ANSYS CFX Solver Theory Guide - [PDF Document] (2022)

  • ANSYS CFX-SolverTheory Guide

    ANSYS CFX Release 11.0

    December 2006

  • ANSYS, Inc.

    Southpointe

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  • Page vANSYS CFX-Solver Theory Guide

    Table of Contents

    Copyright and Trademark Information

    Disclaimer Notice

    U.S. Government Rights

    Third-Party Software

    Basic Solver Capability Theory

    Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

    Documentation Conventions . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . 2Dimensions . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 2List of Symbols . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 2Variable Definitions. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6Mathematical Notation . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .20

    Governing Equations . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .22TransportEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .23Equations of State . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . .24Conjugate Heat Transfer . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . .33

    Buoyancy . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33FullBuoyancy Model . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .34Boussinesq Model . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . .34

    Multicomponent Flow . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . .35MulticomponentNotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . .35Scalar Transport Equation . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . ..35Algebraic Equation for Components . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .37Constraint Equation for Components . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .37MulticomponentFluid Properties . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . .38Energy Equation. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . ..39Multicomponent Energy Diffusion . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .40

  • Table of Contents: Turbulence and Wall Function Theory

    Page vi ANSYS CFX-Solver Theory Guide

    Additional Variables . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .40TransportEquation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .40Diffusive Transport Equation. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..41Poisson Equation . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .41Algebraic Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . .42

    Rotational Forces. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .42AlternateRotation Model . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .43

    Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..43Momentum Sources . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .43General Sources . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .45Mass (Continuity) Sources . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45BulkSources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . .46Radiation Sources . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .46Boundary Sources . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..46

    Boundary Conditions . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .46Inlet(subsonic) . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . .46Inlet (supersonic) . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . .51Outlet (subsonic) . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .51Outlet (supersonic) . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . .54Opening . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .55Wall. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .56Symmetry Plane . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . .58

    Automatic Time Scale Calculation. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .58Fluid Time Scale Estimate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . .59Solid Time Scale Estimate . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .60

    Mesh Adaption. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .61AdaptionCriteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .61Mesh RefinementImplementation in ANSYS CFX . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62MeshAdaption Limitations . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .64

    Flow in Porous Media. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .65Darcy Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .65Directional Loss Model . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . .67

    Turbulence and Wall Function Theory

    Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .69

    Turbulence Models . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .69StatisticalTurbulence Models and the Closure Problem. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..70

    Eddy Viscosity Turbulence Models . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .72The Zero Equation Model inANSYS CFX . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74TwoEquation Turbulence Models. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .75The Eddy Viscosity Transport Model. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . .83

    Reynolds Stress Turbulence Models . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .85The Reynolds Stress Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . .86Omega-Based Reynolds Stress Models . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .89Rotating Frame of Reference forReynolds Stress Models. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .91

    ANSYS CFX Transition Model Formulation . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .92

    Large Eddy Simulation Theory . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .97Smagorinsky Model . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . .98Wall Damping . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..99

  • Table of Contents: GGI and MFR Theory

    ANSYS CFX-Solver Theory Guide Page vii

    Detached Eddy Simulation Theory . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 100SST-DES Formulation Streletset al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .101Zonal SST-DES Formulation in ANSYS CFX . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 101Discretization of the Advection Terms. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 102Boundary Conditions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 103

    Scale-Adaptive Simulation Theory . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 103SAS-SST Model Formulation . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .104

    Modeling Flow Near the Wall . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 107MathematicalFormulation. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 107

    Wall Distance Formulation. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 1171D Illustration ofConcept. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 117Concept Generalized to 3D . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . 118

    GGI and MFR Theory

    Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

    Interface Characteristics . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . 120

    Numerics . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

    Multiphase Flow Theory

    Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

    Multiphase Notation . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . 124Multiphase TotalPressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 124

    The Homogeneous and Inhomogeneous Models . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 125The Inhomogeneous Model . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . 125The HomogeneousModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 127

    Hydrodynamic Equations . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 128InhomogeneousHydrodynamic Equations . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .128Homogeneous Hydrodynamic Equations. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 130

    Multicomponent Multiphase Flow . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 131

    Interphase Momentum Transfer Models. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 131Interphase Drag . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 132Interphase Drag for the Particle Model . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 133Interphase Drag for the MixtureModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .138Interphase Drag for the Free Surface Model . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 139Lift Force . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 139Virtual Mass Force . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . 139Wall LubricationForce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 140Interphase Turbulent Dispersion Force . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . 141

    Solid Particle Collision Models . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 141Solids Stress Tensor .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 142Solids Pressure . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .142Solids Bulk Viscosity . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . 143Solids ShearViscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 144Granular Temperature . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .145

    Interphase Heat Transfer . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . 147Phasic Equations . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 147Inhomogeneous Interphase Heat TransferModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . 148Homogeneous Heat Transfer inMultiphase Flow . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . 152

  • Table of Contents: Particle Transport Theory

    Page viii ANSYS CFX-Solver Theory Guide

    Multiple Size Group (MUSIG) Model . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 152Model Derivation. . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 152Size Group Discretization . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . 155Breakup Models .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 157Coalescence Models . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .158

    The Algebraic Slip Model . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . 159Phasic Equations . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 160Bulk Equations . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .160Drift and Slip Relations . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 161Derivation of theAlgebraic Slip Equation. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 161Turbulence Effects. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . 163Energy Equation. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 163Wall Deposition . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .163

    Turbulence Modeling in Multiphase Flow . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 163Phase-Dependent Turbulence Models . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . 164TurbulenceEnhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 165Homogeneous Turbulence for Multiphase Flow . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 166

    Additional Variables in Multiphase Flow. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 166Additional Variable InterphaseTransfer Models . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . 167HomogeneousAdditional Variables in Multiphase Flow . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

    Sources in Multiphase Flow . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 170Fluid-specificSources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 170Bulk Sources . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 170

    Interphase Mass Transfer . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . 170Secondary Fluxes. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 171User Defined Interphase Mass Transfer . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . 172General Species MassTransfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 172The Thermal Phase Change Model . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 176The Cavitation Model . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 178The Droplet Condensation Model . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 180

    Free Surface Flow . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .184Implementation . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . 184Surface Tension . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 184

    Particle Transport Theory

    Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

    Lagrangian Tracking Implementation . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 187Integration . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 188Interphase Transfer Through Source Terms. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 188

    Momentum Transfer . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . 189Drag Force . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 191Buoyancy Force . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 191Rotation Force . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . 192Virtual orAdded Mass Force . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 192Pressure Gradient Force . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .193Turbulence in Particle Tracking . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 194Turbulent Dispersion . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 195

  • Table of Contents: Combustion Theory

    ANSYS CFX-Solver Theory Guide Page ix

    Heat and Mass Transfer . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . 195Heat Transfer . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 196Simple Mass Transfer . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .197Liquid Evaporation Model . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 197Oil Evaporation/Combustion. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .198Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . 198CoalCombustion . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . 198Hydrocarbon Fuel AnalysisModel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .203

    Basic Erosion Model . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . 205Model ofFinnie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 205Model of Tabakoff and Grant. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .206Overall Erosion Rate and Erosion Output . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 208

    Spray Breakup Models . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . 208PrimaryBreakup/Atomization Models . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 208Secondary Breakup Models . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . 214Dynamic Drag Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 223Dynamic Drag Law Control . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . 224Penetration Depthand Spray Angle. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 225

    Combustion Theory

    Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

    Transport Equations . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . 228

    Chemical Reaction Rate. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . 228

    Fluid Time Scale for Extinction Model . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 229

    The Eddy Dissipation Model . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 229Reactants Limiter . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 229Products Limiter. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .230Maximum Flame Temperature Limiter . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 230

    The Finite Rate Chemistry Model . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 230Third Body Terms. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 231

    The Combined Eddy Dissipation/Finite Rate Chemistry Model . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 232

    Combustion Source Term Linearization . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 232

    The Flamelet Model . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . 233Laminar FlameletModel for Non Premixed Combustion . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . 234Coupling ofLaminar Flamelet with the Turbulent Flow Field . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 237FlameletLibraries . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 239

    Burning Velocity Model (Premixed or Partially Premixed) . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 239Reaction Progress . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .239Weighted Reaction Progress . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 241

    Burning Velocity Model (BVM) . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 242Equivalence Ratio,Stoichiometric Mixture Fraction . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

    Laminar Burning Velocity . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . 244Value . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 245Equivalence Ratio Correlation. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .245

    Turbulent Burning Velocity . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 247Value . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 247Zimont Correlation . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 247Peters Correlation . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . 249MuellerCorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 250

  • Table of Contents: Radiation Theory

    Page x ANSYS CFX-Solver Theory Guide

    Spark Ignition Model . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . 250

    Phasic Combustion . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . 251

    NO Formation Model . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . 252FormationMechanisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 252

    Chemistry Post-Processing . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 258

    Soot Model . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 259SootFormation. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 259Soot Combustion. . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 261Turbulence Effects. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

    Radiation Theory

    Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

    Radiation Transport . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . 263BlackbodyEmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 265Quantities of Interest . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 266Radiation Through Domain Interfaces . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 267

    Rosseland Model . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . 268WallTreatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 269

    The P1 Model . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . 269WallTreatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 270

    Discrete Transfer Model . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . 270

    Monte Carlo Model . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . 271

    Spectral Models . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . 272Gray . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 272Multiband Model . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 272Multigray Model. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .273

    Discretization and Solution Theory

    Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

    Numerical Discretization. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . 277Discretization ofthe Governing Equations. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .277The Coupled System of Equations. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 291

    Solution Strategy - The Coupled Solver. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 292General Solution . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 292Linear Equation Solution. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . 293ResidualNormalization Procedure . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 296

    Discretization Errors . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . 296ControllingError Sources . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 296Controlling Error Propagation. . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . 297

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    ANSYS CFX-Solver Theory Guide

    Basic Solver Capability Theory

    Introduction

    This chapter describes:

    Documentation Conventions (p. 2)

    Governing Equations (p. 22)

    Buoyancy (p. 33)

    Multicomponent Flow (p. 35)

    Additional Variables (p. 40)

    Rotational Forces (p. 42)

    Sources (p. 43)

    Boundary Conditions (p. 46)

    Automatic Time Scale Calculation (p. 58)

    Mesh Adaption (p. 61)

    Flow in Porous Media (p. 65)

    This chapter describes the mathematical equations used to modelfluid flow, heat, and mass

    transfer in ANSYS CFX for single-phase, single andmulti-component flow without

    combustion or radiation. It is designed to be a reference forthose users who desire a more

    detailed understanding of the mathematics underpinning the ANSYSCFX-Solver, and is

    therefore not essential reading. It is not an exhaustive text onCFD mathematics; a reference

    section is provided should you wish to follow up this chapter inmore detail.

    Information on dealing with multiphase flow:

    Multiphase Flow Theory (p. 123)

    Particle Transport Theory (p. 187)

    Information on combustion and radiation theory:

    Combustion Theory (p. 227)

    Radiation Theory (p. 263)

    Recommended books for further reading on CFD and relatedsubjects:

    Further Background Reading (p. 6 in "ANSYS CFXIntroduction")

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    Documentation Conventions

    The topic(s) in this section include:

    Dimensions (p. 2)

    List of Symbols (p. 2)

    Variable Definitions (p. 6)

    Mathematical Notation (p. 20)

    Dimensions

    Throughout this manual, dimensions are given in terms of thefundamental magnitudes of

    length ( ), mass ( ), time ( ), temperature ( ) and chemicalamount ( ).

    List of Symbols

    This section lists symbols used in this chapter, together withtheir meanings, dimensions

    and where applicable, values. Dimensionless quantities aredenoted by 1. The values of

    physical constants (or their default values) are also given.

    More information on the notation used in the multiphase andmulticomponent chapters is

    available.

    Multiphase Notation (p. 124)

    Multicomponent Notation (p. 35).

    L M T A

    Symbol Description Dimensions Value

    linear energy source coefficient

    Stanton number

    linear resistance coefficient

    quadratic resistance coefficient

    - turbulence model constant

    RNG - turbulence model coefficient

    - turbulence model constant

    RNG - turbulence model constant

    - turbulence model constant

    Reynolds Stress model constant

    RNG - turbulence model constant

    fluid speed of sound

    CE M L 1 T 3 1

    Ch 1

    CR1 M L 3 T 1

    CR2 M L 4

    C1 k 1 1.44

    C1RNG k 1 1.42 f h

    C2 k 1 1.92

    C2RNG k 1 1.68

    C k 1 0.09

    CRS 1

    CRNG k 1 0.085

    c L T 1

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    concentration of components A and Bi.e. mass per unit volumeofcomponents A and B (single-phaseflow)

    Reynolds Stress model constant

    specific heat capacity at constantpressure

    specific heat capacity at constantvolume

    Reynolds Stress model constant

    Reynolds Stress model constant

    binary diffusivity of component A incomponent B

    kinematic diffusivity of an additional

    variable,

    distance or length

    constant used for near-wall modeling

    Zero Equation turbulence modelconstant

    RNG- - turbulence model coefficient

    (Video) Tutorial Ansys - How to Make Simulation Fluid Flow by CFX ( Simple for Beginner)

    gravity vector

    specific static (thermodynamic)enthalpy

    For details, see StaticEnthalpy (p. 7).

    heat transfer coefficient

    specific total enthalpy For details, see TotalEnthalpy (p.8).

    turbulence kinetic energy per unit mass

    local Mach number,

    mass flow rate

    shear production of turbulence

    static (thermodynamic) pressure For details, see StaticPressure(p. 6).

    reference pressure For details, seeReference Pressure(p. 6).

    total pressure For details, see TotalPressure (p. 14).

    modified pressure For details, seeModified Pressure(p. 6).

    universal gas constant

    Symbol Description Dimensions Value

    cA cB, M L 3

    cS 1 0.22

    cp L2 T 2 1

    cv L2 T 2 1

    c1 1 1.45

    c2 1 1.9

    DAB L2 T 1

    D

    L2 T 1

    d L

    E 1 9.793

    f 1 0.01

    f h k 1

    g L T 2

    h hstat, L2 T 2

    hc M T 3 1

    htot L2 T 2

    k L2 T 2

    M U c 1m M T 1

    Pk M L 1 T 3

    p pstat, M L 1 T 2

    pref M L 1 T 2

    ptot M L 1 T 2

    p' M L 1 T 2

    R0 L2 T 2 1 8314.5

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    Reynolds number,

    location vector

    volume fraction of phase

    energy source

    momentum source

    mass source

    turbulent Schmidt number,

    mass flow rate from phase to phase

    .

    static (thermodynamic) temperature For details, seeStaticTemperature (p. 9).

    domain temperature For details, seeDomain Temperature(p. 9).

    buoyancy reference temperature usedin the Boussinesqapproximation

    saturation temperature

    total temperature For details, see TotalTemperature (p. 10).

    vector of velocity

    velocity magnitude

    fluctuating velocity component inturbulent flow

    fluid viscous and body force work term

    molecular weight (Ideal Gas fluidmodel)

    mass fraction of component A in thefluid

    used as a subscript to indicate that the

    quantity applies to phase

    used as a subscript to indicate that the

    quantity applies to phase

    coefficient of thermal expansion (for theBoussinesqapproximation)

    RNG - turbulence model constant

    diffusivity

    molecular diffusion coefficient of

    component

    Symbol Description Dimensions Value

    Re rU d m 1r L

    r 1

    SE M L 1 T 3

    SM M L 2 T 2

    SMS M L 3 T 1

    Sct t/t 1

    s

    M T 1

    T T stat,

    Tdom

    T ref

    T sat

    T tot

    U Ux y z, , L T 1

    U L T 1

    u L T 1

    W f M L 1 T 3

    w 1

    Y A

    1

    RNG k 1 0.012 M L 1 T 1

    AA M L

    1 T 1

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    Subscripts Quantities which appear with subscripts , , refer tothat quantity for component ,

    , in a multicomponent fluid.

    Quantities which appear with subscripts , , refer to thatquantity for phase , , in

    a multiphase flow.

    dynamic diffusivity of an additionalvariable

    turbulent diffusivity

    identity matrix or Kronecker Deltafunction

    turbulence dissipation rate

    bulk viscosity

    Von Karman constant

    thermal conductivity

    molecular (dynamic) viscosity

    turbulent viscosity

    effective viscosity,

    density

    laminar Prandtl number,

    turbulent Prandtl number,

    turbulence model constant for theequation

    - turbulence model constant

    - turbulence model constant

    Reynolds Stress model constant

    RNG - turbulence model constant

    RNG - turbulence model constant

    shear stress or sub-grid scale stressmolecular stress tensor

    specific volume

    additional variable (non-reacting scalar)

    general scalar variable

    angular velocity

    Symbol Description Dimensions Value

    M L 1 T 1

    t M L 1 T 1

    1

    L2 T 3

    M L 1 T 1 1 0.41

    M L T 3 1

    M L 1 T 1

    t M L 1 T 1

    eff t+ M L 1 T 1

    M L 3

    Pr cp 1

    Prt cpt t 1

    k k 1 1.0

    k 1 1.3

    k 1 2

    RS 1

    kRNG k 1 0.7179

    RNG k 1 0.7179

    M L 1 T 2

    M 1 L3

    M L 3

    T 1

    A B C A

    B C

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    Such quantities are only used in the chapters describingmulticomponent and multiphase

    flows.

    Multicomponent Flow (p. 35)

    Multiphase Flow Theory (p. 123)

    Variable Definitions

    IsothermalCompressibility

    The isothermal compressibility defines the rate of change of thesystem volume with

    pressure. For details, see Variables Relevant for CompressibleFlow (p. 62 in "ANSYS

    CFX-Solver Manager User's Guide").

    (Eqn. 1)

    IsentropicCompressibility

    Isentropic compressibility is the extent to which a materialreduces its volume when it is

    subjected to compressive stresses at a constant value ofentropy. For details, see Variables

    Relevant for Compressible Flow (p. 62 in "ANSYS CFX-SolverManager User's Guide").

    (Eqn. 2)

    ReferencePressure

    The Reference Pressure (Eqn. 3) is the absolute pressure datumfrom which all other

    pressure values are taken. All relative pressure specificationsin ANSYS CFX are relative to the

    Reference Pressure. For details, see Setting a ReferencePressure (p. 10 in "ANSYS

    CFX-Solver Modeling Guide").

    (Eqn. 3)

    Static Pressure ANSYS CFX solves for the relative StaticPressure (thermodynamic pressure) (Eqn. 4) in

    the flow field, and is related to Absolute Pressure (Eqn.5).

    (Eqn. 4)

    (Eqn. 5)

    ModifiedPressure

    When the - turbulence model is used, the fluctuating velocitycomponents give rise to

    an additional pressure term to give the modified pressure (Eqn.6), where is the turbulent

    kinetic energy. In this case, ANSYS CFX solves for the modifiedpressure. This variable is

    named Pressure in ANSYS CFX.

    (Eqn. 6)

    1---

    ddp------ T

    1---

    ddp------ S

    Pref

    pstat

    pabs pstat pref+=

    k

    k

    p' pstat2k

    3---------+=

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    Static Enthalpy Specific static enthalpy (Eqn. 7) is a measureof the energy contained in a fluid per unit mass.

    Static enthalpy is defined in terms of the internal energy of afluid and the fluid state:

    (Eqn. 7)

    When you use the thermal energy model, the ANSYS CFX-Solverdirectly computes the static

    enthalpy. General changes in enthalpy are also used by thesolver to calculate

    thermodynamic properties such as temperature. To compute thesequantities, you need to

    know how enthalpy varies with changes in both temperature andpressure. These changes

    are given by the general differential relationship (Eqn. 8):

    (Eqn. 8)

    which can be rewritten as (Eqn. 9)

    (Eqn. 9)

    where is specific heat at constant pressure and is density. Formost materials the first

    term always has an effect on enthalpy, and, in some cases, thesecond term drops out or is

    not included. For example, the second term is zero for materialswhich use the Ideal Gas

    equation of state or materials in a solid thermodynamic state.In addition, the second term

    is also dropped for liquids or gases with constant specific heatwhen you run the thermal

    energy equation model.

    Material with Variable Density and Specific HeatIn order tosupport general properties, which are a function of bothtemperature and

    pressure, a table for is generated by integrating Equation 9using the functions

    supplied for and . The enthalpy table is constructed between theupper and lower

    bounds of temperature and pressure (using flow solver internaldefaults or those supplied

    (Video) Ansys CFX Pre

    by the user). For any general change in conditions from to , thechange

    in enthalpy, , is calculated in two steps: first at constantpressure, and then at constant

    temperature using Equation 10.

    (Eqn. 10)

    hstat ustatpstatstat---------+=

    dh hT------ pdT

    hp------ Tdp+=

    dh cpdT1--- 1

    T---

    T------ p+ dp+=

    cp

    h T p,( ) cp

    p1 T1,( ) p2 T2,( )hd

    h2 h1 cpT1

    T2

    dT 1--- 1 T--- T------ p+p1

    p2

    dp+=

  • Basic Solver Capability Theory: Documentation Conventions

    Page 8 ANSYS CFX-Solver Theory Guide. ANSYS CFX Release 11.0.1996-2006 ANSYS Europe, Ltd. All rights reserved.Containsproprietary and confidential information of ANSYS, Inc. and itssubsidiaries and affiliates.

    To successfully integrate Equation 10, the ANSYS CFX-Solver mustbe provided

    thermodynamically consistent values of the equation of state, ,and specific heat capacity,

    . Thermodynamically consistent means that the coefficients ofthe differential terms of

    Equation 9 must satisfy the exact differential propertythat:

    or

    The equation of state derivative within the second integral ofEquation 10 is numerically

    evaluated from the using a two point central difference formula.In addition, the ANSYS

    CFX-Solver uses an adaptive number of interpolation points toconstruct the property table,

    and bases the number of points on an absolute error toleranceestimated using the

    enthalpy and entropy derivatives.

    Total Enthalpy Total enthalpy is expressed in terms of a staticenthalpy and the flow kinetic energy:

    (Eqn. 11)

    cp

    T 2

    2

    h

    p p2

    2

    h

    T=

    T 2

    2

    cp

    p

    1---

    T------ p

    1---

    2T2-------

    T------ p+

    T2-----

    T 2

    2

    p+=

    T

    (p2,T2)(p1,T1)

    p

    h

    21

    htot hstat12-- U U( )+=

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    ANSYS CFX-Solver Theory Guide. ANSYS CFX Release 11.0. 1996-2006ANSYS Europe, Ltd. All rights reserved. Page 9Contains proprietaryand confidential information of ANSYS, Inc. and its subsidiariesand affiliates.

    where is the flow velocity. When you use the total energy modelthe ANSYS CFX-Solver

    directly computes total enthalpy, and static enthalpy is derivedfrom this expression. In

    rotating frames of reference the total enthalpy includes therelative frame kinetic energy.

    For details, see Rotating Frame Quantities (p. 17).

    DomainTemperature

    The domain temperature, , is the absolute temperature at whichan isothermal

    simulation is performed. For details, see Isothermal (p. 8 in"ANSYS CFX-Solver Modeling

    Guide").

    StaticTemperature

    The static temperature, , is the thermodynamic temperature, anddepends on the

    internal energy of the fluid. In ANSYS CFX, depending on theheat transfer model you select,

    the flow solver calculates either total or static enthalpy(corresponding to the total or

    thermal energy equations).

    The static temperature is calculated using static enthalpy andthe constitutive relationship

    for the material under consideration. The constitutive relationsimply tells us how enthalpy

    varies with changes in both temperature and pressure.

    Material with Constant Density and Specific Heat

    In the simplified case where a material has constant and ,temperatures can be

    calculated by integrating a simplified form of the generaldifferential relationship for

    enthalpy:

    (Eqn. 12)

    which is derived from the full differential form for changes instatic enthalpy. The default

    reference state in the ANSYS CFX-Solver is and .

    Ideal Gas or Solid with cp=f(T)

    The enthalpy change for an ideal gas or CHT solid with specificheat as a function of

    temperature is defined by:

    (Eqn. 13)

    When the solver calculates static enthalpy, either directly orfrom total enthalpy, you can

    back static temperature out of this relationship. When varieswith temperature, the

    ANSYS CFX-Solver builds an enthalpy table and static temperatureis backed out by

    inverting the table.

    Material with Variable Density and Specific HeatTo properlyhandle materials with an equation of state and specific heat thatvary as

    functions of temperature and pressure, the ANSYS CFX-Solverneeds to know enthalpy as a

    function of temperature and pressure, .

    U

    Tdom

    T stat

    cp

    hstat href cp T stat T ref( )=

    T ref 0 K[ ]= href 0 J kg( )[ ]=

    hstat href cp T( ) TdT refT stat=

    cp

    h T p,( )

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    can be provided as a table using, for example, an RGP file. If atable is not

    pre-supplied, and the equation of state and specific heat aregiven by CEL expressions or

    CEL user functions, the ANSYS CFX-Solver will calculate byintegrating the full

    differential definition of enthalpy change.

    Given the knowledge of and that the ANSYS CFX-Solver calculatesboth static

    enthalpy and static pressure from the flow solution, you cancalculate static temperature by

    inverting the enthalpy table:

    (Eqn. 14)

    In this case, you know , from solving the flow and you calculateby table

    inversion.

    TotalTemperature

    The total temperature is derived from the concept of totalenthalpy and is computed exactly

    the same way as static temperature, except that total enthalpyis used in the property

    relationships.

    Material with Constant Density and Specific Heat

    If and are constant, then the total temperature and statictemperature are equal

    because incompressible fluids undergo no temperature change dueto addition of kinetic

    energy. This can be illustrated by starting with the totalenthalpy form of the constitutive

    relation:

    (Eqn. 15)

    and substituting expressions for Static Enthalpy and TotalPressure for an incompressible

    fluid:

    (Eqn. 16)

    (Eqn. 17)

    some rearrangement gives the result that:

    (Eqn. 18)

    for this case.

    h T p,( )

    h T p,( )

    h T p,( )

    hstat href h Tstat pstat,( ) h T ref pref,( )=

    hstat pstat T stat

    cp

    htot href cp T tot T ref( )=

    htot hstat12-- U U( )+=

    ptot pstat12-- U U( )+=

    T tot T stat=

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    Ideal Gas with constant cpFor this case, enthalpy is only afunction of temperature and the constitutive relation is:

    (Eqn. 19)

    which, if one substitutes the relation between static and totalenthalpy, yields:

    (Eqn. 20)

    If the turbulence model is employed, turbulence kinetic energy,, is added to the

    total enthalpy, giving the modified total temperature forconstant :

    (Eqn. 21)

    Ideal Gas with cp = f(T)

    The total temperature is evaluated with:

    (Eqn. 22)

    using table inversion to back out the total temperature.

    Material with Variable Density and Specific HeatIn this case,total temperature is calculated in the exact same way as statictemperature

    except that total enthalpy and total pressure are used as inputsinto the enthalpy table:

    (Eqn. 23)

    In this case you know , and you want to calculate , but you donot know . So,

    before calculating the total temperature, you need to computetotal pressure. For details,

    see Total Pressure (p. 14).

    For details, see Rotating Frame Quantities (p. 17).

    Entropy The concept of entropy arises from the second law ofthermodynamics:

    (Eqn. 24)

    h href cp T T ref( )=

    T tot T statU U2cp

    -------------+=

    k k

    cp

    T tot T statU U2cp

    -------------

    kcp-----+ +=

    htot href cp T( ) TdT refT tot=

    htot href h T tot ptot,( ) h T ref pref,( )=

    htot T tot ptot

    Tds dh dp------=

  • Basic Solver Capability Theory: Documentation Conventions

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    which can be rearranged to give:

    (Eqn. 25)

    Depending on the equation of state and the constitutiverelationship for the material, you

    can arrive at various forms for calculating the changes inentropy as a function of

    temperature and pressure.

    Material with Constant Density and Specific HeatIn this case,changes in enthalpy as a function of temperature and pressure aregiven by:

    (Eqn. 26)

    and when this is substituted into the second law gives thefollowing expression for changes

    in entropy as a function of temperature only:

    (Eqn. 27)

    which when integrated gives an analytic formula for changes inentropy:

    (Eqn. 28)

    Ideal Gas with constant cp or cp = f(T)

    For ideal gases changes in entropy are given by the followingequation:

    (Eqn. 29)

    which for general functions for the solver computes an entropytable as a function of

    both temperature and pressure. In the simplified case when is aconstant, then an

    analytic formula is used:

    (Eqn. 30)

    ds dhT------

    dpT-------=

    dh cpdTdp------+=

    ds cpdTT

    -------=

    s sref cpT

    T ref--------- log=

    s srefcp T( )

    T-------------- Td

    T ref

    T R ppref-------- log=cp

    cp

    s sref cpT

    T ref--------- log R ppref--------

    log=

  • Basic Solver Capability Theory: Documentation Conventions

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    Material with Variable Density and Specific HeatThis is the mostgeneral case handled by the ANSYS CFX-Solver. The entropyfunction,

    , is calculated by integrating the full differential form forentropy as a function of

    (Video) ANSYS CFX Workbench - Parameter Optimisation with Response Surface and Design of Experiments (DOE)

    temperature and pressure. Instead of repetitively performingthis integration the ANSYS

    CFX-Solver computes a table of values at a number of temperatureand pressure points. The

    points are chosen such that the approximation error for theentropy function is minimized.

    The error is estimated using derivatives of entropy with respectto temperature and

    pressure. Expressions for the derivatives are found bysubstituting the formula for general

    enthalpy changes into the second law to get the followingexpression for changes in

    entropy with temperature and pressure:

    (Eqn. 31)

    which when compared with the following differential form forchanges in entropy:

    (Eqn. 32)

    gives that:

    (Eqn. 33)

    (Eqn. 34)

    The derivative of entropy with respect to temperature is exactlyevaluated, while the

    derivative with respect to pressure must be computed bynumerically differentiating the

    equation of state. Note that when properties are specified as afunction of temperature and

    pressure using CEL expressions the differential terms inEquation 31 must also satisfy the

    exact differential relationship:

    See the previous section on Static Enthalpy for moredetails.

    s T p,( )

    dscpT-----dT 1

    2-----

    T------ pdp+=

    ds sT------ pdT

    sp------ Tdp+=

    sT------ p

    cpT-----=

    sp------ T

    1

    2-----

    T------ p=

    T 2

    2

    s

    p p2

    2

    s

    T=

  • Basic Solver Capability Theory: Documentation Conventions

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    Unless an externally provided table is supplied, an entropytable is built by using the and

    functions supplied by the user and integrating the generaldifferential form:

    (Eqn. 35)

    To calculate total pressure, you also need to evaluate entropyas a function of enthalpy and

    pressure, rather than temperature and pressure. For details, seeTotal Pressure (p. 14).

    The recipe to do this is essentially the same as for thetemperature and pressure recipe. First,

    you start with differential form for :

    (Eqn. 36)

    and comparing this with a slightly rearranged form of the secondlaw:

    (Eqn. 37)

    you get that:

    (Eqn. 38)

    (Eqn. 39)

    In this case, the entropy derivatives with respect to and can beevaluated from the

    constitutive relationship (getting from and by table inversion)and the

    equation of state . Points in the table are evaluated byperforming the

    following integration:

    (Eqn. 40)

    over a range of and , which are determined to minimize theintegration error.

    Total Pressure The total pressure, , is defined as the pressurethat would exist at a point if the fluid was

    brought instantaneously to rest such that the dynamic energy ofthe flow converted to

    pressure without losses. The following three sections describehow total pressure is

    cp

    s2 s1cpT-----dT

    T1

    T2

    12-----T------

    p1

    p2

    pdp+=

    s h p,( )

    ds sh------ pdh

    sp------ hdp+=

    ds dhT------

    dpT-------=

    sh------ p

    1T---=

    sp------ h

    1T-------=

    h p

    h T p,( ) T h pr T p,( ) s h p,( )

    s sref1T--- hd

    href

    h

    1T-------pref

    p

    dp=

    h p

    ptot

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    computed for a pure component material with constant density,ideal gas equation of state

    and a general equation of state (CEL expression or RGP table).For details, see Multiphase

    Total Pressure (p. 124).

    incompressible FluidsFor incompressible flows such as those ofliquids and low speed gas flows, the total pressure

    is given by Bernoullis equation:

    (Eqn. 41)

    which is the sum of the static and dynamic pressures.

    Ideal GasesWhen the flow is compressible, total pressure iscomputed by starting with the second law

    of thermodynamics assuming that fluid state variations arelocally isentropic so that you get:

    (Eqn. 42)

    The left hand side of this expression is determined by theconstitutive relation and the right

    hand side by the equation of state. For an ideal gas, theconstitutive relation and equation

    of state are:

    (Eqn. 43)

    (Eqn. 44)

    which when substituted into the second law and assuming noentropy variations gives:

    (Eqn. 45)

    where and are the static and total temperatures respectively(calculation of these

    quantities was described in two previous sections, StaticTemperature and Total

    Temperature). If is a constant, then the integral can be exactlyevaluated, but if varies

    with temperature, then the integral is numerically evaluatedusing quadrature.

    For details, see Entropy (p. 11).

    For details, see Ideal Gas Equation of state (p. 25).

    ptot pstat12-- U U( )+=

    dh dp------=

    dh cp T( )dT=

    pRT-------=

    ptot pstat1R---

    cp T( )T

    -------------- TdT stat

    T tot exp=T stat T tot

    cp cp

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    Material with Variable Density and Specific HeatTotal pressurecalculations in this case are somewhat more involved. You need tocalculate

    total pressure given the static enthalpy and static pressure,which the ANSYS CFX-Solver

    solves for. You also want to assume that the local statevariations, within a control volume

    or on a boundary condition, are isentropic.

    Given that you know (from integrating the differential form forenthalpy) and

    , you can compute two entropy functions and . There are two

    options for generating these functions:

    If and are provided by an RGP file then is evaluated by

    interpolation from and tables.

    If CEL expressions have been given for and only, then , and

    are all evaluated by integrating their differential forms.

    Once you have the table, calculated as described in Entropy,computing total

    pressure for a single pure component is a relatively trivialprocedure:

    1. The ANSYS CFX-Solver solves for and

    2. Calculate from

    3. Calculate entropy

    4. Using the isentropic assumption set

    5. Calculate total pressure by inverting

    For details, see Rotating Frame Quantities (p. 17).

    Shear StrainRate

    The strain rate tensor is defined by:

    (Eqn. 46)

    This tensor has three scalar invariants, one of which is oftensimply called the shear strain

    rate:

    (Eqn. 47)

    h T p,( )r T p,( ) s T p,( ) s h p,( )

    s T p,( ) h T p,( ) s h p,( )h T p,( ) s T p,( )

    cp h T p,( ) s T p,( )s h p,( )

    s h p,( )

    htot pstat

    hstat htot

    sstat s hstat pstat,( )=stot sstat=

    stot s htot ptot,( )=

    Sij12--

    U ix j

    ---------

    U jxi

    ---------+ =

    sstrnr 2U ix j

    ---------Sij

    12--

    =

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    with velocity components , , , this expands to:

    (Eqn. 48)

    The viscosity of non-Newtonian fluids is often expressed as afunction of this scalar shear

    strain rate.

    Rotating FrameQuantities

    The velocity in the rotating frame of reference is definedas:

    (Eqn. 49)

    where is the angular velocity, is the local radius vector, andis velocity in the

    stationary frame of reference.

    Incompressible FluidsFor incompressible flows, the totalpressure is defined as:

    (Eqn. 50)

    where is static pressure. The stationary frame total pressure isdefined as:

    (Eqn. 51)

    Ideal GasesFor compressible flows relative total pressure,rotating frame total pressure and stationary

    frame total pressure are computed in the same manner as in TotalPressure. First you start

    with the relative total enthalpy, rothalpy and stationary frametotal enthalpy:

    (Eqn. 52)

    (Eqn. 53)

    (Eqn. 54)

    U x U y U z

    sstrnr 2U xx----------

    2 U yy----------

    2 U zz---------

    2+ +

    =

    U xy----------

    U yx----------+

    2 U xz----------

    U zx---------+

    2 U yz----------

    U zy---------+

    2+ + +

    12--

    Urel Ustn R=

    R Ustn

    ptot pstat12-- Urel Urel R R( )( )+=

    pstat

    ptot,stn pstat12-- Ustn Ustn( )+=

    htot hstat12-- Urel Urel( )+=

    I hstat12-- Urel Urel R R( )( )+=

    htot,stn hstat12-- Ustn Ustn( )+=

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    where is the static enthalpy. In a rotating frame of reference,the ANSYS CFX-Solver

    solves for the total enthalpy, , which includes the relativekinetic energy.

    Important: Rothalpy is not a positive definite quantity. If therotation velocity is large,then the last term can be significantlylarger than the static enthalpy plus the rotating framekineticenergy. In this case, it is possible that total temperature androtating total pressureare undefined and will be clipped atinternal table limits or built in lower bounds. However,this isonly a problem for high angular velocity rotating systems.

    If you again assume an ideal gas equation of state with variablespecific heat capacity you

    can compute relative total temperature, total temperature andstationary frame total

    (Video) LearnCAx Tutorial ANSYS CFX Effect of guide vanes on duct flow in High Turbulence

    temperature using:

    (Eqn. 55)

    and:

    (Eqn. 56)

    and:

    (Eqn. 57)

    where all the total temperature quantities are obtained byinverting the enthalpy table. If

    is constant, then these values are obtained directly from thesedefinitions:

    (Eqn. 58)

    (Eqn. 59)

    (Eqn. 60)

    At this point, given , , , and you can compute relativetotal

    pressure, total pressure or stationary frame total pressureusing the relationship given in the

    section describing total pressure. For details, see TotalPressure (p. 14).

    The names of the various total enthalpies, temperatures, andpressures when visualizing

    results in ANSYS CFX-Post or for use in CEL expressions is asfollows.

    hstat

    htot

    I( )

    htot href cp T( ) TdT refT tot,rel=

    I href cp T( ) TdT refT tot=

    htot,stn href cp T( ) TdT refT tot,stn=

    cp

    T tot,rel T statUrel Urel

    2cp-----------------------+=

    T tot T statUrel Urel R R( )( )

    2cp------------------------------------------------------------------------+=

    T tot,stn T statUstn Ustn

    2cp------------------------+=

    T tot,rel T tot T tot,stn pstat T stat

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    Table 1 Variable naming: Total Enthalpies, temperatures, andpressures

    The Mach Number and stationary frame Mach numbers are definedas:

    (Eqn. 61)

    (Eqn. 62)

    where is the local speed of sound.

    Material with Variable Density and Specific HeatRotating andstationary frame total temperature and pressure are calculated thesame way

    as described in Total Temperature and Total Pressure. The onlychanges in the recipes are

    that rotating frame total pressure and temperature requirerothalpy, , as the starting point

    and stationary frame total pressure and temperature requirestationary frame total

    enthalpy, .

    Courant number The Courant number is of fundamental importancefor transient flows. For a

    one-dimensional grid, it is defined by:

    (Eqn. 63)

    where is the fluid speed, is the timestep and is the mesh size.The Courant number

    calculated in ANSYS CFX is a multidimensional generalization ofthis expression where the

    velocity and length scale are based on the mass flow into thecontrol volume and the

    dimension of the control volume.

    Variable Long Variable Name Short Variable NameTotal Enthalpyhtot

    Rothalpy rothalpy

    Total Enthalpy in StnFrame

    htotstn

    Total Temperature inRel Frame

    Ttotrel

    Total Temperature Ttot

    Total Temperature inStn Frame

    Ttotstn

    Total Pressure in RelFrame

    ptotrel

    Total Pressure ptot

    Total Pressure in StnFrame

    ptotstn

    htotIhtot,stn

    T tot,rel

    T totT tot,stn

    Ptot,rel

    PtotPtot,stn

    MUrel

    c------------=

    MstnUstn

    c-------------=

    c

    I

    htot,stn

    Courant utx---------=

    u t x

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    For explicit CFD methods, the timestep must be chosen such thatthe Courant number is

    sufficiently small. The details depend on the particular scheme,but it is usually of order

    unity. As an implicit code, ANSYS CFX does not require theCourant number to be small for

    stability. However, for some transient calculations (e.g., LES),one may need the Courant

    number to be small in order to accurately resolve transientdetails.

    ANSYS CFX uses the Courant number in a number of ways:

    1. The timestep may be chosen adaptively based on a Courantnumber condition (e.g., toreach RMS or Courant number of 5).

    2. For transient runs, the maximum and RMS Courant numbers arewritten to the outputfile every timestep.

    3. The Courant number field is written to the results file.

    Mathematical Notation

    This section describes the basic notation which is usedthroughout the ANSYS CFX-Solver

    documentation.

    The vectoroperators

    Assume a Cartesian coordinate system in which , and are unitvectors in the three

    coordinate directions. is defined such that:

    (Eqn. 64)

    Gradient operator

    For a general scalar function , the gradient of is definedby:

    (Eqn. 65)

    Divergence operator

    For a vector function where:

    (Eqn. 66)

    the divergence of is defined by:

    (Eqn. 67)

    i j k

    x

    y

    z

    , ,=

    x y z, ,( )

    x------i

    y------ j

    z------k+ +=

    U x y z, ,( )

    UUxU yUz

    =

    U

    U U xx----------

    U yy----------

    U zz---------+ +=

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    Dyadic operator

    The dyadic operator (or tensor product) of two vectors, and , isdefined as:

    (Eqn. 68)

    By using specific tensor notation, the equations relating toeach dimension can be

    combined into a single equation. Thus, in the specific tensornotation:

    (Eqn. 69)

    Matrixtransposition

    The transpose of a matrix is defined by the operator . Forexample, if the matrix is defined

    by:

    (Eqn. 70)

    then:

    (Eqn. 71)

    The IdentityMatrix(KroneckerDelta function)

    The Identity matrix is defined by:

    (Eqn. 72)

    Index notation Although index notation is not generally used inthis documentation, the following may

    help you if you are used to index notation.

    U V

    U VU xV x U xV y U xV zU yV x U yV y U yV zU zV x U zV y U zVz

    =

    U U( )x

    UxUx( ) y U yUx( ) z

    UzUx( )+ +

    x UxU y( ) y

    U yU y( ) z UzU y( )+ +

    x UxUz( ) y

    U yUz( ) z UzUz( )+ +

    =

    T

    x------

    y------

    z------

    =

    T

    x------y------

    z------

    =

    1 0 00 1 00 0 1

    =

  • Basic Solver Capability Theory: Governing Equations

    Page 22 ANSYS CFX-Solver Theory Guide. ANSYS CFX Release 11.0.1996-2006 ANSYS Europe, Ltd. All rights reserved.Containsproprietary and confidential information of ANSYS, Inc. and itssubsidiaries and affiliates.

    In index notation, the divergence operator can be written:

    (Eqn. 73)

    where the summation convention is followed, i.e., the index issummed over the three

    components.

    The quantity can be represented by (when and are vectors), orby

    (when is a vector and is a matrix), and so on.

    Hence, the quantity can be represented by:

    (Eqn. 74)

    Note the convention that the derivatives arising from thedivergence operator are

    derivatives with respect to the same coordinate as the firstlisted vector. That is, the quantity

    is represented by:

    (Eqn. 75)

    and not:

    (Eqn. 76)

    The quantity (when and are matrices) can be written by .

    Governing Equations

    The set of equations solved by ANSYS CFX are the unsteadyNavier-Stokes equations in their

    conservation form.

    If you are new to CFD, review the introduction. For details, seeComputational Fluid

    Dynamics (p. 1 in "ANSYS CFX Introduction").

    A list of recommended books on CFD and related subjects isavailable. For details, see

    Further Background Reading (p. 6 in "ANSYS CFXIntroduction").

    For all the following equations, static (thermodynamic)quantities are given unless

    otherwise stated.

    U U ixi

    ---------=

    i

    U V U iV j U V

    U iV jk U V

    U U( )

    xi U iU j( )

    U U( )

    xi U iU j( )

    x j U iU j( )

    a b a b aijbij

  • Basic Solver Capability Theory: Governing Equations

    ANSYS CFX-Solver Theory Guide. ANSYS CFX Release 11.0. 1996-2006ANSYS Europe, Ltd. All rights reserved. Page 23Contains proprietaryand confidential information of ANSYS, Inc. and its subsidiariesand affiliates.

    Transport Equations

    In this section, the instantaneous equation of mass, momentum,and energy conservation

    are presented. For turbulent flows, the instantaneous equationsare averaged leading to

    additional terms. These terms, together with models for them,are discussed in Turbulence

    and Wall Function Theory (p. 69).

    The instantaneous equations of mass, momentum and energyconservation can be written

    as follows in a stationary frame:

    The ContinuityEquation

    (Eqn. 77)

    The MomentumEquations

    (Eqn. 78)

    Where the stress tensor, , is related to the strain rate by

    (Eqn. 79)

    The TotalEnergy Equation

    (Eqn. 80)

    Where is the total enthalpy, related to the static enthalpyby:

    (Eqn. 81)

    The term represents the work du

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