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ANSYS CFX-Solver Theory Guide
ANSYS, Inc.
Release 14.0
Southpointe
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Canonsburg, PA 15317
November 2011
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Table of Contents
1. Basic Solver Capability Theory
................................................................................................................ 1
1.1. Documentation Conventions ............................................................................................................ 2
1.1.1. Dimensions .............................................................................................................................. 2
1.1.2. List of Symbols ......................................................................................................................... 2
1.1.2.1. Subscripts ....................................................................................................................... 7
1.1.3. Variable Definitions .................................................................................................................. 7
1.1.3.1. Isothermal Compressibility .............................................................................................. 7
1.1.3.2. Isentropic Compressibility ............................................................................................... 7
1.1.3.3. Reference Pressure .......................................................................................................... 7
1.1.3.4. Static Pressure ................................................................................................................. 8
1.1.3.5. Modified Pressure ............................................................................................................ 8
1.1.3.6. Static Enthalpy ................................................................................................................ 8
1.1.3.6.1. Material with Variable Density and Specific Heat ...................................................... 9
1.1.3.7. Total Enthalpy ................................................................................................................ 10
1.1.3.8. Domain Temperature ..................................................................................................... 10
1.1.3.9. Static Temperature ........................................................................................................ 10
1.1.3.9.1. Material with Constant Density and Specific Heat .................................................. 11
1.1.3.9.2. Ideal Gas or Solid with c
p
=f(T) ............................................................................... 11
1.1.3.9.3. Material with Variable Density and Specific Heat .................................................... 11
1.1.3.10. Total Temperature ........................................................................................................ 11
1.1.3.10.1. Material with Constant Density and Specific Heat ................................................ 12
1.1.3.10.2. Ideal Gas with constant c
p
................................................................................... 12
1.1.3.10.3. Ideal Gas with c
p
= f(T) ........................................................................................ 12
1.1.3.10.4. Material with Variable Density and Specific Heat .................................................. 13
1.1.3.11. Entropy ....................................................................................................................... 13
1.1.3.11.1. Material with Constant Density and Specific Heat ................................................ 13
1.1.3.11.2. Ideal Gas with constant c
p
or c
p
= f(T) .................................................................. 14
1.1.3.11.3. Material with Variable Density and Specific Heat .................................................. 14
1.1.3.12. Total Pressure .............................................................................................................. 16
1.1.3.12.1. Incompressible Fluids .......................................................................................... 16
1.1.3.12.2. Ideal Gases .......................................................................................................... 17
1.1.3.12.3. Material with Variable Density and Specific Heat .................................................. 17
1.1.3.13. Shear Strain Rate ......................................................................................................... 18
1.1.3.14. Rotating Frame Quantities ........................................................................................... 18
1.1.3.14.1. Incompressible Fluids .......................................................................................... 19
1.1.3.14.2. Ideal Gases .......................................................................................................... 19
1.1.3.14.3. Material with Variable Density and Specific Heat .................................................. 21
1.1.3.15. Courant Number .......................................................................................................... 21
1.1.4. Mathematical Notation ........................................................................................................... 22
1.1.4.1. The Vector Operators ..................................................................................................... 22
1.1.4.1.1. Gradient Operator ................................................................................................. 22
1.1.4.1.2. Divergence Operator ............................................................................................ 23
1.1.4.1.3. Dyadic Operator ................................................................................................... 23
1.1.4.2. Matrix Transposition ...................................................................................................... 23
1.1.4.3. The Identity Matrix (Kronecker Delta function) ............................................................... 24
1.1.4.4. Index Notation .............................................................................................................. 24
1.2. Governing Equations ...................................................................................................................... 25
1.2.1. Transport Equations ............................................................................................................... 25
1.2.1.1. The Continuity Equation ................................................................................................ 25
1.2.1.2. The Momentum Equations ............................................................................................. 25
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ANSYS CFX-Solver Theory Guide
1.2.1.3. The Total Energy Equation .............................................................................................. 26
1.2.1.4. The Thermal Energy Equation ........................................................................................ 26
1.2.2. Equations of State .................................................................................................................. 27
1.2.2.1. Incompressible Equation of State ................................................................................... 28
1.2.2.2. Ideal Gas Equation of State ............................................................................................ 28
1.2.2.3. Real Gas and Liquid Equations of State ........................................................................... 28
1.2.2.3.1. Real Gas Properties ............................................................................................... 28
1.2.2.3.1.1. Redlich Kwong Models ................................................................................. 29
1.2.2.3.1.1.1. The Standard Redlich Kwong Model ..................................................... 29
1.2.2.3.1.1.2. The Aungier Redlich Kwong Model ....................................................... 29
1.2.2.3.1.1.3. The Soave Redlich Kwong Model .......................................................... 30
1.2.2.3.1.2. Peng Robinson Model .................................................................................. 31
1.2.2.3.1.3. Real Gas Constitutive Relations ..................................................................... 31
1.2.2.3.2. Real Gas Saturated Vapor Properties ...................................................................... 34
1.2.2.3.3. Real Gas Liquid Properties ..................................................................................... 34
1.2.2.3.4. IAPWS Equation of State ........................................................................................ 35
1.2.2.3.5. Metastable Superheated Liquid/Supercooled Vapor States .................................... 37
1.2.2.3.6. Numerical Testing to Delineate Metastable Regions ............................................... 37
1.2.2.3.7. The Acentric Factor ............................................................................................... 37
1.2.2.4. General Equation of State .............................................................................................. 38
1.2.3. Conjugate Heat Transfer ......................................................................................................... 39
1.3. Buoyancy ....................................................................................................................................... 39
1.3.1. Full Buoyancy Model .............................................................................................................. 39
1.3.2. Boussinesq Model .................................................................................................................. 40
1.4. Immersed Solids ............................................................................................................................. 40
1.4.1. Immersed Solid Boundary Details ........................................................................................... 40
1.4.2. Laminar Flow Treatment ......................................................................................................... 42
1.4.3. Turbulent Flow Treatment ....................................................................................................... 42
1.4.3.1. Wall Distance ................................................................................................................. 42
1.4.3.2. SST Model ..................................................................................................................... 43
1.4.3.3. Scalable Wall Function ................................................................................................... 43
1.5. Multicomponent Flow ..................................................................................................................... 44
1.5.1. Multicomponent Notation ...................................................................................................... 44
1.5.2. Scalar Transport Equation ....................................................................................................... 44
1.5.3. Algebraic Equation for Components ....................................................................................... 46
1.5.4. Constraint Equation for Components ...................................................................................... 46
1.5.5. Multicomponent Fluid Properties ........................................................................................... 47
1.5.6. Energy Equation ..................................................................................................................... 48
1.5.7. Multicomponent Energy Diffusion .......................................................................................... 49
1.6. Additional Variables ........................................................................................................................ 51
1.6.1. Transport Equations for Additional Variables ........................................................................... 51
1.6.2. Diffusive Transport Equations for Additional Variables ............................................................. 52
1.6.3. Poisson Equations for Additional Variables .............................................................................. 52
1.6.4. Algebraic Equations for Additional Variables ........................................................................... 53
1.7. Rotational Forces ............................................................................................................................ 53
1.7.1. Alternate Rotation Model ....................................................................................................... 53
1.8. Sources .......................................................................................................................................... 54
1.8.1. Momentum Sources ............................................................................................................... 54
1.8.1.1. Isotropic Loss Model ...................................................................................................... 54
1.8.1.2. Directional Loss Model .................................................................................................. 55
1.8.1.3. General Momentum Sources .......................................................................................... 56
1.8.1.4. Immersed Solid Sources ................................................................................................. 57
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ANSYS CFX-Solver Theory Guide
1.8.2. General Sources ..................................................................................................................... 57
1.8.3. Mass (Continuity) Sources ...................................................................................................... 57
1.8.4. Bulk Sources .......................................................................................................................... 57
1.8.5. Radiation Sources .................................................................................................................. 57
1.8.6. Boundary Sources .................................................................................................................. 58
1.9. Boundary Conditions ...................................................................................................................... 58
1.9.1. Inlet (Subsonic) ...................................................................................................................... 58
1.9.1.1. Mass and Momentum .................................................................................................... 58
1.9.1.1.1. Normal Speed in ................................................................................................... 58
1.9.1.1.2. Cartesian Velocity Components ............................................................................. 58
1.9.1.1.3. Cylindrical Velocity Components ........................................................................... 58
1.9.1.1.4. Total Pressure ........................................................................................................ 59
1.9.1.1.5. Mass Flow Rate ..................................................................................................... 59
1.9.1.2. Turbulence .................................................................................................................... 59
1.9.1.2.1. Default Intensity and Autocompute Length Scale .................................................. 60
1.9.1.2.2. Intensity and Autocompute Length Scale .............................................................. 60
1.9.1.2.3. Intensity and Length Scale .................................................................................... 60
1.9.1.2.4. k and Epsilon ........................................................................................................ 61
1.9.1.3. Heat Transfer ................................................................................................................. 61
1.9.1.3.1. Static Temperature ................................................................................................ 61
1.9.1.3.2. Total Temperature ................................................................................................. 62
1.9.1.4. Additional Variables ....................................................................................................... 62
1.9.2. Inlet (Supersonic) ................................................................................................................... 63
1.9.2.1. Heat Transfer ................................................................................................................. 63
1.9.2.1.1. Static Temperature ................................................................................................ 63
1.9.2.1.2. Total Temperature ................................................................................................. 63
1.9.3. Outlet (Subsonic) ................................................................................................................... 63
1.9.3.1. Mass and Momentum .................................................................................................... 63
1.9.3.1.1. Static Pressure (Uniform) ....................................................................................... 63
1.9.3.1.2. Normal Speed ....................................................................................................... 63
1.9.3.1.3. Cartesian Velocity Components ............................................................................. 63
1.9.3.1.4. Cylindrical Velocity Components ........................................................................... 63
1.9.3.1.5. Average Static Pressure: Over Whole Outlet ............................................................ 64
1.9.3.1.6. Average Static Pressure: Above or Below Specified Radius ...................................... 64
1.9.3.1.7. Average Static Pressure: Circumferential ................................................................ 64
1.9.3.1.8. Mass Flow Rate: Scale Mass Flows .......................................................................... 64
1.9.3.1.9. Mass Flow Rate: Shift Pressure with or without Pressure Profile ............................... 65
1.9.3.1.10. Mass Flow Rate: Shift Pressure with Circumferential Pressure Averaging ................ 66
1.9.3.1.11. Radial Equilibrium ............................................................................................... 66
1.9.3.2. Turbulence, Heat Transfer, Convected Additional Variables, and Other Scalars .................. 67
1.9.4. Outlet (Supersonic) ................................................................................................................ 67
1.9.5. Opening ................................................................................................................................ 67
1.9.5.1. Mass and Momentum .................................................................................................... 67
1.9.5.1.1. Cartesian Velocity Components ............................................................................. 67
1.9.5.1.2. Cylindrical Velocity Components ........................................................................... 67
1.9.5.1.3. Pressure and Direction .......................................................................................... 68
1.9.5.1.3.1. Loss Coefficient ............................................................................................ 68
1.9.5.2. Heat Transfer ................................................................................................................. 68
1.9.5.2.1. Static Temperature ................................................................................................ 68
1.9.5.3. Additional Variables ....................................................................................................... 68
1.9.6. Wall ....................................................................................................................................... 69
1.9.6.1. Mass and Momentum .................................................................................................... 69
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of ANSYS, Inc. and its subsidiaries and affiliates.
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