Dispersion Force ................................................................................................. 575
17.5.9.1. Models for Turbulent Dispersion Force ....................................................................... 575
17.5.9.1.1. Lopez de Bertodano Model ............................................................................... 576
17.5.9.1.2. Simonin Model .................................................................................................. 576
17.5.9.1.3. Burns et al. Model .............................................................................................. 576
17.5.9.1.4. Diffusion in VOF Model ...................................................................................... 576
17.5.9.2. Limiting Functions for the Turbulent Dispersion Force ................................................ 577
17.5.10.Virtual Mass Force ............................................................................................................. 578
17.5.11. Solids Pressure ................................................................................................................. 578
17.5.11.1. Radial Distribution Function ..................................................................................... 580
17.5.12. Maximum Packing Limit in Binary Mixtures ....................................................................... 581
17.5.13. Solids Shear Stresses ......................................................................................................... 581
17.5.13.1. Collisional Viscosity .................................................................................................. 581
17.5.13.2. Kinetic Viscosity ....................................................................................................... 582
17.5.13.3. Bulk Viscosity ........................................................................................................... 582
17.5.13.4. Frictional Viscosity ................................................................................................... 582
17.5.14. Granular Temperature ....................................................................................................... 583
17.5.15. Description of Heat Transfer .............................................................................................. 585
17.5.15.1. The Heat Exchange Coefficient ................................................................................. 585
17.5.15.1.1. Constant ......................................................................................................... 586
17.5.15.1.2. Nusselt Number .............................................................................................. 586
17.5.15.1.3. Ranz-Marshall Model ....................................................................................... 586
17.5.15.1.4.Tomiyama Model ............................................................................................. 586
17.5.15.1.5. Hughmark Model ............................................................................................ 586
17.5.15.1.6. Gunn Model .................................................................................................... 587
17.5.15.1.7. Two-Resistance Model ..................................................................................... 587
17.5.15.1.8. Fixed To Saturation Temperature ...................................................................... 588
17.5.15.1.9. User Defined ................................................................................................... 588
17.5.16. Turbulence Models ........................................................................................................... 588
17.5.16.1. k- ε Turbulence Models ............................................................................................. 589
17.5.16.1.1. k- ε Mixture Turbulence Model ......................................................................... 589
17.5.16.1.2. k- ε Dispersed Turbulence Model ..................................................................... 590
17.5.16.1.2.1. Assumptions .......................................................................................... 590
17.5.16.1.2.2. Turbulence in the Continuous Phase ....................................................... 591
17.5.16.1.2.3.Turbulence in the Dispersed Phase .......................................................... 592
17.5.16.1.3. k- ε Turbulence Model for Each Phase ............................................................... 592
17.5.16.1.3.1.Transport Equations ................................................................................ 592
17.5.16.2. RSM Turbulence Models ........................................................................................... 593
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17.5.16.2.1. RSM Dispersed Turbulence Model .................................................................... 594
17.5.16.2.2. RSM Mixture Turbulence Model ....................................................................... 594
17.5.16.3. Turbulence Interaction Models ................................................................................. 595
17.5.16.3.1. Simonin et al. .................................................................................................. 595
17.5.16.3.1.1. Formulation in Dispersed Turbulence Models .......................................... 595
17.5.16.3.1.1.1. Continuous Phase .......................................................................... 595
17.5.16.3.1.1.2. Dispersed Phases ........................................................................... 596
17.5.16.3.1.2. Formulation in Per Phase Turbulence Models ........................................... 597
17.5.16.3.2. Troshko-Hassan ............................................................................................... 597
17.5.16.3.2.1.Troshko-Hassan Formulation in Mixture Turbulence Models ..................... 597
17.5.16.3.2.2. Troshko-Hassan Formulation in Dispersed Turbulence Models ................. 598
17.5.16.3.2.2.1. Continuous Phase .......................................................................... 598
17.5.16.3.2.2.2. Dispersed Phases ........................................................................... 598
17.5.16.3.2.3.Troshko-Hassan Formulation in Per-Phase Turbulence Models .................. 598
17.5.16.3.2.3.1. Continuous Phase .......................................................................... 598
17.5.16.3.2.3.2. Dispersed Phases ........................................................................... 599
17.5.16.3.3. Sato ................................................................................................................ 599
17.5.16.3.4. None ............................................................................................................... 599
17.5.17. Solution Method in ANSYS Fluent ..................................................................................... 599
17.5.17.1.The Pressure-Correction Equation ............................................................................. 599
17.5.17.2. Volume Fractions ..................................................................................................... 600
17.5.18. Dense Discrete Phase Model ............................................................................................. 600
17.5.18.1. Limitations .............................................................................................................. 601
17.5.18.2. Granular Temperature .............................................................................................. 601
17.5.19. Multi-Fluid VOF Model ...................................................................................................... 602
17.5.20. Wall Boiling Models .......................................................................................................... 603
17.5.20.1. Overview ................................................................................................................. 603
17.5.20.2. RPI Model ................................................................................................................ 604
17.5.20.3. Non-equilibrium Subcooled Boiling .......................................................................... 606
17.5.20.4. Critical Heat Flux ...................................................................................................... 607
17.5.20.4.1.Wall Heat Flux Partition .................................................................................... 607
17.5.20.4.2. Flow Regime Transition ................................................................................... 608
17.5.20.5. Interfacial Momentum Transfer ................................................................................. 609
17.5.20.5.1. Interfacial Area ................................................................................................ 609
17.5.20.5.2. Bubble and Droplet Diameters ........................................................................ 609
17.5.20.5.2.1. Bubble Diameter .................................................................................... 609
17.5.20.5.2.2. Droplet Diameter .................................................................................... 610
17.5.20.5.3. Interfacial Drag Force ...................................................................................... 610
17.5.20.5.4. Interfacial Lift Force ......................................................................................... 610
17.5.20.5.5.Turbulent Dispersion Force .............................................................................. 610
17.5.20.5.6. Wall Lubrication Force ..................................................................................... 610
17.5.20.5.7. Virtual Mass Force ........................................................................................... 610
17.5.20.6. Interfacial Heat Transfer ............................................................................................ 611
17.5.20.6.1. Interface to Liquid Heat Transfer ...................................................................... 611
17.5.20.6.2. Interface to Vapor Heat Transfer ....................................................................... 611
17.5.20.7. Mass Transfer ........................................................................................................... 611
17.5.20.7.1. Mass Transfer From the Wall to Vapor ............................................................... 611
17.5.20.7.2. Interfacial Mass Transfer .................................................................................. 611
17.5.20.8.Turbulence Interactions ............................................................................................ 611
17.6. Wet Steam Model Theory ............................................................................................................ 611
17.6.1. Overview of the Wet Steam Model ...................................................................................... 612
17.6.2. Limitations of the Wet Steam Model .................................................................................... 612
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17.6.3.Wet Steam Flow Equations .................................................................................................. 612
17.6.4. Phase Change Model .......................................................................................................... 613
17.6.5. Built-in Thermodynamic Wet Steam Properties .................................................................... 615
17.6.5.1. Equation of State ....................................................................................................... 615
17.6.5.2. Saturated Vapor Line .................................................................................................. 616
17.6.5.3. Saturated Liquid Line ................................................................................................. 616
17.6.5.4. Mixture Properties ..................................................................................................... 616
17.7. Modeling Mass Transfer in Multiphase Flows ................................................................................ 616
17.7.1. Source Terms due to Mass Transfer ...................................................................................... 617
17.7.1.1. Mass Equation ........................................................................................................... 617
17.7.1.2. Momentum Equation ................................................................................................. 617
17.7.1.3. Energy Equation ........................................................................................................ 617
17.7.1.4. Species Equation ....................................................................................................... 618
17.7.1.5. Other Scalar Equations ............................................................................................... 618
17.7.2. Unidirectional Constant Rate Mass Transfer ......................................................................... 618
17.7.3. UDF-Prescribed Mass Transfer ............................................................................................. 618
17.7.4. Cavitation Models .............................................................................................................. 618
17.7.4.1. Limitations of the Cavitation Models .......................................................................... 619
17.7.4.1.1. Limitations of Cavitation with the VOF Model ..................................................... 620
17.7.4.2.Vapor Transport Equation ........................................................................................... 620
17.7.4.3. Bubble Dynamics Consideration ................................................................................ 621
17.7.4.4. Singhal et al. Model .................................................................................................... 621
17.7.4.5. Zwart-Gerber-Belamri Model ..................................................................................... 623
17.7.4.6. Schnerr and Sauer Model ........................................................................................... 624
17.7.4.7. Turbulence Factor ...................................................................................................... 625
17.7.4.8. Additional Guidelines for the Cavitation Models ......................................................... 626
17.7.4.9. Extended Cavitation Model Capabilities ..................................................................... 628
17.7.4.9.1. Multiphase Cavitation Models ........................................................................... 628
17.7.4.9.2. Multiphase Species Transport Cavitation Model ................................................. 628
17.7.5. Evaporation-Condensation Model ....................................................................................... 629
17.7.5.1. Lee Model ................................................................................................................. 629
17.7.5.2.Thermal Phase Change Model .................................................................................... 631
17.7.6. Interphase Species Mass Transfer ........................................................................................ 632
17.7.6.1. Modeling Approach ................................................................................................... 633
17.7.6.1.1. Equilibrium Model ............................................................................................. 633
17.7.6.1.2.Two-Resistance Model ....................................................................................... 634
17.7.6.2. Species Mass Transfer Models ..................................................................................... 636
17.7.6.2.1. Raoult’s Law ...................................................................................................... 636
17.7.6.2.2. Henry’s Law ...................................................................................................... 637
17.7.6.2.3. Equilibrium Ratio .............................................................................................. 638
17.7.6.3. Mass Transfer Coefficient Models ................................................................................ 638
17.7.6.3.1. Constant ........................................................................................................... 638
17.7.6.3.2. Sherwood Number ............................................................................................ 638
17.7.6.3.3. Ranz-Marshall Model ......................................................................................... 639
17.7.6.3.4. Hughmark Model .............................................................................................. 639
17.7.6.3.5. User-Defined ..................................................................................................... 639
17.8. Modeling Species Transport in Multiphase Flows ......................................................................... 639
17.8.1. Limitations ......................................................................................................................... 640
17.8.2. Mass and Momentum Transfer with Multiphase Species Transport ....................................... 641
17.8.2.1. Source Terms Due to Heterogeneous Reactions .......................................................... 641
17.8.2.1.1. Mass Transfer .................................................................................................... 641
17.8.2.1.2. Momentum Transfer .......................................................................................... 641
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17.8.2.1.3. Species Transfer ................................................................................................ 642
17.8.2.1.4. Heat Transfer ..................................................................................................... 642
17.8.3. The Stiff Chemistry Solver ................................................................................................... 643
17.8.4. Heterogeneous Phase Interaction ....................................................................................... 643
18. Solidification and Melting ................................................................................................................. 645
18.1. Overview .................................................................................................................................... 645
18.2. Limitations .................................................................................................................................. 646
18.3. Introduction ............................................................................................................................... 646
18.4. Energy Equation ......................................................................................................................... 646
18.5. Momentum Equations ................................................................................................................ 647
18.6.Turbulence Equations .................................................................................................................. 648
18.7. Species Equations ....................................................................................................................... 648
18.8. Back Diffusion ............................................................................................................................. 650
18.9. Pull Velocity for Continuous Casting ............................................................................................ 650
18.10. Contact Resistance at Walls ........................................................................................................ 652
18.11.Thermal and Solutal Buoyancy ................................................................................................... 652
19. Eulerian Wall Films ............................................................................................................................ 655
19.1. Introduction ............................................................................................................................... 655
19.2. Mass, Momentum, and Energy Conservation Equations for Wall Film ............................................. 656
19.2.1. Film Sub-Models ................................................................................................................. 657
19.2.1.1. DPM Collection .......................................................................................................... 657
19.2.1.2. Particle-Wall Interaction ............................................................................................. 657
19.2.1.3. Film Separation .......................................................................................................... 657
19.2.1.3.1. Separation Criteria ............................................................................................ 657
19.2.1.3.1.1. Foucart Separation ................................................................................... 658
19.2.1.3.1.2. O’Rourke Separation ................................................................................. 658
19.2.1.3.1.3. Friedrich Separation ................................................................................. 658
19.2.1.4. Film Stripping ............................................................................................................ 659
19.2.1.5. Secondary Phase Accretion ........................................................................................ 660
19.2.1.6. Coupling of Wall Film with Mixture Species Transport ................................................. 661
19.2.2. Boundary Conditions .......................................................................................................... 661
19.2.3. Obtaining Film Velocity Without Solving the Momentum Equations .................................... 662
19.2.3.1. Shear-Driven Film Velocity ......................................................................................... 662
19.2.3.2. Gravity-Driven Film Velocity ....................................................................................... 662
19.3. Passive Scalar Equation for Wall Film ............................................................................................ 663
19.4. Numerical Schemes and Solution Algorithm ................................................................................ 664
19.4.1.Temporal Differencing Schemes .......................................................................................... 664
19.4.1.1. First-Order Explicit Method ........................................................................................ 664
19.4.1.2. First-Order Implicit Method ........................................................................................ 665
19.4.1.3. Second-Order Implicit Method ................................................................................... 665
19.4.2. Spatial Differencing Schemes .............................................................................................. 666
19.4.3. Solution Algorithm ............................................................................................................. 667
19.4.3.1. Steady Flow ............................................................................................................... 667
19.4.3.2. Transient Flow ........................................................................................................... 667
20. Electric Potential ............................................................................................................................... 669
20.1. Overview and Limitations ............................................................................................................ 669
20.2. Electric Potential Equation ........................................................................................................... 669
20.3. Energy Equation Source Term ...................................................................................................... 670
21. Solver Theory .................................................................................................................................... 671
21.1. Overview of Flow Solvers ............................................................................................................ 671
21.1.1. Pressure-Based Solver ......................................................................................................... 672
21.1.1.1. The Pressure-Based Segregated Algorithm ................................................................. 672
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21.1.1.2.The Pressure-Based Coupled Algorithm ...................................................................... 673
21.1.2. Density-Based Solver .......................................................................................................... 674
21.2. General Scalar Transport Equation: Discretization and Solution ..................................................... 676
21.2.1. Solving the Linear System ................................................................................................... 678
21.3. Discretization .............................................................................................................................. 678
21.3.1. Spatial Discretization .......................................................................................................... 678
21.3.1.1. First-Order Upwind Scheme ....................................................................................... 679
21.3.1.2. Power-Law Scheme .................................................................................................... 679
21.3.1.3. Second-Order Upwind Scheme .................................................................................. 680
21.3.1.4. First-to-Higher Order Blending ................................................................................... 681
21.3.1.5. Central-Differencing Scheme ..................................................................................... 681
21.3.1.6. Bounded Central Differencing Scheme ....................................................................... 682
21.3.1.7. QUICK Scheme .......................................................................................................... 682
21.3.1.8.Third-Order MUSCL Scheme ....................................................................................... 683
21.3.1.9. Modified HRIC Scheme .............................................................................................. 683
21.3.1.10. High Order Term Relaxation ..................................................................................... 685
21.3.2. Temporal Discretization ...................................................................................................... 685
21.3.2.1. Implicit Time Integration ............................................................................................ 686
21.3.2.2. Bounded Second Order Implicit Time Integration ....................................................... 686
21.3.2.2.1. Limitations ........................................................................................................ 686
21.3.2.3. Explicit Time Integration ............................................................................................ 687
21.3.3. Evaluation of Gradients and Derivatives .............................................................................. 687
21.3.3.1. Green-Gauss Theorem ............................................................................................... 687
21.3.3.2. Green-Gauss Cell-Based Gradient Evaluation .............................................................. 688
21.3.3.3. Green-Gauss Node-Based Gradient Evaluation ............................................................ 688
21.3.3.4. Least Squares Cell-Based Gradient Evaluation ............................................................. 688
21.3.4. Gradient Limiters ................................................................................................................ 690
21.3.4.1. Standard Limiter ........................................................................................................ 690
21.3.4.2. Multidimensional Limiter ........................................................................................... 691
21.3.4.3. Differentiable Limiter ................................................................................................. 691
21.4. Pressure-Based Solver ................................................................................................................. 691
21.4.1. Discretization of the Momentum Equation .......................................................................... 692
21.4.1.1. Pressure Interpolation Schemes ................................................................................. 692
21.4.2. Discretization of the Continuity Equation ............................................................................ 693
21.4.2.1. Density Interpolation Schemes ................................................................................... 694
21.4.3. Pressure-Velocity Coupling ................................................................................................. 694
21.4.3.1. Segregated Algorithms .............................................................................................. 695
21.4.3.1.1. SIMPLE .............................................................................................................. 695
21.4.3.1.2. SIMPLEC ........................................................................................................... 696
21.4.3.1.2.1. Skewness Correction ................................................................................ 696
21.4.3.1.3. PISO .................................................................................................................. 696
21.4.3.1.3.1. Neighbor Correction ................................................................................. 696
21.4.3.1.3.2. Skewness Correction ................................................................................ 697
21.4.3.1.3.3. Skewness - Neighbor Coupling ................................................................. 697
21.4.3.2. Fractional-Step Method (FSM) .................................................................................... 697
21.4.3.3. Coupled Algorithm .................................................................................................... 697
21.4.3.3.1. Limitations ........................................................................................................ 698
21.4.4. Steady-State Iterative Algorithm ......................................................................................... 699
21.4.4.1. Under-Relaxation of Variables .................................................................................... 699
21.4.4.2. Under-Relaxation of Equations ................................................................................... 699
21.4.5.Time-Advancement Algorithm ............................................................................................ 699
21.4.5.1. Iterative Time-Advancement Scheme ......................................................................... 700
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21.4.5.1.1.The Frozen Flux Formulation .............................................................................. 701
21.4.5.2. Non-Iterative Time-Advancement Scheme .................................................................. 702
21.5. Density-Based Solver ................................................................................................................... 704
21.5.1. Governing Equations in Vector Form ................................................................................... 704
21.5.2. Preconditioning ................................................................................................................. 705
21.5.3. Convective Fluxes ............................................................................................................... 707
21.5.3.1. Roe Flux-Difference Splitting Scheme ......................................................................... 707
21.5.3.2. AUSM  Scheme ......................................................................................................... 707
21.5.3.3. Low Diffusion Roe Flux Difference Splitting Scheme ................................................... 708
21.5.4. Steady-State Flow Solution Methods ................................................................................... 708
21.5.4.1. Explicit Formulation ................................................................................................... 709
21.5.4.1.1. Implicit Residual Smoothing .............................................................................. 709
21.5.4.2. Implicit Formulation .................................................................................................. 710
21.5.4.2.1. Convergence Acceleration for Stretched Meshes ................................................ 710
21.5.5. Unsteady Flows Solution Methods ...................................................................................... 711
21.5.5.1. Explicit Time Stepping ............................................................................................... 711
21.5.5.2. Implicit Time Stepping (Dual-Time Formulation) ......................................................... 711
21.6. Pseudo Transient Under-Relaxation ............................................................................................. 713
21.6.1. Automatic Pseudo Transient Time Step ............................................................................... 713
21.7. Multigrid Method ........................................................................................................................ 715
21.7.1. Approach ........................................................................................................................... 715
21.7.1.1.The Need for Multigrid ............................................................................................... 715
21.7.1.2.The Basic Concept in Multigrid ................................................................................... 716
21.7.1.3. Restriction and Prolongation ...................................................................................... 716
21.7.1.4. Unstructured Multigrid .............................................................................................. 717
21.7.2. Multigrid Cycles .................................................................................................................. 717
21.7.2.1. The V and W Cycles .................................................................................................... 717
21.7.3. Algebraic Multigrid (AMG) .................................................................................................. 721
21.7.3.1. AMG Restriction and Prolongation Operators ............................................................. 721
21.7.3.2. AMG Coarse Level Operator ....................................................................................... 722
21.7.3.3. The F Cycle ................................................................................................................ 722
21.7.3.4. The Flexible Cycle ...................................................................................................... 722
21.7.3.4.1.The Residual Reduction Rate Criteria .................................................................. 723
21.7.3.4.2. The Termination Criteria .................................................................................... 724
21.7.3.5.The Coupled and Scalar AMG Solvers .......................................................................... 724
21.7.3.5.1. Gauss-Seidel ..................................................................................................... 725
21.7.3.5.2. Incomplete Lower Upper (ILU) ........................................................................... 725
21.7.4. Full-Approximation Storage (FAS) Multigrid ......................................................................... 726
21.7.4.1. FAS Restriction and Prolongation Operators ............................................................... 727
21.7.4.2. FAS Coarse Level Operator ......................................................................................... 727
21.8. Hybrid Initialization ..................................................................................................................... 727
21.9. Full Multigrid (FMG) Initialization ................................................................................................. 729
21.9.1. Overview of FMG Initialization ............................................................................................ 729
21.9.2. Limitations of FMG Initialization .......................................................................................... 730
22. Adapting the Mesh ............................................................................................................................ 733
22.1. Static Adaption Process ............................................................................................................... 733
22.1.1. Hanging Node Adaption ..................................................................................................... 733
22.1.1.1. Hanging Node Refinement ......................................................................................... 734
22.1.1.2. Hanging Node Coarsening ......................................................................................... 735
22.2. Boundary Adaption ..................................................................................................................... 735
22.3. Gradient Adaption ...................................................................................................................... 737
22.3.1. Gradient Adaption Approach .............................................................................................. 737
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22.3.2. Example of Steady Gradient Adaption ................................................................................. 739
22.4. Isovalue Adaption ....................................................................................................................... 741
22.5. Region Adaption ......................................................................................................................... 743
22.5.1. Defining a Region ............................................................................................................... 743
22.5.2. Region Adaption Example .................................................................................................. 744
22.6. Volume Adaption ........................................................................................................................ 745
22.6.1.Volume Adaption Approach ................................................................................................ 745
22.6.2.Volume Adaption Example .................................................................................................. 746
22.7.Yplus/Ystar Adaption ................................................................................................................... 747
22.7.1.Yplus/Ystar Adaption Approach .......................................................................................... 747
22.8. Anisotropic Adaption .................................................................................................................. 749
22.9. Geometry-Based Adaption .......................................................................................................... 749
22.9.1. Geometry-Based Adaption Approach .................................................................................. 750
22.9.1.1. Node Projection ......................................................................................................... 750
22.9.1.2. Example of Geometry-Based Adaption ....................................................................... 752
22.10. Registers ................................................................................................................................... 755
22.10.1. Adaption Registers ........................................................................................................... 755
22.10.2. Mask Registers .................................................................................................................. 756
23. Reporting Alphanumeric Data .......................................................................................................... 759
23.1. Fluxes Through Boundaries ......................................................................................................... 759
23.2. Forces on Boundaries .................................................................................................................. 760
23.2.1. Computing Forces, Moments, and the Center of Pressure ..................................................... 760
23.3. Surface Integration ..................................................................................................................... 762
23.3.1. Computing Surface Integrals .............................................................................................. 763
23.3.1.1. Area .......................................................................................................................... 763
23.3.1.2. Integral ...................................................................................................................... 764
23.3.1.3. Area-Weighted Average ............................................................................................. 764
23.3.1.4. Custom Vector Based Flux .......................................................................................... 764
23.3.1.5. Custom Vector Flux .................................................................................................... 764
23.3.1.6. Custom Vector Weighted Average .............................................................................. 764
23.3.1.7. Flow Rate ................................................................................................................... 764
23.3.1.8. Mass Flow Rate .......................................................................................................... 765
23.3.1.9. Mass-Weighted Average ............................................................................................ 765
23.3.1.10. Sum of Field Variable ................................................................................................ 765
23.3.1.11. Facet Average .......................................................................................................... 766
23.3.1.12. Facet Minimum ........................................................................................................ 766
23.3.1.13. Facet Maximum ....................................................................................................... 766
23.3.1.14.Vertex Average ......................................................................................................... 766
23.3.1.15. Vertex Minimum ...................................................................................................... 766
23.3.1.16.Vertex Maximum ...................................................................................................... 766
23.3.1.17. Standard-Deviation .................................................................................................. 766
23.3.1.18. Uniformity Index ...................................................................................................... 767
23.3.1.19. Volume Flow Rate .................................................................................................... 767
23.4. Volume Integration ..................................................................................................................... 768
23.4.1. Computing Volume Integrals .............................................................................................. 768
23.4.1.1.Volume ...................................................................................................................... 768
23.4.1.2. Sum .......................................................................................................................... 769
23.4.1.3. Sum*2Pi .................................................................................................................... 769
23.4.1.4. Volume Integral ......................................................................................................... 769
23.4.1.5.Volume-Weighted Average ......................................................................................... 769
23.4.1.6. Mass-Weighted Integral ............................................................................................. 769
23.4.1.7. Mass .......................................................................................................................... 770
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23.4.1.8. Mass-Weighted Average ............................................................................................ 770
A. Nomenclature ....................................................................................................................................... 771
Bibliography ............................................................................................................................................. 775
Index ........................................................................................................................................................ 807
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