ANSYS Fluent Theory Guide.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 .................................................................................. 4
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 (6DOF) Solver Theory ................................................................................................. 36
3.3. Sliding Mesh Theory ....................................................................................................................... 37
4.Turbulence ............................................................................................................................................. 39
4.1. Underlying Principles of Turbulence Modeling ................................................................................. 39
4.1.1. Reynolds (Ensemble) Averaging .............................................................................................. 39
4.1.2. Filtered Navier-Stokes Equations ............................................................................................. 40
4.1.3. Hybrid RANS-LES Formulations ............................................................................................... 41
4.1.4. Boussinesq Approach vs. Reynolds Stress Transport Models ..................................................... 41
4.2. Spalart-Allmaras Model ................................................................................................................... 42
4.2.1. Overview ............................................................................................................................... 42
4.2.2. Transport Equation for the Spalart-Allmaras Model ................................................................. 43
4.2.3. Modeling the Turbulent Viscosity ............................................................................................ 43
4.2.4. Modeling the Turbulent Production ........................................................................................ 43
4.2.5. Modeling the Turbulent Destruction ....................................................................................... 44
4.2.6. Model Constants .................................................................................................................... 45
4.2.7. Wall Boundary Conditions ...................................................................................................... 45
4.2.8. Convective Heat and Mass Transfer Modeling .......................................................................... 45
4.3. Standard, RNG, and Realizable k-ε Models ........................................................................................ 45
4.3.1. Standard k-ε Model ................................................................................................................ 46
4.3.1.1. Overview ....................................................................................................................... 46
4.3.1.2.Transport Equations for the Standard k-ε Model ............................................................. 46
4.3.1.3. Modeling the Turbulent Viscosity ................................................................................... 47
4.3.1.4. Model Constants ........................................................................................................... 47
4.3.2. RNG k-ε Model ....................................................................................................................... 47
4.3.2.1. Overview ....................................................................................................................... 47
4.3.2.2.Transport Equations for the RNG k-ε Model ..................................................................... 48
4.3.2.3. Modeling the Effective Viscosity ..................................................................................... 48
4.3.2.4. RNG Swirl Modification .................................................................................................. 49
4.3.2.5. Calculating the Inverse Effective Prandtl Numbers .......................................................... 49
4.3.2.6.The R-ε Term in the ε Equation ........................................................................................ 49
4.3.2.7. Model Constants ........................................................................................................... 50
4.3.3. Realizable k-ε Model ............................................................................................................... 50
4.3.3.1. Overview ....................................................................................................................... 50
4.3.3.2.Transport Equations for the Realizable k-ε Model ............................................................ 51
4.3.3.3. Modeling the Turbulent Viscosity ................................................................................... 52
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4.3.3.4. Model Constants ........................................................................................................... 53
4.3.4. Modeling Turbulent Production in the k-ε Models ................................................................... 53
4.3.5. Effects of Buoyancy on Turbulence in the k-ε Models ............................................................... 53
4.3.6. Effects of Compressibility on Turbulence in the k-ε Models ...................................................... 54
4.3.7. Convective Heat and Mass Transfer Modeling in the k-ε Models ............................................... 55
4.4. Standard, BSL, and SST k-ω Models ................................................................................................... 56
4.4.1. Standard k-ω Model ............................................................................................................... 57
4.4.1.1. Overview ....................................................................................................................... 57
4.4.1.2.Transport Equations for the Standard k-ω Model ............................................................. 57
4.4.1.3. Modeling the Effective Diffusivity ................................................................................... 57
4.4.1.3.1. Low-Reynolds Number Correction ......................................................................... 58
4.4.1.4. Modeling the Turbulence Production ............................................................................. 58
4.4.1.4.1. Production of k ..................................................................................................... 58
4.4.1.4.2. Production of ω ..................................................................................................... 58
4.4.1.5. Modeling the Turbulence Dissipation ............................................................................. 59
4.4.1.5.1. Dissipation of k ..................................................................................................... 59
4.4.1.5.2. Dissipation of ω ..................................................................................................... 59
4.4.1.5.3. Compressibility Effects .......................................................................................... 60
4.4.1.6. Model Constants ........................................................................................................... 60
4.4.2. Baseline (BSL) k-ω Model ........................................................................................................ 60
4.4.2.1. Overview ....................................................................................................................... 60
4.4.2.2.Transport Equations for the BSL k-ω Model ..................................................................... 61
4.4.2.3. Modeling the Effective Diffusivity ................................................................................... 61
4.4.2.4. Modeling the Turbulence Production ............................................................................. 61
4.4.2.4.1. Production of k ..................................................................................................... 61
4.4.2.4.2. Production of ω ..................................................................................................... 62
4.4.2.5. Modeling the Turbulence Dissipation ............................................................................. 62
4.4.2.5.1. Dissipation of k ..................................................................................................... 62
4.4.2.5.2. Dissipation of ω ..................................................................................................... 62
4.4.2.6. Cross-Diffusion Modification .......................................................................................... 63
4.4.2.7. Model Constants ........................................................................................................... 63
4.4.3. Shear-Stress Transport (SST) k-ω Model ................................................................................... 63
4.4.3.1. Overview ....................................................................................................................... 63
4.4.3.2. Modeling the Turbulent Viscosity ................................................................................... 63
4.4.3.3. Model Constants ........................................................................................................... 63
4.4.4.Turbulence Damping .............................................................................................................. 64
4.4.5. Wall Boundary Conditions ...................................................................................................... 65
4.5. k-kl-ω Transition Model ................................................................................................................... 65
4.5.1. Overview ............................................................................................................................... 65
4.5.2.Transport Equations for the k-kl-ω Model ................................................................................ 65
4.5.2.1. Model Constants ........................................................................................................... 68
4.6.Transition SST Model ....................................................................................................................... 68
4.6.1. Overview ............................................................................................................................... 69
4.6.2.Transport Equations for the Transition SST Model .................................................................... 69
4.6.2.1. Separation-Induced Transition Correction ...................................................................... 71
4.6.2.2. Coupling the Transition Model and SST Transport Equations ........................................... 72
4.6.2.3.Transition SST and Rough Walls ...................................................................................... 72
4.6.3. Mesh Requirements ............................................................................................................... 73
4.6.4. Specifying Inlet Turbulence Levels .......................................................................................... 76
4.7. Intermittency Transition Model ....................................................................................................... 77
4.7.1. Overview ............................................................................................................................... 77
4.7.2.Transport Equations for the Intermittency Transition Model ..................................................... 78
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4.7.3. Coupling with the Other Models ............................................................................................. 80
4.7.4. Intermittency Transition Model and Rough Walls ..................................................................... 80
4.8.The V2F Model ................................................................................................................................ 80
4.9. Reynolds Stress Model (RSM) ........................................................................................................... 81
4.9.1. Overview ............................................................................................................................... 81
4.9.2. Reynolds Stress Transport Equations ....................................................................................... 82
4.9.3. Modeling Turbulent Diffusive Transport .................................................................................. 83
4.9.4. Modeling the Pressure-Strain Term ......................................................................................... 83
4.9.4.1. Linear Pressure-Strain Model .......................................................................................... 83
4.9.4.2. Low-Re Modifications to the Linear Pressure-Strain Model .............................................. 84
4.9.4.3. Quadratic Pressure-Strain Model .................................................................................... 85
4.9.4.4. Stress-Omega Model ..................................................................................................... 85
4.9.4.5. Stress-BSL Model ........................................................................................................... 87
4.9.5. Effects of Buoyancy on Turbulence ......................................................................................... 87
4.9.6. Modeling the Turbulence Kinetic Energy ................................................................................. 87
4.9.7. Modeling the Dissipation Rate ................................................................................................ 88
4.9.8. Modeling the Turbulent Viscosity ............................................................................................ 88
4.9.9. Wall Boundary Conditions ...................................................................................................... 89
4.9.10. Convective Heat and Mass Transfer Modeling ........................................................................ 89
4.10. Scale-Adaptive Simulation (SAS) Model ......................................................................................... 90
4.10.1. Overview ............................................................................................................................. 90
4.10.2.Transport Equations for the SST-SAS Model ........................................................................... 91
4.10.3. SAS with Other ω-Based Turbulence Models .......................................................................... 93
4.11. Detached Eddy Simulation (DES) ................................................................................................... 93
4.11.1. Overview ............................................................................................................................. 93
4.11.2. DES with the Spalart-Allmaras Model .................................................................................... 94
4.11.3. DES with the Realizable k-ε Model ......................................................................................... 94
4.11.4. DES with the BSL or SST k-ω Model ....................................................................................... 95
4.11.5. DES with the Transition SST Model ........................................................................................ 96
4.11.6. Improved Delayed Detached Eddy Simulation (IDDES) .......................................................... 96
4.11.6.1. Overview of IDDES ....................................................................................................... 96
4.11.6.2. IDDES Model Formulation ............................................................................................ 97
4.12. Shielded Detached Eddy Simulation (SDES) ................................................................................... 97
4.12.1. Shielding Function ............................................................................................................... 98
4.12.2. LES Mode of SDES ................................................................................................................ 99
4.13. Stress-Blended Eddy Simulation (SBES) ........................................................................................ 100
4.13.1. Stress Blending ................................................................................................................... 101
4.13.2. SDES and SBES Example ..................................................................................................... 101
4.14. Large Eddy Simulation (LES) Model .............................................................................................. 102
4.14.1. Overview ........................................................................................................................... 102
4.14.2. Subgrid-Scale Models ......................................................................................................... 103
4.14.2.1. Smagorinsky-Lilly Model ............................................................................................ 104
4.14.2.2. Dynamic Smagorinsky-Lilly Model .............................................................................. 105
4.14.2.3.Wall-Adapting Local Eddy-Viscosity (WALE) Model ...................................................... 106
4.14.2.4. Algebraic Wall-Modeled LES Model (WMLES) .............................................................. 106
4.14.2.4.1. Algebraic WMLES Model Formulation ................................................................ 107
4.14.2.4.1.1. Reynolds Number Scaling ......................................................................... 107
4.14.2.4.2. Algebraic WMLES S-Omega Model Formulation ................................................. 108
4.14.2.5. Dynamic Kinetic Energy Subgrid-Scale Model ............................................................. 109
4.14.3. Inlet Boundary Conditions for the LES Model ....................................................................... 109
4.14.3.1.Vortex Method ........................................................................................................... 110
4.14.3.2. Spectral Synthesizer ................................................................................................... 111
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4.15. Embedded Large Eddy Simulation (ELES) ..................................................................................... 112
4.15.1. Overview ........................................................................................................................... 112
4.15.2. Selecting a Model ............................................................................................................... 112
4.15.3. Interfaces Treatment ........................................................................................................... 113
4.15.3.1. RANS-LES Interface .................................................................................................... 113
4.15.3.2. LES-RANS Interface .................................................................................................... 113
4.15.3.3. Internal Interface Without LES Zone ........................................................................... 114
4.15.3.4. Grid Generation Guidelines ........................................................................................ 114
4.16. Near-Wall Treatments for Wall-Bounded Turbulent Flows .............................................................. 115
4.16.1. Overview ........................................................................................................................... 115
4.16.1.1.Wall Functions vs. Near-Wall Model ............................................................................. 116
4.16.1.2. Wall Functions ........................................................................................................... 118
4.16.2. Standard Wall Functions ..................................................................................................... 118
4.16.2.1. Momentum ............................................................................................................... 118
4.16.2.2. Energy ....................................................................................................................... 119
4.16.2.3. Species ...................................................................................................................... 121
4.16.2.4. Turbulence ................................................................................................................ 121
4.16.3. Scalable Wall Functions ....................................................................................................... 122
4.16.4. Non-Equilibrium Wall Functions .......................................................................................... 122
4.16.4.1. Standard Wall Functions vs. Non-Equilibrium Wall Functions ....................................... 124
4.16.4.2. Limitations of the Wall Function Approach ................................................................. 124
4.16.5. Enhanced Wall Treatment ε-Equation (EWT-ε) ...................................................................... 124
4.16.5.1.Two-Layer Model for Enhanced Wall Treatment ........................................................... 125
4.16.5.2. Enhanced Wall Treatment for Momentum and Energy Equations ................................. 126
4.16.6. Menter-Lechner ε-Equation (ML-ε) ...................................................................................... 128
4.16.6.1. Momentum Equations ............................................................................................... 130
4.16.6.2. k-ε Turbulence Models ............................................................................................... 130
4.16.6.3. Iteration Improvements ............................................................................................. 130
4.16.7. y -Insensitive Wall Treatment ω-Equation ........................................................................... 130
4.16.8. User-Defined Wall Functions ............................................................................................... 131
4.16.9. LES Near-Wall Treatment ..................................................................................................... 131
4.17. Curvature Correction for the Spalart-Allmaras and Two-Equation Models ..................................... 131
4.18. Production Limiters for Two-Equation Models .............................................................................. 133
4.19. Definition of Turbulence Scales .................................................................................................... 135
4.19.1. RANS and Hybrid (SAS, DES, and SDES) Turbulence Models .................................................. 135
4.19.2. Large Eddy Simulation (LES) Models .................................................................................... 135
4.19.3. Stress-Blended Eddy Simulation (SBES) Model ..................................................................... 136
5. Heat Transfer ....................................................................................................................................... 137
5.1. Introduction ................................................................................................................................. 137
5.2. Modeling Conductive and Convective Heat Transfer ...................................................................... 137
5.2.1. Heat Transfer Theory ............................................................................................................. 137
5.2.1.1.The Energy Equation .................................................................................................... 137
5.2.1.2.The Energy Equation in Moving Reference Frames ........................................................ 138
5.2.1.3.The Energy Equation for the Non-Premixed Combustion Model .................................... 138
5.2.1.4. Inclusion of Pressure Work and Kinetic Energy Terms .................................................... 139
5.2.1.5. Inclusion of the Viscous Dissipation Terms .................................................................... 139
5.2.1.6. Inclusion of the Species Diffusion Term ........................................................................ 139
5.2.1.7. Energy Sources Due to Reaction ................................................................................... 140
5.2.1.8. Energy Sources Due To Radiation ................................................................................. 140
5.2.1.9. Energy Source Due To Joule Heating ............................................................................ 140
5.2.1.10. Interphase Energy Sources ......................................................................................... 140
5.2.1.11. Energy Equation in Solid Regions ............................................................................... 140
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5.2.1.12. Anisotropic Conductivity in Solids .............................................................................. 141
5.2.1.13. Diffusion at Inlets ....................................................................................................... 141
5.2.2. Natural Convection and Buoyancy-Driven Flows Theory ........................................................ 141
5.3. Modeling Radiation ...................................................................................................................... 142
5.3.1. Overview and Limitations ..................................................................................................... 142
5.3.1.1. Advantages and Limitations of the DTRM ..................................................................... 143
5.3.1.2. Advantages and Limitations of the P-1 Model ............................................................... 143
5.3.1.3. Advantages and Limitations of the Rosseland Model .................................................... 144
5.3.1.4. Advantages and Limitations of the DO Model ............................................................... 144
5.3.1.5. Advantages and Limitations of the S2S Model .............................................................. 144
5.3.2. Radiative Transfer Equation .................................................................................................. 145
5.3.3. P-1 Radiation Model Theory .................................................................................................. 146
5.3.3.1. The P-1 Model Equations ............................................................................................. 146
5.3.3.2. Anisotropic Scattering ................................................................................................. 148
5.3.3.3. Particulate Effects in the P-1 Model .............................................................................. 148
5.3.3.4. Boundary Condition Treatment for the P-1 Model at Walls ............................................. 149
5.3.3.5. Boundary Condition Treatment for the P-1 Model at Flow Inlets and Exits ...................... 150
5.3.4. Rosseland Radiation Model Theory ....................................................................................... 150
5.3.4.1.The Rosseland Model Equations ................................................................................... 150
5.3.4.2. Anisotropic Scattering ................................................................................................. 151
5.3.4.3. Boundary Condition Treatment at Walls ........................................................................ 151
5.3.4.4. Boundary Condition Treatment at Flow Inlets and Exits ................................................. 151
5.3.5. Discrete Transfer Radiation Model (DTRM) Theory ................................................................. 151
5.3.5.1.The DTRM Equations .................................................................................................... 151
5.3.5.2. Ray Tracing .................................................................................................................. 152
5.3.5.3. Clustering .................................................................................................................... 152
5.3.5.4. Boundary Condition Treatment for the DTRM at Walls ................................................... 153
5.3.5.5. Boundary Condition Treatment for the DTRM at Flow Inlets and Exits ............................ 153
5.3.6. Discrete Ordinates (DO) Radiation Model Theory ................................................................... 154
5.3.6.1. The DO Model Equations ............................................................................................. 154
5.3.6.2. Energy Coupling and the DO Model ............................................................................. 155
5.3.6.2.1. Limitations of DO/Energy Coupling ..................................................................... 156
5.3.6.3. Angular Discretization and Pixelation ........................................................................... 156
5.3.6.4. Anisotropic Scattering ................................................................................................. 159
5.3.6.5. Particulate Effects in the DO Model .............................................................................. 160
5.3.6.6. Boundary and Cell Zone Condition Treatment at Opaque Walls ..................................... 160
5.3.6.6.1. Gray Diffuse Walls ............................................................................................... 162
5.3.6.6.2. Non-Gray Diffuse Walls ........................................................................................ 162
5.3.6.7. Cell Zone and Boundary Condition Treatment at Semi-Transparent Walls ...................... 162
5.3.6.7.1. Semi-Transparent Interior Walls ........................................................................... 163
5.3.6.7.2. Specular Semi-Transparent Walls ......................................................................... 164
5.3.6.7.3. Diffuse Semi-Transparent Walls ............................................................................ 166
5.3.6.7.4. Partially Diffuse Semi-Transparent Walls ............................................................... 167
5.3.6.7.5. Semi-Transparent Exterior Walls ........................................................................... 167
5.3.6.7.6. Limitations .......................................................................................................... 169
5.3.6.7.7. Solid Semi-Transparent Media ............................................................................. 170
5.3.6.8. Boundary Condition Treatment at Specular Walls and Symmetry Boundaries ................. 170
5.3.6.9. Boundary Condition Treatment at Periodic Boundaries ................................................. 170
5.3.6.10. Boundary Condition Treatment at Flow Inlets and Exits ............................................... 170
5.3.7. Surface-to-Surface (S2S) Radiation Model Theory .................................................................. 170
5.3.7.1. Gray-Diffuse Radiation ................................................................................................. 170
5.3.7.2.The S2S Model Equations ............................................................................................. 171
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5.3.7.3. Clustering .................................................................................................................... 172
5.3.7.3.1. Clustering and View Factors ................................................................................ 172
5.3.7.3.2. Clustering and Radiosity ...................................................................................... 172
5.3.8. Radiation in Combusting Flows ............................................................................................ 173
5.3.8.1. The Weighted-Sum-of-Gray-Gases Model ..................................................................... 173
5.3.8.1.1.When the Total (Static) Gas Pressure is Not Equal to 1 atm .................................... 174
5.3.8.2.The Effect of Soot on the Absorption Coefficient ........................................................... 174
5.3.8.3.The Effect of Particles on the Absorption Coefficient ..................................................... 175
5.3.9. Choosing a Radiation Model ................................................................................................. 175
5.3.9.1. External Radiation ....................................................................................................... 176
6. Heat Exchangers .................................................................................................................................. 177
6.1.The Macro Heat Exchanger Models ................................................................................................ 177
6.1.1. Overview of the Macro Heat Exchanger Models .................................................................... 177
6.1.2. Restrictions of the Macro Heat Exchanger Models ................................................................. 179
6.1.3. Macro Heat Exchanger Model Theory .................................................................................... 180
6.1.3.1. Streamwise Pressure Drop ........................................................................................... 181
6.1.3.2. Heat Transfer Effectiveness ........................................................................................... 182
6.1.3.3. Heat Rejection ............................................................................................................. 183
6.1.3.4. Macro Heat Exchanger Group Connectivity .................................................................. 184
6.2. The Dual Cell Model ...................................................................................................................... 185
6.2.1. Overview of the Dual Cell Model ........................................................................................... 185
6.2.2. Restrictions of the Dual Cell Model ........................................................................................ 186
6.2.3. Dual Cell Model Theory ......................................................................................................... 186
6.2.3.1. NTU Relations .............................................................................................................. 187
6.2.3.2. Heat Rejection ............................................................................................................. 187
7. Species Transport and Finite-Rate Chemistry ..................................................................................... 189
7.1. Volumetric Reactions .................................................................................................................... 189
7.1.1. Species Transport Equations ................................................................................................. 189
7.1.1.1. Mass Diffusion in Laminar Flows ................................................................................... 190
7.1.1.2. Mass Diffusion in Turbulent Flows ................................................................................ 190
7.1.1.3.Treatment of Species Transport in the Energy Equation ................................................. 190
7.1.1.4. Diffusion at Inlets ......................................................................................................... 190
7.1.2.The Generalized Finite-Rate Formulation for Reaction Modeling ............................................ 191
7.1.2.1. Direct Use of Finite-Rate Kinetics (no TCI) ...................................................................... 191
7.1.2.2. Pressure-Dependent Reactions .................................................................................... 193
7.1.2.3.The Eddy-Dissipation Model ......................................................................................... 195
7.1.2.4. The Eddy-Dissipation Model for LES ............................................................................. 196
7.1.2.5. The Eddy-Dissipation-Concept (EDC) Model ................................................................. 196
7.1.2.6.The Thickened Flame Model ......................................................................................... 198
7.1.2.7.The Relaxation to Chemical Equilibrium Model ............................................................. 199
7.2.Wall Surface Reactions and Chemical Vapor Deposition .................................................................. 201
7.2.1. Surface Coverage Reaction Rate Modification ....................................................................... 202
7.2.2. Reaction-Diffusion Balance for Surface Chemistry ................................................................. 203
7.2.3. Slip Boundary Formulation for Low-Pressure Gas Systems ..................................................... 203
7.3. Particle Surface Reactions ............................................................................................................. 205
7.3.1. General Description .............................................................................................................. 205
7.3.2. ANSYS Fluent Model Formulation ......................................................................................... 206
7.3.3. Extension for Stoichiometries with Multiple Gas Phase Reactants .......................................... 207
7.3.4. Solid-Solid Reactions ............................................................................................................ 208
7.3.5. Solid Decomposition Reactions ............................................................................................ 208
7.3.6. Solid Deposition Reactions ................................................................................................... 208
7.3.7. Gaseous Solid Catalyzed Reactions on the Particle Surface .................................................... 208
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7.4. Electrochemical Reactions ............................................................................................................. 209
7.4.1. Overview and Limitations ..................................................................................................... 209
7.4.2. Electrochemical Reaction Model Theory ................................................................................ 209
7.5. Reacting Channel Model ............................................................................................................... 212
7.5.1. Overview and Limitations ..................................................................................................... 212
7.5.2. Reacting Channel Model Theory ........................................................................................... 213
7.5.2.1. Flow Inside the Reacting Channel ................................................................................. 213
7.5.2.2. Surface Reactions in the Reacting Channel ................................................................... 214
7.5.2.3. Porous Medium Inside Reacting Channel ...................................................................... 215
7.5.2.4. Outer Flow in the Shell ................................................................................................. 215
7.6. Reactor Network Model ................................................................................................................ 216
7.6.1. Reactor Network Model Theory ............................................................................................ 216
7.6.1.1. Reactor network temperature solution ......................................................................... 217
8. Non-Premixed Combustion ................................................................................................................. 219
8.1. Introduction ................................................................................................................................. 219
8.2. Non-Premixed Combustion and Mixture Fraction Theory ............................................................... 219
8.2.1. Mixture Fraction Theory ....................................................................................................... 220
8.2.1.1. Definition of the Mixture Fraction ................................................................................ 220
8.2.1.2.Transport Equations for the Mixture Fraction ................................................................ 222
8.2.1.3. The Non-Premixed Model for LES ................................................................................. 223
8.2.1.4. Mixture Fraction vs. Equivalence Ratio .......................................................................... 223
8.2.1.5. Relationship of Mixture Fraction to Species Mass Fraction, Density, and Temperature ..... 224
8.2.2. Modeling of Turbulence-Chemistry Interaction ..................................................................... 225
8.2.2.1. Description of the Probability Density Function ............................................................ 225
8.2.2.2. Derivation of Mean Scalar Values from the Instantaneous Mixture Fraction ................... 225
8.2.2.3. The Assumed-Shape PDF ............................................................................................. 226
8.2.2.3.1.The Double Delta Function PDF ........................................................................... 226
8.2.2.3.2.The β-Function PDF ............................................................................................. 227
8.2.3. Non-Adiabatic Extensions of the Non-Premixed Model .......................................................... 228
8.2.4. Chemistry Tabulation ........................................................................................................... 230
8.2.4.1. Look-Up Tables for Adiabatic Systems ........................................................................... 230
8.2.4.2. 3D Look-Up Tables for Non-Adiabatic Systems .............................................................. 232
8.2.4.3. Generating Lookup Tables Through Automated Grid Refinement .................................. 234
8.3. Restrictions and Special Cases for Using the Non-Premixed Model ................................................. 236
8.3.1. Restrictions on the Mixture Fraction Approach ...................................................................... 236
8.3.2. Using the Non-Premixed Model for Liquid Fuel or Coal Combustion ...................................... 239
8.3.3. Using the Non-Premixed Model with Flue Gas Recycle .......................................................... 240
8.3.4. Using the Non-Premixed Model with the Inert Model ............................................................ 240
8.3.4.1. Mixture Composition ................................................................................................... 241
8.3.4.1.1. Property Evaluation ............................................................................................. 242
8.4.The Diffusion Flamelet Models Theory ........................................................................................... 242
8.4.1. Restrictions and Assumptions ............................................................................................... 242
8.4.2.The Flamelet Concept ........................................................................................................... 242
8.4.2.1. Overview ..................................................................................................................... 242
8.4.2.2. Strain Rate and Scalar Dissipation ................................................................................. 244
8.4.2.3. Embedding Diffusion Flamelets in Turbulent Flames ..................................................... 244
8.4.3. Flamelet Generation ............................................................................................................. 245
8.4.4. Flamelet Import ................................................................................................................... 246
8.5. The Steady Diffusion Flamelet Model Theory ................................................................................. 248
8.5.1. Overview ............................................................................................................................. 248
8.5.2. Multiple Steady Flamelet Libraries ........................................................................................ 249
8.5.3. Steady Diffusion Flamelet Automated Grid Refinement ......................................................... 249
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8.5.4. Non-Adiabatic Steady Diffusion Flamelets ............................................................................. 249
8.6. The Unsteady Diffusion Flamelet Model Theory ............................................................................. 250
8.6.1. The Eulerian Unsteady Laminar Flamelet Model .................................................................... 251
8.6.1.1. Liquid Reactions .......................................................................................................... 253
8.6.2. The Diesel Unsteady Laminar Flamelet Model ....................................................................... 253
8.6.3. Multiple Diesel Unsteady Flamelets ....................................................................................... 254
8.6.4. Multiple Diesel Unsteady Flamelets with Flamelet Reset ........................................................ 254
8.6.4.1. Resetting the Flamelets ................................................................................................ 255
9. Premixed Combustion ......................................................................................................................... 257
9.1. Overview and Limitations ............................................................................................................. 257
9.1.1. Overview ............................................................................................................................. 257
9.1.2. Limitations ........................................................................................................................... 258
9.2. C-Equation Model Theory .............................................................................................................. 258
9.2.1. Propagation of the Flame Front ............................................................................................ 258
9.3. G-Equation Model Theory ............................................................................................................. 260
9.3.1. Numerical Solution of the G-equation ................................................................................... 261
9.4. Turbulent Flame Speed Models ..................................................................................................... 261
9.4.1. Zimont Turbulent Flame Speed Closure Model ...................................................................... 261
9.4.1.1. Zimont Turbulent Flame Speed Closure for LES ............................................................. 262
9.4.1.2. Flame Stretch Effect ..................................................................................................... 263
9.4.1.3. Gradient Diffusion ....................................................................................................... 263
9.4.1.4.Wall Damping .............................................................................................................. 264
9.4.2. Peters Flame Speed Model .................................................................................................... 264
9.4.2.1. Peters Flame Speed Model for LES ................................................................................ 265
9.5. Extended Coherent Flamelet Model Theory ................................................................................... 266
9.5.1. Closure for ECFM Source Terms ............................................................................................. 268
9.5.2.Turbulent Flame Speed in ECFM ............................................................................................ 270
9.5.3. LES and ECFM ...................................................................................................................... 270
9.6. Calculation of Properties ............................................................................................................... 272
9.6.1. Calculation of Temperature ................................................................................................... 273
9.6.1.1. Adiabatic Temperature Calculation ............................................................................... 273
9.6.1.2. Non-Adiabatic Temperature Calculation ....................................................................... 273
9.6.2. Calculation of Density .......................................................................................................... 273
9.6.3. Laminar Flame Speed ........................................................................................................... 274
9.6.4. Unburnt Density and Thermal Diffusivity ............................................................................... 274
10. Partially Premixed Combustion ........................................................................................................ 275
10.1. Overview .................................................................................................................................... 275
10.2. Limitations .................................................................................................................................. 275
10.3. Partially Premixed Combustion Theory ........................................................................................ 276
10.3.1. Chemical Equilibrium and Steady Diffusion Flamelet Models ............................................... 276
10.3.2. Flamelet Generated Manifold (FGM) model ......................................................................... 277
10.3.2.1. Premixed FGMs .......................................................................................................... 277
10.3.2.2. Diffusion FGMs .......................................................................................................... 279
10.3.3. FGM Turbulent Closure ....................................................................................................... 279
10.3.4. Calculation of Unburnt Properties ....................................................................................... 281
10.3.5. Laminar Flame Speed ......................................................................................................... 281
11. Composition PDF Transport .............................................................................................................. 283
11.1. Overview and Limitations ............................................................................................................ 283
11.2. Composition PDF Transport Theory ............................................................................................. 283
11.3.The Lagrangian Solution Method ................................................................................................. 284
11.3.1. Particle Convection ............................................................................................................ 285
11.3.2. Particle Mixing ................................................................................................................... 286
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11.3.2.1.The Modified Curl Model ............................................................................................ 286
11.3.2.2.The IEM Model ........................................................................................................... 286
11.3.2.3. The EMST Model ........................................................................................................ 287
11.3.2.4. Liquid Reactions ........................................................................................................ 287
11.3.3. Particle Reaction ................................................................................................................. 287
11.4. The Eulerian Solution Method ..................................................................................................... 288
11.4.1. Reaction ............................................................................................................................. 289
11.4.2. Mixing ................................................................................................................................ 289
11.4.3. Correction .......................................................................................................................... 290
11.4.4. Calculation of Composition Mean and Variance ................................................................... 290
12. Chemistry Acceleration ..................................................................................................................... 291
12.1. Overview and Limitations ............................................................................................................ 291
12.2. In-Situ Adaptive Tabulation (ISAT) ................................................................................................ 291
12.3. Dynamic Mechanism Reduction .................................................................................................. 293
12.3.1. Directed Relation Graph (DRG) Method for Mechanism Reduction ....................................... 294
12.4. Chemistry Agglomeration ........................................................................................................... 295
12.4.1. Binning Algorithm .............................................................................................................. 296
12.5. Chemical Mechanism Dimension Reduction ................................................................................ 298
12.5.1. Selecting the Represented Species ...................................................................................... 298
12.6. Dynamic Cell Clustering with ANSYS CHEMKIN-CFD Solver ........................................................... 299
13. Engine Ignition .................................................................................................................................. 301
13.1. Spark Model ................................................................................................................................ 301
13.1.1. Overview and Limitations ................................................................................................... 301
13.1.2. Spark Model Theory ............................................................................................................ 301
13.1.3. ECFM Spark Model Variants ................................................................................................. 304
13.2. Autoignition Models ................................................................................................................... 305
13.2.1. Model Overview ................................................................................................................. 305
13.2.2. Model Limitations .............................................................................................................. 305
13.2.3. Ignition Model Theory ........................................................................................................ 306
13.2.3.1.Transport of Ignition Species ...................................................................................... 306
13.2.3.2. Knock Modeling ........................................................................................................ 306
13.2.3.2.1. Modeling of the Source Term ............................................................................. 307
13.2.3.2.2. Correlations ...................................................................................................... 307
13.2.3.2.3. Energy Release .................................................................................................. 308
13.2.3.3. Ignition Delay Modeling ............................................................................................. 308
13.2.3.3.1. Modeling of the Source Term ............................................................................. 308
13.2.3.3.2. Correlations ...................................................................................................... 309
13.2.3.3.3. Energy Release .................................................................................................. 309
13.3. Crevice Model ............................................................................................................................. 309
13.3.1. Overview ........................................................................................................................... 309
13.3.1.1. Model Parameters ...................................................................................................... 310
13.3.2. Limitations ......................................................................................................................... 311
13.3.3. Crevice Model Theory ......................................................................................................... 312
14. Pollutant Formation .......................................................................................................................... 313
14.1. NOx Formation ........................................................................................................................... 313
14.1.1. Overview ........................................................................................................................... 313
14.1.1.1. NOx Modeling in ANSYS Fluent .................................................................................. 313
14.1.1.2. NOx Formation and Reduction in Flames .................................................................... 314
14.1.2. Governing Equations for NOx Transport .............................................................................. 314
14.1.3.Thermal NOx Formation ...................................................................................................... 315
14.1.3.1. Thermal NOx Reaction Rates ...................................................................................... 315
14.1.3.2. The Quasi-Steady Assumption for [N] ......................................................................... 315
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14.1.3.3.Thermal NOx Temperature Sensitivity ......................................................................... 316
14.1.3.4. Decoupled Thermal NOx Calculations ......................................................................... 316
14.1.3.5. Approaches for Determining O Radical Concentration ................................................ 316
14.1.3.5.1. Method 1: Equilibrium Approach ....................................................................... 316
14.1.3.5.2. Method 2: Partial Equilibrium Approach ............................................................. 317
14.1.3.5.3. Method 3: Predicted O Approach ....................................................................... 317
14.1.3.6. Approaches for Determining OH Radical Concentration .............................................. 317
14.1.3.6.1. Method 1: Exclusion of OH Approach ................................................................. 317
14.1.3.6.2. Method 2: Partial Equilibrium Approach ............................................................. 317
14.1.3.6.3. Method 3: Predicted OH Approach ..................................................................... 318
14.1.3.7. Summary ................................................................................................................... 318
14.1.4. Prompt NOx Formation ....................................................................................................... 318
14.1.4.1. Prompt NOx Combustion Environments ..................................................................... 318
14.1.4.2. Prompt NOx Mechanism ............................................................................................ 318
14.1.4.3. Prompt NOx Formation Factors .................................................................................. 319
14.1.4.4. Primary Reaction ....................................................................................................... 319
14.1.4.5. Modeling Strategy ..................................................................................................... 319
14.1.4.6. Rate for Most Hydrocarbon Fuels ................................................................................ 320
14.1.4.7. Oxygen Reaction Order .............................................................................................. 320
14.1.5. Fuel NOx Formation ............................................................................................................ 321
14.1.5.1. Fuel-Bound Nitrogen ................................................................................................. 321
14.1.5.2. Reaction Pathways ..................................................................................................... 321
14.1.5.3. Fuel NOx from Gaseous and Liquid Fuels .................................................................... 321
14.1.5.3.1. Fuel NOx from Intermediate Hydrogen Cyanide (HCN) ....................................... 322
14.1.5.3.1.1. HCN Production in a Gaseous Fuel ............................................................ 322
14.1.5.3.1.2. HCN Production in a Liquid Fuel ................................................................ 322
14.1.5.3.1.3. HCN Consumption .................................................................................... 323
14.1.5.3.1.4. HCN Sources in the Transport Equation ..................................................... 323
14.1.5.3.1.5. NOx Sources in the Transport Equation ..................................................... 323
14.1.5.3.2. Fuel NOx from Intermediate Ammonia (NH3) ..................................................... 324
14.1.5.3.2.1. NH3 Production in a Gaseous Fuel ............................................................. 324
14.1.5.3.2.2. NH3 Production in a Liquid Fuel ................................................................ 324
14.1.5.3.2.3. NH3 Consumption .................................................................................... 325
14.1.5.3.2.4. NH3 Sources in the Transport Equation ..................................................... 325
14.1.5.3.2.5. NOx Sources in the Transport Equation ..................................................... 325
14.1.5.3.3. Fuel NOx from Coal ........................................................................................... 326
14.1.5.3.3.1. Nitrogen in Char and in Volatiles ............................................................... 326
14.1.5.3.3.2. Coal Fuel NOx Scheme A ........................................................................... 326
14.1.5.3.3.3. Coal Fuel NOx Scheme B ........................................................................... 326
14.1.5.3.3.4. HCN Scheme Selection ............................................................................. 327
14.1.5.3.3.5. NOx Reduction on Char Surface ................................................................ 327
14.1.5.3.3.5.1. BET Surface Area .............................................................................. 328
14.1.5.3.3.5.2. HCN from Volatiles ........................................................................... 328
14.1.5.3.3.6. Coal Fuel NOx Scheme C ........................................................................... 328
14.1.5.3.3.7. Coal Fuel NOx Scheme D ........................................................................... 329
14.1.5.3.3.8. NH3 Scheme Selection ............................................................................. 330
14.1.5.3.3.8.1. NH3 from Volatiles ........................................................................... 330
14.1.5.3.4. Fuel Nitrogen Partitioning for HCN and NH3 Intermediates ................................ 330
14.1.6. NOx Formation from Intermediate N2O ............................................................................... 331
14.1.6.1. N2O - Intermediate NOx Mechanism .......................................................................... 331
14.1.7. NOx Reduction by Reburning ............................................................................................. 332
14.1.7.1. Instantaneous Approach ............................................................................................ 332
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14.1.7.2. Partial Equilibrium Approach ..................................................................................... 333
14.1.7.2.1. NOx Reduction Mechanism ............................................................................... 333
14.1.8. NOx Reduction by SNCR ..................................................................................................... 335
14.1.8.1. Ammonia Injection .................................................................................................... 335
14.1.8.2. Urea Injection ............................................................................................................ 336
14.1.8.3. Transport Equations for Urea, HNCO, and NCO ............................................................ 337
14.1.8.4. Urea Production due to Reagent Injection .................................................................. 338
14.1.8.5. NH3 Production due to Reagent Injection ................................................................... 338
14.1.8.6. HNCO Production due to Reagent Injection ................................................................ 338
14.1.9. NOx Formation in Turbulent Flows ...................................................................................... 339
14.1.9.1. The Turbulence-Chemistry Interaction Model ............................................................. 339
14.1.9.2. The PDF Approach ..................................................................................................... 340
14.1.9.3.The General Expression for the Mean Reaction Rate .................................................... 340
14.1.9.4.The Mean Reaction Rate Used in ANSYS Fluent ........................................................... 340
14.1.9.5. Statistical Independence ............................................................................................ 340
14.1.9.6.The Beta PDF Option .................................................................................................. 341
14.1.9.7.The Gaussian PDF Option ........................................................................................... 341
14.1.9.8. The Calculation Method for the Variance .................................................................... 341
14.2. SOx Formation ............................................................................................................................ 342
14.2.1. Overview ........................................................................................................................... 342
14.2.1.1.The Formation of SOx ................................................................................................. 342
14.2.2. Governing Equations for SOx Transport ............................................................................... 343
14.2.3. Reaction Mechanisms for Sulfur Oxidation .......................................................................... 344
14.2.4. SO2 and H2S Production in a Gaseous Fuel ......................................................................... 345
14.2.5. SO2 and H2S Production in a Liquid Fuel ............................................................................. 346
14.2.6. SO2 and H2S Production from Coal ..................................................................................... 346
14.2.6.1. SO2 and H2S from Char .............................................................................................. 346
14.2.6.2. SO2 and H2S from Volatiles ........................................................................................ 346
14.2.7. SOx Formation in Turbulent Flows ....................................................................................... 347
14.2.7.1. The Turbulence-Chemistry Interaction Model ............................................................. 347
14.2.7.2. The PDF Approach ..................................................................................................... 347
14.2.7.3.The Mean Reaction Rate ............................................................................................. 347
14.2.7.4.The PDF Options ........................................................................................................ 347
14.3. Soot Formation ........................................................................................................................... 347
14.3.1. Overview and Limitations ................................................................................................... 348
14.3.1.1. Predicting Soot Formation ......................................................................................... 348
14.3.1.2. Restrictions on Soot Modeling ................................................................................... 348
14.3.2. Soot Model Theory ............................................................................................................. 349
14.3.2.1.The One-Step Soot Formation Model .......................................................................... 349
14.3.2.2.The Two-Step Soot Formation Model .......................................................................... 350
14.3.2.2.1. Soot Generation Rate ........................................................................................ 350
14.3.2.2.2. Nuclei Generation Rate ...................................................................................... 351
14.3.2.3. The Moss-Brookes Model ........................................................................................... 351
14.3.2.3.1.The Moss-Brookes-Hall Model ............................................................................ 353
14.3.2.3.2. Soot Formation in Turbulent Flows .................................................................... 354
14.3.2.3.2.1.The Turbulence-Chemistry Interaction Model ............................................ 354
14.3.2.3.2.2.The PDF Approach .................................................................................... 355
14.3.2.3.2.3. The Mean Reaction Rate ........................................................................... 355
14.3.2.3.2.4.The PDF Options ....................................................................................... 355
14.3.2.3.3.The Effect of Soot on the Radiation Absorption Coefficient ................................. 355
14.3.2.4.The Method of Moments Model ................................................................................. 355
14.3.2.4.1. Soot Particle Population Balance ....................................................................... 355
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14.3.2.4.2. Moment Transport Equations ............................................................................ 357
14.3.2.4.3. Nucleation ........................................................................................................ 357
14.3.2.4.4. Coagulation ...................................................................................................... 359
14.3.2.4.5. Surface Growth and Oxidation ........................................................................... 362
14.3.2.4.6. Soot Aggregation .............................................................................................. 364
14.4. Decoupled Detailed Chemistry Model ......................................................................................... 368
14.4.1. Overview ........................................................................................................................... 368
14.4.1.1. Limitations ................................................................................................................ 369
14.4.2. Decoupled Detailed Chemistry Model Theory ..................................................................... 369
15. Aerodynamically Generated Noise ................................................................................................... 371
15.1. Overview .................................................................................................................................... 371
15.1.1. Direct Method .................................................................................................................... 371
15.1.2. Integral Method Based on Acoustic Analogy ....................................................................... 372
15.1.3. Broadband Noise Source Models ........................................................................................ 373
15.2. Acoustics Model Theory .............................................................................................................. 373
15.2.1. The Ffowcs-Williams and Hawkings Model .......................................................................... 373
15.2.2. Broadband Noise Source Models ........................................................................................ 376
15.2.2.1. Proudman’s Formula .................................................................................................. 376
15.2.2.2.The Jet Noise Source Model ........................................................................................ 377
15.2.2.3.The Boundary Layer Noise Source Model .................................................................... 378
15.2.2.4. Source Terms in the Linearized Euler Equations ........................................................... 379
15.2.2.5. Source Terms in Lilley’s Equation ................................................................................ 379
16. Discrete Phase ................................................................................................................................... 381
16.1. Introduction ............................................................................................................................... 381
16.1.1.The Euler-Lagrange Approach ............................................................................................. 381
16.2. Particle Motion Theory ................................................................................................................ 382
16.2.1. Equations of Motion for Particles ........................................................................................ 382
16.2.1.1. Particle Force Balance ................................................................................................ 382
16.2.1.2. Particle Torque Balance .............................................................................................. 382
16.2.1.3. Inclusion of the Gravity Term ...................................................................................... 383
16.2.1.4. Other Forces .............................................................................................................. 383
16.2.1.5. Forces in Moving Reference Frames ............................................................................ 383
16.2.1.6.Thermophoretic Force ................................................................................................ 384
16.2.1.7. Brownian Force .......................................................................................................... 384
16.2.1.8. Saffman’s Lift Force .................................................................................................... 385
16.2.1.9. Magnus Lift Force ...................................................................................................... 385
16.2.2.Turbulent Dispersion of Particles ......................................................................................... 386
16.2.2.1. Stochastic Tracking .................................................................................................... 386
16.2.2.1.1. The Integral Time .............................................................................................. 387
16.2.2.1.2.The Discrete Random Walk Model ...................................................................... 387
16.2.2.1.3. Using the DRW Model ....................................................................................... 388
16.2.2.2. Particle Cloud Tracking ............................................................................................... 389
16.2.2.2.1. Using the Cloud Model ...................................................................................... 391
16.2.3. Integration of Particle Equation of Motion ........................................................................... 391
16.3. Laws for Drag Coefficients ........................................................................................................... 393
16.3.1. Spherical Drag Law ............................................................................................................. 394
16.3.2. Non-spherical Drag Law ..................................................................................................... 394
16.3.3. Stokes-Cunningham Drag Law ............................................................................................ 394
16.3.4. High-Mach-Number Drag Law ............................................................................................ 395
16.3.5. Dynamic Drag Model Theory .............................................................................................. 395
16.3.6. Dense Discrete Phase Model Drag Laws .............................................................................. 395
16.3.7. Rotational Drag Law ........................................................................................................... 396
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16.4. Laws for Heat and Mass Exchange ............................................................................................... 396
16.4.1. Inert Heating or Cooling (Law 1/Law 6) ............................................................................... 397
16.4.2. Droplet Vaporization (Law 2) ............................................................................................... 398
16.4.2.1. Mass Transfer During Law 2—Diffusion Controlled Model ........................................... 399
16.4.2.2. Mass Transfer During Law 2—Convection/Diffusion Controlled Model ........................ 400
16.4.2.3. Defining the Saturation Vapor Pressure and Diffusion Coefficient (or Binary Diffusivity) ......................................................................................................................................... 401
16.4.2.4. Defining the Boiling Point and Latent Heat ................................................................. 401
16.4.2.5. Heat Transfer to the Droplet ....................................................................................... 402
16.4.3. Droplet Boiling (Law 3) ....................................................................................................... 404
16.4.4. Devolatilization (Law 4) ...................................................................................................... 405
16.4.4.1. Choosing the Devolatilization Model .......................................................................... 405
16.4.4.2.The Constant Rate Devolatilization Model ................................................................... 406
16.4.4.3. The Single Kinetic Rate Model .................................................................................... 406
16.4.4.4.The Two Competing Rates (Kobayashi) Model ............................................................. 407
16.4.4.5. The CPD Model .......................................................................................................... 407
16.4.4.5.1. General Description .......................................................................................... 408
16.4.4.5.2. Reaction Rates .................................................................................................. 408
16.4.4.5.3. Mass Conservation ............................................................................................ 409
16.4.4.5.4. Fractional Change in the Coal Mass .................................................................... 409
16.4.4.5.5. CPD Inputs ........................................................................................................ 410
16.4.4.5.6. Particle Swelling During Devolatilization ............................................................ 411
16.4.4.5.7. Heat Transfer to the Particle During Devolatilization ........................................... 412
16.4.5. Surface Combustion (Law 5) ............................................................................................... 412
16.4.5.1.The Diffusion-Limited Surface Reaction Rate Model .................................................... 413
16.4.5.2.The Kinetic/Diffusion Surface Reaction Rate Model ..................................................... 414
16.4.5.3. The Intrinsic Model .................................................................................................... 414
16.4.5.4.The Multiple Surface Reactions Model ........................................................................ 416
16.4.5.4.1. Limitations ........................................................................................................ 416
16.4.5.5. Heat and Mass Transfer During Char Combustion ....................................................... 416
16.4.6. Multicomponent Particle Definition (Law 7) ........................................................................ 417
16.4.6.1. Raoult’s Law .............................................................................................................. 418
16.4.6.2. Peng-Robinson Real Gas Model .................................................................................. 418
16.5.Vapor Liquid Equilibrium Theory .................................................................................................. 419
16.6. Physical Property Averaging ........................................................................................................ 421
16.7.Wall-Particle Reflection Model Theory .......................................................................................... 422
16.7.1. Rough Wall Model .............................................................................................................. 425
16.8.Wall-Jet Model Theory ................................................................................................................. 426
16.9.Wall-Film Model Theory ............................................................................................................... 427
16.9.1. Introduction ....................................................................................................................... 427
16.9.2. Interaction During Impact with a Boundary ......................................................................... 429
16.9.2.1. The Stanton-Rutland Model ....................................................................................... 429
16.9.2.1.1. Regime Definition ............................................................................................. 429
16.9.2.1.2. Splashing .......................................................................................................... 431
16.9.2.2.The Kuhnke Model ..................................................................................................... 434
16.9.2.2.1. Regime definition ............................................................................................. 434
16.9.2.2.2. Splashing .......................................................................................................... 437
16.9.3. Separation Criteria .............................................................................................................. 439
16.9.4. Conservation Equations for Wall-Film Particles .................................................................... 440
16.9.4.1. Momentum ............................................................................................................... 440
16.9.4.2. Mass Transfer from the Film ........................................................................................ 441
16.9.4.3. Energy Transfer from the Film ..................................................................................... 443
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16.10. Particle Erosion and Accretion Theory ........................................................................................ 444
16.11. Particle–Wall Impingement Heat Transfer Theory ....................................................................... 446
16.12. Atomizer Model Theory ............................................................................................................. 447
16.12.1.The Plain-Orifice Atomizer Model ...................................................................................... 448
16.12.1.1. Internal Nozzle State ................................................................................................ 449
16.12.1.2. Coefficient of Discharge ........................................................................................... 450
16.12.1.3. Exit Velocity ............................................................................................................. 452
16.12.1.4. Spray Angle ............................................................................................................. 452
16.12.1.5. Droplet Diameter Distribution .................................................................................. 452
16.12.2. The Pressure-Swirl Atomizer Model ................................................................................... 454
16.12.2.1. Film Formation ........................................................................................................ 454
16.12.2.2. Sheet Breakup and Atomization ............................................................................... 455
16.12.3.The Air-Blast/Air-Assist Atomizer Model ............................................................................. 457
16.12.4.The Flat-Fan Atomizer Model ............................................................................................. 458
16.12.5.The Effervescent Atomizer Model ...................................................................................... 459
16.13. Secondary Breakup Model Theory ............................................................................................. 460
16.13.1.Taylor Analogy Breakup (TAB) Model ................................................................................. 460
16.13.1.1. Introduction ............................................................................................................ 460
16.13.1.2. Use and Limitations ................................................................................................. 461
16.13.1.3. Droplet Distortion .................................................................................................... 461
16.13.1.4. Size of Child Droplets ............................................................................................... 462
16.13.1.5.Velocity of Child Droplets ......................................................................................... 463
16.13.1.6. Droplet Breakup ...................................................................................................... 463
16.13.2.Wave Breakup Model ........................................................................................................ 464
16.13.2.1. Introduction ............................................................................................................ 464
16.13.2.2. Use and Limitations ................................................................................................. 465
16.13.2.3. Jet Stability Analysis ................................................................................................. 465
16.13.2.4. Droplet Breakup ...................................................................................................... 466
16.13.3. KHRT Breakup Model ........................................................................................................ 467
16.13.3.1. Introduction ............................................................................................................ 467
16.13.3.2. Use and Limitations ................................................................................................. 467
16.13.3.3. Liquid Core Length .................................................................................................. 467
16.13.3.4. Rayleigh-Taylor Breakup ........................................................................................... 468
16.13.3.5. Droplet Breakup Within the Liquid Core .................................................................... 469
16.13.3.6. Droplet Breakup Outside the Liquid Core .................................................................. 469
16.13.4. Stochastic Secondary Droplet (SSD) Model ........................................................................ 469
16.13.4.1. Theory ..................................................................................................................... 469
16.14. Collision and Droplet Coalescence Model Theory ....................................................................... 470
16.14.1. Introduction ..................................................................................................................... 470
16.14.2. Use and Limitations .......................................................................................................... 471
16.14.3.Theory .............................................................................................................................. 471
16.14.3.1. Probability of Collision ............................................................................................. 471
16.14.3.2. Collision Outcomes .................................................................................................. 472
16.15. Discrete Element Method Collision Model .................................................................................. 473
16.15.1.Theory .............................................................................................................................. 473
16.15.1.1. The Spring Collision Law .......................................................................................... 474
16.15.1.2. The Spring-Dashpot Collision Law ............................................................................ 475
16.15.1.3. The Hertzian Collision Law ....................................................................................... 475
16.15.1.4. The Hertzian-Dashpot Collision Law ......................................................................... 476
16.15.1.5.The Friction Collision Law ......................................................................................... 476
16.15.1.6. Rolling Friction Collision Law for DEM ....................................................................... 477
16.15.1.7. DEM Parcels ............................................................................................................. 477
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16.15.1.8. Cartesian Collision Mesh .......................................................................................... 478
16.16. One-Way and Two-Way Coupling ............................................................................................... 478
16.16.1. Coupling Between the Discrete and Continuous Phases .................................................... 479
16.16.2. Momentum Exchange ...................................................................................................... 479
16.16.3. Heat Exchange ................................................................................................................. 480
16.16.4. Mass Exchange ................................................................................................................. 481
16.16.5. Under-Relaxation of the Interphase Exchange Terms ......................................................... 481
16.16.6. Interphase Exchange During Stochastic Tracking ............................................................... 482
16.16.7. Interphase Exchange During Cloud Tracking ..................................................................... 483
16.17. Node Based Averaging .............................................................................................................. 483
17. Multiphase Flows .............................................................................................................................. 485
17.1. Introduction ............................................................................................................................... 485
17.1.1. Multiphase Flow Regimes ................................................................................................... 485
17.1.1.1. Gas-Liquid or Liquid-Liquid Flows .............................................................................. 485
17.1.1.2. Gas-Solid Flows .......................................................................................................... 486
17.1.1.3. Liquid-Solid Flows ...................................................................................................... 486
17.1.1.4. Three-Phase Flows ..................................................................................................... 486
17.1.2. Examples of Multiphase Systems ........................................................................................ 487
17.2. Choosing a General Multiphase Model ........................................................................................ 488
17.2.1. Approaches to Multiphase Modeling .................................................................................. 488
17.2.1.1.The Euler-Euler Approach ........................................................................................... 488
17.2.1.1.1.The VOF Model .................................................................................................. 488
17.2.1.1.2. The Mixture Model ............................................................................................ 489
17.2.1.1.3.The Eulerian Model ............................................................................................ 489
17.2.2. Model Comparisons ........................................................................................................... 489
17.2.2.1. Detailed Guidelines ................................................................................................... 490
17.2.2.1.1.The Effect of Particulate Loading ........................................................................ 490
17.2.2.1.2.The Significance of the Stokes Number .............................................................. 491
17.2.2.1.2.1. Examples .................................................................................................. 491
17.2.2.1.3. Other Considerations ........................................................................................ 492
17.2.3.Time Schemes in Multiphase Flow ....................................................................................... 492
17.2.4. Stability and Convergence .................................................................................................. 493
17.3.Volume of Fluid (VOF) Model Theory ............................................................................................ 494
17.3.1. Overview of the VOF Model ................................................................................................ 494
17.3.2. Limitations of the VOF Model .............................................................................................. 494
17.3.3. Steady-State and Transient VOF Calculations ....................................................................... 494
17.3.4.Volume Fraction Equation ................................................................................................... 495
17.3.4.1. The Implicit Formulation ............................................................................................ 495
17.3.4.2.The Explicit Formulation ............................................................................................. 496
17.3.4.3. Interpolation Near the Interface ................................................................................. 497
17.3.4.3.1. The Geometric Reconstruction Scheme ............................................................. 498
17.3.4.3.2.The Donor-Acceptor Scheme ............................................................................. 498
17.3.4.3.3.The Compressive Interface Capturing Scheme for Arbitrary Meshes (CICSAM) ..... 499
17.3.4.3.4.The Compressive Scheme and Interface-Model-based Variants ........................... 499
17.3.4.3.5. Bounded Gradient Maximization (BGM) ............................................................. 500
17.3.5. Material Properties ............................................................................................................. 500
17.3.6. Momentum Equation ......................................................................................................... 500
17.3.7. Energy Equation ................................................................................................................. 501
17.3.8. Additional Scalar Equations ................................................................................................ 501
17.3.9. Surface Tension and Adhesion ............................................................................................ 501
17.3.9.1. Surface Tension ......................................................................................................... 502
17.3.9.1.1. The Continuum Surface Force Model ................................................................. 502
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17.3.9.1.2.The Continuum Surface Stress Model ................................................................. 503
17.3.9.1.3. Comparing the CSS and CSF Methods ................................................................ 503
17.3.9.1.4.When Surface Tension Effects Are Important ...................................................... 504
17.3.9.2.Wall Adhesion ............................................................................................................ 504
17.3.9.3. Jump Adhesion .......................................................................................................... 505
17.3.10. Open Channel Flow .......................................................................................................... 505
17.3.10.1. Upstream Boundary Conditions ............................................................................... 506
17.3.10.1.1. Pressure Inlet .................................................................................................. 506
17.3.10.1.2. Mass Flow Rate ................................................................................................ 507
17.3.10.1.3.Volume Fraction Specification .......................................................................... 507
17.3.10.2. Downstream Boundary Conditions ........................................................................... 507
17.3.10.2.1. Pressure Outlet ................................................................................................ 507
17.3.10.2.2. Outflow Boundary ........................................................................................... 507
17.3.10.2.3. Backflow Volume Fraction Specification ........................................................... 508
17.3.10.3. Numerical Beach Treatment ..................................................................................... 508
17.3.11. Open Channel Wave Boundary Conditions ........................................................................ 509
17.3.11.1. Airy Wave Theory ..................................................................................................... 510
17.3.11.2. Stokes Wave Theories ............................................................................................... 511
17.3.11.3. Cnoidal/Solitary Wave Theory ................................................................................... 512
17.3.11.4. Choosing a Wave Theory .......................................................................................... 513
17.3.11.5. Superposition of Waves ............................................................................................ 516
17.3.11.6. Modeling of Random Waves Using Wave Spectrum ................................................... 516
17.3.11.6.1. Definitions ...................................................................................................... 517
17.3.11.6.2.Wave Spectrum Implementation Theory .......................................................... 517
17.3.11.6.2.1. Long-Crested Random Waves (Unidirectional) ......................................... 517
17.3.11.6.2.1.1. Pierson-Moskowitz Spectrum ......................................................... 517
17.3.11.6.2.1.2. JONSWAP Spectrum ....................................................................... 517
17.3.11.6.2.1.3. TMA Spectrum ............................................................................... 518
17.3.11.6.2.2. Short-Crested Random Waves (Multi-Directional) .................................... 518
17.3.11.6.2.2.1. Cosine-2s Power Function (Frequency Independent) ....................... 519
17.3.11.6.2.2.2. Hyperbolic Function (Frequency Dependent) ................................. 519
17.3.11.6.2.3. Superposition of Individual Wave Components Using the Wave Spectrum ........................................................................................................................... 520
17.3.11.6.3. Choosing a Wave Spectrum and Inputs ............................................................ 521
17.3.11.7. Nomenclature for Open Channel Waves .................................................................... 522
17.3.12. Coupled Level-Set and VOF Model .................................................................................... 524
17.3.12.1. Theory ..................................................................................................................... 524
17.3.12.1.1. Surface Tension Force ...................................................................................... 524
17.3.12.1.2. Re-initialization of the Level-set Function via the Geometrical Method ............. 525
17.3.12.2. Limitations .............................................................................................................. 527
17.4. Mixture Model Theory ................................................................................................................. 527
17.4.1. Overview ........................................................................................................................... 527
17.4.2. Limitations ......................................................................................................................... 528
17.4.3. Continuity Equation ........................................................................................................... 529
17.4.4. Momentum Equation ......................................................................................................... 529
17.4.5. Energy Equation ................................................................................................................. 529
17.4.6. Relative (Slip) Velocity and the Drift Velocity ........................................................................ 530
17.4.7.Volume Fraction Equation for the Secondary Phases ............................................................ 531
17.4.8. Granular Properties ............................................................................................................ 532
17.4.8.1. Collisional Viscosity .................................................................................................... 532
17.4.8.2. Kinetic Viscosity ......................................................................................................... 532
17.4.9. Granular Temperature ......................................................................................................... 532
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17.4.10. Solids Pressure ................................................................................................................. 533
17.4.11. Interfacial Area Concentration .......................................................................................... 533
17.4.11.1. Hibiki-Ishii Model ..................................................................................................... 534
17.4.11.2. Ishii-Kim Model ........................................................................................................ 535
17.4.11.3.Yao-Morel Model ...................................................................................................... 536
17.5. Eulerian Model Theory ................................................................................................................ 537
17.5.1. Overview of the Eulerian Model .......................................................................................... 537
17.5.2. Limitations of the Eulerian Model ........................................................................................ 538
17.5.3.Volume Fraction Equation ................................................................................................... 539
17.5.4. Conservation Equations ...................................................................................................... 539
17.5.4.1. Equations in General Form ......................................................................................... 539
17.5.4.1.1. Conservation of Mass ........................................................................................ 539
17.5.4.1.2. Conservation of Momentum .............................................................................. 540
17.5.4.1.3. Conservation of Energy ..................................................................................... 540
17.5.4.2. Equations Solved by ANSYS Fluent ............................................................................. 541
17.5.4.2.1. Continuity Equation .......................................................................................... 541
17.5.4.2.2. Fluid-Fluid Momentum Equations ...................................................................... 541
17.5.4.2.3. Fluid-Solid Momentum Equations ...................................................................... 541
17.5.4.2.4. Conservation of Energy ..................................................................................... 542
17.5.5. Interfacial Area Concentration ............................................................................................ 542
17.5.6. Interphase Exchange Coefficients ....................................................................................... 543
17.5.6.1. Fluid-Fluid Exchange Coefficient ................................................................................ 543
17.5.6.1.1. Schiller and Naumann Model ............................................................................. 544
17.5.6.1.2. Morsi and Alexander Model ............................................................................... 544
17.5.6.1.3. Symmetric Model .............................................................................................. 545
17.5.6.1.4. Grace et al. Model .............................................................................................. 546
17.5.6.1.5.Tomiyama et al. Model ....................................................................................... 547
17.5.6.1.6. Ishii Model ........................................................................................................ 547
17.5.6.1.7. Universal Drag Laws for Bubble-Liquid and Droplet-Gas Flows ........................... 547
17.5.6.1.7.1. Bubble-Liquid Flow .................................................................................. 548
17.5.6.1.7.2. Droplet-Gas Flow ...................................................................................... 549
17.5.6.2. Fluid-Solid Exchange Coefficient ................................................................................ 549
17.5.6.3. Solid-Solid Exchange Coefficient ................................................................................ 552
17.5.6.4. Drag Modification ...................................................................................................... 552
17.5.6.4.1. Brucato Correlation ........................................................................................... 553
17.5.7. Lift Force ............................................................................................................................ 553
17.5.7.1. Lift Coefficient Models ............................................................................................... 554
17.5.7.1.1. Moraga Lift Force Model .................................................................................... 554
17.5.7.1.2. Saffman-Mei Lift Force Model ............................................................................ 555
17.5.7.1.3. Legendre-Magnaudet Lift Force Model .............................................................. 555
17.5.7.1.4.Tomiyama Lift Force Model ................................................................................ 556
17.5.8. Wall Lubrication Force ........................................................................................................ 556
17.5.8.1.Wall Lubrication Models ............................................................................................. 556
17.5.8.1.1. Antal et al. Model .............................................................................................. 557
17.5.8.1.2.Tomiyama Model ............................................................................................... 557
17.5.8.1.3. Frank Model ...................................................................................................... 558
17.5.8.1.4. Hosokawa Model .............................................................................................. 558
17.5.9. Turbulent Dispersion Force ................................................................................................. 558
17.5.9.1. Models for Turbulent Dispersion Force ....................................................................... 559
17.5.9.1.1. Lopez de Bertodano Model ............................................................................... 559
17.5.9.1.2. Simonin Model .................................................................................................. 559
17.5.9.1.3. Burns et al. Model .............................................................................................. 560
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17.5.9.1.4. Diffusion in VOF Model ...................................................................................... 560
17.5.9.2. Limiting Functions for the Turbulent Dispersion Force ................................................ 560
17.5.10.Virtual Mass Force ............................................................................................................. 561
17.5.11. Solids Pressure ................................................................................................................. 562
17.5.11.1. Radial Distribution Function ..................................................................................... 563
17.5.12. Maximum Packing Limit in Binary Mixtures ....................................................................... 564
17.5.13. Solids Shear Stresses ......................................................................................................... 565
17.5.13.1. Collisional Viscosity .................................................................................................. 565
17.5.13.2. Kinetic Viscosity ....................................................................................................... 565
17.5.13.3. Bulk Viscosity ........................................................................................................... 565
17.5.13.4. Frictional Viscosity ................................................................................................... 565
17.5.14. Granular Temperature ....................................................................................................... 567
17.5.15. Description of Heat Transfer .............................................................................................. 569
17.5.15.1. The Heat Exchange Coefficient ................................................................................. 569
17.5.15.1.1. Constant ......................................................................................................... 569
17.5.15.1.2. Nusselt Number .............................................................................................. 569
17.5.15.1.3. Ranz-Marshall Model ....................................................................................... 570
17.5.15.1.4.Tomiyama Model ............................................................................................. 570
17.5.15.1.5. Hughmark Model ............................................................................................ 570
17.5.15.1.6. Gunn Model .................................................................................................... 570
17.5.15.1.7. Two-Resistance Model ..................................................................................... 570
17.5.15.1.8. Fixed To Saturation Temperature ...................................................................... 571
17.5.15.1.9. User Defined ................................................................................................... 572
17.5.16. Turbulence Models ........................................................................................................... 572
17.5.16.1. k- ε Turbulence Models ............................................................................................. 572
17.5.16.1.1. k- ε Mixture Turbulence Model ......................................................................... 572
17.5.16.1.2. k- ε Dispersed Turbulence Model ..................................................................... 573
17.5.16.1.2.1. Assumptions .......................................................................................... 574
17.5.16.1.2.2. Turbulence in the Continuous Phase ....................................................... 574
17.5.16.1.2.3.Turbulence in the Dispersed Phase .......................................................... 575
17.5.16.1.3. k- ε Turbulence Model for Each Phase ............................................................... 575
17.5.16.1.3.1.Transport Equations ................................................................................ 575
17.5.16.2. RSM Turbulence Models ........................................................................................... 576
17.5.16.2.1. RSM Dispersed Turbulence Model .................................................................... 577
17.5.16.2.2. RSM Mixture Turbulence Model ....................................................................... 578
17.5.16.3. Turbulence Interaction Models ................................................................................. 578
17.5.16.3.1. Simonin et al. .................................................................................................. 579
17.5.16.3.1.1. Formulation in Dispersed Turbulence Models .......................................... 579
17.5.16.3.1.1.1. Continuous Phase .......................................................................... 579
17.5.16.3.1.1.2. Dispersed Phases ........................................................................... 580
17.5.16.3.1.2. Formulation in Per Phase Turbulence Models ........................................... 581
17.5.16.3.2. Troshko-Hassan ............................................................................................... 581
17.5.16.3.2.1.Troshko-Hassan Formulation in Mixture Turbulence Models ..................... 581
17.5.16.3.2.2. Troshko-Hassan Formulation in Dispersed Turbulence Models ................. 581
17.5.16.3.2.2.1. Continuous Phase .......................................................................... 581
17.5.16.3.2.2.2. Dispersed Phases ........................................................................... 582
17.5.16.3.2.3.Troshko-Hassan Formulation in Per-Phase Turbulence Models .................. 582
17.5.16.3.2.3.1. Continuous Phase .......................................................................... 582
17.5.16.3.2.3.2. Dispersed Phases ........................................................................... 582
17.5.16.3.3. Sato ................................................................................................................ 582
17.5.16.3.4. None ............................................................................................................... 583
17.5.17. Solution Method in ANSYS Fluent ..................................................................................... 583
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17.5.17.1.The Pressure-Correction Equation ............................................................................. 583
17.5.17.2. Volume Fractions ..................................................................................................... 583
17.5.18. Dense Discrete Phase Model ............................................................................................. 583
17.5.18.1. Limitations .............................................................................................................. 584
17.5.18.2. Granular Temperature .............................................................................................. 585
17.5.19. Multi-Fluid VOF Model ...................................................................................................... 585
17.5.20. Wall Boiling Models .......................................................................................................... 587
17.5.20.1. Overview ................................................................................................................. 587
17.5.20.2. RPI Model ................................................................................................................ 587
17.5.20.3. Non-equilibrium Subcooled Boiling .......................................................................... 590
17.5.20.4. Critical Heat Flux ...................................................................................................... 590
17.5.20.4.1.Wall Heat Flux Partition .................................................................................... 591
17.5.20.4.2. Flow Regime Transition ................................................................................... 591
17.5.20.5. Interfacial Momentum Transfer ................................................................................. 592
17.5.20.5.1. Interfacial Area ................................................................................................ 592
17.5.20.5.2. Bubble and Droplet Diameters ........................................................................ 593
17.5.20.5.2.1. Bubble Diameter .................................................................................... 593
17.5.20.5.2.2. Droplet Diameter .................................................................................... 593
17.5.20.5.3. Interfacial Drag Force ...................................................................................... 594
17.5.20.5.4. Interfacial Lift Force ......................................................................................... 594
17.5.20.5.5.Turbulent Dispersion Force .............................................................................. 594
17.5.20.5.6. Wall Lubrication Force ..................................................................................... 594
17.5.20.5.7. Virtual Mass Force ........................................................................................... 594
17.5.20.6. Interfacial Heat Transfer ............................................................................................ 594
17.5.20.6.1. Interface to Liquid Heat Transfer ...................................................................... 594
17.5.20.6.2. Interface to Vapor Heat Transfer ....................................................................... 594
17.5.20.7. Mass Transfer ........................................................................................................... 595
17.5.20.7.1. Mass Transfer From the Wall to Vapor ............................................................... 595
17.5.20.7.2. Interfacial Mass Transfer .................................................................................. 595
17.5.20.8.Turbulence Interactions ............................................................................................ 595
17.6. Wet Steam Model Theory ............................................................................................................ 595
17.6.1. Overview of the Wet Steam Model ...................................................................................... 595
17.6.2. Limitations of the Wet Steam Model .................................................................................... 596
17.6.3.Wet Steam Flow Equations .................................................................................................. 596
17.6.4. Phase Change Model .......................................................................................................... 597
17.6.5. Built-in Thermodynamic Wet Steam Properties .................................................................... 598
17.6.5.1. Equation of State ....................................................................................................... 599
17.6.5.2. Saturated Vapor Line .................................................................................................. 600
17.6.5.3. Saturated Liquid Line ................................................................................................. 600
17.6.5.4. Mixture Properties ..................................................................................................... 600
17.7. Modeling Mass Transfer in Multiphase Flows ................................................................................ 600
17.7.1. Source Terms due to Mass Transfer ...................................................................................... 600
17.7.1.1. Mass Equation ........................................................................................................... 601
17.7.1.2. Momentum Equation ................................................................................................. 601
17.7.1.3. Energy Equation ........................................................................................................ 601
17.7.1.4. Species Equation ....................................................................................................... 601
17.7.1.5. Other Scalar Equations ............................................................................................... 601
17.7.2. Unidirectional Constant Rate Mass Transfer ......................................................................... 602
17.7.3. UDF-Prescribed Mass Transfer ............................................................................................. 602
17.7.4. Cavitation Models .............................................................................................................. 602
17.7.4.1. Limitations of the Cavitation Models .......................................................................... 603
17.7.4.1.1. Limitations of Cavitation with the VOF Model ..................................................... 603
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17.7.4.2.Vapor Transport Equation ........................................................................................... 604
17.7.4.3. Bubble Dynamics Consideration ................................................................................ 604
17.7.4.4. Singhal et al. Model .................................................................................................... 605
17.7.4.5. Zwart-Gerber-Belamri Model ..................................................................................... 607
17.7.4.6. Schnerr and Sauer Model ........................................................................................... 608
17.7.4.7. Turbulence Factor ...................................................................................................... 609
17.7.4.8. Additional Guidelines for the Cavitation Models ......................................................... 609
17.7.4.9. Extended Cavitation Model Capabilities ..................................................................... 611
17.7.4.9.1. Multiphase Cavitation Models ........................................................................... 611
17.7.4.9.2. Multiphase Species Transport Cavitation Model ................................................. 612
17.7.5. Evaporation-Condensation Model ....................................................................................... 612
17.7.5.1. Lee Model ................................................................................................................. 612
17.7.5.2.Thermal Phase Change Model .................................................................................... 615
17.7.6. Interphase Species Mass Transfer ........................................................................................ 616
17.7.6.1. Modeling Approach ................................................................................................... 617
17.7.6.1.1. Equilibrium Model ............................................................................................. 617
17.7.6.1.2.Two-Resistance Model ....................................................................................... 618
17.7.6.2. Species Mass Transfer Models ..................................................................................... 620
17.7.6.2.1. Raoult’s Law ...................................................................................................... 620
17.7.6.2.2. Henry’s Law ...................................................................................................... 620
17.7.6.2.3. Equilibrium Ratio .............................................................................................. 621
17.7.6.3. Mass Transfer Coefficient Models ................................................................................ 622
17.7.6.3.1. Constant ........................................................................................................... 622
17.7.6.3.2. Sherwood Number ............................................................................................ 622
17.7.6.3.3. Ranz-Marshall Model ......................................................................................... 622
17.7.6.3.4. Hughmark Model .............................................................................................. 623
17.7.6.3.5. User-Defined ..................................................................................................... 623
17.8. Modeling Species Transport in Multiphase Flows ......................................................................... 623
17.8.1. Limitations ......................................................................................................................... 624
17.8.2. Mass and Momentum Transfer with Multiphase Species Transport ....................................... 624
17.8.2.1. Source Terms Due to Heterogeneous Reactions .......................................................... 625
17.8.2.1.1. Mass Transfer .................................................................................................... 625
17.8.2.1.2. Momentum Transfer .......................................................................................... 625
17.8.2.1.3. Species Transfer ................................................................................................ 626
17.8.2.1.4. Heat Transfer ..................................................................................................... 626
17.8.3. The Stiff Chemistry Solver ................................................................................................... 627
17.8.4. Heterogeneous Phase Interaction ....................................................................................... 627
18. Solidification and Melting ................................................................................................................. 629
18.1. Overview .................................................................................................................................... 629
18.2. Limitations .................................................................................................................................. 630
18.3. Introduction ............................................................................................................................... 630
18.4. Energy Equation ......................................................................................................................... 630
18.5. Momentum Equations ................................................................................................................ 631
18.6.Turbulence Equations .................................................................................................................. 632
18.7. Species Equations ....................................................................................................................... 632
18.8. Back Diffusion ............................................................................................................................. 634
18.9. Pull Velocity for Continuous Casting ............................................................................................ 634
18.10. Contact Resistance at Walls ........................................................................................................ 636
18.11.Thermal and Solutal Buoyancy ................................................................................................... 636
19. Eulerian Wall Films ............................................................................................................................ 639
19.1. Introduction ............................................................................................................................... 639
19.2. Mass, Momentum, and Energy Conservation Equations for Wall Film ............................................. 640
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19.2.1. Film Sub-Models ................................................................................................................. 641
19.2.1.1. DPM Collection .......................................................................................................... 641
19.2.1.2. Splashing .................................................................................................................. 641
19.2.1.3. Film Separation .......................................................................................................... 641
19.2.1.3.1. Separation Criteria ............................................................................................ 641
19.2.1.3.1.1. Foucart Separation ................................................................................... 642
19.2.1.3.1.2. O’Rourke Separation ................................................................................. 642
19.2.1.3.1.3. Friedrich Separation ................................................................................. 642
19.2.1.4. Film Stripping ............................................................................................................ 643
19.2.1.5. Secondary Phase Accretion ........................................................................................ 644
19.2.1.6. Coupling of Wall Film with Mixture Species Transport ................................................. 645
19.2.2. Boundary Conditions .......................................................................................................... 645
19.2.3. Obtaining Film Velocity Without Solving the Momentum Equations .................................... 646
19.2.3.1. Shear-Driven Film Velocity ......................................................................................... 646
19.2.3.2. Gravity-Driven Film Velocity ....................................................................................... 646
19.3. Passive Scalar Equation for Wall Film ............................................................................................ 647
19.4. Numerical Schemes and Solution Algorithm ................................................................................ 648
19.4.1.Temporal Differencing Schemes .......................................................................................... 648
19.4.1.1. First-Order Explicit Method ........................................................................................ 648
19.4.1.2. First-Order Implicit Method ........................................................................................ 649
19.4.1.3. Second-Order Implicit Method ................................................................................... 649
19.4.2. Spatial Differencing Schemes .............................................................................................. 650
19.4.3. Solution Algorithm ............................................................................................................. 651
19.4.3.1. Steady Flow ............................................................................................................... 651
19.4.3.2. Transient Flow ........................................................................................................... 651
20. Electric Potential ............................................................................................................................... 653
20.1. Overview and Limitations ............................................................................................................ 653
20.2. Electric Potential Equation ........................................................................................................... 653
20.3. Energy Equation Source Term ...................................................................................................... 654
21. Solver Theory .................................................................................................................................... 655
21.1. Overview of Flow Solvers ............................................................................................................ 655
21.1.1. Pressure-Based Solver ......................................................................................................... 656
21.1.1.1. The Pressure-Based Segregated Algorithm ................................................................. 656
21.1.1.2.The Pressure-Based Coupled Algorithm ...................................................................... 657
21.1.2. Density-Based Solver .......................................................................................................... 658
21.2. General Scalar Transport Equation: Discretization and Solution ..................................................... 660
21.2.1. Solving the Linear System ................................................................................................... 662
21.3. Discretization .............................................................................................................................. 662
21.3.1. Spatial Discretization .......................................................................................................... 662
21.3.1.1. First-Order Upwind Scheme ....................................................................................... 663
21.3.1.2. Power-Law Scheme .................................................................................................... 663
21.3.1.3. Second-Order Upwind Scheme .................................................................................. 664
21.3.1.4. First-to-Higher Order Blending ................................................................................... 665
21.3.1.5. Central-Differencing Scheme ..................................................................................... 665
21.3.1.6. Bounded Central Differencing Scheme ....................................................................... 666
21.3.1.7. QUICK Scheme .......................................................................................................... 666
21.3.1.8.Third-Order MUSCL Scheme ....................................................................................... 667
21.3.1.9. Modified HRIC Scheme .............................................................................................. 667
21.3.1.10. High Order Term Relaxation ..................................................................................... 669
21.3.2. Temporal Discretization ...................................................................................................... 669
21.3.2.1. Implicit Time Integration ............................................................................................ 670
21.3.2.2. Bounded Second Order Implicit Time Integration ....................................................... 670
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21.3.2.2.1. Limitations ........................................................................................................ 670
21.3.2.3. Explicit Time Integration ............................................................................................ 671
21.3.3. Evaluation of Gradients and Derivatives .............................................................................. 671
21.3.3.1. Green-Gauss Theorem ............................................................................................... 672
21.3.3.2. Green-Gauss Cell-Based Gradient Evaluation .............................................................. 672
21.3.3.3. Green-Gauss Node-Based Gradient Evaluation ............................................................ 672
21.3.3.4. Least Squares Cell-Based Gradient Evaluation ............................................................. 672
21.3.4. Gradient Limiters ................................................................................................................ 674
21.3.4.1. Standard Limiter ........................................................................................................ 674
21.3.4.2. Multidimensional Limiter ........................................................................................... 675
21.3.4.3. Differentiable Limiter ................................................................................................. 675
21.4. Pressure-Based Solver ................................................................................................................. 675
21.4.1. Discretization of the Momentum Equation .......................................................................... 676
21.4.1.1. Pressure Interpolation Schemes ................................................................................. 676
21.4.2. Discretization of the Continuity Equation ............................................................................ 677
21.4.2.1. Density Interpolation Schemes ................................................................................... 678
21.4.3. Pressure-Velocity Coupling ................................................................................................. 678
21.4.3.1. Segregated Algorithms .............................................................................................. 679
21.4.3.1.1. SIMPLE .............................................................................................................. 679
21.4.3.1.2. SIMPLEC ........................................................................................................... 680
21.4.3.1.2.1. Skewness Correction ................................................................................ 680
21.4.3.1.3. PISO .................................................................................................................. 680
21.4.3.1.3.1. Neighbor Correction ................................................................................. 680
21.4.3.1.3.2. Skewness Correction ................................................................................ 681
21.4.3.1.3.3. Skewness - Neighbor Coupling ................................................................. 681
21.4.3.2. Fractional-Step Method (FSM) .................................................................................... 681
21.4.3.3. Coupled Algorithm .................................................................................................... 681
21.4.3.3.1. Limitations ........................................................................................................ 682
21.4.4. Steady-State Iterative Algorithm ......................................................................................... 683
21.4.4.1. Under-Relaxation of Variables .................................................................................... 683
21.4.4.2. Under-Relaxation of Equations ................................................................................... 683
21.4.5.Time-Advancement Algorithm ............................................................................................ 683
21.4.5.1. Iterative Time-Advancement Scheme ......................................................................... 684
21.4.5.1.1.The Frozen Flux Formulation .............................................................................. 685
21.4.5.2. Non-Iterative Time-Advancement Scheme .................................................................. 686
21.5. Density-Based Solver ................................................................................................................... 688
21.5.1. Governing Equations in Vector Form ................................................................................... 688
21.5.2. Preconditioning ................................................................................................................. 689
21.5.3. Convective Fluxes ............................................................................................................... 691
21.5.3.1. Roe Flux-Difference Splitting Scheme ......................................................................... 691
21.5.3.2. AUSM  Scheme ......................................................................................................... 691
21.5.3.3. Low Diffusion Roe Flux Difference Splitting Scheme ................................................... 692
21.5.4. Steady-State Flow Solution Methods ................................................................................... 692
21.5.4.1. Explicit Formulation ................................................................................................... 693
21.5.4.1.1. Implicit Residual Smoothing .............................................................................. 693
21.5.4.2. Implicit Formulation .................................................................................................. 694
21.5.4.2.1. Convergence Acceleration for Stretched Meshes ................................................ 694
21.5.5. Unsteady Flows Solution Methods ...................................................................................... 695
21.5.5.1. Explicit Time Stepping ............................................................................................... 695
21.5.5.2. Implicit Time Stepping (Dual-Time Formulation) ......................................................... 695
21.6. Pseudo Transient Under-Relaxation ............................................................................................. 697
21.6.1. Automatic Pseudo Transient Time Step ............................................................................... 697
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21.7. Multigrid Method ........................................................................................................................ 699
21.7.1. Approach ........................................................................................................................... 699
21.7.1.1.The Need for Multigrid ............................................................................................... 699
21.7.1.2.The Basic Concept in Multigrid ................................................................................... 700
21.7.1.3. Restriction and Prolongation ...................................................................................... 700
21.7.1.4. Unstructured Multigrid .............................................................................................. 701
21.7.2. Multigrid Cycles .................................................................................................................. 701
21.7.2.1. The V and W Cycles .................................................................................................... 701
21.7.3. Algebraic Multigrid (AMG) .................................................................................................. 705
21.7.3.1. AMG Restriction and Prolongation Operators ............................................................. 705
21.7.3.2. AMG Coarse Level Operator ....................................................................................... 706
21.7.3.3. The F Cycle ................................................................................................................ 706
21.7.3.4. The Flexible Cycle ...................................................................................................... 706
21.7.3.4.1.The Residual Reduction Rate Criteria .................................................................. 707
21.7.3.4.2. The Termination Criteria .................................................................................... 708
21.7.3.5.The Coupled and Scalar AMG Solvers .......................................................................... 708
21.7.3.5.1. Gauss-Seidel ..................................................................................................... 709
21.7.3.5.2. Incomplete Lower Upper (ILU) ........................................................................... 709
21.7.4. Full-Approximation Storage (FAS) Multigrid ......................................................................... 710
21.7.4.1. FAS Restriction and Prolongation Operators ............................................................... 711
21.7.4.2. FAS Coarse Level Operator ......................................................................................... 711
21.8. Hybrid Initialization ..................................................................................................................... 711
21.9. Full Multigrid (FMG) Initialization ................................................................................................. 713
21.9.1. Overview of FMG Initialization ............................................................................................ 713
21.9.2. Limitations of FMG Initialization .......................................................................................... 714
22. Adapting the Mesh ............................................................................................................................ 717
22.1. Static Adaption Process ............................................................................................................... 717
22.1.1. Hanging Node Adaption ..................................................................................................... 717
22.1.1.1. Hanging Node Refinement ......................................................................................... 718
22.1.1.2. Hanging Node Coarsening ......................................................................................... 719
22.2. Boundary Adaption ..................................................................................................................... 719
22.3. Gradient Adaption ...................................................................................................................... 721
22.3.1. Gradient Adaption Approach .............................................................................................. 721
22.3.2. Example of Steady Gradient Adaption ................................................................................. 723
22.4. Isovalue Adaption ....................................................................................................................... 725
22.5. Region Adaption ......................................................................................................................... 727
22.5.1. Defining a Region ............................................................................................................... 727
22.5.2. Region Adaption Example .................................................................................................. 728
22.6. Volume Adaption ........................................................................................................................ 729
22.6.1.Volume Adaption Approach ................................................................................................ 729
22.6.2.Volume Adaption Example .................................................................................................. 730
22.7.Yplus/Ystar Adaption ................................................................................................................... 731
22.7.1.Yplus/Ystar Adaption Approach .......................................................................................... 731
22.8. Anisotropic Adaption .................................................................................................................. 733
22.9. Geometry-Based Adaption .......................................................................................................... 733
22.9.1. Geometry-Based Adaption Approach .................................................................................. 734
22.9.1.1. Node Projection ......................................................................................................... 734
22.9.1.2. Example of Geometry-Based Adaption ....................................................................... 736
22.10. Registers ................................................................................................................................... 739
22.10.1. Adaption Registers ........................................................................................................... 739
22.10.2. Mask Registers .................................................................................................................. 740
23. Reporting Alphanumeric Data .......................................................................................................... 743
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23.1. Fluxes Through Boundaries ......................................................................................................... 743
23.2. Forces on Boundaries .................................................................................................................. 744
23.2.1. Computing Forces, Moments, and the Center of Pressure ..................................................... 744
23.3. Surface Integration ..................................................................................................................... 746
23.3.1. Computing Surface Integrals .............................................................................................. 747
23.3.1.1. Area .......................................................................................................................... 747
23.3.1.2. Integral ...................................................................................................................... 748
23.3.1.3. Area-Weighted Average ............................................................................................. 748
23.3.1.4. Custom Vector Based Flux .......................................................................................... 748
23.3.1.5. Custom Vector Flux .................................................................................................... 748
23.3.1.6. Custom Vector Weighted Average .............................................................................. 748
23.3.1.7. Flow Rate ................................................................................................................... 748
23.3.1.8. Mass Flow Rate .......................................................................................................... 749
23.3.1.9. Mass-Weighted Average ............................................................................................ 749
23.3.1.10. Sum of Field Variable ................................................................................................ 749
23.3.1.11. Facet Average .......................................................................................................... 750
23.3.1.12. Facet Minimum ........................................................................................................ 750
23.3.1.13. Facet Maximum ....................................................................................................... 750
23.3.1.14.Vertex Average ......................................................................................................... 750
23.3.1.15. Vertex Minimum ...................................................................................................... 750
23.3.1.16.Vertex Maximum ...................................................................................................... 750
23.3.1.17. Standard-Deviation .................................................................................................. 750
23.3.1.18. Uniformity Index ...................................................................................................... 751
23.3.1.19. Volume Flow Rate .................................................................................................... 751
23.4. Volume Integration ..................................................................................................................... 752
23.4.1. Computing Volume Integrals .............................................................................................. 752
23.4.1.1.Volume ...................................................................................................................... 752
23.4.1.2. Sum .......................................................................................................................... 753
23.4.1.3. Sum*2Pi .................................................................................................................... 753
23.4.1.4. Volume Integral ......................................................................................................... 753
23.4.1.5.Volume-Weighted Average ......................................................................................... 753
23.4.1.6. Mass-Weighted Integral ............................................................................................. 753
23.4.1.7. Mass .......................................................................................................................... 754
23.4.1.8. Mass-Weighted Average ............................................................................................ 754
A. Nomenclature ....................................................................................................................................... 755
Bibliography ............................................................................................................................................. 759
Index ........................................................................................................................................................ 789